# Analysis And Design Of Microwave Communication Antennas Based On Periodic Structures

код для вставкиСкачатьThe Pennsylvania State University The Graduate School College of Engineering ANALYSIS AND DESIGN OF MICROWAVE COMMUNICATION ANTENNAS BASED ON PERIODIC STRUCTURES A Dissertation in Electrical Engineering by Ravi Kumar Arya © 2017 Ravi Kumar Arya Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy August 2017 ProQuest Number: 10666416 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. ProQuest 10666416 Published by ProQuest LLC (2017 ). Copyright of the Dissertation is held by the Author. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code Microform Edition © ProQuest LLC. ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, MI 48106 - 1346 The dissertation of Ravi Kumar Arya was reviewed and approved∗ by the following: Raj Mittra Professor of Electrical Engineering Dissertation Advisor Co–Chair of Committee Ram Narayanan Professor of Electrical Engineering Co–Chair of Committee Victor Pasko Professor of Electrical Engineering Michael T. Lanagan Professor of Engineering Science and Mechanics Kultegin Aydin Professor of Electrical Engineering Head of the Department of Electrical Engineering ∗ Signatures are on file in the Graduate School. ii Abstract Presently, there is considerable interest in the world of Metamaterial (MTM) and Frequency Selective Surfaces (FSSs) in constructing and using crystals with 3D elements. These elements provide the flexibility in controlling the performance characteristics – in regards to frequency bandwidth, polarization sensitivity and angular range. Frequency selective surfaces (FSSs) are periodic structures with a bandpass or a bandstop frequency response dependent upon the geometrical parameters of its elements. Periodic structures are typically modeled as infinite arrays of scatterers, and are commonly analyzed by imposing periodic boundary conditions to a unit cell to reduce the original problem to a manageable size. Analysis of only a single unit cell reduces the computational burden. However, this benefit is marred if the illuminating plane wave has a large incident angle. Large incident angles in numerical simulations require lowering of the step size, which in turn might lead to instability. Even though several Finite-Difference Time-Domain (FDTD)/ Periodic Boundary Condition (PBC) methods are capable of handling the large angle condition in different ways, solving the unit cell remains a computationally intensive problem. The conventional Method of Moments (MoM) approach provides an efficient means to simulating FSSs though given the caveat that the periodic elements are PEC and not inhomogeneous, complex objects, which are more amenable to convenient analysis through the use of Finite Methods (FM). This dissertation starts by presenting a novel approach that is numerically efficient and accurate for the analysis of three–dimensional arbitrarily shaped periodic structures with arbitrary incidence angles. This technique does not suffer from the stability issues encountered in the FDTD/PBC algorithm, which can become unstable and computationally intensive with wide angles of incidence. Next, the design of the flat lenses is explored using modified commercial offthe-shelf (COTS) materials, as opposed to metamaterials (MTMs) that are often required in lens designs based on the Transformation Optics (TO) approach. While lens designs based on Ray Optics (RO) do not suffer from the drawback of having to use metamaterials, they still require dielectric materials that may not be commercially available off-the-shelf. A systematic procedure for realizing the desired iii materials is demonstrated by modifying the COTS types of materials, and illustrates its application with some practical examples. This dissertation investigates the use of 3D–printing for such examples and illustrates its benefits by combining it with the proposed method. Normally, the FSS structure is numerically simulated during the design process and then fabricated to verify if indeed it has the predicted characteristics. It is not unusual to find that there is considerable discrepancy between the simulated and measured results, even when there is only a minor difference between the designed and fabricated structures. This is especially true for Metamaterials used at optical wavelengths, where the difficulties in fabrication almost always introduce small variations in the dimensions of the elements that comprise the “periodic” array. This dissertation explores the resulting effect to the performance of the presence of this type of variation in the unit cell parameters by using the Polynomial Chaos method instead of the traditional Monte Carlo method, which can be extremely expensive in terms of computational resources. Finally, two designs of offset–fed dielectric reflectarray are presented. Both reflectarrays feature broadband designs realized by using dielectric blocks backed by a PEC plane. One of these arrays uses a phase compensating flat lens to reduce the maximum permittivity of the dielectric blocks covering the PEC plane. We compare the performances of both reflectarrays and list their benefits as compared to traditional reflectarrays that use resonant elements for their designs, which render them narrowband. iv Table of Contents List of Figures viii List of Tables xii Acknowledgments Chapter 1 Introduction 1.1 Electromagnetics . . . . . . 1.2 Computational Methods . . 1.3 Motivation . . . . . . . . . . 1.4 Outline of the Dissertation . 1.5 Contributions to Knowledge xiii . . . . . 1 1 2 2 3 4 Chapter 2 Numerical Technique for Efficient Analysis 2.1 Periodic Boundary Conditions in FDTD . 2.2 Structural Transformation . . . . . . . . . 2.3 Numerical Convergence . . . . . . . . . . . 2.4 Numerical Results . . . . . . . . . . . . . . 2.4.1 PEC Spheres . . . . . . . . . . . . 2.4.2 Dielectric Spheres . . . . . . . . . . 2.4.3 Complex 3D Structures . . . . . . . 2.5 Observations and Conclusions . . . . . . . of Periodic Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 6 9 10 12 12 12 14 14 Chapter 3 Lens Design Using Artificially Engineered 3.1 Introduction . . . . . . . . . . . . . . . . . 3.2 Ray-Optics Lens Design . . . . . . . . . . 3.3 3D–Printing Technique . . . . . . . . . . . 3.4 Design of Artificially Engineered Materials Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 19 21 22 24 . . . . . . . . . . v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 3.5 3.6 Designing higher permittivity materials from low permittivity COTS material : Method-1 . . . . . . . . . . . . . . . . . 3.4.2 Designing higher permittivity materials from low permittivity COTS material : Method-2 . . . . . . . . . . . . . . . . . 3.4.3 Designing lower permittivity materials from high permittivity COTS material . . . . . . . . . . . . . . . . . . . . . . . . 3.4.4 Designing lower permittivity materials from high permittivity 3D–printing material . . . . . . . . . . . . . . . . . . . . . Different Lens Designs . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 PLA Lens Design . . . . . . . . . . . . . . . . . . . . . . . 3.5.1.1 Lens Fabrication . . . . . . . . . . . . . . . . . . 3.5.1.2 Lens Measurement . . . . . . . . . . . . . . . . . 3.5.1.3 Results . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 DaD Lens Design . . . . . . . . . . . . . . . . . . . . . . . 3.5.2.1 Lens Fabrication . . . . . . . . . . . . . . . . . . 3.5.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . 3.5.3 ABS Lens Design . . . . . . . . . . . . . . . . . . . . . . . 3.5.3.1 Results . . . . . . . . . . . . . . . . . . . . . . . 3.5.4 Comparison of DaD and ABS Lenses . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 . 25 . 25 . . . . . . . . . . . . . 27 28 30 33 37 37 42 45 45 50 50 55 55 Chapter 4 Uncertainty Management of Periodic Structures 66 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.2 Structures with Uncertain Parameters . . . . . . . . . . . . . . . . . 67 4.2.1 Cross-dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.2.2 3D–printed Lens . . . . . . . . . . . . . . . . . . . . . . . . 68 4.3 The Monte Carlo method . . . . . . . . . . . . . . . . . . . . . . . 70 4.4 Surrogate Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.5 Statistical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.5.1 Polynomical Chaos . . . . . . . . . . . . . . . . . . . . . . . 72 4.5.2 Truncation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.5.2.1 Sparse PC: The Least Angle Regression Truncation 74 4.5.3 Computation of the Coefficients . . . . . . . . . . . . . . . . 75 4.5.4 Quality Assessment of Polynomial Chaos Representations . . 76 4.5.4.1 The Leave-One-Out Cross-Validation . . . . . . . . 77 4.5.4.2 Choice of the “best” meta-model . . . . . . . . . . 77 4.5.5 Senstivity Analysis . . . . . . . . . . . . . . . . . . . . . . . 78 4.6 Illustrative Numerical Results . . . . . . . . . . . . . . . . . . . . . 79 4.6.1 Cross–dipole Analysis . . . . . . . . . . . . . . . . . . . . . . 79 vi 4.6.2 3D–printed Lens Analysis . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 83 Chapter 5 Offset–fed Dielectric Reflectarray Antenna Design 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Different Reflectarray Designs . . . . . . . . . . . . . . . . . . . . . 5.2.1 Offset–fed Dielectric Reflectarray Design . . . . . . . . . . . 5.2.2 Dielectric Reflectarray with a Phase Compensating Flat Lens 5.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 91 92 93 95 96 97 4.7 Chapter 6 Conclusions and Future Work 102 Bibliography 107 vii List of Figures 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 A representative geometry of an infinite doubly periodic array of inhomogeneous 3D elements. . . . . . . . . . . . . . . . . . . . . . . Two-dimensional electromagnetic structure; (a) top view of the structure; (b) top view of the structure with field components for the unit cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Different methods used to apply FDTD with periodic boundary conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modified Waveguide Geometry (the z–axis is normal to the open ends of the waveguide and top-bottom are PEC boundaries). . . . . Conversion technique for doubly–infinite periodic array to a truncated waveguide structure. . . . . . . . . . . . . . . . . . . . . . . . Ey distribution measured along the z–axis: (a) magnitude; and (b) phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Magnitude of the Reflection Coefficient as a function of the number of elements, in the x-direction; solid line: original data; dashed: extrapolated data. . . . . . . . . . . . . . . . . . . . . . . . . . . . Magnitude of Reflection Coefficient for: (a) normal incidence; and (b) 20 degrees incidence, derived by using the present method and compared with those from a commercial FEM (PBC) solver for PEC spheres. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Magnitude of Reflection Coefficient for: (a) normal incidence; and (b) 20 degrees incidence, derived by using the present method and compared with those from a commercial FEM (PBC) solver for dielectric spheres. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geometry of analyzed FSS unit cell: (a) 3D View (b) side view (c) top view. Geometry parameters are: H = 18.96mm, d = 0.508mm, t = 1mm, w = 1mm, s = 2*0.784mm, h = 0.76mm, L = 6.5mm and periodicity Dx = Dy = 24.29mm. . . . . . . . . . . . . . . . . . . . viii 7 8 9 10 11 13 14 15 16 17 2.11 Magnitude of Transmission Coefficient for normal incidence from the present method, a commercial FEM (PBC) solver and measured results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17 3.18 3.19 3.20 3.21 3.22 3.23 3.24 3.25 Sketch of lens design principle. . . . . . . . . . . . . . . . . . . . . . Proposed DaD Lens. . . . . . . . . . . . . . . . . . . . . . . . . . . Unit cell for designing higher permittivity materials from low permittivity COTS material . . . . . . . . . . . . . . . . . . . . . . . . Unit cell for designing higher permittivity materials from low permittivity COTS materials. . . . . . . . . . . . . . . . . . . . . . . . Unit cell for designing lower permittivity materials from high permittivity COTS material. . . . . . . . . . . . . . . . . . . . . . . . Unit cell for designing lower permittivity materials from high permittivity 3D printing material. . . . . . . . . . . . . . . . . . . . . . Effective relative permittivity for the 3D–printed lens as a function of its radius. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulated electric field of the GRIN lens at 15 GHz when fed by a circular Ku–band horn located at the focal point. . . . . . . . . . . Simulated far–field gain pattern of the thick flat GRIN lens at 12, 15 and 18 GHz for: (a) E-plane; and (b) H-plane. . . . . . . . . . . 3D-printed flat PLA lens. . . . . . . . . . . . . . . . . . . . . . . . Sketch of the measurement setup for the 3D–printed lens antenna. . Measurement setup with the 3D–printed lens with a conical feed horn at the focal point of the lens. . . . . . . . . . . . . . . . . . . . Measured broadband gain of 3D-printed lens with different feeding distances. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measured and simulated far–field patterns of the GRIN lens at for: (a) E-plane; and (b) H-plane. . . . . . . . . . . . . . . . . . . . . . Measured patterns of 3D-printed lens with the horn positioned at different off axis distances at 15 GHz. . . . . . . . . . . . . . . . . . Unit cell used for the DaD lens design. . . . . . . . . . . . . . . . . S21 parameter for unit cell of ring 1. . . . . . . . . . . . . . . . . . . S21 parameter for unit cell of ring 2. . . . . . . . . . . . . . . . . . . S21 parameter for unit cell of ring 3. . . . . . . . . . . . . . . . . . . Photographs of the DaD Lens (see Fig. 3.2 for all dimensions). . . . Far-field radiation patterns of DaD Lens at 12 GHz. . . . . . . . . . Far-field radiation patterns of DaD Lens at 15 GHz. . . . . . . . . . Far-field radiation patterns of DaD Lens at 18 GHz. . . . . . . . . . Gain response of the DaD Lens. . . . . . . . . . . . . . . . . . . . . Far-field radiation patterns of 3D–printed ABS Lens at 12 GHz. . . ix 18 22 26 27 28 29 30 32 33 34 36 38 39 40 41 42 44 46 47 48 49 51 52 53 54 57 3.26 3.27 3.28 3.29 3.30 3.31 3.32 3.33 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16 5.1 5.2 5.3 Far-field radiation patterns of 3D–printed ABS Lens at 15 GHz. . Far-field radiation patterns of 3D–printed ABS Lens at 18 GHz. . Gain response of 3D–printed ABS Lens. . . . . . . . . . . . . . . Far-field radiation patterns of DaD and 3D–printed ABS Lens at 12 GHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Far-field radiation patterns of DaD and 3D–printed ABS Lens at 15 GHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Far-field radiation patterns of DaD and 3D–printed ABS Lens at 18 GHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gain response of DaD Lens and 3D–printed ABS Lens. . . . . . . Photograph of the different lenses. . . . . . . . . . . . . . . . . . . . . . 58 59 60 . 61 . 62 . . . 63 64 65 A typical periodic structure and its unit cell . . . . . . . . . . . . . The input parameters of cross-dipole FSS element . . . . . . . . . . 3D printed flat lens and the cross section of a block of the outer ring with 4 holes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Original model and surrogate model. . . . . . . . . . . . . . . . . . Γmag for different number of simulation runs. . . . . . . . . . . . . . Γphase for different number of simulation runs. . . . . . . . . . . . . Q2 of the different meta-models for the Γmag using full PC expansion (in blue) and LAR PC expansion (in red). . . . . . . . . . . . . . . Q2 of the different meta-models for the Γphase using full PC expansion (in blue) and LAR PC expansion (in red). . . . . . . . . . . . . . . Probability Density as a function of Magnitude of Reflection Coefficient. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Probability Density as a function of Phase of Reflection Coefficient. The procedure for estimating ∆ from ∆a: a) the block with holes; b) the homogeneous block. . . . . . . . . . . . . . . . . . . . . . . . The procedure for estimating ∆ from ∆a: a) phase variation for both models; b) the relationship between ∆ and ∆a. . . . . . . . . LOO error for different number of data. . . . . . . . . . . . . . . . . The average gain along the axis of the lens obtained by the PCE and the one calculated with the nominal values. . . . . . . . . . . . The uncertainty of the gain along the axis of the lens. . . . . . . . . Total Sobol’ indices with relative influence of each relative permittivity to the gain along the axis of the lens. . . . . . . . . . . . . . . 68 68 Location of the input and output aperture. . . . . . . . . . . . . . . Dielectric Transmitarray. . . . . . . . . . . . . . . . . . . . . . . . . Offset-fed Dielectric Reflectarray. . . . . . . . . . . . . . . . . . . . x 69 71 80 81 82 83 84 85 85 86 87 88 89 90 94 96 97 5.4 5.5 5.6 Dielectric Reflectarray with phase compensating flat lens. . . . . . . 98 Simulated radiation pattern of the reflectarrays at 15 GHz. . . . . . 100 Simulated gain comparison of reflectarrays. . . . . . . . . . . . . . . 101 xi List of Tables 3.1 3.2 3.3 3.4 3.5 3.6 3.7 COTS dielectric materials. . . . . . . . . . . . . . . . . . . . Design parameters for the PLA lens. . . . . . . . . . . . . . Designed parameters of the 3D–printed lens. . . . . . . . . . Parameters of 3D–printed concentric dielectric rings. . . . . Material Parameters of the DaD Lens. . . . . . . . . . . . . Designed parameters of the DaD lens. . . . . . . . . . . . . . Patch sizes and PLA volume density for different rings of the Lens. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Material Parameters of the DaD Lens. . . . . . . . . . . . . 3.9 Designed parameters of the DaD lens. . . . . . . . . . . . . . 3.10 Parameters of 3D–printed ABS lens dielectric rings. . . . . . . . . . . . 23 31 31 36 42 43 . . . . 45 50 50 54 4.1 4.2 4.3 Input Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effective relative permittivity values of the 3D–printed lens. . . . . Orthogonal polynomials. . . . . . . . . . . . . . . . . . . . . . . . . 67 69 72 5.1 5.2 Permittivity of the Dielectric Blocks for the Reflectarray. . . . . . . Permittivity of the Dielectric Blocks for the Reflectarray with Lens. 95 99 xii . . . . . . . . . . . . . . . . . . DaD . . . . . . . . . . . . Acknowledgments First and foremost, I offer my most sincere gratitude to my thesis supervisor, Dr. Raj Mittra, for his exceptional support and guidance throughout my research work. This dissertation would not have been possible without his persistent pursuit of perfection and incredible dedication in making sure that I comprehend theoretical principles and practical concepts to their entirety. He is not only a great teacher but also has been a source of inspiration for me. I would also like to thank him for all the corrections and revisions of this work. I would like to acknowledge and thank my colleagues in the Electromagnetics Communication Laboratory namely Dr. Kadappan Panayappan, Dr. Chiara Pelletti, Dr. Giacomo Bianconi, Dr. Gu Xiang, Dr. Muhammed Hassan, Dr. Sidharath Jain, Dr. Yuda Zhou, Dr. Hulusi Açıkgöz, Mr. Kapil Sharma, Mr. Mohamed Abdel–Mageed and Mr. Shaileshachandra Pandey, without whom this dissertation would not have been complete. I would also like to thank Dr. Shiyu Zhang, Dr. Will Whittow and Dr. Yiannis Vardaxoglou from Loughborough University for their help with the fabrication and measurement of the prototype in the major part of this work. I would like to sincerely thank all of my committee members, Dr. Ram Narayanan, Dr. Victor Pasko and Dr. Michael T. Lanagan, who set aside their busy schedules to review this work. I am immensely grateful to Ms. SherryDawn Jackson, Ms. MaryAnn Henderson and Ms. Lisa Timko, without whom I could not have made it to this point. Finally, and most importantly, I want to thank my parents, my wife, my children and my brothers for their unconditional support throughout the years. My journey to a Ph.D. degree would not have been possible without the dedication and encouragement of my family. xiii Dedication To my beautiful and precious children, Nikita and Kuvam, for making me better, stronger and more fulfilled than I could have ever imagined. xiv Chapter 1 | Introduction 1.1 Electromagnetics Electromagnetics deals with the theory and applications of generation and manipulation of the electric and magnetic fields. James Clerk Maxwell united all the previous work in this field and expressed it in terms of mathematical equations in 1861 and 1862. These equations are called the Maxwell’s equations and can be expressed in both integral (1.1–1.4) and differential form (1.5–1.8), as given below [1]: ∂B · dS, C S ∂t Z I ∂D J+ H · dl = · dS, ∂tZ C I S I E · dl = − Z D · dS = S (1.1) (1.2) ρ dυ, (1.3) B · dS = 0. (1.4) υ I S ∂B , ∂t ∂D ∇×H =J + , ∂t ∇ · D = ρ, (1.7) ∇ · B = 0. (1.8) ∇×E =− where V E : electric field intensity ( m ) 1 (1.5) (1.6) b B : magnetic flux density ( W ) m2 A H : magnetic field intensity ( m ) J : total current density ( mA2 ) D : electric flux density ( mC2 ) ρ : total charge density ( mC3 ) Today, these equations are the backbone of electromagnetics and lay the very foundation for future research. 1.2 Computational Methods To understand the behavior of different electromagnetic problems, computational methods are typically used to conduct numerical analysis as a convenient tool. These methods can be divided into these broad categories: 1. Finite-difference time-domain (FDTD) The FDTD method is a differential time-domain numerical modeling method. In this method, the time-dependent differential Maxwell’s equations are discretized and solved by a leapfrog scheme. This method is a time domain method. 2. Method of Moments (MoM) Method of moments (MoM) is a computational method that solves linear partial differential equations formulated as integral equations by use of Green’s function. This is a frequency domain method. 3. Finite Element Method (FEM) The Finite Element Method (FEM) is a computational method used to find approximate solutions to partial differential equations. This is a frequency domain method. 1.3 Motivation With theoretical and technical advances in computational methods, there has always been an increasing demand for improvement to the current algorithms in terms of accuracy, efficiency, and computational resources. Historically, research 2 in FSSs and antenna arrays has been a key focus in electromagnetics and recent developments in the field of Metamaterials has garnered considerable academic and engineering interest. Typically comprised of doubly–periodic unit cells of a single, unique element, the modeling of these structures is simplified and reduced by a significant margin by considering the array to be infinitely periodic and analyzing only one unit cell of the periodic structure to simulate the effective outcome for the whole system. For this purpose, Periodic Boundary Conditions (PBCs) are applied to the unit cell. There are different methods for analyzing a unit cell, but all methods either lack simplicity or need large computational resources. Some of these techniques use a transformation of Maxwell’s equations to ease the burden but then lose the simplicity of solving the problem. Therefore, there is a need to simplify a computational method for analyzing periodic structures. Taking this into consideration, we looked into different FDTD/PBC techniques in hopes to simplify the method by reducing computational intensiveness while still being able to model any dielectric/magnetic inhomogeneous element. This technique is versatile and can be used for 3–D elements illuminated with arbitrary incident angles. Armed with a novel FDTD/PBC technique, we look into the case of a flat lens design. We present a new method to artificially engineer the materials that can be used for such lens designs since typical permittivity values these lenses call for are not usually commercially available. Once these lenses are fabricated, there are also issues with levels of mismatch between simulated and fabricated behavior as the fabrication process always introduces some variations in the designs. We study the contributing effect introduced with such variations with statistical methods. We face a similar problem when trying to track down readily available materials for dielectric reflectarrays as the dielectric blocks used in the reflectarray require materials with a wide range of permittivity values. Most of the time, these permittivity values are considerably higher than those produced by using the currently affordable 3D–printing techniques and are, therefore, not a viable option. We design a new dielectric reflectarray that only requires lower and readily available values of permittivity. 1.4 Outline of the Dissertation This dissertation is organized into six chapters. 3 Chapter 2 presents a novel method for analyzing periodic structures in the FDTD method. We analyze the finite, truncated structure to derive the solution for an infinite doubly–periodic problem setup. Chapter 3 introduces the lens design by using artificially engineered materials. We use 3D–printing and Dial–a–Dielectric (DaD) techniques to design three lenses in this chapter and compare their performances. Chapter 4 presents the statistical methods to study the behavior of input variations in the unit cell of periodic structures in general and a 3D–printed lens as a specific case scenario. Chapter 5 introduces the offset–fed dielectric reflectarray antenna designs. We introduce two reflectarrays in this chapter. Both reflectarrays are realized by using dielectric blocks backed by a PEC plane. One of these refectarrays uses a phase compensating lens to reduce the maximum permittivity called by dielectric blocks. The last chapter, Chapter 6, presents a summary and the conclusions of this research. It also explores potential directions for future work. 1.5 Contributions to Knowledge This dissertation has addressed some current problems and challenges in the field of electromagnetics and has shown various approaches to solving them. Several key contributions from this doctoral dissertation can be summarized as follows: 1. An alternate method to addressing the periodic boundary conditions in the FDTD method has been developed, which derives the analytical solution to an infinite, doubly–periodic problem by analyzing a finite, truncated structure. This helped us to calculate the transmission and reflection coefficients for doubly–infinite arbitrary 3D elements by using signal processing techniques. This method was studied for different 3D elements and was published as a book Chapter [2] as well as a full–length Journal paper [3] . 2. We introduced novel lens designs by using artificially engineered materials in Chapter 3. We used 3D–printing and Dial–a–Dielectric (DaD) techniques and designed different lenses in this work. All theses lenses show satisfactory wideband performances and are considerably low–cost to manufacture and make adjustments (should the need arise) as compared to traditional full-dielectric/Fresnel lens designs. The DaD technique is versatile and can be used not only in lenses, but also in 4 other electromagnetic devices where commercial off–the–shelf (COTS) materials are not available. The design of different lenses and their simulated/measured results presented in Chapter 3, have been partially published in the form of a full–length paper in IET Microwaves, Antennas & Propagation [4] and will be complemented in a yet to be published Journal paper. 3. Usually there is considerable discrepancy in the simulated and measured behaviors of Frequency Selective Surfaces (FSSs) due to variabilities during the fabrication process. In Chapter 4, we introduced statistical methods to study the behavior of such variations and list the effect of such fabrication imperfections on the final output behavior of the system. This study will help future lens designers pay more attention to design specifics that have higher potentials for risks. This study was presented as a full–length paper [5] and will be published further as a book chapter in an upcoming IET book [6]. 4. New designs of offset–fed dielectric reflectarray antenna were developed in Chapter 5. Both reflectarray designs are realized by using dielectric blocks backed by a PEC plane. The proposed designs show an original solution to reducing the maximum permittivity that is usually called for by using dielectric blocks and are exceptionally helpful when COTS materials are not readily available. We will present more insights into the designs of dielectric reflectarrays to the wider RF community at 2017 IEEE AP-S Symposium on Antennas and Propagation and USNC-URSI Radio Science Meeting [7]. 5 Chapter 2 | Numerical Technique for Efficient Analysis of Periodic Structures Periodic structures are typically modeled as doubly-infinite arrays of scatterers [8], and are commonly analyzed by imposing the periodic boundary condition on a unit cell to reduce the original problem to a manageable size [1, 8, 9]. The conventional Method of Moments (MoM) is well suited for simulating FSSs comprised of thin planar, PEC elements. However, MoM can become very inefficient when handling inhomogeneous and complex–shaped elements (see Fig. 2.1) that are more amenable to convenient analysis via the use of Finite Methods. In this chapter, we introduce a novel method for calculating the transmission and reflection coefficients of doubly–infinite periodic arrays with arbitrary 3D elements, illuminated by a plane wave incident at an arbitrary angle. The proposed method derives the solution to the infinite doubly-periodic problem by first analyzing a finite, truncated structure. The solution is derived by progressively enlarging the size of the truncated structure and extrapolating its solution via the use of signal processing techniques. The systematic procedure to apply this method is given below after discussing the traditional FDTD method for the periodic structures. 2.1 Periodic Boundary Conditions in FDTD There are multiple artificial boundary conditions used in FDTD to truncate the computational domain. The size of the computational domain determines the speed and storage space of the FDTD simulation. These boundary conditions are placed at a certain distance from the object to truncate the computational domain. There are several different methods for simulating such boundary conditions [9]. 6 Figure 2.1: A representative geometry of an infinite doubly periodic array of inhomogeneous 3D elements. Periodic Boundary Conditions (PBCs) are typically used to analyze periodic structures in FDTD. In this approach, the inherent property of periodicity in the periodic structure is exploited and only one unit cell of the periodic structure is simulated to mimic the behavior of the entire periodic structure. To understand how the basic problems are set up in the FDTD/PBC approach for oblique angles of incidence, let us consider the two–dimensional structure shown in Fig. 2.2 [9, 10]. The array is composed of parallel conducting spheres of diameter “d” separated by a gap “w” and is illuminated by an electromagnetic plane wave with an electric field polarized along z–axis. The propagation vector is directed at an angle Θ with respect to the x–axis. Field equations at the boundary of the unit cell are given by : Hx (x, y = yp + ∆y/2, t) = Hx (x, y = ∆y/2, t − yp sin(Θ)) (2.1a) Ez (x, y = 0, t) = Ez (x, y = yp , t + yp sin(Θ)) (2.1b) 7 (a) (b) Figure 2.2: Two-dimensional electromagnetic structure; (a) top view of the structure; (b) top view of the structure with field components for the unit cell. It is important to note that the electric field at y=0 and time t is equal to the same field at y = yp but at time t = t + yp sin(Θ), which is at a future time. So, it becomes difficult to solve this problem with the traditional FDTD method; and different methods are applied to handle these future values of fields. The methods that are employed to determine the future values can be divided into direct-field and field-transformation methods [9], as shown in Fig. 2.3. 8 Figure 2.3: Different methods used to apply FDTD with periodic boundary conditions. All of the above methods either need special ways to treat the structure, or they transform the conventional FDTD code to solve a periodic problem [11, 12]. Also, the convergence behavior in FDTD requires that the cell size be progressively decreased for increasing angles of incidence, which is computationally inefficient for wide angles. In this work, we develop a new technique to circumvent this shortcoming when handling doubly-infinite periodic structures. 2.2 Structural Transformation To apply this method, the given doubly-infinite periodic structure is first modified to the desired truncated model as shown in Fig. 2.4. Fig. 2.4 also shows the polarization of the incident electric field and the wave propagation for normal incidence; however, the incident angle can be arbitrary. For simplicity, let us consider a doubly–infinite structure, which is periodic in the x–y directions as shown in Fig. 2.5. We truncate the infinite structure as follows. First, we truncate the structure with perfectly matched layers (PML) in the x–direction. Then, to mimic the infinite structure in the y–direction, we insert two perfect electric conductor (PEC) infinite planes in the y–direction as shown in Fig. 2.5. 9 Figure 2.4: Modified Waveguide Geometry (the z–axis is normal to the open ends of the waveguide and top-bottom are PEC boundaries). 2.3 Numerical Convergence After converting the doubly–infinite structure to a modified waveguide structure, we solve this scattering problem by using a Finite Method, e.g., the FDTD or FEM, to compute the scattered fields along the longitudinal directions. We note that the total field on the incident side of the waveguide (z<0) is a summation of the incident and scattered (reflected) fields, while only the transmitted fields exist on the other side (z>0), as shown in Fig. 2.6. Next, for the normal incidence case, we decompose the fields measured along the line z1 − z2 (see Fig. 2.4)) within region z<0, into their incident and reflected components by using the GPOF method [13]. For the oblique incidence case, the fields are measured along specular directions in both the reflection and the transmission regions. The reflection and transmission coefficients are defined as: Γ= E ref l E inc and τ= E trans E inc (2.2) where E ref l and E trans represent the reflected and transmitted fields as discussed above. 10 Figure 2.5: Conversion technique for doubly–infinite periodic array to a truncated waveguide structure. 11 The weights of the transmitted and reflected fields associated with the dominant Floquet harmonic determined by the GPOF algorithm, yield the transmission and reflection coefficients for the truncated array. The computed reflection and transmission coefficients are tracked progressively by increasing the number of elements in the transverse direction (see Fig. 2.4). After calculating the reflection coefficient, Γ, we plot it as a function of the number of cells in the x-direction as shown in Fig. 2.7. Next, the discrete values of Γ are extrapolated to an asymptotic value [3]. We should mention that this procedure needs to be repeated for each frequency, since the reflection coefficient obviously depends on the frequency of interest. 2.4 Numerical Results In this section, we present some representative numerical results for the reflection and transmission characteristics of some canonical 3D structures. We compare our results against those obtained from a commercial FEM solver to validate the accuracy of our method. 2.4.1 PEC Spheres In Fig. 2.8 we show the results for an array of PEC spheres whose diameters are 0.5λ0 , at the operating frequency of 5 GHz. The periodicity in x–axis and y–axis are both 0.75λ0 at the operating frequency. The array is illuminated by a plane wave at normal and 20 degree incidence angles, respectively. We also compare the obtained results against those calculated by using a commercial FEM solver. 2.4.2 Dielectric Spheres As is well known, the FDTD can handle both dielectric and PEC structures, as well as a combination thereof with relative ease. Fig. 2.9 shows the results for an array of dielectric spheres with r = 9 and diameters of 0.5λ0 at the operating frequency of 5 GHz. The periodicities along the x– and y–axes are both 0.75λ0 at the operating frequency. Again, the results have been derived for normal and 20 degree incidence angles, respectively. 12 (a) (b) Figure 2.6: Ey distribution measured along the z–axis: (a) magnitude; and (b) phase. 13 Figure 2.7: Magnitude of the Reflection Coefficient as a function of the number of elements, in the x-direction; solid line: original data; dashed: extrapolated data. 2.4.3 Complex 3D Structures To further illustrate the versatility of this method, we have applied this technique to a complex 3D structure FSS element, as shown in Fig. 2.10. Regarding fabrication and measurements details for this structure, readers can refer to the published work [14]. Fig. 2.11 shows the transmission coefficient of this structure. 2.5 Observations and Conclusions From Figs. 2.8, 2.9 and 2.11, it is evident that there is good agreement between the computed reflection and transmission coefficients derived by using the proposed method and the FEM (PBC) approach. And yet, the use of PBCs has been avoided in our approach, and the difficulties encountered in the FDTD, in terms of 14 (a) (b) Figure 2.8: Magnitude of Reflection Coefficient for: (a) normal incidence; and (b) 20 degrees incidence, derived by using the present method and compared with those from a commercial FEM (PBC) solver for PEC spheres. 15 (a) (b) Figure 2.9: Magnitude of Reflection Coefficient for: (a) normal incidence; and (b) 20 degrees incidence, derived by using the present method and compared with those from a commercial FEM (PBC) solver for dielectric spheres. 16 (a) (b) (c) Figure 2.10: Geometry of analyzed FSS unit cell: (a) 3D View (b) side view (c) top view. Geometry parameters are: H = 18.96mm, d = 0.508mm, t = 1mm, w = 1mm, s = 2*0.784mm, h = 0.76mm, L = 6.5mm and periodicity Dx = Dy = 24.29mm. 17 Figure 2.11: Magnitude of Transmission Coefficient for normal incidence from the present method, a commercial FEM (PBC) solver and measured results. solve–time and the stability behavior, have been totally circumvented. The novel FDTD technique, demonstrated can be used to calculate the transmission and reflection coefficients of doubly–infinite periodic arrays with arbitrary 3D elements, illuminated by a plane wave incident at an arbitrary angle. Unlike the conventional FDTD/PBC algorithms, the proposed method does not suffer from instability problems and performs accurately irrespective of the angle of incidence. Furthermore, it neither calls for reduced time step with an increase in the incident angle as does the traditional FDTD/PBC, nor does it require a modification of the FDTD update equations. In common with the established method [15], the proposed method derives the solution of the periodic problem from that of its truncated counterpart. 18 Chapter 3 | Lens Design Using Artificially Engineered Materials 3.1 Introduction A lens is a commonly used antenna component that transforms a plane wavefront into a spherical one or vice versa. In a conventional lens, the index of refraction is homogeneous within the material. The lens has a variable curvature that gives it its desired behavior. It is possible to manufacture lenses whose index of refraction vary continuously within the material. Such a design is commonly referred to as a gradient–index (GRIN) lens [16]. While a conventional lens is not a suitable candidate for antenna applications due to its curved geometry, a flat GRIN lens has a low–profile, is light–weight and can be easily implemented in proximity to the feed; making it an ideal candidate for antenna applications. Lenses improve the performances of antennas by achieving wide angle scanning capabilities, beamforming and enhancing the overall gain. Besides, lenses generally exhibit broadband behaviors that are not commonly associated with conventional antenna arrays. Recently, there has been significant development in flat lens antennas based on field transformation [17–19], transformation optics [20–22], ray optics [23] and transmit–array approaches [24]. Lenses with radially varying refractive indices are challenging to fabricate. For ease of fabrication, the lenses are often comprised of several zones with different electromagnetic properties in each individual zone. However, it is difficult to source the materials commercially off–the–shelf (COTS) that possess the required EM properties. Another method to produce materials with tailored EM properties is perforating a homogeneous slab with holes of variable diameters or separations 19 with variable distances [25]. Precise machining and tight tolerance control are required for this approach, particularly for high frequency applications. Sometimes the maximum number of holes needs to be limited to prevent the material from physically cracking/breaking apart. On the other hand, there are a number of methodologies which have been proposed for tailoring the EM properties by using synthetic materials [26–30]. The main approach to fabricating these materials for flat lenses has been based on metamaterials [31–34]. However, some of the metamaterial lenses suffer from narrow bandwidth, losses and dispersion, and the fabrication processes of the metamaterials (MTM) based lenses are complicated. In this chapter, we discuss different ways to mitigate some of the problems encountered with MTMs, and present strategies for artificially synthesizing dielectric materials that are broadband as well as low–loss; hence, they are useful for real– world antenna applications involving low–profile flat lenses and reflectarrays, for example. The key to circumventing the difficulties with MTM, which we have identified above, is to steer clear of the common practice of using resonant inclusions or “particles” to achieve extreme material properties, such as 1; 1; negative index; and, zero index. Our strategy is to develop antenna designs that only call for material parameters that are realistic, so that they can either be acquired off–the– shelf, or by slightly tweaking the available materials by embedding small patches or apertures, often referred to as “particles”, whose dimensions are far removed from the resonance range. This obviates the problems of dispersion, narrow bandwidths and losses that plague the MTMs, at least those that fall in the “exotic” category, e.g., the double–negative or DNG type. Although the ray optics approach leads to dielectric–only designs that do not need to use magnetic materials, these designs still typically require dielectric materials that may not be available off–the–shelf. In the following sections, we present several examples to demonstrate the procedure for synthesizing artificial dielectrics for both single–layer and multilayer types–the latter to achieve better performance control including matching. After the procedure for synthesizing artificial dielectrics is presented, we design different lenses and fabricate them using a combination of conventional printed circuit board (PCB) and 3D–printing techniques. The first of these lenses was designed and fabricated using 3D–printing. The second one, a Dial–a–Dielectric (DaD) lens, which used state–of–the–art artificially engineered dielectric materials, was fabricated using 3D–printing and PCB laminate with patches on both sides. The 20 third lens is designed and fabricated in a similar manner to the first lens but with 3D–printing material of higher permittivity. All three designs circumvented the difficulties in finding desired COTS materials for 3D–printed lenses. The lenses comprise of several concentric dielectric rings with bespoke relative permittivities for transforming spherical waves into plane waves and vice versa. These lenses are low–cost and light–weight, but exhibit broadband and high gain performance. Measurement results show that all the lenses show comparable results to simulated designs over the designed frequency band of 12–18 GHz. 3.2 Ray-Optics Lens Design We use the Ray–Optics (RO) methodology to design the lens in the form of concentric rings, with variable permittivity values, as shown in Fig. 3.1. The flat GRIN lens is in the form of a disk and its focal point is located at its axis of symmetry. The refractive index should be a maximum at the center of the lens and gradually decrease to the outermost region [35]. The required variation in the refractive index can be calculated by equating the phase delay of the different rays from the focal point until the point where they exit the lens. It is assumed that the angle between the path where the ray enters the lens and the lens axis is θ (see Fig. 3.1). The relationship between the radially varied r and θ is given in the below equation (3.1), and this becomes the fundamental design equation for the flat GRIN lens. q √ T 2 T (r − sin2 θ) = (r − sin2 θ)( rmax − sec θ + 1) (3.1) F 3 F where T is the thickness of the lens, F is the focal length, rmax is the maximum permittivity at the center of the lens. The design can be simplified by calculating r for several given values of θ and then producing a smooth curve through the plotted points. For practical fabrication, it is considered reasonable to approximate the ideal smooth variation with a step function. Thus the lens can be fabricated from a series of concentric dielectric cylindrical rings with different relative permittivities. The corresponding r values for each ring are designed to ensure that they have the same focal point O in order to convert the spherical wave emanating from the focal point into a plane wave. 21 Figure 3.1: Sketch of lens design principle. Before equation (3.1) can be applied to the lens design, it is necessary to know that the maximum value of θmax is determined by the diameter of the lens (D) and the focal length (F ). In addition, knowing the minimum value of rmin that can be produced by 3D–printing is equally essential. Structural considerations limit the minimum value of r due to the resolution constraints of the 3D–printer and potential damage to structural integrity like cracking. Having determined the values of θmax and rmin , the equation (3.1) was used to find the lens thickness T , hence the r variation. The equation (3.1) can also be applied to find the focal length F for each cylindrical ring. For completeness, we include the dielectric parameters of most commonly available COTS materials in Table 3.1. A quick check shows that many of the desired materials that are derived by using Equation 3.1 are not commercially available from vendors such as Rogers Corporation. 3.3 3D–Printing Technique 3D–printing is an additive manufacturing technique which creates 3D–objects in successive layers. It provides a practical fabrication approach to produce highly customizable structures with the advantages of low–cost and fast, automated repeatable design and manufacturing. The 3D–printing process allows for the 22 Table 3.1: COTS dielectric materials. 1.96 2.17 2.2 2.33 2.5 2.75 2.94 3 3.02 3.2 3.27 3.55 3.6 3.66 4.5 4.7 6 6.15 9.2 9.8 10.2 creation of embedded sub–millimeter internal structures, such as air voids within the 3D–object, in a single process without machining. Compared with perforating a solid material, the design can be easily modified and rapidly prototyped in–house by using low–cost 3D–printing materials, and this is particularly useful for building laboratory prototypes such as lenses as in our case. In this work, a fused deposition modelling (FDM) Makerbot® Replicator™ 2X 3D–printer is used to fabricate the lenses utilizing the Thermoplastic Polylactic Acid (PLA) as well as Acrylonitrile Butadiene Styrene(ABS) based 3D–printing materials. PLA material has r =2.72 with tan(δ)=0.008 whereas ABS material (PREPERM® TP20280) has r =4.4 with tan(δ)=0.004. The heated printer nozzle extrudes the printing material and creates the lens layer–by–layer from the bottom up. This process enables the fabrication of embedded micron–scale particles such as air voids in a single process without machining while keeping the wastage and cost on a low and acceptable level. Although 3D–printing technology is becoming widespread and affordable day– by–day, current 3D–printers can only work with certain materials within a limited range of material parameters. Hence, this current work can be helpful for designing dielectric materials which cannot be directly handled in or realized/produced by 3D– printers. Typically, it is difficult to realize dielectric materials with permittivities that are higher than what is available from 3D–printing materials. In the next section, we explore some of the techniques that we utilize to design artificially engineered materials. 23 3.4 Design of Artificially Engineered Materials RO lenses are realized by using different dielectric materials as explained earlier in Sec. 3.2 where we have pointed out that not all the requisite materials are available commercially, consequently we need to design such artificially engineered materials. Once we have determined the desired dielectric parameters by following the design strategy discussed in Sec. 3.2, we encounter three different possibilities: (i) The desired permittivity value is the same as that of those commercially available. In this scenario, we move forward and use the available COTS material. (ii) The desired dielectric constant is higher than that of the available COTS material. In this case we use the methods presented in Sec. 3.4.1 or 3.4.2. (iii) The desired permittivity value is lower than that of the available COTS material/3D–printing material. In this case we follow the strategies laid out in Sec. 3.4.3 or Sec. 3.4.4 It is important to note that the above methods can be sometimes combined with each other to obtain the desired dielectric permittivity. We have also observed that stacking multiple layers provides considerable flexibility for achieving higher dielectric permittivity. 3.4.1 Designing higher permittivity materials from low permittivity COTS material : Method-1 In this section, we present the technique for engineering the COTS (commercial offthe-shelf) materials to realize the dielectric parameters we desire by implementing the “dial–a–dielectric (DaD)” scheme. In the DaD technique, to tweak the COTS materials, we use square patches (other shapes can be used as well), arranged in a circular/cartesian pattern as shown in Fig. 3.2, and print them on top (bottom or both on top and bottom) of the dielectric rings to realize the desired r values. Alternatively, we can print them on a mylar sheet and then place the sheet above the rings. To carry out the simulation, the concerned ring is discretized in the unit cells of appropriate periodicity (b). The periodicity is determined on the basis of 24 phase value needed to compensate across the unit cell. In our tests, we found that the unit cell with periodicity around λ/10 provides satisfactory results. We use unit cell of COTS material and patch combination (see Fig. 3.3) to realize the artificial dielectric. We start this process by placing a patch with a very small side dimension on the COTS material layer. Phase of S21 for COTS material covered with small patch will be close to COTS–only material layer. After confirming this behavior, we increase patch dimensions. The dimensions of the patches are chosen such that the phase of S21 of a dielectric–only layer, if available would match the S21 of the COTS materials covered by the patch. Since the incremental change is relatively small, the patch-size needed to accomplish this phase shift behavior is such that it is far from its resonance, and this is the key to realizing a wideband low–loss design. 3.4.2 Designing higher permittivity materials from low permittivity COTS material : Method-2 As mentioned in the previous section, there is a limit to the dielectric permittivity value we can achieve by following the approach presented therein. If we find that the required patch size becomes comparable to the local periodicity of the unit cell, insertion loss of the composite material becomes too high to be acceptable. In that scenario, we can modify the above approach as we will now explain. We use two dielectric blocks with one of the blocks of higher permittivity value and the other block of lower permittivity value and place the patch in the middle of the stack as shown in Fig. 3.4. Since commercially available dielectric materials only come with pre–set thicknesses, we can use the patches to fine–tune the effective dielectric constant of the stack. The size of the patch introduces an additional degree of freedom and thus versatility for tuning the permittivity of the unit cell as compared to using just two layers of dielectric blocks. 3.4.3 Designing lower permittivity materials from high permittivity COTS material It is not uncommon to find that the desired value of permittivity is lower than that of the available COTS material and is closest to the desired one. In this event, we 25 (a) Top view. (b) Cross-sectional view. Figure 3.2: Proposed DaD Lens. can use the following approach. Take two COTS dielectrics, one with a lower and the other with a higher permittivity value than the desired one. Stack them and adjust the height of both the dielectric materials (see Fig. 3.5) until the desired phase of S21 is realized. 26 Figure 3.3: Unit cell for designing higher permittivity materials from low permittivity COTS material 3.4.4 Designing lower permittivity materials from high permittivity 3D–printing material The methods discussed in the previous sections, all have conductive patches as structural components and can be easily implemented by using traditional PCB techniques in which metal patches can be printed on the laminate. But most of these PCB laminates have pre–set thicknesses and permittivity values and thus provide limited flexibility to fine–tune the material permittivity. In such cases where the above mentioned PCB technique does not work, we can use 3D printing materials alone to realize the desired value of permittivity. To accomplish this, we insert an air void in the host 3D–printing material which reduces the effective dielectric permittivity of the 3D–printing material. An example of a unit cell of such design is shown in Fig. 3.6. The effective relative permittivity (ref f ) of a 3D–printed unit cell with internal air void volume is given by: 27 Figure 3.4: Unit cell for designing higher permittivity materials from low permittivity COTS materials. ref f = νro + (1 − ν) (3.2) where ro is the relative permittivity of the 100% 3D–printing material and ν is the volume percentage indicated by the ratio of the volume of 3D–printing material to the volume of the entire unit cell. 3.5 Different Lens Designs By using the appropriate techniques for designing artificially engineered materials discussed in the previous section, we present three separate designs of three flat graded–index lens in this section. The following lenses are designed to demonstrate the validity of the design principles of artificially engineered materials: 1. PLA Lens: a 3D–printed lens fabricated by using PLA as the 3D–printing material. In this design, we apply the design methodology where the maximum 28 Figure 3.5: Unit cell for designing lower permittivity materials from high permittivity COTS material. permittivity needed in the lens is less than or equal to the permittivity of the basic 3D–printing material. It is important to note that r =2.72 is the maximum value of r that can be created by using 3D–printing technique with PLA 3D–printing material of r =2.72 using Sec. 3.4.4 exclusively. A more detailed explanation of this lens design is included in Sec. 3.5.1. 2. DaD Lens: a lens design that utilizes the DaD technique. We use PLA as 3D–printing materials and Rogers RO4350B laminate for printing patches on it. This design circumvents the design limitations of the PLA lens. We use the DaD technique in combination with 3D–printing technique to achieve the maximum permittivity of the lens, which is higher than that provided by the PLA 3D–printing material alone. A more detailed explanation of this lens design is included in Sec. 3.5.2. 3. ABS Lens: a 3D–printed lens designed by using ABS as the 3D–printing material. While there was no need for this lens design since the maximum 29 Figure 3.6: Unit cell for designing lower permittivity materials from high permittivity 3D printing material. permittivity value required is lower than that of ABS (r =4.4) and the design can be carried out using the same principles for the PLA lens, we included the design of this lens to compare its performance with that of the DaD lens to show the efficacy of the DaD lens design technique. A more detailed explanation of this lens design is included in Sec. 3.5.3. 3.5.1 PLA Lens Design In this section, we explain in detail the lens design techniques that will be applied to designing the PLA as well as the ABS based lenses. We introduce 3D–printed lens design, its fabrication and measurement of its response to compare it with the simulated design. The flat PLA GRIN lens is in the shape of a cylindrical disk and its focal point is located on the axis of symmetry (see Fig. 3.1). The lens is divided into 6 discrete rings in the radial direction with the width of each ring to be 10mm. The dielectric 30 constants calculated for these rings satisfy the path length condition. The effective permittivity for the outermost ring is calculated to be 1.3, while the centre ring has the highest effective permittivity of 2.72 as shown in Table 3.2 following the principles explained in Sec. 3.2. The derived values of dielectric constant for each ring are designed to ensure that the waves traveling through each ring collimate at the same focal point. Hence, the lens would convert the plane waves entering from one side of the lens to focus at the focal point on the opposite side of the lens. Table 3.2: Design parameters for the PLA lens. r1 2.72 r2 2.60 r3 2.38 r4 2.08 r5 1.71 r6 1.30 The final design parameters are shown in Table 3.3. Fig. 3.7 shows the step function approximation of the ref f versus radial distance across the diameter of the lens. In order to obtain the bespoke ref f , the air volume fraction is gradually increased from the center to the outermost region by decreasing the PLA volume fraction. Increasing the number of rings for a smoother permittivity variation would improve the accuracy of the focal point and increase the gain of the lens. However, narrow rings with small variations of volume fractions are difficult to fabricate accurately due to the resolution of the printer that we are using. Table 3.3: Designed parameters of the 3D–printed lens. Parameter Diameter Focal Length Thickness Value D = 120mm F = 150mm T = 18.5mm A full–wave simulation using CST was carried out to verify the performance of the lens design. In order to simplify the simulation process and reduce the computational complexity, the PLA lens was modeled as six solid concentric cylindrical rings and the internal voids were not considered. Each ring had homogenous dielectric constant and the values of the r follow exactly the step function from the centre to the outermost ring as shown in Fig. 3.7. The dielectric loss tangents for all the rings were set to be the loss tangent of 100% PLA material for simplicity (tan δ = 0.008) in the numerical simulation. Generally, lower PLA volume percentages 31 Figure 3.7: Effective relative permittivity for the 3D–printed lens as a function of its radius. resulted in lower loss tangent values and the correlation of the two was approximately linear [36]. In reality the 3D–printed rings would have lower loss tangent values than the simulation, but since PLA is a low–loss material, the difference in performance is negligible. A Ku–band conical feed horn was placed at the focal point which is 150 mm away from the lens. The simulated electric field of the lens antenna at 15 GHz is shown in Fig. 3.8. It clearly shows that the spherical wavefronts generated from the horn are converted into a planar wavefront in the near–field region of the flat GRIN lens, and the lens in turn will have a highly directive radiation pattern in the far–field region. It is also evident from Fig. 3.8 that not all the energy radiated from the feed antenna is captured by the lens. The middle of the lens has the highest field amplitude, which tapers out in the radial direction. The simulated far–field directivity patterns are shown in Fig. 3.9. A high– directivity beam in the far–field pattern in the z–direction was observed. The gain of the lens antenna and the feed horn composite was 18.0, 21.4 and 24.0 dBi at 12, 15 and 18 GHz respectively. After achieving the desired simulated results, the lens was fabricated as discussed in the next section. 32 Figure 3.8: Simulated electric field of the GRIN lens at 15 GHz when fed by a circular Ku–band horn located at the focal point. 3.5.1.1 Lens Fabrication In this section, thermoplastic polylactic acid (PLA) was used as the print material. The Nicolson–Ross and Weir (NRW) method [37] was used to measure the ref f of the 3D–printed samples with different internal air void volumes. The PLA volume percentage (ν) indicated the ratio of the volume of PLA in the printed structure to the volume of the whole structure. Measurement results by using the NRW method indicated that the 100% solid/homogeneous PLA sample had a relative permittivity of 2.72 and the relative permittivity was reduced to 1.3 with a 18% PLA volume percentage. The measured relative permittivity of 3D–printed samples with different PLA volume percentages ν showed that the relationship between r and ν was approximately linear. The expression for the required PLA volume percentage ν for tailoring the effective permittivity ref f of the 3D–printed dielectrics was extrapolated and is given by equation (3.3) : 33 (a) (b) Figure 3.9: Simulated far–field gain pattern of the thick flat GRIN lens at 12, 15 and 18 GHz for: (a) E-plane; and (b) H-plane. 34 ν= ref f − 1 ro − 1 (3.3) where ro is the relative permittivity of the 100% PLA 3D–printing material (ro = 2.72). The minimum rmin is limited by the 3D–printer resolution and the total volume of the structure. Generally, high resolution printers with small diameter nozzles or structures with larger total volumes can achieve a lower ref f . This equation is used for tailoring the effective permittivity of the 3D–printed GRIN lens. The size of the lens is limited by the maximum printing volume (length × width × height) of the 3D-printer (24.6cm × 15.2cm × 15.5cm). A larger lens could be realised by: i) using a 3D-printer which has a larger printing volume; or ii) printing multiple parts of the lens (for instance, four quarter circles) and then assembling them together. A fused deposition modelling (FDM) Makerbot® Replicator™ 2X 3D–printer was used to fabricate the flat lens utilizing the PLA as the print material. The heated printer nozzle extruded the PLA material and created the lens layer–by– layer from the bottom up. This process enabled the fabrication of embedded micron–scale particles such as air voids in a single process without machining to reduce the wastage and to lower the fabrication cost. A previous research [36] has demonstrated the viability of this approach for rapidly fabricating dielectric materials with different relative permittivities by using 3D–printing. By introducing air voids into the host materials (PLA in this case), the effective permittivity (ref f ) of the mixture was determined by the volume fraction of the host material relative to air. The lens geometry was designed by using computer aided design (CAD) tools for locally changing the infill percentage to tailor the permittivities accordingly. The 3D–printed flat lens with six different PLA volume percentages ν in the concentric cylindrical rings is shown in Fig. 3.10. The matching layer was not introduced in order to minimize the thickness of the lens. Equation (3.3) was used to determine the required ν for each bespoke r . It is worth noting that conventional desktop FDM 3D–printers use “infill density (or infill percentage)” to describe the volume fraction of the thermoplastic to the total volume but excluding the exterior walls of the 3D–object. Generally the minimum wall thickness is equal to the nozzle diameter of the 3D-printer. Therefore, the “infill density” for the 3D–printing CAD 35 software should always be smaller than the PLA volume percentages ν derived from equation (3.3). In this work, the infill density d for each cylindrical ring can be determined by using equation (3.4) and the results for each ring are shown in Table 3.4. Figure 3.10: 3D-printed flat PLA lens. d= νπ(R2 − r2 ) − 2πt(R + r) π(R2 − r2 ) − 2πt(R + r) (3.4) where ν is the PLA volume percentage obtained from equation 3.3 and t is the exterior wall thickness. R and r are the exterior and interior radius of the cylindrical ring respectively and they include the wall thickness t. The Replicator 2X 3D– printer had a minimum wall thickness of 0.4 mm. Table 3.4: Parameters of 3D–printed concentric dielectric rings. Ring No. 1 2 3 4 5 6 ref f 2.72 2.60 2.38 2.08 1.71 1.30 36 ν d 100% 100% 93% 90.8% 80.2% 77.2% 62.8% 58.0% 41.3% 35.1% 18% 10.1% 3.5.1.2 Lens Measurement A 180° azimuth–plane scan measurement was set up for measuring the performance of the 3D–printed lens. A Ku–band pyramid waveguide horn antenna, which was placed at a distance of 1.5m along the lens axis, served as the receiving antenna. This receiving horn was continuously moved from theta = –90° to theta = +90° for azimuth–plane scanning. The sketch of the azimuth scanning system is shown in Fig. 3.11. Note that the 1.5 m distance was slightly shorter than the far–field distance requirement (2D2 /λ) of the lens antenna below 15.6 GHz due to the limited indoor space available in the measuring chamber. A conical feed horn (open end diameter of 22.75 mm) was used as the source for generating spherical wavefronts (see Fig. 3.12a). The 3D–printed lens was held by a foam and was perpendicular to the azimuth–plane. The feed horn was mounted on a slider at the focal point of the lens axis, which was 150 mm away from the 3D–printed lens. The feed horn can be moved along both directions in the azimuth–plane by rotating the knobs for examining the performance off of focal point. The slider with the mounted feed horn is shown in Fig. 3.12b. 3.5.1.3 Results The first step was to place the feed horn on the lens axis with different feed distances to find the optimal focal length. The distance z indicated different distances between the feed horn and the lens surface and it was varied from 130 mm to 170 mm. Fig. 3.13 shows the measured broadband gain results of the lens antenna at boresight (theta = 0°), compared with the simulated gain with the feed horn placed 150 mm away. The gain of the 3D–printed lens antenna increased with frequency as was also found in the simulation results. The highest gain was observed when the feed horn was placed 150 mm away from the lens surface which matched the designed focal length. The lens antenna with far feeding (160 mm and 170 mm) had slightly higher gain compared to the close feeding cases (130 mm and 140 mm), particularly when the frequency was higher than 15 GHz. Moreover, 130 mm feeding had a lower gain compared with 140 mm feeding above 16.2 GHz. However, the measurement results indicated that the small amount of feed distance shifting had insignificant impact on the gain of this 3D–printed GRIN lens. The simulated gain varied from 18 to 24 dBi over the entire 12 to 18 GHz range, whilst the measured gain ranged 37 Figure 3.11: Sketch of the measurement setup for the 3D–printed lens antenna. from approximately 16 to 24 dBi. The conical horn had a gain from 7 to 13 dBi from 12 to 18 GHz; therefore, the lens provided a gain increase of 9 to 11 dB over the frequency of the Ku-band. The difference between simulated and measured gain values was due to the aberrations of the 3D–printed lens. The aberration was mainly due to fabrication tolerance of the infill density which was limited by the resolution of the printer. The interfaces between adjacent rings could also introduce differences between simulated and measured results. The measured gain patterns in the E– and H–plane of the lens, at the frequency of 12, 15 and 18 GHz, are shown in Fig. 3.14, and good agreement can be observed 38 (a) (b) Figure 3.12: Measurement setup with the 3D–printed lens with a conical feed horn at the focal point of the lens. 39 Figure 3.13: Measured broadband gain of 3D-printed lens with different feeding distances. with the simulated patterns. The radiation patterns had a higher directive main beam at higher frequencies. The secondary side–lobes were due to the feed source. At 15 GHz the half–power beamwidth in the H-plane was approximately 11°, and 9.5° at 18 GHz. The peak of the main beam was approximately 13.8 dB higher than the first side lobe level at the center frequency of 15 GHz. The measured aperture efficiency of the fabricated lens was 18% at 12GHz, 33% at 15GHz and 41% at 18GHz. Moving the feed horn off the lens axis resulted in a shift in the direction of the main beam and an increase of the sidelobe levels. Fig. 3.15 shows the measured results of the off–axis experiment at 15 GHz. The feed horn was moved by a distance x in the azimuth–plane. When the feed horn was moved one wavelength (20 mm at 15 GHz) away from the lens axis, an 8° shift of the main lobe was observed on the opposite side of the axis. The side–lobe level increased by approximately 1.5 dB at the same side of the feed horn and reduced by the same amount on the other side of the axis. The main beam was moved 11.5° off–axis after the feed horn was moved 3 cm away from the lens axis. The side–lobe level was increased by 0.8 dB 40 (a) (b) Figure 3.14: Measured and simulated far–field patterns of the GRIN lens at for: (a) E-plane; and (b) H-plane. 41 at the feed horn side. Therefore, illuminating the lens with multiple feeds around the lens axis could achieve greater beam scanning coverage. Figure 3.15: Measured patterns of 3D-printed lens with the horn positioned at different off axis distances at 15 GHz. 3.5.2 DaD Lens Design In this section, we demonstrate the design process of the DaD lens. This new design will have ring dielectric values as shown in Table 3.5 and structural parameters as shown in Table 3.6. We see that the highest dielectric constant of this lens is higher than that provided by PLA 3D–printing material, whose r was 2.72. Table 3.5: Material Parameters of the DaD Lens. 1 3.46 2 3.25 3 2.90 4 2.41 42 5 1.84 6 1.24 Table 3.6: Designed parameters of the DaD lens. Parameter Diameter Focal Length Thickness Value D = 120mm F = 150mm T = 13.08mm A quick search of available materials reveals that not all of these materials are commercially available, off-the-shelf for this lens. Design methodology discussed in Sec. 3.5.1 for PLA lens also can not be applied for the DaD lens as r > 2.72 for the materials of some of the rings. Consequently, we use the Dial-a-Dielectric (DaD) approach, in conjunction with the 3D-printing technique to realize these materials. For this design, we use the PLA 3D–material in combination with Rogers RO4350B (r =3.48 with thickness of 1.52mm) laminate. If the permittivity needed for a particular ring exceeds 2.72, we design the material by using a combination of the DaD approach and the 3D–printing. We see in Table 3.5 that this lens requires higher permittivity values for rings 1–3 and lower values for rings 4–6 than those provided by PLA (r = 2.72). Thus we use the DaD technique for rings 1–3 and pure dielectrics for rings 4–6 which can be designed by using the PLA infill method as explained earlier in Sec. 3.4.4. In the DaD approach we modify the permittivities of the COTS materials, by covering these materials with metallic patches while in the 3D–printing technique, we insert air voids in the 3D–printing material to realize the required permittivity as mentioned in Sec. 3.4. We have found that neither the DaD nor the 3D–printing approaches lead to material realizations that are lossy, dispersive and narrowband, as they would be if we had used resonant metamaterials instead. To tweak the material parameter of the COTS, we use square patches that are distributed in a Cartesian pattern as shown in Fig. 3.2, and print them on both sides of the PCB laminate and place it on the PLA material for the inner 1–3 rings to realize the desired r values. Unlike the previously published synthesis techniques, the presented approach does not rely on resonance properties of patches or apertures to realize the artificial dielectrics; hence it circumvents the problem of losses and narrow bandwidths suffered by metamaterials. For the current DaD unit cell design, we choose the periodicity to be 2mm×2mm and use the patch–dielectrics composite unit cell shown in Fig. 3.16. Design 43 Figure 3.16: Unit cell used for the DaD lens design. parameters for the unit cell are chosen to be t1 =1.52mm, t2 =11.56mm, b=2mm, r1 =3.48, r2 =2.72 and the patch size, a, is variable depending on the desired permittivity value. The dimensions of the patches are chosen such that the phase of S21 of a purely dielectric layer matches that of the COTS materials with patches. Since the incremental change is relatively small, the patch size required to accomplish this is such that it is far from the resonance range, and this is the key to realizing a wideband low–loss design. The full wave simulations are conducted in HFSS for the DaD lens. In Figs. 3.17, 3.18 and 3.19, we show the S21 behavior of this composite unit cell when compared with pure dielectric unit cell for different rings. From these figures, we see that we are able to achieve the comparable S21 phase values for rings 1–3 which means that we are able to achieve the required dielectric values for this patch–dielectrics composite unit cell. We also note that this composite unit cell shows lower transmission coefficient as compared to purely dielectric unit cell. This is due to the large size of the patches that we used in these simulations. We can increase the transmission coefficient by decreasing the size of the patches but we ended up using the large–sized patches as we did not have the facility for 44 printing smaller patches on the PCB laminates. Each dielectric ring is 10mm wide. Each patches–dielectrics composite block is of size 2mm × 2mm so each dielectric ring can fit approximately 5 of such patch–dielectrics blocks in the radial direction. The height of dielectric ring is 13.08mm. Table 3.7 shows the required size of the patches for rings 1–3 where the DaD technique is used. We also show the infill density of 3D–printing material for different rings in Table 3.7. Table 3.7: Patch sizes and PLA volume density for different rings of the DaD Lens. Ring No. 1 2 3 4 5 6 3.5.2.1 a d 1.72mm × 1.72mm 100% 1.51mm × 1.51mm 100% 0.74mm × 0.74mm 100% N/A 84% N/A 52% N/A 17% Lens Fabrication We use the DaD technique only for rings 1–3 as explained earlier and Rogers RO4350B laminate with thickness of 1.52mm is used. The laminate is cut into the same circular shape and patches are printed on both sides as shown in Fig. 3.20. Then we 3D–print the lower part of the DaD lens and cover the patch–etched laminate on the 3D–printed material. 3D–printing technique used for DaD lens is similar to PLA lens and only the height and volume density are different than that of the PLA lens as listed in Table 3.7. 3.5.2.2 Results We use similar measurement methodology for the DaD lens as was used for the PLA lens discussed in Sec. 3.5.1.2. The simulated as well as measured far–field patterns in the H– and E–planes of DaD lens for the lowest operating frequency (12GHz), the center frequency (15GHz) and highest operating frequency (18GHz) are shown in Figs. 3.21, 3.22 and 3.23, respectively. Both simulated and measured results show comparable behavior. 45 0 Uniform dielectric unit cell (εr=3.46) DaD unit cell (εr=3.46) −0.5 S21 (dB) −1 −1.5 −2 −2.5 −3 12 13 14 15 Frequency (GHz) 16 17 18 (a) S21 Magnitude. 20 Uniform dielectric unit cell (εr=3.46) 0 DaD unit cell (εr=3.46) −20 −40 S21 (°) −60 −80 −100 −120 −140 −160 −180 12 13 14 15 Frequency (GHz) 16 17 (b) S21 Phase. Figure 3.17: S21 parameter for unit cell of ring 1. 46 18 0 Uniform dielectric unit cell (ε =3.25) r DaD unit cell (ε =3.25) r −0.5 S21 (dB) −1 −1.5 −2 −2.5 −3 12 13 14 15 Frequency (GHz) 16 17 18 (a) S21 Magnitude. 40 Uniform dielectric unit cell (εr=3.25) 20 DaD unit cell (εr=3.25) 0 −20 S21 (°) −40 −60 −80 −100 −120 −140 −160 12 13 14 15 Frequency (GHz) 16 17 (b) S21 Phase. Figure 3.18: S21 parameter for unit cell of ring 2. 47 18 0 Uniform dielectric unit cell (εr=2.90) −0.2 DaD unit cell (εr=2.90) −0.4 S21 (dB) −0.6 −0.8 −1 −1.2 −1.4 −1.6 −1.8 −2 12 13 14 15 Frequency (GHz) 16 17 18 (a) S21 Magnitude. 60 Uniform dielectric unit cell (εr=2.90) 40 DaD unit cell (εr=2.90) 20 S21 (°) 0 −20 −40 −60 −80 −100 −120 12 13 14 15 Frequency (GHz) 16 17 (b) S21 Phase. Figure 3.19: S21 parameter for unit cell of ring 3. 48 18 (a) PCB printed part of the DaD lens. (b) 3D–printed part of the DaD lens. (c) Assembled DaD lens. Figure 3.20: Photographs of the DaD Lens (see Fig. 3.2 for all dimensions). 49 In Fig. 3.24, we show the gain behavior of the simulated and fabricated DaD lens. It can be observed that the DaD lens shows broadband behavior and its maximum gain increases with frequency as expected. 3.5.3 ABS Lens Design For this lens design, we use the 3D–printing technique to design a lens with ABS as printing material. ABS material (PREPERM® TP20280) has r =4.4 with tan(δ)=0.004. The design parameters of the ABS lens are the same as the DaD lens and are shown again in Tables 3.8 and 3.9 for completeness. The main reason for using the DaD technique for the lens design in the previous section was due to the lack of dielectric material with r >2.72 for 3D–printing. But if we have 3D–printing material with r >2.72 as ABS material does provide, we can directly use 3D–printing for such a lens design. We follow this idea and 3D–printed the whole lens in a single step. Table 3.8: Material Parameters of the DaD Lens. 1 3.46 2 3.25 3 2.90 4 2.41 5 1.84 6 1.24 Table 3.9: Designed parameters of the DaD lens. Parameter Diameter Focal Length Thickness Value D = 120mm F = 150mm T = 13.08mm The ABS lens design methodology is similar to the PLA lens design and we use 3D–printing for both designs, readers can refer to Sec. 3.5.1.1 for more details. The infill density for ABS lens is shown in Table 3.10 3.5.3.1 Results A boresight measurement was set up for the measurements of the performance of the ABS lens. A Ku–band horn was used as the source to generate spherical wavefronts. The source was placed at the focal point of the lens axis, which was 50 0 Simulated: DaD Lens Measured: DaD Lens −5 −10 Gain (dB) −15 −20 −25 −30 −35 −40 −45 −50 −80 −60 −40 −20 0 θ (°) 20 40 60 80 20 40 60 80 (a) H-plane. 0 Simulated: DaD Lens Measured: DaD Lens −5 −10 Gain (dB) −15 −20 −25 −30 −35 −40 −45 −50 −80 −60 −40 −20 0 θ (°) (b) E-plane. Figure 3.21: Far-field radiation patterns of DaD Lens at 12 GHz. 51 0 Simulated: DaD Lens Measured: DaD Lens Gain (dB) −10 −20 −30 −40 −50 −80 −60 −40 −20 0 θ (°) 20 40 60 80 20 40 60 80 (a) H-plane. 0 Simulated: DaD Lens Measured: DaD Lens Gain (dB) −10 −20 −30 −40 −50 −80 −60 −40 −20 0 θ (°) (b) E-plane. Figure 3.22: Far-field radiation patterns of DaD Lens at 15 GHz. 52 0 Simulated: DaD Lens Measured: DaD Lens Gain (dB) −10 −20 −30 −40 −50 −80 −60 −40 −20 0 θ (°) 20 40 60 80 20 40 60 80 (a) H-plane. 0 Simulated: DaD Lens Measured: DaD Lens Gain (dB) −10 −20 −30 −40 −50 −80 −60 −40 −20 0 θ (°) (b) E-plane. Figure 3.23: Far-field radiation patterns of DaD Lens at 18 GHz. 53 24 23 Simulated: DaD Lens Measured: DaD Lens Gain (dB) 22 21 20 19 18 17 16 12 13 14 15 16 Frequency (GHz) 17 18 Figure 3.24: Gain response of the DaD Lens. Table 3.10: Parameters of 3D–printed ABS lens dielectric rings. Ring No. 1 2 3 4 5 6 r 3.46 3.25 2.90 2.41 1.84 1.24 d 72% 66% 56% 41% 25% 7% 150mm away from the ABS lens. The far–field patterns of the lens are measured similar to as mentioned in Sec. 3.5.1.2. We use a conical feed horn (open-end diameter of 22.75 mm) as the feed source. This horn is placed at the focal point (O) and full wave simulation is conducted in HFSS. The simulated and measured far–field pattern in the H– as well as E–plane for the ABS lens are shown in Figs. 3.25, 3.26, 3.27, for 12, 15 and 18GHz, respectively. Fig. 3.28 shows the gain of ABS lens and we see that it shows broadband behavior. We also see that the gain increases with frequency. 54 3.5.4 Comparison of DaD and ABS Lenses It can be observed in the previous sections that the design parameters are the same for the DaD and the ABS lens but the design methodology and material properties required to realize these lenses are different. One uses DaD technique by incorporating traditional PCB and 3D–printing while the other uses 3D–printing alone. Results of both lenses are detailed in previous sections and are shown again in Figs. 3.29, 3.30 and 3.31 for different frequencies for comparison. We also show comparison of the gain response for both lenses in Fig. 3.32. From all these plots, we see that the results are comparable for both lenses. Due to the introduction of patches in the DaD approach, the DaD lens is more lossy as compared to ABS lens. We had also seen similar behavior when we studied the unit cell for the DaD lens as explained earlier. 3.6 Conclusions In this chapter we presented low–cost, light–weight and wideband 3D–printed/DaD flat lenses which can be rapidly prototyped and used for antenna applications (see Fig. 3.33). Two of these are fabricated by using the direct 3D–printing technique while the remaining one is fabricated by using a combination of 3D–printing and PCB–printing techniques. For all three lenses, the dielectric materials are not commercially available and they are realized by using the infill method as well as the Dial-a-Dielectric approach to achieve the desired permittivities for these lenses. The entire PLA lens designed by using PLA as the 3D–printing material is light–weight and the material cost for this lens is less than $15. The 12cm diameter lens is comprised of six concentric rings with different permittivity values. Air voids are created inside the unit cells of the lens during 3D–printing to reduce the permittivity of the 3D–printed material to bespoke values. The entire PLA lens is 3D–printed in a single process without the need for machining or assembling. The lens provides desirable gain over the broad frequency band ranging from 12 to 18 GHz when illuminated by a source feed located on the axis at the focal point. We also demonstrate the design, fabrication and measuring processes of the DaD and the ABS lenses. Both of these lenses have the same design parameters but are fabricated using different techniques. We see that both of these lenses 55 show comparable performance in terms of maximum gain, frequency response and radiation pattern. 56 0 −5 Simulated: 3D−printed Lens Measured: 3D−printed Lens −10 Gain (dB) −15 −20 −25 −30 −35 −40 −45 −50 −80 −60 −40 −20 0 θ (°) 20 40 60 80 20 40 60 80 (a) H-plane. 0 −5 Simulated: 3D−printed Lens Measured: 3D−printed Lens −10 Gain (dB) −15 −20 −25 −30 −35 −40 −45 −50 −80 −60 −40 −20 0 θ (°) (b) E-plane. Figure 3.25: Far-field radiation patterns of 3D–printed ABS Lens at 12 GHz. 57 0 −5 Simulated: 3D−printed Lens Measured: 3D−printed Lens −10 Gain (dB) −15 −20 −25 −30 −35 −40 −45 −50 −80 −60 −40 −20 0 θ (°) 20 40 60 80 20 40 60 80 (a) H-plane. 0 −5 Simulated: 3D−printed Lens Measured: 3D−printed Lens −10 Gain (dB) −15 −20 −25 −30 −35 −40 −45 −50 −80 −60 −40 −20 0 θ (°) (b) E-plane. Figure 3.26: Far-field radiation patterns of 3D–printed ABS Lens at 15 GHz. 58 0 −5 Simulated: 3D−printed Lens Measured: 3D−printed Lens −10 Gain (dB) −15 −20 −25 −30 −35 −40 −45 −50 −80 −60 −40 −20 0 θ (°) 20 40 60 80 20 40 60 80 (a) H-plane. 0 −5 Simulated: 3D−printed Lens Measured: 3D−printed Lens −10 Gain (dB) −15 −20 −25 −30 −35 −40 −45 −50 −80 −60 −40 −20 0 θ (°) (b) E-plane. Figure 3.27: Far-field radiation patterns of 3D–printed ABS Lens at 18 GHz. 59 25 24 Simulated: 3D−printed Lens Measured: 3D−printed Lens 23 Gain (dB) 22 21 20 19 18 17 16 12 13 14 15 16 Frequency (GHz) 17 18 Figure 3.28: Gain response of 3D–printed ABS Lens. 60 0 −5 Simulated: DaD Lens Measured: DaD Lens Simulated: 3D−printed Lens Measured: 3D−printed Lens −10 −15 Gain (dB) −20 −25 −30 −35 −40 −45 −50 −80 −60 −40 −20 0 θ (°) 20 40 60 80 20 40 60 80 (a) H-plane. 0 −5 Simulated: DaD Lens Measured: DaD Lens Simulated: 3D−printed Lens Measured: 3D−printed Lens −10 −15 Gain (dB) −20 −25 −30 −35 −40 −45 −50 −80 −60 −40 −20 0 θ (°) (b) E-plane. Figure 3.29: Far-field radiation patterns of DaD and 3D–printed ABS Lens at 12 GHz. 61 0 −5 Simulated: DaD Lens Measured: DaD Lens Simulated: 3D−printed Lens Measured: 3D−printed Lens −10 −15 Gain (dB) −20 −25 −30 −35 −40 −45 −50 −80 −60 −40 −20 0 θ (°) 20 40 60 80 20 40 60 80 (a) H-plane. 0 −5 Simulated: DaD Lens Measured: DaD Lens Simulated: 3D−printed Lens Measured: 3D−printed Lens −10 −15 Gain (dB) −20 −25 −30 −35 −40 −45 −50 −80 −60 −40 −20 0 θ (°) (b) E-plane. Figure 3.30: Far-field radiation patterns of DaD and 3D–printed ABS Lens at 15 GHz. 62 0 −5 Simulated: DaD Lens Measured: DaD Lens Simulated: 3D−printed Lens Measured: 3D−printed Lens −10 −15 Gain (dB) −20 −25 −30 −35 −40 −45 −50 −80 −60 −40 −20 0 θ (°) 20 40 60 80 20 40 60 80 (a) H-plane. 0 −5 Simulated: DaD Lens Measured: DaD Lens Simulated: 3D−printed Lens Measured: 3D−printed Lens −10 −15 Gain (dB) −20 −25 −30 −35 −40 −45 −50 −80 −60 −40 −20 0 θ (°) (b) E-plane. Figure 3.31: Far-field radiation patterns of DaD and 3D–printed ABS Lens at 18 GHz. 63 25 24 23 Simulated: DaD Lens Measured: DaD Lens Simulated: 3D−printed Lens Measured: 3D−printed Lens Gain (dB) 22 21 20 19 18 17 16 12 13 14 15 16 Frequency (GHz) 17 18 Figure 3.32: Gain response of DaD Lens and 3D–printed ABS Lens. 64 (a) Top View. (b) Isometric View. Figure 3.33: Photograph of the different lenses. 65 Chapter 4 | Uncertainty Management of Periodic Structures 4.1 Introduction Recently, there has been considerable interest in the topics of Metamaterials (MTMs) and Frequency Selective Surfaces (FSSs) [38–40]. The FSS elements act as spatial filters whose passbands and stopbands depend on the geometrical parameters of its elements. Normally, the FSS structure is simulated during the design process and then fabricated to verify if indeed it has the predicted characteristics. It is not unusual to find that there is considerable discrepancy between the simulated and measured results, even when there is only a minor difference between the designed and fabricated structures. This is especially true for Metamaterials used at optical wavelengths, where the difficulties in their fabrication almost always introduce small variations in the dimensions of the elements that comprise the “periodic” array. Typically, the effect of this type of variation in the unit cell parameters are studied by using the Monte Carlo (MC) methods. But if the variations are substantial, the MC method results in a highly computationally intensive problem. A meta-model (or a surrogate model), which was traditionally called a response surface, is intended to mimic the behavior of a computational model (e.g. an FDTD simulation in electromagnetics). In comparison to the original model, which might be very expensive in terms of computational costs (such as computer processing time), the meta-model is inexpensive to evaluate. The main emphasis of this chapter is to show how to use statistical methods to build a meta–model that is able to replace the computer–intensive numerical solver by analytical equations that are as simple as possible and are capable of providing results very quickly but also 66 with the same accuracy. This meta–model can be used to obtain the probability distribution of the quantity of interest (for instance reflection coefficient of FSS unit cell); it can also be used to perform a sensitivity analysis (SA) and characterize the statistical variations of the output when the inputs are varied. In this work, we use the polynomial chaos expansions (PCEs) [41] in preference to the commonly used MC method, to develop a meta-model of the physical model of the FSS by using a polynomial expansion approach. In the following sections, we first present the type of periodic elements we are considering for this study. Next, we demonstrate the method used to investigate the effects of the variations in the periodic elements. Finally, we include some numerical results to illustrate the application of the proposed method. 4.2 Structures with Uncertain Parameters In this section, we will discuss the example problems that we use to show the application of PCE for conducting statistical analysis. 4.2.1 Cross-dipole Periodic structures (see a typical periodic structure in Figure 4.1) are typically modeled as infinite arrays of scatterers, and are commonly analyzed by imposing periodic boundary conditions to a unit cell to reduce the original problem to a manageable size, and to reduce the computational burden. In this study, we use across–dipole [42]– [43] as the FSS element to illustrate the proposed method, which is applicable to an arbitrarily shaped geometry (see Figure 4.2). The cross-dipole has three parameters, namely (i) the periodicity Dx = Dy = D; (ii) length L; and (iii) width W , as shown in Figure 4.2. We assume that these parameters follow normal distributions, and we compute the reflection coefficient for this structure for different values of D, L and W (see Table 4.1). Table 4.1: Input Parameters. D L W Mean 45mm 30mm 6mm Standard Deviation 2.25mm 1.5mm 0.3mm 67 Figure 4.1: A typical periodic structure and its unit cell Figure 4.2: The input parameters of cross-dipole FSS element 4.2.2 3D–printed Lens We discussed the fabrication of lenses by using 3D–printing techniques in the previous chapter. In order to achieve the gradient–index material parameters, 3D–printing provides the capability of building the flat lens from a single material by introducing holes in the material with a single process. By changing the hole/material volume ratio, the desired refractive index or relative permittivity is obtained. Thus, the effective parameters obtained determine the performance of the 68 lens. However, as for all manufacturing processes, 3D–printing technology can only be performed with a limited level/degree of accuracy with respect to the design parameters, for instance the holes dimensions, thickness of the lens, etc. These variabilities are likely to influence the output response of the system. Therefore, key operating parameters of the lens antenna such as the focal length and the gain can be compromised. We will use the PCE analysis technique to assess the effect of such variability on output parameters of the lens. Moreover, PCE analysis allows to determine the effect of each design parameters on the lens output by means of so called Sobol’ indices besides the standard parameters associated with statistical analysis such as, probability density function (PDF), the mean and variance values. The relative contribution of each input parameters to the output of the model is assessed using the Sobol’ indices. The 3D printed lens is shown in Fig. 4.3. It comprises of 6 concentric rings with each of the effective relative permittivity (ref f ) values shown in Table 4.2. Figure 4.3: 3D printed flat lens and the cross section of a block of the outer ring with 4 holes. Table 4.2: Effective relative permittivity values of the 3D–printed lens. 1 2.72 2 2.60 3 2.38 4 2.08 69 5 1.71 6 1.30 4.3 The Monte Carlo method Monte Carlo is a method of generating samples from a random vector. It involves the sampling of the input parameters according to their distribution, and the evaluation of the model response for each realization. In a typical Monte Carlo simulation, the problem of clustering or scarcity sampling might arise in cases where the sample size is small. A large number of simulations are required to ensure a reliable coverage of the entire parameter range. This results in simulations with high computational costs. To circumvent the limitation of the MC method, an alternate way is to build a simpler and inexpensive model called meta–model (or surrogate model), which approximates the behavior of the computational model. 4.4 Surrogate Model Let us consider an electromagnetic problem or system that can be modeled by computational techniques such as the FEM or the FDTD method. This computational model can be represented as: y = M(x) (4.1) where x is a vector composed of the input parameters of the model (x ∈ D ⊂ Rm ) while vector y ∈ RQ is the model response. The computational model, M, is considered as a black box, which is only known point–by–point. The function M here may correspond to the result of an FDTD simulation for the reflection coefficient of the periodic unit cell. A particular set of inputs will produce exactly the same outputs no matter how many times this model is run. In other words, we can say that this model is purely deterministic and provides a unique response for a given set of unique inputs. The computational problem described above involves computer simulations which requires significant computational resources. Similar problems occur when the objective is to characterize the influence of input variations on the statistical distribution of the calculated outputs through simulations. One way to overcome such a limitation is to build simpler approximation models, known as surrogate 70 models, response surfaces or meta-models, which mimic the complex response of the model while at the same time, being inexpensive in terms of computational cost [44]. So, our objective is to build an approximation of the model, which we refer to herein as the surrogate model (see Figure 4.4) such as : ŷ = M̂(x) (4.2) Figure 4.4: Original model and surrogate model. The surrogate model, M̂, does not require the knowledge of the internal operations of the physical system, but only the input–output relationship of the system. Our main objective is to build a surrogate model M̂ using effective and optimum methods. The polynomial chaos expansion (PCE) – also known as “polynomial chaos” (PC) – is one of the methods which builds these meta-models. We will discuss more about PCE in the following sections. 4.5 Statistical Methods As explained, our purpose in this study is to substitute the physical model, which is computationally expensive to evaluate, by a meta-model that is much faster to work with. Once we have derived this meta-model, we can perform an MC analysis on this meta–model to study the propagation of the uncertainty of the input parameters in the physical model. 71 4.5.1 Polynomical Chaos The Polynomial Chaos (PC) is an advanced statistical method, which models the response of a random output variable Y depending on input random parameters X = [x1 · · · xm ], linked by a physical model Y = M(X) [45], where the input parameters are supposed to be independent. Under these conditions we can use the Polynomial Chaos method to construct a meta-model of the physical model by using the polynomial expansion: X Y = M(X) = yα ψα (X) (4.3) α∈Nm where α’s are the multi-indices, ψα ’s are the multivariate orthogonal polynomials and yα ’s are the unknown coefficients of the polynomial expansion. In (4.3), the ψα are multivariate orthogonal polynomials, which constitute a set of basis functions in the probabilistic space. This expansion was originally formulated for Gaussian random input variables by using the Hermite polynomials as the basis, and was termed as the finite-dimensional Wiener polynomial chaos. However, we note that it is possible to have the same formulation for other type of variables; for instance, for the case of uniform input variables spanning the range [-1,1], they are Legendre polynomials [46]. The orthogonal polynomials are chosen in accordance with the probability distributions of input variables following the so-called Askey scheme of polynomial [47]. Table 4.3 shows PDF with their related orthogonal polynomials and ranges for some classical orthogonal polynomials. Table 4.3: Orthogonal polynomials. PDF Distribution Orthogonal polynomials Uniform 1[−1,1] (x)/2 Legendre Pk (x) [-1,+1] Gaussian 2 √1 e−x /2 2π xa e−x 1R+ (x) Hermite Hek (x) (-∞,+∞ ) Laguerre Lak (x) [0,+∞) Gamma Beta 1[−1,1] (x) (1−x)a (1+x)b B(a) B(b) Jacobi Jka,b (x) Support Range [-1,+1] The yα coefficients in (4.3) are the unknowns that are yet to be determined. 72 4.5.2 Truncation The methods that use the PC expansion can be divided into two large categories. On the one hand, the intrusive methods requiring modification of the simulation code of the solver, and on the other hand, there are non-intrusive methods that treat solvers as black boxes. In our work we use non-intrusive method as we need not to know the internal operation of the physical model for this study and we approach complex system as a black box. In the family of non-intrusive methods, there are two approaches for evaluating these coefficients. The first one of these is the projection method [48], which is based on the orthogonality properties of the polynomials. In this approach, we project the expansion on the subspace associated with the coefficient we desire to compute. However, in practice, this approach, though elegant, demands a high computational cost. An alternate approach is to truncate the expansion to set up a regression problem as described in [49], and to use the least square method to compute the coefficients. In our case, we will employ the latter method, which is much less demanding from the point of view of computational cost. We note that the modal expansion defined in equation (4.3) involves an infinite number of polynomials. For practical implementation, a finite–dimensional PCE has to be built. Let us denote Ŷ a truncation of the generalized PCE as: X Ŷ = M̂(X) = yα ψα (X) (4.4) α∈A where A is a finite set of multi-indices of cardinality P . Considering multivariate polynomials, ψk , having M input variables and assuming that p is the total maximum degree of them. The size of this finite-dimensional basis is given as: M + p (M + p)! card A ≡ P = = M ! p! p (4.5) Depending on the input variables and on the maximal polynomial degree retained, the number of polynomials can be significant. However, since we have no information on the impact of the truncation on the quality of the model, this regression approach requires an iterative type of experimental procedure that uses the estimated quality or accuracy of the obtained 73 meta-model. For example, for a given design of experiments with a given number of simulation points, the design of experiment should be improved with new simulation points in order to have a better model accuracy if the quality of the obtained metamodel is not sufficient. In this study, we will use the Latin Hypercube Sampling (LHS) technique [50] to create the experimental designs that we will employ. To add new simulation points to these designs, we will use the Nested Latin Hypercube Sampling (NLHS) technique [45]. 4.5.2.1 Sparse PC: The Least Angle Regression Truncation There are several different types of techniques that we might employ for the truncation of this polynomial expansion presented in equation (4.1). Classically, in the regression technique, one uses the full PC truncation, in which one retains the polynomials {ψα , 0 ≤ |α|≤ p} for a chosen degree p. Thus, the number P of polynomials that are retained in the full-PC truncation depends on the degree p and therefore depends on the number of points in the experimental design, because of conditioning problems of the information matrix arising in the least square estimation. We observe that the number of polynomials increases with the dimensionality M of the input parameters and with the degree p (see equation (4.5)). This is referred to as the curse of dimensionality. Thus, if we want a high degree p, we would need to compute a large number of simulation points in the experimental design. In view of this, we use an optimized truncation, namely the LARS (Least Angle Regression) truncation as described in [51]. Originally, the LARS is a model selection method [52], which selects from the most influential variables in a large panel space. In this work, this technique is used to select the most influential polynomials, as shown in [51]. Our objective is to select these polynomials, from a large number of possible choices, such that they have the most significant impact on the meta-model and, hence, lead to a “sparse” representation of the Polynomial Chaos. The principle is to iteratively choose the polynomials depending on their correlation with the rest of the current model. To begin, we have to determine a large group of polynomials from which we will select the most important ones. In our approach, we choose a full truncation, with a high degree p, and we select most significant polynomial in this full truncation. Once we have our group of possible polynomials we can start the selection with the LARS 74 algorithm. As a first step, we choose the polynomial ψ1 with the highest correlation with the output Y . Then, we move the value of a coefficient γ1 , associated with ψ1 , toward its ordinary least square (OLS) value until some other polynomial ψ2 has the same level of correlation, as does ψ1 , with the current residual kY − γ1 ψ1 k. Thus we have the two most significant set of polynomials among the large full truncation. Then, we move a second coefficient γ2 in the direction defined by the joint least square coefficient of ψ1 and ψ2 , until we find some other polynomial ψ3 which has the same level of correlation with the current residual as does the joint direction. We continue with this procedure of iteratively adding the polynomials until the limit is defined by the number of points in our design of the experiment, on the basis of the condition number of the information matrix (for more details on the LARS selection process, see [52]). For each addition of a new polynomial, we can define a new enriched model. To derive the model corresponding to each new enriched basis of polynomials, we compute the OLS coefficients of the corresponding polynomial expansion. But since these models have been selected from information pertaining to the output (the design of experiments), a model with too many polynomials can be overfitted and they can exhibit poor performance; consequently, we need to assess the quality of these models to select the best one. We will discuss this issue of quality assessment after explaining how to determine the expansion coefficients yα of the PC equation. 4.5.3 Computation of the Coefficients Our next step after choosing the set of candidate polynomials is to determine the expansion coefficients yα of each multivariate polynomial ψα as defined in equation (4.4). The term non-intrusive indicates that the polynomial chaos coefficients are evaluated over a set of input realizations X = {x(1) , ..., x(N ) }, referred to as the experimental design (ED). There are different methods to calibrate PC metamodels such as projection, stochastic collocation and least-square minimization method [53]. If we take Y = {M(x(1) ), ..., M(x(N ) )} as the set of outputs of the computational model M for each point in the ED X , then assume that the expansion coefficients are y = {yα , α ∈ A}. The expansion coefficients y are then computed by minimizing the least square residual of the polynomial approximation 75 over the ED X : N X 1 X (i) ŷ = argmin M x − yα ψα (x(i) ) yα ∈RP N i=1 α∈A !2 . (4.6) The ordinary least-square solution of Eq. (4.6) can be expressed in a matrix notation as follows: ŷ = (AT A)−1 AT Y. (4.7) where A = {Aij = ψj (x(i) ), i = 1, ..., N ; j = 0, ..., P }. in which A is the model matrix that contains the values of all the polynomial basis evaluated at the experimental design points. 4.5.4 Quality Assessment of Polynomial Chaos Representations To assess the quality of the meta-model, the ideal case is to determine the generalization error. For each X, the error can be expressed as Err = E 2 M(X) − M̂(X) (4.8) where M(X) is the physical model, M̂(X) is the meta-model obtained by the Polynomial Chaos and E is the expectation operator. In practice, Err may be estimated by using the empirical error (or training error) defined by: Erremp = N 2 1 X M x(i) − M̂A x(i) N i=1 (4.9) where the x(i) ’s are the points of the experimental design. However, it is well-known that Erremp underpredicts the generalization error. The quantity Erremp is even systematically reduced by increasing the complexity of the PC approximation (i.e. the cardinality of A), whereas Err may increase. This is commonly referred to as the overfitting phenomenon. An error estimate, LeaveOne-Out Cross-Validation (LOOCV), which is known to be much less sensitive to overfitting than Err is investigated in the next section. 76 4.5.4.1 The Leave-One-Out Cross-Validation A good measure of the PCE accuracy is the leave-one-out (LOO) error [54]. The principle is to take one point x (i) out of the experimental design, to compute the meta-model M̂(−i) on the remaining points, and to calculate the prediction error at x (i) as follows: ∆(i) = M x(i) − M̂(−i) x(i) (4.10) By computing ∆(i) for all x(i) in the experimental design, we can derive an estimate of the generalization error of the following Leave-One-Out (LOO) approach by using the expression: N 1 X 2 ErrLOO = ∆(i) (4.11) N i=1 The LOO error is slightly more optimistic than the actual generalization error, but it yields much better error estimate than if we consider a simple empirical mean-squared error. From this LOO error, we can extract a deterministic coefficient Q2 (which is the equivalent to the well-known R2 coefficient): Q2 = 1 − ErrLOO V̂(Y ) (4.12) where V̂(Y ) is the estimated variance of the output Y . The higher is the estimate of the coefficient Q2 , the better is the quality of the meta-model, and we will use this coefficient as a measure of the quality of the meta-model in this study. It has been shown in [55] that the LOOCV method generally performs well in terms of generalization error bias. 4.5.4.2 Choice of the “best” meta-model After we have computed all the possible LARS models (as explained in part 4.5.2.1), we perform a cross-validation for each of the models to determine the most relevant number of LARS polynomials of chaos to use. Thus, we choose the meta-model P with the highest Q2 to derive the “best" meta-model M̂(X) = Pk=1 yk ψk (X) with P LARS polynomials of chaos. Finally, we summarize the entire procedure as follows: • build an initial LHS design of experiments of a chosen number of points 77 • compute each possible LARS PC model from this design of experiments. • compute the corresponding Q2 values for all these models. • only keep the number of polynomials in the truncation with the highest Q2 value. • if the accuracy of this model is not satisfactory, i.e., if the corresponding Q2 value is not sufficiently high, add new simulation points in the design of experiment using NLHS technique. • restart the entire procedure from the second step until the obtained accuracy is satisfactory. 4.5.5 Senstivity Analysis Sensitivity Analysis (SA) methods are invaluable tools to study how the uncertainty in the output of a model can be apportioned to different sources of uncertainty in the model input [56]). SA aims at determining the most contributing input variables out of large set of inputs to an output behavior. There are multiple approaches to perform sensitivity analysis. In contrast to local sensivity analysis, we use “global sensitivity analysis” to consider the whole range of the variation of the inputs. Global sensitivity analysis (also known as variance decomposition techniques) decomposes the variance of the model output in terms of contributions of each single input parameter, or a combination thereof. We look into Sobol’ sensitivity indices which is a global method that takes into account the whole input domain [57]. Using Monte Carlo simulation to compute Sobol’ indices [58] requires a large number of samples but the surrogate model eases this requirement. We use the calculated PCE model to get to the desired Sobol’ indices for sensitivity analysis. Then the surrogate model can be used to compute the sensitivity indices in negligible time. In particular, Sobol’ (resp. total Sobol’) indices can be computed analytically from the PCE coeffcients. From a given PC expansion, the Sobol’ indices can be obtained by a combination of the squares of the coefficients [59]. So, the PC-based estimator of the first-order 78 Sobol’ indices is: X Ŝi = ŷα2 α∈Ai X ŷα2 where Ai = {α ∈ A : αi > 0 , αj6=i = 0} . (4.13) α∈A , α6=0 which leads to the total PC-based Sobol’ indices as: X Ŝitot = ŷα2 α∈Atot i X ŷα2 Atot i = {α ∈ A : αi > 0} . (4.14) α∈A , α6=0 4.6 Illustrative Numerical Results 4.6.1 Cross–dipole Analysis In this study, we are interested in two different output variables, namely the Magnitude (Γmag ) and the Phase (Γphase ) of the Reflection Coefficient for the periodic unit cell. We use three input parameters (see Fig. 4.2) that follow the normal distribution, and their properties are shown in Table 4.1. We choose 3200 simulation points from these distributions and run simulations by using a commercial FEM solver . We run the periodic unit cell simulations at 4.8 GHz where the cross-dipole is nearly resonant. We have plotted Γmag and Γphase in Figs. 4.5 and 4.6, respectively for 3200 simulation runs. We also note from Fig. 4.5 that Γmag is close to 1 for most of the simulations as we would expect from a resonant periodic unit cell. In Figs. 4.7 and 4.8, we plot Q2 for different number of simulation points. With 400 simulation points, we obtained high values of Q2 (> 0.95) for both Γmag and Γphase . By taking the square root of the corresponding ErrLOO (Equation (4.12)), we have an estimation of the general Root Mean Square Error (RMSE) of our models. Then, the RMSE is estimated to be 6.31E-5 and 3.17 respectively for Γmag and Γphase . With these two computationally easy to compute LARS-PC models, we can now perform a classic MC analysis and deduce the distribution of our output variables. The resulting estimated distributions are plotted in Figs. 4.9 and 4.10. In Fig. 4.9, we can see that the Γmag values slightly exceed 1 which is actually 79 1 0.9 0.8 Γmag 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 500 1000 1500 2000 2500 Simulation runs 3000 3500 Figure 4.5: Γmag for different number of simulation runs. impossible. On one hand, it comes from the sweeping window of the function which computes the distribution. Indeed, as there is a great density of values for 1 and theoretically none beyond one, the sweeping window realize a mean distribution in this area, which explains the crossing. On the other hand, it could come from the inaccuracies of the computed LARS–PC model. As a matter of fact, even if we have a very good fit on average, the error of the model can become more important in this region where there is a high density and where the output variable seems to have an asymptotic behavior. This is due to the fact that the polynomial chaos can have some difficulties fitting in an area where the output is locally constant. Despite this, our model has very good performance in term of fitting and we just have to keep in mind that the reality can’t allow values higher than 1 and that these higher values are probably equal to 1. We also see that the LAR truncation performs better than the full PC expansion. The computed meta-models have a satisfying generalization capacity. The conclusions are the same for the Γmag except that the full PC expansion performs worse than it does for the Γphase . For the Γmag , we see the advantage of using LAR expansion, since the full expansion needs 3200 points to have a Q2 coefficient 80 −20 −40 −60 −100 Γ phase −80 −120 −140 −160 −180 0 500 1000 1500 2000 2500 Simulation runs 3000 3500 Figure 4.6: Γphase for different number of simulation runs. higher than 0.9. Thus, for only 200 points in the experimental design we are able to obtain satisfying meta-models with the LAR expansion. 4.6.2 3D–printed Lens Analysis The whole 3D lens with sub-millimeter hole dimensions can be time-consuming in terms of simulation. Therefore, we used the homogenized lens with the effective relative permittivity values given in Table 4.2 for simulations. However, in order to determine the effect of the uncertainty of hole dimensions, a block of the lens comprised of 4 holes is considered. Our first objective is to estimate the uncertainty introduced by the variability of the hole dimension on the effective permittivity and then to determine the overall effect on the output response of the 3D lens by using the homogenized lens. The cross section of the magnified block is shown in Fig. 4.3. In order to estimate the uncertainty introduced into the effective relative permittivity, the unit cell simulation is performed using HFSS software. Both the block with holes (see Fig. 4.11a), with a relative permittivity of r =2.72 for the 81 Figure 4.7: Q2 of the different meta-models for the Γmag using full PC expansion (in blue) and LAR PC expansion (in red). material, and the homogeneous block (Fig. 4.11b) with a relative permittivity of r =1.3 are simulated. For a fixed hole size (or hole/material volume ratio) both structures yield the same electromagnetic response (phase of S21 in this case) to an incident plane wave. By introducing some uncertainties to the hole size (∆X = ∆a for the first case) and to the relative permittivity (∆X = ∆ for the second case) and then by computing the variation of the phase of S21 (see Fig. 4.12a), a relationship between ∆a and ∆ is established. A linear relation (∆=1.7∆a) is obtained and is shown in Fig. 4.12b. In all simulations, the thickness of the 3D lens and that of the block under consideration is assumed to be constant and is equal to 18.5 mm. The PCE analysis is performed with ∆ =8.5%. This corresponds to ∆a=5% for the hole size uncertainty. Different number of data are considered and the one (400 data) with the lowest error is selected after going through Fig. 4.13. The average gain along the axis of the lens obtained by the PCE and calculated with the nominal values given in 82 Figure 4.8: Q2 of the different meta-models for the Γphase using full PC expansion (in blue) and LAR PC expansion (in red). Table 4.2 are plotted in Fig. 4.14. The standard deviation of the estimated gain is also presented. It is shown that close to the lens output surface, the uncertainty introduced in the gain is higher (Fig. 4.15). Finally the relative effect of the effective permittivity of each ring is plotted with respect to the distance to the lens (Fig. 4.16). 4.7 Conclusion In this chapter, we presented a new technique for modeling periodic structures with statistically varying input parameters. The proposed technique yields accurate results for the reflection characteristics with fewer numbers of simulation runs as compared to the full PC method. The computational efficiency is realized by using the truncated LAR technique. After we get good understanding of PCE method by using on periodic case, we also performed a statistical analysis of a 3D printed 83 Probability Density 14 12 10 8 6 4 2 0 0.4 0.5 0.6 0.7 Γmag 0.8 0.9 1 1.1 Figure 4.9: Probability Density as a function of Magnitude of Reflection Coefficient. GRIN lens using the PCE analysis. It has been shown that the variability of the hole size can have a significant effect on the effective permittivity and thus on the output response of the lens. PCE analysis also allows the user to determine the impact of dimensional variations in each individual ring on the performance of the lens. 84 Probability Density 0.02 0.015 0.01 0.005 0 −180 −160 −140 −120 −100 −80 Γphase −60 −40 −20 Figure 4.10: Probability Density as a function of Phase of Reflection Coefficient. (a) (b) Figure 4.11: The procedure for estimating ∆ from ∆a: a) the block with holes; b) the homogeneous block. 85 (a) (b) Figure 4.12: The procedure for estimating ∆ from ∆a: a) phase variation for both models; b) the relationship between ∆ and ∆a. 86 Figure 4.13: LOO error for different number of data. 87 Figure 4.14: The average gain along the axis of the lens obtained by the PCE and the one calculated with the nominal values. 88 Figure 4.15: The uncertainty of the gain along the axis of the lens. 89 Figure 4.16: Total Sobol’ indices with relative influence of each relative permittivity to the gain along the axis of the lens. 90 Chapter 5 | Offset–fed Dielectric Reflectarray Antenna Design 5.1 Introduction High–gain antennas have witnessed an increasing demand for applications such as long distance communication, satellite communication and radar. Traditionally, parabolic reflectors or antenna arrays have been employed to satisfy the need for high gain. Parabolic reflectors have large mass/volume ratios, owing to their curved and electrically large surfaces, which may render them undesirable for certain applications. Fabrication of parabolic antennas designed for high frequency applications also demands close attention, since the surface accuracy must be high for these frequencies. Turning now to the antenna arrays, we note that they have several salutary features such as low profile and scan capability; however, they need bulky and expensive electronic circuitry for beamforming, and these losses can become significant with increasing frequency and, hence, they can become a cause for concern. The reflectarray antenna [60–62], as its name implies, is a hybrid combination of a reflector and an antenna array. It uses a reflecting surface similar to that of a parabolic antenna and controls the phase of the outgoing beam by placing phasing elements on its surface. It is relatively inexpensive to manufacture as compared to a conventional antenna array, because the phasing elements in reflectarray antennas can be printed by using conventional PCB techniques. In a typical reflectarray, a feed horn illuminates the reflecting surface on which multiple reflecting elements are placed. Typically, printed microstrip patches, dipoles, or rings are used in reflectarrays as phase–shifting elements. These elements, placed in a 91 lattice configuration, are designed such that the reflected beam from these elements collimate in the desired direction of radiation, and result in a high–gain performance. The conventional approach [62] to designing the reflectarray is to print microstrip patches of different shapes, sizes and orientations on a dielectric substrate to locally control the phase of the reflected wave when illuminated by an offset–fed horn. Recent developments in 3D printing have spawned interest in dielectric–based reflectarrays [63]. One of the limiting factors for such reflectarrays is the maximum available permittivity of the 3D-printing materials. In this work, we show how we can decrease the required permittivity of the dielectric blocks by placing a lens in front of the feed horn. In this chapter, we present two offset-fed dielectric reflectarray designs. Both reflectarrays feature broadband designs realized by using dielectric blocks backed by a PEC plane. One of these arrays uses a phase compensating flat lens to reduce the maximum required permittivity of the dielectric blocks covering the PEC plane. We compare the results of both reflectarrays and list their benefits as compared to traditional reflectarrays that use resonant elements for their designs, which render them narrowband. 5.2 Different Reflectarray Designs We introduce an alternate reflectarray design in this chapter which is also an extension of the study carried out in the previous chapter on lenses. This reflectarray design invokes image principle to show the principle of the design and the fact that it is similar to that of designing a GRIN lens. We use offset–fed reflectarray design in this study as it is the most common configuration. In this section, we present two offset–fed dielectric reflectarray designs. The first reflectarray is realilzed by using dielectric blocks backed by a PEC plane. We see in this design that it calls for high values of permittivities for the dielectric blocks. To mitigate this constraint, a second reflectarray is proposed which introduces a phase compensating flat lens. 92 5.2.1 Offset–fed Dielectric Reflectarray Design The design specifications chosen for the dielectric reflectarrays are: f (center frequency) = 15GHz; height of feed = 230.64mm; Θ (horn tilt angle) = 36.25◦ ; aperture size = 210mm × 210mm. The design can be appropriately scaled to higher frequencies, if desired. We start the design process by first looking at the design of a transmitarray. Transmitarray design by using the Ray Optics method is straight–forward. The design for the reflectarray can be easily formulated by taking out half of the transmitarray and replacing it with a PEC ground plane following the principles of the image theory, where the two systems are equivalent, and hence have identical properties. As a first step, we assume that the feed horn illuminates the reflectarray surface at an angle Θ as shown in Fig. 5.1. Next, we determine the phase distribution on the top surface of the reflectarray (input aperture) located at z = 10mm, when illuminated by the feed horn. We calculate the desired phase distribution at z = −10mm (exit aperture). By both these phase distributions, we can embed the dielectric blocks between planes z = 10mm and z = −10mm so that the compensating phase introduced by these dielectric blocks enforces the phase distribution at the exit aperture to the desired distribution calculated at the exit plane. Once we place these dielectric blocks, it is similar to the design of a transmitarray except that the output beam is offset by an angle Θ as compared to the normal incidence as is common with a typical transmitarray. We use a Ku–band conical feed horn with an open–end diameter of 22.75 mm at the height of 230.64 mm for this design. Once the phase distribution at the input aperture, Φi , and the phase distribution at the exit aperture, Φo , are known, we can calculate the desired permittivity of the dielectric blocks as follows: 2πt √ rj (5.1) λ where λ is the free space wavelength, t is the thickness of the dielectric block (here, t=20mm), and rj is the dielectric constant of the j th block along the x–axis. We then calculate the values of rj along the x–axis for different values of y. We show the calculated values of r at different locations with respect to x– and y– Φo − Φi = 93 Figure 5.1: Location of the input and output aperture. coordinates in Table 5.1. Each dielectric block is of the size 20mm × 20mm × 20mm, and the Table shows the r for each dielectric block on the x–y plane. The placement of these dielectric blocks completes our transmitarray design and thus forms the transmitarray as shown in Fig. 5.2. The transmitarray works as a preliminary design for the reflectarray. We convert it to the reflectarray by placing a PEC sheet at the x–y plane at z = 0mm and removing the part of the dielectric blocks below z = 0mm. Placement of the PEC sheet at z = 0mm will reflect the incident beam from the feed horn in the specular direction. The final design of the converted reflectarray is shown in Fig. 5.3. This completes our design for the offset–fed dielectric reflectarray. We place 11 × 11 dielectric blocks above the ground plane to provide the desired phase compensation. Each block has the dimensions of 20mm × 20mm and a height of 10 mm. From Table 5.1, we see that the maximum value of rj that we need for this reflectarray is 8.5. 94 Table 5.1: Permittivity of the Dielectric Blocks for the Reflectarray. X Y -100 -80 -60 -40 -20 0 20 40 60 80 100 0 20 40 60 80 100 5.16 6.38 7.5 7.99 8.4 8.5 8.3 8 7.4 6.5 5.4 4.94 6.16 7.12 7.9 8.2 8.3 8.14 7.6 6.9 6.3 5.48 4.25 5.45 6.5 7.2 7.6 7.7 7.5 7.1 6.5 5.76 4.8 3.38 4.44 5.4 6.12 6.6 6.76 6.67 6.2 5.8 5.16 4.44 2.36 3.3 4.13 4.72 5.17 5.35 5.35 5.13 4.75 4.18 3.6 1.34 2.17 2.73 3.3 3.65 3.9 3.85 3.7 3.44 3.0 2.52 A quick check shows that it is difficult to obtain materials that have such a vast expanse of values of permittivities for the reflectarray from commercial off–the–shelf materials. Even advanced techniques like 3D–prininting with air–void embedding [4] are not helpful in this scenario as 3D–printing materials with high r values are not available. To reduce the maximum required value of the permittivity of the dielectric blocks, we propose a modified design by incorporating a lens as explained in Sec. 5.2.2. 5.2.2 Dielectric Reflectarray with a Phase Compensating Flat Lens In this design, we use the same methodology as explained in the previous section. We use the same physical parameters for the second reflectarray but use a phase compensating flat lens between the horn and the array to reduce the phase excursion of the spherical wave impinging upon the dielectric blocks as shown in Fig. 5.4. We show the values of permittivity needed for the reflectarray and the lens in Table 5.2. With the incorporation of the lens, the maximum permittivity needed for the dielectric blocks is reduced to 5.41. The maximum permittivity of the flat lens, at its center, is 5.66. The flat lens is comprised of 20 rings, each ring having a width of 2.5mm and a height of 10mm. The desired values of permittivity in this design can be achieved by using the 95 Figure 5.2: Dielectric Transmitarray. DaD technique as discussed in the previous chapter. We can use ABS as printing material with printed conductive patches to realize desired permittivity values that otherwise cannot be achieved by 3D–printing alone. 5.2.3 Results Fig. 5.5 shows the radiation patterns of reflectarrays. For reference, we have also included radiation pattern of the reflectarray when no dielectric blocks are placed on the PEC sheet. All these patterns show that the reflected beam are all focused in the specular direction. Fig. 5.6 shows the gain comparison of both reflectarrays and demonstrates the effectiveness of the reflectarray design with the phase compensating flat lens. We see a +6dB gain increase with the introduction of the phase compensating flat lens. 96 Figure 5.3: Offset-fed Dielectric Reflectarray. 5.3 Conclusions In this chapter, we presented the designs of two dielectric reflectarrays, both of which utilized dielectric blocks backed by a ground plane. The dielectric blocks with different permittivity values on the ground plane are placed such that the reflected beam is directed in the specular direction. The main difference between the two designs is that one of them utilizes a lens to reduce the required maximum value of the permittivity of the dielectric blocks, which is especially desirable when fabricating the lens using a 3D printer. Additionally, the design may be useful for lowering the profile of the entire reflectarray antenna system. Due to the use of phase compensating flat lens in the latter configuration, the permittivity values required for dielectric blocks need to be changed. This change compensates for the phase changes introduced by the lens so that the reflected beam is still oriented in the desired direction. 97 Figure 5.4: Dielectric Reflectarray with phase compensating flat lens. 98 Table 5.2: Permittivity of the Dielectric Blocks for the Reflectarray with Lens. (a) Permittivity of the Dielectric Blocks for the Reflectarray X Y -100 -80 -60 -40 -20 0 20 40 60 80 100 0 20 40 60 80 100 4.29 4.81 5.18 5.20 5.34 5.41 5.32 5.14 5.1 4.92 4.36 4.09 4.81 5.14 5.12 5.23 5.29 5.27 5.18 5.19 4.97 4.38 3.66 4.51 4.97 5.07 5.06 5.08 5.05 5.07 4.92 4.55 4.06 3.16 3.88 4.47 4.78 4.97 4.98 4.85 4.65 4.36 4.00 3.43 2.3 3.04 3.6 4.01 4.21 4.38 4.37 4.27 4.06 3.7 3.15 1.33 2.03 2.52 2.88 3.13 3.32 3.3 3.18 2.99 2.70 2.30 (b) Permittivity of the Phase Compensating Flat Lens. Ring No. r Ring No. r 1 5.66 11 3.84 2 5.57 12 3.54 3 5.44 13 3.27 4 5.34 14 3.01 5 5.21 15 2.75 99 6 5.10 16 2.41 7 4.91 17 2.00 8 4.71 18 1.61 9 4.46 19 1.30 10 4.15 20 1.12 30 20 Reflectarray with Ground Plane only Dielectric Reflectarray Dielectric Reflectarray with Lens Gain (dB) 10 0 −10 −20 −30 −40 −80 −60 −40 −20 0 θ (°) 20 40 60 80 Figure 5.5: Simulated radiation pattern of the reflectarrays at 15 GHz. 100 26 Gain (dB) 24 Dielectric Reflectarray Dielectric Reflectarray with Lens 22 20 18 16 12 13 14 15 16 Frequency(GHz) 17 Figure 5.6: Simulated gain comparison of reflectarrays. 101 18 Chapter 6 | Conclusions and Future Work This dissertation has presented the results of investigation of some problems in electromagnetics and has shown a number of ways to solve them. Future work can further explore ways for overcoming the shortcomings of the proposed methods. In Chapter 2, we discussed the use of the periodic boundary condition in the FDTD method. We analyzed the finite, truncated structure to derive the solution for infinite doubly–periodic problem. This helped us to calculate the transmission and reflection coefficients for doubly–infinite arbitrary 3D elements by the use of signal processing techniques. This method can be investigated further by studying for the case of grazing angles. Our study shows that the size of the truncated structure has to be increased substantially to study behavior for grazing angles and by doing so we lose the benefit of this method as it takes long time to simulate such truncated structures. We introduced different lens designs by using artificially engineered materials in Chapter 3. We used 3D–printing and Dial–a–Dielectric (DaD) techniques to design three lenses in this work. One of the lenses, namely the DaD lens uses commercial off–the–shelf (COTS) and 3D–printing for its design with radially varying refractive indices. The other two lenses are fabricated by using the same 3D–printing technique but with different 3D–printing materials. All of these lenses show wideband behavior and are low–cost to manufacture. We used comparatively large size patches and that resulted in low transmission in the DaD lens. We have found that if smaller size patches are used in DaD lens, the gain of the lens will increase. We could not use small size patches as we did not have the facility to print such small patches. Future work can include decreasing the size of the patches and studying the behavior of DaD lenses. The DaD technique is versatile not only in lenses but can be used in other electromagnetic problems where COTS materials 102 are not available. Frequency Selective Surfaces (FSSs) are spatial filters whose passbands and stopbands depend on the geometrical parameters of its elements. There may be considerable discrepancy in the simulated and measured behavior of such structures due to fabrication variability in these structures. In Chapter 4, we have used statistical methods to study the behavior of such variations and list the effect of such fabrication imperfections on the final output behavior of the system. We have used Polynomial Chaos (PC) method for such study. We can use other statistical methods in future and compare the advantages and disadvantages of PC method as to other methods. More examples can also be accommodated in this study to verify the functioning of PC method. We conclude this work by introducing offset–fed dielectric reflectarray antenna designs in Chapter 5. We introduce two reflectarrays in this work. Both reflectarrays are realized by using dielectric blocks backed by a PEC plane. One of these refectarrays uses a phase compensating lens to reduce the maximum permittivity called by dielectric blocks. In this way, reflectarray with the lens calls for permittivity values that might be easily available or can be realized by DaD technique. We did not use the DaD technique for this reflectarray but the previous study shows that DaD technique will help to realize such dielectric values. The use of DaD with the reflectarray designs can be investigated to meet the requirements when the COTS materials are not easily available. 103 Publications from this Dissertation Journals 1. S. Zhang, R. K. Arya, S. Pandey, Y. Vardaxoglou, W. Whittow and R. Mittra, “3D-printed planar graded index lenses,” IET Microwaves, Antennas & Propagation, vol. 10, no. 13, pp. 1411–1419, 2016. 2. R. K. Arya, S. Pandey, and R. Mittra, “Flat lens design using artificially engineered materials,” Progress In Electromagnetics Research C, Vol. 64, 71–78, 2016. 3. Ravi Kumar Arya, Pierric Kersaudy, Joe Wiart and Raj Mittra, “Statistical Analysis of Periodic Structures and Frequency Selective Surfaces using the Polynomial Chaos Expansions,” Forum for Electromagnetic Research Methods and Application Technologies (FERMAT), Vol. 12, Nov.–Dec., 2015 4. R. K. Arya, C. Pelletti and R. Mittra, “Numerically Efficient Technique for Metamaterial Modeling,” Progress In Electromagnetics Research, Vol. 140, pp. 263–276, 2013 Book Chapters 1. H. Ackigoz, R. K. Arya, J. Wiart and R. Mittra, “Statistical Electromagnetics for Antennas,” in Developments in Antenna Analysis and Synthesis edited by Raj Mittra, IET, 2018 (in press) 2. C. Pelletti, R. K. Arya, A. Rashidi, H. Mosallaei and R. Mittra, “Numerical Techniques for Efficient Analysis of FSSs, EBGs and Metamaterials,” in Computational Electromagnetics edited by Raj Mittra, Springer, 2013 Conference Proceedings 1. R. K. Arya and Raj Mittra, “Offset-fed Dielectric Reflectarray Antenna Designs,” IEEE APS–URSI, San Diego, California, July 9–14, 2017 (accepted) 104 2. R. K. Arya, Shiyu Zhang, Yiannis Vardaxoglou, Will Whittow and Raj Mittra, “3D-Printed Lens Antenna,” IEEE APS–URSI, San Diego, California, July 9–14, 2017 (accepted) 3. R. K. Arya, Shiyu Zhang, Yiannis Vardaxoglou, Will Whittow and Raj Mittra, “3D-Printed Millimeter Wave Lens Antenna,” Global Symposium on Millimeter-Waves (GSMM 2017), Hong Kong, China, May 24–26, 2017 (accepted) 4. R. K. Arya, S. Pandey and R. Mittra, “A Novel Approach to Designing Phase Shifters for Array Antennas for Satellite Communication by using Reconfigurable FSS Screens” PIERS, Shanghai, China, August 8–11, 2016 5. Hulusi Acikgoz, Ravi Kumar Arya and Raj Mittra, “Statistical Analysis of 3D–Printed Flat GRIN Lenses,” IEEE APS–URSI, Fajardo, Puerto Rico, June 26–July 1, 2016 6. Shaileshachandra Pandey, Ravi Kumar Arya and Raj Mittra, “Flat Lens Design Using Space-qualifiable Multilayer Frequency Selective Surfaces,” IEEE APS–URSI, Fajardo, Puerto Rico, June 26–July 1, 2016 7. Ravi Kumar Arya, Shaileshachandra Pandey and Raj Mittra, “Synthesizing Broadband Low–loss Artificially Engineered Materials (Aka Metamaterials) for Antenna Applications,” International Symposium on Antennas and Propagation (ISAP2015), Hobart, Tasmania, Australia, November 9–12, 2015 8. Raj Mittra, Shaileshachandra Pandey and Ravi Arya, “Low Profile Lens and Reflectarray Design for MM Waves,” IEEE APS–URSI, Vancouver, BC, Canada, July 19–25, 2015 9. Y. Vardaxogolu, R. Mittra, R. K. Arya and S. Pandey, “Techniques for Synthesizing Artificial Dielectrics for Lens and Reflectarray Designs,” Loughborough Antennas and Propagation Conference (LAPC), Nov. 10–11, 2014 10. Ravi Kumar Arya, Shaileshachandra Pandey and Raj Mittra, “A Technique for Designing Flat Lenses Using Artificially Engineered Materials,” IEEE APS–URSI, Memphis, TN, USA, July 6–12, 2014 105 11. Kapil Sharma, Kadappan Panayappan, Ravi Kumar Arya and Raj Mittra, “Combining the FDTD Algorithm with Signal Processing Techniques for Performance Enhancement,” IEEE APS–URSI, Memphis, TN, USA, July 6–12, 2014 12. R. Mittra, C. Pelletti, R. K. Arya, T. Dong and G. Bianconi, “A generalpurpose simulator for metamaterials with three-dimensional elements,” Proceedings of 2013 URSI International Symposium on Electromagnetic Theory (EMTS), Hiroshima, Japan, May 20–24, 2013 13. R. Mittra, R. K. Arya, C. Pelletti and T. Dong, “Efficient and Accurate Analysis of Arbitrary Metamaterials with Three-dimensional Crystal Elements,” 4th Int. Conf. on Metamaterials, Photonic Crystals and Plasmonics, META013, Sharjah, UAE, March 18–22, 2013 14. R. Mittra, R. K. Arya, “A new technique for analyzing periodic structures with arbitrary three-dimensional elements,” Int. Conf. on Electromagnetics in Advanced Applications (ICEAA), Cape Town, South Africa, September 2–7, 2012 15. R. Mittra, R. K. Arya, C. Pelletti, “A new technique for efficient and accurate analysis of arbitrary 3D FSSs, EBGs and Metamaterials,” IEEE APS-URSI, Chicago, Illinois, July 8–12, 2012 16. R. Mittra, C. Pelletti and R. K. Arya, “A new computationally efficient technique for modeling periodic structures with applications to EBG, FSSs and metamaterials,” Int. Conf. on Microwave and Millimeter Wave Technology (ICMMT), Shenzhen, China, May 5–8, 2012 17. R. Mittra, C. Pelletti, R. K. Arya, G. Bianconi, T. McManus, A. Monorchio and N. Tsitsas, “New numerical techniques for efficient and accurate analysis of FSSs, EBGs and Metamaterials,” 6th European Conf. on Antennas and Propagation (EuCAP), Prague, Czech Republic, March 26–30, 2012 18. K. Panayappan, R. Mittra and R. K. 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(2014) “3D printed dielectric reflectarrays: low-cost high-gain antennas at sub-millimeter waves,” IEEE Transactions on Antennas and Propagation, 62(4), pp. 2000–2008. 112 Vita Ravi Kumar Arya Ravi Kumar Arya is a doctoral student in the Department of Electrical Engineering at the Pennsylvania State University under the guidance of Dr. Raj Mittra. Currently, he is involved in the design and analysis of periodic structures with applications to EBG, FSSs and Metamaterials. He received his Master of Technology (M.Tech.) Degree in RF and Microwave Engineering from the Indian Institute of Technology, Kharagpur, India in 2006 and his Bachelor of Engineering (B.E.) Degree from Delhi College of Engineering, Delhi, India in 2003, respectively. Before joining Penn State in August, 2010, he worked with the Electronics Corporation of India Limited, India for 6 months and then with the Centre for Development of Telematics (C-DOT), India from 2006 to 2010. His research interests include RF circuit design, antenna design and the analysis of frequency selective surfaces. He has authored a number of publications in leading IEEE journals and conferences and has published two book chapters. 113

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