Developments in Antenna Analysis and Design presents recent developments in antenna design and modeling techniques for a

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Developments in Antenna Analysis and Design

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Developments in Antenna Analysis and Design Volume 2 Edited by Raj Mittra

The Institution of Engineering and Technology

Published by SciTech Publishing, an imprint of The Institution of Engineering and Technology, London, United Kingdom The Institution of Engineering and Technology is registered as a Charity in England & Wales (no. 211014) and Scotland (no. SC038698). † The Institution of Engineering and Technology 2019 First published 2018 This publication is copyright under the Berne Convention and the Universal Copyright Convention. All rights reserved. Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may be reproduced, stored or transmitted, in any form or by any means, only with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publisher at the undermentioned address: The Institution of Engineering and Technology Michael Faraday House Six Hills Way, Stevenage Herts, SG1 2AY, United Kingdom www.theiet.org While the authors and publisher believe that the information and guidance given in this work are correct, all parties must rely upon their own skill and judgement when making use of them. Neither the authors nor publisher assumes any liability to anyone for any loss or damage caused by any error or omission in the work, whether such an error or omission is the result of negligence or any other cause. Any and all such liability is disclaimed. The moral rights of the authors to be identified as authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.

British Library Cataloguing in Publication Data A catalogue record for this product is available from the British Library ISBN 978-1-78561-890-1 (Hardback Volume 2) ISBN 978-1-78561-891-8 (PDF Volume 2) ISBN 978-1-78561-888-8 (Hardback Volume 1) ISBN 978-1-78561-889-5 (PDF Volume 1) ISBN 978-1-78561-892-5 (Hardback Volumes 1 and 2)

Typeset in India by MPS Limited Printed in the UK by CPI Group (UK) Ltd, Croydon

Contents

Preface

xi

1 Terahertz antennas, metasurfaces and planar devices using graphene Michele Tamagnone, Santiago Capdevila Cascante, and Juan R. Mosig

1

1.1

Introduction 1.1.1 Graphene 1.1.2 Outline 1.2 2D materials in the framework of Maxwell’s equations 1.3 Graphene planar plasmonic antennas 1.3.1 Fixed frequency plasmonic antennas 1.3.2 Frequency-reconfigurable plasmonic antennas 1.3.3 Graphene plasmonic antenna model 1.4 Efficiency upper bounds in graphene tunable and non-reciprocal devices 1.4.1 Generalized electric and magnetic field representation 1.4.2 Demonstration of the upper bound 1.4.3 Graphene figure of merit 1.4.4 Device specific bounds 1.5 Graphene terahertz non-reciprocal isolator 1.5.1 Isolator working principle 1.5.2 Measurements 1.6 Graphene terahertz beam steering reflectarray prototype 1.6.1 Working principle 1.6.2 Design and measurement 1.7 Conclusions Acknowledgments References 2 Millimeter-wave antennas using printed-circuit-board and plated-through-hole technologies Kung Bo Ng, Dian Wang, and Chi Hou Chan 2.1

2.2

Wideband MMW ME dipole antennas 2.1.1 Single feed printed ME dipole antenna 2.1.2 Differential feed printed ME dipole antenna HOM MMW patch antenna 2.2.1 Wideband HOM patch element 2.2.2 Differentially fed HOM patch antenna

1 2 2 3 6 6 9 11 13 14 15 17 19 22 23 24 27 28 29 32 34 34

39 40 40 47 53 53 58

vi

3

4

Developments in antenna analysis and design, volume 2 2.3

Wideband MMW complementary source antennas for 5G 2.3.1 Linearly polarized antenna fed by an SIW 2.3.2 Radiation mechanism of the wideband antenna 2.3.3 Comparison of simulation and measurement results 2.4 Conclusion Acknowledgment References

62 62 63 66 70 71 71

THz photoconductive antennas Mingguang Tuo, Jitao Zhang, and Hao Xin

73

3.1

Introduction of THz technology and photoconductive antenna 3.1.1 Importance of THz technology 3.1.2 THz generation 3.1.3 Pulsed THz generation 3.1.4 Photoconductive antenna 3.1.5 Terahertz time-domain spectroscopy 3.2 Theoretical modeling and numerical simulation 3.2.1 Motivation and challenge 3.2.2 Drude–Lorentz model 3.2.3 Equivalent circuit model 3.2.4 Full-wave model 3.2.5 Simulation examples of full-wave model 3.3 Experimental characterization of PCA component and system 3.3.1 Far-field THz-TDS 3.3.2 THz near-field microscopy 3.4 Summary References

73 73 73 74 75 77 78 78 79 81 83 87 95 95 111 119 120

Optical antennas Chengjun Zou, Withawat Withayachumnankul, Isabelle Staude, and Christophe Fumeaux

127

4.1 4.2 4.3

127 128 129 129 132 134 136 137 137 140 140 141 141 144

Introduction Early history Theory and analysis 4.3.1 Metal properties from microwave to optical frequencies 4.3.2 Plasmonic effects 4.3.3 Mie resonances in nanoscale resonators 4.3.4 Dielectric resonators versus plasmonic resonators 4.4 Nanoantenna fabrication 4.4.1 Top-down approaches 4.4.2 Bottom-up approaches 4.5 Optical characterisation of nanoantennas 4.6 Applications 4.6.1 Localised field enhancement 4.6.2 Sensing

Contents 4.6.3 Integrated photonics 4.6.4 Planar optical components 4.6.5 Photodetection 4.6.6 Selective thermal emission 4.7 Conclusion and outlook References 5 Fundamental bounds and optimization of small antennas Mats Gustafsson, Marius Cismasu, and Doruk Tayli 5.1 5.2

Introduction Stored energies and fundamental bounds for antenna analysis and design 5.2.1 Stored energies 5.2.2 QZ0 computation from current densities 5.2.3 Fundamental bounds 5.3 Antenna optimization 5.3.1 Genetic algorithms 5.3.2 Convex optimization 5.4 Examples 5.4.1 Bent-end simple phone model 5.4.2 Bent-end simple phone model—optimization for QZ0 5.4.3 Wireless terminal antenna placement using optimum currents 5.5 Conclusions References 6 Fast analysis of active antenna systems following the Deep Integration paradigm Rob Maaskant 6.1

Introduction: The Deep Integration paradigm 6.1.1 Potential impact and other integration approaches 6.1.2 Scientific and technological challenges 6.2 Modeling approach and assumptions 6.3 The antennafier array element 6.3.1 Concept 6.3.2 Method-of-moments analysis of a folded dipole antennafier 6.4 Multiscale numerical analysis of an antennafier array 6.4.1 The Characteristic Basis Function Method 6.4.2 Generation of characteristic basis functions 6.4.3 Numerical matrix compression and solution 6.4.4 Active versus passive antenna array results 6.5 Conclusions References

vii 146 147 149 150 151 152 161 161 162 163 165 167 168 169 173 173 173 176 178 181 181

187 187 190 191 191 193 193 194 200 201 203 205 206 209 210

viii 7

Developments in antenna analysis and design, volume 2 Numerically efficient methods for electromagnetic modeling of antenna radiation and scattering problems Yang Su and Raj Mittra 7.1 7.2

8

213

Introduction Numerical analysis of multiple multi-scale objects using CBFM and IEDG 7.2.1 Introduction to CBFM and IEDG 7.2.2 MoM combined with CFIE 7.2.3 Elements of impedance matrix of MoM 7.3 Acceleration of electromagnetic analysis using CBFM 7.3.1 Partition of CBFM 7.3.2 Constructing CBFs by using multiple plane-wave excitation 7.3.3 Generation of reduced matrix equation in the CBFM 7.3.4 Multi-scale discretization using the IEDG method 7.3.5 Numerical results 7.3.6 Summary 7.4 Analysis of scattering from objects embedded in layered media using the CBFM 7.4.1 Introduction to CBFM analysis of the object embedded in layered media 7.4.2 Mixed potential integral equation for objects embedded in layered media 7.4.3 Numerical results 7.4.4 Summary 7.5 CBFM for microwave circuit and antenna problems 7.5.1 Introduction 7.5.2 SVD-based CBFM 7.5.3 Numerical results 7.5.4 Summary 7.6 Conclusions Acknowledgment List of acronyms References

213

Statistical electromagnetics for antennas Hulusi Acikgoz, Ravi Kumar Arya, Joe Wiart, and Raj Mittra

259

8.1 8.2 8.3

259 262 264 264 266 267 271

8.4

Introduction State of the art of variable antennas Statistical methods 8.3.1 General approach and surrogate modeling 8.3.2 MC simulations 8.3.3 Polynomial chaos expansion Case studies

215 215 217 219 221 221 222 224 226 230 234 235 235 237 241 247 248 248 248 251 254 254 254 254 255

Contents 8.4.1 Case I: Split ring resonator 8.4.2 Case II: Wearable textile antenna 8.5 Conclusions References 9 Ultra-wideband arrays Markus H. Novak and John L. Volakis 9.1

Review of current UWB capabilities 9.1.1 Tapered slot 9.1.2 Fragmented aperture 9.1.3 Connected and coupled arrays 9.1.4 Material loading 9.2 Basic model of a UWB TCDA and feed 9.2.1 Modeling infinite coupled arrays 9.2.2 Circuit model of the balun 9.3 Considerations for planar UWB arrays 9.3.1 Feed planarization 9.3.2 Material and process selection 9.3.3 Limitations of PCB processing 9.3.4 Surface waves 9.3.5 Cavity resonance 9.4 Planar UWB arrays for millimeter-waves 9.4.1 Development of a three-pin balun 9.4.2 Sample design for 5G frequencies References

10 Reflectarray antennas Eduardo Carrasco and Jose A. Encinar 10.1 Introduction 10.2 Basic concepts on reflectarray antennas 10.3 Elementary cells in reflectarrays 10.4 Analysis and design of reflectarray antennas 10.4.1 Analysis and design of reflectarray elements 10.4.2 Design and analysis of reflectarray antenna 10.5 Broadband techniques in reflectarrays 10.6 Shaped and multi-beam reflectarrays 10.7 Dual-reflector configurations 10.8 Technological challenges 10.8.1 Deployable and inflatable reflectarrays 10.8.2 Reflectarrays and solar cells 10.8.3 3-D printed reflectarrays 10.8.4 Reflectarrays at terahertz and optical frequencies 10.8.5 Liquid crystal reflectarrays 10.8.6 Reflectarrays using graphene

ix 271 278 282 282 287 288 289 290 291 293 295 295 297 301 301 302 302 303 309 313 313 316 319 323 323 324 325 334 334 334 335 340 342 344 344 345 346 347 349 349

x

Developments in antenna analysis and design, volume 2 10.9 Conclusions Acknowledgement References

11 Novel antenna concepts and developments for CubeSats Yahya Rahmat-Samii, Vignesh Manohar, and Joshua M. Kovitz 11.1 Introduction 11.2 Existing standards for small satellites 11.3 Antenna requirements for CubeSats 11.3.1 Frequency 11.3.2 Antenna radiated power, gain, and radiation pattern 11.3.3 Antenna material 11.4 Representative current antenna concepts for CubeSats 11.5 Ka-band symmetric umbrella reflector antennas (up to 0.5 m) 11.5.1 Antenna configuration 11.5.2 Reflector surface characterization 11.5.3 Deployment strategy 11.6 Ka-band offset reflector antennas (up to 1 m and beyond) 11.6.1 Reflector design and feed development 11.6.2 Proposed deployment strategy 11.7 Reflectarray concept 11.7.1 Deployment and design 11.7.2 Flight model performance 11.8 Patch antennas integrated with solar panels 11.8.1 Transparent (supersolar) patch antennas 11.8.2 Nontransparent (subsolar) patch antennas 11.9 Conclusion Appendix A Characterization of umbrella reflectors A.1 Mathematical representation of the gore surface A.2 Finding the optimum feed location A.3 Gain loss as a function of the number of gores Appendix B Mesh characterization for deployable reflectors B.1 Simple wire grid model B.2 Equivalent wire grid model for complex knits Acknowledgement References Index

351 351 351 361 361 363 364 364 365 366 366 367 369 370 370 371 373 374 376 377 378 380 380 381 384 384 385 385 387 389 389 390 393 393 403

Preface

Antenna design is a mature field, with a long history dating back to Hertz, Marconi and Bose, among others. Of course a wide variety of antennas have been developed over the years since the simple wire antenna configurations were introduced by the pioneers in the field, some examples being microstrip patch antennas (MPAs), reflectors, phased arrays, etc., to name just a few. Not unexpectedly, antenna design methodologies have also evolved over the years to meet the design specifications, which appear to become more and more challenging with time, since the devices utilizing these antennas have progressively increased the number of functionalities they offer, while still placing significant constraints on the real estate available for the platforms upon which these antennas are placed. Additionally, new materials have been developed and these developments continue unabated. The developments have even picked up the pace noticeably with the advent of Metamaterials (MTMs) and recently discovered applications of graphene-based antennas. In light of this background, if one wonders whether there is anything new that is left to do for the antenna design engineers to keep themselves gainfully occupied, the short answer is: ‘‘Definitely, yes.’’ In support of this assertion, we point out that the number of research publications on antennas have been rising progressively in recent years though their focus has shifted to emerging areas. Let us turn now to the problem of antenna analysis, which plays a pivotal role in modern approaches to designing antennas. Before the advent of the computer, the antenna designers relied solely on theoretical approaches for analysis, which were limited in their application to simple configurations that were tractable by using analytical or quasi-analytical techniques. However, all that changed once the computers became sufficiently powerful, and sophisticated computational electromagnetic (CEM) techniques were developed. These techniques included the Method of Moments (MoM), the Finite Element Method (FEM) and the Finite Difference Time Domain (FDTD) method, and commercial codes based on these methods became widely available and affordable to the antenna designers. Hence, one might again ask the question: ‘‘Is there any point in looking for new or more advanced techniques for antenna analysis, given the fact that the commercial codes are already well capable of handling complex antenna configurations comprised of both perfectly conducting, or PEC, as well as dielectric and/or magnetic materials?’’ Once again the short answer is ‘‘yes.’’ One of the reasons is that the time and memory requirements of the commercial codes can be exorbitant for complex antennas mounted on large platforms, and the designers are always looking for EM simulation codes, whose solve times are reasonable so that they can help improve the time for the design cycle, and thereby reduce the ‘‘time to market,’’ which the manufacturers consider to be a very important factor when strategizing the introduction of new antennas in their next-generation products. This is also true when the antennas are integrated with packages and active devices.

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Developments in antenna analysis and design, volume 2

Given this backdrop, the concept of this book, which is titled Developments in Antenna Analysis and Design, was developed to present recent developments in antenna design and modeling techniques for a wide variety of applications. The application areas were chosen because they are contemporary in nature, have been receiving considerable attention in recent years, and also because they are crucial for future developments. The book includes topics such as body-worn antennas that play an important role as sensors for Internet of Things (IoT), and Millimeter Wave antennas that are vitally important for 5G devices. It also covers a wide frequency range that includes Terahertz and Optical frequencies. Additionally, it discusses topics such as theoretical bounds of antennas and aspects of statistical analysis that are not readily found in the existing literature. To ensure that the book covers a wide scope, which includes as many aspects of modern antenna design and analysis as possible, it is comprised of twenty-three chapters, written by authors who are leading experts in their fields. Since we made an editorial judgment to be liberal with the page budget so as not to compromise the quality, readability and usefulness of the presentations, the manuscript grew to become considerably larger than it was originally planned. Hence, a second round of editorial judgment was made, and the decision was to split the book into two parts. The first volume includes chapters covering the topics of Theory of Characteristic Modes (TCM) and Characteristic bases; Wideband antenna element designs; MIMO antennas; Antennas for Wireless Communication; Reconfigurable antennas employing microfluidics; Flexible and Body-worn antennas; and Antennas using Meta–atoms and Artificially Engineered Materials, or Metamaterials (MTMs). The second volume of the book covers the topics of Graphene-based antennas; Millimeter-wave Antennas; Terahertz Antennas; Optical Antennas; Fundamental Bounds of Antennas; Fast and Numerically Efficient techniques for analyzing antennas; Statistical analysis of antennas; Ultra-wideband arrays; Reflectarrays; and antennas for small satellites, viz., CubeSats. We believe that together the two volumes of the book represent a unique combination of topics pertaining to antenna design and analysis, not found elsewhere. It is hoped that the antenna community including designers, students, researchers, faculty engaged in teaching and research of antennas, and the users as well as decision makers would find the book useful and timely, and their feedback and comments are most welcome. They can post these comments on the website for the e-journal FERMAT(www.e-fermat.org). The editor takes this opportunity to thank all the contributors who invested a considerable amount of their precious time to participate in this gargantuan task of preparing their contributions for the book and ensuring the quality of their contributions that helps to make the book so unique. Finally, the editor thankfully acknowledges the help of Ravi Arya, who served as the editorial assistant throughout the entire period of the manuscript preparation, and did such an excellent job of handling all the details that go with the task of putting together such a monumental piece of work. Before closing, I would like to mention that the book often uses the terms S11 and return loss interchangeably. It has recently been pointed out by the IEEE Standards Committee that the two are not the same and that there is a sign difference between the two. So, we want the reader to be aware of this when going through the book so that there is no confusion.

Chapter 1

Terahertz antennas, metasurfaces and planar devices using graphene Michele Tamagnone1, Santiago Capdevila Cascante2, and Juan R. Mosig2

Graphene is expected to be an enabling technology for THz antennas and related devices. This chapter describes the foundations for the theoretical and numerical modeling of graphene devices in the framework of Maxwell’s equations. Subsequently, several designs of graphene planar antennas for terahertz frequencies are proposed showing that high-quality-gated graphene can be used to achieve frequency reconfiguration in resonant plasmonic antennas and beam steering in graphene-based plasmonic reflectarrays. Afterwards, the potential of graphene for non-reciprocal applications is demonstrated experimentally, with the design, fabrication, and measurement of the first terahertz graphene isolator (operating between 1 and 10 THz). Finally, preliminary results concerning the realization of graphene beam steering reflectarray antennas at terahertz frequencies are presented. All of the above takes advantage of a newly developed theoretical ‘‘upper bound’’ which allows one to evaluate the closeness of a given design to the theoretical optimum, and depends uniquely on graphene conductivity.

1.1 Introduction This chapter surveys and summarizes some results obtained in the last five years concerning the use of graphene for terahertz and far-infrared planar optical components and antennas, with particular emphasis on tunable and non-reciprocal devices. More extensive descriptions can be found in the material presented in recent publications [1–7]. Both terahertz technologies and graphene are emerging fields which hold many promises for a number of applications, including ultra-broadband communications, sensing and security. There is a clear set of applications that could benefit from the development of terahertz technologies, but many several technical challenges are still present in terms of very limited availability of materials and components to 1

School of Engineering and Applied Sciences, Harvard University, Cambridge, USA Laboratory of Electromagnetics and Antennas (LEMA), Ecole Polytechnique Fe´de´rale de Lausanne (EPFL), Switzerland

2

2

Developments in antenna analysis and design, volume 2

generate, manipulate and detect terahertz waves. On the other hand, graphene has recently emerged as a very promising material and a very important amount of research is currently devoted to exploring its potential applications and its advantages over existing technologies. The main idea here is to bring these two topics together, to demonstrate that terahertz and far-infrared technologies can greatly benefit from the unique optical and electromagnetic (EM) properties of graphene.

1.1.1 Graphene Graphene is a 2D material, namely, it is a thermodynamically stable assembly of atoms arranged in a periodic bi-dimensional lattice which is kept together by covalent bonds. Graphene, in particular, is formed by carbon atoms organized in a honeycomb structure, and it is naturally found in graphite, which is formed by several layers of graphene stacked one on top of each other and kept together by van der Waals forces (which are much weaker than the covalent bonds in the material). Most 2D materials are also found naturally in similar van der Waals solids, and they can be isolated by exfoliating thin multilayers down to a single monolayer. This mechanical cleaving process produces high-quality single crystal flakes with areas of less than a millimeter [8]. Alternatively, large area graphene can be produced with chemical vapor deposition (CVD) processes; however, this approach results in polycrystalline structures of significantly lower quality in terms of carrier mobility. While graphene is considered one of the best conductors of electrical charge when normalized to its ‘‘volume,’’ a single monolayer of graphene has a relatively low conductivity in the order of 1 mS/& (from DC to microwaves) and hence a sheet impedance in the order of 1 kW/&. Hence, it cannot compete with metals if used simply as material for fixed (non-tunable) antennas at microwave frequencies [9]. However, graphene conductivity has three important properties that set it apart from other materials at terahertz and infrared frequencies, allowing the realization of devices with unprecedented performances in those bands. These properties are, namely: (i)

(ii)

(iii)

The presence of a significant imaginary inductive component of the conductivity at terahertz and infrared frequencies. This is equivalent to a negative real permittivity and allows the existence of ultra-confined plasmons, which can be used to create highly miniaturized plasmonic antennas. The possibility of tuning graphene conductivity via electrostatic field gating. This can be used to create optical modulators, beam steering antennas and other tunable antennas and EM passive devices. The possibility of obtaining magnetostatic-field induced non-reciprocal properties, such as Faraday rotation, which can be harnessed to create non-reciprocal devices like isolators.

This chapter describes devices that have been designed to exploit these unique properties, covering all the three above-presented aspects.

1.1.2 Outline We start describing the modelling and numerical simulation of graphene devices in the framework of Maxwell’s equations. Subsequently, we propose several designs

Terahertz antennas, metasurfaces and planar devices using graphene

3

of graphene planar antennas for terahertz frequencies, where it is shown that high-quality gated graphene can be used to achieve frequency reconfiguration in resonant plasmonic antennas and beam steering in graphene-based plasmonic reflectarrays. These devices exploit both properties (i) and (ii) above. Circuit models are provided as a simple way to understand the behavior of the device. Then, we will show that properties (ii) and (iii) can be used to design metasurfaces that can act as optical modulators and non-reciprocal isolators. Optical modulators at terahertz frequencies are already well known in the literature [10, 11]. Here, we demonstrate that it is possible to obtain a theoretical upper bound on the efficiency and performance of modulators as well as isolators, which depends uniquely on graphene conductivity. More precisely, in this kind of device, it is possible to achieve a targeted functional performance, but at the expense of unavoidable EM losses, which we can quantify from considerations based on Maxwell’s equations. The strength of our approach is that it is independent of the device geometry, thus including waveguide and optical fiber devices as well as tunable and non-reciprocal antennas. This theoretical limit is an important guideline for the design of graphene-based devices, as it can predict the best possible performances prior to any design effort or numerical simulation. It is also demonstrated that devices able to reach the upper bound can be designed, thus obtaining Pareto-optimal devices [12]. The potential of graphene for non-reciprocal applications is then demonstrated experimentally, with the design, fabrication and measurement of the first terahertz isolator (operating between 1 and 10 THz). An isolator is a device that allows the unilateral propagation of EM waves, and for that reason is often called ‘‘optical diode.’’ Our isolator uses graphene immersed in a magnetostatic field and exhibits approximately 7 dB of loss in one direction and more than 25 dB in the other. Furthermore, our device is shown to be quasi-optimum according to the theoretical bound and greatly improved performances should be expected when it will be built using the next generation CVD graphene. Finally, we present preliminary results concerning the realization of graphene beam steering reflectarray antennas at terahertz frequencies. Our device is a metasurface that is able to steer in a desired direction an incoming beam of terahertz radiation. The deviation of the reflected wave from a specular direction is controlled by a DC gating of graphene elements patterned in the reflectarray surface.

1.2 2D materials in the framework of Maxwell’s equations The first step toward the design and realization of the devices presented in this contribution is the formalization of a mathematical model of graphene in the framework of Maxwell’s equations. In the literature, this is done in two alternative ways: 1. 2.

Modelling graphene as a ‘‘very thin’’ 3D material (possibly anisotropic). Modelling graphene as a true 2D object.

The idea of defining an actual graphene thickness is usually ill-posed since at the atomic scale the material is not anymore uniform by definition. This is true in particular for the probability distribution of conduction electrons responsible for the

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Developments in antenna analysis and design, volume 2

graphene EM properties. Conventionally, the thickness is defined by the interlayer spacing of graphene sheets in graphite as 0.34 nm. However, when considering EM problems with monolayer graphene, this number is still an arbitrary choice. Of course, since the properties (e.g. transmissivity) of a thin 3D material depend on the thickness, this means that also the values of the permittivity will depend on the chosen reference thickness. Because of this ambiguity, in the following we use the second strategy, namely, we model graphene as a true 2D entity with an in-plane EM response . This choice is motivated by the fact characterized by a 2D tensorial conductivity s that graphene thickness is several orders of magnitude smaller than the smallest feature in our devices, and it has also the advantage of reducing the complexity in numerical simulations even with commercial packages. relates the tangential electric field Et to the surface current The 2D conductivity s Js on graphene, which induces a discontinuity in the tangential magnetic field: Et ^ n ðH 2 H 1 Þ ¼ J S ¼ s

(1.1)

^ is the unit vector perpendicular to graphene. In the presence of a magnewhere n tostatic field B0, the graphene conductivity takes the form of a 2 2 tensor s (assuming linearity and locality). If no magnetostatic field is applied, a simple model of graphene conductivity valid from DC to visible light is given by the Kubo formula [13], which gives the conductivity (scalar) as a function of the Fermi level mC, frequency w, temperature T and carrier scattering time t: s ¼ sintra þ sinter sintra ¼ sinter

jq2e kB T 2 1

ln 2 þ 2 cosh

mC kB T

pℏ ðw j t Þ ð jq2 ðw j t1 Þ 1 fd ðeÞ fd ðeÞ ¼ e de 2 2 2 pℏ 0 ðw jt1 Þ 4ðe=ℏÞ

(1.2) (1.3) (1.4)

Figure 1.1 represents the conductivity computed with this formula, its reciprocal (sheet or surface impedance) and the ratio between the imaginary and the real part of the conductivity (Q factor). Three distinct frequency ranges can be identified: 1. 2.

3.

The ohmic range (left highlighted region with red background), where graphene conductivity is basically real due to dominant ohmic losses. The plasmonic range (central highlighted region with green background), where graphene conductivity is inductive due to the kinetic inductance associated to the carriers’ inertia. The inter-band range (right highlighted region with blue background), where graphene conductivity is real again due to dominant inter-band transitions.

The first two ranges are described by the intra-band conductivity (which has a form equivalent to a Drude model), while the third one is dominated by the inter-band transitions (occurring for ℏw > 2mC).

Terahertz antennas, metasurfaces and planar devices using graphene

5

× 10-4 12 Re(σ) τ = 10 fs –Im(σ) τ = 10 fs Re(σ) τ = 30 fs –Im(σ) τ = 30 fs Re(σ) τ = 50 fs –Im(σ) τ = 50 fs

Conductivity (S)

10

8 6 4 2 0

4

× 10

6 Re(Z) τ =10 fs –Im(Z) τ =10 fs Re(Z) τ =30 fs –Im(Z) τ =30 fs Re(Z) τ =50 fs –Im(Z) τ =50 fs

Impedance (Ω)

4 2 0 –2 –4

8 6 Q τ = 10 fs Q τ = 30 fs Q τ = 50 fs

Q

4 2 0 –2 10 10

10 11

10 12

10 13

10 14

10 15

Frequency (Hz)

Figure 1.1 Conductivity, sheet impedance and Q factor of graphene calculated with the Kubo formula for mc ¼ 0.2eV, ¼ 300K and various values of t At terahertz frequencies and room temperature, we usually have ℏw 2mC, and therefore, we are either in the ohmic or in the plasmonic range, and inter-band transitions can be neglected. In addition, usually the condition mC kBT holds, allowing for a low-temperature approximation. As such, the final approximated formula is s’

q2e jmC jt ¼ ðR þ jwLÞ1 pℏ ð1 þ jwtÞ

(1.5)

L¼

pℏ2 ; R ¼ t1 L; Q ¼ wt q2e jmC j

(1.6)

2

It is also possible to generalize the Kubo formalism when the presence of a static magnetic field requires a 2D tensorial expression for the conductivity.

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Developments in antenna analysis and design, volume 2

However, a simpler Drude–Lorentz expression exists valid at terahertz frequencies for current graphene properties: sd so ¼ s (1.7) so sd sd ¼ R1

1 þ jwt 2

ðwc tÞ þ ð1 þ jwtÞ

2

; so ¼ R1

wc t 2

2

ðwc tÞ þ ð1 þ jwtÞ

; wc ¼

qe v2f B0 mC (1.8)

where wc is the cyclotron frequency of graphene and vf ’ 106 m/s is the graphene Fermi velocity. This model captures the aforementioned relevant properties of graphene conductivity. First, graphene can be tuned by changing the Fermi level, which can be achieved by gating graphene in a parallel plate capacitor geometry. Second, graphene acts as an inductor in the terahertz-mid-infrared range, and hence it supports plasmonic modes, which are well predicted with the Drude model. Finally, when a magnetic field is applied, the conductivity becomes an antisymmetric tensor, which implies that graphene acts as a non-reciprocal material in this case. As anticipated in the introduction, these properties can be exploited to achieve novel EM devices based on graphene.

1.3 Graphene planar plasmonic antennas 1.3.1

Fixed frequency plasmonic antennas

In the plasmonic region of graphene conductivity (Figure 1.1), graphene is inductive, due to the kinetic inductance of its carriers. In complete analogy to noble metals at optical wavelengths, it is then found that graphene supports surface plasmon polaritons in the terahertz and mid-infrared ranges [14]. Also, similarly to plasmons in gold, surface plasmons in graphene can be used to create highly miniaturized plasmonic antennas, formed by two graphene patches [2] (Figure 1.4). In these antennas, the source is placed between the patches as in planar dipole antennas; however, unlike traditional metallic antennas, the radiating resonant mode is supported by surface plasmon polaritons on graphene. If graphene is placed at the interface between a substrate and a superstrate dielectric layers, the dispersion of surface plasmons is given by [14]: er1 er2 hs pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ 2 2 2 2 k0 gz þ er1 k0 gz þ er2 k0

(1.9)

where s is the graphene conductivity, gz is the propagation constant, k0 is the freespace wavenumber, h is the free-space impedance, er1 is the superstrate permittivity and er2 is the substrate permittivity. Figure 1.2 shows the propagation predicted by this analytical relationship.

Terahertz antennas, metasurfaces and planar devices using graphene

7

x y

G L

z

εr1 εr2

W

Figure 1.2 (Reproduced from [3] with permission) Structure and geometrical parameters of the proposed graphene plasmonic dipole. The central area (in red) is the source (e.g. a terahertz photomixer). G ¼ 2 mm In the high confinement case (namely, high values of gz), the following approximation holds: gz ¼

k0 k0 1 wR þ jw2 L ðer1 þ er2 Þ ðer1 þ er2 Þ ¼ ðR þ jwLÞðer1 þ er2 Þ ¼ hc hs h (1.10)

For more complicated geometries, such as ribbons, no analytical formula is available, and the problem must be solved numerically. Propagation of plasmon along ribbons is particularly important for our case, and it has been studied here with the finite element method. Figure 1.2 shows that the effect of the finite width of the ribbon (in comparison with the infinite plane case) is to increase the confinement of the mode and its effective index. The geometry of the mode is also affected: Figure 1.3 shows the electric field on the ribbon. Importantly, unlike modes in a microstrip line, here graphene alone is sufficient to support the plasmonic mode. By fixing the length of the ribbon, Fabry–Pe´rot resonances are formed, and the antenna (the geometry of which is shown in Figure 1.4) exploits these resonances to couple a lumped source to a radiating dipole mode. The two rectangular patches of graphene are separated by a gap G ¼ 2 mm which hosts the source (e.g. a photomixer). The design is obtained creating this gap in a rectangular W L graphene patch, which can be regarded as a finite length L strip with width W. The length of the antenna is selected using as a criterion the first Fabry–Pe´rot resonance condition. The input impedance of the antenna can then be determined with numerical fullwave simulations. Figure 1.5 shows the simulated input impedance Zin of the antenna for two sets of parameters. At the frequency for which bL ¼ p is satisfied, the antenna resonates, with a maximum of current at its center. This condition corresponds to a low impedance working point (W.P. L in Figure 1.5), where the impedance is real and less than 100 W. If the condition bL ¼ 2p holds instead, then a second resonant mode is excited, with a minimum in the patch surface current

8

Developments in antenna analysis and design, volume 2 12

3 Sheet (theory)

Sheet (sim.) Strip 20 μm Strip 10 μm

2

4

1

α/k0

β/k0

8

0.5

1 Frequency (THz)

1.5

Figure 1.3 (Reproduced from [2] with permission) Simulated propagation constants on an infinite graphene sheet and on ribbons with different widths. Graphene parameters are t ¼ 1 ps, mc ¼ 0.25 eV, T ¼ 300K. In all cases, graphene is placed on a substrate with permittivity of er ¼ 3.8 Graphene ribbon Propagation direction

Vacuum

Port 2

Port 1 Glass

(a)

(b)

(c)

Figure 1.4 (Reproduced from [2] with permission) Numerical simulation of surface plasmon polaritons at terahertz frequencies propagating on graphene ribbons. (a) shows the simulation setup (implemented in Ansys HFSS), (b) is a lateral cross-section of the electric field of the propagating mode and (c) is a longitudinal cross-section at the center. Hence, this condition corresponds to a high impedance working point (W.P. H in Figure 1.5). At both working points, Zin is real, but the high impedance point is particularly interesting since THz sources such as photomixers generally show a very high and real output impedance, and hence this working point would provide an improved impedance matching. The second working point (W.P. H) does not occur for the double of the frequency of the first one, as for thepcase ﬃﬃﬃ of standard metallic dipole antennas, but at a frequency of approximately 2 times larger.

Terahertz antennas, metasurfaces and planar devices using graphene

9

Antenna 1 (RE) Antenna 1 (IM)

900 Input impedance (Ω)

Antenna 2 (RE) 600

Antenna 2 (IM)

Ant1 W.P. H Ant2 W.P. H Ant2 W.P. L

300

Ant1 W.P. L

0 −300 −600 0.5

0.6

0.7

0.8

0.9 1 1.1 1.2 Frequency (THz)

1.3

1.4

1.5

1.6

Figure 1.5 (Reproduced from [2] with permission) Simulated input impedance of the antenna with two different choices of parameters. Antenna 1 has L ¼ 17 mm, W ¼ 10 mm, mc ¼ 0.13 eV. Antenna 2 has L ¼ 23 mm, W ¼ 20 mm, ¼ 0.25 eV. ¼ 1 ps, T ¼ 300 K This is due to the dispersive behavior of b, which is proportional to the square of the frequency, rather than being linear with the frequency, as shown in (1.10). Another difference with respect to normal dipole antennas is that the antennas are electrically small, considering that in these examples they are 17- and 23-mm long and that the wavelength at 1 THz is 300 mm. Yet, with only a l/20 size, they display a behavior akin to one of a normal dipole in terms of radiation pattern and current profile [1].

1.3.2 Frequency-reconfigurable plasmonic antennas A very important consequence of the use of plasmonic modes to achieve the resonance condition in these antennas is the fact that the resonance frequency can be tuned dynamically by changing the gate voltage on graphene. This is possible because the gate voltage affects the Fermi level of graphene, and in turn its conductivity, which alters the propagation constant according to (1.9). Equation (1.10) shows that the guided wavelength is inversely proportional to the kinetic inductance L, and hence proportional to the Fermi level mc. This means that it is possible to dynamically tune the resonance frequency of the antenna, which is proportional to the square root of mc. This trend does not hold for mc < kBT, and in particular for mc ¼ 0, where only thermal carriers contribute to the conductivity. Figure 1.6 shows a proposed implementation of this concept using a stack of two graphene layers separated by a gate dielectric. The dielectric is sufficiently thin to treat the graphene layers as a single one with the double of conductivity. The symmetry of graphene band structure ensures the same upper and lower conductivities (neglecting small residual unwanted doping) for initially undoped graphene.

10

Developments in antenna analysis and design, volume 2 L THz photomixer

Graphene Al2O3 Graphene

Substrate (GaAs) (b) L Substrate

Antenna

W

(GaAs)

S

Si

(a) (c)

H R

Figure 1.6 (Reproduced from [1] with permission) Geometry of the tunable graphene plasmonic dipole. (a) 3D view of the structure, (b) cross section and (c) final system including the silicon lens (S ¼ 160 mm, H ¼ 572 mm, R ¼ 547 mm) μc

0 eV

0 eV

0.2 eV

Impedance (dashes=imaginary, Ω)

0.05 eV 400

0.1 eV 0.15 eV 0.2 eV

200

0

–200

–400 0.5

1

1.5 Frequency (THz)

2

2.5

Figure 1.7 (Reproduced from [1] with permission) Input impedance of the tunable graphene plasmonic antenna upon variation of the Fermi level. t ¼ 1 ps, T ¼ 300 K The final antenna allows then independent biasing of graphene patches as well as the possibility to apply the voltage across the gap, as needed for some terahertz sources such as photomixers. This structure could be integrated with a silicon lens commonly used to increase the directivity of terahertz antennas. This device has been studied theoretically and numerically in [1]. Figure 1.7 illustrates the effect of different Fermi level mc on the antenna input impedance. One can notice a wide tuning range of more than one octave and a very

Terahertz antennas, metasurfaces and planar devices using graphene 0.2 eV

Total efficiency (%) with 10k(Ω) photomixer

Resonance radiation efficiency (%)

20

15 0.15 eV

10

0.1 eV

5 0.05 eV 0 eV

(a)

2.5

2

0.2 eV

μc

2 1.5 1 0.5 0 eV

0 0.5

0 1 1.5 Frequency (THz)

0 eV 0.05 eV 0.1 eV 0.15 eV 0.2 eV

3

11

(b)

1

1.5

2

2.5

Frequency (THz)

Figure 1.8 (Reproduced from [1] with permission) Evaluation of the efficiency of the antenna. (a) Radiation efficiency (radiated power over input power) at the resonance frequency working point. (b) Total efficiency (radiated power over source available power) with a 10 kW photomixer. t ¼ 1 ps, T ¼ 300 K uniform and high impedance peak. The reason for this uniformity is discussed in the circuit model presented below. Figure 1.8 illustrates the radiation efficiency of the antenna and the total efficiency (including also return loss due to source antenna impedance mismatch) with a 10 kW photomixer. Notably, the total efficiency of typical broadband metallic antennas connected to photomixers is below 1%, due to very large impedance mismatch, while resonant antennas can reach higher efficiencies but at a fixed frequency. The proposed antenna is resonant and can reach a total efficiency higher than broadband metal antennas, but at the same time it can be tuned over a wide range. The addition of metallic elements can further increase the total efficiency to more than 8% in the same conditions [15].

1.3.3 Graphene plasmonic antenna model A circuit model of the graphene dipole (Figure 1.9) has been developed [3]. The circuit model allows a complete understanding of the working principles of the antenna as well as providing a tool to scale the antenna for different frequencies and applications. First, a transmission line (TL) model is derived for the plasmons. While the propagation constant is already known from (1.9), the equivalent characteristic impedance of this mode is non-trivial, and its definition has been selected so that two conditions are satisfied: 1.

2.

the total current on the graphene strip must be equal to the current on the equivalent TL (in other words, the current in the equivalent TL model is the so-called natural current). Since losses are localized in graphene, they are modeled by a resistance in series with the inductor of the TL equivalent LC cell.

12

Developments in antenna analysis and design, volume 2 x y

Source

z

+ VG ZS

ZIN

γz , Z C

ε r1

G L

I0

ΓE

ε r2

V IN

CP

IIN I0

γz , Z C ΓE

L

W

(a)

(b)

Figure 1.9 (Reproduced from [3] with permission) Circuit model of the proposed dipole antenna 600

0.1 eV

0 eV

0.2 eV

600

μc

200 0

–200

0.2 eV

μc

200 0

–200 1

(a)

0.1 eV

400 Zin (Ω)

Zin (Ω)

400

0 eV

1.5 2 Frequency (THz)

1 (b)

1.5 2 Frequency (THz)

Figure 1.10 (Reproduced from [3] with permission) Comparison of the antenna input impedance predicted by the TL model (thin lines) and the fullwave simulations (thick lines). Dashed lines represent the imaginary part and continuous lines the real part. (a) model without parasitic elements and (b) model with parasitic elements Following this definition and computing the power associated to the plasmonic mode with a Poynting vector integral, the characteristic impedance is computed as follows: ! Imðgz Þw e0 er1 e0 er2 (1.11) ReðZc Þ ¼ þ 2W jsj2 jgx1 j2 Reðgx1 Þ jgx2 j2 Reðgx2 Þ ImðZc Þ ¼

Reðgz Þ ReðZc Þ Imðgz Þ

(1.12)

The model includes a parasitic capacitor to model the fringing fields in the gap of the antenna, while the two extremities are terminated by a load to model the reflection coefficient of plasmons at the edges. Importantly, the reflection coefficient is constant and independent of graphene properties and antenna geometries and found from simulations to have a phase of approximately p/2. Figure 1.10 shows that the agreement between input impedance and efficiency simulations increases,

Terahertz antennas, metasurfaces and planar devices using graphene

13

especially when the parasitic elements are added to the model. The model confirms the stability of the impedance upon frequency reconfiguration.

1.4 Efficiency upper bounds in graphene tunable and non-reciprocal devices Graphene offers interesting possibilities for tunable and non-reciprocal devices in a very wide frequency spectrum spanning from microwaves to near infrared. However, it is also characterized by optical losses, which can limit the performance of these devices. While developing the concepts presented here, it became soon clear that the issue of losses in graphene has to be tackled by answering very fundamental questions such as ‘‘What is the minimum insertion loss to achieve a given reconfigurability function?’’ or ‘‘How close can we approach an ideal non-reciprocal isolator using graphene?’’ Revisiting a theory developed in [16], this section presents several fundamental limits on non-reciprocal and tunable devices based on graphene. The limits are expressed in the form of upper bounds for several key performances of the selected devices, and typically the outcome of these upper bounds is that there is a minimum amount of insertion loss that must be accepted in order to achieve a given functionality (e.g. 100% modulation depth in a graphene modulator or perfect isolation in a graphene non-reciprocal isolator). Strikingly enough, this minimum amount of loss depends only on graphene conductivity, and it is independent of the particular geometry of the device. The bound is, therefore, a very important tool to estimate the best possible performance prior to any actual design and just depending on graphene parameters. Second, the bound provides important guidelines to the designer, because it reveals how close an actual design is to the best possible performance so that no useless optimizations are attempted once the optimal performance has been reached. The theory proposed here continues a research line started in 1954 by Mason [17], who found out that it is possible to define a value U (the unilateral amplifier gain) for an active two-port device. This value has the following property: if the two-port device is embedded in a lossless reciprocal network to obtain a new transformed two-port device, then the value U does not change in this transformation. If the transformation is such that the transmission coefficient S12 from port 2 to port 1 becomes null, then the transmission in the opposite direction has a magnitude |S21|2 ¼ U. The theory can be applied to amplifiers but also to non-reciprocal passive devices. In this case, U is a figure of merit for isolators, which are devices that ideally transmit power waves perfectly in one direction, while blocking them in the other. Five years later the theory was extended by Shaug-Pettersen and Tonning [16] who elaborated an important mathematical inequality for variable and non-reciprocal networks. Unfortunately, although the inequality is correct, their demonstration in [16] contained some errors, which were corrected in a recent publication [4]. This has been the starting point for the extension of the bound to 2D materials, and its transformation into several theoretical corollary bounds expressed directly as a function of modulators, isolators and antennas figures of merit [5].

14

1.4.1

Developments in antenna analysis and design, volume 2

Generalized electric and magnetic field representation

For an adequate analysis of the EM fields existing in devices based on graphene and related materials, it is convenient to start with the standard constitutive relationships for bianisotropic media, also called Tellegen relations [18, 19]: e x D E ¼ z m (1.13) B H where in addition to the tensorial permittivity e and permeability m, we must introduce the coupling coefficients x and z, related to the reciprocity and chirality parameters of the material under study [19]. However, to highlight the role of the electrical conductivity in our approach, following a path parallel to [16], we will introduce a slightly modified form of (1.13), by collecting all the EM field components in two six-dimensional vectors Q and R, defined as follows: jD jE Q ¼ jw ; R¼ (1.14) B H Then, we have the modified constitutive equation Q ¼ Y R, where the new matrix is given by ! e jx (1.15) Y ¼ jw jz m This matrix will play a role equivalent to that of the Z matrix in [16]. Now, we define a generalized electric conductivity as sEE ¼ jw(e e0) and by the same token, we introduce ‘‘magnetic’’ and ‘‘coupled’’ conductivities as sMM ¼ jw(m m0), sEM ¼ wx, sME ¼ wz. This leads to a compact formulation for our Y matrix, with highlighted generalized conductivities: ! sEM jwe0 þ sEE (1.16) Y ¼ sME jwm0 þ sMM Moreover, this formulation can be directly extended to 2D materials like graphene. In this case, the generalized conductivities are 2D tensors and since the background permittivity and permeability vanish in 2D materials, we are left with a 4 4 matrix: ! sEE sEM (1.17) Y ¼ sME sMM which can be further reduced to a very simple form if the 2D material shows only an electric response (no biasing magnetostatic field) and hence only sEE is non-zero.

Terahertz antennas, metasurfaces and planar devices using graphene

15

1.4.2 Demonstration of the upper bound Let us consider a reconfigurable or non-reciprocal device based on a reconfigurable or non-reciprocal 2D or 3D material such that the device can be described by a passive n n scattering matrix. This representation is suitable for any passive n-mode-guided device, layered surfaces and periodic metasurfaces. We consider the behavior of the device in two distinct situations, A and B, characterized by the considered material matrices YA and YB, the corresponding scattering matrices SA and SB, and arbitrary incident waves aA and aB. We will refer to this tunable and/or non-reciprocal material as the functional material since it enables the device functionality. Let us also assume a general closed surface S that completely surrounds the volume V of the device. If the device is an infinite planar structure, then S is taken as the union of two planes, one on each side of the structure: First, we define W¼ðE A H B E B H A Þ

(1.18)

From the divergence theorem: ððð ððð T T 2aB SA SB aA ¼ ∯ WdS ¼ r WdV RTB YBT YA RA dV S

V

(1.19)

V

The power dissipated in the device in state A and B is given by ððð 1 H PA;B ¼ aH ðI S S Þa ¼ RH ðY þ Y H A;B A;B A;B A;B A;B ÞR A;B dV þ LA;B 2 V A;B A;B

(1.20)

where LA,B accounts for the power dissipated other materials in the device apart from the active one. Let us consider the following quantity, which we call device figure or merit: jaTB S A S TB aA j2 D (1.21) gdev ¼ H H H aA I S H A S A aA aB I S B S B aB Using the relationships above, this quantity can be rewritten as follows: ððð j RTB Y TB Y A RA dV j2 V

ððð

gdev ¼ ððð H H H H RA Y A þ Y A RA dV þ LA RB Y B þ Y B RB dV þ LB V

V

(1.22) Noting that LA and LB are positive, and using the integral absolute value theorem, we have ððð 2 jRTB Y TB Y A RA jdV V

ððð

(1.23) gdev ððð H H H RA Y A þ Y H dV R Y þ Y dV R R A B B A B B V

V

16

Developments in antenna analysis and design, volume 2

Consider now four real positive functions a(r), b(r), c(r), e(r) defined on an arbitrary domain such that a2(r) ¼ b(r)c(r)e(r). Then bounding the integral with the maximum value of e(r) and using the Cauchy–Schwartz inequality, we notice that ð 2 ð ð pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ (1.24) eðrÞbðrÞcðrÞdr max eðrÞ bðrÞdr cðrÞdr r

ð ð

2

aðrÞdr a2 ðrÞ ð max r bðrÞcðrÞ bðrÞdr cðrÞdr

(1.25)

Applying (1.25) to (1.23), we obtain 2 T T RB Y B Y A RA gdev max H V R Y þ Y H R A R H Y þ Y H RB A B A B A B

(1.26)

H Because of passivity, both Y A þ Y H A and Y B þ Y B are Hermitian, therefore they H

H

H can be written as Y A þ Y H A ¼ M A M A and Y B þ Y B ¼ M B M B . Then, defining D

D

KA ¼ MARA and KB¼MBRB we can finally write

gdev max

2 T 1T T K KB M B Y B Y A M 1 A A

V

H KH A KA KB KB

1 2 T 1T T KB M B Y B Y A M A KA ¼ max 2 2 V K A K B (1.27)

This expression can be simplified introducing the normalized vectors ^ B ¼ KB =jKB j. Therefore ^ A ¼ KA =jKA j, K K 2 1 2 ^ T 1T T ^ TB A K ^ A max K ^ A A 2 gdev max K YB YA MA K BM B 2 V

V

(1.28)

with A ¼ M B1T ðY TB Y A ÞM 1 A . Because a similarity transformation does not change the norm of a matrix, we have D gmat ¼ A 2 2 ¼ Largest eigenvalue of AH A ¼ Largest eigenvalue of ðN Þ (1.29)

D

H 1

H N ¼ M 1 A A AM A ¼ Y A þ Y A

H

Y B Y A

T 1

Y B þ Y B

Y TB Y A

(1.30)

Therefore, we finally find gdev gmat

(1.31)

Terahertz antennas, metasurfaces and planar devices using graphene

17

which is referred to as the general scattering upper bound. The bound is immediately extended to 2D materials redefining: 1 1 T D N ¼ sA þ sH sB sH s B þ s TB sB sA A A

(1.32)

Importantly, (1.31) is an inequality relating gdev, which depends on the final device properties only, and gmat, which depends on the material properties only. The fact that device performances are bounded by material properties alone is the main strength of this approach, which holds independently of the device geometry and prior to any device design. In the following sections, it is shown that, in all cases, devices with lower losses (and hence better performances) possess a larger gdev, which therefore is named device figure of merit. However, because the value of gdev is bounded by gmat, a minimum amount of loss is unavoidable, and this loss is determined by gmat which is then called material figure of merit. The two quantities can be used as metrics for the optimality of device and materials, and they take positive real values. The vectors aA and aB are free parameters, namely, this upper bound represents actually an infinite set of upper bounds, each holding for a different choice of aA and aB. It will be shown that an accurate choice of these parameters can lead to specialized upper bounds on relevant figures of merit (such as isolation and insertion loss of an optical isolator). In particular, two families of derived bounds are relevant: 1. 2.

Bounds on tunable devices: the material is characterized by the two values YA and YB which are different but symmetric (reciprocity is assumed here). Bounds on non-reciprocal devices: the material is characterized by a single value of Y which however is non-symmetric (YA ¼ YB ¼ Y ).

1.4.3 Graphene figure of merit While this approach applies to any 3D and 2D material, here we are interested in graphene, both in the tunable case and in the non-reciprocal case. In the tunable case, graphene is tuned changing its Fermi level and takes two scalar values sA and sB in the two states. The modulation figure of merit gm of graphene is then given by 2 2 2 m 2 ð 1 þ w t Þ m c; A c; B jsA sB j ¼ ¼ gM ¼ 4ReðsA ÞReðsB Þ 2mc; A mc; B 2

gmat

(1.33)

where the intra-band approximation given in (1.5) has been used. The figure of merit improves with increasing difference of Fermi levels, increasing frequency and t (once the conductivity is in the plasmonic region, that is, wt > 1). Examples of values of gM computed using the full Kubo formula are shown in Figure 1.11. Notably, the main difference with the intra-band approximation is that the figure of merit drops at visible light frequencies, where graphene conductivity takes a fixed value (universal graphene conductivity) of about 61 mS [20].

T=3 K 10

102

102

μ = 0 eV cA

T = 300 K

2

μcB = 0.8 eV

101

μcA = 0.03eV

μcA = 0.2eV

T = 300 K 10

γM

100

1

γM

γM

T=3 K 10

μcA = 0.07 eV 10

(a)

13

10 10 Frequency (Hz)

14

10

(b)

10−2

13

10 10 Frequency (Hz)

14

1012

(c)

μcA

γM

1

0 =

μc (eV) in state B

0.

02

0.2

0 .7

4

0.8

γ

0.6

M

2

0.0

0 .2

0.4

0 .7

0.2

10

0

τ = 10 fs 10

(d)

1013 1014 Frequency (Hz)

1

2

τ = 100 fs

10

12

1 .6

10

12

μcB = 0.3 eV

10−1

10−2

0

100

11

12

13

10 10 10 Frequency (Hz)

0 0

14

(e)

1.6 4

0.2

0.4 0.6 μc (eV) in state A

0.8

μcB

1

(f)

Figure 1.11 (Reproduced from [4] with permission) Theoretical upper bound gM on the performance of graphene modulators as a function of multiple parameters. In all plots, the quantities that are not swept or otherwise specified have the following default values: f ¼ 1 THz, T ¼ 300 K, mC,A ¼ 0.1eV, ¼ 0.8eV, t ¼ 66fs. (a)–(d) frequency dependence of gM for several values of temperature, Fermi levels and t, (e) parametric level curves of gM for different values of the two Fermi levels

Terahertz antennas, metasurfaces and planar devices using graphene

19

In the non-reciprocal case, using the approximated conductivity in (1.7) and (1.8) as the single, non-symmetrical value of conductivity, we find the nonreciprocity figure of merit gNR of graphene: gmat ¼ gNR

jso j2 ¼ ðwc tÞ2 ¼ ðmB0 Þ2 ¼ ¼ 2 Re ðsd Þ Im2 ðso Þ

qe tv2f B0 mc

!2

(1.34) where m ¼ qe tv2f m1 c is graphene carrier mobility. Higher bias magnetic field and graphene mobility lead to a better non-reciprocal figure of merit. Figure 1.12 shows examples of values of gmat computed for different graphene parameters, using the full Kubo formula.

1.4.4 Device specific bounds Figure 1.13 shows some examples of specific devices for which a corollary bound can be computed starting from (1.21) and (1.31) and with a particular choice of aA and aB. For devices working in reflection (modeled with a single-entry scattering matrix, i.e. a reflection coefficient), the choice is trivially aA ¼ 1 and aB ¼ 1. For devices working in transmission (2 2 scattering matrices), the choice is instead aA ¼ [1,0]T and aB ¼ [0,1]T. For modulators (transmission or reflection coefficients GA and GB), we then find D

gmod ¼

ðGA GB Þ2 gM 1 jGA j2 1 jGB j2

(1.35)

Figure 1.14 represents this bound in the |GA|, |GB|, f space (where f is the phase GA and GB), showing that both amplitude and phase modulation are bounded. For isolators and non-reciprocal polarization rotators, a similar bound can be determined. In particular, for isolators (described as a 2 2 scattering matrix) we find

gisol

jS12 j2 jS21 j2 gNR ¼ 1 jS12 j2 1 jS21 j2 D

(1.36)

In both cases, if a given performance (in terms of modulation efficiency or amount of isolation) is desired, the bound implies a minimum unavoidable insertion loss for the device, based only on the used material figure of merit (gM and gNR for tunable and non-reciprocal devices, respectively). Figure 1.15 shows examples of simulated optimal planar graphene amplitude modulators, i.e. devices for which the bound is reached or almost reached. Similar bounds can be found for tunable antennas [5], while the following section illustrates the design of optimal isolators with graphene, including the experimental validation.

γ

0.3

γ

0.25

0.2

0.4

1.5 4

0.

0.3

0.2

0.8

0.25

μc (eV)

μc (eV)

0.35

1.6

0.2

2.5 0.15

4

3

6

0.1

0.15

10

6

0.1

12

20 30

0.05

0.05 0

(a)

1

2

3

4

5

6

7

γ 500

150

γ

200

250

300

T (K)

NR

0.3

B0

0.45

NR

30

0.4

0.4

0.5

400

0.35

μc (eV)

15

τ (fs)

100

0.5

600

300 8

200

4

100

2 1

0.3

1.4 2 2

1.4

0.15

3 10

Frequency (THz)

31.6

(d)

0

0.3

3.16

0.8

1

0.2

0.1 1

0.6

0.25

0.5

(c)

50

(b)

B0 (T)

0

NR

1

NR 0.1

0.45

3 10

20

30

40

50

60

Frequency (THz)

Figure 1.12 (Reproduced from [4] with permission) Theoretical upper bound gNR to the performance of graphene non-reciprocal devices as a function of multiple parameters. Full magneto-optical conductivity is used. In all contour plots, the quantities that are not swept have the default values of f ¼ 1 THz, T ¼ 300 K, mc ¼ 0.34eV, B0 ¼ 4T, t ¼ 66fs. (a) Temperature vs. Fermi level sweep, (b) bias magnetic field vs. Fermi level sweep, (c) frequency vs. relaxation time sweep and (d) frequency vs. Fermi level sweep

Terahertz antennas, metasurfaces and planar devices using graphene

Reconfigurability

Reflection (a)

Modulation in reflection

21

Transmission (b) Modulation in transmission Port 1

Port 2

Port 1

Isolation, Kerr rotation

Non-reciprocity

(c)

(d)

Isolation Faraday rotation

Port 1

E

Port 2

φ (e)

Multilayer structures

Graphene patterning

Graphene-metal hybrid structures

Design-space dimensions

Figure 1.13 (Reproduced from [4] with permission) Overview of the considered planar graphene devices that exploit either tuneability or nonreciprocity. The devices are grouped by type and by reflection or transmission operation. (a) amplitude and/or phase modulation in reflection, (b) amplitude and/or phase modulation in transmission, (c) isolator in reflection and/or magneto-optical Kerr rotation, (d) isolation and/or magneto-optical Faraday rotation in transmission and (e) features used in the designs, namely, multilayer structures, graphene and/or metallic patterns

A graphene layer on a back metallized dielectric layer (d) can reach optimal performance in a limited range. However, if graphene is patterned in a periodic square array (e) or if an additional dielectric layer is added (f), optimal performances can be reached along the entire boundary curve, including the best possible reflection modulation with 100% modulation depth. (g)–(i) random simulations of different device topologies for transmission modulation. (g) represents random sequences of graphene sheets and dielectric layers. (h) represents random sequences of patterned graphene and dielectric layers. (i) shows an example of complex structure employing polarizers and hybrid graphene metal structures showing near-optimal performances. Structures (d)–(h) are dual polarized, whereas the polarizers in (i) restrict the operation to single linear polarization (with a 90 polarization twist).

22

Developments in antenna analysis and design, volume 2

180 150 ø

120 90 60

0

30 0 0

0.2 0.4 0.2

0.6 0.4

0.6 │ΓB│

0.8 0.8

1

1

│ΓA│

Figure 1.14 Tridimensional representation of the graphene modulation efficiency upper bound. The two planes illustrate that this bound is the generalization of the phase bound (vertical secant plane) and of the amplitude one (horizontal plane)

1.5 Graphene terahertz non-reciprocal isolator The realization of isolators at terahertz frequencies is a very important open challenge due to the intrinsic propagation losses existing in the used non-reciprocal materials. In this section, we present the design, fabrication and measurement of a terahertz non-reciprocal isolator for circularly polarized waves based on a magnetostatically biased monolayer graphene operating in reflection [6]. This is the first terahertz isolator (1–10 THz) with insertion losses lower than 10 dB ever demonstrated experimentally. The device exploits the non-reciprocal optical conductivity of graphene, and despite its simple design, it exhibits almost 20 dB of isolation and only 7.5 dB of insertion loss at 2.9 THz. Operation with linearly polarized light can be achieved using quarter-wave-plates as polarization converters. Several theoretical works have proposed devices, including isolators, based on magnetostatically biased graphene [26–33]. In particular, a narrowband graphene isolator was recently measured at 20 GHz [26, 30]. Ferrite isolators have been demonstrated in the lower THz range [34], showing excellent operational bandwidth and eliminating the requirement for an external biasing magnetic field. However, currently available ferrite isolators are useful only up 0.5 THz and show prohibitive insertion losses in the order of tens of dB beyond this frequency [34]. This intrinsic limit, due to losses in ferrites, has motivated research in graphene and other alternative materials, more promising at higher THz frequencies.

Terahertz antennas, metasurfaces and planar devices using graphene 0

1 γ

γ

γ

=γ

γ

−2

0.8

Ideal modulator Best possible modulator with 100% modulation depth Forbidden areas Example 1 Example 2 Example 3 Example 4 Example 5 Example 6

0

>γ

23

0

0.5

|T A |

Exploded view

0

1

0

(i)

0.5

|T |

1

Metal polarizers

A

Figure 1.15 (Reproduced from [4] with permission) Performances of electrooptical amplitude modulators. (a) Graphical representation of the amplitude modulation inequality in the Cartesian plane G A, G B. The squares represent ideal modulators, and the circles denote the best possible modulators with 100% modulation depth. Forbidden areas (yellow) are delimited by the boundary curve. (b) Same as (a) but using the insertion loss and modulation depth coordinates. (c) Upper bounds for different values of gM. The available designs in the literature are represented by colored symbols, and where possible, the corresponding bound is represented using the same color (examples from 1 to 6 are described in [10, 21–25] respectively). (d)–(f) Simulations of randomly generated reflection modulators. Each red point represents a single simulated device. The frequency considered is 1 THz, and the graphene parameters are T ¼ 300 K, mCA ¼ 0.1eV, mCB ¼ 0.8eV, and t ¼ 66fs (leading to gM ¼ 1.76)

1.5.1 Isolator working principle The proposed graphene terahertz isolator is a planar device, and it is illustrated in Figure 1.16. A number N (for our device N ¼ 3) of graphene sheets are placed on a back-metallized thin silicon layer of thickness d ¼ 10 mm. The sheets are separated by thin poly(methyl methacrylate) (PMMA) layers (approximately 60 nm), while the thickness of the metallization (chromium and platinum) is 200 nm. The whole structure is bonded to a Pyrex wafer which has solely the function of mechanical support. A magnetostatic field B is applied orthogonally to graphene. The device

24

Developments in antenna analysis and design, volume 2 LHCP

RHCP

RHCP

A

y

3

20 AI

High

resis tivity Pt

(c)

3

silic o

n

LHCP

(b)

z

Si

d

Glass support

0

x 10−3

2.5

Re( σ CCW ) at 3 THz Re( σ CW ) at 3 THz

2

Re( σ CCW ) at 8 THz

1.5

Re( σ CW ) at 8 THz η −1

1

−1

(2η )

(3η )−1

0.5 0 0

2

4

6

Magnetostatic bias (T)

8

Refection coefficient (dB)

Graphene equivalent conductivity

(a)

x RHCP

CCW case

Pt reflector

M

3 x PMMA-graphene layers

y x PM

x LHCP

CW case

Al2O3

y z

B

0 −5 −10 −15 −20 −25

10

2

(d)

4

Γ

(CCW case)

Γ

(CW case)

6

8

10

Frequency (THz)

Figure 1.16 (Reproduced from [6] with permission) Simplified TL circuit model of the isolator. (a) Layered structure and (b) equivalent simplified circuit operation is based on reflecting incident LHCP (left-hand circularly polarized) plane waves as RHCP (right-hand circularly polarized) ones, while absorbing RHCP incident waves. The device thus achieves non-reciprocal unidirectional propagation and isolation for circularly polarized waves [34–36]. The principle of the isolator consists of creating for clockwise (CW) rotating waves (incident RHCP or reflected LHCP) a total surface impedance equal to the impedance h of free space (i.e. impedance matching), causing total absorption (reflection coefficient GCW ¼ 0). On the contrary, for counter-clockwise (CCW) ones (incident LHCP or reflected RHCP) the impedance is mismatched, and waves are reflected (GCCW 6¼ 0). This is easily understood inspecting the circuit model in Figure 1.17, which predicts the following reflection coefficients in the CW and CCW cases: GCW;CCW ¼

1 hN sCW;CCW þ jn cotðnk0 d Þ 1 þ hN sCW;CCW jn cotðnk0 d Þ

(1.37)

where h is the free space impedance, n is the refractive index of silicon and k0 is the free space wavenumber. The design condition GCW ¼ 0 requires then NRe (scw) ¼ h1, which can be satisfied with three layers of graphene in our case, as Figure 1.16(c) shows. The thickness instead can be used to select the working frequency (here we targeted 3 THz), as in a common Salisbury screen.

1.5.2 Measurements The device was fabricated using as substrate the device layer of an SOI wafer bonded to a glass support and measured using a Fourier transform infrared

Air

Al2O3

N x PMMAgraphene layers

Pt reflector

Terahertz antennas, metasurfaces and planar devices using graphene

Si

Νσ CW

n–1η

25

Glass support

(a) LHCP RHCP

η

CW case

d

RHCP LHCP

d

(b)

γ iso

CCW case

Νσ CCW n–1η

η

0 γ surf

z

γ =∞

Figure 1.17 (Reproduced from [6] with permission) The proposed graphene terahertz isolator. (a) 3D view of the device. (b) Cross section and schematics of the working principle. (c) Magnetic field induced splitting of equivalent graphene conductivity as a function of the bias B (f ¼ 3 and 8 THz, mc ¼ 0.53 eV, t ¼ 35 fs, T ¼ 290 K) computed using the Kubo formula. The real part of the equivalent conductivity for the CW and CCW cases is shown for monolayer graphene and compared with multiples of the free space impedance h. In yellow the area of interest for the design. (d) Simulation of the reflection coefficients for wave converted from right-handed to left-handed or vice versa, using the simplified model. Two working points are observed; however, the direction of the isolation in the second one is reversed spectrometer connected to a split-coil superconducting magnet. A polarizer is used to create a linearly polarized incident light, while an analyzer (i.e. a second polarizer) is used in front of the detector (see Figure 1.18). The reflected elliptical polarization is mapped by repeating the measurement for different values of the angle q between the two polarizers. The magnetostatic field is normal to the sample surface, while the propagation vector is close to normal. From these polarimetry measures, the isolation and insertion loss of the device can be derived and are plotted in Figure 1.18(d). At both working frequencies, the isolation reaches almost 20 dB (18.8 and 18.5 dB) and the insertion loss is approximately 7.5 dB. The results plotted in Figure 1.18 are fitted with the full-layered model reaching a very good agreement. Conductivity of graphene is computed using the Kubo formula. The fitting improves sensibly if an additional overall loss of 30% is added to the model over the whole bandwidth, which could be attributed to a systematic error due to the non-perfect planarity of the isolator, which caused part of the energy to be reflected out of the detector. Figure 1.19(a) is a Cartesian plot of the isolation versus the insertion loss. The isolator upper bound (1.36) is also represented. For the fitted parameters, it can be shown that the forbidden region is frequency independent in the band from 0 to 20 THz.

26

Developments in antenna analysis and design, volume 2 B

Isolator

1 0

ø E min

Emin Emax

−1

ø (rad)

Emax

Measure Fitting

Einc

(b)

2

3

4

5 6 Frequency (THz)

7

8

9

10

Meas. setup

(c)

Polarizers Source

Polarization (B = 0T) Polarization (B = 7T)

Detector

(a)

Isolation, measured (dB) Insertion loss, measured (dB) Isolation, ﬁt (dB) Insertion loss, ﬁt (dB)

20

dB

15 10 5 0 2

(e)

(d)

3

4

5

6

7

8

9

10

Frequency (THz)

Figure 1.18 (Partially reproduced from [6] with permission) Measured isolator performances. Measures have been performed at T ¼ 290 K. (a) Schematic of the measurement setup configuration and definition of elliptical polarization parameters. (b) measured elliptic polarization parameters (major and minor axis and Kerr rotation angle) as a function of frequency for B ¼ 0 and 7 T. The measures have been fitted (dashed traces) with the full multilayer model and the best fit is obtained for mc ¼ 0.53 eV, t ¼ 35 fs, d ¼ 9.15 mm, additional loss: 30%. (c) Polarization state shown for some representative frequencies. (d) The extracted performances (isolation and insertion loss expressed both as positive dB quantities) of the isolator for circularly polarized waves. (e) Poincare´ sphere showing the evolution of the polarization with frequency

The performance of the device is just 1 dB below the theoretical upper bound which means that the device is near optimal. While the presented isolator operates with circularly polarized light, it is clear that most terahertz applications need components able to handle linearly polarized waves. It is however quite simple to adapt our isolator to linear polarization operation using one or two quarter-wave plates (QWPs) as polarization converters, as shown in Figure 1.19(b) and (c). This simple system can be used to protect a linearly polarized source from harmful reflections, which is one of the most important applications of non-reciprocal isolators [34].

Terahertz antennas, metasurfaces and planar devices using graphene Isolator 0

27

Isolator B

B

−2 LHCP

Insertion loss (dB)

−4 −6

QWP

QWP

Vertical polarizer

−8

QWP

RHCP

Vertical polarizer

Forbidden area (upper bound) Isolator (frequency sweep) First working point (2.9 THz) Second working point (7.6 THz)

−10 −12 0

(a)

5

10 15 Isolation (dB)

20

Output

25

(b)

Protected input

Output

Protected (c) input

Figure 1.19 (Reproduced from [6] with permission) Device optimality and linear polarization operation: (a) representation of the measured device performances for B ¼ 7 T in the Cartesian plane between isolation and insertion loss for different frequencies. On the same plot, the non-reciprocity theoretical upper bound is represented, computed from the fitted graphene parameters. The working frequencies (showing a maximum in the isolation) are highlighted. (b) and (c) Linearly polarized light operation: By combining the proposed isolator for circular polarization with simple quarter-wave plates (QWPs) it is possible to also achieve isolation for linearly polarized waves The main drawbacks of our device are the need for a strong magnetostatic field (7 T) and an insertion loss of more than 7 dB. Both these issues cannot be solved by improving the design since it is already quasi-optimal in this sense. Hence, the only way to alleviate these drawbacks is to use graphene with higher mobility, such as graphene encapsulated in hexagonal boron nitride which can easily reach mobilities in the order of 40,000 cm2 V1 S1 at room temperature. With this mobility value and a biasing field of 1 T (easily generated by rare earth’s permanent magnets), the insertion loss for perfect isolation would be as low as 0.3 dB (according to the upper bound), thus paving the way to commercially relevant devices. These considerations are independent of the carrier density and, even if high mobility is available only for lower carrier density, the design can be adapted using a larger number of graphene layers.

1.6 Graphene terahertz beam steering reflectarray prototype The objective of the work described in this section is the design, implementation and measurement of the first reconfigurable terahertz reflectarray using graphene as tunable material. Our early prototype implementation is based on a beam-steering

28

Developments in antenna analysis and design, volume 2

reflectarray concept; a reflectarray is, in this context, a planar metasurface which reflects an incident beam of THz light (generated by a given illuminating source) in a direction which can be selected electronically through DC control signals biasing the reflectarray.

1.6.1

Working principle

The concept of reflectarray antenna is a very general one which covers several types of devices [37–39]. See also Chapter 10 in this volume for a thorough and upto-date review of reflectarray antennas, their properties and their applications. Recently, many different elaborations of this concept in the range of sub-mm and THz waves have been published [40–56]. The main idea of the reflectarray is to create an EM beam with some given desired properties (e.g. in terms of width, direction, polarization, intensity profile, radiation pattern, etc.) by using a low-profile (flat) metasurface illuminated by a source antenna. The surface is composed by a (quasi)periodical arrangement of cells, where each cell reflects the impinging wave with a certain phase delay. By carefully choosing the phase delay profile of the full surface, the final shape of the reflected beam can be designed precisely. The most interesting property of reflectarrays is that it is possible to include tunable elements in the reflective cells to control the reflective phase of each cell dynamically. An important example of this concept is found in beam steering reflectarrays. In this case, typically, all the elements are identical, but their reflection phase can be controlled individually with separate control signals. The signals can then be selected to obtain the phase profile associated to the desired radiation pattern of the final antenna. Figure 1.20 shows the structure of this device. The size of the cells is subwavelength, and hence, when the same control signal is the same for all the cells, incident light is only reflected in the specular reflection, because of symmetry. Applying a periodic distribution of the control signals, the initial periodicity

(a)

(b)

Figure 1.20 Column addressing scheme to achieve beam steering. (a) Specular reflection occurring when all the columns are driven with the same voltage, (b) beam steering by creating a dynamical phase profile

Terahertz antennas, metasurfaces and planar devices using graphene

29

of the device is broken, and the new period will determine the direction of the reflected beam. By dynamically tuning the signals, beam steering becomes possible.

1.6.2 Design and measurement The design of the cell has been carried out with the numerical tool ANSYS HFSS. First, graphene resistance has been measured in a fabricated gated graphene sample. We observed that the graphene resistance can vary in a range between 800 W and 4000 W. For the used CVD graphene, the imaginary part of the conductivity in this frequency range can be neglected, and the real part is very close to the DC conductivity. This fact makes unfeasible to create a unit cell with a reflection phase changing uniformly with the voltage. Instead, a two-state cell where the reflection coefficient has a phase variation of 180 in the two states is feasible. The unit cell shown in Figure 1.21, based on a resonant cut-wire design with graphene in the gap working at a central frequency of 1.2 THz, has been optimized

100 μm Silicon

Graphene

Ground plane

100 μm

Gold

(a)

(b)

3 μm 3 μm

6 μm

22 μm

π/2

π/2 1.15THz

800 Ω 4000 Ω π

0.2 0.4 0.6 0.8

0

π

1.25 THz

0.2 0.4 0.6 0.8

0

1.12 THz

(c)

3π/2

(d)

3π/2

Figure 1.21 Terahertz beam steering unit cell design and simulation. (a) Unit cell structure (the bottom face of the substrate is metallized). (b) Details of the design parameters. (c) Reflection coefficients computed for different graphene sheet impedance, swept in frequency. (d) as in (c), but here each curve represents a frequency and the sweep is in graphene impedance

30

Developments in antenna analysis and design, volume 2

to provide a phase difference of 180 upon the two extreme values of graphene resistance (800 W and 4000 W). The final layout includes 40 40 cells each having size of 100 mm 100 mm. Each column is piloted with an independent voltage provided by a control unit (an Arduino board connected to a custom-made array of CMOS control transistors) interfaced to a computer. Beam steering can be achieved by illuminating the reflectarray at 45 and then gating the columns with different voltages. For the final reflectarray sample, the Dirac point was found for a gate voltage of 7 V, while 23 V was used to obtain the high conductivity state in graphene. Hence, each column was either gated with 7 (logical 1) or 23 V (logical 0). Patterns of logical 1 s and 0 s can be used to create a super-period in the reflectarray. For example, the string ‘‘000111000111000111’’ shows a super period of six columns. These patterns are referred to in the following as ‘‘Period N,’’ where N is the number of columns of the pattern periodicity. The deflection angle for each pattern can then be estimated according to the new periodicity; the following table shows the list of patterns and corresponding expected deflection angles.

Configuration

Control string

Angle

Period Period Period Period Period

0011001100110011001100110011001100110011 0011000111001100011100110001110011000111 0001110001110001110001110001110001110001 0000111100001111000011110000111100001111 0000011111000001111100000111110000011111

6 13 17 24 28

4 5 6 8 10

The chip was measured by scanning the receiver angle in the available range between 59 and 9 for control strings. The first three plots of Figure 1.22 show the reflected power as a function of the measurement angle and of frequency for three patterns (with periods 4, 6 and 10). The black arrow illustrates the presence of the beam in the central working frequency of 1.23 THz. The remaining ones show differential plots obtained measuring the reflectarray in one configuration and then inverting all the control bits (see the opposite patterns in the table above) and subtracting the radiation patterns in the two cases. This allows removing almost completely the background specular reflection and obtaining a much clearer plot of the beam. The black arrows show clearly that the beam is steering according to the control string. Figure 1.23 shows a plot of the radiation pattern normalized to its maximum value at the central working frequency of 1.23 THz. The curves in the plot are vertical slices of the differential patterns in Figure 1.22, including also the patterns with periodicity 5 and 10. The angles of the obtained maxima are in excellent agreement with the values predicted in the table above. The beams appear very wide (approximately 30 ). This is not a design limitation, but rather the result of the measurement technique, which used focused beams. A more sophisticated setup

Period

1

–15

–15

–20

–20

–20

–25

–25

–25

0.7

–30

–30

0.6

–35

0.5

–40

0.4

–30 –35 –40

Angle (deg)

–15

–35 –40

0.9 0.8

–45

–45

–45

0.3

–50

–50

–50

0.2

–55

–55

–55

0.1

0.6

0.8

1

1.2

1.4

0.4

1.6

0.6

Frequency (THz)

0.8 1.2 1 Frequency (THz)

1.4

1.6

0.4

–10

–10

–15

–15

–15

–20

–20

–20

–25

–25

–25

–30 –35 –40

Angle (deg)

–10

Angle (deg)

Angle (deg)

Period 8 –10

0.4

Differential

6

–10

Angle (deg)

Angle (deg)

Absolute

Period 4 –10

–30 –35 –40

0.6

0.8 1 1.2 Frequency (THz)

1.4

1.6

0 0.3 0.25 0.2

–30 0.15

–35 –40

0.1 –45

–45

–45

–50

–50

–50

–55

–55

–55

0.4

0.6

0.8

1

1.2

Frequency (THz)

1.4

1.6

0.4

0.6

0.8 1 1.2 Frequency (THz)

1.4

1.6

0.05

0.4

0.6

0.8 1 1.2 Frequency (THz)

Figure 1.22 Frequency versus angle dispersion plots. Absolute and differential date is shown

1.4

1.6

0

32

Developments in antenna analysis and design, volume 2 1.4

Period 4 Period 5 Period 6 Period 8 Period 10

Normalized rad patterns

1.2 1 0.8 0.6 0.4 0.2 0

–30

–25 –20 –15 Deflection angle (deg)

–10

Figure 1.23 Reflectarray radiation pattern (normalized to the maximum value) showing beam scanning. Dashed lies are the expected direction of the beam using reflectarray theory

should greatly improve the performances and in particular the directivity. Indeed, this has been clearly shown in a recent experiment, conclusively demonstrating the validity of our approach [57]. Finally, Figure 1.24 represents the process flow for the fabrication of the reflectarray, starting from the substrate developed in [58].

1.7 Conclusions Several main conclusions can be drawn on the future of this technology from the point of view of the applications explored in this chapter. First, research efforts should focus especially on creating CVD graphene with high mobility and strategies to preserve the mobility during device fabrication are very important. This is not the first time that this conclusion is reached, but it is here particularly important and evident, especially in light of the developed graphene figure of merit and its dependence on mobility. Second, graphene is a winning technology for several terahertz applications. Besides the modulation, reflectarray and isolator applications presented here, several works in the literature have pointed out other possibilities in this frequency range. For example, for terahertz detection [59] or to modulate the output of a quantum cascade laser [60]. As such, we believe that graphene is a key technology to close the terahertz gap.

Au

Al2O3 (200nm)

Al2O3 (200 nm)

SOI Si device (10 to 25 mm)

SOI Si device (10 to 25 mm)

Ag (140 nm) + Al (60 nm)

Ag (140 nm) + Al (60 nm)

Ag (140 nm) + Al (60nm)

Pyrex support (525 mm)

Pyrex support (525 mm)

Pyrex support (525 mm)

(a)

SOI Si device (10 to 25 mm)

(c)

(b) Graphene

MMA + PMMA

PMMA

Au

Au

Al2O3 (200 nm)

Al2O3 (200 nm)

Al2O3 (200 nm)

SOI Si device (10 to 25 mm)

SOI Si device (10 to 25 mm)

SOI Si device (10 to 25 mm)

Ag (140 nm) + Al (60 nm)

Pyrex support (525 mm)

(d)

Ag (140 nm) + Al (60 nm)

Ag (140 nm) + Al (60 nm)

Pyrex support (525 mm)

Pyrex support (525 mm)

(f)

(e) Au

Au

Al2O3 (200 nm)

Al2O3 (200 nm)

SOI Si device (10 to 25 mm)

SOI Si device (10 to 25 mm)

Ag (140 nm) + Al (60 nm)

Ag (140 nm) + Al (60 nm)

Pyrex support (525 mm)

Pyrex support (525 mm)

Printed circuit board

(g)

(h)

Figure 1.24 Process flow for graphene terahertz reflectarray. (a) Initial reflection chip (anodic bonding) [57]. (b) ALD deposition of Al2 O3. (c) MMA þ PMMA resist spin coat, e beam exposure and developing. (d) De-scum and gold (100 nm) evaporation and lift-off. (e) Graphene transfer. (f) PMMA coating, e-beam exposure and development. (g) Graphene etching in oxygen plasma and resist strip. (h) Chip gluing on the PCB and wirebonding

34

Developments in antenna analysis and design, volume 2

Acknowledgments The authors acknowledge the support of the EU Graphene Flagship (contract no. CNECT-ICT-604391), the Swiss National Science Foundation (SNSF) under grant nos. 133583 and 168545 and the Hasler Foundation (Project 11149). We gratefully acknowledge our collaborators and the EPFL CMi staff for the useful discussions. We would also like to thank Graphenea Inc. and Prof. Andrea Ferrari’s group at Cambridge University for the graphene used in the experimental results shown in the chapter. We dedicate this work to the memory of Prof. Julien Perruisseau-Carrier.

References [1]

[2]

[3]

[4]

[5]

[6]

[7]

[8] [9]

[10]

[11]

M. Tamagnone, J. S. Gomez-Diaz, J. R. Mosig, and J. Perruisseau-Carrier, ‘‘Reconfigurable terahertz plasmonic antenna concept using a graphene stack,’’ Applied Physics Letters, vol. 101, no. 21, pp. 214 102–4, 2012. M. Tamagnone, J. S. Gomez-Diaz, J. R. Mosig, and J. Perruisseau-Carrier. ‘‘Analysis and design of terahertz antennas based on plasmonic resonant graphene sheets,’’ Journal of Applied Physics. 2012; 112 (11):114915–24. M. Tamagnone and J. Perruisseau-Carrier, ‘‘Predicting input impedance and efficiency of graphene reconfigurable dipoles using a simple circuit model,’’ Antennas and Wireless Propagation Letters, IEEE, vol. 13, pp. 313–316, 2014. M. Tamagnone, A. Fallahi, J. R. Mosig, and J. Perruisseau-Carrier, ‘‘Fundamental limits and near-optimal design of graphene modulators and non-reciprocal devices,’’ Nature Photonics, vol. 8, no. 7, pp. 556–563, 2014. M. Tamagnone and J. Mosig, ‘‘Theoretical limits on the efficiency of reconfigurable and non-reciprocal graphene antennas,’’ IEEE Antennas and Wireless Propagation Letters, vol. 15, pp. 1549–1552, 2016. M. Tamagnone, C. Moldovan, J.-M. Poumirol, et al., ‘‘Near optimal graphene terahertz non-reciprocal isolator,’’ Nature Communications, vol. 7, p. 11216, 2016. M. Tamagnone, ‘‘Theory, Design and Measurement of Near-optimal Graphene Reconfigurable and Non-reciprocal Devices at Terahertz Frequencies,’’ Ph.D. dissertation, STI, Lausanne, 2016. K. S. Novoselov, A. K. Geim, S. V. Morozov, et al., ‘‘Electric field effect in atomically thin carbon films,’’ Science, vol. 306, no. 5696, pp. 666–669, 2004. J. S. Gomez-Diaz and J. Perruisseau-Carrier, ‘‘Microwave to THz properties of graphene and potential antenna applications,’’ in 2012 International Symposium on Antennas and Propagation (ISAP), Nagoya, Japan, Oct. 29–Nov. 2, 2012. B. Sensale-Rodriguez, R. Yan, M. M. Kelly, et al., ‘‘Broadband graphene terahertz modulators enabled by intraband transitions,’’ Nature Communications, vol. 3, p. 780, 2012, 10.1038/ncomms1787. J. S. Gomez-Diaz, C. Moldovan, S. Capdevila, et al., ‘‘Self-biased reconfigurable graphene stacks for terahertz plasmonics,’’ Nature Communications, vol. 6, p. 6334, 2015.

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[12] P. M. Pardalos, A. Migdalas, and L. Pitsoulis, Pareto Optimality, Game Theory and Equilibria. Springer Science & Business Media, 2008, vol. 17. [13] G. W. Hanson, ‘‘Dyadic green’s functions for an anisotropic, non-local model of biased graphene,’’ IEEE Transactions on Antennas and Propagation, vol. 56, no. 3, pp. 747–757, March 2008. [14] M. Jablan, H. Buljan, and M. Soljacic, ‘‘Plasmonics in graphene at infrared frequencies,’’ Physical Review B, vol. 80, no. 24, p. 245435, 2009, pRB. [15] M. Tamagnone, J. S. G. Diaz, J. R. Mosig, and J. Perruisseau-Carrier, ‘‘Hybrid graphene-metal reconfigurable terahertz antenna,’’ in 2013 IEEE MTT-S International Microwave Symposium Digest (MTT), Seattle, WA, 2013. [16] T. Schaug-Pettersen and A. Tonning, ‘‘On the optimum performance of variable and nonreciprocal networks,’’ IRE Transactions on Circuit Theory, vol. 6, no. 2, pp. 150–158, 1959. [17] S. Mason, ‘‘Power gain in feedback amplifier,’’ Transactions of the IRE Professional Group on Circuit Theory, vol. CT-1, no. 2, pp. 20–25, 1954. [18] I. Lindell, A. Sihvola, S. Tretyakov, and A. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media. Artech House, 1994. [19] M. S. Gupta, ‘‘Power gain in feedback amplifiers, a classic revisited,’’ IEEE Transactions on Microwave Theory and Techniques, vol. 40, no. 5, pp. 864–879, 1992. [20] A. B. Kuzmenko, E. van Heumen, F. Carbone, and D. van der Marel, ‘‘Universal optical conductance of graphite,’’ Physical Review Letters, vol. 100, no. 11, p. 117401, 2008, pRL. [21] J. S. Gomez-Diaz and J. Perruisseau-Carrier, ‘‘Graphene-based plasmonic switches at near infrared frequencies,’’ Optics Express, vol. 21, no. 13, pp. 15 490–15 504, 2013. [22] S. J. Koester and M. Li, ‘‘High-speed waveguide-coupled graphene-ongraphene optical modulators,’’ Applied Physics Letters, vol. vol. 100, no. 17, pp. 171 107–4, 2012. [23] S. H. Lee, M. Choi, T.-T. Kim, et al., ‘‘Switching terahertz waves with gatecontrolled active graphene metamaterials,’’ Nature Materials, vol. 11, no. 11, pp. 936–941, 2012, 10.1038/nmat3433. [24] Z. Lu and W. Zhao, ‘‘Nanoscale electro-optic modulators based on graphene-slot waveguides,’’ Journal of the Optical Society of America B, vol. 29, no. 6, pp. 1490–1496, 2012. [25] C. Xu, Y. Jin, L. Yang, J. Yang, and X. Jiang, ‘‘Characteristics of electrorefractive modulating based on graphene-oxide-silicon waveguide,’’ Optics Express, vol. 20, no. 20, pp. 22 398–22 405, 2012. [26] H. S. Skulason, D. L. Sounas, F. Mahvash, et al., ‘‘Field effect tuning of microwave Faraday rotation and isolation with large-area graphene,’’ Applied Physics Letters, vol. 107, no. 9, p. 093106, 2015. [27] D. Sounas, and C. Caloz, Novel Electromagnetic Phenomena in Graphene and Subsequent Microwave Devices Enabled by Multi-Scale Metamaterials. INTECH Open Access Publisher, 2012.

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D. L. Sounas, and C. Caloz., ‘‘Graphene-based non-reciprocal spatial isolator,’’ in 2011 IEEE International Symposium on Antennas and Propagation (APSURSI) (IEEE, 2011), Spokane, WA, pp. 1597–1600, 2011. D. L. Sounas, and C. Caloz. ‘Gyrotropy and nonreciprocity of graphene for microwave applications’. IEEE Transactions on Microwave Theory and Techniques, vol. 60, no. 4, pp. 901–914, 2012. D. L. Sounas, H. S. Skulason, H. V. Nguyen, et al., ‘‘Faraday rotation in magnetically biased graphene at microwave frequencies,’’ Applied Physics Letters, vol. 102, 191901, 2013. I. Crassee, J. Levallois, A. L. Walter, et al., ‘‘Giant Faraday rotation in single-and multilayer graphene,’’ Nature Physics, vol. 7, no. 1, pp. 48–51, 2011, 10.1038/nphys1816. N. Ubrig, I. Crassee, J. Levallois, et al., ‘‘Fabry-Perot enhanced Faraday rotation in graphene,’’ Optics Express, vol. 21, no. 21, pp. 24 736–24 741, 2013. Y. Hadad, A. R. Davoyan, N. Engheta, and B. Z. Steinberg, ‘‘Extreme and quantized magneto-optics with graphene meta-atoms and metasurfaces,’’ ACS Photonics, vol. 1, no. 10, pp. 1068–1073, 2014. M. Shalaby, M. Peccianti, Y. Ozturk, and R. Morandotti, ‘‘A magnetic non-reciprocal isolator for broadband terahertz operation,’’ Nature Communications, vol. 4, p. 1558, 2013, 10.1038/ncomms2572. X. Lin, Z. Wang, F. Gao, B. Zhang, and H. Chen, ‘‘Atomically thin nonreciprocal optical isolation,’’ Science Reports, vol. 4, 2014. O. Morikawa, A. Quema, S. Nashima, H. Sumikura, T. Nagashima, and M. Hangyo, ‘‘Faraday ellipticity and Faraday rotation of a doped-silicon wafer studied by terahertz time-domain spectroscopy,’’ Journal of Applied Physics, vol. 100, no. 3, p. 033105, 2006. E. Carrasco, M. Barba, and J. Encinar, ‘‘Aperture-coupled reflectarray element with wide range of phase delay,’’ Electronics Letters, vol. 42, no. 12, pp. 667–668, 2006. J. Huang and J. Encinar, Reflectarray Antennas, IEEE Press Series on Electromagnetic Wave Theory. (Wiley, 2007). E. Carrasco, M. Barba, and J. A. Encinar, ‘‘Reflectarray element based on aperture-coupled patches with slots and lines of variable length,’’ IEEE Transactions on Antennas and Propagation, vol. 55, no. 3, pp. 820–825, 2007. J. Perruisseau-Carrier and A. K. Skrivervik, ‘‘Monolithic mems-based reflectarray cell digitally reconfigurable over a 360 phase range,’’ Antennas and Wireless Propagation Letters, IEEE, vol. 7, pp. 138–141, 2008. E. Carrasco, J. A. Encinar, and M. Barba, ‘‘Bandwidth improvement in large reflectarrays by using true-time delay,’’ IEEE Transactions on Antennas and Propagation, vol. 56, no. 8, pp. 2496–2503, 2008. E. Carrasco, M. Arrebola, J. A. Encinar, and M. Barba, ‘‘Demonstration of a shaped beam reflectarray using aperture-coupled delay lines for LMDS central station antenna,’’ IEEE Transactions on Antennas and Propagation, vol. 56, no. 10, pp. 3103–3111, 2008.

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[43] C. Guclu, J. Perruisseau-Carrier, and O. A. Civi, ‘‘Dual frequency reflectarray cell using split-ring elements with rf mems switches,’’ in 2010 IEEE Antennas and Propagation Society International Symposium (APSURSI) (IEEE, 2010). p. 1–4. [44] C. Guclu, J. Perruisseau-Carrier, and O. Civi, ‘‘Proof of concept of a dualband circularly-polarized rf mems beam-switching reflectarray,’’ IEEE Transactions on Antennas and Propagation, vol. 60, no. 11, pp. 5451–5455, 2012. [45] E. Carrasco, M. Barba, and J. A. Encinar, ‘‘X-band reflectarray antenna with switching-beam using pin diodes and gathered elements,’’ IEEE Transactions on Antennas and Propagation, vol. 60, no. 12, pp. 5700–5708, 2012. [46] E. Carrasco, M. Barba, B. Reig, C. Dieppedale, and J. A. Encinar, ‘‘Characterization of a reflectarray gathered element with electronic control using ohmic rf mems and patches aperture-coupled to a delay line,’’ IEEE Transactions on Antennas and Propagation, vol. 60, no. 9, pp. 4190–4201, 2012. [47] T. Niu, W. Withayachumnankul, B. S. Y. Ung, et al., ‘‘Experimental demonstration of reflectarray antennas at terahertz frequencies,’’ Optics Express, vol. 21, no. 3, pp. 2875–2889, 2013. [48] D. Rodrigo, L. Jofre, and J. Perruisseau-Carrier, ‘‘Unit cell for frequencytunable beamscanning reflectarrays,’’ IEEE Transactions on Antennas and Propagation, vol. 61, no. 12, pp. 5992–5999, 2013. [49] S. V. Hum and J. Perruisseau-Carrier, ‘‘Reconfigurable reflectarrays and array lenses for dynamic antenna beam control: A review,’’ IEEE Transactions on Antennas and Propagation, vol. 62, no. 1, pp. 183–198, 2014. [50] P. Nayeri, M. Liang, R. A. Sabory-Garcia, et al., ‘‘3d printed dielectric reflectarrays: low-cost high-gain antennas at sub-millimeter waves,’’ IEEE Transactions on Antennas and Propagation, vol. 62, no. 4, pp. 2000–2008, 2014. [51] T. Niu, W. Withayachumnankul, A. Upadhyay, et al., ‘‘Terahertz reflectarray as a polarizing beam splitter,’’ Optics Express, vol. 22, no. 13, pp. 16 148–16 160, 2014. [52] L. Zou, W. Withayachumnankul, C. M. Shah, et al., ‘‘Dielectric resonator nanoantennas at visible frequencies,’’ Optics express, vol. 21, no. 1, pp. 1344–1352, 2013. [53] N. Yu, P. Genevet, M. A. Kats, et al., ‘‘Light propagation with phase discontinuities: Generalized laws of reflection and refraction,’’ Science, vol. 334, no. 6054, pp. 333–337, 2011. [54] N. Yu and F. Capasso, ‘‘Flat optics with designer metasurfaces,’’ Nature Materials, vol. 13, no. 2, pp. 139–150, 2014. [55] B. J. Bohn, M. Schnell, M. A. Kats, F. Aieta, R. Hillenbrand, and F. Capasso, ‘‘Near-field imaging of phased array metasurfaces,’’ Nano Letters, vol. 15, no. 6, pp. 3851–3858, 2015. [56] D. Wintz, P. Genevet, A. Ambrosio, A. Woolf, and F. Capasso, ‘‘Holographic metalens for switchable focusing of surface plasmons,’’ Nano Letters, vol. 15, no. 5, pp. 3585–3589, 2015.

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M. Tamagnone, S. Capdevila, A. Lombardo, et al., ‘‘Graphene reflectarray metasurface for THz beam steering and phase modulation,’’ Submitted for publication, arXiv preprint arXiv:1806.02202, 2018. H. Hasani, M. Tamagnone, S. Capdevila, et al., ‘‘Tri-band, polarizationindependent reflectarray at terahertz frequencies: Design, fabrication, and measurement,’’ IEEE Transactions on Terahertz Science and Technology, vol. 6, no. 2, pp. 268–277, March 2016. L. Vicarelli, M. S. Vitiello, D. Coquillat, et al., ‘‘Graphene field-effect transistors as room-temperature terahertz detectors,’’ Nature Materials, vol. 11, p. 865, 2012. S. Chakraborty, O. P. Marshall, T. G. Folland, Y.-J. Kim, A. N. Grigorenko, and K. S. Novoselov, ‘‘Gain modulation by graphene plasmons in aperiodic lattice lasers,’’ Science, vol. 351, no. 6270, pp. 246–248, 2016.

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Chapter 2

Millimeter-wave antennas using printed-circuit-board and plated-through-hole technologies Kung Bo Ng1, Dian Wang2, and Chi Hou Chan1,3

A new unlicensed high-band spectrum from 64 to 71 GHz has been added to the existing 57–64 GHz by the US Federal Communications Commission (FCC) for the next generation of wireless connectivity in 5 G [1]. To cover 14 GHz absolute or 22% relative bandwidth, it requires wideband antennas at a millimeter-wave (MMW) band. Based on the complementary source technique [2], a magnetoelectric (ME) dipole antenna operating at microwave frequency has been proposed [3]. It has the salient features of wide bandwidth, low cross polarization, low backlobe, symmetrical radiation patterns, and stable gain and radiation pattern across the operating band. The electric dipole is a conventional planar dipole with its width carefully tuned and the equivalent magnetic dipole is realized by a vertically oriented quarter-wave shorted patch. The antenna is fed by a coaxial probe connected to a G-shaped feed placed between the two walls of the vertical shorted patch which in turn couples the energy to feed both the planar electric and the equivalent magnetic dipoles. The first vertical portion of the G-shaped feed forms a 50-W air microstrip transmission line with a vertical wall of the shorted patch. The reactance of the horizontal portion of the G-shaped feed is inductive while that of the remaining open-ended vertical portion is capacitive and their cancellation leads to a wide impedance bandwidth. The G-shaped feed is in fact an air microstrip transmission line connected to a broadband L-probe feed [4]. An implementation of the ME dipole operating at the MMW frequency band has been reported in [5] using printed-circuit-board (PCB) and plated-through-hole (PTH) technologies on a single-layered substrate. The PCB substrate is chosen such that its electrical thickness is about a quarter guided wavelengths at the center frequency. The shorted vertical walls are conveniently implemented with PTHs.

1

State Key Laboratory of Terahertz and Millimeter Waves (City University of Hong Kong) Hong Kong Calterah Semiconductor Technology (Shanghai) Co., Ltd, People’s Republic of China 3 Department of Electronic Engineering, City University of Hong Kong, Hong Kong 2

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A wideband feed is devised in the form of a T-strip feed. The antenna has 27.6% (|S11| 15 dB) impedance bandwidth and retains most of the salient features of its microwave counterpart. The cross-polarization level of the H-plane is about 15 to 20 dB due to the current on the top portion of the T-strip. To reduce the crosspolarization level, a differential feed technique [6] is employed such that the T-probe is removed and the flat dipole is fed by a balanced port rather than a single feed. For the single feed approach, both the measured and simulated crosspolarization levels vary from 19 to 25 dB in the H-plane. On the other hand, the simulated cross-polarization level for the differential feed is kept under 45 dB for both the E- and H-planes while that of the measured results is only better than 25 dB. One reason for this discrepancy is due to the fabrication tolerance of this MMW ME dipole design which requires a precision of 0.01 mm. To alleviate this difficulty, a differentially fed higher-order mode (HOM) patch antenna is proposed in [7] in which a 0.1 mm fabrication accuracy is needed at the cost of a reduced impedance bandwidth of 18% (|S11| 10 dB). At MMW frequency, the pin of the coaxial probe is tiny and the excitation of the antenna cannot be precisely positioned. One approach to eradicate the problem is to employ an aperturecoupled feed using double-layer PCB and substrate-integrated waveguide (SIW), and these other forms of complementary source antenna elements can be easily adapted to design high-gain array antennas [8, 9]. These antenna elements are less susceptible to fabrication tolerance and enjoy a wideband operation covering 14 GHz (or 22%) bandwidth. The aforementioned antennas are discussed in the following sections, addressing the choice of substrate, fabrication tolerance, wideband feeding mechanism, and performance characterization.

2.1 Wideband MMW ME dipole antennas In this section, we present two implementations of MMW ME dipole antennas, one with a single feed as depicted in [5] and the other a differential feed. The latter can be easily integrated with differentially fed RF transceiver ICs [10] and unwanted radiation from the vertical parts of the antenna can be canceled to further suppress the cross-polarization level.

2.1.1

Single feed printed ME dipole antenna

MMW ME dipole cannot be constructed by straightforward frequency scaling of its microwave counterpart due to the small feature size and necessary mechanical support. Innovative design is required to circumvent the limitations arise from electrical properties of the materials and fabrication processes at hand. The geometrical configuration of the MMW ME dipole proposed in [5] is shown in Figure 2.1. The ME dipole mainly consists of three parts: (i) a vertically oriented shorted quarter guided-wavelength patch formed by PTHs, (ii) a wideband T-shaped feed with a vertical ground-signal-ground (G-S-G) transition, and (iii) a flat planar dipole. The dipole is printed on a Duroid 5880 substrate, which has a

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Figure 2.1 Configuration of 60 GHz printed ME dipole antenna: (a) 3-D view and (b) side view. (Reproduced from [5, p. 3130] IEEE 2012.) dielectric constant of 2.2 and a thickness of 0.787 mm. The substrate is about a quarter guided-wavelength thick at 60 GHz. Each of the two vertical conducting walls of the shorted patch is realized by three parallel vias of 0.3 mm in diameter using PTH technology. The 50-W transmission line in [3] is replaced by a groundsignal-ground (G-S-G) transition with the signal pin connected to the T-shaped strip. The two ends of the T-shaped strip are widened in width to prevent the copper peeling off from the substrate. It should be pointed out that the GSG transition is not 50-W but together with the careful design of the T-shaped strip, the input impedance at the coaxial feed point is matched. The resonant frequency of the antenna is dictated by the total length L of the antenna. The antenna gain is stable across the operating band for a fixed separation L2 between the two dipole plates. While the resonant frequency is insensitive to L2, the impedance matching is poor when L2 becomes too small such that there is a strong coupling between the dipole plates and the T-shaped strip. Input impedance of the antenna is sensitive to the width of the dipole plate W when compared with its length L1. In addition, W also seriously affects the antenna gain. In view of these considerations, the final structural dimensions are given in Table 2.1. The fabrication tolerance needed to realize the T-shaped strip in a tight spacing of 0.4 mm

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Table 2.1 Dimensions of the 60 GHz printed ME dipole antenna. (Reproduced from [5, p. 3130] IEEE 2012.) Parameter Unit (mm) lg (guided wavelength)

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(L2) renders our use of the LPKF ProtoLaser S laser circuit structuring system which has a minimum track and gap widths of 50 and 25 mm, respectively.

2.1.1.1

Operating principles of the antenna

The antenna is excited through a G-S-G transition connected to a T-shaped probe printed at the open aperture of the electric dipole. Figure 2.2 shows both the electric dipole and quarter-wave patch with the corresponding simulated electric current and magnetic field distributions. In Figure 2.2(a), the electric fat dipole is constructed using 5 PTHs and a pair of dipole arms. In reference to its electric current, the majority of the currents flowing on both fat plates are in phase. In Figure 2.2(b), the shorted quarter-wave patch formed by parallel via holes is shown together with the T-shaped probe. The volumetric distribution of the magnetic field between the two via walls viewing from the top is also plotted. It should be pointed out that each of the arrows represents the largest magnetic field at a fixed (x, y) coordinates along the z-direction (thickness of the substrate). The direction of the magnetic field also indicates the direction of the magnetic current flow of the magnetic dipole. Figure 2.2(c) shows the G-S-G transition connected to the T-shaped feed together with the simulated current distribution. The cross-polar radiation in the E-plane generated by the upward flowing current is reduced by that of the vertical currents flowing in the opposite directions. On the other hand, the currents on the T-shaped probe along the length of the open aperture contribute to the crosspolarization of the radiation pattern in the H-plane.

2.1.1.2

Measurement and results

To validate the design, a prototype antenna shown in Figure 2.3 was fabricated and measured. The input impedance of the antenna is measured by an MMW band Agilent Network Analyzer (E8361A with N5260-60003 waveguide T/R module). The radiation pattern of the prototype is measured by an in-house far-field MMW antenna manual measurement system as depicted in Figure 2.4. This manual system was upgraded to automatic mode and has been used in radiation pattern measurement for the antenna depicted in Section 2.1.2. The transmitted signal is fed to the antenna under test (AUT) by a WR-15 waveguide-to-coaxial adaptor with a loss of 0.8 dB at 50–75 GHz. A standard horn antenna with a gain of 22.9–24.5 dBi at 50– 75 GHz is used for the receiving antenna at a far-field distance R ¼ 500 mm (100 lo, where lo is the free-space wavelength at 60 GHz). It is connected to a V-band mixer with a spectrum analyzer for detecting and receiving RF power from the AUT. RF absorbers are used around the AUT. Due to the limitation of the

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Figure 2.2 Simulated electric and magnetic current distributions of the ME dipole: (a) Electric current on the electric flat dipole; (b) magnetic field distribution and its equivalent magnetic current; and (c) electric current on the G-S-G transition and T-strip

Figure 2.3 Photo of the prototype ME dipole antenna. (Reproduced from [5, p. 3131] IEEE 2012.)

(a)

OML S 15ms-GA 50–75 GHz MMW source

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Rotary arm for receiving power ranged -75° < θ < +72°

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Figure 2.4 Radiation pattern and reflection coefficient measurement setups: (a) Photo of the measurement platform; (b) block diagram of the radiation measurement setup; and (c) VNA with OML module for reflection coefficient measurement. (Reproduced from [5, p. 3133] IEEE 2012.)

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Figure 2.5 Comparison of the simulated and measured reflection coefficients of the proposed ME dipole. (Reproduced from [5, p. 3134] IEEE 2012.) 10

Antenna gain (dBi)

8 6 4 2 0 50

Simulated Measured

55

60 Frequency (GHz)

65

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Figure 2.6 Comparison of the simulated and measured gains of the proposed ME dipole. (Reproduced from [5, p. 3134] IEEE 2012.)

measurement platform, the radiation pattern can only be measured for q up to 72 . For the gain measurement, two identical standard gain horns from Quinstar QWHVPRR00, with frequency range of 50–75 GHz, are employed for making the direct gain comparison to obtain the antenna gain value of the prototype antenna. Figure 2.5 shows measured results of the reflection coefficient in dB for the prototype. As seen from the curve, the antenna has a wide impedance bandwidth of 43.9% (|S11| 10 dB) from 48 to 75 GHz. The impedance bandwidth is reduced to 27.6% from 53 to 70 GHz if |S11| 15 dB is required. As can be seen in Figure 2.6, the proposed antenna can have an average gain value of ~7.5 dBi across the operating bandwidth from 50 to 70 GHz. Figure 2.7 shows the measured radiation patterns at 55, 60, and 70 GHz for both the E- and H-planes. The broadside

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Figure 2.7 Comparison of the simulated and measured radiation patterns of the proposed ME dipole at 55, 60 and 70 GHz. (Reproduced from [5, p. 3134] IEEE 2012.)

Millimeter-wave antennas using printed-circuit-board

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J Pair of grounded loops

Balanced flat dipole

Figure 2.8 Configuration of the 60 GHz printed differentially fed complementary source antenna with simulated electric current distributions patterns are stable across the operating bandwidth while the cross-polar radiation of the proposed antenna varies between 15 dB to 21 dB. The overall performance of the MMW ME dipole antenna implemented with PCB and PTH technologies is quite similar to its counterpart at the microwave band proposed in [3]. However, its backlobe and cross-polar radiation performances are not as good as the original design. The G-shaped strip feed in [3] allows fine tuning of the phase control between the electric and magnetic dipoles. In the MMW implementation, the tunable parameters are governed by the T-shape feed itself. While it also yields wideband characteristics, the fixed substrate thickness and minimum via-hole dimension limit the freedom of design optimization. The current flowing on the G-shaped strip is always in the xz-plane; but for the T-shaped strip, the current on the strip would flow along the xz-plane (the horizontal strip connected to center via of the G-S-G transition) and the yz-plane (two open ends of the T-shaped strip). This current along the yz-plane would generate the unwanted crosspolar radiation in the H-plane. As a result, the antenna would have a slightly higher cross-polar radiation when compared with the one implemented in [3]. In the following sub-section, we will address the suppression of the cross-polarization level using a differentially fed complementary source antenna. The simulated co-to-cross level can be larger than 40 dB for both the E- and H-planes.

2.1.2 Differential feed printed ME dipole antenna Based on the single feed version described in Section 2.1.1, a differentially fed complementary source antenna is devised as shown in Figure 2.8. The single G-S-G

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Developments in antenna analysis and design, volume 2

Table 2.2 Dimensions of the 60 GHz differentially fed complementary source antenna Parameter Unit (mm) lg (guided wavelength)

G

PL2

PLa

PLb

PH2

PHa

PHb

H2

S

8.00 2.37

0.80 0.24

0.28 0.08

0.24 0.07

1.60 0.47

0.45 0.13

0.60 0.18

0.787 0.23

0.40 0.12

vertical feeding structure is now modified to a differentially fed port. The horizontal T-shaped strip is removed as the flat dipole is now fed by a balanced port. The detailed dimensions of the proposed antenna are shown in Table 2.2.

2.1.2.1

Operating principle of the differentially fed complementary source antenna

The differentially fed complementary source antenna is composed of a planar dipole similar to the single feed version. However, the vertically oriented shorted quarter-wave patch implemented for the magnetic source is changed to a pair of grounded loops. These grounded loops couple energy from the vertical part of the excitation port. Again, the proposed antenna is fabricated using PCB and PTH technologies on a single microwave substrate. Both the electric and magnetic sources are excited simultaneously. Figure 2.9 shows the simulated radiation patterns of the proposed antenna for 55, 60, 65, and 70 GHz. All the cross-polarization levels are maintained lower than 29 dB. At the desired center frequency, the cross-polarization level reaches 42.4 dB. The radiation patterns are very stable across the desired band. The E- and H-plane radiation patterns are similar in shape and the broadside gain can achieve 7 dBi across the operating band. Experiments were conducted to validate the simulated results. Unfortunately, a true differential source at 60 GHz is not available at the time in our laboratory. Therefore, we need to design a single-to-differential feeding network.

2.1.2.2

Design of a waveguide-to-differentially fed port transition

Our measurement equipment at the time could not support a true differentially fed excitation and therefore, a transition from a waveguide (WR-15) to the differential port is required for the antenna measurement. The structure of the transition was designed and shown in Figure 2.10. The design is inspired by the broadband waveguide-to-microstrip finline depicted in [11]. A pair of optimized antipodal finlines are designed and printed on a 0.127-mm-thick double-sided RT/Duroid 5880 substrate which is sandwiched into the WR-15 adaptor with the waveguide as shown in Figure 2.10. The transition transforms energy from the WR-15 waveguide input to the single differential port. The two leads of the single differential port are connected to the two signal pins of the two G-S-G structures in Figure 2.8 to excite the complementary source antenna. Furthermore, in Figure 2.8(a), a series of PTHs are made on both sides of the antipodal finline structure to minimize energy

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Figure 2.9 Simulated radiation patterns of the proposed differentially fed ME dipole antenna at 55, 60, 65, and 70 GHz leakage from the transition through the small gap of the split blocks forming the waveguide. The perspective views of the split blocks of the transition fixture and the associated field distributions are shown in Figure 2.8(a) and (b). The wave propagates upward inside the transition structure and excites the AUT. Detailed geometrical parameters of the waveguide to antipodal finline transition will be given in Section 2.2.

2.1.2.3 Measurement results To verify the simulation results, a prototype antenna shown in Figure 2.11 was fabricated and measured. The measured VSWR and antenna gain are shown in Figure 2.12. The input impedance of the antenna is measured by an MMW Agilent Network Analyzer (E8361A with N5260-60003 waveguide T/R module). The radiation pattern of the prototype is measured by an MMW antenna measurement system shown in Figure 2.13 upgraded from the one shown in Figure 2.4. The VSWR has a 4.1% frequency shift between the measured and simulated

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Developments in antenna analysis and design, volume 2 Differential port

Plated through holes (PTH) array

Plated through holes (PTH) array

Waveguide input port

(a)

AUT

15 mm (b)

Single-layered RT5880 substrate with patterning on both sides

Figure 2.10 Field distributions of waveguide-differential-feed transition and assembly with AUT: (a) testing fixture and the printed antipodal finline transition and (b) E-field distribution from waveguide port to the AUT

results. More than 29% of measured impedance bandwidth can be achieved with VSWR 2. For the antenna gain performance, there is an average of 0.6 dB differences between the simulated and measured gains within the simulation band. This is due to the imperfect alignment and fabrication tolerance of the metal waveguide fixture. In general, the antenna gain is stable across the desired band with an averaged value of 7 dBi. In Figure 2.14, radiation patterns of the single-fed and differentially fed complementary source antennas are compared. For the cross-polarization level observed in Figure 2.14(a), the level is quite high, varying from 19 to 25 dB in the H-plane, in both the simulation and measurement. After applying the differentially fed technique shown in Figure 2.14(b), the simulated crosspolarization level is kept under 45 dB for both the E- and H-planes. However, the measured cross-polarization level is only better than 25 dB. This discrepancy is

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(c)

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Figure 2.11 Geometry of the proposed antenna with waveguide to differential port transition: (a) Magnified view of the proposed antenna; (b) antenna structure simulated with HFSS; and (c) proposed antenna with testing fixture

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Figure 2.12 Comparison of the measured and simulated results of VSWR and antenna gain probably due to the misalignment of the measurement setup and also due to fabrication error. We have demonstrated that MMW ME dipoles, a type of complementary source antennas can be fabricated using PCB and PTH technologies with a wide impedance bandwidth and stable radiation pattern and gain across the operating band. The cross-polarization level can be kept around 20 dB and further

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Developments in antenna analysis and design, volume 2 V band SGH

θ = –90°

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Agilent 11974V preselected millimeter mixer, 50 GHz to 75 GHz

R Quinstar QWH-VPRR00 standard gain horn 50 GHz-75 GHz

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Figure 2.13 Far-field radiation pattern measurement setup

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improvement can be made using a differential feed. Some of the dimensions of the designs, however, require a fabrication accuracy of 0.01 mm. To alleviate this difficulty, we exploit the HOM approach in which the fabrication tolerance is better albeit a narrower operating bandwidth.

2.2 HOM MMW patch antenna For a rectangular patch microstrip antenna, the resonance frequency of the HOM patch is less susceptible to fabrication tolerance than that of the fundamental mode [12]. On the other hand, the bandwidth of the former is narrower than that of the latter together with undesirable radiation patterns [13]. In this section, we demonstrate how we can combine the higher-order and fundamental mode operations to achieve a broad impedance bandwidth using a fabrication tolerance of 0.1 mm. Asymmetric radiation patterns in the E-plane and large cross-polarization levels in the H-plane generated by a single HOM patch are obliterated by feeding two mirrored elements differentially. The designed differentially fed prototype has more than 18% measured impedance bandwidth, stable gain at 9–10 dBi and symmetric and stable radiation patterns across the operating band from 56 to 67 GHz.

2.2.1 Wideband HOM patch element For regular HOM patch antennas, only a few percent bandwidth can be achieved which cannot meet the requirement at 60 GHz band. A wideband HOM patch element has been reported in [7]. In this section, we present its performance and working mechanism, particularly how we can combine the fundamental and higherorder modes for broadband operation.

2.2.1.1 Antenna element performance The geometry of the HOM patch element is shown in Figure 2.15. Duroid 5580 substrate with a thickness of 0.254 mm and dielectric constant of er ¼ 2.2 is employed. The guided wavelength at 60 GHz is lg ¼ 3.37 mm. The patch is excited by a probe feed located at one end of the patch while the other end is shorted with three shorting pins. The patch length is denoted by b and a slot of width e is cut on the patch. The separation between the excited end of the patch to the center of the slot is denoted by c. It should be emphasized that the fabrication precision requirement is relaxed from 0.01 mm of the ME dipole in Section 2.1 to 0.1 mm due to the use of HOM. The design starts from a traditional rectangular patch with a length 1.5 lg working at TM30 mode which has a better fabrication tolerance. However, the conventional HOM patch results in two undesirable sidelobes on the E-plane along the length of the patch where the major current flows. Shorting pins are thus added to reduce this effect and they also reduce the patch to half of its original size (0.86 lg). Comparing with a conventional half-wavelength rectangular patch printed on the same substrate where the length is about 0.46 lg, the antenna is enlarged to 1.87 times along the current direction. With the presence of the shorting pins, the antenna retains similar fabrication tolerance as the original HOM patch (with a length of 1.5 lg).

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Developments in antenna analysis and design, volume 2

G b H

Radiating patch

a e

Shorting pins c z x y

a=2 mm (0.59 λg)

b=2.9 mm (0.86 λg)

c=1.7 mm (0.50 λg)

e=0.3 mm (0.09 λg)

G=15 mm (4.45 λg)

H=0.254 mm (0.08 λg)

Shorting pin diameter=0.5 mm (0.15 λg)

Figure 2.15 Geometry of the wideband higher-order mode patch element. (Reproduced from [7, p. 467] IEEE 2015.) To enhance the bandwidth, a slot near the shorting pins is introduced and the parameters (c and e) need to be carefully tuned so that the added slot does not destroy the original operation mode of the shorted patch while introduces an additional resonance. There are basically two modes working in the range so that the antenna can achieve a wider bandwidth. Figure 2.16(a) shows the current distribution on the shorted HOM patch. The currents flow in the opposite directions along the patch and there is a zero current point at about one-third of the patch from the shorting pins. To introduce another resonance, the slot must be chosen at that zero current point so that the original HOM will not be destroyed. As shown in Figure 2.16(b) after adding the slot at the right position, the current distribution on the patch remains the same as that in Figure 2.16(a) at a higher frequency band (around 61 GHz). At the same time, for the additional resonance mode (around 55 GHz) introduced in Figure 2.16(c), the current is mainly distributed on the portion of the patch without the shorting pins. Thus, it implies another resonance mode (TM10). To validate the proposed wideband patch element, a prototype of this patch antenna was fabricated based on parameters given in Figure 2.15. The reflection coefficient versus frequency is shown in Figure 2.17. The simulated reflection coefficient shows 18% impedance bandwidth from 53.5 to 64 GHz for |S11| 10 dB. Measured result of the fabricated prototype also confirms well with the simulation result. By combining the HOM and fundamental mode operations, bandwidth of the proposed antenna can be greatly improved. The simulated results of the shorted HOM antenna and fundamental mode antenna are compared in the same graph to demonstrate the bandwidth enhancement effect.

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x

y (a)

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Figure 2.16 Current distributions of shorted higher-order mode patch: (a) shorted higher-order mode patch at 61 GHz; (b) shorted higher-order mode patch with the slot at 61 GHz; and (c) shorted higher-order mode patch with the slot at 55 GHz. (Reproduced from [7, p. 467] IEEE 2015.) Figure 2.18 shows the radiation patterns from 55 to 63 GHz. The patterns keep relatively stable across the band. Nevertheless, due to structural asymmetry, they are asymmetric in the E-plane which is the xz-cut in Figure 2.15. The current component going in the y-direction is very small which results in very low crosspolarization level in the E-plane. On the contrary, the currents going through the feeding pin and shorting pins are much stronger. This is the main reason for the higher end-fire cross-polarization level in the H-plane. It is also worth noting that

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Figure 2.17 Reflection coefficients of three types of patches. (Reproduced from [7, p. 468] IEEE 2015.) the larger cross-polarization level at the higher frequency band is due to the stronger currents going through the shorting pins as shown in Figure 2.16(b) comparing with those in Figure 2.16(c). The undesirable cross-polarization level degrades the performance of the antenna.

2.2.1.2

Combining fundamental and higher-order modes for broadband operation

As the antenna structure is rather simple, the critical parameters that dictate its performance are the patch length b, and the slot width e and its position c. Increasing b results in the lowering of the resonance frequency. The width and position of the slot not only have influence on the resonance frequency but also have significant effects on the bandwidth and matching condition. A narrower slot results in two separated resonances while a wider one results in a single narrow band resonance. Similarly, a shorter c will separate two resonances apart and longer c results in merely a single resonance in the operating band. We investigated the resonance of the antenna cavity simulated by HFSS and it gives the relationship of two modes’ eigen-resonance frequencies versus the slot width and position. From Figure 2.19, it is much easier to understand that the position of the slot mainly controls the separation of two modes’ resonances. For the same slot width, when c is around 1.7 mm, resonances of the two modes are closer to each other, which imply a better merging of two modes for widening the bandwidth. Other c values will result in further separation of the resonances of the two modes. At the same time, for the same position of the slot, the width of the slot also has a great impact, especially on the fundamental mode. It is because the slot itself does not destroy HOM much, as analyzed previously. Thus, this parameter is critical for adjusting the modes for better bandwidth performance.

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Developments in antenna analysis and design, volume 2

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e = 0.2 mm e = 0.3 mm e = 0.4 mm e = 0.5 mm

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2.2.2 Differentially fed HOM patch antenna As shown in Figure 2.17, the impedance bandwidth of the HOM patch is around 18% which is enough for covering the typical 60 GHz band. Nevertheless, as one can see from Figure 2.18, the radiation pattern is asymmetric and the crosspolarization level is very large. Furthermore, in many 60 GHz radios, differential signal is adopted for better system performance as mentioned previously. Thus, a differentially fed HOM patch antenna has the advantage of easier integration with active devices and also better radiation characteristics. A waveguide to-differentially fed port fixture depicted in Figure 2.10 is employed to feed the antenna which fully covers the 60 GHz band from 57 to 64 GHz. It can achieve symmetric radiation patterns and lower cross-polarization levels.

2.2.2.1

Antenna geometry and working principle

The antenna structure is shown in Figure 2.20 which consists of two mirrored wideband patch elements of Figure 2.15. The patch element has been tuned to a slightly shorter length at 0.82 lg to shift the working frequency up to 60 GHz band compared with the designs mentioned in Section 2.2.1. All the other parameters are also shown in the figure. The position of the slot also needs to be tuned accordingly to maintain the wideband characteristics. The separation between the two mirrored patches is d. Its effects on the radiation patterns and input impedance will be studied later. The antenna is differentially fed by two pins with each pin through a small aperture on the ground plane connected to each of the two ends of the antipodal finline shown in Figure 2.10. Due to the differential feed design, the antenna not only has the advantage of wideband characteristic of the single element but also eradicates radiation pattern problems of the single element. The design achieves symmetric radiation patterns

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Figure 2.20 Antenna geometry with waveguide transition: (a) top; (b) 3-D; and (3) cross-sectional views. (Reproduced from [7, p. 470] IEEE 2015.) and lower cross-polarization levels. Shown in Figure 2.21, the current on the differentially fed patch is symmetric and 180 out of phase for both modes. As one can observe, currents going through the shorting pins on one patch is comparable with its image counterpart in amplitude, but with different directions. This leads to cancelation of the radiation in the far field in the H-plane and results in a lower cross-polarization comparing with the single element design. Furthermore, the antenna is fabricated on a single-layered substrate by standard PCB process which is cost effective for integration with ICs. In this design, the differential feed can be connected to differential signals from the IC without the waveguide transition. Either flip-chip or bond-wire technology can be adopted for the connection.

2.2.2.2 Simulation and measurement results To examine the performance of the antenna, the prototype was fabricated and shown in Figure 2.22. Reflection coefficient was measured by an MMW Agilent Network Analyzer ranging from DC up to 67 GHz. The measured and simulated reflection coefficients are plotted in Figure 2.22. The proposed antenna covers the

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Developments in antenna analysis and design, volume 2 55 GHz

65 GHz

Figure 2.21 Simulated current distributions on the differentially fed patch for different frequencies. The dark arrows represent the directions of the horizontal and vertical currents. (Reproduced from [7, p. 471] IEEE 2015.) 20 Reflection coefficient (dB)

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Figure 2.22 Comparisons of simulated and measured reflection coefficients and gains. (Reproduced from [7, p. 471] IEEE 2015.) band of 55–66 GHz (|S11| 10 dB). For gain and radiation pattern measurements, the in-house MMW radiation pattern measurement system shown in Figure 2.13 was employed. The gain of the antenna shown in Figure 2.22 is stable across the band of interest which is higher than 9 dBi. The measured radiation patterns at 56, 60, and 64 GHz are shown in Figure 2.23. The measured patterns only show results from 90 to +90 due to the limitation of the facility.

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Figure 2.23 Comparisons of simulated and measured radiation patterns at 56, 60, and 64 GHz. (Reproduced from [7, p. 471] IEEE 2015.)

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Both the E- and H-planes have broadside and symmetric radiation patterns. The measured cross-polarization level is more than 20 dB lower than the copolarization which is much better than that of the single element. The slight differences between simulated and measured results are due to facility sensitivity, leakage of the waveguide fixture, and some minor alignment problems. Nevertheless, they still fit each other quite well. While this differentially fed HOM patch antenna has 18% measured impedance bandwidth and symmetric and stable radiation pattern across the operating band, it does not fulfill the bandwidth requirement of the 5G standard which includes a new band from 64 to 71 GHz on top of the original 57–64 GHz WiGig. Therefore, we need other designs that can cover 14 GHz absolute or 22% fractional bandwidth and yet can be fabricated using commercially available PCB and PTH technologies with 0.1 mm resolution.

2.3 Wideband MMW complementary source antennas for 5G In the 60 GHz antennas presented in Sections 2.1 and 2.3, they are targeted for the operating band from 57 to 64 GHz. The long-awaited 5G standard released by the FCC added a new unlicensed band at 64–71 GHz. It is desirable to design a single antenna that covers the whole 14 GHz bandwidth from 57 to 71 GHz with stable radiation pattern, stable gain, low cross-polarization level, and low backlobe. As demonstrated in Section 2.1, realizations of the ME dipoles using PCB and PTH technologies can be achieved. However, the required fabrication precision is 0.01 mm. To meet commercial fabrication tolerance of 0.1 mm, HOM patch antenna is presented in Section 2.2 with the tradeoff of reduced bandwidth. All the designs in Sections 2.1 and 2.2 only need a single-layered substrate. In this section, we present a two-layer complementary source antenna design that yields all the good features of the 60 GHz ME dipoles but can be fabricated using commercially available processes of 0.1 mm precision.

2.3.1 Linearly polarized antenna fed by an SIW Figure 2.24 shows the geometry of a linearly polarized antenna fed by a substrateintegrated waveguide. In the figure, the ground plane on the top surface of the lower substrate is removed to avoid confusion. At 60 GHz, waveguide-fed antenna is preferable as the diameter of the center pin of the probe feed is only 0.185 mm which is very difficult to place it at the precise location. The complementary antenna has two Duroid 5880 substrates of a thickness 0.787 mm and dielectric constant of 2.2 each. On the bottom surface of the lower substrate, there is a waveguide-to-SIW transition to couple energy from a waveguide to the SIW. The width of the SIW at the transition is designed to match that of the waveguide feed. The width of the later part of the SIW is reduced to prevent generation of the HOM. On the top surface of the lower substrate, there is a slot coupling the energy from the SIW to the cavity in the upper substrate formed by vias using PTHs. There is a via in the lower substrate in proximity to the slot for impedance matching by tuning

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y

z x x

y Matching post

Cavity

Waveguide-to-SIW transition

Figure 2.24 Geometry of the SIW-fed linearly polarized antenna

its diameter and location. In the upper substrate, each of the dipoles is directly fed by two vias residing on the opposite sides of the slot, making it differentially fed. Two dipoles are used to enhance the impedance bandwidth as well as the gain bandwidth. Details of the geometric parameters are provided in Figure 2.25 and Table 2.3. It should be noted that the diameters of the vias are within the precision of 0.1 mm.

2.3.2 Radiation mechanism of the wideband antenna The two dipoles in Figure 2.23 form a composite electric dipole. On the other hand, the slot with cavity forms a magnetic dipole. When the design parameters are tuned such that the electric and magnetic dipoles are excited with proper amplitude and phase, cardiac shape radiation patterns in both the E- and H-planes, as shown in Section 2.3.3, will be achieved. To further understand the effects of each component of the complementary source antenna, we simulated three different scenarios in [14] which will be summarized here. The three cases compared are (i) the proposed design in Figure 2.23, (ii) the cavity structure excited by the slot only without the dipoles, and (iii) the two dipoles fed by the slot without the square cavity. For convenience, all the dimensions are the same as those given in Table 2.3. With the cavity excited by the slot only, the antenna is mismatched when the parameters are used in Table 2.3. With the slot and dipoles only, the antenna is matched from 53 to 75 GHz for VSWR 2. After the cavity is added, the bandwidth is broadened to 50–75 GHz. If VSWR 1.5 is adopted, the antenna with the cavity has a wide bandwidth from 51 to 72 GHz. The cavity also helps to improve the antenna gain by 0.25 dB from 55 to 65 GHz. A square cavity is chosen over the circular one because the overall element size is limited to within one wavelength and also for easy element placement for array applications. The radiating edges of the dipole are rounded to improve the antenna matching as well as broadening the

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H1 H2

y S1

(c)

Figure 2.25 Antenna geometry: (a) top view of the radiating element in Substrate 1; (b) top view of the SIW in Substrate 2; and (c) side view of the antenna

Table 2.3 Dimensions of the proposed wideband antenna (unit: mm) Parameters Values Parameters Values Parameters Values Parameters Values

L1 0.7 L7 2.4 R3 0.15 H1 0.787

L2 1.1 L8 0.3 R4 0.35 H2 0.787

L3 0.6 L9 2.9 S1 1.2 D1 0.7

L4 1.8 L10 1.9 S2 0.8 A1 2.2

L5 1.5 C2 3.6 S3 0.85 B1 0.2

L6 0.6 R1 0.25 W1 2.4

C1 2.7 R2 0.15 W2 4.6

bandwidth slightly over the right-angled edges. Furthermore, the antenna with two dipoles has much wider bandwidth than a single dipole placed at the mid-point of the slot. Figure 2.26 shows the current and electric field distributions in the antenna element on the upper substrate. Figure 2.26(a) shows the current distributions on the electric dipole plates. The currents are mainly flowing in the y-direction. Figure 2.26(b) shows the electric field in each of the gaps between two dipole plates. The electric field is mainly in the negative y-direction which is equivalent to

(a)

(b)

(c)

Figure 2.26 Working mechanism of the SIW-fed antenna element: (a) surface current distribution on the electric dipoles; (b) electric field distributions on the gaps between arms for both dipoles; and (c) electric field distributions over the cavity aperture

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having a magnetic current pointing in the negative x-direction. The electric currents on the dipoles and the electric fields at the gaps are recorded at the same time at 64 GHz. Therefore, the electric and magnetic dipoles are excited with the same phase. Figure 2.26(c) shows the electric field distribution at the aperture of the cavity. It can be seen that the electric field is mainly along the negative y-direction leading to an equivalent magnetic current in the negative x-direction. The majority of the magnetic currents are still at the gaps between the dipole plates while there are some stronger electric fields nearby the right and left edges of the cavity. As illustrated above, the cavity improves the radiation gain because it helps to prevent the propagation of surface wave and contribute to the boresight radiation. To further illustrate the radiation mechanism of the proposed antenna, we show the magnitude of the electric field along two cut planes, A and B, in parallel with the xz- and yz-planes, respectively, in Figure 2.27. Along cut plane A, the electric field is excited at the waveguide aperture and propagates down the SIW. The energy is coupled to the upper substrate through the slot. The electric field at the gap between two dipole plates is also comparatively strong and its equivalent magnetic current there will radiate into the space. Along cut plane B, we can also see that the electric field is coupled into the cavity through the slot and the electric field is pretty much confined in the cavity. The electric field is stronger in the region between the end of the dipole plate and the cavity. As such, there are some equivalent magnetic currents, as indicated in Figure 2.26(c), adding another design freedom for us to tune the magnitude of the equivalent magnetic dipole to achieve a cardiac radiation pattern.

2.3.3

Comparison of simulation and measurement results

The antenna in Figure 2.24 has been fabricated layer by layer using Duroid 5880 substrates. The layers are bonded with Rogers COOLSPAN Thermally and Electrically Conductive Adhesive bonding films with a thickness of 0.05 mm each. The slot in the bonding film is 0.3 mm larger than those on the substrates to allow for fabrication tolerance, alignment error and opening variation of the bonding film during the heating process. Figure 2.28 shows the fabricated prototype after assembly. To facilitate antenna measurements, a wideband SIW to waveguide transition in [15] has been incorporated in Figure 2.24. The average insertion loss of this transition is about 0.2 dB. For VSWR measurement, an Anritsu MS5657B Vector Network Analyzer was used for the frequency band below 70 GHz. The comparisons of the measured and simulated SWRs and gains of the proposed antenna are shown in Figure 2.29. The simulated result for SWR is 38.5% from 50.8 to 75 GHz. For SWR 1.5, the bandwidth reduces slightly to 29.5% from 55 to 74 GHz. The measured results cover 56 to more than 70 GHz. The impedance bandwidth beyond 70 GHz was not measured due to the limitation of the VNA. The radiation patterns were measured by the in-house far-field MMW antenna measurement system shown in Figure 2.13, which works in the V band from 50 to 75 GHz. Due to measurement limitation, only the radiation patterns in the upper hemispherical space were measured. The radiation

E field[V_per_n

y

Cut plane B

x

Cut plane A

8.0000e+004 7.4285e+004 6.8573e+004 6.2859e+004 5.7145e+004 5.1432e+004 4.5719e+004 4.0005e+004 3.4291e+004 2.8578e+004 2.2864e+004 1.7151e+004 1.1437e+004 5.7235e+003 1.0000e+001

Figure 2.27 Magnitude of the electric field distributions of the antenna

Cut plane A

Cut plane B

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Figure 2.28 Fabricated prototype for the wideband SIW-fed linearly polarized antenna: (a) final assembled antenna with the zoomed view of the antenna at the top metal layer of Substrate 1 and (b) the SIW to waveguide transition at the bottom metal layer of Substrate 2

3.5

SWR

3.0 2.5

Simulated Measured

2.0

2 0 –2

Gain (dBi)

10 8 6 4

4.0

–4 –6 1.5 –8 –10 1.0 52 54 56 58 60 62 64 66 68 70 72 74 Frequency (GHz)

Figure 2.29 Comparisons of the measured and simulated SWRs and gains of the proposed antenna gains of the antennas were obtained through a gain comparison method with a standard gain horn. The simulated gain at 64 GHz is 7.6 dBi and the measured gain is 7.7 dBi. It is reported that the V-band 11974 V mixer adopted in the measurement system has a maximum 2 dB error for conversion loss accuracy. Figure 2.30 shows the comparison of the simulated and measured radiation patterns at 54, 64 and 74 GHz. The radiation patterns at 74 GHz are compared to show that the antenna can cover up to 74 GHz even the measured SWR is limited to 70 GHz. Stable and broadside radiations are obtained and the comparisons are in general agree well. The simulated cross-polarization levels in both the E- and H-planes are below 40 dB. The measured cross-polarization levels are better than 25 dB. The simulation results reveal the cardiac shaped pattern which is the signature of the ME-dipole/complementary source antenna. It also implies that the amplitude and phase of the electric and magnetic sources are properly tuned to have excellent cancelation in the backlobe direction.

0

0

330

–10 –20

300

60

–20

–30

–20

–20

240

Simu. Co-pol E plane Simu. Cx-pol E plane Meas. Co-pol E plane Meas. Cx-pol E plane

–10 0

120

330

–20

90

240

Simu. Co-pol E plane Simu. Cx-pol E plane Meas. Co-pol E plane Meas. Cx-pol E plane

120

300

60

–20

–40 270

90

0 330

240

Simu. Co-pol E plane Simu. Cx-pol E plane Meas. Co-pol E plane Meas. Cx-pol E plane

Freq = 56 GHz

120

–10 0

90

240

Simu. Co-pol E plane Simu. Cx-pol E plane Meas. Co-pol E plane Meas. Cx-pol E plane

120

Freq = 74 GHz 0 330

30

–10 300

60

–20

300

60

–30

–40 270

–20

–20

0

30

90 –40 270

–30

–30

60

–40 270

0

Freq = 64 GHz

–30

–30

300

–10

–10

–10

0

60

–30

0

30

30

–30

0

0

0

–10

300

–10

Freq = 56 GHz

330

–10

90 –40 270

–30

0

0

30

–30

–40 270

–20

330

–10

–30

–20

0

0

30

90

–30 240

Simu. Co-pol E plane Simu. Cx-pol E plane Meas. Co-pol E plane Meas. Cx-pol E plane

Freq = 64 GHz

120

–20 –10 0

240

Simu. Co-pol E plane Simu. Cx-pol E plane Meas. Co-pol E plane Meas. Cx-pol E plane

Freq = 74 GHz

Figure 2.30 Comparisons of the measured and simulated radiation patterns at 56, 64, and 74 GHz

120

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Developments in antenna analysis and design, volume 2

2.4 Conclusion In this chapter, we have presented three different design methodologies for MMW antennas targeted to work at the upper bands of the 5G communications, namely, 57–64 GHz and 57–71 GHz, using PCB and PTH technologies. The key design philosophy is to employ available technologies and materials to the farthest extent and then make use of the design to satisfy the required specifications. In the implementation of ME dipole using PCB and PTH technologies, we employed Duroid 5880 with a thickness of 0.787 mm. The substrate thickness is roughly a quarter-wavelength at 60 GHz. We have designed a feeding mechanism making use of a G-S-G transition implemented by vias and a T-shaped strip to feed both the electric dipole and the equivalent magnetic dipole. Through this design, we achieve a wide impedance bandwidth of 27.6%. In addition, it inherits all the salient features of stable radiation patterns, stable gain and low cross-polarization levels. The backlobe is not as low as expected probably due to the imperfect cancelation of the electric and magnetic dipole radiation at the backlobe direction. The T-shaped strip is removed in the second design in which the equivalent magnetic dipole is devised using two current loops. The cross-polarization level is reduced due to the use of a differential feed. In these designs, some of the dimensions require a precision of 0.01 mm which is not suitable for low-cost production. It prompted us to look for the alternative approach using the HOM patch. In the second design methodology using the HOM approach, the precision requirement is relaxed to 0.1 mm. The bandwidth of the antenna is over 18%, covering the 57–64 GHz band sufficiently. In this design, a slot is introduced to a shorted HOM patch such that the element can operate at the fundamental mode at lower frequencies and HOM at higher frequencies. The position and width of the slot are critically tuned to achieve the widest bandwidth. The asymmetric radiation patterns and sidelobes of the element are alleviated using a mirrored pair such that a differentially fed antenna is successfully implemented. While the fabrication tolerance in the first two designs has been mitigated by using the HOM, the achieved bandwidth is not sufficient for covering the whole unlicensed band of 57–71 GHz. The first two design methodologies only require a single layer of the substrate. The third design approach employs two layers with a fabrication precision of 0.1 mm, which can be realized with commercially available low-cost PCB and PTH technologies. The differentially fed antennas presented in Sections 2.1.2 and 2.2 are fed by a waveguide through a waveguide to antipoldal-finline transition only due to the limitation of the available equipment. In the two-layer design, the antenna is fed by a waveguide with an SIW transition. The complementary source design entails a slot and two dipoles together with a square cavity to achieve a wideband design with 14 GHz bandwidth. The antenna radiation pattern and gain are stable with low cross-polarization level. More importantly, we can tune design parameters to achieve excellent cancellation of the radiations due to the magnetic and electric sources at the backlobe direction, resulting in cardiac shaped radiation patterns in both the E- and H-planes.

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Acknowledgment The authors would like to acknowledge the contributions of Miss Qian Zhu for her contributions in the wideband antenna element presented in Section 2.3. Technical discussions with Professor Kwai-man Luk and Dr. Steve Wong, both of City University of Hong Kong, are much appreciated. This work was in part supported by the Hong Kong Research Grants Council (Project No. CityU11200514).

References [1] Fact sheet: Spectrum frontiers proposal to identify, open up vast amounts of new high-band spectrum for next generation (5G) wireless broadband [online]. 2016. Available from https://apps.fcc.gov/edocs_public/attachmatch/DOC-339990A1.pdf [Accessed 29 Oct 2016] [2] Chlavin A. ‘A new antenna feed having equal E- and H-plane patterns’. IRE Trans. Antennas Propag. 1954, vol. 2(3), pp. 113–119. [3] Luk K.-M., and Wong H. ‘A new wideband unidirectional antenna element’. Int. J. Microw. Opt. Technol. 2006, vol. 1(1), pp. 35–44. [4] Luk K.M., Mak C.L., Chow Y.L., and Lee K.F. ‘A novel broadband microstrip patch antenna’. Electron. Lett. 1988, vol. 34, pp. 1442–1443. [5] Ng K.B., Wong H., So K.K., Chan C.H., and Luk K.M. ‘60 GHz plated through hole printed magneto-electric dipole antenna’. IEEE Trans. Antennas Propag. 2012, vol. 60(7), pp. 3129–3136. [6] Xue Q., Zhang X.Y., and Chin C.K. ‘A novel differential fed patch antenna’. IEEE Antennas Wirel. Propag. Lett., 2006, vol. 5, pp. 471–474. [7] Wang D., Ng K.B., Chan C.H., and Wong H. ‘A novel wideband differentiallyfed higher-order mode millimeter-wave patch antenna’. IEEE Trans. Antennas Propag. 2015, vol. 63(2), pp. 466–473. [8] Zhu Q., Ng K.B., and Chan C.H. ‘Printed circularly polarized spiral antenna array for millimeter-wave applications’. IEEE Trans. Antennas Propag. 2017, vol. 65(2), pp. 636–643. [9] Ruan X., Qu S.-W., Zhu Q., Ng K.B., and Chan C.H. ‘A complementary circularly polarized antenna for 60-GHz applications’. IEEE Antennas Wirel. Propag. 2017, vol. 16, pp. 1373–1376. [10] Shamim L.R., Fong N., and Tarr N.G. ‘24 GHz on-chip antennas and balun on bulk Si for air transmission’. IEEE Trans. Antennas Propag. 2008, vol. 56(2), pp. 303–311. [11] Bai R., Dong Y. L., and Xu J. ‘Broadband waveguide-to-microstrip antipodal finline transition without additional resonance preventer’. International Symposium on Microwave, Antennas, Propagation and EMC Technologies for Wireless Communications, Hangzhou, China, Aug. 2007; 2007. pp. 385–388. IEEE. Doi:10.1109/MAPE.2007.4393629.

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[12]

Wang D., Research on Microwave and Millimetre-wave Microstrip Antennas Using Printed and Plated-through-hole Technologies. Ph.D. thesis, City University of Hong Kong, 2015. James J.R., and Hall P.S. Handbook of Microstrip Antennas. London: Peter Peregrinus; 1989. Zhu Q., Ng K.B., Chan C.H., and Luk K.-M. ‘Substrate-integrated-waveguidefed array antenna covering 57–71 GHz band for 5G applications’. IEEE Trans. Antennas Propag. 2017, vol. 65(12), pp. 6298–6306.

[13] [14]

Chapter 3

THz photoconductive antennas Mingguang Tuo1, Jitao Zhang2, and Hao Xin1

3.1 Introduction of THz technology and photoconductive antenna 3.1.1 Importance of THz technology The electromagnetic (EM) spectrum spanning from 0.1 to 10 THz (30–300 cm1 in wave number and 3 mm to 30 mm in wavelength) is usually called the ‘‘terahertz (THz) regime’’, which lies in between optical band and microwave band but was much less explored in history. Until almost 30 years ago, along with the invention of the THz time-domain spectroscopy (THz-TDS), THz technology has experienced significant advances and become a frontier area for research in physics, material science, chemistry, biology and communication. Many interesting physical phenomena, such as resonance of electrons in semiconductors and nano-structures, oscillation of gaseous and solid-state plasma and the vibration of the collective modes of biological proteins, have signatures in the THz regime. In general, THz wave has several attractive features that ensure it being a powerful technique for various applications. Unlike x-ray, THz wave has low photon energy and will not introduce photoionization damage in the biological sample, such as human tissue; thus, it is a safe technique for biomedical applications. In addition, due to its longer wavelength than visible and infrared light, THz wave is transparent to dielectric materials, such as plastic, wood and cloth, which are optically opaque; this makes it highly competitive and promising in nondestructive inspection as well as security screening. Moreover, compared to the microwave spectrum, THz provides more bandwidth which enables a higher data rate for wireless communication and better resolution for wireless sensing.

3.1.2 THz generation The recently blooming advance of THz technology is mainly enabled by the increased availability of THz sources. In general, THz sources can be catalogued into two types: continuous wave (CW) and pulsed sources. The main focus of this chapter is on pulsed THz generation via the photoconducting process. Here, we will only 1 2

Department of Electrical and Computer Engineering, University of Arizona, USA Fischell Department of Bioengineering, University of Maryland, USA

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Developments in antenna analysis and design, volume 2

provide a brief review of CW THz generation, and the reader can find more comprehensive descriptions in the literature elsewhere. CW THz wave can be generated by either up- or down-frequency conversion. Frequency upconversion can be achieved with a microwave source and a multiplier chain [1, 2]. The multiplier can be realized by using Schottky barrier diodes, whose nonlinearity is used to generate harmonics of the input source frequency. A set of multipliers could be cascaded to generate even higher frequencies. Multiplier-based THz sources are mature and commercially available now, and they have been used in many applications of spectroscopy and imaging. On the other hand, frequency downconversion can be achieved by means of photomixing of laser frequencies in nonlinear optical crystals or antenna structure [3]. For example, in 1984, by mixing two frequencies of the CO2 gas laser (has the abundance of laser lines with frequency difference in the THz region) in an open antenna structure, a tunable CW THz emission up to 6 THz can be generated at the power level around 1 mW [4]. One limitation of this method is the very small spectral tunability and fairly low conversion efficiency (in the order of 106). Photomixing can also generate CW THz wave by sending optical laser frequency into a crystal mixer in which the THz frequency component is generated through the second-order nonlinear effect [5]. Various nonlinear materials can be used as mixers at different optical wavelengths, such as LiNbO3, ZnTe, ZnSe, CdS, quartz, GaSe and KTiOPO4. In this way, although the conversion efficiency stays low, the tunable range can be expanded to as large as 10 THz. Much higher and intense THz light can be generated through optically pumped THz lasers that operate from molecular rotational transitions of gas such as CH3OH. Using this method, 100 mW THz signal with very narrow spectral bandwidth (30 dB SNR at peak frequency [22]. Remarkably, it is a time-domain spectrometer, which makes it a powerful tool for time-resolved measurement [23].

3.2 Theoretical modeling and numerical simulation 3.2.1

Motivation and challenge

Although PCA has been developed for many years and are widely used, it still faces challenges. The biggest challenge is the fairly low optics-to-THz conversion efficiency, which is less than 0.1%, and thus the average output THz power is usually less than 100 mW [24]. Most of the input laser power is converted into non-radiated current instead of free-space THz radiation. This will further cause a thermal issue in the semiconductor material which further affects the performance of the PCA devices. On the other hand, it is predicted that the theoretical conversion efficiency of a PCA could reach 100% without any loss [25]. Therefore, it is worth to further study and understand in more depth the THz radiation mechanism of a PCA to improve the realizable conversion efficiency.

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Table 3.1 Key parameters involved in a PCA design Component

Key parameters

Laser source Photoconductor

Wavelength, pulse width, optical power, illumination pattern Carrier lifetime, mobility, optical absorption and reflection, THz transmission, thermal conductivity, breakdown threshold Gap size and geometry, antenna structure and dimension, bias voltage

Electrodes

As described earlier, THz generation of a PCA involves complicated physical processes in which the photon-to-charge carrier conversion process, transient current generation process and EM radiation process all come in and interact with each other. The key design parameters of a PCA are listed in Table 3.1. Although many experimental parametric studies have been conducted to understand the radiation properties of the PCA [26, 27], they usually only involve a limited number of parameters due to the experimental challenges and are usually costly and time-consuming. On the other hand, theoretical modeling and numerical simulation are versatile tools for the PCA study. They are developed based on the fundamental physical principle of the PCA radiation so that all the involved parameters can be studied thoroughly and costeffectively. Theoretical modeling thus can inspire a new PCA design in practice and pave a promising way for improving the optics-to-THz conversion efficiency. In this section, we will introduce three theoretical models that have been experimentally validated and widely used by the THz community: Drude-Lorentz model, equivalent circuit model and full-wave model. We will first explain how the models are built and then discuss the advantages and disadvantages of those models. Based on the recent work on PCA modeling, we will focus on the implementation of the full-wave model and especially its application in studying the radiation properties of different PCAs. Specifically, we will talk about several novel PCAs that emerged recently and showed significant enhancement of conversion efficiency.

3.2.2 Drude–Lorentz model The Drude model was proposed by Paul Drude in 1900 to explain the transport properties of electrons in metals and later has been adopted for semiconductors. In this model, individual carriers are independent with each other, and collision between carriers is considered as an instantaneous event. On the other hand, the Lorentz model was developed to describe the frequency response of dielectric materials using Lorentz oscillators. The combination of them has been used to explicate the carrier transport dynamics within the photoconductor of a PCA once it is excited by a laser pulse [13, 18, 28]. According to the Drude–Lorentz model, the time dependence of the average velocity of photo-excited carriers is given by dvðtÞ v e ¼ þ Emol dt ts m

(3.2)

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where ts is the momentum relaxation time, m* is the effective mass of the carriers and Emol is the electric field at the position of the carriers, which is given by Emol ¼ Ebias

Psc h

(3.3)

where is the permittivity, the geometrical factor h ¼ 3 for an isotropic dielectric materials. Psc is the space-charge polarization due to the separation of the carriers in the field and is determined by dPsc Psc ¼ þ j ðt Þ dt tr

(3.4)

where tr is the recombination lifetime of the carriers, j(t) ¼ nf ev is the current density, nf is the density of free carriers and e is the elementary charge. In general, the recombination lifetime tr is much longer than the momentum relaxation time ts in semiconductor materials. If we insert (3.3) into (3.2) and take the time derivative and further consider (3.4), we can obtain the carrier velocity described by the second-order differential equation: d 2 v 1 dv w2p v ePsc ¼ þ þ h dt2 ts dt m htr

(3.5)

where w2p is the plasma frequency and equal to nf e2/m* . When a biased photoconductor is excited by a laser pulse with photon energies greater than the band gap of the material, free carriers such as electrons and holes are created in the conduction band and valence band, respectively. The time dependence of carrier density is described by the following equation: dnf nf ¼ þ G ðt Þ dt tc

(3.6)

where tc is the trapping time and G(t) is the generation rate of the photo-excited free carriers. By solving (3.5) and (3.6) together, the time-varying photo-excited current j(t) can thus be determined according to the relationship: jðtÞ ¼ nf ev ¼ nf evh nf eve

(3.7)

where vh and ve are the time-varying velocity of holes and electrons, respectively. Since the mobility of holes is typically much smaller than electrons, we can neglect the contribution of holes for simplicity. According to Maxwell’s equations, the time-varying current will lead to EM radiation. The far-field THz radiation thus can be approximated based on the simple Hertzian dipole theory as @jðtÞ (3.8) @t The Drude–Lorentz model described above provides a simplified but effective approach to describe the radiation process of a PCA, which is instructive to understand ETHz

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how a common PCA works. The main focus of this model is to solve the time-varying current originating from the photo-excited carriers by a laser pulse, and thus the effect of the photoconductor and laser pulse on THz radiation can be studied. However, it ignored the contribution of the electrode structure as a radiating antenna (which is actually very important in many realistic cases as to be described later) to the far-field THz radiation. Moreover, it artificially separates the physical processes of photoexcited current generation and EM field radiation into two isolated segments in time sequence. This may lead to inaccurate simulation result because, in fact, these two processes happen simultaneously and interact with each other.

3.2.3 Equivalent circuit model From the perspective of the equivalent circuit model, the THz wave of a PCA can be considered as the result of the radiation of an electrical antenna (electrode structure) that is fed by a time-varying current source. Therefore, one can use lumped element components to describe the radiation mechanism of the PCA [14]–[16]. The structure of a PCA is shown in Figure 3.3(a), and the time-dependent equivalent circuit of a PCA is shown in Figure 3.3(b). It includes a time-varying conductance Gs(t) and a capacitance C(t), a time-dependent voltage-controlled source b(t)Vc(t) that is related to the voltage across the capacitance, a resistance R1 that corresponds to the electrode resistance, the antenna impedance Za that has frequencyindependent resistance and a bias voltage Vbias that is equal to the external bias. The conductance Gs(t) describes the conduction current across the gap due to the photo-excited carriers by the laser pulse. To deduce this source conductance, one first calculate the carrier density from the Drude model, which is similar to (3.6): dnf nf a ¼ þ Il ðr; tÞ dt tc huopt

(3.9)

where h is Planck’s constant, a is the optical absorption coefficient and vopt and I1 (r,t) are the laser frequency and intensity, respectively. Assuming the laser beam has a Gaussian profile, its intensity is given as 2 2r2 2t Il ðr; tÞ ¼ Il ð1 RÞexp 2 exp 2 w0 tl

(3.10)

Laser illumination I (t) y

L

x

+ – 1/Gs (t) = Rs (t)

Vbias

+

– Vc (t)

Vrad (t)

Za THz radiation

C(t)

Biased (a)

β (t)Vc (t) R1

(b)

Figure 3.3 (a) Structure of a PCA and (b) principle of the equivalent circuit model

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where Il is the peak intensity of the laser pulse, R is the reflectivity of the photoconductor, w0 is the beam waist radius at the position where the laser beam hits the device (usually the top surface of the photoconductor) and tl is the laser pulse duration. Solving (3.9) and (3.10), one can get pﬃﬃﬃ pﬃﬃﬃﬃﬃﬃ 2 pﬃﬃﬃ 2t 2tl 2r2 a 2p tl t tl exp nðtÞ ¼ Il exp 2 ð1 RÞ erf þ1 tl 8t2c tc 4tc 4huopt w0 (3.11) Ðx 2 where erf ðxÞ ¼ ð2=pÞ 0 et dt. Further considering the non-uniform optical absorption along the propagation direction and geometrical parameters of the antenna gap, the source conductance Gs(t) is given by pﬃﬃﬃﬃﬃﬃ 2p Weme 2 Gs ðtÞ ¼ Il e ð1 RÞð1 expðaT ÞÞ tl L 4huopt pﬃﬃﬃ 2 pﬃﬃﬃ 2t 2tl tl t exp erf þ 1 tl 8t2c tc 4tc

(3.12)

where me is the electron mobility. W, L and T are the gap width, length and depth at the excitation region, respectively. Kicchoff’s law gives the following: I ðtÞ ¼ Vc ðtÞGs ðtÞ þ C ðtÞ

dVc ðtÞ dC ðtÞ þ V c ðt Þ dt dt

(3.13)

Therefore, the time-dependent gap voltage can be described as dVc ðtÞ 1 1 b ðt Þ Gs ðtÞ Vc ðtÞ dC ðtÞ ¼ V c ðt Þ Vbias V c ðt Þ V c ðt Þ dt Za CðtÞ Za C ðtÞ Za C ðtÞ C ðt Þ C ðtÞ dt (3.14) On the other hand, according to a simplified equivalent circuit model that considers the screening effect, the electric field in the PCA gap is deduced as [16]: dEc ðtÞ 1 1 1 em nðtÞ em Za SnðtÞ Ec ð t Þ e Ebias Ec ðtÞ e E c ðt Þ ¼ z Ltr dt K tr tr eme Za S dnðtÞ Ec ð t Þ L dt

(3.15)

where K ¼ (1 þ (emeZaSn(t)))/L, tr is the recombination time, z is the geometrical factor of the substrate that denotes the screening factor, is the permittivity of the substrate and S is the area of the cross-section of the PCA gap.

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By comparing (3.14) with (3.15), one can deduce the capacitance C(t) and coefficient of the voltage-controlled source b(t) as tr eme Za SnðtÞ em nðtÞtr ; bðtÞ ¼ e C ðt Þ ¼ (3.16) 1þ L Za z Since the current density in the PCA gap is j(t) ¼ en(t)meVc(t)/L, the voltage across the radiating element is given by Vrad ðtÞ ¼ Za J ðtÞS ¼ Za enðtÞme Vc ðtÞS=L

(3.17)

and the radiated THz power can thus be calculated by PTHz ðtÞ ¼

2 Vrad ðtÞ Za

(3.18)

where Rl is ignored by assuming the electrode is good conductor. This equivalent circuit model treats PCA in a similar way as a conventional electrical antenna and thus can study the impacts of the electrode structure and laserinduced current on the THz radiation. The validity of this model is based on the assumption that the gap region is uniformly illuminated, which, however, is not always true in the real case. For example, to get optimized THz radiation, the stripline PCA needs to be partially illuminated at the anode part instead of the whole region. More importantly, this model hypothesizes that the THz radiation only comes from the electrode structure, which, as a matter of fact, is not true either. As what we will show in the next section, the radiation of the photo-excited current within the gap actually has considerable and sometimes dominant contribution to the overall THz radiation.

3.2.4 Full-wave model The full-wave analysis was first proposed by Sano and Shibata in 1990 based on the fact that THz radiation of a PCA is a multi-physical process where laser excitation, carriers’ transport and EM radiation interact with each other [17]. Therefore, an adequate model that couples carriers’ generation and transport equations with fullwave Maxwell’s equations is necessary for PCA simulation. In the context of the full-wave model [17, 29], the THz radiation of a PCA is divided into three phases. In phase I, a static electric field is built due to the externally biased voltage. In phase II, carriers’ transport equations and Maxwell’s equations are solved together, and the near-field THz field is obtained. In phase III, the far-field THz radiation is deduced based on the equivalence principle. In phase I, the Poisson equation and carrier dynamics equations are solved to obtain the bias field in the static state, as given by q * * * r2 V ð r Þ ¼ ðnð r Þ pð r Þ ND þ NA Þ e *

*

*

r J n ð r Þ ¼ qRð r Þ

(3.19) (3.20)

84

Developments in antenna analysis and design, volume 2 *

*

*

r J p ð r Þ ¼ qRð r Þ

(3.21)

*

*

*

*

*

*

*

*

*

*

J n ð r Þ ¼ qmn nð r ÞðrV ð r ÞÞ þ qDn rnð r Þ

(3.22)

J p ð r Þ ¼ qmp pð r ÞðrV ð r ÞÞ qDp rpð r Þ

(3.23)

*

where V ð r Þ is the potential distribution inside the semiconductor, q is the ele* * mentary charge, nð r Þ and pð r Þ are density of electrons and holes in the static state, respectively. ND NA represents the net concentration of the impurities (subscripts *

*

*

*

D and A denote donor and acceptor, respectively). J n ð r Þ and J p ð r Þ are static * current density of electrons and holes, respectively. Rð r Þ is the recombination rate of the mn and mp are mobilities of carriers, and Dn and Dp are diffusion coefficients, which are related to the mobilities by the Einstein relationship: Dn Dp KB T ¼ ¼ q mn mp

(3.24)

where KB is Boltzmann’s constant and T is the semiconductor’s temperature in * * Kelvin. When the steady-state solutions of (3.19)–(3.23) are obtained, E DC ð r Þ can * be easily deduced from V ð r Þ, so as to the static carrier densities (nDC and pDC) and static current (dark current) densities (JnDC and JpDC). In phase II, the coupling between carriers’ transport equations and Maxwell equations is implemented by using the transient current calculated by the former as the excitation source of the latter. As such, the time-varying electric field as well as the photo-excited current can be calculated simultaneously. The involved equations are listed as below * *

@ H ð r ; tÞ r E ð r ; tÞ ¼ m @t * *

(3.25)

* *

* * @ E ð r ; tÞ * * r H ð r ; tÞ ¼ e þ J n ð r ; tÞ þ J p ð r ; tÞ @t * *

(3.26)

*

q

* * @nð r ; tÞ * * ¼ r J n ð r ; tÞ þ qðGð r ; tÞ Rð r ; tÞÞ @t

q

* * @pð r ; tÞ * * ¼ r J p ð r ; tÞ þ qðGð r ; tÞ Rð r ; tÞÞ @t

(3.27)

*

* * J n ð r ; tÞ * * J p ð r ; tÞ

(3.28)

*

*

*

* *

* * *

*

*

*

*

* *

* * *

*

¼ qmn nð r ; tÞðE DC ð r Þ þ E ð r ; tÞÞ þ qmn nDC ð r Þ E ð r ; tÞ þ qDn rnð r ; tÞ (3.29) ¼ qmp pð r ; tÞðE DC ð r Þ þ E ð r ; tÞÞ þ qmp pDC ð r Þ E ð r ; tÞ qDp rpð r ; tÞ (3.30)

Equations (3.25) and (3.26) are Maxwell’s equations, (3.27) and (3.28) are * *

continuity equations and (3.29) and (3.30) are drift-diffusion equations. E ð r ; tÞ and

THz photoconductive antennas

85

* *

H ð r ; tÞ are the time- and spatial-dependent electric and magnetic fields, e and m are *

*

*

*

the permittivity and permeability, J n ð r ; tÞ and J p ð r ; tÞ are deduced by subtracting the dark currents JnDC and JpDC from the total current within the gap of the PCA, * Gð r ; tÞ is the carrier-generation rates and the rest of the notations have the same definitions as before. Assuming a Gaussian distribution of the laser beam in both the spatial and temporal domains, the generation rate is determined by

aI0 h ðx x 0 Þ2 expðaðz z0 ÞÞ exp G r;t ¼ hn s2x ! ðt t0 ðz z0 Þ=vsemi Þ2 exp s2t *

!

ðy y 0 Þ2 exp s2y

!

(3.31)

where a is the absorption coefficient of the semiconductor, I0 is the optical peak intensity of the laser pulse, h is the optical efficiency due to the reflection at the ambient-semiconductor interface, v is the laser frequency. The coordinate is chosen so that the PCA is placed in the xy plane and the laser beam propagates in the z direction. (x0,y0,z0) is the center point of the laser beam on the top surface of the photoconductor. t0 and st represent the temporal peak and duration of the laser pulse, respectively. (sx, sy) is the beam size in the cross-section that is perpendicular to the propagation. vsemi is the speed of light in the photoconductor. Based on the Shockley–Read–Hall process, the recombination rate is given by *

Rð r ; tÞ ¼

*

*

*

nð r ; tÞ pð r ; tÞ n2i ð r Þ *

*

nð r ; tÞ tp þ pð r ; tÞ tn

(3.32)

*

where ni ð r Þ is the intrinsic carrier concentration of the photoconductor, tn and tp are the lifetime of electron and hole, respectively. The carrier mobilities mn and mp are field-dependent in practice and given by mn0 mn ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ * * 1 þ ðmn0 ðE þ E DC Þ=vn;sat Þ2

(3.33)

mp0 ; mp ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ * * 2 1 þ ðmp0 ðE þ E DC Þ=vp;sat Þ

(3.34)

where mn0 and mp0 are the initial electron and hole mobilities without external field, and vn,sat and vp,sat are the saturated electron and hole velocities, respectively. By solving (3.25)–(3.30) together, one can obtain the time-varying EM field originated from the excitation of laser pulse. Considering the computational complexity and efficiency in the reality, the EM field is usually solved in the near-field zone, and the far-field radiation is deduced from near-to-far-field transformation.

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Developments in antenna analysis and design, volume 2

In phase III, the far-field THz radiation is deduced from the near-field result based on the surface equivalent theorem and Green’s theorem. A virtual closed surface that contains the PCA is first created. According to the surface equivalent theorem, the equivalent currents on the surface are determined by !

!

n Hð r ; tÞ Js ð r ; tÞ ¼ ^ !

(3.35)

!

Ms ð r ; tÞ ¼ ^n Eð r ; tÞ

(3.36)

where Js is the magnetic surface current density, Ms is the electric surface current density and ^ n is the unit surface normal vector. In the spherical coordinate, the farfield radiation can then be calculated in the time domain by means of !

Er ð r ; tÞ ﬃ 0

(3.37)

!

!

!

!

!

!

Eq ð r ; tÞ ¼ h0 Wq ð r ; tÞ Uf ð r ; tÞ

(3.38)

Ef ð r ; tÞ ¼ h0 Wf ð r ; tÞ þ Uq ð r ; tÞ (3.39) pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ where h0 ¼ m0 =e0 is the impedance of free space. W and U are vector potentials that are calculated by !

W ð r ; tÞ ¼ !

Uð r ; tÞ ¼

1 @ 4prc @t 1 @ 4prc @t

ð r r0 ^r Js t ds0 c

(3.40)

r r0 ^r Ms t ds0 c

(3.41)

ð

where r is the distance of the far-field point to the origin, r0 is the distance of the near-field point to the origin and r^ is the unit direction vector. The integral is performed on the entire equivalent surface. Equations (3.19)–(3.23) can be numerically solved by the 3D finite-difference method initially. Once the steady-state solutions are obtained, they are used as the initial input parameters of phase II, in which (3.25)–(3.30) are solved by using the 3D finite-difference time-domain (FDTD) method. The full-wave model takes into account the interaction between carriers’ transport and EM radiation and thus makes itself a better fit for the PCA modeling and simulation. As such, almost all of the PCA related parameters that are listed in Table 3.1 can be studied numerically, and the radiation mechanism of the PCA can therefore be understood thoroughly. One challenge of the full-wave model is its complexity of numerical implementation and computational burden of 3D simulations. In the following section, as a demonstration of the versatile capabilities of the full-wave model, we will show a few examples of PCA simulations [30].

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3.2.5 Simulation examples of full-wave model 3.2.5.1 Validation of the model Based on the full-wave model described above, in-house codes on the MATLAB platform were developed, and the THz radiation of a dipole PCA was numerically simulated as a validation of our model. The geometry and dimension of the dipole PCA are shown in Figure 3.4(a). The electrodes have a gap size of 5 mm, and a laser beam with the same diameter illuminates onto the photoconductor within the gap. The parameters used in the simulation are listed as follows. The photoconductor is LT-GaAs, which has carrier lifetimes of 0.1 ps (electron) and 0.4 ps (hole), carrier mobilities of 200 cm2/V s (electron) and 30 cm2/V s (hole) and a permittivity of 12.9; the intrinsic carrier concentration is 2.1 106 cm3; the absorption coefficient of the photoconductor is 1 104 cm1; the laser source has a wavelength of 800 nm, a beam waist of 2.5 mm, a pulse duration of 80 fs and a peak intensity of 1 109 W/cm2; the bias voltage is 60 V. The thickness of the LT-GaAs is 2.2 mm, and the overall structure is digitized into a mesh size of 200 nm. The simulation was carried out for a total time of 2 ps with a time step of 0.33 fs. Figure 3.5 shows the static-state potential distribution when the PCA is biased by an external voltage. Figure 3.5(a) shows the potential distribution on the top surface of the PCA when it was biased at 60 V. Figure 3.5(b) shows the comparison between our method and commercial software COMSOL and Silvaco when it was biased at 120 V. The sub-figures are the potentials along the dashed line of Figure 3.4(a) at depths of (I) 0 mm, (II) 0.8 mm, (III) 1.6 mm and (IV) 2.2 mm. The good agreement of different methods validates the numerical accuracy.

3.2.5.2 Parametric studies of the PCA by simulation The parametric studies of the coplanar stripline PCA shown in Figure 3.4(b) was then carried out using the full-wave model. The laser beam with a diameter of 20 mm was placed close to the anode and LT-GaAs with a thickness of 2.2 mm is x z

5

5

y 5 50 20

50

20

50

(b)

2 2.

(a)

34

2 2.

34

50

Figure 3.4 Structure and dimension of a dipole PCA (a) and coplanar stripline PCA (b). The orange structure and the red solid circle represent the electrodes and laser beam spot, respectively. All the dimensions are in mm

Developments in antenna analysis and design, volume 2 150

20 25

(I)

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0 0

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30

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30 20

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(III) 0

20 40 X position (μm)

60

100 50 0 0

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60

(b)

Figure 3.5 (a) Simulation results of static-state potential of the PCA when it was externally biased. The color bar has a unit of V. (b) Static-state potential at different depth 30

14

Peak THz field (V/m)

12 10

Peak of THz E field (V/m)

DC = 5 V DC = 20 V DC = 50 V

8 6 4 2 0

(a)

0

10

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50

20 15 10 5 0

60 (b)

Laser power = 6 mW Laser power = 20 mW Laser power = 60 mW

25

0

20

40 60 Voltage (V)

80

100

Figure 3.6 Parametric studies of a coplanar stripline PCA simulated by the fullwave model. (a) Laser power dependence at different bias-voltage levels. (b) Bias voltage dependence at different laser powers. The symbols (square, circle and triangle) are simulated data and the solid line follows the guide to the eye used as the photoconductor. All of the rest parameters used in the simulation are the same as before. The dependence of the far-field THz radiation on laser power and bias voltage was studied first. Laser power was varied from 6 to 60 mW at three biasvoltage levels (5, 20 and 50 V) in the study of laser-power dependence, and the bias voltage was varied from 5 to 100 V at three laser-power levels (6,2 and 60 mW) in the study of bias-voltage dependence. The results are shown in Figure 3.6, in which the peak of the time-domain THz field is plotted with regard to the studied parameters. The simulation results reveal that while the THz field linearly increases with regard to the bias voltage, it shows a saturation effect against the increase of the laser power, which is mainly due to the screening effect of the space charges and has been predicted by the scaling rule [31, 32] and observed in the experiment.

THz photoconductive antennas 40 Peak of THz E field (V/m)

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80

DC = 5 V DC = 20 V

35

DC = 50 V

30 25 20 15 10 5 0

(a)

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50

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SI–GaAs, laser power = 6 mW SI–GaAs, laser power = 20 mW SI–GaAs, laser power = 60 mW LT–GaAs, laser power = 60 mW LT–GaAs, laser power = 20 mW LT–GaAs, laser power = 6 mW

70 60 50 40 30 20 10 0

0

89

0

20

40

60

80

100

Voltage (V)

Figure 3.7 Parameter studies of a coplanar stripline PCA using SI-GaAs as the photoconductor. (a) Laser power dependence at different bias-voltage levels. (b) Bias voltage dependence at different laser powers The influence of photoconductor material was also studied. Instead of using LTGaAs, we chose SI-GaAs, which has much higher carrier mobility (electron: 5,000 cm2/V s, hole: 200 cm2/V s) and longer carrier lifetimes (electron: 10 ps, hole: 40 ps) than LT-GaAs. According to (3.29), this will generate larger photoexcited current and thus lead to enhanced THz radiation. We carried out similar parametric studies of SIGaAs-based PCA and the results are plotted in Figure 3.7. The result of LT-GaAs-based PCA was also plotted as a comparison. It is shown that, while the SI-GaAs-based PCA features a similar parameter dependence as that of LT-GaAs, it exhibits highly enhanced THz radiation, which proves that carrier mobility of a photoconductive material is a crucial parameter for THz radiation. As the excitation source, the ultra-fast laser pulse ultimately limits the bandwidth of THz radiation. It is thus important to understand how the pulse width affects the THz radiation of a PCA. For this study, we used a coplanar stripline PCA as an example and varied the pulse width from 20 to 200 fs. The laser power and bias voltage were fixed at 20 mW and 60 V, respectively. The rest of the parameters were the same as the condition of Figure 3.5. The result is shown in Figure 3.8(a). It clearly shows that for a certain photoconductor (LT-GaAs here), the bandwidth of the THz radiation is closely related to the pulse width. For example, when the pulse width is 200 fs, the bandwidth is less than 5 THz; while when it is 20 fs, the bandwidth can exceed 30 THz. In reality, the bandwidth of a PCA is usually within a few THz and not sensitive to the pulse width as long as it is within sub-ps. The reason is that the photoconductive material as well as the electrode structure will affect the bandwidth. For example, experiments have shown that the length of the dipole PCA directly influences the bandwidth of the THz radiation [27]. As a matter of fact, not only the dipole length but also the whole structure of the electrode will affect the THz radiation. Figure 3.8(b) shows one simulation example, where we change the auxiliary electrode of the dipole PCA of Figure 3.4(a) while fixing the length of the dipole The length of the auxiliary electrode is set to 50, 100 and 200 mm. Although the primary peaks of the time-domain signals for the three cases overlap

1

THz field (a.u.)

Developments in antenna analysis and design, volume 2 200 fs 160 fs 80 fs 40 fs 20 fs

0 –1 3.5

4

4.5 5 Time (ps)

5.5

6

1

0 (a)

500

10

20 Frequency (THz)

30

50 μm 100 μm 200 μm

0 –500 3.5

Amplitude (a.u.)

Spectrum (a.u.)

THz field (a.u.)

90

40 (b)

4

4.5

5 5.5 6 Time (ps)

6.5

7

7.5

40 20 0

0

2

4

6

8

10

Frequency (THz)

Figure 3.8 Effect of the laser pulse width (a) and structure of the electrode (b) on the spectrum of THz radiation with each other, the dipole PCA with longer auxiliary electrode shows more secondary peaks afterward, which is due to the reflection at the end of the auxiliary electrode. The influence on the spectrum can be seen more clearly in the frequency domain, as shown in Figure 3.8(b).

3.2.5.3

Design of new PCAs with enhanced THz radiation

One remarkable advantage of the modeling and numerical simulation is that it can be easily applied to inspire novel PCA designs. In this section, we will show two examples of a new PCA design that promise enhanced THz radiation. The first example is a sawtooth PCA with elliptical illumination, as shown in Figure 3.9(a). It has a coplanar stripline structure while one of the electrodes (anode) has periodical sub-triangles like a sawtooth. This structure is illuminated by an elliptical beam (20 mm 40 mm that can cover the whole structure. In the simulation, we use LT-GaAs as the photoconductor, which is biased externally by a 5 V voltage. Unless declared, the rest of the parameters are the same as used in the previous simulations. A regular coplanar stripline of Figure 3.4(b) is also simulated as a comparison. The simulation results are shown in Figure 3.10. Figure 3.10(a)– (d) is the distribution of the bias field on the regular and sawtooth PCAs. It can be seen that the bias field near the apex of the teeth is much higher than other places, which is advantageous for collecting photo-excited electrons. The laser power is set to 60 mW for both the sawtooth and stripline PCAs. The far-field radiations are calculated at the position that is right below the PCA and 200 mm away. The simulation result in Figure 3.10(e) shows that the time-domain peak of the THz field of the sawtooth PCA is 1.4 times stronger than the stripline PCA. In practice, the PCA usually suffers from the saturation effect when the laser power is high enough, as shown in Figure 3.6. The illumination region of the sawtooth PCA is larger than that of the stripline PCA, which indicates that more laser power can be used in sawtooth PCA in case the laser intensities are the same for both.

THz photoconductive antennas x 5

91

Laser beam

y

z w

p

h

50

+V

z

x THz radiation

y 44

(a)

(b)

Figure 3.9 Sketch of the sawtooth PCA. (a) Geometry and dimension of the PCA; the height and width of the tooth are h ¼ 5.6 mm and w ¼ 5.4 mm, respectively. The pitch size p ¼ 7 mm (b) Elliptical illumination of the laser beam. All the numbers have a unit of mm Over time, it has been found that the low conversion efficiency of a typical PCA is probably due to the extremely low quantum efficiency [25]: most of the photoexcited carriers are re-combined within the photoconductor before they can reach the electrodes and contribute to EM radiation in free space. To mitigate this issue, one should make the site of carrier generation and the metallic electrode as close as possible. Therefore, instead of the bare gap used in a common PCA, one can design fine metallic structure (nano-finger) within the gap serving as both a carrier-generation site and a carrier-collection electrode. For example, using 3D plasmonic contact electrodes, researchers have improved the optical-to-THz conversion efficiency ten folds (7.5%) [33]. Here, we propose a new PCA structure called ‘‘nanocrossfinger’’ and compare it with a nano-finger PCA via numerical simulation. The structures are shown in Figure 3.11. In both PCAs, the fingers have a width and separation of 100 mm. Considering the computational memory, the separation of the electrode pads is set to 2 mm, and each pad has a size of 5 mm 5 mm. The substrate is LT-GaAs and has a dimension of 12 mm 5 mm 2.2 mm. The laser power is 6 mW, and the beam diameter is 1 mm. The electrode is biased at 0.5 V. The simulation is run for 1 ps with a time step of 0.033 fs and a mesh size of 20 mm. The far-field THz radiation is calculated along the z-axis and 200 mm away. The static distribution of the DC bias field is shown in Figure 3.12. Compared with nano-finger structure, the nano-crossfinger structure can generate a much stronger DC field with the same biased voltage. The nanometer metallic structure also affects the optical field distribution of the illuminating laser beam. We first use COMSOL to obtain the distribution of the laser beam and then apply it as an input data to FDTD simulation. The simulated transient current on the yz-plane and across the center of the laser beam is shown in Figure 3.13(e), indicating 50% enhancement

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0.2 0.15 0.1 0.05 0 0 (f)

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Figure 3.10 Simulation results of the bias DC field distribution of the sawtooth and coplanar stripline PCAs. (a) and (b) are bias field on the top surface of the photoconductor for the stripline PCA and sawtooth PCA, respectively. (c) and (d) are 1D distribution of the bias DC field along the lines of y ¼ 26 mm and y ¼ 29 mm, respectively. (e) and (f) far-field THz radiation of the sawtooth PCA and coplanar stripline PCA. Both the stripline and sawtooth PCA are simulated at 60 mW laser power

2 μm

2 μm

5 μm

5 μm

1.7 μm

–

+

+

– 1 μm

5 μm

5 μm

y z

x

(a)

(b)

Figure 3.11 Sketch of nano-structure PCA design. (a) Nano-crossfinger PCA. (b) Nano-finger PCA. The red solid circle indicates the laser beam

× 106 5

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2

3

Ex (V/m)

4

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2 3 Z–axis (μm)

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0

5 (f)

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2 3 Z–axis (μm)

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Figure 3.12 Distribution of the DC bias field. (a) and (c) are field components Ez and Ex on the top surface of nano-crossfinger PCA, respectively. (b) and (d) are field components Ez and Ex on the top surface of nano-finger PCA. The color bar has a unit of V/m. (e) and (f) are field distributions along the line that is in the y-axis and across the tips of the fingers

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0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

(e)

0.2

0.4

0.6

0.8

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t (ps)

Figure 3.13 Simulation result of the nano-finger and nano-crossfinger PCA. (a)– (d) are time- and frequency-domain far-field THz radiation. (e) Photo-excited current on the yz-plane and across the center of the laser beam of the peak current. The far-field THz radiations are shown in Figure 3.13(a)–(d). The calculated polarized components Ef and Eq are along the x- and y-axis, respectively. The spectra show that the THz field has moderate enhancement (120%) in the x-axis and has significant enhancement (ten times) in the y-axis.

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3.3 Experimental characterization of PCA component and system 3.3.1 Far-field THz-TDS 3.3.1.1 Introduction and motivation Since PCA has been proposed about three decades ago by Auston et al. [12, 34], it is being commonly used for the THz signal generation. It has the advantages of simple configuration, compact size, broadband property and room temperature operation. By incorporating two PCAs, or PCA with EO crystal as the THz emitter and detector with laser source and additional optical components, the constructed far-field THz-TDS system offers capabilities for extensive applications, such as material characterization, imaging, sensing, security screening and biology [35– 39]. However, this technique is always restricted by its limited THz power radiated from the PCA, which is due to the low optical-to-electrical conversion efficiency. For example, the absolute THz radiation powers from several PCAs were measured by an InSb hot-electron bolometer in [26]. The radiation power level of a dipole antenna on the order of sub-mW was achieved under moderate pump laser power and bias voltage, which translates to a conversion efficiency of 5,000 150–200 30 [34, 44, 45] >100 [46]

107[42] >1018 >1022[34] >51017[47]

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Developments in antenna analysis and design, volume 2

To better understand the photoconductive material effect on the THz radiation properties, the experimental comparison of the LT-GaAs- and SI-GaAs-based PCAs in terms of the time-domain THz field is made based on a butterfly-shaped stripline antenna with an antenna gap of 34 mm (LT-GaAs antenna model: PCA-4434-100-800-h, BATOP GmbH [48], and SI-GaAs antenna is in-house fabricated with identical geometry). The THz detector used in the setup is a butterfly-shaped dipole antenna with a dipole gap of 6 mm (model: PCA-44-06-10-800-h, BATOP GmbH [48]). Due to the large dimension of the butterfly wings, this emitterdetector configuration has a relatively good response in the sub-THz regime. The dependencies of the THz peak field of the time-domain pulsed signal on the DC bias voltage and pump laser power for the two PCAs under anode-illumination (in which the illumination laser spot is in the vicinity of the anode electrode [49, 50] to obtain stronger THz radiation) are compared and plotted in Figure 3.15. A nonlinear relationship between the THz peak field and the DC bias voltage is observed for the SI-GaAs-based PCA, while an almost linear relationship is noticed for the LT-GaAs-based PCA. It is also seen that the SI-GaAs-based PCA actually generates higher THz radiated signals compared to the LT-GaAs-based PCA under moderate bias conditions. However, due to the screening effect caused by larger carrier mobility, the performance of the LT-GaAs-based PCA becomes better in terms of the THz electric field than that of the SI-GaAs-based PCA at higher DC bias voltage and pump laser power. The saturation effect is shown even clearly in the pump laser power dependence curves and can be fitted according to the scaling rule [31, 32]: ETHz ¼ D

P P0 þ P

(3.43)

where ETHz is the THz electric field, P is the pump laser power, D and P0 are the fitting coefficients related to the laser and PCA parameters. The fitted curves validate that the SI-GaAs-based PCA would be ultimately surpassed by the LTGaAs-based PCA under high pump laser power and DC bias voltage. Similar saturation features have been reported before in [26] for a dipole PCA but not in a systematic manner. The silicon-on-sapphire (SoS) substrate has been widely used for the THz pulse generation back in the 1980s [12, 51]. Radiation damage through ion (e.g., Oþ, Arþ) implantation is normally used to drastically reduce the intrinsic-like carrier lifetime to be less than ps [45, 52], which is preferred for broadband generation. On the other hand, the sapphire substrate has an advantage of optical transparency around 800 nm, which is useful for potential THz near-field applications [53–55]. The GaAs material is demonstrated to have higher THz radiation compared to the ion-damaged SoS substrate as reported in [56], based on a 40-mmlong and 10-mm-wide (6 mm slot) dipole emitter antenna, as shown in Figure 3.16. With the development of the MBE technique, the GaAs layer with better photoconductive properties can be directly grown on top of the sapphire substrate (called GaAs-on-sapphire and abbreviated as GoS), avoiding the time-consuming

THz photoconductive antennas 2

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Figure 3.15 Dependence of THz peak fields on DC bias voltage under the pump laser-power level of (a) 2 mW, (b) 6 mW, (c) 10 mW and (d) 14 mW (the solid lines are a guide to the eye) and dependence of THz peak fields on pump laser power at the bias-voltage level of (e) 5 V, (f) 13 V, (g) 21 V and (h) 25 V (the solid lines are the fitted curves based on the scaling rule) and sophisticated fabrication process but sacrificing the surface uniformity due to the crystal mismatch. The far-field THz radiation properties based on this new photoconductive material (GaAs epitaxial layer grown at 600 C with a growth rate of 0.283 nm/s and a total thickness of 1 mm) are also compared to the separately

100

Developments in antenna analysis and design, volume 2

Electric field (a.u.)

5

ELO LT-GaAs SOS

0

–5 0

2 4 Time delay (ps)

6

Figure 3.16 Comparison of the radiated THz electric field between the PCAs made on epitaxial liftoff LT-GaAs-on-sapphire and RD-SOS [56] measured results based on the LT-GaAs substrate as shown in Figure 3.17. The THz emitter antenna structure used in this comparison is still the butterfly-shaped stripline antenna. Clear THz radiation enhancement of the GoS-based PCA compared to the LT-GaAs-based PCA is observed which shows that this GoS substrate is a promising material for THz photoconductive applications. The enhancement is mainly attributed to the large carrier mobility of the GaAs layer, which is related to the growth temperature. The saturation effect under higher pump laser power can also be fitted using the previously mentioned scaling rule but with different fitting coefficients. THz detector is also a critical component in the THz-TDS system. The photoconductive materials used for the THz emitter generally can also be used for the THz detector. Around 1999, a novel material called self-assembled ErAs nano-islands embedded in GaAs (ErAs:GaAs) with promising photoconductive properties and a better degree of control over the carrier lifetime was introduced [57]. Enhanced THz detection is observed using the ErAs:GaAs nano-island superlattices based PCA (30-mm-long dipole) compared to the same antenna fabricated on the LT-GaAs and RD-SoS substrates [58] as shown in Figure 3.18. The observed enhancement is attributed to the greater power coupling efficiency and the involved carrier dynamics in the superlattice for the ErAs:GaAs detector. The antenna structures/geometries could also impact the far-field THz responses. Different antenna structures with different configurations are experimentally studied and compared. The THz emitter with a bowtie structure (with a bowtie length of 160 mm, a bowtie gap of 5 mm and a bowtie angle of 65 ) fabricated on the SIGaAs substrate is tested with the THz detector kept the same, and a parametric study of the THz electric field dependence on the DC bias voltage as plotted in Figure 3.19 (a) shows that the bowtie structure is not as good as the stripline case in terms of the THz radiation field. Moreover, several different emitter–detector configurations are tested in terms of the system bandwidth as plotted in Figure 3.19(b). Both the timedomain pulsed signals and the spectra are normalized to the corresponding maximum value. Those PCAs other than the butterfly-shaped ones are fabricated on the GoS substrate. The stripline antenna used here has a stripline width of 20 mm and a spacing of 50 mm. The dipole antenna used here has an offset structure with a dipole

THz photoconductive antennas 10 LT-GaAs GoS

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Figure 3.17 Dependence of THz peak fields on DC bias voltage under the pump laser-power level of (a) 2 mW, (b) 4 mW, (c) 6 mW and (d) 8 mW and dependence of THz peak fields on pump laser power at the biasvoltage level of (e) 3 V, (f) 7 V, (g) 9 V and (h) 11 V length of 50 mm, a width of 20 mm and a gap of 4 mm [59]. The full-width at half maximum (FWHM) pulse widths of about 2.6, 1.5, 2.5 and 1 ps are obtained and the correspondent peak frequency occurs at about 0.039, 0.078, 0.068 and 0.235 THz for each configuration, respectively. The best obtained bandwidth among the four

Developments in antenna analysis and design, volume 2 25 nm GaAs

Superlattice period

25 nm GaAs GaAs wafer

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Norm. THz power

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silicon lenses optical pulses (a)

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Figure 3.18 (a) TDS setup for the detector characterization with a dipole antenna structure shown in the center and the superlattice side view shown at the top. (b) Normalized terahertz power (circles) from the 30 mm dipole detectors on different photoconductive substrates as a function of incident optical power and model fit (solid curves) to data. (c) Detected power enhancement factor for ErAs:GaAs in comparison to LT-GaAs and RD-SoS [58] 0.6

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Figure 3.19 (a) DC bias dependence of bowtie antenna as the THz emitter under several different pump laser-power levels and (b) the impact of the different antenna structures on THz bandwidth configurations is the ‘‘stripline-dipole’’ one which is up to about 1 THz. This tells that the butterfly-shaped antenna as the THz detector also restricts the whole system bandwidth. The limited THz bandwidth for these configurations is found to be due to the dispersive optical component which widens the laser pulse width illuminating the PCAs. By replacing the optical component with its counterpart with lower dispersion, the system bandwidth can be improved. There are other works reported in the literature with broad THz bandwidth achieved using photoconductive material with long carrier lifetime. For example, an interdigitated PCA structure with small electrode spacing fabricated the on SI-GaAs substrate as shown in Figure 3.20(a) and (b) under compressed 15 fs laser

800 nm 1.0 +

–

500 μm

e-

+

0.6 0.4 0.2 0.0

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Figure 3.20 (a) Side view and (b) top view schematic of the interdigitated PCA structure used as THz emitter. (c) Temporal plot from the THz-TDS setup and (d) the corresponding FFT amplitude as a function of frequency [60]

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Developments in antenna analysis and design, volume 2

pulse excitation is demonstrated to be able to generate THz signal with bandwidth approaching 20 THz as shown in Figure 3.20(d) [60], which further validates the effect of the laser pulse width and antenna structure to the THz bandwidth.

3.3.1.4

Polarization effect and cancellation effect

To better understand the radiation mechanism of a PCA, a 3D full-wave model implemented by the finite-different time-domain (FDTD) algorithm is previously described to estimate the far-field THz radiation properties and used to compare other simplified approaches used in the literature. In the following, the polarization effect and the cancellation effect are experimentally studied based on the butterflyshaped antenna. The schematic of the emitter–detector configuration for the polarization effect measurement and the two different antenna structures for the cancellation effect measurement are described in Figure 3.21. Only the central regions of the butterfly-shaped antennas are drawn in Figure 3.21(a) and (b) for simplicity. The two antennas for the cancellation effect measurement are fabricated on the GoS substrate to eliminate the effect from the photoconductive substrate are shown in Figure 3.21(e) and (f). For the polarization effect measurement, the THz emitter is either oriented in the normal configuration (parallel configuration represented by the combination of Figure 3.21(a) and (b)) or rotated by 90 (perpendicular configuration represented by the combination of Figure 3.21(c) and (d)), with the THz detector kept at fixed orientation. The focused laser spot is scanned along the direction perpendicular to the stripline from the anode towards the cathode as the red arrow shows with a scanning step of 4 mm. The corresponding measured results in terms of THz peak field for both configurations are shown in Figure 3.22. Since electrons have larger carrier mobility than holes, the contribution of holes to the THz generation is usually ignored. The experimental results show that the contribution from the THz field component along the external DC bias field is always dominant because the THz field measured in the parallel configuration is seen to be larger than that measured in the perpendicular configuration for all the laser spot positions. Also, x

0

+

34

2 mm

– y

(a)

(b) y x

(c)

(d)

(e)

(f )

Figure 3.21 Configuration of polarization effect measurement with (a), (c) being the stripline emitters and (b), (d) being the dipole detectors and (e) the whole and (f) half-cut antenna structures used for studying the cancellation effect

THz photoconductive antennas Parallel Perpendicular

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Figure 3.22 Polarization dependence of the THz peak field on the laser spot location. (a) Normalized peak field of the detected THz signals in two configurations with respect to the laser spot location. (b) Ratio of the two curves in (a) the THz signal is observed to be the strongest when the laser spot is in the vicinity of the anode in both configurations since more photo-generated electrons can flow to the electrode and contribute to the THz radiation, as reported in [49, 50]. Since the THz detector is only sensitive to the incoming THz field component polarized along the dipole gap direction (y-direction in Figure 3.21(a)), the detected signal in the perpendicular configuration indicates that there is still fair amount of THz radiation polarized in the y-direction under this configuration. This is a result of the symmetry breaking, which is due to the position offset/misalignment of the laser spot along the y-direction. Moreover, when the laser spot moves away from the anode electrode, the contribution of the current on the electrode gets weaker and the ratio of two polarized components increases as shown in Figure 3.22(b). The numerically different downtrend of the THz peak field for two polarizations suggests that the THz contribution from the gap cannot be ignored. Even though this study is based on the butterfly-shaped antennas, the observations in terms of the contributions of the photo-excited currents both within the antenna gap and on the electrode should be valid for general antenna structures. In the previous experimental studies, the laser beam normally locates in the middle of the stripline along the x-direction as shown in Figure 3.21(a), so that the photo-excited current will be symmetrically distributed on the electrode. Therefore, the radiation along the x-direction should be cancelled out due to the symmetry. To study the cancellation effect when this symmetry is broken, the whole antenna and half-cut antenna as shown in Figure 3.21(e) and (f) are measured for both parallel and perpendicular configurations. During the experiment, the PCA emitter is externally biased at 5 V and illuminated by a 4 mW laser beam with a diameter of 20 mm, and the PCA detector is illuminated by a 5 mW laser beam. The measured results are shown in Figure 3.23, in which the time-domain pulsed signals of the whole antenna structure are arbitrarily delayed for a convenient comparison. For the whole antenna structure, the radiation in the x-direction (Ef) is 5.3 times of that

Developments in antenna analysis and design, volume 2 1

1

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Figure 3.23 Experimental results of the whole and half-cut butterfly-shaped stripline PCAs. (a) Far-field radiated field in the x-direction and (b) corresponding spectra. (c) Far-field radiated field in the y-direction and (d) corresponding spectra in the y-direction (Eq) in terms of the time-domain peak, as shown in Figure 3.23(a) and (c). For the half-cut antenna structure, however, the ratio of the two components (Ef/Eq) drops to 1.6, with the expense that the radiation in the x-direction drops by 25% compared to that of the whole antenna structure as shown in Figure 3.23(a). According to Figure 3.23(c), the radiation of the half-cut antenna structure is enhanced by about 2.5 times in the y-direction compared to the whole antenna structure. The overall enhancement is also obvious according to the spectrum in Figure 3.23(b) and (d). The small decrease of the THz radiation in the x-direction of the half-cut antenna structure is also expected. Because each wing of the butterfly-shaped PCA contributes to the THz radiation in both directions, cutting the wings not only breaks the symmetry in the y-direction but also results in less radiation in the x-direction. This type of measurement validates the existence of the cancellation effect.

3.3.1.5

THz radiation power/efficiency improvement

The anode illumination for conventional PCAs usually provides stronger far-field THz radiation with more photo-induced electrons flowing to the anode electrode. To avoid the electron–hole pairs with short carrier lifetime recombining with each other before reaching the antenna electrodes, nanostructures and plasmonic structures have been experimentally studied in several works recently [25, 33, 61–64] to enhance the THz radiation compared to the conventional antenna structures (e.g., stripline, bowtie, etc.). Some examples of the proposed antenna geometries are shown in Figure 3.24, where the PCA structures in (a) and (b) are adapted as the

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Figure 3.24 Three different proposed PCA structures for THz radiation enhancement [25, 63, 64]

Developments in antenna analysis and design, volume 2 (a) Epump Kpump

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Figure 3.25 Schematic diagram of plasmonic photoconductive emitters with 2D and 3D plasmonic contact electrodes, and their corresponding optical absorption in the LT–GaAs substrate in response to a TMpolarized optical beam at 800 nm wavelength incidence [33]

THz emitter and the PCA structure in (c) is used as the THz detector. The plasmonic structure, for instance, is designed to excite the surface plasmons allowing more optical absorption into the photoconductive substrate, which leads to enhanced quantum efficiency [25]. A PCA structure with 3D plasmonic contact electrodes has been demonstrated to achieve a highest-to-date 7.5% optical-toterahertz conversion efficiency with over 100 mW THz radiation power obtained, which is attributed to the fact that more carriers reach the electrodes and contribute to the far-field THz radiation [33]. Comparisons between the geometries of the 3D plasmonic electrodes and the 2D plasmonic electrodes and their corresponding optical absorption profiles are shown in Figure 3.25, which indicates that a significantly larger fraction of carriers are generated within about 100 nm distance from the 3D plasmonic electrodes and drifted to the electrodes within sub-ps timescale.

3.3.1.6

Applications of the THz-TDS system

Some representative applications of the THz-TDS system are introduced in this section. First, the THz-TDS system can be used to characterize the properties of the bulk materials. High resistivity Si with different thicknesses, MgO and quartz are used as samples to be measured in terms of the transmission coefficient using

THz photoconductive antennas

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Figure 3.26 Averaged THz time-domain pulsed signals (a) and the corresponding extracted real (b) and imaginary (c) part of the refractive index for different bulk dielectric substrates the THz-TDS system, where each sample is located at the focal plane of the THz beam. Every sample is measured multiple times and the averaged time-domain pulsed signals are used for the refractive index extraction afterwards. The raw measurement signals for different bulk materials along with the free-space reference after averaging are plotted in Figure 3.26. The thicknesses for the thick Si, MgO, quartz and thin Si substrates are 480, 200, 360 and 63 mm, respectively. The observed multiple reflections in the time-domain signals need to be truncated to eliminate the Fabry–Pe´rot effect. The refractive index is extracted based on the transmission difference between each sample and the free-space reference signal according to the algorithm described in [65, 66]. The final extracted real and imaginary parts of the refractive index for those bulk dielectric substrates are shown in Figure 3.26 with statistical error bars. The electrical properties (e.g., surface conductivity ss) of the conductive thin films, such as single-walled carbon nanotube (SWCNT) with thickness on the order of several hundreds of nanometers and graphene thin film (2–3 layers) on the substrate, can be characterized using the THz-TDS system and extracted by treating the thin film as a surface boundary condition between the substrate and free

(a)

Developments in antenna analysis and design, volume 2 0.02 0.18 0.16 0.14 0.12 0.01 0.008 0.006 0.004 0.002 00

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Figure 3.27 Extracted real surface conductivities of SWCNT films with different thicknesses (a) and graphene thin film (b) on the glass substrate [20]

(a)

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Figure 3.28 2D THz images of (a) a semiconductor integrated circuit using the transmission magnitude integrated over the 1–3 THz range [67] and (b) a chocolate bar using the transmission magnitude (top) and the transit time (bottom) [68] using the THz-TDS system

space as reported in [20]. The real parts of the extracted surface conductivities of SWCNT and graphene thin films are plotted in Figure 3.27. The extracted graphene surface conductivity at THz frequencies is comparable to the reported values at other frequency regimes, such as DC and optical range. THz imaging is another common application of THz-TDS. An example of semiconductor integrated circuit [67] inspection is shown in Figure 3.28(a). The sample is 2D scanned and the Fourier transformed transmission magnitude through the materials, integrated over a 1–3 THz frequency range is taken to form the image. The plastic package and the metallic traces are fairly distinguishable as shown in Figure 3.28(a) with a rough spatial resolution of 250 mm achieved. The THz images of a chocolate bar in Figure 3.28(b) demonstrate that not only the transmission magnitude (top) but also the phase information (bottom) contained in the THz signals can be used to form the 2D image [68]. The THz-TDS systems have also been explored for medical applications [39, 69, 70], sometimes in the reflection configuration as well.

THz photoconductive antennas

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3.3.2 THz near-field microscopy 3.3.2.1 Motivation The far-field THz-TDS system is demonstrated as a useful tool for various applications as mentioned in the previous section. The traditional far-field THz imaging system, in which the sample under test is in the far-field region of the THz radiation and normally at the focal plane of the THz beam, is hindered by the diffraction limit such that the resolution of the imaging system is on the order of wavelength (e.g., 300 mm at 1 THz). However, a near-field system can overcome this disadvantage to achieve sub-wavelength spatial resolution, which facilitates the detection/imaging of sub-mm or even sub-mm features such as biomolecules inside cells. In contrast to the far-field setup, the sample under test is normally placed in the near-field region of the THz PCA, for example, on the order of microns. Several works on THz near-field imaging, such as dynamic aperture, tapered metal tip and EO near-field emission microscopy [71]–[73], have been done previously to investigate the system properties. Meanwhile, most of the reported THz near-field imaging systems utilize the detection mode [53, 54, 74], in which the sample under test is in close proximity to the THz detector. Emission mode using a single PCA for near-field imaging is reported in [75]. Some of the previously reported configurations are shown in Figure 3.29, where Figure 3.29(a) is the THzTDS setup incorporating a near-field tapered metal tip in contact with the sample and Figure 3.29(b) shows the near-field probe with a small thin film aperture configured in the detection mode. Differently, a PCA array structure can be adapted in the emission mode and near-field configuration with each antenna in the array individually biased. The polarization dependent scanning properties of this THz near-field system are experimentally characterized to reach sub-wavelength spatial resolution. A comprehensive presentation of the available methods to reconstruct the near-field image is given. Several applications utilizing the THz near-field setup are demonstrated at the end of this section as well.

3.3.2.2 THz near-field system setup The schematic of the THz near-field imaging setup in the emission mode is shown in Figure 3.30(a). As can be seen, there is no hyper-hemispherical Si lens attached to the 100 fs

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Figure 3.29 (a) Experimental setup for a tapered metal tip based near-field microscope [72] and (b) schematic of a near-field probe with a small aperture configured in the collection mode [74]

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Developments in antenna analysis and design, volume 2 Beam splitter

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Lock-in detection

(a)

(b)

Antenna array

(c)

Figure 3.30 (a) General simplified schematic of the near-field system setup and (b) microscope image of the central region of the 2 2 emitter array and (c) the schematic showing the relative distance between the critical components (not in scale) emitter compared to the far-field THz-TDS setup because the laser is illuminating the emitter from the backside of the substrate. A 2 2 array (with a pitch size of 500 mm) is used in our preliminary study. Each antenna in the emitter array is biased with a modulated signal instead of constant DC voltage and controlled individually by a series of solid-state relays. The emitter antenna array is fabricated on the opticaltransparent GoS substrate. Each antenna is wire bonded to separated pads on a printed circuit board to provide individual biasing. Only one antenna is drawn in Figure 3.30 (a) for simplicity. A micro-lens array is used in the optical path to split the pump laser beam to excite the photoconductive gap of each antenna in the array. The microscope image of the 2 2 antenna array is shown in Figure 3.30(b), and the schematic showing the relative distance relationship between different components is shown in Figure 3.30(c), with the distance between the micro-lens array and emitter array on the order of mm and the distance between the emitter array and sample on the order of mm.

3.3.2.3

System performance characterization

The near-field system performance is generally characterized in terms of the spatial resolution. Two orthogonal polarizations are experimentally characterized as the vertical polarization; when the edge boundary of the sample is parallel to the radiation polarization of the THz antenna, it is expected to have a few times better resolution than the horizontal polarization as pointed out in [75, 76]. The samples used for the 1D spatial resolution characterization include two structures with a thin metallic film layer partially covering a dielectric substrate, providing a straight metallic-dielectric boundary. The samples are mounted on a 3D automatic moving stage for raster scan. Comparisons of the 2D time-space maps between simulation and experiment (with only single antenna in the array turned on for simplicity) for both scanning polarizations are shown in Figures 3.31 and 3.33 with a scanning step of 20 mm. Specifically, one vertical line in these 2D maps represents a time-domain pulsed signal at a certain scanning position. Reasonable agreement between the simulation and experiment is observed in both figures. Representative normalized time-domain waveforms corresponding to the same position for simulation and experiment for the horizontal scanning case are plotted in Figure 3.32 which shows bipolar feature. It needs to be pointed out that the simulation does not take into account the detector

0 1 2 3 4 5 6 7 8 9 10

(a)

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8 6 4

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Time delay (ps)

THz photoconductive antennas

2 0 –2 –4 –6 –8 0

10

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0.12 0.1 0.08 0.06 0.04 0.02 0 –0.02 –0.04 –0.06 –0.08 1,000 2,000 3,000 4,000 5,000 6,000

0

(b)

Position (μm)

113

Position (μm)

Figure 3.31 2D time-space maps for the horizontal polarization scanning case. (a) Far-field E-field in HFSS simulation at normal direction (q ¼ 0 and f ¼ 0 ) and (b) the experimental received THz signal at the farfield detector side (the color bars of the two figures are with arbitrary unit). Each dot in the figures stands for the THz signal at a certain scanning position and a fixed time delay

Time domain signal (a.u.)

1

Simulation Measurement

Maximum

0.5

0 Maximum position –0.5

Zero crossing position Minimum Minimum position

–1 0

2

4

6

8

10

Time delay (ps)

Figure 3.32 Representative time-domain pulsed signals for simulation and measurement at a certain scanning position effect and the photoconductive material properties used in the simulation model might be different from the experimental case, so that the pulse features of the THz signal in the simulation are not exactly the same as those in the experiment. Two boundaries located around 1,500 and 4,500 mm for the horizontal scanning case and around 400 and 1,200 mm for the vertical scanning case are observed in the 2D maps, which are due to the transmission property changes at the air-to-dielectric interface and dielectric-to-metal interface, respectively. The 1D scanning curves, determined by different criteria based on plotting different time-domain signal values versus scanning positions, can be derived from those 2D time-space maps. A commonly used method in the

Developments in antenna analysis and design, volume 2 0 1 2 3 4 5 6 7 8 9 10

0 1 2 3 4 5 6 7 8 9

8 6 4

Time delay (ps)

Time delay (ps)

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2 0 –2 –4 –6 0

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Position (μm)

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0.1 0.08 0.06 0.04 0.02 0 –0.02 –0.04 –0.04 0

500

1,000

1,500

2,000

Position (μm)

(b)

Figure 3.33 2D time-space maps for the vertical polarization scanning case. (a) Far-field E-field in HFSS simulation at normal direction (q ¼ 0 and f ¼ 0 ) and (b) the experimental received THz signal at the far-field detector side (the color bars of the two figures are with arbitrary unit). Each dot in the figures stands for the THz signal at a certain scanning position and a fixed time delay 1

0.5 0 –0.5 t = 6.2 ps t = 6.4 ps t = 6.6 ps t = 6.8 ps

–1 –1.5 0

(a)

Time domain signal @ fixed delay (a.u.)

Time domain signal @ fixed delay (a.u.)

1

0.6 0.4 0.2 0 –0.2 –0.4 –0.6

1,000 2,000 3,000 4,000 5,000 6,000 Position (μm)

t = 6.4 ps t = 6.6 ps t = 6.8 ps t = 7.0 ps

0.8

0 (b)

400

800

1,200

1,600 2,000

Position (μm)

Figure 3.34 Scanning curves based on normalized experimental THz time-domain signals at several different time delay positions for (a) the horizontal scanning case and (b) the vertical scanning case literature is to choose the time-domain THz signal at a fixed time delay to form the scanning curve [53, 77]. This method is applied to characterize the experimentally obtained spatial resolution first for both polarizations, as shown in Figure 3.34. Several different time-domain delays around the time-domain peak position when the antenna is facing the dielectric region (for example, around t ¼ 6.8 ps for the experimental case) are chosen and the obtained scanning curves are normalized to the corresponding maximum values. The curves clearly define three distinguishable regions, namely, air, dielectric and metal. The spatial resolution is then defined based on the 10–90% criterion by examining the dielectric–metal boundary. Other than the above used method, different methods such as the time-domain maximum signal, minimum signal and peak-to-peak value, different time delay

THz photoconductive antennas 1

0.8 0.6 0.4 0.2 Maximum signal Minimum signal Peak-to-Peak

0 –0.2

Maximum signal Minimum signal Peak-to-Peak

0.8 0.6 0.4 0.2 0 –0.2

0 (a)

Time domain signal (a.u.)

Time domain signal (a.u.)

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115

1,000 2,000 3,000 4,000 5,000 6,000 Position (μm)

0

400

(b)

800

1,200

1,600

2,000

Position (μm)

Figure 3.35 Scanning curves based on normalized experimental THz time-domain maximum signal, minimum signal and peak-to-peak value along different scanning regions for (a) the horizontal scanning case and (b) the vertical scanning case Table 3.3 Spatial resolution characterized in different ways for the horizontal scanning polarization case

Magnitude at time delay of 6.8 ps Magnitude at time delay of 6.6 ps Magnitude at time delay of 6.4 ps Magnitude at time delay of 6.2 ps Time-domain maximum signal Time-domain minimum signal Time-domain peak-to-peak value

Resolution (mm)

Resolution (l)

360 260 200 140 500 400 760

0.476 0.344 0.265 0.185 0.661 0.529 1.005

positions (e.g., maximum position, minimum position and zero crossing position) as labeled in Figure 3.32, and frequency-domain magnitude and phase have also been reported for far-field THz imaging applications in [68, 78, 79]. They could also be applied to characterize the near-field imaging properties, such as the 1D scanning curves plotted in Figure 3.35 using the time-domain signals and reported in [76] using the frequency-domain magnitude. The characterized spatial resolution values defined based on the 10–90% criterion for both polarizations are summarized in Tables 3.3 and 3.4, where the corresponding resolutions with respect to wavelength are calculated based on the peak frequency of 0.397 THz. As can be seen from the summarized table, a spatial resolution of about 100 mm (0.132l) is achieved with this THz near-field system. With the aid of other techniques such as small aperture (e.g., aperture size of 5 mm), sub-10-mm spatial resolution has been demonstrated under the collection mode [74]. However, since no Si lens is adapted for the THz emitter under this near-field configuration as can be seen from Figure 3.30, lower SNR might be an issue deteriorating the system performance. With an emitter array configuration and

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Developments in antenna analysis and design, volume 2 Table 3.4 Spatial resolution characterized in different ways for the vertical scanning polarization case

Magnitude at a time delay of 7.0 Magnitude at a time delay of 6.8 Magnitude at a time delay of 6.6 Magnitude at a time delay of 6.4 Time-domain maximum signal Time-domain minimum signal Time-domain peak-to-peak value

H=

1 1 1 1

1 –1 1 –1

ps ps ps ps

1 –1 –1 1

1 1 –1 –1

Resolution (mm)

Resolution (l)

240 180 160 140 160 100 140

0.317 0.238 0.212 0.185 0.212 0.132 0.185

+1 –1

Figure 3.36 Hadamard matrix for a 22 array antenna and the modulated biasing signals represented by þ1 and –1 in the Hadamard matrix incorporating the Hadamard multiplexing method, the SNR of the THz near-field system can be improved. For a 2 2 array configuration, a 4 4 Hadamard matrix is incorporated to modulate the bias voltage of each antenna. The original 4 4 Hadamard matrix and the bias-voltage signals represented by ‘‘þ1’’ and ‘‘–1’’ in the matrix are shown in Figure 3.36. The two different bias-voltage signals have a phase difference of 180 . Each row in the matrix corresponds to one single configured measurement with four antennas biased independently. By defining the measurement output hi(i ¼ 1, 2, 3, 4) and the single antenna response yi, the single antenna measurement and the Hadamard matrix encoded measurement can be represented by the following two equations, respectively: Single antenna case: 2 6 6 6 4

h1 h2 h3

3

2

7 6 7 6 7¼6 5 4

h4

y1 y2 y3

3

2

7 6 7 6 7þ6 5 4

y4

e1 e2 e3

3 7 7 7 5

(3.44)

e4

Hadamard matrix encoded case: 2

h1

3

2

1

6h 7 61 6 27 6 6 7¼6 4 h3 5 4 1 1 h4

1 1 1 1

1

1

32

y1

6 1 1 7 76 y2 76 1 1 54 y3 1 1 y4

3

2

e1

3

7 6e 7 7 6 27 7þ6 7 5 4 e3 5 e4

(3.45)

THz photoconductive antennas

0.04 0.02 0 –0.02 –0.04 –0.06

(a)

A B C D

0

2

4

6

Time delay (ps)

8

0.06 Time domain signal (a.u.)

Time domain signal (a.u.)

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117

0.02 0 –0.02 –0.04 –0.06

10

A B C D

0.04

0

2

(b)

4 6 Time delay (ps)

8

10

Figure 3.37 The individually measured (a) and the decoded (b) time-domain responses of the four single antennas where ei is the additional noise. The individual antenna response yi for the Hadamard matrix encoded case can be easily obtained based on the combination of the four measured hi. A detailed theoretical analysis of the SNR improvement can be found in [80], which estimates the SNR improvement to be 2 for a 4 4 matrix case. An experimental verification is illustrated with each emitter in the array illuminated by 5 mW laser beam and biased with 10 V modulated at 333 Hz and the detector illuminated by 6 mW laser beam. It is seen that the decoded single antenna responses from the Hadamard multiplexing method as shown in Figure 3.37(b) are less noisy compared to those directly measured ones as shown in Figure 3.37(a) with the same amount of measurement time for both cases. The system SNR improvement is further validated to be 2 for all four individual antennas in the array by comparing the standard deviations based on five repeated measurements of both scenarios, which agrees well with the theoretical prediction.

3.3.2.4 Applications of THz near-field system With the sub-wavelength spatial resolution property, the THz near-field spectroscopy system provides the capability of detecting small-size features. 2D images of two samples with metallic patterns (one is the previous studied butterfly-shaped antenna and the other is made of several square-shaped patterns with different sizes on the substrate) are shown in Figure 3.38. Both images shown here are obtained by choosing the time-domain signal at a fixed time delay position for each pixel. The squared structure with a size of 150 mm in the sample can be easily distinguished, which roughly agrees with the previously characterized spatial resolution result (100 mm). These obtained images can be also quantitatively assessed in terms of the product-moment correlation coefficient r [81] as represented by X X

ðAmn AÞðBmn BÞ m n ﬃ r ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ X X X X ð m n ðAmn AÞ2 Þð m n ðBmn BÞ2 Þ

(3.46)

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200

1

600

1

0.8

500

0.8

400

400

0.6

0.6 300

600

0.4

800

0.2

1,000

0

200

400

600

800 1,000

0.2

100 0

0

(a)

0.4

200

0 100 200 300 400 500 600 700

0

(b)

Figure 3.38 2D THz near-field images of (a) the butterfly-shape antenna and (b) the square patterns with different sizes, the units on the label are mm Table 3.5 Correlation coefficient for assessing the reconstructed image quality Correlation coefficient Butterfly shape antenna Square patterns with different sizes

0.703 0.822

where Amn and Bmn are the signals at each scanning pixel position for the reconstructed image and the original image, A and B are the corresponding averages of the whole scanning area. The correlation coefficient has a range of [0 1], where 1 is the best possible value if the reconstructed image is identical to the original image, which is constructed by defining the signal for the dielectric area to be 1 and that for the metallic area to be 0. The calculated correlation coefficients for both cases are given in Table 3.5, which show reasonably good image quality. Other 2D image demonstrations have been reported in the literature as well, using the THz near-field spectroscopic technique. For example, in [54], the images of a small portion of leaf are formed using the single frequency transmission magnitude and phase at 1.36 THz as plotted in Figure 3.39(a), where the leaf sample is placed in direct proximity to the dipole-shape THz detector (with 5 mm dipole gap) fabricated on the RD-SoS substrate. The leaf is scanned with a spatial step of 50 mm. Another example demonstrated in [72] is based on a tapered metal tip which is in contact with the sample under test as seen in Figure 3.29(a). The near-field image shown in Figure 3.39(b) is obtained by integration in the frequency domain from 0.6 to 2.3 THz with a corresponding average wavelength of 220 mm. The resolution with respect to wavelength for this setup is therefore l/4.4.

THz photoconductive antennas λ = 220 μm

y

THz signal (a.u.) 95.8 – 100 91.5 – 95.8 87.3 – 91.5 83.0 – 87.3 78.8 – 83.0 74.5 – 78.8 70.3 – 74.5 66.0 – 70.3 61.7 – 66.0 57.5 – 61.7 53.3 – 57.5 49.0 – 53.3 44.8 – 49.0 40.5 – 44.8 36.3 – 40.5 32.0 – 36.3

4

ETHz amplitude

phase

y (mm)

x

3 2

119

50 μm

1 1 mm

0 f = 1.36 THz

(a)

(b)

1

2

3

4

x (mm)

Figure 3.39 (a) THz near-field images (represented by THz amplitude and phase at 1.36 THz) of a small portion of the plant leaf (5 mm 5 mm) that was raster scanned in front of the stationary near-field detector [54] and (b) THz near-field image of a test pattern taken with an elliptical aperture in contact with the sample [72]

3.4 Summary In this chapter, the theoretical modeling, numerical simulation and experimental study of PCA are thoroughly discussed. The principles of three representative models, namely, Drude-Lorentz model, equivalent circuit model and full-wave model, were introduced. After summarizing the pros and cons of each model, the full-wave model was chosen for numerical simulation of PCA, as it has least physical assumptions and thus should be the most accurate. The numerical simulation was carried out using in-house codes on the MATLAB platform, which was also verified using commercial software. The radiation properties of a PCA were then thoroughly studied by varying several important parameters, such as laser power, bias voltage, photoconductor material properties and laser pulse width. To demonstrate the application of this model, two new PCAs were designed and simulated, and enhanced THz radiations were predicted for both. The influences of the photoconductive material, antenna structures, etc. on the THz radiation power and bandwidth are systematically investigated to gain a more comprehensive understanding of a PCA. The general radiation mechanism of the PCAs is further studied by implementing the polarization effect and cancellation effect measurements. Recent progresses of the PCA structure development using nanostructure and plasmonic antenna electrodes to improve the THz radiation power/ efficiency are briefly reviewed. In addition, the THz near-field spectroscopic technique based on PCAs is proposed to overcome the resolution limit and achieve sub-wavelength resolution. Specifically, incorporating the Hadamard multiplexing method with an emitter array for the THz near-field configuration, the system SNR is enhanced, agreeing well with theoretical prediction. With more array elements, the system SNR

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can be further improved. Various THz applications are explored utilizing the far-field and near-field THz-TDS setup, including material characterization, imaging and sensing. With the advancement of THz far-field and near-field systems incorporating PCAs in recent years, more innovative and practical THz applications will be enabled.

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[46] J. E. Bjarnason, T. L. J. Chan, A. W. M. Lee, et al., ‘ErAs:GaAs photomixer with two-decade tunability and 12 mW peak output power,’ Appl. Phys. Lett., vol. 85, pp. 3983–3985, Nov. 2004. [47] C. Kadow, J. A. Johnson, K. Kolstad, J. P. Ibbetson, and A. C. Gossard, ‘Growth and microstructure of self-assembled ErAs islands in GaAs,’ J. Vac. Sci. Technol. B, vol. 18, pp. 2197–2203, Aug. 2000. [48] http://batop.com/products/terahertz/photoconductive-antenna/photoconductive-antenna-800nm.html [49] E. Sano and T. Shibata, ‘Mechanism of subpicosecond electrical pulse generation by asymmetric excitation,’ Appl. Phys. Lett., vol. 55, pp. 2748–2750, Dec. 1989. [50] U. D. Keil and D. R. Dykaar, ‘Ultrafast pulse generation in photoconductive switches,’ IEEE J. Quantum Electron., vol. 32, pp. 1664–1671, Sep. 1996. [51] M. B. Ketchen, D. Grischkowsky, T. C. Chen, et al., ‘Generation of subpicosecond electrical pulses on coplanar transmission lines,’ Appl. Phys. Lett., vol. 48, pp. 751–753, Mar. 1986. [52] F. E. Doany, D. Grischkowsky, and C.-C. Chi, ‘Carrier lifetime versus ion-implantation dose in silicon on sapphire,’ Appl. Phys. Lett., vol. 50, pp. 460–462, Feb. 1987. [53] O. Mitrofanov, I. Brener, R. Harel, et al., ‘TeraHertz near-field microscopy based on a collection mode detector,’ Appl. Phys. Lett., vol. 77, pp. 3496–3498, Nov. 2000. [54] A. Bitzer, A. Ortner, and M. Walther, ‘Terahertz near-field microscopy with subwavelength spatial resolution based on photoconductive antennas,’ Appl. Opt., vol. 49, pp. E1–E6, Mar. 2010. [55] M. Tuo, M. Liang, J. Zhang, and H. Xin, ‘Time-domain THz near-field imaging incorporating Hadamard multiplexing method,’ 41st International Conference on Infrared, Millimeter, and Terahertz waves (IRMMW-THz), Sep. 25–30, 2016, IEEE, Copenhagen, Denmark. [56] H.-M. Heiliger, M. Vossebu¨rger, H. G. Roskos, H. Kurz, R. Hey, and K. Ploog, ‘Application of liftoff low-temperature-grown GaAs on transparent substrates for THz signal generation,’ Appl. Phys. Lett., vol. 69, pp. 2903–2905, Nov. 1996. [57] C. Kadow, S. B. Fleischer, J. P. Ibbetson, et al., ‘Self-assembled ErAs islands in GaAs: Growth and subpicosecond carrier dynamics,’ Appl. Phys. Lett., vol. 75, pp. 3548–3550, Nov. 1999. [58] J. F. O’Hara, J. M. Zide, A. C. Gossard, A. J. Taylor, and R. D. Averitt, ‘Enhanced terahertz detection via ErAs:GaAs nanoisland superlattices,’ Appl. Phys. Lett., vol. 88, p. 251119, Jun. 2006. [59] https://www.ekspla.com/product/thz-emitter-and-detector-for-800-nmwavelength-lasers [60] P. J. Hale, J. Madeo, C. Chin, et al., ‘20 THz broadband generation using semi-insulating GaAs interdigitated photoconductive antennas,’ Opt. Express., vol. 22, pp. 26358–26364, Oct. 2014.

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Chapter 4

Optical antennas Chengjun Zou1,2, Withawat Withayachumnankul2, Isabelle Staude1, and Christophe Fumeaux2

4.1 Introduction The function of an antenna is to couple electromagnetic waves propagating in free space with a localised spot in a feed region. This allows conversion to and from guided waves propagating along a transmission line for further processing. One practical way of achieving efficient electromagnetic power transfer between freespace and guided modes is to exploit resonant structures. This is exemplified by classical antenna geometries such as half-wave dipoles or slots, resonant loops, microstrip antennas or dielectric resonator antennas (DRAs). The scalable nature of the time-harmonic Maxwells equations evokes a formalism where all structure dimensions are expressed in relation to the wavelength. Scaling the size of any antenna for operation at a different frequency then becomes straightforward. This suggests an intriguing prospect of creating scaled resonant antennas at any frequency, including nanoscale structures in the optical range. Such a concept of ‘‘optical antennas’’ (also called ‘‘nanoantennas’’) has been conjectured many decades ago and became a reality in the last two decades. ‘‘Antennas for Light’’ [1] have even become a focus of intensive research in nanophotonics in recent years. There are physical and technological challenges that limit the feasibility of optical antennas and preclude a straightforward scaling from their radio-frequency counterparts to the optical regime: ●

●

1 2

The main physical challenges arise from the widely different material properties of metals in the optical regime, as compared to a near-perfect conductor behaviour at radio frequencies. This is best formalised by the plasmonic description of light–metal interactions. The main technological challenges are related to the fabrication of resonant structures at a resolution appropriate to meet resonance conditions. In view of optical wavelengths, precise nanometre-scale fabrication techniques are required for the realisation of optical antennas.

Institute of Applied Physics, Abbe Center of Photonics, Friedrich Schiller University Jena, Germany School of Electrical and Electronic Engineering, The University of Adelaide, Australia

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The scientific developments of the last few decades have progressively eased the difficulties in tackling those challenges. On the one hand, tremendous advances have been made in the physical understanding of plasmonic structures. This progress has been supported by sophisticated electromagnetic numerical solvers running on increasingly powerful computers, which allow today an accurate predictive modelling of optical nanostructures with high complexity. On the other hand, the rapid advances in material sciences and nanofabrication techniques have made practical implementations of precisely patterned nanoantennas a reality in many laboratories worldwide. This has opened new perspectives in the practical applications of optical nanostructures. A prominent functionality of optical antennas is the production of a localised field enhancement. This can be exploited to increase light–matter interaction for sensing or for enhancement of spontaneous emission from fluorescent molecules. Another class of applications considers resonant scattering in nanoantenna arrays with the aim of collectively manipulating optical beams. It can then be argued whether the term antenna is appropriate to characterise these various modes of operation since strictly speaking they usually do not involve transformation from free-space waves to guided modes. Nevertheless, resonant nanostructures, either metallic or dielectric, often take inspiration from corresponding radiofrequency geometries or array configurations. Additionally, they either do produce a localised field enhancement in a well-defined ‘‘feed’’ location or act similarly as reflectarray or transmitarray elements. Therefore, despite obvious differences to radio-frequency devices, optical antennas can be positioned at a crossroad between photonics technology and microwave antenna engineering [2]. As such, their development can benefit from both perspectives. The aim of this chapter is to introduce the fundamental concepts relevant to the understanding of optical antennas and provide a (non-exhaustive) review of recent developments. The authors hope that descriptions in the following sections will be accessible to a large audience in both antenna engineering and photonics communities, who share this scientific research field despite using different terminologies and approaches.

4.2 Early history The concept of engineered optical antennas can be arguably traced back to the end of the 1960s, starting with the observation that metal whiskers used for mixing midinfrared lasers [3] had a polarisation-dependent response. This effect was related to non-resonant antenna theory [4]. Following these conceptual discoveries, attempts to integrate antennas with infrared detectors in planar technology were initiated in the 1970s [5]. However, these initiatives were with severe limitations arising from the available lithographic techniques at the time, which restricted feature sizes to a few micrometres. Progress in planar infrared antenna-coupled detectors resumed in the 1990s, notably in groups at the National Institute of Standards and Technology (NIST) which demonstrated an infrared spiral antenna [6], and at ETH Zurich with various infrared antenna geometries demonstrated in metrology applications [7,8].

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Those attempts mostly targeted infrared applications, but on their basis, the polarised response of tiny lithographic antennas to visible light was then demonstrated at the University of Central Florida in 1999 [9]. In parallel to the activities on infrared antennas, a strong physical correspondence between near-field optical devices (such as the scanning near-field optical microscope – SNOM [10]) and microwave antennas was also noted in the 1990s [11]. All these attempts mostly remained anecdotal until around 2005, where strong optical field enhancements in subwavelength structures were experimentally demonstrated by several research groups (e.g. [12]). Since then, optical antennas have become an increasingly active field of research in physical optics – both theoretically and experimentally. In the last decade, there has been impressive progress both in fabrication techniques and in the understanding of interactions between light and resonant optical nanostructures. Among the various geometries that have been conceptualised, plasmonic structures such as nano-spheres [13], nano-shells [14], Yagi-Uda antennas [15] or reflectarrays [16] can be mentioned. The following sections will review the underlying concepts and applications of optical antennas.

4.3 Theory and analysis The theoretical fundamentals of optical antennas are discussed in this section. Most of the optical antennas are realised based on nanostructures made of metals, which, at optical frequencies, can no longer be approximated as perfect electric conductors (PECs). Thus, in Section 4.3.1, the electromagnetic properties of metals from microwave to optical frequencies are discussed. At optical frequencies, plasmonic effects dominate the electromagnetic responses of metals and play an important role for various applications of optical antennas. Therefore, the basics of plasmonics are introduced in Section 4.3.2. Section 4.3.3 focuses on the Mie theory, which gives rigorous analytical solutions to optical excitations of both metallic and dielectric subwavelength resonators. Recently, Mie-type nanoscale dielectric resonators (DRs) have attracted increasing research interest owing to their high-efficiency and multiple resonance modes compared with plasmonic resonators, and thus a short comparison between DRs and plasmonic resonators is presented in Section 4.3.4.

4.3.1 Metal properties from microwave to optical frequencies Various forms of metallic antennas have been conceived and implemented at microwave frequencies, where metals can be considered in a first approximation as PECs or as good conductors with a finite skin depth. In this frequency range, free electrons closely follow the oscillation of electromagnetic waves, resulting in a rapid exponential decay of the field into metals, and thus the induced currents are restricted to the metal surface. In this case, the skin-depth of a good conductor is classically expressed as sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 d¼ (4.1) wm0 mr s0

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and the corresponding surface impedance is referred to as the first-order Leontovich boundary condition [17]: rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ jwm0 mr ZS ¼ s0

(4.2)

where m0 is the permeability of free space, s0 is the DC conductivity of the metal and mr is the relative permeability of the metal. This boundary condition is appropriate for a sufficiently thick conductor with an extended planar surface. The assumption of a good conductor is generally considered valid up to around 100 GHz. Higher up in frequencies in the terahertz range, between 100 GHz and 10 THz, the electric field penetrates to a greater depth into metals relative to the wavelength. This results in an increased Ohmic loss, which can be incorporated into the surface impedance model in (4.2) as [18]: sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ jwm0 mr ZS ðwÞ ¼ s0 ðwÞ þ jwe0

(4.3)

In this case, the surface impedance is usually expressed in the frequency-dependent form. Equation (4.3) is generally accurate at terahertz frequencies and up to the mid-infrared range. As the frequency approaches optical and visible frequencies, metals can no longer be considered as good conductors. The electron mass causes substantial damping of incident electromagnetic waves. As a result, metals become lossy, an effect widely known in the context of plasmonics [19]. The interaction between optical electromagnetic waves and metals can be described in the classical framework of Maxwell’s equations, and Drude-based dielectric functions are commonly used to describe the electromagnetic responses of metals [20]. In the Drude model, metals are characterised as a cloud of free electrons floating around fixed positive nuclei. Accordingly, the Drude-based complex permittivity e(w) can be expressed based on the time factor e jwt as [20]: em ¼ e1

w2p w2 jgw

(4.4)

where w2p ¼ Ne2 =me0 is the square of the plasma frequency, with N and e being the carrier concentration and the elementary charge, and m being the effective mass of an electron. The variable g ¼ 1/t denotes the collision frequency, with t being the relaxation time, which is needed to describe the damping process via collisions. An example of Drude-based dielectric function according to (4.4) is illustrated in Figure 4.1(a) for silver at optical frequencies [21]. The measured data by Johnson and Christy [22] are shown for comparison, demonstrating the high accuracy of the Drude model over a large frequency range in this case up to 800 THz.

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(a)

(b)

(c)

Figure 4.1 Drude-based optical properties of silver. (a) Dielectric function of silver based on the Drude model [21] and measured data (denoted by subscript J) from Johnson and Christy [22]. (b-c) Corresponding conductivity and wave impedance calculated from the modelled and measured data in (a) At optical frequencies, the conductivity and wave impedance of metal can be calculated from the complex permittivity as sðwÞ ¼ jwe0 ½eðwÞ 1 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ jwm0 mr Z ðw Þ ¼ sðwÞ þ je0 eðwÞ

(4.5) (4.6)

The results for silver are presented in Figure 4.1(b) and 4.1(c). The conductivity of silver at microwave or even lower frequencies is in the order of 107 S/m with a negligible imaginary part. However, as shown in Figure 4.1(b), the conductivity of silver is significantly reduced at optical frequencies, while the imaginary part becomes nearly one order of magnitude larger than the real part. The results demonstrate the

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poor conductivity and lossy features of metal at optical frequencies. The wave impedance of metal remains small, corresponding to the high reflectivity of metals at optical frequencies. However, the value of the wave impedance is much higher than that at microwave frequencies, and the imaginary part becomes significantly larger than the real part. Thus, we can infer that the reflection phase is no longer equal to the p phase due to the increased loss. It should be noted that the Drude-based model does not account for interband transitions of electrons. For this reason, a significant discrepancy between the Drude-based results and the measured data are seen in Figure 4.1(b) and 4.1(c) at the frequency larger than 800 THz. Relevant discussions regarding interband transitions are covered in solid-state physics books [20].

4.3.2

Plasmonic effects

Plasmonic effects arise from the imperfect metal electrical properties at optical frequencies, as discussed in Section 4.3.1. They play an important role in the design and operation of optical antennas [19]. In this subsection, an introduction to the fundamentals of plasmonics is presented. Our discussion focuses on two types of plasmonic effects: surface plasmon polaritons (SPPs) and localised surface plasmon resonances (LSPRs). Both aspects provide an important basis towards further discussion on optical antennas for field enhancement, spectroscopy and integrated photonics. As shown in Figure 4.2(a), SPPs are free electrons collectively coupled with electromagnetic waves propagating along a metal–dielectric interface [19]. Such a surface electromagnetic mode can only be excited by transverse magnetic polarised incident waves. The SPP wavenumber kspp strongly depends on the optical properties of the materials making up the interface and is calculated as [19]:

kspp

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ed em ðwÞ ¼ k0 ed þ em

(4.7)

where ed and em (w) are the relative permittivities of the dielectric and the metal at the two sides of the interface, and k0 is the free-space wavenumber. Figure 4.2(b)

ω ω

εd inc

y

x

spp

(a)

(b)

(c)

Figure 4.2 Concepts of plasmonics. (a) Illustration of SPPs at a metal-dielectric interface. (b) Dispersion diagram of SPPs. (c) Illustration of LSPR on a metallic nanosphere. Figure reproduced from [23] with permission from Nature Publishing Group

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conceptually presents a typical SPP dispersion curve at a metal–dielectric interface. As shown, an important property of SPPs deduced from (4.7) in combination with the Drude model is that at higher frequencies the SPP dispersion curve diverges from the light line of vacuum. In other words, as the frequency increases, the SPP wavelength becomes increasingly shorter than the corresponding free-space wavelength. Such dispersion properties of SPPs lead to a strong field confinement at metal surfaces. It is also because of this phase mismatch that SPPs cannot be directly excited by free-space electromagnetic waves without dedicated coupling structures to compensate the momentum mismatch. Prisms and grating couplers are two conventional coupling techniques [19]. A prism with a relative permittivity er can be used to excite SPPs when kspp ¼ er k0 sin q. The incident angle q is chosen to be equal to the critical angle, so as to ensure a maximum power transfer. For grating coupling, SPPs can be excited whenever the momentum condition [19]: kspp ¼ k0 sin q þ m

2p a

(4.8)

is fulfilled. Here, m denotes the diffraction order and a is the grating period. Differing from SPPs, LSPRs describe the collective oscillation of surface conduction electrons bound to metallic particles when driven by an external electric field at optical frequencies [19]. Metallic particles that support LSPRs are usually much smaller than the excitation wavelength. The small size of the metallic particles provides extra momentum so that LSPRs can be directly excited by free-space electromagnetic waves [24]. Figure 4.2(c) illustrates a subwavelength metallic particle with radius a supporting a dipolar LSPR in response to an external electric field. To analyse the scattering and absorption of this particle, we can consider Laplace’s equation under a quasi-static assumption. This assumption is valid only when the induced electric field across the small particle can be considered spatially invariant. Consequently, the scattering cross section and absorption cross section can be expressed as [19]: Cscat ¼

8p 4 6 em ðwÞ ed 2 k04 2 k0 a jaj ¼ 3 em ðwÞ þ 2ed 6p

Cabs ¼ 4pk0 a3 Im

)

em ðwÞ ed ¼ k0 ImðaÞ em ðwÞ þ 2ed

Cscat / )

a6 ; l4

Cabs /

a3 l

(4.9)

(4.10)

Here, the term a ¼ 4pa3

em ðwÞ ed em ðwÞ þ 2ed

(4.11)

is known as the dipolar polarisability of this particle. Clearly, Cscat and Cabs reach their maxima on resonance, if the polarisability reaches its pole. This occurs when Re½em ðwÞ ¼ 2ed

(4.12)

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where em (w) is the frequency-dependent permittivity of the metallic particle and ed is the relative permittivity of the surrounding dielectric environment. Equation (4.12) is known as the Fro¨hlich condition [19], and the corresponding resonance is the dipolar LSPR on a subwavelength metallic particle. It is emphasised here that the dipolar LSPR is highly dependent on the optical properties of the metal and its dielectric surroundings, rather than the excitation wavelength or the particle size. Particles supporting LSPRs demonstrate strong far-field radiation, local field enhancement and high Ohmic loss. In this way, they can be considered the simplest form of optical antennas. The radiation of such a particle in the near-field and farfield is similar to an infinitesimal electric dipole antenna.

4.3.3

Mie resonances in nanoscale resonators

The basics of plasmonics have been introduced in Section 4.3.2, and it has been discussed that a dipolar LSPR satisfying the Fro¨hlich condition provides strong localised fields and radiates as an infinitesimal dipole antenna. However, this quasistatic analysis is not valid for larger subwavelength particles, where the spatial phase variation can no longer be neglected. To analyse the resonance in a particle of an arbitrary subwavelength size, Mie theory can be applied to provide rigorous solutions of light scattering and absorption. The theory was named after Gustav Mie, who, although not the first to develop the formalism, used it to describe colourful effects in colloidal gold solutions [25]. In general, the central idea of Mie theory is multi-mode expansion, and this method has been successfully used for nano-particles of various shapes. Since the calculations are demanding, many algorithms have been developed to obtain simple computer implementations [26–28]. To understand Mie theory, here we discuss light scattering by a spherical subwavelength particle under a certain excitation condition, such as a plane wave or a Gaussian beam. Such a scenario has been treated rigorously in many textbooks [29,30]. The full mathematical description requires knowledge of special functions such as Bessel functions and associated Legendre function. Our aim here however is to explain, without an extensive mathematical presentation, the general idea of Mie resonances together with its implications for optical antennas applications. Let us consider a plane wave exciting a small sphere with a radius a and refractive index ns embedded in a medium with index of nm. Here, refractive indices are used, rather than permittivities to obtain simpler closed-form expressions. In spherical coordinates, the incident plane wave can be expanded into the linear superposition of vector spherical harmonics, which are linear and demonstrate even and odd properties. The linear property suggests that the whole analysis can be done as multimode expansion, while the parity leads to electric and magnetic resonance modes in the small sphere. Each vector spherical harmonic excites its corresponding normal mode inside the sphere. Therefore, according to the linear property, the scattered field and the internal field of the sphere can be calculated as the superposition of all normal modes with associated coefficients that weight their contributions. It is interesting and important to discuss these coefficients, which underlie the resonance properties of different excitation modes. The scattering coefficients an and bn are

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related to the electric and magnetic resonances, respectively, and are described by the expressions [29]: 0

0

an ¼

myn ðmxÞyn ðxÞ yn ðxÞyn ðmxÞ 0 myn ðmxÞxn ðxÞ xn ðxÞy0n ðmxÞ

bn ¼

yn ðmxÞyn ðxÞ myn ðxÞyn ðmxÞ 0 yn ðmxÞxn ðxÞ mxn ðxÞy0n ðmxÞ

0

(4.13)

0

(4.14)

where m ¼ ns/nm is the relative refractive index of the particle and its surrounding, and x ¼ 2panm/l is the size parameter. Here, the permeability values of the sphere and its host medium are considered to be the same. The Riccati–Bessel functions in (4.14) are given as [29]: yn ð rÞ ¼ rjn ð rÞ; xn ð rÞ ¼ rhðn1Þ ð rÞ

(4.15)

where jn(r) and hð1Þ n ðrÞ are the Bessel and Hankel functions. From the expression of an and bn, it can be inferred that dominant resonances can occur if an and bn reach their poles. The 2n-pole electric resonance dominates when an reaches its nth pole, and the 2n-pole magnetic resonance dominates when bn reaches its nth pole. The dipole resonances belong to the first order with n ¼ 1, while the quadrupole resonances belong to the second order with n ¼ 2. These resonances are largely affected by the size parameter x, which is the reason why Mie resonances are also referred to as morphology-dependent resonances. Other factors that affect excitation of the Mie resonances include the refractive indices, the incident beam profile and the excitation wavelength. It can also be expected that an and bn vanish when m ¼ 1, which is obvious as the sphere and its surrounding medium have the same refractive index. The scattering and extinction cross sections can then be calculated based on the expanded incident field and the scattered field as [29]: sscat ¼ sext ¼

1 2p X 2 2 ð 2n þ 1 Þ ja j þ jb j n n k02 n¼1 1 2p X ð2n þ 1ÞReðan þ bn Þ 2 k0 n¼1

(4.16)

(4.17)

For small spheres with a large index contrast m, the extinction spectrum usually demonstrates distinctive peaks associated with the various orders of Mie resonances. An example is shown in Figure 4.3 where simulated extinction, scattering and absorption cross sections of a 300 nm silver sphere are represented. It is observed that from long to short wavelength, the spectrum exhibits electric dipole, electric quadrupole and electric octupole resonance modes. In this example, the scattering coefficient an dominates, and thus only electric resonances are excited. The coefficient bn are nearly purely imaginary due to the high loss and the corresponding magnetic resonances become virtual modes [29].

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Figure 4.3 Calculated cross-section spectra of a silver sphere with a diameter of 300 nm based on the open source Mie algorithm software ‘MiePlot’ [31]. The resonance modes are verified with full-wave simulation performed with CST Microwave Studio 2016. A plane wave excitation and open boundaries are employed Resonances supported by nonspherical small particles can also be analysed in the framework of Mie theory. For example, Gans theory is developed for spheroidal particles [32]. More complicated nonspherical particles can only be analysed numerically. For nonspherical particles, optical resonances are not only related to the size parameter but also the particle shapes, which can accommodate different modes under different incident polarisations. Such a generalised Mie analysis for optical resonances is important for current nano-optics research since it can offer more degrees of freedom in tuning conditions to explore new tailored resonance properties.

4.3.4

Dielectric resonators versus plasmonic resonators

Dielectric nano-particles operating as optical DRs have recently attracted significant research interests [33,34]. Given the availability of low-loss dielectric materials, DRs can retain their high efficiency even at optical frequencies. Optical excitations on DRs can be conveniently analysed with Mie theory. Different from plasmonic resonators, both the scattering coefficients an and bn can be real, implying efficient electric and magnetic resonances. As exemplified in Figure 4.4, the extinction spectrum of a silicon sphere includes multiple resonant modes: From long to short wavelength, magnetic dipole, electric dipole, magnetic quadrupole and electric quadrupole modes can be clearly identified. The resonance bandwidth is typically narrower than that of a plasmonic resonator due to the low dissipation. Owing to their high efficiency and access to magnetic resonances, DRs offer attractive research prospects for applications in optical metamaterials and efficient integrated optical systems. We further compare the features of plasmonic and DRs and demonstrate their unique advantages in different application areas. Plasmonic resonators can be made of various metals. Based on heavily damped conduction electron oscillations,

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M E

1,000

1,100

1,200

1,300 1,400

Figure 4.4 Simulated extinction spectrum of a silicon sphere with a diameter of 300 nm under a plane wave excitation. Distinct magnetic and electric resonances are observed plasmonic resonators generally have a strong field confined to their surface. This feature makes plasmonic resonators more suitable for such applications as localised field enhancement, sensing and spectroscopy. However, for sensing applications, local heat generated by plasmonic resonators may be detrimental to temperaturesensitive samples, and optical DRs may be a more suitable alternative in this case. The strong fields on resonance of low-loss DRs are typically confined inside their volume in the form of displacement current. Owing to the low dissipation, optical DRs are generally suitable as building blocks for flat optical components and integrated optics where their high efficiency becomes a clear advantage. Moreover, optical DRs with large higher order susceptibilities are promising for efficient nonlinear optics. Their conversion efficiency is several orders of magnitude higher than that based on lossy plasmonic nanostructures. Additionally, dielectric materials such as silicon are CMOS compatible and can be easily integrated with silicon-based photonic platforms. Optical DRs also demonstrate better electromagnetic scalability than plasmonic resonators. Many current optical DR designs are based on materials such as silicon, TiO2 and semiconductors with bandgaps larger than visible frequencies. To further explore the magnetic response of DRs, dielectric materials with high relative permittivities are desirable. This points out to one of the challenges for this area of research since low-loss materials with high relative permittivities at optical frequencies and good processability for nanoscale fabrication are not common.

4.4 Nanoantenna fabrication 4.4.1 Top-down approaches Experimental research in the field of optical nanoantennas would not be at its current state of the art without the capabilities of modern nano-technologies, which enable their precise and reliable top-down fabrication according to specific designs. The common nanofabrication approaches for nanoantennas can be roughly classified into resist-based approaches. Furthermore, the process can be additive or subtractive, or in other words, material can be added or removed in order to define the

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nanoantennas. Which process is most suitable depends on the target material, the design and the sample requirements, as detailed in the following.

4.4.1.1

Resist-based approaches

The by far most flexible technique for the fabrication of nanoantennas by a resistbased process is electron-beam lithography [35], where an electron beam is focused onto a resist, causing it to become more (positive resist) or less (negative resist) soluble to a suitable developer substance. Electron beam lithography (EBL) reliably provides feature sizes down to a few tens of nanometres, and computercontrolled deflection of the electron beam allows for the definition of almost arbitrary two-dimensional geometries. Photolithography or laser beam lithography is less suited for the fabrication of nanoantennas for visible and near infrared radiation since the diffraction limit prevents the definition of patterns with the required feature sizes. While some other lithographic techniques including nano-imprint lithography [36] and nano-sphere lithography [37] were also suggested, EBL is the workhorse of the lithographic techniques used for the fabrication of nanoantennas and thus the following explanations will be based on the example of EBL. Additive resist-based process: A typical additive resist-based process is illustrated in Figure 4.5(a) [35]. The desired wafer is coated with a positive electron-beam resist such as polymethyl methacrylate. Note that for EBL exposure either the wafer has to be conductive by itself, or it has to be made conductive, e.g. by covering it with a layer of indium-tin-oxide (ITO) or by applying a conductive polymer, in order to avoid

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charge accumulation on the sample which can distort the electron beam. After EBL exposure and development, a thin film of the desired nanoantenna material, typically metals such as gold, silver or aluminium, is deposited onto the sample. This is usually done by electron-beam or thermal evaporation, resulting in high-quality polycrystalline metallic structures. Next, a lift-off procedure is performed where the unexposed resist is dissolved, such that only the exposed areas remain to be covered by the evaporated material. The described process and its variations are frequently used for the fabrication of plasmonic nanoantennas but can also be used for certain dielectric structures [38]. An example of a fabricated structure is shown in Figure 4.6(a). Subtractive resist-based process: For a subtractive resist-based process, the lithography step is combined with an etching step, usually reactive-ion etching. The steps performed during a typical process are illustrated in Figure 4.5(b). One can either use a negative resist and directly deploy it as an etch mask or first define an etch mask consisting of another material, which offers higher etching selectivity against the nanoantenna composite material. The latter is usually required for the fabrication of high-aspect ratio structures. Subtractive resist-based processes are mainly used for the fabrication of high-index dielectric or semiconductor nanoantennas [33] since the deposition conditions for the required materials such as high-quality silicon or III–V semiconductor thin films are usually not compatible with a lift-off procedure. An example of a fabricated silicon nanoantenna is shown in Figure 4.6(b).

4.4.1.2 Direct patterning approaches Additive direct process: Direct deposition of various materials from precursor gases is possible by using electron-beam or focused-ion-beam (FIB)-induced deposition. Importantly, this method offers the definition of three-dimensional structures [42] and a few nanometres spatial resolution [43]. Both dielectric and metallic materials can be deposited this way; however, the metal quality remains an issue in the latter

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Figure 4.6 (a) Example of a tapered gold Yagi-Uda nanoantenna [39] fabricated by the electron-beam lithography based process illustrated in Figure 4.5(a). (b) Fano-resonant silicon nanoantenna [40] defined by the subtractive resist-based process shown in Figure 4.5(b). (c) Bowtie nanoantenna with a gap of less than 6 nm created by a combined approach of Gaþ-ion beam milling and subsequent Heþ-ion beam milling of a polycrystalline gold film [41]

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case. Therefore, if plasmonic nanoantennas are desired, deposition of dielectric materials and the subsequent coating of the resulting structures with a metal layer have been suggested as a workaround [44]. Subtractive direct process: Direct subtractive patterning of nanoantennas can be achieved using FIB milling. FIB structuring is based on localised sputtering of material using accelerated ions, which are focused to a small spot on the sample. The focal spot and the sample are scanned with respect to each other to produce a desired pattern [45]. While most commercial FIB systems employ gallium ions and offer a spatial resolution comparable to lithographic approaches, the more recent development of commercial helium-ion FIB systems enables resolutions down to few nanometres at the price of decreased patterning speed. This is particularly important for the realisation of nanoantennas with very small feed gaps [41]. Another important area of application is the realisation of nanoantennas on surfaces which are not compatible with common resist-based processes. The most prominent example is the fabrication of nanoantennas at the tip of scanning probes, such as SNOM (see Section 4.5) tips [46]. Last but not the least, FIB milling has recently proven very successful for the fabrication of plasmonic nanoantennas from chemically synthesised monocrystalline metallic flakes [47], which, compared to polycrystalline nanoantennas, show higher overall structure quality and lower intrinsic losses since roughness and scattering at grain boundaries are avoided.

4.4.2 Bottom-up approaches Apart from top-down methods, bottom-up approaches were also widely employed for the fabrication of various plasmonic and dielectric nanoparticles using chemical synthesis methods, allowing for the creation of single-crystalline nanoparticles with controlled shapes in solution [48]. Furthermore, huge progress was recently made in combining several such nanoparticles to more complex arrangements in a defined manner by DNA origami [49]. Bottom-up approaches are of particular interest for the large-scale production of nanoantennas. However, the approaches are chemistry based, and the controllability of patterns is relatively limited compared with top-down approaches. Thus, the topic is not extensively discussed here.

4.5 Optical characterisation of nanoantennas Commonly employed optical characterization methods for nanoantennas comprise near-field and far-field techniques for both excitation and detection paths. Some representative characterisation techniques are described in this section. For far-field excitation of a single nanoantenna with a focused light beam, the light scattered by the sample is usually very weak and measurement of its extinction or scattering cross section requires a background-free measurement. This is usually realised using a dark-field illumination geometry where the excitation light field does not enter the collection path. For ensemble measurements, such as in the case of nanoantenna reflectarrays, transmitarrays or metasurfaces, the signals are much larger and background-free

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schemes are not required. Instead, conventional spectroscopic techniques can be used, providing a characterisation of the transmittance, reflectance, absorbance or emission of the nano-patterned surface. Polarisation selectivity can be obtained by inserting polarisers into the detection path, and angular dependence of the scattered fields can be measured with advanced setups to obtain the equivalent of classical antenna radiation patterns. Local near-field excitation of a single nanoantenna can be achieved using different modalities that effectively create a broadband local point-dipole source feed. For example, a nanoantenna resonance can be stimulated by coupling it to an emitter such as a fluorescent molecule, and the spectrum or emission pattern [50,51] of the emitted light can be studied. Alternatively, the nanoantenna can be excited with an electron beam. In this latter case, one can then either observe the photons emitted by the nanoantenna after excitation as in cathodoluminescence imaging [52] or investigate the energy loss of the electrons after interaction with the nanoantenna, as in electron-energy loss spectroscopy [53]. Since the electron beam can be focused far below the diffraction limit, these techniques deliver unbeaten spatial resolution down to few nanometres, thus allowing to obtain information about the near-field profile of nanoantennas, with subwavelength resolution. Another method providing subwavelength resolution is SNOM [10], which allows one to image the evanescent electromagnetic near-fields all-optically in the immediate vicinity of the nanoantenna surface. To this end, the nanoantennas are most commonly excited from the far-field by focusing a laser beam onto them and a probe with a tiny (subwavelength) aperture or with a sharp tip is scanned across the sample and guides or scatters the light towards a sensitive detector. Interferometric schemes make it possible to retrieve not only intensity but also phase information. SNOM requires careful control of the distance between the sample and the tip with nanometre precision, making it a rather involved technique.

4.6 Applications Optical antennas enable manipulation of light on the subwavelength scale and can promote enhanced light–matter interaction. This has led to a number of interesting applications [1,54]. Research activities on optical antennas encompass many application aspects including sensing [55,56], photodetection [57], field enhancement [58], imaging [46], directional light scattering [59], integrated optics/photonics [60], optical communications in nano-circuits [61,62] and thermal emission [63,64]. Plasmonic structures have dominated these areas for years. Recently, the realisations of low-loss nanoscale DRAs have led to demonstrations for fluorescence enhancement [65], sensing [66] and efficient wavefront engineering [67,68]. In this section, we will discuss some of these applications with representative examples.

4.6.1 Localised field enhancement Optical antennas are able to localise light within a small volume, resulting in a strongly enhanced field. This process can be characterised by the field enhancement

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factor defined as F ¼ |Eloc|/|E0|, where |Eloc| is the local electric field amplitude and |E0| and is the incident field amplitude [69]. Field enhancement has been extensively demonstrated with metallic nanostructures. As discussed before, metallic particles supporting LSPRs can lead to large F proportional to their polarisability. This is especially true for metallic nanoantennas with a sharp corner that exhibits the socalled lightning-rod effects [70]. Such field enhancement is created due to surface charges being forced to focus into a small volume. One well-known structure that creates the lightning-rod effects [71] is the metallic nano-bowtie with a small gap, where strong field intensity is achieved [58]. For dielectric nanoparticles, various orders of Mie resonances lead to enhanced local fields too. It has also been recently reported that dielectric nano-dimers can create comparable field enhancement as the metallic bowties [72]. Field enhancements have been used in various applications, such as in photoluminescence [58,73], Raman scattering [74] and nonlinear optics [75,76]. In the following, we focus on field enhancement for photoluminescence and briefly discuss other aspects. Photoluminescence is the process of re-emission of light after the absorption of higher energy photons. Photoluminescence is a spontaneous emission process seen in various quantum emitters such as fluorophores and quantum dots. Optical antennas can enhance the photoluminescence process in these quantum emitters, which can be characterised by their intrinsic quantum yield hi ¼ gorad =ðgorad þ gonr Þ and the local field intensity at the resonance frequency [77]. Here, gorad and gonr are the free-space radiative decay rate (i.e. emission) and nonradiative decay rate (i.e. thermalisation), respectively. By adding an optical antenna, the quantum yield of an emitter is modified as [77]: h¼

grad grad þ gonr þ gloss

(4.18)

where grad is the modified radiative decay rate for the emitter at the near field and gloss is the additional nonradiative decay rate introduced by the enhanced local field. The radiative decay rate grad can be enhanced due to an increase in the local density of states that provides more channels of radiation [77]. This effect is especially prominent for emitters with low hi, where their quantum yield and thus their photoluminescence process can be significantly improved. As can be expected, the distance between an emitter and an optical antenna is critical. For very small separation between the pair, nonradiative decay rate, gloss becomes dominant and leads to strong thermal dissipation [78]. In the past decade, many plasmonic nanoantennas have been reported for fluorescence enhancement. One representative example shown in Figure 4.7 is based on gold bowtie arrays, which generate local hotspots to enhance the fluorescence of a low hi single molecule named TPQDIs [58]. The author compares the fluorescence signals of unenhanced and enhanced samples with results presented in Figure 4.7(b). A dramatic photon counts difference is seen between the two cases. The step-like traces correspond to where photobleaching occurred. Based on Figure 4.7(b), the fluorescence enhancement factor fF, which is defined as the ratio between the enhanced rate of photon emission and the unenhanced photon emission rate under the same scaled pumping intensity in this work, is estimated to be 1340.

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Figure 4.7 Fluorescence enhancement of a single molecule by a gold bowtie nanoantenna. (a) Scanning electron microscopy (SEM) image of a gold bowtie nanoantenna and simulated hot spot (electric field) in the bowtie gap. (b) Confocal fluorescence scan and fluorescence timetrace for unenhanced (left) and enhanced (right) emission of single molecules. Figures [58] reproduced with permission from Nature Publishing Group The enhanced quantum yield is reported to be 25%, nearly ten times higher than the intrinsic value of only 2.5% when no antenna is used. One major limitation of such a bowtie nanoantenna is that they significantly increase the Ohmic loss and in turn reduce the total decay lifetime, thus limiting the possibility of achieving a higher quantum yield. To overcome this drawback, optical DRs have been recently proposed for field enhancement with the promise of an ultralow nonradiative decay [65]. Nano-dimers of silicon DRs have been demonstrated for producing polarisation-dependent hot spots [72]. In another work, an array of silicon spheres that supports dark resonance modes was designed to enhance the Purcell factor, which describes the spontaneous emission rate of a fluorophore enhanced by its environment [79]. Different from the plasmonic nanoantennas that create hot spots, the enhancement of the Purcell factor here is achieved by lower intensity dark resonance modes, which weakly couple to the far-field radiations. Optical antennas can also enhance the inelastic light scattering process, such as Raman scattering, which is known to be an extremely weak process [19]. This promotes research in surface-enhanced Raman scattering [74, 80, 81]. Moreover, nonlinear optical processes such as second or third-harmonic generation can benefit from the field enhancement of optical antennas [75,76], to enhance the conversion efficiency with the additional potential benefit of controlling the radiation direction of the generated higher harmonics. Recently, nanoscale DRs have been demonstrated to enhance nonlinear processes [82], where the higher damage threshold and the good spatial overlap of the enhanced near-fields with the nonlinear material supported by DRs promises higher conversion efficiency compared to their plasmonic counterparts.

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Sensing

The resonance of optical antennas can be strongly influenced by a material in close proximity. This property clearly motivates and underlies sensing applications, particularly in view of enhancing the ability to probe tiny amounts of a substance with subwavelength localisation. For plasmonic nanostructures, both SPPs and LSPRs have been used for sensing. The SPP-based sensors are usually based on changes of SPP dispersion relations at metal–dielectric interfaces, while LSPRbased sensors usually are realised with shifts in scattering/absorption spectra. The concepts of sensitivity and figure of merit (FOM) are often used to characterise the sensing performance of LSPR-based devices. Sensitivity is typically defined as the amount of resonance wavelength shift with regard to a unit refractive index change of the (bulk) surrounding environment, i.e. S ¼ DlLSPR/Dnr, with the unit of nm/RIU (unit refractive index), and nr is the corresponding refractive index of ed of the bulk environment. FOM, which is dimensionless, is defined as S/FWHM, i.e. DlLSPR/(Dnr FWHM), where FWHM is the full width of the resonance peak at the half maximum of the corresponding LSPR. FOM compares the amount of resonance shift with the resonance bandwidth, emphasising the importance of a discernible sensing performance. Sensing applications can be separated into two general categories: bulk refractive index sensing and local molecular/biosample sensing [55,69]. Bulk refractive index sensing refers to an observation of the overall dielectric environment around optical antenna sensors. Usually, bulk sensing detection is based on the shift of extinction spectrum for a dense film or on the shift of the scattering spectrum of nanoparticles as measured with dark-field microscopy. Figure 4.8(a) shows such an example of bulk sensing based on LSPR shifting for silver nanoparticles [83]. Here, such nanoparticles with different sizes were immobilised onto a SiO2 wafer, and their scattering spectra were measured with dark-field microscopy. When immersing these nanoparticles in oils with index changing from 1.44 to 1.56, the scattering spectra peaks demonstrate a clear redshift, as shown in Figure 4.8(b). The sensitivity of sensing based on silver nanoparticle can be conveniently evaluated, with an averaged value of around 200 nm/RIU. Such a result suggests that LSPRs can be adopted for ultra-sensitive sensors. Local sensing is usually associated with field enhancement in a deeply subwavelength region created by the mentioned lightning-rod effects. The enhanced field decays rapidly with the distance from the hot spot of an optical antenna, resulting in a localised high-intensity near-field volume. This feature is favourable for molecular or bio-interaction studies. As shown in Figure 4.8(c), Liu et al. reported a hydrogen sensor relying on the hot spot created by the palladium–gold-based nanoantennas [84]. Hydrogen can be absorbed by palladium, forming palladium hydride in a reversible manner. The chemical reaction leads to a change in the local refractive index and size of the palladium nanoparticle that greatly affects the local hot spot. Such a change can be observed in the measured scattering spectrum as the hydrogen concentration increases, as shown in Figure 4.8(c). However, as the distance between the gold triangle and palladium disk increases, the resonance shift becomes smaller. Figure 4.8(d)

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Figure 4.8 Bulk and local sensing. (a) Dark-field image of silver nanoparticles immobilised on SiO2 wafer. The diameter of the nanoparticles is ranging from 40 nm to 120 nm. (b) Measured LSPR peaks of 12 different silver nanoparticles immersed in oils with the refractive index changing from 1.44 to 1.56. Figures [83] reproduced with permission from ACS Publications. (c-e) Measured hydrogen sensing with different hydrogen concentrations. The distance between the gold triangle and palladium disk is (c) 10 nm to (b) 70 nm and (d) 90 nm. Figures [84] reproduced with permission from Nature Publishing Group shows a smaller shift with a gap size of 70 nm due to a lower local field intensity, and in Figure 4.8(e) for a gap size of 90 nm, no clear shift can be observed. Apart from LSPRs, Fano resonances supported by plasmonic optical antennas have also been proposed as a means for sensing [85]. A Fano resonance can be observed in asymmetrical structures that excite both bright and dark modes of resonance, i.e. the modes that can and cannot be directly excited by free-space waves. When the two modes spatially and spectrally overlap, a sharp and sensitive resonance can be created. Hao et al. demonstrated a concentric ring plasmonic structure supporting a Fano resonance for sensing applications [86]. Theoretically, this design demonstrates a high sensitivity of 541 nm/RIU and a corresponding FOM of 5.44. The FOM value is generally limited by the relatively high plasmonic loss. To reduce the plasmonic loss, Fano resonance supported by all-dielectric optical resonators can be explored. Yang et al. have demonstrated a ‘‘101’’ resonator array

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made of silicon that supports Fano resonance [66]. The measured Q-factor is 129, leading to a refractive index sensitivity of 289 nm/RIU and an ultra-high FOM of 103. Similar Fano resonance structures have also been reported based on oligomers of silicon resonators [87,88]. An alternative approach to sensing is not based on the observation of resonance shifting but rather on the monitoring the reflection/transmission amplitude responses at a fixed frequency. Such a response may be created by a resonance shift or a change in resonance strength. This leads to another definition of FOM, namely, the alternative FOM* [89] defined as max |(DI(l)/Dn)/I0(l)|, where DI(l)/Dn describes the intensity change with regard to the unit refractive index change of the surrounding environment and I0(l) is the intensity of a reference signal. An illustrative design is the Au-MgF2-Au plasmonic absorber reported by Liu et al. for sensing refractive index, with the maximum calculated FOM* of 94 [90].

4.6.3 Integrated photonics SPPs have been proposed as carrier waves in photonic integrated circuits with low power consumption [60]. To this end, one important optical component in integrated optics is a unidirectional SPP launcher. A unidirectional SPP launcher couples freespace waves into SPPs propagating in a desired direction along a metal surface. So far, various unidirectional SPP launchers have been demonstrated, aiming at achieving compact and efficient SPP sources. On a metal surface, properly designed nanoslits, grooves or apertures can work as SPP couplers. Nanoslits provide extra momentum to match free-space light with SPPs, while grooves and apertures can create horizontal magnetic currents for SPP coupling under a plane wave excitation. Unidirectional SPP launching has been demonstrated by integrating Bragg reflectors with nanoslits [91] or by using nonuniform arrays of grooves [92] and apertures [93], operating based on a similar principle as a Yagi-Uda antenna of magnetic dipoles [94]. In general, high extinction ratio, i.e. SPP power in the desired direction over that in the opposite direction, can be achieved with such designs. However, the approach does not promise a high coupling efficiency (power coupled into SPPs with regard to the incident power) due to high radiative and dissipative losses. An efficient compact magnetic nanoantenna for SPP launching has been reported in 2013, as shown in Figure 4.9 [95]. In this design, two launchers for opposite directions have been demonstrated. Each launcher is made of two Au–MgF2–Au nanoantennas, similar to patch antennas. Under a plane wave excitation, strong horizontal magnetic dipole resonance can be excited, as shown in the inset of Figure 4.9(b). This horizontal magnetic resonance ensures an efficient coupling from incident free-space waves into SPPs. Nanoantennas of different sizes demonstrate different phase responses, and thus unidirectional SPP launching can be achieved via SPP interferences. Figure 4.9(c) and 4.9(f) shows the measured leakage radiation of the designed leftward and rightward launching, and their corresponding Fourier plane images in Figure 4.9(d) and 4.9(f). The calculated extinction ratio is about 2.69 and coupling efficiency is around 25%, around five times higher than a single nanoslit. To further improve the coupling efficiency, optical DRs supporting horizontal magnetic dipole resonance have been theoretically proposed [96].

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Figure 4.9 Unidirectional SPP launcher made of compact magnetic nanoantennas. (a) leaky-wave measurement of the SPP launcher. (b, e) SEM images of the leftward and the rightward SPP launchers. The separations of the two nanoantennas are 300 nm and 600 nm respectively. (c, f) Leakage radiation microscopy results. (d, g) Respective Fourier plane image of the results in (c) and (f). Figure [95] reproduced with permission from ACS Publications By using low-loss DRs, the resonance is mainly limited in the dielectric part and the overall launcher efficiency can be improved with less dissipation in metal. In more complicated configurations, polarisation and mode selectivity’s can be integrated into directional scatterings [97].

4.6.4 Planar optical components In recent years, optical antennas have been increasingly demonstrated for wavefront engineering [98], towards the creation of planar optical components. Research on optical wavefront shaping is inspired by the long tradition of engineering research on microwave reflectarray/transmitarrays antennas [99]. At optical frequencies, devices such as optical reflectarrays [100], transmittarrays [101], flat lenses [102], waveplates [68,103] and holograms [104] have been extensively reported. These flat optical devices, often called metasurfaces, are foreseen as important components in future optical telecommunication systems and integrated optical systems. Optical metasurfaces are generally composed of subwavelength resonators with spatially varying phase response with 2p coverage for wavefront shaping. As one example presented in Figure 4.10, the V-shaped gold resonators with different designs demonstrate progressive phase responses across the silicon surface [101]. The progressive phase distribution together with the subwavelength inter-element distance results in beam refraction in transmission. However, the transmitted wave is crosspolarised in order to achieve the full 2p phase range. The same principle has been used to create focused beam and vortex beam in the optical range.

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l

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Figure 4.10 Gold V-shaped transmitarray [101]. (a) SEM image of the fabricated V-shaped optical transmitarray. The dashed box denotes a subarray that consists of resonators covering 2 phase range. (b). Illustration of the transmittarray principle based on local phase gradient. Reprinted with permission from The American Association for the Advancement of Science One significant drawback of plasmonic resonators is their high Ohmic loss at optical frequencies. As a result, the efficiency of these metasurfaces is low. Moreover, various orders of magnetic resonances are not conveniently supported with simple plasmonic structures, limiting the functionalities of the devices. These limitations can be overcome with recent research efforts on low-loss dielectric metasurfaces based on Mie resonances [33]. In 2013, Zou et al. reported an optical reflectarray based on optical DRAs operating in their fundamental magnetic dipole mode. The structure demonstrated an improved efficiency compared with plasmonic designs [67]. In another representative work of transmitarray, all-dielectric gradient metasurfaces have been demonstrated for efficient ultra-thin flat lens at visible frequencies [105]. Also in the transmission mode, high-transmission Huygens metasurfaces have been demonstrated based on silicon DRs [106]. The high transmission with 2p phase range [107] is achieved by spatially and spectrally overlapping electric and magnetic dipole resonances with the same polarizabilities, to create localised Huygens sources in a general crossed arrangement of electric/ magnetic dipoles as used in microwave antenna technology [108]. To harness flat optical devices for real-world applications, achieving tunable devices is of critical importance. So far, several mechanisms to realise tunability for either plasmonic or dielectric metasurfaces have been demonstrated. These designs employ phase-change materials [109] such as liquid crystals [110] and germanium antimony telluride [111], mechanical deformation [112,113] and carrier injection in 2D materials [114]. The material phase transition can be achieved with external voltage [115] or temperature control. For mechanical deformation, elastomer substrates such as polydimethylsiloxane (PDMS) can be incorporated into metasurface designs. In Figure 4.11, a uniform array of TiO2 DRs is embedded inside a PDMS matrix. The power transmission spectrum is tuned when the PDMS is deformed by the external strain [112]. For a normally incident plane wave excitation polarised along the y-direction, the excited electric dipole resonance demonstrates redshift and

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Figure 4.11 Mechanically tunable dielectric metasurface [109]. (a) False-colour SEM image of the fabricated sample. Under a y-polarised normally incident plane wave excitation, (b) and (c) present the measured redshift and blueshift of the resonance dip when stretching the PDMS along the x and y-directions blueshift when stretching the PDMS along the x- and y-directions, respectively, with measured results presented in Figure 4.11(b) and 4.11(c).

4.6.5 Photodetection Photodetectors are one of the key optoelectronic devices. Conventional photodetectors operate based on semiconductors, photoelectric effect or thermal effects that convert absorbed photons into detectable electric currents or heat [116]. For faster, more efficient and compact photonic applications, it is desirable to shrink the detectors to much smaller sizes. However, this is generally constrained by the diffraction limit and the required depth for light to be absorbed [69]. Additionally, photocurrent generation is generally limited to photons above the bandgap of the employed semiconductor, limiting the ability for low-frequency detection. In recent years, many plasmonic nanoantennas have been reported to enhance photodetection, aiming at overcoming the aforementioned limitations [57,117,118] by increasing the effective power collection area without increasing the detector size. Plasmonic nanoantennas generate local hot spots that can enhance the photon absorption in a very small region and thus increase photocurrent generation at the hot spot. As an example, [57] in Figure 4.12(a), the nanoantenna consists of an array of gold nanorods on a silicon substrate spaced by a 1-nm adhesion layer of titanium. Gold resonators are separated from each other by SiO2 but electrically connected with ITO. A plane wave polarised along the nanorod axis results in dipolar LSPRs. The resonance largely increases the absorption cross section so that nonradiative plasmon decays into heat. In this structure, the plasmon decay from gold transfer energy to electron–hole pairs and inject hot electrons into the silicon, thus creating detectable photocurrents. Different from conventional photodetectors, the photocurrent here is not limited by the silicon bandgaps, but by the Schottky barrier potential, noting that no bias voltage is required.

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(a)

(b)

Figure 4.12 Plasmonic optical antennas for photodetection [57]. (a) Design illustration. (b) SEM image of the fabricated sample. Figure reproduced with permission from The American Association for the Advancement of Science

4.6.6

Selective thermal emission

Optical antennas have been implemented for coherent or wavelength selective thermal radiation [63,119]. It was widely believed that coherent thermal radiation could not be created since thermal radiation originates from incoherent oscillations of atoms, molecules or lattices within an object. In 2002, however, Greffet et al. experimentally demonstrated that coherent thermal radiation can be achieved by heating up an SiC grating structure [119]. The key point is that coherent surfacephonon polaritons are excited during the heating, thus resulting in coherent thermal radiation. Relevant experimental studies were also conducted around the early 2000s to show that selective thermal emission can be manipulated via micro- or nanostructures on the surface of an object [120]. According to Kirchhoff’s law of thermal radiation [121], the frequency- and angular-dependent absorptivity of an object is equal to its frequency- and angle-dependent emissivity. In recent years, nanofabrication technology has led to increased research activities on controlling thermal emission properties of surfaces by patterning them with specially designed nanostructures. For instance, compact high-efficiency and low-cost light source at mid-infrared frequencies are not easily available, and thus thermal emitters based on optical antennas could be used for this purpose [122]. These sources can be particularly important in detecting common molecular bonds. Another promising application related to selective thermal emission is radiative cooling [123], which is a passive cooling process that does not require external energy. Radiative cooling can be achieved by maximising broadband and wide-angle emission only at the wavelength from 8 to 13 mm, which corresponds to a major atmospheric transparent window in the mid-infrared, thus effectively allowing to radiate heat into the outer space. Different nanostructures have been demonstrated for efficient radiative cooling, including photonic crystals [124], and multilayer pyramid optical antennas [123]. Recently, as shown in Figure 4.13, metal-loaded dielectric optical antennas have been demonstrated for radiative cooling with scalable low-cost fabrication and convenient integration with photonic systems [64].

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Figure 4.13 Metal-loaded dielectric optical antennas for radiative cooling. [64] (a) False-colour SEM image of the fabricated sample. (b) Simulated absorptivity (emissivity) of the optical antenna showing selective absorption (emission) in the 8–13 mm atmospheric window

4.7 Conclusion and outlook Optical antennas are at the crossroads of photonics, material sciences and antenna engineering. Their early developments have stirred a huge interest in various scientific communities, leading in the last decade to the development of a remarkably active field of research. Advances in electromagnetic modelling methods, in material science and in nanofabrication techniques have enabled tremendous achievements by researchers and technologists worldwide. While optical resonant antennas with subwavelength dimensions often take inspiration from their classical radio-frequency counterparts, the scaling of such functionalities to nanometre scales is far from trivial. As elaborated in this chapter, the plasmonic behaviour of metals at optical frequencies requires dedicated considerations and it affects scalability and efficiency of resonant metallic structures. As an alternative to plasmonic resonators, DRs have recently emerged with a promise of a more straightforward scalability, however, with a main limitation arising from scarcity of suitable high-permittivity materials. Both metal and DRs exhibit advantages and weaknesses which will eventually determine their suitability for practical applications. The interest in nanoantennas has been largely fuelled by promises of real-world optical applications for which access to resolving subwavelength features is highly desired. Such applications include localised field enhancement, ultra-fast detection and sub-diffraction sensing. Furthermore, concepts of optical components inspired by reflectarrays, transmitarrays and metasurfaces have emerged to offer fine manipulation of optical beams at a subwavelength scale. As these developments mature and find their ways into future technologies, a new field of applications are emerging. For example, visions of functional smart adaptive optics and wireless communication links at the chip level are in exploratory stages. With no doubt, such visions will open new perspectives in the developments of optical antennas.

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Chapter 5

Fundamental bounds and optimization of small antennas Mats Gustafsson1, Marius Cismasu1, and Doruk Tayli1

5.1 Introduction Mobile wireless standards have been introduced nearly every decade since the early 1980’s. The introduction of a new standard increases performance requirements and expected capabilities of mobile terminal components. For instance, multiple antennas are expected to fit into confined spaces on a terminal. As an antenna’s performance is restricted by its physical size this presents a challenge for antenna designers [1]. Fundamental bounds are a figure of merit that specifies the optimal performance of an antenna with respect to one or several performance parameters [2–17]. In the case of a mobile terminal, the most important parameters are bandwidth, efficiency, and capacity. It has been demonstrated in [13,15] that the fundamental bounds of any arbitrarily shaped antenna can be determined using antenna current optimization. One other widely used performance indicator is the Q-factor computed from the derivative of the input impedance [11]. The latter provides a close estimate of the bandwidth for narrowband antennas. It has recently been shown that the Q-factor can also be computed from the current distribution on a radiating structure at a single frequency [13,18–20]. This method is based on expressing the electric and magnetic energies stored in the fields, and the power radiated by an antenna in terms of the current [18]. The derivation is further explained in [20,21] where it is shown that the derivation is accurate in many situations, especially for electrically small antennas. Optimization is widely used in antenna design to search for optimal geometric, material parameters that satisfy antenna performance requirements. One of the popular optimization classes in antenna design is meta-heuristic algorithms such as genetic algorithm (GA) and particle swarm, and ant colony. In this chapter, a combined GA and MoM with an ant colony optimization technique is used to synthesize antennas [22–24]. Moreover, the optimization procedure uses a Q-factor estimation method to predict antenna performance [25,26]. Antenna current optimization is used 1

Department of Electrical and Information Technology, Lund University, Sweden

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to compute the fundamental bounds that are then compared with the synthesized structures. Examples of planar synthesized structures have been analyzed in [22,25,26]. Here, these are extended to 3D structures that represent simplified models of common mobile wireless terminals [14]. An in-house GA/MoM solver is used to generate/synthesize optimal antennas with minimum Q-factor for these terminals. Furthermore, optimal antenna placement on a terminal is studied using antenna current optimization. Customized bounds and optimum currents are used in the antenna placement study. The objective of this study is to determine the antenna location that maximizes the performance of the device, measured as G/Q-ratio or Q-factor. Single resonance [11,27] and multiple resonance Brune synthesis models [21,28,29] are employed to evaluate the Q-factor of the structures from their input impedance. The results have been validated with simulations from the commercial electromagnetic solver ESI-CEM [30]. The chapter is organized as follows. A background overview is presented in Section 5.2. Stored energies and their computation using the MoM are illustrated in Section 5.2.1. The single frequency QZ0 estimation method is presented in Section 5.2.2. Fundamental bounds on G/Q and the convex optimization formulation are described in Section 5.2.3. Section 5.3 presents antenna optimization and the combined GA/MoM simulation setup used to synthesize antennas, including GA and convex optimization. Sections 5.3.1 and 5.3.2 describe GA and convex optimization, respectively. Section 5.4 presents numerical simulations performed in this chapter and their results. The performance and examples of GA/MoM optimized 3D structures are presented and compared with optimum-current performance in Section 5.4.1. An antenna placement situation using optimum currents and physical limitations is investigated in Section 5.4.3. The chapter ends with conclusions in Section 5.5.

5.2 Stored energies and fundamental bounds for antenna analysis and design Antenna analysis and design is usually performed using numerical techniques that solve differential and/or integral equations describing an electromagnetic problem. Examples and details of numerical techniques for electromagnetics can be found in text books such as [31–34].These techniques are based, in general, on a discretized computation domain. The MoM is a numerical technique where the surface and/or volume of the structure is discretized [34]. When solving an electromagnetic problem with the MoM technique, the current density J excited on the structure is approximated in terms of basis functions yn as J ð rÞ

N X In yn ðrÞ;

(5.1)

n¼1

where r is the position vector, In is the current expansion coefficients, and N is the number of basis functions used to approximate the current. In this chapter, we use a Galerkin-type MoM implementation of the EFIE using surface currents [32–34].

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This gives the following system of equations: ZI ¼ V;

(5.2)

where V is the excitation column matrix, Z is the impedance matrix describing the structure, and I is the column matrix containing the current expansion coefficients. The impedance matrix can be written as Z ¼ R þ jX, where R and X are the real and imaginary parts, respectively. The elements of Z are ð ð ejkR12 Zmn ¼ jh0 kym ðr1 Þ yn ðr2 Þ 4pR12 @V @V (5.3) 1 ejkR12 r1 ym ðr1 Þr2 yn ðr2 Þ dS1 dS2 ; k 4pR12 where h0 is the free space impedance, k ¼ w/c0 is the wave number, c0 is the speed of light in free space, R12 ¼ |r1 r2| is the distance between the source and observation points in the two integration domains, and V is the volume occupied by the antenna, bounded by the surface @V. The derivative of the MoM impedance matrix with respect to the wavenumber is jkR12 ð ð k@Zmn 1 e ¼ j kym ðr1 Þ yn ðr2 Þ þ r1 ym ðr1 Þr2 yn ðr2 Þ k h0 @k 4pR12 @V @V þ ðk 2 ym ðr1 Þ yn ðr2 Þ r1 ym ðr1 Þr2 yn ðr2 ÞÞ

ejkR12 dS1 dS2 4p

(5.4)

which is a straightforward addition to existing MoM software and incurs marginal computational cost. It should be noted that the first term on the right-hand side is identical to (5.3) with an addition instead of a subtraction. Moreover, the second term is a combination of both current and charge terms and does not involve the 1/R singularity of the free-space Green’s function. The Q-factor is the ratio of stored energy to total radiated and dissipated energy in time-harmonic systems [35]. For a lossless antenna, the Q-factor is defined as [11,25]: Q¼

2w max fWe ; Wm g Pr

(5.5)

where We, Wm, Pr are the stored electric energy, stored magnetic energy and radiated power, respectively. This definition is equivalent to that in [35] for resonant antennas. The Q-factor can be used to estimate an antenna’s fractional bandwidth and is a useful parameter in antenna optimization. An overview of expressions for the Q-factor of antennas can be found in [36].

5.2.1 Stored energies The stored energy of radiating systems is an intricate topic, with a far-reaching history in the antenna community. This is the result of the ambiguous definition of the stored energy in many radiating electromagnetic problems. Recently, both the definition and the history of stored energies have been reviewed in [36].

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Stored electric and magnetic energies of the radiating structure presented here are based on the energy expressions derived in [18]. The energies are quadratic forms in terms of the discrete current density matrix I [13]: We

1 H I Xe I and 4w

Wm

1 H I Xm I; 4w

(5.6)

where w is the angular frequency, Xe and Xm are the electric and magnetic reactance matrices, respectively, and the exponent H denotes the Hermitian transpose. The electric and magnetic reactance matrices are obtained from a modified MoM implementation of the impedance matrix Z, and their explicit representations are [13,18]: ð ð cos ðkR12 Þ r1 ym ðr1 Þr2 yn ðr2 Þ Xe;mn ¼ h0 4pkR12 @V @V sin ðkR12 Þ dS1 dS2 (5.7) ðk 2 ym ðr1 Þ yn ðr2 Þ r1 ym ðr1 Þr2 yn ðr2 ÞÞ 8p and ð Xm;mn ¼ h0

ð @V @V

k 2 y m ð r1 Þ y n ðr2 Þ

cos ðkR12 Þ 4pkR12

sin ðkR12 Þ dS1 dS2 : ðk ym ðr1 Þ yn ðr2 Þ r1 ym ðr1 Þr2 yn ðr2 ÞÞ 8p

(5.8)

2

These matrices can also be expressed in terms of the imaginary part of the impedance matrix and its derivative (5.4) with respect to angular frequency w [15,37,38]: w @X X w @X X and Xm ¼ þ : (5.9) Xe ¼ 2 @w w 2 @w w Combining both equations in (5.9), the sum of the stored energies is written as We þ W m

1 H 1 I ðXe þ Xm ÞI ¼ IH X0 I 4w 8

(5.10)

where X0 is the derivative of X with respect to the angular frequency (5.4). The power radiated by an antenna can be written as a quadratic form [18,20,39,40]: 1 Pr IH RI; 2

(5.11)

where R ¼ Re{Z} is the radiation resistance matrix. The Q-factor of a lossless antenna can be expressed in the stored electric and magnetic reactance matrices: 2max IH Xe I; IH Xm I : (5.12) Q¼ IH RI In the case of a self-resonant lossless antenna, (5.12) simplifies to Qres ¼

w IH X0 I : 2 IH RI

(5.13)

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MoM matrices, Z, Xe, etc., are intrinsically suitable for global optimization algorithms such as GA/MoM optimization [23,24] and current optimization [13]. In such algorithms, the optimization time of some antenna parameters reduces, e.g., the bandwidth may be evaluated using the single frequency expression (5.12) for Q.

5.2.2 QZ0 computation from current densities It is also of interest to study another widely used Q-factor definition the QZ0 [11], that is obtained from the derivative of the input impedance of an antenna. Consider an antenna having the input impedance: Zin ðk Þ ¼ Rin ðk Þ þ jXin ðk Þ:

(5.14)

This antenna is tuned to achieve resonance at the wave number k0 using a seriesconnected, ideal, lumped inductor or capacitor, as in [11]. The input impedance of the tuned antenna becomes Zin;t ðk Þ ¼ Zin ðk Þ þ jXt ðk Þ;

(5.15)

where Xt(k) is the tuning term. In the case of a capacitive (Xin(k0) < 0) or inductive (Xin(k0) > 0) input impedance, the tuning term Xt(k) is X t ðk Þ ¼

kXin ðk0 Þ k0

and

X t ðk Þ ¼

k0 Xin ðk0 Þ ; k

(5.16)

respectively. At the resonance frequency, the input impedance is real valued, i.e., Zin;t ðk0 Þ ¼ Rin ðk0 Þ:

(5.17)

The Q-factor, QZ 0 , of the antenna tuned to resonance, is [11]: QZ 0 ðk0 Þ ¼

0 ðk 0 Þ j k0 j Zin;t ; 2Rin ðk0 Þ

(5.18)

where prime denotes the first derivative with respect to wave number. Note the change of variables k ¼ w/c0, performed in order for Zin to be expressed in terms of the same frequency variable as Z, whose elements are (5.3). If the single resonance assumption does not hold, the derivative of the input impedance may approach zero [27], QZ 0 0. Replacing (5.15) and (5.16) in (5.18) gives QZ 0 ðk0 Þ ¼j

k0 Zin0 ðk0 Þ j Xin ðk0 Þ j þj j: 2Rin ðk0 Þ 2Rin ðk0 Þ

(5.19)

An MoM solver provides all quantities needed to evaluate (5.19) except Zin0 . This quantity is traditionally computed using a numerical approximation based on evaluating Zin for two closely spaced frequencies. Here, we present an alternative approach using current densities to compute QZ0 that only requires a single frequency impedance matrix computation.

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The input impedance derivative is expressed in terms of the input admittance. The admittance matrix is Y ¼ Z1. This matrix defines the input impedance of the antenna using a voltage gap model of feeding edge elements: Yin ¼

VT YV ; Vin2

(5.20)

where Vin is the voltage applied across the gap. We assume that the voltage source is real valued and frequency independent, i.e., V0 ¼ 0. The input impedance derivative becomes Zin0 ¼

1 Yin

0

¼

Yin0 ðVT YVÞ0 VT Y0 V ¼ ¼ 2 2 : 2 2 2 Vin Yin Yin Vin Yin

(5.21)

Consider the following identity: 0 ¼ (Z1Z)0 ¼ (Z1)0 Z þ Z1Z. Multiplication from the right by Z1 gives 1 Z (5.22) ¼ Y0 ¼ Z1 Z0 Z1 ¼ YZ0 Y; such that the input impedance derivative is obtained as Zin0 ¼

IT Z0 I ; Vin2 Yin2

(5.23)

using the fact that Z ¼ ZT and Y ¼ YT. Replace (5.23) in (5.19) to obtain: QZ 0 ðk0 Þ j

k0 Zin2 ðk0 ÞIT Z0 I j Xin ðk0 Þ j j; þj 2Rin ðk0 Þ 2Rin ðk0 ÞVin2

(5.24)

the first derivative with respect to the wave number of the impedance matrix, Z0 , is computed for the wave number k0. It should be noted that (5.24) is similar to (5.13), with a transpose in place of the Hermitian transpose. For small antennas, the frequency derivative of the real part R of the impedance matrix Z is negligible ITZ0 I ITX0 I. Therefore the Q-factor for a self-resonant antenna becomes QZ0

w j IT X0 I j w IH X0 I : 2 IH RI 2 IH RI

(5.25)

The inequality holds in the case that X0 is a real-valued symmetric positive-definite matrix and becomes an equality if I is equiphase. This formulation of the impedance derivative speeds up the computational process significantly as it only requires to compute the impedance matrix once unlike using the numerical difference to compute the derivative of the input impedance. The corresponding expression for QZ0 in [41] differs from (5.24) as the former includes frequency derivatives of the current density and complex conjugates. An expression similar to (5.24) can be derived using a parallel tuning susceptance.

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5.2.3 Fundamental bounds The main parameters of interest in small antenna design are the bandwidth, efficiency, input impedance, and SAR [2–13,15]. In many cases, electrically small antenna design becomes a trade-off between the antenna’s physical size and its bandwidth. As the antenna becomes smaller, its bandwidth performance deteriorates. Therefore, it is important to determine the antenna’s physical (fundamental) bounds in order to estimate its optimal performance. One common figure of merit is the Q-factor (5.13) and (5.18), as it is related to the inverse of the fractional bandwidth [11]. Physical bounds of antenna have been extensively investigated since the 1940s. Consequently, a plethora of different techniques are available to compute the physical bounds. These can be grouped as in [14] into circuit models [2,3], mode expansions [4–7], forward-scattering [12], and antenna current optimization [13,20,42]. Antenna current optimization is a powerful approach in which customized bounds are derived without restrictive assumptions, e.g., bounding geometry and electrical size [15]. Antenna current densities are a parameter of interest in antenna current optimization. Using the current densities, the minimization of G/Q and Q can be expressed as optimization problems. For the gain Q-factor quotient, the optimization problem can be solved using convex optimization [27]. In the case of minimizing the Q-factor, the optimization problem is non-convex but a dual problem can be constructed [17]. Once the optimization problem is solved, optimal currents of the antenna are obtained defining an upper bound on the performance of the physical structure. In the gain Q-factor quotient case, the optimal currents can be found using the following convex optimization formulation: minimizeI max IH Xe I; IH Xm I (5.26) subject to FI ¼ j: Alternatively, its dual problem [15] that is maximized over a with the following constraints: minimizeI subject to

IH ðaXe þ ð1 aÞXm Ig FI ¼ j; 0 a 1

(5.27)

For a fixed a, the solution of (5.27) is I¼

jðaXe þ ð1 aÞXm Þ1 FH FðaXe þ ð1 aÞXm Þ1 FH

;

(5.28)

with appropriate scaling of I such that FI is dimensionless. The 1 N matrix F, with the elements ð jkh0 ^ ^e yn ðrÞejk kr dV; Fn ¼ (5.29) 4p V ^ projected on the is used to approximate the far field, F, in the fixed direction, k, ˆ polarization vector, e, as ^ FI: ^e FðkÞ

(5.30)

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Formulation (5.27) minimizes the energy stored in the fields created by a radiating structure for a fixed partial radiation intensity in a specific direction. A more detailed presentation of the optimization formulations (5.26) and (5.27) and the quantities involved in them can be found in [13,15]. Current densities optimum in the sense of (5.27) give the physical limitation on G/Q. However, the Q-factors of these currents (5.5) may not be the global optimum [17]. One advantage of the dual formulation (5.27) and its solution (5.28) is the fact that it can be solved fast and efficiently even for large matrices and multiple directions and polarizations.

5.3 Antenna optimization Antenna performance can be improved by employing various optimization algorithms. These algorithms include different antenna parameters, used as figures of merit, e.g., gain, shape, and impedance, in their optimization goals. In this chapter, we use GA and convex optimization. An evolutionary optimization method, the GA, has been chosen for the examples presented here mainly due to the specifics of the problems considered. GA belongs to the class of global optimization algorithms, for a non-exhaustive introduction to this class and some of its applications see [43–50]. Many antennas embedded in modern devices have a performance that can be predicted with a reasonable accuracy by numerical simulation. Such simulations require a significant amount of computing power. Furthermore, fine details of the structure need to be modeled which translate into a large number of possible solutions. Attempting to study all possible solutions becomes prohibitive from a practical perspective in many situations. However, heuristic methods have been shown to provide reasonable solutions to optimization problems prohibitive for deterministic methods. A few examples of heuristic optimization methods are genetic algorithms, random search, particle swarm optimization, and ant colony [24,51–53]. Table 5.1 explains the procedure to obtain the simulation results. Further details about the items in this table can be found throughout the chapter: general descriptions are given in Section 5.4 for the problems considered there (item 1). The following two paragraphs introduce the in-house MoM solver and commercial Table 5.1 Steps used to obtain the simulation data 1. General description of problem—Definition of the antenna type to be studied, details to be considered, general bounding shape, parameters that apply to later items 2. Generation of mother matrices—An in-house EFIE-based MoM solver computes the matrices Z, Xe, Xm, R, and F that describe the antennas studied, see Section 5.2 3. Performing antenna optimization—The mother matrices are used in GA/MoM [23]; the resulting optimized designs are presented here 4. Computing the bounds—The mother matrices are used for antenna current optimization [13]; these bounds are used as comparison for the optimized designs of item 3 5. Verification using a commercial simulator—The antennas obtained through optimization at item 3 are evaluated using a commercial simulator

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simulator (items 2 and 5). The GA (item 3) is described in more detail in Section 5.3.1. Bounds derived from current optimization (item 4) are the topic of Section 5.2.3. The in-house MoM-solver is based on Galerkin’s method and a mixed-potential EFIE-formulation [31–33]. The basis and testing functions have a ‘‘rooftop’’ profile on pairs of adjacent rectangular mesh elements, i.e., rectangles sharing a common edge [54], as illustrated in Figure 5.4. Such a function has the amplitude linearly increasing toward the common edge and the direction from the first to the second rectangle (numbered according to a fixed mesh element numbering rule). The Green’s function 1/R singularities for self and near-singular terms are integrated using a change of variable [55]. The commercial electromagnetic solver ESI-CEM [30] is used to verify the results obtained with the in-house solver through genetic optimization. This commercial solver uses a triangular mesh to discretize the surfaces. Therefore, the rectangular mesh used in the in-house simulation software is converted to a triangular mesh to be used in the ESI-CEM solver. The antenna feed location is maintained at the same position in both solvers. The ESI-CEM simulation of the GA/ MoM-optimized antenna is used to calculate the cost function. This provides a comparison between the results obtained using the in-house solver and the commercial solver ESI-CEM.

5.3.1 Genetic algorithms Genetic algorithms applied to antenna problems converge with reasonable speed to suboptimal solutions and avoids local extrema [24]. These algorithms mimic human, animal, plant population evolution using genetic principles well-established in genetics—a field of biology. Typically, such principles are concisely contained in the concepts: ‘‘generation,’’ ‘‘individual,’’ ‘‘population,’’ ‘‘gene,’’ ‘‘chromosome,’’ ‘‘breeding,’’ ‘‘offspring,’’ ‘‘crossover,’’ ‘‘mutation,’’ etc. These concepts are used in the context of antenna optimization in the following, for a detailed explanation refer to [24]. The GA used in the examples to search realistic structures with performance close to physical limitations is depicted in Figure 5.1 [25,26]. We developed this algorithm starting from the implementation distributed with PB-FDTD [53]. Population step p + 1

Population step p

...

FC, max “Natural” selection

... ...

Fitness, FC evaluation and ranking

...

...

...

Parent selection Breeding Crossover Mutation

FC, min

Figure 5.1 Illustration of a genetic algorithm implementation for antenna optimization

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Initially-random, 200-individual antennas are improved according to evolutionary principles in steps. At each step, 80 random individuals compete to become one of two breeding parents. The resulting two offsprings are affected by two-point crossover (which happens 80% of the time) and single-gene mutation (with 20% probability). These offsprings are placed in the population, increasing its size by 2 (202 antennas). Antennas of the expanded population are ranked according to their fitness. The two least-fit antennas are removed from the population. Fitness is evaluated as an objective (cost) function that is minimized during optimization. This function is a combination of antenna parameters with different weights. After 300 consecutive steps without population improvement the algorithm enters a phase where the offspring produced always have up to four genes mutated. This phase is meant to reduce the solution time of the GA (however, this time improvement has not been studied). Once improvement is observed, the algorithm returns to ‘‘natural’’ conditions, singlegene mutation with 20% probability. The optimization is stopped after 2 105 steps or when genetic stability during 2 104 steps is observed. As an example consider the over-simplified structure of Figure 5.2(a) as a starting point for antenna optimization using GA/MoM [23,24]. The problem in this example can be formulated as (item 1, Table 5.1): Optimize antennas made of thin PEC. A possible optimal candidate is obtained by placing up to 6 rectangular patches (shaded) in the mesh (black solid lines) of Figure 5.2(a). The mesh assures electrical connection between patches at their edges. Based on this summarized description, the GA/MoM can be set up. The six patches (mesh elements in MoM) are the binary genes describing an antenna ‘‘population.’’ ‘‘Individuals’’ (antennas) have a genotype made of a single chromosome with information about all six genes. The chromosomes are encoded as a 1 6 matrix where 1 denotes a patch placed in the mesh and 0 an empty space in the mesh. I

II 1

w

3

4 6

IV (a)

III

I

2 5

3

7 V

6 VI

IV

7 V

VI

(b)

Figure 5.2 Illustration of a mother structure (a) and an individual derived from it (b). Gray-shaded rectangles—PEC mesh elements/patches. Solid, black lines—mesh edges (they connect patches electrically). Mesh elements are numbered using roman numerals. Mesh edges interior to basis functions domain of definition are numbered with arabic numerals, i.e., there is a surface current perpendicularly across each of these numbered edges corresponding to a basis function

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Figure 5.2(b) depicts an antenna with four patches placed in the mesh as described by the matrix (1,0,0,1,1,1). The interaction between GA and MoM appears during population evolution when the fitness of each individual is evaluated and ranked. This interaction is the method to obtain the MoM solution (5.2), for each individual in the population. We use a method in which the solution is obtained using matrix operations such that computationally intensive calculations, e.g., integration of a Green’s function, are not performed unnecessarily. The ‘‘mother’’ matrices needed for these matrix operations, obtained at item 2 (Table 5.1) are the 7 7 matrices ZM ¼ Rr,M þ j(Xm,M Xe,M) and the 7 1 matrix FM, introduced in Section 5.2. The subscript ‘‘M’’ stands for ‘‘mother.’’ The matrix size is given by the number of edge elements (basis functions) in the problem definition (Figure 5.2). Note that a feed model and position is not necessary for the computation of mother matrices, i.e., ZM, Rr,M, Xm,M, Xe,M and FM depend on geometry alone and are linked with electromagnetics through h0 and k . Mother matrices are pre-computed and then passed to the GA as parameters; i.e., the integration of Green’s functions is performed once outside the main evolutionary algorithm. The electromagnetic solution is obtained during the iterative evolutionary part of the GA using the MoM equation (5.2). The solution we look for is the columnmatrix I of complex surface-current coefficients. All matrices involved in the equations mentioned above have sizes that depend on the ‘‘individual’’ (antenna) for which the solution is derived. In the example of Figure 5.2 the mother impedance matrix is

ð5:31Þ

where rows and columns with indexes 3, 6, and 7 are colored. These indexes result when translating the genetic information of the antenna in Figure 5.2(b) to MoM basis functions. These functions and their interaction with each other are already contained in the mother matrix ZM such that extracting the elements at the intersections of rows 3, 6, and 7 with columns 3, 6, and 7 gives the impedance matrix describing the individual antenna in Figure 5.2(b), i.e., 0 1 z33 z36 z37 (5.32) ZI ¼ @ z63 z66 z67 A; z73 z76 z77 where the subscript I stands for ‘‘individual’’ in an antenna population. The feed model must be defined in the column matrix V now, when calculating the MoM solution. Considering, for example, that we may only feed a unitary voltage gap

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model across the edges of basis functions 1 and 6, we can define a ‘‘mother’’ feed matrix as VM ¼ ð 1

0 0

0

0 1

0 ÞT :

(5.33)

The excitation matrix for the antenna in Figure 5.2(b) is found with the same set of indexes as the impedance matrix, i.e., VI ¼ ð0 1 0ÞT : The MoM solution for the 0 z33 @ z63 V ¼ II ¼ Z1 I I z73

(5.34) individual antenna is 11 0 1 z36 z37 0 z66 z67 A @ 1 A: 0 z76 z77

(5.35)

The same indexes are used to extract individual matrices to compute stored energies or far-fields, e.g., using equations (5.6) or (5.30) where all matrices are replaced with individual matrices, including II calculated above. A GA is neither an exhaustive search of the optimum solution nor an exhaustive evaluation of the characteristics of certain individuals. Such an algorithm uses genetic principles to drive an initially random population toward a suboptimal solution avoiding to some extent local extrema [24]. Genetic principles allow the appearance of unwanted characteristics of offsprings (‘‘malformations’’). Such characteristics may have unpredictable effects on the performance of a fabricated structure [26,56–58]. Two frequent ‘‘malformations’’ are isolated patches and corner connections (Figure 5.3). Isolated patches are active mesh elements (i.e., included in the genotype of an individual) that do not affect the MoM solution because none of the basis functions is defined on such elements. Isolated elements may have no other neighboring active elements or neighboring elements in the corners. These isolated elements are ‘‘purged’’ after each offspring generation in

q II p

II m I n

I

(a)

(b)

Figure 5.3 Example of ‘‘malformations’’ (unwanted characteristics) that may appear in GA-optimized antennas: (a)—isolated mesh elements ‘‘I’’ and ‘‘II’’; (b)—corner connection between mesh elements ‘‘I’’ and ‘‘II’’ that are in the domain of definition of basis functions m, n, p, and q. Metallic regions are depicted in gray shading

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the GA used here. Corner connections are pairs of active mesh elements that are in the domain of definition of two different basis functions and neighbors through a common corner vertex. Different methods can be used to avoid corner connections such as random geometry refinement [56], patch overlapping [57], and faulty-gene purging (used here) [26].

5.3.2 Convex optimization Fundamental bounds presented in Section 5.2.3 are computed with the optimization problems (5.26) and (5.27) that are in fact formulated as convex optimization problems. Convex optimization has a well-developed theory and can be solved efficiently [46]. Moreover, it gives a posteriori error estimates on the solution. Unlike global optimization methods such as GA, it finds the optimal solution; as the local optimum is also the global optimum. Fundamental bounds on the G/Q quotient for an antenna (5.26) can be computed with any convex optimization software package. One example MATLAB library is CVX [59]. The optimization problem (5.26) can be easily written as a CVX model in MATLAB [15]. If the fundamental bounds on the whole structure are of interest, the mother matrices (item 2, Table 5.1) are used directly as inputs of the MATLAB CVX model. In the case of embedded antennas, where only part of the structure should have the controlled currents, a constraint can be added to the optimization problem [13,15]. Once the optimization problem is solved the optimal currents that give the maximum G/Q quotient, the fundamental bounds on G/Q are found. If the dual problem (5.27) is solved with a similar approach, another set of optimal currents giving the same bound is found.

5.4 Examples Combining GA antenna optimization with the fundamental bounds results in a powerful tool to synthesize small antennas. The first two examples illustrate this by finding the optimum antenna for a simplified wireless terminal chassis. The last example demonstrates optimal antenna placement on a wireless terminal. In all examples, the wireless terminal chassis is divided into separate parts: the ground plane and the radiating antenna. The ground plane is kept fixed while the optimization is done on the antenna part. The wireless terminal is modeled as an infinitely thin perfect electrical conductor (PEC) in vacuum.

5.4.1 Bent-end simple phone model The analyzed structures are spatially confined to three rectangular regions connected together as illustrated in Figure 5.4. The first region has the length ‘1 and width w ¼ 7 cm. This region is the fixed ground plane [25,26]. The second and third rectangular regions, with the lengths ‘2 and ‘3 ¼ 0.7 cm, respectively, and width w represent the antenna region [25,26]. The lengths ‘1 and ‘2 are chosen such that ‘1 þ ‘2 ¼ ‘ ¼ 14 cm. The region with the length ‘3 extends in a direction perpendicular to the common plane of the other two regions.

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Antenna

Fixed ground plane

F w l3 l1

l2

Figure 5.4 Illustration of rectangular mesh element discretization and ‘‘rooftop’’ basis function amplitude for a three-dimensional radiating structure. Metal areas are depicted in gray shading. The amplitudes of three of the total 7 3 þ 6 4 4 3 basis functions are depicted in blue, pink, and green shading. The feeding edge is marked F Three cases of the above arrangement are considered. The structures corresponding to these cases have ‘2 ¼ 0.7, 1.4, and 2.8 cm, i.e., 5%, 10%, and 20% of ‘, respectively. To reiterate the whole procedure, the antenna region is used for current optimization, to derive physical limitations (item 4, Table 5.1), and for genetic optimization, to synthesize antennas close to their physical limitations (item 3, Table 5.1). Physical limitations are derived using convex optimization formulation (5.27) for the G/Q -ratio for each situation [13]. Antennas are optimized for minimum Q through the GA/MoM optimization procedure [23,24], see Section 5.3. The mother structure [23,24] corresponding to the arrangement described above consists of three infinitely thin PEC rectangular surfaces with the lengths ‘1, ‘2, and ‘3, and width w arranged as in Figure 5.4. This structure is discretized with a nonuniform mesh, finer in the antenna region than in the ground plane for all cases considered. The first 11.2 cm in the ‘-direction from the left in Figure 5.4 are divided in 40 mesh elements (and 25 in the w-direction). The remaining 2.8 cm in the ‘-direction is divided in 20 mesh elements (and 50 in the w-direction). The bent region is divided in 5 by 50 mesh elements in the ‘3 and w directions, respectively. This particular choice of discretization results in square mesh elements with the side 1.4 mm in the antenna region and 2.8 mm in the ground plane. A row of overlapping basis functions in the ‘-direction at the place of the discontinuity in the mesh size couples electrically the regions with different discretizations. The mother matrices, i.e., the matrices Z, Xe, etc., describing the mother structure, are square with 4,435 rows (item 2, Table 5.1). A block matrix decomposition is applied to these matrices [23]. This decomposition reduces the sizes of the matrices manipulated repetitively during the GA/MoM optimization. These latter matrices are square with 990, 1,485, and 2,475 rows, respectively, for ‘2 ¼ 0.7, 1.4, and 2.8 cm. The genetic optimization of antenna Q has been run for five frequencies, given by ‘/l ¼ 0.1, 0.2, 0.3, 0.4, and 0.5. Five optimized structures have been generated by the GA for each combination of ‘2 and frequency. The smallest optimized-structure

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0.7 cm 1.4 cm 2 = 2.8 cm 2=

102

Bound, rectangle 10

1 0.1

0.2

0.3

0.4

/λ

Figure 5.5 The Q-factors of antennas optimized using a genetic algorithm (‘‘þ’’) compared to corresponding Q-factors of G/Q-optimum current densities [13] (dashed, dash-dotted, and dotted lines), for the bent-end model illustrated in Figure 5.4 with ‘2 ¼ 0.7, 1.4, and 2.8 cm and ‘ ¼ 14 cm. The input impedance of the GA-optimized structures, computed by ESI-CEM [30], has been used to calculate the Q-factors ‘‘’’ using a resonance model [11,27,29]. The physical bound on Q for a rectangular PEC surface 14 7 cm2 [12] is depicted in solid black line

Figure 5.6 Example of GA optimized structures with Q-factors depicted in Figure 5.5 for ‘/l ¼ 0.1 (top row) and ‘/l ¼ 0.5 (bottom row), and ‘2 ¼ 0.7 cm (left column), 1.4 cm (middle column), and 2.8 cm (right column). Gray shading—part of the ground plane, black—antenna region part coplanar with the ground plane, bronze—antenna region part normal to the ground plane. Feeding edges are circled Q-factor (1.5) of the five corresponding to each combination of ‘2 and frequency is labeled ‘‘Pred.’’ in Figure 5.5. The optimized structures with these smallest Q-factors (of which six are depicted in Figure 5.6) have been simulated using the commercial solver ESI-CEM [30] (item 5, Table 5.1). The input impedance of these structures is

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used to obtain the Q-factors labeled ‘‘Sim.’’ in Figure 5.5. These Q-factors agree to a large extent with those obtained using the in-house MoM solver and the discrete expressions (5.6)–(5.5) (less than 6% deviation relative to the former Q values). The single-resonance model described in [11,27], (5.18), is employed to compute the Qfactor for ‘/l ¼ 0.1 and 0.2. The Q-factors for the other frequencies are computed using the multiple-resonance, Brune-synthesis model [21,29]. The single-frequency QZ0 (5.24) has been applied to the structures having the smallest Q-factors mentioned above. The QZ0 values in these cases have less than 5% difference relative to corresponding QZ0 values computed using (5.18). The Q-factors obtained in optimization and simulation are compared to Q-factors given by optimum antenna current distributions, labeled ‘‘Opt.’’ in Figure 5.5. These distributions are obtained using the convex optimization formulation (5.27) for the G/Q-quotient [13]. The matrices involved in these formulations are square with 990, 1,485, and 2,475 rows, respectively, for ‘2 ¼ 0.7, 1.4, and 2.8 cm. These matrices are obtained using a uniform, 1.4-mm-side square mesh element discretization of the mother structure—same mother structure as that considered for GA optimization. The physical bound on the Q-factor of a rectangular PEC region with the dimensions 14 7 cm2, computed using the results in [12], is included for illustration. It is observed in Figure 5.5 that the optimized-structure Q-factors are close to those achieved by optimum antenna currents (less than 13% deviation relative to the optimum-current Q-factors). Note that the current distributions used to compute the curves labeled ‘‘Opt.’’ in Figure 5.5 are optimum in the sense of G/Q . However, the Q-factors computed from these distributions may not be optimum in the sense of the Q-factor. This may result in structures that are on the ‘‘wrong side’’ of the G/Q-optimum current Q-factor, e.g., below the curves in Figure 5.5. The G/Qquotient of such structures is on the ‘‘right side’’ of the physical bound.

5.4.2

Bent-end simple phone model—optimization for QZ0

The bent-end model with ‘1 ¼ 12.6 cm and ‘2 ¼ 1.4 cm, described in Section 5.4.1, has been optimized using the GA for operation between 700 MHz and 960 MHz . This frequency band is divided in two sub-bands with the center frequencies fc,1 ¼ 759.5 MHz and fc,2 ¼ 889.5 MHz. The fractional bandwidths of the two sub-bands are equal, FBW1,2 15.8%. The matrices Z, Xe, Xm, and R are computed for the center frequencies. Two extra impedance matrices are computed for the frequencies 1.001 fc,1,2 in order to evaluate QZ0 at fc,1,2 using (5.18). The cost function minimized by the GA is Q1 Q2 Q1 Q2 FC ¼ aQ;M max ; þ aQ;S þ 7 7 n7 7 o 0 0 (5.36) þ aQZ 0;M max QZ ;1 ; QZ ;2 þ aQZ0 ;S QZ0;1 þ QZ0;2 ; where the indexes 1 and 2 denote the sub-band, Q is the energy-based antenna-Q (5.5), QZ0 is the single-resonance input-impedance-derivative antenna-Q (5.18), and the weights a define the optimization target. The normalization values for Q, 7,

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ensure less than 6 dB reflection coefficient magnitude at the antenna input for the targeted FBW, under the assumption of single-resonance [11]. The QZ0 values are not normalized because some applications target as low QZ0 as possible, i.e., little variation of the input impedance in the operation band. The GA has been run five times for each optimization target whose a-values are listed in Table 5.2. The Q-factors of the four GA-optimized structures depicted in Figure 5.7 (of the total 15 structures) are presented in the same table. The structures corresponding to rows 1, 2, and 3 have the minimum cost function. The structure whose Q-factors are listed on row 4 has been optimized for simultaneous minimum Q and QZ0 , does not have the minimum cost function, but has minimum Q on both sub-bands (out of the total 5 GA-optimized structures with this target). The values for QZ0 listed in Table 5.2 are evaluated with (5.18). These values agree to a large extent with the same values re-evaluated at the center frequencies with (5.24). The four structures of Figure 5.7 have been simulated in ESI-CEM [30]. The magnitudes of the reflection coefficients at the inputs of these structures are depicted in Figure 5.8. Matching networks that yield less than 6 dB reflection coefficient in the entire band have been designed using BetaMatch [60]. These networks are depicted in Figure 5.9 and the resulting S11 magnitudes in Figure 5.8. Real component models of SMD lumped elements, including losses, have been used for matching. The curves in Figure 5.8 offer information about the effort needed to design matching networks for the structures and situation considered. Table 5.2 GA cost function parameters and results for different optimization objectives aQ ,

Target

1 2 3 4

aQZ

0

M

S

M

S

min Q min QZ0

1 0

0.1 0

0 1

0 0.1

min Q and QZ0

1

0.1

1

0.1

1

2

Q1

Q2

QZ0 ,1

QZ0 ,2

4.6 8.2 8.7 6.5

3.7 8.9 6.8 5.5

2.9 0.01 0.08 1.1

0.3 0.01 0.08 1.1

3

4

Figure 5.7 GA-optimized structures whose Q-factors are listed in Table 5.2. Gray shading—part of the ground plane, black—antenna region part coplanar with the ground plane, and bronze—antenna region part normal to the ground plane. Feeding edges are circled

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0.3

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0.5

/λ 1 2 3 4 1m 2m 3m 4m

0 –5 –10 –15 0.5

0.7

1.1 f / GHz

0.9

Figure 5.8 Magnitude of S11 at the input of the structures depicted in Figure 5.7 without matching network, the curves labeled 1, 2, 3, and 4, and with the matching networks sketched in Figure 5.9, the curves labeled 1 m, 2 m, 3 m, and 4 m

2.7 pF 22 nH

1

3.3 pF

2.2 pF

2.2 pF

15 nH

2

4.7 pF

27 nH

3

4

Figure 5.9 Matching networks designed for the structures depicted in Figure 5.7 to yield less than 6 dB reflection coefficient magnitude between 700 . . . 960 MHz (solid curves in Figure 5.8)

5.4.3 Wireless terminal antenna placement using optimum currents Optimum antenna currents can be employed for evaluation and comparison of the performance achievable by a device with antennas placed at different locations. For illustration, we would like to determine the position and shape of the antenna region [25,26] that has the smallest Q-factor in the frequency range of Figure 5.5. The nine 3D simplified models of common hand-held wireless terminals depicted in Figure 5.10 are analyzed. These models have been chosen based on observations of common hand-held devices. The models are limited to a rectangular parallelepiped with dimensions ‘ w h ¼ 14 7 0.7 cm3 (i.e., length width height). Note that limiting the structures to a parallelepiped is introduced for illustration purpose and does not restrict the applicability of the procedure exemplified here. Each model is drawn in Figure 5.10 to scale in three side views from the ‘, w and h -directions (except for Figure 5.10(h) where an h-side view and two sections through the symmetry planes are depicted). Gray and black represent the ground plane and antenna region, respectively. The ground plane [25,26] covers 90% of the area of one ‘ w face of the parallelepiped bounding the antenna. The remaining 10% of that face represents the

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h w a

b

c

d

e

f

g

h

i

Figure 5.10 Nine simplified wireless-device models limited to a parallelepiped, consisting of a planar ground region extending 90% of one length width face, i.e., ‘ w, and an antenna region occupying 10% of the parallelepiped volume. Three side views are depicted for (a)–(g) and i, i.e., structures as seen along the length, width and height. A side view along the height and two sections at the symmetry planes are depicted for h. Gray shading—ground plane; black—antenna region [25,26]

Table 5.3 Dimensions of MoM matrices for the structures of Figure 5.10 Struct.

a

b

c

d

e, f

g

h

i

N NAR

7,584 1,936

8,256 2,608

7,568 1,928

7,584 1,944

8,256 2,616

8,256 2,612

10,830 5,168

7,584 1,992

support of the antenna region, which may be continuous or divided in more subregions. Here, a maximum of two sub-regions have been used. The structures in the antenna regions are limited to infinitely thin PEC sheets placed on faces of the 3D shape of the antenna region. This shape is obtained by translating the 10% of the ‘ w-face area reserved for the antenna region a distance h perpendicular to the ground plane (i.e., by extruding the 10% in the h-direction to the opposed face). The shapes resulting in the antenna region are made of rectangular parallelepipeds. These parallelepipeds are covered with PEC sheets on the four largest area faces (in the case depicted in Figure 10(h) there are four openings adjacent to the ground plane corners in the w h-plane; these are one mesh-element wide and extend the entire h-dimension). The antenna region placement cases introduced above are discretized using a uniform mesh of 1.75 1.75 mm2 rectangular elements. The total number of basis functions, N, resulting in the structures depicted in Figure 5.10 are presented in Table 5.3 (i.e., the number of rows and columns, where applicable, of Z, Z0 , Xe,

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G/Q 1

a b c d e f g h i R

0.1 0.01 10−3 0.1

0.2

0.3

0.4

/λ

Figure 5.11 Physical bounds on G/Q for the structures depicted in Figure 5.10 obtained using the convex optimization formulation (5.27) [13] when only the antenna region (black in Figure 5.10) is optimized. The physical bound on G/Q for a rectangular PEC surface 14 7 cm2 [12] is depicted in solid black line and labeled ‘‘R’’ Xm, R, and F). The same table presents the number of rows and columns, where applicable, NAR, of the blocks [23], corresponding to the 10%-‘ w-area antenna region [25,26]. These blocks are computed for the matrices involved in the convex optimization formulation (5.27). The bounds on G/Q using formulation (5.27) for the simplified models of Figure 5.10 are depicted in Figure 5.11. The formulation has been solved for 46 ‘/l-values between 0.06 and 0.51 (frequency between 128 MHz and 1.092 GHz). Linear polarization along the length and directivity in the direction of the height of the parallelepiped bounding the models are considered. The bound computed using the results in [12]1 for a rectangular, infinitely thin, 14 7 cm2 PEC sheet is labeled ‘‘R’’ in Figure 5.11. The G/Q-optimum current distributions giving the physical bounds in Figure 5.11 are used to compute the Q-factors (5.5) depicted in Figure 5.12. The physical bound on Q for a rectangular 14 7 cm2 PEC sheet [12] is labeled ‘‘R’’ in Figure 5.12. The ring structure depicted in Figure 10(h) outperforms all other structures in the figure in terms of G/Q and Q , except for a frequency region around ‘/l 0.1 where the structure in Figure 5.10(b) has a greater G/Q. We also note that around ‘/l 0.37 a few of the structures in Figure 5.10 reach close to the G/Q bound of a rectangular region and the structure in Figure 5.10(h) has a G/Q value greater than that of a rectangular region. The optimumcurrent Q-factors do not reach as close to the physical bound on Q for a rectangular region as the G/Q-values. Note that, in this example, we have done items 1, 2, and 5 in Table 5.1. The next step would be to run the GA/MoM for the structure in Figure 5.10(h) by assuming a feed location (item 3, Table 5.1). Further on, GA-optimized structures should be verified using a commercial simulator, item 5, Table 5.1.

1

http://www.mathworks.se/matlabcentral/fileexchange/26806-antennaq

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f / GHz, = 14 cm 0.8 1

Q

a b c d e f g h i R

102 10

1 0.1

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/λ

Figure 5.12 The Q-factors (5.5) achieved by the currents that give the optimum G/Q-values depicted in Figure 5.11. The Q-factor of a 14 7 cm2 PEC rectangle [12] is labeled ‘‘R’’

5.5 Conclusions This chapter reviewed the use of fundamental bounds and the Q-factor in small antenna optimization. With the fundamental bounds, the antenna designer can estimate how well the antenna will perform before the design process. This can provide insight if the design specifications can be met with the structure at hand. Moreover, knowledge of the antenna’s bounds can be used in a physical limitationaware optimization, where the optimization process can be terminated once the target is achieved with a certain margin. Formulating the bounds as a convex optimization problem offers the flexibility to add additional ‘‘convex’’ constraints with minor effort. Examples of additional constraints include limitations on efficiency, SAR, and the radiation pattern or optimizing the antenna region of embedded antennas. The antenna designer can investigate numerous situations by adding the previous constraints to the original convex problem. The introduction of a method to estimate QZ0 of antennas from the current distribution computed for a single frequency; the application of fundamental bounds and of the QZ0 single-frequency estimation method to design cases of threedimensional radiating structures has not been considered previously in the literature. The results suggest that customized physical bounds, optimum currents, and single-frequency expressions are tools that are useful for antenna design, e.g., to stop an optimization process, assess realizability of specifications, and assess performance of antenna locations. While the examples in this chapter considered PEC material, the optimization can be extended to antennas consisting of composite materials such as PEC and dielectrics [61].

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[2] L. J. Chu, ‘‘Physical limitations of omni-directional antennas,’’ J. Appl. Phys., vol. 19, pp. 1163–1175, 1948. [3] H. A. Wheeler, ‘‘Fundamental limitations of small antennas,’’ Proc. IRE, vol. 35, no. 12, pp. 1479–1484, 1947. [4] R. F. Harrington, Time Harmonic Electromagnetic Fields. New York, NY: McGraw-Hill, 1961. [5] R. E. Collin and S. Rothschild, ‘‘Evaluation of antenna Q,’’ IEEE Trans. Antennas Propag., vol. 12, pp. 23–27, Jan. 1964. [6] W. Geyi, ‘‘Physical limitations of antenna,’’ IEEE Trans. Antennas Propag., vol. 51, no. 8, pp. 2116–2123, Aug. 2003. [7] A. Karlsson, ‘‘Physical limitations of antennas in a lossy medium,’’ IEEE Trans. Antennas Propag., vol. 52, pp. 2027–2033, 2004. [8] D. M. Pozar, ‘‘New results for minimum Q, maximum gain, and polarization properties of electrically small arbitrary antennas,’’ in 3rd European Conference on Antennas and Propagation, 2009 (EuCAP 2009), Mar. 2009, pp. 1993–1996. [9] J. C.-E. Sten, P. K. Koivisto, and A. Hujanen, ‘‘Limitations for the radiation Q of a small antenna enclosed in a spheroidal volume: axial polarisation,’’ ¨ Int. J. Electron. Commun., vol. 55, no. 3, pp. 198–204, 2001. AEU [10] H. L. Thal, ‘‘New radiation Q limits for spherical wire antennas,’’ IEEE Trans. Antennas Propag., vol. 54, no. 10, pp. 2757–2763, Oct. 2006. [11] A. D. Yaghjian and S. R. Best, ‘‘Impedance, bandwidth, and Q of antennas,’’ IEEE Trans. Antennas Propag., vol. 53, no. 4, pp. 1298–1324, 2005. [12] M. Gustafsson, C. Sohl, and G. Kristensson, ‘‘Illustrations of new physical bounds on linearly polarized antennas,’’ IEEE Trans. Antennas Propag., vol. 57, no. 5, pp. 1319–1327, May 2009. [13] M. Gustafsson and S. Nordebo, ‘‘Optimal antenna currents for Q, superdirectivity, and radiation patterns using convex optimization,’’ IEEE Trans. Antennas Propag., vol. 61, no. 3, pp. 1109–1118, 2013. [14] M. Gustafsson, D. Tayli, and M. Cismasu, Physical bounds of antennas. Handbook of Antenna Technologies Springer-Verlag, 2015, pp. 1–32. [15] M. Gustafsson, D. Tayli, C. Ehrenborg, M. Cismasu, and S. Nordebo, ‘‘Antenna current optimization using MATLAB and CVX,’’ FERMAT. 2016, vol. 15, no. 5, pp. 1–29. [Online]. Available: http://www.e-fermat.org/ articles/gustafsson-art-2016-vol15-may-jun-005/¼0pt [16] L. Jelinek and M. Capek, ‘‘Optimal currents on arbitrarily shaped surfaces,’’ IEEE Trans. Antennas Propag., vol. 65, no. 1, pp. 329–341, 2017. [17] M. Capek, M. Gustafsson, and K. Schab, ‘‘Minimization of antenna quality factor,’’ IEEE Trans. Antennas Propag., vol. 65, no. 8, pp. 4115–4123, 2017. [18] G. A. E. Vandenbosch, ‘‘Reactive energies, impedance, and Q factor of radiating structures,’’ IEEE Trans. Antennas Propag., vol. 58, no. 4, pp. 1112–1127, 2010. [19] M. Gustafsson, M. Cismasu, and B. L. G. Jonsson, ‘‘Physical bounds and optimal currents on antennas,’’ IEEE Trans. Antennas Propag., vol. 60, no. 6, pp. 2672–2681, 2012.

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[20] M. Gustafsson and B. L. G. Jonsson, ‘‘Stored electromagnetic energy and antenna Q,’’ Progress In Electromagnetics Research (PIER), vol. 150, pp. 13–27, 2015. [21] M. Gustafsson and B. L. G. Jonsson, ‘‘Antenna Q and stored energy expressed in the fields, currents, and input impedance,’’ IEEE Trans. Antennas Propag., vol. 63, no. 1, pp. 240–249, 2015. [22] M. Shahpari, D. Thiel, and A. Lewis, ‘‘An investigation into the Gustafsson limit for small planar antennas using optimization,’’ IEEE Trans. Antennas Propag., vol. 62, no. 2, pp. 950–955, Feb. 2014. [23] J. M. Johnson and Y. Rahmat-Samii, ‘‘Genetic algorithms and method of moments GA/MOM for the design of integrated antennas,’’ IEEE Trans. Antennas Propag., vol. 47, no. 10, pp. 1606–1614, Oct. 1999. [24] Y. Rahmat-Samii and E. Michielssen, Electromagnetic Optimization by Genetic Algorithms, ser. Wiley Series in Microwave and Optical Engineering. New York, NY: John Wiley & Sons, 1999. [25] M. Cismasu and M. Gustafsson, ‘‘Antenna bandwidth optimization with single frequency simulation,’’ IEEE Trans. Antennas Propag., vol. 62, no. 3, pp. 1304–1311, 2014. [26] M. Cismasu and M. Gustafsson, ‘‘Multiband antenna Q optimization using stored energy expressions,’’ IEEE Antennas and Wireless Propagation Letters, vol. 13, no. 2014, pp. 646–649, 2014. [27] M. Gustafsson and S. Nordebo, ‘‘Bandwidth, Q-factor, and resonance models of antennas,’’ Prog. Electromagn. Res., vol. 62, pp. 1–20, 2006. [28] O. Brune, ‘‘Synthesis of a finite two-terminal network whose driving-point impedance is a prescribed function of frequency,’’ MIT J. Math. Phys., vol. 10, pp. 191–236, 1931. [29] O. Wing, Classical Circuit Theory. New York: Springer, 2008. [30] ‘‘ESI Group, Paris, France—ESI Group’s computational electromagnetic (CEM) solution,’’ http://www.esi-group.com. [31] R. F. Harrington, Field Computation by Moment Methods. New York, NY: Macmillan, 1968. [32] A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods for Electromagnetics. New York, NY: IEEE Press, 1998. [33] J. M. Jin, Theory and Computation of Electromagnetic Fields. Wiley, 2011. [34] W. C. Gibson, The Method of Moments in Electromagnetics. CRC press, 2014. [35] 145–2013 — IEEE Standard for Definitions of Terms for Antennas, IEEE Std. [36] K. Schab, L. Jelinek, M. Capek, et al., ‘‘Energy stored by radiating systems,’’ IEEE Access, vol. 6, pp. 10553–10568, 2018. [37] M. Gustafsson, D. Tayli, and M. Cismasu, ‘‘Q factors for antennas in dispersive media,’’ Lund University, Department of Electrical and Information Technology and P.O. Box 118 and S-221 00 Lund, Sweden, Tech. Rep. LUTEDX/ (TEAT-7232)/1–24/(2014), 2014, http://www.eit.lth.se. [Online]. Available: http://lup.lub.lu.se/record/4648444¼0pt

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[56] M. Ohira, H. Deguchi, M. Tsuji, and H. Shigesawa, ‘‘Multiband single-layer frequency selective surface designed by combination of genetic algorithm and geometry-refinement technique,’’ IEEE Trans. Antennas Propag., vol. 52, no. 11, pp. 2925–2931, Nov. 2004. [57] M. John and M. Ammann, ‘‘Wideband printed monopole design using a genetic algorithm,’’ Antennas and Wireless Propagation Letters, IEEE, vol. 6, pp. 447–449, 2007. [58] B. G. Xia, J. Meng, D. H. Zhang, and J. S. Zhang, ‘‘PMM-GA method to synthesize quasi-optical frequency selective surface on SiO2 substrate,’’ Prog. Electromagn. Res., vol. 139, pp. 599–610, 2013. [59] M. Grant and S. Boyd, ‘‘CVX: MATLAB software for disciplined convex programming, version 1.21,’’ http://cvxr.com/cvx, Apr. 2011. [Online]. Available: http://cvxr.com/cvx/¼0pt [60] ‘‘MNW Scan, Singapore—BetaMatch, Software for antenna component matching,’’ http://www.mnw-scan.com/. [61] M. Gustafsson and C. Ehrenborg, ‘‘State-space models and stored electromagnetic energy for antennas in dispersive and heterogeneous media,’’ Radio Sci, vol. 52, no. 11, pp. 1325–1343, 2017.

Chapter 6

Fast analysis of active antenna systems following the Deep Integration paradigm Rob Maaskant1

The ultimate form of system integration is arguably to fuse conducting, semiconducting, and non-conducting materials into a single heterogeneous material having the multifunctional properties of a full-blown antenna system. This chapter introduces the so-called Deep Integration paradigm and discusses the fast numerical analysis of these potentially next-generation deeply integrated antenna structures. These multiscale problems are analyzed efficiently through a numerical enhancement technique, called the Characteristic Basis Function Method.

6.1 Introduction: The Deep Integration paradigm Merging wireless subsystems into a single multifunctional RF component has major benefits, particularly at high frequencies, where material losses, spurious radiation from discontinuities and subsystem interconnects, fabrication errors, cross-talk effects, packaging resonances, and so on are more likely to occur. The ultimate form of integration where macroscopic electromagnetic (EM) wave effects and microscopic device physics phenomena interplay throughout a single inhomogeneous distribution of conducting, semiconducting, and insulating materials is called ‘Deep Integration’ [1]. Figure 6.1 shows the various levels of integration that may exist. The classical 50-Ohm isolated subsystem design is depicted on the left, while the middle figure illustrates ‘strong integration’, where (50-Ohm lossy) transmission lines are avoided as much as possible in a co-design approach involving non-standard reference impedances. For both these cases, the distribution of antenna currents, and thus the directivity pattern of the single-port antenna, is not affected by the amplifier characteristics, as opposed to the situation on the right, where the amplifier and antenna functionalities are merged and cannot be separated any longer without affecting the antenna and/or amplifier characteristics. In fact, removing the transistor(s) from the antenna affects the antenna currents and thereby

1

Department of Electrical Engineering (E2), Chalmers University of Technology, Sweden

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Figure 6.1 Levels of integration, from lowest (left) to highest (right)

Classical bottom-up design flow

Deep integration top-down design flow Initial material distribution

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Figure 6.2 Classical bottom-up versus a top-down ‘Deep Integration’ design flow its radiation properties, while removing the antenna removes the parasitic environment to the transistor(s) making up the amplifier. In connection with Figure 6.1, Figure 6.2 (left) illustrates a classical bottom-up design flow, where an active antenna system is designed by optimizing individual components or subsystems in isolation, typically for a fixed port reference impedance. Conversely, in the top-down ‘Deep Integration’ design flow (right), one starts from the required system specifications, after which the optimal material distribution is synthesized through EM wave and material models along with a global optimization technique. In the latter case, the strongly inhomogeneous material may represent a distributed and deeply integrated EM antenna system, where interior subsystems are no longer identifiable; detrimental cross-talk effects disappear since EM field interference effects are naturally exploited; port reference impedance requirements disappear and subsystem interconnection loss and DC/RF

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Figure 6.3 An illustration of a deeply integrated antenna system wiring problems can potentially be overcome. The final optimized interior material distribution will likely be unrecognizable from a classical subsystem point-of-view; yet, the material’s exterior EM response will satisfy the system-level requirements. The sheer complexity of this multi-physics synthesis problem reduces significantly if a spatially extended semiconductor material can be modeled as a distribution of diode-type junctions. That is, local transitions from lower to higher concentrations of negative charge carriers, P to N doped, N to Nþ, an interface of two differently conducting metals, and so on can all be modeled by diode-type junctions. Even single-diode mixers, which are common practice at (sub-)THz frequencies, or single-diode amplifiers (e.g. IMPATT/tunnel diodes) exist with first-order system functional properties. The diode can therefore be considered as the semiconductor building block. Due to the large variety of diodes, it therefore suffices to work mostly with diode-like (stationary and) non-linear I–V characteristics. It is pointed out, however, that in certain specific cases, it may be necessary to consider higher-level two-port non-linear I–V characteristics as well, since these are capable of representing the complex physical behavior inside two-port elementary semiconductor building blocks, such as transistors. An illustration of the material distribution concept is shown in Figure 6.3. With reference to the synthesis procedure in Figure 6.2 (right) and the synthesized structure in Figure 6.3, parts of the inhomogeneous material will be exposed to a static electric and/or magnetic field as a result of which DC currents start flowing throughout the 3-D material. The DC volumetric currents determine the interior operating points along the respective I–V curves. This provides the active material distribution. The small-signal RF field is superimposed on that and undergoes the desired system-level transformation (e.g. reception, amplification, beamforming, etc., all at once1). If the material RF response does not satisfy the system specifications, the material distribution is modified [cf. Figure 6.2(right)], 1 For a large-signal model, the DC-biasing and small-signal RF analysis blocks in Fig. 6.2(right) are replaced by a time-domain – or harmonic balance type – EM field solver. Its discussion is, however, beyond the scope of this chapter.

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etc. The system design problem therefore translates to the major challenge of a fast multiscale EM (non-linear) optimization problem. One can expect to synthesize EM wave-front amplification solutions in conjunction with spatial power distribution/combination implementations to overcome the limited and inefficient power generation of miniaturized devices [2]. The problem of heat dissipation can potentially be incorporated as a constraint in the optimization technique so that fields tend to propagate through lowloss (insulation) regions and will also be spatially distributed to avoid hot spots. Such constraints can also be applied to satisfy certain design rules for integrated circuits, to render it insensitive to mechanical stresses, or to prevent field breakthrough effects (for high power handling). The complexity can be enormously large and is of a multidisciplinary nature if one wants to incorporate all such effects. This chapter will therefore deal with the simplest cases first and elaborates on the multiscale numerical analysis block ‘RF Simulation’ in Figure 6.2 (right) in particular.

6.1.1

Potential impact and other integration approaches

The Deep Integration paradigm leads to a breakthrough in RF system integration if it manages to solve the above-mentioned fundamental bottlenecks and may potentially bridge the microelectronics, electromagnetics, and material science fields. Deep Integration could become a research field complementing bottom-up design methodologies, which is very timely as semiconductors stop shrinking in size in 2024 and will need to be constructed in 3-D, i.e. vertically.2 It is worth pointing out that antenna and waveguide technologies that are capable of providing intrinsic gain have already been reported in the 1960s, where low-frequency off-chip active antennas (called antennafier ¼ antenna þ amplifier) [3] and even diode-loaded waveguides were proposed [4]. The work by Prof. Rutledge and his collaborators in the 1990s on grid amplifiers/oscillators/mixers elaborates on a similar vision [5], but the proposed distributed antenna-amplifier array technology has not found application in on-chip integrated antenna systems as envisioned by the generalizing Deep Integration paradigm. Yet, the unfolding trend on RF system integration was recognized and referred to as integrated-circuit antennas, or simply integrated antennas [6–8]. The research in 2007 by Prof. Rizzoli and his group on material co-simulation strategies incorporate semiconductor devices by separating the computational domains into passive and nonlinear (semiconducting) regions while optimizing the combined system using neural networks. The synthesized material is, however, not truly distributed as the optimized design is constrained to LTCC technology [9]. More recent generalizing ideas consider metamaterials or even meta-atoms [10] but seldom employ nonlinear or linearized active semiconducting materials, which is ultimately essential if the purpose is to synthesize antenna systems that include amplifiers, mixers,

2 IEEE International Roadmap for Devices and Systems, ‘More Moore White Paper,’ 2016 Edition, http://irds.ieee.org/images/files/pdf/2016_MM.pdf.

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harvesting rectannas, and so forth. Varactor diodes, pin diodes, or RF MEMS switches have been considered as a means to reconfigure the antenna structure [11] and used as a means to miniaturize antennas [12]. This could be regarded as a specific realization of Deep Integration. The insertion loss of these components at high frequencies could benefit from compensating negative resistance material, which is obtained after DC biasing the non-linear semiconductor material [cf. Figure 6.2 (right)]. Similarly, spatially distributed positive radiation resistance can be combined with negative resistance material to form active antennas that may partly compensate for Ohmic losses. Finally, with the advent of powerful computers, fast numerical techniques, and future trends in chip designs and material processing technologies, the time is ripe to re-think previously proposed integrated concepts by generalizing it into a concept called Deep Integration.

6.1.2 Scientific and technological challenges The future synthesized inhomogeneous material is not only many wavelengths in size but also exhibits fine geometrical features that are only fractions thereof. The numerical analysis of this so-called multiscale structure poses a serious computational burden, in terms of both memory usage and runtime (Challenge 1: fast numerical analysis). Multiscale device modeling is one of the major future research challenges of the semiconductor industry (cf. Intel [13]) that impedes the development of future integrated wireless systems. Such demands have contributed to the launch of the IEEE Journal on Multiscale and Multiphysics Computational Techniques (2015), which addresses interdisciplinary research challenges from the MTT, AP, and EMC communities. Solving the first challenge speeds-up a single iteration in Figure 6.2(right); however, the synthesis problem is still numerically intractable if many iterations are needed (Challenge 2: fast numerical optimization). To prove that Deep Integration has indeed the potential to overcome several fundamental bottlenecks constitutes another challenge (Challenge 3: proof-ofconcept). The latter includes demonstrating the merits of power distribution implementations in combination with wave-front amplification techniques, showing an increased globally optimized system performance, but also the speed-up that can be achieved by employing numerical efficient algorithms relative to standard approaches (progress on CEM techniques).

6.2 Modeling approach and assumptions Whether it concerns a (small or large signal) frequency or time domain (transient) analysis, the idea is to separate the passive linear part in Figure 6.3 from the nonlinear parts. This is illustrated in Figure 6.4 and is not new in non-linear circuit optimization [9], but in the distributed Deep Integration context of a multiscale EM problem with the chosen elementary non-linear materials (predominantly diodes) it is. The passive linear region is characterized rapidly through dominant-mode EM

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Z Internal + ports

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Figure 6.4 Separating linear from nonlinear materials

wave models, while the active non-linear regions are electrically small and thus accounted for by diode-like I–V characteristics that are embedded locally inside this linear passive material. The ability to incorporate a more complex base material/element, such as a transistor, is also shown in Figure 6.4. This may be important whenever the one-port non-linear diode-type junction turns out inadequate. Technology Computer-Aided Design software could provide us with I–V characteristics, but this is not needed in our approach because we will assume typical (time-independent) diode-type I–V characteristics as a starting point. The superposition principle holds in the linear material; hence, fast Green’s function approaches and spectral domain techniques can be used for the EM part. The ports must be interpreted as generalized (interior) ports, each of which is typically associated with a low-order basis function in a method-of-moments (MoM) formulation. Assuming N basis functions, the N N linear MoM matrix equation ZI ¼ V must hold [for details, see Eq. (6.4)], while at the same time the non-linear vector function V(I) must be satisfied to account for the non-linear I–V characteristics (see the network connection in Figure 6.4). The self-consistent system of non-linear equations ZIV(I) ¼ 0 must be solved iteratively for I, the elements of which are related to the field mode expansion coefficients for the scattered field inside the electrically large material. Physics-based basis functions for modeling the EM fields can reduce the degrees-of-freedom of fields and currents dramatically (typically factor 500 for linear antenna problems, or more, depending on the geometrical feature size [14]). This will be shown in the subsequent sections. The externally applied incident field gives rise to internal sources in Z. In the frequency domain, the currents I are determined for an external static field first, after which the operating points along the I–V curves in the bottom block in Figure 6.4 are determined. Uniqueness of the steady-state DC solution must be studied (beyond the scope of this book chapter). Accordingly, a small signal RF analysis is performed for the linearized system around the operating points of the I–V curves. In the following, we will limit ourselves to the analysis of thin planar material sheets of perfectly conducting metals (PECs) and perform a frequency domain (small-signal) analysis for already DCbiased active materials.

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6.3 The antennafier array element 6.3.1 Concept We will consider a simple antennafier (antenna þ amplifier) array element where the distribution of negative resistance is combined with positive radiation resistance material. The fast numerical analysis of antennafier arrays is treated in the next section, the multiscale analysis of which is the focus of this chapter. The antennafier concept is shown in Figure 6.5. Figure 6.5 (top-left) shows an antenna that is attached to an RF voltage source with amplitude VRF and source resistance RRF. The maximum available power from the RF source is Pav ¼ |VRF|2/(4RRF). The antenna is series connected to a tunnel diode having the typical non-linear I–V characteristics as shown in Figure 6.5 (top-right). The tunnel diode is DC biased through a DC-voltage source in its negative resistance area at VDC. A small-signal RF voltage swing DVd across the diode will give rise to a current swing DId through the diode. Tunnel diode parasitic effects are disregarded for the sake of simplicity. The I–V relation is approximately linear in this area and given by DVd ¼ RdDId, where Rdiode ¼ Rd is the value of negative differential resistance. Considering RF only (DC disregarded), the small-signal equivalent electrical antenna–amplifier circuit is shown in Figure 6.5 (bottom-left). One observes that the positive radiation resistance is reduced by the negative resistance of the tunnel diode, resulting in a larger current Id than when the tunnel diode is absent (shortcircuited). In fact, if RRF þ Rant ? Rd, then |Id| ? ?. This will result in a large radiated power since Prad ¼ 1/2|Id|2 Rrad. However, we should not exceed the voltage swing DVd across the tunnel diode, else distortion of the radiated signal will occur. This limits the maximum achievable output power. The concept of mixing

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Figure 6.6 Gain G as a function of Rrad and Rdiode for RRF ¼ 50W positive and negative resistance materials can be generalized in a 3-D distributed manner as indicated in Figure 6.5(bottom-right) for the 1-D case. Designing the antenna without tunnel diode (Rdiode = 0) would require us to impedance match the antenna such as to achieve RRF ¼ Rrad. In that case Pav ¼ Prad. This could be defined as the reference radiated power so that the system available gain G can be defined with respect to this reference power Pav as G¼

Prad 2Rrad RRF ¼ Pav jRrad þ RRF þ Rdiode j2

(6.1)

The gain G can be larger than unity in case the tunnel diode is attached and properly biased so that the active antennafier radiates larger power than for a passive impedance-matched antenna for which RRF ¼ Rrad and Rdiode ¼ 0. A plot of G for different values of Rrad and Rdiode is shown in Figure 6.6 for the case that RRF ¼ 50W. Note that, for the typical value Rdiode ¼ 50W and Rrad ¼ 50W, G ¼ 2 (i.e. 3 dB). Henceforth, we will assume in the numerical analysis that ● ●

●

●

The system is properly DC biased; that is, only RF behavior is considered. The use of the small-signal model is adequate; that is, large signal distortion effects are assumed negligibly small and therefore not modeled. The parasitic electrodynamic effects of diodes can be ignored; that is, pure negative resistances are used for all frequencies. Only one tunnel diode per antenna array element is employed; that is, the active part of the medium is localized, not arbitrarily distributed in 3-D.

6.3.2 Method-of-moments analysis of a folded dipole antennafier The geometry of the considered antennafier element is shown in Figure 6.7. It is a two-port half wave folded dipole antenna with L3 ¼ l/2 at f ¼ 1 GHz.

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Geometry L3 2 W2 L1 W1

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Figure 6.7 The antennafier element geometry: a two-port folded dipole over ground. External port 1: RF þ DC-biasing; Internal port 2: the active ‘diode port’ At the external port 1, we apply the RF and DC-biasing voltages. The tunnel diode will be connected across the ‘internal port’ 2. Note that the bias voltage applied to port 1 will for DC be directly transferred – and thus be identical – to the voltage across the tunnel diode at port 2. This is not true for RF signals due to the non-negligible radiation properties of the antenna at high frequencies. The antenna impedance and radiation properties are determined numerically through the MoM as described next. In the absence of non-linear components the two-port structure is passive and linear. Hence, we can perform a frequency domain EM analysis. We will further assume perfect electric conductor (PEC) material sheets that are much thinner than the free-space wavelength. The objective is to determine the surface currents for the two port excitations, which in turn enables one to derive the 2 2 antenna input impedance matrix and the two far-field element patterns. This is a sufficientenough characterization of the antenna element for the intended EM/circuit co-simulation purposes. The unknown current distribution J that is yet to be determined is expanded in terms of N known vector basis functions ff n gNn¼1 with unknown expansion coefficients fIn gNn¼1 , i.e. JðrÞ ¼

N X I n f n ðr Þ

(6.2)

n¼1

This current radiates an E-field Es(J) in free-space that is oppositely directed to the incident field Ei to yield zero total E-field tangential to the surface of the conductor to satisfy the PEC boundary condition, that is, Es(J)|tan ¼ Ei|tan. By using the above expansion for the current, the boundary condition becomes [15]: N X In Es ðf n Þ ¼ Ei tan (6.3) n¼1 tan

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Tn

ρn+ rn+

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r

O

Figure 6.8 The Rao–Wilton–Glisson basis function where we used the linearity of the operators that are involved in computing Es. This single boundary condition can, for a set {fn} with N expansion coefficients {In}, in general not be satisfied everywhere on the PEC surface. We therefore test the boundary condition (6.3) in a weak form, i.e. N times using the weight functions ff m gNm¼1 (Galerkin’s approach) to yield the system of N equations with N unknowns: N X Zmn In ¼ Vm for m ¼ 1; . . . ; N

(6.4)

n¼1

where ZÐ mm ¼ hfm,Es(fn)i, Vm ¼ hfm, Eii, and where the symmetric product ha,bi ¼ a bdS. The direct solution of the resulting matrix equation ZI ¼ V requires a run time that scales with OðN 3 Þ, while the memory storage scales as OðN 2 Þ. Too many basis (and test) functions render the solution of the large matrix equation intractable. On the other hand, employing too few basis functions does not allow for enough flexibility in modeling for the large degrees-of-freedom of the current in regions where it is needed. To retain enough flexibility in modeling the geometry and discretizing the current, we triangulate the metal surfaces as shown in Figure 6.1. Each basis function fn is defined on a pair of triangles to represent a spatially localized dipole-type of current distribution. This is known as the Rao-WiltonGlisson (RWG) basis function, which is shown in Figure 6.8, and whose function definition is given by (see also [16]): 8 ‘n þ > > rn ; r 2 Tnþ > > > 2Aþ > < n ‘n (6.5) f n ðr Þ ¼ > r ; r 2 Tn > n > 2A > n > > : 0 otherwise Here, A n designates the area of triangle T with base length ‘n. The vector rn is normalized such that its component normal to the common edge equals unity. Note þ that the magnitude of the vector field rþ n ðrÞ is linearly increasing in Tn when the observation point moves from the corner vertex rþ n towards the common edge, ð Þ r linearly decreases towards the corner vertex of Tn . whereas the magnitude of r n

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Figure 6.9 (a) The input impedance of the antenna at port 1 and (b) gain pattern at 750 MHz. Port 2 is short-circuited The numerical evaluation of the matrix elements and the treatment of singularities in particular (on-diagonal matrix elements) is beyond the scope of this chapter. Interested readers are referred to [17]. The right-hand-side vector V contains two nonzero entries each of which corresponds to the port basis functions modeling the port current across the voltage gap. The gap is formed by the common edge of two triangles that are indicated by the light/dark-shaded (þ/) polarities (cf. Figure 6.7). If element Vp corresponds to the p th RWG modeling the circuit port current, then Vp ¼ hfm,Eii ¼ Vant‘p, where Vant is the applied circuit port voltage and ‘p is the common edge length of the pth RWG [18]. Both the computed antenna port impedance and radiation characteristics for the case that port 2 is short-circuited are shown in Figure 6.9. One can observe a good agreement with the commercial software CST. This validates the MoM code that will serve as the reference solver for the large array simulations described later in Section 6.4. The input (radiation) resistance at 750 MHz is about 86W with almost zero input reactance. The gain of the folded dipole above ground at this resonance frequency of 750 MHz is about 7.9 dBi. The next step is to open port 2. The MoM-computed 2 2 antenna input impedance matrix at 750 MHz is Zant;11 Zant;12 297 815j 58:8 þ 895j Zant ¼ ¼ (6.6) Zant;21 Zant;22 58:8 þ 895j 198 974j The matrix elements Zant;11, Zant;12 ¼ Zant;21 (reciprocity), and Zant;22 can be represented by an equivalent T-network as shown in Figure 6.10. With the tunnel diode and source connected, we can compute the expected power gain as follows. First, we define the source impedance matrix, which is a diagonal matrix, i.e. 0 RRF Zsource ¼ (6.7) 0 Rdiode

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Figure 6.10 Two-port network of antenna with tunnel diode and source connected

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Figure 6.11 Series connection of the source and antenna Z-matrices Next, the two-element vector of voltage excitation sources Vsource ¼ [VRF,0]T, where T is the transposition operator, is connected to Zsource which in turn is connected to Zant. This is visualized in Figure 6.11. From this network, it is readily clear that the time-average radiated power, or dissipated power in Zant (radiation efficiency is 100%), is 1 Prad ¼ Re IH ant Zant Iant 2

(6.8)

where H denotes the Hermitian operator, and because Pav ¼ |VRF|2/(4RRF), the power gain is computed as G¼

Prad 2RRF ¼ Re IH ant Zant Iant 2 Pav jVRF j

(6.9)

where Iant is computed by realizing that Vant ¼ ZantIant and Vant ¼ Vsource ZsourceIant so that Iant ¼ ðZsource þ Zant Þ1 Vsource

(6.10)

A contour plot of the power gain G is illustrated in Figure 6.12 as a function of Rdiode and RRF. One concludes that for RRF ¼ 50W, and a negative diode differential resistance of Rdiode ¼ 86W, the gain G 2, which is about 3 dB. In other words, the antenna radiates twice more power than its passive counterpart, which is assumed to be power-matched to a 50-Ohm source resistance. Two questions remain: (i) what is the Sant;11 at the RF input port and (ii) how does the element pattern deform in the presence of the tunnel diode.

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Figure 6.12 Linear power gain G of two-port antenna with tunnel diode and The´venin source connected The Sant;11 can be computed with the help of Figure 6.10, that is, Sant;11 ¼

Vant;1 RRF Iant;1 5:6ej164 dB Vant;1 þ RRF Iant;1

(6.11)

which is not very well matched; yet, the total absolute radiated power is twice higher than a passive antenna that is power matched to RRF. It is pointed out that no optimization has been performed and no matching network has been designed since the main purpose of this chapter is to introduce the antennafier and to compute large arrays thereof. The radiation pattern in Figure 6.2(b) is affected because the antenna gets, in effect, excited at port 2 by the tunnel diode. The total element pattern g(q,f) is therefore expressed as gðq;fÞ ¼ Vant;1 g1 ðq;fÞ þ Vant;2 g2 ðq;fÞ

(6.12)

where g1(q,f) and g2(q,f) are the far-field pattern functions when exciting the antenna at its two ports by a unit voltage source, respectively, while short-circuiting the other port. This far-field function can be normalized to the total RF input power, see Eq. (6.8), to yield the antenna gain pattern in dBi. For the considered example: Vsource ¼ [1;0]T, which gives rise to Vant;1 ¼ 0.25exp( j16 ) and Vant;2 ¼ 1.19exp(j8 ). This means that although the RF port 1 is weakly excited (25%), the tunnel diode port 2 is strongly excited (119%). Collectively, this results in an increased input power, as discussed above, and affects the antenna gain pattern, albeit in this case only in a minor way. This minor pattern change can be observed in Figure 6.13 when compared to the gain pattern of the passive antenna in Figure 6.9 (b). The pattern has become less directive since the gain has dropped by 0.9 dB, which is due to a redistribution of energy in other directions, i.e. the shape

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Figure 6.13 Overall element power pattern with tunnel diode included of the pattern has slightly changed. However, the overall power that leaves the antenna in an absolute sense has still increased by 3 dB.

6.4 Multiscale numerical analysis of an antennafier array The antennafier that is described in the previous sections can be employed as an antenna array element. One could interpret the antenna array as a very specific material comprised of only PEC metals (conductors), air-dielectrics (insulators), and tunnel diodes (semiconductors). Hence, the search space in terms of possible materials that provide the desired RF system functionality is reduced significantly, which alleviates the material synthesis problem. In fact, we will demonstrate the fast numerical analysis methods for only one realization (i.e. fixed geometry) and do not discuss the synthesis problem. An 11 11 array of antennafiers will be examined as shown in Figure 6.14. The element spacing in both the x- and y-directions is 0.5l at 750 MHz. The array represents a multiscale structure since its overall size measures several wavelengths across, while each array element exhibits fine geometrical features that are only fractions of wavelengths (e.g. the metal strips). Multiscale structures require us to employ a dense mesh, leading to a large number of RWG basis functions and thus a large MoM matrix equation ZI = V, which is difficult to store in memory, let alone to solve it in a direct manner on a standard desktop computer. The example array in Figure 6.14 consists of 121 elements each of which has one external and one internal port and employs 176 RWG basis functions. This amounts to 21,296 RWGs/unknowns in total. A plain MoM approach can still be used to solve this matrix equation, which is useful for validation purposes.

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Figure 6.14 An 11 11 array of antennafiers For the numerical computations, we make use two Intel(R) Xeon(R) E5640 CPUs at 2.67 GHz, each with 4 cores (8 threads), having access to 150 GB of RAM memory. Furthermore, the ‘Linux version 3.7.10-1.45-desktop’ operating system is used. The MoM matrix filling is implemented in Cþþ while the pre- and postprocessing (discretization and visualization) steps are performed in MATLAB.

6.4.1 The Characteristic Basis Function Method The Characteristic Basis Function Method (CBFM) is an enhancement technique for the MoM [19,20]. It belongs to a class of integral-equation-based Domain Decomposition Methods which is capable of compressing the MoM matrix equation to a manageable size so that direct solution methods can be used. The respective steps that are involved will be explained with the help of the above example array of antennafiers. The interested reader is referred to [21] for an extensive overview of numerical enhancement techniques that are closely related to the CBFM. The CBFM subdivides the entire computational domain into M macro subdomains as shown in Figure 6.15. In the Deep Integration context, each subdomain can include both linear and non-linear materials. Note again, as an example, how a transistor can be incorporated as two-port non-linear I–V characteristics whenever one-port diode-type I–V characteristics turn out inadequate. Also, a subdomain can include multiple diodes (and/or transistors), but we will consider only one diode per subdomain or antennafier array element. The moment matrix equation ZI = V can now be block-partitioned as 2 32 3 2 3 Z11 Z12 Z1M I1 V1 6 . 6 7 6 7 .. .. 7 6 .. 6I 7 6V 7 . . 7 6 76 2 7 6 2 7 (6.13) 6 . 6 7¼6 7 .. .. 7 6 . 76 .. 7 6 .. 7 4 . . . 54 . 5 4 . 5 ZM1

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#2

#m

#M

Multiscale EM-part

Z + V1– + V2– I1

I2

+

– Vn–1 In–1

+

– Vn In

+ V– N IN

Non-linear part

Figure 6.15 Domain decomposition method in the Deep Integration context

where the matrix block Zmn is an N N MoM matrix whose entries describe the reaction integral coupling terms between the RWGs on source domain/antennafier n and observation macro subdomain/antennafier m. The vectors Im and Vm are the corresponding segments of I and V, respectively, of macro subdomain/antennafier m. To solve this matrix equation fast and in an iteration-free manner, a set of Characteristic Basis Functions (CBFs) is generated for each subdomain (see Section 6.4.2). Let us denote Jm as the matrix holding the Km CBFs for subdomain m. Each column vector in Jm describes a CBF on macro subdomain m as a set of known expansion coefficients for the Nm ¼ N RWGs on that subdomain. The size of Jm is therefore N Km. Accordingly, the unknown expansion coefficient vector Im for subdomain m can be expanded in terms of the known set of CBFs Jm as (MATLAB notation): Im ¼

Km X am ðk ÞJm ð:; k Þ ¼ Jm am

for

m ¼ 1; 2; . . . ;M

(6.14)

k¼1

Substituting (14) in (13) yields the overdetermined system of equations: 2 32 3 2 3 Z11 J1 Z12 J2 Z1M JM a1 V1 6 . 76 7 6 7 .. .. 6 .. 76 a2 7 6 V2 7 . . 6 76 7 6 7 6 . 76 . 7 ¼ 6 . 7 .. .. 6 . 76 .. 7 6 .. 7 4 . 54 5 4 5 . . ZM1 J1

ZMM JM

aM

(6.15)

VM

which can be solved for the CBF expansion coefficient vector a, for instance, through a pseudo-inverse matrix operation. However, this is a time-consuming step and the matrix is still large so that the memory may be lacking easily. A remedy is to reduce the number of equations by multiplying the mth row in (15)

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by JTm , for m ¼ 1, 2, . . . , M, this yields the reduced system of linear equations ZredIred ¼ Vred, i.e. 2 T 3 2 T 3 J1 Z11 J1 JT1 Z12 J2 JT1 Z1M JM 2 a1 3 J1 V1 6 76 6 T 7 .. .. .. 6 76 a2 7 6 J V2 7 7 2 6 7 6 7 . . . 7 6 76 7 (6.16) 6 . 7¼6 6 7 6 7 . .. .. .. .. 7 6 76 6 .. 7 4 5 . . . 4 5 4 5 T T T aM JM VM J ZMM JM J ZM1 J1 M

M

which can be solved for the unknown CBF expansion coefficient vector a, after which the original RWG expansion coefficient vector I can be computed via (6.14). The compression factor in the number of unknowns can be significant. For instance, if each subdomain employs 100 RWGs, and there are 10 subdomains, then the total number of RWGs is 1,000 so that the MoM matrix Z is of size 1,000 1,000. Then, if each subdomain employs only 10 CBFs, the reduced size matrix Zred is of size 100 100, which can be solved three orders faster and requires two orders of magnitude less random-access memory. The off-diagonal matrix blocks in (6.13) can be constructed in both a time- and memory-efficient manner since these matrices are rank-deficient. A low-rank matrix decomposition can be constructed fast as described in [14], where the CBFM has been hybridized with the Adaptive Cross Approximation (ACA) algorithm. The ACA algorithm is an on-the-fly purely algebraic rank-revealing algorithm that constructs the low-rank matrix decomposition without having to know all the elements of the block matrix in advance. One can also exploit translation symmetry between pairs of CBFs, which is particularly efficient for regularly spaced arrays as, e.g. shown in Figure 6.14. This process has been explained in [22] and even works for electrically connected subdomains. It enables one to construct only some of the matrix blocks Zmn in (6.13), which can then be replicated elsewhere in the MoM matrix in order to fill it fast. For the considered example, only 221 blocks need to be constructed, out of the 121 121 ¼ 14,641 possible combinations (factor 66 speed-up). Finally, the moment matrix is symmetric so that only the upper or lower half of the matrix needs to be computed ( factor 2 speed-up).

6.4.2 Generation of characteristic basis functions The final accuracy of the MoM solution vector hinges on the subdomain partitioning and also on the manner in which the CBFs have been generated. For the regularly spaced array of disjoint antenna elements (see Figure 6.14), the obvious choice is to consider one antennafier element per subdomain. This will give rise to a block-diagonal-dominant matrix in (6.13), which is desired. In fact, when sets of CBFs are going to be generated for perfectly isolated subdomains, they will likely be representative enough as a basis for weakly coupled subdomains. Figure 6.16 shows how the set of CBFs Jm can be generated for the mth subdomain. The first step is to extract element m from the array along with its

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z (m)

0.1 0.05 0 0.2

0.2

0.15 0.1

0.1 0.05

0

0 –0.05 y (m)

–0.1 –0.1 –0.15

–0.2 –0.2

x (m)

–0.3

Figure 6.16 The subarray for generating CBFs on the central element. The arrows indicate the ten excitations that induce ten currents on the center element immediate environment. In this case, a center element is chosen and the radius is chosen such that only its four nearest neighbors are included. The five-element subarray is solved with relative ease for multiple excitation vectors (right-hand sides). We excite the ten ports (i.e. five internal þ five external), one-by-one and retain only those segments of the solution vectors that pertain to the currents for the center element.

CBF generation in the Deep Integration context In the Deep Integration context it is essential that Characteristic Basis Functions are generated by exciting both the externally accessible ports and the internal ports at which non-linear components are to be attached since non-linear components may represent (dependent) sources under appropriate DC-biasing conditions. These ten smaller vectors form the columns of the matrix Um. To arrive at a reduced number of CBFs, Jm for subdomain m, we first perform the singular value decomposition on Um: Um ¼ Qm Dm VH m

(6.17)

where the diagonal matrix Dm holds the ordered singular values (SVs) s1, s2, . . . along its diagonal. The normalized magnitude of this SV-spectrum is shown in Figure 6.17. A typical relative threshold value of 103 on the SV-spectrum is

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100

Singular values Threshold

10–10

n

/ max

10–5

10–15

10–20

2

4

6

8

10

n

Figure 6.17 The singular value spectrum

applied to truncate the number of significant SVs. In this case, we retain five CBFs, which are the first five left-singular vectors in the matrix Qm, i.e. in MATLAB notation: Jm ¼ Qm(:,1: 5). Since all the array elements are identical, we have that J1 ¼ J2 ¼ . . . ¼ JM. Hence, in this specific case, the CBFs are generated only for one extended subdomain (880 RWGs), which is followed by both a spatial truncation (reduced to N ¼ 176 RWGs) and SV-spectrum truncation procedure to end up with only five CBFs. This truncated set is then replicated on all other elements and the reduced matrix equation (6.16) is formed. More information on the generation of CBFs can be found in [23], where also the electrically interconnected subdomain case is treated.

6.4.3 Numerical matrix compression and solution For the array example, we consider 11 11 ¼ 121 antennafier elements, or macro subdomains. After the SVD application, each element employs five CBFs which amounts to 605 CBFs in total. In other words, the reduced matrix in (6.16) is only of size (5 121) (5 121) instead of (880 121) (880 121). This is a reduction factor of 176 in the total number of unknowns, the MoM matrix equation of which solves approximately 1763 times faster and requires 1762 less random-access storage memory than a plain MoM approach would. To compute the solution error, the following metric is defined: e% ¼

CBFM kZMoM kFro ant Zant 100% MoM kZant kFro

(6.18)

where Zant is of size 242 242, since it refers to both the 121 internal and 121 external ports, and where ||A||Fro denotes the Frobenius norm of the matrix A.

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With just five CBFs per antennafier, the CBFM relative error of Zant is only 0.9%. On the contrary, if we limit the subarray in Figure 6.14 to only the central antenna element, so that only two CBFs are generated per antennafier, this relative error increases to 9%. The filling of the MoM matrix took 1 min 44 s, while the ACA-CBFM reduced MoM matrix construction (including the generation of CBFs) took only 4 s. The MoM solution time is 6 min 1 s, while it takes less than a second for the CBFM. The extreme performance gains of CBFM are mostly attributed to the relatively dense mesh that was employed on the antennafier elements and the fact that the array elements are disjoint from one another, both leading to an efficient compression of the original MoM matrix equation.

6.4.4 Active versus passive antenna array results We will now cross-compare the performance of the active versus the passive 121element antenna array. To this end, Figure 6.11 along with Eqs. (6.8)–(6.10) is used, where for large arrays Zsource becomes a block-diagonal matrix, i.e. # 2" 3 RRF 0 0 7 6 6 0 7 Rdiode 6 7 6 7 " # 6 0 RRF .. .. 7 Zsource ¼ 6 (6.19) 7 . . 7 6 6 7 0 R diode 6 7 4 5 .. 0 . Similarly, vector Vsource is now extended to become Vsource ¼ ½ VRF;1 ;

0; VRF;2 ;

0; ; VRF;121 ;

0 T :

(6.20)

Next, the array is configured as a phased array antenna, so that VRF; m ¼ expðjkrm ^r Þ, where k ¼ w/c0 is the free-space wavenumber, rm is the location of element m, and ^r ¼cos ðf0 Þsin ðq0 Þ^x þsin ðf0 Þsin ðq0 Þ^y þcos ðq0 Þ^z is the scan direction expressed in terms of the spherical coordinates (q0, f0). Once again, we assume that RRF ¼ 50W and Rdiode ¼ 86W. Accordingly, the total radiated power is computed by using (6.8) and (6.10), that is, 1 Prad ¼ Re IH ant Zant Iant 2

(6.21)

where Iant ¼ ðZsource þ Zant Þ1 Vsource

(6.22)

The total available input power is computed as Pav ¼

121 X jVRF;m j2 m¼1

4RRF

;

(6.23)

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25 Broadside (θ0=0,ϕ0=0)

20

E-plane (θ0=45,ϕ0=0) H-plane (θ0=45,ϕ0=90)

15

Gain (dB)

10 5 X: –161 Y: 2.954

0 –5 –10 –15 –500

–450

–400

–350

–300

–250

–200

–150

–100

–50

0

Rdiode(Ohm)

Figure 6.18 The power gain G [see Eq. (6.24)] of the active array relative to a power-matched passive array for different negative values of Rdiode and scan angles so that the (available) power gain is finally computed as Prad G ¼ 10 log10 ðdBÞ Pav

(6.24)

The power gain of the antennafier array with DC-biased tunnel diodes is plotted in Figure 6.18 as a function of Rdiode and for three different scan angles. It is observed that the peak maximums in gain for the broadside, E- and H-plane scans occur at different values of Rdiode, which is due to the antenna mutual coupling effects. Large negative resistance values are difficult to realize in practice. As a typical example, the commonly available tunnel diode 1N3712 features a Rdiode of 125W. We will therefore focus on the area on the right of the graph. The slopes are seen to follow each other – and are therefore less scan-angle dependent – down to approximately 161W at which a gain of 3 dB is reached (see the marker), which is significantly different from the optimal 86W value for a single antennafier in isolation. For very large regularly spaced phased array antennas, one can therefore draw the conclusion that it makes sense to optimize an antennafier element in an active unit cell environment through a Floquet-mode infinite array analysis. The array patterns have been computed for both the passive and active array (Rdiode ¼ 0W and 161W, respectively). The normalized gain patterns are shown in Figure 6.19 for the broadside, 45 E-plane, and 45 H-plane scans. It is seen that the side-lobe levels of the far-field patterns increase slightly when the tunnel diodes are connected. Nonetheless, apart from the pattern shape, the unnormalized absolute power level of the active array is about 3 dB higher than the ideally power-matched passive array owing to the connected tunnel diodes that are capable of delivering RF power.

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Developments in antenna analysis and design, volume 2 Broadside scan (θ0=0, ϕ0=0)

Relative gain (dB)

0 R diode = 0

–10

R

diode

(passive antennas)

= –160

(active antennas)

–20 –30 –40

0

10

20

30

40 50 θ (Deg) E-plane (θ0=45, ϕ0=0)

60

70

80

90

60

70

80

90

60

70

80

90

Relative gain (dB)

0 –10 –20 –30

R

–40

R diode = –160

0

10

20

diode

30

=0

(passive antennas) (active antennas)

40 50 θ (Deg) H-plane (θ0=45, ϕ0=90)

Relative gain (dB)

0 –10 –20 –30

R R

–40 0

10

20

diode diode

30

=0

(passive antennas)

= –160

(active antennas)

40 50 θ (Deg)

Figure 6.19 The normalized gain patterns of the 121-element antenna array with and without tunnel diodes

Finally, the so-called power-added efficiency (PAE) is computed, which is typically defined for power amplifiers as PAE ¼

RF PRF out Pin 100% PTotal DC

(6.25)

Since the radiation efficiency hrad for lossless antennas is 100%, the RF radiated output power PRF out ¼ Prad is equal to the total antenna input power delivered to

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both the external and internal ports. Hence, the total RF input power is given by (6.21), i.e. 1 H PRF out ¼ Re Iant Zant Iant : 2

(6.26)

RF The total antennafier RF input power PRF in ¼ Pant;in , which is supplied to the external RF antenna ports only, is equal to (cf. Figure 6.11): RF RF PRF in ¼ Psource;delivered Psource;dissipated H 1 ¼ 1=2Re IH ant Vsource 1=2Re Vsource Zsource Vsource :

(6.27)

Finally, the overall DC input power: PTotal DC ¼ M VDC IDC

(6.28)

where the DC-biasing point (IDC, VDC) is given by the specific I–V characteristics of the tunnel diode as graphically indicated by the marker in Figure 6.5 (top-right). The datasheet values for the 1N3712 tunnel diode will be used as representative values, for which Rdiode ¼ 125W. Furthermore, for this diode VDC ¼ 208 mV and IDC ¼ 0.56 mA so that PTotal DC ¼ 14 mW for the entire array of M ¼ 121 elements. The amplitude of the RF voltage across the diode should not exceed 130 mV to avoid severe distortion effects. This is achieved if we limit the excitation voltage amplitude to |Vsource;m| ¼ 55 mV for all m ¼ 1, . . . , M. In that case, PAE ¼ 31.7% (broadside scan, G ¼ 1.0 dB); PAE ¼ 45.7% (E-plane scan, 45 , G ¼ 2.7 dB); and PAE ¼ 31.5% (H-plane scan, 45 , G ¼ 0.0 dB). These PAE values are merely indicative; yet, they clearly show how the PAE depends on the scan angle, which is due to antenna mutual coupling effects. Also, note the drop in power gains G when Rdiode increases from the previous 161W to 125W, which was already concluded from Figure 6.18.

6.5 Conclusions The CBFM has been applied in the Deep Integration context to rapidly analyze multiscale integrated active antenna arrays. The CBFs are generated by exciting both the externally accessible antenna ports as well as the non-accessible internal ports at which non-linear semiconductor components are connected. In fact, these latter components can also act as (dependent) RF excitation sources when DCbiased properly; DC-biased tunnel diodes can act as negative lumped resistances and are therefore modeled accordingly. Results are shown for arrays of tunneldiode-loaded antennas, so-called antennafier arrays. The achieved available power gain, the power-added efficiency and the effect on the radiation pattern of a scanned 11 11 antennafier array are discussed. Large memory savings are realized and solve times are orders shorter for the ACAenhanced CBFM, which is essential for future studies where the final goal is to synthesize Deeply Integrated active (meta)materials in a time-efficient numerical manner.

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References [1] R. Maaskant, ‘‘Deep integration: A paradigm shift in the synthesis of active antenna systems,’’ in Proc. Antennas and Propagation, San Diego, California, 2017, pp. 1–2. [2] R. Maaskant, W. A. Shah, A. U. Zaman, M. Ivashina, and P.-S. Kildal, ‘‘Spatial power combining and splitting in gap waveguide technology,’’ IEEE MWCL, vol. 26, no. 7, pp. 472–474, 2016. [3] K. Fujimoto, ‘‘Active antennas: Tunnel-diode-loaded dipoles,’’ Proceedings of the IEEE, vol. 53, pp. 556–556, 1965. [4] H. Gerlach. (1967). Waveguide wall-current tunnel diode amplifier and oscillator, US Patent 3,320,550. [5] P. Preventza, B. Dickman, E. Sovero, M. P. D. Lisio, J. J. Rosenberg, and D. B. Rutledge, ‘‘Modelling of quasi-optical arrays,’’ in IEEE MTT-S, Anaheim, CA, 1999, pp. 563–566. [6] K. C. Gupta and P. S. Hall, Analysis and Design of Integrated CircuitAntenna Modules. Hoboken, NJ: John Wiley & Sons, Inc., 1999. [7] S. Gupta, P. K. Nath, A. Agarwal, and B. K. Sarkar, ‘‘Integrated active antennas,’’ IETE Technical Review, vol. 18, no. 2–3, pp. 139–146, Mar. 2001. [8] E. H. Lim and K. W. Leung, Compact Multifunctional Antennas for Wireless Systems. Hoboken, NJ: John Wiley & Sons, Inc., 2012. [9] V. Rizzoli, A. Costanzo, E. Montanari, and P. Spadoni, ‘‘CAD procedures for the nonlinear/electromagnetic co-design of integrated microwave transmitters,’’ in IEEE/MTT-S International Microwave Symposium, Honolulu, Hawaii, 2007, pp. 2031–2034. [10] Y. Vardaxoglou. (2016). Synthesizing 3D METAmaterials for RF, microwave and THz applications (SYMETA) [online]. Available: http://gow. epsrc.ac.uk/NGBOViewGrant.aspx?GrantRef=EP/N010493/1 [11] I. Aryanian, A. Abdipour, and G. Moradi, ‘‘Nonlinear analysis of active aperture coupled reflectarray antenna containing varactor diode,’’ ACES Journal, vol. 30, no. 10, pp. 1102–1108, 2015. [12] L. Huitema and T. Monediere, ‘‘Miniature antenna with frequency agility,’’ in Progress in Compact Antennas, D. L. Huitema, Ed. Intech, 2014 [online]. Available: www.intechopen.com/books/progress-in-compact-antennas/miniature-antenna-with-frequency-agility [13] M. Giles, ‘‘TCAD process/device modeling challenges and opportunities for the next decade,’’ Journal of Computer Electronics, pp. 177–182, 2004. [14] R. Maaskant, R. Mittra, and A. G. Tijhuis, ‘‘Fast analysis of large antenna arrays using the characteristic basis function method and the adaptive cross approximation algorithm,’’ IEEE T-AP, pp. 3440–3451, Nov. 2008. [15] R. F. Harrington, Field Computation by Moment Methods. Hoboken, NJ: Wiley-IEEE Press, 1993. [16] S. Rao, D. Wilton, and A. Glisson, ‘‘Electromagnetic scattering by surfaces of arbitrary shape,’’ IEEE T-AP, vol. 30, no. 3, pp. 409–418, May 1982.

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[17] P. Y. -Oijala and M. Taskinen, ‘‘Calculation of CFIE impedance matrix elements with RWG and n RWG functions,’’ IEEE T-AP, vol. 51, no. 8, pp. 1837–1846, Aug. 2003. [18] R. Maaskant and M. Arts, ‘‘Reconsidering the voltage-gap source model used in moment methods,’’ IEEE Antennas and Propagation Magazine, vol. 52, no. 2, pp. 120–125, April 2010. [19] J. Yeo, V. Prakash, and R. Mittra, ‘‘Efficient analysis of a class of microstrip antennas using the characteristic basis function method (CBFM),’’ Microoptics Technology., vol. 39, pp. 456–464, Dec. 2003. [20] R. Maaskant, ‘‘Analysis of large antenna systems,’’ Ph.D. dissertation, Eindhoven Univ. of Technology, 2010. [21] C. Craeye, J. Laviada, R. Maaskant, and R. Mittra, ‘‘Macro basis function framework for solving Maxwell’s equations in surface-integral-equation form,’’ Forum for Electromagnetic Research Methods and Application Technologies (FERMAT), vol. 3, pp. 1–16, 2014. [22] R. Maaskant, R. Mittra, and A. G. Tijhuis, ‘‘Fast solution of multi-scale antenna problems for the square kilometre array (SKA) radio telescope using the characteristic basis function method (CBFM),’’ Applied Computational Electromagnetics Society (ACES) Journal, vol. 24, no. 2, pp. 174–188, Apr. 2009. [23] R. Maaskant, R. Mittra, and A. G. Tijhuis, ‘‘Application of trapezoidalshaped characteristic basis functions to arrays of electrically interconnected antenna elements,’’ in Proc. ICEAA, Torino, Sep. 2007, pp. 567–571.

Chapter 7

Numerically efficient methods for electromagnetic modeling of antenna radiation and scattering problems Yang Su1 and Raj Mittra2

7.1 Introduction Modern approaches to practical antenna design almost invariably rely heavily on computer simulation of the proposed design as a first step for the initial design. This is often followed by iterative improvements of the initial design via an optimization of the design parameters, also by using computational tools, prior to fabricating and testing the design. A whole host of commercial software modules are now available for the task of computer-aided design, and they are extensively used by the antenna designers for the problem at hand to meet the specifications of the users. Most of these codes are based on either Integral Methods, for example, the method of moments (MoM), on finite methods, for example, finite element method or finite difference time domain. The commercial codes are frequently updated by their developers to keep up with the demands on the part of the users, who have an insatiable thirst to solve increasingly larger and more complex problems that are highly computer-intensive in nature, not only because of their multiscale geometries and/or large sizes at the operating wavelengths but also because their material parameters have complex permittivities (and permeability’s) at the frequencies of interest, which serve to exacerbate the complexity of the problems. While we could list here a few examples of such antenna and platform geometries, it is expedient to refer the interested reader to various chapters in this book, where a plethora of such examples may be found relatively easily. Our goal in this chapter is to present a review of some recent developments in computational electromagnetics (CEM) that are designed to address the type of problems that have a large number of degrees of freedom (DoFs) associated with them, for the reasons alluded to above. The underlying concept upon which the technique presented in this chapter is based may be described as the ‘divide and conquer’ approach, 1 Centre for Intelligent Antenna and Radio Systems, Department of Electrical and Computer Engineering, University of Waterloo, Canada 2 EMC Laboratory, University of Central Florida, USA

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commonly referred to as the ‘domain decomposition’ method. In contrast to most of the commercial codes, which run into problems with memory and CPU runtimes when employed for very large problems, the proposed approach leads to a size-reduction of the associated MoM matrix, accomplished by using a combination of domain decomposition method and macro-basis functions, called the characteristic basis functions (CBFs), as opposed to low-level basis functions such as the RWGs (Rao, Wilton and Glisson) that are almost always employed in the conventional MoM codes. Two contrasting features give the presented method a unique advantage over the commercial codes which rely heavily on the use of iterative techniques for solving large problems. First, the present method is able to use ‘direct’ solvers to handle the reduced matrix generated via the use of CBFs. Such solvers can handle multiple rights hand sides (r.h.s.) efficiently, without having to start the iterative process anew each time the excitation is changed. Second, the reduced matrix generated via the use of CBFs is well conditioned and, hence, it does not require a ‘preconditioner’ as do the iterative schemes, almost always, we might add, not only for large and multi-scale problems but also for those that suffer from ‘internal resonance’ at certain frequencies. Before presenting the details of the method, we should mention that although the examples presented in this chapter frequently deal with scattering problems, that are more relevant when the antenna is operating in the ‘receive’ mode, as opposed to the ‘transmit mode’, we can invoke the reciprocity principle to handle the latter case. We also hasten to point out that the MoM matrix remains essentially unchanged when we switch from the ‘receive’ to the ‘transmit’ mode, although the r.h.s. of the MoM matrix does obviously change. Nonetheless, we can process the MoM matrix the same way by using the technique we describe below, in either of the two cases mentioned above. It is intuitively clear that electromagnetic scattering analysis of a large object, for example, a complex platform, can be made manageable, at least temporarily, by dividing it into parts, or blocks, the number of DoFs associated with each of which is only moderately large. However, a rigorous solution of the original problem requires that we account for the mutual interaction effects rigorously, and this is the bottleneck in conventional methods that can be unsurmountable for large problems unless one resorts to iterative techniques, some of whose drawbacks we have mentioned above. Additionally, the inability to use direct solvers makes it impossible to conveniently reuse the information associated with the blocks that remain unchanged, when one of the domains of the object is modified, for example, by locating the antenna from one area of the platform to another, in order to find an optimum position for the same. Furthermore, if the meshing of the target is changed during the process of domain decomposition, it places an additional burden on the mesh generator, which can be time-consuming as well as inconvenient. In this chapter, we describe a number of different techniques, which include the domain decomposition and multi-level acceleration methods, as well as the multi-scale analysis. Toward this end, we combine the characteristic basis function method (CBFM) [1] with the discontinuous Galerkin method [2] for the analysis of electromagnetic scattering from multi-scale and multiple targets. The combined field surface integral equation is used to calculate the scattered field, and the mono-

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polar RWG functions [3] are used as the low-level basis functions. The use of this method enables us to use discontinuous discretization of a multi-scale target while connecting the constituent blocks of the target. Following this, we use the CBFs to significantly reduce the size of the original MoM matrix, which enables us to employ a direct solver that can be used to efficiently analyze the problem of multiangle excitations and repeatedly reuse the common information associated with different blocks of the multiple targets. Additionally, the process of generating the reduced matrix equation can be parallelized without the requirement for preprocessing, and only one mesh is required for the entire solution process. We show, via illustrative examples, that the presented method is accurate and efficient when dealing with the scattering problems from multi-scale and multiple objects. We have also found the method is not only able to handle large and multiscale problems, it can do so more efficiently than the commercially available codes, especially when solutions to multiple excitation problems are needed. Later, in Section 7.3, we discuss the problem of scattering (or radiation) from objects located in stratified layered media, as opposed to free space, and we describe the multi-level CBFM to analyze the problem at hand. Plane waves impinging upon the lossy layered media are used to generate the multi-level CBFs, and we show how the process of generating the reduced matrix can be parallelized to enhance the computational efficiency of the analysis. Last, in Section 7.4, we discuss an important aspect of CBFM, which is designed for the microwave circuit and antenna problems, as opposed to the scattering problem. What distinguishes the antenna radiation problem from the scattering problem, in the context of CBFM, is the excitation used to construct the CBFs. We detail three options of the excitation and carry out a comparative study of their performances. In the conclusion, we give our recommendation for the preferred choice, which is based on the comparative study.

7.2 Numerical analysis of multiple multi-scale objects using CBFM and IEDG 7.2.1 Introduction to CBFM and IEDG In many real applications, such as electromagnetic scattering problems involving multiple and multi-scale targets, the memory requirement to store the MoM impedance matrix is O(N2), where N is the number of unknowns, and the size of the MoM matrix equation is such that both storage and direct solution are prohibitive [4]. To mitigate these problems, the CBFM has been proposed by Prakash and Mittra [1] as a way to reduce the size of the MoM-associated matrix, and the concept of CBFM for multiple plane wave excitations has been developed in [5], which is especially useful when handling the problem of multiple r.h.s. [6]. For further reducing the associated matrix size, extensions of CBFM for the multi-level case have been described in [7,8]. Since the size of the matrix equation in the CBFM is reduced to a point where one can use a direct solver, it does not rely either upon pre-conditioners or iterative solvers to generate the desired solutions. Besides,

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the CBFM is highly parallelizable in nature, which is an important attribute for accelerating the analysis. Furthermore, when the CBFM is applied to problems involving multiple targets, whose constituent parts separate from each other when the target is moving, both the CBFs and the reduced matrices associated with constituent parts that are unchanged can be used repeatedly [9]; and this, in turn, leads to a significant reduction of the total computation time. It should be recognized that for multiple targets, the MoM discretization at the interface between the constituent parts of the targets is naturally nonconforming, and this situation cannot be handled by the CBFM using a conforming discretization [7,9]. Besides, the CBFM approach using non-uniform volumetric mesh, which has been proposed by Fenni et al. to analyze the electromagnetic scattering in a forested area [10], is not well suited for analyzing conducting objects [11]. For these reasons, it becomes necessary to employ a discontinuous-discretization scheme, in the context of CBFM, when solving the surface integral equations to analyze the problem of scattering from conducting objects. Recently, the integral equation discontinuous Galerkin (IEDG) method has been applied by Peng et al. [12,13] to solve the electromagnetic scattering problems using nonconforming discretization. In this approach, the discontinuous Galerkin method has been used to solve the combined field integral equation (CFIE), where the continuity conditions of the tangential fields are used as interior penalty terms [14]. These conditions are enforced weakly at the boundaries [15] when applying the Galerkin testing method using the nonconforming discretization, and the hyper-singular double contour integrals, introduced in the electric field integral equation (EFIE), can be eliminated. In addition to this, the magnetic field integral equation (MFIE) is used in conjunction with the EFIE to avoid the problem of internal resonance. More recently, the mono-polar RWG functions have been proposed by Ubeda et al. [16] to enhance the accuracy of the MoM-MFIE formulation when dealing with objects with sharp edges and have been employed in the time-domain IEDG method [17], to implement the discontinuous discretization. More recently, Su et al. [18,19] proposed the IEDGbased CBFM to solve the scattering problem for multiple and multi-scale targets using discontinuous discretization. In this chapter, we present the steps for the implementation of the IEDG-based CBFM to solve the scattering problem for multiple and multi-scale targets using discontinuous discretization. The IEDG method is applied to the CFIE-MOM on boundary surfaces to expand the spatial basis functions in the square-integrable L2-space, and the mono-polar RWG basis functions are used to generate the nonconforming discretization while decomposing the object into non-overlapping subdomains. This is done so that each sub-domain can be meshed independently in accordance with its geometrical features. In CBFM, the matrix equations associated with the sub-domains of the targets are solved to derive the original CBFs by using multiple plane-wave excitations. Next, the SVD algorithm is used to down-select the original CBFs to minimize their redundancy. Furthermore, the adaptive cross approximation algorithm (ACA) [20] is employed to accelerate the construction of the reduced matrix. Numerical results show that the entire computation process is naturally parallelizable and that the reduced matrix is considerably smaller than

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that of the original MOM matrix. As a result, both the CBFs and the reduced matrices associated with the unchanged constituent parts of multiple targets are used repeatedly when these parts separate from each other. The results show that both time and memory consumptions are significantly reduced without compromising the accuracy. Furthermore, only a single mesh is sufficient for the geometry since we no longer need to change the discretization of the interfaces of multiple targets.

7.2.2 MoM combined with CFIE 7.2.2.1 CFIE formulations In free space, the magnetic and electric current distributions on the surface can be expressed in terms of the tangential electric and magnetic fields, respectively, as ^ EðrÞ ¼ MðrÞ n

(7.1)

^ HðrÞ ¼ JðrÞ n

(7.2)

^ is the unit normal directed outward from the surface, and J and M are the where n electric and magnetic currents, respectively. E and H are the total electric and magnetic fields, respectively, which can be expressed as follows: EðrÞ ¼ Einc ðrÞ þ Esca ðrÞ

(7.3)

H ðr Þ ¼ H ðr Þ þ H ðr Þ

(7.4)

inc

inc

sca

inc

where E and H are the incident electric and magnetic fields, respectively, and Esca and Hsca are the scattered fields, which can be derived by integrating the electric and magnetic current densities on the surface of the object. At any point in space, the electric and magnetic scattering fields can be expressed as follows: Esca ðrÞ ¼ hLðJðr0 ÞÞ KðMðr0 ÞÞ (7.5) 1 (7.6) Hsca ðrÞ ¼ LðMðr0 ÞÞ þ KðJðr0 ÞÞ h pﬃﬃﬃﬃﬃﬃﬃﬃ where h ¼ m=e is the wave impedance of the homogeneous background medium, e and m are the permittivity and permeability, respectively. Substituting the expression of scattering field into (7.3) and (7.4), we obtain the EFIE and the MFIE, which read: ^ Einc ðrÞ ¼ MðrÞ n ^ ½hLðJðr0 ÞÞ KðMðr0 ÞÞ n 1 inc 0 0 ^ LðMðr ÞÞ þ KðJðr ÞÞ ^ H ðrÞ ¼ JðrÞ n n h The integral operators L and K can be expressed as ð 1 I þ 2 rr0 Xðr0 ÞGðr; r0 ÞdS 0 LðXðr0 ÞÞ ¼ jk k W0 ð KðXðr0 ÞÞ ¼ rGðr; r0 Þ Xðr0 ÞdS 0 W0

(7.7) (7.8)

(7.9) (7.10)

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pﬃﬃﬃﬃﬃ where k ¼ w me is the wavelength of the background medium, w is the angular frequency, I is the unit dyadic vector, W is the domain of EFIE and MFIE, which is on the surface of the object. G is Green’s function in homogeneous medium, which is expressed as follows: 0

G¼

ejkjrr j 4pjr r0 j

(7.11)

where r and r0 are the source and observation points, respectively. If the object is perfectly conducting, the magnetic current on the surface is zero. Substituting M ¼ 0, L and K into (7.7) and (7.8), and then extracting the Cauchy principal value from the K operator, the EFIE and MFIE on the surface of the perfect conducting object can be expressed as ð 1 inc 0 n E ðr Þ ¼ n jwm0 I rr J0 G0 dS 0 (7.12) jwe0 W0 ð JðrÞ inc n P V rG0 ðr; r0 Þ Jðr0 ÞdS 0 (7.13) n H ðr Þ ¼ 2 W0 where P V is the residue term after extracting the Cauchy principle value. When r and r0 overlap, the residue is zero. G0 is the free space Green’s function, pﬃﬃﬃﬃﬃﬃﬃﬃﬃ k0 ¼ w m0 e0 is the wave number of free space, and m0 and e0 are the permittivity and permeability of free space, respectively. A linear combination of the EFIE and MFIE leads us to the expression of the CFIE, which reads: aEFIE þ ð1 aÞh0 MFIE

(7.14)

where the weight a is chosen such that 0 < a < 1.

7.2.2.2

Spatial basis functions

When solving an integral equation by using the MOM, we need to discretize W into a series of sub-domains as follows: W¼

Ne X

Wn

(7.15)

n¼1

where Ne is the number of sub-domains. The current distribution J0 can also be discretized into these sub-domains, expressed as follows: J0 ¼

Ne X

In f n

(7.16)

1

where fn’s are the basis functions, In ’s are the coefficients of the current distribution. In our work, we use the RWG basis functions as fn, which is shown in Figure 7.1.

Numerically efficient methods for electromagnetic modeling T–

T+ r+

219

I +f(r)

I –f(r)

r–

Figure 7.1 RWG basis function The RWG basis function is composed of a pair of triangular patches, which can be expressed as follows: 8 ln > þ þ > > < 2Aþ r rn ; r 2 Tn n f n ðr Þ ¼ (7.17) > ln > > ; r 2 T r r : n n 2A n where r is any point at the domain of basis functions, ln is the length of the common edge of the two triangular patches of the nth basis function, rþ n and rn are the free þ points of the left triangular patch Tn and the right triangular patch Tn , respectively, and Aþ n and An are the patch areas. The RWG basis functions are used to fit the current distributions on the surface of objects, the current is continuous across the common edge. The face divergence which is required in operator L can be calculated as follows: 8 ln > þ > > < Aþ ; r 2 Tn n rs f n ð r Þ ¼ (7.18) > ln > > ; r 2 T : n An

7.2.3 Elements of impedance matrix of MoM Next, we test the EFIE and MFIE with spatial basis functions ^ n f ðrÞ using the Galerkin method, after testing (7.12), we have ð ^ f ðr Þ Þ n ^ Einc ðrÞ dS ðn W 0 1 ð ð (7.19) 1 ^ f ðrÞÞ @n ^ jwm0 I rr0 J0 G0 dS 0 AdS ¼ ðn jwe0 W W0

where we use the vector identities as follows: ^ f ½n ^ F ¼ ðn ^ n ^ Þðf FÞ ½f n ^ ½n ^ F ½n f rF ¼ r ðFf Þ Fr f

(7.20) (7.21)

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^ ¼ 0, and n ^ n ^ ¼ 1, (7.19) can As f is defined on the surface of the object, f n be simplified as follows: ð ð 2 3 jwm0 f ðrÞ J0 G0 ðr; r0 ÞdS 0 dS 6 7 W W0 6 7 ð ð ð 6 7 1 0 6 0 0 0 rs f ðrÞ rs J G0 ðr; r ÞdS dS 7 (7.22) f ðrÞ Einc ðrÞdS ¼ 6 þ 7 jwe0 W 6 7 W0 W 6 7 ð ð 4 5 1 0 rs f ðrÞ rs J0 G0 ðr; r0 ÞdS 0 dS jwe0 W W0 The spatial integral on the surface can be transformed into þ ð rs f ðrÞdS ¼ f ðrÞ ^t dl W

(7.23)

C

where C is the contour of domain W, ^t is the outwards normal unit vector, which is tangential to W and normal to C. Equation (7.22) can be expressed as follows: ð ð 2 3 f ðrÞ J0 G0 ðr; r0 ÞdS 0 dS jwm0 6 7 W W0 6 7 ð ð ð 6 7 1 0 6 7 rs f ðrÞ rs J0 G0 ðr; r0 ÞdS 0 dS 7 (7.24) f ðrÞ Einc ðrÞdS ¼ 6 þ 0 6 7 jwe 0 W W W 6 7 ð þ 4 5 1 0 0 0 0 ^ f ðrÞ t rs J G0 ðr; r ÞdS dl jwe0 C W0 According to the definition of the RWG basis functions, if r is located at the common edge of the two triangular patches, then fðrÞ ^t ¼ 0. This is because the current distribution across the common edge is continuous, the integrals of fðrÞ ^t on the left and right triangular patches can be canceled, and the third term of the r.h.s. of (7.24) becomes zero. Testing (7.13) using f(r), we have ð ^ Hinc ðrÞ dS f ðr Þ n W ð ð ð J ðr Þ (7.25) 0 0 0 ^ dS f ðrÞ n P V : rG0 ðr; r Þ Jðr ÞdS dS ¼ f ðr Þ 0 2 W W W Substituting (7.16) into (7.24), (7.25), and (7.14), we can derive the following expression for the element of the impedance matrix of CFIE: 2 3 ð ð 0 0 0 f m ðr Þ f n ðr ÞG0 ðr; r ÞdS dS ajwm0 6 7 Wm Wn 6 7 6 7 ð ð 6 7 a 0 0 0 0 6 7 rs f m ðrÞ r s f n ðr ÞG0 ðr; r ÞdS dS 6 7 0 jwe0 W W 6 7 Zmn ¼ 6 (7.26) 7 ð 6 7 f n ðr Þ 6 7 ð 1 a Þh dS f ð r Þ 0 6 7 2 W 6 7 7 ð ð 6 4 5 ^ P V rG0 ðr; r0 Þ f n ðr0 ÞdS 0 dS ð1 aÞh0 f ðrÞ n W

W0

Numerically efficient methods for electromagnetic modeling The accompanying excitation vector reads ð ð inc ^ Hinc ðrÞ dS Vmn ¼ a f m ðrÞ E ðrÞdS þ ð1 aÞh0 f m ðrÞ n Wm

221

(7.27)

Wn

By substituting (7.26) and (7.27) into the tested EFIE and MFIE, we can obtain the desired current distribution on the surface of object. When filling the impedance matrix using the RWG basis functions, the integrand of EFIE could be singular when r ¼ r0 . Substituting the expression of RWG basis function into the first two terms of r.h.s. of (7.26) and extracting the singular kernel 1/|r r0 | from the integral, the EFIE part of the element of the impedance matrix can be expressed as follows: EFIE Zmn ¼

2

X X sgnðpÞsgnðqÞl2 n p q A A m n p¼þ q¼þ

3 0 jk0 jrr0 j 1 0 q e dS dS r r rn 6 7 jr r0 j 6 7 W W0 6 7 6 7 ð ð jk0 jrr0 j 6 7 1 e 1 0 6 7 dS dS þ 6 7 0j 0 4pjwe jr r 0 W W 6 7 6 7 6 jwm ð ð 7

1 6 7 0 0 dS r rpm r rqn ðr r0 Þ dS 6þ 7 0 6 16p W W0 7 jr r j 6 7 6 7 ð ð 4 5 1 1 0 dS dS þ 0 4pjwe0 W W0 jr r j jwm0 16p

ð ð

rpm

(7.28)

As for the external integral, we employ numerical methods to compute it. For the inner integral, after extracting the singular kernel 1/|r r0 |, the first two terms of the r.h.s. of (7.28) can be calculated via numerical integration. However, the last two terms of the r.h.s. are still singular, and they can be extracted from the integrals below: ð r r0 dS 0 (7.29) I¼ 0 W0 jr r j ð 1 dS 0 (7.30) I¼ 0 jr r0 j W

7.3 Acceleration of electromagnetic analysis using CBFM 7.3.1 Partition of CBFM To reduce the unknowns of matrix equation and accelerate the electromagnetic analysis of objects of arbitrary shape, and to repeatedly use the common information among different stages when the multiple objects dividing from each other, we now turn to the CBFM. In the CBFM, the original object is divided into a number of small blocks (see Figure 7.2) and higher-level macro basis functions, referred to herein as the CBFs, are employed to ease the computational burden, by reducing the size of the matrix equation significantly in comparison to that needed in the

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Original object

Partition of level 1

Partition of level 2

Figure 7.2 Multi-level partition of the original problem conventional MoM formulation. What is perhaps even more important is the fact that since the CBFM can use direct solvers, the CBFM it can handle multiple r.h.s. more efficiently than can the iterative schemes; furthermore, it does not need any preconditioners that are almost always required by the iterative schemes to speed up their convergence. To implement the multi-level CBFM, which enables us to handle very large problems, we begin by using the Oct-tree to scheme to partition the RWG basis functions into different blocks at different levels. The multi-level partition of the original object is shown in Figure 7.2. In the CBFM, first we generate the CBFs by constructing and solving the matrix equations for every single block, where for the first level, the lower-level basis functions are the RWG basis function while for the second-level CBFs their lower-level basis functions are the first-level CBFs. During the generation of CBFs, to reduce the effect of truncation between different blocks, we can introduce some extended basis function, which can be identified by expanding the scope of blocks a little (usually for 0.1–0.2 wavelength) (see Figure 7.3).

7.3.2

Constructing CBFs by using multiple plane-wave excitation

To address the problem of edge effects that are artificially introduced when truncating the blocks in the process of domain decomposition, we expand them slightly, as shown in Figure 7.3. The extended CBFs are generated as follows: h i h i1 h i l1;ext E l1;ext E l;i l;i ¼ Z V Jl;ext P P El;i El;i i i;i i;i P P Kl1;Spl;i Kl1;Spl;i Kl1;Spl;i N l;pws Kl1;Spl;i N l;pws p p p

p

(7.31) ’s are the extended CBFs associated with the i-th block of the l-th level, where Jl;ext i l1;ext l1;ext and Vi;i are the impedance matrix and excitation vector, respectively, and Zi;i associated with the basis functions at the l 1-th level, that are included in the i-th block of the l-th level. El,i’s denote the number of lower-level basis functions

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223

Extended block

Original block

Figure 7.3 Expanding of block involved in the i-th block (expanded) of the l-th level, and Spl;i denotes their serial number. K is the number of columns of the associated CBF matrix, while the number of rows is the number of lower-level basis functions included by the block. To extract the information, we need for the CBFs of the electric current distributions induced by excitations impinging from different angles, we need to illuminate the basis functions inside and close to the block with plane waves from different angles to obtain the l1;ext , where Nl,pws ¼ Nq Nf is the number of wave planes used to construct the Vi;i CBF. In our study. For the first level, we set Npws ¼ 10 20, which means that there are 10 samples along the direction q and 20 samples along the direction f; for the second level. we set Npws ¼ 20 40. For the extended lower-level basis functions displayed in Figure 7.4. the procedure to generate the expanded CBF is shown as follows: ’s represent the responses due to multiple plane wave excitations, which Jl;ext i includes some redundancy, which depends upon the number of plane wave excitations employed to construct them, but which can be eliminated by applying the SVD algorithm. The decomposed matrices of SVD are expressed as follows: ¼ USVT Jl;ext i

PEl;i

(7.32)

where U and VT is an orthogonal matrices whose size are p Kl1;Spl;i N l;pws and P N l;pws Ep l;i Kl1;Spl;i , respectively. S is a diagonal matrix whose size is Nl,pws Nl,pws

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Figure 7.4 Extended block illuminated by multiple plane waves

and elements along the diagonal are the singular values of Jl;ext , which is listed i from big to small. Next, we construct a new CBF by decomposing the expanded CBF using SVD and discarding the vectors of U associated with the values of S smaller than a threshold to eliminate the redundant information of Jl;ext , the new i CBF is expressed as follows: Jli ¼ U :; 1 : Kl;i (7.33) where Kl,i is the number of columns of the CBF matrix.

7.3.3 Generation of reduced matrix equation in the CBFM The inner product between CBFs can be defined as follows: T hJli ; AJlj i ¼ Jli AJlj

(7.34)

where A is the original or reduced impedance matrix of level l 1. Using the Galerkin method associated with CBFs to test the matrix equations, a new matrix equation of the level l can be expressed as follows: l

l

Z P I ðq; fÞ P ¼ Vl ðq; fÞ P (7.35) Bl Bl Bl Bl P Kl;i Kl;i Kl;i 1 Kl;i 1 i

i

i

i

where Zl, Il, and Vl are the reduced impedance matrix, coefficient of current distribution vector and excitation vector, respectively. Bl is the number of blocks of level l and Kl,i is the number of columns of the i-th block of level l.

Numerically efficient methods for electromagnetic modeling The tested impedance matrix Zl can be written as 2h i h i h i3 Rl1;2 ... Rl1;Bl Rl1;1 6 7 6h i h i h i7 6 7 l l l l 6 R R2;2 ... R2;Bl 7 2;1 Z ¼6 7 6 7 ... ... ... 7 6 ... i h i h i5 4h Bl Bl P RlBl ;2 . . . RlBl ;Bl P RlBl ;1 Kl;i Kl;i i

225

(7.36)

i

where Ri, j’s are the reduced impedance matrix blocks associated with the interaction between the i-th and j-th blocks. They are expressed as l Rli; j ¼ hJli ; Zl1 i; j Jj i

(7.37)

and can be calculated as follows: h i h T i ¼ Jli Rli; j Bl;i P Kl;i Kl; j Kl;i

p

h i Jlj Bl; j P q

Kl1;Spl;i

h i Zl1 Bl;i i; j P p

Bl; j P Kl1;Spl;i Kl1;Sql; j q

(7.38) Kl1;Sql; j Kl; j

Similarly, the reduced excitation vector Vl of level l can be written as h

h i iT Bl Wl2 . . . WlBl;i P Vl ðq; fÞ ¼ Wl1 Kl;i 1

(7.39)

i

where Wli is the reduced excitation vector associated with the i-th block: h T i l

Wi ðq; fÞ Kl;i 1 ¼ Jli Vl1 Bl;i Bl;i i P P Kl;i KSl1 Kl1;Spl;i 1 l;i p

p

(7.40)

p

Next, we substitute (7.36)–(7.39) into (7.35) and solve it to obtain the ‘‘reduced’’ version of the desired current distribution on the surface of the object. To obtain the original current distribution, we need to reverse the procedure of CBFM and express the coefficients of the current distribution of the level l 1, which is associated with the i-th block of level l as follows:

l

l1 l1 Il1 I . . . I ¼ Jli Bl;i Ii Kl;i 1 (7.41) B l;i S1l;i S2l;i SBl;i P l1 P l1 l;i KS l;i 1 KS l;i Kl;i p

p

p

p

By backtracking rom (7.41) to level 0, we can obtain the coefficient vector of the current distribution expressed in terms of the RWG basis functions.

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Before closing this section, we point out that when handling cases in which the object geometry changes dynamically (see Figure 7.6), we need to deal with different stages of modification of the object, whose components would in general have different relative positions at different stages. According to (7.31), if the relative positions of each component of the object remain unchanged, the submatrix of the original MoM matrix associated with the interaction between them would remain unchanged as well, together with the excitation vectors and CBFs associated with these components. This, in turn, implies that the unchanged matrix blocks, the vectors and CBFs can all be saved for repeated use later. Specifically, according to (7.37), if the relative positions of block i and j remain unchanged, the reduced impedance matrix Ri,j remains unaltered. In common with the conventional CBFM, the original MoM impedance matrix can be significantly reduced, and the computational efficiency and memory saving can again be realized as before. When dealing with the multi-level CBFM, we first partition the RWG basis functions into multiple levels, between two adjacent stages where the components of the object may have relative movements. If all the CBFs of the lower-level remain unchanged, the higher-level CBFs that are constructed based on the former CBFs would also remain unchanged and could therefore be saved in the memory for repeated use. However, obviously we would need to update the higher-level CBFs when the interaction among some of the lower-level CBFs changes between the two stages.

7.3.4 Multi-scale discretization using the IEDG method In real applications of analysis of electromagnetic scattering, the discretization of object is an important procedure, especially when dealing with fine features of the object, which needs to be discretized with small-scale elements, and often leading to an ill-conditioned impedance matrix which requires special attention. Furthermore, when dealing with multiple objects whose relative positions change at different stages, the connecting interfaces of different components would change, and they need to be discretized anew, which is inconvenient. To address this issue, we employ the IEDG method to deal with the discretization of connections among the components of the object, and with the associated multi-scale problem, and to enhance the efficiency of the numerical simulation.

7.3.4.1

Mono-polar RWG basis function

At the connection of different components of the object and geometries of different scales, we introduce the mono-polar RWG basis functions for the purpose of discretization (see Figure 7.5). The mono-polar RWG basis functions are constructed with a single triangular patch, which is expressed as 9 8 > = < l n r rþ ; r 2 T þ > n n (7.42) f mono ¼ 2Aþ n n > > ; : 0 else

Numerically efficient methods for electromagnetic modeling

Tn

Tm rm

227

Im fm(r)

In fn (r)

rn

Figure 7.5 Mono-polar RWG basis functions The surface divergence of the mono-polar RWG basis functions is expressed as follows: 8 > < ln ; r 2 T þ n mono rs f n ð r Þ ¼ A þ (7.43) n > : 0 else These expressions would be found useful in what follows.

7.3.4.2 Introducing the penalty term to construct the weak-form integral equation ^ f mono After testing the EFIE equation (7.12) with n , it can be written as m ð ð 2 3 0 0 0 jwm0 f mono ð r Þ J G ð r; r ÞdS dS ð 0 m 6 7 W W0 7 (7.44) f mono ðrÞ Einc ðrÞdS ¼ 6 ð ð m 4 5 1 0 W mono 0 0 0 0 f m ðrÞr r s J G0 ðr; r ÞdS dS jwe0 W W0 where the second term on the r.h.s. of (7.44) can be expressed as follows: ð 2 ð 3 0 mono rs f m Gr s J0 dS 0 dS 6 Wm 7 Wn 6 7 6 ð 7 ð 6 7 6 þ ^t m f mono Grs J0 dS 0 dl 7 m 6 7 ð 7 Cm Wn 1 1 6 0 0 0 6 7 ; rr GJ dS f mono ¼ n ð ð m 6 7 jwe0 jwe0 6 Wn 7 mono 0 0 6 þ rs f m G^t n J dl dS 7 6 7 W C m n 6 7 6 7 ð ð 4 5 mono 0 0 G^t n J dl dl ^t m f m Cm

Cn

(7.45) When testing with the mono-polar RWG basis functions, the contribution of operator rr, which is in the fourth term of (7.45) is found to contain the hypersingularity. Furthermore, as the continuity of the current distribution across the

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edges between triangular patches cannot be satisfied, and we need to enforce the above condition using the penalty term [21]. We choose the penalty term such that it not only eliminates the hype-singularity in (7.45) but also ensures the continuity of the current distributions, which can be expressed as follows: ^t mn Jm ðrÞ ¼ ^t nm Jn ðrÞ;

r 2 Cmn

(7.46)

where ^t mn is the outwards normal vector of the contour Cm, Cmn is the overlapped contour of Cm and Cn. In using the penalty term, we can define the residue of the current and charge across the edge, and test them with ^t m f mono , leading to m 1 ^ ht m f mono ; ^t mn Jm þ ^t nm Jn iCmn ; Cm ; Cn 2 Call m jwe0 ð 1 ^ mono ^t mn Jm þ ^t nm Jn G0 ðr; r0 Þdl0 iC ; ht m f m ; R2 ¼ m jwe0 Cmn R1 ¼

(7.47) Cm ; Cn 2 Call (7.48)

where 1/jwe0 is the scaling coefficient. After testing, we can construct the weakform of the integral equation, which reads ahf m ; EFIEim2Ne þ bR1 aR2 þ ð1 aÞh0 hf m ; MFIEim2Ne ¼ 0

(7.49)

where b is the coefficient of residue of R1, which can be set as l/20, based experience, and set R2 as a. The hyper-singularity in (7.45) is eliminated if we follow this procedure. In our work, we set a ¼ 0.2

7.3.4.3

Filling the element of impedance matrix

The matrix equation of the weak-form equation can be expressed as ! ! ! Zð11Þ Zð12Þ Ið1Þ Vð1Þ ¼ Zð21Þ Zð22Þ Ið2Þ Vð2Þ

(7.50)

where (12) is the testing basis function is the RWG basis function and source basis function is the mono-polar RWG basis function, the elements can be expressed as follows: 2 3 ð ð 0 0 0 f m ðrÞ f n ðr ÞG0 ðr; r ÞdS dS jwm0 6 7 W W0 6 7 ð ð 6 7 6 7 1 0 0 0 6 7 rs f m Grs f n ðr ÞdS dS þ 6 7 jwe 0 Wm W n 6 7 11Þ ¼6 Zðmn 7 (7.51) ð 0 6 7 f n ðr Þ 6 7 dS ð 1 a Þh f ð r Þ m 0 6 7 2 W 6 7 7 ð ð 6 4 5 ^ P V rG0 ðr; r0 Þ f n ðr0 ÞdS 0 dS ð1 aÞh0 f m ðrÞ n W

W0

Numerically efficient methods for electromagnetic modeling 2

229

3 mono 0 0 0 f ð r Þ f ð r ÞG ð r; r ÞdS dS m 0 n 6 7 W W0 6 7 ð ð 6 7 6 7 1 0 mono 0 0 6 7 r f Gr f ð r ÞdS dS þ s m s n 6 7 jwe 0 Wm Wn 6 7 ð ð 6 7 6 7 1 mono 0 0 ^ 6 7 rs f m G t n f n ðr Þdl dS ¼6 7 jwe 0 Wm Cn 6 7 6 7 ð mono 0 6 7 f n ðr Þ 6 7 dS ð 1 a Þh f ð r Þ m 0 6 7 2 W 6 7 7 ð ð 6 4 5 ^ P V rG0 ðr; r0 Þ f mono ðr0 ÞdS 0 dS ð1 aÞh0 f m ðrÞ n n ð ð

jwm0

12Þ Zðmn

W0

W

2

21Þ Zðmn

6 6 6 6 6 6 6 6 6 ¼6 6 6 6 6 6 6 6 6 4

ð ð jwm0

22Þ Zðmn

ð

Wm

ð

1 jwe0

Cm

^t m f mono m

ð ð1 aÞh0

W

jwm0 þ

1 jwe0

W

Wn

0

Wn

Gr s f n ðr0 ÞdS 0 dl

f mono ðr Þ m ð

^ PV n

f mono ðr Þ m ð ð

ð

ð

ð1 aÞh0

2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 ¼6 6 6 6 6 6 6 6 6 6 6 6 6 4

f mono ðrÞ f n ðr0 ÞG0 ðr; r0 ÞdS 0 dS m ð 0 mono rs f m Gr s f n ðr0 ÞdS 0 dS

W W0

1 jwe0

þ

(7.52) 3

f n ðr 0 Þ dS 2 0

W0

0

0

rG0 ðr; r Þ f n ðr ÞdS dS

f mono ðrÞ f mono ðr0 ÞG0 ðr; r0 ÞdS 0 dS m n ð 0 rs f mono Gr s f mono ðr0 ÞdS 0 dS m n

7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 (7.53) 3

W W0

ð

1 jwe0

Wm

ð

Cm

^t m f mono m

ð

Wn

0

Wn

Gr s f mono ðr0 ÞdS 0 dl n

ð ð 1 mono ðr0 Þdl0 dS rs f m G^t n f mono n jwe0 Wm Cn E h D ; ^t mn f mono ðr0 Þ þ ^t nm f mono ðr 0 Þ þ 0 ^t m f mono m m n Cmn jwe0 ð mono 0 f ðr Þ dS ð1 aÞh0 f mono ðr Þ n m 2 W ð ð 0 mono 0 0 ^ ð1 aÞh0 f mono ð r Þ n P V rG ð r; r Þ f ð r ÞdS dS 0 m n W

7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5

W0

(7.54)

230

Developments in antenna analysis and design, volume 2 ð ð1Þ Vmn ¼a ð ð2Þ Vmn ¼a

ð Wm

f m ðrÞ Einc ðrÞdS þ ð1 aÞh0

Wn

f m ðrÞ Hinc ðrÞdS

(7.55)

ð Wm

f mono ðrÞ Einc ðrÞdS þ ð1 aÞh0 m

Wn

f mono ðrÞ Hinc ðrÞdS m

(7.56)

Next, we can follow the CBFM to reduce the size of (7.50) and solve it for the current distribution on the surface of the object by using a direct solver.

7.3.4.4

Computational complexity

By reducing the size of the matrix equation via the CBFM, the computational complexity to implement the direct solver and the memory consumption to store 3 Þ and the reduced matrix of the l-th level of CBFM can be reduced to OðNCBF;l P Bl 2 OðNCBF;l Þ, respectively, where NCBF;l ¼ i¼1 Kl;i is the length of reduced matrix of level l. The computational complexity and memory consumption of conventional 3 2 Þ and OðNRWG Þ, respectively, where NRWG is the number of MoM are OðNRWG conventional RWG and mono-polar RWG basis functions. Since the compression rate of the impedance matrix depends on the effectiveness of the SVD algorithm in (7.2), it is difficult to provide an universal characterization of NCBF,l. To give an example, we employ the coefficients of the reduced matrices generated in the case 0:8707 is length of reduced of qq polarization in Example 2, where NCBF;1 ¼ NRWG matrix of the first level of CBFM. Therefore, the computation complexity and 1:7414 2:6121 Þ and O NRWG Þ, respectively, and for the memory consumption are OðNRWG 2:4054 1:6036 Þ and OðNRWG Þ, respectively. second level they become OðNRWG

7.3.5 Numerical results This section presents two examples to demonstrate the accuracy and efficiency of CBFM/IEDG approach for analyzing the problems of scattering from multiple and multi-scale targets, for which the results for mono-static RCS results for the qq polarization and ff polarization are computed. The computations have been carried out by using MATLAB on an Intel 64-bit Workstation with 4 cores and 8 GB RAM. A PEC target in free space is used for the first example (see Figure 7.6). The target is composed of five parts, namely, a box as its main body and four pyramids that are truncated at the middle. The frequency is 525 MHz, and 16,952 triangular elements are used to discretize the target. Specifically, the body of the target and the upper and lower parts of the pyramid are discretized with elements whose lengths are 0.125, 0.0875 and 0.055 of the wavelength, respectively. The five stages of the target are analyzed, where the pyramids are connected to the body in the first stage, break away from the body one by one in the next four stages. The spatial basis functions are subdivided into 12 blocks for level-1 and 48 blocks for level-2 via the octree algorithm to construct the CBFs. As the target undergoes a transition from the second through fifth stages, the CBFs and the reduced matrices associated

231

1m

1m

Numerically efficient methods for electromagnetic modeling

1m

0.5 m

2m

Figure 7.6 Target analyzed in Example 1

30

(1)

(2)

(3)

(4)

(5)

RCS (dBsm)

20 10 0 –10 Reference 1L-CBFM 2L-CBFM

–20 0

90

180,0 90 180,0 90 180,0 90 180,0 90 Degree of θ

180

Figure 7.7 qq Polarized mono-static RCS of Example 1 obtained by using the CBFM with the blocks that have experienced movements are updated, while the other CBFs and the reduced matrices are saved for reusing. The mono-static RCS results are plotted in Figures 7.7 and 7.8, the RCS calculated by a commercial software is used as a reference since the conventional MOM cannot handle such a large problem by using a direct solver. Using MATLAB, it takes 16.55 h to calculate all the elements of MOM-associated impedance matrix of the first stage. Employing an iterative solver is also not a desirable option because the iterative process including computation of impedance matrix must be repeated anew for each r.h.s. The range of the incidence angle q is from 0 to 180 and the incidence angle f is 0. To construct the CBFs, an extension of 0.1 wavelength is used. For the 2-level CBFM, the tolerances of SVD method are set as 1e 4 for the first level and 1e 3 for the

232

Developments in antenna analysis and design, volume 2 30

(1)

(2)

(3)

(4)

(5)

RCS (dBsm)

20 10 0

Reference 1L-CBFM 2L-CBFM

–10 0

90 180,0 90 180,0 90 180,0 90 180,0 90

180

Degree of θ

Figure 7.8 ff Polarized mono-static RCS of Example 1 obtained by using the CBFM Table 7.1 Unknowns and memory of reduced matrices of Example 1 for one and two levels Example 1

Unknowns

Stage

Method

1-L

1

MOM CBFM-ACA(qq) CBFM-ACA(ff) MOM CBFM-ACA(qq) CBFM-ACA(ff) MOM CBFM-ACA(qq) CBFM-ACA(ff) MOM CBFM-ACA(qq) CBFM-ACA(ff) MOM CBFM-ACA(qq) CBFM-ACA(ff)

23,796 5,348 5,449 24,230 5,268 5,390 24,664 5,189 5,327 25,098 5,185 5,264 25,532 5,087 5,200

2 3 4 5

2-L 2,827 2,038 2,779 2,045 2,731 2,088 2,682 2,117 2,634 2,169

Memory (MB) 1-L 8,640.3 436.4 453.1 8,958.3 423.5 443.3 9,282.1 410.9 433.0 9,611.7 410.2 422.8 9,946.9 394.9 412.6

2-L 121.9 63.4 117.8 63.8 113.8 66.5 109.8 68.4 105.9 71.8

CPU time (h) 1-L

2-L

– 12.04

12.43

– 4.12

4.53

– 6.25

6.67

– 8.01

8.44

– 10.02

10.47

second level, respectively, while for 1-level CBFM, the tolerance is set to 1e 6. As for the tolerance of the ACA algorithm, it is set to 1e 5. For both the pyramids and the body, mono-polar RWG basis functions are used to discretize the connections between pyramids and body of the target, as well as the upper and lower parts of the pyramids, while the other parts of target are discretized with traditional RWG basis functions. Table 7.1 shows that both the results calculated by using 2L-CBFM and 1L-CBFM agree well with the reference for both polarizations. The level of the size reduction and CPU time are presented in Table 7.1, which shows that the CPU time, number of unknowns and the memory usage are reduced significantly.

Numerically efficient methods for electromagnetic modeling

233

In Figures 7.7 and 7.8, the object has four stages that are divided by dashed lines and are noted by using ‘(1)–(4)’, where the horizontal axis denotes the incident angle of the mono-static RCS of each stage (from 0 to 180). For the second example, a PEC propeller in free space is analyzed (see Figure 7.9). The length, width and thickness of the two blades are 10, 0.4, and 0.1 m, respectively. The length and radius of the shaft are 1.6 and 0.15 m, respectively. A box is used as the base, whose length, width and height are 2.5, 1.5, and 0.3 m, respectively. The frequency is 480 MHz, and 16,554 triangular elements with length of 0.1 wavelength are used to discretize the target. The propeller is truncated into two parts at z ¼ 1 m, and their three stages are analyzed. These stages correspond to the blades rotating at the angles 0 , 22.5 , and 45 , respectively, while the base remains stationary. The spatial basis functions are subdivided into 58 blocks for level-1, and 27 blocks for level-2 via the octree algorithm utilized in the CBFM. For the second and third stages, the CBFs and the reduced matrices are associated with the blocks that undergo changes because of the rotation angles are updated, while the others are saved so that they can be reused. The mono-static RCS results are plotted in Figures 7.10 and 7.11, with the RCS calculated by a commercial software serving as the reference. Using MATLAB, it takes 20.36 h to calculate all the elements of MOM-associated impedance matrix of the first stage, and the unknown is too large to be solved by using direct solver. The range of the incidence angle q spans from 0 to 180 , while the incidence angle f is set to 0. A 0.1 wavelength of extension is used to construct the CBFs. For the 2-level CBFM, the tolerances of the SVD method are both set to 1e 4 for the first and second level, while for 1-level CBFM, the tolerance is set as 1e 5. Also, the tolerance of ACA algorithm is chosen to be 1e 5. The mono-polar RWG basis functions are used to discretize the connections between the blades and the base of the target, while the other parts are discretized with the traditional RWG basis functions. Figures 7.10 and 7.11

Figure 7.9 Object considered in Example 2

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Developments in antenna analysis and design, volume 2 (1)

(2)

Reference 1L-CBFM 2L-CBFM

30 RCS (dBsm)

(3)

20 10 0 0

90

180,0

90 180,0 Degree of θ

90

180

Figure 7.10 qq Polarized mono-static RCS of Example 2 obtained by using the CBFM

(1)

(2)

(3)

RCS (dBsm)

30 20 10 0

FEKO 1L-CBFM 2L-CBFM

–10 0

90

180,0

90 180,0 Degree of θ

90

180

Figure 7.11 ff Polarized mono-static RCS of Example 2 obtained by using the CBFM show that we can find that both the results calculated by using 1L-CBFM and 2L-CBFM agree well with those obtained from COMMERCIAL SOFTWARE. The size reduction associated with different levels and the CPU time are shown in Table 7.2, from which we can find that the CPU time, numbers of unknowns and the memory usage are reduced significantly when CBFM is used. In Figures 7.10 and 7.11, the object has three stages, that are divided by dashed lines and are noted by using ‘(1)–(3)’, where the horizontal axis denotes the incident angle of the mono-static RCS of each stage (from 0 to 180).

7.3.6

Summary

In this section, a combination of the CBFM and the IEDG method has been applied to solve the problem of scattering from multiple and multi-scale targets using nonconforming discretization. Multiple excitations have been used to construct the

Numerically efficient methods for electromagnetic modeling

235

Table 7.2 Unknowns and memory of reduced matrices of Example 2 for one and two levels Example 2

Unknowns

Stage

Method

1-L

1

MOM CBFM-ACA(qq) CBFM-ACA(ff) MOM CBFM-ACA(qq) CBFM-ACA(ff) MOM CBFM-ACA(qq) CBFM-ACA(ff)

24,751 6,690 6,012 24,751 6,690 6,010 24,751 6.690 6.012

2 3

2-L 3,334 3,177 3,335 3,179 3.334 3.155

Memory (MB) 1-L 9,347.7 682.9 551.5 9,347.7 682.9 551.1 9,347.7 682.9 551.5

2-L 169.6 154.0 169.7 154.2 169.6 151.9

CPU time (h) 1-L

2-L

– 10.42

10.84

– 2.04

2.47

– 2.07

2.49

CBFs, and the ACA has been employed to accelerate the construction of the reduced matrix. It is shown that the use of CBFM enables us to significantly reduce the size of the associated matrix, the CPU time and the memory, without compromising the accuracy. Furthermore, the reduced matrix equation derived in the context of CBFM can be solved directly without pre-conditioners, that are almost always needed when employing iterative solvers because of their convergence problems—a feature that can be very important when analyzing multiple-excitation problems.

7.4 Analysis of scattering from objects embedded in layered media using the CBFM 7.4.1 Introduction to CBFM analysis of the object embedded in layered media During the last decade various MOM-based CEM algorithms have been developed to analyze a wide variety of electromagnetic radiation and scattering problems, including those involving multi-layered media, such as the remote sensing of objects buried underground [22–24] or located in densely forested environments [25]. The multi-layered environment that we will discuss in this section is also common for antenna problems and, hence, is relevant. Furthermore, the assumption that the conductors are perfect, that is, PEC, is often not valid at the frequencies of interest, and/or for platforms whose materials having finite losses. This topic will also be covered briefly later in this section. To drive the associated MOM impedance matrix for a multi-layered medium, we need to generate the dyadic Green’s functions (DGF) [26], which is not available in closed form; hence, the matrix elements must be calculated numerically by integrating along the Sommerfeld integral path. We employ this procedure in conjunction with the discrete complex image method (DCIM), which was proposed

236

Developments in antenna analysis and design, volume 2

by Fang et al. [27] to express the integrand of the kernels into a series of complex images, and which enable us to evaluate the Sommerfeld integrals [28] analytically. After constructing the impedance matrix, the next step in MOM is to solve the matrix equation. Since the conventional MOM formulations use sub-domain basis functions, whose sizes typically range from to l/20 l/10 [4], the formulation becomes highly computer-intensive when dealing with electrically large objects and one is forced to use iterative solvers, which must be repeated from the start each time the r.h.s. is modified. It is well known that such iterative solvers rely on the availability of preconditioners, though at present there is no reliable or systematic procedure is available for generating such robust pre-conditioners that are universal in nature. To mitigate these problems, a number of methods have been proposed to reduce the CPU time and memory burden in MOM, including the multilevel fast multipole algorithm [29]. Alternatively, the multilevel matrix decomposition algorithm [30] groups the original impedance matrix into blocks to implement a fast matrix-vector multiplication. The wavelet basis functions [31] may help enhance the efficiency of the solution process but they are not universal in nature. The complex multiple beam approach [32] generates the scattered fields by using the beams emanating from a series of complex sources, though it is helpful for reducing the computational loads only for objects with smooth surfaces. The fast inhomogeneous plane wave algorithm [33] proposed as an alternative method to FMA has also been successfully applied for problems involving layered media. However, to implement the methods above, we still need to employ iterative solvers, which is not always desirable for reasons alluded to above. As mentioned earlier in Section 7.2, the CBFM [1] and its extensions [7, 8] are developed to reduce the size of matrix equation of the MOM, leading to the feasibility to employ the direct solver and therefore accelerating the solving process. For the layered medium case, Bianconi et al. have employed the CBFM in the past to analyze microwave circuits printed on a layered medium [34], [35], which is excited by edge ports. However, they have employed a slightly different version of the original CBFM, which is based on the use of a primary basis function, a series of secondary basis functions and only one excitation, which is suited for the analysis of circuits, while cannot handle multiple excitations and therefore is not suited for the problem we are dealing with here. More recently, Fenni et al. [10] have implemented the CBFM to the solution of scattering problems involving objects in a forest environment, in which the tree trunks were modeled as dielectric cylinders that are heterogeneous. In their work, the CBFM has been applied for the analysis of the scattering from objects located in the free space above a half-space environment. However, to solve the problems of scattering from buried objects, it is necessary to model the multi-layered media, especially for the analysis of objects that penetrate multiple layers. To deal with three-dimensional conducting objects, and for the multiple RHS problem, Yang et al. [36] have extended it to the multilevel CBFM to analyze the multiple-incidence problem of scattering from perfectly conducting object embedded in layered media. To construct the multi-level CBFs for objects embedded in multi-layered media, inhomogeneous plane waves are used

Numerically efficient methods for electromagnetic modeling

237

as excitations, whose equi-phase and equi-amplitude surfaces do not coincide with each other. In our case, it can be shown that homogeneous plane waves propagating across a lossy multi-layered stratified medium extending to infinity, will transform into inhomogeneous plane waves [37], in order to satisfy the boundary conditions. To analyze the lossy conducting and composite objects in the layered media, we employ the Poggio–Miller–Chang–Harrington–Wu–Tsai (PMCHWT) equations [38] associated with Mixed Potential Integral Equations for the MOM, and the DCIM is used to generate the DGFs. In contrast to the MOM analysis based on the impedance boundary condition (IBC) [39], the PMCHWT-MPIE-based MOM analysis treats the lossy objects as homogenous, without using approximations that are inherent in the IBC approach. Numerical results show that PMCHWT-MPIE is more robust than the IBC when dealing with highly lossy objects embedded in the layered media. In this section, we implement the ML-CBFM algorithm, to solve the problem of scattering from objects embedded in multi-layered media. The reduced multilevel matrix equations are constructed by using multiple inhomogeneous plane waves as excitations. Next, the SVD is employed to down-select the number of CBFs. The entire computation is naturally parallelizable, and the reduced matrix is considerably smaller in comparison to that of the original MOM matrix; furthermore, the accuracy of the final result is not compromised even as the matrix size is reduced. The PMCHWT-MPIE is employed for the analysis of objects comprising of lossy composite materials embedded in layered media. Compared to the MOM analysis based on the IBC, the PMCHWT-based analysis is more robust when dealing with highly lossy objects embedded in layered media, as may be the case when radar absorbers are used to reduce the RCS of an antenna, which is an important real-world problem.

7.4.2 Mixed potential integral equation for objects embedded in layered media The mixed-potential electric and magnetic integral fields on the surface of the object embedded in the layered media (Figure 7.12) can be expressed as D E D E 1 D VJ 0 0 E AJ ; J0 þ G EM ; M0 r G ; r J þ jwr G jw

(7.57)

E D E D E 1 D VM 0 0 AM ; M0 þ G HJ ; J0 r G ; r M jwr G jw

(7.58)

E ðr Þ ¼ H ðr Þ ¼

where the dyadic vectors in the MPIE can be written as 2 6 AJ =AM ¼ 6 G 4

AJ =AM Gxz

3

AJ =AM Gxx

0

0

AJ =AM Gyy

AJ =AM 7 7 Gyz 5

AJ =AM Gzx

AJ =AM Gzy

AJ =AM Gzz

(7.59)

238

Developments in antenna analysis and design, volume 2 E Z

H k

Zi+1 J

Layer i

Zi

n

M

Z3 Layer 2 Z2 Layer 1

Figure 7.12 Layered media 2

HJ =EM Gxx

6 HJ =EM HJ =EM ¼ 6 G 6 Gyx 4 HJ =EM Gzx

HJ =EM Gxy HJ =EM Gyy HJ =EM Gzy

HJ=EM Gxz

3

7 HJ=EM 7 Gyz 7 5 0

(7.60)

The dyadic vector is constructed with a series of spatial Green’s functions, whose expression may be found in [40], and whose integrands are the combination of the voltages and current induced by voltage and current sources in the layered media. To calculate the spectral-domain Green’s functions, we can transform the layered media into a transmission line model (see Figure 7.13), where zn, kz,n, and of the wave number Znp are the border of the n-th layer, vertical component !p p and the characteristic impedance, respectively. G n and G n are the right-forward and left-forward reflection coefficients of the nth layer, respectively. To obtain Green’s functions for multi-layered media, we need to calculate the voltage and current at z in the m-th layer induced by the voltage and current sources located at z0 in the n-th layer, when m ¼ n, the induced voltage and current can be expressed as follows: p ZnP jkznP jzz0 j 1 ! p P P P 0 e þ P G n ejkzn gn1 þ G n ejkzn gn2 Vi ðzjz Þ ¼ Dn 2

p! p P P jkzn gn3 jkzn gn4 (7.61) þe þ GnGn e

Numerically efficient methods for electromagnetic modeling p

p

Γn

Γn –

p Zn–1

kz,n–1 Zn

239

+ p

p

1V

Zn

Zn+1

1A

kz,n

kz,n+1

Z′

Zn+1

Figure 7.13 Transmission-line mode for layered media

IvP ðzjz0 Þ

p YnP jkznP jzz0 j 1 !p P P e ¼ P G n ejkzn gn1 G n ejkzn gn2 Dn 2

p! p P P jkzn gn3 jkzn gn4 þ GnGn e þe

(7.62)

!p ZnP 1 p P 0 P P kejkzn jzz j P G n ejkzn gn2 G n ejkzn gn1 ¼ Dn 2

p! p Pg Pg jkzn jkzn n4 n3 e þ GnGn e

(7.63)

p YnP 1 ! p P 0 P P kejkzn jzz j P G n ejkzn gn1 G n ejkzn gn2 ¼ Dn 2

p! p P P jkzn gn4 jkzn gn3 e þ GnGn e

(7.64)

VvP ðzjz0 Þ

IiP ðzjz0 Þ

where the first term of (7.61) and (7.63) is the voltage at z which is induced by the voltage and current sources at z0 , and the second term is the voltage of the field at z after times of reflection. The terms appearing in these four equations can be expressed as follows: p! p

DPn ¼ 1 G n G n tnp

(7.65)

0

k ¼ signðz z Þ

(7.66) 0

gn1 ¼ 2znþ1 ðz þ z Þ 0

gn2 ¼ ðz þ z Þ 2zn 0

gn3 ¼ 2dn þ ðz z Þ 0

gn4 ¼ 2dn ðz z Þ

(7.67) (7.68) (7.69) (7.70)

where dn ¼ znþ1 zn is the thickness of the n-th layer. The expressions for the reflection coefficients appear below: p

p Gn

¼

G pn1;n þ G n1 t pn1 p

1 þ G pn1;n G n1 t pn1

(7.71)

240

Developments in antenna analysis and design, volume 2 !

!

p Gn

¼

p

p Gpnþ1;n þ G nþ1 tnþ1 !

(7.72)

p

1 þ G pnþ1;n G nþ1 t pnþ1

where tnp ¼ expð2jkzn dn Þ, and the Fresnel Reflection coefficients are expressed as follows: G pij ¼

ZiP ZjP

(7.73)

ZiP þ ZjP

When m < n, VP and IP can be expressed as follows:

p p n1 1 þ G k ejkzk dk Y p

1 þ G k tkp

k¼mþ1

V ðzÞ ¼ V ðzn Þ P

P

p p G m tm

1þ

p p n1 1 þ G k ejkzk dk Y p

P

1 þ G k tkp

k¼mþ1

I ðzÞ ¼ V ðzn Þ P

1þ

p p G m tm

p p p 1 þ G m ej2kzm ðzzm Þ ejkzm ðzmþ1 zÞ

(7.74)

p p p Ymp 1 G m ej2kzm ðzzm Þ ejkzm ðzmþ1 zÞ (7.75)

where, for a unit current source excitation, we have ViP ðzn Þ ¼ ViP ðzn jz0 Þ. When the excitation is a unit voltage source instead, we have VvP ðzn Þ ¼ VvP ðzn jz0 Þ. Using the reciprocity theorem, we can derive the transformation relationships between ðVip ; Iip Þ and ðVvp ; Ivp Þ: Vip ðzjz0 Þ ¼ Vip ðz0 jzÞ

(7.76)

Ivp ðzjz0 Þ ¼ Ivp ðz0 jzÞ

(7.77)

Vvp ðzjz0 Þ ¼ Iip ðz0 jzÞ

(7.78)

Iip ðzjz0 Þ ¼ Vvp ðz0 jzÞ

(7.79)

When m > n, we can derive the expressions for ðVip ; Iip Þ and ðVvp ; Ivp Þ by replacing the locations of the source and field points. Substituting the expressions of the spectral Green’s functions into the inverse Fourier transform, we can obtain the spatial Green’s functions, where the inverse Fourier transform is an integral over kx and ky from negative infinity to positive infinity. Next, we can transform the inverse Fourier transform into a Sommerfeld Integral (SI) [41]. Toward this end, we define the angle j: j ¼ arctan

y y0 x x0

(7.80)

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241

The Sommerfeld Integrals transformed from the inverse Fourier transform over the cylindrical coordinates (j, r) can be expressed as follows:

(7.81) F 1 ~f kr ¼ S0 ~f kr

F 1 kx ~f kr ¼ j cosðjÞS1 ~f kr (7.82)

(7.83) F 1 ky ~f kr ¼ j sinðjÞS1 ~f kr n h i o

1 (7.84) F 1 kx2 ~f kr ¼ cosð2jÞS2 ~f kr S0 kr2 ~f kr 2 h h i 1 n

i o (7.85) F 1 ky2 ~f kr ¼ cosð2jÞS2 ~f kr þ S0 kr2 ~f kr 2 h i

1 F 1 kx ky ~f kr ¼ sinð2jÞS2 ~f kr (7.86) 2 where ~f ðkr Þ is the spectral Green’s function, Sn[•] is the n-order SI, which is expressed as follows: ð ð

1 1 ~ nþ1 1 1 ~ J kr r f kr kr dkr ¼ H ð2Þ kr r ~f kr krnþ1 dkr Sn f kr ¼ 2p 0 n 4p 1 n (7.87) qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ where r ¼ ðx x0 Þ2 þ ðy y0 Þ2 , Jn is the n-order Bessel function and Hnð2Þ is the n-order Hankel function of the second kind. After obtaining the DGFs, we can use the MPIE to solve for the current distributions on the surface of objects embedded in the layered media via MOM. Using octree to partition the basis functions into different groups of multiple levels, we can use the CBFM to construct the reduced matrix equation.

7.4.3 Numerical results This section demonstrates the accuracy and efficiency of ML-CBFM to analyze the scattering problem from the objects embedded in the multi-layered media via four examples. All the computations have been carried out by using MATLAB. In the first example, we analyze a PEC box which penetrates three layers of the five-layer medium that was used in Example 2 (see Figure 7.14). The length, width, and height of the box are 24, 9 and 4 m, respectively. The frequency is 100 MHz, and the box is discretized by using 29,976 RWG basis functions. The geometry is subdivided into 64 blocks in level 1 and into 8 blocks in level 2 via the octree algorithm to utilize ML-CBFM. DIE denotes the homogeneous medium, the top layer is the air, for the other four layers from top to bottom, the relative permittivities and thicknesses are er1 ¼ 2, er2 ¼ 4, er3 ¼ 6, er4 ¼ 8 and d1 ¼ 2 m, d2 ¼ 2 m, d3 ¼ 2 m, d4 ¼ infinity, respectively, and the loss tangent of these 4 media are all 0.2. The monostatic RCS results have been plotted for the polarizations qq and ff using CBFM and ML-CBFM in Figures 7.15 and 7.16, respectively. The range of the incidence angle q is from 89 to 89 . The incidence angle f is set to be 90 for

Developments in antenna analysis and design, volume 2 Z AIR 24 m

4m

9m

X

DIE 1

DIE 2

DIE 3

DIE 4

Figure 7.14 Multi-layered media of Example 1 θθ

40

RCS (dBsm)

20

φφ

0 –20

Reference 1L-CBFM (Nθ = 3)

–40

1L-CBFM (Nθ = 6) 1L-CBFM (Nθ = 10)

–60 –50

0 Degree of θ

50

Figure 7.15 Mono-static RCS of Example 1 using the CBFM θθ

40 20 RCS (dBsm)

242

φφ

0 –20

Reference 2L-CBFM (Nθ = 3)

–40

2L-CBFM (Nθ = 6) 2L-CBFM (Nθ = 10)

–60 –50

0 Degree of θ

50

Figure 7.16 Mono-static RCS of Example 1 using ML-CBFM

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Table 7.3 Unknowns and memory of reduced matrices of Example 1 Unknowns Polarization MOM CBFM Nq ¼ 3 2L-CBFM CBFM Nq ¼ 6 2L-CBFM CBFM Nq ¼ 10 2L-CBFM

qq 29,976 640 640 2,342 2,178 2,817 2,593

ff 640 640 2,346 2,269 2,848 2,660

Memory (MB) qq 13,710.9 6.2 6.2 83.7 72.4 121.1 102.6

CPU time (h)

ff 6.2 6.2 84.0 78.6 123.8 107.9

– 47.2 47.6 47.5 48.9 48.3 52.0

polarization qq, and 0 for polarization ff. For level 2, the adjacent blocks of level 1 are used to generate the CBFs, while for level 1, 0.1 wavelength of extension is used. The tolerances of SVD method is set to be 1e 3. Different numbers of incidences from 3 to 10 are used. The results generated by a commercial software are used as the reference, while the traditional MOM cannot be used with a direct solver since the number of unknowns is too large. Using MATLAB, it takes about 47 h to calculate all the elements of MOM-associated impedance matrix. Employing an iterative solver is also not a desirable option because the iterative process including computation of impedance matrix must be repeated anew for each RHS. When we use both the CBFM and ML-CBFM, the results for Nq ¼ 3 agree very well with the results calculated by using commercial software modules, whereas the results of Nq ¼ 6 and Nq ¼ 10 agree well with the reference. The level of size reduction is shown in Table 7.3, from which we can see that the numbers of unknowns and memory usage are reduced significantly. For the second example, we use the CBFM to analyze an object which gets divided into parts in the air above the ground (see Figure 7.17). The thickness of the ground is infinity, the relative permittivities of the medium is 8 and the loss tangent is 0.2. The object is located on the interface and is composed of a box and four cylinders. The mono-polar RWG basis functions are not used in this example since there is a gap of 0.1 wavelength between the box and cylinders. The length, width and height of the box are 9, 9, and 2.5 m, respectively, the height and radius of the four cylinders are 3 and 1 m, respectively. The frequency is 100 MHz. This object is discretized into 22,443 RWG basis functions. The RWG basis functions are partitioned into eight blocks, the threshold used in the SVD is 1e 6, and the length of the extension of the block is 0.2 wavelength, and Nq ¼ 10. The mono-static RCS are shown in Figures 7.18 and 7.19, for each one of the four stages, incident angle q is from 89 to 89 , and f is 0 . As the reference, the results calculated using commercial software is included. On MATLAB, for the first stage, it costs 33.7 h to calculate all the elements of the impedance matrix. From the results in Table 7.4, we can see that the CBFM can significantly reduce the size of the matrix equation without compromising the accuracy of the solution, and the total CPU time has been significantly reduced.

Figure 7.17 The object in Example 2 which is dividing into parts

(1)

40

(2)

(3)

(4)

RCS (dBsm)

30 20 10 Reference 1L-CBFM

0 –89

0

89,–89

0

0 89,–89 Degree of θ

89,–89

0

89

Figure 7.18 qq polarized mono-static RCS of Example 2 (1)

40

(2)

(3)

(4)

RCS (dBsm)

30 20 10 0 –89

Reference 1L-CBFM

0

89,–89

0

0 89,–89 Degree of θ

89,–89

0

89

Figure 7.19 ff polarized mono-static RCS of Example 2

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Table 7.4 Unknowns and memory of reduced matrices of Example 3 Stage

Method

Unknowns

Memory (MB)

CPU time (h)

1

MOM CBFM CBFM MOM CBFM CBFM MOM CBFM CBFM MOM CBFM CBFM

22,443 1,140 1,107 22,443 1,120 1,102 22,443 1,100 1,100 22,443 1,083 1,089

7,685.6 19.8 18.6 7,685.6 19.1 18.5 7,685.6 18.4 18.4 7,685.6 17.8 18.1

– 39.0

2 3 4

(qq) (ff) (qq) (ff) (qq) (ff) (qq) (ff)

– 4.7 – 9.1 – 13.4

Figure 7.20 Lossy conducting dipole In Figures 7.18 and 7.19, the stages of the object has four stages, which are divided by a dashed line and noted using ‘(1)—(4)’, where the horizontal axis denotes the incident angle of the mono-static RCS of each stage (from 89 to 89 ). To analyze the lossy conducting and composite objects in the layered media, we employ the PMCHWT associated with mixed potential integral equations for the MOM, and the DCIM is used to generate the DGFs. In the third example, we consider a lossy conducting dipole antenna (see Figure 7.20), whose length, width, and height are 30, 3, and 3 mm, respectively, and which is embedded in the layered media, has been analyzed. The relative permittivity and permeability of the material of the dipole are both 1, and its conductivity is 1e þ 3. The layered medium is 10 mm thick, its relative permittivity is 2, relative permeability is 1, and it is backed by a PEC ground plane. In Figure 7.21, the parameter Z11 calculated by PMCHWT-MPIE is presented; also, the reference results are calculated by using a commercial software, which employs the IBC solver with the thicknesses of the lossy conducting material set at 1, 0.1, and 0.01 mm, respectively. From Figure 7.21 we can see that for highly lossy objects embedded in the layered media, the results of the IBC method is strongly influenced by the

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Developments in antenna analysis and design, volume 2 90 PMCHWT-MPIE 80

IBC (1 mm) IBC (0.1 mm)

70

IBC (0.01 mm)

60 50 40 30

5

6

7

8

9

10

Frequency (GHz)

Figure 7.21 Comparison of Z11

Z/N

X/U Y/V

Figure 7.22 Coated lossy conducting box embedded in layered media thickness of lossy conducting material, while the result of PMCHWT-MPIE is accurate without depending on the setting of thickness. To verify the accuracy of the PMCHWT-MPIE, we consider the example of a lossy conducting box coated with lossy dielectric medium (see Figure 7.22) embedded in layered media. The geometric and electromagnetic parameters of the core are the same as those of the dipole in Example 2, and the length, width, and height of the coated medium are 36, 6, and 6 mm, respectively, the relative permittivity is 4 and conductivity is 0.1. The thickness of the layered medium is 15 mm, the relative permittivity is 2, loss tangent angle is 0.1, and there is a PEC ground. The excitation is plane wave. In Figures 7.23 and 7.24, we present the electric and magnetic current distributions on a line, which is along the direction of

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Magnitudes of the X-Component of Electric Current Distribution 0.01 PMCHWT-MPIE Commercial software

0.005

0 –0.01

200

–0.005 0 0.005 0.01 Coordinate X (mm) Phases of the X-Component of Electric Current Distribution

100 0 –100

–0.01

–0.005 0 0.005 Coordinate X (mm)

0.01

Figure 7.23 Electric current distributions on a line Magnitudes of the Y-Component of Magnetic Current Distribution 0.01

0.05 PMCHWT-MPIE Commercial software

0

–0.01

–0.005 0 0.005 0.01 Coordinate X (mm) Phases of the Y-Component of Magnetic Current Distribution 100 0 –100 –200

–0.01

–0.005 0 0.005 Coordinate X (mm)

0.01

Figure 7.24 Magnetic current distributions on a line length and on the middle of the top surface of the core. The PMCHWT-MPIE results agree well with those obtained from a commercial software. Figures 7.23 and 7.24 show that the two sets of results agree well with each other.

7.4.4 Summary In this work, the ML-CBFM is successfully applied to solve the problems of electromagnetic scattering and radiation from objects embedded in multi-layered

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media. Multilevel CBFs have been employed and have been constructed by using multiple inhomogeneous excitations. The size of the matrix equation as well as the memory used are significantly reduced through the use of ML-CBFM, without compromising the accuracy, and the need to use preconditioners is bypassed; furthermore, multiple-excitation problems are handled numerically efficiently. The PMCHWT-MPIE formulation has been employed for the analysis of objects comprising of lossy composite materials embedded in layered media. The PMCHWTbased analysis is found to be more robust than the MOM analysis based on the IBC, especially when dealing with highly lossy objects embedded in layered media.

7.5 CBFM for microwave circuit and antenna problems 7.5.1 Introduction In this section, we discuss an important aspect of CBFM, which is different from that of the scattering problem that we have presented earlier in this chapter. What distinguishes the antenna radiation problem from the scattering problem, in the context of CBFM, is the excitation used to construct the CBFs, as we show below. Unlike the scattering case, where historically we have always used plane wave type of excitations to generate the CBs, we have several options that we can exercise for CB generation for antenna problems. In the following sections, we detail three options and carry out a comparative study of their performances. Our recommendation for the preferred choice, which is based on the comparative study, is presented in Section 7.5.4, where we summarize our findings to conclude this section.

7.5.2 SVD-based CBFM As mentioned earlier in the Section 7.2 of this chapter, in CBFM, we first partition the object into a series of blocks and generate the CBFs by constructing and solving the matrix equations for the individual blocks, where the lower-level basis functions, such as the RWG basis function, are employed. Also, as mentioned earlier, to reduce the effect of truncation between different blocks during the generation of CBFs, we extend the blocks slightly (usually 0.1–0.2 wavelength, as shown in Figure 7.3). We generate the extended CBFs for each block by solving the submatrix of the impedance matrix of the MOM as follows: ext ext Jext i ¼ Zi Vi

(7.88)

ext ext where Jext i ’s are the extended CBFs associated with the i-th block, and Zi and Vi are the impedance and excitation matrices, respectively. To extract the information, we need for the CBFs of the electric current distributions induced by excitations impinging from different incident angles, we need to illuminate the basis functions inside and close to the block with sources to obtain the Vext i , by filling the elements of ext for each source. J ’s represent the responses due to multiple plane wave the Vext i i excitations, which includes some redundancy, which depends upon the number of

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plane wave excitations employed to construct them, but which can be eliminated by applying the SVD algorithm. The decomposed matrices of SVD are expressed as follows: T Jext i ¼ USV

(7.89)

where the elements along the diagonal are the singular values of Jext i , which are listed from big to small. Next, we construct a new CBF by discarding the vectors of U associated with Ki values of S smaller than a threshold ratio of the largest singular value to eliminate the redundant information of Jext i , the new CBF is expressed as follows: Ji ¼ Uð:; 1 : Ki Þ

(7.90)

We show below how the performance of the CBFM depends on the types of excitations.

7.5.2.1 Plane-wave excitation The first type of source for generating the excitation matrix is the plane wave, which is used, almost always, for RCS problems as may be seen from Section 7.2. However, for microwave circuit and antenna problems, we have other choices, though one of them is still the plane wave excitation, as shown in this section. To illuminate the basis functions inside and close to the block with plane waves from pws different angles to obtain the Vext ¼ Nq Nf as the number of wave i , we set N planes used to construct the CBF. In our study. For the first level, we set Npws ¼ 20 40, which means there are 10 samples along the direction q and 20 samples along the direction f, for both the vertical and horizontal polarizations. The process to generate the excitation matrix is shown as Figure 7.4. This is the same procedure as detailed in Section 7.2.

7.5.2.2 Edge-port excitation For the second type of excitation, we first employ edge ports on the surface of the object to be analyzed. We point that this is the most natural approach to fill the excitation matrix when dealing with the microwave circuit and antenna problems, as opposed to scattering problems. When using the RWG basis functions, the magnitude of tested electric filed generated by the edge port can be expressed as follows: En ¼ sgnðf n Þln

(7.91)

where fn is the n-th basis functions and ln the length of the common edge of the two triangular patches of fn. The value of sgn(fn) depends on the location and orientation of fn, when fn has an edge port set on and the left triangle of fn located at the positive side of the edge port, sgn(fn) ¼ 1; otherwise, sgn(fn) ¼ 1. For the other basis functions which do not have an edge port set on their common edges, we assign sgn(fn) ¼ 0. To generate the CBFs for an antenna geometry, the edge ports used for filling the excitation matrix are shown in Figure 7.25, in this case, the

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Developments in antenna analysis and design, volume 2

1 0.8 0.6 0.4 0.2 0 –0.2 –0.4 –0.6

0.05 N 0 0.05

X

–0.8 –1 0

Y

Figure 7.25 Edge ports on a dipole

z Er Eq

q

r

y

f x

r

Figure 7.26 Dipole moment along the z-axis

geometry of the dipole is partitioned into four blocks, the dark blue faces denote the extended region, and the red lines denote the edge ports which orientation are set along the x-axis. Besides, in an antenna application if we use a single delta-gap source for the excitation, then this source, which is implemented as an edge port, should be considered as one of the sources used to fill the excitation vector.

7.5.2.3

Dipole-moment excitation

A third alternative is to use an excitation source which is neither the far field type, for example, a plane wave, nor located on the body, as was the case with the edgeport excitation. For this case, we choose the dipole moments as the sources in the vicinity of the object to generate the excitation matrix, thereby introducing both the near-field and far-field information into the construction of the CBFs. A dipole moment (DM) oriented along the z-axis is shown in Figure 7.26.

Numerically efficient methods for electromagnetic modeling The electric dipole moment fields can be expressed as Il jkr h 1 Er ¼ þ cos q ¼ er cos q e 2p r2 jwer3 Il jkr jwm h 1 e þ 2þ sin q ¼ eq sin q Eq ¼ 4p r r jwer3

251

(7.92)

where r is the distance between the source and observation points, and the electric DM Il is defined as Il ¼ jE0

4p ðkaÞ3 hk 2

(7.93)

and we set Il ¼ 1, without loss of generality. The dipole moments are located above the object to be analyzed, and for each location, we use both x- and y-oriented dipoles for excitations, separately, for the sake of generality and completeness. The geometry of the DM excitation is shown in Figure 7.27, where the locations of 30 dipole moments are denoted as the red stars, and the extended elements are colored as dark blue.

7.5.3 Numerical results This section considers an example of an antenna problem to compare the accuracy and efficiency of CBFM approach when we use the three types of sources described in the section above, used to fill the excitation matrix. We compute the input impedance (Z11) and compare the results obtained for the three excitations. A PEC dipole in free space is used for this example (see Figure 7.28). When implementing the CBFM, the dipole is partitioned into 4 blocks, along the direction of the length, and a delta gap port is used for the excitation at the center. The

1 0.8 0.6 0.4 0.2 0 –0.2 –0.4 0.1 N 0.05 0

–0.6 –0.8

X

–1 0.05

0

Y

Figure 7.27 Dipole moments above a dipole

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Developments in antenna analysis and design, volume 2

Figure 7.28 Object analyzed in Example 1

Real part of the Z11

150

MoM EP-CBFM PW-CBFM DM-CBFM

100 50 0

3

3.5

4

4.5 5 5.5 Frequency (Hz) Imaginary part of the Z11

6

4.5 5 5.5 Frequency (Hz)

6

6.5

7 × 108

0 –50 –100 –150

3

3.5

4

6.5

7 × 108

Figure 7.29 Z11 of Example 1 obtained by using the CBFM frequency ranges from 300 to 700 MHz, with a step of 50 MHz. A total of 324 triangular elements are used to discretize the antenna, where the lengths of the elements are 0.1-wavelength at 500 MHz. To construct the CBFs, an extension of 0.1 wavelength of 500 MHz is used. The Z11 results are plotted in Figures 7.29 and 7.30, where the results calculated by the conventional MOM are used as the reference. In this example, the thresholds of the SVD down selection, number of samples and unknowns of the reduced matrices, are shown in the Table 7.5, where EP denotes the edge-port, PW denotes the plane-wave and DM denotes the dipolemoment excitation, respectively. Besides, we present the results at frequencies of 800 MHz and 1 GHz in Tables 7.6 and 7.7, respectively. At these two frequencies, the denser discretizations are used, the lengths of the elements of which are 0.1 wavelength. From the results, we see that the EP-CBFM and DM-CBFM algorithms achieve good agreements with the reference work, while comparatively speaking the PW-CBFM is not as accurate. From the results in the tables above, we can see that all three excitations yield comparable results; however, the EP-CBFM does use the least number of unknowns and, hence to, computationally the most efficient.

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Magnitude of the Z11 180 MoM EP-CBFM PW-CBFM DM-CBFM

160

140

120

100

80

60

3

3.5

4

4.5 5 5.5 Frequency (Hz)

6

6.5

7 × 10

8

Figure 7.30 Z11 of Example 1 obtained by using the CBFM Table 7.5 Parameters and results of Example 1 for frequency ¼ 300–700 MHz Method

Threshold

Samples

Min(unknowns)

Max(unknowns)

EP-CBFM PW-CBFM DM-CBFM MOM

0 1e 3 0 0

96 400 120 0

96 110 120 486

96 184 120 486

Table 7.6 Parameters and results of Example 1 for frequency ¼ 800 MHz Method

Threshold

Extension (mm)

Samples

Z11

Unknowns

EP-CBFM PW-CBFM DM-CBFM MOM

0 1e 3 0 0

60 60 60 0

142 400 160 0

35.40545.971i 34.02446.106i 34.48746.033i 32.17345.469i

142 358 160 1,536

Table 7.7 Parameters and results of Example 1 for frequency ¼ 1,000 MHz Method

Threshold

Extension (mm)

Samples

Z11

Unknowns

EP-CBFM PW-CBFM DM-CBFM MOM

0 1e 3 0 0

60 60 60 0

240 400 320 0

17.50541.867i 18.8742.999i 17.74641.57i 17.53541.732i

240 358 320 1,920

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Developments in antenna analysis and design, volume 2

7.5.4 Summary In this section, we have presented three types of sources as excitations for generating the CBFMs in the context of antenna problems, as opposed to RCS. We have shown that the use of CBFM enables us to significantly reduce the size of the associated matrix in comparison to the conventional MOM, and this is true regardless of the excitation, and that the EP-CBFM performs the best. We conclude that the near-filed contents of the EP and DM excitations (invisible range of the spectrum) contribute to the enhancement of the accuracy over that of the PW excitation, which is comprised of the visible spectrum alone.

7.6 Conclusions In this chapter, we have described several numerical techniques for electromagnetic modeling of large and complex antenna problems, which require a large number of DoFs to describe them, and hence, they are CPU time- and memory-intensive when conventional methods are used to tackle them. We have described the CBFM as well as the discontinuous-Galerkin technique IEDG for the handling of these type of problems numerically efficiently. The details of the formulations have been presented and numerical examples have been given to validate them. It is hoped that interested readers who have encountered problems with legacy codes would find these alternative approaches useful for their EM modeling work. For many practical applications, not only the antenna problems are of interest when they operate in either transmit or receive mode but also when we need to design them to have a low RCS, as is frequently the case in practice. Before closing this chapter, we point out that the methods described in this work are quite general and are well suited for all of these cases. Furthermore, they can be used to model antennas with arbitrary material properties, be they lossy or lossless. We also have presented three types of sources as excitations for generating the CBFMs for the microwave circuit and antenna problems, as opposed to RCS. We conclude that the near-filed contents of the EP and DM excitations can enhance the accuracy of the CBFM over the PW excitation.

Acknowledgment Sections 7.2 and 7.3 of this chapter are derived, in part, from two articles published online by Taylor & Francis in Journal of Electromagnetic Waves and Applications, respectively, on 16 May 2017, available at: https://www.tandfonline.com/ [DOI: 10.1080/09205071.2017.1311237], and on 22 Nov 2016, available at: https://www. tandfonline.com/ [DOI: 10.1080/09205071.2016.1250678].

List of acronyms GO PO

Geometrical optics physical optics

Numerically efficient methods for electromagnetic modeling GTD PTD DCIM MOM EFIE MFIE CFIE SVD BOR TDIE FFT FFA LR IE-FFT CG-FFT AIM RAS DDM IEDG CBFM RCS DGF SIP FEM MPIE ACA

255

Geometrical theory of diffraction Physical theory of diffraction Discrete complex image method Method of moments Electric field integral equation Magnetic field integral equation Combined field integral equation Singular value decomposition Bodies of revolution Time domain integral equation Fast Fourier transform Fast multipole algorithm Low rank Integral equation FFT Conjugate gradient FFT Adaptive integral method Random auxiliary sources Domain decomposition method Integral equation discontinuous Galerkin Characteristic basis function method Radar cross section Dyadic Green’s function Sommerfeld integral path Finite element method Mixed potential integral equation Adaptive cross approximation

References [1] Prakash, VVS, and Raj Mittra. ‘‘Characteristic basis function method: A new technique for efficient solution of method of moments matrix equations.’’ Microwave and Optical Technology Letters 36, no. 2 (2003): 95–100. [2] Cockburn, Bernardo, G. E. Karniadakis, and C. W. Shu. Discontinuous Galerkin Methods: Theory, Computation and Applications. Berlin: Springer, 2000. [3] Zhao, Ying, D. Ding, and R. Chen. ‘‘A Discontinuous Galerkin TimeDomain Integral Equation Method for Electromagnetic Scattering From PEC Objects.’’ IEEE Transactions on Antennas and Propagation 64, no. 6 (2016):2410–2417. [4] A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods for Electromagnetics. New York, NY, USA: IEEE Press, 1998.

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[5] Lucente, Eugenio, Agostino Monorchio, and Raj Mittra. ‘‘An iteration-free MoM approach based on excitation independent characteristic basis functions for solving large multiscale electromagnetic scattering problems.’’ IEEE Transactions on Antennas and Propagation, 56, no. 4 (2008): 999–1007. [6] Mittra, Raj. Computational Electromagnetics. Springer, 2003, p. 209. [7] Laviada, Jaime, Fernando Las-Heras, Marcos R. Pino, and Raj Mittra. ‘‘Solution of electrically large problems with multilevel characteristic basis functions.’’ IEEE Transactions on Antennas and Propagation 57, no. 10 (2009): 3189–3198. [8] Delgado, Carlos, Manuel Felipe Ca´tedra, and Raj Mittra. ‘‘Efficient multilevel approach for the generation of characteristic basis functions for large scatters.’’ IEEE Transactions on Antennas and Propagation 56, no. 7 (2008): 2134–2137 [9] Laviada, J., Gutierrez-Meana, J., Pino M.R., et al. ‘‘Analysis of Partial Modifications on Electrically Large Bodies via Characteristic Basis Functions.’’ IEEE Antennas and Wireless Propagation Letters 9, no. 1(2010): 834–837. [10] Fenni, Ines, He´le`ne Roussel, Muriel Darces, and Raj Mittra. ‘‘Fast analysis of large 3-D dielectric scattering problems arising in remote sensing of forest areas using the CBFM.’’ IEEE Transactions on Antennas and Propagation 62, no. 8 (2014): 4282–4291. [11] Nguyen, H., H. Roussel, and W. Tabbara. ‘‘A coherent model of forest scattering and SAR imaging in the VHF and UHF-band.’’ IEEE Transactions on Geoscience and Remote Sensing 44, no. 4(2006):838–848. [12] Peng, Zhen, X. C. Wang, and J. F. Lee. ‘‘Integral equation based domain decomposition method for solving electromagnetic wave scattering from non-penetrable objects.’’ IEEE Transactions on Antennas and Propagation 59, no. 9(2011):3328–3338. [13] Peng, Zhen, K. H. Lim, and J. F. Lee. ‘‘A discontinuous Galerkin surface integral equation method for electromagnetic wave scattering from nonpenetrable targets.’’ IEEE Antennas and Propagation Magazine 61, no. 7 (2013):3617–3628. [14] Arnold, Douglas N. ‘‘An interior penalty finite element method with discontinuous elements.’’ SIAM Journal on Numerical Analysis 19, no. 4 (1982):742–760. [15] Houston, Paul, and D. Scho¨tzau. ‘‘Mixed discontinuous Galerkin approximation of the Maxwell operator.’’ SIAM Journal on Numerical Analysis 42, no. 1(2004):434–459. [16] Ubeda, E., and J. M. Rius. ‘‘Novel monopolar MFIE MoM-discretization for the scattering analysis of small objects.’’ IEEE Transactions on Antennas and Propagation 54, no. 1(2006):50–57. [17] Vechinski, D. A., and S. M. Rao. ‘‘A stable procedure to calculate the transient scattering by conducting surfaces of arbitrary shape.’’ IEEE Transactions on Antennas and Propagation 40, no. 6(1992):661–665.

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[18] Yang, Su. ‘‘the analysis of electromagnetic scattering from targets dividing into parts.’’ PhD diss., Nanjing University of Science and Technology, 2017. [19] Su, Yang, Raj Mittra, and Weixing Sheng. ‘‘Analysis of scattering from multi-scale and multiple targets using characteristic basis function (CBFM) and integral equation discontinuous Galerkin (IEDG) methods.’’ Journal of Electromagnetic Waves and Applications 31, no. 10 (2017): 969–980. [20] Zhao, Kezhong, M. N. Vouvakis, and J. F. Lee. ‘‘The Adaptive cross approximation algorithm for accelerated method of moments computations of EMC problems.’’ IEEE Transactions on Electromagnetic Compatibility 47, no. 4(2005):763–773. [21] Clemens, Markus, Rolf Schuhmann, Ursula van Rienen, and Thomas Weiland. ‘‘Modern Krylov subspace methods in electromagnetic field computation using the finite integration theory.’’ Applied Computational Electromagnetics Society Journal 11, no. 1 (1996): 70–84. [22] Helaly, A., A. Sebak, and L. Shafai. ‘‘Scattering by a buried conducting object of general shape at low frequencies.’’ Microwaves Antennas & Propagation Iee Proceedings H 138, no. 3(1991):213–218. [23] Cui, T. J., and W. C. Chew. ‘‘Fast algorithm for electromagnetic scattering by buried conducting plates of large size.’’ IEEE Transactions on Antennas and Propagation 47, no. 6(1999):1116–1118. [24] Chang, H. S., and K. K. Mei. ‘‘Scattering of electromagnetic waves by buried and partly buried bodies of revolution.’’ IEEE Transactions on Geoscience and Remote Sensing GE-23, no. 4(1985):596–605. [25] Wang, Xiande, D. H. Werner, L. -W. Li, and Y. -B. Gan. ‘‘Electromagnetic scattering from complex targets located in a four-layer medium model of a densely forested environment.’’ In Antennas and Propagation Society International Symposium, Honolulu, Hawaii, 2007: 4829–4832. [26] Aksun M. Irsadi, and Raj Mittra. ‘‘Derivation of closed-form Green’s functions for a general microstrip geometry.’’ IEEE Transactions on Microwave Theory and Techniques 40, no. 11 (1992): 2055–2062. [27] Fang, D. G., J. J. Yang, and G. Y. Delisle. ‘‘Discrete image theory for horizontal electric dipoles in a multilayered medium.’’ In IEE Proceedings H (Microwaves, Antennas and Propagation), vol. 135, no. 5, pp. 297–303. IET Digital Library, 1988. [28] Sommerfeld, Arnold Johannes Wilhelm. Partial differential equations in physics. Academic Press, 1949. [29] Song, Jiming, Cai-Cheng Lu, and Weng Cho Chew. ‘‘Multilevel fast multipole algorithm for electromagnetic scattering by large complex objects.’’ IEEE Transactions on Antennas and Propagation, 45, no. 10 (1997): 1488–1493. [30] Michielssen, Eric, and Amir Boag. ‘‘A multilevel matrix decomposition algorithm for analyzing scattering from large structures.’’ IEEE Transactions on Antennas and Propagation, 44, no. 8 (1996): 1086–1093. [31] Wagner, Robert L., and Weng Cho Chew. ‘‘A study of wavelets for the solution of electromagnetic integral equations.’’ IEEE Transactions on Antennas and Propagation, 43, no. 8 (1995): 802–810.

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Chapter 8

Statistical electromagnetics for antennas Hulusi Acikgoz1, Ravi Kumar Arya2, Joe Wiart3, and Raj Mittra4

8.1 Introduction In a review paper dedicated to the recent trends in antenna engineering published in 1989 [1], the author was saying that ‘‘ . . . antenna technology is a vibrant field that is bursting with activity and likely to remain so into the foreseeable future . . . ’’. Although pioneering works in this area began more than a century ago, the development of new antenna types with various functionalities for specific purposes and applications is still an important research area within the electromagnetic community. Since the invention of wire antenna at the end of the nineteenth century, tremendous work has been done. The discoveries of aperture antennas, antenna arrays, reflectors, and microstrip antennas [2] have been among important milestones in the race for searching the perfection. Furthermore, the last decade has witnessed the breakthrough works on the so-called meta-materials which are when coupled with antennas can substantially improve their performances [3]. Nowadays, electromagnetic antennas are deployed everywhere with various applications spanning from biomedical to defense industry and wireless communication (Figure 8.1(a)). For instance, body-worn antennas for monitoring patients with diseases and for communication with the healthcare specialist are of capital importance (Figure 8.1(b)). With the advent of Internet of Things (IoT) [4], antennas are at the center of electromagnetic engineering activities. Also, nanoantennas are expected to be used in the near future for wireless communication at THz frequencies due to their wide band feature [5]. However, parallel to these cutting-edge researches, the complexity of the antenna and its close environment is continuously increasing. In the context of ‘‘soft electronics’’ (e.g. stretchable, deformable, textile antennas (Figure 8.2)), the complexity attributed to the variability of the intrinsic parameters is the major

1

Electrical and Electronics Engineering Department, KTO Karatay University, Turkey EMC Laboratory, The Pennsylvania State University, USA 3 Chaire C2M, LTCI Telecom ParisTech, Paris Saclay University, France 4 EMC Laboratory, University of Central Florida, USA 2

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(a) Information

Heart rate and respiratory rate

Location (GPS) communication gateway Cell phone network

Emergency

Internet Blu etoo th/Z igB ee/W LA N

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Clinician Intervention (b)

Figure 8.1 (a) Body-worn antenna (Copyright 2017 BAE Systems) and (b) illustration of a remote health monitoring system based on wearable sensors [6]

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Figure 8.2 (a) Flexible compact planar antenna on multilayer rubber polymer composite (reprinted with permission from [7], 2017 Elsevier GmbH) and (b) embroidered patch antenna (courtesy of MDPI AG, Electronics)

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aspect that is influencing the performance of the device and/or system. In fact, due to the very deformable nature of the structure (e.g. bending, stretching, twisting, etc.), the geometry may vary substantially in a certain range of the input parameters and thus lead to an unexpected behavior. Typical other factors affecting the variability of the antenna parameters are the precision of the manufacturing tools and the operator who is performing the task. Indeed, whatever the quality of the machine serving as a manufacturing tool, it can only be as precise as the tolerances given by the manufacturer. Sometimes, the operator needs to perform this task by hand in which case the precision may be even worse. Adding extra materials for gluing different parts of an antenna may also affect its resonance and efficiency. Textile antennas are good examples of this type of antenna manufacturing process. In a more complex environment, an antenna is usually surrounded by different electronic components and other antennas, sensors, etc. (Figure. 8.1(b)). In this case, the antenna is subject not only to its own variability but also to the variability of its neighborhood, close environment such as nearby scatters and supports. Consequently, besides the antenna itself, the close environment needs also to be taken into account for quantifying the variability of the antenna outputs. This section of the book will be devoted to the quantification of the antenna output variabilities using statistical analysis techniques, commonly called uncertainty quantification (UQ). In a generic UQ process, all input variables are represented by an adequate probability density function (PDF) representing a certain distribution over the input space. The aim of the UQ is then to propagate these variabilities through the model and to study their impact on the desired model output such as return loss, efficiency, and gain. More exactly, the quantification of the output variabilities is achieved by characterizing the statistics of the model prediction such as mean value, standard deviation, distributions, correlation between different outputs, and sensitivity analysis. The aim of the sensitivity analysis is then to assess which of the input parameters has the highest impact on the desired output uncertainty. Traditionally, the Monte Carlo (MC) method is used for the UQ. However, this well-known technique has the inconvenience to require a large number of the random samples of the input parameters. Thus, for a complex system having many input variables, the total simulation time for computing these samples may be time consuming and cumbersome. To bypass this computational issue, surrogate modeling techniques have been considered recently [8–13]. These techniques are based on a limited run of the complex model to create the experimental design built-up from the random samples and the corresponding outputs. This chapter is divided into four subsections. After a brief introduction, the next part is on the state of the art of variable antennas where previous works on these antennas are reviewed. It is followed by the introduction of the most common statistical analysis techniques used for the purpose of the uncertainty propagation. The fourth part is on the application of the statistical method described earlier on some real case structures, which are spit ring resonator (SRR) and a wearable antenna.

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8.2 State of the art of variable antennas Recent years have witnessed the extensive applications of electromagnetic antennas in complex environments where their electromagnetic performances may be significantly deteriorated due to the unwanted random disturbances coming out from the nearby elements such as other antennas, scatters, and supports. The complex nature of the supports and their non-planar geometry may require the antenna to be conformal or to have a deformable structure. The deformability of these antennas can be seen as a random process that affects the antenna performances. The randomness of these antennas may also be due to the variability of their intrinsic parameters. Thus, the term ‘‘variable antennas’’ employed in this context is used for both the variability of the antenna itself and the variability of its surroundings. Probably, the most significant application involving such randomness is the health monitoring or body centric applications. Technologies such as wearable antennas developed in the context of health-care system and worn on the body or printed directly on the garment have all the asset to improve the diagnostic and monitoring services provided to patients affected with diseases. Advances in information and wireless communication technologies will certainly help to monitor patients living in rural areas where a few physicians work and there is a deficiency of primary care providers. Follow-up of patients is carried out by antennas deployed according to the application of interest. For instance, for heart failure problems, an antenna measuring the heart rate would be necessary [6]. Due to their high flexibility and low cost, textile antennas have attracted great attention for body centric applications and health-care monitoring. Textile materials allow the antenna to be lightweight, robust, and easily integrated in radio frequency electronics. Wearable antennas based on textile materials that are conformal on the human body are expected to provide uninterrupted body-worn communication in different postures and situations. Most often, microstrip patches are used as a radiating element to send and receive information because they are low profile and they radiate perpendicularly to the planar radiating patch. Besides, the ground plane has a shielding effect against the body tissue. So far, various flexible antenna manufacturing processes have been proposed. The first and common type of manufacturing is by cutting a copper tape manually and adding it on a fabric surface using an adhesive material [14]. This technique has the main drawback to lead to the delamination of the copper tape from the substrate. Conductive ink may also be directly printed on the fabric but in this case, deforming the antenna may lead to the rupture of the ink [15]. Chemical etching is another technique which has gained popularity since its first usage in the 1960s. Highly complex patterns with high accuracy can be obtained by this technique [16]. Due to the repeatability and mass-production capabilities, embroidering techniques attracted attention for manufacturing the textile antenna [17] (Figure 8.2(b)). To our knowledge, one of the first studies on textile antennas is the work of P.J. Massey from Philips Research Laboratories [18]. The fabricated antenna is a

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PIFA antenna made of copper-plated ripstop nylon. This pioneering work has demonstrated the possibility of using fabrics for GSM antenna operating at 900 MHz. Later, Salonen and Hurne developed a fabric WLAN patch antenna fully integrated in clothing [19]. The substrate of the antenna is made of conventional fleece fabric, while the radiating element and the ground plane are made of knitted copper to maintain the antenna’s flexibility. Furthermore, Salonen et al. conducted several considerable works on the effect of different type of material for the development of the substrate and the conductive parts of GPS and WLAN antennas [20,21]. The effect of weave pattern and the orientation of stitches of an embroidered antenna were investigated in [14,22]. Kiourti and Volakis developed a stretchable and flexible wire antenna that can still operate in extreme conditions where the support of the antenna undergoes severe shape deformations. As an application, antenna mounted on a tire to sense its pressure and temperature changes is cited. The antenna is fabricated using conductive fibers (E-fibers). To ensure high stretchability, the radiating embroidered part is embedded into a stretchy polymer (Figure 8.3) [24]. Recently, researchers at The University of North Carolina have developed a sensor for tracking the bioelectric signals such as hydration and temperature of recovering people [24]. A flexible antenna made of silver nanowires is used to send the information collected to a distant doctor. The antenna is made of silver nanowires embedded in a polymer matrix (Figure 8.4). Other antenna types and geometries as rectangular slot, rectangular ring, aperture coupled patch, coplanar patch, asymmetric meandered flare, and electromagnetic band gap backed patch antennas were also investigated. For a comprehensive review on wearable and textile antennas, readers may refer to the review papers [17,25,26]. Parallel to the designing and manufacturing studies for developing the best suitable wearable antenna for a given application, researchers analyzed the effect of the deformation that might be performed on the antenna. One can mention, for instance, the study of Khaleel et al. [16] where strategies for designing, fabricating, and testing flexible antennas are proposed. In this work, a printed monopole antenna is studied, and flexibility tests are performed under different bending conditions for evaluating the antenna performances such as return loss and resonance frequency shift. In [27], a prototype of a UWB textile antenna has been fabricated, and the effect on the return loss of the antenna under bent and wet conditions has been discussed. Boeykens et al. [28] studied the effect of the cylindrical bending of a textile patch antenna by developing an analytical method

Figure 8.3 Stretchable and flexible E-fibers wire antenna. (Reprinted with permission from [23]. Copyright {2014} IEEE)

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Figure 8.4 Flexible antenna for health monitoring: (a) relaxed, (b) bent, (c) twisted, and (d) rolled. (Reprinted with permission from [24]. Copyright {2014} American Chemical Society.) based on the cylindrical cavity model for conformal rigid patch antennas. In this way, it has been showed that it is possible to predict the resonance frequency and the radiation pattern. In [29], same authors proposed the use of the polynomial chaos expansion (PCE) technique to predict the uncertainty in the resonance frequency of a textile antenna due to bending. The variability of the output gain of a graded-index flat lens due to the variability of its geometrical dimension has also been investigated using the PCE [11].

8.3 Statistical methods 8.3.1

General approach and surrogate modeling

In many engineering field, computer simulation codes are used for solving complex engineering problems. The purpose of using such simulation codes is to substitute expensive laboratory experiments and to carry forward the performance of engineered products. Engineering activities involving simulation/computer modeling are commonly referred to as computational engineering. However, computer

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simulations generally require a large amount of computation time to run. Simulations may take many hours, days, and even weeks to run. As a result, computation campaigns necessitating hundreds, even thousands of simulation runs may be quickly cumbersome. To reduce this computational burden and to predict the system performance as accurately as possible, an approximation and replacement of the expensive simulation model with a simpler model is performed. This approximation allows the engineer to mimic the behavior of the physical model, i.e., to predict the relationship between the system inputs and outputs at a lower computational cost. These approximation models are generally called surrogate models (also known as meta-models, replacement models). Surrogate models are statistical or data-driven models emulating the surface response of the model. The main emphasis of this chapter is to show how to use statistical methods to build a meta-model that can replace the computer-intensive numerical solver by analytical equations that are as simple as possible and are capable of providing results very quickly but also with the same accuracy. This meta-model can be used to obtain the probability distribution of the quantity of interest (for instance, the reflection coefficient of an antenna); it can also be used to perform a sensitivity analysis and characterize the statistical variations of the output when the inputs are varied. In this work, we use the PCEs in preference to the commonly used MC method, to develop a meta-model of the physical model by using a polynomial expansion approach. The low-rank tensor approximation [30,31] is an alternative polynomial expansion. In canonical decompositions, a tensor is expressed as a sum of rank-one components. This type of representation constitutes a special case of tensor decompositions, which are typically used to compress information or extract a few relevant modes of a tensor. By exploiting the tensor-product structure of the multivariate polynomial basis, as suggested by [31], canonical decompositions can provide equivalent to PCE representations in highly compressed formats. In the following sections, we first present the commonly used MC method. Next, we present the method used to investigate the effects of the variations in the electromagnetic structure. Finally, we include some numerical results to illustrate the application of the proposed method. Let us consider an electromagnetic problem or system that can be modeled by computational techniques such as the finite element method (FEM) or the finitedifference time-domain (FDTD) method. This computational model can be represented as y ¼ f ðx Þ

(8.1)

where y is a vector composed of the input parameters of the model, x 2 D Rm while the vector y [ RQ is the model response. The computational model, f, is considered as a black box, which is only known point-by-point. The function f here may correspond to the result of an FEM/FDTD simulation for the reflection coefficient of the physical structure. A particular set of inputs will produce exactly the same outputs no matter how many times this model is run. In other words, we can say that this model is purely deterministic and provides a unique response for a given set of unique inputs.

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Physical model

Accurate outputs y

Meta or surrogate model

Approximate outputs ŷ

Inputs: • Size parameters • Frequency • Material parameters

Figure 8.5 The original and the surrogate models The computational problem described above involves computer simulations which requires significant computational resources. Similar problems occur when the objective is to characterize the influence of input variations on the statistical distribution of the calculated outputs through simulations. One way to overcome such a limitation is to build simpler approximation models, known as surrogate models, which mimic the complex response of the model while, at the same time, being inexpensive in terms of computational cost [32]. So, our objective is to build an approximation of the model, which we refer to herein as the surrogate model (Figure 8.5) The surrogate model, yˆ ¼ M(x), does not require the knowledge of the internal operations of the physical system but only the input/output relationship of the system. The main objective is to build a surrogate model yˆ ¼ M(x) using effective and optimum methods. The PCE, also known as ‘‘polynomial chaos’’ (PC), is one of the methods which builds these meta-models. We will discuss more about PCE in the following sections.

8.3.2 MC simulations MC is a method of generating samples from a random vector. It was first coined by Enrico Fermi for the calculation of neutron diffusion in the 1930s [33]. N. Metropolis was the first scientist and ‘‘programmer’’ to use the algorithm in a computer environment [34]. Since that days, it has been used to solve problems in a variety of fields including medicine, engineering, and finance [35–37]. The uncertainty propagation using the MC method is fairly simple and follows the following steps (Figure 8.6): Step 1: A PDF is attributed to each input parameter Step 2: Random variables are generated according to the PDF previously defined (sampling) Step 3: The parametric numerical model is run to generate the corresponding outputs Step 4: The results are analyzed to compute the statistics of interest such as the average, the standard deviation, and the PDF.

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Deterministic numerical model y=f (x)

X={x(1),....,x(n)} Step 1

Step 2

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Figure 8.6 Flowchart of an MC simulation In an MC simulation, the key points are how precisely the model is defined, the PDF defining the range of possible values along with the corresponding probabilities and the number of samples within the limits provided by the PDF of the input parameters. Originally, experiments were used to measure the desired output from a given set of input parameters. These samples are later used in an MC simulation to determine the statistical moments of the desired output. Obviously, although this way of sample generation is closer to the reality, it might be very demanding in terms of resources. Additionally, it might not be possible to consider all the configurations according to the desired statistical distribution. After World War II, the development of computers has fostered the use of simulation models in replacement to the measurement. However, although computer simulations are now very accurate, limitations are quickly attained for problems with high complexity. The sampling of the input parameters according to their probability distribution is crucial in an MC simulation. In a typical MC simulation, the problem of clustering or scarcity sampling might arise in cases where the sample size is small. A large number of simulations are required to ensure a reliable coverage of the entire parameter range. This results in simulations with high computational costs. To circumvent the limitation of the MC method, an alternate way is to build a simpler and inexpensive model called meta-model (or surrogate model), which approximates the behavior of the computational model.

8.3.3 Polynomial chaos expansion The PCE is a probabilistic method which provides solution to a problem where the model parameters are considered to be randomly distributed variables. By means of a statistical PCE, a surrogate model is constructed, then the statistical output response (Y) of the system is computed based on the statistical input set of variables (X). Assuming that the input parameters are all independent, the polynomial expansion for the constructed meta-model is given by [38–40]: X aa ya ðX Þ (8.2) Y ¼ M ðX Þ ¼ a2N m

In (8.2) aa are the unknown coefficients to be determined, ya(X) are the multivariate polynomials, and a is the multi-index that identifies the components of ya(X).

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Two different approaches are proposed for constructing the meta-model of the physical problem: the ‘‘intrusive’’ and ‘‘non-intrusive’’ approaches. In this work, the latter one is preferred. The ‘‘non-intrusive’’ approach is based on the fact that the deterministic numerical model used in the simulation is not modified and is considered as a ‘‘black box’’. Once the computational model is constructed, no additional knowledge of its inner structure is required. For a given set of input parameters, the corresponding output is computed through a few set of models runs. The unknown coefficients of (8.2) are usually determined by truncating the polynomial expansion and thus keeping only a subset of polynomials. The truncated polynomial expansion can be expressed as X b ðX Þ ¼ Y^ ¼ M aa ya ðX Þ (8.3) a2A

In the truncated expression (8.3), A is a subset of finite dimension which contains all the retained multivariate polynomials. The number of polynomials depends on the maximum degree of the kept polynomials and the number of random input parameters. Y^ is the approximated output given by the truncated PCE. The regression approach is then used to minimize the residual e between the output Y of the deterministic numerical model and the approximated solution given by the truncated PCE: Y ¼ Y^ þ e

(8.4)

The ordinary least square resolution is commonly used to compute the coefficients [41]: ^ a ¼ ðYT YÞ1 YT y

(8.5)

where ^ a is the vector containing the estimated coefficients and y is the vector of the exact values of the output of the model. Y is a matrix of all the polynomials and is written as 1 0 ð 0Þ y0 X y1 X ð0Þ . . . yP X ð0Þ B y X ð1Þ y X ð1Þ . . . y X ð1Þ C B 0 C 1 P C Y¼B (8.6) .. .. .. .. B C @ A . . . . y0 X ðnÞ y1 X ðnÞ . . . yP X ðnÞ The classical truncation scheme described so far may be a time-consuming task with the increasing number of input parameters and/or higher order polynomials used to truncate the chaos expansion. To avoid this, the least angle regression selection (LARS) algorithm is usually preferred for the truncation [42]. In this method, the polynomials that have higher influence on the model output are selected. The most influential ones are kept from a full set of truncated polynomials. The accuracy of the obtained meta-model is checked using the so-called leaveone-out (LOO) cross validation algorithm [43]. This algorithm consists of constructing the meta-model with all the experimental design points except one and

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calculating the mean squared error on that taken out data. This procedure is repeated iteratively for each data set of the experimental design. For instance, if the input variable X has N distinct realizations as X ¼ (x(1), . . . , x(N)), the prediction error is calculated on that data point not considered to construct the model as follows: EðLOOÞ ¼

N 2 1X yðxðiÞ Þ ^y r ðxðiÞ Þ N i¼1

(8.7)

In (8.7), y(x(i)) is the output given by the deterministic model for the point xi not considered in the construction of the PCE and yˆr(x(i)) is the output computed at the point xi by the meta-model obtained with N 1 points. Once the meta-model based on the PCE is created, and the model is checked for accuracy, one can estimate the statistical moments such as the mean and the variance of the output variable. In fact, due to the orthogonality of the polynomials, the mean and the variance can be expressed, respectively, as [44]: m ¼ a0 Var ¼ s2 ¼

(8.8) P X

aj

(8.9)

j¼1

In (8.9), P is the number of terms retained in the truncated polynomials. One can notice that these statistical moments depend only on the PCE coefficients. Therefore, their estimation is straightforward. In many applications, the most important goal is to assess which of the input parameters are the most influential on the desired output of the physical process. Thus, it is necessary to have tools to determine quantitatively these effects. The truncated PCE may be used for this purpose [38]. The relative influence of each independent input parameters onto the output of the model is assessed using Sobol’s indices [45]. If the inputs are not independent, then the estimation of global sensitivity indices has to be carried out with methods such as those proposed by Kucherenko et al. [46]. With Sobol’s indices, the sensitivity of the desired output to the input parameter Xi may be estimated by [44]: X a2a a2A SiT ¼ Xi

a2An½0

a2a

(8.10)

The denominator of the expression given in (8.10) is the variance of the output where the mean is excluded, while the numerator is the sum of the squared coefficients of the polynomials for the variable Xi. Sobol’s indices may also be computed using the well-known MC simulations. However, their calculations become cumbersome in the case of complex models necessitating heavy computations. This procedure turns out to be very easy with the use the PCE. Indeed, once the meta-model is constructed by the truncated PCE, because of the orthogonality of

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Outputs are generated Y={y(1),....,y(2)}

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Step 6

Figure 8.7 PCE surrogate model building and sensitivity analysis the polynomials, the variances involved in (8.9) are easily calculated by only taking the sum of the squared coefficients. The statistical analysis procedure using the PCE technique described so far is shown in Figure 8.7. Step 1: Input parameter variables are selected and a PDF is attributed to each of them. Step 2: These parameters are sampled according to the attributed PDF and to an adequate sampling strategy [47]. Different sampling strategies have been proposed for building the experimental design. MC and The Latin Hypercube samplings are the most common ones used for this purpose. The traditional MC techniques use random/pseudo-random numbers to generate samples from the predefined probability distribution. This is an entirely random process, meaning that any generated sample value may fall anywhere within the range of the input distribution. The input parameters distribution can be easily recreated through MC sampling except for a very low number of samples. Indeed, a problem of clustering of data arises when a small number of samples are performed (Figure 8.8(a)). In contrast, Latin Hypercube sampling is a recently developed technique conceived to accurately recreate the input distribution. This is achieved through sampling in fewer iterations when compared with the MC simulation method. Latin Hypercube sampling make use of the stratification of the input probability distributions. It divides the cumulative distribution into equal intervals on the cumulative probability scale. A random sample is then generated in each interval of the input distribution. As a result, clustering of data may be avoided (Figure 8.8(b)). Step 3: The generated random input data are introduced to the deterministic numerical model in order to compute the corresponding output data. Step 4: The input/output data sets are processed, and the PCE coefficients are computed accordingly using the LARS algorithm. Step 5: The statistical output moments (mean, variance, and PDF) are calculated. Step 6: Sobol’s indices are extracted using the estimated coefficients of the PCE.

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Figure 8.8 MC (a) and Latin Hypercube (b) samplings

8.4 Case studies 8.4.1 Case I: Split ring resonator 8.4.1.1 Overview of the split ring resonator After the introduction of the idea of the negative index material by Veselago [48] in 1968, a periodic arrangement of metallic split ring resonator (SRRs) with wires was first proposed by Pendry et al. [49] to show the existence of the negativity of both permeability (m < 0) and permittivity (e < 0) over a certain range of frequency. The ‘‘left-handed’’ term is commonly used for this type of artificial material. Smith et al. [50] demonstrated for the first time the existence of the negative index material at microwave frequencies in 2000. It is well known that the negative permittivity can be obtained by a periodic arrangement of metallic wires at frequencies lower than its plasma frequency. On the other hand, obtaining the negative permeability is not straightforward due to the lack of such magnetic materials. The solution proposed in [49] to circumvent this problem was decisive in the design and fabrication of negative permeability and left-handed meta-materials. In contrast to the lattice of metallic wires for obtaining the negative e, the negative m given by the SRRs appears only at a narrow bandwidth around the magnetic resonance frequency. The SRR is usually formed of two concentric loops of conducting material etched on a substrate. A small portion of the loops is cut off to form a slit situated on opposite sides of the loops. The presence of the slits is of capital importance since they allow the structure to resonate at higher wavelengths (lower frequencies) compared to its physical dimensions. In the literature, the effect of more than one slit is also studied and a shift in the magnetic resonant frequency is observed [51]. The SRR can be considered as an oscillating LC circuit composed of a magnetic coil of inductance L and a capacitance of C. The slits act as parallel plate capacitors while the conducting loops act as inductors. The magnetic resonance of the SRR appears when the incident time-varying electromagnetic field has a magnetic field component penetrating the SRR plane (magnetic field perpendicular to the plane of the SRR). Thus, this transverse

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g

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Figure 8.9 (a) Illustration depicting the SRR array on a substrate and (b) unit cell showing the input parameters electric (TE) polarization mode couples with the capacitance of the loops and induces a current around the SRR. This induced current produces in turn a magnetic field that interacts with the external field and generates the magnetic resonance [52,53]. Several design strategies have been proposed and studied extensively in the literature. One can mention, for instance, works on periodic and aperiodic SRRs [54,55] parametric studies on SRRs [56], homogenization to extract the effective parameters [57], flexible meta-materials [58], etc. Moreover, researcher from all around the world imagined, conceived, and constructed different types of SRRs for different potential applications spanning from microwaves to terahertz and optical regimes. Typical prospective applications are, for instance, microwave filters [59] and bio-sensing [60,61]. Terahertz sensing is one of the most interesting applications of meta-materials. In fact, the well-established terahertz technology and strong respond of a myriad of material at the terahertz regime make it attractive for many applications. THz thin film sensing for biological and chemical sensing could be cited for this purpose [62,63]. The magnetic response of such materials at the THz regime also plays an important role in the development of THz devices such as mirrors and isolators [64].

8.4.1.2

Numerical model and the PCE analysis

An array of SRR consisting of two concentric conducting square loops has been considered. The SRR is of copper with conductivity s ¼ 5.8 102 and it is placed on a quartz substrate with 3.78 as dielectric constant. The SRR array along with the substrate is shown in Figure 8.9(a). The unit cell of the periodic array is investigated using the commercial electromagnetic software HFSS to simulate the magnetic response of the SRR structure (Figure 8.9(b)). The SRR deposited on the quartz substrate is illuminated by a normally incident electromagnetic plane wave. Floquet ports are assigned on top and bottom faces of the unit cell domain for the excitation purpose. Periodic boundary conditions are used to simulate the periodicity of the array structure.

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Table 8.1 Nominal values of the SRR input parameters Variable Nominal values (mm)

a 26

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To identify the resonance behavior of the SRR at THz frequencies, one calculates the S-parameters corresponding to both TE and transverse magnetic (TM) modes. The reflectance quantity which is the ratio between the S11 parameters of both modes (reflectance ¼ S11(TE)/S11(TM)) is considered. The input variable parameters of the SRR and the substrate are shown in Figure 8.9(b) and their nominal values are given in Table 8.1. Three levels of uncertainty, that are 1%, 5%, and 10%, are considered. The input parameters are supposed to have uniformly distributed random values around the nominal values given in Table 8.1. A PCE surrogate model is constructed for each frequency between 0.6 and 1.8 GHz. In other words, randomly selected input parameter data are presented to the deterministic model, and the corresponding random output data is calculated separately for each frequency. Thus, due to the operating frequency, the generated set of input/output data has different output data for the same input data. That is said, by means of the constructed PCE meta-models, a mean and a standard deviation of the reflectance are calculated independently at each frequency. Figure 8.10 shows the reflectance corresponding to each input sample (top plots) computed for each level of uncertainty. The estimated average reflectance obtained with the PCE and the MC technique along with the reflectance given by the nominal values of the inputs is also presented (bottom plots). The results demonstrate how the reflectance can be affected by the variability of the input parameters. It is clearly seen that the resonance frequency is lost while the level of the uncertainty increases. The amplitude of the average reflectance is also low around the resonance compared to the reference reflectance given by the nominal input values. These effects are much more emphasized for a higher uncertainty level. The error bars representing the standard deviation of the output show how much the output parameter is affected by the inputs variability. The accuracy of the meta-model constructed using the PCE technique is assessed by computing the LOO error between the meta-model and the deterministic numerical model outputs. This result is shown in Figure 8.11(a). A few hundreds of data are enough to construct the meta-model and to have the lowest error level. However, it can be seen that the LOO error (Figure 8.11(b)) and the standard deviation (Figure 8.11(c)) are highly dependent on the frequency and they are particularly significant around the resonance. Thus, the high value of the LOO error signifies that the surrogate model does not accurately approximate the output parameter. Indeed, it is difficult to construct the PCE surrogate model directly for the frequency-dependent reflectance. In fact, the non-smooth behavior of the reflectance over the frequency range where rapid changes in the amplitude around the resonance frequency occur and the shift of the resonance frequency due to the input parameter uncertainties make the PCE modeling very challenging. This rapid

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Figure 8.12 Sobol’s indices representing the relative importance of each input variables (a), top and bottom view of the SRR with the magnitude of the surface current density displayed at 1.3 THz (b). Darkest areas correspond to lowest current density changing behavior of the output amplitude is highlighted by plotting the PDFs of the reflectance at three different frequencies. The plots in Figure 8.11(d)–(f) show that the density function has a Gaussian shape for frequency out of the resonance region whereas around the resonance, it becomes elongated, covering a wide range of the amplitude of the reflectance. It is noticed that around the resonance, low amplitude output data having high density are clustered while others with high amplitude also exist. As a result of this phenomenon, the constructed PCE models around the resonance frequency are of higher order with considerable unknown coefficients to be determined. In [65], Yaghoubi et al. employed a method based on the stochastic frequency transformation combined with the principal component analysis to tackle this issue. As our objective is not to solve this problem in this chapter, we refer readers to [65] for a comprehensive description of this method as well as other methods found in the literature. Nevertheless, the average and the standard deviation of the output provided by the PCE surrogate model and the MC technique have similar behavior. The slight difference can be seen in Figure 8.11(c) where the standard deviation of the estimated reflectance against the frequency is plotted. Thus, one can assume that the statistical results provided by the PCE models are reliable. Sobol’s indices that ensure a good estimation of the importance of each input parameter on the output variance of the SRR are calculated. Sobol’s indices versus the frequency are presented in Figure 8.12(a). It can be easily observed that the input parameter having the highest influence on the reflectance is the side length a of the SRR. It has a quasi-stable behavior at all the frequency band. This parameter is followed by the periodicity b of the SRR array and the wire width w, which have a considerable effect around the resonance. All other parameters have a negligible impact. The surface current magnitude displayed at the resonance frequency of

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Figure 8.13 Plot displaying the uncertainty of the outputs (amplitude and the resonance frequency) for two levels of input uncertainty (5% and 10%) 1.3 THz helps to understand this behavior (Figure 8.12(b)). Indeed, the high amplitude of the surface current along the side length a of the SRR and at the wire width w is a sign that mostly these parameters are contributing to the reflectance. So far, the mean and the standard deviation of the reflectance were examined by building a PCE for each frequency. Here, a different strategy has been followed to evaluate the uncertainty of the reflectance. Two different outputs have been examined: the resonance frequency and the corresponding reflectance amplitude at the resonance. Thus, for each set of input data, a characteristic reflectance is calculated using the deterministic model, and the resonance frequency and the resonant amplitude are extracted. Thus, two PCE models are constructed; one for the resonance frequency and another for the amplitude at the resonance. The same set of input/output data set is used to construct the PCEs. The statistical outputs estimated by the PCEs are represented by the square areas (in grey) in Figure 8.13. The squares are centered at the average values and the side length of the squares are determined by calculating the coefficient of variation. In Figure 8.13, Df is the coefficient of variation of the resonance frequency and DR is the coefficient of variation of the amplitude of the reflectance. For example, 10% of variation of the input variables corresponds to 1.25 THz and 14.52 as average values, and 12.5% and 26.5% as coefficient of variation of the resonance frequency and amplitude, respectively. As given by the deterministic model, the resonance frequency and the reflectance amplitude corresponding to nominal values (without uncertainty) are 1.22 THz and 14.48, respectively (black curve). It can be seen that for both level of uncertainty of the input variables (DX), the average values estimated by the PCEs are very close to the deterministic ones. However, the coefficient of variation can be as high as 12.5% and 26% for the resonance frequency and the amplitude, respectively.

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8.4.2 Case II: Wearable textile antenna 8.4.2.1

Description of the manufacturing process

As mentioned before, among others, two major techniques exist for manufacturing flexible textile antennas: laminating and embroidering. Laminating is the process of antenna fabrication consisting of superposing conductive and insulating fabrics and gluing them with a thermal adhesive sheet (Figure 8.14(a)). This fabrication process has mainly two sources of uncertainty. The first one is due to the cutting process which is critical, as the antenna has very thin lines. The second source is from the use of adhesive sheet to glue different parts, such as the radiating patch, the substrate, and the ground plane. It has been shown that the use of an adhesive sheet alters the conductivity of the conducting patch and/or the dielectric constant, especially when ironing is performed for gluing. Embroidering is a promising technique that has gained considerable attention for wearable textile antenna fabrication (Figure 8.14(b)). As the embroidered antennas do not need a cutting or lamination process, it is more suitable for massproduction compared to the lamination. Recently, advanced technologies enabling digital image directly embroidered using computer aided embroidering machine have been used. Several types of embroidered antennas have been proposed, such as the spiral antenna [66], RFID tags [67], and FSS structures [17]. As for the lamination technique, one has to be careful while embroidering the antenna. Indeed, besides the geometrical dimensions that may have impact on the antenna performance, the conductivity of the patches is also a source of uncertainty. In fact, the conductivity is highly dependent on the density of the stitches that are crucial for having a good conductivity of the patch. The effect of the ironing process for gluing the conducting patch on the substrate is shown in Figure 8.15 [14]. It is clearly shown that the magnitude of S11 versus frequency differs from one process to another. For example, ironing with steam and without steam lead to a shift in the resonance frequency and a change in the magnitude of S11. There are also considerable changes between simulation and measurement. This change is certainly due to such uncertainties introduced during the manufacturing process.

Conducting fabric Substrate Thermal adhesive sheet Ground plane (a) (b)

Figure 8.14 Lamination (a) and embroidering techniques for antenna fabrication (Courtesy of MDPI AG, Electronics) (b)

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(a)

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Figure 8.16 SEM images of the cross-section of the antenna: without steam (a) and with steam (b) (Courtesy of MDPI AG, Electronics) The SEM images give an insight into why the ironing process with steam has more impact on the return loss. According to [14], It can be seen from the images that there is a higher compaction of the material when the steam is used for gluing with the adhesive sheet. It is noticed that, in Figure 8.16(a), the adhesive sheet (represented by arrows) is between the path and the substrate, whereas when the steam is applied (as shown in Figure 8.16(b)), it tends to merge with the substrate and slightly with the conductive patch. It is believed that this phenomenon may lead to a decrease in the conductivity, and thus it increases the electrical resistivity. Also, the substrate may have its permittivity changed due to the adhesive sheet. Furthermore, precision of the cutting process is also affecting the performance of the antenna.

8.4.2.2 Numerical model and the PCE analysis PCE analysis is performed to quantify the effect of the antenna input parameters variability on the return loss. Five geometrical dimensions of the patch/substrate, the patch conductivity, and the substrate dielectric constant are investigated. The

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Description Radiating element geometrical dimensions as indicated on the top view of the antenna Substrate thickness Substrate dielectric constant Patch conductivity

Figure 8.17 Top view of the antenna (a) and the nominal values of its variable parameters (b) 8

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Figure 8.18 Return loss estimated by PCE and MC techniques (a) and the corresponding standard deviation (b) nominal values are given in Figure 8.17. An uncertainty of 10% is considered. 100 samples based on a uniform distribution around the nominal values are computed using the deterministic model (HFSS). The obtained input/output set (experimental design) is then used to build the PCE meta-model. In Figure 8.18, the average return loss and the standard deviation are presented. The average return loss estimated by the PCE is also compared with the ones calculated with the nominal values and the MC method. It is demonstrated that the PCE result is similar to that given by MC. However, although the resonant frequency remains quasi unchanged, the return loss magnitude is highly affected. This is especially true around the resonant frequencies where the standard deviation is particularly high. Indeed, the standard deviation representing the variability of the

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Figure 8.19 The surface current magnitude at 900 MHz (a), Sobol’s indices of each input variable versus frequency (b), and the surface current magnitude at 1,800 MHz (c). Darkest areas correspond to lowest current density output is extremely dependent on the frequency (Figure 8.18(b)). It is obvious that this high variability is more crucial at higher frequencies, where a mismatch (10 dB criteria) of the antenna is likely to occur and its performance is compromised. As detailed in the previous section, the truncated PCE can be directly linked to the statistical moments of the output. Sobol’s indices which are direct consequences of those moments are powerful and important tools to estimate the relative effect of each input parameter to the output. In Figure 8.19, Sobol’s indices versus frequency are plotted for each input parameter. It is demonstrated that not all the input parameters have the same effect on the return loss and their relative effect is frequency dependent. Indeed, dependent on the frequency, different input parameters have different levels of importance. For instance, at higher frequencies where the second resonant frequency occurs, only two parameters mainly contribute to the output variability, namely, Wm1 and Wm2 which are the lengths of the two arms of the radiating element (slotted loop). All others have a negligible effect except the patch conductivity and the position of the feed line that have a relatively low impact around 1,800 MHz. On the contrary, it is noticed that the impact of the length of the arms decreases as one goes to lower frequencies. The position of the feed line becomes more important at low frequencies. However, the radiating element arms Wm1 and Wm2, the feed line position Fx and its width Wf, the substrate thickness hsub, and its dielectric constant epsC have a similar impact around the first resonant frequency. To support the estimated Sobol’s indices, surface current distributions at both frequencies 900 MHz (Figure 8.19(a)) and 1,800 MHz (Figure 8.19(c)) are also plotted. These plots partially show the accuracy of the results given by Sobol’s indices. As can be clearly seen, at the higher resonant frequency (1,800 MHz), the magnitude of the surface current is very high at the radiating element arms which is consistent with the high level of Sobol’s indices for Wm1 and Wm2. On the other hand, at the lower resonance, the surface current amplitude is particularly low at the two arms meaning that the arm dimensions are less crucial compared to the higher resonance.

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8.5 Conclusions In this chapter, statistical analysis of some well-known electromagnetic structures has been carried out. The PCE method has been preferred to the commonly used MC simulations as the latter one is computationally more expensive while studying geometrically complex structures. After introducing the variable antennas, and the traditional MC technique and the stochastic PCE method have been briefly described, the PCE has been applied to two widely investigated structures in the literature—that are the SRR and a type of wearable patch antenna, originally intended for electromagnetic energy harvesting. Only the variability of the physical and geometrical parameters due to the manufacturing precision has been considered. In this study, the variability of the performance of these structures due to their deformation has not been considered. The output parameters (reflectance and return loss) variability has been evaluated. It has been shown that uncertainties on the input variables due to the manufacturing process could lead to the deterioration of the performances such as loss of the resonance frequency and mismatch of the antenna. Moreover, input parameters that have the most influence on the outputs are determined by performing sensitivity analysis based on Sobol’s indices. However, although the PCE technique is a powerful tool to propagate uncertainties of the input variables to the output of the antenna, it has limitations when modeling frequency-dependent parameters. Further studies are needed to overcome these difficulties. The use of the stochastic frequency transformation and the principal component analysis could be a good choice for this purpose. In addition, to confirm these numerical results, experimental studies are to be performed in the future.

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[36] Hayakawa C.K., and Spanier J.: ‘Perturbation Monte Carlo methods for the solution of inverse problems’, In: Niederreiter H. (ed), Monte Carlo and QuasiMonte Carlo Methods, Berlin, Heidelberg: Springer, 2002, pp. 227–241. [37] Ljungberg M., Strand S.-E., and King M.A.: ‘Monte Carlo Calculations in Nuclear Medicine’, Applications in Diagnostic Imaging. 2nd edn., Boca Raton, FL: CRC Press, 2012, 357 p. [38] Sudret B., Blatman G., and Berveiller M.: ‘Response surfaces based on polynomial chaos expansions’. In: J. Baroth, F. Schoefs, D. Breysse (eds), Construction Reliability – Safety, Variability and Sustainability, London: ISTE/Wiley, 2013, pp. 147–167. [39] Kersaudy P., Mostarshedi S., Sudret B., Picon O., and Wiart J.: ‘Stochastic analysis of scattered field by building facades using polynomial chaos’, IEEE Trans. Antennas Propag., 2014, 62, (12), pp. 6382–6393. [40] Blatman G., and Sudret B.: ‘Efficient computation of global sensitivity indices using sparse polynomial chaos expansions’, Reliab. Eng. Syst. Saf., 2010, 95, (11), pp. 1216–1229. [41] Blatman G., and Sudret B.: ‘Adaptive sparse polynomial chaos expansion based on least angle regression’, J. Comput. Phys., 2011, 230, (6), pp. 2345–2367. [42] Blatman G., and Sudret B.: ‘Sparse polynomial chaos expansions and adaptive stochastic finite elements using a regression approach’, C. R. Me´c., 2008, 336, (6), pp. 518–523. [43] Molinaro A.M., Simon R., and Pfeiffer R.M.: ‘Prediction error estimation: a comparison of resampling methods’, Bioinformatics, 2005, 21, (15), pp. 3301–3307. [44] Sudret B.: ‘Global sensitivity analysis using polynomial chaos expansions’, Reliab. Eng. Syst. Saf., 2008, 93, (7), pp. 964–979. [45] Sobol I.M.: ‘Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates’, Math. Comput. Simul., 2001, 55, (1), pp. 271–280. [46] Kucherenko S., Tarantola S., and Annoni P.: ‘Estimation of global sensitivity indices for models with dependent variables’, Comput. Phys. Commun., 2012, 183, (4), pp. 937–946. [47] Mckay M.D., Beckman R.J., and Conover W.J.: ‘A comparison of three methods for selecting values of input variables in the analysis of output from a computer code’, Technometrics, 2000, 42, (1), pp. 55–61. [48] Veselago V.G.: ‘The electrodynamics of substances with simultaneously negative values of e and m’, Sov. Phys. Usp., 1968, 10, (4), pp. 509–514. [49] Pendry J.B., Holden A.J., Robbins D.J., and Stewart W.J.: ‘Magnetism from conductors and enhanced nonlinear phenomena’, IEEE Trans. Microw. Theory. Tech., 1999, 47, (11), pp. 2075–2084. [50] Smith D.R., Padilla W.J., Vier D.C., Nemat-Nasser S.C., and Schultz S.: ‘Composite medium with simultaneously negative permeability and permittivity’, Phys. Rev. Lett., 2000, 84, (18), pp. 4184–4187. [51] Penciu R.S., Aydin K., Kafesaki M., et al.: ‘Multi-gap individual and coupled split-ring resonator structures’, Opt. Express., 2008, 16, (22), pp. 18131–18144.

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Developments in antenna analysis and design, volume 2 Katsarakis N., Koschny T., Kafesaki M., Economou E.N., and Soukoulis C.M.: ‘Electric coupling to the magnetic resonance of split ring resonators’, Appl. Phys. Lett., 2004, 84, (15), pp. 2943–2945. Lahiri B.: Split Ring Resonator (SRR) Based Metamaterials, Ph.D. thesis, University of Glasgow, 2010. 170 p. Aydin K., Guven K., Katsarakis N., Soukoulis C.M., and Ozbay E.: ‘Effect of disorder on magnetic resonance band gap of split-ring resonator structures’, Opt. Express., 2004, 12, (24), pp. 5896–5901. Martin F., Falcone F., Bonache J., Marques R., and Sorolla M.: ‘Miniaturized coplanar waveguide stop band filters based on multiple tuned split ring resonators’, IEEE Microwave Compon. Lett., 2003, 13, (12), pp. 511–513. Aydin K., Bulu I., Guven K., Kafesaki M., Soukoulis C.M., and Ozbay E.: ‘Investigation of magnetic resonances for different split-ring resonator parameters and designs’, New J. Phys., 2005, 7, pp. 168–168. Smith D.R., and Pendry J.B.: ‘Homogenization of metamaterials by field averaging (invited paper) ‘, J. Opt. Soc. Am. B., 2006, 23, (3), pp. 391–403. Zhang F., Liu Z., Qiu K., Zhang W., Wu C., and Feng S.: ‘Conductive rubber based flexible metamaterial’, Appl. Phys. Lett., 2015, 106, (6), 061906. Genc A.: ‘Metamaterial-inspired miniaturized multi-band microwave filters and power dividers’, 2010, All Graduate Theses and Dissertations. 700. Available from https://digitalcommons.usu.edu/etd/700 Lee H.J., Lee J.H., Moon H.S., et al.: ‘A planar split-ring resonator-based microwave biosensor for label-free detection of biomolecules’, Sens. Actuators B. Chem., 2012, 169, pp. 26–31. Torun H., Cagri Top F., Dundar G., and Yalcinkaya A.D.: ‘An antennacoupled split-ring resonator for biosensing’, J. Appl. Phys., 2014, 116, (12), 124701. Tao H., Strikwerda A.C., Liu M., et al.: ‘Performance enhancement of terahertz metamaterials on ultrathin substrates for sensing applications’, Appl. Phys. Lett., 2010, 97, (26), 261909. O’Hara J.F., Singh R., Brener I., et al.: ‘Thin-film sensing with planar terahertz metamaterials: sensitivity and limitations’, Opt. Express., 2008, 16, (3), pp. 1786–1795. Kumar N., Strikwerda A.C., Fan K., et al.: ‘THz near-field Faraday imaging in hybrid metamaterials’, Opt. Express., 2012, 20, (10), pp. 11277–11287. Yaghoubi V., Marelli S., Sudret B., and Abrahamsson T.: ‘Sparse polynomial chaos expansions of frequency response functions using stochastic frequency transformation’, Probab. Eng. Mech., 2017, 48, pp. 39–58. Zhang S., Paraskevopoulos A., Luxey C., Pinto J., and Whittow W.: ‘Broadband embroidered spiral antenna for off-body communications’, IET Microw. Antennas Propag., 2016, 10, (13), pp. 1395–1401. Moradi E., Bjorninen T., Ukkonen L., and Rahmat-Samii Y.: ‘Characterization of embroidered dipole-type RFID tag antennas’, IEEE International Conference on RFID-Technologies and Applications (RFID-TA), 2012. Available from http://dx.doi.org/10.1109/rfid-ta.2012.6404522.

Chapter 9

Ultra-wideband arrays Markus H. Novak1 and John L. Volakis2

Many high-performance wireless applications continue to be integrated into increasingly small platforms, such as satellites, UAVs, and handheld devices. The ever-growing need for bandwidth, continually shrinking platforms, and increasingly multi-functional systems are traits observed throughout the defense, scientific, and consumer-electronic communities. Low-profile and ultra-wideband (UWB) antenna arrays have emerged as a potential solution, by allowing many disparate functions to be consolidated into a shared, multi-functional aperture. Many UWB arrays utilize a balanced element; however, signal distribution is often handled in unbalanced lines (for instance, coax or microstrip). This necessitates the integration of compact and wideband baluns to efficiently excite the element. Simultaneously, these feed structures play an important role in impedance matching across frequency and scan angles. However, antenna bandwidth can be undercut by spurious resonances and cross-polarized radiation, both of which are often excited by feed structures. Simultaneously, the demand for high data rate communications has driven these applications to higher frequencies, with many now exploring the use of the Ku-band, Ka-band, and millimeter-wave spectrum. However, existing UWB arrays were predominately designed for the conventional microwave bands below 20 GHz and utilize the above-noted complex feed structures, which cannot scale to these frequencies. Likewise, even at the inherently small scale of millimeter-wave frequencies, arrays must be low profile in order to conform with cellphones and other consumer devices, which themselves are only several millimeters thick. Indeed, the development of wideband millimeter-wave arrays compatible with low-cost commercial fabrication processes will prove critical to enabling these small and highly connected platforms. In this chapter, we focus on these modern challenges faced by wideband array designers, and the substantial advancements which have been made over the past 5 years. Initially, we provide a review of UWB array designs to familiarize the reader with the two most common groups of UWB arrays. These are developed with an overview of their basic principles, practical design considerations, and 1

Novaa Ltd, USA Electrical and Computer Engineering, College of Engineering and Computing, Florida International University, USA 2

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illustrative examples from the literature. Thereafter, we delve into the particular challenges associated with designing these arrays for the emerging applications at millimeter-wave frequencies.

9.1 Review of current UWB capabilities A useful tool for understanding the limitations of UWB arrays is to consider the role of the antenna as that of an impedance transformer, from the 50 W system to a 377 W load, in free space. However, this simple picture is complicated by the nearby presence of a reflective ground plane, which is required to avoid bidirectional radiation. This manifests as a parallel short circuit to the free-space load, as is shown in Figure 9.1. As is shown in Figure 9.2, the parallel short circuit introduces zeros in ZL, which cause total reflection at the input. This fundamentally bounds the achievable impedance bandwidth to the space between two zeros.

Z0 Antenna Z0 ZL

Figure 9.1 A simple circuit model of an antenna acting as impedance transformer to a load, ZL. The load impedance consists of the free-space impedance in parallel with a shorted transmission line, which represents the ground plane

ϕ = 0°

|ZL( jω)|

180°

JANT

ω→0 0

ω0

ω1

JGND

Г = –1 ω = ω0

Figure 9.2 Illustration of the source of fundamental limitations in the impedance bandwidth of an antenna above a ground plane. (a) Impedance response of the circuit depicted in Figure 9.1. (b) Illustration of two zeros in the impedance response, when image currents on the ground plane cancel the aperture current (left) and when reflected radiation is out of phase with forward radiation (right)

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Immediately, we see that increasing the separation from the ground plane serves to increase bandwidth by reducing the rate of change of the groundplane reactance. This inverse relationship to the ground plane height presents a miniaturization challenge, similar to the way the Wheeler and Chu relations [1,2] place a limit on the size of small antennas. Indeed, matching the reactance stemming from the ground plane is the primary challenge in designing low-profile UWB arrays. Naturally, one approach by which to access a larger bandwidth is to simply include multiple narrow-band resonators. Multi-resonant antennas such as the stacked patch antenna combine several independently resonant structures with neighboring resonant frequencies to achieve a wider effective bandwidth. This has been demonstrated in [3] to achieve up to 2:1 bandwidth. A logical extension to this approach is to continue adding resonators at a regular rate of scaling, as in the logperiodic antenna [4]. However, these techniques are inherently dispersive, and the antenna size is determined by the lowest frequency—limiting their use in arrays. Similarly, there exist certain families of UWB antennas which have excellent performance but are not suitable for electronically scanning arrays. This includes antennas which do not radiate normal to the ground plane, such as wideband monopoles [5], as well as electrically large antennas which would exceed the grating lobe spacing, such as spirals [5]. Despite these limitations, various groundplane-backed UWB arrays have been developed with operational bandwidths ranging from an octave up to a decade or more. In this section, we will review the primary UWB array design approaches, providing a brief introduction to their modes of operation as well as relative strengths and weaknesses.

9.1.1 Tapered slot Perhaps the most common and well-understood UWB array element is the exponentially tapered slot, also variously referred to as a ‘‘flared notch’’ or ‘‘Vivaldi’’ antenna. These have been in use for over four decades [6] and have demonstrated over 10:1 bandwidth and scanning up to 60 [7] (see Figure 9.3). Further, the very simple layout of tapered slot antennas makes them easy to modify for specific needs, such as dual-polarizations [8], modularity [9], or power handling [10]. The element consists of a gradually tapered slot line, which serves to significantly reduce the high impedance of the propagating wave. The slot line is 30.35 13.15 9.85

3.4 R = 0.0006

35.75

2.5

17.75 10.1

27.5

200

Figure 9.3 Picture and dimensions of an all-metal, dual-polarized tapered slot array. All values in mm, highest frequency is 5 GHz. Based on the design in [7]

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typically excited by a simple balun. In this way, a relatively low radiation resistance (Rrad) is seen at the feed and prevails over the groundplane inductance (LGP) such that Rrad k LGP is mostly real. Therefore, Vivaldi bandwidth is mostly determined by the bandwidth of the impedance taper from the feed to free space. Just like conventional microstrip impedance transformers [11], this can be accomplished for very wide bandwidths using gradual tapering profiles. However, the primary drawback of tapered slot antennas is their length. Due to the need for wideband impedance tapering, these antennas are typically designed to be a quarter wavelength (0.25 l) at the lowest frequency of operation—translating to 2–3 l at the high frequency. This makes them extremely bulky and heavy for low-frequency applications. Additionally, the vertical currents running along the slot’s length can cause high cross-polarized radiation when scanning. In some cases, cross-polarized radiation even exceeds the co-polarized radiation, at as little as 30 scanning [12]. Elevated cross-polarization can be combated to a certain extent with active compensation algorithms in dual-polarized arrays but are ultimately limited in their angular coverage and bandwidth [12]. Numerous variants of the tapered slot antenna exist [9,13], which seek to address size and cross-polarization issues. These generally utilize a more aggressive tapering profile or truncated fins to reduce height [8]. In particular, the balanced antipodal vivaldi array (BAVA) [13] feeds the flared slotline directly from a stripline input, eliminating the need for a balun, and thereby achieves a lhi/2 profile. The low profile improves cross-polarization performance, however, limits bandwidth to a 4:1 range, and scanning to 45 [8]. Additionally, the asymmetric layout of BAVA elements can lead to asymmetry in the E-plane element pattern [13]. More recently, an unbalanced-fed and capacitively coupled corrugated notch has been demonstrated to achieve up to 7:1 bandwidth with a lhi/2 profile [14].

9.1.2 Fragmented aperture With the advent of accurate and reasonably fast electromagnetic modeling codes in the mid-1990s came a revolution in the ability to analyze complex radiating structures. As a result, the use of evolutionary optimizers became a popular tool for tackling complex electromagnetic problems [15]. Naturally, one such problem was the development of planar, broadband radiators with better gain profiles than canonical elements such as a spiral or bowtie [16]. This would be achieved by treating the aperture as a blank canvas, dividing the available space into small discrete regions which are treated as either conductive or insulating. An iterative genetic algorithm (GA) assigns each region, simulates the electrical performance, and compares the result against a set of goals. As seen in Figure 9.4, the resulting antennas are composed of discrete, visually fragmented scattering structures. Early on, it was determined that a critical feature in achieving broadband performance in such arrays was allowing for electrical contact between neighboring elements [16]. In this way, many elements (electrically small at such low frequencies) could work together to support the necessary radiating currents. However, this also implied a minimum size of the total array in order to support those frequencies. Initial work focused on bidirectional radiators, which were able to produce 10:1 bandwidth

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∆ x = ∆ y = 1.5 mm

291

d = 16.5 mm

Figure 9.4 Schematic of a fragmented aperture element synthesized via genetic algorithm, operating from 1.7 to 4.5 GHz at broadside. Each pixel is 1.5 mm square. Taken from [17] [16]. However, when placed above a ground plane this is significantly reduced, to around 3:1 [17]. Much interest has also been paid to the optimization of multi-layered, frequency-selective absorbing layers, to overcome this limitation. These have been applied to achieve up to 33:1 bandwidth in a planar array above a ground plane, though with a 2.75 lhi profile, and only > 50% efficiency [18,19]. The fragmented aperture approach yields very novel and non-intuitive designs and has also been applied successfully to improve the design of reflectors [20] and frequency-selective surfaces (FSS) [21], among others. However, GA optimization naturally has a high computational burden—the fine resolution designs observed in [16] require access to parallel supercomputing clusters to converge, though lower resolution designs suitable for optimization on a desktop PC have also proven useful [17]. Likewise, this computational expense limits the optimization to planar designs and prevents the co-optimization of feed structures, which may lead to increased bandwidth in low-profile designs. That being said, with the continuing advances in computational power, we expect it is merely a matter of time before one may encounter three-dimensional GA-synthesized designs. Finally, we wish to note blind reliance on optimizers makes it challenging to learn from and meaningfully adapt these designs, without complete re-synthesis.

9.1.3 Connected and coupled arrays Intuitively, the upper frequency bound of a phased array is determined by the onset of grating lobes, which bounds the element spacing to l/2. Likewise, if these elements are isolated from one another (i.e. no mutual coupling), then the current distribution falls to zero at the element edges, preventing the element from supporting significantly lower frequencies. These two conditions result in a limited fractional bandwidth. We have seen that tapered slots overcome this by extending the element orthogonal to the ground plane. Similarly, the effective length of the element can also be expanded horizontally if neighboring elements are connected

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to one another, either electrically or through a strong mutual coupling. In this way, long continuous currents, and thus long wavelengths, can be supported along the aperture across numerous electrically small elements. The consideration of implementing distributed sources to produce an aperture current are seen as early as 1970 [22]. More recently, the use of interconnected elements to expand the low-frequency response of an array was developed by Munk [23]. This approach was inspired by the treatment of a phased array of dipoles as a continuous current sheet, by Wheeler [24] (see Figure 9.5). Munk’s implementation consisted of an array of dipoles, which were capacitively coupled using inter-digitated tips. Called the ‘‘current sheet array’’ (CSA), this array demonstrated nearly 5:1 bandwidth over a ground plane. Arrays utilizing capacitive coupling between element tips are often referred to as tightly coupled dipole arrays (TCDA). Conversely, ‘‘connected arrays’’ utilize a direct electrical connection between radiating elements [25]. Direct connection serves the same purpose of supporting long wavelengths across multiple elements and is capable of very large bandwidths in free space. However, bandwidth is starkly limited in the presence of a ground plane (2:1 bandwidth observed in [25]). This is because the series capacitance observed in TCDAs is useful in partially mitigating the large shunt inductance stemming from the ground plane at low frequencies, substantially improving the low-frequency performance of the array. One drawback to Munk’s CSA was the need for bulky external baluns to provide a balanced excitation to the dipole elements. Due to the electrically small volume of the unit cell across most of the band, integrating a balun above the ground plane is challenging and can significantly reduce the array bandwidth [26]. However, recent work has produced a folded Marchand balun which can be integrated directly within the unit cell volume [27]. Moreover, the balun can be exploited as an additional impedance matching network, expanding the bandwidth of the array. This has been demonstrated in [27] to produce a 7:1 bandwidth in an array only lhi/2 thick, as well as in [28] for a 6:1 bandwidth with scanning to 60 from broadside in all planes (see Figure 9.6). Alternatively, [29] feeds a coupled dipole element directly from the unbalanced source, to achieve a 5:1 bandwidth. z

J

Jy (x, y)

y

x

Figure 9.5 An array of coupled elements (left) showing aperture current direction and magnitude. The quasi-constant current distribution resembles a nominal current sheet (right)

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Figure 9.6 Pictures of a tightly coupled dipole arrays with integrated baluns, capable of 8:1 bandwidth [28] (left) up to 13:1 bandwidth [30] (right) Additionally, several variants and corollaries of connected and coupled arrays have been demonstrated. This includes the magnetic dual of the connected array, the ‘‘long slot array’’ [30], which uses long continuous slots to produce a constant magnetic current along the aperture. Paired with ferrite loading, this has demonstrated 10:1 bandwidth [30]. Similarly, a TCDA with resistive FSS loading is shown in [31], which effectively doubles the array bandwidth (to >13:1) by eliminating the l/2 groundplane resonance. Likewise, the concept of tight coupling between elements has been used to extend the bandwidth of already wideband elements such as spirals. In this way, the interwoven spiral array (ISPA) [32] achieves a 10:1 bandwidth for VSWR < 2. However, the ISPA suffers from high cross-polarization at low frequencies, as well as resonances when scanning.

9.1.4 Material loading Material loading can impact the impedance behavior of the array. In particular, bulk dielectric is often found between the aperture and the ground plane, as the antenna substrate. By reducing the wavelength of the propagating signal, this is an effective tool for reducing the antenna height. However, increasing the dielectric constant of the propagation media results in a reduction of the wave impedance. This in turn increases the reactance of the ground plane (see Figure 9.1), decreasing the achievable bandwidth of the array [33]. Conversely, dielectrics can be placed above the radiating aperture, where they are referred to as a superstrate. Here, they act as an intermediate matching stage, reducing the free-space impedance seen at the aperture. In particular, if the superstrate is l/4 in thickness, it behaves analogous to a quarter-wave impedance transformer. By reducing the radiation impedance, the total load on the antenna Rrad k ZGP is reduced. This increases the impedance bandwidth of the array. Further, wave slowdown in the superstrate reduces the effective angle of incidence of free-space waves, resulting in a more constant impedance behavior across the scanning volume. For this reason, superstrates are often used to improve scanning performance and are sometimes referred to as wide-angle impedance matching layers. However, excessively thick or high dielectric constant superstrates can

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support surface waves, resulting in severe mismatch at the resonant frequency (see Section 9.3.4). Unlike dielectrics, magnetic materials (mr > 1) used in the substrate can increase the bandwidth of low-profile arrays. By increasing the wave impedance, the shunt reactance of the ground plane is suppressed, resulting in a primarily real load across frequency. Unfortunately, magnetic materials are generally heavy and lossy, limiting their use in many applications. The maximum achievable bandwidth for an array above a ground plane when utilizing various dielectric or magnetic substrate materials is shown in Figure 9.7. Similarly, if the perfect electric conductor (PEC) ground plane were replaced with a perfect magnetic conductor (PMC) boundary, in-phase reflection (G ¼ 1) would allow the antenna to sit directly above the ground plane. While these materials do not exist in nature, much ongoing research pertains to the synthesis of artificial magnetic conductors which can emulate this response over a finite band. Generally, this involves the design of sub-wavelength periodic or corrugated surfaces, which can resonate to produce a high surface impedance [34]. However, these materials are only effective over a narrow band. A larger groundplane distance serves to increase bandwidth by reducing the reactive impedance seen at low frequencies. However, this is ultimately limited by the destructive resonance which occurs when the groundplane separation reaches a half wavelength. This resonance can be suppressed, of course, using lossy materials to absorb all backwards radiating energy. In this way, the presence of the ground plane can be negated, resulting in very wide bandwidths [18] but results in a μr εr = 8

16

μr εr = 2

μr εr = 4

μr = εr

BW ratio (ωhi /ωlo)

14 εr μr = 2

12 10 8

εr μr = 4

6

εr μr = 8

4 2 1

1.5

2

2.5

3

3.5

VSWR

Figure 9.7 Achievable bandwidth of an array over a conducting ground plane, for various substrate materials and thicknesses. Notably er > 1 results in a reduction in bandwidth, and mr > 1 an increase

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maximum efficiency of 50%. However, losses can be reduced by confining the lossy material to a relatively narrow-band FSS, paired with the synergistic design of a superstrate. This was demonstrated in [31] to achieve a 13:1 bandwidth above a ground plane, while maintaining efficiency > 70% across the entire band.

9.2 Basic model of a UWB TCDA and feed Of particular interest are UWB arrays which can be implemented conformally above a ground plane, while maintaining a large operating bandwidth and low profile. Based on the above families of UWB arrays, this leads us to look more closely at the tightly coupled dipole array, which has demonstrated substantial bandwidth despite a simple, planar geometry.

9.2.1 Modeling infinite coupled arrays The conventional design approach for phased arrays typically involves the design of an element with good performance in isolation and subsequently mitigating the mutual coupling effects when placed in the array environment. However, for tightly coupled arrays, in which the mutual coupling effects are harnessed to improve the bandwidth of the isolated element, this approach is not effective. Instead, it is necessary to consider the complete array mutual coupling effects from the onset of the design process. This is accomplished through the use of periodic boundary conditions on a single unit cell, emulating an infinite array environment. In this regard, we note that the fields due to an infinite array of radiators are identical to the fields of a single such radiator, inside a waveguide with PEC and PMC boundaries. This configuration is shown in Figure 9.8, superimposed with the virtual array resulting from the image currents on the waveguide walls. The virtual array spacing is equal to the waveguide boundaries, Dy ¼ a and Dx ¼ b. Such a waveguide supports a TEM mode propagating perpendicularly to the plane of the virtual array and has a purely real wave impedance given by (9.1) Rwg ¼ hb=a pﬃﬃﬃﬃﬃﬃﬃﬃ where h ¼ m=e 377 W is the free-space impedance. An antenna receiving from this waveguide acts as a transformer [35], producing a voltage at the terminals proportional to its effective height (h). It therefore encounters a scaled radiation resistance: b h 2 ¼ hh2 =ab (9.2) R¼h a b By reciprocity, (9.2) is true of the transmit case as well. We note that this formulation assumes the nominal waveguide that is fed from one end and thus extends in a single direction. Practically, this implies the array is backed by PMC, or otherwise, PEC at a distance l/4. In the case of bidirectional radiation, two such

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–

–

+

+

–

+

–

+

PMC b –

+

PEC –

+ a

PEC Dy

Virtual array PMC

–

+

–

+

–

+

Dx

Figure 9.8 Schematic layout of a linearly polarized radiator situated inside a waveguide cross-section. The boundary conditions at the PEC and PMC walls match those of an infinite array of identical and aligned radiators waveguides are fed in parallel, extending in opposite directions, such that the radiation resistance in (9.2) is halved. Notably, h ¼ l for a uniform current distribution along a wire of length l. Therefore, a uniform current wire spanning the waveguide length b logically has an effective length h ¼ b, resulting in a radiation resistance R ¼ 377W. Conversely, a resonant half-wavelength dipole has an effective length h ¼ 2l/p; for an array spacing of a ¼ b¼ 0.5l, this produces a radiation resistance R ¼ 153W. Thus, for the quasi-constant current distribution of a TCDA, we expect an R value between these two (i.e. 153W < RTCDA < 377W). Empirical experience has found this value to be 188W [27,28,31]. As was noted before, this high input impedance to the tightly coupled dipole element is challenging to match to 50 W. However, from (9.1), we can see immediately that altering the ratio b/a can decrease this impedance. Significantly increasing a is not feasible due to the resultant onset of grating lobes in the H-plane. Thus, the split unit cell approach halves the value of b, by placing two quarterwavelength radiators inside each 0.5l unit cell. Specifically, halving the length of the periodic cell in the direction of current serves to reduce by half the Floquet mode impedance [24] (i.e. 94 W). By feeding these two miniaturized cells in parallel, the input impedance is further reduced to 47 W and is easily matched to the 50 W system impedance. This formulation is only approximate and assumes that h scales linearly, and that the miniaturized elements are not significantly impacted by increased mutual coupling. Nonetheless, it serves as a useful qualitative tool. Another useful insight stemming from this mode of analysis is an understanding of the array impedance when scanning. As shown in Figure 9.9, the dimensions of the conceptual waveguide are distorted, as the aperture area is projected into the

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θ d2

d1

Figure 9.9 Conceptual diagram of an array radiating at an angle into a waveguide, such as in Figure 9.8. The waveguide width at broadside (d1) is reduced when projected into the scan direction (d2 ¼ d1 cos q). This results in an altered impedance, looking into the array scanning direction. We can see that the length of the aperture is reduced by a factor cos q, in the scanning direction. Adjusting this to account also for azimuth angle, we can modify (9.1) to state more generally: sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ b 1 sin2 q cos2 f Rwg ¼ h (9.3) a 1 sin2 q sin2 f where f ¼ 0 is defined as parallel to the primary current of the radiator (i.e. E-Plane, or TM mode). As such, we observe that the wave impedance is reduced as the array is scanned in the E-plane, but increases as it is scanned in the H-plane (f ¼ 90 , or TE mode).

9.2.2 Circuit model of the balun As discussed in Section 9.1.3, the early TCDAs from Munk [23] and later from Kasemodel [26] operated at higher frequencies (up to 18 GHz), but were limited to 4:1 and 2:1 bandwidths, respectively. Development of a higher-order matching network, in the form of an integrated Marchand balun, enabled the extension of the TCDA bandwidth to 6:1 [28] and 7:1 [27]. However, this wideband performance is achieved at the expense of increased complexity in the feed. This can be seen in [28], where a unit cell design is developed for operation in the UHF band. In particular, we observe a miniaturized Marchand balun, consisting of a stripline input and open circuit stub, with a gap in the shielding at the feedpoint

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of the dipoles. A balanced slotline mode is excited in this gap, both feeding the dipoles and propagating in a guided mode to the ground plane, acting as a parallel short circuit stub. These components of the physical layout are mirrored in the transmission line model, also shown in Figure 9.10. The model shown in Figure 9.10 can be combined with the basic model of an antenna above a ground plane given in Figure 9.1 to form a more general model of a TCDA with integrated balun, as shown in Figure 9.11. In addition to the balun

Superstrate Differential output + –

Coupled dipoles

Zant

Stripline open stub Balun

Zopen Stripline input Zshort

Wilkinson power divider

Zin

Unbalanced feed 0

25

50 (mm)

Figure 9.10 Overview of the TCDA unit cell developed in [28] (left) and detail of the integrated stripline balun (center). Transmission line model of the balun is shown on the right

Zfree Zopen

Zant L

C

Zsuperstrate

Zsubstrate

Zfeed Zin Zshort

Figure 9.11 Transmission line model of a tightly couple dipole array and feeding network. The reactance L is caused by the dipoles inherent inductance, and C is a result of the capacitive coupling between neighboring elements in the array

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circuit, ground plane, and free-space load, this includes a short superstrate layer, Zsuperstrate, as well as the dipole self-inductance L and the capacitive coupling C, characteristic of coupled arrays. This model was originally developed in [27] and is useful to understand the basic components of the antenna and feed. Of course, linearly scaling of previous designs would readily achieve identical bandwidth and beamforming performance at Ku-band. However, fabrication of the scaled array on low-cost printed circuit board (PCB) is not feasible. By way of example, the array in [28] operates up to 4 GHz and requires traces as thin as 0.107 mm (4 mil). Thus, scaling the previous design to 18 GHz would require the manufacture of features on the order of 23.78 mm (0.889 mil), as shown in Figure 9.12. This far exceeds the capability of PCB techniques, which support a minimum feature size of 76 mm (3 mil). In particular, we observe that the smallest feature of the design comprises a high-impedance line within the feeding network of the antenna, labeled Z1 in Figure 9.12. Indeed, the TCDA’s integrated balun requires an approximately 20–200 W; (10:1) range in characteristic impedances (Z0) to maintain wideband performance [36]. Instead, we must produce the requisite range of impedances without decreasing the minimum feature size. To understand how this may be possible, it is useful to consider the lumped-element model of a transmission line, as shown in Figure 9.13. In this approximation, each infinitesimally short length of transmission line is treated as an RLC circuit. The line is assumed to be very long, with no reflections present. The characteristic impedance (Z0) of the transmission

Z1 Z2

0.22:1 scale

Operating band: 2.25–18 GHz Cell spacing: 7.56 mm Min. feature size: 22 μm Substrate thickness: 2.67 mil

Operating band: 0.5– 4 GHz Cell spacing: 34 mm Min. feature size: 100 μm Substrate thickness: 12 mil

Figure 9.12 Proportional scaling of the unit cell design from [28] up to the Ku-band, by reducing all dimensions by a factor of 0.22. The resulting minimum feature size and substrate thickness cannot be realized using commercial processes

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Developments in antenna analysis and design, volume 2 δR

δL δG

δC

δx

Figure 9.13 Distributed element model of a transmission line. Primary constants of the line (R, L, C, G) are assumed to remain constant for the length of the line w h

εr

Substrate integrated waveguide (SIW)

w

h

εr

h

Stripline

h

w εr

Microstrip

εr

Slotline, twinline

Decreasing C, Increasing Z0

Figure 9.14 Cross sections of four common PCB-implemented transmission lines, with key parameters labeled. Representative electric field lines of the primary mode are indicated, and the direction of propagation is perpendicular to the plane of the page. From left to right, C decreases and Z0 increases, for constant values of w, h, and er line can then be calculated as sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ R þ jwL Z0 ¼ G þ jwC

(9.4)

Equation (9.4) provides some insight as to how the four primary transmission line properties can be manipulated in order to achieve the desired characteristic impedance. However, certain practical considerations must be taken into account. In particular, G and R are determined by the conductivity of the substrate and conductor, respectively, and are both minimized to reduce losses. Similarly, L is largely determined by the substrate material’s permeability (mr) and the conductor surface roughness. These are not practical design parameters to control. Therefore, all that remains is an inverse relationship to the capacitance per unit length, C. This relationship explains, for example, why widening a microstrip trace (thus increasing shunt capacitance) decreases its characteristic impedance. Similarly, increasing the height above the ground plane reduces C and therefore increases Z0. Line width and height are useful for tuning, however are limited in their total range. More significant changes in C and thus Z0 can be achieved by altering the configuration of the transmission line. This can be seen in Figure 9.14, wherein several common PCB

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transmission line cross-sections are compared. As noted before, the values w,er and h can be used to adjust Z0. However, for the same values of w,er and h, the four transmission lines shown will exhibit increasing Z0 (from left to right) on account of the decreasing capacitive surface area between the conductors.

9.3 Considerations for planar UWB arrays Whereas the above methodology has been applied to produce as much as 14:1 bandwidth, wide-scanning, and extremely low-profile arrays, these have primarily been accomplished in the lower microwave bands (i.e. ¡ 10 GHz). For the many emerging applications, particularly in the millimeter-wave bands, certain challenges and limitations must be addressed by the antenna designer.

9.3.1 Feed planarization Whereas early TCDAs [23,26] were inherently planar in design, recent TCDAs [27,28,31] have migrated to a non-planar approach. For clarity, we note that herein planar refers to the co-orientation of the fabrication plane with the array plane, as shown in Figure 9.15. The use of vertical PCBs in [27,28,31] is a result of the fact that it is much easier to accommodate complex feeding circuits within the plane of the PCB. Indeed, improvements in bandwidth, size, and scanning over earlier designs have been driven by innovation in the design of the feeding network, including the balun and other matching components. With these improvements, however, comes significant complexity. Simultaneously, vertically oriented PCBs provide a predominately air substrate, largely mitigating surface waves. However, unlike a planar printed array, vertical PCBs require the final physical assembly of separate components to form a complete array. This adds complexity and cost to the fabrication process, and moreover, introduces sources of error and inconsistency in the resulting arrays. In moving to millimeter-wave

z x

y

Figure 9.15 Illustration of planar (left) and non-planar (right) arrays. In both cases, a ground plane resides below the elements in the x, y plane, and broadside is in the ^z -direction. Notably in the planar case, the entire array can be formed from a single PCB, whereas in the nonplanar array several PCBs must be combined to form the array

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frequencies, this trend cannot continue, as subassembly at this scale is not feasible. However, planar fabrication introduces significant challenges in the design and implementation of UWB arrays. As can be seen in Figure 9.15, planar radiating elements (for instance, coupled dipoles) are relatively easily implemented within this framework. However, implementation of the feeding network, which is critical to the wideband operation of the array, is significantly more challenging. This is because planar fabrication techniques such as PCB have little fidelity for complex through-plane structures. As such, the primary building block for the feeding network must be simple metalized vias. For the same reason, through-plane radiating elements such as tapered slots cannot be implemented. As a result, existing planarized UWB arrays must utilize simplified balun structures which significantly limit bandwidth [26,29].

9.3.2 Material and process selection It is important to realize that in planar arrays, the volume of the antenna is buried in dielectric, making material parameters an important consideration. This includes loss tangent as well as dielectric constant, which plays a large role in determining the onset of surface waves (discussed in greater detail in Section 9.3.4). Furthermore, material selection also relates to the choice of fabrication process. Low-temperature co-fired ceramic (LTCC), for instance, is a popular process for millimeter-wave antennas, for its low loss and relatively fine feature sizes. However, the ceramic tapes used in this method have high dielectric constant (er > 6), which can result in surface waves even at broadside. This can be mitigated across a narrow band by making the antenna thin or with EBG structures but is not suitable for wideband applications. Similarly, antenna-on-chip (AoC) designs have been explored, which place the antenna directly on the silicon wafer housing the transceiver circuitry. The silicon substrate, however, has a high dielectric constant (er 11) and high loss tangent at high frequencies (tan d > 0.015), resulting in very low efficiency for these antennas. This has recently been mitigated by isolating the antenna from the silicon and using low-loss polymers as a substrate. Similarly, purely metal–air structures have been demonstrated using micro-fabrication techniques. However, it remains that AoC and micro-fabrication techniques are very expensive. Conversely, PCB fabrication is almost universally available and thus can be sourced at very low cost. Likewise, PCB fabrication is compatible with a wide range of materials, including low loss and low dielectric constant polytetrafluoroethylene (Teflon, er ¼ 2.2). For these reasons, it is desirable to enable PCB fabrication for antenna designs. However, PCB fabrication is also much more limited than LTCC and micro-fabrication, particularly for high frequency and planar antennas.

9.3.3 Limitations of PCB processing Key parameters defined in the basic design rules for advanced PCB processes are summarized in Figure 9.16. In Figure 9.16, the minimum feature size (positive, B, or negative, A) within the plane is limited to 76 mm, which at 90 GHz translates to 2.25% of the wavelength. However, signal paths traveling through plane must

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303

A B

C

D

F

E

Figure 9.16 State of the practice PCB fabrication limits, shown from edge. Key limiting parameters are labeled be formed from vias and are further limited. While laser-drilled holes can reach a similar diameter as the traces (i.e. 76 mm), they are limited to very thin layers. The comparatively thick core substrate must be mechanically drilled, resulting in a minimum diameter, D, of 150 mm. These further require annular catch and landing pads for the metalization process, which increase the surface diameter, E, to 305 mm. Likewise, in-plane traces are required to maintain a 150 mm clearance from all via edges, C. Finally, the minimum edge-to-edge via pitch, F is 254 mm, or 7.6% of the wavelength at 90 GHz. The severe challenge inherent in the available PCB fabrication tolerances is made apparent by briefly considering some scaling examples. Taking, for example [27], and applying the linear scaling principle to shift its operation up to a target frequency of 90 GHz would require fabrication of features down to 5.25 mm in size. Even if we consider the improved high-frequency design described in [37], when scaled to 90 GHz would require features of 41.5 mm. While clearly these dimensions may be achievable using micro-fabrication techniques, they are a long ways from the PCB capabilities outlined above.

9.3.4 Surface waves The large bandwidth of UWB antennas makes them susceptible to spurious narrowband resonances which can occur anywhere inside the operating band. We see this in particular for high-frequency and/or planarized arrays, in which the array is embedded in a dielectric slab as a result of the selected fabrication process. The presence of continuous dielectric elongates the electrical dimensions of the array lattice, lowering the potential resonant frequencies into the operating band. Surface waves can exist in many geometries of dielectric interfaces. In antenna arrays, these can be excited at broadside but are much more commonly observed when scanning (sometimes referred to as ‘‘scan blindness’’). To understand the onset of surface waves in an array setting, it is useful to initially consider a simple waveguide consisting of a grounded dielectric slab, as shown in Figure 9.17. The configuration in Figure 9.17 supports guided TE and TM wave modes, propagating as ejbz. The derivation by Pozar [11] of the TM mode yields the pair

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Developments in antenna analysis and design, volume 2 e 0, μ0

x

er, μr

TM mode d

E H

β

TE mode H E

β z

Groundplane

Figure 9.17 Geometry of a dielectric slab above a ground plane. The dielectric region is assumed to be isotropic, and infinite in y and z. Orientations of the guided TM and TE modes are shown of transcendental equations: ðkc d Þ2 þ ðhd Þ2 ¼ ðer 1Þðk0 d Þ2

(9.5)

kc d tanðkc d Þ ¼ er hd

(9.6)

where h is the rate of attenuation of the evanescent wave in the air region (thus must be >0 for a physical solution), kc is the cutoff wavenumber in the dielectric, and k0 is the free-space wavenumber (k0 ¼ 2p/l0). Derivation for the TE modes yields a similar pair of equations: ðkc d Þ2 þ ðhd Þ2 ¼ ðer 1Þðk0 d Þ2

(9.7)

kc d cotðkc d Þ ¼ hd

(9.8)

These equations must be solved numerically and are plotted for reference in Figure 9.18. Equations (9.5) and (9.7) represent circles in the kcd, hd plane, having a radius: r¼

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ er 1 k0 d

(9.9)

However, this value takes on a more intuitive meaning when put in terms of the guided wavelength, lg: l0 lg ¼ pﬃﬃﬃﬃ er ¼

2p pﬃﬃﬃﬃ k 0 er

which is rewritten as k0 ¼

2p pﬃﬃﬃﬃ lg e r

and inserted into (9.9) to yield pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ er 1 d r ¼ pﬃﬃﬃﬃ 2p lg er

(9.10)

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305

hd

d = 0:756 λg

0.5

0.25

–π

–π 2

π 2

π

π 2

π

kcd

er = 2

hd d = 0:75 λg

0.5

0.25

–π

–π 2

kcd

er = 8

Figure 9.18 Contour plots of equations (9.5)–(9.8) in the kcd, hd plane, for slabs of dielectric constant (a) er ¼ 2, (b) er ¼ 8. Intersections of the curves with the circles represent solutions to the equation pairs. The solid curves correspond to TM modes, whereas dashed curves correspond to TE modes, and the dotted curves represent dielectric slabs of varying thickness These circles are plotted for various values of d/lg in Figure 9.18. Intersection of a circle with the plotted curves represents a valid solution describing a propagating mode. Where multiple tangent curves intersect a circle, the dielectric slab represented by that circle supports multiple modes simultaneously. The propagation factor of the guided wave (bsw) can be determined from the value of kc extracted from Figure 9.18, by kc2 ¼ er k02 b2sw

(9.11)

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Developments in antenna analysis and design, volume 2

Careful observation of Figure 9.18 yields several useful insights. Notably, as the central TM mode includes the origin, any slab of thickness d > 0 and er > 1 supports at least this fundamental mode. As the slab thickness increases, the supported modes progress as TM0, TE1, TM1, . . . , TEn, TMn. Further, as the derivative of (9.6) and (9.8) along the þkcd axis is always positive, and the derivative of the circles (9.5) and (9.7) always negative, the earliest intersection of each new mode occurs as hd ? 0. As such, the cutoff frequencies of any TMn and TEn modes can be calculated as fc;TM ¼

nc pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ; 2d er 1

n ¼ 0; 1; 2; :::

(9.12)

fc;TE ¼

ð2n 1Þc pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ; 4d er 1

n ¼ 1; 2; 3; :::

(9.13)

Clearly, surface waves can be mitigated to a certain extent by using a thin or low dielectric constant substrate and superstrate materials. However, as was discussed in Section 1, even thin UWB arrays such as the TCDA have a thickness above the ground plane of lg/2. It should also be noted that in an array setting, the additional loading of the antenna elements and feed structures will produce a slightly higher effective dielectric constant (eeff > er) [38–40]. Thus, from (9.12) and (9.13), or alternatively from Figure 9.18, we expect to contend with two potential surface wave modes: TM0 and TE1. Critically, the existence of these propagating modes alone does not inherently mean they will become excited in the array. However when excited, energy becomes trapped in the substrate with little or no radiation, resulting in complete reflection at the antenna feed. This condition occurs when the propagation factor of the radiated wave inside the substrate (bsub) matches the propagation factor of a guided wave mode of the same polarization. For an array geometry such as that shown in Figure 9.19, the substrate propagation factor is derived in [38] and is calculated as b2sub ¼ kx2 þ ky2 y

z q b f

a d

er

x

Figure 9.19 Geometry of an infinite, linearly polarized array above a ground plane. Dipole elements are shown to illustrate polarization; however, subsequent derivations assume infinitesimal radiators

Ultra-wideband arrays wherein the tangential wavenumbers Kx,Ky are found to be 2pm þ k0 u kx ¼ a 2pn þ k0 v ky ¼ b

307

(9.14a) (9.14b)

In (9.14), m and n are positive or negative integers corresponding to the Floquet mode, a,b are the element spacing as shown in Figure 9.19, and u,v correspond to the scan angle of the array: u ¼ sinðqÞcosðfÞ

(9.15a)

v ¼ sinðqÞsinðfÞ

(9.15b)

Therefore, we can write the condition of surface wave resonance as 2 2 2 bsw ml0 nl0 þu þ þv ¼ k0 a b

(9.16)

noting again that the polarization condition requires u ¼ 6 0 for TM modes and v 6¼ 0 for TE modes, with the exception u ¼ v ¼ 0. As was discussed before, the surface wave propagation factor is dependent on the thickness and dielectric constant of the substrate. Conversely, (9.16) demonstrates that the propagation factor of the antenna fields in the substrate is dependent only on the element spacing and scan angle. The appropriate value of bsw can be determined from (9.11) in conjunction with Figure 9.18. For convenience, the value of bsw has been solved numerically as a continuous function of d and is given in Figure 9.20. Logically, we see that as the substrate thickness increases, bsw asymptotically approaches the propagation factor of a plane wave in dielectric, √ er

1.4 1.35

er = 8

2.6 TM0

TE1

2.2

1.2

βsw /k0

TE1

1.25

TM0

2.4

1.3

βsw /k0

√er

2.8

er = 2

TM1

1.15

TE2

1.1

TM1

2

TE2

1.8

TM2

1.6

1.05

TE3

1.4

TM2

1.2

TE3

1

1 0

0.25

0.5

0.75

1

d/λg

1.25

1.5

1.75

2

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

d/λg

Figure 9.20 Numerical solutions to the surface wave propagation factor (bsw) as a function of substrate thickness, for (a) er ¼ 2 and (b) er ¼ 8. The first three TE and TM modes are labeled

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Developments in antenna analysis and design, volume 2

pﬃﬃﬃﬃ k0 er . We observe also that increasing the dielectric constant brings the TMn and TEnþ1 modes closer together. Due to the factor er in (9.6), it is in principle possible for the TE mode propagation factor to exceed that of the TM mode. However, this will only occur in the case of very high dielectric constant (e.g. er > 30) and large slab thickness (e.g. d/l0 > 1). Taking as an example an array of thickness 0.5lg with er ¼ 2 substrate, Figure 9.20 yields surface wave propagation factors of 1.3 (TM0) and 1.09 (TE1). In (9.16), these values represent the radii of circles in the u,v plane, as can be seen in Figure 9.21. The centers of these circles are offset by the Floquet mode numbers (m,n) multiplied by the element spacing (a/l0 and b/l0). These surface wave modes will become excited and severely mismatch the array where the plotted circles enter the visible region of the array (u2 þ v2 > 1). Immediately, we see that as bsw increases, the onset of surface waves moves inwards toward broadside, occurring at decreasing values of q. Likewise, increasing the element spacing beyond 0.5l0 draws the center points of the surface wave circles inwards, also exacerbating the onset of these modes. The visible region in Figure 9.21 is plotted in greater detail in Figure 9.22, in order to predict the specific state in which the surface waves will occur. While these will occur for continuous values of f, we consider for simplicity the cardinal scan planes, E-plane or TM mode radiation (f ¼ 0 ) and H-plane or TE mode radiation (f ¼ 90 ). We observe the TM0 curve reaches the closest to the origin

v (H-plane)

TM0

TE1

(0,–1)

(–1,–1)

Visible region (m, n) = (–1,0)

u (E-plane)

Figure 9.21 Solutions of (9.16) plotted in one quadrant of the u, v plane, for an array spacing of a ¼ b ¼ 0.5 l0, on er ¼ 2 dielectric of thickness d ¼ 0.5 lg. Portions of the TE and TM circles which fall inside the visible region (u2 þ v2 < 1) represent scan conditions at which those surface wave modes will become excited

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309

1

0.8 60 50 0.6

v

40 30

0.4

40 30

20 0.2

20 θ = 10

0

0

ϕ = 10 0.2

0.4

0.6

0.8

1

u

Figure 9.22 Detail of the visible region from Figure 9.21, showing azimuth (f) and elevation (q) grids. TM0 and TE1 surface wave modes are plotted (solid and dashed curves, respectively), and will be excited at 44.4 in the E-plane (f ¼ 0 ) and 65.5 in the H-plane (f ¼ 90 ). Results are symmetric about u and v (i.e. for negative values of f,q) (broadside) as it intersects the u axis, corresponding to a q ¼ 44.4 scan angle in the E-plane. The intersection of the same curve with the v axis will not resonate, as the polarizations do not match. As such, the earliest surface wave resonance in the H-plane occurs with the TE1 mode, at 65.5 . This trend can be generalized to state that for any thin and low dielectric constant array, the earliest surface waves will occur in the E-plane, at wide scan angles. Further, as the electrical thickness of the substrate in (9.10) and the element spacing in (9.16) both shrink as the frequency is decreased, we can further state that surface waves will be observed earliest at the high end of the operating bandwidth. We note, however, that surface waves at very shallow scan angles, or even broadside, do not require extreme dielectrics to become excited. For an array spaced at 0.5l0, broadside surface waves can be excited for a value of bsw /k0 ¼ 2, which corresponds to the relatively common dielectric constant of er 4. Thus, with the exception of very thin arrays (d < 0.25 lg) which are typically narrowband, substrate material selection is the most limiting factor with regard to the scanning range in UWB arrays.

9.3.5 Cavity resonance Unlike surface waves, cavity resonances are primarily excited at broadside and dissipate when scanning. These are caused by a net current in the balanced feed of the dipole, compounded by the presence of a resonant cavity within the geometry of the array. At resonance, this induces strong vertical fields beneath the aperture

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Developments in antenna analysis and design, volume 2

which ultimately overwhelm the balanced currents of the differential feed. For this reason, it is sometimes referred to as a ‘‘common-mode’’ resonance. This type of resonance is a characteristic of unbalanced-fed antennas [9,29] and will not occur in an ideal, balanced excitation. However, even with a balun, this condition cannot be guaranteed, and an imbalance in the currents on the differential feed will produce a net vertical current, as shown in Figure 9.23. The unbalanced current causes vertical fields above the ground plane, which under the correct conditions can resonate. In particular, the repeating grounded conductors present in the array baluns form a cavity. At such frequencies that these grounded posts are separated by l/2, the energy fed into the array is stored in these resonant cavities and is not radiated. This condition is illustrated in Figure 9.24.

Figure 9.23 Illustration of unbalanced currents on the differential line of the Marchand balun, resulting in net vertical currents. Unbalanced input feed and open circuit are shown for reference (dashed)

E

I

Figure 9.24 Strong vertical fields at resonance induce common-mode currents in the normally differential feed line. Notably, such resonance will more commonly occur diagonally between rows, rather than in-line as shown here

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311

d

L

Dy

Dx

Figure 9.25 Diagram of a planar array of dipoles, viewed from above. Dipoles are suspended in a vertical dielectric layer. Key values in determining the onset of a cavity resonance are indicated Notably, scanning the array introduces a relative phase difference between the neighboring elements, weakening and ultimately preventing this resonance. Thus, cavity resonance is seen only at or near broadside. As was demonstrated in [9], this resonance can be relatively accurately predicted based on the separation of the grounded posts. In Figure 9.25, we consider a simple linearly polarized array, as viewed from above. If the dipole feeds are imbalanced, as noted above, a cavity resonance is excited along the diagonal length indicated as L. Resonance will occur when this length is a half wavelength: fr ¼

c pﬃﬃﬃﬃﬃﬃﬃ 2L eeff

c ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃpﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 Dx þ D2y ee ff

(9.17)

We note that the value Dx is slightly less than the element spacing (i.e. a in Figure 9.19) due to the width of the feed. Likewise, eeff is the effective dielectric constant and can be approximated for the setup in Figure 9.25 by a simple volumetric average: eeff ¼ 1 þ

d ðer 1Þ Dy

(9.18)

Notably, for 0.5 lhi spaced arrays with > 50% bandwidth, this resonance will occur inside the passband, bifurcating the antenna’s operating bandwidth into two discontinuous bands (see Figure 9.25). Based on (9.17), it is clear that the undesired cavity resonance can be pushed out of band by altering the resonant length, L. With the use of a split unit cell, as in [28,27] this stemmed naturally from a much smaller distance between neighboring baluns. However, a similar effect can be achieved without a split unit cell by introducing a grounded shorting pin between neighboring elements. These run vertically from near the aperture (though not touching the dipoles) to the ground

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Developments in antenna analysis and design, volume 2 Resonance behavior 15

VSWR

10

5

0

4

6

8

10

12

14

16

18

Frequency (GHz)

Figure 9.26 Infinite array input reflection, showing initial spurious cavity resonance (dashed peak at 15 GHz). Introduction of a shorting post shifts the resonance (dashed peak at 16.5 GHz), and finally increasing the dimension ‘‘A’’ (see Figure 9.27) pushes the resonance out of the operating band (solid curve)

Dx A D ꞌx

Shorting post

Figure 9.27 Physical representation of the parameter Dx in Figure 9.25. Addition 0 of a shorting post significantly reduces this value (labeled D x ), as does adjustment of the dimension ‘‘A’’ plane, as shown in Figure 9.27. The response of the array described in [37], with and without the inclusion of shorting pins, is shown in Figure 9.26. However, the new resonant frequency (16.5 GHz) has only slightly increased and still falls within the passband. This is the case because while Dx in (9.17) has decreased, Dy remains constant. The resonant frequency can be further increased and pushed out of the band, by additionally increasing the collar dimension noted as ‘‘A’’ in Figure 9.27 Including this modification, the solid curve in Figure 9.26 demonstrates a resonance-free band, without the use of a split unit cell.

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9.4 Planar UWB arrays for millimeter-waves As was discussed at the start of the chapter, many applications are emerging across the millimeter-wave bands. Numerous antenna designs have been proposed for these applications, primarily operating at the Industrial, Scientific, Medical (ISM) band centered at 60 GHz. AoC arrays [41–43] integrate the array directly above the transceiver circuitry. These are very compact but suffer from low efficiency (10–50%) due to the high loss in the silicon substrate. Conversely, antenna-in-package arrays utilize low-loss substrates to house the radiating elements, typically with a flip–chip interface to the transceiver die. These can be realized in low- and high-temperature co-fired ceramics (LTCC [44,45] and HTCC [46]) as well as standard PCB [47,48]. However, all of the aforementioned arrays utilize narrow-band elements. Given the cost and space required for these arrays, it is not feasible to include multiple narrow-band arrays to cover different functions or bands of operation. As such, a key challenge is the planarization of UWB arrays to enable their fabrication at these small scales, which will allow the consolidation of many different functions into a shared aperture. In this section, we will discuss the development of a simplified feed structure for TCDA elements, which enables the design of planar millimeter-wave UWB arrays, compatible with low-cost PCB fabrication. An optimized design is presented which provides continuous coverage from 24 to 72 GHz.

9.4.1 Development of a three-pin balun Given the limitations outlined in Section 9.3.1, the simplest feeding approach may be a direct or unbalanced feed such as [29]. This has been shown to be a viable option, however, requires extensive matching circuits beneath the ground plane to compensate for the reactive loading of the dipoles. Likewise, the feeding network, including balun, can also be made planar to take advantage of the improved in-plane tolerances [26]. However, it is known that a planar balun is limited in size by the element spacing and thus struggles to achieve a wide impedance ratio, leading to a limited bandwidth (400 NA

NA

[62] Stacked patches

3/0.56

12.0

> 650 NA

NA

[63] Double circular rings 1/0.32 [64] Patch loaded with slot 1/0.70 [65] Coupled structures 1/0.55

9.8 12.5 35.0

20% 360 / NA 4% 673 NA NA

[66] Three parallel dipoles 1/0.50 [67] Double cross rings 1/0.44

300.0 22.0

>400 500 NA NA

[68] Reconfigurable 1/0.54 double square rings

5.4

380 90 for the E, D, and H-planes to the boresight copolar component in dB. The optimized horn was fabricated using a CNC lathe machine. For mechanical stability, the horn profile was bored out of a single piece of solid aluminum cylinder. The horn after fabrication, along with the coax adapter and rectangular to circular adapter are shown in Figure 11.8(a). In order to validate our design, the S11 and the farfield pattern measurements of the optimized horn was carried out. The measured D-plane pattern of the horn is shown in Figure 11.8(b), where cross-polarization is typically high for a horn antenna. The 10 dB beamwidth of the pattern is roughly 63 for all planes (E, D, and H), leading to good illumination. An S11 measurement was also carried out, and S11 20 dB was observed throughout the band of interest. Using the horn radiation pattern, the full reflector radiation patterns were also computed. The simulated results (assuming an ideal reflector) showed that the horn illuminates the reflector properly, providing 50.30 dB directivity and HPBW’s of 0.57 (E-plane) and 0.58 (H-plane). A simplified model of the CubeSat chassis was also included in the simulation to account for any potential degradations in backlobes and sidelobes from the chassis. The radiation pattern results are shown in Figure 11.9, where a large-angle view and a magnified view of the E-plane patterns are shown. The chassis adds minimal signatures on the radiation pattern in this case, primarily because the main beam is pointed away from the chassis. An estimated gain budget is shown in Table 11.3.

11.6.2 Proposed deployment strategy An illustration of the proposed concept is provided in Figure 11.10. The mechanical system uses a backing structure that is a hybrid wrap-rib/perimeter-truss design [103]. A net connected to the truss system supports a reflective mesh with an expected 30 openings-per-inch (OPI). The system is deployed using a tape-spring support structure that expands once released. It is estimated that the system can be

Novel antenna concepts and developments for CubeSats 0

Normalized fields (dB)

Normalized fields (dB)

−20 −30 −40 −50 −60

−80

Without chassis With chassis

−10 −20 −30 −40 −50

−70

(a)

0

Without chassis With chassis

−10

375

0

5

10

15

20

25 θ (deg)

30

35

40

45

50

−60 −2 (b)

−1.5

−1

−0.5

0 θ (deg)

0.5

1

1.5

2

Figure 11.9 Representative simulated E-plane radiation patterns when illuminated by the feed pattern. (a) Wide angle far field patterns and (b) near boresight far field pattern. These results show the minimal effects of the chassis Table 11.3 Simulated gain budget accounting for critical aspects of RF characterization: Mesh loss, aperture illumination, chassis characterization and surface RMS error Ideal directivity Aperture loss (spillover þ taper) Feed mismatch loss Mesh transmission loss Chassis interaction RMS surface errors Final gain

51.46 dB 1.43 dB 0.04 dB 0.3 dB 0–0.1 dB 0.7 dB 48.89–48.99 dB

Figure 11.10 A preview of the deployment sequence for the 1 meter offset mesh reflector (Image from [102]) completely stored within a 3U volume with a mass of 2.5 kg, while finally deploying a 1 m mesh aperture. The RMS surface accuracy target is aimed for 0.28 mm, which translates to l/30 at 35.75 GHz. It is currently speculated that this approach can be scaled to larger apertures than 1 m [102], but more will be known after the final RF tests are conducted. In summary, offset mesh reflector antennas comprise an important design option for future CubeSats, with many exciting results in store for the future. While

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mesh reflectors have been studied well in the literature, packaging such reflector systems in future CubeSats represents the next big challenge for high-gain small satellite systems. It will be interesting to see new scientific opportunities that become available once this technology matures for CubeSats.

11.7 Reflectarray concept A major challenge for high-gain antennas is packaging, where the stowed volume can be cumbersome for big apertures such as deployable mesh reflectors. Reflectarrays are a unique antenna that can offer a good balance of antenna performance and volume requirements. In essence, a reflectarray antenna is a hybrid between a reflector and an antenna array. The reflectarray operates as a ‘‘flat reflector’’, where resonant patch elements on a grounded dielectric are tuned to provide a desired reflection phase. Since the reflection phase can be controlled, one can effectively synthesize any reasonable phase distribution. The flatness of the of the reflectarray is its key advantage over reflector antennas; it can be packed into a very small volume and deployed using hinged panels. For CubeSats, this is very exciting, and such technology could facilitate many interesting applications areas for CubeSats. Recently, NASA has explored the possibility to implement a reflectarray on a 6U CubeSat chassis that facilitates a deep space wireless link from Mars (Figure 11.11). This MarCO mission, as it has been named by NASA, is among the first missions to fly into deep space, where the long distances pose a serious limitation on data rates even with high-gain antennas. The goal of the MarCO mission is to provide a real-time link

X-band transmission

Earth

MarCO spacecraft

UHF transmission

InSight spacecraft

Mars

Figure 11.11 The MarCO bent-pipe Mars-to-Earth telecommunications relay concept [59]

Novel antenna concepts and developments for CubeSats

377

for the InSight mission during InSight’s entry, descent, and landing phase [104]. Furthermore, MarCO will demonstrate that CubeSats can be effectively utilized in deep space. The mission will consist of two identical CubeSats that will receive data from InSight on a UHF link at 8 kbps. The received data is then transmitted through a software-defined radio which transmits through the reflectarrays. The transmit system operates at X-band, and the deep space network (70 m reflector antennas) receives the signal. For a 5 W transmitter, maximum pointing loss of 3 dB (pointing accuracy of 2 ), and 8 kbps data rate, the system would require a 28 dBi antenna gain at X-band [59]. Achieving this level of gain on a 6U CubeSat platform is very challenging, and it has been demonstrated that reflectarrays can meet these demands. The reflectarray approach to achieving high-gain antennas has many key advantages. Reflectarrays can readily be tuned to meet the needs for a diversity of missions in the future. Better yet, reflectarrays fit within the general CubeSat vision to provide good performance on a limited budget and rapid deployment schedule. The folded-panel deployment, as will be discussed next, can be implemented in a variety of different chassis sizes and its simplicity leads to a robust mechanical design. Synthesizing various patterns and beam pointing angles can be accomplished using both the mechanical deployment design and the sizes of the resonant patches. Finally, the low mass and stowage volume are perhaps reflectarrays’ biggest highlights in the context of CubeSats, but certainly these other mentioned advantages make reflectarrays an attractive antenna candidate.

11.7.1 Deployment and design The reflectarray in the MarCO cubesats is composed of three 19.9 33.5 cm2 panels [59]. These panels are attached together using spring-loaded hinges, and the reflectarray deploys as illustrated in Figure 11.12. Before deployment, the panels are folded flush against the cubesat chassis with a stowage volume of

Root hinge

Wing hinges

Root hinges

33 .5

cm

Wing hinges

33.5 cm

Fold

Fold

59.7 cm Spacecraft bus

19.9 cm (a)

Fold

Three panels folded on side of 6U bus

Feed ~22 cm

(b)

Figure 11.12 The MarCO reflectarray in its (a) stowed and (b) deployed positions [59]

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19.9 33.5 1.25 cm3. To deploy, a Nichrome burn wire severs a Vectran tie-down that keeps the panels flush to the chassis. To feed the reflectarray, a 4 2 circularly polarized patch antenna array is used, which is also spring loaded to deploy along with the reflectarray panels. The feed is designed to optimally illuminate the reflectarray providing the maximum aperture efficiency (spilloverþtaper) using the 10 dB edge illumination rule-ofthumb. When stowed, the feed antenna is stored in a small rectangular pocket within the bus. Once deployment has started, the feed will deploy during the stage when the reflectarray ‘‘wing’’ panels are deploying. This overall deployment strategy is both compact and relatively simple to implement. A major advantage is that it only requires a single release mechanism to initiate the entire sequence. Furthermore, the deployment does not significantly increase the overall stowage volume, and all the space is utilized to maximize aperture size.

11.7.2 Flight model performance The actual flight model is shown in Figure 11.13, where both the stowed and deployed antenna phases are illustrated. Clearly, the overall stowage footprint is very compact, allowing for more volume for other important instrumentation, sensors, and potential power sources. This flight model was demonstrated and measured at the Jet Propulsion Laboratory. The feed performance demonstrated the desired illumination for the reflectarray, and the impedance matching was satisfactory. A representative pattern of the entire flight model (feed þ reflectarray) is

Figure 11.13 The MarCO flight model antenna mounted on a test bus: (left) stowed and (right) deployed. [59]

Relative power (dB)

Novel antenna concepts and developments for CubeSats 0 –5 –10 –15 –20 –25 –30 –35 –40 –45 –50 –60 –50 –40 –30 –20 –10

RCP Meas. RCP Calc.

379

LCP Meas. LCP Calc.

0 10 20 30 40 50 60

Azimuth angle (deg)

Figure 11.14 The MarCO flight model reflectarray azimuth plane radiation pattern at 8.425 GHz [59] Table 11.4 Gain budget for the X-band MarCO flight model reflectarray Measured directivity D0 Feed loss Patch dielectric loss Patch conductor loss Mismatch loss Hinge mounting area loss Total loss Gain from D0 þLoss Gain measurement

30.56 0.74 0.25 0.04 0.14 0.15 1.32 29.24 29.20

dB dB dB dB dB dB dB dB dB

shown in Figure 11.14. These models achieved a measured directivity of 30.56 dB and a measured antenna gain of 29.20 dB. A representative gain budget of the losses in this reflectarray design are tabulated in Table 11.4. The details for all the losses provides some good insight to provide some expectations for readers when implementing reflectarrays for CubeSats. Among the various losses, the feed loss is the most significant, although this does include a 4 cm coaxial cable with a measured 0.25 dB insertion loss. Despite the cable, it is important to mention that the losses in feed networks is often quite high, especially when a large array is implemented using a corporate feed network. Reflectarrays avoid these losses for the most part by using a small feed whose radiation illuminates the reflectarray via air (or vacuum in the case of space). The table also shows the estimated gain and the final measured gain. The estimated gain is calculated by Gest ¼ D0 þ L

(11.2)

where D0 is the directivity calculated from the measured radiation patterns and L is the total loss estimate. According to the aperture size of the reflectarray, the estimated

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MarCO gain provides roughly 41.6% efficiency. The authors in [59] also provide further details on losses according to fabrication and deployment errors. These can be critical if the deployment leads to subtle changes in the antenna design; so far their prediction shows reasonable margins can be achieved without expensive, sophisticated mechanical components. Overall, these results are very encouraging for future CubeSats. The measurement of the flight model reflectarray indicated that good aperture efficiency and high gain can be achieved using a planar microstrip reflectarray. The panelþhinge deployment mechanism makes it a straightforward construction project, and the design is also fairly robust to fabrication and deployment errors. It will certainly be exciting to see how future CubeSats use the reflectarray concept to solve the next greatest challenges for small satellite missions.

11.8

Patch antennas integrated with solar panels

Patch antennas are extremely attractive for CubeSat platforms. Patch antennas are low-profile, easy-to-integrate with planar circuits, and do not require deployment for operation (unlike many wire antennas). The only potential issue is the physical area required to mount a patch antenna to a CubeSat chassis face, which can take away precious real-estate for solar cells. Integrating patch antennas into a solar panel, therefore, is an important challenge that has relevance for future CubeSat missions. Currently, there are two design options to integrate patch antennas into solar panels: 1. 2.

Supersolar: transparent patch antennas that go above the solar panels [60] Subsolar: nontransparent patch antennas that go below the solar panels [64,105]

For each strategy, designers must balance antenna performance (e.g. efficiency, cross-polarization, bandwidth) with the solar cell output desired. This section describes several representative designs from each category along with a brief discussion on the various tradeoffs each design strategy faces. Typical frequency bands being tested for both supersolar and subsolar designs include the unlicensed ISM bands, which are commonly used for CubeSats. Among the various ISM bands, integrated patch antennas have been developed and tested at 434 MHz [64] and 2.4 GHz [106]. Using higher frequencies has its advantages, where the patch size gets smaller (as the wavelength gets smaller) and atmospheric effects like Faraday rotation decrease in severity.

11.8.1 Transparent (supersolar) patch antennas In this approach, transparent patch antennas are placed above the solar panels, which can dramatically impact the solar panel performance if good optical transparency is not obtained. Optical transparency can be achieved by using transparent metal, a wire mesh design, or a combination of both. For each of these cases, a transparent substrate such as glass or quartz must also be used. Each case has their own unique fundamental tradeoff between optical transparency and RF performance.

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381

In the transparent metal approach, better optical transparency sacrifices metal conductivity s. Some have investigated clear polyester films with conductive coatings (AgHT-8) [107], while others have investigated ultra-thin metallic layers and multilayer metal-semiconductor hybrids [108]. While the concept of transparent metals sounds convenient and easy to integrate, most investigations at this point have shown that these transparent metals have a relatively high sheet resistance. For example, AgHT-8 has a sheet resistance of 8 W/& and transmittance T > 82%, while a 10 nm film of copper has a sheet resistivity of 8.3 W/& and transmittance T > 45%. Many studies have demonstrated antennas using the transparent metal approach with antenna gains G < 0 dBi, which is too low for many application requirements. There is hope that material research might produce a structure that can nicely balance the tradeoff between sheet resistance and optical transmittance, but implementation in advanced systems is still up for debate. Another potential approach is to create thin mesh structures using opaque materials. In this approach, both the top patch layer and the ground plane are created using mesh structures. Mesh-based patch antennas have demonstrated reasonable RF performance and transparency as high as 95% [64]. It should be noted that transparency in this case is defined as the ratio of the transparent area of the antenna relative to the antenna’s total area as seen from a top view. An important tradeoff in this design approach is higher transparency at cost of higher backlobes (decreased ground-plane isolation). The isolation can be improved by using the solar cell ground plane (rather than a separate ground plane), but detailed knowledge of the solar cell is required to co-design the antenna. A transparent patch antenna example with a meshed patch and meshed ground is seen in Figure 11.15(a). The patch is linearly polarized and was fabricated by depositing silver epoxy onto a quartz substrate [64]. This mesh-based antenna was designed to operate at 1.644 GHz, with the hope to establish communication with the Iridium constellation of low-earth-orbiting satellites. The impedance matching performance and the radiation pattern of the patch antenna are also shown in Figure 11.15. Good impedance matching is achieved with a bandwidth of roughly 2.2%. The measured gain was 4.28 dB at the broadside, although it should be noted that the pattern was slightly changed because of the metallic bracket behind the antenna under test.

11.8.2 Nontransparent (subsolar) patch antennas Nontransparent patch antennas are typically made with substrate and metallic layers that are completely opaque. They can either be placed above or below the solar panels. The optical blockage from the antenna is a strong justification to integrate nontransparent underneath the solar panels (subsolar), rather than place them above the solar panels. In the subsolar case, the design issues have reversed from those of the supersolar: the solar panels block the antenna and can severely degrade the radiation patterns. The challenge in this subsolar approach is to find a configuration in which the RF blockage from the solar panel can be minimized and good radiation patterns are realized.

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Simulation S11 Measurement S11

S Parameters (dB)

–5 –10 –15 –20 –25 1.2 1.3

1.4 1.5 1.6 1.7 1.8 Frequency (GHz)

(b)

(a)

1.9

2

Patterns at 1.64 GHz 00 30 –10

30

–20

60

60

–30 dB 90

90 Measurement Co-Pol, E Plane Measurement Co-Pol, H Plane Simulation Co-Pol, E Plane Simulation Co-Pol, H Plane

120

(c)

150

180

Figure 11.15 (a) A transparent, mesh-based microstrip patch antenna. Both the patch and ground plane are realized using a silver epoxy mesh. (b) The simulated and measured S11 for the transparent patch shown in Figure 11.15a. (c) Simulated and measured radiation patterns [64]

Interestingly, patch antennas have a radiation feature known as radiating edges that can be effective for design integration. In essence, these so-called radiating edges are certain regions near the patch edges that are the primary contributors to the radiation pattern. If these radiating edges are exposed, rather than being covered by the solar panels, then good radiation patterns can be achieved. In general, it is best to avoid covering any open edges of the patch antenna to minimize potential interactions between the solar panel and antenna. In some cases, the solar panel can also impact the impedance matching if these open edges come into close proximity with the solar panel ground plane. Vias can be placed along a given edge to effectively shield these interactions, and designers must take into account the new resonant conditions with the vias.

Novel antenna concepts and developments for CubeSats

(a)

(b)

(c)

(d)

383

Figure 11.16 (a) Configuration of a nontransparent antenna. The square represents the antenna with solar panels above [64]. (b) A top view of the prototype without copper foil. The (c) top and (d) bottom views of the completed prototype

Researchers recently used this approach to develop subsolar patch antennas that could be integrated into solar panels [64]. The integration is illustrated in Figure 11.16(a), where the square represents the patch antenna location below the solar panels. One challenge in this design approach was properly sizing the patch to achieve resonance, especially considering that the solar panels already had fixed dimensions. Both the via arrangement and the substrate permittivity er can be modified to tune the resonance towards the desired frequency. To modify the permittivity, one can take a known substrate and cut holes to effectively decrease er. This is illustrated in Figure 11.16(b), where substrate material was removed from a Rogers Duroid 5880 board in a grid pattern. After cutting the holes, foil is placed to better connect the top layer, and the completed prototype can be seen in Figures 11.16(c)–(d). The coaxial connectors were placed on the 45 lines for the two patch antennas shown above, in order to produce circular polarization when excited with an equimagnitude, quadrature phase power divider (such as a hybrid). Whether antennas are being placed above or below solar panels, integration is a major challenge with big rewards for CubeSats. If good antenna designs can be realized and integrated easily with commercially made solar panels, then real-estate becomes less of an issue for the antenna. Furthermore, the solar panel often represents a large area, and high-gain antennas (e.g. antenna arrays) might be integrated together with solar panels on a regular basis for CubeSats in the future.

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11.9 Conclusion CubeSats provide a small, yet powerful platform for the satellite and space community to economically carry out advanced space missions. The aim of this chapter was to acquaint the readers with the current state of the art for antennas in CubeSats and review the recent developments that can lead to several advanced missions in the future. Following a brief discussion on the developing standards and classifications of the CubeSat standard, general antenna specifications that must be considered while designing antenna systems for CubeSats were highlighted. A comprehensive literature survey is presented that summarizes the recent antenna developments for CubeSats, where one can see the diversity of designs that have been investigated by antenna engineers throughout the world for CubeSats. The next stage in CubeSat research is the integration of high-gain aperture antennas with CubeSats so that high data rate and beyond LEO missions are a possibility. A detailed description of four advanced antenna designs that aim to push the functionality of CubeSat to the next level is provided in this chapter. As we move towards an age where achieving major breakthroughs through small satellites is becoming a reality, we hope that this chapter inspires researchers in continuing research within this emerging area of CubeSats.

Appendix A Characterization of umbrella reflectors Many deployable reflector systems utilize a rib deployable umbrella reflector architecture which can be folded during launch and deployed once in space [19,27,33]. This appendix provides guidelines and representative design curves that aid the design of umbrella reflectors. The umbrella reflector surface consists of a discrete number of parabolic ribs that are connected by surfaces called gores. Each gore surface is a section of a parabolic cylinder, bound between two parabolic ribs, as shown in Figure A.1. In deployable reflector systems, the gore surface can be formed by stretching a mesh between the two ribs. A detailed description of an antenna system utilizing such an architecture was given in Section 11.5.

Gore surface

Parabolic ribs

Figure A.1 A representative example of umbrella reflectors being used for CubeSat applications [54]

Novel antenna concepts and developments for CubeSats

A.1

385

Mathematical representation of the gore surface

Assuming that there are Ng gores, with the rib focal length being Fr, the equation of the gore surface can be expressed as [109]: 0

Fr cos2 ðp=Ng Þ (A.1) 0 fm þ fmþ1 2f 2 cos 2 where fm ¼ (2p/Ng)(m 1) for m ¼ 1,2, . . . ,Ng. The variables r0 and f0 are cylindrical coordinates as defined in Figure A.2. It can be seen from (A.1) that the focal length of the umbrella reflector is a periodic function of the azimuthal angle f0 . Since the umbrella reflector surface does not have a distinct focus, the optimal feed position (or the subreflector position for dual reflector systems) is not obvious. zg ¼

A.2

r2 0 0 ; Fg ðf Þ ¼ 4Fg ðf Þ

Finding the optimum feed location

The basis for finding the optimal feed location can be inferred from Figure A.2. The variation of Fg with f0 causes the total ray length (r0 þ z0 ) to be a periodic function of f0 . This results in phase deviations at the exit aperture. It is intuitive that the optimal feed location (referred to as Fopt) will be the point that minimizes the deviation in this total ray length. Different approaches to accomplish this have been reviewed in [56] and is summarized in Table A.1. It was also noted in [56] that the analytical formulations can be used only if the gain loss due to the surface distortion was under 3 dB. For cases where a lower number of gores are desired, one must resort to manually tuning the feed position through parametric analysis. As the number of gores increases, all techniques converge to the following optimal location: Fopt 2 p 2 ¼1 (A.2) 3 Ng Fr Feed at Fr

z Feed at Fopt Aperture

r'

zg

x'

Fr

z'

zp

ρ' ϕ'

θ' F opt

Umbrella reflector

Ideal Paraboloid x

Figure A.2 Finding the optimal feed position: The aim is to minimize the deviation in r0 þ z0 . The dotted lines show the ray paths for an ideal parabolic reflector [56]

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Developments in antenna analysis and design, volume 2

Table A.1 Summary of two theoretical approaches investigated in the literature to determine the optimal feed location (Fopt /Fr) for the umbrella reflector Technique

Optimal feed location

Parallel ray approximation [109]: Express phase variation as Df(1 þ cos q0 ), where Df ¼ Fopt Fg. The feed point that minimizes Df is the optimal feed position. Best fit approach [110]: Find the best fit parabola to the umbrella reflector through an RMS minimization. The focus of the best fit parabola will be the optimal feed location.

Fopt Ng 2p sin ¼ Fr 2p Ng 2 1 1 þ tan2 f þ tan4 f Fopt 3 5 2 ¼ cos f 1 2 Fr 1 þ tan f 3 p where f ¼ Ng

Optimum feed position (Fopt/Fr)

1 0.95 0.9 0.85

Fopt by best−fit approach

0.8

Fopt by parallel ray approx.

0.75

Fopt = Fr(1−(2/3)(π/Ng)2)

0.7

Manual tuning

0.65 5

10

15

20

25

30

Ng

Figure A.3 Comparison of the optimal feed positions got through various theoretical approaches (refer Table A.1) and manual tuning of feed position for aperture diameter of 1m and Fr /D ¼ 0.5 at 35.75 GHz [56] Results comparing the value of optimal feed location found through different approaches are shown in Figure A.3. A representative far field comparison for the case of D ¼ 1 m, Fr/D ¼ 0.5, and Ng ¼ 10 is shown in Figure A.4 for a frequency of 35.75 GHz. It was observed that the difference between the predictions from the theoretical approaches and manual tuning results in an additional gain loss of approximately 4.5 dB for 10 dB feed taper. It must be noted that in the case of dual reflector systems, the subreflector, and the feed must be suitably positioned so that the virtual focus of the system coincides with the optimal feed location of the main umbrella reflector.

Novel antenna concepts and developments for CubeSats Ideal reflector with feed at focus Feed position:

0 Normalized gain (dB)

387

Fopt = Fr(1−(2/3)(π/Ng)2)

−10

Feed position by parametric tuning

−20 −30 −40 −50 −60

−5

0 θ (deg)

5

Figure A.4 Comparison of far field patterns for Ng ¼ 10 at with an ideal parabolic reflector when the feed is kept at the focal point computed by (A.2) and manual tuning [56]

A.3

Gain loss as a function of the number of gores

As described previously, the gores cause the surface of an umbrella reflector to deviate from an ideal parabolic reflector. This deviation is typically characterized in terms of surface RMS error as described by the equation [110]: sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð ð0 1 ðD=2Þcos f y tan f ðzg zeq Þ2 dx0 dy0 (A.3) Dzrms ¼ Ag 0 0 where 0

zg ¼

r2 ; 4Fg ðf0 Þ

0

zeq ¼

r2 4Fopt

(A.4)

1 where D represents aperture diameter, f ¼ p/Ng and Ag ¼ ðD=2Þ2 sin 2f . The 4 equation for the RMS error in (A.3) can be analytically evaluated to give a closed form expression for the RMS error [110]: D2 cos2 f tan2 f ð1 þ tan2 f Þ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ D zrms ¼ 0:010758 Fr 2 1 ð1 þ tan2 f þ tan4 f Þ 3 5

(A.5)

For a reasonable amount of gores (Ng > 20), the equation for the RMS error can be simplified using the following approximation: 2 tan2 f ð1 þ tan2 f Þ p 2 s ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ cos f (A.6) N g 2 2 1 4 1 þ tan f þ tan f 3 5

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This simplifies the expression for the RMS error significantly to: 2 D p Dzrms ¼ 0:010758 Fr =D Ng

(A.7)

Substituting this into the Ruze’s equation [111] yields 4 Dzrms 2 z2 D2 1 DGðdBÞ ¼ 685:811 z ¼ 7:73 2 (A.8) 2 l l ðFr =DÞ Ng qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ where z ¼ ð4Fr =DÞ ln½1 þ 1=ð4Fr =DÞ2 . This leads to a very interesting result: pﬃﬃﬃﬃ for a given Fr /D, the gain loss scales as the ð D=Ng Þ4 , implying that the number of gores required for the same gain loss scales only as the square root of the diameter. It should be noted that if the amplitude distribution on the aperture of the reflector is tapered, the equation for the RMS error presented in (A.3) must be suitably modified to [112]: vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ u ðD u 1 2 cos f ð y0 tan f t Dzrms ¼ ðzg zeq Þ2 Qðr0 Þdx0 dy0 (A.9) Ag 0 0 where Q(r0 ) and Ag are the taper function and the effective area, respectively, given by [113]: " 0 2 #p r ; Edge taperðET Þ ¼ 20log10 C (A.10) Qð r0 Þ ¼ C þ ð1 CÞ 1 D=2 ð ðD=2Þcos f ð y0 tan f Ag ¼ Qðr0 Þdx0 dy0 (A.11) 0

0

The value of p in (A.10) governs the slope of the taper. Representative simulation results that highlight the dependency of gain loss on the number of gores for various aperture diameters D for X and Ka-band frequencies are shown in Figure A.5. 0

0

−0.4 −0.6 −0.8 −1 20

(a)

D = 0.5m D = 1m D = 2m

−0.2

D = 0.5m D = 1m D = 2m

Gain loss (dB)

Gain loss (dB)

−0.2

−0.4 −0.6 −0.8

30

40

50 Ng

60

70

−1 20

(b)

30

40

50

60

70

Ng

Figure A.5 Gore loss as a function of number of gores when the feed is kept at the optimal feed location for different aperture diameters D and F/D ¼ 0.5. (a) Ka band (35.75 GHz) and (b) X band (7.2 GHz)

Novel antenna concepts and developments for CubeSats

Appendix B

389

Mesh characterization for deployable reflectors

A common feature underlying the design of many high-gain reflector antennas is that the surface is composed of a mesh that is tensioned to maintain the desired profile. Sections 11.5 and 11.6 discussed the development of two such antenna systems in depth. The meshed surface reduces the volume and weight of the antenna system. However, the mesh openings lead to RF transmission losses. Thus, a balance has to be achieved between the density of the mesh surface (typically measured in terms of the number of OPI) and the transmission loss. This appendix provides some useful design curves and guidelines to achieve this tradeoff. Examples of representative knit patterns that are used for reflectors are shown in Figure B.1. It is evident that the knit patterns in general can be very complex in structure, making full-wave simulations unrealistic. This appendix also discusses methodologies to generate simple equivalent models for such complex knits that can be analyzed easily through analytical formulations and/or full-wave solvers.

B.1

Simple wire grid model

The analytical solution to the transmission coefficients of a simple wire grid model (shown in Figure B.2) is known through the formulations by Astrakhan [117,118]. The predictions of Astrakhan’s formulation and a comparison of the formulations with full-wave simulations for various are illustrated in Figure B.3 and Table B.1. Note that the gain loss in this case is defined as DG ¼ 20log10|R|, where R denotes the reflection coefficient. The accuracy of the analytical predictions is evident in these results. This allows engineers to use these analytical formulations to get an initial estimate of the losses that one might expect from a mesh surface without the need of any computer simulations.

1/opi

1/opi

Figure B.1 Representative examples of knit models that are used to form the surface of a mesh deployable reflector [114–116]

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Developments in antenna analysis and design, volume 2 Z

ETM

ETE

θi a ϕi

b X

Y

0

0

−2

−1

−4

−2

∆ G (dB)

∆ G (dB)

Figure B.2 Simple wire-grid model, whose performance can analytically be determined by Astrakhan’s formulations [117,118]. The quantities a and b are measured from the center of one wire to the other. ETE, ETM denote the direction of electric field vector for TE and TM polarization, respectively, for a plane wave incident at an angle qi and fi [55]

Simulated (HFSS) result Astrakhan formulation

−6

(a)

−3 −4

−8 −10

HFSS Astrakhan formulation

10

15

20

25 OPI

30

35

−5

40

(b)

0

20

40

60

80

100

φ (deg)

Figure B.3 Comparison between Astrakhan’s analytical formulations and fullwave simulations at 35.75 GHz for normal incidence. (a) Loss for various OPI [55] and (b) loss as a function of fi for qi ¼ 0 and 40 OPI

B.2

Equivalent wire grid model for complex knits

The aim of this section is to investigate the generation of equivalent simple wire grid models for complex knits, allowing complex knit structures to be easily analyzed. A detailed investigation of the Astrakhan formulations [118] reveals a strong

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Table B.1 Comparison between HFSS and Astrakhan formulations for various diameters for normal incidence (qi ¼ fi ¼ 0 ) and 40 OPI at 35.75 GHz [55] DG (dB)

Diameter

00

0.0008 0.001600 0.00200 0.00400

HFSS

Astrakhan

0.47 0.24 0.17 0.05

0.49 0.25 0.19 0.05

W

1/OPI

1/OPI

(a)

D

(b)

(c)

Figure B.4 A representative complex tricot knit pattern utilized to construct mesh reflectors [55,114]. (a) Mesh structure. (b) Equivalent strip model with strip width W, scaled according to the OPI. (c) Equivalent wire grid model of wire diameter D ¼ W/2 dependency of the mesh transmission loss on the diameter of the constituent wires (see Table B.1) [55]. This implies that while generating the equivalent model of the complex mesh, the equivalent diameter that represents the complex knit surface must be carefully chosen. As a representative example, a complex tricot knit structure that was previously analyzed in [114] is considered. This structure was simulated using floquet analysis through full-wave simulators. The knit structure is shown in Figure B.4(a). The constituent wires of the mesh are assumed to be of 0.000800 diameter, consistent with [54]. The mesh is assumed to be PEC for simplicity. If the OPI is known, the critical step is to find the equivalent diameter. Dense meshes typically consist of intimate strands of wires that cross each other. These sections can be replaced by an equivalent strip since the spacing (in wavelengths) is small enough that the surface currents do not see the difference between the individual strands of wires and the thin strip. This leads to an equivalent strip model with a certain width W. Using this formation of the strip model, the wire grid model can be easily constructed using strip wire equivalence [118–120]. For the knit geometry considered, the process is

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illustrated in Figure B.4. The equivalent strip and wire grid model is illustrated in Figure B.4(b) and (c), respectively. The performance comparison between the mesh loss for normal incidence for the complex knit (through full-wave simulation) and the equivalent simple wire grid model (through analytical Astrakhan formulations) for normal incidence at 35.75 GHz is shown in Table B.2. The importance of choosing the right equivalent diameter can be clearly observed; if an equivalent wire grid model having the same diameter as the constituent wires is used, the difference between the simple model and the actual knit model is significant. Representative cases for oblique incidence for 20 and 40 OPI are shown in Figure B.5. Note that TE and TM refer to electric field and magnetic field being oriented perpendicular to the plane of incidence, respectively. It is evident from the results that the accuracy of the equivalent model increases as the mesh gets denser, consistent with the previous hypothesis. In general, some knit structures could preclude the direct formation of the equivalent Astrakhan model. However, the methodology presented in this section can still be used as a tool Table B.2 Comparison between full-wave simulated tricot knit mesh of wire diameter Dwire and the analytical simple wire grid model with diameter Deq for normal incidence (qi ¼ fi ¼ 0 ) at 35.75 GHz. The parameters D, W, and OPI are as defined in Figure B.4 [55] Gain loss DG (dB)

OPI Tricot knit mesh

Wire grid model

0.56 0.20 0.09

20 30 40

0

Deq ¼ Dwire

Deq ¼ W/2

2.53 1.01 0.47

0.43 0.19 0.11

0

TE

TE −0.5

−1

TM

∆ G (dB)

∆ G (dB)

−0.5

Equivalent wire grid Complex knit

−1.5

TM

−1.5

−2

−2 0

(a)

Equivalent wire grid Complex knit

−1

20

40 θ (deg)

60

80

0

(b)

20

40 θ (deg)

60

80

Figure B.5 Comparisons between the gain loss (DG) of the complex knit mesh surface via full-wave simulations and the equivalent wire grid model for oblique incidence (f ¼ 0 ) using Astrakhan’s formulations (a) 20 OPI. (b) 40 OPI

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to simplify complex knit structures to facilitate quick full-wave simulation of complex knits. Once the transmission coefficients for both polarizations are known for all angles of incidence, it is possible to analyze a curved mesh reflector as done in [121]. Thus, this section provided an analytical basis to simplify the analysis complex mesh surfaces, which is an important consideration for mesh reflectors.

Acknowledgement The authors would like to acknowledge fruitful discussions with colleagues at JPL and Tendeg LLC for Sections 11.5, 11.6, and 11.7.

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Hai FL. The principle error and optimal feed point of umbrella-like parabolic reflector. In: International Symposium on Antennas, Propagation and EM theory (ISAPE); Beijing, China, 2000. p. 697–700. Ruze J. Antenna tolerance theory–A review. Proceedings of the IEEE. 1966 April;54(4):633–640. Rahmat-Samii Y. Effects of deterministic surface distortions on reflector antenna performance. Annales des te´le´communications. 1985;40:350–360. Rahmat-Samii Y. Reflector Antennas. In: Lo YT, Lee SW, editors. Antenna Handbook. New York: Van nostrand reinhold; 1993. Imbriale WA, Galindo-Israel V, and Rahmat-Samii Y. On the reflectivity of complex mesh surfaces. IEEE Transactions on Antennas and Propagation. 1991 Sep;39(9):1352–1365. Li T, and Su J. Electrical properties analysis of wire mesh for mesh reflectors. Acta Astronautica. 2011;69(1):109–117. Sorrell R, Turba FP, and Vanstrum M. Multi-layer highly RF reflective flexible mesh surface and reflector antenna. Google Patents; 2014. US Patent 8,654,033. Astrakhan M. Reflecting and screening properties of plane wire grids. Telecommunications and Radio Engineer-USSR. 1968;(1):76. Rahmat-Samii Y, and Lee SW. Vector diffraction analysis of reflector antennas with mesh surfaces. IEEE Transactions on Antennas and Propagation. 1985 Jan;33(1):76–90. Rajagopalan H, Miura A, and Rahmat-Samii Y. Equivalent strip width for cylindrical wire for mesh reflector antennas: Experiments, waveguide, and plane-wave simulations. IEEE Transactions on Antennas and Propagation. 2006;54(10):2845–2853. Miura A, and Rahmat-Samii Y. RF characteristics of spaceborne antenna mesh reflecting surfaces: Application of periodic method of moments. Microwave and Optical Technology Letters. 2005;47(4):365–370. Miura A, and Rahmat-Samii Y. Spaceborne mesh reflector antennas with complex weaves: Extended PO/Periodic-MoM analysis. IEEE Transactions on Antennas and Propagation. 2007 April;55(4):1022–1029.

Index

active antenna systems, fast analysis of 187 antennafier array element concept 193–4 method-of-moments analysis of folded dipole antennafier 194–200 Deep Integration paradigm 187 potential impact and other integration approaches 190–1 scientific and technological challenges 191 modeling approach and assumptions 191–2 multiscale numerical analysis 200 active versus passive antenna array results 206–9 characteristic Basis Function Method (CBFM) 201–3 characteristic basis functions (CBFs) 203–5 numerical matrix compression and solution 205–6 adaptive cross approximation (ACA) algorithm 203, 216 additive resist-based process 138–9 Anritsu MS5657B Vector Network Analyzer 66 ANSYS HFSS 8, 29 antennafier array, multiscale numerical analysis of 200 active versus passive antenna array results 206–9 characteristic Basis Function Method (CBFM) 201–3 characteristic basis functions (CBFs) 203–5

numerical matrix compression and solution 205–6 antennafier array element concept 193–4 method-of-moments analysis of folded dipole antennafier 194–200 antennafier arrays 209 antenna-on-chip (AoC) designs 302 Antennas for Light 127 antenna under test (AUT) 318 aperture-coupled patches, reflectarray elements based in 330–1 Astrakhan formulations 389–90 Auston switch 76 balanced antipodal Vivaldi array (BAVA) 290 bent-end simple phone model 173–6 optimization for QZ0 176–8 Bessel functions 134 body-worn antenna 260 broadband optimization technique 341 bulk refractive index sensing 144 Cassegrain configuration 369 cathodoluminescence 141 CFIE formulations 217–18 characteristic basis function method (CBFM) 187, 201–3, 214–17 acceleration of electromagnetic analysis using 221 constructing CBFs by using multiple plane-wave excitation 222–4 generation of reduced matrix equation in the CBFM 224–6

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Developments in antenna analysis and design, volume 2

multi-scale discretization using the IEDG method 226–30 numerical results 230–4 partition of CBFM 221–2 analysis of scattering from objects embedded in layered media using 235 CBFM analysis of the object embedded in layered media 235–7 mixed potential integral equation for objects embedded in layered media 237–41 numerical results 241–7 for microwave circuit and antenna problems 248 numerical results 251–3 SVD-based CBFM 248–51 Characteristic Basis Functions (CBFs) 202–5, 214–17 chemical etching 262 circularly-polarized wave (CPW) 328 combined field integral equation (CFIE) 216 MoM combined with 217–19 combined field surface integral equation 214 commercial off-the-shelf (COTS) components 363–4 computational electromagnetics (CEM) 213 COMSOL 87, 91 conductance 81 conductive ink 262 continuous wave (CW) THz generation 74 contoured beam reflectarrays 341 convex optimization 162, 173 copper tape, cutting 262 CubeSats 361 antenna requirements for 364 antenna material 366 antenna radiated power, gain, and radiation pattern 365–6 frequency 364–5

existing standards for small satellites 363–4 Ka-band offset reflector antennas 371 proposed deployment strategy 374–6 reflector design and feed development 373–4 Ka-band symmetric umbrella reflector antennas 367 antenna configuration 369–70 deployment strategy 370–1 reflector surface characterization 370 patch antennas integrated with solar panels 380 nontransparent (subsolar) patch antennas 381–3 transparent (supersolar) patch antennas 380–1 reflectarray concept 376 deployment and design 377–8 flight model performance 378–80 representative current antenna concepts for 366–7 current sheet array (CSA) 292 Deep Integration paradigm 187 potential impact and other integration approaches 190–1 scientific and technological challenges 191 Deep Space Network (DSN) 365, 367 degrees of freedom (DoFs) 213 deployable and inflatable reflectarrays 344–5 deployable reflectors, mesh characterization for 389 equivalent wire grid model for complex knits 390–3 simple wire grid model 389–90 device figure of merit 17 dielectric resonator reflectarrays 348 dielectric resonators versus plasmonic resonators 136–7

Index differentially fed HOM patch antenna 58–62 dipole-moment excitation 250–1 dipole PCA, structure and dimension of 87 Direct Broadcast Satellite (DBS) applications 341 discontinuous Galerkin method 214, 254 discrete complex image method (DCIM) 235, 237, 245 ‘divide and conquer’ approach 213 DM-CBFM 252–3 ‘domain decomposition’ method 202, 214 Drude-based dielectric functions 130 Drude-based optical properties of silver 131 Drude–Lorentz model 76, 79–81 dual-reflector configurations 342–4 Duroid 5880 substrates 40, 48, 62 dyadic Green’s functions (DGF) 235, 245 edge-port excitation 249–50 Einstein relationship 84 electrically small antenna design 167 electric field integral equation (EFIE) 162, 216, 218, 221 electromagnetic analysis, acceleration of using CBFM 221 constructing CBFs by using multiple plane-wave excitation 222–4 generation of reduced matrix equation in the CBFM 224–6 multi-scale discretization using the IEDG method 226–30 numerical results 230–4 partition of CBFM 221–2 electromagnetic spectrum 73 electron beam lithography (EBL) 138–9 electron-energy loss spectroscopy 141 embroidering 278 engineered optical antennas 128

405

Entry, Descent and Landing (EDL) data 345 EP-CBFM 252–4 equivalent circuit model 81–3 fabricated C-band spiraphase reflectarray 328 Fabry–Pe´rot effect 109 Fano resonances 145–6 far-field THz radiation 77, 94 far-field THz-TDS 95 applications of the THz-TDS system 108–10 experimental characterization of PCA 97–104 motivation 95 polarization effect and cancellation effect 104–6 system setup 95–7 THz radiation power/efficiency improvement 106–8 Fast Fourier Transform (FFT) algorithms 335 F/D ratio 369–70 finite-difference time-domain (FDTD) method 86, 104, 265 finite element method (FEM) 265 finite methods 213 first-order Leontovich boundary condition 130 5G frequencies, sample design for 316–19 fixed-beam reflectarray 324, 326, 328 fixed frequency plasmonic antennas 6–9 flared notch 289 flat lenses 147–8 flexible antenna for health monitoring 264 flexible compact planar antenna 260 fluorescence enhancement factor 142 focused-ion-beam (FIB)-induced deposition 139–40

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Developments in antenna analysis and design, volume 2

folded dipole antennafier, method-ofmoments analysis of 194–200 fractal-shaped patches 328 frequency-selective surfaces (FSS) 291 Fro¨hlich condition 134 full-wave electromagnetic analysis 334 full-wave model 83–6 simulation examples of 87–94 design of new PCAs with enhanced THz radiation 90–4 parametric studies of the PCA by simulation 87–90 validation of the model 87 fundamental bounds for antenna analysis and design 167–8 GaAs-on-sapphire (GoS) 98 gain loss as a function of the number of gores 387–8 Galerkin method 169, 214, 216, 224 Gans theory 136 gate-controlled mid-infrared reflectarray 350 general scattering upper bound 17 genetic algorithms (GAs) 161, 162, 168–73, 290 gore surface, mathematical representation of 385 graphene conductivity 5 planar plasmonic antennas 6 fixed frequency plasmonic antennas 6–9 frequency-reconfigurable plasmonic antennas 9–11 graphene plasmonic antenna model 11–13 terahertz beam steering reflectarray prototype 27 design and measurement 29–32 working principle 28–9 terahertz non-reciprocal isolator 22 isolator working principle 23–4 measurements 24–7 tunable devices, efficiency upper bounds in 13

demonstration of the upper bound 15–17 device specific bounds 19–22 generalized electric and magnetic field representation 14 graphene figure of merit 17–19 Green’s functions 169, 171 Hankel function 135 health monitoring, flexible antenna for 264 Hertzian dipole theory 80 heuristic optimization methods 168 higher-order mode millimeter-wave (HOM MMW) patch antenna 53 differentially fed HOM patch antenna 58 antenna geometry and working principle 58–9 simulation and measurement results 59–62 wideband HOM patch element 53 antenna element performance 53–6 higher-order modes for broadband operation 56 high-gain antenna (HGA) 345 high-gain metal-based deployable reflector antennas 366 high-transmission Huygens metasurfaces 148 holograms 147 horn antennas and guided wave structures 367 indium-tin-oxide (ITO) 139, 346 inflatable reflectarrays 344–5 inflatable reflector antennas 345 integral equation discontinuous Galerkin (IEDG) method 215–17 Integral Methods 213 integrated-circuit antennas 190 Integrated Solar Array and Reflectarray Antenna (ISARA) 345

Index inter-band range 4 Intersection Approach technique 340–1 interwoven spiral array (ISPA) 293 isolator 22 Jet Propulsion Laboratory 378 Ka-band 333, 341–2, 345–6 Ka-band offset reflector antennas 371 proposed deployment strategy 374–6 reflector design and feed development 373–4 Ka-band symmetric umbrella reflector antennas 367 antenna configuration 369–70 deployment strategy 370–1 reflector surface characterization 370 Kirchhoff’s law 82 Ku-band 339–40, 344 dual reflectarray antenna in 343 Kubo formula 4–5, 17, 19, 25 lamination technique 278 least angle regression selection (LARS) algorithm 268, 270 leave-one-out (LOO) cross validation algorithm 268, 273 Legendre function 134 lightning-rod effects 142 liquid crystal (LC) reflectarrays 349–50 localised field enhancement 128, 141–3, 151 localised surface plasmon resonances (LSPRs) 132–4, 144 Lorentz model 79 low-temperature co-fired ceramic (LTCC) 190, 302 macroscopic electromagnetic (EM) wave effects 187 magnetic field integral equation (MFIE) 216 Malta cross 328 Marchand balun 292, 297, 310 MarCO CubeSat platform 345–6 MarCO mission 376 material figure of merit 17, 19

407

MATLAB 87, 119, 230–1, 241, 243 Maxwell’s equations 2, 83–4, 130 2D materials in framework of 3–6 membrane antennas 367 mesh-based patch antennas 381 mesh characterization for deployable reflectors 389 equivalent wire grid model for complex knits 390–3 simple wire grid model 389–90 meta-heuristic algorithms 161 metal-only reflectarray antenna 332–3 metasurfaces 147–8 planar 3, 15, 28 method of moments (MoM) 162–3, 165, 213 combined with CFIE 217 elements of impedance matrix of 219–21 formulation 192, 200 MicroSat 363 microstrip patches 326 Mie resonances in nanoscale resonators 134–6 millimeter-wave antennas HOM MMW patch antenna 53 differentially fed HOM patch antenna 58 wideband HOM patch element 53 planar UWB arrays for millimeter-waves 313 wideband MMW complementary source antennas for 5G 62 linearly polarized antenna fed by an SIW 62–3 radiation mechanism of 63–6 simulation and measurement results 66–8 wideband MMW ME dipole antennas 40 differential feed printed ME dipole antenna 47 single feed printed ME dipole antenna 40–2 miniaturized-element frequency selective surfaces (MEFSSs) 339

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Developments in antenna analysis and design, volume 2

MiniSat 363–4 Mixed Potential Integral Equations 237 ML-CBFM algorithm 237 modulation figure of merit 17 molecular beam epitaxy (MBE) technique 97–8 MOM-based CEM algorithms 235 MoMmatrix 214–15 mono-polar RWG basis function 226–7, 233, 243 mono-static RCS 233, 243–4 Monte Carlo (MC) simulations 261, 266–7 morphology-dependent resonances 135 multilayer reflectarray 327, 330 multiplier-based THz sources 74 nanoantennas: see optical antennas nanocrossfinger 91 nano-dimers of silicon DRs 143 NanoSat 363–4 NASA 339, 345, 367, 376 National Institute of Standards and Technology (NIST) 128 natural current 11 nontransparent (subsolar) patch antennas 381–3 ohmic range 4 optical antennas 127 applications 141 integrated photonics 146–7 localised field enhancement 141–3 photodetection 149 planar optical components 147–9 selective thermal emission 150 sensing 144–6 early history 128–9 fabrication 137 bottom-up approaches 140 direct patterning approaches 139–40 resist-based approaches 138–9 top-down approaches 137–40 optical characterisation of 140–1

theory and analysis 129 dielectric resonators versus plasmonic resonators 136–7 metal properties from microwave to optical frequencies 129–32 Mie resonances in nanoscale resonators 134–6 plasmonic effects 132–4 optical diode 3 optical reflectarrays 147–8 optimization algorithms 167–8 convex optimization 173 genetic algorithms 169–73 optimization in antenna design 161 optimum currents, wireless terminal antenna placement using 178–81 optimum feed location, finding 385–7 palladium–gold-based nanoantennas 144 Pareto-optimal devices 3 particle swarm optimization (PSO) 373 patch antennas integrated with solar panels 380 nontransparent (subsolar) patch antennas 381–3 transparent (supersolar) patch antennas 380–1 perfect electric conductor (PEC) 129, 173, 195, 294 perfectly conducting metals 192 perfect magnetic conductor (PMC) 294 phase-only synthesis 323, 340 Phoenix cell 328 photoconductive antenna (PCA) 75–7 experimental characterization of 95 far-field THz-TDS 95–110 THz near-field microscopy 111–20 photodetectors 149 photolithography 138 photoluminescence 142 physical bounds of antenna 167 PicoSat 363–4 planar and patch antennas 367

Index planar infrared antenna-coupled detectors 128 planar UWB arrays 301 cavity resonance 309–12 feed planarization 301–2 material and process selection 302 for millimeter-waves 313 5G frequencies, sample design for 316–19 three-pin balun, development of 313–15 PCB processing, limitations of 302–3 surface waves 303–9 plane-wave excitation 249 plasmonic effects 132–4 plasmonic nanoantennas 139–40, 142–3, 149 plasmonic range 4 plasmonic resonators 136–7 plasmonics 130, 132 PMCHWT-MPIE 237, 245–8 Poggio–Miller–Chang–Harrington– Wu–Tsai (PMCHWT) equations 237, 245, 248 poly(methyl methacrylate) (PMMA) layers 23 polydimethylsiloxane (PDMS) 148–9, 347 polymer-jetting 3-D printing technology 346 polynomial chaos (PC) 266 polynomial chaos expansion (PCE) technique 264–71, 273 Poly-Picosatellite Orbital Deployer (P-POD) 363–4 power-added efficiency (PAE) 208 printed circuit board (PCB) 299 pulsed THz generation 74–5 Pyrex wafer 23 QZ0 computation from current densities 165–6 Q-factor 161–5, 167, 175–6 quantum cascade laser (QCL) 74 quantum yield of an emitter 142

409

quarter-wave plates (QWPs) 26 quasi-dipole unit cells 332–3 Raman scattering 143 Rao-Wilton-Glisson (RWG) basis function 196 reactive reflector antenna 323 reconfigurable reflectarrays 324, 331–2 rectangular metallic patches 327 rectangular printed patches 326 reflectarray antennas 323 basic concepts on 324–5 design and analysis of 334–5 dual-reflector configurations 342–4 technological challenges 344 deployable and inflatable reflectarrays 344–5 liquid crystal (LC) reflectarrays 349–50 reflectarrays and solar cells 345–6 reflectarrays at terahertz and optical frequencies 347–8 3-D printed reflectarrays 346–7 reflectarray concept 376 deployment and design 377–8 flight model performance 378–80 reflectarray elements, analysis and design of 334 reflectarrays 367 broadband techniques in 335–40 deployable and inflatable 344–5 elementary cells in 325–34 inflatable 344–5 liquid crystal 349–50 shaped and multi-beam 340–2 and solar cells 345–6 at terahertz and optical frequencies 347–8 terahertz graphene based 27 3-D printed reflectarrays 346–7 reflective array antenna 323 reflector antennas 366 remote health monitoring system based on wearable sensors 260 resonant nanostructures 128

410

Developments in antenna analysis and design, volume 2

Riccati–Bessel functions 135 RWGs (Rao, Wilton and Glisson) 214, 219 sawtooth PCA 91 S-band communication 366 scan blindness 303 scattering coefficients 134 shaped and multi-beam reflectarrays 340–2 shaped-beam reflectarrays 340 Shockley–Read–Hall process 85 SI-GaAs-based PCA 98 signal-to-noise ratio (SNR) 75, 78, 115–17, 119 silicon-on-sapphire (SoS) substrate 98 Silvaco 87 silver nanowires, antenna made of 263 similarity shaped reflectarray 329 single-layer reflectarray 332 single-walled carbon nanotube (SWCNT) 109–10 SmallSats 361–2 Sobol’s indices 269, 276, 281 solar cells, reflectarrays and 345–6 Sommerfeld integrals 236, 240–1 South Pan-American coverage (PAN-S) 341 spatial basis functions 218–19 spatial Green’s functions 240 Spectral Domain Method of Moments (SDMoM) 334 spectral Green’s functions 240 sphiraphase-type reflectarray element 329 spiraphase array 328 spiraphase-type elements 328 split ring resonator (case study) 271 numerical model and PCE analysis 272–8 overview 271–2 split ring resonator (SRRs) 271 square-integrable L2-space 216 statistical electromagnetics for antennas 259

general approach and surrogate modeling 264–6 Monte Carlo (MC) simulations 266–7 polynomial chaos expansion 267–71 split ring resonator (case study) 271 numerical model and PCE analysis 272–8 overview 271–2 variable antennas, state of the art of 262–4 wearable textile antenna (case study) 278 manufacturing process, description of 278–9 numerical model and PCE analysis 279–81 stored energy of radiating systems 163–5 subreflector, placement of 370 substrate-integrated waveguide (SIW) 62 subtractive direct process 140 subtractive lithographic process 139 surface-enhanced Raman scattering 143 surface plasmon polaritons (SPPs) 132–3, 146 surrogate models 266 SVD algorithm 216, 223–4, 237 SVD-based CBFM 248–51 switched-beam reflectarray antenna 331 Tellegen relations 14 terahertz and optical frequencies, reflectarrays at 347–8 terahertz antennas, metasurfaces and planar devices using graphene 1 2D materials in framework of Maxwell’s equations 3–6 fixed frequency plasmonic antennas 6–9 frequency-reconfigurable plasmonic antennas 9–11 graphene 2 graphene plasmonic antenna model 11–13

Index planar plasmonic antennas 6 terahertz beam steering reflectarray prototype 27 design and measurement 29–32 working principle 28–9 terahertz non-reciprocal isolator 22 isolator working principle 23–4 measurements 24–7 tunable and non-reciprocal devices, efficiency upper bounds in 13 demonstration of the upper bound 15–17 device specific bounds 19–22 generalized electric and magnetic field representation 14 graphene figure of merit 17–19 terahertz time-domain spectroscopy 77–8 three-beam reflectarray antenna in Kaband 342 3D finite-difference time-domain (FDTD) method 86 3-D printed reflectarrays 346–7 three-pin balun, development of 313–15 THz near-field microscopy 111–20 applications of 117–19 motivation 111 system performance characterization 112–17 THz near-field system setup 111–12 THz photoconductive antennas 73 experimental characterization of PCA component and system 95 far-field THz-TDS 95–110 THz near-field microscopy 111–20 importance of 73 photoconductive antenna (PCA) 75–7 pulsed THz generation 74–5 terahertz time-domain spectroscopy 77–8 theoretical modeling and numerical simulation 78 Drude–Lorentz model 79–81 equivalent circuit model 81–3

411

full-wave model 83–94 motivation and challenge 78–9 THz generation 73–4 THz time-domain spectroscopy (THz-TDS) 73, 77–8 tightly coupled dipole arrays (TCDA) 292, 301, 306 time-dependent gap voltage 82 time-harmonic Maxwells equations 127 TPQDIs 142 transformation optics technique 339 transmission line (TL) model 11 transmittarrays 147 transparent (supersolar) patch antennas 380–1 transparent conductive oxide (TCO) 346 transverse electric (TE) polarization mode 271–2 true-time delay (TTD) phase compensation 337–8 truncated polynomial expansion 268 tunable reflectarray cells 331 2D materials in framework of Maxwell’s equations 3–6 Tx-Rx reflectarray antenna 341 ultra-wideband (UWB) arrays 287 current capabilities 288–9 connected and coupled arrays 291–3 fragmented aperture 290–1 material loading 293–5 tapered slot 289–90 planar UWB arrays 301 cavity resonance 309–12 feed planarization 301–2 material and process selection 302 PCB processing, limitations of 302–3 surface waves 303–9 planar UWB arrays for millimeterwaves 313 development of three-pin balun 313–15

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Developments in antenna analysis and design, volume 2

sample design for 5G frequencies 316–19 UWB TCDA and feed 295 circuit model of balun 297–301 modeling infinite coupled arrays 295–7 UWB textile antenna 263 umbrella reflectors, characterization of 384 gain loss as a function of the number of gores 387–8 mathematical representation of gore surface 385 optimum feed location, finding 385–7 uncertainty quantification (UQ) 261 van derWaals forces 2 variable antennas, state of the art of 262–4 variable-sized printed elements, reflectarrays of 326 Vivaldi bandwidth 289–90 waveguide reflectarray 325 wavelet basis functions 236 waveplates 147 wearable sensors, remote health monitoring system based on 260 wearable textile antenna (case study) 278 manufacturing process, description of 278–9

numerical model and PCE analysis 279–81 wideband MMW complementary source antennas for 5G 62 linearly polarized antenna fed by an SIW 62–3 radiation mechanism of 63–6 simulation and measurement results 66–8 wideband MMW ME dipole antennas 40 differential feed printed ME dipole antenna 47 design of waveguide-todifferentially fed port transition 48–9 measurement results 49–53 operating principle of differentially fed complementary source antenna 48 single feed printed ME dipole antenna 40–2 measurement and results 42–7 operating principles of antenna 42 Wide Swath Ocean Altimeter (WSOA) 339 wire antennas 366 wireless terminal antenna placement using optimum currents 178–81 X-band switching-beam reflectarray antenna 331