Adventures in Contemporary Electromagnetic Theory [1st ed. 2023] 3031246160, 9783031246166

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Tom G. Mackay Akhlesh Lakhtakia   Editors

Adventures in Contemporary Electromagnetic Theory

Adventures in Contemporary Electromagnetic Theory

Tom G. Mackay • Akhlesh Lakhtakia Editors

Adventures in Contemporary Electromagnetic Theory

Editors Tom G. Mackay School of Mathematics and Maxwell Institute for Mathematical Sciences University of Edinburgh Edinburgh, UK NanoMM—Nanoengineered Metamaterials Group, Department of Engineering Science and Mechanics The Pennsylvania State University University Park, PA, USA

Akhlesh Lakhtakia NanoMM—Nanoengineered Metamaterials Group, Department of Engineering Science and Mechanics The Pennsylvania State University University Park, PA, USA

ISBN 978-3-031-24616-6 ISBN 978-3-031-24617-3 https://doi.org/10.1007/978-3-031-24617-3

(eBook)

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tom G. Mackay and Akhlesh Lakhtakia

1

2

Our Werner Always Brought Us Joy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Heide Weiglhofer

5

3

Scalar Potentials and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Michael Havrilla

9

4

A Novel Approach to Electromagnetic Constitutive Relations . . . . . . . . Martin W. McCall, Paul Kinsler, and Jonathan Gratus

33

5

On the Anatomy of Voigt Plane Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tom G. Mackay and Akhlesh Lakhtakia

61

6

Electromagnetic Radiation by Finite-Sized Electric and Magnetic Dipoles Embedded in Homogeneous Uniaxial Dielectric Materials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Aamir Hayat and Muhammad Faryad

93

7

Near-Field Microwave Imaging Employing Measured Point-Spread Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 Natalia K. Nikolova, Daniel Tajik, and Romina Kazemivala

8

Electromagnetic Wave Propagation Inside Rectangular Chirowaveguides Using the Coupled Mode Method . . . . . . . . . . . . . . . . . . . . 169 Álvaro Gómez-Gómez, Óscar Fernández, and Gregorio J. Molina-Cuberos

9

On a Steklov Spectrum in Electromagnetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 Francesco Ferraresso, Pier Domenico Lamberti, and Ioannis G. Stratis

10

Using Boundary Conditions with the Ewald–Oseen Extinction Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 Akhlesh Lakhtakia v

vi

Contents

11

Spatial Sampling and Interpolation Techniques in Computational Electromagnetics and Beyond . . . . . . . . . . . . . . . . . . . . . . . 245 Yaniv Brick and Amir Boag

12

Light-Matter Interaction at the Sub-Wavelength Scale: Pathways to Design Nanophotonic Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 M. Pourmand and Pankaj K. Choudhury

13

Integrated Photonics with Near-Zero Index Materials . . . . . . . . . . . . . . . . . 315 Larissa Vertchenko and Andrei V. Lavrinenko

14

Correlated Disorder in Broadband Dielectric Multilayered Reflectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 Vincenzo Fiumara, Paolo Addesso, Francesco Chiadini, and Antonio Scaglione

15

Scattering from Reconfigurable Metasurfaces and Their Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 Mirko Barbuto, Alessio Monti, Davide Ramaccia, Stefano Vellucci, Alessandro Toscano, and Filiberto Bilotti

16

Specular Reflection and Transmission of Electromagnetic Waves by Disordered Metasurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 Kevin Vynck, Armel Pitelet, Louis Bellando, and Philippe Lalanne

17

Continuity of Field Patterns for Exceptional Surface Waves and Exceptional Compound Waves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 Tom G. Mackay, Waleed Iqbal Waseer, and Akhlesh Lakhtakia

18

Cavity Modes and Surface Plasmon Waves Coupling on Nanostructured Surfaces for Enhanced Sensing and Energy Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 Mohammad Abutoama and Ibrahim Abdulhalim

19

Analysis of Diffraction from All-Dielectric Gratings Using Entire-Domain Integral Equation Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . 481 Nikolaos L. Tsitsas

20

Rigorous Coupled-Wave Approach and Transformation Optics . . . . . . 503 Benjamin J. Civiletti, Akhlesh Lakhtakia, and Peter B. Monk

21

Mind the Gap Between Theory and Experiment . . . . . . . . . . . . . . . . . . . . . . . 531 Andrei Kiselev, Jeonghyeon Kim, and Olivier J. F. Martin

22

Theoretical Future: Vision 2030 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549 Amir Boag, Vadim A. Markel, Olivier J. F. Martin, M. Pinar Mengüç, and Kevin Vynck

Chapter 1

Introduction Tom G. Mackay and Akhlesh Lakhtakia

Werner Siegfried Weiglhofer, formerly Professor of Applied Mathematics at the University of Glasgow, died on 12 January 2003 in a mountaineering accident. Born on 25 August 1962, Werner obtained a diploma in Physics and a doctorate in Technical Sciences from the Technical University of Graz. After post-doctoral stints at Graz and the University of Adelaide, Werner joined the Department of Mathematics at the University of Glasgow in 1988 where he remained for the rest of his career. In 1991, he became a Lecturer, then a Senior Lecturer, and then a Reader and was finally elevated to the rank of Professor in September 2002 (Fig. 1.1). Werner’s research interests lay in magnetohydrodynamics and theoretical electromagnetics of complex materials. He authored or co-authored 135 journal papers, in addition to numerous conference publications and presentations. Among his many notable contributions rank the delineation of magnetic instabilities in rotating plasmas; the development of scalar Hertz potentials and Green functions for bianisotropic materials; wave propagation in structurally chiral materials, such as certain sculptured thin films; and homogenization of linear and nonlinear particulate composite materials. His untimely demise was mourned throughout the electromagnetics research community, as recounted shortly thereafter in an anthology on electromagnetic fields in complex mediums [1].

T. G. Mackay () School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, Edinburgh, UK NanoMM—Nanoengineered Metamaterials Group, Department of Engineering Science and Mechanics, The Pennsylvania State University, University Park, PA, USA e-mail: [email protected] A. Lakhtakia NanoMM—Nanoengineered Metamaterials Group, Department of Engineering Science and Mechanics, The Pennsylvania State University, University Park, PA, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 T. G. Mackay, A. Lakhtakia (eds.), Adventures in Contemporary Electromagnetic Theory, https://doi.org/10.1007/978-3-031-24617-3_1

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T. G. Mackay and A. Lakhtakia

Fig. 1.1 Werner S. Weiglhofer at the time of his elevation to Professor at the University of Glasgow

Fig. 1.2 Speakers at the Weiglhofer Symposium on Electromagnetic Theory, photographed by Akhlesh Lakhtakia at the statue of James Clerk Maxwell in George Street, Edinburgh

Had he survived, Werner would be 60 years old in 2022. To focus on current and future developments in the electromagnetic theory of complex materials, the Weiglhofer Symposium on Electromagnetic Theory was convened on 18–19 July 2022. It is most appropriate that the venue for this symposium was the childhood home of the father of electromagnetic theory, namely James Clerk Maxwell, at 14 India Street, Edinburgh. A total of 26 invited attendees, from 14 different countries,

1 Introduction

3

including several of Werner’s collaborators, delivered talks on fields pertaining to the electromagnetic theory of complex materials (Fig. 1.2). The symposium was organized by Tom G. Mackay (Professor of Applied Electromagnetic Theory, School of Mathematics, University of Edinburgh) who was Werner’s only PhD student and Akhlesh Lakhtakia (Evan Pugh University Professor and Charles G. Binder Professor at The Pennsylvania State University) who was Werner’s collaborator-in-chief. This book is based on talks delivered at the symposium. The electromagnetic theory of complex mediums has advanced greatly in recent years. These advances have been partly motivated and partly informed by developments in engineering science and nanotechnology. The collection of chapters provided in this edited book, authored by experts in the field, offers a bird’s eye view of recent progress in electromagnetic theory. A wide range of topics of current interest is spanned, ranging from fundamental issues to applications, involving analytical as well as numerical techniques. The book begins with reminiscences of Werner by his mother Heide and ends with the report of a panel discussion on future developments in electromagnetic theory.

Reference 1. Weiglhofer, W.S., Lakhtakia, A. (eds.): Introduction to Complex Mediums for Optics and Electromagnetics. SPIE, Bellingham (2003)

Chapter 2

Our Werner Always Brought Us Joy Heide Weiglhofer

I worked in an office before my husband Erich and I were blessed with Werner in 1962. I had the great privilege to be a housewife thereafter, and I returned to my office job when Werner started his studies. As an only child, Werner liked to be with other children, which is why he particularly enjoyed kindergarten. We spent a lot of time with his grandparents, who had a big garden. He was their first grandchild and was therefore spoiled by everyone in the family. He looked forward to attending school, especially because of reading many fairy tales and other stories all by himself. At school, his favorite subject was always mathematics. But when the school day would end and all the homework would have been done, the only thing he wanted to do was to play soccer. He was an active member of the swimming club and he played tennis, but when he was allowed to join the soccer club at the age of 10, that was the greatest joy for him. Every year during the summer holidays, he would go with friends to a holiday camp on the shores of one of our beautiful lakes or later by the seaside. Afterward we would spend 2 weeks in the Alps, with one of his grandmothers. Those were fun times. When he got his first camera, he photographed only cows and oxen (Fig. 2.1). All those school years he did not need any help from us with his homework assignments. He was very diligent and also ambitious. Of course, he was always there at school for all the jokes and nonsense, especially so as not to look like a nerd, I think. He finished his secondary education in 1980 by passing the Matura exam with honors. Then he joined the Technical University of Graz to study

Erich Weiglhofer passsed away on December 17, 2022 at his home in Bruck an der Mur. H. Weiglhofer () Bienengasse 11, Bruck an der Mur, Austria e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 T. G. Mackay, A. Lakhtakia (eds.), Adventures in Contemporary Electromagnetic Theory, https://doi.org/10.1007/978-3-031-24617-3_2

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6

H. Weiglhofer

Fig. 2.1 Werner growing up in Austria. (a) With his grandparents in 1964. (b) With cousins in 1974. (c) Practicing soccer in 1979. (d) Receiving an academic prize in 1980. (e) In 1983. (f) Receiving his doctorate in 1986

engineering, and he received only the highest grade in every examination. In 1984, the Diplomingenieur degree was conferred on him. Unfortunately, he injured himself and so could not play soccer anymore. So he started running. He trained every day by running a few kilometers, and since he was

2 Our Werner Always Brought Us Joy

7

always so precise, he also wrote down the distance he would run and the time taken. On the weekends, he liked to go running in the mountains. After his first degree, he took a part-time job at the Technical University of Graz, and now he wrote his doctoral thesis in his free time. In November 1986, at the age of 24, he received the Doctor scientiae technicorum degree with honors. The only foreign scholarship that Austria awarded that year was to Werner. As an Australia–Europe Fellow, he spent a year at the University of Adelaide in Australia. There he lived in a shared apartment and made friends with whom he maintained warm relationships for the rest of his life. After the academic year ended, he stayed on for a few months to visit many beautiful places with his new friends. He was at the Great Barrier Reef. At the summit of Ayers Rock, he met a man from Graz. “The world is so small,” he wrote to Erich and me. Werner traveled in New Zealand in a camping bus. In Tasmania, he took a dangerous rafting trip on River Franklin. He also spent a few days in the islands of Fiji. Then he started to look for employment. Werner signed on as a post-doctoral research scholar in the Department of Mathematics at the University of Glasgow, Scotland. This was for a 3-year project. After that, he got permanent employment in the same department. Staying there for 16 years, he rose from Lecturer to Senior Lecturer to Reader and finally to Professor of Applied Mathematics. His research work was very important to him and he always gave his best. Now he also had to take care of his students. That was something new for him, but he took his time to become acquainted with them and was always there for them. He settled in quickly and, as a newcomer, was supported by everyone and soon felt at home. He was also a personal friend of many of his colleagues. As he really felt like he belonged there, Glasgow almost became his second home, and we visited him there occasionally (Fig. 2.2). During this time, he also rediscovered his love for the mountains. He climbed all of the Scottish peaks collectively called the Munros and then even the Munro Tops and the Corbetts. His next destination was Norway. The mountains in Romsdalen called the Norwegian Alps fascinated him. There he also got to know and love Eva, his long-time girlfriend. He now divided his vacation days between Norway and Austria because our mountains stood up to any comparison. If he spent a weekend in the mountains, the latest photos were already available on the computer on Monday. “Beautiful photos,” I would tell him. He had a good eye when taking photos but, I told him, “sometimes we would love to see you in one of those beautiful pictures too.” From then on, there were also photos of Werner in the mountains. His research was very important to him, which is evident from his many publications. His path also led him to Akhlesh Lakhtakia in USA. Akhlesh and Werner had many joint projects. Their lectures and conferences took them to many countries, and they became best friends and were always there for each other. Werner was also warmly welcomed into Akhlesh’s family. For the last 3 years, he also guided Tom Mackay to his second doctorate—this one in mathematics, the first being in biomedical engineering. Akhlesh helped me a lot during our worst time. He keeps on reporting to me about his research works, all his travels, and his wife Mercedes as well as his dear daughter Natalya. When I read his e-mails, I sometimes forget that Werner is no

8

H. Weiglhofer

Fig. 2.2 Erich, Heide, and Werner at Edinburgh Castle in 2002

longer with us. But Werner lives in our hearts, Erich’s and mine. And he also lives in the hearts of Akhlesh and Tom, and of those scientists who interacted with him as well as those who profit from his research legacy. Erich and I are very glad that so many of his colleagues gathered on July 18 and 19, 2022, for the Weiglhofer Symposium on Theoretical Electromagnetics organized by Tom and Akhlesh. There, at the house of James Clerk Maxwell, they celebrated what would have been Werner’s 60th birthday. And now this commemorative research volume will keep Werner’s research and its outcomes fresh for a few more generations of researchers.

Heide Weiglhofer lives with her husband Erich Weiglhofer in Bruck an der Mur, Austria. They were the parents of Werner Weiglhofer.

Chapter 3

Scalar Potentials and Applications Michael Havrilla

3.1 Introduction In recent years, fabrication capabilities (e.g., 3D printing [1]) have allowed sophisticated materials to be constructed for electromagnetic applications. This construction may be accomplished, for example, via infusion of symmetry into the material design process, leading to generally bianisotropic media [2]. A natural question to ask is: what is the best method for analyzing electromagnetic problems involving complex-media environments? The direct solution of Maxwell equations via computational electromagnetics is certainly a viable option [3] but does have its drawbacks (e.g., convergence criteria, rounding errors, run time, etc.). Potentialbased methods offer an alternative that helps simplify analysis and may offer deeper physical insight. Vector and scalar potentials are discussed in Sect. 3.2. It is demonstrated that the constitutive tensors greatly influence the choice of the potential method and that scalar potentials, pioneered by Weiglhofer [4] for gyrotropic bianisotropic media, are robust in handling complex-media environments. Section 3.3 provides a discussion on aspects of the scalar potential formulation that must be understood in order to apply this method in practice. Section 3.4 gives an example of extracting the electromagnetic material properties of a uniaxial planar sample with the aid of scalar potentials. Final comments and a summary are provided in Sect. 3.5.

M. Havrilla () Air Force Institute of Technology, Wright-Patterson Air Force Base, OH, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 T. G. Mackay, A. Lakhtakia (eds.), Adventures in Contemporary Electromagnetic Theory, https://doi.org/10.1007/978-3-031-24617-3_3

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M. Havrilla

3.2 Potential-Based Methods As discussed earlier, recent capabilities have allowed the fabrication of ever-exotic materials (e.g., nonlinear, bianisotropic, time-varying, etc.). The media considered here are restricted to being linear, bianisotropic, spatially invariant, spatially local, time-invariant, and temporally non-local (unless stated otherwise). In this case, the Maxwell curl equations in the frequency domain (.e−iωt assumed and suppressed) are ∇ × E = −Jh + iωB

(3.1)

∇ × H = Je − iωD.

(3.2)

.

and .

¯¯ Upon using .D = ¯¯ ·E+ ξ¯¯ ·H and .B = ζ¯¯ ·E+ μ·H (see Ref. [5]), Eqs. (3.1) and (3.2) can be written as (∇ × I¯¯ − iωζ¯¯ ) · E = −Jh + iωμ¯¯ · H

(3.3)

(∇ × I¯¯ + iωξ¯¯ ) · H = Je − iω¯¯ · E,

(3.4)

.

and .

ˆ .¯¯ (ω) and where .I¯¯ = I¯¯t + zˆ zˆ is the unit tensor and .I¯¯t = xˆ xˆ + yˆ yˆ = ρˆ ρˆ + φˆ φ; ¯ ¯ ¯¯ ¯ .μ(ω) are the permittivity and permeability tensors; .ξ(ω) and .ζ¯ (ω) are the magnetoelectric tensors; .Je and .Jh are the electric and magnetic current densities; and .E and .H are the electric and magnetic field intensities. The independent variables .r = xˆ x + yˆ y + zˆ z and .ω are omitted for notational convenience (with the constitutive tensors assumed homogeneous here). Since the medium is assumed linear, superposition is often employed in the solution process. If .Jh = 0, solving Eq. (3.3) for .H and insertion into Eq. (3.4) leads to the result H=

.

1 ¯ −1 μ¯ · (∇ × I¯¯ − iωζ¯¯ ) · E iω

(3.5)

and [(∇ × I¯¯ + iωξ¯¯ ) · μ¯¯ −1 · (∇ × I¯¯ − iωζ¯¯ ) − ω2 ¯¯ ] · E = iωJe .

.

If .Je = 0, we have, by duality [6],

(3.6)

3 Scalar Potentials and Applications

E=−

.

11

1 ¯ −1 ¯ · (∇ × I¯¯ + iωξ¯¯ ) · H iω

(3.7)

and ¯¯ · H = iωJh . [(∇ × I¯¯ − iωζ¯¯ ) · ¯¯ −1 · (∇ × I¯¯ + iωξ¯¯ ) − ω2 μ]

.

(3.8)

The Helmholtz-type equations (3.6) and (3.8) may be solved in a variety of manners; however, vector- and scalar-potential-based formulations are explored in this section. It is shown that the type of potential-based formulation utilized is driven by the properties of the constitutive tensors. The mediums considered are: (i) simple, (ii) biisotropic, (iii) general bianisotropic, and (iv) gyrotropic bianisotropic.

3.2.1 Simple Medium ¯¯ The constitutive tensors for a simple medium are .¯¯ (ω) = (ω)I¯¯, .μ(ω) = μ(ω)I¯¯, ¯ ¯ ¯ .ξ(ω) = 0, and .ζ¯ (ω) = 0. In this case, Eqs. (3.3) and (3.4) simplify to ∇ × E = −Jh + iωμH

(3.9)

∇ × H = Je − iωE.

(3.10)

.

and .

The magnetic vector potential .A, first discussed by Neumann (see Refs. [7] and [8]), is readily identified by first setting .Jh = 0 and then taking the divergence of Eq. (3.9) to obtain .∇ · B = 0, which implies .B = μH = ∇ × A, leading to [with .φe being the electric scalar potential, first discussed by Poisson (see Refs. [8] and [9])] H=

.

1 ∇ ×A μ

(3.11)

and, with the aid of Eq. (3.9), E = −∇φe + iωA.

.

(3.12)

Insertion of Eqs. (3.11) and (3.12) into Eq. (3.10) and using .∇ × ∇ × A = ∇∇ · A − ∇ 2 A lead to the Helmholtz equation for .A, namely, ∇ 2 A + k 2 A = −μJe ,

.

where .k 2 = ω2 μ and the Lorenz gauge [10] was invoked, that is,

(3.13)

12

M. Havrilla

∇ · A = iωμφe .

.

(3.14)

The magnetic vector potential .A is often called the Lorentz [11] magnetic vector potential and Eq. (3.14) the Lorentz gauge (named after Hendrik Lorentz [12, p. 194]), but it appears Ludvig Lorenz is deserving of credit (see Refs. [8] and [13]). Setting .Je = 0 leads to, via duality, the electric vector potential formulation for .F as follows: 1 E = − ∇ × F, 

(3.15)

H = −∇φh + iωF,

(3.16)

∇ 2 F + k 2 F = −Jh ,

(3.17)

.

.

.

and (with .φh being the magnetic scalar potential) ∇ · F = iωμφh .

.

(3.18)

If both electric and magnetic current densities exist, then solutions to both .A and .F must be found with the aid of Eqs. (3.13) and (3.17). The fields .E and .H are subsequently recovered from these potentials via superposition as (with the aid of Eqs. (3.14) and (3.18)) .

E=

1 i [∇(∇ · A) + k 2 A] − ∇ × F  ωμ

(3.19)

H=

i 1 [∇(∇ · F) + k 2 F] + ∇ × A. μ ωμ

(3.20)

and .

Substitution of Eq. (3.11) into Eq. (3.10) and of Eq. (3.15) into Eq. (3.9) leads to the equivalent field recovery process, namely, .

E=

1 i [∇ × (∇ × A) − μJe ] − ∇ × F ωμ 

(3.21)

H=

i 1 [∇ × (∇ × F) − Jh ] + ∇ × A. ωμ μ

(3.22)

and .

The vector potential formulation for a simple medium is well known [14]. The major benefit is that the Helmholtz equations for .A and .F are much easier to solve in the presence of sources when compared with the Helmholtz equations for .E and

3 Scalar Potentials and Applications

13

H. The primary restriction is that the medium must be spatially homogeneous. A disadvantage of this vector potential formulation is that, due to the form of field recovery equations, it is more difficult to discern the physical nature of the fields (e.g., are the fields lamellar, rotational, etc.). Also, the fields are needed to satisfy boundary conditions. Alternative potential formulations for simple media have also been explored. Hertz potentials [15], often employed in optics where electric .P (and magnetic .M, by duality) polarization sources are prevalent, are scaled versions of .A and .F, namely (see Refs. [16–18]),

.

.

A = −iωμe

(3.23)

F = −iωμh ,

(3.24)

and .

or .

e =

i A ωμ

(3.25)

h =

i F, ωμ

(3.26)

and .

where .e and .h are the Hertz electric and magnetic vector potentials, respectively. Using Eqs. (3.23) and (3.24), it is seen that the Helmholtz equations (3.13) and (3.17) become 1 ∇ 2 e + k 2 e = − P 

(3.27)

1 ∇ 2 h + k 2 h = − M, μ

(3.28)

.

and .

where .Je = −iωP and .Jh = −iωM have been used. The corresponding field recovery process becomes .

E = ∇(∇ · e ) + k 2 e + iωμ∇ × h

(3.29)

H = ∇(∇ · h ) + k 2 h − iω∇ × e ,

(3.30)

and .

14

M. Havrilla

or, equivalently, 1 E = ∇ × (∇ × e ) − P + iωμ∇ × h 

.

(3.31)

and H = ∇ × (∇ × h ) −

.

1 M − iω∇ × e . μ

(3.32)

The gauge for the Hertz potentials follows immediately from Eqs. (3.14) and (3.18), namely, .

∇ · e = −φe

(3.33)

∇ · h = −φh .

(3.34)

and .

Debye studied the interaction of light with a material sphere in a source-free region with the aid of potential functions (see Refs. [18–21]). The Debye potentials are related to .A and .F via .

A = −iωμ1 r = −iωμˆr1 r

(3.35)

F = −iωμ2 r = −iωμˆr2 r,

(3.36)

and .

where .r = rˆ r, .1 (r, θ, φ) is the type-1 (TM.r ) Debye scalar potential, and 2 (r, θ, φ) is the type-2 (TE.r ) Debye scalar potential. Debye showed that both .1 and .2 satisfy the scalar Helmholtz equation in spherical coordinates, namely [extended here for specialized source regions .P = rˆ Pr and .M = rˆ Mr (see Ref. [22])], .

1 Pr  r

(3.37)

1 Mr , μ r

(3.38)

∇ 2 1 + k 2 1 = −

.

and ∇ 2 2 + k 2 2 = −

.

with the field recovery process given as

3 Scalar Potentials and Applications

15

1 E = ∇ × [∇ × (1 r)] − rˆ Pr + iωμ∇ × (2 r) 

.

(3.39)

and H = ∇ × [∇ × (2 r)] −

.

1 rˆ Mr − iω∇ × (1 r). μ

(3.40)

These above results were obtained with the aid of the Debye gauges ∂(1 r) = −φe ∂r

(3.41)

∂(2 r) = −φh . ∂r

(3.42)

.

and .

Comparison of the Hertz and Debye potentials reveals .e = rˆ 1 r and .h = rˆ 2 r. Thus, the Debye potentials are Hertz potentials that are radially scaled and directed. Although various other potentials exist, one final formulation is discussed due to its similarity with the Hertz and Debye formulations. Bromwich, like Debye, was interested in electromagnetic waves, especially in spherical coordinates. The Bromwich potentials (see Refs. [18, 23–25]) are related to .A and .F via .

A = −iωμˆrU

(3.43)

F = −iωμˆrV .

(3.44)

and .

Comparison with the Debye potentials reveals the simple relationships .U = 1 r and .V = 2 r; thus the field recovery process and gauges are readily identified (and omitted for brevity). The Helmholtz-like equations are altered since the radial scaling is absorbed into U and V , and are given as (extended here for specialized source regions .P = rˆ Pr and .M = rˆ Mr ) ∂ 2U 1 + ∇t2 U + k 2 U = − Pr  ∂r 2

(3.45)

∂ 2V 1 + ∇t2 V + k 2 V = − Mr , 2 μ ∂r

(3.46)

.

and .

where .∇t2 is the transverse Laplacian in spherical coordinates.

16

M. Havrilla

3.2.2 Biisotropic Medium ¯¯ The constitutive tensors for biisotropic media are .¯¯ (ω) = (ω)I¯¯, .μ(ω) = μ(ω)I¯¯, ¯ ¯ ¯ ¯ ¯ .ξ(ω) = ξ(ω)I¯, and .ζ¯ (ω) = ζ (ω)I¯ (see Refs. [26] and [27]). In this case, Eqs. (3.3) and (3.4) become ∇ × E = −Jh + iωζ E + iωμH

(3.47)

∇ × H = Je − iωE − iωξ H.

(3.48)

.

and .

Following along similar lines as in the simple medium case, we have (for .Jh = 0) H=

.

1 (∇ × A − ζ E) μ

(3.49)

and E = −∇φe + iωA.

.

(3.50)

Insertion of Eqs. (3.49) and (3.50) into Eq. (3.48) leads to the result (with . = μ − ξ ζ ) ∇∇ · A − ∇ 2 A − iω(ζ − ξ )∇ × A = μJe + ∇(iω φe ) + ω2 A,

.

(3.51)

where .k 2 = ω2 . Using the gauge ∇ · A = iω φe ,

.

(3.52)

the above Helmholtz-like equation simplifies to (see Refs. [28] and [29]) ∇ 2 A + iω(ζ − ξ )∇ × A + k 2 A = −μJe .

.

(3.53)

Under these conditions, Lakhtakia [28] showed that the potential .A is generally trivalent, having two opposite helical components and one irrotational component, there being three distinct wavenumbers. Lakhtakia [28] also showed that the fields are birefringent. A similar analysis holds for .F but is omitted for brevity. Further simplification occurs if .ξ = ζ , which represents a nonreciprocal biisotropic medium (note, in general, .ξ¯¯ = −ζ¯¯ T is the condition for a reciprocal medium [6]). Under these conditions, the Helmholtz equation for .A becomes [30] ∇ 2 A + k 2 A = −μJe .

.

(3.54)

3 Scalar Potentials and Applications

17

A similar analysis (or using duality) is performed to obtain the bi-isotropic vector potential formulation for .F but is omitted for brevity. As with the simple medium case, the bi-isotropic vector potential formulation has the advantage of relatively simple Helmholtz-type equations. The restrictions are that the medium must be spatially invariant and nonreciprocal for the special case .ξ = ζ . It is important to note that there was some initial debate regarding the physical existence of nonreciprocal biisotropic media due to the Post constraint; see, for example, Lakhtakia and Weiglhofer [31] and Sihvola [32]. Recent research appears to further support the evidence against any significant nonreciprocity in a classical bulk biisotropic medium (see Refs. [33–37]). That is, if a bulk material is biisotropic, it is reciprocal. Finally, it is noted that this formulation reduces to the simple medium case if .ξ = ζ = 0.

3.2.3 General Bianisotropic Medium Using a six-vector and adjoint formalism, Lindell and Olyslager [38] developed a vector-potential-based method for a general bianisotropic medium. The vector potential Helmholtz-type equations developed are .

[(∇ × I¯¯ + iωξ¯¯ ) · μ¯¯ −1 · (∇ × I¯¯ − iωζ¯¯ ) − ω2 ¯¯ ] · A = Je

(3.55)

¯¯ · F = Jh . [(∇ × I¯¯ − iωζ¯¯ ) · ¯¯ −1 · (∇ × I¯¯ + iωξ¯¯ ) − ω2 μ]

(3.56)

and .

However, if these equations are multiplied by .iω and compared to Eqs. (3.6) and (3.8), it is realized that the vector potential Helmholtz-type equations are trivial scaled versions of the Helmholtz-type equations for .E and .H with a Coulomb gauge (.φe = φh = 0) evidently used (upon examination of Eqs. (3.12) and (3.16)). Thus, when solving problems involving a general bianisotropic medium, one is restricted to working directly with Maxwell equations since the vector potential formulation offers no advantage.

3.2.4 Gyrotropic Bianisotropic Media Although the vector potentials of the prior subsection offer no advantage to solving problems involving general bianisotropic media, Weiglhofer [4] pioneered a scalar potential formalism capable of handling a gyrotropic bianisotropic medium with a special class of spatial variation (z-dependence assumed here without significant loss of generality) having constitutive relations

18

M. Havrilla

D = (t I¯¯t + z zˆ zˆ − ig zˆ × I¯¯t ) · E + (ξt I¯¯t + ξz zˆ zˆ − iξg zˆ × I¯¯t ) · H

(3.57)

B = (ζt I¯¯t + ζz zˆ zˆ − iζg zˆ × I¯¯t ) · E + (μt I¯¯t + μz zˆ zˆ − iμg zˆ × I¯¯t ) · H.

(3.58)

.

and .

This is a major contribution, as the prior subsections revealed that, essentially, the vector potential methodology is only capable of handling simple media. That is, as soon as one leaves the realm of isotropy or spatial invariance, the vector potential technique fails (or does not offer substantial simplification). This is not the case for the scalar potential formalism. Following Weiglhofer [39], the first step in the scalar potential formulation is to decompose the Maxwell curl equations Eqs. (3.3) and (3.4) into transverse and longitudinal parts with the aid of Eqs. (3.57) and (3.58), leading to .

−ˆz×∇t Ez +ˆz×

∂Et = −Jht +iωμt Ht +ωμg zˆ ×Ht +iωζt Et +ωζg zˆ ×Et , ∂z ∇t × Et = −Jhz zˆ + iωμz Hz zˆ + ωζz Ez zˆ ,

.

.

∂Ht = Jet −iωt Et −ωg zˆ ×Et −iωξt Ht −ωξg zˆ ×Ht , ∂z

− zˆ ×∇t Hz + zˆ ×

(3.59) (3.60) (3.61)

and ∇t × Ht = Jez zˆ − iωz Ez zˆ − ωξz Hz zˆ .

.

(3.62)

The next critical step is to decompose, with the aid of the 2D Helmholtz theorem [39], the transverse fields and current densities into lamellar (transverse gradient) and rotational (transverse curl) parts, namely, .

Et = ∇t  + ∇t × (ˆzθ ) = ∇t  − zˆ × ∇t θ,

(3.63)

Ht = ∇t π + ∇t × (ˆzψ) = ∇t π − zˆ × ∇t ψ,

(3.64)

.

Jet = ∇t ue + ∇t × (ˆzve ) = ∇t ue − zˆ × ∇t ve ,

(3.65)

Jht = ∇t uh + ∇t × (ˆzvh ) = ∇t uh − zˆ × ∇t vh .

(3.66)

.

and .

Note, in real physical problems, the potentials ., θ, π, and .ψ and current densities ue , ve , uh , and .vh will decay at infinity. Thus, upon Fourier transformation with .∇t → −ikρ , it is seen that the transverse lamellar and transverse rotational parts are orthogonal. Hence, the vector Maxwell equations are scalarized into .

3 Scalar Potentials and Applications

19

transverse lamellar, transverse rotational, and longitudinal (z-directed) components (see Ref. [4]). This is one of the advantages of this scalar potential formulation. The other advantage is that these various components are easily visualized and can thus provide deeper physical insight (see Refs. [14] and [40]). This is especially important in excitation theory [14]. The other obvious advantage is that a wider class of materials can be accommodated, namely, media that are gyrotropic bianisotropic with spatial variation along z. The scalar potential formulation is realized upon substitution of Eqs. (3.63)– (3.66) into Eqs. (3.59)–(3.62) and equating the respective scalarized parts, leading to (see Ref. [39] for details and noting . t = t μt − ξt ζt and . z = z μz − ξz ζz )  .

L1 L2 L3 L4

    ψ s = 1 , s2 θ

(3.67)

where     μz t 2 t ∂ μt ∂ ω t ∂ μt ξg − μg ξt L1 = − ∇ − − μt z t μt ∂z t ∂z μt ∂z

t .   ,  μg (t μg − ξg ζt ) + ζg (μt ξg − μg ξt ) μt ζg − μg ζt ∂ 2 +ω − ω t − μt ∂z μt (3.68)     ξt ∂ ω t ∂ g μt − ξt ζg ξ z t 2 t ∂ − L2 = − ∇t − μt z μt ∂z t ∂z μt ∂z

t .   ,  μg (t ζg − g ζt ) + ζg (g μt − ξt ζg ) t μg − ξt ζg ∂ 2 ζt t −ω −ω − μt ∂z μt μt (3.69)     μt ζg − μg ζt

t ∂ μt μz t ue + ue + ω Jez s1 = − μt μt z μt ∂z t , (3.70) .     t μg − ξt ζg ξt iω t ξ z t

t ∂ uh + ω vh − Jhz + uh + μt μt μt z μt ∂z t    ∂ψ i = − ue ω(μt ξg −. μg ξt )ψ + ω(g μt − ξt ζg )θ + μt ∂z ω t   ∂θ +ξt + uh , ∂z

(3.71)

and Ez =

.

 1  2 ∇t (μz ψ + ξz θ ) + (μz Jez − ξz Jhz ) . iω z

(3.72)

20

M. Havrilla

Note that (.L3 , L4 , s2 , π, Hz ) is the dual of (.L2 , L1 , s1 , , Ez ), respectively, with the duality relations given as (.ψ, θ, , π ↔ θ, ψ, π, ), (.E, H ↔ H, E), and (., μ, ξ, ζ ↔ −μ, −, −ζ, −ξ ). Summarizing, a solution to .ψ and .θ is found, followed by calculation of . and .π , and ending in calculation of .E and .H. Several additional comments, important to practical implementation, are made in the next section.

3.3 Salient Features of the Scalar Potential Formulation As previously mentioned, the scalar potentials simplify the solution process and aid in physical insight due to the lamellar, rotational, and longitudinal decomposition. In addition, this formalism can accommodate gyrotropic bianisotropic media (and any material subclass) having spatial variation along z. In general, Eq. (3.67) represents coupled equations in .ψ and .θ , resulting in fourth-order solutions (dual field sets that are forward and reverse traveling waves). Decoupling into TE.z and TM.z modes occurs if the medium is uniaxial anisotropic (i.e., .ξ¯¯ = ζ¯¯ = 0 and .g = μg = 0) or if the medium is gyrotropic anisotropic with z-invariant solutions (i.e., .ξ¯¯ = ζ¯¯ = 0 and .∂/∂z ≡ 0). In order to use the potentials in practice, it is important to develop scalar-potential boundary conditions, discuss an unexpected depolarization tensor, and demonstrate how to handle the current density terms .ue , uh , ve , and .vh .

3.3.1 Boundary Conditions Once the potentials are computed, the vector fields .E and .H may be found, and vector boundary conditions (e.g., continuity of tangential electric field, etc.) applied in the usual manner. However, the solution process can be simplified (due to scalarization) if boundary conditions are developed in terms of the scalar potentials. As an example, consider continuity of tangential electric and magnetic fields at an infinite planar boundary (with surface unit normal along z), that is, Et1 = ∇t 1 − zˆ × ∇t θ1 = ∇t 2 − zˆ × ∇t θ2 = Et2

(3.73)

Ht1 = ∇t π1 − zˆ × ∇t ψ1 = ∇t π2 − zˆ × ∇t ψ2 = Ht2 .

(3.74)

.

and .

As discussed earlier, the transverse lamellar and transverse rotational components are orthogonal (for physically realizable potentials). In addition, since these boundary conditions must be satisfied for all x and y, it is concluded that  1 = 2 ,

.

(3.75)

3 Scalar Potentials and Applications

21 .

θ1 = θ2 ,

(3.76)

.

π1 = π2 ,

(3.77)

ψ1 = ψ2 .

(3.78)

and .

Similarly, the boundary condition .Etang = 0 at an infinite planar perfect electric conductor (PEC), with surface unit normal along z, results in 1 = 0

(3.79)

θ1 = 0.

(3.80)

.

and .

Other boundary condition relations may be derived similarly but are omitted for brevity.

3.3.2 Depolarization Tensors The vector longitudinal electric field .Ez , with the aid of Eq. (3.72), may be rewritten as Ez = zˆ Ez = zˆ

.

 1  2 ∇t (μz ψ + ξz θ ) + zˆ · (μz Je − ξz Jh ) . iω z

(3.81)

Thus, Eq. (3.81) (and its dual) reveals the expected depolarization tensors (see Ref. [41]), namely, μz zˆ zˆ , L¯¯ ee zz = iω z

(3.82)

ξz zˆ zˆ , L¯¯ eh zz = − iω z

(3.83)

.

.

L¯¯ hh zz =

.

z zˆ zˆ , iω z

(3.84)

and ζz zˆ zˆ . L¯¯ he zz = − iω z

.

(3.85)

22

M. Havrilla

Fig. 3.1 Source region gap and depolarization fields

These terms are present for carefully handling the source-point singularity by excavation of a slice gap (see Fig. 3.1). The current densities .Jez and .Jhz deposit charge at the slice gap, creating an artifact gap field that is absent in the actual source region. The depolarization fields cancel the gap fields, thereby rendering the original continuous source region, leading to mathematically and physically consistent results. However, unexpected depolarization tensors appear for the transverse current densities (which should not deposit charge at the slice gap). To identify these terms, the transverse lamellar electric field is found to be, with the aid of Eq. (3.71), Etl = ∇t  = ∇t n + ∇t d = Etln + Etld ,

.

(3.86)

where   ∂θ ∂ψ i + ξt ω(μt ξg − μg ξt )ψ + ω(g μt − ξt ζg )θ + μt ∂z ∂z ω t (3.87) is the non-depolarization contribution to the transverse lamellar electric field, and 

Etln = ∇t

.

Etld =

.

μt ξt μt ¯¯ ξt ¯¯ It · Jetl − It · Jhtl ∇t ue − ∇t uh = iω t iω t iω t iω t

(3.88)

3 Scalar Potentials and Applications

23

is the unexpected depolarization contribution to the transverse lamellar electric field. Here, .Jetl and .Jhtl are the transverse lamellar electric and magnetic current densities, respectively. Thus, the unexpected transverse depolarization tensors are (with the aid of duality) μt ¯¯ L¯¯ ee It , tt = iω t

(3.89)

ξt ¯¯ It , L¯¯ eh tt = − iω t

(3.90)

.

.

L¯¯ hh tt =

.

t ¯¯ It , iω t

(3.91)

and ζt ¯¯ It . L¯¯ he tt = − iω t

.

(3.92)

An asymptotic evaluation of Eq. (3.67) for large z with .∂/∂z → −ikz via Fourier transformation reveals ∂ψ ∼ ue ∂z

(3.93)

∂θ ∼ −uh . ∂z

(3.94)

.

and .

Inspection of Eq. (3.71) shows that this behavior is exactly what is needed to cancel the unexpected (and nonphysical) transverse depolarization tensors, again leading to mathematically and physically consistent results.

3.3.3 Current Densities The last remaining detail to discuss in implementing scalar potentials in practice is handling of the current densities .ue , .uh , .ve , and .vh . Taking the transverse divergence of the transverse electric current density .Jet leads to the relation (with the aid of Eq. (3.65)): ∇t · Jet = ∇t · Je = ∇t2 ue ,

.

which implies

(3.95)

24

M. Havrilla

−1 ue = ∇t2 ∇t · J e .

.

(3.96)

Note that the relations .∇t · (ˆzJez ) = 0 and .∇t · ∇t × (ˆzve ) = 0 have been used to obtain Eq. (3.96). Fourier transforming Eq. (3.96) and using .∇t → −ikρ lead to the transform-domain relations .

u˜ e =

ikρ ˜ · Je kρ2

(3.97)

u˜ h =

ikρ ˜ · Jh , kρ2

(3.98)

and .

ˆ ρ , .kρ2 = kx2 + ky2 , and duality was used to obtain where .kρ = xk ˆ x + yk ˆ y = ρk .u ˜ h . In a similar manner, taking the transverse curl of Eqs. (3.65) and (3.66) leads, respectively, to the spatial- and transform-domain current density relations

−1 ve = − ∇t2 zˆ × ∇t · Je ,

(3.99)

−1 vh = − ∇t2 zˆ × ∇t · Jh ,

(3.100)

.

.

.

v˜e = −

i zˆ × kρ ˜ · Je , kρ2

(3.101)

v˜h = −

i zˆ × kρ ˜ · Jh . kρ2

(3.102)

and .

The above transverse lamellar and rotational current density representations aid in developing Green functions, as is shown in the next section.

3.4 Application As an example of how scalar potentials are utilized in practice, material characterization of a uniaxial dielectric planar sample is performed using a clamped coaxial waveguide technique (see Ref. [42] for details and Fig. 3.2). A TEM.z wave is incident upon the uniaxial material region (i.e., the parallel-plate region between .z = 0 and .z = d) and scatters from the coaxial apertures causing higher order TM.z modes only (due to .φ-invariance) to be reflected and transmitted

3 Scalar Potentials and Applications

25

Fig. 3.2 Clamped coaxial waveguide fixture and uniaxial sample

in the coaxial regions. A pair of magnetic-field integral equations with coupled equivalent magnetic aperture current densities is formulated (via Love’s equivalence principle [43]) in order to develop theoretical expressions for the reflection and transmission coefficients. The uniaxial material properties are extracted via a rootsearch algorithm that iterates .t and .z until the difference between the experimental and theoretical reflection and transmission coefficients is minimized. The results of deducing the constitutive parameters of a uniaxial material having tetragonal symmetry using a clamped coaxial probe (CCP) are shown in Figs. 3.3 and 3.4 (along with error bars describing uncertainty in thickness and measured scattering parameters, as well as free space focused-beam measurements [44] for comparison). A crucial part of the analysis is obtaining the uniaxial parallel-plate Green function for the equivalent magnetic aperture surface current densities. The desired Green function can be derived using Maxwell equations directly; however, the analysis is greatly simplified via scalar potentials. As mentioned previously for uniaxial anisotropic media, the scalar potentials decouple into TE.z and TM.z field sets. For the TM.z case (required here), the scalar potential formulation reduces to (with .kt2 = ω2 t μt ) .

−

∂2 ∂ue t 2 t ∇t ψ − 2 ψ − kt2 ψ = − + Jez + iωt vh , z ∂z z ∂z   i ∂ψ . = − ue , ωt ∂z

(3.103)

(3.104)

26

M. Havrilla

Fig. 3.3 Real (a) and imaginary (b) relative transverse permittivity .rt = t /0

Fig. 3.4 Real (a) and imaginary (b) relative longitudinal permittivity .rz = z /0

E = ∇t  + zˆ

.

 1 2 ∇t ψ + Jez , iωz

(3.105)

and H = −ˆz × ∇t ψ.

.

(3.106)

The magnetic surface aperture current density, with the aid of Love’s equivalence principle, is given by .Jh = −ˆz × E = −ˆz × ∇t , which is a purely transverse rotational current density. Thus, for this application, only .vh is required (with .ue = 0 and .Jez = 0), and the scalar potential formulation further simplifies to

3 Scalar Potentials and Applications

−

.

27

∂2 t 2 ∇t ψ − 2 ψ − kt2 ψ = iωt vh , z ∂z =

.

i ∂ψ , ωt ∂z

E = ∇t  + zˆ

.

1 ∇ 2 ψ, iωz t

(3.107)

(3.108)

(3.109)

and H = −ˆz × ∇t ψ.

.

(3.110)

The desired tensor magnetic Green function is derived by first solving the Helmholtz equation for .ψ via superposition of principal and scattered solutions, namely, .

−

∂2 t 2 p ∇t ψ − 2 ψ p − kt2 ψ p = iωt vh z ∂z

(3.111)

t 2 s ∂2 ∇t ψ − 2 ψ s − kt2 ψ s = 0. z ∂z

(3.112)

and .

−

The principal and scattered solutions may be found via Fourier transformation and complex-plane analysis, leading to (with the aid of Eq. (3.101)) ψ˜ p (kρ , z) =



.

z

iωt ikzψ |z−z | zˆ × kρ · J˜ h (kρ , z )dz , e 2kzψ kρ2

ψ˜ s (kρ , z) = W + eikzψ z + W − e−ikzψ z ,

.

(3.113) (3.114)

and

kzψ =

.

kt2 −

t 2 k . z ρ

(3.115)

˜ ˜ = 0 and .(d) = 0 using the Enforcing the PEC boundary conditions .(0) transform-domain version of Eq. (3.108) leads to ˜ ρ , z) = .ψ(k where

z

˜ ρ , z|z ) · J˜ h (kρ , z )dz , G(k

(3.116)

28

M. Havrilla 



˜ ρ , z|z ) = −ωt cos[kzψ (d − |z − z |)] + cos[kzψ (d − z − z )] zˆ × kρ . G(k 2kzψ kρ2 sin(kzψ d) (3.117) The desired tensor magnetic Green function due to a transverse rotational magnetic current density (and required for the coupled magnetic field integral equation to ˜ = extract the material properties) is now found with the aid of Eq. (3.110), with .H i zˆ × kρ ψ˜ in the transform domain as follows: .

˜ ρ , z) = .H(k

z

¯¯˜  ˜ h (kρ , z )dz , G(k ρ , z|z ) · J

(3.118)

where cos[kzψ (d − |z − z |)] + cos[kzψ (d − z − z )] ¯¯˜  ˆ × kρ zˆ × kρ . G(k ρ , z|z ) = z i2ω−1 t−1 kzψ kρ2 sin(kzψ d) (3.119) Inverse Fourier transform of Eq. (3.118) leads to the spatial version of the desired Green function, the details of which are omitted for brevity. Thus, it is seen that the desired Green function is obtained with minimal effort when using the scalar potential formulation. In addition, it is clear from Eq. (3.119) (and noting that .zˆ ×kρ is associated with transverse curl in the transform domain) that a transverse rotational magnetic current density maintains a transverse rotational magnetic field, as expected physically. It is also clear from the cosine terms, standing waves will exist along the z-direction, and the sine term set to zero indicates poles for the various parallel-plate modes, again, as physically expected. Finally, the .kρ2 term in the denominator of Eq. (3.119) appears to be nonphysical (.kρ = 0 would suggest no radial propagation, but there should be in a radially unbounded parallel-plate structure). After noting that .kρ = ρk ˆ ρ , it is seen from the numerator terms in Eq. (3.119) that the .kρ2 terms cancel; thus the poles at .kρ = 0 are removable. Accordingly, so long as the analysis is handled carefully, mathematically and physically consistent results will be obtained in practice. .

3.5 Conclusion Field- and potential-based methods of analysis were discussed and compared. It was shown that constitutive tensors are the critical factors that influence the solution process. If a simple medium is considered, one can use a field-based approach (i.e., work with Maxwell equations directly), a vector potential method, or a scalar potential formulation. If a gyrotropic bianisotropic medium with spatial variation along z is considered, then either a field- or scalar-potential-based approach can be used (but not vector potentials). Finally, for general bianisotropic media, only the field-based approach is available.

3 Scalar Potentials and Applications

29

It was shown that the scalar potential formalism, pioneered by Weiglhofer [4], can substantially reduce mathematical effort and also greatly enhance physical insight. Salient features of the scalar potential formulation were discussed to aid in applications. An example of material characterization of a uniaxial sample was given to highlight the simplicity and insight gained by the scalar potential formalism. A common practice in electromagnetics education is to assume simple media, briefly discuss Maxwell equations and plane waves, and introduce vector potentials as quickly as possible (e.g., Refs. [45] and [46]). The problem with this approach is that once one leaves the realm of isotropy or spatial homogeneity, the vector potential method fails (or is of minimal use). A student is then at a big disadvantage and will have a steep learning curve when complex-media environments are encountered. It is the opinion of this author that the scalar potential formalism should be taught early on to better prepare students and researchers for modern electromagnetic applications.

Epilog This author never had the great fortune of meeting or interacting with Professor Werner Weiglhofer. Nevertheless, his pioneering work on scalar potentials had a profound influence on this author’s research. Professor Weiglhofer built a legacy that very few individuals are capable. Disclaimer The views expressed in this article are those of the author and do not reflect the official policy or position of the United States Air Force, Department of Defense, or the U.S. Government.

References 1. Garcia, C.R., Correa, J., Espalin, D., Barton, J.H., Rumpf, R.C., Wicker, R., Gonzalez, V.: 3D printing of anisotropic metamaterials. Progress in Electromagnetics Research Letters 34, 75–82 (2012) 2. Mackay, T.G., Lakhtakia, A.: Electromagnetic Anisotropy and Bianisotropy: A Field Guide. World Scientific, New Jersey (2010) 3. Peterson, A.F., Ray, S.L., Mittra, R.: Computational Methods for Electromagnetics. IEEE Press, New York (1998) 4. Weiglhofer, W.S.: Scalar Hertz potentials for linear bianisotropic mediums. In: Singh, O.N., Lakhtakia, A. (eds.) Electromagnetic Fields in Unconventional Materials and Structures, chap. 1, pp. 1–37. Wiley, New York (2000) 5. Weiglhofer, W.S., Hansen, S.O.: Faraday chiral media revisited. I. Fields and sources. IEEE Trans. Antennas Propag. 47(5), 807–814 (1999) 6. Kong, J.A.: Theorems of bianisotropic media. Proc. IEEE 60(9), 1036–1046 (1972) 7. Neumann, F.E.: Allgemeine gesetze der inducirten elektrischen ströme. Annalen der Physik 143, 31–44 (1846)

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8. Nevels, R., Shin, C.-S.: Lorenz, Lorentz, and the gauge. IEEE Antennas Propag. Mag. 43(3), 70–71 (2001) 9. Poisson, S.D.: Mémoire sur La Théorie du magnétisme. Mémoires de l’Académie Royale des Sciences, Paris (1824) 10. Lorenz, L.V.: XXXVIII. On the identity of the vibrations of light with electrical currents. Philosophical Magazine and Journal of Science 34(230), 287–301 (1867) 11. Lorentz, H.A.: La théorie Électromagnétique de Maxwell et son application aux corps mouvants. Archives Néerlandaises des Sciences Exactes et Naturelles 25, 363–552 (1892) 12. Van Bladel, J.: Electromagnetic Fields. McGraw-Hill Book Company, New York (1964) 13. Van Bladel, J.: Lorenz or Lorentz. IEEE Antennas Propag. Mag. 33(2), 69 (1991) 14. Collin, R.E.: Field Theory of Guided Waves, 2nd edn. IEEE Press, New York (1991) 15. Hertz, H.R.: Die kräfte elektrischer schwingungen behandelt nach der Maxwell’schen theorie. Annalen der Physik und Chemie 36(1), 1–22 (1889) 16. Essex, E.A.: Hertz vector potentials of electromagnetic theory. Am. J. Phys. 45(11), 1099–1101 (1977) 17. Nisbet, A.: Hertzian electromagnetic potentials and associated gauge transformations. Philos. Trans. R. Soc. Lond. Ser. A 231, 250–263 (1957) 18. Lahart, M.J.: Use of electromagnetic scalar potentials in boundary value problems. Am. J. Phys. 72(1), 83–91 (2004) 19. Debye, P.J.W.: Der lichtdruck auf kugeln von beliebigem material. Annalen der Physik 30, 57–136 (1909) 20. Wilcox, C.H.: Debye potentials. Journal of Mathematics and Mechanics 6(2), 167–201 (1957) 21. Gray, C.G., Nickel, B.G.: Debye potential representation of vector fields. Am. J. Phys. 46(7), 735–736 (1978) 22. Nisbet, A.: Source representations for Debye’s electromagnetic potentials. Physica (Utrecht) 21, 799–802 (1955) 23. Bromwich, T.J.I’a.: X. Electromagnetic waves. Philosophical Magazine and Journal of Science 38, 143–164 (1919) 24. Bromwich, T.J.I’a.: V. The scattering of plane electric waves by spheres. Philos. Trans. R. Soc. Lond. A 220, 175–206 (1920) 25. Gouesbet, G., Grehan, G.: Sur la généralisation de la théorie de Lorenz-Mie. J. Opt. (Paris) 13(2), 97–103 (1982) 26. Lindell, I.V., Sihvola, A.H., Tretyakov, S.A., Viitanen, A.J.: Electromagnetic Waves in Chiral and Bi-Isotropic Media. Artech House, Boston (1994) 27. Sihvola, A.H.: Electromagnetic modeling of bi-isotropic media. Prog. Electromagn. Res. 9(1), 45–86 (1994) 28. Lakhtakia, A.: Trirefringent potentials for isotropic birefringent media. Int. J. Appl. Electromagn. Mater. 3, 101–109 (1992) 29. Chambers, LL.G.: Propagation in a gyrational medium. Q. J. Mech. Appl. Math. 9(3), 360–370 (1956) 30. Lindell, I.V.: Methods for Electromagnetic Field Analysis. IEEE Press, New York (1992) 31. Lakhtakia, A., Weiglhofer, W.S.: Are linear, nonreciprocal, biisotropic media forbidden? IEEE Trans. Microw. Theory Tech. 42(9), 1715–1716 (1994) 32. Sihvola, A.H.: Are linear, nonreciprocal, biisotropic media forbidden indeed? IEEE Trans. Microw. Theory Tech. 43(12), 2160–2162 (1995) 33. Lakhtakia, A., Mackay, T.G., Chiadini, F., Diovisalvi, A., Fiumara, V., Scaglione, A.: How much topological insulation does one need? How much can one get? In: Proceedings of the 2017 International Conference on Electromagnetics in Advanced Applications (ICEAA), pp. 729–732 (2017). https://doi.org/10.1109/ICEAA.2017.8065351 34. Krowne, C.M.: Quantum oscillator argument against nonreciprocity in linear bi-isotropic media. Microw. Opt. Technol. Lett. 16(5), 312–315 (1997) 35. Krowne, C.M.: Marginal nonreciprocity in linear bi-isotropic media. Microw. Opt. Technol. Lett. 18(5), 356–359 (1998)

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36. Lakhtakia, A., Mackay, T.G.: Classical electromagnetic model of surface states in topological insulators. J. Nanophotonics 10(3), 1–5 (2016) 37. Schultz, J., Nogueira, F.S., Buchner, B., van den Brink, J., Lubk, A.: Axion Mie theory of electron energy loss spectroscopy in topological insulators. SciPost Physics Core 4, 1–26 (2021) 38. Lindell, I.V., Olyslager, F.: Potentials in bianisotropic media. Journal of Electromagnetic Waves and Applications 15(1), 3–18 (2001) 39. Weiglhofer, W.S.: Scalarisation of Maxwell’s equations in general inhomogeneous bianisotropic media. IEE Proc. H 134(4), 357–360 (1987) 40. Collin, R.E.: Foundations for Microwave Engineering, 2nd edn. Wiley, New Jersey (2001) 41. Hanson, G.W.: A numerical formulation of dyadic Green’s functions for planar bianisotropic media with application to printed transmission lines. IEEE Trans. Microw. Theory Tech. 44(1), 144–151 (1996) 42. Hyde IV, M.W., Havrilla, M.J., Bogle, A.E.: Nondestructive determination of the permittivity tensor of a uniaxial material using a two-port clamped coaxial probe. IEEE Trans. Microw. Theory Tech. 64(1), 239–246 (2016) 43. Harrington, R.F.: Time-Harmonic Electromagnetic Fields. IEEE Press, New York (2001) 44. Ghodgaonkar, D.K., Varadan, V.V., Varadan, V.K.: A free-space method for measurement of dielectric constants and loss tangents at microwave frequencies. IEEE Trans. Instrum. Meas. 37(3), 789–793 (1989) 45. Balanis, C.A.: Advanced Engineering Electromagnetics, 2nd edn. Wiley, New Jersey (2012) 46. Lewin, L.: Theory of Waveguides; Techniques for the Solution of Waveguide Problems. Wiley, Hoboken (1975)

Michael J. Havrilla received B.S. degree in Physics and Mathematics in 1987, the M.S.E.E degree in 1989, and the Ph.D. degree in electrical engineering in 2001 from Michigan State University, East Lansing, MI. He is currently a Professor in the Department of Electrical and Computer Engineering at the Air Force Institute of Technology, Wright-Patterson Air Force Base, OH. He is a Fellow of the Antenna Measurement Techniques Association, and his current research interests include electromagnetic theory, bianisotropic media, and quantum field theory.

Chapter 4

A Novel Approach to Electromagnetic Constitutive Relations Martin W. McCall, Paul Kinsler, and Jonathan Gratus

4.1 Introduction Maxwell’s electrodynamics is, by fairly common consent, the greatest classical theory of all time. Even its successor, quantum electrodynamics, is built heavily on applying quantum field theoretic principles to Maxwell’s equations. So let us start with the most basic statement of the theory, namely Maxwell’s equations in their microscopic form, i.e., ∇ · B = 0,

∇ ×E=−

.

∇ ×B=

1 ∂E + μ0 j c2 ∂t

and

∇ ·E=

∂B , ∂t

.

ρ . 0

(4.1) (4.2)

These equations describe in every classical detail the intimate interplay of the fundamental electric and magnetic fields .E and .B with charges and currents of densities .ρ and .j. For many problems, particularly those involving a small number of charges confined to small volumes, this is all we need. However, most practical situations involve vast numbers of charges distributed over macroscopic volumes, and it is customary to replace Eq. (4.2) with ∇ ×H=

.

∂D + jf ∂t

and

∇ · D = ρf

(4.3)

M. W. McCall () · P. Kinsler Imperial College London, London, UK e-mail: [email protected]; [email protected] J. Gratus Lancaster University, Bailrigg, UK e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 T. G. Mackay, A. Lakhtakia (eds.), Adventures in Contemporary Electromagnetic Theory, https://doi.org/10.1007/978-3-031-24617-3_4

33

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in which the excitation fields1 .D and .H attempt to take account of the medium’s modification in the presence of .E and .B, and the charge and current densities are restricted to free charges and currents. In a dielectric for example, one model is for the .E-field to induce local charge separations that can be accounted for by a bound charge density .ρb . By setting .D = 0 E + P, where .P is the polarization induced by .E, it is readily seen from Eq. (4.3)(b) that Eq. (4.2)(b) is recovered provided .−∇ · P = ρb , and .ρ = ρb + ρf is understood to be the total charge density, bound plus free. Analogously, by setting .H = μ−1 0 B − M, the effect of .B setting up local magnetic dipoles .M inducing a magnetization can be accounted for. Equation (4.2)(a) is recovered from Eq. (4.3)(a) provided the total current density .j = jb + jf , where the bound current density .jb = ∇ × M + ∂P/∂t. Often the next step is to make some assumption as to how .E and .B induce .P and .M. Assuming linearity for example, so that .P = 0 χe E and .M = μ−1 0 χm B (where the constants .χe and .χm are known, respectively, as the electric and magnetic susceptibility), we are led to the simple constitutive relations D = E

.

H = μ−1 B ,

and

(4.4)

where . = 0 (1 + χe ), and .μ−1 = μ−1 0 (1 − χm ). It is important to emphasize that constitutive relations, defined as any relationship linking the fundamental fields .E and .B to the excitation fields .D and .H, are crucial in solving the Maxwell set comprising equations, Eqs. (4.1) and (4.3), which are otherwise under-determined. Constitutive relations come in many guises, the above scalar idealized instantaneous linear response example above being the simplest. Birefringence may be accounted for by replacing . with a tensor, for example. In order to model media that respond non-locally in space and time, Eq. (4.4) must be replaced with convolution integrals. A dielectric medium whose response depends on the history of the stimulating electric field can be modelled via  t .D(t) = (t − τ )E(τ )dτ . (4.5) −∞

Taking the temporal Fourier transform of Eq. (4.5) yields, with the aid of the convolution theorem, ˜ ˜ D(ω) = ˜ (ω)E(ω) ,

.

(4.6)

where the tilde denotes quantities in the temporal frequency (.ω) domain. Equation (4.6) shows that a non-instantaneous response is associated with temporal

1 Whilst .D

is universally known as the displacement field, there does not appear to be a consensus in the designation of .H. This field is variously called ‘magnetic field intensity’ or ‘magnetizing field’ or even just ‘magnetic field’. In collectively designating .D and .H as ‘excitation fields’, we follow the nomenclature of Hehl and Obukhov (cf. [1], p.116).

4 A Novel Approach to Electromagnetic Constitutive Relations

35

dispersion of the constitutive parameters. For an idealized instantaneously responding medium, .(t) = δ(t), and Eq. (4.5) reduces to D(t) = E(t) ,

.

(4.7)

showing that the characterization of Eq. (4.4) is that of an idealized instantaneously responding medium. Similar arguments can be given for media that respond nonlocally in space, giving rise to spatial dispersion. Although an instantaneously, locally responding medium is an idealization, it is a useful one for our consideration of whether a consistent scheme for macroscopic media can be developed solely in terms of .E and .B. Accordingly, in the remainder of this chapter, we consider nondispersive media, unless otherwise stated. The most general linear response of an instantaneously responding medium is given by  .

D H



 =

 α β μ−1

  E , B

(4.8)

in which ., μ−1 , α, and .β are all tensors. The explosive expansion in metamaterials research over the last two decades might usefully be summarized as our ability to manufacture media that access the entire gamut of responses idealized and embraced by Eq. (4.8). It is also important to emphasize that there is considerable flexibility in specifying constitutive relations. For one thing, it is possible for distinct constitutive models to yield identical measurable properties. In Sect. 4.2, we will give a simple example where two distinct constitutive models for a magnetic medium lead to indistinguishable physical results. Moreover, it is noteworthy that Maxwell’s equations give no handle on the measurement of any of the fields .E, .B, .D, and .H, since Maxwell’s equations are invariant under E → E + ∇ EB −

.

∂ EB . ∂t

B → B + ∇ × EB . ∂ DH . ∂t D → D + ∇ × DH ,

H → H − ∇ DH +

(4.9) (4.10) (4.11) (4.12)

where . EB and . DH are arbitrary scalar fields and .EB and .DH are arbitrary vector fields. However, .E and .B are in principle measurable through the Lorentz force law experienced by any charge q moving with velocity .v: F = q (E + v × B) .

.

(4.13)

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Even when charges are moving within media, rendering direct use of the Lorentz force law problematic, the fields .E and .B may be measured non-locally via the Aharonov–Bohm effect [2–4]. Such direct measurability of .E and .B removes the gauge freedom given in Eqs. (4.9) and (4.10) and therefore necessarily . EB = 0 and .EB = 0. There is by contrast no well-established Lorentz force law, or Aharonov–Bohm effect, that enables the direct measurement of the excitation fields .D and .H. Even where these have been proposed [5], the forces specified include those on magnetic monopoles that have yet to be observed. Instead, the constitutive relations offer a conduit to the measurement of .D and .H via measurement of .E and .B. However, as we have seen, constitutive relations are model-dependent, so that the gauge freedom of .D and .H (cf. Eqs. (4.11) and (4.12)) remains for these fields. For these reasons in this chapter, we explore the consequences of regarding .D and .H as fields that are not equal in status to .E and .B, but rather having the status of convenient computational tools. This view is not universally held and indeed the role and meaning of the excitation fields .D and .H has been a subject of debate, with arguments for and against their independent existence and measurability [6– 10]. Jackson’s classic text [11] states that .D and .H are ‘derived fields’, used as a ‘convenience’, but does not discuss their measurability. On the other hand, there is a wide literature that treats all four fields as having the same status, including established textbooks such as [12], pre-metric electrodynamics [1], and others [13]. Some distinguished authors, Feynman for example [14], eschew the use of the excitation fields .D and .H altogether. Given this debate, the question we address in this chapter is whether the introduction of the .D and .H fields along with constitutive models might be dispensed with [15]. A macroscopic Maxwell theory would then consist of replacing Eqs. (4.2) with constitutive relations acting directly on .E and .B. This is precisely the question we explore in Sect. 4.3. How might such a relationship be constrained? There really can only be one answer. The local conservation of charge, i.e., ∇ · jf +

.

∂ρf = 0, ∂t

(4.14)

which is a derivable consequence of Eqs. (4.2), must be so of any potential replacement of Eqs. (4.2). Armed with this new approach to macroscopic electrodynamics, we explore the extent to which metamaterials research is potentially enriched by having a much broader canvas on which to create synthetic electromagnetic responses. We focus on the simplest possibilities which do not involve field derivatives, noting that these are analogous to traditional electromagnetic axions. We show how such responses may be emulated and implemented in media. In Sect. 4.4, we take a most radical step. We argue that the constitutive equations of vacuum (i.e., .D = 0 E and .H = μ−1 0 B) are subject to the same ill-definedness that bedevils the constitutive relations in matter [16, 17]. Then, reasoning much as before, we are led to a Maxwell theory of vacuum that supports a new kind of axion field that we call a topological axion. Such a field has many intriguing

4 A Novel Approach to Electromagnetic Constitutive Relations

37

properties. We focus on just one, namely through its presence, the link between local and global charge conservation can be broken. Such a manoeuvre requires delicate manipulation of the proposed topological axion field on a manifold containing a temporary singularity. We give an explicit electromagnetic solution that demonstrates these features. Equations (4.1), (4.2), and (4.3) are Maxwell’s micro/macro equations expressed in Heaviside 3-vector notation. Interestingly, although this is the ‘textbook’ form of the equations, it is not the simplest form. One might argue that this is a historical accident in that Maxwell’s equations were born before the discovery of the advent of special relativity. Had these discoveries been made in the opposite order, then it is arguable that textbooks would show Maxwell’s equations in their much simpler spacetime symmetric form expressed in the language of differential forms. In form notation, Eqs. (4.1) and (4.2) are expressed as dF = 0

and

.

d F =J ,

(4.15)

where the electric and magnetic fields are subsumed into the so-called Faraday 2form, F : F = E ∧ dt + B ,

.

(4.16)

and the charge and current density are subsumed into the current density 3-form, J : J = −j ∧ dt + ρ˜ .

.

(4.17)

In the context of this chapter, the Hodge star . appearing in Eq. (4.15)(b) can be regarded as representative of a vacuum constitutive relation. In Sect. 4.4, we use form notation as it is the easiest way to formulate the questions of charge conservation on spacetime manifolds. The readers unfamiliar with the use of differential forms in electromagnetism are directed to many standard texts, for example, [18–20] or to [21] for a recent undergraduate text. In addition, we have provided an appendix which gives an introduction to form notation.

4.2 Two Models of a Permanent Magnet The apparent absence of magnetic monopoles exposes the role of constitutive relations quite clearly. Whereas the electrical response of a medium can be intuitively modelled via separation of charges leading to bound dipoles, the absence of isolated magnetic charges implies that the magnetic response can be modelled in different ways, such as via induced magnetic dipoles or via current loops. Here, we consider these two possibilities under the names bulk and surface current models [22]. We will see that these yield distinct models for the magnetic excitation field .H but are experimentally indistinguishable.

38

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Fig. 4.1 The relationship between the component of the bulk .HB field and the magnetic field .B for the bulk model, Eq. (4.18). The red lines show the value of the two fields when no external magnetic field is applied. Reproduced from [22] under the terms of the Creative Commons CC BY licence

4.2.1 Bulk Model Here, the magnetic properties of the material are assumed to be dispersed continuously throughout its volume, along with the coercive field intensity .Hc , defined as the applied field strength at which the magnet’s polarity changes. The magnetic constitutive relation for .H = HB is given by HB =

.

B − Hc , μ

(4.18)

where .μ > 0 is the constant permeability of the magnetic material. From Fig. 4.1, neither the coercive field intensity .HB = Hc nor the remanence field .B = Br is achievable without an externally applied magnetic field. The remanence field, which remains after any applied field is removed, is given by .Br = μHc . Away from other external magnetic fields, the magnitude of .HB can take any value between zero and B .|Hc |. The field within the magnet, .H , is oppositely directed to .B, as shown in Fig. 4.2.

4.2.2 Surface-Current Model In this model, the magnetic properties of the material are once again the result of bulk properties dispersed throughout its volume, but the coercive field intensity .Hc is instead treated as being due to a surface current .σbS . If the bulk permeability is .μ, then the magnetic constitutive relation for .H = HS is given by HS =

.

B and σbS = n × Hc , μ

(4.19)

4 A Novel Approach to Electromagnetic Constitutive Relations

39

Fig. 4.2 Bulk magnetism: a cross section of the magnetic fields inside and outside a permanent magnet. The dashed blue lines are the .B-field, with the direction indicated by the blue arrows. The red lines are the surfaces orthogonal to the .H-field, with the direction of .H indicated by the red arrows. The vacuum .H-field (i.e., .Hvac ) is parallel to the .B-field, whereas in the medium the .H field (i.e., .HB ) is anti-parallel to .B. However, the tangential .HB is continuous as one crosses the boundary of the magnet. Reproduced from [22] under the terms of the Creative Commons CC BY licence

where .n is the outward-pointing normal to the surface of the magnet. In this model, HS is in the same direction as .B, as shown in Fig. 4.3. There is a discontinuity in .HS due to the surface current, i.e.,

.

n × [H] = σbS ,

.

(4.20)

where .[H] = HS − Hvac with .Hvac being the induced magnetic field in vacuum.

4.2.3 Comparison We see on comparing Figs. 4.2 and 4.3 that the bulk and surface current models give distinct predictions. The direction of the induced magnetic field .H in the medium reverses as we switch between models. If we could measure .H in the bulk medium, we could tell which model is physically correct. However, since we can only measure .(E, B), Maxwell’s macroscopic equations (Eqs. (4.1) and (4.3)) are insufficient to enable us to distinguish between these models. Importantly, for the current discussion, the difference in .H between the two models can be bridged by a gauge contribution:

40

M. W. McCall et al.

Fig. 4.3 Surface current magnetism: a cross section of the magnetic fields inside and outside a permanent magnet. The dashed blue lines are the .B-field, with the direction indicated by the blue arrows. The red lines are the surfaces orthogonal to .H-field, with the direction of .H indicated by the red arrows. The vacuum .H-field (i.e., .Hvac ) is parallel to the .B-field, just as in the medium where the .H-field (i.e., .HS ) is parallel to .B. However, there is a discontinuity in tangential .H at the boundary of the magnet, where .Hvac and .HS do not match up. The . and .⊗ indicate the direction of the surface currents, as they either come out of or go into the page. Reproduced from [22] under the terms of the Creative Commons CC BY licence

HB = HS − Hc = HS + ∇ψ ,

.

(4.21)

where .ψ is an inhomogeneous field given by .ψ = −y |H|, y being the coordinate along the axis of the magnet.

4.3 A New Approach to Electromagnetic Constitutive Relations Having posited the non-physicality of the excitation fields and seen a specific example where .H is not uniquely defined, we can start to contemplate an alternative approach to describing the electromagnetic response of media that dispenses with .D and .H altogether [15]. To begin with, we re-write Eqs. (4.3) as j=∇ ×H−

.

∂D ∂t

and

ρ = ∇ · D,

(4.22)

where the f subscripts have been dropped to emphasize that in this approach we will only be dealing with free charges and currents. The strategy now is to note that

4 A Novel Approach to Electromagnetic Constitutive Relations

41

the right-hand sides of both equations consist of first-order operators acting on the excitation fields. Now, with constitutive relations dispensed with, it is proposed to replace the right-hand sides of both equations with first-order operators acting on the fundamental fields .E and .B. Symbolically, .

ρ = ρE E + ρB B

(4.23)

j = jE E + jB B .

(4.24)

and .

The angle brackets indicate a general first-order operator and contain information about the response of the medium. We call Eqs. (4.23) and (4.24) the combined Maxwell and constitutive relation (CMCR) equations. Of course, the first-order operators . ρE , ρB , jE , and . jB are severely constrained in that: A. They must reproduce Eqs. (4.2) in vacuum. B. They must respect local charge conservation, Eq. (4.14). Let us first write down the general first-order operator expression for . ρE E :  i  ij ∂E  0j ∂E j j + ρE

ρE E = ρE Ei + ρE , ∂t ∂x i

.

(4.25)

where we have used the summation convention on repeated indices and assumed spatial coordinates .x j (.j = 1, 2, 3). It turns out that the above form for the firstorder operator is fully equivalent to the following three coordinate-free linearity relations:   E 2 E 2 E . ρ f E = 2f ρ f E − f ρ E , . (4.26)

ρE E1 + E2 = ρE E1 + ρE E2 ,

(4.27)

and

ρE λE = λ ρE E ,

.

(4.28)

where f is a scalar field and .λ is a real constant. The above three equations, taken together, also imply Eq. (4.25). Before any constraints are applied, . ρE and . ρB each have 15 components. This is apparent from Eq. (4.25) where the first term on the right side has 3 components, the second term similarly has 3 components, and the last term has 9 components. Regarding . jE and . jB , these each have 45 components, 15 for each of the components of .j. There is thus a total of .15 + 15 + 45 + 45 = 120 constitutive components. Once the constraints A and B are applied, it turns out, as was shown

42

M. W. McCall et al.

in [15], that the number of independent components is reduced to 55. This is a considerable reduction, although when compared with Eq. (4.8), which can supply up to 36 components before any symmetries are applied, it is clear that the approach developed here potentially provides a greater range of electromagnetic responses.

4.3.1 Traditional Axionic Electromagnetic Response Let us detour for a moment to review the traditional axionic response of an electromagnetic medium. An idealized (i.e., lossless, non-dispersive) axionic medium is described by the constitutive relations D = κax B

and

.

H = −κax E ,

(4.29)

where the axionic response is represented by the scalar .κax . Inserting Eq. (4.29) into Maxwell’s equations, Eqs. (4.1) and (4.3), gives the contributions .ρax and .jax to the charge and current density due to the axionic field as ρax = (∇κax ) · B and

.

jax = −∇κax × E − (∂t κax ) B .

(4.30)

4.3.2 Axionic Response Within the CMCRs Let us now examine the simplest medium response that can be generated using first-order operators. If just the lowest order terms in the CMCRs of Eqs. (4.23) and (4.24) are retained, then Eqs. (4.23) and (4.24) become, in components:  i  i ρax = ρE Ei + ρB Bi

(4.31)

.

and  ij  ij i jax = JE Ej + JB Bj ,

.

i, j = 1, 2, 3.

(4.32)

Now, it turns out (cf. [15]) that when we apply the constraint of local charge conservation, the coefficients appearing in Eqs. (4.31) and (4.32) are distilled into just four numbers, three of which form the components of a vector .ξ , together with a scalar .ξt . Equations (4.31) and (4.32) then reduce to ρax = ξ · B

.

and

(4.33)

4 A Novel Approach to Electromagnetic Constitutive Relations

jax = −ξ × E − ξt B .

.

43

(4.34)

In fact, the charge conservation constraint goes one step further; if the electromagnetic fields are arbitrary, .ξt and .ξ must further satisfy ∇ × ξ = 0 and ∇ξt = ∂t ξ .

.

(4.35)

These further imply that .ξt and .ξ must be derivable from a potential .κax (t, r) satisfying ξ = ∇κax and ξt = ∂t κax ,

.

(4.36)

whence it is clear from comparing Eq. (4.30) with Eqs. (4.33) and (4.34) that the traditional axionic response is recovered. It is important to emphasize that the expressions in Eq. (4.36) are too restrictive for use in many situations, e.g., such as those involving symmetries, where .E and .B may only have specific orientations with non-zero components. In such cases, it may be no longer possible to derive .ξt and .ξ from a single potential, and novel axion-like responses are then possible. We give an example of this in the next section.

4.3.3 Emulating an Axionic Response In this section, we demonstrate how the axionic response considered above could be emulated in the laboratory [15]. Consider the infinite conducting cylinder illustrated in Fig. 4.4. The cylinder, which is charged and thus generates a radial electric field, is surrounded by an array of parallel wires as shown in Fig. 4.4a. With the cylinder and the wires aligned to the z-axis, the proposed axionic response is characterized via ξ=

.

Z0 ˆ θ and ξt = 0 , r

(4.37)

where .Z0 is a constant and is illustrated in Fig. 4.4b. Here, .(r, θ, z) are cylindrical polar coordinates, and .θˆ is a unit vector in the azimuthal direction. The collective medium will be shown to have achieved such a response by generating an axial current from a purely radial electric field, as dictated by Eq. (4.34). A critical point to be made here is the distinction between the above medium response and those achievable via conventional constitutive parameters. For the above response, .ξ cannot be derived from the gradient of a scalar field as in Eq. (4.36). Although locally one can set .κax = Z0 θ , a globally valid .κax cannot be defined, since globally .θ is not single valued. It would therefore be impossible to model this medium response using traditional constitutive relations (Eq. (4.8)) in Maxwell’s equations (Eqs. (4.1) and (4.3)). However, it is possible to emulate

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Fig. 4.4 An axionic response .ξ = ξθ θˆ (arrows) outside a charged conducting cylinder, shown in the cross section of the plane perpendicular to the cylinder. Emulating the .−ξ × E term using a conducting metal cylinder (green) surrounded by a radial array of wires (pink), with a few wires visually emphasized in order to clarify the setup. A series of voltmeters measure the radial electric fields (e.g., .V1 , .V2 , .. . .) near each wire, and the resulting information is used to control a current source that drives currents along those wires (e.g., .|J1 | ∝ V1 /r, .|J2 | ∝ V2 /r, .. . .). As a result, these actively monitored and driven wires will act as a metamaterial, modifying the electromagnetic field as if a constitutive axionic response .ξ were present. Reproduced from [15]

the response using the scheme of Fig. 4.4. A series of voltmeters (e.g., .V1 , .V2 , . . . in Fig. 4.4a) measure the radial electric fields near each wire, and the resulting information is used to control a current source that drives currents along those wires (e.g., .|J1 | ∝ V1 /r, .|J2 | ∝ V2 /r, .. . .). As a result, these monitored and driven wires will act collectively as an active metamaterial, modifying the electromagnetic field as if a constitutive axionic response .ξ were present. Of course were the charge on the cylinder to vary over time, then in order to remain causal, such variation would need to be on a much slower timescale than the reaction time of the active measurement and current generation processes.

.

4.3.4 A Metamechanical Implementation of the Axionic Response Beyond the artificial ‘active’ scheme above, we can next speculate on whether the axionic response considered here can be achieved passively [15]. While it is hard to conceive of such a response being achieved purely electromagnetically (although the possibility is not ruled out), we here show how the response can be approximated opto-mechanically. The basic unit is illustrated in Fig. 4.5 and

4 A Novel Approach to Electromagnetic Constitutive Relations

45

Fig. 4.5 Diagram of an auxetic current-generating axionic response .ξz zˆ , based on a shape whose core stretches (or shrinks) in x whilst simultaneously expanding (or contracting) in y. When an oscillating electric field .Ex is applied, the ‘detector’ charges (blue circles) are pushed together (or pulled apart) horizontally, so that the ‘response’ charges (red squares) are moved together (or apart) vertically. The y-direction current produced by those moving response charges is the axionic response; not shown are the mechanisms that return the system to a default shape when the applied field is removed; for a construction with sufficiently flexible corners, this could be provided by the elastic properties of the material. Reproduced from [15]

consists of a frame with charges embedded as shown. An oscillating electric field applied in the x-direction causes the ‘detector’ charges (blue) to be periodically drawn together and pulled apart. The structure, which effectively has a negative Poisson’s ratio, then expands/contracts in the y-direction, generating a current from the ‘response’ charges (red). This dynamic response, which is suitable only for oscillating fields, generates a side effect current in the x-direction, as well as additional dipolar or quadrupolar fields. However, if the driving field is sufficiently strong and the detector charges sufficiently weaker than the response ones, the system will replicate the desired axionic response. Placing elements of this type radially around a conducting cylinder oriented such that .xˆ is replaced by .rˆ , and ˆ is replaced with .θˆ , produces the configuration shown in Fig. 4.6. Collectively, .y this system produces an axionic response that is a dynamical counterpart to the active, driven scheme discussed in the previous section. Being dynamic, this metamodel will necessarily be dispersive and so can only approach the characterization in terms of (4.37) as an approximation. Nevertheless, the current thought experiment indicates how a material response can be emulated in the laboratory that lies beyond the scope of conventional constitutive relations. Moreover, refined calculations that embrace dispersion within our first-order operator approach are certainly possible.

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Fig. 4.6 Metamaterial axionic response cells arranged radially about a conducting cylinder, to provide an axial axionic response .ξz zˆ , The response is similar to that shown in Fig. 4.4b, except that to achieve the .ξθ θˆ in that case the cells would need to be rotated about the radial axis and reoriented into the .r − z plane. Reproduced from [15]

4.4 The Electromagnetic Response of Vacuum We now discuss how the ideas developed in the foregoing sections might be applied to the simplest electromagnetic ‘medium’, i.e., vacuum. The parallel development is to dispense with the vacuum constitutive relations .D = 0 E and .H = μ−1 0 B, replacing them with first-order operators. Doing so will expose the possibility of vacuum axionic terms. Whereas such terms are not known (or are too weak) to give electromagnetic responses in vacuum akin to those discussed in the previous section, we here point out the topological consequences of such terms when the electromagnetic field propagates in vacuum spacetime with non-trivial topology. We will demonstrate that a careful examination of Stokes’ theorem on such a manifold will sever the link between local and global charge conservation. In discussing Maxwell’s equations on a 4-dimensional manifold, it is most convenient to adopt units where .c = (0 μ0 )−1/2 = 1 and to use form notation. In this way, Eqs. (4.1) and (4.2) may be written as dF = 0

(4.38)

dH = J ,

(4.39)

.

and .

where the electromagnetic 2-form field F was defined in Eq. (4.16), the current density 3-form field J was defined in Eq. (4.17), and H represents the excitation 2-form field. For conventional vacuum, the constitutive relationship between H and F is given by

4 A Novel Approach to Electromagnetic Constitutive Relations

H = F ,

47

(4.40)

.

where . is the Hodge star operator. Although this is the simplest vacuum constitutive relation, it is not unique. Alternative constitutive relations used by particle physicists include for example the weak field Euler–Heisenberg constitutive relations and Bopp–Podolski constitutive relations [23–25] which are given, respectively, by HEH = F −

.

8α 2 [ (F ∧ F ) F + 7 (F ∧ F ) F ] 45m4

(4.41)

and HBP = F + l 2 d d F ,

.

(4.42)

where .α is the fine structure constant, m is the electron mass, and . is a small parameter. In form notation, local charge conservation arises from taking the exterior derivative of Eq. (4.38)(b), i.e., d (dH ) = dJ = 0 ,

.

(4.43)

where the null result is a consequence of taking the exterior derivative twice. It is straightforward to show from the definition of J given in Eq. (4.17) that .dJ = 0 is equivalent to Eq. (4.14) (see the Appendix).

4.4.1 Charge Conservation and Stokes’ Theorem The passage from local charge conservation, .dJ = 0, to global charge conservation is achieved via Stokes’ theorem, which in terms of differential forms may be stated on a manifold .M as (see Eq. (4.92)) 

.

U=∂ N

ω=

N

dω ,

(4.44)

where the p-form field .ω is well defined over the m-dimensional volume .N whose (m − 1)-dimensional boundary is .∂N. It is also assumed that the volume .N is topologically trivial, i.e., that it can be continuously shrunk to a point in .M. Thus, setting .ω = J , the current density 3-form, we have that

.



 .

N

dJ =

U=∂ N

J = 0.

(4.45)

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Fig. 4.7 A closed 3-surface .U in spacetime on which to check conservation of charge. This surface is formed from .U = S ∪ t0 ∪ t1 with the orientations of .S, .t0 , and .t1 given by the blue arrows. Note that we do not necessarily need to consider a 4-volume enclosed by this boundary .U, as can be seen by comparing the topology condition with the gauge-free condition, as discussed in the text. Reproduced from [16] under the terms of the Creative Commons CC BY licence

To see how the latter equality in Eq. (4.45) represents a statement of global charge conservation, consider the spacetime volume .N shown in Fig. 4.7. The boundary of .N is the 3-surface .U = ∂N. This surface is closed, i.e., its boundary .∂U = ∅, and it is therefore topologically equivalent to a 3-sphere. As indicated, .U is decomposed according to .U = S ∪ t0 ∪ t1 , where .t0 and .t1 are bounded regions of space . at times .t0 and .t1 , and .S is the boundary of . between the times .t0 ≤ t ≤ t1 . The orientation of .t1 is outward, whilst those of .t0 and .S are inward. For this case, Eq. (4.45) becomes  .

 U

J =





J− t1

J− t0

S

J = 0,

(4.46)

where the signs take account of the orientations. This is seen to be a statement of global charge conservation: the total charge within a region of 3-space at .t1 is equal to the charge within that region at .t0 , plus the total charge entering the region between those times. There are actually two related ways in which Stokes’ theorem can be used to prove global charge conservation from .dJ = 0: 1. Topological condition: since .U = ∂N is the boundary of a topologically trivial bounded volume .N of spacetime, then one has  .

 U

J =

 ∂N

J =

N

dJ = 0 ,

(4.47)

4 A Novel Approach to Electromagnetic Constitutive Relations

49

since .dJ = 0. 2. Gauge-free condition: integrating Eq. (4.39) over .U and assuming H is well defined over .U, we have that 



 .

U

J =

U

dH =

∂U

H = 0,

(4.48)

since .∂U = ∅. In the second proof, we have emphasized that H must be well defined on .U. If this is not the case (cf. our contention that H can be interpreted as a gauge field for J ), then the middle (Stokes’) equality in Eq. (4.48) is not well defined. We also note that the second proof relies on .U being closed (i.e., having null boundary) but does not require (unlike the first proof) that .U be itself the boundary of a compact 4-volume.

4.4.2 Breaking the Link Between Local and Global Charge Conservation The above considerations suggest how the link between local and global charge conservation may be broken. Importantly, in any scenario where this is achieved, both of the two conditions above must be violated. If either of the conditions prevails, then global charge conservation is assured. To violate the topological condition, the manifold .M must be non-trivial. One such manifold is achieved by taking empty, flat, and vacuum and removing a single point at, say, the coordinate origin. Then, a submanifold .N ∈ M containing the excised point is topologically non-trivial, since then it cannot be shrunk to a point. To violate the second condition requires that we take the position promoted in this chapter, i.e., that the excitation field, H , acts as a gauge field for the current density J . In this view, the fundamental equations are .

dF = 0

(4.49)

dJ = 0 .

(4.50)

and .

Then, Eq. (4.39) must be replaced with a vacuum constitutive relation linking F to J . The first-order operator approach yields the simplest such relation as d F −J =ψ ∧F ,

.

(4.51)

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Fig. 4.8 A bump function used to construct a smooth current density. The function is completely flat for .f > 1/2 and for .ζ near zero. Reproduced from [16] under the terms of the Creative Commons CC BY licence .f (ζ )

where for local charge conservation .d (ψ ∧ F ) = 0. The term on the right posits a vacuum axion-like response analogous to those given for media in Eqs. (4.33) and (4.34) above.

We now present an explicit calculation leading to global charge non-conservation (. U J = 0) for a current distribution J for which charge is locally conserved, i.e., .dJ = 0. We take .M to be flat Minkowski spacetime with a single point excised at the coordinate origin, i.e., .M = R4 0. Denoting .R+ = {r ∈ R|r ≥ 0}, the current density is defined as

J =

.

⎧ ⎨ 0 for t ≤ 0, ⎩

(4.52) J+

for t > 0.

In defining .J + , we first introduce a function .f : R+ → R+ satisfying .f (ζ ) ≥ 0 for .0 ≤ ζ < 1/2 and .f (ζ ) = 0 for .ζ ≥ 1/2 and all derivatives .f (n) (0) = 0 for .n ≥ 1. An example of .f (ζ ) is shown in Fig. 4.8. Such functions are often called bump functions, and the exact form of .f (ζ ) does not need to be specified further. We then have for .J + J+ =

.

1 r  f dx ∧ dy ∧ dz t t3 1 r  − 4f dt ∧ (xdy ∧ dz + ydz ∧ dx + zdx ∧ dy) , t t

(4.53)

where .r = (x 2 + y 2 + z2 )1/2 . The first term on the right of Eq. (4.53) is the charge density, whilst the second term represents the current density. We note, crucially, that as expressed in Cartesian coordinates, J is well defined at the spatial origin for + in spherical polar coordinates .(t, r, θ, φ) gives .t > 0. Expressing .J J

.

+

= sin θf

 r3 dr − 4 dt ∧ dθ ∧ dφ . t3 t

 r   r2 t

(4.54)

4 A Novel Approach to Electromagnetic Constitutive Relations

51

It is then established that .dJ = 0 on all of .M since 1. .dJ + = 0 for .t > 0 and .r > 0 from Eq. (4.54). 2. .dJ + = 0 for .t > 0 and .r = 0 from Eq. (4.53) (n.b. .f (1) = 0). 3. .J = 0 for .t ≤ 0. From a physical standpoint, Eq. (4.54) appears to represent a .δ-function of charge Q appearing at the origin at .t = 0 and then spreading out spatially from the origin into .M where  ∞ .Q = 4π ζ 2 f (ζ )dζ . (4.55) 0

Crucially, however, the origin .0 is not an event in .M. This means that the appearance of Q at .t = 0 does not induce .dJ = 0 at some event in .M. We have successfully constructed a scenario where the total charge is zero for constant time hypersurfaces with .t < 0 but is non-zero (.= Q) for constant time hypersurfaces with .t > 0. Since .M is non-trivial, it is not possible to find a single excitation field H for which .dH = J . However, it is possible to define distinct fields .H + and .H − on − = J respective submanifolds .M+ and .M− , such that .dH + = J in .M+ and .dH − + + − M− , in .M and where .H and .H differ by a gauge on the intersection .M + − i.e., .H = H + dψ, for some scalar field .ψ. Suppose M+ = M\{t < 0} and M\ {t > 0 and r < t/2} ,

.

(4.56)

and define .H + and .H − according to H+ = h

.

=h

r  t r  t

(xdy ∧ dz + ydz ∧ dx + zdx ∧ dy).

(4.57)

(r 3 sin θ dθ ∧ dφ) ,

(4.58)

and H− = 0.

.

(4.59)

The relevant regions of .M together with the respective regions where .H ± are defined are shown in Fig. 4.9. The function .h(ζ ) is defined as h(ζ ) =

.

1 ζ3



ζ

f (ζˆ )ζˆ 2 d ζˆ .

(4.60)

0

This function is smooth about .ζ = 0, and for .ζ > 1/2, we have h(ζ ) =

.

Q . 4π ζ 3

(4.61)

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Fig. 4.9 A spacetime .M in which globally charge is not conserved. The forward cone, lying within the light cone of the excised event at the spacetime origin has non-zero charge, whereas the remainder of the spacetime is uncharged. We show the regions .M+ and .M− where the excitation 2-forms .H + and .H − are defined. Reproduced from [16] under the terms of the Creative Commons CC BY licence

On .M+



M− = {(t, r, θ, φ)|r > t > 0}, we have H+ =

.

Q sin θ dθ ∧ dφ and H − = 0 . 4π

(4.62)

Finally, therefore, we see that on the intersection region we have H+ = H− −

.

Q d (cos θ dφ) , 4π

(4.63)

so that indeed the two excitation forms differ by a gauge, and in this region .dH + = dH − = 0. These considerations lend support to the idea that electromagnetic excitation acts as a gauge field for the current source, even in vacuum.

4.4.3 Reality The above calculation was extremely artificial. A point (event) was excised from a featureless flat Minkowskian manifold, from which we showed how charge, in the form of an axion-like field, could issue from the excised point in a smooth way, such that whilst local charge conservation .dJ = 0 was always obeyed, globally charge

conservation can be violated (. U J = 0). To what extent might these considerations apply to situations that might actually occur? The excision of a point in spacetime could only occur via a spacetime singularity, so laboratory demonstrations would appear to be ruled out. Whilst physicists have taken some time to come to terms with the possibilities of spacetime singularities, they are now generally accepted to occur in black holes on account of the Penrose–Hawking singularity theorems [26]. Still, a singularity at the centre of a black hole in a curved Riemannian geometry is

4 A Novel Approach to Electromagnetic Constitutive Relations

53

still some distance from the simple manifold .M considered above. One step towards merging the two is to deform the excised point into a line and then claim that the modified manifold is representative of a temporary singularity such as occurs with the singularity associated with a black hole that forms and then evaporates due to Hawking radiation. An explicit analytic solution for electromagnetic fields interacting with a topological axion field that violates global charge conservation has recently been constructed [17]. Explicit calculations on realistic spacetime geometries containing a temporary singularity have yet to be done.

4.5 Conclusion In this chapter, we have espoused the view that the electromagnetic excitation fields that couple to the source currents and charges in Maxwell’s equations only become measurable with the introduction of explicit constitutive models. The gauge freedom that exists for the excitation fields motivated a different approach to macroscopic electromagnetism in which the medium response is coupled, in the simplest possible way, to the fundamental and measurable electromagnetic fields .E and .B. In this approach, Faraday’s law, the absence of magnetic monopoles, and local charge conservation are considered fundamental. The lowest-order terms in this approach represent axion-like response terms that can be approximated via laboratory simulation. Our next step was to re-appraise vacuum electromagnetism using the same formalism and to posit an axion-like response to vacuum. By carefully analysing the situations where Stokes’ theorem cannot be applied, we showed how the axion-like vacuum response can lead to a severance of the link between local and global charge conservation. It may be that the consequent failure of global charge demonstrated here is sufficient for many to reject the topological axions in electromagnetic theory as a description of reality. Experiment, the ultimate arbiter, is so far completely inaccessible—at least in the scenarios discussed here. Whether a scenario can be devised to test consequences of vacuum topological axions accessible to a laboratory experiment remains an open question.

Appendix: Maxwell’s Equations in Differential Form Notation This objective of this appendix is to bridge the gap between Maxwell’s equations in conventional vector notation and their expression in terms of differential forms. It is not intended to be mathematically rigorous or general but is rather focussed on the stated objective. Rigour will be left to the many standard texts, for example, [18– 20]. Form notation allows Maxwell’s equations to be expressed in a fully coordinate independent manner. Accordingly, the units of space and time coordinates are rationalized by here taking c, the vacuum speed of light, equal to unity. This also has the effect of making the units of .E and .B identical. Also, by adjusting

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the unit of charge, the traditional response of vacuum to electric field excitation (i.e., .0 ) can be set equal to unity. In these units, .μ0 = 1/(c2 0 ) equals unity as well. The manifold .M hosting the electromagnetic field is presumed to be flat spacetime with a Minkowskian metric. Although one of the beauties of form notation is its coordinate independence, for illustration, we will use a Cartesian basis so that if .{x α } represents the four coordinates of spacetime, then we will take α 0 1 2 3 .{x } = {x = t, x = x, x = y, x = z} and assume orthonormal basis vectors .{et , ex , ey , ez }. Here, orthonormal is with respect to the Minkowski metric, .η, i.e., .η00 = η(et , et ) = −1, ηxx = ηyy = ηzz = 1. Differential forms may be built recursively as different types of function on .M: • A 0-form is simply a function on .M, i.e., .f (p) ∈ R, where .p ∈ M. • A 1-form .α is a function of vectors on .M, i.e., .α(v), where .v is any vector field in .M. • A 2-form .ω may be constructed out of two 1-forms .α and .β as .α ∧ β (the ‘wedge product’ of .α and .β) and is a function of two vectors, .u and .v. The 2-form .α ∧ β acts on the pair .(u, v) anti-symmetrically, i.e., (α ∧ β)(u, v) = α(u)β(v) − β(u)α(v) .

.

(4.64)

• A 3-form .λ = α ∧ β ∧ γ is similarly an anti-symmetric function of three vectors .u, v, and .w: (α ∧ β ∧ γ )(u, v, w) = α(u)β(v)γ (w) + α(w)β(u)γ (v) + α(v)β(w)γ (u)

.

−α(v)β(u)γ (w) − α(w)β(v)γ (u) − α(u)β(w)γ (v) . (4.65) The canonical example of a 1-form is the differential of a function df . For example, if we take the function .x μ = x μ (p), the coordinate of the point .p ∈ M, then we are led to the coordinate 1-form .dx μ . This is defined such that for a coordinate basis μ vector .eν , .dx μ (eν ) = δ ν . The set of co-ordinate 1-forms .{dx μ } constitutes a basis dual to the basis vectors .{eμ }. Forms may be expanded in terms of basis 1-forms, α = αμ dx μ , .

.

1 ωμν dx μ ∧ dx ν , . 2 1 λ = λμνπ dx μ ∧ dx ν ∧ dx π , 3!

ω=

(4.66) (4.67) (4.68)

where the components .ωμν and .λμνπ are completely anti-symmetric. In terms of basis 1-forms:

4 A Novel Approach to Electromagnetic Constitutive Relations

df =

.

∂f dx α , ∂x α

55

(4.69)

where the summation convention is assumed. By extension, differentiation of a 1form produces a 2-form via  dα =

.

∂αμ ∂x α

 dx α ∧ dx μ ,

(4.70)

and differentiation2 of a 2-form produces a 3-form via 

∂ωμν dx λ

dω =

.

 dx λ ∧ dx μ ∧ dx ν .

(4.71)

From this definition and the anti-symmetry of the wedge product, it is easy to show that d(dω) = 0 ,

.

(4.72)

indicating that .dω is closed. We now show that Eq. (4.15)(a) is equivalent to Eq. (4.1). Equation (4.16) states that the Faraday 2-form is given by F = E ∧ dt + B ,

.

(4.73)

where .E = Ex dx + Ey dy + Ez dz is a 1-form representing the electric field, and B = Bx dy ∧ dz + By dz ∧ dx + Bz dx ∧ dy is a 2-form representing the magnetic field. Taking the exterior derivative and presuming the result to vanish yield

.

dF =

.

∂Ey ∂Ex dy ∧ dx ∧ dt + dx ∧ dy ∧ dt + (similar terms) ∂y ∂x ∂Bz dt ∧ dx ∧ dy + (similar terms) ∂t ∂Bx + dx ∧ dy ∧ dz + (similar terms) = 0 . ∂x

+

(4.74)

Anti-symmetry causes terms in the first group to pair up as, e.g.,  .

∂Ey ∂Ex − ∂x ∂y

 dx ∧ dy ∧ dt = (∇ × E)z dx ∧ dy ∧ dt .

(4.75)

These parings are matched with a term in the second group given by .(∂Bz /∂t) dx ∧ dy ∧ dt. Thus, on contracting dF with .(ex , ey , et ), we are left with 2 This

differentiation of forms is referred to as exterior differentiation.

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M. W. McCall et al.

.

(∇ × E)z +

∂Bz = 0, ∂t

which is the z-component of Faraday’s law. The other components are obtained similarly by contracting with .(ey , ez , et ) and .(ez , ex , et ). Contracting with .(ex , ey , ez ) yields .∇ · B = 0. Thus collectively .dF = 0 describes the homogeneous Maxwell equations, Eq. (4.1). In order to understand the form expression for the inhomogeneous Maxwell equations (cf. Eq. (4.15)(b)), the Hodge star operator . must be introduced. The general definition of . maps an n form to a .p − n form, where p is the dimension of .M. In the current discussion, . is only encountered as a map from a 2-form to a .(4 − 2) = 2-form and so a restricted definition of . will suffice. Applied to a basis 2-form, .dx α ∧ dx β , the Hodge star is defined as .

(dx α ∧ dx β ) =

1 αρ βλ η η ρλμν dx μ ∧ dx ν , 2

(4.76)

where the Levi-Civita symbol, .ρλμν , equals .+1(−1) if .ρλμν is an even (odd) permutation of 0123 and is zero otherwise. It is re-iterated that the basis .{eα } is presumed orthonormal. It is readily verified from this definition that .

(dx ∧ dt) = dy ∧ dz,

(dy ∧ dt) = −dx ∧ dz,

(dz ∧ dt) = dx ∧ dy, . (4.77)

(dx ∧ dy) = dt ∧ dz,

(dy ∧ dz) = dt ∧ dx,

(dz ∧ dx) = dt ∧ dy. (4.78)

Now, taking the . of Eq. (4.73), we find .

F = −B (1) ∧ dt + E (2) ,

(4.79)

where .B (1) = Bx dx + By dy + Bz dz is the 1-form representation of the magnetic field, and .E (2) = Ex dy ∧ dz + Ey dz ∧ dx + Ez dx ∧ dy is the 2-form representation of the electric field. The exterior derivative of . F may now be derived similar to that for dF above under the substitution .E → −B (1) , and .B → E (2) . This yields, after contraction with suitable basis vectors, the following components: 

.

  ∂Ex dy ∧ dz ∧ dt = − (∇ × B)x + dy ∧ dz ∧ dt, (4.80). ∂t     ∂Ey ∂Ey ∂Bz ∂Bx − + dz ∧ dx ∧ dt = − (∇ × B)y + dz ∧ dx ∧ dt, (4.81). ∂x ∂z ∂t ∂t     ∂By ∂Bx ∂Ez ∂Ez − + dx ∧ dy ∧ dt = − (∇ × B)z + dx ∧ dy ∧ dt, (4.82). ∂y ∂x ∂t ∂t ∂By ∂Bz ∂Ex − + ∂z ∂y ∂t



4 A Novel Approach to Electromagnetic Constitutive Relations



∂Ey ∂Ex ∂Ez + + ∂x ∂y ∂z

57

 dx ∧ dy ∧ dz = (∇ · E) dx ∧ dy ∧ dz .

(4.83)

These must be matched to the corresponding terms in the current density 3-form (cf. Eq. (4.15)(b)) .J

= −j ∧ dt + ρ˜ ,

(4.84)

where .j = jx dy ∧ dz + jy dz ∧ dx + jz dx ∧ dy is the current density 2-form, and = ρdx ∧ dy ∧ dz is the charge density 3-form. These terms are

.ρ ˜

.

− jx dy ∧ dz ∧ dt , .

(4.85)

−jy dz ∧ dx ∧ dt , .

(4.86)

−jz dx ∧ dy ∧ dt.

(4.87)

and ρdx ∧ dy ∧ dz .

(4.88)

Matching (4.80) with (4.85), etc. yields .∇

×B =

∂E + j. ∂t

(4.89)

and ∇ ·E = ρ,

(4.90)

which are the inhomogeneous microscopic Maxwell equations in units where .c = 1 and .0 = 1. Local charge conservation (Eq. (4.14)) is expressed in form notation (Eq. (4.43)) as .dJ = 0, since .dJ

    = d − jx dy ∧ dz − jy dz ∧ dx − jz dx ∧ dy ∧ dt + ρdx ∧ dy ∧ dz     ∂jy ∂jz ∂ρ ∂jx + + + dx ∧ dy ∧ dz ∧ dt = 0 , = − ∂x ∂y ∂z ∂t

(4.91)

whence equating the terms in square brackets to zero reproduces Eq. (4.14). We conclude this appendix with a brief statement of a generalized Stokes’ theorem: If .M is an orientable m-dimensional topologically trivial manifold with a boundary .∂M, and .ω is an (.m − 1)-form, then   . (4.92) dω = ω. M U=∂ M

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Here, roughly speaking, orientable means a manifold on which it is possible to define a basis of specific handedness at each point.3 Topologically, trivial means that the manifold can be shrunk to a point. We have not formally defined the boundary of a manifold, although for the example in the main text (cf. Fig. 4.7), the boundary accords with the intuitive notion of a surface of a closed volume. We note finally that whilst the most general form of Stokes’ theorem replaces the topologically trivial condition on .M with specifying that .ω has compact support, the above restricted form of Stokes’ theorem is sufficient for the purposes of this chapter.

References 1. Hehl, F.W., Obukhov, Y.N.: Foundations of Classical Electrodynamics: Charge, Flux, and Metric. Birkhäuser, Basel (2003) 2. Ehrenberg, W., Siday, R.E.: The refractive index in electron optics and the principles of dynamics. Proc. Phys. Soc. B 62, 8–21 (1948) 3. Aharonov, Y., Bohm, D.: Significance of electromagnetic potentials in the quantum theory. Phys. Rev. 115, 485–491 (1959) 4. Matteucci, G., Iencinella, D., Beeli, C.: The Aharonov–Bohm phase shift and Boyer’s critical considerations: New experimental result but still an open subject? Found. Phys. 33, 577–590 (2003) 5. Rindler, W.: Relativity and electromagnetism: The force on a magnetic monopole. Am. J. Phys. 57, 993–994 (1989) 6. Heras, J.A., Baez, G.: The covariant formulation of Maxwell’s equations expressed in a form independent of specific units. Eur. J. Phys. 30, 23–33 (2009) 7. Roche, J.J.: B and H, the intensity vectors of magnetism: A new approach to resolving a century-old controversy. Am. J. Phys. 68, 438–449 (2000) 8. Scheler, G., Paulus, G.G.: Measurement of Maxwell’s displacement current. Eur. J. Phys. 36, 055048 (2015) 9. Landini, M.: About the physical reality of “Maxwell’s displacement current” in classical electrodynamics. Prog. Electromagn. Res. 144, 329–343 (2014) 10. Bork, A.M.: Maxwell, displacement current, and symmetry. Am. J. Phys. 31, 854–859 (1963) 11. Jackson, J.D.: Classical Electrodynamics, 3rd edn. Wiley, New York (1999) 12. Reitz, J.R., Milford, F.J., Christy, R.W.: Foundations of Electromagnetic Theory. Pearson Education, India (2009) 13. Kinsler, P, Favaro, A., McCall, M.W.: Four Poynting theorems. Eur. J. Phys. 30, 983–993 (2009) 14. Feynman, R.P., Leighton, R.B., Sands, M.: The Feynman Lectures on Physics, vol. II. AddisonWesley, Boston (1964) 15. Gratus, J., McCall, M.W., Kinsler, P.: Electromagnetism, axions, and topology: a first-order operator approach to constitutive responses provides greater freedom. Phys. Rev. A 101, 043804 (2020) 16. Gratus, J., Kinsler, P., McCall, M.W.: Evaporating black-holes, wormholes, and vacuum polarisation: must they always conserve charge? Found. Phys. 49, 330–350 (2019) 17. Gratus, J., Kinsler, P., McCall, M.W.: Temporary singularities and axions: an analytic solution that challenges charge conservation. Ann. Phys. 533, 2000565 (2021) 18. Flanders, H.: Differential Forms with Applications to the Physical Sciences. Dover Publications Inc., New York (1963) 3A

Möbius strip is a counterexample.

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19. Nakahara, M.: Geometry, topology and physics. CRC Press, Boca Raton (2003) 20. Misner, C.W., Thorne, K.S., Wheeler, J.A.: Gravitation. W. H. Freeman and Company, San Francisco (1973) 21. Altland, A., von Delft, J.: Mathematics for physicists: introductory concepts and methods. Cambridge University, Cambridge (2019) 22. Gratus, J., Kinsler, P., McCall, M.W.: Maxwell’s (D, H) excitation fields: lessons from permanent magnets. Eur. J. Phys. 40, 025203 (2019) 23. Bopp, F.: Eine lineare theorie des elektrons. Ann. Phys. 430, 345–384 (1940) 24. Podolsky, B.: A generalized electrodynamics part I—non-quantum. Phys. Rev. 62, 68–71 (1942) 25. Gratus, J., Perlick, V., Tucker, R.W.: On the self-force in Bopp-Podolsky electrodynamics. J. Phys. A 48, 435401 (2015) 26. Hawking, S.W., Ellis, G.F.R.: The Large Scale Structure of Space-Time, vol. 1. Cambridge University, Cambridge (1973) Martin McCall is Professor of Theoretical Optics at Imperial College London. He began his career working in industry on photorefractive media for real-time holography, the topic of his doctoral thesis. After briefly working at Bath University, where he worked on nonlinear dynamics in semi-conductor lasers, he moved to Imperial College, where he has remained since. His research has ranged over all things optical and optoelectronic, usually involving gratings in some form or other, e.g. temporal gratings, optical back-planes and periodically structurally chiral media. The thematic of ‘complex electromagnetic media’ embraced work on negative refraction and transformation optics, that in 2011 lead to his group extending the concept of object cloaking to so-called spacetime cloaking, demonstrating that time-modulated media could be configured into devices capable of hiding events in spacetime. More recently, McCall has worked on fundamental aspects of Maxwell’s equations, including questioning the role of traditional constitutive relations and solutions on manifolds with singularities. He is the author of an undergraduate text on classical mechanics and relativity. Jonathan Gratus is a senior lecturer in mathematical physics at Lancaster University and a member of the Cockcroft Institute of Accelerator Science. He has a passion for understanding physics from a geometric point of view, in particular looking at geometric objects as pictures. See “A pictorial introduction to differential geometry, leading to Maxwell’s equations as three pictures” (https://arxiv.org/abs/1709.08492), which is based on his MPhys course. He investigates many areas of mathematical physics, which include: the fundamentals of electrodynamics; a geometric approach to the theory of distributional sources of electrodynamics and gravity; novel constitutive relations in media, including spatial dispersion and time dependent media; alternative structures for accelerating particles. He is always looking for potential PhD students.

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M. W. McCall et al. Paul Kinsler has a wide research background, starting with a PhD in quantum optics from the University of Queensland, but also semiconductors, nonlinear optics, and transformation optics and metamaterials. He tends to work with a mix of theory and simulation; with a focus on the fundamental aspects of these areas, perhaps most notably directionality in pulse propagation, causality, and treatments of dispersion; and makes a point of trying to bring a fresh perspective to any research problem. Along with McCall, he also worked on spacetime cloaking, and extended this concept by describing spacetime carpet cloaks. He enjoys creatively disagreeing with both McCall and Gratus on several aspects of physics, which in this case has led to this work challenging the traditional role of constitutive relations in electromagnetism. For some inexplicable reason he has been working on the self-organization of drone swarms in communications- constrained environments for the last year.

Chapter 5

On the Anatomy of Voigt Plane Waves Tom G. Mackay and Akhlesh Lakhtakia

5.1 Introduction Two electromagnetic plane waves with distinct wavenumbers can propagate in any direction inside a homogeneous anisotropic material, provided that the direction of propagation does not coincide with an optic axis of the material [1]. Planewave propagation along an optic axis occurs with only one wavenumber. The optic axes in a nondissipative (and inactive) anisotropic material are called regular, the spatial variation of the transverse components of the electric and magnetic field phasors of the plane wave propagating along a regular optic axis being sinusoidal. Certain dissipative (or active) anisotropic materials, such as uniaxial dielectric materials [2, 3], also possess regular optic axes; however, the spatial variations of the transverse components of the electric and magnetic field phasors of the plane wave propagating along such a regular optic axis are sinusoidal but modulated by an exponential dependency on propagation distance. For other dissipative (or active) anisotropic mediums, such as biaxial dielectric materials [2, 3], the optic axes are called singular. The field components of a plane wave propagating along a singular optic axis have sinusoidal spatial variations modulated by a linear–exponential

T. G. Mackay () School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, Edinburgh, UK NanoMM—Nanoengineered Metamaterials Group, Department of Engineering Science and Mechanics, The Pennsylvania State University, University Park, PA, USA e-mail: [email protected] A. Lakhtakia NanoMM—Nanoengineered Metamaterials Group, Department of Engineering Science and Mechanics, The Pennsylvania State University, University Park, PA, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 T. G. Mackay, A. Lakhtakia (eds.), Adventures in Contemporary Electromagnetic Theory, https://doi.org/10.1007/978-3-031-24617-3_5

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dependency on propagation distance. Plane waves in the last category are called Voigt plane-waves [4–6]. One hundred and twenty years have elapsed since Voigt reported on his eponymous plane waves [4], but they attracted scant attention for much of this period, with some notable exceptions [5–13]. The prospect of realizing engineered composite materials that support Voigt plane-wave propagation, in a controllable way, has prompted some interest in this topic, for both dissipative [14–18] and active [19] regimes. Recent studies have shed light on the physical aspects of Voigt plane-wave propagation in real materials [20, 21], including for lasing applications [22]. However, the singular nature of Voigt plane-wave propagation has remained somewhat obscure. Given that the fields of a Voigt plane-wave have a linear– exponential dependence on the propagation distance, whereas the fields of a plane wave propagating along a direction in the neighborhood of a singular optic axis have only an exponential dependence on the propagation distance, one could expect the spatial variation of a Voigt plane-wave to be distinctive. But that distinction could not be seen in a preliminary analytical investigation of the fields we undertook. So we decided to prosecute the following numerical investigation. The monoclinic material .β-Ga.2 O.3 , whose optical characteristics including singular optic axes have been reported on lately [20, 21], was chosen as the backdrop for our study. The propagation characteristics of Voigt plane-waves were compared with those of plane waves propagating along directions slightly rotated from a singular optic axis. Planewave propagation was characterized in terms of electric field phasors and their polarization states and the time-averaged Poynting vector. In the following, vectors are in boldface; dyadics are double underlined; column vectors are in boldface and enclosed in square brackets; matrixes are double underlined and enclosed in square brackets; the Cartesian unit vectors are denoted by .uˆ x , ˆ y , and .uˆ z ; the complex conjugate is signaled by the superscript .∗ ; the operators .u .Re {·} and .Im {·} deliver the real and imaginary parts of complex quantities; and the superscript .T denotes the transpose.

5.2 Propagation in an Unbounded Biaxial Dielectric Medium Let us begin our investigation of Voigt plane-waves with propagation in an unbounded biaxial dielectric medium.

5.2.1 Theory The sinusoidally time-dependent electromagnetic fields are expressed as ˜ A(r, t) = Re {A(r) exp(−iωt)} ,

.

A ∈ {E, D, H, B} ,

(5.1)

5 On the Anatomy of Voigt Plane Waves

63

√ ˜ ∈ R3 , .A ∈ C3 , t is time, .ω is angular frequency, and .i = −1. Without wherein .A loss of generality, we focus on propagation parallel to the z axis and set A ∈ {E, D, H, B} .

A(r) = Ax (z)uˆ x + Ay (z)uˆ y + Az (z)uˆ z ,

.

(5.2)

The medium is taken to be anisotropic dielectric, as specified by the relative permittivity dyadic ε

.

rel

    = ε11 uˆ x uˆ x + ε22 uˆ y uˆ y + ε33 uˆ z uˆ z + ε12 uˆ x uˆ y + uˆ y uˆ x + ε13 uˆ x uˆ z + uˆ z uˆ x   +ε23 uˆ y uˆ z + uˆ z uˆ y (5.3)

with .εm ∈ C, .( ∈ {1, 2, 3} , m ∈ {1, 2, 3} , m ≥ ). Thus, the constitutive relations can be stated as  D(r) = ε0 ε •E(r) rel (5.4) . , B(r) = μ0 H(r) with .ε0 as the free-space permittivity and .μ0 as the free-space permeability. Insertion of Eqs. (5.3) and (5.4) into the Maxwell curl postulates yields the matrix ordinary differential equation [23] .

  d [f(z)] = ik0 Q • [f(z)] , dz

(5.5)

√ wherein .k0 = ω ε0 μ0 is the free-space wavenumber, and the column 4-vector ⎤ Ex (z) ⎢ Ey (z) ⎥ ⎥ . [f(z)] = ⎢ ⎣ Hx (z) ⎦ Hy (z) ⎡

(5.6)

contains the transverse components of the electric and magnetic field phasors. Containing the scalars

.

the 4 .× 4 matrix

⎫ 2 δ11 = ε11 − ε13 /ε33 ⎬ δ12 = ε12 − ε13 ε23 /ε33 , ⎭ 2 δ22 = ε22 − ε23 /ε33

⎡

0 0

0 0

0   ⎢ −η 0 . Q = ⎢ ⎣ −δ12 /η0 −δ22 /η0 0 δ11 /η0 δ12 /η0 0

(5.7)

⎤ η0 0⎥ ⎥ 0⎦ 0

(5.8)

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is the propagator. The z-directed components of the phasors .E(r) and .H(r) are provided explicitly as

.

Ez (z) = − Hz (z) = 0

5.2.1.1

⎫ ε13 Ex (z) + ε23 Ey (z) ⎬ . ε33 ⎭

(5.9)

Nondegenerate Case

  In general, the matrix . Q has four distinct eigenvalues and four linearly independent eigenvectors. The two eigenvalues with positive imaginary parts are given as

.

k˘a =



⎫ (δ11 + δ22 + ) ⎬ , (δ + δ − ) ⎭

k˘b =

1 2 1 2

=

 2 . (δ11 − δ22 )2 + 4δ12

11

(5.10)

22

wherein the scalar .

(5.11)

Two linearly independent eigenvectors that correspond to the eigenvalues .k˘a and .k˘b are  [va ] = 

.

[vb ] =

δ11 − δ22 +  , 2δ12 δ11 − δ22 −  , 2δ12

1,

k˘a − , η0

1,

k˘b − , η0

T ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ T , ⎪ ⎪ (δ11 − δ22 − ) k˘b ⎪ ⎪ ⎭ 2η0 δ12 (δ11 − δ22 + ) k˘a 2η0 δ12

(5.12)

√ respectively, wherein .η0 = μ0 /ε0 is the intrinsic impedance of free space. Hence, the general solution for propagation in the .+z direction has the form .

    [f(z)] = Ca [va ] exp ik0 k˘a z + Cb [vb ] exp ik0 k˘b z ,

(5.13)

where the coefficients .Ca ∈ C and .Cb ∈ C are determined by the boundary conditions.

5 On the Anatomy of Voigt Plane Waves

5.2.1.2

65

Degenerate Cases

Degeneracy, as signaled by .k˘a = k˘b ≡ k˘s , occurs when . = 0. Thus, from Eq. (5.11), the degeneracy occurs when the condition δ11 − δ22 = ±2iδ12

(5.14)

.

is satisfied. (i) For any nondissipative and inactive medium (including free space), .εm ∈ R for every pair .(, m). The degeneracy condition (5.14) is then √ only satisfied when ˘s = δ11 has two linearly .δ11 = δ22 and .δ12 = 0. The degenerate eigenvalue .k independent eigenvectors, namely

.

√ T  δ11 ,0 [va ] = 0, 1, − η √ 0 T  δ11 [vb ] = 1, 0, 0, η0

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭

(5.15)

.

Accordingly, the general solution for propagation in the .+z direction in a nondissipative and inactive medium has the form .

  [f(z)] = {Ca [va ] + Cb [vb ]} exp ik0 k˘s z .

(5.16)

The general solution (5.16) also holds for dissipative (or active) mediums that are either isotropic dielectric mediums or uniaxial dielectric mediums with their sole optic axis aligned with the z axis, for which .δ11 = δ22 and .δ12 = 0. (ii) For a general dissipative material (and active material), .εm ∈ C and .δ12 = 0.  The matrix . Q then has only one eigenvalue with positive imaginary part, namely  ˘s = .k

1 (δ11 + δ22 ), 2

(5.17)

and only one eigenvector, namely  .

[va ] =

δ11 − δ22 , 2δ12

1,

k˘s − , η0

(δ11 − δ22 ) k˘s 2η0 δ12

T

  In addition, . Q has a generalized eigenvector [24], namely

.

(5.18)

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.

  vg =

√ 2 (δ11 + δ22 ) , δ12

0,

−

1 , η0

3δ11 + δ22 2η0 δ12

T ,

which satisfies the relation       Q − k˘s I • vg = [va ] .

(5.19)

(5.20)

  with . I as the 4 .× 4 identity matrix. Hence, the general solution for propagation in the .+z direction in general dissipative mediums (and active mediums) has the form [11] .

      exp ik0 k˘s z . [f(z)] = Ca [va ] + Cb ik0 z [va ] + vg

(5.21)

The degenerate general solution in Eq. (5.16) represents plane-wave propagation along a regular optic axis for an anisotropic medium, whereas the degenerate general solution in Eq. (5.21) represents Voigt plane-wave propagation along a singular optic axis.

5.2.1.3

Boundary Values for Voigt Plane Waves

On the boundary .z = 0, the Voigt plane-wave solution (5.21) yields .

  [f(0)] = Ca [va ] + Cb vg .

(5.22)

Hence, from the eigenvector (5.18) and the generalized eigenvector (5.19), along with the degeneracy condition (5.14), we find

.

⎫ Ca = Ey (0) ⎬ Ex (0) ∓ iEy (0) . δ12 ⎭ Cb = 2k˘s

(5.23)

The two possible values of .Cb in Eqs. (5.23) give rise to two Voigt plane-wave solutions, which we label with “.±.” The electric field components of .[f(z)] for the Voigt plane-wave solutions may be expressed as [22]  .

Ex (z) Ey (z)

±

     1 δ12 Ex (0) Ey (0) ± iEx (0) + ik0 k˘s z Ey (0) ∓i δ11 + δ22   (5.24) × exp ik0 k˘s z . 

=

5 On the Anatomy of Voigt Plane Waves

67

The natures of the two solutions on the right side of Eq. (5.24) are unveiled by considering the Voigt plane wave to be circularly polarized at .z = 0. I. Suppose, first, the Voigt plane wave is left-circularly polarized (LCP) at .z = 0, i.e.,  .

   Ex (0) 1 = . Ey (0) i

(5.25a)

Correspondingly, the two solutions (5.24) are  .

Ex (z) Ey (z)

+ =

      δ12 1 1 exp ik0 k˘s z − 2k0 k˘s z i δ11 + δ22 −i

(5.25b)

and  .

Ex (z) Ey (z)

− =

    1 exp ik0 k˘s z . i

(5.25c)

Thus, the “.−” solution is simply an LCP plane wave whose field components vary exponentially with z. In contrast, the “.+” solution comprises both LCP and right-circularly polarized (RCP) components, with the LCP component varying exponentially with z and the RCP component varying as the product of a linear function and an exponential function of z. II. Next suppose that the Voigt plane wave is RCP at .z = 0, i.e.,  .

   Ex (0) 1 = . Ey (0) −i

(5.26a)

Then, the two solutions (5.24) are  .

Ex (z) Ey (z)

+

 =

1 −i



  exp ik0 k˘s z

(5.26b)

and  .

Ex (z) Ey (z)

−

 =

     δ12 1 1 exp ik0 k˘s z . + 2k0 k˘s z −i δ11 + δ22 i

(5.26c)

Thus, the “.+” solution is simply an RCP plane wave whose field components vary exponentially with z. In contrast, the “.−” solution comprises both LCP and RCP components, with the RCP component varying exponentially with z and the LCP component varying as the product of a linear function and an exponential function of z.

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5.2.2 Numerical Investigations As a biaxial dielectric material for numerical investigation of the anatomy of Voigt plane-waves, we choose .β-Ga.2 O.3 [20]. At the free-space wavelength of .200 nm, this dielectric material possesses monoclinic symmetry and is characterized by the relative permittivity dyadic [21] ε

.

GaO

= (4.366 + 1.592i)uˆ x uˆ x + (5.016 + 2.15i)uˆ y uˆ y + (4.686 + 1.46i)uˆ z uˆ z +(0.094 + 0.017i)(uˆ x uˆ z + uˆ z uˆ x ).

(5.27)

In order to find those orientations of .β-Ga.2 O.3 that allow Voigt wave propagation along the .+z axis, we use the rotation dyadics .

R z (α) = (uˆ x uˆ x + uˆ y uˆ y ) cos α + (uˆ x uˆ y − uˆ y uˆ x ) sin α + uˆ z uˆ z

(5.28a)

R y (β) = (uˆ x uˆ x + uˆ z uˆ z ) cos β − (uˆ x uˆ z − uˆ z uˆ x ) sin β + uˆ y uˆ y ,

(5.28b)

and .

with Euler angles .α and .β, and set ε

.

rel

= R y (β)•R z (α)•ε

GaO

R Tz (α)•R Ty (β).

•

(5.29)

Our aim is to reveal distinguishing features of Voigt plane-wave propagation along a singular optic axis as compared with plane-wave propagation in the neighborhood of a singular optic axis. We do so by considering the electric field phasor and the energy flow per unit area as provided by the time-averaged Poynting vector [2] ˜ P(r)

=

.

 1  Re E(r) × H∗ (r) . 2

(5.30)

˜ In terms of Cartesian components, we write . P(r)

= Px (r)uˆ x +Py (r)uˆ y +Pz (r)uˆ z . Now, we explore numerically the propagation of Voigt plane-waves along a singular optic axis and plane-wave propagation along neighboring directions, for the material specified by .ε , per Eq. (5.29). There are four singular optic axes for rel the ranges .0 < α < π , .0 < β < π , but only two of these are independent. These are aligned with the z axis for the material orientation angles .(α, β) = (αs1 , βs1 ), ◦ = 180◦ − α , .(αs2 , βs2 ), .(αs3 , βs3 ), and .(αs4 , βs4 ), where .αs1 = 125.193 s4 ◦ ◦ ◦ ◦ .βs1 = 60.168 = 180 − βs4 , .αs2 = 124.360 = 180 − αs3 , and .βs2 = 100.549◦ = 180◦ − βs3 . Let us begin with Voigt plane-wave propagation along the singular optic axis corresponding to .(αs1 , βs1 ) and plane-wave propagation along directions corresponding to the punctured neighborhood of .(αs1 , βs1 ). In Fig. 5.1,

5 On the Anatomy of Voigt Plane Waves

69

Fig. 5.1 Plots of the magnitudes of the components of the electric field phasor and the components of the time-averaged Poynting vector against propagation distance .k0 z for the material . Material orientations: .(α, β) = (αs1 + ξ, βs1 + ξ ) with with relative permittivity dyadic . GaO ◦ ◦ ◦ .ξ = 0 (blue solid curve), .ξ = 0.1 (red short dashed curve), .ξ = −0.1 (red long dashed curve), ◦ ◦ ◦ .ξ = 0.5 (black short dashed curve), .ξ = −0.5 (black long dashed curve), .ξ = 2 (brown short ◦ dashed curve), and .ξ = −2 (brown long dashed curve), with .(αs1 , βs1 ) = (125.193◦ , 60.168◦ ). Normalization: .Ex (0) = 0, .Ey (0) = 1 V m.−1

plots of the magnitudes of the components of the electric field phasor, and the components of the time-averaged Poynting vector, against propagation distance .k0 z are provided for the material orientations: .(α, β) = (αs1 + ξ, βs1 + ξ ) with ◦ ◦ ◦ .ξ = 0 (blue solid curve), .ξ = 0.1 (red short dashed curve), .ξ = −0.1 (red ◦ ◦ long dashed curve), .ξ = 0.5 (black short dashed curve), .ξ = −0.5 (black long dashed curve), .ξ = 2◦ (brown short dashed curve), and .ξ = −2◦ (brown long dashed curve). The field normalization is such that .Ex (0) = 0 and .Ey (0) = 1 V m.−1 . Hence, the fields at .z = 0 are those of a y-polarized plane-wave. For all material orientations considered, the propagation of energy is chiefly along the z axis; energy propagation along the y axis is substantially less and along the x axis is lesser still. At a fixed propagation length, .|Ey |, .|Ez |, .Py , and .Pz vary very little

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Fig. 5.2 As Fig. 5.1 but with .(αs1 , βs1 ) replaced with .(αs2 , βs2 ) = (124.360◦ , 100.549◦ )

as the material orientation changes from .(αs1 ± 2◦ , βs1 ± 2◦ ) to .(αs1 , βs1 ), while smooth variations are observed for .|Ex | and .Px . That is, within the neighborhood of the material orientation .(αs1 , βs1 ), there is no abrupt change in the electric field phasor or the time-averaged Poynting vector as the material orientation tends toward .(αs1 , βs1 ). Next, let us turn to the singular optic axis that is aligned with the z axis when the material orientation is specified by .(αs2 , βs2 ). Plots analogous to those of Fig. 5.1 are displayed in Fig. 5.2. As for Fig. 5.1, in Fig. 5.2, there is very little variation in .|Ey |, .|Ez |, .Py , and .Pz at a fixed propagation length as the material orientation varies from .(αs2 ± 2◦ , βs2 ± 2◦ ) to .(αs1 , βs1 ), while the variations observed for .|Ex | and .Px are smooth. In other words, within the neighborhood .(αs2 + ξ, βs2 + ξ ), there is no abrupt change in the electric field phasor or the time-averaged Poynting vector as the material orientation tends toward .(αs2 , βs2 ). A y-polarized plane-wave is imposed at .z = 0 by the field normalization −1 used for Fig. 5.1 (and Fig. 5.2). The electric .Ex (0) = 0 and .Ey (0) = 1 V m. field profiles that are generated by repeating the computations of Fig. 5.1 for other

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71

Fig. 5.3 As Fig. 5.1 but only the plots of the magnitudes of the components of the electric field phasor for the normalizations: (i) .Ex (0) = 1 V m.−1 , .Ey (0) = 0, (ii) .Ex (0) = 1 V m.−1 , .Ey (0) = i V m.−1 , and (iii) .Ex (0) = 1 V m.−1 , and .Ey (0) = −i V m.−1

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polarization states at .z = 0 are provided in Fig. 5.3. The selected polarization states and field normalizations are: (i) x-polarized plane wave with .Ex (0) = 1 V m.−1 , .Ey (0) = 0. (ii) LCP plane wave with .Ex (0) = 1 V m.−1 , .Ey (0) = i V m.−1 . (iii) RCP plane wave with .Ex (0) = 1 V m.−1 , .Ey (0) = −i V m.−1 . The plot of .|Ey | for the x-polarized plane wave in Fig. 5.3 is very similar to the plot of .|Ex | for the y-polarized plane wave in Fig. 5.1; likewise, plot of .|Ex | for the x-polarized plane wave in Fig. 5.3 is very similar to the plot of .|Ey | for the y-polarized plane wave in Fig. 5.1; and the plots of .|Ez | in Figs. 5.1 and 5.3 are qualitatively similar. The plots of .|Ex | and .|Ey | for the LCP plane wave in Fig. 5.3 are very similar to each other and very similar to the plots of .|Ex | and .|Ey | for the RCP plane wave in Fig. 5.3, and the plots of .|Ez | for the LCP plane wave and the RCP plane wave in Fig. 5.3 are qualitatively similar to each other and qualitatively similar to the plots for .|Ex | and .|Ey | for both the LCP and RCP plane waves in Fig. 5.3. For all three polarization states represented in Fig. 5.3, there is no abrupt change in the electric field phasor as the material orientation tends toward .(αs1 , βs1 ). Next, let us turn to the influence of the polarization state at .z = 0 upon plane waves propagating for the material-orientation neighborhood .(αs2 + ξ, βs2 + ξ ). The profiles of .|Ey | that are generated by repeating the computations of Fig. 5.3 but with the material-orientation neighborhood .(αs1 + ξ, βs1 + ξ ) replaced by materialorientation neighborhood .(αs2 + ξ, βs2 + ξ ) are displayed in Fig. 5.4. As in Fig. 5.3, no abrupt change in .|Ey | is displayed in Fig. 5.4 as the material orientation tends

Fig. 5.4 As Fig. 5.3 but with .(αs1 , βs1 ) replaced with .(αs2 , βs2 ) = (124.360◦ , 100.549◦ ). Only the profiles for .|Ey | are presented

5 On the Anatomy of Voigt Plane Waves

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toward .(αs2 , βs2 ). The same is true for quantities .|Ex | and .|Ez | (and .Px , .Py , and Pz ) that are not presented in Fig. 5.4. The existence of Voigt waves and singular optic axes is intimately connected to material dissipation (or activity). Nondissipative materials do not support Voigt plane-wave propagation—indeed, anisotropic nondissipative materials only have regular optic axes not singular optic axes. In order to explore the role of material dissipation, we consider now  the material specified by the relative permittivity +i γ Im  dyadic .Re  , where the parameter .γ ≥ 0 provides a measure GaO GaO of the degree of dissipation of the material. The case of .γ = 1 has already been dealt with in Figs. 5.1, 5.2, 5.3 and 5.4. For .γ = 0, 0.2, and 5, the magnitudes of the components of the electric field phasor are plotted against .k0 z in Fig. 5.5 for material orientations corresponding to the neighborhood of a regular optic axis for .γ = 0 and singular optic axes for .γ = 0.2 and 5. The regular optic axis considered for .γ = 0 arises at the material orientation .(αs1 , βs1 ) = (138.876◦ , 78.420◦ ), while the singular optic axes considered for .γ = 0.2 and 5 arise at .(αs1 , βs1 ) = (137.661◦ , 71.973◦ ) and .(αs1 , βs1 ) = (105.278◦ , 57.321◦ ), respectively. As in Fig. 5.1, the normalization of fields is such that .Ex (0) = 0 and .Ey (0) = 1 V m.−1 and the neighborhood considered is prescribed by the material orientations: ◦ ◦ .(α, β) = (αs1 + ξ, βs1 + ξ ) with .ξ = 0 (blue solid curve), .ξ = 0.1 (red short ◦ ◦ dashed curve), .ξ = −0.1 (red long dashed curve), .ξ = 0.5 (black short dashed curve), .ξ = −0.5◦ (black long dashed curve), .ξ = 2◦ (brown short dashed curve), and .ξ = −2◦ (brown long dashed curve). For the nondissipative material specified by .γ = 0, the components of the electric field phasor show no decay in Fig. 5.5 as propagation length increases. Indeed, in the case of propagation along the regular optic axis (i.e., .ξ = 0◦ for .γ = 0), .|Ex |, .|Ey |, and .|Ez | do not vary at all as propagation length increases, with .|Ex | being null valued regardless of .k0 z. Furthermore, there is no abrupt change in .|Ex |, .|Ey |, and .|Ez | as the direction of propagation converges upon the direction of the regular optic axis. For the dissipative materials considered, the components of the electric field phasors in Fig. 5.5 decay much more rapidly as propagation length increases for .γ = 5 than they do for .γ = 0.2. Also, there is no abrupt change in .|Ex |, .|Ey |, and .|Ez | as the direction of propagation converges upon the direction of the singular optic axis, for .γ = 0.2 or for .γ = 5. .

5.3 Propagation Through a Slab of a Biaxial Dielectric Material We now consider the more realistic scenario of plane waves incident on a slab of biaxial dielectric material.

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Fig. 5.5 As Fig. 5.1 but only the plots of the magnitudes ofthe components of  the electric field + i γ Im  where (i) phasor for the material with relative permittivity dyadic .Re  GaO GaO ◦ ◦ ◦ ◦ .γ = 0 for .(αs1 , βs1 ) = (138.876 , 78.420 ), (ii) .γ = 0.2 for .(αs1 , βs1 ) = (137.661 , 71.973 ), and (iii) .γ = 5 for .(αs1 , βs1 ) = (105.278◦ , 57.321◦ )

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Fig. 5.6 Schematic representation of a plane wave incident at angle θ on a slab of thickness d

5.3.1 Theory Suppose that a slab of biaxial dielectric material, specified by the relative permittivity dyadic .ε given in Eq. (5.3), occupies the region .0 < z < d, with the half-spaces rel .z < 0 and .z > d being vacuous. As schematically illustrated in Fig. 5.6, a plane wave propagating in the half-space .z < 0 is incident on the plane .z = 0. The direction of propagation of the incident plane is inclined at an angle .θ ∈ [0, π/2) with respect to the z axis and at an angle .ψ ∈ [0, 2π ) with respect to the x axis in the xy plane. The field phasors associated with the incident plane wave are conventionally expressed as [23] .

   Einc (r) = m+ e as , ap  exp (ik0 z cos θ) , Hinc (r) = m+ h as , ap exp (ik0 z cos θ)

z < 0,

(5.31)

wherein the vector functions .

m± e (c, d) = (c s + d p± ) exp {ik0 [(x cos ψ + y sin ψ) sin θ]} −1 m± h (c, d) = η0 (c p± − d s) exp {ik0 [(x cos ψ + y sin ψ) sin θ]}

 (5.32)

are defined using the unit vectors

.

s = −uˆ x sin ψ + uˆ y cos ψ   p± = ∓ uˆ x cos ψ + uˆ y sin ψ cos θ + uˆ z sin θ

 .

(5.33)

The scalars .as and .ap are complex-valued amplitudes of the s and p polarized components, respectively. The presence of the biaxial dielectric slab results in the reflected field phasors

.

   Eref (r) = m− e rs , rp exp (−ik0 z cos θ) ,   Href (r) = m− h rs , rp exp (−ik0 z cos θ)

z < 0,

(5.34)

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and the transmitted field phasors

.

   Etr (r) = m+ e ts , tp exp [ik0 (z − d) cos θ ] ,   Htr (r) = m+ h ts , tp exp [ik0 (z − d) cos θ ]

z > d.

(5.35)

As is comprehensively described elsewhere [23], the reflected phasor amplitudes .rs and .rp and the transmitted phasor amplitudes .ts and .tp can be deduced in terms of .as and .ap by solving the matrix-vector equation ⎡

⎡ ⎤ ⎤ ts as   ⎢t ⎥       ⎢a ⎥ p ⎢ p⎥ ⎥ . K •⎢ ⎣ 0 ⎦ = exp ik0 X d • K • ⎣ rs ⎦ , rp 0

(5.36)

where the 4 .× 4 matrixes ⎡

− sin ψ − cos ψ cos θ − sin ψ   ⎢ cos ψ − sin ψ cos θ cos ψ . K = ⎢ ⎣ −η−1 cos ψ cos θ η−1 sin ψ η−1 cos ψ cos θ 0 0 0 −η0−1 sin ψ cos θ −η0−1 cos ψ η0−1 sin ψ cos θ

⎤ cos ψ cos θ sin ψ sin θ ⎥ ⎥ η0−1 sin ψ ⎦ −η0−1 cos ψ (5.37)

and ⎡

q11   1 ⎢ q ⎢ 21 . X = ε33 ⎣ q31 q41

q12 q22 q32 −q31

q13 q23 q22 −q21

⎤ q14 −q13 ⎥ ⎥ −q12 ⎦ q11

(5.38)

with components

.

q11 q12 q13 q14 q21 q22 q23 q31 q32 q41

⎫ = −ε13 cos ψ sin θ ⎪ ⎪ ⎪ ⎪ = −ε23 cos ψ sin θ ⎪ ⎪ ⎪ 2 ⎪ ⎪ = η0 cos ψ sin ψ sin θ ⎪   ⎪ ⎪ 2 2 ⎪ = η0 ε33 − cos ψ sin θ ⎪ ⎪ ⎬ = −ε13 sin ψ sin θ . ⎪ = −ε23 sin ψ sin θ ⎪   ⎪ ⎪ ⎪ = −η0 ε33 − sin2 ψ sin2 θ ⎪ ⎪   ⎪ −1 2 = −η0 ε33 δ12 + cos ψ sin ψ sin θ ⎪ ⎪ ⎪   ⎪ ⎪ ⎪ = −η0−1 ε33 δ22 − cos2 ψ sin2 θ ⎪   ⎭ −1 2 2 = η0 ε33 δ11 − sin ψ sin θ

    Note that . X = Q when .θ = 0.

(5.39)

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The relations 

.

   ⎫ rs rss rsp as ⎪ ⎪ = ⎬ r r r a  p   ps pp  p ts t t as ⎪ ⎪ ⎭ = ss sp tp tps tpp ap

(5.40)

arise from Eq. (5.36), with linear reflection and linear transmission coefficients being defined as the elements of the 2 .× 2 matrixes therein. Linear reflectances and linear transmittances are defined as the magnitude squared of the corresponding linear reflection and linear transmission coefficients; for example, the linear reflectance corresponding to .rsp is .Rsp = |rsp |2 . Alternatively, the reflection/transmission problem may be formulated in terms of circularly polarized components, as follows. We introduce the vector functions

.

⎫ ! " is − p± is + p± ⎪ ⎪ {ik d) = c − d exp n± cos ψ + y sin ψ) sin θ √ √ (c, [(x ]} ⎬ 0 e 2 2 " ! . is + p± is − p± ⎪ −1 n± +d √ exp {ik0 [(x cos ψ + y sin ψ) sin θ ]} ⎪ c √ ⎭ h (c, d) = −iη0 2 2 (5.41)

   + + Then, in Eqs. (5.31), .m+ e as , ap is replaced by .ne (aL , aR ) and .mh as , ap is replaced by .n+ h (aL , aR ), with the scalars .aL and .aR being complex-valued amplitudes of the LCP and RCP components  of the incident plane wave, respectively. Accordingly, in Eqs. (5.34), .m− rs , rp is replaced by .n− L , −rR ) and e e (−r  − − + r .m , r is replaced by .n (−rL , −rR ), and in Eqs. (5.35), .me ts , tp is replaced s p h h + + by .n+ e (tL , tR ) and .mh ts , tp is replaced by .nh (tL , tR ). The complex-valued scalars .rL and .rR are measures of the strengths of the LCP and RCP components of the reflected plane wave, respectively, and analogously the complex-valued scalars .tL and .tR are measures of the strengths of the LCP and RCP components of the transmitted plane wave, respectively. The reflected phasor amplitudes .rL and .rR and the transmitted phasor amplitudes .tL and .tR can be deduced in terms of .aL and .aR by solving the matrix-vector equation ⎡ ⎡ ⎤ ⎤ i (tL − tR ) i (aL − aR )   ⎢ − (t + t ) ⎥       ⎢ − (a + a ) ⎥ L R ⎥ L R ⎥ ⎢ . K •⎢ ⎣ ⎦ = exp ik0 X d • K • ⎣ −i (rL − rR ) ⎦ . 0 rL + rR 0

(5.42)

Circular reflection and circular transmission coefficients are introduced as elements of the 2 .× 2 matrixes in the following relations:

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.

   ⎫ rL rLL rLR aL ⎪ ⎪ = ⎬ r r r a  R   RL RR  R . tL t t aL ⎪ ⎪ ⎭ = LL LR tR tRL tRR aR

(5.43)

Circular reflectances and circular transmittances are defined as the magnitude squared of the corresponding circular reflection and circular transmission coefficients; for example, the circular reflectance corresponding to .rLR is .RLR = |rLR |2 .

5.3.2 Numerical Investigations Suppose that a slab of thickness .d = 110 nm, made of .β-Ga.2 O.3 as specified by the relative permittivity dyadic in Eq. (5.29), fills the region .0 < z < d. As for Figs. 5.1, 5.2, 5.3 and 5.4, the two independent singular optic axes are aligned with the z axis for the material-orientation angles .(α, β) = (αs1 , βs1 ) and .(αs2 , βs2 ), where .αs1 = 125.193◦ , .βs1 = 60.168◦ , .αs2 = 124.360◦ , and .βs2 = 100.549◦ . We focus on cases of normal incidence, i.e., .θ = 0◦ and .ψ = 0◦ , in Figs. 5.7, 5.8, 5.9, 5.10, 5.11 and 5.12, with oblique incidence being considered in Figs. 5.13 and 5.14. Let us begin with the case in which the material-orientation angles lie in the neighborhood of .(αs1 , βs1 ), i.e., .(α, β) = (αs1 + ξ, βs1 + ζ ) with .−10◦ < ξ < 10◦ and .−10◦ < ζ < 10◦ . Plots of the circular reflectances .RLL , .RLR , and .RRR , and circular transmittances .TLL , .TLR , and .TRL versus .(ξ, ζ ) are displayed in Fig. 5.7. The corresponding plots of .RRL and .TRR are not displayed as they are identical to those for .RLR and .TLL , respectively. The circular reflectance .RLR is much larger than .RLL and .RRR , and the circular transmittance .TLL is much larger than .TLR and .TRL . Each of the quantities .RLL , .RLR , .TLL , and .TRL varies approximately linearly with .ξ and .ζ . Most strikingly, the circular reflectance .RRR and the circular transmittance ◦ ◦ .TLR both exhibit local minimums centered on .(ξ, ζ ) = (0 , 0 ), with both .RRR and ◦ ◦ .TLR being approximately null valued at .(ξ, ζ ) = (0 , 0 ). Thus, for the material orientation .(α, β) = (αs1 , βs1 ): (a) When the incident plane wave is RCP, the reflected plane wave is almost entirely LCP and the transmitted plane wave is almost entirely RCP. (b) When the incident plane wave is LCP, the reflected plane wave is mostly RCP and the transmitted plane wave is mostly LCP. Also, the circular reflectances and circular transmittances exhibit only smooth changes in the limit as the material orientation .(αs1 + ξ, βs1 + ζ ) → (αs1 , βs1 ). Next, we turn to the case in which the material-orientation angles lie in the neighborhood of .(αs2 , βs2 ). Plots of the circular reflectances .RLL , .RLR , and .RRR and circular transmittances .TLL , .TLR , and .TRL versus .(ξ, ζ ), corresponding to those in Fig. 5.7, are presented in Fig. 5.8. As for Fig. 5.7, the plots of .RRL and .TRR are not displayed in Fig. 5.8 because they are identical to those for .RLR and .TLL ,

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Fig. 5.7 Plots of circular reflectances .RLL , .RLR , and .RRR and circular transmittances .TLL , .TLR , and .TRL versus angular orientation coordinates .(ξ, ζ ) for a slab thickness .d = 0.55λ0 and relative . Material orientation: .(α, β) = (αs1 + ξ, βs1 + ζ ) with .(αs1 , βs1 ) = permittivity dyadic . GaO (125.193◦ , 60.168◦ )

respectively. The circular reflectance .RLR dominates over .RLL and .RRR , and the circular transmittance .TLL dominates over .TLR and .TRL . Each of the quantities .RLR , .RRR , .TLL , and .TLR varies approximately linearly with .ξ and .ζ . Conspicuous local minimums centered on .(ξ, ζ ) = (0◦ , 0◦ ) are exhibited by the plots of the circular reflectance .RLL and the circular transmittance .TRL , with both quantities being approximately null valued at .(ξ, ζ ) = (0◦ , 0◦ ). Thus, for the material orientation .(α, β) = (αs2 , βs2 ): (a) When the incident plane wave is LCP, the reflected plane wave is almost entirely RCP and and the transmitted plane wave is almost entirely LCP.

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Fig. 5.8 As Fig. 5.7 but with .(αs1 , βs1 ) replaced with .(αs2 , βs2 ) = (124.360◦ , 100.549◦ )

(b) When the incident plane wave is RCP, the reflected plane wave is mostly LCP and and the transmitted plane wave is mostly RCP. Also, none of the circular reflectances or circular transmittances exhibit abrupt changes in the limit as the material orientation converges toward .(αs2 , βs2 ). Now, let us consider linear reflectances and linear transmittances. As in Fig. 5.7, suppose that the material-orientation angles lie in the neighborhood of .(αs1 , βs1 ). Plots of the linear reflectances .Rss , .Rsp , and .Rpp and linear transmittances .Tss , .Tsp , and .Tpp versus .(ξ, ζ ) are presented in Fig. 5.9. The corresponding plots of .Rps and .Tps are not displayed as they are identical to those for .Rsp and .Tsp , respectively. The co-polarized reflectances .Rss and .Rpp are much larger than the cross-polarized reflectance .Rsp , and the co-polarized transmittances .Tss and .Tpp are much larger than the cross-polarized transmittance .Tsp . Each of the linear reflectances and linear

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Fig. 5.9 As Fig. 5.7 but with plots of linear reflectances .Rss , .Rsp , and .Rpp and linear transmittances .Tss , .Tsp , and .Tpp

transmittances varies approximately linearly with .ξ and .ζ ; i.e., unlike the case for circular reflectances and circular transmittances, plots of the linear reflectances and linear transmittances do not exhibit local minimums. Furthermore, all of the linear reflectances and linear transmittances exhibit only smooth variations in the limit as the material orientation converges toward .(αs1 , βs1 ). Plots of the linear reflectances .Rss , .Rsp , and .Rpp and linear transmittances .Tss , .Tsp , and .Tpp , analogous to those of Fig. 5.9 but for the material-orientation angles lying in the neighborhood of .(αs2 , βs2 ), are provided in Fig. 5.10. As observed for Fig. 5.9, the corresponding plots of .Rps and .Tps are identical to those for .Rsp and .Tsp , respectively, and accordingly are not presented. Also, .Rss Rsp and .Rpp Rsp , and .Tss Tsp and .Tpp Tsp . None of the plots of linear reflectances or linear

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Fig. 5.10 As Fig. 5.8 but with plots of linear reflectances .Rss , .Rsp , and .Rpp and linear transmittances .Tss , .Tsp , and .Tpp

transmittances exhibit local minimums. And, only smooth variations in each of the linear reflectances and linear transmittances are observed in the limit as the material orientation converges toward .(αs2 , βs2 ). As remarked earlier in connection with Fig. 5.5, the existence of Voigt waves and singular optic axes is intimately connected to material dissipation (or activity). Accordingly, it is illuminating here to compare plots of the reflectances and transmittances for the material specified by the relative permittivity dyadic .ε GaO with corresponding plots  for the nondissipative material specified by the relative permittivity dyadic .Re ε . In Fig. 5.11, plots of the circular reflectances .RLL GaO and .RLR and circular transmittances .TLL and .TLR versus .(ξ, ζ ) are provided

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Fig. 5.11 As Fig. 5.7 but plots of circular reflectances .RLL and .RLR and circular  transmittances and .TLR , for the slab material with relative permittivity dyadic .Re  . Material GaO orientation: .(α, β) = (αs1 + ξ, βs1 + ζ ) with .(αs1 , βs1 ) = (138.876◦ , 78.420◦ )

.TLL

 . The materialwhen the slab material has relative permittivity dyadic .Re ε GaO orientation neighborhood .(α, β) = (αs1 + ξ, βs1 + ζ ) is centered on .(αs1 , βs1 ) = (138.876◦ , 78.420◦ ), at which orientation the material’s regular optic axis is aligned with the z axis. The corresponding plots of .RRL , .RRR , .TRL , and .TRR are not displayed as they are identical to those for .RLR , .RLL , .TLR , and .TLL , respectively. In Fig. 5.11, the circular reflectance .RLR RLL and the circular transmittance .TLL TLR . Whereas .RLR and .TLL vary approximately linearly with .ξ and .ζ , the circular reflectance .RLL and the circular transmittance .TLR both exhibit local minimums centered on .(ξ, ζ ) = (0◦ , 0◦ ), with both .RRR and .TLR being approximately null valued at .(ξ, ζ ) = (0◦ , 0◦ ). Thus, for the material orientation .(α, β) = (αs1 , βs1 ): (a) When the incident plane wave is RCP, the reflected plane wave is almost entirely LCP and the transmitted plane wave is almost entirely RCP. (b) When the incident plane wave is LCP, the reflected plane wave is mostly RCP and the transmitted plane wave is mostly LCP. In addition, the circular reflectances and circular transmittances exhibit only smooth changes in the limit as the material orientation .(αs1 + ξ, βs1 + ζ ) → (αs1 , βs1 ). Qualitatively, the chief difference between the circular reflectances and circular

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Fig. 5.12 As Fig. 5.11 but with plots of linear reflectances .Rss , .Rsp , and .Rpp and linear transmittances .Tss , .Tsp , and .Tpp

 transmittances for the slab material with relative permittivity dyadic .Re ε (as GaO represented in Fig. 5.11) and those for the slab material with relative permittivity dyadic .ε (as represented in Fig. 5.7) is that .RLL = RRR and .TLR = TRL for GaO  , whereas .RLL = RRR and for .TLR = TRL for .ε . .Re ε GaO GaO Instead of the circular reflectances and circular transmittances  presented in Fig. 5.11 for the slab material with relative permittivity dyadic .Re ε , let us GaO now consider the corresponding linear reflectances and linear transmittances. Plots of the linear reflectances .Rss , .Rsp , and .Rpp and linear transmittances .Tss , .Tsp , and .Tpp , analogous to those of Fig. 5.11 with the material-orientation angles again lying in the neighborhood of .(αs1 , βs1 ), are displayed in Fig. 5.12. The corresponding

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85

plots of .Rps and .Tps are identical to those for .Rsp and .Tsp , respectively, and are accordingly not displayed. The cross-polarized reflectance .Rsp is much smaller than the co-polarized reflectances .Rss and .Rpp , and the cross-polarized transmittance .Tsp is much smaller than co-polarized transmittances .Tss and .Tpp . None of the plots of linear reflectances or linear transmittances exhibit local minimums. Furthermore, none of the plots of linear reflectances or linear transmittances change abruptly in the limit as the material orientation converges toward .(αs1 , βs1 ). In Figs. 5.7, 5.8, 5.9, 5.10, 5.11, and 5.12, only instances of normal incidence, i.e., .θ = 0◦ and .ψ = 0◦ , were under investigation. Let us now turn to instances of oblique incidence. For the slab material with relative permittivity dyadic .ε , GaO the material orientation is fixed at .(α, β) = (αs1 , βs1 ), where .αs1 = 125.193◦ and ◦ .βs1 = 60.168 . Thereby, a singular optic axis is aligned with the z axis. The angles of incidence are varied over the ranges .0◦ < ψ < 180◦ and .0◦ < θ < 20◦ . Plots of the circular reflectances .RLL , .RLR , .RRL , and .RRR and circular transmittances .TLL , .TLR , and .TRL versus .ψ and .θ are provided in Fig. 5.13. The corresponding plot of .TRR is identical to that for .TLL and is therefore not provided in Fig. 5.13. The circular reflectances .RLR and .RRL are much larger than .RLL and .RRR , and the circular transmittance .TLL is much larger than .TLR and .TRL . All circular reflectances and circular transmittances are insensitive to .ψ at small values of .θ but become increasingly sensitive to .ψ as .θ increases. None of the plots of circular reflectances or circular transmittances exhibit local minimums or abrupt changes as .ψ and .θ vary. Lastly, we consider obliqueincidence for the nondissipative slab material with . The material orientation is fixed at .(α, β) = relative permittivity dyadic .Re ε GaO ◦ ◦ (αs1 , βs1 ) = (138.876 , 78.420 ). Thereby, a regular optic axis is aligned with the z axis. In Fig. 5.14, plots of the circular reflectances .RLL , .RLR , .RRL , and .RRR and circular transmittances .TLL , .TLR , and .TRL versus .ψ and .θ are displayed. The corresponding plot of .TRR is not displayed in Fig. 5.14 as it is identical to that for .TLL . The plots of .RLL and .RRR in Fig. 5.14 look almost the same but in fact there are small differences between the two plots.  Plots for the nondissipative slab material in Fig. 5.14 are qualitatively rather with relative permittivity dyadic .Re ε GaO similar to those for the dissipative slab material with relative permittivity dyadic in Fig. 5.13. As in Fig. 5.13, no local minimums or abrupt changes as .ψ and .ε GaO .θ vary are exhibited in the plots of circular reflectances and circular transmittances in Fig. 5.14. All circular reflectances and circular transmittances in Fig. 5.14 are independent of .ψ at .θ = 0◦ but become increasingly dependent on .ψ as .θ increases.

5.4 Discussion Though Voigt plane waves in certain dissipative anisotropic materials have been known about for 120 years [4], they have not received much attention from the research community and the singular nature of their existence remains obscure. In

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Plots of circular reflectances .RLL , .RLR , .RRL , and .RRR and circular transmittances and .TRL versus angles of incidence .θ and .ψ for oblique incidence upon a slab . Material orientation: material of thickness .d = 0.55λ0 and relative permittivity dyadic . GaO ◦ ◦ .(α, β) = (αs1 , βs1 ) = (125.193 , 60.168 ) .TLL , .TLR ,

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 Fig. 5.14 As Fig. 5.13 but for the slab material with relative permittivity dyadic .Re  . GaO Material orientation: .(α, β) = (αs1 , βs1 ) = (138.876◦ , 78.420◦ )

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particular, from a mathematical perspective, a distinguishing feature of a Voigt plane wave propagating along a singular optic axis is that its amplitude is prescribed by the product of a linear function of propagation distance and an exponential function of propagation distance. In contrast, the amplitudes of plane waves that propagate along directions in the punctured neighborhood of a singular optic axis are prescribed only by exponential functions of propagation distance. A natural question is: how does the amplitude of a plane wave propagating in the neighborhood of a singular optic axis change in the limit as the direction of propagation converges upon the singular optic axis? To address this question, numerical studies were undertaken to dissect the anatomy of Voigt plane waves in dissipative biaxial materials. Both propagation in an unbounded biaxial material and reflectances and transmittances for a slab of biaxial material were investigated. It was found that in the limit as the direction of plane-wave propagation tends toward the direction of a singular optic axis, there is a smooth variation in the electric field phasor and in the time-averaged Poynting vector, regardless of polarization state. Parenthetically, there is also a smooth variation in the components of the magnetic field phasor, though results for .H(r) were not presented here. It was also found that reflectances and transmittances for normal incidence on a biaxial dielectric slab exhibit a smooth variation as the orientation of the slab material varies within a neighborhood centered on an orientation with a singular optic axis aligned with the normal direction. Furthermore, reflectances and transmittances for oblique incidence on a biaxial dielectric slab exhibit a smooth variation as the angles of incidence vary with a singular optic axis of the slab material aligned with the normal direction. Therefore, in terms of amplitude dependency on propagation distance, a Voigt plane wave propagating along a singular optic axis is practically indistinguishable from a plane wave propagating along directions in the immediate neighborhood of the singular optic axis. Indeed, at least in terms of field amplitudes, there is no singularity associated with the singular optic axes.

Epilog on Werner S. Weiglhofer TGM writes: I first met Werner during the academic year 1996/97 when I attended his undergraduate course on mathematical methods in the Department of Mathematics at the University of Glasgow. I subsequently became his first (and only) PhD student. Upon graduating in 2001, I moved to the University of Edinburgh; we continued to work together, mostly on homogenization of complex mediums [25– 30] and especially on the strong-property-fluctuation theory [31–36]. As time went by, our work and non-work lives overlapped more: Edinburgh to Glasgow was a short hop. Shortly before he died, we chatted about Voigt waves and some rough plans were sketched out—I imagine he would be amused to see that after almost 20 years I am still working on them.

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Acknowledgments We thank Alan Brenier for an illuminating discussion on the signature of Voigt plane waves. TGM was supported by EPSRC (grant number EP/V046322/1). AL thanks the Charles Godfrey Binder Endowment at The Pennsylvania State University for partial support of his research endeavors.

References 1. Born, M., Wolf, E.: Principles of Optics, 6th edn. Pergamon Press, Oxford (1980) 2. Chen, H.C.: Theory of Electromagnetic Waves. McGraw–Hill, New York (1983) 3. Mackay, T.G., Lakhtakia, A.: Electromagnetic Anisotropy and Bianisotropy: A Field Guide, 2nd edn. World Scientific, Singapore (2019) 4. Voigt, W.: On the behaviour of pleochroitic crystals along directions in the neighbourhood of an optic axis. Philos. Mag. 4, 90–97 (1902) 5. Pancharatnam, S.: The propagation of light in absorbing biaxial crystals — I. Theoretical. Proc. Ind. Acad. Sci. A 42, 86–109 (1955) 6. Agranovich, V.M., Ginzburg, V.L.: Crystal Optics with Spatial Dispersion, and Excitons. Springer, Berlin (1984) 7. Fedorov, F.I., Goncharenko, A.M.: Propagation of light along the circular optic axes of absorbing crystals. Opt. Spectrosc. 14, 51–53 (1963) 8. Grechushnikov, B.N., Konstantinova, A.F.: Crystal optics of absorbing and gyrotropic media. Comput. Math. Appl. 16, 637–655 (1988) 9. Borzdov, G.N.: Waves with linear, quadratic and cubic coordinate dependence of amplitude in crystals. Pramana J. Phys. 46, 245–257 (1996) 10. Lakhtakia, A.: Anomalous axial propagation in helicoidal bianisotropic media. Opt. Commun. 157, 193–201 (1998) 11. Gerardin, J., Lakhtakia, A.: Conditions for Voigt wave propagation in linear, homogeneous, dielectric mediums. Optik 112, 493–495 (2001) 12. Berry, M.V., Dennis, M.R.: The optical singularities of birefringent dichroic chiral crystals. Proc. R. Soc. Lond. A 459, 1261–1292 (2003) 13. Berry, M.V.: The optical singularities of bianisotropic crystals. Proc. R. Soc. Lond. A 461, 2071–2098 (2005) 14. Mackay, T.G., Lakhtakia, A.: Voigt wave propagation in biaxial composite materials. J. Opt. A: Pure Appl. Opt. 5, 91–95 (2003) 15. Mackay, T.G., Lakhtakia, A.: Correlation length facilitates Voigt wave propagation. Waves Random Media 14, L1–L11 (2004) 16. Mackay, T.G.: Voigt waves in homogenized particulate composites based on isotropic dielectric components. J. Opt. (Bristol) 13, 105702 (2011) 17. Mackay, T.G.: On the sensitivity of directions that support Voigt wave propagation in infiltrated biaxial dielectric materials. J. Nanophoton. 8, 083993 (2014) 18. Mackay, T.G.: Controlling Voigt waves by the Pockels effect. J. Nanophoton. 9, 093599 (2015) 19. Mackay, T.G., Lakhtakia, A.: On the propagation of Voigt waves in energetically active materials. Eur. J. Phys. 37, 064002 (2016) 20. Sturm, C., Grundmann, M.: Singular optic axes in biaxial crystals and analysis of their spectral dispersion effects in β-Ga2 O3 . Phys. Rev. A 93, 053839 (2016) 21. Grundmann, M., Sturm, C., Kranert, C., Richter, S., Schmidt–Grund, R., Deparis, C., Zúñiga– Pérez, J.: Optically anisotropic media: New approaches to the dielectric function, singular axes, microcavity modes and Raman scattering intensities. Phys. Status Solidi RRL 11, 1600295 (2017) 22. Brenier, A.: Voigt wave investigation in the KGd(WO4 )2 : Nd biaxial laser crystal. J. Opt. (Bristol) 17, 075603 (2015)

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23. Mackay, T.G., Lakhtakia, A.: The Transfer-Matrix Method in Electromagnetics and Optics. Morgan and Claypool, San Rafael (2020) 24. Boyce, W.E., DiPrima, R.C.: Elementary Differential Equations and Boundary Value Problems, 9th edn. Wiley, Hoboken (2010) 25. Weiglhofer, W.S., Mackay, T.G.: Numerical studies of the constitutive parameters of a chiroplasma composite medium. Archiv für Elektronik und Übertragungstechnik (Int. J. Electron. Commun.) 54, 259–265 (2000) 26. Mackay, T.G., Weiglhofer, W.S.: Homogenization of biaxial composite materials: dissipative anisotropic properties. J. Opt. A: Pure Appl. Opt. 2, 426–432 (2000) 27. Michel, B., Lakhtakia, A., Weiglhofer, W.S., Mackay, T.G.: Incremental and differential Maxwell Garnett formalisms for bi-anisotropic composites. Compos. Sci. Technol. 61, 13–18 (2001) 28. Mackay, T.G., Weiglhofer, W.S.: Homogenization of biaxial composite materials: bianisotropic properties. J. Opt. A: Pure Appl. Opt. 3, 45–52 (2001) 29. Mackay, T.G., Weiglhofer, W.S.: Homogenization of biaxial composite materials: nondissipative dielectric properties. Electromagnetics 21, 15–26 (2001) 30. Weiglhofer, W.S., Mackay, T.G., Needles and pillboxes in anisotropic mediums. IEEE Trans. Antenn. Propag. 50, 85–86 (2002) 31. Mackay, T.G., Lakhtakia, A., Weiglhofer, W.S.: Third-order implementation and convergence of the strong-property-fluctuation theory in electromagnetic homogenisation. Phys. Rev. E 64, 066616 (2001) 32. Mackay, T.G., Lakhtakia, A., Weiglhofer, W.S.: Homogenisation of similarly oriented, metallic ellipsoidal inclusions using the bilocally-approximated strong-property-fluctuation theory. Opt. Commun. 197, 89–95 (2001) 33. Mackay, T.G., Lakhtakia, A., Weiglhofer, W.S.: Ellipsoidal topology, orientation diversity and correlation length in bianisotropic mediums. Archiv für Elektronik und Übertragungstechnik (Int. J. Electron. Commun.) 55, 259–265 (2001) 34. Mackay, T.G., Lakhtakia, A., Weiglhofer, W.S.: Strong-property-fluctuation theory for homogenization of bianisotropic composites: formulation. Phys. Rev. E 62, 6052–6064 (2000); Erratum 63, 049901 (2001) 35. Mackay, T.G., Lakhtakia, A., Weiglhofer, W.S.: Homogenisation of isotropic, cubically nonlinear, composite mediums by the strong-permittivity-fluctuation theory: third-order considerations. Opt. Commun. 204, 219–228 (2002) 36. Mackay, T.G., Lakhtakia, A., Weiglhofer, W.S.: The strong-property-fluctuation theory for cubically nonlinear, isotropic chiral composite mediums. Electromagnetics 23, 455–479 (2003)

Tom G. Mackay is a professor in the School of Mathematics at the University of Edinburgh and an adjunct professor in the Department of Engineering Science and Mechanics at Pennsylvania State University. He graduated from the Universities of Edinburgh, Glasgow, and Strathclyde. His research has been supported by awards from the Carnegie Trust for The Universities of Scotland, Engineering and Physical Sciences Research Council, Nuffield Foundation, Royal Academy of Engineering/Leverhulme Trust, and Royal Society of Edinburgh/Scottish Executive. He is a fellow of the Institute of Physics (UK) and of SPIE— The International Society for Optics and Photonics. His current research interests include homogenization, complex materials, and surface waves.

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Akhlesh Lakhtakia received the BTech (1979) and DSc (2006) degrees from the Banaras Hindu University and the MS (1981) and PhD (1983) degrees from the University of Utah. In 1983 he joined the Department of Engineering Science and Mechanics at Penn State as a post-doctoral research scholar, where he became a Distinguished Professor in 2003, the Charles Godfrey Binder Professor in 2006, and the Evan Pugh University Professor of Engineering Science and Mechanics in 2018. His current research interests include electromagnetic scattering, surface multiplasmonics, bioreplication, forensic science, solar energy, sculptured thin films, and mimumes. He has been elected a fellow of Optical Society of America (1992), SPIE—The International Society for Optical Engineering (1996), Institute of Physics (UK) (1996), American Association for the Advancement of Science (2010), American Physical Society (2012), Institute of Electrical and Electronics Engineers (2016), Royal Society of Chemistry (2016), and Royal Society of Arts (2017). He has been designated a Distinguished Alumnus of both of his almae matres at the highest level. Awards at Penn State include: Outstanding Research Award (1996), Outstanding Advising Award (2005), Premier Research Award (2008), and Outstanding Teaching Award (2016), and the Faculty Scholar Medal (2005). He received the 2010 SPIE Technical Achievement Award, the 2016 Walston Chubb Award for Innovation from Sigma Xi, the 2022 SPIE Smart Structures and Materials Lifetime Achievement Award, the 2022 Radio Club of America Lifetime Achievement Award, and the 2022 IEEE APS Distinguished Achievement Award. He is a Sigma Xi Distinguished Lecturer (2022–24) and a Jefferson Science Fellow at the US State Department (2022–23).

Chapter 6

Electromagnetic Radiation by Finite-Sized Electric and Magnetic Dipoles Embedded in Homogeneous Uniaxial Dielectric Materials Aamir Hayat and Muhammad Faryad

6.1 Introduction Materials having a single axis of symmetry are called uniaxial materials [1–4]. Two distinct plane waves can propagate in each direction in a uniaxial material except in a direction parallel to .±ˆc, where the unit vector .cˆ is parallel to the axis of symmetry of the uniaxial material. Whenever these waves propagate parallel to .±ˆc, both of these waves have the same phase speed and attenuation rate. The axis that is parallel to .cˆ is called the optic axis [3, 5]. A plane wave that propagates with the same phase speed in all directions in the uniaxial material is called an ordinary plane wave. A plane wave whose phase speed depends upon the direction of propagation and is different from the phase speed of ordinary waves unless the wave is propagating along the optic axis of the material is called an extraordinary plane wave [6]. Uniaxial materials exist in nature [7], e.g., rutile, calcite, and quartz [8, 9], and they can also be fabricated artificially from simple isotropic materials, e.g., a stack of thin films with alternating high and low refractive indexes or by dispersing parallel cylindrical inclusions in an isotropic host [10]. Out of seven crystal systems, the trigonal, tetragonal, and hexagonal crystal systems are uniaxial. Also, many other crystals can be approximated very well as uniaxial materials. Uniaxial materials are ubiquitous in optical components, e.g., birefringent filters, birefringent lenses, waveplates, birefringent interferometers, and nonlinear optical effect generators [11]. Several uniaxial crystals such as quartz have been used

A. Hayat Department of Physics, The University of Lahore, Sargodha, Pakistan e-mail: [email protected] M. Faryad () Department of Physics, Lahore University of Management Sciences, Lahore, Pakistan e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 T. G. Mackay, A. Lakhtakia (eds.), Adventures in Contemporary Electromagnetic Theory, https://doi.org/10.1007/978-3-031-24617-3_6

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to make specialized lenses, windows, and filters used in lasers, microscopes, telescopes, electronic sensors, and scientific instruments [12]. Uniaxial materials occupy an important place in optics because of their varied applications including imaging [13, 14], sensing [15], cloaking [16], waveguiding [17], and simulating space-time phenomena [18]. Artificially prepared uniaxial materials with unconventional material parameter have been used to achieve unprecedented functionality in the control of electromagnetic waves, such as superlensing and negative refraction [19]. The different values of the components of permittivity dyadic of the uniaxial material cause the dispersion relation of uniaxial materials to be an elliptic or a hyperbolic equation. Hyperbolic materials show super-resolution in the far zone through image magnification, negative refraction, and enhanced spontaneous emission (Purcell effect) [20]. Moreover, these materials find applications in nano imaging [21], nano sensing [22], selective resonance [23], efficient absorption [24], and subdiffraction imaging [25]. The application of hyperbolic materials in designing tunable nanophotonic devices is explained in detail in a chapter of this volume [26]. Radiation by canonical sources in unbounded materials forms the bedrock for building solutions for radiation by arbitrary sources and scattering by arbitrary obstacles. Therefore, analytical results for radiation have been worked out for point electric and magnetic dipoles in an unbounded uniaxial material [1, 27, 28]. The point sources are not only important for constructing solutions for finite sources but can also provide approximate solutions for finite sources because finite but small sources of radiation can be modeled as point dipoles as a first-order approximation. This chapter deals with the next order of approximation of finite sources in the uniaxial materials by taking into account their finite size. The size of the radiation source plays a critical role in the directivity of radiation in unbounded isotropic dielectric materials [29, 30]. The effect of size on the directivity could be more important in uniaxial materials because a special direction (optic axis) is already present in the material. To generalize point electric dipole, we considered a finitelength electric dipole because several electrically small sources of electromagnetic waves can be modeled as dipoles of finite length such as quantum dots or quantum wires with uniform current distributions. Furthermore, the finite-length electric dipole is also highly useful for helping to analyze larger radiation sources, which can be subdivided into short sections with uniform currents. To generalize the point magnetic dipole, we considered a uniform current loop of finite radius. The current loop with vanishing radius can model a point magnetic dipole. Radiation by point dipoles placed inside or outside bounded uniaxial materials has also been studied by various authors. Radiation over a layered material with its optic axis lying perpendicular to the plane of stratification has been studied by Tsang et al. [31], Kong [32], Kwon and Wang [33], and Tang [34], while the same problem with a point electric dipole embedded in a stratified material has been studied by Ali and Mahmoud [35]. Radiation emitted by an arbitrarily oriented point electric dipole over or inside two layers of uniaxially anisotropic materials has been treated analytically [36]. However, this chapter is focused on the presentation of the analytical expressions of the electromagnetic fields radiated by finite-sized sources

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in the unbounded uniaxial dielectric materials. The analytical expressions can then be used to construct solutions of the radiation and scattering problems in bounded and unbounded uniaxial materials. Let us note that electromagnetic waves generated by finite sources in a uniaxial material have been tackled numerically [28], and integral equations for generalized electric and magnetic currents have also been obtained [37]. The radiation resistance of an electrically small electric/magnetic dipole has been studied in a cold uniaxial plasma [38]. Also, the current distribution of a thin conducting loop inside a uniaxial material when the optic axis of the material and symmetry axis of the loop are the same has also been studied [39]. Electromagnetic characteristics of loop antennas that are placed over the surface of uniaxial anisotropic cylinder were studied by Kudrin et al. [40]. Radiation by as well as scattering from a microstrip patch antenna on a uniaxial material were studied by Pozar [41]. Furthermore, the problem of determining the current distribution on a loop of a thin strip coiled into a ring was studied by Zaboronkova et al. [39]. However, this chapter is concerned with closedform solutions for the finite-sized electric dipole and magnetic dipole, the latter in the form of a current loop, instead of numerical results for some specialized sources. The plan of this chapter is as follows: The dyadic Green functions and their approximate forms in the near and far zones are presented in Sect. 6.2. The electromagnetic fields and power radiated by the point electric and magnetic dipoles are presented in Sect. 6.3. The detailed derivations and numerical results for the finite-sized electric dipole and a current loop are presented in Sects. 6.4 and 6.5, respectively. Concluding remarks are presented in Sect. 6.6. An .exp(−iωt) time dependence is implicit, where t is the time, .ω is the angular frequency, and .i = √ √ −1. Moreover, .εo , .μo , and .ko = ω εo μo represent the free-space permittivity, permeability, and wavenumber, respectively. Boldface letters represent vectors, and symbols underlined twice represent dyadics [3, Chap. 1]. The dyadic Green functions are represented by upper-case letters that are underlined twice, such as .G, and the scalar Green functions are represented by lower-case letters, for example, g. The identity dyadic is represented as .I .

6.2 Dyadic Green Functions A function that represents the solution of a linear differential equation with only spatial variables for which the source is localized in space is known as the Green function. A source function that is localized in space is called a point source. It is useful in finding the solution for a source that has a finite spatial domain by representing the source as a dense distribution of point sources [2]. A function that maps a vector source into vector solution of a differential equation is known as dyadic Green function. There are several physical processes that require dyadic Green functions, e.g., radiation emitted by a current source in a material and generation of elastodynamic waves by a mechanical source in a material [2, 3].

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Several research articles, books, and chapters have been written on dyadic Green functions in electromagnetism [1–3]. The dyadic Green functions for the uniaxial materials have been derived earlier [1–3, 27] and are reproduced here for completeness. Let us begin with the time-harmonic Maxwell equations [1, 3] ∇ × H(r) + iωD(r) = Je (r) ,

(6.1)

∇ × E(r) − iωB(r) = −Jm (r) ,

(6.2)

.

∇ · D(r) = ρe (r) ,

(6.3)

∇ · B(r) = ρm (r) ,

(6.4)

.

.

and .

where .Je (r), .Jm (r), .ρe (r), and .ρm (r) are the electric current density, magnetic current density, the electric charge density, and magnetic charge density, respectively, and these are related by the continuity equations .∇ · Je (r) − iωρe (r) = 0 and .∇ · Jm (r) − iωρm (r) = 0. The frequency-domain constitutive relations of a uniaxial dielectric material can be written as D(r) = εo ε · E(r) ,

.

r

B(r) = μo μb H(r) ,

(6.5)

with the relative permittivity dyadic defined as ε = εb I + (εa − εb )ˆccˆ .

.

r

(6.6)

Since Eqs. (6.1) and (6.2) are linear in .E and .H, the electric and magnetic field phasors can be written in terms of the dyadic Green functions as    Gee (R) · Je (r ) + Gem (R) · Jm (r ) d 3 r .E(r) =

(6.7)

V

and    Gme (R) · Je (r ) + Gmm (R) · Jm (r ) d 3 r , .H(r) =

(6.8)

V

where .V  is the volume occupied by the current densities and R = r − r .

.

(6.9)

The electric dyadic Green function .Gee (R) and magnetoelectric dyadic Green function .Gme (R) can be found as [2, 3]

6 Electromagnetic Radiation by Finite-Sized Electric and Magnetic Dipoles. . .

  −1 1− .G (R) = iωμo μb ge (R)εa ε ee

r

97

 1 1 − iko no Re (ko no Re )2

 εa2 (ε−1 · R)(ε −1 · R) 3 3 r r − iko no Re (ko no Re )2 Re2  Rgo (R) − Re ge (R) 1 εb go (R) − εa ge (R) K(R) + + εb iko no (R × cˆ ) · (R × cˆ )   × I − cˆ cˆ − 2K(R) (6.10)  −ge (R) 1 −

and Gme (R) =

.

εa (R × cˆ )[R × (R × cˆ )] (1 − iko no Re )ge (R) 2 εb Re (R × cˆ ) · (R × cˆ )   [ˆc × (R × cˆ )](R × cˆ ) + (R × cˆ )[ˆc × (R × cˆ )] + ge (R) − go (R) (R · cˆ ) [(R × cˆ ) · (R × cˆ )]2 −(1 − iko no R)go (R)

[R × (R × cˆ )](R × cˆ ) , R 2 [(R × cˆ ) · (R × cˆ )]

(6.11)

(R × cˆ )(R × cˆ ) , (R × cˆ ) · (R × cˆ )

(6.12)

1 1 1 cˆ cˆ , I− − εa εb εb

(6.13)

where K(R) =

.

ε−1 =

.

r

and no =

.

√ √ εb μb ,

√ ko = ω μo εo .

(6.14)

The scalar Green functions go (R) =

.

exp(iko no R) 4π R

and

ge (R) =

exp(iko no Re ) 4π Re

(6.15)

represent ordinary and extraordinary waves, respectively, and Re =

.

εa (R × cˆ ) · (R × cˆ ) + (R · cˆ )2 . εb

(6.16)

The ordinary waves propagate with the same phase speed in all directions in the uniaxial material, but the extraordinary waves have their phase speed dependent

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upon the direction of propagation. However, when the direction of propagation is along the axis of symmetry (optic axis), both the ordinary and extraordinary waves have the same phase speed [3]. The dyadic Green function .Gmm (R) can be found as [2]   Gmm (R) = iωεo εb go (R) 1 +

 1 i − I ko no R (ko no R)2   3i 3 ˆR ˆ −go (R) 1 + − R ko no R (ko no R)2  Rgo (R) − Re ge (R) 1 εb go (R) − εa ge (R) K(R) − − iko no (R × cˆ ) · (R × cˆ ) εb   × I − cˆ cˆ − 2K(R) , (6.17)

.

and the magnetoelectric dyadic Green function .Gem (R) can be found as Gem (R) =

.

εa [R × (R × cˆ )](R × cˆ ) (1 − iko no Re )ge (R) 2 εb Re (R × cˆ ) · (R × cˆ )   [ˆc × (R × cˆ )](R × cˆ ) + (R × cˆ )[ˆc × (R × cˆ )] + ge (R) − go (R) (R · cˆ ) [(R × cˆ ) · (R × cˆ )]2 −(1 − iko no R)go (R)

(R × cˆ )[R × (R × cˆ )] . R 2 [(R × cˆ ) · (R × cˆ )]

(6.18)

Equation (6.18) can also be obtained by taking the transpose of the right side of Eq. (6.11). Also, an alternative derivation of .Gmm (R) from .Gee (R) is available [42]. Usually, we are interested in the electromagnetic fields in the near and far zones of a source. In the near zone, .ko no R  1 and .ko no Re  1, whereas in the far zone, .ko no R  1 and .ko no Re  1. In the near zone, the dyadic Green function can be approximated by retaining terms proportional to .1/R 3 since we have .ko no R  1 and .ko no Re  1. Therefore, Eqs. (6.10) and (6.11) can be written as ⎡ iεa ee ⎣ .G (R) ≈ 4π ωεo εb

   3εa ε−1 · R ε−1 · R r

r

Re5

−

ε−1 r

Re3

⎤ ⎦

(6.19)

and Gme (R) ≈ 0 ,

.

respectively, in the near zone.

(6.20)

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99

To compute the electromagnetic field in the far zone, we retain the terms proportional to .1/R. So, Eq. (6.10) can be approximated as  εa2 (ε−1 · R)(ε −1 · R) r r Gee (R) ≈ iωμo μb ge (R)εa ε −1 − ge (R) r Re2   1 εb go (R) − εa ge (R) K(R) . + εb

.

(6.21)

After using the identity [2, 3] −1 .ε r

−

εa (ε−1 · R)(ε −1 · R) r

r

Re2

     R × R × cˆ R × R × cˆ 1    − K(R) = , εb εb Re2 R × cˆ · (R × cˆ (6.22)

Eq. (6.21) can be rearranged as        εa R × R × cˆ R × R × cˆ   . Gee (R) = iωμo μb ge (R) (R)K(R) + g o εb Re2 (R × cˆ · (R × cˆ (6.23) Similarly, Eq. (6.11) can be written as .

  (R × cˆ )[R × (R × cˆ )] [R × (R × cˆ )](R × cˆ ) εa − ge (R) .G (R) ≈ iko no go (R) εb Re [(R × cˆ ) · (R × cˆ )] R(R × cˆ ) · (R × cˆ ) (6.24) in the far zone. Since we do not require the approximate forms of the dyadic Green functions mm (R) and .Gem (R), they are not provided here. .G me

6.3 Point Dipoles In this section, the radiated fields of point electric and magnetic dipoles are discussed in the near and far zones for later comparison with finite-sized sources.

6.3.1 Point Electric Dipole Let us consider a point electric dipole having a dipole moment .po and located at the origin. The current densities for the point electric dipole are given by [2, 3] Je (r) = −iωpo δ(r),

.

Jm (r) = 0 .

(6.25)

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As the detailed derivation is given in Ref. [2], we present the final results of the electric and magnetic fields in the near and far zones. The electric field and the magnetic field of a point electric dipole when we are very close to the dipole, i.e., .ko no r  1 and .ko no re  1, are given by [2]   ⎡  ⎤ −1 −1 1 εa ⎣ εa ε r · r ε r · r E(r) = − ε−1 ⎦ · po r 4π εo re3 εb re2 . H(r) = 0

,

(6.26)

,

where re =

.

εa (r × cˆ ) · (r × cˆ ) + (r · cˆ )2 . εb

(6.27)

Equation (6.26) represents the electrostatic field of a point dipole in the uniaxial material since it is independent of .ω. Similarly in the far zone, the electric and magnetic fields are given by [2]     εa r × (r × cˆ ) r × (r × cˆ ) + go (r)K(r) .E(r) = ω μo μb ge (r) εb re2 (r × cˆ ) · (r × cˆ )  rgo (r) − re ge (r)  I − cˆ cˆ − 2K(r) · po , (6.28) + iko no (r × cˆ ) · (r × cˆ ) 2

     r × (r × cˆ ) (r × cˆ ) εa (r × cˆ ) r × (r × cˆ ) +go (r) ·po . .H(r) = ωko no −ge (r) εb re (r × cˆ ) · (r × cˆ ) r(r × cˆ ) · (r × cˆ ) (6.29) Now, suppose that the point electric dipole is oriented parallel to the optic axis, i.e., ˆ , then the electric and magnetic fields become [2] .po = po c 

 i 1 cˆ − ko no re (ko no re )2   ε (ˆc · r)(ε −1 · r)  a 3i 3 r − 1+ ge (r) , − ko no re (ko no re )2 re2

 1+ .E(r) = ω po μo μb 2

(6.30)

and H(r) = iωpo

.

εa (r × cˆ ) (1 − iko no re ) ge (r) , εb re2

(6.31)

6 Electromagnetic Radiation by Finite-Sized Electric and Magnetic Dipoles. . .

101

respectively. As only .ge (r) is present in both Eqs. (6.30) and (6.31), it means that when the electric dipole is aligned parallel to the direction of the optic axis, only extraordinary waves are emitted. The time-averaged power radiated per unit solid angle by the point dipole is given by [29, 30] 1 dP = rˆ · Re(E × H∗ )r 2 , d 2

.

(6.32)

where . denotes the solid angle. When both the optic axis and the point electric dipole are parallel to the z axis, the time-averaged power radiated per unit solid angle can be found by substituting Eqs. (6.28) and (6.29) into Eq. (6.32) as [43] .

k 4 μb no po2 cρε2 dP = o sin2 θ d 32π 2 εo 5

(6.33)

in spherical coordinates, where ρε =

.

εa , εb

 =

cos2 θ + ρε sin2 θ.

(6.34)

When the point dipole is parallel to the z axis and the optic axis is parallel to the x axis, the time-averaged power per unit solid angle can be found in two parts because we have both ordinary and extraordinary waves since the electric field and magnetic field satisfy the orthogonality relations [6] rˆ · (Ee × H∗o ) = 0,

.

rˆ · (Eo × H∗e ) = 0 .

(6.35)

We can find the total radiated power by adding the radiated powers of ordinary and extraordinary waves separately, i.e., dPo dPe dP = + , d d d

(6.36)

1 dPo = rˆ · Re(Eo × H∗o )r 2 d 2

(6.37)

1 dPe = rˆ · Re(Ee × H∗e )r 2 . 2 d

(6.38)

.

where .

and .

Now the time-averaged power radiated per unit solid angle by the point dipole as ordinary waves can be found by substituting the ordinary parts of Eqs. (6.28) and (6.29) into Eq. (6.37) as [43]

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A. Hayat and M. Faryad

.

k 4 μb no po2 c sin2 θ sin2 φ dPo = o d 32π 2 εo sin2 θ sin2 φ + cos2 θ

(6.39)

in spherical coordinates. Similarly, the time-averaged power radiated by the point electric dipole as extraordinary waves can be found by substituting the extraordinary parts of Eqs. (6.28) and (6.29) into Eq. (6.38) and converting into spherical coordinates as .

k 4 μb no po2 cρε2 sin2 θ cos2 φ cos2 θ dPe = o , d 32π 2 εo 5 sin2 θ sin2 φ + cos2 θ

(6.40)

where (θ, φ) =

.

  sin2 θ cos2 φ + ρε sin2 θ sin2 φ + cos2 θ .

(6.41)

The total time-averaged power radiated per unit solid angle by the point dipole can be found by substituting Eqs. (6.39) and (6.40) into Eq. (6.36) as [43] .

k 4 μb no po2 c dP = o d 32π 2 εo



ρε2 cos2 θ cos2 φ + sin2 φ 5



sin2 θ sin2 θ sin2 φ + cos2 θ

. (6.42)

6.3.2 Point Magnetic Dipole Now, let us consider a point magnetic dipole located at the origin with dipole moment .mo with the current densities given by [2] Je (r) = 0,

.

Jm (r) = −iωmo δ(r) .

(6.43)

The electric and magnetic fields in the near zone are given by [2] E(r) = 0 , .

H(r) =

  1 3ˆr(ˆr · mo ) − mo . 3 4π μo μb r

(6.44)

It is clear from Eq. (6.44) that it represents a magnetostatic field because only the magnetic field is present in the absence of .ω. Moreover, when we are very far away from the magnetic dipole, the approximate electric and magnetic fields are given by [2]

6 Electromagnetic Radiation by Finite-Sized Electric and Magnetic Dipoles. . .

103

      (r × cˆ ) r × (r × cˆ ) εa r × (r × cˆ ) (r × cˆ ) · mo , .E(r) = ωko no go (r) − ge (r) εb re (r × cˆ ) · (r × cˆ ) r(r × cˆ ) · (r × cˆ ) (6.45) and     1 2 ˆ rˆ − (6.46) .H(r) = ω εo εb go (r) I − r [εb go (r) − εa go (r)] K(r) · mo . εb Only the terms proportional to .1/r were retained in Eqs. (6.45) and (6.46) under the far-zone approximation. As a special case, when the magnetic dipole is aligned along the direction of the optic axis, i.e., .m = mo cˆ , then the electric and magnetic fields can be written as [2] E(r) = −iωmo (1 − iko no r)

.

(r × cˆ ) go (r) , r2

(6.47)

and  i 1 cˆ − ko no r (ko no r)2    3i 3 ˆ ˆ r (ˆ r · c ) go (r) , − 1+ − ko no r (ko no r)2

 .H(r) = ω mo εo εb 1+ 2

(6.48)

respectively. Equations (6.47) and (6.48) show that only ordinary waves are emitted by a point magnetic dipole because .ge (r) is absent in both equations. If we compare these equations with Eqs. (6.30) and (6.31), we can see that electric and magnetic point dipoles only emit extraordinary and ordinary waves, respectively, when they are aligned parallel to the optic axis. When both the optic axis and the point magnetic dipole are parallel to the z axis, the time-averaged power radiated per unit solid angle can be found by substituting Eqs. (6.45) and (6.46) into Eq. (6.32) as .

k 4 εb no m2o c 2 dP = o sin θ d 32μo π 2

(6.49)

in spherical coordinates. When the point magnetic dipole is parallel to the z axis and the optic axis is parallel to the x axis, the time-averaged power radiated by the point magnetic dipole as ordinary waves can be found by substituting the ordinary parts of Eqs. (6.45) and (6.46) into Eq. (6.37) as [43] .

k 4 εb no m2o c sin2 θ cos2 θ cos2 φ dPo = o . d 32μo π 2 sin2 θ sin2 φ + cos2 θ

(6.50)

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A. Hayat and M. Faryad

Similarly, the time-averaged power radiated by the point magnetic dipole as extraordinary waves can be found by substituting the extraordinary parts of Eqs. (6.45) and (6.46) into Eq. (6.38) as [43] .

sin2 θ sin2 φ dPe k 4 εb no m2o cρε2 . = o d 32μo π 2 3 sin2 θ sin2 φ + cos2 θ

(6.51)

The total time-averaged power radiated per unit solid angle by the point magnetic dipole can be found by substituting Eqs. (6.50) and (6.51) into Eq. (6.36) as   sin2 θ ko4 εb no m2o c dP ρε2 sin2 φ 2 2 = θ cos φ + . . cos d 32μo π 2 3 sin2 θ sin2 φ + cos2 θ (6.52)

6.4 Finite-Sized Electric Dipole We now consider a linear current element of finite length 2L with uniform current on it. This type of dipole not only models radiation sources that have an electrically small size but is also useful for modeling larger linear sources as the latter can be subdivided into short sections having uniform currents. We present the analytical results for two cases of the orientation of the finite-sized electric dipole: (i) when the dipole is parallel to the optic axis and (ii) when the dipole is perpendicular to the optic axis. An arbitrarily oriented dipole can always be broken into two vector components, one along and one perpendicular to the optic axis. Therefore, our formulation can be used to construct the solution for an arbitrarily oriented finite-sized dipole by simple vector superposition.

6.4.1 Dipole and Optic Axis Parallel to x Axis Let us begin with the simpler case when both the electric dipole and the optic axis are parallel to the x axis as shown in Fig. 6.1 with the electric current density Je (r) =

.

 −iωpo δ(y)δ(z)ˆx , 0,

and the magnetic current density .Jm (r) = 0.

|x| ≤ L , |x| > L ,

(6.53)

6 Electromagnetic Radiation by Finite-Sized Electric and Magnetic Dipoles. . .

105

Fig. 6.1 Schematic showing a dipole (thick line) oriented parallel to the optic axis ˆ = xˆ . The field point P is .c located at position vector .r with respect to the origin

6.4.1.1

Near Zone

The electromagnetic fields in the near zone of the electric dipole parallel to the optic axis can be computed by substituting the near-zone approximation of the dyadic Green function (6.19) and the electric current density (6.53) into Eq. (6.7) to get [43, 44] ρε po .E(r) = 4π εo

L  3ε (ε −1 · R )(ε −1 · R ) · xˆ x x a r

r

5 Rex

−L

−

ε−1 · xˆ  r

3 Rex

dx  ,

(6.54)

where Rex =



.

ρε (y 2 + z2 ) + (x − x  )2

(6.55)

and Rx = (x − x  )ˆx + y yˆ + zˆz .

.

(6.56)

Using Eqs. (6.13) and (6.56), we can get the following identities: ε−1 · Rx =

.

r

 1 1   y yˆ + zˆz + (x − x )ˆx , εb εa ε−1 · xˆ =

.

r

1 xˆ . εa

Substituting Eqs. (6.57) and (6.58) into Eq. (6.54), we get [43, 44]

(6.57)

(6.58)

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A. Hayat and M. Faryad

  1 ρε po y yˆ + zˆz  .E(r) = 3 4π εo εb ρε (y 2 + z2 ) + (x − L)2 2  1 − 3 ρε (y 2 + z2 ) + (x + L)2 2   L−x xˆ L+x . + + 3 ρε ρ (y 2 + z2 ) + (x − L)2  32 ρε (y 2 + z2 ) + (x + L)2 2 ε (6.59) Since .H(r) = 0 in the near zone, Eq. (6.59) effectively represents the electrostatic field of a line charge in the uniaxial material parallel to the optic axis.

6.4.1.2

Far Zone

The electric and magnetic fields in the far zone can be found by the substitution of Eqs. (6.21) and (6.53) into Eq. (6.7), to get [43, 44]  L exp(iko no Rex )  ω2 μo μb po xˆ dx .E(r) = 4π Rex −L

L −ρε (y yˆ + zˆz) −L

L −ˆx −L

(x − x  ) exp(iko no Rex )  dx 3 Rex

 (x − x  )2 exp(iko no Rex )  dx , 3 Rex

(6.60)

using Eqs. (6.15), (6.57), and (6.58), where .Rex is given by Eq. (6.55). In the far zone (.re  x  ), we neglect the square and higher-order terms in the binomial expansion of Eq. (6.55) and approximate as Rex re −

.

x  x re

(6.61)

in the exponential factor, where    2 2 2 x + ρε (y + z ) = r sin2 θ cos2 φ + ρε sin2 θ sin2 φ + cos2 θ .re = = r(θ, φ) .

(6.62)

6 Electromagnetic Radiation by Finite-Sized Electric and Magnetic Dipoles. . .

107

In the denominator of Eq. (6.60), we can approximate .Rex re . Thereafter, Eq. (6.60) evaluates to [43, 44]  ko μb po ρε exp(iko no r) (sin2 θ sin2 φ + cos2 θ ) xˆ .E(r) = sin θ cos φ 2π εo no r 2   k Ln sin θ cos φ  o o − sin θ sin φ yˆ − cos θ zˆ sin 

(6.63)

in mixed Cartesian and spherical coordinates. The expression of the magnetic field can be found by using Eqs. (6.15), (6.24), and (6.53) in Eq. (6.8) as

H(r) =

.

ko no ωpo ρε (−y zˆ + zˆy) 4π

L

−L

exp(iko no Rex )  dx . R2ex

(6.64)

Using Eqs. (6.61) in the exponential term and .Rex = re in the denominator of Eq. (6.64), we get   k Ln sin θ cos φ  ko po cρε exp(iko no r)  o o cos θ yˆ − sin θ sin φ zˆ sin 2π r  sin θ cos φ  (6.65) by keeping terms proportional to .1/r only. From Eqs. (6.63) and (6.65), we can see that only extraordinary waves propagate in this case. Furthermore, the electric and magnetic fields are perpendicular to each other, as is usually the case of radiation in the far zone in isotropic materials. After using the expressions for .E and .H from Eqs. (6.63) and (6.65), respectively, in Eq. (6.32), the time-averaged power radiated per unit solid angle by the finitesized dipole is found as H(r) =

.

k 2 μb p2 cρ 2 (sin2 θ sin2 φ + cos2 θ ) 2 dP = o 2 o ε . sin d 8π εo no 3 sin2 θ cos2 φ



ko Lno sin θ cos φ 

 .

(6.66)

When we substitute .ρε = 1 and . = 1, we get the results for the isotropic material [29].

6.4.2 Dipole Parallel to z Axis and Optic Axis Parallel to x Axis Let us now consider the case when the finite-sized dipole is perpendicular to the optics axis, as shown schematically in Fig. 6.2. For this electric dipole, the electric current density is

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A. Hayat and M. Faryad

Fig. 6.2 Schematic showing a dipole (thick vertical line) oriented perpendicular to the optic axis .cˆ = xˆ . The field point P is located at a position vector .r with respect to the center of the dipole

Je (r) =

.

 −iωpo δ(x)δ(y)ˆz,

|z| ≤ L , |z| > L ,

0,

(6.67)

and the magnetic current density is .Jm (r) = 0.

6.4.2.1

Near Zone

In the near zone, the electric field is found by substitution of Eqs. (6.19) and (6.67) into Eq. (6.7) as ρε po .E(r) = 4π εo

L  3ε (ε−1 · R )(ε−1 · R ) · zˆ z z a r

r

5 Rez

−L

−

ε −1 · zˆ  r dz , 3 Rez

(6.68)

where Rz = x xˆ + y yˆ + (z − z )ˆz

.

(6.69)

and Rez =

.

 x 2 + ρε y 2 + ρε (z − z )2 .

After simplification, we get   1 3ρε2 po  .E(r) = x xˆ + ρε y yˆ 3 4π εo εa 2 2 [x + ρε y + ρε (z − L)2 ] 2  1 − 3 [x 2 + ρε y 2 + ρε (z + L)2 ] 2

(6.70)

6 Electromagnetic Radiation by Finite-Sized Electric and Magnetic Dipoles. . .

 (z − L)2 zˆ ρε + 2 (x + ρε y 2 ) [x 2 + ρε y 2 + ρε (z − L)2 ] 23  (z + L)2 − 3 [x 2 + ρε y 2 + ρε (z + L)2 ] 2  zˆ (z − L) + 3(x 2 + ρε y 2 ) [x 2 + ρε y 2 + ρε (z − L)2 ] 12  (z + L) . − 1 [x 2 + ρε y 2 + ρε (z + L)2 ] 2

109

(6.71)

Let us recall that .H(r) ≈ 0 in the near zone. Equation (6.71) represents the electrostatic field of a line charge of length 2L oriented perpendicular to the optic axis of the uniaxial dielectric material.

6.4.2.2

Far-Zone Field Except on the Optic Axis

In the far zone, the electric field can be found by substituting Eqs. (6.23) and (6.67) into Eq. (6.7), and its simplified expression is given by    L x 2 y yˆ − xy 2 xˆ (z − z ) + x 2 zˆ (z − z )2 − x xˆ (z − z )3 ω2 μo μb po ρε .E(r) = 4π y 2 + (z − z )2 −L

exp(iko no Rez )  dz + × 3 Rez

L 

−L

  y 2 zˆ − y(z − z )ˆy exp(iko no Rz )  dz . (6.72) Rz y 2 + (z − z )2

To solve these integrals, we use the far-zone approximation. Using Eq. (6.16) with cˆ = xˆ , we get

.

 1 2 Rez = re2 + ρε z (z − 2z) ,

.

(6.73)

where .re is given by Eq. (6.62). In the far zone, our point of observation is far away from the dipole, i.e., .re  z , so in this limit, we can neglect the higher-order terms in .z /re in the binomial expansion of (6.73) and have Rez re − ρε

.

zz re

(6.74)

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A. Hayat and M. Faryad

in the exponential term, whereas we can approximate .Rez re in the denominator. Similarly,  1 2 Rz = r 2 + z (z − 2z)

.

(6.75)

can be approximated as Rz r −

.

zz r

(6.76)

in the exponential term and .Rz r in the denominator. Furthermore, y 2 + (z − z )2 ∼ y 2 + z2 ,

.

(6.77)

since .y 2 + z2 >> L in the far zone at a point other than the optic axis. Thereafter, Eq. (6.72) becomes [44] E = Eo + Ee ,

.

(6.78)

where  exp(ik n r) sin θ sin φ ko μb po  o o − yˆ + tan θ sin φ zˆ sin(ko no L cos θ ) 2 2π εo no r (sin θ sin2 φ + cos2 θ ) (6.79) represents the ordinary wave and Eo (r) =

.

Ee (r) =

.

ko μb po  − (sin2 θ sin2 φ + cos2 θ )ˆx + sin2 θ cos φ sin φ yˆ 2π εo no r  ρ k n L cos θ   exp(ik n r) sin θ cos φ o o ε o o sin + sin θ cos θ cos φ zˆ 2 2 2 2   (sin θ sin φ + cos θ ) (6.80)

represents the extraordinary wave. Now the magnetic field in the far zone with .cˆ = xˆ can be found by substituting Eqs. (6.24) and (6.67) into Eq. (6.8) as [43, 44] L  [Rz × (Rz × xˆ )](Rz × xˆ ) · zˆ go (Rz ) .H(r) = ko no ωpo Rz (Rz × xˆ ) · (Rz × xˆ ) −L

 (Rz × xˆ )[Rz × (Rz × xˆ )] · zˆ dz . −ρε ge (Rz ) Rez (Rz × xˆ ) · (Rz × xˆ ) Using Eq. (6.15) in Eq. (6.81), we get

(6.81)

6 Electromagnetic Radiation by Finite-Sized Electric and Magnetic Dipoles. . .

H = Ho + He ,

.

111

(6.82)

where .Ho represents the ordinary wave given by Ho (r) =

.

k o po c  (sin2 θ sin2 φ + cos2 θ )ˆx − sin2 θ cos φ sin φ yˆ 2π r  exp(ik n r) sin θ sin φ o o − sin θ cos θ cos φ zˆ sin(ko no L cos θ ) cos θ (sin2 θ sin2 φ + cos2 θ ) (6.83)

and .He represents the extraordinary wave given by He (r) =

.

 exp(ik n r) sin θ cos φ k o po c  o o − cos θ yˆ + sin θ sin φ zˆ 2π r (sin2 θ sin2 φ + cos2 θ )  ρ k n L cos θ  ε o o . (6.84) × sin 

Substituting the expression for .Eo and .Ho from Eqs. (6.79) and (6.83) into Eq. (6.37) and converting into spherical coordinates, we get .

k 2 μb p2 c dPo sin2 θ sin2 φ = o 2 o sin2 (ko no L cos θ ) . d 8π εo no cos2 θ (sin2 θ sin2 φ + cos2 θ )

(6.85)

Similarly, by substituting the expression for .Ee and .He from Eqs. (6.80) and (6.84) into Eq. (6.38) and converting into spherical coordinates, we get .

  dPe sin2 θ cos2 φ k 2 μb p2 c 2 ρε ko Lno cos θ . sin = o 2 o d  8π εo no 3 (sin2 θ sin2 φ + cos2 θ )

(6.86)

The total time-averaged power radiated per unit solid angle by the dipole is found by substituting Eqs. (6.85) and (6.86) into Eq. (6.36) as [43, 44] .

 2  ρ k Ln cos θ  cos φ k 2 μb p2 c dP sin2 θ ε o o = o 2 o sin2 2 2 3  d 8π εo no (sin θ sin φ + cos2 θ )   sin2 φ 2 + 2 sin (ko no L cos θ ) . (6.87) cos θ

When we substitute .ρε = 1 (giving . = 1), Eq. (6.87) reduces to the same result as obtained directly for the isotropic dielectric material [29].

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A. Hayat and M. Faryad

6.4.2.3

Far-Zone Field on the Optic Axis

On the optic axis (x axis), .R = R xˆ = Rˆc (.θ = π/2, .φ = 0, π ), and the fields and the power have to be computed carefully using the limiting procedure [2]. The dyadic Green functions at the optic axis in the far zone are given by [2] Gee (R) = iωμo μb

.

    ˆR ˆ + εb − εa I − cˆ cˆ go (R) εa ε −1 − R r 2εb

(6.88)

iko no (ρε + 1) go (R)ˆc × I . 2

(6.89)

and Gme (R) =

.

ˆ = xˆ in The electric field can be found by substituting Eqs. (6.67) and (6.88) with .R Eq. (6.7) and is given by E(r) =

.

exp(iko no r) ko2 μb po L zˆ . (ρε + 1) 4π εo r

(6.90)

Similarly, the magnetic field can be found by substituting Eqs. (6.67) and (6.89) in Eq. (6.8) and is given by H(r) = −

.

exp(iko no r) ko2 no po cL yˆ . (ρε + 1) r 4π

(6.91)

Now by substituting Eqs. (6.90) and (6.91) into Eq. (6.32), the time-averaged power radiated per unit solid angle by the dipole is given by [43, 44] .

k 4 μb no po2 cL2 dP = o (ρε + 1)2 . d 32π 2 εo

(6.92)

6.4.3 Dipole and Optic Axis Parallel to z Axis The results derived in Sects. 6.4.1 and 6.4.2 are sufficient to construct solutions for an arbitrarily oriented finite-sized dipole. In this section, we present the results for a finite-sized electric dipole when both the dipole and the optic axis are parallel to the z axis. This will help us make the consistency check because the electromagnetic fields of the finite-sized electric dipole in the uniaxial material should reduce to that of a finite-sized dipole in the isotropic material when .ρε = 1, since the results for finite-sized dipoles are usually provided in the textbooks when the dipole is parallel to the z axis [29]. When the dipole and the optic axis are parallel to the z axis as shown in Fig. 6.3, the electric field is given by [44]

6 Electromagnetic Radiation by Finite-Sized Electric and Magnetic Dipoles. . .

113

Fig. 6.3 Schematic showing a dipole (thick vertical line) oriented parallel to the optic axis .cˆ = zˆ . The field point P is located at a position vector .r with respect to the center of the dipole

E(r) =

.

  ko μb po ρε exp(iko no r) sin2 θ ˆ ˆ ˆ z − sin θ cos φ x − sin θ sin φ y 2π εo no r cos θ 2   ko Lno cos θ (6.93) , × sin 

and the magnetic field is given by    ko po cρε exp(iko no r)  ko Lno cos θ .H(r) = −sin θ cos φ yˆ +sin θ sin φ xˆ sin . 2π r  cos θ  (6.94) The time-averaged power then is given by [44] dP k 2 μb p2 cρ 2 sin2 θ sin2 . = o 2 o ε 3 d 8π εo no  cos2 θ



ko Lno cos θ 

 .

(6.95)

The result is independent of .φ as should be the case since the dipole and the optic axis are both along the z axis and the problem has azimuthal symmetry. The far-zone field for the isotropic material can be obtained by setting .ρε = 1 and . = 1. These far-zone fields agree with the fields calculated directly for the isotropic material [29]. Furthermore, Eqs. (6.95) and (6.87) reduce to exactly the same result when .ρε = 1.

6.4.4 Radiation Patterns The spatial profile of time-averaged power per unit solid angle of radiation emitted by an antenna in the far zone is generally referred to as the radiation pattern of that antenna. The radiation patterns of the finite-sized electric dipole can be plotted using Eqs. (6.66), (6.85)–(6.87), (6.92), and (6.95). To demonstrate the role of the finite length of the dipole, we present representative radiation patterns of the finite-

114

A. Hayat and M. Faryad

y 0.04

40

y

0.03

35 30 25

0.02

Z

0.01

20

Z

15 10

0.00

x

x

5 0

Fig. 6.4 Radiation pattern .dP /d (in W sr.−1 ) of an electric dipole given by Eq. (6.95), which is oriented parallel to the optic axis (z axis) and lying in a uniaxial material (rutile) with .p0 = 1/ω, .εa = 8.427, .εb = 6.843, .μb = 1, .λo = 0.584 µm [6], and (left) .L = 0.01λo , (right) .L = 0.3λo . The plot is given for .0 ≤ θ ≤ π and .π/2 ≤ φ ≤ 3π/2

y 0.04 0.03 0.02

Z

0.01

x

0.00

40

y

30 20

Z

10

x

0

Fig. 6.5 Radiation pattern .dP /d (in W sr.−1 ) for ordinary waves given by Eq. (6.85) when the dipole is oriented along z axis and the optic axis along x axis for a uniaxial material (rutile) with .εa = 8.427, .εb = 6.843, .μb = 1, .λo = 0.584 µm [6], and (left) .L = 0.01λo , (right) .L = 0.3λo . The plot is given for .0 ≤ θ ≤ π and .π/2 ≤ φ ≤ 3π/2. The pattern is symmetric about xz plane

sized electric dipole embedded in rutile with .εa = 8.427, .εb = 6.843, .μb = 1 at λo = 0.584 μm [6] for a dipole with .p0 = 1/ω and length .L = 0.01λo and .0.3λo . The radiation patterns of extraordinary waves emitted by a finite-sized electric dipole aligned with the optic axis (z axis) are shown in Fig. 6.4. The plot is given only for .0 ≤ θ ≤ π and .π/2 ≤ φ ≤ 3π/2 for easier visualization since the pattern is independent of .φ, as can be seen from Eq. (6.95). The figure shows that the pattern is like that of a dipole in an isotropic material [29], and its directivity increases as the length of the dipole increases. When the dipole is perpendicular to the optic axis, the radiation pattern of ordinary waves is shown in Fig. 6.5 for the same material with .L = 0.01λo and .L = 0.3λo . Again, the plot is provided only for .0 ≤ θ ≤ π and .π/2 ≤ φ ≤ 3π/2 since the pattern as given by Eq. (6.85) is symmetric around xz plane as it should be since the dipole and the optic axis define this plane. It is clear from the figure that the radiation in the direction of optic axis is suppressed, though not zero. When we .

6 Electromagnetic Radiation by Finite-Sized Electric and Magnetic Dipoles. . .

y

x

115

0.07 0.06 0.05

Z

0.04

y

60

x

50 40

0.03

30

0.02 0.01 0.00

Z

20 10 0

Fig. 6.6 Same as Fig. 6.5 except that the pattern is that for extraordinary waves given by Eq. (6.86)

increase the length of dipole, the radiation pattern becomes more pronounced in a direction perpendicular to the dipole. The radiation pattern for extraordinary waves when the dipole is perpendicular to the optic axis is shown in Fig. 6.6. The figure shows that the radiation is highly directive along the optic axis, though the radiation is suppressed along the optic axis in the plane perpendicular to the dipole. Furthermore, there is no radiation emitted along the y axis, a direction perpendicular to both the dipole and the optic axis. When the length of the dipole is increased, the directivity of the radiation pattern increases much more than the previous two cases. The figure shows strong dependence of radiation pattern on the length of the dipole even when it is a fraction of the free-space wavelength. Since .L = 0.01λo effectively represents a point dipole, Figs. 6.4, 6.5, and 6.6 show that the length of the dipole has a strong effect on the directivity of the radiation pattern as compared with the point dipole.

6.5 Finite-Sized Current Loop We now turn to the radiation characteristics of a uniform current loop with arbitrary radius in the near and far zone. We present closed-form results of electromagnetic fields for the finite-sized loop when the loop axis of symmetry is either parallel or perpendicular to the optic axis of the uniaxial material [43, 45].

6.5.1 Loop Axis and Optic Axis Parallel to z Axis Consider a uniform current loop of radius a shown in Fig. 6.7. The loop lies wholly in the xy plane and carries constant current .I0 . Since the optic axis is along the z axis, .cˆ = zˆ . The current densities in the spherical coordinates are given by [43, 45]

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A. Hayat and M. Faryad

Fig. 6.7 Schematic showing a current loop with its axis aligned parallel to the optic axis with .cˆ = zˆ

Je (r) =

.

I0 π δ(r − a)δ(θ − )φˆ , a 2

Jm (r) = 0 .

(6.96)

The electric field in the far zone can be computed by substituting Eqs. (6.23) and (6.96) into Eq. (6.7) to get [43, 45] 2π  E(r) = iωμo μb I0 a

.

0

˜ ˜ ˆ ˜ (R × zˆ )(R × zˆ ) · φ go (R) ˜ × zˆ ) · (R ˜ × zˆ ) (R

      ˜ × R ˜ × zˆ ˜ × R ˜ × zˆ · φˆ   ρε R R ˜     + ge (R) dφ  , 2 ˜ ˜ Rez R × zˆ · R × zˆ

(6.97)

where φˆ  = − sin φ  xˆ + cos φ  yˆ

(6.98)

˜ = (x − a cos φ  )ˆx + (y − a sin φ  )ˆy + zˆz , R

(6.99)

.

and .

after we replaced the primed Cartesian coordinates with their expressions in spherical coordinates. The expression for extraordinary part of the far-zone electric field from Eq. (6.97) can be written as [43, 45] iaI0 ωμo μb ρε .Ee (r) = 4π

2π 0

 exp(iko no Rez ) z(x sin φ  − y cos φ  )ˆz 3 rez

6 Electromagnetic Radiation by Finite-Sized Electric and Magnetic Dipoles. . .

117

z2 (xy cos φ  − x 2 sin φ  − ay cos2 φ  + ax cos φ  sin φ  )ˆx (r × zˆ )2  z2 (y 2 cos φ  − xy sin φ  − ay cos φ  sin φ  + ax sin2 φ  )ˆy dφ  , + (r × zˆ )2 (6.100)

+

˜ with .r, and .Rez with .rez . However, in the where in the denominator we replaced .R exponential term, we must approximate [45] Rez ≈ rez −

.

ρε ra sin θ cos(φ − φ  ) , rez

(6.101)

where rez

.

 = r = r cos2 θ + ρε sin2 θ .

(6.102)

Since the electric field must be independent of .φ due to the symmetry of current distribution about the z axis, let us determine it by setting .φ = 0. By substituting Eq. (6.101) and .φ = 0 in Eq. (6.100), we get [45] Ee (r) =

.

   ko no aρε sin θ  cos2 θ iI0 aμo μb c   yˆ . exp(ik n r ) J o o ez 1    2n0 2 r 2 sin2 θ

(6.103)

Since the final expression contains .1/r 2 terms only, Ee (r) ≈ 0

.

(6.104)

in the far zone. The part of Eq. (6.97) representing the ordinary wave is given by

Eo (r) =

.

iωμo μb I0 a 4π

2π 0

˜ × zˆ )(R ˜ × zˆ ) · φˆ  ˜ (R exp(iko no R) dφ  , r (r × zˆ )2

(6.105)

for .r  a by approximating .R˜ as r in the denominator. But, Eq. (6.99) is approximated as R˜ = r − a sin θ cos φ 

.

(6.106)

and is used in the exponential term in the far zone. After solving integrals in Eq. (6.105) and retaining the terms proportional to .1/r, we get

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Eo (r) =

.

ωμo μb I0 a exp(iko no r) J1 (ko no a sin θ ) yˆ . 2 r

(6.107)

The magnetic field can be obtained by substituting Eqs. (6.24) and (6.96) in Eq. (6.8) as [45] iko no Io a .H(r) = 4π

2π  0

−

˜ × (R ˜ × zˆ )](R ˜ × zˆ ) · φˆ ˜ [R exp(iko no R) ˜ × zˆ ) · (R ˜ × zˆ ) ˜ R R˜ R(

˜ × zˆ )[R ˜ × (R ˜ × zˆ )] · φˆ  ρε exp(iko no Rez ) (R dφ  . ˜ ˜ Rez Rez [(R × zˆ ) · (R × zˆ )]

(6.108)

With the substitutions of Eqs. (6.98), (6.99), and (6.101) in the numerator with .φ = 0 and .Rez ≈ rez and ˜ × zˆ ) · (R ˜ × zˆ ) = x 2 (R

.

(6.109)

in the denominator of Eq. (6.108), the extraordinary part of the magnetic field is given by   2π iko no Io aρε exp(iko no rez ) 2    axz xˆ . sin φ exp(−is cos φ )dφ .He (r) = 2 x2 4π rez 0

(6.110) After performing the integration and converting into spherical coordinates, we get    ko no aρε sin θ    xˆ . .He (r) = exp(iko no rez )J1    2r 2 2 sin2 θ iIo a cos θ

(6.111)

Since the final expression contains terms with .1/r 2 , we have He (r) ≈ 0

.

(6.112)

in the far zone. With the use of Eqs. (6.98) and (6.99) in the numerator, Eq. (6.106) in the exponential term, and .R˜ ≈ r and Eq. (6.109) in the denominator of Eq. (6.108) with .φ = 0, we get [45] Ho (r) =

.

iko no Io a exp(iko no r) x2 4π r 2 2π × 0

   x 3 + a 2 x + 2a 3 x

cos φ  exp(−iτ cos φ  )dφ 

6 Electromagnetic Radiation by Finite-Sized Electric and Magnetic Dipoles. . .



− a +a x 4

2 2

 2π

119

exp(−iτ cos φ  )dφ 

0

2π −2ax









cos φ exp(−iτ cos φ )dφ zˆ

2

2

0

  2π 2 3 − x z+a z cos φ  exp(−iτ cos φ  )dφ  0

2π −a xz 2

exp(−iτ cos φ  )dφ 

0

2π −axz

  cos2 φ  exp(−iτ cos φ  )dφ  xˆ .

(6.113)

0

By finding the solution of the integrals, transforming the Cartesian coordinates to the spherical coordinates and retaining the terms proportional to .1/r only, we get Ho (r) =

.

  ko no Io a exp(iko no r) sin θ zˆ − cos θ xˆ J1 (ko no a sin θ ) . 2r

(6.114)

From Eqs. (6.107) and (6.114), it is clear that only the ordinary wave is radiated when the axis of the loop and the optic axis are parallel to each other. Furthermore, both the fields are perpendicular to each other, as is the case of radiation in the far zone in free space. The time-averaged power radiated per unit solid angle by the current loop can be found by substituting Eqs. (6.107) and (6.114) in Eq. (6.37) as .

k 2 no I 2 a 2 μo μb c 2 dPo J1 (ko no a sin θ ) . = o o 8 d

(6.115)

The above result is independent of .εa since the ordinary waves are polarized perpendicular to the optic axis, as is also evident from Eq. (6.107).

6.5.2 Loop Axis Parallel to z Axis and Optic Axis Parallel to x Axis Let us now consider the case when the axis of the current loop and the optic axis are perpendicular to each other, as shown schematically in Fig. 6.8. For this current

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Fig. 6.8 Schematic showing a current loop with its axis oriented perpendicular to the optic axis with .cˆ = xˆ

loop, the current densities are the same as given in Eqs. (6.96) but .cˆ = xˆ for the uniaxial dielectric material. We are interested in finding the fields at point P that is far away from the loop. The electric field can be computed by substituting Eqs. (6.15), (6.23), and (6.96) in Eq. (6.7) as [45] 

     ˜ × R ˜ × xˆ ˜ × R ˜ × xˆ · φˆ  R R ρε exp(iko no Rex ) iωμo μb I0 a     .E(r) = 4π Rex 2 R ˜ ˜ ˆ ˆ R × x · R × x ex 0    ˜ × xˆ R ˜ × xˆ · φˆ   ˜ R exp(iko no R)     dφ  , (6.116) + ˜ ˜ R˜ R × xˆ · R × xˆ 2π 

where .Rex is given by Rex =

.

 ˜ × xˆ ) · (R ˜ × xˆ ) + (R ˜ · xˆ )2 . ρε (R

(6.117)

For .R  a, the expression of .Rex in the spherical coordinates can be approximated from Eq. (6.117) and is given by Rex = rex −

.

ax cos φ  + aρε y sin φ  , rex

(6.118)

where    rex = r = r sin2 θ cos2 φ + ρε sin2 θ sin2 φ + cos2 θ .

.

(6.119)

In order to get Eq. (6.118) in a simplified form, we introduce the following change of variables

6 Electromagnetic Radiation by Finite-Sized Electric and Magnetic Dipoles. . .

x = η cos α sin β,

.

121

yρε = η sin α sin β .

(6.120)

Hence, Eq. (6.118) is given by   Rex = rex − s1 cos α − φ  ,

(6.121)

.

where s1 =

.

ko no aη sin β . r

(6.122)

By substituting Eqs. (6.98) and (6.99) in the numerator and Eq. (6.121) in the exponential term with .Rex ≈ rex and ˜ × xˆ ) · (R ˜ × xˆ ) = y 2 + z2 (R

(6.123)

.

in the denominator of Eq. (6.116), the extraordinary part of the electric field is −iωμo μb I0 aρε exp(iko no rex ) .Ee (r) = 3 rex 4π(y 2 + z2 ) 2π ×

   xy 3 + xyz2

cos φ  exp[−is1 cos(α − φ  )]dφ 

0

2π 2 2









sin φ exp[−is1 cos(α − φ )]dφ xˆ

+(y + 2y z + z ) 4

4

0

 − x2y2

2π

cos φ  exp[−is1 cos(α − φ  )]dφ  + (xy 3 + xyz2 )

0

2π ×

 sin φ exp[−is1 cos(α − φ )]dφ yˆ 





0

 2π 2 − x yz cos φ  exp[−is1 cos(α − φ  )]dφ  0

2π +(xy 2 z + xz3 ) 0

  sin φ  exp[−is1 cos(α − φ  )]dφ  zˆ ,

(6.124)

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A. Hayat and M. Faryad

where we have retained only those terms that are proportional to .1/r and ignored higher-order terms in the far zone. After finding the solution of integrals and using Eq. (6.120), we get Ee (r) =

.

  ωμo μb I0 aρε y exp(iko no r) 2 2 ˆ − (y + z )ˆ x + xy y + xzˆ z r 2  sin θρd 2(y 2 + z2 )   ko no aρd sin θ (6.125) , × J1 

with ρd =

.

 cos2 φ + ρε2 sin2 φ .

(6.126)

To evaluate the ordinary part of the electric field from Eq. (6.116), we can approximate .R˜ from Eq. (6.99) for the exponential term as   R˜ = r − a sin θ cos φ − φ  ,

.

(6.127)

and for denominator, it is approximated as .R˜ ≈ r. Substituting Eqs. (6.98) and (6.99) in the numerator, Eq. (6.127) in the exponential term, and Eq. (6.123) with .R˜ ≈ r in the denominator of Eq. (6.116), and retaining only the terms proportional to .1/r, we get [45] iωμo μb I0 a exp(iko no r) .Eo (r) = r 4π(y 2 + z2 ) 2π +az



2π − yz

cos φ  exp[−iτ cos(φ − φ  )]dφ 

0

 sin φ  cos φ  exp[−iτ cos(φ − φ  )]dφ  zˆ

0

2π +z

2

 cos φ exp[−iτ cos(φ − φ )]dφ yˆ . 





(6.128)

0

Solving integrals and further simplification gives us  ωμo μb I0 a exp(iko no r)  2 ˆ −yzˆ z + z y cos φJ1 (ko no a sin θ ) . r 2(y 2 + z2 ) (6.129) Now the magnetic field can be obtained by substituting Eqs. (6.15), (6.24), and (6.96) in Eq. (6.8) as [45] Eo (r) =

.

6 Electromagnetic Radiation by Finite-Sized Electric and Magnetic Dipoles. . .

iko no Io a .H(r) = 4π

2π  0

123

˜ × (R ˜ × xˆ )](R ˜ × xˆ ) · φˆ ρε exp(iko no Rex ) ˜ [R exp(iko no R) − ˜ × xˆ ) · (R ˜ × xˆ ) ˜ R Rex R˜ R(

˜ × xˆ )[R ˜ × (R ˜ × xˆ )] · φˆ  (R dφ  . × ˜ × xˆ ) · (R ˜ × xˆ ) Rex (R

(6.130)

The extraordinary part of the magnetic field can be extracted from Eq. (6.130) by substituting Eqs. (6.98) and (6.99) in the numerator, Eq. (6.127) in the exponential term, and Eq. (6.123) with .Rex ≈ rex in the denominator, and retaining only the terms proportional to .1/r as  ko no I0 aρε y exp(iko no r)  y zˆ − zˆy J1 .He (r) = 2(y 2 + z2 ) rρd sin θ



ko no aρd sin θ 

 .

(6.131)

Similarly, the ordinary part of the magnetic field can be extracted from Eq. (6.130) by substituting Eqs. (6.98) and (6.99) in the numerator, Eq. (6.127) in the exponential term, and Eq. (6.123) with .R˜ ≈ r in the denominator, and retaining only the terms proportional to .1/r as  ko no I0 a z exp(iko no r)  2 2 ˆ −(y +z )ˆ x +xy y +xzˆ z cos φJ1 (ko no a sin θ ) . r2 2(y 2 + z2 ) (6.132) Hence, we see that both the ordinary and extraordinary waves are radiated when the axis of the loop is perpendicular to the optic axis. Since the electric and magnetic fields satisfy the orthogonality relations given in Eq. (6.35), we can find the total radiated power by adding the radiated powers of ordinary and extraordinary wave separately as given in Eq. (6.36). Substituting the expressions for .Eo and .Ho from Eqs. (6.129) and (6.132) into Eq. (6.37) and converting into spherical coordinates, we get Ho (r) =

.

.

k 2 μo μb no I02 a 2 c cos2 θ cos2 φ dPo = o J12 (ko no a sin θ ) . 8 d (sin2 θ sin2 φ + cos2 θ )

(6.133)

Similarly, by substituting the expressions for .Ee and .He from Eqs. (6.125) and (6.131) in Eq. (6.38) and converting into spherical coordinates, we get .

k 2 μo μb no I02 a 2 cρε2 sin2 φ dPe = o J12 2 2 d 8ρd (sin θ sin2 φ + cos2 θ )



ko no aρd sin θ 

 .

(6.134) The total time-averaged power radiated per unit solid angle by the current loop is given by substituting Eqs. (6.133) and (6.134) into Eq. (6.36) as

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A. Hayat and M. Faryad

.

 ko2 μo μb no I02 a 2 c dP cos2 θ cos2 φJ12 (ko no a sin θ ) = d 8(sin2 θ sin2 φ + cos2 θ )   ko no aρd sin θ ρ 2 sin2 φ . + ε 2 J12  ρd

(6.135)

When we substitute .ρε = 1 (giving . = 1 and .ρd = 1), Eqs. (6.115) and (6.135) reduce to the exactly same results of circular current loop in isotropic material [2].

6.5.3

Electrically Small Current Loop (Point Magnetic Dipole)

An electrically small current loop is effectively a point magnetic dipole. For an electrically small loop, the radius a is small enough that .|ko no a|  1. Then .J1 (ko no a sin θ ) can be approximated by .(ko no a/2) sin θ so that Eq. (6.107) simplifies to [45] E(r) = ko2 no μo μb π I0 a 2 cgo (r) sin θ yˆ ,

.

(6.136)

and Eq. (6.114) can be simplified as   H(r) = ko2 n2o Io π a 2 go (r) sin θ zˆ − cos θ xˆ sin θ .

.

(6.137)

From Eqs. (6.136) and (6.137), we see that the electric and magnetic fields are independent of .φ, and these have the same spatial dependence as given for point magnetic dipole parallel to the optic axis in [2]. Similarly, the expression from Eq. (6.115) can be simplified as [45] .

k 4 n3 I 2 a 4 μo μb c 2 dP = o o o sin θ . d 32

(6.138)

Also for small radius of the loop, .|ko no a|  1, .J1 (ko no aρd sin θ/) can be approximated by .ko no aρd sin θ/2 so that Eqs. (6.125), (6.129), (6.131), and (6.132) become   y −(y 2 + z2 )ˆx + xy yˆ + xzˆz ko2 no μo μb π I0 a 2 ρε c ge (r) , (6.139) .Ee (r) = r y 2 + z2 Eo (r) =

.

  ko2 no μo μb π I0 a 2 c go (r) −yzˆz + z2 yˆ sin θ cos φ , 2 2 y +z He (r) =

.

  ko2 n2o I0 π a 2 ρε ge (r)y y zˆ − zˆy , 2 2 y +z

(6.140)

(6.141)

6 Electromagnetic Radiation by Finite-Sized Electric and Magnetic Dipoles. . .

125

and Ho (r) =

.

  ko2 n2o I0 π a 2 2 2 ˆ g (r) − (y + z )ˆ x + xy y + xzˆ z z sin θ cos φ , o r(y 2 + z2 )

(6.142)

respectively. In the far zone, the expression for the time-averaged Poynting vector of ordinary wave given in Eq. (6.133) can be simplified as .

k 4 μo μb n3o I02 a 4 c cos2 θ cos2 φ dPo = o sin2 θ , 32 d (sin2 θ sin2 φ + cos2 θ )

(6.143)

and the expression for extraordinary wave can be written as .

k 4 μo μb n3o I02 a 4 cρε2 sin2 φ sin2 θ dPe = o , 3 2 d 32 (sin θ sin2 φ + cos2 θ )

(6.144)

while the total power radiated by the point magnetic dipole can be approximated from Eq. (6.135) as   ko4 μo μb n3o I02 a 4 c dP ρε2 sin2 φ 2 2 = . cos θ cos φ + d 32 3 ×

sin2 θ (sin2 θ sin2 φ + cos2 θ )

.

(6.145)

The far-zone results in Eqs. (6.143) and (6.144) agree with those derived directly for the point magnetic dipole and given in Eqs. (6.50) and (6.51), respectively. Furthermore, when we substitute .ρε = 1 (giving . = 1), Eqs. (6.138) and (6.145) reduce to exactly the same results, which is also the result for the point magnetic dipole in the isotropic material.

6.5.4 Radiation Patterns The closed-form results given in Eqs. (6.115) and (6.133)–(6.135) can be used to find radiation patterns of the finite-sized current loop. To illustrate the radiation characteristics, we present representative radiation patterns for radiation by current loops of different sizes in rutile, with .I0 = 0.1 A, .εa = 8.427, .εb = 6.843, .μb = 1, and .λo = 0.584 µm [6] for a current loop with radius .a = 0.01λo and .a = 0.3λo . The radiation patterns for ordinary waves emitted by a current loop with its axis aligned with the optic axis (z axis) are shown in Fig. 6.9. We have plotted Eq. (6.115) only for .0 ≤ θ ≤ π and .π/2 ≤ φ ≤ 3π/2 because it is easier to visualize the pattern without loss of any information since the pattern is independent of .φ. The

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A. Hayat and M. Faryad

x y 1.4

y

1.2

0.00003

1.0

0.000025

0.8 0.6

0.00002 0.000015

Z

0.00001 5. ¥ 10

x

Z

0.4 0.2 0.0

-6

0

Fig. 6.9 Radiation pattern .dP /d (in W sr.−1 ) using Eq. (6.115) for ordinary waves emitted by a current loop with its axis parallel to the optic axis (z axis) in rutile. The plot is given for .0 ≤ θ ≤ π and .π/2 ≤ φ ≤ 3π/2 with .I0 = 0.1A, .εa = 8.427, .εb = 6.843, .μb = 1, .λo = 0.584 µm [6], (left) .a = 0.01λo , and (right) .a = 0.3λo

figure shows that the pattern is like that of a dipole in an isotropic material when a = 0.01λo . With the increase in the radius of the loop (.a = 0.3λo ), the radiation pattern compresses around the loop and becomes more directive along the axis of the loop. Let us note that no radiation exists along the optic axis that also happens to be the axis of the loop in this case. When the axis of the current loop is perpendicular to the optic axis, the radiation pattern for the ordinary waves is shown in Fig. 6.10 when .a = 0.01λo and .a = 0.3λo . Again we have plotted Eq. (6.133) for .0 ≤ θ ≤ π and .π/2 ≤ φ ≤ 3π/2 because the pattern is symmetric about yz plane. It is clear from the figure that the radiation in the direction of optic axis is suppressed, though not zero. The radiation pattern for the extraordinary wave, given by Eq. (6.134), for a current loop that has its axis perpendicular to the optic axis is shown in Fig. 6.11 when .a = 0.01λo and .a = 0.3λo . When the loop radius is small, the radiation is directed perpendicular to both the optic axis and the axis of the loop. But, when the dipole size is large, the radiation is suppressed completely along the axis of the loop and the plane perpendicular to this axis with the suppression strongest along a direction perpendicular to both the optic axis and the axis of the loop. We chose .a = 0.01λo because such an electrically small loop represents a point magnetic dipole. A comparison of radiation patterns of a point magnetic dipole and a finite-sized loop in Figs. 6.9, 6.10, and 6.11 shows that the increase in the size of the loop not just changes the directivity but also changes the radiation pattern.

.

6 Electromagnetic Radiation by Finite-Sized Electric and Magnetic Dipoles. . .

y y

127

x

1.4

x

1.2

0.00003

1.0

0.000025

0.8

0.00002

0.6

Z

0.4

0.000015

0.2 0.0

0.00001

Z

5. ¥ 10

-6

0

Fig. 6.10 Radiation pattern .dP /d (in W sr.−1 ) for ordinary waves given by Eq. (6.133) when the axis of the loop is oriented along z axis and the optic axis along x axis for a uniaxial material (rutile) with .εa = 8.427, .εb = 6.843, .μb = 1, .λo = 0.584 µm [6], (left) .a = 0.01λo , (right) .a = 0.3λo , and .Io = 0.1A. The plot is given for .0 ≤ θ ≤ π and .π/2 ≤ φ ≤ 3π/2. The plot is symmetric about yz plane

x y 0.6 0.5

y

0.4 0.00004 0.00003

0.3

Z

0.2 0.1

Z

0.00002

0.0

0.00001

x

0

Fig. 6.11 Same as Fig. 6.10 except that the pattern is that for extraordinary waves given by Eq. (6.134) with (left) .a = 0.01λo and (right) .a = 0.3λo

6.6 Concluding Remarks The electromagnetic fields of finite-sized electric and magnetic dipoles in uniaxial dielectric material were analytically derived in the near and far zones, and the

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A. Hayat and M. Faryad

radiation patterns were computed. A current loop was considered to model the finite-sized magnetic dipole. The dipoles were taken to be along and perpendicular to the optic axis so that fields for arbitrary orientations can be constructed when required using vector addition. When the dipoles were placed parallel to the optic axis, either only extraordinary or only ordinary waves were emitted in the far zone. When the dipoles were taken perpendicular to the optic axis, both ordinary and extraordinary waves were emitted, but radiation along the optic axis was suppressed for both the ordinary and extraordinary waves. For all the cases, the directivity of the radiation pattern increases significantly with the increase in the size of the dipole. The analytical results for finite-sized electric and magnetic dipoles in uniaxial materials can be used to solve radiation and scattering problems involving more complicated geometries because bigger sources and scatterers can be assumed as assemblies of smaller parts. Furthermore, scattering from impurities present in uniaxial metamaterials can also be studied using the results in this chapter. The radiation pattern of the finite-sized electric dipole reduces to that of the point electric dipole as .L → 0. When the dipole is parallel to the optics axis, the radiation pattern of finite-sized dipole in Eq. (6.95) reduces to that of the point electric dipole in Eq. (6.33). Similarly, when the dipole is perpendicular to the optic axis, the radiation pattern of the finite-sized electric dipole in Eq. (6.87) reduces to that of the point electric dipole in Eq. (6.42) when .L → 0 in Eq. (6.87). Similar to the case of electric dipole, the radiation pattern of finite-size current loop reduces to that of point magnetic dipole as the radius of the loop vanishes. When the loop axis is parallel to the optic axis, the radiation pattern of the current loop in Eq. (6.115) reduces to that of point magnetic dipole in Eq. (6.49) when .a → 0. Similarly, when the loop axis is perpendicular to the optic axis, the radiation pattern of the current loop in Eq. (6.135) reduces to that of point magnetic dipole in Eq. (6.52) when .a → 0. Let us note that the closed-form expressions of the electromagnetic fields in the near and far zones presented in this chapter can also be used when the uniaxial medium is a hyperbolic medium [46]. Care has to be taken while evaluating the fields because the relative permittivity scalars can assume negative values or become zero [43, 46].

References 1. Chen, H.C.: Dyadic Green’s function and radiation in a uniaxially anisotropic medium. Int. J. Electron. 35, 633–640 (1973) 2. Faryad, M., Lakhtakia, A.: Infinite-Space Dyadic Green Functions in Electromagnetism. Morgan and Claypool, San Rafael (2018) 3. Chen, H.C.: Theory of Electromagnetic Waves: A Coordinate-Free Approach. McGraw-Hill, New York (1983) 4. Felsen, L.B., Marcuvitz, N.: Radiation and Scattering of Waves. IEEE Press, Piscataway (1994) 5. Mackay, T.G., Lakhtakia, A.: Electromagnetic Anisotropy and Bianisotropy: A Field Guide, 2nd edn. World Scientific, Singapore (2020)

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6. Yariv, A., Yeh, P.: Optical Waves in Crystals: Propagation and Control of Laser Radiation. Wiley, Hoboken (1984) 7. Gribble, D.C., Hall, J.A.: Optical Mineralogy: Principles and Practice. Springer, New York (1992) 8. Korzeb, K., Gajc, M., Pawlak, D. A.: Compendium of natural hyperbolic materials. Opt. Express 23, 25406–25424 (2015) 9. Poddubny, A., Iorsh, I., Belov, P., Kivshar, Y.: Hyperbolic metamaterials. Nat. Photon. 7, 948– 957 (2013) 10. Born, M., Wolf, E.: Principles of Optics, 4th edn. Pergamon Press, Oxford (1970) 11. Veiras, F.E., Perez, L.I., Garea, M.T.: Phase shift formulas in uniaxial media: an application to waveplates. Appl. Opt. 49, 2769–2777 (2010) 12. Mason, W.P.: Applications and production of quartz crystals. Nature 168, 1098–1098 (1951) 13. Milton, G.W., Nicolae-Alexandru P.N., Ross C.M., Viktor A.P.: A proof of superlensing in the quasistatic regime, and limitations of superlenses in this regime due to anomalous localized resonance. Proc. R. Soc. Lond. A 461, 3999–4034 (2005) 14. Jacob, Z., Alekseyev, L.V., Narimanov, E.: Optical hyperlens: far-field imaging beyond the diffraction limit. Opt. Express 14, 8247–8256 (2006) 15. Kabashin, A.V., Evans, P., Pastkovsky, S., Hendren, W., Wurtz, G.A., Atkinson, R., Zayats, A.V.: Plasmonic nanorod metamaterials for biosensing. Nat. Mater. 8, 867–871 (2009) 16. Schurig, D., Mock, J.J., Justice, B.J., Cummer, S.A., Pendry, J.B., Starr, A.F., Smith, D.R.: Metamaterial electromagnetic cloak at microwave frequencies. Science 314, 977–980 (2006) 17. Govyadinov, A.A., Podolskiy, V.A.: Metamaterial photonic funnels for subdiffraction light compression and propagation. Phys. Rev. B 73, 155108 (2006) 18. Smolyaninov, I.I., Hung, Y.J.: Modeling of time with metamaterials. J. Opt. Soc. Am. B 28, 1591–1595 (2011) 19. Zhang, X., Wu, Y.: Effective medium theory for anisotropic metamaterials. Sci. Rep. 5, 7892 (2015) 20. Purcell, M.E.: Spontaneous emission probabilities at radio frequencies. Phys. Rev. 69, 681 (1946) 21. Balmain, K.G., Luttgen, A.A.E., Kremer, P.C.: Resonance cone formation, reflection, refraction, and focusing in a planar anisotropic metamaterial. IEEE Antennas Wirel. Propag. Lett. 1, 146–149 (2002) 22. Taubner, T., Korobkin, D., Urzhumov, Y., Shvets, G., Hillenbrand, R.: Near-field microscopy through a SiC superlens. Science 313, 1595–1595 (2006) 23. Yao, J., Yang, X., Yin, X., Bartal, G., Zhang, X.: Three-dimensional nanometer-scale optical cavities of indefinite medium. Proc. Nat. Acad. Sci. 108, 11327–11331 (2011) 24. Tumkur, T.U., Gu, L., Kitur, J.K., Narimanov, E.E., Noginov, M.A.: Control of absorption with hyperbolic metamaterials. Appl. Phys. Lett. 100, 161103 (2012) 25. Liu, Z., Lee, H., Xiong, Y., Sun, C., Zhang, X.: Far-field optical hyperlens magnifying subdiffraction-limited objects. Science 315, 1686–1686 (2007) 26. Pourmand, M., Choudhury, P.K.: Light-matter interaction at the sub-wavelength scale: pathways to design nanophotonic devices. In: Mackay, T.G., Lakhtakia, A. (eds.) Adventures in Contemporary Electromagnetic Theory, pp. 281–314. Springer, Cham (2023) 27. Lakhtakia, A., Varadan, V.K., Varadan, V.V.: Radiation and canonical sources in uniaxial dielectric media. Int. J. Electron. 65, 1171–1175 (1988) 28. Alexeyeva, L.A., Kanymgaziyeva, I.A., Sautbekov, S.S.: Generalized solutions of Maxwell equations for crystals with electric and magnetic anisotropy. J. Electromagn. Waves Appl. 28, 1974–1984 (2014) 29. Jackson, D.J.: Classical Electrodynamics, 3rd edn. Wiley, Hoboken (2001) 30. Zangwill, A.: Modern Electrodynamics. Cambridge University Press, Cambridge (2012) 31. Tsang, L., Njoku, E., Kong, J.A.: Microwave thermal emission from a stratified medium with nonuniform temperature distribution. J. Appl. Phys. 46, 5127–5133 (1975) 32. Kong, J.A.: Electromagnetic fields due to dipole antennas over stratified anisotropic media. Geophysics 37, 985–996 (1972)

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33. Kwon, Y.S., Wang, J.J.: Computation of Hertzian dipole radiation in stratified uniaxial anisotropic media. Radio Sci. 21, 891–902 (1986) 34. Tang, C.M.: Electromagnetic fields due to dipole antennas embedded in stratified anisotropic media. IEEE Trans. Antennas Propag. 27, 665–670 (1979) 35. Ali, S., Mahmoud, S.: Electromagnetic fields of buried sources in stratified anisotropic media. IEEE Trans. Antennas Propag. 27, 671–678 (1979) 36. Eroglu, A., Lee, J.K.: Far field radiation from an arbitrarily oriented Hertzian dipole in the presence of a layered anisotropic medium. IEEE Trans. Antennas Propag. 53, 3963–3973 (2005) 37. Bellyustin, N.S., Dokuchaev, V.P.: Generation of electromagnetic waves by distributed currents in an anisotropic medium. Radiophys. Quantum Electron. 18, 10–17 (1975) 38. Wang, T., Wang, T.L.: Radiation resistance of small electric and magnetic antennas in a cold uniaxial plasma. IEEE Trans. Antennas Propag. 20, 796–798 (1972) 39. Zaboronkova, T.M., Kudrin, A.V., Petrov, E.Y.: Toward the theory of a loop antenna in an anisotropic plasma. Radiophys. Quantum Electron. 41, 236–246 (1998) 40. Kudrin, A.V., Zaboronkova, T.M., Zaitseva, A.S., Krafft, C.: Electrodynamic characteristics of a loop antenna located on the surface of a uniaxial anisotropic cylinder. In: 2016 Days on Diffraction, pp. 253–258. IEEE (2016). https://doi.org/10.1109/DD.2016.7756852 41. Pozar, D.M.: Radiation and scattering from a microstrip patch on a uniaxial substrate. IEEE Trans. Antennas Propag. 35, 613–621 (1987) 42. Monzon, J.C., Lakhtakia, A.: Alternative approach for the derivation of the magnetic Green’s dyadic for uniaxial dielectrics. Int. J. Electron. 67, 243–244 (1989) 43. Hayat, A.: Radiation by Finite-Sized Sources in Uniaxial Materials. PhD Thesis, Lahore University of Management Sciences (2020) 44. Hayat, A., Faryad, M.: On the radiation from a Hertzian dipole of a finite length in the uniaxial dielectric medium. OSA Continuum 2, 1411–1429 (2019); Erratum: 2, 2855–2855 (2019) 45. Hayat, A., Faryad, M.: Closed-form expressions for electromagnetic waves generated by a current loop in a uniaxial dielectric medium in the far zone. J. Opt. Soc. Am. B 36, F9–F17 (2019) 46. Hayat, A., Faryad, M.: Radiation by a finite-length electric dipole in the hyperbolic media. Phys. Rev. A 101, 013832 (2020)

Aamir Hayat is an assistant professor of physics at the University of Lahore, Sargodha Campus, Sargodha, since 2020. He obtained his M. Sc. and M. Phil. degrees in physics from Quaid-i-Azam University in 2013 and 2015, respectively, and his Ph. D. degree in physics from the Lahore University of Management Sciences in 2020. His research is focused on electromagnetics of anisotropic materials.

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Muhammad Faryad is an associate professor of physics at the Lahore University of Management Sciences since 2014. Before that, he was a postdoctoral research scholar at The Pennsylvania State University from 2012 to 2014. He obtained his M. Sc. and M. Phil. degrees in electronics from Quaid-i-Azam University in 2006 and 2008, respectively, and his Ph. D. degree in engineering science and mechanics from The Pennsylvania State University in 2012. His research interests include electromagnetics of anisotropic materials, plasmonics, and quantum computing. He is a senior member of SPIE and Optica. He was awarded the Gallieno Denardo award by the Abdus Salam International Center of Theoretical Physics (ICTP) in 2019 and the Early Career Achievement Award by the Department of Engineering Science and Mechanics at The Pennsylvania State University in 2021.

Chapter 7

Near-Field Microwave Imaging Employing Measured Point-Spread Functions Natalia K. Nikolova, Daniel Tajik, and Romina Kazemivala

7.1 Introduction Microwave radiation has been utilized for the imaging of optically obscured targets for decades. Applications in concealed-weapon detection, through-the-wall imaging, and nondestructive testing have all seen commercial success, and clinical trials of biomedical applications are underway [1–11]. The benefits of utilizing microwave radiation stem from its ability to penetrate through many optically opaque materials without ionization effects or health hazards. Here, we focus on the imaging of penetrable dielectric objects, which is the most common scenario encountered in practice. We also employ a number of assumptions, which simplify the inverse-scattering problem and which are common in microwave and millimeterwave imaging. We assume that the imaged dielectric object is isotropic and is thus represented by a scalar complex-valued permittivity distribution. Further, we neglect the frequency dispersion of the object’s permittivity in the frequency bandwidth of the imaging system. In effect, this leads to images that depict a permittivity distribution averaged over this bandwidth. Spectroscopic microwave imaging is, in principle, possible. It allows for frequency-dependent image reconstruction. But this is a research field in its infancy, and it is not discussed here. Utilizing microwave radiation in close-range imaging applications can be very beneficial. The image quality benefits from the relatively high signal strength and the improved spatial resolution, which is due to the increased viewing angles and the potential to exploit evanescent-field information. However, there are significant challenges due to operating in the near-field zone of the antennas and the scattering object. In this zone, neither the incident electromagnetic (EM) field nor the Green

N. K. Nikolova () · D. Tajik · R. Kazemivala McMaster University, Hamilton, ON, Canada e-mail: [email protected]; [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 T. G. Mackay, A. Lakhtakia (eds.), Adventures in Contemporary Electromagnetic Theory, https://doi.org/10.1007/978-3-031-24617-3_7

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function can be accurately predicted using analytical plane- or spherical-wave approximations [9]. At the same time, it is these two field distributions, which determine the resolvent kernel in the integral equation of scattering. Thus, methods to accurately derive this kernel are needed. In the case where the measurements are frequency-dependent scalar responses such as scattering parameters or voltages, the resolvent kernel is proportional to the product of the total internal electric field due to the transmitting (Tx) antenna and the vector Green function, which is shown to be equal to the incident electric field due to the receiving (Rx) antenna, if this antenna were to transmit [9, 12]. Here arises the opportunity to determine the kernel via calibration measurements, which are independent of the imaged object. If the scattering object is electrically small (a point scatterer), its scattering is weak, allowing for Born’s 0-th-order approximation of the total internal field, i.e., its substitution with the incident field of the Tx antenna. Unlike the total internal field, which depends on the imaged object’s contrast function, the incident internal field depends only on the measurement setup, including the background medium and the antennas. The measurement of a point-like scattering probe produces the system pointspread function (PSF), which captures the specifics of the measurement setup, including the near-field behavior of the antennas. Mathematically, the PSF can be viewed as the transfer function of the measurement system, i.e., the measurement system’s response to a scatterer, which is a three-dimensional (3D) .δ-function of the spatial coordinates. Thus, the PSF is proportional to the resolvent kernel of the integral equation of scattering describing the specific imaging system. The coefficient of proportionality is known as it is determined by the known permittivity contrast and volume of the scattering probe. Here, we demonstrate the advantages of using measured system PSFs in fast (real-time) image reconstruction, which exploits a linearized model of scattering. Such image reconstruction methods are referred to as linear or direct. Direct image reconstruction methods are fast, but the accuracy of the employed linearized model of scattering is limited to weak scattering [9, 13]. Thus, when imaging strongly heterogeneous objects under test (OUTs), image artifacts are expected, especially in regions where the total internal field is very different from the assumed incident-field distribution. Strong heterogeneity leads to inherently nonlinear scattering effects, e.g., multiple scattering and mutual coupling, which are unaccounted for. Despite this fundamental limitation, direct image reconstruction methods remain the workhorse of the microwave and millimeter-wave imaging radars. This is why the research on methods and approaches improving their performance has been and remains highly relevant. Here, we present the recent advances in casting the linearized frequency-domain scattering model in terms of PSFs and demonstrate the remarkable improvement in image quality in near-field imaging scenarios.

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7.2 Direct Reconstruction Algorithms Employing Point-Spread Functions Image reconstruction algorithms invert a forward scattering model that maps the microwave data to the dielectric properties of the OUT. This EM scattering model is inherently nonlinear. Nonlinear reconstruction methods aim at inverting it without linearizing approximations, which demands a solution not only for the dielectric profile of the OUT but also for the total internal field distribution, which must satisfy Maxwell’s equations. This necessitates the use of EM simulations, leading to significant computational time. The advantage is that the images are quantitative, i.e., the real and imaginary parts of the object’s permittivity can be recovered, subject to the fidelity of the simulations. Examples of such methods include the Born iterative and distorted Born iterative methods [14, 15] and the modelbased optimization methods [10, 16], which are common in biomedical microwave imaging. In contrast, the direct (or linear) methods employ the linearizing approximation of the forward-model kernel to drastically accelerate the reconstruction. Since the approximation assumes weak scattering, if the object violates the assumption, its image may contain significant artifacts. In the past, linear methods have been able to produce only qualitative images.1 However, recent developments have shown that these methods are in fact capable of generating quantitative images. As detailed next, there are two critical developments that enable this capability. First, quantitatively accurate data equations have become available [9, 12]. These map the measured scattering parameters (or voltages and currents) to the dielectric properties of the OUT. Second, a calibration method has been developed [17–19], which enables the extraction of the resolvent kernel from the measured system PSF. Here, two recent quantitative real-time microwave imaging methods are reviewed that employ the above concepts: quantitative microwave holography (QMH) [19– 21] and scattered-power mapping (SPM) [17, 22, 23]. QMH builds on the far-field qualitative image reconstruction methods known as microwave holography [1, 2] and range migration [24–26]. It emerged from an effort to reconstruct images from near-zone measurements using holographic principles [18, 27, 28]. It became apparent that the analytical far-field approximations used in microwave holography and synthetic aperture radar [29–31] are inadequate for near-field measurements. Even attempts to use simulated incident-field distributions (instead of the analytical approximations) proved to be unreliable in practice [18] due to modeling errors.2 To

1 A qualitative image depicts only the normalized reflectivity of the imaged object, i.e., the distribution of the strength of the scattering sources within its volume. Thus, it provides structural information about the object’s interior but no information about its EM properties. In contrast, a quantitative image depicts the property distribution of the imaged object, i.e., the distribution of its permittivity in its real and imaginary parts. 2 The modeling errors are due to the inability to represent accurately every detail and electrical property of the measurement setup in the input to the EM simulator.

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overcome this problem, in Ref. [18], a calibration measurement of the system PSF is proposed, from which a qualitative estimate of the resolvent kernel is derived for the specific imaging system. This measurement is done on a calibration object.3 Later, in Ref. [19], a derivation is provided of the quantitative estimate of the resolvent kernel from calibration-object measurements, leading to a quantitatively accurate system-specific data equation and the QMH reconstruction algorithm. Recent experimental results demonstrate that QMH is capable of reconstructing complex tissue phantoms—an extremely challenging task due to the high dielectric contrast and significant heterogeneity [20, 32]. Similarly to QMH, SPM builds on a prior qualitative linear-inversion algorithm (sensitivity mapping) [33–35]. Sensitivity mapping can be viewed as a projection reconstruction approach where the projection is done in the space of the system responses to a point scatterer (the PSFs). From a mathematical standpoint, what distinguishes the sensitivity mapping from microwave holography is that holography performs data matching at each point in the Fourier (or wavenumber k) space, whereas the sensitivity maps minimize the .2 norm of the data error in the entire data space [9]. Like QMH, the quantitative capability of SPM stems from the system-specific data equation derived from the calibration measurement of the PSF. Unlike holography, however, SPM can perform reconstruction using data acquired with any configuration of observation points [17], i.e., it does not require apertures of canonical shapes (e.g., plane or cylinder), which holography needs in order to employ fast Fourier transforms (FFTs). Still, SPM benefits greatly if the data are measured on such apertures because it can then employ deconvolution in Fourier space, which reduces the computational time by at least an order of magnitude [9, 20]. Fourier-space SPM and QMH have very similar computational costs, and they execute in a matter of seconds on personal computers even if the imaged volume consists of millions of voxels. This review starts with a description of the calibration measurements needed to extract the incident-field responses and the PSF of the specific measurement system. The scattering data equations are then derived in the case of the Born and Rytov linearizing approximations. Section 7.5 describes QMH and SPM as examples of real-time quantitative reconstruction algorithms from two distinct classes of directinversion methods. An approach to combine the Born and Rytov models in the case of QMH is described and shown to significantly improve the reconstructed images. Examples based on simulated and measured data illustrate the importance of the system-specific PSF for successful near-field imaging and demonstrate the quantitative reconstruction of the complex permittivity of objects under challenging near-field scenarios.

3 The typical calibration object is made of an electrically small scattering probe immersed in a homogeneous background medium. The electrical properties of the probe and the background are chosen in accordance with the OUTs, which are to be inspected after the system calibration is completed [9, 17].

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7.3 Calibration Measurements In this work, a planar acquisition setup is considered. The principle remains the same for cylindrical and hemispherical apertures [36]. To extract the scattering PSF of the system, two objects are measured: the reference object (RO) (see Fig. 7.1), which represents the setup in the absence of an OUT, and the calibration object (CO) (Fig. 7.2), which represents the setup measuring a scattering probe (SP), which emulates a point scatterer. It is common to choose the position of the scattering probe at the center of the imaged volume, i.e., .rsp = 0. The RO comprises the components of the measurement setup, antennas included, along with a uniform background medium made of a material that is the same as or electrically similar to the material in which the OUT is to be embedded. Examples of typical embedding media include air or coupling liquids such as those used in biomedical imaging [20]. The RO measurement produces baseline responses, which are the system-specific incident-field responses. They are needed to extract the scattered-field portion of the total measured OUT responses and to suppress the background clutter produced by the acquisition setup. The CO is the same as the RO except for the small SP embedded in the center of the imaged volume. The CO measurement produces the total-field system responses of a point scatterer. Together with the RO data, the CO data allow for the extraction of the system-specific PSF, which can be viewed as the impulse response of the imaging system. Note that the PSF is commonly used in medical imaging for quality control and assurance [37–39]. Also, it should be clarified that the system (or data) PSF differs from the image PSF. The system PSF is the point scatterer data set

Fig. 7.1 Illustration of the setup for measuring the reference object (RO). It captures the incidentfield responses, which depend on the antennas and the background environment. An object under test (OUT) is absent in this measurement. The positions of the transmitting (Tx) and receiving (Rx) antennas are denoted as .rTx and .rRx , respectively

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Fig. 7.2 Illustration of the setup for measuring the calibration object (CO). It captures the totalfield responses due to a scattering probe (SP) immersed in the background environment. The SP is electrically small in order to emulate a point scatterer. Its size and permittivity are known. The position of the probe is denoted as .rsp . These CO responses are used to obtain the system PSFs

acquired with the specific imaging hardware. Thus, it describes the performance of the hardware. In contrast, the image PSF is the image reconstructed by an inversion algorithm from the PSF data. It describes the performance of the algorithm. For optimal image reconstruction, the size and permittivity of the SP must be chosen carefully. The probe must be electrically small, i.e., smaller than .λmin /4 (.λmin being the shortest wavelength in the background), in order to represent a point scatterer. This allows for an assumption that the incident field is uniform within the volume of the probe, which simplifies the forward model of scattering (detailed later). On the other hand, the probe cannot be too small since its scattered-field signal should be well above the measurement noise. Also, the strength of the probe’s scattered field increases with its contrast with the background medium. However, both the size and the contrast of the SP should respect the limits of the first-order Born approximation for the external field [9, 40]: 2a|ks (r , ω) − kb (ω)|max < π .

.

(7.1)

Here, .ks (r , ω) is the wavenumber inside the scatterer at position .r and at angular frequency .ω, .kb is the background wavenumber, and a is the radius of the smallest sphere circumscribing the scatterer. In the case of the probe, .ks is constant in its volume. The selection of the permittivity of the probe is important as well. For the best quantitative result, it is shown in Ref. [17] that either of the following conditions should be observed:

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(r ; ω) ≈ sp (ω) ,

(7.2)

     (r ; ω) , sp (ω)  |b (ω)| .

(7.3)

.

or .

Here, .(r ; ω) is the complex permittivity of the OUT at .r , whereas .sp (ω) and .b (ω) are those of the uniform probe and background medium, respectively. From Eq. (7.2), it follows that the probe’s permittivity should be selected as close as possible to that of the OUT. With a strongly heterogeneous OUT, this choice is dictated by the structural components that are of the greatest interest. For example, in breast-cancer imaging, the probe permittivity is selected to match closely the permittivity of cancerous tissue [19, 20]. The condition (7.3) points to another option, namely, using a background with a significantly higher dielectric constant than that of the OUT and the probe. This option, however, has an unfavorable influence on the reconstruction process. First, the sharp contrast between the OUT and its embedding medium leads to multiple scattering effects, which are not handled well by the linear reconstruction methods. Second, in cases such as tissue imaging, the OUT permittivity is already high, making it difficult to find a coupling medium with even higher permittivity.

7.4 Forward Model of Scattering Consider the model of scattering (data equation) in terms of S-parameters [12], Sjsck (rRx , rTx ; ω) =

.

−iω0 2aj ak

 V

   tot   r (r ) Einc (r , r ; ω) · E (r , r ; ω;  ) Rx Tx r dr , j k (7.4)

where .Sjsck is the scattered portion of the measured S-parameter with the receiving (Rx) antenna at port j and the transmitting (Tx) antenna at port k, .rRx = (xRx , yRx , zRx ) is the Rx antenna position, .rTx = (xTx , yTx , zTx ) is the Tx antenna position, .0 is the permittivity of vacuum, .aξ is the incoming root-power wave at  the .ξ -th port (.ξ = j, k),4 .V  is the imaged volume, .Einc j (r , rRx ; ω) is the incident  electric field that would be produced by the Rx antenna at .r if it were operating in a  transmitting mode excited by .aj , and .Etot k (r , rTx ; ω; r ) is the total electric field at  .r generated by the Tx antenna. The complex relative-permittivity contrast between the OUT and the background medium is defined as .r (r ) = r (r ) − r,b (r ), 4 The root-power wave is the square root of the incident power at a port if the respective field is expressed as a root-mean-square phasor. In general, the root-power wave is a complex quantity, the phase of which equals that of the electric field vector of the traveling wave. At excitation ports, this phase is usually assumed equal to zero.

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where .r (r ) and .r,b (r ) are the complex relative-permittivity functions of the OUT and the background, respectively. Note that at each position pair .(rRx , rTx ), several responses can be measured, denoted by the subscripts j and k, .j, k = 1, . . . , Np , where .Np is the number of the system ports. The E-field dot product in Eq. (7.4) defines the resolvent kernel of the inversescattering problem. Since this kernel depends on the unknown contrast .r (r ) through .Etot k , the inverse-scattering problem is nonlinear. It is linearized by Born’s (or Rytov’s) 0-th-order approximation of the total internal field [9, 13]5  inc  Etot k (r , rTx ; ω; r ) ≈ Ek (r , rTx ; ω).

.

(7.5)

This approximation is subject to limitations in the size and contrast of the OUT [9]. Exceeding these limitations is the fundamental reason for the appearance of artifacts in the images produced by the linear-inversion methods. With this linearization, the forward model is stated as    −iω0 sc  inc   r (r ) Einc .Sj k (rRx , rTx ; ω) ≈ j (r , rRx ; ω) · Ek (r , rTx ; ω) dr . 2aj ak V

(7.6)

The resolvent kernel is now independent of the contrast and entirely determined by the incident-field distributions of the specific antennas and measurement setup. This suggests that the system-specific kernel can be determined through a calibration measurement independently of the OUT. As shown next, this measurement provides the system response to a point scatterer, i.e., its system (or data) PSF. As discussed in Sect. 7.3, the CO consists of a scattering probe (SP) embedded in a uniform background medium. Since the probe is electrically small, it is assumed inc that both .Einc j and .Ek are constant within its volume . sp . Let the probe be at   .rsp ≡ r . In this case, Eq. (7.6) predicts a CO scattered-field response in the form Sjsc,CO (rRx , rTx ; r ; ω) ≡ Hj k (rRx , rTx ; r ; ω) k   −iω0  inc  ≈ r,sp Einc j (r , rRx ; ω) · Ek (r , rTx ; ω) sp . 2aj ak (7.7)

.

This is the system PSF, which is denoted as .Hj k hereafter. Here, .r,sp = r,sp − r,b is the known relative-permittivity contrast of the probe, whereas .r,sp is its complex relative permittivity. The resolvent kernel is now extracted from the measured PSF as

5 Both Born’s and Rytov’s 0-th-order approximations of the total internal field result in simply replacing the total field with the incident field.

7 Near-Field Microwave Imaging Employing Measured Point-Spread Functions inc  inc  .Ej (r , rRx ; ω) · Ek (r , rTx ; ω)

 ≈

2aj ak −iω0



Hj k (rRx , rTx ; r ; ω) . r,sp sp

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(7.8)

Substituting Eq. (7.8) in Eq. (7.6) yields Sjsck (rRx , rTx ; ω) ≈

.

1 r,sp sp



r (r )Hj k (rRx , rTx ; r ; ω)dr .

(7.9)

V

This is the forward model of scattering in terms of the system PSF. The PSF .Hj k depends on the position of the probe .r , which implies the need to perform a very large number of calibration measurements with the probe being placed at each voxel of the imaged volume. This is not practical, and it is unnecessary if the background medium is uniform. Let the Rx and Tx antennas scan along x and y in the planar acquisition shown in Fig. 7.3. If the background is uniform and infinite along x and y, then .Hj k is translationally invariant along x and y, meaning that what matters is the relative position between the Tx/Rx antenna pair and the SP. Mathematically, Hj k (xRx , yRx , zRx ; xTx , yTx , zTx ; x  , y  , z ; ω)

.

= Hj k (xRx − x  , yRx − y  , zRx ; xTx − x  , yTx − y  , zTx ; z ; ω) .

(7.10)

As illustrated in Fig. 7.2, the PSF is acquired from the measurement of the SP when it resides at the lateral center of an imaged slice, .(x  , y  ) = (0, 0), with .z being constant, i.e., the CO measurement provides experimentally the PSF distribution

Fig. 7.3 Illustration of the setup for measuring the object under test (OUT). It captures the OUT total-field responses. The position in the imaged volume is denoted as .r . The OUT total-field responses are used to obtain the OUT scattered-field responses

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Hj k (xRx , yRx , zRx ; xTx , yTx , zTx ; z ; ω). Now, the data equation (7.9) for any OUT can be expressed in terms of the PSF as

.

 Tx Sjsck (rRx xy , zRx ; rxy ; zTx ; ω) ≈

.

ρ(r )

V  Tx      Hj k (rRx xy − rxy , zRx ; rxy − rxy , zTx ; z ; ω)dx dy dz , (7.11) Tx    where .rRx xy ≡ (xRx , yRx ), .rxy ≡ (xTx , yTx ), .rxy ≡ (x , y ), and

ρ(r ) =

.

r (r ) r,sp sp

(7.12)

is the complex reflectivity function to be reconstructed. It is evident that the integration over .x  and .y  within the volume integral of Tx Eq. (7.11) appears in the form of a double (for .rRx xy and .rxy ) two-dimensional (2D) convolution of the reflectivity function and the system PSF. This observation is important because it suggests that an image in each slice of constant .z can be obtained through deconvolution. It is worth noting that the forward model in Eq. (7.11) defaults to the classical models of real-time imaging (e.g., synthetic aperture radar) when the system PSF is expressed with analytical far-field approximations. For example, the analytical PSF H (rRx , rTx ; r ; ω) =

.

e−ikb RRx e−ikb RTx 4π RRx RTx

(7.13)

is employed in the case of multiple-input multiple-output close-range radars [26].6 Here, .RRx = |rRx − r | and .RTx = |rTx − r |. The expression in Eq. (7.13) assumes that each incident-field term in the resolvent kernel (see the left side of Eq. (7.8)) is a scalar spherical wave centered on the respective antenna. Notice that Eq. (7.13) takes into account the amplitude decay of the field. This decay is often ignored in far-field imaging. For instance, the PSF H (rRx , rTx ; r ; ω) = e−ikb 2R

.

(7.14)

is used in far-field monostatic radar [1, 2], where .R = |r − r | and .r ≡ rRx = rTx . With analytical kernels such as Eqs. (7.13) and (7.14), the reconstruction is only qualitative because these kernels carry no information about the Tx and Rx antennas (e.g., their gain patterns) and the radar circuits. They cannot be related to

6 The imaging scenario is referred to as “close-range” when the extent of the acquisition aperture is comparable to the distance to the target, and the free-space wave attenuation cannot be ignored.

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the sensitivity of the system to a particular contrast either. In contrast, the kernel Hj k in Eq. (7.11), when extracted from calibration measurements, contains all the information required for quantitative image reconstruction.  It is also worth noting that simulated incident-field distributions, .Einc j (r , rRx ; ω)  and .Einc k (r , rTx ; ω), have been used to approximate the system PSF too [28]. While such approximations may be more accurate than the analytical expressions, the computational requirements of the simulations of realistic close-range measurement setups are often prohibitive, and the results are susceptible to modeling errors [18]. The forward model (7.11) operates on the scattered portion of the measured , and with the OUT (the data), .Sjsck . responses with the CO (the PSFs), .Sjsc,CO k

.

and .Sjtotk , which However, the measurements acquire the total responses, .Sjtot,CO k include not only the scattering from the respective object but also the background (or RO) responses, which are measured in the absence of a scatterer; refer to Fig. 7.1. It is thus necessary to de-embed the RO responses from the measurements so that the scattered portions of the PSF and the OUT data are extracted as accurately as possible. Two main strategies exist for this extraction, namely, Born’s and Rytov’s first-order approximations [9, 13, 40].

7.4.1 Born’s Approximation Born’s first-order approximation assumes that the total measured response is a superposition of the incident and scattered responses. The scattered portion of the response can therefore be extracted as Sjsck (r; ω) ≈ Sjtotk (r; ω) − Sjinc k (r; ω) ,

.

(7.15)

RO where .Sjinc k ≡ Sj k is the incident-field response from the RO measurement (see sc Fig. 7.1), .Sj k is represented by the linearized scattering model in (7.11), and .r is the observation position. The subtraction in Eq. (7.15) is applied to the total-field . responses of both the OUT, .Sjtotk , and the CO, .Sjtot,CO k The limitation (7.1) of Born’s approximation in the size and permittivity contrast of the scattering object must be kept in mind. This limitation is satisfied in the case of the CO as long as the SP is chosen properly; see Sect. 7.3. However, it may not always be satisfied by the OUT, which leads to image artifacts in regions where second-order scattering effects exist such as mutual coupling and multiple scattering. Note that in near-field measurements, such second-order effects may exist not only within the OUT but also between the OUT and the antennas.

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7.4.2 Rytov’s Approximation Rytov’s first-order approximation assumes that the total field is a complex-phase correction to the incident field, where the phase correction is the scattered field normalized by the incident field. This can be rearranged to extract the scatteredfield response as [19]  sc .Sj k (r; ω)

≈

Sjinc k (r; ω) ln

Sjtotk (r; ω) Sjinc k (r; ω)

(7.16)

.

Unlike Born’s approximation, Rytov’s approximation is limited only by the permittivity contrast of the object, and not its size:7 .



   ks (r ) − kb /kb  0, .E  . are the electric and magnetic flux densities, .J is the electric current density, and  is the density of the (externally impressed) electric charge.  = D(  E,  H ) and .B  = Constitutive relations, i.e., relations of the form .D    B(E, H ), must accompany Eqs. (9.1). The constitutive relations for an electromagnetic medium reflect the physics that governs electromagnetic phenomena and

F. Ferraresso School of Mathematics, Cardiff University, Cardiff, United Kingdom e-mail: [email protected] P. D. Lamberti Dipartimento di Tecnica e Gestione dei Sistemi Industriali (DTG), Università degli Studi di Padova, Vicenza, Italy e-mail: [email protected] I. G. Stratis () Department of Mathematics, National and Kapodistrian University of Athens, Panepistimiopolis, Zographou, Greece e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 T. G. Mackay, A. Lakhtakia (eds.), Adventures in Contemporary Electromagnetic Theory, https://doi.org/10.1007/978-3-031-24617-3_9

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are expected to comply with the fundamental physical laws, which play the role of physical hypotheses, or postulates, concerning the properties of the material inside the considered domain. Some constitutive equations are simply phenomenological; others are derived from first principles.

9.1.1 Time-Harmonic Maxwell’s Equations in a Linear Homogeneous Isotropic Dielectric Medium We consider electromagnetic wave propagation in a linear homogeneous isotropic dielectric medium in .R3 , with permittivity .ε, and permeability .μ; hence, .J ≡ 0 and  . ≡ 0. Following Section 1.2 of Ref. [1], we assume that the electromagnetic wave is propagating at a specific frequency .ω > 0; in mathematical terms, this means that , E,  D  and .B,  must be of the form: the vector fields .H    (x, t) = Re U(x)e−iωt , U

.

(9.2)

where .U stands for appropriate time-independent vector fields .H, E, D, B, respectively. By using Eq. (9.2) in the first two equations of Maxwell’s system (9.1), we obtain .

curlH(x) = −iωD(x), curlE(x) = iωB(x).

(9.3)

Due to our assumptions on the medium, the spatial constitutive relations1 are D = εE,

.

B = μH,

(9.4)

for real constants .ε and .μ. Equations (9.3) and (9.4) now imply .

curlH(x) = −iωεE(x), curlE(x) = iωμH(x).

(9.5)

1 These relations can also be stated in the time domain; for instance, for a linear medium with causal response, the constitutive relations are  ∞  ∞  t) =  t − τ ) dτ, B(x,  t) = (x, t − τ ) dτ. .D(x, ε(τ ) E(x, μ(τ ) H 0

0

9 On a Steklov Spectrum in Electromagnetics

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Fig. 9.1 A bounded domain in .R3 , its (smooth) boundary . := ∂, and the unit outer normal vector .ν to .

ν

.

Ω

Γ

The normalization 1 E = √ E, ε

.

1 H= √ H μ

(9.6)

finally gives the time-harmonic Maxwell’s equations for the field phasors .E and .H curlH = − ikE, curlE = ikH ,

.

(9.7)

√ where .ω > 0 is the angular frequency of the electromagnetic wave and .k := ω με is the wave number. Let us note that since .ε and .μ (and therefore k) are uniform in space, both .E and .H are divergence-free: divE = divH = 0.

.

(9.8)

Let . be a bounded domain (i.e., a bounded connected open set) in .R3 with smooth boundary . (Fig. 9.1). We consider the following boundary-value problem:  .

curlE − ikH = 0 , curlH + ikE = 0, in , ν × E = m, on ,

(9.9)

where .ν denotes the unit outer normal to . and .m is a given tangential field. By eliminating .H , we obtain  .

curl curlE − k 2 E = 0, in , ν × E = m, on .

(9.10)

Instead of the standard interior Calderón operator (see e.g., Refs. [2] and [3]), in what follows we consider its variant where2 .m → (ν × H ) × ν, i.e., ν×E →  −

.

i (ν × curlE) × ν, on . ωμ

(9.11)

symbol .→ denotes a mapping between two sets, while .→ denotes to which element of the second set is mapped each element of the first set.

2 The

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Calderón operators are also called Poincaré–Steklov, or impedance, or capacity operators.

9.1.2 The Classical Steklov Eigenvalue Problem We recall that the classical Steklov3 eigenvalue problem for the Laplacian on a bounded domain . of .Rn , .n ≥ 2, with boundary . := ∂ is the problem ⎧ ⎨ u = 0, in , . ∂u ⎩ = λu, on , ∂ν

(9.12)

in the unknowns u (the eigenfunction) and .λ (the eigenvalue). The domain . is assumed to be sufficiently regular (usually one requires that the boundary . is at least Lipschitz continuous), and the (scalar) harmonic function u is required to belong to the standard Sobolev space .H 1 (). By .L2 (), .H 1 (), .H01 (), .L2 (), 1/2 (), H −1/2 (), .H 3/2 (), we denote the standard Lebesgue and Sobolev .H spaces, see e.g., Ref. [5]. Problem (9.12) can be considered as the eigenvalue problem for the celebrated Dirichlet-to-Neumann map defined as follows, see Ref. [6]. Given the solution .u ∈ H 1 () to the Dirichlet problem  .

u = 0, in , u = f, on ,

(9.13)

with datum .f ∈ H 1/2 (), one can consider the normal derivative . ∂u ∂ν of u as an −1/2 1/2 element of .H (), where .H () is the standard Sobolev space defined on −1/2 () is its dual. This allows to define the map .D from .H 1/2 () to . and .H −1/2 () by setting .H Df =

.

∂u . ∂ν

(9.14)

The map .D is called Dirichlet-to-Neumann map, and its eigenpairs .(f, λ) correspond to the eigenpairs .(u, λ) of problem (9.12), f being the trace of u on ..

3 Vladimir Andreevich Steklov (1864–1926) was not only an outstanding mathematician, who made many important contributions to Applied Mathematics, but also had an unusually bright personality. The Mathematical Institute of the Russian Academy of Sciences in Moscow bears his name. On his life and work, see the very interesting paper by Kuznetsov et al. [4].

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9.1.3 The Electromagnetic Steklov Eigenvalue Problem The natural analogue in electromagnetics of the classical Steklov problem (9.12) in .R3 can be defined as the eigenvalue problem for the (rescaled) interior Calderón operator defined as the map ν×E →  −(ν × curlE) × ν.

.

(9.15)

Therefore, one looks for values .λ such that .(ν × curlE) × ν = −λν × E, or equivalently (by taking another cross product by .ν) ν × curlE = λE T ,

.

(9.16)

where .E satisfies the equation .curlcurlE − k 2 E = 0 and .E T := (ν × E) × ν is the tangential component of .E. In conclusion, the Steklov eigenvalue problem for Maxwell’s equations is  .

curl curlE − k 2 E = 0, in , on . ν × curlE = λE T ,

(9.17)

To the best of our knowledge, Problem (9.17) was introduced for .k > 0 by Camaño, Lackner and Monk in Ref. [7], where it was pointed out that the spectrum of this problem is not discrete.4 In particular, for the case of the unit ball in .R3 , it turns out that the eigenvalues consist of two infinite sequences, one of which is divergent and the other is convergent to zero. To overcome this issue, in that paper, a modified problem, having discrete spectrum, was considered and then used to study an inverse scattering problem. On the other hand, in Ref. [9], two of the authors of this chapter have analyzed Problem (9.17) only for tangential vector fields .E, in which case the problem can be written in the form  curl curlE − k 2 E = 0, in , . (9.18) ν × curlE = λE, on . Note that the boundary condition in problem (9.18) automatically implies that .E is tangential, that is .E · ν = 0 on .. Because of this restriction, the null sequence of eigenvalues disappears and the spectrum turns out to be discrete. For .k = 0, Problem (9.18) has been considered in Refs. [10] and [11] in the analysis of Steklov eigenvalues for the Hodge Laplacian on differential forms.

4 Here

“discrete” is understood in the sense of Spectral Theory, namely, .κ belongs to the discrete spectrum of an operator .K if it is an isolated eigenvalue of finite (geometric) multiplicity, see, e.g., Ref. [8], p. 73.

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Steklov eigenproblems have been and are extensively studied for a variety of differential operators, linear and nonlinear, mostly in the scalar case. On the contrary, there are not so many publications devoted to analogous problems for Maxwell’s equations: apart from the two papers mentioned above, we are aware of the ones by Cakoni, Cogar and Monk (Ref. [12]); Cogar (Refs. [13], [14], and [15]); Cogar, Colton and Monk (Ref. [16]); and Halla (Refs. [17] and [18]). In this chapter, after presenting several comments and results for the classical Steklov eigenproblem (see Sect. 9.2), we discuss the approach introduced in Ref. [9] (see Sect. 9.3), and we include explicit derivations of formulae for the eigenvalues and eigenvectors of the related problems in the case of the unit ball in .R3 (see Sect. 9.4).

9.2 On the Classical Steklov Eigenvalue Problem 9.2.1 Some Indicative Applications Problem (9.12) has a long history that goes back to a paper (Ref. [19]) written by Steklov himself. It is not easy to give a complete account of all possible applications of the problem and its variants. Here we briefly mention three main fields of investigation: • The sloshing problem This consists of the study of small oscillations of a liquid in a finite basin that can be thought of as a bounded container (a tank, a mug, a snifter, etc.). The basin is represented by a bounded domain . in .R3 with boundary . = 1 ∪ 2 , where .1 is a two-dimensional domain representing the horizontal (free) surface of the liquid at rest, and .2 represents the bottom of the basin. The Steklov boundary condition . ∂u ∂ν = λu is imposed only on .1 , while the Neumann condition . ∂u is imposed on .2 . The gradients .grad u(x, y, z) ∂ν of solutions u represent the (stationary) velocity fields of the oscillations and √ . λ the corresponding frequencies. In particular, .u(x, y, 0) is proportional to the elevation of the free surface, and the so-called high-spots correspond to its maxima. We refer to Refs. [4, 20, 21], and [22] for more details on classical and more recent aspects of the problem. • Electrical prospection The study of the Dirichlet-to-Neumann map received a big impulse from the seminal paper (Ref. [23]) by Calderón. It poses the inverse problem of recovering the electric conductivity .γ of an electric body . from the knowledge of (the energy form associated with) the voltage-to-current map .Dγ defined in the same way as Dγ f = γ

.

where .uγ is the solution to the problem

∂uγ , ∂ν

(9.19)

9 On a Steklov Spectrum in Electromagnetics

 .

201

div(γ graduγ ) = 0, in , on . uγ = f,

(9.20)

The problem of Calderón was solved paper Ref. [24], where

in a fundamental

it was proved that the map .f →  f Dγ f dσ (=  γ |graduγ |2 dx) uniquely identifies .γ (for conductivities .γ of class .C ∞ ()). • Vibrating membranes Problem (9.12) can be used in a linear elasticity to model the vibrations of a free membrane . in .R2 with mass concentrated at the boundary. Recall that the normal modes of a free membrane with mass density .ρ are the solutions to the Neumann eigenvalue problem ⎧ ⎨ − u = λρu, in , . ∂u ⎩ = 0, on , ∂ν

(9.21)

see Ref. [25]. The total mass of the membrane is given by .M =  ρ(x) dx. If we consider a family of mass densities .ρ for . > 0, such that the support of .ρ is contained in a neighborhood of the boundary . of radius . (with .ρ constant therein) and such that the total mass .M = M does not depend on ., then the solutions of Problem (9.21) converge to the solutions of ⎧ ⎨ u = 0, in , . ∂u ⎩ = λρu, on , ∂ν

(9.22)

where .ρ = M/|| and .|| is the perimeter of .. Thus, Problem (9.22) can be considered as a limiting/critical case of a family of Neumann eigenvalue problems. We refer to Refs. [26, 27], and [28] for further details, in particular for an asymptotic analysis. Finally, we mention that the Steklov problem has been recently used in Ref. [29] for a mathematical model related to the study of information transmission in the neural network of the human brain.

9.2.1.1

Details on the Formulation and Its Connections to Trace Theory

For any .n ≥ 2, the weak (variational) formulation of Problem (9.12) is easily obtained by multiplying the equation . u = 0 by a test function .ϕ and integrating by parts over .. This leads to the equality 

 gradu · gradϕ dx = λ

.



uϕ dσ, 

(9.23)

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which needs to be satisfied for all functions .ϕ ∈ H 1 () and can be taken as the formulation of the classical problem (9.12) in .H 1 (). The advantage of formulation (9.23) is evident since it easily allows to apply standard tools from functional analysis and calculus of variations to prove existence of solutions. Indeed, one can prove that the spectrum of Problem (9.23) is discrete and consists of a divergent sequence of eigenvalues 0 = λ1 ≤ λ2 ≤ · · · ≤ λk ≤ . . . ,

(9.24)

.

where it is assumed that each eigenvalue is repeated as many times as its multiplicity, which is finite. The corresponding eigenfunctions .uj define a complete orthogonal 1 system for the subspace .H() of harmonic can

functions in .H (). Note that .H() 1 be described as .H() = {u ∈ H () :  gradu · gradϕ dx = 0, ∀ϕ ∈ Cc∞ ()}. In fact, the following decomposition holds H 1 () = H01 () ⊕ H() .

(9.25)

.

We describe one straightforward way to prove these results, since this method will be applied to the case

of Maxwell’s equations. By adding the term .  uϕ dσ to both sides of Eq. (9.23) and setting .μ = λ + 1, one gets 





gradu · gradϕ dx +

.



uϕ dσ = μ 

uϕ dσ,

(9.26)



where the quadratic form associated to the left side, namely 



Q(u) :=

|gradu| dx +

|u|2 dσ,

2

.



(9.27)



is coercive in .H 1 () and in particular defines a norm .Q(u)1/2 equivalent to the Sobolev norm of .H 1 (). Thus, the operator .L from .H 1 () to its dual defined by the pairing 

Lu, ϕ =

 gradu · gradϕ dx +

.



uϕ dσ

(9.28)



is invertible. Then we can consider the operator T from .H 1 () to itself defined by −1 .T = L ◦ J where J is the operator from .H 1 () to its dual defined by the pairing 

J u, ϕ =

uϕ dσ.

.

(9.29)



The operator T is self-adjoint with respect to the scalar product associated with the quadratic form Q above (see e.g., Ref. [30] for an analogous Steklov problem).

9 On a Steklov Spectrum in Electromagnetics

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Moreover, since the standard trace operator .Tr from .H 1 () to .L2 () is compact, it follows that T is also compact; hence, its spectrum consists of zero and a decreasing divergent sequence of positive eigenvalues .μj . Thus the eigenvalues .λj above can be defined by the equality .μj = (λj + 1)−1 . Moreover, since the kernel of T is exactly .H01 () and its orthogonal is .H(), the decomposition (9.25) immediately follows. If . is the ball of radius R centered at zero, the eigenvalues are given by all numbers of the form lj =

.

j , j ∈ N0 . R

(9.30)

The corresponding eigenfunctions are the homogeneous polynomials of degree j and can be written in spherical coordinates in the form u(r, ξ ) = r j Yj (ξ )

.

(9.31)

for .r = |x| ≥ 0 and .ξ = x/|x| ∈ S n−1 (the .(n − 1)-dimensional unit sphere), where .Yj is any spherical harmonic of degree j . In particular, the multiplicity of .lj is .(2j + n − 2)(j + n − 3)!/(j !(n − 2)!), and only .l0 is simple, the corresponding eigenfunctions being the constant functions. Note that the enumeration .lj , .j ∈ N0 is different from the enumeration .λj , .j ∈ N discussed above since it does not take into account the multiplicity of the eigenvalues. We note en passant that the eigenvalues of the Laplace–Beltrami operator on the .(n − 1)-dimensional sphere of radius R are given by the formula .σl = l(l + n − 2)/R 2 (see, e.g., Ref. [31]) and coincide with the squares of the .lj for .n = 2 (the corresponding eigenfunctions are given by the restrictions of the corresponding Steklov eigenfunctions). We refer to Ref. [32] for further discussions. We now describe the method of Auchmuty (see Ref. [33]) for the spectral representation of the trace space .H 1/2 (), since the same method will be used in the vectorial case (in which case the Steklov eigenvectors for Maxwell’s equations will be used). By exploiting an argument similar to the one discussed above (with an operator analogous to T defined on .L2 () rather than on .H 1 ()), one can actually see that the traces .Tr(uj ) of the eigenfunctions .uj on . define a complete orthogonal system for .L2 (). Assume that those eigenfunctions are normalized in .L2 (), i.e.,

 2 . by Eq. (9.23), it follows that .uj / λj + 1 is normalized  |Tr(uj )| dσ = 1. Then,   in .H 1 (), that is .Q uj / λj + 1 = 1. Thus, .H() can be described as follows: ⎫ ∞ ⎬ uj |cj |2 < ∞ . .H() = cj  : ⎩ ⎭ λj + 1 j =1 j =1 ⎧ ∞ ⎨

(9.32)

Recall that the trace space .Tr(H 1 ()) coincides with the standard fractional Sobolev space .H 1/2 () and note that .Tr(H 1 ()) = Tr(H()) by Eq. (9.25). This,

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combined with Eq. (9.32), yields ⎧ ∞ ⎨

H 1/2 () = Tr(H()) =

.

⎩

j =1

⎧ ∞ ⎨

=

⎩

j =1

⎫ ∞ ⎬  cj Tr(uj ) λj + 1 : |cj |2 < ∞ ⎭ j =1

⎫ ∞ ⎬ cj Tr(uj ) : (λj + 1)|cj |2 < ∞ . (9.33) ⎭ j =1

If . is sufficiently smooth, then Weyl’s law, describing the asymptotic behavior of the eigenvalues, is 1

λj ∼ cj n−1 , as j → ∞,

.

(9.34)

where c is an explicitly known constant. It follows that the space .H 1/2 () can be described as follows: ⎧ ⎫ ∞ ∞ ⎨ ⎬ 1 1/2 .H () = cj Tr(uj ) : j n−1 |cj |2 < ∞ . (9.35) ⎩ ⎭ j =1

j =1

Thus the condition on the Fourier coefficients .cj is that the sequence 1

j 2(n−1) cj , j ∈ N,

.

(9.36)

belongs to the space .2 of square summable sequences. We note that the appearance of the factor .1/2 at the exponent in Eq. (9.36) is not coincidental and corresponds to the exponent of the space .H 1/2 (). In fact, an analogous representation was found in Ref. [34] for the space .H 3/2 () where the exponent .3/2 naturally appears by using the Weyl asymptotic for a biharmonic Steklov eigenvalue problem.

9.3 On the Electromagnetic Steklov Eigenproblem In this section, we briefly present some of the results in Ref. [9] for Problem (9.18). In the sequel, . will denote a bounded domain in .R3 with sufficiently smooth boundary, say of class .C 1,1 (see, e.g., Definition 1 in Ref. [35]). As was done in Ref. [36] for analogous problems, we introduce a penalty term .θ grad divu in the equation, where .θ can be any positive number, in order to guarantee the coercivity of the quadratic form associated with the corresponding differential operator. Namely, we consider the eigenvalue problem

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⎧ ⎨ curl curlE − k 2 E − θ grad divE = 0, in , . ν × curlE = λE, on , ⎩ E · ν = 0, on ,

(9.37)

where .E is the unknown vector field. In this section we allow .k 2 ∈ R to be not necessarily positive. Note that the second boundary condition above is in fact embodied in the first one, but we prefer to emphasize it since we need to include it in the definition of the energy space. By .L2 (), .H 1 (), .H01 (), .L2 (), .H 1/2 (), H −1/2 (), we denote the standard Lebesgue and Sobolev spaces. Further, recall the definitions of the following function spaces appearing in the mathematical theory of electromagnetism (for details see Refs. [1, 2, 5, 37–40]): • .H (curl, ) = {u ∈ (L2 ())3 : curlu ∈ (L2 ())3 } , 1/2  with norm: .uH (curl,) = u2(L2 ())3 + curlu2(L2 ())3 • .H (div, ) = {u ∈ (L2 ())3 : divu ∈ L2 ()} , 1/2  with norm: .uH (div,) = u2(L2 ())3 + divu2L2 () • .H0 (div, ) = {u ∈ H (div, ) : ν · u = 0 on } • .XT () = H (curl, ) ∩ H0 (div, ) , with norm: 1/2  2 2 2 .uH (curl,)∩H (div,) = u 2 + curlu + divu 3 2 3 2 (L ()) (L ()) L () It is important to note that since we have assumed . to be of class .C 1,1 , the space 1 3 .XT () is continuously embedded in .(H ()) , and there exists .c > 0 such that the Gaffney inequality   u(H 1 ())3 ≤ c uL2 ()3 + curluL2 ()3 + divuL2 () ,

(9.38)

.

holds for all .u ∈ XT (). Problem (9.37) has to be interpreted in the weak sense as follows: find .E ∈ XT () such that     . curlE · curlϕ dx − k 2 E · ϕ dx + θ divE divϕ dx = −λ E · ϕ dσ , 







(9.39) for all .ϕ ∈ XT (). The above formulation is obtained from Problem (9.37) by a standard procedure: for a smooth solution .E of Problem (9.37), we multiply both sides of the first equation in problem (9.37) by .ϕ ∈ XT (), integrate by parts, and use the following standard Green-type formula: 





curlE · curlϕ dx =

.



curl curlE · ϕ dx − 

(ν × curlE) · ϕ dσ . 

(9.40)

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Conversely, by the fundamental lemma of the calculus of variations (see, e.g., Ref. [41]), one can see that if .E is a smooth solution of .(9.39), then it is also a solution of problem .(9.37) in the classical sense. Note that the weak formulation allows to avoid assuming additional regularity assumptions on ., see, e.g., Ref. [42]. In order to study our eigenvalue problem, we need to assume that .k 2 does not coincide with an eigenvalue A of the problem ⎧ ⎨ curl curlE − θ grad divE = AE, in , . ν × E = 0, on , ⎩ E · ν = 0, on .

(9.41)

Clearly, the two boundary conditions in this problem are equivalent to the Dirichlet condition .E = 0 on .. We note that Problem (9.41) has a discrete spectrum that consists of a sequence .An , .n ∈ N of positive eigenvalues of finite multiplicity, the first one being

A1 =

.

min

 |curlϕ|

ϕ∈(H01 ())3 ϕ=0

+ θ  |divϕ|2 dx

> 0. 2  |ϕ| dx

2 dx

(9.42)

For the sake of brevity, we assume in the sequel that .k 2 < A1 . See Ref. [9] for details on the more general case An < k 2 < An+1 .

(9.43)

.

The key result in the case being considered is the following. Theorem 1 Let .k 2 < A1 and .θ > 0. The eigenvalues of problem (9.37) are real, have finite multiplicity, and can be represented by a sequence .λn , n ∈ N, divergent to .−∞. Moreover, the following min-max representation holds:

 λn = −

.

min

max



V ⊂XT () ϕ∈V \(H 1 ())3 0 dimV =n

 |curlϕ|2 − k 2 |ϕ|2 + θ|divϕ|2 dx

. 2  |ϕ| dx

(9.44)

To prove this result, we follow the strategy described in Sect. 9.2.1.1. Namely, by

adding the term .η  E · ϕ dσ to both sides of Eq. (9.39), we obtain 

 curlE · curlϕ dx − k

.



 E · ϕ dx + θ

2 

divE divϕ dx 

 +η



E · ϕ dσ = γ 

E · ϕ dσ,

(9.45)



where .γ = −λ + η. Under our assumptions, it is proved in Theorem 3.1 of Ref. [9] that if .η is big enough, then the quadratic form associated with the left side of Eq. (9.45), that is

9 On a Steklov Spectrum in Electromagnetics



207





Q(E) :=

|curlE|2 dx − k 2

.





|E|2 dx + θ 

|divE|2 dx + η 

|E|2 dσ, 

(9.46) is coercive in .XT (); hence, .(Q(E))1/2 defines a norm equivalent to that of .XT (). Thus, the operator .Lη from .XT () to its dual defined by the pairing 



L E, ϕ :=

.

curlE · curlϕ dx −k

η







E · ϕ dx +θ

2 

divE divϕ dx +η 

E · ϕ dσ 

(9.47)

is invertible. Then we can consider the operator .T from .XT () to itself, defined by T = (Lη )−1 ◦ J,

(9.48)

.

where .J is the operator from .XT () to its dual defined by the pairing 

JE, ϕ =

E · ϕ dσ

.

(9.49)



for all .E, ϕ ∈ XT (). As in the case described in Sect. 9.2.1.1, it is not difficult to prove that .T is a self-adjoint operator with respect to the scalar product associated with the quadratic form .Q above. Again, since the trace operator is compact, it follows that .T is also compact; hence, its spectrum consists of zero and a decreasing divergent sequence of positive eigenvalues .γj . Thus the eigenvalues .γj above can be defined by the equality .γj = (−λj + η)−1 . Then the characterization in Eq. (9.44) follows by the classical min-max principle applied to the operator .T. It follows by the previous results that the space .XT () can be decomposed as an orthogonal sum with respect to the scalar product associated with the form .Q, namely XT () = KerT ⊕ (KerT)⊥ = (H01 ())3 ⊕ H() ,

(9.50)

.

where   H() := (KerT)⊥ = E ∈ XT () : curlE · curlϕ dx

.

−k





 E · ϕ dx + θ

2 

divE divϕ dx = 0, ∀ϕ ∈ 

 (H01 ())3

. (9.51)

Note that .E ∈ H() if and only if .E is a weak solution in .(H 1 ())3 of the problem  .

curl curlE − k 2 E − θ grad divE = 0, in , ν · E = 0, on .

(9.52)

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Solutions to Problem (9.52) play the same role as the harmonic functions in H() used in Sect. 9.2.1.1, and similarly, the eigenfunctions associated with the eigenvalues .γn define a complete orthonormal system of .H(). Let . = {λn : n ∈ N}. It is important to know whether .0 ∈ . This condition can be clarified as follows. We consider two auxiliary eigenproblems. The first is the classical eigenvalue problem for the Neumann Laplacian

.

⎧ ⎨ − φ = λφ, in , . ∂φ ⎩ = 0, on ∂ , ∂ν

(9.53)

which admits a divergent sequence .λN n , .n ∈ N, of non-negative eigenvalues of finite N multiplicity, with .λ1 = 0, see, e.g., Ref. [25]. The second is the eigenproblem ⎧ curl curlψ = λψ, ⎪ ⎪ ⎨ divψ = 0 . ⎪ ν × curlψ = 0, ⎪ ⎩ ψ · ν = 0,

in , in , on , on ,

(9.54)

which admits a divergent sequence .λM n , .n ∈ N, of non-negative eigenvalues of finite multiplicity, see, e.g., Ref. [36]. In the following result from Theorem 3.10 of Ref. [9], one has actually to better assume the condition .k = 0. On the other hand, it is straightforward that if . is simply connected and .k = 0, then .0 ∈ /  because the corresponding eigenfunctions would have zero div and zero curl (see Eq. (9.44)); hence, being tangential, they would be identically zero, see Prop. 2, p. 219 of Ref. [5] for more information; see also the more recent paper Ref. [43]. Theorem 2 Assume that .k = 0 and .θ > 0. We have that .0 ∈  if and only if k 2 ∈ {θ λnN : n ∈ N} ∪ {λM n : n ∈ N}.

.

9.3.1 Remarks on Trace Problems and Steklov Expansions We denote by .E n , .n ∈ N, an orthonormal sequence of eigenvectors associated with the eigenvalues .λn of Problem (9.37), where it is understood that they are normalized with respect to the quadratic from .Q. Let .πT denote the tangential components trace operator from .XT () to .T L2 (), where T L2 () = {u ∈ (L2 ())3 : ν · u = 0 on }.

.

By setting

(9.55)

9 On a Steklov Spectrum in Electromagnetics

E n :=



.

209

|λn − η| πT E n ,

(9.56)

one can prove, in the spirit of Sect. 9.2.1.1, that .E n , .n ∈ N, is an orthonormal basis of .T L2 (). Such bases can be used to represent the solutions of the following problem:  .

curl curlU − k 2 U − θ grad divU = 0, in , ν × curlU = f , on ,

(9.57)

where .f ∈ T L2 (). Let .f have the following representation: f =

∞

.

cn E n ,

(9.58)

n=1

with .(cn )n∈N ∈ 2 . It is proved in Theorem 4.1 of Ref. [9] that if .0 ∈ / , then the solution .U of Problem (9.57) can be expanded in terms of the above basis as follows: U=

.

∞ √ |λn − η| n=1

λn

 cn E n .

(9.59)

Finally, under our assumptions, we can represent the trace space of .XT () as π (XT ()) = πT (H()) =

. T

⎧ ∞ ⎨ ⎩

j =1

cj E j :

∞ j =1

⎫ ⎬ |λj − η||cj |2 < ∞ , ⎭

(9.60)

which is the counterpart of the representation (9.33) for our problem.

9.4 The Case Where  Is the Unit Ball In this section, we consider Problem (9.37) in . = B, where B is the unit ball in R3 centered at zero, and we compute explicitly its eigenvalues and eigenvectors. In particular, we prove that there exist two families of eigenvectors, one of which is not divergence-free. Throughout this section, it will be understood that the parameter k in Eq. (9.37) is a non-zero real number. The case .k = 0 is briefly discussed in Remark 4. We proceed to define the vector spherical harmonics, following the notation of Ref. [44]. Recall that the (scalar) spherical harmonics are given by

.

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F. Ferraresso et al.

 Yσ ml (ϑ, ϕ) =

.

 εm 2π

(2l + 1)(l − m)! m Pl (cos ϑ)Fσ (ϕ), 2(l + m)!

(9.61)

where .ϑ ∈ [0, π ], .ϕ ∈ [0, 2π ), .σ ∈ {e, o}, .l ∈ N ∪ {0}, .m ∈ N ∪ {0}, .m ≤ l, .Plm is the Legendre function associated with the Legendre polynomial .Pl , .εm = 2 − δm0 , where .δmm is the Kronecker delta, and Fe = cos mϕ, Fo = sin mϕ.

.

(9.62)

Following p. 627 of Ref. [44], we use the multi-index notation Yn := Yσ ml ,

.

(9.63)

and we denote by .O the set of triple indices .σ ml where .σ ∈ {e, o}, .l ∈ N ∪ {0} and m ∈ {0, . . . , l}. For .n ∈ O, let

.

.

A1n (ξ ) = √

1 gradξ Yn (ξ ) × ξ , . l(l + 1)

(9.64)

A2n (ξ ) = √

1 gradξ Yn (ξ ), . l(l + 1)

(9.65)

A3n (ξ ) = Yn (ξ )ξ ,

(9.66)

for all .ξ ∈ ∂B, where .Yn = Yσ ml is defined as in Eq. (9.61), see p. 350 of Ref. [44]. We extend this definition to points .x ∈ B \ {0} by setting     x x |x| ,. × gradx Yn .A1n (x) = √ |x| |x| l(l + 1)   x |x| gradx Yn , A2n (x) = √ |x| l(l + 1)

(9.67) (9.68)

and  A3n (x) = Yn

 x x .. |x| |x|

(9.69)

By definition .A1σ 00 = A2σ 00 = 0. The family .{Aτ n : τ ∈ {1, 2, 3}, n ∈ O} is a complete orthonormal system in .L2 (∂B)3 . Therefore, we can expand any vector field .E ∈ L2 (B)3 as follows:   1 2 3 En (r)A1n (ξ ) + En (r)A2n (ξ ) + En (r)A3n (ξ ) . .E(r, ξ ) = n∈O

(9.70)

9 On a Steklov Spectrum in Electromagnetics

211

We note immediately that since we are interested in solutions of Problem (9.37), the boundary condition .E · ν = 0 is equivalent to .En3 (1) = 0, since one easily checks that .A2n (ξ ) · ξ = 0 and .A1n (ξ ) · ξ = 0. Recall that the vector Laplacian acts in the following way:  (f (r)V (ξ )) =

.

  1 ∂ 2 ∂f (r) r V (ξ ) + f (r) V (ξ ). ∂r r 2 ∂r

(9.71)

We have the following formulae for the vector Laplacian of the vector spherical harmonics .An : 1 l(l + 1)A1n , r2 1 2 = 2 (l(l + 1))A3n − 2 l(l + 1)A2n , r r √ 1 2 l(l + 1) = − 2 (2 + l(l + 1))A3n + A2n . r r2

A1n = − A2n

.

A3n

(9.72)

Moreover, one can compute (see also Eq. (C.14) in Ref. [44]) divE(r, ξ ) =

 ∂E 3 (r) n

.

n∈O

∂r

2 + En3 (r) − r

√

 l(l + 1) 2 En (r) Yn (ξ ). r

(9.73)

Let us define  (r) :=

.

∂En3 (r) 2 3 + En (r) − r ∂r

√  l(l + 1) 2 En (r) . r

(9.74)

Note that .divE = 0 is equivalent to . = 0. Remark 1 In Ref. [7], the authors find two families of solutions to the equation curlcurl u − k 2 u = 0: one is given by .M n = curl(xjl (k|x|)Ylm (x/|x|)), and the other one by .N n = curlM n .√ Here .jl is the spherical Bessel function of the first kind of order l, namely .jl (z) = π/(2z)Jl+1/2 (z). These two families are divergencefree by definition. Indeed, the solutions .M n have .E j = 0, .j = 2, 3, while the solutions .N n satisfy .(r) = 0 for .E 2 , .E 3 not identically zero. The function .N n = curlM n (x) in Ref. [7], for .n ∈ O, is given by

.

     x x × curlM n (x) = curl jl (k|x|) gradx Yn |x| |x| .    jl (k|x|)l(l + 1) 1  A2n . = A3n + l(l + 1) jl (k|x|)k + jl (k|x|) |x| |x| (9.75) This is consistent with the fact that .divcurlM n = 0.

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From Eq. (9.73), it is easy to compute

grad(divE)(r, ξ ) =

.

 (r)A3n (ξ ) +

n∈O

(r)  l(l + 1)A2n (ξ ). r

(9.76)

n∈O

Due to Eqs. (9.71), (9.72), and (9.76), we then have that − E + (1 − θ )graddivE =  ⎛  ⎞ En1 l(l+1) ∂En1 ∂ 1 2 − 2 ∂r r ∂r − ⎞ ⎜ ⎟ ⎛ r r2 ⎟  A1n √ ⎜ 2 2 ⎜ ⎟ √ ⎟ ⎜ − E 3 2 l(l+1) − 1 ∂ r 2 ∂En + En l(l+1) + (1 − θ) l(l + 1)  ⎟ · ⎜ n ⎜ r ⎟ ⎝A2n ⎠ . ∂r r2 r 2 ∂r r2 ⎜ ⎟   √ n∈O ⎝ A3n 3 3 ⎠ ∂ r 2 ∂En + (2+l(l+1))En − E 2 2 l(l+1) + (1 − θ) − 12 ∂r 2 2 n ∂r .

r

r

r

(9.77)

Consider now .− E + (1 − θ )graddivE − k 2 E = 0 in B. It is clear that the first equation provided by the first row in Eq. (9.77) above can be solved independently of the other two, by means of separation of variables and classical Sturm–Liouville theory, yielding En1 (r) = El1 (r) = jl (kr),

(9.78)

.

where .jl is the spherical Bessel function of the first kind of order l as above. The other two equations form a coupled system of Sturm–Liouville equations. For the sake of clarity, we first solve the system for .θ = 1. Case .θ = 1. We need to solve the two-parameter family of ODE systems ⎧ 2 1 ∂ 2 ∂En ⎪ ⎪ ⎨− r 2 ∂r r ∂r + .

3 ∂ 2 ∂En r ∂r − r12 ∂r ⎪ ⎪ 3 ⎩ En (1) = 0.

√ l(l+1)En2 2 l(l+1)En3 − − k 2 En2 = 0, 2 2 r r√ (2+l(l+1))En3 2 l(l+1)En2 − − k 2 En3 = r2 r2

+

in (0, 1), 0,

in (0, 1),

(9.79)

Recall the spherical Bessel equation .

∂ ∂r

  ∂ r 2 f + (k 2 r 2 − l(l + 1))f = 0. ∂r

(9.80)

Define the spherical Bessel operator of indices .k ∈ R and .l ∈ Z as follows: Lk,l (f ) =

.

∂ ∂r

 r2

∂ f ∂r

 + (k 2 r 2 − l(l + 1))f.

(9.81)

9 On a Steklov Spectrum in Electromagnetics

213

We require .f ∈ H 2 (0, 1), which is equivalent to imposing that the solution f to (9.80) is not singular in zero. System (9.79) can be then rewritten in the simpler form ⎧ √ 2 ⎪ 1)E 3 , ⎪ ⎨ Lk,l (E ) = −2 l(l + √ 3 3 (9.82) . Lk,l (E ) = 2E − 2 l(l + 1)E 2 , ⎪ ⎪ ⎩ E 3 (1) = 0, where we have omitted the dependence on n in .E 2 and .E 3 . Remark 2 Note that, upon replacing .E 2 in the first equation, by means of the 2E 3 −Lk,l (E 3 ) equality . 2√l(l+1) = E 2 , we obtain the fourth-order equation .

 2 Lk,l (E 3 ) − 2Lk,l (E 3 ) = 4l(l + 1)E 3 ,

(9.83)

with the boundary condition .E 3 (1) = 0. In view of Remark 1, a solution to the differential equation in Problem (9.82) is given by E 2 (r) =

.



  jl (kr) , l(l + 1) jl (kr)k + r

E 3 (r) =

jl (kr)l(l + 1) ; r

(9.84)

note however that the condition .E 3 (1) = 0 is not satisfied for general k. To overcome this problem, it is convenient to first find another explicit solution of problem (9.82). We claim that the functions E2 (r) :=

.



l(l + 1)

jl (kr) , r

E3 (r) := kjl (kr),

(9.85)

are solutions of the system of differential equations (9.82). In the sequel, we use the abbreviated notation .L := Lk,l . Let us define L (k jl (k r)) = k 3 r 2 jl (k r)+2k 2 r jl (k r)+k (k 2 r 2 −l(l +1)) jl (k r),

.

(9.86)

where prime denotes differentiation with respect to the argument. Recall that .jl (·) satisfies Eq. (9.80). Differentiation of Eq. (9.80) with .f (r) = jl (kr) gives k 3 r 2 jl (k r) + 4k 2 r jl (k r) + k(k 2 r 2 + 2 − l(l + 1)) jl (kr) + 2k 2 r jl (k r) = 0. (9.87) Equation (9.87) implies that we can rewrite Eq. (9.86) as .

(3)

L(kjl (kr)) = −2k 2 rjl (kr) − 2kjl (kr) − 2k 2 rjl (kr).

.

(9.88)

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F. Ferraresso et al.

The spherical Bessel equation .L(jl ) = 0, see Eq. (9.80), again implies that   jl (kr) k 2 r jl (kr) = −2kjl (kr) − k 2 r 2 − l(l + 1) . r

.

(9.89)

Hence, Eq. (9.88) can be rewritten as       jl (kr) L k jl (kr) = 2 2kjl (kr) + k 2 r 2 − l(l + 1) − 2kjl (kr) − 2k 2 r jl (kr) r .  2l(l + 1) = 2kjl (kr) − jl (kr) = 2E3 − 2 l(l + 1)E2 , r (9.90) √ so that .L(E3 ) = 2E3 − 2 l(l + 1)E2 . We note that the functions .E 2 and .E 3 defined in Eq. (9.84) satisfy the equalities   E2 = E 3 / l(l + 1) and L(E 3 ) = 2E 3 − 2 l(l + 1)E 2 .

.

(9.91)

√ Then it is immediate to check that .L(E2 ) = −2 l(l + 1)E3 , which implies that the couple .(E2 , E3 )t solves the two equations in problem (9.82), as claimed. Note that the divergence of the function E(r, ξ ) =

.

  E2n (r)A2n (ξ ) + E3n (r)A3n (ξ )

(9.92)

n∈O

is non-trivial unless .k = 0. Indeed, √  l(l + 1) 2 E (r) Yn (ξ ) divE(r, ξ ) = ∂r r n∈O   2  l(l + 1) jl (kr)  2 . jl (kr)k + jl (kr)k − Yn (ξ ) = r r r  ∂E3 (r)

n∈O

= −k 2



2 + E3 (r) − r

(9.93)

jl (kr)Yn (ξ ) = 0,

n∈O

where in the last equality we used Eq. (9.80) again. We note en passant that the previous formula for the divergence, namely  .

∂E3 (r) 2 3 + E (r) − r ∂r

√

 l(l + 1) 2 E (r) = −k 2 jl (kr) r

(9.94)

gives the equality r .E (r) = √ l(l + 1) 2

! 1 ∂(r 2 E3 (r)) 2 + k jl (kr) . ∂r r2

(9.95)

9 On a Steklov Spectrum in Electromagnetics

215

A similar derivation for the functions .E 2 and .E 3 defined in Eqs. (9.84) gives E 2 (r) = √

.

∂(r 2 E 3 (r)) 1 ∂r l(l + 1) r

(9.96)

in agreement with Formula (7.4) in Ref. [44]. For the boundary condition to be satisfied, we then choose a linear combination of .(E 2 , E 3 )t and .(E2 , E3 )t . If .jl (k) = 0, then .(E2 , E3 )t is already a solution of problem (9.82) with non-trivial divergence. Otherwise, we just set   jl (kr) (l(l + 1))3/2 jl (k) jl (kr)   + l(l + 1) + kjl (kr) F (r) := − r r kjl (k) .   jl (k)  jl (kr) 3 , F (r) := −l(l + 1)  jl (kr) − jl (k) r 2

(9.97)

and then .F := (F 2 , F 3 )t solves Problem (9.82) with the right boundary condition. For subsequent use, note that F 2 (1) = .

=



   l(l + 1)jl (k) jl (k) + l(l + 1)kjl (k) l(l + 1) 1 −  kjl (k)



jl (k)jl (k)k − l(l + 1)jl (k)2 + k 2 (jl (k))2 l(l + 1) kjl (k)

(9.98)

for all .l ≥ 1. Let us also note that     j (kr) jl (kr) l(l + 1)jl (k) d 2 1 1 − − + k 2 jl (kr) F (r) = k l √ r kjl (k) r2 l(l + 1) dr .     jl (kr) l(l + 1)jl (k) l(l + 1) − 1 (9.80) jl (kr) l(l + 1)jl (k) 2 − k − k 1 + = + kjl (k) r kjl (k) r2 r2 (9.99) yields 

    d 1 jl (k)2 l(l + 1) 2  F (r) |r=1 = l(l + 1) − (k + 1)jl (k) − kjl (k) + . . dr kjl (k) (9.100) Note that the solution .E of the problem ⎧ 2 ⎪ in B, ⎪ ⎨curl curlE − k E = 0, . ν × curlE − λ(ν × E × ν) = 0, on ∂B, ⎪ ⎪ ⎩ν · E = 0, on ∂B,

(9.101)

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F. Ferraresso et al.

can be found in the form .E = an (Fn2 A2n + Fn3 A3n ) + bn El1 A1n ,

(9.102)

n∈O

where .Fn2 and .Fn3 are defined in Eq. (9.97), .En1 is defined in Eq. (9.78), and .an , bn ∈ C. In what follows, it is useful to recall the formulae (see also p. 633 of Ref. [44])   √  l(l + 1) 1 ∂ f (r) A3n + (rf (r)) A2n , r ∂r r   1 ∂ (rf (r)) A1n , . curl(f (r)A2n ) = − r ∂r  √ l(l + 1) f (r) A1n . curl(f (r)A3n ) = r curl(f (r)A1n ) =

(9.103)

In particular, we have that (ξ × curl(Fn2 A2n + Fn3 A3n ))|r=1 .

 = −ξ × ((Fn2 ) (1) + Fn2 (1))A1n + ξ × ( l(l + 1)Fn3 (1)A1n )

(9.104)

= −((Fn2 ) (1) + Fn2 (1))A2n . Imposing the Steklov condition and taking into account that .ν · E = 0, we get 0 = (ν × curlE − λ(ν × E × ν))|r=1 =− an ((Fn2 ) (1) + (1 + λ)Fn2 (1))A2n n∈O

.

−



(9.105)

" # bn jl (k) + jl (k)k + λjl (k) A1n .

n∈O

Hence, the eigenvalues .λn of Problem (9.37) are given by two families. The first one is obtained by imposing 0 = ((Fn2 ) (1) + (1 + λ)Fn2 (1)) .

⇒ λ(1) l =

k 2 k jl (k) jl (k) , + 1)(jl (k))2 + k 2 (jl (k))2

(9.106)

kjl (k)jl (k) − l(l

for all .l ≥ 1. The second one is obtained by imposing that the coefficient of .A1n vanishes for some .n ∈ O. We then obtain

9 On a Steklov Spectrum in Electromagnetics

(2)

λl

.

=−

217

jl (k) + jl (k)k . jl (k)

(9.107)

l→+∞

It is not difficult to check that the eigenvalues λ(2) ∼ −l . l (1) Regarding .λl , we note that, due to the recurrence formulae for the derivatives of Bessel functions, .

kjl (k)jl (k) − l(l + 1)(jl (k))2 + k 2 (jl (k))2 = −jl−1 (k)jl+1 (k), k2

(9.108)

we get (1)

λl

.

=−

kjl (k)jl (k) jl+1 (k)jl−1 (k)

(9.109)

$ π for .l ≥ 1. Now, recalling that .jl (z) = 2z Jl+1/2 (z), where .Js is the Bessel function of the first kind of index .s ∈ R, and that we have the following large index asymptotic formula: Js (z)

s→+∞

∼

√

1 2π s

 ez s 2s

(9.110)

for all .z ∈ C, we deduce that l→∞

jl (z) ∼

el+1/2 zl 2l+3/2 (l + 1/2)l+1

(9.111)

for .l ≥ 1. By using Eqs. (9.109) and (9.111), we obtain (1) l→∞

λl

∼ −l;

(9.112)

therefore, the eigenvalues of the first family are diverging to .−∞ as .l → +∞ (as expected) (Fig. 9.2). General case .0 < θ < +∞. Since we are interested in solutions of curl curlE − θ grad divE − k 2 E = 0

.

(9.113)

with non-trivial divergence, it is convenient to find an equation for .divE. The following discussion applies to a general domain . (not necessarily a ball). Let 2 .ϕ ∈ H () be an arbitrary function. Multiply Eq. (9.113) by .grad ϕ and integrate in . to obtain  . (curl curlE · grad ϕ − θ grad divE · grad ϕ − k 2 E · grad ϕ) dx = 0. (9.114) 

218

F. Ferraresso et al.

(1)

Fig. 9.2 Eigenvalues .λl , .l = 1, . . . , 10, as a function of .k 2 ∈ (−100, 100), for .θ = 1

Due to the Steklov boundary condition, the identity .curl gradϕ = 0 and standard integration by parts yield 

 curl curlE · gradϕ dx = 





.

(ν × curlE) · grad ϕ dσ 

E · gradϕ dσ = λ

=λ 

(9.115) (div E) ϕ dσ,



where .grad and .div denote the standard tangential gradient and tangential divergence, respectively. Moreover,  .



−θ



grad divE·gradϕ dx = θ

divEϕ dx −θ







∂ (divE)ϕ dσ, ∂ν

(9.116)

as well as  .

−k

 E · gradϕ dx = k

2 

 divEϕ dx − k

2 

2 

(E · ν) ϕ dσ ; % &' (

(9.117)

=0

hence,    ∂ λdiv E + θ (divE) ϕ dσ, (θ divE + k divE)ϕ dx = ∂ν  

 .

2

which implies that .divE satisfies the following boundary-value problem:

(9.118)

9 On a Steklov Spectrum in Electromagnetics

) .

219

k2 θ divE = − λθ div E,

− divE − ∂ ∂ν divE

=

0,

in ,

(9.119)

on .

Returning to the case of the ball, the equation in . = B has the explicit solutions  divE n (r, ξ ) = ajl

.

 k √ r Yn (ξ ), θ

n ∈ O, a ∈ C.

(9.120)

Therefore, arguing as in Eq. (9.93), we have the equality

.

∂E3 (r) 2 3 + E (r) − r ∂r

√

!   k l(l + 1) 2 E (r) = ajl √ r , r θ

(9.121)

which leads to the ansatz 1 ∂(r 2 E3 (r)) − ajl ∂r r2

r .E (r) = √ l(l + 1) 2



k √ r θ

! .

(9.122)

According to the previous derivations and keeping in mind the case .θ = 1, in view of Eq. (9.85), we claim that

E2 (r) =

.



 l(l + 1)

jl

√k r θ

r

 ,

k E3 (r) = √ jl θ



 k √ r , θ

(9.123)

verify Eq. (9.122) for .a = −k 2 /θ , and they are solutions of the system (9.79). Let us show that Eq. (9.122) is satisfied. Note that ∂E3 (r) 2 3 1 ∂(r 2 E3 (r)) = + E (r) ∂r ∂r r r2     k k 2  k 2 k = jl √ r + √ jl √ r . r θ θ θ θ     2 k l(l + 1) k jl √ r , = − + θ r2 θ

(9.124)

√ where the last equality is deduced using Eq. (9.80) with k replaced by .k/ θ, so the right side of Eq. (9.122) becomes r .√ l(l + 1)



    k l(l + 1) k2 jl √ r , −a + − θ r2 θ

(9.125)

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√

jl

√k r θ



for .a = −k 2 /θ , as claimed. In a similar which equals .E (r) = l(l + 1) r way, one can check that the couple .(E2 , E3 ) satisfies the system 2

√ ⎧ 2 l(l + 1)En3 l(l + 1)En2 1 ∂ 2 ∂En2 ⎪ ⎪ − − k 2 En2 r + − ⎪ ⎪ ⎪ ∂r r2 r2 r 2 ∂r ⎪ ⎪ √ √ ⎪   ⎪ ⎪ (1 − θ) l(l + 1) ∂En3 (r) l(l + 1)En2 (r) 2 ⎪ ⎪ = 0, + + En3 (r) − ⎨ r ∂r √ r r . 2 3 3 ⎪ ⎪ − 1 ∂ r 2 ∂En + (2 + l(l + 1))En − 2 l(l + 1)En − k 2 E 3 ⎪ ⎪ n ⎪ ∂r r 2 ∂r r2 r2 ⎪ ⎪ ⎪ √   ⎪ ⎪ ∂ ∂En3 (r) l(l + 1)En2 (r) 2 ⎪ ⎪ ⎩ + (1 − θ) + En3 (r) − = 0, ∂r ∂r r r

in (0, 1),

in (0, 1), (9.126)

which replaces the system (9.79) used in the case .θ = 1. Clearly, another solution (which is divergence-free) is given by .(En2 , En3 ) as defined in Eq. (9.84). F 2  E 2  E2  As in the case .θ = 1, we consider the linear combination . Fn3 := a En3 + b n3 , En n n and we impose the boundary condition .Fn3 (1) = 0, corresponding to .E · ν = 0 on .∂B. We obtain   jl √k √k θ θ ; (9.127) .a = −b jl (k)l(l + 1) hence, up to a constant factor, 

    jl √k r j (kr) l θ 2 + jl (kr)k + l(l + 1) .Fn (r) := − √ r r jl (k) l(l + 1) jl

√k θ





√k θ

(9.128)

and Fn3 (r) := −

.

jl



√k θ



jl (k)

√k θ

jl (kr) + jl r



k √ r θ



k √ . θ

(9.129)

As in the case .θ = 1, we deduce that there are two families of eigenvalues (2) diverging to .−∞: the first one coincides with .λn defined in Eq. (9.107), and it is associated with eigenfunctions in the form .E n (r, ξ ) = jl (kr)A1n (ξ ); the second one is obtained as in the case .θ = 1 by imposing the Steklov boundary conditions, and its members are given explicitly as solutions of the equation λ(1) l =−

.

Fl2 (1) + (Fl2 ) (1) Fl2 (1)

.

(9.130)

9 On a Steklov Spectrum in Electromagnetics

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  We proceed therefore to finding .Fl2 (1) and . Fl2 (1). We easily obtain  Fl2 (1) =

jl

.

√k θ



jl (k) l(l + 1) − jl



√k θ

jl (k)

√



2

jl (k) √k − jl θ

l(l + 1)



√k θ



jl (k) √k

θ

.

(9.131)

Next, we find that 



 kjl (kr) jl (kr)  2 − + jl (kr)k r r2      √k r  √k r  j j  l l k θ θ + l(l + 1) √ − r r2 θ    jl √k √k  kj  (kr) j (kr) (9.80) l θ θ l 2 2 + − (l(l + 1) − k r − 1) = − √ r r2 jl (k) l(l + 1)      √k r   √k r j j  l l k 1 θ θ + ; − l(l +1) √ l(l + 1)jl (k) √ 2 r r jl (k) l(l + 1) θ (9.132) jl

√k θ

k

√ ∂ 2 θ Fl (r) = − √ ∂r jl (k) l(l + 1)

.



hence,  .

Fl2



(1) =

jl



√k θ



√k θ

     kjl (k) + jl (k)(k 2 + 1) − jl √k jl (k)l(l + 1) θ . √ jl (k) l(l + 1) (9.133)

Therefore, (1)

λl .

=−

Fl2 (1) + (Fl2 ) (1) Fl2 (1) jl



√k θ



jl (k) √k k 2 θ     =−   , 2 k k   √ √ jl (k) l(l + 1) − jl jl (k) √k − jl √k jl (k) √k jl θ θ θ θ θ (9.134) which is in agreement with the case .θ = 1. We summarize the previous discussion in the following theorem where it is understood that the values of k are such that the denominators in Eqs. (9.135) and (9.136) do not vanish (in the remark below, we explain the meaning of this condition). Theorem 3 If .k = 0, then the eigenvalues and eigenfunctions of the Steklov problem (9.37) in the unit ball B of .R3 are given, for .n ∈ O, by the following two families:

222

F. Ferraresso et al.   ⎧ jl √k jl (k) √k k 2 ⎪ (1) θ ⎨ θ    λn = −  k  , k2 −jl √k jl (k) √k jl √ jl (k) l(l+1)−jl √k jl (k) √ . θ θ θ θ θ ⎪ ⎩ F n = Fn2 A2n + Fn3 A3n ,

(9.135)

and ) .

(2)

λn = −

jl (k)+jl (k)k , jl (k)

E n = En1 A1n ,

(9.136)

where .Aτ n , .τ = 1, 2, 3, .n ∈ O, are the vector spherical harmonics defined in Eq. (9.64), the functions .Fn2 , .Fn3 are defined, respectively, in Eqs. (9.128) and (9.129), and .En1 is defined in Eq. (9.78). (1) (2) Moreover, .λn , λn → −∞, as .l → ∞. Additionally, .divF n = 0 and .divE n = 0. Remark 3 The squared values of .k = 0 for which the denominators in Eqs. (9.135) and (9.136) vanish are the eigenvalues A of the Dirichlet problem (9.41). Indeed, the derivations in this section show that the solutions of the partial differential equation (9.37), which are tangential at the boundary of B, are given by the two families of vector fields .F n = Fn2 A2n + Fn3 A3n and .E n = En1 A1n . Note that .E n is tangential because .A1n is tangential, while .F n is tangential because .Fn3 (1) = 0. Consider now the condition En1 (1) = 0 .

.

(9.137)

The values of k for which Eq. (9.137) is satisfied are exactly the values of k for which the denominator on the right side of first relation in Eq. (9.136) vanishes: in this case, the vector field .E n vanishes at the boundary of B, so that .E n is a solution of (9.41) with .A = k 2 . Assume now that the denominator in Eq. (9.136) does not vanish. In this case, the function .Fn2 is well-defined, and the solution .F n is therefore well-defined as well. Under this assumption, consider the condition Fn2 (1) = 0 .

.

(9.138)

The values of k for which Eq. (9.138) is satisfied are exactly the values of k for which the denominator on the right side of the first relation in Eq. (9.135) vanishes: in this case, the vector field .F n vanishes at the boundary of B, so that .F n is a solution of (9.41) with .A = k 2 . Thus, assuming that the denominators on the right side of the first relation in Eqs. (9.135) and (9.136) do not vanish corresponds to our assumption (9.43), which is at the base of the analysis carried out in Ref. [9].

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We note en passant that the standard eigenvalues of Maxwell’s equations in a cavity are given by two families of positive numbers, one of which is the family of the squares of the zeros of the equation .jl (k) = 0, see Ref. [45] and the Appendix of Ref. [35], for more details. Remark 4 The case where .k = 0 and .θ = 1 has been considered in Ref. [10]. We note that if we let .k → 0 in Eq. (9.135), (9.136), by using the well-known series expansion jl (z) = 2l

.

∞ (−1)m (m + l)!z2m+l , l!(2m + 2l + 1)!

(9.139)

m=0

we obtain the following formulae: .

lim λ(1) n =−

k→0

l(2l + 3)θ l(θ + 1) + 1

(9.140)

and .

lim λ(2) n = −(l + 1),

(9.141)

k→0

which for .θ = 1 agree with the values computed in Theorem 2 of Ref. [10] (note that in the notation of Theorem 2 of Ref. [10], one obtains our case when .p = 1, while the eigenvalues in that work are by definition opposite in sign to ours). The formulae in the right sides of Eqs. (9.140) and (9.141) can also be obtained by the method that we have used for .k = 0: it suffices to replace the spherical Bessel functions .jl by the appropriate power-type functions that solve systems (9.77) (for the case .k = 0, .θ = 1) and (9.126) (for the case .k = 0 and arbitrary positive .θ ). Namely, for .k = 0, one has to replace the functions in Eq. (9.84) by E 2 (r) =

.

√ l + 1 r l−1 ,

E 3 (r) =

√

l r l−1 ,

(9.142)

and replace the functions in Eq. (9.123) by E2 (r) =

.

√ l+1 lr ,

E3 (r) =

√ l+1

2 + (1 − θ )l r l+1 . −2 + (1 − θ )(l + 3)

(9.143)

Now arguing as in the proof of Theorem 3, the eigenfunctions of Problem (9.37) in the ball . = B are linear combinations of the functions in Eqs. (9.142) and (9.143), yielding the family of eigenvalues λ(1) n =−

.

l(2l + 3)θ . l(θ + 1) + 1

(9.144)

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Note that for .l = 0 one finds that the bounded solutions .(F 2 , F 3 ) satisfying the boundary condition .F3 (1) = 0 are .F 2 ≡ F 3 ≡ 0, and .λ = 0 is not an eigenvalue; here, it is crucial to recall that the eigenvectors satisfy .ν · E = 0 on .∂B. In the same way, one has to replace the function in Eq. (9.78) by .E 1 (r) = r l : this provides the eigenvectors E n = En1 A1n = r l A1n

.

(9.145)

associated with the eigenvalues λ(2) n = −(l + 1),

.

(9.146)

defined also for .l = 0. In physical terms, the above discussion corresponds to the limit in which the electric permittivity .ε of a given material is zero at a specific frequency. So the magnetic field is curl-free, and wave propagation in this material can happen only with phase velocity being infinitely large (satisfying the “static-like” equation . E = 0). Such metamaterials are called “Epsilon-Near-Zero” (ENZ) and have very interesting applications, see, e.g., Ref. [46].

9.5 Summary In this chapter, we first presented various concepts and results concerning the classical Steklov eigenproblem. Further—out of this problem’s numerous applications in different areas—we discussed the sloshing problem, the problem of electrical prospection, and the problem of vibrating membranes. Next, we gave some details on the formulation of the classical Steklov eigenproblem (i.e., for the scalar Laplacian) and its connections to trace theory. We then presented an electromagnetic analogue of the former; in particular, we considered the timeharmonic Maxwell’s equations in a cavity, and we performed a spectral analysis of the Steklov eigenproblem in an appropriate energy space. To this end, we considered a modified Steklov eigenproblem for the .curlcurl operator, whose coercivity was obtained by introducing a suitably selected penalty term. Characterizations of the naturally associated trace spaces in terms of the eigenpairs of this problem were then established. Finally, we found explicit formulae for the eigenvalues and eigenfunctions of this problem in the unit ball of .R3 , by using classical vector spherical harmonics. Two families of eigenvalues diverging to .−∞ were derived, corresponding, respectively, to divergence-free and nondivergence-free eigenfunctions.

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Epilog Nature does not play fair! This was my first thought, along with the tremendous shock to me—as for sure it was to all of Werner’s relatives, friends and colleagues— when I learned the terrible news: he died on January 12, 2003, struck down by an avalanche in Norway. I received the news a couple of days later, by an email from Akhlesh. It was Akhlesh, with whom we had scientific contacts already since the mid-1990s without having met, who introduced me to Werner, in early June 1997, on the day before the beginning of that unforgettable conference (“Bianisotropics 97”) in Glasgow. I knew of a part of their individual and collaborative research work, but it was a real pleasure meeting both of them in person and immediately feeling close to them in human terms. This is not the place to refer to Werner’s internationally leading role in the field of electromagnetic theory of complex materials; our friendship developed rapidly and since then we remained in contact. In May 1998 I invited him to Athens; among scientific and other activities, we had long walks and talks about the Athenian Democracy of classical antiquity, a topic Werner was indeed interested in. Of course he brought his gear with him! He wanted to climb Mount Olympus (the highest mountain in Greece, which rises to 2917 m), the home of the Gods according to Greek Mythology. So after his visit in my department, he went to Olympus alone and then we met in Thessaloniki for the “URSI International Symposium on Electromagnetic Theory”. Our last meeting was in Glasgow again, in May 2000. Ioannis Acknowledgments The authors are thankful to the Departments of Mathematics of the University of Padova and of the National and Kapodistrian University of Athens for the kind hospitality. In addition, they acknowledge financial support from the research project BIRD191739/19 “Sensitivity analysis of partial differential equations in the mathematical theory of electromagnetism” of the University of Padova. The first named author (FF) is grateful for the received support to the UK Engineering and Physical Sciences Research Council through grant EP/T000902/1. The second named author (PDL) is a member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). Finally, the authors thank Dr. Luigi Provenzano for bringing to their attention Refs. [10] and [11].

References 1. Monk, P. B.: Finite Element Methods for Maxwell’s Equations. Clarendon Press, Oxford (2003) 2. Cessenat, M.: Mathematical Methods in Electromagnetics: Linear Theory and Applications. World Scientific, Singapore (1996) 3. Kristensson, G., Stratis, I. G., Wellander, N., Yannacopoulos, A. N.: The exterior Calderón operator for non-spherical objects. SN Partial Differential Equations and Applications 1(6), 32 pp. (2020) 4. Kuznetsov, N., Kulczycki, T., Kwa´snicki, M., Nazarov, A., Poborchi, S., Polterovich, I., Siudeja, B.: The legacy of Vladimir Andreevich Steklov. Not. Am. Math. Soc. 61, 9–22 (2014)

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5. Dautray, R., Lions, J.-L.: Mathematical Analysis and Numerical Methods for Science and Technology, vol. 3. Spectral Theory and Applications. Springer, Berlin (1990) 6. Sylvester, J., Uhlmann, G.: The Dirichlet to Neumann map and applications. In: Inverse problems in Partial Differential Equations (Arcata, CA, 1989), pp. 101–139. SIAM, Philadelphia, PA (1990) 7. Camaño, J., Lackner, C., Monk, P.: Electromagnetic Stekloff eigenvalues in inverse scattering. SIAM J. Math. Anal. 49, 4376–4401 (2017) 8. Davies, E. B.: Spectral Theory and Differential Operators. Cambridge University, Cambridge (1995) 9. Lamberti, P. D., Stratis, I. G.: On an interior Calderón operator and a related Steklov eigenproblem for Maxwell’s equations. SIAM J. Math. Anal. 52, 4140–4160 (2020) 10. Raulot, S., Savo, A.: On the spectrum of the Dirichlet-to-Neumann operator acting on forms of a Euclidean domain. J. Geom. Phys. 77, 1–12 (2014) 11. Savo, A.: Hodge-Laplace eigenvalues of convex bodies. Trans. Am. Math. Soc. 363, 1789– 1804 (2011) 12. Cakoni, F., Cogar, S., Monk, P.: A spectral approach to nondestructive testing via electromagnetic waves. IEEE Trans. Antennas Propag. 69, 8689–8697 (2021) 13. Cogar, S.: Analysis of a trace class Stekloff eigenvalue problem arising in inverse scattering. SIAM J. Appl. Math. 80, 881–905 (2020) 14. Cogar, S.: Existence and stability of electromagnetic Stekloff eigenvalues with a trace class modification. Inverse Probl. Imaging 15, 723–744 (2021) 15. Cogar, S.: Asymptotic expansions of Stekloff eigenvalues for perturbations of inhomogeneous media. Res. Math. Sci. 9, 7 (2022) 16. Cogar, S., Colton, D., Monk, P.: Eigenvalue problems in inverse electromagnetic scattering theory. In: Langer, U., Pauly, D., Repin, S. (eds.) Maxwell’s Equations: Analysis and Numerics, pp. 145–169. De Gruyter, Berlin (2019) 17. Halla, M.: Electromagnetic Stekloff eigenvalues: existence and behavior in the selfadjoint case. arXiv:1909.01983v1 [math.SP] 18. Halla, M.: Electromagnetic Stekloff eigenvalues: approximation analysis (ESAIM). Math. Model. Numer. Anal. 55, 57–76 (2021) 19. Stekloff, W.: Sur les problémes fondamentaux de la physique mathématique (suite et fin). Annales Scientifiques de l’École Normale Supérieure 19, 455–490 (1902) 20. Canavati, J. A., Minzoni, A. A.: A discontinuous Steklov problem with an application to water waves. J. Math. Anal. Appl. 69, 540–558 (1979) 21. Chiadò Piat, V., Nazarov, S. A., Ruotsalainen, K. M.: Spectral gaps and non-Bragg resonances in a water channel. Rendiconti Lincei. Matematica e Applicazioni 29, 321–342 (2018) 22. Levitin, M., Parnovski, L., Polterovich, I., Sher D. A.: Sloshing, Steklov and corners: Asymptotics of sloshing eigenvalues. Journal d’Analyse Mathématique 146, 65–125 (2022) (2021) 23. Calderón, A-P.: On an inverse boundary value problem, Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980), pp. 65–73. Sociedade Brasileira de Matemática, Rio de Janeiro (1980) 24. Sylvester, J., Uhlmann, G.: A global uniqueness theorem for an inverse boundary value problem. Ann. Math. 125, 153–169 (1987) 25. Courant, R., Hilbert, D.: Methods of Mathematical Physics, vol. I. Wiley, New York (1953) 26. Dalla Riva, M., Provenzano, L.: On vibrating thin membranes with mass concentrated near the boundary: an asymptotic analysis. SIAM J. Math. Anal. 50, 2928–2967 (2018) 27. Lamberti, P. D., Provenzano, L.: Viewing the Steklov eigenvalues of the Laplace operator as critical Neumann eigenvalues. In: Mityushev V. V., Ruzhansky M. V. (eds.) Current Trends in Analysis and its Applications, pp. 171–178. Birkhäuser/Springer, Cham (2015) 28. Lamberti, P. D., Provenzano, L.: Neumann to Steklov eigenvalues: asymptotic and monotonicity results. Proc. R. Soc. Edinburgh 147, 429–447 (2017) 29. Heyden, S., Ortiz, M.: Functional optimality of the sulcus pattern of the human brain. Math. Med. Bio. 36, 207–221 (2019)

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30. Lamberti, P. D.: Steklov-type eigenvalues associated with best Sobolev trace constants: domain perturbation and overdetermined system. Complex Var. Elliptic Equations 59, 309–323 (2014) 31. Chavel, I.: Eigenvalues in Riemannian Geometry, 2nd ed. Academic Press, Orlando (1984) 32. Provenzano, L., Stubbe, J.: Weyl-type bounds for Steklov eigenvalues. J. Spectral Theory 9, 349–377 (2019) 33. Auchmuty, G.: Spectral characterization of the trace spaces H s (∂). SIAM J. Math. Anal. 38, 894–905 (2006) 34. Lamberti, P. D., Provenzano, L.: On the explicit representation of the trace space H 3/2 and of the solutions to biharmonic Dirichlet problems on Lipschitz domains via multi-parameter Steklov problems. Revista Matemática Complutense 35, 53–88 (2022) 35. Lamberti, P. D., Zaccaron, M.: Spectral stability of the curlcurl operator via uniform Gaffney inequalities on perturbed electromagnetic cavities. Math. Eng. 5, 1–31 (2022) 36. Costabel, M., Dauge, M.: Maxwell and Lamé eigenvalues on polyhedra. Math. Methods Appl. Sci. 22, 243–258 (1999) 37. Assous, F., Ciarlet, P., Labrunie, S.: Mathematical Foundations of Computational Electromagnetism. Springer, Cham (2018) 38. Girault, V., Raviart P.-A.: Finite Element Approximation of the Navier-Stokes Equations. Springer, Berlin (1981) 39. Kirsch, A., Hettlich, F.: The Mathematical Theory of Time-Harmonic Maxwell’s Equations: Expansion-, Integral-, and Variational Methods. Springer, Cham (2015) 40. Roach, G. F., Stratis, I. G., Yannacopoulos, A. N.: Mathematical Analysis of Deterministic and Stochastic Problems in Complex Media Electromagnetics. Princeton University Press, Princeton (2012) 41. Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, New York (2011) 42. Weber, C.: Regularity theorems for Maxwell’s equations. Math. Methods Appl. Sci. 3, 523–536 (1981) 43. Ciarlet, P.G., Ciarlet, P. Jr., Geymonat, G., Krasucki, F.: Characterization of the kernel of the operator CURL CURL. C. R. Math. Acad. Sci. Paris 344, 305–308 (2007) 44. Kristensson, G.: Scattering of Electromagnetic Waves by Obstacles. SciTech Publishing, Edison (2016) 45. Costabel, M., Dauge, M.: Maxwell eigenmodes in product domains. In: Langer, U., Pauly, D., Repin, S. (eds.) Maxwell’s Equations: Analysis and Numerics, pp. 171–198. De Gruyter, Berlin (2019) 46. Li, Y., Zhou, Z., He, Y., Li, H.: Epsilon-Near-Zero Metamaterials. Cambridge University Press, Cambridge (2022)

Francesco Ferraresso is a Research Associate at the School of Mathematics, Cardiff University, UK, collaborating with M. Marletta on the EPSRC project “A new paradigm for spectral localization of operator pencils and analytic operator-valued functions.” He obtained his PhD from the University of Padova, Italy, in 2018 with a thesis entitled “On the spectral stability of polyharmonic operators on singularly perturbed domains,” supervised by P. D. Lamberti. He held a post-doctoral position at the University of Bern, Switzerland, from 2018 to 2020, where he collaborated with C. Tretter. He has been recently invited to present his research at Spectral Geometry in the Clouds and Durham Days of Analysis and PDE 2022.

228

F. Ferraresso et al. Pier Domenico Lamberti is a full professor at the Dipartimento di Tecnica e Gestione dei Sistemi Industriali of the University of Padova, Italy, from where he got a PhD in Mathematics in 2003 under the supervision of Prof. M. Lanza de Cristoforis. His scientific interests lie in partial differential equations, spectral theory, theory of functions spaces, functional analysis, calculus of variations, and homogenization theory, with special attention to spectral perturbation problems. He has co-authored about fifty papers in international journals and the problem book 100+1 Problems in Advanced Calculus (2022, with G. Drago and P. Toni) published by Springer. Four PhD students have defended their theses under his supervision. Since 2006 he is one of the organizers of the annual international summer school Minicourses in Mathematical Analysis at the University of Padova.

Ioannis G. Stratis is an emeritus professor at the Department of Mathematics of the National and Kapodistrian University of Athens, Greece. He is the co-author of Mathematical Analysis of Deterministic and Stochastic Problems in Complex Media Electromagnetics (Princeton U. P., 2012) with G. F. Roach and A. N. Yannacopoulos. He has published 6 sets of lecture notes, in Greek. Hellenic Academic Open Textbooks will publish An Introduction to Mathematical Biology (2023, co-authored with V. Bitsouni and N. Gialelis, in Greek). He is the co-translator, with V. Dougalis and D. Mitsoudis, of Applied Mathematics by J. D. Logan (published, in Greek, by Crete University Press, 2002). He is the author or co-author of more than 110 research papers in international journals and edited volumes. He has supervised 6 PhD theses.

Chapter 10

Using Boundary Conditions with the Ewald–Oseen Extinction Theorem Akhlesh Lakhtakia

Prolog At the beginning of his career, Werner S. Weiglhofer preferred potentials to fields. This is easy to see from his first five journal papers, all published in 1987 [1–5]. As he pointed out next year in a paper on scalar Hertz potentials in an isotropic chiral medium [6], his fondness for potentials contrasted with my fondness for fields [7]. During the 1990s and later, he continued to work sporadically on potentials [8– 14]. Apart from a triplet of forays [15–17], two with Werner, I stuck close to the church of fields. Our difference led to not only a close collaboration but also a close friendship that ended only with his untimely death at the age of 40 years in 2003 [18]. Werner’s work on scalar potentials bore fruit subsequently [19, 20].

10.1 Introduction Smitten by the Ewald–Oseen extinction theorem from the beginning of my graduate studies, I have loved that theorem for longer than I have known my wife. According to this theorem, the incident electric and magnetic field phasors are annulled throughout the interior of a scatterer by the electric and magnetic field phasors produced by the electric and magnetic surface current densities induced on the surface of the scatterer. This theorem was initially formulated, in the mid-1910s, for scattering by a dielectric object in free space [21, 22]. The ambient free space can be replaced by a

A. Lakhtakia () NanoMM—Nanoengineered Metamaterials Group, Department of Engineering Science and Mechanics, The Pennsylvania State University, University Park, PA, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 T. G. Mackay, A. Lakhtakia (eds.), Adventures in Contemporary Electromagnetic Theory, https://doi.org/10.1007/978-3-031-24617-3_10

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homogeneous isotropic dielectric–magnetic medium without significantly changing the mathematical expression of the Ewald–Oseen extinction theorem. About 75 years after its inception, it was extended to encompass the homogeneous external medium being either isotropic chiral [23] or biisotropic [24]. About 5 years ago, its scope was enlarged so that the homogeneous external medium is a very general, but not the most general, linear bianisotropic medium [25]. The most common application of this theorem is in the formulation of the extended boundary condition method (EBCM) [26–28] for frequency-domain scattering problems. However, emerging from the Huygens principle [29], this theorem also underlies numerical techniques in which a surface integral equation is solved [30–32]. In these applications, the tangential components of the electric and magnetic field phasors are assumed to be continuous across the surface of the scatterer. In this chapter, I show that more complicated boundary conditions can be patched on to the Ewald–Oseen extinction theorem. These boundary conditions encompass topologically insulating surface states [33, 34], charged surfaces [35–37], graphene and other 2D materials [38], and the impedance boundary condition [39, 40]. An .exp(−iωt) time dependence is implicit in this chapter, with .ω being the angular √ frequency, t the time, and .i = −1. Single-underlined letters denoted vectors, with .0 denoting the null vector, .r the position vector, and the caret (.ˆ) identifying unit vectors. Dyadics [41] are underlined twice, with .I denoting the identity dyadic and T .0 the null dyadic. The superscript . denotes the transpose.

10.2 Before Boundary Conditions 10.2.1 Constitutive Relations Suppose that all space .V is divided into two mutually disjoint regions: (i) .Ve bounded by the surfaces .S∞ as well as .S (ii) .Vi enclosed by the surface .S as shown in Fig. 10.1. The source current density phasors are confined to the bounded region .Vs ⊂ Ve , with .Vs assumed to be far away from .Vi . Furthermore, the surface .S is described by a continuous and once-differentiable function, .S∞ is the surface of a sphere of radius .r∞ , and we eventually assume that .r∞ → ∞. The region .Ve is occupied by a linear homogeneous bianisotropic medium with the frequency-domain constitutive relations

.

  ⎫   D(r) = ε • E(r) + ξ + K e − Γ e × I • H (r) ⎬ e   e ,   B(r) = μ • H (r) − ξ − K e + Γ e × I • E(r) ⎭ e

e

(10.1)

10 Using Boundary Conditions with the Ewald–Oseen Extinction Theorem

231

Fig. 10.1 Schematic showing all space .V divided into: (i) .Ve bounded by the surfaces .S∞ as well as .S and (ii) .Vi enclosed by the surface .S. The source current density phasors are confined to the region .Vs ⊂ Ve

where the arbitrary vectors .K e and .Γ e as well as the symmetric dyadics .ε = ε T , e e .μ = μT , and .ξ = ξ T are implicit functions of the angular frequency .ω. When e e e e .Γ e = 0, Eqs. (10.1) describe the most general linear, homogeneous, bianisotropic medium that is Lorentz reciprocal [42, 43]. When .Γ e = 0, Eqs. (10.1) encompass not only the simplest Lorentz nonreciprocal medium [44, 45] but also mediums inspired by gravitationally affected vacuum [46–48]. The identity of the medium occupying .Vi is irrelevant to the Ewald–Oseen extinction theorem.

10.2.2 Dyadic Green Functions The linearity of the constitutive relations (10.1) allows the prescription of Green functions [49] for the medium occupying .Ve [29]. The electric dyadic Green function .Gee (r, r  ) satisfies the differential equation e



.

    ∇ × I + iω ξ + K e − Γ e × I • μ−1 e

•

e



  ∇ × I + iω ξ − K e + Γ e × I e

= iωI δ(r − r  ) ,



− ω2 ε

e

• Gee (r, r  ) e

(10.2)

where .δ(•) is the Dirac delta. The magnetic dyadic Green function .Gmm (r, r  ) is the e solution of the differential equation

232

A. Lakhtakia



    ∇ × I + iω ξ − K e + Γ e × I • ε −1

.

e

e

•

    ∇ × I + iω ξ + K e − Γ e × I − ω2 μ • Gmm (r, r  ) e

e

e

= iωI δ(r − r  ).

(10.3)

Although both dyadic Green functions are known in closed form for only a few special cases [46, 50–53], their spectral forms can be obtained in general by using the spatial Fourier transform [42, 46, 54]. In addition, there are two magnetoelectric dyadic Green functions Gme (r, r  ) =

.

e

  1 −1

μ • ∇ × I + iω ξ − (K e + Γ e ) × I • Gee (r, r  ) e e iω e (10.4)

and   1 −1

ε • ∇ × I + iω ξ + (K e − Γ e ) × I • Gmm (r, r  ) . e e e iω e (10.5) These are, of course, derivable from .Gee (r, r  ) and .Gmm (r, r  ) [29]. e e The dyadic Green functions obey the symmetries [25, 46] Gem (r, r  ) = −

.

⎫ T   

 ⎪ Gee (r, r  ) exp 2iωΓ e • r  = Gee (r  , r) exp 2iωΓ e • r ⎪ ⎪ e e ⎪ ⎬

T     mm mm    . G (r, r ) exp 2iωΓ e • r = G (r , r) exp 2iωΓ e • r e e ⎪ ⎪

 ⎪ T  ⎪   Gem (r, r  ) exp 2iωΓ e • r  = − Gme (r  , r) exp 2iωΓ e • r ⎭

.

e

(10.6)

e

    The factors .exp 2iωΓ e • r and .exp 2iωΓ e • r  in Eqs. (10.6) arise from the Lorentz nonreciprocity inherent in the term .Γ e × I contained in Eqs. (10.1).

10.2.3 Source Fields The electric source current density phasor .J e and the magnetic source current density phasor .J m are wholly confined to the bounded region .Vs ⊂ Ve , as shown in Fig. 10.1. When .Vi is also occupied by the medium with constitutive relations (10.1), these sources are responsible for the source field phasors [46]. E s (r) =

.

     Gee (r, r  ) • J e (r  ) + Gem (r, r  ) • J m (r  ) d 3 r  , e

e

r∈V,

Vs

(10.7)

10 Using Boundary Conditions with the Ewald–Oseen Extinction Theorem

233

and H s (r) =

.

     Gme (r, r  ) • J e (r  ) + Gmm (r, r  ) • J m (r  ) d 3 r  ,

r∈V.

e

e

Vs

(10.8) With .J e (r) and .J m (r) known for all .r ∈ Vs , .E s (r) and .H s (r) can be calculated straightforwardly for any .r ∈ / Vs .

10.2.4 Ewald–Oseen Extinction Theorem Of course, when the region .Vi is occupied by a medium that is different from the medium occupying the region .Ve , scattering takes place, and the actual field phasors, denoted by .E(r) and .H (r)), differ from the source field phasors .E s (r) and .H s (r). As shown elsewhere [25, 46], application of the Huygens principle yields E(r)

.

 = E s (r) +

0 

 

   n(r ˆ ) × E + (r  ) • Gme (r  , r) e

S





+ n(r ˆ ) × H + (r )



• Gee (r  , r) e



   exp 2iωΓ e • r − r  d 2 r  ,



r ∈ Ve r ∈ Vi

, (10.9)

and .

H (r) 0

 = H s (r) −

 

   n(r ˆ ) × H + (r  ) • Gem (r  , r) e

S







+ n(r ˆ ) × E + (r )



• Gmm (r  , r) e



   exp 2iωΓ e • r − r  d 2 r  ,



r ∈ Ve r ∈ Vi

,

(10.10) is the unit normal to .S pointing into .Ve at .r ∈ S and the Sommerfeld ˆ where .n(r) radiation conditions [55, Sec. 5.2.2] have been assumed to hold as .r∞ → ∞. Also, .n ˆ × E + and .nˆ × H + are tangential components of the electric and magnetic field phasors, respectively, on the exterior side of .S. In passing, when the region .Ve is vacuous, Eq. (10.9) could be attributed to Waterman [56, Eq. (2)], but I think that both equations were anticipated by Faxén [57, Eqs. (23) and (24)]. My knowledge of the German language has not improved

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A. Lakhtakia

during the past two decades, and I still miss Werner when reading old German papers. The surface equivalence principle [58, Sec. 3-5] postulates the equivalent surface current density phasors

.

surf

(r) = n(r) ˆ × H + (r)

surf

(r) = −n(r) ˆ × E + (r)

Je

Jm

 r ∈S.

,

(10.11)

Therefore, Eq. (10.9) may be recast as

.

E(r)

 = E s (r) +

0

    surf  • ee   • me  −J surf , r) , r) + J (r ) G (r (r ) G (r e m e

e

S

  × exp 2iωΓ e • r − r  d 2 r  , 



r ∈ Ve

(10.12)

,

r ∈ Vi

and Eq. (10.10) as H (r)

.



0

= H s (r) +

     mm  −J surf (r  ) • Gem (r  , r) + J surf (r , r) e m (r ) • G e

e

S

   × exp 2iωΓ e • r − r  d 2 r  ,



r ∈ Ve r ∈ Vi

(10.13)

.

Finally, the use of Eqs. (10.6) leads to

.

E(r)



      surf  ee Gem (r, r  ) • J surf (r ) d 2 r  , = E s (r) + m (r ) + G (r, r ) • J e e

0

e

S



r ∈ Ve r ∈ Vi

(10.14)

,

and .

H (r) 0



     2  Gme (r, r  )• J surf (r  ) + Gmm (r, r  )• J surf = H s (r) + e m (r ) d r , e

S

e

10 Using Boundary Conditions with the Ewald–Oseen Extinction Theorem



r ∈ Ve r ∈ Vi

235

(10.15)

.

For .r ∈ Vi , Eqs. (10.14) and (10.15) encapsulate the cancelation throughout .Vi of the electric and magnetic field phasors due to the sources contained in .Vs by the electric and magnetic field phasors radiated jointly by the two equivalent surface current density phasors postulated on the exterior side of .S. Equations (10.14).2 and (10.15).2 thus constitute the Ewald–Oseen extinction theorem. Originally derived for free space [21, 22] (and therefore easily extended to homogeneous dielectric– magnetic mediums) and later for homogeneous biisotropic mediums [23, 24], this theorem was extended in 2017 [46] for homogeneous bianisotropic mediums described by Eqs. (10.1). The Ewald–Oseen extinction theorem holds regardless of the nature of the medium occupying .Vi : it can be homogeneous or nonhomogeneous, and it can be isotropic or biisotropic or anisotropic or bianisotropic. That medium can even be free space. All that are needed in Eqs. (10.14) and (10.15) for .r ∈ Vi are the two surface current density phasors on the exterior side of .S. surf surf and .J m , After Eqs. (10.14) and (10.15) with .r ∈ Vi have been solved for .J e the same equations can be used to determine the actual electric and magnetic field phasors at any .r ∈ Ve . Thus, the Ewald–Oseen extinction theorem is a cornerstone of electromagnetic-scattering analysis.

10.3 Incorporation of Boundary Conditions 10.3.1 Scattering Problem The incident field in a scattering problem is the field that exists everywhere when the region occupied by the scatterer is filled with the same medium as the one occupying the region outside the scatterer. Therefore, .E s (r) and .H s (r) are the incident electric and magnetic field phasors, respectively. Since the scattered field in .Ve must be the difference of the actual field and the incident field, the scattered field phasors are given by

.

E sca (r) = E(r) − E s (r) H sca (r) = H (r) − H s (r)

 ,

r ∈ Ve .

(10.16)

At this stage, it is convenient to substitute Eqs. (10.11) and (10.16) in Eqs. (10.14) and (10.15) to get

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A. Lakhtakia

.

E sca (r)

 =±

E s (r)

 

   −Gem (r, r  ) • n(r ˆ ) × E + (r  ) e

S

+Gee (r, r  ) • e



   n(r ˆ ) × H + (r  ) d 2 r  ,

r ∈ Ve r ∈ Vi

, (10.17)

and .

H sca (r) H s (r)

 =±

 

   Gme (r, r  ) • n(r ˆ ) × H + (r  ) e

S

−Gmm (r, r  ) • e

   n(r ˆ ) × E + (r  ) d 2 r  ,



r ∈ Ve r ∈ Vi

.

(10.18)

10.3.2 Boundary Conditions Usually, the continuity of tangential components of the electric and magnetic field phasors across .S is assumed [25]. However, let us define a boundary function .α(r), .r ∈ S, and set .

ˆ × E − (r) n(r) ˆ × E + (r) = n(r)

  n(r) ˆ ˆ × H + (r) = n(r) × E − (r) ˆ × H − (r) − α(r) • n(r)

 ,

r ∈S,

(10.19) where .n×E ˆ and . n×H ˆ are the tangential components of the electric and magnetic − − field phasors, respectively, on the interior side of .S. Although various models could emerge for .α, a physically plausible one is   ˆ α(r) = γ + σ b • n(r) ×I .

.

b

(10.20)

Topologically insulating scatterers can be accommodated by setting .γ = γb I b and .σ b = 0 [34], with .γb as the surface admittance. Charged scatterers can be accommodated by setting .γ = 0 and .σ b = σb I [35, Sec. 5], with .σb as the surface b conductivity, and likewise for graphene-covered scatterers [38]. Also, both .γ and b .σ can be functions of .r ∈ S. With symmetry in mind, we can set b

10 Using Boundary Conditions with the Ewald–Oseen Extinction Theorem

.

237

  ˆ n(r) ˆ × E + (r) = n(r) ˆ × E − (r) − β(r) • n(r) × H − (r)   , n(r) ˆ ˆ × E − (r) ˆ × H − (r) − α(r) • n(r) × H + (r) = n(r)

r ∈S,

(10.21) instead of Eqs. (10.19), with .β as another boundary function. Since the field phasors on the interior side of .S must be the internal field phasors induced in .Vi and evaluated on .S, it follows that .

  ˆ × E + (r) = n(r) ˆ × E int (r) − β(r) • n(r) × H int (r) n(r) ˆ   , n(r) ˆ ˆ × H + (r) = n(r) × E int (r) ˆ × H int (r) − α(r) • n(r)

r ∈S,

(10.22) where .E int and .H int are the internal electric and magnetic field phasors, respectively. The foregoing considerations allow the transformation of Eqs. (10.17) and (10.18) into .



E sca (r)

=±

E inc (r)

 

   ¯ em (r, r  ) • n(r −G ˆ ) × E int (r  ) e

S

¯ ee (r, r  ) • +G e



   n(r ˆ ) × H int (r  ) d 2 r  ,

r ∈ Ve r ∈ Vi

, (10.23)

and .

H sca (r)



H inc (r)

=±

 

   ¯ me (r, r  ) • n(r G ˆ ) × H int (r  ) e

S

¯ mm (r, r  ) • −G e



   n(r ˆ ) × E int (r  ) d 2 r  ,

r ∈ Ve r ∈ Vi

, (10.24)

respectively, where ¯ ee (r, r  ) = Gee (r, r  ) + Gem (r, r  ) • β(r  ) G .

⎫

⎪ ⎪ e e e ⎪ ⎪ mm mm me     ¯ ⎬ G (r, r ) = G (r, r ) + G (r, r ) • α(r ) ⎪ e e e em em ee ¯ (r, r  ) = G (r, r  ) + G (r, r  ) • α(r  ) ⎪ G ⎪ ⎪ e e e ⎪ ⎪ me me ee     ¯ G (r, r ) = G (r, r ) + G (r, r ) • β(r ) ⎭ e e e

.

(10.25)

Equations (10.23).2 and (10.24).2 are a pair of coupled integral equations that must ˆ  ) × E int (r  ) and .n(r ˆ  ) × H int (r  ) be solved for all .r ∈ Vi in order to determine .n(r

238

A. Lakhtakia

at all .r  ∈ S. Once these quantities have been determined, they can be used on the right sides of Eqs. (10.23).1 and (10.24).1 to calculate .E sca (r) and .H sca (r) at any point .r ∈ Ve . Suppose that a homogeneous medium described by the constitutive relations

.

  ⎫   D(r) = ε • E(r) + ξ + K i − Γ i × I • H (r) ⎬ i   i   • B(r) = μ H (r) − ξ − K i + Γ i × I • E(r) ⎭ i

(10.26)

i

occupies .Vi , with symmetric dyadics .ε , .μ , and .ξ . If bilinear expansions of i

i

i

the dyadic Green functions for that medium (denoted by .Gee (r, r  ), etc.) are i known [25], then series representations of the electric and magnetic field phasors in .Vi become available [25, 46]. In that case, the expansion coefficients in the series representations of .E int and .H int can be determined after the solution of Eqs. (10.23).2 and (10.24).2 . This is the process adopted in the extended boundary condition method (EBCM) [23, 24, 27, 53, 59–61]. However, as explained elsewhere in detail [25], EBCM also requires bilinear expansions of the dyadic Green functions ee  .G (r, r ), etc., for series representations of .E s , .H s , .E sca , and .H sca . e If .Vi is occupied by either a homogeneous medium for which the bilinear expansions of the dyadic Green functions are not known or a nonhomogeneous medium, local basis functions can be adequate to represent .E int and .H int . Such local basis functions are used in the method of moments [62] and the finitedifference method [63]. Parenthetically, when a homogeneous medium described by Eqs. (10.26) occupies .Vi , the Huygens principle delivers .

 

   Gem (r, r  ) • n(r ˆ ) × E − (r  ) E int (r) = i

S

   −Gee (r, r  ) • n(r ˆ ) × H − (r  ) d 2 r  , i

r ∈ Vi , (10.27)

and .

H int (r) = −

 

   Gme (r, r  ) • n(r ˆ ) × H − (r  ) i

S

   −Gmm (r, r  ) • n(r ˆ ) × E − (r  ) d 2 r  , i

r ∈ Vi . (10.28)

These two equations can sometimes be useful for calculating .E int and .H int , if Gee (r, r  ), etc., are known.

.

i

10 Using Boundary Conditions with the Ewald–Oseen Extinction Theorem

239

10.3.3 Impedance Boundary Condition For some scattering problems, it is appropriate to use the impedance boundary condition [40]     ˆ ˆ n(r) ˆ × E + (r) = Z(r) • n(r) × I • n(r) × H + (r) ,

r ∈S,

.

(10.29)

where .Z is the surface impedance dyadic. Equations (10.17) and (10.18) then become    

r ∈ Ve E sca (r)    2  ee  •  H e (r, r ) n(r ˆ ) × H + (r ) d r , . , =± E s (r) r ∈ Vi S

(10.30) and .

H sca (r)



H s (r)

 

   2   •  H me n(r ˆ r ) ) × H (r ) d r , (r, =± + e



r ∈ Ve r ∈ Vi

S

,

(10.31) where

.

  ⎫  ) = Gee (r, r  ) − Gem (r, r  ) • Z(r  ) • n(r ) × I ⎬ ˆ H ee (r, r e e e  .  H me ˆ ) × I ⎭ (r, r  ) = Gme (r, r  ) − Gmm (r, r  ) • Z(r  ) • n(r e e

(10.32)

e

Equation (10.11).1 may then be used to formulate .

E sca (r)



E s (r)

     • surf  H ee ) J ) d 2r  , (r r (r, =± e e S



r ∈ Ve r ∈ Vi

,

(10.33)

.

(10.34)

and .

H sca (r) H s (r)



     • surf  H me ) J (r ) d 2r  , (r, r =± e e S



r ∈ Ve r ∈ Vi

Equations (10.33).2 and (10.34).2 are a pair of coupled integral equations that surf must be solved for all .r ∈ Vi in order to determine .J e (r  ) at all .r  ∈ S. Once that quantity has been determined, the right sides of Eqs. (10.33).1 and (10.34).1 can be used to calculate .E sca (r) and .H sca (r) at any point .r ∈ Ve . Bilinear expansions

240

A. Lakhtakia

Fig. 10.2 Svein O. Hansen enjoying retirement in Austria. (Photo courtesy of Heide Weiglhofer.)

of the dyadic Green functions for the homogeneous medium occupying .Ve help represent .E s , .H s , .E sca , and .H sca . Similar expansions can be used for a series surf surf representation of .J e only at [26]; alternatively, one can choose to calculate .J e a multitude of points on .S [64, 65]. Note that there is no interest in calculating the electric and magnetic field phasors in .Vi , when Eq. (10.29) is pressed into service.

10.4 Closing Remarks A veritable cornucopia, the Ewald–Oseen extinction theorem keeps on giving. In the past decades, it had been used to formulate the scattering responses of perfect electrically conducting objects [26] as well of scatterers with electrically neutral surfaces [25]. In this chapter, it has been applied for scatterers whose surfaces are electrically and/or magnetically non-neutral for a variety of reasons. Time will show what other treasures lie inside this theorem.

Epilog I was successful in enticing Werner from potentials to fields—mostly but not completely. As Werner’s confidante and even one-time research collaborator [66] named Svein Oskar Hansen recently reminded me by e-mail ([email protected]), Werner had returned to potentials during his last 2 years with a loud bang [12– 14]. Some may recall Svein as the stuffed toy rabbit who accompanied Werner on numerous trips from about 1994 [18]. A native of Molde, Norway, Svein is now comfortably retired (Fig. 10.2) in Bruck an der Mur, Austria, where in November 2019 I had the pleasure of meeting him after a lapse of some 17 years.

10 Using Boundary Conditions with the Ewald–Oseen Extinction Theorem

241

Acknowledgments I am grateful to the Charles Godfrey Binder Endowment at Penn State for ongoing support of my research activities.

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21. Ewald, P.P.: Zur Begründung der Kristalloptik. Ann. Phys. (4th Series) 49, 117–143 (1916) 22. Oseen, C.W.: Über die Wechselwirkung zwischen zwei elektrischen Dipolen und über die Drehung der Polarisationsebene in Kristallen und Flüssigkeiten. Ann. Phys. (4th Series) 48, 1–56 (1915) 23. Lakhtakia, A.: The extended boundary condition method for scattering by a chiral scatterer in a chiral medium. Optik 86, 155–161 (1991) 24. Lakhtakia, A.: On the Huygens’s principles and the Ewald–Oseen extinction theorems for, and the scattering of, Beltrami fields. Optik 91, 35–40 (1992) 25. Lakhtakia, A.: The Ewald–Oseen extinction theorem and the extended boundary condition method. In: Lakhtakia, A., Furse, C.M. (eds.) The World of Applied Electromagnetics—In Appreciation of Magdy Fahmy Iskander, pp. 481–513. Springer, Cham (2018) 26. Waterman, P.C.: Matrix formulation of electromagnetic scattering. Proc. IEEE 53, 805–812 (1965) 27. Waterman, P.C.: Scattering by dielectric obstacles. Alta Frequenza (Speciale) 38, 348–352 (1969) 28. Barber, P.W., Yeh, C.: Scattering of electromagnetic waves by arbitrarily shaped dielectric bodies. Appl. Opt. 14, 2864–2872 (1975) 29. Faryad, M., Lakhtakia, A.: On the Huygens principle for bianisotropic mediums with symmetric permittivity and permeability dyadics. Phys. Lett. A 381, 742–746 (2017). errata: 381, 2136 (2017) 30. Wu, T.K., Tsai, L.L.: Numerical analysis of electromagnetic fields in biological tissues. Proc. IEEE 62, 1167–1168 (1974) 31. Wu, T.-K., Tsai, L.L.: Scattering from arbitrarily-shaped lossy dielectric bodies of revolution. Radio Sci. 12, 709–718 (1979) 32. Tsai, C.-C., Wu, S.-T.: A new single surface integral equation for light scattering by circular dielectric cylinders. Opt. Commun. 277, 247–250 (2007) 33. Ando, Y.: Topological insulator materials. J. Phys. Soc. Jpn. 82, 102001 (2013) 34. Lakhtakia, A., Mackay, T.G.: Classical electromagnetic model of surface states in topological insulators. J. Nanophotonics 10, 033004 (2016) 35. Stratton, J.A., Chu, L.J.: Diffraction theory of electromagnetic waves. Phys. Rev. 56, 99–107 (1939) 36. Garrigos, R., Kofman, R., Richard, J.: Phenomenological interpretation of electroreflectance in gold. Solid State Commun. 14, 1029–1031 (1974) 37. Bohren, C.F., Hunt, A.J.: Scattering of electromagnetic waves by a charged sphere. Can. J. Phys. 55, 1930–1935 (1977) 38. Chiadini, F., Scaglione, A., Fiumara, V., Shuba, M.V., Lakhtakia, A.: Effect of chemical potential on Dyakonov–Tamm waves guided by a graphene-coated structurally chiral medium. J. Opt. (Bristol) 21, 055002 (2019). corrigendum: 21, 079501 (2019) 39. Mohsen, A.: On the impedance boundary condition. Appl. Math. Model. 6, 405–407 (1982) 40. Hoppe, D.J., Rahmat-Samii, R.: Impedance Boundary Conditions in Electromagnetics. CRC Press, Boca Raton (1995) 41. Chen, H.C.: Theory of Electromagnetic Waves: A Coordinate-Free Approach. McGraw–Hill, New York (1983) 42. Kong, J.A.: Theorems of bianisotropic media. Proc. IEEE 60, 1036–1046 (1972) 43. Krowne, C.M.: Electromagnetic theorems for complex anisotropic media. IEEE Trans. Antennas Propag. 32, 1224–1230 (1984) 44. Lakhtakia, A.: Planewave response of a simple Lorentz-nonreciprocal medium with magnetoelectric gyrotropy. Optik 182, 372–381 (2019) 45. Alkhoori, H.M., Lakhtakia, A., Breakall, J.K., Bohren, C.F.: Scattering by a three-dimensional object composed of the simplest Lorentz-nonreciprocal medium. J. Opt. Soc. Am. A 35, 2026– 2034 (2018)

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46. Faryad, M., Lakhtakia, A.: Infinite-Space Dyadic Green Functions in Electromagnetism. Morgan and Claypool, San Rafael (2018) 47. Plébanski, J.: Electromagnetic waves in gravitational fields. Phys. Rev. 118, 1396–1408 (1960) 48. Lakhtakia, A., Mackay, T.G.: Dyadic Green function for an electromagnetic medium inspired by general relativity. Chin. Phys. Lett. 23, 832–833 (2006). errata: 29, 019902 (2012) 49. Stakgold, I.: Green’s Functions and Boundary Value Problems, 2nd edn. Wiley, New York (1998) 50. Weiglhofer, W.S.: Analytic methods and free-space dyadic Green’s functions. Radio Sci. 28, 847–857 (1993) 51. Olyslager, F., Lindell, I.V.: Electromagnetics and exotic media: A quest for the holy grail. IEEE Antennas Propag. Mag. 44(2), 48–58 (2002) 52. Mackay, T.G., Lakhtakia, A.: The Huygens principle for a uniaxial dielectric-magnetic medium with gyrotropic-like magnetoelectric properties. Electromagnetics 29, 143–150 (2009) 53. Lakhtakia, A., Mackay, T.G.: Vector spherical wavefunctions for orthorhombic dielectricmagnetic material with gyrotropic-like magnetoelectric properties. J. Opt. (India) 41, 201–213 (2012) 54. Ogg, N.R.: A Huygen’s principle for anisotropic media. J. Phys. A: Gen. Phys. 4, 382–388 (1971) 55. Rothwell, E.J., Cloud, M.J.: Electromagnetics. CRC Press, Boca Raton (2001) 56. Waterman, P.C.: Symmetry, unitarity, and geometry in electromagnetic scattering. Phys. Rev. D 3, 825–839 (1971) 57. Faxén, H.: Der Zusammenhang zwischen den Maxwellschen Gleichungen für Dielektrika und den atomistischen Ansätzen von H. A. Lorentz u. a. Zeitschrift für Physik 2, 218–229 (1920) 58. Harrington, R.F.: Time-Harmonic Electromagnetic Fields. McGraw–Hill, New York (1961) 59. Alkhoori, H.M., Lakhtakia, A., Breakall, J.K., Bohren, C.F.: Plane-wave scattering by an ellipsoid composed of an orthorhombic dielectric-magnetic material. J. Opt. Soc. Am. A 35, 1549–1559 (2018) 60. Alkhoori, H.M., Lakhtakia, A., Breakall, J.K., Bohren, C.F.: Plane-wave scattering by an ellipsoid composed of an orthorhombic dielectric-magnetic material with arbitrarily oriented constitutive principal axes. J. Opt. Soc. Am. B 36, F60–F71 (2019) 61. Alkhoori, H.M., Lakhtakia, A., Breakall, J.K., Bohren, C.F.: Sufficient conditions for zero backscattering by a uniaxial dielectric-magnetic scatterer endowed with magnetoelectric gyrotropy. IEEE Trans. Antennas Propag. 68, 1023–1030 (2020) 62. Gibson, W.C.: The Method of Moments in Electromagnetics, 2nd edn. CRC Press, Boca Raton (2015) 63. Monk, P.B.: Finite Element Methods for Maxwell’s Equations. Oxford University Press, Oxford (2003) 64. Bates, R.H.T.: Rayleigh hypothesis, the extended-boundary condition and point matching. Electron. Lett. 5, 654–655 (1969) 65. Lewin, L.: On the restricted validity of point-matching techniques. IEEE Trans. Microw. Theory Tech. 18, 1041–1047 (1970) 66. Weiglhofer, W.S., Hansen, S.O.: Faraday chiral media revisited–I. Fields and sources. IEEE Trans. Antennas Propag. 47, 807–814 (1999)

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Akhlesh Lakhtakia received the BTech (1979) and DSc (2006) degrees from the Banaras Hindu University and the MS (1981) and PhD (1983) degrees from the University of Utah. In 1983, he joined the Department of Engineering Science and Mechanics at Penn State as a post-doctoral research scholar, where he became a Distinguished Professor in 2003, the Charles Godfrey Binder Professor in 2006, and the Evan Pugh University Professor of Engineering Science and Mechanics in 2018. His current research interests include electromagnetic scattering, surface multiplasmonics, bioreplication, forensic science, solar energy, sculptured thin films, and mimumes. He has been elected a fellow of Optical Society of America (1992), SPIE—The International Society for Optical Engineering (1996), Institute of Physics (UK) (1996), American Association for the Advancement of Science (2010), American Physical Society (2012), Institute of Electrical and Electronics Engineers (2016), Royal Society of Chemistry (2016), and Royal Society of Arts (2017). He has been designated a Distinguished Alumnus of both of his almae matres at the highest level. Awards at Penn State include: Outstanding Research Award (1996), Outstanding Advising Award (2005), Premier Research Award (2008), and Outstanding Teaching Award (2016), and the Faculty Scholar Medal (2005). He received the 2010 Technical Achievement Award from SPIE, the 2016 Walston Chubb Award for Innovation, the 2022 Smart Structures and Materials Lifetime Achievement Award, the 2022 Radio Club of America Lifetime Achievement Award, and the 2022 IEEE Antennas and Propagation Society Distinguished Achievement Award. He is presently a Sigma Xi Distinguished Lecturer (2022–24) and a Jefferson Science Fellow at the US State Department (2022–23).

Chapter 11

Spatial Sampling and Interpolation Techniques in Computational Electromagnetics and Beyond Yaniv Brick and Amir Boag

11.1 Introduction Computational electromagnetics (CEM) [1, 2] and acoustics [3] seek accurate solutions to problems involving interactions of fields with various material structures of practical interest while striving to minimize the required resources, primarily, the processing time and data storage. The development of computational techniques passed through several stages in diverse disciplines, ranging from signal integrity, through antennas, and all the way to optics. Early on, it has become clear that there is no single computational approach suitable for all problems of interest. Depending on the problem dimensions relative to the characteristic wavelength, three distinct classes of methods were developed almost independently, namely: (i) low-frequency quasi-static approximations, (ii) numerically rigorous techniques for the “resonant regime,” and (iii) high-frequency asymptotic approximations, for different applications. Furthermore, frequency and time domain formulations and solvers for each regime, have been proposed. In spite of the variety, bruteforce implementations of many of these techniques are characterized by a common feature, namely, high computational complexity. Such high complexity effectively precludes their straightforward application to large problems of practical interest due to the limited computational resources. In turn, these computational bottlenecks stimulated the development of so-called “fast algorithms,” generally enabling the

Y. Brick School of Electrical and Computer Engineering, Ben-Gurion University of the Negev, Be’er Sheva, Israel e-mail: [email protected] A. Boag () School of Electrical Engineering, Tel Aviv University, Tel Aviv, Israel e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 T. G. Mackay, A. Lakhtakia (eds.), Adventures in Contemporary Electromagnetic Theory, https://doi.org/10.1007/978-3-031-24617-3_11

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accomplishment of given tasks at considerably lower computational costs, at the expense of slight approximations but with controllable accuracy. In this chapter, we review various computational tasks that can be efficiently performed using fast algorithms based on the spatial sampling and interpolation approach. Quite often, substantially different tasks can be accelerated by essentially the same algorithm. Thus, in each subsection of the introduction, we discuss a specific computational task while directing the reader to an appropriate algorithm detailed in one of the following sections or prior publications. In particular, integral eq. (IE) solvers for quasi-static and dynamic problems are reviewed in Sects. 11.1.1 and 11.1.2, respectively. Subsequently, algorithms for approximate solutions in the high-frequency regime and rigorous IE solvers in the time domain are discussed in Sects. 11.1.3 and 11.1.4, respectively. Interestingly, the sampling and interpolation approach can be applied, beyond the conventional CEM or computational acoustics, to problems involving integral operators characterized by high computational complexity. For example, it gives rise to efficient algorithms for various imaging modalities, including synthetic aperture radar and tomographic imaging, as discussed in Sect. 11.1.5. Following the review of the computational tasks for various applications in Sect. 11.1, the reader will find a more technical description of the approach. It begins with the general development of spatial field sampling criteria and reconstruction techniques (Sect. 11.2). Then, “core” algorithms utilizing a hierarchical sampling and interpolation paradigm are presented (Sects. 11.3, 11.4, and 11.5). These include algorithms for the computation of far region fields in the frequency domain (Sect. 11.3), more generalized ones that treat also the near region fields (Sect. 11.4), and methods for computing algebraically compressed representation of moment matrices (Sect. 11.5). Section 11.6 concludes the chapter and discusses future research directions and prospects.

11.1.1 Integral Equation Treatment of Static and Quasi-Static Problems Both low-frequency and full-wave models for wave problems are conventionally formulated as differential or IEs, which are subsequently discretized and solved over pertinent volume or surface domains. Although differential equations are often preferred for problems involving highly inhomogeneous volumetric materials, here we focus on IE-based techniques. In fact, IE-based methods have several important advantages as compared to their differential equation-based counterparts, in particular, for problems defined in unbounded domains [1–3]. In cases involving impenetrable and/or piecewise homogeneous materials, the IEs can be formulated over surface domains only, thus greatly reducing the numbers of unknowns relative to the volumetric discretization. For large problems, discretized IEs are often solved iteratively, thus requiring repeated evaluation of the pertinent integral operators, which can be interpreted as field computations due to given source distributions [2].

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For example, the capacitance extraction problem is naturally formulated in the form of an IE for the electrostatic potential. An algorithm based on hierarchical sampling and interpolation between non-uniform grid (NGs), for electrostatic potential evaluation due to given charge distributions, has been developed in Ref. [4]. This algorithm reduces the computational complexity from O(N 2 ) of the straightforward computation to O(N), which is the same as that of the multilevel fast multipole method (FMM) [5, 6]. Indeed, this static version of the NG algorithm can be considered a spatial counterpart of the FMM. The same algorithm has also been employed in magneto-quasi-static applications [7]. A special version designed for elongated geometries of interest in quantum calculations using the density functional theory has been presented in Ref. [8]. Its simple structure was found to be advantageous for parallelization [9–11]. The key principles enabling these savings are reviewed in Sect. 11.2.1.

11.1.2 Numerically Rigorous Analysis in the Frequency Domain Frequency domain IEs provide a convenient framework for the full-wave analysis of EM and acoustics problems with time-harmonic excitation. The discretization of IEs leads to large systems of linear equations that can be solved either directly or iteratively. Currently, fast iterative solvers are the only viable means for treating very large systems and, thus, they will be discussed first. As in the static case, the iterative solvers involve repeated field evaluations, via the integral operators, for given source distributions. As described in Sect. 11.2, fields radiated by finite source domains can be effectively represented by their samples on coarse and radially non-uniform grids, following phase- and amplitude-compensation. The fast algorithms in Sect. 11.3 evaluate the field integrals hierarchically. They employ a partitioning of the source domain and compute the fields through hierarchical interpolation between the gradually refined grids, corresponding to gradually increasing source subdomains. For well-separated sources and observers, as in the cases covered by Sect. 11.3, this approach is structurally related to the butterfly-based multilevel field evaluation algorithms [12, 13]. In the context of fast iterative IE solvers, the evaluation of nearzone contribution to the field is added to the algorithms, as described in Sect. 11.4. Two-dimensional versions of the algorithm were first described in Refs. [14, 15] and later extended to three dimensions in acoustics and electromagnetics in Refs. [16–19]. A special version providing a reduced complexity for highly elongated structures has been reported in [20]. Fast field evaluation is also important for acceleration of differential equation solvers that make use of global absorbing boundary conditions for non-convex domains [21, 22]. While being generally more powerful for very large problems, iterative solvers may suffer from poor and unpredictable convergence for resonant problems. Also, if analysis is required for a large number of excitations, it might be advantageous to perform a factorization of the system matrix that would facilitate a low-cost solution

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for each excitation. Unfortunately, the O(N 3 ) complexity of the conventional direct solvers, such as the LU-factorization, precludes their use for large systems of equations. Often, the O(N 2 ) storage requirements do not even allow one to compute the whole matrix. In such cases, one has to replace the matrix with some compressed form, for which a compressed factorized form can be computed efficiently. In many cases, low-rank approximations of off-diagonal matrix blocks of hierarchically structured matrices can provide the desired savings if efficiently computed by exploiting the particular properties of the integral operators they represent. Fast direct solvers, powered by the NG field representations for the fast compression of blocks, were proposed in Refs. [23–27]. The algorithmic principles used for their design are reviewed in Sect. 11.5. To not dwell on the algebra of fast direct solvers, Sect. 11.5 focuses on the NG-based algorithms in Refs. [28, 29] for the fast low-rank approximation of single off-diagonal matrix blocks – a task that is the cornerstone of many direct solvers. As the development of fast direct solvers is a highly active field, ongoing research efforts are expected to provide further advances and improvements to these algorithms already in the near future.

11.1.3 High-Frequency Radiation and Scattering The physical optics (PO) approach is one of the best-known high-frequency approaches, allowing for robust calculations for arbitrary shaped geometries comprising surfaces that are large and smooth on the wavelength scale. For the PO approach to be valid, the radii of curvature of all surfaces are also required to be large as compared to the wavelength. In the PO approximation, the surface fields are computed locally, using the incident field, while replacing the actual surface by its tangential plane. Subsequently, the scattered or radiated fields are evaluated via Kirchhoff type integration of surface quantities. Though conceptually straightforward, such computations can be highly time consuming and can benefit from algorithmic acceleration. Consider, for example, the analysis of radiation by a reflector or lens antenna. Here, the incident field is that of the primary feed, whose radiation pattern is assumed to be known. The antenna radiation pattern computation entails evaluation of the far-field Chu-Stratton integral for an angular domain of interest. This computation can be accelerated by the far-field version of the two-level or, even better, multilevel sampling and interpolation algorithm described in Sect. 11.3 and detailed in Refs. [30, 31]. Interestingly, this algorithm resembles the up-tree part of the multilevel fast multipole algorithm [32]. Furthermore, the diffraction effects in reflector antennas can be accounted for by using incremental length diffraction coefficients. This involves the evaluation of a line integral, which is accelerated, adding a negligible computational cost [33]. More complex dual and multi-reflector antennas involve reflector-to-reflector propagation and, thus, evaluation of near-zone Chu-Stratton integral. To that end, an algorithm for acceleration of the reflectorto-reflector propagation has been developed in Ref. [34]. This algorithm relies

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on an analytic description of the reflector surfaces for improved performance, as compared to the non-uniform grid (NG) algorithm designed for general near-zone field evaluation, to be described in Sect. 11.4, mainly, in the context of fast IE solvers. Furthermore, a near-zone back-propagation version optimized for antenna diagnostics was presented in Ref. [35]. Turning to high-frequency scattering, a natural application of the algorithms in Sect. 11.3 is the radar cross section (RCS) computation for convex shapes. In the PO approximation, the incident plane wave induces a surface field only over the lit part of the scatterer. For a given incident field, the far scattered field, or bi-static RCS, is again obtained via surface integration, as in Ref. [36]. For the monostatic RCS, examination of the resulting integral operator reveals that it is similar to the radiation one discussed above [30, 31], but with a doubled wavenumber, obviously stemming from the two-way propagation characteristic for monostatic scattering. Consequently, this computation is accelerated by just a slightly modified version of the algorithm presented in Sect. 11.3 [31]. Certain refinement of the algorithm is required where the lit part of the scatterer surface changes abruptly versus observation angles [37]. For non-convex scatterers, one has to account for multiple scattering and mutual shadowing effects. A special highfrequency RCS computation algorithm accelerating the double-bounce computation has been developed in Ref. [38]. Both single- and double-bounce fast algorithms have been generalized for near-zone computations in Refs. [39, 40]. Naturally, these algorithms can also be applied, as a post-processing step for scattered field or RCS computation, to distributions computed in a numerically rigorous (e.g., using the algorithm in Sect. 11.4) or another, approximate, manner. For example, in Refs. [41, 42], the algorithm in Ref. [30] was extended to the case of multi-static scattering by inhomogeneous volumetric objects, under the Born approximation assumptions. The solution can also be sought in a multibounce or iterative PO manner, employing the NG algorithm in Sect. 11.4 for the acceleration of its iterative steps [15]. This algorithm for high-frequency scattering from impenetrable bodies was extended in Ref. [43] to account also for complex mutual visibility (or self-shadowing) in complex impenetrable objects, through nested shadow radiation iterations, also accelerated by the NG algorithm. The shadow radiation mechanism provides smoother and physically appealing shadow boundaries, thus producing more accurate results.

11.1.4 Time-Domain Radiation and Scattering Time-domain analysis is truly necessary in situations involving non-linear or time varying material properties. It might also be more efficient numerically for ultrawideband excitations, in particular, of very short temporal duration. To that end, the frequency domain IEs discussed in Sect. 11.1.2 can be transformed to the time domain. Similarly, the high-frequency scattering and radiation problems can be formulated in the time domain. It appears natural, therefore, to extend the

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spatial sampling and interpolation-based fast algorithms to the time domain. Explicit time dependence entails that the solution process is performed by marching on in time instead of the iterative solution in the frequency domain. Consequently, one needs to store the time history of each source, thus greatly increasing the memory requirements. Two different versions of the time domain algorithm have been developed. In Ref. [44], the frequency domain algorithm of Sect. 11.4 has been transformed to the time domain, while a similar transformation of the far-field algorithm of Sect. 11.3 was presented in Refs. [45, 46]. These algorithms reduce the computational complexity from O(N 2 ) to O(NlogN) per time step. A substantially different version computing pairwise interactions between boxes was presented in Ref. [47]. The latter algorithm’s complexity is slightly higher, being O(Nlog2 N) per time step, and, structurally, it is close to the plane wave time domain (PWTD) technique [48]. Finally, a special version of fast TD field evaluation for elongated geometries has been presented in Ref. [49].

11.1.5 Imaging Synthetic aperture radar (SAR) and sonar systems collect multidimensional scattering data. For example, the mono-static scattered field can be measured versus the frequency and azimuthal angle. Subsequently, an image is computed using one of the SAR imaging algorithms, which can be interpreted as back-propagation of the scattered field followed by various approximations. The need for approximations stems from the high complexity of the straightforward implementation of the backpropagation. Assuming an N × N frequency-azimuth data, direct computation of an N × N image would incur an O(N 4 ) computational cost. To that end, SAR processing often employs fast Fourier transforms (FFT) to reduce the complexity to O(N 2 logN). The adaptation of the back-propagation to the FFT compatible form is achieved, however, at the expense of approximations that tend to spoil the resulting image quality. The image distortions and smearing might be particularly severe, in near field, wideband and wide angular aperture scenarios. On the other hand, the spatial sampling and interpolation approach allows for accurate evaluation of the back-propagation operator, without introducing algorithm-produced errors beyond those of the multilevel interpolation operations. In Ref. [50], this approach has been applied to a simple inverse SAR configuration producing complexity comparable to that of the FFT-based techniques, but without significant approximations. The backprojection imaging can be extended to more complex cases, for example, involving moving targets. Recently, it has been shown the spatial sampling and interpolation approach reduces the computational complexity of detecting and imaging moving targets embedded in static environment as described in Ref. [51]. A substantially different task of X-ray computer tomography involves reconstruction of the body absorption image based on its line integrals. The forward problem here is known as Radon transform, while the imaging amounts to the computation of the inverse Radon transform. Imaging based on sampling and

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interpolation in the spatial domain (of the image) has been shown to produce excellent results even in high-contrast configurations [52]. Interestingly, here too, the conventional FFT-based imaging algorithms suffer from significant artifacts. Due to their wide variety of and limited space, these algorithms are not presented in this chapters. They do, however, make use of the field sampling criteria and multilevel algorithm principles in Sects. 11.2, 11.3, 11.4, and 11.5.

11.2 Spatial Sampling and Interpolation The key principle on which the class of methods covered by this chapter relies is the efficient sampling of fields, produced by finite size source distributions, in a manner that enables their reconstruction, with controllable error, via interpolation. The sampling rates are dictated by the limited effective spatial bandwidth of the fields, which is revealed by simple phase- and amplitude-compensation operations. These can be viewed as demodulation and compression of the field’s spatial spectrum. In this section, we provide the theoretical foundation for optimal sampling of phaseand amplitude-compensated fields.

11.2.1 Optimal Sampling of Static and Quasi-Static Fields It is convenient to first discuss the static case, for which the theory of accurate field representation using a finite number of field objects, namely, multipoles, is well established [5, 6]. Consider the potential φ(r) produced by a charge density σ (r) distributed on a surface S contained in a sphere of radius R. The potential can be written as a superposition integral, convolving σ (r) with the homogeneous   medium’s static Green’s function, .G0 r, r = 1/|r − r | (up to constant a factor). It is of interest to compute the potential for observers r = (r, θ , ϕ) located at r > R for some  > 1, i.e., to evaluate      .φ (r) = G0 r, r σ r dr , r > R. (11.1) S

For such r values, the multipole expansion of Eq. (11.1) can be truncated, with a relative error , to include only P terms, with P ≥ logr/R (1/) − 1, such that 1 .φ (r) ≈ (α/R)n r P

n=0



P 

 Mnm Ynm (θ, ϕ)

, r > R,

(11.2)

m=−P

where α = R/r , .Mnm are the expansion coefficients, and .Ynm (θ, ϕ) are the spherical harmonics. For given (θ , ϕ), Eq. (11.2) is a polynomial of degree P(, ) in α. Accounting for the 1/r dependence analytically, Eq. (11.2) can be recovered from

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Fig. 11.1 (a) Non-uniform grid for static kernel fields produced by a source domain S – the angular sampling becomes coarser with r. (b) Rotated coordinate system used for derivation of sampling criteria for phase-compensated fields in the dynamic case. (c) Uniform angular sampling, at a rate that is independent of r, and non-uniform radial sampling, for dynamic kernel fields produced by a source domain S

uniform α-samples via polynomial interpolation of order P + 1 in α ∈ [R/D, 1/], with D being the largest distance of interest. This uniform sampling in α translates into a non-uniform sampling in r – hence the name “non-uniform grid.” As for the angular sampling, fixing r = rp (for the pth sampling radius) and ϕ, as the angular harmonics are Legendre functions .Pnm (cos θ ) with n ≤ Pp = P(rp /R, ), Eq. (11.2) can be viewed as a combination of cos(mθ ) and sin(mθ ) terms with non-negative integer frequencies m ≤ Pp . This implies that Pp + 1 uniform angular samples are sufficient for recovery of φ(rp , θ , ϕ) to the prescribed relative error via trigonometric interpolation [53]. Similarly, 2Pp + 1 samples are required in the ϕ coordinate. The angular sampling becomes coarser with the distance, suggesting that φ(r) can be first reconstructed for the entire (θ , ϕ) range for each rp and then interpolated in the α coordinate. It is worth noting that the number of sampling points is independent of R. This is a key property in the design of the fast algorithms in Refs. [4, 7, 8]. An illustration of the NG for this case is shown in Fig. 11.1a.

11.2.2 Sampling Radiated Fields in the Spatial-Baseband For problems of large dimensions compared to the wavelength, the kernels are oscillatory and the sampling criteria derived for the quasi-static case are no longer valid. As the EM field can be described in a mixed-potential manner, it is sufficient to consider the computation of a single scalar field, produced by a scalar source distribution on S, which can represent the scalar or any of the vector potential components, and naturally fits problems in computation acoustics. The integral expressions, written for the time-harmonic case, with an e−iωt dependence assumed and suppressed, take the general form of

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     F r, r eik |r−r | σ r dr , r > R.

φ (r, θ, ϕ) =

.

(11.3)

S

Here, F(r, r ) is a slowly varying function of r, which may include both radial (e.g., |r − r |−p ) and angular slow variations. Unlike in the static case, here, it appears that the oscillations dictate that the radial sampling should be at a rate proportional to the wavenumber, k. However, at a given r, the phase associated with the contributions of each r contains a large common component, which is of the order of kr. Only a small phase variation bounded by kR differs between the contributions. The common component can be viewed as a “carrier” for the spatial amplitude-modulation of the source’s contributions. Its removal, via multiplication by a phase-compensation factor, .e−ik r˜ , can effectively demodulate the field spatially and enable its sampling in “spatial baseband,” at much lower sampling rates dictated by its actual spatial bandwidth. Additional compression of the base-band field’s spectral content can be achieved using an  amplitude compensation factor

−1 (r). We will next show that the choice .r˜ = r 2 + R 2 /2 minimizes the spatial bandwidth, for which convenient expressions for the sampling rates in a spherical coordinate system can be deduced. Let .φ˜ (r, θ, ϕ) = −1 (r)e−ik r˜ φ (r, θ, ϕ) denote the phase- and amplitudecompensated field. It can be considered a superposition of exponential terms −ik (r˜ (r)−|r−r |) . The coordinate system can be rotated so that a given source point .e r is located on the positive z-axis as shown in Fig. 11.1b. In the rotated coordinates, .r, θˆ , ϕ, ˆ the source is at (r , 0, 0) and the corresponding exponential term can be approximated as:

e

.

−ik [r˜ (r)−|r−r |]

≈e

−ik



R2 4r

   2

1−cos2 θˆ +r  cos θˆ − r2r

(11.4)

. 

The main angular variations in Eq. (11.4) are due to the term .e−ikr cos θˆ . This term can be represented by the Anger-Jacobi expansion [54] e−ikr

.

 cos θˆ

∞      = J0 kr  + 2 (−i)n Jn kr  cos nθˆ ,

(11.5)

n=1

where Jn (·) denotes the Bessel function of order n. As Bessel functions decay faster than exponentially for orders higher than the argument, for kr ≤ kR, the series can ˆ be truncated uniformly. This implies that the .θ-bandwidth of .φ˜ is kR and that it can be sampled in the .θˆ coordinate at a rate fθ ≥ θ kR/π , where θ > 1 is an angular oversampling ratio. This rate is used in practice for both θ and ϕ coordinates in the un-rotated coordinate system. If F(r, r ) also exhibits an angular dependence, ˆ fθ can be modified to account for that slow variation, such that such as .cosm θ, fθ ≥ θ (kR + m)/π . Note that, unlike in the quasi-static case, this sampling rate is dependent on R and independent of r. As such, it remains valid even for very

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Fig. 11.2 (a) Real part of the phase- and amplitude-compensation and non-compensated field produced by an elemental (dipole) source that is shifted to the edges of the circumscribing sphere. (b) Convergence of the angular and radial interpolation errors with θ and α

large r, including for the sampling of the corresponding far-field radiation pattern. However, it does not reduce to that derived in Sect. 11.2.1 when kR → 0. As for the radial dependence, each of the terms of the form (11.4)  can be

r 2 ) 1 − cos2 θˆ . For expressed as .e−iωα α (α = R/r), where .ωα = k ( R4 ) − ( 2R sources within the sphere of radius R, |ωα | is bounded by kR/4. This implies that uniform α-sampling (non-uniform radial sampling), at a rate fα = α kR/4π (with an oversampling ratio α > 1), allows for the reconstruction of φ, with controllable error. Here too, in contrast to the static case, the rate (or number of sampling points) is dependent on R. An illustration of the radially non-uniform grid for this case is shown in Fig. 11.1c. The example in Fig. 11.2 demonstrates the effectiveness of the phase- and amplitude-compensation in enabling the recovery of the field from samples on sparse grids. Figure 11.2a shows the smoothening influence of the compensation on the field, which enables its coarse sampling, using the case of a single elemental (dipole) source, restricted to a sphere of radius R = λ and shifted from its center. In Fig. 11.2b, the maximum relative interpolation error for radial (for fixed θ , ϕ) and angular sampling (at r = R), computed for a source located within the sphere of radius R = 2λ and  = 3, is shown to converge as a function of α and θ at rates corresponding to the interpolation order.

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11.2.3 Optimal Sampling Fields of More Complicated Structures The compensation factor and sampling rates in Sect. 11.2.2, developed for capturing the fields by sources radiating in free-space, suffice for the understanding and implementation of the algorithms presented in the remainder of this chapter. Yet, it is worth to mention other tasks that require the efficient computation of fields that have more complex spatial characteristics. One such example is the efficient reconstruction of the Green’s function for a layered medium from its tabulated samples [55], where the radial sampling is dictated by the modal wave number and the vertical sampling is designed to accurately capture the modal function. Another case of Green’s functions with multiple modalities is that of bi-anisotropic materials. In Ref. [56], the bi-anisotropic Green’s function was interpolated in boxes that increase in size with the distance to the source. At the absence of simple closed form expressions for a phase compensation term, the sampling strategy is suboptimal for electrically large observation domains. Yet another case is that of evaluating the modified Green’s function (MGF) of a source in the presence of simple scatterers. These classical problems in wave theory have recently become of particular relevance in the CEM and computational acoustics context of advanced IE formulations, which enable enhanced compression and computational savings in their numerical solution [25, 26, 57]. These generalized IE formulations make use of simple objects, on which auxiliary sources can be induced, as “shields” that direct the fields of elemental sources in desired orientations. Considering these shields as part of the source distribution that gives rise to the elemental source’s MGF benefits the IE in certain aspects while significantly complicates the computation of field integrals. Each call to the MGF computation involves the integration over a larger source distribution. The works in Refs. [27, 58] make use of non-uniform sampling grids tailored to capture the MGF in order to tabulate it and enable its reconstruction upon request from the set of samples. In Ref. [58], the MGF is that of the elemental source in the presence of an impenetrable smooth scatterer. Depending on the observer’s location with respect to the source and the scatterer, either the reflected contribution to the MGF or the total MGF is computed. The phase compensation for the total MGF for observers in the scatterer’s shadow region, with respect to the source, accounts for the path of an eigen-ray from the source to the surface, its creeping along the surface, and its departure from the surface and propagation to the observer. For observers in the source’s line of sight, the direct contribution can be computed analytically whereas the scattered component is sampled after phase compensation that accounts for the eigenray exhibiting specular reflection. In Refs. [27, 59], absorbing arc (in 2-D) and dome (in 3-D) shields are used. In these shields’ shadow region, the total MGF can be attributed to the shields’ edge diffraction. As such, it can be sampled efficiently if its phase is compensated with respect to the phase-center and size of the contributing part of the edge. In Refs. [25, 27, 59], non-uniform sampling was also used for the economical representation of field integrals that make use of the MGF

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kernel. These grids were inspired by those used for capturing the MGF (in Ref. [25]) and heuristically designed (in Ref. [26]). The design of optimal non-uniform field sampling techniques for arbitrary kernels remains an active area of research.

11.3 Fast Far-Field Computation: Multilevel Interpolation and Aggregation In this section, the sampling criteria developed in Sect. 11.2 are utilized for the fast computation of fields radiated to finite distance from the source region S beyond the sphere of radius R to which it is confined. The sampling criteria suggest that the larger the source distribution’s support is, the denser the sampling rate required for accurate reconstruction of the field via interpolation. As an alternative to the dense sampling of the field produced by a very large source region, one can consider the coarser sampling of partial contributions by sub-distributions confined to smaller domains. For example, assuming N = O[(kR)2 ] sampling points are describing the distribution σ on S, O(N) sub-distributions of very small support provide O(N) partial contributions that may be sampled at O(1) points each. However, this does not reduce the overall cost of field integrations, as interpolation of the resulting O(N) coarsely sampled fields is required for each of the many observation points of interest, whose number is dictated by the size of S. In order to take advantage of the optimal sampling, the reconstruction of fields via interpolation must be performed hierarchically, as will be explained next. Consider, for example, the evaluation of a scalar potential produced by a singlelayer source distribution σ on S, at points r = (r, θ , ϕ) ∈ , where r > R, i.e.,  φ (r) =

.

    σ r G r, r dr , r ∈ .

(11.6)

S

       Here, .G r, r = Gk r, r = eik |r−r | / r − r  is the free-space’s scalar Green’s function. As a particular case, for a that is very far from S , Eq. (11.6) can also be deduced from its far-field (radiation) pattern ˘ (θ, φ) = lim re−ikr φ (r) = .φ r→∞



   σ r e−ik·r dr ,

(11.7)

S

where k = k(sinθ cos ϕ, sinθ sin ϕ, cosθ ). The far-field integral (11.7) can be accurately reconstructed from O[(kR)2 ] samples on a uniform θ − ϕ grid, at a total cost of O[(kR)4 ] operations. The cost of computing (11.6) and (11.7) can be reduced by adopting a hierarchical computation scheme based on the partitioning S into an Llevel hierarchy of sub-surfaces defined via child-parent relations. Each sub-surface .S p,l of level l ∈ [0, L − 1], centered at .rp,l and circumscribed by a sphere of radius Rl , is partitioned into non-overlapping child sub-surfaces .S q,l+1 , q ∈Qp,l , centered at .rq,l+1 circumscribed by spheres of radii Rl + 1 , such that .S p,l = q∈Qp,l S q,l+1 . The set .Qp,l contains all .S p,l ’s children. At the top of the hierarchy, .S 1,0 = S.

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11.3.1 Fast Far-Field Pattern Computation Starting from the somewhat simpler far-field case in Eq. (11.7), for levels l ∈ [0, L − 1], partial contributions .φ˘ p,l , associated .S p,l , to the integral (11.7), can be represented as the aggregation of corresponding child contributions, .φ˘ q,l+1 , i.e.      ˘ p,l (θ, φ) = σ r e−ik·(r −rp,l ) dr = .φ eik·(rq,l+1 −rp,l ) φ˘ q,l+1 . (11.8) S p,l

q∈Qp,l

In other words, .φ˘ p,l is obtained by weighted summation of child contributions ˘ q,l+1 , with weights .e−ik·(rp,l −rq,l+1 ) . However, assuming that Rl + 1 ∼ Rl /2, the .φ parent partial contributions .φ˘ p,l should be sampled at rates twice as large as those of the child ones .φ˘ q,l+1 . Therefore, prior to multiplication by the phase weight, interpolation from the set of sampling points sufficient for capturing .φ˘ q,l+1 to that sufficient for capturing .φ˘ p,l is required. Only at the bottom level, L, the partial .φ˘ q,L must be computed directly, via integration over the corresponding .S q,L . Terminating the partitioning at level L such that RL = Rc ∼ λ/2 the corresponding .φ˘ q,L need to be evaluated at only O(1) points. This suggests the following multilevel far-field (MLFF) algorithm (written recursively) for the computation of φ: Algorithm MLFF(input: , ̅ , , output: ̆ if ( ̅

,


1 the near-region observers’ set of S. The subset .Fp,l = S\Np,l denotes the points on S not in .Np,l (as show in Fig. 11.5). At the points of .Fp,l , the partial contribution

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Fig. 11.5 Left: A subdomain .S p,l of S. Middle: Near and far regions of S defined with respect to a 2-level setting, the field is computed directly at grid points .Gp,l and near observers in .Np,l and interpolated to far observers in .Fp,l . Right: Parent and child subdomains. The observers .ℐq,l+1 are defined on the parts of S within the shaded region

.S p,l .For

 φ p,l (r) =

    σ r G r, r dr , r ∈ S

.

(11.12)

S p,l

can be reconstructed from its phase- and amplitude-compensated samples at a set of points .Gp,l of a radially non-uniform grid, with the compensation factors in Eq. (11.9) and sampling rates dictated by .R p,l , as in Sect. 11.3. For .Np,l , direct integration should be carried. For bottom-level subdomains, .S q,L , with parent .S p,L−1 , the contribution to the field at points .r ∈ Nq,L is computed by direct integration. For points in .r ∈ Fq,L , the contribution can be computed by interpolation from the samples of its phaseand amplitude-compensated version, .φ˜ p,l at the points .Gq,L . However, it is only recovered this way if .r ∈ Np,L−1 . Otherwise, the contribution will be combined with those of .S q,L ’s sibling domains and recovered from sampling points on higher level grids. We denote .ℐq,l+1 the set of points in the far region .Fq,l+1 of subdomain .S q,l+1 that are in its parent’s near region .Np,l . For points .r ∈ ℐq,l+1 , the contribution of .S q,l+1 is computed by interpolation from the corresponding set of points .Gq,l+1 , followed by phase and amplitude restoration. This process is described, as a recursive scheme in Algorithm MLNG below. In the algorithm, the N and . G are used to represent field integration on .S notations .p,l p,l evaluated p,l G

N at points of .Np,l and .Gp,l , respectively. The operators .Iq,l+1 and .Iq,l+1 indicate the interpolation from a child grid .Gq,l+1 to points in .Nq,l+1 and its corresponding parent grid .Gp,l , respectively.

11 Spatial Sampling and Interpolation Techniques in Computational. . .

Algorithm MLNG(input: , ̅ , , global: , output: ̃ ̅

)

:= radius( ̅ , )

,

if ( ̅


0), Type I HMM (  > 0,  ⊥ < 0), and Type II HMM (  < 0,  ⊥ > 0) mediums; (b) isofrequency contour for an array of silver nanorods embedded in Si substrate (hyperbolic curves in blue), and dielectric medium (circle in grey); the black arrows represent the wavevector and Poynting vector in the dielectric medium, and the blue and red arrows, respectively, show the wavevector and Poynting vector at the interface of dielectric and HMM mediums; (c) the refraction angle of silver nanorods as a function of wavelength [43]

the dissipation can be ignored [45]. The behavior of light dramatically changes as one or two components of the relative permittivity tensor have a different signature; meaning the medium becomes highly anisotropic, thereby causing its dispersion relation to be hyperbolic in nature. In Eq. (12.10), if either of   and  ⊥ is negative, the equation becomes that of a hyperbola, which essentially results in a hyperboloidal isofrequency surface for the extraordinary waves. The medium, therefore, is said to possess hyperbolic dispersion, thereby exhibiting a number of unconventional properties. HMMs can be classified based on the properties of the constitutive parameters. Considering the signs of relative permittivity components   and  ⊥ assuming  .I  = 0, I (⊥ ) = 0, in the case of   > 0 and  ⊥ < 0, both the s- and ppolarized waves would exist at the same time – the superlattice is designated to be the Type I HMM. On the other hand, when   < 0 and  ⊥ > 0, the p-polarized wave

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only exists, and the material is categorized as the Type II HMM. This constitutes the basis of hyperbolic mediums – the conceptual understanding which can be exploited in the design of HMMs. If all the permittivity components are negative, the medium becomes metallic, and the all positive-valued permittivity components correspond to the case of dielectric medium. It is interesting to note that the Type II HMM is highly reflective in nature because it exhibits the properties more like a metal than the Type I counterpart [46, 47]. Yet, the Type I HMMs show less reflection and high transmission. In this context, it is worth mentioning the two orthogonal linear polarization states, namely the p- and s-polarizations, which refer to the polarized light that has an electric field in the direction parallel and perpendicular, respectively, to the plane of incidence. The polarization states of light greatly govern the reflection and transmission characteristics of the optical device on to which the light waves impinge. In HMMs, the high-magnitude wavevectors of light dramatically enhance the local density of states (LDOS) for photons within a broad range of wavelengths [48, 49], leading to efficient enhancement in light-matter interactions (in the HMM). The radiative decay of light emitters directly depends on the photonic density of states (PDOS). According to Fermi’s golden rule [50], a high PDOS corresponds to a larger number of radiative decay channels for an excited atom, thereby enhancing the emission rate for a particular mode. When a dipole-like emitter is located in the near field of an HMM (distance d  λ), the decay rate is dominated by the contribution of the high-k states [51];  ≈

∞

.

 I rp (kx )2 e−2kx d dkx ,

(12.11)

kx

 where .I rp is the imaginary part of the reflection coefficient for p-polarized wave with wavenumber kx and d is the distance of the dipole from the HMM. It is shown that the wavenumber (kx ) dependence of the reflection coefficient would be dropped for kx k0 . Considering the case of Type II HMM ( < 0, ⊥ > 0), the imaginary part of rp for can be written as [51, 52]       ⊥ 2    . .I r p = 1 +     ⊥

(12.12)

Interestingly, the unbounded isofrequency surface of HMMs promises anomalously large PDOS, which causes an enhanced emission rate [47, 53, 54, 55]. It is worth noting that the enhanced emission rate of HMMs could open up applications in on-chip light sources with dramatically increased photon extractions [55, 56]. It becomes clear from the foregoing discussion that the nature of relative permittivity components plays a determining role in the design of HMM so that the hyperbolic dispersion can be achieved. The property of hyperbolicity demands metallic behavior in one direction and dielectric behavior in the other. As such, an

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HMM is generally composed of both the metal and dielectric mediums, and the high-wavenumber modes in HMM are the Bloch modes [47] of the metal-dielectric superlattice structure. HMMs can be designed using two different geometries of the unit cell, viz. the layered and arrayed configurations of the meta-atoms, as Fig. 12.2 illustrates. The former type relates to a thin-film multilayer structure composed of sub-wavelength thick layers of metal and dielectric mediums to achieve extreme anisotropy [57]. Such a constraint on the layer thickness is due to the homogenizedmedium theory (which will be discussed at a later stage) to be valid. On the other hand, the arrayed HMMs are composed of periodically arranged metallic nanowires in a dielectric medium host [58, 59]; the periodicity being sub-wavelength, in order to fulfil the medium homogenization requirements [47].

12.2.2 Response of HMMs As discussed above, HMMs can assume the periodically arranged layered (Fig. 12.2a) and (nanowire) arrayed (Fig. 12.2b) configurations of metal-dielectric superlattice structures. Upon homogenization, the effective relative permittivity values of HMMs can be evaluated. The existence of poles and zeros in the plots of effective = relative permittivity tensor .  eff (against wavelength λ or frequency f ) remains = interesting to note, in order to characterize the resonance properties. In the .  eff λ plot, zeros and poles refer to the specific wavelengths at which a component of the dielectric tensor of the metamaterial either passes through zero (epsilon-nearzero, ENZ) [60] or has a resonant pole (epsilon-near-pole, ENP), respectively [61]. The ENZ and ENP resonances change the reflection and transmission spectra of the two types (i.e., layered and arrayed) of HMMs, which essentially depends on the directions they occur. The resonant ENZ takes place parallel to the thin film of the layered configuration or along the length of the nanowire in the arrayed configuration. In contrast, the resonant ENP happens in the direction perpendicular

Fig. 12.2 Schematic of HMMs in the forms of (a) multilayered metal-dielectric thin films and (b) metallic nanowires arranged periodically inside a dielectric medium

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to the thin film in the layered HMM structure and perpendicular to the nanowires in the arrayed structure [47, 61]. HMMs can be characterized by the reflection and transmission spectra these exhibit. Both the layered and arrayed HMMs can exhibit the Type I behavior, which includes the properties of conventional dielectric mediums. Also, the Type I HMMs are highly transmissive. Conversely, the Type II HMMs are highly reflective, and possess the properties common to conventional metals. It must be added at this point that it is easier to achieve the Type I behavior using the arrayed configuration of HMM, whereas the layered HMMs mostly exhibit the Type II behavior.

12.2.3 Negative Refraction in HMMs HMMs can exhibit either negative or positive refraction of light, which results in numerous exotic optical phenomena [53], such as abnormal propagation [62–65], near-perfect absorption [66–68], collimation [69], abnormal scattering [70–73], and directional propagation [74, 75]. It has been shown that the emission pattern in hyperbolic mediums has a characteristic of a cross-like shape in which the waves propagate inside a certain cone and decay outside the cone [76]. This property indicates that light can propagate through a hyperbolic medium in certain directions, and the fields orthogonal to hyperbolic asymptotes are much stronger than elsewhere due to larger LDOS [77]. To elucidate this, we assume the light incident from an isotropic material (e.g., air) to a Type I HMM at an angle θ i , marked with a black arrow ki in Fig. 12.1b. The tangential component of the wavevector of the incident light (kx ) is conserved at the interface of an isotropic medium and an HMM. The wavenumber kx is also equal to the tangential component of wavevector of the reflected plane wave kr at the side of HMM, as depicted in Fig. 12.1b with a dashed black line. In an isotropic medium, light can propagate in all directions with the wavevector .k parallel to the direction of energy flow rate .S (the time-averaged Poynting vector) ∗ where .< S > = 12 E × H is orthogonal to the isofrequency surfaces [78, 79]. Here ∗ .H is the complex conjugate of .H . In order to maintain the causality rule [10], the energy must flow away from the interface. Therefore, the appropriate branch of hyperbola must be designated. The perpendicular and tangential components of .S are given by Sz =

kz H02  2ω0

(12.13)

Sx =

kx H02 , ⊥ 2ω0

(12.14)

.

and .

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respectively. It can be readily shown from Eq. (12.13) that in the case of Type I HMM with   > 0 and  ⊥ < 0, the outgoing energy flow rate Sz at the interface takes positive values for kz > 0; thereby, the right-hand hyperbola is satisfying the causality law. Further, it can be obtained from Eq. (12.14) that the situation of  ⊥ < 0 and kx > 0 leads to negative refraction in the hyperbolic region (the wavelength region at which the medium shows the hyperbolic behavior) as the tangential component Sx takes negative values. Figure 12.1c shows the angle of refraction at the interface of air with the silver gratings with the height of 80 nm, the width of 90 nm and the pitch of 150 nm as a function of wavelength [43]. As shown in Fig. 12.1c, the angle of refraction takes negative values for λ < 540 nm, which is the wavelength of transition from the elliptical to the hyperbolic medium. Interestingly, the transient wavelength can be tuned by modifying geometrical parameters, such as the pitch, rod length, and the surrounding dielectric medium [43].

12.3 Designing HMMs It is essential to understand the realization methods of HMMs to optimize the design for operating at the target wavelength with desired properties. The individual atomic planes of the atomic layers of bulk materials are bonded by van der Waals attraction (vdW). The weak vdW forces between the atomic planes (out-of-plane forces) and strong Coulomb interaction along the atomic sheets result in rich anisotropy in vdW materials [47, 55, 80–82]. This class of materials includes graphene and other twodimensional (2D) crystals, such as SiC, Bi2 Se3 , MgB2 , hexagonal boron nitride (hBN), α-MoO3 , α-V2 O5 [83, 84]. Among all known materials, vdW materials exhibit the highest degree of confinement – the property that offers tunability by electrical stimuli [85]. However, their operating wavelength is intrinsically depending on the crystal structure [85]. To overcome the shortcoming of the naturally existing mediums, artificial structures can be realized in which the optical properties can be adjusted by varying the operational parameters. The recent advances in nanofabrication processes allow the realization of structures of the nanoscale size to operate in a wide range of electromagnetic spectrum from the UV to THz. As stated before, the superlattices made of metal and dielectric mediums can exhibit hyperbolic dispersion (Fig. 12.2). The optical response of such composites can be evaluated by the effective medium theory (EMT) – a qualitative approach to evaluate the optical response by homogenizing a sub-wavelength-sized unit cell in an infinite periodic composite [86]. The method follows a generalized Maxwell Garnett approach to obtain the   and  ⊥ components of the effective relative permittivity tensor. However, it leads to errors for the structures having a small finite number of periods [40, 43]. The effective relative permittivity components in the parallel (i.e.,   ) and perpendicular (i.e.,  ⊥ ) directions for a layered stack are determined as [86, 87]

12 Light-Matter Interaction at the Sub-Wavelength Scale: Pathways to Design. . .

 = ρm + (1 − ρ) d and ⊥ =

.

m d , ρd + (1 − ρ) m

291

(12.15)

respectively. Here ρ is the filling fraction of metal, given as ρ = tm /tz , with tz (=tm + td ) as the total thickness of the stack, and tm and td as the respective thickness values of the metallic and dielectric medium layers. Also,  m and  d are their respective relative permittivity values. Another type of HMM can be realized by embedding a nanowire array in the host dielectric medium, as Fig. 12.2b demonstrates [88, 89]. The corresponding relative permittivity tensor components parallel and perpendicular to the optic axis can be defined as [90, 91]  =

.

[(1 + ρ) m + (1 − ρ) d ] d and ⊥ = ρm + (1 − ρ) d , [(1 − ρ) m + (1 + ρ) d ]

(12.16)

respectively. Here the filling fraction is ρ = A/A0 , with A and A0 as the respective cross-sectional areas of metal wires and the host dielectric medium in the unit cell. Surprisingly, the real part of metal relative permittivity is negative for frequencies lower than the plasma frequency, and this yields the Type I HMM behavior of nanowires in the optical regime. As the negative permittivity has been attained without optical resonances, a high transmission at a broad spectrum with a comparably low loss is attainable by modifying the structural parameters [47]. In practice, the material loss cannot be avoided. The realistic metals have nonzero imaginary parts of the relative permittivity leading to degrading of the overall performance of the HMM. The wavenumber for propagation wave will be as  kz =

.

 k02 − 

kx2 + ky2 ⊥

= β + iα,

(12.17)

with β and α as the phase and the attenuation constants, respectively. The attenuation constant can reduce the design performance by degrading the intensity of the travelling wave and also impacting the bandwidth of the device. This issue can be tackled by choosing the proper constituent materials considering the operating frequency and the structural designs [92, 93]. We discussed before the reflection and transmission characteristics of HMMs. However, apart from the reflection and transmission, absorption is another important feature that can be exploited to design HMMs for useful applications, such as solarenergy harvesters. In this chapter, we touch upon a few different types of HMMs that incorporate phase-change mediums (PCMs). In particular, our focus is on designing modulators in the visible and THz regions of the electromagnetic spectrum. As such, we first discuss in brief the essence of using certain PCMs in the construction of metamaterials.

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12.4 Phase-Change Mediums in Metamaterials PCMs have received increased attention owing to the possible added feature of tunability of the device where these are exploited [94–98]. These are the class of materials with the capability of manipulating the optical and electrical properties by applying external stimuli (e.g., ambient temperature, applied voltage or ultrafast optical pulse), thereby allowing the metamaterials to be programmable in nature. Also, the use of such materials facilitates multicontrollable feature to the metasurface-based device [99]. This extraordinary adjustment ability gives rise to a scalable and cost-effective scheme to devise photonic components for diverse technological applications [100]. A few different kinds of PCMs are vanadium dioxide (VO2 ), GeSbTe (germanium-antimony-tellurium or GST), strontium titanate (SrTiO3 ), and certain perovskites, such as methyl ammonium lead iodide (MAPbI3 ). Tuning the electrical property of materials started way back when the permanent resistance-change property of MoS2 (from high to low resistance) was observed [101]. Moreover, the work was followed by another paper that reported the reversible resistance change of As-Te-Br or -I ternary glasses in 1962 [102]. But, Ovshinsky was the first to propose the idea of utilizing PCMs for memory switch-based applications in 1968 [101, 102]. Later in the 1990s, advances in lithography technology made it possible to reduce the minimum feature size in the semiconductor industry, which brought the phase-change memories into prominence again. Huge R&D investments by the military and aerospace industries led to the rapid development of phase-change memory technology [103]. In the 1990s, the development of phase-change memories gave rise to re-recordable optical disk memories comprising chalcogenide PCMs. Many PCMs were investigated in the progress of realizing PCM-based memories. G2 Se2 Te5 (GST) and Ag5 In5 Sb60 Te30 (AgInSbTe) were found to perform excellently in terms of transition speed and sustainability [101]. Current investigations on phase-change memories are not solely restricted to conventional electrical and optical storage technologies. Instead, the developments of the novel nanophotonic devices, such as multi-level photonic memories [103], photonic synapses for neuromorphic photonic devices [104], nonvolatile reflective displays [97], reflective modulators and switches [105, 106] are gaining technological attention. The phase-transition mechanisms differ in metal-oxides, chalcogenide PCMs and perovskites. The chalcogen-based PCMs (such as GST) are volatile in nature, whereas the optical property of metal-oxides and perovskites is robust. The phase transition in these materials is usually reversible; the material turns to the initial state without special sample treatment when the temperature is decreased to a degree below the transition temperature [107]. Hence, these “phase-transition materials” are useful for reconfigurable and nonvolatile applications. This chapter emphasizes designing programmable HMM-based nanostructures using the PCMs as the dielectric medium. The following sections discuss two different types of HMM-based configurations for optical modulator application in different spectral regimes, namely the visible light and THz spans.

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12.5 HMM as Programmable Reflection Modulator in Visible Light Programmable photonic devices are highly desirable because most realized nanodevices suffer from post-process tuning [108]. This drawback can be eliminated by exploiting tunable materials as the key constituent component. VO2 is a common PCM that exhibits a relatively low dielectric-to-metal transition temperature Tc that varies gradually as the temperature is raised from 331 K to 345 K, and a fast switching speed [109, 110]. However, the considerable amount of absorption by VO2 is not acceptable for photonic integrated circuit applications [103]. As pointed out earlier, GST is a common PCM used in programable optical devices, nonvolatile optical memories, and programmable displays [104]. It undergoes relatively large refractive index change (n = 3.56) at low temperatures in a broad wavelength span. However, GST exhibits fairly large absorption in the visible regime due to a considerably large imaginary part η of refractive index in both the material phases. In particular, GST shows η = 1.42 in the crystalline state and η = 0.12 in the amorphous state [111, 112]. Stibnite (Sb2 S3 ) is another chalcogenide PCM, which exhibits large refractive index shifts and low loss in the visible light regime, thereby making it promising for solar cells and photodetectors [112]. Moreover, the amorphous and crystalline phases of Sb2 S3 show a better match with that of silicon. As such, the use of Sb2 S3 remains convenient owing to the possible conventional techniques of deposition processes. In the following section, we touch upon a specially designed layered HMM configuration comprising periodically arranged silver (Ag) and Sb2 S3 layers [113]. Our focus remains on designing a tunable reflection filter well-suited to operate in the visible regime. We also attempt to demonstrate the use of the structure as a programmable reflection modulator.

12.5.1 Structural Details of the HMM We consider a structure comprising two similar HMMs with Sb2 S3 |Ag unit cell surrounding a defect layer of Sb2 S3 (having thickness dD ) embedded on a heating system made of bilayer graphene. Figure 12.3 illustrates the schematic of the configuration considered as (SiO2 )(Sb2 S3 |Ag)N (Sb2 S3 |Graphene |SiO2 ) over (Sb2 S3 |Ag)N (SiO2 ), with N being the number of unit cells. We use a capping SiO2 layer (of thickness 100 nm, i.e., ds1 ; Fig. 12.3) to protect the Sb2 S3 PCM from evaporation. Another SiO2 layer (of 10 nm thickness, i.e., ds2 ; Fig. 12.3) incorporated under graphene serves two-fold – it is an adherent for graphene and it protects the bottom PCM layer from heat (originated by graphene). In Fig. 12.3, dPCM and dm are, respectively, the thicknesses of the Sb2 S3 and Ag layers. Also, θ i is the angle of incidence of a plane wave impinging on the top SiO2 layer.

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Fig. 12.3 Schematic of the layered HMM configuration with Sb2 S3 defect layer

12.5.2 Constitutive Properties and Spectral Features To investigate the constitutive properties of the layered HMM configuration, we use the filling fraction ρ (of the Sb2 S3 PCM layer) as the geometrical parameter to determine the different kinds of structure. In our work, we take the value of ρ to be 0.5, and obtain the optical properties of the same composite under different operational conditions in the spectral range of 0.2–1.6 μm. For this purpose, we first deduce the effective relative permittivity components   and  ⊥ in the stated wavelength range using Eq. (12.15). Figure 12.4a illustrates the obtained dispersion plots that show the wavelength dependence of the real ( .R) and imaginary ( .I) parts of   and  ⊥ corresponding to the homogenized (Sb2 S3 |Ag) unit cell. This assists in determining the metallic, Type I HMM and Type II HMM kinds of behavior of the periodic stack depicted in Fig. 12.3. We then evaluate the optical response of the homogenized (Sb2 S3 |Ag) bilayer stack using the transfer matrix method (TMM) [114]; Fig. 12.4b shows the obtained results for the wavelength dependence of reflectance R, transmittance T, and absorptance A. As can be seen in this figure, the absorption is higher than transmission and reflection in the metallic regime (marked by the light blue color area), which essentially happens due to the better impedance matching with the free

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Fig. 12.4 (a) Dispersion plot of the layered HMM configuration and (b) spectral response of the (Sb2 S3 |Ag) bilayer stack

space. It is evident from Fig. 12.4a that the periodic structure behaves as a Type I HMM in the wavelength regime of 0.34–0.58 μm (falling under the blue color area in Fig. 12.4a, b), where the effective relative permittivity components are as .R (⊥ ) < 0 and .R  > 0, thereby determining higher transmission in this regime  [47, 55]. The Type I HMM region ends at 0.58 μm where .R  undergoes epsilonnear-zero (ENZ), and the composite begins to behave as Type II HMM. Figure 12.4a illustrates this using a circle, whereas Fig. 12.4b exhibits the same by a dashed line. The Type II HMM behavior of the structure remains  in the 0.58–1.6 μm region of operating wavelength, where .R (⊥ ) > 0 and .R  < 0. It was stated before that the Type II HMMs exhibit high reflection property, which Fig. 12.4b essentially confirms as the unit cell shows large reflection in the stated range of wavelength. Interestingly, the hyperbolic transition wavelength (i.e., the ENZ determining the type of HMM) can be governed by modifying the effective relative permittivity tensor components. The transition can be achieved by altering either the ρ-value or the relative permittivity of the constitutive materials. Herein, we vary ρ from 0 (pure Sb2 S3 ) to 1 (pure Ag) to adjust the ENZ wavelength, and consequently, the reflection magnitude. Figure 12.5 illustrates the results in terms of wavelength as a function of ρ organized into the different  areas according to the signs of the permittivity components .R (⊥ ) and .R  . The results explicitly reveal that ρ = 0.5 leads to the maximum hyperbolic dispersion – the value of ρ that we consider in further investigations of the spectral response.

12.5.3 Effect of Unit Cells and Defect Layer on the Reflection Response The number of alternating unit cells (i.e., the Sb2 S3 |Ag) in the design also remains greatly important to determine the reflection characteristics of the HMM. As explained earlier, EMT is based on indefinite repetitions of unit cells along the

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Fig. 12.5 Dispersion properties of the structure corresponding to different ρ-values

Fig. 12.6 Map of reflection peak values vs. wavelength and N

optic axis [86]. As such, it is essential to study the effect of the number N of unit cells on the spectral response. As such, we vary the number of periodic Sb2 S3 |Ag bilayers from 01 to 08, and obtain the reflection spectra under the excitation due to the normally incident p-polarized wave; Fig. 12.6 illustrates the obtained results. A frequently used term for the p-polarized wave is transverse magnetic (TM) polarization, whereas the s-polarization is also called transverse electric (TE) polarization. It is obvious from this figure that the increase in the number of layers has negligible impact on the reflection characteristics. The reflection peak shows a small improvement corresponding to N = 2, while the spectral characteristics remain almost unchanged for the larger values of N. The defect layer thickness has a significant effect on the resonance modes generated by multiple reflections that take up different phases as propagating through the layers. In the structure of Fig. 12.3, the defect layer (of thickness dD ) of Sb2 S3 at the mid-point (of the configuration) is embedded to excite resonances in the visible regime as the periodicity of the overall HMM configuration is perturbed. As mentioned in the previous section, the ENZ wavelength can also be tuned by changing the relative permittivity of the constitutive components. We aim to tune

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Fig. 12.7 Reflection spectra for different values of dD while the other operational and parametric conditions remain fixed as before

the resonance reflection mode using the refractive index alterations (of Sb2 S3 ) of the defect layer by controlling the ambient temperature. This we achieve by applying voltage pulses to the electro-thermal system consisting of the bilayer graphene microheater placed underneath the defect layer. It is worth noting that the graphene microheater leaves no significant effect on the overall optical response in the visible regime [115]. To evaluate the effect of dD on the spectral response, we consider increasing its value from 1 × dPCM to 2 × dPCM ; Fig. 12.7 depicts the computed results. This figure reveals that the reflection valley undergoes a small red-shift, and becomes sharper in the visible span, whereas it remains almost unaffected in the UV regime. We take dD = 1.8 × dPCM as the corresponding resonance has the smallest minimum. The use of a defect medium offers extra tunability feature owing to it being a PCM. This is because the constitutive properties of PCM can be altered depending on the external stimuli. We now investigate the effect of phase transition of Sb2 S3 layer from the a- to c-state on the reflection spectrum. For this purpose, we consider excitation due to Joule heating to vary the refractive index of Sb2 S3 layer by applying electrical pulses to the microheater (Fig. 12.3). Since the level of crystallinity (the ratio of crystalline to amorphous mass) is governed by the duty cycles of the melting and cooling periods, the refractive index of partially crystalline PCM can be defined by the ratio m of the a- to c-state [103, 115]. The effective permittivity  eff (λ) of the defect layer for a medium with crystallinity ratio m can be defined by the Maxwell Garnett formula [111, 115] .

eff (λ) − 1 c (λ) − 1 a (λ) − 1 =m + (1 − m) , eff (λ) + 2 c (λ) + 2 a (λ) + 2

(12.18)

where m can take values from 0 to 1, corresponding to the pure amorphous (i.e., m = 0) and completely crystalized (i.e., m = 1) states, respectively. In Eq. (12.18),  a (λ) and  c (λ) are, respectively, the wavelength-dependent relative permittivity of Sb2 S3 layer in the a- and c-states.

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Fig. 12.8 Reflection spectra for different incidence angles for the a- (a, b) and c- (c, d) states of Sb2 S3 , and also, the p- (a, c) and s-polarized (b, d) incidence

12.5.4 Effect of Oblique Incidence The oblique of the incident plane wave remains another operational factor of great significance. We investigate the effect of oblique incidence of the s- and p-polarized waves on the reflection characteristics of the structure with the amorphous a- and crystalline c-states of the Sb2 S3 defect layer. Figure 12.8 depicts the obtained results wherein Fig. 12.8a, b correspond to the reflectance spectra in the a-state, whereas Fig. 12.8c, d represent those in the c-state. Also, Fig. 12.8a, c show the reflectance spectra in the case of p-polarized illumination, whereas Fig. 12.8b, d exhibit the same for the s-polarized incidence. We notice in this figure that the amorphous state (i.e., m = 0) of Sb2 S3 gives the minimum reflection at 0.56 μm for both polarization states, which undergoes red-shifts for increasing incidence angles for the p-polarized light (Fig. 12.8a, b). Moreover, a Brewster angle of 62◦ in the UV regime is evident in the a-state of Sb2 S3 ; the p-polarized light shows a local reflectance minimum, whereas the spolarized light is reflected. This phenomenon also happens for the Sb2 S3 defect layer in the c-state in the range of incidence angle 68◦ –85◦ . We observe in Fig. 12.8a, c that the reflection minimum for the p-polarized incidence undergoes a shift

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from 0.5723 μm to 0.639 μm upon fully switching the state due to the larger value of the real part of refractive index of Sb2 S3 in the c-state. In Fig. 12.8c, we also notice that the progressive increase of oblique incidence up to 62◦ results in the lowest reflectance. This trend is partially followed by the s-polarization, where the reflection minimum vanishes upon reaching the oblique incidence of 48◦ .

12.5.5 Application of the HMM as Modulator The spectral characteristics in Fig. 12.8 reveal the structure can be used as a reflection modulator in two distinct regimes – a Brewster modulator in the UV regime operating at an oblique incidence of 62◦ , and also, a reflection modulator in the visible regime. It is worth mentioning at this point that Brewster angle determines vanishing reflectivity of the p-polarized incident light. This is why we address the proposed structure as Brewster modulator. Here, we focus on an application of the proposed structure – a reflection amplitude modulator in the visible span, which can operate in the fully driven states (a- and c- states) or the intermediate states of the PCM layer. In view of this, the figure of merit (FOM) of the modulator is defined by the contrast ratio C given by C=

.

Rmax , Rmin

(12.19)

where Rmax and Rmin are the maximum and the minimum of the reflection, respectively, at a given wavelength. For the proposed structure (Fig. 12.3) operating in the c-state (m = 1), we can compare the contrast ratio C corresponding to the two different illumination angles, namely 0◦ and 62◦ ; Fig. 12.9a shows the computed results. This figure yields the contrast ratio for the incidence angle of 62◦ giving the value of C = 3.29 × 107 , which is significantly higher than that given for the normal counterpart (which provides C = 49.5) at the wavelength of 0.538 μm. Further, we also define the “8-bit bandwidth” of the device with C = 256 as a threshold—the contrast between the maximum and minimum amplitudes in an 8-bit grayscale level. The obtained contrast ratio is much higher than C = 256 in 11.7 nm bandwidth (i.e., 0.6283–0.64 μm), thereby promising great capability as a reflection modulator. Another FOM attributed to the contrast ratio is the modulation depth D, which can be obtained for the reflected wave by using the relation [116] 

D = 1 − C −2 × 100%,

.

(12.20)

with C being the contrast ratio. We evaluate the modulation depths of the ppolarized light with the incidence angle of 62◦ using different values of crystallinity ratios; Fig. 12.9b depicts the obtained results. We observe that a wide range of

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Fig. 12.9 (a) The reflection contrast ratio corresponding to the incidence angles 0◦ (red line) and 62◦ (blue line); the dashed black line represents the threshold (C = 256) of the 8-bit grayscale encoding. (b) Percentage modulation depth for different values of m (with 0 ≤ m ≤ 1) for the p-polarized incidence light at the angle of 62◦

intensity modulation can be achieved by phase transition from amorphous to fully crystalline states and the volatile intermediate states. As such, the HMM-based device comprising Sb2 S3 and silver thin films shows promise for use as a reflection modulator in the UV and visible regimes of the electromagnetic spectrum.

12.6 HMM as Broadband THz Brewster Modulator THz waves are technologically significant for cutting-edge applications, such as wireless high-speed communications, biomedical diagnostics, security imaging, and signal processing [117–121]. THz modulators are of great interest for various applications owing to the high intensity and phase tunability. Different mechanisms implementing the thermal, optical, electrical, or mechanical actuation have been used to enhance the light-matter interaction in THz metamaterials. As becomes clear from the foregoing discussions, the use of tunable dielectric mediums in metamaterial-based devices allows refractive index modification in the constituent materials, such as superconductors, perovskites, liquid crystals and PCMs by alterations in temperature [122–124]. Herein, we numerically demonstrate an intensity and phase modulator at the THz regime. The results may be of potential in the realization of THz devices such as modulators, phase shifters, polarization switches, so forth.

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12.6.1 HMM Structure, Constitutive Properties, and Spectral Response Figure 12.10 illustrates the proposed multilayered structure comprising periodically arranged tri-layer unit cell of ITO, SiO2, and SrTiO3 (STO) deposited over the Si substrate. The effective permittivity of ITO|STO|SiO2 unit cell can be defined by adopting EMT. We keep the thickness values of the different layers (ITO, dm ; STO, db; SiO2 , ds) fixed as 50 nm, 450 nm, and 1000 nm, respectively; the thickness of the bottom Si layer is 100 μm and the ambient temperature is 300 K. The parallel and perpendicular uniaxial relative permittivity components (of the relative permittivity tensor) are given by  = f1 m + f2 STO + f3 SiO2

.

(12.21)

and ⊥ =

.

f1 STO SiO2

m STO SiO2 = + f2 m SiO2 + f3 m STO



f2 f3 f1 + + m STO SiO2

−1 , (12.22)

Fig. 12.10 The schematic of the HMM structure comprises the periodic ITO|STO|SiO2 unit cells

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 respectively. Here (f1 ,  m ), (f2 ,  STO ) and . f3 , SiO2 are the respective filling fractions and the complex permittivity of ITO, STO, and SiO2 layers (with f1 + f2 + f3 = 1). The relative permittivity of SiO2 is taken from the experimental results [125]. The dielectric characteristics of ITO, without considering the thermal effect, can be obtained by the Drude model as  (ω) = ∞ − ωp2 /ω (ω + j ) ,

.

(12.23)

with  ∞ , ωp and  being the high-frequency bulk relative permittivity, the plasma frequency and the collision frequency, respectively, with their respective values chosen to be 4.4, 1.6 × 1015 rad/s and 1.05 × 1015 rad/s [126]. The relative permittivity of STO changes due to temperature and frequency, according to [127, 128]  (ω) = ∞ +

.

ν ω02

− ω2 − j ωγ

,

(12.24)

where  ∞ = 9.6 and ν = 2.6 × 106 cm−2 are the bulk high-frequency relative permittivity and the oscillator strength, respectively. The soft-mode frequency ω0 and damping constant γ are the functions of temperature, and defined by [128] .

 ω0 (T ) cm−1 = 31.2 (T − 42.5)

(12.25)

γ (T ) cm−1 = −3.3 + 0.094 × T ,

(12.26)

and .

respectively, with T being the temperature (K). In Eqs. (12.25) and (12.26), the quantities ω0 and γ are stated in the form of wavenumber. Figure 12.11a illustrates the frequency dependence of the computed real and imaginary parts of the  and  ⊥ permittivity components upon exploiting the EMT. We observe that .R  takes positive values upon reaching 2.65 THz where it turns to the negative values upon crossing zero.  This figure clearly shows that the ENZ occurs at the same frequency where .I  undergoes a Lorentzian peak. Yet, the real and imaginary parts of the  ⊥ -components are undergoing trivial changes and keeping the positive values as 5 and 0.01, respectively. The medium exhibits  elliptical dispersion corresponding to frequencies below the ENZ owing to .R  > 0 and .R (⊥ ) > 0. Further, the unit cell exhibits theType II HMM behavior in the range of frequencies above 2.65 THz where .R  < 0 and .R (⊥ ) > 0. The optical responses of such unit cell are evaluated by using the TMM, and compared with those obtained by the EMT; Fig. 12.11b depicts the obtained results. As the figure shows, the TMM results (represented by solid lines) are in good

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Fig. 12.11 Plots of the (a) real parts and imaginary parts of   and  ⊥ of an ITO-STO-SiO2 trilayer stack, and (b) transmittance T, reflectance R, and absorptance A characteristics of the same, wherein the solid lines correspond to the TMM results and dotted lines to the EMT results

agreement with the EMT results (represented by dotted lines). We clearly see in this figure that the reflectance rises to a local maximum and the transmittance drops to a minimum at the ENZ frequency, thereby confirming the transition frequency from the dielectric medium to HMM Type II at 2.65 THz frequency.

12.6.2 Effect of Geometrical Properties We now touch upon the spectral response of the proposed structure under different parametric conditions. The number of unit cells is considered to be 05 for the rest of this study. First, we use the parametric values (of the structure) as before, excluding the filling fraction f2 (of STO) which varies from 0.3 to 0.045 as the STO layer thickness varies from 450 nm to 50 nm. Figure 12.12 shows the computed results of the plots of   (the solid lines at the left axis) and  ⊥ (the dotted lines at the right axis) components of effective relative permittivity for different values of db. As can be clearly seen in Fig. 12.12a that the real and imaginary parts of  ⊥ are not significantly affected by changing the STO layer thickness, whereas the   counterparts are drastically altered in the orders of magnitude. However, the ENZ frequency is not affected. We choose  the STO layer thickness as 450 nm to maintain the maximum magnitude of .R  .

12.6.3 Thermal Effect on the Spectral Response As the relative permittivity of STO has a thermal dependence, this feature can provide tunability to the proposed THz modulator. Figure 12.13 exhibits the

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Fig. 12.12 The real (a) and imaginary (b) parts of   and  ⊥ components of the ITO|STO|SiO2 unit cell under varying STO layer thickness where the temperature is fixed at 300 K and the thicknesses of the ITO layer and SiO2 are, respectively, fixed to 50 nm and 1000 nm

Fig. 12.13 Plots of the (a) real and (b) imaginary parts of   and  ⊥ at different temperatures, while the other operational and geometrical conditions are kept the same

obtained results (for the configuration in Fig. 12.10) for different ambient temperatures in a range of 223 K to 400 K. This figure shows that the increase in temperature gradually shifts the ENZ and the Lorentzian peak of .I  toward higher frequencies, thereby providing expansion in the dielectric region through red-shifts. However, the perpendicular counterpart remains almost constant.

12.6.4 Application of HMM as Brewster Modulator  It has been shown that the Type I HMM (i.e., .R  > 0 and .R (⊥ ) < 0) can exhibit Brewster angle corresponding to which the structure exhibits vanishing reflectance for the p-polarized incident light [124]. However, the dielectric mediums can also exhibit closely similar properties [124]. The Type II HMM cannot support such a vanishing reflection because the propagation of the p-polarized wave is

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Fig. 12.14 Reflection spectra for different values of incidence angle considering the (a) p- and (b) s-polarized waves using the TMM

restricted in this case owing to the negative   [47]. Herein, we show that the dielectric region can exhibit Brewster angle at a certain value of oblique incidence where both the intensity and phase of THz light can be modulated by varying the ambient temperature. To be more concise, the change in temperature shifts the (ENZ) frequency of transition from the dielectric region to the Type II HMM, in order to modulate both the intensity and the phase of the reflected wave. We attempt to evaluate the reflection spectra of the Brewster modulator by using different values of incidence angle of the s- and p-polarized THz wave. Figure 12.14 illustrates the computed results of reflection characteristics obtained by changing the incidence angle from 40◦ to 89◦ in a frequency range of 0.1–6 THz. The figure shows that there is an extended area inside the elliptical phase of HMM from 0.6–2.6 THz at which the reflectance of the p-polarized wave is almost zero (Fig. 12.14a), but that of the s-polarized wave is 100% (Fig. 12.14b). The choice of Brewster angle of 82◦ and 1.7 THz frequency seems to be more practical as the higher incidence angles correspond to almost grazing angle. The in-depth analysis of Brewster modulator requires calculation of ellipsometry parameters, given by  = arctan

  r p  |rs |

(12.27)

 = φp − φs ,

(12.28)

.

and .

  where .rp = rp  ej p and .rs = |rs | ej s are the theoretical expressions obtained by the TMM for the p- and s-polarization states, respectively. At the Brewster angle,  vanishes, i.e., |rp | goes to zero, and the phase difference  undergoes an abrupt shift from 0◦ to 180◦ [129, 130].

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Fig. 12.15 The ellipsometry parameters (a)  and (b)  obtained for the incidence angle of 82◦ and under the thermal conditions of 223 K, 273 K, and 300 K. The other operational and geometrical parameters are kept the same

Figure 12.15a, b illustrate the obtained results corresponding to an oblique incidence of 82◦ . At this value,  approaches the local minimum of 8◦ at 1.7 THz frequency, and  undergoes an abrupt phase shift from −17.5◦ to 162.5◦ (i.e., 180◦ phase shift at 1.83 THz). This demonstrates almost vanishing reflection at the incidence angle of 82◦ . Figure 12.15 also elucidates the thermal dependence of the Brewster angle considering different values of ambient temperature, namely 223 K, 273 K, and 300 K. We observe that the variation in temperature shifts the  minimum values toward higher frequencies due to the shift in ENZ. This demonstrates the potential of the structure to be used as a tunable modulator in the THz regime. To evaluate the efficacy of the Brewster modulator, we numerically investigate the reflectance Rp = |rp |2 and modulation depth [131]  D = 1−

.

|Rmin |2 |Rmax |2

 × 100%

(12.29)

of the p-polarized light at the Brewster angle of 82◦ for different thermal conditions. Here, Rmin and Rmax are the minimum value of Rp in the off-state and the maximum value of Rp in the on-state, respectively. Figure 12.16a depicts the obtained spectra for p-polarization in the temperature range of 223 K to 400 K. The results demonstrate that the minimum reflectance remains unchanged for all temperatures, whereas the resonance frequency undergoes blue-shifts from 1.37 THz to 2.41 THz. We assume the off-state at 400 K (i.e., |Rmin |) and the on-state at 223 K (i.e., |Rmax |) to proceed. This leads to the maximum modulation depth of 99.8% at the resonance frequency of 2.28 THz while the incidence angle is fixed at 82◦ . Figure 12.16b illustrates the obtained modulation depth for the p-polarized reflectance intensity at 2.28 THz. It clearly shows that the reflectance takes different values in a range of 0 to 1 upon choosing the temperature range of 223 K to 400 K. Further, the insertion

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Fig. 12.16 Temperature dependence of the (a) p-polarized reflectance and (b) modulation depth at Brewster angle

loss of the device, given by log10 (|Rmax |2 ), is also evaluated to be as −1.65 dB, which is much lower [124, 131]. This study reveals that the periodic stack of ITO|STO|SiO2 unit cells exhibits the HMM behavior in the THz regime and can be used to design Brewster modulator at the transition frequency from the elliptical dispersion to the Type II kind of HMM. The operating temperature also plays a significant role in enhancing the tunability of the composite. The structure yields phase modulation of the reflected light up to 100% with low insertion loss of −1.65 dB. The device can be used in THz applications.

12.7 Conclusion Extraordinary electromagnetic properties of HMMs can be exploited for versatile photonic applications. The chapter briefly touches upon the unique characteristics of HMMs, followed by the application of certain newly conceptualized layered HMMs that exploit PCMs in designing optical modulators. In particular, STO and Sb2 S3 PCMs have been used in the formation of two different types of HMMs. The results reveal that both the configurations exhibit the Type I and Type II kinds of behavior which essentially depends on the transition wavelength. The optical response of the structures has been retrieved, and the effects of parametric and operational conditions have been evaluated; the latter basically emphasizes on the thermal situations of the ambience. The resonance conditions could be tuned by suitably altering the temperature, as achieved by graphene and/or ITO micro-heaters embedded into the structure. This tuning property of the reported HMM configurations supports the possible usage of them as modulators in different wavelength spans, such as the visible and THz regimes of the electromagnetic spectrum.

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M. Pourmand received his B.Sc. degree in electrical engineering from the Khajeh Nasirodin Toosi University of Technology (Iran) in 2004. During 2004–2020, he worked as an electronics engineer in industry and he also studied for the M.Sc. degree in electrical engineering at the Tafresh University (Iran), and received the degree in 2013. He received a doctoral degree from the Institute of Microengineering and Nanoelectronics, Universiti Kebangsaan Malaysia (The National University of Malaysia, Malaysia). His research interests lie in metamaterials, photonic crystals and plasmonics from the fundamental as well as practical perspectives.

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Pankaj K. Choudhury received a Ph.D. degree in Physics in 1992. He held academic positions in India, Canada, Japan, Malaysia and China. During 2003−2009, he was a professor at the Faculty of Engineering, Multimedia University (Cyberjaya, Malaysia). Thereafter, he became a professor at the Institute of Microengineering and Nanoelectronics, Universiti Kebangsaan Malaysia (Malaysia). Also, he served the Telekom Research and Development (TMR&D, Malaysia) as a consultant for projects on optical devices. Currently he is a professor at the ZJUUIUC Joint Research Center, Zhejiang University (China). His research interests lie in the theory of optical waveguides, which include complex mediums, fiber optic devices, optical sensors and metamaterial properties. He has published over 270 research papers, contributed chapters to 22 books, and edited and coedited 9 research level books. He is a reviewer for over fifty research journals. Also, he is the Section Editor of Optik – International Journal for Light and Electron Optics (Elsevier, The Netherlands) and the Editor-in-Chief of the Journal of Electromagnetic Waves and Applications (Taylor & Francis, UK). He is a Fellow of IET and SPIE, and a Senior Member of IEEE and Optica.

Chapter 13

Integrated Photonics with Near-Zero Index Materials Larissa Vertchenko and Andrei V. Lavrinenko

13.1 Electromagnetism with Near-Zero Index Materials A new class of materials with refractive indices having extremely low values, so-called near-zero index (NZI) materials, have great potential for applications with photonic chips. Such materials exhibit exotic phenomena such as boost of the electric field and huge enlargement of the wavelength, resulting in better transmission properties in distorted channels [1]. Moreover, they enable pronounce enhancement of nonlinearities due to simplified phase-matching conditions and high internal fields [2, 3]. Additionally, their constant phase property gives radiation a very long coherence length, making them a promising platform to integrate with quantum emitters for quantum optics purposes [4]. We start by describing the principal electromagnetic characteristics of the NZI materials. It is well known that mathematically the refractive index n is related to two other characteristics, namely the relative permittivity .ε and relative permeability .μ [5]. Both represent how the charges of a medium respond to electromagnetic radiation, i.e., how well they get polarized. These characteristics are connected √ to the refractive index by .n = εμ. For the refractive index of a material to be near zero, either .ε or .μ or both of them must be very small. To differentiate such cases, we refer to these materials as epsilon-near-zero (ENZ), mu-near-zero (MNZ), and epsilon-mu-near zero (EMNZ), respectively [6]. However, realistic materials have intrinsic losses mostly due to electron scattering, adding imaginary terms to such quantities. For example, ENZ materials are easily found in nature as gold (Au), silver (Ag), and many other plasmonic materials, where the real part of their permittivity .ε crosses zero at the plasma frequency [7, 8]. These intrinsic

L. Vertchenko · A. V. Lavrinenko () Department of Electric and Photonics Engineering, Technical University of Denmark, Kongens Lyngby, Denmark e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 T. G. Mackay, A. Lakhtakia (eds.), Adventures in Contemporary Electromagnetic Theory, https://doi.org/10.1007/978-3-031-24617-3_13

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losses, part of the permittivity .ε , make the refractive index  √ related to the imaginary   .n = iε = (1 + i) ε /2 differ from zero even if the real part of the permittivity is exactly zero, detaining these materials to be considered NZI. As a consequence, most optical effects related to a low-index materials are completely annihilated. Thus, the aforementioned classification is important because it helps to make a clear distinction between NZI and other near-zero classes of materials. In electromagnetism, we can describe both electric (.E) and magnetic (.B) fields of propagating waves in the form e, . E(r, t) = E0 ei(k·r−ωt)

(13.1a)

E0 i(k·r−ωt)  (k × e), e c

(13.1b)

.

B(r, t) =

where .r is the spatial coordinate vector, .E0 is the field amplitude, c is the speed of light, .k is the wave vector, and  .e is the unitary polarization vector. These equations describe a plane wave propagating in free space with .k = k0 = ω/c as the free space wave number. To simplify the notation, the parenthesis with temporal and spatial dependencies of the fields will be omitted throughout this chapter. In order to account for the effects of matter, we must use the following constitutive relations for the electric displacement vector .D, magnetic field .H, and current density .J: D = ε0 εE, .

(13.2a)

B ,. μ0 μ

(13.2b)

.

H=

J = σ E,

(13.2c)

where .ε0 and .μ0 are the vacuum permittivity and permeability, respectively, and σ is the conductivity. The dynamics of these fields considering the surrounding environment properties is described by Maxwell’s equations in the form

.

∇ · D = ρf , .

(13.3a)

∇ · B = 0, .

(13.3b)

∇ × E = iωB, .

(13.3c)

∇ × H = J − iωD,

(13.3d)

.

where .ρf represents the free charge density. In the absence of free charges in a homogeneous medium, Maxwell’s equations can be combined into the so-called wave equation, which describes a propagating wave by one field ∇ 2E +

.

εμω2 E = 0. c2

(13.4)

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Dependence of the wave propagation on the material properties is brought by functions .ε and .μ. Assigning direction of propagation with the x-axis, the solution to Eq. (13.4) is a plane wave with exponential part related to the permittivity by E = E0 ei

.

√

εμk0 x

= E0 eikx x ,

(13.5)

√ where .kx = εμk0 . By mapping light’s behavior with the real part of the relative permittivity (.ε ), we can classify materials into two categories: conductors (.ε < 0) and dielectrics (.ε > 0). In Fig. 13.1, we show qualitatively the transition region between these two categories: where in conductors, such as metals, oxides, and heavily doped semiconductors the field experiences an exponential decay resulting in absorption and reflection, whereas in dielectrics the wave maintains its oscillatory behavior for a long distance. The near-zero material classes correspond to the regions, where .|ε |  1 (epsilon-near-zero) and .|μ |  1 (mu-near-zero). When both parameters are much smaller than one, we refer to such materials as epsilonmu-near-zero (EMNZ). If additionally the imaginary parts .ε and .μ are negligibly small, they will also be referred to as NZI materials. By looking at the wave equation (13.4), when .ε and .μ become small, the second term becomes negligible, reducing Eq. (13.4) to Laplace’s equation .∇ 2 E = 0 which governs electrostatics. This results in the field having a constant amplitude in space while oscillating in time [9]. Thus, there is an obvious decoupling between spacial and temporal parts that describe the electromagnetic field dynamics. This fact has important implications in the development of novel resonant cavities. It is well known that the solution of the wave equation for a certain cavity depends on the

Fig. 13.1 Transition between conductive dielectric materials with its respective propagation properties, where the NZI region is highlighted by yellow with .ε between .−1 and 1 and .μ less than 1

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boundary conditions. This means that the eigenfrequency values will depend on the geometry of the cavity. Thus, resonant cavities operate with a fixed frequency for a particular shape and size. By decoupling spatial and temporal parts of the electromagnetic field, it is possible to find the same resonant frequency values for different cavity configurations. Such frequency invariance with respect to geometrical transformations is well suited for applications with deformable cavities and flexible photonics [10]. As an example, such cavities can have their eigenfrequency overlapping atomic transitions of quantum emitters embedded within them. As demonstrated by Liberal et al., the interaction between a two-level system and the near-zero index background becomes equivalent to a single-mode cavity [11]. In this scenario, the population of the emitter exhibits a reversible dynamics with the Rabi frequency defined by the size of the cavity, without detuning its resonance frequency. In conventional systems, the deformation of a cavity would result in the weakening of the Rabi oscillations due to a shift in the resonance. Such results bring novel possibilities to manipulate quantum states and their decay dynamics [12]. Spatial–temporal decoupling is not the only one which happens in NZI materials. Another interesting decoupling occurs between the electric and magnetic fields. Consider an incoming wave with constant magnetic field (TM) pointing in the zdirection .Hz ez . From Eq. (13.3d), the electric field inside ENZ and EMNZ materials takes the form .E = (1/ − iωεε0 )∇H × ez , where .H = |H|. Since the electric field must take a finite value, .∇Hz × ez must vanish, which results in .Hz being also constant in space and not depending on E. It is important to stress that the .Hz component of the magnetic field must remain constant when applying boundary conditions, as ENZ and MNZ materials are selective toward polarization, as further analyzed in Sect. 13.2. As long as the boundary condition maintains orthogonality between magnetic and electric fields, i.e., PEC for the TM polarization and PMC for the TE, the electric and magnetic field distributions can be manipulated independently. The near-zero refractive index has also strong impact on the wavelength. As electromagnetic radiation travels inside a material, its wavelength changes according to .λ = λ0 /n, where .λ0 is the wavelength in free space (i.e., vacuum). From this equation, we arrive at one of the most fundamental properties of NZI materials, the enlargement of the wavelength. In Fig. 13.2, we illustrate the evolution of a sinusoidal wave in the medium described by .ε. If .ε = 0, the wavelength inside such material becomes infinite, and the absolute phase, which is considered to be responsible for the temporal evolution between the wavefronts, becomes constant. At first, by looking at the wavelength stretching inside the NZI material, one would assume that the phase velocity, given by .v = λf , would be superluminal and approach infinity in the limit where .n = 0, thus allowing instantaneous signal transmission. However, a perfectly monochromatic wave is unrealistic. Any optical information must be encoded with pulses, which consists of monochromatic waves of different frequencies. Except for vacuum all media are dispersive. By definition, dispersion occurs when the optical properties of the medium, such as .ε and .μ, depend on frequency. In Fig. 13.3, we exemplify dispersion by dependence of both permittivity and refractive index of a titanium nitride (TiN) film on relative to the

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Fig. 13.2 Illustration of the difference between the wavelength of a trivial dielectric material with > 1 and zero-index material, where the wavelength becomes infinite when .ε = 0. The amplitude is not on scale giving a simplified geometric vision of the transition between air and an ENZ material

.ε

Fig. 13.3 Real (.ε , .n ) and imaginary (.ε , .n ) parts of the permittivity (a) and the refractive index (b) for a 100 nm thin film of titanium nitride as a function of the wavelength, exemplifying the dispersion properties of such medium. Red dashed lines are relative to the spectral positions of .Re(ε) = 0

wavelength. The data were retrieved from ellipsometry measurements of a 100 nm thick film of TiN [13, 14]. As such dispersion guarantees that each and every pulse composed of different wavelengths even for very small linewidths will have the group velocity with a finite value [15].

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Another important condition for information transfer in the NZI channels is that the imaginary part of the refractive index must have very small value. From the Kramers–Kronig relations, this means rather flat dispersion [16]. As the imaginary part of the refractive index is related to absorption, big fluctuations in a short frequency interval will distort too much the signal resulting in loss of information. These two conditions are responsible for making information transfer in NZI channels achievable. This observation is fundamental for the explanation of the supercoupling effect, described with more details in the next section.

13.2 The Supercoupling Effect Waveguides are structures able to confine light in a finite space and control its propagation direction. The electromagnetic modes supported inside a waveguide are subordinate to its constituent materials and geometry. The transmission properties of light traveling inside a waveguide will depend on the impedance. Impedance is a physical constant that relates the electric and magnetic fields through the equation Z=

.

E = H



μ0 μ ε0 ε

(13.6)

and is associated with a resistance to the propagation of waves, in analogy with the impedance of a circuit as a ratio between voltage and current .Z = V /I . When connected waveguides have different cross sections, the incoming modes in the first waveguide suffer considerable back reflection due to the impedance mismatch. A counter-intuitive phenomenon occurs when very distorted waveguides are filled with near-zero index materials. Normally, light would be highly reflected inside crooked channels and abrupt bends. Nonetheless, when waveguides are filled in with NZI materials, modes are able to tunnel through the channels and junctions. This phenomenon is known as the supercoupling effect. Intuitively, we could argue that the wavelength within the NZI material is extremely large, so no diffraction nor reflection from abrupt bends is sensed by the mode [17]. A strict explanation lies within the matching impedance conditions. As illustrated in Fig. 13.4, the mode will match both channels’ impedance, thus effectively tunneling through the shrunken part. As discussed in the previous section, optical effects related to NZI, such as the supercoupling effect, appear when the imaginary part of permittivity and permeability is negligible, so in this chapter terms ENZ and NZI can be used interchangeably. In this system, an incoming electric field polarized in the y direction (see Fig. 13.4), .Ey propagates first through a hollow waveguide entering an ENZ waveguide with variable cross section and length. We use Faraday’s law . E · dl = − ∂B ∂t to calculate the electric field at both ends of the narrow channel of length d and width b [17, 18]. Knowing that .Hz is constant inside the ENZ material, we arrive at

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Fig. 13.4 Illustration of the impedance difference at the ports of a waveguide with the ENZ channel of length .d + L1 + L2 , coded in yellow. The blue color represents a hollow waveguide. The channel is surrounded by perfect electric conductors (PECs). The red dashed line represents the integration region for Faraday’s law

Ey (L1 + d)b − Ey (L1 )b = iωμHz db, .

(13.7a)

Ey (L1 + d) Ey (L1 ) = iωμd. − Hz Hz

(13.7b)

.

The wave impedance for the √ transverse mode is defined as .Z = Ey /Hz , and the free space impedance is .η0 = μ0 /ε0 . Combining both equations in Eq. (13.7) and √ substituting .ω with .k0 / ε0 μ0 , we get the impedance difference between the ends of the narrow channel filled with an ENZ material as .

Zout Zin = iμk0 d, − η0 η0

(13.8)

where .Zin and .Zout are the impedances at the beginning and end of the channel, respectively. From this equation, we have no information regarding how the ENZ channel may decrease the impedance difference and culminate in full transmission. For that, we need to relate the impedance calculated at the end of the channel to the impedance at the waveguide ends. In the limit where .k0 L1  1, the transversal impedance of the waveguide before the channel is proportional to the taper ratio  Zin ≈

.

b Zin , l

(13.9)

and similarly the impedance after the channel is  Zout ≈

.

b Zout , l

(13.10)

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where l is the width of the waveguide before and after the channel. Substituting .Zin  and .Z  and .Zout by .Zin out in Eq. (13.8) results in   Zout − Zin ≈

.

iμk0 η0 db . l

(13.11)

Thus, in order for the difference of impedances between the input and output waveguides to be small, at least one of the dimensions of the channel, either d or b, has to be reduced. In on-chip photonics, radiation must be able to not only transmit signals but also implement logic operations. If a channel is too short, other devices, such as optical cavities, have no space to interact with the system, so longer channels are often preferred. Thus, a narrow channel would be the suitable option for the supercoupling effect to happen. For waveguides of arbitrary cross sections (.l1 and .l2 ), a general formula for the reflection coefficient at their interface is given by r=

.

(l1 − l2 ) + iko μA , (l1 + l2 ) − iko μA

(13.12)

where .A = (L1 + L2 )l + bd is the cross-sectional area of the ENZ waveguide. From this equation, it is clear that to minimize reflection both waveguides should have the same cross section l at both ends, as well as minimal area, as predicted before. The rigorous derivation of this formula can be found in Ref. [17]. The supercoupling effect is also possible even in channels with several fabrication imperfections due to ability of modes in the ENZ material to overcome diffraction. Indeed, diffraction is an optical effect related to the deviation of a wave due to physical barriers that have dimensions close to the wavelength. As a result of the great enlargement of the wavelength, the wave will not experience the channels’ deformities, therefore exhibiting efficient transmission properties inside deformed waveguides [19]. We performed simulations for the supercoupling effect using the commercially available software COMSOL Multiphysics [20], based on the finite element method. In Fig. 13.5, we depict our results for tapered waveguides, composed of broad parts connected though a thin channel filled with an ENZ material of refractive index .n = 0.001. A source of electromagnetic radiation of wavelength 1500 nm is positioned to the right of the PML in the begining of the hollow waveguide. Perfectly electric conductors are used as boundary conditions around the whole waveguide. Figures 13.5a and 13.5b represent the results for ENZ channels of .b = 20 nm and .b = 300 nm with reflectance .R = 0.04 and .R = 0.39, respectively. Similar simulations were also performed for the very distorted channel in Fig. 13.5c, where almost the same transmission was observed as for the straight channel of same width. Reflectance in the waveguide system with a low index channel of variable width .(20 nm < b < 300 nm) and fixed length .d = 2 μm is shown in the plot of Fig. 13.6, with the lowest value of 4 .% corresponding to the narrowest channel. The

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Fig. 13.5 (a) Supercoupling effect observed in a hollow waveguide with a narrow connecting channel filled with an ENZ material of width 20 nm and 2000 nm length. We use a wavelength of 1500 nm for the incoming source of radiation. The boundaries around the waveguides are made of perfectly electric conductors. For an incident electric field polarized in the y direction, 4.% of reflectance was measured, while for a thicker channel of 300 nm, depicted in (b), the reflectance was 39.%. In (c), we show that for the narrowest channel, even big deformities would not increase reflection

Fig. 13.6 Plot of the reflectance for an ENZ waveguide with variable cross section b and fixed length .d = 2 μm, where the red circles represent retrieved data from the simulation and the blue curve is the analytical solution obtained through the reflection coefficient from Eq. (13.12)

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blue curve represents the analytical results for the back reflectance taken from the reflection coefficient in Eq. (13.12). A small deviation from complete transmission is predicted because the channel area would have to approach zero thickness, which would make no sense since the waveguide is used to transmit the wave. Therefore, a small impedance mismatch is expected. A way to get around this is to also have .μ close to zero, where a very small value of permeability would be able to match the impedance for any channel thickness. The supercoupling effect seems very promising for enhancing transmission properties of waveguides, but it also holds big limitations. A realistic ENZ material will always have a permittivity with a non-zero imaginary part. Consequently, the refractive index will also be a complex number, where its imaginary part .n is directly related to losses in the system. In Fig. 13.7, we calculated, for the same system of Fig. 13.5a, reflectance for different values of the imaginary part .n , while keeping the real part .n = 0. It is evident that even fluctuations in .n as small as 0.1 will degrade the signal and extinguish the supercoupling effect. For the case where the channel would be filled with an MNZ material, the field configuration in Fig. 13.4 would be changed to an incoming magnetic field polarized in the y direction. In such system, the resulting equations for impedance mismatch would be complementary to the ENZ case, and instead of decreasing the channel area, the tunneling would be improved for an increased area of the MNZ material [6]. To summarize, the supercoupling effect brings broad prospects to enhancing signal transmission inside waveguides, being a great solution for the propagation losses encountered in on-chip devices. However, challenges also arise regarding material properties as intrinsic losses related to the imaginary part of permittivity will effectively suppress this effect. Fig. 13.7 Plot of the reflectance for an ENZ waveguide with variable value of the imaginary part .n of the refractive index, while the real part is kept constant and equal to 0

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13.3 Coherence in Low-index Materials A fundamental property of light when propagating inside an NZI material is its constant phase. The phase is a property related to variation of a wave cycle. In an electromagnetic wave, the phase is given by the argument of the field’s function, (.k·r−ωt +φ), where .φ is a relative phase term that influences the spatial distribution of the wave. When the phase difference between waves is kept constant, we say that such waves are coherent, as sketched in Fig. 13.8. However, during interaction with matter, due to scattering, reflections, and intrinsic losses, the material imposes additional random phase changes that modulate the signal. We motivate our discussion starting from the well-known analysis of Young’s double slit experiment, illustrated in Fig. 13.9. Light from a monochromatic source reaches the apparatus and each slit will behave as a source of the incident radiation, according to the Huygens–Fresnel principle [21]. At point .r on the observation plane, a detector measures the average light intensity. So if we write the resulting field from slits 1 and 2 as E(r, t) = A1 E(r1 , t1 ) + A2 E(r2 , t2 ),

.

(13.13)

where .A1 and .A2 are geometric factors dependent on the slits sizes, the intensity is I (r) = |E(r, t)|2  .

.

(13.14)

Here, the bracket symbol denotes the time average. In order to simplify our analysis and remembering that NZI materials are selective toward the polarization, the fields will be treated as scalars by assuming that they have constant polarization. So, the total intensity becomes

Fig. 13.8 (a) Illustration of coherent waves where the relative phase .φ is kept constant through space. (b) Incoherent waves with changeable phase difference .φ due to wavelength change in parts of the propagating wave

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Fig. 13.9 Sketch of Young’s double slit experiment, where a light source illuminates two slits and a detector captures the intensity values in different positions r in the observation plane

I (r) = |A1 |2 |E(r1 , t1 )|2 +|A2 |2 |E(r2 , t2 )|2 +2Re(A∗1 A2 E ∗ (r1 , t1 )E(r2 , t2 )). (13.15)

.

The first two terms on the right side of Eq. (13.15) are the intensities from each of the slits, while the last term is responsible for the interference effect [22]. From this term, we introduce the definition of spatial coherence by the expression for the first-order normalized mutual coherence function E ∗ (r1 , t1 )E(r2 , t2 ) γ12 =  . |E(r1 , t1 )|2  |E(r2 , t2 )|2 

.

(13.16)

When .|γ12 | = 1, we define the fields as completely coherent; with .0 < |γ12 | < 1, they are partially coherent and for .|γ12 | = 0 incoherent. Physically, this means that at the plane of the detector we would observe interference fringes with highest visibility for the completely coherent case, while no interference fringes would be observed for the incoherent one. If we look at the interference term in Eq. (13.15), for conventional material such as air, this term will be proportional to .cos(k1 · r1 − k2 · r2 + Δφ), where .Δφ is the relative phase difference. It is the oscillatory character of this function that gives rise to interference fringes. Now, let us suppose that we place two electromagnetic sources in a lossless NZI material. The enlargement of the wavelength results in the wave number .k ≈ 0, which means that the wave has a constant field inside the whole material and, therefore, a constant phase. In this manner the fields in the mutual coherence function in Eq. (13.16) for the two positions will be the same and .|γ12 | = 1. Therefore, inside the ideal NZI material, radiation is perfectly coherent [23]. For two synchronized sources in an NZI material, both .k1 and .k2 are zero, and if their relative phases are the same, the argument of this cosine function becomes zero. This means that for any number of sources placed in the NZI material, the waves will always interfere uniformly. We illustrate such results through numerical simulations as shown in Fig. 13.10, where for convenience the material is modeled as a lossless

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Fig. 13.10 In the first and the third columns, we depict simulation results of a single electric dipole source embedded in a material with variable .ε, delimited by the square, and surrounded by air. The second and fourth columns show interference effects between two dipoles placed within the square with different materials

with different .ε and .μ = 1. As most quantum emitters, such as quantum dots, emit in the visible, we arbitrarily chose a wavelength of 500 nm for the modeling. The first and the third columns represent simulation results for a single electric dipole embedded in a material, delimited by the square region. The surrounding material is considered to be air. As the relative permittivity .ε increases, we observe the decrease in the wavelength. The second and the fourth columns show the interference effects for two electric dipoles placed inside the square patch with different materials. In the case of lowest .ε, the square is completely filled with only one color, denoting a complete and uniform field map. As .ε increases, peaks and dips appear, as well as pronounced reflections at the interface between two materials, resulting in a messier field profile. Such results bring important consequences to the field of quantum information. For example, Dicke superradiance [24] is an effect in which spontaneous emission

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is enhanced by a collection of quantum emitters. Such cooperative phenomenon is a result of near field interactions between the emitters which must be placed within one wavelength from each other. When excited, the emitters will radiate coherently and the emissions will constructively interfere, increasing the spontaneous emission rate. Normally, the distance between the emitters is extremely short and random, thus making observation of superradiance challenging. However, if placed in an NZI material, the wavelength will be greatly enlarged and consequently conditions for observation of superradiance in such array would be significantly facilitated [25, 26]. Up until this point, we analyzed the spatial coherence property inside NZI materials, meaning that any separation between sources would result in interference with a uniform amplitude. Another important coherence property belong to a single source that emits pulses with temporal separation, as opposed to the previous situation of two sources with spatial separation. We may now advance to understanding how the characteristics of a single source of radiation are related to the degree of coherence. When we superpose a wave at the same position .r but at different times t and .t + τ , as illustrated by the Michelson interferometer in Fig. 13.11, the fields are described by E(r, t) = E0 ei(k·r−ωt) , .

.

(13.17a)

Fig. 13.11 A light beam is directed toward a beamsplitter and interferes with a time-delayed version of itself at the detector. Such delay is caused by a longer distance in one of the arms of the interferometer

13 Integrated Photonics with Near-Zero Index Materials

E(r, t + τ ) = E0 ei(k·r−ω(t+τ )) .

329

(13.17b)

The autocorrelation function, also known as the temporal coherence function [27], is defined by G(τ ) = E ∗ (r, t)E(r, t + τ ) = E02 e−iωτ .

.

(13.18)

Since the wave number has negligible value in the NZI regime, we reduce the .G(τ ) expression to G(τ ) = E02 e−iωτ .

.

(13.19)

The degree of temporal coherence is given by g(τ ) =

.

E02 e−iωτ E ∗ (r, t)E(r, t + τ ) = = e−iωτ . E ∗ (r, t)E(r, t) E02

(13.20)

Normally, when we consider fluctuations of an electromagnetic source on a time scale, there will be certain intervals that might have a well-defined phase difference until it changes by some random process. These intervals will determine the coherence time of a light source as they carry information about both intensity and degree of coherence. For the case of sources in NZI materials, .|g(τ )| = 1 for any time .τ , meaning that the radiation is characterized by an infinite coherence time [21], just as perfectly monochromatic light sources. A major challenge of application of NZI materials within quantum platforms is their intrinsic losses. In a realistic approach for all NZI effects, we must take into account the imaginary part of permittivity. Consequently, the wave number cannot be negligible anymore and the coherence time converges to a finite value. In the next section, we show an alternative way to reduce the material losses, i.e., to implement an all-dielectric material platform.

13.4 Dirac’s Triple Point and Near-Zero-Index Materials As a result of intrinsic losses inherent to conductors, wave manipulation within homogeneous ENZ materials turns into a great challenge, and the use of alldielectric platforms becomes highly desirable. The latter approach is based on a specific feature of photonic crystals: dispersion bands degeneracy close to the .Γ point [28]. From the previous sections, we recall that the permittivity is a measure of how the charges of a material will become polarized in the presence of an electric field, where each atom would behave as a tiny dipole. Considering the case of discrete objects, such as nano-spheres or pillars, these dipoles not only get polarized but

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collectively yield scattered radiation, behaving as nano-scale resonators. The fields reemitted by the dipoles interfere to form specific radiation patterns, also known as Mie resonances [29, 30], which depend both on size and shape of the structure. The full description of nano-particle excitations dependent on frequency is given by the Mie scattering theory. By adjusting structural parameters of the photonic crystal unit cell, it is possible to tune Mie resonances for the desired frequencies with mode profiles associated with a monopole, a dipole, and other higher order multipoles. When the energy of electromagnetic modes is linearly proportional to the wave vector, the band structure gains a cone-like shape, usually referred to as the Dirac cones. The intersection between two cones is called the Dirac point [31]. A wellknown system that supports such behavior is the electric band structure of graphene, as illustrated in Fig. 13.12. Dirac-like cone dispersion profiles are also found in photonic crystals and other periodic systems [32, 33]. When a homogeneous material has both .ε(ω0 ) = 0 and .μ(ω0 ) = 0, the dispersion near .ω0 becomes linear and is associated with Dirac cones just like in graphene, with the exception of the Dirac point being at .Γ instead of at the Brillouin zone boundaries [34]. For periodic structures, such as photonic crystals, the Dirac point must have an additional mode [35]. In this particular case, a triple degenerate state is formed and the system may effectively behave as an NZI material. However, some considerations regarding the nature of excited modes must be taken into account regarding the structure. In order for a system of dielectric pillars to exhibit near-zero index properties, the three Mie resonances supported by the system must correspond to two dipoles and one monopole, while in the case of perforated membranes, a quadrupole, a dipole, and a monopole or hexapole must coexist [36, 37].

kx

π

-2 -1 0

a 1

2

15 10

5 E (eV) k 0

Dirac point

-5

-2

ky

0 π

2

Dirac cone

a

Fig. 13.12 Band structure of graphene indicating both the Dirac cone and Dirac point [14]

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Fig. 13.13 (a) Band structure for TM polarization of a square lattice of Si (.εr = 12.5) pillars with = 0.2a. The red circle indicates the triple degeneracy at .Γ . (b) Mode profiles close to .Γ point (.k = 0.002) along the .Γ −X direction, corresponding to one monopole and two dipoles. The black line indicates the geometry of the rod within the unit cell. The simulations were performed using the free software MIT Photonic Bands

.r

Considering the simple case of a square lattice of silicon cylinders (.ε = 12.5) of a radius .r = 0.2a suspended in air, where a is the lattice constant, the band diagram for TM polarization (electric field along the cylinders axis) is depicted in Fig. 13.13a. The red circle indicates the triple degeneracy point at .Γ , where three modes converge. These electromagnetic modes correspond to three Mie resonances shown in Fig. 13.13b. The images certify the presence of two dipoles and one monopole [38, 39]. In the case of cylinders close to the point of mode degeneracy, the effective medium theory can be applied to the material. At the triple point frequency, the effective constitutive parameters are .ε(ω0 ) = 0 and .μ(ω0 ) = 0, so the medium can be assigned with the effective zero index [35]. However, by looking at the dipole modes excited in this system along the .Γ − X direction, we notice that they have in-plane magnetic field components in the same direction of k. This means that

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longitudinal modes are also excited within the system. These modes are able to disturb the phase of radiation inside the zero-index material losing its uniformity property. Hence, one solution is to excite only the monopole resonance, shifting the frequency a bit away from the exact triple point. By such action, instead of a zero index, we would have an NZI material, which is sufficient to manifest low index phenomena. We validate this statement by demonstration of the supercoupling effect with the uniform phase in the full photonic crystal patch with design shifted from the .Γ point at .k = 0.002, corresponding to the wavelength of .1530 nm, as depicted in Fig. 13.14. The radiation from a line source placed inside one of the pillars in the left cavity propagates through a narrow channel to other side of the waveguide exhibiting the uniform phase distribution. The single row of cylinders in such narrow channel cannot be described by an effective approach, and due to the complexity of such system, analytical solutions become unrealistic. Thus, to handle such system, a solution is to reach the same monopole resonance field profile as the one from the triple point by adjusting the size of the cylinders through extensive numerical computations. Inside the narrow channel, we notice a stronger electric field, as predicted for the supercoupling effect due to energy conservation. The system is surrounded by another photonic crystal designed to work as a mirror with .r = 121 nm and .a = 367 nm. The wavelength of the line source was situated in the band gap of the ambient photonic crystal, shown by the red stripe in Fig. 13.15. Such a system can be potentially employed for several applications such as photonic circuitry and optical sensors [40]. Although intrinsic losses are overcome by employing dielectric photonic crystals, in reality light still experiences out-of-plane radiative losses, i.e., the modes excited in the system easily couple to those in the surrounding environment causing leakage of radiation [41]. In order to circumvent these types of losses, one can explore the possibility of combining a near-zero index photonic crystal with a class of special resonances named bound states in the continuum (BIC) [42, 43]. These states were originally proposed mathematically in 1929 [44] with solutions of the Schrödinger equation describing a particle trapped in a modulated potential well (Fig. 13.16), where destructive interference from the oscillating potential would suppress radiative leakage. Nonetheless, fabrication limitations at that time put this theory on hold. Later in 2008, the theory of BICs was resurrected and experimentally verified in photonic systems [45]. BICs can be classified into two types: symmetryprotected and accidental or trapped BICs [46]. Symmetry-protected BICs occur at the .Γ point and result from a mismatch between the symmetry of modes in free space and in the system. The trapped BICs are a combined effect of Mie resonances in the system that collectively result in destructive interference, preventing leakage of radiation to the environment [47, 48]. Thus, such bound states are nonradiative, having ideally an infinite Q-factor [36]. It is, however, challenging to achieve perfect confinement since the quality of the fabricated structures, as well as its finite size will affect the performance of both NZI effects and BICs [49]. As is well known, implementing devices with NZI effects in all-dielectric structures can be hampered by fabrication imperfections. For instance, we can mention the case of inverted opals, which were considered as photonic crystals having complete 3D photonic

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a) x108 4.5 4 3.5 3 2.5 2

E

1.5 1 0.5 0

b)

p

j

-p Fig. 13.14 (a) Supercoupling effect for a photonic crystal platform made of Si pillars (.εr = 12.5) where the frequency of operation is close to the triple .Γ −point (.k = 0.002). The electric field is emitted by a line source placed in the middle of the left cavity (green arrow) at the wavelength of .1530 nm. The structure is surrounded by another photonic crystal with .r = 121 nm and .a = 367 nm, providing the mirror effect as its band gap covers the frequency of the source. (b) Phase distribution of the same system

band gaps. However, as shown in Ref. [50], such band gaps are extremely fragile and can be eradicated by rather low level of imperfections. Stability of NZI structures toward random deviations of parameters in fabrication routines were studied in Ref. [49]. While the performance of the whole device is not distorted, as validated in experiments, the effect of imperfections on propagation losses was not examined. It is clear that this question is complicated and requires much more attention to prepare a clear path to implementations. Nonetheless, in the case of NZI materials, the small slope of the band corresponding to the monopole mode ensures that low refractive indices can still be reached even rather away from the .Γ = 0 frequency

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frequency(ω a /2π c)

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

Γ

X

k(2π/a)

M

Γ

Fig. 13.15 Band structure of the photonic crystal with .r = 0.24a, designed to operate as a mirror for the supercoupling effect. The red region corresponds to the band gap where we situate the dipole’s frequency

Fig. 13.16 Illustration of a uniform potential well with discrete levels of energy (left) and a BIC in a modulated potential well with resonance in the continuum (right)

[37]. Additionally, systems designed to exhibit BICs can achieve extremely large Q-factors in spite of fabrication imperfections, such as in the order of .105 reported by Jin et al. by minimizing the radiative leakage to the environment [51].

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13.5 Conclusion Unconventional properties of ENZ materials were proclaimed and studied more than a dozen years ago. Extraordinary phenomena enabled by ENZ materials, such as the supercoupling effect in ultra-thin and distorted waveguides, quasi-static phase mapping, and decoupling between electric and magnetic fields during wave propagation boosted interest in their application. However, realization of ENZ materials through conventional metals around the wavelength where the permittivity crosses zero encountered serious limitations through inevitable losses, especially at the optical frequencies. In most cases, extraordinary ENZ effects expected theoretically when disregarding losses are completely washed out if realistic material properties are considered. Therefore, the ENZ research has further advanced to the idea of implementation of the all-dielectric platforms, where such materials are classified by an effective near-zero index. The nontrivial part of such transformation is the appearance of Dirac points in band structures of arrays of dielectric elements, also known as photonic crystals. In this chapter, we start by classifying materials accordingly to their electromagnetic properties and which conditions give the ENZ, MNZ, and EMNZ cases. Each of these examples can be directly projected on the NZI case by neglecting the imaginary parts of the corresponding dielectric and magnetic functions. We consider such virtual situations and show how the efficient transmission of the tapered waveguides can be reached despite the huge difference in the linear dimensions of the waveguides’ parts. We also analyze the advantages in coherence properties of radiation from quantum emitters propagating in an NZI material. Quasi-static phase distribution leads to uniform interference at all points occupied by an NZI space. We illustrate the appearance of the conventional interference pattern from two coherent sources with deviation from the NZI (ENZ) regime toward high-index dielectric case. In the last section, we show results, which prove the appearance of the effective NZI configuration in a conventional silicon photonic crystal. As expected, even with some deviations from the exact Dirac point, a propagating mode still can be characterized by a refractive index close to zero and as such exhibit supercoupling effect illustrated in brute force simulations. We anticipate that all-dielectric platforms can create a reliable configuration for NZI modes, which will be implemented in diverse situations with all-optical processing of quantum and classical information directly on-chip. Acknowledgments We sincerely appreciate Prof. Tom Mackay and Prof. Akhlesh Lakhtakia for the invitation to participate in the Weiglhofer Symposium on Electromagnetic Theory. We have not been acquainted with Prof. Weiglhofer personally, but many years ago one of us (AVL) was greatly inspired by Weiglhofer’s seminal works and bright ideas in bianisotropy, which actually can be considered as being an essential prerequisite for the concept of fine-structured materials or simply metamaterials as we see it now.

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27. Saleh, B., Teich, M.: Fundamentals of Photonics. Wiley, New York (2019) 28. Chan, C.T., Hang, Z., Huang, X.: Dirac dispersion in two-dimensional photonic crystals. Modern Trends. Metamaterial Applics. 2012, 313984 (2012) 29. Zhao, Q., Zhou, J., Zhang, F., Lippens, D.: Mie resonance-based dielectric metamaterials. Mater. Today 12, 60–69 (2009) 30. Hergert, W., Wriedt, T.: The Mie Theory: Basics and Applications. Springer, Heidelberg (2012) 31. Novoselov, E.: Two-dimensional gas of massless Dirac fermions in graphene. Nature 438, 197– 200 (2005) 32. Diem, M., Koschny, T., Soukoulis, C.: Transmission in the vicinity of the Dirac point in hexagonal photonic crystals. Phys. B: Condens. Matter 405, 2990–2995 (2010) 33. Raghu, S., Haldane, F.: Analogs of quantum-Hall-effect edge states in photonic crystals. Phys. Rev. A 78, 033834 (2008) 34. Wang, L., Wang, Z., Zhang, J. Zhu, S.: Realization of Dirac point with double cones in optics. Opt. Lett. 34, 1510–1512 (2009) 35. Huang, X., Lai, Y., Hang, Z., Zheng, H., Chan, C.T.: Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials. Nat. Mater. 10, 582–586 (2011) 36. Minkov, M., Williamson, I., Xiao, M., Fan, S.: Zero-index bound states in the continuum. Phys. Rev. Lett. 121, 263901 (2018) 37. Vertchenko, L., DeVault, C., Malureanu, R., Mazur, E., Lavrinenko, A.: Near-zero index photonic crystals with directive bound states in the continuum. Laser Photon. Rev. 15, 2000559 (2021) 38. Li, Y., Chan, C.T., Mazur, E.: Dirac-like cone-based electromagnetic zero-index metamaterials. Light: Sci. Appl. 10, 203 (2021) 39. Vulis, D., Reshef, O., Camayd-Muñoz, P., Mazur, E.: Manipulating the flow of light using Dirac-cone zero-index metamaterials. Rep. Progr. Phys. 82, 012001 (2018) 40. Alu, A., Engheta, N.: Dielectric sensing in epsilon-near-zero narrow waveguide channels. Phys. Rev. B 78, 045102 (2008) 41. Sadrieva, Z., Sinev, I., Koshelev, K., Samusev, A., Iorsh, I., Takayama, O., Malureanu, R., Bogdanov, A., Lavrinenko, A.: Transition from Optical Bound States in the Continuum to Leaky Resonances: Role of Substrate and Roughness. ACS Photon. 4, 723–727 (2017) 42. Hsu, C., Zhen, B., Stone, A., Joannopoulos, J., Soljaˇci´c, M.: Bound states in the continuum. Nat. Rev. Mater. 1, 16048 (2016) 43. Stillinger, F., Herrick, D.: Bound states in the continuum. Phys. Rev. A 11, 446 (1975) 44. Neumann, J., Wigner, E.: Über merkwürdige diskrete Eigenwerte. In: The Collected Works Of Eugene Paul Wigner, pp. 291–293 (1993) 45. Marinica, D., Borisov, A., Shabanov, S.: Bound states in the continuum in photonics. Phys. Rev. Lett. 100, 183902 (2008) 46. Kami´nski, P., Taghizadeh, A., Breinbjerg, O., Mørk, J., Arslanagi´c, S.: Control of exceptional points in photonic crystal slabs. Opt. Lett. 42, 2866–2869 (2017) 47. Overvig, A., Malek, S., Carter, M., Shrestha, S., Yu, N.: Selection rules for symmetry-protected bound states in the continuum. ArXiv Preprint ArXiv:1903.11125 (2019) 48. Sadrieva, Z., Frizyuk, K., Petrov, M., Kivshar, Y., Bogdanov, A.: Multipolar origin of bound states in the continuum. Phys. Rev. B 100, 115303 (2019) 49. Kita, S., Li, Y., Camayd-Muñoz, P., Reshef, O., Vulis, D., Day, R., Mazur, E., Lonˇcar, M.: Onchip all-dielectric fabrication-tolerant zero-index metamaterials. Opt. Express 25, 8326–8334 (2017) 50. Lavrinenko, A., Wohlleben, W., Leyrer, R.: Influence of imperfections on the photonic insulating and guiding properties of finite Si-inverted opal crystals. Opt. Express 17, 747–760 (2009) 51. Jin, J., Yin, X., Ni, L., Soljaˇci´c, M., Zhen, B., Peng, C.: Topologically enabled ultrahigh-Q guided resonances robust to out-of-plane scattering. Nature 574, 501–504 (2019)

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Larissa Vertchenko received her B.Sc. and M.Sc. degrees in physics from the Federal University of Minas Gerais, Brazil, under the supervision of Prof. C. Monken in 2016. She received her Ph.D. degree in Photonics Engineering from the Technical University of Denmark within the field of metamaterials in 2020 under the supervision of Prof. A. V. Lavrinenko. She is currently a researcher with Sparrow Quantum at University of Copenhagen. Her work is centered in the area of nanophotonic devices including design and characterization of photonic structures for both classical and quantum communication.

Andrei V. Lavrinenko received the M.S. and Ph.D. degrees from the Belarusian State University (BSU), Minsk, Belarus, under supervision of Prof. L. M. Barkovski and Prof. G. N. Borzdov in 1982 and 1989, respectively. In 2004 he received the Doctor of Science degree in optics. He was an Assistant Professor and Associate Professor with the Department of Physics, BSU, from 1990 to 2004. Since 2004, he has been an Associate Professor with the Department of Photonics Engineering, (now Department of Electrical and Photonic Engineering), Technical University of Denmark, Kongens Lyngby, Denmark. Since 2008, he has been leading the Metamaterials Group. He is the author of five textbooks, ten book chapters, and more than 220 journal papers. His research interests include metamaterials, plasmonics, photonic crystals, quasicrystals and photonic circuits, slow light, and numerical methods in electromagnetics and photonics.

Chapter 14

Correlated Disorder in Broadband Dielectric Multilayered Reflectors Vincenzo Fiumara, Paolo Addesso, Francesco Chiadini, and Antonio Scaglione

14.1 Introduction Electromagnetic propagation in one-dimensional photonic structures consisting of a stack of alternating layers of two (or more) dielectric materials with different refractive indices is a thoroughly investigated topic in classical electrodynamics [1]. Such structures are widely used in various optical devices: filters [2–4], high-reflection coatings [5, 6], and low-reflection coatings [7]. As an example, high-reflection coatings realized as dielectric layered media play a very important role in determining the sensitivity and the visibility distance of gravitational-wave interferometric detectors [8, 9]. Here, we consider disordered multilayers consisting of two lossless dielectric materials with different refractive indices that alternate in the stack with random thicknesses. Nature suggests us to use these structures as broadband reflectors [10]. Indeed, disordered multilayers can be found in the animal kingdom in various organisms whose skin exhibits a broadband high-reflection characteristic. This is the case of various silvery fishes, which, due to apparently random stacks of guanine and cytoplasm layers in their skin, exhibit high reflectance in the visible and nearinfrared wavelength band [11].

V. Fiumara () School of Engineering, University of Basilicata, Potenza, Italy e-mail: [email protected] P. Addesso Department of Information Engineering, Electrical Engineering and Applied Mathematics, University of Salerno, Fisciano (SA), Italy e-mail: [email protected] F. Chiadini · A. Scaglione Department of Industrial Engineering, University of Salerno, Fisciano (SA), Italy e-mail: [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 T. G. Mackay, A. Lakhtakia (eds.), Adventures in Contemporary Electromagnetic Theory, https://doi.org/10.1007/978-3-031-24617-3_14

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Generally, a realization of a random multilayer exhibits a wide reflection wavelength range, in which, however, transmission notches that break the reflectance band continuity may be present due to stochastic resonances. The number, depth, spectral position, and width of these notches depend on the specific realization of the disordered structure. However, configurations of disordered multilayers, which do not suffer from these notches and function as high-performance broadband reflectors, can be designed by numerical procedures. Here, we illustrate two design strategies: (i) random inspection and (ii) search by a dedicated genetic algorithm. Statistical analysis of the configurations selected by using these procedures sheds light on the characteristics of a disordered multilayer that make it a good broadband reflector. The characterization of both distribution and autocorrelation properties of high-performance realizations reveals that correlated disorder (i.e., thickness sequence presenting a non-negligible degree of autocorrelation) plays a key role in determining the reflector performance.

14.2 Disordered One-Dimensional Photonic Structures A disordered one-dimensional photonic structure can be realized as a dielectric multilayer stack consisting of two materials with different refractive indices .nh and .nl (.nh > nl ) that alternate in the structure with random thicknesses. An example of such a stack is shown in Fig. 14.1. In this section, we consider disordered photonic structures that are periodic on average, i.e., the thickness sequence is a periodic sequence perturbed by zero-mean random noise [12]. In this case, the thickness .di (.i = 1, 2, ..., N) of the ith layer can be written as  h d (1 + vi ), i odd .di = (14.1) , d l (1 + vi ), i even

incident wave

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where N is the total number of layers in the stack, .d h and .d l are the thicknesses of the high- and low-refractive-index materials in the unit cell of the unperturbed periodic structure, and .vi is a random variable with zero mean. The first question that arises is the following: how does the added noise change the optical characteristics of the disordered structure compared to the unperturbed periodic one? We focus here on the analysis of the reflectance spectrum for normal incidence. As an example helping to analyze the effect of disorder, we consider an unperturbed periodic structure whose layers are quarter-wave at 600 nm wavelength, i.e., .d h = 600/(4nh ) nm and .d l = 600/(4nl ) nm, and assume .vi as random variables with identical uniform distribution in the range .(−1/3, 1/3). In other words, the layer thicknesses .di are assumed to be random variables with uniform distribution in the range from the quarter-wave value at 400 nm to the quarter-wave value at 800 nm.

14.2.1 Reflectance Spectra The dielectric multilayers, whose reflectance spectra are discussed in the following, consist of .N = 40 layers of lossless materials with refractive indices .nh = 2.60 and .nl = 1.45. Titanium dioxide (TiO.2 ) and silicon dioxide (SiO.2 ) can be high-index and low-index candidate materials, respectively [13]. A uniform plane wave impinges normally on the top layer of the structure, which lies on a substrate with refractive index .ns = 1.51 (BK7 glass), as illustrated in Fig. 14.1. Given the multilayer configuration, i.e., the thickness sequence in the stack, the reflectance spectrum can be calculated by using the well-known characteristic matrix method [14]. All of the spectra reported and discussed here were calculated with a wavelength resolution of .0.5 nm. First, let us observe Fig. 14.2, where the reflectance of the unperturbed periodic structure, which is quarter-wave at 600 nm, is reported as a function of the wavelength .λ in the range .λ ∈ [300, 900] nm. This spectrum, as is well known, exhibits a forbidden-transmission band (about .510 − 740 nm) encompassing .λ = 600 nm. 1.0 0.8

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Fig. 14.2 Reflectance spectrum of the unperturbed periodic structure that is quarter-wave at 600 nm

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In order to obtain the realizations of the disordered structure described by Eq. (14.1), a numerical generator can be used to produce a string of random numbers .vi (.i = 1, 2, ..., N) with a standard uniform distribution in the interval .(−1/3, 1/3). As examples, the thickness sequences of four realizations are reported in the bar graphs in Fig. 14.3, where the thickness values are normalized with respect to the quarter-wave length at .λ = 600 nm, i.e., .d h for layers with high refractive index (odd positions in stack) and .d l for layers with low refractive index (even positions in the stack). With this normalization, a realization of the disordered multilayer is unambiguously represented by a random sequence of size N, say .w = (w1 , · · · , wN ), where each element .wi is uniformly distributed in the interval .[2/3, 4/3]. The reflectance spectra of the multilayer configurations in Fig. 14.3 are shown in Fig. 14.4. They are representative of the reflection characteristics of periodic on-average disordered multilayers. Generally, these structures exhibit a reflection band wider than that of the unperturbed periodic structure, in which, however, transmission notches may be present to break the reflectance band continuity. The number, depth, spectral position, and width of these notches depend on the specific realization of the disordered structure. Since the layer thicknesses range from the quarter-wave value at 400 nm to the quarter-wave value at 800 nm, we focus on the reflection properties in the band . = [400, 800] nm. The structure in Fig. 14.3d behaves as a good reflector in .: the spectrum reported in Fig. 14.4d shows that the reflectance in this band is higher than .0.931. On the other hand, the reflectance spectra of the realizations in Fig. 14.3a– c (see Fig. 14.4a–c) show several transmission notches in . with significant width and depth that affect the reflection performance. In order to explore the potential of disordered multilayers to behave as broadband high-performance reflectors, a simple random search can be performed. A large number of random sequences .w can be generated, and the reflection spectra of the corresponding multilayer configurations calculated. In this connection, in order to characterize the reflection performance by using a unique parameter, we choose the minimum value of the reflectance in the band . as the figure of merit (FOM) of a multilayer configuration. The configurations in Fig. 14.3 have the following values of FOM: (a) .0.079, (b) .0.375, (c) .0.045, and (d) .0.931. Following this descriptive statistical approach, we generated .M = 106 thickness sequences, each of which engenders a one-dimensional disordered structure, and calculated the corresponding FOMs. In order to perform a simple statistical characterization of the FOM, its theoretical cumulative distribution function (CDF) should be known. Unfortunately, this function is unknown, but it can be effectively approximated by computing the empirical CDF (ECDF) of the data [15]. The results of this analysis are summarized in Fig. 14.5 where, for sake of readability, the FOM complementary CDF (CCDF) is reported (.CCDF = 1 − CDF). Configurations having FOM .0.80 are .0.015% (the CCDF in .0.80 is equal to .1.50 · 10−4 ). Only 13 configurations have FOM greater than .0.90, and only one configuration has FOM greater than .0.95.

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Thus, the probability that a disordered one-dimensional photonic structure designed as described above has high FOM is very low. However, the analysis of the high-performance configurations can shed light on the characteristics of a disordered multilayer that make it a good broadband reflector. To this end, we selected the .Mb = 50 best-performing sequences (lowest FOM .= 0.856, highest FOM .= 0.964) and conducted a quantitative statistical analysis aimed at the characterization of both their distribution and autocorrelation properties.

14.2.2 First-Order Statistical Analysis: Distribution Properties In this subsection, we analyze whether the .Mb = 50 best-performing sequences (say .wb , with .b ∈ (1, · · · , Mb )) are characterized by a different sample distribution of the normalized thicknesses as compared to the original uniform distribution.

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As expected, the ECDF of the normalized thicknesses of .Mr randomly chosen sequences is very close to the theoretical uniform CDF, as shown in Fig. 14.6 for .Mr = 50 (red curve). On the other hand, the ECDF of the normalized thicknesses of the best-performing .Mb sequences slightly differs from the theoretical uniform CDF (see dashed blue curve in Fig. 14.6). Is this difference significant or is it a statistical artifact? To further investigate the relationship between the FOM values and the deviation from the uniform distribution, a statistical analysis, based on the well-known Kolmogorov–Smirnov (K–S) goodness-of-fit test, can be carried out. A micro-primer on this subject is reported in Sect. 14.4 for the benefit of the reader with no previous knowledge of the matter. Here, we only note that, given a data vector .w = (w1 , · · · , wN ), in which the elements .wi are identically distributed random variables with CDF .F (u, w), the K–S test compares two hypotheses regarding these variables: (i) the null hypothesis .H0 : .F (u, w) = F0 (u), where .F0 (u) is the CDF of a given distribution; and (ii) the negation of the null hypothesis, say .H0 : .F (u, w) = F0 (u). In the present case, the vector .w is a sequence of normalized thicknesses, and .F0 (u) is the CDF of the uniform distribution in .[2/3, 4/3]. The test result can be inferred by computing the p-value .p (w) (see Sect. 14.4 for details) that, under the null hypothesis, assumes random values uniformly distributed in the interval .[0, 1]. As a consequence, when the null hypothesis is true, the average p-value is .0.5. In view of these considerations, we performed the K–S test for each of the .Mb = 50 best-performing sequences (say .wb ) and computed the corresponding p-values .p (wb ). Then, we calculated the average p-value . p b , according to  pb =

.

Mb 1  p (wb ). Mb

(14.2)

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The obtained value for . p b is .0.457, whereas the .95% confidence interval is [0.368, 0.546]. These data indicate that the deviation from the uniform CDF observed in Fig. 14.6 is not significant. For the sake of completeness, we performed this analysis also on the whole group of .M = 106 sequences, according to the following procedure:

.

• Compute the p-value for each realization among the .M = 106 sequences of the normalized thicknesses. • Sort the realizations for increasing values of FOM and group them in S equally spaced bins: in general, each bin will contain a different number .Ms of realizations, with .s ∈ (1, · · · , S). • Compute the average p-value . p s for each bin. The results are shown in Fig. 14.7, where the FOM data have been grouped in S = 10 bins of size .0.1. It is evident that for low and medium FOM values, the average p-value is very close to .0.5, as expected when the null hypothesis .H0 is true. Only for higher values of FOM, a slight decrease is observed, but, also in this case, it does not appear to be very significant, because .0.5 is well within the .95%

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Fig. 14.6 Empirical cumulative distribution functions (ECDFs) of the normalized thickness w.r.t. the theoretical uniform cumulative distribution function (CDF). The dotted black curve (labeled as “CDF Uniform”) is the theoretical CDF of a random variable uniformly distributed in [2/3,4/3]; the continuous red curve (labeled as “ECDF Random”) is the ECDF computed by using .Mr = 50 randomly chosen sequences; the dashed blue curve (labeled as “ECDF Best Random”) is the ECDF computed by using the .Mb = 50 random sequences that exhibit the best FOMs

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Fig. 14.7 Kolmogorov– Smirnov statistical goodness-of-fit test: average p-value vs. FOM. Vertical bars indicate .95% confidence interval

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confidence interval of the estimated average p-value. Finally, it is worth noting that the number .Ms of realizations for each bin s decreases for increasing values of the FOM, as expected from Fig. 14.5. That is why the confidence interval is wider for increasing values of the FOM.

14.2.3 Second-Order Statistical Analysis: Autocorrelation Properties The second step of the statistical analysis consists in investigating whether the thickness values in the best-performing .Mb sequences are correlated or not. Let us consider the sample autocorrelation function (ACF) of the zero-mean residuals

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computed at lag k, i.e., N 

ρk (w) =

.

(wi −  μw )(wi−k −  μw )

i=k+1 N 

, (wi −  μw )

(14.3)

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where . μw is the sample mean of the current thickness sequence .w. As expected, the ACF averaged over .Mr = 50 randomly chosen sequences keeps close to zero for the lag .k = 0 (see red curve with squares in Fig. 14.8). On the other hand, the average ACF of the best-performing .Mb = 50 sequences is noticeably greater than zero for low and medium values of the lag, as shown in Fig. 14.8 (see blue dashed curve with crosses). In order to investigate the statistical significance of this result, the Ljung–Box (LB) goodness-of-fit test can be conducted [16]. In this case, the different hypotheses are: – .H0 (null hypothesis): The thickness sequences are not autocorrelated. – .H0 (complementary hypothesis): The thickness sequences are autocorrelated. The L-B test involves the weighted sum of the autocorrelation functions computed for different lags up to a maximum lag value .kmax . Details are given in Sect. 14.4. First, we performed the L-B test for the best-performing .Mb sequences. The average p-values and the .95% confidence interval for different values of .kmax ∈ 1

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Fig. 14.8 Average ACF of the normalized thickness sequences vs. lag k. The dotted black curve with circles (labeled as “Not Correlated”) is the theoretical ACF of an uncorrelated random sequence; the continuous red curve with squares (labeled as “Random”) is the averaged ACF computed by using .Mr = 50 randomly chosen sequences; and the dashed blue curve with crosses (labeled as “Best Random”) is the averaged ACF computed by using the .Mb = 50 random sequences that exhibit the best FOMs

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Table 14.1 Average p-values . p b of the L-B test performed on the best-performing .Mb = 50 sequences, and their .95% confidence interval (CI) computed for different values of the maximum lag .kmax .kmax

=2

. p b

.0.192

CI

.[0.127, 0.257]

=5 0.200 .[0.134, 0.266] .kmax

= 10 0.247 .[0.175, 0.319] .kmax

= 20 0.267 .[0.173, 0.360] .kmax

{2, 5, 10, 20} are reported in Table 14.1. For all considered values of .kmax , the average p-value is lower than .0.5, which is the expected value when the null hypothesis .H0 is true. From a statistical point of view, the extent of the deviation from 0.5, which decreases by increasing .kmax , is measurable even if its significance is not very strong. Finally, we performed the L-B test on the whole group of .M = 106 sequences, by following the same steps described in Sect. 14.2.2. The results calculated for .kmax ∈ {2, 5, 10, 20} and .S = 10 bins (each of size .0.1) are shown in Fig. 14.9, where the average p-value for each bin of FOM and its .95% confidence interval (the vertical bar) are plotted. We see that, in all four cases, the average p-value is close to .0.5 for the first five bins (.FOM < 0.5), as expected when the null hypothesis is verified. For the successive bins, the average p-value drops significantly reaching its lowest value in the last bin. These results clearly show that multilayer configurations with FOM up to .0.5 consist of uncorrelated thickness sequences, while for .FOM > 0.5 the null hypothesis gradually weakens as the FOM increases.

14.3 Searching for High-Performance Disordered Mirrors by Genetic Algorithm: Methods and Results The random search described in the previous section is characterized by a low probability of selecting multilayer configurations functioning as high-performance broadband reflectors. Recently, we proposed a method based on a genetic algorithm to find disordered one-dimensional photonic structures that exhibit high reflection in a wide band [17].

14.3.1 Genetic Algorithms: Short Description Genetic algorithms (GAs) are iterative stochastic algorithms inspired by nature, particularly useful when dealing with optimization problems with a large number of parameters [18]. A GA, starting from an initial population, searches for highly performing individuals by mimicking the process of biological evolution. In the present case, an individual is a multilayer of two alternating materials with given refractive indices .nh and .nl , the total number of layers being N . The parameters to

14 Correlated Disorder in Broadband Dielectric Multilayered Reflectors

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.95%

be optimized are the layer thicknesses, which are encoded in a binary string named the chromosome of the individual. The goal of the algorithm is to select near-optimal thickness sequences exhibiting very high FOM. At each iteration, the GA computes the FOM of the individuals of the current population, selects the best-performance individuals, and combines their chromosomes to generate the next population. The algorithm stops when an individual with FOM greater than a given value is found, or it can be stopped after a predetermined number of iterations. The details of the GA are given elsewhere [17]. Here we focus on the analysis of the high-performance thickness sequences that can be selected by using the GA. To this end, we fixed the multilayer parameters as in Sect. 14.2.1: .N = 40, .nh = 2.60, and .nl = 1.45. The multilayer lies on a substrate with refractive index .ns = 1.51, and a uniform plane wave impinges normally on the top layer. The layer thicknesses can range from the quarter-wave value at 400 nm to the quarter-wave value at 800 nm. The reflectance spectra are calculated with a wavelength resolution of .0.5 nm, and the FOM is defined as in Sect. 14.2.1. Each individual of the initial population consists of a random thickness

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1.4 1.2 1.0 0.8 0.6 0.4 0.2 0

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0

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Fig. 14.10 Normalized thicknesses of two different multilayer configurations selected by the GA

1.0

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0 300 400 500 600 700 800 900 Wavelength (nm) (a)

0 300 400 500 600 700 800 900 Wavelength (nm) (b)

Fig. 14.11 Reflectance spectra of the multilayer configurations in Fig. 14.10. The dashed red lines delimit the band .. FOM: (a) .0.977, (b) .0.992

sequence generated in a similar way as in the random search described in Sect. 14.2: the normalized thicknesses can assume .28 discrete values uniformly distributed in the range [2/3, 4/3]. As a consequence, the chromosome of each individual is represented by a string of N bytes, which unambiguously defines the multilayer configuration. By using the GA, we selected .Mg = 50 disordered-thickness sequences exhibiting FOM greater than .0.95. As examples, the normalized values of two thickness sequences are reported in Fig. 14.10. The corresponding reflectance spectra are shown in Fig. 14.11, the two FOMs being .= 0.977 and .= 0.992. It is worth emphasizing that while all the .Mg sequences selected by the GA have FOM greater than .0.95, only one of the .M = 106 sequences selected by the random search has FOM greater than .0.95. Thus, the multilayer configurations

14 Correlated Disorder in Broadband Dielectric Multilayered Reflectors 1

CDF

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CDF Uniform ECDF Best Random ECDF Genetic

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Fig. 14.12 Empirical cumulative distribution functions (ECDFs) of the normalized thickness w.r.t. the theoretical uniform cumulative distribution function (CDF). The dotted black curve (labeled as “CDF Uniform”) is the CDF of a random variable uniformly distributed in [2/3,4,3]; the dashed blue curve (labeled as “ECDF Best Random”) is the ECDF computed by using the .Mb = 50 random sequences that exhibits the best FOMs; and the dash-dotted magenta curve (labeled as “ECDF Genetic”) is the ECDF computed by using .Mg = 50 sequences generated by GA

selected by using the GA allowed to perform the same statistical analysis, described in Sects. 14.2.2 and 14.2.3, on a set of thickness sequences characterized by FOM values higher than the ones exhibited by the sequences obtained by the random search.

14.3.2 First-Order Statistical Analysis: Distribution Properties The ECDF of the normalized thicknesses of the .Mg sequences selected by the GA is reported in Fig. 14.12 (dash-dotted magenta line). It is evident that the difference from the theoretical uniform CDF is higher for the ECDF of the .Mg = 50 GA sequences than that for the ECDF of the .Mb = 50 best random sequences (dashed blue curve). As illustrated in Sects. 14.2.2 and 14.4.1, the analysis of the statistical significance of the deviation from the theoretical uniform CDF can be performed by conducting the K–S test on the thickness values of the GA sequences. The average p-value is . pg = 0.247, whereas the .95% confidence interval is .[0.168, 0.325]. These values indicate that the distribution of the thicknesses of the configurations selected by the GA deviates from the uniform distribution in a measurable way, even if the statistical significance of the deviation is not very strong.

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1

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0.6 0.4 0.2 0 -0.2 -10

-5

0

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5

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Fig. 14.13 Averaged ACF of the normalized thickness sequences vs. lag. The dotted black curve with circles (labeled as “Not Correlated”) is the theoretical ACF of an uncorrelated random sequence; the dashed blue curve with crosses (labeled as “Best Random”) is the averaged ACF computed by using the .Mb = 50 random sequences that exhibits the best FOMs; and the dashdotted magenta curve with plus signs (labeled as “Genetic”) is the averaged ACF computed by using the .Mg = 50 GA sequences

14.3.3 Second-Order Statistical Analysis: Autocorrelation Properties In this section, the analysis of the autocorrelation structure of the GA sequences is presented. The ACF averaged over the .Mg = 50 sequences selected by the GA is reported in Fig. 14.13 (magenta curve with plus signs) where the ACF of the .Mb = 50 best random sequences is also plotted (blue curve with crosses). It is evident that the GA sequence ACF is noticeably greater than the best-random-sequence ACF for low and medium values of the lag k. This result seems to indicate that the GA sequences that exhibit the highest FOM values also exhibit the highest degree of autocorrelation. To further investigate this aspect, the L-B test was conducted. The results of the test for different values of the maximum lag .kmax are reported in Table 14.2. The average p-values appear to be significantly low, indicating a significant deviation from the null hypothesis. This implies that the hypothesis that the thickness sequences are autocorrelated can be accepted with a sufficient level of statistical significance. Comparing the results in Figs. 14.9 and 14.13, and Tables 14.1 and 14.2, we conclude that the higher the FOM value is, the higher is the degree of autocorrelation of a disordered sequence. In order to confirm this conclusion, we calculated the scatter plots of .1 − FOMh vs. L-B test p-value .p (wh ) for the GA sequences .wh .(h ∈ {1, · · · , Mg }). The plots obtained for the maximum lag .kmax ∈ {2, 5, 10, 20} are reported in Fig. 14.14. A clear trend is found between .FOMh and .p (wh ): the higher the FOM, the lower the p-value, i.e., the higher the degree of autocorrelation of the thickness

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Table 14.2 Average p-values . p g of the L-B test performed on the .Mg = 50 GA sequences, and their .95% confidence interval (CI) computed for different values of the maximum lag .kmax .kmax

=2

=5 0.0408 .[0.0141, 0.0675]

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Fig. 14.14 Scatter plot of the p-values of the LBT test vs. 1-FOM for each multilayer configuration generated by the genetic algorithm. The values of the maximum lag .kmax are: (a) 2, (b) 5, (c) 10, (d) 20

sequence. The relationship between the L-B test p-value and the FOM is confirmed by the moderate/high values of the correlation coefficients .rg (see definition and interpretation details in Sect. 14.4), computed between .log10 ( p(wh )) and .log10 (1 − FOMh ), reported as insets in Fig. 14.14. Finally, we conclude that the high-performance multilayer configurations selected by the GA are characterized by correlated disorder. The layer thicknesses are distributed according to a probability distribution only moderately different from the uniform one, while the main characteristic appears to be the significant degree of autocorrelation exhibited by the thickness sequences. The relationship

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between the degree of autocorrelation and performance can be seen, albeit with less statistical significance, also from the analysis of the less-performing configurations generated by the random search procedure.

14.4 Statistical Tools This section provides details about the statistical tools used in this chapter.

14.4.1 Statistical Tests and p-Values In this subsection, some simple concepts about statistical tests and p-values are presented [15]. Let us start by considering a vector .W = (W1 , · · · , WN ), in which each element .Wi is a random variable, and let .w = (w1 , · · · , wN ) be the observed realization of .W, i.e., the data vector. A binary statistical test uses the observed data vector .w in order to compare two different hypotheses: – .H0 , i.e., the null hypothesis, in which .W has a distribution1 .FW (u, H0 ). – .H1 , i.e., the alternative hypothesis, in which .W has a distribution .FW (u, H1 ). We remark that the test cannot establish whether .H0 or .H1 is true, but only whether the observed data support the acceptance of .H0 or its rejection (i.e., the acceptance of .H1 ). Typically, the test is performed by computing a test statistic, say .T = g(W), which is a function of the vector .W and is a random variable itself. The decision about which of the two hypotheses should be accepted is taken by comparing the actual value of the statistic .t = g(w) w.r.t. a threshold .γ . For example: H1 g(w) > < γ, H0

.

(14.4)

i.e., if .g(w) < γ , the null hypothesis .H0 is accepted; otherwise, it is rejected in favor of the alternative hypothesis .H1 . The value of the threshold .γ is chosen in order to guarantee a certain significance level .α, arbitrarily chosen in advance, such that α = P r {T > γ | H0 } ,

.

(14.5)

1 In this section, we use the word distribution as synonym of the cumulative distribution function (CDF).

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where .P r {T > γ | H0 } indicates the probability of the event .{T > γ } when the null hypothesis .H0 holds true. It is evident that the significance level .α is the probability of wrongly rejecting the null hypothesis .H0 when it is actually true. It is possible in many applications to remove the arbitrary choice of the significance level .α by resorting to the concept of p-value [15], which gives a measure of how strongly the observed data .w contradict the null hypothesis .H0 . In more detail, if the distribution of the test statistic T is known when the null hypothesis .H0 is true, say .FT (u | H0 ), the p-value can be computed as [15] .

p (w) = P r {T > g(w) | H0 } = 1 − FT (g(w) | H0 ).

(14.6)

For the sake of simplicity, let us further assume that T has a continuous distribution. In this scenario, the main properties of the p-value can be summarized as follows: • The p-value is defined as the probability of a particular event; therefore, .p (w) ∈ (0, 1). • The lower the p-value, the higher the possibility that the null hypothesis is false. • The null hypothesis is rejected with a significance level .α if .p (w) < α. • Under the null hypothesis, .p (w) is distributed as a random variable uniformly distributed in .[0, 1]: therefore, its expected value is .0.5.

14.4.2 Kolmogorov–Smirnov Test Let us consider the problem of comparing two different hypotheses .H0 and .H1 regarding the distribution of a data set .w. The two hypotheses are defined as: – .H0 (null hypothesis): The data .w are distributed according to a known distribution .F0 (u). – .H1 (alternative hypothesis): The data .w are not distributed according to the distribution .F0 (u), i.e., they follow another unknown distribution (in this scenario, we can also define the alternative hypothesis as .H1 = H0 ). There are several goodness-of-fit tests designed to face this problem. Among them, the Kolmogorov–Smirnov (K–S) test is effective and widely used. The K–S statistic reads   (u, w) − F0 (u)| , d(w) = sup |F

.

(14.7)

u

(u, w) is the ECDF computed for the data vector .w. When the null where .F hypothesis holds true, the statistic .d(w) has a known behavior, and it is distributed according to the Kolmogorov distribution, say .FD (u) [15, 19]. Thus, it is possible to compute the p-value .p (w), as described in Sect. 14.4.1.

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14.4.3 Ljung–Box Test The Ljung–Box (L-B) test [16] is tailored for testing the degree of the autocorrelation of a data sample by comparing the different hypotheses: – .H0 (null hypothesis): The data .x are not autocorrelated. – .H1 (alternative hypothesis): The data .x are autocorrelated (also in this scenario, we can define the alternative hypothesis as .H1 = H0 ). The test statistic used by the L-B test is qkmax (w) = N(N + 2)

k max

.

k=1

ρk2 (w) , N −k

(14.8)

where .ρk (w) is the sample autocorrelation function defined in Eq. (14.3), and .kmax is the maximum used lag. The degree of correlation of the sample is measured by using the weighted sum of the autocorrelations computed for different lags. When the null hypothesis .H0 holds true (i.e., the elements .wi of the data sample .w are completely uncorrelated), the statistic .qkmax (w) has an asymptotically known behavior, and (in our scenario) it is distributed according to a .χk2max −1 distribution [16]. Thus, also in this case, it is possible to compute the p-value .p (w), as described in Sect. 14.4.1.

14.4.4 Correlation Coefficient: A Simple Rule-of-Thumb Now we introduce some general considerations and a simple rule-of-thumb about the sample correlation coefficient (CC) [20]. The CC between two random vectors .a and .b of the same size .M∗ is defined as M∗ 

(am −  μa )(bm −  μb )

m=1

r(a, b) =  ,  M∗ M∗   (am −  μa )2 (bm −  μb )2

.

m=1

(14.9)

m=1

where . μa and . μb are the sample means of the vectors .a and .b, respectively. The CC, denoted by r, is a normalized statistic, i.e., .r ∈ [−1, 1]. Its absolute value measures the strength of the linear relationship between the variables under investigation. The higher the correlation coefficient’s absolute value is, the stronger is the relationship, which entails that high values of .am are inclined to be associated with high values of .bm (positive correlation) for .r > 0, while high values of .am are associated with low values of .bm (negative correlation) for .r < 0.

14 Correlated Disorder in Broadband Dielectric Multilayered Reflectors Table 14.3 Rule-of-thumb scale for evaluating the strength of a linear relationship

Value of .|r| .|r| ∈ [0.9, 1.0] .|r| ∈ [0.7, 0.9) .|r| ∈ [0.5, 0.7) .|r| ∈ [0.3, 0.5) .|r| ∈ [0.0, 0.3)

357 Interpretation Very high correlation High correlation Moderate correlation Low correlation Negligible correlation

In experimental sciences, it is very common to use rule-of-thumb scales [21, 22] for evaluating the correlation coefficient’s strength .|r|, even if they are affected by some limitations [21]. In any case, in order to simplify the interpretation of r, a rule-of-thumb scale is provided in Table 14.3. Finally, we remark that the CC is tailored to measure the strength of a linear relationship between the vectors .a and .b. If the relationship is nonlinear, a preliminary linearization is mandatory. For example, in this work, we set .am = log10 [ p(wm )] and .bm = log10 (1− FOM.m ), where the logarithmic transformation appears to be well-suited to linearize the data relationship, as shown in Fig. 14.14.

14.5 Conclusions A disordered one-dimensional photonic structure can be realized as a dielectric multilayer consisting of two lossless materials with different refractive indices that alternate in the structure with random thicknesses. If the thickness distribution ranges from the quarter-wave length at .λ1 to the quarter-wave length at .λ2 , these structures generally exhibit a quite high reflectance in the wavelength band . = [λ1 , λ2 ] except for transmission notches that break the reflectance band continuity. The number, depth, spectral position, and width of these notches depend on the specific realization of the disordered structure. We illustrated two numerical procedures (random inspection and search by dedicated genetic algorithm) to select disordered-multilayer configurations that do not suffer from these notches and function as high-performance reflectors in all of .. As an example of a broad band, we considered herein . = [400, 800] nm. A first relevant result is that disordered high-performance broadband dielectric layered reflectors can be designed. Moreover, a statistical analysis of the thickness sequences of the high-performance multilayer configurations was carried out to shed light on the characteristics of a disordered multilayer that make it a good broadband reflector. In particular, distribution and autocorrelation properties of these thickness sequences were analyzed. Our analysis shows that the high-performance multilayer configurations are characterized by correlated disorder. Their thickness sequences are distributed according to a probability distribution only moderately different from the uniform one, while the main characteristic appears to be the significant degree of autocorrelation they exhibit. Our results allow us to conclude that the

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correlated disorder plays a key role in determining the reflector performance. As a consequence, correlated disorder can be the starting point to define new families of broadband dielectric multilayered reflectors, which are currently under investigation.

References 1. Yeh, P.: Electromagnetic Optical Waves in Layered Media. Wiley, Hoboken (2005) 2. Chiadini, F., Fiumara, V., Ilaria Gallina, Pinto, I.M., Antonio Scaglione, A.: Filtering properties of defect-bearing periodic and triadic cantor multilayers. Opt. Commun. 281, 633–639 (2008) 3. Zhukovsky, S., Lavrinenko, A., Gaponenko, S.V.: Optical filters based on fractal and aperiodic multilayers. In: Dal Negro, L. (ed.) Optics of Aperiodic Structures: Fundamentals and Device Applications, pp. 91–142. Pan Stanford Publishing, Singapore (2013) 4. King, T.-C., Wu, C.-J.: Design of multichannel filters based on the use of periodic Cantor dielectric multilayers. Appl. Opt. 53, 6749–6755 (2014) 5. Fink, Y., Winn, J.N., Fan, S., Chen, C., Michel, J., Joannopoulos, J.D., Thomas, E.L.: A dielectric omnidirectional reflector. Science 282, 1679–1682 (1998) 6. Southwell, W. H.: Omnidirectional mirror design with quarter-wave dielectric stacks. Appl. Opt. 38, 5464–5467 (1999) 7. Chiadini, F., Fiumara, V., Scaglione, A.: Synthesis method for N-band multilayer antireflection coatings. J. Nanophotonics 7, 073097 (2013) 8. Pierro, V., Fiumara, V., Chiadini, F., Bobba, F., Carapella, G., Di Giorgio, C., Durante, O., Fittipaldi, R., Mejuto Villa, E., Neilson, J., Principe, M., Pinto, I.M.: On the performance limits of coatings for gravitational wave detectors made of alternating layers of two materials. Opt. Mater. 96, 109269 (2019) 9. Pierro, V. Fiumara, V., Chiadini, F., Granata, V., Durante, O., Neilson, J., Di Giorgio, C., Fittipaldi, R., Carapella, G., Bobba, F., Principe, M., Pinto, I.M.: Ternary quarter wavelength coatings for gravitational wave detector mirrors: Design optimization via exhaustive search. Phys. Rev. Res. 3, 023172 (2021) 10. Parker, A.R., Mckenzie, D.R., Large, M.C.J.: Multilayer reflectors in animals using green and gold beetles as contrasting examples. J. Exp. Biol. 201, 1307–1313 (1998) 11. McKenzie, D.R., Yin, Y., McFall, W.D.: Silvery fish skin as an example of a chaotic reflector. Proc. R. Soc. Lond. A 451, 579–584 (1995) 12. Kashdan, E., Kuritz, N., Karpovski, M., Makarov, N.M.: Correlated disorder: a novel approach to filter design. J. Opt. (Bristol) 17, 055001 (2015) 13. Wakaki, M., Kudo, K., Shibuya, T.: Physical Properties and Data of Optical Materials. CRC Press, Boca Raton (2007) 14. Born, M., Wolf, E.: Principles of Optics, 6th edn. Cambridge University, Cambridge (1980) 15. Lehmann, E.L., Romano, J.P.: Testing Statistical Hypotheses. Springer, New York (2005) 16. Ljung, G., Box, G.: On a measure of lack of fit in time series models. Biometrika 65, 297–303 (1978) 17. Fiumara, V., Addesso, P., Chiadini, F., Scaglione, A.: Broadband reflectors with a disordered layered structure: statistical properties of high performing configurations selected via genetic algorithm. J. Opt. (Bristol) 24, 0356101 (2022) 18. Banzhaf, W.: Evolutionary computation and genetic programming. In: Lakhtakia, A., MartínPalma, R.J. (eds.) Engineered Biomimicry, pp. 429–447. Elsevier, Waltham (2013) 19. Kolmogorov, A.N.: Sulla determinazione empirica di una legge di distribuzione. Giornale dell’Istituto Italiano degli Attuari 4, 83–91 (1933) 20. Mood, A.M., Graybill, F.A., Boes, D.C.: Introduction to the Theory of Statistics. McGraw-Hill, New York (1974)

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21. Monroe, W.S., Stuit, D.B.: The interpretation of the coefficient of correlation. J. Exp. Educ. 1, 186–203 (1933) 22. Asuero, A.G., Sayago, A., González, A.G.: The correlation coefficient: an overview. Crit. Rev. Anal. Chem. 36, 41–59 (2006) Vincenzo Fiumara received the Laurea degree in electrical engineering (summa cum laude) and the Ph.D. degree in applied electromagnetics from the University of Salerno, Italy, in 1993 and 1997, respectively. In 1998, he was awarded a CNR (Italian National Research Council) fellowship. He became an Assistant Professor of Electromagnetics in 1999 and an Associate Professor in 2005. From 1999 to 2005, he was with the Department of Information Engineering and Electrical Engineering of the University of Salerno. From 2005, he is with the School of Engineering of the University of Basilicata, Italy. His current research interests include electromagnetic surface waves, mirror coatings for gravitational-wave detectors, and disordered photonic structures. He is affiliated with INFN (Italian National Institute for Nuclear Physics). He has been elected a Senior Member of SPIE. He is a Member of the Virgo Collaboration. Paolo Addesso received the Laurea degree in electronic engineering (cum laude) and the Ph.D. degree in information engineering from the University of Salerno, Italy, in 2000 and 2005, respectively. He is currently an Assistant Professor with the University of Salerno. His main research interests are about statistical methods in signal processing, with special reference to remote sensing, gravitational waves, and data networks.

Francesco Chiadini is an Associate Professor of Electromagnetics in the Department of Industrial Engineering at the University of Salerno, Italy, where he leads the Microwave and Optical Technology Lab. He received the Laurea degree in electronic engineering and the Ph.D. degree in information engineering from the University of Salerno. He has co-authored about 200 scientific papers of which more than 100 are articles in peerreviewed international journals. He is currently serving as an Associate Editor of Microwave and Optical Technology Letters and as a regular reviewer for many journals, conferences, and funding agencies. His research interests encompass electromagnetic surface waves, anti-reflection coatings, mirrors for gravitational interferometry, and bioinspired structures. He is a Senior Member of IEEE, Optica, and SPIE; he is also a member of the Virgo Collaboration.

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V. Fiumara et al. Antonio Scaglione was born in Napoli, Italy. He received the Laurea degree (summa cum laude) in electrical engineering and the Ph.D. degree in electrical engineering from the University of Napoli, in 1982 and 1986, respectively. Winner of national competitions, he was appointed as an Assistant Professor of Electromagnetics in 1986 and Associate Professor of Electromagnetics in 1998 at the University of Salerno, where he has been teaching electromagnetics and optics. His main fields of interest include bioelectromagnetism, microwave industrial applications, optical materials, and optical components.

Chapter 15

Scattering from Reconfigurable Metasurfaces and Their Applications Mirko Barbuto, Alessio Monti, Davide Ramaccia, Stefano Vellucci, Alessandro Toscano, and Filiberto Bilotti

In this contribution, the design of reconfigurable metasurfaces for applications to wireless systems is discussed. It is shown that the potentialities of static metasurfaces in manipulating electromagnetic fields can be enhanced by loading the structure with electronic elements, improving their adaptive capability to external stimuli and making them a key enabling technology for innovative, adaptive, and even cognitive next generation antenna systems. Here, in Sect. 15.1, the analytical modelling of passive and time-invariant metasurfaces is reported and the design formulas of some simple yet widespread structures are discussed. In Sect. 15.2, the designs of metasurfaces made self-reconfigurable with respect to specific characteristics of the signal such as the level of power or its pulse width are presented. Antenna systems characterized by unconventional properties exploiting these augmented metasurfaces are also discussed. Then, Sect. 15.3 reports the possibilities offered by loaded metasurfaces in designing compact radiating systems equipped with dynamic frequency reconfigurability. Finally, in Sect. 15.4, the generation and control of the frequency harmonics scattered by a time-varying metasurface are presented and discussed, deriving the general time-dependent relations between the incident and the reflected fields and between the reflection coefficient and surface admittance of the metasurface in the case of analogue and digital modulation.

M. Barbuto Niccolò Cusano University, Rome, Italy A. Monti · D. Ramaccia · S. Vellucci · A. Toscano · F. Bilotti () ROMA TRE University, Rome, Italy e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 T. G. Mackay, A. Lakhtakia (eds.), Adventures in Contemporary Electromagnetic Theory, https://doi.org/10.1007/978-3-031-24617-3_15

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15.1 Modelling of Metasurfaces Metasurfaces (MTS) can be defined as artificial surfaces textured on a subwavelength scale and exhibiting unusual wave-matter interaction. The electrically small separation distance between their constitutive elements allows the homogenization of the metasurfaces in the surface plane: in other terms, the behavior of a metasurface, which in general depends on its geometry, can be described macroscopically using appropriate homogenized parameters [1, 2]. In this context, several different possibilities have been explored over the years. One of the most used homogenization approaches is the one based on the so-called surface impedances, which link the induced surface current densities with the averaged electric and magnetic fields on the two sides of a metasurface [2]. According to this approach, the description of the complex response of a metasurface requires, in the general case, the definition of four surface impedance matrices without a clear physical meaning. However, there are several simpler cases, that are still relevant for applications, for which metasurfaces can be effectively homogenized through a few parameters with a simple physical interpretation. A particularly relevant example is the case of metasurfaces with zero thickness and that do not involve magnetic materials: in this scenario, indeed, only electric surface currents may be excited in the plane of the metasurface. By using the boundary conditions, the induced electric currents introduce a discontinuity of the tangential component of the magnetic field on two two-side of the surface .(Ht + and .Ht − , respectively) that can be quantified in the following form: Et = Zs · nˆ × (Ht + − Ht − ) ,

.

(15.1)

where Et is the (continuous) tangential component of the electric field on the surface, .nˆ is the unit vector normal to the surface, and .Zs is the bi-dimensional surface impedance matrix of the metasurface, whose elements depend on both the frequency and angle of incidence. In general, each element of the surface impedance matrix is a complex quantity: the real part (surface resistance) relates to the Ohmic losses of the structure, while the imaginary one (surface reactance) to its reactive behavior. The surface impedance matrix reduces to a scalar quantity for a symmetric metasurface, whose response is the same for both polarizations and is characterized by zero polarization coupling. This family of metasurfaces has been widely investigated in the literature since the 50 s, even though the term “metasurface” has been introduced more recently as the 2D counterpart of metamaterials. One of the first geometries considered is the one shown in Fig. 15.1a, also referred to as grid metasurface. It consists of an array of metallic strips placed on a dielectric material, which is assumed to have a relative permittivity εr and to extend throughout the entire lower z half-space (z < 0). We assume that both the metal and the dielectric substrate are lossless, which implies that the surface resistance is zero. The equivalent surface reactance of the infinitely periodic structure has been studied by different authors [1–11] through different

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Fig. 15.1 (a) Geometry of a grid metasurface. It is assumed that the backing dielectric is infinitely extended throughout the lower z half-space; (b) TM surface reactance of a grid metasurface for different values of its geometrical parameters; (c) TE surface reactance of a grid metasurface for different values of its geometrical parameters

approaches. The first model was obtained in Ref. [1] and generalized, in a semiempirical way, by Ulrich [4]. More accurate results have been obtained in Ref. [8] in which a transmission-line model was used to obtain the equivalent circuit of the grid. The same expression has been later obtained in Ref. [9] using the average boundary approach. In particular, for electrically dense arrays of lossless strips, the grid,TM surface impedance value .Zs exhibited by the structure for a TM (i.e., electric field parallel to the strips) normally incident plane wave is purely reactive (i.e., grid,TM grid,TM grid,TM .Zs = j Xs , being .Xs the surface reactance and j the imaginary unit) can be expressed as [10] grid,TM

Zs

.

grid,TM

= j Xs

=j

ηeff keff a   π w  , ln csc 2a 2π

(15.2)

with keff the effective wave number, ηeff the effective wave impedance, and w the width of the strip. The definition of the geometrical parameters w and a can be found in Fig. 15.1a. Note that, as the dielectric is assumed to extend throughout the entire lower z half-space, the effective relative dielectric permittivity can be expressed as εeff = (εr + 1)/2, assuming that the incident medium is the vacuum (see Ref. [9] for more details). To study the response of the structure of Fig. 15.1a for a TE (i.e., electric field perpendicular to strips) normally incidence plane wave, we can apply the formulas for its complementary structure. Typically, they are obtained using Babinet’s principle in its modified form to consider the presence of a dielectric discontinuity at the interface [6]. Therefore, the surface impedance of the grid for TE incidence is grid,TE

Zs

.

grid,TE

= j Xs

=

η02 2 , (εr + 1) 4Zsgrid,TM

(15.3)

with η0 being the free-space impedance. We note here that for TE incidence, the grid,TM is now replaced by a-w. It is worth noticing term w in the expression of .Zs

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that when the strips are made by a lossy metal with finite conductivity (15.2, 15.3), also have a real part corresponding to a surface resistance. However, for the majority of the metals used to realize microwave printed-circuit-board, the value of the surface resistance is significantly lower than the imaginary part and, for most of the applications, can be safely neglected (more details can be found in Ref. [12–17]). In Figs. 15.1b and 15.1c, we report the grid surface reactance for different values of the geometrical parameters w and a for the TM and TE polarization, respectively. As it can be appreciated, the response for the TM polarization is always positive, corresponding to an inductive behavior. On the contrary, the TE surface reactance is always negative, corresponding to a capacitive response. These results can be easily understood by considering the geometry of the grid metasurface: indeed, when the electric field is parallel to the strips, it induces electric currents flowing along the y-direction. The self-inductance of the strips originates from the weak inductive behavior of the metasurface. The use of meandered strips would allow increasing the value of the inductance, as it will be shown in the next Sections. On the contrary, when the impinging electric field is TE-polarized, a displacement of charges is induced in each strip along the x-direction. Therefore, each pair of adjacent strips behaves as the plates of a capacitor and the overall response of the structure is capacitive. As a further observation, from Figs. 15.1b and 15.1c, it is also possible to appreciate that the surface reactance approaches zero when w approaches a. Indeed, in the limit case w = a, the grid metasurface reduces to a metal plate that, in the lossless assumption, exhibits a zero-surface impedance. Despite their effectiveness in ideal conditions, the homogenization model discussed above needs to be properly corrected in realistic scenarios. In particular, the presence of a finite-thickness substrate and/or of objects in close proximity may significantly affect the surface reactance of the grid metasurface compared to one predicted by the homogenization model (15.2, 15.3). In these cases, it is possible to retrieve the surface impedance from full-wave simulations [12] using an appropriate transmission-line model. For instance, in Fig. 15.2, we report the relevant case of a grid metasurface without a supporting substrate placed at a distance d1 from a perturbing object modelled as a finite-thickness dielectric. Once the reflection coefficient of this structure is evaluated frequency-byfrequency through numerical simulations, it is possible to retrieve the surface reactance in the real scenario. In particular, once defined Zd as the line-impedance evaluated before the metasurface section in the transmission-line model and ¦ as the reflection coefficient at its input port (see Fig. 15.2b for more details), the following formula can be used to retrieve the surface impedance for each polarization n: Zsn =

.

with

Zd η0 (1 + n ) , Zd (1 − n ) − η0 (1 + n )

n = TM, TE,

(15.4)

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Fig. 15.2 (a) Grid metasurface placed in proximity of a perturbing object; (b) Equivalent transmission model of the structure

        √  √ 2π d1 εd εd η0 tan 2πλ0d2 − j + η0 1 + j εd tan 2πλ0d2 tan λ0      . .Zd = √  √   2π d2  2π d1 εd εd tan λ0 − j + εd + j tan 2πλ0d2 tan λ0 (15.5) In the above expressions, d2 is the separation distance between the metasurface and the perturbing object, εd is its relative permittivity, λ0 is the free-space wavelength for which the retrieval is applied. Equations (15.4, 15.5) can be directly included in a numerical optimization routine and allow finding the optimal metasurface geometry able to return the desired value of its surface reactance in a given realistic environment. The family of metasurfaces discussed above does not allow to control reflection and transmission independently; nonetheless, they have found extensive applications in several application fields, such as electromagnetic cloaking [13–23], lenses [24, 25], polarization transformers [26], wide-angle impedance matching layers [27], spatial filters [28], just to name a few. It should be clear that different unit-cell geometries would result in a different behavior of the metasurface. An exhaustive summary of some of the most used unit-cells with the relative homogenization model can be found in Ref. [29].

15.2 Self-Reconfigurable Metasurfaces for Powerand Wave-Form-Dependent Effects In modern wireless systems, the capability to react to external perturbations or to adapt to different operative conditions is a key parameter for judging the effectiveness and the robustness of the overall system. This capability is increasingly requested for the RF-microwave modules of a transceiver, being the radiating

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structures required to adapt their main radiating properties in real-time. In addition, new applicative scenarios for antennas that are no longer limited to passively transmit/receive the useful signals but also performing preliminary signal manipulations, have been envisioned for reducing the virtualization of hardware functions at the software level and, thus, satisfying the key performance indicators (KPIs) of future wireless systems [30]. In this framework, the most used approach for designing reconfigurable radiating structures is based on antenna arrays [31], which consist of several antenna elements with a proper separation distance and excitation conditions. Although they have been successfully applied in several practical applications, their reconfigurable properties are limited by the wavelength-order separation distance between the elements, which typically leads to possible grating- and sidelobes. In order to overcome this inherent limitation, the use of metasurfaces, i.e., planar structures made by electrically small elements placed at a sub-wavelength distance, has been proposed [32]. Indeed, the electromagnetic properties of metasurfaces can be locally controlled to tailor the overall reflected/transmitted field almost at will. Still, this approach is limited by the huge number of constituting elements that, for adding reconfigurability, should be independently controlled in real-time. In particular, for tuning the response of a metasurface, each element should be loaded with active elements (e.g., transistors, varactors or pin diodes), and a complex control network should be implemented for tuning their operation states. As a possible solution for reducing the system complexity, the design of all-passive metasurfaces with self-reconfigurable properties has been proposed. In this case, the constituting elements of the metasurface are still loaded by lumped electronic elements that, however, do not require biasing circuits and/or controllers but just self-reconfigure their behavior according to the characteristics of the incident field. In particular, both the power and the waveform of the incoming signal have been used for this purpose [33–42]. Some relevant examples of this kind of structure will be presented in the next subsections.

15.2.1 Non-Linear Metasurfaces for Power-Dependent Radiating Structures As first investigated in Refs. [33, 34], the electromagnetic power of an incoming signal can be exploited to externally control the response of metamaterials and metasurfaces, whose constituting elements are loaded by non-linear circuits. In particular, the non-linear current-voltage characteristics of electronic diodes can be engineered for switching the metasurface response between two different states. Indeed, as shown in Fig. 15.3 for the specific case of a strip-based metasurface, the metasurface structure can be loaded with antiparallel diode pairs that, depending on the impinging power, behave as an open- or short-circuit condition, leading to a dramatic change of the metasurface response. In particular, for low-power signals, the diodes are turned off and the metasurface response is only partially

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Fig. 15.3 Example of a strip-based metasurface loaded with non-linear elements, whose response depends on the power level of the impinging signal

affected by the parasitic elements of the diodes (i.e., the parallel combination of a low capacitance with a high resistance). On the contrary, high-power incoming signals can induce voltages between the strips higher than the threshold voltage of the diodes that, being in the conducting state, severely alter the metasurface geometry and, thus, its surface impedance. This physical mechanism of operation can be extended to other metasurface geometries and properly engineered to design different radiating structures with self-reconfigurable properties. This approach has been exploited in Ref. [34] to design a horn antenna with a self-filtering notched band, which is effective only when high-power signals are transmitted or received by the horn itself. A similar approach has been exploited in Ref. [37] for implementing power-dependent mantle cloaking devices. This kind of metasurface covers are typically implemented for reducing the overall scattering of passive structural elements or linear antennas. However, their applicability is limited by the inherent reciprocity of passive and linear metasurfaces. On the contrary, by loading the cover with diode pairs, metasurfaces, whose cloaking behavior can be turned on and off by acting on the power level of the impinging electromagnetic field, have been implemented. These cloaks have been exploited for designing

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Fig. 15.4 Examples of innovative radiating structures based on non-linear metasurfaces: (a) a self-filtering horn antenna exhibiting a drop of the realized gain around a specific frequency only for low-power signals; (b) a dipole antenna array that, for high-power levels, focuses power in a specific direction while, for low-power signals, exhibits an omnidirectional radiation pattern; (c) a Yagi-Uda antenna array exhibiting different radiation patterns depending on the level of the input power

power-dependent antenna arrays whose radiation pattern can be tuned by changing the input power of the antenna itself. Moreover, non-linear metasurfaces have been envisioned as a possible solution for circumventing fundamental limitations of mantle cloaking for antennas [38]. Exploiting this approach, new degrees of freedom have been introduced for designing complex antenna systems in which both the electric and the radiation properties may be made selective with respect to the input power. In particular, a phased array able to transmit towards an arbitrary pointing direction and receive with an omnidirectional radiation pattern has been proposed. We remark here that more details on the modeling of non-linear metasurfaces can be found in Ref. [37], while some of the aforementioned applications are schematically presented in Fig. 15.4.

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Fig. 15.5 Example of a strip-based metasurface loaded with non-linear elements, whose response depends on the waveform of the impinging signal

15.2.2 Non-Linear Metasurfaces for Waveform-Dependent Radiating Structures In addition to the power level, the waveform of an electromagnetic field can also be exploited for designing self-reconfigurable metasurface-based systems [39–42]. By loading metasurfaces with lumped elements exhibiting a proper time constant, the response of the metasurface itself can be automatically adapted to the time-domain behavior of the impinging signal [40]. For this purpose, as schematically shown in Fig. 15.5, a possible solution is to load the metasurface with waveform-selective circuits composed of a diode bridge rectifier and passive elements exhibiting a time constant (i.e., with both storing and dissipating capabilities). In this way, the field impinging onto the metasurface induces a voltage across the circuit that, thanks to the presence of the diode bridge, is rectified and almost converted to zero frequency (i.e., DC) and applied to the lumped elements. Thanks to the different time-domain responses of the RLC elements, the circuit behaves differently in presence of a shortpulsed waveform signal (PW) or a long-pulse/continuous waveform (CW). This time-dependent behavior has been exploited in Ref. [41] for designing a radiating structure exhibiting both frequency- and time-domain filtering functionalities. In particular, a standard waveguide aperture antenna has been capped with a metasurface-inspired iris, which has been loaded with the circuits reported in Fig. 15.5. The overall structure, thus, can receive/transmit only signals with specific characteristics in the frequency and time domain. The same circuits have been also applied to cloaking structures in Ref. [42]. A waveform-dependent cloaking has been used to cover a dipole antenna that, thus, is able to automatically hide/show itself depending on the waveform of the received/transmitted signal without any external control unit. The aforementioned applications are schematically reported in Fig. 15.6. Please note that, in all the aforementioned designs, losses of both substrate materials and lumped elements have been fully taken into account. Moreover, we

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Fig. 15.6 Examples of innovative radiating structures based on metasurfaces loaded with timedependent circuits: (a) a self-filtering waveguide antenna that exhibits both frequency- and timedomain selectivity with radiation properties that depend on the pulse width of the impinging signal; (b) a dipole antenna coated with a time-dependent cloaking structure: the antenna is visible and effective for CW signals while is invisible and, thus, do not receive/transmit efficiently, for PW signals

remark here that the discussed designs of both power- and waveform-dependent radiating structures are only some possible examples of a new generation of intelligent and self-reconfigurable antenna systems, which expand the growing field of metasurfaces and enable advanced and unconventional functionalities of microwave systems.

15.3 Reconfigurable Metasurfaces for Frequency Tunability of Antennas With the latest advancement in wireless communication technology, there has been an exponential increase in the number of wireless services which are expected to further grow with next-generation communications systems [30]. Consequently, the issue of spectrum congestion will be even more urgent, and the quest for approaches able to mitigate it will experience a greater push. Among them, the possibility to allocate in real-time the different services within prescribed frequency bands depending on the frequency spectrum usage has proved its great potential, leading to the definition of so-called cognitive radio systems. Here, following the scan of the frequency space occupancy, the service is allocated within the available sub-band for operation, avoiding the possible interferences with the other services in the neighboring frequencies. Therefore, in these systems, the frequency reconfigurability of the antenna is a key enabling element.

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Although several approaches can be exploited to design a frequency reconfigurable radiating element, such as signal-processing algorithms, wideband/multiple antenna systems, or artificial intelligence routines exploiting microwave filters to select the desired sub-band of operation [43, 44], still the possibility to reconFig. an antenna at the physical level presents some inherent advantages (such as the speed of the operation, the low latency, etc.) [44]. In this frame, we report the potentialities of reconfigurable metasurfaces for conceiving frequency tunable radiating devices where the antenna resonance is modified according to the specific properties of the metasurface. Albeit the use of reconfigurable metasurfaces for manipulating at will the properties of a radiated electromagnetic field has been deeply explored in the literature [45], very few studies report on the use of conformal ultra-thin metasurface wrapping the antenna systems to achieve frequency reconfigurability. Here, first, we report on the general possibilities of coating metasurface for tailoring the input impedance of wire antennas, i.e., to shift within a quite broad frequency range the natural antenna resonance, while then we focus on the design of a frequency reconfigurable radiating element by considering many fabrication challenges and reporting the possible solutions to overcome them.

15.3.1 Antenna Impedance Tuning Through Metasurface Coatings Conformal metasurfaces wrapped around wire antennas are well known to offer the possibility to manipulate the scattering characteristics of the element [46]. Here, through the engineering of the surface impedance, the currents flowing onto the metasurface are tailored to activate an interfering effect with the field radiated by the coated element and achieve manipulation of the overall scattered field. However, by a judicious design of the metasurface, i.e., by a proper choice of its surface impedance value and geometrical characteristics, the coating metasurface can be used to partially reflect the radiated field and induced secondary currents onto the antenna which summed with its natural mode and frequency shifts its resonance. The scenario considered is reported in Fig. 15.7a. A conventional halfwavelength dipole antenna is surrounded by a conformal cylindrical metasurface with a radius ac > a, where a is the radius of the wire radiating element. Assuming that the metasurface is lossless, homogeneous, isotropic, and time-invariant, its behavior can be described by a scalar value of the surface impedance Zs = jXs . Indeed, as can be observed in Fig. 15.7b, by properly tuning both the radius and the value of the surface impedance the input impedance of the antenna can be tailored in both the imaginary and real part once loaded to a conventional 50  source. For ac >> a and for low capacitive values of the metasurface Xs (i.e., |Xs | → 0), the zero value of the imaginary part of the Zin is slightly shifted toward high frequency, compared to the original case. On the contrary, by reducing the metasurface radius

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Fig. 15.7 (a) Example of a linear antenna wrapped by a conformal cylindrical metasurface. (b) Magnitude of the reflection coefficient of the antenna in the bare scenario and the coated one, when varying the metasurface radius and surface impedance values (black curve: ac = 13a, Xs = −49 /sq.; magenta curve: ac = 7.5a, Xs = −15 /sq.; blue curve: ac = 5a, Xs = −5 /sq). (c) Antenna input impedance in the bare scenario and the coated one, when varying the metasurface radius and surface impedance values (red curve: bare antenna; magenta curve: ac = 13a, Xs = −49 /sq.; blue curve: ac = 5a, Xs = −5 /sq)

(i.e., for ac → a) and for strong capacitive Xs , the zero value of the imaginary part is shifted at a higher frequency. More importantly, at the same time, the real part of the Zin is flattened towards the value of the load within a wide frequency range, guaranteeing a good matching with the source. Arguably, the metasurface acts as an equivalent complex matching transmission-line network that can be tuned by modifying its surface impedance and its diameter. This behavior is enabled by the distinct characteristics of electromagnetic cloaking applied to antenna systems. It is worth noticing that, because of the tuning of the Zin , the resonant frequency of the antenna is shifted accordingly, as shown in Fig. 15.7c. The possibility of tailoring both the real and imaginary parts of the antenna input impedance is a significant advantage compared to solutions where the antenna resonance is modified by exploiting reactive tuning elements. As well-known, in this case, severe constraints on the operational bandwidth of the antenna would affect the final performance [31]. We emphasize that, as is shown in the following, the surface impedance value of the metasurface can be easily controlled by loading it with reactive components such as the varactor diodes but, compared to a conventional antenna tuner, also the real part of the Zin can be controlled.

15.3.2 Design of Varactor-Loaded Reconfigurable Metasurface To enable the tuning of the resonant frequency of the coated dipole, both the radius and surface impedance value of the metasurface should be controlled. While the latter can be controlled by loading the metasurface unit cells with reactive elements, the tuning of the metasurface radius would require a mechanical modifier, inevitably increasing the difficulty of the design. To avoid this complication, the single-layer

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Fig. 15.8 (a) Frequency reconfigurable dipole antenna coated by the varactor-loaded metasurface coat. In the insets, equivalent circuit schematic of the loaded metasurface unit cell. (b) Reflection coefficient of the antenna coated for different combinations of the varactors junction capacitances on the three layers CA , CB , CC , from inner to outer. Full-wave simulation results

metasurface structure can be replaced with a multi-layered design, as shown in Fig. 15.8a, with a fixed value of the radius for each of the layers characterized by the surface reactance (i.e, XA , XB , XC ). Dynamic frequency reconfigurability can be achieved by controlling the values of XA , XB , and XC , switching their behavior from the one where the Xs assume a very large value to the value required to achieve the resonance shift. In fact, from an equivalent transmission-line point of view, the large impedances of the metasurfaces eventually appear in parallel to the antenna input impedance and, thus, do not affect the antenna reflection coefficient [38]. To control the surface impedance values of the layers a capacitive metasurface like the one reported in Fig. 15.8a can be exploited. The metasurface is made of metallic rings placed on a dielectric cylindrical support able to synthesize a negative large value of the Xs , as discussed in the first section. The gap between the rings is then loaded with varactor diodes in order to control the metasurface response. From a circuit schematic point of view, the equivalent impedance of the varactor Zvar can be seen in shunt connection to the equivalent surface impedance of the metasurface Zs (insets of Fig. 15.8a). The latter depends on the width (w) and periodicity (d) of the metallic rings and can be approximated as (15.2), whilst the equivalent lumped impedance of the varactor can be expressed as .

1 1 + j ωCpkg , Zseries = Rs + j ωLpkg + 1/j ωCj , = Zseries Zv

(15.6)

where Cj is the value of the variable varactor junction capacitance, whilst Rs , Lpkg , and Cpkg are the internal parasitic loads accounting for the reactive and resistive effect. Finally, the total surface impedance of the loaded metasurface reads as

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.

1 1 1 + . = Zs Zv Ztot

(15.7)

Hence, by assuming negligible the values of the parasitic reactance of the varactor, a close form expression for the Ztot can be derived as [47] Ztot = −j

.

  Rs − j/ωCj

 , Cmts Rs − j/ωCj − ωCmts

(15.8)

where Cmts is the value of the equivalent capacitance of the static metasurface, i.e., 1/jωZs . Once the relation between the Ztot and the Cj is known, it is possible to properly dimension the varactors. The full-wave numerical results for the scenario of a half-wavelength dipole antenna with a radius a = λ0 /100 and resonating at 1 GHz, coated by a three-layered coating metasurface of radius from outer to inner equal to ac = 13.3/7.5/5a are reported. Here, the metasurface layers are printed on a cylindrical dielectric substrate of thickness t = 0.003 λ0 , and with εr = 2.9 and a dissipation factor of 0.0025. The loading varactors considered is the GC15006 Microsemi, with the values of the parasitic reactance and inner resistance available from the datasheet by the producer [48]. As can be appreciated from Fig. 15.8b, by tuning the values of the varactors’ capacitance on the different layers a broad frequency shift of the antenna resonance can be achieved. Remarkably, continuous sub-bands of operations can be defined, making this reconfigurable antenna particularly appealing for applications in cognitive radio systems [47]. We finally report that as table omnidirectional paper can be observed at all the different frequencies of operation of the antenna, as well as overall efficiency of 0.8 [49].

15.4 Dynamic Metasurfaces for Frequency Harmonic Generation and Control Since their beginning, metamaterials and metasurfaces have demonstrated to be a breakthrough technology in a number of applications [50–56], spanning from microwave to optical frequency ranges [57], thanks to their capabilities in manipulating the amplitude, phase, and spatial spectrum of the incident field, leaving the frequency spectrum unaltered. The first attempts to also alter the response over time have been done with reconfigurable metamaterials and metasurfaces, exhibiting a variation over time several orders slower than the periodicity of the interactive wave. Therefore, such dynamic behavior in reconfigurable systems is studied in the quasi-static case, where temporal variations can be treated as a tuning parameter of the structure’s static response. Only in the very last few years, it has been experimentally demonstrated the effects of time modulation beyond the quasi-static regime, revamping the topic of time-varying artificial media [58, 59]

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envisioned by Morgentaler in the mid of the last century [60]. In Ref. [61], it has been demonstrated that medium/surface modulation over time enables several exotic space-time scattering phenomena never observed before. This has been possible thanks to the advancement of technology that can be used for extremely fast reconfiguring schemes: pin diodes [73], varactors [74–76], MEMS [77], graphene [78], and liquid crystal [79] are the most common ones since they are all reacting to the presence of a simple voltage signal, both analog [69–71] or digital [80– 83]. In this scenario, it is clear that the possibilities offered by the dynamic fast modulated metasurfaces are unprecedented: among them, we mention here the possibility to break the reciprocity constraint and generate and control frequency harmonics in the scattered field [62–72]. Both effects are enabled by the frequency modulation induced by the metasurface on the electromagnetic field interacting with it. A time-modulated reflective metasurface producing a frequency shift with respect to the impinging radiation is proposed in [69]: the reflection phase of the whole metasurface changes linearly with time and an artificial Doppler effect for a nonmoving electrically thin structure is realized. In Refs. [83] and [84], 2-bit and 3-bit time-varying phases are employed to match the linear phase change, respectively, and a frequency shift can be induced for different higher-order harmonics. In all cases, the metasurface provides an electrically controlled phase of the reflection coefficient that covers 360◦ . In the following, we discuss the generation and control of the frequency harmonics scattered by a time-varying metasurface in the case of analog and digital modulation schemes. In particular, the aim of this section is to define the design rules for a time-varying fully reflective metasurface, whose modulating signal is a continuous analog or a discrete digital voltage signal. Exploiting the analytical expression of the reflection coefficient in the case of normal incidence, we derive the general time-dependent relations between the incident and the reflected fields, and between the reflection coefficient and the surface admittance, which are the fundamental quantities to design the needed temporal profile that enable a controlled harmonic generation into the scattered wave.

15.4.1 Analog Dynamic Metasurfaces for Harmonic Generation Dynamic metasurfaces modulated by using an analog signal exhibit an electromagnetic response that varies continuously over time. This can be achieved if the surface admittance of the metasurface is properly engineered to cover the entire range of values that allows the harmonic generation by interaction. The metasurface is composed by an array of electrically small elements that can be modeled through an effective surface admittance Ys . In a limited frequency band centered at the operative frequency of the metasurface, we can assume that it exhibits a slow frequency dispersion. This would relax the analysis as if the metasurface were frequency

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Fig. 15.9 (a) Analog dynamic metasurface based on high-impedance surface for harmonic generation. (b) Amplitude of the scattered harmonics for the up-converting (blue bars) or downconverting (red bars) process taking place at metasurface level

dispersion-less. To satisfy this constraint, also, the modulation frequency must be much smaller than the illumination frequency so that the generated harmonics are still within the aforementioned band. Under these assumptions, the surface admittance can be described as only a function of time, i.e., Ys = Ys (t), as shown in Fig. 15.9a. If the metasurface is fully reflective, no transmitted field is present beyond the metasurface and the total field under normal incidence is described by  Ei (z, t) = Re E 0.i e−j k i z ej ωi t ,

.

 Er (z, t) = Re E 0.r e+j k r z ej ωr t , (15.9)

where .E0.i(r) , ki(r) , and ωi(r) are the complex electric field amplitude, free-space wavevector of the wave propagating along z-direction and angular frequency, respectively, and Re{•} denotes the real part of the complex quantity in the brackets. At the metasurface location, i.e., z = 0, the incident and reflected electric fields in the time domain are related as Er (z = 0, t) = (t)Ei (z = 0, t) ,

.

(15.10)

where (t) = Y0 − Ys /Y0 + Ys is the time-varying complex reflection coefficient and, in turn, Y0 = 1/120π is the free-space admittance. Equation (15.10) can now be used to derive the necessary temporal profile of the surface impedance to achieve a perfect red- or blue-shift of the incident monochromatic plane wave at the frequency ωi to the reflected one at frequency ωr :

15 Scattering from Reconfigurable Metasurfaces and Their Applications

 (ωi , t) =

.

E E r (z = 0, t) = 0.r ej (ωr −ωi )t . E i (z = 0, t) E 0.i

377

(15.11)

In the lossless case and for very small differences between the incident and reflected frequencies, the amplitude of the reflection coefficient approaches unity according to the Manley-Rowe relations [85]. Therefore, the reflection coefficient in Eq. (15.11) turns to be a simple time-varying phase, spanning the 2π range with angular frequency ωr − ωi . We note that such a frequency corresponds to the modulation frequency ωm of the metasurface, i.e., ωm = |ωr − ωi |. Details for the design of the metasurface that implements such a time-varying reflection coefficient can be found in Ref. [69]. Figure 15.9b shows the amplitude of the scattered harmonics for the upconverting (blue bars) or down-converting (red bars) process that is taking place at the metasurface level according to the design of the modulating signal applied to it. The metasurface is, therefore, to suppress the back radiation of the fundamental mode and couples the most part of the impinging energy to the ±first-harmonics. Other higher-order harmonics are generated during the process due to the not ideal response of the realistic metasurface implementing the frequency conversion [69].

15.4.2 Digital Dynamic Metasurfaces for Harmonic Generation In contrast with the analog modulation, dynamic metasurfaces under digital modulation exhibit an electromagnetic response that abruptly changes over time following a square wave modulating profile of the surface properties. In this section, we report the key design steps for proper modulating a metasurface with a digital signal to have a single sideband modulation. In the previous section, Eqs. (15.9–15.11) demonstrated that a single sideband modulation can be achieved if the reflection coefficient (t) introduces the frequency shift exp[±jωm t]. Considering the positive sign case, we can rewrite the frequency shift in the form of i-phase (I) and quadrature (Q) terms:

e

.

j 2πfp t

t = cos (ωm t) + j sin (ωm t) = cos 2π Tm



 t + j sin 2π , Tm

(15.12)

where cos(·) denotes the I term, sin(·) denotes the Q term, and Tm is the time period related to ωm . The two reflective contributions are realized by two adjacent portions of the metasurface as shown in Fig. 15.10a. The interference between the scattered fields by each portion will realize the desired scattering response in the far-field. The complex time-varying reflection coefficient in Eq. (15.12) must be digitalized and we used rectangular pulse signals, whose amplitudes “+1” and “−1” correspond to the phase “0” or “π,” respectively, as shown in Fig. 15.10b, c. To maximize the amplitude of only the ±first-order harmonics, proper time sequences of the

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Fig. 15.10 (a) Digital dynamic metasurface based on the I/Q path technique for harmonic generation. (b–c) Digitalized versions (blue curve) of the complex time varying reflection coefficient: (b) real part, namely I path, and (c) imaginary part, namely Q path

rectangular pulse must be engineered, by controlling the amplitude of the scattered harmonics. For the first time-period Tm starting at t = 0, the rectangular time sequence can be expressed as ⎧ ⎨ +1, .i(t) = − 1, ⎩ 0,

⎧ t 1 ≤ t ≤ t1 + τ t 1 ≤ t ≤ t1 + τ ⎨ −1, t2 ≤ t ≤ t2 + τ ; q(t) = + 1, t2 ≤ t ≤ t 2 + τ . ⎩ others within [0; Tm ] 0, others within [0; Tm ] (15.13)

The periodic time sequence is decomposed by using the Fourier series to highlight the contribution of the different frequency harmonics: ah =

.

   2j sin (hπ τ˜ ) sin hπ t˜2 − t˜1 e−j hπ (t˜2 +t˜1 +τ˜ ) , hπ

(15.14)

where h is the harmonic order (h = 0)and thesymbol “~” denotes normalized time (t/Tm ). Considering the term .sin hπ t˜2 − t˜1 , when .t˜2 − t˜1 = 0.5, all the evenorder harmonics vanish, i.e., |ah | = 0 for h = 2k, k ∈ Z. Moreover, considering the term:.sin (hπ τ˜ ), setting .τ˜ = 1/3, the hth -order harmonics with h = 3 k, k∈Z also vanish. In this case, only a few harmonics far from the ±1st ones survive, e.g., the ±5th , ±7th , ±11th , and so on. This configuration ensures the highest isolation between the first order harmonic and the other order harmonics achievable with a

15 Scattering from Reconfigurable Metasurfaces and Their Applications Normalized reflected power _Erif (f )_2 0

Normalized reflected power _Erif (f )_2 0

analytical result numerical result

Blue-shift

–2

–4

–4

–6

–6

–8

–8

–10

–10

–12

–12

–14

–14

–16

–16

–20 3.

4 14

14

3.

3.

3.

13 3.

analytical result numerical result

Red-shift

14 6 3. 14 8 3. 15 3. 15 2 3. 15 4 3. 15 6 3. 15 8 3. 16 3. 16 2 3. 13 8 3. 14 3. 14 2 3. 14 4 3. 14 6 3. 14 8 3. 15 3. 15 2 3. 15 4 3. 15 6 3. 15 8 3. 16 3. 16 2

–18

–20 2

–18 8 14

Logarithmic scale (dB)

–2

379

Frequency (GHz)

Frequency (GHz)

(a)

(b)

Fig. 15.11 Amplitude of the scattered harmonics generated by the digital dynamic metasurface: (a) in case of up-converting scheme (blue bars) and (b) in case of down-converting scheme (red bars)

binary time modulated metasurface. However, it is worth noticing that to exactly match the temporal profile shown in Fig. 15.10b, c, an absorption state is required between two consecutive opposite pulses. To further simplify the implementation, we remove the possibility to have an absorption state, setting .τ˜ = 0.5. This leads to a time-varying reflection coefficient that can exhibit only “+1” and “-1” reflection states, which corresponds to the reflection phases “0” and “π ,” respectively, as shown in Fig. 15.4a, over a timeperiod Tp . Consequently, the presence of the hth -order harmonics with h = 3 k, k∈Z, is restored. The frequency spectra of the normally reflected fields are shown in Fig. 15.11. As expected, the ±1st -order harmonics are fully excited according to the used modulation scheme: the +first-harmonic is excited in case of upconverting (blueshift, Fig. 15.11a), whereas the -first-harmonic in case of downconverting (redshift, Fig. 15.11b). In both cases, the excited harmonic has an amplitude 18 dB higher than the 0th-order harmonic of the incident field. As expected, the ±3rd -order harmonics are present, but their amplitude are still much lower than the desired ±1st -order harmonic.

15.5 Closing Remarks In this chapter, we have offered an overview of some of the recent applications of reconfigurable metasurfaces, both as stand-alone devices for scattering manipulation and as add-on device for antenna systems. After a general introduction to the

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working principle of static and passive metasurfaces, we focused on the design of non-linear, waveform-dependent, frequency reconfigurable, and time-modulated devices. With the aid of practical examples, we have shown that the metasurfaces with reconfigurable properties can be one of the most suitable technology for conceiving innovative radiating systems with cognitive and adaptive capabilities according to the current operative condition perceived from the environment. Such a new design possibility represents a major step forward in the definition of future wireless communications equipped with dynamic reconfigurable characteristics, allowing to overcome some of the existing limitations in current telecommunication technology, such as the high latency, low speed of signal processing, and high jitter. Although the presence of electronic elements in the metasurface design inevitably introduced some challenges related to the increased fabrication complexity and possible stability issues, especially when compared to the maturity of passive metasurface design, we believe that the recent effort of the community in this framework is quickly boosting the technology, opening great prospective for a robust establishment of these solutions in next-generation wireless systems. Acknowledgments The authors wish to thank Dazhi Ding, Xinyu Fang, and Mengmeng Li for their precious support and the fruitful discussions on analog and digital dynamic metasurfaces for harmonic generation.

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Mirko Barbuto received the B.S., M.S. and Ph.D. degrees from Roma Tre University, Rome, Italy, in 2008, 2010 and 2015, respectively. Since September 2013, he is with Niccolò Cusano University, Rome, Italy, where he currently serves as an Associate Professor of electromagnetic field theory, as the Director of the Applied Electromagnetic Laboratory, as the Coordinator of the Bachelor’s degree in Electronic and Computer Engineering, and as a member of the Doctoral Board in Industrial and Civil Engineering. His main research interests are in the framework of applied electromagnetics, with an emphasis on innovative antennas and components at RF and microwaves enabled by metamaterials, metasurfaces, or topological properties of vortex fields. He serves as Associate Editor for IEEE AWPL (since 2019) and, for this role, he has been awarded for exceptional performances in 2021 and 2022. He is a member of the Editorial Board of the Radioengineering Journal (since 2019), of the Technical Program Committee of the International Congress on Artificial Materials for Novel Wave Phenomena (since 2017), and of secretarial office of the International Association METAMORPHOSE VI (the Virtual Institute for Artificial Electromagnetic Materials and Metamaterials). Currently, he is the author of more than 100 papers in international journals and conference proceedings.

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Alessio Monti was born in Rome, Italy, in 1987. He received the B.S. degree (summa cum laude) in electronic engineering, the M.S. degree (summa cum laude) in telecommunications engineering, and the Ph.D. degree in biomedical electronics, electromagnetics, and telecommunications engineering from ROMA TRE University, Rome, in 2008, 2010, and 2015, respectively. Since November 2021, he has been with Roma Tre University, where he serves as an Associate Professor of Electromagnetic Field Theory. His research activities resulted in 100+ papers published in international journals, conference proceedings, and book chapters. His research interests include varied theoretical and application-oriented aspects of metamaterials and metasurfaces at microwave and optical frequencies, the design of functionalized covers and invisibility devices for antennas and antenna arrays, and the electromagnetic modeling of micro- and nano-structured artificial surfaces. Dr. Monti was a recipient of several national and international awards and recognitions, including the URSI Young Scientist Award in 2019, the Outstanding Associate Editor of the IEEE Transactions on Antennas and Propagation in 2019, 2020, 2021 and 2022, the Finmeccanica Group Innovation Award for young people in 2015, and the 2nd Place at the Student Paper Competition of the Conference Metamaterials in 2012. Davide Ramaccia received the B.S. and M.S. (both summa cum laude) degrees in electronic and ICT engineering and the Ph.D. degree in electronic engineering from Roma Tre University, Rome, Italy, in 2007, 2009, and 2013, respectively. Since 2013, he has been with the Department of Industrial, Electronic, and Mechanical Engineering (2021), at Roma Tre University. His main research interests are in the modelling and design of (space)time-varying metamaterials and metasurfaces, and their applications to microwave components and antennas, and the analysis of anomalous scattering effects in temporal metamaterials. He has coauthored more than 100 articles in international journals, conference proceedings, book chapters, and holds one patent. Davide Ramaccia has been serving the scientific community, by playing roles in the management of scientific societies, in the editorial board of international journals, and in the organization of conferences and courses. He was the recipient of a number of awards and recognitions, including The Electromagnetics Academy Young Scientist Award (2019) seven Outstanding Reviewer Awards by the IEEE Transactions on Antennas and Propagation (2013–2021), the IET Best Poster Prize (2013) and IET Best Poster Award (2011).

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M. Barbuto et al. Stefano Vellucci received the B.S. and M.S. (summa cum laude) degrees in Electronic Engineering, and the Ph.D. degree in Applied Electronics from Roma Tre University, Rome, Italy, in 2012, 2015, and 2019, respectively. He is currently a Postdoctoral Research Fellow with the Department of Industrial, Electronic and Mechanical Engineering, at Roma Tre University. His current research interests include the design and applications of artificially engineered materials and metamaterials to RF and microwave components, non-linear and reconfigurable circuit-loaded metasurfaces for radiating structures, analysis and design of metasurface-based cloaking devices for the antennas. Dr. Vellucci was the recipient of some national and international awards, including the URSI Young Scientist Award (2022), the IEEE AP-S Award of the Central-Southern Italy Chapter (2019), the Leonardo-Finmeccanica Innovation Award for Young Students (2015), the Outstanding Reviewers Award assigned by the IEEE Transactions on Antennas and Propagation (20182019-2020-2021), and was finalist in the Telespazio Technology Contest (2021). He has been serving as a Technical Reviewer of many high-level international journals and conferences related to electromagnetic field theory, metamaterial, and metasurfaces. Alessandro Toscano graduated in Electronic Engineering from Sapienza University of Rome in 1988 and received his Ph.D. in 1993. Since 2011, he has been Full Professor of Electromagnetic Fields at the Engineering Department of Roma Tre University. He carries out an intense academic and scientific activity, both nationally and internationally. From April 2013 to January 2018, he was a member of Roma Tre University Academic Senate. From October 2016 to October 2018, he is a member of the National Commission which enables National Scientific Qualifications to Full and Associate Professors in the tender sector 09/F1 – Electromagnetic fields. Since 23rd January 2018 he has been Vice-Rector for Innovation and Technology Transfer. Alessandro Toscano’s scientific research has as ultimate objective the conceiving, designing, and manufacturing of innovative electromagnetic components with a high technological content that show enhanced performance compared to those obtained with traditional technologies and that respond to the need for environment and human health protection. He is the author of more than 100 publications in international journals indexed in ISI or Scopus; of these on a worldwide scale, three are in the top 0.1 percentile, five in the top 1 percentile and twentyfive in the top 5 percentile in terms of number of citations and journal quality.

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Filiberto Bilotti received the Laurea and Ph.D. degrees in electronic engineering from ROMA TRE University, Rome, Italy, in 1998 and 2002, respectively. Since 2002, he has been with the Faculty of Engineering (2002–2012) and, then, with the Department of Engineering (2013-now), at ROMA TRE University, where he serves as a Full Professor of electromagnetic field theory (2014) and the Director of the Antennas and Metamaterials Research Laboratory (2012). His main research contributions are in the analysis and design of microwave antennas and arrays, analytical modelling of artificial electromagnetic materials, metamaterials, and metasurfaces, including their applications at both microwave and optical frequencies. The research activities developed in the last 20 years (1999–2019) has resulted in more than 500 papers in international journals, conference proceedings, book chapters, and 3 patents. Prof. Bilotti was a founding member of the Virtual Institute for Artificial Electromagnetic Materials and Metamaterials – METAMORPHOSE VI in 2007, and was elected as a member of the Board of Directors of the same society for two terms (2007– 2013) and as the President for two terms (2013–2019). Currently, he serves the METAMORPHOSE VI as the Vice President and the Executive Director (2019-now). Prof. Bilotti was the recipient of a number of awards and recognitions, including the elevation to the IEEE Fellow grade for contributions to metamaterials for electromagnetic and antenna applications (2017). Outstanding Associate Editor of the IEEE Transactions on Antennas and Propagation (2016), NATO SET Panel Excellence Award (2016), Finmeccanica Group Innovation Prize (2014), Finmeccanica Corporate Innovation Prize (2014), IET Best Poster Paper Award (Metamaterials 2013 and Metamaterials 2011), and Raj Mittra Travel Grant Senior Researcher Award (2007).

Chapter 16

Specular Reflection and Transmission of Electromagnetic Waves by Disordered Metasurfaces Kevin Vynck, Armel Pitelet, Louis Bellando, and Philippe Lalanne

16.1 Introduction The phenomenon of wave scattering by random rough surfaces is encountered in many physical settings and techniques. These include radar remote sensing and imaging of planetary surfaces [1], optical characterization of surfaces [2] and LiDAR [3], underwater acoustics [4], wireless communications [5], seismology [6], to cite only a few. A typical problem of interest is a coherent wave impinging on a surface with random heterogeneities (e.g., random height variations, supported particles at random positions, etc.). The intensity of the reflected signal generally takes the form of a speckle, showing bright and dark spots due to constructive and destructive interferences between waves coming from different positions of the sample. As might be expected, the statistical properties of the measured signal contain statistical information on the surface topology. Considerable advances have been made in the past decades on establishing this relationship, as discussed in several excellent textbooks [7–10]. This period has also been marked by the exploration of novel strategies to control the emission, propagation, and confinement of light with nanostructures— a discipline nowadays known as “nanophotonics” [11, 12]. Research has been greatly stimulated by advances in nanofabrication techniques, which enabled the

K. Vynck () Université Claude Bernard Lyon 1, CNRS, iLM, Villeurbanne, France e-mail: [email protected] A. Pitelet · P. Lalanne Institut d’Optique Graduate School, CNRS, LP2N, Université de Bordeaux, Talence, France e-mail: [email protected] L. Bellando Université de Bordeaux, CNRS, LOMA, Talence, France © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 T. G. Mackay, A. Lakhtakia (eds.), Adventures in Contemporary Electromagnetic Theory, https://doi.org/10.1007/978-3-031-24617-3_16

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realization of a large variety of dielectric and metallic subwavelength structures with nanometer-scale resolution, in one, two, and three dimensions [13, 14]. One major outcome has been to unveil the potential of finely engineered high-index resonant nano-objects to exhibit exotic properties, such as a light bending effect [15], a strong scattering anisotropy [16], or a side-dependent coupling to waveguides [17]. By optimizing the composition, size, and shape of the nano-objects and placing them on a substrate, one creates a so-called metasurface that can deviate, focus, or alter the polarization state of an incident beam. The topic has become extremely popular and is covered by many review and expert opinion articles [18–21]. While in most situations, the nano-objects are ordered, i.e., arranged deterministically on a lattice, and the surfaces are realized by top-down fabrication techniques (e.g., lithography), increasing efforts are being made to realize metasurfaces by bottom-up techniques, such as colloidal chemistry and self-assembly. A major motivation is the expected benefit in terms of scalability and manufacturing cost. The resulting structures are usually disordered, even though some degree of control over structural correlations is possible [22]. Light interaction with disordered metasurfaces is receiving growing attention since a few years, with a focus on functionalities that do not require a precise control over the position of the inclusions, such as light absorption for photodetection, sensing and photovoltaics [23–26], light extraction for organic LEDs [27, 28], light scattering for augmented reality displays [29], and visual appearance design [30]. Disordered metasurfaces may thus be seen as a special type of random scattering surface, and naturally, one is led to the question of whether (and how well) previously established theoretical models can be used to describe their scattering properties. The present chapter aims at answering, at least in part, to this question. In modeling studies, the intensity scattered by a surface is generally decomposed into two components: a so-called coherent component, which corresponds to the specularly reflected and transmitted waves,1 and a so-called incoherent component, which corresponds to the diffuse (non-specular) intensity. Formally, as will be shown later in this chapter, the former is given by the ensemble-averaged electric field produced by the system, leading to the definition of complex reflection and transmission coefficients, whereas the latter is given by the fluctuations of the electric field around its average value, leading to the definition of an angle-resolved scattering diagram. The theoretical modeling of the diffuse intensity created by planar, disordered assemblies of resonant particles in layered media remains, to our knowledge, an open problem, as models are in general limited to particle monolayers embedded in a uniform background [31] or to particles much smaller than the wavelength [32]. As a first journey in the optics of disordered metasurfaces, we will focus here on their specular response. The problem of determining the reflection and transmission coefficients of particle monolayers is closely related to that of the electromagnetic homogenization. For

1 The term “specular” is used here independently for both reflected and transmitted waves, but one may also use the term “ballistic” for the transmitted waves.

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particles that are very small compared to the wavelength in the embedding medium (i.e., in the quasi-static regime), the system may be seen as a homogeneous layer with an effective permittivity tensor,2 which may be obtained from the Maxwell Garnett mixing rule [32], the Yamaguchi model [33], or its extensions [34, 35]. An alternative approach, still applicable to monolayers of very small particles only, proposed by Bedeaux and Vlieger [36, 37] and implemented in the software GranFilm [38], relies on the notions of excess fields and surface susceptibilities to solve the electromagnetic problem. However, for larger particles possibly exhibiting high-order resonances, one needs to resort to the electromagnetic scattering theory by discrete media [39], an approach that has notably been followed by GarcíaValenzuela and colleagues [40, 41]. In light of the increasing attention given to disordered metasurfaces in recent years, we propose in this chapter to give some keys to understand the physical origin, underlying approximations, and range of validity of known analytical expressions for the reflection and transmission coefficients of particle monolayers in layered media. The chapter is decomposed as follows. In Sect. 16.2, starting from the simple case of an individual, finite-size heterogeneity in a uniform medium (see Fig. 16.1a), we introduce the basic theoretical concepts in electromagnetic scattering, namely the wave propagation equation and the transition operator, the dyadic Green function and its angular spectrum representation, and the scattering amplitude. We reach an expression of the field scattered by a particle in a planewave

Fig. 16.1 Illustration of the different problems of interest in this chapter and definition of certain variables. A discrete medium composed of particles is illuminated by a planewave .Eb (r) = Eb eˆ i exp[iki · r], where .Eb , .eˆ i , and .ki are the wave amplitude, unit polarization vector, and wavevector, respectively. (a) Electromagnetic scattering by an isolated particle in a uniform medium. In Sect. 16.2, we will derive an expression of the scattered field in a planewave basis. Here, .eˆ s and .ks are the unit polarization vector and wavevector of the scattered planewave. (b) Specular reflection and transmission by a planar, disordered assembly of identical particles in a uniform medium. Section 16.3 aims mainly at presenting two models for the complex reflection and transmission coefficients of particle monolayers in uniform media, .rcoh and .tcoh , respectively. (c) Specular reflection and transmission by a disordered metasurface made of identical particles on a layered substrate. We will explain, at the end of Sect. 16.3, how the models can be used to predict the specular reflection, described by the coefficient .rst , of particle monolayers on layered substrates. In Sect. 16.4, we will test those models using full-wave multiple-scattering computations

2 The

effective permittivity of a monolayer of particles is not a scalar in general.

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basis. In Sect. 16.3, we treat the problem of the coherent intensity scattered by a monolayer of identical particles in a layered background. After introducing the multiple-scattering equations and formally defining the coherent and incoherent intensities, we present two models for the specular reflection and transmission by (infinite) particle monolayers in a uniform medium (see Fig. 16.1b). We conclude this section with a brief explanation of how the results can be generalized to handle particle monolayers above a layered substrate (see Fig. 16.1c). In Sect. 16.4, finally, we test the validity of these two analytical models on different systems, made of either metallic or dielectric particles, on a bare semi-infinite or layered substrate, by comparing the analytical predictions with those from rigorous, fullwave computations using an in-house multiple-scattering code.

16.2 Basics of Electromagnetic Scattering by Particles 16.2.1 Wave Equations We consider a finite region of space filled with a non-magnetic material (relative permeability .μ(r) = 1), described by a relative permittivity tensor .(r), in a uniform host medium with relative scalar permittivity .b . The permittivity variation .δ(r) ≡ (r) − b I, with .I the unit tensor, defines a compact heterogeneity, which will be our particle later on, though no assumption on the particle composition, size, or shape is in fact necessary at this stage. We consider harmonic fields at frequency .ω with the −iωt convention and drop hereafter the explicit dependence in the permittivities, .e fields, etc., for simplicity. We start from the macroscopic Maxwell’s (curl) equations for the electric and magnetic fields, .E and .H, at angular frequency .ω with a current density source .J, as follows: ∇ × E(r) = iωμ0 H(r)

(16.1)

∇ × H(r) = −iω0 (r)E(r) + J(r).

(16.2)

.

and .

Taking the curl of Eq. (16.1) and inserting Eq. (16.2) in the resulting expression lead to a vector wave propagation equation for the electric field ∇ × ∇ × E(r) −

.

ω2 (r)E(r) = iωμ0 J(r), c2

(16.3)

where .c2 = (0 μ0 )−1 , c being the speed of electromagnetic waves in vacuum.

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In a scattering problem, it is convenient to decompose the field as the sum of a background field and a scattered field, .E = Eb + Es . The background field .Eb is the solution of the wave propagation equation with the source term but without the heterogeneity, ∇ × ∇ × Eb (r) − kb2 Eb (r) = iωμ0 J(r),

.

(16.4)

with .kb2 = k02 b and .k0 = ω/c. From Eqs. (16.3) and (16.4), one therefore reaches a wave equation for the scattered field only, ∇ × ∇ × Es (r) − kb2 Es (r) = k02 δ(r)E(r).

.

(16.5)

The scattered field in the background medium is thus generated by the total field in the heterogeneity volume, which is nothing but the polarization density .P(r)/0 = δ(r)E(r).

16.2.2 Lippmann–Schwinger Equation The electromagnetic problem described by Eq. (16.5) can conveniently be reformulated by introducing the notion of dyadic Green function [42]. The dyadic Green function .G(r, r ) of a medium describes the electric field produced at point .r by a radiating point electric dipole at point .r . While it can be defined for an arbitrary environment, we are interested here in the dyadic Green function in the background medium, .Gb , which is then the solution of   ∇ × ∇ × Gb (r, r ) − kb2 Gb (r, r ) = Iδ(r − r ).

.

(16.6)

Multiplying both sides by .k02 δ(r )E(r ), integrating over .r , and using Eq. (16.5) lead to  2 .Es (r) = k0 (16.7) Gb (r, r )δ(r )E(r )dr . Only the field in the heterogeneity volume contributes to the scattered field. Thus, ones reaches the so-called Lippmann–Schwinger equation for the total field:  E(r) = Eb (r) + k02

.

Gb (r, r )δ(r )E(r )dr .

(16.8)

Note that the total field at .r depends explicitly on the total field at all points .r in the heterogeneity.

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16.2.3 Transition Operator To reach more practical expressions, we can express the scattered field in Eq. (16.7) by successive iterations as a series of scattering events,  Es (r) = k02

.

Gb (r, r )δ(r )Eb (r )dr 

+

k04

Gb (r, r )δ(r )Gb (r , r )δ(r )Eb (r )dr dr

+ ...

(16.9)

The first integral term gives the scattered field due to one interaction within the heterogeneity, the second to two interactions within the heterogeneity, etc. The scattered field is now expressed in terms of the background field only. Eventually, all multiple-scattering orders can be incorporated into a dyadic tensor .T, hereafter called the T operator, such that  Es (r) =

.

Gb (r, r )T(r , r )Eb (r )dr dr

(16.10)

with    T(r, r ) = k02 δ(r) δ(r − r ) + Gb (r, r )T(r , r )dr .

.

(16.11)

Equation (16.10) can be physically interpreted as follows: a background field .Eb arriving a point .r in the heterogeneity induces, via the T operator .T, a dipole moment at position .r , which then radiates in the background medium toward point .r via the dyadic Green function .Gb . Note that .T is spatially non-local in general, as it contains all multiple-scattering events relating two points within the heterogeneity. It is a complicated quantity, which varies with frequency, but, once known, it allows predicting the scattered field for any background field. In the case of an individual, finite-size heterogeneity (i.e., a particle), it is convenient to expand the incident and scattered fields in a basis of vector spherical wave functions. The relation between the corresponding expansion coefficients is then formally established via a T matrix, which is nothing but the T operator expressed in a specific (numerically truncated) basis [43]. The T matrix can be obtained analytically using Mie theory for simple objects (e.g., particles with spherical symmetry) [44] and numerically by solving Maxwell’s equations otherwise [43]. For assemblies of particles, one can define a T matrix of the whole system, which can be expressed in terms of the T matrix of the individual particles. The computation can be achieved by solving the multiple-scattering problem using the T-matrix method [43].

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16.2.4 Dyadic Green Function in a Homogeneous Medium To proceed further, we need to define more properly the dyadic Green function above. We will skip the derivation, which can be found elsewhere [42, 45–47]. In translationally invariant and isotropic media, the dyadic Green function .Gb (r, r ) ≡ Gb (r − r ) in Cartesian coordinates is given by   ikb |r − r | − 1 exp[ikb |r − r |] I−u⊗u+ .Gb (r − r ) = PV (I − 3u ⊗ u) 4π |r − r | (kb |r − r |)2 



−

δ(r − r ) I, 3kb2

(16.12)

where .PV stands for principal value, .⊗ denotes the tensor product, .u = (r − r )/|r − r | is a unit vector, and .δ(.) is the Dirac delta function. The first term (PV) corresponds to the non-local part of the dyadic Green function, which contains evanescent and propagating components, and the second term to its singular part at the origin. In the far field, .kb |r − r |  1, only propagating waves remain and the expression reduces to Gb (r − r ) ∼

.

exp[ikb |r − r |] [I − u ⊗ u] . 4π |r − r |

(16.13)

Equation (16.13) shows that the field radiated by a compact heterogeneity behaves at large distances as a transverse, outgoing spherical wave, where the transverse nature is due to the last term between brackets. The representation in outgoing spherical waves is well suited to scattering problems in volumes, but when the problem involves planar geometries (e.g., particle monolayers, interfaces), it is more convenient to express the waves as a linear combination of planewaves. This is the so-called angular spectrum representation, a.k.a. Weyl expansion or Weyl identity [48], and that we will now apply to the dyadic Green function. For this, let us start by expressing the dyadic Green function in real space in terms of its Fourier transform,

   1 1 k ⊗ k  .Gb (r − r ) = I− exp ik · (r − r ) dk. (16.14) 2 2 2 (2π )3 k − kb kb Writing the vectors in terms of their in-plane and normal components,  respectively,     as .r = r , z and .k = k , kz , and defining .k = |k | and .γ = kb2 − k2 with .Re[γ ] > 0 and .Im[γ ] > 0, Eq. (16.14) can then be rewritten as Gb (r − r ) =

.

1 (2π )2





gb (k , z, z ) exp ik · (r − r ) dk ,

(16.15)

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with 1 .gb (k , z, z ) = 2π 



  k⊗k 1 I− exp ikz (z − z ) dkz . (16.16) 2 2 2 kz − γ kb



The solution of the integral over .kz in Eq. (16.16) is evaluated taking care of the poles by applying Cauchy’s residue theorem to the contour integrals, leading to  gb. (k , z, z ) = PVzz

i 1 2γ

I−

κ± ⊗ κ± kb2



   δ(z − z )  δ zz , (16.17) exp iγ |z − z | − kb2

where .κ ± = [k , ±γ ] with .± = sign(z − z ), and .δ ij is the Kronecker delta tensor. Inserting this expression into Eq. (16.15) and keeping the non-singular part only (i.e., .z = z ), we obtain the following angular spectrum representation of the dyadic Green function:

 ± ⊗ κ±   i 1 κ  .Gb (r − r ) = exp iκ ± · (r − r ) dk .(16.18) I− 2 γ 2(2π )2 kb Note that the integration is performed over the entire (infinite) reciprocal plane  described by .k . Because .γ = kb2 − k2 , wavevectors fulfilling .k2 ≤ kb2 or .k2 > kb2 correspond to propagating or evanescent waves, respectively.

16.2.5 Scattering of a Planewave by a Particle To complete this section, we will now derive an expression for the scattered field Es produced by an individual particle in the angular spectrum representation. Let us first consider that the particle is centered at position .rj and is described by a T operator .Tj . The relative position of an individual particle in space does not affect its scattering properties, of course, but the relative position between particles in an assembly does. This step is therefore important for our purpose, as will be shown later on. Equation (16.10) may then be rewritten as

.

 Es (r) =

.

Gb (r, r )Tj (r − rj , r − rj )Eb (r )dr dr .

(16.19)

The T operator can be expressed in terms of its Fourier transform as Tj (r. − rj , r − rj ) =

1 (2π )6



  exp ip · (r − rj )

  × Tj (p , p ) exp −ip · (r − rj ) dp dp .

(16.20)

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Let us then consider that the background field .Eb is a planewave with amplitude Eb , wavevector .ki = kb kˆ i , and polarization .eˆ i ,

.

Eb (r) = Eb eˆ i exp [iki · r] .

.

(16.21)

Using Eqs. (16.20) and (16.18), we can rewrite Eq. (16.19) as

    iEb 1 κ± ⊗ κ± .Es (r) = Tj (p , p )ˆei exp iκ ± · (r − r ) I− 8 2 γ 2(2π ) kb        × exp ip · (r − rj ) exp −ip · (r − rj ) exp iki · r × dr dr dp dp dk .

(16.22)

 Having that . exp[iqr]dr = (2π )3 δ(q), the integrals over .r and .r , and then .p and   ± and .p = k . After simplification, we thus obtain .p , lead to .p = κ i

   1 iEb κ± ⊗ κ± exp iki · rj .Es (r) = Tj (κ ± , ki )ˆei I− γ 2(2π )2 kb2   × exp iκ ± · (r − rj ) dk . (16.23) Similarly to Eq. (16.15), we can define .Es (k , z, zj ) such that Es (r) =

.

1 (2π )2



  Es (k , z, zj ) exp ik · (r − rj, ) dk ,

(16.24)

leading to

iEb exp[iγ |z − zj |] κ± ⊗ κ± .Es (k , z, zj ) = Tj (κ ± , ki )ˆei exp[iki · rj ]. I− 2 γ kb2 (16.25) Note that the wavevector .k still covers the entire reciprocal space, so that both evanescent and propagating waves are considered. As we are more interested here in waves propagating in the far field, let us select a propagating wave with wavevector ˆ s and consider a plane above the particle, .z > zj . We therefore use .κ + = .ks = kb k [ks, , +ks,z ]. For waves propagating in the far field, we can further introduce the vector scattering amplitude, a classical tensor quantity in scattering theory [47],3

3 Important quantities, notably the extinction cross-section and differential scattering cross-section, can be calculated directly from the scattering amplitude.

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fj (kˆ s , kˆ i ) =

.

1 I − kˆ s ⊗ kˆ s Tj (kb kˆ s , kb kˆ i ). 4π

(16.26)

This leads to Es (ks, , z, zj ) = 2iπ Eb

.

exp[iks,z (z − zj )] ˆ ˆ fj (ks , ki )ˆei exp[iki · rj ], (16.27) ks,z

where .ks,z = kb cos(θi ), with .θi the angle of incidence with respect to the zaxis. Equation (16.27) formally expresses the planewave decomposition of the field scattered by a particle at .rj in the far field in terms of the particle scattering amplitude. Similar steps will later be made to derive expressions for the field scattered by a monolayer of particles and the resulting reflection and transmission coefficients.

16.3 Specular Reflection and Transmission by Particle Monolayers 16.3.1 Multiple Scattering by Discrete Media In the previous section, we considered the electromagnetic scattering problem for an individual particle, centered at point .rj , described by a T operator .Tj , and illuminated by a background field .Eb . In the present section, we will consider scattering by an ensemble of such particles. In the general case, each particle may have a different composition, size, and shape, and therefore a different T operator. Following Eq. (16.19), the field scattered by a set of N particles is now simply the sum of the fields scattered by each of them Es (r) =

N  

.

Gb (r, r )Tj (r − rj , r − rj )Eexc (r )dr dr . j

(16.28)

j =1 j

We define .Eexc as the field exciting the j -th particle, which, very importantly, can differ from the background field .Eb . Indeed, quite intuitively, this exciting field for the j -th particle should be the sum of the background field and the field scattered by all other particles, .l = j , as j .Eexc (r)

= Eb (r) +

N  

Gb (r, r )Tl (r − rl , r − rl )Elexc (r )dr dr . (16.29)

l=j

Equations (16.28) and (16.29) define the multiple-scattering problem. The excitj ing field .Eexc depends on the positions and properties of all other particles and is

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generally unknown. This set of equations, expressed in the basis of vector spherical wave functions, can be solved numerically on finite ensembles of particles using the T-matrix method [43], as mentioned above. Illuminating a given disordered configuration of particles typically leads to a specific speckle pattern.4 Here, we are interested, instead, in the statistical properties of the scattering response, such as the average field or intensity, intensity fluctuations, field–field or intensity– intensity correlations, etc. [47]. Theoretically, the problem is tackled by taking the configurational average of the relevant quantity (the field, intensity, etc.) from Eqs. (16.28) and (16.29) and making certain approximations on the interaction between particles.

16.3.2 Coherent and Incoherent Intensity The intensity scattered by a medium, which is the quantity that is usually measured experimentally, can formally be decomposed into two components, denoted as coherent and incoherent. To show this, let us write the scattered field as the sum of its average value and a fluctuating part, Es (r) = Es (r) + δEs (r),

.

with δEs (r) = 0.

(16.30)

The configurational average (formally defined below) is written here with angle brackets, . ·. Calculating the configurational average of the electric field norm  squared, . |E(r)|2 , which is directly proportional to the average intensity, shows that it can be decomposed into two terms

|E|2  = |Eb |2 + 2Re[Eb · E∗s ] + | Es |2 + |δEs |2 ,

.

= | E|2 + |δEs |2 .

(16.31)

The first term, .| E|2 = |Eb |2 +2Re[Eb · E∗s ]+| Es |2 , is known as the coherent intensity.5 In volume scattering, this component defines the extinction coefficient of the medium, which describes the attenuation rate of an incident wave due to scattering and absorption. In surface scattering, this leads to specularly reflected and transmitted waves defined by their reflection and transmission coefficients. The second term, . |δEs |2  = |E|2  − | E|2 = |Es |2  − | Es |2 , is known as the incoherent intensity. It corresponds to the diffuse intensity, created by field fluctuations from realization to realization and characterized by a differential scattering cross-section. Note that the term “incoherent” may be misleading since 4 The study of wave scattering in “frozen” systems, which is then a deterministic process, is very relevant in the framework of wavefront shaping techniques [49], for instance. 5 .| E|2 contains an interference term between the background and scattered fields, hence the term “coherent.”

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the diffuse intensity can be impacted by interference between scattered waves, as in the case of correlated disorder [50]. As stated previously, we are interested here in the specular reflection and transmission of a monolayer of particles and will therefore focus on theory for the average field. For further details on the theory for the intensity, we recommend the textbooks from Tsang and Kong [39] and from Carminati and Schotland [47].

16.3.3 Average Scattered Field We start by defining the configurational average for discrete media. Our main ingredient here is the N -dimensional probability density function .p(R) of finding N particles in the configuration .R = [r1 , r2 , · · · , rN ]. The configurational average of a variable .f (r) is defined as 

f (r) =

.

f (r, R)p(R)dR,

(16.32)

with .f (r, R) the variable evaluated at .r for a specific configuration .R. Let us then proceed by calculating the configurational average of the scattered field, Eq. (16.28), considering a set of N identical particles described by a unique T operator, .Tj (r − rj , r − rj ) ≡ T(r − rj , r − rj ). We thus have

Es (r) =

N  

.

  j Gb (r, r ) T(r − rj , r − rj )Eexc (r ) dr dr , (16.33)

j =1

with  .

  j j T(r − rj , r − rj )Eexc (r ) = T(r − rj , r − rj )Eexc (r , R)p(R)dR, (16.34)

following Eq. (16.32). Using .p(R) = p(R|rj )p(rj ) with .p(R|rj ), the conditional probability density function for .R having .rj fixed, and .dR = dr1 dr2 · · · drN , we obtain    j j    . T(r − rj , r − rj )Eexc (r ) = T(r − rj , r − rj ) Eexc j (r , rj )p(rj )drj , (16.35) j

j

where . Eexc j is the average exciting field .Eexc having the particle j fixed at .rj , given by

16 Specular Reflection and Transmission of Electromagnetic Waves by. . .

Eexc j (r , rj ) =

.

j



 ···

401

Eexc (r , R)p(R|rj )dr1 · · · drj −1 drj +1 · · · drN . j

(16.36) Inserting Eq. (16.35) into Eq. (16.33), we finally obtain

Es (r) =

N  

.

Gb (r, r )T(r − rj , r − rj ) Eexc j (r , rj )p(rj )drj dr dr . j

j =1

(16.37) j

The difficulty at this stage is to express . Eexc j in a suitable manner as to solve j Eq. (16.37). Without any approximation, . Eexc j can be expressed, in a similar manner as done above using Eq. (16.29), in terms of the average exciting field with two particles j and l fixed, and so on. Different levels of approximations are obtained depending on the order at which the truncation is made [39]. Hence, the lowest-order approximation, known as independent scattering approximation (ISA), completely j neglects the interaction between particles, assuming then . Eexc j (r , rj ) Eb (r ). Instead, the so-called effective field (or Foldy’s) approximation (EFA) considers that j   . Eexc j (r , rj ) E(r ), whereas the so-called quasi-crystalline approximation j j (QCA) assumes that . Eexc j l (r , rj , rl ) Eexc j (r , rj ). We will treat below the ISA and EFA models. The QCA was used by García-Valenzuela et al. [41] and constitutes, to our knowledge, the state-of-the-art on multiple-scattering models for the coherent intensity from particle monolayers.

16.3.4 Independent Scattering Approximation (ISA) The ISA assumes, as the name indicates, that the particles behave independently from each other; an electromagnetic wave interacts only once with each particle, and the mutual interactions between particles are therefore neglected. The exciting field (see Eq. (16.29)) is then given by the background field only. Evidently, this approximation should only be valid for very dilute systems. Assuming then

Eexc j (r , rj ) Eb (r ),

.

j

(16.38)

Eq. (16.37) becomes

Es (r) =

N  

.

Gb (r, r )T(r − rj , r − rj )Eb (r )p(rj )drj dr dr . (16.39)

j =1

 and (16.21), and having . exp[i(p − κ ± ) · r ]dr = Using Eqs. (16.18), (16.20),  (2π )3 δ(p − κ ± ) and . exp[i(ki − p ) · r ]dr = (2π )3 δ(ki − p ), leads to

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iEb . Es (r) = 2(2π )2

1 κ± ⊗ κ± T(κ ± , ki ) exp[iκ ± · r]ˆei I− γ kb2 ⎧ ⎫ N  ⎨ ⎬ × exp[i(ki − κ ± ) · rj ]p(rj )drj dk . (16.40) ⎩ ⎭ 

j =1

Considering that the particles are randomly distributed on a surface of area S at z = 0 (for simplicity) leads to .p(rj ) = δ(zj )/S. In addition, since all particles obey the same statistics, the sum reduces to a prefactor N. Then, we consider the limit of an infinite surface, .limN,S→∞ (N/S) = ρ, with .ρ the particle surface density. In this limit, the term between curled brackets in Eq. (16.40) becomes

.

 .

lim

N,S→∞

N S



±



exp[i(ki − κ ) · rj ]δ(zj )drj

= (2π )2 ρδ(ki, − k ), (16.41)

thereby imposing the conservation of the parallel wavevector between the incident and scattered waves. Two solutions appear, .κ ± = [ki, , ±ki,z ] ≡ kt/r , which correspond to the specularly reflected and transmitted planewaves. Having .ki,z = kb cos(θi ) and using Eq. (16.26) for the scattering amplitude, we reach .

lim Es (r) = Eb ρ

N,S→∞

2iπ f(kˆ t/r , kˆ i )ˆei exp [ikt/r · r] , . kb cos(θi )

≡ Es (kˆ t/r ) exp [ikt/r · r] .

(16.42) (16.43)

Note the similarity with Eq. (16.27) obtained for an individual particle, which makes sense considering the approximation of independent scattering made here. Indeed, independent scattered waves emerging from random (statistically uniform) positions on a surface add up coherently to form a planewave whose amplitude depends only on the scattering properties of the individual particle (i.e., the source of the scattered field) and the particle density. Having now the average scattered field as a planewave with parallel wavevector .ki, either in reflection or in transmission, we may define the complex reflection and transmission coefficients, .rcoh and .tcoh , of the particle monolayer, as   rcoh Eb = eˆ r · Es (kˆ r ) ,

(16.44)

  tcoh Eb = Eb + eˆ t · Es (kˆ t ) ,

(16.45)

.

and .

with .eˆ r and .eˆ t the polarization vectors of the specularly reflected and transmitted planewaves, respectively. Noting that .kt = ki , we obtain

16 Specular Reflection and Transmission of Electromagnetic Waves by. . .

rcoh = ρ

.

  2iπ eˆ r · f(kˆ r , kˆ i )ˆei kb cos(θi )

403

(16.46)

and tcoh = 1 + ρ

.

  2iπ eˆ t · f(kˆ i , kˆ i )ˆei . kb cos(θi )

(16.47)

For spherical particles, for which the scattering response depends on the scattering angle rather than the incident and scattered angles, and producing no polarization conversion (.eˆ t = eˆ i ), these expressions can further be simplified using the following definitions: .

f(kˆ i , kˆ i )ˆei ≡ f (0)ˆei , .

(16.48)

f(kˆ r , kˆ r )ˆer ≡ f (0)ˆer , .

(16.49)

f(kˆ r , kˆ i )ˆei ≡ fp (π − 2θi )ˆer ,

(16.50)

f(kˆ i , kˆ r )ˆer ≡ fp (π − 2θi )ˆei ,

(16.51)

and .

where .f (0) is the forward scattering amplitude and .fp (π − 2θi ) is the polarizationdependent scattering amplitude in the direction of specular reflection. Equations (16.46) and (16.47) thus become rcoh = ρ

.

2iπ fp (π − 2θi ) kb cos(θi )

(16.52)

and tcoh = 1 + ρ

.

2iπ f (0). kb cos(θi )

(16.53)

We have therefore derived the first set of equations for the specular reflection and transmission coefficients by particle monolayers, assuming here that the particles do not interact with each other. These expressions are often used to in theoretical studies on metasurfaces. Let us emphasize, however, that the coefficients diverge at grazing angles (.θi approaching .π/2). Intuitively, indeed, the interaction between particles should become more significant at oblique incidence, as the particles start to “shadow” each other. As we will show in Sect. 16.4, it can nevertheless be a good approximation for very dilute media (1% surface coverage).

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16.3.5 Effective Field Approximation (EFA) The EFA is a higher-order approximation than the ISA, in the sense that the average exciting field on particle j (with particle j fixed) is approximated not only by the background field but also the average scattered field, which partly accounts for the interaction between particles. After assuming

Eexc j (r , rj ) E(r ),

.

j

(16.54)

Eq. (16.37) reads

Es (r) =

N  

.

Gb (r, r )T(r − rj , r − rj ) E(r )p(rj )drj dr dr . (16.55)

j =1

Having . E = Eb + Es , this brings us to a similar problem as that with the Lippmann–Schwinger equation, Eq. (16.8), namely that the field at a point .r, depends on the field at another point .r . As previously, this can be solved by successive iterations, thereby expressing the problem as a series of scattering events between particles (not within a single heterogeneity as in the derivation of the T operator). A similar procedure as done above for the ISA can be followed and applied to the resulting infinite sum, leading to .

2iπ f(kˆ t/r , kˆ i )ˆei exp [ikt/r · r] kb cos(θi )  2 2iπ +Eb ρ f(kˆ t/r , kˆ t/r )f(kˆ t/r , kˆ i )ˆei exp [ikt/r · r] kb cos(θi )

lim Es (r) = Eb ρ

N,S→∞

+···

(16.56)

The first integral term of the sum involves one vector scattering amplitude and is identical to Eq. (16.42), the second integral term involves two vector scattering amplitudes (note the different input and output wavevector directions), etc. The sum takes the form of a geometric series, which, using Eqs. (16.44) and (16.45), leads to new expressions for the reflection and transmission coefficients 

rcoh

.

2iπ f(kˆ r , kˆ r ) = eˆ r · I − ρ kb cos(θi )

−1  ρ

 2iπ f(kˆ r , kˆ i )ˆei (16.57) kb cos(θi )

and  tcoh = eˆ t · I − ρ

.

2iπ f(kˆ i , kˆ i ) kb cos(θi )

−1

eˆ i ,

(16.58)

16 Specular Reflection and Transmission of Electromagnetic Waves by. . .

405

where .[·]−1 denotes the dyadic inverse. For spherical particles, we can use the equalities in Eqs. (16.48)–(16.51) to reach rcoh =

.

2iπ ρ kb cos(θ fp (π − 2θi ) i) 2iπ f (0) 1 − ρ kb cos(θ i)

(16.59)

and tcoh =

.

1 2iπ 1 − ρ kb cos(θ f (0) i)

.

(16.60)

We have thus reached a second set of equations for the reflection and transmission coefficients of particle monolayers, which considers the interaction between particles in a mean-field sense. Importantly, unlike in the ISA, the coefficients now behave correctly at grazing angles. Taking the limit .θi → π/2 leads to a reflection coefficient .rcoh that approaches .−1 (and thus, a reflectance .|rcoh |2 approaching 1) since .fp (π − 2θi ) → f (0), and a transmission coefficient .tcoh approaching 0. We will see in Sect. 16.4 that the EFA leads to physically sound and quantitatively accurate predictions for moderately dense systems (typically, 10% surface coverage) even at relatively large incident angles (.60◦ ). As a final remark, note that these equations have been derived heuristically in Ref. [40] by supposing that the exciting field is the average field transmitted through the monolayer. Instead, we show here that this result stems from a well-handled approximation that naturally appears in the multiple-scattering expansion.

16.3.6 Generalization to Particle Monolayers on Layered Substrates We now move to the last step of the derivation, which is to consider the impact of a layered environment on the specular reflection and transmission of a particle monolayer. A first possible approach would be to start from Maxwell’s equations and solve the electromagnetic scattering problem using the layered medium as a background. The dyadic Green function should then be the one of the layered geometries [51]. While this could be done for very small particles behaving as electric dipoles [32], in which case scattering by an individual particle is not described by a non-local T operator but by a local polarizability, it turns out to be much more challenging for large particles. A more practical approach is to consider that the interaction with the layered geometry is mediated by the average scattered field from the particle monolayer in the uniform background [41]. In other words, the particle monolayer is treated as planar interface with reflection and transmission coefficients given by .rcoh and .tcoh , and the reflection and transmission coefficients of the entire structure are

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determined by classical formulas of electromagnetic waves in layered media [52]. In addition to the approximations made at the level of the particle monolayer only, the mutual interaction between particles via the interfaces of the environment is therefore discarded in this approach, which may be problematic in certain situations, for instance, when incident planewaves couple efficiently to guided (photonic or plasmonic) modes in the layered medium. In the framework of the latter approach and using classical recursive relations for wave propagation in thin films [52], the reflection coefficient .rst of a particle monolayer above a layered substrate (made of isotropic materials, to disregard polarization conversion6 ) is simply given by rst = rcoh +

.

2 exp[2ik n a cos(θ )] rsub tcoh 0 b i , 1 − rcoh rsub exp[2ik0 nb a cos(θi )]

(16.61)

where .rcoh and .tcoh are the reflection and transmission coefficients of the particle monolayer—determined with any preferred model (ISA, EFA, QCA, ...), .rsub √ is the reflection coefficient of the substrate alone, .nb = b is the refractive index of the medium in which the particles are embedded, and a is the height separating the monolayer and the first substrate interface (a equals the particle radius for spherical particles). Importantly, .rsub already takes into account all multiple reflections in between the interfaces if several layers are considered. This quantity can be calculated using, for instance, the transfer matrix [55] or the scattering matrix method [56]. In the next section, we will test the ISA and EFA models for the specular reflection of particle monolayers on layered substrates in several experimentally relevant situations.

16.4 Numerical Validation of Theoretical Predictions To verify the validity of the ISA and EFA models for the problem of interest, we use an in-house multiple-scattering code developed by Jean-Paul Hugonin at Laboratoire Charles Fabry (Institut d’Optique Graduate School, CNRS, Université Paris Saclay). The numerical method belongs to the broad family of T-matrix methods [43], that is probably the most adapted to solve wave scattering problems by discrete media. T-matrix methods benefit from the high degree of analyticity of the functions describing the wave propagation between particles, typically dyadic Green functions, or vector spherical wave functions, depending on the problem of interest and the specific implementation. With the recent emergence of

6 Multilayered substrates with anisotropic materials could be considered as well, in which case the reflection and transmission coefficients should be written as .2 × 2 matrices for TE and TM (or s and p) polarizations. Several publicly available codes can be used for the purpose [53, 54].

16 Specular Reflection and Transmission of Electromagnetic Waves by. . .

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Fig. 16.2 Sketch of the supercell approach used to compute the specular reflectance from particle monolayers on layered substrates. Non-overlapping particles are distributed, using a random sequential addition algorithm, at random positions in a square of side . with periodic boundary conditions. The unwanted effects of this artificial periodicity on the scattering properties are expected to vanish as . increases

disordered media in photonics [50], the modeling of multiple scattering by particles in layered media has gained considerable attention, leading to the development of new methods [57] and powerful publicly available tools [58]. One of the major difficulties in numerical studies of multiple scattering in disordered media comes from the fact that the phenomena of interest may take place on mesoscopic scales, much larger than the wavelength, implying that care should be taken to avoid finite-size effects. This issue is often circumvented partially by simulating larger systems illuminated by (smaller) collimated beams, but this strategy generally loses reliability for studies at large incident angles. In addition to being capable of modeling large particle assemblies incorporated in an arbitrary layered medium, made of isotropic or anisotropic materials, the in-house code used here implements the supercell method, which is an artificial periodization of the electromagnetic problem, wherein each supercell can contain a large number of particles (see Fig. 16.2) to cope with finite-size effects. The supercell method has been widely used in photonics for the study of photonic crystal cavities and waveguides [59] as well as more complex, aperiodic (e.g., quasiperiodic) photonic structures [60]. Technical details on the implementation used here can be found in Ref. [61]. The numerical code has been used successfully in several recent studies [27, 30, 57, 62]. To test the validity of the ISA and EFA models, we propose here to compute the specular reflectance spectra of several systems under planewave illumination using the supercell approach. The artificial periodicity translates the scattering problem as a diffraction problem where the radiated power is distributed in several diffraction orders, with the 0-th order corresponding to the specular component and all the others to the diffuse component. One expects that, upon increasing the size of the supercell at constant particle density, the power radiated in the 0-th order converges toward a stable value. Let us then start our numerical study by testing the convergence of the method. We consider two systems for this convergence test, monolayers of either 10-

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Fig. 16.3 Convergence of the supercell method for the specular reflectance with increasing particle number. The convergence study is done in the same conditions as for the spectra calculations (incident angles and polarizations, the number of disorder configurations). One expects to converge to stable values as the particle number (or equivalently at fixed density, the system size) increases. The average specular reflectance (blue squares) is computed from a set of 10 independent disorder configurations and the error bars correspond to the standard deviation. The black dashed line serves as a guide to the eye. The T matrix of the particles is truncated at the quadrupole order. (Left) For a monolayer of spherical Au nanoparticles (NPs) of radius .a = 10 nm on a SiO.2 substrate at a surface coverage .f = 0.10 and a wavelength .λ = 500 nm .(nAu = 0.9698 + 1.8568i). (Right) For a monolayer of spherical Si NPs of radius .a = 50 nm on a SiO.2 substrate at a surface coverage .f = 0.10 and at a wavelength .λ = 470 nm .(nSi = 4.4910 + 0.0643i)

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nm-radius spherical gold (Au) particles or 50-nm-radius spherical silicon (Si) particles, deposited at a surface coverage of 10% (.f = 0.10) on a semi-infinite silica (SiO.2 ) substrate. The particle monolayers are illuminated by a planewave at ◦ ◦ .λ = 500 and 470 nm, respectively, and at either .θi = 0 or .θi = 60 in either TE or TM polarization. The simulations are repeated on 10 independent disorder configurations for systems up to .N = 100 particles, and the average and standard deviation of the specular reflectance are computed. The results given in Fig. 16.3 show some stabilization around a fixed value (the black dashed lines serve as guides to the eye), although some oscillations and fluctuations remain. Predictions appear reasonably accurate for .N ≤ 50. We thus proceed fixing .N = 50 for all systems and compute the specular reflectance spectra in the visible range for different systems, composed of spherical Au or Si particles at different surface coverages on different substrates. The results are given in Figs. 16.4, 16.5, 16.6, and 16.7. The numerical predictions are compared with those from the ISA model (Eqs. (16.52), (16.53), and (16.61)) and the EFA model (Eqs. (16.59), (16.60), and (16.61)). For the gold particles at low surface coverage (.f = 0.01), as presented in Fig. 16.4, all model curves superimpose with the numerical data, showing clearly that the assumption of independent scattering is fully justified here. The peak observed in the spectrum corresponds to the plasmon resonance of individual particles. The situation is different for the same system at higher surface coverage (.f = 0.10), as presented in Fig. 16.5, where the agreement between the models and the numerics is only moderately satisfactory. Remarkably, the EFA does not yield significantly better predictions compared to the ISA, suggesting that the mutual interaction between particles is more intricate than assumed. A reason may be the strong near-field interaction, accompanied by the formation of “hot spots,” which are expected in dense heterogeneous metallic nanostructures. The spectra of high-index Si particles at the same surface coverage (.f = 0.10), as presented in Fig. 16.6, exhibit sharp spectral resonances due to the Mie resonances of the individual particles. The strengths of these resonances are strongly overestimated by the ISA model, especially at grazing angles, where values up to about 4 times the numerical value are reached. This is likely being due to the unphysical divergence of the coefficients at large angles in the ISA. By comparison, the EFA model performs very well: despite some inaccurate predictions near resonance wavelengths, the spectral features are overall very well reproduced. Even more impressive are the results presented in Fig. 16.7 for the same particle monolayer on a layered substrate composed of a 500-nm-thick SiO.2 intermediate layer on top of a semi-infinite Si substrate. The multiple reflections between the monolayer and the various interfaces lead to strong spectral variations, which are well captured by the EFA, unlike the ISA. All in all, our full-wave computations show that, whereas the use of the ISA should be restricted to very dilute systems, the EFA model can be quantitatively accurate for monolayers of dielectric particles with surface coverages of about 10% even at large angles of incidence.

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Fig. 16.4 Specular reflectance spectra of a monolayer of 10-nm-radius gold (Au) nanoparticles (NPs) deposited at a surface coverage (or filling fraction) .f = 0.01 on top of a SiO.2 substrate. The three panels correspond to three different incident angles and polarizations: .0◦ (polarization independent), .60◦ in TE polarization and in TM polarization. The numerical predictions obtained by full-wave multiple-scattering computations (square markers, results averaged over 10 disorder configurations, error bar = standard deviation) are compared to the predictions from the ISA model (dotted–dashed line) and the EFA model (solid line)

16.5 Concluding Remarks To summarize, we have derived, step by step and from Maxwell’s equations, analytical expressions for the specular reflection and transmission coefficients from disordered monolayers of identical particles in layered media under the ISA and EFA models. The expressions are straightforward to use and very intuitive, in the sense that the specular intensity is primarily driven by the scattering properties of the individual particles (described via the scattering amplitude) and the particle density, in addition to the reflection and transmission properties of the layered substrate, if any.

Fig. 16.5 Same as Fig. 16.4 for a higher surface coverage .f = 0.10

Fig. 16.6 Same as Fig. 16.5 for 50-nm-radius silicon (Si) particles. The particles exhibit strong Mie resonances in the blue part of the spectrum

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Fig. 16.7 Same as Fig. 16.6 for a layered substrate composed of a 500-nm-thick SiO.2 layer on top of a semi-infinite Si substrate

Our full-wave numerical simulations allowed us to test these two models in few practical cases. This helps us identifying some limitations of the models, though we cannot be quantitative on their validity range (e.g., up to which filling fraction or particle density the models are accurate). The ISA model, which neglects the electromagnetic interaction between particles, should be restricted to very dilute systems (on the order of 1% and below) and near normal incidence only, due to the unphysical divergence of the coefficients at grazing angles, independently of the polarization. The EFA model, in which the effect of neighboring particles on scattering is treated in the mean-field sense, is expected to yield better and physically sound predictions for denser systems for all incident angles. Our numerical simulations show in particular that the EFA model performs very well for resonant dielectric particles at surface coverage around 10%, even when on top of a layered substrate.

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Acknowledgments We are grateful to Jean-Paul Hugonin (Laboratoire Charles Fabry, Palaiseau, France) for providing the multiple-scattering code used in this work. This work has received financial support from the PSA group and from the French National Agency for Research (ANR) under the projects “NanoMiX” (ANR-16-CE30-0008) and “NANO-APPEARANCE” (ANR-19CE09-0014).

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41. García-Valenzuela, A., Gutiérrez-Reyes, E., Barrera, R.G.: Multiple-scattering model for the coherent reflection and transmission of light from a disordered monolayer of particles. J. Opt. Soc. Am. A 29, 1161–1179 (2012) 42. Tai, C.T.: Dyadic Green Functions in Electromagnetic Theory. IEEE Press, New York (1994) 43. Mishchenko, M.I., Travis, L.D., Mackowski, D.W.: T-matrix computations of light scattering by nonspherical particles: a review. J. Quant. Spectrosc. Radiat. Transf. 55, 535–575 (1996) 44. Bohren, C.F., Huffman, D.R.: Absorption and Scattering of Light by Small Particles. Wiley, New York (2008) 45. Jackson, J.D.: Classical Electrodynamics, 3rd edn. Wiley, New York (1999) 46. Novotny, L., Hecht, B.: Principles of Nano-Optics. Cambridge University Press, Cambridge (2012) 47. Carminati, R., Schotland, J.C.: Principles of Scattering and Transport of Light. Cambridge University Press, Cambridge (2022) 48. Weyl, H.: Ausbreitung elektromagnetischer wellen über einem ebenen leiter. Ann. Phys. 365, 481–500 (1919) 49. Gigan, S., Katz, O., de Aguiar, H.B., Andresen, E., Aubry, A., Bertolotti, J., Bossy, E., Bouchet, D., Brake, J., Brasselet, S., et al.: Roadmap on wavefront shaping and deep imaging in complex media. J. Phys.: Photonics 4, 042501 (2022) 50. Vynck, K., Pierrat, R., Carminati, R., Froufe-Pérez, L.S., Scheffold, F., Sapienza, R., Vignolini, S., Sáenz, J.J.: Light in Correlated Disordered Media. arXiv:2106.13892 (2021) 51. Paulus, M., Gay-Balmaz, P., Martin, O.J.: Accurate and efficient computation of the Green’s tensor for stratified media. Phys. Rev. E 62, 5797–5807 (2000) 52. Yeh, P.: Optical Waves in Layered Media, 2nd edn. Wiley, New York (2005) 53. Hugonin, J.P., Lalanne, P.: RETICOLO software for grating analysis. arXiv:2101.00901 (2021) 54. Bay, M.M., Vignolini, S., Vynck, K.: PyLlama: A stable and versatile Python toolkit for the electromagnetic modelling of multilayered anisotropic media. Comput. Phys. Commun. 273, 108256 (2022) 55. Mackay, T.G., Lakhtakia, A.: The Transfer-Matrix Method in Electromagnetics and Optics. Morgan and Claypool, San Rafael (2020) 56. Ko, D.Y.K., Sambles, J.: Scattering matrix method for propagation of radiation in stratified media: attenuated total reflection studies of liquid crystals. J. Opt. Soc. Am. A 5, 1863–1866 (1988) 57. Bertrand, M., Devilez, A., Hugonin, J.P., Lalanne, P., Vynck, K.: Global polarizability matrix method for efficient modeling of light scattering by dense ensembles of non-spherical particles in stratified media. J. Opt. Soc. Am. A 37, 70–83 (2020) 58. Egel, A., Czajkowski, K.M., Theobald, D., Ladutenko, K., Kuznetsov, A.S., Pattelli, L.: SMUTHI: A python package for the simulation of light scattering by multiple particles near or between planar interfaces. J. Quant. Spectrosc. Radiat. Transf. 273, 107846 (2021) 59. Joannopoulos, J.D., Johnson, S.D., Winn, J.N., Meade, R.D.: Photonic Crystals: Molding the Flow of Light, 2nd edn. Cambridge University Press, Cambridge (2008) 60. Dal Negro, L.: Optics of aperiodic structures: fundamentals and device applications. CRC Press, New York (2013) 61. Langlais, M., Hugonin, J.P., Besbes, M., Ben-Abdallah, P.: Cooperative electromagnetic interactions between nanoparticles for solar energy harvesting. Opt. Express 22, A577–A588 (2014) 62. Blanchard, C., Hugonin, J.P., Nzie, A., Meneses, D.D.S.: Multipolar scattering of subwavelength interacting particles: Extraction of effective properties between transverse and longitudinal optical modes. Phys. Rev. B 102, 064209 (2020)

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K. Vynck et al. Kevin Vynck is a CNRS Researcher at the Institute of Light and Matter (iLM) in Lyon (France), specialized in the theoretical and numerical modeling of light scattering by complex nanostructures. He received his PhD from the University of Montpellier (France) in November 2008 and was a post-doctoral fellow at LENS in Florence (Italy) and at Institut Langevin in Paris (France). Between 2013 and 2021, he was a CNRS researcher at the Laboratory for Photonics Numerics and Nanosciences (LP2N) at the Institut d’Optique in Bordeaux. With his colleagues, he was among the first to propose using resonant silicon nanostructures for metamaterial applications, to exploit correlated disorder in planar photonic structures for light trapping in thin films, and to investigate the potential of disordered metasurfaces for visual appearance design. In 2019, he was awarded the CNRS Bronze Medal. Armel Pitelet obtained his PhD in 2018 at Université ClermontAuvergne after working on theoretical and computational nanophotonics. He joined the “Light in Complex Nanostructures” group at LP2N in 2020 as a post-doctoral researcher to work on numerical simulation of disordered metasurfaces. He is now a software engineer at the IT company Apside Clermont-Ferrand.

Louis Bellando received his PhD in 2013 from the University of Nice Sophia Antipolis under the supervision of Robin Kaiser. During his PhD, he studied both theoretically and experimentally light localization and cooperative effects in cold atomic clouds. He was then a post-doctoral researcher in the “Light in Complex Nanostructures” group led by Philippe Lalanne at LP2N to study coherent light scattering by disordered metasurfaces. In 2018, he entered the “NanoOptics” group at LOMA to study ultrafast Brownian motion with an optical tweezer in liquids and out-ofequilibrium rotational dynamics of an optically trapped nanoparticle in vacuum, in collaboration with Yann Louyer, Yacine Amarouchene, and Julien Burgin. More recently, he joined the “Singular” group led by Etienne Brasselet at LOMA to work on spin and orbital angular momentum transfer between light and a mechanical resonator. His main research interests are light–matter interaction, the study of coherent effects in the multiple-scattering regime, and optical manipulation of atomic vapor, micro-, and nanosized particles.

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Philippe Lalanne is a CNRS research scientist (Centre National de la Recherche Scientifique), working at Institut d’Optique d’Aquitaine. He received his PhD from Université Paris-Sud in 1989 under the supervision of Pierre Chavel. He is an expert in nanoscale electrodynamics. His current research is devoted to understanding how light interacts with subwavelength structures to demonstrate novel optical functionalities. He has developed new modal theories in nanophotonics and has pioneered the development of large-NA metalenses with high-index nanostructures in the 1990s. He has received several honors. He is currently the Director of GDR Ondes, a lab without walls that gathers the French community working on electromagnetic and acoustic waves. He is a Fellow of the Optical Society of America, SPIE, and the Institute of Physics.

Chapter 17

Continuity of Field Patterns for Exceptional Surface Waves and Exceptional Compound Waves Tom G. Mackay, Waleed Iqbal Waseer, and Akhlesh Lakhtakia

17.1 Introduction Surface waves propagate at the planar interface of two dissimilar mediums [1, 2]. A variety of different surface waves have been identified, depending upon the constitutive properties of the two partnering mediums. For example, surfaceplasmon-polariton (SPP) waves are guided by the planar interface of a dielectric medium and a plasmonic medium [3, 4], whereas Dyakonov surface waves are guided by the planar interface of two dielectric mediums, at least one of which is anisotropic [5–7]. Surface waves are classified as either unexceptional or exceptional [8], according to the mathematical nature of the localization of their electric and magnetic fields to the interface. In the case of unexceptional surface waves, the field localization in each partnering medium is governed by functions that decay T. G. Mackay () School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, Edinburgh, UK NanoMM—Nanoengineered Metamaterials Group, Department of Engineering Science and Mechanics, The Pennsylvania State University, University Park, PA, USA e-mail: [email protected] W. I. Waseer NanoMM—Nanoengineered Metamaterials Group, Department of Engineering Science and Mechanics, The Pennsylvania State University, University Park, PA, USA Department of Electronics, Quaid-i-Azam University Islamabad, Islamabad, Pakistan Department of Electrical and Computer Engineering, COMSATS University Islamabad, Islamabad, Pakistan A. Lakhtakia NanoMM—Nanoengineered Metamaterials Group, Department of Engineering Science and Mechanics, The Pennsylvania State University, University Park, PA, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 T. G. Mackay, A. Lakhtakia (eds.), Adventures in Contemporary Electromagnetic Theory, https://doi.org/10.1007/978-3-031-24617-3_17

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exponentially with distance from the interface. When both partnering mediums are isotropic, only surface waves that are unexceptional can propagate. But, when one of the two partnering mediums is anisotropic, then the possibility of exceptional surface-wave propagation arises, depending on the constitutive parameters of both partnering mediums. In the case of exceptional surface waves, the localization in the anisotropic partnering medium is governed by the product of a linear function and an exponential function of the distance from the interface. Furthermore, surface waves that are doubly exceptional can exist if both partnering mediums are anisotropic, in which case the field localization in both partnering mediums is governed by functions that decay as a linear–exponential product of the distance from the interface [9]. Exceptional surface waves can propagate only in one or more isolated directions in each quadrant of the interface plane [10], whereas unexceptional surface waves propagate in one or more sectors of directions in each quadrant of the interface plane. For certain partnering mediums, the sector of propagation directions for unexceptional surface waves can extend to the entire quadrant, while for other partnering mediums the sector may extend to just a few degrees (or even less than a degree [7]). In several studies, we have noted that the isolated propagation direction for an exceptional surface wave lies within a sector of propagation directions for the corresponding unexceptional surface waves. Given the differences between unexceptional and exceptional surface waves in terms of their mathematical representations [8, 9], a natural question to ask is: how do the spatial profiles of the electric and magnetic fields of an exceptional surface wave compare to those of the corresponding unexceptional surface wave? More specifically, suppose that an exceptional surface wave propagates at an angle ◦ ◦ .ψe ∈ [0 , 90 ] in the interface plane for a given pair of partnering mediums. How do the spatial profiles of the electric and magnetic fields of this exceptional surface wave compare to those of the corresponding unexceptional surface waves that propagate in the angular sector .[ψe − δ, ψe + δ], where .δ > 0◦ is a suitably small angular increment? This matter is explored in this chapter by numerically solving the corresponding canonical boundary-value problems, for both Dyakonov surface waves and SPP waves. Instead of the planar interface of two dissimilar mediums, suppose we consider a pair of parallel planar interfaces formed by a stack of three mediums with the middle medium being different from the top and bottom mediums. If the middle medium is sufficiently thick, then the upper and lower interfaces may independently guide surface waves [11–13]. However, if the middle medium is sufficiently thin, then mixing of fields at the two interfaces can give rise to the guided propagation of compound waves [14–16]. In an analogous manner to surface waves, compound waves can be classified as unexceptional or exceptional [17– 19], according to the mathematical nature of the spatial profiles of their electric and magnetic fields at the two interfaces, with exceptional compound waves being characterized by field localization governed by functions that decay as a linear– exponential product of the distance from an interface. And, as in the case of surface waves, a natural question to ask is: how do the spatial profiles of the electric

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and magnetic fields of an exceptional compound wave compare to those of the corresponding unexceptional compound waves? This matter is also explored in this chapter by numerically solving the corresponding canonical boundary-value problem for compound-plasmon-polariton (CPP) waves. In the remainder of this chapter, vectors are underlined, dyadics are double underlined, column vectors are underlined and enclosed in square brackets, matrixes are double underlined and enclosed in square brackets, the Cartesian unit vectors are denoted by .uˆ x , .uˆ y , and .uˆ z , the complex conjugate is signaled by the superscript ∗ . , the operators .Re {·} and .Im {·} deliver the real and imaginary parts of complex quantities, and the superscript .T denotes the transpose. The 3 .× 3 identity   dyadic is ˆ x uˆ x + uˆ y uˆ y + uˆ z uˆ z , while the 4 .× 4 identity matrix is written as . I . The free.I = u space permittivity, free-space permeability, and free-space wavelength are denoted √ as .ε0 , .μ0 , and .λ0 , respectively, and the free-space wavenumber is .k0 = ω ε0 μ0 with .ω being the angular frequency. An .exp(−iωt) dependency on time t is implicit throughout.

17.2

Unexceptional and Exceptional Surface Waves

17.2.1 Theory: Canonical Boundary-Value Problem 17.2.1.1

Preliminaries

Let us begin by considering surface waves guided by the planar interface of material A that occupies the half-space .z > 0 and material .B that occupies the half-space .z < 0. As described in greater detail elsewhere [20, 21], material .A is taken as the uniaxial dielectric material specified by the constitutive relations [22] .

D(r) = ε0  •E(r) A B(r) = μ0 H(r)

.

 ,

(17.1)

wherein the relative permittivity dyadic 

.

A

s t s = A I + (A − A ) uˆ x uˆ x .

(17.2)

Material .B is the isotropic dielectric material specified by the constitutive relations .

 D(r) = ε0 B E(r) . B(r) = μ0 H(r)

(17.3)

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Fig. 17.1 A schematic representation of the canonical boundary-value problem for surface wave propagation. Angle .ψ indicates the propagation direction parallel to the xy plane, relative to the x axis

For a surface wave propagating with wavenumber q, at an angle .ψ relative to the x axis in the xy plane, the electromagnetic field phasors for all .z ∈ {−∞, ∞} are written as     E(r) = ex (z) uˆ x + ey (z) uˆ y + ez (z) uˆ z exp iq uˆ prop •r   ,  . (17.4) H(r) = hx (z) uˆ x + hy (z) uˆ y + hz (z) uˆ z exp iq uˆ prop •r with the unit vector in the direction of propagation being uˆ prop = uˆ x cos ψ + uˆ y sin ψ .

.

(17.5)

A schematic diagram is provided in Fig. 17.1. We focus our attention on the spatial profiles of the electric and magnetic fields in the z direction. In view of Eqs. (17.1) and (17.3), the Maxwell curl postulates may be cast in the matrix ordinary differential equation (ODE) form [23–25] ⎧   ⎨ i P • [f(z)] , z > 0, d  A . [f(z)] = ⎩ i P • [f(z)] , z < 0, dz B

(17.6)

with the column 4-vector .

T  [f(z)] = ex (z), ey (z), hx (z), hy (z)

(17.7)

  and the 4 .× 4 matrix . P  depending upon q, .ψ, and the constitutive parameters of material . ∈ {A, B}.

17 Fields of Exceptional Surface and Compound Waves

17.2.1.2

423

Half-Space z > 0

The tangentially directed components of the electric and magnetic field phasors for z > 0 are governed by the 4 .× 4 matrix

.

.

  PA =

⎡

s − q 2 cos2 ψ ⎤ k02 A q 2 cos ψ sin ψ ⎢ ⎥ s s ωε0 A ωε0 A ⎢ ⎥ ⎢ ⎥ s + q 2 sin2 ψ 2 ⎢ ⎥ −k02 A cos ψ sin ψ q ⎢ ⎥ 0 0 − ⎢ ⎥ s s ωε0 A ωε0 A ⎢ ⎥, ⎢ ⎥ s 2 2 2 2 ⎢ q cos ψ sin ψ −k0 A + q cos ψ ⎥ ⎢− ⎥ 0 0 ⎢ ⎥ ωμ ωμ 0 0 ⎢ 2 t ⎥ 2 2 2 ⎣ k0  − q sin ψ ⎦ q cos ψ sin ψ A 0 0 ωμ0 ωμ0

0

0

(17.8) while the normally directed components of the same phasors are given explicitly as

.

hx (z) sin ψ − hy (z) cos ψ ez (z) = q s ωε0 A ey (z) cos ψ − ex (z) sin ψ hz (z) = q ωμ0

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

(17.9)

.

(a) Unexceptional Case   In the case of unexceptional surface waves, the matrix . P A has four distinct eigenvalues, namely .±αA1 and .±αA2 , with

.

 s αA1 = i q 2 − k02 A    s +  t ) − ( s −  t ) cos 2ψ − 2k 2  s  t q 2 (A 0 A A A A A αA2 = i s 2A

⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭

.

(17.10)

In the half-space .z > 0, surface waves must decay in the limit .z → ∞. Consequently,  the values of .αA1 and .αA2 must be selected such that .Im αA1 > 0  and .Im αA2 > 0. Furthermore, as only one independent eigenvector is associated with each eigenvalue, the eigenvectors associated with .αA1 and .αA2 are

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.

   T   2k02 As cot 2ψ k0 αA1 csc 2ψ −1 , vA1 = 0, 2 + 1− , η0 η0 η0 q sin ψ cos ψ q2  T   αA2 q 2 (cos 2ψ + 1) q 2 cos ψ sin ψ vA2 = 1 − , 0, ,− s s ωμ0 k02 A 2k02 A

⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭

,

(17.11)

respectively. Hence, the unexceptional surface wave solution to the matrix ODE (17.6).1 for .z > 0 is expressed as .

      [f(z)] = CA1 vA1 exp iαA1 z + CA2 vA2 exp iαA2 z .

(17.12)

By the imposition of boundary conditions at .z = 0, the constants .CA1 and .CA2 may be determined.

(b) Exceptional Case   In the case of exceptional surface waves, the matrix . P A has only two distinct eigenvalues, namely .±αA , with  s tan ψ. αA = iσ k0 A

.

(17.13)

  The value of .αA is selected such that .Im αA > 0, in order to conform to a surface-wave solution for .z > 0. The non-semisimple degeneracy of eigenvalues arises when  s k0 A . .q = σ (17.14) cos ψ Herein, .σ = +1 for .0◦ < ψ < 90◦ and .σ = −1 for .90◦ < ψ < 180◦ , the existence of an exceptional surface wave being impossible for .ψ ∈ {90◦ , 180◦ }. Only one independent eigenvector is associated with .αA , namely ⎡ ⎤T   iσ −1 ⎦ ⎣ . v ; A = 0,  s , 0, η0 A the corresponding generalized eigenvector [26] is obtained as

(17.15)

17 Fields of Exceptional Surface and Compound Waves



  1 . w A =k 0

2 tan ψ t − s , s A A A

 cot ψ − 2 2

 ⎤T s 2iσ A  , 0⎦ , t − s η0 A A

425 s −  t cot2 ψ A A s − t A A

 ,

(17.16)

    with . vA and . wA being related per .

    P A − αA I

•

    wA = vA .

(17.17)

Hence, the exceptional surface-wave solution to the matrix ODE (17.6).1 for .z > 0 is expressed as .

          [f(z)] = CA1 vA + CA2 iz vA + wA exp iαA z ,

(17.18)

with the constants .CA1 and .CA2 being determined via the boundary conditions at z = 0.

.

17.2.1.3

Half-Space z < 0

The tangentially directed phasor components for .z < 0 are governed by the 4 .× 4 matrix .

  = P B ⎡

⎤ k02 B − q 2 cos2 ψ q 2 cos ψ sin ψ 0 0 ⎢ ⎥ ωε0 B ωε0 B ⎢ ⎥ ⎢ 2 2 2 2 −k0 B + q sin ψ q cos ψ sin ψ ⎥ ⎢ ⎥ ⎢ ⎥ 0 0 − ⎢ ⎥ ωε  ωε  0 B 0 B ⎢ ⎥, 2 2 2 2 ⎢ q cos ψ sin ψ −k0 B + q cos ψ ⎥ ⎢− ⎥ 0 0 ⎢ ⎥ ωμ ωμ 0 0 ⎢ ⎥ 2 ⎣ k 2  − q 2 sin2 ψ ⎦ q cos ψ sin ψ 0 B 0 0 ωμ0 ωμ0

(17.19) while the normally directed phasor components are given explicitly as hx (z) sin ψ − hy (z) cos ψ ωε0 B ey (z) cos ψ − ex (z) sin ψ hz (z) = q ωμ0 ez (z) = q

.

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

.

(17.20)

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  The matrix . P B has two distinct eigenvalues, namely .±αB , with 

αB = −i q 2 − k02 B .

(17.21)

.

In the half-space .z < 0, surface waves must decay in the limit  .z → −∞. Consequently, the value of .αB must be selected such that .Im αB < 0. The following two independent eigenvectors are associated with .αB :

.

 T   αB q 2 cos2 ψ q 2 cos ψ sin ψ vB1 = 1 − , 0, ,− ωμ0 k02 B k02 B  T   αB q 2 cos ψ sin ψ q 2 sin2 ψ vB2 = ,0 − 1, , ωμ0 k02 B k02 B

⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭

.

(17.22)

Hence, the surface-wave solution to the matrix ODE (17.6).2 for .z < 0 is expressed as .

      [f(z)] = CB1 vB1 + CB2 vB2 exp iαB z ,

(17.23)

wherein the constants .CB1 and .CB2 may be determined by imposing boundary conditions at .z = 0.

17.2.1.4

Boundary Conditions

The tangential components of the electric and magnetic field phasors must be continuous at the .z = 0 interface, for both unexceptional and exceptional surface waves. That is, the boundary condition .

 +   −  f(0 ) = f(0 )

(17.24)

must be satisfied. By the imposition of this condition, Eqs. (17.12), (17.18), and (17.23) give rise to .

   T M · CA1 , CA2 , CB1 , CB2 = [0, 0, 0, 0]T .

(17.25)

  For surface-wave propagation, the 4 .× 4 matrix . M must be singular. Thus, we arrive at dispersion equation .

! ! ! ! ! M ! = 0.

(17.26)

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(a) Unexceptional Case In the case of unexceptional surface waves, the dispersion equation (17.26) is equivalent to   2 s s α −ε α εA k02 εA B B A1 αA2 − αB tan ψ . = 1,   s 2 αB αA2 − εB αA αA1 αB − αA1 εA 1

(17.27)

from which the wavenumber q can be extracted by numerical means. We infer from Eq. (17.27) that if an unexceptional surface wave propagates in the direction given by .ψ = ψ † , then unexceptional surface waves also propagate in the directions given by .ψ = −ψ † and .ψ = 180◦ ± ψ † .

(b) Exceptional Case In the case of exceptional surface waves, the dispersion equation (17.26) is equivalent to  s 2   s t s  + s A + A cot ψ − 2A B − A B A . = 1.    s  s s s +  2ψ +  −  cot  2 A B B A A A

(17.28)

There is no need to extract the wavenumber q for exceptional surface waves from the dispersion equation as it is already known from Eq. (17.14). The symmetries of Eq. (17.28) for exceptional surface waves are equivalent to those of Eq. (17.27) for unexceptional surface waves; that is, the existence of an exceptional surface wave propagating in the direction given by .ψ = ψ † implies the existence of exceptional surface waves propagating in the directions given by .ψ = −ψ † and .ψ = 180◦ ±ψ † .

17.2.2 Numerical Studies 17.2.2.1

Dyakonov Surface Waves

We begin our numerical studies of surface waves with the case in which materials .A and .B are both non-dissipative dielectric materials. In this case, the unexceptional surface waves are called Dyakonov surface waves and the exceptional surface waves are called Dyakonov–Voigt surface waves [20]. s = 2, . t = 6.5, and . The relative permittivity parameters .A B = 2.15 are A selected. In this case, the angular existence domain for Dyakonov surface waves (in the first quadrant of the interface plane) is .15.086◦ < ψ < 24.559◦ , while the propagation direction for the Dyakonov–Voigt surface wave is given by .ψ =

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ψe = 15.320◦ . Let us compare the spatial profiles of the electric and magnetic field phasors for the exceptional surface wave propagating in the direction of .ψe with those spatial profiles for the unexceptional surface waves propagating in directions close to .ψe , i.e., for .ψ ∈ {ψ1 , ψ2 , ψ3 }, where .ψ1 = 15.30◦ , .ψ2 = 15.33◦ , and .ψ3 = 15.34◦ , with .ψ1 < ψe < ψ2 < ψ3 . The magnitudes of the Cartesian components of the electric and magnetic field phasors are plotted against .z/λ0 in Fig. 17.2 for .ψ ∈ {ψ1 , ψe , ψ2 , ψ3 }. Also plotted are the Cartesian components of the time-averaged Poynting vector [27] 1    P(r) = Px uˆ x + Py uˆ y + Pz uˆ z = Re E(r) × H∗ (r) . 2

.

(17.29)

For these computations, .CB1 = 1 V m.−1 . In Fig. 17.2, a smooth transition is observed in the spatial profiles of the electric and magnetic field phasors, as well as in the spatial profiles of the time-averaged Poynting vector, as the propagation direction increases from .ψ1 , through .ψe , to .ψ3 . That is, there is no discontinuity apparent in the plotted spatial profiles as .ψ tends toward .ψe , from above or below. 17.2.2.2

Surface-Plasmon-Polariton Waves

Next, we consider the case involving the planar interface of a dielectric material and a plasmonic material. In this case, the unexceptional surface waves are called SPP waves and the exceptional surface waves are called SPP–Voigt surface waves [28]. We begin with the instance in which material .A is a plasmonic material specified s = −1 + 0.1i and . t = −2.85 + 0.015i, by the relative permittivity parameters .A A while material .B is a dielectric material specified by .B = 5. The angular existence domain for SPP waves extends over the entire interface plane, while the propagation angle for the SPP–Voigt surface wave (in the first quadrant of the interface plane) is ◦ .ψ = ψe = 40 . Here, we compare the spatial profiles of the electric and magnetic field phasors, and the time-averaged Poynting vector, for the exceptional surface wave propagating in the direction given by .ψe with those spatial profiles for the unexceptional surface waves propagating in directions close to .ψe , i.e., for .ψ ∈ {ψ1 , ψ2 , ψ3 }, where .ψ1 = 39◦ , .ψ2 = 41◦ , and .ψ3 = 42◦ , with .ψ1 < ψe < ψ2 < ψ3 . For these computations, .CB1 = 1 V m.−1 . Plots of .|Ex,y,z |, .|Hx,y,z |, and .Px,y,z versus .z/λ0 are provided in Fig. 17.3. The spatial profiles in Fig. 17.3 for SPP waves are quite different, qualitatively and quantitatively, from those presented in Fig. 17.2 for Dyakonov surface waves. However, as in Fig. 17.2, a smooth transition is observed in Fig. 17.3 in the spatial profiles of the electric and magnetic field phasors, as well as in the spatial profiles of the time-averaged Poynting vector, as the propagation direction increases from .ψ1 , through .ψe , to .ψ3 . That is, there is no discontinuity apparent in the plotted spatial profiles as .ψ tends toward .ψe , from above or below.

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Fig. 17.2 Spatial profiles of the magnitudes of the Cartesian components of the electric and magnetic field phasors and the time-averaged Poynting vector, for Dyakonov surface waves propagating in the directions given by .ψ = ψ1 = 15.30◦ , .ψ2 = 15.33◦ , and .ψ3 = 15.34◦ and for the Dyakonov–Voigt surface wave propagating in the direction given by .ψ = ψe = 15.320◦ (Profiles for .Pz are absent since .Pz is null valued)

Finally, turn to the instance in which material .A is a dissipative dielectric material s = 2 + i and . t = 8.93 + 0.94i, specified by the relative permittivity parameters .A A while material .B is a plasmonic material specified by the relative permittivity .B = −16.07 + 0.44i (which is the relative permittivity of silver at .λ0 = 633 nm [29]). The angular existence domain for SPP waves extends over the entire interface plane, while the propagation angle for the SPP–Voigt surface wave (in the first quadrant of the interface plane) is .ψ = ψe = 40◦ . As in Fig. 17.3, let us compare the spatial profiles of the electric and magnetic field phasors, and the time-averaged Poynting

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Fig. 17.3 Spatial profiles of the magnitudes of the Cartesian components of the electric and magnetic field phasors and the time-averaged Poynting vector, for SPP waves propagating in the directions of .ψ = ψ1 = 39◦ , .ψ2 = 41◦ , and .ψ3 = 42◦ and for the SPP–Voigt surface wave propagating in the direction given by .ψ = ψe = 40◦ . Material .A is an anisotropic plasmonic material, while material .B is an isotropic dielectric material

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vector, for the exceptional surface wave propagating in the direction of .ψe with those spatial profiles for the unexceptional surface waves propagating in directions prescribed by .ψ ∈ {ψ1 , ψ2 , ψ3 }, where .ψ1 = 35◦ , .ψ2 = 45◦ , and .ψ3 = 50◦ , with −1 .ψ1 < ψe < ψ2 < ψ3 . As for Figs. 17.2 and 17.3, .C B1 = 1 V m. . In Fig. 17.4, the quantities .|Ex,y,z |, .|Hx,y,z |, and .Px,y,z are plotted against .z/λ0 . The spatial profiles in Fig. 17.4 for SPP waves guided by the anisotropic dielectric/isotropic plasmonic material interface are quite different, qualitatively and quantitatively, from those presented in Fig. 17.3 for SPP waves guided by the isotropic dielectric/anisotropic plasmonic material interface. But, as in both Figs. 17.2 and 17.3, the spatial profiles in Fig. 17.4 of the electric and magnetic field phasors, as well as the spatial profiles of the time-averaged Poynting vector, all vary smoothly as the propagation direction increases from .ψ1 , through .ψe , to .ψ3 . Again, as .ψ tends toward .ψe , from above or below, there is no discontinuity apparent in the plotted spatial profiles.

17.3 Unexceptional and Exceptional Compound Waves 17.3.1 Theory: Canonical Boundary-Value Problem 17.3.1.1

Preliminaries

Now, let us turn to the canonical boundary-value problem for CPP waves guided by two planar interfaces [17]. Suppose that material .A occupies the half-spaces .z > D and .z < 0, while material .B occupies the bounded region .0 < z < D. As in Sect. 17.2.1, the constitutive relations for materials .A and .B are those in Eqs. (17.1) and (17.3), respectively. The electric and magnetic field phasors are given by Eqs. (17.4), but here the quantity q represents the CPP wavenumber. Schematically, the canonical boundary-value problem is represented in Fig. 17.5. As in Sect. 17.2.1, our attention is focused upon the spatial profiles of the electric and magnetic fields in the z direction. In view of Eqs. (17.1) and (17.3), the Maxwell curl postulates are cast in the matrix ODE form ⎧   ⎨ i P • [f(z)] , d  A . [f(z)] = ⎩ i P • [f(z)] , dz B

z>D

and

z < 0,

0 < z < D,

(17.30)

    wherein the 4 .× 4 matrixes . P A and . P B are as in Eqs. (17.8) and (17.19), respectively. The phasor components .ez (z) and .hz (z) are provided by Eqs. (17.9) for .z > D and .z < 0 and provided by Eqs. (17.20) for .0 < z < D.

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Fig. 17.4 As Fig. 17.3 but for the case in which material .A is an anisotropic dielectric material while material .B is an isotropic plasmonic material. Here, .ψ1 = 35◦ , .ψ2 = 45◦ , and .ψ3 = 50◦

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Fig. 17.5 A schematic representation of the canonical boundary-value problem for CPP-wave propagation. Angle .ψ indicates the propagation direction parallel to the xy plane, relative to the x axis

17.3.1.2

Half-Spaces z > D and z < 0

  The eigenanalysis of . P A is similar to that described in Sect. 17.2.1 for surface waves.

(a) Unexceptional Case The unexceptional CPP wave solution to the matrix ODE (17.30).1 for .z > D and z < 0 is expressed as

.

⎧ ↑   ↑ ↑ ↑ ⎪ ⎨ CA1 [ vA1 ] exp iαA1 (z − D) + CA 2 [ vA2 ]  .[ f(z) ] = z > D, × exp iαA2 (z − D) ,    ⎪ ⎩ C ↓ [ v↓ ] exp −iα z + C ↓ [v↓ ] exp −iα z , z < 0. A1 A2 A1 A1 A2 A 2 (17.31) Herein, the eigenvalues .αA1 and .αA2 are given in Eqs. (17.10); the eigenvectors ↑ ↑ [ vA1 ] and .[ vA1 ] are identical with those in Eqs. (17.11), while the eigenvectors

.

↓

↓

[ vA1 ] and .[vA1 ] are the same as those in Eqs. (17.11) but with the terms .αA1 and .αA2 therein replaced by .−αA1 and .−αA2 . By the imposition of boundary ↑ ↑ conditions at .z = 0, the constants .CA1 and .CA2 may be determined; likewise,

.

↓

↓

the constants .CA1 and .CA2 may be determined via the imposition of boundary conditions at .z = D.

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(b) Exceptional Case The exceptional CPP-wave solution to the matrix ODE (17.30).1 for .z > D and z < 0 is expressed as

.

" # ⎧ ↑ ↑ ↑ ↑ ↑ ⎪ ⎪ ⎨ CA1 [vA ] + k0 CA2 i(z − D) [vA] + [wA ]  .[ f(z) ] = × exp iαA , z > D,  # (z − D) " ⎪   ⎪ ⎩ C ↓ [v↓ ] + k0 C ↓ iz [v↓ ] + [w↓ ] exp −iα z , z < 0. A A1 A A2 A A (17.32) ↑

Herein, the eigenvalue .αA is given in Eq. (17.13); the eigenvector .[ vA ] and ↑ generalized eigenvector .[ wA ] are identical with those in Eqs. (17.15) and (17.16), ↓

↓

respectively, while the eigenvector .[vA ] and generalized eigenvector .[ wA ] are the same as those in Eqs. (17.15) and (17.16), respectively, but with the term .σ therein ↑ replaced by .−σ . Boundary conditions at .z = 0 and .z = D allow the constants .CA1 , ↑

↓

↓

CA2 , .CA1 , and .CA2 to be determined.

.

17.3.1.3

Bounded Region 0 < z < D

The CPP-wave solution to the matrix ODE (17.30).2 for .0 < z < D is expressed as # " [ f(z) ] = exp i[ P B ]z • [f(0+ ) ] .

.

(17.33)

Accordingly, the tangential components of the electric and magnetic field phasors at .z = 0+ and .z = D − are related as # " • − ]D .[ f(D ) ] = exp i[ P [f(0+ ) ] . (17.34) B 17.3.1.4

Boundary Conditions

The standard boundary conditions at the interfacial planes .z = 0 and .z = D are represented as .

 [f(0+ ) ] = [ f(0− ) ] , [f(D + ) ] = [f(D − ) ]

(17.35)

which when combined with Eq. (17.34) provide [2] # " [ f(D + ) ] = exp i[ P B ]D • [f(0− ) ] .

.

(17.36)

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For the case of unexceptional CPP waves, Eqs. (17.31) yield ↓

.

↓

↓

↓

[f(0− ) ] = CA1 [vA1 ] + CA2 [vA2 ] ↑

↑

↑

 ;

↑

[f(D + ) ] = CA1 [vA1 ] + CA2 [vA2 ]

(17.37)

consequently, Eq. (17.36) can be written as # " # " ↑ ↑ ↑ ↑ ↓ ↓ ↓ ↓ CA1 [vA1 ] + CA2 [vA2 ] = exp i[ P B ]D • CA1 [ vA1 ] + CA2 [ vA2 ] . (17.38) For the case of exceptional CPP waves, Eqs. (17.32) yield .

↓

.

↓

↓

↑

↑

↑



↓

[ f(0− ) ] = CA1 [ vA ] + k0 CA2 [wA ] ↑

[ f(D + ) ] = CA1 [vA ] + k0 CA2 [ wA ]

(17.39)

,

leading to # " # " ↓ ↓ ↓ ↓ ↑ ↑ ↑ ↑ CA1 [vA ] + k0 CA2 [wA ] = exp i[ P B ]D • CA1 [ vA ] + k0 CA2 [ wA ] . (17.40) Equations (17.38) and (17.40) can both be formulated as

.

 ↑ [ Y ]• CA1 ,

.

↑

CA 2 ,

↓

↓

CA1 ,

CA 2

T

= [ 0,

0,

0,

0 ]T .

(17.41)

  For CPP-wave propagation, the 4 .× 4 matrix . Y must be singular. Thus, we arrive at the dispersion equation .

! ! ! ! ! Y ! = 0.

(17.42)

Notice that the dispersion equation (17.42) is invariant under the transformations ψ → −ψ and .ψ → 180◦ ± ψ. Accordingly, the existence of a CPP wave propagating in the direction given by .ψ = ψ † implies the existence of CPP waves propagating in the directions given by .ψ = −ψ † and .ψ = 180◦ ± ψ † .

.

17.3.2 Numerical Studies In order to numerically explore the spatial profiles of field phasors for CPP waves, s = 1.5 + 0.5i, . t = we select the following relative permittivity parameters: .A A 3.5401 + 0.0842i, and .B = −16.07 + 0.44i, as in Ref. [17]. Notice that .B = −16.07 + 0.44i is the relative permittivity of silver at .λ0 = 633 nm. The thickness

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of material .B is set at .D = 60 nm; by way of comparison, the skin depth of silver at λ = 633 nm is 20.11 nm [30]. An exceptional CPP wave propagates in the direction given by .ψ = ψe = 30◦ . We compare the spatial profiles of the electric and magnetic field phasors and of the time-averaged Poynting vector, for the exceptional CPP wave propagating in the direction given by .ψe with those spatial profiles for the unexceptional CPP waves propagating in directions prescribed by .ψ ∈ {ψ1 , ψ2 , ψ3 }, where .ψ1 = 25◦ , .ψ2 = 35◦ , and .ψ3 = 40◦ , with .ψ1 < ψe < ψ2 < ψ3 . The normalization .|E(D uˆ z )• uˆ s | = 1 V m.−1 is set, wherein the unit vector

. 0

uˆ s = −uˆ x sin ψ + uˆ y cos ψ.

.

(17.43)

  With respect to the basis vectors . uˆ prop , uˆ s , uˆ z , we express the electric and magnetic field phasors, as well as the time-averaged Poynting vector, as A = Aprop uˆ prop + As uˆ s + Az uˆ z ,

.

A ∈ {E, H, P } .

(17.44)

The spatial profiles of .|Eprop,s,z |, .|Hprop,s,z |, and .Pprop,s,z are plotted against .z/λ0 in Fig. 17.6 for .ψ ∈ {ψ1 , ψe , ψ2 , ψ3 }. The plots of electric and magnetic field phasors in Fig. 17.6 are symmetric with respect to reflection about .z = D/2, due to the symmetry of the material .A/material .B/material .A configuration being studied. A smooth transition is observed in Fig. 17.6 in the spatial profiles of the electric and magnetic field phasors, as well as in the spatial profiles of the time-averaged Poynting vector, as .ψ increases from .ψ1 , through .ψe , to .ψ3 . That is, there is no discontinuity apparent in the plotted spatial profiles as .ψ tends toward .ψe , from above or below. In this respect, the spatial profiles in Fig. 17.6 are wholly consistent with those presented in Figs. 17.2, 17.3, and 17.4.

17.4 Discussion Exceptional surface waves were first described in 2019, firstly as Dyakonov– Voigt surface waves [20] and secondly as SPP–Voigt waves [28]. Subsequently, the notion encapsulated by exceptional surface waves was extended to exceptional compound waves [17]. A key distinguishing feature of exceptional surface waves and exceptional compound waves lies in their localization characteristics: the localization of exceptional surface and compound waves is governed by functions that decay in a linear–exponential product of distance from the interface, whereas the localization of unexceptional surface and compound waves is governed by functions that decay only in an exponential manner with distance from the interface. That being the case, the following question naturally arises: how do the spatial profiles of the electric and magnetic fields of an exceptional surface wave or

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Fig. 17.6 Spatial profiles of .|Eprop,s,z |, .|Hprop,s,z |, and .Pprop,s,z for unexceptional CPP waves propagating in the directions of .ψ = ψ1 = 25◦ , .ψ2 = 35◦ , and .ψ3 = 40◦ and for the exceptional CPP wave propagating in the direction of .ψ = ψe = 30◦ . The two vertical black lines represent the planar interfaces .z = 0 and .z = D

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exceptional compound wave compare to those of the corresponding unexceptional surface wave or unexceptional compound wave? We undertook a series of numerical studies based on the corresponding canonical boundary-value problems to answer this question. We found that, in the limit as the direction of unexceptional surface/compound wave propagation tends toward the direction of propagation for the corresponding exceptional surface/compound wave, there is a smooth variation in the electric field phasor and the magnetic field phasor, as well as in the time-averaged Poynting vector. Therefore, in terms of localization, an exceptional surface/compound wave is practically indistinguishable from a corresponding unexceptional surface/compound wave propagating along directions in the immediate neighborhood of the exceptional wave. Furthermore, at least in terms of localization and amplitude of field phasors, there is no singularity associated with the transition from an unexceptional surface/compound wave to its exceptional counterpart. Acknowledgments TGM was supported by EPSRC (grant number EP/V046322/1). WIW thanks the Higher Education Commission of Pakistan for an IRSIP Scholarship that enabled him to spend six months at the Department of Engineering Science and Mechanics at The Pennsylvania State University. AL was supported by the US National Science Foundation (grant number DMS1619901) as well as the Charles Godfrey Binder Endowment at Penn State.

References 1. Boardman, A.D. (ed.): Electromagnetic Surface Modes. Wiley, Chicester (1982) 2. Polo, Jr., J.A., Mackay, T.G., Lakhtakia, A.: Electromagnetic Surface Waves: A Modern Perspective. Elsevier, Waltham (2013) 3. Homola, J. (ed.): Surface Plasmon Resonance Based Sensors. Springer, Berlin (2006) 4. Maier, S.A.: Plasmonics: Fundamentals and Applications. Springer, New York (2007) 5. Marchevski˘ı, F.N., Strizhevski˘ı, V.L., Strizhevski˘ı, S.V.: Singular electromagnetic waves in bounded anisotropic media. Soviet Phys. Solid State 26, 911–912 (1984) 6. D’yakonov, M.I.: New type of electromagnetic wave propagating at an interface. Soviet Phys. JETP 67, 714–716 (1988) 7. Takayama, O., Crasovan, L.C., Johansen, S.K., Mihalache, D., Artigas, D., Torner, L.: Dyakonov surface waves: a review. Electromagnetics 28, 126–145 (2008) 8. Lakhtakia, A., Mackay, T.G., Zhou, C.: Electromagnetic surface waves at exceptional points. Eur. J. Phys. 42, 015302 (2020) 9. Lakhtakia, A., Mackay, T.G.: From unexceptional to doubly exceptional surface waves. J. Opt. Soc. Amer. B 37, 2444–2451 (2020) 10. Zhou, C., Mackay, T.G., Lakhtakia, A.: Singular existence of a Dyakonov–Voigt surface wave: Proof. Results Phys. 24, 104140 (2021) 11. Turbadar, T.: Complete absorption of light by thin metal films. Proc. Phys. Soc. 73, 40–44 (1959) 12. Turbadar, T.: Complete absorption of plane polarized light by thin metal films. Optica Acta 11, 207–210 (1964) 13. Chiadini, F., Fiumara, V., Scaglione, A., Lakhtakia, A.: Compound guided waves that mix characteristics of surface-plasmon-polariton, Tamm, Dyakonov–Tamm, and Uller–Zenneck waves. J. Opt. Soc. Am. B 33, 1197–1206 (2016) 14. Economou, E.N.: Surface plasmons in thin films. Phys. Rev. Lett. 182, 539–554 (1969)

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15. Yang, F., Sambles, J.R., Bradberry, G.W.: Long-range surface modes supported by thin films. Phys. Rev. B 44, 5855–5872 (1991) 16. Faryad, M., Lakhtakia, A.: Surface-plasmon-polariton wave propagation guided by a metal slab in a sculptured nematic thin film. J. Opt. (Bristol) 12, 085102 (2010) 17. Zhou, C., Mackay, T.G., Lakhtakia, A.: Exceptional compound plasmon-polariton waves guided by a metal film embedded in a uniaxial dielectric material. Opt. Commun. 483, 126628 (2021) 18. Lakhtakia, A., Zhou, C., Mackay, T.G.: Exceptional compound plasmon-polariton waves. OSA Continuum 4, 748–761 (2021) 19. Zhou, C., Mackay, T.G., Lakhtakia, A.: A multiplicity of exceptional compound plasmonpolariton waves. J. Modern Opt. 68, 284–294 (2021) 20. Mackay, T.G., Zhou, C., Lakhtakia, A.: Dyakonov–Voigt surface waves. Proc. R. Soc. Lond. A 475, 20190317 (2019) 21. Zhou, C., Mackay, T.G., Lakhtakia, A.: On Dyakonov–Voigt surface waves guided by the planar interface of dissipative materials. J. Opt. Soc. Am. B 36, 3218–3225 (2019) 22. Mackay, T.G., Lakhtakia, A.: Electromagnetic Anisotropy and Bianisotropy: A Field Guide, 2nd edn. World Scientific, Singapore (2019) 23. Keller, H.B., Keller, J.B.: Exponential-like solutions of systems of linear ordinary differential equations. J. Soc. Ind. Appl. Math. 10, 246–259 (1962) 24. Berreman, D.W.: Optics in stratified and anisotropic media: 4×4-matrix formulation. J. Opt. Soc. Am. 62, 502–510 (1972) 25. Mackay, T.G., Lakhtakia, A.: The Transfer-Matrix Method in Electromagnetics and Optics. Morgan and Claypool, San Rafael (2020) 26. Boyce, W.E., DiPrima, R.C.: Elementary Differential Equations and Boundary Value Problems, 9th Edn. Wiley, Hoboken (2010) 27. Chen, H.C.: Theory of Electromagnetic Waves. McGraw–Hill, New York (1983) 28. Zhou, C., Mackay, T.G., Lakhtakia, A.: Surface-plasmon-polariton wave propagation supported by anisotropic materials: multiple modes and mixed exponential and linear localization characteristics. Phys. Rev. A 100, 033809 (2019) 29. Johnson, P.B., Christy, R.W.: Optical constants of the noble metals. Phys. Rev. B 6, 4370–4379 (1972) 30. Iskander, M.F.: Electromagnetic Fields and Waves. Waveland Press, Long Grove (2013) Tom G. Mackay is a professor in the School of Mathematics at the University of Edinburgh and also an adjunct professor in the Department of Engineering Science and Mechanics at The Pennsylvania State University. He graduated from the Universities of Edinburgh, Glasgow, and Strathclyde. His research has been supported by awards from the Carnegie Trust for The Universities of Scotland, Engineering and Physical Sciences Research Council, Nuffield Foundation, Royal Academy of Engineering/Leverhulme Trust, and Royal Society of Edinburgh/Scottish Executive. He is a fellow of the Institute of Physics (UK) and of SPIE— The International Society for Optics and Photonics. His current research interests include homogenization, complex materials, and surface waves.

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Waleed Iqbal Waseer received the BS degree in telecommunication engineering from Iqra University in 2015 and the MSc degree in electrical engineering from COMSATS Institute of Information Technology (CIIT) in 2017. Currently, he is with the Department of Electrical and Computer Engineering, COMSATS University Islamabad, pursuing his PhD in electrical engineering. In January 2022 he was awarded IRSIP scholarship to conduct part of his PhD research in the Department of Engineering Science and Mechanics at The Pennsylvania State University. His research interests include surface waves, nanoscale heat transfer, and topological photonics.

Akhlesh Lakhtakia received the BTech (1979) and DSc (2006) degrees from the Banaras Hindu University and the MS (1981) and PhD (1983) degrees from the University of Utah. In 1983 he joined the Department of Engineering Science and Mechanics at Penn State as a post-doctoral research scholar, where he became a Distinguished Professor in 2003, the Charles Godfrey Binder Professor in 2006, and the Evan Pugh University Professor of Engineering Science and Mechanics in 2018. His current research interests include electromagnetic scattering, surface multiplasmonics, bioreplication, forensic science, solar energy, sculptured thin films, and mimumes. He has been elected a fellow of Optical Society of America (1992), SPIE—The International Society for Optical Engineering (1996), Institute of Physics (UK) (1996), American Association for the Advancement of Science (2010), American Physical Society (2012), Institute of Electrical and Electronics Engineers (2016), Royal Society of Chemistry (2016), and Royal Society of Arts (2017). He has been designated a Distinguished Alumnus of both of his almae matres at the highest level. Awards at Penn State include: Outstanding Research Award (1996), Outstanding Advising Award (2005), Premier Research Award (2008), and Outstanding Teaching Award (2016), and the Faculty Scholar Medal (2005). He received the 2010 Technical Achievement Award from SPIE, the 2016 Walston Chubb Award for Innovation from Sigma Xi, the 2022 Smart Structures and Materials Lifetime Achievement Award from SPIE, the 2022 Radio Club of America Lifetime Achievement Award, and the 2022 Distinguished Achievement Award from the IEEE Antennas and Propagation Society. He is a Sigma Xi Distinguished Lecturer (2022–24) and a Jefferson Science Fellow at the US State Department (2022–23).

Chapter 18

Cavity Modes and Surface Plasmon Waves Coupling on Nanostructured Surfaces for Enhanced Sensing and Energy Applications Mohammad Abutoama and Ibrahim Abdulhalim

18.1 Introduction The strong localization of surface plasmon waves to the plasmonic surface where they are excited makes them potential candidates to be used in a wide field of applications in which field enhancement strategies are required, such as sensing, surface-enhanced spectroscopic signals, and energy harvesting. There are two main types of surface plasmon phenomena: the first one is the extended or propagating surface plasmon (ESP) wave, while the second one is the localized surface plasmon (LSP). The ESP is a longitudinal electromagnetic surface wave excited at and propagating along the interface between a metal and a dielectric material and decays evanescently along the normal to the interface. The ESP is excited with transverse magnetic (TM) polarization at a specific wavelength and an incidence angle when the resonance condition of momentum matching along the surface is satisfied, causing a sharp reflection dip. The second phenomenon is the localized (non-propagating) surface plasmon excited on the surface of metallic nanoparticles

M. Abutoama () Department of Electrooptics and Photonics Engineering and the Ilse Katz Institute for Nanoscale Science and Technology, School of Electrical and Computer Engineering, Ben Gurion University, Beer Sheva, Israel DTU Electro, Technical University of Denmark, Lyngby, Denmark NanoPhoton - Center for Nanophotonics, Lyngby, Denmark e-mail: [email protected]; [email protected] I. Abdulhalim Department of Electrooptics and Photonics Engineering and the Ilse Katz Institute for Nanoscale Science and Technology, School of Electrical and Computer Engineering, Ben Gurion University, Beer Sheva, Israel e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 T. G. Mackay, A. Lakhtakia (eds.), Adventures in Contemporary Electromagnetic Theory, https://doi.org/10.1007/978-3-031-24617-3_18

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(MNPs) and nanostructures with dimensions less than half the wavelength of the exciting electromagnetic wave. A considerable amount of the designs and configurations discussed in the present chapter are based on dielectric and metallic gratings [1]. First, a novel grating coupling geometry based on the combination of a thin dielectric grating with a thin metal film is discussed [2–4] and was shown to exhibit the excitation of two optical modes, where one of them is a long-range surface plasmon wave. The main application of the discussed geometry involves the design of a compact self-referenced biosensor that makes the measurement stable and less sensitive to temperature fluctuations and optomechanical drifts. The sensor was designed to work in three operation modes (spectral (normal incidence), angular, and intensity). Moreover, by simple tuning of the grating parameters, the spectral range in which the phenomenon occurs can be easily tuned between the visible and infrared regimes. Depending on the grating parameters, metallic gratings can support the excitation of ESP modes, cavity modes and, more importantly, coupling of the modes is possible depending mainly on the grating parameters [4–6]. The resonant modes were shown to be spread over a wide range of wavelengths, which makes the discussed geometry a potential nanostructure for simultaneously fulfilling different purposes, such as sensing (in the visible and infra-red ranges), surface-enhanced Raman scattering (SERS), surface-enhanced fluorescence (SEF), and energy harvesting. Moreover, when MNPs are involved and dispersed on top of the metallic grating, coupling between the ESPs and cavity modes to the LSPs of the nanoparticles (NPs) might take place and lead to the generation of ultrahigh field enhancement on the NPs’ surface up to three orders of magnitude higher than that obtained using freespace excitation of LSPs. More importantly, the existence of the different optical modes supported by the discussed coupling geometry based on MNPs on a metallic grating was demonstrated to allow the generation of ultrahigh fields over a wide spectral range by simply tuning the grating and MNP parameters. This is critical for solar energy harvesting and improving the efficiency of infra-red optoelectronic devices. Besides grating-based geometries, many efforts were made to investigate MNPs on metal film (MNPs-MF) geometry. However, to the best of our knowledge, the relationship between the splitting of the energy spectrum and field enhancement in the MNPs−MF geometry has remained unknown. Therefore, the main aim of Ref. [7] was to characterize this relationship to facilitate performance optimization by the MNPs−MF geometry regarding field enhancement for both free-space and prism excitation. The conclusions in this work are critical from both theoretical and practical aspects. For example, if a specific NP density is used to generate ultrahigh field enhancement at a desired wavelength and if this density falls in the regime of energy spectrum splitting generation under either free-space or prism excitation, then the new resonances (new hybrid modes) are shifted from the desired wavelength, thus decreasing the field enhancement at the selected wavelength. Throughout the work, field distribution calculations were performed and confirmed the expected physical behavior of the field associated with each geometry

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shown above. In the second and third geometries mentioned above, experimental results validated the theoretical predictions. Before going into details, we start with a brief introduction on the nature and properties of the ESP and LSP phenomena, as well as their coupling. We also briefly discuss the field enhancement mechanism associated with the ESP and LSP. Finally, we introduce the simulation tools and analytical methods used to treat the optical behavior of the discussed nanostructures.

18.2 Theoretical Background 18.2.1 Bulk Surface Plasmons and the Single Interface Problem The ESP is characterized as a longitudinal charge density distribution generated at the interface on the metal side (see Fig. 18.1). Therefore, to induce the charge distribution, TM polarized light, in which the magnetic field is oriented in the y direction, is required. The ESP dispersion relation can be derived by solving the boundary problem of Maxwell’s equations when the boundary is between the metal and the dielectric and is expressed as:  kSP = kx = (ω/c) εm εd / (εm + εd ),

.

(18.1)

where kSP is the ESP wave constant, ω and c are the angular frequency and the speed of light in free space, respectively, εm and εd are the dielectric functions of the metal and dielectric, respectively.

18.2.2 Extended Surface Plasmon in the Single Thin Metal Film Two points should be addressed about the bulk ESP we discussed in Sect. 18.2.1. First, from the practical point of view, the use of the geometry in Fig. 18.1 is not Fig. 18.1 Surface plasmon wave at the interface between semi-infinite metal and dielectric

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Fig. 18.2 Multiple transmissions and reflections from thin metal film with thickness dm

possible without a coupling medium; therefore, a thin metal film can be used instead. Secondly, the ESP cannot be excited from air, and a coupling medium is required, as will be shown later. Two approaches can be applied to understand the possible ESP modes that can be excited in the single thin metal film embedded between the coupling and the dielectric media (Fig. 18.2). The first approach is based on the Airy formula, while the other is based on the Abeles matrix method dealing with multilayer systems [8], which we introduce in the simulation methods section below. The ESP is excited under the total internal reflection condition; therefore, the dispersion relation can be calculated using the waveguide condition (finding the poles of the reflection function) [9]. Since the z-component of the wavevector in the metal film (kzm ) is mainly imaginary, and if the metal film is thick enough, the following dispersion relation is achieved:  .

kzm kzc + εc εm



kzd kzm + εm εd

 = 0,

(18.2)

where εc is the dielectric function of the top semi-infinite medium, kzc and kzd are the z-components of the wavevector in the top and the bottom semi-infinite medium, respectively. By equating each of the brackets to zero, we get two dispersion equations (like the bulk plasmon case) that describe the excitation conditions of two ESPs – one at each boundary of the metal film. It is important to note that the case of using a thick metal film is less practical, and the metal films used are around 40–50 nm thick. The metal film thickness (dm ) plays a critical role in generating or exciting the ESP at the metal boundaries. When the metal film becomes thin, the medium above the metal starts to play an important role in determining the reflectivity of the structure and causes shift of the ESP wave constant, meaning shift of the resonance location. The conclusion is that there is an optimal thickness of the metal film at which we get a minimum of the reflectivity; physically, this happens when totally destructive interference occurs between the reflected wave and the wave released by the plasmons. The shift of the ESP resonance wave constant is given by [9]

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    εm εd 3/2 1 ω . Re {kx } = Re 2rcm exp (2ikzm dm ) , c εm + εd εd − εm

(18.3)

where rcm is the TM Fresnel coefficient at the metal boundary with the top semi-infinite medium. Note that using different metal thicknesses (not the optimal thickness) causes shift of the resonance location, as well as degrading of the contrast (the minimum point of the reflectivity will not be zero).

18.2.3 Propagation Length and Penetration Depth of the ESP The fact that the ESP wave has a propagating field in the x direction while in the z direction the field is decaying makes it a potential candidate to be used in integrated photonics as well as sensing applications, respectively. One of the serious problems that limits the ESP performance is the limited penetration depth, which limits its use to sensing of only small entities existing near the metal film surface. Hence, sensing large entities, such as cells and bacteria, using the conventional ESP geometry is not possible, and other configurations are needed, such as the long-range surface plasmon (LRSP) configuration that will be shown later. The propagation length of the ESP depends on the imaginary part of the propagation constant kx [8]. The real (kxr ) and imaginary (kxi ) parts of kx are expressed as:  kxr = (ω/c)

.

εmr εd ; εmr + εd

 kxi = (ω/c)

εmr εd εmr + εd

3/2 

εmi 2εmr 2

 .

(18.4)

The propagation length (Lx ) of the ESP wave along the metal surface is expressed  λ εmr +εd 3/2 εmr 2  as: .Lx = 2|k1xi | = 2π where λ is the optical wavelength in εmr εd εmi vacuum, εmr and εmi are the real and imaginary part of the dielectric function of the metal film, respectively. As can be seen, Lx is limited by εmi ; otherwise, the ESP wave might theoretically propagate to infinity. On the other hand, the ESP penetration depth (δ) can be calculated using: δ = 1/|kz |. Hence, the ESP penetration depths inside the metal (δ m ) and the dielectric (the analyte in the sensing applications) (δ d ) are given by:  δm =

.

λ 2π



εmr + εd ; −εmr 2

 δd =

λ 2π



εmr + εd . −εd 2

(18.5)

Figure 18.3a shows the ESP penetration depth in vacuum (dashed curves) and in metal (solid curves) for silver (blue) and gold (red). As can observed, δ d is slightly larger for silver than for gold, but the values are very comparable while δ m is larger for gold than for silver.

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Fig. 18.3 (a) ESP wave penetration depths in vacuum (dashed curves) and in metal (solid curves) for silver (blue) and gold (red); (b) ESP wave propagation lengths along silver/vacuum (blue) and gold/vacuum (red) interfaces; (c) ESP wave penetration depths in metal (solid curves) for silver and for different types of analyte mediums: vacuum (blue), water (red), and 1.6 refractive index (RI) (green); (d) ESP wave propagation lengths along silver/analyte interface for different types of analytes: vacuum (blue), water (red), and 1.6 RI (green). The dispersion of water was considered in the calculations. (Reproduced with permission from Ref. [4])

In other words, the absorption is larger in case of gold than in silver, resulting in lower quality factor of the ESP resonance. It can be also observed that δ d is much larger in the near infra-red than in the visible wavelengths; this is a result of the larger ratio between the real and imaginary parts of the dielectric function as the wavelength increases while δ m is larger for the visible wavelengths than the near infra-red ones as expected. Figure 18.3b shows the Lx along silver/vacuum (blue) and gold/vacuum (red) interfaces where a larger propagation length is observed for silver in the given spectral range. Figure 18.3c shows the behavior of δ d (dashed curves) and δ m (solid curves) for silver and for different types of analytes: vacuum (blue), water (red), and 1.6 refractive index (RI) (green). As observed, δ d and δ m decrease when the analyte RI increases in the whole spectral range. Figure 18.3d shows Lx along silver/analyte interface for different types of analyte mediums: vacuum (blue), water (red), and (1.6 RI). It can be observed that Lx decreases when the analyte RI increases for the whole wavelengths. In addition, there is an inverse correlation between the penetration depth in the dielectric and the propagation length because the increase in the penetration depth means most of the electromagnetic energy exists in the dielectric and as a result, there is less dissipation in the metal, hence, the wave can propagate for a longer distance along the interface. To conclude, for sensing

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operation, silver has larger penetration depth and propagation length than gold, on the other hand silver can be easily oxidized and a protection method is needed.

18.2.4 Excitation Techniques of the ESP The ESP dispersion relation can be obtained by substituting the simple Drude model for the√dielectric function of the metal, εm = 1 − (ωp 2 /ω2 ), in .kSP = kx = (ω/c) εm εd / (εm + εd ): ⎡ 1⎣ 2 2 .ω = ωp + kx 2 c2 2



1 1+ εd



 −

ωp 4 + 2kx 2 c2 ωp 2

⎤     1 1 2⎦ 4 4 1− + kx c 1 + . εd εd

(18.6) Dividing Eq. (18.6) by ωp 2 (where ωp is the plasma frequency) and defining the normalized parameters ωn = ω/ωp and kn = kSP /kp = kx /kp , where kp = ωp /c, we can present the dispersion relation in Eq. (18.6) in the normalized form as follows: ωn 2

.

⎤ ⎡      2  1 1 1 1⎣ ⎦. − 1 + 2kn 2 1 − + kn 4 1 + = 1 + kn 2 1 + 2 εd εd εd (18.7)

It should be noted that the ESP dispersion in Eq. (18.7) is valid only when using the simplified Drude model which is a crude approximation for the case of noble metals and for frequency approaching the plasma frequency. For simplicity, this model assumes ideal metal without losses and damping to the oscillations of the free electrons within the metal. Figure 18.4 shows three different dispersion relations: the first dispersion relation is for incident light from a vacuum (purple line) at an incidence angle equal to 60◦ , while the green curve is for an ESP at the interface between metal and water, which behaves according to Eq. (18.7). No intersection is observed between the light line and the ESP curve, which indicates that the ESP cannot be excited from air and a coupling medium is required. This is a result of the missing momentum needed for the incident photons to support the ESP excitation. In general, the required excitation angle can be calculated from the relation kSP = kx = k0 ni sin θ i , where k0 = ω/c, θ i is the incidence angle, and ni is the RI of the medium where the incident light comes from. In the case where the incoming medium is air (ni = 1), we get kSP = k0 sin θ i < k0 , which √ is in contradiction to the ESP excitation condition and leads to .kSP = k0 εd / (1 − |εd /εm |) > k0 . The same conclusion (as that concluded from Fig. 18.4) can also be obtained from the last analysis. The conclusion above is that it is not possible to excite the ESP when the light is coming from air or from another

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Fig. 18.4 Dispersion curves for: incident light into the metal from vacuum at incidence angle equal to 60◦ (Light line-purple), ESP at the interface between metal and water (ESP curve-green), and incident light emerging from coupling medium with 1.77 RI (prism coupling line-blue)

medium with a RI smaller than the RI of the second medium bounding the metal film, and this is the reason for the need of the coupling medium.

18.2.4.1

Prism Coupling

There are two main coupling methods or techniques that provide the additional momentum along the surface kx for the incident photons to excite the ESP. The first technique proposes the use of a prism to couple the incident light to the ESPs (called the Kretschmann-Raether configuration, see Fig. 18.5a), while the second one is using grating coupling, where the missing momentum is provided by the diffracted light from the grating as will be shown below. Previously, another prism coupling configuration was proposed by Otto [10] where a dielectric gap separates between the prism and a bulk metal which is problematic from the practical point of view. The purple line (light line) in Fig. 18.4 satisfies the relation ωn = kn /(nair sin 60◦ ). The blue line is for incident light from the coupling medium (SF11 prism in this case) with 1.77 RI at an incidence angle equals to 60◦ , which satisfies the relation ωn = kn /(nc sin θ c ) = kn (1.77 sin 60◦ ), where nc and θ c are the coupling medium RI and angle, respectively. In this case, the blue line intersects the green curve at the plasmon frequency (named excitation point in Fig. 18.4). LRSP One of the most interesting parameters in the biosensing field using evanescent waves is the penetration depth inside the analyte (dielectric medium of relative permittivity εd in Fig. 18.5a), which can be enhanced using what is called LRSP geometry. The LRSP penetrates for a larger penetration depth in the analyte compared to the conventional ESP configuration. This means that the absorption caused by the metal film in the case of the LRSP is smaller than in the case of the short-range surface plasmon resonance, leading the LRSP to propagate for a longer propagation length at the surface. The advantage of using the LRSP in biosensing is the ability to detect large bioentities, such as cells and bacteria, while the disadvantage lies in degrading the specificity of detecting small bioentities and molecules. The coupling in the Kretschmann configuration (Fig. 18.5a) is

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Fig. 18.5 (a) Prism coupling for the ESP excitation (Kretschmann-Raether configuration). (b) Configuration of double mode ESP sensor dielectric layer with a low RI, this geometry supports the LRSP

based on the total internal reflection that occurs at the prism-metal interface, where an evanescent wave is generated. The evanescent wave tunnels to the metalanalyte interface and excites the ESP at this interface. This means that using this configuration it is not possible to couple the incident light to the surface plasmons at the bottom boundary of the metal film. To excite two different ESPs, one at each metal boundary, first the metal has to be thin enough, and then one can insert a dielectric film with a low RI – less than that of the prism (to satisfy the total internal reflection condition) – in the Kretschmann configuration between the prism and the metal film (see Fig. 18.5b). Otherwise, this dielectric film will act as a continuation of the prism medium, and an additional ESP cannot be excited. In this case, the evanescent wave (generated at the prism-dielectric film interface) will reach the two boundaries of the metal film and excite two different ESPs. Note that the requirement of using a thin metal film is to allow the evanescent field to reach the two boundaries of the metal film. Assuming a symmetric structure surrounding the metal film, the solution to the above-mentioned dispersion relations of the two ESPs of the two metal film interfaces is given by [8, 9]: .

εm kzd + εd kzm tanh (kzm dm /2j ) = 0, εm kzd + εd kzm coth (kzm dm /2j ) = 0.

(18.8)

Note that, when using a thick enough metal film, Eqs. (18.8) reduce to εm kzd + εd kzm = 0. Using the known relations: kzm =

.

we get

 εm k0 2 − kx 2 and

 kzd =

εd k0 2 − kx 2 ,

(18.9)

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 kSP = kx = k0

.

εm εd . εm + εd

(18.10)

Note that we got the same dispersion relation using two methods: the first one is by solving Maxwell’s equations of the boundary problem while the second one is by using the Airy function and the multiple interference effects. For the case where the structure is symmetric, we get two identical ESPs. Figure 18.6a shows a TM reflection simulation performed using GSolver software for the configuration in Fig. 18.5b; these results are reproduced results of the simulation shown in Fig. 7.7 in Ref. [9]. The prism is SF11, the RI and the thickness of the dielectric film are 1.49 (poly methyl methacrylate-PMMA) and 480 nm, respectively. The wavelength of the incident light is 788 nm, the metal is silver with a thickness of 45 nm and the analyte is water. Without using the dielectric film between the metal film and the prism, only one dip (at θ res1 ) is observed in the reflection spectrum (blue curve). On the other hand, with the insertion of the dielectric film between the metal film and the prism and optimizing the parameters, two dips (at θ res1 and θ res2 ) are observed with very good contrast (red and green curves). Due to variation of the analyte RI from 1.33 to 1.34, the left-side dip location of the red curve shifts (see the green curve), meaning that the ESP associated with this dip is excited at the metal-analyte interface and is the LRSP, since its full width at half maximum (FWHM) is narrower than that of the right-side dip, which is excited at the top dielectric-metal interface and is a short-range surface plasmon. On the other hand, Fig. 18.6b shows the reflection spectra due to variation of the dielectric film RI (poly(methyl methacrylate) (PMMA) film). In this case, the right-side dip is the one that is sensitive to variations in the RI of the PMMA film, while the left-side dip is not. This means that the plasmon associated with the right-side dip is excited at the dielectric-metal interface. It should be noted that the incidence angles shown in the simulations are the angles inside the prism. It is important to note that one of the plasmons is in an asymmetric mode around the center of the metal film, meaning that this ESP has zero energy at the center of the metal film (see Fig. 18.7a). The other ESP is symmetric and has its maximum energy at the center of the metal film, meaning that this ESP cannot propagate for long distances since it will be absorbed quickly by the metal film [9] and is called a short-range surface plasmon). On the other hand, the asymmetric ESP can propagate for long distances along the surface and penetrates for a deeper penetration depth and LRSP. Figure 18.7b shows the field distribution and was performed using Comsol software at the two resonance angles θ res1 and θ res2 . The ESP excited at the metal-analyte interface (see Fig. 18.7b, left-hand side) has a larger penetration depth inside the analyte than that of the ESP excited at the dielectric-metal interface inside the dielectric layer (Fig. 18.7b, right-hand side). This means that the left-side dip in the reflection spectra is the LRSP, while the right-side one is the short-range surface plasmon. Another nice observation from the field distribution in Fig. 18.7b is the correlation between the sensitivity and the field distribution; as we showed in

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Fig. 18.6 (a) TM reflection simulation of single and double ESP (investigation of the effect of using dielectric film between the metal film and the prism). (b) Reflection variation due to variation of the PMMA film RI. (Reproduced with permission from Ref. [9])

Fig. 18.6a, the left-side dip is the one that is sensitive to variations in the RI of the analyte, while the right-side dip is not. The field distribution in Fig. 18.7b (left-hand side) demonstrates this fact because the field is located mainly in the analyte region, while there is no field in the dielectric film. On the other hand, the right-side dip is sensitive to variations in the RI of the bottom dielectric film (see Fig. 18.6b), which explains the fact that the field is located mainly in the dielectric film.

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Fig. 18.7 (a) Hy intensity enhancement factor in the double mode (asymmetric and symmetric) ESP configuration. (Calculated using the algorithm developed by Shalabney and Abdulhalim [11] based on the 2 × 2 Abeles matrix formalism. Reproduced with permission from Ref. [9]). (b) Field distribution at (a) θ res1 , (b) θ res2 of the double mode ESP configuration

Field Enhancement Mechanism in the ESP Case As the issue of the field enhancement is a major part of this work, we will not close this section before discussing the essence of the field enhancement in the ESP case. Field confinement near the metal surface arises from the nature of the ESP, for which in the parallel direction (x direction) to the metal surface the ESP propagates, but decays after a certain distance because of the absorption of the metal film, while in the perpendicular direction (z direction) it decays with a maximum at the metal surface. Both facts lead to strong confinement of the ESP wave near the metal surface, which then, due to energy conservation, imposes high concentration of the energy near the surface and thus leads to enhancing the field near the metal surface. The field distribution in multilayered structures can be calculated using the algorithm developed by Shalabney and Abdulhalim [11]; therefore, the ESP field distribution in several prism coupling cases was also investigated based on the mentioned algorithm, showing a direct correlation between the ESP resonance sensitivity to the analyte RI and the field enhancement. The main conclusions in the mentioned works were the following: first, when using a lower index prism or adding a nanolayer of high index material on top of the metal, the ESP resonance angle increases and, more importantly, the field enhancement also increases [12]. Simplified analytic expressions for the field enhancement were given for the ESP case under several approximations [13, 14] and show that the field enhancement is approximately proportional to |εmr /εmi |; this can be improved by using metal with lower absorption and also by choosing the incident wavelength, as both εmr and εmi are dispersive. The most important conclusion is that in the ESP case excited in the prism coupling geometry, enhancement of ~1–2 orders of magnitude can be achieved depending on the specific design and parameters.

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As a final remark, the reader should be referred to the new studies on the nature of the ESP excitation in which the conventional resonance theory of Kretschmann and Otto configurations was revised [15–17]. According to the conventional theory, the excitation of the ESP mode is associated with both the observation of reflection dip and enhanced electromagnetic field at the metal film surface. As demonstrated using the rigorous mathematical analysis in the new studies on the subject, the reflection dip and enhanced electromagnetic field are two independent physical phenomena that occur under different conditions, and more importantly, both do not coincide with the ESP excitation condition [17], but rather correspond to the perfectly absorbing mode, which may occur in thin metal films [15]. For a deeper understanding, we believe that an additional and very careful study is required. It is worth mentioning the use of the 4 × 4 propagation matrix approach to treat the ESP excitation when anisotropic films are involved [18] or when the metal film itself is anisotropic, for example, nano-sculptured thin films (nSTFs), which are columnar with certain porosity. The theoretical treatment of these films is done using homogenization approach to find the structure effective dielectric tensor and the 4 × 4 propagation matrix approach to calculate the reflectance and field distribution [19–21]. The sensitivity and field enhancement are larger depending on the porosity. However, for porosities larger than 35–45%, the resonance becomes very broad, indicating localization effects start to be dominant. The homogenization approach becomes less valid with such large porosities and the start of localization. 18.2.4.2

ESP in the Grating Coupling Geometry

The second coupling method is grating coupling, in which the additional momentum needed to couple the incident light to the ESP wave is provided by the diffraction grating. In grating coupling, there are two main possibilities: first the grating itself is metallic, while in the second, the grating is dielectric on top of a thick metal film. In this sense, we discussed a new configuration composed of a thin dielectric grating combined with a thin metal film [2–4], in which two optical modes are excited. The derivation of the ESP dispersion relation in the case of the grating coupling case is similar to that for prism coupling. Considering the analyte material in the case of sensing applications to be existed in the grating spaces and on top of the grating (superstrate), the equation for calculating the spectral position of the ESP resonant mode is called the plasmon momentum matching equation and is given by kxi + kG = ± kSP , where kxi is the k component of the incident light in the polarization direction (x-axis), kG is the k constant of the grating required to provide the additional momentum kG = (2π / )m to excite the ESP. The plasmon momentum matching equation can be written as:  k0 nd sin θi +

.

2π



 m = ±k0

εm εd , εm + εd

(18.11)

where nd is the dielectric medium RI, is the grating period, m is the diffraction order (integer), the sign ‘+’ in the right side of the equation corresponds to (m > 0),

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while the sign ‘−’ corresponds to (m < 0). At normal incidence (θ = 0◦ ), the excitation wavelength of the ESP can be calculated using: λexc =

.

m



εm εd . εm + εd

(18.12)

This means, at normal incidence, only the momentum supported by the grating (2π m/ ) can lead to excite the ESP, and hence, the condition |m| > 0 should be satisfied. In addition, at normal incidence, the excitation wavelength for the positive and negative diffraction orders is equal. As done for the prism case, for the sake of simplicity, the simple Drude model for the dielectric function of the metal εm = 1 − (ωp 2 /ω2 ) is substituted into Eq. (18.11), and hence, the dispersion relation for the ESP excitation in the case of the grating coupling can be derived and leads to the following relation:    1 1 2 ω = c 1+ (kxi + kG )2 + ωp 2 2 εd .   2 1  2 2 2 4 2 2 − − 4εd c ωp (kxi + kG ) −c (1 + εd ) (kxi + kG ) − εd ωp εd (18.13) 2

After algebraic manipulation of Eq. (18.13), we get:    1 1 2 2 2 ω = ωp + (kxi + kG ) c 1 + εd 2 ⎤  .   2  1 1 ⎦ + (kxi + kG )4 c4 1 + − ωp 4 + 2(kxi + kG )2 c2 ωp 2 1 − εd εd 2

(18.14) Similar to the normalization that was done in the case of the prism coupling, we can again normalize Eq. (18.14) (note that in this case the normalized plasmon wave constant is given by kn = kSP /kp = (kxi + kG )/kp ). Note that the normalized equation of the dispersion relation for the grating coupling is the same as Eq. (18.7) (prism case) with a shift of the wave constant kxi by the grating wave constant kG . The dispersion relation of the ESP in the grating geometry was plotted in Fig. 18.8 for four RIs of analytes and for the diffraction orders (m = −1, 0, 1) to show the variation in the ESP excitation condition as the analyte changes. The solid curves in Fig. 18.8 are for air (nd = 1), the dashed curves are for nd = 1.33 (close to the water RI in the visible wavelengths), the dotted curves are for nd = 1.39 (close to the glycerol RI), and the dashed-dotted curves are for nd = 1.6 (close to their RI of other organic materials). This demonstrates the fact that in general, the ESP in the grating geometry can be used for sensing of different types of analytes such as gases, water, organic materials, and others. Assuming fixed normalized momentum kn , the

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Fig. 18.8 Dispersion relation for ESP in the grating coupling geometry at different diffraction orders and analyte RIs. (Reproduced with permission from Ref. [4])

normalized frequency ωn decreases as the anlyte RI increases as can be observed from Fig. 18.8. Moreover, the sensitivity becomes larger as kn becomes away from the exact matching excitation condition.

18.2.5 The Nature of the Resonant Modes Supported by Thick Metallic Grating Due to the importance of a thick metallic grating [4–6], the optical behavior of this geometry will be briefly and separately addressed in this section. The optical behavior of the metallic grating is determined by several parameters. The role of the grating thickness (h) and Λ in supporting the excitation of ESP modes and cavity modes excited within the slits of the grating is highlighted in Refs [4, 6]. Starting with the ESP excitation, in which in the limit of shallow gratings (thin grating in comparison to the wavelength), the spectral position of the resonance can be simply determined by the RIs of the grating and the surrounding materials, as well as by Λ, without much dependence on h (assuming fixed incidence angle). In this case, the spectral position of the resonance can be estimated using the above-mentioned plasmon momentum matching equation. As h increases and becomes larger than or in the order of the incident wavelength, the issue becomes more complicated, and several points should be highlighted. First, the plasmon momentum matching

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equation no longer accurately predicts the ESP’s resonance position. Secondly, the grating is now able to support new electromagnetic modes (cavity modes). Finally, an additional scenario may take place in which at certain conditions the ESPs and cavity modes can couple to generate a new resonant mode (ESP-cavity modes). The coupling, as well as the field distribution associated with each case, was discussed in detail in Refs [4, 6].

18.2.6 Localized Surface Plasmon (LSP) The second type of surface plasmon phenomenon is the LSP excited on the surface of NPs and nanostructures that have become very popular during the last two decades. Due to their wide variety of applications, such as in biosensing, nowadays many theoretical and experimental research groups are active in the investigation of nanoplasmonic metallic structures [22–26]. The LSP can be excited in metallic structures in which the dimensions of the structures are less than half the wavelength of the exciting electromagnetic wave. In the general case, the LSP can be excited in different nanostructures with different shapes such as nanospheres, nanobipyramids, and nanoellipsoids and can be made of different metals such as silver and gold. In each shape, the structure still exhibits scattering and absorption properties, but the difference may be in the location and the number of the structure resonances [27]. Due to the curvature of the NPs’ surface (non-planar surface as in the case of the conventional metal film), the plasmons can be excited by direct illumination without the requirement for coupling medium, as in the ESP case. The incident field displaces the free-electron cloud and produces a charge distribution near the NPs’ surface, as shown in Fig. 18.9. The LSP resonant modes and the resonance condition arise naturally from the scattering problem, which deals with the interaction between the sub-wavelength NPs and the electromagnetic waves in the quasi-static approximation, which is valid since the dimensions of the structure are less than half the wavelength of the exciting electromagnetic wave. Hence, Maxwell’s equations in the quasi-static approximation can be solved [28], where in this case, the magnitude of the electric field seems static around the NP. Fig. 18.9 Different shapes of NPs supporting the excitation of LSP

18 Cavity Modes and Surface Plasmon Waves Coupling on Nanostructured. . .

18.2.6.1

457

The Physics of the LSP

Assuming a homogeneous and isotropic sphere of radius a and dielectric function εm (ω) located at the origin, the surrounding medium is isotropic and non-absorbing with a dielectric constant εd . We consider the case of spherical NP located in a static and uniform electric field; the electrostatic polarizability is then given by: α = 4π a 3

.

εm (ω) − εd . εm (ω) + 2εd

(18.15)

According to this equation, α has a resonance when εm (ω) = − 2εd (called Fröhlich resonance [28]), which is defined as the resonance condition of the LSP. This illustrates the sensitivity of the LSP to the surrounding medium around the NP, a fact that makes the LSP a candidate to be used in sensing applications. In many cases, there is an interest in using NPs of radius much smaller than the incident wavelength. Under this assumption, Maxwell’s equations can be solved in the quasi-static approximation in which the electric field seems static around the NP. A simple expression of the electric field outside the sphere is then given by [27] out (x, y, z) = E0 zˆ − a 3 .E



εm (ω) − εd εm (ω) + 2εd



 E0

 zˆ z  − 5 x xˆ + y yˆ + zˆz , r3 r (18.16)

where E0 is the amplitude of the applied field (polarized in the z direction in this case). We notice that Eout also resonates when εm (ω) = − 2εd . As a result of the enhancement in the polarizability when the resonance condition is satisfied, resonance in the internal and dipolar electric fields occurs and leads to enhancing the electric field on the surface of the NP and starts to decrease when getting far from the surface, as can be seen in Eq. (18.16). Considering the case of an ellipsoid, the polarizability becomes a tensor with its principal components given by αj =

.

εm (ω) − εd 4π abc . 3 Lj [εm (ω) − εd ] + εd

(18.17)

This equation shows that the polarizability depends on the axis, and Lj is the geometrical factor of the specific axis j when the applied electric field polarized along this axis leads to polarizability α j . The extinction cross section (Cext ) is given by Cext = Cabs + Csca , where Cabs is the absorption cross section, while Csca is the scattering cross section. The absorption cross section depends on the axis as Cabs = k Im (α j ) [28], k is the wave number in the surrounding material around the   4  2 and the extinction NP. The scattering cross section is given by .Csct = 8π 3 k αj cross section is then given by Cext = Cabs + Csca ∼ = 4π k Im (α j ).

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18.2.6.2

Field Enhancement Mechanism in the LSP Case

As in the ESP case, the field enhancement in the LSP case depends on |εmr /εmi | as shown below. Besides the material effect, a geometrical factor Lj (mentioned above) which depends on the NP shape plays an important role in determining the enhancement factor, as predicted by the anisotropic Mie theory for a single MNP. Due to the nanometric vicinity in which the field is enhanced in the case of the LSP excitation, the local field enhancement can be enhanced by a factor of ~1000, mainly depending on the NP shape. As shown above, the LSP modes originate from the resonant behavior of the polarizability, which is proportional to the local field. To derive the relation between the enhancement factor and the above-mentioned factors, we may write the field in the vicinity of a small NP surface as the sum of the incident field and the one from the contribution of the polarization induced in the NP, which is a type of Mossotti equation in a tensorial form [12]:   ∼ ∼ E = Einc + 4π 1 − L α Einc .

(18.18)

.

∼

∼

Here, .L is the geometrical factor tensor while .α is the polarizability tensor per unit volume. Then, the field intensity enhancement factor along any one of the axes j of an ellipsoid (considering the general case of a NP shape, as mentioned above) may then be written as [12]: η Ej .

  Ej =  E

inc

2   

    = 1 + 1 − Lj

2 2     εm (ω) εm (ω) − εd  =  ,   Lj [εm (ω) − εd ] + εd Lj (εm (ω) − εd ) + εd  (18.19)

Expressing the real and imaginary parts of the metal as εm = εmr + iεmi , the following relation can be written as: εmr 2 + εmi 2 η Ej =     . Lj (εmr − εd ) + εd 2 + Lj εmi 2

.

(18.20)

At the LSP resonance, |Lj (εmr − εd ) + εd | = 0 is satisfied, which leads to the final expression of the maximum intensity enhancement factor: η Ej

.

max

εmr 2 + εmi 2 =   ≈ Lj εmi 2



εmr Lj εmi

2 .

(18.21)

18 Cavity Modes and Surface Plasmon Waves Coupling on Nanostructured. . .

459

Fig. 18.10 (a) Extinction cross section (far field calculation) of gold (Au) and silver (Ag) NPs embedded in air, and the radius of the NPs is 50 nm. The incident light is a plane wave polarized along the z-axis. (b) Near field distribution at the LSP resonant wavelengths of the Au and Ag NPs

The approximation on the rightmost side of Eq. (18.21) is valid only at wavelengths range in which the imaginary part of the dielectric function of the metal is much smaller than its real part. As seen from the expression of the electric field outside the NP under the quasistatic approximation (Eq. (18.16)), and since the field is highly localized to the NP surface and exponentially decays in all directions within 10–100 nm, the LSP enhancement phenomenon can be treated as localizing the electric field of a dipole field located at the center of the spherical NP. In other words, the NP acts as a nanoantenna which enhances the scattered light intensity [9]. Figure 18.10a shows the extinction cross sections of gold (Au) and silver (Ag) NPs of 50 nm radius excited with a plane wave polarized in the z-direction, where the surrounding medium around the NPs is air. As observed, under the same conditions, the LSP resonant wavelength in the case of the Au NP is observed at longer wavelength (520 nm) than in the case of the Ag NP (observed at 410 nm). Figure 18.10b shows the near field distribution at the LSP resonant wavelengths of the Au and Ag NPs. The calculation of the cross sections (far field) and the field distribution calculations (near field) was performed using Comsol software (with 5 nm steps

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in the wavelength just to qualitatively calculate the LSP resonance position) with defining the equation of the scattering and the absorption cross sections according to Ref. [29] and then the extinction cross section can be obtained by the sum of the scattering and absorption cross sections. We should note that the field enhancement at the resonant wavelength of the Ag NP is larger than in the case of the Au NP. This is partially true since the absolute value of the real part of the Ag dielectric function is larger than that of the Au. The values of the field enhancement in Fig. 18.10b are in good agreement with the approximate theoretical value calculated using Eq. (18.21) for the field enhancement when taking the special case of a spherical NP having Lj = 1/3. Finally, we highlight the fact that the reported LSP resonances are due to the first LSP mode excitation (dipolar mode). In general, increasing the NP size causes the appearance of higher LSP modes.

18.2.7 ESP-LSP Coupling Geometry, the Concept, and the Rabi Splitting Effect As concluded in Sect. 18.2.6 that deals with LSP field enhancement, the material and shape of the NP play a key role in determining the field enhancement factor. In this section, we show that the LSP excitation methodology also has a critical influence on the field enhancement factor, for which ~1–3 orders of magnitude higher than the case of free-space excitation of the LSP can be achieved by exciting the LSPs via an extended surface electromagnetic wave (ESP in this case) and not from free-space. The basic concept behind the ESP-LSP [12] coupling is illustrated in Fig. 18.11.

Fig. 18.11 Schematic of the LSP excitation. Left-hand side: excitation from free space. Righthand side: excitation using ESP waves generated on the thin metal film surface

18 Cavity Modes and Surface Plasmon Waves Coupling on Nanostructured. . .

461

On the left-hand side, the LSPs are excited from free space, while on the righthand side, they are excited through the ESPs generated on the surface of the thin metal film. Considering the exact matching of the ESP and LSP excitation conditions, the field enhancement factor can be roughly determined by the product of the two separate enhancement factors: FESP − LSP = FESP FLSP , where FESP and FLSP are associated with the ESP and LSP enhancement factors, respectively. In reality, even higher enhancement than the product of the two factors was achieved, which is believed to be due to the fact that when the NP is located close to the metal surface, its response changes due to the mirror charge. Generally speaking, the extended surface electromagnetic wave can also be generated [30] using other geometries, such as the periodic structures supporting the excitation of a Bloch surface wave and Tamm waves, absorbing layers (Zenneck waves), and Dyakonov waves at the interface between isotropic and anisotropic materials. Besides the prism, the coupling medium can be a waveguide, fiber, or grating. Recently, the coupled oscillator model, a pure classical model [31], showed success in analytically describing many light-matter interaction phenomena [32, 33]. In particular, the ESP-LSP coupling observed in the MNP-MF geometry can be explained using this model, which facilitates understanding of the resonant behavior of such structures and the behavioral dependence on structural parameters [34, 35]. In classical physics, the Eigen-frequencies of oscillators with masses m1, 2 and  spring constants k1, 2 are .ω1,2 = k1,2 /m1,2 . Then the equations of motion can be written as: .

q¨1 (t) + γ1 q˙1 (t) + ω1 2 q1 (t) + κ q˙2 (t) = ηE(t), q¨2 (t) + γ2 q˙2 (t) + ω2 2 q2 (t) − κ q˙1 (t) = 0,

(18.22)

where γ 1, 2 are the damping constants and κ is the coupling constant. The incident time harmonic optical field generates the driving force for the first oscillator, which is given as f (t) = ηE(t) = ηE0 e−iωt , when η is an efficiency factor. When exact coupling occurs (ω1 = ω2 = ω0 ), the two mode solutions can be easily extracted by subtracting and adding the two motion equations in Eq. (18.22) and using the same damping constant to get [12]: .

q¨+ (t) + γ q˙+ (t) + ω0 2 q+ (t) + κ q˙− (t) = ηE(t), q¨− (t) + γ q˙− (t) + ω0 2 q− (t) − κ q˙+ (t) = ηE(t),

(18.23)

where q± = q1 ± q2 . Let us now talk in ESP, LSP frequencies terms: Fig. 18.12 describes the ESP-LSP interaction using the coupled-oscillator model. The ESP and LSP modes are separately described as classical harmonic oscillators, while the ESP-LSP coupling process is described using an oscillator that links the two. The hot spots at the edges of the NP and the metal film in the coupling direction indicate that both the ESP and LSP modes are excited and the oscillator linking them indicates that there is strong mode interaction. Therefore, two new oscillation modes appear in the new coupled system at one of two new frequencies, either oscillating at the same phase or at anti-phase.

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Fig. 18.12 Left-hand side: schematic of ESP-LSP coupling on the NP-MF geometry using the classical coupled-oscillator model. Red color indicates hot spots and strong coupling. Right-hand side: schematic of the Rabi splitting occurring because of the ESP-LSP coupling. (Reproduced with permission from Ref. [7])

The new hybrid modes without considering damping in the system can be classically obtained [32] using: ω± =

.

 1 ωLSP ωESP ± + A + (ωESP − ωLSP )2 . 2 2 2

(18.24)

Therefore, at resonance (ωESP = ωLSP ), the Rabi splitting which is the signature of the strong ESP-LSP coupling can be expressed as: √ A= . = ω+ − ω− =



N e , √ V ε0 m

(18.25)

where ω± are the frequencies of higher and lower modes of the new coupled system; ωESP and ωLSP , the resonance frequencies of the ESP and LSP modes, respectively; N/V, the nanobipyramid number density of nanobipyramid number N per unit volume V; ε0 , the vacuum dielectric constant; and e and m, the electron charge and mass, respectively. As can be seen, the parameter A is proportional to the NPs’ density (N/V) and, more importantly, the Rabi splitting  is determined by √ . A (assuming that the NPs don’t couple with each other, and the coupling occurs only between the NPs and the metal film). The volume in the 2D geometry is defined by the area of the unit cell of the periodic structure in the xy plane multiplied by the vertical dimension of the particle (along z). After incorporating the damping γ , Eq. (18.24) becomes:    iγ 1 ωLSP iγ 2 ωESP − ± + A + ωESP − ωLSP + . .ω± = 2 4 2 2 2

(18.26)

18 Cavity Modes and Surface Plasmon Waves Coupling on Nanostructured. . .

463

The Rabi  splitting at resonance (ωESP = ωLSP ) can be obtained using . = ω+ − 2 ω− = A − γ4 . Understanding the relation between the field enhancement and Rabi splitting effects in such configurations, which support the excitation of different optical modes such as LSP and ESP, is critical for generating ultrahigh field enhancement at any desired wavelength. For example, if a specific NP density is used to generate ultrahigh field enhancement at a desired wavelength and this density falls in the regime of Rabi splitting generation under either free-space or prism excitation, then the new structural resonances (new hybrid modes) are shifted from the desired wavelength, thus decreasing the field enhancement at the selected wavelength [7].

18.2.8 Spectroscopic Signal Enhancement As discussed, the ESP and LSP are strongly localized to the surface and considerable enhancement was observed in both the ESP and LSP cases. Much higher enhancement was reported for the ESP-LSP coupling geometry. The enhancement in the field causes enhancement in several optical phenomena, such as SEF and SERS (defined below), in which the output intensity depends on the intensity of the exciting field to some power. In the SERS and SEF cases, the intensity is usually enhanced as the fourth and second power of the field (|E|4 and |E|2 , respectively).

18.2.8.1

Surface-Enhanced Raman Scattering (SERS)

When an electromagnetic wave interacts with molecule, part of the energy excites a vibrational mode of the molecule while the remained part is emitted as new photons in what is called Raman scattering. Usually, the main problem of Raman scattering is the very weak signal which makes it difficult to be experimentally detected. Two main accepted mechanisms are used to explain SERS [9]. The first one is called chemical enhancement, which might occur as a result of a charge transfer effect or chemical bond formation between the surface of the metal and the molecules of the analyte and leads to enhancement of their polarizability. The second mechanism is the one that stands at the basis of our SERS samples design and is due to the field enhancement at the surface of the metal; it occurs when the incident light wavelength coincides with the LSP one. Hence, the geometry plays a crucial role in enhancing Raman scattering. For example, NPs with sharp edges were found to yield strong SERS. Moreover, as seen in Refs. [7, 36], different geometries involving ESP and LSP excitation were proposed and showed strong SERS in what is called an ESP-LSP coupling configuration. A simplified model for Raman scattering enhancement considering field enhancement near a single isolated particle surface was shown in Ref. [37]. According to this model, the total average enhancement factor of Raman scattering can be obtained by the multiplication of

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the incident and scattered field enhancements. An example of the calculation of the SERS enhancement factor per molecule is given in the supporting information of Ref. [36]. A recent review on the present and future of the SERS is given in Ref. [38].

18.2.8.2

Surface-Enhanced Fluorescence (SEF)

Fluorescence occurs in molecules with fluorescent chemical structure, such as fluorophore, where the exciting and emitted photons are of different wavelengths. When a molecule is excited, there are several processes that will return the molecule to its ground state, such as radiationless energy loss, intersystem crossing through the triplet state, and fluorescent emission of photons [9]. The mentioned processes determine the fluorescence emission rate which is proportional to the inverse of the average lifetime of the excited state. When fluorophores are in the proximity of metallic nanostructures [7, 39], their radiative properties are modified, the spontaneous emission rate, as well as the fluorescent signal are enhanced, and the lifetime of the fluorescence is decreased [40–43]. It is important to note that the fluorophore emission depends on the fluorophore-metal distance. At distances less than a few nanometers, the emission is strongly quenched, while at intermediate distances, the fluorescence enhancement begins to dominate.

18.2.9 Simulation Methods Used in the Simulations Besides the analytical approaches such as the Airy formula, the Abeles matrix approach which is a very useful approach for solving the case of light propagation in multi-layer structures (isotropic) [44] and the 4 × 4 matrix approach for solving the case of light propagation in anisotropic multilayered medium [45]. We mention below a few useful numerical methods for treating optical structures.

18.2.9.1

Simulations Methods for Periodic Structures

RCWA When the medium has lateral variations in the lateral directions, such as in the case of gratings, then the above analytic matrix treatments are not applicable and semi-analytic or pure numerical approaches are used. Rigorous electromagnetic simulations of diffraction gratings are used in our simulations based on GSolver (see http://www.gsolver.com/), which is based on the rigorous coupled-wave analysis (RCWA) method [46, 47]. RCWA is a Fourier-space method, in which the grating is defined as a piecewise-constant, periodic, lamellar structure at the boundary between the two semi-infinite media (superstrate and substrate). The main concept

18 Cavity Modes and Surface Plasmon Waves Coupling on Nanostructured. . .

465

is to divide the structure into uniform layers in the z direction so the propagating electromagnetic modes in each layer can be calculated. The overall problem is solved by matching boundary conditions at each of the interfaces between the layers using a technique like scattering matrices. In the simulations performed by GSolver [48], the grating is illuminated by an incident plane wave with the following parameters: wavelength, two angles determine the direction of the wave travelling from the superstrate to the substrate media, and two angles determine the polarization. It is important to note that the incident electric field has unit magnitude. GSolver computes the fields inside each layer and explicitly at the boundaries. To compute the reflected and transmitted diffracted fields, an elimination for all internal fields will be done and then solving for the fields at the top and bottom interface. As we mentioned before, in RCWA method, the grating is divided into layers, the field inside each layer is computed by solving Maxwell’s equations with the permittivity expanded as a Fourier series. The result is getting a solution of the field equations with a number of unknown amplitude coefficients. To find these coefficients, an application of the boundary conditions for the tangential components of the electric and magnetic fields is done. After ordering the equations, all internal fields are eliminated, thus making the solution simpler and allowing it to proceed in an iterated manner.

FDTD The finite-difference time-domain (FDTD) method is a very useful numerical tool used for modeling physical systems. This method was developed in 1966 by Yee [49]. In this method, the structure is divided into small physical regions which are cubic cells (also called Yee cells) meaning to solve the discretized time-dependent Maxwell’s equations in an iterative manner. The calculation of the fields at the edges and the faces of the Yee cells is based on the finite difference version of the Maxwell’s equations. Lumerical is one of the most important and strong simulation tools used to simulate plasmonic structures which is based on FDTD method. This tool is considered as one of the most easy and fast simulation tools. Part of the advantages of FDTD method are [50]: (i) the simplicity of the implementation, (ii) the direct use of Maxwell’s equations without approximations, (iii) the ability to simulate both periodic and non-periodic structures, (iv) simple algorithms and large number of built-in simulations are available. On the other hand, the disadvantages in using these methods are: (i) the dielectric function analytic definition affects the computational stability, (ii) time-consuming for spectral simulations.

FEM The finite elements method (FEM) is a very useful numerical tool that helps to find approximate solutions for partial differential equations with boundary conditions

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[51]. As we mentioned before that in the case of FDTD method, the calculation of the fields is based on the finite difference version of the Maxwell’s equations, while in the case of FEM, it is based on either the Rayleigh Ritz or the Galerkin method. Like the case of the FDTD method, in the case of the FEM method, the structure is also divided into small regions called finite elements. Finite elements analysis (FEA) is the practical application of the FEM, and it is a very useful numerical tool that helps to find approximate solutions for large number of physical systems represented by the partial differential equations with boundary conditions. FEA is useful to simulate complicated structures (complicated geometry). Comsol multiphysics is one of the most important and strong simulation tools used to simulate physical structures in different scales which is based on FEA. It is possible to calculate both the near and far field both in the time and frequency domains using Comsol simulation [50]. Briefly, in Comsol multiphysics, the problem is solved by performing the following steps: (i) define the geometry of the structure, (ii) define the material in each region, (iii) define the physical equation in each region, (iv) define the boundary conditions, (v) choose the power source, (vi) use symmetry conditions if possible, and (vii) define the mesh size. Some of the advantages of FEM method are [50] (i) the ability to simulate arbitrary shapes and dielectric function, (ii) the ability to simulate periodic structures, and (iii) the ability to permit coupling between physical quantities such as fields. On the other hand, following are the disadvantages of these methods: (i) time-consuming for the simulations and (ii) complex structures require large memory.

18.3 Applications of Plasmonic-Based Nanostructures 18.3.1 Thin Dielectric Grating on Thin Metal Film (TDGTMF)-Based Self-referenced Sensor The idea behind the self-referenced configuration based on a thin dielectric grating ( d3 + d2 , ⎪ ⎪ ⎪ ⎪ ⎪ ik x −ik x ⎪ ⎪ A2 e x2 + B2 e x2 , d3 < x < d3 + d2 , ⎪ ⎨ .0 (x) = A3 eikx3 x + B3 e−ikx3 x , −d3 < x < d3 , ⎪ ⎪ ⎪ ⎪ ⎪ A4 eikx4 x + B4 e−ikx4 x , −d3 − d4 < x < −d3 , ⎪ ⎪ ⎪ ⎪ ⎩ T eikx5 x , x < −d3 − d4 ,

(19.6)

with .kx1 = k0 n1 sin θ , .kxj = k0 nj sin θj , for .j = 2, 3, 4, 5, and .θj are the (real) angles formed between the plane waves and the z-axis. The explicit expressions of the coefficients .R, Aj , Bj and T are given in Appendix A of Ref. [44]. Moreover, the Green functions .G1 and .G2 are expressed as Fourier integrals as G1,2 (x, z; x  , z ) =

.

1 4π



+∞ −∞



γ1,2 (λ, x, x  )e−iλ(z−z ) dλ,

(19.7)

with the explicit expressions of the kernels .γ1 and .γ2 given in Appendix A of Ref. [42].

19.2.3 Formulation and Solution of the Integral Equation Since the infinitely periodic structure of Fig. 19.1 is invariant under translations .z → z + m (m ∈ Z) and also since the excitation is due to a plane wave, the solutions of the Maxwell equations are also invariant up to phase factors under the same translations [51]. Thus, according to the Floquet–Bloch theorem [52], the electric field satisfies the Bloch property (pseudoperiodicity condition) [52, 53] as follows: (x, z + m) = e−ik0 n1 cos θm (x, z), m ∈ Z.

.

(19.8)

This implies that (x, z) = e−ik0 n1 z cos θ u(x, z),

.

(19.9)

where .u(x, z) is a .-periodic function of z. The integrals on .Sd1 and .Sd2 of Eq. (19.4) are reduced by means of suitable  transformations to integrals on the basic unit cells .S01 = M i=1 [−d3 , −d3 − w1 ] ×  M [ai , ai + si ] and .S02 = [d , d + w ] × [b , b + l 2 i i i ] of the two gratings, i=1 3 3 respectively. Next, substituting the Fourier integral expressions (19.7) in Eq. (19.4) and employing the Poisson summation formula for the Dirac function [54], we obtain

19 Entire-Domain Integral Equation Analysis of All-Dielectric Gratings

487

u(x, z) = 0 (x)

.

+∞ k02 (n27 − n24 )  −i 2πp z + e  2 p=−∞

 2πp  2πp , x, x  dx  dz × u1 (x  , z )ei  z γ1 k0 n1 cos θ +  S01 +∞ k02 (n26 − n22 )  −i 2πp z e  2 p=−∞  2πp  × u2 (x  , z )ei  z

+

S02

× γ2

2πp  k0 n1 cos θ + , x, x dx  dz , 

(x, z) ∈ R2 .

(19.10)

Then, the functions .u1 and .u2 are expanded in Fourier series with respect to z as +∞ 

u1 (x, z) =

.

φ1n (x)e−i(2π n/)z , (x, z) ∈ S01 ,

(19.11)

φ2n (x)e−i(2π n/)z , (x, z) ∈ S02 ,

(19.12)

n=−∞

and +∞ 

u2 (x, z) =

.

n=−∞

with φ1n (x) = cn1+ eg7n

.



x+d3 +

w1 2

+ cn1− e−g7n



x+d3 +

w1 2

,

(19.13)

x−d3 −

w2 2

,

(19.14)

and φ2n (x) = cn2+ eg6n

.

where .gj (λ) =

 w x−d3 − 22

+ cn2− e−g6n



λ2 − k02 n2j , and

gj n = gj

.

=

2π n k0 n1 cos θ + 



[k0 n1 cos θ + (2π n/)]2 − k02 n2j , j = 1, . . . , 7,

while .cn1± and .cn2± are unknown coefficients

(19.15)

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N. L. Tsitsas

Now, combining Eq. (19.10) with Eqs. (19.11) and (19.12) gives the following expression of .u(x, z) in all regions: u(x, z) = 0 (x)

.

+

+∞ +∞  k02 (n27 − n24 )   1 −i 2πp z  1+ 1+ cn Qnp (x) + cn1− Q1− Jp−n e  (x) np 2 p=−∞ n=−∞

+∞ +∞ k02 (n26 − n22 )   2 −i 2πp z Jp−n e  2 p=−∞ n=−∞   2− 2− × cn2+ Q2+ (x) + c Q (x) , (x, z) ∈ R2 , np n np

+

(19.16)

where Jq1 =

.

Jq2 =

⎧ ⎨

1 i2π q

⎩ ⎧ ⎨ ⎩

 M  i2π q ai  i2π q si   −1 e , q = 0 i=1 e ,  M 1 q=0 j =1 sj ,  (19.17)

1 i2π q

 M  i2π q bi  i2π q li   −1 e , q = 0 i=1 e ,  M 1 l , q = 0 j j =1 

1,2± while .Qnp (x) are given in Appendix B of Ref. [42]. Next, an entire-domain Galerkin procedure is applied for the determination of the unknown coefficients .cn1± and .cn2± in Eq. (19.16). This procedure is described as follows:

• The observation vector .(x, z) in Eq. (19.16) is restricted to either .Sd1 or .Sd2 . • The inner products of both sides of Eq. (19.16) are taken with the test functions   w w ±g6m x−d3 − 22 i 2πm z ±g7m x+d3 + 21 i 2πm z or .e e , respectively. e .e • All of the involved integrations are carried out analytically. • An infinite square nonhomogeneous linear system of coupled algebraic equations is obtained with respect to the unknown coefficients .cn1± and .cn2± . This system has a .2 × 2 block-matrix form with infinite-size blocks. • For the numerical solution of the infinite system, the terms of the expansions in Eqs. (19.11) and (19.12) as well as the test functions in the inner products are considered with maximum block truncation order N. • The respective truncated .(8N + 4) × (8N + 4) nonhomogeneous linear system is obtained. This truncated system has the following form:

19 Entire-Domain Integral Equation Analysis of All-Dielectric Gratings

⎡

+− ++ +− A++ 11 A11 A12 A12

⎤⎡

c1+

⎤

⎡

b1+

⎤

⎢ −+ −− −+ −− ⎥ ⎢ 1− ⎥ ⎢ 1− ⎥ ⎢ A11 A11 A12 A12 ⎥ ⎢ c ⎥ ⎢ b ⎥ ⎥⎢ ⎥ ⎢ ⎥ .⎢ ⎢ ++ +− ++ +− ⎥ ⎢ 2+ ⎥ = ⎢ 2+ ⎥ , ⎣ A21 A21 A22 A22 ⎦ ⎣ c ⎦ ⎣ b ⎦ −− −+ −− A−+ 21 A21 A22 A22

c2−

489

(19.18)

b2−

where .c1± and .c2± are .2N + 1 column vectors of the coefficients .cn1± and .cn2± , .A±± ij (.i, j = 1, 2) are .(2N + 1) × (2N + 1) matrices with elements given by Eq. (27) of Ref. [42], and .b1± and .b2± are .2N + 1 column vectors with elements given by Eqs. (24) and (25) of Ref. [44]. • The coefficients .cn1± and .cn2± , for .n = −N, . . . , N, are finally determined by the numerical solution of the linear system (19.18).

19.2.4 Computation of the Diffracted Fields and Powers Once the coefficients .cn1± and .cn2± have been determined, the fields in the layered grating structure are computed by means of the basic representation (19.16). Particularly, the electric fields reflected and transmitted (refracted) by the layered grating structure are found through  r (x, z) =

+∞ 

rp e−i(kxp x+kzp z) , x > d3 + d2 , z ∈ R,

(19.19)

tp e−i(kxp x+kzp z) , x < −d3 − d4 , z ∈ R,

(19.20)

r

.

p=−∞

and  t (x, z) =

+∞ 

.

t

p=−∞

where 2πp ,. kzp = k0 n1 cos θ + 

r kxp = −i (kzp )2 − k02 n21 = −ig1p ,

(19.22)

t 2 2 kxp . = i (kzp )2 − k n = ig5p . 0 5

(19.23)

.

(19.21)

and

In Eqs. (19.19) and (19.20), .rp and .tp are the complex amplitudes of the p-reflected and p-transmitted (refracted) diffracted orders, respectively. They are given, for .p ∈ Z, by

490

N. L. Tsitsas +∞  k02 (n27 − n24 )   1 1+ 1− Jp−n (cn1+ ρnp + cn1− ρnp ) .rp = δp0 R + 2 n=−∞ +∞  k02 (n26 − n22 )   2 2+ 2− Jp−n (cn2+ ρnp + cn2− ρnp ) 2 n=−∞

+

(19.24)

and tp = δp0 T +

.

+

+∞  k02 (n27 − n24 )   1 1+ 1− Jp−n (cn1+ τnp + cn1− τnp ) 2 n=−∞

+∞  k02 (n26 − n22 )   2 2+ 2− Jp−n (cn2+ τnp + cn2− τnp ) , 2 n=−∞

(19.25)

where .δp0 the Kronecker symbol (namely, .δ00 = 1 and .δ0p = 0, p = 0), while 1± , .ρ 2± , .τ 1± , and .τ 2± are given in Appendix B of Ref. [44]. functions .ρnp np np np In Eqs. (19.24) and (19.25), the terms involving R and T correspond to the reflection and transmission from the non-periodic (homogeneous) structure. Distinguishing these terms is feasible due to the integral equation method employed and can be useful in optimization schemes where one needs to isolate the effects of the background layers from those of the two gratings so that the layered grating structure is designed to yield a desired overall response. The conditions that need to be fulfilled so that the p-reflected and p-transmitted orders are propagating are .

  n1 (1 + cos θ ) < p < n1 (1 − cos θ ) λ0 λ0

(19.26)

  (n5 + n1 cos θ ) < p < (n5 − n1 cos θ ). λ0 λ0

(19.27)

−

and .

−

The power densities corresponding to the p-reflected and -transmitted orders are defined as [55] 

2 |r | λ0 2 p r 1 − cos θ + p .Pp = sin θ n1 

(19.28)

and t .Pp

|tp |2 = sin θ



λ0 2 − cos θ + p . n1  n21

n25

(19.29)

19 Entire-Domain Integral Equation Analysis of All-Dielectric Gratings

491

The reflectance/transmittance of the layered structure is defined as the finite sum of the power densities of all Floquet harmonics departing from the structure and propagating in the reflection/transmission half-space.

19.3 Evaluation of the Integral Equation Method The main beneficial characteristics of the presented volume integral equation Galerkin method with entire-domain basis functions concern the guaranteed convergence, the high numerical stability and efficiency, and the controllable accuracy of the derived semi-analytical solutions. Particularly, the attained computational efficiency is mainly due to the fact that the unknown electric field and the entiredomain expansion functions satisfy the same physical laws, namely the Maxwell equations. The required truncation order in the Fourier series (19.11) and (19.12) is determined by applying a convergence control to the solutions for an increasing number N of the expansion functions. One of the basic advantages of the integral equation method is that relatively small values of N provide sufficient (for most of the applications) accuracy. This assures, further, speed in execution time and economy in computer memory. A representative convergence pattern of the method showing the normalized truncation error in the .2 -norm as a function of N is depicted in Fig. 3 of Ref. [44]. Moreover, the utilized layered medium (background) Green function is expressed analytically in the Fourier domain by means of Eq. (19.7). For this reason, it is feasible all the involved integrations to be carried out analytically. Besides, this Green function provides a compact formulation inherently satisfying the transmission conditions of the associated boundary-value problem in the non-grating structure. Hence, no discretization of the involved integral equation is required, and the sole approximation in the derived solutions concerns the final truncation of the expansion function sets. On the contrary, in other developed integral equation methods treating similar grating problems, such as, e.g., the ones in Refs. [28] and [32], the freespace Green function is employed, and the obtained accuracy depends strongly on the discretization in boundary elements forming the supports of the considered subdomain basis functions. Validations of the numerical code developed to implement the presented volume integral equation method with respective results obtained by other methods are included in Refs. [42–44]. Numerical comparisons with the results of other methods (such as, e.g., those of Refs. [56] and [57] for the case of a single grating) are included in Refs. [42] and [43] and reveal that the presented entire-domain integral equation method requires a significantly reduced number of terms to compute accurately the diffracted fields.

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19.4 Numerical Results and Discussion This section contains some representative numerical results, derived by the presented volume-integral-equation entire-domain Galerkin method, concerning frequency-selective operation of a layered structure and manipulation of the diffracted beams to propagate in prescribed directions.

19.4.1 Two Shifted Gratings Consider the dielectric structure composed of the two free-standing shifted gratings shown in the top panel of Fig. 19.2. This structure is obtained from the layered grating structure of Fig. 19.1 for .n1 = n2 = n3 = n4 = n5 = 1, .n26 = n27 = 2.59 (Plexiglas), .w1 = w2 = 1.29 cm, .d3 = 0, . = 3 cm, .M = 1, . ≡ s1 = l1 = 0.5, and .a1 = 0. By varying the parameter .b1 , a different shift . is obtained between the two gratings. In the bottom panel of Fig. 19.2, the frequency dependence of the reflectance is depicted for different values of the grating’s normalized shift ./; the case . = 0 corresponds to the two gratings stacked vertically exactly one on top of the other. The reflectance results of Fig. 19.2 for . = 0 are in excellent agreement with those presented in Ref. [7] and derived by means of the rigorous coupled-wave analysis [30]. The spatial shift of the gratings by . causes some interesting effects in the resonance frequency of such a reflection filter. Precisely, as ./ increases, the resonance frequency decreases, while the bandwidth of the resonance exhibits a significantly increasing broadening.

19.4.2 Array of Free-Standing Rectangular Bars Now, we examine single-layer all-dielectric metasurfaces supporting two propagating diffracted orders (the 0- and the .−1-order) and steering the major part of their diffracted powers in the .−1-order. Structures of this type were investigated extensively in Refs. [58] and [59] aiming to determine the optimal metasurface’s parameters for generating the so-called anomalous reflection and refraction effects, respectively, namely to guide significant portion of the diffracted powers in directions different than the specular ones, i.e., different than the 0-order ones in the reflection and refraction regions. Consider the array of free-standing rectangular hafnium dioxide (.HfO2 ) bars, with refractive index .n = 2.12, depicted in the top panel of Fig. 19.3. The period of the array is . = 678 nm, and its width is .s = 0.34. The plane wave exciting the metasurface has an angle of incidence .θ = 66o . The free-space wavelength is .λ = 530 nm and corresponds to the green color of the visible range. For these values of parameters, the bars array supports only the 0- and .−1-propagating diffracted orders

19 Entire-Domain Integral Equation Analysis of All-Dielectric Gratings

493

1 0.9 0.8

reflectance

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 8.3

8.4

8.5

8.6

8.7

8.8

8.9

frequency (GHz) Fig. 19.2 Top panel: shifted two-gratings structure composed of Plexiglas dielectric bars with thickness 1.29 cm, width 1.5 cm, and period 3 cm and excited by a plane wave under normal incidence. The common width of the two gratings is . = 0.5, while the spatial shift . varies. Bottom panel: frequency variations of the reflectance from the two-grating structure for . = 0 and ./ = 0.03, 0.06, 0.09

in reflection and refraction. The bottom panel of Fig. 19.3 shows the variations of r , and .P t of the propagating reflected and refracted the power densities .P0r , .P0t , .P−1 −1 orders as functions of the thickness w of the rectangular bars. t attains a maximum of 92% It is observed that the refracted power density .P−1 at thickness .w = 174 nm. Moreover, for an overall range of 70 nm to both sides t of this thickness, i.e., from .w = 150 nm to .w = 220 nm, the power density .P−1

494

N. L. Tsitsas

Fig. 19.3 Top panel: .-periodic array of free-space-standing bars composed of hafnium dioxide (of refractive index .n = 2.12) with thickness w and duty cycle s and excited by a plane wave under oblique incidence. Bottom panel: variations of the propagating diffracted orders power r , and .P t versus the thickness w for . = 678 nm and .s = 0.34, with angle densities .P0r , .P0t , .P−1 −1 of incidence .θ = 66o and free-space wavelength .λ = 530 nm corresponding to the green color of the visible range

.HfO2

remains above 80%. This fact can provide robustness in experimental realizations since the (negative) steering toward the direction of the .−1-refracted order will not be affected substantially by manufacturing imperfections.

19.4.3 Array Composed of Two Elements Per Unit Cell In different wavelengths, more than one element per unit cell of the array of freestanding cylinders may be required to obtain enhanced refraction in the .−1-order. A paradigm system for the case of the violet color (.λ = 420 nm) is depicted in the top

19 Entire-Domain Integral Equation Analysis of All-Dielectric Gratings

495

Fig. 19.4 Top panel: .-periodic free-space-standing array of alternating bars composed of hafnium dioxide .HfO2 (of refractive index .nb = 2.15) and magnesium fluoride (.MgF2 ) (of refractive index .na = 1.38) with thickness w and duty cycle s and excited by a plane wave under oblique incidence. Bottom panel: variations of the propagating diffracted orders power densities t t r r .P0 , .P0 , .P−1 , and .P−1 versus the thickness w for . = 449 nm and .s = 0.28, with angle of incidence .θ = 62o and free-space wavelength .λ = 420 nm corresponding to the violet color of the visible range

panel of Fig. 19.4. The considered array has period . = 449 nm and comprises two elements per unit cell, precisely: hafnium dioxide (.HfO2 ) bars of width .0.28 and refractive index .nb = 2.15 and magnesium fluoride (.MgF2 ) bars of width .0.72 and refractive index .na = 1.38. The angle of incidence is .θ = 62o . The bottom panel r , and .P t of the propagating of Fig. 19.4 depicts the power densities .P0r , .P0t , .P−1 −1 reflected and refracted orders versus the thickness w of the rectangular bars forming the two-element array. t in this case is 86% and The maximum obtained refracted power density .P−1 t remains attained at .w = 166 nm. As in Fig. 19.3, also here, the power density .P−1 above 80% for a significant interval of thicknesses being from .w = 143 nm to .w = 180 nm.

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N. L. Tsitsas

19.4.4 Equal-Power Splitter Finally, in this section, it is demonstrated that arrays of free-standing cylinders can also be used as power splitters/dividers. The top panel of Fig. 19.5 shows an array of rectangular bars composed of hafnium dioxide (.HfO2 ), with refractive index .n = 2.11, considered at free-space wavelength .λ = 610 nm corresponding to the orange color of the visible range. The array has period . = 781 nm and width .s = 0.13; hence, it is quite sparse. The plane wave impinges on the array at an angle r , and .P t of incidence .θ = 66o . The variations of the power densities .P0r , .P0t , .P−1 −1 of the propagating reflected and refracted orders as functions of the thickness w of the rectangular bars are depicted in the bottom panel of Fig. 19.5. Similarly to the arrays of Figs. 19.3 and 19.4, the array of Fig. 19.5 also exhibits significant negative refraction. Precisely, from .w = 265 nm to .w = 400 nm t is remarkably large, (excluding two very narrow intervals), the power density .P−1 t i.e., .P−1 is larger than 90% and it even reaches exactly 100%. Importantly, here a new phenomenon is observed (not observed in the previously examined arrays above): at .w1 = 334 nm and .w2 = 357 nm, an equal-power splitting occurs because then all diffracted powers are almost equal to .25%. Thus, if the array is viewed as a four-port network, at .w1 and .w2 , the overall diffracted power exits the network equally shared/divided between the four output ports. For .w ∈ (w1 , w2 ), almost t is from 98.5% to 100% meaning perfect negative refraction is observed, since .P−1 that almost all the incident field’s power is steered to the .−1-refracted order.

19.5 Conclusions and Prospects Over the past decades, several different methodologies have been developed for the analysis of guiding and diffraction by all-dielectric grating-assisted slab waveguides. Still, emerging microwave and optical applications concerning the manipulation of waves diffracted by all-dielectric metasurfaces continue to motivate the systematic analysis of grating configurations. To this end, optimizing efficiently the parameters of all-dielectric grating structures to steer diffracted electromagnetic waves to prescribed directions becomes necessary. Hence, in this context, application of a fast and robust numerical method for the solution of the associated boundary-value problem proves particularly useful for implementing efficient optimization schemes. This chapter analyzed a volume-integral-equation Galerkin method for the electromagnetic modeling of diffraction and guidance by all-dielectric planar-layered structures incorporating two dielectric gratings in two different layers. A Fredholm integral equation of the second kind was employed for the unknown electric fields in the periodic gratings. In the integrals’ kernels, the Green function was employed corresponding to line-source excitation of the layered medium in the absence of both gratings. This Fredholm integral equation was solved by applying an entire-

19 Entire-Domain Integral Equation Analysis of All-Dielectric Gratings

497

Fig. 19.5 Top panel: .-periodic array of free-space-standing bars composed of hafnium dioxide (of refractive index .n = 2.11) with thickness w and duty cycle s and excited by a plane wave under oblique incidence. Bottom panel: variations of the propagating diffracted orders power r , and .P t versus the thickness w for . = 781 nm and .s = 0.13, with angle densities .P0r , .P0t , .P−1 −1 of incidence .θ = 66o and free-space wavelength .λ = 610 nm corresponding to the orange color of the visible range

.HfO2

domain Galerkin technique, based on a Fourier-series expansion of the electric field in the gratings’ domains. In this way, a linear system was derived, the solution of which provides the powers of the propagating reflection and refraction orders by the layered structure. The method is fast, accurate, and numerically efficient, and hence, it is particularly suited for optimizations such as the ones discussed above. The analysis concerned the case of E-polarized waves. It is challenging to analyze also the case of H -polarized waves by modifying and extending the developed integral equation techniques. Future work directions may concern the application of the presented method to inverse design problems concerning the channeling of the incident field’s power

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to specific single-diffracted orders [60] as well as the optimization of the optical wavefront over a spectrum of frequencies and angles of incidence [61]. Also, applications related to employing phase-change materials in grating structures [62] as well as to breaking transmission symmetry by using all-dielectric metagratings can be considered [63]. Besides, it is interesting to study the effects on the scattering performances when the gratings are composed of materials with low permittivity [64]. Finally, this method can be used as an appropriate optimization platform assisting deep-learning algorithms in the design of all-dielectric metasurfaces [65, 66]. Acknowledgments The author thanks sincerely Akhlesh Lakhtakia and Tom Mackay for the invitation to the special symposium honoring the memory and the legacy of Werner Weiglhofer.

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Nikolaos L. Tsitsas was born in Athens, Greece, in 1979. He received the Diploma and Ph.D. degree in electrical engineering from the National Technical University of Athens (NTUA) in 2002 and 2006, respectively, and the M.Sc. degree in applied mathematics from the National and Kapodistrian University of Athens in 2005. From 2008 to 2011, he was an Adjunct Lecturer with the School of Applied Mathematical and Physical Sciences, NTUA. From 2009 to 2011, he was an Adjunct Lecturer with the Hellenic Army Academy. Since 2012, he has been with the School of Informatics, Aristotle University of Thessaloniki, Greece, where he is currently an Associate Professor. He is the author or coauthor of 80 papers in scientific journals and over 80 papers in conference proceedings. His research is focused on methodologies of applied mathematics in direct and inverse wave scattering and propagation theory. He is a member of the American Mathematical Society, a senior member of the Institute of Electrical and Electronics Engineers (IEEE), a senior member of the Optica (formerly Optical Society of America) and a senior member of the Union Radio-Scientifique Internationale (URSI). He was the Guest Editor of the special issue Analytical Methods in Wave Scattering and Diffraction in Mathematics (MDPI) journal, while he has also organized four special sessions in international conferences on computational electromagnetics and photonics.

Chapter 20

Rigorous Coupled-Wave Approach and Transformation Optics Benjamin J. Civiletti, Akhlesh Lakhtakia, and Peter B. Monk

20.1 Introduction We are interested in the optimal design of thin-film photovoltaic solar cells [1] based on predicting the power-conversion efficiency of different designs. The first step in predicting the efficiency is to determine the electromagnetic field everywhere inside the solar cell due to impinging solar radiation and hence predict the electron– hole generation rate in the semiconductor layers of the device. This must be done at wavelengths ranging from approximately 400–1200 nm in 1–2-nm increments. Furthermore, as the design process is iterative, many widely different designs usually need to be analyzed. Thus the algorithm for solving the frequency-domain Maxwell equations in this application needs to be efficient, and it also must be versatile to accommodate photon absorption in diverse semiconductors. Many techniques are available to solve this problem. The finite-element method (FEM), together with the perfectly matched layer domain truncation above and below the solar cell (see, e.g., Ref. [2]), can solve the underlying electromagnetic problem with graded index of refraction and complex geometry. Indeed, we use numerical results computed using NGSpy [3] FEM software to provide accurate solutions to determine the errors for the algorithms to be discussed later in this chapter. The main disadvantages of the FEM are: (i) the long solution time when using direct solution methods and (ii) the need to remesh whenever the interfacial regions between different materials change.

B. J. Civiletti · P. B. Monk () Department of Mathematical Sciences, University of Delaware, Newark, DE, USA e-mail: [email protected]; [email protected] A. Lakhtakia NanoMM—Nanoengineered Metamaterials Group, Department of Engineering Science and Mechanics, The Pennsylvania State University, University Park, PA, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 T. G. Mackay, A. Lakhtakia (eds.), Adventures in Contemporary Electromagnetic Theory, https://doi.org/10.1007/978-3-031-24617-3_20

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Creating a full mesh can be avoided if integral-equation techniques are used. Such methods can be very fast and flexible [4, 5]. Interest in using integral-equation solvers has also recently resulted in advances in computing the quasi-periodic fundamental solution rapidly near Rayleigh–Wood anomalies [6]. However, integralequations are not well suited to solving problems featuring graded-index materials that are central to our interest in designing efficient solar cells. Since we wish to avoid creating FEM meshes for different designs, one attractive and frequently used technique is the rigorous coupled-wave approach (RCWA) that is based on the use of Fourier series and a special fast solution technique. RCWA was first proposed by Gaylord et al. [7]. An important stabilized fast solution technique was developed subsequently [8]. Further progress included the best choice of Fourier expansions for the relative permittivity for incident s- and p-polarized light [9, 10]. The resulting algorithm is described in detail in Ref. [11] for s- and p-polarized incidence conditions for two-dimensional (2D) geometries as well as more generally for three-dimensional (3D) geometries. Our experience with RCWA is positive [12], so we decided to analyze convergence of the technique, starting with s-polarized incidence conditions in 2D geometries, all materials taken to be describable by a scalar frequency-dependent relative permittivity [13]. Since the Cartesian components of the electric field for spolarized light have two weak derivatives, we were able to obtain a satisfactory convergence theory under the assumption of a non-trapping structure (which, essentially, means a monotonic increasing or decreasing relative permittivity). We shall not discuss that case more here. Continuing our study, we moved on to p-polarized incidence conditions in 2D geometries [14]. Convergence of RCWA for this problem is more difficult to analyze, since the governing Helmholtz equation involves the relative permittivity in the principal part of the differential operator. Still under the non-trapping assumption, we were able to prove convergence although the expected rate of convergence is less than that for s-polarized incidence conditions. The generalization of this theory to allow less restrictive assumptions on the relative permittivity is still an open problem. We discuss the p-polarization case further in Sect. 20.3. Because of the difficulties inherent in RCWA for the p-polarization case, we then studied a variant of the C-method of Chandezon and co-workers [15, 16]. This method, which is restricted to structures in which boundaries between dissimilar materials are graphs of single-valued functions, uses transformation optics to flatten bimedium interfaces at the cost of introducing anisotropic material parameters. Our version [17] uses a different transformation to the original work [15, 16] so that multiple layers can be handled easily, and we suggest the use of RCWA on the transformed structure to solve the transformed boundary-value problem. The solution in the actual spatial domain can then be computed by inverting the transformation. Both numerical analysis and computational experiments attest to the rapid convergence of this method. We term our version C-RCWA and discuss it in Sect. 20.4. A particularly interesting observation is that C-RCWA can be applied to the Maxwell equations in biperiodic media with little or no change. If RCWA is used

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directly, double Fourier series are used to expand the components of the electric and magnetic fields [11, 18]. At bimedium interfaces across which the relative permittivity jumps, the normal component of the electric field is discontinuous. Thus we expect a slow convergence rate due to the Gibbs phenomenon. While modifications to the basic RCWA have been suggested (see, e.g., Ref. [19]), these complicate the implementation considerably (and changes in interfaces need special processing to identify normal components) [20, 21]. However, if the structure to be analyzed has interfaces that are graphs of single-valued functions, transformation optics can again be applied and the transformed problem solved by RCWA. In this chapter, we summarize the development of C-RCWA [17]. First, we discuss to what extent our existing proof of convergence of RCWA for p-polarized light [14] can be extended to C-RCWA. We note that the main difficulty with the analysis of RCWA is to prove mesh-independent stability of an intermediate problem arising in our analysis, and we show how C-RCWA removes this difficulty. We then describe a few salient features of C-RCWA, some √ of which are new. We now summarize our notation. We denote by .i = −1. Vector quantities and function spaces are identified using a boldface font. We use several function spaces without introduction. In particular for a suitable open set .S ⊂ Rn (.n = 2, 3) with boundary .∂S, we denote by .L2 (S) (resp., .L2 (T )) the space of square integrable functions on S (resp., .T ⊂ ∂S). The corresponding inner products are denoted by  (u, v)S =

.

u · v dx

(20.1a)

u · v dS,

(20.1b)

S

and  u, vT =

.

T

respectively, where the overline indicates complex conjugation. The corresponding L2 norm is denoted  (u, u)S . (20.2) .u =

.

Then, H 1 (S) = {p ∈ L2 (S) | ∇p ∈ L2 (S)}

.

(20.3)

with norm uH 1 (S) =

.



(∇u, ∇u)S + (u, u)S .

(20.4)

The energy space for the Maxwell equations is H(curl; S) = {u ∈ L2 (S) | ∇ × u ∈ L2 (S)}.

.

(20.5)

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20.2 The Maxwell Equations In order to handle dispersive media, we solve the time-harmonic Maxwell equations at a fixed angular frequency .ω assuming that the time dependence is given by .exp(−iωt) with t as time. We denote by .ε(x) the spatially dependent relative permittivity at a point .x ∈ R3 and assume that the relative permeability is unity (consistent with the materials used in a solar cell). Once we apply transformation optics, we encounter a relative permittivity tensor and a relative permeability tensor. √ The wave number in air is defined by .κ = ω ε0 μ0 , where .ε0 and .μ0 are the permittivity and permeability of free space, respectively. Then, after denoting by ˜ ˜ ∈ C3 and .H(x) ∈ C3 the electric and magnetic field phasors, respectively, the .E(x) Maxwell curl equations can be written as .

˜ = −iκεE˜ ∇× H

(20.6a)

˜ ∇ × E˜ = iκ H,

(20.6b)

and .

√ where we have scaled the actual physical electric and magnetic phasors by . ε0 √ and . μ0 , respectively, to remove the impedance of free space from subsequent ˜ to obtain equations. We can eliminate .H .

  ∇ × ∇ × E˜ − κ 2 εE˜ = 0,

(20.7)

and this form is used in the remainder of this chapter. We use a discrete form of (20.6) to compute solutions. In our applications, .ε(x) ∈ C, where .x = (x1 , x2 , x3 )T . If .Im[ε(x)] = 0, then we assume that .Re[ε(x)] is strictly positive. If .Im[ε(x)] > 0, then .Re[ε(x)] can be positive (as in semiconductors) or negative (as in certain metals). In all cases, .ε is piecewise smooth. In order to avoid loss of uniqueness in the solution, .Im[ε(x)] > 0 is assumed in at least one sub-region of the solar cell. This in turn guarantees existence of a solution for all .κ provided anomalous frequencies are avoided (see after Eq. (20.14) for the definition of these frequencies). A defining characteristic of thin-film solar cells is that they are of bounded height. So we assume that there is a parameter .H > 0 such that .ε(x) = 1 for .|x3 | > H . The solar cell is illuminated from above by a plane wave, due to the Sun, propagating in the direction .d = (d1 , d2 , d3 )T ∈ R3 (where .d = 1 and .d3 < 0) having polarization .p ∈ R3 , .p = 0. In order to ensure a divergence-free solution, we set .d · p = 0. This incident plane wave, which satisfies the Maxwell equations above the device, is specified by

20 RCWA and Transformation Optics

.

E˜ i (x) = p exp(iκd · x) ˜ i (x) = (iκ)−1 ∇ × Ei (x) H

507

 (20.8)

.

We now describe the full 3D problem and the restricted 2D problem to be studied here.

20.2.1 Biperiodic Solar Cell in 3D Modeling the solar cell as a biperiodic structure, we assume that it is periodic in .x1 with period .L1 and periodic in .x2 with period .L2 . Thus for any .x, ε(x1 + nL1 , x2 + mL2 , x3 ) = ε(x1 , x2 , x3 ) , ∀n ∈ Z, m ∈ Z .

.

(20.9)

The incident field is not generally periodic unless the incidence direction .d = (0, 0, ±1)T . Using Floquet–Bloch theory [11], we insist that the electromagnetic field in the cell be quasi-biperiodic so that

.

⎫ ˜ 1 , x2 , x3 )⎬ ˜ 1 + nL1 , x2 + mL2 , x3 ) = exp[iκ(nL1 d1 + mL2 d2 )] E(x E(x ˜ 1 , x2 , x3 )⎭ ˜ 1 + nL1 , x2 + mL2 , x3 ) = exp[iκ(nL1 d1 + mL2 d2 )] H(x H(x

, ∀n ∈ Z, m ∈ Z .

(20.10) As a result of imposing quasi-periodicity, it suffices to solve for the electromagnetic field in the semi-infinite cylinder 3D = (−L1 /2, L1 /2) × (−L2 /2, L2 /2) × (−H, ∞)

.

(20.11)

with quasi-periodic boundary conditions. For simplicity, we also assume the lower boundary .x3 = −H of the solar cell to be perfectly electrically conducting (PEC), so that E˜ 1 (x1 , x2 , −H ) = E˜ 2 (x1 , x2 , −H ) = 0 .

.

(20.12)

A suitable upward-propagating radiation condition needs to be imposed above ˜ s = E˜ − E˜ i . It remains the device. In order to do this, we define the scattered field .E s ˜ for us to specify a radiation condition for .E when .x3 > H . This can be done using a Dirichlet-to-Neumann map on the surface .x3 = H to enforce that the scattered field is a combination of upward-propagating modes, together with exponentially decaying evanescent modes [20–23]. To this end, we follow Ref. [22] and note that the electromagnetic field satisfies the Maxwell equations for a homogeneous medium together with quasi-periodic boundary conditions in the domain + 3D = {x | − L1 /2 < x1 < L1 /2, −L2 /2 < x2 < L2 /2, x3 > H }.

.

(20.13)

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Using the method of separation of variables, we expand .E˜ s as a Fourier series where each component satisfies the Helmholtz equation. Therefore,

˜ s (x) = E

.

An exp [i (α n · x + βn x3 )] ,

(20.14)

n∈Z

2

where .{An }n∈Z2 are constant vectors of Fourier coefficients. Furthermore, with .n = (n1 , n2 ), we have .α n = (κd1 + 2π n1 /L1 , κd2 + 2π n2 /L2 , 0)T and   2 2 2 2 κ − |α n | , if κ > |α n | , .βn = 2 2 2 i |α n | − κ , if κ < |α n |2 .

(20.15)

It is necessary to assume that .κ is such that .κ 2 = |α n |2 for any .n to avoid anomalous frequencies, and this excludes resonant cases. The sign of the term .βn in the above expansion is chosen so that the scattered field is either outgoing (if .βn is real so that mode is propagating) or evanescent (if .βn is imaginary so that the mode is exponentially decaying and non-propagating). Note also that only the first two components of .An are independent, since the third component can be determined from the first two via the divergence-free condition .∇ · E˜ s = 0 in .+ 3D that gives βn An,3 + α n · An = 0.

.

(20.16)

Then, setting .e3 = (0, 0, 1)T , we can explicitly calculate .

  ∇ × E˜ s × e3 = Bκ (EsT |x3 =H ) ,

(20.17)

where the operator .Bκ is specified as Bκ f = i

.

1 (βn )2 (fn,1 , fn,2 , 0)T + (α n · fn )α n exp(iα n · x) βn 2 n∈Z

(20.18)

with fn = (fn,1 , fn,2 , fn,3 )T = (L1 L2 )−1



L1



L2

.

0

f(x) exp(−α n · x) dx2 dx1 .

0

(20.19)

We also define the computational domain H 3D = {x ∈ 3D | − H < x3 < H }

.

(20.20)

and its upper boundary 3D H = {x | − L1 /2 ≤ x1 ≤ L1 /2, −L2 /2 ≤ x2 ≤ L2 /2, x3 = H }.

.

(20.21)

20 RCWA and Transformation Optics

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Now we use the usual strategy for determining a Galerkin variational problem ˜ We take the dot product of Eq. (20.7) with the complex conjugate of a smooth for .E. vector test function .v, integrate over .H 3D , and integrate the curl–curl term by parts. After slightly modifying a result from Ref. [22] to allow for a lower PEC boundary condition, the problem of solving the boundary-value problem is then to seek the ˜ ∈ X such that total electric field phasor .E ˜ ∇ × v) H − κ 2 (εE, ˜ v) H − Bκ E˜ T , v 3D = F(v) ∀v ∈ X , (∇ × E,  

.

3D

3D

H

(20.22)

where .E˜ T = (E˜ 1 , E˜ 2 , 0)T is the tangential component of .E˜ and X = {v ∈ H (curl; H 3D ) | v is quasi-biperiodic and vT = 0 when x3 = −H }. (20.23) The data functional .F is given from the incident field by .

 F(v) =

.

H 3D

[(∇ × E˜ i ) × e3 − Bκ E˜ iT ] · vT dS.

(20.24)

˜ with a nonUsing (20.22), we can solve the Maxwell equations for .E˜ and .H homogeneous boundary condition on the plane .x3 = H . Equivalently, we can ˜ s using homogeneous modify the variational formulation and solve for .E˜ s and .H boundary conditions and a suitable source current density arising from the incident field. Existence and uniqueness of a solution for any non-resonant .κ > 0 can be verified; see Refs. [22] and [23].

20.2.2 Periodic Solar Cell in 2D Often solar cells are translationally invariant in one direction. Suppose that .ε is ˜ can be chosen independent of independent of .x2 . If .d2 = 0 in addition, .E˜ and .H .x2 . Furthermore, assuming that the incident plane wave is p-polarized, we have ˜ = (E˜ 1 , 0, E˜ 3 )T and .H ˜ = (0, H˜ 2 , 0)T , and the Maxwell curl equations simplify to .E ˜

−iκεE˜ 1 = − ∂∂xH32 −iκεE˜ 3 =

.

iκ H˜ 2 =

∂ H˜ 2 ∂x1

∂ E˜ 1 ∂x3

−

∂ E˜ 3 ∂x1

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭

.

(20.25)

Obviously, we can eliminate .E˜ 1 as well as .E˜ 2 and define .u = H˜ 2 to obtain the scalar Helmholtz equation ∇ · ε−1 ∇u + κ 2 u = 0.

.

(20.26)

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The PEC boundary condition for the total field on the plane .x3 = −H reduces to E˜ 1 = 0, which becomes

.

ε−1

.

∂u = 0 on x3 = −H ∂x3

(20.27)

after using the first of Eqs. (20.25). As in Sect. 20.2.1, the incident plane wave propagates down onto the cell. In particular, the incident plane wave becomes ui (x1 , x3 ) = exp[iκ(x1 d1 + x3 d3 )],

.

(20.28)

where .d12 + d32 = 1 and .d3 < 0. The total field u is then given as the sum of the incident field and an unknown scattered field us = u − ui

.

(20.29)

that is composed of outgoing travelling waves and evanescent waves above the device. As in Sect. 20.2.1, the quasi-periodicity of the incident field and Floquet–Bloch theory requires u to be quasi-periodic, i.e., u(x1 + nL1 , x3 ) = exp(iκnL1 d1 ) u(x1 , x3 ) , ∀n ∈ Z,

.

(20.30)

for all .x1 and .x3 . This quasi-periodicity of the solution allows us to restrict the computational domain to the region .2D = {(x1 , x3 ) | − L1 /2 < x1 < L1 /2, x3 > −H }, with the quasi-periodicity condition prevailing on .x1 = ±L1 /2 and the PEC boundary condition on the plane .x3 = −H . We also still need an upward-propagating radiation condition. Moreover, from the point of view of computation (and theory), it is convenient to reduce the problem further to one posed on the following bounded domain containing one period of the solar cell: H 2D = {(x1 , x3 ) | − L1 /2 < x1 < L1 /2, −H < x3 < H }.

.

(20.31)

As in the 3D case formulated in Sect. 20.2.1, we need to provide a boundary condition on the upper boundary 2D H = {(x1 , x3 ) | − L1 /2 ≤ x1 ≤ L1 /2, x3 = H }.

.

(20.32)

In particular, we use a standard Dirichlet-to-Neumann map on .x3 = H [14]. For a function .φ defined on .x3 = H with Fourier expansion

20 RCWA and Transformation Optics

511

φ(x1 ) =



.

φn exp(iαn x1 ),

(20.33)

n∈Z

where .αn = κd1 + 2π n/L1 . After defining (Tκ2D φ)(x1 ) = i



.

 φn κ 2 − αn2 exp(iαn x1 ),

(20.34)

n∈Z

we get .

∂us |x =H = Tκ2D (us |x3 =H ) ; ∂x3 3

(20.35)

note the similarity of the right side of Eq. (20.34) to the first term in the Dirichletto-Neumann map (20.18) for the 3D problem. Just as in Sect. 20.2.1, we can summarize the model using a weak formulation. For this, we need the space 1 1 H Hqp (H 2D ) = {u ∈ H (2D ) | u is quasi-periodic in x1 }

(20.36)

.

of quasi-periodic functions. Multiplying (20.26) by the complex conjugate of a smooth test function v and integrating over .H 2D , and then integrating by parts and 1 (H ) using the boundary and quasi-periodicity conditions, we see that .u ∈ Hqp 2D satisfies   −1 1 .(ε ∇u, ∇v)H − κ 2 (u, v)H − TH2D (u|x3 =H ), v 2D = F (v) ∀v ∈ Hqp (H 2D ), 2D

2D

H

(20.37)

where 

∂ui .F (v) = − Tκ2D (ui |x3 =H ), v ∂x3

 (20.38)

. 2D H

Equivalently, we could pose the problem for the unknown scattered field .us with homogeneous boundary conditions and a distributed source computed from the incident field [14]. The problem has a unique solution for any .κ > 0 under the stated conditions on .ε, details being available in Ref. [14]. For ease of use later, we define the sesquilinear form   Bε (u, v) = (ε−1 ∇u, ∇v)H − κ 2 (u, v)H − TH2D (u|x3 =H ), v

.

2D

2D

2D H

.

(20.39)

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20.3 RCWA for p-polarization Case From a mathematical point of view, the development of the RCWA to compute an approximate solution to the p-polarized total field satisfying Eq. (20.37) proceeds in two steps: 1. Approximation of the relative permittivity .ε by a function .εh . This approximation is chosen to be piecewise constant in .x3 . 2. Approximation of the solution of the perturbed problem using a finite Fourier expansion of the solution in .x1 . An approximation analysis of the problem can then be carried out by estimating the error introduced by each of these discretization steps. The computational implementation of the method does not use the Galerkin approach [24]. We give some general comments on the computational problem later in this section, but for full details we refer to Ref. [11]. Now we discuss each of the two steps in the algorithm stated in the previous paragraph.

20.3.1 1st Step: Discretization of Relative Permittivity The relative permittivity .ε is replaced by an approximation denoted by .εh . This is constructed via a mesh in .x3 . We use a uniform mesh of size .h > 0 so that the h = j h − H , .j = 0, · · · , N , where .h = 2H /N . On each strip mesh points are .x3,j 3 3 h h .Sj = (−L1 /2, L1 /2) × (x 3,j , x3,j +1 ), .j = 0, · · · , N3 , we assume that .εh is the interpolant of .ε on the midline of the strip so that for .(x1 , x3 ) ∈ Sj we have h h εh (x1 , x3 ) = ε(x1 , (x3,j + x3,j +1 )/2).

.

(20.40)

1 (H ) that Using this approximation, we define an intermediate solution .uh ∈ Hqp 2D satisfies 1 Bεh (uh , v) = F (v) ∀v ∈ Hqp (H 2D ),

.

(20.41)

where .Bεh is defined as for .Bε in Eq. (20.39) but with .ε replaced by .εh . Note that .uh is only used for theoretical analysis and is not computed in practice. Since this problem is of the same type as our original problem (and .εh still satisfies the same general assumptions as for the relative permittivity .ε), the existence and uniqueness of .uh are not in doubt. Furthermore, uh H 1 (H ) ≤ C(h)F L2 (H ) ,

.

2D

2D

(20.42)

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where .C(h) < ∞ is a positive constant depending on the mesh size h. The issue is that the continuous-dependence constant .C(h) depends on h, so a precise order estimate of the error in this procedure is not immediate. In Ref. [14], we analyzed the error under the assumption that the material in the solar cell is non-trapping (exactly not what we want in practice!). This allowed us to assert that Eq. (20.41) cannot become ill-behaved as h decreases in the sense that .C(h) is bounded independent of h. In general, this desired result seems difficult to prove. One possible approach to proving convergence under more general assumptions for .ε is to use an idea from Ref. [17] to estimate the difference between u and .uh . This uses operator theory and does not require the non-trapping hypothesis. We will prove: Theorem 1 Assume that .maxx∈H |ε(x) − εh (x)| → 0 as .h → 0. Then: 2D

1. For all h small enough, Problem (20.41) has a unique solution, and the solution depends continuously on the data with continuity constant independent of h. Furthermore, .u − uh H 1 (H ) → 0 as .h → 0. 2D 2. If .maxx∈H |ε(x) − εh (x)| ≤ Chα for some constant C and exponent .α, then 2D α .u − uh  1 H (H ) = O(h ) as .h → 0. 2D

1 (H ) → H 1 (H ) and .T : H 1 (H ) → Proof Define operators .T : Hqp h qp qp 2D 2D 2D 1 (H ) by requiring that Hqp 2D

A(Tw, v) = Bε (w, v)

.

A(Th w, v) = Bεh (w, v)

 1 (H ∀v ∈ Hqp 2D ),

(20.43)

1 (H ), with .A(·, ·) denoting the .H 1 inner product given by for .w ∈ Hqp 2D

A(u, v) = (∇w, ∇v)H + (w, v)H .

.

2D

2D

(20.44)

Then A(Tw − Th w, v) = (ε−1 ∇w − εh−1 ∇w, ∇v)H

.

2D

≤ max |ε

−1

x∈H 2D

(x) − εh−1 (x)| wH 1 (H ) vH 1 (H ) . 2D 2D

(20.45)

So, under our assumption on the pointwise convergence of .εh to .ε, .Th converges to T in the operator norm as .h → 0. Then, using Corollary 10 of Ref. [25], we know that .Th is invertible with uniformly bounded inverse. We have the error estimate

.

u − uh H 1

.

H qp (2D )

≤ C(T − Th )uH 1

H qp (2D )

,

(20.46)

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for some constant C independent of h. This proves the first result because of the uniform bound on the norm of .uh and the convergence of .Th to .T. The second bound just follows using the order estimate for .ε − εh . This theorem begs the following question: When does .εh converge to .ε pointwise? There are two practical cases when this can occur: 1. If discontinuities in .ε follow paths made of segments that are either entirely vertical (along lines where .x1 is constant) or entirely horizontal (along lines where .x3 is constant), and the horizontal discontinuities occur at mesh points, then on each mesh slice .ε = εh . This occurs often in practice where profiles are given by stair steps. 2. If .ε is differentiable on each mesh strip .Sj (i.e., piecewise differentiable with discontinuities allowed at mesh points), then interpolation via Eq. (20.40) converges pointwise with error .O(h). Hence, it is desirable to avoid discontinuities inside mesh slices, and this is the observation behind the use of the C-method in Sect. 20.4. The approach taken in Ref. [14] is to verify the well-posedness of Eq. (20.41) by assuming a special structure (non-trapping in that case). Then the poor behavior of the error .ε − εh near discontinuities in .ε can be traded against smoothness of u to provide convergence estimates even for a general class of piecewise smooth .ε. Another case in which RCWA is provably convergent is when .ε has a nonzero imaginary part in all of .H 2D (i.e., the device is made entirely of dissipative materials). In this case, the sesquilinear form .Bεh (·, ·) is coercive so there is no question about the well-posedness of Eq. (20.41). Note that all materials used in solar cells are not necessarily dissipative. As a last comment, a better analysis would estimate .u − uh,M directly without introducing .uh , but this has yet to be accomplished.

20.3.2 2nd Step: Fourier Discretization in x1 Having discretized .ε by using .εh and having obtained .uh , we now want to use a Fourier approximation to .uh in order to obtain a fully discrete and implementable scheme. We note that the solutions u and .uh are both quasi-periodic and therefore have Fourier expansions with respect to .x1 . Motivated by this observation, we can choose an integer order M and define the space  1 XM = wh,M ∈ Hqp (H 2D ) | wM =

M

.

 wm (x3 ) exp(iαm x1 ) .

(20.47)

m=−M

Here the functions .{wm (x3 )}M m=−M are unknown coefficient functions. We then seek .uh,M ∈ XM such that

20 RCWA and Transformation Optics

515

Bεh (uh,M , v) = F (v) ∀v ∈ XM .

(20.48)

.

Although the space .XM is a little non-standard since functions are only expanded as Fourier expansions in .x1 , whereas the .x3 dependence is a general function, this fits within the standard Galerkin framework of partial differential equations [24]. Nonetheless, in general, .Bεh is not coercive, and we have to use analysis appropriate for this case. We suppose now that h is fixed and that it is small enough that Problem (20.41) has a unique solution. Schatz’s method [26] for analyzing problems with a noncoercive sesquilinear form requires us to consider the adjoint problem of finding H ) such that 1 .w ∈ Hqp ( 2D  Bεh (v, w) =

.

H 2D

1 gv dx ∀v ∈ Hqp (H 2D ),

(20.49)

where .g ∈ L2 (H 2D ) is a general function, although chosen finally to be .uh − uh,M . The existence of w follows since the foregoing adjoint problem can be related to Problem (20.41) that has already been proved to be well-posed. Then Schatz’s argument shows that the following theorem [14] holds even in the absence of a nontrapping condition. Theorem 2 Assume that h is chosen small enough that Problem (20.41) has a unique solution. Then there is an index .β > 0 such that for M large enough, uh − uh,M H 1 (H ) ≤ C(h)M −β

.

(20.50)

2D

for some constant .C(h) depending on h, but independent of u and M. The value of .β in this theorem is dictated by the Sobolev regularity of .uh . The precise value is difficult to determine a priori. This is discussed in Ref. [14]. In order to use Theorems 1 and 2 to obtain a convergence result, it suffices to use the triangle inequality u − uh,M H 1 (H ) ≤ u − uh H 1 (H ) + uh − uh,M H 1 (H ) .

.

2D

2D

2D

(20.51)

The first term on the right side can be analyzed using Theorem 1. To do this, we must first chose an appropriate h such that Problem (20.41) is well-posed and .u − uh is small enough. Once h is fixed, we can use Theorem 2 to bound the second term, and we can choose M large enough that the error due to Fourier discretization is negligible. A disadvantage of this result is that both the constant .C(h) and the choice of M needed to ensure this convergence estimate depend on h. In Ref. [14], the assumption of a special structure where the device is nontrapping allowed us to state convergence as .h → 0 and .M → ∞ simultaneously, but at the cost of a very special assumption.

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We have seen that the key issue is the well-posedness of Problem (20.41) for h sufficiently small and that for p-polarized incident conditions, this is a delicate issue requiring, in our simplified analysis, pointwise convergence of the approximate relative permittivity .εh to .ε. This difficulty motivated us to propose the C-RCWA method described in Sect. 20.4.

20.3.3 A Numerical Example for RCWA Although the convergence analysis of RCWA presents interesting challenges, the method has several attractive features when used to design solar cells. These include: 1. There is no need to use an elaborate meshing strategy. The only mesh used is a simple uniform mesh along .x3 . However, it is important that horizontal bimedium interfaces occur at mesh points. Even so, when the design is changed, remeshing is simple. 2. Changes in the relative permittivity, including changes to bimedium interfaces between different materials, can be accomplished at low cost. 3. There is a remarkably efficient computational approach for finding .uh,M [8]. We discuss this a little more next. The discrete solution .uh,M is given by the Fourier series uh,M (x1 , x3 ) =

M

.

uh,M n (x3 ) exp(iαn x1 ),

(20.52)

n=−M M with the coefficient functions .{uh,M n (x3 )}m=−M determined from the variational Problem (20.48). Reference [14] proves that this is equivalent to requiring that on M each slice .Sj the coefficients .{uh,M m (x3 )}m=−M in Eq. (20.52) satisfy the following second-order system of ordinary differential equations in .x3 : M

.

n=−M

−1 εh,n−m

M

d 2 uh,M m −1 2 h,M = α εh,n−m αm uh,M n m − κ un . dx32 m=−M

(20.53)

−1 Here, .εh,n−m is the .(n, m) entry of the Toeplitz matrix formed from the coefficients of .1/εh [11]. There is no source term since we are computing with the total field. Equation (20.53) being an ordinary differential equation with constant coefficients, its general solution can be determined by computing the eigenvalues and eigenvectors of a system matrix (assuming that the Toeplitz matrix is non-degenerate). The global solution can then be determined by enforcing transmission conditions at boundaries between strips, the PEC condition at .x3 = −H , and the coupling conditions implied by the Dirichlet-to-Neumann map at .x3 = H . This would still be

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Fig. 20.1 Cartoon of the computational region for the RCWA example including the choice of .ε. For this device, .H = 1400 nm, the bottom of the device is at .x3 /H = −1, the lowest point of the triangle is at .x3 /H = −1/2, and the triangle has non-dimensional height of .1/7. The upper interface between .ε ≈ 1 and .ε ≈ 4 is at .x3 /H = 1/2. The top of the air layer is at .x3 /H = 1

quite an expensive algorithm, but following Ref. [27] the solution can be computed via a two-pass, non-iterative, and stable solution strategy. Details of this strategy and a complete description of the implementation of RCWA can be found in Ref. [11]. Here we present a simple numerical example showing the behavior of the error in the RCWA solution. In particular, we investigate a problem in which the interface between the different materials making up the device is not parallel to the coordinate axis so that .εh cannot converge to .ε pointwise. The example chosen here features an asymmetric triangular grating profile so that .εh = ε. In addition, the real part of .ε is non-monotonic (a small imaginary part is used to stabilize the fast solution algorithm). The entire device has .H = 1400 nm (this includes an air layer above as well as the layers of the device) and .L1 = 500 nm. The height of the triangle is 200 nm. A cartoon of the device with the values of .ε is presented in Fig. 20.1, together with details of the geometry. Spatial maps of .|u| computed for the free-space wavelength .λo = 2π/κ = 600 nm are presented in Fig. 20.2. RCWA calculations were made using .M = 10 and .h = 1 nm, whereas Netgen [3] was used for FEM calculations with a highly refined grid. Visually, the two maps look quite similar. However, a plot of the magnitude of the difference in Fig. 20.3 reveals characteristic high-frequency error near the triangular interface. To further investigate the error in the RCWA algorithm, we show convergence curves in Fig. 20.4. These are computed using the reference FEM solution denoted by .uFEM . Here .M = 10 is fixed but h is varied from 5 nm to 100 nm in the left panel, .h = 1 nm is fixed but M is varied from 2 to 10 in the right panel. Convergence is seen in the left panel with decrease of h although saturation for .h < 20 nm is most likely due to the fact that discretization in M is limiting overall accuracy. In the right panel, saturation is seen at roughly 1% relative error for .M > 8. As commented by Shuba et al. [12], RCWA can attain a reasonable error (say 1% relative error) without an excessive computational burden. But the slow convergence in M and h requires a careful choice of parameters. Motivated by the goal to improve convergence rates, we developed a new version of the C-method [17] discussed next.

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Fig. 20.2 Spatial maps of .|uh,M | and .|uFEM | in the unit cell of the device shown in Fig. 20.1, when .λo = 600 nm. Left: RCWA result. Right: FEM result. The interfaces between different materials are shown as a thick white lines Fig. 20.3 Spatial map of − uFEM | in the unit cell of the device shown in Fig. 20.1, when .λo = 600 nm. The error is largest near the triangular interface and has a characteristic oscillation there

.|uh,M

20.4 C-RCWA for the p-polarization Case In Sect. 20.3, we outlined some theory for the standard RCWA and demonstrated with an example that the observed convergence rate with respect to the Fourier parameter M is slow. The obstruction to a simple theoretical treatment is the fact that .εh does not in general converge to .ε pointwise, even though step changes in .ε at horizontal interfaces cause no difficulties, provided these changes occur at vertical mesh points. If the interfaces between two dissimilar materials are described by the graph of a function, we can apply the C-method [15, 16]. In Ref. [17], we proposed two modifications of the C-method to create the C-RCWA and provided a mathematical analysis of convergence. To compare the

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519

10 0

0.035 0.03

10

-1

10

-2

10

-3

10

-4

0.025 0.02 0.015

0.01 10 1

10 2

0

2

4

6

8

10

Fig. 20.4 Convergence curves for RCWA. Values of the relative .L2 (H 2D ) norm against the discretization parameters h and M are shown. Left: .M = 10, but h varies from 5 nm to 100 nm. Right: .h = 1 nm, but M varies from 2 to 10

original C-method and our suggested modifications, let us consider a simple surface relief grating as in Ref. [10] illuminated with p-polarized monochromatic light. The extension to a more complex multilayered structure is relatively straightforward. Suppose the grating interface is given as the graph of an .L1 -periodic function .g(x1 ) so the equation for the interface is x3 = g(x1 ),

.

−L1 /2 ≤ x1 ≤ L1 /2 ,

(20.54)

where we assume that g is a twice continuously differentiable function and .−H < g(x1 ) < H so that the grating occupies .H 2D as before. Then ε(x1 , x3 ) =

.

⎧ ⎨

1, ε+ (x1 , x3 ), ⎩ ε− (x1 , x3 ),

|x3 | > H, g(x1 ) < x3 < H, −H < x3 < g(x1 ),

(20.55)

where .ε+ and .ε− are real-valued, strictly positive, periodic, and smooth functions of position. Allowing .ε± to be functions allows handling of graded-index materials. The central idea of the C-method is to use transformation optics to flatten the boundary at the expense of more complex anisotropic material parameters [10], just as is done commonly to track photons in a gravitational scenario [28, 29]. This makes discontinuities in the transformed coefficient corresponding to .ε (denoted .εˆ ) occur on horizontal lines, which, as we have seen in the previous section on RCWA, is favorable to methods based on the use of Fourier series in the horizontal variable. Our two modifications of the fundamental C-method are then as follows: 1. We use a more complex transformation than in the original C-method to maintain the upper and lower edges of the grating fixed. This also allows for multiple layered devices since we can simply use a different transformation for

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each interface; furthermore, since the transformed domains do not overlap, the transformed problem simply has multiple flat interfaces with anisotropic media in between. 2. We suggest using RCWA to compute the solution of the transformed problem. With flat interfaces and smooth coefficients in between, this is well suited to RCWA and, as we have commented before, RCWA is computationally efficient. Details of the C-RCWA and an error analysis for non-trapping domains being provided elsewhere by us [17], our main contribution here is to extend the error analysis for this method to trapping domains and provide a new numerical example. We next describe the transformation used by us and the resulting transformed problem. Then we summarize the known analytical results and, finally, we provide a simple test problem where C-RCWA exhibits rapid convergence.

20.4.1 Transformation Optics H = (−L1 /2, L1 /2) × (−H, H ) having coordinates .xˆ = (xˆ1 , xˆ3 )T . We want Let . 2D H → H that transforms a flat interface in an invertible transformation .GS :  2D 2D H H  . to the true interface in . . There are many possible choices, but we make the 2D 2D simple choice .GS (xˆ1 , xˆ3 ) = (x1 , x3 ), where .



x1 = xˆ1 x3 = S(xˆ3 )g(xˆ1 ) + xˆ3

,

(20.56)

and the piecewise cubic function [17] ⎧ 3 ⎪ 1 − 2 xˆ32 − ⎪ ⎪ ⎪ H ⎨ .S(x ˆ3 ) = 1 − 3 xˆ 2 + ⎪ ⎪ H2 3 ⎪ ⎪ ⎩ 0,

2 3 xˆ , H3 3 2 3 xˆ , H3 3

−H < xˆ3 < 0, 0 < xˆ3 < H,

(20.57)

|xˆ3 | > H,

is designed so that S(−H ) = S(H ) = S (−H ) = S (H ) = S (0) = 0

.

(20.58)

and .S(0) = 1. This choice ensures that .xˆ3 = 0 is transformed to the physical interface, and the transformation does not interfere with transmission or boundary conditions at the top (.xˆ3 = H ) and bottom (.xˆ3 = −H ) of the device. However, this choice does impose a constraint on g since we need .GS to be invertible, which requires .xˆ3 + S(xˆ3 )g(xˆ1 ) to be an increasing function of .xˆ3 . As .|S (xˆ3 )| ≤ 9/(2H ) follows from Eq. (20.57), we must impose the constraint

20 RCWA and Transformation Optics

|g(xˆ1 )| < 2H /9,

.

521

0 ≤ xˆ1 ≤ L1 .

(20.59)

ˆ H could be considered to improve Other choices of .S or even other choices of . 2D this bound on g. H with u on .H via .u(G (ˆx)) = u(ˆ We now associate the solution .uˆ on . ˆ x). S 2D 2D Using the Jacobian .D(ˆx) of the transformation .GS , it is well known (see, for example, Ref. [24]) that u(ˆ ∇u(Gs xˆ ) = D −T (ˆx)∇ ˆ x),

(20.60)

.

 denotes the gradient with respect to “hat” variables. If .xˆ = (xˆ1 , xˆ3 )T , we where .∇ have   1 0 , .D(ˆ x) = (20.61) S(xˆ3 )g  (xˆ1 ) g(xˆ1 )S (xˆ3 ) + 1 where the prime indicates differentiation with respect to the argument. Note that the determinant of .D(ˆx) equals .1 + S (xˆ3 )g(xˆ1 ) > 0 by our assumption on the bounds of g in Eq. (20.59). Changing variables in Eq. (20.37), we obtain the following variational formula1 ( H ) such that tion for the transformed variables: we seek .uˆ ∈ Hqp 2D   2D u,  ˆ H − κ 2 (nˆ u, (ˆε−1 ∇ ˆ v) − T ( u| ˆ ), v ˆ ∇v) H ˆ x ˆ =H H 3  

.

2D

2D

2D H

1 ˆH = F (v) ∀v ∈ Hqp (2D ),

(20.62)

where .

εˆ (ˆx) = | det(D)(ˆx)|−1 D(ˆx)ε(GS (ˆx))D T (ˆx) n(ˆ ˆ x) = | det(D)(ˆx)|

 .

(20.63)

The relative permittivity .εˆ (ˆx) in the transformed problem is a 3.×3 matrix function of position that is both symmetric and positive definite when .ε is real and strictly positive. The function .F (v) on the right side of Eq. (20.37) appears unchanged on the right side of Eq. (20.62), because the transformation preserves the upper boundary (at .x3 = H ). Indeed, the function values, and the gradient, on the upper boundary also do not change. We thus have to solve a problem for .uˆ that is the same form as in Sect. 20.3, but with the added complication of a tensor coefficient .εˆ . Under condition (20.59), the transformation .GS is invertible and, hence, Problems (20.37) and (20.62) are simultaneously well-posed. It remains now to discretize and solve Eq. (20.62). For this, we apply RCWA to the transformed problem as we did in Sect. 20.3. We note that if we apply the FEM to Eq. (20.62), we can change both the interface function g and the relative permittivity .ε without having to remesh. We would have to recompute (or reassemble) a relevant system matrix and then solve the discrete

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problem. The RCWA also has to recompute the relevant matrices and eigenvalue expansions, but the stabilized solution strategy does not involve solving a large sparse system.

20.4.2 C-RCWA and Analysis of the Transformed Problem Discretization of the transformed problem, i.e., Eq. (20.62), follows the pattern for ˆH RCWA outlined in Sect. 20.3. We introduce a mesh in the vertical direction for . 2D such that the transformed interface at .xˆ3 = 0 lies on a node of the mesh. Each vertical mesh interval divides the domain into horizontal slices. Then we replace .εˆ and .nˆ by their respective interpolants on the mid line of each mesh slice to obtain .εˆ h and .nˆ h . These approximations are then used to replace .εˆ and .nˆ in Eq. (20.62), and we need to know the well-posedness of this problem. But the perturbed problem using .εˆ h and .nˆ h is no longer directly related to a problem in the physical domain, so we proceed via perturbation analysis. By virtue of the transformation and our choice to put the interface on slice boundaries, we know that .εˆ is at least twice continuously differentiable on each slice. Hence, on the ith slice .Si , we have ⎫ maxxˆ ∈Si ⊂ˆ H |ˆε(ˆx) − εˆ h (ˆx)| = O(h) ⎬ 2D . (20.64) . maxxˆ ∈Si ⊂ˆ H |n(ˆ ˆ x) − nˆ h (ˆx)| = O(h) ⎭ 2D

Now a slight extension of Theorem 2 to allow for the approximation of .nˆ by .nˆ h shows that for h small enough, the discrete problem is well-posed, and the solution of this problem is stable with a continuity constant independent of h. Thus, if .uˆ h solves Eq. (20.62) with .εˆ h and .nˆ h replacing .εˆ and .n, ˆ we know that uˆ − uˆ h H 1 (ˆ H ) = O(h).

.

2D

(20.65)

We now consider h to be small enough that the error above is below a desired tolerance. We then proceed to discretize in .xˆ1 by expanding the fully discrete solution denoted .uˆ h,M as a Fourier series in .xˆ1 : uˆ h,M (x1 , x3 ) =

M

.

uˆ h,M n (xˆ 3 ) exp(iαn xˆ 1 ),

(20.66)

n=−M

and using such a finite Fourier series as trial and test functions in the perturbed variational formulation. This Galerkin approximation is analyzed in Ref. [17] where it is shown that

20 RCWA and Transformation Optics

523

uˆ h − uˆ h,M  ≤ C(M −2s + hM −s )

.

(20.67)

for some constant C. This constant is bounded independent of h for h small enough because the solution of the discrete problem converges to the original solution as .h → 0. Putting the two estimates together, we obtain a slight generalization of Theorem 8.3 of Ref. [17] as follows. Theorem 3 Suppose that the transformation .GS is invertible, and suppose that Eq. (20.62) has a unique solution. Then for .h > 0 small enough and .M > 0 large enough, there are constants .C < ∞ and .s0 ∈ (0, 1/2) independent of h and M such that uˆ − uˆ h,M H 1 (ˆ H ) ≤ C(M −s + h) ,

.

2D

(20.68)

where .s ∈ (0, s0 ).

20.4.3 Numerical Example of C-RCWA Convergence Here we give a numerical example for C-RCWA similar to one in Ref. [17] to demonstrate that C-RCWA can exhibit very rapid convergence as M increases. We choose .H = 700 nm and the period .L1 = 500 nm, along with g(x1 ) = 100 cos(2π x1 /L1 )2

.

(20.69)

for the example depicted in Fig. 20.5. Above the grating interface, we take .ε+ = 1+ 10−6 i (air with a small imaginary component to stabilize the marching algorithm), while below the grating interface .ε− = −15 + 4i (plasmonic metal). As a result, the electromagnetic field does not penetrate far into the lower region. In Fig. 20.6, we show numerical results for this grating illuminated by a normally incident plane wave (i.e., .d = (0, −1)T ) of free-space wavelength .λo = 600 nm. ˆ H yielded by C-RCWA In the left panel, we show the spatial map of .|u(ˆ ˆ x)| on . 2D Fig. 20.5 Unit cell of the grating used to test C-RCWA

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Fig. 20.6 Spatial maps of (left) .|uˆ h,M (ˆx)| and (right) .|uh,M (x)| in the transformed unit cell and original unit cell of the grating shown in Fig. 20.5, respectively, with .uh,M computed from .uˆ h,M by inverting the transformation .GS . White lines show the bimedium interface in the transformed and original domains, respectively

10

10

-1

10

0

-1

10 -2

10

10

-2

10

-3

10

-4

10

-5

-3

50

100

150 200 250

350

0

2

4

6

8

10

Fig. 20.7 Convergence curves for C-RCWA. Values of the relative .L2 (H 2D ) norm against the discretization parameters h and M are shown. Left: .M = 10, but h varies from 50 nm to 350 nm. Right: .h = 1 nm, but M varies from 1 to 10

after the grating has been flattened. The anisotropic relative permittivity .εˆ (ˆx) causes the field above the grating interface to depend on .xˆ1 . The C-RCWA results can then be transformed using .G−1 S to obtain .|u(x)| in physical space, as shown in the right panel. As we do not have an analytical solution for this problem, in order to test the convergence of C-RCWA, we can compare the C-RCWA to an FEM result computed using Netgen [3] with a highly refined grid. This is done in transformed space on ˆ H so that FEM does not suffer from any geometric error approximating a curved . 2D interface (as there would be in physical space). In the left panel of Fig. 20.7, we show convergence results as the mesh size h is varied from a very coarse mesh with .h = 350 nm down to .h = 50 nm, with

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M = 10 fixed. The log–log plot suggests a convergence rate of .O(h2 ) in the relative 2 .L norm. This is surprisingly fast (we predicted .O(h) in Theorem 3). In the right panel of Fig. 20.7, we show results of convergence in the truncation number M of Fourier series when the mesh size .h = 1 nm. In this case, the log-linear plot of relative error against M suggests exponential convergence that is faster than our predicted polynomial rate of convergence (see Theorem 3). As might be expected from the symmetry of the chosen grating, odd-numbered terms in the Fourier series do not enhance convergence. Of course for a more general grating, all terms in the Fourier series would be needed. .

20.5 Extension of C-RCWA to 3D The direct application of RCWA to solve the Maxwell equations in 3D problems is described in, e.g., Refs. [11] and [18]. However, the normal component of the electric field is discontinuous at interfaces across which .ε jumps. Thus the direct use of RCWA implies approximating discontinuous functions by Fourier series with the attendant issue of the Gibbs phenomenon. If a solar cell has layers with interfaces that are graphs of single-valued functions, we can again use C-RCWA to make the interfaces horizontal in a transformed problem. For concreteness, let us suppose, as in Sect. 20.4, that the device has just two layers separated by the interface x3 = g(x1 , x2 )

(20.70)

.

for some twice continuously differentiable biperiodic function g. In 3D, we assume that ⎧ 1, |x3 | > H, ⎨ .ε(x1 , x2 , x3 ) = (20.71) ε+ (x1 , x2 , x3 ), g(x1 , x2 ) < x3 < H, ⎩ ε− (x1 , x2 , x3 ), −H < x3 < g(x1 , x2 ), where .ε+ and .ε− are real-valued, strictly positive, biperiodic, and smooth functions of .x = (x1 , x2 , x3 )T . Now we choose .GS (xˆ1 , xˆ2 , xˆ3 ) = (x1 , x2 , x3 ), such that x1 = xˆ1 .

x2 = xˆ2 x3 = S(xˆ3 )g(xˆ1 , xˆ2 ) + xˆ3

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

.

(20.72)

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In this case, the Jacobian .D(ˆx) is given by ⎞ 0 ⎟ ⎜ 0 .D(ˆ x) = ⎝ ⎠, ∂g ∂g  ( x ˆ , x ˆ ) S( x ˆ ) ( x ˆ , x ˆ ) g( x ˆ , x ˆ )S ( x ˆ ) + 1 S(xˆ3 ) ∂x 1 2 3 1 2 1 2 3 ∂x2 1 (20.73) with .xˆ = (xˆ1 , xˆ2 , xˆ3 )T . As in Sect. 20.4, the determinant of .D(ˆx) equals .1 + S (xˆ3 )g(xˆ1 ) > 0 by our assumption on the bounds of g in Eq. (20.59) that still must hold. We now relate the electric field phasor in the transformed and untransformed variables by ⎛

1 0

0 1

ˆ˜ −T ˜ E(G S (ˆx)) = D (ˆx)E(ˆx).

(20.74)

.

In this case, the curl transforms like a magnetic field to give .

1 ˆ˜ ˜ ˆ ∇ × E(G S (ˆx)) = det(D)(ˆx) D(ˆx)∇ × E(ˆx).

(20.75)

Thus, on transforming Eq. (20.22) from the physical domain .H 3D to the compuˆ H , we obtain the problem of finding the transformed total field tational domain . 3D ˆ˜ ∈ xˆ such that .E ˆ˜ , v ˆ˜ ∇ˆ × v) 2 ˆ˜ (μˆ −1 ∇ˆ × E, εE, v)ˆ H − Bκ E T ˆ H − κ (ˆ 3D = F(v) 

.

3D

3D

H

where .Eˆ˜ T = (Eˆ˜ 1 , Eˆ˜ 2 , 0)T and

.

⎫ εˆ (ˆx) = | det(D)(ˆx)|D −1 (ˆx)ε(GS (ˆx))D −T (ˆx) ⎪ ⎪ ⎬ −1 −T . μ(ˆ ˆ x) = | det(D)(ˆx)|D (ˆx)D (ˆx) ⎪ ⎪ ⎭ n(ˆ ˆ x) = | det(D)(ˆx)|

∀v ∈ xˆ , (20.76)

(20.77)

The transformed relative permittivity .εˆ (ˆx) and the transformed relative permeability μ(ˆ ˆ x) in the transformed problem above are 3.×3 matrix functions of position; both are symmetric as well as positive definite, when .ε is real and positive. Similarly to the 2D problem in Sect. 20.4, the boundary operator remains unchanged as also the right side of Eq. (20.76). Now the way to implement the RCWA for the transformed problem is clear [11]. We introduce the one-dimensional mesh in .x3 used in Sects. 20.3 and 20.4, and we approximate .εˆ and .μˆ by .εˆ h and .μˆ h , respectively, that are piecewise constant in .xˆ3 on this mesh. As for the 2D C-RCWA, the important point is that we are approximating twice continuously differentiable matrix functions and so are guaranteed first-order convergence pointwise in the mesh parameter h.

.

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527

Once .εˆ h and .μˆ h are determined, we can then use the fact that the fields are quasiperiodic to develop a Fourier series approximation [11] for the electric field phasor as



˜ h,M (ˆx) = ˆ ), .E (20.78) E˜ h,M n (xˆ 3 ) exp(iα n · x |n1 |≤M1 |n2 |≤M2

˜ h,M where .M = (M1 , M2 )T is a vector of positive integers and .{E n } are .3(2M1 + 1)(2M2 + 1) functions of .xˆ3 . Similarly, ˜ h,M (ˆx) = H





.

˜ h,M ˆ) H n (xˆ 3 ) exp(iα n · x

(20.79)

|n1 |≤M1 |n2 |≤M2

˜ h,M for the magnetic field phasor with .{H n } as .3(2M1 + 1)(2M2 + 1) functions of .xˆ 3 . ˆ ˆ Both .E˜ 3 and .H˜ 3 can be eliminated, since they satisfy algebraic equations. We are left to find the remaining .4(2M1 + 1)(2M2 + 1) coefficients in Eqs. (20.78) and (20.79). As for RCWA and C-RCWA, this can be done by noting that the unknown coefficients satisfy a second-order differential equation in .xˆ3 in each layer of the vertical mesh with coefficients that are independent of .xˆ3 . Thus, using the eigenvalues and eigenvectors of the propagation matrix for each slice, we can develop the solution in each slice using the matrix exponential. Enforcing transmission conditions across slices, as well as initial and final conditions from the Dirichlet-to-Neumann map and the PEC boundary, we may utilize the stabilized marching scheme for the RCWA again to solve this problem. We remark that the 3D problem is substantially more challenging than the 2D case. The propagation matrices for all slices are derived from the Maxwell equations for anisotropic dielectric–magnetic media [11], but, more importantly, in each slice we must find all eigenvalues of a .4(2M1 + 1)(2M2 + 1) × 4(2M1 + 1)(2M2 + 1) matrix, and this must be done .N3 times. Clearly, investigation of how to diminish the number of unknown functions in each slice would be very useful.

20.6 Conclusion In this chapter, we have presented some mathematical underpinnings for the RCWA method and examined the general case when a non-trapping assumption does not hold. In the special case when the periodic structures making up the solar cell or grating are given by the graph of a function, we have then described the use of transformation optics to obtain an artificial modified problem with anisotropic material properties that is amenable to solution by RCWA. This version converges rapidly when the structures are smooth, and there is no need for special assumptions on the media. Finally, we have outlined how this technique could be extended to the Maxwell equations.

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The state of convergence theory for RCWA, C-RCWA, and Maxwell analogues still has many open problems. Foremost of these is to extend the analysis to general gratings in the case of RCWA. Technically, our analysis proceeds in two steps by defining an intermediate solution of a perturbed problem. Hence, our analysis suffers from requiring both the convergence of the intermediate solution and convergence of the RCWA solution to this intermediate solution. Thus our convergence analysis produces a worst-case error analysis (as we have seen, the real convergence rate is often much faster than our prediction). A better convergence theory would compare the RCWA solution directly to the true solution. With regard to C-RCWA, it could be useful to investigate more general transforˆH mations between .H 2D and .2D in order to improve the accuracy of the method, as well as allow for more general grating profiles. Acknowledgments This research was supported by the US National Science Foundation under grants numbered DMS-1619901 and DMS-1619904 as well as by the Charles Godfrey Binder Endowment at Penn State. P.B. Monk is partially supported by AFOSR grant FA9550-20-1-0124.

References 1. Anderson, T.H., Civiletti, B.J., Monk, P.B., Lakhtakia, A.: Coupled optoelectronic simulation and optimization of thin-film photovoltaic solar cells. J. Comput. Phys. 407, 109242 (2020); Corrections: 418, 109561 (2020) 2. Chen, Z., Wu, H.: An adaptive finite element method with perfectly matched absorbing layers for the wave scattering by periodic structures. SIAM J. Numer. Anal. 41, 799–826 (2003) 3. Schöberl, J.: Netgen/NGSolve (2022). https://www.ngsolve.org 4. Liu, Y., Barnett, A.H.: Efficient numerical solution of acoustic scattering from doubly-periodic arrays of axisymmetric objects. J. Comput. Phys. 324, 226–245 (2016) 5. Barnett, A.H.: Efficient high-order accurate Fresnel diffraction via areal quadrature and the nonuniform fast Fourier transform. J. Astron. Telesc. Instrum. Syst. 7, 021211 (2021) 6. Bruno, O.P., Fernandez-Lado, A.G.: On the evaluation of quasi-periodic Green functions and wave-scattering at and around Rayleigh–Wood anomalies. J. Comput. Phys. 410, 109352 (2020) 7. Moharam, M.G., Gaylord, T.K.: Rigorous coupled-wave analysis of planar grating diffraction. J. Opt. Soc. Am. 71, 811–818 (1981) 8. Moharam, M.G., Pommet, D.A., Grann, E.B., Gaylord, T.K.: Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach. J. Opt. Soc. Am. A 12, 1077–1086 (1995) 9. Lalanne, P., Morris, G.M.: Highly improved convergence of the coupled-wave method for TM polarization. J. Opt. Soc. Am. A 13, 779–784 (1996) 10. Li, L., Chandezon, J., Granet, G., Plumey, J.-P.: Rigorous and efficient grating-analysis method made easy for optical engineers. Appl. Opt. 38, 304–313 (1999) 11. Polo, A., Mackay, T.G., Lakhtakia, A.: Electromagnetic Surface Waves: A Modern Perspective. Elsevier, Waltham (2013) 12. Shuba, M.V., Faryad, M., Solano, M.E., Monk, P.B., Lakhtakia, A.: Adequacy of the rigorous coupled-wave approach for thin-film silicon solar cells with periodically corrugated metallic backreflectors: spectral analysis. J. Opt. Soc. Am. A 32, 1222–1230 (2015) 13. Civiletti, B.J., Lakhtakia, A., Monk, P.B.: Analysis of the rigorous coupled wave approach for s-polarized light in gratings. J. Comput. Appl. Math. 368, 112478 (2020)

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14. Civiletti, B.J., Lakhtakia, A., Monk, P.B.: Analysis of the rigorous coupled wave approach for p-polarized light in gratings. J. Comput. Appl. Math. 386, 113235 (2021) 15. Chandezon, J., Raoult, G., Maystre, D.: A new theoretical method for diffraction gratings and its numerical application. J. Opt. (Paris) 11, 235–241 (1980) 16. Chandezon, J., Dupuis, M.T., Cornet, G., Maystre, D.: Multicoated gratings: a differential formalism applicable in the entire optical region. J. Opt. Soc. Am. 72, 839–846 (1982) 17. Civiletti, B.J., Lakhtakia, A., Monk, P.B.: Hybridization of the rigorous coupled-wave approach with transformation optics for electromagnetic scattering by a surface-relief grating. J. Comput. Appl. Math. 412, 114338 (2022) 18. Bräuer, R., Bryngdahl, O.: Electromagnetic diffraction analysis of two-dimensional gratings. Opt. Commun. 100, 1–5 (1992) 19. Götz, P., Schuster, T., Frenner, K., Rafler, S., Osten, W.: Normal vector method for the RCWA with automated vector field generation. Opt. Express 16, 17295–17301 (2008) 20. Ahmad, F., Anderson, T.H., Civiletti, B.J., Monk, P.B., Lakhtakia, A.: On optical absorption peaks in a nonhomogeneous thin-film solar cell with a two-dimensional periodically corrugated metallic backreflector. J. Nanophoton. 12, 016017 (2018) 21. Civiletti, B.J., Anderson, T.H., Ahmad, F., Monk, P.B., Lakhtakia, A.: Optimization approach for optical absorption in three-dimensional structures including solar cells. Opt. Eng. 57, 057101 (2018) 22. Ammari, H., Bao, G.: Maxwell’s equations in chiral periodic structures. Mathematische Nachrichten 251, 3–18 (2003) 23. Haddar, H., Lechleiter, A.: Electromagnetic wave scattering from rough penetrable layers. SIAM J. Math. Anal. 43, 2418–2443 (2011) 24. Monk, P.B.: Finite Element Methods for Maxwell’s Equations. Oxford University Press, Oxford (2003) 25. Kress, R.: Linear Integral Equations. Springer, New York (1999) 26. Schatz, A.H.: An observation concerning Ritz–Galerkin methods with indefinite bilinear forms. Math. Comput. 28, 959–962 (1974) 27. Moharam, M.G., Grann, E.B., Pommet, D.A., Gaylord, T.K.: Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings. J. Opt. Soc. Am. A 12, 1068–1076 (1995) 28. Plébanski, J.: Electromagnetic waves in gravitational fields. Phys. Rev. 118, 1396–1408 (1960) 29. Mackay, T.G., Lakhtakia, A.: Toward the construction of parts of the universe on tabletops. In: Lakhtakia, A., Furse, C.M. (eds.) The World of Applied Electromagnetics—In Appreciation of Magdy Fahmy Iskander, pp. 631–654. Springer, Cham (2018)

Benjamin Civiletti is a postdoctoral researcher at the University of Delaware. He received his B.A. (2011) and M.A. (2014) degrees in mathematics from The College of New Jersey and Villanova University, respectively. He received his M.S. (2016) and Ph.D. (2020) degrees in mathematics from the University of Delaware. His research interests include the development and analysis of computational methods for the Maxwell equations.

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Akhlesh Lakhtakia is Evan Pugh University Professor and Charles Godfrey Binder (Endowed) Professor of Engineering Science and Mechanics at The Pennsylvania State University. He received his B.Tech. (1979) and D.Sc. (2006) in electronics engineering from the Institute of Technology, Banaras Hindu University, and his M.S. (1981) and Ph.D. (1983) in electrical engineering from The University of Utah. He has been elected a Fellow of the American Association for the Advancement of Sciences, American Physical Society, Institute of Physics (UK), Optica, SPIE, IEEE, Royal Society of Chemistry, and Royal Society of Arts. He was the Editor-in-Chief of SPIE’s online Journal of Nanophotonics from its inception in 2007 through 2013. His current research interests include: electromagnetic fields in complex mediums, sculptured thin films, mimumes, surface multiplasmonics, electromagnetic surface waves, thinfilm solar cells, forensic science, engineered biomimicry, and biologically inspired design.

Peter B. Monk is a Unidel Professor in the Department of Mathematical Sciences at the University of Delaware. He received his B.A. in mathematics from Cambridge University, UK, in 1978 and his Ph.D. in Mathematics from Rutgers University, NJ, USA, in 1983. Since 1982, he has worked at the University of Delaware where he served as the Chair of the department from 2007–2010. He is a Fellow of the Institute for Mathematics and its Applications in the UK, and SIAM in the USA. He is the author of Finite Element Methods for Maxwell’s Equations (Oxford, 2003) and co-author with F. Cakoni and D. Colton of The Linear Sampling Method in Inverse Electromagnetic Scattering (SIAM, 2011). His current research interests are the modeling and optimization of thin-film solar cells, inverse electromagnetic scattering, and finite element methods.

Chapter 21

Mind the Gap Between Theory and Experiment Andrei Kiselev, Jeonghyeon Kim, and Olivier J. F. Martin

21.1 Introduction We did not have the pleasure to meet Werner S. Weiglhofer and only know of him through his scientific publications. In spite of his too short career, they are extremely numerous, diverse and impactful. Following the Web of Science categories, one notices that these contributions do not only cover optics and electromagnetics but also reach out to applied physics, materials sciences and—of course—mathematics. They are very well cited: his works on demystified negative index of refraction [1] and that on light-propagation in helicoidal bianisotropic media [2], at the top of his list of citations. Working at the Department of Mathematics of the University of Glasgow, it is not surprising that Werner’s publications have a strong theoretical flavour and have inspired many theoretical works. Yet, analysing their citations further indicates that these theoretical developments inspired numerous experimental projects. As an example, among the citations of his work on lightpropagation in helicoidal bianisotropic media [2], a third of the citing articles report experiments. This illustrates how well Werner succeeded in bridging the gap between theory and experiments. Obviously, theory is very important and progress within the realm of theoretical physics is often fascinating in itself. Evidently, new theories are often the driver behind new experimental work. This chapter, however, focuses on the inverse process, where experimental work requires numerical support as close as possible to the experimental situation. After briefly presenting the numerical technique we have developed for over a decade to solve Maxwell’s equations, we discuss three different experimental situations where we attempted to model the real experiment as closely as possible.

A. Kiselev · J. Kim · O. J. F. Martin () Nanophotonics and Metrology Laboratory, Swiss Federal Institute of Technology Lausanne (EPFL), EPFL-STI-NAM, Lausanne, Switzerland e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 T. G. Mackay, A. Lakhtakia (eds.), Adventures in Contemporary Electromagnetic Theory, https://doi.org/10.1007/978-3-031-24617-3_21

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21.2 Computational Electromagnetics At the onset of studying a given experimental situation lies the fundamental question of the choice of the most appropriate numerical method. It is fair to say that there is not one single numerical technique that is fit for all situations, and even for the narrow field of plasmonics, which is the focus of our work, numerous approaches exist as illustrated in a recent review article [3]. Furthermore, each numerical method can be put to good use as long as it is utilized wisely and carefully. Especially, sufficient efforts must be undertaken to characterize the algorithm beforehand, to make sure that it will converge well for the problem at hand and is free from spurious behaviours. This task is especially thankless, but of paramount importance if the numerical results are to be trusted. Note that it does not only apply to home-developed numerical codes but should be equally undertaken with commercial packages that should never be trusted blindly, even if they produce beautiful and colourful images. To assess the accuracy of a numerical technique and obtain a metric to quantify it, one usually resorts to canonical problems. Unfortunately, there are essentially only two such problems for which a reference solution exists (the quasi-analytical Mie solution): the scattering by a sphere for three-dimensional (3D) problems or by a cylinder for two-dimensional (2D) geometries [4]. Figure 21.1 illustrates this approach for a 2D solution obtained with a volumetric Green’s tensor approach [5]. In this case, two different incident polarizations must be considered, with

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the electric field either perpendicular to the cylinder axis (transverse electric or TE field) or parallel to the cylinder axis (transverse magnetic or TM field). The differential cross section can be computed as a function of the scattering angle and compared with the quasi-analytical Mie solution, Fig. 21.1a. This panel indicates that many features exist in that response, which need to be reproduced accurately with the numerical method. A more quantitative metric is obtained by integrating the difference between this cross section and the Mie solution over all scattering angles and repeating the calculations with an increasing number of discretized elements, Fig. 21.1b. In principle, the error should decrease as the number of elements increases. However, this behaviour is far from monotonous since it includes different facets of the numerical problem: on the one hand, a finer mesh approximates the scatterer better and should provide a more accurate solution, and on the other hand, it requires a larger numerical matrix to be solved, which is more difficult, especially when the matrix condition number increases, as is the case here [6, 7]. Consequently, plateaus appear in the convergence curve, Fig. 21.1b. We also notice that the polarization influences the solution accuracy, reminiscent that in electromagnetics all field components do not behave in the same way: some are continuous across materials’ boundaries, and others are not [8]. Experimental situations are usually much more complicated than a sphere or a cylinder, and we will show in Sect. 21.3.2 that it is possible to use reciprocity to assess the accuracy of numerical results produced for complex geometries. In this chapter, we focus on the surface integral equation (SIE) method for the numerical solution of Maxwell’s equations. An interesting feature of such a formulation is that the boundary conditions at the edge of the computation window are included in the equations and need not be taken care of by using ad hoc prescriptions, such as absorbing boundary conditions or perfectly matched layers [9]. Indeed, these boundary conditions are already included in the kernel of the integral equation and can take different forms, like infinite homogeneous space [10], surfaces or stratified media [11, 12], or waveguide cavities [13]. There is of course a price to pay for this: except for infinite homogeneous space where the kernel is known analytically [10], it must be evaluated numerically, usually by resorting to plane waves or eigenmode expansions [11, 14]. The SIE is constructed from the combination of an equation for the electric field and one for the magnetic field; different weighted combinations can be used here [15]. The volume integral form of Maxwell’s equations is transformed into a surface equation using Gauss’ theorem, and the solution is computed from unknown electric and magnetic currents defined only on the surface of the scatterer [16]. This is very advantageous since only that surface needs to be discretized; on the other hand, a limitation of this approach is that the resulting matrix is dense since each mesh is connected to all the other meshes in the system, different, e.g., from the finite difference time domain method, where only nearest neighbours are connected [17]. The resulting system of linear equations is constructed through a Galerkin procedure, where Rao–Wilton–Glisson functions are used as both basis and test functions [18]. The accuracy of the method strongly depends on the order used for the quadrature in the Galerkin scheme [19]. Once the surface currents are known,

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different observables can be computed, from the near-field to the different cross sections [20] or even the force and torque produced by the incident light on the nanostructure [21, 22].

21.3 Approaching Experimental Situations Having settled for the numerical technique, we wish to address the question of modelling the geometry of a real experiment as accurately as possible and will do that in the context of three different plasmonic systems. This field of research studies the interaction of light with coinage metals, like gold, silver, aluminium or heavily doped semiconductors [23]. When light impinges on a nanostructure made from such a metal, it resonantly excites the free electrons in the metal, producing a very strong near-field at the vicinity of the nanostructure [24, 25]. It is quite remarkable that nanostructures much smaller than the wavelength can exhibit such strong resonances, the reason being the localisation of the free charges in a specific pattern associated with each optical resonance [26, 27].

21.3.1 Fano-Resonant Systems In principle, any resonant system has an optical response with a Lorentzian shape [4]. This is also true for a plasmonic nanostructure, as long as only one single resonance is excited like in a small particle or a dipole antenna [28]. On the other hand, as soon as more than one resonance is present, the lineshape can become very complicated with several different peaks. A prominent family of such irregular responses is the so-called Fano lineshape, following the name of Ugo Fano who discovered them while interpreting atomic spectroscopy experiments [29]. In the context of plasmonics, Fano resonances occur when two modes are present in the system, often a bright mode (a mode that radiates into the near-field, like a dipole) and a dark mode (a mode that does not radiate into the far-field, like a quadrupole) [30]. In plasmonics, the intrinsic losses associated with the metal make the resonances relatively broad [31], such that several modes can overlap and interact, even when their exact resonance frequencies are different. The bright mode is excited by the incoming excitation and produces some near-field that can in turn excite the dark mode [32]. The latter will also produce a near-field that affects the bright mode. Depending whether both responses are in- or out-of-phase (i.e., depending on the excitation wavelength), they will interfere constructively or destructively, producing the asymmetric lineshape. The interest for plasmonic Fano-resonant systems lies in the fact that they exhibit very narrow spectral features, in spite of the significant losses inherent to plasmonic metals. This is useful for sensing, where the quality factor (i.e., the resonance width)

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Fig. 21.2 Optical trapping with an hexamer. (a) Six gold nanostructures are deposited on a substrate and produce a strong optical field under linear polarized illumination, which can trap a gold nanosphere. The mesh for an ideal structure is shown. (b) (from left to right) Realization of a realistic mesh based on an SEM image of an effectively realized nanostructure (scalebar 100 nm), which outline is determined using the Canny edge detector from the Scikit-image Python package. The realistic mesh is built in Blender from this outline by inspection and comparison with the SEM image. (c) Comparison of the scattering cross sections for the ideal and realistic meshes as a function of the wavelength in vacuum .λ. (d) Multipolar decomposition of the scattering cross section obtained from the realistic mesh into Cartesian multipoles: electric/magnetic dipoles (ED/MD), electric/magnetic quadrupoles (EQ/MQ), and electric/magnetic octupoles (EO/MO)

determines the sensitivity and several experiments have been performed along those lines [33–40]. Here, we are rather interested in the very strong optical near-field generated by Fano-resonant nanostructure, as illustrated in Fig. 21.2a. Six gold nanostructures are positioned on a circular ring forming a hexamer and illuminated with linear polarized light. These structures produce a strong near-field gradient that will exert a force on a nearby gold sphere (the experiment is performed in water), which will become trapped at the centre of the structure. Once the sphere is trapped, the structure becomes a heptamer and its spectrum changes. In order to guide this experiment, it is important to have an accurate description of its spectral response, especially to choose the best excitation wavelength. To this end, the experimentally realized hexamer is used to build a finite element mesh for the SIE calculations, Fig. 21.2b: first, the edge of the structure is obtained from the scanning electron microscope (SEM) image using the edge detector in the Scikit-image Python package with a standard deviation .σ = 7 pixels, which corresponds to the

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spatial extent for that edge detection [41]. This 2D outline is imported into Blender version 3.2 [42] and extruded to form a 3D object, whose sharp edges are smoothed using the bevel function, and sculpted until it mimics the shape inferred from the SEM image. A more accurate approach would consist in using tomography in an electron microscope [43], e.g., high-angle annular dark-field scanning transmission electron microscopy, which provides amazing 3D reconstructions of plasmonic nanostructures [44, 45]. Inspecting the scattering cross sections (SCSs, which correspond to the power flow scattered by the structure in the far-field) for the ideal and realistic meshes in Fig. 21.2c, we observe some differences in the Fano lineshapes, especially the magnitudes of the different peaks. Based on the realistic meshes, the SCS can be decomposed into different Cartesian multipoles [46]. Interestingly, although the electric dipole dominates the response of the system, we also observe a rather important magnetic quadrupole around .λ = 700 nm, Fig. 21.2d. The Fano resonance proceeds from the interaction between the electric dipole (bright mode) and the magnetic quadrupole (dark mode).

21.3.2 Near-Field of Plasmonic Antennae The near-field distribution at the vicinity of plasmonic nanostructures is the driver for all interactions with molecular or atomic systems, such as fluorescence [47–51] or surface enhanced Raman spectroscopy (SERS) [52–54]. Computing an absolute value of this field enhancement is an extremely difficult task, which certainly still deserves important research developments. In this section, we wish to show that the exact nanostructure geometry can play a significant role for the computed nearfield. To this end, we consider a dipole antenna made from gold with two 100 nm long arms separated with a 25 nm gap. It is possible to fabricate such a nanostructure fairly accurately with a high-resolution electron beam system and ion etching [55]. However, it can happen that the produced nanostructure resembles more a pair of potatoes than two perfect parallelepipeds, as shown in the SEM image in Fig. 21.3a. Using a similar approach as that described in Sect. 21.3.1, it is possible to infer from the SEM image a finite element mesh for that structure and use it to compute the optical response of the realistic particle. Interestingly, in the far-field, both the perfect rectangular and the realistic one have the same spectral response with a resonance at .λ = 630 nm and a modest magnitude difference. Considering both antennae as electromagnetic objects and drawing the electric field lines produce also quite similar impressions, with maybe slightly denser field lines for the real antenna, Fig. 21.3c. A very different behaviour is observed in the near-field of both antennae, as shown in Fig. 21.4, where we compute the near-field intensity enhancement around the left arm of each antenna at the resonance wavelength .λ = 630 nm. Let us first focus on the field at the close vicinity of the metal (2 nm), unwrapped like for a geography map. The incident field has unit intensity and the intensity is enhanced

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by three orders of magnitude for both antennae. The near-field distribution is very different for each geometry, with most of the enhancement at the vicinity of the corners for the ideal structure and a broader and smoother field distribution for the realistic structure. Using the ideal geometry to compute how molecules would be driven by that antenna produces very different results compared to those obtained with the realistic structure. Especially, those “hot-spots” at the antenna corners are very unlikely to appear in an experiment, where fabricated metal nanostructures always exhibit significant roughness. We will come back to this issue of roughness in Sect. 21.3.3 and show that its importance strongly depends on the physical situation at hand. At larger distances from the surface, 10 nm in Fig. 21.4, both field distributions become much more similar, without any noticeable “hot-spots” near the rectangular corners. The resemblance of a numerical model with the experimental reality is therefore especially important in the ultimate near-field, the region where, for example, molecules interacting with the structure, would be located. Sadly, many simple numerical models used to study near-field interactions with plasmonic nanostructures rely on ideal, parallelepiped structures. The previous observations on the near-field distribution at the vicinity of a plasmonic nanostructure prompt us to make a short discussion on reciprocity and its use to validate numerical models. Reciprocity is a complicated concept that can be easily misused, and we refer the interested reader to the excellent review article

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Fig. 21.4 Comparison of the near-field intensity enhancement computed around the ideal rectangular antenna and the real one deduced from the SEM image. The equirectangular projection coordinate system used to map the field intensity is shown at the top for the real antenna. The intensity maps are shown for the left arm of each antenna (rectangular or real) at two different separation distances from the metal: 2 nm and 10 nm. Adapted from Ref. [56] with permission, copyright American Chemical Society 2011

by Caloz et al. where it is discussed in detail [57]. Briefly, in an optical experiment with a source and a detector, reciprocity requires that the system responds in a similar way when the source and the detector are exchanged. This can be illustrated with the plasmonic antennas considered in this section. Figure 21.5 shows the field enhancement for the ideal dipole antenna (top row) and the realistic antenna (bottom row). The solid lines show the enhancement of the light radiated by a dipole source detected in the far-field at (x;y;z) = (0;0;106 nm). Three different dipole locations indicated by the black dots are considered, as well as two different dipole orientations: parallel to the antenna axis (red lines) and normal to the antenna axis (green lines). When the dipole is in the gap of the antenna, the intensity enhancement resembles the scattering cross sections shown in Fig. 21.3b, with again a slightly larger enhancement for the realistic antenna. This coupling between the dipole and the antenna strongly depends on the dipole orientation, and no field enhancement is observed when the dipole is normal to the antenna axis (green lines). As soon as the dipole is displaced away from the antenna centre, the enhancement decreases significantly (note the different vertical axis ranges for the second and third columns in Fig. 21.5). In this case, the ideal antenna still exhibits a Lorentzian response with a single spectral feature, indicating that this geometry supports only one electric dipole resonance. This is not the case for the realistic geometry, where several Cartesian multipoles interact to produce a more complicated response. These data indicate that molecules spin-coated on a dipole antenna will experience very different radiation enhancements, depending on their location on the nanostructure, with the molecules located close to the gap benefiting most from that enhancement.

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Fig. 21.5 Solid lines show the enhancement of the intensity radiated to the far-field by a dipole source in the proximity of an ideal rectangular (top row) or realistic (bottom row) nanoantenna. The source is located in the x-z-plane (the same axes as in Fig. 21.3 as indicated by the dots in the insets, oriented in x-direction along the antenna axis (red curves) or in z-direction normal to the antenna axis (green curves), while far-field detection is at (x;y;z)=(0;0;106 nm) and in xpolarization. Symbols represent the reverse process with source and detection points exchanged: the x-polarized source is located at (x;y;z) = (0;0;106 nm) and the detection is in the near-field in x-polarization (crosses) or z-polarization (circles). Adapted from Ref. [56] with permission, copyright American Chemical Society 2011

The crosses and dots shown in Fig. 21.5 are the results of separate calculations where a dipolar source was used in the far-field at the location (x;y;z) = (0;0;106 nm) and the field intensity was computed near the antenna at the location marked with the black dot. It should be noted that reciprocity applies to the field components, not the total intensity. Hence, for the first calculations with the dipolar source close to the antenna, only the x-polarization of the far-field was computed; for the second calculations, the dipole located in the far-field was oriented in the same x-direction, while the intensity of only the x-component (respectively, z-component) of the electric field at the black dot was computed for the red (respectively, green) lines. Altogether, this procedure corresponds to exchanging the source and the detector, and the perfect agreement between the lines and the symbols in Fig. 21.5 indicates that those numerical results fulfil reciprocity. This provides a useful way of checking the accuracy of numerical results beyond the comparison with a reference solution on a very simple, canonical geometry discussed in Sect. 21.2. This check is very easy to perform when the numerical method at hand can handle infinite geometries, as is the case for algorithms based on the integral form of Maxwell’s equations. To conclude this section, let us note that in Fig. 21.4 we dared to compute the electromagnetic field at a very short distance from the metal: 2 nm. Whether a pure classical electromagnetic approach is sufficient to do so is of course an intricate question. There appears however to be a consensus that down to that distance, it is still reasonable to do so: for shorter distances, one should resort to more

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sophisticated models, beginning first by including non-local effects [58], while even shorter distances require more complicated approaches such as time-dependent density functional theories [59] or advanced quantum (or quantum-corrected) models [60–62].

21.3.3 Hybrid Nanostructures So far, we have considered the strong optical resonances that can be excited in plasmonic metals and are associated with their very high density of free electrons [23]. Surprisingly, strong resonances can also be observed in high refractive index dielectric nanostructures by virtue of so-called Mie resonances [63]. In short, one can say that plasmonic nanostructures have essentially a fundamental resonance with an electric dipolar character, while dielectric structures have a magnetic dipolar fundamental resonance. An interesting question is whether combining these two materials can open up a new field of investigations where electric and magnetic resonances are combined? Such hybrid nanostructures are beginning to emerge, with only a few experimental demonstrations so far [64, 65], including one from our group [66]. Theoretical studies have demonstrated that indeed coupling metals and dielectrics produce a rich spectrum with some very narrow features like anapoles [67]. Figure 21.6a shows the geometry for such a hybrid nanostructure we realized for sensing applications [66]. Nominally, it is composed of a 220 nm thick Si cylinder base with a 60 nm thick Al disc cap, separated with a 75 nm .SiO2 spacer; the overall structure has a diameter of 470 nm. The dielectric spacer thickness can be adjusted to control the coupling between the modes in the dielectric Si cylinder and those in the Al plasmonic disc, providing control on the spectral response of the system, as studied in detail in Ref. [68] and illustrated in the calculations shown in Fig. 21.6c. Note that due to absorption of Si, this structure’s response is mainly in the near infrared range of the optical spectrum. Note also that the fundamental mode of the structure, around .λ = 1600 nm, is magnetic dipolar. The effectively fabricated nanostructure realized by reactive ion etching, shown in Fig. 21.6b, has dimensions that are quite close to the ideal structure with a 239 nm thick Si cylinder, a 84 nm spacer and a 57 nm thick Al disc. Their utilization for bulk refractive index sensing was tested experimentally, yielding a rather modest sensitivity of 208 nm/RIU [66], about half the best value obtained for bulk refractive index sensing with pure plasmonic nanostructures. The reason for that disappointing sensitivity can be well understood from numerical simulations. It is known that the spatial overlap between the near-field produced by the sensing nanostructure and the analyte is key for the sensitivity [69–71]. Unfortunately, the field distribution computed for the ideal structure, shown in Fig. 21.7a, indicates that most of the electric field remains within the dielectric spacer. This observation prompted the idea to etch away some of this spacer, sufficiently to expel some of the electric field into the background, but not too much to compromise the nanostructure stability.

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This can be performed with an additional wet etch step with hydrofluoric acid [68]. The resulting nanostructures are shown with false colours SEM images in Fig. 21.7c with the Si cylinder in blue, the recessed .SiO2 spacer in green and the Al disc in grey. Figure 21.7b,c indicates that this treatment increases indeed the electric field in the background. Experimentally, the sensitivity increased from 208 nm/RIE to 245 nm/RIE [68]. Further insights into this modest experimental improvement are provided in Fig. 21.8a, which shows the maximum of the near-field enhancement computed in the background for the different distributions shown in Fig. 21.7. First, we notice that the strongest enhancement is obtained for only lightly etched spacers and located close to the sharp metal edge in the metal (see Fig. 21.7b); we are again facing an issue related to sharp unrealistic geometrical features as in Sect. 21.3.2. Furthermore, there is a wavelength shift for the main resonance as the spacer diameter decreases, Fig. 21.8a. In that context, it is interesting to notice that the second resonance around .λ = 1100 nm is quite prominent for the more realistic nanostructures; to the extent that the field enhancement is almost as strong as that provided by the fundamental resonance at .λ = 1600 nm. This observation has important implications for the experiment, and it would have been unnoticed if more realistic nanostructures had not been simulated. Figure 21.8b indicates that, on the

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other hand, the scattering cross section of the nanostructure is not very sensitive to the exact geometry, as was already the case in Sect. 21.3.2. Finally, the fabricated hybrid nanostructures have an extremely rough Al top surface caused by the morphological growth of this metal, Fig. 21.6c. Even with advanced nanofabrication techniques, such roughness cannot be avoided [72], and an interesting question is whether it disturbs the nanostructure optical response. Since the SIE relies on a triangular mesh, it is possible to build models for rough nanostructures, as shown in Fig. 21.7d and f. This roughness was simply created

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21.4 Summary and Outlook In summary, we have discussed some examples where numerical simulations based on effectively fabricated nanostructures can provide additional insights into an experiment. While calculations are often used at the inception of a project, closing the loop and redoing calculations from the experimental data is very rewarding and one should probably perform the full cycle “simulations .→ nanofabrication .→ characterization .→ measurements .→ simulations” several times to gain additional insights into the underlying phenomena.

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For the specific case of plasmonics considered here, we have noticed that the same experimental detail can play a very different role, depending on the type of physical observable. For example, roughness can significantly influence the nearfield but be totally unnoticed in the far-field. It can affect molecules adsorbed on the surface, while refractive index sensing can be fully immune to such roughness. As scientists, we often have a taste for “perfection” and wish to fabricate nanostructures that have ideal shapes. Obviously, this is not possible since materials have their own minds and will not submit to our square-headed epitome. Knowing how far to go (or not) towards perfection can save a lot of time and efforts. Approaching the experimental situation as closely as possible is certainly a challenging task. An important issue is to build numerical models that mimic the effectively realized nanostructures. Here, we have used a very simple approach based on SEM images. Tomography in an electron microscope provides a more sophisticated way forward to retrieve accurate 3D representations of nanostructures [43–45]. At the same time, it is clear that each individual nanostructure will be different and efforts should also be invested in building statistics to determine the details that really matter for the optical response. Another key issue that we have not addressed in this chapter is the dielectric function used for the simulations. Even for plasmonic metals, one finds many different values in the literature, which can produce quite different optical responses. In addition, it is very unlikely that a metal deposited in a specific machine will exactly match those values from the literature. In principle, one should characterize each material with ellipsometry to retrieve its exact dielectric function, which is quite tedious and might not even provide a more accurate solution: ellipsometry requires thick metal films (at least 100 nm thick), while plasmonic nanostructures are often much thinner and have a different roughness, which can influence at least the absorption. Altogether, bridging the gap between theory and experiment is not such a trivial task. However, some of the simple steps illustrated in this chapter can help build numerical models that match the experiment better. In any case, the very first step in that endeavour is to check the convergence and stability of the numerical method at hand.

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56. Kern, A.M., Martin, O.J.F.: Excitation and reemission of molecules near realistic plasmonic nanostructures. Nano Lett. 11, 482–487 (2011) 57. Caloz, C., Alù, A., Tretyakov, S., Sounas, D., Achouri, K., Deck-Léger, Z.-L.: Electromagnetic nonreciprocity. Phys. Rev. Appl. 10, 047001 (2018) 58. García de Abajo, F.J.: Non-local effects in the plasmons of strongly interacting nanoparticles, dimers, and waveguides. J. Phys. Chem. C 112, 17983–17987 (2008) 59. Zuloaga, J., Prodan, E., Nordlander, P.: Quantum description of the plasmon resonances of a nanoparticle dimer. Nano Lett. 9, 887–891 (2009) 60. Esteban, R., Borisov, A.G., Nordlander, P., Aizpurua, J.: Bridging quantum and classical plasmonics with a quantum-corrected model. Nat. Commun. 3, 825 (2012) 61. Savage, K.J., Hawkeye, M.M., Esteban, R., Borisov, A.G., Aizpurua, J., Baumberg, J.J.: Revealing the quantum regime in tunnelling plasmonics. Nature 491, 574–577 (2012) 62. Zhu, W.Q., Esteban, R., Borisov, A.G., Baumberg, J.J., Nordlander, P., Lezec, H.J., Aizpurua, J., Crozier, K.B.: Quantum mechanical effects in plasmonic structures with subnanometre gaps. Nat. Commun. 7, 11495 (2016) 63. Baranov, D.G., Zuev, D.A., Lepeshov, S.I., Kotov, O.V., Krasnok, A.E., Evlyukhin, A.B., Chichkov, B.N.: All-dielectric nanophotonics: the quest for better materials and fabrication techniques. Optica 4, 814–827 (2017) 64. Guo, R., Rusak, E., Staude, I., Dominguez, J., Decker, M., Rockstuhl, C., Brener, I., Neshev, D.N., Kivshar, Y.S.: Multipolar coupling in hybrid metal-dielectric metasurfaces. ACS Photonics 3, 349–353 (2016) 65. Yang, J.-H., Chen, K.-P.: J. Hybridization of plasmonic and dielectric metasurfaces with asymmetric absorption enhancement. Appl. Phys. 128, 133101 (2020) 66. Ray, D., Raziman, T.V., Santschi, C., Etezadi, D., Altug, H., Martin, O.J.F.: Hybrid metaldielectric metasurfaces for refractive index sensing. Nano Lett. 20, 8752–8759 (2020) 67. Gurvitz, E.A., Ladutenko, K.S., Dergachev, P.A., Evlyukhin, A.B., Miroshnichenko, A.E., Shalin, A.S.: The high-order toroidal moments and anapole states in all-dielectric photonics. Laser Photonics Rev. 13, 1800266 (2019) 68. Ray, D., Kiselev, A., Martin, O.J.F.: Multipolar scattering analysis of hybrid metal-dielectric nanostructures. Opt. Express 29, 24056–24067 (2021) 69. Vollmer, F., Arnold, S., Keng, D.: Single virus detection from the reactive shift of a whisperinggallery mode. Proc. Nat. Acad. Sci. 105, 20701–20704 (2008) 70. Santiago-Cordoba, M.A., Boriskina, S.V., Vollmer, F., Demirel, M.C.: Nanoparticle-based protein detection by optical shift of a resonant microcavity. Appl. Phys. Lett. 99, 073701 (2011) 71. Zhang, W., Martin, O.J.F.: A universal law for plasmon resonance shift in biosensing. ACS Photonics 2, 144–150 (2015). 72. Thyagarajan, K., Santschi, C., Langlet, P., Martin, O.J.F.: Highly improved fabrication of Ag and Al nanostructures for UV and nonlinear plasmonics. Adv. Opt. Mater. 4, 871–876 (2016) Andrei Kiselev is a 4th year Ph.D. student in the Laboratory of Nanophotonics and Metrology of the Swiss Federal Institute of Technology in Lausanne (EPFL, Switzerland). Andrei defended his bachelor and master theses at Lomonosov Moscow State University, where his research was focused on photonic crystals and metamaterials based on bulk Dirac semimetals. Andrei has interests in BioNanophotonics, a field he was studying during his internship in the laboratory of Prof. Aleksandra Radenovic with the goal of developing a portable DNA sequencer. Current Andrei’s field of research focuses on the exploration of optical interactions of nanoparticles with the goal to control the optical forces between them and eventually become able to create a laserdriven nanofactory-on-chip capable of assembling nanostructures on demand.

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A. Kiselev et al. Jeonghyeon (Jenna) Kim is a Ph.D. student in Photonics at the École Polytechnique Fédérale de Lausanne (EPFL). She received a bachelor’s degree in Electrical Engineering and a master’s degree in Integrated Technology from Yonsei University, South Korea. She joined the Nanophotonics and Metrology Laboratory (NAM) at EPFL in 2017, where she is carrying her Ph.D. thesis on optical trapping of gold nanoparticles. Her current research interests include nanophotonics, optical manipulation, colloids, statistical physics, and interface science.

Olivier J. F. Martin studied physics at the Swiss Federal Institute of Technology Lausanne (EPFL) and conducted his Ph.D. research at IBM Zurich Research Laboratory, where he studied semiconductor physics. After a stay at the University of California in San Diego, he became Assistant Professor at the Swiss Federal Institute of Technology Zurich (ETHZ). In 2003 he was appointed at the EPFL, where he is currently Full Professor of Nanophotonics and Optical Signal Processing, Director of the Nanophotonics and Metrology Laboratory. Between 2005 and 2017 he was Director of the EPFL Doctoral Program in Photonics (approx. 100 Ph.D. students) and between 2016 and 2020 he was Director of the EPFL Microengineering Section (1,000 students). In that latter capacity he conducted an in-depth reform of the study plan and introduced the new EPFL Master in Robotics. Dr. Martin conducts a comprehensive research that combines the development of numerical techniques for the solution of Maxwell’s equations with advanced nanofabrication and experiments on plasmonic systems. Applications of his research include optical antennae, metasurfaces, nonlinear optics, optical nano-manipulations, heterogeneous catalysis, security features and optical forces at the nanoscale. Dr. Martin has authored over 300 journal articles and holds a handful of patents and invention disclosures. He received in 2016 an Advanced Grant from the European Research Council on the utilization of plasmonic forces to fabricate nanostructures; he is a Fellow of the Optical Society of America and Associate Editor of both Advanced Photonics and Frontiers in Physics.

Chapter 22

Theoretical Future: Vision 2030 Amir Boag, Vadim A. Markel, Olivier J. F. Martin, M. Pinar Mengüç, and Kevin Vynck

A round table discussion on the topic “Theoretical future—Vision 2030” was organized during the Weiglhofer Symposium on Electromagnetic Theory. The panel included Amir Boag from Tel Aviv University, Vadim A. Markel from the University of Pennsylvania, M. Pinar Mengüç from Ozyegin University, and Kevin Vynck from the University of Lyon, and it was chaired by Olivier J.F. Martin from the Swiss Federal Institute of Technology in Lausanne (EPFL). All participants to the symposium contributed actively to the discussion (Fig. 22.1). With their opening statements, the panelists emphasized the key role that Maxwell’s equations are playing in our society today. Electromagnetism is the driver behind so many indispensable technologies: from automotive to imaging, from telecommunications to remote sensing, from electronics to energy harvesting, with application frequencies that cover at least fifteen orders of magnitude. Industrial cycles are becoming shorter as the time between an initial idea and its application in a product is shrinking. This calls for novel design approaches, possibly beginning from materials and their functions, then their characterization and integration into a complete system—metamaterials represent a good example where this approach is quite successful.

A. Boag Tel Aviv University, Tel Aviv-Yafo, Israel V. A. Markel University of Pennsylvania, Philadelphia, PA, USA O. J. F. Martin () Swiss Federal Institute of Technology Lausanne (EPFL), Lausanne, Switzerland e-mail: [email protected] M. P. Mengüç Ozyegin University, Istanbul, Turkey K. Vynck Université de Lyon, Lyon, France © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 T. G. Mackay, A. Lakhtakia (eds.), Adventures in Contemporary Electromagnetic Theory, https://doi.org/10.1007/978-3-031-24617-3_22

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Fig. 22.1 The panelists (from left to right): A. Boag, V. A. Markel, M.P. Mengüç, K. Vynck, and O.J.F. Martin (photo P.B. Monk)

The rapid acceleration that we have witnessed over the last couple of decades in the utilization of Maxwell’s equations also brings about novel societal responsibilities. With the Internet of Things, the number of radiators in a room is increasing at a rapid pace and one may wonder whether all that connectivity is really needed, a societal debate that extends much beyond electromagnetics. The influence of electromagnetic fields on humans and animals is also a topic that requires multifaceted competences and should be investigated considering not only possible adverse effects but also potential electricity-enabled medical treatments. Many technologies based on electromagnetics are quite energy-greedy and we should endeavor that our research does not hurt the environment and be always aware of its impact, including the gray energy it generates. At the same time, it was eloquently explained that electromagnetics also bears the cure against some of its excesses and research topics like radiative heat transfer, thermal radiation, or passive cooling fall perfectly within the realm of Maxwell’s equations. Overall, inspiration from the natural world can help us address some of those challenges. Although computational electromagnetics is a very mature field of research, there are still quite a few challenges that cannot be solved with commercial programs. These include complex, multiscale, three-dimensional geometries, or systems with internal resonances, which remain difficult to handle. Large disordered systems

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as found in nature are also beyond the current state of the art. Sometimes, the numerical approaches are excessive in what they produce and it would be efficient to limit calculations to what is really needed, e.g., the scattering cross section or the optical force acting on an object. Additional research efforts should be invested into direct solvers and irregular grids. Multiphysics approaches are becoming increasingly important and so are techniques that can bridge the gap between classical electromagnetics and quantum systems, although they are still in their infancy. Machine learning has certainly emerged as a very popular approach, and one should attempt to seamlessly integrate Maxwell’s equations into those new techniques, rather than applying them by brute force. Some participants emphasized also that some mathematical tools, like the complete characterization of an algebraic variety, are essential to gain insights in the response of complex electromagnetic systems, like three-dimensional photonic crystals. The panel was unanimous to note that the widespread availability of commercial and free software for the solution of Maxwell’s equations is a mixed blessing. On the one side, it supports the central position of electromagnetics in many modern technologies and allows facile simulations to illustrate many physical situations. On the other side, there are numerous pitfalls in the utilization of these commercial and free software it seems that a significant portion of the user community is satisfied with a colorful image of the field distribution and rarely question the validity of their numerical results or carefully check the convergence of their simulation. This observation echoes a topic that emerged strongly in the course of the discussion: how to educate students in electromagnetics? The panel noted that good students interested in that subject are difficult to find. Some panelists regretted that this field of study had shifted from the physics departments to electrical engineering departments. There was also a consensus that electromagnetics is taught in a rather old-fashioned way, with “modern” textbooks merely duplicating previous works. All participants to the symposium shared a vivid enthusiasm for Maxwell’s equations and the amazing construction he established over 150 years ago. There was a general feeling that this enthusiasm should be better conveyed to students and used to tease their intellectual curiosity and inspire them. Some disruptive approaches to diversify teaching were also proposed, like replacing the seemingly complicated vector calculus with matrices and numerical calculations or using graphical examples to visualize abstract concepts: showing, for example, to the students what the nabla operator does to the electric field. The ubiquity of Maxwell’s equations in today’s technology suggests that Centers might be more appropriate for electromagnetics research than Departments. This proposition lends itself well to more inter- and multi-disciplinary approaches, where teams of students from different backgrounds work together on a specific project. There was however a call for caution with interdisciplinary projects introduced too early in the curriculum, which boils down to the least common multiple of available competences. It is essential that students master their own discipline well, before contributing meaningfully to a complex, multifaceted project. Yet, learning to communicate, collaborate, and work in a team are also key assets that need to be practiced.

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Although not specific to electromagnetics, it was noted that scientific integrity and awareness of the societal impact of technology should be prominent throughout the entire curriculum. Finally, the participants wished for a better gender balance in hard sciences at large and electromagnetics in particular.