optimization and physical bounds for passive and non

Optimization and Physical Bounds for Passive and Non-passive Systems









Linnaeus University Dissertations No 373/2019

OPTIMIZATION AND PHYSICAL BOUNDS FOR PASSIVE AND NON-PASSIVE SYSTEMS

YEVHEN IVANENKO

LINNAEUS UNIVERSITY PRESS



YEVHEN IVANENKO




YEVHEN IVANENKO




YEVHEN IVANENKO




YEVHEN IVANENKO




YEVHEN IVANENKO




YEVHEN IVANENKO




YEVHEN IVANENKO




YEVHEN IVANENKO


Optimization and Physical Bounds for Passive and Non-passive Systems: Doctoral Dissertation, Department of Physics and Electrical Engineering, Linnaeus University, Växjö, 2019 ISBN: 978-91-89081-23-9 (print), 978-91-89081-24-6 (pdf) Published by: Linnaeus University Press, 351 95 Växjö Printed by: Holmbergs, 2019









Abstract Ivanenko, Yevhen (2019). Optimization and Physical Bounds for Passive and Non-passive Systems, Linnaeus University Dissertations No 373/2019, ISBN: 978-91-89081-23-9 (print), 978-91-89081-24-6 (pdf). Physical bounds in electromagnetic field theory have been of interest for more than a decade. Considering electromagnetic structures from the system theory perspective, as systems satisfying linearity, time-invariance, causality and passivity, it is possible to characterize their transfer functions via Herglotz functions. Herglotz functions are useful in modeling of passive systems with applications in mathematical physics, engineering, and modeling of wave phenomena in materials and scattering. Physical bounds on passive systems can be derived in the form of sum rules, which are based on low- and high-frequency asymptotics of the corresponding Herglotz functions. These bounds provide an insight into factors limiting the performance of a given system, as well as the knowledge about possibilities to improve a desired system from a design point of view. However, the asymptotics of the Herglotz functions do not always exist for a given system, and thus a new method for determination of physical bounds is required. In Papers I–II of this thesis, a rigorous mathematical framework for a convex optimization approach based on general weighted L -norms, 1 ∞ , is introduced. The developed framework is used to approximate a desired system response, and to determine an optimal performance in realization of a system satisfying the target requirement. The approximation is carried out using Herglotz functions, B-splines, and convex optimization. Papers III–IV of this thesis concern modeling and determination of optimal performance bounds for causal, but not passive systems. To model them, a new class of functions, the quasi-Herglotz functions, is introduced. The new functions are defined as differences of two Herglotz functions and preserve the majority of the properties of Herglotz functions useful for the mathematical framework based on convex optimization. We consider modeling of gain media with desired properties as a causal system, which can be active over certain frequencies or frequency intervals. Here, sum rules can also be used under certain assumptions. In Papers V–VII of this thesis, the optical theorem for scatterers immersed in lossy media is revisited. Two versions of the optical theorem are derived: one based on internal equivalent currents and the other based on external fields in terms of a T-matrix formalism, respectively. The theorems are exploited to derive fundamental bounds on absorption by using elementary optimization techniques. The theory has a potential impact in applications where the surrounding losses cannot be neglected, e.g., in medicine, plasmonic photothermal therapy, radio frequency absorption of gold nanoparticle suspensions, etc. In addition to this, a new method for detection of electrophoretic resonances in a material with Drude-type of dispersion, which is placed in a straight waveguide, is proposed. Keywords: Convex optimization, physical bounds, Herglotz functions, quasi-Herglotz functions, passive systems, non-passive systems, approximation, absorption in lossy media


Abstract Ivanenko, Yevhen (2019). Optimization and Physical Bounds for Passive and Non-passive Systems, Linnaeus University Dissertations No 373/2019, ISBN: 978-91-89081-23-9 (print), 978-91-89081-24-6 (pdf). Physical bounds in electromagnetic field theory have been of interest for more than a decade. Considering electromagnetic structures from the system theory perspective, as systems satisfying linearity, time-invariance, causality and passivity, it is possible to characterize their transfer functions via Herglotz functions. Herglotz functions are useful in modeling of passive systems with applications in mathematical physics, engineering, and modeling of wave phenomena in materials and scattering. Physical bounds on passive systems can be derived in the form of sum rules, which are based on low- and high-frequency asymptotics of the corresponding Herglotz functions. These bounds provide an insight into factors limiting the performance of a given system, as well as the knowledge about possibilities to improve a desired system from a design point of view. However, the asymptotics of the Herglotz functions do not always exist for a given system, and thus a new method for determination of physical bounds is required. In Papers I–II of this thesis, a rigorous mathematical framework for a convex optimization approach based on general weighted L-norms, 1∞,is introduced. The developed framework is used to approximate a desired system response, and to determine an optimal performance in realization of a system satisfying the target requirement. The approximation is carried out using Herglotz functions, B-splines, and convex optimization. Papers III–IV of this thesis concern modeling and determination of optimal performance bounds for causal, but not passive systems. To model them, a new class of functions, the quasi-Herglotz functions, is introduced. The new functions are defined as differences of two Herglotz functions and preserve the majority of the properties of Herglotz functions useful for the mathematical framework based on convex optimization. We consider modeling of gain media with desired properties as a causal system, which can be active over certain frequencies or frequency intervals. Here, sum rules can also be used under certain assumptions. In Papers V–VII of this thesis, the optical theorem for scatterers immersed in lossy media is revisited. Two versions of the optical theorem are derived: one based on internal equivalent currents and the other based on external fields in terms of a T-matrix formalism, respectively. The theorems are exploited to derive fundamental bounds on absorption by using elementary optimization techniques. The theory has a potential impact in applications where the surrounding losses cannot be neglected, e.g., in medicine, plasmonic photothermal therapy, radio frequency absorption of gold nanoparticle suspensions, etc. In addition to this, a new method for detection of electrophoretic resonances in a material with Drude-type of dispersion, which is placed in a straight waveguide, is proposed. Keywords: Convex optimization, physical bounds, Herglotz functions, quasi-Herglotz functions, passive systems, non-passive systems, approximation, absorption in lossy media







Sammanfattning Fundamentala begränsningar inom elektromagnetisk fältteori har varit intressanta i mer än ett decennium. Om man betraktar elektromagnetiska strukturer ur systemteoriperspektiv, som linjära, tidsinvarianta, kausala och passiva system, är det möjligt att karakterisera deras överföringsfunktioner via Herglotzfunktioner. Herglotzfunktioner är användbara vid modellering av passiva system med tillämpningar inom matematisk fysik, teknik och modellering av vågfenomen i material och spridning. Begränsningarna för passiva system kan härledas i form av summationsregler som baseras på låg- och högfrekvensasymptoter för deras motsvarande Herglotzfunktioner. Dessa begränsningar ger en inblick i de faktorer som begränsar prestandan för ett givet system, såväl som kunskap om möjligheterna att förbättra ett önskat system ur en designsynpunkt. Det är dock inte alltid som Herglotzfunktionernas asymptoter existerar för ett givet system. Då kan summationsregler inte härledas, och därför krävs en ny metod för att bestämma de fysikaliska begränsningarna. I artiklarna I–II i denna avhandling introduceras ett rigoröst matematiskt ramverk för en konvex optimeringsmetod baserad på viktade L -normer, 1 ∞ . Ramverket används för att approximera en önskad systemrespons och för att bestämma en optimal prestanda vid realiseringen av ett system som uppfyller givna specifikationer. Approximationen utförs med hjälp av Herglotzfunktioner, B-splines och konvex optimering. Artiklarna III–IV i denna avhandling handlar om modellering och bestämning av optimala prestandagränser för kausala, men icke-passiva system. För att modellera dessa introduceras en ny funktionsklass, s.k. kvasi-Herglotzfunktioner. De nya funktionerna definieras som differensen mellan två Herglotzfunktioner och bevarar viktiga egenskaper för Herglotzfunktioner. Detta gör dem användbara i det matematiska ramverket baserat på konvex optimering. Förstärkande, eller s.k. aktiva media med vissa önskade egenskaper modelleras som ett kausalt system som kan vara aktivt för vissa frekvenser eller frekvensintervall. En summationsregel kan också användas under vissa antaganden. Artiklarna V–VI i den här avhandlingen behandlar det optiska teoremet för spridare i förlustmedia. Två versioner av det optiska teoremet härleds fram. Den första versionen bygger på interna ekvivalenta strömmar medan den andra bygger på externa fält i termer av T-matrisformalism. Teoremen utnyttjas för att härleda fundamentala begränsningar för absorption med hjälp av optimeringstekniker. Denna teori är relevant för tillämpningar där de omgivande förlusterna inte kan försummas, t.ex. inom medicin, plasmonisk fototermisk terapi, radiofrekvensabsorption av guldnanopartikel-suspensioner, etc. Utöver detta, föreslås en ny metod för detektering av elektroforetiska resonanser i ett material med dispersion av Drude-typ, som placerats i en rak vågledare.


Sammanfattning Fundamentala begränsningar inom elektromagnetisk fältteori har varit intressanta i mer än ett decennium. Om man betraktar elektromagnetiska strukturer ur systemteoriperspektiv, som linjära, tidsinvarianta, kausala och passiva system, är det möjligt att karakterisera deras överföringsfunktioner via Herglotzfunktioner. Herglotzfunktioner är användbara vid modellering av passiva system med tillämpningar inom matematisk fysik, teknik och modellering av vågfenomen i material och spridning. Begränsningarna för passiva system kan härledas i form av summationsregler som baseras på låg- och högfrekvensasymptoter för deras motsvarande Herglotzfunktioner. Dessa begränsningar ger en inblick i de faktorer som begränsar prestandan för ett givet system, såväl som kunskap om möjligheterna att förbättra ett önskat system ur en designsynpunkt. Det är dock inte alltid som Herglotzfunktionernas asymptoter existerar för ett givet system. Då kan summationsregler inte härledas, och därför krävs en ny metod för att bestämma de fysikaliska begränsningarna. I artiklarna I–II i denna avhandling introduceras ett rigoröst matematiskt ramverk för en konvex optimeringsmetod baserad på viktade L-normer, 1∞. Ramverket används för att approximera en önskad systemrespons och för att bestämma en optimal prestanda vid realiseringen av ett system som uppfyller givna specifikationer. Approximationen utförs med hjälp av Herglotzfunktioner, B-splines och konvex optimering. Artiklarna III–IV i denna avhandling handlar om modellering och bestämning av optimala prestandagränser för kausala, men icke-passiva system. För att modellera dessa introduceras en ny funktionsklass, s.k. kvasi-Herglotzfunktioner. De nya funktionerna definieras som differensen mellan två Herglotzfunktioner och bevarar viktiga egenskaper för Herglotzfunktioner. Detta gör dem användbara i det matematiska ramverket baserat på konvex optimering. Förstärkande, eller s.k. aktiva media med vissa önskade egenskaper modelleras som ett kausalt system som kan vara aktivt för vissa frekvenser eller frekvensintervall. En summationsregel kan också användas under vissa antaganden. Artiklarna V–VI i den här avhandlingen behandlar det optiska teoremet för spridare i förlustmedia. Två versioner av det optiska teoremet härleds fram. Den första versionen bygger på interna ekvivalenta strömmar medan den andra bygger på externa fält i termer av T-matrisformalism. Teoremen utnyttjas för att härleda fundamentala begränsningar för absorption med hjälp av optimeringstekniker. Denna teori är relevant för tillämpningar där de omgivande förlusterna inte kan försummas, t.ex. inom medicin, plasmonisk fototermisk terapi, radiofrekvensabsorption av guldnanopartikel-suspensioner, etc. Utöver detta, föreslås en ny metod för detektering av elektroforetiska resonanser i ett material med dispersion av Drude-typ, som placerats i en rak vågledare.







Dedicated to my family and friends, those who are alive and passed away...

AbstractPhysical bounds in electromagnetic field theory have been of interest for morethan a decade. Considering electromagnetic structures from the system theoryperspective, as systems satisfying linearity, time-invariance, causality and passivity,it is possible to characterize their transfer functions via Herglotz functions.Herglotz functions are useful in modeling of passive systems with applicationsin mathematical physics, engineering, and modeling of wave phenomena inmaterials and scattering. Physical bounds on passive systems can be derived inthe form of sum rules, which are based on low- and high-frequency asymptoticsof the corresponding Herglotz functions. These bounds provide an insight intofactors limiting the performance of a given system, as well as the knowledgeabout possibilities to improve a desired system from a design point of view.However, the asymptotics of the Herglotz functions do not always exist for agiven system, and thus a new method for determination of physical bounds isrequired. In Papers I–II of this thesis, a rigorous mathematical framework for aconvex optimization approach based on general weighted Lp-norms, 1 p 1,is introduced. The developed framework is used to approximate a desiredsystem response, and to determine an optimal performance in realization of asystem satisfying the target requirement. The approximation is carried out usingHerglotz functions, B-splines, and convex optimization.

Papers III–IV of this thesis concern modeling and determination of optimalperformance bounds for causal, but not passive systems. To model them, a newclass of functions, the quasi-Herglotz functions, is introduced. The new functionsare defined as differences of two Herglotz functions and preserve the majorityof the properties of Herglotz functions useful for the mathematical frameworkbased on convex optimization. We consider modeling of gain media with desiredproperties as a causal system, which can be active over certain frequencies orfrequency intervals. Here, sum rules can also be used under certain assumptions.

In Papers V–VII of this thesis, the optical theorem for scatterers immersed inlossy media is revisited. Two versions of the optical theorem are derived: onebased on internal equivalent currents and the other based on external fields interms of a T-matrix formalism, respectively. The theorems are exploited to derivefundamental bounds on absorption by using elementary optimization techniques.The theory has a potential impact in applications where the surrounding lossescannot be neglected, e.g., in medicine, plasmonic photothermal therapy, radiofrequency absorption of gold nanoparticle suspensions, etc. In addition to this,a new method for detection of electrophoretic resonances in a material withDrude-type of dispersion, which is placed in a straight waveguide, is proposed.

Keywords: Convex optimization, physical bounds, Herglotz functions, quasi-Herglotz functions, passive systems, non-passive systems, approximation, absorp-tion in lossy media

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SammanfattningFundamentala begränsningar inom elektromagnetisk fältteori har varit intressantai mer än ett decennium. Om man betraktar elektromagnetiska strukturer ursystemteoriperspektiv, som linjära, tidsinvarianta, kausala och passiva system,är det möjligt att karakterisera deras överföringsfunktioner via Herglotzfunk-tioner. Herglotzfunktioner är användbara vid modellering av passiva system medtilllämpningar inom matematisk fysik, teknik och modellering av vågfenomeni material och spridning. Begränsningarna för passiva system kan härledas iform av summationsregler som baseras på låg- och högfrekvensasymptoter förderas motsvarande Herglotzfunktioner. Dessa begränsningar ger en inblick i defaktorer som begränsar prestandan för ett givet system, såväl som kunskap ommöjligheterna att förbättra ett önskat system ur en designsynpunkt. Det är dockinte alltid som Herglotzfunktionernas asymptoter existerar för ett givet system.Då kan summationsregler inte härledas, och därför krävs en ny metod för attbestämma de fysikaliska begränsningarna. I artiklarna I–II i denna avhandlingintroduceras ett rigoröst matematiskt ramverk för en konvex optimeringsmetodbaserad på viktade Lp-normer, 1 p 1. Ramverket används för att approx-imera en önskad systemrespons och för att bestämma en optimal prestanda vidrealiseringen av ett system som uppfyller givna specifikationer. Approximationenutförs med hjälp av Herglotzfunktioner, B-splines och konvex optimering.

Artiklarna III–IV i denna avhandling handlar om modellering och bestämningav optimala prestandagränser för kausala, men icke-passiva system. För attmodellera dessa introduceras en ny funktionsklass, s.k. kvasi-Herglotzfunktioner.De nya funktionerna definieras som differensen mellan två Herglotzfunktioneroch bevarar viktiga egenskaper för Herglotzfunktioner. Detta gör dem användbarai det matematiska ramverket baserat på konvex optimering. Förstärkande, ellers.k. aktiva media med vissa önskade egenskaper modelleras som ett kausaltsystem som kan vara aktivt för vissa frekvenser eller frekvensintervall. Ensummationsregel kan också användas under vissa antaganden.

Artiklarna V–VI i den här avhandlingen behandlar det optiska teoremetför spridare i förlustmedia. Två versioner av det optiska teoremet härledsfram. Den första versionen bygger på interna ekvivalenta strömmar medanden andra bygger på externa fält i termer av T-matrisformalism. Teoremenutnyttjas för att härleda fundamentala begränsningar för absorption med hjälp avoptimeringstekniker. Denna teori är relevant för tillämpningar där de omgivandeförlusterna inte kan försummas, t.ex. inom medicin, plasmonisk fototermiskterapi, radiofrekvensabsorption av guldnanopartikel-suspensioner, etc. Utöverdetta, föreslås en ny metod för detektering av elektroforetiska resonanser i ettmaterial med dispersion av Drude-typ, som placerats i en rak vågledare.

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Acknowledgments

It has been a great privilege to work with and be surrounded by a lot of amazingindividuals.

First of all, I want to express my deep gratitude to my head supervisor,Prof. Sven Nordebo, for inspiration, guidance, motivation to become a goodresearcher, and support during my postgraduate studies. I am sincerely thankfulto my co-supervisor, Prof. Börje Nilsson, for wise counsel, collaboration, andguidance on postgraduate courses. I am grateful to my co-supervisor, Prof.Mats Gustafsson, for inspirational discussions on Herglotz functions and passivesystems, valuable suggestions, especially the ones related to grant applicationsand international research visits, as well as for the opportunity to visit and stayat the Department of Electrical and Information Technology at Lund University. Iwould like to thank my co-supervisor, Dr. Mariana Dalarsson, for collaborationand her valuable recommendations on my licentiate and doctoral theses.

It has been a luck to be a part of the SSF-research project Complex analysisand convex optimization for EM design. I am grateful to all the project members,especially Prof. Lars Jonsson, Prof. Annemarie Luger, Prof. Daniel Sjöberg, Dr.Casimir Ehrenborg, Dr. Mitja Nedic, Dr. Andrei Osipov, and Dr. Dale Frymark.Thank you all for the experience obtained during my doctoral studies, as well asfor extremely fruitful collaboration and other activities within the SSF-researchproject: workshops, summer school, paper discussions, extraordinary researchfocus weeks, etc. I gratefully acknowledge Prof. Joachim Toft for his valuableadvices on mathematical issues and for his contribution in the paper on passiveapproximation and optimization with B-splines.

I want to thank the Swedish Foundation for Strategic Research (SSF) andABB AB for their financial support of my doctoral studies which has made thisdoctoral thesis possible.

I am thankful to my colleagues from the Department of Physics and ElectricalEngineering at Linnæus University, especially Dr. Matz Lenells, Prof. Sven-ErikSandström, Dr. Pieternella Cĳvat, Dr. Pieter Kuiper, Dr. Anna Pertsova, and Dr.Magnus Perninge for their helpfulness. The same goes to the colleagues fromother departments of the Faculty of Technology at Linnæus University, especiallyDr. Anna Wingkvist, Prof. Andrei Khrennikov, Dr. Christian Engström, Dr.Patrik Wahlberg, and Dr. Karl-Olof Lindahl. Many thanks go to my colleaguesand friends at Linnæus University for their help, support and fruitful discussions,including Dr. Kostiantyn Kucher, Alisa Lincke, Dr. Ekaterina Yurova-Axelsson,Dr. Davood Khodadad, Dr. Dmitry Prokhorov, and all the others. Also, I want tothank Roland Lindholm, who has been my teacher in Swedish during the lastyears, and helped to improve my language skills a lot.

I would also like to acknowledge the Electromagnetic Theory group at LundUniversity, especially Prof. Anders Karlsson, Prof. Gerhard Kristensson, Dr.

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Doruk Tayli, Dr. Andreas Ericsson, Dr. Jakob Helander, and Johan Lundgren,who also helped a lot to make my PhD studies such a great experience.

I am extremely grateful to Prof. Richard Bayford from the Department ofNatural Sciences at Middlesex University in London, for the possibility to havea training on measurements and experimentation with gold nanoparticles atthe department’s laboratory, as well as for the acquired knowledge. The samegoes to the Department of Natural Sciences at Middlesex University in London,especially Dr. Nazanin Neshatvar, Rui Damaso, Dr. Serena De Gelidi, and Dr.Nima Seifnaraghi. This research training visit has been supported by ErasmusStaff Mobility for Training grant under the European Commission’s Programmefor education, training, youth and sport (Erasmus+).

Looking one step back, I would like to thank the Department of ElectronicDevices and Information Computer Technologies at Odessa National PolytechnicUniversity, especially Dr. Oleg Tsyganov, Dr. Valeriy Skonechnyi, Prof. LeonidPanov, Prof. Halyna Shcherbakova, Prof. Anatoliy Yefimenko, Dr. Oleksiy Pavlov,Dr. Oleksandr Tynynyka, as well as Prof. Porfiriy Baranov, Viktoriya Kuzmina,and Prof. Vira Lyubchenko.

Last, but not the least, I would like to express my deepest gratitude myfamily and relatives, especially to my parents Hennadii and Nataliia, my lategrandparents Aleksey, Zoya, Viacheslav, and Tamara, for believing in me andtheir infinite support. I want to thank my partner Liuba for being by my sideand supportive, and my friends for always being there.

Växjö, Sweden2019

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Table of Contents

Abstract xi

Sammanfattning xiii

Acknowledgments xv

Table of Contents xvii

List of Symbols and Notation Used xix

Preface xxi

Part I: Introduction and Research Overview 11 Background, Motivation, and Goals . . . . . . . . . . . . . . . . . . 32 Passive and Causal Systems . . . . . . . . . . . . . . . . . . . . . . 73 Integral Representation of Passive and Causal Systems . . . . . . . 124 Optimization Techniques . . . . . . . . . . . . . . . . . . . . . . . . 175 Approximation of Passive and Non-passive Systems . . . . . . . . 226 Material Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 Optical Theorem in Lossy Media . . . . . . . . . . . . . . . . . . . 328 Research Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . 399 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4310 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Bibliography 44

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List of Symbols and Notation Used

R, C the sets of real numbers and complex numbers, respec-tively

C+ the upper half-plane, C+ ⇤ {z 2 C | Im{z} > 0}C+ the right half-plane, C+ ⇤ {z 2 C | Re{z} > 0}i imaginary unit, i2 ⇤ �1z ⇤ x + iy complex number with x ⇤ Re{z} and y ⇤ Im{z}z⇤ complex conjugate, (x + iy)⇤ ⇤ x � iy! non-tangential limitp.v.

Ø(·) the Cauchy principal value integral

(·)+ highlights the fact that parameters with the correspond-ing underscore represent a function with non-negativeimaginary part

↵ Hölder exponentcn B-spline expansion coefficient or optimization variableh a Herglotz functionq a quasi-Herglotz functionF the target functionpn B-spline basis functionpn the (negative) Hilbert transform of B-spline basis functionpi magnitude of a point-massw weight functionw(t) the impulse responseC space of continuous functionsCk space of k times continuously differentiable functionsC0,↵ space of Hölder continuous functions, 0 < ↵ < 1C1

0 space of smooth functions with compact supportk · kLp (w) the norm corresponding to the weighted Lebesgue Lp

space, w > 0, 1 p 1⌦ the approximation domainO open neighborhood of ⌦Q set of all quasi-Herglotz functionsQsym set of all symmetric quasi-Herglotz functions, Qsym ✓ Q

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Preface

This doctoral dissertation summarizes the research I have carried out at theDepartment of Physics and Electrical Engineering at Linnæus University duringthe last five years. The first introductory part provides a review on causaland passive systems, their integral representations, electromagnetic models, aswell as approximation and optimization tools for the determination of optimalrealizations and physical bounds used in Papers I–VII. The second part presentsthe Papers I–VII, which have been published or are submitted to peer reviewedscientific journals and conference proceedings. The papers are the outcome ofcollaboration within the project Complex analysis and convex optimization forEM design supported by the Swedish Foundation for Strategic Research (SSF),grant no. AM13-0011 under the program Applied Mathematics.

List of Included PapersI. Y. Ivanenko, M. Gustafsson, B. L. G. Jonsson, A. Luger, B. Nilsson, S.

Nordebo, and J. Toft. Passive approximation and optimization using B-splines. SIAM J. Appl. Math., 79(1): 436–458, 2019.

Contributions of the author: The author of this thesis has been leadingthe writing of this paper, which has been conducted as a subproject withinthe larger SSF project mentioned above. The first author has collectedinformation, comments and suggestions from the co-authors and has beenresponsible for the numerical examples.

II. Y. Ivanenko and S. Nordebo. Passive Approximation with High-OrderB-Splines. In Analysis, Probability, Applications, and Computation. Edited byK.-O. Lindahl et al., pages 83–94. Springer Nature Switzerland AG, 2019.

Contributions of the author: The author of this thesis has been re-sponsible for all of the analysis, numerical examples and writing of thispaper.

III. Y. Ivanenko, M. Nedic, M. Gustafsson, B. L. G. Jonsson, A. Luger, and S.Nordebo. Quasi-Herglotz functions and convex optimization. Royal Soc.Open Sci, submitted for publication, arXiv preprint arXiv:1812.08319, 2019.

Contributions of the author: The author of this thesis has been leading thewriting of this paper, which has been conducted as a subproject within the

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http://www.eit.lth.se/index.php?puid=175&projectpage=projektfakta

http://www.eit.lth.se/index.php?puid=175&projectpage=projektfakta

http://www.stratresearch.se/

larger SSF project mentioned above. The first author has been responsiblefor the parts concerning convex optimization whereas the second authorhas been responsible for the mathematical analysis and theorems. Thefirst author has collected information, comments and suggestions from theco-authors, and has been responsible for the numerical examples.

IV. Y. Ivanenko and S. Nordebo. Non-passive approximation as a tool tostudy the realizability of amplifying media. In Proceedings of 2019 URSICommission B International Symposium on Electromagnetic Theory (EMTS).IEEE, 2019.�


V. S. Nordebo, M. Dalarsson, Y. Ivanenko, D. Sjöberg, and R. Bayford. On thephysical limitations for radio frequency absorption in gold nanoparticlesuspensions. J. Phys. D: Appl. Phys., 50(15): 155401, 2017.

Contributions of the author: The author of this thesis has contributedwith comments and suggestions on theory as well as on writing, andnumerical experiments.

VI. Y. Ivanenko, M. Dalarsson, S. Nordebo, and R. Bayford. On the plasmonicresonances in a layered waveguide structure. In 12th International Congresson Artificial Materials for Novel Wave Phenomena (Metamaterials 2018), pages188–190. IEEE, 2018.


VII. Y. Ivanenko, M. Gustafsson, and S. Nordebo. Optical theorems and physicalbounds on absorption in lossy media. Opt. Express, 27(23): 34323–34342,2019.

Contributions of the author: The author of this thesis has been leadingthe writing of this paper, which has been conducted as a subproject withinthe larger SSF project mentioned above. The first author has collectedinformation, comments and suggestions from the co-authors and has beenresponsible for the numerical examples.

�Awarded with the URSI Young Scientist Award at 2019 URSI Commission B InternationalSymposium on Electromagnetic Theory (EMTS) in San Diego CA, United States.

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Other Publications by the AuthorThe author of this dissertation is also the author or co-author of the followingpublications which are related to but not part of the dissertation:

VIII. Y. Ivanenko. Estimation of electromagnetic material properties with ap-plication to high-voltage power cables. Licentiate thesis, Department ofPhysics and Electrical Engineering, Linnæus University, Växjö, Sweden.Linnæus University Press, 2017.

IX. Y. Ivanenko and S. Nordebo. Measurements and estimation of the complexvalued permeability of magnetic steel. Linnæus University, Department ofPhysics and Electrical Engineering, 351 95, Växjö, Sweden, Tech. Rep. URI:urn:nbn:se:lnu:diva-48445, 2015.

X. Y. Ivanenko and S. Nordebo. Estimation of complex valued permeability ofcable armour steel. In Proceedings of 2016 URSI International Symposium onElectromagnetic Theory (EMTS), pages 830–833, IEEE, 2016.

XI. B. Nilsson, A. Ioannidis, Y. Ivanenko, and S. Nordebo. On absence ofsources at infinity and uniqueness for waveguide solutions. In Proceedingsof 2016 International Conference on Electromagnetics in Advanced Applications(ICEAA), pages 780–783, IEEE, 2016.

XII. Y. Ivanenko and S. Nordebo. Approximation of dielectric spectroscopydata with Herglotz functions on the real line and convex optimization. InProceedings of 2016 International Conference on Electromagnetics in AdvancedApplications (ICEAA), pages 863–866, IEEE, 2016.

XIII. S. Nordebo, M. Gustafsson, Y. Ivanenko, B. Nilsson, and D. Sjöberg. Cylin-drical multipole expansion for periodic sources with applications for three-phase power cables, Mathematical Methods in the Applied Sciences, 41(3):959–965, 2018.

XIV. A. Ludvig-Osipov, J. Lundgren, C. Ehrenborg, Y. Ivanenko, A. Erics-son, M. Gustafsson, B. L. G. Jonsson, and D. Sjöberg. Fundamentalbounds on transmission through periodically perforated metal screenswith experimental validation, IEEE Trans. Antennas Propagat., 2019.doi:10.1109/TAP.2019.2943430

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https://doi.org/10.1109/TAP.2019.2943430

XV. M. Nedic, C. Ehrenborg, Y. Ivanenko, A. Ludvig-Osipov, S. Nordebo, A.Luger and B. L. G. Jonsson, D. Sjöberg, and M. Gustafsson. Herglotzfunctions and applications in electromagnetics. In Advances in MathematicalMethods for Electromagnetics, Edited by K. Kobayashi and P. Smith, 22 pages.IET, 2019.

XVI. S. Nordebo, Y. Ivanenko, and R. Bayford. On the optimal plasmonic reso-nances in gold nanospheres embedded in dispersive media. In Proceedingsof 2019 URSI Commission B International Symposium on Electromagnetic Theory(EMTS). IEEE, 2019.

xxiv

Part I

Introduction and Research Overview

Yevhen Ivanenko

�. BACKGROUND, MOTIVATION, AND GOALS 3

1 Background, Motivation, and Goals

Holomorphic mappings between certain half planes have taken an importantplace in modeling electromagnetic structures from a system viewpoint in thelast decades. Such mappings are characteristic for passive systems [95,98–100],which intuitively can be defined as systems that do not produce energy. However,for a physical system to be passive from a system point of view, it has to besingle-valued, linear, continuous, time-translationally invariant, and have aninput-output relation in convolution form. An important property implied bypassivity is causality [100]. Causality associates the real and imaginary parts of asignal in the frequency domain in the form of the Hilbert transform [3,48]. Thecorresponding relations are also known as Kramers-Kronig relations [45,48,52,62],which play an important role in Titchmarsh’s theorem [48,72]. However, passivityadds more severe constraints on bandwidth of a signal than causality [100]. Theimpulse response of a passive system in the frequency domain is given by aHerglotz function. Herglotz functions (also known as Nevanlinna and R-functions)or Positive Real (PR) functions [1,4,5,30,47,72,100] are analytic functions mappingcomplex half planes to their corresponding closures, and provide a completeinformation about a passive system via their integral representation formula. Thelow- and/or high-frequency asymptotic properties of Herglotz functions are usedin the derivation of sum rules, which are related to physical bounds for givenpassive systems [13].

Sum rules are useful in the determination of physical bounds in manyphysical and engineering branches such as dielectric constants [52], matchingnetworks [28], antennas [34,37,38,46], high-impedance surfaces [36], absorbers [76],periodic structures [39], extraordinary transmission [56], reflection coefficients [33],broadband metamaterials [35], as well as for interaction between the waves andscatterers [11,83]. They relate dynamic parameter values of a passive system toits asymptotics, and provide theoretical insight on improvement of a designedsystem. Also, several other limitations, such as speed-of-light limitations inpassive linear media, have been derived by using general Herglotz functiontheory [94]. However, in some cases, the presence of losses in a considered systemaffects the small- and large-argument asymptotic expansions of a representingHerglotz function. An example of such a system is the reflection coefficient froma passive metal-backed dielctric slab, where the metal is finitely conductive; seee.g., [69]. This results in an impossibility to derive the physical bound based onsum rules, and hence for systems that inhibit asymptotic properties of Herglotzfunctions, an alternative approach is required.

To derive a physical bound on a given system, the problem has to be reformu-lated with respect to the following question: “What is a best/optimal realization of alinear system with a desired target response over a finite frequency band subject to givensystem constraints?" Existing methods for solving such a problem are related to

4 Part I� Introduction and Research Overview

boundary values of analytic functions on the real axis, where linear systems canbe approximated via the Routh approximation method, i.e., by linear systemsof lower order [42], using minimum energy functions [49], Laguerre and Kautzfunctions [91], or Hardy approximations [8]. However, for approximation theory,a different route is followed, where passive approximation based on convexoptimization is employed [69].

The first goal of this thesis is to develop a rigorous mathematical tool forstudying physical bounds on given passive systems, which is supported byconstructive proofs. This framework is supposed to be equipped with weightedLp-norms, and involves the convex cone of approximating Herglotz functions,which are Hölder continuously extended to an open neighborhood of the realaxis and have a measure given as a finite B-spline expansion of a fixed arbitraryorder. In Papers I–II of this thesis, we incorporate this framework with convexoptimization approach, which allows us to add a priori knowledge about adesired system, such as measurement data or asymptotic properties in the formof a sum rule, as a constraint in the optimization problem to be considered.

Papers III–IV of this thesis are focused on modeling of causal systems,which do not satisfy the passivity requirement. Such systems can e.g., beassociated with the use of gain media. One well known example of gainmedia are fluorescent dyes. The fluorescent dye is a causal non-linear materialwhich can be linearized as a four-level system for pumping energy [19], and itscorresponding dispersion relation is given by the Lorentz model for non-passivematerials [19,53,78,93,96]. Gain media are used for light localization in plasmonics,extraordinary transmission, nanoantennas, plasmon waveguides, cloaking, andother applications; see e.g., [6,19,20,31,53,55,58,81] with references. These causalsystems can be treated on the real axis via Kramers-Kronig relations in the Hilbertspace [48, 72], or in the sense of distributions [10, 100]. The bounds on dispersionrelations for causal systems over a finite frequency intervals has been proposedin [62], where the method is based on finite Kramers-Kronig relations. Thereexist several studies and methods for approximation of given causal systemsusing Hardy approximants [7] for Lp-functions, and solving bounded externalproblems with pointwise constraints [9]. However, these methods do not relatethe dynamic parameters of causal systems to their low- and/or high-frequencyasymptotics, and the causal systems cannot be described by Herglotz functions.

The problem we would like to solve in Papers III–IV of this thesis is based onthe question: “What is an optimal realization of a causal system with a given targetresponse over a finite frequency band, which may have amplification properties overcertain frequencies or frequency intervals, and is passively constrained outside of theseintervals?” Hereby, the second goal of this thesis is to extend the existing class ofadmittance passive systems, which will involve a subclass of non-passive systemscharacterized by a certain set of analytic functions in the frequency domain. Asa result, a new set of functions, the quasi-Herglotz functions, is proposed. These

�. BACKGROUND, MOTIVATION, AND GOALS 5

functions are defined as differences of two Herglotz functions, and preserve mostof the properties of Herglotz functions: the integral representation formula, theboundary-value representation, and for some cases admit the sum-rule identities.We restrict the set of functions to those, which have a Hölder continuous extensionto an open neighborhood of the approximation domain on the real axis, andwhere the generating measure is defined as a finite-order B-spline expansion.The proposed approximants are suitable for incorporation with the convexoptimization approach formulated in Papers I–II of this thesis. In some cases,it is also possible to correlate a priori known asymptotic properties of a desiredcausal system with its dynamic parameters using sum-rule constraints in theoptimization problem under consideration.

Papers V–VII of this thesis are focused on physical bounds on absorption ofpower by scatterers immersed in lossy media. The bounds on scattering andabsorption have been derived for many physical and engineering applications suchas small dipole scatterers [88], radar absorbers [76], high-impedance surfaces [36],passive metamaterials [81, 84], absorption and scattering [61, 65, 80], etc. TheHerglotz function theory has been used in [83] to derive the physical bound onthe extinction cross section, which depends on the geometry of a consideredscatterer. However, these bounds have been derived under condition thatthe medium surrounding the scattering object is lossless. It has been shownin [14,27, 29, 54, 63, 64, 70, 71, 85, 86, 97] that these bounds will be nullified in thepresence of losses, and thus, the power absorbed by the surrounding mediumshould be included in the power balance for the optical theorem [15].

Physical bounds on absorption in lossy media will bring new insight on thescattering properties of objects both from the theoretical and from the designperspective in application areas, where losses cannot be neglected. One suchexample from medicine, is shown in Figure 1. Figure 1 shows the uptake of goldnanoparticles by cancer cells. This can enable the cancer cells to be heated inthe radio frequency and microwave frequency ranges, without damaging thesurrounding normal tissue; see the references [21–23, 68, 79]. The cancer cellwith inserted gold nanoparticles is henceforth referred to in this thesis as agold nanoparticle suspension. The phenomenon of heating of gold nanoparticlesuspensions is not completely understood [68, 79] and requires additional studiesfrom the physical and design perspectives. In the optical frequency range, theapplications include the use of gold nanoparticles in plasmonic photothermaltherapy for cancer treatment [41], and are concerned with light penetrationin biological tissue [26]. Beyond cancer, scattering and absorption parametersare important in medical telemetry [60, 82], where the objective is to reducethe amount of absorbed power, and to design a reliable communication link fortransmission of power from an antenna implanted in the human body to a receiver.Yet other applications include photonics with z-dotted PMMA materials [2], and


in short-wave communications at 60GHz [40, 59, 74, 90, 92], where the absorptionband of oxygen is.

Figure 1: Uptake of 5nm glutathione coated gold nanoparticles in colorectalcancer cells: a) after 90 minutes; b) after 15 hours. Reprinted from Biomaterials, 32,T. Lund, M. F. Callaghan, P. Williams, M. Turmaine, C. Bachmann, T. Rademacher,I. M. Roitt, and R. Bayford, The influence of ligand organization on the rate ofuptake of gold nanoparticles by colorectal cancer cells, 9776–9784, 2011, withpermission from Elsevier.

Physical bounds on absorption in lossy media cannot be determined in theform of a sum rule due to the presence of losses in the considered system,which has been discussed in Papers I–II of this thesis described above. Hence, adifferent approach based on optimization techniques is proposed. We presenttwo versions of an optical theorem in lossy media. The first version is formulatedin terms of the internal fields, i.e., it is based on equivalent currents and servesas an extension of the work in [61]. The second version is based on externalfields, where the T-matrix method [52, Eq. (7.34)] is used, and generalizes theresults in [70]. The two versions are incorporated then with the Method ofLagrange Multipliers [17, 57], which is an elementary optimization techniqueused to determine the physical bounds on absorption in lossy media for arbitraryscatterers made of arbitrary materials.

1.1 Thesis OutlineThis thesis consists of two parts and is organized as follows. Section 2 providesan overview on one-port causal and passive systems, and the correspondingintegral representations of these systems are discussed in Section 3. Section 4introduces optimization techniques used in Part II for determination of optimalrealizations and physical bounds for passive and a subclass of non-passivesystems. Section 5 contains theory for the developed approximation frameworkincorporated with the numerical convex optimization approach described inSection 4.1. Section 6 reviews the dispersion models characterizing linear passiveand non-passive materials, and the corresponding constructed Herglotz and

�. PASSIVE AND CAUSAL SYSTEMS 7

quasi-Herglotz functions, respectively, describing the systems based on thesemodels. Section 7 presents physical bounds on absorption in lossy media derivedusing the analytical optimization technique discussed in Section 4.2. Section 8gives an overview on the contributions included in Part II of this thesis andhighlights the results achieved in this manuscripts. Part I ends with conclusionsand discussions on future work in Sections 9 and 10, respectively. Part II includesthe scientific papers, as listed in the Preface.

1.2 Notation and Conventions

The electric and magnetic field intensities EEE and HHH are given in SI-units [45]and the time convention for time harmonic fields (phasors) is given by e�i!t ,where ! is the angular frequency and t the time. Let µ0, ✏0, ⌘0 and c0 denotethe permeability, the permittivity, the wave impedance and the speed of light invacuum, respectively, and where ⌘0 ⇤

pµ0/✏0 and c0 ⇤ 1/pµ0✏0. The wave number

of vacuum is given by k0 ⇤ !pµ0✏0, and hence !µ0 ⇤ k0⌘0 and !✏0 ⇤ k0⌘�1

0 .The definition of spherical vector waves [3, 15, 16, 45, 52, 67] is summarized inAppendix B of Paper VII. The field expansions based on the spherical vectorwaves [3, 15, 16, 45, 52, 67] are presented in Section 7. In particular, the regularspherical Bessel functions, the Neumann functions, the spherical Hankel functionsof the first kind and the corresponding Riccati-Bessel functions [52] are denotedby jl(z), yl(z), h(1)

l (z) ⇤ jl(z)+ iyl(z), l(z) ⇤ zjl(z) and ⇠l(z) ⇤ zh(1)l (z), respectively,

all of order l, where the complex-valued argument z 2 C is written z ⇤ x+ iy withx , y 2 R. The real and imaginary parts and the complex conjugate of a complexnumber ⇣ are denoted by Re {⇣}, Im {⇣} and ⇣⇤, respectively. For dyadics, thenotation (·)† denotes the Hermitian transpose.

2 Passive and Causal Systems

In this section, we give an overview on the basic properties of passive and causalsystems, which are within the scope of this thesis.

Passive systems can intuitively be thought of as physical objects that donot produce energy. However, from a system point of view, the mathematicalissue is with definition of such a system and its corresponding input-outputrelation. Here, we consider systems from macroscopic point of view, i.e., asystem considered as a “black box” having one input and one correspondingoutput, where the signals can be measured, see Figure 2(a), and all its dynamicproperties can be described mathematically by the representation formula inintegral form. To define passive and causal systems, we restrict the systems underconsideration to satisfy the criteria of linearity, time-invariance, continuity andhaving input-output relation in convolution form.


System

inputa)

output

b)

b)

Ru(t) v(t)

Figure 2: Illustration of a one-port system: a) Structural representation; b)Mathematical representation.

For the definitions given below, let R denote the system’s operator as shownin Figure 2(b).

Definition 2.1 A system with input u(t) and output v(t) in the time domain is linearif

R{↵1u1(t)+↵2u2(t)} ⇤ ↵1R{u1(t)}+↵2R{u2(t)} (1)

for ↵1 ,↵2 2 R or C, implying that the output is given as a linear function of theinput [100, Sec. 5.8]. ⇤

Definition 2.2 A system with input u(t) and output v(t) in the time domain istime-translationally invariant if

v(t) ⇤ R{u(·)}) v(t + t0) ⇤ R{u(·+ t0)} (2)

for all t0, implying that the system does not explicitly depend on time, by producing ashifted output v(t + t0) for a given shifted input u(t + t0) [100, Sec. 5.8]. ⇤

Definition 2.3 A system with input u(t) and output v(t) in the time domain iscontinuous if [100, Sec. 5.8]

un ! u ) R{un}! R{u}. (3)

⇤

Intuitively, this can be interpreted as a small perturbation in the input signal to acontinuous system only leads to a small perturbation in the output signal [12].

The properties given in Definitions 2.1 through 2.3 characterize systems inconvolution form [100, Sec. 5.8].

Definition 2.4 A system with input u(t) and output v(t) in the time domain is inconvolution form if

v(t) ⇤ (w ⇤ u)(t) :⇤πR

w(⌧)u(t � ⌧)d⌧, (4)

where w(t) is the impulse response [66]. ⇤


Systems satisfying the criteria given in Definitions 2.1 through 2.4 are calledone-ports. In this thesis, we restrict ourselves to real-valued one-port systems.

To define a causal system, a one-port system should be restricted as follows.

Definition 2.5 A one-port system with convolution operator R{·} in the time domainis causal if

w(t) ⇤ 0 for t < 0, (5)

where w is the impulse response of the one-port system [100, Sec. 10.3]. ⇤

The Definition 2.5 implies that the output of a one-port causal system cannotprecede the given input. It is also applicable to the input signals u(t), which canbe causal as well. The corresponding holomorphic Fourier transform of a causalsignal is given by

U(!) ⇤πR+

u(t)ei!t dt (6)

for Im{!} > 0. Note that if the function U(⌫), ⌫ 2 R, is square integrable, i.e., thatU(⌫) is in the Hilbert space (or L2-space) [3], and has its inverse Fourier transformu(t) vanishing for t < 0, i.e., u(t) ⇤ 0 for t < 0 as in (5), then by Titchmarsh’stheorem [72, Thm. 1.6.1], it can be shown that the function is also square integrableon any straight line parallel to the real axis as

πR|U(!)|2 d⌫ < C, (7)

where ! ⇤ ⌫ + i Im{!}, U(!) is holomorphic for Im{!} > 0, and U(⌫) ⇤limIm{!}!0+ U(⌫ + i Im{!}) for almost all ⌫ 2 R, and C is a constant. Further,from Titchmarsh’s theorem [72, Thm. 1.6.1], the real and imaginary parts of acausal signal U in the Hilbert space can be determined by employing Sokhotski-Plemelj formulas [72] as

Re{U(⌫)} ⇤ 1⇡

p.v.πR

Im{U(⇠)}⇠� ⌫ d⇠, (8)

andIm{U(⌫)} ⇤ � 1

⇡p.v.

πR

Re{U(⇠)}⇠� ⌫ d⇠, (9)

respectively, for ⌫ 2 R. The relations (8) and (9) are well known as Kramers-Kronig relations [45, 52]. The symmetry property of these relations implies thatthey are the Hilbert transforms [48] of each other, which allows reconstructionof the whole signal when either its real or imaginary part is given, such asU ⇤Re{U}� iH Re{U} or U ⇤H Im{U}+ i Im{U}, respectively, where H denotesthe Hilbert transform operator.

Passive systems can be considered as a subclass of causal systems additionallysatisfying certain requirements. There exist two ways to define passivity for


different types of systems: admittance passivity developed by Zemanian [99, 100]and scattering passivity developed by Youla et al. [98]. Note that it is possible toconstruct a new admittance passive system from a given scattering passive viathe Cayley transform, and vice versa; see [95, Thm. 5], as well as [89, 98, 99].

Definition 2.6 Consider a convolution system with input signal u(t) and output signalv(t), both of which in general can be complex valued. The system is called admittance-passive if

Wadm(T) :⇤ Reπ T

�1v(t)u(t)⇤ dt � 0 (10)

for all T 2 R and all u 2 C10 (i.e., smooth functions with compact support). ⇤

Here, Wadm(T) denotes the energy absorbed by the system until the time T. Byrequiring this quantity to be non-negative, we say that the system absorbs moreenergy than it emits, and hereby, it does not produce energy. Passivity alsoimplies that the system is causal [100].

Consider an example of an admittance-passive system in electromagnetics.

Example 2.7 Passive material modelsConsider a linear, isotropic and time-translationally invariant material having

the response to an applied electric field intensity EEE(t) in terms of a time domainconstitutive relation

DDD(t) ⇤ ✏0✏r(t) ⇤EEE(t) ⇤ ✏0✏1EEE(t)+ ✏0

πR�(t � ⌧)EEE(⌧)d⌧, (11)

where DDD(t) is the electric flux density, ✏r(t) the real-valued relative permittivityconvolution kernel, ✏0 the permittivity of free space, ✏1 > 0 the instantaneousresponse, and the susceptibility � ⇤ 0 for t < 0; see, e.g., [45, 52]. Based onPoynting’s theorem [45,52] and according to the Definition 2.6, the material (ormore precisely, the convolution operator ✏r(t)) is passive ifπ T

�1EEE(t) · @DDD

@tdt ⇤ ✏0

π T

�1

πR

EEE(t) · @@t

(✏1�(t � ⌧)+�(t � ⌧))EEE(⌧)d⌧dt � 0, (12)

for all T, and for all electric fields EEE(t) given as testing functions with compactsupport. If the passive convolution operator is an operator of slow growth,then the energy expression (12) is valid also for the testing functions of rapiddescent [100]. It can be shown that this passivity condition is equivalent to thecondition

h(!) ⇤ !✏(!), (13)

where h(!) is a function characterizing passive systems in the frequency domain,known as a Herglotz function; see details in Section 3. Here, ✏(!) is theholomorphic Fourier transform of the convolution kernel ✏r(t)

✏(!) ⇤πR+

✏r(t)ei!t dt (14)


in the upper complex half plane for Im{!} > 0; see e.g., [35, 100].

Let us now consider the second definition of passivity, scattering passivity.

Definition 2.8 Consider a convolution system with input signal u(t) and output signalv(t). The system is called scattering-passive if

Wscat(T) :⇤π T

�1

�|u(t)|2 � |v(t)|2

�dt � 0 (15)

for all T 2 R and all u 2 C10 . ⇤

Here, Wscat(T) corresponds to the energy balance of a given system until thetime T. Wscat is non-negative if the system is passive, implying that the energyof its output signal is always less than that of its input signal. It can be shownthat the transfer function Wscat(s) of a scattering-passive system satisfies therelation |Wscat(s)| 1 for s 2 C+, i.e., s 2 C for Re{s} > 0 [13,100]. Then a suitablerational (Cayley) transformation of the transfer function, namely the functions 7! i(1+Wscat(�is))/(1�Wscat(�is)) where the Laplace parameter s ⇤ �i! forIm{!} > 0, is a Herglotz function, described in Section 3.

Consider an example of a scattering-passive system in electromagnetics.

Example 2.9 Reflection from an isotropic slab placed above a ground planeConsider the reflection coefficient � from an isotropic non-magnetic slab of

thickness d characterized by passive permittivity ✏, and placed above a perfectelectric conducting plane (PEC), see Figure 3. The reflection coefficient can beused for describing the given system as a scattering-passive system. Let theincident wave EEEi be the input signal and the reflected wave EEEr the output signalof the system. Assume the reference plane is placed in front of the slab, i.e., x 0.

✏ PEC

Ei

Er

x

0d

Figure 3: Illustration of a non-magnetic dielectric slab placed above the groundplane. The reference plane is chosen at x ⇤ 0.

The reflection coefficient for an isotropic slab corresponds to the reflectedwave EEEr ⇤ �EEEi and is calculated as

�(k) ⇤ �0 �e2iknd

1� �0e2iknd, (16)


where �0 ⇤ (⌘�1)/(⌘+1) is the reflection coefficient at the air-slab interface, k thewavenumber, n ⇤

p✏ the refractive index of the slab, and ⌘ ⇤

p1/✏ the relative

wave impedance of the slab. This function can be shown to map the open upperhalf of complex plane to the closed unit disk.

3 Integral Representation of Passive and CausalSystems

The impulse response of a passive real-valued one-port system [95,100] can berepresented in the time domain as [100]:

w(t) ⇤ b�0(t)+H(t)πR

cos(⇠t)d�+(⇠), (17)

where b � 0, �0 denotes the first derivative of the Dirac delta distribution, H theHeaviside step function, and �+ the positive Borel measure satisfying the growthcondition

ØR

d�+(⇠)/(1+ ⇠2) <1; see also [66].Borel measures can be defined as follows; see for details [87, Chs. 8.2 and 8.4].

Definition 3.1 A measure � defined on all open sets of a topological space is called aBorel measure, if �(K) <1 for every compact set K. ⇤

By application of the Fourier–Laplace transform [77] to iw(t), where w(t)is given by (17), the impulse response of the passive system in the frequencydomain gets the representation similar to the integral representation of Herglotzfunctions [13,30] satisfying the symmetry property to be described in this section.This result demonstrates a crucial connection between passive systems andHerglotz functions, which have properties useful for determination of physicalbounds.

In this section, we give a review information about Herglotz functions andtheir properties related to bounds on passive systems. Further, a class of quasi-Herglotz functions is introduced, which is useful for modelling a subclass ofcausal systems that do not fulfil admittance-passivity requirement (10) (seeDefinition 2.6), but still preserving the integral representation typical for passivesystems.

3.1 Herglotz FunctionsThroughout of this section and till the end of Part I, let the following open halfplanes be defined as:

C+ ⇤ {z 2 C | Im{z} > 0}, (18)C+ ⇤ {z 2 C | Re{z} > 0}, (19)

where C+ and C+ denote the open upper and right half planes, respectively.

�. INTEGRAL REPRESENTATION OF PASSIVE AND CAUSAL SYSTEMS 13

Definition 3.2 A function h :C+ !C is called a Herglotz function if it is holomorphicwith Im{h(z)} � 0 for any z 2 C+. ⇤

One of the properties of Herglotz functions is their integral representationformula, given by

h(z) ⇤ a+ + b+z +πR

1+ ⇠z⇠� z

d�+(⇠) (20)

for z 2 C+, where a+ 2 R, b+ � 0, and �+ is a finite positive Borel measure; see,e.g., [1, 5, 10, 30, 47, 72]. Let �+ denote the positive Borel measure satisfyingthe growth condition

ØR

d�+(⇠)/(1+ ⇠2) <1; then the integral representation ofHerglotz functions becomes

h(z) ⇤ a+ + b+z +πR

✓1

⇠� z� ⇠

1+ ⇠2

◆d�+(⇠). (21)

In the representations (20) and (21), the real-valued parameter a+ is given bya+ ⇤ Re{h(i)}, and b+ can be obtained as b+ ⇤ limz!1 h(z)/z, where ! denotes anon-tangential limit, and hence z!1 means that |z | !1 in the Stolz cone

' arg z ⇡�' (22)

for any ' 2 (0,⇡/2].

z

Re

Im

Im{z} > 0

h(z)

Re

Im

Im{h(z)} � 0

h

Figure 4: Illustration of a Herglotz function h mapping the open upper half ofcomplex plane C+ to the closed upper half of complex plane C+[R.

As mentioned in this section above, the crucial property of symmetric Herglotzfunctions satisfying the symmetry condition

h(z) ⇤ �h(�z⇤)⇤ , (23)

is that they are related to real-valued one-port passive systems and used indetermination of physical bounds.

Definition 3.3 A Herglotz function h satisfying the additional condition (23) is calledsymmetric. ⇤


The symmetric Herglotz function can be represented as

h(z) ⇤ b+z +p.v.πR

1+ ⇠2

⇠� zd�+(⇠), (24)

for z 2 C+, and where the measure �+ is symmetric, and the integral is the Cauchyprincipal value (p.v.) integral at infinity. Consequently, symmetric Herglotzfunctions have integral representation in terms of the measure �+:

h(z) ⇤ b+z +p.v.πR

1⇠� z

d�+(⇠), (25)

where the measure �+ is symmetric, i.e., d�+(⇠)⇤ d�+(�⇠). Note that the measure�+ can be uniquely determined by the Herglotz function h from the Stieltjesinversion formula [47] as

�+ ((x1 , x2))+12 �+ ({x1})+

12 �+ ({x2}) ⇤ lim

y!0+

1⇡

π x2

x1

Im{h(⇠+ iy)}d⇠, (26)

including a possibility of having point masses xi on the real axis, i.e., xi 2 R. Forabsolutely continuous measures on R, the following notation can be introduced:

d�+(⇠) ⇤ �0+(⇠)d⇠ ⇤1⇡

Im{h(⇠+ i0)}d⇠, (27)

implying the relation between the imaginary part of the Herglotz function andits corresponding measure Im{h} ⇤ ⇡�0+; see [30, 47].

Assume that a symmetric Herglotz function admits the following small- andlarge-argument asymptotic expansions:

h(z) ⇤(

a�1z�1 + a1z + · · ·+ a2N0�1z2N0�1 + o(z2N0�1) as z!0,

b1z + b�1z�1 + · · ·+ b1�2N1 z1�2N1 + o(z1�2N1 ) as z!1,(28)

respectively, for all real-valued expansion coefficients, where ! denotes thenon-tangential limit such that z!0 and z!1 denote |z | ! 0 and |z | !1 in theStoltz cone (22), respectively. Here, a�1 0, b1 � 0 and coincides with b+ in theintegral representation (25), N0 and N1 are non-negative such that 1�N1 N0.

By employing the integral representation (25), the following integral identities,known as sum rules, can be derived:

2⇡

π 1

0+

Im{h(x)}x2k

dx def⇤ lim

"!0+lim

y!0+

2⇡

π 1/"

"

Im{h(x + iy)}x2k

dx ⇤ a2k�1 � b2k�1 (29)

for k ⇤ 1�N1 , . . . ,N0; see e.g., [1, 13].

�. INTEGRAL REPRESENTATION OF PASSIVE AND CAUSAL SYSTEMS 15

3.2 Quasi-Herglotz Functions

Here, a class of analytic functions on the upper half of complex plane useful formodeling a subclass of non-passive systems is introduced.

Definition 3.4 An analytic function q : C+ ! C is called a quasi-Herglotz functionif there exist two Herglotz functions h1 and h2 such that

q(z) ⇤ h1(z)� h2(z) (30)

for any z 2 C+. The set of all quasi-Herglotz functions is denoted by Q. ⇤

Any quasi-Herglotz function q 2Q admits the following integral representation

q(z) ⇤ a + bz +πR

1+ ⇠z⇠� z

d�(⇠) (31)

for z 2 C+, which is inherited from the properties of Herglotz functions. Here, theparameters a , b 2 R are defined as a ⇤ a+,1 � a+,2 and b ⇤ b+,1 � b+,2, respectively,and � ⇤ �+,1��+,2 is the signed measure, where for j ⇤ 1,2, the triple of parameters(a+, j , b+, j ,�+, j) represents the corresponding Herglotz function hj in the sense ofrepresentation (20).

z

Re

Im

Im{z} > 0

q(z)

Re

Im

Im{q(z)} 2 R

q

Figure 5: Illustration of a quasi-Herglotz function q mapping the upper half ofcomplex plane C+ to the entire complex plane C.

Symmetric quasi-Herglotz functions can be defined in a similar way as follows.

Definition 3.5 An analytic function q :C+ !C is called a symmetric quasi-Herglotzfunction if there exist two symmetric Herglotz functions h1 and h2 satisfying (23) suchthat equality (30) holds for any z 2 C+. The set of all symmetric quasi-Herglotz functionsis denoted by Qsym. ⇤


Any symmetric quasi-Herglotz function q 2 Qsym admits the following integralrepresentation:

q(z) ⇤ bz +p.v.πR

1+ ⇠2

⇠� zd�(⇠) (32)

for z 2 C+, and where the signed measure � is symmetric. In representations(31) and (32), the parameter b can be defined as b ⇤ limz!1 q(z)/z, where z!1denotes that |z | !1 in the Stolz cone (22) for any ' 2 (0,�/2].

Further, it has to be noted that any Herglotz or symmetric Herglotz functioncan be considered as a quasi-Herglotz or symmetric quasi-Herglotz function,respectively, assuming that h2(z) ⇤ 0 in equality (30). There is also an issue ofnon-uniqueness in Definition 3.4: for any Herglotz (symmetric Herglotz) functionsh1 and h2 in (30), the corresponding quasi-Herglotz (symmetric quasi-Herglotz,respectively) function can also be defined

q(z) ⇤ (h1 + h3)(z)� (h2 + h3)(z) (33)

in terms of other Herglotz (symmetric Herglotz, respectively) function h3.The representation of quasi-Herglotz functions (31) can be also written in

terms of the measure � as

q(z) ⇤ a + bz +πR

✓1

⇠� z� ⇠

1+ ⇠2

◆d�(⇠), (34)

for example, when the measure � from representation (31) has compact support.Then the measure � may be defined as d�(⇠)⇤ (1+⇠2)d�(⇠). Similarly, symmetricquasi-Herglotz functions may be given in terms of the symmetric measure � as

q(z) ⇤ bz +p.v.πR

1⇠� z

d�(⇠), (35)

such that d�(⇠) ⇤ d�(�⇠).Suppose that a symmetric quasi-Herglotz function admits the following

asymptotic expansions:

q(z) ⇤(

a�1z�1 + a1z + · · ·+ a2N0�1z2N0�1 + o(z2N0�1) as z!0,

b1z + b�1z�1 + · · ·+ b1�2N1 z1�2N1 + o(z1�2N1 ) as z!1,(36)

for all real-valued expansion coefficients at z ⇤ 0 and z ⇤1, respectively, andthat at least one of the symmetric Herglotz functions h1 and h2 admits thecorresponding asymptotic expansion there, respectively. Here, b1 coincides withb given in the integral representation (35), N0 and N1 are non-negative such that1�N1 N0, and similarly as for Herglotz functions, ! means the non-tangentiallimit. Then, it can be shown that the following sum-rule identities hold:

2⇡

π 1

0+

Im{q(x)}x2k

dx def⇤ lim

"!0+lim

y!0+

2⇡

π 1/"

"

Im{q(x + iy)}x2k

dx ⇤ a2k�1 � b2k�1 (37)

for k ⇤ 1�N1 , . . . ,N0; see e.g., [44, Sec. 3].

�. OPTIMIZATION TECHNIQUES 17

4 Optimization TechniquesIn this section, we describe the optimization techniques used Papers I–V andVII to determine physical bounds and optimal realizations of given propertiesof passive and non-passive systems. In Section 4.1, we give a summary onconvex optimization and on the framework, where the optimization procedure isinvolved. Section 4.2 is focused on analytical optimization techniques, which hasbeen used for determination of physical bounds described in Paper VII.

4.1 Convex Optimization

Convex optimization has been employed as a part of mathematical frameworkdeveloped in Papers I and III for modeling and determination of optimal perfor-mance bounds for passive and non-passive systems, respectively. In this section,we introduce the central concepts in convex optimization, as well as the propertiesused in formulation of convex optimization problems. For complete informationon convex optimization and related algorithms, see [17].

Definition 4.1 A set S ⇢ Rn is convex if

x1 ,x2 2 S ) (1� t)x1 + tx2 2 S, (38)

for all 0 < t < 1 [17, p. 23]. ⇤

Geometrically this means that the straight line between any two points in a convexset remains in the set, see Figure 6 for an illustration of the two-dimensional case.

a) A convex set b) A non-convex set

x1

x2

x1

x2

Figure 6: Illustration of sets in R2: a) a convex set; b) a non-convex set.

Theorem 4.2 The intersection \↵S↵ of convex sets S↵ is a convex set [17, p. 36].

Definition 4.3 Let f be a function defined on the convex set S ⇢ Rn . The function f issaid to be a convex function if

x1 , x2 2 S ) f ((1� t)x1 + tx2) (1� t) f (x1)+ t f (x2), (39)

for all 0 < t < 1 [17, p. 67]. ⇤


x

f(x)

a) A convex function b) A non-convex function

x1 x1

f(x1)

x2 x2

f(x2)

x

g(x)

g(x1)

g(x2)

Figure 7: Illustration of functions in R: a) a convex function; b) a non-convexfunction.

Geometrically this means that a convex function f is always less then or equalto a corresponding linear (or affine) function passing through the values f (x1)and f (x2) where x1 and x2 are any two points in the convex set S, see Figure 7for an illustration of the one-dimensional case.

Definition 4.4 If the function f satisfies a strict inequality in (39), it is said to be astrictly convex function [17, p. 67]. ⇤

Definition 4.5 A function f is said to be concave if � f is convex [17, p. 67]. ⇤

Theorem 4.6 Let f1 and f2 be convex functions defined on a convex set S. Then thepositive linear combination

f ⇤ ↵1 f1 +↵2 f2 (40)

(with ↵1 > 0 and ↵2 > 0) is a convex function. If any one of f1 and f2 is strictly convex,then the function f is strictly convex [17, p. 79].

Theorem 4.7 Let f be a two times differentiable continuous function defined on theopen convex set S ⇢ Rn ( f 2 C2(S)). It can then be shown that f is a convex function ifand only if the Hessian Hi j(x) ⇤ rxirxj f (x) is a positively semidefinite matrix for allx 2 S, i.e.,

H(x) � 0 8x 2 S , f convex.

If the Hessian H(x) is positively definite for all x 2 S, then f is strictly convex, i.e.,

H(x) > 0 8x 2 S ) f strictly convex;

see e.g., [17, p. 71].

The converse for positive definite case is not true: take e.g., f (x)⇤ x4, see e.g., [17].


Theorem 4.8 Let f (x) ⇤ 12 xTAx+bTx+ c be a quadratic form where A is an n ⇥ n

matrix, x and b are n⇥1 vectors, c a scalar and ( · )T denotes the transpose. The functionf is convex if and only if the Hessian matrix A is positively semidefinite, i.e.,

A � 0 , f convex. (41)

The quadratic form f is strictly convex if and only if the Hessian matrix A is positivelydefinite, i.e.,

A > 0 , f strictly convex; (42)

see [17, p. 71].

Theorem 4.9 Let f be a convex function defined on the convex set S ⇢ Rn . Then f is acontinuous function on the interior of S [17, p. 68].

Definition 4.10 A convex optimization problem, in general, is a problem of the form

minimize f (x)subject to x 2 S,

(43)

where f is a convex function defined on the convex set S. It is common that the convexset S is given in the form S ⇤ {x 2 Rn |gi(x) 0} where {gi(x)}M

i⇤1 is a set of convexfunctions representing the convex constraints [17, p. 127]. ⇤

Definition 4.11 A point x⇤ 2⌦ is said to be a local minimum point of f over⌦ if thereis an " > 0 such that f (x) � f (x⇤) for all x 2⌦ within a distance |x�x⇤ | < " [57, p. 184].⇤

Definition 4.12 A point x 2⌦ is said to be a global minimum point of f over ⌦ iff (x) � f (x⇤) for all x 2⌦ [57, p. 184]. ⇤

One of the most important properties of a convex optimization problem is thefollowing.

Theorem 4.13 Any local minimum in convex set S is also a global minimum1 [57,p. 197].

Theorem 4.14 Any norm f (x) ⇤ kxk is a convex function on Rn since by the triangleinequality we have

f ((1� t)x1 + tx2) ⇤ k(1� t)x1 + tx2k (1� t)kx1k+ tkx2k⇤ (1� t) f (x1)+ t f (x2), (44)

where 0 < t < 1 [17, pp. 72–73].1Here the term “minimum” refer to the minimizing point x and the term “minimum value” to the

corresponding minimum value f (x).


Theorem 4.15 The norm of a linear (or affine) form is a convex function on Rn [17,pp. 72–73].

Example 4.16 Consider the function f (x) ⇤ kAx�bk, where A is an M⇥N matrix, xan N ⇥1 vector and b an M⇥1 vector. The function f (x) is convex since by the triangleinequality

f ((1� t)x1 + tx2) ⇤ kA((1� t)x1 + tx2)�bk⇤ k(1� t)(Ax1 �b)+ t(Ax2 �b)k (1� t)kAx1 �bk+ tkAx2 �bk

⇤ (1� t) f (x1)+ t f (x2);

see also Definitions 4.1 and 4.3.

The following example is related to the quasi-Herglotz function theory intro-duced in Section 3.2. In this example, the associated approximation theory to bediscussed in Section 5 is used in the applications described in Papers III–IV.

Example 4.17 Consider the following convex optimization problem:

minimize kq �FkLp (w ,⌦)subject to cn 0,

b1 � 0,(45)

where k · kLp (w ,⌦), 1 p 1, denotes a suitable weighted Lebesgue norm [51]on the approximation domain ⌦ ⇢ R, where w is a positive continuous weightfunction on ⌦, q the approximating symmetric quasi-Herglotz function, whichhas a representation on ⌦ to be discussed in Section 5, and F the symmetrictarget function. The convex optimization formulation (45) can be represented inmatrix form as

minimize kb1x+Hc+ iPc� fk`p (w ,⌦)subject to c 0,

b1 � 0,(46)

where k · k`p (w ,⌦) denotes the norm of a weighted sequence space [51]. Here,w and ⌦ are sampled versions of the weight function and the approximationdomain, respectively, corresponding to (45), c is the N ⇥1 vector of optimizationvariables cn for n ⇤ 1, . . . ,N , and f the corresponding column vector representingthe target function F. The imaginary and real parts of q can be expressed ina semi-infinite matrix notation as Im{q} ⇤ Pc and Re{q} ⇤ b1x+Hc, where xis a column vector corresponding to the approximation domain, xi 2 ⌦, andthe columns of the matrices P and H are similarly given by the B-spline basisfunctions pn(x) and their corresponding Hilbert transform pn(x), respectively.The corresponding numerical problem (45) can be solved efficiently by using theCVX MATLAB software for disciplined convex programming [32].


4.2 Method of Lagrange MultipliersIn this section, we introduce some general definitions and conditions necessaryfor solving problems with the described method. A more detailed descriptioncan be found in [17, 57].

The method of Lagrange multipliers is an optimization technique used todetermine extreme points of function f subject to the constrained vector-valuedfunctions g and h and is applied to the problems of the form:

minimize f (x)subject to gi(x) 0,

hj(x) ⇤ 0,(47)

where x 2 Rn ,�

gi(x) m

i⇤1 and�

hj(x) p

j⇤1 are sets of ineaquality and equalityconstraint functions representing the convex and active constraints, respectively,and m , p n. Note that if functions gi(x), i ⇤ 1, . . . ,m, are convex, and functionshj(x), i ⇤ 1, . . . , p, are affine, then (47) is a convex optimization problem [17,Eq. (4.15)].

Definition 4.18 Let x⇤ be the point satisfying the constraints

g(x⇤) 0, h(x⇤) ⇤ 0, (48)

and let I be the set of indices i for which gi(x⇤) ⇤ 0. Then x⇤ is said to be a regular pointof the constraints (48) if the gradient vectors rgi(x), i 2 I, and rhj(x), j ⇤ 1, . . . , p, arelinearly independent [57, p. 342]. ⇤

Karush-Kuhn-Tucker conditions [57, p. 342]: Let { f , g, h} 2 C1, where g and hare the m- and p-dimensional (m , p n) vector-valued functions, respectively, and let x⇤be a local minimum point for the problem

minimize f (x)subject to g(x) 0,

h(x) ⇤ 0,x 2⌦ ⇢ Rn ,

(49)

where ⌦ denotes the constrained set, and suppose that x⇤ is a regular point for theconstraints. Then, there is a vector �� 2 Rm with �� 0 and vector ⌫⌫⌫ 2 Rp such that

r f (x⇤)+m’

i⇤1�irgi(x⇤)+

p’j⇤1⌫ jrhj(x⇤) ⇤ 0, (50)

m’i⇤1�i gi(x⇤) ⇤ 0. (51)

⇤


The corresponding Lagrangian associated with the problem (49) is

L(x,��, ⌫⌫⌫) ⇤ f (x)+m’

i⇤1�i gi(x)+

p’j⇤1⌫ j h j(x), (52)

where �� and ⌫⌫⌫ are the Lagrange multiplier vectors associated with the inequalityand equality constraints, respectively, and for which the Karush-Kuhn-Tuckerconditions (48), (50) and (51) for �� 0 are necessary. Note that if the problem(49) is convex, then the Karush-Kuhn-Tucker conditions for �� 0 are not onlynecessary, but also sufficient [17, p. 244].

5 Approximation of Passive and Non-passiveSystems

In this section, we describe the mathematical framework developed for approx-imation and identification of passive and a subclass of non-passive systemswhich can be characterized, in general, by the class of quasi-Herglotz functionsdescribed in Section 3.2. The framework is based on the integral representationof quasi-Herglotz functions (31) used for approximation of causal and passiveone-port systems in convolution form; see Definition 2.4 in Section 2. Notethat the theory and formulation given in this section are applicable to passivesystems given in terms of Herglotz functions, because a Herglotz function can beconsidered as a quasi-Herglotz function as in (30), with h2 ⇤ 0; see Section 3.2 fordetails.

To facilitate the formulation of a convex optimization problem, it is necessaryto first put a priori constraints on the class of quasi-Herglotz functions to be usedas approximants, as well as on the target function characterizing the system ofinterest. In particular, the interest is in approximating functions having continuousreal and imaginary parts. Thus, quasi-Herglotz functions are restricted to have alocal Hölder continuous extension to some given intervals on the real axis. Inaddition, the target response of the system of interest F 2 C(⌦) can be given byan arbitrary complex-valued continuous function over the approximation domain⌦ ⇢ R consisting of a finite union of closed and bounded intervals on the realaxis. The norms used in approximation problems are weighted Lp-norms [77],where 1 p 1, and are denoted by k · kLp (w ,⌦), where the weight w is a positivecontinuous function on ⌦.

Let O denote some open neighborhood of the approximation domain ⌦ suchthat ⌦ ⇢ O ⇢ R, as illustrated in Figure 8. Further, let q denote the approximatingfunction which is Hölder continuous on ⌦ and which coincides with a quasi-Herglotz function having a Hölder continuous extension to the closure of O. Thus,the approximating quasi-Herglotz function can be generated by an absolutelycontinuous measure � having a Hölder continuous density �0 on the closure

�. APPROXIMATION OF PASSIVE AND NON-PASSIVE SYSTEMS 23

O �⌦. This implies that the function �0 belongs to the Hölder space C0,↵(O) withHölder exponent ↵, meaning that

��0(&1)� �0(&2)�� C |&1 � &2 |↵ for all &1 , &2 2 O

and fixed 0 < ↵ < 1, and where C > 0 is an arbitrary constant; see e.g., [50, pp. 94-104]. The Hölder conitnuity requirement is necessary in order to show thatthe Hilbert transform operator H is a bounded operator H : C0,↵(O)! C0,↵(⌦);cf., [50, Thm. 7.6 and Cor. 7.7] and [48]. Since the imaginary part of theapproximating quasi-Herglotz function is related to the measure � as Im{q} ⇤ ⇡�0on O (cf., [47, p. 7]) and, thus the Hölder continuity of the density �0 on O impliesthat the real part of q given by the associated Hilbert transform [50] is continuous.

⌦

OR

x

Figure 8: Illustration of the approximation domain ⌦ and the closure of someopen neighborhood O of the approximation domain, such that ⌦ ⇢ O ⇢ R forx 2⌦.

Let us formulate an approximation problem of interest. The greatest lowerbound on the approximation error can be defined by

d :⇤ infqkq �FkLp (w ,⌦) , (53)

where the infimum is taken over all quasi-Herglotz functions q generated by ameasure having a Hölder continuous density on O, and where where the normsare well-defined for 1 p 1 due to continuity of q.

A best approximation in problem (53) can be achieved by solving a finite-dimensional approximation problem. This approach is suitable to incorporatewith numerical optimization algorithms [32], where B-spline expansions [25] ofan arbitrary order can be utilized for discretization of approximating functions.

B-spline basis functions of a fixed polynomial order m � 2 are compactlysupported positive basis functions that are m�2 times continuously differentiable[24,25], and have m+1 break points and a continuous (negative) Hilbert transform[43]. Here, the B-spline basis functions are defined as pn(x), where n ⇤ 1, . . . ,N isthe number of spline functions used for a finite-dimensional discretization of thefunction �0, and pn(x) denotes their corresponding (negative) Hilbert transform.The explicit formulas on B-splines functions and their Hilbert transforms aregiven in [43].


Let the approximating function q can be represented on the closure O as:

q(x) ⇤ a + bx +

M’i⇤1

pi

⇠i � x+p.v.

πR

✓1

⇠� x� ⇠

1+ ⇠2

◆�0(⇠)d⇠+ i⇡�0(x) (54)

⇤ a + bx +

M’i⇤1

pi

⇠i � x+p.v.

πR

1⇠� x

�0(⇠)d⇠+ i⇡�0(x) (55)

for x 2 ⌦, and where a ⇤ a �p.v.ØR

⇠1+⇠2 �0(⇠)d⇠. In (54) and (55) the density �0

is assumed to be given by a finite uniform B-spline expansion of a fixed order,and a finite number of point masses at ⇠i <⌦ with real-valued amplitudes pi ,i ⇤ 1, . . . ,M, have also been included.

A discretization problem of (53) on the finite partition of ⌦ can be formulatedas follows. Let qN denote approximating functions represented as in (55), andhence

Im{qN (x)} ⇤ ⇡�0(x) ⇤N’

n⇤1cn pn(x), (56)

and

Re{qN (x)} ⇤ a + bx +

M’i⇤1

pi

⇠i � x+

N’n⇤1

cn pn(x), (57)

for x 2 ⌦, and where cn are the corresponding B-spline expansion coefficients.Note that all the parameters a, b, {pi}M

i⇤1 and {cn}Nn⇤1, as well as the break-points

of the B-splines defined above depend on N .Consider the following optimization problem

minimize kq �FkLp (w ,⌦)

subject to blower(x) �0(x) bupper(x),(58)

where the upper and lower bounds on density �0 are included to regularize thephysical properties of a given system inside and outside of the approximationdomain ⌦. Further, these constraints are useful for prevention of non-physicaloscillatory behavior and for regularization of small- and large-argument propertiesof a given system. In practice, we solve the following discretized problem

minimize kqN �FkLp (w ,⌦)

subject to ✓lower, j ✓j ✓upper, j , j 2 J,(59)

for fixed N and finite index set J, and where the minimization is over the vectors✓j , each of them consisting of the parameters ✓j 2 {a , b , p1 , . . . , pM , c1 , . . . , cN },for j 2 J. Further, it should be noted that the sum-rule identities (29) and (37) canbe discretized and used as convex constraints for a given optimization problem.

Finally, numerical implementations of convex optimization problems given by(59) can be obtained by using, e.g., the CVX MATLAB software for disciplined

�. MATERIAL MODELS 25

convex programming [32], where the calculation of the norm above must beapproximated based on a finite set of sample points in ⌦. However, due to theuniform continuity of all functions involved, this can, in principle, be done withinarbitrary numerical accuracy.

6 Material ModelsIn this section, we describe the dielectric permittivity models used in PapersI–VII included in this thesis. The corresponding Herglotz and quasi-Herglotzfunctions are constructed and their asymptotic expansions are derived. Note thatthese functions do have an analytic extension to some open neighborhood of thereal axis; however, this is not true for all physical systems in general. Here, forconvenient representation, the complex-valued frequency ! (in rad/s) is used asan argument of Herglotz and quasi-Herglotz functions, respectively. Furthermore,the time convention e�i!t for time harmonic fields (phasors) is used, as stated inSection 1.2.

6.1 Debye ModelThe Debye model is used in modeling of dielectric responses of dispersivematerials and has the following representation in the frequency domain:

✏(!) ⇤ ✏1+✏s � ✏11� i!⌧ , (60)

where ✏1, ✏s, and ⌧ > 0 denote the instantaneous response (at ! ⇤1), the staticresponse (at ! ⇤ 0), and the relaxation time, respectively. The correspondingresponse of the Debye model in the time domain is given by

✏r(t) ⇤ ✏1�(t)+✏s � ✏1⌧

e�t/⌧H(t), (61)

where �(t) is the Dirac delta function, and H(t) the Heaviside unit step function.Using (60), the corresponding Herglotz function h(!)⇤ !✏(!) can be constructed,which has the following small- and large-argument asymptotics:

h(!) ⇤ !✏1+!✏s � ✏11� i!⌧ ⇤

(!✏s + o(!), !!0,

!✏1+ o(!), !!1.(62)

The function (62) can be extended to the meromorphic function in C\{�i/⌧} witha pole at ! ⇤ �i/⌧. From (62), the spectral measure Im{h(⇠)} ⇤ ⇡�0+(⇠) is givenby

Im{h(⇠)} ⇤ ⇠2⌧(✏s � ✏1)1+ ⇠2⌧2 (63)

for ⇠ ⇤ Re{!} 2 R. Note that for ✏s ⇤ ✏1, the spectral measure Im{h(⇠)} ⇤ 0 overall ⇠ 2 R, and hence the material is lossless.


For this model, there exists a sum rule given as

2⇡

π 1

0+

Im{h(⇠)}⇠2 d⇠ ⇤

2⇡

π 1

0+

Im{✏(⇠)}⇠

d⇠ ⇤ ✏s � ✏1 (64)

for k ⇤ 1, and where a1 ⇤ ✏s, and b1 ⇤ ✏1, respectively. Note that the sum rule(64) is valid for all the passive materials having the same asymptotic expansionsas in (62).

6.2 Conductivity ModelThe conductivity model is used for materials with an electric conductivitysatisfying the Ohm’s law

JJJ ⇤ �EEE, (65)

and is given in the frequency domain as

✏(!) ⇤ ✏1+ i �!✏0, (66)

where ✏1, ✏0 ⇡ 8.854 · 10�12 F/m, and � > 0 denote the instantaneous response,the permittivity of free space, and the static conductivity, respectively. Thecorresponding time-domain representation of this model is

✏r(t) ⇤ ✏1�(t)+�✏0

H(t), (67)

where H(t) is the Heaviside unit step function. Note that the Fourier transformof (67) is a distribution [100]

✏(⇠) ⇤ ✏1+�✏0

i 1⇠+⇡�(⇠)

�, as Im{!}! 0+, (68)

and ⇠ ⇤ Re{!} 2 R. The corresponding Herglotz function is constructed ash(!) ⇤ !✏(!) and has the following small- and large-argument asymptotics:

h(!) ⇤ !✏1+ i �✏0

⇤

(o(!�1), !!0,

!✏1+ o(!), !!1.(69)

The Herglotz function (69) can be extended to an entire function, i.e., the functionis analytic in the entire complex plane. For this function, the spectral measureIm{h(⇠)} ⇤ ⇡�0+(⇠) is given by

Im{h(⇠)} ⇤ �✏0. (70)

Note that for such a Herglotz function (69), no sum rule can be constructed.In practice, the conductivity model can be combined with the Debye model

described in Section 6.1 to characterize materials with conductive characteristics


such as saline water, metals, etc. The resulting permittivity function for this caseis known as the modified Debye model, which is given by

✏(!) ⇤ ✏1+✏s � ✏11� i!⌧ + i �

!✏0. (71)

The resulting Herglotz function h(!) ⇤ !✏(!) has the same asymptotic propertiesas in (62).

6.3 Havriliak–Negami ModelThe Havriliak–Negami model is an empirical relaxation model that is used tocharacterize conductive materials with non-Debye dispersive behavior. Thefrequency-domain representation of this model is given by

✏(!) ⇤ ✏1+✏s � ✏1

(1+ (�i!⌧)↵)� + i �✏0

(72)

for 0 < ↵ 1 and 0 < � 1, and where ✏s and ✏1 denote the static and the instanta-neous responses, respectively, ⌧ > 0 the relaxation time, ✏0 ⇡ 8.854 ·10�12 F/m thepermittivity of free space, and � > 0 the static conductivity. Using De Moivre’sformula [3], the real and imaginary parts of the Havriliak–Negami model can bedetermined as:

Re{✏(!)} ⇤ ✏1+(✏s � ✏1)cos(��)⇥

1+2(!⌧)↵ sin� ⇡

2 (1+↵)�+ (!⌧)2↵

⇤�/2 , (73)

andIm{✏(!)} ⇤ (✏s � ✏1)sin(��)⇥

1+2(!⌧)↵ sin� ⇡

2 (1+↵)�+ (!⌧)2↵

⇤�/2 +�!✏0, (74)

respectively, where

� ⇤ arctan

((!⌧)↵ cos

� ⇡2 (1+↵)

�1+ (!⌧)↵ sin

� ⇡2 (1+↵)

�)

(75)

for ! 2 C+. The corresponding Herglotz function can be constructed as:

h(!) ⇤ !✏(!) ⇤ !✏1+!(✏s � ✏1)

(1+ (�i!⌧)↵)� + i �!✏0

(76)

for ! 2 C+, and which can be used e.g., in the approximation problem withhigh-order B-splines described in Paper II.

6.4 Lorentz’ ModelThe Lorentz’ model is used to model the response of a plasma and is given by

✏(!) ⇤ ✏1�!2

p

!2 + i!⌫�!20, (77)


where ✏1 is the optical repsonse, !p > 0 the plasma frequency, !0 > 0 theresonance frequency, and ⌫ > 0 the collision frequency. The correspondingtime-domain response is given by

✏r(t) ⇤ ✏1�(t)+!2

p

⌫0e�⌫t/2 sin(⌫0t)H(t), (78)

where ⌫0 ⇤q!2

0 � ⌫2/4 and !0 � 2. The corresponding Herglotz function h(!) ⇤!✏(!) can be constructed and has the following asymptotic expansions:

h(!) ⇤ !✏1�!!2

p

!2 + i!⌫�!20⇤

(!✏s + o(!), !!0,

!✏1�!�1!2p + o(!�1), !!1,

(79)

where the static permittivity ✏s ⇤ ✏1+!2p/!2

0 and ! , 0. The Herglotz function(79) can be extended to a function meromorphic in C\{!1 ,!2} with two poles at!1,2 ⇤ �i⌫/2±

q⌫2/4+!2

0. The spectral measure Im{h(⇠)} ⇤ ⇡�0+(⇠) is given by

Im{h(⇠)} ⇤⇠2!2

p⌫

(⇠2 �!20)2 + ⇠2⌫2

(80)

for ⇠ ⇤ Re{!} 2 R. For !0 , 0, there exists a sum rule for k ⇤ 1, the same as in(64). However, for !0 ⇤ 0, there is another sum rule derived for k ⇤ 0, which isgiven by:

2⇡

π 1

0+Im{h(⇠)}d⇠ ⇤ !2

p , (81)

which is also valid for the Drude model; see Section 6.5.The Lorentz’ model is also useful in modeling of linear non-passive gain

media, i.e., the media having Im{✏} < 0 over some frequency intervals. Note that therepresentation of Lorentz’ model for gain media is different in comparison withthe model for passive media (77), and is given by

✏(!) ⇤ ✏1+!2

p

!2 + i!⌫�!20. (82)

The corresponding representation of this model in the time domain can beobtained as:

✏r(t) ⇤ ✏1�(t)�!2

p

⌫0e�⌫t/2 sin(⌫0t)H(t), (83)

where ⌫0 ⇤q!2

0 � ⌫2/4. The corresponding quasi-Herglotz function q(!) ⇤ !✏(!)can be constructed and has the following asymptotic expansions:

q(!) ⇤ !✏1+!!2

p

!2 + i!⌫�!20⇤

(!✏s + o(!), !!0,

!✏1+!�1!2p + o(!�1), !!1,

(84)


where the static permittivity ✏s ⇤ ✏1 �!2p/!2

0 and ! , 0. Consequently, thequasi-Herglotz function based on the negative Lorentz’ model (82) admits thesum rules for k ⇤ 1 as in (64) for passive media characterized by the Debye modelwhen !0 , 0. However for k ⇤ 0, the sum rule based on the quasi-Herglotzfunction (84) is

2⇡

π 1

0+Im{q(⇠)}d⇠ ⇤ �!2

p , (85)

which is valid when !0 ⇤ 0.In practical examples, such as modeling of laser dyes (e.g., Rhodamine 6G

and Rhodamine R800 laser dye molecules [19]) considered as a four-level atomicsystem, the Lorentz model for linear gain media is given by:

✏(!) ⇤ ✏1+1✏0

�a

!2 + i!�!a �!2a

(⌧21 � ⌧10)�pump

1+ (⌧32 + ⌧21 + ⌧10)�pumpN0 , (86)

where �a is the coupling strength of the polarization density at the emissionfrequency band to the electric field, �!a the bandwidth of the dye transitionat the emitting angular frequency !a, ⌧i j the lifetime for transition from state ito state j, �pump the pumping rate from level 0 to level 3, and N0 the total dyeconcentration [19].

6.5 Drude ModelThe Drude model is used to model the condition of charges in metals and is aspecial case of the Lorentz’ model with !0 ⇤ 0:

✏(!) ⇤ ✏1�!2

p

!(!+ i⌫) ⇤ ✏1+ i �0!✏0

11� i!/⌫ , (87)

where ✏1 denotes the instantaneous response, !p > 0 the plasma frequency, ⌫ > 0the collision frequency, ✏0 ⇡ 8.854 ·10�12 F/m the permittivity of free space, and�0 ⇤ !2

p✏0/⌫ the static conductivity. The corresponding representation in the timedomain is given as:

✏r(t) ⇤ ✏1�(t)+�0✏0

�1�e�⌫t � H(t). (88)

The corresponding Herglotz function h(!) ⇤ !✏(!) can be constructed, whichhas the following asymptotic expansions:

h(!) ⇤ !✏1+ i �✏0

11� i!⌧ ⇤

(i�0/✏0 + o(1) ⇤ o(!�1), !!0,

!✏1�!�1!2p + o(!�1), !!1,

(89)

where !2p ⇤ �0⌫/✏0. The Herglotz function (89) can be extended to a function

meromorphic in C\{�i⌫} with a single pole at ! ⇤ �i⌫. The spectral measureIm{h(⇠)} ⇤ ⇡�0+(⇠) is given by

Im{h(⇠)} ⇤ �0✏0

11+ ⇠2/⌫2 (90)


for ⇠ ⇤ Re{!} 2 R. There exists a sum rule for k ⇤ 0 given as

2⇡

π 1

0+Im{h(⇠)}d⇠ ⇤

2⇡

π 1

0+⇠ Im{✏(⇠)}d⇠ ⇤ !2

p. (91)

It can be concluded that a priori knowledge of the static conductivity as well asthe mean value (the first moment) of Im{✏(⇠)} gives us a possibility to determinethe modified Debye model via (91); the collision frequency can be determined as⌫ ⇤ !2

p✏0/�0. Note that the sum rule for k ⇤ 1 as in (64) does not exist.The Drude model is related to the modified Debye model (71) described

in Section 6.2. To show this relation, first we have apply the partial fractiondecomposition to (87), and thus, the representation of the Drude model becomes

✏(!) ⇤ ✏1�!2

p/⌫2

1� i!/⌫ + i!2

p/⌫!. (92)

Now, the Drude model can be considered as a partial case of the modified Debyemodel, where 8>>>><

>>>>:

✏s ⇤ ✏1�!2p/⌫2 ,

⌧ ⇤ 1/⌫,

� ⇤ ✏0!2p/⌫.

(93)

Note that the parameters of the Debye model can be given in terms of theparameters of the Drude model as in (93) if the condition

✏s � ✏1+�⌧✏0

⇤ 0 (94)

is satisfied [52]. Further note that the standard conductivity model

✏(!) ⇤ ✏(!)+ i �!✏0

(95)

is a special case of the Drude and the modified Debye models, respectively, where✏(!) is regular at the origin [52].

One of the practical applications of the Drude model is within characterizationof gold nanoparticle suspensions, where the electric current is governed by anelectrophoretic mechanism. A realistic electrophoretic Drude model depends onthe physical and chemical properties of nanoparticle’s components. The mass ofone gold nanoparticle can be obtained as m ⇤ ⇢Au

⇣4⇡a3

Au/3⌘+⇢L

⇣4⇡(a3 � a3

Au)/3⌘,

where ⇢Au and ⇢L are the mass densities of gold and ligands, respectively, a andaAu are the radii of one nanoparticle and its core, respectively. Additionally to thediscussed parameters, the net charge of a gold nanoparticle, the friction constant,and the number of particles per unit volume have to be taken into account. Thenet charge of the gold nanoparticle depends on the radius of core aAu, as well ason the electron count nL, and can be determined as q ⇤

⇣3aAu +0.5a2

Au

⌘e0 + nLe0,


where e0 ⇤ 1.6 ·10�19 C denotes the electron charge [68, 79]. The friction constant� ⇤ 6⇡µfa is obtained from Stoke’s law, where µf is the dynamic shear viscosityof the host medium [68, 79]. The number of particles per unit volume can beobtained as N ⇤ �/(4⇡a3/3), where � is the volume fraction of gold nanoparticlesin the spherical suspension [68, 79]. Finally, the corresponding static conductivityand the relaxation time of electrophoretic Drude model can be determined as�0 ⇤Nq2/� and ⌧ ⇤ m/�, respectively.

6.6 Brendel–Bormann ModelThe Brendel–Bormann model is used for modeling dielectric properties of ma-terials, especially metals, in the optical frequency range, and has the followingrepresentation in the frequency domain as:

✏(!) ⇤ ✏1�⌦2

p

!(!� i�0)+

k’j⇤1� j(!), (96)

where ✏1 is the instantaneous response, ⌦p ⇤p

f0!p the plasma frequencyassociated with interband transitions with oscillator strength f0 and dampingconstant �0 [75]. Here, k is the number of Gaussian-line-shape oscillators � j(!)defined as:

� j(!) ⇤1p

2⇡� j

π 1

�1exp

�(x �! j)2

2�2j

!f j!2

p

(x2 �!2)+ i!� jdx , (97)

where � j is the Gaussian broadening parameter. The analytical solution of (97) isgiven by

� j(!) ⇤ip⇡ f j!2

p

2p

2a j� j

"w

a j �! jp

2� j

!+w

a j +! jp

2� j

!#(98)

for a j ⇤

q!2 � i!� j and Im{a j} > 0; see [75] for details. Here, w(z) denotes the

error function integral

w(z) ⇤ e�z2✓1+ i2

⇡

π z

0et2 dt

◆⇤ e�z2 erfc (iz) (99)

for Im{z} > 0.In [75], an alternative solution of (97) based on confluent hypergeometric

functions, the Kummer functions of the second kind U(a , b , z), is proposed. As aresult, the Gaussian oscillators � j(!) can be represented as:

� j(!) ⇤i f j!2

p

2p

2a j� j

8>><>>:

U26666412 ,

12 ,�

a j �! jp

2� j

!2377775+U

26666412 ,

12 ,�

a j +! jp

2� j

!23777759>>=>>;, (100)


where the relation between the Kummer and error functions

U✓12 ,

12 , z

2◆⇤p⇡ez2 erfc (z) (101)

has been employed [75].The Brendel–Bormann model has demonstrated more accurate results in

fitting the experimental data of materials in the optical frequency range incomparison with the Drude–Lorentz model, which is based on the Lorentz-line-shape oscillator [18]. In Paper IV, the Brendel–Bormann model of gold [75] isused to determine the optimal plasmonic singularity in terms of the non-passivegain surrounding medium. In Paper VII, the same model of gold is used tocharacterize the dielectric properties of homogeneous and core-shell spheres.However in 2018, it turned out that this model does not satisfy the Kramers-Kronig relations (8) and (9), and thus, it is impossible to achieve an impulseresponse in the time domain of a dielectric material characterized by this modelin the frequency domain; see [73]. To satisfy the symmetry requirement of theKramers-Kronig relations, a singularity cancellation procedure at the origin inthe Gaussian-line-shape oscillator is required, which is described in [73].

7 Optical Theorem in Lossy Media

In this section, we review optical theorems for scatterers immersed in surroundingabsorptive media described in Paper VII.

Consider a scatterer of a volume V made of a linear general bianisotropicmaterial and immersed in a linear isotropic passive background medium, asshown in Figure 9. The incident and scattered fields in the exterior region R3\Vare given by Maxwell’s equations

(r⇥EEE{i,s} ⇤ ik0⌘0µbHHH{i,s} ,

r⇥HHH{i,s} ⇤ �ik0⌘�10 ✏bEEE{i,s} ,

(102)

where EEE and HHH are electric and magnetic field intensities, the subscripts “i”and “s” correspond to the incident and scattered fields, respectively, µb withIm{µb} � 0 and ✏b with Im{✏b} � 0 denote the permeability and permittivity ofthe passive background medium, respectively, and the total fields are given byEEE ⇤ EEEi +EEEs and HHH ⇤ HHHi +HHHs. Note that the incident fields EEEi and HHHi are valid inthe whole R3.

The power balance at the external boundary surface @V surrounding thescatterer in a lossy medium can be obtained by employment of the Poynting’stheorem

Pa ⇤ �Ps +Pt +Pi , (103)

�. OPTICAL THEOREM IN LOSSY MEDIA 33

✏b, µb

✏,�em

�me,µ

V

Ei,Hi

n

-� 2a

Figure 9: Problem setup. Here, ✏b and µb denote the relative permittivity and per-meability of the background medium, respectively, ✏✏✏ and µµµ the permittivity andpermeability dyadics, respectively, ��em and ��me the dimensionless susceptibilitydyadics, and nnn the outward unit vector.

where Pa denotes the absorbed power, Ps the scattered power, Pt the total (extinct)power, and Pi the incident power absorbed by the surrounding medium. Here,the powers are given by

Pa ⇤ �12 Re

⇢π@V

EEE⇥HHH⇤ · nnn dS�, (104)

Ps ⇤12 Re

⇢π@V

EEEs ⇥HHH⇤s · nnn dS

�, (105)

Pt ⇤ �12 Re

⇢π@V

�EEEi ⇥HHH⇤

s +EEEs ⇥HHH⇤i�· nnn dS

�, (106)

and

Pi ⇤ �12 Re

⇢π@V

EEEi ⇥HHH⇤i · nnn dS

�, (107)

where the integrals are defined with the outward unit normal n, see also [15,Eq. (3.19)].

7.1 Physical bound on absorption based on interior-field formu-lation

Let the interior scattering region V be characterized by the following constitutiverelations for a general bianisotropic material:(

DDD ⇤ ✏0✏✏✏ ·EEE+ 1c0��em ·HHH ,

BBB ⇤ 1c0��me ·EEE+µ0µµµ ·HHH ,

(108)


where DDD is the electric flux density, BBB the magnetic flux density, ✏✏✏ ⇤ ✏bIII+��ee andµµµ ⇤ µbIII +��mm are the permittivity and permeability dyadics, where ��ee, ��em,��me, and ��mm are the dimensionless susceptibility dyadics. Then, the Maxwell’sequations for the interior region V

(r⇥EEE ⇤ ik0��me ·EEE+ ik0⌘0µµµ ·HHH ,

r⇥HHH ⇤ �ik0⌘�10 ✏✏✏ ·EEE� ik0��em ·HHH ,

(109)

can be reformulated in terms of the background medium by using the volumeequivalent principle as:

(r⇥EEE ⇤ ik0⌘0µbHHH � JJJm ,

r⇥HHH ⇤ �ik0⌘�10 ✏bEEE+ JJJe ,

(110)

where (JJJe ⇤ �ik0⌘�1

0 ��ee ·EEE� ik0��em ·HHH ,

JJJm ⇤ �ik0��me ·EEE� ik0⌘0��mm ·HHH .(111)

are the equivalent electric and magnetic contrast currents.By employing the boundary conditions at the surface @V

(nnn ⇥ (EEEi +EEEs) ⇤ nnn ⇥EEE,

nnn ⇥ (HHHi +HHHs) ⇤ nnn ⇥HHH ,(112)

together with the divergence theorem and the vector identity n · (XXX ⇥ YYY) ⇤n ⇥ (XXX ·YYY), it is possible to show that the absorbed, total, and incident pow-ers involved in the optical theorem for the lossy background (103) can be givenin the form based on the interior fields

Pa ⇤k0

2⌘0Im

⇢πV

FFF⇤ ·MMMa · FFF dv�, (113)

Pt ⇤k0

2⌘0Im

⇢πV

FFF⇤i ·MMMt · FFF dv

��2Pi , (114)

Pi ⇤k0

2⌘0Im

⇢πV

FFF⇤i ·MMMb · FFFi dv

�, (115)

where the field vectors are

FFF ⇤

✓EEE⌘0HHH

◆and FFFi ⇤

✓EEEi⌘0HHHi

◆. (116)

Here, the material dyadics are given by

MMMa ⇤

✓✏✏✏ ��em��me µµµ

◆⇤ ��+MMMb , (117)


where

�� ⇤

✓��ee ��em��me ��mm

◆, MMMb ⇤

✓✏bIII 000000 µbIII

◆, (118)

and

MMMt ⇤

✓✏✏✏� ✏⇤bIII ��em��me µµµ�µ⇤bIII

◆⇤ ��+ i2 Im{MMMb}. (119)

Note that the absorbed power Pa in (113) is given by a positive definite (strictlyconvex) quadratic form, and the total power Pt is given by an affine form in thefield quantities.

The optimization problem of interest can now be formulated as:

maximize Pasubject to Ps ⇤ �Pa +Pt +Pi � 0, (120)

which is a convex maximization problem formulated in terms of the interiorfields FFF and constrained with a non-negative scattered power. The Lagrangianfor this optimization problem is given by

L(FFF,�) ⇤ (1��) Im⇢π

VFFF⇤ ·MMMa · FFF dv

�+� Im

⇢πV

FFF⇤i ·MMMt · FFF dv

�

�� Im⇢π

VFFF⇤

i ·MMMb · FFFi dv�, (121)

where � is the Lagrange multiplier. Taking the first variation of (121)

�L(FFF,�) ⇤ Im⇢π

V�FFF⇤ ·

h(1��)

⇣MMMa �MMM†

a

⌘· FFF��MMM†

t · FFFi

�dv

�, (122)

a stationary solution (�L(FFF,�) ⇤ 0) for the field quantity FFF can be found as

FFF ⇤↵2i (Im{MMMa})�1 ·MMM†

t · FFFi , (123)

where ( · )† denotes the Hermitian transpose, ↵ ⇤ �/(1� �), and Im{MMMa} ⇤⇣MMMa �MMM†

a

⌘/2i.

The optimal absorbed power can be determined by inserting the stationarysolution into the expression for absorbed power (113), which results as

Popta ⇤

k0↵2

8⌘0

πV

FFF⇤i ·MMMt · (Im{MMMa})�1 ·MMM†

t · FFFi dv. (124)

The parameter ↵ can be determined by putting the solution (123) into the activeconstraint in optimization (120) and resulting from the equation

↵2+2↵ ⇤ q , (125)


with the maximizing root determined as

↵ ⇤ �1�p

1� q , (126)

where

q ⇤

4π

VFFF⇤

i · Im {MMMb} · FFFi dvπV

FFF⇤i ·MMMt · (Im{MMMa})�1 ·MMM†

t · FFFi dv(127)

in the range 0 q 1. The lower bound of this range can be approached forIm{MMMb} ⇤ 0, i.e., in the case when the surrounding medium is lossless. Todetermine the upper bound of q, observe that its denominator is convex in MMMa forIm{MMMa} > 0. By minimization of denominator, it can be proved that max{q} ⇤ 1for MMMa ⇤ MMMb.

7.2 Physical bound on absorption based on exterior-field formu-lation

Let the electromagnetic field be expanded in spherical vector waves as:

8>>>>><>>>>>:

EEE(rrr) ⇤’⌧,m ,l

a⌧mlv⌧ml(krrr)+ f⌧mlu⌧ml(krrr),

HHH(rrr) ⇤ 1i⌘0⌘

’⌧,m ,l

a⌧mlv⌧ml(krrr)+ f⌧mlu⌧ml(krrr),(128)

where v⌧ml(krrr) and u⌧ml(krrr) are the incident (regular) and the scattered (outgoing)spherical vector waves, respectively, having the properties described in AppendixB of Paper VII, and ai

⌧ml and f⌧ml the corresponding multipole coefficients,l ⇤ 1,2, . . . , is the multipole order, m ⇤ �l , . . . , l, the azimuthal index, and ⌧ ⇤ 1,2corresponds to a transverse electric (TE) magnetic multipole (with ⌧ ⇤ 1) and atransverse magnetic (TM) electric multipole (with ⌧ ⇤ 2); see e.g., [3,15,16,45,52,67].Here, ⌧ is the dual index, i.e., 1 ⇤ 2 and 2 ⇤ 1.

Let us consider an arbitrary linear scatterer, which may consist of a generalbianisotropic linear material. The scatterer is circumscribed by a spherical volumeof radius a, which is surrounded by a linear isotropic lossy medium; see Figure 9.The power balance to be used for this problem is the same as in (103), and involvesthe absorbed, scattered, total, and the incident powers defined by (104) through


(107), respectively. By employing the orthogonality properties of spherical vectorwaves

π@Va

w⌧ml(krrr)⇥z⇤⌧m0l0(krrr) · rrr dS

⇤ a2�mm0�ll0

8>>>>><>>>>>:

wl(ka)✓(kazl(ka))0

ka

◆⇤, ⌧ ⇤ 1,

�✓(kawl(ka))0

ka

◆z⇤l (ka), ⌧ ⇤ 2,

(129)

and π@Va

w⌧ml(krrr)⇥z⇤⌧m0l0(krrr) · rrr dS ⇤ 0, (130)

for ⌧ ⇤ 1,2 on the spherical surface @Va , the scattered, total, and incident powerscan be given by

Ps ⇤Re{p✏b}2 |kb |2 ⌘0

’⌧,m ,l

A⌧l�� f⌧ml

��2 , (131)

Pt ⇤Re{p✏b}2 |kb |2 ⌘0

’⌧,m ,l

2Re{B⌧l ai⇤⌧ml f⌧ml}, (132)

and

Pi ⇤Re{p✏b}2 |kb |2 ⌘0

’⌧,m ,l

C⌧l��ai⌧ml

��2 , (133)

respectively, where

A⌧l ⇤1

Re{kb}

(� Im{k⇤b⇠l⇠0⇤l }, ⌧ ⇤ 1,

Im{k⇤b⇠0l⇠

⇤l }, ⌧ ⇤ 2,

(134)

B⌧l ⇤1

i2Re{kb}

(k⇤b⇠l 0⇤

l � kb ⇤l⇠

0l , ⌧ ⇤ 1,

�k⇤b⇠0l

⇤l + kb 0⇤

l ⇠l , ⌧ ⇤ 2,(135)

C⌧l ⇤1

Re{kb}

(Im{k⇤b l 0⇤

l }, ⌧ ⇤ 1,

� Im{k⇤b 0l

⇤l }, ⌧ ⇤ 2,

(136)

for ⌧ ⇤ 1,2 and l ⇤ 1, . . . ,1, and the arguments of the Ricatti-Bessel ( l) andRicatti-Hankel functions (⇠l) are z ⇤ kba. The coefficients A⌧l > 0 and C⌧l � 0,which follows from the application of the divergence theorem to (105) and (107)in the case when the surrounding medium is passive implying that Ps � 0 andPi � 0, respectively. The coefficient B⌧l is complex-valued.


The physical bound on absorption in terms of the exterior fields can be derivedas follows. First, consider the contribution to the absorbed power from a singlepartial wave obtained from the power balance relation (103)

Pa,⌧ml ⇤Re{p✏b}2 |kb |2 ⌘0

h�A⌧l

�� f⌧ml��2 +2Re{B⌧l ai⇤

⌧ml f⌧ml}+C⌧l��ai⌧ml

��2i , (137)

for a given multi-index (⌧,m , l), which is based on the expressions for the scattered,total, and incident powers (131) through (133), respectively. Then, we define thescattering coefficients f⌧ml in terms of T-matrix for an arbitrary linear scatterer,which is circumscribed by a spherical volume Va , as

fn ⇤

’n0

Tn ,n0ain0 , (138)

where the multi-index notation n ⇤ (⌧,m , l) has been introduced. It is observedthat (137) is a concave function of the complex-valued variables Tn ,n0 with respectto the primed index n0. This allows to derive the stationary condition

A⌧l ain0

’n00

ai⇤n00T⇤

n ,n00 ⇤ B⌧l ai⇤n ai

n0 (139)

by differentiating (137) with respect to Tn ,n0 for fixed n, from which the pseudo-inverse solution to the T-matrix can be obtained as

Tn ,n0 ⇤B⇤⌧l

A⌧l gai

n ai⇤n0 , (140)

where g ⇤Õ⌧,m ,l

��ai⌧ml

��2 is the corresponding matrix norm. Using the results (137)and (140), and by completing the squares, the contribution from a single wave tothe absorbed power can be given by

Pa,⌧ml ⇤Re{p✏b}2 |kb |2 ⌘0

(�A⌧l

��’⌧0m0l0

T⌧ml ,⌧0m0l0 �

B⇤⌧l a

i⌧ml a

i⇤⌧0m0l0

A⌧l g

!ai⌧0m0l0

��2

+

✓|B⌧l |2A⌧l

+C⌧l

◆ ��ai⌧ml

��2� , (141)

which is concave since A⌧l > 0. It is observed that (141) reaches its maximumwhen the first term equals to 0. Hereby, summing over all the indices (⌧,m , l), thephysical bound on absorption formulated in terms of exterior fields is given by

Popta ⇤

Re{p✏b}2 |kb |2 ⌘0

’⌧,m ,l

✓|B⌧l |2A⌧l

+C⌧l

◆ ��ai⌧ml

��2 , (142)

which is independent of the T-matrix, and thus is valid for scatterers of anarbitrary shape circumscribed by a spherical surface @Va and made of arbitrary

�. RESEARCH CONTRIBUTION 39

linear materials (including general bianisotropic). It should be noted that theinfinite dimensional matrix equation in (139) in general is related to an unboundedoperator, where the series given by g in (140) does not converge because thecorresponding matrix norm does not exist. However, this is merely a mathematicalsubtlety that does not pose any real problem here because in presence of a lossybackground, the T-matrix (138) can be truncated to a finite size L with l , l0 L,for which the solution converges. Hereby, the obtained physical bound (142) canalso be interpreted as the optimal absorption with respect to all incident andscattered fields up to multipole order L, as L !1.

8 Research Contribution

Paper I. Passive approximation and optimization using B-splines

Paper I provides a method for approximation of linear, time-translationallyinvariant passive systems. The proposed method is based on Herglotz functionswith Hölder continuous extension to the real axis, finite B-spline expansion ofan arbitrary order, and convex optimization. Here, the objective is to develop amathematical framework that allows determination of an optimal realization ofa passive system, characterized by a given continuous target function over theapproximation domain. Such a system must satisfy the Kramers-Kronig relations,and thus the real and imaginary parts of the system function are related to eachother via the Hilbert transform. The method exploits the fact that the Hilberttransform is a bounded operator on Hölder spaces.

In this paper, one of the main results is that we have proved that the convexcone consisting of approximating functions, whose measure is given by a finiteuniform B-spline expansion of a fixed arbitrary order, is dense in the convex coneof Herglotz functions which are Hölder continuous in an open neighborhoodof the approximation domain, as mentioned above; see Theorem 3.4 in Paper I.We have also proved that the greatest lower bound can be approached withinarbitrary accuracy by using a finite B-spline expansion of an arbitrary orderas a generating measure for the approximating function; see Theorem 3.7 inPaper I. We have also derived a new physical bound on realization of passivelossy metamaterials with constant permittivity over a finite frequency interval. Inthe numerical examples, we treat the passive realization of lossy metamaterialswith a fixed negative-permittivity property over various frequency bandwidths,including the passive realization of an optimal plasmonic resonance (polesingularity) of a dielectric sphere immersed in vacuum. Here, the resultingapproximating Herglotz functions have been generated by a linear B-splineexpansion.


Paper II. Passive Approximation with High-Order B-Splines

Paper II is focused on application of the mathematical framework developedin Paper I. Here, we summarize the explicit results concerning the Hilberttransform of general B-splines and the sum rule, which can be expanded withB-spline basis functions of an arbitrary order. The developed framework isapplied to a system with a non-trivial response over a large frequency bandwidth.A numerical example of a power-engineering application is presented. Theexample concerns the estimation of the static conductivity of power-cableinsulation material. The obtained approximation results show that high-order,cubic, B-splines are efficient in solving convex optimization problems for givensystems with non-trivial response and non-uniform measurement data.

Paper III. Quasi-Herglotz functions and convex optimization

In Paper III, we introduce the class of quasi-Herglotz functions suitable formodeling a subclass of non-passive systems. The new class of functions is astraightforward extension of the convex cone of Herglotz functions. We definequasi-Herglotz functions as differences of two Herglotz functions, and thus, thenew functions preserve the integral representation similar to the representationof Herglotz functions. However, not all the quasi-Herglotz functions admitthe sum-rule indentities, which are based on the small- and large-argumentasymptotic properties of this functions; see Section 3 in Paper III.

The new functions can also be restricted to be Hölder continuous on theopen neighborhood of the subset of the real axis, which is suitable for theapproximation theory developed in Paper I. We also prove that the subspaceof quasi-Herglotz functions generated by finite B-spline expansions of anarbitrary order is dense in the space of quasi-Herglotz functions, which areHölder continuous on the open neighborhood of the approximation intervalon the real axis; see Theorem 4.5 in Paper III. It is also proved that we canapproach the greatest lower bound with an arbitrary accuracy by using afixed-order B-spline expansion and point masses as generating measures forthe approximating function; see Theorem 4.7 and Corollary 4.8 in Paper III. Inthe numerical examples, we employ the non-passive approximation frameworkto determine optimal realizations of non-passive metamaterials with a fixednegative-permittivity property over a finite frequency interval. Interestingly, thesupport of the measure of the optimized permittivity function is concentrated atthe outermost points of frequency sets, where the measure is restricted to benon-positive. Using this observation, it has been discovered that the desiredpermittivity property can be similarly realized by a function, which has ameasure generated only by point masses inside the active (gain) frequencyregions. Also in these examples, it is noted that the approximating measure is

�. RESEARCH CONTRIBUTION 41

typically zero inside the approximation interval itself. We have also shown thatby using the sum rules, it is possible to construct the optimization problem forrealization of both the negative-permittivity and asymptotic properties of thedesired material.

Paper IV. Non-passive approximation as a tool to study the realizability ofamplifying media

Paper IV presents an application of the non-passive approximation frameworkas a tool for realization of amplifying media. As a physical application problem,we study the dipole absorption of a dielectric sphere immersed in a hypotheticalgain medium. Hereby, we formulate Mie theory for amplifying media andderive the plasmonic singularity of sphere with a given dielectric property interms of amplifying background medium; see Sections 2 and 3 in Paper IV,respectively. In the numerical example, we study an optimal realization ofa background medium with the desired amplifying properties over a givenfinite frequency interval. It has been investigated that for accurate realizationsof such materials, it is necessary to restrict the density of the measure to benon-negative over finite frequency intervals surrounding the approximationdomain. Note that the accuracy of realization depends on the width of theseintervals. Finally, we have studied the electric-dipole absorption of a goldsphere immersed in an amplifying background medium. We have observedthat when the sphere is embedded in the optimal amplifying medium, theelectric-dipole absorption level of the sphere is three orders of magnitudehigher than the physical bound on electric-dipole absorption for sphere in vacuum.

Paper V. On the physical limitations for radio frequency absorption in goldnanoparticle suspensions

Paper V focuses on determination of physical limitations on absorption in goldnanoparticle (GNP) suspensions in the MHz and GHz frequency ranges. In thispaper, as a problem setup, a spherical geometry consisting of GNP suspensionimmersed in a weak electrolyte solution is considered. The suspension ischaracterized by electrophoretic Drude model, and the electrolyte solution isrepresented via the Debye model with parameters relatively close to the ones thatare used in the model of saline water [52]. For study of absorption properties, ageneralized Mie theory in lossy media has been used. In this generalization, oneneeds to take into account the power absorbed by a surrounding lossy medium.It has been investigated that maximal absorption in GNP suspensions can beachieved when the permittivity characterizing the suspension is a conjugatematch with respect to the permittivity of the surrounding medium. Hereby, wehave studied narrowband and wideband realizabilities of the conjugate match.


The narrowband optimization has been achieved by "tuning" the Drude modelby using the suitable parameters with respect to the desired frequency. Thewideband realization is with the metamaterial problem, i.e., with the achievementof the desired property over a given frequency bandwidth. Hereby, we employthe passive approximation framework developed in Paper I to determine anoptimal realization of the desired property.

Paper VI. On the plasmonic resonances in a layered waveguide structure

Paper VI presents an investigation on transmission and reflection coefficientsthough a thin layer made of a composite material and which is placed in arectangular straight waveguide working in TM-mode. The material of the layerhas a Drude type of dispersion, which is suitable for study of electrophoreticmicrowave heating of gold nanoparticles described in Paper V and for obtainingplasmonic resonances. Through the asymptotic analysis of expressions for thetransmission and reflection coefficients, we show that it is possible to obtain aresonance in a plasmonic thin layer when the permittivity of the layer tends tozero, while the layer’s thickness is fixed. Furthermore, we derive the Fröhlichresonance condition, which indicates the resonance frequency of a thin layerunder certain assumptions. The numerical example presented in this paperillustrates the derived theoretical results.

Paper VII. Optical theorems and physical bounds on absorption in lossy media

Paper VII introduces two versions of an optical theorem for scatterersimmersed in a lossy surrounding medium. The corresponding upper bounds onabsorption are derived using conventional analytical optimization techniques.The two versions are based on interior and exterior fields, respectively: thefirst version of the optical theorem is derived in terms of equivalent currentsinside a scattering object, while the second version is formulated in terms ofT-matrix parameters of a linear scatterer circumscribed by a spherical volume.The first upper bound on absorption is valid for scattering objects of an arbitraryshape with a given material property. The second upper bound on absorptioncorresponds to the absorption inside a given spherical inclusion for a givenmaterial property of surrounding lossy medium. This bound is valid for linearscatterers that can be placed inside the spherical inclusion. The shape of such ascatterer can be arbitrary, as well as its material properties (including generalbianisotropic). Numerical examples with homogeneous and core-shell spheresimmersed in lossy surrounding media demonstrate that the corresponding upperbounds on absorption provide complementary information in given scatteringproblems.

�. CONCLUSIONS 43

9 ConclusionsThis thesis has presented an overview on the determination of physical limitationsof linear, time-translationally invariant, causal passive and non-passive systemsbased on their integral representation properties, as well as numerical andanalytical optimization techniques for passive and non-passive approximation.The following results have been achieved:

• The mathematical and numerical frameworks for determination of physicalbounds for passive systems based on Herglotz functions, convex optimiza-tion, and B-splines has been developed.

• The new class of functions, i.e., the quasi-Herglotz functions, based ondifferences of two Herglotz functions, has been introduced for modelingadmittance non-passive systems.

• The mathematical and numerical frameworks for determination of optimalrealizations of admittance non-passive systems based on the set of approxi-mating quasi-Herglotz functions, convex optimization, point masses andB-splines has been developed.

• A method for detection of electrophoretic resonances in sub-wavelengthparticles placed in straight waveguides through measurements of thereflection coefficient has been proposed.

• The optical theorem for scattering objects immersed in lossy media has beenrevisited. Physical bounds on absorption of an arbitrary scatterer madeof arbitrary materials, which is immersed in a lossy isotropic medium,have been derived by employing the method of Lagrange multipliers.The two physical bounds on absorption derived in terms of internal andexternal fields, respectively, provide complementary information, and serveas a tool for understanding the limits of power that can be absorbed byhomogeneous, composed, or homogenized scatterers immersed in lossymedia in a given frequency range.

10 Future WorkThe developed theory for mathematical frameworks presented in Papers I andIII is quite general and leaves possibilities to consider other electromagnetic andengineering applications. Several tracks remain for future work.

The first track is to extend the developed theory to matrix-valued operators,i.e., to involve the matrix-valued Herglotz and quasi-Herglotz functions forcharacterization of multiport passive and non-passive systems. The second trackis to apply the theory developed in Paper III to studying bounds on optimal


realization of non-passive electric-circuit components with desired properties,which is of interest in antenna and communication systems. The third track isto determine physical bounds on scattering and extinction in lossy media basedon the methods employed in Paper VII, which is of interest in antenna designand telemetry applications. The last track is to utilize the physical bounds onabsorption derived in Paper VII for understanding the electrophoretic heating ofgold nanoparticle suspensions, and improving their corresponding design withrespect to the desired characteristics and properties.

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