Manipulation of Surface Waves through Metasurfaces [1st ed.] 978-3-030-14033-5, 978-3-030-14034-2

Table of contents :
Front Matter ....Pages i-xix
Introduction (Mario Junior Mencagli)....Pages 1-6
Surface Wave Dispersion for Anisotropic Metasurfaces Constituted by Elliptical Patches (Mario Junior Mencagli)....Pages 7-25
Closed-Form Representation of Metasurface Reactance and Isofrequency Dispersion Curve (Mario Junior Mencagli)....Pages 27-54
Flat Optics for Surface Waves (Mario Junior Mencagli)....Pages 55-82
Basic Properties of Checkerboard Metasurfaces (Mario Junior Mencagli)....Pages 83-91
Conclusion (Mario Junior Mencagli)....Pages 93-96
Back Matter ....Pages 97-106

Citation preview

Springer Theses Recognizing Outstanding Ph.D. Research

Mario Junior Mencagli

Manipulation of Surface Waves through Metasurfaces

Springer Theses Recognizing Outstanding Ph.D. Research

Aims and Scope The series “Springer Theses” brings together a selection of the very best Ph.D. theses from around the world and across the physical sciences. Nominated and endorsed by two recognized specialists, each published volume has been selected for its scientific excellence and the high impact of its contents for the pertinent field of research. For greater accessibility to non-specialists, the published versions include an extended introduction, as well as a foreword by the student’s supervisor explaining the special relevance of the work for the field. As a whole, the series will provide a valuable resource both for newcomers to the research fields described, and for other scientists seeking detailed background information on special questions. Finally, it provides an accredited documentation of the valuable contributions made by today’s younger generation of scientists.

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Mario Junior Mencagli

Manipulation of Surface Waves through Metasurfaces Doctoral Thesis accepted by the University of Siena, Italy

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Author Dr. Mario Junior Mencagli Dipartimento di Ingegneria dell’Informazione e Scienze Matematiche University of Siena Siena, Italy

Supervisor Prof. Stefano Maci Dipartimento di Ingegneria dell’Informazione e Scienze Matematiche University of Siena Siena, Italy

ISSN 2190-5053 ISSN 2190-5061 (electronic) Springer Theses ISBN 978-3-030-14033-5 ISBN 978-3-030-14034-2 (eBook) https://doi.org/10.1007/978-3-030-14034-2 Library of Congress Control Number: 2019932618 © Springer Nature Switzerland AG 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To my parents and my wife.

Supervisor’s Foreword

It is my great pleasure to introduce Dr. Mario Junior Mencagli’s thesis work, accepted for publication within Springer Theses Book Series and awarded with a prize for outstanding original work. I have known Dr. Mencagli for 6 years: first as a student in my course on Antennas and Propagation at the University of Siena, next during his master thesis, and last as a Ph.D. student under my supervision. His position has been financed by Thales Research and Technology, Paris, France. Dr. Mencagli completed his doctoral studies with a final viva voce defense on December 14, 2016, obtaining the highest rating (cum laude). From the beginning, I was very impressed with his eagerness and capability to learn new things quickly and to apply them to find innovative solutions. Thanks to his brilliant research qualities, combining scientific rigor, creativity, and enthusiasm, he obtained outstanding theoretical and applicative results, as testified by his excellent curriculum vitae, collecting several papers published in international scientific journals or books. Dr. Mencagli’s thesis theoretically and numerically examines the prospects of modulated metasurface (MTS) as a new platform for planar transformation optics devices, which allow one to manipulate the propagation of the supported surface waves. Such devices can be used as a part of flat lens antennas producing a fan-beam radiation pattern at microwaves but can also be employed as components of optical circuits. The first part of this dissertation describes two accurate and efficient approaches for the analysis of MTSs consisting of metallic patches printed on a grounded slab. The first technique is restricted to metallic patches of the elliptical shape. It resorts to an analytical expression of the currents excited on the elements, and then a spectral MoM formulation is used to build the matrix of a periodic MoM formulation with only two/three basis functions. This procedure can be used for the design of planar lenses, transformation optics devices, and leaky wave antennas. Unfortunately, this kind of MTS is not able to provide a high impedance value and a high range of anisotropy. Therefore, in MTS-based applications that require these characteristics, constitutive elements of different geometries are needed. To this vii

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end, we present a second approach for the analytical characterization of anisotropic MTS, which is valid for constitutive elements that possess at least two orthogonal axes. It relies on a simple analytical form of the isofrequency dispersion curve which depends on two parameters only. These parameters are the equivalent quasi-static capacitances along the symmetry axes of the element, which can be derived by the expressions based on equivalent gap capacitance (for simple shapes) or by extracting them from a single run of a full-wave eigensolver. Also, a closed-form representation of the group velocity and the limits of the validity of the formulation are derived. This approach has been found useful for the design of surface wave devices based on modulated MTSs and simplifies the design of modulated MTS antennas. The second part of this thesis introduces the concept of Flat Optics for surface waves, denoting the manipulations of surface waves through modulated surface impedance boundary conditions. All aspects relevant to ray-optics, such as ray-tracing, transport of energy, and ray-velocity, are rigorously treated for both isotropic and anisotropic surface impedances. We would like to emphasize that this formulation allows one to solve an issue which has not been treated in the literature, namely, the estimation of the amplitude of the field distribution over a surface that supports an anisotropic non-uniform, impenetrable impedance tensor. The theory, presented in this part, results in an elegant formulation which leads to closed-form analysis of planar operational devices based on modulated MTS. We provide various examples, suggesting that modulated MTSs can be a good platform for planar wave-guiding structures and transformation optics devices. Although the presented examples are focused in the microwave range, the proposed theory is also applicable in the terahertz and infrared regions, as well as at optical frequencies provided that one possesses the right technology for MTS implementation. The last part of this manuscript is devoted to the theoretical description of a type of MTS that is especially well suited for its integration with photoconductive switches. The MTS consists of a chessboard-type layout, made of electrically small complementary metallic patches and apertures on a grounded substrate. Depending on whether the patch vertexes are interconnected or not, the structure may support a propagating quasi-TEM mode. This offers the possibility of designing arbitrary transmission line paths on the MTS by dynamically changing the vertex connections. Moreover, one may combine several rows of patches to tailor the characteristic impedance. The structure has been analyzed to determine its equivalent circuit, which allows us to predict the dispersion equation of the transmission lines and their characteristic impedance. The proposed model has been verified using full-wave simulations. Both analytical and simulated results are in excellent agreement with measurements done on structures manufactured using an ideal switch, i.e., a small metalization. It is important to notice that the proposed MTS-based transmission line has a constant characteristic impedance regardless

Supervisor’s Foreword

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of the dimension of the vertex. This has been exploited to use micrometer-scale gaps that allowed us to confine the optical beam in a small area, hence reducing the losses due to the presence of the photoconductive switches. Preliminary results are shown. Siena, Italy October 2018

Prof. Stefano Maci

Parts of this thesis have been published in the following articles and chapters Journal Paper J1. M. Jr. Mencagli, C. Della Giovampaola, and S. Maci, “A Closed-form Representation of Isofrequency Dispersion Curve and Group Velocity for Surface Waves Supported by Anisotropic and Spatially Dispersive Metasurfaces,”, vol. 64, no. 6, pp. 2319–2327, 2016. J2. M. Jr. Mencagli, E. Martini, and S. Maci, “Transition Function for Closed-Form Representation of Metasurface Reactance,” IEEE Trans. Antennas Propag., vol. 64, no. 1, pp. 136–145, 2016. J3. E. Martini, M. Jr. Mencagli, D. González-Ovejero and S. Maci, “Flat Optics for Surface Waves,” IEEE Trans. Antennas Propag., vol. 64, no. 1, pp. 155–166, 2016. J4. E. Martini, M. Jr. Mencagli and S. Maci, “Metasurface Transformation for Surface Wave Control,” Phil. Trans. R. Soc. A, vol. 373, no. 2049, 2015. J5. D. González-Ovejero, E. Martini, B. Loiseaux, C. Tripon-Canseliet, M. Jr. Mencagli, J. Chazelas and S. Maci, “Basic Properties of Checkerboard Metasurfaces,” IEEE Antennas Wireless Propag. Lett., vol. 14, pp. 406–409, 2015. J6. M. Jr. Mencagli, E. Martini and S. Maci, “Surface Wave Dispersion for Anisotropic Metasurfaces Constituted by Elliptical Patches,” IEEE Trans. Antennas Propag., vol. 63, no. 7, pp. 2992–3003, 2015. J7. M. Jr. Mencagli, E. Martini, D. González-Ovejero and S. Maci, “Metasurfing by Transformation Electromagnetic,” IEEE Antennas Wireless Propag. Lett., vol. 13, pp. 1767–1770, 2014. J8. M. Jr. Mencagli, E. Martini, D. González-Ovejero and S. Maci, “Metasurface Transformation Optics,” Journal of Optics, vol. 16, pp. 125106–125114, 2014.

Book Chapter B1. G. Minatti, M. Faenzi, M. Jr. Mencagli, F. Caminita, D. González-Ovejero, C. Della Giovampaola, A. Benini, E. Martini, M. Sabbadini and S. Maci, “Metasurface Antennas,” in Aperture for mm Wave and sub-mm Wave Applications, Eds. A. Boriskin and R. Sauleau, Springer, in press.

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Conference Proceedings C1. C. Della Giovampaola, C. Tripon-Canseliet, J. Chazelas, M. Jr. Mencagli, G. Minatti, and S. Maci, “Digitizing Metasurface Antennas,” 2016 IEEE International Symposium on Antennas and Propagation, Puerto Rico, 2016. C2. M. Jr. Mencagli, E. Martini, C. Della Giovampaola and S. Maci, “Ray-Optics and Transformation Optics for Surface Waves,” 2016 IEEE International Symposium on Antennas and Propagation, Puerto Rico, 2016. C3. M. Jr. Mencagli, E. Martini, C. Della Giovampaola and S. Maci, “Transition Function for SW Metasurface-Dispersion,” 2016 IEEE International Symposium on Antennas and Propagation, Puerto Rico, 2016. C4. M. Jr. Mencagli, E. Martini, and S. Maci, “Transition Function for Describing Metasurface Dispersion,” 10th European Conference on Antennas and Propagation (EuCAP 2016), Davos, 2016. C5. M. Jr. Mencagli, E. Martini, D. González-Ovejero and S. Maci, “Two-dimensional Optics for Surface Waves,” 10th European Conference on Antennas and Propagation (EuCAP 2016), Davos, Switzerland, 2016. C6. M. Jr. Mencagli, E. Martini, D. González-Ovejero and S. Maci, “Surface Wave Control by Transformation Optics,” 2015 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting, Vancouver, BC, 2015. C7. M. Jr. Mencagli, D. González-Ovejero, E. Martini, B. Loiseaux, C. Tripon-Canseliet, J. Chazelas and S. Maci, “Optically Reconfigurable Metacheckerboard,” 2015 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting, Vancouver, BC, 2015. C8. M. Jr. Mencagli, E. Martini, D. González-Ovejero and S. Maci, “Transformation Optics SW-Based Devices,” 2015 9th European Conference on Antennas and Propagation (EuCAP), Lisbon, 2015. C9. M. Jr. Mencagli, D. González-Ovejero, E. Martini, B. Loiseaux, C. Tripon-Canseliet, J. Chazelas and S. Maci, “Optically Reconfigurable Metacheckerboard,” 2015 9th European Conference on Antennas and Propagation (EuCAP), Lisbon, 2015. C10. M. Faenzi, M. Jr. Mencagli, E. Martini, D. González-Ovejero and S. Maci, “Design of Modulated Metasurface Antennas based on Elliptical Patches,” 2015 9th European Conference on Antennas and Propagation (EuCAP), Lisbon, 2015. C11. M. Jr. Mencagli, E. Martini, F. Caminita, M. Faenzi and S. Maci, “Fast approach to the design of planar devices and antennas based on elliptical patch metasurfaces,” 8th International Congress on Advanced Electromagnetic Materials in Microwaves and Optics (Metamaterials 2014), Copenhagen, 2014.

Parts of this thesis have been published in the following articles and chapters

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C12. M. Jr. Mencagli, E. Martini, D. González-Ovejero, F. Caminita and S. Maci, “Efficient design of Transformation Optics Devices based on Anisotropic Metasurfaces,” General Assembly and Scientific Symposium (URSI GASS), 2014 XXXIth URSI, Beijing, 2014. C13. M. Jr. Mencagli, M. Faenzi, E. Martini and S. Maci, “Efficient Design of Electromagnetic Devices Using Modulated Elliptical Patch Metasurfaces,” Radio Science Meeting (Joint with AP-S Symposium), 2014 USNC-URSI, Memphis, TN, 2014. C14. M. Jr. Mencagli, S. Sørensen, E. Martini, C. Cappellin, R. Jørgensen, M. Zhou, M. Sabbadini and S. Maci, “Metasurface Approach for Contoured Beam Reflectors,” The 8th European Conference on Antennas and Propagation (EuCAP 2014), The Hague, 2014. C15. M. Jr. Mencagli, E. Martini, D. González-Ovejero and S. Maci, “Transformation Optics for Anisotropic Metasurfaces,” The 8th European Conference on Antennas and Propagation (EuCAP 2014), The Hague, 2014. C16. M. Jr. Mencagli, D. González-Ovejero, E. Martini, F. Caminita and S. Maci, “Closed-form Characteristic Basis Functions for Metasurfaces,” Radio Science Meeting (Joint with AP-S Symposium), 2013 USNC-URSI, Lake Buena Vista, FL, 2013. C17. E. Martini, M. Jr. Mencagli and S. Maci, “Addressing Surface Waves on Modulated Metasurfaces,” 2013 IEEE Antennas and Propagation Society International Symposium (APSURSI), Orlando, Florida, 2013, (invited).

Acknowledgements

This dissertation would not have been possible without the help of several individuals who in one way or another contributed in the preparation and completion of this study. First and foremost, my utmost gratitude to my supervisor, Prof. Stefano Maci, and my co-advisors Dr. Enrica Martini and Dr. Brigitte Loiseaux. Prof. Maci, I want to thank you for support, invaluable guidance, and mentorship. Your passion for research and positive attitude are extremely contagious. Enrica, thank you for the great skills and knowledge you have imparted on me. Thanks for being extremely patient with me and for clarifying all my doubts. I will always be grateful to you for your help, support, and kindness. Brigitte, thanks for giving me the opportunity to spend a period training in one of the most laboratories advanced of Europe, like that of Thales Research & Technology. I would also like to thank the members of the jury (Prof. Giuseppe Vecchi, Prof. Giuliano Manara, and Prof. Alessandro Galli) for their constructive comments and careful reading of the manuscript. I am grateful to my good friend, Dr. David González-Ovejero, who helped me a great deal with scientific research. His dedication to work has truly inspired me. His kindness will forever be of immense value to me. To all current and previous LEA members, thank you for making my time at the laboratory very enjoyable. Particularly, I am grateful to Alice Benini, Santi Concetto Pavone, Marco Faenzi, Valentina Sozio, Gabriele Minatti, Cristian Della Giovampaola, Francesco Caminita, Federico Puggelli, Giovanni Maria Sardi, Davide Rossi, and Stefano Frigerio. Thanks to all the people around me, especially to those who always supported me, telling me to never give up during hard times. My gratitude must be extended to everyone in my family, to those who left and to those who are still here, from aunts and uncles (Azelio, Marco, Guido, Corradina, Nada, Anna Maria) to cousins (Simone, Vanessa). Their love and encouragement have been invaluable. The most special appreciation is for my parents, Mario and Donatella, and my sister, Kelly, for their constant support and patience.

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Last but not least, I truly and deeply thank my wife, Eleonora, for her love and endless patience, and for being by my side, even when I was irritable and restless. I love you!

Contents

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2 Surface Wave Dispersion for Anisotropic Metasurfaces Constituted by Elliptical Patches . . . . . . . . . . . . . . . . . . . . 2.1 Analytic Basis Functions . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Problem Geometry . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Circular Patches . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Elliptical Patches . . . . . . . . . . . . . . . . . . . . . . . 2.2 Spectral Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 MoM Matrix Entries . . . . . . . . . . . . . . . . . . . . 2.2.2 Anisotropic MTS Impedance . . . . . . . . . . . . . . 2.2.3 Homogenized MTS-Impedance Approximation . 2.2.4 Dispersion Equation of the Dominant SW Mode 2.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Closed-Form Representation of Metasurface Reactance and Isofrequency Dispersion Curve . . . . . . . . . . . . . . . . . . . . 3.1 Dispersion Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Equivalent Capacitance . . . . . . . . . . . . . . . . . . . . . 3.1.2 Closed Form Formulas for Quasi-static Capacitance . 3.1.3 Frequency (Phasings) Regions . . . . . . . . . . . . . . . . 3.2 Low Frequency and Transition Regions . . . . . . . . . . . . . . . 3.2.1 Low Frequency Approximation . . . . . . . . . . . . . . . 3.2.2 Cardano’s Transition Function . . . . . . . . . . . . . . . .

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1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Background on Electromagnetic Metasurfaces 1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3.2.3 Simple Non-uniform Approximation . . . . . . . . . . . . . 3.2.4 Identification of the Limits of the Frequency Regions 3.3 FB Region Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 TE Dispersion Close to Its Cut-Off Frequency . . . . . . . . . . . 3.5 Comparison with Full-Wave Analysis . . . . . . . . . . . . . . . . . 3.6 Extension to Anisotropic MTS . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Approximation of the IDCs . . . . . . . . . . . . . . . . . . . 3.6.2 Phase and Group Velocities . . . . . . . . . . . . . . . . . . . 3.7 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Basic Properties of Checkerboard Metasurfaces . . . . . . . . . 5.1 Connected Patch Transmission Lines . . . . . . . . . . . . . . . 5.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 CBMS Transmission Lines . . . . . . . . . . . . . . . . . 5.2.2 CBMS Patch Antennas . . . . . . . . . . . . . . . . . . . 5.2.3 Preliminary Results for Optically Tunable CBMS 5.3 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 Flat Optics for Surface Waves . . . . . . . . . . . . . . . . . 4.1 Flat Optics for Isotropic MTSs . . . . . . . . . . . . . . 4.1.1 Eikonal Equation . . . . . . . . . . . . . . . . . . . 4.1.2 Ray Tracing and Phase Velocity . . . . . . . . 4.1.3 Poynting Vector . . . . . . . . . . . . . . . . . . . 4.1.4 Transport of Energy Equation . . . . . . . . . . 4.2 Flat Optics for Anisotropic MTSs . . . . . . . . . . . . 4.2.1 Eikonal Equation . . . . . . . . . . . . . . . . . . . 4.2.2 Poynting Vector . . . . . . . . . . . . . . . . . . . 4.2.3 Ray Tracing and Ray Velocity . . . . . . . . . 4.2.4 Transport of Energy Equation . . . . . . . . . . 4.3 Flat Transformation Optics . . . . . . . . . . . . . . . . . 4.3.1 Rays and Wavefronts . . . . . . . . . . . . . . . . 4.3.2 Conformal and Quasi-conformal Mappings 4.4 Comparison with Full-Wave Analysis . . . . . . . . . 4.5 Synthesis of Modulated IBC . . . . . . . . . . . . . . . . 4.5.1 Luneburg and Maxwell’s Fish-Eye Lens . . 4.5.2 Modified Luneburg Lens . . . . . . . . . . . . . 4.5.3 Beam Bender . . . . . . . . . . . . . . . . . . . . . 4.5.4 Beam Splitter . . . . . . . . . . . . . . . . . . . . . 4.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Author Biography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Appendix A: Asymptotic Evaluation of Grounded Slab Green’s Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6 Conclusion . . . . . . . . . . . . . . . 6.1 Summary of Contributions 6.2 Future Directions . . . . . . . References . . . . . . . . . . . . . . . .

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Appendix B: Derivation of the Validity Conditions for Homogenization of the MTS-Impedance . . . . . . . . . . . . 101 Appendix C: Solution of Third-Degree Algebraic Equation Relevant to Impenetrable Impedance of Isotropic MTS . . . . . . . . . . 103 Appendix D: MFE and LL Differential Geometry . . . . . . . . . . . . . . . . . . 105

Chapter 1

Introduction

1.1 Background on Electromagnetic Metasurfaces Metamaterials (MTMs) are artificial materials that are structured at a subwavelength scale to exhibit desired effective constitutive parameters. Achievable material parameters can include those not found in nature [1–3]. MTMs can be formed by periodic arrangements of many small inclusions in a dielectric host environment, so that the resulting effective medium possesses desired bulk properties. Metasurfaces (MTSs) [4] can be regarded as the two-dimensional equivalent of MTMs. Their increasing popularity is due, to a large extent, to the technological simplification that they offer with respect to volumetric MTMs. Namely, MTSs are planar structures, and thus more compact and exhibit lower losses than three-dimensional MTMs. In addition, one can find a vast variety of MTS applications, which cover all the regions of the electromagnetic spectrum [5–8]. At microwave frequencies, MTSs can be realized by printing a dense periodic texture of electrically small patch elements on a (grounded or ungrounded) dielectric slab. The basic assumption in the conventional analysis of MTSs is the periodical distribution of their constituent elements, with a small period compared to the wavelength. Under this condition, the MTS can be macroscopically described through impedance boundary conditions (IBCs) which relate the tangential components of the average electric and magnetic fields. Two types of MTSs can be distinguished, depending on the presence or absence of a ground plane below the dielectric slab. The case in which the ground plane is present is referred to as impenetrable MTS. In such case, the IBCs impose a relationship between the average tangential electric and magnetic fields at the interface with the free space [4, 9]. In absence of the ground plane one deals with penetrable MTSs, and the IBCs impose an impedance-type relationship between the electric and magnetic tangential fields on the two sides of the MTS [4, 10]. The impenetrable MTS is also known as Tensor Impedance Surface [11–13] and it is the configuration we will study in the remainder of this thesis. When the shape of the constituent elements is regular enough, the impedance is a scalar and the effect of the IBC is isotropic with respect to the direction of the supported mode’s © Springer Nature Switzerland AG 2019 M. J. Mencagli, Manipulation of Surface Waves through Metasurfaces, Springer Theses, https://doi.org/10.1007/978-3-030-14034-2_1

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1 Introduction

wavevector. When the shape contains asymmetric features, like slots, grooves or cuts [13] the impedance is a tensor, thus introducing an effect equivalent to anisotropy in volumetric media. Although many applications deal with periodic MTSs, aperiodic (modulated) MTSs have recently emerged as flexible and relatively simple solutions for: (1) the design of leaky-wave antennas [5, 14–18]; (2) the control of field transmission [19]; and (3) the manipulation of surface-waves (SWs) [20]. This thesis focuses mainly on the latter application. The first part of this dissertation presents an effective method to analyze MTSs constituted by metallic texture with certain geometries. This method can be applied to the design of MTS antennas, which are currently being developed for deep space communications [21], and other planar microwave devices that will be shown in this manuscript. In the second part of this thesis, a general methodology is developed for analyzing flat devices realized by using modulated MTSs, which opens new design possibilities for a large number of microwave devices based on the manipulation of SWs. Finally, a novel approach of reconfigurability, which is based on a class of checkerboard MTS, is explored.

1.2 Motivation In recent years, the desire of integrating electromagnetic devices onto the surfaces of vehicles, aircrafts, satellites and other existing platforms has attracted a lot of attention in modulated MTSs. To date, the most popular applications of modulated MTSs consists of designing low profile, high gain, planar antennas [5, 14, 16–18]. However, in this manuscript we will deal with another cutting-edge application of modulated MTSs to the design of a class of planar devices. Great strides have been made in the design of devices based on modulated MTSs, but some of the issues that arise in the design process can be further improved. Modulated MTSs are obtained by gradually changing the geometry and/or size of the elements in contiguous cells, while maintaining the period unchanged and very small in terms of a wavelength. The local value of the equivalent impedance is assumed equal to that of a periodic arrangement which matches the local geometry. For this reason, the availability of a fast and accurate method for the characterization of the periodic MTS is a key aspect in the design procedure. For an arbitrary MTS, a rigorous analysis can be performed through a spectral Method of Moments (MoM), as shown in [22]. For particular geometries of the constituent elements, approximate closedform expressions are available. This is the case for MTS consisting of square patches, inductive grids [23] or circular patches [24]. However, all these structures provide a quite isotropic equivalent impedance, whereas an anisotropic impedance tensor is required for a large number of applications [5, 14, 20, 25]. A simple anisotropic MTS is obtained by periodically printing small elliptical patches on a grounded dielectric slab. In this thesis, an efficient approach is presented for the accurate characterization of this kind of MTS. Although a printed element of elliptical shape represents a valid

1.2 Motivation

3

geometry for patterning anisotropic MTSs, it has some limitations. Namely, it is not able to provide an high range of anisotropy and high impedance values. Therefore, for MTS-based devices in which these characteristics are required, one needs to use another geometry for the constitutive element. To overcome this limitation, this thesis presents an analytical characterization of a generic anisotropic, spatially and frequency dispersive MTS, which is valid for constituent elements that possess at least two orthogonal symmetry axes. Although the above-mentioned formulations are an excellent tool for the design of leaky-wave antennas based on MTSs [13, 14, 16, 21], in this manuscript, they have been used to design modulated MTSs which allow manipulation the SW propagation. In the last years, several works on SW propagation on modulated MTSs have been published. However, a rigorous treatment of all the aspects relevant to SW 2D optics, like ray tracing, transport of energy, and ray velocity, has not been provided yet. This thesis discusses and compares two approaches for the analysis of SW propagation on modulated impedance surfaces. The first one is an extension of geometrical optics (GO) description of plane waves propagation in graded index materials to SWs supported by modulated, impenetrable, isotropic or anisotropic impedance boundary conditions. The second one is a flat version of transformation optics (TO). The first approach is quite general, and it extends to SWs the basic concepts of GO (raypath, ray-velocity, transport of energy), thus, resulting in an elegant formulation which enables a closed-form analysis of planar operational devices. On the other hand, the second approach is only applicable when the impedance modulation can be described through an appropriate coordinate transformation, and in these cases it allows one to conveniently determine ray paths without resorting to ray tracing. Devices such as polarization splitters [26, 27], beams-shifters [26, 28], collimators [26], beam-benders [26, 28] and cloaks [29, 30] based on TO concepts have been realized with metamaterial transmission lines or inside parallel plate wave guides by using anisotropic material parameters. Similar devices have been designed by using an impedance sheet [20]. Here, as a further step, the tensor impedance is also implemented using a MTS consisting of small patches printed on a grounded slab. The last part of this thesis focuses on a novel approach of reconfigurability. This feature is a fundamental asset for modern communication systems, which often seek to offer several functionalities in a confined volume. MTSs are especially attractive as support for reconfigurable devices, provided that they can be designed for controlling the propagation path of SW or for leaking SW power into space harmonics. This thesis presents an experimental demonstration of a reconfigurable transmission line (TxL) using a checkerboard MTSs (CBMS), which consists of a checkerboard-type layout, made of electrically small complementary metallic patches and apertures on a grounded slab. This structure offers the possibility of designing arbitrary TxL paths on the CBMS by dynamically changing the vertex connections. The CBMS TxL presents two important advantages over other slow-wave structures. First, the supported mode exhibits a very low dispersion over a very large bandwidth. Second, the vertices’ dimensions can be reduced to the micrometer scale without impacting the CBMS TxL characteristic impedance. Therefore, the propagation path of the supported mode can be directed by acting on an extremely small region at the metallic

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1 Introduction

patches’ vertices. This can be efficiently done by locally changing the effective conductivity of a photo-sensitive substrate by focusing a limited level of optical power on the area of interest. The use of photo-conductive switches is advantageous with respect to electronic switches, since they do not require bias lines. Preliminary results for optically tunable CBMS are presented.

1.3 Thesis Outline This thesis is organized in four main chapters. This introductory chapter has briefly introduced metasurfaces and it has outlined the main motivations and goals of this thesis. Chapter 2 studies a class of anisotropic MTS which consists of elliptical patches printed on a grounded slab, in order to provide an effective approach for the derivation of the 2D frequency wavenumber dispersion surface. These MTSs are important in the design of leaky-wave antennas and TO SW-based devices. The formulation resorts to an analytical expression of the currents excited on the constitutive element to define a reduced spectral MoM procedure with only three basis functions. An exact compact formula, which links the MoM matrix to the homogenized equivalent anisotropic impedance of the MTS, is derived. The formulation presented in this chapter has been found accurate and useful for designing MTS antennas and TO devices. Chapter 3 introduces a general representation of isotropic frequency dependent reactance which is valid along the dispersion curve of the relevant TM SW. This representation is written in terms of a transition function derived from a manipulation of the Cardano’s formula for 3rd degree algebraic equations. Throughout a large portion of the dispersion curve, this transition function depends on one parameter only, which is identified as an equivalent quasi-static capacitance. Approaching the Floquet–Bloch region, where many higher order Floquet-modes significantly interact with the ground plane, two additional parameters should be extracted from the fullwave data in order to complete the transitional representation of the reactance up to the upper boundary of the Brillouin region. This formulation is restricted to a generic isotropic reactance and for an anisotropic reactance when the direction of propagation is along a symmetry axis of the unit cell element. However, a generalization of the previous solution to arbitrary direction of propagation is shown. It results in a closedform representation of the isofrequency dispersion curves and group velocity with the only limitation that the printed elements possess at least two symmetry axes. This formulation is very useful to simplify the design of MTS antennas and flat TO devices. Chapter 4 presents an extension of the Flat Optics theory. The name Flat Optics (FO) has been introduced in a recent paper by Capasso’s group for denoting lightwave manipulations through a general type of penetrable or impenetrable MTSs. There, the attention was focused on plane waves, while here we treat SWs excited on impenetrable impedance surfaces. The space variability of the boundary conditions imposes a deformation of the SW wavefront, which addresses the local wavevector

1.3 Thesis Outline

5

along not-rectilinear paths. The ray paths are subjected to an eikonal equation analogous to the one for Geometrical Optics rays in graded index materials. For the first time, we present the basic relations between ray-paths, ray velocity and transport of energy for both isotropic and anisotropic boundary conditions. This leads to an elegant formulation which enables a closed form analysis of flat operational devices (lenses or beam formers), giving a new guise to old concepts. It is shown that when an appropriate transformation is found, the ray paths can be conveniently controlled without the use of ray tracing, thus simplifying the problem and leading to a flat version of TO, which is framed here in the general Flat Optics theory. Chapter 5 presents a new type of MTS, which consists of a checkerboard-type layout, made of electrically small complementary metallic patches and apertures. Depending on whether the patches’ vertexes are interconnected or not, the structure may support or not a propagating quasi-TEM mode. This feature offers the possibility of designing arbitrary transmission line paths on the CBMS by dynamically changing the vertex connections. An equivalent network model is proposed for the CBMS transmission line, and validated through simulated and experimental results. Furthermore, a preliminary experiment with a CBMS transmission line printed on a semiconductor substrate shows the possibility of reconfiguring the metasurface by photon injection. Chapter 6 draws conclusions and presents future research lines.

References 1. Veselago VG (1968) The electrodynamics of substances with simultaneously negative values of  and μ. Sov Phys Usp 10(4):509 2. Pendry JB (2000) Negative refraction makes a perfect lens. Phys Rev Lett 85:3966–3969 3. Shelby RA, Smith DR, Schultz S (2001) Experimental verification of a negative index of refraction. Science 292(5514):77–79 4. Holloway CL, Kuester EF, Gordon JA, O’Hara J, Booth J, Smith DR (2012) An overview of the theory and applications of metasurfaces: the two-dimensional equivalents of metamaterials. IEEE Antennas Propag Mag 54(2):10–35 5. Minatti G, Faenzi M, Martini E, Caminita F, Vita PD, González-Ovejero D, Sabbadini M, Maci S (2015) Modulated metasurface antennas for space: synthesis, analysis and realizations. IEEE Trans Antennas Propag 63(4):1288–1300 6. Kildishev AV, Boltasseva A, Shalaev VM (2013) Planar photonics with metasurfaces. Science 339(6125) 7. Chen P-Y, Soric J, Padooru YR, Bernety HM, Yakovlev AB, Alú A (2013) Nanostructured graphene metasurface for tunable terahertz cloaking. New J Phys 15(12):123029 8. Yu N, Aieta F, Genevet P, Kats MA, Gaburro Z, Capasso F (2012) A broadband, backgroundfree quarter-wave plate based on plasmonic metasurfaces. Nano Lett 12(12):6328–6333 9. Maci S, Minatti G, Casaletti M, Bosiljevac M (2011) Metasurfing: addressing waves on impenetrable metasurfaces. IEEE Antennas Wirel Propag Lett 10:1499–1502 10. Patel AM, Grbic A (2013) Effective surface impedance of a printed-circuit tensor impedance surface (PCTIS). IEEE Trans Microw Theory Tech 61(4):1403–1413 11. Patel AM, Grbic A (2013) Modeling and analysis of printed-circuit tensor impedance surfaces. IEEE Trans Antennas Propag 61(1):211–220

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12. Patel AM, Grbic A (2014) The effects of spatial dispersion on power flow along a printed-circuit tensor impedance surface. IEEE Trans Antennas Propag 62(3):1464–1469 13. Fong BH, Colburn JS, Ottusch JJ, Visher JL, Sievenpiper DF (2010) Scalar and tensor holographic artificial impedance surfaces. IEEE Trans Antennas Propag 58(10):3212–3221 14. Minatti G, Maci S, Vita PD, Freni A, Sabbadini M (2012) A circularly-polarized isoflux antenna based on anisotropic metasurface. IEEE Trans Antennas Propag 60(11):4998–5009 15. Minatti G, Caminita F, Casaletti M, Maci S (2011) Spiral leaky-wave antennas based on modulated surface impedance. IEEE Trans Antennas Propag 59(12):4436–4444 16. Faenzi M, Caminita F, Martini E, Vita PD, Minatti G, Sabbadini M, Maci S (2016) Realization and measurement of broadside beam modulated metasurface antennas. IEEE Antennas Wirel Propag Lett 15:610–613 17. Minatti G, Caminita F, Martini E, Maci S (2016) Flat optics for leaky-waves on modulated metasurfaces: adiabatic floquet-wave analysis. IEEE Trans Antennas Propag 64(9):3896–3906 18. Minatti G, Caminita F, Martini E, Sabbadini M, Maci S (2016) Synthesis of modulatedmetasurface antennas with amplitude, phase, and polarization control. IEEE Trans Antennas Propag 64(9):3907–3919 19. Pfeiffer C, Grbic A (2013) Millimeter-wave transmitarrays for wavefront and polarization control. IEEE Trans Microw Theory Tech 61(12):4407–4417 20. Patel AM, Grbic A (2014) Transformation electromagnetics devices based on printed-circuit tensor impedance surfaces. IEEE Trans Microw Theory Tech 62(5):1102–1111 21. Caminita F, Martini E, Minatti G, Maci S (2016) Fast integral equation method for metasurface antennas. In: 2016 URSI international symposium on electromagnetic theory (EMTS), pp 480–483 22. Maci S, Cucini A (2006) FSS-based EBG surfaces. In: Engheta N, Ziolkowski R (eds) Electromagnetic metamaterials: physics and engineering aspects. IEEE-Wiley, New York 23. Luukkonen O, Simovski C, Granet G, Goussetis G, Lioubtchenko D, Raisanen AV, Tretyakov SA (2008) Simple and accurate analytical model of planar grids and high-impedance surfaces comprising metal strips or patches. IEEE Trans Antennas Propag 56(6):1624–1632 24. Ramaccia D, Toscano A, Bilotti F (2011) A new accurate model of high-impedance surfaces consisting of circular patches. Prog Electromagn Res M 21:1–17 25. Martini E, Maci S (2014) Metasurface transformation theory. In: Werner DH, Kwon D-H (eds) Transformation electromagnetics and metamaterials. Springer, London, pp 83–116 26. Kwon DH, Werner DH (2010) Transformation electromagnetics: an overview of the theory and applications. IEEE Antennas Propag Mag 52(1):24–46 27. Kwon D-H, Werner DH (2008) Polarization splitter and polarization rotator designs based on transformation optics. Opt. Express 16(23):18731–18738 28. Rahm M, Cummer SA, Schurig D, Pendry JB, Smith DR (2008) Optical design of reflectionless complex media by finite embedded coordinate transformations. Phys Rev Lett 100:063903 29. Schurig D, Mock JJ, Justice BJ, Cummer SA, Pendry JB, Starr AF, Smith DR (2006) Metamaterial electromagnetic cloak at microwave frequencies. Science 314(5801):977–980 30. Edwards B, Alù A, Silveirinha MG, Engheta N (2009) Experimental verification of plasmonic cloaking at microwave frequencies with metamaterials. Phys Rev Lett 103:153901

Chapter 2

Surface Wave Dispersion for Anisotropic Metasurfaces Constituted by Elliptical Patches

As mentioned in the previous chapter, a simple anisotropic MTS can be obtained by periodically printing small elliptical patches on a grounded dielectric slab. Rotation of the ellipses with respect to the direction of SW propagation may provide control of the field polarization in circularly-polarized leaky-wave antennas or in TO SW-based devices. The analysis of periodic MTSs is based on the estimate of the dispersion equation for the dominant SW mode. In the low-frequency regime, this mode exhibits a dominantly Transverse Magnetic (TM with respect to the surface normal) field structure. MoM can be adopted for estimating the local dispersion. However, the conventional MoM approach may be not reliable in the low-frequency limit due to the deterioration of the MoM matrix condition number when the basis functions become too small in terms of a wavelength. This issue is addressed in literature by the use of a loop-star decomposition of basis functions [1]. The proposed approach relies on the consideration that the patch currents can be accurately reconstructed by properly superimposing few contributions which depend very weakly on frequency and wavenumber. On this basis, we define a reduced spectral MoM procedure with only three basis functions which are independent of frequency and phasing. In particular, these functions are inspired by the currents obtained from a Rao–Wilton–Glisson (RWG) MoM analysis [2] of circular patches; the generalization to elliptical shapes is performed by applying a proper stretching along one axis. Since these basis functions exhibit analytical expressions in both spatial and spectral domain, we will denote them as “analytic”. The proposed formulation uses the Green’s function (GF) of the grounded slab. This allows one to rigorously take into account the interaction with the ground plane, contrary to other approaches which provide simpler formulas whose accuracy, however, may decrease when the thickness of the slab is decreased and/or the magnitude of the transverse wavenumber is increased. As a result, the proposed approach provides an excellent accuracy in the description of the MTS dispersion characteristics all over the irreducible Brillouin region. At the same time, it provides intuitive physical insight and practical design-oriented equivalent circuits. The equivalent circuit derivation is performed here with a new formulation which, under © Springer Nature Switzerland AG 2019 M. J. Mencagli, Manipulation of Surface Waves through Metasurfaces, Springer Theses, https://doi.org/10.1007/978-3-030-14034-2_2

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“homogenization” conditions, allows one to extract the dominant quasi-static capacitances and inductances of the MTS. This chapter is articulated as follows. Section 2.1 presents the analytical expression of the basis functions for elliptical patches. Section 2.2 describes an effective spectral domain procedure for the derivation of the 2D dispersion surface of the MTS and formulates the extraction of the dominant capacitances and inductances at low frequency. Section 2.3 presents numerical results. Finally, Sect. 2.4 draws the conclusion of this chapter.

2.1 Analytic Basis Functions 2.1.1 Problem Geometry The geometry of the elementary cell of the MTS is shown in Fig. 2.1. A rectangular (x, y, z) reference system is assumed with the z axis orthogonal to the MTS and the origin at the MTS level. The MTS consists of a periodic arrangement of metallic elliptical patches printed on a grounded dielectric slab characterized by thickness h and relative permittivity εr . Since we are focused on microwave applications, we assume that the metal can be treated as a perfect electric conductor. Losses in the dielectric will be also neglected in the present analysis, but they may be included quite simply in the formulation. The unit cell of the texture possesses equal periods d in the two principal directions. Periodic boundary conditions are imposed in the two x and y directions with phasing k x d and k y d, respectively. The imposed phasing corresponds to a plane wave which is √ either propagating (k x2 + k 2y < ω 2 ε0 μ0 = k 2 ) or evanescent (k x2 + k 2y > k 2 ) along z, in order to treat reflection as well as dispersion phenomena. Although a circular patch can be seen as a particular case of an elliptical patch, for the sake of convenience

Fig. 2.1 Geometry of circular (a) and elliptical patches (b)

2.1 Analytic Basis Functions

9

we will treat the circular case first. The corresponding geometry is shown in Fig. 2.1a for completeness.

2.1.2 Circular Patches Examples of currents on a circular patch are illustrated in Fig. 2.2a, c, e. The currents have been obtained by applying the spectral-domain formulation presented in [3] with RWG basis functions. A small phase shift has been imposed across the unit cell along the x direction and the polarization has been imposed by forcing the tangential electric field. The current in Fig. 2.2a is obtained by forcing a TE mode on a circular patch MTS floating in free space; the analysis is done at a very low frequency λ = 10d. This current exhibits a loop-type (solenoidal) behavior. The currents in Fig. 2.2c, e are obtained by analyzing a circular patch MTS in the presence of the grounded slab at low frequency and they are relevant to the excitation of the dominant TM and TE Floquet mode, respectively. Both the TM-excited and the TE-excited currents actually present an irrotational and a solenoidal component and appear to be rotated by 90◦ with respect to each other. The three currents in Fig. 2.2a, c, e exhibit a singularity at the patch edge, described within the limit of discrimination of the RWG basis functions. The electric current distribution on the patch for an arbitrary excitation is assumed to be a linear combination of three frequency/wavenumber-independent basis functions, i.e., j (ρ) = I0 j0 (ρ) + I1 j1 (ρ) + I2 j2 (ρ) with, for ρ ≤ a j0 (ρ) =

√

ρ

ˆ φ

j1 (ρ) =

a 3 1−ρ2 /a 2  1 1 − ρ2 /a 2 a2

j2 (ρ) =

ρ2 /a 2

 1

a2

1−

cos (φ) ρˆ − sin (φ) ρˆ +

a2 a2

√sin(φ)2

1−ρ /a 2 cos(φ)

√

1−ρ2 /a

ˆ φ

(2.1)

ˆ φ 2

The three functions are zero for ρ > a. In (2.1), a is the patch radius, ρ = x xˆ + y yˆ = ρ(cos φˆx + sin φˆy) and ρ, φ are the coordinates of a polar reference system with the origin at the center of the patch (Fig. 2.1). It is noticed that the ρˆ component of the ˆ component exhibits a functions in (2.1) vanishes at the patch edge ρ = a, while the φ square root type singularity. The behavior of these functions is illustrated in Fig. 2.2b, d, f. As can be seen, they mimic well the behaviour of the numerical currents obtained from the MoM analysis with a large number of RWG basis functions, reported in Fig. 2.2a, c, e. We note that the loop function j0 (ρ) is important for the analysis of MTS suspended in free space or printed on ungrounded slab. On the other hand, it plays a minor role in the presence of the grounded slab. The Fourier spectra of the functions in (2.1) are expressed in the variable k = k x xˆ + k y yˆ = kρ (cos αˆx + sin αˆy)

(2.2)

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Fig. 2.2 Currents on a circular patch from a full wave analysis (a), (c), (e) and relevant closed form counterparts (b), (d), (f). a TE case in free space; c TM case on a grounded slab; e TE case on a grounded slab (the white arrow represents the direction of the phasing); b j0 (ρ); d j1 (ρ); f j2 (ρ)

  Notice that we have indicated with (ρ, φ) and kρ , α the polar coordinates in the spatial and spectral domain, respectively. The spectra of the functions in (2.1) can be written in closed form as

2.1 Analytic Basis Functions

11

Jn (k) = JT M,n (k) kˆ + JT E,n (k) α ˆ

(2.3)

α ˆ = zˆ × kˆ

(2.4)

JT M,0 (k) = 0, JT E,0 (k) = − j 21 kρ a f (kρ ) JT M,1 (k) = − cos (α) f (kρ ), JT E,1 (k) = − sin (α) g(kρ ) JT M,2 (k) = − sin (α) f (kρ ), JT E,2 (k) = cos (α) g(kρ )

(2.5)

where n = 0, 1, 2 and

with f (kρ ) = g(kρ ) =



sin(ak ) 4π cos(akρ ) − akρ ρ a 2 kρ2 2π sin(ak ) 1 f (kρ ) + akρ ρ 2

 (2.6)

We note that all the contributions are continuous at kρ = 0. The inverse Fourier transforms of JT M,n (k) kˆ = Jn (k) · kˆ kˆ and JT E,n (k) α ˆ = ˆα ˆ for n = 0, 1, 2 identify the spatial-domain irrotational (TM) and Jn (k) · α solenoidal (TE) components of jn (ρ). Furthermore, since j0 (ρ) is solenoidal, we have JT M,0 (k) = 0. For large kρ , the spectral-TM and TE components decay as kρ −2 and kρ −1 , respectively. Figure 2.3 shows a comparison between the spectral TE and TM components of J1 (k) and the corresponding numerical results. These latter are obtained by the following two steps: first, the problem of the MTS printed on a grounded slab is solved, for TM excitation, by applying the periodic spectral MoM formulation in [3] with RWG basis functions. Then, the current spectrum is constructed by using the resulting basis function coefficients to combine the spectra of the RWG basis functions [4]. Notice that when we talk about “TM excitation” we refer to the nature of the dominant mode; of course the higher order modes of the resulting current (that must fit the patch shape) will contain in general both TM and TE components. These latter are obtained by projection onto the unit vectors kˆ and α, ˆ respectively. The agreement between numerical and analytical currents is quite good, except for the TE component for large values of kρ . This discrepancy may be attributed to the fact that the numerical current reconstructed as a superimposition of RWG basis functions fails to correctly describe the rapid variation of the tangential current at the patch edge. This smoothing of the spatial current corresponds to a low-pass filtering of its spectrum. On the contrary, the analytical function possesses a square root-type singularity at the edge, which is the behavior expected on the basis of theoretical considerations, and corresponds to a wider spectrum.

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Fig. 2.3 Comparison between the current spectra provided by numerical simulations of a MTS (a)– (b) and the proposed analytical expression J1 (k) (c)–(d). a, c TM component; b, d TE component

2.1.3 Elliptical Patches The previous analytical functions defined on a circular domain can be used to obtain basis functions for elliptical patches. In the most general case, the elliptical patch will be characterized by the orientation angle ψ and the axes ratio η (see Fig. 2.1b); the (ψ,η) relevant basis functions will be therefore denoted as jn (ρ). We note that circular patches are included in this notation as the particular case ψ = 0, η = 1, namely j(0,1) (ρ) ≡ jn (ρ). n Consider first the case where the two axes of the ellipse are aligned with the principal axes of the lattice, x and y, with semi-axes a and b = ηa, respectively η < 1. This corresponds to the case ψ = 0 in Fig. 2.1b. It is observed from numerical simulations that the currents on an elliptical patch look like a stretched version of the currents on a circular patch, with a compression of a factor η along the y axis. Denoting the circular domain currents in (2.1) by their Cartesian components jx,n (x, y) , j y,n (x, y) , n = 0, 1, 2, the elliptical domain currents are given by

2.1 Analytic Basis Functions

13

Fig. 2.4 Elliptical domain functions for an ellipse of axial ratio η = 0.5, a = 0.45d and ψ = 30◦ . (ψ,η) (ψ,η) (ψ,η) (ψ,η) a j0 (ρ); b j2 (ρ); c j1 (ρ); d TM component of the spectrum JT M,1 (k)

j(0,η) (ρ) = jx,n (x, y/η) xˆ + η j y,n (x, y/η) yˆ n

(2.7)

with n = 0, 1, 2. The spectra of the functions in (2.7) are obtained by rewriting (2.3)   in terms k , Jy,n (k) = , k , i.e., J , k = J of rectangular spectral coordinates k (k) x y x,n x,n x y   Jy,n k x , k y , and then stretching the k y axis of a factor η:       Jn (0,η) (k) = η Jx,n k x , ηk y xˆ + η Jy,n k x , ηk y yˆ

(2.8)

The rotation of the ellipse of an angleψ is next accounted  for by introducing the rotation tensor R(ψ) = cos ψ xˆ xˆ + yˆ yˆ + sin ψ xˆ yˆ − xˆ yˆ ; thus, leading to the following expressions

R(ψ) · ρ (2.9) j(ψ,η) (ρ) = R(−ψ) · j(0,η) n n

Jn(ψ,η) (k) = R(−ψ) · Jn(0,η) R(ψ) · k

(2.10)

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2 Surface Wave Dispersion for Anisotropic Metasurfaces …

The spectral functions Jn (ψ,η) (k) can be rewritten in terms of spectral-TM (spatially irrotational) and spectral TE (spatially solenoidal) components by projection onto the unit vectors kˆ and α; ˆ namely (ψ,η)

ˆ JT M,n (k) = Jn (ψ,η) (k) · k,

(ψ,η)

JT E,n (k) = Jn (ψ,η) (k) · α ˆ

(2.11)

 (ψ,η) Figure 2.4 shows, for the case ψ = 30◦ η = 1 2, the representation of jn (ρ) and of (ψ,η) the TM component of JT M,1 (k) as provided by (2.9) and (2.10)–(2.11), respectively.

2.2 Spectral Analysis 2.2.1 MoM Matrix Entries Due to the periodicity of the problem, the numerical analysis is reduced to a single periodic cell by resorting to the Floquet theorem. The field is therefore expanded in series of Floquet waves (FWs) with wave vectors located at the “reciprocal lattice” nodes kq = k0 +

2πq y 2πqx xˆ + yˆ d d

(2.12)

where k0 · xˆ d and k0 · yˆ d are the phase shifts across the two parallel sides of the cell along x and y, respectively. The single index q denotes the couple of FW indices  qx ; q y = (0, ±1, ±2, . . . ; 0, ±1, ±2, . . .) and it is defined so that q = 0 corresponds to the dominant FW. We introduce the following unit vectors kq kˆ q =  ; α ˆ q = zˆ × kˆ q kq · kq

(2.13)

so that kˆ 0 identifies the direction of propagation of the dominant FW. Using a Galerkin spectral approach and the Poisson summation formula [3], the MoM mutual impedance takes the form MoM Z mn (k0 , ω) =

M 

    TM TE Z mn kq , ω + Z mn kq , ω

(2.14)

q=0

with (ψ,η)

(ψ,η)∗

TM Z mn (k, ω) = Z GT M F (k, ω) JT M,n (k) JT M,m (k) (ψ,η) (ψ,η)∗ TE TE Z mn (k, ω) = Z G F (k, ω) JT E,n (k) JT E,m (k)

(2.15)

2.2 Spectral Analysis

15

where m, n = 0, 1, 2, . . . , M + 1 is the number of FWs considered in the impleE are the TM and TE spectral components mentation of the MoM procedure, Z GT M,T F (ψ,η) (ψ,η) of the GF defined in Appendix A, JT M,n , JT E,n are defined in (2.10)–(2.11) and the asterisk denotes complex-conjugate. Equation (2.14) expresses, for any phasing k0 d, the entries of the 3 × 3 spatial domain MoM matrix in terms of a summation T M,T E of samples of the functions Z mn (k, ω) at the spectral points kq . The number of spectral points retained in the summation in (2.14) depends on the asymptotic decay T M,T E T M,T E of Z mn (k, ω) for kρ large. It is seen that Z mn (k, ω) for kρ decays as kρ −3 for kρ large for both the TM and the TE case. For the TM case, this behavior arises from the combination of the asymptotic orders kρ −2 and kρ of the spectral currents and GF spectra, respectively; for the TE case, it arises from the order kρ −1 of the spectral currents and kρ −1 of the GF spectrum. The computation of (2.14) requires the evaluation of M + 1 terms for any frequency. In order to speed up the calculation of the MoM entries it is convenient to T M,T E∞ extract the GF asymptotic spectral value Z mn (k, ω), thus obtaining the following function X X X,∞ ζmn (k, ω) = Z mn (k, ω) − Z mn (k, ω)

(2.16)

where X = T M, T E and the various symbols are defined in Appendix A. Based on this definition, the following decomposition of the MoM matrix is introduced   [Z MoM ] = [Z slab ] + Z dyn + [Z L F ]

(2.17)



TM TE [Z slab ] = Z mn (k0 , ω) + Z mn (k0 , ω) n,m=0,1,2

(2.18)

where



Z dyn



⎧ ⎫ M ⎨ ⎬  TM     TE = ζmn kq , ω + ζmn kq , ω ⎩ ⎭ q=1

n,m=0,1,2

 [Z L F ] =

(2.19)



1 jωcmn (k0 )

+ jωlmn (k0 )

(2.20) n,m=0,1,2

with ε0 (εr +1)/2 cmn (k0 )

=

lmn (k0 ) μ0

1 2

=

1 2

M 

M   (ψ,η)   (ψ,η)∗   kq · kq JT M,n kq JT M,m kq

q=1

q=1

JT E,n (kq ) JT E,m (kq ) (ψ,η)

√

(ψ,η)∗

kq ·kq

(2.21)

16

2 Surface Wave Dispersion for Anisotropic Metasurfaces …

T M (a) and ζ T M (b) for the case of a Fig. 2.5 Log-scale amplitude of the spectral functions Z 11 11 circular patch with a = b = 0.45d

We note that the extraction of the asymptotic contribution is performed for all spectral samples except for q = 0. The low frequency contribution [Z L F ] comes from the spectral asymptotic (spatial quasi-static) GF, and its extraction allows one to define the quantities cmn and lmn that are related to the capacitances and inductances of the quasi static equivalent network of the MTS, as will be shown in the next section. The term [Z slab ] is the one relevant to the dominant FW; it is directly related to the GF of the environment and therefore it √contains the possible singularities due to the grounded slab GF, which impact when k0 · k0 approaches  the mode wavenumber of the bare grounded slab. The residual  contribution Z dyn does not contain grounded-slab induced singularities and may impact only at high frequency close to the resonance of the MTS element, namely when d approaches a half effective wavelength.  It plays a fundamental role close T M,T E inside Z dyn asymptotically decay as kρ −5 , to the band-gap. The functions ζmn T M,T E while the functions Z mn decay as kρ −3 . For this reason, a reduced number of spectral samples (M) is required to evaluate the summation in (2.19) with respect to the one in (2.14). As an example, Fig. 2.5 shows the logarithmic scale amplitude of TM TM Z 11 (k, ω) and ζ11 (k, ω) at 16 GHz. The white dots denote the reciprocal lattice TM points kq for k0 = 0. As expected, ζ11 (k, ω) possesses a reduced spectral bandwidth, and it is evident that only a few points correspond to a non-negligible value of TM the function ζ11 (k, ω). In particular, for the evaluation of the summation in (2.19) the inclusion of 16 spectral points is sufficient to reduce the relative error to the same value obtained in the evaluation of (2.14) by considering around 170 spectral points. The computation of the quantities cmn (k0 ) , lmn (k0 ) in Eq. (2.21) requires approxTM imately the same number of samples as the computation of Z mn (k, ω); however, these quantities are frequency independent and therefore can be calculated once for all.

2.2 Spectral Analysis

17

Fig. 2.6 Two-port network model for the MTS

2.2.2 Anisotropic MTS Impedance From the MoM formulation, we can derive the Floquet wave-based two-port network in Fig. 2.6 [3]. In this network, the shunt load characterized by the 2 × 2 impedance matrix represents the anisotropic impedance and the section of transmission line terminated in a short circuit represents the grounded slab. This anisotropic impedance establishes the tensorial relationship between an average transverse electric field and the jump-discontinuity of the average transverse magnetic field at the MTS. We will see here that this matrix assumes a particularly appealing form in the range of frequency where the MTS can be homogenized. In order to derive [Z M T S ] as a function of [Z MoM ] it is convenient to project the MoM currents onto the dominant (q = 0) TM and TE Floquet modes through the following projection matrix   (ψ,η) (ψ,η)

 0 JT M,1 (k0 ) JT M,2 (k0 ) = Q i, j i=1,2 (2.22) [Q] = (ψ,η) (ψ,η) (ψ,η) j=0,1,2 JT E,0 (k0 ) JT E,1 (k0 ) JT E,2 (k0 ) It can be seen [3] that −1  − [Z G F ] [Z M T S ] = [Q] [Z MoM ]−1 [Q] H where H stands for transpose conjugate and   TM 0 Z G F (k0 , ω) [Z G F ] = 0 Z GT EF (k0 , ω)

(2.23)

(2.24)

18

2 Surface Wave Dispersion for Anisotropic Metasurfaces …

Equation (2.23) arises from the identification, through the network in Fig. 2.6, of the field radiated by the MoM currents in presence of the slab with the sum of the field radiated by the MTS modal currents and the field scattered by the grounded dielectric slab. Equation (2.23) can be inverted via the Moore–Penrose pseudoinverse of [Q]; thus avoiding the direct inversion of [Z MoM ], which is ill conditioned close to the dispersion curve.

2.2.3 Homogenized MTS-Impedance Approximation If we are just interested in studying the modes supported by the elliptical MTS printed on the grounded slab, it is possible to consider just the two basis functions j1 and j2 . In fact, in this case, the contribution of the function j0 is negligible close to the dispersion curve. ˜ Under this assumption,   we can consider in (2.23) the 2 × 2 matrices [ Z MoM ] =

MoM  ˜ Z nm n,m=1,2 and Q = Q i, j i j=1,2 obtained by suppressing the 0-indexed entries in (2.17) and (2.22), respectively. Then, after distributing the inverse operator, the following expression is obtained   −1    −1 H Z˜ MoM Q˜ − [Z G F ] [Z M T S ] = Q˜

(2.25)

Notice that Eq. (2.25), contrary to (2.23), does not require the inversion of the MoM matrix, and therefore can be directly applied also on the dispersion curve. It can be shown (see Appendix B) that, when only the dominant (q = 0) Floquet modes are accessible to the ground plane, [Z M T S ] may be approximated by the  “homogeneized MTS” matrix Z hom M T S defined as follows   −1   H −1   ˜ Z˜ L F Q˜ [Z M T S ] ≈ Z hom MT S = Q

(2.26)

  where Z˜ L F is the 2 × 2 matrix obtained from [Z L F ] by suppressing the 0-indexed row and column. We notice that the equivalent impedance in (2.23) depends on the slab thickness h through the GF impedances (see Eq. A.1 in Appendix A). On the contrary, h does not appear in Eq. (2.26). This is due to the fact that this latter expression has been derived under the hypothesis that only the dominant mode interacts with the ground plane, so that the influence of the slab thickness can be taken into account by incorporating the MTS impedance into the equivalent network of Fig. 2.6. The approximation in (2.26) can be straightforwardly associated with a circuit network involving quasi-static capacitances and inductances by using (2.21)    hom  1 + jωL l,k (k0 ) (2.27) Z MT S = jωCl,k (k0 ) l,k=1,2

2.2 Spectral Analysis

19

where the expressions of Cl,k (k0 ) , L l,k (k0 ) as a function of cn,m (k0 ), ln,m (k0 ) in (2.21) are reported in Appendix B. In the same Appendix,  it isshown that the range of validity of (2.27) can be determined by imposing [Z dyn ]  [Z L F ] , which, for λsw > λ/2, leads to the following conditions d (1 + δ) < λ

for h/d > τ

d < λ for h/d < τ (2 − 0.23d/ h)

(2.28) (2.29)

 where λ is the free-space wavelength, δ = π (εr 2 + 1)/[2 (εr + 1)],  and τ = 0.23 (1 + δ) / (1 + 2δ). These conditions are obtained by imposing [Z dyn ] 

[Z L F ] . δ ranges from 2.2 to 6.7 when ranges from 1 to 10. Equations (2.28), (2.29) establish conditions for which the MTS can be homogenized through the use of the capacitances and inductances in (2.27). The validity of these conditions goes beyond the particular patch geometry we are dealing with and it is one of the main achievements of this work. In fact, in practical MTS application (2.28), (2.29) are often fulfilled. However, this approximation is not valid when approaching the bandgap. In fact, there neither [Z dyn ] nor the zero-order function j0 can be neglected and the exact version (2.23) should be used.

2.2.4 Dispersion Equation of the Dominant SW Mode The SW dispersion equation ω = ω(k0 ) can be obtained by setting to zero the determinant of the MoM matrix, namely by imposing det[Z MoM (k0 , ω)] = 0

(2.30)

We notice that for k x2 + k 2y > k 2 all the FWs are attenuated along z and the dominant one (q = (0, 0)) is identified as the SW mode supported by the MTS. In most of the cases, more than one solution ω = ωi (k0 ) (i = 0, 1, . . .) can be obtained from (2.30). The low frequency solution ω = ω0 (k0 ) is relevant to a TM-dominant mode, which does not have a cut-off frequency. In practical engineering applications involving MTS [5–7] the focus is concentrated on the isofrequency dispersion curve (IDC) of this quasi-TM mode. This curve is the locus described by the tip of k0 when the angle of propagation α is changed (see Fig. 2.7). It is found value of  by setting a √ 2 2 ω and finding, for different angles α the value of kρsw = k x0 + k y0 = k0 · k0 which makes zero the determinant in (2.30). For any given value of frequency, the isofrequency curve is defined in polar coordinates by the relation kρsw = kρsw (α). An equivalent form of the dispersion equation is obtained by imposing the resonance  of the circuit in Fig. 2.6. Since the matrix Z G F represents the impedance seen

20

2 Surface Wave Dispersion for Anisotropic Metasurfaces …

Fig. 2.7 IDC (dashed line) for the dominant SW quasi-TM mode propagating on the MTS

by the  MTS level, the resonance condition can be written as  dominant FW at the det [Z M T S ]−1 + [Z G F ]−1 = 0 or, in an equivalent form, as det ([Z M T S ] + [Z G F ]) = 0

(2.31)

 It can be seen from (2.23) that the condition above is equivalent to det [Q] −1 = 0, and hence to (2.30), since the rank of [Q] is equal to 2. [Z MoM ]−1 [Q] H When (2.28) are satisfied, the use of the approximate expression (2.26) in (2.31) may significantly speed up the numerical computation since the frequency dependence in (2.26) is given in explicit form. When the conditions in (2.28), (2.29) are not satisfied, dispersion curves obtained from the approximate expression may be used as an initial guess for the exact dispersion equation (2.30) or (2.31).

2.3 Numerical Results In this section, numerical results are presented to verify the effectiveness of the proposed formulation as well as to investigate the electromagnetic behavior of elliptical patch metasurfaces. In all the following results, d is 3 mm, and a = 0.45d; when not otherwise stated, the dielectric slab has thickness h = 1 mm and relative permittivity εr = 9.8. Although the emphasis in this thesis is on SWs, we first show that the proposed formulation can be conveniently used also to predict the scattering from an elliptical patch MTS. This has been implemented by using (2.23) along with the network representation in Fig. 2.6. Figure 2.8 presents a comparison between the results obtained with the proposed approach and those provided by the commercial software CST [8] for the phase of the TM-TE reflection coefficients. In Fig. 2.9, the dispersion curves for the dominant SW mode obtained by solving (2.31), with given by (2.25), have been successfully compared with the results of the CST eigensolver. We note that convergence of CST results requires upper boundary conditions sufficiently far away from the MTS and a sufficiently fine mesh. This significantly increases the calculation time. On the other hand, after pre-calculating

2.3 Numerical Results

21

Fig. 2.8 Phase of the reflection coefficient from a MTS consisting of circular patches, as a function of the plane wave incidence angle at different frequencies: comparison between CST results (squares) and the proposed formulation (continuous and dashed lines)

and interpolating in k-space the quantities in (2.27), our solutions are obtained in less than 10 s. Without pre-interpolation, it requires less than two minutes. Figure 2.9a represents the dispersion diagram for different eccentricities of the ellipse, starting from a dipole-like patch and ending to a circular patch. The SW propagation direction is in all the cases parallel to the ellipse’s major axis. The dispersion curves smoothly move toward lower frequencies when the thin dipole becomes a circular patch. It is noted that the accuracy of the proposed method is remarkable also for the thin dipole case. Figure 2.9b shows the results obtained fordifferent angles α of propagation of the SW, for an ellipse characterized by η = 1 2 with the major axis aligned with the x-axis. Figure 2.9b shows instead results for different angles ψ of ellipse rotation for a SW propagating along the x-axis. We observe that the dispersion curves in the two cases are quite similar to each other; that is ω(kρsw , α, ψ) ≈ ω(kρsw , 0, ψ − α). The small difference is due to the influence of the rectangular lattice. This effect is particularly clear when looking at the IDCs as those in Fig. 2.10. This kind of representation illustrates, for different frequencies, the dependency of the dispersion curve on the propagation direction and it is very important for the characterization of anisotropic MTS to be used in the design of antennas or microwave devices [9, 10]. Figure 2.10a, b correspond to ellipses characterized by ψ = 0◦ and ψ = 30◦ , respectively. At low frequencies, the dispersion curves are quite similar (except for the 30◦ rotation, since the symmetry axes of the dispersion curves are aligned with the axes of the ellipse) and the wavenumber kρ is larger propagation along the major axis of the ellipse. At higher frequency, the dispersion curve is deformed by the influence of the square lattice. A very thin ellipse is considered in Fig. 2.10c; it shows a qualitative behavior similar to the one in Fig. 2.10a. These plots also provide information on the direction of power flow, which is orthogonal to the IDCs. However, this aspect will be properly addressed in the next chapter. Basically, the deviation of

22

2 Surface Wave Dispersion for Anisotropic Metasurfaces …

Fig. 2.9 Comparison between the dispersion curves provided by the solution of (2.30) and the numerical results of CST (dots). a Various eccentricities of the ellipse and fixed direction of phasing (along x); b ellipse with b = a/2 and major axis aligned with the x-axis for several propagation angles α, c ellipse with b = a/2 rotated at different angles ψ and phasing along x

these curves from the circular shape is a measure of the anisotropy of the MTS: As can be seen, in all the three cases the MTS is approximately isotropic at low frequency, but the dependency on the direction becomes significant at higher frequencies.

2.3 Numerical Results

23

Fig. 2.10 IDCs for different elliptical metasurfaces. Colormaps represent, for any spectral point, the frequency of the dominant SW. a η = 0.5 and ψ = 0◦ ; b η = 0.5 and ψ = 30◦ ; c η = 0.1 and ψ = 0◦

Figure 2.11 shows a comparison between the IDCs obtained by using in (2.31) the matrix [Z M T S ] from (2.31) and the ones obtained by using its approximate form [Z hom M T S ] in (2.27). Failure of conditions (2.28)–(2.29) deteriorates the accuracy of the approximation. Note that condition (2.28) for the case under consideration, (characterized by d = 3 mm, r = 9.8 and h = 1 mm) implies λ < 2.3 cm, i.e. f < 13 GHz. The dispersion curves in Figs. 2.10 and 2.11 provide comprehensive information on the phase and group velocity of the SW supported by the MTS; however, they do not provide any information on the polarization. The polarization nature of the dominant SW is illustrated in the colored map in Fig. 2.12 which represents the log-scale ratio between the amplitude of the TE and TM components of the average tangential electric field of the dominant SW mode. This coincides with the ratio between the TE and TM components of the eigenvector of [Z M T S ]−1 + [Z G F ]−1 associated with the relevant vanishing eigenvalue. In the plot, the blue and red colors correspond to a dominantly TM and dominantly TE mode, respectively. We observe that, when the homogenization conditions (2.28), (2.29) are respected ( f < 13 GHz) and also

24

2 Surface Wave Dispersion for Anisotropic Metasurfaces …

Fig. 2.11 Comparison between the IDCs obtained by using Eq. (2.30) (continuous line) and  hom  Z M T S in Eq. (2.30) (dots); d = 3 mm, h = 2 mm, the ellipse has η = 0.5 and ψ = 0◦

Fig. 2.12 Colormap of the spectral plane representing the ratio between the TE and the TM components of the average tangential electric field of the dominant SW. The blue and red color are referred to the dominance of TM and TE components, respectively. The ellipse is characterized by η = 0.5 and ψ = 0◦ . White lines replicates the isofrequency dispersion lines in Fig. 2.10a

beyond, the polarization is dominantly TM for any propagation direction. At higher frequencies the mode becomes hybrid (neither purely TM not purely TE) outside the symmetry planes. In the symmetry planes the MTS matrix becomes diagonal and the mode cannot be hybrid. In the case considered in Fig. 2.12 for α = 90◦ the mode changes its nature from TM to TE around the frequency of 15 GHz. This particular behavior has been also verified by looking at the field distribution of the dominant eigenmode in CST. We emphasize that the frequency range of interest for MTS applications (TO devices or holographic antennas) is typically where the SW mode is dominantly TM, and it satisfies the conditions in (2.28), (2.29).

2.4 Chapter Summary

25

2.4 Chapter Summary An effective approach has been presented for the characterization of metasurfaces consisting of elliptical patches printed on a grounded dielectric slab. It relies on an analytic expression of the currents flowing on the patches, and provides a compact closed form expression of the equivalent surface impedance of the MTS. An expression of the low-frequency capacitances and inductances has also been provided, along with the relevant range of applicability. These expressions can be extended to more general cases, since they derive from a general relationship between the MTS equivalent network representation and the MoM matrix. Numerical results have demonstrated the accuracy of the proposed formulation in the prediction of the dispersion equation of the MTS.

References 1. Vecchi G (1999) Loop-star decomposition of basis functions in the discretization of the EFIE. IEEE Trans Antennas Propag 47(2):339–346 2. Rao S, Wilton D, Glisson A (1982) Electromagnetic scattering by surfaces of arbitrary shape. IEEE Trans Antennas Propag 30(3):409–418 3. Maci S, Cucini A (2006) FSS-based EBG surfaces. In: Engheta N, Ziolkowski R (eds) Electromagnetic metamaterials: physics and engineering aspects. IEEE-Wiley, New York 4. Vipiana F, Polemi A, Maci S, Vecchi G (2008) A mesh-adapted closed-form regular kernel for 3d singular integral equations. IEEE Trans Antennas Propag 56(6):1687–1698 5. Minatti G, Caminita F, Casaletti M, Maci S (2011) Spiral leaky-wave antennas based on modulated surface impedance. IEEE Trans Antennas Propag 59(12):4436–4444 6. Minatti G, Maci S, Vita PD, Freni A, Sabbadini M (2012) A circularly-polarized isoflux antenna based on anisotropic metasurface. IEEE Trans Antennas Propag 60(11):4998–5009 7. Patel AM, Grbic A (2014) Transformation electromagnetics devices based on printed-circuit tensor impedance surfaces. IEEE Trans Microw Theory Tech 62(5):1102–1111 8. Computer Simulation Technology (2012) CST microwave studio, Darmstadt, Germany 9. Patel AM, Grbic A (2013) Modeling and analysis of printed-circuit tensor impedance surfaces. IEEE Trans Antennas Propag 61(1):211–220 10. Patel AM, Grbic A (2013) Effective surface impedance of a printed-circuit tensor impedance surface (PCTIS). IEEE Trans Microw Theory Tech 61(4):1403–1413

Chapter 3

Closed-Form Representation of Metasurface Reactance and Isofrequency Dispersion Curve

The first part of this chapter proposes a formulation for the description of the equivalent reactance of MTS constituted by periodic small patches printed on a grounded dielectric slab. As already mentioned in the previous chapters, the printed elements are assimilated to homogenized scalar boundary conditions that can be represented by the circuit in Fig. 3.1. This scalar description is valid for printed elements with symmetric shapes (e.g. circular or square patches) and also for more complex elements with two axes of symmetry, provided the direction of propagation matches one of the symmetry axes of the printed elements (Fig. 3.1). A systematic way to extract a circuit description by a full-wave analysis is given by the pole-zero matching technique [1], which requires the use of a Foster type expansion of the patch reactance. Derivation of a quasi-analytical representation of the circuit elements as a function of phasing is shown in the previous chapter for elliptical patches. For a large variety of patches with different shapes, the final dispersion curves are surprisingly similar each other, inducing the idea that the frequency behavior of the equivalent impedance can be synthesized by means of few equivalent parameters throughout the entire first  Brillouin zone. The component Cd kρ in the equivalent circuit in Fig. 3.1 is much simpler than the circuit expansion associated with the Floquet modes and is not intended to be as the capacitance associated with the dominant term of the Floquet mode expansion. It is instead an equivalent fictitious component, which accounts for the actual boundary conditions imposed by the metallic patches in a synthetic way, or, more appropriately, for the reactive energy stored in the evanescent Floquet modes, also when they interact with the ground plane. For small shifts across  phase  the periodic cell, the wavenumber-dependent capacitance Cd kρ should coincide with the quasi-static capacitance of the layer of distributed patches. We will see in the next that increasing the phasing across the unit cell, this equivalent capacitance remains significantly constant in a large portion of the first Brillouin zone. Close to the Floquet–Bloch (FB) region, namely where the SW half-wavelength is close to the periodicity, the higher order Floquet modes become significant all  suddenly  together, and one should expect a drastic increase of Cd kρ . This chapter initially formulates a transition function that patches up the asymptotic behavior of the equivalent reactance along the various characteristic regions of the dispersion curve, from © Springer Nature Switzerland AG 2019 M. J. Mencagli, Manipulation of Surface Waves through Metasurfaces, Springer Theses, https://doi.org/10.1007/978-3-030-14034-2_3

27

28

3 Closed-Form Representation of Metasurface Reactance …

Fig. 3.1 Description of the dominant TM SW propagation. a Geometry of the physical structure, b equivalent network

the low-frequency region to the FB region. We will see that, in a large part of the dispersion diagram, this transition function can be obtained from the knowledge of the characteristics of the substrate (permittivity and thickness), and of just one parameter independent of phasing, which is the quasi-static capacitance. It will not be presented here a general way to express the key quasi-static capacitance as a function of the geometry of the patches, since it can be obtained from a single run of a full wave eigensolver. However, for simple geometries one can also approximate it by a closed form expression [2–4]. The impact of the present formulation is quite important in practice, since in most of the applications concerning isotropic MTSs (see in particular [5]) one needs accurate expression of the equivalent impedance as a function of frequency, especially in a transition range where the low frequency approximation is invalid and most of the available formulas fail. In fact, in this range, the dependence of the reactance on the geometrical parameters and on frequency is sensitive enough for allowing design flexibility, but not so fast (like close to the FB region) to render the design difficult to control. The last part of this chapter generalizes the above-mentioned formulation to anisotropic and spatially dispersive elements and to any direction of SW propagation, with the only limitation that the printed elements possess at least two symmetry axes. The generalization allows in a closed-form representation of the isofrequency dispersion curve and of a group velocity as a function of two parameters only, which are the equivalent quasi-static capacitances along the symmetry directions of the geometry. This chapter is articulated as follows. Section 3.1 sets up  the  dispersion equation and discusses the dynamic behavior of the element Cd kρ . Sections 3.2 and 3.3 derive the solution for low frequency region and Floquet–Bloch region. In particular, Sect. 3.2 presents a single parameter transition function, formulated on the basis of the Cardano’s solution formula for 3rd degree algebraic equations. Section 3.3 uses the same transition function to cover the Floquet–Bloch region by introducing two additional parameters. Section 3.4 presents a very simple formula to estimate the cutoff frequency of the higher order TE mode. Section 3.5 shows the accuracy of the formulas in several examples. Section 3.6 extends the approach to arbitrary directions

3 Closed-Form Representation of Metasurface Reactance …

29

of propagation and provides the expression of phase and group velocities. Section 3.7 presents numerical results relevant to four types of patches with two symmetry axes. Section 3.8 concludes the chapter.

3.1 Dispersion Equation Let us assume that the patches are printed on a grounded slab of thickness h, and relative permittivity r . Although the following analysis is valid for any relative permittivity larger than unity, the examples will be mostly relevant to high values of permittivity, as they are the most useful in practice. In the following, we denote with kρ the propagation constant of the SW. The transverse resonance equation relevant to the dominant TM mode derived by the circuit in Fig. 3.1 is −

  εr k 1 + cot (k z1 h) − ωCd kρ = 0 X ζk z1

(3.1)

  In (3.1), k z1 = k εr − kρ2 /k 2 = k εr − 1 − (X/ζ)2 is the z-propagation constant in the dielectric and  kρ2 − 1 = −X T M (3.2) X =ζ k2 is the resonant reactance, which is equal in amplitude an opposite in sign to the TM characteristic reactance X T M . Equation (3.1) is the implicit form of the dispersion equation of the dominant SW. The second term in (3.1) is the contribution of the ground plane seen at the interface through a short piece of transmission line with characteristic impedance given by k z1 ζ/(εr k). It is understood and implied throughout the chapter that once X is determined, the dispersion equation kρ = kρ (ω) is found from (3.2) as   2 X kρ = k 1 + (3.3) ζ

3.1.1 Equivalent Capacitance The quasi static capacitance Cd (0) = C0 that the patches exhibit at zero frequency and wavenumber is conveniently expressed through the following expression C 0 = ν0 ε 0 d

(εr + 1) 2

(3.4)

30

3 Closed-Form Representation of Metasurface Reactance …

where d represents the period of the lattice and ν0 is a scalar non-dimensional “shape factor” whose value mainly depends on the shape of the patches and only weakly on the substrate. Indeed, ν0 becomes actually independent of the substrate when the thickness becomes comparable with the periodicity, namely when only the dominant Floquet mode interacts with the ground plane. Typical values of ν0 range from few tenths to one. This parameter remains quite stable in a large portion of the dispersion curve. Approaching the frequency at which kρ d = π, the capacitance changes drastically following the behavior   C d kρ =

1−

C0 n kρ d γπ

(3.5)

where γ is a dimensionless parameter close to one. This phenomenon is due to the global resonance of the lattice, that renders the higher order Floquet modes significantly interacting with  the ground plane. In (3.5), the exponent n is a real number that depends on the ratio d h, and its value should be identified by numerical fitting of the dispersion curve. Figure 3.2a shows the dynamic capacitance as a function of the phasing across the periodic cell kρ d ∈ (0, π) for elliptical patches with different ratios between the semi-axes printed over a dielectric slab with relative permittivity equal to εr = 9.8 and thickness h = 1 mm with periodicity d = 3 mm. The continuous lines represent the value of νd = ν0 Cd /C0 obtained from a full-wave MoM analysis discussed in Chap. 2, where C0 is given in (3.4). Dotted lines represent instead the value obtained from (3.5) with n = 4 and γ = 1.09. Note that we have used the same value for these two parameters. Figure 3.2b shows the same quantities as a function of frequency, where the relationship between frequency and phasing is given by the numerical dispersion curve. The curves appear even more flat on a large frequency range, therefore, the reactance of the printed elements appears to be like a low pass filter of order 12 with cut-off frequency at γ  ω0 , where ω0 is the angular frequency at which kρ d = π and γ  = γ 3/4 = 1.067; i.e., the behavior of the dynamic capacitance 

12 −1 is of the type Cd = C0 [1 − ω/ γ  ω0 ] . The previous example evidences the steep slope of the dynamic capacitance. This preliminary example is propaedeutic to the next formulation, and will be furtherly investigated in Sect. 3.3. Before proceeding further, we emphasize that, for a given slab permittivity, the cell-phasing at which the increase  of the dynamic capacitance starts to become significant depends on C0 and on d h, but it is always contained in the range 120◦ –160◦ .

3.1.2 Closed Form Formulas for Quasi-static Capacitance Forsomesimplegeometries,(i.e.squarepatches,circularpatches,self-complementary geometries), the quasi-static capacitance can be approximated through (3.4) with

3.1 Dispersion Equation

31

Fig. 3.2 Normalized capacitance as a function of a phasing across the unit cell and b frequency for printed ellipses with various axial ratios a/b = η and h = 1 mm, εr = 9.8, d = 3 mm, a = 0.9d

ν0 =

2 ln 1/ sin(πge f f /(2d)) π

(3.6)

and ge f f is an equivalent gap in the direction of propagation [3]. For rectangular patches, the effective gap is equal to the physical gap between adjacent patch edges. In other cases one can determine the effective gap by defining an equivalent rectangular gap of equal area [2, 4]. Equation (3.6) loses accuracy for values of ν0 less than 0.6. (Note that ν0 ranges from 1.18 to 0.6 when ge f f /d ranges from 0.1 to 0.25.) The approximate formula in (3.6) does not account for the interaction of  higher order Floquet modes with the ground plane, so it is not accurate for h < d 2.

32

3 Closed-Form Representation of Metasurface Reactance …

3.1.3 Frequency (Phasings) Regions The dispersion equation in (3.1) is composed by three contributions: (i) (ii) (iii)

−1/ X

εr k/ (ζk  z1) cot (k z1 h) −ωCd kρ

(SW mode contribution) (ground plane contribution) (patch contribution)

The global resonance of these three contributions gives the dispersion equation. It is instructive to analyze how these three terms contribute to the resonance balancing along the dispersion curve. To this end, Fig. 3.3 shows the amplitude of the three

Fig. 3.3 Normalized capacitance as a function of a phasing across the unit cell and b frequency for printed ellipses with various axial ratios a/b = η and h = 1 mm, εr = 9.8, d = 3 mm, a = 0.9d

3.1 Dispersion Equation

33

contributions as a function of cell-phasing (Fig. 3.3a) and frequency (Fig. 3.3b) along the dispersion curve. For convenience, the horizontal axis in Fig. 3.3a, b is also rescaled in period and the surface wave wave  terms of the ratio between the lattice length (d λsw ) (free space wavelength (d λ)). The results are relevant to circular patches (η = 1 in Fig. 3.3); however, the behavior is qualitatively the same also for different patch geometries. We can identify three regions: (a) “low-frequency” region (b) “transition” region (c) “Floquet–Bloch (FB)” region

(dominated by i and ii) (all terms contribute) (dominated by ii and iii)

These regions correspond to those defined in [6] for general metamaterials. It is seen that at low frequency the resonance is essentially dominated by the balance of the SW mode contribution and the ground plane contribution, namely the patch contribution plays a minor role. This is what in metamaterial theory is defined as “effective medium” region. In the transition region, all the three contributions are significant. This means that the patches play a role, due to the linear increase of ωC0 , still being the capacitance equal to its quasi-static value (Fig. 3.3b). Therefore, one can approximate X , and therefore the dispersion curve, from the low frequency region to the beginning of the FB region with only one parameter, that is the normalize quasi-static capacitance ν0 . Note that, if the elements are designed in such a way to resonate for dimension very small in terms of wavelength—such as for instance split rings—an additional lumped inductance should be introduced, that changes the dynamic behavior before the FB region. Although this additional element can be incorporated in our scheme, we will not include it for simplifying the analytical treatment, still remaining the present formulation of practical use in most cases. It is anyway worth noting that extremely miniaturized elements require the extraction of more parameters from the full-wave analysis. This issue is the subject of a future work. In the FB region, the lattice resonates, and higher order Floquet modes become significant in the global effect. In our model, this is described by a capacitance that becomes highly non-local, namely wavenumber-dependent. In this region, the energy balance is dominated by the patch-lattice/ground plane interaction (contributions ii and iii), and some more parameters are needed for representing the resonant reactance. We emphasize that the more important region to exploit the potential of the MTS in antenna applications is the transition region, while the FB region is concerned with possible occurrence of bandgaps.

3.2 Low Frequency and Transition Regions In this section, we will illustrate how to approximate the resonant reactance X in a uniform fashion up to the end of the transition region by means of the only knowledge of the low-frequency capacitance. To this end, we first note that the cotangent function

34

3 Closed-Form Representation of Metasurface Reactance …

∼ 1 − z , with an error less than 4% for in (3.1) can be approximated as cot (z) = z 3 k z1 h < 1. This approximation is found very robust for solving the dispersion equation throughout the whole dispersion curve in all the practical cases of interest (the first order expansion is indeed definitely insufficient). Once inserted in (3.1), the cotangent approximation leads to the following equation εr k 1 − + X ζk z1



1

k z1 h − k z1 h 3

 − ωCd = 0

(3.7)

Expressing k z1 as a function X , (3.7) is rewritten as    2 −1  εr 1 X ε0 εr h + Cd = 0 − + −ω εr − 1 − X ωμ0 h ζ 3

(3.8)

3.2.1 Low Frequency Approximation In the low frequency regime, the solution of (3.8) is obtained by neglecting (X/ζ)2 in the second term, and approximating Cd ≈ C0 ; this leads to ωL eq  X ≈ XLF =  2 1 − ω 2 /ωeq where





ωeq = L eq and L eq =

ε0 εr h + C0 3 εr − 1 hμ0 εr

(3.9)

−1/2 (3.10)

(3.11)

The equivalent circuit of the single-pole Foster-type reactance in (3.9) is given in Fig. 3.4.

3.2.2 Cardano’s Transition Function Approaching ωeq , the term (X/ζ)2 in the second contribution of (3.8) cannot be neglected, so that one should solve Eq. (3.8) exactly. This is possible, since it can be reduced to a third degree algebraic equation, whose solution is well known by Cardano’s formula [7]. The result is found in Appendix C, and algebraically rearranged as

3.2 Low Frequency and Transition Regions

35

Fig. 3.4 Equivalence circuit for resonant reactance at low frequency for the dominant TM mode

ωL eq  W (ξ) X (ω) =  2 1 − ω 2 /ωeq

(3.12)

where the transition function W (ξ) is given by     3 2 2 ξ +1+ξ − ξ +1−ξ → ξ = r eal          ξ= j |ξ| 2 2 −3 3 3 W (ξ) = |ξ| |ξ| |ξ| |ξ| + − − 1 + − 1 → 2|ξ| ⎪ ⎪  |ξ| > 1 ⎪ ⎪ 1 −1 ⎪ ξ= j |ξ| −3 ⎪ ⎩ 2|ξ| cos 3 cos |ξ| → |ξ| < 1 (3.13) The frequency dependent parameter ξ = ξ (ω) is defined as ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

3 2ξ

  3

 2 ω/ωeq ξ (ω) = σ   2 3/2 1 − ω/ωeq where σ= εr 3



3/2 εr 3

+

ν0 dh (εr 2+1)



(3.14)

(3.15)

and ωeq can be rewritten from (3.10) as ωeq = σ

2εr c √ h 3 3 (εr − 1)

(3.16)

where c is the speed of light in free space. For ω < ωeq , ξ (ω) is purely real, and for ω > ωeq it is purely imaginary. Therefore, the change of definition in the transition function occurs at ω = ωeq . These definitions eliminate any ambiguity due to the choice of the branch of the roots; in other terms, all the roots in (3.13) are algebraic roots. Since it comes from a manipulation of a third degree solution [7], we will refer to W (ξ(ω)) as “Cardano’s transition function”. The dependence of the parameters

36

3 Closed-Form Representation of Metasurface Reactance …

Fig. 3.5 a Parameters σ and b normalized ωeq as function of ν0 d/ h for values of relative permittivity ranging from 4 to 11 with step of 1

ωeq normalized to c/ h and σ on ν0 d/ h is shown in Fig. 3.5 (we note that the angular frequency c/ h is the one at which the substrate thickness is 1/2π of free-space wavelength). It is seen that both ωeq and σ decrease (and then the slope of the transition function increases) when the permittivity and the quasi static capacitance increase, as expected. We see that in all practical cases σ < 1. Figure 3.6 shows the Cardano’s transition function W as a function of ω/ωeq for various values of σ ∈ (0.1, . . . , 1) with steps of 0.1. The transition function is equal to one and very flat at zero frequency and next it goes to zero at ω = ωeq with steeper slope for lower values of σ. The zero of W at ω = ωeq compensates the apparent singularity of the first factor in (3.12). We note that the condition ω = ωeq approximately occurs where the contribution of the cotangent function in Fig. 3.3 has a minimum. Since σ and ωeq only depend on ν0 (for a given substrate), matching the transition function to practical cases requires the extraction of the quasi-static capacitance only. This can be done in general through a single run of full-wave analysis. In our experience, the best point of the dispersion diagram to extract this parameter is when the cell-phasing is 40◦ . The formula in (3.12) is uniformly valid up to the beginning of the FB region, namely for cell-phasing approximately up to 130◦ .

3.2 Low Frequency and Transition Regions

37

Fig. 3.6 Cardano’s transition function for various values of σ

3.2.3 Simple Non-uniform Approximation A good approximation of the Cardano’s transition function (3.12) is given by the following expression  (10/σ)2/3 W (ξ (ω)) ≈ 1 − ω/ωeq (3.17) which has a maximum error of about 0.7% in the range ω < 0.7ωeq and for 0.2 < σ < 0.9. This expression has been found by applying a least mean square approximation with respect to the two-parameters (γ, χ) using the family of functions (γ/σ)χ  1 − ω/ωeq . Equation (3.12) is useful to estimate the error on the quasi-static approximation (3.9). For ω ≈ ωeq , W (ξ) can be approximated as   W (ξ) = −3 ±|(2ξ)|−2/3 + (|2ξ|)−4/3

ω/ωeq ≈ 1

(3.18)

where the upper and lower sign applies to ω > ωeq and ω < ωeq , respectively. This approximation is obtained by assuming ξ large in (3.10). A straightforward error analysis conducted on the above formula shows that the relative

win√ error on the dowing function is smaller than 3% when ω/ωeq ∈ 1 − 0.19 σ, 1 + 1.24σ . Using (3.17) and (3.18) in (3.12), leads to a non-uniform approximation   (10/σ)2/3  ωL eq 1 − ω/ωeq √ ω   < 1 − 0.19 σ X≈ for 2 2 ωeq 1 − ω /ωeq X≈



2 −2/3

−2/3  1 − 2σω 1− 2 ωeq

√ 1 − 0.19 σ, 1 + 1.24σ

2 3ωL eq 2σω 2 ωeq for ωωeq ∈

ω2 2 ωeq

(3.19)

 (3.20)

38

3 Closed-Form Representation of Metasurface Reactance …

Fig. 3.7 Normalized resonant reactance for different values of quasi-static capacitance. Uniform approximation (3.12) (continuous line), transition function W (dotted-line), non-uniform approximation (3.19)–(3.20) (dashed line). The latter is stopped at the limit indicated by (3.20). The jump in the non-uniform approximation is very weakly visible in the adopted scale

 √ which produces a small jump at ω ωeq = 1 − 0.19 σ with relative value smaller than 3% in the range 0.2 < σ < 0.9. Removing the factor (3.17) from (3.19) (namely substituting value in √ (3.9)) and changing the boundary   (3.19) with its √ low frequency from ω ωeq = 1 − 0.19 σ to ω ωeq = 1 − 0.31 σ produces a jump of 14% with respect to the uniform solution (3.12) at the boundary. The approximation (3.20) is important since it is valid in the central part of the transition region, where one operates in many applications based on MTSs. Figure 3.7 compares the uniform (3.12) and non-uniform (3.19), (3.20) approximations of X for different values of ν0 d/ h for the case εr = 9.8.

3.2.4 Identification of the Limits of the Frequency Regions It is seen that the low-frequency approximation possesses √ a maximum error of 2% for < 1 − 0.5 σ. We can therefore establish σ in the range 0.2 < σ < 0.9 when ω/ω eq  √  that the frequency ω L F = ωeq 1 − 0.5 σ is the limit of the low frequency region. The uniform approximation in (3.12) is valid up to the FB region. Finding a precise definition of this region is not an easy task; however, we have seen that, in most of the practical cases, this region starts when the cell-phasing is contained between 120◦ and 160◦ (closer to 120◦ for higher values of quasi-static capacitance). Precautionary, we can establish this limit to 100◦ (1, 74 rad). Under this assumption, the FB angular frequency ω B is found by solving the equation  d 1, 74 = ω B c

1+



X (ω B ) ζ

2 (3.21)

3.2 Low Frequency and Transition Regions

39

Fig. 3.8 Normalized ω B and ω L F as a function of ν0 d/ h for values of relative permittivity ranging from 5 to 11 with steps of 2. The grey region denotes the transition region for εr = 11

where X is obtained by (3.12). It can be seen that a good approximation of the above curve for 0.1 < σ < 0.8 is ω B = 0.52σ −1/2 ωeq . We have therefore identified the limits of the three regions defined in Sect. 3.1.3: ω < ωL F ωL F ≤ ω ≤ ω B ω B ≤ ω ≤ ω0

(a) “low-frequency” region (b) “transition” region (c) “Floquet–Bloch” region where, using (3.16), we have ωL F ≈

2εr c σ(1 − 0.5σ 1/2 ) √ h 3 3 (εr − 1)

(3.22)

2εr c 0.52σ 1/2 √ h 3 3 (εr − 1)

(3.23)

ωB ≈

and ω0 is the angular frequency at which the phasing across the unit-cell is π. Figure 3.8 presents the two angular frequencies ω L F and ω B normalized to c/ h as a function of ν0 d/ h for different values of permittivity. We emphasize that the two frequencies are identified by using only the parameter ν0 . We see from Fig. 3.8 that the transition region becomes smaller for lower values of permittivity.

3.3 FB Region Correction The single parameter uniform approximation described in the previous section is very accurate until the FB frequency ω B . Afterwards, we can improve the representation   by introducing the dispersive variation of the equivalent capacitance Cd kρ . To this end, as anticipated in Sect. 3.1, more information should be extracted by a full wave

40

3 Closed-Form Representation of Metasurface Reactance …

  analysis. The dynamic term Cd kρ can be written as a function of frequency by using the expression  Cd =

C0

α C0 ω−ω 1− δ ω −ωB 0

f or ω ≤ ω B f or ω > ω B

(3.24)

B

where ω0 is the frequency at which the phasing across the unit cell is 180◦ . The full wave data are normally given in terms of dispersion curve kρ = kρ (ω). Thus, in order to provide the data to fit (3.24) one should substitute the numerical data kρ = kρ (ω) in (3.1). The parameters α and δ can then be obtained by fitting full wave data for ω > ω B with the expression in (3.24) (fitting can be performed e.g., by using the curve fitting toolbox of Matlab). Examples are shown in Fig. 3.9, where the normalized dynamic capacitance has been calculated for various elliptical and rectangular patches. The table in the inset

Fig. 3.9 Normalized dynamic capacitance as a function of frequency for a elliptical and b rectangular patches; a = 2.7 mm, εr = 9.8, d = 3 mm, h = 1 mm. The various shapes are shown in the inset

3.3 FB Region Correction

41

Fig. 3.10 FB transition function as a function of ω/ω B for different values of ν0 h/d and α. The curves are stopped at ω0 /ω B

shows the values of the parameters which fit the different curves. The parameters α and δ depend on the geometry and on the periodicity (besides the slab parameters). Once one has determined the frequency dependent approximation of Cd , this value is used to find frequency dependent value of ωeq by replacing ν0 with νd = ν0 Cd /C0 in (3.14)–(3.15) and apply (3.12) again. This leads to an extremely accurate approximation of the impedance throughout the whole dispersion curve, including the FB region. The above procedure can be synthesized in the formula

where

ωL eq  W (ξ) W B (ω) X (ω) =  2 1 − ω 2 /ωeq

(3.25)

 2  (d)  W ξ 1 − ω/ωeq W B (ω) = 

2  (d) 1 − ω/ωeq W (ξ)

(3.26)

(d) is denoted as FB-transition function. In (3.26), ωeq and ξ (d) are defined as ωeq and ξ in (3.10) and (3.14), respectively, except that the quasi-static capacitance C0 is substituted by the dynamic capacitance Cd in (3.24). The FB-transition functions W B (ω) is equal to one for ω < ω B and it increases for ω > ω B . Figure 3.10 shows an example for various values of the parameter α. The FB function is very simple to be implemented, since it requires the extraction of two additional parameters only; however, in many practical applications of MTS, the operating frequency is in the range ω < ω B , where W B (ω) ≈ 1.

42

3 Closed-Form Representation of Metasurface Reactance …

3.4 TE Dispersion Close to Its Cut-Off Frequency In practical applications, one should work with TM mode only, avoiding possible excitation of TE mode. To this end, it is important to know the cut-off frequency of the TE mode and possibly an approximation of its dispersion equation. Proceeding as for TM mode, it is found that the resonant impedance χ of the TE mode is given by −ωμ0 h ω > ωT E (3.27) χ ≈ 2 ω − 1 ωT E where ωT E is the TE cut-off frequency given by 

ωT E

  −1/2 2 2 = μ0 h ε0 h (εr − 1) + 0.94C T E π

(3.28)

where C T E is the quasi static capacitance for TE polarization. The approximation in (3.27)permits the calculation of the dispersion curve close to the TE cut-off as kρT E = k 1 + (ζ/χ)2 . The expression in (3.27) allows for finding the mono-modal region and the possible occurrence of a bandgap. The bandgap will be indeed present if ω0 < ωT E , where ω0 is the frequency at which the cell-phasing of the TM SW-mode is 180◦ .

3.5 Comparison with Full-Wave Analysis Comparison with full-wave analysis is shown in Fig. 3.11 for the case of elliptical and square patches with different slab thicknesses. Full-wave results obtained with a periodic MoM code are compared with the results from this formulation with and without the FB-transition function. It can be seen that the FB transition function changes the value of the admittance only in the narrow region between ω B and ω0 . In deriving C0 , the full-wave analysis has been calculated at a phasing across the unit cell equal to 40◦ . The values of α and δ have been found by matching (3.24) with full-wave results. The value of the parameters is shown in the inset table. We note that the values of ν0 for the square as calculated from the approximation in (3.6) is ν0 = 0.7 for the ellipse and ν0 = 1.18 for the square patch. These values are quite close to the values extracted by the full wave analysis. Figure 3.12 shows the dispersion curves relevant to the same cases as those in Fig. 3.11. We notice that the small error in the reactance has an even smaller impact on the accuracy of the dispersion curve. In Fig. 3.12b, the dashed line denotes the curves obtained using the value given by the closed form (3.6) for the static capacitance.

3.6 Extension to Anisotropic MTS

43

Fig. 3.11 Resonant reactance as a function of normalized frequency for different values of h/d (a = 2.7 mm, εr = 9.8, d = 3 mm). a Ellipses with axial ratio equal to 0.75; b square patches. In the tables: values of parameters extracted by full wave analysis. Continuous line: this formulation, including FB transition function correction; dashed line: this formulation without FB correction; dots: full wave analysis

3.6 Extension to Anisotropic MTS The initially part of this chapter presented a formulation for representing a closedform dispersion equation for isotropic elements, or for directions of propagation aligned with a symmetry axis. This section generalizes the previous formulation to anisotropic elements and to any direction of SW propagation, with only the limitation that the printed elements possess at least two symmetry axes. Here, we limit our investigation to a region for which the phasing of the cell is about 90◦ and the generalization is achieved by combining the solutions obtained for the two principal directions and reutilizing only the two quasi-static capacitances of the individual circuits (Fig. 3.13). We denote by Ci and X i (i = 1, 2) the quasi-static capacitances and the “opaque” principal reactance, respectively.

44

3 Closed-Form Representation of Metasurface Reactance …

Fig. 3.12 Dispersion curves obtained by using the resonant impedance in Fig. 3.11. a Ellipses with axial ratio equal to 0.75; b square patch and different values of h/d as those in Fig. 3.11 (a = 2.7 mm, εr = 9.8, d = 3 mm). MoM (dots), approximation including FB transition function (continuous line). b Dashed lines denote the curves obtained using (3.6) for the quasi-static capacitance

Fig. 3.13 MTS constituted by periodic elements printed on a grounded dielectric slab with two symmetry axes within the periodic cell. a Equivalent networks for dominant TM SW propagation for direction of propagation along the principal axes of the cell; b relevant geometry of the MTS

3.6 Extension to Anisotropic MTS

45

Fig. 3.14 Dispersion surface of the dominant TM surface wave and isofrequency curves for two different values of the dielectric thickness

In TO [8, 9] applications, important design parameters associated with the dominant TM SW are the IDC and the group velocity. As already mentioned in Chap. 2, for any given angular frequency ω0 , any IDC represents the cut at ω = ω0 of the dispersion surface ω = ω(kρ , α) sketched in Fig. 3.14, which is always below the “light cone” describedby kt · kt = ω 2 /c2 , with kt = kρ (α, ω0 ) cos αˆx + kρ (α, ω0 ) cos αˆy and α = cos−1 kt · xˆ /kρ . The transverse gradient ∇kt ω(kt ) = vg represents the group velocity of a SW bundle with spectrum centered around (ω, kt ). The determination of the IDCs all over the irreducible Brillouin region by brute-force full-wave analysis is not a trivial task, since it requires the construction of a 3D database in the (ω, kt ) space. Here, the database for any element is constructed by the use of the two parameters Ci which can be calculated in analytical form for simple geometries or extracted from a single run of a full-wave analysis for more complex cases. It is also worth noting that an analytical definition of the local IDC is useful in describing the variable boundary conditions in integral equations [10, 11]. Figure 3.15 shows the qualitative shapes of the IDCs in the frequency regions described in Sect. 3.1.3. In this case, low-frequency/Floquet–Bloch limit can be identified by taking the minimum among the two limits obtained for the two principal directions (ω L F = min ω L Fi and ω B = min ω Bi ). In the low frequency region, the i=1,2

i=1,2

IDCs are almost circles (Fig. 3.15a) since the patch contribution ωCi plays a minor role. In the transition region, the patches play a role due to the linear increase of ωCi , still being the capacitance equal to its quasi-static value (Fig. 3.15b). Until the limit ω B of this region, one can approximate X i , and therefore the IDCs with only two parameters, namely the normalized capacitances ν0i . We note that, across the entire transition region, is spatially dispersive due to the presence of the substrate,

46

3 Closed-Form Representation of Metasurface Reactance …

Fig. 3.15 Qualitative shape of the IDCs in the various frequency regions. e1 and e2 denote the direction of the symmetry axes of the geometry of the periodic cell

although the capacitances are constant. The two-term approximation is found to be very robust for solving the dispersion equation throughout the entire dispersion curve (also beyond 100◦ phasing) up to the cut-off frequency of the first TE mode (see Eq. (3.28)). In the FB region (Fig. 3.15c), the lattice resonates and higher order Floquet modes become significant and more than two parameters are needed for representing the IDC. Here we disregard an IDC closed-form approximation in this region.

3.6.1 Approximation of the IDCs The IDCs can be approximated by their values on the principal plane kρi (ω) by using a “two-circle approximation”. Let us assume that kρ1 > kρ2 and the amplitude √ vector wavenumber kt · kt = kρ (ω, α) depends on both the angular frequency and α, which is the direction of propagation with respect to the axis e1 . The isofrequency dispersion curve can be approximated as (Fig. 3.16) 

(kt ∓ e1 kc ) · (kt ∓ e1 kc ) = K

kc (ω) =  where kρi = k 1 +



X i (ω) ζ

2 2 kρ1 − kρ2

2kρ1

;

K (ω) =

2 2 kρ1 + kρ2

2kρ1

(3.29) (3.30)

2 with X i given by (3.10). In (3.29), “−” and “+” sign

apply to α ∈ (−π/2, π/2) and α ∈ (π/2, 3π/2), respectively. Solving (3.29) for √ kt · kt = kρ (ω, α), one has

3.6 Extension to Anisotropic MTS

47

Fig. 3.16 Approximation of IDC at a certain frequency

kρ (ω, α) = kc |cos α| +

 kc 2 cos2 α + kρ2 2

(3.31)

From (3.10) it is seen that, at low frequency, one has X 1 ≈ X 2 and therefore kρ (ω, α) → kρ1 ≈ kρ2 , namely the element becomes isotropic. This approximation only requires the knowledge of the two principal capacitances ν0i and it is found to be much more accurate than any other two-point approximation like the one using ellipses in the range ω < min [ω Bi ,ωT Ei ]. It is clear than the a priori knowledge of i=1,2

more than two parameters can improve the approximation of the dispersion curve. However, the approximation in (3.31) turns to be very accurate, within clearly identified limits, and it may be used in most of the practical applications of antennas and transformation optics.

3.6.2 Phase and Group Velocities The phase velocity associated to the dispersion diagram can be calculated as v (ω, α) =

ω kˆ t ω kˆ t  = kρ kc |cos α| + kc 2 cos2 α + kρ2 2

(3.32)

where kˆ t = kt /kρ is the unit vector along the direction ρ. The group velocity can be determined by  ∇kρ ,α F(kρ , α, ω)ω  vg (ω, α) = ∇kρ ,α ω(kρ , α) = − ∂ F(kρ ,α,ω)   ∂ω (kρ ,α)

(3.33)

48

3 Closed-Form Representation of Metasurface Reactance …

where F(kρ , α, ω) = kρ 2 − 2kρ kc cos α − kρ2 2 = 0 denotes the dispersion equation in (3.29). From (3.33), one has vg (ω, α) =

  ˆ kρ − kc cos α kˆ æ ± (kc sin α) α c (ω) −kρ ∂k∂ω cos α − kρ2 (ω)

∂kρ2 (ω) ∂ω

(3.34)

where the upper (lower) sign applies to kˆ t · eˆ 1 ≥ 0 (< 0).

3.7 Numerical Results Figure 3.17 shows various printed elements which have been considered to test the accuracy of the formulation: (i) elliptical patch; (ii) rectangular patch, (iii) elliptical ring; (iv) circular patch with rectangular slot. The basic periodic cell is always taken with period d = 3 mm in both directions. The substrate has εr = 9.8 and two different thicknesses (h = 1 mm and h = 3 mm) for each configuration. The geometries are summarized in Fig. 3.17. The Floquet–Bloch frequency from (3.23) and the cut-off frequency of the TE mode in (3.22) (the minimum between i = 1 and i = 2) have been reported in the same figure.

Fig. 3.17 Investigated geometries; ν0 normalized capacitance, f F B Floquet–Bloch frequency (3.23) and f T E TE mode cut-off frequency (3.22)

3.7 Numerical Results

49

Fig. 3.18 IDCs for the elliptical patch. a h = d/3, b d = h. MoM: solid lines, analytical: dashed lines. Labels in GHz

Fig. 3.19 IDCs for the rectangular patch. a h = d/3, b d = h. MoM: solid lines, analytical: dashed lines. Labels in GHz

The normalized capacitances along the two principal directions are listed in the table for the two cases h = d/3 and h = d. In the first case the normalized capacitances are larger due to the effect of the higher-order Floquet modes accessible to the ground plane. These modes (dominantly TM) increment the storage of electric energy in the substrate. It is also evident that the capacitance along the x axis is always higher than the one along the y axis, due to the smaller effective gaps between the contiguous edges, as predicted by (3.6). The value of normalized capacitance can be scaled with εr and d through (3.4). Therefore, one can construct the dispersion diagram everywhere with only two numbers per geometry independently on the periodicity and substrate. Figures 3.18, 3.19, 3.20 and 3.21 show the isofrequency dispersion diagrams of the geometries presented in Fig. 3.17. Our two-parameter dependent analytical formulation (dashed lines) is compared with the one obtained from a full-wave analysis (solid lines). The agreement is good up to the FB frequency, as expected. From the comparison among the various geometries, the following general considerations arise.

50

3 Closed-Form Representation of Metasurface Reactance …

Fig. 3.20 IDCs for the elliptical ring. a h = d/3, b d = h. MoM: solid lines, analytical: dashed lines. Labels in GHz

Fig. 3.21 IDCs for the slotted circular patch. a h = d/3, b d = h. MoM: solid lines, analytical: dashed lines. Labels in GHz

i. Decreasing the thickness h allows for a better accuracy until higher frequencies, since the FB frequency (3.23) increases. ii. Increasing h implies a shifting at low frequencies of the effects of anisotropy and spatial dispersion. This is evident from (3.22). From practical point of view, it implies a better “homogenization” of the boundary conditions. iii. The slotted circular patch and the rectangular patch exhibit, among the other geometries, larger phasing across the cell (shorter SW wavelength) for the same substrate thickness. The slotted circular patch has more regular IDCs. It is also important to investigate the effect of the rotation of the element inside the perimeter of the cell. The results in Fig. 3.22 are relevant to the elliptical patch in Fig. 3.17 with both d = h/3 and d = h. The axes of the printed ellipses are rotated of 0◦ , 30◦ , 70◦ with respect to the x axis (see insets (Fig. 3.22)). The labels denote the frequency in GHz. The full-wave results (solid line) are compared with those from our formulation (dashed line) until the frequency where the latter is applicable. It is apparent that in the low-frequency region the IDCs are circles. In the transition region the curves are quasi elliptical and follow the orientation of the elliptical patches in the lattice. In FB region the IDCs are deformed by the geometry of the lattice especially for the thinner substrate. Figure 3.23 shows the value of the quasi static normalized capacitances for the three rotation angles in Fig. 3.22. It is seen that the

3.7 Numerical Results

51

Fig. 3.22 IDCs for MTS constituted by elliptical patches (Fig. 3.17) with periodicity d = 3 mm printed on a grounded slab with and εr = 9.8, and several angle of rotation of the ellipse axes w.r.t. the lattice (see insets). The labels denote the frequency in GHz. LHS: d = h/3, RHS: d = h. Solid line: MoM, dashed line: analytical solution (Eq. (3.29))

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3 Closed-Form Representation of Metasurface Reactance …

Fig. 3.23 Quasi-static capacitance versus rotation angle Fig. 3.24 Comparison of IDCs of elliptical rings with different filling fraction. Solid line: MoM, dashed line: analytical solution

principal capacitance are quite stable, and the small discrepancy do not affect the approximation. In general, it can be affirmed that up to the FB frequency, one can approximate the solution with the simple 2-parameter closed-form for any rotation of the element inside the lattice. Figure 3.24 shows the comparison between IDCs of an elliptical patch and those for elliptical rings with different width at the given frequency of 13 GHz and for h = d/3. As expected, rings support slower phase velocity (larger wavenumber) for any direction. This effect can be used in practice for enlarging the dynamic range of feasible impedances. Finally, Fig. 3.25 presents results for phase and group velocities for elliptical patches (aligned with the lattice axes), for two substrates (d = h and d = h/3) and for three angles of propagation (0◦ , 45◦ , 90◦ ). As expected, the phase velocity is always larger than the group velocity and both of them significantly decrease when increasing the substrate at the same frequency.

3.8 Chapter Summary

53

Fig. 3.25 Phase and group velocity for the elliptical patch. Solid line: commercial software (CST), dashed line: analytical solution (Eq. (3.34))

3.8 Chapter Summary In the first part of this chapter, a simple and elegant analytical form of the resonant reactance of isotropic MTS which supports a TM surface wave has been presented. The solution has been provided in terms of a single parameter transition function, which extends the quasi-static “Foster-type” solution till the beginning of the Floquet–Bloch region, where an additional correction has to be used. The entire curve up to FB transition can be derived by approximating the quasi-static capacitance with an equivalent gap (for simple shapes), or by extracting it from a single run of a full-wave eigensolver. The availability of the closed form solution allows for defining with a precise error control the limit of the low frequency and of the FB region, as well as of the mono-modal region defined by the cut-off frequency of the TE mode. Next, an extension of the above formulation to anisotropic and spatially dispersive MTS has been presented, which allows a closed form representation of IDC and group velocity as a function of two parameters only up to the limit of the FB region. Thanks to this formulation, the numerical construction of a database for many geometries for solving complex problems like those in [12–16] is accurate and efficient.

References 1. Maci S, Caiazzo M, Cucini A, Casaletti M (2005) A pole-zero matching method for EBG surfaces composed of a dipole FSS printed on a grounded dielectric slab. IEEE Trans Antennas Propag 53(1):70–81 2. Ramaccia D, Toscano A, Bilotti F (2011) A new accurate model of high-impedance surfaces consisting of circular patches. Prog Electromagn Res M 21:1–17

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3. Luukkonen O, Simovski C, Granet G, Goussetis G, Lioubtchenko D, Raisanen AV, Tretyakov SA (2008) Simple and accurate analytical model of planar grids and high-impedance surfaces comprising metal strips or patches. IEEE Trans Antennas Propag 56(6):1624–1632 4. González-Ovejero D, Martini E, Maci S (2015) Surface waves supported by metasurfaces with self-complementary geometries. IEEE Trans Antennas Propag 63(1):250–260 5. Maci S, Minatti G, Casaletti M, Bosiljevac M (2011) Metasurfing: addressing waves on impenetrable metasurfaces. IEEE Antennas Wirel Propag Lett 10:1499–1502 6. Holloway CL, Kuester EF, Gordon JA, O’Hara J, Booth J, Smith DR (2012) An overview of the theory and applications of metasurfaces: the two-dimensional equivalents of metamaterials. IEEE Antennas Propag Mag 54(2):10–35 7. Cardano G (1968) Artis magnae sive de regvlis algebraicis (English translation: The great art or the rules of algebra). MIT Press, Cambridge 8. Martini E, Maci S (2014) Metasurface transformation theory. In: Werner DH, Kwon D-H (eds) Transformation electromagnetics and metamaterials. Springer, London, pp 83–116 9. Patel AM, Grbic A (2014) Transformation electromagnetics devices based on printed-circuit tensor impedance surfaces. IEEE Trans Microw Theory Tech 62(5):1102–1111 10. Francavilla MA, Martini E, Maci S, Vecchi G (2015) On the numerical simulation of metasurfaces with impedance boundary condition integral equations. IEEE Trans Antennas Propag 63(5):2153–2161 11. González-Ovejero D, Maci S (2015) Gaussian ring basis functions for the analysis of modulated metasurface antennas. IEEE Trans Antennas Propag 63(9):3982–3993 12. Minatti G, Faenzi M, Martini E, Caminita F, Vita PD, González-Ovejero D, Sabbadini M, Maci S (2015) Modulated metasurface antennas for space: synthesis, analysis and realizations. IEEE Trans Antennas Propag 63(4):1288–1300 13. Minatti G, Caminita F, Casaletti M, Maci S (2011) Spiral leaky-wave antennas based on modulated surface impedance. IEEE Trans Antennas Propag 59(12):4436–4444 14. Minatti G, Maci S, Vita PD, Freni A, Sabbadini M (2012) A circularly-polarized isoflux antenna based on anisotropic metasurface. IEEE Trans Antennas Propag 60(11):4998–5009 15. Faenzi M, Caminita F, Martini E, Vita PD, Minatti G, Sabbadini M, Maci S (2016) Realization and measurement of broadside beam modulated metasurface antennas. IEEE Antennas Wirel Propag Lett 15:610–613 16. Minatti G, Caminita F, Martini E, Sabbadini M, Maci S (2016) Synthesis of modulatedmetasurface antennas with amplitude, phase, and polarization control. IEEE Trans Antennas Propag 64(9):3907–3919

Chapter 4

Flat Optics for Surface Waves

As already mentioned in Chap. 1, a modulated MTS may be obtained by gradually varying the geometry of the elements in contiguous cells, while maintaining the period unchanged and electrically small. Macroscopically, this results in a modulation of the IBC that, due to the small dimensions of the unit cell, can be assumed to be almost continuous and locally treated by assuming each element as immersed in a periodic environment. The spatial variability of the IBC imposes a deformation of the SW wavefront, which addresses the local wavevector along non-rectilinear paths. A simple example is illustrated in Fig. 4.1. It shows a MTS consisting of a regular lattice of printed square patches modulated in size along x, with larger sizes for increasing x. Assume that this MTS is excited at y = 0 with a SW beam, represented in Fig. 4.1 by parallel equi-length wavevectors k t . As the wave progresses along y, the increasing value of reactance along x imposes a decrease of the local phase velocity, thus producing a bending of the wavefront towards higher levels of impedance, consistently with the dispersion equation determined by the local reactance value. This phenomenon is very similar to the one described in geometrical optics (GO) for graded index materials, and leads to a poorly explored branch of wave theory, which some scientists have referred to as Flat Optics (FO). In our knowledge, the name FO has been introduced in a recent paper by Capasso’s group [1]; in the latter, however, more emphasis is given to space waves manipulations through MTSs, while here we use this terminology with reference to SW manipulation. Several works have been published on SW propagation on modulated MTSs [2–7], some of them addressing these phenomena in the framework of TO [3–6]. However, a rigorous treatment of all aspects relevant to SW FO, like ray tracing, transport of energy, and ray velocity, have not been treated in literature in our knowledge, and are rigorously derived here for both cases of isotropic and anisotropic IBCs. In the isotropic case, the formulation can be seen as an adaptation to MTSs of the GO for evanescent waves introduced by Felsen in [8]. The formulation presented here opens new design possibilities for a large number of microwave devices based on SWs. In particular, whenever the objective of the design can be described through a mathematical transformation, FO can be treated trough Flat Transformation Optics. This extension of the more popular volumetric transformation optics [9] has been © Springer Nature Switzerland AG 2019 M. J. Mencagli, Manipulation of Surface Waves through Metasurfaces, Springer Theses, https://doi.org/10.1007/978-3-030-14034-2_4

55

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Fig. 4.1 Example of curved-wavefront SW supported by a modulated IBC obtained by changing the patch dimensions

introduced in [3, 4], and it is here framed in the general Flat Optics theory. It is shown here as Flat Transformation Optics constitutes a valid alternative to the solution of the differential equation for ray-path. In [3, 4], TO for surface waves is introduced by using a Poynting vector, looking similar to what we present in Sect. 4.2; however, we introduce here the transport of energy formula that allows for treating analytically the global distribution of power density on the surface, which is untreated in literature. The chapter is organized as follows. Section 4.1 presents FO for isotropic IBC. Section 4.2 extends the theory to anisotropic IBC. Section 4.3 outlines the basic principles of Flat Transformation Optics. Section 4.4 presents the comparison with full-wave analysis. Section 4.5 presents the synthesis of modulated IBC through modulated MTSs and some practical examples. Chapter conclusions are drawn in Sect. 4.6.

4.1 Flat Optics for Isotropic MTSs Let us assume that impenetrable, scalar, continuous, lossless IBCs are imposed on a planar surface on the plane z = 0 of a Cartesian reference system (x, y, z) with unit vectors xˆ , yˆ , zˆ    (4.1) Et z=0+ = j X (ρ) · zˆ × Ht z=0+ Free-space is assumed in the half-space z > 0. In (4.1), Et and Ht are the transverse electric and magnetic fields. The IBC in (4.1) is characterized by a lossless, reactive, continuous impedance Z (ρ) = j X (ρ) where the reactance X (ρ) is a scalar function of the observation point ρ = x xˆ + y yˆ on the plane z = 0. We assume that the functional variability with ρ is smooth. In the following, we will assume that the boundary conditions (4.1) are weakly space dispersive. Although not strictly necessary, we assume that a vertical elementary electric dipole is placed on the surface at the origin of the reference system. This vertical dipole launches a SW on the MTS. Since in practical cases the impedance is created by sub-wavelength patches printed on a grounded slab, X is inductive, and therefore the SW is TM with respect to z. The goal is to find an asymptotic structure of the SW fields “far enough” from the dipole source, that in practice means at a distance of at least one free-space wavelength from the dipole.

4.1 Flat Optics for Isotropic MTSs

57

4.1.1 Eikonal Equation At a distance larger than a free-space wavelength from the exciting dipole, we assume that a z-directed magnetic potential is of the form A z (ρ, z) = I (ρ, z) exp (− jk (ρ, z)) where I (ρ, z) and  (ρ, z) are complex functions of the space variable ρ assumed weakly dependent on k, as normally assumed √ in GO (see, e.g.√[10], p. 111). This weak variability should be quantify as ∂/∂k > 1 and kz < 1, the quantities f /k and g/k 2 are negligible in (4.4), and the wave equation in (4.3) is satisfied when   g = zk∇t 2 ξ − k 2 z 2 ∇t ξ · ∇t ξ

(4.7)

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The electromagnetic field can be derived from the magnetic potential in (4.2) as 

 ∂ Az ∂2 E = − jkζ ∇t 2 + 1 + 2 2 A z zˆ k ∂z k ∂z

(4.8)

H = ∇t A z × zˆ

(4.9)

Under the asymptotic assumptions ks >> 1 and kz < 1, one has     E ∼ − jkζ j∇t sξ + 1 + ξ 2 zˆ A z

(4.10)

H ∼ − jk∇t s × zˆ A z

(4.11)

Applying the boundary conditions in (4.1) implies      Et z=0+ = jξζ zˆ × Ht z=0+ = j X zˆ × Ht z=0+

(4.12)

which leads to X/ζ = ξ and then, from (4.7), to |∇t s| =



. 1 + (X (ρ) /ζ)2 = n eq (ρ)

(4.13)

The quantity n eq > 1 plays the same role of the refractive index in GO. Therefore, (4.7) can be referred to as “eikonal equation”, in analogy with GO [8]. We note that since n eq > 1, the SW satisfies the local non-radiative condition assumed at the beginning. Also, the weak phase-variability assumption c[∂/∂ω]z=0+ = √ c [∂s/∂ω] 0 and a “real” one, whose points are identified by an unprimed vector r = x xˆ + y yˆ + z zˆ = ρ + z zˆ , z > 0 (see Fig. 4.5). The two vectors are expressed in Cartesian coordinates and unit vectors of their respective spaces. Both half-spaces z > 0 and z > 0 are filled by free space, but with two different boundary conditions at z = 0 and z = 0. We assume that the virtual space possesses boundary conditions described by a scalar reactive impenetrable uniform reactance and refraction index X related each other by the eikonal relation (4.13). In the real space, instead, we define at z = 0 two complementary surfaces S and S ∗ , separated by a continuous line boundary ∂S, whose summation covers the

Fig. 4.5 Geometry for MTS transformation and representation of the local isofrequency dispersion curves in: a virtual space (circle), and b real space (ellipse)

4.3 Flat Transformation Optics

69

entire plane (see Fig. 4.5). In S ∗ , we assume the same impedance boundary condition described by the surface reactance X . Next, coordinate transformations are defined, which map the real space into the virtual space and viceversa, leaving unchanged the z coordinate. In particular, the transformation  maps univocally the virtual space into the real space and its inverse transformation  maps the real space into the virtual space. We also impose that the transformations are identities in the region of space S ∗ , where S ∗ also includes a circular region close to the source (see Fig. 4.5),  ρ=

ρ =

  ! " ! "  ρ : x , y → x(x , y ), y(x , y ) in S ρ in S ∗

(4.65)

" !  (ρ) : {x, y} → x (x, y), y (x, y) in S ρ in S ∗

(4.66)



Since the coordinate transformation  is the identity in S ∗ , the primed space is coincident with the real space there. The Jacobian of the transformation M and its = M can be expressed in covariant and contravariant bases as inverse M−1  M = xˆ g x + yˆ g y = γ x xˆ + γ y yˆ M = gx xˆ + g y yˆ = xˆ γ x + yˆ γ y

(4.67)

where gx = ∂ρ/∂x , g x = ∇t x , γ x = ∂ρ /∂x, γ x = ∇t x, with analogous definitions for g y , g y , γ y and γ y . Consider a TM-SW launched by a dipole source in the virtual space. Since the reactive impedance j X is uniform in space, the surface wave is a cylindrical wave, phase centered at the source. dis  At a certain tance from the source, the asymptotic phase is of type exp − jkn eq ρ , where ρ = x 2 + y 2 and n eq satisfies the eikonal equation n eq = 1 + (X /ζ)2 . The SW in the transformed space is obtained by applying the transformation to the phase  . factor exp − jkn eq ρ = exp − jkn eq (ρ) = exp (− jks(ρ)). Hence, k∇t s (ρ) = kn eq kˆ t is the local wavevector of the SW in the transformed space. Therefore, one has (4.68) n eq kˆ t = n eq MT · kˆ t   where kˆ t = ρˆ = x xˆ + y yˆ /ρ . Pre-multiplying both sides of (4.68) by M−1 =  M and equalizing the amplitude of the result, leads to the following eikonal equation  2  n 2eq = n eq / kˆ t · αt · kˆ t

(4.69)

where  2  2   T = xˆ xˆ γ x + yˆ yˆ γ y + xˆ yˆ + yˆ xˆ γ x · γ y αt = M · M

(4.70)

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The eikonal equation (4.69) represents ellipses with axes aligned with the eigenvectors of αt . By denoting these eigenvectors as eˆ i = eˆ i (æ), (i = 1, 2), one can write αt in the diagonal form αt = eˆ 1 eˆ 1 σ1 + eˆ 2 eˆ 2 σ2 , where σi are the eigenvalues of αt .This allows for rewriting (4.69) in the form  2  −1 n 2eq = n eq cos 2 ϕσ1 + sin2 ϕσ2

(4.71)

where cos ϕ = eˆ 1 · kˆ t . In the polar plane (n eq , ϕ), (4.71) represents an ellipse with √ semi-axes n eq / σi (see Fig. 4.5). When compared with (4.49), (4.71) can be interpreted as the eikonal equation associated with the following anisotropic IBC            n eq 2  n eq 2   X (ρ) = eˆ 1 eˆ 1 ζ − 1 + eˆ 2 eˆ 2 ζ −1 σ1 (ρ) σ2 (ρ)

(4.72)

Therefore, in order to implement the desired transformation, one should use a variable impedance locally satisfying (4.72), which is fully defined by the transformation. We underlined that the representation is not exact, since it is based on the approximation (4.49) for the eikonal equation. Note that, although we have considered here a point source, the above procedure is general and valid for any type of wavefront in the virtual space.

4.3.1 Rays and Wavefronts The transformation of phase through  (ρ) identifies the ray-paths as mapping of the radial lines emerging from the point source. The radial line associated with ρˆ  in a certain direction φ 0 (which traces a ray from the source) is parameterized as x =  cos φ 0 , y =  sin φ 0 with  ranging from 0 to infinity. Mapping this straight-line with  onto the real plane leads to the parametric curve x = x( cos φ 0 ,  sin φ 0 ), y = y( cos φφ 0 ,  sin φφ 0 ), which represents the curvilinear ray-path starting from the dipole in direction φ 0 (see Fig. 4.5b). Similarly, the wave-front at ρ 0 , which is defined in the virtual plane by x = ρ 0 cos t, y = ρ 0 sin t, with t ∈ (0, 2π) is transformed in the curved wavefront with the parametric form x = x(ρ 0 cos t, ρ 0 sin t), y = y(ρ 0 cos t, ρ 0 sin t), where t ranges from 0 to 2π. This allows for avoiding the sometimes complex issue of ray tracing.

4.3.2 Conformal and Quasi-conformal Mappings Particular cases of transformations are conformal mappings. These mappings satisfy the condition that angles of intersection between two lines are maintained after mapping. If we assume that the transformations  and  are conformal, they satisfy the

4.3 Flat Transformation Optics

71

  Cauchy–Riemann (CR) conditions. For the transformation  æ , these conditions are ∂x/∂x = ∂ y/∂ y , ∂x/∂ y = −∂ y/∂x which imply γ x · γ y = 0, |γ x | = |γ y |   2 2  2 and ∇t x  = ∇t y  . Therefore αt = ∇t x  I where I is the identity dyad, and from   (4.71) one has n eq = n eq / ∇t x , namely, the impedance is isotropic, and therefore realizable through geometrically regular printed elements. If the mapping is quasi  conformal, one can have a good approximation by using n eq = n eq / ∇ t x × ∇ t y . We note that in this case the Poynting vector is orthogonal to the wavefronts.

4.4 Comparison with Full-Wave Analysis We will consider here examples in which the equivalent refraction index and the ray paths are known in analytical form. We show a comparison between the present theory and a realistic model that uses subwavelength printed patches. The details of this latter design, for the provided examples and others, are given in the next section in which the process of synthesis of a modulated IBC is addressed. We consider first a Maxwell’s fish-eye (MFE) lens [17] and the Luneburg lens (LL) [2, 18]. These lenses are defined by the following equivalent refraction index  n eq (ρ) =

2n 0 /(1 + (ρ/R)2 ) (MFE) 1/2 n 0 [2 − (ρ/R)2 ] (LL)

(4.73)

for 0 < ρ < R and n eq = n 0 for ρ > R, where R is the lens radius. The geometrical and analytical details concerning the ray paths are given in Appendix D, for both MFE and LL.  The z-component of the electric field for an impedance reactance surface X = ζ 1 − n 2eq illuminated by a point source placed at x = −R, y = 0 is given by (4.15), (4.17) and (4.26) as  E z (ρ) = E z0 n eq exp

ρ ρ0

    ∇t2 s 2 − jkn eq − d − zk 1 − n eq 2n eq

(4.74)

where E z0 is a constant, ρ0 ≡ (x = −a, y = 0) is the position of the source point, and the integral is performed along the ray-path. Caustic points are avoided just stopping the integration close to them. The other components of the fields can be derived from (4.15) and (4.16). For both MFE and LL, the ray path and ∇t2 s can be evaluated in analytical form as shown in Appendix D. The MFE lens is a circular lens that focuses the rays coming from a point source placed on the lens rim on the diametrically opposite point [17]. Numerical results are presented in Fig. 4.6 for a single x-directed dipole. In the analytical calculations, in order to simulate the single dipole in FO framework, the rays emerging from the source point have been angularly weighted by cos γ (Fig. 4.6) where is the angle formed by the rays launched by the source with the x axis (see Appendix D). A small region around the focusing point is excluded from the FO calculation, since the FO

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Fig. 4.6 Maps of the real part of the z-component of the electric field for the MFE lens at 7.5 GHz, with radius R = 125 mm (3λ), observed at a distance of 5 mm (0.125λ) from the interface and calculated by (4.72) with cos(γ) tapering and an ideal reactance X (top picture), and compared with full-wave results of a modulated MTS obtained with circular patches and slotted circular patches in the capture. The circumference delimits the printed region. Excitation is provided by an x-directed dipole placed on the left hand side

model is not valid there. The results obtained by using (4.74) are compared with numerical results obtained through full-wave simulation of a MTS lens implemented by using printed patches (Fig. 4.6). Figure 4.7 shows a comparison between FO and full-wave analysis for a LL. The details of the geometry are given in the caption. We note that, while in the LL simple circular patches are used, the MFE lens requires the use of circular patches with a hole inside in order to increase the dynamic range of reactance. The agreement between the results obtained by the proposed formulation and full-wave analysis is very good for both MFE and LL. Consider now results relevant to a point source placed on top of an anisotropic uniform MTS with principal axes forming an angle δ = 41◦ with the x axis and ratio of the eigenvalues equal to X 1 = X 2 = 2.22. The effect of the anisotropy can be also seen as a linear coordinate transformation in Cartesian coordinates x = x ,  y = − tan θ0 x + y , where ρ ≡ x , y is the virtual plane and ρ ≡ (x, y) is the real plane. The angle θ0 of the transformation depends on δ, X 1 / X 2 and on n eq . For n eq = 1.26 one has θ0 = 15◦ . The effect of the linear Cartesian transformation on the cylindrical coordinates is shown in Fig. 4.8. The circles of the virtual plane are transformed into ellipses with major axis rotated by an angle δ = 41◦ with respect to the x axis. The radial lines are transformed in radial straight lines. This means, in our interpretation, that the circular wavefronts become elliptical wavefronts with Poynting vectors still along radial lines. In the same figure, the eikonal ellipses,

4.4 Comparison with Full-Wave Analysis Fig. 4.7 Maps of the real part of the z-component of the electric field for the Luneburg lens at 7.5 GHz, with radius R = 125 mm (3λ), observed at a distance of 5 mm (0.125λ) from the interface and calculated by (4.72) with cos(γ) tapering and an ideal reactance X (top picture), and compared with full-wave analysis of a MTS realized through circular patches in the capture. The circumference delimits the printed region. Excitation is provided by an x-directed dipole placed on the left hand side

Fig. 4.8 Virtual isotropic plane (x < 0) and plane transformed by the Cartesian linear transformation x = x , y = − tan (15◦ ) x + y (x > 0). The circles in the virtual plane (extended ideally for x > 0) are transformed into ellipses. The eikonal ellipses (inclined of an angle δ = 41◦ from the x-axes) are depicted, as well as the Poynting vector direction kˆ r and the normal to the wavefront kˆ t

73

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Fig. 4.9 Real part of the vertical electric field excited by a horizontal electric dipole at the MTS interface at f = 9 GHz. Upper figure, analytical FO results with a constant anisotropic impedance (X 1 = 0.47ζ, X 2 = 1.04ζ, eˆ 1 = cos δ xˆ + sin δ yˆ , eˆ 2 = − sin δ xˆ + cos δ yˆ , δ = 41◦ ). Lower figure: method of moments results relevant to a MTS obtained by printing the central zone with elliptical metallic dipoles shown in the caption

resulting from (4.71), are plotted in a lattice of intersection points among rays and wavefronts. Since the anisotropy is uniform, all the eikonal ellipses are oriented in the same direction δ. In the same picture, the normal to the wavefront kˆ t and the normal to the eikonal ellipses are also depicted. It is visible that these latter vectors ˆ as expected. Figure 4.9 are coincident with the radial unit vector, namely kˆ r ≡ ρ,

4.4 Comparison with Full-Wave Analysis

75

presents the real part of the vertical SW electric field excited by a horizontal elementary dipole. The result obtained with FO is compared with the one obtained from a full-wave method of moments analysis (Fig. 4.9). The latter is relevant to printed elements on a substrate with r = 14. The printed elements are subwavelength elliptical printed patches all rotated by 41◦ with respect x axes (see inset and caption for the exact dimensions in Fig. 4.9). The full wave results have been obtained by horizontal

Fig. 4.10 Geometry like in Fig. 4.9, with excitation given by seven half-wavelength spaced dipoles. The amplitude excitations of the dipoles are weighted by a cosine functions with nulls at halfwavelengths from the final dipoles

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4 Flat Optics for Surface Waves

(along x) printed dipoles. The analytical FO has been modified by weighting the rays emerging from the source point by cos(γ), as for the results in Figs. 4.6 and 4.7. The wavefront of the SW is elliptical as predicted in Fig. 4.8. The operational frequency is 9 GHz; the anisotropic-element region is located in a strip of width 30 cm; outside this strip, the substrate is unprinted; the height h = 1.55 mm is such as to realize an equivalent refractive index n eq = 1.26 as the one for the virtual plane. In both cases the field is observed a height of 5 mm, introducing in FO the field attenuation factor exp −hk X eq /ζ . The agreement is found excellent; the very small interference found in MoM is due to the second interface with the bare slab, not present in the FO model. Figure 4.10 shows the effect of an alignment of seven x-directed dipoles separated by a half-wavelength and weighted by a cosine function. The global effect—well replicated by our theoretical results (see Fig. 4.10)—is to orient the beam in direction θ0 = 15◦ while preserving the local wavefront flat. Actually, the angular deviation of the beam is due to the Huygens-like envelope of the elliptical wavefronts of each dipole contribution.

4.5 Synthesis of Modulated IBC As mention at the beginning of this chapter, in MTSs consisting of an arrangement of electrically small elements periodically printed on a grounded slab, a modulated IBC can be obtained by gradually varying the geometry of the elements in contiguous cells. The local dispersion equation in correspondence of each constituent element can be obtained by analyzing a periodic texture that locally matches the unit cell geometry. The local dispersion equation in correspondence of each constituent element can be obtained by analyzing a periodic texture that locally matches the unit cell geometry. A unit cell with rotational symmetry order higher than two (circular patch, square patch), leads to an approximately isotropic MTS and can be described by an equivalent refractive index. In order to obtain an anisotropic MTS one needs to synthesize a different response on the two principal axes. If the unit cell possesses two orthogonal symmetry axes, those principal directions can be identified, in the low frequency regime, with the principal axes of the isofrequency dispersion ellipse (see Fig. 4.3).

4.5.1 Luneburg and Maxwell’s Fish-Eye Lens As previously mentioned, the equivalent refraction indexes of the MFE lens and LL are known in analytical form (see Eq. (4.73)). The MTS yielding the local dispersion characteristics that correspond to (4.73) has been obtained by means of two types of elements (see Fig. 4.11) printed on a grounded slab with r = 9.8 and thickness h = 1.575 mm. The circular and slotted circular patches MTSs have been analyzed by using the approaches discussed in Chaps. 2 and 3, respectively. In both lenses,

4.5 Synthesis of Modulated IBC

77

Fig. 4.11 Wavenumber as a function of patch radius a for circular patches (dashed line) and as a function of the square slot side b for a slotted circular patch with radius a equal to 2.45 mm (continuous line)

the printed elements are arranged in a periodic lattice of square unit cells with side d = 5 mm (λ/8). The range of needed kt is [165, 270] for the LL and [165, 330] for the MFE lens and as observed in Fig. 4.11, they can be synthesized by using circular patches and combining both types of patches, respectively. The square slot allows one to increase the patch loading by extending the path of the currents. A full-wave analysis of the overall structures have been performed by using ADF [19]. The simulated structures consist of an infinite grounded dielectric slab with patches printed inside a circular area with radius R = 12.5 cm. A horizontal elementary monopole placed at x = 13 cm, y = 0 is used as excitation. The results are shown in Figs. 4.6 and 4.7. As expected, the SW rays launched by the dipole are all focused on the opposite side of the circumference in the MFE lens and transformed into parallel rays at the exit from LL.

4.5.2 Modified Luneburg Lens In the modified Luneburg lens, unlike in the classical LL, the focus is not at the periphery, but inside the lens. It is characterized by an equivalent refractive index profile given by R 2 + f 2 − ρ2 (4.75) n eq = n 0 f where ρ = x 2 + y 2 is the distance from the lens’ center, R is the lens’ radius, n 0 is the refractive index of the background and f is the distance of the focus from the center. The local refractive index (4.75) has been synthesized at 7.5 GHz by using both circular and square patches printed on a grounded slab with permittivity 9.8 and thickness 1.575 mm. Two cases, corresponding to f = R/2 (Fig. 4.12a) and f = 2R/3 (Fig. 4.12b), have been implemented with printed elements and analyzed with ADF [19]. In both cases, the printed elements are arranged in a periodic lattice with square unit cell of 5 mm size. A horizontal short monopole has been placed at the focal point. In the case f = R/2, the refractive index in (4.75) requires lower

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Fig. 4.12 Electric field distribution for a 26 cm diameter modified Luneburg’s lens at 7.5 GHz excited by a horizontal dipole at the focal point. a f = 8.6 cm, designed by circular patches, b f = 6.5 cm, designed by square patches. The circumference delimits the printed region; the insets show the local shapes of sub-wavelength patches used in the simulation

value of reactance, compatible with the use of circular patches. When the focal point is located more inside the lens, the increased level of reactance prevents the use of circular patches and imposes the use of square patches.

4.5.3 Beam Bender This subsection presents a planar (BB) defined through the coordinate transformation given in [14]. The BB is a device that rotates the direction of the beam propagation by 90◦ . Its MTS implementation has been designed at the operational frequency of 8.5 GHz, through circular patches printed on a grounded slab with a thickness of 1.55 mm and relative permittivity r = 14, within cells of size 5 mm × 5 mm. The dimensions of the patches, depicted in the inset of Fig. 4.13, have been set so as to match the local wavenumber along ρ-constant arcs of the coordinate transformation. In order to verify the proposed design, the beam bending device has been analyzed with the full-wave commercial software ADF [19]. The simulated structure consists of a grounded dielectric slab parallel to the x y plane over which a periodic arrangement of circular patches is printed in a quarter-section of the hollow cylindrical area, with inner and outer radii equal to 8 and 25 cm, respectively. A SW beam is used as excitation. The result is shown in Fig. 4.13. As expected, the beam is bent by 90◦ upon encountering the modulated MTS.

4.5 Synthesis of Modulated IBC

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Fig. 4.13 Snapshot of the electric field distribution for SW beam-bending device. The continuous lines delimit the printed area, and the inset shows the local shape of subwavelength patches used in the simulation

4.5.4 Beam Splitter The goal here is to design a MTS device that allows for splitting a SW beam in two directions with specular symmetry with respect to x axis, while leaving unchanged the wavefront direction. To this end, we consider the coordinate transformation presented in [14] for bending the Poynting vector of a SW beam by an angle θ0 without changing the wavefront direction. A SW beam shifter based on this transformation was analyzed in [3] by using an ideal tensor IBC, while the results shown below are synthesized with printed patches. The Jacobian of the required transformation is a constant matrix given by [14]: M (θ0 ) =

1 0 tan θ0 1

(4.76)

namely, the coordinate transformation is linear. Therefore, the MTS needed to implement the transformation is uniform. After applying (4.70) we find the eigenvalues of αt as tan θ0 1 4 + tan2 θ0 (4.77) σi = 1 + tan2 θ0 + (−1)i 2 2 with i = 1, 2 and its eigenvectors as eˆ i =

1 (σ3−i − 1)2 + tan2 θ0



σ3−i − 1 tan θ0

(4.78)

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Therefore, to design a MTS beam splitter one may simply use M(θ0 ) in the region x ∈ [−A, A], y ∈ [0, B] and M(−θ0 ) in the region x ∈ [−A, A], y ∈ [0, −B]. The designed beam splitter operates at 9 GHz and spans a rectangular surface with dimensions A = 30 cm, B = 35 cm, while the deflection angle is θ0 = 15◦ . In order to synthesize the required IBC, we use elliptical patches printed in 5 mm × 5 mm cells on a grounded slab with a thickness of 1.55 mm and relative permittivity r = 14. The elliptical patches, depicted in the inset of Fig. 4.14, possess two symmetry axes and they are characterized by three parameters: (a) the orientation angle ψ, which primarily affects the orientation of the principal axes of the reactance tensor, (b) the elliptical patch’s major axis a, which controls the magnitude of the larger principal value of the reactance tensor, and (c) a non-dimensional parameter b/a representing the ratio between the minor and the major axes of the ellipse, which controls the anisotropy of the reactance tensor. On the basis of periodic MoM results (proposed in Chap. 2), the values of the geometrical parameters of the unit cell have been set so as to approximate the desired elliptical dispersion curve in the low frequency regime. The elliptical patch major and minor axis lengths are equal to 4 mm and 0.7 mm, respectively, and the rotation angle of the elliptical patch is ψ = 41◦ . In the upper region, the elliptical patches are oriented along the opposite direction and have identical dimensions. The IDC obtained from the full-wave simulation is compared in Fig. 4.14 with the objective isofrequency dispersion ellipse retrieved from the transformation. Excellent agreement is observed among these curves. Figure 4.15 presents the field distribution provided by the full wave simulation of the whole structure performed with ADF [19]. As expected, the SW is splitted in two beams shifted by 15◦ upon encountering the anisotropic MTS, while the wavevector direction is almost unchanged. Due to anisotropy, over the elliptical patch MTS the Poynting vector is not orthogonal to the wavefront.

Fig. 4.14 Comparison between the IDC provided by the coordinate transformation (solid line) and the one obtained for the MTS (dots). The MTS consists in the elliptical patches shown in the inset

4.6 Chapter Summary

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Fig. 4.15 Snapshot of the electric field distribution for a SW beam-splitting device. The overall dimensions of the anisotropic MTS are 60 × 70 cm

4.6 Chapter Summary A general methodology has been presented for analysing flat devices realized using MTSs. In these devices, the wavefront of the SW is controlled by designing boundary conditions. A general Flat Optics theory describing ray-paths, ray velocity, Poynting vector and transport of energy is derived for both isotropic and anisotropic IBCs. In particular, it is shown, for both isotropic and anisotropic case, that the ray paths are governed by eikonal equations analogous to the one for Geometrical Optics rays. The link with flat Transformation Optics is also illustrated. Finally, practical realizations of modulated isotropic/anisotropic MTSs, which allow to manipulate the SWs by control of the IBCs, have been shown. In all the cases, MTSs consisting of small patches printed on a grounded slab have been used. The performances of the devices have been verified through full-wave simulations.

References 1. Yu N, Genevet P, Aieta F, Kats MA, Blanchard R, Aoust G, Tetienne JP, Gaburro Z, Capasso F (2013) Flat optics: controlling wavefronts with optical antenna metasurfaces. IEEE J Sel Top Quantum Electron 19(3):4700423–4700423 2. Maci S, Minatti G, Casaletti M, Bosiljevac M (2011) Metasurfing: addressing waves on impenetrable metasurfaces. IEEE Antennas Wirel Propag Lett 10:1499–1502 3. Patel AM, Grbic A (2014) Transformation electromagnetics devices based on printed-circuit tensor impedance surfaces. IEEE Trans Microw Theory Tech 62(5):1102–1111 4. Martini E, Maci S (2014) Metasurface transformation theory. In: Werner DH, Kwon D-H (eds) Transformation electromagnetics and metamaterials. Springer, London, pp 83–116

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5. Vakil A, Engheta N (2011) Transformation optics using graphene. Science 332(6035):1291– 1294 6. Tang W, Argyropoulos C, Kallos E, Song W, Hao Y (2010) Discrete coordinate transformation for designing all-dielectric flat antennas. IEEE Trans Antennas Propag 58(12):3795–3804 7. Bosiljevac M, Casaletti M, Caminita F, Sipus Z, Maci S (2012) Non-uniform metasurface Luneburg lens antenna design. IEEE Trans Antennas Propag 60(9):4065–4073 8. Felsen LB (1976) Evanescent waves. J Opt Soc Am 66(8):751–760 9. Pendry JB (2000) Negative refraction makes a perfect lens. Phys Rev Lett 85:3966–3969 10. Born M, Wolf E (1980) Principles of optics, 6th (corrected) edn. Pergamon, Oxford 11. Sharma A, Kumar DV, Ghatak AK (1982) Tracing rays through graded-index media: a new method. Appl Opt 21(6):984–987 12. Jenkins C, Bingham R, Moore K, Love GD (2007) Ray equation for a spatially variable uniaxial crystal and its use in the optical design of liquid-crystal lenses. J Opt Soc Am A 24(7):2089–2096 13. Leonhardt U (2006) Optical conformal mapping. Science 312(5781):1777–1780 14. Kwon DH, Werner DH (2010) Transformation electromagnetics: an overview of the theory and applications. IEEE Antennas Propag Mag 52(1):24–46 15. Yaghjian AD, Maci S (2008) Alternative derivation of electromagnetic cloaks and concentrators. New J Phys 10(11):115022 16. Schurig D, Mock JJ, Justice BJ, Cummer SA, Pendry JB, Starr AF, Smith DR (2006) Metamaterial electromagnetic cloak at microwave frequencies. Science 314(5801):977–980 17. Born M, Wolf E (1980) Foundation of geometrical optics. In: Born M, Wolf E (eds) Principles of optics, 6th (corrected) edn. Pergamon, Oxford, pp 109–132 18. Pfeiffer C, Grbic A (2010) A printed, broadband Luneburg lens antenna. IEEE Trans Antennas Propag 58(9):3055–3059 19. ADF-EMS: antenna design framework and electromagnetic satellite. IDS S.p.A., Pisa, Italy. www.idscorporation.com

Chapter 5

Basic Properties of Checkerboard Metasurfaces

Checkerboard metasurface (CBMS) consists of an infinitesimally thin layer of electrically small complementary square metallic patches and apertures (Fig. 5.1). The name MTS is adopted here because the periodicity d of the checkerboard lattice is small in terms of the wavelength so that the global effect is the same as that of a continuous impedance surface. Propagation of waves driven by impedance discontinuity has been investigated in [1]. The present solution can be viewed as an implementation of this concept. In fact, the behavior of the CBMS changes depending on whether the square patches’ vertexes are connected by short-circuits or separated by a small gap. The current distribution on the connected patches is also shown in the inset in Fig. 5.1. In the following, it is shown that a checkerboard layer printed on a grounded slab may support a mode confined in the region where patches are connected. The main advantages of the present configuration over other slow-wave structures are the following: (i) the supported mode exhibits a very low dispersion over a very large bandwidth; (ii) the propagation path of the supported mode can be directed by acting on an extremely small region at the vertexes. This can be efficiently done by locally changing the effective conductivity of a photo-sensitive substrate [2–4] (e.g. Si) through highly-focused light spot beams. Therefore, the CBMSs’ properties offer the possibility of defining transmission lines embedded in the CBMS by locally changing the connection status of the patches’ vertexes. This may allow one to dynamically control beam-forming networks or to change the resonant frequency of planar antennas [5]. This chapter illustrates the basic operating principle of this novel metasurface, and its application to the design of transmission lines and planar antennas. A preliminary experiment with a CBMS transmission line printed on a semiconductor substrate shows the potential of using photo-conductive switches for reconfiguring the CBMS.

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5.1 Connected Patch Transmission Lines First, let us explore the CBMS capability of supporting a propagating mode along one line of diamond-oriented patches connected through their vertexes (see Fig. 5.1). The structure used to illustrate this phenomenon consists of a CBMS where the square metallic patches (b = 0.84 mm) are printed on a grounded substrate with relative permittivity r = 10.2 and thickness h = 0.635 mm. The dispersion analysis of the proposed transmission line has been carried out with the eigenmode solver of CST Microwave Studio [6]. The structure is composed of 7 disconnected patches along y and it is infinitely periodic along x, with only the central patch electrically connected to the adjacent cells. Numerical results confirm the expected presence of a mode without cut-off frequency and confined in the region of connected patches. The dispersion diagram of the dominant mode is shown in Fig. 5.2 and compared with the dispersion diagram of a microstrip line with the same characteristic impedance (width a = 0.60 mm). It is observed that the dispersion characteristic of the CBMS transmission line is closer to the light line in the dielectric than the one of the microstrip line and it is almost non-dispersive in a large frequency band. Only at higher frequencies does the curve diverge from the dielectric light line.

Fig. 5.1 Checkerboard metasurface, where the metallic patches (dark grey) are all disconnected except along the central line. The insets schematically show the behavior of the currents for connected patches in presence of an external source Fig. 5.2 Dispersion curves for the CBMS transmission line, and for a microstrip line of width a = 0.60 mm. The continuous line has been obtained using (5.5) and the circuit model in Sect. 5.1, whereas the white dots on top of it represent the results from the CST analysis

5.1 Connected Patch Transmission Lines

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Fig. 5.3 Equivalent circuit for a unit cell of the CBMS transmission line

On the other hand, the supported mode is quasi-TEM, like in a conventional microstrip line. It is important to note that, since the modal fields are extremely localized around the connected patches, the effect of the presence of the surrounding disconnected elements is practically negligible. This legitimates the definition of a model based on an isolated line of connected patches. The inspection of the geometry of the periodic unit cell suggests an equivalent circuit as the one shown in Fig. 5.3. The elements of this circuit can be found as

L C2 B

√

 √ L C1 B = μ0 h 2 2ε0 εr b2 h   2π h  1 − L C2 B L C1 B 1 1 CB CB  = 2π μ0 wlog w C2 = 2 C1 1 + L C2 B L C1 B

C1C B =

(5.1)

where w is the width of the small connection between adjacent patches (see Fig. 5.3). C1C B and L C1 B represent the capacitance and inductance, respectively, of a square √ portion of a parallel plate waveguide, with a 2 geometrical shape factor. L C2 B represents the inductance of the small section of microstrip line used to connect two adjacent cells. The latter is calculated as suggested in [7]. The capacitance C2C B accounts for the fringing phenomenon and its expression is justified afterwards. The CBMS transmission line is obtained by periodically cascading the unit cell of Fig. 5.3. The resulting structure can be analyzed applying Bloch theory. To this end, the ABCD matrix [8], which relates the voltage and current input variables Vn ,In to the output ones Vn+1 ,In+1 (with the convention shown in Fig. 5.3 for the current direction) is found as 1 A = D = 1 − ω2 (C1C B + 2C2C B )(L C1 B + L C2 B ) 2  1  1 + ω4 C1C B L C1 B C2C B L C2 B + C1C B (L C1 B )2 C2C B + L C1 B (C2C B )2 L C2 B 2 2 1 6 CB CB 2 CB 2 CB − ω C1 (L 1 ) (C2 ) L 2 8 (5.2) j CB CB 2 CB CB CB B = − ω(2L 1 + 2L 2 − ω L 1 C2 L 2 ) (5.3)  2 16C B C B ω (2C1 (L 1 + L C2 B ) + 4C2C B L C2 B ) − ω4 C1C B L C1 B C2C B L C2 B − 8   j  C = − ω ω2 L C1 B C2C B − 2 −ω2 C1C B L C1 B C2C B + 2C1C B + 4C2C B 4

(5.4)

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5 Basic Properties of Checkerboard Metasurfaces

Fig. 5.4 Scattering parameters for a single-row and a double-row CBMS transmission line; b = 0.84 mm, h = 0.635 mm, r = 10.2

Since the network in Fig. 5.3 is reciprocal and symmetric we obtain the following dispersion equation cos(βd) = (A + D)/2 = A (5.5) where d is the periodicity in the Bloch analysis. The numerical results reported in Fig. 5.2 suggest that this equations should closely match the dispersion line for the dielectric medium. The enforcement of this condition on the second order ω-expansion of (5.5) leads to the identity (C1C B + 2C2C B )(L C1 B + L C2 B ) = μ0 0 r d 2 = 2C1C B L C1 B , which implies that C2C B can be written as in (5.1). Solving (5.5) for ω leads to the solid black line in Fig. 5.2, which shows a very good agreement with CST’s full-wave results (white dots). Moreover, since AD − BC = 1 and A = D, the√Bloch characteristic impedance at the reference plane in Fig. 5.3 is given by Z B = B/C [8]. Therefore, the characteristic impedance at low frequency is obtained as  



L C2 B B ω→0 L C1 B + L C2 B L C1 B 1 + (5.6) −−→ = ZB = C C1C B + 2C2C B 2C1C B L C1 B Regarding CBMS transmission lines with more than one row of connected patches, their equivalent circuit consists of a parallel connection of N circuits like the one in Fig. 5.3, with N being the number of rows in the CBMS line. Hence, for an N rows CBMS transmission line, the characteristic impedance will be Z B /N , where Z B is obtained from (5.6). Moreover, one can verify through a Bloch analysis of

5.1 Connected Patch Transmission Lines

87

this new equivalent circuit that the dispersion characteristic for an N -rows CBMS transmission line is identical to the one-row case. Two CBMS transmission lines have been analyzed with the time domain solver of CST. Both lines are printed on the same substrate, with r = 10.2 and h = 0.635 mm. The first CBMS transmission line is made of a single row of connected patches with b = 0.84 mm, which yields a 50  characteristic impedance. The second line consists of two rows of connected patches with identical dimensions, thus, its characteristic impedance is halved to 25 . In the simulated structure, each CBMS transmission line branch is connected at the two endings to a microstrip line which width has been computed [8, (3.165)] to match the impedance values of 50 and 25  (a = 0.60 mm and a = 1.84 mm, respectively). Figure 5.4 shows the reflection S11 and the transmission S21 coefficients at the microstrip ports as a function of frequency.

5.2 Experimental Results 5.2.1 CBMS Transmission Lines Figure 5.5 shows the S21 measurements of four prototypes, labeled as A, B, C, and D. Prototype A is a 50  microstrip line, with a total length of 224 mm, and printed on a Rogers 6010 substrate (h = 0.635 mm, r = 10.2 ± 0.25, a = 0.64 mm, tan δ = 0.023 at 10 GHz). Prototype B consists of a single-row CBMS transmission line printed on the same substrate, and connected at its end points to two short sections of microstrip line. The length of the CBMS transmission line is 194 mm, and the two sections of microstrip line are 15 mm long. The dimensions of the checkerboard and the width of the microstrip are the same as for the 50  lines in Sect. 5.1. Prototype C is a microstrip L-type bend, and prototype D is its equivalent CBMS version. The total length of the line is 87 mm for both C and D. The propagation effect in the CBMS transmission line is evident in both the rectilinear and the bent cases, which demonstrates the validity of the concept. However, it appears that the checkerboard lines exhibit larger losses with respect to the equivalent microstrip lines. The main difference is to be attributed to the fact that the effective permittivity of a checkerboard transmission line is very close to r , whereas the one of the microstrip is closer to (r + 1)/2. The additional losses with respect to an equivalent microstrip are estimated (for tan δ = 0.023 at 10 GHz) at about 0.0025 dB/cm at 2 GHz and 0.005 dB/cm at 6 GHz.

5.2.2 CBMS Patch Antennas Figure 5.6 presents a comparison between measurements on a conventional patch antenna at 1.75 GHz, and on an equivalent patch antenna realized by connecting the vertexes inside a CBMS. Both prototypes have been printed on the same substrate

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5 Basic Properties of Checkerboard Metasurfaces

Fig. 5.5 Measured S21 for an ordinary microstrip transmission line compared to a single-row CBMS transmission line for the prototypes shown on top

used for the circuits in Fig. 5.5, and both antennas are excited by a 50  transmission line with a matching inset feed. The contour of the area inside which the checkerboard-patches are connected is depicted in the photo of the CBMS antenna (inset of Fig. 5.6a, right side), and it is equal to the dimension of the conventional patch antenna (left side). Note that the transmission line with the feed inset is also realized by properly connecting the vertexes of the CBMS patches. The CBMS antenna presents a shift of the resonant frequency equal to 35 MHz (2%) with respect to the conventional patch. The bandwidths of the two antennas (which are narrow due to the high permittivity substrate) are also comparable (around 1 and 1.3% at −10 dB matching level). Figure 5.6b also shows the measured results for the radiation pattern. It is found that the radiation patterns of the two prototypes are very similar in both E and H planes.

5.2.3 Preliminary Results for Optically Tunable CBMS In the previous section, different microwave functions have been demonstrated with the same CBMS. Hence, once each vertex on the CBMS can be individually controlled, the CBMS will become fully reconfigurable. In this way, a CBMS printed

5.2 Experimental Results

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Fig. 5.6 Comparison between a conventional patch antenna and an antenna realized by connecting patch vertexes in a CBMS. a: reflection coefficient (left scale), and gain (right scale). The contour profile of the patch is drawn in the picture of the CBMS antenna. Enlarged detail of the inset-feed is shown on the right hand side. b Normalized radiation pattern in the E-plane (left) and H-plane (right)

on a high-resistivity semiconductor substrate will allow one to exploit the photoconductive effect for changing the electrical status of the vertexes from disconnected to connected. Figure 5.7a shows a Si-based CBMS in an optoelectronic experimental environment, where a 32.8 mW laser source at wavelength of 0.8 µm is focused on a spot of size 31.6 µm to ensure the electrical connection between two patches using the photoconductive properties of the Silicon. The experimental results, reported in Fig. 5.7b, show the S21 when the optical switch is illuminated with different levels of optical

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Fig. 5.7 a Experimental setup for the optical control of a CBMS transmission line. b Measured S21 as a function of optical power of the laser focused on a spot of size 31.6 µm. Note the different scale for laser off curve

power. The gap between disconnected vertexes is 15 µm. The Silicon Oxide layer (used for passivation) has been removed in the gap zone. One can observe that the S21 increases by more than 34 dB at 1 GHz under 32.8 mW illumination. At 10 GHz the increase is about 14 dB for the same optical power.

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5.3 Chapter Summary A new metasurface consisting of a checkerboard type structure printed on a grounded slab has been introduced. This metasurface supports propagation of a quasi-TEM mode when the vertexes of contiguous patches are connected. Starting from an equivalent network and a Bloch analysis, we have designed and measured several elementary RF circuits and one patch antenna in the GHz range. This first demonstration of devices based on checkerboard metasurface allows one to envision new functionalities such as tunable antennas or dynamically reconfigurable transmission lines in beam forming networks. This is related to the possibility of locally changing the microwave behavior of the metasurface by an optical beam focused on a very small area around disconnected vertexes. Preliminary experimental results show promising perspective in this regard.

References 1. Quarfoth R, Sievenpiper D (2013) Artificial tensor impedance surface waveguides. IEEE Trans Antennas Propag 61(7):3597–3606 2. Tripon-Canseliet C, Faci S, Pagies A, Magnin V, Formont S, Decoster D, Chazelas J (2012) Microwave on/off ratio enhancement of GaAs photoconductive switches at nanometer scale. J Lightwave Technol 30(23):3576–3579 3. Panagamuwa C, Chauraya A, Vardaxoglou J (2006) Frequency and beam reconfigurable antenna using photoconducting switches. IEEE Trans Antennas Propag 54(2):449–454 4. Haupt RL, Flemish J, Aten D (2009) Broadband linear array with photoconductive weights. IEEE Antennas Wirel Propag Lett 8:1288–1290 5. Christodoulou C, Tawk Y, Lane S, Erwin S (2012) Reconfigurable antennas for wireless and space applications. Proc IEEE 100(7):2250–2261 6. Computer Simulation Technology (2012) CST microwave studio, Darmstadt, Germany 7. Kim KH, Aine JES (2008) Analysis and modeling of hybrid planar-type electromagneticbandgap structures and feasibility study on power distribution network applications. IEEE Trans Microw Theory Tech 56(1):178–186 8. Collin RE (1992) Foundations for microwave engineering, 2nd edn. McGraw-Hill, New York

Chapter 6

Conclusion

6.1 Summary of Contributions In this thesis we have theoretically and numerically examined the prospects of modulated metasurface (MTS) as a new platform for planar transformation optics devices, which allow one to manipulate the propagation of the supported surface waves (SWs). Such devices can be used as a part of flat lens antennas producing a fan-beam radiation pattern at microwaves, but can be also employed as components of optical circuits. In Chaps. 2 and 3 we have described two accurate and efficient approaches for the analysis of MTSs consisting of metallic patches printed on a grounded slab. The first technique is restricted to metallic patches of elliptical shape. It resorts to an analytical expression of the currents excited on the elements, then a spectral MoM formulation is used to build the matrix of a periodic MoM formulation with only two/three basis functions. This procedure can be used for the design of planar lenses, transformation optics devices (see Chap. 4) and leaky wave antennas [1]. Unfortunately, this kind of MTS is not able to provide an high impedance value and an high range of anisotropy. Therefore, in MTS-based applications that require these characteristics, constitutive elements of different geometry are needed. To this end, we have presented a second approach for the analytical characterization of anisotropic MTS, which is valid for constitutive elements that possess at least two orthogonal axis. It relies on a simple analytical form of the isofrequency dispersion curve which depends on two parameters only. These parameters are the equivalent quasi-static capacitances along the symmetry axes of the element, which can be derived by the expressions based on equivalent gap capacitance (for simple shapes) or by extracting them from a single run of a full-wave eigensolver. In addition, a closed-form representation of the group velocity and the limits of the validity of the formulation have been derived. This approach has been found useful for the design of SW devices based on modulated MTSs (see Chap. 4) and simplifies the design of modulated MTS antennas [2–8].

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6 Conclusion

Chapter 4 introduced the concept of Flat Optics for surface waves (SWs), denoting the manipulations of SWs through modulated surface impedance boundary conditions. All aspects relevant to ray-optics, such as ray-tracing, transport of energy, and ray-velocity, have been rigorously treated for both isotropic and anisotropic surface impedances. We would like to emphasize that this formulation allows one to solve an issue which has not been treated in the literature, namely, the estimation of the amplitude of the field distribution over a surface that supports an anisotropic non-uniform, impenetrable impedance tensor. The theory, presented in this chapter, results in an elegant formulation which leads to closed form analysis of planar operational devices based on modulated MTS. We provided various examples, suggesting that modulated MTSs can be a good platform for planar wave-guiding structures and transformation optics devices. Although the presented examples are focused in the microwave range, the proposed theory is also applicable in the terahertz and infrared regions, as well as at optical frequencies provided that one possesses the right technology for MTS implementation. Chapter 5 is devoted to the theoretical description of a type of MTS that is specially well suited for its integration with photo-conductive switches. The MTS consists of a chessboard-type layout, made of electrically small complementary metallic patches and apertures on a grounded substrate. Depending on whether the patch vertexes are interconnected or not, the structure may support a propagating quasi-TEM mode. This offers the possibility of designing arbitrary transmission line paths on the MTS by dynamically changing the vertex connections. Moreover, one may combine several rows of patches to tailor the characteristic impedance. The structure has been analyzed to determine its equivalent circuit, which allows us to predict the dispersion equation of the transmission lines and their characteristic impedance. The proposed model has been verified using full wave simulations. Both analytical and simulated results are in excellent agreement with measurements done on structures manufactured using an ideal switch, i.e., a small metalization. Finally, it is important to notice that the proposed MTS-based transmission line has a constant characteristic impedance regardless of the dimension of the vertex. This has been exploited to use micrometer scale gaps that allowed us to confine the optical beam in a small area, hence reducing the losses due to the presence of the photo-conductive switches. Preliminary results have been shown.

6.2 Future Directions One of the aims of this thesis has been to provide an improved understanding of the guidance characteristics of isotropic and anisotropic modulated surface impedances, paving the way for the design of a large number of SW devices based on modulated MTSs. In addition, we have developed methods which make easier the design of wave-guiding MTSs. Another goal in this thesis has been the design of SW devices based on modulated MTS as well as the study of a novel approach to reconfigurability based on checkerboard MTS. Moreover, there are just various directions to pursue

6.2 Future Directions

95

on the topics we introduced in this thesis. In the following we refer to some of the research lines that could be followed in the future. Chapter 3 presented an analytical form of the resonant reactance and of the isofrequency dispersion curve for isotropic and anisotropic MTSs, respectively. The presented formulations were restricted to MTSs constituted by patch elements. Therefore, future work could include extending of the presented formulation to slot-type MTSs. In Chapter 4, a method for designing planar and transformation optics devices using MTSs has been introduced. In this thesis, we have presented the design of Luneburg lens, Maxwell’s fish-eye lens, modified Luneburg lens, a beam shifter, a beam splitter and a beam bender. Some work for the near future could include fabricating these devices and measuring the field distribution on the non-uniform MTS. As pointed out in Chap. 5, a checkerboard MTS is an excellent host for optically reconfigurable devices at microwave frequencies. However, the optical switch is still too lossy. The obtained S21 (−5 dB) can be further reduced (i) with a smaller vertex gap [9], (ii) increasing the optical power [10], (iii) using a single mode optical fiber to couple energy from the laser source, or (iiii) using doped semiconductors [11, 12]. This could be the subject of future investigations as well as the development of an accurate and efficient formulation which allows to account the losses due to the optical switch because commercial softwares usually fail to predict the measured losses. Once these issues have been addressed, checkerboard MTS will allow one to envision new functionalities such as tunable antennas or dynamically reconfigurable transmission lines in beam forming networks.

References 1. Caminita F, Martini E, Minatti G, Maci S (2016) Fast integral equation method for metasurface antennas. In: 2016 URSI international symposium on electromagnetic theory (EMTS), pp 480–483 2. Tellechea A, Caminita F, Martini E, Ederra I, Iriarte JC, Gonzalo R, Maci S (2016) Theoretical design considerations for dual circularly-polarized broadside beam metasurface antenna. In: Proceedings of the 10th European conference on antennas and propagation, pp 1–3 3. Fong BH, Colburn JS, Ottusch JJ, Visher JL, Sievenpiper DF (2010) Scalar and tensor holographic artificial impedance surfaces. IEEE Trans Antennas Propag 58(10):3212–3221 4. Minatti G, Caminita F, Martini E, Sabbadini M, Maci S (2016) Synthesis of modulatedmetasurface antennas with amplitude, phase, and polarization control. IEEE Trans Antennas Propag 64(9):3907–3919 5. Minatti G, Caminita F, Martini E, Maci S (2016) Flat optics for leaky-waves on modulated metasurfaces: adiabatic floquet-wave analysis. IEEE Trans Antennas Propag 64(9):3896–3906 6. Minatti G, Maci S, Vita PD, Freni A, Sabbadini M (2012) A circularly-polarized isoflux antenna based on anisotropic metasurface. IEEE Trans Antennas Propag 60(11):4998–5009 7. Faenzi M, Caminita F, Martini E, Vita PD, Minatti G, Sabbadini M, Maci S (2016) Realization and measurement of broadside beam modulated metasurface antennas. IEEE Antennas Wirel Propag Lett 15:610–613

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6 Conclusion

8. Minatti G, Faenzi M, Martini E, Caminita F, Vita PD, González-Ovejero D, Sabbadini M, Maci S (2015) Modulated metasurface antennas for space: synthesis, analysis and realizations. IEEE Trans Antennas Propag 63(4):1288–1300 9. Tripon-Canseliet C, Faci S, Pagies A, Magnin V, Formont S, Decoster D, Chazelas J (2012) Microwave on/off ratio enhancement of GaAs photoconductive switches at nanometer scale. J Lightwave Technol 30(23):3576–3579 10. Panagamuwa C, Chauraya A, Vardaxoglou J (2006) Frequency and beam reconfigurable antenna using photoconducting switches. IEEE Trans Antennas Propag 54(2):449–454 11. Döhler GH, Ruden PP (1984) Theory of absorption in doping superlattices. Phys Rev B 30:5932–5944 12. Döhler GH (1990) Electro-optical and opto-optical devices based on n-i-p-i doping superlattices. Superlattices Microstruct 8(1):49–58

Author Biography

Mario Junior Mencagli received the B.Sc. and M.Sc. degree (cum laude) in Telecommunications Engineering from University of Siena, Italy, in 2008 and 2013, respectively. In 2016, he received the Ph.D. degree (cum laude) in electromagnetics from University of Siena, Italy, under the supervision of Prof. Stefano Maci. His Ph.D. program has been sponsored by Thales Research and Technology, Paris, France, where he spent a few months as a visiting student researcher in 2015. Dr. Mencagli is serving as a reviewer for several journals and international conferences. Since January 2017, Dr. Mencagli is a Postdoctoral Researcher in Prof. Nader Engheta’s group at University of Pennsylvania, Philadelphia, USA. Dr. Mencagli has been working on reconfigurable metasurface, periodic structures, circuit implementation, numerical methods for electromagnetic problems, high-frequency techniques for electromagnetic scattering, metamaterials for both microwave and optical regimes, transformation optics, metatronic, filter at optical and UV frequencies, and analog computing. He was also involved in measurements of optically reconfigurable transmission lines based on checkerboard metasurfaces.

© Springer Nature Switzerland AG 2019 M. J. Mencagli, Manipulation of Surface Waves through Metasurfaces, Springer Theses, https://doi.org/10.1007/978-3-030-14034-2

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Appendix A

Asymptotic Evaluation of Grounded Slab Green’s Function

The TM and TE GF impedances of the grounded slab can be expressed in the spectral domain as E Z GT M,T (k, ω) = F

1 YGT FM,T E

where Y0T M (k) = Y0T E (k) =

=

1 Z 0T M

(k)

1 Z 0T E

(k)

1 Y0T M,T E

(k) −

jY1T M,T E

(k) cot (k z1 h)

(A.1)

=

ωε0 1 ωε0 εr ; Y1T M (k) = T M = kz k z1 Z 1 (k)

(A.2)

=

1 kz k z1 ; Y1T E (k) = T E = ωμ0 ωμ0 Z 1 (k)

(A.3)

are the modal z-transmission line TM/TE characteristic admittances relevant to the free-space (subscript 0) and to the dielectric (subscript 1) regions, respectively. In (A.2)–(A.3) thefunctional dependence on ω is understood and suppressed. Furthermore, k z = k 2 − k x2 − k 2y , k z1 = εr k 2 − k x2 − k 2y , where k is the free-space wavenumber. For large values of kρ we have k z → − jkρ and k z1 → − jkρ , the function − j cot (k z1 h) in (A.1) tends to unity, thus we have TM Z GT M F (k, ω) → Z ∞ (k, ω) =

kρ jωε0 (εr + 1)

(A.4)

jωμ0 2kρ

(A.5)

TE Z GT EF (k, ω) → Z ∞ (k, ω) =

Therefore, in (2.16) we have (X = T M, T E)   (ψ,η) (ψ,η)∗ X X ζmn (k, ω) = Z GX F (k, ω) − Z ∞ (k, ω) J X,n (k) J X,m (k)

(A.6)

from which one gets the decomposition in (2.17)–(2.21). © Springer Nature Switzerland AG 2019 M. J. Mencagli, Manipulation of Surface Waves through Metasurfaces, Springer Theses, https://doi.org/10.1007/978-3-030-14034-2

99

Appendix B

Derivation of the Validity Conditions for Homogenization of the MTS-Impedance

When only the dominant (q = (0; 0)) modes are accessible at the ground plane, the asymptotic expressions in (A.4) and (A.5) provide a good low frequency approximation of the GF admittance for all q except for q = (0, 0). This implies 

     Z˜ slab = Q˜ H Z G F Q˜

(B.1)



 TE ˜ where Z G F = diag Z GT M F (k0 , ω) , Z G F (k0 , ω) and [ Z slab ] is like [Z slab ] with suppressed 0-indexed row and column. After inserting (B.1) in (2.17) and using (2.25), we obtain (2.26). By exploiting Eq. (2.26), [Z M T S ] can be approximated as in (2.27) with −1  −1  {lmn (k0 )}m,n=1,2 Q˜ {L lk (k0 )}l,k=1,2 = Q˜ H  −1   −1  1 1 Q˜ = Q˜ H Clk (k0 ) l,k=1,2

(B.2)

cmn (k0 ) m,n=1,2

 

(ψ,η) (ψ,η) with Q˜ = Q i j i, j=1,2 , Q 1, j = JT M, j (k0 ) , Q 2, j = JT E, j (k0 ).

The accuracy of the approximation (2.27) is controlled by imposing [Z dyn ]  [Z L F ]. This condition is fulfilled if  X  X √   Z G F (k, ω) − Z ∞ (k, ω)  

ln λsw d d 2h λsw (εr + 1)  where λsw = 2π/kρsw and δ = π (εr 2 + 1)/[2 (εr + 1)]. Observing that in the frequency range of interest for practical applications of the MTS λ/2 < λsw < λ and that εr / (εr + 1) > 1/2, from (A.6) one obtains that sufficient conditions for (B.5) to hold are h λ (B.6) δ+1< ; d < d 0.23 + 2h λ The first (second) condition is more restrictive for h/d > τ , (h/d < τ ) where τ = 0.23 (1 + δ) / (1 + 2δ). It can also be shown that  TE   TE   ZG F − Z∞ k2   ≈  1    4  Z GT EF kρsw −



 2π 2 d

(1 + εr ) −

  2π    −2h kρsw −   d e 



(B.7)

from which one can see that also the error on the TE component of the GF is controlled by the conditions in (B.5).

Appendix C

Solution of Third-Degree Algebraic Equation Relevant to Impenetrable Impedance of Isotropic MTS

Equation (3.8) can be reduced to Ax 3 + Bx 2 + C x + D = 0

(C.1)

where x = X/ζ and A = ωCeq ζ, B = 1, C = −ωCeq (εr − 1) ζ +

εr ζ , D = 1 − εr ωμ0 h

(C.2)

It is seen that the quadratic term is always negligible in the solution, thus allowing rewriting the previous equation as x 3 + px + q = 0 with

  2 ωeq C = (εr − 1) p= −1 A ω2 q=

D (εr − 1)  = − A ωCeq ζ

(C.3)

(C.4)

(C.5)

The solution of (C.3) is given by the Cardano’s formula [1]           1 1 4 4 3 3 p3 +  p3 x = −q + q 2 + −q − q 2 + 2 27 2 27

(C.6)

√ By posing ξ = (|q| / p) 27/(4 p) one can find the transition function by an appropriate choice of the branch of the cubic root in (C.6) and by observing that, for low frequency, the normalized reactance can be approximated by x ≈ |q|/ p. © Springer Nature Switzerland AG 2019 M. J. Mencagli, Manipulation of Surface Waves through Metasurfaces, Springer Theses, https://doi.org/10.1007/978-3-030-14034-2

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104

Appendix C: Solution of Third-Degree Algebraic Equation Relevant …

Reference 1. Cardano G (1968) Artis magnae sive de regvlis algebraicis (English translation: The great art or the rules of algebra). MIT Press, Cambridge

Appendix D

MFE and LL Differential Geometry

The ray-paths for a MFE lens and a LL, are given by F(x, y, b) = 0 where  F(x, y, b) =

2 2 2 x2 + (|y| +  b) − R − b 2 MFE 2 2 b ρ − R − 2x |y| b + 2y LL 2

(D.1)

 where R is the radius and ρ = x 2 + y 2 . The parameter b is related to the angle γ between the ray and the x axis at the source point (see Fig. D.1) as  γ=

tan−1 (R/b) MFE LL tan−1 b

(D.2)

For the MFE lens, the ray-path are portion of circles centred at (x, y) = (0, −b); for the LL they are portions of ellipses centred at the origin of the reference system ˆ  of an angle γ/2. The unit tangent to the ray path is given by kt (b) = and rotated ∇t F × zˆ / |∇t F| where b is kept constant in making the gradient. The expression

Fig. D.1 Geometrical construction of the ray paths (red lines) of Maxwell’s fish-eye (a) and Luneburg lens (b) © Springer Nature Switzerland AG 2019 M. J. Mencagli, Manipulation of Surface Waves through Metasurfaces, Springer Theses, https://doi.org/10.1007/978-3-030-14034-2

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106

Appendix D: MFE and LL Differential Geometry

of kˆ t as a function of x and y is obtained by expressing b as a function of (x, y) through F(x, y, b) = 0, and substituting it in kˆ t (b) (this is found with the following choice  for  the branches: b = (R 2 − ρ2 )/2 |y| for MFE lens and     b = |y| −x + 2 R 2 − y 2 + x 2 / R 2 − ρ2 for LL). This allows for calculat  ing ∇t 2 s = ∇t 2 n eq kˆ t in analytical form.