Surface Electromagnetics: With Applications in Antenna, Microwave, and Optical Engineering [Hardcover ed.] 1108470262, 9781108470261

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Surface Electromagnetics: With Applications in Antenna, Microwave, and Optical Engineering [Hardcover ed.]
 1108470262, 9781108470261

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Surface Electromagnetics Written by the leading experts in the field, this text provides systematic coverage of the theory, physics, functional designs, and engineering applications of advanced engineered electromagnetic surfaces. All the essential topics are included, from the fundamental theorems of surface electromagnetics to analytical models, general sheet transmission conditions (GSTC), metasurface synthesis, and quasi-periodic analysis. A plethora of examples throughout illustrate the practical applications of surface electromagnetics, including gap waveguides, modulated metasurface antennas, transmit arrays, microwave imaging, cloaking, and orbital angular momentum (OAM) beam generation, allowing readers to develop their own surface electromagneticsbased devices and systems. Enabling a fully comprehensive understanding of surface electromagnetics, this is an invaluable text for researchers, practicing engineers, and students working in electromagnetics, antennas, metasurfaces, and optics. Fan Yang is Professor in the Department of Electronic Engineering and Director of the Microwave and Antenna Institute at Tsinghua University. He is the coauthor of Electromagnetic Band Gap Structures in Antenna Engineering (Cambridge, 2008) and Reflectarray Antennas: Theory Designs and Applications (2018) and is a Fellow of the IEEE and ACES. Yahya Rahmat-Samii is Distinguished Professor and the holder of the Northrop Grumman Chair in Electromagnetics in the Department of Electrical and Computer Engineering at the University of California, Los Angeles. He is the coauthor of Electromagnetic Band Gap Structures in Antenna Engineering (Cambridge, 2008); a Fellow of the IEEE, AMTA, URSI, EMA, and ACES; and a member of the US National Academy of Engineering.

Surface Electromagnetics With Applications in Antenna, Microwave, and Optical Engineering Edited by

FA N YA N G Tsinghua University, Beijing

YA H YA R A H M AT- S A M I I University of California, Los Angeles

University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 79 Anson Road, #06–04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781108470261 DOI: 10.1017/9781108556477 © Cambridge University Press 2019 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2019 Printed in the United Kingdom by TJ International Ltd. Padstow Cornwall A catalogue record for this publication is available from the British Library. Library of Congress Cataloging-in-Publication Data Names: Yang, Fan, 1975– editor. | Rahmat-Samii, Yahya, editor. Title: Surface electromagnetics : with applications in antenna, microwave, and optical engineering / edited by Fan Yang (Tsinghua University, Beijing), Yahya Rahmat-Samii (University of California, Los Angeles). Description: Cambridge, United Kingdom ; New York, NY : Cambridge University Press, 2019. Identifiers: LCCN 2018051219 | ISBN 9781108470261 (hardback) | ISBN 1108470262 (hardback) Subjects: LCSH: Electromagnetism. | Surfaces (Technology) | Antennas (Electronics)–Design and construction. | Electromagnetic devices. Classification: LCC TK454.4.E5 S87 2019 | DDC 621.3–dc23 LC record available at https://lccn.loc.gov/2018051219 ISBN 978-1-108-47026-1 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet Web sites referred to in this publication and does not guarantee that any content on such Web sites is, or will remain, accurate or appropriate.

We dedicate this book to our family, mentors, and past, present, and future students. May surface electromagnetics enlighten all of us!

Contents

Contributors Acknowledgments 1

Introduction to Surface Electromagnetics

page xvi xix 1

Fan Yang and Yahya Rahmat-Samii

2

1.1 What Is Surface Electromagnetics? 1.2 Development of Electromagnetic Surfaces 1.2.1 Classical Uniform EM Surfaces 1.2.2 Periodic EM Surfaces: Frequency Selective Surface 1.2.3 Periodic EM Surfaces: Soft/Hard Surface and EBG Surface 1.2.4 Recent Progress on Quasi-periodic EM Surfaces 1.3 Importance of Surface Electromagnetics 1.3.1 Surface Equivalence Theorem 1.3.2 Prominent Features of EM Surfaces 1.3.3 Comparisons with Related Sciences and Technologies 1.4 Research Frontiers of Surface Electromagnetics 1.4.1 Surface Electromagnetic Theory 1.4.2 Artificial Surface Designs with Novel Properties 1.4.3 SEM-Based Engineering Applications 1.5 Contents of This Book 1.5.1 Models, Analysis, and Synthesis of EM Surfaces 1.5.2 Guided Wave, Leaky Wave, and Plane Wave Properties of EM Surfaces 1.5.3 Applications of Surface Electromagnetics References

1 3 3 5 7 9 12 13 16 19 20 21 22 22 23 24 24 26 27

Analytical Modeling of Electromagnetic Surfaces

30

Viktar Asadchy, Ana Díaz-Rubio, Do-Hoon Kwon, and Sergei Tretyakov

2.1 Introduction: Definitions, Basic Classification, Main Functionalities of Metasurfaces 2.2 Metasurfaces versus Thin Slabs of Homogeneous Materials and Other Artificial Periodic Surfaces

30 33 ix

x

3

Contents

2.3 Comparison of Possible Functionalities 2.4 Homogenization Models of Metasurfaces 2.4.1 Polarizability-Based Model 2.4.2 Susceptibility-Based Model 2.4.3 Model Based on Equivalent Impedance Matrix 2.5 Bi-anisotropy and Nonreciprocity: Definitions and Enabled Functionalities 2.5.1 Bi-anisotropy 2.5.2 Nonreciprocity 2.5.3 Enabled Functionalities 2.6 Metasurfaces for Shaping Transmitted Fields and Reflected Fields 2.6.1 Control of Wave Propagation Direction in Transmission 2.6.2 Control of Wave Propagation Direction in Reflection 2.6.3 Control of Polarization in Reflection 2.6.4 Control of Multiple Waves in Reflection References

36 37 40 42 43 46 46 49 52 54 54 56 57 60 61

Using Generalized Sheet Transition Conditions (GSTCs) in the Analysis of Metasurfaces

66

Christopher L. Holloway and Edward F. Kuester

3.1 3.2 3.3 3.4

Introduction and Definitions of Metasurfaces Metasurfaces versus Frequency-Selective Surfaces Characterization of Metasurfaces: Surface versus Bulk Properties Generalized Sheet Transition Conditions (GSTCs) 3.4.1 GSTCs for a Metafilm 3.4.2 GSTCs for a Metascreen 3.4.3 GSTCs for a Metagrating 3.5 Reflection and Transmission Coefficients 3.5.1 Metafilms 3.5.2 Metascreens 3.5.3 Metagratings 3.6 Determining the Surface Parameters 3.6.1 Retrieval Expressions for Metafilms 3.6.2 Retrieval Expressions for Metascreens 3.6.3 Retrieval Expressions for Metagratings 3.7 Some Applications of GSTCs 3.7.1 Guided Waves on a Single Metasurface 3.7.2 Resonator Size Reduction 3.7.3 Waveguides 3.7.4 Controllable Reflections and Transmissions 3.8 Impedance-Type Boundary Conditions 3.9 Isolated Scatterers and One-Dimensional Arrays 3.10 Summary References

66 68 73 78 78 79 80 81 81 87 90 92 93 95 98 99 99 103 108 110 114 114 115 115

Contents

4

Electromagnetic Metasurface Synthesis, Analysis, and Applications

xi

124

Karim Achouri, Yousef Vahabzadeh, and Christophe Caloz

5

4.1 Introduction 4.2 Mathematical Synthesis 4.2.1 Metasurface Boundary Conditions 4.2.2 Synthesis Procedure 4.3 Numerical Analysis 4.3.1 Metasurface Analysis 4.3.2 Two-Dimensional Finite-Difference Frequency-Domain Method 4.3.3 Two-Dimensional Finite-Difference Time-Domain Method 4.3.4 Finite-Difference Time-Domain Scheme for Dispersive Metasurface 4.3.5 One-Dimensional Analysis of Nonlinear Second-Order Metasurfaces 4.4 Illustrative Examples 4.4.1 Negative Refraction Metasurface 4.4.2 Nongyrotropic Nonreciprocal Metasurface 4.4.3 Time-Varying Half-Wave Absorber 4.5 Practical Realization 4.5.1 Relation with Scattering Parameters 4.5.2 Implementation of the Scattering Particles 4.6 Conclusion 4.7 Conditions of Reciprocity, Passivity, and Loss References

124 125 126 128 133 133

Analysis and Modeling of Quasi-periodic Structures

165

136 138 141 142 144 144 145 146 147 147 152 159 160 161

Maokun Li

5.1 Introduction 5.2 Study of Quasi-periodic Effect 5.2.1 Calculation of Element Reflection Phase 5.2.2 Study of Quasi-periodic Effect 5.2.3 Phase Adjustment in Reflectarray Antennas 5.3 Full-Wave Modeling of Quasi-periodic Structures Using Reduced Basis Method 5.3.1 Formulation 5.3.2 Numerical Results 5.4 Summary and Outlook References

165 168 168 172 174 181 181 189 193 194

xii

6

Contents

Gap Waveguide Technology

198

Eva Rajo-Iglesias, Zvonimir Sipus, and Ashraf Uz Zaman

6.1 Origin of Gap Waveguide Technology 6.2 Approximate Method of Analysis of Parallel-Plate Waveguides Containing EBG Surfaces 6.2.1 Plane Wave Spectral Domain Approach 6.2.2 Homogenization Using Spectral Surface Admittance 6.3 Design of Stop Bands for Parallel Plate Structures 6.4 Application to RF Packaging 6.5 Evaluation of Losses 6.6 Gap Waveguide Antennas 6.6.1 High-Gain Antennas Designed in Ridge Gap Waveguide and Groove Gap Waveguide Geometries 6.6.2 High-Gain Antennas Designed in Inverted Microstrip Gap Waveguide and Printed Gap Waveguide Geometry 6.6.3 Single-Layer Antennas Based on Gap Waveguide Technology 6.6.4 Frequency Scanning Antenna Based on Gap Waveguide Technology 6.7 Conclusion References 7

Modulated Metasurface Antennas

198 203 203 204 208 211 213 214 216 220 222 223 225 226 231

Gabriele Minatti, David González-Ovejero, Enrica Martini, and Stefano Maci

7.1 Introduction 7.2 Adiabatic Floquet Waves for Curvilinear Locally Periodic Boundary Conditions 7.2.1 Constant Average Non-uniform Reactances 7.2.2 Adiabatic Floquet Wave Expansion 7.3 Design of Modulated MTS Antennas 7.3.1 Continuous Reactance Synthesis 7.3.2 Pixel Modeling and Detailed Layout 7.4 Analysis of Modulated Metasurface Antennas 7.4.1 Full-Wave Homogenized Impedance Analysis 7.5 Efficiency and Bandwidth of Modulated Metasurface Antennas 7.5.1 Efficiency of Metasurface Antennas 7.5.2 Bandwidth of Gain 7.6 Examples of Antenna Design 7.6.1 Shaped Beam Antenna 7.6.2 Multibeam Modulated Metasurface Antennas 7.7 Discussion and Future Outlook References

231 234 235 236 238 240 241 245 246 250 251 255 258 258 259 266 268

Contents

8

Transmission Surfaces and Transmitarray Antennas

xiii

272

Fan Yang and Shenheng Xu

9

8.1 Introduction 8.2 Phase Limits of Transmission Surfaces 8.2.1 Phase Limit of a Single-Layer Transmission Surface 8.2.2 Phase Limits of Multilayer Transmission Surfaces 8.2.3 Discussion on Nonidentical Layers, Wire Coupling, and Cross-Polarization 8.3 Transmission Surface Designs 8.3.1 A Quad-Layer Transmission Surface Using E-Shaped Elements 8.3.2 A Double-Layer Transmission Surface Using Malta-Cross Elements with Vias 8.3.3 A Single-Layer Transmission Surface Using Cross-Polarized Fields 8.3.4 Other Transmission Surface Designs 8.4 Reconfigurable Transmission Surface Designs 8.4.1 FSS-Type Reconfigurable Design 8.4.2 R/T-Type Reconfigurable Design 8.5 Transmitarray Antennas 8.5.1 Concept and Design Procedure of Transmitarray Antennas 8.5.2 A Transmitarray Design Example References

272 274 274 276

Coding and Programmable Metasurfaces

301

279 280 280 283 284 287 287 288 290 293 293 296 298

Shuo Liu and Tie Jun Cui

10

9.1 Introduction 9.2 Coding Metasurfaces and Their Controls to EM Waves 9.3 Programmable Metasurfaces and Imaging Applications 9.3.1 Programmable Metasurface 9.3.2 Programmable Metasurface under Point Source Excitation 9.3.3 Transmission-Type Programmable Metasurface for Imaging Applications 9.4 Summary and Outlook References

301 304 307 307 311

Metamaterial and Metasurface Cloaking: Principles and Applications

325

315 319 320

Giuseppe Labate, Ladislau Matekovits, and Andrea Alù

10.1 Introduction 10.2 Non-uniqueness of the Scattering Problem: Non-radiating Sources and Cloaking Devices

325 325

xiv

Contents

10.3 Scattering Theory: Harmonic Field Series and Field Integral Equation Representations 10.4 Harmonic Field Series Representation: Cloaking Design with Mie Solutions 10.4.1 Plasmonic Cloaking: A Volumetric Metamaterial Coating 10.4.2 Mantle Cloaking: A Thin Metasurface Coating 10.4.3 Parity-Time Symmetry Cloaking: A Balanced Loss-Gain Coating 10.5 Field Integral Equation Representation: Cloaking Design with Non-radiating Sources 10.5.1 The Strong Solution: Impedance Matching and Transformation Optics 10.5.2 The Weak Solution: Kirchhoff’s Current Law and General Scattering Cancellation 10.6 Bounds on Cloaking: Causality, Passivity, Time Invariance, Linearity 10.6.1 Directionality Issue 10.6.2 Bandwidth Issue 10.7 Conclusions References 11

Orbital Angular Momentum Beam Generation Using Textured Surfaces

328 329 331 333 335 339 340 344 349 351 353 355 357 363

Mehdi Veysi, Caner Guclu, Filippo Capolino, and Yahya Rahmat-Samii

11.1 OAM Beams: Concept and Historical Background 11.1.1 Bessel–Gaussian Beams 11.1.2 Laguerre–Gaussian (Helical) Beams 11.2 Near-Field Applications of OAM Beams 11.2.1 Generating Cylindrical Vector Beams 11.2.2 Increasing Channel Capacity of Wireless Communication Systems 11.3 Potential Far-Field Applications of OAM Beams 11.4 Far-Field Characteristics of OAM Beams 11.4.1 Bessel–Gaussian Beams 11.4.2 Laguerre–Gaussian (Helical) Beams 11.5 OAM Beam Generation Using Reflectarray Antennas 11.5.1 Rotational Phase Control Principle 11.5.2 Double Split-Ring Element 11.6 Reflectarrays with Cone-Shaped Patterns 11.6.1 Bessel-Beam Reflectarray 11.6.2 Helical (Laguerre–Gaussian) Beam Reflectarray 11.7 Reflectarrays Radiating Multiple Azimuthally Distributed Pencil Beams 11.8 Conclusions and Observations References

363 364 365 366 366 367 368 369 369 371 372 372 374 376 376 381 383 386 387

Contents

12

Applications of Metasurfaces in the Microwave, Terahertz, and Optical Regimes

xv

393

Daniel Binion, Lei Kang, Zhi Hao Jiang, Shengyuan Chang, Xingjie Ni, and Douglas H. Werner

Appendix

12.1 Applications of Metasurfaces in the Microwave and Millimeter-Wave Regimes 12.1.1 Ultrathin Electromagnetic Absorbers 12.1.2 Polarization Control Surfaces 12.1.3 Artificial Grounds for Low-Profile Antennas 12.1.4 Antenna Superstrates and Coatings 12.1.5 Modulated Metasurfaces for Leaky Wave Radiation 12.1.6 Scattering Signature Control 12.1.7 Reflect-/Transmit-Arrays 12.2 Applications of Metasurfaces in the Terahertz Regime 12.2.1 History of Terahertz Metasurfaces 12.2.2 Recent Developments in Terahertz Metamaterial Technology 12.2.3 Future Developments 12.3 Applications of Metasurfaces in the Optical Regime 12.3.1 Generalized Snell’s Law 12.3.2 Metalenses 12.3.3 OAM Beam Generation 12.3.4 Holography 12.3.5 Optical Invisibility Cloak 12.4 Conclusion References

393 394 394 396 398 399 400 402 403 403 406 417 418 418 420 421 422 423 424 424

Representative Literature Review on Surface Electromagnetics

438

Fan Yang, Yahya Rahmat-Samii, Xibi Chen, Xingliang Zhang, and Hongjing Xu

Index Color plate section to be found between pages 242 and 243

466

Contributors

Karim Achouri École Polytechnique Fédérale de Lausanne Andrea Alù City University of New York Viktar Asadchy Aalto University Daniel Binion The Pennsylvania State University Christophe Caloz Polytechnique Montréal Filippo Capolino University of California, Irvine Shengyuan Chang The Pennsylvania State University Xibi Chen Tsinghua University Tie Jun Cui Southeast University Ana Díaz-Rubio Aalto University David González-Ovejero Université de Rennes I

xvi

Contributors

Caner Guclu University of California, Irvine Christopher L. Holloway National Institute of Standards and Technology Zhi Hao Jiang Southeast University Lei Kang The Pennsylvania State University Edward F. Kuester University of Colorado Do-Hoon Kwon University of Massachusetts, Amherst Guiseppe Labate Politecnico de Torino Maokun Li Tsinghua University Shuo Liu Southeast University Stefano Maci University of Siena Enrica Martini University of Siena Ladislau Matekovits Politecnico de Torino Gabriele Minatti Wave Up S.r.l Xingjie Ni The Pennsylvania State University Yahya Rahmat-Samii University of California, Los Angeles

xvii

xviii

Contributors

Eva Rajo-Iglesias Carlos III University of Madrid Zvonimir Sipus University of Zagreb Sergei A. Tretyakov Aalto University Yousef Vahabzadeh Polytechnique Montréal Mehdi Veysi University of California, Irvine Douglas Werner The Pennsylvania State University Hongjing Xu Tsinghua University Shenheng Xu Tsinghua University Fan Yang Tsinghua University Ashraf Uz Zaman Chalmers University of Technology Xingliang Zhang Tsinghua University

Acknowledgments

The editors would like to sincerely thank the chapter contributors for their timely perpetration of their chapters and for supporting us throughout the development of this book. The editors would also like to acknowledge the support they received from their organizations during the preparation of this book. Cambridge University Press staff were most patient and collaborative in helping us to publish an outstanding book that we strongly believe will serve the scientific and engineering community for many years to come. Particularly, David Liu, Sarah Strange, and Jade Taylor-Salazar from Cambridge University Press are thanked for their almost daily interaction with us to finalize the book. In addition, the editors would like to thank Dr. Xiao Liu from Tsinghua University for his help in rendering the illustrations in Chapter 1.

xix

1

Introduction to Surface Electromagnetics Fan Yang and Yahya Rahmat-Samii

1.1

What Is Surface Electromagnetics? The essence of a thing is hidden in its interior, while the (misleading) sensate qualities are caused by the surface. Democritus, 460–365 BC

Most of the time, people start to comprehend the essence of a material from its outside appearance, i.e., the surface features. Surfaces, surface phenomena, and surface processes widely exist in nature and our daily lives. In general, surface science and surface engineering explore these phenomena and processes and utilize the acquired knowledge to improve the lives of human beings [1–3]. For example, when people discovered the surface tension effect from water beading on a leaf, waterproof cloth was invented accordingly. In chemistry, it is critical to understand chemical reactions on surfaces, either for desirable reactions, such as in heterogeneous catalysis, or undesirable ones, such as in corrosion chemistry. In modern physics, scientists study the specific arrangement of atoms on the surface layer of matter through advanced experimental equipment such as X-ray photoelectron spectroscopy and scanning tunneling microscopy, which help us to see and to change the micro-world. Novel surface-type materials, such as graphene [4], have been invented recently and have attracted growing interest in science and engineering communities. In brief, surface science and engineering have broad impacts on modern industries such as oil, metallurgy, microelectronics, lubrication, and adhesion. They are also closely related to human health because of their impacts on biology and pharmaceuticals. This book focuses on “surface electromagnetics” (SEM), a subdiscipline in electromagnetics with an abundance of specialized and exciting knowledge yet to be explored. As we know, electromagnetics is a fundamental science discipline that describes the temporal and spatial behaviors of electric and magnetic fields [5]. From the temporal viewpoint, electromagnetics is usually classified into different categories according to the field’s oscillation frequency, such as direct current (DC), radio frequency (RF), microwaves, terahertz (THz), optics, X-rays, and beyond. This begs the question of how we categorize electromagnetics from the spatial viewpoint. To answer this, serious and in-depth contemplation is required.

1

2

Introduction to Surface Electromagnetics

Figure 1.1 Classifications of electromagnetic phenomena in space domain based on the

three-dimensional electric sizes. A black and white version of this figure will appear in some formats. For the color version, please refer to the plate section.

Figure 1.1 illustrates our preliminary contemplation on this question. As a unified and fundamental theorem, Maxwell’s equations are used to describe all macroscopic electromagnetic phenomena, where three-dimensional spatial variations are considered. Thus, the spatial dimensions of the field oscillation can be used as a measure to categorize different electromagnetic (EM) phenomena. For each group of electromagnetic phenomena, corresponding theorems have been developed based on Maxwell’s equations. •





General 3-D EM phenomena. When field variations are comparable to wavelength in all three dimensions (Lx , Ly , Lz ~ λ), it belongs to the three dimensional (3-D) phenomena. General electromagnetic theory has experienced a long time of development to deal with these 3-D phenomena. Due to the complexity, analytical solutions exist only for several canonical geometries. Recently, with the progress of computing capability, people are able to solve Maxwell’s equations numerically [6] in complex media with certain (ε, μ), general shapes, and various boundary conditions. Of course, computing time and memory usage are still limiting factors. 0-D EM phenomena. When the spacial variations of a device or an EM phenomenon are much smaller than wavelength in all three dimensions (Lx , Ly , Lz ! λ), it belongs to the zero-dimensional (0-D) phenomena. Circuit theory is proved to be an accurate and efficient approach for 0-D problems, which is a good simplification of Maxwell’s equations. Resistor (R), inductor (L), and capacitor (C) are typical components in electronic circuits with lumped voltage and current sources [7]. 1-D EM phenomena. In microwave circuits and optical waveguides, their transverse dimensions are much smaller than wavelength, but the longitudinal

1.2 Development of Electromagnetic Surfaces



3

dimension is comparable to wavelength (Lx , Ly ! λ, Lz ~ λ). Thus they are categorized as one-dimensional (1-D) phenomena. Circuit theory is no longer valid, while general EM theory is relatively complex for the analysis. Hence, transmission line theory, which has become a fundamental pillar stone of microwave engineering, was developed [8]. The characteristic impedance (Z0 ) and the propagation constant (β) are critical parameters to describe these onedimensional (1-D) EM phenomena. 2-D EM phenomena. Reviewing Figure 1.1, it is interesting to note that surface electromagnetic theory, which deals with two-dimensional (2-D) EM phenomena (Lx , Ly ~ λ, Lz « λ), has not been fully developed yet. The eigen-parameters of general surfaces need to be defined, and the simplified theorems from Maxwell’s equations need to be derived. Furthermore, application of appropriate SEM theorems to analyze and design advanced EM surfaces is still challenging.

By focusing on the 2-D EM phenomena, this book provides a comprehensive presentation of surface electromagnetics by editing the most state-of-the-art concepts, physics, engineering designs, and exciting applications. It covers the fundamental theorems of SEM, including the analytical surface models, general sheet transmission condition (GSTC), and efficient numerical algorithms. A plethora of applications are presented in this book to demonstrate the broad impact of SEM, including gap waveguides, leaky wave antennas, transmitarrays, orbital angular momentum (OAM) beam generation, and mantle cloaking. These SEM-based applications span widely from microwave to terahertz to optical spectrums.

1.2

Development of Electromagnetic Surfaces Surface electromagnetics is not a completely new topic. Instead, it has a rich history of fascinating discoveries. For example, light bends at the interface between air and water, and light focused by a parabolic reflector can ignite a paper [9]. On the other hand, surface electromagnetics is also an exciting frontier full of new breakthroughs. Anomalous reflection and refraction have been recently discovered [10], and metalenses of nano-thickness have been successfully designed [11], which opens up a whole new way to design optical devices. This section provides a historical review on the development of electromagnetic surfaces and interesting phenomena associated with them. It is expected that this history will stimulate and inspire new ideas in the future.

1.2.1

Classical Uniform EM Surfaces It might be argued that the earliest electromagnetic surfaces people encountered were the natural surfaces that resided between two different media, for example, the surface of earth, an interface between air and water. Reflection occurs on these surfaces, and refraction also occurs when the second medium is a dielectric. Usually, the reflection

4

Introduction to Surface Electromagnetics

(b)

(a)

Figure 1.2 Reflection and refraction on a uniform surface: (a) an interface between air and a

conductor; (b) an interface between air and a dielectric.

angle equals to the incident angle, which agrees with people’s intuition. Meanwhile, the refraction angle is different from the incident angle (Figure 1.2b), which has fascinated and puzzled people for years. In 1621, Dutch astronomer Willebrord Snellius (Snell) derived a mathematical formula for the incident, reflection, and refraction angles [12]: ?

0 μ0 sin θi =

? ? 0 μ0 sin θr = d μd sin θt

(1.1)

This equation is well known as Snell’s law, which provides a quantitative relationship of the three angles. It agrees well with measurement results and has been widely used to design electromagnetic devices or to measure the properties of an unknown medium. Based on this equation, many interesting phenomena have been discovered, such as the critical angle for total reflection and Brewster’s angle for total transmission. Optical fiber, a fundamental device in the modern communication world, is designed based on the total reflection phenomenon. It is important to point out that these natural surfaces are uniform surfaces. Along the normal direction (n), medium properties change. Along the tangential directions (t1 , t2 ), the surface property does not change. Hence, the following relation exists: B B B = 0, =0 ‰ 0, Bn Bt1 Bt2

(1.2)

Besides planar surfaces, curved surfaces have also been studied. Cylindrical surfaces, spherical surfaces, and many other canonical surfaces have been studied comprehensively. They have been broadly used to design electromagnetic devices. For example, a

1.2 Development of Electromagnetic Surfaces

5

parabolic metallic surface has been used to design reflector antennas, both for ground and space applications. A lens antenna is designed using a certain dielectric material with a combination of two curved surfaces. Furthermore, coating metallic ground with a layer of dielectric substrate is a popular technique in the electromagnetic community. Impedance boundary conditions (IBC) have been proposed to characterize these hybrid surfaces. Various analytical and numerical methods have been developed to analyze these surfaces [13]. Although the surfaces are uniform, this research area is enriched tremendously when considering surface curvatures, surface combinations, dielectric properties, frequency responses, incident angles, and incident polarizations.

1.2.2

Periodic EM Surfaces: Frequency Selective Surface A giant step in surface electromagnetics was the invention of frequency selective surface (FSS) [14]. In the optics community, it is also known as optical grating. The demand for FSS originated in the early 1950s with the development of radar technology, which was to find a radar absorbing material with a narrow-band frequency window for antennas to operate. This demand has propelled the FSS field of study, resulting in many interesting artificial surfaces. In contrast to conventional uniform surfaces, these artificial surfaces now have variations along the tangential directions,   B B B , ‰ 0, ‰0 (1.3) Bn Bt1 Bt2 In particular, these variations are periodic, either one-dimensional or two-dimensional. Figure 1.3 shows two typical geometries of frequency selective surfaces. Basically, an FSS consists of an array of identical unit cells arranged in a periodic lattice. Various scatterers have been used as the unit cells, and they can be classified into two main categories: dipole-type FSS and slot-type FSS. In the former group, conductive scatterers are arranged in a plane, and they are not connected with each other. In the latter group, periodic slots are perforated on a conductive sheet.

Figure 1.3 Frequency selective surfaces (FSS) and the band pass / band stop features.

6

Introduction to Surface Electromagnetics

The basic function of the FSS, as its name suggests, is to provide a certain frequency response for a plane wave to transmit through the surface. To be specific, the wave can pass through the surface in a certain frequency band and is rejected at other frequencies, or vice versa. As shown in Figure 1.3, a dipole-type FSS usually provides a band stop feature, while a slot-type FSS gives a band pass feature. Therefore, an FSS is also known as a spatial filter. The FSS properties can be studied from both the circuit viewpoint and the field viewpoint. From the circuit viewpoint, it is analogous to an LC filter, and the resonant curve is of particular interest. The resonant frequency and the operation bandwidth are the most important features. Furthermore, wideband and multiband operations are always demanded. Meanwhile, unlike an LC filter that deals with a voltage or a current, an FSS deals with a plane wave. Hence, the field viewpoint is also critical. For example, angular stability and polarization stability are important considerations for FSS designs, as well as the cross-polarization levels. Various FSSs have been developed to fulfill diversified design requirements. Different shapes have been studied, from simple cross dipoles, loops, and circular disks to advanced Jerusalem crosses and slotted patches and to their combinations. Dielectric loading is also found to be an effective approach, not only to provide mechanical support for scatterers but also to minimize their sizes. Multilayer FSSs are popularly designed, because they can cascade transmission zeroes and poles to shape the resonant curves, such as a flat top or a fast roll off. Furthermore, active FSSs have also been developed for reconfigurable frequency, beam control, grid oscillation and mixers, and spatial power combination [15]. To characterize the performance of these FSSs, various analysis methods and measurement techniques have been developed. For the analysis methods, the equivalent circuit approach provides a simple and straightforward understanding of the FSS’s operating mechanism, while the mode-matching approach based on the Floquet theory provides an accurate result [16]. More recently, full-wave numerical techniques, such as the periodic method of moments (MoM) [17], the finite difference time domain (FDTD) method with periodic boundary conditions (PBC) [18], and the finite element method (FEM) [19], have also been used for FSS analysis with general applicability and high accuracy. Nowadays, frequency-selective surfaces have been widely used in many civilian and defense applications. For example, it is used to design the door of a microwave oven, where the microwave cannot penetrate through but light can pass through. In contrast, the reverse function of the FSS is used to design a sun shield in a spacecraft, where a microwave signal can pass through for communication purpose, but light is reflected back to avoid overheating. FSSs are also used in a radome design for airplanes or a subreflector design for radio telescopes. In optics, it is a critical component in a laser source. As a summary of this FSS section, it is worthwhile to emphasize that electromagnetic surfaces have evolved from a uniform surface to a periodic surface. An interesting analog is the evolution in circuit theory: from a direct current (DC) circuit to an alternating current (AC) circuit, which had a profound impact in circuits. The switch from DC to

1.2 Development of Electromagnetic Surfaces

7

AC is an evolution in the time domain, while the switch from a uniform surface to a periodic surface is an evolution in the space domain. This switch also has a profound impact in surface electromagnetics.

1.2.3

Periodic EM Surfaces: Soft/Hard Surface and EBG Surface It is an important milestone in surface electromagnetics that a surface evolves from uniform to periodic. Besides the FSS designs, periodic surfaces have been used to design other functional surfaces, and many exciting phenomena have been discovered. This section will review some representative progress in this aspect. As discussed before, periodic surfaces can be studied from both the circuit viewpoint and the field viewpoint. From the circuit viewpoint, FSS research mainly focuses on the resonant curve of the transmission magnitude. The transmission coefficient is usually a complex number with both the magnitude and the phase. When people explore the phase curve of a surface, interesting observations can be made such as an artificial magnetic conductor (AMC). From the field viewpoint, FSS research is limited within the plane wave incidence. When the incident wave is a surface wave propagating along the surface, novel surfaces and the corresponding features are discovered. Here, the soft/hard surfaces and high impedance surfaces are most prominent examples. Figure 1.4a shows the geometry of a typical soft/hard surface. Basically, it is a corrugated surface with periodic metal walls on a ground plane [19,20]. The height of the corrugation wall is usually around quarter wavelength. When a surface wave propagates along the x direction, a low TE surface impedance and a high TM surface impedance are obtained, which represents a soft operation that stops the wave propagation. When a surface wave propagates along the y direction, a high TE surface impedance and a low TM surface impedance are obtained, which indicates a hard operation that allows the wave to propagate. The above corrugated surface is periodic along the x direction but uniform along the y direction. Later on, a mushroom-like surface was proposed in [21], which is periodic in both x and y directions. It provides a high surface impedance for both x- and y-directed TM surface waves, which are usually the fundamental and lowest surface wave mode on a ground. Hence, it is also called a high impedance surface that stops the surface wave propagation. A dispersion diagram of surface wave modes is plotted in Figure 1.5a. It is observed that a distinct band gap exists where no surface wave can propagate on the surface regardless of its direction. Therefore, this type of surface is named an electromagnetic band gap surface [22], which resembles the energy band gap in semiconductors. The EBG surface exhibits many interesting and unconventional features that do not occur or may not be readily available in nature. Thus, together with double-negative composites, it was put under the broad terminology of “metamaterials” [23]. No matter what we call them, we are more interested in their features. Besides the surface wave band gap, another exciting feature of the EBG surface is the in-phase reflection property. This time, we look back at the plane wave response of the EBG surface. However, instead of the magnitude curve in FSS, now we focus on the phase response of the

8

Introduction to Surface Electromagnetics

(a)

(b)

Figure 1.4 Geometries of (a) a soft and hard surface and (b) an electromagnetic band gap (EBG)

surface.

surface. Figure 1.5b shows the reflection phase variation of an EBG surface versus frequency. At low frequency and high frequency, the reflection phase of the EBG surface is similar to a perfect electric conductor (PEC). At the center frequency, the reflection phase of the EBG surface is 0 degrees, which is the same as for a perfect magnetic conductor (PMC) that does not exist in nature. Since the EBG surface can mimic a nonexistent PMC surface, it is also called an artificial magnetic conductor (AMC) in some applications. There is an abundance of exciting research going on when the surface evolves from uniform to periodic. There are multitude dimensions that the research can dig into. For example, besides the magnitude and phase, polarization is another important measure of an electromagnetic wave. A lot of interesting surfaces have been designed to control the wave polarization, such as from horizontal polarization to vertical polarization or from linear polarization to circular polarization [22]. A common feature of these designs is the thinness, which is usually much smaller than the wavelength. This active surface research has led to many new device and system designs. For example, the soft/hard surfaces have been used in waveguide designs to support TEM wave propagation. They are also used in horn antennas to obtain a symmetric beam in E- and H-planes. The EBG surfaces have been used to reduce mutual coupling between array elements and in the IC packaging. They have also been used as ground planes of wire antennas to achieve a low profile configuration. Polarizer grids are also widely

1.2 Development of Electromagnetic Surfaces

9

(a)

(b)

Figure 1.5 Properties of an EBG surface: (a) a frequency band gap for surface waves;

(b) reflection phase varies with frequency.

used in microwave and optic systems to enhance the polarization purity. In summary, periodic surfaces are active and exciting research topics with broad impacts.

1.2.4

Recent Progress on Quasi-periodic EM Surfaces When people are excited in exploring the realm of periodic surfaces and discovering interesting new phenomena, an even larger research tide is coming in. The core concept of this huge tide is quasi-periodic surfaces, which demonstrate more versatile and exceptional capabilities to manipulate electromagnetic waves. A broader area has opened to

10

Introduction to Surface Electromagnetics

the scientists and engineers in the electromagnetic community! A similar breakthrough occurred in materials science, where Dan Shechtman observed unusual diffractograms from certain aluminum-manganese alloys in 1982. In contrast to the conventional periodic crystals, these types of materials are called quasi-periodic crystals or quasi-crystals, for which Shechtman was awarded the Nobel Prize in Chemistry in 2011 [24]. In a quasi-periodic surface, unit cells are arranged in a periodic lattice, and each cell differs from the others with some geometrical variations, such as scatterer size or orientation angle. Figure 1.6 shows two representative quasi-periodic surfaces developed by Prof. Fan Yang’s group: one in the microwave band and the other in the optic region. Although the frequencies are vastly different, the operation principle behind them is identical. Figure 1.6a shows a reflectarray surface operating at 32 GHz, which consists of 9,636 square patches of varying sizes. The reflection phase changes with the patch size, similar to that in Figure 1.5b. When the patches are arranged to realize a Fresnel zone–type phase distribution, the surface can effectively focus the energy from a feed horn to generate a radiation beam with 43 dB high gain. Figure 1.6b illustrates an ultrathin (240-nm thickness) beam splitter operating at red light (632 nm), which consists of 640,000 circular disks of two diameters: 110 nm and 200 nm, respectively. Two columns of each type of disk are alternatively arranged, forming a phase distribution with a 180˝ phase variation. As a result, an incident red light will be split into two red lights upon the reflection from this ultrathin quasi-periodic surface. From these examples, we can briefly summarize the operation principle of the quasiperiodic surfaces. First, each cell manipulates the incident wave locally and individually. The wave control and design method are similar to those in periodic surfaces discussed in the previous two sections. Next, the spatial distributions of these wave manipulations work together to change the wavefront of the incident wave, leading to extraordinary electromagnetic responses. The above operation principle of quasi-periodic surfaces has enabled fascinating phenomena and designs that cannot be achieved conventionally, both in optics and in microwaves. An important breakthrough in optics is the generalized law of reflection and refraction [10]. By adding a gradient phase distribution of the surface into the conventional Snell’s law, electromagnetic waves can now be reflected and refracted to any anomalous angle. Further research on the engineered phase gradient on the surface has allowed a negative-angle refraction in a broad wavelength range [25]. When this phase gradient is anisotropic, polarization conversion is observed together with the anomalous refraction at a subwavelength thickness [26]. Furthermore, if the phase gradient value is large enough, the surface can act as a bridge linking a plane wave with a surface wave [27]. Moreover, with careful design of surface phase distribution, various planar optical lenses for focusing and waveplates have been successfully designed [28,29], as have high-resolution holograms [30–34] and vortex beams [35,36]. By manipulating the phase of a propagating wave locally, metasurfaces were elaborately designed to function as an ultrathin invisibility skin cloak for visible light [37,38]. In the microwave region, quasi-periodic surfaces progress even further, since there are vigorous demands from communications and radars, and the fabrication technique is also readily available, which has led to practical devices and systems. The most prominent examples are reflectarray antennas (RA) [39] and transmitarray antennas

1.2 Development of Electromagnetic Surfaces

11

(a)

(b)

Figure 1.6 Prototypes of quasi-periodic surfaces: (a) a Ka-band reflectarray surface consisting of

square patches of varying sizes; (b) an optic beam splitter surface consisting of circular disks of different diameters. A black and white version of this figure will appear in some formats. For the color version, please refer to the plate section.

(TA) [41]. Using a quasi-periodic surface with a certain phase distribution, a focused beam with a high gain can be realized by a thin planar surface instead of a curved parabolic surface or a heavy bulky dielectric lens. Various broadband RA, multiband RA, shaped-beam RA, and beam-scanning RA have been designed and tested [41]. It is also worthwhile to point out that a new type of antenna, the modulated surface wave

12

Introduction to Surface Electromagnetics

Figure 1.7 A historical review on the development of surface electromagnetics (SEM): from

uniform surfaces to periodic surfaces and then to quasi-periodic surfaces. This evolution in space variation is analogous to the study of signals in circuits, from DC signals to AC signals and then to modulated signals. A black and white version of this figure will appear in some formats. For the color version, please refer to the plate section.

antenna, has recently been proposed [42]. It is truly low profile, since the feed structure is also integrated into the surface. By modulating the surface wave impedance over the quasi-periodic surface, the radiation beam direction, wave polarization, and antenna gain can be readily controlled. To conclude this section on surface electromagnetic (SEM) development, Figure 1.7 provides a brief historical review. SEM starts with uniform surfaces, and a milestone is Snell’s law, which was formulated in the 1600s. The demand for periodic surfaces originated in the 1950s, and various periodic surfaces, including frequency-selective surfaces, soft/hard surfaces, and electromagnetic band gap surfaces, have been proposed with exciting features. Around the 2000s, the tide for quasi-periodic surfaces arrived, and surface electromagnetics (SEM) entered a new era full of great challenges and grand opportunities. It is also interesting to compare the evolution of surface properties to the evolution of signals in circuits. The electromagnetic surface evolved from uniform to periodic, and then to quasi-periodic, which occurs in the space domain. The circuit signals also evolved from direct current (DC) signals to alternating current (AC) signals, and to modulated signals (also quasi-periodic), which happens in the time domain. Having witnessed the prosperous and profound research in the circuit signals, we believe in and are expecting a similarly promising research progression in surface electromagnetics.

1.3

Importance of Surface Electromagnetics As a subcategory of electromagnetics, surface electromagnetics also follows Maxwell’s equations, where a tremendous amount of knowledge is hidden:

1.3 Importance of Surface Electromagnetics

13

Ñ Ñ

Ñ

BB ∇ ˆ E = ´M ´ Bt

(1.4)

Ñ Ñ

Ñ

BD ∇ ˆH = J + Bt

(1.5)

Ñ

∇¨D =ρ

(1.6)

Ñ

∇ ¨ B = ρm

(1.7)

Together with Maxwell’s equations, there are constitutive relations that describe material properties and boundary conditions that describe the field discontinuity across a surface: Ñ

Ñ

Ñ

Ñ

D = εE

B = μH  Ñ Ñ Ñ nˆ ˆ E2 ´ E1 = ´MS   Ñ Ñ Ñ nˆ ˆ H2 ´ H1 = J S 

(1.8) (1.9) (1.10) (1.11)

In particular, surface electromagnetics focuses on the study of boundary conditions. Meanwhile, some other research, such as on metamaterials, focuses on the constitutive relations, namely, the tensor forms of the equivalent permittivity and permeability.

1.3.1

Surface Equivalence Theorem A very important and fundamental theorem in electromagnetics is the surface equivalence theorem, which can be regarded as the physical foundation of surface electromagnetics. The basic idea of the surface equivalence theorem originates from Huygens’ principle, which states [43] that “each point on a primary wavefront can be considered to be a new source of a secondary spherical wave, and that a secondary wavefront can be constructed as the envelope of these secondary spherical waves.” It is also based on the uniqueness theorem, and a particular situation is stated as follows [44]: “if the tangential electric and magnetic fields are completely known over a closed surface, the fields in the source-free region can be determined.” Figure 1.8 illustrates the implementation of the surface equivalence theorem. Figure 1.8a shows the original and actual problem: the electromagnetic fields in space are radiated from an electric current source and/or a magnetic current source. These sources can be an impressed current like that on an antenna, or an induced source like that on a scattering object; they can be an actual source or an equivalent source. An imaginary surface S is introduced to enclose all current sources. Figure 1.8b shows an equivalent problem that has the same electromagnetic fields outside the surface S but arbitrary fields inside the surface S. To satisfy the boundary conditions, suitable electrical surface currents and magnetic surface currents need to be introduced on the surface S. For the majority of applications, the fields outside of S are of interest, the

14

Introduction to Surface Electromagnetics

(b)

(a)

Figure 1.8 Surface equivalence theorem: (a) the actual problem, (b) the equivalent problem with

suitable surface currents that satisfy the boundary conditions. The two problems share the same electromagnetic fields outside the surface S.

fields inside of S are selected based on practical problems, and the key point is to determine and construct the equivalent surface currents on S. There are different variations of the surface equivalence theorem, where the internal fields are set up differently. For example, a simple setup is to assume the fields within S equal to zero, which is known as Love’s equivalence principle [44]. Furthermore, the properties of the medium within S are changed under the zero field assumption; for example, it is replaced by a perfect electric conductor (PEC). The electric current density is short-circuited by the PEC, hence the equivalent problem becomes the radiation of a magnetic current density in the presence of the PEC that will generate the same fields outside S. This is an effective approach to analyzing certain problems, such as radiation from a waveguide aperture mounted on an infinite flat electric ground plane. A duality of the PEC setup is to replace the medium within S by a perfect magnetic conductor (PMC). A more general idea of the surface equivalence theorem could be expressed as follows: • • •

The internal fields are the fields generated by a group of sources A. The external fields are the desired fields generated by another group of sources B. Suitable surface currents are constructed on S that satisfy the boundary conditions and transform the internal fields to external fields.

Hence, the surface is an engineered surface with field transformation capabilities. Analyses, designs, and applications of these artificial engineered surfaces are main research efforts in modern surface electromagnetics. The surface equivalence theorem has a profound impact in electromagnetics which leads to many practical applications – both in electromagnetic computation and measurement. Figure 1.9a shows a computation model in the finite difference time domain (FDTD) method [18], where an imaginary closed surface is introduced to calculate the

1.3 Importance of Surface Electromagnetics

(a)

(b)

Figure 1.9 Applications of surface equivalence theorem: (a) a computation model in the FDTD

method; (b) a bipolar near-field chamber at UCLA’s antenna lab.

15

16

Introduction to Surface Electromagnetics

far-field radiation properties. Usually, the far-field radiation can be directly calculated from the electric and magnetic currents on an antenna and/or a scattering object. However, for a general program that is capable of analyzing arbitrary shapes and materials, a feasible technique is to construct a common surface to enclose these objects. Based on the surface equivalence theorem, if the tangential fields on this common surface are obtained, the far-field radiation can be determined accordingly. This technique avoids the complexity arising from dealing with different geometries of specific antennas or scatterers and becomes a standard procedure in FDTD-based program and software. Figure 1.9b shows another application of the surface equivalence theorem which is the antenna measurement in a bipolar near-field chamber at Prof. Yahya RahmatSamii’s antenna lab at the University of California, Los Angeles (UCLA) [45]. In highgain antenna measurement, it is usually required to put a receiving antenna far from the antenna undergoing testing (AUT) in order to satisfy the far-field condition. The cost to build an anechoic chamber with a large enough distance becomes prohibitively high, hence the near-field measurement technique is proposed based on the surface equivalence principle. The electric and/or magnetic fields are measured by a probe moving along a certain surface, which can be a planar surface, a cylindrical surface, or a spherical surface. The far-field radiation properties can then be determined from the near-field measured data. The near-field chamber is much smaller and more economic than the far-field chamber. Furthermore, it can collect more information than far-field measurements, and an antenna can be effectively diagnosed using the near-field holographic technique. Nowadays, various near-field measurement systems have been developed and play an important role in antenna design. The previous two examples demonstrate the applications of the surface equivalence theorem, where the fields over a certain surface are either computed or measured to derive the far-field radiation. No field transformation is acted on the surface yet. In the SEM research and the contents to be presented in this book, various field transformations are witnessed. The abundance of design flexibilities and the versatility of surface functions make surface electromagnetics an exciting, vigorous, and attractive research frontier.

1.3.2

Prominent Features of EM Surfaces As an active research frontier, various terminologies appear in literature about surface electromagnetics. These have been used by different researchers with different electromagnetic features. They are briefly reviewed and categorized here according to their basic functionalities, as shown in Fig. 1.10. •

Focus on magnitude response of a plane wave. –



Frequency selective surface (FSS). FSS acts as a spatial filter, where a plane wave can pass through or be rejected within a certain frequency band. It is a well-developed yet active research topic. Optical grating. Essentially, it is the same as FSS, but the terminology is popularly used in the optics community.

1.3 Importance of Surface Electromagnetics

17

Figure 1.10 Classifications of various SEM terminologies based on their functionalities.





Focus on phase response of a plane wave. –







Absorber. Instead of being reflected or transmitted, energy of a plane wave will be absorbed into a lossy surface. It plays an important role in stealth designs.

Artificial magnetic conductor (AMC). It reflects a plane wave with zerodegree reflection phase, resembling a perfect magnetic conductor (PMC) that is not found in nature. Phase shifting surface (PSS). The function of a PSS is to provide a certain phase shift when a plane wave reflects from or transmits through the surface. The current challenge is to provide a full 360˝ phase shift with minimum loss. Holographic surface. Originating from optics, this surface has a certain phase distribution, which is capable of regenerating a corresponding image in space. In antenna terms, it can generate a specific pattern in space including pencil beams, shaped beams, and contour beams.

Focus on polarization response of a plane wave. –



Polarization grid or polarizer. It is also a spatial filter but depends on polarization instead of frequency. For example, a specific design can allow a horizontally polarized wave to pass through while reflecting a vertically polarized wave. Polarization converter. It converts the polarization of an incident wave upon reflection or transmission. For example, it can convert from a ver-

18

Introduction to Surface Electromagnetics

tically polarized wave to a horizontally polarized wave, from a linearly polarized wave to a circularly polarized wave, or from a left-handed circular polarization to a right-handed circular polarization. •

Focus on surface wave features. –

– –



General terminologies. –









Soft/hard surface. As a 1-D corrugated surface, it allows the surface wave propagation in one direction (hard operation) while suppressing the surface wave propagation in the orthogonal direction (soft operation). High impedance surface (HIS). It provides a high surface impedance for the fundamental TM mode, thus stopping the surface wave’s propagation. Electromagnetic band gap (EBG) surface. A distinct frequency band gap is observed in this surface, where surface waves are prohibited in all directions.

Huygens’ surface. As the name indicates, it manipulates the wavefront of an incident wave, so a secondary wavefront with desired features can be obtained. Metasurface. This name evolves from metamaterials and is considered as two-dimensional equivalence of metamaterials. It is a general term, where any artificial surfaces with unconventional features may be claimed as metasurfaces. Metafilm and metascreen. They are two subclasses of metasurfaces according to their geometrical features. They have complementary geometries, i.e., isolated scatterers in a film and isolated apertures in a screen, respectively. Coding metasurface. It is a digital-type metasurface that manipulates electromagnetic waves through a predesigned coding sequence. Usually, each coding particle is characterized by digital states of “0” or “1.”

Focus on active functions. –



Reconfigurable surface and programmable metasurface. Active control devices such as PIN diodes, MEMS switches, or varactors are integrated into the surface to provide real-time control of the surface properties. Grid arrays. Following the design approaches of optical systems, arrays of amplifiers, oscillators, and mixers are spatially distributed in a planar grid to build a quasi-optic system with greater power and performance in microwave, millimeter wave, and THz frequency ranges.

It is worthwhile to point out that the above discussion is a brief classification based on the fundamental electromagnetic features. Some advanced functions may not be included within this classification. For example, a special surface is proposed in [27] to convert a plane wave to a surface wave. Furthermore, a certain surface may have two or more functionalities discussed above. For example, an EBG surface exhibits both a

1.3 Importance of Surface Electromagnetics

19

surface wave band gap and an AMC-type reflection phase feature. Hence, researchers are reminded to pay more attention to the surface features rather than the surface names. Reviewing all these interesting surfaces and their development, two prominent features are summarized. The first one is to discretize the surface from a continuous surface to a discrete array with the goal of enabling localized and distributed wave control. According to Nyquist’s sampling principle, a continuous function can be discretized into a series of data with a proper sampling rate, and the information integrity is maintained. In surface electromagnetics, when the surface is discretized with a spacing smaller than half a wavelength, the discrete surface can have the same spectral performance as the corresponding continuous surface. The discretization process converts the surface to a periodic lattice. Each element in this lattice can be the same (leading to a periodic surface) or different (leading to a quasi-periodic surface). Nevertheless, the discretization enables each element to manipulate electromagnetic waves locally and individually. The interaction between adjacent elements may still exist but can be controlled or minimized. These individual elements are then arranged on the surface with a certain distribution such as uniform distribution, gradient distribution, or hologram distribution. Consequently, all individual wave controls accumulate together, which manipulates the wavefront as desired, such as a mirror reflection, an anomalous reflection, or a reflection with a specific focused beam. The second prominent feature of advanced electromagnetic surfaces is the capability to completely control electromagnetic waves. Early surface designs only focus on a specific aspect of the wave, such as magnitude, phase, or polarization. Recent surface designs are required to control the magnitude, phase, and polarization simultaneously. Furthermore, anisotropic surfaces with tensor characters and nondispersive surfaces with wide bandwidth are of great interest among researchers. Nonlinear surfaces and nonreciprocal surfaces also emerge in literature. Meanwhile, advanced surfaces with real-time control by integration of circuit components are both interesting to researchers and useful to engineers which can be considered as a combination of the spatial variation by the surface and the time variation by the circuit. It is believed the surface capability to control electromagnetic waves will become more and more powerful with the development of surface electromagnetics.

1.3.3

Comparisons with Related Sciences and Technologies The importance of surface electromagnetics (SEM) can also be appreciated through comparisons with related sciences and technologies. Here, we compare SEM with transmission line theory, with terahertz research, and with metamaterials, respectively. Some interesting observations will be made that will propel SEM research forward. As discussed in the beginning of the book, transmission line theory deals with 1-D electromagnetic phenomena, while surface electromagnetics deals with 2-D electromagnetic phenomena. In a transmission line, the transverse dimensions are much smaller than the wavelength, but the longitudinal dimension is comparable to the wavelength (Lx , Ly « λ, Lz ~ λ). In a transmission surface, the transverse dimensions are comparable to the wavelength, but the longitudinal dimension is much smaller than the

20

Introduction to Surface Electromagnetics

wavelength (Lx , Ly ~ λ, Lz « λ). We can see a clear and beautiful duality between a transmission line and a transmission surface. Many concepts and methods in transmission line theory can be extended to surface electromagnetics. Since transmission line theory is a fundamental cornerstone of microwave engineering, the importance of SEM, as its dual, can’t be overestimated. It is also interesting to compare surface electromagnetics with terahertz (THz) research. THz is a spectrum between microwaves and optics that has not been fully studied before. Today it attracts growing interest among scientists and engineers, and the new IEEE Transactions was established in 2011 to publicize the THz research progress. Meanwhile, SEM is a discipline between 1-D transmission line theory and 3-D general electromagnetic theory, and only limited research was carried out before. Today fascinating phenomena are continually emerging in SEM, and new devices are being developed accordingly, with unprecedented performance. If THz research can be regarded as a new frontier of electromagnetics in the time domain, then SEM research can be considered as a new frontier of electromagnetics in the space domain. An interesting remark is that the synergy of THz and SEM brings exciting scientific discoveries and feasible engineering solutions. Finally, let’s compare surface electromagnetics with metamaterials. The latter has grown rapidly since 2000, and it has propelled the development of surface electromagnetics to some extent. For example, electromagnetic band gap (EBG) surfaces were considered as a subcategory of metamaterials, and the name metasurfaces also evolved from metamaterials. However, there exist several distinct differences between the two. First, metamaterials focus on three-dimensional volumetric structures, but surface electromagnetics focuses on two-dimensional surfaces. Second, metamaterials study the constitutive relations of materials as described in Eqs. (1.8) and (1.9), whereas SEM mainly deals with the boundary conditions in Eqs. (1.10) and (1.11), which leads to very different analysis methodologies. Finally, a surface is more preferable than a volumetric structure in engineering applications, because of the desirable features such as a low profile, light weight, low cost, and easy integration with active devices. In summary, we have sufficient reason to believe in a prosperous future of surface electromagnetics. The research on surface electromagnetics will greatly enrich our knowledge and will have a broad influence on our society through engineering innovations.

1.4

Research Frontiers of Surface Electromagnetics God made solids, but surfaces were the work of the devil. Wolfgang Pauli, 1900–1958

The SEM research is flourishing, as witnessed by the numerous terminologies and fascinating phenomena discussed previously. It includes ongoing development of existing techniques, such as frequency selective surfaces and electromagnetic band gap surfaces.

1.4 Research Frontiers of Surface Electromagnetics

21

It is also enriched by exotic discoveries, such as anomalous reflection/refraction and mantle cloaking. Some of them are driven by purely scientific curiosities, while some are urgently demanded by engineering applications. It is a very important yet quite challenging task to establish a unified framework for surface electromagnetics research. This framework is aimed to organize the exciting and diversified research progress with a logical order and a hierarchy structure. It will help to unveil hidden knowledge, to explore new frontiers, and to guide engineering innovations. For example, it can provide answers to the following questions: • • • •

Which analysis techniques should be used to characterize metasurfaces accurately and efficiently? What new functionalities can be discovered from a quasi-periodic surface? How do we design feasible devices from an artificial surface? What are the future directions of surface electromagnetic research?

Literature review on microwaves, optics, and surface physics can give some hints and guidelines to building such a framework for SEM. This edited book contributes our preliminary effort toward this SEM framework. At this moment, SEM research is mainly divided into three areas, namely, the SEM theorems, various artificial surface designs, and SEM-based engineering applications.

1.4.1

Surface Electromagnetic Theory A basic question in SEM is how to characterize an artificial surface. In circuit theory, the fundamental parameters are resistance (R), inductance (L), and capacitance (C). In transmission line theory, the fundamental parameters are characteristic impedance (Z0 ), transmission line length (l), and propagation constant (β). In general electromagnetic theory, the fundamental parameters are permittivity (ε), permeability (ε), and conductivity (σ ). In surface electromagnetics, surface impedance (ZS ) and surface admittance (YS ) are popularly used. However, they not only depend on the surface properties but also depend on the frequency, polarization, and incident angle of any incoming wave. Thus, they are not eigen-parameters of the surface itself. Recently, it was found that surface susceptibility (χ ) and surface porosity (π ) are desirable eigen-parameters. Hence, how to extract their values for a specific surface with certain geometry and how to design an artificial surface to achieve certain (χ, π ) become fundamental problems in SEM theory. The next question is on how to efficiently analyze the interaction of an electromagnetic wave with an artificial surface. Basically, all analysis methods can be derived from Maxwell’s equations, and they are simplified approaches to deal with specific groups of electromagnetic phenomena. In circuit theory, Kirchhoff current law (KCL) and Kirchhoff voltage law (KVL) are fundamental equations that determine the circuit functions. In transmission line theory, the telegraph equations are used to compute the properties of transmission line circuits. In surface electromagnetics, general sheet transition conditions (GSTC), a more rigorous formulation of boundary conditions, are found to be fundamental equations that govern various electromagnetic behavior. How

22

Introduction to Surface Electromagnetics

to apply GSTC to various complex surfaces, how to determine the surface functions from GSTC, and how to implement GSTC in numerical algorithms are theoretical problems for researchers to explore.

1.4.2

Artificial Surface Designs with Novel Properties It is an exciting and rewarding research experience to invent new artificial surfaces with unprecedented properties. In fact, a plethora of artificial surfaces have been proposed in microwave, THz, and optical regions. Advanced materials and cutting-edge fabrication technologies greatly propel the development of these artificial surfaces, especially at the nanometer scale. Figure 1.10 summarizes the recent progress on artificial surface designs. A hybrid control of the magnitude, phase, and polarization of an incoming wave is demanded for new artificial surfaces. Bi-anisotropic, nonreciprocity, and nonlinear surfaces are more challenging to design. With the increased design freedom, they may open opportunities to more advanced features. Furthermore, integration of semiconductor layers or semiconductor devices into artificial surfaces is also an exciting research topic. Realtime control and active response become new goals for artificial surface designs. It is worthwhile to remark that a new artificial surface may appear when translating a microwave concept to optics, or vice versa.

1.4.3

SEM-Based Engineering Applications The demand from engineering applications is a major driving force for the surface electromagnetics development; on the other hand, the SEM development also introduces new concepts and approaches to designing unconventional devices and systems. Some representative devices include ultrathin absorbers, mantle cloaking, reflectarrays/transmitarrays, and metalenses. When these devices are applied in practical systems, such as stealth airplanes, radio telescopes, and smartphones, groundbreaking performance and results can be expected. There are a number of devices based on SEM knowledge; here we select reflectarray antennas as an example to illustrate its relation with SEM. As a new generation of high-gain antennas, reflectarrays (RA) combine the unique features of parabolic reflectors and microstrip patch arrays, offering desirable advantages such as high efficiency, low profile, light weight, low cost, and easy integration with RF circuitry. The core component of a RA is a quasi-periodic surface with a predesigned phase distribution realized by patches of variable sizes or of different rotation angles. This surface converts a spherical wavefront from a feed to a planar wavefront, hence a high-gain pencil beam can be obtained. Although the RA concept was first introduced in 1963, the vast interest in RA did not come until the late 1980s, with the development of lowprofile microstrip antennas. In other words, RA research bloomed when surfaces with wavefront transformation capabilities became available – although people at that time might not have been aware about RA’s relation with SEM. After the 2000s, study of the reflection phase of an EBG surface introduced a new approach to designing RAs, which

1.5 Contents of This Book

(a)

23

(b)

Figure 1.11 SEM-based high-gain reflectarrays (RA): (a) a Ku-band RA consisting of 76,176 elements of variable sizes, forming a quasi-periodic surface with a Fresnel-type phase distribution; (b) an X-band RA consisting of 10,240 reconfigurable elements, forming an active surface for 2-D beam scanning and beam shaping. A black and white version of this figure will appear in some formats. For the color version, please refer to the plate section.

achieved a broader bandwidth using subwavelength elements. Later on, the relation between RA and SEM became more and more close: the knowledge accumulated in RA development has been used to design artificial surfaces in THz and optics, and the new discoveries in SEM have been applied in RA designs for better performance. Figure 1.11 shows two SEM-based reflectarrays recently developed by Prof. Fan Yang’s group that achieve superior performance at an extremely low cost. These antennas, especially the one with beam-scanning capability, have great potential in 5G wireless communications, radar systems for unmanned vehicles, and millimeter-wave imaging systems for security checks. In summary, the reflectarray research field has become an active and indispensable part of surface electromagnetics.

1.5

Contents of This Book This edited book aims to provide a comprehensive presentation of surface electromagnetics by editing the most state-of-the-art concepts, physics, engineering designs, and exciting applications in this area. This book contains chapters by world-prominent scientists who are pioneers in the development of SEM, and it covers the most cutting-edge discoveries and progress. To warrant the novelty of the book, some related works that have already been published in previous books, such as general impedance boundary

24

Introduction to Surface Electromagnetics

conditions [13], frequency selective surfaces [14], EBG surfaces [22], and reflectarray antennas [41], will not be included in this book.

1.5.1

Models, Analysis, and Synthesis of EM Surfaces Figure 1.12 shows the organization of the book contents. It starts with an introduction of surface electromagnetics (SEM), which focuses on electromagnetic phenomena occurring on two-dimensional surfaces. The development of various electromagnetic surfaces is reviewed, and a clear clue from uniform surfaces to periodic surfaces and then to quasi-periodic surfaces can be observed. The importance of SEM research is discussed in this chapter, and it includes a brief summary of SEM research topics. Next, fundamental SEM theorems are presented. Chapter 2 is devoted to analytical models of electromagnetic surfaces, including the polarizability-based model, the susceptibility-based model, and the equivalent impedance matrix model. Surfaces with general properties, such as bi-anisotropy and nonreciprocity, are discussed, and surfaces with different functionalities are analytically derived and analyzed. Chapter 3 discusses the generalized sheet transition conditions (GSTCs), which are found to be the most appropriate way to analyze artificial surfaces. The GSTCs are applied to different surfaces, including metafilms, metascreens, and metagratings. Using the GSTCs, the reflection/transmission coefficients can be derived, and surface parameters can also be retrieved effectively. Using the GSTC, Chapter 4 discusses the mathematical synthesis and numerical analysis of artificial surfaces. The synthesis procedure leads to the determination of susceptibilities corresponding to the field transformation specifications, while the numerical analysis computes the exact scattered fields for various excitation conditions based on novel frequency- and time-domain method schemes. Furthermore, the geometry of scattering particles forming physical metasurfaces is determined via scattering parameter mapping. Since many new artificial surfaces are quasi-periodic, Chapter 5 presents a detailed study on quasi-periodic analysis. It compares the deviation of element phase in a quasiperiodic array from an infinitely periodic array and reveals that elements with strong resonance are more easily affected by their surroundings – causing them to suffer from a large error margin due to periodic boundary conditions (PBC). Moreover, a numerical scheme is developed for modeling quasi-periodic arrays using the reduced basis method (RBM), which includes an offline training procedure and an online analysis procedure. Numerical examples verify the accuracy and efficiency of this method.

1.5.2

Guided Wave, Leaky Wave, and Plane Wave Properties of EM Surfaces Chapters 6–9 present artificial surface designs from different viewpoints, each with a specific functionality and design approach. Chapter 6 studies the guided wave property in an emerging gap waveguide. It is based on the use of engineered surfaces inside the parallel-plate waveguide with the aim to direct electromagnetic waves in the desired direction and to prevent propagation in other directions. Different versions of gap

1.5 Contents of This Book

Figure 1.12 SEM research contents organized in this book. A black and white version of this figure will appear in some formats. For the color version, please refer to the plate section.

25

26

Introduction to Surface Electromagnetics

waveguides are analyzed, and their applications to millimeter-wave antennas are also covered, showing an enormous potential. The transformation from a surface wave (SW) into a leaky wave (LW) by a modulated metasurface (MTS) antenna is presented in Chapter 7. The amplitude, phase, and polarization of the aperture field are fully controlled through the imposition of proper impedance boundary conditions (BC) seen by the SW. Hence, MTS antennas are highly customizable in terms of performance simply by changing the MTS layout. Readers are guided through the design process – from the determination of the continuous impedance BCs required for reproducing a desired aperture field to the implementation of the BCs with electrically small metallic patches printed on a regular lattice. Chapter 8 focuses on the interaction of artificial surfaces with plane waves, and a special emphasis is placed on the transmission phase control due to its essential role in wave phenomena. The theoretical phase limits for various transmission surfaces are derived, based on which several specific transmission surfaces are presented with the consideration of low-profile geometry, angular stability, and cross-polarization levels. Furthermore, reconfigurable transmission surfaces with dynamic phase control properties are presented. As an application, the transmitarray antenna concept, design procedure, and practical examples are presented. Chapter 9 introduces the information theory into metasurface designs, leading to the so-called coding metasurfaces and programmable metasurfaces. Several illustrative examples are presented to show their flexible manipulation of EM waves, including anomalous reflection, beam splitting, and diffuse scattering. Moreover, an interesting single-sensor single-frequency imaging system is demonstrated, which is built with a two-bit transmission-type programmable metasurface capable of generating domically changing random radiation patterns.

1.5.3

Applications of Surface Electromagnetics Engineering applications are not only the direct outcomes of SEM progress but also the main forces that drive SEM development. Chapters 10–12 are oriented more toward SEM applications. Chapter 10 focuses on the exciting topic of electromagnetic cloaking, where metamaterials and metasurfaces are used to reduce the scattering of electromagnetic waves from passive objects. Several popular approaches, including transformation optics and scattering cancellation techniques, are presented under a unified formulation, shedding light on the unusual responses associated with metamaterial cloaks. Chapter 11 revisits the concepts and unique features of orbital angular momentum (OAM) beams, which have a great potential in wireless communication systems in both near-field range and far-field range. Textured electromagnetic surfaces are proposed as an efficient apparatus to generate such OAM beams. A general guideline of designing textured surfaces is provided with interesting examples. Chapter 12 provides a general review of SEM applications in microwave, THz, and optics. In microwaves, metasurfaces are designed to enhance antenna performance, such as artificial grounds, antenna superstrates, and scattering control metasurfaces. In THz, graphene and vanadium dioxide (VO2 ) are exploited to design tunable metasurfaces,

References

27

which enable electrical and thermal modulation of terahertz radiation. At optical wavelengths, metasurfaces are beginning to revolutionize lens technology, to advance optical cloaking, and to effectively generate 3-D images. In summary, this book provides a comprehensive presentation of the state of the art in surface electromagnetics. It tries to establish a common ground for SEM researchers from various aspects of theoretical analysis, functional designs, and engineering applications. It is our sincere hope that this book will inspire innovative scientific discoveries and influential engineering applications in the future.

References [1] [2] [3] [4] [5] [6] [7] [8] [9]

[10]

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J. C. Vickerman (ed.), Surface Analysis – The Principal Techniques, Wiley, 1998. M. Prutton, Surface Physics, 2nd edition, Oxford University Press, 1984. F. Bechstedt, Principle of Surface Physics, Springer, 2003. A. K. Geim, “Graphene: status and prospects,” Science, vol. 324 (5934), pp. 1530–1534, 2009. D. K. Chang, Field and Wave Electromagnetics, 2nd edition, Addison-Wesley, 1992. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference TimeDomain Method, 2nd edition, Artech House, 2000. R. L. Boylestad and L. Nashelsky, Electronic Devices and Circuit Theory, 11th edition, Prentice Hall, 2012. L. N. Dworsky, Modern Transmission Line Theory and Applications, 1st edition, Wiley, 1980. Y. Rahmat-Samii and R. Haupt, “Reflector antenna developments: a perspective on the past, present and future,” IEEE Antennas and Propagation Magazine, vol. 57 (2), pp. 85–95, 2015. N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science, vol. 334, pp. 333–337, 2011. M. Khorasaninejad et al., “Metalenses at visible wavelengths: Diffraction-limited focusing and subwavelength resolution imaging,” Science, vol. 352 (6290), pp. 1190–1194, 2016. A. Kwan, J. Dudley, and E. Lantz, “Who really discovered Snell’s law?,” Physics World vol. 15 (4), p. 64, 2002. D. Hoppe and Y. Rahmat-Samii, Impedance Boundary Conditions in Electromagnetics, CRC Press, 1995. B. A. Munk, Frequency Selective Surfaces: Theory and Designs, Wiley-Interscience, 2000. T. K. Wu (ed.), Frequency Selective Surface and Grid Array, Wiley-Interscience, 1995. R. Mittra, C. H. Chan, and T. Cwik, “Techniques for analyzing frequency selective surfacesa review,” Proceedings of the IEEE, vol. 76, pp. 1593–1615, 1988. J.-M. Jin, “The method of moments,” in Theory and Computation of Electromagnetic Fields, Wiley-IEEE Press, 2010. K. ElMahgoub, F. Yang, and A Z. Elsherbeni, Scattering Analysis of Periodic Structures Using Finite-Difference Time-Domain Method, Morgan & Claypool, 2012. P. S. Kildal, “Artificially soft and hard surfaces in electromagnetics,” IEEE Transactions on Antennas and Propagation, vol. 38 (10), pp. 1537–1544, 1990.

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[20] P. S. Kildal, A. A. Kishk, and S. Maci. “Special issue on artificial magnetic conductors, soft/hard surfaces, and other complex surfaces,” IEEE Transactions on Antennas & Propagation, vol. 53 (1), pp. 2–7, 2005. [21] D. Sievenpiper, L. Zhang, R. Broas, N. Alexopolous, and E. Yablonovitch, “Highimpedance electromagnetic surfaces with a forbidden frequency band,” IEEE Transactions on Microwave Theory and Techniques, vol. 47, pp. 2059–2074, 1999. [22] F. Yang and Y Rahmat-Samii, Electromagnetic Band Gap Structures in Antenna Engineering, Cambridge University Press, 2009. [23] N. Engheta and W. R. Ziolkowski (eds.), Metamaterials: Physics and Engineering Explorations, Wiley, 2006. [24] “The Nobel Prize in Chemistry 2011,” Nobelprize.org. Retrieved 2011-10-06. [25] X. Ni, N. K. Emani, A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Broadband light bending with plasmonic nanoantennas,” Science, vol. 335, pp. 427–427, 2012. [26] N. K. Grady, J. E. Heyes, D. R. Chowdhury, Y. Zeng, M. T. Reiten, A. K. Azad, A. J. Taylor, D. A. R. Dalvit, and H. Chen, “Terahertz metamaterials for linear polarization conversion and anomalous refraction,” Science, vol. 340, pp. 1304–1307, 2013. [27] S. Sun, Q. He, S. Xiao, Q. Xu, X. Li, and L. Zhou, “Gradient-index metasurfaces as a bridge linking propagating waves and surface waves,” Nature Materials, vol. 11, pp. 426–431, 2012. [28] D. Lin, P. Fan, E. Hasman, and M. L. Brongersma, “Dielectric gradient metasurface optical elements,” Science, vol. 345, pp. 298–302, 2014. [29] N. Yu, F. Aieta, P. Genevet, M. A. Kats, Z. Gaburro, and F. Capasso, “A broadband, background-free quarter-wave plate based on plasmonic metasurfaces,” Nano Letters, vol. 12, pp. 6328–6333, 2012. [30] X. Ni, A. V. Kildishev, and V. M. Shalaev, “Metasurface holograms for visible light,” Nature Communications, vol. 4, p. 2807, 2013. [31] L. Huang, X. Chen, H. Mühlenbernd, H. Zhang, S. Chen, B. Bai, Q. Tan, G. Jin, K. W. Cheah, and C. Qiu, “Three-dimensional optical holography using a plasmonic metasurface,” Nature Communications, vol. 4, 2013. [32] G. X. Zheng, H. Muhlenbernd, M. Kenney, G. X. Li, T. Zentgraf, and S. Zhang, “Metasurface holograms reaching 80% efficiency,” Nature Nanotechnology, vol. 10, pp. 308–312, 2015. [33] B. Cai, Y. Li, W. Jiang, Q. Cheng, and T. Cui, “Generation of spatial Bessel beams using holographic metasurface,” Optics Express, vol. 23, pp. 7593–7601, 2015. [34] Y. Li, B. Cai, Q. Cheng, and T. Cui, “Isotropic holographic metasurfaces for dualfunctional radiations without mutual interferences,” Advances Functional Materials, vol. 26, pp. 29–35, 2016. [35] L. Huang et al., “Dispersionless phase discontinuities for controlling light propagation,” Nano Letters, vol. 12, pp. 5750–5755, 2012. [36] A. Arbabi, Y. Horie, M. Bagheri, and A. Faraon, “Dielectric metasurfaces for complete control of phase and polarization with subwavelength spatial resolution and high transmission,” Nature Nanotechnology, vol. 10, pp. 937–943, 2015. [37] X. Ni, Z. Wong, M. Mrejen, Y. Wang, and X. Zhang, “An ultrathin invisibility skin cloak for visible light,” Science, vol. 349, pp. 1310–1314, 2015. [38] A. Cho, “Skintight invisibility cloak radiates deception,” Science, vol. 349, p. 1269, 2015. [39] J. Huang and J. A. Encinar, Reflectarray Antennas, Wiley-IEEE Press, 2007.

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[40] A. H. Abdelrahman, F. Yang, A. Z. Elsherbeni, and P Nayeri, Analysis and Design of Transmitarray Antennas, Morgan & Claypool, 2017. [41] P. Nayeri, F. Yang, and A. Z. Elsherbeni, Reflectarray Antennas: Theory, Designs and Applications, Wiley-IEEE Press, 2018. [42] G. Minatti, M. Faenzi, E. Martini, F. Caminita, P. D. Vita, D. Gonzlez-Ovejero, M. Sabbadini, and S. Maci, “Modulated metasurface antennas for space: Synthesis, analysis and realizations,” IEEE Transactions in Antennas Propagation, vol. 63 (4), pp. 1288–1300, 2015. [43] J. D. Kraus and K. R. Carver, Electromagnetics, 2nd edition, McGraw-Hill, 1973. [44] C. A. Balanis, Advanced Engineering Electromagnetics, John Wiley, 2012. [45] Y. Rahmat-Samii, L. I. Williams, and R. G. Yaccarino, “The UCLA bi-polar planarnear-field antenna-measurement and diagnostics range,” IEEE Antennas and Propagation Magazine, vol. 37 (6), pp. 16–35, 1995.

2

Analytical Modeling of Electromagnetic Surfaces Viktar Asadchy, Ana Díaz-Rubio, Do-Hoon Kwon, and Sergei Tretyakov

2.1

Introduction: Definitions, Basic Classification, Main Functionalities of Metasurfaces Natural materials and substances possess a rich variety of electromagnetic properties over the entire electromagnetic spectrum. Despite this diversity, nature does not equip us with a full set of possible material tools for controlling electromagnetic waves and realizing all physically possible effects. Only the use of artificial, engineered substances can give us full control over electromagnetic properties of materials. Such materials, namely, metamaterials, consist of inclusions whose dimensions are small but comparable with the operating wavelength, which enables existence of spatial dispersion effects in them, such as artificial magnetism and bi-anisotropy. Unique properties of metamaterials have led to numerous applications in applied physics, filling the gaps of electromagnetic response of natural materials, alloys, and chemical compounds. Nevertheless, manufacturing of volumetric metamaterials is often a serious challenge even with the present-day technologies, especially when it comes to three-dimensional optical metamaterials. Since most metamaterial applications imply wave propagation through electrically thick volumes, there are significant dissipation losses that can severely distort the device operation. On the other hand, small thickness of metamaterial-based devices can be critical, e.g., for nanophotonics and microwave applications. Finally, designing a volumetric metamaterial, one must often engineer proper electromagnetic response (e.g., polarizability tensors of the inclusions) in all three dimensions. Fortunately, it is not always necessary to design a bulk metamaterial and analogous response can be achieved with an electrically thin counterpart representing a single-layer metamaterial, the so-called metasurface. This fact can be proven based on the Huygens’ principle and its generalization (Huygens 1690, Lindell, Tretyakov & Nikoskinen 2000). Let us consider a volume V filled with arbitrary sources of electromagnetic radiation, electric charges qi , and currents Ji (see the left side of Figure 2.1a). These sources create electric field E and magnetic induction B in the area outside the volume. According to the Huygens’ principle, this system of scatterers can be always replaced by an arbitrarily thin layer of specific electric JeV and magnetic JmV currents enclosing the volume V . The thickness of the layer can be electrically small but nonzero since magnetic currents can be generated only via loops of electric currents with finite thickness. Importantly, the equivalent currents JeV and JmV (surface currents if the layer thickness is electrically

30

2.1 Introduction: Definitions, Basic Classification, Main Functionalities of Metasurfaces

eV , mV

,

,

a

b Figure 2.1 (a) Illustration of the Huygens’ principle applied to scattering from volumetric

electromagnetic sources. (b) Application of the Huygens’ principle to the concept of metamaterials. Electromagnetic response from any volumetric material (or metamaterial) in principle can be always reproduced with an arbitrarily shaped closed two-dimensional layer carrying electric and magnetic currents. A black and white version of this figure will appear in some formats. For the color version, please refer to the plate section.

negligible) scatter electromagnetic fields only outward of volume V , and these fields are the same as those created by the original system of sources, i.e., E and B (see the right side of Figure 2.1a). Such a system of currents, which scatter only into one side, are called Huygens’ surfaces or Huygens’ sources. This concept has important implications for the metamaterials paradigm. Let us consider an arbitrary volumetric metamaterial sample (shown on the left side of Figure 2.1b) excited by an arbitrary external wave with the wavevector ki . The external wave induces some charges qi and currents Ji in the inclusions of the sample which reradiate secondary fields Eout and Bout into the space outside volume V which encloses the sample. According to the previously described principle, one can replace the bulky metamaterial sample with induced polarization charges and currents by equivalent surface currents JeV and JmV which would scatter the same fields Eout and Bout outside volume V . Next, knowing these equivalent currents, one can in principle determine appropriate topologies of meta-atoms (placed at the surface of volume V ) which under excitation by the external wave with ki would generate the same currents JeV and JmV (see the right side of Figure 2.1b). Such an arrangement of meta-atoms, namely, metasurface, would possess identical electromagnetic response to that of the original metamaterial sample for the given excitation ki . For other excitations, the metasurface generally would not imitate the same bulky sample. Thus, one can conclude that in the cases when specific electromagnetic response is required for particular excitation, volumetric metamaterials can be replaced by electrically thin and in general curved metasurfaces.

31

32

Analytical Modeling of Electromagnetic Surfaces

During the last decade, planar metasurface devices have successfully substituted their bulky counterparts for various applications: negative refraction (Yu, Genevet, Kats, Aieta, Tetienne, Capasso & Gaburro 2011), scattering cancellation cloaking (Alù 2009), reciprocal (Niemi et al. 2013) and nonreciprocal (Ra’di, Asadchy & Tretyakov 2014a) optical activity, absorption (Ra’di, Asadchy & Tretyakov 2013), multichannel surfaces (Asadchy, Albooyeh, Tcvetkova, Díaz-Rubio, Ra’di & Tretyakov 2016), general reflection and transmission control (Ra’di, Asadchy & Tretyakov 2014b; DíazRubio, Asadchy, Elsakka & Tretyakov 2017; Holloway, Kuester, Gordon, O’Hara, Booth & Smith 2012; Yu & Capasso 2014; Glybovski, Tretyakov, Belov, Kivshar & Simovski 2016; Asadchy, Ra’di, Vehmas & Tretyakov 2015; Asadchy, Albooyeh & Tretyakov 2016; Asadchy, Faniayeu, Ra’di, Khakhomov, Semchenko & Tretyakov 2015; Asadchy, Albooyeh, Tcvetkova, Díaz-Rubio, Ra’di & Tretyakov 2016; Asadchy, Wickberg, Díaz-Rubio & Wegener 2017), etc. Although all metasurfaces share the same feature of subwavelength unit cell sizes (as well as subwavelength distances between the cells), they generally can have distinctly different electromagnetic responses. Therefore, it is important to classify metasurfaces based on most important criteria. First of all, they can have periodic (Glybovski et al. 2016) and nonperiodic (or “amorphous”) (Albooyeh, Kruk, Menzel, Helgert, Kroll, Krysinski, Decker, Neshev, Pertsch, Etrich, Rockstuhl, Tretyakov, Simovski & Kivshar 2014; Andryieuski, Lavrinenko, Petrov & Tretyakov 2016; Bilotti & Tretyakov 2013) structure. In both cases, the lattice feature is smaller than half of the wavelength in surrounding medium (to ensure homogenization). Metasurface can have a planar or curved shape. The latter scenario is less studied in the literature; however, it is very important for applications where designed metasurface should be conformal to the object on which it is installed. An important classification is based on polarization response. Metasurfaces can incorporate only electrically (Yu et al. 2011) or only magnetically (Bilotti, Toscano, Alici, Ozbay & Vegni 2011) polarizable inclusions, as well as inclusions with both polarization properties (Pfeiffer & Grbic 2013). In a more general case, the inclusions can be also bi-anisotropic (Niemi et al. 2013), i.e., acquire electric polarization due to external magnetic field, and vice versa. Furthermore, metasurfaces can incorporate a ground plane (blocking transmission at all frequencies) (Sun, Yang, Wang, Juan, Chen, Liao, He, Xiao, Kung, Guo, Zhou & Tsai 2012) or not (enabling frequency selective response in transmission) (Asadchy, Faniayeu, Ra’di, Khakhomov, Semchenko & Tretyakov 2015). Finally, in different applications, metasurface inclusions can be resonant and nonresonant. It should be mentioned that recently the definition of metasurfaces has been extended to the case of nonuniform surfaces. In such structures, the distances between adjacent inclusions as well as their sizes are below the half-wavelength scale, while the structural periodicity exceeds this value. Due to this periodicity, the surface might be not homogenizable, and its response should be described using Bloch–Floquet analysis (based on expansion of the field into spatial harmonics) (Sakoda 2005). Probably, the most appropriate name for these nonuniform metasurfaces could be “metasurface-based” gratings, by analogy with conventional diffraction gratings.

2.2 Metasurfaces, Homogeneous Slabs, and FSS

33

Owing to their two-dimensional structure, the main functionalities of metasurfaces refer to general manipulation of incident electromagnetic radiation and its characteristics: polarization, amplitude, phase, and even wavefront shape. Metasurfaces can arbitrarily (within the limits dictated by the energy conservation and reciprocity) modify all of these wave attributes both in reflection and transmission scenarios. This property makes metasurfaces ideal candidates for such applications as absorbers, phase shifters, polarization transformers, frequency filters, focusing lenses, steering antennas, holograms, etc.

2.2

Metasurfaces versus Thin Slabs of Homogeneous Materials and Other Artificial Periodic Surfaces It is important to discuss the place of metasurfaces in the framework of general periodic electrically thin structures. All infinite planar periodic structures are characterized by the spatial period D (in the surface plane) which determines if they exhibit diffraction effects or not. Let us consider a two-dimensional planar periodic structure illuminated by a plane wave at an angle θi , as shown in Figure 2.2. The structure is located between two homogeneous media with refractive indices n1 and n2 . Calculating the optical length difference between two rays (which pass through the structure at two points separated by D) accumulated during their propagation, one can find the condition of diffraction maximum in the first medium at an angle θr (or, alternatively, in the second medium at an angle θt ). These classical diffraction conditions for reflection and transmission scenarios read n1 D(sin θi ´ sin θr ) = mr λ0,

D(n1 sin θi ´ n2 sin θt ) = mt λ0,

(2.1)

where λ0 is the wavelength in vacuum and mr and mt are some integer numbers defining the diffraction orders. As it is seen from these conditions, diffraction occurs only when the period D is comparable with or larger than the wavelengths in both media. Assuming for simplicity that n1 = n2 = 1, relations (2.1) can be written as D mr,t = . λ0 sin θi ´ sin θr,t

(2.2)

From this relation, two well-known implications follow. First, for normally incident waves (θi = 0˝ ), diffraction occurs only if the periodicity of the structure is D ě λ0

i

r 1 2

Figure 2.2 Diffraction of electromagnetic waves on periodic planar structures.

34

Analytical Modeling of Electromagnetic Surfaces

Figure 2.3 Classification of different types of periodical planar structures with typical topologies

of each type operating at a wavelength λop . The red and green regions denote nonresonant and resonant types of structures, respectively. A black and white version of this figure will appear in some formats. For the color version, please refer to the plate section.

(mr,t { sin θr,t ě 1 for any mr,t ‰ 0). Second, for other incidence angles (θi ‰ 0˝ ), diffraction may appear if D ě λ0 {2. These two important conditions are often used, for example, in the antenna theory, where the periodicity is chosen based on them to avoid propagating diffraction modes (lobes). It is convenient to classify different periodical structures into several groups based on the distance between adjacent inclusions/elements d and the type of electromagnetic response (resonant or nonresonant). For clarity, only uniform structures (D = d) are considered. Figure 2.3 presents the classification of different types of periodical planar structures with typical topologies of each type operating at a wavelength λop . Naturally, for other wavelengths the same planar topology may behave differently: e.g., metasurfaces at the short-wavelength limit behave as diffraction gratings. The first group of periodical structures is formed by sheets of usual materials. In this scenario, the array period is extremely small compared to the wavelength. We can consider a small (compared to the wavelength) area of the sheet, and for every molecule within this small piece the environment would look the same as if the sheet would be infinite and uniformly excited. Indeed, the incident field is the same for millions of molecules around any reference molecule. The structure can be easily homogenized. In this situation we can characterize the sheet by conventional macroscopic parameters, such as resistance per square or, more generally, the sheet impedance. This parameter defines the ratio of the tangential electric field (this field is equal to the voltage between the two sides of a 1 ˆ 1 m square sample of this sheet) and the surface current density (equal to the current in this square), and it does not depend on how the sheet is illuminated and even on the sheet shape (it may be curved). Due to the small dimensions of

2.2 Metasurfaces, Homogeneous Slabs, and FSS

35

atoms and molecules of natural materials, at frequencies below the infrared range they are nonresonant (at the level of atoms and molecules). One can also refer dense two-dimensional arrays of electrically long conducting wires to the first group of periodical surfaces (see Figure 2.3). Such structures usually are designed for achieving a full reflection regime and are used as light-weight reflectors and screens in various microwave applications (e.g., meshes in microwave ovens). To ensure high reflection level, the periodicity of wire arrays is usually made small compared to the wavelength in free space d ! λop {2. An incident wave with the electric field oscillating along the wires induces electric current in them, which radiates symmetrically (in the forward and backward directions) waves with the phase opposite to that of the incident wave. Therefore, in the forward direction, the scattered and incident waves interfere destructively, resulting in low transmission. Probably the first study on dense arrays of wires appeared in 1898 (Lamb 1897). Being a two-dimensional analogue of the wire medium, dense arrays of wires also exhibit effects of strong spatial dispersion (Kontorovich, Petrunkin, Yesepkina & Astrakhan 1962). Dense arrays of wires are nonresonant structures and, due to subwavelength distances between the wires, can be homogenized in the plane. The second group of periodical planar structures is commonly referred to as frequency-selective surfaces or FSS (Munk 2000). These structures are usually formed by periodically arranged patches (on a dielectric substrate) or slots (in a metal sheet) of various shapes (Munk 2000) (see illustration in Figure 2.3). In contrast to inductive arrays of wires which usually operate at nonresonant frequencies, FSSs are resonant structures (resonance of single inclusions) due to the presence of both inductive and capacitive properties (because of the gaps in the direction of the electric field between adjacent elements). Hence, FSSs can be designed to resonate at frequencies where the element size is comparable to the half-wavelength. Due to the intermediate sizes of the elements (λop {2 ď d ă λop ), FSSs do not have diffraction lobes only for some specified incidence angles and cannot be homogenized to an equivalent continuous impedance sheet. FSSs are widely used as microwave reflection/transmission filters and low-profile antennas. Nonuniform nonperiodic FSSs elements called reflectarrays and transmitarrays are used mainly as passive antenna arrays. At the end of the twentieth century, two important periodical structures were proposed based on the so-called Jerusalem crosses (Anderson 1975; Yang, Ma, Qian & Itoh 1999). They comprise the third group in the present classification. These structures with subwavelength inclusions (which can be homogenized) resonate due to the capacitive coupling between adjacent elements (so-called distributed resonance). According to classification in some literature (Munk 2000), these structures can be referred as FSSs with subwavelength inclusions. So-called mushroom-type high-impedance surfaces (HIS) introduced in Sievenpiper, Zhang, Broas, Alexopolous, and Yablonovitch (1999) can be attributed to the same type of resonant periodic structures. Typically, sizes of their elements are very small compared to the wavelength (d „ λop {10) due to additional inductance caused by a dielectric slab covered with a ground plane. Diffraction gratings (Palmer & Loewen 2005) (see illustration in Figure 2.3) and phase array antennas representing the fourth group in the present classification have the

36

Analytical Modeling of Electromagnetic Surfaces

longest history: probably the first grating was made in 1785 using strung hairs between two finely threaded screws (Hopkinson & Rittenhouse 1786). In contrast to the previous groups of periodic structures, gratings are intentionally designed to reflect/transmit incident waves mostly into a diffraction mode; therefore, their periodicity is always comparable to or larger than the wavelength d ą λop {2. Gratings which diffract incident energy toward only one direction are called blazed gratings (e.g., Destouches, Tishchenko, Pommier, Reynaud, Parriaux, Tonchev & Ahmed 2005). Usually, gratings are designed for the optical range and represent a two-dimensional version of photonic crystals. Metasurfaces are homogenizable (λop {30 ă d ă λop {2) single-layer composites of inclusions, which can be referred to both the first (nonresonant) and third (resonant) groups in the present classification. Recently, the metasurface concept has been extended to nonuniform structures, yielding the so-called metasurface-inspired diffraction gratings (Yu et al. 2011). The distinctive feature of such gratings is the subwavelength distance between the elements over one period (d ă λop {2, while D ą λop {2). In this case, the grating can be described by the surface-averaged impedance. Metasurfaces can exhibit spatial dispersion effects of the first (bi-anisotropy) and second (artificial magnetism) orders. Thus, on the one hand, the recently proposed concept of metasurfaces resembles and combines previously known planar structures (wire arrays, FSS with subwavelength elements, reflect- and transmitarrays, mushroom-type surfaces), while, on the other hand, it generalizes the theory of periodical structures and enables novel, previously unattainable devices: wavefront transformers and holograms (Yu & Capasso 2014; Zheng, Mhlenbernd, Kenney, Li, Zentgraf & Zhang 2015), nonreciprocal sheets (Ra’di et al. 2014a), shadow-free mirrors and absorbers (Asadchy, Ra’di, Vehmas & Tretyakov 2015; Asadchy, Albooyeh & Tretyakov 2016; Asadchy, Faniayeu, Ra’di, Khakhomov, Semchenko & Tretyakov 2015), cascaded antennas (Elsakka, Asadchy, Faniayeu, Tcvetkova & Tretyakov 2016), 100% efficient (Díaz-Rubio et al. 2017; Wong, Epstein & Eleftheriades 2016; Epstein & Eleftheriades 2016; Asadchy et al. 2017) and multichannel (Asadchy, Albooyeh, Tcvetkova, Díaz-Rubio, Ra’di & Tretyakov 2016) gratings, etc.

2.3

Comparison of Possible Functionalities Here we compare attainable functionalities of thin layers of different fundamental classes, if the goal is to control (in a tunable fashion) reflected and transmitted waves. •

Suppose that the required functionality of a tunable thin-sheet device is such that the “input field” is a single plane wave and the “output field” is another plane wave traveling in the same direction or reflected specularly (possibly having zero amplitude in the case of perfect absorbers). To realize such response, the surface should be uniform at the wavelength scale. If it is an array of unit cells, then the period should be smaller than λ{2, and all the cells should be the same, at least

2.4 Homogenization Models of Metasurfaces



2.4

37

when the response is averaged over any wavelength-size area. Here we assume that the array size is electrically large and neglect the edge effects. To realize such simple functions, one can use either of the three classes of tunable surfaces: material sheets (controllable by external stimuli, e.g. stretchable) or phase arrays or metasurfaces. The choice can be made based on availability, simplicity, and cost of the control elements (just one control signal is needed). It is perhaps advantageous to use conventional phased-array antennas (reflectarrays and transmittarrays), just because here the number of unit cells is minimized. On the other hand, since the control voltage is the same for all units, in some situations, increasing the number of unit cells may not necessarily increase the overall complexity and cost. From the practical point of view it may be perhaps better to have several units per wavelength because it can be easier to set the average properties to the correct values having several step-wise controlled units rather than just one. If the required functionality of a tunable surface is more complex, so that the surface should change the wave propagation direction, then the surface should be nonuniform at the wavelength scale. Here, homogeneous material sheets become practically problematic, as it is not easy to control material properties in a wide range and maintain good spatial resolution, especially at high frequencies. Thus, antenna arrays or metasurfaces are used. If the desired function requires deflection of waves by moderate angles and focusing with a large focal distance, both conventional antenna arrays and metasurfaces can do the job well, but antenna arrays exhibit more significant discretization errors of the required phase distribution over the surface, since the unit cell is larger. For moderate deflection angles, the effects of this error are pretty small and usually can be neglected. However, if the functionality is more demanding and quasi-arbitrary (like anomalous reflection into arbitrary directions, from small to very steep angles, or focusing at a small, comparable to λ, focal distance), metasurfaces offer possibilities that antenna arrays cannot provide (Díaz-Rubio et al. 2017).

Homogenization Models of Metasurfaces As it was discussed above, owing to their subwavelength periodicity, metasurfaces can be homogenized. In other words, their electromagnetic properties can be fully described by some set of macroscopic parameters, on the analogy how volumetric materials are described by effective permittivity and permeability. Knowledge of these macroscopic parameters of a metasurface allows us to predict its electromagnetic response for arbitrary excitation (with arbitrary wavefront, incidence angle, and polarization). In contrast, it is impossible to introduce effective macroscopic parameters to inhomogenizable thin structures such as conventional and metasurface-based diffraction gratings. The first attempts to homogenization of electrically thin surfaces were based on measuring the reflection and transmission coefficient from the layer. Using these data and considering the surface as an effectively homogeneous material slab, one can

38

Analytical Modeling of Electromagnetic Surfaces

Figure 2.4 (Left) A metasurface consisting of arbitrary subwavelength inclusions. An incident

wave excites certain currents in the metasurface. (Right) Modeling the actual metasurface by a continuous sheet of averaged over the unit cell currents. For clarity, only tangential (to the metasurface plane) current components are depicted.

calculate its effective permittivity and permeability (Nicolson & Ross 1970; Weir 1974; Smith, Schultz, Marko & Soukoulis 2002). However, this approach, when applied to metasurfaces, led to nonphysical results (Smith et al. 2002). Therefore, the subsequent studies tried to solve the problem of metasurface homogenization using different approaches. Let us consider a metasurface consisting of an array of arbitrary subwavelength inclusions with the periodicity d ! λ, shown on the left side of Figure 2.4. Since the inclusions and the periodicity are sufficiently small compared to the wavelength of incident waves, one can assume that the incident fields are homogeneous (uniform over the unit cell size). An incident wave induces electric and magnetic polarization currents in the inclusions. Obviously, the total fields in the metasurface plane may change very fast, taking very high values in the immediate vicinity of the polarized inclusions and low values in between them. To simplify the analysis, one can replace the coordinatedependent electric and magnetic polarizations of each unit cell by averaged quantities. In this case, the problem of scattering from an array of finite-size inclusions evolves into a problem of scattering from a continuous sheet of electric and magnetic currents (see the right-hand side of Figure 2.4). The assumptions of homogenized incident fields and induced currents at the metasurface plane enable relatively simple introduction of macroscopic parameters. One can consider a metasurface as an infinitely thin planar sheet of electric and magnetic polarization surface densities Ps and Ms (these can be related to the corresponding current surface densities as Je = BPs {Bt and Jm = μ0 BMs {Bt). For simplicity, let us assume that the metasurface is located in a vacuum at position z = 0 (xy-plane), as is shown in Figure 2.5. An incident wave with arbitrary polarization and wavefront (provided that the incident fields do not significantly change over the unit cell) impinges on it. It is convenient to decompose the incident electric and magnetic fields (E, H) into tangential (Et , Ht ) and normal (En n, Hn n) to the metasurface plane components, where n is the unit normal vector. Analogous decomposition can be applied to the surface densities Ps = Pt + Pn n and Ms = Mt + Mn n, as well as to the nabla operator ∇ = ∇t + BBz n. Currents induced in the metasurface plane scatter, in general, electromagnetic waves in both half-spaces below and above the metasurface. After solving a

2.4 Homogenization Models of Metasurfaces

39

i

i i

s

s

Figure 2.5 Geometry of the boundary problem. A metasurface is positioned at z = 0 plane. An

incident wave induces electric and magnetic surface polarizations.

boundary problem at the interface between the two half-spaces, one can relate the jumps of the total fields across the metasurface to the polarization surface densities (Senior 1985; Idemen 1988; Senior & Volakis 1995; Kuester, Mohamed, Piket-May & Holloway 2003):   Pn ´ ´ , n ˆ H+ = j ωPt ´ n ˆ ∇t Mn, (2.3) E+ t ´ Et = j ωμ0 n ˆ Mt ´ ∇t t ´ Ht 0 n ¨ (E+ ´ E´ ) = ´∇ ¨

Pt , 0

n ¨ (H+ ´ H´ ) = ´∇ ¨ Mt,

(2.4)

where “+” and “´” superscripts denote the fields at z Ñ 0 in the +z and ´z half-spaces, respectively. Here, the time-harmonic dependency in the form ej ωt is assumed. Boundary conditions (2.3) and (2.4) are referred to in the literature as Generalized Sheet Transition Conditions (GSTCs). Commonly, relations (2.4) are not considered in analysis of specific problems because they can be derived from (2.3) using Maxwell’s equations. The field discontinuities in the left sides of Eqs. (2.3) can be referred to as effective ´ ´ + total electric and magnetic currents, i.e., n ˆ (H+ t ´ Ht ) = Jtot,e and n ˆ (Et ´ Et ) = ´Jtot,m . Therefore, such effective currents are given by Jtot,e = j ωPt ´ n ˆ ∇t Mn,

Jtot,m = j ωμ0 Mt + n ˆ ∇t

Pn . 0

(2.5)

It is seen from these relations that effective magnetic polarization current can be created in a metasurface without magnetically polarizable inclusions. Indeed, properly tuning the normal component of electric polarization (creating its variations over the tangential plane), one can achieve the same response as with magnetic tangential polarization (Albooyeh, Kwon, Capolino & Tretyakov 2017). Boundary conditions (2.3) allow one to determine what polarization properties of the metasurface are required to provide a specific electromagnetic response to an incident wave. In other words, one can study transmission and reflection from the metasurface, knowing the induced polarizations, and vice versa. Note that these relations do not describe the properties of the metasurfaces; they only determine the tangential field jumps if the surface polarizations are known. In order to solve electromagnetic problems, they have to be complemented by relations between the polarizations and the fields, which model the metasurface properties. We discuss such relations next,

40

Analytical Modeling of Electromagnetic Surfaces

in Sections 2.4.1–2.4.3. There are several conventional models for characterization of metasurface response to incident fields in terms of its macroscopic parameters: models based on polarizabilities (Simovski, Kondratjev, Belov & Tretyakov 1999; Tretyakov 2003; Albooyeh, Tretyakov & Simovski 2016) and susceptibilities (Kuester et al. 2003; Achouri, Salem & Caloz 2015; Zhao, Liu & Al 2014) of metasurface inclusions, the equivalent impedance matrix model (Yatsenko, Maslovski, Tretyakov, Prosvirnin & Zouhdi 2003; Zhao, Engheta & Al 2011; Pfeiffer & Grbic 2013; Selvanayagam & Eleftheriades 2013; Epstein & Eleftheriades 2014; Asadchy, Albooyeh, Tcvetkova, Díaz-Rubio, Ra’di & Tretyakov 2016), and the diffractive interface model (Roberts, Inampudi & Podolskiy 2015).

2.4.1

Polarizability-Based Model Polarizabilities relate polarizations in a metasurface with the local fields which cause these polarizations. The local field at the position of a particular metasurface inclusion is a sum of the incident field and the scattered fields from all other inclusions, measured at the position of this particular inclusion (Tretyakov 2003): Eloc = Einc + β e ¨ p,

Hloc = Hinc + β m ¨ m.

(2.6)

Here β e and β m are the interaction constant dyadics that describe the effect of the entire metasurface on a single inclusion, p = SPs and m = μ0 SMs are the electric and magnetic dipole moments of a single inclusion (S is the unit cell area). The approximate expressions for the interaction constant dyadics can be found in (Tretyakov 2003). Thus, the polarization surface densities in (2.3) for a case of a general bi-anisotropic metasurface can be written as Ps =

α ee α em ¨ Eloc + ¨ Hloc, S S

μ0 Ms =

α me α mm ¨ Eloc + ¨ Hloc . S S

(2.7)

Here α ee , α mm , α em , and α me are the electric, magnetic, electromagnetic, and magnetoelectric (responsible for bi-anisotropy) polarizabilities of an individual inclusion, respectively. These individual polarizabilities depend only on the physical properties of the inclusion, such as its shape, size, and material composition. The polarizabilities do not depend on the external excitation. In most situations, characterization of metasurfaces in terms of the individual polarizabilities does not provide all necessary tools for the design. Since metasurfaces represent dense arrays of subwavelength inclusions, usually with resonant responses, the local fields entering into (2.7) may drastically differ from the incident fields and depend on the inclusions and the array period. Therefore, it is beneficial to express the polarization surface densities in terms of known incident fields Ei and Hi . Substituting (2.6) into (2.7), one can express polarization densities Ps and Ms as functions of fields Ei and Hi : pee pem pme pmm α α α α μ0 Ms = (2.8) ¨ Ei + ¨ Hi, ¨ Ei + ¨ Hi, S S S S where the dyadics with hats denote effective or collective polarizabilities which characterize not only the properties of individual inclusions but also their interactions in the Ps =

2.4 Homogenization Models of Metasurfaces

41

lattice. The general expressions for the effective polarizabilities as functions of the individual ones and the interaction constants of the array can be found in Niemi et al. (2013). Now we have all necessary tools for designing a metasurface with desired electromagnetic response for given incidence. First, one should substitute the desired reflected and transmitted fields together with given incident ones into the total fields in the left sides of Eqs. (2.3). Next, one can find the tangential and normal components of the required polarization surface densities (note that generally the system of equations is undetermined and one can implement metasurface with only tangential or only normal polarizations; Albooyeh et al. 2017). Finally, one determines from (2.8) what effective and individual polarizabilities are required to produce the desired response. The reverse approach is also possible. If one knows the individual polarizabilities of the inclusions and the lattice periodicity, it is possible to calculate the effective polarizabilities and, subsequently, the reflected and transmitted fields from the metasurface. Although it is possible to write general formulas relating reflected and transmitted fields from an arbitrary metasurface and its polarizability components, these formulas have complex structure and do not provide clear physical insight. Therefore, next we consider a special case which is very important for many practical metasurface realizations. Here we assume that a metasurface is illuminated by a normally incident plane wave and no normal polarization is induced. We also assume that the metasurface has uniaxial symmetry along the normal vector n, i.e., it works identically for any polarization of the incident wave. The geometry of the problem is shown in Figure 2.6. In this case, the polarizability dyadics can be decomposed onto symmetric (“co” terms) and antisymmetric (“cross” terms) parts: co I + α cr J , pee pee pee = α α t t co cr pem pem pem = α It + α J t, α

co cr pmm pmm pmm = α It + α J t, α co cr pme pme pme = α It + α J t. α

(2.9)

Here I t = I ´ zz = xx + yy is the transverse unit dyadic and J t = z ˆ I t = yx ´ xy is the vector-product operator. Under the aforementioned assumptions the reflected Er and transmitted Etr fields can be expressed in terms of the effective polarizabilities in a simple and elegant form (Niemi et al. 2013):

i i

e

r

r

m tr

tr Figure 2.6 Scattering from a uniform metasurface illuminated normally by a plane wave.

42

Analytical Modeling of Electromagnetic Surfaces

j "„ jω 1 co co cr cr pem ˘ α pme ´ α pee ˘ α p =´ It η0 α 2S η0 mm j * „ 1 cr cr co co pem pme pee pmm J t ¨ Ei, + η0 α ¯α ¯α ´ α η0 "„  j jω 1 co ˘ co cr cr pem ¯ α pme + α pee ˘ α p Etr = 1´ It η0 α 2S η0 mm j * „ jω 1 cr cr co co pem pme pee p ´ ¯α ˘α + α J t ¨ Ei, η0 α 2S η0 mm E˘ r

(2.10)

(2.11)

where η0 is the wave impedance of free space and the top and bottom signs refer to the scenarios when ki ÒÓ n (illumination from the upper half-space) and ki ÒÒ n (from the bottom half-space), respectively. These expressions provide a straightforward way for the analysis of metasurfaces formed by general bi-anisotropic inclusions, reciprocal and nonreciprocal (see discussion in the next section). One important implication of these equations is that the zeroreflection regime for normal incidence is not achievable with metasurfaces possessing solely electric or magnetic properties. To eliminate reflection, the metasurface should have either balanced electric and magnetic responses (so-called Huygens’ condition co = α co {η ) or proper bi-anisotropic response [so that all the expressions in pmm pee η0 α 0 the square brackets of (2.10) become zero]. Other important implications of expressions (2.10) and (2.11) are mentioned in Section 2.5. To summarize, the homogenization model based on polarizabilities provides a straightforward way to determine what kind of constituents and their properties a metasurface should incorporate in order to obtain specific electromagnetic response. Indeed, looking at Eqs. (2.10) and (2.11), we can immediately see how each type of polarization (electric, magnetic, or bi-anisotropic) affects the scattered fields. Another advantage of this model is the possibility to calculate required polarizabilities of the individual inclusion (sitting in free space, not interacting with other inclusions), which allows rigorous engineering of the whole metasurface. On the other hand, the use of the polarizability-based model has two disadvantages: values of collective polarizabilities do not directly indicate if a metasurface with given collective polarizabilities is lossless, lossy, or active, and it loses its meaning for metasurfaces whose unit cells cannot be viewed as separate inclusions (such as metal-backed metasurfaces or connected meshes).

2.4.2

Susceptibility-Based Model Another homogenization model is based on the macroscopic surface susceptibility tensors. It relates the averaged (in the metasurface plane) polarization surface densities Ps and Ms to electric Eav and magnetic Hav fields averaged over the two sides of the metasurface (Kuester et al. 2003; Achouri et al. 2015; Zhao et al. 2014): ? Ps = 0 φ ee ¨ Eav + μ0 0 φ em ¨ Hav, c (2.12) 0 φ me ¨ Eav, Ms = φ mm ¨ Hav + μ0

2.4 Homogenization Models of Metasurfaces

43

where dyadic φ denotes macroscopic surface-averaged susceptibilities (SI units of meters). The averaged fields can be expressed in terms of the incident, reflected, and transmitted fields as Ei + Er + Et Hi + Hr + Ht , Hav = . (2.13) Eav = 2 2 Note that all these fields are surface averaged over unit cell areas, so that only propagating Floquet harmonics contribute to (2.13). One can express the susceptibility dyadics in terms of the effective polarizability dyadics. Assuming the absence of normal polarizations in the metasurface and using relations (2.3), (2.8), (2.12), and (2.13), after some derivations we obtain ff «  ´1 j ω 1 j ω pee + pem ¨ S I t ´ α pmm pme , φ ee = (p )´1 ¨ α ¨α α 0 2η0 2η0 ff «  ´1 j ω 1 j ω pem + pem ¨ S I t ´ α pmm pmm , φ em = ? (p )´1 ¨ α ¨α α μ0 0 2η0 2η0 (2.14) ff «   j ωη0 ´1 1 j ωη0 ´1 pme + pme ¨ S I t ´ α pee pee , φ me = ? (m ) ¨ α ¨α α μ0 0 2 2 ff «   j ωη0 ´1 1 j ωη0 ´1 pmm + pme ¨ S I t ´ α pee pem , φ mm = (m ) ¨ α ¨α α μ0 2 2 where

  j ωη0 j ω ´1 ω2 pem ¨ S I t ´ α pmm pme, ¨α + α 2 4 2η0   jω j ωη0 ´1 ω2 pmm pme ¨ S I t ´ α pee pem . + ¨α α m = S I t ´ α 2η0 4 2 pee p = S I t ´ α

(2.15)

The main advantage of the homogenization model based on the macroscopic surface susceptibility tensors is that, knowing the required tensors for some specific electromagnetic response, one can immediately see if the metasurface inclusions must be lossy (tφ ij u ă 0) or possess some gain (tφ ij u ą 0). On the other side, the knowledge of the required susceptibility dyadics does not give us a straightforward way to determine the inclusions properties. The geometry of the inclusions is usually adjusted based on secondary quantities (such as scattering parameters) which are calculated for an ideal metasurface modeled by the required susceptibilities.

2.4.3

Model Based on Equivalent Impedance Matrix An alternative homogenization model for metasurfaces can be established based on the analogy between plane-wave propagation in free space and signal propagation in a transmission line (see, e.g., Tretyakov 2003; Yatsenko et al. 2003; Selvanayagam & Eleftheriades 2013). In this scenario, the electric and magnetic fields of a wave propagating in free space can be referred to voltages and currents of a signal propagating in

44

Analytical Modeling of Electromagnetic Surfaces

+ t

− t

+ t

− t



+

+

e



tot,e

Figure 2.7 A metasurface with effective electric response (left) and its equivalent

transmission-line model (right).

the equivalent transmission line. Components of the equivalent impedance matrix of the metasurface play the role of its macroscopic parameters. Let us consider a metasurface which under illumination by incident waves carries only electric effective currents Jtot,e (conduction currents or polarization currents) as the one shown in Figure 2.7 on the left. For simplicity we assume that no polarization transformations occur in the metasurface. Using (2.3) and (2.5), the total tangential fields at both sides of the metasurface simplify to   ´ ´ + ´ E = 0, n ˆ H ´ H (2.16) = Jtot,e . E+ t t t t Such a metasurface in the transmission line representation is modeled by a shunt impedance Ze as shown on the right side of Figure 2.7. Indeed, the voltages at both sides of the impedance are equal, while the currents experience a jump v+ ´ v´ = 0,

i + + i ´ = v+ {Ze,

(2.17)

expressing a response equivalent to that described by (2.16). Relating the electric fields ´ ´ + + ´ E+ t , Et to the voltages v , v , and the magnetic fields n ˆ Ht , n ˆ Ht to the currents + ´ i , ´i , one can express the shunt impedance as Ze Jtot,e = E+ t .

(2.18)

This expression is analogous to the definition of sheet or grid (for grids of inclusions) impedance (Tretyakov 2003). Thus, electromagnetic response of the considered metasurface can be completely described by its shunt (or sheet) impedance. It should be noted that in the general case, when the metasurface is anisotropic, its equivalent shunt impedance becomes a dyadic. Likewise, we can find the equivalent transmission line model for a metasurface with only magnetic response (Jtot,e = 0, Jtot,m ‰ 0) (see the left side of Figure 2.8). From (2.3) and (2.5), the total tangential fields at both sides of the metasurface are simplified to   ´ ´ n ˆ H+ = 0. (2.19) E+ t ´ Et = n ˆ Jtot,m, t ´ Ht The equivalent transmission line of the metasurface consists of a series impedance Zm (series admittance Ym ) as shown on the right side of Figure 2.8. The voltages and currents at both sides of the impedance are given by

2.4 Homogenization Models of Metasurfaces

45

Figure 2.8 A metasurface with effective magnetic response (left) and its equivalent

transmission-line model (right).

Figure 2.9 A general metasurface with electric and magnetic responses (left) and its equivalent

transmission-line model (right).

v+ ´ v´ = i + Zm = i + {Ym,

i + + i ´ = 0.

(2.20)

Following the same procedure as for the previous example, we find the relation between the series admittance and the effective magnetic current density: Ym Jtot,m = H+ t .

(2.21)

In the case of a general metasurface possessing both electric and magnetic polarization responses (Jtot,e ‰ 0, Jtot,m ‰ 0), the fields at both sides are written as (see the left side of Figure 2.9)   ´ ´ E+ n ˆ H+ (2.22) = Jtot,e . t ´ Et = n ˆ Jtot,m, t ´ Ht Such a metasurface can be modeled by equivalent - or T -circuits. Figure 2.9 (the right side) depicts the T -circuit transmission-line representation. The voltages at both sides of the metasurface can be written in terms of the currents v+ = (Z1 + Z3 )i + + Z3 i ´, v´ = Z3 i + + (Z2 + Z3 )i ´ . These relations can be written in a matrix form:  +     +  v Z11 Z12 i = ¨ , v´ Z21 Z22 i´

(2.23)

(2.24)

where Z11 = Z1 + Z3 , Z12 = Z21 = Z3 , and Z22 = Z2 + Z3 . Alternatively, we can express the impedances as Z1 = Z11 ´ Z12 , Z2 = Z22 ´ Z21 , and Z3 = Z12 . The matrix in (2.24) is called an equivalent impedance matrix of the metasurface. By analogy

46

Analytical Modeling of Electromagnetic Surfaces

with the transmission-line model, we can write similar relations for the total tangential fields:     +   n ˆ H+ Et Z11 Z12 t = ¨ . (2.25) Z21 Z22 E´ ´n ˆ H´ t t In the case of anisotropic metasurfaces, the components of the impedance matrix become dyadics. The equivalent impedance matrix representation is a powerful tool for designing metasurfaces of arbitrary complexity. It has several advantages. First of all, when the fields at both sides of the metasurface are specified, one can easily calculate the impedances in (2.25) and, based on this knowledge, extract the following information: whether metasurface should be capacitive ([Zij ] ă 0) or inductive ([Zij ] ą 0), lossy ([Zij ] ą 0) or active ([Zij ] ă 0), reciprocal (Z12 = Z21 ) or nonreciprocal (Z12 ‰ Z21 ), symmetrical (Z11 = Z22 ) or nonsymmetrical (Z11 ‰ Z22 ), etc. Second, based on the transmission-matrix approach, one can easily calculate reflection/transmission properties from a cascade of different metasurfaces with known equivalent impedance matrices. Alternatively, a metasurface with known equivalent impedance matrix can be replaced by a cascade of basic metasurfaces. The latter feature was exploited in multiple works (see, e.g., Wong et al. 2016) for designing metasurfaces with complex equivalent impedance matrix using a cascade of three shunt impedances (metasurfaces with only electric response) whose implementation is straightforward. Third, impenetrable metasurfaces (with a ground plane) can be equally easily characterized by the impedance matrix homogenization model. In this case, the metasurface design is simplified because the impedance matrix degenerates into a single scalar (or dyadic for anisotropic structure), called surface impedance Zs . The total tangential fields are written as + E+ t = Zs n ˆ Ht .

2.5

Bi-anisotropy and Nonreciprocity: Definitions and Enabled Functionalities

2.5.1

Bi-anisotropy The concept of metamaterials is strongly associated with the concept of spatial dispersion. When the size of inclusions or the distances between them become comparable to the wavelength, the composite constructed from them possesses nonlocal polarization response and generally cannot be described in terms of the permittivity and permeability quantities solely (electric and magnetic polarizabilities/susceptibilities in the case of metasurfaces). In this case, it is not enough to know the local electric field at one point in order to find the induced dipole moment in the inclusion. Information about the electric field in the entire volume occupied by the inclusion is required. Equivalently, it is enough to know the electric field vector and all its spatial derivatives at one point of the inclusion (e.g., its geometrical center at r0 ). The electric field in other points with coordinates r can be expressed via a Taylor series at the central point: 1 E(r + r0 ) = E(r0 ) + ∇j E(r0 )rj + ∇j ∇k E(r0 )rj rk + ¨ ¨ ¨ , 2

(2.26)

2.5 Bi-anisotropy and Nonreciprocity: Definitions and Enabled Functionalities

47

Figure 2.10 Illustration of an isotropic chiral slab comprising a nonperiodic array of randomly oriented metallic helices embedded in supporting dielectric material.

where ∇j denotes spatial derivative with respect to rj and the repeating indices imply summation according to the conventional tensor notations. Bi-anisotropy (or reciprocal chirality) is commonly referred to spatial dispersion effects of the first order (so-called weak dispersion), i.e., when the term with firstorder derivative in series (2.26) cannot be neglected. Here the name “weak dispersion effect” implies that the constitutive relations for materials with such spatial dispersion do not include field derivatives in explicit form and therefore “appear” to be local. In contrast, strong spatial dispersion stands for the case when higher-order derivatives in series (2.26) cannot be neglected and, therefore, the constitutive relations will include spatial derivatives of the fields. In this case, no effective macroscopic parameters can be introduced for the material. A classical example of a bi-anisotropic medium is an artificial composite comprising a three-dimensional arrangement of metal helices inside a supporting dielectric material matrix (see Figure 2.10). Let us assume that the composite sample is illuminated by an incident wave whose wavelength is comparable to the size of each helix. In this case, the incident electric field moves free electrons along the helical inclusion, inducing an effective magnetic moment (circulating current) in the helix. Multiple induced magnetic moments in the inclusions correspond to macroscopic magnetic polarization of the composite. On the other hand, the incident electric field also generates an electric dipole moment in the inclusion. This electric dipole moment is caused by two different effects: ordinary polarization in external electric field (local effect) and additional polarization due to the finite size of the helix and nonuniform circulating electric field in form of ∇ ˆ E (nonlocal effect). Constitutive relations of a bi-anisotropic medium are usually written in the following form (Serdyukov, Semchenko, Tretyakov & Sihvola 2001): ? D =  ¨ E ´ j 0 μ0 κ ¨ H,

? T B = μ ¨ H + j 0 μ0 κ ¨ E,

(2.27)

where T denotes the transpose operator and κ is the chirality dyadic. Constitutive relations (2.27) can be transformed to the following microscopic form (for single inclusions):

48

Analytical Modeling of Electromagnetic Surfaces

ext

ext

a

ext

b

c

Figure 2.11 Topologies of basic reciprocal bi-anisotropic inclusions. External electric field Eext induces magnetic moments m in the inclusions. (a) A right-handed chiral canonical helix. (b) An omega inclusion with shape of the  letter. (c) A twisted omega inclusion.

Ps =

α ee α em ¨ Eloc + ¨ Hloc, S S

(2.28) α me α mm μ 0 Ms = ¨ Eloc + ¨ Hloc, S S which corresponds to relations (2.7). These relations express that in a spatially dispersive material, each inclusion can be polarized electrically and magnetically by both 1 electric and magnetic fields. The electromagnetic α em and magnetoelectric α me polarizabilities characterize the strength and type of bi-anisotropic response of the inclusions. It is important to introduce a basic classification of bi-anisotropic reciprocal inclusions. A typical inclusion with strong electric polarizability αee is a resonant straight metal wire of about λ{2 length. A double split-ring resonator exhibits large magnetic polarizability αmm (Pendry, Holden, Robbins & Stewart 1999). Naturally, an inclusion with magnetoelectric polarizability αem should in a sense combine characteristics of these two elements. As is seen from (2.28), there are two basic scenarios of magnetoelectric coupling depending on the mutual orientation of the field and the dipole moment which is induced by this field. The first scenario, where the moment and the field vectors are collinear, can be realized with a canonical metallic three-dimensional helix (Jaggard, Mickelson & Papas 1979) shown in Figure 2.11a. Under excitation by vertically oriented electric fields, the current induced in the wire forms a loop corresponding to a magnetic moment along the external electric field. The direction of the magnetic moment as well as the sign of the magnetoelectric polarizability depends on the helicity state of the helical inclusion. In the second scenario, the induced moment and the field vectors are orthogonal. This can be realized by orienting the loop of the helix into another plane, as shown in Figure 2.11b. This planar geometry, often referred to as the omega shape (Simovski, Tretyakov, Sochava, Sauviac, Mariotte & Kharina 1997) (after Greek letter ), provides magnetoelectric polarization orthogonal to the exciting

1

Indeed, the actual mechanism of polarization is due not to magnetic field but to spatial dispersion; however, the description of polarization in terms of local fields significantly simplifies theoretical analysis.

2.5 Bi-anisotropy and Nonreciprocity: Definitions and Enabled Functionalities

49

field. The sign of the magnetoelectric polarizability can be reversed by twisting the loop of the inclusion, as shown in Figure 2.11c.

2.5.2

Nonreciprocity The concept of nonreciprocity is closely related to time inversion of electromagnetic processes occurring inside matter. Let us consider time inversion in electromagnetics (hereafter, time inversion implies inversion of the time sign as well as time flow direction). Maxwell’s equations are symmetric with respect to time inversion t Ñ ´t (which implies also dt Ñ ´dt), i.e., all electromagnetic processes taking place in a closed lossless system (without any external sources or fields) are reversible in time. This statement holds for both microscopic and macroscopic Maxwell’s equations under the assumption of linear medium (note it is not applicable to “active” media since they imply the existence of external to the system sources which break the time symmetry). When the time direction is reversed dt Ñ ´dt, the electric charge does not change (an even quantity under time inversion), therefore, the electric current changes sign (an odd quantity under time inversion) since it is the time derivative of the charge. Equivalent statements hold also for the corresponding density quantities ρ and J. Since the Lorentz force acting on electric charge q (in general moving with the linear speed v) F = qE + qv ˆ B

(2.29)

is even under time inversion (acceleration and mass are even parameters), the electric field E is invariant and the magnetic induction B changes sign. Now the invariance of macroscopic Maxwell’s equations under time reversal is obvious: from Gauss’s law it follows that the electric induction D is an even parameter under time reversal, while from Ampere’s law, the magnetic field H is a time-odd parameter. If an electromagnetic system under consideration is not closed so that there are some external perturbations acting on it and they are odd with respect to time inversion, timereversal symmetry of Maxwell’s equations in the system becomes broken. Such perturbations can be of nonelectromagnetic as well as electromagnetic nature (in this case they should be external to the system). A good example is an external static magnetic field bias. Materials such as metals and ferromagnetics placed in this field would exhibit different responses for different directions of time flow in the system. Here it is assumed that the external magnetic field is invariant to the time flow since it is external to the considered system. Materials that possess different electromagnetic response under time reversal are called nonreciprocal. Let us consider a classical optical example which illustrates the difference between reciprocal and nonreciprocal materials. It is well known that chiral materials, such as sugar solutions, exhibit different refractive indices for electromagnetic waves with leftand right-handed polarizations. The reason for this effect is that in chiral materials, right-handed and left-handed inclusions are in different proportions. Let us consider the case when a chiral material slab is illuminated by a linearly polarized wave, as shown in Figure 2.12a. The incident wave can be represented as a sum of two waves with

50

Analytical Modeling of Electromagnetic Surfaces

a

b

c

d

Figure 2.12 Wave propagation in opposite directions through (a–b) a reciprocal chiral slab and (c–d) a nonreciprocal magneto-optical slab. The transparent cylinders depict the slabs. The red and blue arrows denote the electric fields of left and right circularly polarized waves, respectively. A black and white version of this figure will appear in some formats. For the color version, please refer to the plate section.

opposite circular polarizations and equal phases. These waves of different handedness travel through the material with different speeds, which results in accumulation of the phase difference between them. At the second interface, the superposition of the two circularly polarized waves with different phases is a linearly polarized wave with a tilted polarization axis. Thus, chiral materials rotate polarization plane of waves propagating through them. This effect is called the optical activity or isotropic gyrotropy effect. Since chiral materials are reciprocal (no external perturbations are required for its operation), the incident wave, which has passed the material, under time reversal would return back along the same way, and its field vectors would form the same traces in space (see Figure 2.12b). A similar effect of polarization plane rotation is observed for waves traveling through magneto-optical (gyrotropic) materials placed in external bias field H0 (an external time-odd parameter). Polarization rotation in such materials occurs due to anisotropic permeability tensor and is called the Faraday effect (shown in Figure 2.12c). It can be observed, for example, for light incident on a slab of a magneto-optical material. In contrast to the previously considered reciprocal scenario, the incident wave under time reversal experiences a different response from the magnetized magneto-optical material and the traces of the field vectors do not coincide (see Figure 2.12d). This effect is nonreciprocal and is the basis of very important electromagnetic devices, such as isolators and circulators.

2.5 Bi-anisotropy and Nonreciprocity: Definitions and Enabled Functionalities

51

It appears that nonreciprocal materials break time-reversal symmetry, and Maxwell’s equations become not reversible with respect to time. This happens because the system under consideration is not closed. If one adds also the source of the magnetic bias field into the system, it becomes closed and time-inversion symmetry will hold. Indeed, any source of static magnetic field must include some circulating direct currents which maintain this field: it can be an electromagnet with a solenoid coil or natural magnets consisting of ordered microscopic current loops (magnetic dipole moments) formed by rotating electrons. Thus, under time reversal in this electrodynamic system, the currents which form the static magnetic field will change their directions and the magnetization field will be reversed. In this case, the wave propagation inside the ferromagnetic material becomes reversible in time. Other examples of nonreciprocal materials are magnetized plasma and magnetized graphene. Nonreciprocal response can be achieved also by other means: materials moving with some speed, magnetless active materials mimicking electron spin precession of natural magnets (Kodera, Sounas & Caloz 2011a; Kodera, Sounas & Caloz 2011b), and nonlinear materials (Mahmoud, Davoyan & Engheta 2015). Nonreciprocal materials (namely, materials biased by some external time-odd parameter) can be used to construct nonreciprocal bi-anisotropic composites. Constitutive relations for such composites (assuming no spatial dispersion of the first order) are written as (Serdyukov et al. 2001) ? D =  ¨ E + 0 μ0 (χ s + Va ˆ I ) ¨ H,

? B = μ ¨ H + 0 μ0 (χ s ´ Va ˆ I ) ¨ E, (2.30)

where χ s and Va ˆ I are the symmetric and anti-symmetric parts of the general nonreciprocity coupling dyadic. Based on the time-reversal properties of these constitutive relations, it is seen that both parts of the nonreciprocity dyadic are time-odd, or in other words, the material response depends on an external time-odd parameter (e.g., magnetic bias field). Due to the spatial-symmetry properties, construction of a nonreciprocal material with a symmetric bi-anisotropic dyadic χ s requires that its internal structure should be mirror-symmetric (Serdyukov et al. 2001). Such materials or media were named after Bernard Tellegen who suggested the idea of an electromagnetic gyrator as a building block of these media (Tellegen 1948). Nonreciprocal bi-anisotropic materials with antisymmetric dyadic Va ˆ I , in contrast, must have a structure which cannot be superposed onto its mirror image. This class of media was named an artificial “moving” medium. Although it is in fact at rest, its electromagnetic response is analogous to that of an ordinary material which is truly moving with some speed v = Va (a true vector) (Serdyukov et al. 2001; Sihvola & Lindell 1995). Figure 2.13 demonstrates topologies of nonreciprocal bi-anisotropic inclusions that can be used as building blocks in composite materials of the mentioned two types. Both inclusions consist of a ferrite sphere (here, ferrite is chosen since it is nonconductive in contrast to other magnetic alloys) magnetized by external magnetic field H0 and located in the proximity of metal wires of a specific shape. The electric field of an incident wave excites the electric current along the wires which, in turn, excites alternating magnetic field around the wires. This magnetic field induces a magnetic

52

Analytical Modeling of Electromagnetic Surfaces

i

i

0 0

a

b

Figure 2.13 Topologies of nonreciprocal bi-anisotropic inclusions. (a) Uniaxial Tellegen inclusion. (b) Uniaxial artificial moving inclusion. The ferrite sphere is depicted in the center of each figure. The inclusions are excited by an incident electric field Ei . For clarity, magnetic moments m induced due to only nonreciprocal effects are shown.

moment in the ferrite sphere. Likewise, the incident magnetic field excites an electric dipole moment in the wires through magnetization of the sphere. The induced moment in the inclusion shown in Figure 2.13a is co-directed with the electric field which caused it. This inclusion was firstly proposed in Tretyakov (1998) and experimentally tested in Tretyakov, Maslovski, Nefedov, Viitanen, Belov, and Sanmartin (2003). It should be noted that in addition to the nonreciprocal Tellegen response, the inclusion exhibits also reciprocal bi-anisotropic properties of a uniaxial omega inclusion. The inclusion shown in Figure 2.13b, sometimes named as an artificial moving element (Tretyakov 1998), in addition to the nonreciprocal bi-anisotropic response, exhibits reciprocal chiral effects due to its mirror-asymmetric shape. The analytical polarizabilities of the nonreciprocal inclusions shown in Figure 2.13 were reported in Mirmoosa, Ra’di, Asadchy, Simovski, and Tretyakov (2014). It is important to mention that nonreciprocal bi-anisotropic coupling is not an effect of spatial dispersion. In contrast to reciprocal spatially dispersive inclusions where magnetoelectric and magnetic response can occur only due to their finite sizes, nonreciprocal inclusions exhibit these responses even in locally uniform external fields (when the particle size is negligibly small compared to the wavelength). For example, in the considered ferrite-based inclusions, a uniform electric field excites electric current in the wires, which in turn induces magnetic moments in the ferrite sphere. Thus, generally, bi-anisotropic properties of a medium can be caused by reciprocal spatial dispersion effects or nonreciprocal magnetoelectric coupling.

2.5.3

Enabled Functionalities Let us summarize all basic functionalities available with metasurfaces possessing reciprocal or nonreciprocal bi-anisotropic couplings. We consider for simplicity the case of uniaxial metasurfaces whose reflection and transmission properties are described by

2.5 Bi-anisotropy and Nonreciprocity: Definitions and Enabled Functionalities

53

expressions (2.10) and (2.11). There are both co- and cross-components of electroco , α cr , α co , and α cr entering these expressions. From the pem pme pme pem magnetic polarizabilities α Onsager–Casimir relations (Serdyukov et al. 2001; Landau & Lifshitz 2013) T

α ee (H0 ) = α ee (´H0 ),

T

α mm (H0 ) = α mm (´H0 ),

T

α me (H0 ) = ´α em (´H0 ), (2.31)

we can separate the polarizability components responsible for four different types of T bi-anisotropy (here construction of type α ij (´H0 ) denotes a transposed polarizability dyadic for the same medium when all the external nonreciprocal parameters switch cr = signs). For reciprocal chiral inclusions the magnetoelectric dyadic is symmetric (p αem co co pem = ´p αme . Omega inclusions are reciprocal and asym0) and we have condition α cr = α cr . The polarizabilities of nonreciprocal inclusions, namely, pme pem metric, therefore, α co = α co and α cr = ´p cr , pme pem pem αme Tellegen and artificial moving inclusions, obey conditions α respectively (note that in the case of nonreciprocal inclusions α em (´H0 ) = ´α em (H0 )). Based on this separation of magnetoelectric polarizability components, we can now analyze what new functionalities in metasurfaces each of them provides. Table 2.1 presents these available functionalities together with possible realizations. Metasurfaces with reciprocal and nonreciprocal bi-anisotropic inclusions became subjects of a variety of publications (see, e.g., Elsakka et al. 2016; Ra’di et al. 2013; Ra’di et al. 2014b;

Table 2.1 Typical response of reciprocal and nonreciprocal bi-anisotropic uniaxial metasurfaces for plane-wave illumination

Electromagnetic response

Possible realizations

Polarization rotation in

1 Metasurface with non-reciprocal uniaxial Tellegen inclusions (Ra’di et al. 2014a) 2 Nonreciprocal materials such as magnetized plasmas and ferrites (Lindell, Sihvola, Tretyakov & Viitanen 1994)

reflection (Er = a0 J t ¨ Ei , where a0 is a complex coefficient and 0 ă |a0 | ď 1)

Required collective polarizabilities co = α co ‰ 0 pme pem α

cr ´ α cr {η ‰ 0 pee pmm η0 α 0

Polarization rotation in transmission (Etr = a0 J t ¨ Ei )

1 Metasurface with reciprocal uniaxial chiral inclusions (Niemi et al. 2013) (optical activity) 2 Nonreciprocal materials such as magnetized plasmas and ferrites (the Faraday effect)

co = ´p co ‰ 0 pem αme α

Reflection asymmetry for illuminations from opposite ´ sides (E+ r ‰ Er )

Metasurface with reciprocal uniaxial omega inclusions (Ra’di et al. 2014b)

cr = α cr ‰ 0 pme pem α

Transmission asymmetry for illuminations from ´ opposite sides (E+ tr ‰ Etr )

Metasurface with nonreciprocal uniaxial artificial moving inclusions (Ra’di et al. 2014a)

cr = ´p cr ‰ 0 pem αme α

cr + α cr {η ‰ 0 pee pmm η0 α 0

54

Analytical Modeling of Electromagnetic Surfaces

Ra’di et al. 2014a; Niemi et al. 2013). Recent advances on this topic were summarized in review paper (Asadchy, Díaz-Rubio & Tretyakov 2018).

2.6

Metasurfaces for Shaping Transmitted Fields and Reflected Fields The analytical models presented in this chapter allow synthesizing metasurfaces for numerous applications, which include the control of the phase, direction, or propagation of the scattered fields. In what follows, we provide a set of examples based on the equivalent impedance matrix model which show the route for full control of the refraction and reflection. The impedance matrix method has been applied in many scenarios and has shown several advantages for modeling metasurfaces in comparison with polarizabilities/susceptibilities methods (especially for the reflection regime). First, we will study the manipulation of the propagation direction, both in the transmission and reflection scenarios. Second, we will introduce the impedance tensor required for controlling the polarization of the reflected field. Finally, we will present the next step toward the implementation of more complex functionalities where multiple plane waves are controlled.

2.6.1

Control of Wave Propagation Direction in Transmission The most simple and illustrative example of the possibilities that metasurfaces offer for controlling refraction is the arbitrary manipulation of the direction of propagation of a transmitted wave, also called anomalous refraction. An illustration of this functionality is shown in Figure 2.14a. For this example, we consider TE-polarization, i.e., the electric field polarized along the y-direction. As explained in Section 2.4.3, the first step in the design process is to fully define the fields at both sides of the metasurface: ´j k1 sin θi x , E+ t = Ei e ´j k2 sin θt x , E´ t = T Ei e

n ˆ H+ t =

cos θi Ei e´j k1 sin θi x , η1

n ˆ H´ t =T

cos θt Ei e´j k2 sin θt x , η2

(2.32) (2.33)

where η1 and η2 are the wave impedances of the media above and below the metasurface and T is the global transmission coefficient which relates the amplitudes of the incident and refracted plane waves, T = |T |ej φt ,

(2.34)

where φt represents an arbitrary phase shift. The amplitude of the global reflection coefficient can be calculated by requiring the energy conservation over a unit cell, which imposes the continuity of the normal component of the Poynting vector at each point of the metasurface:  1  ´ 1  + ´˚ ˚ = ˆ H  Et ˆ H+  E . (2.35) t t t 2 2

2.6 Metasurfaces for Shaping Transmitted Fields and Reflected Fields

i

55

i r

i i

i

i

1

i

1 2

t 2

t

t

a i

b i

i

i i

i

r

r

r

i r

r

r r

r1 r1

r

r −

r2

c

r

r

r

d

Figure 2.14 Schematic of metasurfaces for controlling the wave propagation direction in (a) transmission and (b) reflection, (c) with additional polarization control in reflection and (d) multiple waves in reflection.

This condition defines a relation between the incident and the reflected fields, which reads d c cos θi η1 |T | = . (2.36) cos θt η2 Knowing the expression for the incident and reflected plane waves, we just need to apply Eq. (2.25) and calculate the values of the impedance matrix. For reciprocal metasurfaces, i.e., Z12 = Z21 , these values can be written as Z11 = j

η1 cot t (x), cos θi

Z22 = j ?

η η

2 Z12 = Z21 = j ?cos θ1 cos θ i

t

η2 cot t (x), cos θt

(2.37)

1 sin t (x) ,

where t (x) = k1 sin θi x ´k2 sin θt x. Looking to the solution, we can see that the values of the impedance matrix are purely imaginary, meaning that a lossless implementation is possible. Figure 2.15 represents the components of the impedance matrix when θi = 0 to θr = 70˝ . We can see that Eqs. (2.37) are periodic functions whose period can be calculated as D = λ{|sin θi ´ sin θt |. The required properties of the metasurface can be understood from the equivalent circuit shown in Figure 2.9 (on the right). The corresponding values for each component of the equivalent circuit are represented in Figure 2.15, where we can see that the circuit

56

Analytical Modeling of Electromagnetic Surfaces

10

10

X1

X11 X12

5

X2

5

X3

0

X/

X/

1

1

X22

-5

-10 -0.5

0

-5

-0.25

0

0.25

0.5

-10 -0.5

-0.25

0

0.25

0.5

x/

x/

Figure 2.15 Impedance profile for anomalous refraction metasurface from θi = 0 to θr = 70˝ :

(a) impedance matrix; (b) equivalent circuit (see Figure 2.9).

becomes asymmetric because Z1 ‰ Z2 . As a consequence, the metasurface has to be bi-anisotropic of the omega type. Appropriate topologies include arrays of -shaped inclusions, arrays of split rings, double arrays of patches (patches on the opposite sides of the substrate must be different to ensure proper magnetoelectric coupling) (Chen, Abdo-Sanchez, Epstein & Eleftheriades 2017; Lavigne, Achouri, Asadchy, Tretyakov & Caloz 2018), etc.

2.6.2

Control of Wave Propagation Direction in Reflection Similar analysis can be made for the manipulation of the direction of reflected waves. In this case, we will consider reflective metasurfaces assuming that the tangential fields ´ behind the metasurface are zero, E´ t = 0 and n ˆ Ht = 0. This assumption allows us to model the metasurface with the surface impedance defined as Zs =

E+ t . n ˆ H+ t

(2.38)

Tangential components of the electric and magnetic fields in medium 1 can be written as ” ı ´j k1 sin θi x E+ + Re´j k1 sin θr x , t = Ei e n ˆ H+ t =

ı Ei ” cos θi e´j k1 sin θi x ´ R cos θr e´j k1 sin θr x , η1

(2.39)

where R is the global reflection coefficient which relates the amplitudes of the incident and reflected plane waves. This coefficient can be written as R = |R|ej φr ,

(2.40)

with φr being an arbitrary phase shift. As in the transmission scenario, the amplitude of the reflection coefficient can be extracted by imposing the energy conservation. The power carried by the incident plane wave Pi = carried by the reflected wave Pr =

|Er |2 2η1

|Ei |2 2η1

cos θi has to be equal to the power

cos θr . Equating these two expressions, we

2.6 Metasurfaces for Shaping Transmitted Fields and Reflected Fields

57

8

Rs

6

Xs

Zs/

1

4 2 0 -2 -4

-0.5

-0.25

0

0.25

0.5

x/ Figure 2.16 Surface impedance profile for an anomalous reflector from θi = 0 to θr = 70˝ .

can find the relation between the amplitudes of the incident and reflected waves. This relation reads d c cos θi η1 . (2.41) |R| = cos θr η2 Knowing the expression for the desired field, we can calculate the surface impedance by using Eq. (2.38), which reads ? ? cos θr + cos θi ej r (x) η1 a a Zs (x) = a , (2.42) cos θi cos θr cos θi ´ cos θr ej r (x) where r (x) = k1 (sin θi ´ sin θr )x. The value of the surface impedance is represented in Figure 2.16 when θi = 0 to θr = 70˝ . Interestingly, this impedance is a complex number, meaning that in some regions of the metasurface, the z-component of the total Poynting vector (normal to the surface) must be positive, and in others, it should be negative. The positive values of the real part of the surface impedance can be interpreted as losses (normal component of the Poynting vector entering in the metasurface) and negative values as gain (power emerging from the metasurface). However, the required surface impedance indicates nonlocal properties, which can be realized by excitation of additional auxiliary evanescent fields or by carefully engineering the surface reactance profile. Realization of this surface impedance profile can be achieved based on a nonuniform array of metal patches separated by a dielectric layer from a ground plane (DíazRubio et al. 2017).

2.6.3

Control of Polarization in Reflection During a reflection transformation, the polarization of the reflected wave can be made different from the incident wave. For an oblique illumination by a TE-mode plane wave,

58

Analytical Modeling of Electromagnetic Surfaces

an anomalously reflected plane wave in the TM mode is illustrated in Figure 2.14c. In this case, a surface impedance treatment is applicable, but the surface impedance becomes a tensor, Z s , relating the tangential electric and magnetic fields via + E+ t = Z s n ˆ Ht .

(2.43)

At z = 0+ , let the tangential field components be written as ˆ Ei0 Rxy e´j k1 x sin θr + yˆ Ei0 e´j k1 x sin θi , E+ t =x Ei0 Ei0 ˆ H+ cos θi e´j k1 x sin θi + yˆ Rxy e´j k1 x sin θr , t =x η1 η1 cos θr

(2.44)

where Rxy = xˆ ¨ E+ y ¨ E+ t (x = 0){ˆ t (x = 0) is the cross-coupled reflection coefficient between the y-directed incident and x-directed reflected electric field amplitudes evaluated at the coordinate origin. Let the reflection coefficient be written as Rxy = |Rxy |ej φr . For power-conserving reflections, there should be no real power penetrating the metasurface. Setting the z-component of the Poynting vector, Sz+ = zˆ ¨(1{2)tE+ t ˆ ˚ u, equal to zero, we find H+ t a |Rxy | = cos θi cos θr . (2.45) The phase angle φr can be arbitrary. In order to characterize the metasurface as a reactive surface, let Z s be written in terms of a reactance tensor X s as j „ Xxx Xxy , (2.46) Z s = j Xs = j Xyx Xyy where all four reactance elements are real-valued. In terms of the reactance elements, the impedance boundary condition (2.43) can be written in a matrix form as ff j „ j« „ Ei0 ´j k x sin θr Xxx Xxy ´ η1 cos θr Rxy e 1 Ei0 Rxy e´j k1 x sin θr =j . (2.47) Ei0 ´j k1 x sin θi Ei0 e´j k1 x sin θi Xyx Xyy η1 cos θi e There are four real-valued equations in (2.47), so the four reactance values can be uniquely determined. They are found to be ff „ j « η1 |Rxy | η1 cos θr cot tot Xxx Xxy cos θi sin tot = , (2.48) η1 cos θr η1 Xyx Xyy cos θ cot tot |R | sin  xy

tot

i

where tot = r + φr = k1 x(sin θi ´ sin θr ) + φr . For lossless cases, the reactance tensor becomes a Hermitian tensor. For a reactance tensor of the form (2.46), Xs becomes a Hermitian tensor if the two off-diagonal terms are equal to each other. It is seen that this condition leads to (2.45). There are a couple of observations regarding (2.48) that are worth mentioning. First, it is noted that (2.48) is a special case of a Hermitian tensor in that the off-diagonal elements have no imaginary parts. In general, a lossless surface impedance that satisfies the boundary condition (2.43) is not unique. Second, under the condition of (2.45), (2.48) represents a lossless, reactive surface impedance for all x. This contrasts with the

2.6 Metasurfaces for Shaping Transmitted Fields and Reflected Fields

59

Figure 2.17 A polarization-converting anomalous reflector from θi = 0 to θr = 70˝ . The incident

plane wave is TE-polarized (with the electric field along the y-axis direction), and the reflected plane wave is TM-polarized. (a) The normalized reactance tensor elements over a unit cell period. (b) A snapshot of the y-component of the electric field. (c) A snapshot of the y-component of the magnetic field.

scalar surface impedance (2.42) for polarization-preserving anomalous reflectors, where the surface impedance alternates between active and passive ranges. This pointwise lossless characteristic is due to the orthogonality between the incident and reflected wave polarizations, resulting in no interference in the local Poynting vector. As an example, anomalous reflection from normal incidence (θi = 0) to a θr = +70˝ reflection is designed and numerically analyzed. Choosing an in-phase reflection (φr = 0), the cross-coupled reflection coefficient is found to be Rxy = 0.585 from (2.45). The four reactance tensor elements are plotted in Figure 2.17a, over a period of D = λ{| sin θi ´ sin θr | = 1.064λ. The two diagonal elements, Xxx and Xyy , have a period of D{2, while the off-diagonal elements, Xxy and Xyx , have a period of D. The off-diagonal terms are equal to each other, appropriately for the lossless design. Upon illumination by a unit-amplitude, y-polarized plane wave at normal incidence, the scattering characteristics are simulated using COMSOL Multiphysics. A snapshot of the y-component of the total electric field is plotted in Figure 2.17b. Only the incident field appears because there is no co-polarized reflection. A snapshot of the y-component of the magnetic field plotted in Figure 2.17c shows a single TM-mode plane wave propagating in the +70˝ direction as desired. In order to realize the polarization-converting anomalous reflector, one can utilize various anisotropic metallic patterns (crosses, split rings, etc.) over a metal-backed dielectric layer (Grady, Heyes, Chowdhury, Zeng, Reiten, Azad, Taylor, Dalvit & Chen 2013; Cheng, Withayachumnankul, Upadhyay, Headland, Nie, Gong, Bhaskaran, Sriram & Abbott 2014; Ma, Wang, Kong & Cui 2014).

60

Analytical Modeling of Electromagnetic Surfaces

2.6.4

Control of Multiple Waves in Reflection An example of controlling more than one reflected plane waves is illustrated in Figure 2.14d, where a reflectionless split of a normally incident plane wave into two plane waves propagating in the ´θr and +θr directions is shown. This polarization-preserving wave splitting transformation can be described by a scalar surface impedance Zs as in (2.38). Let the complex amplitudes of the incident, ´θr -reflected, and +θr -reflected waves be denoted by Ei0 , Er1 , and Er2 , respectively. Then, the total tangential fields are given by   ˆ Ei0 + Er1 e´j k1 x sin θr + Er2 ej k1 x sin θr , E+ t =y   (2.49) r1 ´j k1 x sin θr r2 j k1 x sin θr ˆ EZi0i ´ E H+ ´E , t =x Zr1 e Zr2 e where Zi = η1 is the wave impedance of the incident wave and Zr1 = Zr2 = η1 { cos θr denotes the wave impedance for the two reflected plane waves with respect to the z-axis (surface normal). Defining the relative magnitudes (a1 and a2 ) and phases (δ1 and δ2 ) of the reflected field amplitudes with respect to that of the incident field, let us write Er1 = Ei0 a1 ej δ1 , Er2 = Ei0 a2 ej δ2 .

(2.50)

Consider a lossless power split with equal amounts (i.e., one-half) of power into the ˘θr directions. From |Ei0 |2 cos θi = (|Er1 |2 + |Er2 |2 ) cos θr and |Er1 | = |Er2 |, we obtain d a = a1 = a2 =

Zr = 2Zi

d

cos θi , 2 cos θr

(2.51)

where the same relative magnitude is denoted by a. From (2.38), the surface impedance is expressed as ‰ “ 1 + a e´j (k1 x sin θr ´δ1 ) + ej (k1 x sin θr +δ2 ) “ ‰. Zs = η1 1 ´ a cos θr e´j (k1 x sin θr ´δ1 ) + ej (k1 x sin θr +δ2 )

(2.52)

Let us consider an equal-power split into ˘70˝ (θr = 70˝ ). The impedance is a periodic function with a length of the unit cell given by D = λ{ sin θr = 1.064λ. Over the period ´D{2 ă x ă D{2, the surface impedance profiles for two different reflection phase combinations are shown in Figure 2.18. Figure 2.18a plots Zs normalized by η1 for an in-phase split case where the reflected fields have the same phase as the incident field at the metasurface, i.e., δ1 = δ2 = 0. The surface resistance swings over a large range from ´0.78η1 at x = ˘D{2 to 19.8η1 at x = 0, while the surface reactance is constant at zero. Relatively small ranges of resistance and reactance values are observed in Figure 2.18b, where an out-of-phase split is considered with parameters δ1 = π and δ2 = 0. Both the resistance and reactance alternate between positive and negative values. The possible realization of the surface impedance is based on nonuniform array of metal patches over a ground plane (Díaz-Rubio et al. 2017).

References

61

Figure 2.18 Surface impedance profile for an equal-power plane-wave splitter. (a) An in-phase splitter with δ1 = δ2 = 0. (b) An out-of-phase splitter with δ1 = π and δ2 = 0. The polarization for both the incident and reflected waves is TE (the electric field along the y-axis direction).

References Achouri, K., Salem, M. A. & Caloz, C. (2015), ‘General metasurface synthesis based on susceptibility tensors’, IEEE Transactions on Antennas and Propagation 63(7), 2977–2991. Albooyeh, M., Kruk, S., Menzel, C., Helgert, C., Kroll, M., Krysinski, A., Decker, M., Neshev, D. N., Pertsch, T., Etrich, C., Rockstuhl, C., Tretyakov, S. A., Simovski, C. R. & Kivshar, Y. S. (2014), ‘Resonant metasurfaces at oblique incidence: Interplay of order and disorder’, Scientific Reports 4, 04484. Albooyeh, M., Kwon, D.-H., Capolino, F. & Tretyakov, S. A. (2017), ‘Equivalent realizations of reciprocal metasurfaces: Role of tangential and normal polarization’, Physical Review B 95(11), 115435. Albooyeh, M., Tretyakov, S. & Simovski, C. (2016), ‘Electromagnetic characterization of bianisotropic metasurfaces on refractive substrates: General theoretical framework’, Annalen der Physik 528(9–10), 721–737. Alù, A. (2009), ‘Mantle cloak: Invisibility induced by a surface’, Physical Review B 80(24), 245115. Anderson, I. (1975), ‘On the theory of self-resonant grids’, The Bell System Technical Journal 54(10), 1725–1731. Andryieuski, A., Lavrinenko, A.V., Petrov, M. & Tretyakov, S. (2016), ‘Homogenization of metasurfaces formed by random resonant particles in periodical lattices’, Physical Review B 93, 205127. Asadchy, V., Albooyeh, M. & Tretyakov, S. (2016), ‘Optical metamirror: All-dielectric frequencyselective mirror with fully controllable reflection phase’, JOSA B 33(2), A16–A20. Asadchy, V. S., Albooyeh, M., Tcvetkova, S. N., Díaz-Rubio, A., Ra’di, Y. & Tretyakov, S. A. (2016), ‘Perfect control of reflection and refraction using spatially dispersive metasurfaces’, Physical Review B 94(7), 075142. Asadchy, V. S., Díaz-Rubio, A. & Tretyakov, S. A. (2018), ‘Bianisotropic metasurfaces: physics and applications’, Nanophotonics 7(6), 1069–1094.

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Asadchy, V. S., Faniayeu, I. A., Ra’di, Y., Khakhomov, S. A., Semchenko, I. V. & Tretyakov, S. A. (2015), ‘Broadband reflectionless metasheets: Frequency-selective transmission and perfect absorption’, Physical Review X 5(3), 031005. Asadchy, V. S., Ra’di, Y., Vehmas, J. & Tretyakov, S. A. (2015), ‘Functional metamirrors using bianisotropic elements’, Physical Review Letters 114(9), 095503. Asadchy, V. S., Wickberg, A., Díaz-Rubio, A. & Wegener, M. (2017), ‘Eliminating scattering loss in anomalously reflecting optical metasurfaces’, ACS Photonics 4(5), 1264–1270. Bilotti, F., Toscano, A., Alici, K. B., Ozbay, E. & Vegni, L. (2011), ‘Design of miniaturized narrowband absorbers based on resonant-magnetic inclusions’, IEEE Transactions on Electromagnetic Compatibility 53(1), 63–72. Bilotti, F. & Tretyakov, S. A. (2013), Amorphous metamaterials and potential nanophotonics applications, in C. Rockstuhl & T. Scharf, eds, ‘Amorphous Nanophotonics’, Springer Verlag, pp. 39–66. Chen, M., Abdo-Sanchez, E., Epstein, A. & Eleftheriades, G. V. (2017), ‘Experimental verification of reflectionless wide-angle refraction via a bianisotropic Huygens’ metasurface’, 2017 XXXIInd General Assembly and Scientific Symposium of the International Union of Radio Science (URSI GASS) pp. 1–4. Cheng, Y. Z., Withayachumnankul, W., Upadhyay, A., Headland, D., Nie, Y., Gong, R. Z., Bhaskaran, M., Sriram, S. & Abbott, D. (2014), ‘Ultrabroadband reflective polarization convertor for terahertz waves’, Applied Physics Letters 105(18), 181111. Destouches, N., Tishchenko, A. V., Pommier, J. C., Reynaud, S., Parriaux, O., Tonchev, S. & Ahmed, M. A. (2005), ‘99% efficiency measured in the -1st order of a resonant grating’, Optics Express 13(9), 3230–3235. Díaz-Rubio, A., Asadchy, V., Elsakka, A. & Tretyakov, S. (2017), ‘From the generalized reflection law to the realization of perfect anomalous reflectors’, Science Advances 3(8), e1602714. Elsakka, A. A., Asadchy, V. S., Faniayeu, I. A., Tcvetkova, S. N. & Tretyakov, S. A. (2016), ‘Multifunctional cascaded metamaterials: Integrated transmitarrays’, IEEE Transactions on Antennas and Propagation 64(10), 4266–4276. Epstein, A. & Eleftheriades, G. V. (2014), ‘Passive lossless Huygens’ metasurfaces for conversion of arbitrary source field to directive radiation’, IEEE Transactions on Antennas and Propagation 62(11), 5680–5695. Epstein, A. & Eleftheriades, G. V. (2016), ‘Synthesis of passive lossless metasurfaces using auxiliary fields for reflectionless beam splitting and perfect reflection’, Physical Review Letters 117(25), 256103. Glybovski, S. B., Tretyakov, S. A., Belov, P. A., Kivshar, Y. S. & Simovski, C. R. (2016), ‘Metasurfaces: From microwaves to visible’, Physics Reports 634, 1–72. Grady, N. K., Heyes, J. E., Chowdhury, D. R., Zeng, Y., Reiten, M. T., Azad, A. K., Taylor, A. J., Dalvit, D. A. R. & Chen, H.-T. (2013), ‘Terahertz metamaterials for linear polarization conversion and anomalous refraction’, Science 340, 1304–1307. Holloway, C. L., Kuester, E. F., Gordon, J. A., O’Hara, J., Booth, J. & Smith, D. R. (2012), ‘An overview of the theory and applications of metasurfaces: The two-dimensional equivalents of metamaterials’, IEEE Antennas and Propagation Magazine 54(2), 10–35. Hopkinson, F. & Rittenhouse, D. (1786), ‘An optical problem, proposed by Mr. Hopkinson, and solved by Mr. Rittenhouse’, Transactions of the American Philosophical Society 2, 201–206. Huygens, C. (1690), Traité de la Lumière, Pieter van der Aa, Leiden. Idemen, M. (1988), ‘Straightforward derivation of boundary conditions on sheet simulating an anisotropic thin layer’, Electronics Letters 24(11), 663–665.

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3

Using Generalized Sheet Transition Conditions (GSTCs) in the Analysis of Metasurfaces Christopher L. Holloway and Edward F. Kuester

3.1

Introduction and Definitions of Metasurfaces In recent years, there has been a great deal of attention devoted to metastructures, which include metamaterials and metasurfaces. The prefix meta is a Greek preposition meaning (among other things) “beyond.” In the context of metamaterials or metasurfaces, this refers to a metastructure that has some type of exotic property that normally does not occur in nature. Although in this chapter we will discuss metasurfaces and the use of generalized sheet transmission conditions (GSTCs), we begin our discussion with threedimensional metamaterials. Metamaterials are novel, synthetic materials engineered to achieve unique properties, i.e., materials beyond those occurring naturally. Metamaterials are often engineered by arranging a set of small scatterers (e.g., metallic rings and rods, spherical magnetodielectric particles, or other arbitrarily shaped inclusions) in a regular array throughout a region of three-dimensional space, in order to obtain some desirable bulk behavior [1–10]. The term metamaterial does not refer to classical periodic structures, such as what are now called photonic band gap (PBG) structures, or to frequency-selective surfaces (FSS). The term metamaterial refers to a material or structure with more exotic properties than artificial dielectrics, but which can still be described by bulk material parameters in the same way that natural materials can. One example is a so-called double-negative (DNG) material [11–26], also known as negative-index material (NIM), a backward-wave (BW) medium, or left-handed material (LHM). This type of material has the property that its effective permittivity and effective permeability are simultaneously negative in a given frequency band. A second example is near-zero refractive index material, with either the permittivity or permeability designed to have its real part close to zero. Materials with unique properties such as these have a wide range of potential applications in electromagnetics at frequencies ranging from the low microwaves to optical, including shielding, low-reflection materials, novel substrates, antennas, electronic switches, devices, perfect lenses, resonators, and of course cloaking, among many other possibilities. The concept of three-dimensional metamaterials can be extended by placing the scatterers (or apertures) in a two-dimensional arrangement on a surface. This surface version of a metamaterial is called a metasurface, and includes metafilms, metascreens, and metagratings as special cases [27–34]. Metasurfaces are an attractive alternative to three-dimensional metamaterials because of their simplicity and relative ease of

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3.1 Introduction and Definitions of Metasurfaces

67

fabrication. Throughout the literature, there are several applications where metamaterials are replaced with metasurfaces. Metasurfaces have the advantage of taking up less physical space than do full 3D metamaterial structures; as a consequence they can also offer the possibility of lower losses. The application of metasurfaces at frequencies from microwave to optical has attracted great interest in recent years [27–84]. In addition to the applications mentioned above for metamaterials, metasurfaces allow for controllable smart surfaces, miniaturized cavity resonators, novel waveguiding structures, compact and wide-angle absorbers, impedance matching surfaces, biomedical devices, wavefront tailoring, polarization conversion, antennas, and high-speed switching devices, to name only a few. In general, a metasurface is any periodic two-dimensional structure whose thickness and periodicity are small compared to a wavelength in the surrounding media. Metasurfaces should not be confused with classical frequency-selective surfaces (FSS); the important distinction between the two is discussed below. We can identify two important subclasses (metafilms and metascreens) within this general designation of metasurfaces. These two subclasses are distinguished by the type of topology that constitutes the metasurface. Metafilms (as coined in [29]) are metasurfaces that have a “cermet” topology, which refers to an array of isolated (nontouching) scatterers (see Figure 3.1). Metascreens are metasurfaces with a “fishnet” structure (see Figure 3.2), which are characterized by periodically spaced apertures in an otherwise relatively impenetrable surface [31]. There are other types of meta-structures that lie somewhere between a metafilm and a metascreen. For example, a grating of parallel coated wires (a metagrating; see Figure 3.3) behaves like a metafilm to electric fields perpendicular to the wire axes and like a metascreen for electric fields parallel to the wire axes [32]. It is important to note that the individual scatterers constituting a metafilm (or apertures constituting a metascreen) are not necessarily of zero thickness (or even small compared to the lattice constants); they may be of arbitrary shape, and their dimensions are required to be small only in comparison to a wavelength in the surrounding medium, a fortiori because the lattice constant has been assumed small compared to a wavelength. It has been shown that a type of boundary conditions known as generalized sheet transition conditions (GSTCs) is the most appropriate way to model metasurfaces [29–34]. While the functional form of the GSTCs may be different depending on the type of metasurface, the GSTC description allows the metasurface to be replaced by an interface, as shown in Figure 3.4. The interaction of the electric (E) and magnetic (H) fields on either side of the metasurface is taken care of through the GSTCs applied at that interface. In this model, all the information about the metasurface (geometry of the scatterers or apertures: shape, size, material properties, etc.) is incorporated into the effective surface parameters that appear explicitly in the GSTCs. These surface parameters (effective electrical and magnetic surface susceptibilities and surface porosities) that appear explicitly in the GSTCs are uniquely defined and as such serve as the physical quantities that most appropriately characterize the metascreen. The effective surface parameters for any given metasurface together with the GSTCs are all that are required to model its interaction with an EM field at the macroscopic level. In this chapter, we present the GSTCs needed to analyze the three main types of metasurfaces: metafilms, metascreens, and metagratings. From these GSTCs we derive

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Using Generalized Sheet Transition Conditions (GSTCs) in the Analysis of Metasurfaces

x

y

z

(a) array of arbitrarily shaped scatterers

2a e2, m2

p e1 m1

(b) metafilm: array of spherical particles Figure 3.1 Illustration of a metafilm, which consists of an array of arbitrarily shaped scatterers

placed on the xz-plane: (a) array of arbitrarily spaced scatterers and (b) array of spherical particles.

for each of these metasurfaces the plane-wave reflection (R) and transmission (T ) coefficients. They are expressed in terms of the surface parameters that characterize the metasurface. These coefficients are then used to develop a retrieval approach for determining from measured or simulated data the uniquely defined effective surface parameters (i.e., the electric and magnetic surface susceptibilities and surface porosities) that characterize each of the metasurfaces. Finally, in the last section, we present various other applications of the GSTCs.

3.2

Metasurfaces versus Frequency-Selective Surfaces Before we introduce and discuss the different types of GSTCs for the various metasurfaces, we need to make the distinction between metasurfaces and frequency-selective surfaces. Depending on the wavelength and the periodicity of the inclusions that make up an engineered periodic structure, it may or may not be possible to model it as an effective medium (as for the case of a metamaterial) or using effective surface parameters (as for the case of a metasurface). In fact, the electromagnetic field interaction with these types of engineered structures can be divided into three separate regions of

3.2 Metasurfaces versus Frequency-Selective Surfaces

y

69

x

z

(a) array of arbitrarily shaped apertures in a conducting screen

(b) metafilm: array of square apertures Figure 3.2 Illustration of a metascreen, which consists of an array of arbitrarily shaped apertures

in a conducting screen located in the xz-plane: (a) array of arbitrarily shaped apertures and (b) array of square apertures.

Figure 3.3 Illustration of a metagrating, which consists of an array of coated and arbitrarily

shaped wires.

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Using Generalized Sheet Transition Conditions (GSTCs) in the Analysis of Metasurfaces

(a)

(b)

(c)

(d) Figure 3.4 (continued on next page) Reference planes for GSTC models: (a) a metafilm with arbitrarily shaped scatterers, (b) reference plane for a metafilm at which the GSTCs are applied, (c) a metascreen with arbitrarily shaped apertures, (d) reference plane for a metascreen at which the GSTCs are applied, (e) a metagrating with arbitrarily shaped coated wire grating, and (f) reference plane for a metagrating at which the GSTCs are applied. Note that there are three different types of GCTSs in (b), (d), and (f).

3.2 Metasurfaces versus Frequency-Selective Surfaces

71

B (e)

B

(f) Figure 3.4 (Cont.)

operation (see Figure 3.5), with distinctive behaviors in each region. It is important to be aware of this and to understand the behavior in each region when either performing measurements or analyzing metastructures at different length scales and/or frequencies, as will be described below. In addressing this, we will comment on (1) the difference between a metamaterial and a conventional photonic band gap (PBG) or electromagnetic band gap (EBG) structure and, in turn, (2) the difference between a metasurface and a conventional frequency-selective surface (FSS). Consider first three-dimensional metamaterials; these ideas will be extended below to metasurfaces. The behavior of such a composite material is qualitatively different in each of the three distinct regions shown in Figure 3.5. Region 1 corresponds to the quasistatic region, which implies low frequencies; more specifically, frequencies at which the wavelength is much larger than the period of the structure (that is, the periodicity of the scatterers that compose the composite medium) and at which the scatterers are not resonant. These scatterers could have induced or permanent dipole moments, as is the case for atoms or molecules for classical materials, or these scatterers could be generic in shape and placed in a host matrix to obtain a man-made composite material designed

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Using Generalized Sheet Transition Conditions (GSTCs) in the Analysis of Metasurfaces

Figure 3.5 Three characteristic regions of composite material or metastructure behavior.

to achieve some specified behavior. In this region, classical mixing formulas are used to obtain equivalent effective-material properties (permittivity , permeability μ, etc.). The determination of effective-medium properties and the modeling of electromagnetic response to inclusions embedded in a host material is a problem with a long history going back to Maxwell and Rayleigh, and to Poisson, Clausius, and Mossotti before that. Much work has been done since then to compute the effective properties of homogeneous composite materials, a survey of which can be found in [85–88]. Before we discuss the behavior in Region 2, let us examine Region 3. Here, the wavelength becomes comparable to or smaller than the period of the structure, and the fields no longer “see” the composite as an effective medium. At these frequencies, a more complicated field behavior exists and more elaborate techniques to analyze the EM field interaction with the composite periodic structures must be used (i.e., fullwave approaches). The classical analytical approach for this is the Floquet–Bloch mode expansion [88–95] of spatially modulated plane waves propagating in various directions. As the wavelength becomes smaller than the period, higher-order Floquet–Bloch modes must be considered. These higher-order modes then interfere with the fundamental wave propagating through the composite, so that stop bands and pass bands will develop in the structure (in this case, the structures are often referred to as EBG or PBG structures [88, 89, 94, 95]). The constructive and destructive interference of the various modes as the wave propagates through the composite are what give rise to the unique characteristics of EBG and PBG structures. It is important to note that such effects cannot be represented or captured by a conventional effective medium theory. These stop band and pass band effects in EBG and PBG structures are caused by resonances associated with the periodicity of the structure. Resonances associated with the scatterers themselves can cause interesting and unexpected effects in the effective material properties of a composite. Region 2 in Figure 3.5 corresponds to a region where the period of the structure is still small compared to a wavelength, but the individual scatterers are designed in such a manner (either via their shape or their constitutive properties) that the scatterers themselves can resonate. When this occurs, a new class of engineered materials (metamaterials) is realized, making

3.3 Characterization of Metasurfaces: Surface versus Bulk Properties

73

possible a broad range of unique behavior not commonly found in nature (DNG or near-zero index materials, for example). Region 2, where the scatters resonate (but not the lattice, as is the case in Region 3), is where metamaterials lie. The medium is dispersive here, but we may still characterize its electromagnetic behavior with the effective parameters  and μ. Similar to metamaterials, depending on the wavelength-to-period spacing, three regions of behavior will occur for electromagnetic interactions with a metasurface. For a two-dimensional lattice of scatterers or apertures, Region 1 in Figure 3.5 corresponds to classical thin-film materials, while Region 3 in Figure 3.5 corresponds to resonances associated with the periodicity of the scatterers or apertures. The conventional FSS and PBG surfaces [89, 94, 95] fall into this third region. On the other hand, when we talk about a metasurface, we are referring to an array of scatterers or apertures that lies in Region 2 (or even Region 1). Resonances of the surface may be associated with the resonances of the scatterers or apertures, but not with the periodicity of the array. Ordinary FSSs are sometimes operated in this regime, but the distinction between this type of operation and that of Region 3 is not always made clear. We emphasize that the characteristic behavior of Region 2 in Figure 3.5 may not always occur for a given metamaterial or metasurface. The scatterers or apertures need to be properly designed, such that the scatterers’ (or apertures’) resonances occur at a frequency well below that where the next higher-order Floquet–Bloch mode can propagate. For example, if the constituent materials or the sizes or shapes of the scatterers used in the material were not properly chosen [23], the scatterers’ resonances could be pushed toward the Floquet–Bloch-mode region, and in this case an effective-medium model would not adequately describe the behavior of the composite material. In summary, in Regions 1 and 2 of Figure 3.5 the interaction of an electromagnetic field with a metasurface is described by effective surface parameters of some kind, as will be discussed below. In Region 1 (analogous to the classical mixing theory region for the case of a metamaterial), the effective surface parameters are not frequencydependent (except insofar as the constituent bulk properties have a frequency dependence). In Region 2 (the scatterers’ or apertures’ resonant region), the metasurface is still modeled by effective surface properties, which now may possess an inherent frequency dependence that makes interesting resonant behaviors possible. In the last region (Region 3 in Figure 3.5), the electromagnetic field’s interaction with the periodic array is very involved. We may no longer think of the surface as behaving like an interface with effective surface parameters. When the wavelength is comparable to the period, higher-order Floquet–Bloch modes must be considered, and one typically does not refer to these materials as metamaterials or metasurfaces in this region, but refers to them as EBG, PBG, or FSS structures.

3.3

Characterization of Metasurfaces: Surface versus Bulk Properties Like a metamaterial, the behavior of a metafilm is determined by the electric and magnetic polarizabilities of its constituent scatterers (or for a metascreen, of its constituent

74

Using Generalized Sheet Transition Conditions (GSTCs) in the Analysis of Metasurfaces

apertures). The traditional and most convenient method by which to model metamaterials is with effective-medium theory. Attempts to use a similar bulk-parameter analysis for metasurfaces have been less successful (see [27], [33], and [34] for a detailed discussion on this point). Indeed, some previous metafilm studies have modeled the film as a single-layer metamaterial in which effective bulk material properties of the metasurface are obtained by forcing the introduction of an arbitrary non-zero thickness parameter into the analysis. As we will demonstrate, several problems arise from the physically artificial character of this parameter; the bulk material property characterization of a metasurface is incorrect at a fundamental level. It is known that localized effects arise near the boundary of a metamaterial that can significantly modify effects such as planewave reflection and transmission [29–31, 42, 43, 96]; these are analogous to the effect of cutoff modes near the junction between two different waveguides. Classical algorithms for bulk parameter extraction do not account for these boundary effects, and indeed a metamaterial sample that is only one or a few layers thick should be expected to behave as “all boundary” without exhibiting a uniquely identifiable bulk behavior at all. To illustrate this point, consider the equivalent-bulk layer representation of a metasurface shown in Figure 3.6. The problem we face is that the thickness of a metasurface cannot be uniquely defined, nor can the effective material properties. In [29–34], it is shown that the effective surface parameters of a metafilm or metascreen are unique properties of the structure and thus are the most appropriate way to characterize it. We will see below that the surface parameters for a metafilm are what we will call effective surface susceptibilities (defined below as χMS and χES – the magnetic and electric surface susceptibilities, respectively) [27, 30], and for a metascreen, the so-called surface porosities (defined below as πMS and πES – the magnetic and electric surface porosities, respectively) [27, 31]. Techniques for retrieving the surface parameters for a given metasurface based on reflection and transmission measurements (or simulations) are presented in [27, 33, 34], [38], and [39] and will be discussed below. The problems that arise when we try to represent a metasurface as a material with a bulk effective permittivity and permeability as in Figure 3.6 are demonstrated in Figure 3.7a, where retrieved values of r for an array of lossy spherical particles, as shown in Figure 3.1b, are plotted for different values of the assumed thickness d. These results were obtained by computing numerically simulated values of the reflection and transmission coefficients for this array of spheres and then using the modified

Figure 3.6 Representation of a metafilm as a bulk effective medium with thickness d.

3.3 Characterization of Metasurfaces: Surface versus Bulk Properties

75

(a)

(b) Figure 3.7 Results for an array of lossy spheres with radius a = 10 mm, period p = 25.59 mm, yy r2 = 2, μr2 = 900, and loss tangent tan δ = 0.04: (a) retrieved r and (b) d(r ´ 1) and χES .

Nicolson–Ross–Weir (NRW) method [42, 97–102] for determining the effective r of the slab (see [34] for details; there is also an effective μR which is not shown here). As we expect, these results show a functional dependence of r on d. Figure 3.7b shows results for d(r ´1) for different values of d. We have also plotted the retrieved values of yy the surface susceptibility χES for this array that appears in (3.84) below (also obtained from using retrieval algorithms and the simulated numerical values of reflection and

76

Using Generalized Sheet Transition Conditions (GSTCs) in the Analysis of Metasurfaces

(a)

(c)

(b)

(d)

yy zz Figure 3.8 Surface electric susceptibilities for two metafilms: (a) χES and (b) χES for a metafilm

of spherical particles (a = 10 mm, p = 25.59 mm, r = 2, μr = 900, and tan δ = 0.04); (c) zz for the metafilm array of thin metallic scatterers shown in (d) surface electric susceptibility χES with t = 3 μm, A = 40 μm, p = 54 μm, and l = 12 μm.

yy

transmission coefficients [34]; see also Section 3.6 below). The retrieved values for χES are the same as the analytical values given in [33]. The results shown in Figure 3.7b illustrate that for sufficiently low frequencies, the product d(r ´ 1) is independent of yy d and identical to χES . Although the connection between surface susceptibilities and the effective bulk properties of a slab was not discussed explicitly there, Smith et al. [21] do allude to the fact that the product of the slab thickness and the effective material properties of the slab should be constant. Further results for the surface susceptibilities of this same metafilm are shown in Figures 3.8a and 3.8b, while results for an array of thin metallic scatterers (Figure 3.8d) are shown in Figure 3.8c.

3.3 Characterization of Metasurfaces: Surface versus Bulk Properties

(a)

77

(b)

(c) Figure 3.9 Surface susceptibilities and surface porosities for a metascreen composed of an array zz zz and πMS for square aperture of apertures : (a) array of square apertures of length l, (b) χMS yy yy array (h = 10 mm and p = 100 mm), and (c) χES and πES for circular aperture array

(h = 5 mm and p = 100 mm).

As we will see in Section 3.4.2, a metascreen requires both surface susceptibilities and surface porosities for its complete characterization. Figure 3.9 shows plots of these quantities for a metascreen composed of an array of square apertures – again, these are unique and do not require the postulation of an arbitrary equivalent slab thickness. When all is said and done, we would argue that a model for a metafilm that uses uniquely specified quantities (i.e., χMS or χES as defined below [29, 30]) is more natural than an approach that involves two arbitrary quantities (d and r ). Likewise, for a metascreen, we should use both surface susceptibilities (χES and χMS ) and surface porosities (πES and πMS ) as defined below [27, 31]. Even though it has been shown

78

Using Generalized Sheet Transition Conditions (GSTCs) in the Analysis of Metasurfaces

that the electric and magnetic surface parameters are the most appropriate manner to characterize metasurfaces, some researchers have continued to characterize them in terms of bulk effective material properties. If one insists on characterizing a metasurface as a thin material slab with bulk effective material properties and a thickness d, the only meaningful (and unique) parameters will be products such as d(r ´ 1) and d(μr ´ 1), if the slab is centered at the plane containing the metafilm. A retrieval approach that gives unique quantities like χMS , χES , πMS , and πES is more natural than one that merely gives products of otherwise undetermined quantities [27, 33, 34, 38]. The effective surface parameters (surface susceptibilities and surface porosities) of the metasurface appear in effective boundary conditions called generalized sheet transition conditions (GSTCs) [27, 29–31], and these parameters are all that are required to model the macroscopic interaction of any given metasurface with an electromagnetic field. The GSTCs allow this surface distribution of scatterers (or of apertures) to be replaced with boundary conditions that are applied at an infinitely thin equivalent surface (hence the name metasurface, metafilm or metascreen), as indicated in Figure 3.4. The size, shape, and spacing of the scatterers (or apertures) are incorporated into the GSTCs through the effective surface susceptibilities and surface porosities.

3.4

Generalized Sheet Transition Conditions (GSTCs) The GSTCs for a metasurface take on distinct functional forms depending on whether it is a metafilm, a metascreen, or a metagrating. In this section, we present the GSTCs required to analyze these three kinds of metastructure.

3.4.1

GSTCs for a Metafilm As stated above, a metafilm is a metasurface that has a “cermet” topology, which refers to an array of isolated (nontouching) scatterers (see Figure 3.1). For a metafilm, the GSTCs specify the jumps in the tangential components of the electric (E) and magnetic (H) fields across the metafilm (see Figures 3.4a and 3.4c) and take on the following form [27–30]: ı ” Ø    ˜ av ´ ay ˆ ∇t ay ¨ Ø ˜ av = ´j ωμ0 χ MS ¨ H χ ES ¨ E (3.1) ay ˆ EA ´ EB y=0

and ı ” ay ˆ HA ´ HB

y=0

t

Ø    ˜ av ´ ay ˆ ∇t ay ¨ Ø ˜ av , = j ω0 χ ES ¨ E χ MS ¨ H t

(3.2)

where the subscript “t” represents the x and z components and the average fields are defined by   1 A 1 A Eav = E + EB H + HB . and Hav = (3.3) 2 2 The superscripts A and B correspond to the regions below and above the reference plane of the metafilm. The parameters in the GSTCs in Eqs. (3.1) and (3.2) are surface susceptibility dyadics and are defined as

3.4 Generalized Sheet Transition Conditions (GSTCs)

xy

xz xx χ ES = χES ax ax + χES ax ay + χES ax az

Ø

χ MS =

Ø

yx yy yz +χES ay ax + χES ay ay + χES ay az zy zx zz +χES ax ax + χES az ay + χES az az, xy xz xx χMS ax ax + χMS ax ay + χMS ax az yx yy yz +χMS ay ax + χMS ay ay + χMS ay az zy zx zz +χMS ax ax + χMS az ay + χMS az az .

79

(3.4)

(3.5)

The surface susceptibilities have units of length. Note that the coordinate system used here is the same as that used in [30], but different from the one used in [29], [33], and [34].

3.4.2

GSTCs for a Metascreen Metascreens are metasurfaces with a “fishnet” structure (see Figure 3.2), which are characterized by periodically spaced apertures in an otherwise relatively impenetrable surface. This fundamentally different topology requires a different functional form of GSTCs than what applies to a metafilm: a set of BCs for the jumps in both tangential E and H fields at its reference surface. As it turns out, GSTCs can still be used for a metascreen, but the topology demands a different form for them, for the following reason. For a problem involving two regions separated by a boundary surface, one needs at least two effective boundary conditions (EBCs) to constrain the tangential E- and/or H -fields at that boundary. A metascreen differs from a metafilm in that there is the possibility of having tangential surface currents (flowing on the surface of the screen along the z and x directions). Typically these currents would only be known once the tangential components of the H -field are known. This concept can be illustrated by the case of an electromagnetic field at the surface of a perfect electric conductor (PEC). For a PEC, only the BC for the tangential E-field at the PEC (i.e., Et = 0 on the PEC) is used in solving boundary problems for the field. The tangential H -field at the PEC is related to the surface current flowing on the PEC and this current is only known once the H -field has been determined, so this second boundary condition is not used to solve for the fields. It is useful, therefore, to classify EBCs either as essential for the solution of an electromagnetic boundary problem, or applicable only a posteriori when quantities such as surface current or charge density are to be computed from the fields. For a metascreen, any EBC for the tangential H field is an a posteriori BC and can only be used once the fields have been solved for. Thus, the required essential BCs for metascreen should constrain only tangential E, and should be expressed as conditions on the jump in the tangential E-field and on the sum (twice the average) of the tangential E-fields, i.e., on ExA ´ ExB



ExA + ExB



“ “

y=0 y=0

‰ “ ; EzA ´ EzB y=0 ‰ “ ; EzA + EzB y=0 ,

(3.6)

80

Using Generalized Sheet Transition Conditions (GSTCs) in the Analysis of Metasurfaces

where the superscripts A and B correspond to the regions above and below the reference plane of the metascreen, respectively. We add that these types of GSTCs (on both the jump in, and the average of, the E-field at the interface) also appear in the analysis of a wire grating (or metagrating) [32], as we will see in the next subsection. The required GSTCs for a metascreen are given by [27, 31] ” ı “ ‰ Ayy Byy ay ˆ EA (ro ) ´ EB (r0 ) = ´ay ˆ χES ∇t EyA (ro ) + χES ∇t EyB (ro ) “ Axx A Bxx H B (r ) ´ax j ωμ0 χMS Hx (ro ) + χMS xı o Axz A Bxz B +χMS Hz (ro ) + χMS Hz (ro ) ” Azx A Bzx B Hx (ro ) + χMS Hx (ro ) ´az j ωμ0 χMS ı Azz A Bzz B +χMS Hz (ro ) + χMS Hz (ro )

(3.7)

” ı “ ‰ Ayy Byy ay ˆ EA (ro ) + EB (r0 ) = ´ay ˆ πES ∇t EyA (ro ) ´ πES ∇t EyB (ro ) “ Axx A Bxx H B (r ) ´ax j ωμ0 πMS Hx (ro ) ´ πMS xı o Axz A Bxz B +πMS Hz (ro ) ´ πMS Hz (ro ) ” Azx A Bzx B Hx (ro ) ´ πMS Hx (ro ) ´az j ωμ0 πMS ı Azz A Bzz B +πMS Hz (ro ) ´ πMS Hz (ro ) .

(3.8)

and

As before, χES and χMS are effective electric and magnetic surface susceptibilities, respectively, while πES and πMS are interpreted as effective electric and magnetic surface porosities of the metascreen. Like the surface susceptibilities, the surface porosities have units of length.

3.4.3

GSTCs for a Metagrating A metagrating [32] is a metasurface composed of a grating of arbitrarily-shaped parallel material-coated wires as shown in Figure 3.3. A metagrating behaves like a metafilm to electric fields perpendicular to the wire axes and like a metascreen for electric fields parallel those axes. Thus, the form of GSTCs required for a metagrating is a combination of those of the metafilm and metascreen. The GSTCs for the metagrating are given by [32] “ ‰ xy zz Hz,av (ro ) az ´ j ω χMS By,av (ro ) ax ay ˆ EA (ro ) ´ EB (ro ) = ´j ωμ0 χMS “ Axx A ‰ Bxx B B (r ) a +j ω χMS Bx (ro ) + χMS x x o “ yx ‰ ´ay ˆ χES ∇Ex,av (ro ) ” ı Ayy Byy ´ay ˆ χES ∇EyA (ro ) + χES ∇EyB (ro ) ,

(3.9)

3.5 Reflection and Transmission Coefficients

xx E HzA (ro ) ´ HzB (ro ) = j ω0 χES (r ) ” x,av o ı Axy A Bxy +j ω0 χES Ey (ro ) + χES EyB (ro ) yy B B

(r )

o ´ μ10 χMS y,av Bz ” ı Ayx B BxA (ro ) Byx B BxB (ro ) , ´ μ10 χMS + χ MS Bz Bz

81

(3.10)

and xy

EzA (ro ) + EzB (ro ) = ´j ωπMS By,av (ro ) ‰ “ Axx A Bxx B B (r ) Bx (ro ) ´ πMS ´j ω πMS x o xy B Ex,av (ro ) Bz A B Ayy B Ey (ro ) Byy B Ey (ro ) ´πES ´ π ES Bz Bz ,

´πES

(3.11)

where certain average fields at the interface have been defined as ExA (ro ) + ExB (ro ) , 2 ByA (ro ) + ByB (ro ) , (3.12) By,av (ro ) = 2 H A (ro ) + HzB (ro ) Hz,av (ro ) = z . 2 The coefficients χES and χMS are effective electric and magnetic surface susceptibilities of the metagrating, while πMS,ES are effective electric and magnetic surface porosities of the metagrating; both have units of length. Ex,av (ro ) =

3.5

Reflection and Transmission Coefficients The plane-wave reflection and transmission coefficients for three types of metasurfaces are derived in this section. We present results for both a transverse electric (TE) and transverse magnetic (TM) polarized plane wave incident onto the metasurface.

3.5.1

Metafilms As seen from the GSTCs (3.1) and (3.2), the electric and magnetic surface susceptibilizy,yz xy,yx xz,zx , χES,MS , and χES,MS . ties may have cross-polarization (or off-diagonal) terms χES,MS These terms can result in coupling between TE and TM fields. Initially, we will assume for simplicity that the scatterers are sufficiently symmetric that these off-diagonal terms are zero. These assumptions correspond to many metafilms that are encountered in practice; the more general case will be treated later. The symmetry of the scatterers results in zy,yz

xy,yx

xz,zx = χES,MS = χES,MS ” 0. χES,MS

(3.13)

82

Using Generalized Sheet Transition Conditions (GSTCs) in the Analysis of Metasurfaces

Figure 3.10 TE polarized plane-wave incident onto a metastructure.

Under this condition, the surface susceptibility dyadics reduce to Ø χ Ø χ

ES

MS

yy

xx a a + χ a a + χ zz a a = χES x x ES y y ES z z xx a a + χ yy a a + χ zz a a . = χMS x x MS z z MS y y

(3.14)

TE Polarization for a Metafilm Let a metafilm be located in the plane y = 0 in free space. Assume that a TE polarized plane wave is incident onto the metafilm as shown in Figure 3.10, such that the total E-field in region A (y ą 0) is given by E = Ei + Er , where the incident and reflected fields are Ei = az E0 e´j ki ¨r Er = az RT E E0 e´j kr ¨r .

(3.15)

The transmitted field in region B (y ă 0) is given by Et = az TT E E0 e´j kt ¨r,

(3.16)

kt = ki = tax sin θ ´ ay cos θ u k0, kr = tax sin θ + ay cos θ u k0, r = xax + yay + +zaz,

(3.17)

where

? and k0 = ω μ0 0 is the wavenumber of free space. In (3.15) and (3.16), RT E and TT E are the reflection and transmission coefficient, respectively. From Maxwell’s equations, the incident, reflected and transmitted H -fields are given by   0 E0 t´ax cos θ ´ ay sin θu e´j ki ¨r, Hi = kωμ   0 RT E (3.18) Hr = k0 Eωμ tax cos θ ´ ay sin θu e´j kr ¨r,   0 TT E tax cos θ + ay sin θu e´j kt ¨r . Ht = ´ k0 Eωμ

3.5 Reflection and Transmission Coefficients

83

Substituting the electric and magnetic field components given in (3.15), (3.16), and (3.18), into the GSTCs (3.1) and (3.2) results in RT E (θ ) =

 1´

k0 2

2

 k0  zz yy xx 2 2 ´j 2 cos θ χES +χMS sin θ ´χMS cos θ     zz yy xx χ zz +χ yy sin2 θ +j k0 xx 2 2 χMS MS ES 2 cos θ χES +χMS cos θ+χMS sin θ

(3.19)

and 

TT E (θ ) =

 1´

1+

 k0 2 xx  zz yy χMS χES +χMS 2

 k0 2 xx  zz yy χMS χES +χMS 2

sin2 θ



 . k0  zz yy xx 2 2 sin2 θ +j 2 cos θ χES +χMS cos θ+χMS sin θ

(3.20)

TM Polarization for a Metafilm Assume that a TM polarized H -field plane wave is incident onto the metafilm shown in Figure 3.11, such that the H -field components of the incident, reflected, and transmitted plane waves are given by Hi = az Eζ00 e´j ki ¨r,

Hr = ´az RT M Eζ00 e´j kr ¨r, az TT M Eζ00 e´j kt ¨r,

(3.21)

= a where ζ0 = μ0 {0 is the wave impedance of free space. From Maxwell’s equations, the incident, reflected, and transmitted E-fields are given by   Ei = E0 ax cos θ + ay sin θ e´j ki ¨r,   (3.22) Er = E0 ax cos θ ´ ay sin θ RT M e´j kr ¨r,   t ´ j k ¨ r t . E = E0 ax cos θ + ay sin θ TT M e Ht

Figure 3.11 TM polarized plane-wave incident onto a metastructure.

84

Using Generalized Sheet Transition Conditions (GSTCs) in the Analysis of Metasurfaces

Substituting these expressions into the GSTCs given in (3.1) and (3.2), we obtain RT M (θ ) =

 1´

k0 2

2

 k0  xx yy zz 2 2 ´j 2 cos θ χES cos θ ´χMS ´χES sin θ     zz yy xx χ zz +χ yy sin2 θ +j k0 xx 2 2 χES ES MS 2 cos θ χMS +χES cos θ+χES sin θ

(3.23)

and 

TT M (θ ) =

 1´

k0 2

2

1+

 k0 2 xx  zz yy χES χMS +χES 2

sin2 θ



  zz  . yy xx χ zz +χ yy sin2 θ +j k0 xx 2 2 χES MS ES 2 cos θ χMS +χES cos θ+χES sin θ

(3.24)

Cross-Polarized Terms When the full surface susceptibility dyadics are included, the reflection and transmission coefficients are more complicated. In fact, the cross-polarization surface susceptibilities result in coupling (or mode conversion) between TE and TM modes when a plane-wave is incident onto a metafilm. For an asymmetric metafilm, an incident TE plane wave can result in both TE and TM waves as defined in Figure 3.12. To account for the possibility of polarization conversion, the reflected and transmitted fields [for the TE incident wave as indicated in Eq. (3.15)] are expressed as Er = az E0 RT E e´j kr ¨r   M a cos θ ´ a sin θ e´j kr ¨r +E0 RTT E x y

(3.25)

Et = az E0 TT E e´j kt ¨r   +E0 TTTEM ax cos θ + ay sin θ e´j kt ¨r,

(3.26)

and

Figure 3.12 TE plane wave incident onto an asymmetric metastructure.

3.5 Reflection and Transmission Coefficients

85

where kt = ki and kr = tax sin θ + ay cos θ u k0

(3.27)

and RT E and TT E are the (co-polarized) reflection and transmission coefficients of M and T T M are the (cross-polarized) reflection and transmission a TE wave, and RTT E TE coefficients of a TM wave for an incident TE wave. Upon substituting these expressions for the fields into the GSTCs, the reflection and transmission coefficients for a TE incident plane-wave are found to be [103, 104] yx

RT E (θ ) = ´

yx

xy

yx

xy

2(A C + k02 χMS χMS )

+



xy

2(A C + k02 χMS χMS )

=

j k0 χMS (B ´ C) yx

xy

2(A C + k02 χMS χMS )

yx

xy

yx

xy

2(D E + k02 χES χES )

+

yx

TTTEM (θ )

xy

,

(3.28)

(3.29)

,

xy

j k0 χMS (B ´ C) yx

yx

E N ´ k02 χES χES

yx

M (θ ) RTT E

xy

2(D E + k02 χES χES )

xy

A B ´ k02 χMS χMS

yx

E N ´ k02 χES χES

+

2(A C + k02 χMS χMS ) yx

TT E (θ ) =

xy

A B ´ k02 χMS χMS

j k0 χES (N ´ D) yx

xy

2(D E + k02 χES χES )

,

(3.30)

xy

+

j k0 χES (N ´ D) yx

xy

2(D E + k02 χES χES )

,

(3.31)

where A=2+

j k0 yy cos θ χMS

+

j k0 zz cos θ χES

xx cos θ, sin2 θ ; B = 2 ´ j k0 χMS

(3.32)

N =2´

j k0 yy cos θ χES

´

j k0 zz cos θ χMS

xx cos θ, sin2 θ ; E = 2 + j k0 χES

(3.33)

D =2+

j k0 yy cos θ χES

+

j k0 zz cos θ χMS

xx cos θ . sin2 θ ; C = 2 + j k0 χMS

(3.34)

and

From these expressions, we see that due to the anisotropic terms in the GSTCs, polarization conversion will occur, in that an incident TE wave will generate TM waves in M and addition to the expected TE waves. If the cross-coupling terms are zero, then RTT E T M TT E are also zero, and we are left with a pure TE reflection and transmitted field. For an asymmetric metafilm, a TM plane wave can also result in both TE and TM waves. Assume that a TM polarized plane wave is incident onto the metafilm as shown in Figure 3.13, such that the incident electric field is given by

86

Using Generalized Sheet Transition Conditions (GSTCs) in the Analysis of Metasurfaces

Figure 3.13 TM plane wave incident onto an asymmetric metastructure.

  Ei = E0 ax cos θ + ay sin θ e´j ki ¨r .

(3.35)

Similar to the derivation above, to account for the possibility of the metafilm converting the incident TM wave into both TE and TM waves (see Figure 3.13), the reflected and transmitted fields are expressed as E e ´ j kr ¨ r Er = az E0 RTT M   +E0 RT M ax cos θ ´ ay sin θ e´j kr ¨r

(3.36)

Et = az E0 TTTME e´j kt ¨r  +E0 TT M ax cos θ + ay sin θ e´j kt ¨r,

(3.37)

and

where RT M and TT M are the (co-polarized) reflection and transmission coefficients of E and T T E are the (cross-polarized) reflection and transmission the TM wave, and RTT M TM coefficients of a TE wave for an incident TM wave. Upon substituting the expressions above into the GSTCs, the reflection and transmission coefficients are found to be [103, 104] yx

TT E (θ ) =

xy

yx

C M ´ k02 χMS χMS

RT E (θ ) = ´

2(A C

yx xy + k02 χMS χMS ) yx

xy

yx

xy

C M ´ k02 χMS χMS 2(A C + k02 χMS χMS )

+

+



j k0 χMS (M + A) yx

xy

2(A C + k02 χMS χMS )

2(D E

yx xy , + k02 χES χES ) yx

xy

yx

xy

D F ´ k02 χES χES

2(D E + k02 χES χES )

yx

M (θ ) RTT E

xy

D F ´ k02 χES χES

(3.38)

(3.39)

,

xy

+

j k0 χES (E ´ D) yx

xy

2(D E + k02 χES χES )

,

(3.40)

3.5 Reflection and Transmission Coefficients

yx

TTTEM (θ ) =

87

xy

j k0 χMS (B + A)

j k0 χES (E + F)

,

(3.41)

xx sin2 θ ; F = 2 ´ j k0 χES cos θ,

(3.42)

yx

xy

2(A C + k02 χMS χMS )

´

yx

xy

2(D E + k02 χES χES )

where M=2´

j k0 MS cos θ χES

´

j k0 zz cos θ χES

and A, B, C, D, N , E are given in Eqs. (3.32)–(3.34). From these expressions, we see that due to the anisotropic terms in the GSTC mode conversion will occur in that an yx,xy incident TM mode will generate a TE mode. If the cross-coupling terms (χMS and yx,xy E = 0 and T T E = 0, and R χES ) are zero, then RTT M T M and TT M reduce to the pure TM TM reflected and transmitted field.

3.5.2

Metascreens As seen from the GSTCs in (3.7) and (3.8), the magnetic surface parameters for a xz zx xz , πMS , χMS , and metascreen may have cross-polarization (or off-diagonal) terms πMS zx . As for a metafilm, these terms can result in coupling between TE and TM fields. χMS Initially, we will assume for simplicity that the apertures are sufficiently symmetric that these off-diagonal terms are zero. We will also assume that regions A and B are both free space, and that the screen possesses mirror symmetry about the reference plane y = 0. These assumptions correspond to many metascreens that are encountered in practice; the more general case will be treated below. Symmetry of the apertures results in xz zx xz zx = πMS = χMS = χMS ” 0, πMS

(3.43)

and the symmetry with respect to either side of the metascreen results in Ayy

Byy

yy

Ayy

Byy

yy

πES = πES ” 2πES ; χES = χES ” 12 χES , Axx = π Bxx ” 2π xx ; χ Axx = χ Bxx ” 1 χ xx , πMS MS MS MS MS 2 MS Azz πMS

=

Bzz πMS



zz 2πMS

;

Azz χMS

=

Bzz χMS



(3.44)

1 zz 2 χMS .

Under these conditions, the GSTCs become “ “ ‰ ‰ zz xx H ay ˆ EA (ro ) ´ EB (ro ) = ´j ωμ0 ax χMS x,av (ro ) + az χMS Hz,av (ro ) , (3.45) yy ´χES ay ˆ ∇t Ey,av (ro ) and  “ “ ‰ ‰( xx H A (r ) ´ H B (r ) + a π zz H A (r ) ´ H B (r ) ay ˆ Eav (ro ) = ´j ωμ0 ax πMS o z MS o o x x ıo z z ” yy ´πES ay ˆ ∇t EyA (ro ) ´ EyB (ro ) , (3.46) where ı ı 1” A 1” A Hx,av (ro ) = and Eav (ro ) = Hx (ro ) + HxB (ro ) E (ro ) + EB (ro ) (3.47) 2 2 represent the average of the fields on the two sides of the reference plane at y = 0.

88

Using Generalized Sheet Transition Conditions (GSTCs) in the Analysis of Metasurfaces

TE Polarization for a Metascreen Let a metascreen be located in the plane y = 0 in free space. If we assume a TE polarized plane wave is incident onto the metastructure shown in Figure 3.10, then the reflected and transmitted E and H fields are given in Eqs. (3.15), (3.16), and (3.18). Substituting the fields into the GSTCs given in Eqs. (3.45) and (3.46), we obtain the reflection and transmission coefficients for a TE incident field [38]: RT E (θ ) =

xx π xx cos2 θ 1 + k02 χMS MS  k0  xx xx cos θ ´ 1 2 cos θ ´ j 2 χMS + 4πMS

xx π xx k02 χMS MS

(3.48)

and TT E (θ ) =

‰ i k0 cos θ “ xx xx χMS ´ 4πMS 2   xx π xx cos2 θ ´ j k0 χ xx + 4π xx cos θ k02 χMS MS MS MS 2

´1

.

(3.49)

xx and π xx ) are needed Note that for this polarization only two surface parameters (χMS MS to determine RT E (θ ) and TT E (θ ). This is in contrast to a metafilm, where three different surface parameters are needed to determine these coefficients [27, 33–35].

TM Polarization for a Metascreen Assume that a TM polarized H -field plane wave is incident onto the metastructure shown in Figure 3.11, such that the E and H -field components of the incident, reflected, and transmitted plane waves are given by Eqs. (3.21) and (3.22). Substituting the field into the GSTCs (3.45) and (3.46), the reflection and transmission coefficients for a TM incident field are found to be [38] k02 AB cos2 θ k0 + 2jcos θ (A

´1 ´

RT M (θ ) = 1´

k02 AB cos2 θ

(3.50) + 4 B)

and j k0 cos θ

TT M (θ ) = ´ 1´

k02 AB cos2 θ

rA ´ 4Bs +

j k0 2 cos θ

,

(3.51)

(A + 4 B)

where yy

zz A = χMS + χES sin2 θ yy zz + πES sin2 θ. B = πMS

(3.52)

Equation (3.52) applies only to (3.50)–(3.51). Note that for this polarization four surface yy yy zz zz , πMS , χES , and πES ) are needed to determine RT M (θ ) and TT M (θ ). parameters (χMS This is in contrast to a metafilm, where only three different surface parameters are needed to determine these quantities [27, 33–35].

Cross-Polarized Terms As for the metafilm, cross-polarization terms in the GSTCs result in coupling between TE and TM fields (i.e., TE polarized fields will generate TM fields, and vice versa). Here we will assume the apertures are symmetric with respect to the plane y = 0 and that

3.5 Reflection and Transmission Coefficients

89

the material properties in region A and region B are the same. These two assumptions correspond to most metascreens encountered in practice and result in Axz Bxz xz πMS = πMS = πMS

Azx Bzx zx and πMS = πMS = πMS ,

Axz Bxz xz χMS = χMS = χMS

Azx Bzx zx and χMS = χMS = χMS ,

Ayy

Byy

yy

Ayy

Byy

yy

πES = πES = πES Axx = π Bxx = π xx πMS MS MS

and χES = χES = χES , Axx = χ Bxx = χ xx , and χMS MS MS

Azz Bzz zz πMS = πMS = πMS

Azz Bzz zz and χMS = χMS = χMS ,

(3.53)

Note that these definitions do not include the factors of 2 and 12 that are seen in (3.44). Under these conditions, the GSTCs reduce to a simpler form: “ ‰ ay ˆ EA (ro ) ´ EB (r ) 0  xx “ A “ A ‰ ‰( xz Hx (ro ) + HxB (ro ) + χMS Hz (ro ) + HzB (ro ) = ´ax j ωμ0 χMS  zz “ A “ ‰ ‰( zx HxA (ro ) + HxB (ro ) ´az j ωμ0 χMS Hz (ro ) + HzB (ro ) + χMS ” ı yy ´ay ˆ χES ∇t EyA (ro ) + ∇t EyB (ro ) (3.54) and “ ‰ ) ay ˆ EA (ro ) + EB (r 0  xx “ A “ A ‰ ‰( xz H (ro ) ´ HxB (ro ) + πMS Hz (ro ) ´ HzB (ro ) = ´ax j ωμ0 πMS  zz “ xA “ ‰ ‰( zx HxA (ro ) ´ HxB (ro ) ´az j ωμ0 πMS Hz (ro ) ´ HzB (ro ) + πMS ” ı yy ´ay ˆ πES ∇t EyA (ro ) ´ ∇t EyB (ro ) . (3.55) Using the incident, reflected, and transmitted fields as defined in Eqs. (3.25) and (3.26) in the GSTCs (3.54) and (3.55), the reflection and transmission coefficients (as defined in Figure 3.12) for a TE incident plane wave are obtained [39]: RT E (θ ) = 1 ´

TT E (θ ) = ´ +

M RTT E (θ ) = ´



A

X A+



1 V 4 X

V 4X



+

X A+ 4VX xx cos θ j 2k0 πMS Y

B





Y B+



4W Y

Y B+ 4W Y

´

j

4W Y

,

(3.56)

,

(3.57)



k0 xx 2 χMS

cos θ

X

zx zx cos θ j k20 χMS j 2k0 πMS cos θ   , ´   4W X A + 4VX Y B+ Y

(3.58)

and TTTEM (θ ) =

zx zx cos θ j k20 χMS j 2k0 πMS cos θ   , ´   V X A + 4X Y B + 4W Y

(3.59)

90

Using Generalized Sheet Transition Conditions (GSTCs) in the Analysis of Metasurfaces

where A = cos θ + j

ko zz ko yy χ + j χES sin2 θ 2 MS 2

(3.60)

and yy

zz + j 2 k0 πES sin2 θ, B = cos θ + j 2 k0 πMS

X =1+j

k0 xx xx cos θ , Y = 1 + j 2k0 πMS cos θ, χ 2 MS

xz zx xz zx V = k02 χMS χMS cos θ , W = k02 πMS πMS cos θ .

(3.61) (3.62) (3.63)

Equations (3.60) and (3.61) apply only to (3.56)–(3.59). From these expressions, we xz,zx xz,zx and πMS ), polarization see that due to the anisotropic terms in the GSTCs (χMS conversion will occur, so that an incident TE wave will generate a TM wave. If the xz,zx xz,zx M = 0 and T T M = 0, = 0, πMS = 0), then RTT E anisotropic terms are zero (i.e., χMS TE and RT E and TT E reduce to those given above [38]. In this case, we are left with pure TE reflected and transmitted fields, as defined above. Using the incident, reflected, and transmitted fields as defined in Eqs. (3.35)–(3.37) in the same GSTCs as for the incident TE case, the reflection and transmission coefficients (as defined in Figure 3.13) for a TM incident plane wave are found to be [39] RT M (θ ) = 1 ´ TT M (θ ) =

E (θ ) = ´ RTT M

cos θ A+

V 4X

cos θ A+

V 4X

´

´

cos θ B+

4W Y

cos θ B+

4W Y

,

,

xz xz cos θ j k2o χMS cos θ j ko πMS ı, ” ´ ‰ “ V X A + 4X Y B + 4YW

(3.64)

(3.65)

(3.66)

and TTTME (θ ) =

xz xz cos θ j k2o χMS cos θ j ko πMS ı. ” ´ ‰ “ V X A + 4X Y B + 4YW

(3.67)

We see that due to the anisotropic terms in the GSTCs, polarization conversion will occur, and an incident TM wave will generate a TE wave. If the anisotropic terms are xz,zx xz,zx E = 0 and T T E = 0; R zero (i.e., χMS = 0, πMS = 0), then RTT M T M and TT M reduce TM to those given above [38], and we are left with pure TM reflected and transmitted fields, as defined above.

3.5.3

Metagratings The GSTCs for the metagrating can be separated into two different sets of expressions that apply to either a TE or a TM incident plane wave. For the TE polarized case, the GSTCs reduce to [32]

3.5 Reflection and Transmission Coefficients

91

“ ‰ ‰ Axx H A (r ) + μ μ χ Bxx H B (r ) (3.68) EzA (ro ) ´ EzB (ro ) = ´j ωμ0 μ0 μA χMS o 0 B MS o x x



“ A1 ‰ “ ‰ Axx H A (r ) ´ μ π Bxx H B (r ) , Ez (ro ) + EzB1 (ro ) = ´j ω μA πMS o B MS o x x

(3.69)

while for the TM polarized case, they reduce to [32] ‰ “ A zz Hz,av (ro ) Ex (ro ) ´ ExB (ro ) = +j ω χMS yx

+χES BBx Ex,av (ro ) ” ı Ayy Byy + BBx χES EyA (ro ) + χES EyB (ro ) ”

ı 1 1 xx E (ro ) ´ HzB (ro ) = j ω0 χES HzA x,av (ro ) ” ı Axy A Bxy +j ω0 χES Ey (ro ) + χES EyB (ro ) .

(3.70)

(3.71)

The forms of these reduced conditions are such that no polarization conversion will occur so long as the incident wave is either TE or TM to the z-direction.

TE Polarization for a Metagrating Let a metagrating be located in the plane y = 0 in free space. If we assume a TE polarized plane wave is incident onto the metagrating shown in Figure 3.10, then reflected and transmission E and H field are given in Eqs. (3.15), (3.16), and (3.18). Substituting the fields into the GSTCs (3.68) and (3.69) gives the reflection and transmission coefficients for a TE incident field: RT E =

 Axx    Bxx + π Axx ´ π Bxx cos θ + k 2 χ Bxx π Axx + χ Axx π Bxx cos2 θ 2 ´ j k0 χMS ´ χMS MS MS MS MS MS MS 0  Axx    Bxx + π Axx + π Bxx cos θ + k 2 χ Bxx π Axx + χ Axx π Bxx cos2 θ ´2 ´ j k0 χMS + χMS MS MS MS MS MS MS 0 (3.72)

and TT E =



Axx + χ Bxx ´2 ´ j k0 χMS MS

 Axx  Axx cos θ j 2k0 χMS ´ πMS    . Axx + π Bxx cos θ + k 2 χ Bxx π Axx + χ Axx π Bxx cos2 θ + πMS MS MS MS MS MS 0 (3.73)

For a symmetric wire grating, Axx = χ Bxx = χ xx χMS MS MS Axx = π Bxx = π xx , πMS MS MS

(3.74)

and in that case the reflection and transmission coefficients reduce to RT E =

TT E

xx π xx cos2 θ 1 + k02 χMS  MS xx + χ xx cos θ + k 2 χ xx π xx cos2 θ ´1 ´ j k0 χMS MS 0 MS MS



 xx  xx cos θ j k0 χMS ´ πMS  xx  = . xx cos θ + k 2 χ xx π xx cos2 θ ´1 ´ j k0 χMS + χMS 0 MS MS

(3.75)

(3.76)

These have the same functional form as those given by Wainstein for an uncoated wire grating [105, Eq. (2.30)].

92

Using Generalized Sheet Transition Conditions (GSTCs) in the Analysis of Metasurfaces

TM Polarization for a Metagrating To simplify the formulas in the TM case, we will assume that no coupling terms are present: Axy

Bxy

yx

χES = χES = χES ” 0.

(3.77)

The more general case can be derived (following a similar procedure) if desired. Assume that a TM polarized plane wave is incident onto the metascreen shown in Figure 3.11, such that the E and H -field components of the incident, reflected, and transmitted plane waves are given by Eqs. (3.22) and (3.21). Substituting the field into the GSTCs (3.70) and (3.71), the reflection coefficient for a TM incident field is found to be NR ” ı ,   RT M = ´   Ayy Byy Bzz xx ´2j + k0 χES cos θ ´2j cos θ + k0 χMS ´ χES + χES sin2 θ (3.78) where ”   Ayy Byy xx xx NR = k0 ´2j χES cos2 θ + k0 χSE χES ´ χES cos θ sin2 θ  ı   Ayy Byy Bzz ´ χES + χES sin2 θ . (3.79) +2j χMS The transmission coefficient for a TM incident field is given by   Bzz xx xx χ Ayy sin2 θ cos θ 4 + k02 χMS χES ´ 2k02 χES ES ” ı .   TT M =   Ayy Byy Bzz xx ´2j + k0 χES cos θ ´2j cos θ + k0 χMS ´ χES + χES sin2 θ (3.80) For the case of a symmetric metagrating, Ayy

Byy

yy

χSE = χSE = χSE Axy

Bxy

yx

χSE = χSE = χSE ” 0, and the reflection and transmission coefficients reduce to  zz  yy xx cos2 θ ´2j k0 χMS ´ 2χES sin2 θ ´ χES RT M =     xx cos θ ´2j cos θ + k0 χ zz ´ 2χ yy sin2 θ 2 ´2j + k0 χES ES MS and TT M

  zz xx ´ 2k 2 χ xx χ yy sin2 θ χES cos θ 4 + k02 χMS 0 SE ES =  .   xx cos θ ´2j cos θ + k0 χ zz ´ 2χ yy sin2 θ 2 ´2j + k0 χES MS SE

(3.81)

(3.82)

(3.83)

These have the same functional form as Wainstein’s for an uncoated wire grating [105, Eq. (2.31)].

3.6

Determining the Surface Parameters The surface parameters that appear explicitly in the various boundary conditions for different types of metasurfaces are uniquely defined and are their best quantitative characterization when their microscopic scale is small compared to a wavelength. At

3.6 Determining the Surface Parameters

93

times, these surface parameters can be difficult to determine. For specific geometries, there are various approximate expressions to be found throughout the literature that are based on dipole interactions. However, these approximate expressions are limited in their range of application. The electric and magnetic susceptibilities for an array of spherical particles are given in [33] and [35]. The surface susceptibilities for an array of square patches are given in [37] and [106], and the surface susceptibility of an array of circular wires is given in [32]. The electric and magnetic surface porosities for arrays of circular apertures or square apertures are given in [107] and [108]. The magnetic surface porosity for an array of wires is given in [32]. The surface susceptibilities and porosities can also be determined by solving the static field problems defined in [30–32], giving a framework for calculating the surface parameters for metafilms, metascreens, and metagratings. As shown in these references, the solution of these static field problems can be computationally challenging for arbitrarily shaped structures. Alternatively to using approximate expressions or solving the static field problems, we can derive expressions that allow for the surface parameters to be retrieved from measured or simulated values of R and T . This method is analogous to the modified Nicolson–Ross–Weir (NRW) approach used for retrieving the effective permeability and permittivity of a metamaterial [42, 97–102], which has been extended to the retrieval of the surface parameters for metafilms and metascreens [33, 34, 38, 39]. In this section, the GSTCs for the various metasurfaces presented above will be used here to develop retrieval techniques for the surface parameters of metafilms, metascreens, and metagratings.

3.6.1

Retrieval Expressions for Metafilms Once the reflection and transmission coefficients are obtained (either from measurements or from numerical calculations), the surface susceptibilities can be determined. For the case of scatterers that have diagonal surface susceptibility dyadics, two different sets of R and T data are required (one at normal incidence and one at oblique incidence) for each polarization. For the TE polarized wave, the three unknown surface susceptibilities are determined from Eqs. (3.56) and (3.57) to give xx = χMS

2j RT E (0) ´ TT E (0) + 1 2j RT E (0) + TT E (0) ´ 1 zz = , χES k0 RT E (0) ´ TT E (0) ´ 1 k0 RT E (0) + TT E (0) + 1

(3.84)

and yy

yy

χMS = ´

χES 2

sin (θ )

+

2j cos(θ ) RT E (θ ) + TT E (θ ) ´ 1 . k0 sin2 (θ ) RT E (θ ) + TT E (θ ) + 1

(3.85)

In these expressions, RT E (0) and TT E (0) are the reflection and transmission coefficients at normal incidence (θ = 0˝ ), while RT E (θ ) and TT E (θ ) are the reflection and transmission coefficients at some oblique incidence angle, sufficiently different from θ = 0. If experimental data are used, two angles other than θ = 0 may be required because measuring the reflection coefficients at θ = 0 can be difficult; this leads to a different set of equations than those presented above.

94

Using Generalized Sheet Transition Conditions (GSTCs) in the Analysis of Metasurfaces

(a) RT M (0)

(b) TT M (0)

Figure 3.14 Numerically computed RT M (0) and TT M (0) (referred to y = 0) for the array of metallic scatterers shown in Figure 3.8d.

For the TM polarized wave, the three unknown surface susceptibilities are determined from Eqs. (3.64) and (3.65) to give [33, 34] xx χES =

2j RT M (0) + TT M (0) ´ 1 2j RT M (0) ´ TT M (0) + 1 zz = , χMS k0 RT M (0) + TT M (0) + 1 k0 RT M (0) ´ TT M (0) ´ 1

(3.86)

and yy

yy

χES = ´

χMS 2

sin (θ )

+

2j cos(θ ) TT M (θ ) ´ 1 ´ RT M (θ ) . k0 sin2 (θ ) TT M (θ ) + 1 ´ RT M (θ )

(3.87)

Similar to the TE case, RT M (0) and TT M (0) are the reflection and transmission coefficients at normal incidence, while RT M (θ ) and TT M (θ ) are the reflection and transmission coefficients at some oblique incidence angle, sufficiently different from θ = 0. In the retrieval approaches for both polarizations, it is important to realize that the reference plane for RT E,T M (0), TT E,T M (0), RT M (θ ), and TT M (θ ) is required to be located at y = 0. This is a consequence of how the GSTCs were derived in [29] and [30]. The GSTCs (and the surface parameters) would need to be modified for different choices of reference plane location [105, 109, 110]. In order to illustrate the validity of these expressions for retrieving the surface susceptibilities of a metafilm, we will next consider some examples. We first use the retrieval method to determine the surface susceptibilities of a metafilm composed of the metallic scatterers shown in Figure 3.8d. The TM mode reflection (RT M (0)) and transmission coefficients (TT M (0)) at an incidence angle of zero degrees for this structure were obtained from a commercial finite-element EM field solver. These numerical results are shown in Figure 3.14, for the E-field polarized along the x-axis and the reference plane of RT M (0) and TT M (0) located at y = 0. These values are used in xx , a plot of which was shown in Figure 3.8c. Eq. (3.86) to obtain χES

3.6 Determining the Surface Parameters

(a) RT M (0) and RT M (30˝ )

95

(b) TT M (0) and TT M (30˝ )

Figure 3.15 Numerically computed RT M (0) and TT M (0) (referred to y = 0) for the array of spherical particles shown in Figure 3.1b.

In the next example, we consider a metafilm composed of a square array of spherical particles, as shown in Figure 3.1b. For this metafilm, a = 10 mm, p = 25.59 mm, r = 2, μr = 900, tan δ = 0.04. The TE reflection and transmission coefficients at incidence angles of 0˝ and 30˝ for this structure were obtained from a commercial finiteelement EM field solver and are shown in Figure 3.15. These values were then used in xx and χ yy , which are shown in Figures 3.8a and 3.8b. Eqs. (3.84) and (3.85) to obtain χES ES In [35] and [33], it is shown that if the scatterers are spaced sufficiently far from each other, the components of the electric and magnetic surface susceptibility dyadics can be determined by means of “sparse-array” approximation formulas. These approximate analytical values are also plotted in Figures 3.8a and 3.8b. From this comparison, we see that the surface susceptibilities obtained from the retrieval approach are virtually identical to those obtained from the approximate analytical expressions. Retrieval expressions for anisotropic metafilms are given in [104, 105].

3.6.2

Retrieval Expressions for Metascreens For a metascreen, we see from (3.48) and (3.49) that only two surface parameters xx and π xx ) determine R (χMS T E (θ ) and TT E (θ ). Using these two expressions, the two MS unknown surface parameters are determined from [38] xx πMS =

j RT E (0) + TT E (0) + 1 2k0 RT E (0) + TT E (0) ´ 1

(3.88)

xx = χMS

2j RT E (0) ´ TT E (0) + 1 , k0 RT E (0) ´ TT E (0) ´ 1

(3.89)

and

where RT E (0) and TT E (0) are the reflection and transmission coefficients at normal incidence (θ = 0˝ ). Any angle for RT E (θ ) and TT E (θ ) in (3.88) and (3.89) could have been used, but θ = 0˝ is a convenient choice. If RT E (0) and TT E (0) are known, either

96

Using Generalized Sheet Transition Conditions (GSTCs) in the Analysis of Metasurfaces

(a)

(b)

Figure 3.16 Metascreen: (a) circular apertures and (b) magnetic surface susceptibility and surface porosity for an array of circular apertures: p = 100 mm and h = 10 mm.

from experimental measurements or from a numerical simulation of the metascreen, xx and π xx . then (3.88) and (3.89) can be used to retrieve the values of χMSS MS From (3.50) and (3.51) we see that R(θ) and T (θ) depend on four of the surface parameters. Thus, unlike the TE polarization case, two sets of RT M (θ ) and TT M (θ ) data are required to determine all four unknowns for the TM polarized wave [38]: zz = πMS

j RT M (0) + TT M (0) + 1 , 2k0 RT M (0) + TT M (0) ´ 1

(3.90)

zz χMS =

2j RT M (0) ´ TT M (0) + 1 , k0 RT M (0) ´ TT M (0) ´ 1

(3.91)

zz πMS

yy

πES = ´

2

sin θ

+

j cos θ RT M (θ ) + TT M (θ ) + 1 , 2k0 sin2 θ RT M (θ ) + TT M (θ ) ´ 1

(3.92)

+

2j cos θ RT M (θ ) ´ TT M (θ ) + 1 , k0 sin2 θ RT M (θ ) ´ TT M (θ ) ´ 1

(3.93)

and yy

χES = ´

zz χMS 2

sin θ

where RT M (0) and TT M (0) are the reflection and transmission coefficients at normal incidence (θ = 0˝ ), while RT M (θ ) and TT M (θ ) are the reflection and transmission coefficients at some oblique incidence angle, sufficiently different from θ = 0˝ . With RT M (0), TT M (0), RT M (θ ), and TT M (θ ) available from either simulation or experiment, zz zz , πMS , (3.90)–(3.93) can be used to retrieve the four unknown surface parameters (χMS yy yy χES , and πES ). Here we consider two examples for retrieving the surface parameters of a metascreen. We first consider an array of circular apertures in a perfect conductor of thickness h, as shown in Figure 3.16a. For this metascreen, p = 100 mm, the thickness h = 10 mm, and

3.6 Determining the Surface Parameters

(a)

97

(b)

Figure 3.17 Metascreen: (a) square apertures and (b) magnetic surface susceptibility and surface porosity for an array of circular apertures: p = 100 mm and h = 10 mm.

a is the radius of the apertures (which will be varied). The reflection and transmission coefficients for the metascreen were determined numerically from the finite-element software HFSS (mentioning this product does not imply an endorsement, but serves to clarify the numerical program used) for both θ = 0˝ and θ = 30˝ at a frequency of 500 MHz and a{p ranging from 0 to 0.5. The numerical values for RT M (0), TT M (0), RT M (θ ), and TT M (θ ) at the reference plane y = 0 were used in (3.90) and (3.91) to zz zz and πMS ). Figure 3.16b shows determine the two magnetic surface parameters (χMS the retrieved magnetic surface parameters for a range of a. Using dipole-interaction approximations, closed-form results for these surface porosities and susceptibilities have been obtained in [107] for an array of circular apertures. These approximations are also shown in Figure 3.16b. The surface susceptibilities and porosities can also be determined by solving the static field problem as defined in [31]. These homogenizationbased surface parameters are also shown in Figure 3.16b. From the comparison, we see that the retrieved surface magnetic susceptibility and surface magnetic porosity correlate very well with the other two approaches for the whole range of a{p = 0 to a{p = 0.49. Next, we consider an array of square apertures in a perfectly conducting plate as shown in Figure 3.17(a). For this array, p = 100 mm, the thickness is h = 10 mm, and l is the length of the side of the square (which will be varied). The numerical values for RT M (0), TT M (0), RT M (θ ), and TT M (θ ) at the reference plane y = 0, were zz and used in (3.90)–(3.93) to determine the two magnetic surface parameters (χMS

98

Using Generalized Sheet Transition Conditions (GSTCs) in the Analysis of Metasurfaces

zz πMS ). Figure 3.17(b) shows the retrieved magnetic surface parameters for a range of l. In this figure, we also show the dipole-approximation from [107] and results from the homogenization model [31]. From the comparison, we see that the retrieved surface magnetic susceptibility and surface magnetic porosity correlate very well with the other two approaches for the whole range of l{p, except that the dipole approximation for zz becomes somewhat inaccurate for l{p ą 0.5. Retrieved results for other apertures πMS are found in [38]. Retrieval expressions for anisotropic metascreens are given in [39].

3.6.3

Retrieval Expressions for Metagratings As was the case for a TE incident wave onto a metascreen, we see from (3.75) and xx and π xx ) determine R (3.76) that only two surface parameters (χSM T E (θ ) and TT E (θ ) MS for a metagrating. Using these two expressions, the two unknown surface parameters are determined from xx πMS =

j RT E (0) + TT E (0) + 1 k0 RT E (0) + TT E (0) ´ 1

(3.94)

xx χMS =

j RT E (0) ´ TT E (0) + 1 . k0 RT E (0) ´ TT E (0) ´ 1

(3.95)

and

From (3.82) and (3.83) we see that RT M (θ ) and TT M (θ ) for the symmetric case depend on three surface parameters. Thus, unlike the TE polarization case, two sets of RT M (θ ) and TT M (θ ) data are required to determine all three unknowns for the TM polarized wave:

zz χMS =

2j RT M (0) ´ TT M (0) + 1 , k0 RT M (0) ´ TT M (0) ´ 1

(3.96)

xx = χES

2j RT M (0) + TT M (0) ´ 1 , k0 RT M (0) + TT M (0) + 1

(3.97)

and yy

χES =

zz χMS

2 sin2 θ

+

j k0 xx 2 χES r1 + RT M (θ )s cos θ . xx R 1 ´ RT M (θ ) ´ j 2k0 χES T M (θ ) cos θ

j cos θ RT M (θ ) + k0 sin2 θ

(3.98)

As an example, we consider the uncoated circular wire grating shown in Figure 3.18a. xx , χ xx , and π xx ) [32]. This metagrating has only three nonzero surface parameters (χMS ES MS The retrieved surface parameters are shown in Figure 3.18b for a range of wire radius a, together with Wainstein’s analytical results [105] and results from homogenization [32] for comparison.

3.7 Some Applications of GSTCs

(a) wire grating

99

(b) surface parameters

Figure 3.18 Metagrating: (a) uncoated wire grating and (b) magnetic and electric surface susceptibility and surface porosity for an array of wires of radius a. Also shown in this figure are results from [105].

3.7

Some Applications of GSTCs However they are obtained, we have seen how the surface susceptibilities and porosities together with the GSTCs allow the plane-wave reflection and transmission coefficients to be computed. In this section we will discuss the use of GSTCs to the modeling of metasurfaces in a number of other applications.

3.7.1

Guided Waves on a Single Metasurface As is the case for an ordinary dielectric slab, a metasurface may also support surface waves under suitable conditions. However, unlike a conventional dielectric slab, forward and backward surface waves as well as complex modes may be excited simultaneously on the metasurface; this is a direct consequence of engineering the properties of the constituent scatterers or apertures. In fact, the scatterers that compose a metafilm can be judiciously chosen such that surface waves and/or complex modes will be present only at certain desired frequencies. Conditions for guided waves that can be supported on a metafilm are given in detail in [36]. Here we will merely summarize the use of GSTCs in this analysis, which could be extended to other metasurfaces such as metascreens or metagratings if desired. Consider an electric line source oriented in the z-direction, at a distance d above and parallel to a metafilm of infinite extent in the xy-plane (see Figure 3.19, and note the change in coordinate system compared to that used in [36]). The scatterers of the metafilm are assumed to be of rather arbitrary shape, but with enough symmetry that

100

Using Generalized Sheet Transition Conditions (GSTCs) in the Analysis of Metasurfaces

Figure 3.19 Line source above a metafilm. The line source is either an electric or magnetic current source located at z = d, and the metafilm is represented by the dashed line at z = 0.

(3.13)–(3.14) hold. If the regions above and below the metafilm consist of the same dielectric material, the problem space can be divided into three sections. Region I (y ą d) extends to infinity in the +y-direction, while region III (y ă 0) extends to infinity in the ´y-direction. Region II (0 ă y ă d) is the area between the plane of the line source and the metafilm. Using a spectral-domain approach and applying the GSTCs at y = 0 and the appropriate jump conditions at the source location (i.e., y = d), the scattered field for y ą 0 is given by [36] ż ωμ0 8 e´jρ (y+d) e´j ν x (3.99) RST E (ν) dν, Es = ´ 4π ´8 ρ where „

RST E (ν) =

´j  xx 1´χMS

k02 zz ρ xx ν 2 yy 2ρ χES + 2ρ χMS ´ 2 χMS

k02 zz ν 2 yy 4 χES + 4 χMS



+j



j

k02 zz ρ xx ν 2 yy 2ρ χES + 2ρ χMS + 2 χMS

j

(3.100)

and ρ and ν are associated with an incidence angle θ by ρ = k0 cos θ and ν = k0 sin θ .

(3.101)

The poles of the reflection coefficient RST E are functions of the surface susceptibilities of the metafilm. Once these poles are determined, the propagation constant of a surface wave along the metafilm (i.e., along the x-axis) for this guided-wave pole is given by b (3.102) ν = k 2 ´ ρs2, where ρs is given in Table 3.1 under various conditions for an electric line source; values for a magnetic line source are given in Table 3.2 [36]. When the surface susceptibilities of the metasurface meet certain of these conditions, surface waves or complex modes can be excited. As an example, consider a metafilm composed of spherical particles that have a radius of a = 10 mm and are placed in an array with a period of p = 25.67 mm (see Figure 3.1b). The material properties of these particles were chosen to be r = 100, μr = 1, and tan δ = 1 ˆ 10´4 – representative of some commercially available

3.7 Some Applications of GSTCs

101

Table 3.1 Conditions for surface waves and guided modes for an electric-line source (as well as expressions for the propagation constant) Number of guided waves Even-class poles

Constraints on χES and χMS yy χMS ą 0

yy zz ě 1 χES yy ´ χMS k 2 χMS

2

ρE1,2 =

c   zz +χ yy χ yy ´1 ´j ˘ k 2 χES MS MS

2

— ρE1,2 = ´j – »

zz ď ´χ yy χES MS

1

— ρE1 = ´j –

c fi   zz +χ yy χ yy 1˘ 1´k 2 χES MS MS

2

c fi   zz +χ yy χ yy 1+ 1´k 2 χES MS MS

yy

χMS = 0



c

MS

zz ą ´χ yy χES MS

1

zz ą 0 χES

1

— ρE1 = ´j –

ffi fl

yy χMS

2 yy — 1˘ 1´k χMS ρE1,2 = ´j – yy χ

» yy χMS ă 0

ffi fl

yy χMS

»

zz = 0 χES

complex mode

yy χMS

» yy yy zz ď 1 ´χMS ă χES yy ´ χMS k 2 χMS

Type of guided wave

2 zz ρE1 = ´j2k χES

surface wave

2 fi

surface wave

ffi fl

c fi   zz +χ yy χ yy 1´ 1´k 2 χES MS MS yy χMS

surface wave

ffi fl

surface wave

surface wave

These results are valid for pure real surface susceptibilities. Note that the total number of guided waves increases by one if xx ) ă 0. Re(χMS

yy,xx

yy,xx

zz zz ceramics at microwave frequencies. Values for χMS , χES , χES χMS as functions of frequency for this array are given in [36]. The symmetry of the spherical particles means zz xx and χ zz = χ xx . Figure 3.20 shows the magnitudes of the electric = χES that χES MS MS fields for an electric line-source excitation for this array at two different frequencies (1.5 GHz and 2.0 GHz), as computed by a commercial finite-element program, with the line source placed 45.59 mm above the metafilm. This figure shows the presence of one surface wave and a pair of complex modes at the lower frequency (Figure 3.20a), while Figure 3.20b illustrates what happens at the higher frequency when two surface waves are present. Figure 3.21 shows the magnitude of the E-field for a magnetic line source placed above the same array. Figure 3.21a illustrates the presence of one surface wave, while Figure 3.21b illustrates the existence of a complex mode. By varying the properties of the scatterers, the surface susceptibilities will change, and hence, from the results in Tables 3.1 and 3.2, it is possible to have a surface wave and/or complex mode exist at any desired frequency, i.e., to realize a frequency-agile guided-wave structure. Leaky waves are also possible for this metafilm. The angle at which energy propagates away from the surface for a leaky wave is given by [139]

 θLEAKY = arcsin

Re[ν] k0

 (3.103)

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Using Generalized Sheet Transition Conditions (GSTCs) in the Analysis of Metasurfaces

Table 3.2 Conditions for surface waves and guided modes for a magnetic-line source (as well as expressions for the propagation constant) Number of guided waves

Constraints on χES and χMS yy χES ą 0

yy zz ě 1 χMS yy ´ χES k 2 χES

2

ρE1,2 =

c   zz +χ yy χ yy ´1 ´j ˘ k 2 χMS ES ES

2

— ρE1,2 = ´j – »

zz ď ´χ yy χMS ES

1

— ρE1 = ´j –

c fi   zz +χ yy χ yy 1˘ 1´k 2 χMS ES ES

2

c fi   zz +χ yy χ yy 1+ 1´k 2 χMS ES ES

yy

χES = 0

ffi fl

yy χES 

c

2 yy — 1˘ 1´k χES ρE1,2 = ´j – yy χ ES

» yy χES ă 0

zz ą ´χ yy χMS ES

1

zz ą 0 χMS

1

— ρE1 = ´j –

ffi fl

yy χES

»

zz = 0 χMS

complex mode

yy χES

» yy yy zz ď 1 ´χES ă χMS yy ´ χES k 2 χES

Type of guided wave

Even-class poles

surface wave

surface wave

2 fi

surface wave

ffi fl

c fi   zz +χ yy χ yy 1´ 1´k 2 χMS ES ES yy χES

2 zz ρE1 = ´j2k χMS

ffi fl

surface wave

surface wave

These results are valid for pure real surface susceptibilities. Note that the total number of guided waves increases by one if xx ) ă 0. Re(χES

(a) f=1.50 GHz

(b) f=2.0 GHz

Figure 3.20 Magnitude of the E-field from an electric-line source placed 45.49 mm above an array of spherical particles: (a) f = 1.50 GHz (one surface wave and a complex mode pair) and (b) f = 2.0 GHz (two surface waves). For an array of spherical particles consisting of a = 10 mm, p = 25.67 mm, r = 100, μr = 1, and tan δ = 1 ˆ 10´4 . A black and white version of this figure will appear in some formats. For the color version, please refer to the plate section.

3.7 Some Applications of GSTCs

(a) f=1.42 GHz

103

(b) f=1.5 GHz

Figure 3.21 Magnitude of the E-field (on a linear scale) from a magnetic line source placed 45.49 mm above an array of spherical particles: (a) f=1.42 GHz: one surface wave, and (b) f=1.5 GHz: complex mode. A black and white version of this figure will appear in some formats. For the color version, please refer to the plate section.

under the restriction that Re[ν] ă k0 .

(3.104)

Figure 3.22 shows θLEAKY as a function of frequency for the same array of spherical particles as that used above. These results were obtained by using (3.103) and the surface parameters for the array given in [36]. To validate these results, Figure 3.23 shows the magnitudes of the E-field for frequencies of 1.70 GHz, 1.66 GHz, and 1.64 GHz. These results were obtained with a commercial finite-element program for the same array as used in Figure 3.22. These results illustrate that little energy is flowing along the x-axis, while energy is propagating at some angle away from the metasurface as a leaky wave. The angles at which the energy propagates away from the surface correspond to those given in Figure 3.22.

3.7.2

Resonator Size Reduction Engheta [140, 141] has demonstrated that the classical lower bound on the size of a resonant structure can be reduced if a cavity is partially filled with a negative-index material. Extending this idea, it was shown in [37, 142] that the same thing could be accomplished with a metasurface. The advantage of a metasurface is that because it requires less physical space than a 3D metamaterial, a cavity or resonator with a metasurface can in principle be made smaller than those that use 3D metamaterials. Consider the resonator shown in Figure 3.24. This resonator consists of perfectly conducting metallic walls at y = d1 and y = ´d2 with a metafilm placed at y = 0. The resonator is divided into two regions (labeled A and B). It will be assumed for simplicity that both regions are filled with the same homogeneous medium. An x-polarized electric

104

Using Generalized Sheet Transition Conditions (GSTCs) in the Analysis of Metasurfaces

Figure 3.22 Leaky-wave angle versus frequency for an array of spherical particles with a = 10 mm, p = 25.67 mm, r = 100, μr = 1, and tan δ = 1 ˆ 10´4 .

(a) f=1.70 GHz: θl = 55˝ ,

(b) f=1.66 GHz: θl = 30˝ ,

(c) f=1.64 GHz: θl = 20˝

Figure 3.23 Magnitude of E-field and Poynting’s vector for an electric-line source along the y-axis placed 45.49 mm above an array of spherical particles: (a) f=1.70 GHz, (b) f=1.66 GHz, and (c) f=1.64 GHz.

3.7 Some Applications of GSTCs

105

Figure 3.24 Resonator: a metafilm placed between two parallel metal plates.

field, independent of x and z, is assumed. The electric field in region A is then given by ” ı E¯ A = a¯ x E1 e´jβy + E2 ejβy (3.105) and in region B by ” ı E¯ B = a¯ x E3 e´jβy + E4 ejβy ,

(3.106)

? where β = ω μ = 2π{λ is the wavenumber in the resonator medium, a¯ x is a unit vector, and E1 , E2 , E3 , and E4 are constants to be determined from the boundary conditions. Applying the boundary condition that the total electric field on the two perfect conductors at y = d1 and y = ´d2 must be zero, we obtain relationships between E1 and E2 from the y = d1 plane boundary condition and between E4 and E3 from the y = ´d2 plane boundary condition. The GSTCs for the metafilm are then applied at y = 0 in order to obtain the phase-matching condition for the resonator [37]. The phase-matching conditions required at resonance for the separation distance d1 = d2 = d between two metal plates with a metasurface placed in the center are given by [37] ”  ı d = λ2 1 + nπ ´ π2 tan´1 πλ χES ” for n = 0,1,2,3,..., (3.107)  ı d = λ2 2nπ ´ π2 tan´1 πλ χMS where n = 0 is not allowed if χMS ě 0 [37]. From these expressions it is seen that by judiciously choosing the metasurface, it is possible to have a resonator that overcomes the λ{2 size limit. Various cases for different conditions on the surface parameters are given in [37]. As an example, consider a metafilm composed of thin square perfectly conducting patches placed between two metal plates. The square-patch metafilm is shown in

106

Using Generalized Sheet Transition Conditions (GSTCs) in the Analysis of Metasurfaces

(a)

(b) Figure 3.25 Resonant frequency of air-filled parallel-plate resonator loaded with square-patch capacitive metafilm: (a) array of square conducting patches and (b) resonant frequency.

Figure 3.25a. Figure 3.25b shows the resonant frequency (fr ) versus l{p (where p is the period and l is the length of the side of one of the squares) for three different plate separations. For reference, the classical result d = λ{2 occurs when l{p = 0. To verify these theoretical results, this resonator was analyzed with the commercial finite-element (FEM) software code HFSS from Ansoft Corporation (mentioning this product does not imply an endorsement, but serves to clarify the numerical program used). It is seen that the capacitive metasurface can significantly reduce the resonant frequency for a given resonator size d, or equivalently reduce the size of the resonator needed to obtain a desired resonant frequency. The reduction in resonant frequency of the resonator for the simple square-patch metafilm resonator is shown in Table 3.3 for p = 500 μm. In this table, the percentage reduction (which is also a measure of size reduction possible for a given resonant frequency) is listed for different values of l{p. The results in this table show that a square-patch metafilm placed at the center of a resonator can reduce the size by as much as 56%. If metasurfaces are designed with scatterers having more elaborate

3.7 Some Applications of GSTCs

107

Table 3.3 Resonant frequency reduction as a function of l{p for a parallel-plate resonator with a metafilm consisting of square metal patches with p = 500 μm Reduction in fr l{p

d1 + d2 = 0.5 mm

d1 + d2 = 1.0 mm

d1 + d2 = 2.0 mm

0 0.4 0.6 0.8 0.96

0% 7.3 % 20.9 % 38.4 % 56.1 %

0% 3.8 % 11.9 % 25.2 % 42.6 %

0% 1.9 % 6.4 % 14.9 % 29.0 %

εr

l p

2a

Figure 3.26 A surface array of cylindrical dielectric pucks with relative permittivity r , radius a and height l.

polarizability characteristics (e.g., judiciously chosen resonant behavior), it should in principle be possible to achieve even greater reduction in size. In fact, by controlling the properties of the metasurface, a frequency-agile resonator could be realized. Losses in the materials making up the metafilm will have a limiting effect on the performance of these resonators. Losses show up in the GSTC formulation as complex values of the effective surface susceptibilities. To investigate losses, we consider a nonmagnetic dielectric cylinder with radius a, height l (oriented along the y-axis), and permittivity r relative to the surrounding medium, see Figure 3.26. If l ! a, this is a thin dielectric disk. Analytical formulas for the surface parameters for an array of dielectric cylinders, which can be applied to the lossy case simply by making the cylinder’s permittivity complex, are given in [37]. The resonant frequencies obtained by solution of the GSTC-based equation were compared with FEM results. The results are summarized in Tables 3.4 and 3.5. Agreement between the GSTC model and the FEM results is good. For reference, recall that this resonator will have a resonant frequency of fr = 10 GHz with no metafilm present. Its lowest resonant frequency will be fr = 2.887 GHz when completely filled with the dielectric medium used for the pucks, with a quality factor Q = 25 given the loss tangent of 0.04. We observe that there is a trade-off between the reduction in resonant frequency (or resonator size) that we can achieve, and the value of Q. This is quite natural from a physical point of view, since as the

108

Using Generalized Sheet Transition Conditions (GSTCs) in the Analysis of Metasurfaces

Table 3.4 Resonant frequency and Q as functions of l{2a for a parallel-plate resonator with a metafilm consisting of cylindrical dielectric pucks with r = 12, tan δ = 0.04, d1 = d2 = 7.5 mm, p = 10 mm, and a = 4.6 mm GSTC

FEM

l{2a

fr , GHz

Q

fr , GHz

Q

0.05 0.1 0.2 0.3

8.756 8.116 7.297 6.718

147 120 106 100

8.626 7.919 7.058 6.533

116 93 81 78

Table 3.5 Resonant frequency and Q as functions of l{2a for a parallel-plate resonator with a metafilm consisting of cylindrical dielectric pucks with r = 12, tan δ = 0.04, d1 = d2 = 7.5 mm, p = 10 mm, and a = 4.0 mm GSTC

FEM

l{2a

fr , GHz

Q

fr , GHz

Q

0.05 0.1 0.2 0.3

9.160 8.707 8.100 7.652

221 178 155 144

9.077 8.593 7.986 7.580

179 144 129 124

polarizability of the metafilm scatterers increases, the proportion of field energy located in the lossy region increases.

3.7.3

Waveguides Because metasurfaces can be designed to give total reflection of an incident plane wave, it should be possible to trap and guide electromagnetic energy in a region between two metasurfaces [40]. This class of metasurface waveguides is illustrated in Figure 3.27 (note the change in coordinate system compared to that used in [40]). This waveguide consists of a region of space 0 ă y ă d bounded by two identical parallel metafilms. All regions are assumed to be air, so we may take their material properties to be those of free space. Assume that we have a TE mode polarized in the x-direction, propagating in the zdirection, and with a time dependence ej ωt . The field between the metafilms is given by the superposition of two plane waves incident at the angle θ to the ˘y-directions, as shown in Figure 3.2:   0 ă y ă d, (3.108) Ex = E1 e´j ky y + E2 e+j ky y e´jβz where b β = k0 sin θ is the unknown propagation constant, which must be determined, ? k02 ´ β 2 = k0 cos θ, and k0 = ω μ0 0 is the wave number of free space. ky = Outside this region, the field is written as outgoing (or attenuating) waves:

3.7 Some Applications of GSTCs

109

Figure 3.27 An illustration of a waveguide composed of two metafilms separated by a distance d.

Ex = E3 e´j ky y e´jβz

yąd

(3.109)

+j ky y ´jβz

y ă 0.

(3.110)

Ex = E4 e

e

Equations (3.56) and (3.57) show that, in order for total reflection to occur, the normalized propagation constant must obey g f yy xx f χMS χES + 42 β k0 e = (3.111) ne = yy xx k0 ´χMS χMS [see also Eq. (3.114)]. This quantity will generally be complex for leaky modes. Once the scatterers that compose the metafilm are chosen to meet the above criterion, and β is determined, the transverse wavenumber in the x direction is given by b b kx = k02 ´ β 2 = k0 1 ´ n2e , (3.112) and the separation distance d between the two metasurfaces is given by  yy    kx χMS 1 ´ 1 n + 2 π + 2 tan 2 for n = 1,2,3,.... d= kx

(3.113)

It is desirable to find conditions for which the imaginary part of ne is as small as possible, and we need to ensure that Im(ne ) ă 0 and 0 ă Re(ne ) ă 1 (the latter because otherwise the mode will basically be a surface wave localized near the two metasurfaces and will likely suffer increased attenuation as a result). Similar sets of expressions for the TM modes along with an example are given in [40]. This waveguide can be compact, with low material and radiation losses. If the metafilms were constructed of a polymer type of material, it would be possible to have a flexible waveguiding structure. It is also possible to control the surface properties of the scatterers (and in turn the surface parameters), which would result in controllable or smart, frequency-agile waveguiding structures.

110

Using Generalized Sheet Transition Conditions (GSTCs) in the Analysis of Metasurfaces

3.7.4

Controllable Reflections and Transmissions Given a generic metasurface, one could use any of a number of commercial computational codes to analyze the interaction of an electromagnetic field with that metasurface. However, as we saw in Section 3.5, the GSTCs allow us to obtain closed-form expressions for the plane-wave reflection and transmission coefficients, the advantage of which is that one can get some physical insight into the relationship of the surface parameters to the reflection and transmission behavior. Expressions given in Section 3.5 illustrate that if the surface parameters (surface susceptibilities and/or porosities) can be changed, it will be possible to control the reflection and transmission behavior of the surface.

Metafilm From Eqs. (3.19), (3.20), (3.23), and (3.24) one can write down the relationship between the electrical and magnetic surface susceptibilities needed to have either total reflection or total transmission for a metafilm. For total reflection, the following conditions must be satisfied:   xx χ zz + χ yy sin2 θ = ´4 for TE k02 χMS ES MS   (3.114) xx χ zz + χ yy sin2 θ = ´4 for TM, k02 χES ES MS while for total transmission, the required conditions are yy

zz xx cos2 θ = 0 for TE χES + χMS sin2 θ ´ χMS yy zz xx cos2 θ = 0 for TM. ´χMS ´ χES sin2 θ + χES

(3.115)

Metascreen Using Eqs. (3.48)–(3.51), for a metascreen the following conditions must be satisfied for total reflection: zz χMS

yy + χES

xx = 4π xx χMS for TE “ MS ‰ yy zz 2 sin θ = 4 πMS ´ πES sin2 θ for TM,

(3.116)

while for total transmission, the required conditions are 

yy zz + χES χMS

xx π xx cos2 θ = ´1 k02 χMS for TE MS    2 yy zz sin2 θ πMS ´ πES sin2 θ = ´ cos2 θ for TM.

(3.117)

k0

Metagrating Using Eqs. (3.75)–(3.83), for a metagrating the following conditions must be satisfied for total reflection: xx = π xx χMS MS zz xx ko2 χMS χES

for TE

yy

2 xx χ ´ 2k02 χES ES sin θ = ´4 for TM,

(3.118)

while for total transmission, the required conditions are xx π xx cos2 θ = ´1 k02 χMS MS zz χSM

yy

for TE

xx cos2 θ = 0 TM. ´ 2χSE sin2 θ ´ χSE

(3.119)

In principle, control of the reflection and transmission behavior can be done in a number of ways, e.g., (1) by changing the electrical or magnetic surface parameters or (2) by

3.7 Some Applications of GSTCs

111

Figure 3.28 Controlling the transmission properties of a surface with an external DC magnetic field (see [41]).

changing the properties (either the material properties or the geometry) of the substrate on which the metasurfaces lie. For example, a metafilm made up of spherical magnetic particles can be controlled. Such a controllable surface has been realized by using a metafilm of spherical YIG particles and controlling the surface parameters (the magnetic surface susceptibilities) with a DC magnetic bias field [41]. Figure 3.28 shows the transmission behavior of such a metafilm as a function of the external DC magnetic field. Other approaches have been used to control the surface behavior of a metasurface, and a significant amount of research is ongoing in the area of dynamically controllable metasurfaces, as we discuss in Sections VII and IX of [27]. For example, the highly resonant nature of metasurfaces provides a means for tuning the frequency response of these structures. Many metasurfaces are constructed from metallic inclusions that have plasma resonances governed largely by the choice of their geometry. Aside from the specific geometry, the resonances of these metallic inclusions may also be controlled by affecting the capacitive and or inductive properties that dictate the plasma resonances. The electric coupled resonator is a structure that provides a means for directly perturbing the capacitive response by altering the electrical properties of the material occupying the electrical gap. One way to accomplish this is to cause different fluids to flow in a channel over the gap; for details see [44]. Such a structure is shown in Figure 3.29 and other fluid-tunable structures are considered in [44].

Angularly Independent Behavior and Anisotropic Metastructures The use of metasurfaces with anisotropic parameters can achieve a number of useful results not possible with isotropic metasurfaces. In particular, if the surface parameters are varied along the surface, interesting properties might be achieved. For example, it is

112

Using Generalized Sheet Transition Conditions (GSTCs) in the Analysis of Metasurfaces

Figure 3.29 A fluid-tunable metafilm composed of a 36-element array of metallic resonators fabricated from gold on glass with PDMS (polydomethykiloxane) fluid-channel sections (see [44]). A black and white version of this figure will appear in some formats. For the color version, please refer to the plate section.

possible to design a metasurface so as to focus an EM plane wave to a desired region in space, much like a focusing antenna array. If the surface parameters of the metasurface were designed in a manner such that they could be changed at will, one could as a result have a metasurface capable of changing the direction and frequency where the energy is focused, i.e., a frequency- and space-agile surface. One can also design a metasurface that converts one type of polarization to another. These concepts are currently under investigation. Another example deals with how to obtain (at least in some parameter range) angularly independent behavior of the reflection and transmission [45]. Consider a metafilm. For a given incidence angle, Eq. (3.114) gives the relationships between the electric and magnetic surface susceptibilities required to obtain total reflection. This suggests a mechanism by which one could achieve approximate angular independence yy zz " χMS , the angular of such total reflection. For instance, in the TE case when χES dependence becomes less significant, because the first term in the parentheses of the first equation in (3.114) dominates over the second angularly dependent term. Similarly, yy zz " χES , the angular dependence likewise becomes weak. in the TM case, when χMS These conditions could occur, for example, at certain resonance frequencies of the scatterers that constitute the metafilm. When the indicated terms are dominant, the total reflection conditions become yy

4 xx χ χMS ES = ´ 2 for TE yy

k0

4 xx χ χES MS = ´ 2 for TM.

(3.120)

k0

Similar expressions can be derived for other types of metastructures. Thus, if a metasurface is designed such that the transverse components of the surface susceptibilities are highly resonant, as compared to the normal components, then the metasurface may exhibit angle-independent behavior. Figure 3.30 shows one example of this, where we

3.7 Some Applications of GSTCs

113

0 –10 q = 0° q=10° q = 20° q = 30° q = 40° q = 50° q = 60° q = 70° q = 80°

|R| (dB)

–20 –30 –40 –50 –60 0

5

10

15

20

Frequency (GHz)

Figure 3.30 The reflection coefficient for a TE-polarized incident plane wave for an electrical resonator metasurface structure (see [45]).

have plotted the reflection coefficient of a metasurface composed of the metallic structure shown in Figure 1b in [45]. In this figure we see that at about 14 GHz the surface exhibits a nearly angle-independent behavior up to incidence angles as large as 60˝ . One can extend this concept to many other types of structures. For example, it can be shown that highly anisotropic slabs of materials can exhibit angle-independent behavior as well. This can be seen by examining the reflection properties of an inhomogeneous anisotropic slab, as discussed in [45, Eqs. (31)–(38)]. Additionally, the mathematically constructed perfectly matched layer (PML) [83] that has been introduced into computational electromagnetics for reducing non-physical reflections at radiation boundaries also requires such angle independence. Such PMLs may be physically realizable by use of the concept of metasurfaces. The idea of designing a metasurface structure to use as PML in computational codes is under investigation. This mechanism offers a unique way to match lossy materials for the purpose of developing compact electromagnetic absorbers and make possible unique designs of impedance-matching surfaces. Compact absorbers based on such ideas have recently begun to appear in the literature [46–51]. These structures typically are designed with a metasurface on the front side of a lossy substrate backed by a metal plate. These structures may be narrowband, but are quite compact in size. An additional benefit of using the metasurface is the fact that these structures show substantial angle independence (as discussed above). These structures have been demonstrated both numerically and experimentally [46, 48, 50]. Various groups have also investigated the use of metasurfaces to develop impedancematching surfaces, a wide-angle impedance-matching surface having been fabricated [52]. This structure has highly anisotropic properties, which as we saw above can enable angularly independent behavior. There is a great deal of attention of using metasurfaces for tailoring wave fronts and for polarization conversion [74–81]. Other applications of metasurfaces include biomedical, microwave-assisted chemistry, and terahertz devices to name a few (see [27] for a list of references). In fact, new applications for metasurfaces are emerging almost daily as researchers understand their potential in all fields of physics and engineering.

114

Using Generalized Sheet Transition Conditions (GSTCs) in the Analysis of Metasurfaces

3.8

Impedance-Type Boundary Conditions Some researchers prefer to express the GSTCs in the form of impedance-type boundary conditions [110, 143–146]. For plane wavefields, whose variation parallel to a metafilm is of the form e´j k¨rt ,

(3.121)

k = kx ax + ky ay and rt = xax + yay ,

(3.122)

where

we can use Maxwell’s equations to write the GSTCs for the metafilm given in Eqs. (3.1) and (3.2) as ı ” Ø = ´Z MS ¨ Ht,av (3.123) ay ˆ EA ´ EB y=0 ı ” Ø ay ˆ HA ´ HB = Y ES ¨ Et,av . (3.124) y=0

Here, the spatially dispersive (k-dependent) surface transfer admittance and transfer impedance dyadics are given by Ø

Ø

Y

Ø

ES

= j ωχ ES +

j χ MS (az ˆ k)(az ˆ k) ωμ

(3.125)

Ø

j χ ES (3.126) (az ˆ k)(az ˆ k). ω Boundary conditions of this form can also be interpreted as lumped elements in equivalent transmission-line circuits [144]. The GSTCs for metascreens and the metagratings can also be cast in impedance boundary condition forms, if desired. Ø

Ø

Z MS = j ωχ MS +

3.9

Isolated Scatterers and One-Dimensional Arrays Can metastructures also be realized with one-dimensional structures or single isolated scatterers? The two-dimensional metamaterial (i.e., metasurface) concept can be extended even further to the concept of using only a linear array rather than a surface array. Furthermore, one can use only a single subwavelength resonant structure for some desired effect or behavior. In fact, we have already begun to see a few applications of this concept. One in particular is the use of a unit cell in the design of electrically small antennas. In antenna applications, the unit cell acts like a parasitic element to the radiating element of the antenna and serves as a means to match the electrically small radiating element to both (1) the feeding transmission line and (2) free space. Such designs have been shown to enable efficient electrically small antennas [111–118]. Nanoparticles have also been used for tuning so-called optical nanoantennas [119]. An additional example is the use of a one-dimensional unit cell as a tuning structure for planar transmission lines [120]. Another emerging area of application is the use of one-dimensional chains of nanoparticles as waveguides supporting surface waves, of which examples can be found in [121–127].

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115

Summary The recent development of various engineered materials (3-D metamaterials, 2-D metasurfaces, single arrays, and single particles) is bringing us closer to realizing the exciting predictions made over 100 years ago by the work of Lamb (1904), Schuster (1904), and Pocklington (1905) [128–130]. Although early investigators in the fledgling area of metamaterials attributed the first study of such media to Veselago [4] in 1967, the aforementioned authors anticipated some of his work by many decades, and L. I. Mandel’shtam (1945) [131, 132], Malyuzhinets (1951) [133], and Sivukhin (1957) [134] had all previously discussed the properties of wave propagation in backward-wave media. Some other historical (or “prehistorical”) surveys have been given in [135–138]. In this chapter, we have discussed two-dimensional metamaterials, referred to as metasurfaces. We have defined three major types of metasurfaces: metafilms, metascreens, and metagratings, each having a distinct surface topology. We introduced the GSTCs required to accurately model these three different types of metasurface. From these GSTCs, we derived the reflection and transmission coefficients for each metasurface. These coefficients are expressed in terms of the surface parameters that characterize the metasurface. The coefficients are then used to develop a retrieval approach for determining the uniquely defined effective surface parameters that characterize each of the metasurfaces from measured or simulated data. We concluded the chapter with a presentation of various other applications of the GSTCs. The application of metasurfaces at frequencies from microwave to optical and beyond has attracted great interest in recent years [27, 28, 35–84]. Metasurfaces allow for controllable “smart” surfaces, miniaturized cavity resonators, novel waveguiding structures, compact and wide-angle absorbers, impedance matching surfaces, biomedical devices, tailoring wave fronts, polarization conversion, antennas, and high-speed switching devices, to name only a few. While there is still much work to be done in the understanding, analysis, design, and fabrication of these engineered materials, the potential of these materials has forever changed the landscape of RF, microwaves, optics, and photonics for the future. It seems that nearly every day, new applications of these metasurfaces emerge as researchers begin to tap into their potential and understand how to design and utilize them.

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4

Electromagnetic Metasurface Synthesis, Analysis, and Applications Karim Achouri, Yousef Vahabzadeh, and Christophe Caloz

4.1

Introduction An electromagnetic metasurface is a two-dimensional structure that is thin with respect to the wavelength of operation and that may be used to control the scattering of electromagnetic waves. Such a structure is conventionally composed of a periodic, or quasiperiodic, arrangement of engineered subwavelength scattering particles that enables one to control the amplitude, phase, polarization, and direction of propagation of the fields reflected and transmitted by the metasurface, when the latter is illuminated by a specific incident field [1–4]. While the idea of controlling light with thin surfaces has been around for a very long time, the mathematical and physical understanding as well as the technical capabilities required to realize such complex structures, especially those that perform advanced control of the fields, have only been available since the last decade. However, there has been a crucial lack of rigorous and universal synthesis techniques that would apply to any field specification. To overcome this issue, we have developed a general synthesis framework for the mathematical modelization, numerical analysis, and practical implementation of metasurfaces, irrespectively of the prescribed electromagnetic transformations. The chapter presents an overview of this metasurface synthesis framework, which is based on rigorous zero-thickness sheet transition conditions. The synthesis procedure is an inverse problem that yields the metasurfaces’ susceptibilities in terms of the fields corresponding to the specified electromagnetic transformations. We are considering the very general case of fully bianisotropic metasurfaces with both tangential and normal susceptibility components. We will also present the case of second-order nonlinear metasurfaces. Based on the same rigorous boundary conditions, a general numerical analysis framework is also developed. It includes several different simulation schemes for frequency and time-domain simulations for nonlinear, time-varying, and dispersive metasurfaces. The mathematical synthesis and the numerical analysis of metasurfaces is theoretically and numerically illustrated with several examples, which include the case of metasurfaces performing negative refraction, nongyrotropic nonreciprocity, and timevarying absorption. Following the mathematical developments of the synthesis, the practical realization of metasurfaces is then addressed. This chapter discusses two different approaches for the implementation of the metasurface scattering particles, based on cascaded metallic

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125

layers and on dielectric resonators, respectively. These two technologies exhibit specific advantages and disadvantages in terms of loss, fabrication complexity, degrees of freedom, and sizes.

4.2

Mathematical Synthesis In this chapter, we present an extensive discussion on the mathematical synthesis of metasurfaces [5–8]. We define a metasurface as a two-dimensional electromagnetic discontinuity of subwavelength thickness, t ! λ. The metasurface synthesis problem is illustrated in Figure 4.1, where the metasurface lies in the xy-plane at z = 0 and is of finite size with dimensions Lx ˆ Ly . Figure 4.1 shows a typical (nonuniform) metasurface made of a periodic arrangement of distinct scattering particles (here made of planar metallic crosses) that transforms an incident wave into specified reflected and transmitted waves. The objective of the synthesis procedure is to obtain the metasurface material parameters leading to a specified monochromatic transformation prescribed in terms of an arbitrary incident wave, ψ i (r), an arbitrary reflected wave, ψ r (r), and an arbitrary transmitted wave, ψ t (r). The synthesis procedure provides a solution that is expressed in terms of the bianisotropic surface susceptibility tensors, χ ee (ρ), χ mm (ρ), χ em (ρ) and χ me (ρ), which correspond to the electric, magnetic, electromagnetic, and magnetoelecˆ tric, susceptibility tensors, respectively, and where ρ = x xˆ + y y. Although the synthesis procedure always yields a mathematical solution, the resulting susceptibilities may not necessarily correspond to physically realizable scattering

Figure 4.1 Metasurface synthesis problem. The metasurface to be synthesized lies in the xy-plane

at z = 0. The synthesis procedure consists in finding the susceptibility tensors, χ (ρ), for specified arbitrary incident, ψ i (r), reflected, ψ r (r), and transmitted, ψ t (r), waves.

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particles. The difficulty in realizing the metasurface may stem from spatial variations of the susceptibilities larger than the size of the scattering particles or the presence of electric/magnetic gain/loss or nonreciprocal features that may be beyond the reach of the specific intended technology. Nevertheless, in many cases, it is possible to sacrifice some of the specified design constraints so as to relax the requirements on the susceptibilities and thus synthesize a structure that is physically implementable. In this chapter, we will present and discuss the mathematical synthesis of metasurfaces, which is only the first step in the broader framework of metasurface synthesis. Upon this mathematical basis, we will address, in the forthcoming sections, the second main step of the synthesis, which consists in finding the appropriate geometries of the scattering particles. Note that the time harmonic dependence ej ωt is implicitly assumed throughout the chapter.

4.2.1

Metasurface Boundary Conditions Let us consider the metasurface depicted in Figure 4.1 and ask the question, how can one obtain its susceptibilities in terms of the specified fields? The operation corresponding to this question corresponds to solving an inverse problem, which consists in finding the material parameters as functions of space so as to produce specific scattered fields under a specified illumination. As such, this inverse problem has an infinite number of solutions. Additionally, the metasurface in Figure 4.1 has a certain thickness and, even if it is deeply subwavelength, it generally still consists of a thin slab of material with two distinct interfaces. This makes the problem even more complicated since continuity at both interfaces must be ensured. In order to simplify this complicated problem, we make the assumption that the metasurface is an electromagnetic discontinuity of exactly zero thickness and thus consists only of a single interface. This assumption of zero thickness is supported by the fact that the metasurface is electromagnetically thin, but comes at the cost of a discrepancy between the scattering response of the zero-thickness model and that of the thin metasurface slab, to which we will come back later. At this point, the synthesis problem reduces to finding the susceptibilities of a polarizable zero-thickness sheet. Now that we have decided to model the metasurface as a zero-thickness electromagnetic discontinuity, we can again ask the question, how can one obtain the susceptibilities, in terms of the specified fields, of an interface inducing both electric and magnetic field discontinuities? It turns out that even the conventional boundary conditions found in most electromagnetic textbooks, which may allow relating the fields to the presence of the metasurface, do not rigourously apply to this kind of polarizable interface, as pointed out by Schelkunoff [9]. To illustrate this point, let us look, without loss of generality, at the specific example of the discontinuity of the displacement vector D at an interface between two media. Gauss law stipulates that ∇ ¨ D = ρ,

(4.1)

4.2 Mathematical Synthesis

127

where ρ is the charge density per unit volume. Applying Gauss theorem to (4.1), by considering a volume V enclosed by a surface S around the interface, leads to ¡ £ ¡ ∇ ¨ D dV = D ¨ nˆ dS = ρ dV = ρs , (4.2) V

S

V

where nˆ is the unit vector normal to S and ρs is the surface charge density. After simplifying the surface contour integral of D, Eq. (4.2) reduces to +

zˆ ¨ D|0z=0´ = ρs ,

(4.3)

which is the conventional textbook boundary condition for the discontinuity of the displacement vector D in the presence of an impressed surface charge density. However, Eq. (4.3) is not rigorous for the following reasons: the first one lies in the fact that the application of Gauss theorem in (4.2) is valid only if D is continuous inside the volume V , which is obviously not the case when ρs ‰ 0. Note that (4.3) is correct away from the interface, i.e., up to z = 0˘ , which is why it can be safely applied in most practical situations, but fails to describe the behavior of the field at z = 0. The second reason is the incompleteness of (4.3), which implies that D is perfectly continuous in the absence of impressed surface charges but completely fails to consider the contribution of excitable dipole or higher-order multipole moments. Therefore, the conventional boundary conditions are clearly not capable of correctly accounting for the effects of the metasurface. Fortunately, rigorous boundary conditions, which apply to zero-thickness sheets, first developed by Idemen [10] and later applied to metasurfaces by Kuester et al. [11], are available. These boundary conditions are conventionally referred to as the Generalized Sheet Transition Conditions (GSTCs). In the absence of impressed sources, the GSTCs read zˆ ˆ H = j ωP  ´ zˆ ˆ ∇ Mz,   Pz E ˆ zˆ = j ωμ0 M  ´ ∇ ˆ zˆ, 0

(4.4a) (4.4b)

zˆ ¨ D = ´∇ ¨ P ,

(4.4c)

zˆ ¨ B = ´μ0 ∇ ¨ M ,

(4.4d)

where the terms on the left-hand sides of the equations correspond to the differences of the fields on both sides of the metasurface, which may be expressed as ˇ0+ ˇ u = uˆ ¨  ˇ

z=0´

= u,t ´ (u,i + u,r ), u = x,y,z,

(4.5)

where  represents any of the fields H , E, D, or B and where the subscripts i, r, and t denote incident, reflected, and transmitted fields and P and M are the electric and magnetic surface polarization densities, respectively. In the general case of a bianisotropic metasurface, these polarization densities are related to the acting (or local) fields, E act and H act , by [12, 13]

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Electromagnetic Metasurface Synthesis, Analysis, and Applications

1 Nα em ¨ H act, c0 1 M = Nα mm ¨ H act + Nα mm ¨ E act, η0 P = 0 Nα ee ¨ E act +

(4.6a) (4.6b)

where the α ab terms represent the polarizabilities of a given scatterer, N is the number of scatterers per unit area, c0 is the speed of light in vacuum, and η0 is the vacuum impedance. This is a microscopic description of the metasurface response which requires an appropriate definition of the averaging operation as well as the coupling between adjacent scattering particles. In this work, we use the concept of susceptibilities rather than the polarizabilities to provide a macroscopic description of the metasurface, which allows a direct connection of the total fields with material parameters such as  r and μr . Bringing about the susceptibilities, relations (4.6) can be transformed by noting that the acting fields, at the position of a scattering particle, can be defined as the average fields from which the scattered field of the considered scattering particles has been removed [11], i.e., E act = E av ´ E scat . The contributions of the scattering particle may be expressed by considering the particle as a combination of electric and magnetic dipoles contained within a small disk. Then, the scattered fields from this disk can be related to P and M by taking into account the coupling with adjacent scattering particles. Therefore, the acting fields are functions of the average fields and the polarization densities. Upon substitution of this definition of the acting fields in (4.6), the resulting expressions of the polarization densities become 1 χ ¨ H av, c0 em 1 M = χ mm ¨ H av + χ me ¨ E av, η0 P = 0 χ ee ¨ E av +

(4.7a) (4.7b)

where the average fields are defined as u,t + (u,i + u,r ) , u = x,y,z, (4.8) 2 where  corresponds to E or H . The constitutive relations with the convention used in (4.7) are consequently given by u,av = uˆ ¨  av =

1 χ ¨ H av, c0 em 1 B = μ0 (H + M) = μ0 (I + χ mm ) ¨ H av + χ me ¨ E av, c0

D = 0 E + P = 0 (I + χ ee ) ¨ E av +

(4.9a) (4.9b)

where I is the identity matrix. In the general case of a volumetric medium, the susceptibilities in (4.9) are dimensionless quantities. However, the surface susceptibilities that apply to zero-thickness metasurfaces are measured in meters.

4.2.2

Synthesis Procedure The metasurface synthesis procedure consists in solving the GSTCs relations in (4.4) to obtain the susceptibilities in (4.7), which are the unknown of this inverse problem,

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129

so that the metasurface performs a desired electromagnetic transformation specified in terms of incident, reflected, and transmitted fields. According to the uniqueness theorem, Eqs. (4.4c) and (4.4d) are redundant relations in the absence of impressed sources since the transverse components of the fields are sufficient to completely describe the electromagnetic problem. Therefore, we usually only consider relations (4.4a) and (4.4b) for the synthesis of metasurfaces. As they are, the GSTCs form a set of coupled inhomogenous partial differential equations whose differential nature resides in the spatial derivatives of the normal components of the polarization densities in relations (4.4a) and (4.4b). As a consequence, solving the inverse problem, in the most general case where all susceptibility components are considered, is nontrivial and may require involved numerical analyses. It is therefore convenient to assume that the metasurface does not possess normal susceptibility components such that Pz = Mz = 0 irrespectively of the illumination. In this chapter, we will therefore restrict our attention to cases where Pz = Mz = 0, which leads to closed-form solutions of the synthesized susceptibilities. The more general case of nonzero normal susceptibilities is notably discussed in [5]. Enforcing Pz = Mz = 0 may a priori seem an important restriction but, as we will see, still corresponds to a very broad class of metasurfaces. This restriction mostly affects the realization of the scattering particles that may be polarizable in the normal direction, which may ultimately alter the scattering response of the structure. However, it should be noted that in the particular case where all the specified waves are propagating normally with respect to the metasurface, the normal polarization densities do not contribute to the scattering. This is because, for such specified normally propagating waves, the metasurface is perfectly spatially uniform, and thus the spatial derivatives in (4.4) vanish. In the case of oblique wave propagation, the assumption that Pz = Mz = 0 breaks down as the angle of propagation increases due to the excitation of normal susceptibility components. However, in the small angle limit, this approximation is valid, as the tangential susceptibility components dominate over the normal ones. Let us now simplify the GSTCs so as to obtain the final form of the synthesis relations. Substituting (4.7) into (4.4a) and (4.4b), and dropping the spatial derivatives, leads to zˆ ˆ H = j ω0 χ ee ¨ E av + j k0 χ em ¨ H av,

(4.10a)

E ˆ zˆ = j ωμ0 χ mm ¨ H av + j k0 χ me ¨ E av,

(4.10b)

where k0 is the free-space wavenumber and where the susceptibility tensors only contain the tangential susceptibility components. This system can also be written in matrix form to simplify the synthesis procedure. The matrix equivalent of (4.10) is given by ⎞ ⎛ xx χree Hy yx ⎜Hx ⎟ ⎜χree ⎟=⎜ ⎜ ⎝Ey ⎠ ⎝χrxx ⎛

Ex

me yx χrme

xy

χree yy χree xy χrme yy χrme

xx χrem yx χrem xx χrmm yx χrmm

⎞ xy ⎞ ⎛ Ex,av χrem yy ⎜ ⎟ χrem ⎟ ⎟ ⎜Ey,av ⎟ , xy ⎠ ¨ ⎝ Hx,av ⎠ χrmm yy χrmm

Hy,av

(4.11)

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Electromagnetic Metasurface Synthesis, Analysis, and Applications

where the tilde symbol indicates that the susceptibilities in (4.10) have been normalized. The relationship between the susceptibilities in (4.10) and those in (4.11) is ⎛

xx χee ⎜χ yx ⎜ ee ⎜ xx ⎝χme

χee yy χee xy χme

xy

xx χem yx χem xx χmm

yx

χme

yy

χmm

χme

yx

⎛ j xx xy ⎞ ree χem ω χ ⎜ j0 yx yy ⎟ ´ χr χem ⎟ ⎜ ⎜ ω0 ee xy ⎟ = ⎜ xx χmm ⎠ ⎝ ´ kj χrme yy

χmm

0

j yx rme k0 χ

xy j ree ω0 χ yy ´ ωj 0 χree xy ´ kj0 χrme j yy rme k0 χ

j xx rem k0 χ j yx ´ k0 χrem j ´ ωμ χrxx 0 mm yx j rmm ωμ0 χ

j xy ⎞ rem k0 χ j yy ⎟ ´ k0 χrem ⎟ ⎟ xy ⎟ . j r ´ ωμ χ mm ⎠ 0 yy j rmm ωμ0 χ

(4.12)

As it stands, the system (4.11) contains 16 unknown susceptibilities for only 4 equations, which means that it is heavily underdetermined and thus cannot be solved directly. This calls for two fundamental considerations. The first one is the fact that, to solve the system (4.11), the number of independent unknowns should match the number of equations. Accordingly, the number of independent unknowns must be reduced to four so as to have a full-rank system. Since many different sets of four susceptibility components may be considered as valid candidates to solve the system, we can assert that different combinations of susceptibilities produce the exact same scattered fields. The second consideration is the fact that one can increase the number of transformations instead of reducing the number of unknowns to four. This means that the metasurface has the capability to simultaneously transform several sets of incident, reflected, and transmitted waves, providing that they are independent from each other. Consequently, three main methods may be considered to solve the inverse synthesis problem: reducing the number of independent unknowns, increasing the number of transformations, or a combination of these two methods. In order to reduce the number of unknowns, one may, for instance, enforce conditions on the susceptibilities so that they depend on each other. These conditions may include the reciprocity conditions in (4.67) or the passivity and losslessness conditions in (4.70). If these conditions are imposed on the susceptibilities, then the number of independent variables is reduced since several susceptibility components are related to each other. For instance, the reciprocity conditions reduce the number of independent tangential susceptibilities from 16 to 10. However, this method may not be the most appropriate since the conditions that are imposed on the susceptibilities may not be compatible with the specified transformation. In general, the most appropriate approach to solve the system (4.11) is to match the number of unknown susceptibilities to the number of specified transformations. In many cases, only one transformation is required, and thus only four susceptibilities are used to synthesize the metasurface. With 4 susceptibilities to select out of 16, the number of different possible combinations is very large. However, most of these combinations of susceptibilities lead to nonphysical or unpractical designs. It is then obvious that the choice of susceptibilities depends on the requirements of the specified problem. Note that these considerations are naturally extendable to the cases where more than one transformation is desired. In the forthcoming discussions, we will only present a limited number of combinations of susceptibilities for the sake of conciseness but without loss of generality. First, we will start by considering the synthesis of a birefringent

4.2 Mathematical Synthesis

131

metasurface, which is one of the most commonly used types of structure. Then, we will present an illustrative example of multiple transformations. Note that a single transformation between specified incident, reflected, and transmitted waves requires four susceptibilities only in the general case where these waves exhibit both x and y polarization states. However, if only one of these two polarization states is considered, then the system (4.11) reduces to two equations. In this scenario, only two susceptibilities instead of four are required to synthesize the metasurface.

Synthesis of Birefringent Metasurfaces We now consider the most simple and conventional case of metasurface synthesis. It consists in synthesizing a monoanisotropic (χ em ” χ me = 0) metasurface possessing only diagonal nonzero susceptibility components, which corresponds to a birefringent structure [14]. For such a metasurface, the system (4.11) reduces to ⎞ ⎛ xx ⎞ ⎛ ⎞ ⎛ 0 0 0 χree Ex,av Hy yy ⎜ ⎟ ⎜Hx ⎟ ⎜ 0 0 0 ⎟ χree ⎟ ⎜ ⎟ ¨ ⎜Ey,av ⎟ , ⎜ (4.13) xx ⎠ ⎝ ⎝Ey ⎠ = ⎝ 0 0 Hx,av ⎠ 0 χrmm yy Ex Hy,av 0 0 0 χrmm which may be straightforwardly solved and yields, using (4.12), the following simple relations for the four susceptibilities: ´Hy , j ω0 Ex,av Hx = , j ω0 Ey,av Ey = , j ωμ0 Hx,av ´Ex = , j ωμ0 Hy,av

xx = χee

(4.14a)

yy

(4.14b)

χee

xx χmm yy

χmm

(4.14c) (4.14d)

where, according to (4.5) and (4.8), Hy = Hy,t ´ (Hy,i + Hy,r ), Ex,av = (Ex,t + Ex,i + Ex,r ){2, and so on. By synthesis, a metasurface with the susceptibilities in (4.14) will exactly produce the specified reflected and transmitted transverse components of the fields when the metasurface is illuminated by the specified incident field. Since the longitudinal fields are completely determined from the transverse components, according to the uniqueness theorem, the complete specified electromagnetic fields are exactly generated by the metasurface. Due to the orthogonality between x- and y-polarized waves, the susceptibilities in (4.14) can be separated into two subsets corresponding to Eqs. (4.14a) and (4.14d) and Eqs. (4.14b) and (4.14c), respectively. These two sets of susceptibilities are able to independently and simultaneously transform x- and y-polarized waves. Consequently, each subset allows one to perform the simplest example of single transformation. If the two electric and the two magnetic susceptibilities in (4.14) are equal to each other xx = χ yy and χ xx = χ yy ), then the metasurface is monoisotropic and accordingly (χee ee mm mm performs the same operation for both x- and y-polarized waves. If this is not the case, then the metasurface is monoanisotropic and can perform the simplest case of double

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transformation (birefringence). Note that we will see in the next section a more general case of multiple transformation that is not based on the orthogonal separation of x- and y-polarized waves as it is the case here. The system in (4.13) represents the most conventional way of synthesizing metasurfaces performing single (or double by birefringence) transformation, but it is obviously not the only one possible. One may for instance imagine a monoanisotropic metasurface with nonzero off-diagonal components, which would be solved in the exact same fashion while performing a different kind of electromagnetic transformation since it would correspond to a gyrotropic structure.

Multiple Transformations We have just seen how a metasurface can be synthesized to perform a single transformation, if the structure is monoisotropic, or a very particular case of double transformations, if the structure is monoanisotropic. However, as previously mentioned, the general system of Eqs. (4.11) has the capability to perform multiple transformations given its large number of degrees of freedom, i.e., its 16 susceptibility components. Here, we will see how the system (4.11) can be solved for several transformations including incident waves coming from one side only or both sides of the metasurface. To accommodate for the additional degrees of freedom, three additional wave transformations are added, so that (4.11) transforms to ⎞ ⎛ Hy1 Hy2 Hy3 Hy4 ⎜Hx1 Hx2 Hx3 Hx4 ⎟ ⎟ ⎜ ⎝Ey1 Ey2 Ey3 Ey4 ⎠ Ex1 Ex2 Ex3 Ex4 ⎛ xx xy xy ⎞ ⎛ xx χree χree χrem Ex1,av χrem yx yy yx yy ⎟ ⎜ ⎜χree r r r E χ χ χ ee em em ⎟ ⎜ y1,av =⎜ xy xy ⎠ ¨ ⎝ xx ⎝χrxx χrme r r Hx1,av χ χ mm me mm yx yy yx yy Hy1,av χrme χrme χrmm χrmm

Ex2,av Ey2,av Hx2,av Hy2,av

Ex3,av Ey3,av Hx3,av Hy3,av

⎞ Ex4,av Ey4,av ⎟ ⎟, Hx4,av ⎠

(4.15)

Hy4,av

where the subscripts 1, 2, 3, and 4 indicate the electromagnetic fields corresponding to four distinct and independent sets of waves. As previously done, the susceptibilities can be obtained by matrix inversion conjointly with (4.12). In order to illustrate the concept, we now consider a specific example of double transformations, which consists of synthesizing a monoanisotropic metasurface with full electric and magnetic tangential susceptibility tensors. The corresponding system is, from (4.15), given by ⎛ ⎞ ⎛ xx ⎞ ⎛ ⎞ xy Hy1 Hy2 0 0 χree χree Ex1,av Ex2,av yx yy ⎜Hx1 Hx2 ⎟ ⎜χree χree ⎜ ⎟ 0 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜Ey1,av Ey2,av ⎟ . (4.16) xy ⎠ ¨ ⎝ xx ⎝Ey1 Ey2 ⎠ = ⎝ 0 Hx1,av Hx2,av ⎠ 0 χrmm χrmm yx yy Ex1 Ex2 0 0 χrmm χrmm Hy1,av Hy2,av We assume that the two transformations that we are considering possess fields with both x and y polarizations. The solution of (4.16) is readily found by matrix inversion and, after using (4.12), the resulting susceptibilities are

4.3 Numerical Analysis

133

j (Ey1,av Hy2 ´ Ey2,av Hy1 ) , 0 ω (Ex2,av Ey1,av ´ Ex1,av Ey2,av )

(4.17a)

xy

j (Ex2,av Hy1 ´ Ex1,av Hy2 ) , 0 ω (Ex2,av Ey1,av ´ Ex1,av Ey2,av )

(4.17b)

yx

j (Ey2,av Hx1 ´ Ey1,av Hx2 ) , 0 ω (Ex2,av Ey1,av ´ Ex1,av Ey2,av )

(4.17c)

yy

j (Ex1,av Hx2 ´ Ex2,av Hx1 ) , 0 ω (Ex2,av Ey1,av ´ Ex1,av Ey2,av )

(4.17d)

j (Hy2,av Ey1 ´ Hy1,av Ey2 ) , μ0 ω (Hx2,av Hy1,av ´ Hx1,av Hy2,av )

(4.17e)

xy

j (Hx1,av Ey2 ´ Hx2,av Ey1 ) , μ0 ω (Hx2,av Hy1,av ´ Hx1,av Hy2,av )

(4.17f)

yx

j (Hy1,av Ex2 ´ Hy2,av Ex1 ) , μ0 ω (Hx2,av Hy1,av ´ Hx1,av Hy2,av )

(4.17g)

yy

j (Hx2,av Ex1 ´ Hx1,av Ex2 ) , μ0 ω (Hx2,av Hy1,av ´ Hx1,av Hy2,av )

(4.17h)

xx χee =

χee = χee = χee = xx = χmm

χmm = χmm = χmm =

where the subscripts 1 and 2 stand for the first and the second wave set transformation, respectively. Applying the conditions (4.67) and (4.70) to (4.17) indicates that in most cases, the metasurface will not be only active/lossy but also nonreciprocal. The same argument applies to the more general case of the fully bianisotropic metasurface described by the susceptibilities in (4.15), which may, depending on the choice of transformations, be nonreciprocal and active/lossy. Note that the choice of using the susceptibility tensors χ ee and χ mm in (4.16) was arbitrary and other sets of susceptibilities, for instance, including bianisotropic components, may be better suited to performing the desired transformations.

4.3

Numerical Analysis

4.3.1

Metasurface Analysis In the previous section, we have seen how an electromagnetic metasurface can be mathematically synthesized, in a rigorous fashion, using GSTCs. We will now address the numerical analysis of metasurfaces. The analysis allows one to verify the functionality of the designed metasurface before attempting to find the geometry of the particle composing it. It also allows one to study the behaviour of the designed metasurface for out-of-specification functionalities, such as different polarizations, frequencies, and directions of propagation.

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In what follows, we consider, for simplicity, as mentioned in Section 4.2, that the metasurface does not support any normal electric and magnetic polarization currents, so that the corresponding polarization densities are zero, i.e., Pz = 0 and Mz = 0, (4.1). The corresponding general system of equations is thus the one given in (4.10). In these equations, the metasurface is considered as a zero-thickness sheet discontinuity. Such types of general sheets, where the electric and magnetic fields can both be discontinuous, cannot be analyzed using conventional computational techniques. This is because in these techniques, the metasurface is placed at the nodes of the simulation grid, which necessarily implies that the metasurface thickness is equal to the size of the mesh discretization, which is incompatible with the assumption of zero-thickness required to apply the GSTCs. It follows that metasurfaces simulated with this approach suffer from spurious numerical effects, such as erroneous scattering and loss. We will now study the discrepancies between the ideal zero-thickness metasurface model and the simulation results assuming thin metasurfaces. We consider the case of a reflectionless metasurface that absorbs (with transmission coefficient T ) the power of a normally incident plane wave [15]. From the synthesis technique presented in Section 4.2, the susceptibilities for y-polarized normally propagating plane waves are straightforwardly found, from (4.14), to be yy

xx = χ = χee = χmm

2j (T ´ 1) . k0 (T + 1)

(4.18)

Next, 10 simulations are performed where the transmission coefficient, T , is varied from 1 to 0, corresponding to full transmission and full absorption, respectively. The exact simulation results, for T = 0, are plotted in Figures 4.2a and 4.2b. These simulations were performed using a GSTCs-based finite-difference frequency-domain (FDFDGSTCs) scheme, which will be explained in the next section. As a comparison, we now simulate the same metasurface in COMSOL assuming a thickness of d = λ0 {100. This yy yy thin metasurface slab is implemented in COMSOL by considering that r « 1+χee {d xx and μxx r « 1 + χmm {d, as discussed in [10, 15]. The COMSOL simulated result, for T = 0, is shown in Figure 4.2c, where we can clearly see that part of the incident power still transmits through the metasurface despite the specification of full absorption (T = 0). The discrepancy, due to the thin slab approximation, is further detailed in the plot of Figure 4.2d. The inaccuracy of the COMSOL simulations can be explained by considering the propagation of the wave through the metasurface slab. Since the metasurface is matched, no reflection occurs. The amplitude of the transmission wave, at the output side of the slab is thus simply given, using (4.18), by „ j (T ´ 1) ´j k0 n2 d ´j k0 d ´j k0 χ | = |e ||e | = exp 2 , (4.19) |Ttrs | = |e (T + 1) b yy where n2 is the refractive index of the metasurface slab and is given by n2 = r μxx r . With T = 0, |Ttrs | = e´2 « 13%, which exactly corresponds to the simulated transmission coefficient in Figure 4.2d.

4.3 Numerical Analysis

135

a |Hx | simulated by GSTCs-FDFD.

b Im(Hx ) simulated by GSTCs-FDFD.

c |Hx | simulated by COMSOL.

d Result discrepancy in COMSOL for different specified transmittance.

Figure 4.2 Comparison of exact FDFD-GSTCs and COMSOL results for a fully absorptive (R = T = 0) metasurface, indicated by dashed white line and illuminated from the top. (a) Expected absolute value of Hx computed using a GSTCs-FDFD scheme. (b) Imaginary part. (c) COMSOL result, with about 13% of spurious field transmission through the metasurface. (d) Parametric study of COMSOL inaccuracy versus transmission coefficient, T .

The results presented in Figure 4.2 illustrate, in the simple case of an absorber, the issue of simulating metasurfaces assuming a nonzero thickness. More complicated phenomena appear when the metasurface is nonuniform, which is the case of refractive metasurfaces for instance. In that case, in addition to the discrepancies of the amplitude of the transmission coefficient, the simulations lead to undesired diffraction orders, which would not be present with the exact zero-thickness model. The results of this section should not come up as a surprise if one considers that a slab, which is a two-interface structure, is an entity that is fundamentally different from a metasurface, which is a single interface structure. For instance, the former may support Fabry–Perot resonances, whereas the latter cannot. So, no matter how deeply subwavelength the slab is, it will never exactly model the behavior of a real metasurface. Agreements as observed in the left part of Figure 4.2d are a priori more of a lucky coincidence than a real validation of the synthesized metasurface. Thus, thin slabs with diluted volume parameters can generally not be used to analyze metasurfaces.

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Electromagnetic Metasurface Synthesis, Analysis, and Applications

In order to properly simulate a zero-thickness sheet discontinuity, the simulation scheme must be directly derived from the GSTCs. Accordingly, many simulation techniques have been recently developed for the simulation of metasurfaces [16–23]. In what follows, we will discuss GSTCs-FDFD and GSTCs-FDTD (finite-difference timedomain) techniques [16, 22, 23], as they are general, are fully numerical, provide better physical insight, and are easy to implement.

4.3.2

Two-Dimensional Finite-Difference Frequency-Domain Method The 2D standard FDFD equations [24], in the absence of any metasurface and for the case of TEx (Ex = Hz = Hy = 0) polarization, read j +1,k

Ez

j,k+1

j,k

´ Ez y

´

j,k

´ Ey z

Ey

j,k

j,k

= j ωμ0 μxx Hx

or

y

D e E z ´ D ze E y = μxx H x , (4.20a)

´

j,k Hx

j ´1,k ´ Hx

y j,k

Hx

j,k

j,k

or

´ D h H x =  zz E z,

(4.20b)

j,k

j,k

or

D zh H x =  yy E y ,

(4.20c)

= j ω0 zz Ez

y

j,k ´1

´ Hx z

= j ω0 yy Ey

where D ul with l = e, h and u = x,y is the derivative operator in the u-direction for the electric (l = e) or magnetic (l = h) field, E and H are electric and magnetic field column vectors, and μ and  are relative permittivity and permeability matrices, respectively. Details on the forms of these vectors and matrices are given in [24]. In what follows, we consider the numerical analysis of two-dimensional bianisotropic metasurfaces. For the simplicity of the analysis and without loss of generality, we only develop the equations for the TEx case, the TMx case being analogous. For the TEx case, relations (4.10) reduce to yy

yx

Hx = j ω0 χee Ey,av + j k0 χem Hx,av, Ey =

xx j ωμ0 χmm Hx,av

xy + j k0 χme Ey,av .

(4.21a) (4.21b)

To be properly taken into account, the metasurface should be placed in an appropriate position within the FDFD staggered Yee grid. Placing the metasurface on the E-field nodes or H-field nodes is always inappropriate. If the metasurface is considered to have a width corresponding to the grid cell size, then it will suffer from the spurious thin slab issues described in Section 4.3.1. If it is considered to have zero thickness, at it should, it presents electric current magnetic field discontinuity when placed at an H-field node or magnetic current electric field discontinuity when placed at an E-field node. Based on the above considerations, a solution would be to place the metasurface between the grid nodes as shown in Figure 4.3, namely, between the electric (right facing arrows) and the magnetic (circles) nodes [25], which may be considered as virtual mesh points as they are not directly on the Yee grid structure. In this scheme, the conventional standard FDFD equations in (4.20) can be used for the nodes that are away from the metasurface discontinuity. However, for the grid nodes that are surrounding

4.3 Numerical Analysis

137

Figure 4.3 Metasurface virtual node position in the 2D staggered Yee grid, between the Hx and Ey field nodes at k = d and j = nb : nl in the y-direction. The z axis is normal to the metasurface. The numbers in parentheses refer to the cell numbers. For instance, Ey (n,d) represents the dth and nth cell in the z and y direction, respectively.

the discontinuity, these conventional FDFD equations cannot be used since there is a discontinuity between the nodes which is not taken into account in these equations. We therefore need to modify the conventional FDFD equations so as to take into account the presence of the metasurface. In reference to Figure 4.3, one sees that the update equation of the magnetic fields right before the metasurface (i.e. Hx (nb : nl ,d)), according to (4.20a), involves Ey (nb : nl ,d) and Ey (nb : nl ,d + 1). The field Ey (nb : nl ,d + 1) is located at the other side of the discontinuity with respect to Hx (nb : nl ,d), and if we simply use Ey (nb : nl ,d + 1) in the update equation, the discontinuity is missed. Thus, to properly take into account the effect of the discontinuity, the solution proposed in [25] is to use a discretized version of GSTCs equation (4.21b) instead of (4.20a). After discretizing (4.21b) and grouping similar components, one obtains the following relation: Ey (nb : nl ,d+1)αe+ ´Ey (nb : nl ,d)αe´ = xy

xx j ωμ0 χmm (Hx (nb : nl ,d)+Hx (nb : nl ,d+1)), 2 (4.22)

where αe˘ = 1 ˘ j k02χme . y Implementing (4.22) requires adjusting the operators D e and D ze as well as the matey rial matrix μ to D e (i, :) = 0, D ze (i,i) = ´αe´ , D ze (i,i + 1) = αe+ , and μxx (i,i) = xx j ωμ0 χmm , where i = d + (j ´ 1)Nz with j = nb : nl and Nz is number μxx (i,i + 1) = 2 of cells in the z-direction. From (4.20c), the update equations for Ey (nb : nl ,d + 1) involve Hx (nb : nl ,d) and Hx (nb : nl ,d + 1), with the former being on the other side of the discontinuity. Thus, using the conventional standard FDFD equation, the discontinuity is ignored. To avoid this issue, we proceed similarly as for the previous case; the proposed solution in [25] is to employ (4.21a) instead of (4.20c), which yields Hx (nb : nl ,d+1)αh+ ´Hx (nb : nl ,d)αh´ =

yy

j ω0 χee rEy (nb : nl ,d) + Ey (nb : nl ,d+1)s, 2 (4.23)

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Electromagnetic Metasurface Synthesis, Analysis, and Applications

yx

where αh˘ = 1 ˘ j k02χem . The implementation of (4.23) requires changing the operators D zh and the material matrix yy . The corresponding changes are D zh (i,i) = ´αh+ , yy

D zh (i,i ´ 1) = αh´ and  yy (i,i) =  yy (i,i ´ 1) = j ω20 χee where i = d + 1 + (j ´ 1)Nz with j = nb : nl and Nz is number of the cells in the z-direction.

4.3.3

Two-Dimensional Finite-Difference Time-Domain Method We will now discuss the simulation of metasurfaces in the time domain, which will allow us to simulate time-varying and nonlinear metasurfaces. In this scenario, the metasurface susceptibility components are explicitly given as functions of time and space. This analysis technique is applicable to specific families of dispersive and nonlinear metasurfaces, which will be addressed thereafter. For the simplicity of the analysis, we consider the case of the 2D TEx problem, as before. Moreover, the metasurface is considered to be uniaxial (diagonal susceptibility tensors only) and monoanisotropic (χ em = χ me = 0). The extension of the simulation scheme to the most general case of 3D bianisotropic metasurfaces is tedious but straightforward and is thus omitted here. From (4.10), the time-domain GSTCs equations can d , which yields be obtained by replacing j ω with the time derivative operator dt ‰ “ yy d χee Ey,av Hx = 0 , (4.24a) dt xx d rχmm Hx,av s . (4.24b) Ey = μ0 dt Note that these relations assume nondispersive susceptibilities, in which case the convolution products of the time-domain formulation disappear. These equations involve discontinuity on the Hx and Ey components of the fields. From [26], the conventional FDTD update equation for the Ey field of a 2D TEx problem reads „ j t n´ 1 n´ 1 Eyn (j,k) = Eyn´1 (j,k) + Hx 2 (j,k) ´ Hx 2 (j,k ´ 1) , (4.25) 0 z and that for the Hx component reads n+ 21

Hx

n´ 21

(j,k) = Hx ´

ı t ” n Ey (j,k + 1) ´ Eyn (j,k) μ0 z ‰ “ n Ez (j + 1,k) ´ Ezn (j,k) .

(j,k) +

t μ0 y

(4.26)

As in the frequency-domain analysis, the metasurface cannot be positioned on the Yee grid cell nodes. A solution, proposed in [27], is to place the metasurface between the FDTD grid nodes, as shown in Figure 4.4. For all the nodes nonadjacent to the metasurface, the standard FDTD algorithm is applicable since the metasurface does not introduce any discontinuity of the fields there. However, for the nodes around the discontinuity, the standard FDTD equations are not applicable since they do not take into account the presence of the metasurface. This problem is solved by the concept of auxiliary (virtual) nodes [27]. These virtual nodes are positioned right before and after the metasurface, between Yee grid nodes, as shown in Figure 4.4, where the green and red dots, respectively, represent the electric and magnetic virtual node.

4.3 Numerical Analysis

139

Figure 4.4 Position of the metasurface and the virtual nodes in the FDTD grid nodes. The filled

green and red circles represent electric and magnetic virtual nodes just before and just after the metasurface, i.e., at z = 0´ and z = 0+ , respectively. The incident wave propagates in the zy-plane. A black and white version of this figure will appear in some formats. For the color version, please refer to the plate section.

The update equation for the magnetic field in (4.26) at the position k = kd , requires the knowledge of the electric field at kd and kd +1. However, the latter electric field node is positioned at the other side of the discontinuity. It can therefore not be simply used in the update equation since this would miss the effect of the metasurface. Therefore, instead of using Ey (j,kd + 1), we consider the virtual node Ey (nl : nh,0´ ), which transforms the update Eq. (4.26) into ı t ” n  ´  n+ 1 n´ 1 Hx 2 (j,kd ) = Hx 2 (j,kd ) + Ey j,0 ´ Eyn (j,kd ) μ0 z ‰ “ t (4.27) ´ Ezn (j + 1,kd ) ´ Ezn (j,kd ) . μ0 y Now, to find Ey (j,0´ ), we employ Eq. (4.24b), which, upon discretization and simplification, becomes j „  ´ n+ 1  xx  n´ 1 μ0  xx n n 2 2 Ey j,0 = Ey (j,kd + 1) ´ . (4.28) ´ χmm Hx,av χmm Hx,av t Substituting this expression for the electric field at the position of the electric virtual node into (4.27) and grouping similar components, we obtain ı t ” n n+ 1 x,n+ 1 n´ 1 x,n´ 1 Ey (j,kd + 1) ´ Eyn (j,kd ) Hx 2 (j,kd ) Am 2 =Hx 2 (j,kd ) Am 2 + μ0 z ‰ t “ n (4.29a) ´ Ez (j + 1,kd ) ´ Ezn (j,kd ) μ0 y xx,n+ 1

xx,n´ 1

χmm 2 n+ 12 χmm 2 n´ 21 ´ Hx Hx (j,kd + 1) + (j,kd + 1) , 2z 2z

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Electromagnetic Metasurface Synthesis, Analysis, and Applications

with Axx,n mm = 1 +

xx,n χmm . 2z

(4.29b)

Now consider updating the electric field at k = kd + 1. Since, from (4.25), the magnetic field at k = kd and k = kd + 1 are involved with the former nodes positioned before the discontinuity, instead of using Hx (j,kd ) in the update equation for Ey (j,kd + 1), we use the magnetic field Hx (j,0+ ) at the position of the virtual magnetic nodes. This transforms (4.25) into j „  t n´ 21 n´ 21  n n´1 + . j,0 Hx Ey (j,kd + 1) = Ey (j,kd + 1) + (j,kd + 1) ´ Hx 0 z (4.30) In this equation, the magnetic field Hx (j,0+ ) is unknown. However, its value can be computed from the GSTCs equation (4.24a), which becomes, upon discretization and simplification,  n  yy n´1 ı 0 ” yy n´ 1  n´ 1 . (4.31) Hx 2 j,0+ = Hx 2 (j,kd ) + χee Ey,av ´ χee Ey,av t Substituting this equation into (4.30) yields the update equation for the electric field on the nodes right after the metasurface: yy,n

Eyn (j,kd + 1) Aee

yy,n´1

= Eyn´1 (j,kd + 1) Aee (4.32a) „ j t n´ 1 n´ 1 Hx 2 (j,kd + 1) ´ Hx 2 (j,kd ) + 0 z ı 1 ” yy,n n yy,n´1 n´1 Ey (j,kd ) , χee Ey (j,kd ) ´ χee ´ 2z

with yy,n

χee . (4.32b) 2z Equation (4.32a), as (4.29a), has an interesting physical interpretation. First, consider the case where χ = 0, corresponding to the absence of metasurface. In this case, we expect to retrieve the conventional FDTD equation in free space. With the assumption yy,n that χ = 0 and resulting Aee = 1, the above equation reduces to „ j t n´ 1 n´ 1 Hx 2 (j,kd + 1) ´ Hx 2 (j,kd ) , Eyn (j,kd + 1) = Eyn´1 (j,kd + 1) + 0 z (4.33) yy,n

Aee

=1+

which is the conventional FDTD equation for the free space, as expected. yy,n yy yy,n Second, consider the case of time-invariant metasurface, χee = χee , and Aee = yy χee r = 1 + 2z . Dividing both sides of (4.32a) by r gives „ j t n´ 21 n´ 21 n n´1 Ey (j,kd + 1) = Ey (j,kd + 1) + Hx (j,kd + 1) ´ Hx (j,kd ) 0 r z (4.34) yy ” ı χee Eyn (j,kd ) ´ Eyn´1 (j,kd ) . ´ 2zr

4.3 Numerical Analysis

141

Without the last two terms, this equation corresponds to the conventional FDTD equation for the simulation of a metasurface slab of thickness 2z and volume permittivity r . The last two terms correspond therefore to a correction that collapses the slab metasurface of thickness 2z into a zero-thickness sheet.

4.3.4

Finite-Difference Time-Domain Scheme for Dispersive Metasurface The aforementioned scheme that was developed for the analysis of time-varying metasurfaces is also applicable to a specific family of dispersive metasurfaces. This family consists of those metasurfaces which, after substitution of the frequency-domain susceptibilities into (4.10) and conversion into time domain, are described by a finitedifference discretizable equation. An example of such a dispersive metasurface is that with susceptibilities of the form jκω , where κ is a constant, corresponding to a partially absorbing metasurface. The assumption that the susceptibilities take this form, with the factor κ constant, is obviously not valid at all frequencies. However, it is a good approximation in the case of narrow-band structures. In the most general case of dispersive metasurfaces, where this approximation does not hold anymore, one should rather make use of auxiliary functions in the calculations of the FDTD field update equations, as discussed in [28]. In order to analyze metasurfaces represented by susceptibilities of the form jκω , the first step is to obtain the corresponding time-domain equation relating the fields at the two sides of the metasurface. Upon substitution of this susceptibility form into (4.10) and assuming TMz polarization and isotropy, we obtain Hx = 0 κEy,av,

(4.35)

Ey = μ0 κHx,av .

(4.36)

The corresponding time-domain discretized GSTCs equations are then simply  0 κ  n n´ 1 n´1 Hx 2 = Ey,av + Ey,av , (4.37a) 2   μ0 κ n+ 1 n´ 1 Eyn = Hx,av2 + Hx,av2 . (4.37b) 2 Now, exactly the same procedure as in Section 4.3.3 can be applied for the virtual nodes surrounding the metasurface. This allows us to obtain the appropriate update equation for the fields around the metasurface, where (4.37a) and (4.37b) should be used to express the fields at the virtual nodes. The resulting equations are ı t ” n n+ 1 n´ 1 Ey (j,kd + 1) ´ Eyn (j,kd ) Hx 2 (j,kd ) F + = Hx 2 (j,kd ) F ´ + μ0 z ‰ t “ n ´ E (j + 1,kd ) ´ Ezn (j,kd ) μ0 y z j „ κt n+ 12 n´ 21 ´ Hx (j,kd + 1) + Hx (j,kd + 1) , 4z (4.38)

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Electromagnetic Metasurface Synthesis, Analysis, and Applications

with F ˘ = 1 ˘ Eyn (j,kd

4.3.5

+ 1) F

κt 4z , +

and

j „ t n´ 21 n´ 21 = + 1) F + Hx (j,kd + 1) ´ Hx (j,kd s 0 z ı κt ” n (4.39) ´ Ey (j,kd ) + Eyn´1 (j,kd ) . 4z Eyn´1 (j,kd

´

One-Dimensional Analysis of Nonlinear Second-Order Metasurfaces So far, we have been interested only in purely linear metasurfaces, i.e., metasurfaces whose polarization densities are linear functions of the electric and magnetic fields. Let us now investigate the case of a metasurface with nonzero second-order nonlinear electric and magnetic susceptibility tensors [8, 29–32]. Second-order nonlinearity is a weak effect that is present in noncentrosymmetric structures. For instance, we may find electric nonlinearities in some crystals [33], while nonnegligible magnetic nonlinearities may be found in some ferrofluids [34]. In the case considered here, the polarization densities in (4.7), for a linear bianisotropic medium, transform into (1)

(2)

P = 0 χ ee ¨ E av + 0 χ ee : E av E av, M= (1)

(1) χ mm

(2) ¨ H av + χ mm

: H av H av,

(4.40a) (4.40b)

(2)

where χ and χ correspond to the first-order (linear) and second-order (nonlinear) susceptibility tensors, respectively. In order to simplify the numerical analysis of the nonlinear metasurfaces, we assume that they are isotropic and that their susceptibility tensors in (4.40) are of rank zero. We now show how the previously developed metasurface FDTD simulation scheme may be modified to account for nonlinear susceptibility components. For simplicity, we consider a 1D problem case. As before, the FDTD scheme consists in using traditional FDTD update equations everywhere on the Yee simulation grid except at the nodes that are just before and after the metasurface. For these nodes, the update equations are modified, using the GSTCs relations, to take into account the effect of the metasurface. The conventional FDTD 1D equations, for the Yee grid, are given by  t  n n+ 1 n´ 1 (4.41a) Ey (i + 1) ´ Eyn (i) , Hx 2 (i) = Hx 2 (i) + μ0 z   t n´ 1 n´ 1 (4.41b) Eyn (i) = Eyn´1 (i) + Hx 2 (i) ´ Hx 2 (i ´ 1) , 0 z where i and n correspond to the cell number and time coordinates and z and t are their respective position and time step. In the FDTD scheme, the metasurface is inserted at a virtual node between the Yee nodes i = nd and i = nd + 1, corresponding to a position between an electric and a magnetic node, as depicted in Figure 4.5. To take into account the effect of the metasurface, an electric virtual node is placed right before the metasurface (at i = 0´ ) and a magnetic virtual node is placed right after the metasurface

4.3 Numerical Analysis

143

Figure 4.5 1D FDTD Yee grid with a metasurface placed in between the nodes. The two small

circles, before and after the metasurface, represent the electric and magnetic virtual node, respectively. Original image from [35]. A black and white version of this figure will appear in some formats. For the color version, please refer to the plate section.

n+ 1

(at i = 0+ ). From (4.41), the update equations for Hx 2 (nd ) and Eyn (nd + 1) are connected to these virtual nodes via the following relations  t  n ´ Ey (0 ) ´ Eyn (nd ) , μ0 z   t n´ 21 n´ 21 + n n´1 Ey (nd + 1) = Ey (nd + 1) + Hx (nd + 1) ´ Hx (0 ) , 0 z n+ 12

Hx

n´ 21

(nd ) = Hx

(nd ) +

(4.42a) (4.42b)

where the value of the electric and magnetic fields at these nodes are obtained from the GSTCs relations as B (2) B 2 Ey,av + 0 χee E , Bt Bt y,av (1) B (2) B Ey = μ0 χmm Hx,av + μ0 χmm H2 . Bt Bt x,av

(1) Hx = 0 χee

(4.43a) (4.43b)

Using (4.43), the expression of the electric and magnetic fields at the virtual nodes in (4.42) read (1)   χ (2)   0 χee  n 0 ee n´1 n n´1 2 )2 ´ (Ey,av ) , Ey,av ´ Ey,av (Ey,av + t t (4.44a)     (1) (2) μ0 χmm μ0 χmm n+ 21 n´ 21 n+ 21 2 n´ 21 2 n ´ n Ey (0 ) = Ey (nd + 1) ´ Hx,av ´ Hx,av ´ (Hx,av ) ´ (Hx,av ) , t t (4.44b) n´ 21

Hx

n´ 21

(0+ ) = Hx

(nd ) +

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Electromagnetic Metasurface Synthesis, Analysis, and Applications

where the average electric field is defined as n Ey,av =

Eyn (nd ) + Eyn (nd + 1)

(4.45) 2 and the average magnetic field is defined similarly. Substituting (4.44) along with (4.45) into (4.42) leads to two quadratic equations that may be independently solved to obtain the final update equations. Each of them yields two possible solutions but only one of the two corresponds to a physical response. The two solutions that produce physical results are ? n+ 1 (1) (2) + Hx 2 (nd + 1)χee )μ0 + h ´(2dz + χee n+ 12 , (4.46a) Hx (nd ) = (1) μ0 χee ? (1) (2) ´(2dz + χmm + Eyn (nd )χmm )0 + e n , (4.46b) Ey (nd + 1) = (1) 0 χmm where the discriminant h is given by ! ” (2) (1) h = μ0 4t (Eyn (nd + 1) ´ Eyn (nd ))χee + μ0 4z2 + (χee n´ 21

+ (Hx

n+ 12

(nd ) + Hx

(2) 2 (1) (nd + 1))χee ) + 4z(χee ı) n´ 1 n+ 1 (2) + (Hx 2 (nd ) + Hx 2 (nd + 1))χee )

(4.47)

and the discriminant e is given by ! ” n´ 1 n´ 1 (2) (1) + 0 4z2 + (χmm e = 0 4t (Hx 2 (nd + 1) ´ Hx 2 (nd ))χmm (2) 2 (1) + (Eyn´1 (nd ) + Eyn´1 (nd + 1))χmm ) + 4z(χmm ı) (2) + (Eyn (nd ) + Eyn´1 (nd + 1))χmm ) .

(4.48)

Because of the square roots in (4.46), the update equations may lead to nonphysical responses depending on the values of the two discriminants. This limits the range of allowable values that the susceptibilities and the amplitude of the incident field may take.

4.4

Illustrative Examples In this section, we will present different examples of metasurface simulations using frequency- and time-domain finite-difference techniques. For the frequency-domain simulations, the results are compared with COMSOL simulation results, where the metasurface is simulated as a thin (d = λ0 {100) slab.

4.4.1

Negative Refraction Metasurface The first example is a reflectionless metasurface performing negative refraction. The metasurface is assumed to be isotropic and the susceptibilities are synthesized, using (4.14), as

4.4 Illustrative Examples

a Re(Hx )

b Im(Hx )

145

c |Hx |

Figure 4.6 GSTC-FDFD simulation result of the refractionless negative refraction metasurface whose position is indicated by the dashed white line. yy

1 Hx , j ω0 Ey,av 1 Ey = . j ωμ0 Hx,av

χee = xx χmm

(4.49) (4.50)

The metasurface is specified to refract at a 45˝ refraction angle a wave incident at an angle of 30˝ . The incident wave has the Gaussian profile Hxinc = e+j k0 cos θinc z+j k0 sin θinc y e´y , 2

(4.51)

2πf c0

where k0 = with c0 being the light speed in free space and f = 1 GHz the operation frequency. The surrounding medium is assumed to be free space. The simulation results for this example are shown in Figure 4.6. In this simulation, the resolution, or number of cells per wavelength, is 40, and the metasurface is positioned at y = 0, as shown by the black dashed line. As may be seen from the simulation results, the metasurface successfully refracts the incident beam at an angle of 45˝ while the reflection is zero.

4.4.2

Nongyrotropic Nonreciprocal Metasurface The second example is a nonreciprocal metasurface that acts as a one-way screen without altering the polarization of the wave (nongyrotropic). In other words, it behaves as a full absorber when excited in +z-direction, while being perfectly transparent when excited in the ´z-direction. For this specific transformation, the metasurface is bianisotropic with the susceptibilities [36]     j 1 0 j 1 0 χ ee = ´ , χ mm = ´ , (4.52a) k0 0 1 k0 0 1 χ em =

j k0



 0 1 , ´1 0

χ me =

j k0



0 1

 ´1 . 0

(4.52b)

The corresponding simulation results are shown in Figure 4.7. As can be seen, they agree well with the specifications. In these simulations, the resolution is 80 and the metasurface is positioned at z = 6λ0 . When the metasurface is excited from the right, it

Electromagnetic Metasurface Synthesis, Analysis, and Applications

1

Re(Hx)

2

Re(Hx)

146

0

0 -1

-2 4

6

8

Im(H x)

1 0

4

6

8

10

2

4

6

8

10

2

4

6

8

10

2 0 -2

-1 2

4

6

8

10

2

|Hx|

2

|Hx|

2 10

Im(H x)

2

1

1 0

0 2

4

6

8

z/ 0

10

0

a Illumination from the right.

b Illumination from the left.

Figure 4.7 GSTCs-FDFD simulation results for the bianisotropic nonreciprocal metasurface. The metasurface is positioned at λz = 6, as indicated by an asterisk. 0

transmits all the incident wave. In contrast, when it is illuminated from the other side, all the incident wave is absorbed.

4.4.3

Time-Varying Half-Wave Absorber We now present an example of a one-dimensional half-wave absorbing metasurface, corresponding to a 2D-FDTD problem. The metasurface varies in time in such a way as being every 120 time steps switched between a fully transparent screen (χ = 0) and 2c0 a half-power absorber (χ = 3j ω ). In this simulation, since the switching is performed abruptly (time discontinuity), a high temporal resolution is required. The metasurface is illuminated by a plane wave of unit amplitude at the operation frequency of 2 GHz. The resolution and time step are set to 50 and t = 6.8718 ps, respectively, and the metasurface is positioned at z = 4.17λ0 . The corresponding simulation results are shown in Figure 4.8. The periodic behavior of the metasurface, varying between full transparency and half-power transmission, can be visualized in Figures 4.8a and 4.8b. The plot shown in Figure 4.8b corresponds to the real part of the electric field (Hx ) measured at y = 0 in Figure 4.8a. The amplitudes 1 and 12 correspond to the times when the metasurface was transparent and half-power transmitting, respectively. From these results, we can infer that the GSTCs-FDTD simulation scheme can efficiently simulate this kind of time-varying metasurface. Note that because of the limited size of the metasurface, edge diffraction is unavoidable. This phenomenon manifests as a standing wave in the transmitted and reflected field regions, explaining the ripples visible in the simulation results. The reader is referred to [27] for more details and examples.

4.5 Practical Realization

1.5

1

1

5

0.5

0

0

E (V/m)

z/ 0

147

0.5 0 -0.5

-0.5

-5

-1 -1

-5

0 y/

5

-1.5

-5

0

5

0

a Hx field at t = 1200dt

b Hx at y = 0.

Figure 4.8 GSTCs-FDTD simulation result for the time-varying metasurface. The metasurface is positioned at λz = 4.17, as indicated by a white dashed line. In (b), the metasurface is indicated 0

by a dot. The metasurface is illuminated by a plane wave propagating in the negative z-direction.

4.5

Practical Realization In Section 4.2, we were only interested in the mathematical synthesis of metasurfaces, which consists in finding the susceptibilities in terms of specified fields. We shall now investigate how the synthesized susceptibilities may be related to the geometry of the scattering particles that will constitute the metasurfaces to be realized. We will start by presenting the mathematical expressions that relate the susceptibilities to the scattering particles, which will be followed by a more in-depth discussion on the realization of metasurface scattering particles.

4.5.1

Relation with Scattering Parameters The conventional method for relating the scattering particle geometry to equivalent susceptibilities (or material parameters) is based on homogenization techniques. In the case of metamaterials, these techniques may be used to relate homogenized material parameters to the scattering parameters of the scatterers. From a general perspective, a single isolated scatterer is not sufficient to describe an homogenized medium. Instead, we shall rather consider a periodic array of scatterers, so as to take into account the interactions and coupling between adjacent scatterers and hence attain a more accurate description of a “medium” compared to a single scatterer. The susceptibilities, which describe the macroscopic responses of a medium, are thus naturally well suited to describing the homogenized material parameters of metasurfaces. It follows that the equivalent susceptibilities of a scattering particle may be related to the corresponding scattering parameters, conventionally obtained via full-wave simulations, of a periodic array made of an infinite repetition of that scattering particle [37–40]. Because the periodic array of scatterers is uniform with subwavelength periodicity, the scattered fields obey Snell’s law. More specifically, if the incident wave propagates normally with

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Electromagnetic Metasurface Synthesis, Analysis, and Applications

respect to the array, then the reflected and transmitted waves also propagate normally. In most cases, the periodic array of scattering particles is indeed excited with normally propagating waves. This allows one to rigorously obtain the 16 tangential susceptibility components in (4.15). However, this does not provide any information on the normal susceptibility components of the scattering particles. This is because normally propagating waves do not excite the normal susceptibilities due to the purely tangential nature of their electromagnetic fields. Nevertheless, this method allows one to match the tangential susceptibilities of the scattering particle to the susceptibilities found from the metasurface synthesis performed following the procedure in Section 4.2 and precisely yielding the ideal tangential susceptibility components. It is clear that the scattering particles may, in addition to their tangential susceptibilities, possess nonzero normal susceptibility components. In that case, the scattering response of the metasurface, when illuminated with obliquely propagating waves, will differ from the expected ideal behavior prescribed in the synthesis. Consequently, the homogenization technique only serves as an initial guess to describe the scattering behavior of the metasurface. Note that it is possible to obtain all 36 susceptibility components (4 susceptibility tensors, each with 9 components) of a scattering particle, but this would require solving the 4 GSTCs relations for 9 independent sets of incident, reflected, and transmitted waves, which is particularity tedious and is thus avoided. We will now derive the explicit expressions relating the tangential susceptibilities to the scattering parameters in the general case of a fully bianisotropic uniform metasurface surrounded by different media and excited by normally incident plane waves. Let us first write the system (4.15) in the following compact form: r ¨A , =χ v

(4.53)

r , and A correspond to the field differences, the normalized where the matrices , χ v susceptibilities, and the field averages, respectively. In order to obtain the 16 tangential susceptibility components in (4.15), we will now define 4 transformations by specifying the fields on both sides of the metasurface. Let us consider that the metasurface is illuminated from the left with an x-polarized normally incident plane wave. The corresponding incident, reflected, and transmitted electromagnetic fields read ˆ E i = x,

Hi =

1 ˆ y, η1

yx

xx ˆ E r = S11 xˆ + S11 y,

Hr =

1 yx xx ˆ (S xˆ ´ S11 y), η1 11

yx

xx ˆ E t = S21 xˆ + S21 y,

Ht =

1 yx xx ˆ (´S21 xˆ + S21 y), η2

(4.54a)

(4.54b)

uv , with a,b = t1,2u and u,v = tx,yu, are the scattering parameters, where the terms Sab with ports 1 and 2 corresponding to the left and right sides of the metasurface, respectively. The medium of the left of the metasurface has the intrinsic impedance η1 , while the medium on the right has the intrinsic impedance η2 . In addition to (4.54), three other cases have to be considered, i.e., y-polarized excitation incident from the left (port 1)

4.5 Practical Realization

149

and x- and y-polarized excitations incident from the right (port 2). Inserting these fields into (4.15) leads to the following matrix :  =

´N 2 {η1 + N 2 ¨ S 11 {η1 + N 2 ¨ S 21 {η2



´N 2 {η2 + N 2 ¨ S 12 {η1 + N 2 ¨ S 22 {η2

´N 1 ¨ N 2 ´ N 1 ¨ N 2 ¨ S 11 + N 1 ¨ N 2 ¨ S 21

, N 1 ¨ N 2 ´ N 1 ¨ N 2 ¨ S 12 + N 1 ¨ N 2 ¨ S 22 (4.55)

and, similarly, the matrix Av reads Av =

1 2





I + S 11 + S 21

I + S 12 + S 22

N 1 {η1 ´ N 1 ¨ S 11 {η1 + N 1 ¨ S 21 {η2

´N 1 {η2 ´ N 1 ¨ S 12 {η1 + N 1 ¨ S 22 {η2

where the matrices S ab , I , N 1 , and N 2 are defined by  xx    xy  Sab Sab 1 0 0 S ab = I= , N1 = yx yy , 0 1 1 Sab Sab

 ´1 , 0

(4.56)

,

 1 N2 = 0

 0 . ´1 (4.57) Now, the procedure to determine the susceptibilities of a given scattering particle is as follows. First, the scattering particle is simulated with periodic boundary conditions and normal excitation. Second, the resulting scattering parameters are used to form the matrices in (4.55) and (4.56). Finally, the corresponding susceptibilities are obtained by matrix inversion of (4.53). Alternatively, it is possible to express the scattering parameters (for normal wave propagation) of a uniform metasurface with known susceptibilities by solving (4.53) for the scattering parameters. This leads to the matrix equation. ´1

S = M 1 ¨ M 2,

(4.58)

where the scattering parameter matrix, S, is defined as   S 11 S 12 S= S 21 S 22

(4.59)

and the matrices M 1 and M 2 are obtained from (4.53), (4.55), and (4.56) by expressing the scattering parameters in terms of the normalized susceptibility tensors. The resulting matrix M 1 reads 

r {2 + χ r ¨ N {(2η ) N 2 {η1 ´ χ 1 1 ee em M1 = r {2 + χ r ´N 1 ¨ N 2 ´ χ me mm ¨ N 1 {(2η1 )

 r {2 ´ χ r ¨ N {(2η ) N 2 {η2 ´ χ 1 2 ee em , r {2 ´ χ r N ¨N ´χ ¨ N {(2η ) 1

2

me

mm

1

2

(4.60)

and the matrix M 2 reads  r {2 + N {η + χ r ¨ N {(2η ) χ 2 1 1 1 em M 2 = r ee r χ {2 + N ¨ N + χ ¨ N {(2η ) me

1

2

mm

1

1

 r {2 + N {η ´ χ r ¨ N {(2η ) χ 2 2 1 2 ee em . r {2 ´ N ¨ N ´ χ r χ ¨ N {(2η ) me

1

2

mm

1

2

(4.61) We now provide the expressions relating the susceptibilities to the scattering parameters (and vice versa) in the particular case of the monoanisotropic diagonal meta-

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Electromagnetic Metasurface Synthesis, Analysis, and Applications

surface discussed earlier. We assume that the media on both sides of the metasurface are the same and correspond to vacuum, i.e., η1 = η2 = η0 . By definition, we xy know that this metasurface is nongyrotropic and reciprocal. Therefore, we have Sab = yx

T

Sab = 0 and that S 21 = S 12 . Solving (4.53) along with (4.12) leads to the following susceptibilities: 2j (Tx + Rx ´ 1) , k0 (Tx + Rx + 1)   2j Ty + Ry ´ 1  , = k0 Ty + Ry + 1   2j Ty ´ Ry ´ 1  , = k 0 Ty ´ R y + 1 2j (Tx ´ Rx ´ 1) = , k0 (Tx ´ Rx + 1)

xx χee =

(4.62a)

yy

(4.62b)

χee

xx χmm yy

χmm

(4.62c) (4.62d)

xx and R = S xx , and so on. Reversing where, for convenience, we have Tx = S21 x 11 these relations so as to express the scattering parameters in terms of the susceptibilities leads to yy

Tx =

2 xx χ 4 + χee mm k0

, xx )(2 + j k χ yy ) (2 + j k0 χee 0 mm  yy  xx 2j k0 χmm ´ χee   Rx =  xx 2 + j k χ yy 2 + j k0 χee 0 mm

(4.63a) (4.63b)

for x-polarized waves and to yy

Ty =

xx k 2 4 + χee χmm 0

, yy xx ) (2 + j k0 χee )(2 + j k0 χmm  xx yy  2j k0 χmm ´ χee  Ry =  yy   xx 2 + j k0 χee 2 + j k0 χmm

(4.64a) (4.64b)

for y-polarized waves. Equations (4.63b) and (4.64b) reveal that the conditions for no yy xx and χ xx = χ yy , which corresponds to orthogonal electric reflection are χmm = χee ee mm and magnetic dipoles radiating with equal strength and opposite phase in the incident side and with same phase in the transmission side. At this point, one may wonder whether the expressions obtained above apply only to metasurfaces synthesized to transform incident, reflected, and transmitted waves that are normally propagating with respect to the metasurface, or whether they apply to arbitrary field transformations, which generally involves nonuniform metasurfaces? To answer this question, let us consider the following four cases: (1) the metasurface is synthesized only for normally propagating waves; (2) the metasurface is synthesized for obliquely propagating waves but without changing the direction of wave propagation, i.e., the metasurface is uniform; (3) the metasurface is synthesized to change the direction of wave propagation (e.g., refraction, collimation) but, at least, one of the specified waves propagates normally to the metasurface; and (4) the metasurface is synthesized

4.5 Practical Realization

151

to change the direction of wave propagation but none of the specified waves propagate normally (e.g., negative refraction). Case 1. The expressions derived above perfectly apply and the realized metasurface response will be in exact agreement with the specified one. Case 2. For illustration, let us consider the synthesis of a reflectionless uniform metasurface that rotates the polarization of the incident wave impinging on the metasurface at an angle θi = θt = 30˝ from broadside. If the expressions above are used and the scattering particles simulated with normal wave propagation, then the response of the realized metasurface for the specified incidence angle of 30˝ will not correspond to the expected result. Indeed, as mentioned above, simulating all the scattering parameters with normally propagating waves and solving (4.53) yields the exact tangential susceptibility components but does not provide any information on the normal susceptibility components. Therefore, an obliquely impinging wave may excite these normal polarizations, thus affecting the scattering response of the metasurface. The second reason is due to nonnegligible coupling between adjacent scattering particles, which depends on the incidence angle of the excitation (spatial dispersion). This coupling is thus different for normal and oblique wave propagation. In order to correctly synthesize this metasurface, one should change relations (4.54) so as to include the angle of wave propagation. Accordingly, the scattering particles must be simulated with the same specified angle of wave propagation. Only then would the metasurface yield the expected scattering response. Case 3. For illustration, let us consider the synthesis of a reflectionless metasurface that refracts a normally incident plane wave at a given refraction angle θt . In that case, the metasurface is nonuniform and the susceptibilities are thus functions of x and y on the metasurface. Inserting these susceptibilities into the relations above will yield scattering parameters that are themselves spatially varying. It is important to understand that these scattering parameters do not represent the overall scattering behavior of the metasurface but rather correspond to the local scattering parameters. Due to the nonuniformity of the structure, several different scattering particles have to be realized, each exhibiting the local scattering parameters corresponding to their position (x,y) on the metasurface. This is achieved by simulating each scattering particle individually with periodic boundary conditions (PBC). Once they are all implemented and combined together to form the final metasurface, the coupling between them differs from when they were simulated with PBC. This leads to a scattering response that is different from the expected one, irrespectively of whether the relations above (assuming normal wave propagation) are used or any other one (assuming oblique wave propagation). In that case, the relations between the susceptibilities and the scattering parameters only serve as an initial guess to realize the scattering particles, which then usually require to be further optimized so that the final metasurface performs the expected response. In addition to this, the realized scattering particles may also exhibit nonzero normal susceptibilities, which further degrades the scattering behavior of the metasurface. Since, in the example that is considered here, the incident wave is normally impinging on the metasurface, the relations derived above provide a relatively good initial guess to realize the metasurface scattering particles because the excitation of the normal susceptibility components is limited.

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Electromagnetic Metasurface Synthesis, Analysis, and Applications

Case 4. This case is a combination of the two previous ones since all the waves propagate obliquely (and at different angles) with respect to the metasurface that is therefore nonuniform. If the specified incident, reflected, and transmitted fields are all plane waves, then it would be adequate to adjust the direction of wave propagation to that of the specified incident wave, in (4.54) as well as in the full-wave simulations. However, if the specified incident, reflected, and transmitted fields are not plane waves but arbitrary fields, then the relations derived above are sufficient to obtain an initial approximation of the specified scattering behavior that may require further optimization. This is because metasurfaces are usually synthesized by assuming that the normal susceptibility components are zero, as was done in Section 4.2.2. Therefore simulating the scattering particles with normally propagating waves ensures that their resulting tangential susceptibilities correspond to the desired ones obtained from the synthesis. Then, the discrepancies in the final metasurface response may only come from the presence of nonzero normal polarizations and coupling with adjacent unit cells. But at least their tangential polarizations are the expected ones, which minimizes the errors in the scattering response of the metasurface.

4.5.2

Implementation of the Scattering Particles We will now discuss the physical realization of metasurfaces, which is the second step of metasurface synthesis [6, 41]. The conventional procedure to realize a metasurface is not trivial. It may be decomposed into four main steps. First, the metasurface susceptibilities are synthesized from the initial specified fields. They are then usually simulated using the FDFD/FDTD methods discussed in Sections 4.3.2 and 4.3.3, respectively, to verify that they perform the expected transformation. In some cases, it is required to adjust the susceptibilities by changing the initial specified fields so as to achieve an optimal transformation. Once the optimal susceptibilities have been obtained, the resulting scattering parameters are computed using (4.63) and (4.64) for birefringent metasurfaces or (4.53) in a more general case. Second, the scattering parameter functions (which are continuous functions) are spatially discretized using a square lattice in the xy-plane with subwavelength resolution and where each discrete point corresponds to a scattering particle (or unit cell) to be implemented. This step requires to either choose a specific unit cell size, which is generally in a range between λ0 {2 and λ0 {10, or to know a priori the lateral size of the unit cells, which may be the case when one already has a database of scattering particle 1 responses. In all cases, it is important that the unit cell lateral size be less than λ0 to avoid generating undesired diffraction orders [14], ensuring the homogeneity of the metasurface and also properly sampling the scattering parameter functions according to the Nyquist criterion. Ideally, the smaller the unit cells, the better. But the smaller they are, the less they interact with the exciting field, which limits their minimum size to about λ0 {10. Below that limit, it is difficult to achieve resonance, and the control of the 1

For instance, we have been using the same type of unit cell (with λ0 {5 lateral size) for several different metasurfaces.

4.5 Practical Realization

153

electromagnetic field is therefore limited because of the limited range of values that its material parameters may take. Third, full-wave simulations of unit cells, whose shapes will be discussed later, are performed using commercial software, which yield their scattering parameters. For each lattice site, the shapes of the unit cells are tuned until their scattering parameters match those found in the previous step. These simulations are done assuming periodic boundary conditions (PBC) even if the metasurface to be implemented is nonuniform (scattering parameters are varying functions of x and/or y). In this latter case, local periodicity is assumed, meaning that the metasurface must exhibit slowly varying spatial features and that, consequently, the coupling between adjacent unit cells in the final structure is comparable to that in the perfectly periodic environment of the simulation software. However, when the metasurface exhibits rapid spatial variations compared to the freespace wavelength, the assumption of local periodicity is not valid anymore and the coupling between adjacent unit cells may be very different when the unit cell is inserted into the final nonuniform metasurface compared to when it is simulated with PBC. In such a case, the electromagnetic response of the realized metasurface will not correspond to the expected result. Therefore, it is in general difficult (or even impossible) to realize metasurfaces with very rapid spatial variations compared to the size of the unit cells. Finally, once all the unit cells have been obtained from the full-wave simulations, one can simulate the entire structure (or only a part of it, if the scattering parameters are periodic functions) to analyze the electromagnetic response of the metasurface. In most cases, the response will slightly differ from the expected result, which is likely to be due to the error induced because of the local periodicity assumption. In such a case, the unit cells may be further tuned via parametric analysis or standard optimization techniques to improve the response of the metasurface and achieve a better agreement with the initial specifications. In the following two sections, we present two different unit cell structures that may be used to realize metasurfaces. The first one consists in cascaded metallic layers, while the second one is based on dielectric resonators.

Scattering Particles Based on Cascaded Metallic Layers In what follows, we will discuss the design of unit cells based on cascaded metallic layers. It must be noted that the following developments are based on well-known microwave theory concepts already used since the 1950s, notably in the implementation of frequency selective surfaces (FSS) [42]. More recently, these concepts have been adopted for the realization of metasurfaces [43–49]. Most metasurfaces designed for the microwave regime are composed of one or several layers of metallic patterns separated by dielectric substrates. This technology may also be considered at optical frequencies, but the additional plasmonic loss due to the metallic parts as well as the complexity of fabrication make it less attractive compared to other alternatives, like the fully dielectric unit cell structure, that will be discussed in the next section. The condition to realize a fully functional metasurface is that its unit cell scattering parameters cover the 2π -phase range while being able to maintain a constant transmission (or reflection) amplitude, which is usually equal to 1 to maximize

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efficiency. However, it was shown that metasurfaces using one single metallic layer cannot achieve full transmission and 2π -phase coverage [50]. This limitation is due to the limited number of degrees of freedom provided by a single-layer structure. It naturally follows that increasing the number of metallic layers increases the number of degrees of freedom, which may contribute to improve the bandwidth as well as the achievable transmission or reflection phase coverage. A remarkable and important case is that of a three metallic layer structure, and more specifically the particular case where the two outer layers are the same. This three-layer structure is the simplest 2 one that provides a full transmission and a 2π -phase coverage [43–49]. This kind of structure is symmetric with respect to the longitudinal direction and is therefore limited to the implementation of monoanisotropic metasurfaces. Interestingly, a unit cell with three different layers would be asymmetric and thus would allow the realization of bianisotropic metasurfaces, at the cost of a more complicated design. Note that the overall thickness of the three-layer structure remains deeply subwavelength. For most practical applications, this kind of three metallic layer structure is sufficient but more layers may be considered as a means to increase the bandwidth [42] at the cost of extra loss, weight, and implementation complexity. It can be easily verified, using microwave theory concepts, that cascading three metallic layers (with similar outer layers) is sufficient to realize a transmission coefficient of 0 ď |T | ď 1 with a 2π-phase coverage. For a given polarization, each metallic layer can be described by an impedance layer and each of the two dielectric spacers by an equivalent transmission line. The entire structure can then be analyzed using the ABCD matrix technique [51], as such: 

 A B C D           cos(βd) j ηd sin(βd) cos(βd) j ηd sin(βd) 1 0 1 0 1 0 ¨ ¨ ¨ j sin(βd) ¨ j sin(βd) , = Y2 1 Y1 1 Y1 1 cos(βd) cos(βd) ηd ηd (4.65)

where β is the propagation constant along the z-direction and d and ηd are the thickness and the impedance of the dielectric substrates, respectively. The terms Y1 and Y2 correspond to the admittances of the outer layers and the middle layer, respectively. Finally, the ABCD matrix (4.65) can be converted into scattering parameters using the following relation [51]     1 B{η0 ´ Cη0 S11 S12 2 = . (4.66) S21 S22 2 B{η0 ´ Cη0 2A + B{η0 + Cη0 To illustrate how the transmission coefficient, S21 , changes as function of Y1 and Y2 , we plot it in Figure 4.9. These two figures are obtained by substituting (4.65) into (4.66) and plotting the amplitude and phase of S21 versus the imaginary parts of Y1 and Y2 for arbitrary values of β, ηd , and d. We assume here that the unit cell is perfectly lossless and passive, thus Re(Y1 ) = Re(Y2 ) = 0. These figures reveal that it is possible to cover 2

Similar considerations naturally apply for the reflection coefficient if the metasurface is used in reflection.

4.5 Practical Realization

−0.08 0.9

−0.06

0.8

−0.04

0.7 0.6

0

0.5

0.02

0.4

Im(Y2 )

−0.02

0.3

0.04

0.2

0.06

0.1 −0.01

0

0.01

Im(Y1 )

a

0.02

350

−0.06

300

−0.04 250 −0.02

Im(Y2 )

−0.08

155

200 0 150

0.02

100

0.04

50

0.06 −0.01

0

0.01

Im(Y1 )

0.02

b

Figure 4.9 (a) Transmitted power (|S21 |2 ) and (b) phase for the three cascaded metallic layers

versus the imaginary parts of the impedance of Y1 and Y2 . The black line indicates that full transmission can be achieved and that the corresponding phase varies between 0 and 2π . A black and white version of this figure will appear in some formats. For the color version, please refer to the plate section.

a 2π -phase range while maintaining full transmission (see the solid black line in the two figures). The question that arises now is, what kind of shape should the metallic layers have to realize the unit cells? Several different shapes have been investigated over time [42], but the one we have used for the vast majority of our metasurfaces is the Jerusalem cross. A generic Jerusalem cross with all its tunable dimensions is shown in Figure 4.10a, while an illustration of a unit cell made with three cascaded crosses is shown in Figure 4.10b. The characteristic shape of the Jerusalem cross provides a relatively independent control of x and y polarizations, which allows for relatively easy design. More specifically, independent control of the x and y polarizations is achieved by considering that tuning the parameters Bx , Wx , and Ly mostly affects y-polarized waves and tuning parameters By , Wy , and Lx mostly affects x-polarized waves. With this kind of geometry, there is an important capacitive coupling between adjacent unit cells. This coupling has the effect of increasing the response of the scattering particles, thus enabling the implementation of unit cells with smaller transverse and longitudinal dimensions with respect to the operating wavelength [42, 52]. However, this comes at the cost of a more complex structure due to the large number of tunable parameters. The coupling between adjacent unit cells also has the disadvantage of making the implementation of nonuniform metasurfaces more difficult because the overall response of the metasurface is more sensitive to the variations of shape of adjacent unit cells. Note that the discrepancy due to the coupling may generally be lowered by decreasing the relative permittivity of the dielectric substrate and/or by increasing its thickness. We are now interested in finding the exact dimensions, shown in Figure 4.10a, of the Jerusalem cross of the outer and middle layers, to which we will now refer as layers 1 and 2, respectively. Two main methods are considered here to design these layers. The first method consists in inserting the required scattering parameters found from

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a

b

Figure 4.10 Representations of (a) a generic Jerusalem cross with the dimensions that can be modified and (b) a unit cell with three metallic (PEC) Jerusalem crosses separated by dielectric slabs with identical outer layers.

the synthesis into Eqs. (4.65) and (4.66) and solving these relations so as to obtain the corresponding required admittance of each layer. However, even if it is possible to obtain the admittance of each individual layer, there is unfortunately no direct and analytical method to relate the dimensions of layers 1 and 2 to the admittances Y1 and Y2 . The conventional method consists in simulating each layer individually using PBC to obtain its corresponding scattering parameters. From the simulated scattering parameters, the corresponding admittance of the layer can be found using (4.66) with [A,B;C,D] = [1,0;Y,1] and solving for Y . Then, by iterative tuning, the Jerusalem cross dimensions are modified until the layer admittance matches the required one found from the previous steps. While this method is useful because it simplifies the realization of the unit cells by allowing one to design each layer separately, it is rigorous only in absence of longitudinal evanescent coupling between the metallic layers since the ABCD matrix method only takes into account zeroth-order propagating 3 waves . And, unfortunately, the unit cell total thickness is in the order of λ0 {10, which means that the longitudinal coupling between the layers is nonnegligible. Nevertheless, this method may still be used to obtain an initial structure, which would then require further numerical optimizations to achieve the specified response. Instead of trying to realize the unit cells by using the admittances of individual layers, we may alternatively consider the method consisting in simulating the entire three3

Higher diffraction orders or evanescent waves are not considered in the ABCD method.

4.5 Practical Realization

157

layer structure and tuning the dimensions of the layers until the required scattering parameters, found from the synthesis, are obtained. From an electromagnetic perspective, we can consider that the interactions between the exciting wave and the three metallic layers generate equivalent electric and magnetic responses. In the frequency range of interest, the unit cell is small compared to the wavelength and thus exhibits only dipolar electric and magnetic resonances. Higher-order multipolar resonances occur at higher frequencies where the unit cell size approaches that of the wavelength. Thus, the metasurface can be seen as an equivalent array of electric and magnetic dipole moments. The complete control of the transmission amplitude and phase that the unit cell provides is achieved by the superposition of the fields scattered by the electric and magnetic dipolar resonances. In the case of the three-layer unit cell, the structure is symmetric in the longitudinal direction. We can thus analyze the behavior of the structure using even/odd mode analysis. The odd mode resonance is achieved when the surface currents induced by the incident wave are equal and opposite on the outer layers. In that case, the total current is zero on the middle layer. Such a current distribution corresponds to an equivalent magnetic dipole moment. Since the current is null on the middle layer, this layer does not play a role in the magnetic response of the structure [49]. The even mode resonance is achieved when the same current distribution is excited on the three layers, thus corresponding to an equivalent electric dipole moment. Practically speaking, this means that layers 1 are modified first, to tune the magnetic response, and then layer 2 is modified to adjust the electric response of the structure. Usually, a few optimization iterations are required to design one unit cell.

Scattering Particles Based on Dielectric Resonators In addition to the aforementioned metallic unit cells, the realization of the metasurface scattering particles may be achieved by considering purely dielectric resonators without any metallic part. As previously discussed, one essential requirement to achieve full transmission (or reflection) and a 2π-phase coverage is the presence of both electric and magnetic resonances, which are naturally occurring in dielectric resonators [53, 54]. For this reason, dielectric resonators have been widely used to realize metasurfaces [55–57]. A dielectric metasurface is, as before, a uniform (or nonuniform) two-dimensional array of dielectric resonators, where the thickness of the metasurface and the period of the unit cells in the array are both subwavelength. Dielectric unit cells are particularly attractive in the optical regime, where they are more easily realizable compared to the cascaded metallic layer structure discussed above and also exhibit no plasmonic loss thanks to the absence of metallic inclusions. A typical unit cell is shown in Figure 4.11a where the resonator is a dielectric cylinder of circular cross section with permittivity r,1 placed on a dielectric slab with permittivity r,2 (usually silica) for mechanical support [56, 58–61]. Other types of particle shapes are naturally also possible. This includes 90˝ symmetric shapes like squares, which (like cylinders) present the same behavior for x- and y-polarized waves, but also 90˝ asymmetric shapes, such as ellipses and rectangles, which allow a complete and independent control of the two orthogonal polarizations [56]. An additional advantage of dielectric unit cells is that they have many fewer physical parameters to adjust compared to the

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Electromagnetic Metasurface Synthesis, Analysis, and Applications

a

b

Figure 4.11 Representations of two dielectric metasurface unit cell designs. In (a), the conventional optical regime design, where a dielectric resonator of permittivity r,1 is placed on a dielectric substrate of permittivity r,2 . In (b), our unit cell design for the microwave region, where the resonators are held together by dielectric connections.

Jerusalem cross, which effectively simplifies the optimization procedure to achieve a specified response but comes at the cost of less degrees of freedom. We have also proposed an alternative implementation of dielectric metasurfaces, which is designed to be used in the microwave regime [62, 63]. In the proposed design, the resonators are not placed on a substrate; rather, they are held together with dielectric connections, as depicted in Figure 4.11b. This design has the advantage of being easy to fabricate since the fabrication process essentially consists in laser cutting a pattern mask. An additional advantage of this structure is that it is immune to the presence of multiple reflections occurring inside the supporting slab, as it is the case of the structure in Figure 4.11a. In general, and for simplicity, we assume that all resonators have the same thickness, which can be achieved by gluing together several slabs of commercially available dielectric substrates. An example of such structure is depicted in Figure 4.12, which shows a fabricated metasurface made of several patterned dielectric slabs. The disadvantage of the dielectric connections is that they affect the electromagnetic behavior of the resonators. In order to minimize these effects, these connections are made very small with respect to the operating wavelength. In that case, their presence has negligible effect on the polarization that is orthogonal to them, e.g., Ex in Figure 4.11b. However, their effects on the polarization that is parallel to them cannot be ignored but can be compensated by numerical optimizations. For both structures in Figure 4.11, the physical dimensions of the unit cells are obtained using the same approach as for the cascaded metallic layer structure discussed above. Each unit cell is simulated assuming PBC. Its dimensions are tuned until the desired scattering parameters are obtained, which in general requires a few optimization iterations. The various advantages of dielectric metasurfaces may play an important role in the development of metasurfaces designed to work at multiple frequencies or over

4.6 Conclusion

159

Figure 4.12 Example of a fabricated dielectric metasurface with interconnected resonators made of four stacked dielectric slabs glued together.

extended frequency bands. Traditionally, the bandwidth of transmit arrays or FSS may be increased by cascading several layers [42] at the cost of having thicker, more complex and more lossy structures. A potential alternative lies in the use of highly coupled dielectric resonators, which are easier to design and less lossy compared to metallic structures. The idea consists in placing resonating particles close to each other so as to increase mutual coupling and consequently generate new resonant frequencies. Thus, the coupling between the resonators can be used as an additional degree of freedom to tune the frequency response of metasurfaces [64]. A disadvantage of dielectric resonators is that, to achieve a small unit cell lateral size and thickness, a high contrast between the dielectric permittivity of the particle and that of the surrounding medium is typically required. This means that metasurfaces with rapid spatial variations compared to the free-space wavelength can only be realized using high permittivity resonators, which may not be easily available. As a comparison, the typical lateral size and thickness of metallic unit cells are about λ0 {5 and λ0 {10, respectively; while the dielectric unit cell counterparts are about λ0 {1.1 and λ0 {3.3, respectively, for resonators of relative permittivity r = 10.2 surrounded by air.

4.6

Conclusion We have presented an overview of the general metasurface synthesis framework that we have developed. This chapter was specifically written so as to provide the reader with the necessary tools required to mathematically synthesize, numerically analyze, and practically realize almost any metasurface. The mathematical synthesis is based on rigorous generalized sheet transition conditions that allow a complete description of the electromagnetic properties of the metasurfaces. In particular, the presence of tangential and normal polarization densities including fully bianisotropic susceptibility components as well as nonlinear terms are fully taken into account by our model.

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Similarly, the numerical analysis schemes are all based on these rigorous boundary conditions, thus ensuring the most accurate and complete metasurface simulations.

4.7

Conditions of Reciprocity, Passivity, and Loss In this appendix, we review the properties of reciprocity, passivity, and loss applying to time-invariant linear metasurfaces. These conditions are essential to the mathematical synthesis procedure discussed in the previous sections. From a general perspective, the reciprocity of a given system can be evaluated by considering the scattering interactions between an arbitrary emitter and receiver and that system. If the position of the emitter and the receiver can be interchanged without affecting the transmission between the two, then the system is reciprocal. The Lorentz reciprocity theorem is the most conventional method to derive the reciprocity conditions of an electromagnetic system. If the system consists of a bianisotropic medium, the reciprocity conditions, that result from the Lorentz theorem, read [12, 13] T

χ ee = χ ee,

T

χ mm = χ mm,

T

χ me = ´χ em,

(4.67)

where the superscript T denotes the matrix transpose operation. We will now investigate how the metasurface susceptibility values can be related to gain or to dissipation. As seen previously, the synthesized metasurface susceptibilities are generally complex values, whose real and imaginary parts may represent gain or loss. In order to precisely determine whether the metasurface is active or lossy, or even a combination of both, we need to explicitly compute the bianisotropic Poynting theorem. Assuming the convention ej ωt , the time-average bianisotropic Poynting theorem is given by [12] ∇ ¨ xSy = ´xIJe y ´ xIJm y ´ xIP y ´ xIM y,

(4.68)

where x ¨ y denotes the time-average operation, S is the Poynting vector and IJe , IJm , IP and IM are loss (or gain) contributions emerging from the electric currents, magnetic currents, electric polarization, and magnetic polarization, respectively. If the metasurface is surrounded by vacuum on both sides, then the terms in (4.68) read 1 (4.69a) Re(E ˆ H ˚ ), 2   1 ˚ (4.69b) xIJe y = Re j ω0 E ˚ ¨ (χ ee ´ χ ee ) ¨ E , 4   1 ˚ (4.69c) xIJm y = Re j ωμ0 H ˚ ¨ (χ mm ´ χ mm ) ¨ H , 4 ı ” 1 † (4.69d) xIP y = Re j ω0 (E ˚ ¨ (χ ee ´ χ ee ) ¨ E + 2η0 E ˚ ¨ χ em ¨ H ) , 4 ı ” 1 † † (4.69e) xIM y = Re j ωμ0 (H ˚ ¨ (χ mm ´ χ mm ) ¨ H ´ 2E ˚ ¨ χ me ¨ H {η0 ) , 4 where † is the conjugate transpose operator. The fields E and H in (4.69) are the fields acting on the metasurface and can be replaced by the corresponding average fields. xSy =

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The term on the left-hand side of (4.68) corresponds to the divergence of the power flow through an arbitrary volume, which, in our case, surrounds the metasurface. If this term is zero, then the amplitude of a wave going through the metasurface remains the same, which may a priori indicate that the metasurface is passive and lossless. However, this is not necessarily the case since the terms on the right-hand side of (4.68) may compensate each other such that gain and loss perfectly cancel out. Therefore, it is in general necessary to compute all the terms from (4.69b) to (4.69e) to determine whether the metasurface is active, which is the case if these terms are negative, or lossy, in the case where they are positive. If all these terms are zero, then the metasurface is passive and lossless. By combining relations (4.69b) to (4.69e) and conditions (4.67), one obtains the conditions that make a metasurface simultaneously passive, lossless, and reciprocal. These conditions read T

˚

χ ee = χ ee,

T

˚

χ mm = χ mm,

T

˚

χ me = χ em .

(4.70)

We see that the conditions in (4.67) and (4.70) establish relations between different susceptibility components, thus decreasing the number of degrees of freedom available to a metasurface to control electromagnetic fields. Accordingly, this has the effect of reducing the diversity of metasurface wave transformations.

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[52] J.-S. G. Hong and M. J. Lancaster, Microstrip Filters for RF/Microwave Applications. John Wiley, 2004. [53] G. Mie, “Beiträge zur optik trüber medien, speziell kolloidaler metallösungen,” Ann. Phys., vol. 330, no. 3, pp. 377–445, 1908. [54] M. Kerker, The Scattering of Light and Other Electromagnetic Radiation. New York: Academic Press, 2013. [55] A. Krasnok, S. Makarov, M. Petrov, R. Savelev, P. Belov, and Y. Kivshar, “Towards alldielectric metamaterials and nanophotonics,” Proc. SPIE, vol. 9502, 950 203, 2015. [56] A. Arbabi, Y. Horie, M. Bagheri, and A. Faraon, “Dielectric metasurfaces for complete control of phase and polarization with subwavelength spatial resolution and high transmission,” Nat. Nanotechnol., vol. 10, pp. 937–943, 2015. [57] D. Lin, P. Fan, E. Hasman, and M. L. Brongersma, “Dielectric gradient metasurface optical elements,” Science, vol. 345, no. 6194, pp. 298–302, 2014. [58] Y. Yang, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “All-dielectric metasurface analogue of electromagnetically induced transparency,” Nat. Commun., vol. 5, no. 5753, pp. 366–368, 2014. [59] J. Sautter, I. Staude, M. Decker, E. Rusak, D. N. Neshev, I. Brener, and Y. S. Kivshar, “Active tuning of all-dielectric metasurfaces,” ACS Nano., vol. 9, no. 4, pp. 4308–4315, 2015. [60] M. Decker, I. Staude, M. Falkner, J. Dominguez, D. N. Neshev, I. Brener, T. Pertsch, and Y. S. Kivshar, “High-efficiency dielectric Huygens surfaces,” Adv. Opt. Mater., vol. 3, no. 6, pp. 813–820, 2015. [61] M. I. Shalaev, J. Sun, A. Tsukernik, A. Pandey, K. Nikolskiy, and N. M. Litchinitser, “Highefficiency all-dielectric metasurfaces for ultracompact beam manipulation in transmission mode,” Nano Lett., vol. 15, no. 9, pp. 6261–6266, 2015. [62] S. Gupta, K. Achouri, and C. Caloz, “All-pass metasurfaces based on interconnected dielectric resonators as a spatial phaser for real-time analog signal processing,” in 2015 IEEE Conference on Antenna Measurements Applications (CAMA), Nov 2015, pp. 1–3. [63] K. Achouri, A. Yahyaoui, S. Gupta, H. Rmili, and C. Caloz, “Dielectric resonator metasurface for dispersion engineering,” IEEE Trans. Antennas Propag., vol. 65, no. 2, pp. 673–680, 2017. [64] F. Aieta, M. A. Kats, P. Genevet, and F. Capasso, “Multiwavelength achromatic metasurfaces by dispersive phase compensation,” Science, vol. 347, no. 6228, pp. 1342–1345, 2015.

5

Analysis and Modeling of Quasi-periodic Structures Maokun Li

5.1

Introduction With the development of electromagnetic research and engineering, many devices are studied to better control the propagation of electromagnetic waves. They include metasurfaces [1], reflectarray antennas [2], nano particle arrays [3], etc. These devices can be modeled as quasi-periodic arrays, in which similar elements with some varying geometrical parameters are positioned on a periodic lattice. A simple example of 1D quasi-periodic array is shown in Figure 5.1. The dashed line shows the periodic lattice of the quasi-periodic array. These devices have been widely used in radar, remote sensing, satellite communication, on-chip optical lenses and circuits. With the development of fabrication technology, quasi-periodic arrays will gain more and more applications in electromagnetic and optical engineering. In the design of quasi-periodic arrays, such as reflectarray antennas and metasurfaces, reflection phase of individual array element is first computed, then the array is synthesized from the element phase based on array theory. The performance characters can then be calculated [4–7]. In this design procedure, the accuracy of the element reflection phase is crucial to correctly predicting the practical array performance. In the early days, the reflection phase of every element was simply calculated individually and the effects of surrounding elements were ignored [8–10]. This scheme is easy to use, but it introduces design errors because the practical reflection phase in an antenna could deviate from the designed phase. Hence the array performance cannot meet the design expectation. Therefore, in recent design of quasi-periodic arrays, the element reflection phase with different geometrical parameters is computed under periodic boundary conditions, assuming the surrounding elements are identical and infinite [5,11,12]. This is usually considered as a very reasonable assumption because the element sizes vary slowly on the surface of quasi-periodic arrays, except for a sharp change due to phase wrapping (phase jump from ´180˝ to 180˝ ). The array performance is also much closer to the design expectation compared with the previous design methodology. It is generally believed that the practical phase of an element is very close to the design value. Hence, in design of quasi-periodic arrays, the element phase is mostly computed based on the assumption of periodicity. However, using periodic assumption to compute the element reflection phase still cannot take into account the exact surroundings of an element in a real quasi-periodic array. Therefore, the practical element phase in a quasi-periodic array may not be the 165

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Analysis and Modeling of Quasi-periodic Structures

Figure 5.1 An example of a 1D quasi-periodic array. © 2018 IEEE. Reprinted, with permission,

from Xunwang Dang 2017.

same as the design value. With the development of computational electromagnetic tools, it is now feasible to compute the practical element phase and study the design error due to quasi-periodic effect. Many researchers have investigated this problem. For example, Neyeri et al. [13] and Vita et al. [14] simulated mid-size reflectarray antennas using electromagnetic full-wave solvers and calculated the practical element phase on an antenna. Hani and Laurin [15,16] found that the specular reflection in a reflectarray antenna radiating in off-specular directions can be due to a periodic phase error led by the foregoing quasi-periodic effect. Moreover, Nayeri et al. [17] added random phase errors into each element in the array and studied the mechanism of performance degradation due to quasi-periodic effect. They found that the radiation pattern does get much closer to the practical arrays. Furthermore, by studying a large unit cell consisting of 3 ˆ 3 transmitarray elements, they also show that periodic approximation appears to be the primary reason for the error in the transmission coefficient of the elements. In addition to the above studies based on numerical simulations, theoretical analysis has also been carried out for analysis of quasi-periodic arrays [18,19]. Besides the above analysis of quasi-periodic effect, some researchers also investigated methods to improve the design of quasi-periodic arrays, especially reflectarray antennas, by taking the quasi-periodic effect into consideration. For example, Milon et al. [20] developed the “surrounded elements” approach to compute the element phase together with its quasi-periodic neighbors. They observe difference between the element reflection phase with periodic boundaries and the one surrounded by elements of different type. Yoon et al. [21] studied the reflection phase error on a practical antenna and proposed to use a combination of three types of elements to improve the antenna performance. Tenuti et al. [22] studied the design method for conformal reflectarray antennas by taking the non-radiating currents on the reflecting surface into account. In this work, we study the quasi-periodic effect in quasi-periodic arrays. We use reflectarray antennas as an example to quantify the phase error between the practical and designed element phase due to periodic assumption and also look for methods to further improve antenna performance by correction of this phase deviation. To achieve this goal, we first apply the Floquet theory [23,24] to calculate the element reflection phase from the electrical field on the surface of an element computed by numerical simulation. This scheme allows us to calculate not only the reflection phase of a single element but an element on a practical antenna. Then we analyze the reflection phase of a 1D quasi-periodic array constructed by square patches. We observe that resonant patch elements are more sensitive to quasi-periodic effects and their practical phase could be very different from the design value. These resonant elements usually have a reflection phase close to zero degrees. This finding is against our intuitive conjecture that large phase error may exist where neighboring elements have significant difference in their geometrical sizes.

5.1 Introduction

167

Based on the above analysis, we develop two schemes to reduce phase deviation and improve performance of quasi-periodic arrays. They include the phase correction and the adjustment of global reference phase. The first scheme is based on the idea of reducing the element phase deviation between practical and design values, while the second one is to weaken the impact of quasi-periodic effect by adjusting the quantity and position of resonant unit cells in a quasi-periodic array. Numerical design experiments of a thin dielectric substrate antenna of square patches show that these two schemes together could improve both the gain and aperture efficiency of the antenna. In the design of quasi-periodic arrays, another challenge is the fast and accurate modeling of quasi-periodic arrays. This is important in their design and optimization. However, due to the large number of elements and multi-scale features in the array, direct full-wave modeling methods are still not efficient enough, and their usage is limited in the verification process. Therefore, traditional design and optimization rely on some approximation to simplify the problem. For example, a periodic boundary condition (PBC) [5,25–27] can be used for element simulation, as we discussed above. If the period of the array is only a small fraction of wavelength, the computation can be accelerated by homogenizing the elements into a generalized impedance boundary condition [28–31] that requires many fewer unknowns to model. Some analytical methods [18,32] can also be used to compute the dispersion diagram of modulated quasi-periodic arrays. For certain element shapes and placement, special basis functions have also been developed [33] to reduce the number of unknowns. Furthermore, model-order reduction techniques also show a good potential in acceleration of full-wave modeling of quasiperiodic arrays [34,35]. In this study, we investigate modeling of quasi-periodic arrays using an integral equation formulation accelerated by the reduced basis method (RBM) [36–38]. This scheme is based on the fact that although each element in a quasi-periodic array is different, they share some geometrical similarities. It leads to similarities of currents on array elements, i.e., the currents are sparse and compressible in the space spanned by geometrical parameters. RBM can construct a basis set in this space and reduce computational complexity by using many fewer unknowns for element currents compared with the original problem. In this work, the reduced basis functions are macro basis functions constructed from Rao–Wilton–Glisson (RWG) basis functions on one element [39] through an offline process. To improve the convergence of modeling quasi-periodic arrays using reduced basis, we set up the reduced basis set from current solutions on elemental arrays so that mutual coupling is taken into account. Then the whole array can be solved through a fast online process. This technique has some similarities with the construction of the characteristic basis function (CBF) [40]. Both CBF and RBM can accelerate the simulation under different directions of incident waves. In addition, RBM can model other types of control parameters, such as element sizes, orientation angles or element heights. RBM has been widely used in fast analysis of mechanical systems [41], photonic crystals [42] and many other systems and devices. One of the challenges in applying RBM to method of moments (MoM) is that numerical construction of matrix equations depends on geometrical parameters of every element. Direct construction requires calculation of the entire matrix in method

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Analysis and Modeling of Quasi-periodic Structures

of moments that can be inefficient for large arrays. To accelerate the construction of impedance matrix, we apply the empirical interpolation method (EIM). Furthermore, a linear transform is applied to reduce the sampling in interpolation from two to one dimension. Numerical examples show that both the computation and memory efficiency are improved using RBM compared with direct MoM. This chapter is organized into two sections, as follows: in Section 5.2, we will focus on analysis of design error due to the neglect of quasi-periodic effect. First we describe the scheme for calculating element reflection phase from numerical simulation; then we analyze the phase deviation in a 1D quasi-periodic array and a practical reflectarray antenna constructed by square patches; after that, we discuss two schemes to improve the antenna performance in the design phase. In Section 5.3, we will formulate an integral equation to model quasi-periodic arrays. Both RBM and EIM are introduced to reduce the number of unknowns and accelerate the construction of matrix equations. Reduced basis from an elemental array is applied to improve the convergence of modeling a quasi-periodic array. The performance of this algorithm will then be benchmarked using numerical examples of 1D and 2D quasi-periodic arrays.

5.2

Study of Quasi-periodic Effect

5.2.1

Calculation of Element Reflection Phase Formulation Based on Floquet Theory The reflection phase of an element can be computed from the field on the surface of the element based on Floquet theory [23]. For simplicity, we first consider a periodic scalar field E(x,y) on the xy-plane (as shown in Figure 5.2) that can be described as E(x,y) =

+ 8 ÿ

+ 8 ÿ

f (x ´ xmn,y ´ ymn ) exp(´j kxi xmn ´ j kyi ymn ) ,

(5.1)

m=´8 n=´8

where f (x,y) is a complex function on the xy-plane; (xmn,ymn ) represents periodic grid points; kxi and kyi are x- and y-components of incident wave vector ki . Here we

Figure 5.2 Calculation of reflection phase based on Floquet theory.

5.2 Study of Quasi-periodic Effect

169

use rectangular grids, i.e., xmn = ma and ymn = nb, where m and n are integers. The Fourier transform of E(x,y) is defined as ż +8 ˜ x ,ky ) = 1 E(k E(x,y) exp(j kx x + j ky y)dxdy . (5.2) 4π 2 ´8 Substituting E(x,y) in Eq. (5.1) into Eq. (5.2), we obtain ˜ x ,ky ) = f˜(kx ,ky ) E(k

+ 8 ÿ

+ 8 ÿ

exp [j xmn (kx ´ kxi ) + jymn (ky ´ kyi )] ,

(5.3)

m=´8 m=´8

where f˜(kx ,ky ) is the two-dimensional Fourier transform of f (x,y). As we know that   + 8 +8 ÿ 2π ÿ 2mπ exp(j mkx a) = δ kx ´ , (5.4) a m=´8 a m=´8 and similarly, + 8 ÿ n=´8

exp(j nky b) =

2π b

  2nπ δ ky ´ , b m=´8 + 8 ÿ

(5.5)

substitute Eq. (5.4) and Eq. (5.5) into Eq. (5.3), and we have + 8 + 8 2 ÿ ÿ ˜ x ,ky ) = 4π f˜(kx ,ky ) δ(kxim )δ(kyin ) , E(k ab m=´8 n=´8

(5.6)

where kxim = kx ´ kxi ´ 2π m{a, kyin = ky ´ kyi ´ 2π n{b. Applying inverse Fourier transform to Eq. (5.6), E(x,y) can be expressed as E(x,y) =

4π 2 exp (´j kxi x ´ j kyi y) ab   + 8 + 8 ÿ ÿ 2π m 2π n x´j y , f˜(kxmn,kymn ) exp ´j a b m=´8 n=´8

(5.7)

where kxmn = kxi + 2πam , kymn = kyi + 2πb n and f˜(kxmn,kymn ) represents the coefficient of Floquet mode (m,n), which can be computed as ż 1 E(x,y) exp (j kxi x + j kyi y) f˜(kxmn,kymn ) = 4π 2 D   2π m 2π n exp j x+j y dxdy . (5.8) a b Using Eq. (5.8), we can compute the reflection coefficient of Floquet mode (m,n) as mn = S11

ref f˜mn , inc f˜mn

(5.9)

inc are the coefficient of Floquet mode (m,n) for the reflected and inciwhere f˜mn and f˜mn dent wave, respectively. In design of quasi-periodic arrays, such as reflectarray antennas, we usually only consider the reflection coefficient of the fundamental mode (0,0). ref

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Analysis and Modeling of Quasi-periodic Structures

Figure 5.3 Model of the square patch unit cell, grid period a = 11.19 mm, thickness of substrate

h = 0.787 mm, dielectric constant of substrate r = 2.2.

a Phase

b Magnitude

Figure 5.4 Calculation of element reflection phase under normal incidence: a reflection phase, b reflection magnitude. The solid line shows the results computed using HFSS, and the dotted line shows the results computed using Eq. (5.10).

Therefore, Eq. (5.9) can be simplified as ş ref E sca (x,y) exp (j kxi x + j kyi y)dxdy f˜00 00 S11 = inc = şD inc , (x,y) exp (j kxi x + j kyi y)dxdy f˜ DE

(5.10)

00

00 . and the element reflection phase is the phase angle of S11

Numerical Experiments In this subsection, we will validate the formulation derived in the previous subsection using numerical experiments. In these experiments, we calculate the reflection coefficients S11 of array elements using Eq. (5.10). For simplicity, a square patch element is chosen as unit cell whose dimension is the same as the one used in [43]. The element grid size is 11.19 mm, 0.5λ at operating frequency 13.4 GHz. The patch size L varies from 3 to 10 mm. The substrate (RT Duroid 5880) has a thickness of 0.787 mm and a dielectric constant 2.20. Figure 5.3 shows the model of one unit cell plotted in the High Frequency Structure Simulator (HFSS). Simulated using HFSS with periodic boundary conditions, the element reflection phase ranges from ´179˝ to 143˝ , as shown in the solid curve in Figure 5.4.

5.2 Study of Quasi-periodic Effect

a Phase

171

b Magnitude

Figure 5.5 Calculation of element reflection phase and magnitude under oblique incidence. The labels and notations in the figures are the same as in Figure 5.4.

We can also apply Eq. (5.10) to calculate the reflection characteristics of this unit cell. The procedure includes the following steps: 1.

2. 3. 4.

Simulate the unit cell using a full-wave simulator. Set the observation plane to be 0.1 mm above the patch (shown in Figure 5.2), and compute the electric field on the observation plane. Export the electric field data, denoted as Et . Remove the element and keep the observation plane unchanged, simulate the unitcell and export the electric field on the observation plane, denoted as Ei . Calculate the scattered field by subtracting Ei from Et , denoted as Es . Apply Eq. (5.10) to calculate the reflection coefficients S11 of the element using Ei and Es .

Figure 5.4 compares the reflection phase and magnitude under a normal incident plane wave computed using both HFSS (solid lines) and the above procedure (dotted lines). Figure 5.5 does the same comparison under an oblique incident plane wave with an incident angle of θ = 30˝ and φ = 0˝ . From both these experiments, we can observe that the results computed using the two methods agree with each other well. They help us to validate the process of reflection phase calculation using Eq. (5.10). We can also apply the above procedure to calculate element reflection phase in a practical array. Figure 5.6a shows an example of calculating the reflection phase of the central element in a finite size array. The elements are the same as the previous examples. Radiation boundary condition is applied to truncate the air region outside the antenna. Figure 5.6b shows that the phase curves of the central element are very close to that of infinite periodic array when the array size is larger than 5 ˆ 5, i.e., unit cells in finite arrays of identical elements perform similarly to the elements with infinite periodic boundary. This result is consistent with our common sense and further validates the procedure of calculating reflection phase of elements in an array. In the rest of this chapter, we will use this procedure to study the influence of quasi-periodic effect on the performance of reflectarray antennas.

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Analysis and Modeling of Quasi-periodic Structures

150

infinite array 5´5 array 9´9 array

100

Phase (deg)

50 0 –50 –100 –150 –200 3

4

5

6

7

8

9

10

Patch size (mm)

a

b

Figure 5.6 Calculation of element reflection phase in a finite array: a plot of a 5 ˆ 5 array in

which the pale square indicates the region of electric field to calculate the element phase and the dash lines indicate the radiation boundary; b comparison of reflection phase of central element for 5 ˆ 5, 9 ˆ 9, and infinite array.

Figure 5.7 One-dimensional quasi-periodic array of square patch elements. © 2018 IEEE.

Reprinted, with permission, from Tong Liu 2017.

5.2.2

Study of Quasi-periodic Effect Quasi-periodic Effect in 1D Arrays To study the quasi-periodic effect, we first conduct numerical experiments on a 1-D quasi-periodic array. The array we study in this subsection is finite in one direction and infinitely periodic in the other direction, as shown in Figure 5.7. The array elements are the same as the one used in previous examples. This 1-D array contains nine squarepatch elements, with radiation boundary condition set along the x-axis direction, and periodic boundary condition set along the y-axis direction. From the study in the previous section, we know that the reflection phase of the central element in a nine-element array is close to the one under periodic boundary conditions. In the numerical experiment, we vary the patch size of the central element and the other eight elements independently, then calculate the reflection phase of the central element. The difference in the element phase compared with the one under periodic assumption is shown in Figure 5.8. In this figure, the vertical axis represents the patch size of the central element, and the horizontal axis represents the patch size of the surrounded elements. The color of each pixel in the figure represents the difference between the practical element phase and the one computed under periodic boundary condition. We observe from the figure that (1) central elements with resonant neighbors (patch size

5.2 Study of Quasi-periodic Effect

173

Figure 5.8 Difference between practical reflection phase of the central element and the one

computed under periodic boundary condition. © 2018 IEEE. Reprinted, with permission, from Tong Liu 2018. A black and white version of this figure will appear in some formats. For the color version, please refer to the plate section.

Figure 5.9 The geometry of a 665-element center-feed reflectarray antenna. © 2018 IEEE.

Reprinted, with permission, from Tong Liu 2019.

near 6.7 mm) have larger phase difference as seen from horizontal rows and (2) central elements in resonance (patch size near 6.7 mm) suffer larger phase deviation as seen from vertical columns. These two observations show that the phase error introduced by quasi-periodic effect is more prominent among the resonant elements, i.e., they are more sensitive to quasi-periodic effects.

Reflection Phase Analysis on Reflectarray Antennas Using the procedure in the previous subsection, we can also analyze the phase error in practical reflectarray antennas. For this purpose, we design a reflectarray antenna using the square patch element discussed in the previous sections. The reflecting surface is illuminated by a corrugated horn antenna with gain about 14 dB, forming a symmetric center-fed system, as shown in Figure 5.9. The designed main beam direction is at 0˝ .

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Analysis and Modeling of Quasi-periodic Structures

a

b

Figure 5.10 Phase analysis of the reflectarray antenna shown in Figure 5.9: a designed phase

distribution; b deviation of practical phase from designed values. © 2018 IEEE. Reprinted, with permission, from Tong Liu 2020.

The diameter of the array is about 15 wavelengths, and the operating frequency is 13.4 GHz. The optimal ratio between focal length and diameter (F/D) is 0.75 in this case. With these parameters, the required reflection phase distribution can be calculated as   (5.11) φmn = k ¨ rf mn ´ uˆ 0 ¨ rmn + φ , where φmn is the required reflection phase of the mnth element in the array, k is the wave vector, rf mn is the distance between the feed center and the mnth element, uˆ 0 is the unit vector of the main beam direction and rmn is the position vector from the array center to the mnth element. φ is a constant phase added to every element. Figure 5.10a shows the designed reflection phase distribution. We can build up an antenna using square patch elements with their phase curve in Figure 5.4. This curve maps the reflection phase to an element with certain patch size. After the antenna is designed, we can also calculate the practical reflection phase of elements in the antenna. These phase values reveal the actual performance of each element. Figure 5.10b shows the phase deviation between the practical and designed value. It is observed that significant phase difference occurs on the strong resonant elements, whose reflection phase is around zero degree. This observation agrees with our previous analysis on the 1-D quasi-periodic array. It also gives us a hint on improving the antenna performance by correcting this phase deviation.

5.2.3

Phase Adjustment in Reflectarray Antennas Based on the above analysis, we further investigate two methods to improve the performance of reflectarray antennas by reducing phase deviation from designed values caused by neglecting the quasi-periodic effect.

Element Phase Correction Using full-wave simulation, we can calculate the deviation of the practical element phase from the designed value. Furthermore, we can also adjust the element phase

5.2 Study of Quasi-periodic Effect

175

Table 5.1 Performance comparison of square patch reflectarrays Array Original Revised

Gain (dB)

Aperture efficiency

SLL (dB)

28.8 30.0

34.2% 45.0%

´14.2 ´15.7

Figure 5.11 Phase deviation in the updated design of the reflectarray antenna. © 2018 IEEE. Reprinted, with permission, from Tong Liu 2021.

based on full-wave simulation to reduce the phase deviation and improve the antenna performance. The formula for phase adjustment can be written as φnew = φ0 ´ γ 0 ,

(5.12)

where φnew is the corrected element phase, φ0 is the original designed phase, 0 is the phase deviation of the element as shown in Figure 5.10 and γ is an empirical parameter that is set as 0.5 in this case. After this adjustment, we can simulate the new reflectarray and check the phase deviation. Figure 5.11 shows the distribution of phase deviation in the new antenna. We can observe that the phase deviation is significantly reduced. The average phase deviation reduces from 96˝ to 44˝ . This phase adjustment scheme also improves the performance of the reflectarray antenna. Figure 5.12 compares antenna radiation pattern before and after the phase adjustment. The antenna gain increases by 1.2 dB, the aperture efficiency was improved by about 10.7%, and the side lobe level is reduced by 1.5 dB, as summarized in Table 5.1. This example clearly shows that the element phase correction scheme can improve the antenna performance.

Adjustment of the Global Reference Phase From the analysis of phase deviation in quasi-periodic arrays, we observe that the reflection phase of resonating elements is more sensitive to its surrounding elements. Therefore, in a practical reflectarray, these elements may generate larger phase error compared with the designed phase. This observation suggests that reducing the number of resonating elements may reduce the average phase error in the entire array. It can be achieved by adjusting the global reference phase in Eq. (5.11).

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Analysis and Modeling of Quasi-periodic Structures

Figure 5.12 Comparison of radiation pattern of the square patch array, before and after the phase

correction.

a ´145˝

b ´90˝

c 0˝

d ´145˝

e ´90˝

f 0˝

Figure 5.13 Design phase distribution (upper row) and practical phase deviation (lower row) for different global reference phase φ. A black and white version of this figure will appear in some formats. For the color version, please refer to the plate section.

According to array theory, adding a constant phase to all the elements has no effect to the performance of antenna array. However, the global reference phase φ changes the absolute phase of an element, hence modifying the position and location of the resonating elements impacts the overall performance of the reflectarray antenna. Figure 5.13 shows a comparison of the phase deviation on the 665-element reflectarray antenna with different global reference phases. Figure 5.14 compares radiation pattern of these three antennas, and their performance is further tabulated in Table 5.2. We observe that the distribution of phase deviation changes with the global reference phase and large

5.2 Study of Quasi-periodic Effect

177

Table 5.2 Performance comparison of square patch reflectarrays with different global reference phase φ (degree)

´145

´90

´30

0

75

140

Gain (dB) Aperture efficiency (%)

28.4 31.2

28.8 34.2

29.9 44.0

29.9 44.0

29.8 43.0

29.0 35.8

Figure 5.14 Comparison of radiation patterns of square patch array for different global reference

phase φ.

phase difference occurs on and near the resonant elements, deteriorating the antenna performance. We can further study the correlation between the phase deviation and the global reference phase as shown in Figure 5.15. In this figure, we define the weighted phase error as M N 1 ÿ ÿ |φmn ´ φ0,mn |Amn , (5.13) Ew = MN m=1 n=1 where φmn and φ0,mn are the practical and designed reflection phase of Element (m,n). M and N represent the number of rows and columns in the array, respectively. The weighting factor Amn represents the normalized extraction amplitude of Element (m,n) on the array aperture and is used to take into account the influence of element location. We can further observe from Figure 5.15 that the global reference phase will affect the weighted averaged phase error, hence the performance of the antenna. Therefore, optimizing the global reference phase could possibly improve the performance of the antenna. The above analysis is based on results from full-wave simulation. In the design process of reflectarray antennas, we choose the following formula to approximate the error caused by quasi-periodic effect as Qe =

M N 1 ÿ ÿ S(m,n)Amn , MN m=1 n=1

(5.14)

where M and N represent the number of rows and columns in the array. Amn is a weighting factor proportional to the normalized extraction amplitude of Element (m,n)

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Analysis and Modeling of Quasi-periodic Structures

Figure 5.15 Correlation between averaged phase error and the global reference phase φ to the antenna gain.

on the array aperture. S(m,n) is an indicator whether an element is close to resonance and can be defined as " 1, |φmn | ă a , S(m,n) = (5.15) 0, else . Here a is a threshold number. In our following analysis, we consider patch size ranges from 6.5 to 6.95 mm as resonating elements, and a is 40˝ . This formula only requires the design phase of each element and does not ask for full-wave simulation. Therefore, we can minimize this value by adjusting the global reference phase during the design process. A smaller Qe will indicate better antenna performance. Figure 5.16 shows an example of the antenna used in this section. As we can see, the weighted number of resonant elements is correlated with the antenna gain while adjusting the global reference phase φ. Therefore, we can search for the best φ to reduce Qe during the design process to achieve a better performance of the antenna without running full-wave simulation. In our study of global reference phase, we also notice that φ = 0 usually leads to a small weighted phase error and gives a good antenna performance. This is consistent with our previous experience that φ = 0 can be a good choice especially for elements with less resonance. The reason for this is still under investigation. However, we also notice that φ = 0 may not always be a good choice all the time [44,45]. Therefore, it will be necessary to check the weighted resonant element number when designing a reflectarray antenna. Moreover, this process is inexpensive to compute.

Optimization of Antenna Performance Combining the two schemes discussed in the previous section, we can update the design workflow of reflectarray antennas as shown in Figure 5.17. In addition to the traditional steps including element design, calculation of array configuration, full-wave simulation, fabrication and measurements, we can further correct element phase by adjustment of global reference phase φ and element phase correction. The former approach does not require full-wave simulation and hence can be carried out together with calculation of

5.2 Study of Quasi-periodic Effect

179

Table 5.3 Comparison of antenna gain and aperture efficiency for elements with strong resonance. Design Original Optimized

Gain (dB)

Aperture efficiency (%)

SLL (dB)

28.8 30.7

34.2 53.0

–13.1 –18.9

Figure 5.16 Correlation between weighted number of resonant elements and practical gain with different global reference phase φ.

Figure 5.17 Design workflow of a reflectarray antenna.

array configuration, while the latter technique will help to further improve the antenna performance based on the result of full-wave simulation. A numerical experiment is conducted on the 665-element reflectarray made up of square patches. Both of the two techniques are used in the design process. Compared with the original design procedure, in which φ is set randomly to ´145˝ , the gain of the antenna increases about 1.9 dB, and the aperture efficiency improves about 19%, as shown in Table 5.3. A comparison of the radiation pattern is shown in Figure 5.18. We can observe that the new design has lower sidelobe level compared with the original

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Analysis and Modeling of Quasi-periodic Structures

Table 5.4 Comparison of antenna gain and aperture efficiency for elements with low resonance. Design Original Optimized

Gain (dB)

Aperture efficiency (%)

SLL (dB)

31.1 31.8

58.0 68.2

–17.4 –19.9

Figure 5.18 Comparison of radiation pattern before and after the phase adjustment.

Figure 5.19 Comparison of radiation pattern before and after the phase adjustment using elements with thicker substrate.

design. This result shows that for high Q elements such as the one in this chapter, the quasi-periodic effect can be strong, and these techniques can significantly improve the antenna performance.

5.3 Full-Wave Modeling of Quasi-periodic Structures Using Reduced Basis Method

181

Another design of the same antenna using elements with less resonance is also tested. Here we increase the thickness of the substrate to 1.58 mm. Therefore, the resonant elements will have a lower Q factor. Using the above workflow, we can improve the antenna performance as shown in Table 5.4 and Figure 5.19. The improvements are not as significant as the previous design because the quasi-periodic effect is mitigated with the increase of substrate thickness. But the result still shows that reducing the error due to quasi-periodic effect could improve the performance of the reflectarray antenna.

5.3

Full-Wave Modeling of Quasi-periodic Structures Using Reduced Basis Method

5.3.1

Formulation EFIE Formulation for Quasi-periodic Arrays In this study, we applied the Electric Field Integral Equation (EFIE) to model metallic quasi-periodic arrays. EFIE can be written as [46] η0 nˆ ˆ L(J) = ´nˆ ˆ Einc,

(5.16)

where L(X) = j k

ż

g(r,r1 )XdS 1 ´ S1

j k

ż S1

∇g(r,r1 )∇ 1 ¨ XdS 1

(5.17)

and e´j k |r´r | . 4π |r ´ r1 | 1

g(r,r1 ) =

(5.18)

In the above equation, r and r1 represent the observation point and the source point, respectively; k and η0 are the wave number and wave impedance in free space, and g(r,r1 ) is the Green’s function acting as the kernel function in the integral. Using MoM and the Galerkin testing procedure, we can derive a matrix equation of the discrete current coefficients. Given N0 basis functions tfi ui=1,...,N0 , the entities of impedance matrix zmn are [39] zmn = xfm,η0 L,fn y ż ż g(r,r1 )fn (r1 )dS 1 = j kη0 fm (r) S S1 ż ż j η0 ´ ∇ ¨ fm (r) g(r,r1 )∇ 1 ¨ fn (r1 )dS 1 . 1 k S S

(5.19)

For a quasi-periodic array as shown in Figure 5.1, the entire impedance matrix equation is shown as Eq. (5.20). In this equation, Zmn is the mutual impedance matrix

182

Analysis and Modeling of Quasi-periodic Structures

between the mth and nth elements. Entries in Jn are the current coefficients on the nth element. Entries in En are the incident electric field coefficients on the nth element: fi » Z11 Z12 . . . Z1N » J1 fi » E fi — Z21 Z22 . . . Z2N ffi — ffi — 1 ffi ffi J2 ffi — E2 ffi — (5.20) — . .. .. ffi — . – fl = – ... fl . . . – . . . . fl ... JN EN ZN1 ZN2 . . . ZN N

Reduced Basis Method (RBM) It has been pointed out in [47] that the reduced basis method assumes the existence of a low-dimensional basis set representing the solutions of a range of control parameters. It is suitable when the solutions change continuously with control parameters. RBM contains offline and online processes. The offline process will construct the reduced basis set, and the online process will solve the matrix equation of current spanned by the reduced basis.

Offline Process Given a sample of a parameter space, we can use either singular value decomposition (SVD) or greedy algorithm referred in Algorithm 1 to set up the space WRB spanned by Nr reduced basis. As the similarity in the elements leads to similar currents on the array elements, Nr can be much smaller than the original number of unknowns N0 on each element. The subscript “RB” stands for reduced basis. These reduced basis functions are computed from solutions of Eq. (5.16) for one element with different parameters [47]. In order to take the coupling into account, we extract reduced basis from the current by many incident wave excitations, as the weak couplings can be approximated by a few superpositions of planar waves. A straightforward approach is to construct reduced-basis functions from current solutions of wave interaction with a single element of different sizes, as shown in Figure 5.20a [42,48]. This reduced basis set is a rough approximation of the element currents in quasi-periodic arrays, because mutual coupling among different kinds of elements are not considered while setting up the reduced basis set. In this study, the current samples are solutions of small elemental arrays, as shown in Figure 5.20b. These currents consist of effects of mutual coupling and are closer to the real current in the array. Therefore this new reduced basis can represent the current on different kinds of elements more accurately. This scheme is similar to the ones used in [49–51]. In the new reduced space, the element current is sparser, i.e., we need less of a reduced basis to represent the element current. Figure 5.20b shows one element with eight surrounding elements as an elemental array, and the reduced basis is constructed from sample current Jsurf 1 , Jsurf 2 and Jsurf 3 in this 3 ˆ 3 subarray. We can also use a 5 ˆ 5 elemental array for the same computation. In general, a larger elemental array can provide more information about mutual coupling. The most optimal elemental array would be the entire array, but it will be too

5.3 Full-Wave Modeling of Quasi-periodic Structures Using Reduced Basis Method

183

Algorithm 1 Reduced Basis Method: Greedy Algorithm for the Offline Process Require: The sampling points of μ in Ensure: Reduced basis WRB for n = 1 to Nr do for all μ in do the relative approximation error n (μ) is n (μ) = }J(μ) ´ Jn (μ)}2 where J(μ) is the solution of (5.16) for a given parameter set μ, Jn (μ) is the projection of J(μ) on the reduced basis space WRB,n´1 spanned by n ´ 1 basis. end for choose μn = arg sup |n (μ)| μP Then, WRB,n = spantWRB,n´1,J(μn )u. end for Orthogonalize WRB,Nr to get WRB .

a Conventional reduced basis, WRB is extracted from the set of tJsurf u in the offline process

b Reduced basis from elemental array, WRB is extracted from the set of tJsurf 1,Jsurf 2,Jsurf 3 u in the offline process

Figure 5.20 Construction of reduced basis (RB) space by two methods. © 2018 IEEE. Reprinted, with permission, from Xunwang Dang 2017.

costly to compute, and we will lose the computational efficiency of this method. As a numerical validation, we can plot the normalized singular values of the current on a single element or an elemental array in Figure 5.22a. The singular values of current decrease in a similar fashion for 3 ˆ 3 and 5 ˆ 5 elemental arrays above 10´4 . We can choose the size of elemental array by the singular value distribution. Then we can

Analysis and Modeling of Quasi-periodic Structures

Figure 5.21 Illustration of the 10 ˆ 10 array. © 2018 IEEE. Reprinted, with permission, from

Xunwang Dang 2018.

100

100

3´3 elemental array 5´5 elemental array RB from one element

–2

10

Projection Error e

Normalized Singular Value

184

10–6 10–6 10–8 –10

10

50

Order of Singular Value

a Singular values

10–2 10–3 10–4

100 150 200 250 300

3´3 elemental array 5´5 elemental array RB from one element

10–1

50 50 100 150 200 250 300 Number of Reduced Basis

b Convergence of current projection using different reduced basis

Figure 5.22 The singular value distribution of current and the current projection error. © 2018 IEEE. Reprinted, with permission, from Xunwang Dang 2019.

project the current of a quasi-periodic array in Figure 5.21 onto different reduced basis sets WRB and compare the projection error , as shown in Figure 5.22b. The reduced basis sets are extracted from 1 ˆ 1, 3 ˆ 3 and 5 ˆ 5 elemental arrays, respectively. The projection process can be written as Eq. (5.21), and the total projection error  of all the elements in the array is indicated as Eq. (5.22): H J˜ n = WRB WRB Jn

=

2 ˜ n=1 ||Jn ´ Jn || . řN 2 n=1 ||Jn ||

(5.21)

řN

(5.22)

In Eq. (5.21), the superscript H represents the Hermitian of a matrix. Figure 5.22b shows the error convergence with the number of reduced basis functions. We observe that the reduced basis from elemental arrays converges faster compared with conventional reduced basis from single element. And the reduced basis from the 3ˆ3 elemental array has a similar performance with the 5 ˆ 5 one.

5.3 Full-Wave Modeling of Quasi-periodic Structures Using Reduced Basis Method

185

Online Process Using RBM, Eq. (5.20) can be converted into a reduced matrix equation with N Nr unknowns as » rb rb rb fi Z11 Z12 . . . Z1N »Jrb fi »Erb fi 1 1 — rb rb rb ffi — Z21 Z22 . . . Z2N ffi —Jrb ffi —Erb ffi 2 — ffi— ffi — 2 ffi (5.23) .. .. ffi – ... fl = – ... fl . — .. .. . – . . . fl rb JN Erb rb rb rb N ZN1 ZN2 . . . ZN N rb

In the above equation, each subblock Zmn indicates the interaction between the mth and nth elements under the reduced basis WRB , and Erb n represents the excitation vector. They can be written as rb

H

Zmn = WRB Zmn WRB H

Erb n = WRB En .

(5.24)

(5.25)

For a new parameter value μ, we can compose the reduced matrix using EIM in Section 5.3.1 to avoid the construction of the full impedance matrix Zmn . It reduces the memory 2 N 2 ). cost of matrix storage from O(N02 N 2 ) to O(Nrb

Empirical Interpolation Method (EIM) In a quasi-periodic array, the geometry of the element, such as the element’s shape, orientation or position, often changes with certain control parameters. Hence, we cannot reduce the entire impedance matrix Z into one or a few subimpedance matrices because the value of the Green’s function and the domain of integration in the impedance matrices will both change across elements. For example, in reflectarray, we use different element size to achieve a 360˝ phase range, and μ can be the size a of an element. Hence, Zmn in Eq. (5.20) will be a function of μ of the mth and nth elements and can be written as Zmn (μm,μn ). For large arrays, direct construction of all Zmn (μm,μn ) can be very expensive. In order to acclerate the construction of impedance matrices, we apply the empirical interpolation method (EIM) to interpolate the Green’s function g(x,x1 ;μ) on some known points and save the matrix assembling time of Zmn (μm,μn ), using the fast matrix assembling method shown in Section 5.3.1.

EIM for Green’s Function The details of EIM can be found in [47]. A function g(x,μ) for x P S, which depends on parameter μ P D, can be approximated in an affine way with EIM as

g(x,μ) «

Mg ÿ n=1

n (μ)qn (x) ,

(5.26)

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Analysis and Modeling of Quasi-periodic Structures

where qn (x) is the interpolation function, which is a linear combination of g(x,μk ) as n ÿ

qn (x) =

ank g(x,μk ).

(5.27)

k=1

n (μ) can be computed by solving Eq. (5.28) as Mg ÿ

qj (xk )j (μ) = g(xk,μ),k = 1,...,Mg .

(5.28)

j =1

The above equation consists of Mg linear equations using Mg interpolation points xk P TM and interpolation functions qj (μ). Here TM = txi ui=1...Mg represents the set of interpolation points. Both TM and qn (x) can be computed using Algorithm 2 [47]. We can also write g(x,μ) in a more direct form in Eq. (5.29), as a linear combination of tg(x,μk )u: g(x,μ) «

Mg ÿ

n (μ)

n=1

=

Mg ÿ

=

Mg ÿ

ank g(x,μk )

k=1

»

Mg ÿ

– k=1

Mg ÿ

fi n (μ)ank fl g(x,μk )

(5.29)

n=1

αk (μ)gk (x,μk ).

k=1

In modeling quasi-periodic arrays, the setup of EIM requires samples from the twodimensional parameter space of (x,x1 ). Note that the Green’s function depends only on the distance between the source and observer, therefore the setup time of EIM can be saved by converting the sampling variables to g(|x ´ x1 |,μ) instead of g(x,x1,μ). For example, Ganesh et al. [52] utilized the translational invariant property of Green’s function to reduce g(x,x1,μ) to g(x,μ). Here we use another parameter sampling scheme to accelerate the setup of EIM in modeling 2D quasi-periodic arrays. As shown in Figure 5.23, w refers to the side length of the reference patch, dimensionless variables a1 and a2 represent the scale factor of the patches (assume a1 ě a2 ); dx and dy represent the distance between patch centers along the x- and y-axes, respectively. Here we assume dx = dy for simplicity; x and x1 represent coordinates of points on Patch 1 and Patch 2, respectively. The setup of EIM requires samples from the parameter space of (x,x1 ). As the Green’s function g(x,x1 ) depends on |x ´ x1 |, Algorithm 2 requires sampling all the |x ´ x1 | between the two patches. The maximum and minimum distance of |x ´ x1 | are d     a1 + a2 2 a1 + a2 2 1 + dy + w (5.34) dx + w |x ´ x |max = 2 2 and

5.3 Full-Wave Modeling of Quasi-periodic Structures Using Reduced Basis Method

187

Algorithm 2 Empirical Interpolation Method Require: function g(x,μ)to interpolate, point set X, parameter set samples of D, order of EIM Mg Ensure: interpolation point set TM = txi ui=1...Mg , interpolation functions QM = tqi (x)ui=1...Mg for p = 1 to Mg do for all μ in do calculate j (μ) from p interpolation point p ÿ

qj (xk ) j (μ) = g(xk,μ),k = 1,...,p

(5.30)

j =1

where qj (xk ) is in the k-th row and j -th column Define the error function e(μ,¨) e(¨,μ) = g(x,μ) ´

m ÿ

(5.31)

j (μ)qj (x)

j =1

end for Search for the μm as following μm = arg min }e(¨,μ)}L8 (5.32) μP then get the m + 1’s interpolation function and interpolation point as (5.33) qm+1 (x) =

e(¨,μm ) }e(¨,μm )}L8

(5.33)

xm+1 = arg max |e(¨,μm )| x PX

then we can update the interpolation points and functions as Tm+1 spantTm,xm+1 u, and Qm+1 = spantQm,qm+1 (x)u end for d  |x ´ x |min = 1

a1 + a2 dx ´ w 2

2



a1 + a2 + dy ´ w 2

=

2 .

(5.35)

In this implementation, we fix x1 at the farthest corner P as in Figure 5.23 and only sample x. To guarantee complete coverage of all possible |x ´ x1 |, we extend the large patch (Patch 1) to 2a1 w as shown in the dotted line in Figure 5.23. The minimum value |x ´ x1 |min can then be reached because the minimum distance between points in Extended Patch 1 and point P is larger or equal to |x ´ x1 |min under the assumption

188

Analysis and Modeling of Quasi-periodic Structures

Figure 5.23 Illustration of reducing g(x,x1,a) to g(x,a;x0 ) by replicating the larger patch.

© 2018 IEEE. Reprinted, with permission, from Xunwang Dang 2020.

a1 ě a2 , as shown in Eq. (5.36):   3a1 ´ a2 2 + dy ´ w 2 d 2    a1 + a2 a1 + a2 2 ď + dy ´ w dx ´ w 2 2

d 

dx ´ w

3a1 ´ a2 2

2

(5.36)

= |x ´ x1 |min . This scheme reduces two-dimensional sampling of both x and x1 to one-dimensional sampling of x. Hence it significantly saves the time in setting up EIM.

Fast Matrix Assembling As the impedance matrix relies on linear operations of Green’s function, interpolation schemes of Green’s function can also be used to approximate the impedance matrix. Therefore, an impedance matrix representing interactions between Elements m and n with parameters μm and μn can be approximated as Eq. (5.37), where Zmn,p (μ1,p,μ2,p ) is the interpolating impedance matrix with parameters μ1,p and μ2,p between Elements m and n. This equation indicates that we only need prepare Mg matrices Zp (μ1,p,μ2,p ) in the offline process, then we can construct any new Zmn (μm,μn ) online. Therefore, the entire impedance matrix Z can be efficiently constructed without computing numerical integration of every matrix element. The Mg 2 N ) that matrices Zp (μ1,p,μ2,p ) in the offline process have a memory cost of O(Mg Nrb increases linearly with N:

Zmn (μm,μn ) =

Mg ÿ p=1

αpZ (μm,μn )Zmn,p (μ1,p,μ2,p ).

(5.37)

5.3 Full-Wave Modeling of Quasi-periodic Structures Using Reduced Basis Method

189

The excitation vector on the nth element En (μn ) in the matrix equation can also be approximated in a similar form using Me interpolation functions as En (μn ) =

Me ÿ

αpE (μn )En,p (μp ) ,

(5.38)

p=1

where En,p (μp ) is the excitation on the nth element for the pth interpolation parameter μp . It represents the pth interpolating vector to construct En (μn ). Furthermore, the interpolation matrices and vectors can be projected to the reduced basis space in the rb offline process. After setting up Zmn and Erb m in the online process, the entire reduced matrix equation (5.23) is then constructed. Using RBM and EIM, we avoid the direct numerical integration in method of moments, and the matrix equation can be set up efficiently.

5.3.2

Numerical Results Arrays with Conventional Reduced Basis First we simulate the scattering of a single square patch expanded from a reference patch with side length 1 meter. The parameter μ here is the ratio a between the side length of the target patch and that of the reference patch. Assume the direction of the incident wave is fixed to θ inc = 45˝,φ inc = 45˝ and polarized along θ direction. The parameterized Green’s function and excitation are e´j ka |xˆ ´xˆ | 1

g(x,x ,a) = 1

4π a|ˆx ´ xˆ 1 |

E(x,a) = e´j k¨(a xˆ ),

(5.39) (5.40)

where x and x1 are points on the target patch, xˆ and xˆ 1 are points on the reference patch, hence x = a xˆ , x1 = a xˆ 1 ; g(x,x1,a) is the Green’s function, and E(x,a) represents planar wave excitation; xˆ and xˆ 1 are position vectors on the reference patch with x and y components in the range of [´0.5 ,0.5] meters; and g(x,x1,a) is only related with |x ´ x1 |, so the value of g(x,x1,a) is equivalent to g(x,a;x0 ), where x0 is a fixed point on the corner of the patch. This can reduce the traversing time of g(x,x1,a). We take Mg = 10, Me = 10 for EIM. The error of the interpolation is shown in Figure 5.24b. The original mesh has 328 unknowns. The patch size is taken as a = 1.456, for example. When we increase the number of reduced basis, the error of electric current on the patch, the radar cross section (RCS) at φ = 0˝ plane and the theoretically estimated error are shown together in Figure 5.24c. With the increase of reduced basis, these errors all converge, which validates the application of RBM to MoM on a single patch. The time to assemble the original MoM matrix is 0.54 s, while it is 0.015 s using RBM. This example shows RBM can reduce the dimension of the problem to solve. The computational complexity of the reduced basis method is O(M 2 ), where M is the number of reduced basis functions. However, the value of M and the ratio of acceleration (N{M) is problem dependent, where N is the number of

190

Analysis and Modeling of Quasi-periodic Structures

a Mesh of the reference patch

b Error of using EIM to approximate function Eq. (5.39) and Eq. (5.40)

c Error convergence with the increased number of reduced basis

Figure 5.24 Error convergence of EIM and RBM. © 2018 IEEE. Reprinted, with permission, from Xunwang Dang 2021.

Figure 5.25 Illustration of a 20-element array. © 2018 IEEE. Reprinted, with permission, from Xunwang Dang 2022.

original unknowns. This is because the reduced basis is from a set of existing numerical solutions. The more similarities in the current solution, the less number of reduced basis M is reduced and hence a larger acceleration ratio. We can also simulate a quasi-periodic square patch array. We take a 1D quasi-periodic patch array as an example, as shown in Figure 5.25. Sizes of the square patches range from 1 to 2 m. The distance between two neighboring patches is 3 m. The frequency of simulation is at 100 MHz. The reference patch has a length of 1 m, and the array is aligned along the x-axis. The angle of incident wave is θ inc = 45˝ , φ inc = 45˝ . We compute two examples of 20-element and 100-element arrays, respectively. The RCS in the φ = 0˝ plane computed by MoM, RBM-MoM and commercial simulation software CST are shown in Figures 5.26a and 5.26b for the two cases, respectively. These results match well with each other. The CPU times of MoM and RBM are shown in Table 5.5. From these results we can observe that the computation using RBM is significantly faster than using MoM directly.

5.3 Full-Wave Modeling of Quasi-periodic Structures Using Reduced Basis Method

191

Table 5.5 Comparison of MoM and RBM in CPU time and unknown numbers Number of array elements

20

100

Method

MoM

RBM

MoM

RBM

Time to fill the matrix (sec) Time to solve the matrix (sec) Unknown number

58.84 10.60 6560

0.3317 0.0038 280

1528 2628 32800

5.269 0.489 1400

a Case 1 with 20 elements

b Case 2 with 100 elements

Figure 5.26 Simulation results for the two 1D quasi-periodic arrays. © 2018 IEEE. Reprinted, with permission, from Xunwang Dang 2023.

Next, we take a 10 ˆ 10 2D quasi-periodic patch array shown in Figure 5.21 as an example. Size of the square patches ranges from 1 to 2 m, and the distance between two neighboring patch centers is 3 m. There are 328 unknowns on each element. In the offline process, we sample 31 points uniformly from 1 to 2 m under 900 incident wave directions. Each simulation costs 0.7 s, and it can be done in a highly parallel fashion. We approximate the impedance matrix using EIM with 961 (μ1,μ2 ) interpolating points. The time for setting up and solving the matrix equation using traditional MoM is 3373 s, while it only takes 80 s in the online process using RBM with 50 reduced basis functions. The radar cross section (RCS) computed by MoM, RBM and commercial simulation software CST is shown in Figure 5.27. These results match well with each other.

Arrays with Reduced Basis from Elemental Array Figure 5.22b shows the convergence of projection error as the number of reduced basis functions increases on a single patch. The convergence of the current solution on the quasi-periodic array with the number of reduced basis functions is shown in Figure 5.28. The solid line shows the convergence using conventional reduced basis from single element, and the asterisk marker shows the convergence using reduced basis from the elemental array. It can be observed that the reduced basis from an elemental array performs better than the conventional reduced basis. Therefore, we can use a fewer

192

Analysis and Modeling of Quasi-periodic Structures

a RCS on φ = 0˝ plane

b RCS on φ = 90˝ plane

Figure 5.27 RCS for the 10 ˆ 10 patch array. © 2018 IEEE. Reprinted, with permission, from Xunwang Dang 2024.

Figure 5.28 RBM with different reduced basis. © 2018 IEEE. Reprinted, with permission, from Xunwang Dang 2025.

number of reduced basis functions from the elemental array, thus less computation time, to get the same level of accuracy as the reduced basis from a single element.

Simulation of 20 ˆ 20 Malta Cross Array We also compute a larger 20 ˆ 20 array with Malta cross elements as shown in Figure 5.29a using reduced basis from elemental array. Size of the elements ranges from 1 to 2 m, and the distance between two neighboring patch centers is 3 m. There are 1128 unknowns on each element, and it is hard to solve using direct MoM when the array is large. The reference results are computed by Multi-Level Fast Multipole Algorithm (MLFMA) [53] with diagonal preconditioner. The comparison of RCS on φ = 45˝ and φ = 135˝ plane between RBM and MLFMA are shown in Figures 5.29b and 5.29c. These results agree with each other well. RBM costs about 9000 s, while MLFMA costs about 4 ˆ 104 s to reach the same tolerance of the iterative method.

5.4 Summary and Outlook

193

a An array with Malta cross elements

b RCS on φ = 45˝ plane

c RCS on φ = 135˝ plane

Figure 5.29 RCS of the 20 ˆ 20 Malta cross array. © 2018 IEEE. Reprinted, with permission, from Xunwang Dang 2026.

5.4

Summary and Outlook In this work, we studied two problems in designing and modeling quasi-periodic arrays. First, we analyze the deviation of the element phase in a quasi-periodic array from infinite periodic assumption. We observe that elements with strong resonance are more easily affected by their surroundings, and their phase differs more from the design value calculated under periodic boundary conditions. This phenomenon could introduce design error in quasi-periodic arrays, such as reflectarray antennas. Based on this observation, we introduce two techniques: element phase correction and adjustment of global reference phase, to reduce the design error due to quasi-periodic effect and optimize the antenna performance in the design. Numerical experiments show that the performance of the antenna could be improved significantly when elements with strong resonance are used. For elements with less resonance, slight improvements can still be observed in antenna gain, aperture efficiency or side lobe level. This scheme can also be applied to analyze and optimize other quasi-periodic structures, such as transmitarray antennas and metasurfaces. Moreover, we also model electromagnetic wave interaction with quasi-periodic arrays using the reduced basis method. RBM includes the offline and online processes. The offline process constructs the reduced basis and the interpolating matrices. The online

194

Analysis and Modeling of Quasi-periodic Structures

process constructs and solves the matrix equation of reduced basis for quasi-periodic structures in an efficient way. Numerical examples verify the accuracy and efficiency of this method. This method has potential to efficiently model quasi-periodic arrays, such as metasurfaces, reflectarray antennas and nano particle arrays.

Acknowledgements The author would like to thank Mr. Tong Liu for his help in studying the quasi-periodic effect and Mr. Xunwang Dang for his help in full-wave modeling of quasi-periodic arrays using the reduced basis method. He also thanks Prof. Fan Yang and Prof. Shenheng Xu for fruitful discussion and collaboration on this study. They are all from Department of Electronic Engineering, Tsinghua University. He also appreciates the advice on applying RBM to model quasi-periodic arrays from Prof. Weng Cho Chew in the Department of Electrical and Computering Engineering, Purdue University. The authors would like to thank the support from the National Science Foundation of China (61571264) and the National Key R&D Program of China (2018YFC0603604); the Guangzhou Science and Technology Plan (201804010266); the Beijing Innovation Center for Future Chip; and the Research Institute of Tsinghua, Pearl River Delta.

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6

Gap Waveguide Technology Eva Rajo-Iglesias, Zvonimir Sipus, and Ashraf Uz Zaman

6.1

Origin of Gap Waveguide Technology A gap waveguide is a modified parallel-plate waveguide where the bottom metallic plate is replaced by a textured surface with periodic elements, usually metallic pins. The gap waveguiding concept is based on directing the electromagnetic waves in desired directions inside a parallel-plate waveguide. The operation is inspired by soft and EBG surfaces which stop the propagation of the electromagnetic waves along the surface in one or two directions, respectively. The concept of soft and hard surfaces was introduced to the electromagnetic community by Kildal in 1988 [1, 2]. If we consider a wave propagating along a surface, the soft surface is defined in a way that both transverse components of the electric field (tangential and normal to the surface) are zero along the soft surface. In the case of a hard surface, both components have derivatives in the normal direction equal to zero. In principal, the power density at the considered surface is zero for the soft surface and has a maximum for the hard surface. It can be easily shown that soft and hard surfaces act as a perfect electric conductor (PEC) for one polarization and as a perfect magnetic conductor (PMC) for the other. Therefore, a physical model of an ideal soft and hard surface for plane wave incidence is a grid with parallel and infinitely narrow PEC and PMC strips (see Figure 6.1 taken from [3] and [4]). For the plane wave propagating perpendicular to the strips, the PEC/PMC strip surface acts as a soft surface, and for a wave propagating along the strips, the PEC/PMC surface acts as a hard surface (Figure 6.1a). All good conductors are usually treated as a PEC in field analysis, but there are no materials in nature which sufficiently well approximate PMC for microwave frequencies. One way to artificially realize a PMC is to use a quarter-wavelength transformer with a short circuit (PEC shunt) at the end of the transformer, since the PEC shunt is transformed to a PMC which corresponds to an open circuit. In practice, this can be done by using, e.g. corrugations (Figure 6.1b) or a strip-loaded grounded dielectric slab, and these are the two most commonly used realizations of soft and hard surfaces. In 1999, a new type of electromagnetic surface, called mushroom structure, was proposed to act as a high-impedance surface for normal incidence (practical realization of a PMC surface) and as a soft surface isotropically in any direction along the surface [5]. Thus, a new abbreviation was introduced: electromagnetic band gap (EBG) structure [5, 6], taken from the analogy with photonic band gap (PBG) crystals. In principle, a periodic pin structure (also called bed-of-nails) or a mushroom structure (low-profile version

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6.1 Origin of Gap Waveguide Technology

199

(a)

(b) Figure 6.1 (a) The concept of soft and hard surfaces: PEC/PMC strip grid; (b) practical realization

using corrugations. A black and white version of this figure will appear in some formats. For the color version, please refer to the plate section.

of pin structure) is a two-dimensional version of corrugations. The EBG structures are intensively used in the design of low-profile antennas and in the design of antennas with improved radiation characteristic (lower mutual coupling between antenna elements, no diffraction from the edges of the antenna, smaller back-side radiation). In 2007 Valero-Nogueira et al. started to investigate oversized parallel plate waveguides in which the lower plate is realized as a hard surface (e.g. using longitudinal corrugations; see Figure 6.2 and [7, 8, 9]). By this the waves are directed in the oversized parallel-plate waveguide to propagate along the corrugations, i.e. there is no propagation in the orthogonal (to corrugations) lateral direction. Consequently, a simple slotted waveguide array with steering beam possibility can be obtained (Figure 6.2b). The fact that it is possible to direct the waves inside a parallel plate waveguide without placing lateral walls resulted in the new concept proposed by Kildal: gap waveguide technology [10, 11]. The idea behind this is the fact that there is no electromagnetic wave propagation (i.e. no propagating waveguide modes) in a parallel-plate waveguide thinner than λ{4 and with one wall replaced with a high-impedance surface (ideally PMC; see Figure 6.3). The high-impedance surface can be realized in various ways; most commonly, it is realized using a periodic pin (bed-of-nails) structure, like that shown in Figure 6.4a. Figure 6.4b shows a typical dispersion diagram, i.e. the obtained stop band commonly has a bandwidth larger than 2:1. However, in order to be able to guide the EM waves in the desired direction, one needs to introduce a perturbation in

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(a)

(b)

Figure 6.2 (a) Ideal PEC/PMC interpretation of the single-hard-wall parallel-plate waveguide.

© 2018 IEEE. Reprinted, with permission, from Alejandro Valero-Nogueira 2009. (b) Slotted waveguide array with steering beam possibilities. Reprinted with permission from Valero-Nogueira et al. Copyright © 2007 Wiley Periodicals, Inc.

Figure 6.3 The gap waveguide concept.

a structure in a form of a ridge, a groove or a strip to force the waves to propagate in desired directions along the introduced perturbation (see Figure 6.5). When comparing different realizations of gap waveguide transmission lines, the ridge and the groove (Figures 6.5a and 6.5b) have the lowest losses. Depending on the considered component, one would select either the groove technology, which is equivalent to a conventional rectangular waveguide (suitable, e.g., for filters, resonators, leakywave antennas, high power handling capability), or the ridge technology (suitable, e.g., for distribution networks for antenna arrays, implementation of active components). However, sometimes one would like to reduce the cost (and time) of production, even though it causes larger losses to the component. In that case it is possible to select an inverted microstrip transmission line (Figure 6.5c) or to realize the complete component using printed circuit board (PCB) technology (Figure 6.5d). In the first case the designed circuit is printed on a thin dielectric slab; instead of a ground plane, the general periodic bed-of-nails structure (without any guiding structure inclusion) is used. This approach has important advantages in terms of cost since the bed-of-nails can be reused, it is uniform, and the prototyping stage becomes shorter [12, 13]. When comparing with the classical microstrip technology, note that in both cases (Figures 6.5c and 6.5d), most of the electromagnetic field is located in the gap between the PCB and the upper PPW plate, i.e. in the air, and consequently the losses are smaller compared to an equivalent component realized in microstrip technology [14].

6.1 Origin of Gap Waveguide Technology

201

(a)

(b) Figure 6.4 (a) The periodic pin (bed-of-nails) realization of the gap waveguide structure and (b) the corresponding dispersion diagram.

The gap waveguide technology is in particular interesting in the frequency range above 30 GHz since there are manufacturing problems due to the small dimensions of the classical rectangular waveguide-based components. The hollow structure can be manufactured in two parts that are joined together, but then there are mechanical complexities with ensuring good electrical contact in the joints (see e.g. [15]). Even typical production tolerances in plate flatness (and other manufacturing errors) will cause small gaps when these two parts are mounted together. Therefore, there is a need for new types of waveguides or transmission lines operating at high frequencies, in particular above 30 GHz. The developed gap waveguide technology, in particular the properties of components and devices realized in this technology, has ensured that one of the main candidates for future mm-wave components is actually the gap waveguide concept. Recently, a complementary technology to gap waveguide technology has been introduced – a holey glide-symmetric structure (see Figure 6.6). The unit cell of the proposed EBG walls of the parallel-plate waveguide consists of glide-symmetric holes. Like in the gap waveguide components, the presence of the holey structure ensures the presence of the stop band in the lateral walls of a parallel-plate structure. In principle, glidesymmetric holey structures are cheaper and easier to produce compared to the pin

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Figure 6.5 Different realizations of a transmission line embedded into the bed-of-nails structure:

(i) ridge, (ii) groove, (iii) inverted microstrip, (iv) microstrip ridge.

Figure 6.6 Sketch of the groove transmission line realized in glide-symmetric holey technology.

© 2018 IEEE. Reprinted, with permission, from Mahsa Ebrahimpouri 2018.

structure at the same working frequency. However, until now, only the transmission lines with no lateral leakage of EM waves and ultra wideband two-dimensional lenses were proposed [16, 17, 18]. This chapter is organized as follows. Section 6.2 analyzes the basic concepts of gap waveguide technology using approximate boundary conditions, thereby avoiding the complexity of the Floquet-mode expansion. Design of the stop band for parallel plate structures is discussed in Section 6.3; this design is essential as it dictates some of the key performance parameters of the gap waveguide components. The stop band properties are exploited further in Section 6.4, which shows the gap-waveguide packaging solutions which can be used in RF modules. Losses in gap waveguide structures are studied in Section 6.5 since their evaluation is critical when selecting a suitable technology for building microwave and mm-wave components. Finally, Section 6.6 presents different types of antennas designed in gap waveguide technology. Due to the low loss properties of gap waveguides, high-efficiency and high-gain planar antenna arrays can be designed with this technology.

6.2 Approximate Method of Analysis of Parallel-Plate Waveguides

6.2

203

Approximate Method of Analysis of Parallel-Plate Waveguides Containing EBG Surfaces In this section we will illustrate how the Green’s function analysis approach together with the approximate boundary conditions can simplify the analysis of gap waveguide structures, in particular in the first step of the design process, where the basic parameters and properties of the considered structures need to be determined. The rigorous analysis of periodic structures can be performed by expanding the electromagnetic field into Floquet modes and then by using some general numerical approach (e.g. MoM, FEM or FDTD) to determine the fields and/or currents in a unit cell, but this is still very time consuming if the source excites a spectrum of plane waves, such as it does in our Green’s function case (i.e. when the excitation is a point source). However, when the periodic structure is asymptotically taken into account, i.e. if the period of the structure is very small compared to wavelength, we can greatly simplify the analysis by replacing the periodic structure with suitable approximate boundary conditions [19]. This homogenizes the surface and captures the main physical phenomena. The proposed method is suitable for an initial design of the considered structure. The final design will be made using a general EM solver or a specialized solver for analyzing gap waveguide components based, e.g., on the mode matching or moment method (see, e.g., [20]).

6.2.1

Plane Wave Spectral Domain Approach The derivation of the Green’s functions will be obtained in the plane wave spectral domain. The method was previously commonly used in the analysis of microstrip circuits and antennas (see, e.g., [21, 22, 23]), where [23] represents the formulation and algorithm implementation of it used in the present chapter. The plane wave spectral domain is defined by (the coordinate system is defined in Figure 6.7) ż8 ż8 ˜ x ,ky ,z) = E(x,y,z)ej kx x ej ky y dxdy (6.1) E(k ´8 ´8

where ˜ denotes the two-dimensional Fourier transformation with kx and ky the spectral variables, and correspondingly for the H-field. The spatially varying field is obtained by the corresponding inverse transformation. In order to determine the field solutions in the spectral domain (without single separate ridges, grooves or strips), we assume that the

Figure 6.7 Single-ridge type of gap waveguide; the periodic texture is the bed-of-nails structure.

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spectral solutions of the Helmholtz differential equation for Ez and Hz components of the electromagnetic field have the form E˜ z (kx ,ky ,z) = A cos(kz z) + B sin(kz z)

H˜ z (kx ,ky ,z) = C cos(kz z) + D sin(kz z) (6.2)

where bthe exp(´j (kx x + ky y)) variation is understood and suppressed, kz = k02 ´ kx2 ´ ky2 , k0 is the free space wave number, and A, B, C and D are the unknowns which are determined by fulfilling the boundary conditions on each of the two surfaces for the x- and y-components of the fields. It is important to mention that it is sufficient to determine only the z-components of the electromagnetic field, while all the other components are given from these by Eq. (12) in [21].

6.2.2

Homogenization Using Spectral Surface Admittance The general idea is to use some approximate boundary condition replacing the periodic texture in the plane wave spectral domain and to derive the Green’s functions for this simpler waveguide. The approximate boundary condition will for the considered surfaces be conveniently represented by an homogenized surface impedance in the plane wave spectral domain. This will then have an angle of incidence variation (spectral kx and ky variation), through which the depth d of the surface is accounted for, and to be general, we need to allow the surface impedance to be anisotropic (it is isotropic when considering bed-of-nails types of structures, but it is anisotropic for a corrugated surface). Thus, we will finally use the general surface admittance components Y˜xy and Y˜yx defined by (the periodic surface coincides with the xy-plane) Y˜yx =

H˜y , E˜x

H˜x Y˜xy = ´ E˜y

(6.3)

The spectral Green’s functions for a z-directed current source located just below the upper waveguide plate are equal: „ j   ky sin kz z cos kz z 1 sin kz z cos kz z ˜¯¯ H J C1 kx + + C2 ky + C3 (6.4) Gxz = 2 sin kz h cos kz h sin kz h kz sin kz h β Dsw ˜¯¯ H J G yz =

„ j   cos kz z kx sin kz z J˜z sin kz z cos kz z C k k + ´ C ´ C (6.5) 1 y 2 x 3 sin kz h cos kz h sin kz h kz sin kz h β 2 Dsw

where C1 = j kx ky kz (Y˜yx ´ Y˜xy )

(6.6)

C2 = ´β 2 η0 k0 tan(kz h)Y˜xy Y˜yx ´ j kz (kx2 Y˜yx + ky2 Y˜xy )

(6.7)

C3 = j k02 tan(kz h)(kx2 Y˜xy + ky2 Y˜yx ) ´ kz β 2

(6.8)

„ j 1 cot(kz h) (6.9) Dsw = ´Y˜yx (k02 ´kx2 )´Y˜xy (k02 ´ky2 )+j k0 kz η0 tan(kz h)Y˜xy Y˜yx ´ η0

6.2 Approximate Method of Analysis of Parallel-Plate Waveguides

205

The PMC is a special case of the above general surface impedance case, giving Y˜yx = Y˜xy = 0 Dsw =

1 cot(kz h) η0

(6.10) (6.11)

The bed-of-nails is a most-often used surface for realization of gap waveguide components. It is two-dimensional periodic and can therefore exhibit cutoff properties in all directions of wave propagation between the two plates. The topology of the surface is shown in Figure 6.7. The homogenization method in [24] is based on modeling the pin structure as a uniaxial medium whose permittivity may be characterized by the permittivity tensor  = 0 r (xˆ xˆ + yˆ yˆ + zz (λ,kz,P ,rw )ˆzzˆ )

(6.12)

Note that the z-component of the permittivity tensor depends on the direction of the incoming plane wave, on the periodicity of the structure P and on the radius of the wires rw [24]. The model predicts that such a wire medium can support three different types of modes: transverse electromagnetic (TEM), transverse magnetic (TM) and transverse electric (TE) modes (relative to the direction of the pins, i.e. the z-direction). Using the TE-TM decomposition of the electric point source and by imposing the appropriate boundary conditions at all the interfaces, it is possible to obtain the reflection coefficients for both polarizations. For the TM case the result [24] is T M = ´

kdie kp2 tan(kdie d) ´ β 2 γT M tanh(γT M d) + r γ0 (kp2 + β 2 ) κdie kp2 tan(kdie d) ´ β 2 γT M tanh(γT M d) ´ r γ0 (kp2 + β 2 )

(6.13)

Here kdie is the wave number in the pin medium with permittivity r , kp is the plasma wave number of the pin lattice (kp2 = (1{a 2 )(2π{(ln(a){(2π rw )) + 0.5275)), β 2 = b b 2 and γ = β 2 ´ k02 ; a and b are the periodicities in kx2 + ky2 , γT M = kp2 + β 2 ´ kdie 0 the x- and y-directions (in the considered cases a = b = P ). In the TE case, the electric field is perpendicular to the pins and not affected by them. Consequently, we are left only with the dielectric slab, and the reflection coefficient can be found to be b b b 2 ´ β 2 ´ j k 2 ´ β 2 tan(d k 2 ´ β 2 ) kdie die 0 b b (6.14) T E = ´ b 2 ´ β 2 + j k 2 ´ β 2 tan(d k 2 ´ β 2 ) kdie die 0 The surface impedances of the bed-of-nails structure for these two polarizations are now found to be k0 1 +  T M TM TM = Y˜yx = ¨ (6.15) Y˜xy η0 kz 1 ´  T M kz 1 ´ T E TE TE = Y˜yx = ¨ Y˜xy η0 k0 1 +  T E

(6.16)

The ridge/strip itself is introduced into the analysis via an infinite transmission line current of the form Jy (x 1 ) exp(´j keff y 1 ), where keff is the unknown propagation constant, y 1 is the position at the ridge or strip and Jy (x 1 ) is an assumed expression for

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Gap Waveguide Technology

the transverse distribution across the ridge or strip of the longitudinal current [25]. Thus, we assume that the current is entirely longitudinal, with a known current distribution. The simplest assumption for Jy (x 1 ) is to assume that it is constant over the width, but it will give better results if the known singularities of the current at the sharp strip edges or 90 ridge wedges are included in Jy (x 1 ) [26]. By transforming Jy (x 1 ) to the spectral domain and inserting it into the inverse transformation expression for the ydirected electric field, it is possible to find the field on the ridge itself. Enforcing the boundary condition that the Jy (x 1 )-weighted electric field is zero across the ridge (this corresponds to Galerkin’s method when applying method of moments), the following characteristic equation is finally obtained: ż8 ˜¯¯ EJ ˜2 (6.17) G yy (kx ,keff ) ¨ Jy (kx ) ¨ dkx = 0 ´8

˜¯¯ EJ Here G yy is the spectral domain Green’s function for a y-directed source located just above the periodic texture in our waveguide. J˜y (kx ) is the Fourier transform of Jy (x). From this equation, it is possible to determine the propagation constant keff of the ridge current, and thereafter we can evaluate the modal field of the gap waveguide. The characteristic impedance Z0 of the ridge/strip type of transmission line can be calculated using the following definition: Z0 = 2P {I 2 . The power and the current are obtained by integrating the product Ez ¨ Hx˚ and Hx field component in the observation plane, respectively: ż ż żhż8 1 h 8 ˜ ˜˚ (Ez Hx˚ ´ Ex Hz˚ )dxdz = (Ez Hx ´ E˜ x H˜ z˚ )dkx dz (6.18) P = 2π 0 ´8 0 ´8 As an example, let us consider a bed-of-nails structure of thickness d = 8.66 mm and with the gap height h = 3.5 mm. The period of the metallic pins is P = 3.75 mm, the diameter of the pins is 2rw = 0.375 mm and the permittivity of the dielectric slab is r = 1 (i.e. the structure uses no dielectric filling). As expected, the dispersion plot in Figure 6.8

Figure 6.8 Dispersion diagram for the parallel-plate waveguide with bed-of-nails structure

inserted.

6.2 Approximate Method of Analysis of Parallel-Plate Waveguides

207

Figure 6.9 Propagation constant of the fundamental mode of the example ridge gap waveguide,

computed with the present method and with CST Microwave Studio.

Figure 6.10 Ridge gap waveguide characteristic impedance.

shows that surface wave propagation is possible in all directions for frequencies outside the stop band, which is in this case between 8.6 and 11.1 GHz. Using CST Microwave Studio to model the actual non-homogeneous pin surface, the predicted band gap is between 8.4 and 11.1 GHz. In order to obtain a transmission line, a ridge of width 7.5 mm and of the same height as the surrounding pins was inserted into the bed-ofnails structure. The calculated propagation constant of the fundamental mode is plotted in Figure 6.9 and is compared with results obtained by the general EM solver (CST Microwave Studio). The plotted dispersion diagram reveals the presence of confined quasi-TEM mode within the stop band of the periodic parallel-plate structure. The accuracy of calculating the characteristic impedance was checked by considering the results given in [27] (see Figure 6.10). The analyzed ridge gap waveguide has the following dimensions: periodicity of the bed-of-nails P = 2 mm, pin height d = 7.5 mm, pin diameter 2rw = 1 mm, gap height h = 1 mm and ridge width 5 mm. The structure uses no dielectric filling. The obtained results are in good agreement with CST calculations.

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Alternatively, the ridge/strip type of transmission line can be analyzed using the mode matching approach. The PPW is divided into three regions: region above the ridge/strip and two lateral regions above the high-impedance surface (e.g. bed-of-nails surface). In the lateral regions the field is described by exponentially decaying variation (in direction normal to the ridge/strip), while in the region above the strip the EM field has cosine cross-section variation. By matching the field amplitudes at the boundary between the regions, the propagating mode together with its parameters (propagation constant, characteristic impedance) can be determined. Details about the mode matching approach can be found in [27] and [28]. The groove transmission line has a large similarity with an ordinary rectangular waveguide. However, the propagation constant cannot be accurately approximated with the one of a rectangular waveguide with effective width since this effective width varies with frequency. An accurate analysis can be performed using the transverse transmission line method. Here, first the effective impedance of the lateral walls is determined by considering one row of gap waveguide structure (with periodic boundary conditions in the longitudinal direction), and then the parameters of a rectangular waveguide with impedance sidewalls are easily determined. Details about the transverse transmission line method can be found in [29].

6.3

Design of Stop Bands for Parallel Plate Structures After the introduction of the technology and the analytical study, which provides intuition on the fundamentals of the global idea under this technology, it is clear that there is always an artificial surface made of a periodic structure that must provide a condition of AMC that, combined with the upper metal lid, will produce the cut-off of all parallel plate modes. The initial step when designing any device in this technology is the selection of the periodic structure and the calculation of its dimensions. Considerations about the required bandwidth but also manufacturing limitations have to be taken into account at this point. As seen in the previous sections, for simple cases, as periodic structures made of pins or mushrooms, analytical calculations can give fast and accurate results. In any case, the easiest way to do a first calculation of the stop band is the calculation of the dispersion diagram of the unit cell with periodic boundary conditions, then assuming the ideal infinite structure. Along these years developing this technology, the periodic structure that has been mainly used is the well-known bed-of-nails [24] because it is wideband, low loss because it is metallic and compact in periodicity; even more, the relatively long vertical dimension of the pins is not a problem for millimeter wave frequencies. The typical bandwidth of the stop band created by this structure is of at least 2 to 1. An example of a dispersion diagram achieved with this unit cell is included in Figure 6.11 with the same dimensions used in [11]. The different parameters of the bed-of-nails have effects on the size of the stop band. A detailed study on how the parameters affect the size and position of the stop band for the case of the bed-of-nails as well as for corrugations and mushroom-type EBG structure is presented in [30]. In all cases, a reduction of the gap implies an increase on

6.3 Design of Stop Bands for Parallel Plate Structures

209

Figure 6.11 Example of dispersion diagram for the unit cell of the bed-of-nails. Reprinted from Kildal, P. S., Zaman, A. U., Rajo-Iglesias, E., et al.: ‘Design and experimental verification of ridge gap waveguide in bed of nails for parallel-plate mode suppression’, IET Microwaves, Antennas & Propagation, 2010, 5, (3), pp. 262–270.

Figure 6.12 Example of dispersion diagram for a ridge gap waveguide implemented with the bed-of-nails of Figure 6.11. Reprinted from Kildal, P. S., Zaman, A. U., Rajo-Iglesias, E., et al.: ‘Design and experimental verification of ridge gap waveguide in bed of nails for parallel-plate mode suppression’, IET Microwaves, Antennas & Propagation, 2010, 5, (3), pp. 262–270.

the stop band due to the capacitive effect of the top of these structures with the top metal lid, which reduces the starting frequency of the stop band. The theoretical analysis in the case of the bed-of-nails suggests a starting stop band when the pin’s height is λ/4 and ending when the height is λ/2. However, the periodicity and the pin’s width also play a role in defining the stop band. If now any of the three versions of the gap waveguide described in Figure 6.5 is used with a given periodic structure, the dispersion diagram of a semi-infinite structure (only using periodic boundary conditions in the direction of propagation) can be calculated. In this case, the propagating mode along the line (a quasi-TEM for ridge or inverted versions or a TE when dealing with the groove) must be clearly observed within the stop band. An example for the ridge version is presented in Figure 6.12. In this situation,

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(a)

(b)

Figure 6.13 Example of dispersion diagram for a ridge gap waveguide implemented with

mushrooms. © 2018 IEEE. Reprinted, with permission, from Elena Pucci et al. 2012. A black and white version of this figure will appear in some formats. For the color version, please refer to the plate section.

(a)

(b)

Figure 6.14 Dispersion diagram of the proposed unit cell based on glide symmetric holes. Figure (b) is courtesy of Rhiannon Mitchell.

sometimes the monomode band is limited by other factors related with the line itself, as for instance the excitation of the odd mode. As mentioned, most of the examples in the literature use either the pins or printed mushrooms [31–34], as shown in Figure 6.5, but recently, a new structure based on the use of holes in a glide-symmetric disposition (Figure 6.14) has been proposed to be combined with the groove version. The complete understanding of the structure for design is presented in [17] and [18]. Here the central frequency of the stop band is determined by the periodicity of holes, and it requires two textured surfaces. Still, this structure is wideband, as the ones made with pins.

6.4 Application to RF Packaging

6.4

211

Application to RF Packaging One straightforward application of the parallel plate cutoff property is its use as a packaging solution for conventional printed circuit boards. The fundamentals are the same as for gap waveguide technology: whenever a PMC and a PEC are located together, separated by a distance smaller than λ/4, no modes can propagate. PCBs have always a metal ground plane that we can consider as an ideal PEC; consequently, if we locate a periodic structure providing an AMC boundary condition on top of the PCB with a small gap, we will avoid radiation from discontinuities and bends, any kind of leakage, the excitation of cavity modes and consequently any coupling or cross-talk problem. This idea is behind the different examples of the use of the gap waveguide concept for packaging. The first example of application of packaging was designed using the bed-of-nails and published in [35]. The methodology for designing the periodic structure is the same as described before, i.e. the calculation of the dispersion diagram of the unit cell, although in this case taking into account the dielectric material of the PCB. As it can be seen in the example presented in Figure 6.15, the solution is wideband. A simple way to visualize what happens inside the cavity with pins and without the pins but with a smooth metal lid is presented in Figure 6.16. In the example, the excitation of the cavity modes (when no packaging solution is implemented) is clearly observed in different frequencies, and how these cavity modes are removed with the top lid made of pins is demonstrated. An example of the use of this technique with this particular periodic structure to improve coupled-lines microstrip filters has been published in [36].

Figure 6.15 Example of dispersion diagram for the unit cell of the bed-of-nails to be used as packaging solution. © 2018 IEEE. Reprinted, with permission, from Eva Rajo-Iglesias et al. 2010.

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Figure 6.16 Two-dimensional color plots showing the field in packaged microstrip line with a double 90o bend: (a) first doing the packaging with a smooth metal lid and (b) afterwards with a bed-of-nails. © 2018 IEEE. Reprinted, with permission, from Eva Rajo-Iglesias et al. 2010. A black and white version of this figure will appear in some formats. For the color version, please refer to the plate section.

Studies on how to extend the same procedure for lower frequencies (the typical microwave frequencies) can be also found in the literature. As the bed-of-nails can be considered as 2D corrugations, it is clear that they need to have a height of approximately λ/4, and as consequence, they are bulky for low frequencies. To reduce the height of the unit cell, springs, zigzag or pyramidal pins have been proposed respectively in [37], [38] and [39]. Most of the studies about the packaging performance are made considering a passive circuit to be packaged. A classical way of demonstrating the effectiveness of the proposed packaging methodology is the use of a double bent (90o bends) microstrip line in a PCB with an electrical size big enough to support cavity modes. When no package is in use or when a simple metallic lid at a certain distance is used, the cavity modes are strongly excited. This is the approach used in [35], [37] and [38]. However, the packaging of active circuits, including complete RF modules, can also be accomplished with this same methodology, as it has been demonstrated in [39]. In this study, an exhaustive comparison with a classical way of doing packaging by using both vias in the substrate and separating metallic walls is presented for passive circuits. In all the situations the packaging with a bed-of-nails outperforms the classical solutions. Finally, chains of amplifiers are checked as a critical example for packaging. To this aim, a maximum stable gain test and self-resonance test are performed, and the

6.5 Evaluation of Losses

213

comparison of using the bed-of-nails as solution, instead of a conventional absorber, has proven again to give better performance for RF packaging.

6.5

Evaluation of Losses The fundamentals of gap waveguide technology include the propagation of the EM signal in air. Among the three most common versions (ridge, groove and inverted microstrip), in two of them, no dielectric is required. For these reasons, low losses are expected and claimed as one of the main advantages of this technology. As the arrangements of guiding structures and propagating modes are different in these gap waveguide structures, the attenuation will also be different. That is why it is important to investigate the attenuation and losses accurately for all four categories shown in Figure 6.5 of gap waveguide structures. The evaluation of losses has been initially made by means of the design of resonators fed by uncoupled connectors [40]. These first evaluations for the three most common versions of the technology were made at relatively low frequency (in X band) for simplicity in manufacturing and measurement. The loss study, even if made at low frequencies, proves the low loss characteristic compared to printed technology, where even at 10 GHz the version with more losses (inverted microstrip, version c of Figure 6.5) has half the losses of a conventional microstrip line. The groove version exhibits losses at a level very close to the conventional rectangular waveguide, and the ridge version has lower losses than the inverted microstrip case but more than the groove case. The technology is proposed to be used at millimeter frequency range; consequently, it is vital also to do a comparative study of losses with other available millimeter wave technology solutions. In order to do that, several sufficiently long gap waveguide prototypes (with 20λ length) have been manufactured and the attenuation constant α is calculated from the S21 parameter directly measured with conventional VNA to determine the losses. A careful consideration on any transition used in the measurement is also mandatory. The first example of this kind of evaluation together with a comparison with the losses of other types of waveguides was presented in [41] for the inverted microstrip version. The comparison with conventional microstrip technology using the same material evidenced that the inverted microstrip gap waveguide has 10 times fewer losses than a conventional microstrip at 60 GHz. The propagation of Quasi-TE10 and Quasi-TEM mode in the all-metal groove gap waveguide and ridge gap waveguide is affected mainly by ohmic losses as the leakage loss is negligible (compared to conductor losses) within the parallel plate stop band of the periodic structures. If compared with a classical all-metal waveguide, this conductor loss is expected to be slightly higher due to higher surface current on the square pins having sharp edges. The inverted microstrip gap waveguide and microstripridge gap waveguide geometries suffer from the same conductor losses. In addition, dielectric losses due to the presence of substrate also occur in these two PCB versions of gap waveguide geometries. In Table 6.1 we present the losses in different gap

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Table 6.1 Losses for the different versions of gap waveguide in the 60 GHz band Simulated loss (dB/cm)

Measured min-max loss (dB/cm)

0.0136

0.0295–0.042

0.019

0.03–0.0442

0.036

0.05–0.058

0.0373

0.058–0.0705

0.0805

0.162–0.23

0.0934

0.21–0.288

LCP : 0.23

0.62–0.77

Rogers 4003 : 0.271

0.7055

0.0621

0.0615

0.1327

0.2172

Prototype (frequency) Rect. Waveguide (50–75 GHz) VER-pol Groove (50–75 GHz) HOR-pol Groove (56–75 GHz) Ridge gap (50–75 GHz) Microstrip-ridge gap (56–68 GHz) Invert-micro gap (56–72 GHz) Microstrip line (50–75 GHz) 0.127–0.2mm substrate Microstrip line (50–75 GHz) 0.127–0.2mm substrate Air-filled SIW (50 GHz) Rogers-5880, SIW (50 GHz)

waveguide geometries at the 60 GHz frequency range, and thereby, we can gather relevant information about the efficiency when dealing with gap waveguide antennas in the next section.

6.6

Gap Waveguide Antennas With the demanding system requirements for the fifth-generation (5G) wireless communications and the severe spectrum shortage at conventional cellular frequencies, antenna systems operating in the millimeter-wave frequency bands have attracted a lot of research interest. From the system point of view, the antenna is one key component which is responsible to a great extent for the noise performances and power budget of RF systems. This is because noise characteristics of systems are mainly determined by the first building blocks. At millimeter-wave frequencies, the noise issues are much more pronounced, which are closely related to the loss, efficiency of the antenna and front-end structures. Also, the power budget of a RF system could be adjusted by the gain of its antenna, and therefore, the use of a high-gain

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215

antenna scheme would lower the burden on the power amplifier (for the transmit case) and LNA (for the receive case). Moreover, high-gain antennas can potentially relax the requirements on highly selective band pass filters, which is particularly important in the case of full-duplex point-to-point communication systems. Although high antenna efficiency can be obtained by using dielectric lens antennas or reflector antennas, it is difficult to realize planar profiles with such configurations, as they essentially need focal spatial length. The conventional low-cost planar structures, such as microstrip lines and coplanar waveguides (CPW), which are common for the design of circuits and antennas, cannot be readily used for millimeter-wave and sub-millimeter wave applications. This is because they present some drawbacks: (1) such semi-open planar structures generally exhibit high transmission losses because of the presence of dielectric material, narrow conductor lines and spurious radiation losses, thus lowering the antenna efficiency; (2) design and implementation of such planar structures are usually subject to complicated packaging issues, crosstalk, cavity modes etc.; (3) power handling and thermal management become tedious and complicated because of their transverse electromagnetic (TEM) mode type of propagation requiring separate conductors; and (4) fabrication tolerances are more tricky at millimeter-wave range (compared to the microwave range), requiring advanced processing techniques for making fine details of the antenna and thereby increasing the cost of the antenna. On the other hand, the conventional nonplanar structures, such as hollow metallic waveguides, have been widely used for the development of millimeter-wave components even though they are non-integratable and bulky. Slotted hollow waveguide-based planar array antennas are free from large feed-network loss and can be applied to design high-efficiency, high-gain or moderate-gain antennas. Other attractive characteristics of the waveguide structures are their high-power handling capability and completely shielded characteristics. However, the production cost of waveguide-based antennas is generally very high in the millimeter wave range because they usually consist of metal blocks with complicated three-dimensional structures with very strict electrical contact and assembly requirements. Consequently, conventional waveguide slot arrays have not been used commercially until today, with the exception of a few military and space applications. Apart from the conventional microstrip antennas and waveguide slot antennas, substrate integrated waveguide (SIW) based planar array antennas are also proposed as a promising platform for the development of low-cost and highperformance millimeter-wave planar antennas. The SIW technology enables integration of active circuits together with the antennas. The losses in SIW are better than microstrip and coplanar structures. Still, losses may be of real concern, especially for highefficiency and high-gain (above 28–30 dBi) configurations, due to the presence of dielectric material. Based on the above considerations, it is implied that existing millimeter-wave antenna technologies have limitations with respect to manufacturing cost, efficiency, bandwidth and mechanical simplicity, giving an opportunity for more research work on new technologies, such as gap waveguide. In this section, we have discussed several gap waveguide antenna geometries working in microwave and millimeter-wave frequency range.

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6.6.1

High-Gain Antennas Designed in Ridge Gap Waveguide and Groove Gap Waveguide Geometries According to [42] and [43], for the millimeter wave frequency band, antenna arrays with a gain higher than 30 dBi would be used when the communication distance is larger than 100 m. Another important issue for the system cost reduction is the antenna robustness and ease of integration with other components in a wireless system [44]. In this section, we present several high-gain and high-efficiency planar antenna arrays designed in gap waveguide technology. The gap waveguide-based slot arrays suffer neither from dielectric nor from the radiation losses, and these antennas operate over more than 15% relative bandwidth. However, wideband waveguide slot arrays require corporate feed networks and therefore require more space to design the feed network. Usually, this is done by placing an intermediate cavity layer between the radiating slot layer and the feeding waveguide layer. The concept of the cavity-backed subarray is shown in Figure 6.17. Apart from the problem of metal contact, the antennas need a very thin slot layer (100 μm) which raises mechanical challenges during assembly due to bending of these very thin slot layers. To overcome this problem and also decrease grating lobes, some corrugations are created in the radiating layer. The width and depth of these corrugations are optimized to improve both the S11 and the radiation pattern of the antenna. Also, the radiating slot layer could be designed with some flare angle to increase the thickness of the slot layer. The flared section can also be optimized for S11 and radiation patterns. This is also shown in Figure 6.17. The simulated reflection coefficients for such cavity-backed subarrays are presented in Figure 6.18. The results show about 15% of reflection coefficient bandwidth (|S11 | < ´10 dB) in the frequency band of 57–66 GHz for these types of cavity-backed subarrays fed with gap waveguide feeding networks.

(a)

(b)

Figure 6.17 Cavity backed slot array antenna subarray with ridge gap feed network and groove

gap feed network.

6.6 Gap Waveguide Antennas

217

0 Ridge gap Groove gap

-5

|S11 | (dB)

-10 -15 -20 -25 -30 54

56

58

60

62

64

66

68

Frequency (GHz)

Figure 6.18 Simulated S11 for the cavity backed subarray with ridge gap and groove gap feed

network.

Figure 6.19 Sixteen-way power divider designed in ridge gap waveguide technology; top metal plate not shown.

In order to expand the designed subarray to a bigger structure, we need to use a corporate feed network including power dividers and T-junctions. The T-junctions can also be designed both in ridge gap waveguide and groove gap waveguide geometries. The pictures of such T-junctions and 16-way power divider designed in a ridge gap waveguide are shown in Figure 6.19. Once the corporate feeding network has been designed, the complete antenna design can be completed easily by coupling the subarray at the output of each feed line. In such a way, a 16ˆ16 slot array antenna has been designed combining the ridge gap and groove gap waveguide technology. Notice that due to the restrictions in the fabrication process, all sharp corners and edges have been rounded with the smallest radius of 0.2 mm. Thus, according to this matter, the antenna structures, including feed layer, cavity layer and radiating layer, have been optimized again. The complete array antennas (dimensions of aperture are 70.4 ˆ 64 mm2 ) and the different metal layers have been fabricated in aluminum by CNC milling, and the metal

Gap Waveguide Technology

(a)

(b)

Figure 6.20 (a) Measured S11 for the ridge gap and groove gap waveguide-based 16 ˆ 16 slot array antenna; (b) the manufactured ridge gap and groove gap waveguide-based 16 ˆ 16 slot array antenna. A black and white version of this figure will appear in some formats. For the color version, please refer to the plate section. 35

% 100 % 90

34

Gain (dBi)

218

% 80 % 70

33 32

Ridge Meas. Groove Meas.

31 30 56

57

58

59

60

61

62

63

64

65

66

67

Frequency (GHz) Figure 6.21 Gain versus frequency for the manufactured ridge gap and groove gap waveguide 16ˆ16 slot antenna.

layers were assembled by four simple screws. The measured S11 results in comparison with the simulated results are plotted in Figure 6.20a, and the pictures of the final prototypes are shown in Figure 6.20b. Also, the gain vs frequency plot with the indicated efficiency is plotted in Figure 6.21 and the measured radiation pattern for the ridge gap slot array is shown in Figure 6.22. The radiation patterns for the groove gap slot array are very similar to the presented groove gap slot array. The only difference between the groove and ridge gap slot arrays is the efficiency. As expected, the groove gap slot array has higher total efficiency compared to the ridge gap version. The obtained gain is more than 32.5 dBi with a measured total efficiency higher than 70% for the ridge gap waveguide-based slot array. Slightly more gain has been measured for the groove gap waveguide-based antenna. For both the antennas, the gain response is quite flat, with a gain variation of 0.7 dB over the 56–66 GHz bandwidth. Also, the measured radiation patterns for the ridge waveguide-based slot array shown in Figure 6.22 are symmetrical, and the first side-

6.6 Gap Waveguide Antennas

(a)

219

(b)

Figure 6.22 Measured radiation patterns for the groove gap waveguide-based 16ˆ16 slot antenna. © 2018 IEEE. Reprinted, with permission, from Ali Farahbakhsh 2017.

(a)

(b)

Figure 6.23 Schematic of the horn array fed with ridge gap waveguide and the S11 results for the antenna array.

lobe levels in both E- and H-planes are around ´13 dB. The measured side lobe levels of the fabricated array in the 45˝ plane are below ´25 dB over the frequency band of interest. More detailed information about these two high-gain, high-efficiency antennas can be found in [45] and [46]. Also a 3-D printed horn array at E-band frequency has been presented recently in [47]. The schematic of the array is shown in Figure 6.23. In this array, the proposed horn unit cell dimensions are larger than one wavelength both in E-plane and H-plane. As a result, the grating lobe problem is evident, and this problem has been solved by inserting a septum inside the horn in E-plane. The horn unit cell is excited by a coupling aperture from a ridge gap waveguide distribution network on the back side of the same plate, i.e. it is a two-layer design. A 4ˆ4 horn array prototype was fabricated by DMLS 3-D printing technique and CNC milling technique. The total size of the antenna is 32ˆ32 mm2 ; however, the effective array aperture is

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(a)

(b)

Figure 6.24 Measured E-plane and H-plane pattern for the proposed horn array at 81 GHz.

24 ˆ 24.4 mm2 . The proposed antenna shows a very wide impedance bandwidth with reflection coefficient below ´10 dB over the 69–88 GHz frequency band. The schematic of the proposed horn array, the measured S11 and gain vs frequency curve are presented in Figure 6.23. The simulated and measured normalized radiation far-field patterns of the antenna in E- and H-planes at 81 GHz are shown in Figure 6.24. The computed radiation patterns of the antenna without the inserted septum are also presented in the same figure. We see that an inserted septum improves the radiation pattern and suppresses grating lobes in the H-plane. The measured cross-polar level of the proposed antenna is below ´40 dB within the band of interest.

6.6.2

High-Gain Antennas Designed in Inverted Microstrip Gap Waveguide and Printed Gap Waveguide Geometry As shown in Figure 6.5, the inverted microstrip gap waveguide technology is based on the use of a thin substrate for guiding the EM field (which can be used as a feed network), and this substrate is placed over a periodic pin pattern. This periodic metal pin layer constitutes an AMC surface and, in combination with the upper metallic lid, prohibits any wave propagation along the lateral directions. On the other hand, in case of the printed ridge gap waveguide (PRGW), the ridge is a microstrip line that is grounded by connecting it by metallized via holes along the line, and it is on the same substrate of the mushroom texture surrounding it. Recently, several millimeter wave antennas have been designed based on these two types of gap waveguide technology. The antenna subarrays for different configurations of antennas are presented in Figure 6.25. The perspective view of the 2ˆ2-element subarrays in Figure 6.25a briefly shows the radiating slot layer, cavity layer, PCB microstrip layer and bed-of-nails. Here the groove gap waveguide cavity is partitioned into four spaces by two sets of metallic

6.6 Gap Waveguide Antennas

(a)

221

(b)

Figure 6.25 (a) Inverted microstrip gap waveguide based antenna subarray; (b) printed ridge waveguide-based antenna subarray [48].

walls extending in the x and y directions. The PCB microstrip layer feeds the groove waveguide cavity with identical phase and amplitude by the middle coupling hole. On the other hand, the other subarray shown in Figure 6.25b is made of two double-sided PCB layers. In this case, the bottom PCB is used for the design of the corporate feed network based on printed gap waveguide technology. The upper PCB is a SIW cavity layer with internal walls made up of grounded vias. The ground plane of the upper PCB holds a coupling slot which is used to couple energy from the printed gap waveguide feed network to the SIW cavity layer. After that, the energy is radiated from the SIW cavity to the air with the help of the radiating slots which are etched on the top metal layer of the upper PCB. The details of these antennas are described in [48] and [49]. We present the radiation patterns for 16ˆ16 and 8ˆ8 slot array antennas designed with inverted microstrip gap waveguide technology and printed gap waveguide technology in Figures 6.26 and 6.27, respectively. Also, another interesting slot array and magneto-electric dipole array antenna based on printed gap waveguide technology have been reported in [33] and [50]. Both these two antennas were designed at W-band and Ka-band. Also, an L-probe excited cavity backed 8 ˆ 8 antenna array feed by a LTCC based gap waveguide transmission line is presented in [51]. In this work, the radiation layer is composed of a substrate integrated cavity and an L-probe. The array feed network was realized by LTCC-based gap waveguide technology. The LTCC-based gap waveguide transmission lines exhibit the same low-loss characteristics as the traditional printed gap waveguide, as well as offering higher integration ability. The overall structure of the proposed antenna array was composed of 13 layers of the LTCC substrate and consisted of the radiation layers, feeding network layers and rectangular waveguide–gap waveguide transition layers. The proposed antenna is fabricated using the Ferro A6-M LTCC tape with a fired tape thickness of 0.096 mm in each layer. In this case, the r and tanδ are 6 and

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Figure 6.26 Measured E-plane and H-plane patterns for the inverted gap waveguide-based slot array at 60 GHz. © 2018 IEEE. Reprinted, with permission, from Jinlin Liu 2017.

Figure 6.27 Measured E-plane and H-plane radiation patterns for the printed gap waveguide-based slot array.

0.002, respectively. The conductor used for the metallization and vias is gold, and the metallization thickness used was 8 μm. The detailed design process and the relevant parameters are presented in [51].

6.6.3

Single-Layer Antennas Based on Gap Waveguide Technology Apart from the different multi-layered antennas mentioned in the previous sections, some single-layer antennas based on ridge and groove gap waveguide antennas have also been recently designed. One such antenna is presented in Figure 6.29. In this work, the slots are fed through a groove gap corporate feed network with E-plane splitters, and also the slots are backed by coaxial cavities, which are more compact than conventional rectangular cavities. The coaxial cavity is integrated in the same pin layer of the horizontally polarized groove gap waveguide feed network. It can be observed that the 2ˆ2 unit cells in the bottom half of the antenna are mirrored versions of those in the upper part to counteract the 180˝ phase shift originating from the E-plane splitter

6.6 Gap Waveguide Antennas

(a)

223

(b)

Figure 6.28 Schematic of the L-probe radiating antenna with LTCC-based gap waveguide transmission line. © 2018 IEEE. Reprinted, with permission, from Baolin Cao 2015.

of the corporate feed networks. The novel cavity concept introduced in this antenna is based on a coaxial resonance between top and bottom layers by shortening one of the nails used in the GGW, leading to a compact design. Great potential is foreseen for the proposed concept since it does not increase manufacturing time or complexity. The details of this antenna are described in [52]. A similar approach to a single-layer slot array concept based on a ridge gap waveguide is also possible to design. The ridge gap waveguide can be also as compact as the previously mentioned horizontally polarized groove gap waveguide-based slot array in [53]. In this work, two test antennas, a 4ˆ1 linear slot array and a 2ˆ2 planar slot array, have been presented to demonstrate the potential of ridge gap waveguide-based slot array antennas. The radiating slot element, the feeding network and the transitions used in both antennas have been designed to suppress the reflection coefficient, and the results show more than 20% relative bandwidth of the whole arrays. The schematic of the antennas and the measured results are shown in Figure 6.30. The presented 2ˆ2 slot array element can be used as a subarray to build up much larger antennas. The element spacing of 17.5 mm corresponds to 0.875λ, which is small enough to avoid grating lobes for fixed-beam applications. The linear array in this case was excited by a coaxial probe-to-gap waveguide transition [54], and the planar array was fed by a rectangular to waveguide transition section [52].

6.6.4

Frequency Scanning Antenna Based on Gap Waveguide Technology Apart from the fixed beam antennas, several beam scanning antennas have been also designed recently based on gap waveguide technology. A wideband circularly polarized (CP) frequency scanning slot antenna array design based on ridge gap waveg-

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Gap Waveguide Technology

(a)

(b)

(c)

(d)

Figure 6.29 Single-layer groove gap waveguide-based slot antenna array, measured S11 and E-plane pattern. © 2016 IEEE. Reprinted, with permission, from Jiménez Sáez 2016.

(a)

(b)

Figure 6.30 The manufactured prototype of the 4ˆ1 linear array and the 2ˆ2 planar array. A black and white version of this figure will appear in some formats. For the color version, please refer to the plate section.

uide (RGW) technology is presented in [55]. The LHCP antenna array is designed and fabricated using seven sets of inclined radiating slots with one guided wavelength separation at a center frequency of 14 GHz. The slots are etched on the copper layer of

6.7 Conclusion

225

Figure 6.31 (a) Manufactured prototype and (b) gain and axial ratio (dashed line: measured; solid line: simulated) of a leaky wave antenna. © 2018 IEEE. Reprinted, with permission, from Mohamed Al Sharkawy 2014. A black and white version of this figure will appear in some formats. For the color version, please refer to the plate section.

an RT-5880 substrate, and this substrate is placed on top of the ridge gap waveguide. Each set consists of two inclined radiating slots of spacing λg /4 from center to center with equal slot length. The two slots in each set are orthogonal to each other. The angle of inclination φ= 38˝ is found to provide the best AR performance within the 3 dB AR level. The array also exhibits a continuous frequency beam-scanning capability from backward to forward directions including broadside radiation, covering the entire band of interest starting with a scanning angle of ˘15˝ . The prototype of the array is shown in Figure 6.31. Also, the measured axial ratio and gain are shown in the same figure. Another simple leaky wave antenna implemented in a groove version was presented in [56]. In this case the pins of one of the sides of a groove gap waveguide are modified to leak, and by controlling the height and periodicity of the pins, total control over the attenuation and propagation constant is achieved.

6.7

Conclusion The coming years will eventually bring new applications of wireless communications at higher frequencies (30 GHz and above). Modern wireless technologies like massive MIMO, multi-user MIMO and gigabit transmission will become reality. The industrial winners will be the companies that can provide the millimeter wave hardware at the lowest cost. This requires new waveguide technologies that are more economical than conventional rectangular waveguide technology and more power-efficient (lower losses) than PCB-based microstrip and coplanar waveguides. The gap waveguide technology presented in this chapter is a new technology with lots of potential to be used at millimeter wave frequency range. It is much lower loss than microstrip or CPW and has very flexible mechanical assembly compared to metal waveguides, which allows lowcost fabrication of waveguide components. The basis of this technology is the use

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of an artificial surface with AMC behavior to create a stop band for parallel-plate modes within an oversized parallel plate structure. The combination of this artificial surface with some metallic trail or path can allow wave propagation inside the oversized structure without being leaked in the unwanted direction. The present chapter gives the historical background of gap waveguide technology since its invention in 2008, and how it has evolved since then. The chapter deals with parallel-plate stop band design and RF packaging solutions for PCB circuit boards. The design of the stop band is quite important, as it dictates the performance of the gap waveguide components and the packaging performance. Typically, a unit cell of the periodic structure is analyzed with the eigenmode solver of commercial software to determine the parallel-plate stop band, and this type of study consists of parametric sweeps of all parameters associated with the periodic structure. A part of the chapter also contains an overview of analytical approaches used so far in the prediction of propagation performance of groove gap waveguides and ridge gap waveguides. One of the analytical approaches is the spectral domain approach in which the Green’s functions of the considered structure are determined. The last part of the chapter presents a summary on how to use this technology to design efficient antennas in the millimeter wave frequency range. Due to the low loss properties of gap waveguides, high-efficiency and high-gain planar antennas can be designed with this technology. Single-layer slot arrays, double-layer slot arrays based on ridge and groove gap waveguide technology, PCB-based slot arrays and horn arrays fed by gap waveguide technology have been discussed in detail. However, many of these antennas presented here are designed and measured to validate the concept of the gap waveguide and have been manufactured using traditional CNC milling or metal sawing techniques. Still, the inherent mechanical flexibility of gap waveguide technology can allow low-cost manufacturing techniques, such as plastic injection molding, metal die casting or polymer micromachining, for millimeter wave high-efficiency antennas.

Acknowledgement The authors of this chapter would like to dedicate it to the memory of their beloved mentor Prof. Per-Simon Kildal.

References [1] P. S. Kildal, “Definition of artificially soft and hard surfaces for electromagnetic waves,” Electronics Letters, vol. 24, pp. 168–170, 1988. [2] P. S. Kildal, “Artificially soft and hard surfaces in electromagnetics,” IEEE Transactions on Antennas and Propagation, vol. 38, pp. 1537–1544, 1990. [3] Z. Sipus, H. Merkel, and P. S. Kildal, “Green’s functions for planar soft and hard surfaces derived by asymptotic boundary conditions,” IEE Proceedings – Microwaves, Antennas and Propagation, vol. 144, pp. 321–328, Oct. 1997. [4] P. S. Kildal and A. A. Kishk, “EM modeling of surfaces with STOP or GO characteristics – artificial magnetic conductors and soft and hard surfaces,” Applied Computational Electromagnetics Society Journal, vol. 18, pp. 31–40, 2003.

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[21] D. Pozar, “Radiation and scattering from a microstrip patch on a uniaxial substrate,” IEEE Transactions on Antennas and Propagation, vol. 35, pp. 613–621, 1987. [22] N. K. Das and D. M. Pozar, “A generalized spectral-domain Green’s function for multilayer dielectric substrates with application to multilayer transmission lines,” IEEE Transactions on Microwave Theory and Techniques, vol. 35, pp. 326–335, 1987. [23] Z. Sipus, P. Kildal, R. Leijon, and M. Johansson, “An algorithm for calculating Green’s functions of planar, circular cylindrical and spherical multilayer substrates,” Applied Computational Electromagnetics Society Journal, vol. 13, pp. 243–254, 1998. [24] M. Silveirinha, C. Fernandes, and J. Costa, “Electromagnetic characterization of textured surfaces formed by metallic pins,” IEEE Transactions on Antennas and Propagation, vol. 56, pp. 2695–2700, 2008. [25] R. W. Jackson and D. M. Pozar, “Full-wave analysis of microstrip open-end and gap discontinuities,” IEEE Transactions on Microwave Theory and Techniques, vol. 33, pp. 1036–1042, 1985. [26] D. Jones, The Theory of Electromagnetism. Pergamon Press, 1964. [27] A. Polemi, S. Maci, and P. S. Kildal, “Dispersion characteristics of a metamaterial-based parallel-plate ridge gap waveguide realized by bed of nails,” IEEE Transactions on Antennas and Propagation, vol. 59, pp. 904–913, 2011. [28] A. Polemi and S. Maci, “Closed form expressions for the modal dispersion equations and for the characteristic impedance of a metamaterial-based gap waveguide,” IET Microwaves, Antennas and Propagation, vol. 4, pp. 1073–1080, 2010. [29] A. Berenguer, V. Fusco, D. E. Zelenchuk, D. Sánchez-Escuderos, M. Baquero-Escudero, and V. E. Boria-Esbert, “Propagation characteristics of groove gap waveguide below and above cutoff,” IEEE Transactions on Microwave Theory and Techniques, vol. 64, pp. 27–36, 2016. [30] E. Rajo-Iglesias and P.-S. Kildal, “Numerical studies of bandwidth of parallel-plate cut-off realised by a bed of nails, corrugations and mushroom-type electromagnetic bandgap for use in gap waveguides,” IET Microwaves, Antennas and Propagation, vol. 5, pp. 282–289, 2011. [31] E. Pucci, E. Rajo-Iglesias, and P. S. Kildal, “New microstrip gap waveguide on mushroomtype EBG for packaging of microwave components,” IEEE Microwave and Wireless Components Letters, vol. 22, pp. 129–131, 2012. [32] M. S. Sorkherizi and A. A. Kishk, “Fully printed gap waveguide with facilitated design properties,” IEEE Microwave and Wireless Components Letters, vol. 26, pp. 657–659, 2016. [33] M. S. Sorkherizi, A. Dadgarpour, and A. A. Kishk, “Planar high-efficiency antenna array using new printed ridge gap waveguide technology,” IEEE Transactions on Antennas and Propagation, vol. 65, pp. 3772–3776, 2017. [34] S. I. Shams and A. A. Kishk, “Printed texture with triangle flat pins for bandwidth enhancement of the ridge gap waveguide,” IEEE Transactions on Microwave Theory and Techniques, vol. 65, pp. 2093–2100, 2017. [35] E. Rajo-Iglesias, A. U. Zaman, and P. S. Kildal, “Parallel plate cavity mode suppression in microstrip circuit packages using a lid of nails,” IEEE Microwave and Wireless Components Letters, vol. 20, pp. 31–33, 2010. [36] A. A. Brazalez, A. U. Zaman, and P. S. Kildal, “Improved microstrip filters using pmc packaging by lid of nails,” IEEE Transactions on Components, Packaging and Manufacturing Technology, vol. 2, pp. 1075–1084, 2012.

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[37] E. Rajo-Iglesias, E. Pucci, A. A. Kishk, and P. S. Kildal, “Suppression of parallel plate modes in low frequency microstrip circuit packages using lid of printed zigzag wires,” IEEE Microwave and Wireless Components Letters, vol. 23, pp. 359–361, 2013. [38] E. Rajo-Iglesias, P. S. Kildal, A. U. Zaman, and A. Kishk, “Bed of springs for packaging of microstrip circuits in the microwave frequency range,” IEEE Transactions on Components, Packaging and Manufacturing Technology, vol. 2, pp. 1623–1628, 2012. [39] A. U. Zaman, M. Alexanderson, T. Vukusic, and P. S. Kildal, “Gap waveguide PMC packaging for improved isolation of circuit components in high-frequency microwave modules,” IEEE Transactions on Components, Packaging and Manufacturing Technology, vol. 4, pp. 16–25, 2014. [40] E. Pucci, A. U. Zaman, E. Rajo-Iglesias, P. S. Kildal, and A. Kishk, “Study of Q-factors of ridge and groove gap waveguide resonators,” IET Microwaves, Antennas and Propagation, vol. 7, pp. 900–908, 2013. [41] A. A. Brazález, E. Rajo-Iglesias, J. L. Vázquez-Roy, A. Vosoogh, and P. S. Kildal, “Design and validation of microstrip gap waveguides and their transitions to rectangular waveguide, for millimeter-wave applications,” IEEE Transactions on Microwave Theory and Techniques, vol. 63, pp. 4035–4050, 2015. [42] J. Ala-Laurinaho, J. Aurinsalo, A. Karttunen, M. Kaunisto, A. Lamminen, J. Nurmiharju, A. V. Räisänen, J. Säily, and P. Wainio, “2-D beam-steerable integrated lens antenna system for 5G E -band access and backhaul,” IEEE Transactions on Microwave Theory and Techniques, vol. 64, pp. 2244–2255, 2016. [43] J. Zhang, X. Ge, Q. Li, M. Guizani, and Y. Zhang, “5G millimeter-wave antenna array: Design and challenges,” IEEE Wireless Communications, vol. 24, pp. 106–112, 2017. [44] A. Zaman and P. Kildal, “GAP Waveguides,” in Handbook of Antenna Technologies. Springer 2016. [45] D. Zarifi, A. Farahbakhsh, A. U. Zaman, and P. S. Kildal, “Design and fabrication of a highgain 60-GHz corrugated slot antenna array with ridge gap waveguide distribution layer,” IEEE Transactions on Antennas and Propagation, vol. 64, pp. 2905–2913, 2016. [46] A. Farahbakhsh, D. Zarifi, and A. U. Zaman, “60-GHz groove gap waveguide based wideband h -plane power dividers and transitions: For use in high-gain slot array antenna,” IEEE Transactions on Microwave Theory and Techniques, vol. 65, pp. 4111–4121, 2017. [47] A. Vosoogh, P. S. Kildal, V. Vassilev, A. U. Zaman, and S. Carlsson, “E-band 3-D metal printed wideband planar horn array antenna,” in 2016 International Symposium on Antennas and Propagation (ISAP), pp. 304–305, Oct. 2016. [48] J. Liu, A. Vosoogh, A. U. Zaman, and J. Yang, “Design and fabrication of a high-gain 60GHz cavity-backed slot antenna array fed by inverted microstrip gap waveguide,” IEEE Transactions on Antennas and Propagation, vol. 65, pp. 2117–2122, 2017. [49] S. A. Razavi, P. S. Kildal, L. Xiang, E. A. Alós, and H. Chen, “2ˆ 2-slot element for 60-GHz planar array antenna realized on two doubled-sided PCBs using SIW cavity and EBG-type soft surface fed by microstrip-ridge gap waveguide,” IEEE Transactions on Antennas and Propagation, vol. 62, pp. 4564–4573, 2014. [50] B. Cao, H. Wang, and Y. Huang, “W-band high-gain TE220 -mode slot antenna array with gap waveguide feeding network,” IEEE Antennas and Wireless Propagation Letters, vol. 15, pp. 988–991, 2016. [51] B. Cao, H. Wang, Y. Huang, and J. Zheng, “High-gain L-probe excited substrate integrated cavity antenna array with LTCC-based gap waveguide feeding network for W-band application,” IEEE Transactions on Antennas and Propagation, vol. 63, pp. 5465–5474, 2015.

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[52] A. J. Sáez, A. Valero-Nogueira, J. I. Herranz, and B. Bernardo, “Single-layer cavity-backed slot array fed by groove gap waveguide,” IEEE Antennas and Wireless Propagation Letters, vol. 15, pp. 1402–1405, 2016. [53] A. U. Zaman and P. S. Kildal, “Wide-band slot antenna arrays with single-layer corporatefeed network in ridge gap waveguide technology,” IEEE Transactions on Antennas and Propagation, vol. 62, pp. 2992–3001, 2014. [54] A. U. Zaman, E. Rajo-Iglesias, E. Alfonso, and P. S. Kildal, “Design of transition from coaxial line to ridge gap waveguide,” in 2009 IEEE Antennas and Propagation Society International Symposium, pp. 1–4, June 2009. [55] M. A. Sharkawy and A. A. Kishk, “Wideband beam-scanning circularly polarized inclined slots using ridge gap waveguide,” IEEE Antennas and Wireless Propagation Letters, vol. 13, pp. 1187–1190, 2014. [56] M. Vukomanovic, J. L. Vazquez-Roy, O. Quevedo-Teruel, E. Rajo-Iglesias, and Z. Sipus, “Gap waveguide leaky-wave antenna,” IEEE Transactions on Antennas and Propagation, vol. 64, pp. 2055–2060, 2016.

7

Modulated Metasurface Antennas Gabriele Minatti, David González-Ovejero, Enrica Martini, and Stefano Maci

7.1

Introduction Metasurfaces (MTSs), the 2D equivalent of metamaterials, are thin artificial layers characterized by unusual reflection and transmission properties of plane waves and/or dispersion properties of surface/guided waves. These artificial materials can be obtained by periodically arranging many electrically small inclusions in a dielectric host environment, and they are conceived to achieve macroscopic electromagnetic or optical properties that cannot be found in nature. MTSs are sometimes distinguished as penetrable and impenetrable: small periodic elements in a very thin dielectric host medium constitute a penetrable MTS (or metafilm) [1]. A dense texture of small metallic elements printed on a grounded slab (with or without shorting vias) at the nodes of a regular lattice constitutes instead an impenetrable MTS. Impenetrable MTSs can also be simply constituted by a dense distribution of pins [2] on a ground plane, a solution usually adopted in the millimeter and submillimeter wave range to overcome the issue of losses in conventional dielectric substrates [3]. On the other hand, high-resistivity silicon and polymers are dielectric alternatives at mm and sub-mm waves [4]. The common denominator in all MTS structures is the presence of a periodic lattice consisting of sub-wavelength unit cells. When the constitutive elements in the unit cells are equal one to each other [5], the MTS is referred to as uniform. When the geometrical parameters of the elements gradually change from cell to cell, the MTS is referred to as non-uniform or modulated. Non-uniform MTSs, the ones considered in this chapter, allow one to tailor the phase/group velocity and/or propagation path of the guided wave sustained by the MTS. Modulated MTSs have been applied to realize aperture antennas (the ones addressed in this chapter) [6–16], to control surface-waves (SWs) wavefront [17–20], to implement screens for the control of field transmission or reflection [21–23], and to realize electromagnetic band gap (EBG) structures [24]. Several recent papers [6–39] address the subject of SW wavefront control by means of MTS, some of them in the framework of Transformation Optics (see [18, 27–33]). Another class of papers deal with the control of leaky wave (LW) radiation through the modulation of impedance boundary conditions (IBCs), the basis of a new class of MTS antennas [6–11] (in [8] these antennas are referred to as MoMetAs, which stands for Modulated Metasurface Antennas). These antennas transform a bounded SW into a radiative LW by exploiting the interaction of the SW with the modulated MTS. 231

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At the microwave frequencies, modulated MTS antennas are typically constituted by subwavelength patches of different shapes, printed on a grounded dielectric substrate [6–13]. For applications at millimeter and submillimeter waves, the patch textured surface can be conveniently replaced by a textured metallic surface [3]. Among the key features of MTS antennas, it is worth mentioning •

• •

• • • • •

A unique feature of decoupling of electrical and thermo-mechanical design, due to the fact that the general structure is independent of the specific radiation pattern characteristics. This leads to mechanical design re-usability and unique support for late electrical performance refinement. A unique capability of low-complexity pattern shape control, amenable to dynamic reconfigurability (shaping and scanning). Simple and complete control of aperture field for medium-to-large size antennas. In particular, the operating principle of modulated MTS antennas offers an enormous flexibility with respect to conventional LW antennas, thanks to the possibility of a 2D control of the leakage parameter and consequently of the apertures’ tapering. Simple low-cost manufacturing. Implementation amenable to very different technologies. Low losses thanks to the absence of a feeding network. Low mass and low envelope. Furthermore, these antennas do not require external protruding or backing feed arrangements or (sub-) reflectors, since the feeding structure is embedded in the MTS plane: this is an advantage, with respect to other types of printed antennas, like reflectarrays, in terms of reduced complexity, especially for space applications [8].

A couple of prototypes are shown in Figure 7.1 as examples of modulated MTS antennas. The elements that implement the MTS are electrically small (typically, between one-fifth and one-tenth of the wavelength at the operative frequency) so that the SW practically sees a continuous IBC. These elements behave like pixels in a black-andwhite printed image, whose grey scale is realized by changing their geometry. Also, one can control the polarization by rotating the pixel elements. The modeling of these antennas is inherently multi-scale (see Figure 7.2), and this characteristic drives the entire design process. By adopting a terminology often found in physics, we denote the scales as (1) “macroscopic scale”, (2) “mesoscopic scale”, (3) “molecular scale” and (4) “atomic scale” (Figure 7.2). At the macroscopic scale, we have the whole antenna whose largest dimension can be of several wavelengths (up to 50λ). At a closer sight, at the mesoscopic scale, the antenna and its model are dominated by the local period of the global MTS modulation which is comparable with the wavelength (Figure 7.2b). At molecular scale, one sees the lattice of the texture, consisting of pixel elements with a size in the range of λ/10´λ/5 (Figure 7.2c). Finally, at the atomic scale, specific features, possibly added to the patches to control the polarization (e.g., the slots), become notable. These features can have a size of the order of λ/50-λ/100, which is approximately the side of the discretization triangles required in a method

7.1 Introduction

233

(a)

(b)

Figure 7.1 Examples of realized prototypes of MTS antennas. The SW is excited by a single

probe (right inset of (a)) [12] or by four probes (right inset of (b)) [10]. In both cases, the feeding region is located at the center of the MTS. A black and white version of this figure will appear in some formats. For the color version, please refer to the plate section.

(a)

(b)

(c)

(d)

Figure 7.2 Multiscale features of MTS antennas: (a) “macroscopic” scale, (b) “mesoscopic”

scale, (c) “molecular” scale, (d) “atomic” scale.

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of moments mesh (Figure 7.2d). It is hence clear that a conventional global full-wave analysis of the overall structure could require the solution of a problem with millions of unknowns. For this reason, a single model cannot cover all the scales, and the synthesis procedure cannot be solely based on a direct full-wave solver. In this chapter, we face the problem of the analysis and design of modulated MTS antennas. First, in Section 7.2, we introduce an adiabatic Floquet wave (FW) model for curvilinear locally periodic IBCs. The model comes from the rigorous FW expansion of 1D periodic problems, and it is asymptotically extended to the case of locally periodic 2D IBCs. Even if the actual IBCs are not strictly periodic, the FW expansion still gives a good description of the fields and currents associated with the local periodicity of the surface, and can be successfully exploited to analyze and design MTS antennas. Next, Section 7.3 treats the design process. After an initial global picture, the paragraph goes into detail for each part of the design process, from the synthesis of the continuous impedance surface to the implementation of the final antenna layout. Section 7.4 deals with the analysis of modulated MTS antennas for the numerical prediction of the performance: a full-wave solver for continuous IBCs is described. Right after, Section 7.5 discusses the performance of modulated MTS antennas in terms of efficiency and bandwidth, focusing on the case of broadside, circularly polarized antennas, which is of special interest for space applications. Section 7.6 presents two examples of antenna design: shaped beam and multibeam antennas. Finally, a recap of the chapter and a discussion on future perspectives for MTS antennas are provided in Section 7.7.

7.2

Adiabatic Floquet Waves for Curvilinear Locally Periodic Boundary Conditions The adiabatic Floquet wave expansion is a theoretical model developed to describe currents and fields on locally periodic IBCs, as the ones formed by the patch texture in modulated MTS antennas. The model, that we succinctly denote as “Flat Optics” (FO), has been thoroughly discussed in [38, 39] and will be summarized in this section. In the following, we will refer to the configuration shown in Figure 7.3, where the MTS is defined on a circular area of radius a, and  it is centered at the origin of a ˆ ϕ ˆ . The observation point is on polar reference system (ρ,ϕ), with unit vectors ρ, the MTS, at the interface between the metal cladding and free space, and it is indicated by the position vector ρ = ρ cos ϕˆx + ρ sin ϕˆy. Bold characters denote vectors and bold characters underlined by a double bar denote tensors; k and ζ identify the free-space wavenumber and impedance, respectively. Transverse magnetic (TM) and transverse electric (TE) modes are referred to the normal zˆ to the surface. Our analysis is restricted to MTSs consisting of a lossless grounded dielectric slab of relative permittivity εr and thickness h, printed with perfectly electric conducting subwavelength patches. The metallic cladding constitutes a penetrable IBC with modulated capacitive reactance. We assume that the SW is excited by an electric dipole located at the center of the reference system. The feeder shall be designed to maximize the amount of SW power and to minimize the amount of power directly radiated in the free space. Therefore, the feed structure can be more complex than a simple small vertical dipole in order to maximize

7.2 Adiabatic Floquet Waves for Curvilinear Locally Periodic Boundary Conditions

235

Figure 7.3 Geometry of the MTS antenna: the subwavelength printed patches are modeled by an

anisotropic penetrable tensor reactance, on a grounded dielectric slab (inset on the left corner). At the top right corner, the inset sketches the local periodic problem for the definition of the adiabatic Floquet modes. A black and white version of this figure will appear in some formats. For the color version, please refer to the plate section.

the SW launch efficiency. Nonetheless, the excited SW is still azimuthally symmetric, as the one launched by a vertical dipole, and this is enough for giving validity to our simplified dipole assumption.

7.2.1

Constant Average Non-uniform Reactances We model the metallic cladding constituting the MTS with an anisotropic “penetrable” IBC [38, 40] identified by   (7.1) Et = j X ¨ zˆ ˆ Ht |0+ ´ Ht |0´ = j X ¨ J In (7.1) X is the penetrable reactance of the metallic cladding and J is the electric current flowing in it. X is a Hermitian tensor since the model is defined in absence of losses [33], which implies that its eigenvectors eˆ 1, eˆ 2 are orthogonal and the eigenvalues are real. The class of transparent lossless reactance tensors of our interest is referred to as “constant-average non-uniform reactances”, and is given by

X(¯1) =

X = X(0) + X(´1) + X(+1)

(7.2)

ˆϕ ˆ X(0) = X¯ ρ ρˆ ρˆ + X¯ ϕ ϕ

(7.3)

  1 ˘j Ks(ρ) ” ˆϕ ˆ e˘j ρ (ρ) mρ (ρ) X¯ ρ ρˆ ρˆ ´ X¯ ϕ ϕ e 2 ı   ˆ +ϕ ˆ ρˆ e˘j ϕ (ρ) +mϕ (ρ) X¯ ρ ρˆ ϕ

(7.4)

Where the space independent quantities X¯ ρ and X¯ ϕ are negative (i.e. capacitive), K is a large ρ-independent constant such that K |∇t s (ρ)| ąą |∇t ρ,ϕ (ρ)| (with the transverse gradient operating in ρ), and s(ρ) has unit peak value. The above conditions identify reactances with ρ-independent average reactance with principal axes

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  ˆ ϕ ˆ , and a rapidly oscillating part described by X(¯1) which contains the aligned with ρ, modulation functions mϕ,ρ (ρ). The oscillating part is written in terms of exponentials, which combined together form cosine functions. Within the entries of X(¯1) one can identify three fundamental factors: (1) a rapidly varying phase factor exp (˘j Ks(ρ)), responsible for the main interaction with the exciting SW; (2) a slowly varying phase factor exp ˘j ρ,ϕ (ρ) , mainly controlling the polarization; and (3) the modulation index mϕ,ρ (ρ), mostly determining the amplitude field. The modulation indexes are taken small enough so as to avoid local changes of the nature of the transparent reactance from capacitive to inductive (namely mρ,ϕ (ρ) ă 1), which could excite a higher order, TE-dominant, SW mode. Finally, it is noted that the aforementioned class of reactances is constituted by symmetric tensors, i.e. the subclass of Hermitian tensors which is representative of MTS constituted by elements with two orthogonal symmetry axes.

7.2.2

Adiabatic Floquet Wave Expansion This subsection introduces the key points of the FO model: the reader interested in further details is referred to [38]. One of the main properties of the constant-average reactance X in (7.2) is that the average current flowing in it is quite similar to the SW-current that the dipole source would excite in X(0) . Hence, the 0-index mode of an adiabatic FW expansion can be obtained by an appropriate modification of these currents. Since X(0) is ρ-independent, the SW excited on it by a monopole is a purely TM cylindrical wave whose asymptotic expression is given by (2)

J0 = J0 H1 (βsw ρ) ρˆ

(7.5)

where H1(2) is the Hankel function of second kind and first order. The propagation constant βsw , solution of a dispersion equation affected by the X¯ ρ component only, can be also obtained by the accurate closed-form approximation provided in [36]. In presence of the modulation, the 0-indexed FW mode (simply denoted as 0-mode) is obtained from (7.5) by locally transforming the real unperturbed wavenumber βsw into the complex wavenumber k (0) (ρ) = βsw +β (ρ)´j α(ρ), where the parameters β (ρ) and α(ρ) are a function of the X entries. Hence, it is implicitly assumed that the 0-mode has a cylindrical wavefront and it is radially attenuated with a local attenuation parameter α(ρ) which accounts for the transfer of energy along the propagation path from the   (0) ˜ 0-mode to the ´1 (leaky) mode. By integrating the local relation B k (ρ) ρ {Bρ = k0 (ρ), from the reference point at the origin, we obtain the argument of the Hankel function or global phase k˜ (0) (ρ) ρ, namely, ˜ (0)

k

1 (ρ) = ρ

żρ k

(0)

1 dρ = βsw + ρ 1

0

żρ ”

   ı β ρˆ 1 ´ j α ρˆ 1 dρ 1

(7.6)

0

This gives the following global adiabatic FW expansion for the current: ÿ J(n) J« n (2)

J(n) = j(n) e´j nKs(ρ) H1



k˜ (0) ρ

(7.7)

 (7.8)

7.2 Adiabatic Floquet Waves for Curvilinear Locally Periodic Boundary Conditions

237

where Ks (ρ) is the same as in the definition of the reactances. Despite the adiabatic expansion having an infinite number of terms, when only the ´1 mode falls into the visible spectral range, only the first three terms (n= ´1,0,+1) are sufficient to describe the current. This is the approach we will adopt in the following. The complex terms ˆ + Jρ(n) ρˆ in (7.8) are the slowly varying part of the nth current mode and j(n) = Jϕ(n) ϕ they are the unknowns of the problem. For what concerns the 0-indexed mode, this mode has a dominant TM component Jρ(0) with amplitude much larger than that of the TE component Jϕ(0) and also much larger than both the components of the (˘1) indexed modes. This dominant behaviour is more evident for small modulation indexes: indeed, in the limit case of vanishing modulation, Jρ(0) should be the only current component and it should coincide with J0 in (7.5). From the asymptotic form of the Hankel function for large argument it is seen that each mode in (7.8) propagates with local n-indexed FW wavevector ” ı (7.9) β(n) = Re∇t k˜ (0) ρ + nKs(ρ) = (βsw + β ) ρ + nK∇t s (ρ) and with a local attenuation parameter α(ρ) which does not depend on the modal index n. Each FW current mode has a curvilinear wavefront given by βsw ρ+nKs (ρ) = const. Through the spectral Green’s function (GF) of the grounded slab in (7.9), the adiabatic FW electric fields are given by ÿ ÿ Et (ρ) = E(n) « Z(n) ¨ J(n) (7.10) GF n

n

where Z(n) is the grounded-slab spectral GF evaluated at the local wavevector β(n) (see GF

is purely imaginary for any n ‰ ´1 and it is complex appendix in [38]). Note that Z(n) GF for n= ´1. Typically, the ´1 mode is the only mode of interest to design a modulated impedance that provides a desired radiation pattern. We point out that the basis on which fields and currents have been expanded does not rigorously satisfy Maxwell’s equations, since it is based on an asymptotic, adiabatic approximation which is valid far from the point source. However, this basis locally recovers a Floquet wave expansion and may be used for obtaining an adiabatic solution by using (7.10) in the transparent IBCs (7.2). Retaining only three current modes, we get 1 ÿ

    (+1) (0) (´1) (n) (0) (´1) (+1) Z(n) ¨ J = j X + X + X + J + J ¨ J GF

(7.11)

n=´1

This allows one to find an analytical solution and a local ρ-dependent adiabatic dispersion equation. To this end, the terms in (7.11) with the same rapid phase variation are equalized and a set of equations is obtained. The latter is solved by substitutions, thus leading to a homogeneous system of type j χ (ρ) ¨ J(0) = 0. The latter admits a nonı ” trivial solution only for det χ (ρ) = 0, which is the local dispersion equation. This equation allows to determine the complex value of the 0-mode local wavenumber k (0) . The final solution is individuated in terms of the two TE/TM components of the 0-mode coefficient J(0) (namely, Jϕ(0) and Jρ(0) ), still to be determined. While their ratio is found

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(0)

by using k (0) in the linear system j χ (ρ) ¨ J(0) = 0, the actual value of Jρ can be found by some considerations based on power conservation [38, 39]. Indeed, within a zeroorder approximation, the average current flowing in the constant-average reactance X can be identified with the SW-currents flowing in X(0) in absence of modulation. Thus, (0)

Jρ is estimated by means of the radial Poynting vector associated with the surface wave excited on the uniform MTS [38].

7.3

Design of Modulated MTS Antennas The global synthesis scheme, shown in Figure 7.4 [39], is composed of three main blocks, each of them working on a different model scale. The first block synthesizes the continuous impedance surface, which, when excited by a reference SW, produces the objective aperture field. Then, within the second block, the continuous impedance surface is discretized and implemented through a dense texture of small patches. Finally, the third block analyzes the textured antenna layout through a global full-wave solver. Since the number of pixels is very large (typically several thousands), the global solver must be engineered with smart numerical strategies to overcome the computational complexity arising from the huge number of unknowns of the linear systems arising, for instance, in the solution of integral equations. The design process starts assuming that the radiative components of the objective aperture field are known. We do not treat here the process of synthesizing a continuous

Figure 7.4 Block diagram of the synthesis process (see [39] for further details).

7.3 Design of Modulated MTS Antennas

239

aperture field from the pattern requirements, since this is the subject of a rich literature, (see, e.g., [41] and the references therein). In the first block, the synthesis of the IBCs is performed at “mesoscopic” level (see the left-hand side of Figure 7.4). To that end, one assumes that the patterned metallic cladding is represented by a continuous, spacedependent reactance tensor X, which models the continuous anisotropic IBC related to the discontinuity of the magnetic field. The continuous reactance synthesis block, the core of this task, relies on the analytical FO method detailed in [38–39] and will be summarized in the next paragraph. The output from the continuous reactance synthesis block is the modulated reactance X, generated in just a few seconds using a good notebook, independently of the antenna size. After the synthesis has been performed, a full-wave analysis, still based on the continuous IBC approach, is used to check the accuracy of the synthesis. This step is performed with an extremely fast formulation detailed in [42] and described in the next section, which requires less than ten seconds for antennas with diameter of 20 wavelengths. If the first check with this solver is successful, one proceeds to the synthesis of the obtained IBC by means of sub-wavelength printed patches. Notice that the synthesis procedure could be also extended to different impedance implementation (e.g. metallic pins or equivalent fully metallic elements) by considering impenetrable impedances instead of penetrable ones. The element synthesis (represented by the “pixel modeling” block in Figure 7.4) is carried out at “atomic” level using a full-wave code with Rao–Wilton–Glisson (RWG) basis functions [43]. A local “micro-periodicity” concept is used in this phase assuming the local element immersed in a periodic environment of equal elements with the periodicity of the lattice. Such assumption allows one to apply periodic IBCs on the elemental cell boundaries and to use the periodic Green’s function in the integral equation formulation, thus reducing the computational effort to that of a single unit cell. The analysis of the periodic structure is therefore extremely fast and it is repeated several times to construct a database associating the entries of the anisotropic reactance tensor to the corresponding element geometry. The database can be built before the entire synthesis process starts and can be re-used for other MTS syntheses. Actually, only a few samples of the parameter-space are directly calculated and a denser database is created by interpolation. A few tens of minutes are required to construct a 2D database (i.e. a database with two varying geometrical parameters) and a few hours for a 3D database (i.e. with three varying geometrical parameters). Usually, the database is constructed at the antenna operative frequency: its extension around the operative frequency can be important when the antenna is required to operate over a large bandwidth: at this purpose, the pole-zero matching method [44] or the analytical model presented in [36] can be used. The output of the “atomic” part of the design is a complete layout of the antenna, where the dimensions and the orientation of any single pixel-element are defined. The final phase of the design process is a detailed analysis of the layout, which requires a global full-wave integral equation model implemented at “molecular level”. In case the final check is not successful, a repetition of the continuous model synthesis may be required. Actually, the analytic core of the synthesis, i.e. the “continuous reactance synthesis” block in Figure 7.4, is powerful and accurate, and the overall process rarely

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requires feedback, which is typically needed only for very demanding requirements, like a large bandwidth or a very high efficiency.

7.3.1

Continuous Reactance Synthesis This subsection describes the synthesis process of the continuous reactance based on the FO model introduced in Section 7.2.2. The reader interested in further details is referred to [39]. Using the theoretical framework introduced in Section 7.2, it is possible to set up the synthesis process of the IBCs, namely the continuous reactance synthesis block in Figure 7.4. The objective of the design is to determine the expression of X in (7.2) on the circular antenna aperture, which creates the aperture fields required to radiate the target far field pattern. We describe the latter aperture field by the rather general form ” ı ˆ UA (7.12) EA = E0 e´j k(ρ) eρ (ρ) ej γρ (ρ) ρˆ + eϕ (ρ) ej γϕ (ρ) ϕ where UA is a unit step function having unitary value inside a circle of radius a and zero elsewhere. In (7.12), γρ (ρ), γϕ (ρ), eρ (ρ), eϕ (ρ),  (ρ) are real functions weakly variable with ρ. In order to determine X, we identify EA with E(´1) , namely with field associated with the adiabatic LW field. By doing so, we implicitly assume that the difference between the objective aperture field EA and the Et field in (7.10) is a dominantly reactive field contribution, with spectrum mainly concentrated outside the visible region. This assumption can be verified once the design procedure is completed and the aperture fields are assessed. In order to identify EA with E(´1) , we first use (7.8) in (7.10) to obtain a modal approximation of the electric field through the use of the spectral Green’s function. Next, we identify the electric field modes with the electric field coming from the imposition of the boundary conditions (7.11). At the end, one can determine the LW field E(´1) as a function of the modulation and the 0-mode current as ı´1 ”   (2) ´1) (´1) E(´1) = j Z(GF ¨ ZGF ´ j X(0) ¨ X(´1) ¨ j(0) H1 k˜ (0) ρ (7.13) Identifying EA with E(´1) , therefore using (7.12) in (7.13), allows one to determine amplitude and phase of the modulation to reproduce the objective field EA through an iterative process. By equating the fast varying parts of (7.12) to (7.13), after few simple mathematical operations, the dominant phase term Ks(ρ) of the modulation is found as Ks (ρ) = βsw ρ +

żρ

  β ρ1 dρ 1 ´ k (ρ)

(7.14)

0

Also, by equating (7.12) and (7.13), it possible to determine modulation indexes functions of the transparent reactance in (7.2)–(7.4) through the following formulas: m = σ Q´ 1 ¨ P ¨ e

(7.15)

7.3 Design of Modulated MTS Antennas

241

Figure 7.5 Block diagram for synthesizing the continuous reactance X from the aperture field EA . The initial values are set by using aperture field, aperture size, substrate parameter and frequency as input parameters.

with σ =

 ´1 2E0 ˇ ˇ ; P = 1 ´ j X(0) ¨ Z(´1)  GF ˇ ˇ j Jρ(0) ˇH1(2) k˜ (0) ρ ˇ

ˆ e = eρ ej γρ ρˆ + eϕ ej γϕ ϕ ˆ m = mρ ej ρ ρˆ + mϕ ej ϕ ϕ;    χρϕ  ˆ ´ X¯ ϕ ϕ ˆ ρˆ ˆϕ ˆ ´ Q = X¯ ρ ρˆ ρˆ + ϕ X¯ ρ ρˆ ϕ χϕϕ

(7.16) (7.17) (7.18)

The determination of X involves the iterative process shown in Figure 7.5 and thoroughly described in [39]. The process starts setting the aperture field, aperture size, substrate parameters and operational frequency as input parameters. An initial guess of the modulation parameters (namely, Ks(ρ), mρ,ϕ (ρ), ρ,ϕ (ρ)) is found taking the 0-mode current and wavenumber as the ones in the uniform MTS with average reactance X(0) . Next, this first initial guess of the MTS is analyzed to get the effective values of the complex displacement β (ρ) ´ j α(ρ) on the modulated MTS. The complex displacement is then used to refine the estimation of the modulation parameters in the next iterative step. When the modulation parameters vary less than a chosen threshold, the process is stopped and X is retrieved.

7.3.2

Pixel Modeling and Detailed Layout At this point of the design process, the continuous reactance X that produces the objective aperture field is completely known on the entire aperture domain. The next design step consists in sampling it and implementing it by a dense texture of electrically small patches. Typically, the reactance is discretized in a Cartesian lattice with square cells;

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Modulated Metasurface Antennas

(b)

(a)

(c)

(d)

Figure 7.6 Examples of patch geometries for the synthesis of an anisotropic surface impedance. Each geometry possesses two specific non-dimensional parameters a{a 1 and ψ that are

considered for constructing the reactance database. © 2018 IEEE. Reprinted, with permission, from Gabriele Minatti 2012.

as a rule of thumb, the side of such unit cells ranges from λ/10 to λ/5. However, in principle, other lattices with non-square cells could also be used. The periodicity of the lattice is always taken as a constant on the aperture, and the variation of impedance is achieved by changing the geometrical parameters. Different geometrical shapes can be used for the pixel elements to produce anisotropic impedances, with features to change the electromagnetic properties of SWs propagating along different directions. The solutions of Figure 7.6b and Figure 7.6d are the ones suggested in [9] and [10], respectively. The elliptical shape in Figure 7.6c can be analyzed with the quasi-analytical method proposed in [34]. All the proposed solutions exhibit two non-dimensional parameters a{a 1 and ψ. The reactance values of a given geometry can be recovered from the geometrical parameters by resorting to a local periodicity concept, that is by exploiting periodic boundary conditions applied to a unit cell with parameters a{a 1 and ψ. The reactance values are collected in a database, which is next interpolated defining two continuous functions of the parameters a{a 1 and ψ. The values required in (7.2) are obtained pixel by pixel using a best fit process. This process is typically very fast after the database construction has been performed.

Figure 1.1 Classifications of electromagnetic phenomena in space domain based on the

three-dimensional electric sizes. A black and white version of this figure will appear in some formats.

Figure 1.7 A historical review on the development of surface electromagnetics (SEM): from

uniform surfaces to periodic surfaces and then to quasi-periodic surfaces. This evolution in space variation is analogous to the study of signals in circuits, from DC signals to AC signals and then to modulated signals. A black and white version of this figure will appear in some formats.

Figure 1.6 Prototypes of quasi-periodic surfaces: (a) a Ka-band reflectarray surface consisting of

square patches of varying sizes; (b) an optic beam splitter surface consisting of circular disks of different diameters. A black and white version of this figure will appear in some formats.

Figure 1.11 SEM-based high-gain reflectarrays (RA): (a) a Ku-band RA consisting of 76,176 elements of variable sizes, forming a quasi-periodic surface with a Fresnel-type phase distribution; (b) an X-band RA consisting of 10,240 reconfigurable elements, forming an active surface for 2-D beam scanning and beam shaping. A black and white version of this figure will appear in some formats.

Figure 1.12 SEM research contents organized in this book. A black and white version of this figure will appear in some formats.

eV , mV

,

,

a

b

Figure 2.1 (a) Illustration of the Huygens principle applied to scattering from volumetric

electromagnetic sources. (b) Application of the Huygens principle to the concept of metamaterials. Electromagnetic response from any volumetric material (or metamaterial) in principle can be always reproduced with an arbitrarily shaped closed two-dimensional layer carrying electric and magnetic currents. A black and white version of this figure will appear in some formats.

Figure 2.3 Classification of different types of periodical planar structures with typical topologies

of each type operating at a wavelength λop . The red and green regions denote nonresonant and resonant types of structures, respectively. A black and white version of this figure will appear in some formats.

Figure 2.12 Wave propagation in opposite directions through (a–b) a reciprocal chiral slab and (c–d) a nonreciprocal magneto-optical slab. The transparent cylinders depict the slabs. The red and blue arrows denote the electric fields of left and right circularly polarized waves, respectively. A black and white version of this figure will appear in some formats.

Figure 3.20 Magnitude of the E-field from an electric-line source placed 45.49 mm above an array of spherical particles: (a) f = 1.50 GHz (one surface wave and a complex mode pair) and (b) f = 2.0 GHz (two surface waves). For an array of spherical particles consisting of a = 10 mm, p = 25.67 mm, r = 100, μr = 1, and tan δ = 1 ˆ 10´4 . A black and white version of this figure will appear in some formats.

Figure 3.21 Magnitude of the E-field (on a linear scale) from a magnetic line source placed 45.49 mm above an array of spherical particles: (a) f=1.42 GHz: one surface wave, and (b) f=1.5 GHz: complex mode. A black and white version of this figure will appear in some formats.

Figure 3.29 A fluid-tunable metafilm composed of a 36-element array of metallic resonators fabricated from gold on glass with PDMS (polydomethykiloxane) fluid-channel sections (see [44]). A black and white version of this figure will appear in some formats.

Figure 4.5 1D FDTD Yee grid with a metasurface placed in between the nodes. The two small

circles, before and after the metasurface, represent the electric and magnetic virtual node, respectively. Original image from [35]. A black and white version of this figure will appear in some formats.

Figure 4.4 Position of the metasurface and the virtual nodes in the FDTD grid nodes. The filled

green and red circles represent electric and magnetic virtual nodes just before and just after the metasurface, i.e., at z = 0´ and z = 0+ , respectively. The incident wave propagates in the zy-plane. A black and white version of this figure will appear in some formats.

a

b

Figure 4.9 (a) Transmitted power (|S21 |2 ) and (b) phase for the three cascaded metallic layers

versus the imaginary parts of the impedance of Y1 and Y2 . The black line indicates that full transmission can be achieved and that the corresponding phase varies between 0 and 2π . A black and white version of this figure will appear in some formats.

Figure 5.8 Difference between practical reflection phase of the central element and the one

computed under periodic boundary condition. © 2018 IEEE. Reprinted, with permission, from Tong Liu 2018. A black and white version of this figure will appear in some formats.

a −145◦

b −90◦

c 0◦

d −145◦

e −90◦

f 0◦

Figure 5.13 Design phase distribution (upper row) and practical phase deviation (lower row) for different global reference phase φ. A black and white version of this figure will appear in some formats.

Figure 6.1 (a) The concept of soft and hard surfaces: PEC/PMC strip grid; (b) practical realization

using corrugations. A black and white version of this figure will appear in some formats.

Figure 6.16 Two-dimensional color plots showing the field in packaged microstrip line with a double 90o bend: (a) first doing the packaging with a smooth metal lid and (b) afterwards with a bed-of-nails. © 2018 IEEE. Reprinted, with permission, from Eva Rajo-Iglesias et al. 2010. A black and white version of this figure will appear in some formats.

Figure 6.13 (b) and Figure 6.31 (a) (b) Example of dispersion diagram for a ridge gap waveguide implemented with mushrooms, and (a) manufactured prototype of a leaky wave antenna. © 2018 IEEE. Reprinted, with permission, from Elena Pucci et al. 2012, and Mohamed Al Sharkawy 2014. A black and white version of this figure will appear in some formats.

Figure 6.30 The manufactured prototype of the 4 ˆ 1 linear array and the 2 ˆ 2 planar array. A

black and white version of this figure will appear in some formats.

Figure 6.20 (b) the manufactured ridge gap and groove gap waveguide-based 16 ˆ 16 slot array antenna. A black and white version of this figure will appear in some formats.

Figure 7.1 Examples of realized prototypes of MTS antennas. The SW is excited by a single

probe (right inset of (a)) [12] or by four probes (right inset of (b)) [10]. In both cases, the feeding region is located at the center of the MTS. A black and white version of this figure will appear in some formats.

Figure 7.3 Geometry of the MTS antenna: the subwavelength printed patches are modelled by an

anisotropic penetrable tensor reactance, on a grounded dielectric slab (inset on the left corner). At the top right corner, the inset sketches the local periodic problem for the definition of the adiabatic Floquet modes. A black and white version of this figure will appear in some formats.

Figure 8.9 (a) A quad-layer transmission surface using E-shaped elements. (b) Simulated

transmission coefficient of the proposed surface. A black and white version of this figure will appear in some formats.

Figure 8.10 (a) Geometry of a double-layer transmission surface using Malta-cross elements with vertical vias. © 2017 IEEE. Reprinted, with permission, from Wenxing An, 2017. (b) Simulated transmission coefficients of the surfaces with and without vias. A black and white version of this figure will appear in some formats.

Figure 8.21 Photograph of a transmitarray prototype consisting of quad-layer slotted E-shaped elements, measured by a near-field scanning system in an anechoic chamber. A black and white version of this figure will appear in some formats.

Figure 9.1 (a) Schematic of the 1-bit coding metasurface. Schematic illustrations of the

coding metasurface with different coding sequences: (b) “101010 . . .” coding sequence and (c) chessboard coding sequence. A black and white version of this figure will appear in some formats.

Figure 9.4 (c) A flow chart of a 1D-programmable metasurface under the control of a FPGA.

Adapted from [46] with the permission of Nature Publishing Group. A black and white version of this figure will appear in some formats.

Figure 9.9 Schematic and unit cell design of the proposed single-sensor and single-frequency

microwave imaging system. (c) The 2-bit programmable coding metasurface, in which each unit is controlled by an FPGA with binary codes (0 or 3.3V voltage) on rows (green) and columns (red). (d) The schematic of the proposed single-sensor and single-frequency microwave imaging system. Adapted from [67] with the permission of Nature Publishing Group. A black and white version of this figure will appear in some formats.

Figure 10.1 The non-uniqueness of the Scattering Problem, as related to the scattering operator L.

A black and white version of this figure will appear in some formats.

Figure 10.15 Active cloaking with non-Foster metasurfaces, implemented with Negative

Impedance Converters (NIC). Reprinted with permission from: Pai-Yen Chen et al., Broadening the Cloaking Bandwidth with Non-Foster Metasurfaces, 10.1103/PhysRevLett.111.233001 and 2018 Copyright 2018 by the American Physical Society. A black and white version of this figure will appear in some formats.

Figure 10.16 (left) Physical bounds on cloaking: trade-off between scattering reduction and

fractional bandwidth, according to Bode-Fano matching theory. (right) Frequency spectra in terms of cloaking techniques used. Reprinted, with permission, from “Invisibility and cloaking structures as weak or strong solutions of Devaney-Wolf theorem”, The Optical Society. Francesco Monticone and Andrea Alù, “Invisibility exposed: physical bounds on passive cloaking,” Optica 3, 718–724 (2016). A black and white version of this figure will appear in some formats.

Figure 11.3 Schematic of a GEO satellite-based navigation and guidance system serving moving vehicles. © 2018 IEEE. Reprinted, with permission, from Mehdi Veysi 2018. A black and white version of this figure will appear in some formats.

Figure 11.8 Ideal (a) phase and (b) amplitude of the RHCP field on the Bessel-beam reflectarray’s cells upon reflection from reflectarray surface. The reflectarray diameter is D = 7.5λ. © 2018 IEEE. Reprinted, with permission, from Mehdi Veysi 2018. A black and white version of this figure will appear in some formats.

Figure 11.16 Fourier-transform RHCP radiation pattern (in dB) of an azimuthal multi-beam

reflectarray composed of two concentric annular segments (as shown in Figure 11.15a) for different (l1,l2 ) combinations at 30 GHz. The inner segment radius and azimuthal index number are set at rin = 1.5λ and l1 = +1, respectively, while the outer segment radius and azimuthal index number are rout = 3.75λ (4λ) and l2 = ˘3(˘4), respectively, where λ = 10 mm. © 2018 IEEE. Reprinted, with permission, from Mehdi Veysi 2018. A black and white version of this figure will appear in some formats. Figure 11.17 Full-wave RHCP radiation pattern of a quad-beam reflectarray with rin = 1.5λ,

rout = 3.75λ (where λ = 10 mm), l1 = 1, and l2 = ´3 at 31 GHz. © 2018 IEEE. Reprinted, with permission, from Mehdi Veysi 2018. A black and white version of this figure will appear in some formats.

Figure 12.2 (a) A photograph of a wideband linear-to-circular polarization rotator. (c) Photograph of a metasurface-coated chiral waveguide. A black and white version of this figure will appear in some formats.

Figure 12.3 (a) A photograph of a flexible CP wearable antenna with a metasurface ground plane. A black and white version of this figure will appear in some formats.

Figure 12.3 (c) A photograph of the miniaturized reconfigurable UHF CP antenna with a metasurface ground plane. A black and white version of this figure will appear in some formats.

Figure 12.7 (a) A photograph of a millimeter-wave metasurface transmit-array. Simulated and measured (d) A photograph of a multi-beam antenna enabled by a metasurface-based transmit-array. A black and white version of this figure will appear in some formats.

Figure 12.8 THz modulators. (a) Schematic (c) Scanning electron microscopy images and (e) Schematic of the VO2 integrated memory device that consists of a gold SRR array on a VO2 film, and the measured persistent electrical tuning behavior is shown in (f). A black and white version of this figure will appear in some formats.

(a) 1

2

3

4

5

6

7

8

(b) x (E)

x (E) y (H)

y (H)

θt

Figure 12.9 THz wavefront and polarization controllers. (a) Schematic of a three-layered metasurface structure as a broadband wavefront controller and the corresponding measured cross-polarized transmittance at 1.4 THz under normal incidence as a function of angle. From Grady et al., 2013, Terahertz Metamaterials for Linear Polarization Conversion and Anomalous Refraction, Science. Reprinted with permission of AAAS. (b) Schematic and measured polarization conversion (in transmission) results of a THz metasurface polarization converter. (Reprint from [180].) A black and white version of this figure will appear in some formats.

7.3 Design of Modulated MTS Antennas

243

(a)

(b)

Figure 7.7 Example of impedance maps for the patch geometry of Figure 7.6d ((a) Xρρ , (c) Xρϕ ,

(e) Xϕϕ ) and for the patch geometry of Figure 7.6c. ((b) Xρρ , (d) Xρϕ , (f) Xϕϕ ). Maps (a), (b), (e) and (f) have been normalized to X¯ s = 300. All the geometries are in a lattice with cell size λ{13 on a dielectric with thickness λ{23 and εr = 13.

Figure 7.7 shows examples of databases, namely the impedance maps, relevant to the patch geometry of Figure 7.6d and Figure 7.6c, respectively, obtained through a periodic full-wave MoM solver. All of the examples are evaluated for a periodic square cell with side a « λ/13 on a dielectric slab with εr =13 and thickness λ/23. The maps show the components of the opaque reactance tensor Xρρ ,Xρϕ ,Xϕϕ normalized to an impedance X¯ s = 300 (for the Hermitian property of the reactance tensor it results Xρϕ = Xϕρ ).

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(c)

(d)

Figure 7.7 (Cont.)

Although a systematic way to build the database is given by full-wave analysis [36, 40], the alternative approach presented in [37] can also be used. By using such a technique, one can construct an extensive database for various shapes, while minimizing the number of parameters to be stored and reducing memory requirements. The formulation is however restricted to elements with at least two symmetry axes and it is valid for the dominant TM SW until the limit of the Floquet–Bloch region [36] (i.e. the region where higher order Floquet modes become significant).

7.4 Analysis of Modulated Metasurface Antennas

245

(e)

(f)

Figure 7.7 (Cont.)

7.4

Analysis of Modulated Metasurface Antennas The analysis of modulated MTS antennas appears at two different stages of the design process and involves the use of two different approaches. Indeed, such analysis can be performed either on the homogenized IBCs to check the accuracy of the synthesis process or on the whole pixelated layout, as “virtual prototyping” of the antenna structure. In both cases, as modulated MTS antennas are usually electrically large, some strategies have to be adopted to reduce the unknowns of the problem and its computational burden. The global full-wave analysis of the pixelated layout, usually the last step of the design, is a cumbersome task, since the final layout is composed of several thousands of patches with different shapes. The problem to be solved may consist of several millions

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of unknowns in a conventional MoM procedure, and even more with other techniques such as FDTD or FEM. Furthermore, the discretization of the problem generally results in an ill-conditioned linear system. This issue can be faced by using basis functions obtained by grouping RWG functions on a singular value decomposition basis [45]. Alternatively, the MTS can be textured with suitable patch geometries so that entire domain basis functions can be used: if the currents on the patches are represented by the combination of few entire domain basis functions, the number of unknowns of the problem can be significantly reduced. For instance, currents on elliptical patches can be effectively represented using the entire domain basis functions defined in [34]. Also, the MoM procedure can be furthermore engineered using an algorithm based on the Fast Multiple Method (FMM) to accelerate the solution [46–48]. On the other hand, the analysis of the homogenized impedance, despite being performed on the simpler model of the continuous IBCs, is still effective for the prediction of the antenna performance. In the next subparagraph we will focus on a MoM solver for IBCs rather than on the global full-wave solver for the sake of brevity. In particular, we will discuss the analysis of the homogenized impedance with a MoM technique with Gaussian ring basis functions. The method based on such basis functions, detailed in [42], allows us to reduce the computational time thanks to the analytical closed form of the reaction integrals.

7.4.1

Full-Wave Homogenized Impedance Analysis The continuous reactance obtained in Section 7.3.1 is verified at this step, as explained in the block diagram of Figure 7.4, by using a full-wave solver that supports a spacedependent IBC for planar apertures. In general, the synthesized apertures are electrically large, and an efficient analysis is thus necessary to complete the “mesoscopic” (homogenized impedance) block of the scheme in Figure 7.4. One can model the problem at hand by imposing a transparent IBC on the air–dielectric interface of a grounded dielectric slab. To that end, we assume the substrate is infinite in the x-y plane, and we exploit the appropriate Green’s functions. Therefore, after imposing the IBC on the interface, it is possible to reduce the unknowns to horizontal electric currents on the IBC plane. The resulting integral equation (IE) reads as fi » ij (7.19) zˆ ˆ – GEJ (ρ,ρ1 ) ¨ J(ρ1 ) dS 1 ´ j X(ρ1 ) ¨ J(ρ1 ) fl = ´ˆz ˆ Einc S1

with ρˆ and ρˆ 1 being the source and observation points, respectively. GEJ stands for the electric dyadic Green’s function, Einc is the incident electric field imposed by the excitation, and J represents the unknown surface current density. It is important to notice that using a formulation with the transparent reactance X instead of the opaque one obeys to a double purpose. On the one hand, as detailed in [40], this approach is helpful to overcome ill-conditioning issues often encountered in IEs based on opaque reactance for typical values of the surface reactance found in MTS antennas and transformation optics devices. On the other hand, we can easily combine the transparent IBC with

7.4 Analysis of Modulated Metasurface Antennas

247

the layered medium Green’s functions, which allows one to efficiently account for the effect of the grounded slab, since the unknowns in (7.19) will be just the electric currents J. Therefore, it is possible to include in the analysis the spectral dispersion due to the thickness of the grounded dielectric slab [35] without increasing the number of unknowns. In addition to the advantages of the IE formulation described above, it is important to note that, provided that the IBC is modulated at a mesoscopic scale, the evolution of the currents on the aperture will be mesoscopic too, i.e., with variations of the order of one wavelength. Therefore, it is possible to describe such currents with a number of unknowns drastically lower than the ones needed for classical discretization, like the ones based on RWG basis [43]. To that end, entire-domain basis functions have been recently proposed to reduce the number of unknowns by two orders of magnitude. One can find two families of functions for the analysis of circular MTS antennas, namely, Fourier–Bessel basis functions (FBBFs) [49] and Gaussian ring basis functions (GRBFs) [42]. The former are div-conforming and orthogonal on the aperture, while the latter are just quasi-orthogonal. However, GRBFs admit closed-form expressions for the reaction integrals. Each family has its own advantages, but in the remainder of this chapter we will present GRBFs only for the sake of brevity. GRBFs constitute a particularly convenient choice for representing J on circular apertures, and they can be written in the space-domain as ρ ρ  1 ´ (ρ´ρ2m )2 m 4σ e ϒ (7.20) fm,n (ρ,ϕ) = e´jnϕ m,n (ρ); m,n (ρ) = n 2σ 2 2σ 2 where ϒn (z) = In (z) e´|Retzu| , with In (z) being the modified Bessel function of the first kind and order n. Equation (7.20) corresponds to a Gaussian-type ring with linear azimuthal phase. GRBFs peak at ρm and their half-maximum beam-width (HMBW) is ? values of σ . One can 4 ln 2σ . Figure 7.8a shows the shape of fm,n (ρ,0) for different ? also fix σ to a constant value and choose ρm = ρ0 + m 4 ln 2σ (m = 0,1,2,3 . . .) for numbering the basis functions in a progressive sequence of GRBFs that cross each other at half of their maximum (see Figure 7.9a). Since the dependence of m,n (ρ) on n (Bessel function’s order) is extremely weak for this particular choice of ρm , changing n maintains the sequence of functions unchanged. On the other hand, the spectral-domain counterpart of (7.20) presents an analytical form: 2 2

m,n (kρ ) = e´σ kρ Jn (ρm kρ ) (7.21) b where kρ = kx xˆ + ky yˆ = kρ cos α xˆ + kρ sin α yˆ , with kρ = kx2 + ky2 and kx , ky being the spectral variables of the classical Fourier domain. Equation (7.21) can be used for obtaining closed-form expressions of the MoM impedance matrix elements. Examples of GRBFs spectra are shown in Figures 7.8b and 7.9b. The surface current density on the MTS plane can be expanded using the proposed basis functions as Fm,n (kρ ) = e´jnα 2πjn m,n (kρ );

J(ρ) =

Mÿ ´1

{2 N ÿ

m=0 n=´N {2

y

y

x im,n fxm,n (ρ)xˆ + im,n fm,n (ρ)yˆ

(7.22)

Modulated Metasurface Antennas

(a)

ψm0(ρ) normalized to maximum

1 4 ln2 σ / λ0 = 0.1 4 ln2 σ / λ0 = 0.15

0.75

4 ln2 σ / λ0 = 0.2 4 ln2 σ / λ0 = 0.25

0.5

0.25

0

0

0.25

0.5

0.75

1 ρ / λ0

1.25

1.5

1.75

2

(b) 1

φm0(kρ) normalized to maximum

248

0.8

4 ln2 σ / λ0 = 0.1

0.6

4 ln2 σ / λ0 = 0.15

0.4

4 ln2 σ / λ0 = 0.2 4 ln2 σ / λ0 = 0.25

0.2 0

–0.2 –0.4 0

2

4

kρ / k0

6

8

10

Figure 7.8 Visualization of the GRBF (a) in the space-domain as? a function of ρ{λ0 and (b) in the spectral-domain as a function of kρ {k0 , for different values of 4 ln 2σ (ρm {λ0 = 1, n = 0). © 2018 IEEE. Reprinted, with permission, from David González-Ovejero 2015.

where M and N/2 are the number of basis functions along ρand the highest order of the azimuthal harmonics, respectively. In (7.22), x- and y-directed components are required to account for the vector nature of the current. After testing with the complex conjugate of the basis functions and arranging the unknowns in the system of equations according to their direction, one arrives to an algebraic system Zi = v, where Z = rZxx ,Zxy ;Zyx ,Zyy s, with Zxx being an M(N + 1) ˆ M(N + 1) matrix. Similarly, i = rix ;iy s v = rvx ; vy s with ix and vx being M(N + 1) ˆ 1 vectors. A generic entry of the Zxx sub-matrix can be written as the sum of ż ż ˇ ˇ  1 1 ‹ 1ˇ x ˇ (ρ) GEJ (7.23) Gxx = fx, xx ( ρ ´ ρ ) fm1,n1 ρ dS dS m,n S

S1

7.4 Analysis of Modulated Metasurface Antennas

249

(a) ym0(r) normalized to maximum

1

m=0 m=1 m=2 m=3 m=4

0.75

0.5

0.25

(b)

1

φm0(kρ) normalized to maximum

0

0.8

0

0.25

0.5

0.75

1 r / l0

1.25

1.5

1.75

2

m=0 m=1 m=2

0.6

m=3 m=4

0.4 0.2 0 –0.2 –0.4

0

1

3

2

4

5

kρ / k0

Figure 7.9 Visualization of the GRBF (a) in the space-domain as a function of ρ{λ ? 0 and (b) in the + m4 ln 2σ spectral-domain as a function of kρ {k0 , for different values of ρm = ρ0 ? (m = 0,1,2,3 . . .) with ρ0 = 0.5λ0 and m = 0,1, . . . ,4 (σ = 0.25λ0 {(4 ln 2), n = 0). © 2018 IEEE. Reprinted, with permission, from David González-Ovejero 2015.

and, χ

xx



ż

‹ xx x fx, m,n (ρ)jX (ρ)fm1,n1 (ρ)dS

(7.24)

S

  Both quantities depend implicitly on m,n,m1,n1 . In (7.23) and (7.24), the asterisk EJ

xx denotes the complex conjugate and GEJ xx is the xx component of G . The term G xx involves the relevant grounded slab Green’s function, whereas χ is related to the impedance boundary condition. It can be shown [42] that the integral in (7.23) can be written in closed-form after combining the spectral-domain approach and rational function fitting, whereas the use of an asymptotic expansion allows one to evaluate (7.24) without resorting to any numerical integration. The other sub-matrices can be

250

Modulated Metasurface Antennas

found by using formal substitution of superscripts. Thus, the proposed basis functions allow one to write the MoM impedance matrix entries in a closed form. Figure 7.10 shows a schematic structure of the 2M(N + 1) ˆ 2M(N + 1) MoM impedance matrix obtained using GRBFs. The Green’s function contribution to the MoM impedance matrix and the IBC one present different structures. Figure 7.10a shows that the Green’s function contribution (Gxx ) consists of (N + 1) ˆ (N + 1) blocks; for instance, the white dashed line marks the particular case m = m1 = 0 and n,n1 P [´N{2,N {2]. Such (N + 1) ˆ (N + 1) blocks are square tri-diagonal matrices, with null entries for |n ´ n1 | ą 2. It is also important to notice that the Gxx contribution does not depend on X and should be calculated only once in an optimization procedure. On the other hand, the IBC contribution is a dominantly blockdiagonal matrix, which means that only the few off-diagonal (N + 1) ˆ (N + 1) blocks with |m ´ m1 | ď 2 contain elements with significant magnitudes. The structure of the IBC matrix reflects the fact that χ xx is non-vanishing only for overlapping GRBFs. Having a limited number of entries with non-negligible magnitude allows one to reduce the computation time of each step in an iterative optimization process of the profile of X. Moreover, the magnitude of the entries in the IBC contribution is substantially larger than that of the corresponding entries in the Green’s function. Indeed, the colour-bar values in Figure 7.10b are 10 times larger than those of the colour-bar in Figure 7.10a. This feature suggests that IBCxx can be used as preconditioner for the iterative solution of very large problems (larger than 30λ). We have shown how the introduction of the MTS as a transparent IBC in the integral equation, and the closed-form spectra of the GRBFs enable an efficient computation of the MoM impedance matrix using the spectral-domain approach and an asymptotic expansion. More importantly, these basis functions represent the global evolution of the surface current density in an effective manner, which results in a significant reduction of the number of unknowns, if compared with sub-entire domain basis functions, like Rao–Wilton–Glisson functions defined on triangular domains.

7.5

Efficiency and Bandwidth of Modulated Metasurface Antennas Modulated MTS antennas are based on a SW to LW transformation phenomenon by means of the SW interaction with the modulated MTS, which controls the aperture field. Indeed, the non-uniform modulation of the MTS determines phase, amplitude and polarization of the aperture field, thus fully controlling the antenna performance. For a given structure and surface, it is possible to shape the beam and choose the field polarization, simply by changing the surface impedance pattern. The impedance pattern determines also efficiency and bandwidth of this class of antennas. Efficiency is defined here as the ratio between the gain of the antenna and the maximum directivity of an ideal, lossless, uniformly illuminated aperture with the same area. The other performance indicator is the gain bandwidth, namely, the frequency range in which the gain is reduced less than a given threshold. In this section we will discuss the efficiency and bandwidth of modulated MTS antennas, paying special attention to the case of broadside, circularly polarized beams.

7.5 Efficiency and Bandwidth of Modulated Metasurface Antennas

251

(a)

(b)

Figure 7.10 Magnitude of the MoM impedance matrix elements for a typical structure with M = 48 and N = 16. For simplicity, the first 51 rows and columns are shown for (a) the matrix with the Green’s function contribution and (b) the matrix with IBC contribution. © 2018 IEEE. Reprinted, with permission, from David González-Ovejero 2015.

7.5.1

Efficiency of Metasurface Antennas In order to quantify the efficiency achievable by MTS antennas it is necessary to consider several aspects related to the radiation mechanism [50]. These are sketched in Figure 7.11: assuming a power Pin at the input port of the feeder, part of it is directly radiated in free space (Pf eed ), while the remaining power is transformed into a SW (Psw ). The SW power is partly lost in the antenna structure due to losses (P ), partly radiated as a LW (Plw ), while the remaining part reaches the antenna rim. This latter gives rise to diffraction at the edge (Pdiff ) and reflection, which produces an

252

Modulated Metasurface Antennas

Figure 7.11 Sketch of the time average power contributions characterizing the MTS antenna efficiency. Pin : input time-averaged power. Psw : power transformed by the feeder into a SW. Pf eed : power directly radiated by the feed as space wave. Pdiff : power diffracted at the edge. Pswb : SW power transported by edge-excited FWs. Plw : power transferred from SW and high order FWs to LW and radiated in free space. Plwb : power transferred from edge-excited SW/HO-FW to LW. P : power dissipated by ohmic and dielectric losses.

inward SW carrying the power Pswb . In practical designs, Pswb doesn’t reach the input port in a significant quantity, so that one may assume that all the reflected power is reradiated (Plwb ). All the phenomena relevant to the introduced wave-mechanism are characterized through the definitions of relevant efficiencies that will be discussed in the following.

Feed Efficiency The feed efficiency εf eed is defined as the ratio between the input power and the power delivered to the SW, i.e. εf eed = Psw {Pin = 1 ´ Pf eed {Pin

(7.25)

In an ideal design, all the input power at the feeder of a MTS antenna is delivered to the SW, thus eliminating Pf eed . However, a real feeder will always have a residual Pf eed , directly radiated in free space. The optimum excitation of surface waves for this kind of antenna has been studied in detail in [50], where it is shown that high values of εf eed (around 90%) can be reached with a SW launcher as simple as a coaxially fed monopole on a circular patch, provided the size of the patch, the substrate and the average opaque impedance are properly chosen (Figure 7.12a). However, more complex feeding solution can be sought for obtaining higher values of εf eed once the substrate and the opaque impedance have been chosen. Practical TM feeds can be realized by (1) a coaxially fed monopole on a circular metallic disc (Figure 7.12a); (2) a circular waveguide fed by a TM01 (coax-type) mode, coupled to the MTS with a circular slot on the ground (Figure 7.12b); and (3) a corrugated top cover fed by a probe (Figure 7.12c). However, such a discussion is beyond the scope of this chapter and the interested reader is referred to [50–58] for further insight.

7.5 Efficiency and Bandwidth of Modulated Metasurface Antennas

253

(a)

(b)

(c)

Figure 7.12 Example of standard feed configurations for MTS antennas (top row): (a) a probe with a annular patch; (b) slot ring etched on the ground plane; (c) corrugated hat above the MTS. Bottom row sketches the simplified models used for the analysis of the efficiency of the feeds.

Ohmic Efficiency The ohmic efficiency ε is defined by only considering the losses in the dielectric. This is a reasonable assumption in the microwave regime where the dielectric losses dominate on the metal losses. We define the ohmic efficiency as the ratio between the

254

Modulated Metasurface Antennas

radiated power Plw and the SW power Psw in presence of losses and the same ratio in absence of losses Plw {Psw (7.26) ε = Plw |P =0 {Psw In presence of ohmic-losses the propagation constant βsw on a non-modulated MTS becomes slightly complex, namely βsw Ñ β sw = βsw ´ j αp , with the phase parameter βsw not significantly affected. The attenuation constant αp can be estimated by solving the dispersion equation with a perturbative approach.

Conversion and Tapering Efficiency Conversion efficiency εconv represents the fraction of radiated LW power with respect to the SW power. In absence of losses, i.e. when P = 0, the conversion efficiency is L εconv = Plw |P =0 Psw (7.27) The tapering efficiency εtap is related to the directivity loss of a given aperture illumination with respect to a uniform distribution. For broadside beams, it is quantified as ˇť ? ˇť ˇ2 ˇ2 ˇ ˇ ˇ ˇ A Et dA Ať S (ρ)dA = εtap = ť (7.28) A A S (ρ) dA A |Et |2 dA A

where Et is the aperture field distribution and S(ρ) is the power density distribution associated with Et . The right-hand side of (7.28) has been obtained by using the relation ? Et « 2ζ S (ρ) valid for broadside beam antennas. The quantities in (7.27) and (7.28) are controlled by the attenuation parameter α, which, in turn, is controlled by the amplitude of the modulation [39].

Overall Efficiency Figure 7.13 shows a plot of the overall efficiency, obtained as the product of all the efficiencies previously introduced, for broadside, circularly polarized, MTS antennas with different sizes. We have considered a simple feeder, realized with a vertical electric monopole top loaded with a circular patch (Figure 7.12a) and designed to maximize the power of the excited SW. Two different lossy substrates have been considered with relative permittivity 9.8 and 4.5, respectively, and with identical dissipation factor tan δ = 0.002. The considered power distribution on the aperture is given by $ ρλ ď 0.5 sin2 (πρλ ) ; S (ρλ ) & = (7.29) 0.5 ă ρλ ď aλ ´ 2 ‰ “ 1; Smax % sin2 π4 (aλ ´ ρλ ) ; aλ ´ 2 ă ρλ ď aλ so that tapering and conversion efficiencies approximately follow the relation b εconv = εtap = aλ { (aλ + 2)

(7.30)

where aλ is the aperture radius normalized to the free space wavelength. The overall efficiency is plotted in Figure 7.13 for several values of the opaque (impenetrable) impedance. A design that makes use of standard substrates, a non-uniform modulation and a simple feeder consisting of a vertical electric dipole with a metallic disc printed

7.5 Efficiency and Bandwidth of Modulated Metasurface Antennas

255

Figure 7.13 Overall efficiency for MTS antennas of several sizes, radiating a broadside, circularly polarized beam. The curves are obtained for substrates with thickness h such that kh = 0.2, with loss tangent of 0.002, with different values of the dielectric permittivity. Several values of the normalized opaque impedance X¯ op {ζ have been considered.

on the MTS (see the inset on top-right corner of Figure 7.13) can easily reach 75% efficiency. The values shown by Figure 7.13 should be considered as typical overall efficiency figures for a good design and can be improved with a more refined design. On the other side, as for any other antenna typology, efficiency may be adversely affected by additional requirements on the antenna pattern, like a very low side lobes level or cross-polar radiation.

7.5.2

Bandwidth of Gain The rate at which the antenna loses gain when changing frequency is related to the dispersion of the MTS [59]. In fact, this dispersion introduces a variation βsw Ñ βsw + βsw and α Ñ α + α which implies a phase mismatch between the SW-wavelength and the period of the modulation, as well as a distortion of the amplitude of the aperture field. The main detrimental effect on the gain is due to the phase error; therefore, only the effect of βsw will be considered hereinafter. From the first order Taylor expansion of βsw (ω) at the center frequency, we can obtain the value of βsw . βsw «

Bβsw 1 ω = ω Bω vg

(7.31)

where vg is the group velocity of the SW and ω is the unilateral angular-frequency bandwidth. We refer to the antenna bandwidth as the frequency range in which the gain is less than 3dB lower than the gain at the operating frequency. The antenna gain is defined as G = (ka)2 ε

(7.32)

256

Modulated Metasurface Antennas

where (ka)2 is the directivity of a uniformly illuminated circular aperture of radius a and ε is the antenna efficiency. From (7.32), the reduction of gain with frequency is translated into a decrease of the antenna efficiency. Naming ε the efficiency at the design working frequency, and ε1 the efficiency when a shift βsw arises on the wavenumber due to the frequency change, the unilateral bandwidth is obtained when the condition ε1 = 0.5ε is met.

Bandwidth of a Highly Efficient MTS Antenna We discuss here the bandwidth of a highly efficient MTS antenna radiating a broadside beam. On the antenna surface, we consider the power distribution given in (7.29), which provides a good tapering efficiency. We also assume that the major contribution to the reduction of ε comes from the reduction of the tapering efficiency εtap , which is coherent with the premise that the phase error is the main detrimental effect on the gain. Therefore, the unilateral bandwidth is given by the condition: ˇ2 ˇ2 ˇż a ˇż a ˇ ˇ ˇ a ˇ a ´j βsw ρ ˇ = 1ˇ ˇ ˇ S ρdρ S (7.33) (ρ)e (ρ)ρdρ ˇ ˇ ˇ ˇ 2 0 0 To estimate the antenna bandwidth, we use (7.29) into (7.33) and solve this latter numerically for finding βsw . The numerical solution is well approximated by βsw « 3.77{a. Using the latter expression in (7.31) gives a bilateral relative bandwidth B=2

vg 1 ω 1 ă 1.2 « 1.2 ω c aλ aλ

(7.34)

where c is the speed of light in free space. Figure 7.14 summarizes efficiency and bandwidth of broadside MTS antennas as a function of the antenna radius, for several values of the opaque impedance. The results in Figure 7.14 have been obtained for a lossless substrate with permittivity εr = 10 and thickness h = 0.2{k, and implementing the power distribution in (7.29). Efficiency plots assume a feed efficiency as the one of a vertical electric dipole with a metallic disc printed on the MTS (as in Figure 7.13). One can conclude from Figure 7.14 that higher values of the opaque impedance increase the antenna efficiency at the cost of a smaller bandwidth. A representation useful in practice relates the antenna gain in dBi for εf eed ε = 1 to the percentage bandwidth 100B = B% . To this purpose, we evaluate the product between the antenna gain in (7.32) and the half-power relative bandwidth in (7.34) when εf eed ε is assumed equal to unity and εconv εtap comes from (7.30): „ j vg aλ (7.35) aλ ă 47.37aλ GB « 47.37 c (aλ + 2) where the term in square brackets can be approximated to unity for large aλ . Using again Eq. (7.34) and taking the logarithm leads to:     1 B% B% G[dB] « 57 ´ 10 log 1 + ´ 20 log (7.36) 60 γ γ where γ = vg {c is the group velocity normalized to the speed of light. Figure 7.15 plots (7.36) as function of the percentage bandwidth for γ ranging from 0.1 to 1. Also, the

7.5 Efficiency and Bandwidth of Modulated Metasurface Antennas

257

Figure 7.14 Efficiency and bandwidth plots of a broadside MTS antenna, realized on a lossless substrate with permittivity ε = 10 and thickness h = 0.2{k, and implementing the power distribution in (7.29). Efficiency curves have been obtained assuming a feeder constituted by vertical electric dipole top loaded with an annular patch. Circular dots on bandwidth curves come from the FO analysis [38].

Figure 7.15 Gain as a function of relative bandwidth for the tapered distribution in (7.29) (dotted line) and a uniform amplitude distribution. Plots are for values of the relative group velocity γ = vg {c ranging from 0.1 to 1.

258

Modulated Metasurface Antennas

same picture shows the theoretical bandwidth of a MTS antenna designed for having a uniform illumination. Although this case is not physically feasible, it is useful as a theoretical limit for the bandwidth of a MTS antenna designed to maximize the antenna efficiency.

7.6

Examples of Antenna Design

7.6.1

Shaped Beam Antenna Antennas on board of satellite platforms for Earth observation used for data download are usually required to radiate a circularly polarized field with an isoflux pattern. Such a shaped beam provides a uniform power-flux density (hence the name “isoflux”) over the visible portion of the Earth’s surface. Hence, the antenna compensates the rather large differential path loss between Nadir and grazing incidence on the Earth with a higher gain at grazing. Standard isoflux antennas provide Earth coverage with a conical isoflux beam [10]. However, higher gains, enabling higher data rate transmissions, can be obtained with a sector isoflux beam: in this case, an azimuthal mechanical rotation is provided to the antenna to keep the beam pointing toward the ground station while the satellite moves along the orbit. In this section we provide an example of MTS sector isoflux beam antenna operating at the frequency of 26.25 GHz, inside the Ka-band. The antenna is designed with printed patches on a grounded dielectric substrate having relative permittivity of 9.8 and 0.5 mm thickness. The radius of the circular aperture is 9λ, and the antenna converts 90% of the input SW power into LW power. The target aperture distribution can be found through standard techniques. In this example, we have extended the method presented in [60] to polarized aperture fields. Figures 7.16a and 7.16b show the amplitude and the phase, respectively, of the objective aperture field distribution for the radial component, whereas Figures 7.16c and 7.16d depict the amplitude and the phase of the azimuthal component. Once the input parameters for the continuous reactance synthesis block in Figure 7.4 have been determined, the surface impedance is synthesized through the process described in Section 7.3.1, assuming the impedance surface excited by a small vertical electric dipole at the center of the surface. Figure 7.17 shows the obtained values of X on the whole aperture for the example presented in this section. The calculated surface impedance is next analyzed through the continuous impedance boundary condition full-wave solver (IBC-MoM) [42]. If the result for IBC-MoM analysis is judged satisfactory, the metasurface is implemented by a tight texture of electrically small patches. In this example, we have chosen elliptical patches as the one depicted in Figure 7.6c: this type of patches allows to use the entire domain basis functions described in [34] to create the database maps and to analyze the final layout with a global full-wave solver. The final layout for this example is shown in Figure 7.18 and it is composed of 8936 elliptical patches with geometrical parameters gradually variable with the position on the surface. The final structure is next analyzed using a global full-wave solver: the current on each patch is described by two entire domain basis functions and a fast multiple algorithm is used to speed up the MoM procedure [48]. Directivity patterns obtained from the Flat Optics solution (Section 7.2),

7.6 Examples of Antenna Design

(a)

(c)

259

(b)

(d)

Figure 7.16 Target aperture field distributions used for the example of the isoflux shaped beam antenna: (a) amplitude and (b) phase of the radial component. Amplitude and phase of the azimuthal component are, respectively, in (c) and (d).

the IBC-MoM solution (Section 7.4.1) and the MoM solution of the textured layout are shown and compared in Figure 7.19. Finally, Figure 7.20 shows the directivity pattern obtained with the IBC-MoM analysis on the whole visible spectral u-v plane.

7.6.2

Multibeam Modulated Metasurface Antennas In this section we explore three different possibilities for obtaining a multibeam radiation pattern with a single metasurface (MTS) aperture. Two single-source and one multisource feeding schemes are considered. The problem has been described in detail in [61] and [62]: here we report just the main results, which show the potential of MTS antennas for multibeam applications. The presented designs consist in MTS antennas radiating four right-handed circularly polarized (RHCP) beams, tilted 30˝ from the z

260

Modulated Metasurface Antennas

(a)

(b)

(c)

(d)

Figure 7.17 Impedance maps of the components of X resulting from the synthesis process: (a) Xρρ , (b) Xρϕ , (c) Xϕρ and (d) Xϕϕ .

Figure 7.18 Layout of the antenna implemented by elliptical pixel elements with details of the patch texture close to the center of the antenna.

7.6 Examples of Antenna Design

261

Figure 7.19 Copolar (RHCP) directivity diagrams coming from different analysis tools are compared. The dashed line is from the objective aperture distribution in Figure 7.16. The thick solid line is the directivity diagram radiated by the currents from the Flat Optics (FO) [38, 39] analysis. The pale solid line is the result of the IBC-MoM analysis of the synthesized continuous impedance surface [42]. Finally, the black solid line is obtained by the FMM-MoM analysis of the textured layout of the antenna [48].

axis. Nonetheless, the presented approach is also valid for beams with different pointing angles and polarizations [61]. The frequency of operation and substrate parameters in the following examples are the same as in Section 7.6.1, whereas the radius of the antennas is 12λ. First, we will describe the two single-source configurations. The first one consists in dividing the MTS aperture in several angular sectors, as shown in Figure 7.21a, each one devoted to the formation of a beam in a given direction. The directivity patterns obtained with this approach are shown in Figures 7.21b and 7.21c. In the second single-source approach, the whole aperture is shared by a superposition of individual modulations, which correspond to those required to obtain beams in the desired set of directions. In this case, instead of dividing the aperture in independent angular sectors, the individual modulation patterns for every beam are added up at each point of the aperture, as shown in Figure 7.22a. The aperture sharing solution provides beams with a higher directivity (see Figures 7.22b and 7.22c), than the aperture partition solution. This is expected, given that each beam is formed by the entire aperture instead of by just one sector. In the single-source configurations, exciting the feeding point meant that the four beams appear at the same time, which can be interesting for transmitting antennas. However, on receive, waves coming from the four principal lobes will add up in the backend and it will not be possible to process the received signals individually. The configuration based on a multi-source feeding scheme allows to process separately the incoming waves. Also, the multi-source scheme offers the possibility of having one independent

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Modulated Metasurface Antennas

(a)

(b)

Figure 7.20 Directive diagrams in the spectral u-v plane obtained by analyzing the continuous

synthesized impedance surface with a full-wave MoM solver for continuous IBCs [42]: (a) LHCP and (b) RHCP components.

beam at a time, as opposed to the single-source case where all the beams coexist at the same time. Dividing the aperture in angular sectors would require a complex feeding system in which each source illuminates only its corresponding sector. Nevertheless, one can still apply the superposition of modulation patterns to obtain one independent

7.6 Examples of Antenna Design

263

(a)

(b) MoM vs Flat Optics: directive patterns comparison, j=0 25

MoM: RHCP MoM: LHCP Flat Optics: RHCP Flat Optics: LHCP

Directivity [dBi]

20 15 10 5 0 –5 –10 –90

–60

–30

0

30

60

90

q [deg]

(c) MoM - RHCP Directive Pattern (dBi) 1

25 20

0.5

15

v

10 0

5 0 –5

–0.5

–10 –1 –1

–0.5

0

0.5

1

–15

u Figure 7.21 Multibeam MTS antenna realized by dividing the aperture in sectors, each of them producing a beam: (a) Xρρ component of the tensor obtained for a single point source MTS antenna with aperture partition; (b) comparison between the directivity patterns obtained with IBC-MoM analysis [42] and Flat Optics model [38] in the ϕ = 0 plane; (c) directivity diagram resulting from IBC-MoM in the u-v spectral plane. © 2018 IEEE. Reprinted, with permission, from David González-Ovejero 2017.

Modulated Metasurface Antennas

(a)

(b) RHCP - j = 0 LHCP - j = 90 RHCP - j = 90 LHCP - j = 0

Directivity [dBi]

25 20 15 10 5 0 –5 –10 –90

–60

–30

0 q [deg]

30

60

90

(c) MoM - RHCP Directive Pattern (dBi)

v

264

1

30

0.5

20

0

10

0

–0.5

–1 –1

–10 –0.5

0

0.5

1

u Figure 7.22 Multibeam MTS antenna with a single source and aperture sharing: (a) Xρρ

component of the impedance tensor obtained by a superposition of modulated impedance patterns, each modulation provides a beam in one of the desired directions; (b) directivity patterns from IBC-MoM analysis [42] in the ϕ = 0 and ϕ = 90 planes; (c) directivity patterns obtained with IBC-MoM in the u-v spectral plane. © 2018 IEEE. Reprinted, with permission, from David González-Ovejero 2017.

beam per source. The main difference with respect to the previous configuration is that each of the superimposed impedance patterns will present a different reference point in the aperture, as shown in Figure 7.23a. Such reference will be the position of the source

7.6 Examples of Antenna Design

265

(a)

(b) MoM - RHCP Gain [dBi] 1

30 25

0.5

20

v

15 0 10 5

–0.5

0 –1 –1

–0.5

0 u

0.5

1

–5

(c) MoM - LHCP Gain [dBi] 1

30 25

0.5

20

v

15 0 10 5

–0.5

0 –1 –1

–0.5

0 u

0.5

1

–5

Figure 7.23 Multi-source multibeam MTS antenna: (a) Xρρ component of the impedance tensor

obtained for four sources and aperture sharing by superposition of four modulations; (b) RHCP and (c) LHCP directivity patterns in the u-v spectral plane obtained with IBC-MoM analysis when the VED at x = 1.5λ, y = 0 is excited; (d) RHCP and LHCP directivity patterns obtained exciting each of the VED at a time in the ϕ = 0 and ϕ = π{2 planes. © 2018 IEEE. Reprinted, with permission, from David González-Ovejero 2017.

266

Modulated Metasurface Antennas

(d) 30

VEDI, RHCP, f = 0 VED2, RHCP, f = 0 VED3, RHCP, f = p/2 VED4, RHCP, f = p/2

25

Directivity [dBi]

20 15 10 5 0 –5 –90

–60

–30

0

30

60

90

J [deg] Figure 7.23 (Cont.)

we need to excite in order to generate the beam associate to the relevant modulation. In Figure 7.23 we present the result of a MTS antenna with four vertical electric dipoles (VED) located at distance 1.5λ from the antenna center, along the positive and negative x and y axes. The antenna can produce four independent RHCP beams, depending on which VED is activated. Figures 7.23b and 7.23c represent the RHCP and LHCP directivity patterns, respectively, obtained with IBC-MoM in the u ´ v plane when the VED at x = 1.5λ, y = 0 is excited. Finally, Figure 7.23c depicts two cuts of the RHCP far-fields obtained with IBC-MoM along the u and v spectral lines when exciting each of the four VEDs at a time.

7.7

Discussion and Future Outlook Modulated MTS antennas represent a significant improvement from traditional antenna solutions since, rather than shaping a fixed boundary condition (mostly PEC) to achieve desired radiation characteristics, they change the boundary condition itself to achieve the desired performance, without affecting the global shape of the antenna. In fact, all the design examples herein shown are relevant to antennas which differ only in the MTS layout and feeding details. The full control of the aperture field (and hence of the far-field properties like beam shape, pointing angle and polarization) is strictly related to the capability of accurately designing the IBC imposed by the MTS. Here, an effective synthesis method has been introduced which permits the control of amplitude, phase, and polarization of the aperture field by adjusting the boundary conditions imposed by the MTS. The complete procedure is described in [39]. Broadly speaking, the IBC anisotropy controls the field

7.7 Discussion and Future Outlook

267

polarization, and the shape and periodicity of the modulation control the phase of the aperture field, while field amplitude is controlled by the amplitude of the modulation. The adiabatic Floquet-wave expansion of currents and fields over the surface introduced in [38] allows to set up an effective synthesis process whose main advantages with respect to the previously proposed methods are the following: • •



A systematic way is introduced to synthesize an aperture field with analytical formulas and effective validation tools are also developed as sketched in Figure 7.4. New formulas based on energy conservation have been introduced for controlling the aperture field amplitude and designing antennas with higher efficiency than before. Transparent boundary conditions have been used in place of the opaque one (as used in previous schemes), applied within a rigorous Green’s function scheme and accounting for space dispersion of the grounded slab in practical implementations.

Future research trends span from the design of conformal modulated MTS antennas to the design of a reliable dynamic impedance pattern. Conformal MTSs are interesting in that they would allow to further reduce the encumbrance and the profile of these antennas by perfectly matching them to curved surfaces. For instance, one may use conformal MTSs matched to the fuselage of airplanes or of any other fast moving vehicle, thus reducing the impact of the antenna on the aerodynamics. Dynamic control of the impedance patterns is of importance for tracking applications that require to hold on a link between moving platforms, as for instance radars, data connection between satellite and ground stations or data connections on board of moving vehicles. The dynamic control of MTS antennas is absolutely appealing for the telecom-market. Indeed, in modulated MTS antennas, the surface impedance pattern completely controls the beam shape and polarization. Therefore, dynamically controlled impedance surfaces virtually allow to address all the degrees of freedom of the radiated wave, with conceptually very simple means and amenable to attractive implementations. At a glance, possible solutions for reconfigurable dynamic MTSs span from mechanical stretching of a proper dielectric material, to reconfigurations by electronic devices. A mechanical reconfigurable surface seems to be particularly suitable whenever antennas are required to produce pencil beams, steerable in one direction. At the opposite end of the spectrum of solutions, an electronically reconfigurable surface could exploit all the potentialities of MTS antennas in terms of flexibility of radiative characteristics. Materials having characteristics similar to liquid crystals at RF or based on arrays of MEMS as well as electronic devices like PiN diodes, are also good candidates for adaptive surfaces. The final goal is to realize a device able to radiate a wide range of arbitrarily shaped beams, circularly or linearly polarized by simply modifying the surface pattern impedance, without changing the in-plane feeder. Enlargement of the pattern-bandwidth of modulated MTS antennas is another important research topic, since the limited pattern bandwidth is currently their most significant limitation: in some recent works [63–64] the use of cells with glide symmetry has been proposed for realizing planar MTS lenses with ultra wide band.

268

Modulated Metasurface Antennas

References [1] C. L. Holloway, A. Dienstfrey, E. F. Kuester, J. F. O’Hara, A. K. Azad and A. J. Taylor, “A discussion on the interpretation and characterization of metafilms/metasurfaces: The two dimensional equivalent of metamaterials,” Metamaterials, vol. 3, no. 2, pp. 100–112, 2009. [2] M. G. Silverinha, C. A. Fernandes and J. R. Costa, “Electromagnetic characterization of textured surfaces formed by metallic pins,” IEEE Trans. Antennas and Propag., vol. 56, no. 2, pp. 405–415, 2008. [3] D. González-Ovejero, T. J. Reck, C. D. Jung-Kubiak, M. Alonso-DelPino and G. Chattopadhyay, “A class of silicon micromachined metasurface for the design of high-gain terahertz antennas,” IEEE Int. Symp. Antennas Propag. (APSURSI), Fajardo, 2016, pp. 1191–1192. [4] A. Arbabi, Y. Horie, M. Bagheri and A. Faraon, “Dielectric metasurfaces for complete control of phase and polarization with subwavelength spatial resolution and high transmission”, Nature Nanotechnol., vol. 10, pp. 937–943, 2015. [5] C. L. Holloway, E. F. Kuester, J. A. Gordon, J. O’Hara, J. Booth and D. R. Smith, “An overview of the theory and applications of metasurfaces: the two-dimensional equivalents of metamaterials,” IEEE Antennas Propag. Mag., vol. 54, no. 2, pp. 10–35, 2012. [6] G. Minatti, F. Caminita, M. Casaletti and S. Maci, “Spiral leaky-wave antennas based on modulated surface impedance,” IEEE Trans Antennas Propag., vol. 59, no. 12, pp. 4436– 4444, 2011. [7] A. M. Patel and A. Grbic, “A printed leaky-wave antenna based on a sinusoidally-modulated reactance surface,” IEEE Trans. Antennas Propag., vol. 59, no. 6, pp. 2087–2096, 2011. [8] G. Minatti, M. Faenzi, E. Martini, F. Caminita, P. De Vita, D. Gonzalez-Ovejero, M. Sabbadini and S. Maci, “Modulated metasurface antennas for space: synthesis, analysis and realizations,” IEEE Trans. Antennas Propag., vol. 63, no. 4, pp. 1288–1300, 2015. [9] B. H. Fong, J. S. Colburn, J. J. Ottusch, J. L Visher and D. F. Sievenpiper, “Scalar and tensor holographic artificial impedance surfaces” IEEE Trans. Antennas Propag., vol. 58, no. 10, pp. 3212–3221, 2010. [10] G. Minatti, S. Maci, P. De Vita, A. Freni and M. Sabbadini, “A circularly-polarized isoflux antenna based on anisotropic metasurface”, IEEE Trans. Antennas Propag., vol. 60, no. 11, pp. 4998–5009, 2012. [11] M. Faenzi, F. Caminita, E. Martini, P. De Vita, G. Minatti, M. Sabbadini and S. Maci, “Realization and measurement of broadside beam modulated metasurface antennas,” IEEE Antennas Wireless Propag. Lett., 2016. [12] F. Caminita, G. Minatti, E. Martini, M. Sabbadini, F. De Paolis and S. Maci, “Very high gain metasurface VSAT antenna”, ESA Antenna Workshop on Innovative Antenna Systems and Technologies for Future Space Missions 2017, Noordwijk, Netherlands. [13] S. Pandi, C. A. Balanis and C. R. Birtcher, “Design of scalar impedance holographic metasurfaces for antenna beam formation with desired polarization,” IEEE Trans. Antennas Propag., vol. 63, no. 7, pp. 3016–3024, 2015. ´ [14] M. Casaletti, M. Smierzchalski, M. Ettorre, R. Sauleau and N. Capet, “Polarized beams using scalar metasurfaces,” IEEE Trans. Antennas Propag., vol. 64, no. 8, pp. 3391–3400, 2016. [15] A. Tellechea, F. Caminita, E. Martini, I. Ederra, J. C. Iriarte, R. Gonzalo and S. Maci, “Dual circularly-polarized broadside beam metasurface antenna,” IEEE Trans. Antennas Propag., vol. 64, no. 7, pp. 2944–2953, 2016.

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[16] M. Sabbadini, G. Minatti, S. Maci and P. De Vita, “Method for designing a modulated metasurface antenna structure,” Patent WO 2015090351 A1, 25 June 2015. [17] S. Maci, G. Minatti, M. Casaletti and M. Bosiljevac, “Metasurfing: addressing waves on impenetrable metasurfaces,” IEEE Antennas Wireless Propag. Lett., vol. 10, pp. 1499–1502, 2011. [18] E. Martini and S. Maci, “Metasurface transformation theory,” in Transformation Electromagnetics and Metamaterials, D. H. Werner and D. H. Know, editors, pp. 83–116, Springer, 2013. [19] C. Pfeiffer and A. Grbic, “A printed, broadband Luneburg lens antenna,” IEEE Trans. Antennas Propag., vol. 58, no. 9, pp. 3055–3059, 2010. [20] M. Bosiljevac, M. Casaletti, F. Caminita, Z. Sipus and S. Maci, “Non-uniform metasurface Luneburg lens antenna design,” IEEE Trans. Antennas Propag., vol. 60, no. 9, pp. 4065– 4073, 2012. [21] C. Pfeiffer and A. Grbic, “Metamaterial Huygens’ surfaces: tailoring wave fronts with reflectionless sheets,” Phys. Rev. Lett., vol. 110, no. 19, 197401, 2013. [22] M. Selvanayagam and G. Eleftheriades, “Discontinuous electromagnetic fields using orthogonal electric and magnetic currents for wavefront manipulation,” Opt. Express, vol. 21, no. 12, pp. 14409–14429, 2013. [23] N. Yu, P. Genevet, F. Aieta, M. A. Kats, R. Blanchard, G. Aoust, J.-P. Tetienne, Z. Gaburro and F. Capasso, “Flat optics: controlling wavefronts with optical antenna metasurfaces,” IEEE J. Sel. Topics Quantum Electron., vol. 19, no. 3, 4700423, 2013. [24] P. S. Kildal, E. Alfonso, A. Valero-Nogueira and E. Rajo-Iglesias, “Local metamaterialbased waveguides in gaps between parallel metal plates,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 84–87, 2009. [25] A. M. Patel and A. Grbic, “Transformation electromagnetics devices based on printed-circuit tensor impedance surfaces,” IEEE Trans. Microw. Theory Tech., vol. 62, no. 5, pp. 1102– 1111, 2014. [26] R. Quarfoth and D. Sievenpiper, “Surface wave scattering reduction using beam shifters,” IEEE Antennas Wireless Propag. Lett., vol. 13, pp. 963–966, 2014. [27] A. Vakil and N. Engheta, “Transformation optics using graphene,” Science, vol. 332, pp. 1291–1294, 2011. [28] R. Yang and Y. Hao “An accurate control of the surface wave using transformation optics,” Opt. Express, vol. 20, no. 9, pp. 9341–9350, 2012. [29] R. Yang, W. Tang and Y. Hao, “Wideband beam-steerable flat reflectors via transformation optics,” IEEE Antennas Wireless Propag. Lett., vol. 10, pp. 1290–1294, 2011. [30] W. Tang, C. Argyropoulos, E. Kallos, S. Wei and Y. Hao, “Discrete coordinate transformation for designing all-dielectric flat antennas,” IEEE Trans. Antennas Propag., vol. 58, no. 12, pp. 3795–3804, 2010. [31] M. Mencagli Jr, E. Martini, D. González-Ovejero and S. Maci, “Metasurface transformation optics,” J. Optics, vol. 16, 125106, 2014. [32] M. Mencagli Jr, E. Martini, D. González-Ovejero and S. Maci, “Metasurfing by transformation electromagnetics,” IEEE Antennas Wireless Propag. Lett., vol. 13, pp. 1767–1770, 2014. [33] E. Martini, M. Mencagli and S. Maci, “Metasurface transformation for surface wave control”, Philos. Trans. R. Soc. A, vol. 373, 20140355, 2015.

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8

Transmission Surfaces and Transmitarray Antennas Fan Yang and Shenheng Xu

8.1

Introduction Surface electromagnetics studies the interactions of electromagnetic waves with various natural or engineered surfaces. These interactions can be categorized into two groups: surface wave interaction and plane wave interaction. The former group focuses on the wave propagation along surfaces, and two representative examples in this group are the modulated meta surface antennas and the gap waveguide technology of previous chapters. The latter group mainly studies the reflection and transmission of a plane wave when it impinges on a specific surface. This chapter will present the designs and applications of some transmission surfaces that belong to the latter group. In general, an engineered surface is designed to convert an incident wave into a scattered wave with the desired magnitude, phase, and polarization [1, 2]. According to the surface equivalence principle, if we enclose actual sources, such as antennas or scatterers, with an imaginary surface, the field outside that surface can be determined by the equivalent surface electric and magnetic fields that satisfy the boundary conditions [3]. On one hand, if we can obtain the surface electric and magnetic fields from a source either through computation or measurement, we will be able to determine the radiation field outside the surface. On the other hand, if we know the source and expect to obtain a radiation field that is different to the direct radiation field from the source, we can design an engineered surface to enclose the original source. This is then capable of changing the incident field from the source to a transmitted field that will generate the desired radiation pattern. Some well-known designs, such as a parabolic reflector or a lens antenna, can be considered as representative examples, since they convert a broad radiation beam directly from a source to a narrow focused beam from the surface [4]. Recently, many engineered surfaces have been proposed with advanced and dynamic capabilities to manipulate the electromagnetic waves impinging on them. These surfaces have exciting applications in antennas, microwaves, and optics. Analog to the reflector and lens, these engineered surfaces can also be grouped into reflection-type [1] and transmission-type [2], and this chapter focuses on the transmission-type engineered surfaces, so called transmission surfaces. To manipulate an electromagnetic wave passing through a transmission surface, an effective approach is to discretize the continuous surface into an array of scattering elements. According to the Nyquist sampling theory, as long as the element spacing is smaller than half wavelength, it will have the same far-field angular performance as

272

8.1 Introduction

273

a corresponding continuous surface. After discretization, two issues need to be considered: the local feature that studies the scattering property of each element, and the global feature that studies the distribution of the elements’ scattering properties over the surface. For the first issue, various structures are designed to satisfy different requirements. Numerous frequency selective surfaces (FSSs) [5] are designed to achieve the magnitude control. For example, periodic dipole elements work as a band stop filter, and periodic slot elements work as a band pass filter [5]. Phase shifting surfaces (PSSs) [2, 6] are designed to realize the phase control while polarization grids [7, 8] and meander-line polarizers [9, 10] are designed to achieve the polarization control. Recently, advanced transmission surfaces are designed to achieve hybrid control of magnitude, phase, and polarization altogether [11, 12]. Furthermore, when control devices such as PIN diodes or varactor diodes are integrated into surface elements, reconfigurable control of electromagnetic waves can be achieved, which will lead to many new functionalities and applications [12–20]. Once the local feature of each individual transmission element is determined, the next task is to determine the distribution of these elements over the surface. Conventional FSSs [5] use identical elements over the surface, hence forming a periodic grid. Recently, quasi-periodic surfaces, which consist of similar elements on a periodic lattice, attract more attention since they exhibit comprehensive capability to manipulate an electromagnetic wavefront [1, 2]. For example, a transmission surface with a gradient phase variation can realize anomalous reflection and transmission [21]. Hologram surfaces are used in optics to generate three dimensional images in space [22, 23]. Transmitarray antennas with Fresnel-zone distributions are designed to radiate a high gain focused beam [24]. Conventional array theory [2, 25–27] has been used to analyze the performance of these surfaces, and various optimization algorithms including the alternating projection method (APM) [26–28] and the particle swarm optimization (PSO) algorithm [27, 29, 30] have been utilized to synthesize these surfaces. Transmission surfaces have found exciting applications in microwave, terahertz (THz), and optical frequency ranges. Compared to conventional devices, such as reflectors and lenses, transmission surfaces have thinner profiles, lighter weight, and lower fabrication costs [2]. It can be built to conform with installation platforms. Furthermore, since each element can be individually designed, transmission surfaces provide much more functionalities than conventional curved surfaces. In addition, they are easy to be integrated with semiconductor devices in order to provide real-time dynamic control. In microwaves, they are used to design novel transmitarrays and radomes with high efficiency and advanced performance [31–37]. In optics, they are used to design metalens with nanometer-scale thickness [38, 39], which is desirable for integrated optical systems. With the research progress of transmission surfaces, novel electrical and optical devices will keep on emerging that will propel the development of future communications, radar, and imaging systems. This chapter presents some of the latest progress on transmission surfaces, and special emphasis will be put on the transmission phase control due to its essential role in wave phenomena. In Section 8.2, we derive the theoretical phase limits for various transmission surfaces, which can be used as general guidelines to design transmission

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surfaces. Based on this discovery, several specific transmission surfaces are presented in Section 8.3 to demonstrate their capabilities for phase control, where the low profile requirement, angular stability, and cross-polarization levels are considered for practical designs. Furthermore, reconfigurable transmission surfaces with dynamic phase control capabilities are presented in Section 8.4. Finally, a representative application of transmission surfaces, namely, the transmitarray antenna, is presented in Section 8.5. The antenna concept, design procedure, and a design example are illustrated in details.

8.2

Phase Limits of Transmission Surfaces In this section, phase limits of various transmission surfaces are analytically derived based on fundamental physical principles, namely, the energy conservation law and the field continuity condition on surfaces. Hence, the phase limit results are general for surfaces with arbitrary geometries. It is revealed that the phase limits of transmission surfaces are mainly determined by the number of layers, the substrate property, and the separation between layers. The phase limits of single, double, and multilayer transmission surfaces are studied in detail, and their validity is demonstrated through full-wave simulations of representative surface examples. Furthermore, discussions on nonidentical layers, cross-polarizations, and vertical vias are presented for further development of the phase limit study.

8.2.1

Phase Limit of a Single-Layer Transmission Surface When an electromagnetic (EM) wave impinges on a natural or engineered surface, reflection and transmission occur, as shown in Figure 8.1. The reflection coefficient (R) and transmission coefficient (T ) can be calculated as R=

E1´

+,

E1

T =

E2+

E1+

(8.1)

Note that two assumptions need to be considered when using the above formulas •

The surface is assumed to be uniform or periodic so that only the fundamental modes are concerned. For higher-order spatial modes, the corresponding coefficients can be obtained through the Floquet analysis.

Figure 8.1 Transmission and reflection of an electromagnetic wave incident on an infinitesimally

thin surface.

8.2 Phase Limits of Transmission Surfaces



275

The transmitted field and reflected field are assumed to be co-polarized with the incident field. If cross-polarized fields are generated by the surface, the corresponding coefficients can be defined and calculated.

For a single-layer transmission surface whose thickness is infinitesimally thin compared to the operating wavelength, the following equation can be obtained from the boundary condition of the electric field continuity on the dielectric or perfect electric conductor (PEC) interface: 1+R =T

(8.2)

Furthermore, if the surface is passive and lossless, the following equation can be obtained based on the energy conservation law: |R|2 + |T |2 = 1

(8.3)

A complex transmission coefficient (T ) can be expressed in terms of the magnitude (t) and phase (ϕ) as T = t ¨ ej ϕ

(8.4)

R = t ¨ ej ϕ ´ 1

(8.5)

Substituting (8.4) into (8.2),

Substituting (8.4) and (8.5) and after some basic mathematical manipulation, t = cos(ϕ)

(8.6)

Equation (8.6) reveals the general relation between the transmission magnitude and the transmission phase, which is independent of the type of element on the surface. Figure 8.2 shows this relationship in a polar diagram, where the transmission coefficient lies on a circle centered at (x = 0.5, y = 0) with a radius of 0.5. The transmission phase is between [´90˝ , +90˝ ]. The maximum transmission (t = 1 = 0 dB) is achieved only at 0˝ phase. In practice, we could allow the transmission coefficient to a certain level, such as ´1 dB or ´3 dB. As shown in Figure 8.2, for ´1 dB criterion, the corresponding phases are ˘ 27˝ , and the transmission phase range is then 54˝ . For ´3 dB criterion, the corresponding phases are ˘ 45˝ , and the transmission phase range is 90˝ .

Figure 8.2 Transmission coefficient of a single-layer surface in a polar diagram. © 2018 IEEE.

Reprinted, with permission, from Ahmed H. Abdelrahman 2014.

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Transmission Surfaces and Transmitarray Antennas

(a)

(b)

(c)

Figure 8.3 Various single-layer surfaces with different geometries: (a) cross dipole, (b) cross slot,

and (c) double square loop. © 2018 IEEE. Reprinted, with permission, from Ahmed H. Abdelrahman 2014.

To demonstrate the validity of the above analytical derivation, several different transmission surfaces are analyzed using full-wave simulation software. Figure 8.3 shows the unit cells of three representative transmission surfaces: • • •

A cross dipole element in Figure 8.3a that has a band stop feature A complementary cross slot element in Figure 8.3b that has a band pass feature A double square loop element in Figure 8.3c that has a multi-resonant feature

The operation frequency is 8.4 GHz, and the unit periodicity is half-wavelength (P = 17.86 mm). The element width is 1 mm, and the element length is varied to realize different transmission coefficients. The conductor is assumed to be PEC while its thickness is assumed to be infinite thin. The simulated transmission coefficients of these three surfaces are plotted in Figure 8.4, as well as the analytical results. Although they have different element geometries and different resonant features, the simulation results align very well with the analytical ones, which demonstrate the generality of the theoretical analysis. Their differences are manifested on the different portions they occupy on the analytical circle. The cross dipole occupies the lower left region of the circle, and a total blockage (t = 0) is observed when L = 17 mm. The cross slot occupies the upper right region of the circle, and a total transmission (t = 1) is observed when L = 16.9 mm. The double square loop has a double-resonance feature, so it occupies the entire circle and rotates along it twice. To summarize, the transmission magnitude and phase relation of a single-layer transmission surface has been analytically derived [2, 27, 40], and it has been verified by full-wave simulations of representative surfaces. Although the surfaces can be different, they follow the same analytical circle on the polar diagram, hence the same transmission phase limit: 54˝ for transmission above ´1 dB and 90˝ for transmission above ´3 dB.

8.2.2

Phase Limits of Multilayer Transmission Surfaces As discussed in the previous section, a single-layer surface has a limited transmission phase range that is usually insufficient for electromagnetic wave manipulation. To

8.2 Phase Limits of Transmission Surfaces

(a)

277

(b)

(c)

Figure 8.4 Transmission coefficients of different single-layer surfaces: (a) cross dipole, (b) cross

slot, and (c) double square loop. © 2018 IEEE. Reprinted, with permission, from Ahmed H. Abdelrahman 2014.

Figure 8.5 Geometry of a double-layer transmission surface. © 2018 IEEE. Reprinted, with

permission, from Ahmed H. Abdelrahman 2014.

enlarge the phase range, multilayer surfaces are widely used. The scattering matrix of multilayer transmission surfaces can be analytically derived by cascading the S matrices of each layer, such as the metal layer, the dielectric substrate layer, etc. Thus, the theoretical study in the previous section can be extended to multilayer transmission surfaces [2, 27, 40]. A double-layer transmission surface, as shown in Figure 8.5, can be considered as three cascaded sections: the top conductor layer, the dielectric substrate, and the bottom conductor layer. For simplicity, assume that both conductor layers are identical, so they have the same scattering matrix: j „ j „ R T cos(ϕ)ej ϕ ´ 1 cos(ϕ)ej ϕ (8.7) = rS1 s = rS3 s = cos(ϕ)ej ϕ cos(ϕ)ej ϕ ´ 1 T R

278

Transmission Surfaces and Transmitarray Antennas

(b)

(a)

Figure 8.6 Transmission coefficients of different double-layer transmission surfaces: (a) different dielectric constants of the substrate (βLd = 90˝ ) and (b) different thicknesses of the substrate (εr = 1). © 2018 IEEE. Reprinted, with permission, from Ahmed H. Abdelrahman 2014.

Apparently, the scattering matrix of the conductor layer is only a function of the transmission phase ϕ. For the dielectric layer, the scattering matrix is » fi (1 ´ )e´j 2βLd (1 ´ e´j 2βLd ) — ffi 1 ´ 1 ´ e´j 2βLd ffi — 1 ´  2 e´j 2βLd rS2 s = — (8.8) ffi – (1 ´  2 )e´j 2βLd (1 ´ e´j 2βLd ) fl 1 ´  2 e´j 2βLd 1 ´  2 e´j 2βLd where =

? 1 ´ εr ? , 1 + εr

β=

? 2π εr λ0

(8.9)

It is clear that the scattering matrix S2 depends both on the substrate’s permittivity (εr ) and the substrate’s thickness (Ld ). After cascading the three S matrices together, the transmission coefficient of the double-layer surface is analytically derived and plotted in Figure 8.6. The transmission coefficient exhibits a heart shape in the polar diagram. By changing the dielectric constant (εr ) or substrate thickness (Ld ), the phase range could be expanded in regions B and C, at the cost of magnitude reduction at point A. After a parametric study, it is revealed that the maximum phase range of a double-layer transmission surface is 170˝ for a ´1 dB transmission threshold and 229˝ for a ´3 dB transmission threshold. This phase range is much wider than that of a single-layer surface; however, it is not enough for a full 360˝ phase range yet. To further enhance the phase range, an effective approach is to stack more layers. Using the same cascading approach as the double-layer surface analysis, the transmission coefficients for the triple-layer and quad-layer surfaces are obtained (shown in Figure 8.7). For simplicity, we also assume that the conductor layers are identical in the analysis here and that the layer separation is quarter-wavelength. Figure 8.7a presents the transmission coefficients of triple-layer surfaces with different substrate permittivities. It is noticed that the transmission coefficient curves are

8.2 Phase Limits of Transmission Surfaces

(a)

279

(b)

Figure 8.7 Transmission coefficients of different transmission surfaces: (a) triple-layer

configuration and (b) quad-layer configuration. © 2018 IEEE. Reprinted, with permission, from Ahmed H. Abdelrahman 2014.

symmetric with respect to the horizontal axis of the polar diagram, and a maximum transmission (t = 1 = 0 dB) is obtained with a phase of 180˝ . The case of εr = 1 has the smallest phase range. Increasing the substrate permittivity enhances the transmission magnitude at certain phase regions (C, D, and E), and reduces it in other regions (A and B). After a parametric study, it is revealed that the maximum phase range of a triplelayer transmission surface is 308˝ for a ´1 dB transmission threshold and full 360˝ for a ´3 dB transmission threshold. Figure 8.7b presents the transmission coefficient of a quad-layer surface. It is found that a transmission phase range of full 360˝ can be readily achieved with a transmission loss smaller than 1 dB. In this case, air substrate (εr = 1) is used, and the layer separation is quarter-wavelength. To verify the effectiveness of this analysis, a full-wave simulation of a quad-layer surface consisting of the double square loop (Figure 8.3c) is carried out, and the results are also plotted in Figure 8.7b. It is observed that the simulation results agree well with the analytical results, and a full 360˝ phase range is obtained for this specific configuration.

8.2.3

Discussion on Nonidentical Layers, Wire Coupling, and Cross-Polarization The phase limit study here provides general guidelines to design transmission surfaces. The maximum phase range is mainly determined by the number of layers, the separations between these layers, and substrate permittivity, which is generally applicable to various surface geometries. It should be pointed out that several assumptions are considered here to simplify the analytical derivation. For example, the conductor layers are assumed to be identical in the multilayer study. In practice, if the layers are different, more design freedom can be utilized, and the transmission phase range will be different. Some detailed analysis can be found in [27]. It is noticed that for double-layer surfaces, nonidentical layers always have a narrower phase range. However, for triple-layer surfaces, nonidentical layers could result in a wider phase range, even a full 360˝ for certain designs.

280

Transmission Surfaces and Transmitarray Antennas

Another important assumption is used in the coupling analysis between different conductor layers. When cascading the S matrices of different layers, only the scattering coefficients of fundamental modes are used while the higher-order modes are ignored. When the layers are very close to each other, the coupling effects of higher-order modes need to be included. Furthermore, only the space coupling is considered in the cascading analysis. If different layers are connected with conductive wires, electric currents can flow directly from one layer to the other, leading to a strong direct coupling. This strong wire coupling will also change the transmission phase range significantly. One more interesting assumption is the co-polarized assumption in the analysis. The electromagnetic field is a vector field, and in the plane wave case, there are two orthogonal components perpendicular to the wave propagation direction. Although many devices use only co-polarized fields, cross-polarized fields can lead to lots of interesting phenomena. In fact, if the cross-polarized field is considered, Eqs. (8.1) to (8.6) need to be updated accordingly, and the transmission phase range of crosspolarized field is no longer limited to [´90˝ , +90˝ ]. More details will be discussed in the next section.

8.3

Transmission Surface Designs The previous section provides general guidelines to design transmission surfaces with certain phase ranges. In this section, several transmission surfaces consisting of specific elements, including a quad-layer E-shaped element [41, 42], a double-layer Maltacross element with vias [37], and a single-layer defected slot element [43], will be presented. Practical design considerations, such as oblique incidence effect, low profile requirement, and cross-polarization consideration, will be discussed in details.

8.3.1

A Quad-Layer Transmission Surface Using E-Shaped Elements According to the phase limits discussed previously, a quad-layer transmission surface can provide full 360˝ phase range with transmission loss smaller than 1 dB. Hence, we start with a quad-layer transmission surface consisting of a dipole element, as shown in Figure 8.8a. The dipole element is the most fundamental geometry in electromagnetic scatterers. By varying the dipole length (L), the resonant feature will change accordingly that results in a variation of the transmission phase. However, due to the single-resonant band-stop feature, a dipole element cannot cover the full range of the transmission circle, as previously shown in Figure 8.4a. To increase the phase coverage, a dual-resonant element needs to be considered. A direct idea is to extend the length of the dipole to reach the second resonance. Since each element is arranged in a periodic lattice with the size of a half-wavelength, a long dipole needs to be bent in order to fit into this lattice. To maintain the same polarization of the dipole, an asymmetric U-shaped element is designed, as shown in Figure 8.8b. This element has two horizontal strips and a short vertical connecting strip, and the equivalent length (EL) is approximately the sum of the two strip lengths. When EL equals to half-

8.3 Transmission Surface Designs

281

Figure 8.8 Evolution of a transmission surface element: (a) dipole element, (b) asymmetric

U-shaped element, (c) E-shaped element, and (d) slotted E-shaped element.

wavelength and one-wavelength, two resonances can be obtained. Hence, it can provide the expected full phase coverage. The asymmetric U-shaped element achieves the full phase range, but it suffers from a relatively high cross-polarization. This is due to the unwanted radiation from the orthogonal current flowing on the vertical connecting strip. To suppress the cross polarization, an E-shaped element is evolved subsequently, as shown in Figure 8.8c. The E-shaped element can be considered as a combination of two asymmetric U-shaped elements, which are mirrored with respect to each other. Hence, the orthogonal currents on the two connecting strips have opposite directions, and their radiation is cancelled, which leads to a low cross-polarization. Another important consideration in transmission surface designs is angular stability. In practice, electromagnetic waves impinging on a transmission surface may have different incident angles. It is observed from simulations that when a TE wave is obliquely incident on the E-shaped element, the currents on the top and bottom arms are no longer symmetric because the mirror condition is broken by the oblique incidence. As a consequence, the cross-polarization field increases. To enforce the current symmetry, a slot is added in the middle arm of the E-shaped element, as shown in Figure 8.8d. It eliminates the direct current flow from the top arm to the bottom arm and successfully suppresses the cross-polarization field. Through the analysis above, a slotted E-shaped element is evolved from a fundamental dipole element in order to achieve sufficient phase range, low cross polarization, and stable angular performance. The scattering performance of the quad-layer transmission surface consisting of the proposed element is obtained from a full-wave simulation. Periodic boundary condition (PBC) is applied in simulation to account for the coupling between adjacent elements. Two Floquet ports are set at the top and bottom surfaces of the unit cell to illuminate and receive plane waves. The design frequency is 20 GHz, and the element periodicity is 7 mm. Figure 8.9a shows the unit cell model of a practical design. A dielectric substrate (Taconic TLX-8, εr = 2.55, 0.79 mm thickness) is used to support the E-shaped metal element in practical fabrication, and the layer separation is set to be 3 + 0.79 mm which is close to a quarter wavelength in free space. The other parameters are as follows: L = 5 mm, W = 0.4 mm, Lst = 0.4 mm, Ls = 2 mm, and W s = 0.2 mm.

282

Transmission Surfaces and Transmitarray Antennas

(a)

(b)

Figure 8.9 (a) A quad-layer transmission surface using E-shaped elements. (b) Simulated

transmission coefficient of the proposed surface. A black and white version of this figure will appear in some formats. For the color version, please refer to the plate section.

As shown in Figure 8.9a, the quad-layer element has the capability to cover the full 360˝ phase range with transmission loss lower than 1 dB. This ´1 dB transmission range is achieved when the side arm length Le varies within [0.75 mm, 3.55 mm]. Meanwhile, the proposed element also has a broad frequency bandwidth. For the case Le = 2.5 mm, the ´1 dB transmission bandwidth is 38% which is also due to the double resonance feature of the E-shaped element [41, 42].

8.3 Transmission Surface Designs

8.3.2

283

A Double-Layer Transmission Surface Using Malta-Cross Elements with Vias In the previous section, a transmission surface consisting of E-shaped elements is proposed to provide a full 360˝ phase range with low cross polarization, stable angular performance, and broad bandwidth. However, it has four layers, which increases surface thickness and fabrication complexity. A challenging question arises: can we design a transmission surface with fewer layers? Triple-layer transmission surfaces have been proposed in several papers [36, 44], and in this section, a double-layer transmission surface is studied. From the theoretical study in Section 8.2.2, the phase limit for a conventional doublelayer transmission surface is around 180˝ , far less than the required 360˝ phase range. To break the phase limit, we need to introduce a revolutionary change in surface design that has not been considered in the theoretical study. After reviewing the derivation in Section 8.2.2, we notice that the cascading approach only considers the space coupling between layers. Hence, we introduce vertical vias in transmission surface designs [37]. These vertical vias connect different layers, and electric currents can flow through these vias from one layer to the other, which leads to more design freedom and a broader phase range. Based on the idea above, a double-layer transmission surface is proposed, as shown in Figure 8.10a. It consists of two Malta-cross patches printed on the opposite sides of a dielectric substrate. It is worthwhile to point out that four symmetrical vias are employed to connect the two patches for direct coupling. The proposed element has abundant freedom over the independent design variable, namely, the patch length, the slot length and width, and the position of the vias. A comprehensive parametric study has been performed to optimize both its phase and magnitude performance. The operating frequency is set to be 20 GHz for satellite communications. A laminate of Arlon AD255C with thickness of 1.575 mm, relative permittivity of 2.55, and loss tangent of 0.0014 is used as the dialectric substrate. The unit periodicity is 6.5 mm (0.43 λ0 ), and other parameters are optimized to be: slot width W = 0.3 mm, slot separation S = 0.79 mm, and vias position V = L{4.1. The proposed element is analyzed using the infinite array method, similar to that discussed in the previous section. The simulated transmission phase and magnitude are plotted in Figure 8.10b. It is observed that a phase shift from ´170˝ to 135˝ is realized with the change of patch length L from 4.15 mm to 6 mm. The total achievable phase shift range is 305˝ , which is adequate for some practical applications. The maximum magnitude loss is 1.75 dB. Within the phase range of [´160˝ , ´90˝ ] and [´40˝ , 135˝ ] (about 80% of the achievable phase shift range), the magnitude loss is less than 1 dB. To illustrate the importance of vias, the same element configuration without vias is also simulated and presented in Figure 8.10b for comparison. When the patch length L is large, the transmission performance of both designs are similar. However, when the patch length is small, its phase shift can only cover up to 40˝ with a severe transmission loss that is over 8 dB. The total phase range with a 1 dB transmission loss is 175˝ , which agrees well with the analytical results in Section 8.2.2. Further study shows that the proposed element has two different working modes. When the patch length L is large, the element functions mainly at the FSS mode, and the space coupling is strong for energy transmission from the top patch to the bottom. When the patch length L is small, the element functions mainly at the receiver/transmitter

284

Transmission Surfaces and Transmitarray Antennas

(a)

(b)

Figure 8.10 (a) Geometry of a double-layer transmission surface using Malta-cross elements with vertical vias. © 2017 IEEE. Reprinted, with permission, from Wenxing An, 2017. (b) Simulated transmission coefficients of the surfaces with and without vias. A black and white version of this figure will appear in some formats. For the color version, please refer to the plate section.

mode, and the vias become essential for direct current flow from the top patch to the bottom [37]. In summary, an improved transmission performance is obtained with the proposed double-layer element using vertical vias.

8.3.3

A Single-Layer Transmission Surface Using Cross-Polarized Fields Using vertical vias, a double-layer transmission surface can cover a phase shift range over 360˝ . An interesting yet challenging question is: can we design a single-layer

8.3 Transmission Surface Designs

285

transmission surface to cover the full 360˝ phase range? From the analysis in Section 8.2.1, we learned that the transmission coefficient circle is located on the right part of the polar diagram, and the transmission phase is between [´90˝ , +90˝ ]. However, that analysis only considers the co-polarized field, while the electromagnetic field is a vector field consisting of both co- and cross-polarized fields. In this section, we introduce the cross-polarized field into the analysis and reveal the interesting outcome of this new design freedom [43]. If the cross-polarized field is considered, Eqs. (8.1) to (8.6) need to be updated accordingly. First, there will be two transmission coefficients and two reflection coefficients as below: ´ + + E´ E2C E1X E2X , R = , T = , T = (8.10) RC = 1C X C X E1+ E1+ E1+ E1+ Considering these four coefficients, the boundary conditions become 1 + RC = TC ,

RX = T X ,

(8.11)

|RC | + |RX | + |TC | + |TX | = 1. 2

2

2

2

(8.12)

After some mathematical manipulations, the following equation can be derived: tC2 ´ tC cos ϕC + tX2 = 0,

(8.13)

where tC and tX are the magnitudes of the co- and cross-polarized transmission coefficients, and ϕC is the phase of the co-polarized transmission coefficient. It is very important to point out that the phase of the cross-polarized transmission coefficient ϕX is not included in this equation. When tX equals to zero, (8.13) will return to (8.6). However, if tX does not equal to zero, new phenomena will occur. Figure 8.11 shows both the co- and cross-polarized transmission coefficients with different tX values. The co-polarized transmission coefficient is represented by a circle with its center at (x = 0.5, y = 0), and the crosspolarized transmission coefficient is also represented by a circle with its center at (x = 0, y = 0). When tX increases, the radius of the TC circle reduces and the radius of the TX circle increases. When the TX circle reaches its maximum radius of 0.5, the TC circle shrinks to a point (x = 0.5, y = 0). At this time, the cross-polarized transmission coefficient can achieve a full 360˝ phase range with a constant magnitude of 0.5 (´6 dB), while the co-polarized transmission coefficient constantly has a phase of 0˝ with a magnitude of 0.5. From the above analysis, a full 360˝ phase range can be achieved on a single-layer transmission surface if the cross-polarized field is used. To verify this idea, a defected square slot is designed on a metallic sheet, as shown in Figure 8.12a. The operation frequency is chosen at 10 GHz. The element spacing P is 15 mm (equal to 0.5 λ0 ) and the slot width W is fixed to be 0.5 mm. By changing the slot length L and the defect size d, different transmission coefficients can be obtained. The electromagnetic performance is characterized using a full-wave simulation software with periodic boundary condition such as CST Microwave Studio [45]. The computed transmission coefficient for the cross-polarization (TX ) is shown in Figure 8.12b. After careful design

286

Transmission Surfaces and Transmitarray Antennas

(a)

(b)

Figure 8.11 Analytical results of (a) co-polarized transmission coefficient and (b) cross-polarized transmission coefficient.

(a) (b)

Figure 8.12 (a) Geometry of a defected square slot element on a metal sheet. (b) Simulated transmission coefficient of the cross-polarized field.

of slot parameters (L, d) and using mirror elements, a full 360˝ phase range is obtained and the magnitude is near the maximum value of 0.5, which matches very well with the analytical results. This proposed design has been experimentally demonstrated in [43].

8.4 Reconfigurable Transmission Surface Designs

8.3.4

287

Other Transmission Surface Designs In previous discussions, several representative transmission surfaces are studied, which include the quad-layer E-shaped element, the double-layer Malta-cross element with vias, and the single-layer defected slot element. These designs belong to a general category similar to FSS, where the element scattering property is of particular interest. Meanwhile, there are some other approaches to design transmission surfaces, namely, the receiver/transmitter (R/T) design approach and the metamaterial transformation approach. An R/T type transmission surface [32, 33] usually consists of a receiving antenna array, a transmitting antenna array, and transmission lines between them. The receiving array receives electromagnetic waves from space, then the transmission line transfers energy from the receiving array to the transmitting array, which finally radiates waves into free space. Instead of conventional horn antennas and coaxial cables, microstrip patches and microstrip lines are popularly used to form a planar surface. In this design approach, the transmission magnitude is determined by the antenna matching, and the transmission phase is determined by the transmission line length. The operation principle of this approach is quite straightforward. However, the number of substrate layers is usually large, leading to complexity in design and fabrication. For the metamaterial transformation approach, the core idea is to design a dielectric substrate with specific effective permittivity and permeability using a metamaterial configuration [46–48]. The transmission phase is controlled by the effective constitutive parameters and the thickness of the substrate; the transmission magnitude is determined by a properly designed matching layer [48]. Various designs have been proposed in literature, such as periodic hole perforations, dielectric resonators, split ring resonators, etc. The insertion loss and the layer thickness are of particular concern in this design approach.

8.4

Reconfigurable Transmission Surface Designs Various transmission surfaces have been discussed in the previous section with specific transmission performance. Element geometries and dimensions are critical to determine their performance. Once the elements are fabricated, their characteristics are fixed. In some advanced applications such as phased array antennas, however, it is required that the element characteristics can be dynamically tuned. Hence, designs of transmission surfaces with reconfigurable characteristics have become an important yet challenging research topic. In order to realize the reconfigurable characteristics, one has to integrate certain tuning techniques into the transmission surface designs. Various technologies have been studied, including mechanical actuators, electronic control devices, and functional materials [17, 49]. Their operation mechanisms are briefly discussed below, as well as their advantages and limitations: •

Mechanical actuators. In this approach, reconfigurable transmissions are achieved by mechanical actuation such as an actuator or a micromotor that

288

Transmission Surfaces and Transmitarray Antennas





changes the physical dimension or orientation of a transmission element. Usually, this approach can handle high RF power and has a small insertion loss. However, the reconfigurable speed is limited, and it usually requires high driven power. Electronic control devices. Electronic devices can provide faster control speed and consume less power. Hence, they are more popular in reconfigurable designs. Furthermore, due to the advancement of the semiconductor industry, miniaturized sizes are preferable to be integrated into transmission surface designs. A variety of electronic devices have been utilized, such as PIN diodes [16, 18–20], varactor diodes [12, 15, 50], and micro-electromechanical system (MEMS) switches [14], etc. However, power handling capability remains to be a concern. Functional materials. Recently, progress in materials science provides new opportunities in reconfigurable transmission surface designs. Liquid crystals (LC), thin film ferroelectrics, graphene, and other novel materials have been explored. These materials demonstrate a great potential, especially for millimeter wave or THz applications. However, research in this area is at early stages, and many practical issues such as response time, insertion loss, etc., still need to be carefully solved for practical engineering applications.

As a new research frontier, reconfigurable transmission surfaces have attracted a lot of interest. A 2-bit antenna-filter-antenna element is presented in [13, 14], which achieves larger than 2 dB loss using MEMS switches. A 5-layer varactor-loaded FSS type slot element with 360˝ phase range is presented in [50]. However, it is difficult to place bias lines into an array design. The bridge-T phase shifter is also widely used [15], but its complexity and loss are relatively high. In summary, it is challenging to design a reconfigurable transmission surface with low insertion loss and fine phase resolution, while still having an affordable system complexity. In this section, two reconfigurable transmission surfaces are presented, one using the FSS approach [19] and the other using the R/T approach [20]. The goal is to simultaneously control transmission magnitude and phase with a minimum number of diodes and a simple biasing circuit. It is also worthwhile to point out that 1-bit design is discussed here for its structural simplicity and functional capability in a large transmission surface consisting of hundreds or thousands of elements [51].

8.4.1

FSS-Type Reconfigurable Design The FSS approach is widely used to design a transmission surface, and a major effort is made to reduce the number of layers while still covering the required phase range. For the 1-bit reconfigurable transmission surface element, the two phase states are 0˝ and 180˝ , hence the required phase range is not 360˝ but 180˝ . Therefore, it is possible to design an FSS-type element with fewer layers. According to the analysis in Section 8.2, the theoretical transmission coefficient of a double-layer surface is computed and plotted in Figure 8.13 [19]. Two identical metal sheets are etched on opposite sides of a substrate layer with thickness Hs. It is noticed that a double-layer element is sufficient for 1-bit reconfigurable transmission surface

8.4 Reconfigurable Transmission Surface Designs

(a)

289

(b)

Figure 8.13 The double-layer FSS-type transmitarray element. (a) The general geometry model. (b) The polar plot of theoretical transmission coefficients with different dielectric thicknesses. © 2016 IEEE. Reprinted, with permission, from Jun Luo 2016.

designs. The transmission magnitudes at 0˝ and 180˝ vary with the thickness Hs. For an air substrate (εr = 1), the transmission loss is 2.9 dB for quarter-wavelength thickness (βH s = 90˝ ). The transmission loss decreases when the substrate thickness reduces. When βH s = 10˝ , the transmission loss is less than 0.2 dB, which is acceptable for many applications. An interesting observation of Figure 8.13 is that the element with a thin thickness has a denser distribution around phase 0˝ and a sparser distribution at phase 180˝ . This means that the sensitivity at 180˝ is higher than that at 0˝ . So for an element designed at phase 180˝ , a slight size variation or fabrication error will result in a large phase shift from 180˝ . As a trade-off between magnitude loss and element sensitivity, the dielectric thickness needs to be carefully designed. Another important consideration is the element type selection. It is found that the ring-type element has a lower sensitivity around phase 0˝ but a higher sensitivity around phase 180˝ . As a complementary geometry, the slot-type element has the opposite performance, namely, a higher sensitivity around phase 0˝ but a lower sensitivity around phase 180˝ . Considering the previous phase distribution and the element property, the slot structure is chosen to design the reconfigurable element. Figure 8.14 shows the top-layer geometry of the proposed element. The square slot element (W s = g = 0.6 mm) is placed in a periodic lattice with periodicity P of 11.43 mm, close to half wavelength at 12.5 GHz. It is printed on both sides of a Taconic TLX-8 substrate (εr = 2.55) with thickness of 1.58 mm (βH s = 37.8˝ ). For the ideal OFF state, no metal strip exists on the slot. For the ideal ON state, two metal strips are placed on the slot at position Dp. The operation principle of this element is analog to a reconfigurable patch antenna, which works at half-wavelength mode: at OFF state and at quarter-wavelength mode at ON state. By optimizing Ws and Dp, the phase difference between ON and OFF states is 180˝ and the transmission loss is minimized to be around 1 dB.

290

Transmission Surfaces and Transmitarray Antennas

(a)

(b)

Figure 8.14 The square slot geometry of the double-layer 1-bit transmitarray element. (a) The square slot element (ideal OFF state). (b) The slot element with PEC connecting strips (ideal ON state).

Two prototypes, one for the ideal OFF state and the other for the ideal ON state, are fabricated to demonstrate the design’s feasibility. They are measured in a waveguide and calibrated using a standard calibration kit. The measurement results are plotted in Figure 8.15 with comparison to the simulations. It is observed that at the OFF state, the transmission magnitude is ´1.2 dB with a phase of ´177˝ . At the ON state, the transmission magnitude is ´1.0 dB with a phase of ´13˝ . The measured results agree well with the simulation results, which validate the proposed design. Some further issues need to be discussed here. First, a metal strip is used to model the ON state of a PIN diode or a MEMS switch, which will cause some discrepancies. In practice, actual diodes need to be implemented in reconfigurable designs. Second, to control the state of diodes, a proper biasing circuit needs to be designed, and its influence on the transmission property should be minimized. Lastly, there are two diodes on each layer and four diodes in total for each element. The element’s geometry may be further optimized to reduce the number of diodes so that the cost and power consumption can be reduced.

8.4.2

R/T-Type Reconfigurable Design Besides the FSS approach, the receiver/transmitter (R/T) approach is another effective method to design a transmission surface. Integrating a reconfigurable coupling line between the receiving element and the transmitting element, one can design a transmission surface with dynamic performance. For the reconfigurable coupling line, the traditional way is to add a phase shifter into the line [12]. For 1-bit phase resolution, an alternative and effective approach is to use the current reversal technique [16, 20]. When the induced current has an opposite flow direction, the re-radiating field will also be reversed, which is equivalent to a 180˝ phase change. The current reversal technique has a broad operation bandwidth, and is a desirable technique for 1-bit transmission surface designs. The structure of the proposed transmission element is illustrated in Figure 8.16a [20]. It consists of two modified slots that act as the receiving (Rx) and transmitting (Tx)

8.4 Reconfigurable Transmission Surface Designs

291

(a)

(b)

Figure 8.15 Comparison of simulated and measured transmission coefficients of the double-layer 1-bit element: (a) ideal OFF state and (b) ideal ON state.

layers respectively. They are orthogonally placed to isolate direct coupling. A U-shaped coupling line is designed for power transmission between them, on which two PIN diodes are integrated. When a positive or negative bias voltage is added in the middle, one of the two PIN diodes is turned ON while the other is turned OFF. Depending on the diodes’ states, the coupling current from the Rx slot excites the Tx slot with opposite directions. Hence, two transmission phase states with a phase difference of 180˝ is achieved. As shown in Figure 8.16b, the Rx slot is etched on the top surface of substrate 1a, which has dielectric constant of 2.2, loss tangent of 0.0009, and a thickness of 0.78 mm. The coupling line is printed on the bottom surface of substrate 1b. A layer of DC biasing line is inserted in between to provide the control signal. The Tx slot is fabricated on a 0.3-mm thick stainless steel sheet. It is suspended 0.9 mm from the transmission

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(b) (a)

Figure 8.16 An R/T-type 1-bit reconfigurable transmission element: (a) exploded element structure and (b) DC bias line design. © 2017 IEEE. Reprinted, with permission, from Min Wang 2017. (a)

(b)

Figure 8.17 Simulated transmission performance of the proposed element under normal and 30˝

oblique incident angles: (a) magnitudes and (b) phases.

line layer by plastic spacers so that the PIN diodes are accessible for fabrication and maintenance purpose. The proposed element is simulated using Ansys HFSS [52]. The element is placed in periodic boundary conditions with Floquet mode excitation. The size of the element is 8 mm ˆ 8 mm, equal to 0.33 λ0 at the design frequency of 12.5 GHz. The parameters of the two modified slots and the coupling transmission line are optimized to ensure good matching between the slots and the coupling line, as well as a stable performance under oblique incidences. The PIN diodes are represented as parallel lumped RLC elements, R = 7.8, L = 35 pH for the ON state, and C = 25 fF, L = 35 pH for the OFF state. The simulated transmission coefficients of the proposed element under normal and 30˝ oblique incident angles are plotted in Figure 8.17. Under normal incidence, the element exhibits 0.8 dB insertion loss at 12.5 GHz, and the reflection coefficient is below ´30 dB. The 3-dB transmission bandwidth is 10%. Under a 30˝ oblique incidence, the transmission coefficient curve slightly shifts towards a higher frequency, and the

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293

insertion loss at 12.5 GHz increases to 1.1 dB. Because of the current reversal design, the transmission phases of the two states exhibit a stable 180˝ phase difference within a wide frequency range and over a broad incident angle range. The R/T approach discussed here may be combined together with the FSS approach to design a transmission surface with better performance. For example, a 2-bit transmission surface can be designed as below: the first bit (180˝ ) is achieved using the current reversal technique of the R/T approach, and the second bit (90˝ ) is achieved using the FSS approach. As a result, the design can be relatively simplified with low insertion loss.

8.5

Transmitarray Antennas A transmission surface can effectively manipulate electromagnetic waves passing through it, and thus it has a wide range of applications from microwaves to optics. It opens up a new arena to design electronic devices. Among them, transmitarray antennas represent an emerging generation of high gain antennas, which have great potential in wireless communications, radar systems, earth remote sensing, etc. [2]. In this section, a brief overview of transmitarray antennas will be presented, and their design using transmission surfaces will be illustrated.

8.5.1

Concept and Design Procedure of Transmitarray Antennas A high gain antenna is usually designed through two approaches: the optic theory and the array method. The former approach manipulates the surface curvature of an antenna to focus the radiation beam, such as a parabolic reflector or a lens antenna. The latter controls the interference of radiating elements properly to generate a specific radiation pattern. Representative examples include waveguide-slot arrays and printed microstrip antenna arrays. As an emerging concept, the transmitarray antenna combines the favorable features of optic theory and array method, leading to a low profile conformal design with high radiation efficiency and versatile radiation performance. As shown in Figure 8.18, a transmitarray antenna consists of an illuminating feed source and a thin transmission surface [2, 27]. The feed source is employed to radiate or receive electromagnetic waves in the space. A typical selection is a horn antenna located on the equivalent focal point of the transmission surface. The transmission surface is an array of scattering elements, whose transmission coefficients are individually designed to convert the original radiation from the feed into a specified radiation pattern. For example, it can convert a wide radiation beam into a narrow pencil beam with high gain, or it can convert a pencil beam into a contoured shaped beam. The transmitarray concept evolves from lens antennas and reflectarray antennas, and has its own desirable features. Lens antennas are commonly used from high frequencies all the way up to optic ranges. However, the curved surface of the lens increases the fabrication complexity, and the bulky volume limits its applications. In contrast, a thin transmission surface can be fabricated using the standard low-cost printed circuit board (PCB) technique and can be installed on various conformal platforms. Furthermore,

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Figure 8.18 Geometry of a transmitarray antenna.

Figure 8.19 General design procedure for a transmitarray antenna.

since each transmission element is individually designed, the radiation characteristics of a transmitarray antenna is much more flexible than its ancestor lens antenna. Compared to a reflectarray antenna, the prominent feature of transmitarray antennas is that there is no feed blockage, so that near-field feeds can be used to achieve a lowprofile configuration. However, the transmitarray encounters a great design challenge: both magnitude and phase control of the array element. In reflectarrays, the reflection magnitude is close to 1 (0 dB) due to the existence of a metal ground plane; thus, one only needs to control the element reflection phase. In transmitarrays, besides the phase control, the magnitude of the transmission coefficient needs to be close to 1 (0 dB) to ensure high efficiency. Figure 8.19 summarizes a general design procedure for a transmitarray antenna [2, 27]. It consists of two major tasks: transmission element design and antenna system

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295

design. In the transmission element design, one needs to select a proper element type and determine the element dimensions for certain transmission phase and magnitude requirements. This has already been discussed in detail in previous sections, so it will not be presented here. In the antenna system design, the aperture size and feed location can be first determined for a certain gain requirement through the gain and efficiency analysis (GEA). When associated with the beam direction or radiation pattern specifications, the transmission phase compensation over the aperture can be calculated. From the phase compensation, the transmission element geometry can be decided for prototype fabrication and measurement. Also from the phase compensation, both the aperture field and far field of a transmitarray antenna can be computed through the radiation analysis (RA). Finally, a comparison between simulation and measurement is made to validate the transmitarray antenna design, and the antenna characteristics such as radiation patterns, directivity, gain, axial ratio, bandwidth, etc., can be obtained. Two critical issues during this procedure are the transmission phase compensation over the aperture and the radiation analysis of the transmitarray. The required transmission phase of each transmission element is designed to compensate the spatial phase delay from the feed to that element so that a certain phase distribution can be realized for a required radiation pattern. For example, if a focused beam is desired in direction rˆ0 , the transmission phase on the i th element can be calculated as follows using the ray tracing method [2, 27]: ψi = k(Ri ´ ri ¨ rˆo ) + ψ0,

(8.14)

where k is the propagation constant in free space, Ri is the distance from the feed to the i th element, ri is the position vector of the i th element, and ψ0 is a constant reference phase. Figure 8.20 presents the required phase distribution of a circular transmitarray antenna of 30 ˆ 30 elements with half-wavelength unit cell. The focal point is centered with an F {D = 0.8, and a pencil beam is required in the broadside direction. To calculate the radiation performance of a transmitarray, different radiation analysis (RA) techniques have been developed [2, 27]. A simple and straightforward method is

Figure 8.20 Required phase compensation in a circular aperture transmitarray.

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the array theory approach. Each transmission element has an excitation magnitude and phase. The magnitude is determined by the feed pattern and the distance between the feed and the element. The phase is the sum of the incident wave phase and the element transmission phase. Once the excitation magnitude and phase are determined for each element, the far-field radiation pattern can be determined through array summation. This method provides a fast calculation of the radiation pattern and is quite accurate around the main beam region. To obtain a more rigorous result, more advanced approaches can be used, such as the aperture field method and the full-wave simulation of the entire antenna [53]. Once the radiation pattern is obtained, some characteristics parameters of the transmitarray antenna can be computed, such as gain, efficiency, beamwidth, sidelobe levels, and cross-polarization levels.

8.5.2

A Transmitarray Design Example Following the previous design procedure, a transmitarray antenna is successfully fabricated and tested. To focus a radiation beam, a transmission surface consisting of quad-layer slotted E-shaped elements is used here. The element performance has been presented in Section 8.3.1 where a full 360˝ phase range is achieved with a transmission loss less than 1 dB. A circular transmitarray with a diameter of 340 mm is designed at Ka band [41, 42]. A linearly polarized corrugated horn with a gain of 15.6 dBi is used as a prime-focus feed, and its q factor is around 10 using the cosq (θ ) model. The phase center of the horn is optimized to be 320 mm from the aperture through the gain and efficiency analysis (GEA). Hence, the corresponding F /D ratio is 0.9, and the largest incident angle of the elements is 29˝ . An antenna prototype is built, and photos are shown in Figure 8.21. There are 1788 slotted E-shaped elements within the array aperture. These elements’ sizes vary according to the transmission phase distribution on the aperture. To keep the required layer spacing and alleviate the deformation of substrates, 88 holes are drilled on each of the 4 layers, and plastic screws are placed through the holes with nylon spacers inserted between adjacent layers. The array and the feed horn are then assembled on an aluminum framework. The radiation performance of the transmitarray is calculated using the array theory method, and is also measured by a planar near-field scanning system in an anechoic chamber. The calculated and measured radiation patterns of the antenna at 20 GHz are plotted in Figure 8.22. The measured main beam agrees well with the calculated beam. The half-power beamwidths (HPBW) are 2.8˝ and 2.9˝ in the E- and H-planes, respectively. The measured sidelobe levels are ´24.1 dB (E-plane) and ´25.7 dB (Hplane). The cross polarization is below ´28 dB within the half-power beamwidths in both E- and H-planes. At the design frequency of 20 GHz, the measured gain is 33.0 dBi and the aperture efficiency is 40%. The measured gains are also plotted versus frequency in Figure 8.22a. The maximum measured gain is 33.4 dBi at 20.75 GHz. A broad ´1 dB gain bandwidth of 15% (3.0 GHz, from 19.4 GHz to 22.4 GHz) is observed in the measurement.

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297

Figure 8.21 Photograph of a transmitarray prototype consisting of quad-layer slotted E-shaped elements, measured by a near-field scanning system in an anechoic chamber. A black and white version of this figure will appear in some formats. For the color version, please refer to the plate section.

Figure 8.22 Radiation performance of the transmitarray prototype: (a) measured gain bandwidth; (b) E-plane and (c) H-plane patterns at 20 GHz.

In this transmitarray design, a corrugated horn is used as the feeding structure of the antenna. The distance between the horn and the transmission surface (320 mm) is relatively large. To achieve a low profile configuration, near-field feeding structures can also be used in transmitarray designs because no feed blockage exists. Combining a conventional array as a feed and specific transmission surfaces, many novel transmitarrays can be designed with advanced radiation performance and attractive low profiles. For example, a traditional waveguide slot array usually radiates a pencil beam; however, a contoured shaped beam can be achieved when it is covered with a transmission surface with a carefully designed phase distribution. Furthermore, if a reconfigurable transmission surface is covered on top of the slot array, a beam-scanning antenna can be realized. In addition, if multiple transmission surfaces are utilized, many new features can be discovered. For example, when two transmission surfaces with gradient phase distributions are put above a planar array, both the azimuth and elevation angles can be adjusted just by planar rotations of these two surfaces [54]. Operating like double prisms in optics, when the two transmission surfaces are rotated in the same direction, the azimuth beam direction changes; when rotated in opposite directions, the elevation beam direction changes. This low-profile antenna with high gain and variable inclination

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angle is desirable in various mobile satellite communications platforms such as airplanes and vehicles. In summary, development of transmission surfaces has broad impacts on antennas and electromagnetics. It is believed that new antennas and electromagnetic devices will keep on emerging. In turn, these devices will propel the research of transmission surfaces to more advanced levels.

References [1] J. Huang and J. A. Encinar, Reflectarray Antennas, John Wiley, 2008. [2] A. H. Abdelrahman, F. Yang, A. Z. Elsherbeni, and P. Nayeri, Analysis and Design of Transmitarray Antennas, Morgan and Claypool, 2017. [3] C. A. Balanis, Advanced Engineering Electromagnetics, 2nd ed., John Wiley, 2012. [4] J. L. Volakis, Antenna Engineering Handbook, 4th ed., McGraw-Hill, 2007. [5] B. A. Munk, Frequency Selective Surfaces: Theory and Design, John Wiley, 2000. [6] N. Gagnon, A. Petosa, and D. A. MaNamara, “Research and development on phase-shifting surfaces (PSSs),” IEEE Antennas and Propagation Magazine, vol. 55, no. 2, pp. 29–48, 2013. [7] S. M. Abadi and N. Behdad, “A broadband, circular-polarization selective surface,” Journal of Applied Physics, vol. 119, 244901, 2016. [8] M. Hosseini and S. V. Hum, “A circuit-driven design methodology for a circular polarizer based on modified Jerusalem cross grids,” IEEE Transactions on Antennas and Propagation, vol. 65, no. 10, pp. 5322–5331, 2017. [9] L. Young, L. A. Robinson, and C. A. Hacking, “Meander-line polarizer,” IEEE Transactions on Antennas and Propagation, vol. 21, pp. 376–378, 1973. [10] R.-S. Chu and K.-M. Lee, “Analytical model of a multilayered meander-line polarizer plate with normal and oblique plane-wave incidence,” IEEE Transactions on Antennas and Propagation, vol. AP-35, no. 6, pp. 652–661, 1987. [11] N. Gagnon, A. Petosa, and D. A. McNamara, “Hybrid printed lens antenna,” IEEE Transactions on Antennas and Propagation, vol. 60, no. 5, pp. 2514–2518, 2012. [12] W. Pan, C. Huang, X. Ma, and X. Luo, “An amplifying tunable transmitarray element,” IEEE Antennas and Wireless Propagation Letters, vol. 13, pp. 702–705, 2014. [13] C. C. Cheng and A. Abbaspour-Tamijani, “Study of 2-bit antenna-filter-antenna elements for reconfigurable millimeter-wave lens arrays,” IEEE Transactions on Microwave Theory and Techniques, vol. 54, no. 12, pp. 4498–4506, 2006. [14] C. C. Cheng, B. Lakshminarayanan, and A. Abbaspour-Tamijani, “A programmable lensarray antenna with monolithically integrated MEMS switches,” IEEE Transactions on Microwave Theory and Techniques, vol. 57, no. 8, pp. 1874–1884, 2009. [15] J. Y. Lau and S. V. Hum, “A wideband reconfigurable transmitarray element,” IEEE Transactions on Antennas and Propagation, vol. 60, no. 3, pp. 1303–1311, 2012. [16] A. Clemente et al., “Wideband 400-element electronically reconfigurable transmitarray in X band,” IEEE Transactions on Antennas and Propagation, vol. 61, no. 10, pp. 5017–5027, 2013. [17] 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.

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[37] W. An, S. Xu, F. Yang, and M. Li, “A double-layer transmitarray antenna using Malta crosses with vias,” IEEE Transactions on Antennas and Propagation, vol. 64, no. 3, pp. 1120–1125, 2016. [38] F. Aieta et al., “Aberration-free ultrathin flat lenses and axicons at telecom wavelengths based on plasmonic metasurfaces,” Nano Letters, vol. 12, no. 9, pp. 4932–4936, 2012. [39] M. Khorasaninejad et al., “Metalenses at visible wavelengths: Diffraction-limited focusing and subwavelength resolution imaging,” Science, vol. 352, no. 6290, pp. 1190–1194, 2016. [40] A. H. Abdelrahman, F. Yang, and A. Z. Elsherbeni, “Transmission phase limit of multilayer frequency-selective surfaces for transmitarray designs,” IEEE Transactions on Antennas and Propagation, vol. 62, no. 2, pp. 690–697, 2014. [41] J. Luo, F. Yang, and S. Xu, “E-shaped element design for linearly polarized transmitarray antennas,” in Proceedings of 2014 International Symposium on Antennas and Propagation (ISAP), pp. 269–270, Dec. 2014. [42] J. Luo, F. Yang, S. Xu, M. Li, and S. Gu, “A high gain broadband transmitarray antenna using dual-resonant E-shaped element,” Microwave and Optical Technology Letters, in press. [43] F. Yang, R. Deng, S. Xu, and M. Li, “Design and experiment of a near-zero-thickness highgain transmit-reflect-array antenna,” IEEE Transactions on Antennas and Propagation, in press. [44] A. Aziz, F. Yang, S. Xu, and M. Li, “An efficient dual-band orthogonally-polarized transmitarray design using 3-dipole elements,” IEEE Antennas and Wireless Propagation Letters, in press. [45] CST Microwave Studio, Computer Simulation Technology, Dassault Systems, 64289 Darmstadt, Germany. [46] Q. Cheng, H. F. Ma, and T. J. Cui, “Broadband planar Luneburg lens based on complementary metamaterials,” Applied Physics Letters, vol. 95, 181901, 2009. [47] Y. Zhang, R. Mittra, and W. Hong, “On the synthesis of a flat lens using a wideband low reflection gradient-index metamaterial,” Journal of Electromagnetic Waves and Applications, vol. 25, no. 16, pp. 2178–2187, 2012. [48] M. Wang, S. Xu, F. Yang, and M. Li, “Design of a Ku-band triple-layer perforated dielectric transmitarray antenna,” in Proceedings of 2016 IEEE International Symposium on Antennas and Propagation, pp. 1381–1382, 2016. [49] P. Nayeri, F. Yang, and A. Z. Elsherbeni, “Beam-scanning reflectarray antennas: a technical overview and state of the art,” IEEE Antennas and Propagation Magazine, vol. 57, no. 4, pp. 32–47, 2015. [50] L. Boccia, I. Russo, G. Amendola, and G. Di Massa, “Multilayer antenna-filter antenna for beam-steering transmit-array applications,” IEEE Transactions on Microwave Theory and Techniques, vol. 60, no. 7, pp. 2287–2300, 2012. [51] H. Yang et al., “A 1600-element dual-frequency electronically reconfigurable reflectarray at X/Ku-band,” IEEE Transactions on Antennas and Propagation, vol. 65, no. 6, pp. 3024– 3032, 2017. [52] Ansys HFSS, ANSYS Inc., Canonsburg, PA. [53] P. Nayeri, F. Yang, and A. Z. Elsherbeni, Reflectarray Antennas: Theory, Designs, and Applications, Wiley-IEEE Press, 2018. [54] Chinese patent 201510890728.0.

9

Coding and Programmable Metasurfaces Shuo Liu and Tie Jun Cui

9.1

Introduction The emergence of metamaterials has greatly extended the attainable range of medium parameters and enhanced the manipulation of electromagnetic (EM) wave in the past 20 years [1–5]. While the EM properties of natural materials are determined by the spatial arrangement of atoms in molecules, the medium parameters of metamaterials can be artificially tailored by designing different unit cell structures and adopting different arrangements of them, far beyond what could be obtained from natural materials. Negative permittivity [6], negative permeability [7], negative refractive index [3–5, 8] and zero refractive index [9] had been experimentally demonstrated at the turn of the century, leading to many exotic phenomena such as negative refraction [8,10], perfect lens [11], inverse Doppler effect [12], reversed Cherenkov radiation [13], etc., as well as a lot of interesting EM devices, including high-directivity antennas [14–16], invisible cloaks [17,18], perfect absorbers [19–23], etc. Nowadays, metamaterials have fundamentally changed the way people manipulate EM wave, bringing many new devices with new functionalities and improving the performance of conventional devices. However, a non-negligible thickness is always required by metamaterials because they manipulate EM wave through the phase accumulation along the optical path [14–18], which inevitably results in issues such as extra loss, bulky volume and heavy weight. Researches have to face great cost for the fabrication of such 3D metamaterials at microwave or millimeter wave regimes. At terahertz, infrared or visible light spectra, it is very challenging, or almost impractical, to fabricate such 3D metamaterials due to the limitation of current micro-fabrication technique, which has seriously hindered the development of metamaterials at higher frequencies. To overcome the abovementioned disadvantages, researches attempted to build an ultrathin metamaterial on a two-dimensional surface, which is called the metasurface [24]. Unlike 3D metamaterials, the amplitude and phase distributions of EM wave are manipulated through the abrupt EM response changes across the ultrathin metasurface. The excellent flexibility of metasurfaces allows them to be bent or twisted according to shape of objects, which could be very applicable in applications that require devices or materials to be conformal with the equipment. In addition, the ultrathin thickness of metasurfaces offers them lower material loss as well as lower weight, leading to miniaturized devices such as Huygen’s surface [25] and conformal spoof SPP waveguides and 301

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components [26–29], etc. These flexible metasurfaces may have promising applications in flexible circuits, conformal antennas and wearable devices. Early works on negative refraction requires metamaterials to have simultaneous negative permittivity and permeability [13,30], which commonly occurs at the resonant frequency of the unit cell structure, and thus associated with high material loss. Due to the deep subwavelength thickness of metasurfaces, traditional effective medium theory cannot be directly applied to achieve negative reflection. In 2001, Yu et al. proposed the concept of generalized Snell’s Law and realized negative refraction using a single layer of V-shaped antennas with different orientations, which could tailor the phase of the cross-polarized component of refracted (reflected) wave across a 2π phase span [31]. Such an ultrathin metasurface with abrupt phase distribution aroused great interests of researchers from both physics and engineering communities in recent years, and have been extensively applied in the design of wavefront manipulation devices [31–42]. Some milestone works include the spatially propagating wave to surface wave conversion using H-shaped structure [40], linear-to-circular polarization conversion with arbitrarily polarized incidences using double-row V-shaped antennas [35], independent manipulations of right-handed and left-handed circularly polarized wave using differently rotated anisotropic structures [34, 36, 37], ultrathin invisible cloak at optical frequency based on phase compensation technique [41]. Although the abovementioned metasurfaces show powerful manipulations to the reflected wave, they are faced with the same problem of low efficiency in controlling the transmitted wave. This is due to the fact that most of the above mentioned metasurfaces contain only one-layer metallic structure, and part of the incident energy will be reflected back according to the reciprocity of a passive two port network. This issue was explicitly pointed out by Prof. Alù in 2013 by making a rigorous theoretical analysis of a passive, single-layered, non-magnetic metasurface. They concluded that it is impossible to obtain 2π phase with unity amplitude for controlling the co-polarized component with such metasurfaces due to the trade-off relationship between the attainable phase range and amplitude of the transmitted wave [43]. While for the cross-polarized component, the maximum normalized transmission is still limited by 0.5. To realize 360˝ phase coverage without sacrificing the transmitted amplitude, they developed a high-efficient triple-layer metasurface working in the visible light spectrum by arranging aluminum-doped zinc oxide (AZO) and silicon (Si) in a shunt configuration, in which the effective surface impedance can be arbitrarily designed by changing the ratio between them. In the same year, Prof. Grbic’s group proposed the concept of Huygens’ surface with simultaneous electric and magnetic responses, which could manipulate the amplitude and phase of the transmitted wave at will with nearly 100% efficiency [25]. Such a concept was extended to the case of point source excitation to realize nearly 100% aperture efficiency [44,45]. It is clear from the abovementioned devices that metasurfaces, with much thinner thickness and easier fabrication, have more versatile functionalities and superior performance than 3D metamaterials. More recently, Prof. Capasso’s group reported a focusing lens at visible light frequency by fabricating an array of

9.1 Introduction

303

600-nm-thick cylindrical TiO2 , which is about 100,000 times thinner than conventional optical lens. Due to its outstanding optical performance, it can be viewed as the most successful application of metasurface at optical frequency, and may replace conventional optical lens in the near future. Similar to metamaterials, the unit cell of metasurfaces, homogeneous or inhomogeneous, is still characterized with continuous value. Hence, they can be considered as an analogy to the analogue circuit from the circuit perspective. To implement digitaltype metasurfaces, we proposed the concept of coding, digital, and programmable metasurfaces in 2014, which could manipulate the EM wave through pre-designed coding sequences [46]. Although there have been several methods on the parameter retrieval of metasurfaces such as the generalized sheet transition conditions (GSTCs) proposed by Prof. Holloway [47], they are still more complex compared to the coding metasurface, in which the response of each coding particle is characterized by digital states “0” or “1”. Due to the binary states of coding particle, an active coding particle can be readily implemented by biasing a pin-diode at the “ON” and “OFF” state, leading to the programmable metasurface which could make real-time control of EM waves [46]. The digitalized description of coding and programmable metasurfaces have significantly simplified the design procedure and increased the degree of freedom in the manipulation of EM wave. Beam splitting, random diffusion, anomalous reflection can be conveniently realized with coding metasurfaces at microwave [46] or terahertz [48] frequencies. By designing a dumb-bell shaped coding particle, a polarizationdependent anisotropic coding metasurface with independent functionality under the xand y-polarized incidences was designed [49]. Dual-functional coding metasurface as well as free-standing transmission-type coding metasurfaces were also reported recently at terahertz frequencies with excellent performance [50]. Most importantly, coding metasurfaces have become a bridge linking metasurfaces with information science, making it possible to combine many existing algorithms in digital signal processing with coding metasurfaces. For instance, the convolution theorem was initially introduced to metasurface to realize an ideal rotation of radiation patterns [51], which was later utilized to design a controllable random surface [52] and a metasurface antenna with arbitrary cone-shaped radiation patterns [53]. The Shannon entropy was also employed to predict the amount of information carried by a coding metasurface [54]. This chapter will review the working mechanism and design strategies of coding metasurfaces, showing the powerful manipulation of programmable metasurfaces to EM wave with a single-sensor single-frequency imaging application. This chapter is organized as follows. Section 9.2 presents the basic design and working principles of coding metasurfaces by presenting several coding patterns with different functions. In Section 9.3, the programmable metasurface under the normal illumination is firstly introduced. Then, we discuss how programmable metasurface should be modified to work under oblique incidence. At the end of this section, a 2-bit transmission-type coding metasurface is presented for realizing single-sensor single-frequency imaging system.

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9.2

Coding Metasurfaces and Their Controls to EM Waves As mentioned in the introduction, coding metasurface is a digital-type metasurface due to the digital description of each unit cell, which is schematically illustrated in Figure 9.1a. For the simplest 1-bit coding metasurface, each coding particle has only two different states “0” and “1” (see Figure 9.1a), representing reflection phases of 0˝ and 180˝ , respectively. Here, we note that only the phase difference between the two states are needed to be maintained at a certain value over the working bandwidth, whereas the absolute phase of each state has no influence on the performance. Although 1-bit coding metasurface only has two types of coding particles with opposite reflection phases, there are a huge number of possible combinations of these particles if we arrange a bunch of them on a two-dimensional plane. These combinations of coding particles, as we called coding patterns, produce a variety of radiation patterns. Two classical examples are given in Figures 9.1b and 9.1c, which are the “101010. . . ” coding sequence and chessboard coding sequence, respectively. When a plane wave is normally incident

Figure 9.1 (a) Schematic of the 1-bit coding metasurface. Schematic illustrations of the coding

metasurface with different coding sequences: (b) “101010 . . .” coding sequence and (c) chessboard coding sequence. (d) Unit cell design of the coding metasurface. (e) Phase responses for the “0” and “1” elements. Adapted from [46] with the permission of Nature Publishing Group. A black and white version of this figure will appear in some formats. For the color version, please refer to the plate section.

9.2 Coding Metasurfaces and Their Controls to EM Waves

305

on these encoded metasurfaces, it will be reflected to two symmetrical directions (see Figure 9.1b) and four symmetrical directions (see Figure 9.1c) in the upper-half space. Note that the coding particle is not necessarily limited to reflection-type, but can be extended to transmission-type as well (refer to Section 9.3.3 for details). One of the reasons for why the reflection-type coding particle was chosen in the initial proposal of coding metasurface is due to the difficulty in designing transmission-type coding particle, as will be discussed in detail in Section 9.3.3. To implement the aforementioned 1-bit coding metasurface with real structure, a unit cell shown in Figure 9.1d is designed, which is composed of a grounded dielectric substrate having period a and thickness h and a metal square patch with length w. By changing the side length w, the reflection phases experience a phase shift, which can be utilized to build the desired “0” and “1” coding particles. The reflection phases of the two coding particles “0” and “1” are numerically simulated with the frequency-domain solver in CST Microwave Studio, and are given in Figure 9.1e from 7 to 14 GHz. It can be observed that the phase difference between the two states fluctuates from 60˝ to 200˝ , reaching exactly 180˝ at 8.7 GHz and 11.5 GHz. Although the bandwidth to realize an ideal 1-bit coding metasurface (i.e. with 180˝ phase difference and equal amplitude) is quite narrow for the current design, the performance may be acceptable in a relatively broad bandwidth, in which the phase difference fluctuates in a certain range, for instance, from 160˝ to 200˝ . In addition, one may achieve an ultra-broadband 180˝ phase difference by introducing active elements to the coding particle, such as varactor diode and transistor. Three different coding patterns, each composed of 42 ˆ 42 coding particles, are simulated and their radiation patterns are shown in Figures 9.2a–9.2c. The first example is all “0” coding pattern, which apparently results in a single beam radiation pointing in the backscattering direction (0˝ with respect to the z-axis). As the coding pattern becomes periodic “010101. . . ” coding sequence, the normal incident wave is equally redirected to two directions, being symmetrical to the normal axis. Note that each digit in the coding sequence represents a super unit cell, which is composed of N ˆ N identical coding particles. The advantages of super unit cell are two-fold. On one hand, the phase responses rely not only on the resonant nature of the structure itself, but also on the EM coupling between its neighboring ones. When the adjacent coding particles have different geometries, the reflection phases may deviate from the designed value, which is obtained by simulating a single coding particle using the unit cell boundary condition. To avoid the influence of the adjacent coding particles on the phase responses, the single coding particle is replicated to form a unit cell array (i.e., super unit cell). On the other hand, because the size of coding particle is typically 1/8 to 1/4 free-space wavelength, the period of coding sequence (for instance, “010101 . . .”) is smaller than one free-space wavelength, which cannot radiate any energy into the upper-half space. Super unit cell is able to increase such a period, thus allowing the reflected wave to appear in the visible angle range of the upper-half space (θ ă 90˝ ). Furthermore, the radiation direction can be manipulated by changing the size of super unit cell. The third example is encoded with a chessboard coding sequence (i.e., 010101 . . . {101010 . . . {010101 . . . {101010 . . .), which will result in four symmet-

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Figure 9.2 Numerical simulated radiation patterns with different coding sequences. (a–c) The all

zero, “010101. . . ” and chessboard distribution coding sequences, respectively. (d–f) With the same random coding sequences at 8, 10, and 11.5 GHz, respectively. Results are calculated with the time-domain solver in CST Microwave Studio. Adapted from [46] with the permission of Nature Publishing Group.

rically oriented beams, as shown in Figure 9.2c. These are the three most classical periodic coding patterns for the 1-bit coding metasurface, and of course, there are a variety of them, each corresponding to a different radiation pattern. Here, a function is also provided to analytically calculate the radiation pattern of a given coding pattern [55,56] Dir(θ,ϕ) =

4π |f (θ,ϕ)|2 2π ş πş{2 0

f (θ,ϕ) = fe (θ,ϕ)

N ÿ

(9.1)

2

|f (θ,ϕ)| sin θ dθ dϕ

0

! ! ” ı))    exp ´i ϕ(m,n) + kD sin θ m ´ 1{2 cos ϕ + n ´ 1{2 sin ϕ

n=1

(9.2)

Due to the deep-subwavelength nature of the coding particle, the scattering function fe (θ,ϕ) of a single coding particle is neglected. While the periodic coding pattern typically generates a single or limited number of beams pointing in definite directions, random coding patterns scatter the illuminating wave to multiple directions, leading to diffused radiation patterns, as can be observed from Figures 9.2d–9.2f, which are numerically obtained from a random coding pattern at 8, 10 and 11.5 GHz, respectively. Because the EM energy of the impinging wave is randomly scattered to multiple directions, the RCS (Radar Cross Section) will have a significant reduction in the backscattering direction under the normal illumination, or in the specular direction for the oblique incidence. To quantitatively evaluate the level of

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RCS reduction, the RCS of a same-sized metallic plate is selected as the reference. The following function is given to calculate the RCS reduction of a coding metasurface with a random coding pattern, RCS reduction =

λ2 Max rDir (θ,ϕ)s 4πN 2 D 2

(9.3)

in which λ is the free-space wavelength, D the size of super unit cell, N the lattice number of super unit cells in either the row or column. Note that better RCS reduction is achieved for larger lattice number N . This is due to fact that the back scattering intensity of the bare PEC board increases faster with respect to its aperture size than that of the coding metasurface with random codes. Because the coding particle shown in Figure 9.1d could experience nearly 300˝ as we change the side length w, 2-bit coding metasurface can be readily realized if the 360˝ phase span is equally divided into four values, for example, 0˝ , 90˝ , 180˝ , and 270˝ . Apparently, an n-bit coding metasurface can be implemented by using 2n coding particles with 360˝ {2n phase step. Due to the 180˝ phase difference of the coding particle for the 1-bit coding metasurface, the radiation pattern is always symmetrical with respect to the normal axis (z-axis in Figure 9.1), which this is not the case for the 2-bit coding metasurface. Much more functionalities can be realized with 2-bit coding metasurface, including anomalous beam reflection and refraction. Figure 9.2a shows the radiation pattern of a 2-bit coding sequence “0001101100011011. . .”, with super unit cell size of 6 ˆ 6. The normal incidence is redirected to an anomalous direction due to the gradient phase distribution. The electric-field distribution is given in Figure 9.3b for a direct view of the anomalously reflected wave. The perturbance observed in the wavefront of reflection is mainly due to the limited size of the encoded coding metasurface, which only consists of two complete periods of the above coding sequence. As the far-field radiation pattern is the Fourier transform of the near-field distribution, i.e. the coding pattern, it is obvious that more side lobes would appear if the coding pattern contains less complete period of the gradient coding sequence. The 2-bit coding particles can also be used to realize better RCS reduction, as is shown in Figure 9.3d, which has the same size with the 1-bit samples (Figures 9.2d–9.2f). Figure 9.3d shows its monostatic RCS reduction (normal direction) in broad bandwidth, which is lower than ´10 dB from 7.5 to 15 GHz. It is numerically found that one could achieve better performance (i.e. larger RCS reduction in broader bandwidth) with 2-bit random coding patterns.

9.3

Programmable Metasurfaces and Imaging Applications

9.3.1

Programmable Metasurface In Section 9.2, we have introduced the concept and some basic functions of the coding metasurfaces. However, the building blocks of coding metasurface are passive, which, in other words, cannot be altered once it is designed. Owing to the quantization charac-

Coding and Programmable Metasurfaces

z

(a)

3.86 2.61 1.36 0.11 –1.14 –2.39 –3.64 –4.89 –6.14 –7.39 –8.64 –9.89 –11.1 –12.4 –13.6 –14.9

Theta

(b)

y

Phi x z y x

(c)

(d) RCS Reduction (dB)

308

y

0 –5

–10 –15 –20

x

7

8

9

10

11 12 Frequency (GHz)

13

14

15

16

Figure 9.3 (a,b) The simulated near-electric-field distribution and far-field radiation pattern of a

coding metasurface with 2-bit coding sequence “00 01 10 11 00 01 10 11. . .”, respectively. (c) Coding pattern of 2-bit random coding sequence. (d) Broadband RCS reduction in the backscattering direction of the random coding patterns in (c). Results are calculated with the time-domain solver in CST Microwave Studio. Adapted from [46] with the permission of Nature Publishing Group.

teristic of coding particles, one could easily realize a tunable coding particle using some basic logic elements that have been widely used in digital circuits, such as diode. A simple implementation of a 1-bit tunable coding particle is demonstrated in Figure 9.4a, in which a pin-diode (Skyworks, SMP-1320) is mounted between two hollow conductors. Two separate metal strips at the back of the dielectric substrate (F4B, εr = 2.65, δ = 0.001) are connected to the top conductors through two metal vias and serve as the bias line. By switching the applied DC voltage, the pin-diode can be biased at the “ON” and “OFF” states, corresponding to the “0” and “1” digital states of the coding metasurface, respectively. Figure 9.4b shows the numerically optimized reflection phases from 7 to 10 GHz when the diode is biased “ON” and “OFF”. It can be observed that the phase difference is approximately 180˝ from 8.3 to 8.9 GHz and reaches exactly 180˝ at 8.6 GHz. One should notice that the phase difference changes significantly around the operational frequency. Therefore, the working bandwidth should depend on the requirement on the level of deviation from the optimum value 180˝ . By arranging these tunable coding particles in a 30ˆ30 array, with 5ˆ30 subarray as a control line, a programmable metasurface shown in the rightmost photo of Figure 9.4c can be implemented. As very little current is required to switch on each pin-diode, a field programmable gate array (FPGA) is able to provide sufficient current supply to drive the entire programmable metasurface and dynamically control it with pre-set coding sequences. Figure 9.5 shows the 3D simulated radiation patterns and 2D measured radiation patterns (in the anechoic chamber) when the programmable metasurface is

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309

Figure 9.4 (a) The structure design of the programmable metasurface. (b) Phase response for the

designed unit cell at the “ON” and “OFF” states. (c) A flow chart of a 1D-programmable metasurface under the control of a FPGA. Adapted from [46] with the permission of Nature Publishing Group. A black and white version of this figure will appear in some formats. For the color version, please refer to the plate section.

Figure 9.5 Numerical simulated radiation patterns at 8.3 GHz for the 1D programmable

metasurface with different coding sequences: (a) “000000”, (b) “010101” and (c) “001011”. Measured radiation patterns at 8.3 GHz for the 1D programmable metasurface with different coding sequences: (a) “000000”, (b) “010101” and (c) “001011”. Figures (d–f) are in dB scale and are normalized to the maximum radiation intensity of the case with “000000” coding sequence. Adapted from [46] with the permission of Nature Publishing Group.

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input with three different coding sequences, “000000”, “010101” and “001011”. For the first all “0” coding pattern, it is obvious that the normal incident wave should be reflected back to the backscattering direction, resulting in a single beam radiation in the normal direction (see Figures 9.5a and 9.5d). When the coding sequence changes to “010101”, the incoming wave splits into two beams (see Figures 9.5b and 9.5e), similar to the results in Figure 9.1b. For the third coding sequence 001011, multiple beams appear in the observation plane (see Figures 9.5c and 9.5f), resulting in lower energy radiation in each direction. One may notice from Figures 9.5d–9.5f that the radiation patterns are not symmetrical with respect to the normal axis. This is mainly attributed to the inequality of the reflection amplitude of the tunable coding particle biased at the “ON” and “OFF” states, which can be improved by a careful optimization of the unit cell structure design. In the initial proposal of the programmable metasurface, only 1D programmable metasurface is presented as a proof of principle, in which all coding particles are controlled by either row or column. Of course, more versatile coding patterns as well as radiation patterns will be obtained if each coding particle can be independently controlled, leading to a 2D programmable metasurface. However, some engineering problems need to be carefully considered when designing the layout of bias line of a large-scale programmable metasurface, which may inevitably influence the phase responses. For the above presented 1D example, all six control lines are directly connected to six output pins of the FPGA, while for the 2D programmable metasurface, the number of control lines dramatically increases to N 2 , which is far beyond the number of available output pins of a single FPGA or any other control unit. Some techniques, such as the strategy of “line-by-line” scan and holding circuit [58], must be employed to reduce the required number of control lines. We remark that the coding metasurfaces and phased-array antennas [59–61] have fundamental differences. First, the unit cell of coding metasurfaces is typically in deep subwavelength scale, with size between 1/8 to 1/4 free-space wavelengths. This is in contrast to that of the conventional array antennas, which is normally over 1/2 wavelength. Due to the unpredictable EM couplings between neighboring unit cells that have different geometrical parameters, the real reflection response of each unit cell must be different from the ideal reflection under the periodic boundary condition (PBC). Fortunately, such EM couplings can be suppressed by adopting the strategy of super unit cell, which consists of an array of N ˆ N identical unit cells. Those unit cells inside the super unit cell could exhibit accurate phase as what is obtained under the PBC. The smaller the unit cell is, the more unit cells can be included in the same area, and the more accurate phase responses are expected. However, the reflect-array antennas will suffer more serious EM couplings than the coding metasurfaces because of the larger unit cell [55, 65, 66]. Second, owing to the smaller size of unit cells, coding metasurfaces could provide more versatile functionalities than conventional array antennas. For example, a 2-bit coding metasurface could be used to realize the conversion from spatially propagating waves (PWs) to surface waves (SWs) with a gradient coding sequence ‘01230123. . . ’, by setting the coding period smaller than one free-space wavelength. However, such

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311

PW-SW conversion cannot be realized with the reflect-array antennas because the 2π gradient phase period is always longer than one free-space wavelength, which cannot provide the required wave momentum for PW-SW conversion. Third, the main purpose of introducing coding metasurfaces is not only limited to generate a single or multiple scattering beam patterns with controllable shape and direction, but also to establish a bridge between the physical EM fields and the digital information. Owing to the Fourier transform relation between the coding pattern and the far-field pattern proposed in this article, the digital characterization of the coding metasurface provides us the possibility to study metasurfaces and their scattering patterns from a fully digital perspective. We could therefore apply many existing theorems from the information theory directly to the design of coding metasurfaces to realize more versatile controls of EM waves, and to explore new relations that may further facilitate the designs. For example, we use Shannon entropy to estimate the information capacity carried by a coding metasurface by analyzing the entropy of its coding pattern [54]. We reveal the proportional relationship between the geometrical entropy (the entropy of coding pattern) and physical entropy (the entropy of the radiation pattern). Based on this new finding, we could generate a number of coding patterns that have roughly the same physical entropy, which may have potential usage of designing different antennas. For instance, in long-range or high-sensitivity radar detection applications, small physical entropy (with single or fewer beams) is always desired; while large physical entropy (with many beams distributed as randomly as possible) is required for single-sensor single frequency imaging that will be introduced in the following section. Furthermore, with the digital characterization of coding metasurface, we can also perform digital signal processing on coding metasurfaces to enable more freedoms in controlling the EM waves. Borrowing the concept of convolution theorem from signal processing, we proposed the principle of scattering pattern shift, which could steer the scattering beam to arbitrary direction with negligible distortion [51]. This new coding scheme solves, from the fundamental level, the problem of discrete scanning angle limited by the previous coding strategy. A continuous scan of the single-beam radiation in the upper-half space is allowed using a 2-bit coding metasurface which consists of only four coding states. It can be expected that many other digital signal processing algorithms will be exploited in the near future to realize more exotic physical phenomena.

9.3.2

Programmable Metasurface under Point Source Excitation In the previous section, we have reviewed the real-time control of radiation pattern for programmable metasurface. However, all the examples are considered under the normally incident plane-wave. In real applications, it is impossible, and also not wise, to provide a plane-wave illumination for the programmable metasurface, due to the limited antenna aperture of the feeding antenna. In addition, smaller feeding antenna is mostly preferred in practical applications to minimize the blockage effect of the normal radiation, which inevitably results in a distortion of the radiation pattern. In view of the above

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Coding and Programmable Metasurfaces

reasons, we should consider the scenario when there is a non-planar phase distribution on the programmable metasurface. As a proof of concept, [57] takes the point source excitation as an example to demonstrate how the coding pattern of programmable metasurface should be modified under such a spherical wavefront illumination. Figure 9.6a shows the schematic of a programmable metasurface under the point source illumination. Due to the fact that the impinging wavefront from the point source is non-planar, an additional phase variation is required for each coding particle of the programmable metasurface to compensate the spherical wavefront. In this case, the scattering electric field intensity in the far-field of each lattice (i.e., a super unit cell) can be expressed as Em,n = Km,n m,n fm,n (θ,ϕ)

1 ) exp(´j k0 rm,n ) exp(´j k0 rm,n 1 rm,n rm,n

(9.4)

in which Km,n is the scale coefficient, rm,n the distance between the lattice and the observation point, fm,n (θ,ϕ) the scattering pattern function of each lattice. For the item m,n = am,n exp(j φm,n ), am,n and φm,n are the amplitude and phase of the reflection coefficient, respectively. We should note that the last item in Eq. (9.4) is the contribution of the spherical wavefront of the point source. Due to the subwavelength nature of the coding particle, the detailed radiation characteristic of each coding particle becomes vague in the far-field region. Therefore, it is safe to neglect the effect of Km,n and fm,n (θ,ϕ) from Eq. (9.4) in calculating the radiation pattern of the entire programmable metasurface: F (θ,ϕ) =

M ÿ N ÿ

“ exptj ϕ 1 (m,n) + kDx (m ´ 1{2) sin θ cos ϕ

m=1 n=1

+kDy (n ´ 1{2) sin θ sin ϕsu

(9.5)

1 . It is clear that the phase is no longer discrete, in which ϕ 1 (m,n) = ϕ(m,n) ´ krm,n but becomes continuous in value, under the point source excitation. A strategy has to be found to compensate the spherical wavefront and keep the binary phase feature of the coding particle. Wan et al. propose to change the geometrical parameter of every coding particle according to their position on the programmable metasurface [57]. Since the position of point source is fixed relative to the modified programmable metasurface, it can function correctly for any coding patterns. A 1-bit unit cell is designed for the programmable metasurface, as shown in Figure 9.6b. A pin-diode with the same model type (SMP 1320 from SKYWORKS) is mounted to each unit cell, and is biased by two feeding lines connected to the back layer through two metallic vias. Numerical simulations show that the optimized unit cell works at 8.9 GHz, with the “ON” and “OFF” states having opposite reflection phases and identical reflection amplitudes. The authors present full-wave numerical simulations for the proposed programmable metasurface, which consists of 20 ˆ 20 coding particles. As comparisons, two different chessboard coding patterns are firstly simulated under the normal illumination. The first one includes 5 ˆ 5 control units, each consisting of 4 ˆ 4 unit cells biased with the same DC voltage, as is shown in Figure 9.6c, in which the black and white lattices

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313

Figure 9.6 Unit cell design and simulated radiation patterns under plane wave illumination. (a)

Schematic of a programmable metasurface under point source illumination. (b) Unit cell design for the programmable metasurface. (c,e) The chessboard coding pattern with lattice size of 5 ˆ 5 and 10 ˆ 5, respectively. (d,f) The simulated radiation patterns of the coding patterns in (c) and (e), respectively. Adapted from [57] with the permission of Nature Publishing Group.

(i.e., control units) represent digital states “1” and “0”, respectively. The simulated far-field radiation pattern is presented in Figure 9.6d, where four beams occur at four angles being symmetrical to the normal axis. As the control unit becomes rectangular shape consisting of 10 ˆ 5 unit cells (see Figure 9.6e), the radiation pattern changes accordingly, with the four beams pointing at (32.6˝ , 63.4˝ ), (32.6˝ , 116.6˝ ), (32.6˝ , ´116.6˝ ), and (32.6˝ , ´63.4˝ ). Using the least-square algorithm, the optimized values of S and W for each unit cell (see Figure 9.6b) are obtained, which are displayed in Figures 9.7a and 9.7b, respectively. As expected, the structure parameters of each unit cell on the programmable metasurface presents a ring-shaped distribution, which could compensate the spherical wavefront of the point source excitation and maintains 180˝ phase difference for the digital states “0” and “1”. To verify the effectiveness of the proposed compensation technique, full-wave numerical simulations are performed under the point source excitation for the same coding patterns as in Figures 9.6c and 9.6e. A rectangular horn antenna is employed as the point source and is placed 250 mm above the metasurface. The simulated radiation patterns of the two coding patterns under the point source illumination are shown in Figures 9.7c and 9.7d, which is in excellent agreements with the results under normal illumination (see Figures 9.6d and 9.6f). The small discrepancy should be attributed to the non-uniform amplitude distribution of the impinging wave on the programmable metasurface.

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Coding and Programmable Metasurfaces

Figure 9.7 (a,b) The distributions of the optimized geometrical parameters S and W , respectively, under the point source illumination. (c,d) Far-field radiation patterns of the coding patterns in Figures 9.7c and 9.7e, respectively, under the point source illumination. Adapted from [57] with the permission of Nature Publishing Group.

A prototype of the point source excited programmable metasurface is displayed in Figure 9.8a, in which both the programmable metasurface and rectangular waveguide are installed on a trestle. The programmable metasurface is composed of 20 ˆ 20 unit cells which are divided into 5 ˆ 5 independent control units (see Figure 9.8b). An FPGA written with the pre-designed coding patterns is employed to dynamically control the programmable metasurface. Figures 9.8c and 9.8d show the measured and simulated radiation patterns for such two coding patterns. Good agreements can be observed between the measured and simulated results, demonstrating the effectiveness of the proposed design. Since the 1-bit programmable metasurface will always produce more than one beam, the authors discuss some practical issues of the programmable metasurface in radar detection applications. They remark that the inevitable grating lobes could help position the target faster if prior knowledge about rough directions of targets is known. In addition, the multi-beam capability can reduce the position time as well as the computation complexity in the post-processing of radar location.

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315

Figure 9.8 Experimental configuration and measured results for the programmable metasurface

under point source excitation. (a) The experimental setup. (b) Fabricated sample of the programmable metasurface. (c) The simulated and measured gains when the azimuth angle is 63.4˝ (d) The simulated and measured gains when the azimuth angle is 45˝ Adapted from [57] with the permission of Nature Publishing Group.

9.3.3

Transmission-Type Programmable Metasurface for Imaging Applications In the previous sections, we have reviewed the huge potential of programmable metasurface in making real-time manipulations to EM wave. However, it should be noted that all the previous designs are aimed at controlling the reflected wave, which brings two major disadvantages in practical applications. First, the radiation pattern, even for the offset feeding configuration, will be affected by the shielding effect of the feeding antenna. Such distortions to the radiation pattern cannot be avoided, or compensated. Second, the physical space taken by the reflection-type programmable metasurface is typically larger than the transmission-type programmable metasurface, because the metasurface can be attached tightly to the aperture of horn antenna for the transmission-type programmable metasurface. For the above reasons, Li et al. attempted to design a 2-bit programmable metasurface at microwave frequency, which is illustrated in Figure 9.9a [67]. A smart design of the transmission-type unit cell structure is proposed by adopting a two-layered configuration, in which the unit cell can be controlled by both row and column. Two F4B substrates with thickness of 1mm are separated by an air gap of 3mm. Two pin-diodes (SMP 1320-079LF) are mounted on the top side of the top substrate and the bottom side of the bottom substrate, respectively. To conveniently control each diode, a metallic frame is added to the edge of unit cell to serve as the common ground. The central metal patch, connected to the bias line in the middle layer, serves as the positive bias line. When the

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Coding and Programmable Metasurfaces

Figure 9.9 Schematic and unit cell design of the proposed single-sensor and single-frequency

microwave imaging system. (a) The structure of the 2-bit coding unit cell, which is composed of two layers separated by an air gap of 3 mm. (b) The simulated transmission amplitudes and phases of the 2-bit coding unit cell at different states. (c) The 2-bit programmable coding metasurface, in which each unit is controlled by an FPGA with binary codes (0 or 3.3V voltage) on rows (green) and columns (red). (d) The schematic of the proposed single-sensor and single-frequency microwave imaging system. Adapted from [67] with the permission of Nature Publishing Group. A black and white version of this figure will appear in some formats. For the color version, please refer to the plate section.

two diodes are independently switched “ON” and “OFF”, there will be four different digital states “11”, “01”, “10”, and “00”, as indicated in Figure 9.9a. Figure 9.9 shows the simulated amplitudes and phases of transmission of the four digital states from 8 to 12 GHz. Although the phase span of the four digital states cannot cover the required 360˝ range, the structure can provide sufficient amplitude and phase shifts in the considered frequency band (9–10 GHz) for the single-sensor single-frequency imaging. Figure 9.9c shows a transmission-type programmable metasurface which consists of 5 ˆ 5 unit cells. As we can see, the diodes on the top substrate and bottom substrate can be controlled by rows and columns, respectively. It should be noted that the current feeding configuration does not allow an independent control of each unit cell. However, such a “line and column” feeding strategy is capable of generating different random coding patterns required by the imaging application. The transmission-type programmable metasurface featuring various random radiation patterns can be used to design a single-sensor single-frequency imaging system, which mainly involves the wavefront reconstruction or inverse scattering problem in the imaging plane. Unlike the conventional imaging technologies which commonly employ an array of detectors, the single-sensor imaging technique only requires one detecting source and one transmitting antenna to generate random radiation patterns. One of the

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317

key technologies behind the single-sensor imaging technique is the random modulators or masks [62–64]. Some attempts have been made on single-sensor imaging, including dispersive metamaterials [68–70], metamaterials absorbers [71,72], holographic metasurfaces [73,74]. Figure 9.9d shows the schematic of a single-sensor single-frequency microwave imaging system based on the 2-bit transmission-type programmable metasurface. The working mechanism is described as follows. The programmable metasurface, illuminated by a horn antenna and electrically controlled by an FPGA with random “row and column” coding sequences, could generate a series of random radiation patterns. Part of the radiation is reflected back to the horn antenna and is recorded by the VNA. As there are 5ˆ5 = 25 controlled elements in the programmable metasurface, a total number of 25 measurements, each time with a different random radiation pattern, is required to build up the matrix equation. Let us define the measured signal for a (p) given radiation pattern as V = (Vi ), p = 1,2, . . . ,P , which is related to the original object-area vector σ = (σi ), i = 1,2, . . . ,N, through the generalized system response matrix G = (Gpj ), p = 1,2, . . . ,P ,j = 1,2, . . . ,N . The following equation describes the relationship among V ,G and σ : » — — — –

V (1) V (2) .. . V (P )



»

G11 ffi — G21 ffi — ffi=— fl – GP 1

G12 G22

¨¨¨ ..

G1N

fi» ffi— ffi— ffi— fl–

. GP N

σ1 σ2 .. .

fi ffi ffi ffi fl

(9.6)

σN

For the current design, P and N are equal to 5 ˆ 5 = 25, resulting in a matrix G with 25 ˆ 25 = 625 elements. Our goal is to solve σ = G´1 V with the 25 measurement data (V ). One of the key points is to determine the generalized system response matrix G, which will be discussed in details in the following. To experimentally demonstrate the single-sensor single-frequency imaging system, the authors fabricate transmission-type programmable metasurface which includes 10 ˆ 10 = 100 elements. To simplify the feeding network complexity as well as the calibration process, the metasurface is divided into 5 ˆ 5 = 25 control units, each independently biased by an FPGA. Accordingly, the object is divided into 5 ˆ 5 = 25 sub-areas, each has an area of 20 ˆ 20 mm2 (about λ{2 at 9 GHz). Figure 9.10 presents the experimental setup of the imaging system. A computer with Labview software, connected to both the VNA (Agilent N5230C) and FPGA, is employed to send the required random coding sequences, and record the reflected signals automatically. We remark that a careful calibration should be performed for the generalized system response matrix before the imaging experiment. It can be accomplished by making a point-to-point scanning by placing a small metallic object at every sub-area on the imaging plane. Some of the calibration results of the generalized system response matrix under different coding patterns are presented in Figure 9.10b. An irregular variety can be observed for different curves, which is beneficial for better object reconstruction. Three different objects, a horizontal bar, a T-type object, and a T-type object rotated

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Figure 9.10 Experimental verification of the single-sensor single-frequency imaging system using transmission-type programmable metasurface. (a) Experiment setup for the single-sensor and single-frequency imaging system. (b) The measured information of the generalized system response matrix (G11 , G15 , G33 , G51 and G55 ) calibrated under the 25 different radiation patterns. Photographs of three metallic objects: (c) the horizontal bar, (d) the T-type object, and (e) the T-type object rotated by 90 degrees. (f–h) The imaging results of the three metallic objects at 9.2 GHz. Adapted from [67] with the permission of Nature Publishing Group.

by 90 degrees, are selected as the objects to be imaged in the experiment, which are shown in Figures 9.10c–9.10e, respectively. The experimentally reconstructed images (at 9.2 GHz) of the three objects are shown in Figures 9.10f–9.10h, from which the shape of objects can be clearly observed, validating the performance of the proposed transmission-type programmable metasurface based imaging system. The authors note

9.4 Summary and Outlook

319

that the proposed method can be extended to terahertz frequency, and further studied for near-field imaging.

9.4

Summary and Outlook In this chapter, we firstly review the basic concept, common functionality, working principle, and design strategy of coding and programmable metasurfaces. Some classical functions are presented with both 1-bit and 2-bit coding metasurfaces to demonstrate their powerful manipulations to EM waves, including beam splitting, anomalous beam reflection, and random diffusion. An analytical function is also provided for the fast calculation of the radiation pattern of a given coding pattern. Then we review a programmable metasurface design that is composed of six control lines, each independently controlled by an FPGA. Three different coding sequences that are pre-stored in the FPGA can be easily switched to generate different radiation patterns, which clearly demonstrate the reconfigurable behavior of the programmable metasurface. We also discuss the difficulties in designing the layout of bias line of a large-scale programmable metasurface, and provided a possible solution to reduce the number of control lines by adopting “line-by-line” scanning technique, which is widely applied in the liquid crystal display (LCD). In view of the similarity between coding metasurfaces and traditional phased-array antennas, we distinguish them in terms of the electrical size, EM interference between neighboring unit cells, functionalities and capabilities, especially emphasizing on the digital description of coding metasurface which links the physical metasurface with the field of information science. Next, we review a subsequent work of the initial programmable metasurface, which considers a more realistic feeding scenario of point source excitation. To allow each coding particle in the programmable metasurface to have the correct phase responses, a modification should be performed to the geometry of every coding particle to compensate the spherical wavefront of the point source. Both simulations and experiments show that the compensated programmable metasurface under point source excitation can generate almost the same radiation patterns as what are obtained under the plane wave excitation. Such a practical technique makes programmable metasurfaces more adaptable in engineering applications. In the end, we review a 2-bit transmission-type programmable metasurface, which is employed to realize a single-sensor single-frequency imaging system. To simplify the feeding line design, a “row and column” strategy is proposed for generating different radiation patterns. A prototype of the programmable metasurface based imaging system is fabricated to demonstrate the excellent imaging performance at a single frequency with only one horn antenna. There is currently a growing demand on the compact, beam-forming, energy efficient antennas. Programmable metasurfaces, with outstanding performance in radiating desired patterns in real-time, could be further exploited to realize the high-level functionality of adaptive beamforming. We believe, in the near future, these advanced programmable metasurfaces could detect and analyze the surrounding EM environ-

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ment, then transmit these data to network processing unit for optimization, and finally generate the optimized radiation pattern which could provide the best signal for all terminals located at different positions. Owing to the digital representation of programmable metasurface, it can be envisioned that more and more researches in the field of information theory, cybernetics and artificial intelligence theory can be combined with programmable metasurfaces to enable more intelligent programmable metasurface with some exotic behaviors like machine-learning, cloud computing, and data mining.

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10

Metamaterial and Metasurface Cloaking: Principles and Applications Giuseppe Labate, Ladislau Matekovits, and Andrea Alù

10.1

Introduction In this chapter, we review the use of metamaterials and metasurfaces to induce an exotic electromagnetic response: a zero (or quasi-zero) scattered field outside the domain where the metamaterial or metasurface structures are localized. When an object or a collection of scatterers establishes the zero scattering condition, it forms an example of non-radiating secondary sources whereas if the process can be tuned through a coating layer, properly designed, a cloaking device is realized. These electromagnetic field distributions are generally described in this chapter in a compact fashion. Starting from the non-uniqueness of the inverse scattering problem, the Harmonic Field Series (HFS) representation and the Field Integral Equation (FIE) formulation are introduced for the scattering analysis: for the synthesis process, the HFS design gives rise to plasmonic cloaking, mantle cloaking and parity-time cloaking, whereas the FIE design leads to the concepts, for a non-radiating or cloaking problem, of the strong and weak solutions. Similarities of and differences between these design techniques are discussed, with both theoretical formulations and experimental examples. Physical bounds and limitations due to material dispersion, time-invariance, linearity and passivity are pointed out, with impact on the robustness against the change in the direction of incidence of the incoming wave and against the change in the frequency of the impinging field. Future paths towards practical non-radiating structures and cloaking devices are finally discussed, together with an outlook on the field.

10.2

Non-uniqueness of the Scattering Problem: Non-radiating Sources and Cloaking Devices The propagation of electromagnetic waves is naturally perturbed when an arbitrary object is placed along the propagation path: such perturbation gives rise to the scattering from – dielectric or conductive – objects when excited by an external electromagnetic field. This chapter addresses the question of whether tailored metamaterial and metasurface covers can avoid or suppress the scattering and under which conditions – related to the incoming wavelength and direction of arrival of the incident wave – this effect may take place. 325

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Even if the mantle of invisibility has fascinated the curiosity of the humankind for centuries, the topic has been mainly relegated to the non-scientific literature, influenced also by science fiction writers and movie directors. According to [1], one of the first scientific works on invisibility took place in 1902, when the physicist R. W. Wood reported that “a transparent body, no matter what its shape, disappears when immersed in a medium with the same refractive index and dispersion” [2]. In the literature of inverse source and scattering problems [3–7], the invisibility effect can be compactly reduced to investigating the scattering operator, which takes into account shape boundaries, constitutive parameters and location in space of an arbitrary object or scatterer. No matter how complicated and exotic the interactions between the wave and objects are, the rigorous solution that guarantees exact zero scattering response is – by definition of a generic scattering operator L – within the subdomain whose image is zero. We are interested in solutions to be found within the non null-space of the kernel of the operator L. If this subdomain exists the inverse scattering operator gives in general a non-unique solution: it could go back to the true solution (that we do not know a priori), or on the other hand, it could give a set of solutions that are a linear combination of the initial scattering problem plus its zero-scattering contributions, independently and randomly located within the kernel subdomain. This is a fundamental issue for imaging problems, where the final goal is to extract – as accurately as possible – useful information content from scattered waves in order to image or sense a domain of interest: for biomedical applications, the reconstruction process [8–10] could be related to unknown scatterers as tumoral or cancer cells and the non-uniqueness feature of the problem has to be handled seriously in order to avoid false positive or false negative test results. However, the non-uniqueness of the scattering problem represents an additional degree of freedom to be explored for synthesis problems, since the solution has not to be sensed or reconstructed, but it is completely assembled within the kernel itself: from another perspective, this research area has been extensively investigated in the literature for zero scattering responses during the years [11–14]. As reported in Figure 10.1, the scattering from a generic object – here represented by a square – gives field values in the observation domain that correspond to a certain group of scattering measurements. Since there are some other shapes or objects – symbolized by a star and a thunder shape – that have zero scattered fields, in the source domain these objects can be combined with the initial solution in order to create other possible solutions – symbolized by a star-square and a thunder-square shape, which provide the same identical scattering response. As shown in the next sections, invisible objects can be categorized in terms of their induced equivalent electromagnetic sources, responsible for any scattering process and all the field perturbations. Specific and tunable transparency responses can be supported by suitably tailored artificial materials (metamaterials [15]) or artificial thin impedance boundaries (metasurfaces [16]). Considering the electromagnetic response of a dielectric or metallic particle in a background scenario, the zero-scattering effect can be achieved by (1) locally shaping its boundaries, or (2) forming novel architecture schemes, enabling non-radiating sources [17–20], or (3) adding an external coating – or cloak – to the bare object kept in its original state, forming cloaking devices [21–23].

10.2 Non-uniqueness of the Scattering Problem

327

The final effect remains to emulate the absence of objects, i.e. to restore wave propagation in the background. Since any complicated electromagnetic device possessing a zero or quasi-zero scattering response tends to behave equivalently to the background material (or neutral inclusion [24]), it is possible to derive design equations for these non-scattering sources or cloaking conditions. Due to the fact that any combination of scatterers can be studied by substituting their constitutive parameters with the background ones (thus, complete free-space scenario) while considering induced equivalent electromagnetic sources (thus, not actual physical sources), the invisibility problem can be recast by exploiting the volume and surface equivalence principles [25–27], and the kernel of the scattering operator can be categorized in subdomains in order to indicate how these zero or quasi-zero values of such equivalent sources are imposed. In this chapter, we first present two general approaches to analyze a generic scattering event (Section 10.3), based on the harmonic expansion or Mie theory [28] (Section 10.4) and based on a weighted summation of point-wise elementary responses or FIE [29] (Section 10.5). Within each of these two scattering techniques, we consider how it is possible to pass from the analysis of a generic scattering process to the synthesis of a zero-scattering phenomenon. Having in mind Figure 10.1 throughout this chapter, we will focus our attention on two kind of solutions based on the FIE interestingly related to several cloaking techniques in the literature and grouped as a function of the zero or quasi-zero values of their induced electromagnetic equivalent sources, namely the strong solution and the weak solution [30]. Primarily, Section 10.5.1 provides insights of the transformation optics method [31–39], as the family of impedance matching-based techniques and the second kind of solution, in Section 10.5.2, gives insight into the scattering cancellation method, as the family of plasmonic [40–50], mantle [51–65] and parity-time symmetry [66–69] cloaking. After having completed this broad section of overview on cloaking methods and the relation among them, bounds and implications of these design formulas (Section 10.6), in particular focusing on the ability to achieve invisibility from several impinging directions, namely the directionality issue (Section 10.6.1) and the bandwidth issue (Section 10.6.2), will be discussed. Future trends and opportunities are highlighted in the conclusions.

Figure 10.1 The non-uniqueness of the Scattering Problem, as related to the scattering operator L.

A black and white version of this figure will appear in some formats. For the color version, please refer to the plate section.

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10.3

Scattering Theory: Harmonic Field Series and Field Integral Equation Representations Electromagnetic wave scattering is of crucial importance for many imaging, monitoring and sensing applications: detection of breast cancer in human tissues, finding objects in the subsoil (mines, oil or gas, among others) or through-the-wall imaging. For this reason, for a good modeling of the scattering problem, it is important to understand how all the mutual interactions between the incoming fields and the illuminated objects occur. In particular, as a function of the data and unknowns of a scattering process, once the incident illumination is set in the scenario two kind of problems can be defined: (1) scattering analysis or the Forward Scattering Problem (FSP) [70–75], where all the data about the object – shape and material properties – are known and the electromagnetic scattered field is the unknown response of the system to compute, and (2) scattering sensing or the Inverse Scattering Problem (ISP) [4–10], where the scattered field response is probed or assumed to be known, and the object characteristics – shape and material properties – are inferred from these measurements. Consistent with Figure 10.1, the FSP implies moving from the source to the observation domain, whereas the ISP is moving in the opposite direction: due to the non-uniqueness of the scattering problem, it is more challenging to solve an ISP rather than a FSP and further regularizations are needed in solving the ISP with good accuracy. Including the scattering responses of natural existing objects, the synthesis process can be guided by the ideal scattering response for an invisible system, that is, pˆ ¨ Fs (r) = 0 with r P  where pˆ is a unit polarization vector and Fs is the scattered field in the region , where the observation domain is located. For non-radiating and cloaking problems, the  domain can be in the near- or in the far-field of the object to be hidden and the scattering problem reduces to find an inverse strategy to compute the device (bare object with tunable size/shape or fixed object plus a tunable cloak) that achieves such invisibility condition. A possible timeline of the invisibility problem according to this mathematical definition is sketched in Figure 10.2. Since the first experimental proof of Wood in 1902 [2], the research area about invisibility has taken advantage of the main breakthroughs achieved in the area of artificial dielectrics, plasmonics/metamaterials and thin metasurfaces. Here, they are roughly grouped in three main milestones with related papers (even if the list is not including many other exciting developments in the area): the works of Brown and Rotman about less-than-one refractive index in artificial dielectrics in 1953 and 1962 [74, 75]; the work of Veselago on the theoretical investigation of simultaneously negative constitutive parameters in 1968 [76]; and the one of Kildal on hard surfaces for the reduction of scattering levels in 1996 [77]. The inherent challenges in this problem stimulated by the exotic propagation within complex volumetric and/or surface metamaterial structures has fascinated researchers in pursuing the ultimate control of electromagnetic fields through invisibility, pushing the technology beyond. A significant step towards practical cloaking devices has been made in 2005–2006 [31–33, 40], when the first theoretical cloaking techniques based on metamaterials have been put forward in the literature. In the next subsections, we

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329

Figure 10.2 Timeline of the Invisibility Problem, from first experimental proof to recent works.

describe two different scattering models: the harmonic-based Mie theory (Section 10.4) and the field integral equation (Section 10.5) representations, and their connection with these cloaking techniques.

10.4

Harmonic Field Series Representation: Cloaking Design with Mie Solutions In 1908 Gustav Mie wrote a paper about the color of gold colloids [78], introducing the Mie solution of the spherical scattering problem, the first rigorous dynamic solution for the scattering of electromagnetic waves by a finite 3D object. Electric and magnetic fields are expanded, separately in their radial and angular variables, according to a proper set of harmonic basis functions. It is important to notice that this method can be applied to particles considered to be center-symmetric objects like spheres or cylinders. Despite the fact that this kind of solution is almost 100 years old, it was only around 2005 [40] that it has been used to investigate the conditions to design metamaterial coatings able to ensure that the scattering coefficients become identically zero in the observation domain . In this chapter, we will consider a 2D scenario for ease of treatment and consistency with the next chapters, assuming canonical cylinders for the scattering objects to be immersed in a homogeneous background material and illuminated by an incident electromagnetic plane wave. A straightforward generalization can be pursued for spherical objects by using harmonic spherical waves, as already detailed in the literature [21, 23]. The nature of the object to be hidden gives rise to specific boundary conditions at the outer interface: in this chapter, we will consider first the scattering problem for a dielectric object, with internal fields induced within the particle itself. The metallic case can be treated as the limiting case for which the electromagnetic fields are exactly zero inside the object. For a cloaked cylinder with homogeneous dielectric permittivity ε, the electric and magnetic fields can be expanded as a sum of known cylindrical harmonic basis functions

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Metamaterial and Metasurface Cloaking: Principles and Applications

and unknown coefficients for each polarization [25, 28]. Assuming, throughout this chapter, transverse-magnetic (TMz ) incident fields – where zˆ is along the cylinder’s axis – in order to derive the cloak design formulas, it is possible to write Et[1] (r,ϕ) =

n=+ ÿ8

j ´n f (r)ej nϕ

for r ď a

(10.1)

for a ď r ď b

(10.2)

for r ě b

(10.3)

n=´8

Et[2] (r,ϕ) =

n=+ ÿ8

j ´n g(r)ej nϕ

n=´8

Et[b] (r,ϕ) =

n=+ ÿ8

j ´n h(r)ej nϕ

n=´8

Ht[1] (r,ϕ) = ´ Ht[2] (r,ϕ) = ´

j ωμb

j ωμb

Ht[b] (r,ϕ) = ´

n=+ ÿ8

1

j ´n f (r)ej nϕ

for r ď a

(10.4)

for a ď r ď b

(10.5)

for r ě b

(10.6)

n=´8

n=+ ÿ8

1

j ´n g (r)ej nϕ

n=´8

j ωμb

n=+ ÿ8

1

j ´n h (r)ej nϕ

n=´8

where the radial functions (and accordingly, their derivatives with respect to r) are grouped as f (r) = an Jn (k1 r)

(10.7)

g(r) = bn Jn (k2 r) + dn Yn (k2 r)

(10.8)

h(r) = Jn (kb r) + cn Hn(2) (kb r)

(10.9)

where k1 , k2 and kb are the wave vectors in the core (r ď a), cladding (a ď r ď b) and background (r ě b) region, respectively. For this 2D scattering problem, the basis (2) functions are Bessel Jn (¨), Neumann Yn (¨) and Hankel Hn (¨) functions (the latter of the second order) [79] with their unknown expansion coefficients an , bn , dn and cn [40– 46]. From the boundary conditions, by imposing the continuity of tangential electric and magnetic total fields in Eqs. (10.1)–(10.6) at r = a and r = b it is possible to determine all the unknown coefficients of the harmonic series. In the following section, plasmonic cloaking [40–50], mantle cloaking [51–65] and parity-time cloaking [66–69] approaches will be reviewed, aiming to select a proper shell that suppresses the scattering from the core cylinder. It is worth noting that all these techniques belong to the scattering cancellation family and are categorized as a function of the method used to ensure the continuity in the field functions while canceling the scattering of one (or more) harmonic wave(s). In order to solve for the synthesis process, one needs to enforce an ideal invisibility condition, which – by looking at Eq. (10.9) – is always achieved if cn fi

Un =0 Un + j Vn

@n

(10.10)

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331

for all the relevant scattering harmonic waves, where Un and Vn are determinants that can be computed applying Cramer’s rule [41]. For the invisibility condition, Un = 0 is equivalent to Eq. (10.10), whereas for the resonant (maximum) scattering, i.e. cn = 1, the condition Vn = 0 is to be fulfilled [40].

10.4.1

Plasmonic Cloaking: A Volumetric Metamaterial Coating For a dielectric object with homogeneous dielectric permittivity ε1 and radius a, coated by a homogeneous dielectric material ε2 of thickness d = b´a, the boundary conditions can be written as f (a) = g(a)

(10.11)

f (a) = g (a)

(10.12)

g(b) = h(b)

(10.13)

1

1

1

1

g (b) = h (b)

(10.14)

Since the system is linear, we can solve it using a matrix formalism [40]: fi » fi » fi » Jn (k1 a) ´Jn (k2 a) ´Yn (k2 a) 0 0 an 1 1 1 ffi —?εr J (k1 a) ´?εr J (k2 a) ´?εr Y (k2 a) 0 ffi 0 1 n 2 n 2 n ffi — bn ffi — — ffi ffi– fl = — — (2) – Jn (kb b) fl 0 Jn (k2 b) Yn (k2 b) ´Hn (kb b) fl dn – 1 1 1 ? ? ? (2)1 cn Jn (kb b) εr2 Jn (k2 b) εr2 Yn (k2 b) ´ εr2 Hn (k2 b) 0 (10.15)

Since, for the invisibility condition, Eq. (10.10) or Un = 0 has to be enforced, for this case, for the cloaking condition, one gets ˇ ˇ ˇ ˇ Jn (k1 a) ´Jn (k2 a) ´Yn (k2 a) 0 ˇ ˇ 1 1 1 ? ? ˇ ˇ ? 0 ˇ ˇ εr1 Jn (k1 a) ´ εr2 Jn (k2 a) ´ εr2 Yn (k2 a) Un = ˇ ˇ = 0 (10.16) ˇ Yn (k2 b) Jn (kb b) ˇ 0 Jn (k2 b) ˇ ˇ 1 1 1 ? ? ˇ εr2 Jn (k2 b) εr2 Yn (k2 b) Jn (kb b) ˇ 0 If a non-radiating condition with a stand-alone dielectric cylinder has to be achieved, the material shell with k2 can be removed (mathematically, it corresponds to bn = dn = 0) and the compact result is [65] ˇ ˇ ˇ J (k a) 1 1 Jn (kb a) ˇˇ ? ˇ n 1 Un = ˇ ? ˇ = 0 or Jn (k1 a)Jn (kb a)´Jn (kb a) εr1 Jn (k1 a) = 0 1 1 ˇ εr1 Jn (k1 a) Jn (kb a) ˇ (10.17) However, it should be noticed that for a bare cylinder, this condition cannot be achieved for k1 significantly higher than kb without having other scattering orders much larger than zero, since this bare cylinder with high contrast (ε1 " εb ) would not meet the invisibility condition for all the relevant scattering orders n [40]. Using a metamaterial shell, however, condition (10.16) can be met for the relevant scattering orders, realizing a cloaked object for the dominant scattering harmonic waves. A physical insight into this condition is provided in the limit max(k1 a,k2 a,k2 b,kb b) ! 1

(10.18)

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Metamaterial and Metasurface Cloaking: Principles and Applications

Figure 10.3 Design plot for 2D cloaking condition for a core-shell particle in quasi-static regime. Physical insight on plasmonic cloaking: polarizability compensation effect, with opposite value within the core and shell region. Colors represent the ratio γ = a{b P [0,1]. Reprinted, with permission, from: Andrea Alù et al., Achieving transparency with plasmonic and metamaterial coatings, 10.1103/PhysRevE.72.016623 and 2018 Copyright 2018 by the American Physical Society.

and this condition can be useful to synthesize cloaking devices that are in the subwavelength limit or quasi-static regime. In this simplified case, the relevant scattering orders are one or two, and therefore the invisibility condition is easily achievable. In this limit, Eq. (10.18) for the main dominant harmonic wave of index n = 0 can be written as [40] c ε2 ´ εb γ = (10.19) ε2 ´ ε1 where εb , ε1 and ε2 are the absolute permittivity of the background, object and cloak, respectively, with the radii’s ratio γ = a{b. The physical interpretation of the simplified Eq. (10.19) is reported in Figure 10.3 (central panel), and it highlights the mechanism behind plasmonic cloaking: in this frequency regime, once defined, the background permittivity all the other dielectrics are normalized with, it is only possible to combine a natural–artificial dielectric pair (II and IV quadrant of the chart), with proper γ ratio, in order to achieve a cloaking condition. As sketched in the side panels, the polarization induced in the core region can be coupled to the one of opposite sign induced in the cladding region, giving rise to an overall system with a net zero polarization. Since the concept of plasmonic cloaking was introduced back in 2005, other interesting

10.4 Harmonic Field Series Representation: Cloaking Design with Mie Solutions

333

studies have been performed with further analysis and additional investigations of this technique [41–50]. The first experimental verification of plasmonic cloaking came in 2012 [50], when the cloaking device was assembled with metallic implants in a dielectric material, mimicking propagation and scattering as in an artificial dielectric with less-than-one permittivity value. Figure 10.4 reports near-field maps of the electric field distribution as shown in [5]. It is possible to observe that, at the designed frequency of 3.1 GHz, the deepest reduction of the scattering levels occurs, whereas the phase fronts of the total fields are restored very similarly as for the free-space case, as expected. The additional row at 3.3 GHz highlights the limits of the cloaking bandwidth, with deteriorating and poor performances at 2.7 and 3.8 GHz, far from the central cloaking frequency. In the right panel, the details about the experimental sample of the object plus its coating cloak, as realized at microwave frequencies [50], are shown.

10.4.2

Mantle Cloaking: A Thin Metasurface Coating For a dielectric object with homogeneous dielectric permittivity ε1 and radius a, now coated by a surface impedance Zs or surface admittance Ys , loaded on top of a homogeneous substrate, with dielectric permittivity ε2 and thickness d = b ´ a, the boundary conditions can be written as in Eqs. (10.11)–(10.13), but the last one i.e. Eq. (10.14), changes into [51] ı 1 ” 1 1 h (b) ´ g (b) (10.20) Ys g(b) = ´j ωμb due to the novel admittance/impedance boundary condition. Please notice that the absence of the surface admittance (Ys = 0) leads to the same boundary condition as in Eq. (10.14). As for plasmonic cloaking since the system is linear it is common in the literature to solve this problem in matrix formalism; for details, the reader is referred to [51–64]. Since the invisibility condition always leads to Eq. (10.10), the solution can be found as a one-to-one correspondence between the surface admittance/impedance and the related harmonic index n to be canceled. As it was demonstrated in [51], approximations in the quasi-static regime, similarly to the previous discussion on plasmonic cloaking lead to specific dispersive values of the surface impedance (or admittance) cloak and the result is   ωεb εr1 ´ 1 γ a 2   or Ys (ω) = ´j (10.21) Zs (ω) = +j 2 ωεb εr1 ´ 1 γ a As reported in Figure 10.5, the impedance/admittance boundary condition can be supported by loading the dielectric substrates – of crucial importance for conductive core objects – with metallic patterns: square patches, Jerusalem crosses and square-crosses [55] are only a few examples of them.

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Metamaterial and Metasurface Cloaking: Principles and Applications

Figure 10.4 Details on the experimental sample at microwaves for plasmonic cloaking (top). Electric field distribution for near-field mapping of the uncloaked, cloaked and free-space dielectric cases (bottom). D Rainwater et al., “Experimental verification of three-dimensional plasmonic cloaking in free-space”, New Journal of Physics, Volume 14, January 2012 © Deutsche Physikalische Gesellschaft. Reproduced by permission of IOP Publishing. CC BY-NCSA.

First proposed in 2009, mantle cloaking [51] has led to many other investigations that have been – even recently – pursued in the literature for several applications and insights [52–65]. The main difference with respect to plasmonic cloaking is related to the external boundary condition, but physically, the compensation mechanism, based on

10.4 Harmonic Field Series Representation: Cloaking Design with Mie Solutions

335

Figure 10.5 Thin impedance coatings for mantle cloaking dielectric cylinders against TM polarized fields: square patch (left), Jerusalem cross (center), cross (right). Reprinted from Yashwanth R. Padooru et al. ‘Analytical modeling of conformal mantle cloaks for cylindrical objects using sub-wavelength printed and slotted arrays’ 2012, with the permission of AIP Publishing.

the mutual interference between the core and cladding region, remains the main guideline of this cloaking technique. The main advantage is that now the cloak is no more volumetric, with the complications of installing plasmonic or metamaterial particles within a specific volume, but instead the cloak region is confined to a thin conformal surface, like a mantle wrapping the object to be hidden. As reported in Figure 10.6, for a finite dielectric rod loaded by a mesh of metallic strips, the scattering gain levels are lower than the 0 dB line, representing the object without any cloak. At the central frequency, for which the cover has been designed, a deep reduction of 9 dB is achieved for both full-wave and analytical analysis of such coating device. An experimental setup has been realized [60], where one antenna is transmitting an electromagnetic wave and another one is receiving the radiation (bi-static configuration), not perturbed if the cloak is correctly working (bottom panel).

10.4.3

Parity-Time Symmetry Cloaking: A Balanced Loss-Gain Coating In all the previous approaches, the boundary conditions are imposed separately for each n harmonic wave. However, in the case of arbitrarily shaped contours, the boundary value problem becomes more intricate. In all the non-radiating and cloaking problems, in any case, it is of interest to investigate the simultaneous suppression of all – or a finite number N – of harmonic waves [66]. This can be performed in a compact fashion by considering the Mie solution as a whole, not only with its radial functions f (r), g(r) and h(r) but – this is the key point – also considering the angular dependence (in 2D, only azimuthally). This gives the following boundary conditions:

Ys (ϕ)Et[2] (r

Et[1] (r = a,ϕ) = Et[2] (r = a,ϕ)

(10.22)

Ht[1] (r Et[2] (r

(10.23)

= a,ϕ) =

Ht[2] (r = a,ϕ) Et[b] (r = b,ϕ)

= b,ϕ) = ” ı = b,ϕ) = Ht[b] (r = b,ϕ) ´ Ht[2] (r = b,ϕ)

(10.24) (10.25)

In the case of metallic cylinders, Eqs. (10.22) and (10.23) show a zero in the left side, thus compactly yielding a single independent boundary condition. In the following

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Metamaterial and Metasurface Cloaking: Principles and Applications

Figure 10.6 Full-wave simulation and experimental setup for mantle cloaking. Reprinted, with permission, under CreATIVE Commones License 3.0 (https://creativecommons.org/licenses/by/ 3.0/) from [66].

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337

section, due to the simplicity of treatment, computations will be performed in this case, when a short-circuit condition is imposed at r = a (metal boundary). The invisibility condition is achieved when Eq. (10.10) is verified – the zeros of the external scattering coefficients cn – but in this approach, more emphasis is put on the fact that this is simultaneously achieved for all the infinite (or a subset N ) harmonic waves. In this scenario, by exploiting Eq. (10.24) with only the incident fields expanded in cylindrical waves, the surface admittance boundary condition can be rewritten as ff « Ht[b] (r = b,ϕ) ´ Ht[2] (r = b,ϕ) Ys (ϕ) = Et[b] (r = b,ϕ) n=+8 ı j ´j kb cos(ϕ) ÿ ´n ” 1 1 h (b) ´ g (b) ej nϕ =´ e j (10.26) ωμb n=´8 By asking for all harmonic coefficients to be set to zero, Eq. (10.26) reduces to Ys (ϕ) = ´j YB e´j kb cos(ϕ)

n=+ ÿ8

” 1 ı 1 1 j ´n Jn (kb b) ´ bn Jn (kb) ´ dn Yn (kb) ej nϕ

n=´8

(10.27) where YB = kb {ωμb is the free-space admittance and k is the wavenumber in the dielectric spacer. From the first boundary conditions in Eq. (10.22) and the third one in Eq. (10.24), rewritten for a metallic cylinder as 0 = bn Jn (ka) + dn Yn (ka)

(10.28)

bn Jn (kb) + dn Yn (kb) = Jn (kb b)

(10.29)

it is possible to compute the unknown coefficients bn and dn . Please notice that Eq. (10.29) has been written by considering that all cn coefficients are zero, thus ensuring both the tangential continuity and the cloaking conditions. By exploiting the Bessel and Neumann property [79] 2 (10.30) πx the derivatives disappear in Eq. (10.27), and the final azimuthal surface admittance reads „ j n=+8 Jn (ka) 2 ´j kb cos(ϕ) ÿ ´n j e ej nϕ Ys (ϕ) = ´j YB π kb J (kb)Y (ka) ´ J (ka)Y (kb) n n n n n=´8 (10.31) This mathematical function, representing the continuous admittance curve that yields perfect invisibility, shows the following interesting property: * " +Re rYs (π ´ ϕ)s = ´Re rYs (ϕ)s ˚ (10.32) Ys (π ´ ϕ) = ´Ys (ϕ) = +Im rYs (π ´ ϕ)s = +Im rYs (ϕ)s 1

1

Jn (x)Yn (x) ´ Jn (x)Yn (x) =

which is defined as parity-time symmetry condition. As reported in Figure 10.7, this directly translates into a coating with a complex impedance surface: considering the symmetry at π, the real part has positive and negative values, indicating a cloak with loss and gain [66].

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Metamaterial and Metasurface Cloaking: Principles and Applications

Figure 10.7 (left) Basic principle of parity-time symmetry cloaking: front absorption and back-lasing. (right) Possible implementation with finite loading admittances. Reprinted, with permission, from: Dimitrios L. Sounas et al. ‘Unidirectional Cloaking Based on Metasurfaces with Balanced Loss and Gain’, 10.1103/PhysRevApplied.4.014005 Copyright 2018 by the American Physical Society.

Figure 10.8 (left) Simulated fields for parity-time symmetry cloaking: bare metallic cylinder, (center) cloak with only loss side, and (right) cloak with both loss-gain sides. Reprinted, with permission, from: Dimitrios L. Sounas et al. ‘Unidirectional Cloaking Based on Metasurfaces with Balanced Loss and Gain’, 10.1103/PhysRevApplied.4.014005 and 2018 Copyright 2018 by the American Physical Society.

In particular, the cloaking mechanism is based on a dual absorption-emission effect: in the first half of the admittance coating, the incoming energy is absorbed due to the lossy coating loads, whereas in the second half of the coating, the admittance loads turn out to be active, and synthesized in such a way that the same energy is released without affecting the external fields, as the object was not present at all in the scenario. If the dielectric spacer, useful to avoid a direct short-circuit between the impedance surface and the metallic cylinder, is very thin and the condition kd ! 1 holds true (with d = b ´ a), the function that depends on the variable b can be expanded in Taylor series around the point r = a , which gives Re tYs (ϕ)u = ´YB cos(ϕ) „ j 1 1 Im tYs (ϕ)u = ´YB ´ k(b ´ a) 2ka

(10.33) (10.34)

i.e. a cosine-type functional dependence for the real part and an azimuthally constant imaginary part of such complex surface admittance. In Figure 10.8, simulation results are reported for the uncloaked object (left), object with only a lossy coating (center) and the object with its proper loss and gain cloak (right) [66]. The results show that the energy flow is completely re-established as the object would not be present in this scenario, with the phase front completely flat as of the incoming fields. Due to this azimuthal variation that ensures a general scattering cancellation, this technique can be applied also to non-canonical supports with structures as rhomboids

10.5 Field Integral Equation Representation

339

or other arbitrary shaped devices as detailed in [66]. The physical principle behind this mechanism can be inferred by the reflection coefficient: at the gain side, an infinite value of the local reflection coefficient is formed and, with an impinging infinitesimal energy, a finite scattered power is released. Additional investigations on the stability have been performed [66], showing a trade-off between the achievable bandwidth and the deepest reduction on scattering levels.

10.5

Field Integral Equation Representation: Cloaking Design with Non-radiating Sources In this section, we focus our attention on 2D problems formulated according to the Volume and Surface Equivalence Principles [25]. The equivalent sources that sustain the scattered energy are mathematical functions that help solving complicated electromagnetic problems by removing all the scatterers of arbitrary shape in a certain background scenario and substituting them with specific sources in suitable domains of definition. A dielectric object is considered a volumetric scatterer, because it permits fields within its localized region, and thus it can be equivalently substituted by a volumetric source [25] Jv (r) = j ωεb χ (r)Et (r) with r P

(10.35)

where Jv (r) is a volumetric current density [Am´2 ] that replaces the dielectric, embedded in a homogeneous background with absolute permittivity εb , in its spatial domain , where the object is localized with its electric susceptibility χ (r) ” ε(r) ´ 1. A metallic object is considered a surface scatterer, because it enforces to zero the fields in its entire volume, shielding any wave from entering in its domain of definition, and thus it can be equivalently substituted by a surface source at its boundary as [25] Js (r) = nˆ ˆ [Ht (r + ) ´ Ht (r ´ )]

with r P 

(10.36)

where Js (r) is a surface current density [Am1 ] that completely replaces the metallic object at its contour boundary . The scattering problem is thus the sum of all the elementary responses of multiple current density elements that interact with each other, also considering their reciprocal spatial distance. If the scattered field is measured, as 1 reported for example in Figure 10.2, in the external observation points r P  with respect to the defined volume of source points r P V where all the scatterers are placed, the equation for the non-radiating condition is [30] ij 1 F rJeq s fi Jeq (r)G(r 1,r)dV = 0 Ñ @r P  (10.37) V

where F r¨s is the external operator: it maps the generic equivalent surface and/or 1 volumetric source Jeq weighted by G(r ,r), the Green’s function that takes into account the intensity of all the mutual interactions that all source points r are able to induce in 1 the particular observation point r . If condition (10.37) is verified in the entire external domain , the source is non-radiating by definition. This is a FIE representation due to the fact that each equivalent source depends on the total fields (electric or magnetic),

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Metamaterial and Metasurface Cloaking: Principles and Applications

without any expansion. Please notice that the formulation of the invisibility or nonradiating problem in terms of FIE representation, in this sense, is different with respect to the HFS formulation. The first technique is based on computing the scattered fields as a continuous sum of multiple spatial sources with their continuous frequency behavior, whereas the second one was an infinite sum of multiple harmonic waves with their continuous spatial behavior. However, despite the fact that FIE acts in the spatial domain and HFS operates in the frequency domain, both representations identify the origin of invisibility in a scattering interference due to exotic objects’ locations, shape and constitutive parameters. If the observations points are placed in the near-field, the Green’s function of a homogeneous background is known to represent outgoing scattered waves of a point-like source. Mathematically, this means that Eq. (10.37) in the near-field becomes ij Jeq (r)H0(2) (kb |r 1 ´ r|)dV = 0 (10.38) V 1

where G(r ,r) is now the Hankel function of zero order (a point is, by definition, perfectly in quasi-static regime and it always scatters like the lowest order harmonic wave) and second kind (outgoing waves), as employed in the ISP [4, 6]. On the other hand, if the scattering process is seen in the far-field, the elementary response seen at the location of each observation point passes from the original outgoing cylindrical wave in Eq. (10.38) to an outgoing plane wave, giving ij 1 Jeq (r)e´j kb r dV = 0 (10.39) V

which is nothing more than looking for the zeros of the Fourier transform for the equivalent sources. Quite interestingly, this result has been found in 1973 by Devaney and Wolf as the Theorem III for non-radiating sources [11]. Despite the fact that the FIE and HFS representations are different, the results obtained in the cloaking literature are expected to be included as well within this integral equation formalism with some possible generalizations and practical extensions. In the next subsections, we will find which kind of solutions satisfies the general Eq. (10.37), which turns out to describe the ultimate equation we ask in Figure 10.1 for the scattering operator: in the observation domain we have the scattered fields – and we are looking for their zeros (invisibility, non-radiating or cloaking problem) – whereas in the source domain we have located the equivalent sources. The general Eq. (10.37) turns out to refer to the equation describing the kernel of the scattering operator.

10.5.1

The Strong Solution: Impedance Matching and Transformation Optics We are now close to exploring the solutions that we can obtain from Eq. (10.37), which represents a general formulation of Theorem III as derived by Devaney and Wolf in the far-field [11]. One solution that appears to be trivial is, by simple inspection, Ñ Ý Ñ Ý pp ¨ J eq (r) = 0 and pp ¨ M eq (r) = 0 (10.40)

10.5 Field Integral Equation Representation

341

Ñ Ý Ñ Ý where the (equivalent) electric source J eq (r) and magnetic source M eq (r) are locally p to be exactly zero in their domain considered, along their own unit polarization vector p, of localization (if the scattering process, in a general vector formalism, is formulated accordingly with the presence of both electromagnetic sources). Since there are no sources, no scattering will be generated: the question is which kind of invisibility devices satisfying Eq. (10.40) can be designed that do not correspond to a trivial solution of the problem, like the empty background. Since any scattering event can be represented at the external domain of a boundary, according to the Surface Equivalence Principle as in Eq. (10.36), the conditions in Eq. (10.40) can be rewritten as follows [65]: “ ‰ " Ñ Ý pˆ ¨ J eq (r) = nˆ ˆ “H ( + ) ´ H ( ´ )‰ = 0 (10.41) Ñ Ý pˆ ¨ M eq (r) = nˆ ˆ E( ´ ) ´ E( + ) = 0 The final result is a compact condition on the ratio between magnetic and electric fields, defined in terms of admittance contrast χY ”

H ( + ) H ( ´ ) ´ =0 E( + ) E( ´ )

(10.42)

This kind of solution for the scattering problem as formulated by a FIE and defined in Eq. (10.40) in terms of equivalent sources or in Eq. (10.42) in terms of admittance (or impedance) functions has been referred as the strong solution [30]. The origin of the name stems from considering the zeros as locally enforced at any given boundary , thus forming an equivalent admittance (or impedance) matching that ensures transmission without reflection or any other perturbation. With this simple principle, it is possible, for example, to cloak small objects within a mesh of matched elements using classical transmission-line networks’ methodology [80–83]. Examples of non-scattering devices that are able to reproduce, with their own limitations, a strong solution behavior are (1) the non-scattering waveguide for a fixed direction of incidence, (2) mantle cloaking for a single harmonic wave and (3) transformation optics for a single frequency point, as detailed in the following. The first example is the non-scattering waveguide, a device presented in 2015 [84] and a good example of metasurface platform with a straightforward realization and double functionality: its main characteristics can be compactly analyzed in terms of strong solution or zero admittance/impedance contrast, as defined in Eq. (10.42). As reported in Figure 10.9 (left), such a device is a grounded dielectric slab loaded with rectangular patches in the central (or core) region and with square patches in the side (or cladding) zones. Analogously to an optical fiber, when an incident wave is impinging with its k-wavevector in the yˆ direction, the metasurface behaves as a waveguide, confining the fields in the core region where the rectangular patches are placed. If the mode confinement is due to the impedance contrast in the yˆ direction, as named in [84], the anisotropy on the other side is chosen in order to achieve a zero impedance contrast for the other orthogonal direction. If an incident wave is impinging with its k-wavevector in the xˆ direction, the response of the structure, as detailed in Figure 10.9 (left), depends on the impedance contrast dictated by the amount of metallization transverse to the propagation itself. Since the shorter side of the rectangular patches is the same as the

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Metamaterial and Metasurface Cloaking: Principles and Applications

Figure 10.9 Examples of strong solution. (left) Non-scattering waveguide. Reprinted, with permission, from “Invisibility and cloaking structures as weak or strong solutions of Devaney-Wolf theorem”, The Optical Society. (right) Exact-harmonic mantle cloaking. Reprinted, with permission, from: Giuseppe Labate et al., Surface-admittance equivalence principle for nonradiating and cloaking problems, 10.1103/PhysRevA.95.063841 and 2018 Copyright 2018 by the American Physical Society.

side of the squared ones, the impedance contrast is zero, thus no equivalent sources are formed and no scattering is generated: this occurs only at this particular direction. In all the other cases where the wavevector is slightly not parallel to the xˆ direction, fields are transmitted but with partial non-zero reflections. For more details and experimental validation, the reader can look at [84]. The second example of strong solution is mantle cloaking when referred to a specific single harmonic wave. In this case, Eq. (10.42) can be written as a proper difference between normalized admittances at the external boundary : χy ” Y˜b ( + ) ´ Y˜ns ( ´ ) = 0

(10.43)

where Y˜b is the normalized background admittance as seen from above the  domain ( + ) and Y˜ns is the non-scattering device admittance as seen from below the  domain ( ´ ). In the case of cloaking systems, where the non-radiating device is made up of a fixed object (dielectric load admittance Y˜d , for example) and a coating layer (surface cloak admittance Y˜s ), the overall condition reads Y˜s ( ´ ) = Y˜b ( + ) ´ Y˜d ( ´ )

(10.44)

from which a closed-form expression can be computed for the surface admittance cloak as in the sketch reported in Figure 10.9 (right) [65]. It realizes the surface cloak, when the contrast between object and background is not zero in terms of lumped admittances, that is needed in order to ensure such admittance matching or contrast to be zero. For 2D objects as dielectric cylinders, the generic expression for a normalized admittance is [85, 86] 1

? J (ka) Y˜ = ´j εr n Jn (ka)

(10.45)

10.5 Field Integral Equation Representation

343

1

whereJn (¨) and Jn (¨) are the Bessel function and its derivative with respect to the ? argument, which is the product between the specific wavenumber k = εr kb and a, the cylinder’s radius. As reported in [65], the final closed-form expression for the surface admittance cloak is « 1 ff 1 ? Jn (ka) (k a) J b n ´ εr (10.46) Y˜s = ´j Jn (kb a) Jn (ka) as the result of the difference, for the electromagnetic wave with n-harmonic index, between background and uncloaked object. This result is directly provided in terms of the n-harmonic wave to be cancelled, and also as a function of the frequency dispersion, due to the intrinsic dependence from the frequency regime λ¯ ” kb a, defined in [65] as the product between the background wavevector and the cylinder’s radius. In the quasi-static regime, when λ¯ ! 1, the formula gives the result already found in the literature for mantle cloaking [52] in the form « j „ ? ff λ¯ ? λ¯ εr (εr ´ 1) ¯ ˜ ´ εr = ´j λ (10.47) Ys = ´j 2 2 2 which is the same result as obtained in Eq. (10.21) without normalization. The third example is transformation optics, one of the earliest and most popular cloaking techniques as first proposed in 2006 [31–39]. The main novelty of this methodology resides in the insertion of a volumetric inhomogeneous cloak able to guide the fields all around a predefined region without any perturbation: in the ray tracing technique, this means stretching the directions of waves according to a conformal path, and since the wavenumbers locally depend on the constitutive parameters, this directly translates into anisotropic coatings with high and low values of both permittivity and permeability. In this chapter, we will not focus on all the kinds of transformations that can be pursued; for further details, the reader can look at [31–39] and [87]. However, as also highlighted in [34], there is a basic property satisfied by each transformation optics cloak in terms of admittance contrast. If, in the near-field, the admittances in Eq. (10.46) depend on the ratio of magnetic (derivative of Bessel functions) and electric fields (Bessel functions), the shape of such cylindrical waves is lost in the far-field and the admittance (or impedance) itself is a simple ratio that only depends on the constitutive parameters as c εr ˜ Y = (10.48) μr where εr and μr are the relative dielectric permittivity and relative magnetic permeability of the object under investigation. Any transformation (linear, quadratic, etc.) performed by the coordinate transformation method keeps as invariant the ratio of the permittivity and permeability, from point to point, thus preserving the original admittance of the complete background scenario [87]. Since also in this case, with artificial inhomogeneous coatings supporting the non-scattering process, the device contrast is locally kept to zero for the far-field evolution, the final result is somewhat expected to be without perturbations, as shown in Figure 10.10, no matter what is inside the region to be hidden.

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Metamaterial and Metasurface Cloaking: Principles and Applications

Figure 10.10 Transformation optics: simulation of the electric field for cloaking effect. Reprinted,

with permission, from: Steven A. Cummer et al., Full-wave simulations of electromagnetic cloaking structures, 10.1103/PhysRevE.74.036621 and 2018 Copyright 2018 by the American Physical Society.

The importance of the impedance concept, for matching purpose or cloaking effects – but also for problems of shielding and power absorption – has been highlighted in 1938 [88] and, as shown in [65], it can be combined with the surface equivalence principle in the form of Schelkunoff [89] in order to achieve compact design formulas in the general framework of the strong solution, the first compact class, within the kernel of the scattering operator, where invisible, non-radiating and cloaking structures can be categorized.

10.5.2

The Weak Solution: Kirchhoff’s Current Law and General Scattering Cancellation The general problem as formulated in Eq. (10.37) strongly depends on the choice of the Green’s function of interest: if the non-scattering process is observed in the farfield, the Green’s function is a complex exponential function, representing the Fourier transform of the (equivalent or physical) sources as derived by Devaney and Wolf [11]. From the strong solution, it is possible to derive another solution, multiplying both terms 1 in Eq. (10.40) by an unknown function g(r ,r) – that depends also on the observation points – and integrating both sides in the domain V supporting the non-scattering process. The solution appears to be as ij Ñ Ý J eq (r)g(r 1,r)dV = 0 pˆ ¨ (10.49) V

which has been formed by applying the mathematical definition of weak solution: for this reason, this is also the name for this second class of solutions for non-scattering and cloaking devices. By comparing Eq. (10.49) with Eq. (10.37), the analogy between the 1 unknown function g(r ,r) and the Green’s function itself is evident. In the simplified 1 case, when g(r ,r) is constant in the considered volume V , this reflects a quasi-static condition in the Green’s function, being so compact in the domain V that Eq. (10.49) reduces to ij Ñ Ý J eq (r)dV = 0 (10.50) pˆ ¨ V

10.5 Field Integral Equation Representation

345

Figure 10.11 Kirchhoff’s Current Law: from DC for electrical circuits to quasi-static regime for

invisibility devices. Reprinted from Labate, G., Matekovits, L.: ‘Kirchhoff’s current law as local cloaking condition: theory and applications’, Electronics Letters, 2016, 52, (21), pp. 1749–1751.

According to the equivalence principles for an area in Eq. (10.35) or for a contour  in Eq. (10.36), the operation remains a double integral in the first case (2D surface) or a single integral in the second case (1D contour). Since each equivalent source is integrated in its domain of definition, a simple dimensional analysis of Eq. (10.50) indicates that the sum of all the currents (left-hand side) has to be set to zero amperes: this may be interpreted as nothing more than a generalization of the Kirchhoff’s current law (KCL) in the static regime [90]. For this reason, this result is expected to be valid in the quasi-static regime for invisibility, non-radiating and cloaking devices. As reported in Figure 10.11, it can be argued that, while a node in a circuit (left side) ensures exact zero ampere current value in its infinitesimal domain of definition, a general system – to be invisible in the quasi-static regime – has to ensure the average value of all its current flowing to be zero amperes as well. Without considering this kind of application, but only reasoning about current conservation in the static regime, Gustav R. Kirchhoff reported for the first time this result in 1845, winning a university prize for this and other related results when he was a student of 21 years old [91]. This result is now of interest not only for the formulation of a general invisible system in quasi-static regime but also for the design of a cloaking device made up of a fixed desired object and a proper coating layer. Let us consider a homogeneous dielectric cylinder, having relative permittivity εr1 and arbitrary cross section area 1 , to be covered by a homogeneous dielectric with arbitrary area cross section area 2 and relative permittivity εr2 . By substituting the volume equivalence principle in their proper domain of definitions, the condition by applying the KCL reads as     (10.51) j ωεb εr1 ´ 1 E˜ t1 1 + j ωεb εr2 ´ 1 E˜ t2 2 = 0 where E˜ t1 and E˜ t2 are the average total fields in the region 1 and 2 , respectively. Please notice that the choice of a cloak with the same material characteristics as the object to be hidden (volumetric-volumetric system) leads us to remove the ω-dependence, and since a quasi-static regime is assumed to hold, the total fields within the object and cloak domains can be considered to be the same constant value. The final condition is obtained as [90] d εr2 ´ 1 a (10.52) = χ1 1 + χ2 2 = 0 Ñ b εr2 ´ εr1

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Metamaterial and Metasurface Cloaking: Principles and Applications

where the definition of contrast is exploited, or here in this case for the permittivity values, electric susceptibility χ ” (εr ´1). Please notice that the first side of Eq. (10.52) is valid for dielectric scatterers of arbitrary shape, since only the areas are important in the quasi-static regime, whereas the second side of Eq. (10.52) is particularized to the case of dielectric objects with circular cross section, having 1 = π a 2 and 2 2 1 = π(b ´ a ). By comparing Eq. (10.19) with Eq. (10.52), the KCL for volumetricvolumetric cloaking systems confirms the result as obtained for plasmonic cloaking [40]. If we consider now the same dielectric object (to be hidden) covered by an arbitrary contour 2 , able to sustain a magnetic field jump as described by Eq. (10.36), we can mix the volume and the surface equivalence principle in the KCL, obtaining the condition [90]   j ωεb εr1 ´ 1 E˜ t1 1 zˆ + nˆ ˆ ϕˆ H˜ t2 2 = 0 (10.53) where now H˜ t2 is the average residual magnetic field provided at the surface contour. As an additional proof, in the simplified case of a circular contour 2 = 2π b, the normal unit vector nˆ is coincident with the radial unit vector rˆ and the vector product with ϕˆ gives the current element in the cloak contour directed along the polarization axis of the incident TM wave (ˆz = rˆ ˆ ϕ). ˆ The final result gives Zs (ω) = +j

ωεb

2π b  εr1 ´ 1



(10.54) 1

since the ratio between average electric and magnetic fields is – by definition – the surface impedance Zs . Please notice that now the ω-dependence is explicitly left in Eq. (10.54), and it is for this reason that the surface impedance shows a dispersive behavior also in the quasi-static limit. More details about the frequency dispersion aspect will be given in Section 10.6.2, where we will focus on the bandwidth issue. If the object to be cloaked is not of arbitrary shape but it has a circular external boundary as well, i.e. 2 1 = π a , the result is 2b   (10.55) Zs (ω) = +j ωεb εr1 ´ 1 a 2 which is the same result as in Eq. (10.55) with γ = a{b, confirming that, in this volumetric-surface cloaking system, the KCL becomes mantle cloaking for the scattering cancellation of the lowest order mode in 2D scenarios [52]. Since we are interested in solving the weak solution in its complete form without any approximation in the involved frequency regime, we remind ourselves that this means solving a restricted inverse scattering problem, where we already know that the scattered field values should be zero. Such strategy named as inverse design is quite common in the literature and it has been first introduced for cloaking in the framework of topology and other optimization techniques [93–99]. For metallic cylinders, this has been successfully achieved by obtaining volumetric cloaks beyond the quasi-static regime with all-positive dielectric materials [96]. This experimental confirmation has been used as the main hypothetical guideline for cloaking dielectric cylinders – instead of metallic objects – beyond the quasi-static regime by all-positive dielectric cloaks. If, for a sensing process, the non-uniqueness of

10.5 Field Integral Equation Representation

347

Figure 10.12 Scattering cancellation: in quasi-static regime, homogeneous plasmonic cloaks are

locally enforced, whereas beyond the sub-wavelength limit, it is possible to enforce inhomogeneous all-positive dielectric cloaks for the same global scattering cancellation. Reprinted, with permission, under Creative Commons license (http://creativecommons.org/ licenses/by/4.0/) [100].

the inverse problem is an obstacle that prevents finding a unique solution for imaging existing scatterers, for the synthesis process the non-uniqueness is of great value due to the fact that we can look for specific solutions in the kernel domain that can be feasibly realized with a 3D printer, as for example pursued in [100]. The reasons behind this assumption are explained in Figure 10.12. In the subwavelength limit (or quasi-static regime), the object is very small with respect to the incoming wavelength, implying that the entire weak solution is affected by the local material properties: in order to achieve the cancellation effect for scattered fields, we need to design block pairs of contrast values with negative (solid) and positive (cross hatch) signs. However, if we are ready to renounce to this simplified case, thus taking into account also the total field interactions within the overall volume itself, we can exploit the values (magnitude and phase) of the internal fields to achieve the same cancellation effect but with all-positive dielectrics. The problem can be recast as the minimization process [100] › › › M › ÿ › J ν ´ χ Eiν ´ χ FV rJ ν s ›2 › F rJ ν s ›2 › › › › (χ,J ) = › › + › Eν › Eν ν=1

i

(10.56)

i

where the first term is related to the internal scattering operator FV r¨s, which maps the internal total fields as the sum of the incident fields plus the scattered waves (properly normalized). The second term is related to the external scattering operator F r¨s, that addresses the zeros – when achievable – of its argument, providing the non-radiating topology of the currents J ν as a function of a specific direction of arrival ν [6]. Due to the fact that the ISP is related to an imaging process, where it is of interest to detect or reveal inhomogeneous scatterers in a given scenario, the direction of arrival highly affects the way we see this inhomogeneity in space, and for this reason, the direction of arrival is usually referred as the view [6]. Please notice that the second term in Eq. (10.22) is properly normalized in order to minimize the radar cross section [101]

Metamaterial and Metasurface Cloaking: Principles and Applications

–0.005

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Figure 10.13 General scattering cancellation beyond quasi-static regime: (a) uncloaked case, (b)

plasmonic cloaking and (c–h) related all-dielectric configurations as cloaking system made up of all-dielectric materials. Reprinted, with permission, under Creative Commons license (http:// creativecommons.org/licenses/by/4.0/) [100].

or echo width [70]. As a result of the minimization procedure all-positive dielectric cloaks can be synthesized for a collection of scatterers or a single object [100]. As reported in Figure 10.13, the synthesized cloaks according to the cost function in Eq. (10.56) are shown with their proper permittivity quantization levels: the plasmonic cloak (b) has a negative homogeneous permittivity value whereas other inhomogeneous coatings, designed beyond the quasi-static assumption, are able to support a scattering cancellation effect, with respect to the reference uncloaked case (a), with all-positive dielectric materials (c)–(h). In particular, the cloaks (c), (e) and (g) have a radius of 1λ, whereas the coatings (d), (f) and (g) have a radius of 2λ. Despite the change in size, the couple (c)–(d) has been designed to be cloaked for two antiparallel incident directions, coming at 0 and at πangles with respect to the x-axis; the couple (e)–(f) has been designed to be cloaked for four incident directions, and the couple (g)–(h) has been designed to be cloaked for eight incident directions. How many lines of symmetry can be enforced in a cloaking device and how this directly reflects its cloaking effectiveness from different directions will be discussed in Section 10.6.1. Related to this non-radiating mechanism in volumetric structures such as collections of dielectric objects, it is a specific electromagnetic field distribution dubbed anapole [102–108]: literally it means ‘without pole’ since the poles of the local reflection coefficient – as also observed in the case of parity-time symmetry cloaking [66] – are related to a radiation or emission mechanism. As stressed by all three main cloaking approaches related to the scattering cancellation family, it is possible to combine also different electromagnetic responses in order to obtain a destructive interference in the far-field zone, thus providing nontrivial non-radiating distributions [105, 107]. In the general framework described in Figure 10.1, within the kernel of the scattering operator, the dynamic anapole can be regarded as the basic element which an arbitrary non-radiating source can be composed of or decomposed into. With respect to the pre-

10.6 Bounds on Cloaking: Causality, Passivity, Time Invariance, Linearity

349

viously discussed techniques, such as plasmonic cloaking, which combines the electric dipole in the object (P1 ) with the additional dipolar response of the cloak (P2 ) such that a global cancellation is achieved (P1 + P2 = 0), as highlighted in Figure 10.3, dynamic anapoles are characterized by the same global scattering cancellation purpose, but combined with T , an additional toroidal response (P1 + T = 0) [105]. Preliminary verifications of such non-radiating process came in recent works [102–107] where, for example, a metamolecule device was fabricated with metallic split ring resonators [108], whereas also structures made up of all-dielectric materials, with a collection (or cluster) of scatterers, have been investigated when excited by an electric dipole antenna at the center [104]. As also highlighted by the KCL interpretation of plasmonic cloaking and mantle cloaking in terms of resonant circuits [90], the non-radiating process in the HFS is related – as stressed above – to the zeros of the scattering coefficients cn but, on the other hand, also simultaneously related to a dramatic increase of the values in the internal an expansion coefficients: in this way, the global response is almost zero or poorly scattering because the fields get confined within the device itself. This physical mechanism of concentrating the field energy is a resonance process inside the object and cloak particle, while achieving scattering reduction at the outside [86]. As highlighted in the KCL interpretation [90], the object and the cloak regions are exactly approximated as a LC circuit in the quasi-static regime, able to achieve very high Q-factor [108] or – from a cloaking perspective – deep scattering cancellation levels in the resonance condition. This physical process of strongly concentrated fields in a small volume is also behind the non-radiating operation of the anapole mode, which behaves like a resonator with high Q-factor [108].

10.6

Bounds on Cloaking: Causality, Passivity, Time Invariance, Linearity Non-radiating theories and cloaking techniques cannot be treated without considering the electromagnetic response provided by each material component. One of the fundamental limitations is related to causality, which limits the temporal window over which the materials involved can respond to an excitation. When the background is vacuum or free space, causality and Kramers-Kronig relations [109] ensure that for a passive scatterer ” ı + ż8 Im ε(ω1 ) 2 1 dω (10.57) χε (ω = 0) ” Re rε(ω = 0)s ´ 1 = 1 π ω 0

the static value χε (ω” = 0) is ı always greater than zero, due to the fact that a passive scat1

terer has always Im ε(ω ) ě 0 at any frequency. If the scatterers within a background region are passive and linear, the value of the spectrum of the scattering cross section (SCS) is constant, and its value + ż8 0

“ ‰ SCS(λ)dλ = C χε (ω = 0),χμ (ω = 0)

(10.58)

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Metamaterial and Metasurface Cloaking: Principles and Applications

depends on the constant value C[i], which can be computed as a function of two variables: the static values of the electric and magnetic susceptibilities (or alternatively, the electric and magnetic polarizabilities). If the area formed by integrating the scattering cross section (SCS) along the frequency axis depends only on the static values of the constitutive parameters, it is possible to study what happens in static conditions when an object is wrapped with a certain cloak (plasmonic, mantle or transformation optics coating layer) and infer how much and in which conditions the introduction of additional loading elements affects the integral in Eq. (10.58). It can be demonstrated [110, 111] that adding materials with relative constitutive parameters larger than the background always leads to an increase in the overall static values of the polarizabilities. It has been reported that, with respect to a reference uncloaked case which shows a constant value Cun , the mantle (MC), plasmonic (PC) and transformation optics (TO) cloaking techniques achieve SCS integral spectra performances as [112] CMC = 1.75Cun

(10.59)

CP C = 3.15Cun

(10.60)

CT O = 17.5Cun

(10.61)

This approach turns out to be useful to categorize and evaluate different cloaking approaches in terms of residual scattering outside the desired frequency bandwidth. The limits as computed by Eq. (10.58) share the same minimum for all the cloaking approaches at the central frequency of the design [112]. With reference to the uncloaked case, it is observed how the scattering is increased outside the bandwidth operation, degrading the scattering cancellation effect and instead enhancing the out-of-band visibility of the object with the additional cover. The material properties affect cloaking also in terms of absorption. In the above discussions excluding parity-time symmetry cloaking where they play a crucial role, losses have not been considered in realistic cloaking design. If we admit that the involved materials have loss, however, we cannot achieve perfect zeros for the scattered fields [109]. As it can be inferred by looking at Figure 10.8 (central panel), a system with material losses creates a shadow in the forward direction when illuminated from one side, thus making the device itself highly detectable. This is consistent with the optical or forward scattering theorem [25, 101], which states that the extinction cross section – the sum of absorption and scattering cross section as defined in the literature [101] – is directly proportional to the scattering towards the forward scattering direction. If the absorption is non-zero, the scattering will follow: this is the reason why parity-time cloaking systems need gain in order to compensate such shadow effect [66]. If causality and passivity – with intrinsic small losses – limit the performances of cloaking device, other two main assumptions – tacitly assumed – should be mentioned: time invariance and linearity. For all these reasons, the introduction of novel devices for enlarging the cloaking bandwidth has been considered with active inclusions (breaking the passivity limit), with non-linear components (breaking the linearity limit) and with time-dependent elements (breaking the time-invariance limit). In the following sections, we will discuss the issues that these novel components can solve and what remains

10.6 Bounds on Cloaking: Causality, Passivity, Time Invariance, Linearity

351

to be performed in order to realize non-radiating and cloaking devices in their full potentialities.

10.6.1

Directionality Issue One of the main challenges associated with any invisibility cloak is related to the direction of arrival for the incident fields: in some applications, it can be fixed and unique whereas in others, it can be unknown a priori. In the latter case, the specification required to the overall cloaking system is to scatter zero or quasi-zero values regardless of the incoming direction of arrival. This is not a problem in the quasi-static regime, since we have shown how the weak solution that reduces to the KCL does not feel the influence of the total fields in its domain of definition, because field distributions are constant over the object and cloak regions: no matter where the incidence comes from, a subwavelength object is intrinsically ominidrectional [100]. Problems arise when we are beyond the quasi-static regime since the proper shape (corners, asymmetries, etc.) and local constitutive parameters affect the scattered fields as a function of the incidence direction. Parity-time symmetry cloaks are inherently asymmetric [66], and therefore they are sensitive to this issue. As reported in Figure 10.15a, the cloaking effect is achieved when Ñ Ý the incident fields have the wavevector parallel to the k b used in the synthesis process. If a large deviation from the optimal direction is expected or – in the worst case – if the cloak is excited from the active side as in Figure 10.15b, the scattered field is no more zero but it is increased since the mechanism of absorption-gain is completely broken and the fields are directly reflected without being absorbed. One solution to this issue is to consider locally tunable metasurfaces that are able to change – or to program – their loss/gain nature as a function of the incident direction of arrival. Despite the problem of passivity, it is possible to achieve cloaking for a finite number of directions of arrival as reported in [100]. For sure, a relaxed specification about cloaking from just a single direction of incidence alleviates the design problem for what concerns the cloak region and the more the direction we ask simultaneously a non-radiating or cloaking effect, the more complicated the coating structure would be in order to accomplish for all the different incident directions. This is shown in Figures 10.15c and 10.15d, where the minima of the SCS [101] indicates the scattering levels when the angular direction of the incoming wave changes. The number of directions of arrival (or views ν) reported here on the x-axis clearly shows the deviation from the optimal incident direction to the worst case where the SCS increases, when the cloaked object scatters much less (minima) or much more (maxima) than the bare case. The cloaking effect clearly changes as a function of the angular direction of incidence, and this process is highly sensitive when the cloaking specifications include a higher number of directions of arrival (from four to eight) for which we ask a non-radiating or transparency effect. When the overall size of the external cloak passes from 1λ to 2λ, the electromagnetic response remains quite sensitive to the angular deviation when the dimensions increase: however, a slight independence for the case with eight incident directions and radius b = 1λ can be observed. It is worthwhile mentioning that, as

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Figure 10.14 (a, b) Directionality issue: parity-time cloaking with unidirectional constraint. (c, d)

All-positive dielectric passive cloaking with N-finite directional specification. Adapted with permission [66] and reprinted, with permission, under Creative Commons license (http:// creativecommons.org/licenses/by/4.0/) [100].

Figure 10.15 Active cloaking with non-Foster metasurfaces, implemented with Negative

Impedance Converters (NIC). Reprinted, with permission, from: Pai-Yen Chen et al., Broadening the Cloaking Bandwidth with Non-Foster Metasurfaces, 10.1103/PhysRevLett.111.233001 and 2018 Copyright 2018 by the American Physical Society. A black and white version of this figure will appear in some formats. For the color version, please refer to the plate section.

10.6 Bounds on Cloaking: Causality, Passivity, Time Invariance, Linearity

353

represented in Figure 10.1, the observation region  consists of infinite points where the scattered fields can be evaluated. However, a well-known result in the literature of ISP states that the number of degrees of freedom for the scattered field is not infinite: it has a finite spatial bandwidth [113], thus a finite number of independent expansion coefficients, or degrees of freedom [114], where the scattered fields are represented by basis functions. If the scattering system can be encircled in a circumference with maximum radius equal to b, the number of degrees of freedom is finite, and it is approximately equal to M « 2kb b. It indicates the minimum nonredundant number of points that we can use to sample the scattered fields in the observation region  by considering only a circumference, with a certain radius R located in the far-field zone [115], with these Mangularly equispaced points along it [113, 114]. Interestingly, this rule of thumb has also been reported when referring to the relevant number of harmonic series in Mie theory [28], and it is related to the same reasoning: the maximum number Mof degrees of freedom for the scattered fields is useful to estimate the number of harmonic waves (one or more) to suppress, thus choosing to apply one of the approaches from the scattering cancellation family [40, 51, 66]. For more details, the reader is referred to [100] and other works on topology and related optimization schemes [93–99]: it can be demonstrated that if N is the number of directions of arrival we ask the cloaking device to be hidden from, we achieve N planes of symmetry when the materials are arranged in our final cloaked structure [94, 95] as it is clearly visible, for example, from the coating layers in Figures 10.13c–10.13h.

10.6.2

Bandwidth Issue An important aspect of all the cloaking techniques discussed above is their achievable frequency bandwidth over which they can be effective. Plasmonic cloaking requires – in the quasi-static regime – the cloak region to provide less-than-unity permittivity with respect to a defined background. It is interesting to notice that, when the background is not free space, relative “plasmonic” values can be achieved by natural dielectric materials that induce a negative polarization when embedded in a medium which is denser: as result, the bandwidth appears to be large [92], because one does not need to comply with the bandwidth limitations of plasmonic materials, which are necessarily dispersive. On the contrary, in the case of a free-space background, the natural dispersion of plasmonic materials will fundamentally limit the bandwidth of operation. In the mantle cloaking approach, the dependence of the surface impedance/ admittance on frequency is explicit, as for example in the quasi-static regime. Interestingly, Eq. (10.21) violates the dispersion as dictated by Foster’s theorem for passive materials [116]. This theorem on reactance networks has been formulated by Robert Foster in 1924 [116], and it states that, no matter how complicated a purely reactive network is, it is always possible to demonstrate that any equivalent reactance X(ω) has to monotonically increase as a function of frequency dβ(ω) dX(ω) ą 0 or ą0 dω dω

(10.62)

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Metamaterial and Metasurface Cloaking: Principles and Applications

Figure 10.16 (left) Physical bounds on cloaking: trade-off between scattering reduction and

fractional bandwidth, according to Bode-Fano matching theory. (right) Frequency spectra in terms of cloaking techniques used. Reprinted, with permission, from “Invisibility and cloaking structures as weak or strong solutions of Devaney-Wolf theorem”, The Optical Society. Francesco Monticone and Andrea Alù, “Invisibility exposed: physical bounds on passive cloaking,” Optica 3, 718–724 (2016). A black and white version of this figure will appear in some formats. For the color version, please refer to the plate section.

This poses limits on the bandwidth of operation if the loading surface impedance/ admittance is realized with common inductive or capacitive elements: in order to broaden the cloaking bandwidth, non-Foster elements should be inserted [117]. As shown in Figure 10.16, a negative impedance converter (NIC) can be employed between the metallic patches, in order to change the capacitance load into a negative one: in this manner, the dispersive behavior of the surface impedance/admittance can be tuned in order to follow the proper cloaking design over broad bandwidths. For what concerns implants in a generic cloaking device, plasmonic structures can be emulated without the use of negative constitutive materials but by realizing composite devices possessing metal-dielectric boundaries. This is equivalent to fabricating plasmonics without negative dielectrics, as stated in [118], and the frequency dispersion still complies with similar restrictions in terms of bandwidth performance. Again, to go around the passivity constraints on bandwidth, one may employ metamaterials with active inclusions. Recent advances on invisibility for all passive scatterers have been highlighted in [119], in order to derive a general theory to investigate the origin and define the limitations between scattering cancellation and attainable frequency bandwidth. In this framework, and in order to obtain preliminary results for simplified cases, it is instructive to analyze the case of a planar device, a dielectric slab of thickness d having relative permittivity εr . At the central frequency fc , the product between the fractional bandwidth B = f {fc and the inverse of the reflection coefficient is a constant value, and according to Bode–Fano theory on matching circuits [120, 121], for a planar object of thickness d and relative permittivity εr , this constant is bound to B ln

  c0 1 ď  d (εr ´ 1) fc

(10.63)

10.7 Conclusions

355

whereas for a metallic sphere, the relation between the SCS and the fractional band is [119] SCS ´ 1 2π ď 2e B kb a (10.64) π a2 Both Eqs. (10.63) and (10.64) demonstrate how there is a trade-off between achievable frequency bandwidth and scattering levels reduction as reported in Figure 10.16. Physical bounds are computed in Figure 10.16a, in terms of the trade-off between the maximum scattering reduction and the fractional bandwidth where this effect can be supported. As it can be noticed, the more the reduction the narrower the bandwidth for all passive scatterers, with an ultimate red curve representing the main contour that can be approached before entering in the forbidden zone: in this region prohibited to allpassive cloaking techniques, performances could be achieved only by devices that break the initial passivity hypothesis, as active materials or non-linear loading elements in the coating layer. In Figure 10.16b, it is shown an example of a spherical region, cloaked according to plasmonic or transformation optics approach, with the reference uncloaked scenario and the physical bounds, as dictated by passivity. If plasmonic cloaking is increasing the scattering levels both in the low- and high-frequency spectrums, while maintaining an absolute minimum of –3dB, the transformation optics approach keeps its bandwidth very narrow, diverging rapidly in the out-of-band zones.

10.7

Conclusions In this chapter, we have discussed in a compact framework the various recent approaches to realize invisible devices, non-radiating sources and cloaking structures based on metamaterials and metasurfaces. No matter how complicated a general non-scattering system is, its final goal is to stay as close as possible to design specifications that ask the scattered fields to be zero outside its domain of localization: as a consequence, invisible particles, non-radiating sources and cloaking devices have to stay, as close as possible, in the kernel region of the scattering operator. Since the first experimental work of Wood [2], the advances on metamaterials and metasurfaces have made the dream of invisibility into a solid design project, with its advantages and disadvantages according to the chosen approach. The scattering problem has been detailed and described according to two main representations: one based on the harmonic field series, related to the early theoretical work of Mie [78], and a second one based on the field integral equation, related to an extensive literature on inverse scattering problems [3–10]. With the Mie theory formulation, several approaches have been reviewed: plasmonic cloaking (2005) [40], mantle cloaking (2009) [51] and parity-time symmetry cloaking (2015) [66]. All of them have been highlighted as a function of their compensation effects on scattered fields with their own cloaking layer: a plasmonic or metamaterialbased coat for plasmonic cloaking [40–50], a thin impedance surface loaded with capacitive or inductive elements for mantle cloaking [51–65] and a thin metasurface loaded with lossy/gain elements for parity-time cloaking and related approaches [66–69].

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With the field integral equation formulation, a compact equation for the zeros of the scattering operator has been identified: consistently with the Devaney–Wolf theorem [11], a necessary and sufficient non-radiating condition has been analyzed and two main classes of solutions have been pinpointed [30]. The first one has been named the strong solution, since it locally enforces the electromagnetic sources to be zero, thus turning off the scattered fields in the observation region, both in the near- and in the far-field. This is the case for invisible theories and devices such as the non-scattering waveguide (limited to a fixed direction of incidence) [84], mantle cloaking (limited to a single harmonic wave) [65] and transformation optics (limited to a single frequency point) [87]. The second class, which has been identified in the kernel of the scattering operator, has been named the weak solution since it enforces globally (with their mean value) the electromagnetic sources to be zero, thus limiting the scattered fields to be zero in the observation region only for some frequency values. In particular, in the subwavelength limit, this kind of solution has been explicitly derived to be nothing but plasmonic cloaking [40] and mantle cloaking [51] for the lowest order harmonic mode cancellation. In addition, the physical dimension of the formula describing the weak solution indicates how the sum of currents in amperes (current density multiplied by the corresponding area or surface) is closely related to Kirchhoff’s current law, generalized from DC [91] to the quasi-static regime [90]. Beyond the quasi-static regime, the zeros of the scattering operator have been solved without any approximation in order to perform scattering cancellation with an additional constraint: asking for a volumetric cloak without the need of plasmonic or metamaterial materials [100]. These efforts have been justified by a distributed system and complicated architecture schemes of scatterers, able to compensate the scattered field of the bare object without the need of lumped elements with opposite sign in the susceptibility – or permittivity contrast – as for plasmonic cloaking [40]. In the last section, general considerations about causality, passivity, linearity and time invariance have been marked in order to explore the limitations in terms of sensitivity when the direction of arrival or the frequency of the incident wave change. It has also been highlighted that physical bounds exist for all passive cloaking systems [119] and that these constraints, while putting several limits in terms of performances, indicate novel scenarios in order to go beyond these intrinsic limitations dictated by causal, linear, passive and time-invariant scatterers. Recent advances have shown how mantle cloaking can be enhanced by breaking the Foster theorem for passive metasurfaces [116], thus enlarging the frequency bandwidth with active elements [117]. As also pointed out by parity-time cloaking with compensated losses [66], active loading may be an important path towards a solid design of an invisible system with enlarged bandwidth. Other original ideas could involve the use of non-linear or time-dependent materials, in order to break the limitations of current cloaking techniques and to provide new exciting solutions within the kernel of the scattering operator.

Acknowledgements This work has been partially supported by the National Science Foundation and the Air Force Office of Scientific Research.

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[86] R. G. Quarfoth, D. F. Sievenpiper, “Nonscattering waveguides based on tensor impedance surfaces,” IEEE Trans. Antennas Propag., vol. 63, no. 4, pp. 1746–1755, 2015. [87] N. Marcuvitz, Waveguide Handbook, Electromagnetic Wave Series 21, IET, 1986. [88] J. C. Sureau, “Reduction of scattering cross section of dielectric cylinder by metallic core loading,” IEEE Trans. Antennas Propag., vol. 15, no. 5, pp. 657–662, 1967. [89] U. Leonhardt, T. G. Philbin, Geometry and Light: The Science of Invisibility, Mineola, NY: Dover, Mineola, 2010. [90] S. A. Schelkunoff, “The impedance concept and its application to problems of reflection, refraction, shielding and power absorption”, Bells Labs Technol. J., vol. 17, no. 1, pp. 17– 48, 1938. [91] S. A. Schelkunoff, “Some equivalence theorems of electromagnetics and their application to radiation problems”, Bell Syst. Technol. J., vol. 15, p. 92, 1936. [92] G. Labate, L. Matekovits, “Kirchhoff’s current law as local cloaking condition: Theory and applications”, Electr. Lett., vol. 52, no. 21, pp. 1749–1751, 2016. [93] G. Kirchhoff, “Ueber den Dunchgang eines elektrischen Stromes durch eine Ebene, insbesondere durch eine kreisformige”, Ann. Phys., vol. 140, no. 4, pp. 497–514, 1845. [94] A. Alù, N. Engheta, “Effects of size and frequency dispersion in plasmonic cloaking”, Phys. Rev. E, vol. 78, no. 4, 045602(R), 2008. [95] J. Valentine, J. Li, T. Zentgraf, G. Bartal, X. Zhang, “An optical cloak made of dielectrics”, Nat. Mater., vol. 8, no. 7, pp. 568–571, 2009. [96] J. Andkjær, O. Sigmund, “Topology optimized low-contrast all-dielectric optical cloak”, Appl. Phys. Lett., vol. 98, no. 2, 021112, 2011. [97] J. Andkjær, N. Asger Mortensen, O. Sigmund, “Towards all-dielectric, polarizationindependent optical cloaks”, Appl. Phys. Lett., vol. 100, no. 10, 101106, 2012. [98] Y. Urzhumov, N. Landy, T. Driscoll, D. Basov, D. R. Smith, “Thin low-loss dielectric coatings for free-space cloaking”, Opt. Lett., vol. 38, no. 10, pp. 1606–1608, 2013. [99] K. Ladutenko, O. Peña-Rodríguez, I. Melchakova, I. Yagupov, P. Belov, “Reduction of scattering using thin all-dielectric shells designed by stochastic optimizer”, J. Appl. Phys., vol. 116, no. 18, 184508, 2014. [100] A. Mirzaei, A. E. Miroshnichenko, I. V. Shadrivov, Y. S. Kivshar, “All-dielectric multilayer cylindrical structures for invisibility cloaking”, Sci. Rep., vol. 5, p. 9574, 2015. [101] N. Heo, J. Yoo, “Dielectric structure design for microwave cloaking considering material properties”, J. Appl. Phys., vol. 119, no. 1, 014102, 2016. [102] L. Di Donato, T. Isernia, G. Labate, L. Matekovits, “Towards printable natural dielectric cloaks via inverse scattering techniques”, Sci. Rep., vol. 7, 3680, 2017. [103] E. Knott, J. Shaeffer, M. Tuley, Radar Cross Section, 2nd ed., London: Artech House, 1993. [104] A. E. Miroshnichenko, A. B. Evlyukhin, Y. Feng Yu, R. M. Bakker, A. Chipouline, A. I. Kuznetsov, B. Luk’yanchuk, B. N. Chichkov, Y. S. Kivshar, “Nonradiating anapole modes in dielectric nanoparticles”, Nat. Commun., vol. 6, p. 8069, 2015. [105] L. Wei, Z. Xi, N. Bhattacharya, H. P. Urbach, “Excitation of the radiationless anapole mode”, Optica. vol. 3, pp. 799–802, 2016. [106] N. A. Nemkov, I. V. Stenishchev, A. A. Basharin, “Nontrivial nonradiating all dielectric anapole”, Sci. Rep., 2017. [107] N. A. Nemkov, A. A. Basharin, V. A. Fedotov, “Nonradiating sources, dynamic anapole, and Aharonov-Bohm effect”, Phys. Rev. B, vol. 95, 165134, 2017.

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[108] B. Luk’yanchuk, R. Paniagua-Domínguez, A. I. Kuznetsov, A. E. Miroshnichenko, Y. S. Kivshar “Hybrid anapole modes of high-index dielectric nanoparticles”, Phys. Rev. A, vol. 95, 063820, 2017. [109] T. Raybould, V. Fedotov, N. Papasimakis, I. Youngs, N. Zheludev, “Exciting electromagnetic anapoles with Flying Doughnut pulses”, Appl. Phys. Lett., vol. 111, 081104, 2017. [110] A. A. Basharin, V. Chuguevsky, N. Volsky, M. Kafesaki, E. N. Economou, “Extremely high Q-factor metamaterials due to anapole excitation”, Phys. Rev. B, vol. 95, no. 3, 035104 2017. [111] L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (translated by J. B. Sykes and J. S. Bell), Elmsford, NY: Pergamon, 1960. [112] D. S. Jones, “Scattering by inhomogeneous dielectric particles”, Q. J. Mech. Appl. Math., vol. 38, p. 135, 1985. [113] D. Sjöberg, “Variational principles for the static electric, and magnetic polarizabilities of anisotropic media with perfect electric conductor inclusions”, J. Phys. A, vol. 42, 335403, 2009. [114] F. Monticone, A. Alù, “Do cloaked objects really scatter less?”, Phys. Rev. X, vol. 3, 041005, 2013. [115] O. M. Bucci, G. Franceschetti, “On the spatial bandwidth of the scattered fields”, IEEE Trans. Antennas Propag., vol. 35, no. 12, pp. 1445–1455, 1987. [116] O. M. Bucci, G. Franceschetti, “On the degrees of freedom of scattered fields”, IEEE Trans. Antennas Propag., vol. 37, no. 7, pp. 918–926, 1989. [117] R. Bansal, “The far-field: How far is far enough?”, Appl. Microwave Wireless, vol. 11, pp. 59–60, 1999. [118] R. M. Foster, “A reactance theorem”, Bell Syst. Tech. J., vol. 3, p. 259, 1924. [119] P.-Y. Chen, C. Argyropoulos, A. Alù, “Broadening the cloaking bandwidth with non-foster metasurfaces”, Phys. Rev. Lett., vol. 111, 233001, 2013. [120] C. Della Giovampaola, N. Engheta, “Plasmonics without negative dielectrics”, Phys. Rev. B, vol. 93, 195152, 2016. [121] F. Monticone, A. Alù, “Invisibility exposed: Physical bounds on passive cloaking”, Optica, vol. 3, p. 718, 2016. [122] H. Bode, Network Analysis and Feedback Amplifier Design, New York: David Van Nostrand, 1945. [123] R. M. Fano, “Theoretical limitations on the broadband matching of arbitrary impedances”, J. Franklin Inst., vol. 249, pp. 57–83, 1950.

11

Orbital Angular Momentum Beam Generation Using Textured Surfaces Mehdi Veysi, Caner Guclu, Filippo Capolino, and Yahya Rahmat-Samii

The concept of orbital angular momentum (OAM) of light has captured the attention of the optics and electromagnetics communities in both physics and engineering and in particular has been instrumental in various wireless communication systems [1–8]. In this chapter, we first briefly revisit the OAM beam concept by focusing on its unique and compelling near-field and far-field characteristics and applications. In particular, we demonstrate both theoretically and numerically that OAM beams radiate cone-shaped patterns with a high azimuthal symmetry in far-field with potential applications in wireless and satellite communication systems. We introduce the textured electromagnetic surfaces as efficient apparatuses to generate and manipulate such beams at both nearfield and far-field ranges.

11.1

OAM Beams: Concept and Historical Background It is well known in classical electrodynamics that optical beams can carry angular momentum in addition to the linear momentum [9, 10]. The angular momentum of a beam is calculated by integrating the cross product of the wave Poynting vector and the position vector (i.e., r) over a free-space volume V as [9, 11, 12] ż (11.1) J = ε0 r ˆ Re tE ˆ B˚ u dV . Here ε0 is the free space permittivity and the asterix (˚) denotes complex conjugation. The total angular momentum of a beam is composed of two key components: (1) intrinsic, also named spin angular momentum whose mode number is represented by s [10], and (2) extrinsic or OAM whose mode number is denoted by l [1]. The spin angular momentum of a beam is related to its circular/elliptical polarization, while the OAM of a beam is associated only with its field’s spatial phase distribution and not its polarization. Therefore, the total angular momentum mode number is j = l + s, and is explicitly found by [9, 11, 12] j = Jz {W,

(11.2)

where W is the beam energy in the volume V . A fully transverse plane wave has a Poynting vector in its propagation direction (e.g., z-direction) and therefore does not have a component of angular momentum in the z-direction (i.e., Jz = 0). However, 363

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Orbital Angular Momentum Beam Generation Using Textured Surfaces

Figure 11.1 Three known families of optical beams with twisted wavefronts and representative schematics of their generation from a fundamental Gaussian beam at its minimum waist. © 2018 IEEE. Reprinted, with permission, from Mehdi Veysi 2018.

there is a class of laser beams with helical wavefronts (so-called helical beams) whose Poynting vector spirals around their propagation axis, giving rise to an OAM along their propagation axis [1, 7]. An optical beam with a helical phase term exp (´j lϕ)(i.e., its phase linearly changes with the azimuth angle as lϕ where ϕ denotes the azimuth angle and l is an integer, so-called OAM number or azimuthal index number) has photons that carry an OAM of l (per photon) where  is the reduced Planck constant [1]. Due to the helical phase term exp (´j lϕ), such beams possess a phase vortex around their beam axis (i.e., the z-axis) and thus they are sometimes referred to as vortex beams. The important properties of the three well-known categories of optical beams with twisted wavefronts (Laguerre–Gaussian beams, fundamental (zero-order) and higherorder Bessel–Gaussian beams) are summarized in Figure 11.1 [13, 14]. Among them the Laguerre–Gaussian beams and higher-order Bessel–Gaussian beams possess a helical phase term exp (´j lϕ) and therefore carry OAM. Yet, they have different radial phase distributions. In the following subsections, we briefly introduce such beams.

11.1.1

Bessel–Gaussian Beams Let us consider a circularly polarized (CP) ideal Bessel beam in cylindrical coordinate system (see Figure 11.1) which is an exact solution of the Helmholtz equation and its electric field distribution is expressed as [14, 16]      ? 1 1 EBB ρ 1,ϕ 1,z1 = E0BB Jl kρ ρ 1 e´j lϕ e´j kz z xˆ 1 ˘ j yˆ 1 { 2;

(11.3)

11.1 OAM Beams: Concept and Historical Background

365

here Jl (¨) is the Bessel function of first kind of lth order [17], l is an integer  number (socalled OAM number), E0BB is a complex amplitude coefficient, ρ 1,ϕ 1,z1 are cylindrical b kρ2 + kz2 are radial, longitudinal and free-space coordinates, and kρ , kz , and k0 = wave vectors, respectively. Furthermore, the minus and plus signs in Eq. (11.3) represent right-hand circularly polarized (RHCP) and left-hand circularly polarized (LHCP) waves, respectively. Throughout this chapter, bold letters are reserved for vectors, carets (^) denote unit vectors, and time harmonic dependence is considered as exp (j ωt) which is suppressed in notation for convenience. Note that Eq. (11.3) with a noninteger value of l can be expressed in terms of a Fourier series superposition of orthogonal OAM states (integer l numbers) [18, 19]. Note that in the special case when l is set to zero, Eq. (11.3) represents the field distribution of a fundamental (zero-order) Bessel beam which does not possess OAM (refer to Figure 11.1). Since ideal Bessel beams expressed in Eq. (11.3) carry infinite power, similar to uniform plane waves, they are in fact nonphysical [20]. In this regard, Bessel–Gaussian beams with limited power have been investigated and experimentally generated from a Gaussian beam by using an axicon phase plate as shown Figure 11.1. The field distribution of a general CP Bessel–Gaussian beam (which satisfies the wave equation under paraxial approximation) in transverse plane z1 = 0 of minimum waist is expressed as follows [16]:  1   1 1 1   ˆ ˘ j yˆ 1 BG BG 1  ´ρ 12 {w2g ´j lϕ 1 x ? , (11.4) ρ ,ϕ ,z = 0 = E0 Jl kρ ρ e e E 2 where wg is the beam waist of the Gaussian term controlling the transverse extent of the Bessel beam and E0BG is a complex amplitude coefficient.

11.1.2

Laguerre–Gaussian (Helical) Beams The electric field distribution for a general CP Laguerre–Gaussian beam is given in cylindrical coordinate system as [6, 7]  1 1 1  1 1 1  1  ? LG (11.5) xˆ ˘ j yˆ 1 { 2, ELG l,p ρ ,φ ,z = E0 ul,p ρ ,φ ,z where E0LG is a complex amplitude coefficient, l and p are integer numbers (so-called azimuthal and radial mode numbers, respectively), and the complex scalar function  ul,p ρ 1,φ 1,z1 is a solution of wave equation under paraxial approximation which in cylindrical coordinates reduces to [6] j „     1 B B 1 B2 1 B ul,p ρ 1,φ 1,z1 = 0. (11.6) ´ 2j k ρ + 0 1 1 1 1 1 2 1 2 ρ Bρ Bρ Bz ρ Bϕ   The solutions ul,p ρ 1,φ 1,z1 are Laguerre–Gaussian beams that possess an azimuthal phase factor exp(´j lϕ 1 ) and carry an OAM of l per photon [1]. Here,  is reduced Planck’s constant and the parameter l is called OAM number or l-number. Such Laguerre–Gaussian mode solutions are expressed as [1]  ? |l | d    1 1 1 1 2p! ρ1 2 12 2 |l |   ul,p ρ ,φ ,z = e´ρ {w Lp 2ρ 12 {w2 w π p + |l| ! w (11.7) 12

k0 ρ 1 j (2p+|l |+1) tan´1 ˆ e´j lϕ e´j 2R e



z1 zR



1

e´j k0 z ,

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Orbital Angular Momentum Beam Generation Using Textured Surfaces

where ul,p has a unit norm and w = wg

b

2  1 + z1 {zR ,

 2   R = z1 1 + zR {z1 .

(11.8)

Llp (¨) is also the associate Laguerre polynomial given as [17] (|l |)

Lp



2ρ 12 w2

 =

   12 m (´1)m p + |l| ! 2ρ   . w2 ´ m) ! |l| + m ! m! (p m=0 p ÿ

(11.9)

Also, zR = π w2g {λ0 is the Rayleigh range, and k0 = 2π {λ0 and λ0 are the wavenumber and wavelength in the host medium, respectively. The polarization of the Laguerre– Gaussian beam in Eq. (11.5) is circular, but the analysis provided in this chapter can be straightforwardly extended to the cases with linear polarizations too.

11.2

Near-Field Applications of OAM Beams Since the detection of OAM in helical beams in 1992 [1], OAM-carrying laser beams have attracted extensive attention and academic interest in the optics community and have given rise to important developments in optical communication systems [2–7, 21–24]. In the following subsections, we briefly discuss two important near-field applications of the OAM beams in generating cylindrical vector beams and increasing channel capacity of near-field communication links.

11.2.1

Generating Cylindrical Vector Beams One of the most exciting applications of OAM-carrying beams is their use in generating cylindrical vector beams, such as radially and azimuthally polarized beams. In particular, it is demonstrated in [6] that an azimuthally/radially polarized beam can be efficiently constructed by superposition of the RHCP and LHCP Laguerre–Gaussian beams carrying OAM with l = ˘1 as ?  j 2 u´1,0 eˆ RH ¯ u1,0 eˆ LH e´j k0 z, E= 2

(11.10)

  ?   ? where eˆ RH = xˆ ´ j yˆ { 2 and eˆ LH = xˆ + j yˆ { 2 are, respectively, right and left hand circularly polarized unit vectors, and u˘1,0 is provided in Eq. (11.7). In Eq. (11.10), the minus sign results in an azimuthally polarized beam (see Figure 11.2) while the plus sign results in a radially polarized beam. The azimuthally polarized beams with their unique circulating electric fields [Eq. (11.10) with minus sign] have a salient feature that they contain a magnetic-dominant region within which the total electric field is ideally null while the longitudinal (i.e., z-directed) magnetic field is

11.2 Near-Field Applications of OAM Beams

367

Figure 11.2 Generation of the azimuthally polarized beam, as a superposition of two CP helical beams with l = ˘1, from a linearly polarized fundamental Gaussian beam, by using a textured metasurface.

maximum [6, 23–27]. Therefore, they are especially valuable in optics and microscopy for boosting the weak photoinduced magnetic response and suppressing relatively stronger electric response of a sample matter [24, 27]. The access to the photoinduced magnetic response in sample matter would add an additional degree of freedom to future optical photoinduced force microscopy and spectroscopy systems based on the excitation of photoinduced magnetic dipolar transitions, as thoroughly discussed in [24]. As a proof of concept, an electromagnetic textured surface (so-called metasurface [28]) is designed in [6] to generate an azimuthally polarized beam from a linearly polarized fundamental Gaussian beam, as schematically shown in Figure 11.2. Note that the radially polarized beams [whose field distributions are given in Eq. (11.10) with plus sign] when focused by a lens possess focal regions with very long depth of focus and narrow lateral full width at half maximum [29]. Thus, they provide a longer interaction range between the focused beam and the specimen in optical systems and therefore remedy the problems caused by the specimen being out of focus [30].

11.2.2

Increasing Channel Capacity of Wireless Communication Systems Another interesting characteristic of OAM-carrying beams is that they form a complete orthogonal modal basis set [1, 5, 7], and thus establish a new set of data carriers, which does not rely solely on polarization or frequency. Therefore, they offer the potential for increasing the spectral efficiency and capacity of wireless communication links at both optical [2, 5, 31] and RF ranges [32–34]. Yet, there have been intense back-and-forth scientific discussions and controversies concerning the advantages of the OAM-carrying beams in increasing the spectral efficiency and capacity of RF wireless communication links [11, 32, 33, 35–44]. In particular, since such beams inherently diverge with increasing distance from the transmitter, a very large (compared to wavelength) antenna or a combination of a few antennas is required on the receiver side to perceive the azimuthal phase variation of the field in the far-field rather than sampling only the local field phase.

368

Orbital Angular Momentum Beam Generation Using Textured Surfaces

On the transmitter side, various methods have been proposed to generate OAMcarrying beams at optical frequency ranges, such as spiral phase plates [38], liquidcrystal based spatial light modulators (SLMs) [2, 5], plasmonic metasurfaces [6, 45], and holograms [46]; and at RF frequencies, such as twisted parabolic reflectors [32, 47, 48], circular arrays of antennas [11, 12, 43, 49–52], spiral phase plates [33, 53–55], dielectric resonator antennas [56, 57], circular [58] and elliptical [59] patch antennas, ring cavity resonators [8], leaky wave antennas [60], slot-array antenna [61], and using phased array [43] and transmitarray [62, 63] concepts. On the receiver side, a fundamental challenge is how to sense the OAM information in the far-field of an OAM-carrying beam transmitter. Recall that the OAM information is stored in the azimuthal phase variations of the field and is not a local field property. One way to sense the OAM azimuthal phase variations is to use an electrically large receiving antenna whose size linearly scales with the distance from the transmitter and covers a sufficiently large solid angle or arc of the generated OAM beam. Therefore, the potential applications of the OAM to enhance the wireless communication and MIMO (multiple-in-multiple-out) systems in RF range are mostly considered in the radiating near-field region (i.e., the Fresnel region) rather than the far-field one. Note that this may be less of an issue in optical range with a communication link distance of a few meters, where the transmitter and receiver aperture diameters are usually very large compared to wavelength (in the order of several hundred wavelengths) [5, 64, 65]. In this chapter, we mainly investigate the far-field features and applications of the OAM-carrying beams rather than the detection of OAM information.

11.3

Potential Far-Field Applications of OAM Beams One exotic feature of the OAM-carrying beams is that their field vanishes on the beam axis (so-called phase vortex), and therefore they have an annular-shaped intensity map in near-field region with potential applications in optical trapping [13, 66, 67]. Accordingly, the OAM-carrying beams have a cone-shaped pattern in the far-field, as we show both theoretically and numerically in the next sections. Such a cone-shaped pattern is of crucial importance for local satellite-based navigation and guidance systems serving moving vehicles. For such applications, satellites are on the geostationary orbit (GEO) and use a contoured-beam antenna to cover a certain geographical region on earth. In most cases, distinct and fixed location of the satellite with respect to the geographical region requires a CP conical-scanned beam antenna on the mobile vehicles moving in the geographical region. In other words, the pencil beam pattern of the mobile vehicle’s antenna constantly scans mechanically/electronically a cone in space to capture the satellite signal. However, such an option is very complex and expensive and not suitable for large-scale and customer-driven markets. An alternative choice for the mobile vehicle’s antenna is an antenna with a fixed CP cone-shaped pattern with high azimuthal symmetry. The peak of the antenna’s cone-shaped pattern points toward the satellite at the elevation angle θc (so-called radiation cone angle), as shown in Figure 11.3. Various antennas with cone-shaped patterns have been developed

11.4 Far-Field Characteristics of OAM Beams

369

Figure 11.3 Schematic of a GEO satellite-based navigation and guidance system serving moving vehicles. © 2018 IEEE. Reprinted, with permission, from Mehdi Veysi 2018. A black and white version of this figure will appear in some formats. For the color version, please refer to the plate section.

in the literature, such as circular patch antennas operating at higher order modes [68], phased arrays [69], bird-nest antennas [70], crosspatch fed surface wave antennas [71, 72], leaky-wave antennas [73], and arrays of antennas generating cone-shaped patterns [74, 75], to name a few. However, the aforementioned antennas usually suffer from limited bandwidth or from low gain. In the following sections, first a detailed theoretical analysis of the far-field radiation of the beams with twisted wavefronts (see Figure 11.1) is presented by using the aperture field method. Inspired by the well-defined cone-shaped far-field radiation pattern of such beams, which is suited particularly well for satellite communications with mobile vehicles, next, we examine the generation of such beams at Ka-band (28–32 GHz). We demonstrate that the textured surfaces such as reflectarray antennas are powerful machines to systematically generate complex beams with twisted wavefronts and coneshaped patterns. In addition, at the end of this chapter, we further extend the OAM beam concept to generate multiple simultaneous pencil beams azimuthally distributed in space with potential applications in single-point to multi-point communications, diversity, and MIMO concepts.

11.4

Far-Field Characteristics of OAM Beams Optical beams with twisted wavefronts, such as Laguerre–Gaussian (helical) beams, fundamental, and higher-order Bessel–Gaussian beams, have been widely investigated within their Fresnel region [4, 13, 76]; however, to the authors’ knowledge, their characteristics in the far-field region have not been investigated, and their far-field applications are yet explored. In this section, we theoretically examine the far-field characteristics of the beams introduced in Figure 11.1 by using the aperture field method, i.e., by taking the Fourier transform of the aperture field [77].

11.4.1

Bessel–Gaussian Beams The field distribution of the Bessel–Gaussian beam at the transverse z1 = 0 minimum waist plane is given in Eq. (11.4). Using the aperture field method, the far-field of the

370

Orbital Angular Momentum Beam Generation Using Textured Surfaces

Bessel–Gaussian beam is then calculated in the cylindrical coordinate system as (see [77], Chapter 12, p. 711) EBG ff (r,θ,ϕ)

»Q

BG

ż8

    Jl kρ ρ 1 exp ´ρ 12 {w2g ρ 1 dρ 1

0

ż2π e

ˆ

(11.11) ´j lϕ 1

“  ‰ dϕ 1, exp j k0 ρ 1 sin θ cos ϕ ´ ϕ 1 

0

where QBG = E0BG j k0 e´j k0 r {(2π r) e˘j ϕ



 ? ˆ { 2. θˆ ˘ j cos θ ϕ

(11.12)

Note that Eq. (11.11) in fact illustrates the 2D Fourier transform of the transverse aperture field given in Eq. (11.4). By using the integral identities given in [15, 16] [refer to Eq. (5) in [15]], the far-field expression for the Bessel–Gaussian beam given in Eq. (11.4) is calculated as   BG EBG πw2g (´j )l e´j lϕ Il w2g kρ k0 sin θ{2 ff (r,θ,ϕ) » Q ” ı   (11.13) ˆ exp ´w2g kρ2 + (k0 sin θ)2 {4 . Here Il (¨) is the modified Bessel function of first kind of order l [17]. In particular, for beams with a large beam waist wg (i.e., a large aperture) and at large elevation angles θ (away from the beam axis, i.e., the θ ‰ 0 axis), the argument of the modified Bessel function (i.e., 12 w2g kρ k0 sin θ ) is large. Therefore, by using the asymptotic form of the modified Bessel function for a large argument [see Eq. (7) in [15]], the far-field expression for the Bessel–Gaussian beam in Eq. (11.13) is rewritten as [15] b   BG l π { kρ k0 wg e´j lϕ „ Q EBG (r,θ,ϕ) (´j ) ff ” (11.14)  2 ı ? exp ´w2g kρ ´ k0 sin θ {4 { sin θ. Note that the far-field expression in Eq. (11.14) is similar to the equation representing the focal-plane field distribution of a conventional lens illuminated by a Bessel– Gaussian beam (calculated using Fresnel integral) as in [16]. The far-field equation in Eq. (11.14) clearly embodies a CP cone-shaped pattern with a complete azimuthal symmetry (note that the amplitude of the far-field expression does not depend on azimuthal angle ϕ). Furthermore, the cone-shaped pattern peaks at the elevation angle of [15]   θrBG = sin´1 kρ {k0 .

(11.15)

Note that the radiation cone angle of the Bessel–Gaussian beam’s far-field cone expressed in Eq. (11.15) only depends on the radial wavenumber kρ and is independent of the beam waist wg . The axial ratio (defined as the ratio of the magnitudes of Eθ and

11.4 Far-Field Characteristics of OAM Beams

371

Eϕ components) of the ideal theoretical far-field equation in Eq. (11.14) is calculated at the radiation cone angle θrBG as [15] AR = 1{cos θrBG .

(11.16)

It is observed from Eq. (11.16) that the minimum theoretically achievable axial ratio at the radiation cone angle increases as the cone angle θrBG increases. In order to generate the Bessel–Gaussian beams examined above using textured electromagnetic surfaces, we need to simultaneously impose both the phase profile of an axicon phase plate (which contains a radial-phase term; see Figure 11.1) and that of a spiral phase plate (which contains an azimuthal-phase term; see Figure 11.1) onto the incident Gaussian beam upon its reflection/transmission from the surface [78]. Note that the equivalent beam waist of the Bessel–Gaussian distribution (i.e., wg ) is defined as the half-width of the normalized aperture field amplitude at 1{e. The generated Bessel–Gaussian field distribution radiates a cone-shaped far-field pattern [as given in Eq. (11.14)] with a radiation cone angle of θrBG , which is merely dictated by the aperture field’s radial phase gradient (i.e., the kρ parameter). Note that the aperture field’s azimuthal phase gradient (i.e., l parameter) results in a phase vortex in the boresight direction (i.e., θ = 0 axis) [due to the presence of the phase term exp (´j lϕ) in Eq. (11.14)], and therefore forming a null at boresight (note that the field is ideally null on the phase vortex).

11.4.2

Laguerre–Gaussian (Helical) Beams The electric field distribution of the Laguerre–Gaussian beam given in Eq. (11.5) is rewritten at the transverse z1 = 0 minimum waist plane as  ? |l | d  1 1 1  E0LG 12 2 2p! ρ1 2 LG   E ρ ,φ ,z = 0 = e´ρ {wg wg wg π p + |l| ! (11.17)   ? |l | 1 12 2 ´j lϕ 1  1 xˆ ˘ j yˆ { 2. ˆ Lp 2ρ {wg e By using the aperture field method, the far-field of the Laguerre–Gaussian field, whose transverse field distribution is given in Eq. (11.17), is found as d ELG ff (r,θ,ϕ)

»Q

LG

p!   π p + |l| !

? |l|+1 ˆ 2{wg

ż2π e

ż8

  12 2 |l | e´ρ {wg Lp 2ρ 12 {w2g ρ 1|l |+1 dρ 1

0

´j lϕ 1

(11.18) ej k0

ρ 1 sin θ rcos

(

ϕ ´ϕ 1

)s dϕ 1,

0

where QLG is given in Eq. (11.12) with E0BG substituted by E0LG . By using the integral identities given in [23] [see Eq. (A4) in [23]], the far-field equation for the Laguerre– Gaussian beam with the transverse field distribution given in Eq. (11.17) is then calculated as

372

Orbital Angular Momentum Beam Generation Using Textured Surfaces

b   LG p l ELG w 2πp!{ p + |l| ! » Q (r,θ,ϕ) (´1) (´j ) g ff  ? |l | ´  2 {4 |l | ” 2 ı ´j lφ ˆ sgn(l){ 2 e Lp  {2 e ,

(11.19)

where  = wg k0 sin θ. For the case of Laguerre–Gaussian beam generation using textured surfaces, such as reflectarrays (i.e., implying the required azimuthal phase on the incident Gaussian beam, as is the case in this chapter), the Laguerre–Gaussian mode with p = 0 is the dominant mode, and therefore the far-field expression in Eq. (11.19) is reduced to   d | l | ´  2  2π sgn(l) 4 LG l ? e e´j lφ . (11.20) ELG ff (r,θ,ϕ) » Q wg (´j ) |l| ! 2 Note that the above far-field equation embodies a CP cone-shaped radiation pattern with complete azimuthal symmetry and its radiation cone angle is at the elevation angle of θrLG given as [15] „b j   θrLG = sin´1 (11.21) 2 |l|{ k0 wg . It is observed from Eq. (11.21) that for a constant beam waist of the Laguerre–Gaussian beam (i.e., constant aperture diameter), the radiation cone angle decreases as the l parameter increases. On the other hand, for a constant l parameter, the radiation cone angle decreases with increasing the beam waist of the Laguerre–Gaussian beam (as we numerically show in the following sections).

11.5

OAM Beam Generation Using Reflectarray Antennas The textured electromagnetic surfaces are planar surfaces populated on one side (or both sides) by antenna elements of varying sizes or orientations. Such surfaces can be designed in transmission mode, such as transmitarrays [79] or transmission metasurfaces [28], or reflection mode (i.e., backed on one side by a metallic ground plane), such as reflectarrays [79] and reflection metasurfaces [80]. The sizes/orientations of the antenna elements on the surface are locally adjusted to impose the required phase profile on the illuminated beam upon transmission or reflection. In this chapter, we particularly focus on reflectarrays and show that such surfaces are efficient vehicles to realize the required azimuthal and radial phases for the generation of Bessel–Gaussian and Laguerre–Gaussian beams.

11.5.1

Rotational Phase Control Principle Let us first show the rotational phase control principle for tailoring the transverse phase profile of a CP wave illuminating a textured reflective surface. We assume that a RHCP incident wave, Einc rh , propagating in the negative z-direction, normally impinges upon the reflective surface populated by antennas with unit cell shown in Figure 11.4a.

11.5 OAM Beam Generation Using Reflectarray Antennas

373

Figure 11.4 Top view of a reflective unit cell with a tilted antenna element (in general other antenna shapes can be chosen depending on performance requirements such as bandwidth). The antenna element is backed by a metallic ground plane on the other side. (a) Reference unit cell with zero rotation angle. Local (primed) and global (unprimed) coordinate systems coincide with each other. (b) The same unit cell rotated by ψ degrees.

Figure 11.4b shows the general case of the reference unit cell rotated by angle ψ around the propagation axis (z or z1 -axis, i.e., z ” z1 ). Note that the local (i.e., the primed) coordinate system is attached to the rotating unit cell, while its longitudinal axis coincides with that of the global (i.e., the unprimed) coordinate system; in other words, we have that z ” z1 . Let us assume that the reflection matrix in the local (primed) coordinate system (which is independent of the rotation angle ψ) is known for the unit cell. The reflection matrix in the global coordinate system is then calculated using the coordinate transformation forth and back as [6, 81]       inc   ref Ex cos ψ sin ψ cos ψ ´ sin ψ Rx 1 Ex 0 , = ref Eyinc ´ sin ψ cos ψ sin ψ cos ψ 0 Ry 1 Ey (11.22) where the subscripts x and y represent, respectively, the x- and y-components of the electric field and the superscripts ref and inc denote reflected and incident fields, respectively. The diagonal entries Rx 1 and Ry 1 denote the reflection coefficients of the unit cell along the x 1 -axis and y 1 -axis in the local coordinate system, respectively. Note that the off-diagonal entries of the reflection matrix in local coordinate system are ideally assumed null. The RHCP incident wave illuminating the surface is expressed as   ˆ + j yˆ j kz inc inc x ? e , (11.23) E = Erh 2 inc is a complex amplitude coefficient. Accordingly, after the matrix multipliwhere Erh cation in (11.22), the reflected electric field is expressed as «    ff xˆ ´ j yˆ 2j ψ xˆ + j yˆ 1 inc ref e e´j kz, +B ? (11.24) E = Erh A ? 2 2 2

where A = Rx 1 ´ Ry 1 ,

B = Rx 1 + Ry 1 .

(11.25)

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Orbital Angular Momentum Beam Generation Using Textured Surfaces

The first term in Eq. (11.24) represents the co-polarized wave, while the second term shows the cross-polarized wave. Note that when Ry 1 = ´Rx 1 (B = 0 and A = 2), the reflected wave is purely RHCP (as the incident wave), and its phase, upon reflection from the reflective surface, is progressed by twice the rotation angle ψ of the unit cell. Therefore, the phase profile of the incident RHCP wave after reflection can be locally manipulated by rotating the reflectarray antennas on the surface about their own axes. This approach is accurate when the mutual coupling between the various antennas of the reflectarray is negligible. If the antenna elements on the textured surface do not change rapidly along the surface, the local reflection properties of the surface can be inferred by resorting to the concept of local periodicity, and the reflection matrix of the reference unit cell can be characterized in a 2D infinitely periodic setup in full-wave simulations. In the following subsection, we introduce a proper CP reflectarray element for Ka-band (28–32 GHz), i.e., Ry 1 « ´Rx 1 within the antenna bandwidth, as a building block of the reflectarray surface. The reflectarray surface is backed by a metallic ground plane on one side, while is populated with proper antenna elements on the other side. The orientations of the antenna elements systematically change over the reflectarray surface using the rotational phase control principle described above to provide the required local phase shift on the illuminating beam generated by a feed horn antenna.

11.5.2

Double Split-Ring Element A proper CP reflectarray element must highly suppress cross-polarized reflection with respect to the incident CP sense [i.e., Ry 1 « ´Rx 1 within the antenna bandwidth; see Eq. (11.24)]. In order to control the phase of the CP wave reflected from the reflectarray surface, while maintaining the same CP sense as the incident wave, we use the rotational phase control principle [6, 79, 82, 83] as described above. In other words, the antenna element dimensions are fixed over the surface and the required spatial phases are achieved by locally rotating the CP element around its center. Various CP reflectarray elements have been proposed in the literature, such as square patches with attached microstrip phase-delay lines [82], crossed dipoles [84], and single split-ring elements [85]. In this chapter, in order to obtain a wideband response, we consider a double split-ring element [81], whose unit cell geometry is shown in Fig. 11.5. The reflectarray element is characterized in a 2-D periodic-array configuration under a normally incident plane wave using the frequency domain solver based on the finite element method implemented in CST Microwave Studio. The physical parameters of the double split-ring reflectarray element (see Fig. 11.5) are tuned such that the phase difference between the x- and y-polarized reflection coefficients [see Fig. 11.6a] is around 180˝ (i.e., Ry 1 « ´Rx 1 ) within the operating frequency range. Therefore, when a RHCP wave impinges onto the reflectarray surface, the reflected wave remains mainly RHCP (i.e., the polarization sense is preserved upon reflection) [see Eq. (11.24) and also [86], Chapter 4, p. 100]. The optimized dimensions of the reflectarray element are provided in the caption of Fig. 11.5. The magnitude of the CP reflection coefficients for the reflectarray element for both the co- and cross- (X-) polarized components are shown in Figure 11.6b. It is observed

11.5 OAM Beam Generation Using Reflectarray Antennas

375

Figure 11.5 Unit cell geometry of a double split-ring reflectarray element. The optimized dimensions of the reflectarray element at Ka-band are h = 0.81mm, p = 5 mm, g1 = 0.85 mm, g2 = 0.2 mm, r1 = 1 mm, r2 = 1.4 mm. The ring width is also set at 0.2 mm. © 2018 IEEE. Reprinted, with permission, from Mehdi Veysi 2018.

Figure 11.6 Schematic of the CP reflectarray element unit cell and its rotation around the z-axis. (b) The amplitude of the co- and cross- (X-) polarized reflection coefficients when the unit cell is illuminated by a normally incident CP wave. (c) The reflection phase of the co-polarized component versus the angular orientation of the CP element for RHCP incident wave at 30 GHz. © 2018 IEEE. Reprinted, with permission, from Mehdi Veysi 2018.

ˇ ˇ that the cross-polarized suppression band (defined as ˇX´pol ˇ ď ´10dB) is wide ranging from 27.2 to 32.8 GHz. It should be noted that the magnitude of the CP reflection coefficients are independent of the element’s angular orientation ψ and the sense of the incident circular polarization. The full-wave simulation results for the co-polarized reflection phase of the element as a function of its angular orientation ψ is shown at f = 30 GHz in Figure 11.6c for the case that the element is illuminated by a normally incident RHCP wave. It is observed that the reflection phase of the co-polarized component linearly scales up by twice of the element’s angular orientation as is expected

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Orbital Angular Momentum Beam Generation Using Textured Surfaces

from the rotational phase control principle [see Eq. (11.24)]. Note that for an incident LHCP wave, the co-polarized reflection phase linearly decreases by twice the element’s angular orientation, which is not discussed here for brevity. In addition, we observe from Figure 11.6c that the co-polarized reflection phase contains the full 360˝ phase span necessary to fully manipulate the incident beam’s wavefront. In the following section, we systematically decorate the reflectarray surface with the proposed CP elements with fixed physical parameters (provided in the caption of Figure 11.5) but different orientation angles ψ (as in Figure 11.6a) to imitate the phase functions of helical axicon and spiral phase plates (see Figure 11.1).

11.6

Reflectarrays with Cone-Shaped Patterns In this section, we develop two different categories of cone-shaped beam reflectarrays, namely Bessel-beam and helical-beam reflectarrays, by making use of the rotational phase control technique and the CP reflectarray element introduced in the previous section.

11.6.1

Bessel-Beam Reflectarray In order for a reflectarray antenna to radiate a CP Bessel–Gaussian beam, the required local phase shift χr,i on the reflectarray surface achieved by the ith antenna element must vary by the element location as χr,i = k0 di ´ ρi k0 sin θr ´ lϕi ,

(11.26)

where k0 is the free-space wavenumber, di is the distance between the phase center of the illuminating feed horn and the center of the ith cell on the reflectarray surface (as shown in Figure 11.7), θr is the radiation cone elevation angle [see Eq. (11.26)], and (ρi ,ϕi ) is the cylindrical coordinate of the ith cell’s center (see Figure 11.7). Here, we assume that the l parameter is an integer; however, in general it can take any real value. Note that the first term on the right-hand side of Eq. (11.26) compensates for the phase delay over the path from the illuminating feed horn to the ith cell’s center on the reflectarray surface. On the other hand, the required radial and the azimuthal phase gradients for generating the Bessel–Gaussian beam are provided by the second and the third phase terms on the right-hand side of Eq. (11.26), respectively. Note that to form a phase vortex around the z-axis, the azimuthal phase gradient in Eq. (11.26) must be nonzero (i.e., l ‰ 0). Let us assume an ideal cos-q shaped RHCP incident beam is impinging upon the reflectarray surface. The q-value is selected such that the incident field magnitude drops by 12dB at the reflectarray edge with respect to its maximum at the reflectarray center (in other words the reflectarray’s edge taper is ´12dB) [79, 87, 88]. The reflectarray is center-fed and has a circular-shaped aperture with a diameter of D = 7.5λ and a focal length to diameter ratio of F {D = 0.7, where λ = 10 mm. Accordingly, the q-parameter is found to be 5.7 in order to obtain a ´12dB edge taper

11.6 Reflectarrays with Cone-Shaped Patterns

377

Figure 11.7 Schematic of a reflectarray generating an OAM-carrying cone-shaped pattern. © 2018 IEEE. Reprinted, with permission, from Mehdi Veysi 2018.

for the given F /D ratio. The goal is to design a RHCP reflectarray radiating a coneshaped pattern with a radiation cone angle of θc = 15˝ (see Figure 11.3) and a phase vortex in the boresight direction (l ‰ 0). The required local reflection phase shifts on the Bessel-beam reflectarray surface are accordingly found from Eq. (11.26) (by setting θc = 15˝ and l = 1). In order to numerically calculate the far-field pattern of the reflectarray, the reflected field on the reflectarray surface is assumed to be step-wise approximated. In other words, the reflected electric field at each reflectarray unit cell is uniform and is calculated by multiplying the  incident field at each cell’s center by the  required exponential phase term, exp j χr,i , with χr,i calculated via Eq. (11.26). The phase and the magnitude of ideal step-sized approximated reflected RHCP field on the reflectarray surface are shown in Figure 11.8. We clearly observe that the reflected RHCP field on the reflectarray surface has a spiral-shaped phase profile which is analogous to the phase profile imposed by a helical axicon phase plate upon transmission of a linearly polarized incident beam (see Figure 11.1). The RHCP far-field radiation pattern of the reflectarray is subsequently calculated by numerically taking the Fourier transform of the reflected field on the reflectarray surface and is plotted in Figure 11.9. In addition, we also simulate the combined horn-reflectarray system with a center-fed configuration using the finite element method in CST Microwave Studio. The double split-ring elements on the reflectarray surface are systematically rotated [based on Figure 11.6c and Eq. (11.26)] such that the reflected wave has the spiral-shaped phase profile over the reflectarray surface (as shown in Figure 11.8a). The RHCP feed horn is pointing toward the reflectarray center and has a boresight gain of about 13.6 dB which is necessary to obtain a ´12 dB reflectarray edge taper. The full-wave simulation also considers undesirable effects due to the feed horn blockage and the reflectarray’s edge diffraction. Figure 11.9 compares the Fourier transform and the full-wave simulation results for the reflectarray’s RHCP radiation patterns at two orthogonal planes at 31 GHz. It is observed that the Fourier transform and the full-wave simulation results are in good agreement, especially

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Figure 11.8 Ideal (a) phase and (b) amplitude of the RHCP field on the Bessel-beam reflectarray’s cells upon reflection from reflectarray surface. The reflectarray diameter is D = 7.5λ. © 2018 IEEE. Reprinted, with permission, from Mehdi Veysi 2018. A black and white version of this figure will appear in some formats. For the color version, please refer to the plate section.

Figure 11.9 Comparison between full-wave (FW) and Fourier-transform (FT) results for the RHCP radiation patterns of the Bessel-beam reflectarray (with D = 7.5λ) at 31 GHz on two different elevation planes: (a) ϕ = 0˝ and (b) ϕ = 90˝ . © 2018 IEEE. Reprinted, with permission, from Mehdi Veysi 2018.

around the radiation cone angle. In addition, the radiation cone angle for both the Fourier transform and the full-wave radiation patterns is around θc = 15˝ which matches well with the theoretical expectation [i.e., setting θr = 15˝ in Eq. (11.26)]. Note that the feed horn blockage, which is only accounted for in the full-wave simulation, does not significantly change the radiation pattern even for the case of a cone-shaped pattern with a radiation cone angle as small as 15˝ thanks to the horn’s position (center-fed configuration) on the cone’s phase vortex. Figure 11.10 shows the full-wave co-polarized (RHCP) and cross-polarized (LHCP) radiation patterns of the designed Bessel-beam reflectarray at different frequencies within its bandwidth. The maximum antenna directivity in the radiation cone angle (i.e., at the elevation angle of θc = 15˝ ) is „14.6 dB and occurs at 31 GHz. In addition, we observe from Figure 11.10 that the cone-shaped pattern of the reflectarray is fairly well preserved within the antenna bandwidth (28 to 33 GHz) while the crosspolarized radiation (whose phase is not controlled by the rotational phase control method) is significant at 33 GHz. This is mainly ascribed to the high X-polarized reflection coefficient magnitude of the element at 33 GHz (see Figure 11.6b).

11.6 Reflectarrays with Cone-Shaped Patterns

379

Figure 11.10 Full-wave co-polarized (RHCP) and cross-polarized (LHCP) radiation patterns (in dB) of the Bessel-beam reflectarray (with D = 7.5λ) in ϕ = 0˝ elevation plane at (a) 28GHz, (b)

29GHz, (c) 30GHz, (d) 31GHz, (e) 32GHz, and (f) 33GHz. © 2018 IEEE. Reprinted, with permission, from Mehdi Veysi 2018.

Figure 11.11 also shows the RHCP radiation pattern of the Bessel-beam reflectarray on the θ = θc = 15˝ azimuth cone for different frequencies within its bandwidth, demonstrating a fairly good azimuthal symmetry for the cone-shaped radiation pattern. The maximum variation of the antenna directivity on the azimuth cone within the antenna bandwidth is about 2 dB and occurs at 32 GHz. Note that since the Fourier transform results agree well with the full-wave results (as shown in Figure 11.9), for the sake of simplicity, in the following, we perform the parametric analyses based on the Fourier transform method only. We first investigate the effect of the azimuthal phase gradient [third term on the righthand side of Eq. (11.26)] on the Bessel-beam reflectarray’s radiation pattern. In order to increase the accuracy of the Fourier transform method, we increase the reflectarray

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Figure 11.11 Full-wave RHCP radiation pattern (in dB) of the Bessel-beam reflectarray (with D = 7.5λ) in θ = 15˝ azimuth cone at various frequencies. © 2018 IEEE. Reprinted, with

permission, from Mehdi Veysi 2018.

Figure 11.12 Fourier-transform radiation pattern (in dB) of the Bessel-beam reflectarray (with D = 30λ) in ϕ = 0˝ elevation plane with (l = 1) and without (l = 0) azimuthal phase gradient

at 31 GHz. © 2018 IEEE. Reprinted, with permission, from Mehdi Veysi 2018.

diameter from 7.5λ to 30λ (λ = 10 mm) while keeping the edge taper, the F /D ratio, and θr parameter fixed as in the previous case. The RHCP radiation patterns for the Bessel-beam reflectarrays with [l = 1 in Eq. (11.26)] and without [l = 0 in Eq. (11.26)] azimuthal phase gradient are compared in Figure 11.12. The first and foremost observation is that the presence of the azimuthal phase gradient has no significant effect on the radiation cone angle (i.e., θc = θr = 15˝ ) and the radiation pattern around the radiation cone angle. Furthermore, it is observed that the presence of the azimuthal phase gradient (in addition to the radial phase gradient) features a phase vortex on the beam’s axis, which in turn results in a deep null in the boresight direction as compared to the Bessel-beam reflectarray without the azimuthal phase gradient (i.e., the case with l = 0). Note that such a phase vortex is a result of the phase term exp(j lϕ) in the far-field equation of a higher-order Bessel–Gaussian beam given in Eq. (11.14). In addition, increasing the l number from 1 does not provide any significant advantage over the case with l = 1 in terms of the boresight null depth, and therefore is not examined here for brevity.

11.6 Reflectarrays with Cone-Shaped Patterns

381

Figure 11.13 The effect of (a) the diameter of the cone-shaped Bessel-beam reflectarray (with l = 1 and θr = 15˝ ) and (b) the radial phase gradient amount (i.e., θr ) in Eq. (11.15) on the cone

angle. © 2018 IEEE. Reprinted, with permission, from Mehdi Veysi 2018.

Next, we investigate the effects of the reflectarray diameter and the θr parameter on the reflectarray’s cone-shaped pattern. In each case, we only change the corresponding parameter and keep the F {D = 0.7, l = 1, and the edge taper (´12 dB) fixed as in the previous design. We observe from Figure 11.13a that when the reflectarray’s diameter is doubled, the antenna directivity at the radiation cone angle (θc = 15˝ ) increases by about 2.9 dB. In addition, Figure 11.13b shows the effect of the radial phase gradient [i.e., θr in Eq. (11.26)] on the radiation cone angle. We observe that the radiation cone angle takes a wide range of angles [the results for radiation cone angle as large as 45˝ are shown in Figure 11.13b). Note that Bessel-beam reflectarrays with cone-shaped beam scanning capability can be also developed by integrating RF MEMS switches or varactor diodes into the reflectarray elements to dynamically vary the local elements’ angular orientations [80] and thus the radial phase gradient on the reflectarray surface.

11.6.2

Helical (Laguerre–Gaussian) Beam Reflectarray In this subsection, we show some features of helical-beam reflectarrays which radiate OAM-carrying helical beams (i.e., Laguerre–Gaussian beams as theoretically investigated in Section 11.3.2). Such beams are generated by imposing only the azimuthal phase gradient [i.e., removing the radial phase gradient in Eq. (11.26) by setting θr = 0] onto the beam illuminating the reflectarray surface, to emulate the phase function of a spiral phase plate (see Figure 11.1). Despite θr = 0, helical beams still possess coneshaped radiation patterns (see Section 11.3.2). The RHCP radiation pattern of a helicalbeam reflectarray (with diameter D = 7.5λ where λ = 10 mm) is plotted in Figure 11.14a as a function of azimuthal index number l in ϕ = 0˝ elevation plane. We observe that as the l number increases from 1 to 2 to 3, the radiation cone angle (i.e., θc ) shifts from 7.5˝ to 11˝ to 14˝ , respectively. However, these simulated radiation cone angles are different from the theoretical radiation cone angles evaluated by Eq. (11.21) as 4.14˝ , 5.86˝ , and 7.17˝ , respectively. The equivalent beam waist in Eq. (11.21) is equal to wg = 0.415D for the helical-beam reflectarray with F {D = 0.7. Recall that the equivalent beam waist (i.e., wg ) of the

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Orbital Angular Momentum Beam Generation Using Textured Surfaces

Figure 11.14 RHCP radiation pattern (in dB) of an OAM-carrying cone-shaped helical-beam

reflectarray designed based on only azimuthal phase gradient versus (a) azimuthal index number l (D = 7.5λ) and (b) reflectarray diameter (l = ˘3) in ϕ = 0˝ elevation plane. Patterns evaluated by using Fourier transform of the aperture field. © 2018 IEEE. Reprinted, with permission, from Mehdi Veysi 2018.

Laguerre–Gaussian beam generated by the helical-beam reflectarray is evaluated as the half width of the normalized aperture field amplitude on the reflectarray surface at 1{e. Since the theoretical radiation cone angle in Eq. (11.21) is calculated using the paraxial approximation it is only accurate for the beams with large waists (i.e., reflectarrays with large diameters or small radiation cone angles). Therefore, as the reflectarray diameter increases the simulated radiation cone angle must approach to the theoretical one, as we demonstrate in the following. In addition, we observe from Figure 11.14a that the HPBW solid angle of the cone-shaped pattern increases as the azimuthal index number l increases (the radiation cone angle increases), which in turn decreases the maximum directivity of the reflectarray at its radiation cone angle. It should be also pointed out that the azimuthal index numbers larger than 3 represent a large azimuthal phase gradient that may not be successfully resolved with a reflectarray with D = 7.5λ and cell size as large as λ{2. This would in turn ruin the azimuthal symmetry of the reflectarray’s cone-shaped pattern and thus is not shown here. Note that the radiation patterns for helical-beam reflectarrays with azimuthal index numbers –l and +l are completely identical (see Figure 11.14a). Although a helical-beam reflectarray with a radiation cone angle as big as θc = 14˝ is developed in this section by just using the azimuthal phase gradient (l = ˘3), the radiation cone angle shifts toward smaller angles as the reflectarray diameter (wg = 0.415D) increases for high-gain applications. The variation of the radiation cone angle with the diameter of the helical-beam reflectarray is shown in Figure 11.14b where the azimuthal index number is fixed at l = ˘3. It is observed that the radiation cone angle shifts toward smaller angles from about 3.5˝ to 1.72˝ to 0.9˝ as the reflectarray diameter increases from 15λ to 30λ to 60λ. Such simulated radiation cone angles also agree well with the theoretical radiation cone angles evaluated by Eq. (11.21), i.e., 3.58˝ , 1.79˝ , and 0.9˝ , respectively. In general, the range of the radiation cone angle obtained by the helical-beam reflectarray is much narrower than that obtained by the Bessel-beam reflectarray, in particular when a high gain reflectarray is required. It is also evident from Eq. (11.21) that the radiation cone angle decreases as the reflectarray

11.7 Reflectarrays Radiating Multiple Azimuthally Distributed Pencil Beams

383

diameter increases (here wg = 0.415D). In the following section, we employ the helical beam concept to generate multiple azimuthally-distributed pencil beams in space.

11.7

Reflectarrays Radiating Multiple Azimuthally Distributed Pencil Beams Let us assume that two OAM-carrying helical beams with azimuthal index numbers l1 and l2 are superimposed. The electric field of the interference beam is evaluated as the vectorial summation of the electric fields associated with the superimposed beams as E = E1 + E2 . Subsequently, the electric field intensity of the interference beam (i.e., I (θ,ϕ) = |E (θ,ϕ)|2 ) is found as I (θ,ϕ) = I1 (θ ) + I2 (θ ) + 2

a

I1 (θ) I2 (θ) cos β (θ,ϕ) ,

(11.27)

where I1 and I2 are, respectively, the field intensities of the helical beams with azimuthal index numbers of l1 and l2 . The phase of a helical beam is generally composed of two main terms: (i) the azimuthal phase term, and (ii) the elevation phase term. In Eq. (11.27), the phase parameter β (θ,ϕ) = (l1 ´ l2 ) ϕ + α (θ ) represents the phase difference between the two helical beams’ fields, where ϕ P r0,2π s and α is the phase difference between the θ -dependent phase terms of the two superimposed helical beams. Let’s assume that the two superimposed helical beams have equal radiation cone angles and their field intensities at their radiation cone angles are of comparable magnitude [i.e., I1 (θ = θc ) « I2 (θ = θc )]. Subsequently, the intensity of the interference beam at far-field given by Eq. (11.27) is simplified around the radiation cone angle as   I 91 + cos r(l1 ´ l2 ) ϕ + αs = 2 cos2 r(l1 ´ l2 ) ϕ + αs{2 .

(11.28)

Equation (11.28) shows that the interference beam possesses a cosine-shaped pattern versus ϕ with total number of |l1 ´ l2 | maxima equally spaced on a ring whose central axis is aligned with boresight direction. In the following, we numerically demonstrate that the interference of two OAM-carrying helical beams with different azimuthal index numbers results in multiple azimuthally-distributed pencil beams in far-field, which is potentially useful for sectorization of the 360˝ azimuthal coverage of the space, single point to multipoint communication, diversity, and MIMO concepts. Various techniques have been proposed in the literature to generate multiple simultaneous pencil beams, such as using reflectors with feed horn clusters [88], phased array concept [89], and single-feed reflectarrays designed based on iterative optimization techniques [90, 91]. In the following, we design a single-feed azimuthal multi-beam reflectarray antenna based on a direct, yet efficient, technique rather than time consuming optimization techniques. As a proof of concept, we design a single-feed azimuthal multibeam reflectarray radiating multiple pencil beams, all pointing toward the elevation angle of 15˝ . To this goal, we divide the reflectarray surface into two concentric areas, as shown in Figure 11.15a, where each area radiates a helical beam with a certain azimuthal index number. Here, we keep the azimuthal index number of the inner segment (i.e., l1 ) fixed at +1

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Orbital Angular Momentum Beam Generation Using Textured Surfaces

Figure 11.15 (a) Schematic of a reflectarray antenna with two concentric annular segments

featuring different azimuthal index numbers. (b) Fourier-transform RHCP radiation pattern radiated only from the inner segment of the reflectarray in (a) with l1 = ˘1 in ϕ = 0 elevation plane. Also, the directivity patterns radiated only from the outer segments with l2 = ˘3 and l2 = ˘4 are plotted. The radius of the inner segment is fixed at rin = 1.5λ, and the outer segment radii for l2 = ˘3 and l2 = ˘4 are set as rout = 3.75λ and rout = 4λ, respectively, where λ = 10 mm. © 2018 IEEE. Reprinted, with permission, from Mehdi Veysi 2018.

and only change the azimuthal index number of the outer segment (i.e., l2 ) such that we generate various numbers of pencil beams (i.e., |l1 ´ l2 |) as set by the design. The radius of the inner segment (rin ) is 1.5λ such that we have a cone-shaped pattern with l1 = 1 and radiation cone angle of 15˝ . The required reflection phase from each reflectarray cell on each reflectarray segment with a certain azimuthal index number is calculated from Eq. (11.26) with θr = 0. The directivity pattern radiated solely from the inner reflectarray segment is plotted in Figure 11.15b. The radius of the outer reflectarray segment (i.e., rout ) for the azimuthal index number l2 = ˘3 (l2 = ˘4) is also found to be 3.75λ (4λ), to generate a cone-shaped pattern with a radiation cone angle of 15˝ . Accordingly, the directivity pattern radiated solely by the outer (annular) reflectarray segment is calculated by setting the field magnitude on the inner reflectarray segment to zero and is plotted in Figure 11.15b for two different azimuthal index numbers. As shown in Figure 11.15a, the directivity patterns for all three cases have cone-shapes with equal radiation cone angles and comparable directivity values around their cone angles. The RHCP radiation pattern of the azimuthal multi-beam reflectarray (see Figure 11.15a) is plotted in the u ´ v plane in Figure 11.16. for various pairs of inner and outer segments’ azimuthal index numbers (l1,l2 ). It is observed from Figure 11.16 that the total number of the pencil beams for each case is |l1 ´ l2 | and that they are uniformly distributed around the boresight direction and well separated by an azimuth angle of 360˝ {|l2 ´ l1 |, which agree well with the conclusions drawn from Eq. (11.28). As a representative example, we perform full-wave simulations for a CP quad-beam reflectarray (with l1 = 1, l2 = ´3, rin = 1.5λ, and rout = 3.75λ) and report its 3D RHCP radiation pattern in Figure 11.17. Full-wave 2D RHCP radiation patterns and the axial ratios of the quad-beam reflectarray at θ = 15˝ azimuth cone are also plotted at different frequencies within the antenna bandwidth in Figure 11.18. It is observed

11.7 Reflectarrays Radiating Multiple Azimuthally Distributed Pencil Beams

385

Figure 11.16 Fourier-transform RHCP radiation pattern (in dB) of an azimuthal multi-beam

reflectarray composed of two concentric annular segments (as shown in Figure 11.15a) for different (l1,l2 ) combinations at 30 GHz. The inner segment radius and azimuthal index number are set at rin = 1.5λ and l1 = +1, respectively, while the outer segment radius and azimuthal index number are rout = 3.75λ (4λ) and l2 = ˘3(˘4), respectively, where λ = 10 mm. © 2018 IEEE. Reprinted, with permission, from Mehdi Veysi 2018. A black and white version of this figure will appear in some formats. For the color version, please refer to the plate section.

Figure 11.17 Full-wave RHCP radiation pattern of a quad-beam reflectarray with rin = 1.5λ,

rout = 3.75λ (where λ = 10 mm), l1 = 1, and l2 = ´3 at 31 GHz. © 2018 IEEE. Reprinted, with permission, from Mehdi Veysi 2018. A black and white version of this figure will appear in some formats. For the color version, please refer to the plate section.

that the maximum directivity of the quad-beam reflectarray varies by about 1.5 dB within the antenna bandwidth (the maximum directivity is 16.4, 16.9, 17.6, 17.9, 18, and 18dB, at 28, 29, 30, 31, 32, and 33 GHz, respectively). In addition, the maximum deviations between the maximum directivities of the four pencil beams are about 0.9, 0.4, 0.6, 0.3, 0.6, and 0.4 dB at 28, 29, 30, 31, 32, and 33 GHz, respectively. Thus, the directivity difference between the four pencil beams at each frequency within the antenna bandwidth is negligible. It is also observed from Figure 11.18 that the designed quad-beam reflectarray radiates four well-separated RHCP pencil beams whose axial ratios are well below 3 dB around the main beams’ directions (i.e., θ = 15˝ ) within the antenna bandwidth. Note

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Orbital Angular Momentum Beam Generation Using Textured Surfaces

Figure 11.18 Full-wave RHCP radiation pattern and axial ratio for the quad-beam reflectarray with diameter of 7.5λ in θ = 15˝ azimuth cone at various frequencies. © 2018 IEEE. Reprinted,

with permission, from Mehdi Veysi 2018.

that the axial ratio slightly exceeds 3 dB at 33 GHz which is mainly attributed to the element’s high cross-polarized reflection coefficient magnitude (refer to Figure 11.16b). Recall that, the minimum theoretically achievable axial ratio in the elevation angle of 15˝ can be calculated by using Eq. (11.16) and is found to be „0.3 dB.

11.8

Conclusions and Observations In this chapter, we revisited the concept of the orbital angular momentum (OAM) carrying beams and their wide range of potential applications in wireless communication systems, from near-field region to far-field region, and from RF range to optical range. In particular, we have shown both theoretically and numerically that such beams radiate cone-shaped patterns in far-field which are potentially beneficial for satellite communications with moving vehicles, such as cars and airplanes, in a certain geographical area. In order to further investigate the unique features of the OAM beams in far-field region, engineered textured electromagnetic surfaces are employed to conceive such beams at RF ranges. The used textured electromagnetic surfaces are designed in reflection mode and are commonly known as reflectarrays. Various OAM reflectarray antennas with circularly polarized cone-shaped patterns of high azimuthal symmetry and a deep null in boresight direction are developed such as Bessel-beam and helical-beam reflectarrays. While the helical-beam reflectarray generates a cone-shaped pattern with a small radiation cone angle, the Bessel-beam reflectarray is a great candidate for generating coneshaped patterns with high gain and radiation cone angles as large as 45˝ . At the end of the chapter, we examined the interference of two heli