# Analytical Modeling of Waveguide-fed Metasurfaces for Microwave Imaging and Beamforming

код для вставкиСкачатьAnalytical Modeling of Waveguide-fed Metasurfaces for Microwave Imaging and Beamforming by Laura Maria Pulido Mancera Department of Electrical and Computer Engineering Duke University Date: Approved: David R. Smith, Supervisor Steven Cummer William Joines Henry D. Pfister Willie Padilla Dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Electrical and Computer Engineering in the Graduate School of Duke University 2018 ProQuest Number: 10750592 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 10750592 Published by ProQuest LLC (2018 ). 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 Abstract Analytical Modeling of Waveguide-fed Metasurfaces for Microwave Imaging and Beamforming by Laura Maria Pulido Mancera Department of Electrical and Computer Engineering Duke University Date: Approved: David R. Smith, Supervisor Steven Cummer William Joines Henry D. Pfister Willie Padilla An abstract of a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Electrical and Computer Engineering in the Graduate School of Duke University 2018 c 2018 by Laura Maria Pulido Mancera Copyright All rights reserved except the rights granted by the Creative Commons Attribution-Noncommercial License Abstract A waveguide-fed metasurface consists of an array of metamaterial elements excited by a guided mode. When the metamaterial elements are excited, they in turn leak out a portion of the energy traveling through the waveguide to free space. As such, a waveguide-fed metasurface acts as an antenna. These antennas possess a planar form factor that offers tremendous dexterity in forming prescribed radiation patterns; a capability that has led to revolutionary advances in antenna engineering, microwave imaging, flat optics, among others. Yet, the common approach to model and design such metasurfaces relies on effective surface properties, a methodology that is inspired by initial metamaterial designs. This methodology is only applicable to periodic arrangements of elements, and the assumption that the neighboring elements are identical. In the scenarios where the metasurface consists of an aperiodic array, or the neighboring elements are significantly different, or the coupling to the waveguide structure changes; the aforementioned approaches cannot predict the electromagnetic response of the waveguide-fed metasurface. In this thesis, I have implemented a robust technique to model waveguide-fed metasurfaces without any assumption on the metamaterial elements’ geometry or arrangement. The only assumption is that the metamaterial elements can be modeled as effective dipoles, which is usually the case given the subwavelength size of metamaterial elements. Throughout this document, the simulation tool will be referred to Dipole Model. iv In this framework, the total response of each dipole, representing a metamaterial element, depends on the mutual interaction between elements, as well as the perturbation of the guided mode. Both effects are taken into account and, by using full-wave simulations, I have confirmed the validity of the model and the ability to predict radiation patterns that can be used for beamforming as well as for microwave imaging. Once the capabilities of the dipole model are compared with full wave simulations of both traditional antenna designs as well as more elaborated waveguide-fed metasurfaces, I develop an analysis on the use of these metasurfaces for microwave imaging systems. These systems are used to form images of buried objects, which is crucial in security screening and synthetic aperture radar (SAR). Traditionally, the hardware needed for many imaging techniques is cumbersome, including large arrays of antennas or bulky, moving parts. However, one attractive alternative to overcome these problems is to use dynamic metasurface antennas. By quickly varying the radiation patterns generated by these antennas, enough diverse measurements can be made in order to produce high quality images in a fraction of the time. The compact size and speed come with a trade-off: a computationally intensive optical inverse problem has to be solved, which has so far prohibited these antennas from enjoying widespread use. I address this problem by reformulating the problem to make it similar to a SAR scenario, for which fast image reconstruction algorithms already exist. By adapting an algorithm known as the Range Migration to be compatible with these metasurfaces, I can cut down on real-time computation significantly. The computer simulations performed are highly promising for the field of microwave imaging, since it is demonstrated that diffraction-limited images can be acquired in a fraction of the time, in comparison with other imaging techniques. v To my family: Rosa, Isaias, Angela, Eduardo and Ines, who have given me all the love and strength in life, and to whom I owe so much. vi Contents Abstract iv List of Tables x List of Figures xi Acknowledgements xiv 1 Introduction 1 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2 Dipole Model for Waveguide-fed Metasurfaces 11 2.1 Fundamentals of the Dipole Model . . . . . . . . . . . . . . . . . . . 13 2.2 Polarizability Extraction Techniques . . . . . . . . . . . . . . . . . . 16 2.2.1 Method 1: Surface Equivalence Principle . . . . . . . . . . . . 19 2.2.2 Method 2: Coupled Mode Theory and Scattering Parameters . 21 2.2.3 Simulated Results in Rectangular Waveguides . . . . . . . . . 27 2.2.4 Coupled-mode theory with Cylindrical waves . . . . . . . . . . 32 2.2.5 Simulated Results in Planar Waveguides . . . . . . . . . . . . 34 2.3 Effective Dipole Moments . . . . . . . . . . . . . . . . . . . . . . . . 39 2.4 Mutual Interaction Between Metamaterial Elements . . . . . . . . . . 41 2.4.1 Waveguide-fed metasurface with arbitrary element arrangement 45 2.4.2 Mutual Interactions Outside the Waveguide . . . . . . . . . . vii 49 2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Waveguide-fed Metasurfaces for Microwave Imaging 53 57 3.1 Microwave Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.2 Dynamic Metasurface Antenna . . . . . . . . . . . . . . . . . . . . . 64 3.2.1 Imaging with the DMA . . . . . . . . . . . . . . . . . . . . . . 67 Range Migration Algorithm . . . . . . . . . . . . . . . . . . . . . . . 71 3.3.1 Stationary Case . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.3.2 Moving the DMA in the y direction . . . . . . . . . . . . . . . 74 3.3.3 Moving the DMA in the z direction . . . . . . . . . . . . . . . 75 3.3.4 Moving the DMA in the y-z plane . . . . . . . . . . . . . . . . 77 Algorithm Implementation and Simulated Results . . . . . . . . . . . 79 3.4.1 Pseudo-inversion of the aperture field . . . . . . . . . . . . . . 81 3.4.2 Algorithm Implementation . . . . . . . . . . . . . . . . . . . . 83 3.4.3 Simulated Results . . . . . . . . . . . . . . . . . . . . . . . . . 85 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 3.5.1 Stationary DMA . . . . . . . . . . . . . . . . . . . . . . . . . 92 3.5.2 Moving the DMA in the y direction . . . . . . . . . . . . . . . 93 3.5.3 Moving the DMA in the z direction . . . . . . . . . . . . . . . 94 3.5.4 Moving the DMA in the y-z plane . . . . . . . . . . . . . . . . 95 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 96 3.3 3.4 3.5 3.6 4 Waveguide-fed Metasurfaces for Beamforming 99 4.1 Beamforming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.2 Principles of Operation for Beamforming . . . . . . . . . . . . . . . . 103 4.3 Example I: Slotted Waveguide Antennas . . . . . . . . . . . . . . . . 107 4.3.1 Directivity Enhancement of SWAs with Parasitic Elements . . 111 viii 4.3.2 Dipole Model for Unusually Tapered SWAs . . . . . . . . . . . 115 4.4 Example II: Kymeta mTenna . . . . . . . . . . . . . . . . . . . . . . 122 4.5 Example III: Waveguide-fed Metasurfaces for Computational Microwave Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 4.6 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 132 5 Conclusions 134 A Scattered fields in a parallel plate waveguide 139 B Formulation of the Range Migration Algorithm 146 B.1 RMA for motion in the y- direction . . . . . . . . . . . . . . . . . . . 149 B.2 RMA for motion in the z-direction . . . . . . . . . . . . . . . . . . . 153 C Dipole Model Inversion Technique 158 Bibliography 161 Biography 179 ix List of Tables 3.1 Image Reconstruction Time with the Adapted RMA . . . . . . . . . . 4.1 Modulation Techniques for Beamforming . . . . . . . . . . . . . . . . 107 x 89 List of Figures 1.1 Metamaterials, Metasurfaces and Waveguide-fed Metasurfaces . . . . 3 1.2 Examples of waveguide-fed metasurfaces . . . . . . . . . . . . . . . . 5 1.3 State of the art on metasurface’s modeling . . . . . . . . . . . . . . . 7 1.4 Dipole model adapted to waveguide-fed metasurfaces. . . . . . . . . . 8 2.1 Dipole Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Extraction of the Polarizability Tensor by means of Surface Equivalence Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3 Polarizability Extraction using Coupled Mode Theory . . . . . . . . . 21 2.4 Effective Polarizability of small elliptical irises . . . . . . . . . . . . . 30 2.5 Effective Polarizability of small metamaterial apertures . . . . . . . . 31 2.6 Polarizability Extraction from Coupled Mode Theory in a Planar Waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.7 Amplitude Coefficients of the scattered field Ez . . . . . . . . . . . . 35 2.8 Effective magnetic polarizabilities calculated for the ELC embedded in the planar waveguide . . . . . . . . . . . . . . . . . . . . . . . . . 37 Real part of the scattered field Ez pV {mq at different frequencies . . . 38 2.10 Extraction of the polarizability tensor for two different complementary metamaterial elements . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.11 Magnetic dipole moment at location x0 . Comparison between the magnetic dipole model as predicted in Eq.2.37, and full-wave simulation. a) Diagram of the problem. b) Magnetic dipole moment for the elliptical iris. c) Magnetic dipole moment for the ELC. While my is smaller, good agreement between the two methodologies is observed. . 40 2.9 xi 2.12 Mutual interaction between metamaterial elements . . . . . . . . . . 43 2.13 Comparison between the scattered fields predicted from full wave simulation and the dipole model. . . . . . . . . . . . . . . . . . . . . . . 45 2.14 Relative error between the scattered field from full wave simulation and the dipole model using 12 elliptical irises. . . . . . . . . . . . . . 49 2.15 Relative error between the scattered field from full wave simulation and the dipole model using 12 ELC resonators. . . . . . . . . . . . . 50 2.16 Far-field of the waveguide-fed metasurface composed of ELC resonators. 51 2.17 E-plane and H-plane far-fields of waveguide-fed metasurface composed of ELC resonators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.18 Method of Images for Complementary Metamaterials in Waveguides. 55 3.1 Microwave Imaging: Range and Crossrange. . . . . . . . . . . . . . . 59 3.2 Microwave Imaging Techniques . . . . . . . . . . . . . . . . . . . . . 60 3.3 Microwave imaging with a Dynamic Metasurface Antenna. . . . . . . 65 3.4 Diagram of two different imaging scenarios with a single receiving antenna. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.5 Moving the DMA along they direction. . . . . . . . . . . . . . . . . . 74 3.6 Moving the DMA along the z -direction. . . . . . . . . . . . . . . . . . 76 3.7 Moving the DMA along y-z directions . . . . . . . . . . . . . . . . . 78 3.8 Block diagram of the adapted RMA . . . . . . . . . . . . . . . . . . . 79 3.9 Singular value spectra for different sets of masks applied . . . . . . . 83 3.10 Transformation of the signal measured with the dynamic metasurface to apply the RMA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3.11 Steps of the RMA for a PSF . . . . . . . . . . . . . . . . . . . . . . . 85 3.12 Point Spread Function . . . . . . . . . . . . . . . . . . . . . . . . . . 87 3.13 Image reconstruction in 2D . . . . . . . . . . . . . . . . . . . . . . . . 87 3.14 Image reconstruction in 3D . . . . . . . . . . . . . . . . . . . . . . . . 88 3.15 Experimental setup for microwave imaging with the DMA . . . . . . 91 xii 3.16 Near-field scans of the DMA . . . . . . . . . . . . . . . . . . . . . . . 92 3.17 Stationary DMA Experiment . . . . . . . . . . . . . . . . . . . . . . 93 3.18 Moving DMA along the y-direction. Experiment . . . . . . . . . . . . 94 3.19 Moving DMA along the z-direction. Experiment . . . . . . . . . . . . 95 3.20 Moving DMA along the y-z-direction. Experiment . . . . . . . . . . . 96 4.1 Phased Arrays vs. Waveguide-fed Metasurfaces . . . . . . . . . . . . 100 4.2 Beamforming based on Holographic Techniques . . . . . . . . . . . . 103 4.3 Slotted Waveguide Antenna . . . . . . . . . . . . . . . . . . . . . . . 108 4.4 Various elements used to improve the directivity of the SWA . . . . . 112 4.5 Results for a SWA with parasitic elements . . . . . . . . . . . . . . . 113 4.6 Farfields of the SWA . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 4.7 dipole model for the SWA . . . . . . . . . . . . . . . . . . . . . . . . 115 4.8 One dimensional waveguide antenna structures . . . . . . . . . . . . . 118 4.9 Uniform SWA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 4.10 Tapered SWA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 4.11 Kymeta Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 4.12 Using the dipole model on the mTenna . . . . . . . . . . . . . . . . . 125 4.13 Different modulation patterns applied on the mTenna . . . . . . . . . 126 4.14 Directivity of the mTenna. Comparison between dipole model and HFSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 4.15 Frequency diverse metasurface antenna. . . . . . . . . . . . . . . . . . 128 4.16 Normalized |m|. a) with and b) without mutual element interaction. . 130 4.17 Imaging system configuration and Singular Value Decomposition Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 5.1 Implementation of the dipole model in a set of Modules. . . . . . . . 137 A.1 Multiple Images of an element embedded in a parallel plate waveguide. 140 xiii Acknowledgements I am immensely grateful for the opportunity of pursuing my graduate degree at Duke University. At times it has been challenging and overwhelming, but always exciting and worthwhile. When I applied to Duke University I cited Clarke’s third law “Any sufficiently advanced technology is indistinguishable from magic”. At the time, Smith’s group was very famous for their works on invisibility cloaks. During my years at Duke, I replaced the magic with deep analytical thinking, becoming the pillar for my professional career. I would like to thank my advisor Prof. David R. Smith, for encouraging and supporting me through numerous challenges and drastically widening my vision and understanding of metamaterials. I am still fascinated by the wide array of novel technologies and techniques developed in our group, and I am very grateful to be part of this cutting-edge scientific adventure. I would also like to thank the members of my committee, Prof. William Joines, Prof. Steven Cummer, Prof. Willie Padilla and Prof. Henry Pfister, who have shown a great interest in my research. Especially, Dr. Nathan Kundtz has been tremendously influential in motivating me to develop simulation tools for real-life applications of metamaterials. I have been inspired by his work at Kymeta and it has been fascinating to learn about bringing metamaterials to market. I especially want to thank Dr. Mohammadreza F. Imani, I could not have hoped for a better colleague and friend. In my career development, Dr. Imani has been xiv one of the researchers from whom I have learned the most. Not only do I appreciate our stimulating academic conversations, but he also thought me to become a collaborative researcher while practicing systematic research approaches. I will always be thankful for the support Dr. Imani provided during all the times I felt frustrated and discouraged. I would also like to thank Timothy Sleasman, Michael Boyarsky, Jonah Gollub and Aaron Diebold, for their friendship and excellent academic partnership; and specifically for their careful reading and feedback on this manuscript. Additionally, I would like to thank my colleagues and friends in the Pratt School of Engineering for their intellectual support and significant friendship throughout the PhD process. Although it would be impossible to properly thank all of these people, I would like to particularly thank the students of Smith’s Group and my close friend Daniela Cruz. Finally, I would like to thank my family; my Latin-dancing friends; and my friends from Colombia. I’m grateful to Duke University and in general the city of Durham, not only for allowing me to pursue my dreams of becoming a researcher, but also for providing me with opportunity to pursue my passions. xv 1 Introduction It is always challenging, and to some extent, controversial, to assign a starting point for any revolutionary scientific discovery. Considering only the “revolutionary” aspect, one can argue that metamaterials, as a transforming concept, started by the experimental work of Prof. David Smith and his colleagues at UCSD, where they demonstrated the first left-handed metamaterial at microwave frequencies. While theoretically predicted by Victor Veselago thirty years prior to that experiment, the groundbreaking nature of this experiment is attributed to the fact that no naturally occurring material or compound with a negative refractive index had ever been reported. This experiment transformed our perception of material properties, and altered our approach for devising new technologies and devices. Metamaterials can be considered as a design tool for creating devices that control waves in a particular way. These devices are composed of periodic or aperiodic inclusions or building blocks, such that, when they interact with an incoming wave or signal, they transform it as prescribed for. The underlying philosophy of the metamaterial paradigm is that the behavior of waves propagating within a large (many wavelengths) metamaterial composite medium can be understood from the properties 1 of its building blocks—each subwavelength in dimensions—and their mutual interactions. In other words, in the same way that atoms and molecules govern the response of naturally occurring materials, subwavelength resonant building blocks govern the response of a metamaterial. In fact, the notion of metamaterials and the ability to sculpt desired responses that may not naturally occur; has had a profound impact across numerous scientific fields, including electromagnetic [1–4], acoustic wave phenomena [5–8], materials science [9], chemistry and nanoscience [10–12]. In particular, electromagnetic metamaterials research has provided a venue to tailor material properties in ways not feasible with conventional materials [13–15], opening the door to unique and often exotic wave phenomena such as negative and near-zero refractive index materials [16–19], as well as invisibility cloaks [20] and superlenses [21–23]. The analogy between conventional materials and their constituent atoms to metamaterials and its building blocks, has been the primary approach to design and model metamaterials. This analogy has been formulated in the form of retrieval efforts where the metamaterial is replaced by a conventional material whose characteristic are effectively determined by the metamaterial inclusions. The metamaterial retrieval procedure links the underlying “microscopic” physics of the metamaterial elements to the macroscopic response (in the form of an effective medium) of a structure comprising many (hundreds or thousands) of such inclusions. While the effective constitutive parameters obtained by numerical retrieval methods must be applied with considerable caution, they have nevertheless been used with success in the design of many metamaterial structures [24–26]. By replacing the details of an artificial medium with effective constitutive parameters, we facilitate, the numerical simulations and optimization cycles, vastly reducing the computational requirements to predict the response of any metamaterial device [27]. Complex metamaterial devices have been designed and demonstrated by this technique, including the transformation optical structures that rely on precise variations of material properties throughout a vol2 Metamaterial Metasurface Waveguide-fed Metasurface ce our al S Unit cell Plane W ave Plane Wave ric lind Cy Figure 1.1: Metamaterials, Metasurfaces and Waveguide-fed Metasurfaces ume [28–31]. Despite the compelling features of volumetric metamaterials, as shown in Fig.1.1a, and their unique properties, their applications have been limited. This limitation stems from the fact that most intriguing properties of metamaterials occur near the element’s resonance, which often impose bandwidth limitations and produce large resistive losses [25]. Thus, waves propagating through any significant volume (even just a few wavelengths) of a volumetric metamaterial can be heavily attenuated. The difficulties associated with volumetric metamaterials are considerably reduced for structures consisting of just a single or a few layers of elements—also known as metasurfaces [32–34], as shown in Fig.1.1b. Being easier to design, model and implement [35–37], metasurfaces have rapidly gained traction as a major subfield in metamaterials research [38]. As quasi-optical devices [37], metasurfaces provide control of reflection and transmission across the spectrum [39], paving the way for advanced components such as flat lenses [37, 40], thin polarizers [41, 42], spatial or frequency filters [43], and holographic and diffractive elements [29, 44, 45]. Used as coatings, metasurfaces can control the absorbance and emissivity of a surface, and thus have relevance to thermophotovoltaics [46], detectors and sources [47–53]. Following the same line of thought as volumetric metamaterials, the scattering properties of a metasurface are usually characterized by a set of effective sur3 face constitutive properties, which homogenize—or average over—the properties of many identical, discrete, and periodic metamaterial elements [33, 54]. These effective medium properties are related to the discontinuity of the fields across the metasurface—approximated as having infinitesimal thickness—and they are encapsulated in a set of generalized boundary conditions [40,55–60]. Given the capabilities of metasurfaces to control waves, but without many of the limitations of volumetric metamaterials, metasurfaces have proven to be a good match for commercialization efforts, with many serious applications now being pursued on antenna technology. For such applications, instead of a single-layer metasurface, the concept of waveguidefed metasurface has been introduced. A waveguide-fed metasurface consist of one or two dimensional arrays of metamaterial elements embedded into the surface of a waveguide [38,61,62], as shown in Fig.1.1c. These metamaterial elements are excited by the guided wave inside the structure and leak portion of the guided wave to free space [63]. Waveguide-fed metasurface’s capabilities can be significantly enhanced (while maintaining hardware simplicity) by introducing dynamic components into each element and addressing them individually [59,60,64–66]. For example, a waveguide-fed metasurface can be tuned to generate a highly directive beam. Meanwhile, with a different tuning configuration, the same metasurface can generate a diverse radiation pattern, which is desired in computational microwave imaging [67–69]. This diverse set of functionalities can be accomplished without using complex RF circuitry (such as phase shifters), or any moving parts, an advantageous trait with extensive application in any technology with need for wavefront shaping. Such capabilities have garnered much attention from all scientific and industrial communities [70], especially for antenna engineering applications. For example, waveguide-fed metasurfaces have been tremendously successful in terrestrial and satellite communications [71,72], millimeter wave imaging [73–75] as well as synthetic aperture radar [76, 77]. Examples 4 a) b) c) d) e) f) Figure 1.2: a) Artificial impedance surfaces. Photo taken from [78]. b) Sinusoudally-modulated reactance surface. Photo taken from [79]. c) Holographic Metasurface Antenna for Radar (Echodyne). d) Modulated metasurface antennas for space. Photo taken from [80]. e) Metasurface Antenna for Satellite Communications (Kymeta). f) Frequency diverse metasurface antenna for millimiter wawe computational imaging. Photo taken from [73] of these applications are shown in Fig.1.2. 1.1 Motivation The state of the art on the modeling techniques for the waveguide-fed metasurfaces is relatively limited. The most prominent approach to tackle this problem is based on modulated surface impedances [59,62,72,78] as shown in Fig.1.3a. In this framework, the metasurface consists of a periodic array of metamaterial elements patterned on a grounded dielectric substrate [80, 81]. The elements are modeled as an effective surface impedance with a periodic modulation [59, 79, 82–85]. This interpretation of metasurface was inspired by the original works on homogenization of metamaterials, as well as by the initial works of Oliner and Jackson [83,86,87,87,88]. They predicted that a modulated impedance surface can support leaky or surface waves [59, 72, 84], as shown in Fig.1.3b. 5 Surface retrieval methods have been utilized along with holographic design principles, leading to tremendous success in the area of antenna engineering. They have served as a useful starting point to harness the enormous palette of material response available from metamaterial elements. While more elaborated techniques can provide a rigorous description of scattering behavior, the application of such techniques tends to obviate the advantage of the metamaterial approach–which is the simplification of the design of complex electromagnetic media [89]. One of the inconvenient assumptions of the retrieval methods include periodicity and no magneto-electric coupling between metamaterial elements. The validity of such retrieval methods can thus become suspect when gradient or transformation optical media are being designed, since the metamaterial element’s distribution is not necessarily periodic. Nevertheless, it is always possible to determine the exact electromagnetic response of a collection of metamaterial elements through a full-wave simulation, whose domain includes all of the elements arranged in the configuration of interest. However, for the waveguide-fed metasurfaces, full-wave simulation methodologies easily become intractable given the large contrast between the overall size of the metasurface (many wavelengths) and the metamaterial element’s size (sub-wavelength), as shown in Fig.1.3c. In order to overcome the disadvantages of the current modeling techniques, it is crucial to understand the principles behind the operation of a waveguide-fed metasurface. An element size of one-tenth to one-fifth of a wavelength is typical for many of these structures, which leads to the key simplifying assumption that the interaction between metamaterial elements is predominantly dipolar. While it is not difficult to violate this assumption in a metamaterial with arbitrary geometry, the validity of the dipole approximation is critical for the successful application of common retrieval methods. When the higher order multipoles of nearby elements begin to couple more significantly, the effective constitutive parameters will fail to accurately describe the 6 a) b) c) Figure 1.3: State of the art on metasurface’s modeling. a) Surface’s homogeneization [33] b) Full-wave simulation (CST Microwave Studio).c) Dipole Model response of the waveguide-fed metasurface [62, 80, 81]. In addition to the dipolar response of the elements, the waveguide-fed metasurface can be analyzed by dividing the structure into two domains. The first domain is the one outside of the waveguide, where the desired response is formed by realizing a required aperture distribution; and the domain inside the waveguide, where interaction of the metamaterial elements and the guided wave gives rise to the realization of the aforementioned aperture distribution. Given that the dipolar response is the key interaction between metamaterial elements, a hybrid modeling approach in which the electric and magnetic dipole responses associated with an element are first determined, and then the properties of the overall waveguide-fed metasurface is then computed self-consistently. The dipolar approximation for a metamaterial element has been discussed for applications in free-space [89], but scarce research has been presented for metamaterial-elements embedded in waveguides [90]. The goal here is to extend the dipole model as an analytical tool for waveguide-fed metasurfaces. Unlike the free-standing metasurface or volumetric metamaterial– for which each metamaterial element can be reduced to a free space dipole– an individual metamaterial element patterned in a waveguide also interacts with the waveguide structure, as shown in Fig.1.4. By assigning an effective polarizability to a metamaterial element rather than treating the metamaterial or metasurface as a 7 Waveguide-fed Metasurface Collection of Effective Dipoles Aperture Plane Incident Field Ground Plane Figure 1.4: Dipole model adapted to waveguide-fed metasurfaces. continuous medium with constitutive parameters, it is possible to predict the overall response of the metamaterial element in the waveguide structure. The combination of polarizability extraction and the dipole representation forms an alternative, powerful modeling platform for metasurfaces and metamaterials. The perspective enabled by the dipole model has in fact important implications in modeling and designing new metasurface structures. To better illustrate this point, we should note that full potentials of metasurfaces can only be realized when we can alter each element’s response independently, rather than being limited to a periodic arrangement [64, 70]. For instance, if the desired radiation pattern is a beam in a prescribed direction, the waveguide-fed metasurface resembles a holographic leaky wave antenna (LWA) and can be used in terrestrial and satellite communications as well as in radar. Meanwhile, if the desired radiation pattern consists of an arbitrary collection of beams (resembling a speckle), then the waveguide-fed metasurface acts like a diffuser and can be used for computational imaging. The outstanding feature of a waveguide-fed metasurface is that both operations can be possible by the same hardware, if one can predict all interactions between metamaterial elements accurately, and use that to realize the required aperture field distribution. Assuming an averaged value for the surface impedance implies losing the detailed 8 control (at the individual element’s level) expected for metasurface antennas. Abandoning homogenized periodic structures and approaching the metamaterial design –in the more general sense of an array of perturbations with arbitrary spacing and shaping – has also gained traction, partially due to the extra degrees of freedom that aperiodic structures offer. In such structures, interpreting metasurface as an array of individual effective dipoles, instead of homogenization techniques, can be more effective. [91–95]. The dipolar model developed in this thesis can provide the unique platform to design and model any arrangement of metamaterial elements in an efficient, yet accurate manner. 1.2 Outline The contents of this thesis are divided as follows: I introduce the formalism of the dipole model in the context of metamaterials in chapter 2. In this chapter, the intricate steps of implementation of the model for different waveguide structures are illustrated. Special attention is devoted to the metamaterial element’s characterization in waveguides by means of polarizability extraction methods. The techniques presented in this chapter are of significant importance to differentiate the dipole model from the well-established Discrete Dipole Approximation (DDA), commonly used for predicting the scattering response of arbitrarily shaped objects. In this chapter, the model is used to simulate a planar waveguide composed of an arbitrary arrangement of tunable metamaterial elements. A full analysis of the importance of the mutual element interactions is provided, and direct comparisons between the dipole model and full-wave simulations are demonstrated. Once the fundamental processes and techniques are described, in chapter 3, I provide an application of the dipole model to be used in conjunction with microwave imaging techniques. The integration of this type of waveguide-fed metasurfaces in an imaging system is described with a basic description of the common image re9 construction techniques when using metamaterial antennas. More explicitly, this chapter focuses on the specific algorithms used for accelerated image reconstruction. I develop a novel image reconstruction technique based on the Range Migration Algorithm (RMA) to accelerate the imaging process using metamaterial apertures. The combination of the dipole model and adapted RMA provides a tremendously powerful tool for simulating and optimizing metamaterial imaging systems that would be impossible to simulate with full wave-simulations due to their large domain. Finally, in chapter 4, different examples of the use of the dipole model on waveguide-fed metasurfaces for beamforming and other microwave imaging scenarios are provided. The results presented in this chapter have direct application on the design of highly directive antennas for satellite and terrestrial communications, as well as on the design of metamaterial apertures for microwave imaging applications. Each of the aforementioned chapters includes a section summarizing the presented material, highlighting the novel contributions, and offering my outlook for the model advantages and limitations, as well as the wide range of applications where the implementation of this model as a software tool would have a profound impact. 10 2 Dipole Model for Waveguide-fed Metasurfaces In this chapter, I describe a fast, accurate and robust technique known as Dipole Model to predict the electromagnetic response of waveguide-fed metasurfaces. With this technique, the metasurface is modeled as a collection of effective dipoles that are accurately characterized by means of the polarizability, and that interact with each other as well as with the guided wave. My specific contributions to this topic are as follows: • I provide a numerical technique to characterize the response of metamaterial elements embedded in waveguide-fed metasurfaces. This technique is known as the Polarizability Extraction. Excerpts of the this discussion are taken from the paper L Pulido-Mancera, PT Bowen, MF Imani, N Kundtz, DR Smith. Polarizability extraction of complementary metamaterial elements in waveguides for aperture modeling Physical Review B, 96, 235402, 2017. More explicitly, I present two methods to extract this polarizability: The first 11 method invokes surface equivalence principles, averaging over the effective surface currents and charges induced in the elements surface in order to obtain the effective dipole moments and in turn the polarizability. The second method is based in the coupled-mode theory (CMT), from which a direct relationship between the polarizability and the amplitude coefficients of the scattered waves can be deduced. While the polarizability extraction methods based on CMT has been previously derived for periodic metasurfaces [90, 96–98], they cannot be used directly for complementary metamaterials in waveguide-fed metasurfaces. Therefore, the presented methods are novel and crucial for the numerical implementation of the dipole model. I provide examples of these numerical methods on several variants of waveguide-fed metasurface elements, as well as in different types of waveguides, finding excellent agreement between the two, as well as with the analytical expressions derived for circular and elliptical irises. • I provide a complete description of the dipole model applied to a complex waveguide-fed metasurface which could not be modeled using effective medium techniques. In particular I consider a parallel-plate waveguide with arbitrarily arranged, complementary metamaterial elements patterned into one of the conducting surfaces. Each metamaterial is modeled as a polarizable dipole that accounts for the mutual interactions through the guided and radiated fields. By using full-wave simulations, I confirm the validity of the dipole model and demonstrate the ability to predict the scattered fields inside the waveguide, as well as the radiation patterns of arbitrarily arranged elements. Excerpts of the this discussion are taken from the paper in progress L Pulido-Mancera, MF Imani, PT Bowen, DR Smith. Analytical Modeling of 2D Waveguide-fed Metasurfaces. 12 2.1 Fundamentals of the Dipole Model In 1909, Lorentz showed that the dielectric properties of a substance could be directly related to the polarizabilities of the individual atoms of which it was composed by using the Clausius-Mossotti (or Lorentz-Lorenz) relation, when the atoms are located on a cubic lattice. This demonstration provides the physical inspiration for the Discrete Dipole Approximation (DDA). The basic idea of the DDA was introduced in 1964 by DeVoe [99], who applied it to study the optical properties of molecular aggregates. Since retardation effects were not included, De Voe’s treatment was limited to aggregates of particles that were deeply subwavelength and non-resonant. Later, the DDA was proposed by Purcell and Pennypacker for modeling light scattering produced by interstellar dust grains (inhomogeneous and anisotropic objects), ranging from nanometers to microns in size. In their work, the DDA was also referred to as the coupled dipole method (CDM). This method was later popularized by Draine and Flatau [99, 100] who released a FORTRAN software package with the DDA algorithm, and many authors have since contributed to this DDA algorithm for modeling the scattering of complex structures. The DDA has now been applied to a multitude of problems, including calculations of scattering and absorption by rough and porous particles, interstellar graphite particles, inhomogeneous particles, and scatterers placed on reflective surfaces. In the DDA approach, an arbitrarily shaped object is discretized in a set of dipoles as an initial approximation to describe the object’s scattering properties [101]. Each one of these dipoles possesses a characteristic polarizability determined by the Clausius-Mossotti (CM) relation [99]. Furthermore, the overall scattering of the object is determined by the sum of the scattered fields generated by each dipole, which in turn is determined by the Green’s function. One fundamental assumption made when modeling metamaterials is that the ele13 a) b) h x a c) Metasurface’s Farfield H tot H0 z y Dipole’s Farfield H tot z Cylindrical Source y z x y x Figure 2.1: a)The electric and magnetic fields are generated by a cylindrical source inside the waveguide, in the fundamental T M z mode. b) The incident fields excite each metamaterial element and these elements in turn scatter as dipoles. c) The overall radiation pattern of the metasurface is computed as the sum of the scattered fields by each dipole, taking into account their mutual interactions. ments that compose the structure are electrically small, and therefore their response can be modeled out the element’s multiple components. A few number of multipoles per element could be used as the basis of an accurate description of a metamaterial structure, but since the important contribution of many designs is restricted to the dipole term, then the entire metamaterial structure can be simplified as a collection of interacting dipoles. When the dipolar approximation is applied, the computational burden of complex metamaterials is reduced, while retaining the highest level of accuracy [90, 102]. When this concept is applied to waveguide-fed metasurfaces, it is important to analyze the different environments in which the metamaterial elements are embedded. In these scenarios, the concept of polarizability as established in the ClausiusMossotti relation does not hold, and other forms of extracting the polarizability are necessary. In order to explain how the dipole model is adapted to waveguide fed metasurfaces, a careful description of the waveguide structure is provided: Figure 2.1a shows a general example of a parallel plate waveguide. The waveguide is the host of the metamaterial elements and the fields generated by a cylindrical source. When a single metamaterial element is etched on the surface of the waveguide, the element acts as a dipole that perturbs the incident field inside the waveguide, and 14 leaks a portion of the energy to free-space. The energy leaked in the form of radiation, follows the radiation pattern of a dipole [88], as shown in Fig.2.1b. The total effective dipole moments correspond to the incident fields evaluated at the element’s location, multiplied by a coupling coefficient termed the dynamic polarizability. More explicitly, p “ αe pωqE loc and m “ αe pωqH loc [96,98,103–107]. When many elements are etched on the waveguide’s surface the local electromagnetic fields incident on the i´th dipole (arranged in ri locations) can be written as the incident fields E0 ,and H0 plus the scattered fields Esc ,and Hsc . These scattered fields correspond to the sum of the fields radiated by all the dipoles as Eloc pri q “ E0 pri q` ÿ Gee pri , rj qpprj q ` Gem pri , rj qmprj q (2.1a) j Hloc pri q “ H0 pri q` ÿ Gme pri , rj qpprj q ` Gmm pri , rj qmprj q (2.1b) j where Gee , Gem , Gme and Gmm correspond to the electric/magnetic components of the dyadic Green’s function, respectively, and E0 and H0 correspond to the incident fields. Examining the expressions in Eq.(2.1), it can be seen that these coupled equations capture the interaction of the incident wave with each of the metamaterial elements (through the j “ 0 terms) as well as interaction between different elements (the summation term). Recalling that Eloc “ αe´1 p and Hloc “ αe´1 m, and moving the latter term in Eq.(2.1) to the left side of the equation, it is possible to re-write Eq.(2.1) as ˆ„ „ ˙ ˆ ˙ ˆ 0 ˙ ¯ e´1 0 ᾱ Gee Gem p E “ ´1 ´ ¯ 0 ᾱm Gme Gmm m H0 15 (2.2) Equation (2.2) represents a matrix system that can be solved for the total electric and magnetic dipole moments representing each metamaterial elements, taking into account their mutual interactions. This collection of effective dipole moments resemble the surface current over the top of the waveguide-fed metasurface, as shown in Fig.2.1c. In the following sections, special attention is devoted to the polarizability matrix, as well as to the Green’s function matrix, taking into account that their definitions change for waveguide-fed metasurfaces, in contrast with the expressions known from previous DDA approaches. 2.2 Polarizability Extraction Techniques When a dipole is placed in a waveguide environment, it emits a field that interacts with the environment, as well as with the dipole itself. This phenomenon of selfinteraction– or radiation reaction–occurs even when a dipole radiates in free space, where it exerts a force on itself that opposes its oscillation and decays its amplitude in time [108]. This is the mechanism by which the dipole loses energy in accordance with its radiation losses. The self-interaction of a dipole is commonly represented by the Green’s function at the location of the dipole, where special care is taken due to singularity of the Green’s function at the origin [97, 108]. If a dipole is placed in free space, then a Taylor series expansion of the Green’s function shows that ImtGpr0 , r0 qu “ k 3 {6π [109, 110], and this yields the radiation reaction to the polarizability of a dipole in free space αmm “ α̃mm , 1 ` iα̃mm k 3 {6π (2.3) where α̃m is the intrinsic polarizability of the element – which depends on its geometry only–αm is the effective polarizability – which accounts for the fact that the element is embedded in a particular environment; and k is the free-space wavenumber. The 16 expression in Eq.(2.3) is often known as the radiation reaction correction or the SipeKranendonk relation [109,111,112]. In the particular scenario that a dipole is placed on the surface of a rectangular or planar waveguide, the dipoles fields interact with the walls of the waveguide, and then exert a force that interacts with the dipole. By means of the method of images, the element embedded in a waveguide structure can be seen as a dipole embedded in a lattice of dipoles due to its self-images formed by the walls of the waveguide. In this manner, the self-interaction of a dipole in a waveguide environment can again be represented by the Green’s function of the self-images array. [113–115]. The fact that the dipole is embedded in such lattice is the physical phenomenon that differentiates intrinsic polarizability α̃ (if the dipole did not have any images) and the effective polarizability α (accounting for the multiple self-images of the dipole). In these scenarios, the self-image of the dipoles dipoles and their mutual interaction with the dipole itself must be taken into account. This idea has been the basis of previous works where the effective polarizability of an element in an array is calculated based on the intrinsic polarizability of a metamaterial element in free space [116], and the detailed algebraic and mathematical computation of the interaction constant in the array [96, 98, 108]. Rather than working through this complication, we can instead apply a numerical polarizability extraction procedure using the waveguide modes, arriving at an effective polarizability that also captures the waveguide interactions. Toward this goal, we note that an effective polarizability might be ascribed to any arrangement of dipoles where there is sufficient symmetry in the system and the incident field exciting all the dipole moments is equal. Under these assumptions the total field incident on the i´th dipole (arranged in ai locations) can be written as the incident field plus the sum of the fields radiated by all the dipoles in the space 17 as ¯ ´1 mpa q Hloc pai q “ α̃ i mm ÿ ¯ pa ´ a qmpa q, “ H0 pai q ` Ḡ i j j (2.4) j ¯ ¯ where α̃ mm is the intrinsic polarizability in its tensor form, Ḡpai ´ aj q represents the Green’s function and the j “ i terms in the sum represents the self-interaction of the dipole. All the mpai q terms may be collected leading to ¯ p0qqmpa q “ H0 pa q ` ¯ ´1 ´ Ḡ pα̃ i i mm ÿ ¯ pa ´ a qmpa q, Ḡ i j j (2.5) j‰i moreover, if it is known by the symmetry of the problem that mpaj q “ mpai q for all j, then the previous equation may be written in the form ¯ ´1 δ ´ pα̃ mm i,j ÿ ¯ pa ´ a qqmpa q “ H0 pa q. Ḡ i j j i (2.6) j ¯ (in The quantity that multiplies mpaj q becomes the effective polarizability ᾱ its tensor form), and the infinite sum per the Green’s function is defined as the interaction constant, which changes according to the dipole array, i.e. such definition of effective polarizability is nonlocal. It is important highlight that, while the effective polarizability changes depending on the environment in which the metamaterial element is embedded, it does not change with the incident field. For example, when a single metamaterial element is embedded in a planar waveguide–located at x0 – its effective polarizability can be found as α “ mpx0 q{H0 px0 q and this polarizability is equal to the extracted polarizability when the element is placed at any arbitrary location x1 in the planar waveguide, despite the incident field exciting the element is different, α “ mpx1 q{H0 px1 q. 18 a) b) n n z y z x y x Figure 2.2: Extraction of the Polarizability Tensor by means of Surface Equivalence Principle. Given the location of the source and the location of the element, all components of the polarizability tensor can be found following Eqs. (2.8). In the following subsections, two novel methodologies developed to extract the polarizability of complementary metamaterial elements are presented: the first method invokes surface equivalence principles, averaging over the effective surface currents and charges induced in the element’s surface in order to obtain the effective dipole moments, i.e. the polarizability. The second method is based on the coupled mode theory, from which a direct relationship between the effective polarizability and the amplitude coefficients of the scattered waves inside the waveguide can be deduced. It is demonstrated that these methods find excellent agreement between the two, as well as with the analytical expressions derived for circular and elliptical irises. For the sake of demonstration, the effective polarizability of different metamaterial elements embedded in rectangular waveguides as well as in parallel plate waveguides, is presented. The polarizability extraction techniques presented here should be performed for a single metamaterial element in the absence of any other element on the waveguide. 2.2.1 Method 1: Surface Equivalence Principle To begin, we characterize the response of each complementary metamaterial element [117] using the surface equivalence principle [63]: This principle states that the 19 electric field on the boundary of a domain is equivalent to a magnetic surface current Km “ E ˆ n̂, where E is the total electric field on the surface of the waveguide and n̂ is the normal to this surface. Knowing Km and the corresponding Green’s functions, one can determine the electromagnetic fields inside the waveguide [88,116]. Applying this principle to the metamaterial elements shown in Fig. 2.2, we observe that the tangential electric field is zero everywhere on the waveguide surface except over the void regions defining the metamaterial element (surface highlighted in red); if the element is deeply subwavelength, then the field scattered into the far-field may be approximated merely by the first term of the multipole expansions of Km . In the first order approximation, the magnetic dipole moment representing the metamaterial element can be calculated as ż 1 n̂ ˆ Etan da m“ iµω ż p “ x̂ ¨ Etan da. (2.7) The integration is performed over the surface of the element, n̂ “ ẑ is the vector normal to the top surface, and Etan corresponds to the tangential field at the surface of the iris. Notice that the magnetic dipole moment m can be decomposed into mx and my . These values are related to the incident magnetic field through a dynamic polarizability, defined as tensor. Placing the metamaterial element in a position p, it is where the incident magnetic field has only one component, i.e. H0 “ H0 x possible to simplify our calculations. As shown in Fig. 2.2, there are two different scenarios required to find the the four components of the polarizability tensor. In the first scenario, the element’s main axis is oriented perpendicular with respect to H0x in our coordinate system. When the element is excited by the incident field, it induces a magnetic dipole moment m ~ “ tmx , my u, which when divided by H0x , yields the two first components of the polarizability tensor as αxx “ mx {H0x and 20 p Port 1 m Port 1 y ∆ z x Figure 2.3: Metamaterial element effectively acts as a electric and magnetic dipole that scatters inside the waveguide. αxy “ my {H0x . The other components of the polarizability tensor can be found from a second scenario, where the element is placed parallel to H0x . To further simplify our calculations, we also utilize the elements symmetry. If the element possesses mirror symmetry, the extracted polarizabilities αxy “ αyx should be equal to zero. These considerations can be summarized for the final expressions for the polarizability as m αxx m αyx 1 “ iµωH0x ż ´1 “ iµωH0x ż e αzz Eytan da m αxy AK Extan da m αyy Ak 1 “ E0z ´1 “ iµωH0x ż 1 “ iµωH0x ż Extan da AK Eytan da (2.8) Ak ż xExtan ` yEytan da Ak where the subindex Ak corresponds to the element’s main axis orientation parallel to the incident field, and AK corresponds to the element’s rotated 90 degrees. 2.2.2 Method 2: Coupled Mode Theory and Scattering Parameters While calculating the polarizability of a metamaterial element by means of Eq.(2.7) provides a physically accurate characterization, the integration over the surface of the element can be complicated to compute for all desired frequency points and for arbitrary geometries. For example, this integration may also be subject to numerical 21 inaccuracies due to singularities near edges or coarse meshing, as it is especially the case for resonant elements such as those examined in subsection 2.2.3. Instead of the direct integration, in this subsection we consider the extraction of the polarizabilities from the fields scattered by the element into the waveguide. For this calculation, we apply coupled mode theory to determine the coupling of the element embedded in a waveguide to the forward and backward scattered fields. The fields inside the waveguide at any plane of constant z (along the propagation direction as shown in 2.3) can be expanded as a discrete sum of orthogonal modes. These modes are defined as [63] ´iβn z E` n “ pEnt px, yq ` Enz px, yqq e (2.9a) ´iβn z H` n “ pHnt px, yq ` Hnz px, yqq e (2.9b) iβn z E´ n “ pEnt px, yq ´ Enz px, yqq e (2.9c) iβn z H´ n “ p´Hnt px, yq ` Hnz px, yqq e (2.9d) ` where E´ n and En are respectively the waveguide modes traveling in the backwards and forwards directions. The subscript “t” refers to the component of the fields that are transverse to the direction of propagation, and βn is the propagation constant of the nth mode. The mode normalization used in Eq.(2.9) is defined from the integral over the cross section of the waveguide, such that ż En ¨ Em da “ δmn , (2.10) where δmn is 1 for n “ m and 0 otherwise. Furthermore, the magnetic fields are normalized as [63] ż Hn ¨ Hm da “ δmn {Zn2 , 22 (2.11) where the wave impedance Zn is defined as a normalization constant for each mode as Zn “ ş 1 . En ˆ Hn ¨ n̂da (2.12) In Eq.(2.12) the integration is over the cross sectional surface of the waveguide, i.e. the surface representing Port 1 in Fig. 2.3. Consider a metamaterial element placed at the center of the top plate of the waveguide. We assume that the incident field is the forward-propagating fundamental mode—coming from Port 1 and de-embedded a distance ∆—with unit amplitude E0` , as shown in Fig. 2.3. When the metamaterial element is present, it couples and scatters to all modes. While the element has a finite size, for points inside the waveguide that are few wavelengths away, the element is well-approximated as a point dipole. In the absence of the metamaterial element, the total field is simply the incident field, which is identical in both the forward and backward directions. As a result, we express the modal decomposition of the total (both incident and scattered) fields into backwards propagating modes at Port 1 (z “ ´∆) as E´ “ E` 0 ` ÿ ´ A´ n En , (2.13) n where A´ n are the amplitudes of the modes scattered by the element in the backwards direction, and n is the mode number. Similarly, a modal decomposition of the fields in the plane of z “ `∆ into forward propagating modes yields ÿ ` E` “ E` A` 0 ` n En (2.14) n where A` n are likewise the mode amplitude coefficients of the scattered field by the element in the forward direction, and the incident field E` 0 has been written as a separate term. In these calculations, ∆ can be any distance as long as it is larger than the size of metamaterial element. 23 It is first considered a volume within the waveguide that encompasses the metamaterial element (bounded by the two ports). The incident field impinging on the element will induce a set of fields that we denote as E and H. Within the coupled mode formulation, these fields can be related to the waveguide modes through Poynting’s theorem, or p˘q p˘q p˘q ∇ ¨ pE ˆ Hp˘q n ´ En ˆ Hq “ Je ¨ En ´ Jm ¨ Hn . (2.15) Integrating Eq.(2.15) over the volume V the portion of the waveguide between the two ports, and applying the divergence theorem, Eq.(2.15) becomes ż p˘q pE ˆ Hp˘q n ´ En ˆ Hq ¨ n̂da “ S ż ˘ Je ¨ E ˘ n ´ Jm ¨ Hn dV (2.16) V where S is the closed surface that encloses V and n̂ is an outwardly directed normal. Since the waveguide walls are assumed to be perfectly conducting, the only nonzero contributions to the surface integrals arise from the surfaces representing Port 1 and Port 2 (depicted in Fig.2.3), and the surface of the metamaterial element. Since the field E can be written as a fictitious magnetic surface current through Km “ E ˆ n̂ and H can be related to a fictitious electric surface current in the same way, then the surface integral in Eq.(2.16) indicates the manner in which the effective dipoles representing the metamaterial element couple to each of the waveguide modes, as might be expected from Lorentz reciprocity. p˘q The amplitude coefficients An determine the perturbation of the fundamental mode, and therefore, they have direct relationship with the effective dipole moment. p˘q In order to obtain the amplitude coefficients An , we assume there are no current sources in the volume, implying that the volume integral in Eq.(2.16) vanishes. Substituting the expansions of the fields in Eq.(2.14) and Eq.(2.13) into Eq.(2.16) and 24 using the orthogonality relations in Eq.(2.10) and Eq.(2.11), we obtain the amplitude coefficients as an overlap integral of the waveguide mode fields with the total field taken over the surface of the iris. More explicitly, the amplitude coefficients can be found as Ap˘q n Zn “ 2 ż p¯q pE ˆ Hp¯q n ´ En ˆ Hq ¨ nda. (2.17) element In an alternative approach, the electric field in the aperture could be considered zero and replaced by an equivalent electric and magnetic surface current. In this case, the surface integral vanishes everywhere except over the surfaces of the ports, but the volume integral over the metamaterial element becomes a surface integral of the equivalent surface currents Km “ E ˆ n̂ and Ke “ ´H ˆ n̂. Using Eq.(2.13) and Eq.(2.14) and invoking orthogonality, we obtain Ap˘q n Zn “ 2 ż p¯q pKe ¨ Ep¯q n ´ Km ¨ Hn qda. (2.18) element Since the metamaterial element is deeply subwavelength, the fields of the waveguide modes can be expanded in a Taylor series around the center of the element. The lowest order term is constant over the surface of the element, yielding « ż Z n p˘q p¯q An “ En px0 q ¨ Ke da 2 element ff ż ´ Hp¯q n px0 q Km da . ¨ (2.19) element As previously stated in Eq.(2.7), the two integrals in Eq.(2.19) are proportional to the electric and magnetic dipole moments, p and m. Therefore, the final expression for the amplitude coefficients in terms of these dipole moments is given by ˘ iωZn ` ´ A` E n ¨ p ´ µ0 H ´ (2.20a) n “ n ¨m 2 A´ n “ ˘ iωZn ` ` E n ¨ p ´ µ0 H` ¨ m . n 2 25 (2.20b) Equation (2.7) shows that the dipole moments are related to the incident fields, which in turn can be expanded in terms of eigenmodes. Since the incident field is the fundamental mode, the polarizability is defined by ¯ ee E` p “ ᾱ 0 (2.21a) ¯ mm H` m “ ᾱ 0. (2.21b) Due to the symmetry of the fields in the rectangular waveguide, the αzmm component cannot be excited by the z-component of the magnetic field. Hence, Eq.(2.20) reduces to two coupled equations with two unknowns: αey and αmx , which can be recast as ˘ iωZn ` ` ´ ` ´ αey E0y Eny ´ µ0 αmx H0x Hnx 2 (2.22a) ˘ iωZn ` ` ` ` ` αey E0y Eny ´ µ0 αmx H0x Hnx . 2 (2.22b) A` n “ A´ n “ Considering the orthogonality of the eigenmodes and the symmetry properties of the electromagnetic fields—the transverse components of the electric field are symmetric ` ´ ), while the magnetic field is antisymmetric “ Eny under a flip of direction (Eny ` ´ )—we can solve Eq.(2.22) in order to find the polarizabilities as “ ´Hnx (Hnx αey “ αmx “ ´ 2 pA` 0 ` A0 q ` 2 iωZn pE0y q (2.23a) ´ 2 pA` 0 ´ A0 q ` 2 . iωZn µpH0x q (2.23b) For the fundamental mode (monomode propagation), the normalized fields and impedance at the dipole location are given by ` 2 |E0y | “ 4 ab ` 2 |H0x | “ 2 4β10 abZ02 k 2 Z0 “ ηk{β10 (2.24) where η is the vacuum impedance. Furthermore, the amplitude coefficients A` 0 and A´ 0 correspond to the amplitude terms for the fundamental mode of the scattered 26 fields in the forward and backward directions. Therefore they are directly related to the scattering parameters with respect to each port: the reflected field, related to A´ 0 is proportional to the reflection coefficient i.e. S11 , while the transmitted field related to A` 0 is proportional to the transmission coefficient S21 and the incident field in the forward direction. More explicitly, these relationships are expressed by A´ 0 “ S11 A` 0 “ S21 ´ 1. (2.25) Taking into account Eq.(2.25) in conjunction with Eq.(2.23) and Eq.(2.24) it is possible to find the final expression for the polarizabilities as αey “ αmx “ ´iabβ10 ` ´iabβ10 pA0 ` A´ pS21 ` S11 ´ 1q 0q “ 2 2k 2k 2 (2.26a) ´iab ` ´iab pA0 ´ A´ pS21 ´ S11 ´ 1q. 0q “ 2β10 2β10 (2.26b) Equation (2.26) provides the polarizabilities of any metamaterial element embedded in a rectangular waveguide in terms of the scattering parameters, which can be obtained from direct measurement or full-wave simulation. Another important point to note is that we have assumed ports which only excite/represent single mode. This condition should be applied when simulating these structures in numerical solvers. More importantly, since it is cumbersome in experiment to excite purely the fundamental mode, the ports should be placed at least one wavelength away from the metamaterial element to ensure the non-propagating higher order modes have decayed. Equations(2.26) are similar to the expressions found for the effective polarizabilities of metamaterial elements in periodic metasurfaces [96, 98]. 2.2.3 Simulated Results in Rectangular Waveguides By using full-wave simulation, it is possible to extract the effective polarizability of arbitrary metamaterial elements patterned into rectangular waveguides from the two different approaches described in the previous subsections. For both extraction 27 techniques, a single full-wave simulation in CST Microwave Studio is performed assuming a waveguide designed to operate over frequencies in the X-band (8-12 GHz). The waveguide dimensions are a “ 21.94 mm, b “ 5 mm, and L “ 22.7 mm. We perform this simulation for several different metamaterial element geometries: circular iris, elliptical iris, iris-coupled patch antenna, and the cELC resonator [34, 118, 119]. One important characteristic of the ELC is that it induces two orthogonal polarizabilities at different resonance frequencies. As such, the appropriate selection of its geometrical parameters permits the independent control of it’s polarization. In addition to the methods described above, the dipole moments of simple geometries, such as an elliptical iris may be also obtained from the static dipole moments of general ellipsoidal dielectric and permeable magnetic bodies. Consider an elliptically shaped aperture with the major axis along the x´direction and minor axis along the z´direction. Let the major radius be l1 and minor radius l2 . In the static limit, the intrinsic polarizabilities of such an elliptical iris (static case) are given by [116] 4πl13 e2 3rEpeq ´ Kpeqs (2.27a) 4πl13 e2 p1 ´ e2 q 3rEpeq ´ p1 ´ e2 qKpeqs (2.27b) 4πl13 p1 ´ e2 q , 3Epeq (2.27c) α̃mx “ α̃mz “ α̃ey “ ´ where e “ a 1 ´ pl2 {l1 q2 (assuming l1 ą l2 ) is the eccentricity of the ellipse, and Kpeq and Epeq are the complete elliptic integrals of the first and second kind, respectively. If e “ 0 these expressions reduce to the static polarizabilities of circular irises [63]. Equation (2.27) corresponds to the intrinsic polarizability of the irises in free space. If instead of having a dipole in free space, the dipole is placed just above an infinite ground plane, the dipole radiates twice as much energy, and so the interaction constant that accounts for the Sipe-Kranendonk relation is ImtGpr0 , r0 qu “ k 3 {3π. 28 If the dipole is a complementary metamaterial element embedded in a waveguide wall, then it will radiate both into the upper half space and into the waveguide, and so the radiation reaction correction would need to take into account both scattered fields. Considering that this correction must account for half of the radiation in freespace, as in Eq.(2.3), and half of the radiation inside the waveguide, the corrected polarizability has the form αm “ α̃m , 1 ` iα̃m pk 3 {3π ` k{abq (2.28) where a and b correspond to the rectangular waveguide dimensions. Equation (2.28) has significant implications on any polarizability extraction method that deals with waveguide integrated metamaterial elements. For example, if the static polarizability of an element is calculated with Eq.(2.27), then the radiation reaction correction will be different depending on whether that element is placed in a rectangular or a planar waveguide or a cavity, and so the proper interaction constant will need to be applied in each environment. Figure 2.4 shows the polarizability of simple circular and elliptical irises computed using the two methods described in this paper as well as the theoretical methods, denoted by Bethe Theory and given by Eq.(2.27) in conjunction with Eq.(2.28). Equivalence Principle plots correspond to the polarizability extracted from Eq.(2.8) and Coupled Mode Theory plots correspond to the polarizability extracted from Eq.(2.26). As shown, excellent agreement between the analytical expressions and the numerical extractions is obtained, verifying the proposed methods. Since the circular iris considered here does not possess a resonance, it is expected that the effective polarizabilities extracted numerically are well-approximated by the theoretical expressions. Next, we examine the case of an elliptical iris, as shown in Fig.2.4b. The effective polarizabilities computed using the two numerical extraction methods of previous section exhibit excellent agreement. However, as the frequency increases, the nu29 a) y 0.5 0 -0.5 -1 b) a l1 10 b z b L x y x 10-8 1 R z 1.5 a 8 9 10 11 12 10-8 5 0 L -5 8 9 10 11 12 Figure 2.4: Effective Polarizability of small apertures. a) Circular iris. Dimensions are R “ 2 mm. b) Elliptical iris. Dimensions are l1 “ 5 mm and l2 “ 0.5 mm. merical extraction methods differ from the analytical expression from Bethe Theory. This is expected since the elliptical iris supports a resonance over the frequency band of interest, which is not captured in the analytical expressions derived for the static field. This case further highlights the need for a precise numerical method to compute the polarizability of a metamaterial element. While the circular and the elliptical irises may be analyzed using analytical expressions for the polarizabilities derived in the static limit, such closed-form expressions are not available for most metamaterial designs. For example, an element of potential interest in the design of metasurface antennas is the iris-fed patch, shown in Fig.2.5a [70]. The inclusion of the metallic patch above the iris enhances the resonant response of the element, as exemplified by the narrower and stronger resonant response [103]. Another common metamaterial element is the complementary electric LC resonator (cELC), shown in Fig.2.5b, commonly used in metasurface antenna designs [120]. The resonant response of the cELC is highly susceptible to variations 30 a) a l1 b z 2 y L x 10-7 0 -2 -4 8 9 10 11 12 b) 4 a 10-7 l1 z y x b l2 2 0 -2 L -4 8 9 10 11 12 Figure 2.5: Effective Polarizability of small apertures. a) Iris-fed patch. For this example, The iris dimensions are same as in Fig.2.4b, and the square patch size is lp “ 5 mm . b) cELC resonator. Its dimensions are l1 “ 5 mm and l2 “ 5 mm. The element’s width is w “ 1 mm in its geometry [118,120,121]. For both elements, we observe excellent agreement between the two numerical methods: Equivalence Principle based on equation Eq.(2.8), and Coupled Mode Theory, based on equation Eq.(2.26), as shown in Fig. 2.4 and Fig. 2.5. The geometry of metamaterial elements can be quite complicated, such that the numerical integrals in Eq.(2.8) are likely to yield inaccuracies. For this reason, the extraction based on computing the waveguide scattering parameters is more reliable and easier to implement. In fact, this method can also be used in measurements on fabricated samples since the scattering parameters can be measured directly. It is worth noting that in all of the results presented in Fig.2.4 and Fig. 2.5 the electric polarizability is much smaller than the magnetic polarizability—in fact, three orders of magnitude smaller. This phenomenon is expected considering that the geometry under study corresponds to a small opening in a metallic wall. 31 y L z x h Dipole Source Figure 2.6: Planar waveguide with a cELC etched at the center. The traveling wave is excited by an electric dipole oriented along the z´direction. The waveguide dimensions are L “ 100mm, h “ 1.27mm. Dipole source location 0.45L 2.2.4 Coupled-mode theory with Cylindrical waves In the case of a parallel plate waveguide the polarizability extraction method based on the Surface Equivalence Principle holds, but the nature of the Coupled Mode Theory changes from the formulation above. We begin by considering that a metamaterial element scattering into a waveguide can be described in terms of a sum of waveguide modes. Because the element is placed in the upper surface of the waveguide, the boundary condition dictates the tangential electric field and the normal magnetic field to be zero and the element can only couple to the transverse magnetic (TM) modes. Since the natural symmetry of the system is cylindrical, mode decomposition is simpler if we use cylindrical coordinates (r, θ). Setting the origin of the coordinate system to the center of the metamaterial element, the z-components of the scattered electric field—for the TM modes characterized by the pm, nq indices— are given by sc Ez,c “ βm p2q H pβm rq cospnθq k n (2.29a) sc Ez,s “ βm p2q H pβm rq sinpnθq k n (2.29b) 32 where the subscripts “c” and “s” refer to modes that have angular dependence cospnθq and sinpnθq, respectively. The propagation constant is given by βm “ a k 2 ´ pmπ{hq2 , where h is the height of the waveguide. Invoking the superposition principle, the total solution for the z-component of the scattered electric field can be expressed as Ez “ ÿÿ n mn mn Asmn Ez,s ` Acmn Ez,c . (2.30) m When h ă π{k, only the m “ 0 mode is propagating, and in this case the electric field at all points where r " h{π is dominated by the m “ 0 mode. Therefore we can reduce Eq.(2.30) to Ez “ ÿ 0n 0n Asn Ez,s ` Acn Ez,c . (2.31) n The m “ 0 modes are given by 0n Ez,c “ Hnp2q pkrq cospnθq (2.32a) 0n Ez,s “ Hnp2q pkrq sinpnθq. (2.32b) The amplitude coefficients, An , can be found from the scattered electric field Ez using the orthogonality of the tsinpθq, cospθqu basis. By integrating over a circle of radius r centered at the origin of the metamaterial element, as shown in Fig.2.2c-d, it is possible to define the amplitude coefficients as Asn Acn 1 “ lim 2 pkrq rÑ8 πHn ż 2π Ez pr, θq sinpnθqdθ (2.33a) 0 1 “ lim 2 pkrq rÑ8 πp1 ` δn0 qHn ż 2π Ez pr, θq cospnθqdθ. (2.33b) 0 The cylindrical wave propagating through the waveguide may excite the two tangential components of the magnetic polarizability, which leads to a more complete 33 characterization of the polarizability tensor. The scattered fields generated by the metamaterial element can be represented as the sum of the moments of the surface current Jnm multiplied by the different eigenmodes of the scattered fields shown in Eq.(2.32). More explicitly, this relationship is given by Ez “ mx Z0 k 2 01 my Z0 k 2 01 ´ipz k 2 00 Ez,s ` Ez,c ` E . 4h 4h 4h0 z,c (2.34) A direct mapping between Eq.(2.34) and Eq.(2.33) demonstrates that the first three amplitude coefficients, tAc0 , Ac1 , As2 u are directly related to the three dominant dipole moments as [97] mx “ As1 4h Z0 k 2 my “ Ac1 4h Z0 k 2 pz “ Ac0 i4h0 . k2 (2.35) The effective polarizabilities given the incident wave due to the line source, can be directly obtained from their corresponding dipole moments given in Eq. (2.35) as p αzz “ pz {Ez0 m αxy “ my {Hx0 m αxx “ mx {Hx0 . (2.36) For clarification, the double indices on the polarizabilities represent the entry in the polarizability tensor; for example, αxy represents the component of the polarizability that generates a dipole moment oriented in y, due to the x component of the incident field. In order to find all three components of this tensor, it is necessary to rotate the metamaterial element by π{2, and perform the same extraction technique. 2.2.5 Simulated Results in Planar Waveguides To better illustrate the utility of Eq. (2.33), we consider a lossless parallel plate waveguide, fed by a cylindrical source oriented along the z´direction, as shown in Fig. 2.6. The source is placed far enough from the metamaterial element to avoid evanescent coupling. The metamaterial element is a cELC, with the same 34 Figure 2.7: Amplitude Coefficients of the scattered field Ez . The scattered fields from the metamaterial element are added to the incident field produced by an electric current source. geometrical parameters as the one used in the previous subsection. Since the fullwave simulation domain represents the total field instead of the scattered field, this structure is simulated with and without the cELC, and the difference of the two simulation results are taken to obtain the scattered field due to the metamaterial element, such that Ezsc “ Eztot ´ Ez0 at the plane z “ h{2. Once the scattered field is computed, the integration outlined in Eq. (2.33) is performed to find the amplitude coefficients. The integration radius is selected electrically large enough so that the evanescent modes have decayed —it is also ensured the integration curve does not contain the cylindrical source. Figure 2.7 shows the magnitude of the amplitude coefficients for the scattered fields computed for the metamaterial element shown in Fig.2.6. To better illustrate the physics behind these coefficients, we apply Poynting’s theorem Eq. (2.15) which directly links the amplitude coefficients to the dominant dipole moments of the metamaterial element. In contrast to the rectangular waveguide (1D-waveguide) 35 examined in section 2.2, the amplitude coefficients are not related to the scattering parameters, but rather to the scattered fields, by means of Eq. (2.33). Moreover, while the location of the metamaterial element in the rectangular waveguide limits the calculation of a single component of the magnetic polarizability, such limitation disappears in the case of the planar waveguide. As shown in Fig. 2.7, the predominant amplitude mode is As1 , which is directly associated with mx , while the amplitude of the modes Ac0 and Ac1 , associated with pz and my , are significantly smaller—by two orders of magnitude. The electric polarizability and two of the components of the magnetic polarizability tensor are thereby obtained and shown in Fig.2.8a. As shown, excellent agreement for the magnetic polarizabilities is obtained between the two numerical polarizability extraction methods. For this particular example, note m that the polarizability αyy has a resonant response out of the X-band, but its geome- try can be modified such that it has both resonances appear in the same band [121]. In the case of the electric polarizability (Fig.2.8b), the numerical values obtained are significantly smaller, which makes it more susceptible to numerical inaccuracies when the integration in Eq. 2.33 is performed. It is important to highlight in this example that the term As3 is associated with the quadrupole moment. This term has been traditionally neglected in most metamaterial design strategy. The example at hand provides a useful framework to examine the contribution of the quadrupole term and the error introduced by neglecting it. To study the impact of the quadrupole term, we compared the simulated scattered field (shown in Fig. 2.9 first row) with its theoretical expression Eq. 2.31 up to only the dipolar contribution, as shown in the second row of Fig.2.9. We observe excellent agreement between the two rows, confirming the assumption that the main contribution of the scattered field is dipolar. The physical implications of this result can be understood by calculating the difference between the scattered fields from a full-wave simulation and from the analytical expression in (Eq. 2.31). As shown in 36 a) b) z z z z z Figure 2.8: Effective magnetic polarizabilities calculated for the ELC embedded in the planar waveguide. The metamaterial element size is enlarged in the figure to clarify its orientation with respect to the incident field. the third row of Fig. 2.9, the error due to assuming the dominant dipolar term is several orders of magnitude smaller than the amplitude of the scattered field, and the largest discrepancy is observed within the close vicinity of the metamaterial element. This result, in conjunction with the amplitude coefficients shown in Fig.2.7 also demonstrates that most of the radiation is associated with the dipolar term and higher order modes can be ignored. However, if the elements are placed at distances where these higher order modes have not decayed, these modes can alter the cou- 37 Figure 2.9: Real part of the scattered field Ez pV {mq at different frequencies. (Top row) Full-wave simulation in CST Microwave Studio. (Middle Row) Analytic expression from Eq. 2.31 up to the dipolar term only. The difference is shown in the bottom row. pling between the two meta atoms and change the total scattered fields inside the waveguide. Extending the polarizability extraction methods to a tensor form, all the components of the tensor are extracted based on Eq.(2.8), for two different elements: an elliptical iris, and an ELC resonator. Since a numerical integral over the element’s opening is necessary to extract the polarizability via Surface Equivalence Principle, a fine mesh over the structure is necessary. The values of the polarizability tensor across the X-band are calculated of the aforementioned elements, as shown in Fig. 2.10. As expected, the elliptical iris (Fig. 2.10a possesses only one significant component of the magnetic polarizability, αxx . Meanwhile, the ELC shown in Fig. 2.10b exhibits a narrower linewidth, due to its resonant nature. The elements’ geometry in this example is selected such that one resonance (αxx ) appears in the X band and the resonance associated with the other polarization (αyy ) appears outside the band 38 a) l1 b) l1 g l2 l2 l3 w Figure 2.10: Extraction of the polarizability tensor for two different complementary metamaterial elements: a) Elliptical iris. Its dimensions are l1 “ 14.17mm, l2 “ 1.52mm b) ELC-resonator. Its dimensions are l1 “ 5.26mm, l2 “ 3.5mm, l3 “ 2mm,g “ 0.3mm, w “ 0.51mm. of operation. 2.3 Effective Dipole Moments The polarizability extraction technique presented above was performed with the element located at the origin of the coordinate system. As a first test to demonstrate the utility of the proposed model, we place the element at a different location and compute its equivalent dipole in two manners: 1) use full wave simulation and equivalent principle, as given by Eq. (2.37). 2) we calculate the equivalent dipoles using the computed polarizability and the incident magnetic field at the element’s location. The total magnetic dipole moment in any arbitrary location r of the element is given by mx “ αxx H0x prq ` αxy H0y prq (2.37) my “ αyx H0x prq ` αyy H0y prq In the parallel plate waveguide, the excitation is defined as an electric line source 39 a) z y b) x r0 z y x r0 Figure 2.11: Magnetic dipole moment at location x0 . Comparison between the magnetic dipole model as predicted in Eq.2.37, and full-wave simulation. a) Diagram of the problem. b) Magnetic dipole moment for the elliptical iris. c) Magnetic dipole moment for the ELC. While my is smaller, good agreement between the two methodologies is observed. of amplitude Ie “ 1A, from the bottom to the top plates of the waveguide. The incident magnetic field generated by this source is given by the analytical expressions H0x “ H0y iIe βm p2q H1 pβm |ρ ´ ρ1 |q sin Ψ 4 (2.38) ´iIe βm p2q “ H1 pβm |ρ ´ ρ1 |q cos Ψ 4 where the propagation constant is given by βm “ a p2πf {cq2 ´ pmπ{hq2 , |ρ ´ ρ1 | is the distance from the source to the observation point in the plane of z “ 0, and Ψ is the circumferential angle around the source. Figure 2.11 presents a comparison between the magnetic dipole moments obtained from the two different methods. 40 As shown, when the element is shifted to the position r0 “ px0 , y0 , 0q “ p20, 20, 0q mm, excellent agreement between these methods is achieved, demonstrating that the extracted polarizability accounts for the element’s response in any arbitrary location. For the particular elements presented here, both elements exhibit possesses a strong magnetic response along the x´direction, while the ELC element also exhibit tangible resonance in the other direction. In these examples, the element was only translated from the origin, however, it is important to highlight that if the element is rotated, a rotation matrix should be used over the polarizability tensor before it is replaced in Eq. 2.37, in order to find the total dipole moment. 2.4 Mutual Interaction Between Metamaterial Elements In mathematics, a Green’s function is the impulse response of an inhomogeneous differential equation defined on a domain, with specified initial conditions or boundary conditions. By using the superposition principle, the convolution of a Green’s function with an arbitrary function f pxq on that domain is the solution to the inhomogeneous differential equation for f pxq. In electromagnetics, for instance, it is of interest to solve Maxwells equations for the electric field given some driving polarization P or M ∇ ˆ ∇ ˆ E ´ k 2 E “ ω 2 µ0 P (2.39) ∇ ˆ ∇ ˆ H ´ k 2 H “ ω 2 ε0 M (2.40) When Eq.(2.39) is solved in a discrete domain, the left hand side can be considered as a matrix operator acting on the electric field and magnetic fields, more explicitly, ¯ as a matrix equal to it is possible to define the operator L̄ ¯ ” r∇ ˆ ∇ ˆ L̄ 41 ´k 2 s, (2.41) ¯ as (this is analogous for the then Eq.(2.39) can be solved simply by inverting L̄ magnetic field) ¯ E “ ω 2 µ P ð E “ ω 2 µ L̄ ¯ ´1 P L̄ 0 0 (2.42) ¯ L̄ ¯ ´1 “ I¯ the Considering that the inverse of a matrix is the matrix defined such as L̄ inverse matrix must be a function that satisfies ∇ ˆ ∇ ˆ Gpr, r1 q ´ k 2 Gpr, r1 q “ Iδpr, r1 q (2.43) The term Gpr, r1 q is the so called Dyadic Greens function. For any given problem, the Greens function can take various forms and can sometimes be found in closed form or series form. The closed form can be derived using the Sturm-Liouville Theorem [87]. However, since it is our interest to develop a general description of the dipole model, it is more convenient to describe the solution of the Dyadic Green’s in terms of the modal solutions inside the particular waveguide used. Such solution is given by Gee pr, r1 q “ ÿ Gem pr, r1 q “ ř Hm,σ prqEm,σ pr1 q Gmm pr, r1 q “ ř Em,σ prqEm,σ pr1 q m Em,σ prqHm,σ pr1 q (2.44) Hm,σ prqHm,σ pr1 q (2.45) m Gme pr, r1 q “ ÿ m m where the eigenmode solutions for the field are given by " ` Em prq Em,σ prq “ ´ Em prq " ` 1 Em pr q 1 Em,σ pr q “ ´ 1 Em pr q z ą z1 z ă z1 (2.46) z1 ą z z1 ă z (2.47) It is worth noting that Eq.(2.47) is equivalent to Eq.(2.9) for the rectangular waveguide, and to Eq.(2.38) for the planar waveguide. In the previous section we demonstrated that if the element is shifted to an arbitrary location, its total magnetic dipole moment is proportional to its polarizability 42 p q Figure 2.12: Mutual interaction between metamaterial elements. The mutual interaction must account for the propagation of the scattered fields through freespace. and the incident field at the element’s location. However, in the presence of multiple elements the total field that excites each element is the sum of the incident field plus the scattered field produced by all other dipoles. As shown in Eq.2.2, we can model this interaction in a matrix equation, by appropriately defining the matrices corresponding to the element’s polarizability and the Green’s function. With the Green’s function technique, a solution of Helmholtz equation is obtained using an impulse, Dirac-delta as the driving function. For a given problem, the Green’s function can take various forms, according to the specified boundary conditions. Considering that the metamaterial elements are patterned at the boundary of two different domains, i.e. different boundary conditions, their mutual interaction, characterized ¯ must account for the analytical expressions of by the Green’s function matrix Ḡ ij ¯ W G , and outside the the Green’s function in both domains: inside the waveguide Ḡ ij ¯ F S . Inside the waveguide we can describe the components of the Green’s waveguide Ḡ ij 43 function as (See appendix for details): G GW xx “ ´ik 2 p2q pH0 pβ|ρp ´ ρq |q 8h (2.48a) p2q ´ cosp2Ψp,q qH2 pβ|ρp ´ ρq |qq G GW xy “ ´ik 2 p2q sinp2Ψp,q qH2 pβ|ρp ´ ρq |q 8h (2.48b) G GW yx “ ´ik 2 p2q sinp2Ψp,q qH2 pβ|ρp ´ ρq |q 8h (2.48c) ´ik 2 p2q pH0 pβ|ρp ´ ρq |q 8h (2.48d) G GW yy “ p2q ´ cosp2Ψp,q qH2 pβ|ρp ´ ρq |qq. where h is the height of the parallel plate waveguide, and |ρp ´ρq | “ a pxp ´ xq q2 ` pyp ´ yq q2 ´yq is the distance between the two dipoles, and Ψp,q “ tan´1 p xypp ´x q. q Outside the waveguide, the Green’s function in free-space is given by ˆ ˙ 3 3i FS 1 ¯ Ḡ pr, r q “ ´ ´ 1 gpRqr̂r̂ k 2 R2 kR ˙ ˆ 1 i ´ gpRqI¯ ` 1` kR k 2 R2 (2.49a) ikR where R is R “ |r ´ r1 |, I¯ corresponds to the identity matrix, and gpRq “ 2 e R . The factor of 2 in gpRq accounts for the fact that when the elements are seen from the outside of the waveguide, the magnetic dipoles representing the elements are backed by the ground plane and the dipoles have their corresponding self-image. Given the expressions for the Dyadic Green’s function in Eq. (2.48) and Eq. (2.49), it is possible to compute the total magnetic dipole moments by using Eq. (2.2) and in turn, compute the scattered fields inside and outside the waveguide. 44 CST Microwave Studio Element’s Locations a) b) V/m Non-interacting Dipoles Coupled Dipoles c) V/m d) V/m f) V/m g) V/m 40 y (mm) 20 0 e) -20 V/m -40 -40 -20 0 20 40 x (mm) Figure 2.13: Comparison between the scattered fields predicted from full wave simulation and the dipole model. a) Antenna array composed of 12 ELC resonators, red dot represents the source. b) Full wave simulation. c) Dipole model without the mutual element’s interaction. d) Dipole model with the mutual element’s interaction. 2.4.1 Waveguide-fed metasurface with arbitrary element arrangement To demonstrate the utility of the model formulated above, we consider a parallel plate waveguide composed of 12 metamaterial elements placed at random locations as shown in Fig.2.13. It is worth emphasizing such an arbitrary arrangement of strongly resonant metamaterial elements cannot be modeled using any of the conventional methodologies such as modulated surface impedance models [85]. However, by using the dipole model presented here, it is possible to predict with good accuracy the electromagnetic response of such an arrangement both inside and outside the waveguide. In addition, it is worth noting that this is not a contrived configuration. In fact, such a configuration of arbitrary metamaterial elements have shown great promise in generating frequency diverse patterns used in computational microwave imaging. The model proposed here paves the way for complete analytical treatment of such structures [68, 75]. To better illustrate this point, let us examine this interaction and its role in determining each magnetic dipole complex amplitude more explicitly. The electric 45 field along the z´direction, generated by a magnetic dipole moment oriented along the x´direction, requires the Green’s function Gem zx . For any arbitrary orientation of the elements in the waveguide–which might induce both components of the magnetic dipole moments–the electric field along the z´direction is given by EzDM pρq “ ÿ em mx pρi qGem zy pρi ´ ρq ` my pρi qGzy pρi ´ ρq, (2.50) i where, as shown in the appendix ı ´iβ 2 Z0 ” p2q “ H1 pβ|ρi ´ ρ|q sin Ψi 4h ı iβ 2 Z0 ” p2q “ Gem H pβ|ρ ´ ρ|q cos Ψ i i . 1 yz 4h Gem xz (2.51) where Z0 corresponds to the free-space impedance. While the expressions for Gem remain unchanged, the magnetic dipole moment depends on the mutual interactions. Figure 2.13b-d depicts the electric field along the z´ direction, Ez , generated by all the effective magnetic dipoles when the metamaterial element used correspond to the elliptical iris shown in Fig.2.10a. Likewise, Fig.2.13e-g depicts Ez when the element corresponds to the ELC shown in Fig.2.10b. Figure 2.13b and e correspond to the field obtained from full wave simulation in CST Microwave Studio, EzCST used here for the model’s validation. To highlight the importance of element-element interactions and the utility of the proposed model, we compute Ez using the proposed dipole model in two different ways: first, we ignore the mutual interactions as shown in Fig.2.13c and f. Next, we include all the mutual interactions, as shown in and Fig.2.13d and g. Examining Figs. 2.137 b,c, and d with each other, as well as Figs.2.13e, f, and g with each other, it is noticeable that the inclusion of the mutual interactions between the metamaterial elements is crucial to obtain a precise model for waveguide-fed metasurfaces. 46 In this example, the metamaterial elements have been placed far enough from each other, such that their coupling is only through the fundamental mode of the waveguide. However, in cases where the elements are closer to each other, they may also couple to each other through higher order modes (evanescent modes). We would like to emphasize the framework proposed in this work can also capture such interactions, taking into account that the Greens function can be expressed as the modal sum of the tensor product of eigenmodes. Therefore, Eq. 2.48 can be recast to take into account evanescent guided modes for the mutual interactions. Another important fact to note here is that our model only relies on full-wave simulation of one single element one time only, and we can use that to build a complete model for such a complex metasurface. The presented example of a waveguide-fed metasurface is only 4λ length, for which full-wave simulation is possible, however, for many applications, much larger aperture with many frequency points need to be examined, and full-wave simulations are not an option. However, our method provides a simple, low cost, and yet effective method to model such complex structures. While the results presented on Fig.2.13 demonstrate in a qualitative manner the importance of the mutual interaction, we analyze the relative error between full-wave simulation (EzCST ) and the dipole model (EzDM ). More explicitly this error is defined as “ |EzCST ´ EzDM | and it is computed in four different scenarios: • Non-interacting dipoles: In this case, the magnetic dipole moments used in ¯ “ 0. The relative error is shown in Fig.2.14a Eq.2.50 result from setting Ḡ ij and Fig.2.15a for the elliptical irises and the ELC resonators respectively. As shown, the relative error is large across the waveguide’s domain. • Interacting dipoles inside the waveguide: The magnetic dipole moments used in ¯ “ Ḡ ¯ W G , as detailed in Eq. (2.48). The relative Eq.2.50 result from setting Ḡ ij error is shown in Fig.2.14b for the elliptical irises and Fig.2.15b for the ELC 47 resonators respectively. It can be noticed that the error decreases significantly when this mutual interaction is included. Therefore, the mutual element’s interactions through the waveguide play a significant role in predicting the perturbation of the fields in the waveguide, due to the presence of the dipoles. • Interacting dipoles in free space: The magnetic dipole moments used in Eq.2.50 ¯ “ Ḡ ¯ F S as detailed in Eq. 2.49. The relative result from Eq.2.2 when Ḡ pq error is shown in Fig.2.14c for the elliptical irises and Fig.2.15c for the ELC resonators respectively. It can be noticed that the error is reduced compared to the case of no interaction, but can be concluded that it is not the main source of error for the configuration at hand. Therefore, while the mutual element’s interaction through free-space is important, it is not as significant as the interaction through the waveguide in the geometries examined here. • Interacting dipoles across the surface: The magnetic dipole moments used in ¯ “ pḠ ¯ W G ` Ḡ ¯ F S q. The relative error is Eq.2.50 result from Eq.2.2 when Ḡ pq shown in Fig.2.14d for the elliptical irises and Fig.2.15d for the ELC resonators respectively. It can be noticed that the relative error is minimum when both interactions are taken into account. Once both mutual interactions are taken into account the lowest error is obtained. Furthermore, since EzCST is the sum of all eigenmodes and EzDM corresponds to the dipolar contribution only, can also reveal the role of higher order modes. Figure 2.14a-b shows the relative error for the same array of metamaterial elements, using elliptical irises and ELC irises respectively. As shown in Fig.2.14d, the higher order modes are only noticeable in the close proximity of each element’s location and decay rapidly away from the element, demonstrating that the dipolar approximation for the ellipse is quite accurate. However, dipolar assumption is only a good approximation for the metamaterial elements. While this approximation is sufficient for 48 a) b) c) d) Figure 2.14: Relative error between the scattered field from full wave simulation and the dipole model using 12 elliptical irises. a) Non-interacting dipoles. b) Interacting dipoles inside the waveguide. c) Interacting dipoles in free-space. d) Interacting dipoles across the surface. Arrows represent the elliptical iris’ size. most applications, our future works intend to fully characterize such interactions, including higher order modes. 2.4.2 Mutual Interactions Outside the Waveguide In order to predict the radiated fields from the waveguide-fed metasurface, we take into account the specific magnetic dipole moment distribution over the aperture plane, from the solution of the matrix problem depicted in Eq.2.2. In the presented dipole model, it is assumed that the points of observation are in the far-field relative a to each dipole. By using the Fraunhofer approximation, |~r ´ ~r1 | « r 1 ´ 2~r ¨ ~r1 {r2 , where r1 is the dipole’s location and r the distance of observation, it is possible to separate the radial and the angular dependences of the far-fields. In particular, the radial dependence becomes a simple pre-factor, with the angular distribution of the 49 a) b) d) c) Figure 2.15: Relative error between the scattered field from full wave simulation and the dipole model using 12 ELC resonators. a) Non-interacting dipoles. b) Interacting dipoles inside the waveguide. c) Interacting dipoles in free-space. d) Interacting dipoles across the surface. Arrows represent the ELC’s size electric field given by Er « 0 Hr « 0 Eθ « ´jωηFφ Eφ « jωηFθ (2.52a) Hθ « jωFθ (2.52b) Hφ « ´jωFφ (2.52c) where Fθ “ εe´jkr ÿ pmx,i cos θ cos φ ` my,i cos θ sin φq ˆ AF p~ri q 4jωπr i (2.53a) εe´jkr ÿ p´mx,i sin φ ` my,i cos φq ˆ AF p~ri q. 4jωπr i (2.53b) Fφ “ The term AF p~ri q corresponds to the Array Factor, which is the factor by which the directivity function of the individual dipoles is multiplied to get the directivity of the entire waveguide-fed metasurface. Figure 2.16 shows a diagram of the angular 50 a) b) z θ r φ x c) dBi d) y dBi Figure 2.16: Far-field of the waveguide-fed metasurface composed of ELC resonators. a) Waveguide-fed Metasurface. b) Equivalent dipoles on the aperture plane to calculate the Array Factor. c) Farfields obtained through full wave simulation. d) Fardields obtained with the dipole model. components θ and φ with respect to the metasurface’s orientation. For the coordinate system presented here, the array factor (AF) is given by 1 AF “ ejkr cosψ , r1 cosψ “ xi sin θ cos φ ` yi sin θ sin φ. (2.54) where pxi , yi q correspond to the element’s location at the aperture plane, and k corresponds to free-space wavenumber. In this section, we compute the predicted radiation pattern for the arbitrary arrangement of Fig. 2.13a. For comparison purposes, we also have computed the radiation pattern in full-wave simulation.The results are shown in Fig. 2.16 where excellent agreement is observed for the radiation pattern at all directions. In addition, in order to analyze the effects of the mutual interaction for the calculation of the farfield, we have compared the directivity at specific planes, 51 a) 20 10 0 -10 -20 -80 -60 -40 -20 0 20 40 60 80 -80 -60 -40 -20 0 20 40 60 80 b) 20 10 0 -10 -20 c) 20 10 0 -10 -20 d) -80 -60 -40 -20 0 20 40 60 80 -80 -60 -40 -20 0 20 40 60 80 20 10 0 -10 -20 Figure 2.17: Far-field of the waveguide-fed metasurface in the planes a) φ “ 0 and b) φ “ 90. 52 i.e. φ “ 0 and φ “ 90. As shown in Fig. 2.17, when both mutual interactions are taken into account, the best agreement between full wave simulation and the dipole model is obtained. The results in Fig. 2.17 once again highlight the capabilities of the proposed method to compute the electromagnetic response of waveguide-metasurface in an accurate, yet efficient process. Here, we did not solve for complex dispersion equations, as is the case in most conventional techniques. Instead, we employ a simple analytical formulation to relate the field of each metamaterial element to the overall response. 2.5 Summary In this chapter, I have fully described the dipole model as a modeling tool for waveguide-fed metasurfaces. It has been discussed the importance of the appropriate element’s characterization in the waveguide’s aperture, as well as the importance of the mutual dipole interactions. In the following a review of the presented contibutions is provided, including possible avenues for future research: On the polarizability extraction The cornerstone of the presented dipole model, and its main distinction with respect to previous models such as the Discrete Dipole Approximation, relies on the effective polarizability of complementary metamaterial elements embedded in waveguides. I have presented two comprehensive methods for extracting these effective polarizabilities on rectangular and parallel plate waveguides. The first method consists of direct extraction of the tangential components of the scattered fields, to find the effective dipole moment and therefore its polarizability. Meanwhile, the second method consists in using the S-parameters—along with the knowledge of the normalized fields inside the waveguide. Excellent agreement between the two methods was demonstrated and its applicability in different types of waveguide structures 53 was discussed. For example, while the method based on the scattering parameters of a rectangular waveguide can be used to characterize polarizabilities experimentally, this methodology is limited to the type of waveguide used. It is important to highlight that while the presented technique can be computationally expensive due to the fine mesh, this simulation is performed only one time for a single metamaterial element, and the simulation of the overall antenna is performed by means of the proposed Coupled dipole model described in equations 2.2. Moreover, it has been demonstrated that, while the effective polarizability changes according to the waveguide structure in which the element is embedded, it is a nonlocal property. As a non-local property, the polarizability of a single element can be calculated at any location on the waveguide and it will have the same electromagnetic response with respect to the frequency. In particular it is important to highlight that for many metamaterial elements the polarizability fits a Lorentzian response, characterized by its resonance frequency ω0 and quality factor Q. From the results presented in subsections 2.2.3 and 2.2.5, it can be observed that Q is different when the element is embedded in a rectangular waveguide or a planar waveguide, despite the geometry is equal. The physics behind this particular difference in Q can be explained by means of the Method of Images [88]. As previously described, the surface equivalence principle can be used to describe a metamaterial element as a collection of co-located electric and magnetic dipoles. Since these effective dipoles rely on the waveguide’s domain, they exist in the presence of the waveguide’s walls, which introduce the corresponding self-images of the dipole. The presence of these self images changes the total local field that excites each effective dipole, and this change can be calculated by computing the lattice interaction constants [108]. While the calculation of these interaction constants are not simple, some of them can be found in [90, 108]. As shown in Fig. 2.18 the lattice of dipole’s self-images is different between a rectangular waveguide 54 a) n z y x b) n Figure 2.18: Method of Images for Complementary Metamaterials in Waveguides. and a planar waveguide, which explains the difference in the lattice and in turn, the difference in the quality factor. On the dipolar response of metamaterial elements I have shown that the dipole modes predominate the scattering from metamaterial elements, since all higher-order multipole fields decay more rapidly with distance. Nevertheless, it should be highlighted that the dipole approximation will not always characterize effectively the response of a single metamaterial element. For example, preliminary studies have demonstrated that the metamaterial element presented in [121] does not scatter as a point dipole. In these scenarios, the coupled mode theory is particularly helpful. The method described in section 2.2.5 can be used in conjunction with Eq.(2.33) to find the amplitude of higher order modes and, instead of defining an effective polarizability, it would be possible to find a quadrupole term. As shown in Fig.2.14d, the higher order modes are only noticeable in the close proximity of each element’s location and decay rapidly away of the element, demonstrating that the dipolar approximation for the ellipse is accurate. However, dipolar assumption is only a good approximation for the metamaterial elements. 55 On the mutual interactions between dipoles I have presented a robust tool for modeling a waveguide-fed metasurface without any restrictions on the metamaterial element or its arrangement. We provided a set of analytical expressions that rigorously captures the interactions between metamaterial elements. The only assumption is that each metamaterial-element composing the metasurface is subwavelength and can be modeled as a dipole. This formulation requires only one full wave simulation of a single element to compute polarizability of each metamaterial element. Using the polarizability, along with analytically derived Greens functions, we can accurately model the response of a waveguide-fed metasurface of arbitrary size or arrangement. Is it important to highlight that the presented calculation of the scattered fields inside and outside the waveguide is computationally inexpensive and it allows for the prediction of the arbitrary radiation patterns as well as well-defined directive beams. This capability of the dipole model is particularly interesting when the antenna is composed of highly resonant elements that are addressed independently. In particular I have demonstrated that an accurate prediction of a waveguidefed metasurface requires capturing the interaction of metamaterial elements through both the waveguide and free space. This fact distinguishes this work from previous works where they relied on weak scattering of the metamaterial elements. Overall, the proposed model, offers a simple, systematic, and computationally low cost method to predict the response of a waveguide-fed metasurface. This exciting capability opens many exciting opportunities: the proposed model can be used to optimize metasurfaces used in computational microwave imaging and replace experimental trial and errors. It can be used to design and optimize electronically steerable metasurface antennas for communication and imaging purposes. These exciting outlook is topic of the upcoming chapters. 56 3 Waveguide-fed Metasurfaces for Microwave Imaging This chapter is devoted to the use of waveguide-fed metasurfaces in the area of Microwave Imaging. In particular I discuss one type of waveguide-fed metasurfaces: Dynamic Metasurface Antennas (DMAs) [64]. Since these antennas exhibit strong frequency dispersion and produce rapidly varying radiation patterns, they present unique challenges for integration with conventional imaging algorithms. I have studied the use of DMAs in computational imaging and I have focused on the image reconstruction techniques. My contributions to this topic are as follows: • I have adapted a technique known as Range Migration Algorithm (RMA) to be compatible with DMAs and accelerate the image reconstruction process. In particular, I propose a pre-processing step that transforms the measurements taken with the DMA, ultimately allowing for the processing in the spatial frequency domain. In this domain, the fast Fourier transform can efficiently reconstruct the scene. Numerical studies compare the imaging performance using the adapted RMA and previous methods used in the context of compu57 tational imaging, demonstrating that the adapted RMA can reconstruct images with comparable quality in a fraction of the time. Excerpts of this discussion are taken from the paper L Pulido-Mancera, T Fromenteze, T Sleasman, M Boyarsky, MF Imani, MS Reynolds, DR Smith. Application of range migration algorithms to imaging with a dynamic metasurface antenna JOSA B 33,10, 2610-2623 2016. • I have extended the use of the adapted RMA for the scenarios in which the DMA is moving, which allows the DMA to be used for Synthetic Aperture Radar (SAR). More explicitly, I have derived the expressions that account for the dispersion relation in different moving scenarios, and I have provided computational and experimental demonstrations. Excerpts of the this discussion are taken from the paper AV Diebold and L Pulido-Mancera, T Sleasman, M Boyarsky, MF Imani, DR Smith. Generalized range migration algorithm for synthetic aperture radar image reconstruction of metasurface antenna measurements JOSA B 34,12, 2610-2623 2017. 3.1 Microwave Imaging Microwave imaging has historically found application in non-destructive testing [122], security screening [123–125], through-wall imaging [126, 127] and subsurface detection [128]. Particularly on Radio Frequency (RF) imaging, it is appealing since microwaves and millimeter waves can penetrate optically-opaque objects. Typical high resolution RF imaging techniques make use of a large frequency bandwidth coupled with a large aperture. The frequency bandwidth provides information about the distance in dept or range, based on the echo of the signal reflected by an object. 58 Range Cross-range Figure 3.1: Microwave Imaging: Range and Crossrange. Meanwhile, a large aperture uses the angular information of the object’s reflected signal to determine the actual extent of the object itself, as shown in Fig.3.1 Some applications, such as security screening and through-wall imaging [126,129], use an array imaging scheme where multiple transmitters and receivers (MIMO) are spatially distributed to cover the observation area, as shown in Fig. 3.2a. When an enormous area is under examination—as might be the case in many geophysical related scenarios— a technique known as Synthetic Aperture Radar (SAR) is employed, in which an antenna in motion collects data over one or more paths [130,131]. In fact, SAR uses the motion of the radar antenna over a target region to provide finer spatial resolution than conventional beam-scanning radars. SAR is typically mounted on a moving platform, such as an aircraft or spacecraft, as shown in Fig.3.2b. Because it relies on radio or microwaves rather than visible light, it can see through haze, clouds, and sometimes even walls. For that reason, it has become the go-to technique for Earth sensing, security screening, and state-sponsored spying. While SAR has proven to be a reliable technique to obtain high-resolution images, it possesses practical disadvantages. Long acquisition times, complex architectures, and expensive hardware, have hindered the expansion of SAR systems and innovative platforms to simplify hardware are highly sought-after. The large effective aperture 59 a) b) c) Figure 3.2: Microwave Imaging Techniques. a) Multiple input multiple output (MIMO) antennas for security screening. b) Synthetic Aperture Radar (SAR) for landscape imaging. c) Waveguide-fed metasurfaces for computational imaging size allows SAR systems to capture the reflectivity of an object from a wide range of angles, in a manner not possible with stationary imaging systems. For the long distances considered in SAR, highly directive antennas are required. These antennas usually take the form of large phased arrays or mechanically actuated gimbal dishes. While these methods have shown excellent performance, phased array antennas are expensive, heavy, and usually consume large amounts of power since they require complex control circuitry and a large number of RF components (such as filters, amplifiers, phase shifters, etc.) [132–135]. These drawbacks have limited the development of SAR systems for more extensive scenarios. In terms of the image reconstruction, for both MIMO and SAR cases, frequency sampling of the operational spectrum and spatial sampling of the physical aperture should satisfy the Nyquist theorem to retrieve images without aliasing effects [136]. Sampling the aperture fields at the Nyquist limit provides a definitive path to diffraction-limited image reconstruction, but meeting this requirement generally results in cumbersome hardware architectures. More explicitly, Nyquist theorem requires sampling at half-wavelength of the center frequency of the operating band. However, the effective synthetic aperture path can be hundreds or thousands of wavelengths, for which either a large number of antennas, and good memory requirements 60 are necessary. One means of alleviating this hardware burden is to consider new perspectives developed in the context of computational imaging, where a multitude of imaging architectures can be considered so long as the backscattered fields on the aperture can be reasonably estimated from some number of measurements, as shown in Fig.3.2c [137–142]. Sampling at the Nyquist rate ensures that the transmitted and received fields can sample at a rate that captures all the information from a continuous-time signal of finite bandwidth. Therefore it ensures that the signal in the Fourier domain (frequency-wavenumber) forms a complete set, such that a discrete Fourier transform can be defined over the aperture field. When combined with knowledge of the free space propagator and a scattering model for objects in the scene, this Fourier transform relationship can be used as the foundation for efficient image reconstruction algorithms. From this information, the fields can be propagated from the transmitters to the object and back-propagated to the receivers, ultimately enabling reconstruction of the scene’s reflectivity distribution. For example, modern computational imaging schemes, such as coded apertures and single-pixel techniques, have been applied to achieve scene estimates with relaxed hardware constraints [143, 144]. In particular, it has been demonstrated that a frequency diverse metasurface can produce a sequence of arbitrary field patterns with low spatial correlation that can be used to acquire scene information [145–147]. An alternative platform for computational imaging is a dynamic metasurface antenna (DMA), which similarly produces spatially-diverse patterns, as described in [64, 65, 148], but by dynamically reconfiguring the properties of the metamaterial elements. The DMA is one of the examples of waveguide-fed metasurfaces shown in the previous chapter. The DMA possesses strong spatial dispersion and generates different radiation patterns at different frequencies, as such, it is an excellent candidate to be simulated by means of the dipole model. 61 The use of metasurface antennas in imaging systems simplifies the physical hardware and can increase the data acquisition rate. However, they do not generate uniform radiation patterns, and they do not provide information that can be directly transformed in the Fourier basis. Unlike the traditional transceivers used in SAR, where the signal is transmitted from a single source and received at the same single source (representing a single direction in the Fourier domain), the DMA is simultaneously illuminating from all metamaterial elements composing the structure, and the received signal can not be represented as a single Fourier component. In other words, the advantages of metasurface antennas come with a trade-off: a more computationally intensive image reconstruction problem must now be solved. The situation becomes increasingly daunting when a metasurface aperture is physically translated, as in SAR imaging, leading to a larger region of interest and larger datasets. For conventional SAR imaging, which routinely deals with large datasets, a wealth of fast and high-fidelity reconstruction algorithms have been developed [149–153]. However, SAR systems tend to be big, power hungry, and mechanically complex when they have steering mechanisms for beamforming, making them very expensive. It is the main reason why SAR is used mainly by the kind of military and government organizations that can afford it. So any way to make these SAR systems smaller, cheaper, and more efficient would be hugely significant. In the farfield approximation, in which an antenna with a constant phase center moves along a synthetic aperture path, the spatially-varying distance from the transceiver to a particular target is approximately quadratic, a phenomenon termed range migration. The analogous effect in near-field microwave imaging is the inherent curvature of the probing wavefronts. In both cases, accurate reconstruction requires that this curvature be accounted for in processing. The range migration algorithm (RMA), which will be explained in detail in the following sections, compensates for this curvature efficiently by processing the measured signal in the 62 frequency-wavenumber domain and performing frequency-dependent phase shifts, as determined by the free-space dispersion relation [154] in a process known as the Stolt Interpolation. The RMA takes advantage of efficient fast Fourier transforms (FFTs) instead of solving a matrix system, accelerating the reconstruction process and eases computational demands. However, the RMA is not compatible with the fields measured with a metasurface antenna and therefore, additional processing to ensure compatibility with Fourier transform approaches is required. This chapter addresses two important questions: • How can metasurface antennas be used in different microwave imaging scenarios? • How to adapt the image reconstruction algorithms to be compatible with these metasurface antennas? To this purpose, in Section 3.2 I study an antenna created by Sleasman et. al [64] and developed for microwave imaging called Dynamic Metasurface Antenna (DMA) and how it is used for microwave imaging. Then, in Section 3.3 an efficient method to reconstruct images obtained using a dynamic metasurface antenna is presented. Specifically, I show that the RMA can be made compatible with the metasurface architecture, which allows FFTs to be used in the image reconstruction steps. The key modification proposed consists in separating the role of the dynamic metasurface from the back-propagation step. In addition, a detailed analysis of the RMA for different scenarios of in DMA motion is presented. The derivations of the RMA for different moving scenarios are presented in Appendix B. Finally, in Sections 3.4 and 3.5, I describe the computational and experimental configuration for testing the DMA. While the results presented here use a specific DMA design, the derivations presented lay the foundation for using the RMA with many other metasurface designs, as well as other translated-array imaging configurations. 63 3.2 Dynamic Metasurface Antenna The DMA [65] consists of a microstrip waveguide with complementary electric LC (cELC) [34, 118, 119] elements patterned at a subwavelength distance into the upper conductor as shown in Fig. 3.3. Two diodes are used to modify the resonance frequency of each cELC with an applied voltage. As such, each cELC can be switched on or off by an external stimulus, creating a collection of mask that produces a variety of spatially distinct radiation patterns. At a given frequency, a number of measurements can be made by cycling through multiple masks. The operation is similar to coded aperture imaging systems, which encode a series of scene measurements onto distinct aperture masks using structured illumination [144,155–157]. Due to the large amount of degrees of freedom for the DMA (2N , where N is the number of cELCs), it possesses excellent flexibility to create numerous radiation patterns, either directive beams, or multiple beams. In the case of a DMA system, a correspondence is made between each random mask and its associated radiation pattern. The DMA can thus operate with a reduced spectral bandwidth and fast (real-time) acquisition rates [65]. The DMA can be considered a passive device in the sense that no gain components are required for its operation. A great simplification over the phased array design is thus achieved, in that no phase shifter or amplifiers are utilized [135]. However, independent control over the magnitude and phase is not achievable with most dynamic metasurfaces, because these quantities are jointly determined by the incident field and the characteristics of the resonators. Nonetheless, extraordinary control can be achieved over the collective aperture to generate the desire patterns, chiefly through the dense sampling of the guided wave enabled by the subwavelength placement of the elements. For example, in [158], steerable, highly-directive beams were generated by modeling the aperture with the dipole model in conjunction with holographic ap64 z y Rx Tx Rr Rt Scene s On Off- g State n i Tun Figure 3.3: The DMA acts as a transmitter (Tx) while a single rectangular waveguide probe (Rx) acts as a receiver. proaches. In [120], the metamaterial radiators were turned on/off to create random, spatially-diverse radiation patterns. The flexibility offered by dynamic metasurfaces may be used to steer directive beams for enhanced signal strength, create nulls in the pattern to avoid jamming, probe a large region of interest with a wide beam, or even interrogate multiple positions at once with a collection of beams. In the present context, I will develop a method that is applicable to generic radiation patterns from a dynamic metasurface—illustrated in Fig. 3.3– which is described by the following criteria: 1. The dynamic metasurface consists of a one-dimensional waveguide (e.q., rectangular waveguide, SIW, microstrip). 2. N metamaterial irises are embedded within the waveguide. Each of these elements acts as a linearly-polarized magnetic dipole, excited by the magnetic field associated with the waveguide mode. 3. The dipole response can be modified in any way, including the most complete case of individual addressability. Each tuning state is referred to as a mask. 65 To obtain a working model of the DMA, it is assumed that the magnetic field along the center of the waveguide can be approximated as Hg pyq “ H0 e´jβy (3.1) where β is the propagation constant of the waveguide, which β is real, ignoring the attenuation due to propagation of the mode down the waveguide. Each metamaterial can be represented as a magnetic polarizability with the Lorentzian form of Eq.4.1. The magnetic dipole moment induced in each metamaterial element can be found as mpyi q “ αpyi qHg pyi q “ αpyi qH0 e´jβyi . (3.2) Since each iris radiates as a magnetic dipole, the radiated field from the aperture can be found by summing the dipole contributions from each metamaterial element as U p~r; f q9 ÿ Z0 kωmpyi q i 4π|~r ´ yi ŷ| e´jk|~r´yi ŷ| sin θ (3.3) where Z0 is the vacuum impedance, θ is the angle between mpyi q and ~r, and k “ ω{c is the wavenumber in free space. In the far field, Eq.(3.3) reduces to an array factorlike calculation, or U p~r; f q9 Z0 kω ÿ αpyi qe´jβyi e´jkyi sin θ . 4π|~r| i (3.4) If it is assumed that the polarizabilities can be switched between two states by a control voltage, then the set of values tαpyi qu constitutes a mask, where each polarizability may have a value of α0 or zero. A binary case is considered, though it is possible for the polarizability to have a continuous range of magnitude or phase values depending on the tuning implementation. Equations (3.3) or (3.4)—depending 66 on the approximation—can then be used to compute the field everywhere in the scene. While the approximations used in the above equations are simplistic, they are nevertheless sufficient to model the dynamic aperture for the imaging scenarios presented here, where the focus is on the algorithm development. For an actual imaging scenario, the fields from the aperture could be measured experimentally or possibly predicted using more refined models for the aperture. 3.2.1 Imaging with the DMA As the driving frequency or the voltage tuning changes, the radiation pattern from the DMA illuminating the scene changes, as shown in Fig. 3.4a. Objects within the scene scatter the incident fields, producing a backscattered field that can be detected at the aperture plane. In a transceiver configuration, the backscattered fields are detected by the same aperture which produced the incident field. However, transceiver architectures generally require more complicated radio design, so here an alternative configuration is considered. In this configuration, a single low-gain waveguide probe is used to measure the backscattered field. The low-gain antenna ensures all backscattered radiation is collected from the scene. The aperture-to-antenna configuration can be viewed as a single-input-multiple-output (SIMO) scheme, as shown in Fig. 3.4c and discussed below. When modeling the propagation in the imaging configuration shown in Fig.3.4, the first Born approximation is assumed, which states that the total field is equal to the incident field. Mathematically, this may be represented as ż Spyt , yr ; f q “ Gpyt , ~r; f qσp~rqGpyr , ~r; f qdV (3.5) V where σp~rq is the reflectivity of the object. If the antennas can be modeled as point- 67 b) a) = Collection of Magnetic Dipoles z c) z Rx Rx Tx Tx Figure 3.4: a) DMA is the transmitter and generates diverse patterns through different masks. c) Single dipole antenna moving along a synthetic aperture path. The measurements taken in a) can be transformed to c) by properly characterizing the DMA, as shown in b). like dipoles, the fields can be expressed as 2πf 1 ej c |~r´~r | . Gp~r, ~r ; f q “ |~r ´ ~r1 | 1 (3.6) If the transmit and receive antennas are more complicated and can not be modeled as single dipoles, a more complex mathematical description of the signal is required, which requires Eq.(3.3) ż gpf q “ Ut p~r; f qσp~rqUr p~r; f qdV. (3.7) V Note that we no longer have an unambiguous source location for either the transmit or the receive antenna, so the symmetry contained in a propagator such as in 3.6 is lost. Therefore, one can only indicate the fields as coming from either the transmit or receive antenna using the subscripts as shown. Figure 3.4 represents the analogy between the two scenarios: when we have a collection of independent antennas, we measure the signal by using Eq.(3.5); while employing the dynamic metasurface, we 68 measure the signal by using Eq.(3.7). If we were able to rely solely on frequency measurements, as is possible for frequency-diverse imaging systems [68, 159], then the set of measurements indexed by frequency would have the form ż gpf q “ Hp~r; f qσp~rqdV (3.8) V where the integral is taken over the reconstruction region in the scene. Assuming that the measurements are sampled at a finite number of frequencies and the reflectivity distribution is discretized into a finite number of voxels, Eq.(3.8) can be written as a matrix equation as g “ Hσ ` n (3.9) where g is a vector of measurements, H is defined as the sensing matrix, σ is the scene reflectivity vector (to be estimated), and n is an additive noise term included for generality. Within the approximations leading to Eq.(3.8), the elements of H correspond to the product of the transmit and receive fields, so that each row is a measurement mode interrogating the scene. In the absence of any additional information regarding the propagator, Eq.(3.9) can form the basis for a reconstruction process in which the reflectivity of the scene σ is estimated using methods such as matched filtering or the Moore-Penrose pseudo-inverse. The direct solution of Eq.(3.9) represents a general approach for reconstructing a scene using arbitrary field patterns and unconventional antennas. Such an approach has been successfully applied in computational imaging systems [69, 159]. Furthermore, the form of Eq.(3.9) also works well with compressive imaging algorithms that make use of iterative methods to refine the estimation of under sampled scenes. However, calculation time and storage requirements increase rapidly as the dimensionality of the sensing matrix H grows. It is therefore desirable to either avoid or simplify the calculation of H and apply imaging processing techniques directly to g. 69 Our primary goal here is to develop a preprocessing step in which the measurements provided by a dynamic metasurface can be transformed into a set of equivalent spatial measurements emanating from a collection of effective dipole sources. While this transformation will require knowledge of the radiated fields from the dynamic metasurface, it will not require using H during image reconstruction and instead will allow the use of the RMA. To develop this preprocessing transformation, measurements recorded as a function of tuning mask must be moved from the mask domain to the spatial domain. From Eq.(3.3), the field over the aperture for a given mask is approximated as Um pyi ; f q9Z0 H0 αm pyi ; f qe´jβpf qyi . (3.10) The index m has been introduced to indicate each mask, which is a specific pattern of susceptibilities that, for our purposes, take values of either zero or unity. In addition, the measured signal at the aperture plane can be expanded in terms of the fields associated with all masks as Spyi ; f q “ ÿ gm pf qUm pyi ; f q “ ÿ Φm pyi ; f qgm pf q, (3.11) m m where the field at the aperture Um pyi ; f q has been re-named as Φm pyi ; f q9Z0 H0 αm pyi ; f qe´jβpf qyi . (3.12) and resembles the effective magnetic dipoles at the aperture plane, as shown in Fig. 3.4b. If we can assume or determine a set of aperture modes that exhibit some degree of orthonormality and completeness, then we may write ÿ Φm pyi ; f qΦm1 ` pyi ; f q “ i δm,m1 |Z0 H0 |2 (3.13) where |Z0 H0 |2 accounts for a normalization factor such that Ut p~r; f q has the same units as Gpyt , ~r; f q. If Eq.(3.13) holds, then we can invert Eq.(3.11) as by first 70 multiplying both sides by the pseudo inverse of Φm and then integrating over the dipoles along the aperture domain ÿ Φm1 ` pyi ; f qSpyi ; f q “ i ÿÿ i gm pf qΦm pyi ; f qΦm1 ` pyi ; f q (3.14) m Then, using Eq.(3.13), we find gm1 “ ÿ Φm1 ` pyi ; f qSpyi ; f q. (3.15) i If we now consider Φm pyi ; f q as a matrix Φ in which each row corresponds to the mask function along the aperture coordinate, then we can write Eq.(3.15) as Φg “ S (3.16) ΦΦ` “ I (3.17) where, as described by 3.13, Equation (3.16) shows that from a collection of measurements taken with a dynamic metasurface g, it is possible to estimate the field at the aperture plane at an equally spaced number of points that can be considered effective dipole sources, also called S. The Green’s function in Eq.( 3.6) can then be used for each of these sources, and the estimation of the scene reflectivity can proceed using the RMA as explained is the next section. 3.3 Range Migration Algorithm In this section I develop a range migration method based on bistatic detection of the fields, following closely the path presented by Zhuge and Yarovoy [160]. Nevertheless, the RMA can be extended to bi-static [161] and multi-static [151, 162, 163] imaging configurations. Details of this derivation are included in the Appendices B. In the specific case of a physically moving linear array [164], the RMA can similarly be 71 implemented under this framework with knowledge of the transmit and receive coordinates. This configuration involves neither a single, static beam aimed at broadside, as in stripmap SAR, nor a steered beam, as in spotlight SAR. The image reconstruction can be successfully handled using an appropriately formulated RMA. It is worth noting that, as shown in [67, 76], DMAs are also capable of imaging in spotlight or stripmap imaging modalities. The treatment presented in this section can be applied to those cases following the described transformation, as they are special cases of the general field patterns produced by the DMA. For a stationary DMA, our coordinate system has the aperture oriented along the horizontal y-direction, located at z “ 0 and centered at y “ 0, as shown in Fig.3.4a. After the signal transformation described in Eq. (3.16), the received signal is given by ż Spyt , f q “ V σpx, y, zq ´jkRt ´jkRr e e dV 16π 2 Rt Rr (3.18) where σpx, y, zq is the reflectivity of the object, while the terms Rt and Rr are defined as a Rt “ x2 ` py ´ yt q2 ` pz ´ zt q2 (3.19) a Rr “ x2 ` py ´ yr q2 ` pz ´ zr q2 where yr , zr and yt , zt correspond to the positions of the transmitters and receivers, respectively. The locations of the transmitters and receivers change in accordance with the imaging scenario, which leads to a unique derivation of the RMA for each scenario. In the following, I extend the above concept to imaging scenarios employing a stationary and a moving (virtual) SIMO configuration. The RMA is derived for the cases of vertical motion, horizontal motion, and combined vertical-horizontal aperture motion, with the resulting dispersion relations specifically identified for each configuration. 72 3.3.1 Stationary Case In the particular case that the DMA is not moving, it can be represented as a 1D array of elements illuminating the region of interest; therefore, the image reconstruction is performed in the z “ 0 plane and zr “ zt “ 0. Since this configuration consists of a single probe at the origin as the receiver, yr “ 0 and the location of each effective dipole—after the appropriate signal transformation—corresponds to yt . Given these constraints, and taking Fourier transforms of this signal along the yt coordinate, the signal representation in the wavenumber domain is obtained, which upon simplifying as detailed in [77] yields ż ż Apky , kqσpx, y, zqe´jkx x e´jky y dydx Ŝpky , kx q “ x (3.20) y where ky “ kyt , b kx “ k 2 ´ ky2 ` k , (3.21) and Apky , kq corresponds to an amplitude term that oes not impact the quality of the image reconstruction significantly, but it has been considered here for completeness [165]. The reflectivity σpx, yq is finally obtained upon taking the 2D inverse Fourier transform of the transformed signal given by Eq.(3.20). Equation (3.21) is the dispersion relation which depends strongly on the imaging configuration. In particular, this equation differs from the traditional dispersion relation used in monostatic SAR scenarios [150]. In fact it is specifically derived for a single-input multiple-output (SIMO) configuration, with the multiple antennas arranged in a uniform linear array. Notice that the DMA platform can be viewed as a virtual SIMO system after the preprocessing step in Eq.(3.16). 73 z y ys ya Rr Rt Scene x Figure 3.5: Dynamic metasurface antenna imaging system with the antenna moving along the y direction. 3.3.2 Moving the DMA in the y direction While the DMA is an electrically large aperture capable of achieving good resolution in cross-range, it can be displaced along the y direction in order to improve the resolution and survey a larger scene. In this case the measured signal is also taken in the z “ 0 plane, while the locations of the transmitters and receivers are modified. As shown in Fig. 3.5 the effective dipoles are located at ya with respect to the center of the antenna, which is moving along with the receiver in the coordinate ys . As such, the transmitter and receiver locations are given by zr “ zt “ 0 (3.22a) yr “ ys (3.22b) yt “ ys ` ya Substituting Eq.(3.22) into Eq.(3.19), which in turn is inserted into the measured signal given by Eq.(3.18), one can obtain the signal in the wavenumber domain. Taking the Fourier transform in this case requires a careful separation of variables, 74 detailed in Appendix C. The signal in the wavenumber-domain as ż ż ż Apkys , kya , kqσpx, y, zqe Ŝpkys , kya , kx q “ ? ˆ e´j k2 ´pk ys ´kya x y q2 x e´jkya y e´jkys y ejkya y dzdydx ´j ? 2 x k2 ´kya z (3.23) The exponential expressions in (3.23) allow us to define the dispersion relation of the imaging system as b b kx “ 2 ` k 2 ´ kya k 2 ´ pkys ´ kya q2 (3.24) ky “ kys Thus, (3.23) can be recast as Ŝpkz , ky , kx q “ ż ż Apkys , kya , kqσpx, y, zqe´jkx x e´jky y dydx x (3.25) y where Apkys , kya , kq is given by Apkys , kya , kq “ j 8πx 1 ˆ 2 qpk 2 ´ pk 2 1{4 rpk 2 ´ kya ya ´ kys q qs (3.26) Implementing the above transformations and computing the 2D IFFT (in kx and ky ) will result in the reconstructed image, σest . 3.3.3 Moving the DMA in the z direction In this subsection, I consider the case of the DMA moved along the z-dimension. In this modality, the DMA is capable of volumetric (3D) imaging. The spectral bandwidth provides range resolution and the aperture provides spatial diversity for cross-range resolution. By translating the DMA along the z-direction, as shown in 75 z y zs ya Rr Rt x Scene Figure 3.6: Dynamic metasurface antenna imaging system moving along the z direction Fig.3.6, additional information can be obtained to achieve cross range resolution in the other dimension. The locations of the transmitters and receivers in this scenario are given by zr “ zs zt “ za (3.27a) yr “ 0 yt “ ya (3.27b) Replacing these positions in Eq.(3.19) and Eq.(3.18), and carefully performing the Fourier transform across the coordinates zs , za and ya , as shown in Appendix B, the 76 measured signal in the wavenumber domain is given by ?2 2 2 σpx, y, zqe´jx k ´ky ´pkz {2q ż ż ż Ŝpkz , ky , kx q9 x ? ˆ e´j x2 `y 2 y ? z (3.28) k2 ´pkz {2q2 e´jky y e´jkz z dzdydx The exponential expressions in (3.28) allow us to define a dispersion relation as kx “ b a k 2 ´ ky2 ´ pkz {2q2 ` k 2 ´ pkz {2q2 where the paraxial approximation x « (3.29) a x2 ` y 2 has been used. Thus, (3.28) can be recast as Ŝpkz , ky , kx q “ ż ż ż Apky , kz , kqσpx, y, zqe´jkx x e´jky y e´jkz z dzdydx x y (3.30) z where Apky , kz , kq is given by ? 2πe´jπ{4 Apky , kz , kq “ j8πpx2 ` y 2 q1{4 (3.31) kx ˆ 2 2 1{4 pk ´ pkz {2q q pk 2 ´ ky2 ´ pkz {2q2 q Taking a 3D inverse Fourier transform of the signal given by Eq.(3.30) finally yields σpx, y, zq. 3.3.4 Moving the DMA in the y-z plane Finally, following the same approach as in the previous sections, I consider the case that the aperture is moving in both the y and z directions, such that large scenes can be reconstructed with high resolution in 3D. As shown in Fig. 3.7, the locations of transmitters and receivers are given by zr “ zs zt “ za (3.32a) yr “ ys yt “ ys ` ya . (3.32b) 77 z y Scene Rr x Rt Figure 3.7: Dynamic metasurface antenna imaging system moving in both the y and z directions. and the measured signal in the wavenumber domain is given by ż Apkys ,kya , kzs , kzr , kqσpx, y, zq Ŝpkys , kya , kzs , kzr , kq “ V (3.33) ˆ e´jkx x e´jky y e´jkz z dV Performing the appropriate variable separation across each dimension, the dispersion relation for this scenario is given by b b 2 2 2 kx “ k ´ kya ´ pkz {2q ` k 2 ´ pkys ´ kya q2 ´ pkz {2q2 (3.34) ky “ kys and the amplitude term Apkys , kya , kzs , kzr , kq is given by Apkys , kya , kzs , kzr , kq “ ´k 2 x2 2 ´ pk {2q2 qpk 2 ´ pk ´ k q2 ´ pk {2q2 q 4pk 2 ´ kya z ys ya z 78 (3.35) Raw dataset from the dynamic metasurface antenna Dataset transformed to aperture coordinate ya n-dimensional FFT with respect to transmitter and receiver coordinates Filtering and backpropagation Stolt interpolation 2D/3D inverse Fourier transform Image reflectivity Figure 3.8: Block diagram of the adapted RMA applied to the dynamic metasurface antenna. Finally, a 3D inverse Fourier transform on Eq.(3.33) returns the desired scene reconstruction σpx, y, zq. Given these signal transformations and dispersion relations, it is possible to perform microwave imaging with the DMA in a fast and accurate way. 3.4 Algorithm Implementation and Simulated Results This section illustrates the practical implementation of the adapted RMA formulation described in the previous section. Numerical implementation of the RMA was performed following Fig.3.8. Once the received signal has been transformed into the dipole coordinate basis, the frequency-domain signal is zero padded in order to prevent aliasing in the spatial wavenumber domain. An n-dimensional FFT is then performed on the signal with respect to the transmit and/or receive coordinates, 79 where the dimensionality of the FFT depends on the imaging configuration used. The signal thus transformed into the wavenumber domain is assigned to k values according to the spatial sampling ∆i,t , ∆i,r of the transmitters and receivers [150]: ki,t “ r´ ki,r “ r´ π π , s ∆i,t ∆i,t π π , s, ∆i,r ∆i,r (3.36) i “ y, z. At this point, the dispersion relations derived in Section 3.3 may be applied in order to specify kx values corresponding to each signal value. Filtering is then performed on the signal by multiplying by the appropriate amplitude term, and back-propagation to the scene center proceeds by changing the phase of each data point according to its kx value, achieving a focus of the image. As noted in [151], the spatialcoordinate (x,y,z ) dependence of the amplitude terms cannot be handled readily in the wavenumber domain, but due to their weak influence on the final reconstruction, these terms may be ignored. The present signal is now distributed over an irregular grid of wavenumber values. In order to recover the final image, an n-dimensional IFFT must be performed on the modified signal. Implementing this operation using uniform FFTs requires an evenly-sampled dataset, which is obtained by interpolating our data onto a uniform grid. This interpolation step, i.e. the Stolt interpolation, achieves focusing of the image by enforcing the dispersion relation and correcting for the wavefront curvature. The Stolt interpolation step plays a fundamental role in the signal reconstruction as it ensures that the signal is properly mapped onto the k-space. This step is also the most computationally burdensome [166]. In this section, a scattered bilinear interpolator is used to perform the transformation efficiently. Once the signal is 80 defined on an even grid of wavenumber values, the reconstructed image may finally be obtained by performing an n-dimensional IFFT. For all the simulation results presented throughout this paper, the frequency band of operation is 17.522.0 GHz, sampled with nf “ 51 uniformly spaced points. The inter-element spacing of elements composing the dynamic metasurface is 6.8 mm, corresponding to the half-wavelength at 22 GHz in free-space, satisfying the Nyquist sampling requirement. The total number of metamaterial elements is ny “ 105, corresponding to a 71.4 cm overall aperture. The imaging scene is the domain described as x P r0.75, 1.25sm , y P r´0.25, 0.25sm. This region is discretized using 61 by 94 pixels, resulting into 5354 pixels. A noiseless scenario is considered throughout this paper to focus on the implementation of the proposed algorithm. The numerical results are obtained using a workstation equipped with a 64-bit 3.5 GHz CPU. 3.4.1 Pseudo-inversion of the aperture field Considering the fact that not all masks applied to the DMA result in some degrees of orthonormality and completeness, it is not granted that the exact inverse of Φ can be obtained. Therefore, in order to satisfy Eq.(3.13), one must calculate a highfidelity estimate of the pseudo inverse Φ` . To this purpose, a study of the matrix ill-conditioning is provided, through the Singular Value Decomposition (SVD) [143] of Φ. At each frequency, the SVD factorizes Φ as Φ “ UΣV: (3.37) where .: stand for the complex conjugate operator, and U and V are unitary matrices made of an orthonormal set of bases, weighted by the singular values s1 , , sN , ordered from largest to smallest in the quasi-diagonal matrix Σ. Under this factorization, the pseudo inverse matrix as described in Eq. (3.37) is Φ` “ VΣ` U: 81 (3.38) where Σ` is a diagonal matrix given by »1 s1 — Σ` “ – : 0 ¨¨¨ ... ¨¨¨ 0 fi (3.39) ffi : fl 1 s ny The conditioning of Φ is defined as the ratio between the smallest and the largest singular values s1 s ny . When all the singular values are equal, the vectors of the or- thogonal basis U and V are equally weighted in the signal reconstruction, indicating that for each mask, the radiation patterns generated by the collection of dipoles in Φ are spatially independent. As such,the SV spectrum is flat and s1 snt “ 1. An example of such scenario is shown in Fig. 3.9a, where a set of masks with only one element on was used, this set of masks is called identity. In practice, the weak coupling between the guided wave and each metamaterial element in the dynamic metasurface implies a small radiated power per element and therefore many more elements should be turned on. An example of this case is shown in Fig. 3.9b, where a random distribution of on/off elements has been used. In this case, more energy is being radiated towards the scene which results in a system more robust to noise. This comes at the cost of correlation among the resulting radiation patternswhich can be seen from the condition number due to the fact that the dynamic metasurface does not provide full, independent control of the magnitude and phase of each dipole. In our particular simulation, 105 metamaterial elements compose the dynamic metasurface, 105 masks have been applied, although this number does not necessarily needs to be the same, and for each random mask half of the elements are turned on. Since half of the elements are turned on, the expected normalized power radiated is 0.49, and this value matches in high fidelity with s1 {105 “ 0.50, [143] as shown in Fig. 3.9c. In order to retrieve high fidelity images, a regularization parameter β must be introduced to avoid singular values which contribute to correlated 82 Figure 3.9: Singular value spectra for different sets of masks applied. a) Φ for Identity masks b) Φ for Random masks c) SV spectrum. measurement bases. This regularization parameter ensures a proper reconstruction of the data, depending on the intrinsic ill-conditioning of Φ , by removing the lower singular values of the SV spectrum, such that Σ` moves to t s11 , , s1β , , 0, .., 0u [167]. 3.4.2 Algorithm Implementation The signal measured with a collection of independent antennas, as shown in Fig. (3.4a) and computed using 3.20, is Sny ˆnf , where nf is the number of frequencies and ny is the number antennas. The signal measured with the dynamic metasurface, as shown in Fig(3.4b) and computed as in Eq.(3.7), becomes gnm ˆnf , where nm is the number of masks. Finally, the transformed signal which satisfies Eq.(3.16), will be called Ŝny ˆnf . Given Φ` we multiply it by g in order to find the signal at the aperture plane, which is compatible with the RMA. Figure. 3.10a, corresponds to the backscattered signal measured with an independent array of antennas, satisfying 3.20, while Fig.(3.10b) corresponds to the backscattered signal measured with the dynamic metasurface, satisfying 3.9. As shown in Fig. 3.10c, the transformed signal Ŝ “ Φg appears comparable to the signal shown in Fig. (3.10a). Therefore, I have separated 83 Figure 3.10: Transformation of the signal measured with the dynamic metasurface to apply the RMA. a) Sny ˆnf , b) gnm ˆnf c) Ŝny ˆnf . the role of the dynamic metasurface and the backpropagation of the received signal. In this example, a single point-like scatterer located at px0 , 0, 0q with x0 “ 1 m is used as a target, resulting in a point spread function (PSF). With the transformed signal shown in Fig. 3.10c, I walk through each step of RMA flowchart depicted in Fig.3.8. The next step is the plane-wave decomposition of the signal by taking the Fourier transform of Ŝny ˆnf , as shown in Fig.(3.11a) . Then, the signal is backpropagated to the center of the region of interest in range x0 , multiplying the signal by ejkx x0 , where kx satisfies the dispersion relation of Eq.( 3.21). Then, the signal is also re-sampled onto the kx vector, as shown in Fig.3.11b. Notice that the component of the wave vector along the optical range must be real in order to get a propagating mode. Therefore, the range of values where 3.21 is valid, reduces to k 2 ě ky2 . The values that do not satisfy this inequality, corresponding to the evanescent fields, are 84 Figure 3.11: Steps of the range migration algorithm for a PSF. a) Ŝny ˆnf b) FFT in cross range c) Stolt Interpolation d) 2D-IFFT ignored. Finally, given the signal in the wavenumber domain, a 2D inverse Fourier transform is performed in order to find the reflectivity of the object, as shown in Fig. 3.11c. 3.4.3 Simulated Results Given the appropriate regularization parameter for the reconstruction, it is our interest to compare the presented RMA with the previous computational imaging techniques used for the dynamic metasurface. The main advantage of the RMA described above relies in the use of Fast Fourier transforms to back-propagate the transformed signal in the wavenumber domain and inverse Fast Fourier Transforms to find the signal in the spatial domain. To validate the adapted RMA, I compare its results to those obtained with computational imaging techniques, which have been previously used with dynamic metamaterial antennas [64]. For such techniques the measured signal is found following 3.9, and the simplest method to achieve the image reconstruction is σest “ H: g 85 (3.40) where σest is an estimate of the imaged scene, and H` is the pseudo inverse of the sensing matrix. An initial approximate calculation of the pseudo inverse is H` “ H : where : represents the conjugated transpose, usually referred to as matched filter (MF). In order to reconstruct better quality images, it is possible to use iterative techniques, such as least squares or the generalized minimal residual method (GMRES) to invert 3.9. The GMRES algorithm takes the matched filter reconstruction as an initial estimate - considering that MF applies only a phase compensation- and iterates towards the best solution including the magnitude information [168]. For the point spread function shown in Fig. 3.11, it is expected that the resolution in range and cross-range to follow the diffraction limit as c 2B (3.41) λc x0 . L (3.42) δr “ δcr “ where B is the frequency bandwidth, λc is the center wavelength, x0 the range distance of the target, and L is the physical length of the dynamic metasurface. For the simulated parameters described in the previous section, the expected values for 3.42 are δr “ 3.33 cm and δcr “ 2.14 cm. As shown in Fig.(3.12), these values correspond to the expected values for all the different reconstruction methods, where δr “ 3.38 cm and δcr “ 2.15 cm respectively. Figure 3.13 shows the image reconstruction of a complex target by using different reconstruction methods. Interestingly, the image reconstructed with the RMA has better quality than matched filter; this result can be associated with the fact that before the RMA is performed, the signal has been transformed with a more accurate calculation of Φ` with the aforementioned truncated SVD, while in the matched filter technique, Φ` “ Φ: , implicitly computed in H: . Moreover, the signal is better 86 Figure 3.12: Cross sections of the PSF in (a) range and (b) cross range. Different colors correspond to different reconstruction methods Figure 3.13: Image reconstruction of a complex target with different reconstruction methods. a) Object b)Matched Filter, c) GMRES d) RMA. reconstructed with the GMRES technique, considering that this iterative approach provides a more accurate calculation of H` , while the RMA technique implies the a approximation x2 ` y 2 « x. The adapted RMA can be used for imaging systems in which a platform moves to synthesize a large aperture and obtain higher resolution. This case can be encountered especially in earth observation. One form of SAR imaging involves translating the aperture along the z direction, with the aim of retrieving 3D images. In this approach, range information is obtained from frequency, height information is 87 Figure 3.14: 3D image reconstruction by moving the dynamic metasurface antenna along the z direction. a) target locations. b) reconstructed images obtained by moving along the z axis, and cross range information is obtained by illuminating the scene with different radiation patterns. The transformation of the signal remains the same as in the stationary case and the reflectivity can be found using equations 3.30 - 3.29. To demonstrate the utility of this approach, the dynamic metasurface and the receiving probe are used to reconstruct an image five points spread over the region of interest, as shown in Fig. 3.14a. In this scenario, the aperture was moved at 6.8mm increments along z axis. For further comparison with other reconstruction techniques, the aperture and the domain size have been reduced. At each z position, 68 masks were used to illuminate the scene, and the total number of effective dipoles in the 2D synthetic aperture is 45 ˆ 45 “ 2025, and the number of frequencies was reduced to 15. The reconstructed image is shown in Fig.(3.14b), which exhibit great similarity with the target under test. This scenario, which involves 15 by 45 by 45 “ 30375 voxels defined in the region of interest 88 Table 3.1: Precomputing time (PT) and Reconstruction time (RT) using different techniques. 2D domain corresponds to the image in Fig.(3.13), 3D domain corresponds to the image in Fig.(3.14) Domain 2D – PT . Matched Filter 3.65s GMRES 3.65s RMA 0.25s 3D PT 1.32 ˆ 103 s 1.32 ˆ 103 s 0.25s 2D RT 0.05s 30.41s 0.03s 3D RT 1.11s ´´ 0.01s x P r0.1 ´ 0.9s, y P r´0.15 ´ 0.15s, z P r´0.15 ´ 0.15s, further highlights major advantages of the presented algorithm. The aforementioned methods, based in computing the sensing matrix H, might become intractable as the region of interest increases, as shown in Table 3.1. While matched filter and RMA methods may produce images of similar quality, the presented RMA offers significant advantages both in pre-computation and reconstruction time for the stationary and moving case scenarios. For the MF and GMRES reconstructions, the pre-computing time is the time required to compute H: , while for the RMA the pre-computing time is the time required to compute Φ` . Therefore, despite the fact that reconstruction time via MF is faster, the pre-computing time is considerably larger and requires a predefined scene, while the RMA only requires a knowledge of the average distance in range for the center of the region of interest. Note that the pre-computing time for the 3D reconstruction appears to be smaller than the 2D reconstruction. Lets recall than in the 3D scenario the aperture size and number of masks has been reduced in order to make simulations comparable with other reconstruction methods and even in this reduced case, the reconstruction with the GMRES technique cannot be performed due to the limitations of the computer hardware used. 89 3.5 Experimental Results To verify the utility of the proposed reconstruction algorithm, I consider the imaging configuration investigated in [67]. In this configuration, shown in Fig. 3.15, a 40cm-long DMA, described in [65], operates as transmitter, while a single open-ended waveguide situated above the center of the DMA acts as a receiver. Port 1 of an Agilent E8364c vector network analyzer (VNA) is connected to the feed positioned at the center of the DMA, and port 2 connects to the receiving open-ended waveguide. Consistent with our derivations in the previous section, the location of the receiving probe as the aperture position ys is defined. For simplicity, the scene is translated relative to the DMA using a Newmark D-slide 2D stage, resulting in an inverse SAR configuration that is completely analogous to standard SAR. Though inverse SAR is often developed in a spotlight modality (i.e. the radar beam tracks the target) [169], it is important to highlight that, in contrast, the DMA does not necessarily direct a beam toward the moving scene, but rather uses complex radiation patterns. As noted earlier, our formulation can also accommodate the spotlight and stripmap cases by performing the described transformation, as these are special cases of the complex fields produced by the DMA.The VNA, translation stage, and DMA are controlled through MATLAB. The design principles and specific fabrication considerations of the DMA are detailed in [65]. A brief review of the important characteristics of the DMA is provided. The DMA was designed and fabricated to operate at lower K band frequencies (17.5 to 22 GHz), with a bandwidth of 4.5 GHz. Each metamaterial element is a cELC resonator which couples portions of the guided wave into free space. The cELCs are designed and positioned such that alternating elements resonate at 17.8 GHz and 18.9 GHz, with an element spacing of 3.33 mm and a total of 112 elements. To dynamically alter the resonant response of the elements, two PIN diodes are placed 90 Moving Stage Rx Scene Tx Figure 3.15: Experimental setup for microwave imaging with the DMA, including the transmitting metamaterial aperture and the receiving open-ended waveguide. across the gap of each cELC. An Arduino micro-controller is then used to apply DC bias to each element independently by way of bias patterned in the resonant structures [120]. This allows each element to be turned on or off by external control of the bias voltages. Changing the sequence of on and off elements results in a different radiation pattern. Due to the interactions between the metamaterial elements and the waveguide, one cannot accurately predict the fields generated by the DMA by analytical methods. The characterization of the DMA’s response (and thereby its Φ) is thus performed by using experimental near-field scans [170]. The fields are measured over a discrete collection of aperture domain locations ya , for all frequencies and mask states, using an open ended waveguide. From surface equivalence principles, these measurements may then serve as the effective dipoles characterizing our DMA. Fig. 3.16a shows a subset of the experimental measurements of the Φ matrix for a single frequency, when the collection of tuning states is random. As shown, the magnitude of the effective dipole moments is stronger towards the center and decays as the dipoles are farther from the feed. Since the tuning states are random the radiated fields are not strictly orthogonal and therefore pseudo-inversion techniques 91 20 Mask Index Mask Index 20 40 60 80 40 60 80 100 100 20 40 60 80 100 20 Effective Dipoles 40 60 80 100 Mask Index Figure 3.16: a) Experimental near field scan measurements stored in the Φ matrix. b.) Product between Φ and its pseudo-inverse Φ` . are required [167]. In our particular scenario, the pseudo-inverse was calculated performing an SVD decomposition, which factorizes the matrix as Φ “ UΣV: . Here U and V are unitary matrices composed of an orthonormal set of vectors, weighted by the singular values s1 ; ...; sN , ordered from largest to smallest in the diagonal matrix Σ. Under this factorization, the pseudo-inverse matrix is given by Φ` “ VΣ´1 U: , where Σ´1 “ diag t1{s1 , ..., 1{sN u. Fig. 3.16b shows the product of Φ shown in Fig.3.16a with its pseudo-inverse, demonstrating its close approximation to the identity matrix. For each synthetic aperture position and tuning state, the VNA takes measurements as it sweeps through a chosen set of frequencies (17.5 to 22 GHz sampled at 90 MHz intervals, for a total of 51 frequency points). A set of 25 tuning states is chosen from the set of binary sequences available. This procedure yields a total of 1275 measurements for each aperture location. 3.5.1 Stationary DMA Using the acquisition procedure outlined in Section 3.5, I begin by demonstrating experimentally the case of imaging with no relative motion between the DMA and the scene. In this configuration, range and cross-range information are gathered according to the bandwidth and physical length of the aperture, respectively. The target is a set of three metallic cylinders oriented along the z-direction, placed ap- 92 Figure 3.17: Imaging three metallic cylinders in the xy plane. The DMA is stationary. proximately 40 cm from the aperture plane and spaced approximately 5 cm apart. Each cylinder is 3 mm in diameter („λmin {5). The reconstructed image is shown in Fig. 3.17. This result demonstrates that the transformed DMA dataset is indeed compatible with the RMA as derived. As the stationary case has been illustrated with good image quality, the process may now be extended to more sophisticated configurations involving movement of the antenna. 3.5.2 Moving the DMA in the y direction For the case of a DMA that is physically translated, the reflected signals are collected as the receiving probe and transmitting aperture are moved in tandem along the length direction of the DMA. As in a traditional SAR scenario, the motion of the DMA allows synthesis of an effective aperture much larger than the physical antenna used. Fig. 3.18 illustrates a set of five metallic spheres reconstructed in the xy plane using the dispersion relation calculated in Eq.(3.24). The collection is placed 93 Figure 3.18: Imaging five metallic spheres in the xy plane. The DMA is moving along the y direction. at an average range of approximately 62 cm, with an object spacing of about 4 cm. Each sphere is 1.5 cm in diameter („ λmin ). Here, our 40 cm DMA is translated in 5 mm increments to synthesize an 88 cm transmit aperture. This configuration results in an expected cross-range resolution of approximately 6.8 mm (λmin {2), versus an expected range resolution of 3.33 cm, resulting in the oblate shape of the reconstructed images. The resolution achieved in the reconstruction is 1.25 cm in the cross-range direction and 4.19 cm in the range direction. The obtained crossrange resolution is less than the diameter of the object due to the fact that the backscattered signal is dominated by specular reflection. 3.5.3 Moving the DMA in the z direction Results obtained by moving the aperture along a 30 cm path in the z direction, and reconstructed using the dispersion relation developed in Eq. (3.29), are shown in Fig. 3.19. This type of array motion synthesizes a 2D array which, when paired with frequency information, allows three-dimensional scene reconstruction. The first 94 a) b) Figure 3.19: Imaging (a) metallic cylinders of 3mm diameter and (b) a foil smiley face in all three dimensions. The DMA is moving along the z-direction. example consists of three 3-mm-diameter metallic cylinders placed 40 cm from the aperture and spaced approximately 5 cm apart, as in the stationary case above. Since the predicted cross-range resolution is 1.36 cm in the y direction, the objects are clearly resolved in all three dimensions. The second object consists of aluminum foil patterned in the shape of a smiley face, placed a distance 35 cm in range from the DMA. Qualitative agreement is observed in this example as well. 3.5.4 Moving the DMA in the y-z plane In order to synthesize an extended, two-dimensional aperture, the DMA is translated in both the y and z directions, in a meandering path, as shown in Fig.(3.7). This type of motion is essentially a combination of the previous two cases and allows threedimensional reconstruction with improved cross-range resolution. The reconstructed images of metal wires in the shape of the letters ”D U K E” are shown in Fig.3.20. Each letter was placed a distance 45 cm from the aperture plane and imaged inde95 Figure 3.20: Imaging metallic letters in 3D. The DMA is moving along the crossrange and the elevation directions. pendently. This result indicates that general array motion may be accommodated by the RMA in a SAR scenario, obtaining high resolution and low-noise images. 3.6 Summary and Conclusions In this chapter, I have demonstrated the capabilities of a dynamic metasurface antenna in performing high-speed image reconstruction of metallic objects at microwave frequencies. While the use of metasurface antennas for computational imaging has been previously demonstrated, the most important contribution on this topic is the use of the Range Migration Algorithm, and its adaptation to be compatible with this type of antennas. In analyzing this algorithm, I have confirmed dispersion relations that are used on SAR and MIMO imaging configurations; and I have summarized these relationships as applied to the DMA in motion. In summary, I have provided a signal processing technique that can be used with different types of metasurface antennas and moving SIMO arrays, and I have demonstrated its success in image reconstructions in both computational and experimental way. 96 As discussed in section 3.5, use of the RMA is motivated in part by the computational advantages afforded by Fourier transform. When the metamaterial antennas are translated and the scene size increases, intermediate calculations employed in computational imaging techniques require the generation of the fields on the region of interest–stored on the sensing matrix H– which quickly exceeds available memory. When performing 3D imaging, the storage requirements and number of operations scale up significantly, and techniques that do not require the previous knowledge of H are highly sought after. For example, for each distinct measurement with the DMA (frequency, tuning state, aperture location), a unique volume of complex fields must be calculated and stored in the sensing matrix H for matched filter (MF) and least squares (GMRES) reconstruction. The memory required for such allocation quickly becomes excessive even for moderate scene sizes. For example, reconstructing a 3D scene of size 0.027 m3 , discretized into 91,125 voxels, requires 8.75 GB of memory at our frequencies of interest, assuming only storage of the H matrix and the dataset. The RMA developed in this chapter becomes a particularly appealing alternative. In contrast, the RMA demands roughly enough memory allocation to accommodate the a single set of the aperture fields and the measurement set (determined by the number of measurements and sample aperture positions), and the values involved in the FFT operations. Knowledge of the fields at the scene locations is not necessary, so there is no need to pre-compute the H matrix, offering a significant reduction in storage. The size of the resulting reconstruction can be adaptive, as it is determined by the sampling and interpolation steps performed in the algorithm. Most importantly, the RMA involves performing operations on a single measurement array, so that it avoids the multiplicative scaling with scene size observed in MF. As a result, the memory required for reconstructing a scene of the size mentioned above is only 1.4 GB when using the RMA. Computation times using the RMA are largely determined by the Stolt interpolation step, as discussed in [166]. This is because the FFT steps 97 are known to require a relatively small number of operations (OpNa logNa ), where Na is the number of aperture positions). Reconstruction by MF is straightforward, yet also increases in complexity with scene size, requiring OpM N q operations. Here, M is the number of measurements and N is the number of scene voxels. GMRES reconstruction time depends on the number of iterations necessary for satisfactory image quality, and can be excessive for large H matrices. The development of the adapter RMA in conjunction to the flexibility offered by dynamic metasurfaces has lead to an extensive collection of important contributions across the entire field of microwave sensing. Examples of this success include single-frequency imaging [171], single-frequency SAR imaging [76], and through-wall imaging [75]. 98 4 Waveguide-fed Metasurfaces for Beamforming This chapter is dedicated to the modeling techniques for beamforming, a technique commonly used in array technology for highly directional signal transmission and reception. My contributions to this topic are as follows: • I have used the dipole model to predict the electromagnetic response of slotted waveguide antennas (SWAs). In particular, I have extended the work of Johnson et. al. [172] to SWAs with parasitic elements and tapering. Excerpts of the this discussion are taken from the papers L Pulido-Mancera, T Zvolensky, MF Imani, PT Bowen, M Valayil, DR Smith. Discrete dipole approximation applied to highly directive slotted waveguide antennas IEEE Antennas and Wireless Propagation Letters, 15, 1823-1826 2016. L Pulido-Mancera, MF Imani, DR Smith Discrete dipole approximation for simulation of unusually tapered leaky wave antennas 2017 IEEE MTT-S International Microwave Symposium 1 409-412 2017. 99 a) c) b) Figure 4.1: a) Phased Array b)Slotted Waveguide Antennas c) Waveguide-fed Metasurface • I have used the dipole model to predict the radiation patterns of a Frequency Diverse Metasurface Antenna (FDMA) to be used in microwave imaging. Excerpts of the this discussion are taken from the paper in progress L Pulido-Mancera, MF Imani, DR Smith. Dipolar Model for Metamaterial Imaging Systems. 4.1 Beamforming An antenna beamforming array consists of two or more antenna elements that are spatially arranged and electrically interconnected to produce a directional radiation pattern. The interconnection between elements can provide a fixed phase to each element and form a phased array. In the optimal scenario, the magnitude and phase of the feed network are adjusted to optimize the radiation pattern characteristics, as shown in Fig. 4.1a. The geometry of each element–as well as its orientation and polarization– influences the performance of the array, such that the generated beams can be made to have high gain, low sidelobes and controlled beamwidth [87]. Highly directive antennas with steering capabilities are usually developed as large phased arrays or mechanically actuated gimbal dishes. In typical array antennas, radiating antenna modules such as horn-antennas or dipoles, tile the aperture with roughly a half-wavelength spacing, with control over the phase introduced by active 100 phase shift circuits. While phased array antennas have shown excellent performance, they are expensive, heavy, and usually consume large amounts of power. In order to overcome these drawbacks, leaky-wave antennas (LWA) have been used for their simplified hardware and frequency beam-steering performance [83,86,173–175]. Given these interesting attributes, they have found application in a wide array of technologies from collision avoidance radars to wireless power transfer [175–177]. A LWA consist of a waveguide that supports a slow (non-radiating) wave that has been periodically modulated, as shown in Fig.4.1b. The periodic perturbation causes a gradual leakage of the non-radiating mode, resulting in collimated beams. The propagation constant controls the beam angle (and thus can be varied with frequency), while the attenuation constant controls the beamwidth. The most common example of LWA is the slotted waveguide antenna (SWA), which offers clear advantages in terms of design simplicity, weight, volume, power handling, directivity, and efficiency [72, 174, 178, 179]. The SWA illustrates the basic properties common to all uniform LWAs: The fundamental T E10 mode of a rectangub lar waveguide is a fast wave, with a propagation constant given by β “ k02 ´ p πa q2 , where k0 is the wavenumber in free-space, and a is the waveguide’s width. By etching irises on the upper wall of the waveguide with a period, p, smaller than the guided wavelength, λg , the energy leaks out to free space. The phase advance that reaches each iris generates a beam in a particular direction φ0 . The ratio between the wavenumbers k0 and β yields the direction of the main beam as β k0 “ λ0 λg « sin φ0 . While SWAs posses less hardware constraints in comparison with phased arrays, the aperture sampling and beam control is limited. The waveguide-fed metasurface presents an alternative architecture as compared with that of the phased array or traditional LWAs [55–60]. In this case, a dielectricfilled planar waveguide serves to excite an array of complementary metamaterial 101 elements patterned into the upper wall of the waveguide. The elements have dimensions and spacing significantly smaller than both the free space wavelength λ0 and the guided wavelength λg . In this example, the design of a waveguide-fed metasurface can be thought of as being in between the discrete sampling of the aperture used in array antennas, and the continuous sampling that would motivate holographic methods. Unlike phased arrays, the scale and spacing (d) between the radiating metamaterial elements are significantly smaller than the typical λ0 {2 spacing, though often practically limited to dimensions on the order of λ0 {10 ă d ă λ0 {5 [61]. Because independent and complete control over the phase at each element is not possible, the subwavelength sampling of the aperture is crucial to obtain the best performance of waveguide-fed metasurfaces. Analogous to a LWA, the metasurface antenna leverages the phase advance of the waveguide mode, avoiding the need for phase-shifting circuits that can add cost and complexity to the system. The absence of complete control over phase can be compensated, at least partially, by sampling the aperture as finely as is feasible, enabling a holographic design methodology to be pursued [55,56,180] as shown in Fig.4.1c. Even in cases where little or no additional phase shift is added from each metamaterial element, waveguide-fed metasurfaces have demonstrated high-quality beam forming and other wavefront shaping functionality by sampling the phase of the reference wave, often rivaling the performance of more advanced active antenna systems [62, 81]. In this chapter, the goal is to analyze and demonstrate the beamforming operation of different examples of waveguide-fed metasurfaces, under a set of assumptions that enable relatively simple expressions to be found for the key antenna features. A brief description of the dipole model in conjunction to various antenna metrics is provided in Section 4.2. In addition, closed-form analytical results are provided in order to illustrate the dependencies of these metrics on the antenna parameters for beamforming. More explicitly, a brief description on the modulation of waveguide102 a) Beamsplitter b) Illumination wave z Object θ Object wave Coherent light beam φ y Mirror Reference wave x Aperture plate Figure 4.2: Beamforming based on Holographic Techniques fed metasurfaces is presented. It is important to highlight that, while the initial expressions to determine the metasurface’s modulation rely in a Non-interacting Dipole Model, the full simulation of the waveguide-fed metasurfaces presented here includes the mutual interactions between elements. Section 4.3 demonstrates the beamforming capabilities of a SWA and it is shown how the dipole model can me used to predict its performance. Demonstrating that the dipole model is capable to predict the radiation characteristics of the SWA is crucial for the creation of more elaborated structures. An example of this model used on tapered SWAs is shown in subsection 4.3.2. The capabilities of the dipole model are further extended to the planar waveguide scenario using an antenna created at Kymeta Corporation. This structure offers a planar antenna technology for satellite communications on mobile platforms. The details and results of the dipole model applied for this particular antenna are described in Section 4.4. Finally, an example of a planar waveguide antenna for microwave imaging is presented in Section 4.5. 4.2 Principles of Operation for Beamforming The fundamental idea of beamforming with waveguide-fed metasurfaces is based on holography. Typically, a hologram is a recording of the interference pattern between a reference wave and an object wave. It is commonly used to display a fully 3D image 103 of the object, which can be observed without any other focusing device, as shown in Fig.4.2a. For a waveguide-fed metasurface it is possible to encode the aperture plane with the interference pattern of the desired field that would generate a beam, and the guided wave. In this manner, when the antenna is excited with the same guided wave, the desired beam is formed. However, the perturbation of the guided wave due to the element’s patterning, and the element’s mutual coupling usually results in a complicated design process [79, 81, 181–183]. An example of a waveguide-fed metasurface for beamforming is illustrated in Fig. 4.2b. In order to generate a beam in a desired direction pφ0 , θ0 q, it is necessary to find the specific polarizability distribution needed for the waveguide-fed metasurface antenna to form such a beam. In other words, determine the polarizability of each metamaterial element that must be chosen such that the waves from each of the radiators are in phase and interfere constructively in the chosen direction. For the sake of a demonstration, here we briefly describe the methodology of Smith, et. al. [61]. Let us consider a 1D guiding structure (not limited to a rectangular waveguide), with metamaterial elements etched along a line. If the metamaterial elements are patterned periodically along the propagation direction, they can be thought of as sampling the reference wave and aperture at locations designated by xi “ id, where i is an integer and d is the distance between adjacent elements. As previously demonstrated in Section 2.2, the metamaterial elements posses a strong resonant response that can be modeled by the Lorentzian form αm pωq “ F ω2 . ω02 ´ ω 2 ` jΓω (4.1) where F is the resonator strength, ω0 is the resonance frequency and Γ is the damping factor. All of these parameters can be varied via external control, and waveguide-fed metasurfaces can be reconfigured dynamically [64]. The polarizability in Eq. (4.1) connects the induced magnetic dipole moment on the metamaterial element, m, ~ with 104 ~ ref , which we refer to here the local magnetic field of the feeding waveguide mode, H as the reference wave, as shown in Fig. 4.1b. In the absence of interactions between the metamaterial elements, the total magnetic dipole moment is ~ ri q. m ~ i “ αm,i pωqHp~ (4.2) From Eqs.(4.1) and (4.2) it can be seen that the field radiated from the element ~ ref multiplied by has an amplitude and phase determined by the reference wave H the polarizability of the metamaterial element, which also introduces an additional amplitude and phase advance to the incident reference wave. Given m ~ i it is possible to obtain the far-field patterns following the array factor calculations depicted in Eqs.(2.52) and (2.53) [184]. At the plane φ “ 0, Eq.(2.52) can be simplified, and the magnetic field is given by 2 ÿ ~ rad “ ´ ω cos θ mx,i ejkxi sinθ . H 4πr i (4.3) In addition, the reference wave is ~ ref “ H0 e´jβg x x̂, H (4.4) which is assumed to not to be perturbed by the scattering from the metamaterial elements. The far-field radiation pattern from the metasurface antenna can be approximated by superposing the fields sourced by all of the elements. Following the holographic techniques, Hrad is given by the interference pattern between the reference wave and the desired wave: N ÿ ω2 ~ Hrad “ ´H0 cos θ αm,i pωqe´jβxi e´jk|~r´xi x̂| θ̂. 4πr i“1 (4.5) The field distribution for a plane wave propagating in the direction pφ0 , θ0 q has the form ~ pw “ Hpw e´jpkx x`kz zq θ̂. H 105 (4.6) As shown in Eq.(4.6), kx and kz denote the wavenumbers in the x- and z-axes respectively (let us recall that the plane φ “ 0 is the same as the xz´plane). At broadside, (z “ 0), the desired polarizabilities required to obtain such a field distribution must be αm,i pωq “ ejβxi ejkxi sin φ0 . (4.7) In this idealized approach to find the polarizabilities, it is possible to observe that the polarizabilities are chosen to compensate for the propagation of the waveguide mode, then to add the phase and amplitude distribution required to generate the directed beam. Substituting Eq. (4.7) into the array factor calculation in Eq. (2.54), a simplified expression is obtained as AF pφ, θq “ cos θ N ÿ e´jkxi psin φ´sin φ0 q . (4.8) i“1 The array factor of Eq.(4.8) predicts a radiation pattern highly peaked in the θ0 direction, with a series of side lobes that fall off away from the central peak. The polarizabilities– determined by Eq.(4.7)– would require full control over the phase (with the amplitude being constant) of the transmitted radiation at each position xi along the waveguide-fed metasurface, which is generally not feasible given the constraints of the metamaterial elements. It is at this point that it is necessary to move away from conventional phased array design methodology, and seek alternative modulation functions that will enable the same beam forming capabilities with the metasurface architecture. Table 4.1 shows the mathematical expressions for the polarizability under different modulation schemes, which can be easily added to the dipole model for beamforming. A detailed analysis of these modulation techniques can be found in [61]. 106 Table 4.1: Modulation Techniques for Beamforming Amplitude-only Hologram αm,i pωq “ Xi ` Mi cospβxi ` kxi sin φ0 q Binary-Amplitude Hologram αm,i pωq “ Xi ` Mi ΘH rcospβxi ` kxi sin φ0 qs jΨi Lorentzian-constrained Phase Hologram αm,i “ j`e2 Euclidean αm,i “ minpαDesired ´ αLorentzian q Traditional antenna designs such as Slotted Waveguide Antennas (SWAs) can be considered as special cases of the aforementioned modulation techniques. More explicity, SWAs can be thought of an amplitude-only hologram with a spacing of d “ λg {2. Once the modulation is applied, it is possible to compute the radiated fields shown in Eq.(2.52), and in turn compute main characteristics of the beam such as directivity, gain, sidelobe level, etc [184]. For future reference, the directivity is given by Dtotal “ b Eφ2 ` Eθ2 Prad (4.9) Dφ “ }Eφ2 } Prad (4.10) Dθ “ }Eθ2 } . Prad (4.11) where Prad corresponds to the total power radiated. Equation (4.11) will be used for all the results presented in the following sections. The gain correspond to the product of the efficiency and the directivity, and the sidelobe level is the difference between the two largest peaks of the radiation pattern [87]. 4.3 Example I: Slotted Waveguide Antennas A slotted waveguide antenna (SWA) consists of a collection of periodic λ{2 spaced radiating slots etched on the top wall of a rectangular waveguide. Without loss of generality, SWAs exhibit high power handling capabilities, high efficiency, and the 107 Figure 4.3: Schematic of SWA divided in a set of unit cells. Dimensions are: a = 21.94 mm, b = 5.08 mm, p = 23 mm, slot width = 2.54 mm, slot length = 12.32 mm, slot position centered at a/4 and p/2. The SWA is designed for X-band and the center frequency is 9.5 GHz. potential for beam-steering [83,185]. These advantages make SWAs suitable in many application venues, including SAR, as well as maritime and aerospace communications. For such applications the directivity is of crucial importance for achieving high signal-to-noise ratios, prevention of eavesdropping, and resistance to jamming. As shown in Fig.4.3, a SWA consists of a set of longitudinal slots located in the broad face of a rectangular waveguide. The SWA can be divided into a set of subwavelength elliptical irises with spacing p, each approximated as an effective magnetic dipole oriented along the propagation direction. As such, the SWA is a type of antenna that can be well approximated by the dipole model. This model can be used to quickly and efficiently perform parametric studies a necessity in the design process without the need for costly full-wave simulations. When the fundamental mode of the rectangular waveguide excites each effective magnetic dipole, the scattered fields produced by each dipole within the waveguide interact with all the other magnetic dipoles. The interaction of these dipolar-elements is expressed as a Greens function summation in 1D (using the Green’s function for the rectangular waveguide). It should be emphasized that when considering SWAs the dipole model closely 108 resembles methods developed earlier by Stevenson and Elliot [173, 178]. In these works, the equivalent circuit model for the fundamental mode of the waveguide is used to describe the excitation of the slots, while each slot is modeled by means of an active impedance. Furthermore, this impedance is optimized to achieve the appropriate radiation pattern. In comparison to the dipole model, the self-impedance is analogous to the polarizability of the slot and the mutual impedance is analogous to the Greens function of the waveguide. While sharing similarities, the dipole model offers a different view of the antenna, which can be easily combined with metamaterial phenomena, such as graded index, transformation optics, frequency-diverse metamaterial apertures, or holographic metamaterial antennas. In this particular scenario, since the propagation is along a single dimension and a single component of the magnetic field is exciting the elements along the SWA, Eq.(2.2) can be simplified in scalar form as H loc pyi q “ H 0 pyi q ` k2 ÿ Gmm mj µ0 j‰i (4.12) where H loc is the total magnetic field at the location of the yi iris. k is the propagation constant in free space, mj is the effective magnetic dipole, and Gmm corresponds to a single component of the Green’s function inside a rectangular waveguide, which in this case is given by Gmm px, y, z, x1 , y 1 , z 1 q “ 8 8 2j ÿ ÿ 1 mπx1 mπx nπz 1 mπz 1 sinp qsinp qcosp qcosp qˆejβy |y´y | ab m“1 n“1 βy a a b b (4.13) where βy is the propagation constant inside the waveguide. The Greens function Gmm px, y, z, x1 , y 1 , z 1 q is calculated as the sum of the eigenmodes inside the rectangular waveguide. In Eq.(4.12), the total magnetic field corresponds to the fundamental T E10 mode of the rectangular waveguide, plus the scattered field produced by all 109 other magnetic dipoles. Equation (4.12), cam be simplified as „ mπxi mπxj jβy |yi ´yj | 2k 2 αi H loc pyj q “ H 0 pyi q. sinp qsinp qe δij ´ abβy µ0 a a (4.14) The second term in Eq.(4.14) is proportional to the scattering parameters S11 and S21 of the iris, or [63] S11 “ mπxi mπxj 2k 2 αi sinp qsinp q abβy µ0 a a (4.15) reducing Eq.(4.14) to “ ‰ δij ´ S11 ejβy |yi ´yj | H loc pyj q “ H 0 pyi q. (4.16) Equation (4.16) is used for matched ended SWAs where only one incident wave is propagating along the waveguide in the positive y-direction. However, if the SWA is shorted - which is the most common case for practical implementations of SWAs - a reflected wave H r pyi q and the dipolar interaction between the apertures and their corresponding images, located at yi1 must be included in Eq.(4.16). Thus, for shorted-ended SWAs the matrix equation of the dipole model is rδij ´ Cij s H loc pyj q “ H 0 pyi q ` H r pyi q. (4.17) where the coefficients Cij are equal to " 1 pS21 ´ 1qpejβy |yi ´yj | ´ ejβy |yi ´yj | q yi ě yj Cij “ 1 S11 pejβy |yi ´yj | ´ ejβy |yi ´yj | q yi ă yj (4.18) It is important to note that when the SWA termination is shorted and the dipolar interaction of the images is included, the number of apertures in Eq.(4.16) does not increase; rather, the Greens function is modified. Once the matrix system is solved and the total magnetic field is obtained, the magnetic dipole is simply mi “ 110 αi H loc pyj q. Being αi the polarizability of the iris as found in section 2.2. Although the scattering parameters can be computed theoretically, here these parameters are computed through full-wave simulation which account for the radiation accurately. Notice that this simulation is made for a single slot only, while Eq.(4.17) is used to compute the entire response of the SWA. 4.3.1 Directivity Enhancement of SWAs with Parasitic Elements The directivity can be improved by increasing the aperture size, or by placing parasitic elements close to the radiating apertures of an SWA. In fact, parasitic elements are especially attractive since they prevent the need for larger aperture antenna structures. They have been shown to increase the directivity of an SWA or modify its polarization in [186, 187]. Recently, a directivity increase of 1.35 dB was demonstrated when a collection of metamaterial elements, known as split ring resonators (SRRs, were placed on top of the radiating slots of a SWA [188]. However, it is not clear in [188] why SRR or I-shaped dipoles enhance the directivity; how their orientation changes the radiation of the antenna; or what is the appropriate distance to place the SRR near the slots. To obtain a better understanding of the full-array performance of the SWA with such parasitic elements, the conventional method is to rely on full-wave simulations. However, parasitic elements are usually placed in close proximity to the aperture, where the coupling between elements is complicated. As a result, full-wave simulations become computationally expensive, due to the large contrast between the feature sizes of the parasitic elements and the total size of the SWA. Such problems can be avoided by using the dipole model, to obtain an accurate yet fast estimate of the performance of the aggregate apertures with different parasitic elements, thereby facilitating the design process. For example, in order to enhance the directivity of the SWA shown in the previous subsection, parasitic elements can be placed close on top of each elliptical iris, as 111 Figure 4.4: Various elements used to improve the directivity of the SWA shown in Fig. 4.4. The parasitic elements considered here are not resonant, and can be considered as simple electric dipoles with approximately similar phases as the magnetic dipole associated with the slots. By including the corresponding images of the dipoles over the ground plane, one can notice how such configuration improves the directivity of unit cell of the SWA. There is little leverage in investigating a large number of parasitic elements of various shapes; therefore, we limit the investigation to parasitic elements with simple geometry. For simplicity, the repeated cell of the SWA and the parasitic elements were simulated as PEC to focus on their effect on the slot radiation characteristics (Fig. 4.4A). The parasitic element shown in Fig.4.4B corresponds to the simplest electric dipole and is therefore an intuitive design choice. The element in Fig. 4.4C is based on a modified version of the split ring resonator (SRR) presented in [188] and whose excitation is electric in this case. Finally, in order to study the effect of several parasitic elements, the element shown in Fig. 4.4D is proposed. This unit cell can be analyzed as two electric dipoles near the effective magnetic dipole. For each design, the radiation characteristics, simulated in CST Microwave Studio, are compared in Table 4.5 a. As shown, all the parasitic element designs increase the directivity; however, there is a noticeable difference in the power radiated and return loss per unit cell. This difference is associated with the transverse location of the parasitic element: while the parasitic elements of B and C are located in the middle of the iris, the parasitic elements of D are located near its edges. Considering that the maximum power is 112 a) b) Figure 4.5: a) Table I. Comparison of various parasitic elements with respect to the maximum directivity, radiated power and return loss. Input power of 0.5 W. b)Comparison between the full wave simulation and the discrete dipole approximation for different scenarios. (Top) Magnitude and phase of S11 parameter for a SWA with two different terminations matched-ended and shorted-ended. (Bottom) Magnitude and phase of S11 parameter for a SWA with and without parasitic elements. radiated from the center of the aperture, when the parasitic element is located at the center, most of the radiated energy is reflected back to the waveguide. The radiation efficiency, defined as the ratio between the power accepted and the input power pPin “ 0.5W q, is computed for each design and corresponds to « 0.99 for all the parasitic elements used, which implies that most of the energy is being transmitted to the next element. The differences in the radiation properties for each design cannot predict the response of the aggregate array and the dipole model is considered. In order to verify the utility of the model, a comparison with full wave simulation of a SWA composed of four ellipses is provided. A comparison between the dipole model and CST simulations in terms of the scattering parameters is presented for different scenarios: the SWA with and without parasitic elements, and the SWA with a matched and shorted termination. As shown in Fig.4.5, the reflection coefficients obtained from both methods are in close agreement in all cases. The results presented in Fig.4.5 113 a) b) Figure 4.6: a) Table II. Comparison between full-wave simulation and dipole model b) Comparison of the directivity for the SWA shown in Fig 1. (top) Without parasitic elements, (bottom) With parasitic elements shown in Fig 2D. are necessary for computing the radiative properties of the aggregate SWA. Figure 4.6 shows the directivity of the SWA in the e-plane and h-plane. These results are presented with and without parasitic elements. As shown, the presence of parasitic elements improves the performance of the SWA. This increase is further highlighted in Table 4.6.a which lists the main characteristics of the radiation patterns depicted in Fig. 4.6b. The dipole model exhibits close agreement with full wave simulation, thereby verifying utility of the dipole model in obtaining an accurate representation of the SWA. It is worth noting that the dipole model obtains excellent agreement with full-wave simulation with much lower computational cost. For instance, the code of the algorithm and the full wave simulations have been implemented using a workstation equipped with a 64-bit 3.5 GHz CPU. The full-wave simulations developed for a simple array of 16 slots take 146 s as the SWA is discretized into 442.680 mesh cells in the time domain solver, whereas the dipole model created in Matlab takes 0.34s. Having confirmed the accuracy of the dipole model for SWAs, we take advantage of its speed to study more complicated scenarios. For example, we examine the effects of adding more cells to the SWA and track the maximum directivity that can 114 a) b) Figure 4.7: a) Table III. Comparison between the realized gain (dB) for different SWA size and different parasitic elements. A,C, and D refer to Fig. 4.4. b) Directivity of the 4 array SWA (matched ended) at different frequencies. As shown, the direction of the main lobe changes uniformly with frequency. be achieved, in addition to the return loss of the entire SWA. Table III summarizes different simulations of the dipole model for SWAs of different sizes and parasitic elements. As presented in this table, by using parasitic elements in a SWA with 4 unit cells, it is possible to achieve the same realized gain as with a SWA composed of 16 unit cells. Lastly, we study the beam-steering capabilities of the SWA with the dipole model, as shown in Fig.4.7. We find good agreement between the directivity computed from dipole model as an array of dipoles, and the full wave simulation at different frequencies. Moreover, as expected from the SWA, the main beam continuously changes its orientation in the forward direction as the frequency increases. 4.3.2 Dipole Model for Unusually Tapered SWAs An important factor determining a SWA performance is the beamwidth and sidelobe level (SLL) of radiated patterns. Akin to most antenna designs, the LWA can be tapered (TLWA) in order to improve the generated beams in terms of beamwidth and SLL [179]. For SLL reduction, the cosine tapering has been commonly used, while for broad-beam antennas, chirp-like tapering has been proposed [189–191]. In the case of SWAs, the taper can be controlled by means of the slots’ size or the slots’ displacements from the waveguide centerline. For a specified number of identical 115 longitudinal slots, different procedures, based in Elliot’s design equations [178]; allow one to find the slots’ lengths, widths, and locations along the waveguide to reduce the SLL at a single frequency. However, these design techniques are limited to SWAs with simple slot geometry and smooth variation, conditions that do not always hold true. Unlike the previous section, the polarizability of each element along the waveguide is subject to change due to the tapering. Therefore, the formulation of the dipole model is sightly different. As an example, let us consider an array of arbitrarily shaped iris (for example, an elliptical slot, or a metamaterial element) etched on the top plate of a rectangular waveguide. The waveguide is excited by a T E10 port at one end. The coordinate system is chosen so that the propagation direction is in the z-direction, a corresponds to the width of the waveguide along the x-axis, and b corresponds to the height along the y-axis, as shown in Fig.(4.8). If the iris is sufficiently small in dimensions compared to the guided wavelength, it can be approximated by its lowest order multipole moments, usually a combination of both electric and magnetic dipoles. In our particular scenario, each iris can be characterized by a magnetic dipole moment tangential to the waveguide’s surface. The scattered field produced by each dipole within the waveguide interacts with all the other fields produced by the dipoles as well as the incident wave, such that the total field within the waveguide, Htot pri q, is given by Htot pri q “ H0 pri q ` ÿ` Gmm pri , rj q ¨ mef f pri q ˘ (4.19) j Where Gmm correspond to the magnetic dyadic Green’s function inside a rectangular waveguide [116], H0 pri q is the incident field, and ri is the position of the i ´ th iris. The dyadic Green’s function can be expressed in terms of the product of eigenmodes which correspond to the solution for the fields inside the waveguide without irises. The dipole moments mef f denotes the magnetic dipole of the i´th element, and 116 they are proportional to the local field at the center of the element with a coupling ¯ mm,i Htot pri q. After coefficient, called polarizability α [96, 98], such that mef f pri q “ ᾱ some algebraic manipulations, the general dipole model equation can be rewritten in a matrix form as ` ˘ ef f ´1 Iij ´ Gmm mj “ H0i αmm,i ij (4.20) where Iij corresponds to ij’th element of the identity matrix and the sub-indices i, j correspond to the iris’ locations. Moreover, by solving Eq.(4.20) we can find the dipole moment mef f representing each element while accounting for all the dipoledipole interactions. The magnetic dipoles calculated in this manner can be used as the weighting coefficients in an array factor (AF) calculation, in order to obtain an accurate calculation of the far-fields. Two different antennas are examined in this subsection, each consists of an airfilled X-band WR-90 waveguide with dimensions a “ 21.94 mm and b “ 5.08 mm. The radiating iris in one antenna is an elliptical iris and a CELC resonator in the other one, as shown in Fig.4.8. Both illustrative examples have a matched termination to absorb the guided wave and prevent it from being reflected. Both antennas are designed to operate at 9 GHz frequency. The longitudinal positions of the irises on the broadwall are chosen at half of the guided wavelength λg {2 along the wave propagation direction (z-axis), and alternating across the xdirection between λg {4 and 3λg {4. The first step toward characterizing the response of these antennas is to compute the polarizability representing the radiating elements. Then, the Green’s function Gmm can be found theoretically from the eigenmodes inside a rectangular wavegij uide [86]. This expression, in addition to the Green’s function in free space, can be introduced in Eq.(4.20) to find the effective dipole moments representing each tapered iris. With uniform irises, both the SWA and the metasurface antenna are 117 Figure 4.8: . a) Slotted waveguide antenna (SWA). the slot dimensions are slength “ 16.33 mm, swidth “ 1.52 mm. b) Metasurface antenna. The iris is called CELC resonator, and its dimensions are l “ 9 mm, w “ 4.8 mm, g “ 2.1 mm, the CELC thickness is t “ 1 mm. similar to the case of antenna arrays with discrete elements having equal excitation, which results in an SLL around « 13 dB and a maximum directivity of 14 dB, as shown in Fig. (4.9), which represents the φ´component of the directivity Dφ . As shown, excellent agreement between the full-wave simulation and the dipole model is obtained. In addition, the dipole model can reconstruct the desired response for different frequencies, highlighting the beam-steering capabilities of the SWA and the metasurface antenna. Lower SLLs are obtained upon using non-uniform irises, and different tapers can be used to generate the distribution coefficients such as the Chebyshev method, well known in antenna array designs [87]. Since the iris’ geometry used for metasurface antennas is complex, the previous design equations [178] cannot be used. Instead, we outline a design process based on the dipole model formulations: uni 1. Calculate the magnetic polarizability αmm of the uniform metamaterial ele- ments at the desired operating frequency. 118 Figure 4.9: Uniform LWA: Comparison of the φ´component of the directivity in the H-Plane between full wave simulation and the dipole model at different frequencies. a) SWA. b) Metasurface antenna . 119 2. Calculate the required distribution coefficients an using the Chebyshev method for the given number of elements and desired SLL (in our example the number of elements is 10 and the desired SLL is 20 dB). 3. Normalize these coefficients such that ř n an “ 1. uni tap . “ an αmm 4. Set the polarizability for the tapered metasurface αmm 5. Simulate a single iris and calculate its effective αmm at the desired frequency, as a function of a single geometrical parameter,(in our example, the iris’ length l was considered) and create a function (αmm plq), as shown in Fig.(4.10a). tap 6. Find the values of l that satisfy αmm using the function developed in the pre- vious step. The directivity for the examples of a tapered metasurface antenna are shown in Fig.4.10a-b. The SLL is drastically reduced to 19.5 dB and 17.6 dB respectively, while the maximum directivity was slightly reduced to 13.6 dB,as expected from the Chebyshev tapering. The results presented in this subsection demonstrate the efficiency of the dipole model to predict the response of uniform and tapered LWAs. As such, the model also provides a platform that does not require a smooth variation of the unit cell geometry, and includes the coupling between the radiating elements. Therefore, this technique can be extended to combine antenna design with metamaterial concepts, and is especially well suited for novel guiding structures, such as waveguide-fed metasurfaces. The details on the iris’ characterization can be also extended to different dipole moment orientations as well as different waveguide structures, a two dimensional scenario is shown in the next section. 120 a) SLL=19.5 dB b) SLL=17.6 dB Figure 4.10: a) Polarizability of the meta-atom as a function of the geometric parameter l. The points represent the coefficients of the Chebychev distribution, which are used to taper the metasurface antenna. b) Comparison of the directivity in the H-Plane between Full wave simulation and the dipole model. Figure 4.11: Kymeta Antenna. Images taken from www.kymetacorp.com 121 4.4 Example II: Kymeta mTenna Broadband satellite communications in mobile platforms require a scanning antenna. Based on metasurface design principles, Kymeta has developed a novel electronicallyscanned antenna (mTenna) which is capable of large-angle beam scanning with benefits in cost, size, weight, and power; as compared with traditional dish antennas and phased arrays, as shown in Fig. 4.11. The enhanced capabilities of this antenna are possible by the nature of the Metamaterial Surface Antenna Technology (MSAT). MSA-T is based on the idea that a radio frequency signal is fed into a guided mode which propagates along the surface covered with controllable metamaterial elements. By using different mechanisms of tunability (such as liquid crystals for example), a specific pattern set at the aperture plane is applied in order to generate a beam in the desired direction. The mTenna uses the principles of holography described in section 4.2 to direct its radiated energy primarily in the direction of the desired satellite. The guided wave, Ψref corresponds to the fundamental mode of a dielectric-filled planar waveguide. The desired image wave, Ψobj , is the plane wave propagating to/from a satellite. In the mTenna, the interference pattern is recorded on the surface of the array by tuning each metamaterial element at the mTenna’s aperture plane. The task to address at the aperture plane can be posed as: At each element’s location r, what is the best tuning state that would resemble the desired interference pattern?. Considering that the elements are not patterned in a periodic fashion on the mTenna’s surface, the Euclidean Modulation was used, as shown in Table 4.1. Approximating the product of the reference wave and the object wave in terms of their respective propagation constants we get Ψobj “ e´jkpθ0 ,φ0 q¨r (4.21) Ψref « e´jβ0 ¨r (4.22) 122 where kpθ0 , φ0 q is the desired directional complex propagation vector in free space, β0 is the complex propagation vector of the reference wave, and r is a coordinate on the recording surface. As an initial approximation, the interference pattern to be recorded is given by Ψhol “ Ψobj Ψref ˚ (4.23) Equation (4.23) determines the desired complex phase required on the surface of the mTenna. In practice, the antenna is composed of a collection of metamaterial elements that only can achieve a discrete set of states or resonance frequencies, as shown in 4.12a. To begin, each possible tuning state of the metamaterial elements is characterized by means of the polarizability extraction technique developed in section 2.2. Since the desired beam is selected at a single operating frequency, the object wave can be characterized by the wavenumber kpθ0 , φ0 q, and the polarizabilities are extracted as a function of the tuning states, as shown in Fig.4.12b. This function defines the set of “available states” that will be used to map onto the “desired states” –which resemble Ψhol by means of the Euclidean Modulation. This modulation consists into mapping the set of available tuning states to the set of desired tuning states using the shortest distance –defined in an Euclidean norm– in the complex plane. As such, for each element at the aperture plane, as shown in Fig.4.12c, there exists a unique state for the polarizability, yielding a distribution as shown in Fig.4.12d. The modulation determines the tuning state, i.e. the polarizability, that each metamaterial element will have at a specific frequency, and for a particular direction of the desired beam. For the specific example presented here, the desired beam is selected at pφ0 , θ0 q “ p0, π{6q. Details of this modulation technique can be found in [61]. Once the polarizability is set for each element, then the mutual interaction between elements is included. By using the expressions for the Green’s function in a planar waveguide (See chapter 2, section 2.4), it is possible to account for the perturbation 123 of the guided mode, as well as for the interaction between elements. By solving Eq.(2.2) for the particular case of the mTenna, it is possible to find each component of the effective dipole moments, as shown in Figs.4.12e-f. Given the mathematical tools presented in the previous sections, it is possible to model different beams created by the mTenna. The calculation of the effective dipole moments can be used to find the total directivity, following Eqs. (4.11). As shown in Fig. 4.13 the overall radiation pattern of the mTenna can be calculated in a fraction of the time with the dipole model in comparison with full wave simulations, yet obtaining excellent agreement between the two methodologies. In any beamforming scheme, an accurate description of forward model (i.e. the modulation and the calculation of the radiation pattern) is crucial for reliable image reconstruction as well as for estimating the imaging capabilities. Figure 4.14 correspond to different comparisons between the results presented by using the dipole model and a full wave simulation using HFSS. As shown, a good comparison is obtained between the two presented simulations. The differences in the presented method are shown for the secondary lobes, which in average correspond to a sidelobe level below 15dB. Moreover, for the particular example shown in Fig.4.12, for a linearly polarized beam at p0, π{6q; the maximum directivity corresponds to 25.75 dBi. It is worth noting that the dipole model obtains excellent agreement with fullwave simulation with much lower computational cost. For instance, the code of the algorithm and the full wave simulations have been implemented using a workstation equipped with a 64-bit 3.5 GHz CPU. The full-wave simulations developed for the mTenna simulation in HFSS takes an approximate time of 8 hours at a single frequency, whereas the dipole model created in Matlab takes approximately 12 seconds. As such, an improvement of 2400 times of accelerated computing time is achieved. 124 a) b) c) d) e) f) Figure 4.12: a) Polarizability of the meta-atom as a function of the frequency for all possible tuning states. b) Polarizability of the meta-atom as a function of the tuning states at an specific operating frequency. c) Ditribution of the metamaterial elements at the aperture plane d) Distribution of the polarizability by means of the Euclidean Modulation. 125 a) b) c) d) Dipole Model HFSS Dipole Model HFSS Figure 4.13: a) Effective Magnetic dipole moments for a beam at θ0 “ π{6. b) Effective Magnetic dipole moments for a beam at θ0 “ π{3. c) 4.5 Example III: Waveguide-fed Metasurfaces for Computational Microwave Imaging As a different example on the simulation of waveguide-fed metasurface antennas, in this section a comprehensive simulation platform for computational microwave imaging is presented. This particular case corresponds to a computational imaging system is based on frequency-diverse metasurface antennas (FDMAs). FDMAs consist of a waveguide patterned with complementary metamaterial elements with resonant frequencies selected randomly from a band of operation, enabling the generation of distinct frequency-indexed radiation patterns. By accurately modeling the 126 a) b) c) dBi dBi Figure 4.14: 3D comparison of the Directivity obtained with Full Wave Simulation (HFSS) and the dipole model. a) Total Directivity. b) Dφ c) Dθ . Modulation applied at 30 deg. scan angle. fields produced by the FDMA using the dipole model, it is possible to predict the capabilities of the imaging system in a fast and reliable manner. All the mutual interactions between metamaterial elements are included, and it is demonstrated that these interactions are crucial to a better understanding of the FDMAs’ capabilities, e.g. effective aperture area and correlation of radiation patterns. The simplicity and accuracy of the proposed model permits the simulation of different metasurfaces for computational microwave imaging, where traditional antenna design– and metamaterial modeling–are prohibitively costly. In any computational imaging scheme, an accurate description of forward model is crucial for reliable image reconstruction as well as for estimating the imaging capabilities. Formulating an accurate forward model, especially for FDMAs, is not an easy task, since they are composed of arbitrarily arranged metamaterial elements. Common techniques for modeling such structures often requires a smooth varia127 Figure 4.15: Frequency diverse metasurface antenna. tion of the metamaterial’s geometry over the surface [78]– which cannot be assumed for FDMAs. On the other hand, full-wave simulations are intractable due to FDMAs electrically large size. An alternative approach was proposed in [69] where the FDMA was modeled as a collection of effective dipoles [69] with their complex amplitude calculated as the product of the guided mode and the element’s polarizability. This technique ignored the mutual interaction between metamaterial elements or any phase perturbation of the guided wave that the elements may cause. As a result, the technique of [69] could only capture the salient features of an FDMA and could not replace experimental trial and errors. In this section, the dipole model is extended to include the mutual interaction between effective dipoles in the FDMA as well as their interaction with the guided wave, following the expressions shown in Chapter 2. Using this enhanced technique, we capture many interesting physics behind FDMAs operation. The proposed model can be used to inform design of FDMAs for future generations of microwave imaging systems [73, 75]. The dipole model is analyzed starting from a square FDMA operating at the K-band where the center frequency is 22 GHz (corresponding to λ “ 1.3 cm). The FDMA’s length is L “ 20λ and the waveguide mode is excited from the center of the metasurface. The top plate of the FDMA is patterned with 376 metamaterial 128 elements, arranged periodically (for simplicity, not required for the model) and separated d “ 1.05λ. Each metamaterial element scatters as an effective magnetic dipole m– which can characterized by its polarizability α [103] and it is proportional to the total field at the metamaterial element’s location, m “ αHtot . The total field Htot is superposition of the guided mode and the scattered field by all the other dipoles. This modeling technique leads to a matrix system in which the unknowns are the m, while the polarizability [103] and the Green’s function (which accounts for the mutual interaction between dipoles) are known [116]. For brevity, we consider the metamaterial elements with a polarizability that follows a Lorentzian response. A larger magnitude of the polarizability means the dipole moment is stronger and therefore, the perturbation of the guided mode is more pronounced. Using this model, and assuming all elements have the same resonant frequencies, the resulting dipoles are calculated with and without the mutual element interactions, as shown in Fig.4.16. When the mutual element’s interaction is ignored, the magnetic dipole moments follow a similar functional form as the cylindrical guided mode. Since the decay of the guided wave due to radiation from elements is not included, the effective area of the FDMA appears large. In contrast, when the mutual interactions between elements are included, the effective area is reduced and the magnitude of the magnetic dipoles does not follow the waveguide distribution. Fig. 4.16 clearly demonstrates that by ignoring the element-element interaction, many interesting physical phenomenon inside an FDMA are not modeled, and may result in misleading estimates of the FDMA performance. Furthermore, an imaging system with one transmitter and receiver is considered. A FDMA with randomly selected resonant metamaterial elements that can generate frequency diverse patterns, is used to illuminate a scene, while a single low-gain probe placed next to it, is used as the receiver, as shown in Fig.4.17a. Given the magnetic dipoles computed using the proposed method, the electric fields at each 129 a) b) 20 20 20 20 Figure 4.16: Normalized |m|. a) with and b) without mutual element interaction. pixel in the scene are calculated. Assuming first Born approximation and diffraction ¯ , whose entries are the inner prodlimited imaging, we populate a sensing matrix H̄ uct of the transmitted and received fields (See Chapter 3). In the imaging scheme at hand, the most crucial factor is the diversity of frequency-indexed radiation patterns (examples of the frequency-indexed patterns are plotted in Fig.4.17c). To assess this ¯ , as discussed in subfactor, we compute the singular value (SV) decomposition of H̄ section 3.4.1. The SVs identify the number of measurement modes that contribute significant information to the image reconstruction. Figure 4.17b shows the SVs of ¯ associated with the imaging system shown in Fig. 4.17a, for different values of the H̄ coupling factor F . It is important to highlight that when the mutual element interactions are ignored (dotted line), change of coupling factor does not affect imaging performance, a conclusion that is not correct from physics perspective. When the mutual element interaction is included, we gain insight into the complex relationship between coupling factor and pattern diversity. Small values of F represent a dipole moment that does not perturb the guided mode significantly, and the only mechanism contributing to the frequency-diversity is the resonant nature of the elements. As F increases, the dipole perturbation is more significant the guided wave scrambles around within the waveguide, further increasing the frequency diversity, as shown in 130 Fig.4.17b for F “ 0.3mm3 . However, for strongly interacting dipoles (F “ 0.4mm3 and F “ 0.5mm3 ), the SV performance is poor, since most of the energy coupling leaks into free-space and the effective area of the FDMA is decreased. Notice that the dipole model provides an efficient and simple method to analyze FDMAs, and can find application in rethinking the hardware design of future microwave imaging systems. Fully leveraging the power of the dipole model for FDMA could lead to the implementation metamaterial imaging systems that can rival imaging performance of current security screening systems, but from a much simple and lower cost hardware. The computational advantages rely of the implementation of the dipole model (for the antenna design) in conjunction with the adapted RMA (for the image reconstruction). a) b) 0 F=2 10 F=3 F=4 F=5 SV Rx FDMA scene -2 10 0 c) 18.8 GHz 21.4 GHz 20 40 60 80 Mode index 24.2 GHz 100 dB Figure 4.17: a) Imaging system configuration. b) SV decomposition of the modes of the imaging system shown in a). Dotted line corresponds to the FDMA with noninteracting dipoles and solid lines correspond to the case including coupled dipoles for different values of coupling F . c) Radiation patterns at different frequencies. 131 4.6 Summary and Conclusions On the dipole model for SWAs The dipole model provides a simple, low-cost, yet perfectly compatible platform to combine antenna design with metamaterial concepts, and is especially well suited for 1D-waveguide-fed metasurfaces. Let us recall that the only requirement of the dipole model is that the radiative elements are electrically small and have sufficiently separation to avoid higher-order mode coupling, requirements that are satisfied for the particular case of SWAs. It is important to note that when the SWA termination is shorted and the dipolar interaction of the images is included, the number of apertures in Eq.(4.16) does not increase; rather, the Greens function is modified. Once the matrix system is solved and the total magnetic field is obtained, the magnetic dipole is simply mi “ αi H loc pyj q. Being αi the polarizability of the iris as found in section 2.2. Although the scattering parameters can be computed theoretically, here these parameters are computed through full-wave simulation which account for the radiation accurately. Notice that this simulation is made for a single slot only, while Eq.(4.17) is used to compute the entire response of the SWA. In addition, it has been demonstrated that the presence of parasitic elements improves the performance of the SWA significantly. This increase is further highlighted in Table II which list the main characteristics of the radiation patterns depicted in Fig. 4.6. The dipole model model exhibits close agreement with full wave simulation, thereby verifying utility of the dipole model in obtaining an accurate representation of the SWA. It is worth noting that the dipole model obtains excellent agreement with full-wave simulation with much lower computational cost. For instance, the code of the algorithm and the full wave simulations have been implemented using a 132 workstation equipped with a 64-bit 3.5 GHz CPU. The full-wave simulations developed for a simple array of 16 slots take 146 s as the SWA is discretized into 442.680 mesh cells in the time domain solver, whereas the dipole model created in Matlab takes 0.34s. On the dipole model for metamaterial antennas The dipole model applied for the mTenna, though it may be elaborated upon, has been demonstrated to be successful in predicting the radiation pattern from an aperiodic array of tunable metamaterial elements. The benefits of this model are numerous: It is much faster than the full-wave simulations simulated in HFSS. These simulations take an approximate time of 8 hours at a single frequency, whereas the dipole model created in Matlab takes approximately 12 seconds. As such, an improvement of 2400 times of accelerated computing time is achieved. The polarizability extraction from accurate full-wave simulation is straightforward and performed in a small simulation domain for all possible tuning states. On the dipole model for frequency diverse apertures The dipole model provides an efficient and simple method to analyze FDMAs, and can find application in rethinking the hardware design of future microwave imaging systems. Fully leveraging the power of the dipole model for FDMA could lead to the implementation metamaterial imaging systems that can rival imaging performance of current security screening systems, but from a much simple and lower cost hardware. The computational advantages rely of the implementation of the dipole model (for the antenna’s aperture design) in conjunction with the adapted RMA (for the image reconstruction). 133 5 Conclusions The concepts presented in this thesis pave the way for a simple and efficient approach to the analysis of metasurface antennas. The combination of polarizability extraction techniques with the dipole model provides an inherently multiscale modeling tool that interprets metasurfaces as array of dipoles with given polarizabilities, a powerful framework to design and characterize waveguide-fed metasurfaces without any limitation on the element geometry or periodicity assumptions common to other homogenization techniques. The results in Fig. 2.17 once again highlights our proposed method to compute the electromagnetic response of waveguide-metasurface in an accurate, yet efficient process. Here, the model did not solve for complex dispersion equations–as is the case in most conventional techniques on waveguide-fed metasurfaces. Instead, we employ a simple analytical formulation to relate the field of each metamaterial element to the overall response. The dipole model only relies on full-wave simulation of the polarizability extraction of a single element. Then it is possible to use this element’s characterization to build a complete model for the overall waveguide-fed metasurface. As such, this model can be considered in between the analytical ex134 pressions and numerical simulations. In particular, full-wave simulations discretize the entire structure as a collection of “patches” for which Maxwell’s equations are solved using a matrix formulation based on finite element methods. The structure’s meshing is deeply subwavelength but the structure’s size can be many wavelengths in size. Thus, the solution for the structure’s electromagnetic response is found after solving a matrix problem of thousands (or even millions) of patches or mesh cells. Instead, the dipole model reduces the overall structure to a collection of effective dipoles, creating a matrix problem which is as big as the number of metamaterial elements only. As shown in chapter 4, for different antenna structures the computation time is drastically reduced when the dipole model is used. In fact, while full-wave simulation is a possibility for the waveguide-fed metasurface shown in the previous section, for many applications; full-wave simulations is not an option. However, our method provides a simple, low cost, and yet effective method to model such complex structures. In the particular example presented in chapter 2, full wave simulation takes approximately 180 seconds and solves for 85,537 mesh cells. Meanwhile, the dipole model takes approximately 0.22 seconds and solves the problem for 12 effective dipoles. Therefore, dipole model leads to an improvement around 800 times of computing time. The dipole model presented in the aforementioned examples stands alone as a technique to improve the performance of waveguide-fed metasurfaces, but it may also be used as the starting point for further design and optimization techniques. Moreover, the notion of effective dipoles, both electric and magnetic, and the concept of a waveguide-fed metasurface as a collection of properly arranged and suitably arranged metamaterial elements, has facilitated the modeling and enabled modularity in designing complex antenna structures with unprecedented functionalities. Figure 5.1 shows the different modules for the current dipole model implementation that 135 are considered throughout this document: 1. We begin with the module shown in Fig.5.1a and we select the application of interest, which can be divided between a.) beamforming: to model and design highly directive antennas for satellite and terrestrial communications as well as SAR. And b.) microwave imaging: to design apertures to be used in security screening, through wall imaging, computational imaging, among others. 2. Next, the domain over which the metamaterial elements will be etched is chosen. The type of waveguide structure depends on the direction of propagation of the guided wave, which can be one-dimensional such as rectangular waveguides, transmission lines, microstrip lines, etc; or two-dimensional, such a parallel plate waveguide; as shown in Fig.5.1b. 3. Once the domain is chosen, i.e. once the incident fields are defined inside the waveguide structure, it is possible to define the distribution of the metamaterial elements over the waveguide’s surface. The element’s distribution is closely correlated to the modulation and tuning state assigned to each metamaterial element. The individual element’s response is determined by its polarizability, whose details will be described in section 2.2. Some examples of this response is shown in Fig. 5.1c and some examples of the element’s distribution is shown in Fig.5.1d. 4. Given the polarizability distribution, it is possible to choose between a fast model known as non-interacting dipoles in which the effective dipole moments are a direct product between the polarizabilities and the incident fields. A more accurate model known as interacting dipoles solves the dipole model Eq.(2.2) to determine the effective dipole moments including their mutual interactions, as shown in Fig.5.1e. 136 a) Application b) Waveguide Structure c) Polarizability Beamforming 1D X- band Microwave Imaging 2D K- band d) Element’s Distribution e) Dipole Model f) Scattered Fields Slotted Antenna Non-interacting dipoles Waveguide Domain Dynamic Metasurface Random Periodic Interacting Dipoles Far-field Domain Figure 5.1: Implementation of the dipole model in a set of Modules. 5. Finally, given the solution for the effective dipole model, the scattered fields inside and outside the waveguide-domain can be calculated, as shown in Fig.5.1f. Overall, the proposed model, offers a simple, systematic, and computationally low cost method to predict the response of a waveguide-fed metasurface. This exciting capability opens many exciting opportunities: the proposed model can be used to optimize metasurfaces used in computational microwave imaging and replace experimental trial and errors [73]. It can be used to design and optimize electronically 137 steerable metasurface antennas for communication and imaging purposes. It can be used analyze aperiodic structures, recently garnered much attention at optical frequencies. 138 Appendix A Scattered fields in a parallel plate waveguide Let us consider a single metamaterial element patterned on the top plate of a parallel plate waveguide. By means of the surface equivalence principle, the element can be represented as a magnetic surface current Mprq. Taking a Taylor expansion over this surface current yields an effective dipole moment given by 1 m“ jωµ ż MprqdV, (A.1) where V represents the effective volume of the metamaterial element embedded in the waveguide. The effective dipole moment–when observed from the waveguide’s domain– has multiple images along the z´axis, as shown in Fig.A.1a, due to the presence of the plates of the waveguide. When the spacing between the plates is subwavelength, the stack of dipoles can be approximated as an infinite line of magnetic surface current density with infinite-small width given by M “ M δpρ ´ ρs qx̂, where ρs is the element’s location. As such, the problem of a single metamaterial element embedded in a parallel plate waveguide can be approximately replaced by the problem of an infinite ribbon of magnetic current density with infinite small width, 139 a) mx z h x y b) c) M rs r my z y x Figure A.1: Multiple Images of an element embedded in a parallel plate waveguide. under the particular approximation that the spacing between the metallic plates is significantly smaller in comparison with the wavelength, as shown in Fig.A.1b. The amplitude of this magnetic current density is given by mx “ 2M . 2jhωµ (A.2) In the following, we derive the field scattered by this magnetic current density. To do that, we note that the vector potential F associated with this driving source satisfies Helmholtz equation as [192] ∇2 F ` β 2 F “ εM. (A.3) Due to the symmetry of the problem, the solution of Eq. (A.3) is given in terms of 140 cylindrical functions as F“ M p2q H pk|ρ ´ ρs |qx̂ 4j 0 (A.4) where |ρ| is the distance between the location of the dipole pxs , ys q and any point in the plane along the waveguide (x,y), i.e. ρ “ px ´ xs qx̂ ` py ´ ys qŷ. Given the solution for the magnetic vector potential, the scattered magnetic field is H “ ´jωF ´ j 1 ∇p∇ ¨ Fq ωµε (A.5) The divergence of the vector potential is p2q M BH0 pkp|ρ ´ ρs |q ∇ ¨ F “ Bx Fx “ 4j Bx (A.6) “ M 4j p2q BH0 puq Bu Bu Bx where u is defined as u“k a px ´ xs q2 ` py ´ ys q2 Bu kpx ´ xs q px ´ xs q “a “ k cos θ. “ k2 Bx u px ´ xs q2 ` py ´ ys q2 p2q (A.7) p2q Taking into account the recurrence relations on Bessel functions, Bu H0 puq “ 1{2pH´1 puq´ p2q p2q H1 puqq “ ´H1 puq, and replacing Eq.(A.7) into Eq. (A.6), we get the divergence of F as ∇¨F“ ´M k p2q cos θH1 pkp|ρ ´ ρs |q 4j (A.8) Furthermore, the gradient over ∇ ¨ F is given by ˙ ´M k 2 px ´ xs q p2q H1 puq x̂ ∇p∇ ¨ Fq “ Bx 4j u ˆ ˙ ´M k 2 px ´ xs q p2q `By H1 puq ŷ. 4j u ˆ 141 (A.9) The first derivative Bx in Eq.(A.9) is given by ˆ ˙ ´M k 2 px ´ xs q p2q Bx H1 puq “ 4j u « ˜ ¸ ˜ ¸ff p2q p2q ´M k 2 H1 puq H1 puq px ´ xs qBx ` . 4j u u ´ Considering that p2q H1 puq u ¯ p2q (A.10) p2q “ 1{2pH0 puq ` H2 puqq, the first derivative in Eq.(A.10) is equal to ˜ Bx p2q H1 puq u ¸ 1 p2q p2q Bu “ Bu pH0 ` H2 q . 2 Bx Taking into account both recurrence relations 1 p2q p2q p2q Bu H2 “ pH1 puq ´ H3 puqq 2 p2q H3 “ 4 p2q p2q H puq ´ H1 puq, u 2 (A.11) (A.12a) (A.12b) and replacing Eqs. (A.12) and Eq.(A.7) in Eq.(A.11), we get ˜ Bx p2q H1 puq u ¸ p2q “´ k 2 H2 puqpx ´ xs q . u2 (A.13) Replacing Eq.(A.13) into Eq.(A.10) we get « ff p2q p2q ´M k 2 H puq H puq p2q “ ´H2 puq cos2 θ ` 0 ` 2 4j 2 2 (A.14) 2 ı ´M k ” p2q p2q “ H0 puq ´ cos 2θH2 puq 8j Following a similar procedure for the partial derivative By in Eq.(A.9), we get ˆ ˙ ´M k 2 px ´ xs q p2q H1 puq “ By 4j u ˜ ¸ p2q ´M k 2 H1 puq px ´ xs qBy 4j u 142 (A.15) where ˜ By p2q H1 puq u ¸ p2q “´ k 2 H2 puqpy ´ ys q . u2 (A.16) Replacing Eq.(A.16) into Eq.(A.15) we get “´ M k2 p2q sin 2θH2 puq. 8j (A.17) Replacing Eq.(A.14) and Eq.(A.17) in Eq.(A.9), in addition to the vector potential defined in Eq.(A.4), it is possible to obtain a complete expression for the magnetic field inside the waveguide. Following Eq.(A.5), the magnetic field is given by H“ ı ´M ω ” p2q p2q H0 pkp|ρ ´ ρs |q ´ cos 2θH2 pkp|ρ ´ ρs |q x̂` 8 (A.18) ´M ω p2q sin 2θH2 pkp|ρ ´ ρs |qŷ. 8 In order to compute the electric field related to the magnetic vector potential we have 1 E“´ ∇ˆF ε (A.19) The curl of the magnetic vector potential is given by ∇ˆF“ ı M” p2q p2q Bz pH0 puqqŷ ´ By pH0 puqqẑ 4j (A.20) Following similar procedures, the partial derivatives are given by p2q p2q By pH0 puqq “ ´H1 puq Bu p2q “ ´k sin θH1 puq By (A.21) Replacing Eq.(A.21) in Eq.(A.20), we can find the electric field as E“ ´M k p2q H pkp|ρ ´ ρs |q sin θẑ. 4jε 1 143 (A.22) For completion, the expressions for the scattered fields can also be derived for the case when the magnetic dipole is rotated π{2, i.e. if the surface current is oriented along the ŷ direction. In this case, the scattered fields are given by H: “ ´M ω p2q sin 2θH2 pk|ρ ´ ρs |qx̂` 8 (A.23) ı M ω ” p2q p2q H0 pk|ρ ´ ρs |q ´ cos 2θH2 pk|ρ ´ ρs |q ŷ, 8 where .: indicates that the source is oriented along the ŷ direction. Analogously, the electric field is E: “ M k p2q H pk|ρ ´ ρs |q cos θẑ. 4jε 1 (A.24) Once the scattered fields are calculated for two different orientations of the dipole, it is possible to determine all the components of the Green’s function as follows: mm H “ tHx , Hy u “ mx Gmm xx , mx Gxy ( ( ( mm H: “ Hx: , Hy: “ my Gmm , yx , my Gyy (A.25) E “ Ez “ mx Gem xz E: “ Ez: “ mx Gem xz where mx and my correspond to the magnetic dipole moments that result from the integration of Eq.(A.1), over the differential volume of the element in the waveguide. Replacing these values of the magnetic dipole moments into Eqs. (A.5), (A.19), (A.23) and (A.24); it is possible to calculate the components of the Dyadic Green’s 144 function as Gmm xx “ ı ´jk 2 ” p2q p2q H0 pk|ρ ´ ρs |q ´ cos 2θH2 pk|ρ ´ ρs |q 8h (A.26a) ´jk 2 p2q sin 2θH2 pk|ρ ´ ρs |q 8h (A.26b) Gmm xy “ ´jk 2 p2q sin 2θH2 pk|ρ ´ ρs |q 8h ı ´jk 2 ” p2q p2q “ H0 pk|ρ ´ ρs |q ´ cos 2θH2 pk|ρ ´ ρs |q . 8h Gmm yx “ Gmm yy (A.26c) (A.26d) For clarification, the super-indices in Eq.(A.26) represent the component of the Green’s function that relates the magnetic field due to a magnetic source, and the sub-indices represent the component of the magnetic dipole and magnetic field respectively. For example Gxy mm corresponds to the Green’s function associated to the Hx due to a magnetic dipole oriented along the y´direction. Analogously, the electro-magnetic Green’s function tensor–relating the electric field due to magnetic sources– is given by ı ´jk 2 Z0 ” p2q “ H1 pk|ρ ´ ρs |q sin θ 4h ı jk 2 Z0 ” p2q H pk|ρ ´ ρ |q cos θ Gem “ s 1 yz 4h Gem xz (A.27a) (A.27b) Equations (A.26) and (A.27) are in two particular scenarios: Eq. (A.26) is used ¯ W G and accounts for the mutual interactions between metamaterial to represent Ḡ elements inside the waveguide’s domain. Moreover, Eq.(A.27) is used to find the scattered fields inside the waveguide, once the effective magnetic dipole moments have been found by means of Eq.2.2. These fields are further compared with the scattered fields obtained with full-wave simulation, in order to validate our proposed model. 145 Appendix B Formulation of the Range Migration Algorithm In this section, the development of the range migration method is presented. This work was performed in co-authorship with Aaron Diebold. In the particular imaging scenario considered here, a single receiver is located at the origin and a set of transmitters are uniformly distributed along an axis, as shown in Fig. (3.3a). Though we will later discretize these equations with respect to the coordinates, it is convenient in deriving the basic reconstruction approach to assume the coordinates are continuous variables. We begin by writing the field at the receive aperture in terms of contributions from an object in the scene. Starting with Eq.(3.5) and Eq.(3.6), we have ż Spyt , f q “ V σpx, y, zq ´jkRt ´jkRr e e δpyr qδpzr qδpzt qdV 16π 2 Rt Rr (B.1) The use of the Dirac delta functions is justified as we will be integrating over the receive antenna location, which is fixed in the aperture plane. We imagine the field at coordinate yt to be a Hertzian dipole. Then, the distance from each of the effective dipole sources to a scattering point in the scene is 146 Rt “ a x2 ` py ´ yt q2 ` pz ´ zt q2 . (B.2) Although we will fix the receiver position at the origin of the coordinate system, we can generally write the distance from a receive point to the object as Rr “ a x2 ` py ´ yr q2 ` pz ´ zr q2 . (B.3) We now assume that the transmit and receive aperture fields are continuous and that a Fourier transform can be established over those fields. Thus, taking the Fourier transform of both sides of B.1 over the aperture coordinates yields ż ż Spyt , kqe´jkyt yt e´jkzt zt e´jkyr yr e´jkzr zr dAt dAr Ŝpky , kq “ Ar At ż “ V σpx, y, zq 16π 2 ż ˆ Ar ż At e´jkRt ´jkyt yt ´jkzt zt e e δpzt qdAt Rt (B.4) e´jkRr ´jkyr yr ´jkzr zr e e δpzr qδpyr qdAr dV Rr The integral over Ar is ż Ar (B.5) a x2 ` y 2 ` z 2 . The integral over At can be simplified as where R “ ż At where Rt “ functions e´jkRr ´jkyr yr ´jkzr zr e´jkR e e δpzr qδpyr qdAr “ Rr R ejkRpyq Rpyq e´jkRt ´jkyt yt ´jkzt zt e e δpzt qdAt “ Rt ż yt 1 e´jkR ´jkyt yt e dyt R1 (B.6) a x2 ` py ´ yt q2 ` z 2 . Notice that B.6 is the convolution between the and e´jkyt y , and can be rewritten using the convolution theorem as 147 ż yt 1 e´jkR ´jkyt yt e dyt “ R1 ż 1 1 1 1 qejkyt y dkyt qE2 pkyt E1 pkyt (B.7) K where E1 pky1 q can be solved by using the method of stationary phase as [151] ż 1 E1 pkyt q e´jkR ´jkyt 1 y e dy R y ? ? 2 12 ? 2 2 j 2π «b e´j k ´kyt x `z 12 k 2 ´ kyt “ (B.8) Equation B.8 is an approximation considering that the magnitude term 1{R1 has 1 q can be been ignored as in other RMA derivations [151, 160]. The function E2 pkyt easily evaluated as ż 1 E2 pkyt q 1 1 e´jkyt y e´jkyt y dy “ δpkyt ` kyt q. “ (B.9) y Inserting B.8 and B.9 into B.7 yields ż e ´jkRt1 ´jkyt yt e yt ? ? 2 12 ? 2 2 j 2π e´j k ´kyt x `z e´jkyt y . dyt « b 2 k 2 ´ kyt (B.10) Finally, inserting B.5 and B.10 into B.4 yields ż Spky , kq “ V ? σpx, y, zq j 2π b ˆ 16π 2 2 2 k ´ kyt ? e´jk x2 `y 2 `z 2 ´j e (B.11) ? ? 2 k2 ´kyt x2 `z 2 ´jkyt y e dV. In the absence of a kz component, the object cannot be resolved along the zdirection and therefore, it is preferable in this scenario to have an object lying in the 148 xy-plane, such that z=0 everywhere. Additionally, if x " y, making the approximaa tion x2 ` y 2 « x allows B.11 to be rewritten as ? ż ż j 2π σpx, y, zq ´jkx x ´jky y a 2 Spky , kx q “ e e dydx 2 16π x y k ´ ky2 (B.12) where ky “ kyt (B.13) b kx “ k 2 ´ ky2 ` k. From B.12, it can be seen that Spky , kx q is the Fourier transform of the reflectivity of the signal and therefore, in order to retrieve the reflectivity, an inverse Fourier ? transform over Spky , kx q must be performed. In addition, the term ?jk22π acts as a ´k2 y filter of the signal in the Fourier domain. Equation B.13 will be referred as the dispersion relation. The mapping of the signal from Spky , kq to Spky , kx q is known as Stolt Interpolation, which compensates the range curvature of all scatterers by an appropriate warping of the backscattered data. Additionally, Stolt Interpolation resamples S onto a uniform grid of ky and kx so that Fast Fourier Transforms can readily operate on S. In the present context, a this dispersion relation arises as the result of an approximation that x2 ` y 2 « x, which means that an image will only be reconstructed with high fidelity when the object is near the center of the aperture. B.1 RMA for motion in the y- direction For the case of a 1-D aperture oriented horizontally along the y direction (at z “ 0) and moving along the y´direction, we first denote the aperture/receiver location by ys . We may then define the dipole locations along the aperture relative to the center, 149 with coordinate ya (absolute position yt “ ys ` ya ). The received signal is then ż Spys , ya , f q “ V σpx, y, zq e´jkRt e´jkRr dV, 16π 2 Rt Rr (B.14) where a Rr “ x2 ` py ´ ys q2 ` z 2 (B.15) a Rt “ x2 ` py ´ ys ´ ya q2 ` z 2 . Taking the Fourier transform of both sides gives ż Ŝpkys ,kya , kq “ V σpx, y, zq 16π 2 ż ys e´jkRr ´jkys ys e ˆ Rr (B.16) ż ya e´jkRt ´jkya ya e dya dys dV. Rt We may first modify the integral over ya by applying the substitution y 1 “ y ´ ys so that ż ya e´jkRt ´jkya ya e dya “ Rt ?2 1 2 2 e´jk x `py ´ya q `z a e´jkya ya dya , 2 1 2 2 x ` py ´ ya q ` z ya ż which is the convolution between the functions e´jkR R1 1 (B.17) 1 and e´jkya y , where a R 1 “ x2 ` y 1 2 ` z 2 (B.18) According to the convolution theorem, this integral is therefore evaluated as ż ya e´jkRt ´jkya ya e dya “ Rt ż 1 1 1 1 1 E1 pkya qE2 pkya qejkya y dkya , (B.19) K 1 where E1 pkya q is the Fourier transform of e´jkR R1 1 1 of e´jkya y . We first evaluate E1 pkya q: 150 1 1 and E2 pkya q is the Fourier transform 1 ż 1 q E1 pkya “ y1 e´jkR ´jkya 1 y1 e dy 1 . 1 R (B.20) This integral may be evaluated by the method of stationary phase. By the method of stationary phase, rapid fluctuation of the argument of the above integral causes cancellation of terms except at the point of stationary phase, i.e. the point where the derivative of this phase term is zero. The integral may then be approximated by evaluating its argument at this point. The phase term is given by φpy 1 q “ ´k a 1 x2 ` y 1 2 ` z 2 ´ kya y1 (B.21) so that Bφ ´ky 1 1 a ´ kya “ “0 12 1 2 2 By x `y `z (B.22) The point of stationary phase is then given by ? 1 x2 ` z 2 k ya y01 “ ˘ b 12 k 2 ´ kya (B.23) and the resulting phase term evaluated at this point (taking the negative solution) is given by b ? 12 φpy01 q “ ´ k 2 ´ kya x2 ` z 2 (B.24) Upon substituting this expression as the phase term, we have the solution 1 E1 pkya q ? ? 12 ? 2 2π ´jπ{4 ´j k2 ´kya x `z 2 « e e 1 2 2 2 2 1{4 rpk ´ kya qpx ` z qs (B.25) 1 We may next evaluate E2 pkya qby exploiting the orthogonality of the exponential function ż 1 E2 pkya q “ 1 1 1 1 q e´jkya y e´jkya y dy 1 “ δpkya ` kya 151 (B.26) The integral over ya is therefore evaluated as ż ya ?2 2?2 2 e´jkRt ´jkya ya e dya “ A1 e´jkya py´ys q e´j k ´kya x `z Rt (B.27) where we have replaced y’ with py ´ ys q according to the change of variable made above, and A1 is an amplitude term given by ? 2π A1 “ e´jπ{4 . 2 2 rpk ´ kya qpx2 ` z 2 qs1{4 (B.28) For clarification, the amplitude coefficients A shown in this document were explicitly calculated by my colleague Aaron Diebold. After substituting and rearranging, we obtain ? A11 σpx, y, zq ´jkya y ´j ?k2 ´kya 2 x2 `z 2 ˆ e e 2 16π V ż ´jkRr e e´jpkys ´kya qys dys R r ys ż Ŝpkys , kya , kq “ (B.29) As reasoned similarly above, the integral over ys is the convolution between the functions e´jkR R and e´jpkys ´kya qy . By using the convolution theorem ż ys e´jkRr ´jpkys ´kya qys e dys “ Rr ż 1 1 1 1 E3 pkys qE4 pkys qejkys y dkys (B.30) K 1 1 The function E3 pkys q is evaluated as in the case of E1 pkya q by the method of stationary 1 phase, and E4 pkys q by orthogonality ż 1 E3 pkys q e´jkR ´jkys 1 y e dy R y ? ? 12 ? 2 2π ´jπ{4 ´j k2 ´kys x `z 2 e e “ 12 2 2 2 1{4 rpk ´ kys qpx ` z qs « ż 1 E4 pkys q 1 1 e´jpkys ´kya qy e´jkys y dy “ δpkys ` pkys ´ kya qq. “ y 152 (B.31) The integral over ys is therefore evaluated as ż ys e´jkRr ´jpkys ´kya qys e dys Rr ?2 ? 2 2 2 “ A2 e´j k ´pkya ´kys q x `z e´jpkys ´kya qy , (B.32) where the amplitude term is given by ? 2π A2 “ e´jπ{4 . rpk 2 ´ pkya ´ kys q2 qpx2 ` z 2 qs1{4 (B.33) Finally, upon substituting these expressions and evaluating at z “ 0, we have the result ż Aσpx, y, zqe´jkx x e´jky y dV, Ŝpkys , kya , kq “ (B.34) V where A“ 8πjxrpk 2 ´ ´1 ´ pkya ´ kys q2 qs1{4 2 qpk 2 kya b b 2 2 kx “ k ´ kya ` k 2 ´ pkya ´ kys q2 (B.35) ky “ kys . B.2 RMA for motion in the z-direction We wish to derive the RMA in the case of a 1-D aperture oriented horizontally (along the y-direction) and moving along the vertical z -direction. In this case the received signal is given by ż Spyr , yt , zr , zt , f q “ V σpx, y, zq e´jkRt e´jkRr dV, 16π 2 Rt Rr 153 (B.36) where the distances from the scene location to each transmitter and receiver are given respectively as a Rt “ x2 ` py ´ yt q2 ` pz ´ zt q2 (B.37) a Rr “ x2 ` py ´ yr q2 ` pz ´ zr q2 The variables yt , zt , yr and zr represent the horizontal and vertical positions of the transmitting dipoles and the receiving probe, respectively. The 4-D Fourier transform of this signal is given by ż Ŝpkyr , kyt ,kzr , kzt , kq “ V σpx, y, zq 16π 2 ż ż yt zt e´jkRt ´jkyt yt ´jkzt zt e e ˆ Rt (B.38) ż ż yr zr e´jkRr ´jkyr yr ´jkzr zr e e δpyr qdyt dzt dyr dzr dV Rr Evaluating the integral over yr leaves the following integral over the receive locations: ż zr 1 e´jkRr ´jkzr zr e dzr Rr1 (B.39) where a Rr1 “ x2 ` y 2 ` pz ´ zr q2 This integral is the convolution between the functions (B.40) e´jkR R and e´jkzr z , with a R “ x2 ` y 2 ` z 2 (B.41) By the convolution theorem, the solution is therefore given by ĳ 1 1 1 1 E1 pkzr qE2 pkzr qejkzr z dkzr (B.42) K 1 where E1 pkzr q is the Fourier transform of e´jkR R 1 and E2 pkzr q is the Fourier transform of e´jkzr z . We then have ż 1 q E1 pkzr “ z e´jkR ´jkz1 z r dz e R 154 (B.43) which may be evaluated by the method of stationary phase. The phase term is given by a 1 z φpzq “ ´k x2 ` y 2 ` z 2 ´ kzr (B.44) The derivative of the phase is given by Bφ ´kz 1 ´ kzr “0 “a Bz x2 ` y 2 ` z 2 (B.45) so that the point of stationary phase is a 1 kzr x2 ` y 2 z0 “ ˘ a 12 k 2 ´ kzr (B.46) Taking the negative solution and inserting this point into the phase term, we find the solution to our integral is ? ? 12 2 2 2 1 E1 pkzr q « A1 e´j k ´kzr x `y (B.47) where A1 is an amplitude term given by ? 2π e´jπ{4 A1 “ 12 2 2 rpx ` y qpk 2 ´ kzr qs1{4 (B.48) 1 q proceeds according to the orthogonality of the exponential The evaluation of E2 pkzr function: ż 1 E2 pkzr q 1 1 e´jkzr z e´jkzr z dz “ δpkzr ` kzr q “ (B.49) z It follows that the integral over zr is found to be ? ´jkzr z ´j A1 e e 2 k2 ´kzr ? x2 `y 2 (B.50) Upon substitution, our Fourier integral is now ? A1 σpx, y, zq ´jkzr z ´j?k2 ´kzr 2 x2 `y 2 e e ˆ 2 16π V ż ż ´jkRt e e´jkyt yt e´jkzt zt dyt dzt dV R t yt zt ż Ŝpkyt , kzr , kzt , kq “ 155 (B.51) The double integral over yt and zt can be solved through a similar process used above. We see that this integral is the convolution between the functions e´jkR R and e´jpkzt z`kyt yq . By the convolution theorem, the solution is given by ĳ 1 1 1 1 1 1 1 1 E3 pkyt , kzt qE4 pkyt , kzt qejpkzt z`kyt yq dkyt dkzt (B.52) K 1 1 where E3 pkyt , kzt q is the 2-D Fourier transform of e´jkR R 1 1 and E4 pkyt , kzt q is the 2-D 1 1 Fourier transform of e´jpkzt z`kyt yq . We evaluate E3 pkyt , kzt q again by the method of stationary phase, where we now take the point of stationary phase to be the point at which both partial derivatives of the phase function are simultaneously zero. In this case, the phase function is given by a 1 1 φpy, zq “ ´k x2 ` y 2 ` z 2 ´ kyt y ´ kzt z (B.53) with partial derivatives ´ky Bφ 1 “a ´ kyt 2 By x ` y2 ` z2 ´kz Bφ 1 “a ´ kzt 2 Bz x ` y2 ` z2 (B.54) These partial derivatives are found to vanish simultaneously at the point 1 kyt x 1 kzt x z0 “ ˘b 1 1 k 2 ´ kyt2 ´ kzt2 y0 “ ˘b 1 1 k 2 ´ kyt2 ´ kzt2 (B.55) Evaluating the phase at this point yields the solution 1 1 E3 pkyt , kzt q « A2 e b 12 1 ´kzt2 ´x k2 ´kyt with A2 “ jpk 2 2πkx 1 1 ´ kyt2 ´ kzt2 q (B.56) Also, we have ż ż 1 1 q E4 pkyt , kzt 1 1 e´jpkzt z`kyt yq e´jpkzt z`kyt yq dydz “ y z (B.57) 1 1 “ δpkzt ` kzt , kyt ` kyt q 156 so that our the Fourier transform of our signal takes the form ż Aσpx, y, zqe´jpkzr `kzt qz ˆ Ŝpkyt , kzr , kzt , kq “ V ? e´j 2 k2 ´kzr If we may use the approximation ? x2 `y 2 ? e´x 2 ´k 2 k2 ´kyt zt (B.58) e´jkyt y dV a x2 ` y 2 « x, i.e. if the distance to the scene is significantly larger than the aperture size, then this expression is in the form ż Aσpx, y, zqe´jkx x e´jky y e´jkz z dV Ŝpkyt , kzr , kzt , kq “ (B.59) V where A“ A1 A2 16π 2 (B.60) and the dispersion relations are given as b a 2 2 2 kx “ k 2 ´ kyt ´ kzt ` k 2 ´ kzr (B.61) ky “ kyt kz “ kzr ` kzt In the special case where the vertical positions of the receiver and transmitters coincide, we have kzr “ kzt “ kz {2 157 (B.62) Appendix C Dipole Model Inversion Technique It is possible to use either traditional holographic beamforming techniques–or the Gerchberg Saxton (GS) algorithm– to determine the set of desired dipole moments that generate any sculpted radiation pattern. As shown in chapter 4, different modulation techniques determine the collection of polarizabilities that would resemble the desired response. However, these modulation techniques ignore the mutual coupling that exists between the elements. In this appendix it is our interest to find the set of metamaterial elements that represent those exact dipole moments, considering that the elements are embedded in a waveguide, and that these elements are interacting with each other. Let us consider a simplified scenario: a one dimensional waveguide with a set of metamaterial elements patterned on top of the waveguide that can be approximated ~ tot that as magnetic dipoles, as discussed in chapter 2. The total field magnetic H excites dipole is given by ~ tot p~ri q “ H ~ 0 p~ri q ` H ~ sc p~ri q H 158 (C.1) ~ 0 and H ~ sc are the incident field and the scattered field from all the other where H dipoles, respectively. It is possible to rewrite C.1 in terms of the total dipole moment as: ´1 ¯ m ~ 0 ` Ḡ ¯ ij ᾱ m ~j “H ij ~ j i (C.2) which gives us a matrix equation for the total dipole moment given by ´1 ¯ qm ~0 ¯ ij ´ Ḡ pᾱ ij ~ j “ Hi . (C.3) ¯ p~r ´ ~ 0 p~ From C.3, it is known that the incident field H ri q, the Green’s function Ḡ i ~rj q, and the total dipole moment that we would like to achieve m ~ j . We can determine m ~ j from the holographic approach and determine the polarizability for each element. Therefore,it is needed to ensure that an equal number of unknowns as the number of equations is obtained. Let us consider an example where we have N elements, and only one component of the polarizability tensor per element. Equation (C.3) can be re-defined as »¨ ˛´1 ¨ α1 0 0 0 0 G12 ¨ ¨ ¨ — ˚ 0 α2 0 ‹ ˚ 0 0 ‹ —˚ ˚ G21 ´ ‹ ˚ —˚ . . . .. .. 0 ‚ –˝ 0 0 ˝ .. 0 GN 1 GN 2 ¨ ¨ ¨ 0 0 0 αN ˛fi ¨ ˛ ¨ ˛ m1 G1N H10 ffi ˚ ‹ ˚ 0‹ G2N ‹ ‹ffi ˚ m2 ‹ ˚ H2 ‹ .. ‹ffi ˚ .. ‹ “ ˚ .. ‹ . ‚fl ˝ . ‚ ˝ . ‚ 0 mN (C.4) HN0 which can be solved for each value of α as ¨ ˛ ¨ 1{α1 0 ˚ 1{α2 ‹ ˚ G21 {m2 ˚ ‹ ˚ ˚ .. ‹ “ ˚ .. ˝ . ‚ ˝ . 1{αN G12 {m1 0 .. . GN 1 {mN GN 2 {mN ¨¨¨ 0 ¨¨¨ ˛¨ ˛ ¨ ˛ G1N {m1 m1 H10 {m1 ˚ ‹ ˚ 0 ‹ G2N {m2 ‹ ‹ ˚ m2 ‹ ˚ H2 {m2 ‹ ‹ ˚ .. ‹ ` ˚ ‹ .. .. ‚˝ . ‚ ˝ ‚ . . 0 mN (C.5) HN0 {mN Once we find the desired polarizabilities at the aperture plane– notice that these would include the dipolar interaction–then it is possible to use the modulation tech159 niques shown in Table 4.1 to apply the mapping onto the available polarizabilities per tuning state. With this formulation, there is no need to consider an iterative approach for the Dipole Model. 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JOSA B 34,12, 2610-2623 2017. 2. L Pulido-Mancera, PT Bowen, MF Imani, N Kundtz, DR Smith. Polarizability extraction of complementary metamaterial elements in waveguides for aperture modeling. Physical Review B, 96, 235402, 2017. 3. M Boyarsky, T Sleasman, L Pulido-Mancera, A Diebold, MF Imani, DR Smith. Single-Frequency 3D Synthetic Aperture Imaging with Dynamic Metasurface Antennas. Applied Optics, 2017. 4. T Sleasman, M Boyarsky, L Pulido-Mancera, T Fromenteze, MF Imani, MS Reynolds,DR Smith. Experimental synthetic aperture radar with dynamic metasurfaces. IEEE Transactions on Antennas and Propagation 65,12, 6864-6877 2017 5. DR Smith, O Yurduseven,L Pulido-Mancera, PT Bowen, NB Kundtz. Analysis of a Waveguide-Fed Metasurface Antenna. Physical Review Applied, 8, 054048, 2017. 6. L Pulido-Mancera, T Fromenteze, T Sleasman, M Boyarsky, MF Imani, MS Reynolds, DR Smith. Application of range migration algorithms to imaging with a dynamic metasurface antenna JOSA B 33,10, 2610-2623 2016. 7. M Boyarsky, T Sleasman, L Pulido-Mancera, T Fromenteze, A Pedross-Engel, CM Watts, MF Imani, MS Reynolds, DR Smith. Synthetic aperture radar with dynamic metasurface antennas: a conceptual development. JOSA A 34,5, 22-36 2017. 8. L Pulido-Mancera, T Zvolensky, MF Imani, PT Bowen, M Valayil, DR Smith. Discrete dipole approximation applied to highly directive slotted waveguide antennas. IEEE Antennas and Wireless Propagation Letters, 15, 1823-1826 2016. 179 9. L Pulido-Mancera, JC Gonzalez, A Avila, JD Baena. Measurements of Permittivity Based in Microstrip Technology. MOMENTO-Revista de Fsica 47 68-76 2013. 10. L Pulido-Mancera, MF Imani, DR Smith Discrete dipole approximation for simulation of unusually tapered leaky wave antennas 2017. IEEE MTT-S 1 409-412 2017 11. L Pulido-Mancera, MF Imani, P Bowen, DR Smith. Analytical Modeling of 2D Waveguide-fed Metasurfaces. Physical Review X In progress 12. L Pulido-Mancera, MF Imani, DR Smith. Dipolar Model for Metamaterial Imaging Systems. JOSA B In progress 13. L Pulido-Mancera, MF Imani, DR SMith. Limitations of the Dipole Model for Metamaterial Modeling. Physical Review B In progress 14. L Pulido-Mancera, MF Imani PT Bowen DR Smith Extracting polarizability of complementary metamaterial elements using equivalence principles. 2017 11th International Congress on Engineered Materials Platforms for Novel Wave Phenomena (Metamaterials) 1 - 2017. 15. L Pulido-Mancera, T Fromenteze, T Sleasman, M Boyarsky, MF Imani, MS Reynolds, DR Smith Adapting range migration techniques for imaging with metasurface antennas: analysis and limitations.Passive and Active Millimeter-Wave Imaging XX, 10201-102010D 2017. 16. L Pulido-Mancera, MF Imani, DR Smith Discrete dipole approximation for the simulation of edge effects on metasurfaces. 2016 IEEE International Symposium on Antennas and Propagation (APSURSI), 1 107-108 2016. 17. L Pulido-Mancera, JD Baena Theoretical constraints on reflection and transmission through metasurfaces.2015 9th International Congress on Advanced Electromagnetic Materials in Microwaves and Optics (METAMATERIALS), 1 37-39 2015 18. L Pulido-Mancera, JD Baena Waveguide model for thick complementary split ring resonators.2014 IEEE International Symposium on Antennas and Propagation (APSURSI), 1 2086-2087 2014. 19. L Pulido-Mancera, JD Baena Equivalent circuit model for thick split ring resonators and thick spiral resonators. 2014 IEEE International Symposium on Antennas and Propagation (APSURSI), 1 2086-2087 2014. 20. L Pulido-Mancera, JD Baena JL Araque Quijano Thickness effects on the resonance of metasurfaces made of SRRs and C-SRRs. 2013 IEEE International Symposium on Antennas and Propagation (APSURSI), 1 2314-315 2014. Patents 1. L Pulido-Mancera MF Imani PT Bowen “DDA- Inversion Technique” Duke University. IDF 567. 180

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