kth.diva-portal.org1386207/... · 2020. 1. 16. · Abstract Inthisthesis,theoptimalbandwidthperformanceofperiodicelectromag-neticradiatorsandscatterersisstudied. Themainfocusisonthedevelopment

Fundamental Bounds on Performance ofPeriodic Electromagnetic Radiators and Scatterers

ANDREI OSIPOV

Doctoral Thesis in Electrical EngineeringSchool of Electrical Engineering and Computer Science

KTH Royal Institute of TechnologyStockholm, Sweden, 2020

TRITA-EECS-AVL-2020:8ISBN 978-91-7873-413-9

KTH Royal Institute of TechnologySchool of Electrical Engineering

and Computer ScienceSE-114 28 Stockholm

SWEDEN

Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan framläggestill offentlig granskning för avläggande av teknologie doktorsexamen fredagen den7:e februari 2020 klockan 13.00 i Kollegiesalen, Brinellvägen 8, Kungl Tekniskahögskolan, Stockholm.

© Andrei Osipov, February 2020

Tryck: Universitetsservice US AB

Abstract

In this thesis, the optimal bandwidth performance of periodic electromag-netic radiators and scatterers is studied. The main focus is on the developmentand application of methods to obtain fundamental physical bounds, relatinggeometrical parameters, frequency bandwidth, efficiency and radiation char-acteristics of periodic electromagnetic structures.

Increasing demand on the performance of wireless electromagnetic systemsin the modern world requires miniaturization, high data rates, high efficiency,and reliability in harsh electromagnetic environments. Attempts to improveall these design metrics at once confront the inevitable physical limitations.For example, an antenna’s size is fundamentally bounded with bandwidthperformance, and attempts to decrease size result in reduced performancecapabilities. Knowledge of such physical bounds is vital to achieve high per-formance: to gain an understanding of the trade-off between parameters andrequirements, or to evaluate how optimal the realized design is.

Periodic structures are indispensable components in many wireless sys-tems. As antenna arrays, they are in base stations of mobile phone networks,in radio astronomy, in navigation systems. As functional structures, they areused as frequency-selective filters, polarizers and metamaterials.

In this thesis, methods to construct fundamental bounds on Q-factor –a quantity inversely proportional to bandwidth – are presented for periodicstructures. First, the Q-factor representation is derived in terms of the electriccurrent density in a unit cell. Then, the bounds are obtained by minimizingthe Q-factor over all current densities, that are supported in a specified spatialsubset of a unit cell, with possibly additional constraints (e.g. on conductivelosses, or on polarization) imposed.

Moreover, an alternative approach for obtaining fundamental bandwidthbounds is investigated – the sum rules, that are based on representing aphysical phenomenon as a passive input-output system. Transmission of aplane wave through a periodically perforated metal screen is described by apassive system, and the sum rule bounds the transmission bandwidth withthe static polarizability of the unit cell. Such a bound is shown to be tightfor simulated and measured perforated screens.

iv

Sammanfattning

Den här avhandlingen undersöker den optimala prestandan av elektro-magnetiska periodiska radiatorer och spridare. Huvudinriktningen är utveck-ling och tillämpning av metoder för att erhålla fundamentala fysikaliska be-gränsningar, som relaterar geometriska parametrar, bandbredd, verknings-grad/effektivitet och strålningsegenskaper av periodiska elektromagnetiskastrukturer.

Ökande krav på prestanda av trådlösa elektromagnetiska system dri-ver fram miniatyrisering, hög datahastighet och hög tillförlitlighet i robustaelektromagnetiska miljöer. Försök att förbättra alla dessa designegenskaperpå en och samma gång möter oundvikliga fysikaliska begränsningar. För an-tenner är deras bandbredd begränsad av antennens elektriska storlek, ochförsök att minska storleken resulterar i minskad prestanda. Kunskap om så-dana fysikaliska relationer är avgörande för att uppnå hög prestanda: att ökaförståelsen för kompromisser mellan olika parametrar, eller att avgöra huroptimal konstruktionen är.

Periodiska strukturer är viktiga komponenter i många trådlösa system.Till exempel gruppantenner, som finns i basstationer för mobiltelefonnätverk,i radioastronomi och i navigationssystem. Ytterligare exempel är funktionellastrukturer som används som frekvensselektiva filter och metamaterial.

I denna avhandling presenteras metoder för att erhålla begränsningar avQ-faktorn, en storhet omvänt proportionell mot bandbredden för periodiskastrukturer. Först bestäms Q-faktorn i termer av ytströmstätheten i en en-hetscell. Sedan bestäms begränsningar genom att minimera Q-faktorn överalla möjliga strömstätheter i en delmängd av en enhetscell, med möjligtvisytterligare restriktioner (t. ex. resistiva förluster).

I denna avhandling kommer även ett alternativt förhållningssätt för attuppnå fundamentala bandbredds begränsningar att undersökas – summareg-ler, baserade på att framställa ett fysikaliskt fenomen som ett passivt input-outputsystem. En överföring av en våg genom en periodiskt perforerad me-tallskärm beskrivs av ett passivt system, och summareglen begränsar band-bredden med enheltscellens statiska polariserbarhet. En sådan begränsningvisar sig vara skarp för några simulerade och uppmätta perforerade skärmar.

v

Preface

This thesis is in partial fulfillment for the Doctor of Philosophy degree at KTH RoyalInstitute of Technology, Stockholm, Sweden. The work presented in this thesiswas performed at the Department of Electromagnetic Engineering at School ofElectrical Engineering and Computer Science, KTH Royal Institute of Technologyfrom January 2015 till January 2020. Professor Lars Jonsson has supervised thework presented in this thesis.

The thesis was supported by the Swedish Foundation for Strategic Research bythe project “Complex Analysis and Convex Optimization for EM design” (SSF/AM13-0011), and the Swedish Governmental Agency for Innovation Systems (Vinnova)through the ChaseOn project “iAA” (ChaseOn/iAA).

vi

Acknowledgments

First and foremost, I would like to express my deepest gratitude to Prof. LarsJonsson, my supervisor, whose insightful advice, guidance and support have beeninvaluable for me throughout the last five years. Thank you for being a greatteacher, colleague and friend.

I would like to thank the SSF group: Prof. Mats Gustafsson, Prof. DanielSjöberg, Prof. Sven Nordebo, Prof. Annemarie Luger, Dr. Casimir Ehrenborg,Dr. Yevhen Ivanenko, Dr. Mitja Nedic, Dr. Dale Frymark, as well as Dr. AndreasEricsson, Johan Lundgren and Dr. Shuai Shi for our fruitful collaborations. Manythanks to Prof. Marianna Ivashina for an encouraging research environment withinthe ChaseOn/iAA project. Special thanks to Dr. Jari-Matti Hannula, who hasrecently joined our division, for having my back and being a colleague to rely on.

I would like to acknowledge Prof. Martin Norgren for the review of my thesisand our discussions on electrodynamics, Dr. Nathaniel Taylor for supporting theLinux server for computations, and Assoc. Prof. Oscar Quevedo-Teruel for hiscontinuous energy in our antenna group. I send my appreciation to Rajeev Thot-tappillil for his administrative and strategic support as the head of our division, toBrigitt Högberg, Emmy Axén and Carin Norberg for administration and to PeterLönn for maintaining computers and software. I would like to thank Prof. AndersForsgren with the Department of Mathematics at KTH for his excellent coursesin optimization techniques, and Prof. Thomas Rylander at Chalmers for a greatcourse in computational electromagnetics.

I am grateful to my friends and colleagues at the division of ElectromagneticEngineering for their companionship. These include Fatemeh, Jan-Henning, Mari-ana, Mahsa, Sanja, Kateryna, Wadih, Janne, Henrik, Lei, Christos, Kun, Mengni,Elena, Qingbi, Qiao, Leila, Sylvie, Patrick, Oskar, Pilar, David, Roya, Patrik, Za-karia, Tanbhir, Per, Ahmad, Mauricio, Mrunal, Cong-Toan, among many others.

Prior to pursuing a doctoral degree, I have been blessed to encounter manyinspiring and outstanding teachers. For my education during school years, I amthankful to Nelly Udaltsova, Ludmila Frolova, Igor Starobogatov and Elena Ko-zlova. During my undergraduate studies, I learned a lot from Prof. AlexanderSergienko, Dr. Mikhail Sugak, Prof. Valery Ipatov, Prof. Viktor Ushakov, andProf. Valery Stepanov.

I feel fortunate to have many friends here in Stockholm and elsewhere in theworld, with whom I shared many wonderful moments. Natalya, Christina P.,Camille, Christina S., Pavel, Juliette, Artyem, Mikhail, Alexander and Paulina,thank you for keeping me inspired in many different ways.

Last but not least, I am forever grateful to my parents Lyubov and Vladislavfor always believing in me, and for their endless love and support.

Andrei OsipovStockholm, January 2020

vii

List of publicationsThis thesis is based on the following journal papers:

I. A. Ludvig-Osipov, B. L. G. Jonsson, “Stored energies and Q-factor of two-dimensionally periodic antenna arrays,” under review, 2019.Contributions of the author: I performed derivations, wrote numerical im-plementation, done simulations, prepared figures and wrote the manuscript.B.L.G.J. suggested the overall topic, participated in derivations and per-formed derivations in Appendix C. All authors reviewed and edited themanuscript.

II. A. Ludvig-Osipov, B. L. G. Jonsson, “Q-factor and bandwidth of peri-odic antenna arrays over ground plane,” Early Access at IEEE Antennas andWireless Propagation Letters, 2019.Contributions of the author: I performed derivations, wrote numericalimplementation, done simulations, prepared figures and wrote the manuscript.B.L.G.J. participated in derivations. All authors reviewed and edited themanuscript.

III. A. Ludvig-Osipov, J.-M. Hannula, B. L. G. Jonsson, “Physical limitationsof phased antenna arrays,” manuscript, 2020.Contributions of the author: I wrote the numerical implementation, per-formed some of the numerical examples and wrote majority of the manuscript.J.-M.H. performed most of the numerical examples and wrote most of Sec-tion 4. B.L.G.J. proposed the optimization approaches. All authors reviewedand edited the manuscript.

IV. A. Ludvig-Osipov, J. Lundgren, C. Ehrenborg, Y. Ivanenko, A. Ericsson,M. Gustafsson, B. L. G. Jonsson, and D. Sjöberg, “Fundamental bounds ontransmission through periodically perforated metal screens with experimentalvalidation,” Early Access at IEEE Transactions on Antennas and Propaga-tion, 2019.Contributions of the author: I wrote the manuscript, performed prelim-inary studies, some of the simulations, and took part in manufacturing andmeasurement of the sample. J.L. performed majority of simulations, and donepost-processing of measured data. J.L., C.E., Y.I. and A.E. participated inmanufacturing and measurement process. C.E. and Y.I. performed some ofthe simulations. M.G. and D.S. derived the sum rule. All authors reviewedand edited the manuscript.

V. A. Ludvig-Osipov, B. L. G. Jonsson, “Evaluation of the Electric Polariz-ability for Planar Frequency-Selective Arrays,” IEEE Antennas and WirelessPropagation Letters, vol. 17, no. 7, pp. 1158–1161, July 2018.

viii

Contributions of the author: I performed derivations, wrote numericalimplementation, done simulations, prepared figures and wrote the manuscript.All authors reviewed and edited the manuscript.

Other publications related to but not included in this thesis:

VI. M. Nedic, C. Ehrenborg, Y. Ivanenko, A. Ludvig-Osipov, S. Nordebo,A.Luger, B. L. G. Jonsson, D. Sjöberg, M. Gustafsson “Herglotz functionsand applications in electromagnetics,” In Advances in Mathematical Methodsfor Electromagnetics, Edited by K. Kobayashi and P. Smith. IET, 2020.

VII. S. Shi, A. Ludvig-Osipov, and B. L. G. Jonsson. “Examination of band-width bounds of antennas using sum rules for the reflection coefficient.” 2016IEEE International Conference on Microwave and Millimeter Wave Technol-ogy (ICMMT), Beijing, China, 2016.

Parts of this thesis have been presented in the following peer-reviewedconference papers or workshops:

VIII. A. Ludvig-Osipov, J. Lundgren, C. Ehrenborg, Y. Ivanenko, A. Ericsson,M. Gustafsson, B. L. G. Jonsson, and D. Sjöberg, “Fundamental bound on ex-traordinary transmission through periodically perforated screens,” in ISAAC(The International Society for Analysis, its Applications and Computation)2017, Växjö, Sweden, 2017, abstract.

IX. A. Ludvig-Osipov, B. L. G. Jonsson, “Electric Polarizability Estimation ForPlanar Frequency Selective Arrays,” in 32nd URSI GASS, Montreal, Canada,2017, abstract.

X. A. Ludvig-Osipov, B. L. G. Jonsson, “On Q-factors for thin periodic an-tenna and scatterer arrays,” in 2nd URSI AT-RASC, Gran Canaria, Spain,2018, abstract.

XI. A. Ludvig-Osipov, B. L. G. Jonsson, “On the Q-factor of periodic antennaarrays over ground plane,” in 2019 International Conference on Electromag-netics in Advanced Applications (ICEAA), Granada, Spain, 2019, abstract.

Acronyms and abbreviations

2D two-dimensional3D three-dimensional5G fifth generation mobile networksCVX ® Matlab software for disciplined convex programmingco-pol co-polarizedcx-pol cross-polarizedEFIE electric field integral equationEM electromagneticFS functional structureFSS frequency-selective structureFT Fourier transformKKT Karush-Kuhn-TuckerMoM method of momentsPEC perfect electric conductorPMC perfect magnetic conductorradar radio detection and rangingRWG Rao-Wilton-GlissonSDR semidefinite relaxationQCQP quadratically constrained quadratic program

ix

Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiSammanfattning (in Swedish) . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiAcronyms and abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . ix

Contents x

Part I: Introduction and research overview 1

1 Introduction 31.1 Background and motivation . . . . . . . . . . . . . . . . . . . . . . . 31.2 Papers summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Electromagnetism in periodic setting 92.1 Electromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Periodic geometry and fields . . . . . . . . . . . . . . . . . . . . . . . 112.3 On propagating and evanescent modes . . . . . . . . . . . . . . . . . 122.4 Ground plane, method of images . . . . . . . . . . . . . . . . . . . . 142.5 Numerical treatment of periodic structures . . . . . . . . . . . . . . 142.6 Infinite structures analysis . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Q-factor, bandwidth and stored energies 213.1 Frequency bands and resonances . . . . . . . . . . . . . . . . . . . . 213.2 Q-factor and bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . 223.3 Radiation efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.4 Stored energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.5 Numerical implementation, matrix form . . . . . . . . . . . . . . . . 32

4 Fundamental bounds from Q-factor minimization 354.1 Optimization methods . . . . . . . . . . . . . . . . . . . . . . . . . . 35

x

CONTENTS xi

4.2 Optimal arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5 Fundamental bounds from sum rules 455.1 Sum rules and Herglotz functions . . . . . . . . . . . . . . . . . . . . 455.2 Application of the sum rule . . . . . . . . . . . . . . . . . . . . . . . 49

6 Conclusions, outlook and sustainability discussion 516.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516.2 Future work and extensions . . . . . . . . . . . . . . . . . . . . . . . 526.3 Discussion on the sustainability . . . . . . . . . . . . . . . . . . . . . 53

Bibliography 55

Part II: Included papers 65Paper I: Stored energies and Q-factor of two-dimensionally periodic an-

tenna arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67Paper II: Q-factor and bandwidth of periodic antenna arrays over ground

plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83Paper III: Physical limitations of phased antenna arrays . . . . . . . . . . 91Paper IV: Fundamental bounds on transmission through periodically per-

forated metal screens with experimental validation . . . . . . . . . . 111Paper V: Evaluation of the electric polarizability for planar frequency-

selective arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

Part II is not available in the online version.

Part I: Introduction andresearch overview

Chapter 1

Introduction

1.1 Background and motivation

Electromagnetic waves are essential in the world around us. Natural sources, suchas sun, stars and fire produce electromagnetic radiation in the form of light, re-ceived by our eyes which provides us vision, and in the form of heat, received byour bodies. Human-made devices and systems, employing electromagnetic wavesto obtain and transmit information, or to transfer energy, are vital in modern soci-ety. Electromagnetic waves are used in our day-to-day life – e.g. in mobile phonenetworks [1] and navigation systems [2], as well as in specific applications, such asmedical imaging [3] or exploring the Universe [4].

An important class of electromagnetic components in such systems are periodicstructures. Periodicity introduces remarkable performance opportunities, such ashigh directive radiation, electronic beam-scanning, materials with unusual proper-ties [5]. At the same time, it brings unique challenges in design and applications.This dissertation is an investigation of the physical limitations of the performanceopportunities of periodic structures, with the goal of addressing the design chal-lenges. The trade-offs between bandwidth, size, radiation efficiency and radiationproperties are derived and discussed.

Periodic EM structures in modern worldPeriodic structures for emitting, receiving and processing electromagnetic wavesare indispensable components of wireless communication, navigation and imagingsystems. In wireless systems, antennas play crucial role – they radiate and receiveelectromagnetic waves, translating those to electric signals. If an antenna is pe-riodic or has some repeatable element, it is usually called an antenna array. Theapplications of periodic structures as antenna arrays range from base stations in mo-bile phone networks and airplane navigation radars to radiofrequency astrophysics.Plane-wave filters and polarizers are typically realized as periodic structures. Meta-materials and metasurfaces, comprised of repeated sub-wavelength elements, have

3

4 CHAPTER 1. INTRODUCTION

s1+s2

s1ejφ1+s2e

jφ2

s1ej2φ1+s2e

j2φ2

s1ej3φ1+s2e

j3φ2

s1

s2

(a)

f3

f2

f1

f2

(b)

Figure 1.1: Examples of applications of periodic structures. (a) Phased array trans-mitting two signals, s1 and s2, in two different directions, shown here by mainbeams of radiation patterns. The signals s1 and s2 are fed to array elements withtwo different inter-element phase shifts φ1 and φ2 respectively (represented by thecomplex-exponents factors). The inter-element phase shifts determine the direc-tions of main beams. (b) Plane wave filter, transmitting a wave of frequency f2and blocking waves with frequencies f1 and f3.

bulk electromagnetic properties, not readily available in nature in a form convenientfor applications [6]. These unusual properties, for example a negative refractive in-dex, can be used in applications such as electromagnetic lenses and cloaking.

A tendency in modern wireless communication standards – the fifth generationmobile networks (5G) [7] and beyond – is in employing frequency bands abovethe spectrum used by previous generations of wireless devices. Together with thedirect advantages of this choice – access to new, largely unused spectrum rangeand higher data rates – comes the challenge: high path losses. A proposed solutionto mitigate such a problem and to increase the system’s power efficiency is touse highly directive antennas [8]. Phased array antennas are suitable candidatesfor such role: they can be highly-directive and the beam direction is adjustedelectronically. The electronic beam scanning allows e.g. multiple channels pointingin different directions and real-time adjustment of beam directions. The electronicbeam scanning is illustrated by Figure 1.1(a).

Phased arrays have been used in radars as early as during the Second WorldWar [9]. The phased-array radars found many civil and defense applications [10].They do not require mechanically moving parts, and thus are commonly placedon aircraft, ships and vehicles for navigation purposes. Early-warning radars usemulti-beam scanning properties to track several targets at a time. In space-basedapplications, arrays are used for satellite communications and space probe missioncommunication (e.g. on the MESSENGER spacecraft). In the latter, the fault

1.1. BACKGROUND AND MOTIVATION 5

tolerance is an important issue – phased arrays can gracefully degrade, i.e. to stillfunction properly with malfunctioning of some of the array elements [11]. At opticalfrequencies, the beam-scanning capabilities of phased arrays are used in lens-lesscameras, image scanners, and image projectors [12].

Another application of periodic arrays is functional structures (FS) [13] andmetamaterials [6]. FS are used as parts of antennas, filters (see Figure 1.1(b)),polarizers and directional couplers, and as radiation-absorbing materials for stealthtechnologies. Metamaterials allow new possibilities in electromagnetic design suchas lens antennas.

The ever increasing demands on the performance of wireless electromagnetic sys-tems require miniaturization, low cost, low weight, high data rates and operationalspeed, low energy consumption, low losses and high reliability. This motivates de-velopment of advanced tools for design of electromagnetic periodic structures, thatare crucial components in such systems.

Electromagnetic design and fundamental bounds

Even nowadays, when numerical electromagnetic simulation tools are effective andaccessible, antenna-array and functional-structures design are the disciplines farfrom being trivial. Usually, a well-investigated design template, suiting the appli-cation, is taken, finely adjusted and possibly extended to meet the requirements andspecifications. However, not always the performance demands can be attained byexisting templates. In such a case, a new class of array or FS design is required to bedeveloped. The design development process then turns into a labor-intensive trialand error procedure, guided by advanced design tools, such as circuit models, ma-trix representations, genetic and other non-deterministic optimization algorithmsand, of course, by knowledge, experience and intuition of the designer.

Fundamental physical bounds are a powerful class of such advanced designtools, indispensably used in electromagnetic design since 1947 [14,15]. Fundamentalbounds received renewed interest and many fruitful results in the last decade [16,17].Physical bounds indicate how far the bandwidth performance of the given designfrom optimal within the given constraints, and can be used as figures of merit. Theyprovide the information on a trade-off between the best attainable bandwidth andother parameters, such as size and medium losses (losses in the constituent mediaof the antenna array). Nowadays, physical bounds can be obtained for many typesof electromagnetic devices. Two main approaches for deriving fundamental boundsare:

• By analyzing a Q-factor representation [16]. This approach, apart from thebound, often gives, e.g. electric current distribution or radiation mode co-efficients that realize the optimal performance. Such information can oftenprovide ideas to develop optimal designs. Examples: small [14,15,18,19] andembedded [20] antennas, small antennas over ground plane [21]. In Paper III,the Q-factor bounds for periodic structures are presented.


• By considering an input-output passive system representation [17]. The boundis typically represented as a sum rule, that singles out the quantity which lim-its the performance. An advantage over the Q-factor representation boundsis that the sum rules are valid for wideband multi-resonant cases. Examples:frequency selective structures (FSS) [22], metamaterials [23], high-impedancesurfaces [24], wideband arrays [25]. In Paper IV, the sum-rule bound forperiodically perforated thin metal screens is investigated.

Optimal periodic structures, optimal currentsFor many systems, the frequency bandwidth plays a crucial role. For example,in communication networks bandwidth is proportional to the data transmissionrate. In radio-frequency astronomy, the bandwidth defines the range of frequencies,at which electromagnetic waves can be detected and observed. For radars, thebandwidth defines how many sub-bands there are to switch between when somefrequency range gets jammed or overloaded.

That is, in this thesis, optimal periodic structures are defined as the ones thathave as large bandwidth as possible under certain constraints. It can be periodicelement geometry or form-factor constraints, or requirements on efficiency, beam-steering characteristics and polarization.

In the Q-factor minimization approach, considered in this thesis, optimization isperformed over electric current densities in a specified volume/surface. Additionalconstraints (for example, on efficiency or polarization mismatch) can be included.An optimal solution is a current density minimizing the Q-factor under specifiedconstraints. It is referred to as optimal currents, or optimal current density [26].

An example of the optimal current density1 on a plate with size l × 0.5l in anarray with unit cell size 1.7l× 1.7l is shown in Figure 1.2. There are no additionalconstraints imposed except from the form-factor and size, defined by the plateregion where currents are allowed. These currents represent a best possible arrayelement design, bounded by the plate region.

Fundamental bounds from current density optimization can be seen as a steptowards automated electromagnetic design [26, 27]. Apart from using the boundsas stopping criteria or as a reference [28], the obtained optimal currents can bemimicked by e.g. multiport feeding, see initial investigations in [27, Section 5.2].

1.2 Papers summary

Papers I-III establish fundamental bounds and trade-off relations on Q-factor fornarrow-band periodic electromagnetic structures. The first two papers derive theQ-factor representation in terms of current density within a unit cell. Paper Iconsiders the free-space case, and in Paper II the periodic structure is placed above

1It is optimal for the given mesh representation of the plate, see discussion on meshing andlower bound of Q in the end of Section 4.1

1.3. THESIS OUTLINE 7

−20

−10

0

Normalized

currentdensity,dB

Figure 1.2: Optimal current density, minimizing Q-factor of an array of rectangularplates. The plates’ size is l× 0.5l, the array is on a square periodic grid with a unitcell size 1.7l × 1.7l, and the frequency corresponds to the wavenumber k = 2.7/l.

an infinite ground plane. The derived representations are applied in Paper III to findfundamental bounds and optimal currents by using convex optimization techniques.The trade-off relations between the Q-factor and design characteristics, such asgeometrical parameters, efficiency and polarization are numerically obtained there.

In Paper IV, the sum rule for the transmission through a periodically perfo-rated metal screen is investigated. The sum-rule-based bound ties the transmissionbandwidth with the static polarizability of a unit cell. A critical issue for physicalbounds is whether or not a bound is ‘tight’, i.e. if the predicted best performancehas shown to be attained by some designs. One of the focuses in Paper IV ison demonstrating that the considered bound can actually be reached not only bythe ideal numerically simulated structures, but that it is also relatively tight fora manufactured sample. A method to evaluate static polarizability in a periodicstructure, based on a transmission line model, is proposed in Paper V.

1.3 Thesis outline

This thesis contains six chapters.

• Chapter 1 discusses background and applications of periodic electromagneticstructures and provides motivation of this thesis. A brief summary of theenclosed papers is included.

• In Chapter 2, the two-dimensionally (2D) periodic structures, considered inthis thesis, are defined. The fundamentals of electromagnetic theory, appliedto periodic structures, are reviewed. Potentials and fields are defined in termsof 2D periodic Green’s function and electric current density in a unit cell.


The method of moments (MoM) – a numerical approach to find currents ona periodic structure – is reviewed.

• Chapter 3 starts with a discussion on relations between the Q-factor, band-width and stored energies, and a brief literature review on Q-factor boundsand stored energies. The expressions for the stored energies, radiated power,conductive losses and polarization are provided in terms of electric currentdensities for arrays. The numerical implementation of the expressions is given,along with the numerical validation of the Q-factor, which is obtained fromthe stored-energies and radiated-power expressions. The results from PapersI-II are partially reproduced there.

• Two approaches for efficient solution of Q-factor minimization problems withor without constraints are presented in Chapter 4. The optimization is per-formed using the Q-factor expressions given in Chapter 3 with electric currentdensity being an optimization variable. Examples of trade-off relations be-tween the Q-factor, geometrical parameters and efficiency are demonstrated.The results from Paper III are partially reproduced there.

• In Chapter 5, a theoretical background for sum-rules and Herglotz functionsis given, and the sum rule for transmission through perforated screens isdiscussed. The results from Papers IV-V are partially reproduced there.

• Finally, in Chapter 6, the summary, conclusions and possible continuationsof the presented work, and a brief discussion regarding the sustainability aregiven.

Chapter 2

Electromagnetism in periodicsetting

In this chapter, the fundamentals of electromagnetic theory are reviewed in theperiodic setting, starting from Maxwell’s equations. The periodic array configura-tion is given with the corresponding periodic Green’s function. To include arraysover the ground plane into consideration, the method of images is briefly reviewed.The method of moments approach to solve the electric field integral equation in aperiodic setting is described. The chapter ends with a discussion on how the resultsof infinite structures’ analysis are applied for finite arrays.

2.1 Electromagnetism

Analysis in frequency domain and Fourier transform

The analysis in this thesis and the appended papers is performed in the frequencydomain. A time-dependent function can be expressed as a continuous sum (integral)of harmonic components via Fourier transform (FT)

f(t) =∫ ∞−∞

F (ω)ejωtdω. (2.1)

The transformed function F (ω) – the spectrum of f(t) – is complex-valued. Due tothe linearity of the integration operator, each harmonic component F (ω)ejωt can beconsidered separately. For linear physical systems, the factor ejωt can be omitted,and time derivatives and integrals are replaced as d

dt → jω and∫

(...)dt → 1jω .

Hence, in this thesis, complex amplitude (phasor) functions, such as F (ω), areconsidered. For vector fields (such as electric and magnetic fields, current density)the corresponding complex amplitude functions are vector-valued with complex-valued components.

9

10 CHAPTER 2. ELECTROMAGNETISM IN PERIODIC SETTING

Maxwell’s equationsThe electromagnetic phenomena considered in this thesis are governed by Maxwell’sequations. For time-harmonic fields with time-dependency ejωt, the Maxwell’s equa-tions in differential form are

∇×E(r, ω) = −jωB(r, ω), (2.2)∇×H(r, ω) = J(r, ω) + jωD(r, ω), (2.3)∇ ·D(r, ω) = %(r, ω), (2.4)∇ ·B(r, ω) = 0. (2.5)

Here, E is an electric field, D = εE is a displacement field, H is a magneticfield, and B = µH is a magnetic flux density, % and J are electric charge andcurrent densities respectively, ω is an angular frequency, t is a time variable. Thepermittivity and permeability are respectively denoted by ε and µ. In this thesis,ε and µ are assumed homogeneous, and frequency- and time-independent, whichis the case for Papers I-IV, where periodic structures in free space are considered.In Paper V, the inclusion of dielectric slabs is accounted for by a transmission linemodel.

Helmholtz equations and potentialsThe Maxwell’s equations (2.2)-(2.5) can be reformulated into inhomogeneous Helmholtzequations for fields due to the sources % and J

(∇2 + k2)E = 1ε∇%+ jωµJ , (2.6)

(∇2 + k2)H = −∇× J , (2.7)where k = √µεω is the wavenumber.

Throughout this thesis, the Lorenz gauge condition for electric scalar φ andmagnetic vector A potentials is adopted

∇ ·A+ jωµεφ = 0. (2.8)The potentials are related to fields via

E = −∇φ− jωA, (2.9)

H = 1µ∇×A. (2.10)

The Helmholtz equations for fields can be equivalently replaced by Helmholtz equa-tions for potentials

(∇2 + k2)φ(r) = −%(r)/ε, (2.11)(∇2 + k2)A(r) = −µJ(r). (2.12)

The continuity equation, that follows from (2.3) and (2.4), relates the electric chargeand current densities

jω% = −∇ · J . (2.13)

2.2. PERIODIC GEOMETRY AND FIELDS 11

z

x

y0

a b

U

...

(a)

y

x

zE(t)

E(i)

E(r)

(b)

Figure 2.1: Examples of periodic structures, considered in this thesis: (a) peri-odic array of disconnected elements on rectangular grid, (b) periodically perforatedscreen. (© 2019 IEEE)

2.2 Periodic geometry and fields

Periodic array geometryIn this thesis, two electromagnetic phenomena are considered: radiation and scat-tering; both in a periodic setting. The examples of geometries with 2D periodicityon a rectangular grid are shown in Figure 2.1. The structure is an array of losslessperfectly electrically conducting (PEC) elements of arbitrary geometry. The unitcell here is defined as

U = r = (x, y, z) ∈ R3 : x ∈ [0, a], y ∈ [0, b], z ∈ R, a > 0, b > 0. (2.14)

Both the phased array setting and the scattering of an electromagnetic planewave implies that electric currents on the structure, regardless of their origin, havea phase-shift periodicity relation:

J(r + ζmn) = J(r)ejkt00·ζmn . (2.15)

Here r ∈ R3, ζmn = amx + bny; m,n ∈ Z, and kt00 is a coordinate-independenttransverse phasing wave-vector. Note that due to the continuity equation (2.13),the same relation applies to the charge density

%(r + ζmn) = %(r)ejkt00·ζmn . (2.16)

Periodic Green’s functionThe Helmholtz operator is a linear differential operator, and thus instead of solvingthe inhomogenious Helmholtz equations (2.11), (2.12) for each current or charge


density, we introduce a Green’s function G(r) – a solution to the equation

(∇2 + k2)G(r) = −∑

(m,n)∈Z2

δ(r − ζmn)e−jkt00·ζmn , (2.17)

where δ(r) is a Dirac delta-function. The non-homogeneous potentials solution arethen found as convolutions of the Green’s function with a corresponding source

φ(r1) = 1ε

∫Ω%(r2)G(r1 − r2)dv2, (2.18)

A(r1) = µ

∫ΩJ(r2)G(r1 − r2)dv2. (2.19)

Here, Ω is the spatial support of J and % in a unit cell, dv2 is volume element,and integration is performed over coordinate vector r2. The solution of (2.17) is atwo-dimensionally periodic Green’s function, which in spatial form reads as [29,30]

G(r) =∑

(m,n)∈Z2

G0(r − ζmn)e−jkt00·ζmn , (2.20)

where G0(r) = e−jk|r|/(4π|r|) is a free space Green’s function. In Papers I-III thespectral form of the periodic Green’s function is used [29]

G(r1, r2) = G(r1 − r2) = 12jS

∑(m,n)∈Z2

1kzmn

e−jktmn·(ρ1−ρ2)e−jkzmn|z1−z2|, (2.21)

where ktmn = kt00 + 2πma x + 2π nb y is the transverse wavenumber for mn-thmode, kzmn =

√k2 − ktmn · ktmn is the longitudial wavenumber, S = ab, and

the coordinate vectors are expanded into transverse and longitudinal componentsri = ρi + zzi, i = 1, 2. Whenever (k2 − ktmn · ktmn) is negative, we choosethe negative branch of the square root to evaluate kzmn, so that e−jkzmn|z1−z2| isexponentially damped.

2.3 On propagating and evanescent modes

The fields due to the electric currents on the periodic structure can be separatedinto modes. The direct substitution of the scalar potential (2.18) and the vectorpotential (2.19) into the electric field expression (2.9) gives

E(r1) = jηk

∫Ω

(∇1∇2 − k21

)G(r1, r2)

· J(r2)dv2, (2.22)

where ∇1 denotes a Jacobian with respect to r1, and η =√µ/ε is the wave

impedance. Consider the contribution to the electric field due to the (m,n)-th

2.3. ON PROPAGATING AND EVANESCENT MODES 13

term of the Green’s function (2.21)

Emn(r1) =η

2kS

∫Ω

(∇1∇2 − k21

) 1kzmn

e−jktmn·(ρ1−ρ2)e−jkzmn|z1−z2|· J(r2)dv2.

(2.23)

Evaluation of the dyadic-valued operator acting on the Green’s function term gives(for the case z1 ≥ z2) (

∇1∇2 − k21) 1kkzmn

e−jktmn·(ρ1−ρ2)e−jkzmn(z1−z2) =

1kkzmn

k2xm − k2 kxmkyn kxmkzmnkxmkyn k2

yn − k2 kynkzmnkxmkzmn kynkzmn k2

zmn − k2

e−jktmn·(ρ1−ρ2)e−jkzmn(z1−z2) =

K+e−jktmn·(ρ1−ρ2)e−jkzmn(z1−z2).

(2.24)

Here, kxmx + kyny = ktmn. For z1 above the spatial support Ω of the currentdensity, i.e. for z1 ≥ z2, r2 ∈ Ω, we have

Emn(r1) = η

2SK+ ·∫

Ωejktmn·ρ2ejkzmnz2J(r2)dv2e−jktmn·ρ1e−jkzmnz1 =

η

2SK+ · J(ktmn, kzmn)e−jktmn·ρ1e−jkzmnz1 ,

(2.25)

where J(ktmn, kzmn) denotes a three-dimensional (3D) spatial FT of J(r2). Thevector-valued coefficients K+ · J(ktmn, kzmn) in front of the exponential factorsin (2.25) are independent of the coordinates, and hence such a vector coefficientcan be considered an expansion coefficient. The expansion in basis functionse−jktmn·ρ1−jkzmnz1 is known in electromagnetics as Floquet expansion [31].1 Wesee that whenever longitudinal wavenumber is imaginary kzmn = −j|kzmn| (notethat we pick the negative branch of a square root), the field in (2.25) is a planewave, exponentially damped with the growth of z1 towards +∞. Whenever kzmnis real-valued, Emn is a non-decaying plane wave propagating towards infinity.

Similarly, for z1 below Ω, Emn is a plane wave that for imaginary-valued kzmnexponentially decays as z1 goes towards −∞, and for real-valued kzmn propagatestowards minus infinity.

In this thesis and in Papers I-III we refer to individual terms Emn as in (2.23)as modes, i.e. an (m,n)-th mode here is the contribution to the total electricfield E (2.22) due to (m,n)-th term in the spectral form of Green’s function G.Depending on whether kzmn is real- or imaginary-valued, these modes can be eitherpropagating or evanescent respectively.

1Floquet considered linear differential equations with periodic coefficients in 1883 [32,33], LordRayleigh was the first to use the expansion for a physical phenomenon (acoustic waves scatteredby a periodic grating) in 1907 [34]. Bloch used the expansion to describe electrons in a crystal in1929 [35]. Hence the same expansion is referred to as Rayleigh waves or Bloch waves [36].


J(r) J(r)

J i(r)

Figure 2.2: Current density above a ground plane (left) and the correspondingimage problem (right)

2.4 Ground plane, method of images

The presence of the ground plane can be accounted for by the image method [37],illustrated by Figure 2.2. The image problem is constructed by replacing the infiniteground plane by an image current J i(r) = −1z · J(ri). Here 1z = xx + yy − zzis a dyadic tensor inverting the sign of z-component of a vector, and the imagecoordinate is ri = 1z · r. The total current density in the image problem is givenby

Jtot(r) = J(r)− 1z · J(ri). (2.26)

This allows us to express potentials above the ground plane in terms of the currentJ on the array element only

A(r1) = µ

∫Ω

[G(r1, r2)1−G(r1, r2i)1z] · J(r2)dv2, (2.27)

φ(r1) = −jωε

∫Ω

(∇2[G(r1, r2)−G(r1, r2i)]) · J(r2)dv2. (2.28)

The associated fields are determined by (2.9) and (2.10).

2.5 Numerical treatment of periodic structures

Electric field integral equation (EFIE)

An electric field, incident on a PEC structure, can be related to the induced currentsthrough boundary condition on a PEC surface. Consider an electric field Ei due toa given source in absence of a PEC scatterer. If the PEC scatterer with a surfaceS is placed in this field, a current density J is induced on S. The current densitygenerates the scattered electric field Es. This scattered field, and hence the currentdensity, should be such that the boundary condition for the tangential component

2.5. NUMERICAL TREATMENT OF PERIODIC STRUCTURES 15

of the total electric field Ei +Es on the PEC surface S is satisfied

n× (Ei(r) +Es(r)) = 0 at r ∈ S. (2.29)

Here, n is the outward unit normal of the surface S. For the free space case, thescattered field is expressed via potentials (2.9), which are related to currents asin (2.18) and (2.19) for free space. PEC objects only have currents on the surface(it is also a good approximation for real conductors) [38], and the volume integralsreduce to surface integrals in (2.18) and (2.19)

Es(r1) =− jωε∇1

∫S

G(r1, r2)∇2 · J(r2)dS2

− jωµ∫S

G(r1, r2)J(r2)dS2 = −(LJ)(r1).(2.30)

Here, the volume current densities are replaced with surface current densities, andthe symbol J(r2) is reused to denote surface current density. The linear opera-tor L, introduced here, represents the integral relation. The EFIE follows fromsubstitution of this relation in the boundary condition (2.29)

n×Ei(r1) = n× (LJ)(r1)

= n×(

jωε∇1

∫S

G(r1, r2)∇2 · J(r2)dS2 + jωµ∫S

G(r1, r2)J(r2)dS2

).

(2.31)In this integral equation, the unknown is the current density J .

For the ground plane case, the potentials (2.28) and (2.27) are used, which leadsto the following EFIE

n×Ei(r1) = n× (LgpJ)(r1)

= n×(

jωε∇1

∫S

[G(r1, r2)−G(r1, r2i)]∇2 · J(r2)dS2

+jωµ∫S

[G(r1, r2)1−G(r1, r2i)1z] · J(r2)dS2

).

(2.32)

Integral equationsIn EFIEs (2.31) and (2.32), the unknown is under the integral sign. These areFredholm equations of the first kind, which in a general form are represented as [39]

g(y) =∫ b

a

K(x, y)f(x)dx. (2.33)

Here, the function g(y) and the kernel K(x, y) are known, and f(x) is the unknownfunction. It is known that Fredholm equations of the first kind are ill-posed when-ever the kernel is smooth [39,40]. A common approach to find a numerical solutionof the first kind integral equations is the method of moments, which transforms theintegral equation to a linear system of algebraic equations [41].


Method of moments

The most common and efficient way to solve EFIE is by the method of moments.The unknown current density is approximately represented by a set of linearlyindependent basis functions fn(r)Nn=1

J(r) ≈N∑n=1

Infn(r), (2.34)

where InNn=1 are unknown current coefficients, and the basis functions fn areknown. Since L is a linear operator, the substitution of the expansion into EFIEgives

n×Ei(r1) ≈ n×N∑n=1

In(Lfn(r2))(r1). (2.35)

The inner product with a test function tm(r1), defined to be tangential on S, [42]yields an algebraic equation

N∑n=1〈tm(r1), (Lfn)(r1)〉 In ≈ 〈tm,Ei(r1)〉 . (2.36)

The angle brackets denote the inner product defined as

〈t(r),f(r)〉 =∫S

t∗(r) · f(r)dS, (2.37)

where (.)∗ denotes the complex conjugate. The procedure of taking inner prod-uct (2.36) in the context of integral equations is called taking moments, hence thename of the approach – the method of moments. All the functions in inner prod-ucts in (2.36) are known, and thus the inner products can be directly computed.To find all the unknown current coefficients, N linearly independent test functionstm are used, generating N linearly independent equations of the form (2.36). Theresulting system of equations can be represented in the matrix form

ZI ≈ V. (2.38)

Here, the unknown current coefficients are grouped in the column vector I ∈ CN ,and Z ∈ CN×N is the impedance matrix with elements

Zmn = 〈tm(r1), (Lfn(r2))(r1)〉 . (2.39)

The excitation vector is a column V ∈ CN with elements

Vm = 〈tm,Ei(r1)〉 . (2.40)

2.6. INFINITE STRUCTURES ANALYSIS 17

In the free space case the impedance matrix elements are

Zmn = − jωε

∫S

∫S

G(r1, r2)∇1 · tm(r1)∇2 · fn(r2)dS1dS2

+ jωµ∫S

∫S

G(r1, r2)tm(r1) · fn(r2)dS1dS2.

(2.41)

In the ground plane case the impedance matrix elements are given by

Zmn = − jωε

∫S

∫S

[G(r1, r2)−G(r1, r2i)]∇1 · tm(r1)∇2 · fn(r2)dv1dv2

+ jωµ∫S

∫S

tm(r1) · [G(r1, r2)1−G(r1, r2i)1z] · fn(r2)dv1dv2.

(2.42)

On basis and test functionsIn Papers I-III of this thesis, an in-house MoM code for periodic structures is usedto compute impedance matrices, and also matrices representing stored energies(quantities required to compute Q-factor, see Section 3.4). Periodic elements arenumerically represented by a triangular mesh. For both the basis functions and thetest functions, Rao-Wilton-Glisson (RWG) edge elements [43] are employed – theseare local basis functions, convenient for current density optimization. In fact, thesets tmNm=1 and fnNn=1 are chosen identical – in that case, the MoM approachis called the Galerkin method [1] (on the side note: the method was discoveredby W. Ritz [44, 45], which was credited by Galerkin in his work on elasticity [46]).There are several advantages of having the sets of test and basis functions identical,apart from good numerical accuracy. First, the resulting impedance matrix Zis symmetric, which is convenient in solving matrix equations. Second, as it isshown in Section 3.4, the stored energies and the radiated power are expressedas quadratic forms of current density. Under the finite basis approximation, theycan be represented with symmetric matrices. In Section 3.5, it is noted that theresult of the impedance matrix computation can be reused in calculating the stored-energies and radiated-power matrices if the test functions are chosen equal to thebasis functions.

2.6 Infinite structures analysis

In this thesis, the analysis of periodic structures is performed in an infinite setting,with the exception of experimental results in Paper IV, where a finite FSS has beenmanufactured and measured. Analysis of infinite periodic structures has been oneof the central tools in antenna array and FSS design [5, 31, 47, 48]. For finite peri-odic arrays and scatterers, a rigorous numerical analysis becomes computationallydifficult with the growth of elements number, and consideration of infinite struc-tures can greatly reduce the computational efforts. In the latter case, it is sufficient


(a)

l

a

w1

w2

(b)

Figure 2.3: A periodically perforated metal screen: (a) finite manufactured sampleand (b) unit-cell of infinitely periodic numerical model (© 2019 IEEE)

to consider a unit cell of the array with periodic boundary conditions. Anotheradvantage of considering infinite periodic structures is the ability to separate theperformance of the array element in the periodic environment from the edge effects(behavior of array elements close to edges of a finite periodic structure). Many ap-proaches for finite array analysis using the infinte array results has been publishedin research articles [49–52] and chapters in array textbooks [31,47,48].

As an example, in Paper IV we demonstrate an excellent agreement of plane-wave transmission coefficients between a finite manufactured sample and a simu-lated infinitely periodic structure. The transmission of a plane wave through a thinmetal screen with periodic perforations is considered. The manufactured sample(Figure 2.3(a)) consists of 35 × 45 = 1530 apertures. The simulated infinitely-periodic model (unit cell shown in Figure 2.3(b)) is set with the nominal thicknessof the manufactured sample, the same element geometrical parameters and the cor-responding material model (aluminum). The considered transmission coefficient isthe co-polarized transmission coefficient of a fundamental mode. The measurementsetup is shown in Figure 2.4(a), and only the central part of the perforated sheet inbetween of the two horns determines the transmission – the multipath propagatingand the edge effects are filtered out by time-gating [53,54] in the time domain. Anexcellent agreement of processed (time-gated) measurement result and simulatedtransmission coefficient in Figure 2.4(b) indicates the adequacy of infinite-periodic-structure analysis.

2.6. INFINITE STRUCTURES ANALYSIS 19

(a)

10 12 14 16 180

0.2

0.4

0.6

0.8

1

f , GHz

|T |2

Measured (processed) Simulated

(b)

Figure 2.4: Transmission through periodically perforated metal screen: (a)measurement setup and (b) measured and simulated transmission coefficients.(© 2019 IEEE)

Chapter 3

Q-factor, bandwidth and storedenergies

3.1 Frequency bands and resonances

Physical objects and systems respond differently to external excitations at differentfrequencies. Grass appears green, because it absorbs visible light at all wavelengths,except from ones in range 500− 565 nm, which are perceived as green by our eyes.A filtering circuit in an analog radio picks out the signal at the frequency of aradio station, chosen by a user, and sends it to the amplifier and the loudspeaker;other frequencies are suppressed. From a practical standpoint, a system can becharacterized by its frequency band(s) – the interval(s) of frequencies, at whichthe system performs satisfactorily with respect to some metric. For radiating andscattering systems, a reflection coefficient Γ (fraction of external excitation reflectedby a system) is commonly used in such a metric.

Figure 3.1 shows an example of the reflection coefficient magnitude |Γ| as afunction of angular frequency ω. We define satisfactory performance as when |Γ|is no larger than a chosen threshold Γ0. The frequency band is centered at ω0 andhas width ∆ω. The fractional bandwidth is defined by

B = ∆ωω0

. (3.1)

The frequency behavior of physical objects and systems is closely related tothe notion of resonance. A resonance occurs at frequencies where oscillations aresupported by an object or a system. If an external excitation at such resonantfrequency is applied to the system, the oscillations in the system accumulate exter-nal energy and grow in amplitude. An antenna at its resonance frequency admitsall power supplied to it, i.e. has zero reflection coefficient. The Q-factor model isinstrumental in describing resonances.

21

22 CHAPTER 3. Q-FACTOR, BANDWIDTH AND STORED ENERGIES

ω0

Γ0

1∆ω

B = ∆ωω0

ω

|Γ|

Figure 3.1: Reflection coefficient magnitude as a function of angular frequency

3.2 Q-factor and bandwidth

The Q-factor, or quality factor,1 of an oscillating system, by definition, is propor-tional to the ratio between the stored energy and the dissipated energy per cycle.Thus, the Q-factor indicates the rate of energy dissipation, and estimates the decayof oscillations, not supported by external energy [40]. For radiating structures, andalso resonators, circuits and circuit elements, the Q-factor is indirectly related tothe bandwidth at which an external excitation can be applied efficiently. For nottoo wideband systems (i.e. Q ≥ 5), the Q-factor gives an accurate prediction ofthe fractional bandwidth [57]:

B ≈ 2Γ0

Q√

1− Γ20. (3.2)

Here, Γ0 is the absolute value of the reflection coefficient threshold with respect towhich the bandwidth is considered; and B is a fractional bandwidth, see Figure 3.1.In Figure 3.2, it is demonstrated that the Q-factor model accurately approximatesthe main-band resonance in the array of loops tuned at angular frequency ω0.

The concept of the Q-factor originates from using it to evaluate quality of coils,as proposed by K. S. Johnson [58]. Back then, it was defined as a ratio of reactanceto effective resistance in an inductor. Johnson picked the letter ‘Q’ to denote thequantity only because all the other symbols, at the time, were already overused [59].The Q-factor, as defined for a coil, was found to be applicable to resonant circuits,where an equivalent parameter, ‘sharpness of resonance’, has been already used [60,61]. The concept of Q-factor was then generalized, by defining it via stored anddissipated energies, to resonant systems, ranging from transmission lines, materials,piezoelectric resonators, atoms and molecules to tuning forks, bouncing balls androtating bodies [61].

For some systems, such as a laser and a clock, a high Q-factor is desirable.For antennas, on the opposite, the lower Q-factor means larger bandwidth, thatis, better transmission capacity. From the point of view of stored and dissipated

1Q-factor is also referred to as Q, and, in a case when it describes radiating structures, asantenna Q [55] or radiation Q [56].

3.2. Q-FACTOR AND BANDWIDTH 23

0 ω0 2ω0

0

−10

−20

Q-model

B ≈ 12% at loop arrayΓ0 = −10dB

Angular frequency ω

Refl

ecti

onco

effici

ent

mag

nit

ud

e|Γ|,

dB

l/2

l

p

ph

w

Figure 3.2: Reflection coefficient magnitude of a loops array above a ground plane(unit cell shown in the inset) and of a Q-factor model of the resonance at ω0. Thearray is at h = 0.75l above the ground plane, with period p = 1.2l, width w = 0.1l,and the central angular frequency is ω0 ≈ 1.7

(l√εµ)−1. The feeding is placed in

the middle of the shorter side of the loop (marked with the red dot).

energies, whenever Q-factor is high, a larger part of the supplied energy is storedaround the structure, rather than being dissipated. For radiating structures, theradiated energy is a part of dissipated energy. Hence, to be a good radiator, thestructure should have low Q-factor, i.e. dissipate energy through radiation, ratherthan store it.

The relation between the tuned fractional bandwidth and Q-factor (3.2) forantennas, investigated in the seminal work of Yaghjian and Best [57], has beenvalidated for dipole arrays by Kwon and Pozar [62]. In Papers I-II of this thesis,the validation was extended to a number of other antenna array geometries.

Tuned bandwidth

An input impedance of an antenna can be tuned to have a resonance at a desiredfrequency by one inductive or capacitive element. Such a tuning procedure is calledimpedance matching.

The tuned bandwidth of an antenna at a central frequency ω0 is determined asfollows in [57]. An antenna with unmatched input impedance Z(ω) is tuned withan element of impedance jXs(ω), connected in series such that the imaginary partof the total impedance vanishes at ω0, i.e. ImZ(ω0) + Xs(ω0) = 0. The serieselement is either an inductance L or capacitor C, with the impedance Xs(ω) =


ωL or Xs(ω) = −1/(ωC) respectively. The characteristic impedance is assignedas a constant Zch = ReZ(ω0), the overall input impedance after the tuning isZ0(ω) = Z(ω) + jXs(ω). The bandwidth ∆ω is estimated from reflection coefficientΓ(ω) = (Z0−Zch)/(Z0+Zch) at the threshold level Γ0, see Figure 3.1. The obtainedfractional bandwidth is defined as B = ∆ω

ω0.

Q-factor bounds in literature

The relation of the Q-factor both to field quantities, such as stored energies, andto bandwidth allows to bound the performance of a radiating structure by its ge-ometrical and material properties, such as size and losses. One of the classicalQ-factor-results, dating back to 1948, is Chu bound for electrically small antennas,derived from a circuit model of a spherical waves expansion of the fields [14]. Itbounds the Q-factor by antenna’s electrical size ka [55]

Q ≥ 1(ka)3 + 1

ka, ka 1. (3.3)

Chu bound provides a simple intuitive rule – the larger (electrically) an antennais, the better is its performance. Antennas, reaching this bound, are typically of aspherical shape [63] which are practical only to a limited range of applications. TheChu’s approach of circuit models for spherical waves was extended by Thal [64].Collin and Rothschild proposed the extraction of radiating spherical and cylindri-cal modes at the field level [55]. Fante proposes Q-factor for arbitrary shapes ofradiators by using Collin and Rothschild’s approach to extract radiation around anantenna [65].

A decade ago, Vandenbosch proposed the Q-factor expression for radiatingstructures in terms of the current density on the structure [66] (the correction termsintroduced in [19, 67]). Representation of the Q-factor as a functional of currentdensities allowed obtaining bounds for arbitrarily shaped structures [18,20,68–71].In such case, optimization is performed over all possible currents confined to thegiven volume or surface [26]. A distinctive feature compared to other methods isthat the Q-factor minimization problem can be formulated [71] as a convex opti-mization problem (or equivalent), ensuring that the result is a global minimum. Anumber of additional constraints, such as for medium losses or for radiation pattern,can be introduced in the optimization problem.

The expression by Vandenbosch [66] is applicable to finite radiating objects.For large periodic structures, which can be described accurately and effectively bya unit cell representation, the non-periodic finite approach [66] is impractical froma computational point of view. The main goal for Papers I-III is to derive the Q-factor for periodic structures in terms of currents in a unit cell only, and to obtainfundamental bounds for arbitrary shapes by optimizing the derived expression overthe possible current distributions.

3.2. Q-FACTOR AND BANDWIDTH 25

The Q-factor representations for radiating structuresThe importance of the Q-factor for radiating structures is in its relation to the frac-tional bandwidth (3.2). From a practical standpoint, the end goal of optimization,in most cases, is the maximization of bandwidth. For a given or known fractionalbandwidth B, its equivalent Q-factor is defined using (3.2)

QB = 2Γ0

B√

1− Γ20. (3.4)

This bandwidth-equivalent Q-factor QB is used as a reference in Papers I-II tovalidate the proposed Q-factor, i.e. to show how well the proposed expressionpredicts the true fractional bandwidth.

Another expression to estimate an antenna Q-factor was proposed by Yaghjianand Best, and was shown to be very accurate in many cases [57]. For inputimpedance Z(ω) = R(ω) + jX(ω), the formula reads

QZ(ω) = ω|Γ′(ω)| = ω|Z ′0(ω)|R

=ω√

[R′(ω)]2 + [X ′(ω) + |X(ω)|/ω]22R(ω) . (3.5)

Here, Z0(ω) is tuned input impedance, and (.)′ denotes a derivative with respectto ω. This approach is effectively a Q-factor of an RLC-circuit, obtained by a Padéapproximation of the tuned input impedance around the central frequency ω0 [72].In Papers I-II and also in [62] it is shown that the QZ-method is applicable tophased arrays.

The Q-factor expression, obtained directly from its definition [73] is

Q = 2ωmaxWe,WmPdiss

. (3.6)

Here, We and Wm are electric and magnetic stored energies respectively, Pdiss isthe power dissipated by the system. From the standpoint of a radiating structure,the dissipated power consists of the radiated power and medium losses. In the caseof fully PEC structure, the dissipated power is equal to the radiated power.

The expression (3.6) is a starting point for a whole family of Q-factor calculationmethods (including the Q-factor expressions proposed in Papers I-II). In radiatingsystems, both the radiated power and the medium losses are typically well-defined.However, the definition of stored energy is ambiguous for radiating systems: thereis no consistent and universally accepted method of dividing the energy into storedand propagating parts. Hence, many methods to calculate stored energy wereproposed, each with their advantages and drawbacks [16]. A brief overview of thoseapproaches is presented in Section 3.4 with an emphasis on the approach adoptedin Papers I-III.

Another way, however not discussed in enclosed Papers, to determine the Q-factor of a radiating system is by finding the Q-factor of the broadband circuit modelof the input impedance. The Q-factor of the circuit is found by evaluating stored


energies of inductors and capacitors, and the resistive losses. The circuit modelcan be synthesized in an automated way by Brune synthesis [74, 75]. A detailedcomparative analysis of different Q-factor representations for finite structures canbe found in [72].

3.3 Radiation efficiency

In radiating structures, there are two mechanisms of power dissipation: by radiationand by medium losses, and the dissipated power can be represented as a sum ofthese:

Pdiss = Pr + POhm. (3.7)

From the Q-factor definition (3.6) it then follows that the low Q-factor (i.e. largebandwidth) can be achieved by increasing medium losses POhm. This situation isundesirable from a practical perspective, and hence the allowed losses should belimited. A straightforward way to do so is to constrain radiation efficiency, definedas

ηrad = Pr

Pr + POhm. (3.8)

The radiation Q-factor is then Qrad = Q/ηrad.

3.4 Stored energies

The electric and magnetic stored energies of lumped-elements circuits, resonators,and other closed systems are typically well-defined. For distributed and radiatingstructures, however, the definition of stored energies is more difficult [16] due toambiguity in separating energy between stored and propagating, as compared toe.g. lumped elements such as a capacitor, for which stored energy is confined insidethe element.

The approaches for defining stored energy for radiating structures can be dividedinto three groups, depending whether it is expressed in terms of electromagneticfields [55, 65, 76–78], in terms of currents [66, 79–84], or in system-level quantities(port or feed) [14,85,86]. The features that are desirable from each method (howeverthere is no method satisfying all of them) are:

• Coordinate independence, i.e. the resulting stored energy is independent of achoice of origin. This is a reasonable requirement, as the stored energy of asystem should not depend on the observer.

• Positive semi-definiteness, i.e. the stored energies are positive.

• Finite region over which a data set is required (some methods require e.g.knowledge of electromagnetic fields in all space). This is often crucial propertyfrom the standpoint of numerical implementation.

3.4. STORED ENERGIES 27

• Applicability to current-density optimization, which is instrumental in obtain-ing fundamental bounds and optimal currents.

A comprehensive review on stored energies in finite radiating systems is presentedin [16].

Stored energies in periodic structuresOne of the main results of this thesis, the stored energies representation for periodicstructures, has been preceded and built upon the excellent work of other authorson energy storage in antenna arrays, which is summarized below.

Edelberg and Oliner (1960) in their paper on mutual coupling effects in largearrays [87], consider planar arrays of slots and, by Babinet’s principle extended theirresults on arrays of dipoles [88]. By using waveguide-type analysis of an infiniteperiodic structure, they derive stored and radiated powers per unit cell, and discusshow Q-factor depends on the geometrical parameters. They provide a qualitativeand quantitative analysis of stored and radiated powers on scanning angle, and takeinto account higher-order modes.

Tomasic and Steyskal are first, to the author’s knowledge, to propose the lowerQ-factor bound result for arrays [89,90]. They derived an expression for the minimalQ-factor of an element in a phased array, by considering a canonical problem – anarray of infinite cylinders with magnetic [89] and electric [90] currents in free spaceand over a PEC or PMC ground. To estimate the lower bound, they account foronly the dominant cylindrical mode in the analysis. The stored energies are definedvia electric or magnetic fields in a unit cell, excluding the fields corresponding tothe propagating Floquet mode. The fields are given via vector potentials, whichare expressed in terms of Floquet modes. The integration over infinite unit cellvolume is performed numerically (in [90] it is proposed that volume integral abovecylinders can be evaluated analytically).

The expression for a Q-factor of an array of rectangular dipoles, derived byKwon and Pozar, is given in terms of Fourier transform of electric current pro-file [62]. Their work extends the result of Edelberg and Oliner by also consideringdipole arrays above a ground plane and above a grounded dielectric substrate.2 Ad-ditionally, Kwon and Pozar’s article focuses more on the Q-factor. In that paper,stored energies are expressed in fields, and the fields are expanded into Floquetmodes. The amplitudes of Floquet modes are expressed in terms of the Fouriertransform of the current density on the dipoles. There are two assumptions taken:only one Floquet mode propagates, and currents are only in the direction alongthe dipole. The integration over the infinite unit cell volume is performed analyt-ically, and only a knowledge of the electric current density is required to evaluatethe stored energies as an infinite sum of Floquet-modes contributions, given asclosed-form expressions.

2Interesting to note that Kwon and Pozar, as it seems, derived their result independently ofthe earlier Edelberg and Oliner’s work.


Papers I-II, appended to this thesis, present the array Q-factor and stored ener-gies, expressed in terms of a current density in a unit cell, in free space and above aground plane. The array elements can be of arbitrary shape and support arbitrarilydirected currents. Beam steering and multiple propagating modes are permitted.The two major advantages of this method are direct applicability to current den-sity optimization, and the finiteness of the domain over which the integration isperformed.

Stored energies in periodic setting: current densityrepresentation, free space caseThe stored energy is typically understood as the energy around the structure, thatdoes not contribute to radiation, nor dissipates in the antenna’s material. Singlingout the energy dissipated in the medium is straightforward (e.g. as conductivelosses), hence we consider a lossless structure for a moment. In the lossless case, thestored energy can intuitively be thought of as the total energy minus the radiatedenergy. The cornerstone of this reasoning is how to define the radiated energy, whichcan be done in many ways, see a review on this in [16] for finite structure case. InPapers I-II, we extract the energy associated with the propagating-Floquet-modesfields from the total energy to obtain the stored energies

We = ε

4

∫U

(|E|2 − |Ep|2

)dv, (3.9)

Wm = µ

4

∫U

(|H|2 − |Hp|2

)dv. (3.10)

Here, U is the unit cell as defined in (2.14), and Ep and Hp are electric andmagnetic fields associated with propagating Floquet-modes. These fields are foundfrom the propagating-modes scalar φp and vector Ap potentials as

Ep = −∇φp − jωAp, (3.11)

Hp = 1µ∇×Ap. (3.12)

The potentials here correspond to propagating Floquet-modes

φp(r1) = 1ε

∫Ω%(r2)Gp(r1, r2)dv2,

Ap(r1) = µ

∫ΩJ(r2)Gp(r1, r2)dv2,

(3.13)

with spatial support Ω of current density (see Section 2.2), and the propagating-modes terms of the periodic Green’s function

Gp(r1, r2) = 12jS

∑(m,n)∈P

1kzmn

e−jktmn·(ρ1−ρ2)e−jkzmn|z1−z2|. (3.14)


The summation is done over the set of propagating modes P = (m,n) : k2 −ktmn · ktmn ≥ 0, i.e. modes with real-valued longitudinal wavenumber. The totalelectric E and magnetic H fields are found from (2.9)-(2.10).

After the derivations, desctibed in detail in Paper I, the stored energies arerepresented by the current density:

We = µ

4k2

∫Ω

∫Ω

(∇1 · J∗(r1))[ReG(r1, r2)− k2g(r1, r2)

]∇2 · J(r2)dv2dv1

+ µk2

4

∫Ω

∫Ωg(r1, r2)J∗(r1) · J(r2)dv1dv2,

(3.15)

and

Wm =− µk2

4

∫Ω

∫Ω

(∇1 · J∗(r1))g(r1, r2)∇2 · J(r2)dv2dv1

+ µ

4

∫Ω

∫Ω

[ReG(r1, r2) + k2g(r1, r2)

]J∗(r1) · J(r2)dv1dv2.

(3.16)

The function g is emerged from the analytical treatment of the integral over theunit cell in the derivation (where Gp is involved) and is given by

g(r1, r2) = 14S

∑(m,n)∈Z2\P

1|kzmn|2

ejktmn·(ρ1−ρ2)e−|kzmn||z2−z1|(

1|kzmn|

+ |z1 − z2|).

(3.17)

By using vector calculus identities, the stored energies can be reformulated asquadratic forms of the current densities in a unit cell

We =∫

Ω

∫ΩJ∗(r1) ·Ke(r1, r2) · J(r2)dv1dv2, (3.18)

Wm =∫

Ω

∫ΩJ∗(r1) ·Km(r1, r2) · J(r2)dv1dv2. (3.19)

The dyadic kernels here are

Ke(r1, r2) = µ

4k2 Re∇1∇2G(r1, r2)

+ µ

4

(k21−∇1∇2

)g(r1, r2), (3.20)

Km(r1, r2) = µ

4 ReG(r1, r2)1

+ µ

4

(k21−∇1∇2

)g(r1, r2), (3.21)

with ∇1 a Jacobian with respect to r1, and 1 a unit dyadic.


Stored energies in periodic setting: current densityrepresentation, ground plane case

For the ground plane case, the same definition (3.9)-(3.10) of stored energies canbe used. There are no fields below the ground plane, thus integration is performedonly over the part of the unit cell U above the ground plane. The fields are relatedto potentials via (2.9)-(2.10) and (3.11)-(3.12). The total potentials in the halfspaceabove the ground plane are found from the image problem (see Section 2.4) as (2.27)and (2.28). The propagating-modes part is given by

Ap(r1) = µ

∫Ω

[Gp(r1, r2)1−Gp(r1, r2i)1z] · J(r2)dv2, (3.22)

φp(r1) = −jωε

∫Ω

(∇2[Gp(r1, r2)−Gp(r1, r2i)]) · J(r2)dv2. (3.23)

The derivation of the current-density representation of stored energies is given indetail in Paper II. The stored energies are quadratic forms, as in (3.18)-(3.19) withthe dyadic kernels

Ke(r1, r2) = µ

4k2 Re∇1∇2[G(r1, r2)−G(r1, r2i)]

+ µ

4

(k21−∇1∇2

)g(r1, r2)− µ

4

(k21z −∇1∇2

)g(r1i, r2),

(3.24)

Km(r1, r2) = µ

4 ReG(r1, r2)1−G(r1, r2i)1z

+ µ

4

(k21−∇1∇2

)g(r1, r2)− µ

4

(k21z −∇1∇2

)g(r1i, r2).

(3.25)

Radiated power in free space and ground plane cases

Radiated power can be expressed as the total power flow (real part of Poynting’svector) through the unit cell boundary δU

Pr = 12 Re

∫δU

[E(r)×H∗(r)] · n(r)dS. (3.26)

Here, n is the outward-normal unit vector of δU . From Poynting theorem fortime-harmonic fields, which reads as [38]

−∇ · (E ×H∗) + jω(ε|E|2 − µ|H|2) = E · J∗, (3.27)

we find the radiated power in terms of electric field and current density

Pr = Re−1

2

∫ΩE · J∗dv

. (3.28)


Substitution of the electric field with (2.9) and the corresponding potentials (from (2.18)and (2.19) for free space, and from (2.28) and (2.27) for the ground plane case) yieldsthe current-density representation of radiated energies as a quadratic form

Pr = 12

∫Ω

∫ΩJ∗(r1) ·Kr(r1, r2) · J(r2)dv1dv2. (3.29)

The dyadic kernel for free space is

Kr(r1, r2) = η

kIm(k21−∇1∇2

)G(r1, r2)

, (3.30)

and for the ground plane case

Kr(r1, r2) = η

kIm(k21−∇1∇2

)G(r1, r2)−

(k21z −∇1∇2

)G(r1, r2i)

.

(3.31)

The details of this derivation are given in Papers I-II.

Conductive lossesTo estimate the conductive losses in a unit cell, the surface resistivity model is used

POhm = 12

∫ΩRs(r1)|Js(r1)|2dS1. (3.32)

Here, Rs(r1) is the surface resistivity, Js(r1) is the surface current density. InPaper III, the surface resistivity is evaluated based on the skin-depth model Rs =√ωµ/2σ [38], with σ being conductivity.

Polarization, free space caseTo examine the polarization characteristics of an array, the vector-valued coordinate-independent coefficients in (2.25) are considered

Fmn+ = K+ · J(ktmn, kzmn) = K+ ·∫

Ωejktmn·ρ2ejkzmnz2J(r2)dv2. (3.33)

This is the far-field coefficient for (m,n)-th mode above an array. The far-fieldcoefficient in the region below an array is

Fmn− = K− ·∫

Ωejktmn·ρ2e−jkzmnz2J(r2)dv2, (3.34)

where

K− = 1kkzmn

k2xm − k2 kxmkyn −kxmkzmnkxmkyn k2

yn − k2 −kynkzmn−kxmkzmn −kynkzmn k2

zmn − k2

. (3.35)


The cross-polarized (cx-pol) and co-polarized (co-pol) components of the far-field are [91]

Fcx,mn± = ecx · Fmn± and Fco,mn± = eref · Fmn± (3.36)

respectively, where ecx and eref are chosen cross- and reference-polarization unitvectors. Polarization mismatch from the reference polarization is studied in PaperIII by considering the ratio between the cx-pol and co-pol components of the farfield.

Polarization, ground plane caseIn the ground plane case, there is no radiation below the ground plane, hence

Fmn− = 0. (3.37)

Above the array, the far-field coefficients are

Fmn+ = K+ ·∫

Ωejktmn·ρ2ejkzmnz2J(r2)dv2 −Kim ·

∫Ω

ejktmn·ρ2e−jkzmnz2J(r2)dv2,

(3.38)where

Kim = η

2kSkzmn

k2xm − k2 kxmkyn −kxmkzmnkxmkyn k2

yn − k2 −kynkzmnkxmkzmn kynkzmn k2 − k2

zmn

. (3.39)

The cx-pol and co-pol components are found as in (3.36).

3.5 Numerical implementation, matrix form

For numerical implementation, where the currents are approximated by a finite basisas in (2.34), the stored energies and the radiated power kernels can be representedby matrices with elements

W(u,w)e

W(u,w)m

R(u,w)

=∫

Ω

∫Ωfu(r1) ·

Ke(r1, r2)Km(r1, r2)Kr(r1, r2)

· fw(r2)dv1dv2. (3.40)

Here, the upper indices (u,w) denote the indices of an element in a matrix. Thestored energies and radiated power are then approximated by quadratic forms

We ≈ IHWeI, Wm ≈ IHWmI, Pr ≈12IHRI. (3.41)

Note the similarity of expression (3.40) with the expression for impedance-matrixelements (2.41) (or (2.42) in the ground plane case) when the sets of test func-tions and basis functions are chosen identical. In fact, results of the impedance-matrix computation can be reused in evaluating the radiation-power matrix, and

3.5. NUMERICAL IMPLEMENTATION, MATRIX FORM 33

contributions to the stored-energies matrices, corresponding to the first terms inkernels (3.20) and (3.21) (or (3.24) and (3.25) in the ground plane case). That is,to evaluate stored energies and radiated power only the contributions, associatedwith the function g in stored energies kernels, need to be additionally computed.

The conductive losses are represented by a quadratic form of currents with Grammatrix Ψ acting as a kernel [92].

POhm = Rs

2 IHΨI. (3.42)

The cx-pol and co-pol components of far-field coefficients can be represented byvectors with elements (here demonstrated for the free space case)

F(u)cx,mn±

F(u)co,mn±

=ecxeref

·K± ·

∫Ω

ejktmn·ρ2e±jkzmnz2fu(r2)dv2. (3.43)

Here, the upper index (u) denotes the index of an element in a vector. The far-fieldcomponents are approximated as linear forms of the electric current

Fcx,mn± ≈ Fcx,mn±I and Fco,mn± ≈ Fco,mn±I. (3.44)

Numerical validation of the Q-factor expressionFor the derived stored energies and Q-factor expressions to be useful and reliable inthe current density optimization to obtain fundamental bounds, they should be val-idated by comparison with alternative methods. Figures 3.3 and 3.4 demonstrate,for an array in free space and over a ground plane respectively, a good agreementof the derived Q-factor with the true-bandwidth-based QB and input-impedance-based QZ. Differences in the curves, still within a range of reasonable estimation,are observed at long electrical lengths kl and in the regions where the Q-factor islow and near the grating lobes. More validation examples are shown in Papers I-II.


0 2 4 6 8 10100

101

102

103

kl

Q

QB

Q

h l

p

l/2

QZ

x

y

Figure 3.3: Capped dipole array Q-factor evaluated from stored energies, inputimpedance and tuned fractional bandwidth as a function of electrical size

0 2 4 6 8

100

101

102

kl

QQZ

QB

Q

l

p

x

y dl

p

x

z

Figure 3.4: Q-factor evaluated from stored energies, input impedance and tunedfractional bandwidth as a function of electrical size for an array of tilted dipolesover a ground plane

Chapter 4

Fundamental bounds from Q-factorminimization

4.1 Optimization methods

The optimization methods, considered in this section are used for obtaining the Q-factor bounds and trade-off relations in Paper III. They give the global minimum ofthe considered optimization problems. An important class of optimization problemsare convex problems – there exist many efficient algorithms to solve such problemswith obtaining a global minimum.

Convex functions and sets, convex optimization

A function f(x) : Rn → R is called a convex function if it satisfies the inequality [93]

f(αx + (1− α)y) ≤ αf(x) + (1− α)f(y), (4.1)

for all x,y ∈ Rn and all α ∈ R : 0 ≤ α ≤ 1. A local minimum for a convexfunction is always a global minimum. This crucial property of convex functions is areason to reformulate optimization problems as convex optimization problems. Forexample, consider the minimization of the functions in Figure 4.1. In the case of aconvex function in Figure 4.1(a), once a local minimum is found, the search can beterminated, as this is the global minimum. In case of a non-convex function, seeFigure 4.1(b), once a local minimum is found, the search should still be carried on,because there might be another local minimum with a lower function value.

A set C is a convex set if for any x,y ∈ C and any α ∈ [0, 1] it holds that

αx + (1− α)y ∈ C. (4.2)

A convex optimization problem has a convex cost function, and the constraints

35

36CHAPTER 4. FUNDAMENTAL BOUNDS FROM Q-FACTOR

MINIMIZATION

f(x)

x

global minimum

convex

(a)

f(x)

xglobal minimum

local minimum

non-convex

(b)

Figure 4.1: Examples of (a) convex and (b) non-convex functions of one variable.

define a convex set. A convex optimization problem in a standard form is [93]

minimize f0(x)subject to fi(x) ≤ 0, i = 1, 2, ...,M

hj(x) = 0, j = 1, 2, ...,N ,(4.3)

where f0(x) and fi(x) are convex functions, and hj(x) are affine functions. An affinefunction is a linear function plus a constant, i.e. of the form hj(x) = aT

j x+bj whereaj ∈ Cn is a column vector, bj ∈ R is a constant, and (.)T transpose operator.

Convex problems have a wast number of efficient optimization methods andtheir solutions are global solutions [93–95]. Hence, it is important to be able toidentify convex problems, and to be able to reformulate optimization problems asconvex programs. Below, it is shown how a non-convex problem is relaxed to aconvex program.

Semidefinite programming

A semidefinite program in a standard form is

minimize tr(BX)subject to tr(CiX) = bi, i = 1, 2, ..., p

X 0.(4.4)

Here, the variable X ∈ CN×N is a matrix, and B,Ci ∈ CN×N are positive semidef-inite matrices. Semidefinite programs are a subclass of convex optimization prob-lems, and have many efficient optimization algorithms [95].

4.1. OPTIMIZATION METHODS 37

Minimum Q problemA simple Q-factor minimization problem without any additional constraints isgiven, from the definition of the Q-factor (3.6) as

minimizeJ

2ωmax(We(J),Wm(J))Pr(J) (4.5)

This minimization problem has infinite number of solutions, i.e. if J∗ is a solutionof the optimization problem, then aJ∗ is a solution as well for all a ∈ C \ 0.To pick one of these solutions, radiated power is typically normalized, yielding thefollowing optimization program

minimizeJ

2ωmax(We(J),Wm(J))

subject to Pr(J) = 1.(4.6)

For numerical implementation, the matrix representations of stored energies andradiated power (3.41) are substituted

minimizeI

2ωmax(IHWeI, IHWmI)

subject to 12IHRI = 1.

(4.7)

This problem falls into the class called quadratically constrained quadratic pro-grams (QCQP). It is non-convex, as the equality constraint includes a quadraticfunction, which is non-affine. The direct solution of such a problem can be madeby a nonlinear or heuristic solver, however, a minimum obtained this way mightbe a local minimum. Alternatively, the QCQP can be relaxed into a convex prob-lem. Two powerful methods, discussed here and used in Paper III, are semidefiniterelaxation (SDR) and the eigenvalue-based approach.

Semidefinite relaxationThe program (4.7) is reformulated as

minimizeI

Q

subject to 2ωIHWeI ≤ Q2ωIHWmI ≤ Q12IHRI = 1

Note that it is still non-convex due to the non-affine equality constraint. For anysymmetric matrix A ∈ RN×N the following statement holds

IHAI = tr(IHAI) = tr(AIIH) = tr(AX), (4.8)


MINIMIZATION

where IIH = X ∈ CN×N is a Hermitian matrix. The matrices We,Wm,R aresymmetric, see (3.40). Then an equivalent formulation of (4.8) is

minimize04X∈H

Q

subject to 2ω tr(WeX) ≤ Q2ω tr(WmX) ≤ Qtr(RX) = 2rank X = 1,

(4.9)

where H denotes the class of Hermitian matrices. The only non-convex constraintin this program is the rank constraint. This problem can be relaxed into a convexsemidefinite program by dropping the rank constraint

minimize04X∈H

Q

subject to 2ω tr(WeX) ≤ Q2ω tr(WmX) ≤ Qtr(RX) = 2.

(4.10)

This problem is of a polynomial complexity, and there exist efficient methods tosolve this type of problems [96]. The solution of this problem is a lower boundfor the program (4.9) (and consequently to the program (4.7)). Whenever theoptimal solution X∗ is of rank one, the optimal solution I∗ of (4.7) is found fromI∗IH∗ = X∗. If rank X∗ > 1, a vector I∗, feasible for (4.7), should be extracted

from X∗, however in this case (4.10) still gives the lower bound on Q. There areproblem-specific, and often intuitively reasoned heuristic methods to extract I∗ thatare used, see discussion and examples in [97,98]. It has been shown [99, §5] that asemidefinite program with three constraints (which is the case in (4.10)) always hasa rank one solution. One should note, however, that there might be other solutionsof a higher rank, and hence it is not guaranteed that the solution found by a solverfrom (4.10) is of rank one.

Inclusion of additional quadratic constraints can be straightforwardly done us-ing (4.8). For example, the efficiency can be studied by considering constrainton the conductive losses IHROhmI = δ which can be included in the semidefiniteprogram as

tr (ROhmX) = δ. (4.11)The constraint on the polarization mismatch is

|Fcx,mn±||Fco,mn±|

= ξ. (4.12)

It can be equivalently represented in a quadratic form

IHFHcx,mn±Fcx,mn±I = ξ2IHFH

co,mn±Fco,mn±I. (4.13)


The polarization-mismatch constraint can be included in the semidefinite programas

tr(

FHcx,mn±Fcx,mn± − ξ2FH

co,mn±Fco,mn±)X

= 0. (4.14)Both conductive-losses and polarization-mismatch constraints are considered in Pa-per III.

The eigenvalue methodAlthough SDR allows to solve a QCQP problem (4.7) in possibly a polynomial time,it is memory demanding: the number of elements in unknown X grows as a squareof the number of basis functions N . To circumvent this problem, an alternativeapproach is proposed, – the eigenvalue-based method. It has similarities to theeigenvalue methods in [100, 101] and utilizes the fact that the radiation matrix Rhas low rank.

The objective function in (4.7) is bounded from below by a convex combinationof the electric and magnetic stored energies

max(IHWeI, IHWmI) ≥ αIHWeI + (1− α)IHWmI = IHWαI, (4.15)

where α ∈ [0, 1] and Wα denotes the convex combination. Hence, the problem canbe relaxed as (the 2ω factor is intentionally omitted here)

maxα

minI

IHWαI

subject to 12IHRI = 1.

(4.16)

The Lagrangian with Lagrange multiplier λ associated with this problem is

L(α, λ) = IHWαI + λ

(1− 1

2IHRI). (4.17)

By the Karush-Kuhn-Tucker (KKT) conditions [93] the optimal I should satisfythe following conditions

∇IL(α, λ) = 2WαI− λRI = 0 (4.18)

1− 12IHRI = 0. (4.19)

Here, ∇I stands for a variational derivative with respect to I. The solutions of thesystem of equations (4.18)-(4.19) are critical points to the minimization problemin (4.16). The critical point that minimizes the cost function IHWαI is the solutionof minimization in (4.16). The maximization over parameter α is performed by asweep or a search algorithm.

The difficulty in solving the system (4.18)-(4.19) is in the generalized eigenvalueproblem

2WαI = λRI, (4.20)


MINIMIZATION

emerging from (4.18). This eigenvalue problem is ill-conditioned, whenever rank Ris low. This is usually the case in practice, as the rank of radiation matrix corre-sponds to the number of radiating modes. However, the low rank can be used togreatly simplify the eigenvalue problem and circumvent the ill-conditioning.

The radiated power matrix is symmetric and positive semidefinite (has onlynon-negative eigenvalues), and thus can be represented as R = URUH by eigen-decomposition. Here U is a unitary matrix, composed column-wise of eigenvectorsof R, and R is

R = UHRU =

d1. . .

dr0

. . .0

=[R11 0

0 0

]. (4.21)

Here, d1, d2, ..., dr are non-zero eigenvalues of R, and r = rank R. The sub-matrixR11 represents the non-zero part of R. Given the matrix U the currents vector canbe represented as a product of U with a vector I ∈ CN :

I = UI (4.22)

Multiplication of the generalized eigenvalue problem (4.20) by UH from the leftyields

2UHWαUI = λUHRUI. (4.23)The matrix product in the left hand side is denoted as

Wα = UHWαU =[W11 W12W21 W22

]. (4.24)

The partition is performed accordingly to the partition in (4.21): W11 is the topleft r× r segment of W. After the partition, the generalized eigenvalue problem isrepresented by a system of equations

W11I1 + W12I2 = λR11I1

W21I1 + W22I2 = 0(4.25)

Here, the current vector was partitioned as

I =[I1I2

](4.26)

with I1 ∈ Cr and I2 ∈ C(N−r). The second line of (4.25) relates the two parts of I

I2 = −W−122 W21I1. (4.27)


Direct substitution into the first line gives a new generalized eigenvalue problem

(W11 − W12W−122 W21)I1 = λR11I1, (4.28)

which is of size r × r. From the obtained eigenvalues and eigenvectors, the low-est eigenvalue and its corresponding eigenvector are picked as solutions λ and I1.The solution I of the original generalized eigenvalue problem (4.20) is found bybacktracking the relations (4.27), (4.26) and (4.22). As the eigenvalue problem isinvariant to a multiplication of its solution by a complex scalar, the second KKToptimality condition (4.19) is satisfied by rescaling the solution I→ ξI. This solvesthe minimization problem in (4.16) for a given α.

Note the drastic reduction of complexity here: instead of solving an N × Nill-conditioned generalized eigenvalue problem (4.20) for each iteration of searchover α, one only requires the solution of r×r generalized eigenvalue problem (4.28)on each search iteration. The eigenvalue problem for R to find R and U is onlyrequired to be performed once for the whole search. In practical cases N r.

The program (4.16) is a dual of the original minimum-Q-factor problem (4.7). Inthis specific case, a duality gap may appear, which here can be resolved by a linearcombination of degenerate eigencurrents (i.e. different eigencurrents with equaleigenvalues), minimizing the Q-factor, see discussion in [102]. Such degeneraciesare typically caused by symmetries in the structure on which currents are permitted,resulting in symmetric eigencurrents.

Additional quadratic constraints (e.g. on medium losses or on polarization) canbe introduced in the optimization problem and then included in the Lagrangian (4.17)similarly to the radiated-power constraint.

On meshing and lower boundThe numerical formulation (4.7) is an approximation of the Q-factor optimizationprogram (4.6). The optimization domain of the approximate problem (4.7) is theset of currents represented by the basis approximation (2.34), i.e. the set

Ja =Ja(r) : Ω→ R3,Ja(r) =

N∑n=1

Infn(r), In ∈ C

. (4.29)

This set is clearly a subset of the domain of the original program (4.6) whichincludes all the current densities in the region Ω. Hence, the original problem can bethought of as a relaxation of the approximate problem, and thus the optimal valueof the approximate problem cannot be lower than the optimal value of the originalproblem. In other words, the current density Ja(r) =

∑Nn=1 Infn(r) obtained from

the solution I of (4.7) and substituted into the objective function of (4.6) gives anupper approximation of the lower bound.

Refining the mesh allows to approximate currents finer. Hence, the solution ofthe approximate problem can be closer to the solution of the original problem.


MINIMIZATION

Mag

nitu

de in

dB

Optimal currents at 𝜆𝜆 = 1.5

Figure 4.2: Lower Q-factor bound for an array of rectangular geometries. The totalarea of a rectangle is 1/9 of the total area of the unit cell.

4.2 Optimal arrays

Optimization of the Q-factor over current densities by using SDR or eigenvaluetechniques allows to obtain fundamental bounds on the Q-factor. These funda-mental bounds are helpful to answer very practical questions in array design, andcan be obtained for specific geometries. For example, the results displayed in Fig-ure 4.2 address the following question: If we are to design a flat planar arrayelement bounded by a rectangle of a fixed surface area, should we look into squareor oblong shapes? We see that the best current density in the square region givessignificantly higher Q-factor than the best current density in the long narrow rect-angle region. These are the lower bounds of Q-factors for the respective regions forany array element within them. Hence, to obtain an array with wider bandwidthwe should design an oblong element, approximately realizing the optimal currents.Approaches for designing antennas close to the bound for non-periodic cases wereproposed in [69] where a genetic algorithm to shape antenna geometry was used,and in [70], where the placement of excitations was optimized to approximate theoptimal current density.

Another interesting example is a trade-off between the radiation Q-factor Qradand the radiation efficiency ηrad (see definitions in Section 3.3), shown in Fig-ure 4.3 for rectangular plates array with lx = 2ly and p2 = 9lxly, and surfaceresistance 1 Ohm/. With respect to this trade-off, the best radiation efficiencyvalue is around ηrad ' 95% – the radiation Q-factor reaches its minimum there, andabruptly increases for higher efficiency values. Interesting to note that a decreasein efficiency does not give improvements with respect to the bandwidth: Q-factorgradually increases as ηrad becomes lower.

4.2. OPTIMAL ARRAYS 43

0.5 0.6 0.7 0.8 0.9 10

20

40

60

ηrad

Qra

d

λ = 1.5p λ = 2p

λ = 3p λ = 5p

Figure 4.3: Trade-off between efficiency and Q-factor for an array of rectangularplates.

More numerical examples of trade-off relations of the Q-factor with geometricalparameters and polarization are given in Paper III.

Chapter 5

Fundamental bounds from sumrules

Another class of fundamental bounds is based on the sum rules: the relationsbetween a dynamic behavior of a system and its low- and high-frequency asymp-totes [103]. One of the powerful approaches to derive sum-rule-based fundamentalbounds for passive systems is by modeling them as a certain class of analytic func-tions of a complex variable, – Herglotz funtions [17,104,105].

5.1 Sum rules and Herglotz functions

The focus of this section is on the class of Herglotz functions and its applications.Herglotz functions is a class of holomorphic functions that map the open uppercomplex half-plane to the closed upper complex half-plane. The following subsec-tion provides the basics of complex analysis of functions of one variable, helpful inunderstanding Herglotz functions.

PrerequisitesThe open upper complex half-plane is denoted and defined as

C+ , z ∈ C : Im z > 0 . (5.1)

The closed upper complex half-plane is C+ ∪ R, i.e. it also includes the real line,which is the limit points of C+.

To define holomorphicity (also called analyticity), a notion of a complex deriva-tive should be introduced. A complex derivative of a function f(z) : Θ → C (withΘ ⊆ C) at point z0 ∈ Θ is defined as

f ′(z0) = limz→z0

f(z)− f(z0)z − z0

, (5.2)

45

46 CHAPTER 5. FUNDAMENTAL BOUNDS FROM SUM RULES

if this limit exists. If the limit does not exist, it is said that function does not have acomplex derivative at point z0. A holomorpic function on an open set is a functionthat has a complex derivative at any point of this open set. A holomorpic functionon a (possibly) non-open set V is a function that is holomorphic on some open setcontaining set V .

Herglotz functions basicsHerglotz function is a holomorphic function h : C+ → C such that Im h(z) ≥ 0for all z ∈ C+ [17]. Other names for this (or equivalent) type of functions in theliterature are Nevanlinna, Herglotz-Nevanlinna, R-, Pick and positive real functions.The following functions are simple examples of Herglotz functions

h1(z) = 1, h2(z) = j, h3(z) = z, h4(z) = −z−1,

h5(z) = tan z, h6(z) = ln z.(5.3)

To construct new Herglotz functions, the two following properties are useful.

1. Any positive linear combination of any Herglotz functions is a Herglotz func-tion.

2. For any two Herglotz functions f(z) and g(z), with f(z) not attaining a realvalue, the composition g(f(z)) is a Herglotz function.

Constructing new Herglotz functions is instrumental in obtaining sum rules, see forexample Paper IV. It is typically desired to have such a Herglotz function, whoseimaginary-part integral over a subset of R is a dynamic quantity of the interest;and whose asymptotics are simply obtained quantities.

It is said that a Herglotz function h admits an asymptotic expansion of orderK ≥ −1 at z =∞ if the real numbers b1, b0, b−1, ..., b−K exist such that

h(z) = b1z + b0 + b−1z−1 + ...+ b−Kz

−K + o(z−K

)as z→∞. (5.4)

Here, → is the limit in a cone α ≤ arg(z) ≤ (π − α), for some α > 0.Similarly, it is said that a Herglotz function h admits an asymptotic expansion

of order K ≥ −1 at z = 0 if the real numbers a−1, a0, a1, ..., aK exist such that

h(z) = a−1z + a0 + a1z + ...+ aKzK + o

(zK)

as z→0. (5.5)

Integral identities for Herglotz functionsThe next two theorems introduce the integral identities, the central results in thetheory of Herglotz functions from a practical standpoint.Theorem 1. [103] For a Herglotz function h and some integer N∞ ≥ 0, the limit

limε→0+

limy→0+

∫ε<|x|< 1

ε

x2N∞ Im h(x+ jy)dx (5.6)

5.1. SUM RULES AND HERGLOTZ FUNCTIONS 47

exists as a finite number if and only if the funtion h admits an asymptotic expansionof order 2N∞ + 1 at z =∞. Then the integral identity

limε→0+

limy→0+

1π

∫ε<|x|< 1

ε

xn Im h(x+ jy)dx =a−1 − b−1, n = 0−b−n−1, 0 < n ≤ 2N∞

(5.7)

holds.Theorem 2. [103] For a Herglotz function h and some integer N0 ≥ 0, the limit

limε→0+

limy→0+

∫ε<|x|< 1

ε

Im h(x+ jy)x2N0

dx (5.8)

exists as a finite number if and only if the funtion h admits an asymptotic expansionof order 2N0 + 1 at z = 0. Then the integral identity

limε→0+

limy→0+

1π

∫ε<|x|< 1

ε

Im h(x+ jy)xn

dx =a1 − b1, n = 2an−1, 2 < n ≤ 2N0

(5.9)

holds.The proof of both theorems follow from the representation theorem for Herglotz

functions. It is remarkable that by requiring just a few conditions – holomorphicityand restrictions on a range of a function – such powerful relations hold: a dynamicbehavior of a function along the real line is described by one or two expansioncoefficients.

In Paper IV, the integral identity (5.9) for n = 2 is used.

Transmission through a periodic screen as a passive systemConsider a causal, passive system in a convolution form (which implies linearity,continuity and time-invariance) with an impulse response w. A FT of w in such asystem has been shown to be related with a Herglotz function [106,107]. Identifyinga system with these properties within a physical object, phenomenon or system isan essential step to obtain a sum rule.

In Paper IV, as a linear, continuous, time-invariant, causal, passive system, weconsider the co-polarized transmission of a fundamental mode through a negligi-bly thin periodically perforated screen. In the spectral (i.e. Fourier transformed)domain, the function representing such a system is the co-polarized transmissioncoefficient T (k) of the fundamental mode. Here, k ∈ C+ is the wavenumber andT (k) is holomorphic [106,108].

From the fact that the screen is negligibly thin, and also from the conservation ofpower principle, it is found that the transmission coefficient is bounded by |T (k)−1/2| ≤ 1/2. The transmission coefficient T (k) is then a holomorphic mapping fromC+ to the closed disc D of radius 1/2 and centered in 1/2.

48 CHAPTER 5. FUNDAMENTAL BOUNDS FROM SUM RULES

1 2 3 4 50

0.2

0.4

0.6

0.8

1

T 20

λ/l

|T |2 |T (λ)|2Imh∆(m( ))

l

b

aw

T (λ)

Figure 5.1: Transmission coefficient and pulse function, representing transmissionbands for an array of cross-potent-shaped perforations. (© 2019 IEEE)

To represent it as a Herglotz function, T (k) is composed with a Möbius trans-formm [109], that retains holomorphicity. The resulting composed functionm(T (k))is a Herglotz function. However, we compose it with another Herglotz function– the pulse function h∆, whose imaginary part is unity whenever |T | > T0 andzero otherwise. Finally, the composed function h∆(m(T (k))) is a Herglotz func-tion with imaginary part that indicates whether the transmission is higher than agiven threshold T0, see Figure 5.1 (here, the horizontal axis is given in normalizedwavelengths λ/l, which are inversely proportional to k).

Application of the integral identity (5.9) for n = 2 allows to quantify the totallength of wavelength intervals, where |T | > T0. The right-hand side of the identityis solely dependent on the low-frequency expansion coefficient of T , which is pro-portional to the unit cell’s static polarizability. For a single transmission band, thesum rule gives an upper limit on fractional bandwidth around a central wavelengthλ0

B ≤ πγ

Aλ0

√1− T 2

0T 2

0. (5.10)

Here, A is the surface area of a unit cell, and γ is a static polarizability. The detailsof the derivation are in Paper IV. Paper V presents a method for evaluating thestatic polarizability of planar periodic structures.

5.2. APPLICATION OF THE SUM RULE 49

1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

T 20

λ/l

|T |2

l

a

w1

w2

96%

Figure 5.2: Transmission coefficient for an array of horseshoe-shaped perforations.The main band attains 96% of fundamentally available bandwidth. (© 2019 IEEE)

5.2 Application of the sum rule

One of the critical properties of fundamental bounds is whether the bound is tight.A bound, for which there exist examples reaching the limiting performance, hasmuch more practical use than a non-reachable bound. Thus it is important todemonstrate for each given bound that it is possible to attain the limiting perfor-mance at least in some cases. Then the bound would have much more credibilityto serve as a figure of merit.

In Paper IV, we demonstrate that a simulated structure can reach 96% of thebandwidth, predicted by the sum-rule’s upper bound (5.10), see Figure 5.2. Itis also shown that the seemingly idealistic assumptions (PEC screen with negligi-ble thickness and infinite periodicity) used in the sum-rule derivation are a goodapproximation of the real structures. The sum rule can be used in design and opti-mization problems in several ways – as a figure of merit or as an optimization goal,or to find the more promising perforation shape by picking the one with the largestpolarizability.

Chapter 6

Conclusions, outlook andsustainability discussion

6.1 Conclusions

To keep up with overall technological progress in the field of wireless systems,the electromagnetic design tools need to be continuously improved and extended.Currently, full-wave electromagnetic simulation tools are easily available and pro-vide accurate prediction for real electromagnetic structures; non-deterministic op-timization algorithms, such as genetic methods, are typically built-in into softwarepackages [110–112]. The non-deterministic optimization methods can significantlyimprove a design (e.g. in Paper IV we use a genetic algorithm in CST MW Stu-dio [110] to increase transmission bandwidth), the result is usually a local optimum,and there is no information about the global optimum available in such methods.

A novel and seminal trend in research on electromagnetic design tools are fun-damental bounds, previously found mainly for finite antennas [18, 21] by usingcurrent density optimization [26] and for various structures (e.g. absorbers [113],metamaterials [23]) by using the sum-rules approach [103].

This trend is addressed by one of the central results of this thesis – the pro-posed method to derive fundamental Q-factor bounds for periodic electromagneticstructures. The main building blocks of the proposed method are the following.

• The representation of the Q-factor for periodic structures in terms of electriccurrent densities. It was derived analytically, and the numerical implementa-tion, compatible with MoM-type codes is proposed. The representation takesinto account the exact current density distribution on a periodic element,and does not require a specified excitation or port to evaluate the Q-factor.The key advantage of the representation is its applicability to current densityoptimization.

• Electric current density optimization using the semidefinite-relaxation and the

51

52CHAPTER 6. CONCLUSIONS, OUTLOOK AND SUSTAINABILITY

DISCUSSION

eigenvalue-based optimization methods. The minimal values of the Q-factor,obtained by these optimization methods are global minima, which meansthat they can serve as fundamental lower bounds for the Q-factor. Optimalcurrents can be, in most cases, extracted from solutions of the optimizationproblems. Optimal current densities can be used to gain an understanding ofthe mechanism behind optimal radiation.

• The obtained fundamental bounds can be formulated in terms of the trade-offrelations, bounding Q-factor at a given frequency with size parameters, scan-ning, efficiency or polarization. Such trade-off relations give vital information:for example, in Paper III, we find that attempts to increase efficiency beyonda certain limiting value lead to an abrupt increase in the Q-factor.

Methods to obtain fundamental physical bounds and optimal currents are inti-mately related to the efforts toward automated electromagnetic design, discussedin the next section.

The fundamental bounds for periodic structures are also considered from a per-spective of sum rules. The fundamental limitations on the bandwidth of trans-mission through a thin periodically perforated screen are investigated. The ad-vantage of this relation is that it is simple and straightforward to use: it boundsthe attainable bandwidth by static polarizability from above, giving an insight onthe connection of dynamic performance (bandwidth) and static-charge-separationproperties.

6.2 Future work and extensions

The work, presented in this thesis, can be continued in the several directions, sug-gested below.

• Extension of the array Q-factor optimization. Adding dielectric inclusions isof practical interest – many array antennas have dielectric layers (a substrate)next to the ground plane. Considering types of periodicity grids other thanrectangular – e.g. triangular, hexagonal, or askew – can help investigatehow different kinds of periodicity affect array bandwidth performance. Bothof these extensions require modification of Green’s functions to the case of amultilayered dielectric environment [114] and to non-rectangular lattices [115].Another direction of extensions is to introduce additional constraints, forexample on an embedded radiation pattern. This would yield new trade-offcurves, useful from a practical perspective.

• Finding arrays close to the bounds. Demonstrating array designs that performoptimally with respect to the bound is important for showing that the boundis tight. For the small non-periodic antennas, designs approaching the boundshave been investigated in [21,28,100].

6.3. DISCUSSION ON THE SUSTAINABILITY 53

• Automated array design. The ultimate goal in the development of electromag-netic design tools is the automation of the design process, i.e. , creating analgorithm that would synthesize an optimal antenna, with respect to a givenmetric, or constraints. Despite the significant progress in antenna theory,numerical tools, optimization methods and an ever-increasing computationalpower, such an algorithm, synthesizing optimal antennas, does not as of yetexist. The prominent works, pursuing the goal of automated antenna design,have employed genetic algorithms [69], feed positioning [27], and topologysensitivity [116]. Optimization of the electric-current-density representationof the Q-factor is one of the key elements in these works [27,69,116]. Hence,the Q-factor for periodic structures, presented in this thesis, is a steppingstone for the automated design of array antennas.

6.3 Discussion on the sustainability

In the modern world, human-made technologies significantly affect our planet. Thediscussion of sustainability within each branch of technologies is thus of vital im-portance. The global information and communications technology, which includeswireless systems, is responsible for an amount of carbon dioxide emissions equivalentto that of global aviation [117].

The impact of the work, presented in this thesis, on sustainable development ofwireless systems is twofold. Firstly, the trade-off results for antenna arrays allow tobalance between the bandwidth and efficiency. The proposed tools are useful to findmore optimal array designs, reducing energy consumption by an array. In mobilecommunication networks, the base stations consume the majority of the power(up to 80%) [118] – larger part of which is used by amplifiers and antennas [117].Phased arrays are a common choice for base station antennas, and the improvedenergy efficiency in arrays means a significant overall improvement in the mobilecommunication networks in general. Optimal designs would also have longer life-cycle, as they cannot be replaced solely because a better design was developed.Secondly, the here reported results are a step towards automated antenna arraydesign. This could shift the focus of antenna engineers’ efforts from electromagneticdesign to better integration on a system level for improved energy efficiency.

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kth.diva-portal.org1386207/... · 2020. 1. 16. · Abstract Inthisthesis,theoptimalbandwidthperformanceofperiodicelectromag-neticradiatorsandscatterersisstudied. Themainfocusisonthedevelopment

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