Optical Waveguide Theory: Mathematical Models, Spectral Theory and Numerical Analysis (Springer Series in Optical Sciences, 237) 9811905835, 9789811905834

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Springer Series in Optical Sciences 237

Yury Shestopalov Yury Smirnov Eugene Smolkin

Optical Waveguide Theory Mathematical Models, Spectral Theory and Numerical Analysis

Springer Series in Optical Sciences Founding Editor H.K.V. Lotsch

Volume 237

Editor-in-Chief William T. Rhodes, Florida Atlantic University, Boca Raton, FL, USA Series Editors Ali Adibi, School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA, USA Toshimitsu Asakura, Toyohira-ku, Hokkai-Gakuen University, Sapporo, Hokkaido, Japan Theodor W. Hänsch, Max Planck Institute of Quantum Optics, Garching b. München, Bayern, Germany Ferenc Krausz, Max Planck Institute of Quantum Optics, Garching b. München, Bayern, Germany Barry R. Masters, Cambridge, MA, USA Katsumi Midorikawa, Laser Tech Lab, RIKEN Advanced Science Institute, Saitama, Japan Herbert Venghaus, Fraunhofer Institute for Telecommunications, Berlin, Germany Horst Weber, Berlin, Germany Harald Weinfurter, München, Germany Kazuya Kobayashi, Dept. EECE, Chuo University, Bunkyo-ku, Tokyo, Japan Vadim Markel, Department of Radiology, University of Pennsylvania, Philadelphia, PA, USA

Springer Series in Optical Sciences is led by Editor-in-Chief William T. Rhodes, Florida Atlantic University, USA, and provides an expanding selection of research monographs in all major areas of optics: • • • • • • • •

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Yury Shestopalov · Yury Smirnov · Eugene Smolkin

Optical Waveguide Theory Mathematical Models, Spectral Theory and Numerical Analysis

Yury Shestopalov Faculty of Engineering and Sustainable Development University of Gävle Gävle, Sweden

Yury Smirnov Department of Mathematical Modeling and Supercomputer Penza State University Penza, Russia

Eugene Smolkin Department of Mathematical Modeling and Supercomputer Penza State University Penza, Russia

ISSN 0342-4111 ISSN 1556-1534 (electronic) Springer Series in Optical Sciences ISBN 978-981-19-0583-4 ISBN 978-981-19-0584-1 (eBook) https://doi.org/10.1007/978-981-19-0584-1 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 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 Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Foreword

The book addresses the most advanced to date mathematical methods in electromagnetic field theory and offers a universal approach applicable to the analysis of virtually all known problems related to the wave propagation in diverse waveguide families. The developed methods and techniques are applied towards the analysis of the waves in the following types of guiding structures: shielded and open metal-dielectric waveguides of arbitrary cross section, planar and circular waveguides filled with inhomogeneous dielectrics, metamaterials, chiral media, anisotropic media, and layered media with absorption. Spectral properties of the problems of wave propagation for the considered waveguide families are investigated which constitutes a core issue of the book. Definitions of various types of waves are formulated, the existence and distribution of the wave spectra are studied. Several relevant mathematical techniques are developed including elements of the spectral theory of non-self-adjoint (OVFs) of one or several complex variables. A collection of rigorous proofs of the existence of various types of waves is presented; this issue constitutes an item of key significance which distinguishes this book from other publications. Numerical methods are described; they are constructed on the basis of mathematical approaches developed in the book. The results of numerical modeling of various structures are presented and discussed. The primary target readers for the book are pure and applied mathematicians working in electromagnetic field theory, researchers and engineers with deeper knowledge in mathematics, as well as graduate, post-graduate, and PhD students of universities, technical universities, and other higher educational institutions worldwide specializing in electrical engineering, mathematics, applied mathematics, physics, optics, acoustics, and related disciplines. Another definite target group are those who want to study a subject independently or/and in absentia, so that the book can be used online as a ‘self-educator’ and as a source material for different learning platforms like Canvas. The book may be a good supplementary text for upper undergraduate students who will improve their knowledge in mathematical methods for electromagnetics v

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Foreword

and are interested in learning more advanced topics. It is suitable both for students of mathematical specialities studying electrical engineering and electromagnetics as part of or a complement to the main speciality, and major engineering specialities, including electrical and civil engineering, when the students plan to master electromagnetic field theory, optics, or acoustics. The main attractive and competitive features of the book are – it offers, from a universal viewpoint, the most complete to date mathematical theory of the wave propagation in all known types of shielded and open waveguides; – unlike most of the available books on the electromagnetic field theory and mathematical methods, the book provides recently obtained rigorous proofs of the existence of waves in all considered types of waveguides and the description of the distribution of the wave spectra for all known types of waves in these waveguides (complex, leaky, etc.); – the book envelopes all recent findings under one cover and in a compact form; – the reported studies address, contrary to many other books on this subject, virtually all types of open and shielded metal-dielectric waveguides due to the universal character of the developed mathematical techniques; – the book summarizes many important results on the waveguide theory published in the former Soviet Union (mostly in Russian) and therefore not available to the Western reader; – the book may be used as an advanced textbook in mathematical methods for electromagnetic field theory, wave propagation, and waveguides. Yury Smirnov Penza State University Penza, Russia

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 3

2 Shielded Waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Method of Operator Pencils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Propagation of Normal Waves in Waveguides . . . . . . . . . . 2.1.2 Eigenvalue Problem for Operator Pencil . . . . . . . . . . . . . . . 2.1.3 Property of the Spectrum of Pencil . . . . . . . . . . . . . . . . . . . 2.1.4 Theorems of Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Properties of the System of Eigenwaves and Associated Waves of a Waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Eigenwaves and Associated Waves . . . . . . . . . . . . . . . . . . . 2.2.2 Completeness of the System of Transversal Components of Eigenwaves and Associated Waves . . . . . 2.2.3 Biorthogonal Property for Eigenwaves and Associated Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Basis Property of the System of Eigenwaves and Associated Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 7 8 14 19 26

3 Planar Waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 TE-Polarized Waves in a Partially Shielded Dielectric Layer . . . . . 3.1.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Surface Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Leaky Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 TE-Polarized Waves in an Inhomogeneous Partially Shielded Dielectric Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Sobolev Spaces and Variational Relation . . . . . . . . . . . . . . 3.2.3 Properties of the Operator Pencil . . . . . . . . . . . . . . . . . . . . . 3.2.4 Completeness of System of Eigenvectors and Associated Vectors of Operator Pencil . . . . . . . . . . . . .

36 37 42 50 52 56 59 61 61 64 66 69 69 71 72 74 vii

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3.3

TE-Polarized Waves in an Inhomogeneous Metamaterial Partially Shielded Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Sobolev Spaces and Variational Relation . . . . . . . . . . . . . . 3.3.3 Properties of the Operator Pencil . . . . . . . . . . . . . . . . . . . . . 3.3.4 Completeness of the System of Eigenvectors and Associated Vectors of Operator Pencil . . . . . . . . . . . . . 3.4 TE-Polarized Waves in an Inhomogeneous Partially Shielded Lossy Dielectric Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Sobolev Spaces and Variational Relation . . . . . . . . . . . . . . 3.4.3 Properties of the Operator Pencil . . . . . . . . . . . . . . . . . . . . . 3.4.4 Completeness of System of Eigenvectors and Associated Vectors of Operator Pencil . . . . . . . . . . . . . 3.5 TM-Polarized Waves in an Inhomogeneous Partially Shielded Dielectric Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Sobolev Spaces and Variational Relation . . . . . . . . . . . . . . 3.5.3 Properties of the Operator Pencil . . . . . . . . . . . . . . . . . . . . . 3.5.4 Completeness of System of Eigenvectors and Associated Vectors of Operator Pencil . . . . . . . . . . . . . 3.6 TM-Polarized Waves in an Inhomogeneous Metamaterial Partially Shielded Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2 Sobolev Spaces and Variational Relation . . . . . . . . . . . . . . 3.6.3 Properties of the Operator Pencil . . . . . . . . . . . . . . . . . . . . . 3.6.4 Completeness of System of Eigenvectors and Associated Vectors of Operator Pencil . . . . . . . . . . . . . 3.7 TM-Polarized Waves in an Inhomogeneous Partially Shielded Lossy Dielectric Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.2 Sobolev Spaces and Variational Relation . . . . . . . . . . . . . . 3.7.3 Properties of the Operator Pencil . . . . . . . . . . . . . . . . . . . . . 3.7.4 Completeness of System of Eigenvectors and Associated Vectors of Operator Pencil . . . . . . . . . . . . . 3.8 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.2 Numerical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Waveguides of Circular Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Surface Waves in Inhomogeneous Waveguides . . . . . . . . . . . . . . . . . 4.1.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Sobolev Spaces and Variational Relation . . . . . . . . . . . . . .

77 77 78 80 81 83 83 85 86 87 89 89 92 93 95 97 97 99 100 101 103 103 105 107 108 110 110 112 114 120 123 124 124 128

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4.1.3 Spectrum of the OVF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.4 Properties of the Spectrum of OVF . . . . . . . . . . . . . . . . . . . 4.2 Surface Waves in a Waveguide Filled with Metamaterial Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Sobolev Spaces and Variational Relation . . . . . . . . . . . . . . 4.2.3 Spectrum of the OVF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Properties of the Spectrum of OVF . . . . . . . . . . . . . . . . . . . 4.3 Surface Waves in a Waveguide Filled with Lossy Medium . . . . . . . 4.3.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Sobolev Spaces and Variational Relation . . . . . . . . . . . . . . 4.3.3 Spectrum of the OVF of the Problem . . . . . . . . . . . . . . . . . 4.3.4 Properties of the Spectrum of the OVF . . . . . . . . . . . . . . . . 4.4 Surface Waves in a Waveguide Filled with Anisotropic Media . . . 4.4.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Sobolev Spaces and Variational Relation . . . . . . . . . . . . . . 4.4.3 Spectrum of the OVF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.4 Properties of the Spectrum of OVF . . . . . . . . . . . . . . . . . . . 4.5 Surface Waves in a Waveguide Filled with Inhomogeneous Chiral Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Sobolev Spaces and Variational Relation . . . . . . . . . . . . . . 4.5.3 Properties of the OVF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Leaky Waves in an Inhomogeneous Waveguide . . . . . . . . . . . . . . . . 4.6.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2 Sobolev Spaces and Variational Relation . . . . . . . . . . . . . . 4.6.3 Spectrum of the OVF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.4 Properties of the Spectrum of the OVF . . . . . . . . . . . . . . . . 4.7 Leaky Waves in a Waveguide Filled with Metamaterial Media . . . 4.7.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.2 Sobolev Spaces and Variational Relation . . . . . . . . . . . . . . 4.7.3 Spectrum of the OVF of the Problem . . . . . . . . . . . . . . . . . 4.7.4 Properties of the Spectrum of the OVF . . . . . . . . . . . . . . . . 4.8 Leaky Waves in a Waveguide Filled with Lossy Medium . . . . . . . . 4.8.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.2 Sobolev Spaces and Variational Relation . . . . . . . . . . . . . . 4.8.3 Spectrum of the OVF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.4 Properties of the Spectrum of the OVF . . . . . . . . . . . . . . . . 4.9 Leaky Waves in a Waveguide Filled with Anisotropic Media . . . . . 4.9.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9.2 Sobolev Spaces and Variational Relation . . . . . . . . . . . . . . 4.9.3 Spectrum of the OVF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9.4 Properties of the Spectrum of the OVF . . . . . . . . . . . . . . . . 4.10 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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131 132 137 137 139 141 142 145 145 147 149 150 152 152 155 157 158 160 160 165 168 169 172 172 176 178 180 183 183 185 187 188 190 190 193 194 195 198 198 200 202 203 206

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4.10.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.10.2 Numerical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.10.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

206 208 210 218

5 Open Waveguides of Arbitrary Cross Section . . . . . . . . . . . . . . . . . . . . . 5.1 Normal Waves in an Open Metal-Dielectric Waveguide . . . . . . . . . 5.1.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Variational Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Spectrum of the OVF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.4 Properties of OVF N(γ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Normal Waves in an Open Inhomogeneous Metal-Dielectric Waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Variational Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Spectrum of the OVF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.4 The Properties of OVF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

221 224 224 227 234 237 241 241 243 247 250 256

6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

Chapter 1

Introduction

Investigations of electromagnetic waves are nowadays one of the major topics in optics, electromagnetics, acoustics, material science, and beyond. A huge flow of publications including several hundred books and many thousand articles focuses on these studies. One can conditionally divide the set of available books on mathematical methods in electromagnetic field theory and wave propagation (to name the few) into, roughly, four categories: (a) with a larger component of rigorous mathematics containing many (but not all) necessary rigorous proofs, [1–9]; (b) a successful mixture of mathematical methods and applications, however without rigorous proofs, [10–17]; (c) focusing on electromagnetic and physical applications, [18–24]; and (d) having a character of textbooks oriented towards learning and studying electromagnetic field theory and optics, [3, 5, 9–11]. Another possible division is into the works addressing different specific waveguide families: Dielectric waveguides [21, 24], planar or circular waveguides [23], layered media [14, 21, 23], and the like. The proposed book unites the properties of the categories (a) and (b) being essentially close to (a). However, still, there is a lack of background results, specifically for structures with nonhomogeneous dielectric filling that provide basic knowledge on running waves. On the other hand, availability of the closed-form solutions to the wave diffraction problems in coordinate domains [12, 19, 25] opens a possibility to create a theoretical background proceeding from and employing rigorous proofs, facts, and results and ending up with clear mathematical description and the justified physical analysis.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 Y. Shestopalov et al., Optical Waveguide Theory, Springer Series in Optical Sciences 237, https://doi.org/10.1007/978-981-19-0584-1_1

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1 Introduction

A central task of the spectral theory of open structures [2], in particular, and mathematical theory of diffraction, in general, is rigorous proofs of existence of complex eigenvalues of the spectral problems related to the analysis of running waves and description of their distribution in the complex domain. Spectra of running waves in open waveguides are studied in the mathematical theory of open structures using the reduction to spectral problems that describe singularities of the analytical continuation of the solution to the problems with sources to the domain of complex values of a parameter (frequency, longitudinal wavenumber) [26]. The resulting solution may not have direct physical meaning. Namely, a diffraction problem considered initially at real frequencies can be continued (extended) analytically to a certain multi-sheet Riemann surface H where the chosen spectral parameter is varied. Next, the (nonhomogeneous) diffraction problem is uniquely solvable everywhere in H except for a discrete set of eigenfrequencies (eigenvalues of the corresponding homogeneous (spectral) problem) forming a countable set of complex points with the only accumulation point at infinity; these points are finite-multiplicity poles of the analytical continuation of the operator of the initial diffraction problem and its Green function. This mathematically abstract approach demonstrates its efficiency in some canonical problems when it becomes possible to prove the existence and describe distribution of the spectra of eigenwaves or eigenfrequencies in the complex plane using closed-form solutions to forward scattering problems and analysis of operators as functions of several complex variables. Examples of such problems are multi-layered parallel-plane dielectric inclusions in waveguides and free space [27] and shielded metal-dielectric waveguides of complicated cross-sectional structure [28] and open dielectric waveguides of circular cross section with layered cover [29]. Recent achievements related to applications of the non-self-adjoint operator spectral theory in electromagnetics are the discoveries of complex resonances and waves [30–32] in shielded [33] and open guiding structures filled with linear3−5 or nonlinear [34] media. A core objective of the book is to provide the reader with detailed statement of the mathematical apparatus and problems that lead to rigorous proofs of the existence and localization on the complex plane of the spectra of normal waves for the considered families of waveguides. Let us present the structure of the book. Introduction provides a survey by placing what the authors have to say in a historical context. Chapter 2. Shielded waveguides of arbitrary cross section. This chapter is devoted to the analysis of the wave propagation in shielded waveguides of arbitrary cross section filled with inhomogeneous dielectrics, metamaterials, chiral media, anisotropic media, and media with absorption. Spectral properties of the problems of wave propagation for the considered waveguide family are investigated. Definitions of various types of waves are formulated, the existence and distribution of the wave spectra are studied.

1 Introduction

3

Chapter 3. Planar waveguides. This chapter addresses waves in plane waveguides filled with inhomogeneous dielectrics, metamaterials, and media with absorption. Spectral properties of the problems of wave propagation for this family of waveguides are investigated in detail. Chapter 4. Waveguides of circular cross section. This chapter is devoted to the analysis of wave propagation in circular waveguides filled with inhomogeneous dielectrics, metamaterials, chiral media, and media with absorption. The notions, results, and methods developed in Chap. 3 are applied and concretized for this family of waveguides. The existence of real and complex normal waves and analysis of the distribution of the wave spectra are backed by a variety of numerical results. Chapter 5. Open regular waveguides of arbitrary cross section. In this chapter, open waveguides of arbitrary cross section are considered; the material filling consists of inhomogeneous dielectric. The problems on normal waves are formulated with the conditions at infinity that enable one to take into account all types of waves, including complex and leaky. Spectral properties of the problems of wave propagation in open waveguides are investigated using the specially developed extensions of the spectral theory and particularly the operator-pencil approach.

References 1. E. Sanchez-Palencia, Non-Homogeneous Media and Vibration Theory (Springer, Berlin, 1980), p. 409 2. V. Shestopalov, Y.V. Shestopalov, Spectral Theory and Excitation of Open Structures (IET, London, 1996), p. 412 3. A.S. Ilinskii, V.V. Kravtsov, A.G. Sveshnikov, Mathematical Models of Electrodynamics (Vysshaya Shkola, Moscow, 1991), p. 321 4. A.S. Zilbergleit, Y.I. Kopilevich, Spectral Theory of Regular Waveguides (Leningrad: Izd-vo Fiz.-tech. in-ta., 1983) (in Russian) 5. G. Hanson, Yakovlev A Operator Theory for Electromagnetics, 2nd edn. (Springer, Berlin, 2002), p. 653 6. M. Mrozowski, Guided Electromagnetic Waves: Properties and Analysis (Wiley, Chichester, 1997), p. 370 7. A.S. Rayevskii, S.B. Rayevskii, Nonlinear Guiding Structures Described by Nonselfadjoint Operators (Radiotekhnika, Moscow, 2004), p. 112 8. Y.G. Smirnov, D.V. Valovik, Electromagnetic Wave Propagation in Nonlinear Layered Waveguide (PSU Press, Penza, 2011) 9. Y.G. Smirnov, Mathematical Methods for Studying Problems of Electrodynamics (PSU Press, Penza, 2009). ((in Russian)) 10. J. Franklin, Mathematical Methods for Oscillations and Waves (Cambridge University Press, 2020), p. 272 11. M. Cessenat, Mathematical Methods in Electromagnetism, Linear Theory and Applications (World Scientific, Singapore, 1996), p. 396 12. J. van Bladel, Electromagnetic Fields IEEE Press Series on Electromagnetic Wave Theory. (Wiley, Piscataway, 2007) 13. A.S. Rayevskii, S.B. Rayevskii, Complex Waves (Radiotekhnika, Moscow, 2010) 14. L. Brekhovskikh, Waves in Layered Media, 2nd edn. (Academic, New York, 2012), p. 503 15. E. Kuester, Theory of Waveguides and Transmission Lines (CRC Press, Boca Raton, 2020), p. 610

4

1 Introduction

16. L. Vainstein, Open Resonators and Open Waveguides (CO Golem Press, Boulder, 1969), p. 423 17. L. Lewin, Advanced Theory of Waveguides, Wireless Engineer by Iliffe (1951), p. 192 18. R. Mittra, S.W. Lee, Analytical Techniques in the Theory of Guided Waves (MacMillan, New York, 1971), p. 325 19. R.W.P. King, T. Wu, The Scattering and Diffraction of Waves (Harvard University Press, Cambridge, 1959) 20. R. Collin, Field Theory of Guided Waves (IEEE Press, Piscataway, 1991), p. 852 21. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974) 22. L.B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (Prentice Hall, Englewood Cliffs, 1973) 23. H.-G. Unger, Planar Optical Waveguides and Fibers (Oxford University Press, Oxford, 1993), p. 751 24. A. Snyder, J. Love, Optical Waveguide Theory (Springer, New York, 1983), p. 735 25. R. James, Wait Scattering of a plane wave from a circular dielectric cylinder at oblique incidence. Canadian J. Phys. 33(5), 189–195 (1955) 26. Y.V. Shestopalov, Resonant states in forward and inverse waveguide scattering problems, in Proceedings of the International Conference Electromagnetics in Advanced Applications ICEAA, Torino, Italy, September 7–11 (2015), p. 31 27. Y. Shestopalov, Y. Smirnov, E. Derevyanchuk, Permittivity Reconstruction of Layered Dielectrics in a Rectangular Waveguide from the Transmission Coefficient at Different Frequencies. Springer Proceedings in Mathematics, vol. 52 (Springer, Berlin, 2013), pp. 169–183 28. Y. Shestopalov, Y. Smirnov, Eigenwaves in waveguides with dielectric inclusions: spectrum. Appl. Anal. 24(2), 408–427 (2014) 29. E.Y. Smolkin, Y.V. Shestopalov, Numerical study of multilayered nonlinear inhomogeneous waveguides in the case of the TE polarisation, in Proceedings of the 2016 European Conference on Antennas and Propagation EuCAP 30. Y.V. Shestopalov, Resonant states in waveguide transmission problems. PIER B 64, 119–143 (2015) 31. Y.V. Shestopalov, Singularities of the transmission coefficient and anomalous scattering by a dielectric slab. J. Math. Phys. 59 (2018) 32. Y.V. Shestopalov, E. Kuzmina, On a rigorous proof of the existence of complex waves in a dielectric waveguide of circular cross section. PIER B 82, 137–164 (2018) 33. Y. Shestopalov, Y. Smirnov, Eigenwaves in waveguides with dielectric inclusions: spectrum. App. Anal. 93(2), 408–427 (2014) 34. Y.G. Smirnov, E. Smolkin, Y.V. Shestopalov, On the existence of non-polarized azimuthalsymmetric electromagnetic waves in circular metal-dielectric waveguide filled with nonlinear radially inhomogeneous medium. J. Electromagn. Waves Appl. 32(11), 1389–1408 (2018)

Chapter 2

Shielded Waveguide

Analysis of the wave propagation in complicated shielded waveguides with inclusions constitutes an important class of vector electromagnetic problems. There has been an increasing interest during the last 10–15 years to study the processes of electromagnetic wave propagation in guiding systems with nonhomogeneous filling. Although different types of them are widely used in various practical applications for more than 30 years, and a lot of their physical properties have been established the interest in developing rigorous mathematical technique in this field of electromagnetics remains. A main reason is wide application of modern guiding systems such as microstrip and slot transmission lines, which produces new types of problems to be solved numerically and requires elaborating specific methods of the spectral theory of (OVFs) and (OPs). Another driving force to accomplish this study is the absence of rigorous mathematical proofs of the existence and distribution of the wave spectra on the complex plane for a wider family of waveguides with arbitrary inclusions and dielectric filling. The typical settings applied in mathematical models of the wave propagation employ non-self-adjoint boundary eigenvalue problems for the systems of Helmholtz equations with piecewise constant coefficients and transmission conditions containing the spectral parameter. On the discontinuity lines (surfaces), the additional conditions are stated, called the transmission conditions. In standard settings applied for hollow and homogeneously filled waveguides, the spectral parameter enters only the equations and not the transmission conditions, resulting in eigenvalue problems for self-adjoint operators. Sometimes, however, the spectral parameter enters both the equations and the transmission conditions, and often in a nonlinear manner; in addition, one obtains non-self-adjoint problems. There was an enormous amount of investigations related to these problems. The main attention was paid to the results most important in practical applications: determination of spectral properties of the waveguide dominant modes not backed by rigorous mathematical verification. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 Y. Shestopalov et al., Optical Waveguide Theory, Springer Series in Optical Sciences 237, https://doi.org/10.1007/978-981-19-0584-1_2

5

6

2 Shielded Waveguide

Numerical methods for the determination of parameters of various guiding systems are presented in [1–7]. Theory of the electromagnetic wave propagation in waveguides with homogeneous filling was elaborated in classical works by Tikhonov and Samarsky [8–10]. They verified the existence of the TE and TM waves in hollow and homogeneously filled waveguides; the wave spectra formed by eigenvalues of the corresponding Dirichlet and Neumann problems for the Laplacian are located on the real (finitely many) and imaginary (infinite number) axes on the complex plane of the spectral parameter (longitudinal wavenumber). Note that here, the core settings of the theory are scalar self-adjoint problems rather than vector ones. For nonhomogeneous waveguides, the well-known results, including explicit dispersion relations for eigenvalues, are obtained for rectangular [3, 11] and circular [6] waveguides with partial dielectric filling. However, in these cases, there were no results concerning the existence and distribution of the wave spectra on the complex plane as well as completeness and basis property of the system of normal waves. The propagation of normal waves in waveguides with nonhomogeneous filling constitutes essentially vector and non-self-adjoint problems. In nonhomogeneous waveguides, “complex” waves may exist that correspond to eigenvalues, which do not lie on real or imaginary axis. This phenomenon was discovered and studied in [12, 13]. The existence of eigenvalues of multiplicity greater than 1 was discussed in [14, 15]. Important contribution to the mathematical theory of the electromagnetic wave propagation in complicated waveguides with inclusions was made by Ilyinsky and Shestopalov [5, 7, 16–19]. They proposed the reduction of the waveguide problem of normal waves to the investigation of nonlinear meromorphic (OVFs) depending on the spectral parameter. OVFs are defined by a system of boundary integral equations with logarithmic singularity of the kernels. The integral equations are considered on the lines of the permittivity discontinuities. This approach was developed by Chernokozhin [20] and the authors [21–23]. Using the technique of (OVFs), the discreteness of the wave spectrum was proved for a wide family of waveguides with nonhomogeneous filling. For slot transmission lines, the existence of eigenvalues was established by Ilyinsky and Shestopalov [5] using the generalized Rouche theorem. The localization of eigenvalues on the complex plane was studied in [5, 16, 17, 20]. One has to mention that it is not generally possible to obtain the statements of completeness and basis property of the system of the waveguide normal waves by these methods. Another approach based on the reduction to the eigenvalue problems for operator pencil in Sobolev spaces was developed by the authors. General theory of OPs was sufficiently well-elaborated beginning from the fundamental work by Keldysh [24] devoted to the analysis of non-self-adjoint polynomial (OVFs) (pencils) and advanced then in [25–34]. Note that theory of OPs are very close to theory of nonself-adjoint operators [25, 26]. It allows us to apply in this book the most powerful to date methods of this theory and perform a comprehensive analysis of the wave propagation in virtually all known types of waveguides. Note that OPs were applied to the investigation of electromagnetic problems in [35–37].

2.1 Method of Operator Pencils

7

2.1 Method of Operator Pencils For the investigation of the propagation of normal waves in waveguides of arbitrary cross section and dielectric filling, the method of OPs is a natural and effective approach. The reduction of the governing boundary eigenvalue problem to an eigenvalue problem for an operator pencil (namely, to the determination of characteristic numbers of the pencil) allows one to apply well-developed methods of functional analysis in order to study the spectral properties of the pencil. Methods of analyzing spectral properties of (OPs) were elaborated in many details (see, for example, [38] and references therein). Note that another approach where the methods of integral (OVFs) are applied yields less interesting theoretical results [5]. However, in general, these two approaches efficiently complement each other. We describe a wide class of guiding systems and formulate the problems of normal waves for the homogeneous Maxwell equations. The setting is reduced to boundary eigenvalue problems for the longitudinal components of the electromagnetic field. We consider (weak) solutions to the boundary eigenvalue problem using variational relations in Sobolev spaces. The boundary eigenvalue problem is equivalent to an eigenvalue problem for an operator pencil L (γ) of the fourth order. First, we establish the properties of the spectrum of pencil L (γ). Then we prove the spectrum discreteness and describe localization of eigenvalues of the pencil on the complex plane. Furthermore, we consider the properties of the system of eigenvectors and associated vectors of pencil L (γ). Double completeness of this system is established with finite defect or without defect under certain additional conditions imposed on the pencil parameters. The results are obtained using perturbation of a simple pencil and pencil factorization. Section 2.1.4 is devoted to investigation of the properties of the system of eigenwaves and associated waves of the waveguide: completeness, basis property, and biorthogonal property of the system of eigenwaves and associated waves. These properties are important when the excitation waveguides are studied. We consider only the case ε1 = ε2 , which leads to a vector problem; for ε1 = ε2 the problem of normal waves is reduced to two scalar problems for the Helmholtz equations. First, the definition is formulated of eigenwaves and associated waves of a waveguide via eigenvectors and associated vectors of pencil L (γ). We show that this definition is equivalent to the standard one employing solutions to the Maxwell equations. Next, we prove a basic theorem stating the completeness of the system of the transversal components of eigenwaves and associated waves in L 42 (). This fundamental result allows us to apply the theorems of Sect. 2.1 for the analysis of the completeness of the transversal wave components. Finally, we establish the biorthogonal relations for the transversal components of eigenwaves and associated waves. Biorthogonal relations allow us to construct a system biorthogonal with respect to the system of transversal components of eigenwaves and associated waves in L 42 (). Thus we establish “minimality” of this system. It

8

2 Shielded Waveguide

is demonstrated, however, that this system is not a Schauder basis in L 42 () in the general case.

2.1.1 Propagation of Normal Waves in Waveguides Assume that Q ⊂ R 2 = {x3 = 0} is a bounded domain on the plane O x1 x2 with boundary ∂ Q. Let l ⊂ Q be a simple closed or unclosed C ∞ -smooth curve without points of intersection dividing Q into domains 1 and 2 ; Q = 1 ∪ 2 ∪ l. If l is an unclosed curve, then the points ∂l do not coincide and belong to ∂ Q: ∂l ⊂ ∂ Q. We will assume also that boundaries ∂ Q, ∂1 , ∂2 of domains Q, 1 , 2 are simple closed piecewise-smooth curves formed of finite number of C ∞ -smooth arcs intersecting at nonvanishing angles. Let Pi ∈ l be the arbitrary 2N points Pi = P j dividing l into the parts  and   such that  = l\  ,   = l\,  ∪   ∪ Pi = l ( ∩   = ∅). If N = 0 then  = l, i

  = ∅. Let us denote also  = 1 ∪ 2 ∪ , 0 = ∂ Q ∪   . In the general case, boundary ∂ of domain  contains the points with inner angles 0 < α ≤ 2π. Such point for α = 2π is called “edge”. Domain  satisfies the cone property [39, 40], which allows one to apply the embedding and trace theorems in Sobolev spaces [39, 40]. We will consider the problem of normal waves in a shielded waveguide where the transversal (with respect to O x3 ) cross section is formed by domain Q. We will assume that waveguide’s filling contains two isotropic media with relative permittivity ε j in domain  j ; ε j ≥ 1, Im ε j = 0, μ j = 1, ( j = 1, 2). 0 is the projection of the surface of the infinitely thin and perfectly conducting shields and  is the projection of the dielectric surfaces. The considered geometry of the waveguides contains all types of shielded transmission lines: cylindrical and rectangular waveguides with partially filling, slot and strip lines [4] (Fig. 2.1).

Fig. 2.1 Two types of waveguides

2.1 Method of Operator Pencils

9

Propagation of electromagnetic waves in guiding systems is described by the homogeneous system of Maxwell equations with a dependence eiγx3 on the longitudinal coordinate x3 [5]: curl E = −iH, X = (x1 , x2 , x3 ) ∈ , curl H = iεE, x = (x1 , x2 ) , E (X ) = (E 1 (x) e1 + E 2 (x) e2 + E 3 (x) e3 ) eiγx3 , H (X ) = (H1 (x) e1 + H2 (x) e2 + H3 (x) e3 ) eiγx3 ,

(2.1.1)

with the boundary conditions for the tangential electric field components on the perfectly conducting surfaces (2.1.2) Et | M = 0, transmission conditions for the tangential electric and magnetic field components on the surfaces of discontinuity (“jump”) of the permittivity [Et ] L = 0, [Ht ] L = 0,

(2.1.3)

and the finite energy condition 

  ε |E|2 + |H|2 d X < ∞.

(2.1.4)

V

Here, M = {X : x ∈ 0 } is the shielded part of the boundary, L = {X : x ∈ } is the boundary of discontinuity, V ⊂  = {X : x ∈ } is an arbitrary bounded domain. System of Maxwell equations (2.1.1) is written in the normalized √ form. We use the dimensionless variables and parameters [23, 41]: k0 x → x, μ0 /ε0 H → H, E → E; k02 = ε0 μ0 ω 2 , where ε0 and μ0 are the permittivity and permeability of vacuum (time-dependent factor eiωt is omitted). Problem of normal waves is an eigenvalue problem for the system of Maxwell equations with respect to spectral parameter γ. Eigenfunctions corresponding to certain complex values of longitudinal wavenumber γ are usually called normal waves of the waveguide. Let us write system of Maxwell equations (2.1.1) in the form ∂ E3 ∂ H3 ∂ H3 − iγ H2 = iεE 1 , − iγ E 2 = −i H1 , iγ H1 − = iεE 2 , ∂x2 ∂x2 ∂x1 iγ E 1 −

∂ E3 ∂ H2 ∂ H1 ∂ E2 ∂ E1 = −i H2 , − = iεE 3 , − = −i H3 , ∂x1 ∂x1 ∂x2 ∂x1 ∂x2

and express functions E 1 , H1 , E 2 , H2 via E 3 and H3 from the first, second, fourth, and fifth equations

10

2 Shielded Waveguide

    ∂ E3 ∂ E3 ∂ H3 i ∂ H3 γ , E2 = γ , (2.1.5) − + ∂x1 ∂x2 ∂x2 ∂x1 k˜ 2     i ∂ H3 i ∂ H3 ∂ E3 ∂ E3 H1 = +γ +γ ε , H2 = −ε ; k˜ 2 = ε − γ 2 . ˜k 2 ˜k 2 ∂x2 ∂x1 ∂x1 ∂x2 E1 =

i k˜ 2

It is possible if γ 2 = ε1 , γ 2 = ε2 . It follows from (2.1.2) that the field of a normal wave can be represented via two scalar functions  (x1 , x2 ) = E 3 (x1 , x2 ) ,  (x1 , x2 ) = H3 (x1 , x2 ) . Thus the determination of normal waves is reduced to the problem for functions  and . For functions  and  from (2.1.1) to (2.1.2), we have the following eigenvalue problem: find γ ∈ C, called eigenvalues such that there exist nontrivial solutions of the Helmholtz equations  + k˜ 2  = 0, x = (x1 , x2 ) ∈ 1 ∪ 2 ,  + k˜ 2  = 0, k˜ 2 = k˜ 2j = ε j − γ 2 ,

(2.1.6)

satisfying the boundary conditions on 0 |0 = 0,  ∂  = 0, ∂n 0

(2.1.7)

transmission conditions on  [] = 0, [] = 0, 

1 ∂ γ ˜k 2 ∂τ 

1 ∂ γ ˜k 2 ∂τ







ε ∂ + ˜k 2 ∂n 



1 ∂ − ˜k 2 ∂n



= 0,



= 0,

(2.1.8)

and the energy (“edge”) condition  (|∇|2 + |∇|2 + ||2 + ||2 )d x < ∞. 

(2.1.9)

2.1 Method of Operator Pencils

11

Here, n and τ denote the (exterior to 2 ) normal and tangental unit vectors such that x1 × x2 = τ × n. Square brackets [ f ] = f 2 | − f 1 | denote the difference of limiting values of a function on  in domains 2 and 1 . Conditions (2.1.7) are to be satisfied on both sides of the part of the boundary   . In order to obtain (2.1.6)–(2.1.9), we used formulas (2.1.5). Conditions (2.1.7)– (2.1.9) are another form of conditions (2.1.2)–(2.1.4). Thus longitudinal components of a normal wave satisfy (2.1.6)–(2.1.9). The inverse statement is true. If  and  is a solution of problem (2.1.6)–(2.1.9) then the transversal components can be determined by (2.1.5). The field E, H will satisfy all conditions (2.1.1), (2.1.2)–(2.1.4). The equivalence of the reduction to problem (2.1.6)–(2.1.9) is not valid only for γ 2 = ε1 or γ 2 = ε2 ; in this case, it is necessary to study system (2.1.1) directly. System of equations (2.1.6) with boundary conditions (2.1.7), transmission conditions (2.1.8), and condition (2.1.9) constitutes a boundary eigenvalue problem. Note that coefficient ε is not continuous and the transmission conditions contain spectral parameter γ. Moreover, boundary  has “edges” in the general case. Let us formulate the definition of the solution to problem (2.1.6)–(2.1.9). We look for the solutions of (2.1.6)–(2.1.9) in Sobolev spaces [39, 40]  f |0 = 0 ,

H01 () = f : f ∈ H 1 () , 1

H () =

⎧ ⎨ ⎩

 f dx = 0

f : f ∈ H 1 () , 

⎫ ⎬ ⎭

with the inner product and the norm  ∇ f ∇ gd ¯ x,  f 21 = ( f, f )1 .

( f, g)1 = 

1

The seminorm  · 1 in H 1 () is a norm in H01 () and H () because the sesquilinear form ( f, g)1 in these spaces is coercive [1] (note that it is sufficient to use boundedness of  to prove the coercive property of the form in H01 () while it is necessary 1

1

to use cone property to prove this statement in H ()). Spaces H01 () and H () can be defined as a supplement of spaces of infinitely smooth functions C0∞ () and 1

C ∞ () with respect to the norm  · 1 (under the condition  f 1 < ∞). H () is a subspace of functions from H 1 () which are orthogonal to the unit function. Under the above assumptions, domain  satisfies cone property: there is a cone

 K 0 = x : 0 ≤ x1 ≤ b, x22 ≤ ax12 ; a > 0, b > 0 such that any point P ∈  can be the vertex of cone K p which is equal to K 0 and the cone belongs to , K p ⊂ . This property allows us to apply the Sobolev trace

12

2 Shielded Waveguide

theorem [39] and consider the trace of function f ∈ H 1 () on  as an element of space H 1/2 (). Due to the trace theorem f |0 = 0 means that the function is equal to zero in H 1/2 (0 ). For any function f ∈ H 1 (), we have [ f ] = 0 in the sense of space H 1/2 (). And vice versa, if [ f ] = 0, f |1 ∈ H 1 (1 ), f |2 ∈ H 1 (2 ), then f ∈ H 1 (). On the part of the boundary   ⊂ 0 the trace theorem should be applied on both sides of   ; in this case functions f ∈ H 1 () have, in general, different traces on different sides of   . Note that the embeddings H01 () ⊂ H01 (Q) ⊂ H 1 (Q) ⊂ H 1 () , hold; however, all embeddings are not dense if   = ∅. 1

Let  ∈ H01 (),  ∈ H (). Condition (2.1.6) is fulfilled in 1 and 2 in terms of distributions [42]. Moreover, we have for the boundary condition on 0  j

0 ∩∂ j

∈H

1/2

   ∂ j   0 ∩ ∂ j , ∂n 

0 ∩∂ j

  ∈ H −1/2 0 ∩ ∂ j .

For the transmission condition on  | ∈ H 1/2 () , | ∈ H 1/2 ()     ∂ j  ∂ j  ∂  ∂  −1/2 , ∈H , ∈ H −1/2 () , () , ∂n  ∂n  ∂τ  ∂τ  where  j and  j are restrictions of  and  on  j . Let us give variational formulation of problem (2.1.3)–(2.1.9). Multiply 1

(2.1.3) and (2.1.4) by arbitrary test functions u¯ ∈ H01 () and v¯ ∈ H () (we can ¯ 2 because ¯ 1 and  assume that these functions are continuously differentiable in  1

they form a dense set in H01 () and H ()) and apply Green’s formula [43] for each domain  j separately. The possibility of application of Green’s formula for these functions is proved in [43]. We have 

∇∇ ud ¯ x − k˜ 2j

j

 j

∇∇ vd ¯ x − k˜ 2j

(2.1.10)

 ∂  vdτ ¯ ; j = 1, 2. ∂n ∂ j

(2.1.11)

 ud ¯ x = (−1)

j



j ∂ j

  vd ¯ x = (−1) j

j

 ∂  udτ ¯ , ∂n ∂ j



∂ j

Then, substituting the normal derivatives from (2.1.7) and (2.1.8) to (2.1.10), (2.1.11) we obtain the variational relation

2.1 Method of Operator Pencils

 

1 k˜ 2

13

 ¯ d x − (εu¯ +  v) ¯ dx (ε∇∇ u¯ + ∇∇ v)      ∂ −γ k˜12 v¯ − ∂ u¯ dτ = 0 ∂τ ∂τ

(2.1.12)



which is derived for smooth functions u, v. Further, we will prove the continuity of the sesquilinear forms defined by integrals in (2.1.12). Hence, relation (2.1.12) can be extended to arbitrary functions  1 u ∈ H01 (), v ∈ H (). Here and below, the f in f dτ is understood to be the 

trace of a function on . For v ≡ 1, u ≡ 0 we get in a similar way  

 d x = − 



1 ∂ k˜ 2 ∂n





1 dτ = −γ k˜ 2 

 

 ∂  dτ = 0; ∂τ 

(2.1.13)

1

consequently, the choice of space H () does not contract the space of solutions of problem (2.1.6)–(2.1.9). In (2.1.13) we used the condition   |l ∈ H˜ 1/2 ¯ , supp |l ⊂ ¯ since  ∈ H01 ()   and, hence, set of functions C0∞ () is dense in H˜ 1/2 ¯ . Definition 2.1.1 The pair of functions 1

 ∈ H01 () ,  ∈ H () (1 + 1 = 0) is called the eigenvector of problem (2.1.6)–(2.1.9) corresponding to eigenvalue γ0 1

if the variational relation given by (2.1.12) holds for u ∈ H01 (), v ∈ H (). 1

Thus, if  ∈ H01 () and  ∈ H () and (2.1.6)–(2.1.9) are fulfilled then variational relation (2.1.12) is fulfilled too. The inverse assertion is true. Choosing u and v with a support in  j we have that (2.1.6) are fulfilled in terms of distributions. The first conditions in (2.1.7) and (2.1.8) and condition (2.1.9) are fulfilled by the 1

definition of spaces H01 () and H (). If we choose u ≡ 0 and the support of v contains the part 1 of boundary 0 then from (2.1.12) and Green’s formula we find [43] that the second condition in (2.1.7) is fulfilled in term of distributions. Choosing arbitrary u and v on  in (2.1.12) and applying formulas (2.1.10), (2.1.11), we obtain the relation

14

2 Shielded Waveguide

          1 ∂ 1 ∂ ε ∂ 1 ∂ γ udτ ¯ + γ vdτ ¯ = 0, + − k˜ 2 ∂τ  k˜ 2 ∂n  k˜ 2 ∂τ  k˜ 2 ∂n 





which means that the second and third conditions in (2.1.8) are fulfilled in terms of distributions. Give some remarks about the smoothness of eigenvectors of problem (2.1.6)– (2.1.9). It is well known [44, 45] that solutions  and  of the homogeneous Helmholtz equations (2.1.6) are infinitely smooth in 1 and 2 : ,  ∈ C ∞ (1 ∪2 ); consequently, (2.1.6) are satisfied in the “classical” sense. In the vicinity of an arbitrary smooth part 1 of boundary 0 conditions (2.1.7) are fulfilled as well in the “classical” sense and functions  and  are infinitely smooth up to the boundary. The behavior of  and  in the vicinity of “angle” points was analyzed in [46]. Note, however, that the properties of smoothness of functions  and  will not be used in what follows.

2.1.2 Eigenvalue Problem for Operator Pencil Multiplying (2.1.12) by k˜12 k˜22 we rewrite it in the form γ4

 

¯ d x + γ2 (εu¯ +  v)

 

¯ dx (ε∇∇ u¯ + ∇∇ v)



   ∂  v¯ − ∂ u¯ dτ ¯ d x + (ε1 − ε2 ) γ − (ε1 + ε2 ) (εu¯ +  v) ∂τ ∂τ        1 ¯ d x − ∇∇ u¯ + ε ∇∇ v¯ d x = 0, +ε1 ε2 (εu¯ +  v) 

∀u ∈

H01

1

(2.1.14)



() , v ∈ H () . 1

Let H = H01 () × H () be the Cartesian product of the Hilbert spaces with the inner product and the norm ( f, g) = ( f 1 , g1 )1 + ( f 2 , g2 )1 ,  f 2 =  f 1 21 +  f 2 21 ; where 1

f, g ∈ H, f = ( f 1 , f 2 )T , g = (g1 , g2 )T , f 1 , g1 ∈ H01 () , f 2 , g2 ∈ H () . Then the integrals in (2.1.14) can be considered as sesquilinear forms on C defined in H with respect to the vector functions f = (, )T , g = (u, v)T .

2.1 Method of Operator Pencils

15

These forms (if they are bounded) define, in accordance with the results of [47], linear bounded operators T : H → H t ( f, g) = (T f, g) , ∀g ∈ H.

(2.1.15)

Linear property follows from linear property of the form with respect to the first argument and continuity follows from the estimates T f 2 = t ( f, T f ) ≤ C  f  T f  . Let us consider the following quadratic forms and the corresponding operators  (ε∇ f 1 ∇ g¯1 + ∇ f 2 ∇ g¯2 ) d x = (A1 f, g) , ∀g ∈ H,

a1 ( f, g) := 

a2 ( f, g) :=

   1 ∇ f 1 ∇ g¯1 + ∇ f 2 ∇ g¯2 d x = (A2 f, g) , ∀g ∈ H, ε



 (ε f 1 g¯1 + f 2 g¯2 ) d x = (K f, g) , ∀g ∈ H,

k ( f, g) := 

  s ( f, g) := 

 ∂ f1 ∂ f2 g¯2 − g¯1 dτ = (S f, g) , ∀g ∈ H. ∂τ ∂τ

(2.1.16)

It is easy to see that forms a1 ( f, g) and a2 ( f, g) are bounded. The same property for k ( f, g) follows from Poincare’s inequality [39]. Prove   that1 form   s ( f, g) is bounded too. Assume that functions f 1 , f 2 , g1 , g2 ∈ ¯1 ∩C  ¯ 2 . Then C1    

    ∂ f1 ξ ∂ f 1 ∂ g¯ 2 ∂ f2 ∂ f 1 ∂ g¯ 2 ∂ f 2 ∂ g¯ 1 ∂ f 2 ∂ g¯ 1 g¯ 2 − g¯ 1 dτ = d x, − + − ∂τ ∂τ 2 ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x2 ∂x2 ∂x1 



where ξ=

1, x ∈ 1 ; −1, x ∈ 2

using the Schwartz inequality, we finally obtain |s ( f, g)| ≤

1  f  g . 2

(2.1.17)

The required property of the form may be easily obtained if to extend the estimate given by (2.1.17) for arbitrary functions f, g ∈ H using the continuity.

16

2 Shielded Waveguide R

In the expression

  

g1 , g2 , ∂∂τf1 , ∂∂τf2

 ∂ f1 ∂ f2 g¯ 2 − g¯ 1 dτ ∂τ ∂τ

mean restriction of the traces

   ∂ f1 ∂ f2     ∈ H −1/2 ∂ j g1 , g2 |∂ j ∈ H 1/2 ∂ j , ,  ∂τ ∂τ ∂ j on  [42]. Since on l ¯ supp g1 ⊂ , ¯ f 1 |  = 0, g1 |  = 0, supp f 1 ⊂ , then the following formulas of integration by parts hold     ∂ f1 ∂ g¯ 2 ∂ f2 ∂ g¯ 1 g¯ 2 dτ = − f 1 dτ , g¯ 1 dτ = − f 2 dτ . ∂τ ∂τ ∂τ ∂τ 





(2.1.18)



Now the variational problem given by (2.1.14) may be written in the operator form (L (γ) f, g) = 0, ∀g ∈ H which is equivalent to the following equation for the operator-valued pencil L (γ) f = 0, L (γ) : H → H, L (γ) := γ 4 K + γ 2 (A1 − (ε1 + ε2 ) K ) + (ε1 − ε2 ) γ S + ε1 ε2 (K − A2 ) , (2.1.19) where all operators are bounded. Equation (2.1.19) is another form of variational relation (2.1.14). Eigenvalues and eigenvectors of the pencil coincide with eigenvalues and eigenfunctions of problem (2.1.6)–(2.1.9) for γ 2 = ε1 , γ 2 = ε2 by the definition. Thus the problem of normal waves is reduced to the eigenvalue problem for pencil L (γ). Following this setting, we consider the properties of the operators in (2.1.16). Lemma 2.1.1 Operators A1 , A2 are uniformly positive: I ≤ A1 ≤ εmax I, ε−1 max I ≤ A2 ≤ I,

(2.1.20)

where εmax = max (ε1 , ε2 ), I is the unit operator in H . Proof The proof of the lemma is reduced to verification of simple inequalities  f 2 ≤ (A1 f, f ) ≤ εmax  f 2 , 2 2 ε−1 max  f  ≤ (A2 f, f ) ≤  f  .



Lemma 2.1.2 Operator S is self-adjoint, S = S ∗ , and the following inequalities hold

2.1 Method of Operator Pencils

17

−

1 1 I ≤ S ≤ I. 2 2

(2.1.21)

Proof Self-adjoint property of operator S follows from the equality   

    ∂ f1 ∂ g¯1 ∂ f2 ∂ g¯2 g¯2 − g¯1 dτ = f2 − f 1 dτ , ∂τ ∂τ ∂τ ∂τ 

which follows in its turn from the remark and formulas (2.1.18). Inequalities (2.1.21) follow from (2.1.17).  Lemma 2.1.3 Operator K is positive, K > 0 and compact. The following estimate holds for its eigenvalues   λn (K ) = O n −1 , n → ∞. Proof It is easy to see that (K f, f ) > 0 for f = 0 since

 

(2.1.22)  ε | f 1 |2 + | f 2 |2 d x = 0 is

fulfilled only for f 1 = 0, f 2 = 0 (in H 1 ()). Let us prove that K is compact. Assume that f n = (u n , vn )T → 0 weakly in H . It follows from the theorem of compactness of embedding H 1 () into L 2 () [39] that f n → 0 strongly in L 2 (). Using the Schwartz inequality, we have            K f n 2 =  εu n K u n d x + vn K vn d x                      ≤  εu n K u n d x  +  vn K vn d x        ⎛ ⎞1/2 ⎛ ⎞1/2 ⎛ ⎞1/2 ⎛ ⎞1/2     2 2 2 2 ≤ εmax ⎝ |u n | d x ⎠ ⎝ |K u n | d x ⎠ + ⎝ |vn | d x ⎠ ⎝ |K vn | d x ⎠ 

 ≤ εmax 



  |u n |2 + |vn |2 d x ·







  |K u n |2 + |K vn |2 d x



≤ εmax  f n  L 2 K f n  , K f n  ≤ εmax  f n  L 2 → 0, n → 0,

and the compactness of the operator is proved. Proof of asymptotic behavior (2.1.22) is based on the Courant variational principle. From the inequality  

  ε | f 1 |2 + | f 2 |2 d x ≤ εmax

 

  | f 1 |2 + | f 2 |2 d x

18

2 Shielded Waveguide

we obtain [27] that λn (K ) ≤ εmax λn (K H ) , n ≥ 1, where λn (K H ) are eigenvalues of the operator induced by the sesquilinear form  q ( f, g) :=



 f 1 g 1 + f 2 g 2 d x = (K H f, g) , ∀g ∈ H.

(2.1.23)



Thus it is sufficient to consider operator K H . Let W = H01 (Q) × H 1 (1 ∪ 2 ) . Then H ⊂ W . Consider operator K w : W → W defined by form (2.1.23) on space W (with the same inner product and the norm). Taking into account, the variational principle [27, 45] we have inf sup λ j+1 (K H ) =   u1, . . . , u j f ∈H u i ∈ L 2 (Q) q ( f, u i ) = 0

q ( f, f )  f 2

≤  inf sup  u1, . . . , u j f ∈W u i ∈ L 2 (Q) q ( f, u i ) = 0

q ( f, f ) , j ≥ 0,  f 2

since sup on the right-hand side of the inequality is considered on a wider set of functions. Eigenvalues of operator K w correspond to the boundary eigenvalue problem for the Laplace operator − = μ, x ∈ Q; |∂ Q = 0;   ∂  ∂  − = μ, x ∈ 1 ∪ 2 ; = = 0; ∂n ∂1 ∂n ∂2 μ = λ−1 (K w ) , μn = λ−1 n (K w ) , in domains with the piecewise smooth boundary; the well-known asymptotic behavior [48] yields μn ∼

4πn 2πn , n → ∞; = mesQ + mes1 + mes2 mesQ

hence, λn (K w ) ∼

mes Q , n → ∞, 2πn

2.1 Method of Operator Pencils

19

λn (K ) ≤ cn −1 , n → ∞; c > εmax

mesQ . 2π 

The statement is proved.

Thus all operators A1 , A2 , K , S are self-adjoint and KerK = {0}. There exist 1/2 −1/2 : H → H ; these operbounded inverse operators A−1 j : H → H and A j , A j ators are uniformly positive. Note that in this case condition B > 0 yields B = B ∗ since Hilbert space H is considered on complex-valued functions. Using these lemmas, we obtain Corollary 2.1.1 Operator pencil L (γ) is self-adjoint: L ∗ (γ) = L (γ) ¯ .

(2.1.24)

From variational relation (2.1.14), we have Corollary 2.1.2 Let P be the orthogonal projection such that P ( f 1 , f 2 )T = (− f 1 , f 2 )T Then A1 = P A1 P, A2 = P A2 P, K = P K P, S = −P S P and the following representation holds L (−γ) = P L (γ) P.

(2.1.25)

The proof of this assertion directly follows from the explicit form of variational relation (2.1.14). Note that S is not a Fredholm operator since dim Ker S = ∞. Indeed, all functions f = ( f 1 , f 2 )T with additional conditions f 1 | = 0, f 2 | = 0 belong to the kernel of operator S.

2.1.3 Property of the Spectrum of Pencil We will denote by ρ (L) the resolvent set of L (γ) (consisting of all complex values of γ where there exists a bounded inverse operator L −1 (γ)) and by σ (L) = C\ρ (L) the spectrum of L (γ). Let us formulate the definitions [24, 27, 28]. Definition 2.1.2 OVF A (γ) : H → H , γ ∈ C is called finite-meromorphic at a point γ0 if there exists a vicinity of γ0 where the expansion

20

2 Shielded Waveguide

A (γ) =

∞ 

(γ − γ0 ) j A j (γ0 ) ,

(2.1.26)

j=−n

holds in the vicinity of γ0 . In the expansion A−k (γ0 ) (k = 1, . . . , n) are finitedimensional operators. If n = 0, then operator-function A(γ) is called holomorphic at a point γ0 . If A(γ) is finite-meromorphic (holomorphic) at any point of domain G, then A(γ) is finite-meromorphic (holomorphic) in G. Finite-meromorphic OVF A(γ) is called Fredholm at a point γ0 (in domain G) if A0 (γ0 ) (for any γ0 ∈ G) is a Fredholm operator with the index zero in expansion (2.1.26). Let OVF A (γ) is holomorphic in G. γ0 is called eigenvalue of A (γ) if equation A (γ0 ) ϕ0 = 0 has a nontrivial solution ϕ0 = 0. This solution is called eigenvector of A (γ). Vectors ϕ0 , ϕ1 , . . . , ϕk form the chain of associated vectors if the following equalities A (γ0 ) ϕ p +

1 ∂ A (γ0 ) 1 ∂ p A (γ0 ) ϕ p−1 + · · · + ϕ0 = 0; ( p = 1, . . . , k) 1! ∂γ0 p! ∂γ0p (2.1.27)

hold. Number k + 1 is called the length of the chain. The length can be finite or infinite. Eigenvector ϕ0 has a finite rank r if the chain with the maximum length has length r . Definition 2.1.3 The system of vectors (k) (k) ϕ(k) 0 , ϕ1 , . . . , ϕm k , k = 1, 2, . . .

is called the canonical system of eigenvectors and associated vectors at γ = γ0 if the following properties hold: (1) vector ϕ(1) 0 is an eigenvector with maximum rank m 1 + 1; (k) (2) vector ϕ0 is an eigenvector with maximum rank m k + 1, which is not a linear (k) (k) combination of vectors ϕ(k) 0 , ϕ0 , . . . , ϕ 0 ; (k) (k) (3) vectors ϕ0 , ϕ1 , . . . , ϕ(k) m k is the chain of associated vectors;   (k) (4) vectors ϕ0 is a basis of space Ker A (γ0 ). Number m 1 + 1 + m 2 + 1 + · · · is called algebraic multiplicity of eigenvalue γ0 . Definition 2.1.4 System of eigenvectors and associated vectors of OVF A (γ) is called complete with power n if any set of n vectors f 0 , f 1 , . . . , f n−1 can be represented as a limit with respect to the norm of the linear combination of the elements of the system N   (k,v) a (k) , v = 0, 1, . . . , n − 1, (2.1.28) f v,N = p,N ϕ p k=1

p

2.1 Method of Operator Pencils

21

where the coefficients do not depend on v, ϕ(k,v) p

  p d v  (k) t (k) t γk t (k) ϕ p + ϕ p−1 + · · · + ϕ0 , = v e dt t=0 1! p!

and γk are eigenvalues of OVF A (γ). For n = 1, the definition coincides with a standard definition of the completeness of eigenvectors and associated vectors. If the multiplicity of all eigenvectors is equal to 1 we have N  v (k) a (k) f v,N = N γk ϕ0 . k=1

We will consider OVFs A (γ) which have eigenvalues with finite algebraic multiplicity. Let us study the spectrum of pencil L (γ). It is more convenient for us to consider the regularized pencil   −1/2 −1/2 = γ 4 K˜ + γ 2 I − (ε1 + ε2 ) K˜ L˜ (γ) := A1 L (γ) A1   , + (ε1 − ε2 ) γ S˜ + ε1 ε2 K˜ − A˜ 2

(2.1.29)

−1/2 −1/2 −1/2 −1/2 −1/2 −1/2 where K˜ = A1 K A1 , S˜ = A1 S A1 , A˜ 2 = A1 A2 A1 . It is easy to see that σ (L) = σ L˜ , and the following relations hold for eigenvectors and associated vectors   −1/2 (2.1.30) ϕ j (L) = A1 ϕ j L˜ .

˜ and A˜ 2 keep all properties of operators K , S, and A2 given in Operators K˜ , S, Lemmas 2.1.1–2.1.3 with the estimates −

1 1 ˜ I ≤ S˜ ≤ I, ε−2 max I ≤ A2 ≤ I. 2 2

(2.1.31)

Properties of the spectrum of pencil L (γ) are summarized in the following theorems. Theorem 2.1.3 σ (L) ⊂ l = {γ : |Re γ| < l} , i.e., for a certain l > 0, the spectrum of pencil L (γ) lies in the strip l . √ Proof In order to prove this theorem, assume that l > ε1 + ε2 and consider the OVF −1  L˜ (γ) = γ 2 K˜ + I + γ −1 T (γ) , F (γ) := γ 2 − (ε1 + ε2 )

(2.1.32)

22

2 Shielded Waveguide

in the domain D0 = {γ : |γ| > l}, where    −1  × (ε1 + ε2 ) I + (ε1 − ε2 ) γ S˜ + ε1 ε2 K˜ − A˜ 2 T (γ) = γ γ 2 − (ε1 + ε2 ) is a holomorphic and OVF in the domain D0 : T (γ) ≤ T0 for γ ∈ D0 .  bounded  ˜ One can see that σ L ∩ D0 = σ (F) ∩ D0 . If |Reγ| > l, there exists a bounded operator −1  R (γ) = γ 2 K˜ + I and the estimates [26] hold for its norm  −2  γ 

|γ|2 1 = R (γ) ≤  =    I mγ −2  2 |γ | |γ | 2

     γ   γ  + |γ  | |γ  |

  |γ| 1 1+ ≤ 2 l

        when γ   > γ   and R (γ) = 1 for γ   ≤ γ   where γ = γ  + iγ  . Choosing the value l > T0 we obtain   ! ! −1 !γ T (γ) R (γ)! ≤ 1 1 + |γ| T0 < 1, |γ| 2 l hence there exists a bounded operator −1  , F −1 (γ) = R (γ) I + γ −1 T (γ) R (γ) what yields the existence of bounded operators L˜ −1 (γ) and L −1 (γ) for γ ∈ l = {γ : |Re γ| < l} outside the strip l . The theorem is proved.  Corollary 2.1.4 Resolvent set of pencil L (γ) is not empty C\l ⊂ ρ (L) . Theorem 2.1.5 The spectrum of pencil L (γ) is symmetric with respect to the real and imaginary axes: σ (L) = σ (L) = −σ (L) . If γ0 is an eigenvalue of pencil L (γ) corresponding to the eigenvector f 1 = (, )T then −γ0 , γ¯ 0 and −γ¯ 0 are also eigenvalues of L (γ) corresponding to the eigenvectors     ¯  ¯ T and f 4 = −, ¯  ¯ T with the same multiplicity. f 2 = (−, )T , f 3 = , Proof The first assertion of Theorem 2.1.5 follows from (2.1.24)–(2.1.25). Proof of the second assertion is simple verification of variational relation (2.1.14). Note that associated vectors at γ¯ 0 can be obtained by complex conjugation of the associated  vectors corresponding to γ0 .

2.1 Method of Operator Pencils

23

Theorem 2.1.6 Let δ = (ε2 − ε1 ) /2, "

# 1/2 1/2  2  2 − |δ| + |δ| δ + 4ε1 δ + 4ε2 ≤ |γ| ≤ . I0 = γ : Im γ = 0, 2 2 In domain C\I0 the spectrum of pencil σ (L) consists of isolated set of eigenvalues √ with finite algebraic multiplicity.   γ j = ± εi (i = 1, 2) are the degeneration points of pencil L (γ) : dim ker L γ j = ∞. Let γ = γ  + iγ  , γ  = 0. Then  Im

1 γ 

  ε1 ε 2 ε1 ε2 γ A1 − 2δS − = A1 + A2 ≥ I, A2 γ |γ|2

hence [26] the operator L 0 (γ) := γ 2 A1 − 2γδS − ε1 ε2 A2 has a bounded inverse, L (γ) is a Fredholm pencil, and ind L (γ) = 0. Introduce operators A1 : H → H where A1 is determined by the form a1

 ( f, g) :=

  ξ (ε∇ f 1 ∇ g¯1 + ∇ f 2 ∇ g¯2 ) d x = A1 f, g , ∀g ∈ H,

(2.1.33)



and

−1/2 −1/2 A˜ 1 = A1 A1 A1 .

For these operators, the following estimates hold − εmax I ≤ A1 ≤ εmax I, −I ≤ A˜ 1 ≤ I.

(2.1.34)

Denote p = ((ε2 + ε1 ) /2)1/2 .   For real γ ∈ / I0 the estimate γ 2 − p 2  > |δ| (1 + |γ|) holds; consequently, the   −1/2 −1/2 = γ 2 − p 2 I − 2γδ S˜ + δ A˜ 1 and operator operators L˜ 0 (γ) := A1 L 0 (γ) A1 L 0 (γ) have bounded inverse. L (γ) is a Fredholm operator with index zero. Here, we used estimates (2.1.31) and (2.1.34). The second statement of the theorem follows from variational relation (2.1.14) for γ = γ j with the functions such that   ¯0 , ,  ∈ C0∞ 

 d x = 0, 

¯ 0 ⊂ 1 and  ¯ 0 ⊂ 2 . for 

24

2 Shielded Waveguide

From the physical point of view, it is interesting to consider real and pure imaginary points of spectrum σ (L), which correspond to the propagating and decreasing waves. However, there may be “complex” waves [12, 49] for γ0 ∈ σ (L), γ0 · γ0 = 0 (γ0 = γ0 + iγ0 ). Note that in the general case, strip l cannot be replaced by the set 0 = {γ : (Re γ) · (Im γ) = 0} in Theorem 2.1.3. From Theorem 2.1.5, it follows that “complex” waves arise in “fours”. If a waveguide has homogeneous filling (ε1 = ε2 ) then there are no “complex” waves. The spectrum existence does not follow from Theorem 2.1.6 (with the exception √ for γ j = ± εi ). Proof of the existence of a countable set of eigenvalues of L (γ) with the accumulation point an infinity will be given below. Note that the equivalence of the reduction of the boundary eigenvalue problem of √ normal waves to an eigenvalue problem for the pencil at the points γ j = ± εi is not valid (see [39]). Hence, eigenvectors of pencil L (γ) corresponding to eigenvalues γ j should be excluded. Using the methods of potential theory [5], we can prove that L (γ) is a Fredholm pencil for other real points γ. Emphasize here that there are no other degeneration points and finite accumulation points. Let us prove the existence of the discrete spectrum of pencil L (γ). Prove first the following statement.   Lemma 2.1.4 If the vector-function ϕ (γ) = F −1 (γ) γ −1 f 0 + f 1 , f 0 , f 1 ∈ H , is holomorphic for |γ| ≥ R with a certain R > 0 then the vector-function is uniformly positive (with respect to the norm) on this domain. Proof Let be the angle   $    3π  π    γ  . We also have R (γ) = 1,     under the condition γ   ≥ γ  . Thus, R (γ) ≤ 1 + ctgθ, γ ∈ / .

(2.1.35)

2.1 Method of Operator Pencils

25

Let |γ| > R1 > T0 (1 + ctgθ) and γ ∈ / . The inequalities  −1 ! −1 ! 1 + ctgθ ! F (γ)! ≤ R (γ) 1 − T0 R (γ) ≤ R1 1 − RT01 (1 + ctgθ)

(2.1.36)



follow from (2.1.35). Moreover, if vector-function ϕ (γ) := γϕ (γ) = F −1 (γ) ( f 0 + γ f 1 ), f 0 , f 1 ∈ H , is holomorphic for |γ| = r > R1 then (according to Lemma 1.3 in [33]) 

ln  ϕ (γ) ≤ c1 ln r + c2

c3 r 2

n(t, K˜ ) dt, t

0

    where n t, K˜ is the number of s-values of operator K˜ on interval t −1 , ∞ .         Since λn (K ) = O n −1 , we have [26] λn K˜ = O n −1 , n → ∞, and n t, K˜ = O (t), t → ∞. The following inequality holds ! ! ! ! ln !ϕ (γ)! ≤ c4 |γ|2 .

(2.1.37)

Choose θ < π8 , R1 > T0 (1 + ctgθ). Estimates (2.1.36) and (2.1.37) allow us to apply Phragmen–Lindeloeff principle [50]. This principle establishes that the boundedness of functions ϕ (γ) on the sides of angle |γ| > R1 and holomorphic property for ϕ (γ) yield the boundedness of ϕ (γ) for all |γ| > R1 (including the points inside angle ) and the following inequality holds  ϕ (γ) ≤ max

  (1 + cotθ)  f 0  R1−1 +  f 1  1−

T0 R1

(1 + cotθ)

, max ϕ (γ) . |γ|=R1

(2.1.38)

Thus vector-function ϕ (γ) is uniformly bounded with respect to the norm in domain |γ| > R.  Theorem 2.1.7 In domain C\I0 the spectrum of pencil L (γ) forms a countable set of isolated eigenvalues of finite algebraic multiplicity with the accumulation point at infinity. Proof It is sufficient to prove that spectrum of L (γ) (or F (γ)) is not empty in domain |γ| > R for any R (see Theorem 2.1.6). Assume that this assumption is not valid. Then the vector-function ϕ (γ) = F −1 (γ) √ f is holomorphic in domain |γ| > R for any f ∈ H . We can assume that R > ε1 + ε2 and OVF F (γ) has a bounded inverse on circumference |γ| = R. From estimate (2.1.38), it follows that

26

2 Shielded Waveguide

ϕ (γ) ≤ c  f  , |γ| ≥ R, ∀ f ∈ H, and

! −1 ! ! F (γ)! ≤ c, |γ| ≥ R.

Let us integrate the equality  −1 −1 1  γ 2 K˜ + I T (γ) F −1 (γ) f, f − F −1 (γ) f = γ 2 K˜ + I γ on contour k , which contains only one eigenvalue γk of OVF γ 2 K˜ + I . In this case, the integral of F −1 (γ) f is equal to zero and integrals of other terms are equal to the residuals at point γk . For resolvent R (γ), it is known [47] the expansion in the vicinity of γk  −1 1 1 γ 2 K˜ + I = Pk + Sk (γ) 2γk γ − γk (Sk (γ) is holomorphic in the vicinity of γk and Pk is an “eigenprojector” corresponding to eigenvalue γk ). The expansion yields   Pk I − γk−1 T (γk ) F −1 (γk ) f = 0. ! ! Note that we have the estimates !T (γ) F −1 (γ)! ≤ T0 c, |γ| ≥ R. Hence, for sufficiently large γk (such γk can be chosen since eigenvalues of compact operator K˜ > 0 have the accumulation point at infinity) operator I − γk−1 T (γk ) F −1 (γk ) has a bounded inverse. Since f is arbitrary we obtain Pk f˜ = 0 for all f˜ ∈ H , which  is not possible because Pk = 0. This contradiction proves the theorem. Figure 2.2 shows the distribution of the spectrum of pencil L (γ) on the complex plane.

2.1.4 Theorems of Completeness We propose two approaches for the investigation of completeness of the system of eigenvectors and associated vectors of pencil L (γ). In the first approach, we consider L (γ) as a perturbation of a certain simple pencil. Two cases will be addressed. In the first case, the initial pencil is represented as a perturbation of a Keldysh pencil by a holomorphic OVF; we will not set additional conditions and limit ourselves to the proof of double completeness with finite defect. In the second case, we prove double

2.1 Method of Operator Pencils

27

Fig. 2.2 ◦ – the degeneration points L (γ); × – eigenvalues of pencil L (γ) which are not equal to √ ± εi

completeness of the system of eigenvectors and associated vectors of pencil L (γ) assuming additionally that the parameter δ = (ε2 − ε1 ) /2 is sufficiently small. The second approach is based on factorization of pencil L (γ) with respect to the special contour on the complex plane. We prove double completeness of the system of eigenvectors and associated vectors of L (γ) corresponding to eigenvalues located outside a certain contour. In the proof, an essential additional condition imposed on the pencil parameters will be applied. Note that pencil L (γ) does not belong to any well-known family of pencils (Keldysh pencils, hyperbolic pencils, and so on). However, spectral properties of the pencil can be studied using the approach developed in this study. Consider pencil L (γ) in√domain Dη = {γ : |γ| > η} where η is an arbitrary positive number such that η > ε1 + ε2 . Completeness of the system of eigenvectors and associated vectors of L (γ) corresponding to eigenvalues located in Dη is equivalent to the completeness of the system of eigenvectors and associated vectors of pencil F (γ) corresponding to eigenvalues located in Dη . Indeed, the spectrum of pencils in Dη are the same one, and eigenvectors and associated vectors satisfy the relation

28

2 Shielded Waveguide −1/2

ϕ(k) j (L) = A1

ϕ(k) j (F) ,

  which directly leads to the equivalence of the completeness of systems ϕ(k) j (L)   and ϕ(k) . (F) j Theorem 2.1.8 System of eigenvectors and associated vectors of pencil L (γ) corresponding to eigenvalues located in domain |γ| ≥ η is double complete with finite defect in H × H :     (k,1) < ∞, dim coker < ∞; dim coker L ϕ(k,0) L ϕ p p   where η ≥ 0 is an arbitrary nonnegative number and L ϕ(k,v) denotes the closure p

(k,v)  of linear combinations of vectors ϕ p . Proof It is sufficient to prove the statement of the theorem for pencil F (γ) under the √ condition η > ε1 + ε2 . F (γ) is considered as a perturbation of pencil γ 2 K˜ + I by the OVF T1 (γ) = γ −1 T (γ), T1 (∞) = 0 which is holomorphic in Dη . In this case, K˜ > 0 is a Hilbert–Schmidt operator; consequently, all conditions of Theorem 1 in [51] are fulfilled. This theorem implies double completeness of the system of eigenvectors and associated vectors of pencil F (γ) (and L (γ)) with finite defect in H × H , the closure of linear combinations of the vectors  (k,0) (k,1) T ϕp , ϕp ∈ H × H, has finite defect in H × H , γk ∈ Dη , where vectors ϕ(k,v) are determined by (2.1.28). p  If we increase the value of η then the dimension of the defect subspace increases as well. From the other side, it is necessary to exclude eigenvectors and associated √ vectors corresponding to eigenvalues ± εi . However, the dimension of the defect subspace is not known. In various applications, the statements are important that provide completeness of the system of eigenvectors and associated vectors of the pencil without defect. Below we formulate the corresponding theorem which will be proved under the condition that the parameter δ = (ε2 − ε1 ) /2 is sufficiently small. Write L (γ) in the form  L (γ) = γ 2 −

   2  ε1 +ε2 2 2 2 K − ε1 −ε K + γ 2 − ε1 +ε A1 2 2  2  2 . +γ (ε1 − ε2 ) S + ε1 +ε A − ε ε A 1 1 2 2 2

It is easy to verify that ε1 + ε 2 ε 1 − ε2  A 1 − ε1 ε 2 A 2 = A1 , 2 2

(2.1.39)

2.1 Method of Operator Pencils

29

where bounded self-adjoint operator A1 is determined by formula (2.1.33). The expression for pencil L˜ (γ) has the form    2   p − γ 2 K˜ − I + δ B (γ) , L˜ (γ) = p 2 − γ 2 %

where B (γ) := −2γ S˜ − δ K˜ − A˜ 1 ; p =

(2.1.40)

ε 1 + ε2 . 2

The spectrum of L (γ) coincides with the spectrum of pencil L˜ (γ). Eigenvectors and associated vectors of pencils satisfy formula (2.1.30). Hence, the completeness of the system of eigenvectors and associated vectors of L (γ) is equivalent to that of pencil L˜ (γ). Theorem 2.1.9 Let M > 1 be an arbitrary number. Then there exists δ∗ = δ∗ (M; ) such that for any ε j , 1 ≤ ε j ≤ M, the system of eigenvectors and asso√ ciated vectors of pencil L(γ) corresponding to eigenvalues γn = ± εi , i = 1, 2 is double complete in H × H under the condition |δ| < δ∗ . Proof It is sufficient to prove the assertion of the theorem for L˜ (γ) considered as a perturbation of a simple pencil     2    p − γ 2 K˜ − I = p 2 − γ 2 F˜ (γ) , F0 (γ) = p 2 − γ 2 where

  F˜ (γ) = p 2 − γ 2 K˜ − I,

(2.1.41)

(2.1.42)

by OVF δ B (γ).  Spectrum σ F˜ is located on the real and imaginary axes. The spectrum consists of eigenvalues of finite algebraic multiplicity with the accumulation point at infinity. Eigenvectors of pencil F˜ (γ) form an orthonormal basis in H (by the Hilbert–Schmidt theorem [52]). Since K˜ > 0 the eigenvalues γ˜ n of pencil F˜ (γ) satisfy the estimate 1 γ˜ n2 ≤ p 2 − ! ! , n = ±1, ±2, . . . , ! ˜! !K ! where γ˜ −n = −γ˜ n and the numeration of eigenvalues decreases with respect to the values of γ˜ n2 . Under the conditions imposed on coefficients ε j , there exists an M0 < 1 (which does not depend on ε j ) such that p − γ˜ 1 ≥ 3M0 . Let |δ| ≤ M0 , introduce the circles 

√ ± = γ : |γ ∓ p| = r ; r = p + M0 − εmin and consider pencil L˜ (γ) in the domain (Fig. 2.3)

30

2 Shielded Waveguide

Fig. 2.3 x—eigenvalues of OVF F˜ (γ)

D = {γ : |γ − p| > r, |γ + p| > r } . The domain contains all eigenvalues of pencil F˜ (γ). L˜ (γ) is a Frehholm pencil in  √ D by Theorem 2.1.6. If γ0 ∈ 0 , 0 = + ∪ − then |γ0 !− γ˜ n | ≥ !M0 , γn ± εi  ≥ M0 . Hence F0 (γ) has a bounded inverse one on 0 and ! F0−1 (γ)! ≤ C0 , where C0 −1 −1 ˜ does not depend on ε j , γ. Moreover, B (γ) ! ≤ B0 on ! 0 . If |δ| < B0 C0 then L (γ) ! ˜ −1 ! also has bounded inverse one on 0 and ! L (γ)! ≤ C1 uniformly with respect to γ ∈ 0 , 1 ≤ ε j ≤ M. According to [51] in order to prove the double completeness of the system of eigenvectors and associated vectors of pencil L˜ (γ) corresponding to eigenvalues located in D, we will establish that the holomorphic property of the vector-function f (γ) = L˜ −1 (γ) ( f 0 + γ f 1 ) in D implies f 0 = f 1 = 0 for any f 0 , f 1 ∈ H . Let f (γ) be holomorphic in D. It follows from Lemma 2.1.1 that the operatorvalued function  γ 2 − ε1 − ε2 −1  F −1 (γ) γ −1 f 0 + f 1 = L˜ (γ) ( f 0 + γ f 1 ) γ is bounded at infinity and the following expansion holds L˜ −1 (γ) ( f 0 + γ f 1 ) =

∞  k=1

gk γ −k , |γ| > R.

2.1 Method of Operator Pencils

31

From the equality ∞   f 0 + γ f 1 = L˜ (γ) L˜ −1 (γ) ( f 0 + γ f 1 ) = γ 4 K˜ + · · · gk γ −k k=1

and the property K˜ > 0 we have g1 = g2 = 0; consequently f (γ) has zero at infinity of order no less than 3. Integrate the equalities γ F0−1 (γ) ( f 0 + γ f 1 ) − γ f (γ) = γ F0−1 (γ) δ B (γ) f (γ) , F0−1 (γ) ( f 0 + γ f 1 ) − f (γ) = F0−1 (γ) δ B (γ) f (γ) with respect to 0 . Taking into account the properties of f (γ) at infinity and that L˜ (γ) and F0 (γ) are continuously invertible on 0 we obtain δ 2πi

f0 = f1 =

δ 2πi

 0

γ F0−1 (γ) B (γ) f (γ) dγ,

 0

F0−1 (γ) B (γ) f (γ) dγ.

Hence, there exists c > 0 (which does not depend on ε j , γ) such that  f 0  ≤ |δ| C ( f 0  +  f 1 ) ,  f 1  ≤ |δ| C ( f 0  +  f 1 ) . If |δ| < (2C)−1 then  f 0  =  f 1  = 0 and f 0 = f 1 = 0. In order to complete the proof of the theorem, we can choose   δ∗ < min (2C)−1 , M0 , B0−1 C0−1 . 

The theorem is proved.

Let us prove the completeness of the system of eigenvectors and associated vectors of pencil L (γ) using the factorization method. Theorem 2.1.10 System of eigenvectors and associated vectors of pencil L (γ) cor√ responding to eigenvalues γn = ± εi , i = 1, 2 is double complete in H × H under the conditions εmax < 9εmin and

(2.1.43)

32

2 Shielded Waveguide

 

  1 ε ||2 + ||2 d x ≤ 2

   1 |∇|2 + |∇|2 d x, ε



1

∀ ∈ H01 (),  ∈ H ().

(2.1.44)

Proof For the factorization of pencil L (γ) let us determine domains on the plane of variable γ where the equation (L (γ) f, f ) = 0,

(2.1.45)

has a definite number of roots with respect to γ for any f = 0. First let us prove two estimates. Using Green’s formula and Schwartz inequality, we obtain:              ¯ ¯ ¯ ¯   ∂ ∂ ∂ ∂  ∂ ∂  ∂ ∂  ∂ ∂    ¯ − ¯ dτ  =    dx − + −     ∂τ ∂τ ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x 2 1 1 2 1 2 2 1     

j

              ¯ ¯ ¯ ¯ ∂ ∂  ∂ ∂  ∂ ∂  ∂ ∂      = 2  Re dx ≤ 2  dx − −     ∂x2 ∂x1 ∂x1 ∂x2 ∂x2 ∂x1 ∂x1 ∂x2  j  j   ⎛ ⎜ ≤ 2⎝



⎞1/2 ⎛ ⎟ |∇|2 d x ⎠

j

⎜ ⎝

⎞1/2



⎟ |∇|2 d x ⎠

; j = 1, 2.

j

Thus      2     ∂ ∂  ¯ − ¯ dτ  ≤ 4 |∇|2 d x |∇|2 d x.     ∂τ ∂τ j   

(2.1.46)

j

Consider the equation with respect to

√

εj P + Q ±

εj:

√

ε j R = 0,

where      ∂ ∂ 2 2 ¯ ¯ |∇| |∇| d x (≥ 0) , Q = d x (≥ 0) , R = P= −  dτ . ∂τ ∂τ j

j

Using estimate (2.1.46) we have



2.1 Method of Operator Pencils

33

R 2 − 4P Q ≤ 0 and for any

√ εj

εj P + Q ±

√ εj R ≥ 0

which is equivalent to the inequality 

  √ ε j |∇|2 + |∇|2 d x ± ε j

j

  

 ∂ ∂ ¯ ¯ −  dτ ≥ 0, j = 1, 2. ∂τ ∂τ (2.1.47)

We will assume that ε2 ≥ ε1 . Denote       ∂ ∂ ¯ − ¯ dτ , K := S :=   ε ||2 + ||2 d x, ∂τ ∂τ 

a ( j) :=





  ε j |∇|2 + |∇|2 d x, a2 :=

j

   1 |∇|2 + |∇|2 d x. ε



Estimates (2.1.47) have the form a (1) ± a (2) ±

√

ε1 s ≥ 0,

√

(2.1.48)

ε2 s ≥ 0.

According to (2.1.12), (2.1.45) can be represented in the form f (γ) :=

a (1) + γs a (2) − γs + = k, γ 2 = ε j , ε1 − γ 2 ε2 − γ 2

where a (1) ≥ 0, a (2) ≥0, k > 0. √ √ If γ ∈ − ε1 , ε1 then √ √ a (1) + ε1 s a (1) − ε1 s √ √ √ √ + 2 ε1 ( ε1 −γ ) 2 ε1 ( ε1 +γ )   (1) (1) (1) min √ε a√ε −γ , √ε a√ε +γ > a2ε1 , a (1) ) ) 1( 1 1( 1 a (1) +γs ε1 −γ 2

≥

=

 = 0 .

 √ √  If γ ∈ − ε2 , ε2 then √ √ a (2) − ε2 s a (2) + ε2 s √ √ + 2√ε √ε +γ 2 ε ε −γ ( ) ( ) 2 2 2 2    (2) a (2) a (2) a (2) √ √ √ √ min ε ε +γ , ε ε −γ > 2ε2 , a ) ) 2( 2 2( 2 a (2) −γs ε2 −γ 2

≥

=

 = 0 .

(2.1.49)

34

2 Shielded Waveguide

 √ √  Thus for γ ∈ − ε1 , ε1 we have the estimate f (γ) >

1 2



a (1) a (2) + ε1 ε2

 =

a2 . 2

From the estimate and (2.1.43) (2.1.44), it follows that (2.1.48) (and (2.1.45))  √ and √  has no real roots for γ ∈ − ε1 , ε1 . From the representation √ √ a (1) + ε1 s a (1) − ε1 s √ √ + 2√ε √ε +γ 2 ε1 ( ε1 −γ ) ) 1( 1 √ √ a (2) − ε2 s a (2) + ε2 s + 2√ε √ε +γ , γ 2 = ε j , + 2√ε √ε −γ ) ) 2( 2 2( 2

f (γ) =

we obtain the following property. If the signin inequalities (2.1.48) “>”, then √ is √ √ √ and ε , − ε ε , ε2 , and (2.1.49) has at least one root on each interval − 2 1  √  1√ there are no roots of the equation for γ ∈ −∞, − ε2 ∪ ε2 , +∞ . Here, we took into account the properties lim √ γ→ ε j ∓0

f (γ) =

lim √ γ→− ε j ±0

f (γ) = ±∞, ( j = 1, 2)

and f (γ) < 0, |γ| >

√

ε2 .

Let γ1 , γ2 be the roots of equation (2.1.45). We can calculate other two roots of equation (2.1.45) by Vieta formula: γ 4 K + γ 2 (a1 − (ε1 + ε2 ) K ) + γ√(ε1 − ε2 ) S + ε1 ε2 (K − a2 ) = 0, 2) ±i γ3,4 = − (γ1 +γ 2

where θ :=

4ε1 ε2 θ/|γ1 γ2 |−(γ1 +γ2 )2 , 2

(2.1.50)

a2 − 1(1 ≤ θ < +∞). k

√ √ Using the inequalities |γ11γ2 | ≥ ε12 and |γ1 + γ2 | ≤ ε2 − ε1 , it is easy to verify that there exists δ˜ > 0 (which depends on ε j ) such that for ε2 < 9ε1 we have   γ3,4 − p  > r˜ ; r˜ =

√ √ √ √ ε2 − ε1 ˜ p = ε1 + ε2 . + δ, 2 2

(2.1.51)

Thus in the domains {γ : | p − γ| < r˜ } , {γ : | p + γ| < r˜ } ,  √ √  (2.1.44) has only one real root on each interval − ε2 , − ε1 and Equation √ √  ε1 , ε2 , and the equation has two roots in the domain

2.1 Method of Operator Pencils

35

{γ : | p ± γ| > r˜ } . Taking into account the continuity of (roots of equation (2.1.50) with) respect (√ √ √ √ ) to coefficients, we establish that γ1 ∈ − ε2 , − ε1 , γ2 ∈ ε1 , ε2 , γ3,4 ∈ {γ : | p ± γ| ≥ r˜ } under the conditions (2.1.48). √ √ ε2 − ε1 δ˜ Choose δ0 = 2 and denote r = + δ0 , 2 σ+ = {γ : | p − γ| < r } , σ− = {γ : | p + γ| < r } , σ1 = C\ (σ+ ∪ σ− ) , σ2 = σ+ ∪ σ− .

We proved that domains σ+ , σ− contain only one root of equation (2.1.44) while domain σ1 contains two roots of equation (2.1.45). The spectrum of pencil L (γ) is divided into three domains σ+ , σ− and σ1 . We have, in addition to this, ( √ (√ √ ) √ ) ε1 , ε2 . σ (L) ∩ σ− ⊂ − ε2 , − ε1 , σ (L) ∩ σ+ ⊂ Let ± = {γ : |γ ∓ p| = r } , 1 = + ∪ − . Equation (2.1.45) has no roots on 1 and inf

 f =1, γ∈1

|(L (γ) f, f )| > 0.

(2.1.52)

Estimate (2.1.52) and the properties of the spectrum of pencil L (γ) are valid for pencil L˜ (γ). In this case, L˜ (γ) admits the factorization with respect to contour 1 : L˜ (γ) = L 1 (γ) L 2 (γ) , where

(2.1.53)

  L 1 (γ) = γ 2 K˜ + γ K˜ B1 + I + K˜ (ε1 + ε2 ) I + B2 − B12 , L 2 (γ) = γ 2 I + γ B1 + B2 ,

B1 , B2 are bounded operators, and σ (L 1 ) ⊂ σ1 , σ (L 2 ) ⊂ σ2 . This assertion follows from [34] (in order to use Theorems 1 and 2 in [34] it is necessary to introduce a new variable t = ( p − γ)−1 ). Pencil L 2 (γ) has a bounded inverse on σ1 . The system of eigenvectors and associated vectors of pencil L 1 (γ) is double complete in H × H by the Keldysh theorem [53] and, consequently, the system of eigenvectors and associated vectors of pencil L˜ (γ) corresponding to the eigenvalues γn ∈ σ1 is double complete in H × H too.

36

2 Shielded Waveguide

The completeness of system of eigenvectors and associated vectors of L (γ) is equivalent to the completeness of the system of eigenvectors and associated vectors of pencil L˜ (γ).  Note that the equivalence of the reduction of the boundary eigenvalue problem √ of normal waves to the eigenvalue problem of the pencil at the points γ j = ± εi is not valid. In Theorems 2.1.9 and 2.1.10, we formulate the conditions which separate √ degeneration points ± εi of pencil L (γ). We will not consider the system of eigen√ vectors and associated vectors of pencil L (γ) corresponding to eigenvalues ± εi . Theorem 2.1.8 shows that this system corresponding to the eigenvalues satisfying the condition |γ| > η for arbitrary η > 0, are sufficiently broad: in fact, we can add only a finite set of elements in order to obtain a double complete system in H × H . Below we will show the importance of the double completeness of eigenvectors and associated vectors of pencil L (γ) for the solution to the boundary eigenvalue problem of normal waves. More statements concerning the completeness of the system of eigenvectors and associated vectors of L (γ) can be found in [54, 55].

2.2 Properties of the System of Eigenwaves and Associated Waves of a Waveguide This section is devoted to the analysis of the system of eigenwaves and associated waves of a waveguide described in Sect. 2.1. The addressed properties of the system are: completeness, basis property, and biorthogonality. These properties are important for solving the problems of the waveguide excitation [14]. Without the knowledge about the completeness and basis property of the system of eigenwaves and associated waves, the expansions of solutions [41] are not correct. We will focus on the basic case ε1 = ε2 , which leads to a vector problem. For ε1 = ε2 , the problem of normal waves is reduced to two well-known scalar problems [8–10]. In Sect. 2.2.1, we formulate the definition of eigenwaves and associated waves of waveguide via eigenvectors and associated vectors of pencil L (γ). We show that this definition is equivalent to usual definition based on solutions of Maxwell equations. The aim of new definition is that associated waves are defined only by longitudinal components  p ,  p . This allows us to study only pencil L (γ). Section 2.2.2 deals with the proofs of the completeness in L 42 () of the system involving transversal components of eigenwaves and associated waves. It is important that the double completeness (according to Keldysh) of the system of eigenvectors and associated vectors of pencil L (γ) in H × H yields the completeness (in a usual sense) of the system of transversal components of eigenwaves and associated waves in L 42 (). This result allows one to apply the theorems of Sect. 2.1 for the analysis of completeness of the transversal components of eigenwaves and associated waves.

2.2 Properties of the System of Eigenwaves and Associated Waves of a Waveguide

37

In Sect. 2.2.3, we establish the biorthogonal relations for transversal components of eigenwaves and associated waves. For homogeneous waveguides (ε1 = ε2 ) with a piecewise smooth boundary of domain  and when associated waves are absent, similar relations are well known [16]. The obtained biorthogonal relations allow one to construct a system biorthogonal with respect to the system of transversal components of eigenwaves and associated waves in L 42 (). As a result, we establish “minimality” of the system of transversal components of eigenwaves and associated waves. Section 2.2.4 is devoted to the proof of an important fact that, in the general case, this system is not a Schauder basis in L 42 (). The results of the section were mainly published in [55–57].

2.2.1 Eigenwaves and Associated Waves Below we will use the notations of Sect. 2.1. Let f 0 , f 1 , . . . , f m ∈ H be the chain of eigenvectors and associated vectors of pencil L (γ) corresponding to the eigenvalues γ (γ 2 = εi , i = 1, 2). Using vectors  T f p =  p ,  p , we define a system of functions on : ( p)

E1

( p)

E2

( p)

H2 ( p)

E3

   ∂ p i ∂ p ( p−1) ( p−1) − , − i E1 − i H2 ∂x1 k˜ 2 ∂x2     i ∂ p iγ ∂ p ( p−1) ( p−1) + , = − i E2 − i H1 k˜ 2 ∂x2 k˜ 2 ∂x1 =

iγ k˜ 2

( p) H1

iε = k˜ 2

=−

iε k˜ 2

( p)

=  p , H3







   ∂ p iγ ∂ p ( p−1) ( p−1) + , − i E2 − i H1 ∂x2 k˜ 2 ∂x1

   ∂ p iγ ∂ p ( p−1) ( p−1) + , − i E1 − i H2 ∂x1 k˜ 2 ∂x2

(2.2.54)

=  p ; E ( p) ≡ H ( p) ≡ 0 for p < 0; ε = ε j in  j ; j = 1, 2.

Definition 2.2.1 T  ( p) ( p) ( p) ( p) ( p) ( p) W˜ ( p) = V˜ ( p) exp (iγx3 ) , V˜ ( p) = E 1 , E 2 , E 3 , H1 , H2 , H3 is called the eigenwave for p = 0 or the associated wave for p ≥ 1 corresponding to eigenvalue γ. Vector V˜ ( p) will be considered as an element of the space

38

2 Shielded Waveguide 1

H˜ = L 2 () × L 2 () × H01 () × L 2 () × L 2 () × H () equipped with the standard inner product and norm of the product of spaces. From the results of Sect. 2.1, it follows that x3 -components of vector V˜ (0) are eigenfunctions of the problem of normal waves of a waveguide. Consequently, the above definition of eigenwaves coincides with the standard one. However, the definition of associated wave ( p ≥ 1) is not standard. In fact, we define associated wave in terms of associated vectors of pencil L (γ) and the definition is not generally connected directly with setting (2.1.1)–(2.1.4). The standard definition may be accomplished as follows. System (2.1.1) of the Maxwell equations may be considered as an eigenvalue problem for the linear pencil M (γ) = M1 + γ M2 , and eigenwaves and associated waves can be introduced as solutions of the boundary value problems (M1 + γ M2 ) V˙ (0) = 0, (M1 + γ M2 ) V˙ ( p) + M2 V˙ ( p−1) = 0, p ≥ 1, with the corresponding boundary and transmission conditions. These problems have the form ( p) ∂ H˙ 3 ( p) ( p) ( p−1) − iγ H˙ 2 − iε E˙ 1 = i H˙ 2 , ∂x2 ( p)

∂ H˙ 3 ( p) ( p) ( p−1) − iε E˙ 2 = −i H˙ 1 , iγ H˙ 1 − ∂x1 ( p) ( p) ∂ H˙ 2 ∂ H˙ 1 ( p) − − iε E˙ 3 = 0, ∂x1 ∂x2 ( p)

∂ E˙ 3 ( p) ( p) ( p−1) − iγ E˙ 2 + i H˙ 1 = i E˙ 2 , ∂x2

(2.2.55)

( p)

∂ E˙ 3 ( p) ( p) ( p−1) + i H˙ 2 = −i E˙ 1 , iγ E˙ 1 − ∂x1 ( p) ( p) ∂ E˙ 1 ∂ E˙ 2 ( p) − + i H˙ 3 = 0; ∂x1 ∂x2  E˙ τ( p) 0 = 0;

(

E˙ τ( p)

) 

) ( = H˙ τ( p)  = 0.

(2.2.56) (2.2.57)

2.2 Properties of the System of Eigenwaves and Associated Waves of a Waveguide

39

The point over functions shows that we use an alternative definition of eigenwaves and associated waves (not in the sense of Definition 2.2.1). Here, it is assumed that E˙ ( p) ≡ H˙ ( p) ≡ 0, p < 0.

(2.2.58)

Prove the equivalence of both definitions for sufficiently smooth functions  p ,  p ; i.e., let us prove the equalities ( p)

E1

( p) ( p) ( p) = E˙ 1 , . . . , H3 = H˙ 3 .

Assume that conditions (2.2.55)–(2.2.58) hold. Then from (2.2.55), we have ( p) E˙ 1

( p) H˙ 2

iγ = k˜ 2

iε =− k˜ 2

iγ ( p) E˙ 2 = k˜ 2 ( p) H˙ 1



iε = k˜ 2









( p)

∂ E˙ 3 i ( p−1) − i E˙ 1 − ˜k 2 ∂x1



( p)

∂ E˙ 3 iγ ( p−1) − i E˙ 1 + ∂x1 k˜ 2



( p)

∂ E˙ 3 i ( p−1) − i E˙ 2 + ∂x2 k˜ 2

( p)

∂ E˙ 3 iγ ( p−1) − i E˙ 2 + ∂x2 k˜ 2



( p)

∂ H˙ 3 ( p−1) − i H˙ 2 , ∂x2 ( p)

∂ H˙ 3 ( p−1) − i H˙ 2 , ∂x2 ( p)

∂ H˙ 3 ( p−1) − i H˙ 1 , ∂x1 ( p)

∂ H˙ 3 ( p−1) − i H˙ 1 . ∂x1

(2.2.59)

Substituting representation (2.2.59) into (2.2.55), we obtain ( p) ( p) ( p−1) ( p−2)  E˙ 3 + k˜ 2 E˙ 3 − 2γ E˙ 3 − E˙ 3 = 0, ( p)

 H˙ 3

( p)

( p−1)

+ k˜ 2 H˙ 3

( p−2)

− 2γ H˙ 3

− H˙ 3

= 0.

(2.2.60)

From (2.2.56), (2.2.57), it follows that 

( p)  E˙ 3 



Moreover, from the condition

we find using (2.2.59)

0

( p) E˙ 3

 ( p) ∂ H˙ 3  = 0,  ∂n   



( p) = H˙ 3

= 0; 0

 

 E˙ τ( p) 0 = 0,

(2.2.61)

= 0.

(2.2.62)

40

2 Shielded Waveguide



1 k˜ 2

−1 *

+

( p)

1 ∂ H˙ 3 k˜ 2 ∂n

   

( p) 



∂ E˙ −γ 3 ∂τ



 * * + + ( p−1) ( p−1)  ( p−2) 2γ ∂ H˙ 3 ∂ E˙ 3 1 ∂ H˙ 3  + − = 0.  + ∂n ∂τ  ∂n [ε] [ε] 



(2.2.63)



Similarly, from the condition (

H˙ τ( p)

) 

=0

we obtain 

1 ˜k 2

−1 *

( p) ε ∂ E˙ 3 k˜ 2 ∂n

+ 

 + * ( p) ( p−1) ∂ H˙ 3  2γ ∂ E˙ 3 +γ + ε  + ∂τ  ∂n [ε] 



 + * ( p−1)  ( p−2) ∂ H˙ 3 1 ∂ E˙ 3  + = 0. ε  + ∂τ  ∂n [ε] 

(2.2.64)



T  From the other side, the associated vectors f p =  p ,  p of pencil L (γ) satisfy the variational relation    ε 1 ∇ p ∇ u¯ + ∇ p ∇ v¯ − ε p u¯ −  p v¯ d x −k˜12 k˜22 k˜ 2 k˜ 2 

  − [ε] γ 

    ∂ p ∂ p−1 ∂ p ∂ p−1 v¯ − u¯ dτ − [ε] v¯ − u¯ dτ ∂τ ∂τ ∂τ ∂τ 

     ε∇ p−1 ∇ u¯ + ∇ p−1 ∇ v¯ − k˜12 + k˜22 ε p−1 u¯ +  p−1 v¯ d x +2γ 

+

     ε∇ p−2 ∇ u¯ + ∇ p−2 ∇ v¯ + 4γ 2 − k˜12 − k˜22 ε p−2 u¯ +  p−2 v¯ d x 

 +4γ 

  ε p−3 u¯ +  p−3 v¯ d x +



  ε p−4 u¯ +  p−4 v¯ d x = 0,



∀ (u, v)T ∈ H ; p = 0, . . . , m. Assuming that  p ≡  p ≡ 0 for p < 0 choose

(2.2.65)

2.2 Properties of the System of Eigenwaves and Associated Waves of a Waveguide

41

  u, v ∈ H01 j , ¯  j ⊂  j , and apply to (2.2.65) Green’s formula to obtain       k˜12 k˜22 k˜12  p +  p − 2γ  p−1 + k˜12 + k˜22  p−1     −  p−2 + k˜12 + k˜22 − 4γ 2  p−2 + 4γ p−3 +  p−4 = 0 in  j and the same equation for  p . From both equations (using induction with respect to p), we obtain  p + k˜ 2  p − 2γ p−1 −  p−2 = 0,  p + k˜ 2  p − 2γ p−1 −  p−2 = 0 in  j ;

(2.2.66)

p = 0, 1, . . . , m. Varying u and v on 0 and , we find  ∂ p  = 0, p = 0, 1, . . . , m; ∂n 0 and



+

1 k˜ 2

−1 

ε ∂ p k˜ 2 ∂n



+

(2.2.67)

 2γ ∂ p−1 ε ∂n [ε] 

   ∂ p−2 ∂ p  ∂ p−1  1 ε +γ + = 0; ∂n ∂τ  ∂τ  [ε]  

1 k˜ 2 

+

−1 

1 ∂ p k˜ 2 ∂n

1 ∂ p−2 ∂n [ε]





−γ



+

 2γ ∂ p−1 ∂n [ε] 

  ∂ p  ∂ p−1  − = 0; ∂τ  ∂τ 

(2.2.68)

p = 0, 1, . . . , m.  ( ) ( ) The boundary,  p 0 = 0, and transmission,  p  =  p  = 0 conditions together with the condition providing the finiteness of energy in  are satisfied 1 ( p) by the definition of spaces H01 () and H (). Note that for functions H˙ 3 , we  ( p)  1 have H˙ 3 d x = 0; hence condition d x = 0 (for functions  in H ()) does 

not restrict the space of solutions.



42

2 Shielded Waveguide

Taking into account (2.2.60)–(2.2.64) and (2.2.66)–(2.2.68), we get ( p) ( p) E˙ 3 =  p , H˙ 3 =  p , p = 0, 1, . . . , m.

Using (2.2.54) and (2.2.59), we have ( p) ( p) ( p) ( p) E˙ 1 = E 1 , . . . , H˙ 3 = H3 for all p ≥ 0.

R

(2.2.69)

Assume that the multiplicity of eigenvalue γ is greater than 1. In this case, we ( p) ( p) choose E˙ 3 , H˙ 3 in terms of functions  p ,  p . If this requirement is not fulfilled then (2.2.69) is not valid in the general case; however, the subspaces of eigenwaves and associated waves corresponding to eigenvalue γ will be the same.

Thus, we have established the equivalence of both definitions of associated waves for sufficiently smooth functions  p ,  p . Note that we choose (2.2.54) in accordance with (2.2.59). The equalities ( p) ( p) E˙ 3 =  p , H˙ 3 =  p

are not trivial. We can study smoothness of functions  p ,  p in domains  j by the methods of Sect. 2.1. Let us underline that associated waves (2.2.54) are defined solely by longitudinal components  p ,  p . This makes it possible to apply pencil L (γ) in our analysis, which is an objective of new Definition 2.2.1.

2.2.2 Completeness of the System of Transversal Components of Eigenwaves and Associated Waves T T   ( p) ( p) ( p) ( p) ( p) ( p) Consider the transversal components E t = E 1 , E 2 , Ht = H1 , H2 and denote the longitudinal components by  p ,  p ; they are related to an eigenwave and associated wave corresponding to eigenvalue γ. Introduce the differential operators ∂f ∂f ∂f ∂f e1 + e2 , ∇  f = e1 − e2 . ∇f = ∂x1 ∂x2 ∂x2 ∂x1 Our aim is to prove the following formulas: −i

  

( p) Et







−∇ g¯ +

( p) Ht

      ¯ ∇ f dx = ε p f¯ +  p g¯ d x; 

2.2 Properties of the System of Eigenwaves and Associated Waves of a Waveguide 1

p = 0, 1, . . . , m, ∀ f ∈ H01 () , g ∈ H () ;

−i

43

(2.2.70)

    ( p)  ( p) εE t ∇ f¯ + Ht (∇ g) ¯ dx 

 =γ 



 ε p f¯ +  p g¯ d x +



  ε p−1 f¯ +  p−1 g¯ d x;

 1

p = 0, 1, . . . , m, ∀ f ∈ H01 () , g ∈ H () . ( p)

(2.2.71)

( p)

Components E t , Ht are defined by (2.2.54). Using (2.2.66)–(2.2.68) and the ( p) ( p) method of induction with respect to p, it is easy to verify that E t and Ht satisfy (2.2.55). In particular,  ( p−1) ( p) ( p) ( p−1) ∂ E2 ∂ E1 γ ∂ E2 ∂ E1 ( p) − + i H3 = − ∂x1 ∂x2 ∂x1 ∂x2 k˜ 2  ( p−1) ( p−1)   1 ∂ H1 ∂ H2 i  i   p + k˜ 2  p +  p + k˜ 2  p + + = ∂x1 ∂x2 k˜ 2 k˜ 2 k˜ 2  ( p−1)  ( p−1) ( p−2) ( p−2) 2γ ∂ E 2 ∂ E1 ∂ E1 1 ∂ E2 + − − + ∂x1 ∂x2 ∂x1 ∂x2 k˜ 2 k˜ 2   i  p + k˜ 2  p − 2γ p−1 −  p−2 = 0; = k˜ 2 ( p) ( p)  ∂ H2 ∂ H1 iε  ( p)  p + k˜ 2  p − − iεE 3 = − ∂x ∂x k˜ 2  1 ( p−1) 2 ( p−1)  ( p−1) ( p−1) ∂ E2 ∂ H1 ε ∂ E1 γ ∂ H2 + − − + ∂x1 ∂x2 ∂x1 ∂x2 k˜ 2 k˜ 2  ( p−1) ( p−1)   2γ ∂ H2 ∂ H1 iε  p + k˜ 2  p + − =− ∂x1 ∂x2 k˜ 2 k˜ 2  ( p−2) ( p−2)  ∂ H1 1 ∂ H2 iε   p + k˜ 2  p − 2γ p−1 −  p−2 = 0. − + =− ∂x1 ∂x2 k˜ 2 k˜ 2 Since  p and  p are solutions of the Helmholtz equations (2.2.66) with smooth right-hand sides, these functions are infinitely smooth in 1 and 2 . Then functions ( p) ( p) E t , Ht are infinitely smooth in 1 and 2 as well. The verification of (2.2.55) is not complicated. Thus,

44

2 Shielded Waveguide ( p)

( p)

∂ E2 ∂ E1 − = −i p ; ∂x1 ∂x2 ( p)

(2.2.72)

( p)

∂ H2 ∂ H1 − = iε p ; p = 0, 1, . . . , m. ∂x1 ∂x2

(2.2.73)

  ∂ p   p 0 = 0, = 0, ∂n 0

Since

we can check using (2.2.54) that   E τ( p) 0 = 0, Hn( p) 0 = 0; p = 0, 1, . . . , m.

(2.2.74)

Let us verify the validity of the transmission conditions ( (

Hτ( p)

) 

E τ( p)

)

= 0;

(2.2.75)

= 0; p = 0, 1, . . . , m.

(2.2.76)



Formulas (2.2.75) and (2.2.76) for eigenfunctions ( p = 0) directly follow from conditions (2.2.71). Assume that (2.2.75) and (2.2.76) hold for the functions with the indices p = 0, 1, . . . , q − 1 and prove that these formulas are valid for p = q. From (2.2.55) we obtain [ε]

[ε]



Hτ( p) 



E τ( p) 



∂ p = −i ∂n



;

 ∂ p =i ε , p = 0, 1, . . . , q − 1, ∂n 

note that these formulas follows from (1). Then,     ( (q) ) 1 ∂q  1 E τ  = iγ +γ E τ(q−1)   k˜ 2 ∂τ  k˜ 2    1 ∂q 1 Hn(q−1)  −i − ˜k 2 ∂n  ˜k 2       ∂q−1  1 2γ ∂q−1 1 ∂q−2 = −i − − ∂τ  [ε] ∂n  [ε] ∂n  k˜ 2  +

   1  γ E τ(q−1)  − Hn(q−1)  k˜ 2

(2.2.77)

(2.2.78)

2.2 Properties of the System of Eigenwaves and Associated Waves of a Waveguide

 = −i 

1 + ˜k 2



1 k˜ 2

(

    ∂q−1  2γ ∂q−1 1 ∂q−2 − − ∂τ  [ε] ∂n  [ε] ∂n 



2iγ − [ε]



)



1 = −i ˜k 2

∂q−1 ∂n

Hτ(q)  +





45

1 k˜ 2





   ∂q−1  i ∂q−2 +i − = 0; ∂τ  [ε] ∂n  

ε =i k˜ 2



   ∂q  1 ∂q  + iγ ∂n  k˜ 2 ∂τ 

 (q−1)   +γ εE n 



1 k˜ 2



 Hτ(q−1) 

    ∂q−1  ∂q−2 2γ ∂q−1 1 ε ε + + ∂τ  [ε] ∂n ∂n [ε]     1   (q−1)  εE n + + γ Hτ(q−1)   ˜k 2 



1 = −i k˜ 2  +

1 k˜ 2



    ∂q−1  ∂q−2 2γ ∂q−1 1 + + ε ε ∂τ  [ε] ∂n ∂n [ε]  

     ∂q−1  ∂q−2 2iγ ∂q−1 i = 0. ε + + i ε ∂τ  ∂n ∂n [ε]  

Now we can prove formulas (2.2.70) and (2.2.71). Applying Green’s formula and using (2.2.72)–(2.2.76), we obtain  

( p)

Et



  ( p)   ¯ −∇  g¯ + Ht ∇ f dx



 =

⎛⎛ ⎝⎝−



  ( p) ∂ g¯ E 1 ∂x2

⎞   ( p) ( p) ( p) ∂ g¯ E 2 ⎠ + ∂ E1 − ∂ E2 + ∂x1 ∂x2 ∂x1

 ⎞   ⎛  ( p) ( p) ( p) ( p) ∂ f¯ H1 ∂ f¯ H2 ⎠ + ∂ H2 − ∂ H1 +⎝ − ∂x2 ∂x1 ∂x1 ∂x2   = 

( p)



( p)

∂ E1 ∂ E2 − ∂x2 ∂x1

g¯ +

( p)

( p)

∂ H2 ∂ H1 − ∂x1 ∂x2

g¯

⎞ f¯⎠ d x

f¯ d x

46

2 Shielded Waveguide

 =i

  ε p f¯ +  p g¯ d x,



which means that formula (2.2.70) is true. Using (2.2.54) we have −i

   ( p) ( p) εE t ∇ f¯ + Ht ∇ g¯ d x 

  = 

 +

ε ∂ p γ ∂ p + k˜ 2 ∂x2 k˜ 2 ∂x1

  −i 

 +

εγ ∂ p ε ∂ p − k˜ 2 ∂x1 k˜ 2 ∂x2



∂ f¯ + ∂x1



εγ ∂ p ε ∂ p + k˜ 2 ∂x2 k˜ 2 ∂x1



∂ f¯ ∂x2

   ∂ g¯ ε ∂ p γ ∂ p ∂ g¯ + − + dx ∂x1 k˜ 2 ∂x1 k˜ 2 ∂x2 ∂x2



εγ ( p−1) ε ( p−1) E − H2 ˜k 2 1 ˜k 2



∂ f¯ + ∂x1



εγ ( p−1) ε ( p−1) E + H1 ˜k 2 2 ˜k 2



∂ f¯ ∂x2

    ε ( p−1) γ ( p−1) ∂ g¯ ε ( p−1) γ ( p−1) ∂ g¯ E2 + H1 + − E1 + H2 dx ∂x1 ∂x2 k˜ 2 k˜ 2 k˜ 2 k˜ 2     ( p)  ( p)   ¯ E t −∇  g¯ + Ht ∇ f dx = −iγ 

−i

 

( p−1)

Et

   ( p−1)   ¯ −∇  g¯ + Ht ∇ f dx



  + 

 ∂ p ∂ g¯ ∂ p ∂ g¯ ∂ p ∂ f¯ ∂ p ∂ f¯ dx − + − ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x2 ∂x2 ∂x1 = −iγ

 

( p)

Et

   ( p)   ¯ −∇  g¯ + Ht ∇ f dx



−i

 

( p−1)

Et

   ( p−1)   ¯ −∇  g¯ + Ht ∇ f d x,



so that formulas (2.2.71) are valid as well. Let L 22 () be the Cartesian product of two copies of space L 2 ().

2.2 Properties of the System of Eigenwaves and Associated Waves of a Waveguide

47

Lemma 2.2.1 For any element u ∈ L 22 () the decomposition u = ε∇ f + ∇  g 1

holds for certain functions f ∈ H01 () and g ∈ H (). Define function f ∈ H01 () by the variational relation 

 u∇ ϕd ¯ x= 

ε∇ f ∇ ϕd ¯ x, ∀ϕ ∈ H01 () .

(2.2.79)



By the Riesz theorem [58], there exists unique element f . Denote v := u − ε∇ f ; v ∈ L 22 () . Thus, for any ϕ ∈ C0∞ () ⊂ H01 (), j = 1, 2, by the definition of generalized derivatives,     ¯ d x = − u∇ ϕd ¯ x + ε∇ f ∇ ϕd ¯ x = 0, ¯ x = − v (∇ ϕ) (div v) ϕd 







  and hence, divv = 0 in  j , j = 1, 2 in terms of distributions. Since v ∈ L 22  j , divv = 0 in  j then [59] the trace of the vector normal component exists on piecewise smooth boundary ∂ j :   v · n|∂ j ∈ H −1/2 ∂ j , j = 1, 2. From formula (2.2.79), we find that element f ∈ H01 () solves the problem "

ε f = div u 2  j , j = 1, 2; f |0 = 0, ε ∂∂nf − u · n = 0. 

The second transmission condition is equivalent to the condition [v · n] = 0. Then for any ϕ ∈ C0∞ () ⊂ H01 (), we have 

 ¯ x =− (div v) ϕd 





 ¯ =− [v · n] ϕdτ

v (∇ ϕ) ¯ dx + 

 u∇ ϕd ¯ x+



i.e., div v = 0 in  as a distribution. 1

We define an element g ∈ H () by the variational relation

ε∇ f ∇ ϕd ¯ x = 0, 

48

2 Shielded Waveguide



¯ x= v∇  hd





¯ x, ∀h ∈ P, ∇g∇ hd

(2.2.80)



where P :=

⎧ ⎨ ⎩

h : h| j

  ¯ j , j = 1, 2, [h] = 0, ∈ C1 

 

⎫ ⎬

hd x = 0 . ⎭ 1

From the Riesz theorem, it follows that there exists unique element g ∈ H () 1

since the left-hand side of (2.2.80) is an antilinear continuous functional on H () 1

and set P is dense in H (). Every element w ∈ Q,   

¯ j , j = 1, 2, [w] = 0 , Q := w : w ∈ C 1 

 can be represented in the form w = ∇ p + ∇  h, where p ∈ p : p ∈ P, p|0 = 0 , h ∈ P. Since functions p and h are smooth, this relation can be proved in a standard manner using curve integrals [60]. Taking into account the condition div v = 0, it is easy to show that (2.2.80) is equivalent to the variation relation 

 v · wd ¯ x= 

∇  g · wd ¯ x, ∀w ∈ Q.

(2.2.81)



However, Q is dense in L 22 (); hence v = ∇  g, and the lemma is proved. # Lemma 2.2.2 For any element u ∈ L 22 (), the decomposition u = ∇  f + ∇g, 1

holds for a certain f ∈ H01 (), g ∈ H (). Proof The proof is similar to that of Lemma 2.2.1. Element f ∈ H01 () is defined by the variational relation  

If to set

u∇  ϕd ¯ x=

 ∇ f ∇ ϕd ¯ x, ∀ϕ ∈ H01 () . 

v := u − ∇  f ; v ∈ L 22 () ,

(2.2.82)

2.2 Properties of the System of Eigenwaves and Associated Waves of a Waveguide

49

∂v1 ∂v2 we find that ∂x − ∂x = 0 in  as a distribution. 2 1 1 Element g ∈ H0 () is defined by the relation



 v · ∇ pd ¯ x= 

 ∇g∇ pd ¯ x, ∀ p ∈ p : p ∈ P, p|0 = 0 ,



which is equivalent to 

 v · wd ¯ x=



∇g · wd ¯ x, ∀w ∈ Q, 

so that v = ∇g.



By L 42 (), we denote the Cartesian product of four copies of space L 2 (). Denote T  ( p) ( p) the transversal components of eigenwaves and associated waves by E n,t , Hn,t corresponding to eigenvalues γn ; p = 0, 1, . . . , m n . By A we denote the set of indices for n, n ∈ A. We assume, in addition, that different eigenvectors ϕ(n) 0 have different indices n and the case γn = γm for n = m is possible. 

Theorem 2.2.1 Let ε1 = ε2 . If the system of eigen- and associated vectors ϕ(n) p ( p = 0, 1, . . . , m n ) of pencil L (γ), corresponding to eigenvalues  γn , n ∈ A, $is T ( p) ( p) , double complete in H × H , then the system of vector-functions E n,t , Hn,t n ∈ A, p = 0, 1, . . . , m n , is complete in L 42 (). Proof Using Lemmas 2.2.1 and 2.2.2, we represent arbitrary element u ∈ L 42 () in the form   f j ∈ H01 () , ε∇ f 2 − ∇  g1 u= 1 ∇  f 1 + ∇g2 g j ∈ H () ; j = 1, 2. For the proof of the theorem, it is sufficient to show that the conditions  

  ( p)  ( p)  E n,t ε∇ f¯2 − ∇  g¯1 + Hn,t ∇  f¯1 + ∇ g¯2 d x = 0,

(2.2.83)



n ∈ A, p = 0, 1, . . . , m n , yield u = 0. By formulas (2.2.70) and (2.2.71), one can reduce (2.2.83) to the equations  

 (n)  ε p f¯1 +  (n) ¯1 d x + γn p g

 

 (n)  ε p f¯2 +  (n) ¯2 d x p g

50

2 Shielded Waveguide

+

 

 ¯ ε(n) d x = 0, n ∈ A, p = 0, 1, . . . , m n , +  g ¯ f 2 p−1 2 p−1

(2.2.84)



or, in the operator form, 

   (n) (n) ˜ ˜ K ϕ(n) p , f 0 + K ϕ p + γn K ϕ p−1 , f 1 = 0, n ∈ A, p = 0, 1, . . . , m n , (2.2.85)  (n) (n) T T ˜ T (n) ˜ , f 0 = ( f 1 , g1 ) , f 1 = ( f 2 , g2 ) . where ϕ p =  p ,  p Taking into account (2.2.81), we show that (2.2.85) is equivalent to the equations 

 ˜ K ϕ(n,v) , f v = 0; v = 0, 1; n ∈ A, p = 0, 1, . . . , m n , p

(n) (n,1) where ϕ(n,0) = ϕ(0) = γn ϕ(n) . Taking into account that K > 0 and p p , ϕp p + ϕ p−1

(n)  using the double completeness of system ϕ p in H × H we obtain that



 ˜ = 0, n ∈ A, p = 0, 1, . . . , m n , ϕ(n,v) , K f v p

and K f˜v = 0, f˜v = 0; v = 1, 2.



By this theorem, verifying completeness of the system of transversal components of eigenwaves and associated waves in L 42 () is reduced to proving double completeness of the system of eigenvectors and associated vectors of pencil L (γ) in H × H . This very problem was considered in Sect. 2.2.1. We established sufficient conditions providing double completeness of the system of eigenvectors and associated vectors of pencil L (γ) in H × H . Thus, under the conditions of Theorems 2.1.9 and 2.1.10, the system of transversal components of eigenwaves and associated waves is complete in L 42 (). Note that in Theorem 2.2.1, we used the assumption ε1 = ε2 . If ε1 = ε2 then the problem of normal waves is reduced to two well-studied scalar Dirichlet and Neumann problems for the Helmholtz equation [8–10].

2.2.3 Biorthogonal Property for Eigenwaves and Associated Waves Introduce the following notations. Let T T   ( p) ( p) ( p) ( p) ( p) ( p) Vn := E n,t , Hn,t and Wn := Hn,t × e3 , e3 × E n,t be the transversal components of aneigenwave ( p = 0) or associated wave ( p ≥ 1) for the “direct” wave and “conjugate” wave, respectively (corresponding to eigenvalue γn ). Brackets ·, · denote the inner product in L 42 ():

2.2 Properties of the System of Eigenwaves and Associated Waves of a Waveguide

 V, W  =

51

V · W¯ d x.



Let us prove first the following basic formula: - , - , , (γn − γm ) Vn( p) , Wm(q) = Vn( p) , Wm(q−1) − Vn( p−1) , Wm(q) ; p ≥ 0, q ≥ 0, (2.2.86) ( p) (q) where Vn ≡ 0, Wm ≡ 0 for p < 0, q < 0. ( p) ( p) ( p) (q) Components E n,t , Hn,t satisfy (2.2.55). Expressing γn Vn , γm Wm from these equations and using (2.2.70) and (2.2.71) we obtain 

  γn Vn( p) · W¯ m(q) d x −



= −i

 



  Vn( p) · γm W¯ m(q) d x



        (q) (q) ( p) ( p) (n) (n) (m) (m) Hm,t −∇   p + E m,t ∇  ψ p + E n,t −∇  ψq + Hn,t ∇  q dx



 +

Vn( p) · W¯ m(q−1) d x −



 =−



Vn( p−1) · W¯ m(q) d x



 (m) (n)  εq  p + ψq(m) ψ (n) dx + p





 (n) (m)  (m) ε p q + ψ (n) dx p ψq



, - , - , - , + Vn( p) , Wm(q−1) − Vn( p−1) , Wm(q) = Vn( p) , Wm(q−1) − Vn( p−1) , Wm(q) . By formula (2.2.86) we have that for p ≥ 0, q ≥ 0 the following relations hold: ,

Vn( p) , Wm(q) = 0 for γn = γm .

(2.2.87)

Note that eigenvalues with different indices may coincide: γn = γm for n = m. Theorem 2.2.2 that ε1 = ε2 and the system of eigenvectors and associ

Assume p = 0, 1, . . . , m n ) of pencil L (γ) corresponding to eigenvalated vectors ϕ(n) ( p , n ∈ A, is double complete in H × H . Then the system of vector-functions ues γ  n T $ ( p) ( p) E n,t , Hn,t , n ∈ A, p = 0, 1, . . . , m n , is complete, satisfies the “minimality” property in L 42 (), and there exists the unique system biorthogonal with respect to the system of the vector-functions.   ( p) was proved in Theorem 2.2.1. The “miniProof Completeness of system Vn mality” follows from the existence of a biorthogonal system [61].

52

2 Shielded Waveguide

Renumber sequence γn so that every eigenvalue does not coincide with any other ˜ Introduce the systems of functions one: γ˜ k , k ∈ A. . Vn( p) , dimL(γ˜ k ) = rk ; (γ˜ k ) := n, p:γn =γ˜ k ,0≤ p≤m n

.

Q (γ˜ k ) :=

Wn( p) , dimQ(γ˜ k ) = rk .

n, p:γn =γ˜ k ,0≤ p≤m n

  ( p) Taking into account (2.2.87), for a system biorthogonal to Vn , n ∈ A, p = 0, 1, . . . , m n , it is sufficient to form finite systems biorthogonal to (γ˜ k ) for a fixed K using Q (γ˜ k ), and then combine these systems. We will construct the elements of a system biorthogonal to (γ˜ k ) by linear combinations of the elements of system Q (γ˜ k ). Let v1 , . . . , vr and w1 , . . . , wr be elements of systems (γ˜ k ) and Q (γ˜ k ), respectively. Let us determine uq =

r 

a¯ pq w p , q = 1, . . . , r,

p=1

from the conditions

,

v p , u q = δ pq , p, q = 1, . . . , r,

(2.2.88)

, -

 which are equivalent to a matrix equation GA=I where A := a pq , G := v p , wq ( p)

is the matrix of order r × r . Since system Vn

is complete in L 42 () the determi-

nant of matrix G is not equals to zero. Thus, there exists the unique matrix A = G −1 . The theorem is proved.  For homogeneous waveguides (ε1 = ε2 ) with a piecewise smooth boundary of domain  and when associated waves are absent, the relations similar to (2.2.87) are well known [41]. However, the setting is not a vector problem in this case.

2.2.4 Basis Property of the System of Eigenwaves and Associated Waves   ( p) posTaking into account Theorem 2.2.2, a natural question arises if system Vn sesses the basis property of in L 42 (). We address this issue below searching for a Schauder basis [62]. Let us prove the following lemmas.

2.2 Properties of the System of Eigenwaves and Associated Waves of a Waveguide

53

Lemma 2.2.3

Let  {ϕi } be a complete normal system in the Hilbert space H(ϕi  =1) and system ψ j satisfy the conditions   ϕi , ψ j = N j δi j and

! ! 0 < C1 ≤ !ψ j ! ≤ C2 .

If there exists a subsequence N jk of sequence N j such that N jk → 0 for jk → ∞, then system {ϕi } is not a basis in H. Proof Since system {ϕi } is complete in H then every N j = 0. Let{ϕi } bea basis in H. Form the system ψ j = ψ j /N j which is biorthogonal to {ϕi }: ϕi , ψ j = δi j .     Then ψ j is also a basis in H. If basis {ϕi } is normal, then basis ψ j is almost ! ! ! ! ψ  normal [62]. We have !ψ jk ! = N jjk → ∞, i.e., N jk → 0, for jk → ∞. Thus, we k   have obtained a contradiction to the condition concerning the normalization of ψ j . The lemma is proved.  2.2.4 Let the system {ϕi } be a basis in Hilbert space H. Then the system

Lemma  ϕi , ϕi = ϕi / ϕi  is also basis in H. Proof Since {ϕi } is a basis in H, then for any ϕ the following (unique) expansion holds  ϕ= Ci ϕi , i

and the series converges with respect to the norm. Thus ϕ=



Ci ϕi 

i

 ϕi Ci ϕi , = ϕi  i

where Ci = Ci ϕi . The latter series also converges with respect to the norm of space H since ! ! ! ! N N ! ! ! !   ! ! !  ! Ci ϕi ! = !ϕ − Ci ϕi ! → 0, N → ∞. !ϕ − ! ! ! ! i

i

Moreover, if ϕ=

 i

Ci ϕi and ϕ =



Ci ϕi

i

and both series converge with respect to the norm, we have

54

2 Shielded Waveguide



ϕ=

Ci ϕi and ϕ =

i



Ci ϕi / ϕi  .

i

From unique expansion {ϕi }, we obtain Ci = Ci / ϕi , or Ci = Ci ϕi  = Ci . Hence, the series expansion with respect to system ϕi is unique. Consequently,  system ϕi is a basis in space H. Lemma is proved. , (0) Consider eigenvalue γ of multiplicity 1 and compute inner product V , W (0) for γ = 0 using (2.2.54) and (2.2.70): 

 γ V, W  = 2γ

(E 1 H2 − E 2 H1 ) d x = − 

 2  εE t + Ht2 d x



 + (E 1 (εE 1 + γ H2 ) + E 2 (εE 2 − γ H1 ) + H1 (H1 − γ E 2 ) + H2 (H2 + γ E 1 )) d x 

 =−

 2  εE t + Ht2 d x + i







    E t −∇  ψ + Ht ∇   d x



 =−

 2  εE t + Ht2 d x −





 2  ε + ψ 2 d x.



Here V ≡ V (0) , W ≡ W (0) . Thus for eigenwaves corresponding to eigenvalues γn = 0 of multiplicity 1 we have ,

=−

1 γn

Vn(0) , Wn(0)

-

      2  2  2  2  1 (0) (0) (n) ε E n,t dx − ε (n) d x. + Hn,t + ψ 0 0 γn





(2.2.89) Similarly, for eigenwaves of multiplicity 1 corresponding to γ such that γ = γ, ¯ we obtain  

 ε |E t |2 + |Ht |2 d x





=



        E 1 ε E¯ 1 + γ¯ H¯ 2 + E 2 ε E¯ 2 − γ¯ H¯ 1 + H1 H¯ 1 − γ¯ E¯ 2 + H2 H¯ 2 + γ¯ E¯ 1 d x



 − γ¯ 



 E 1 H¯ 2 − E 2 H¯ 1 − H1 E¯ 2 + H2 E¯ 1 d x

2.2 Properties of the System of Eigenwaves and Associated Waves of a Waveguide

 = −i



  ,   ¯ d x − γ¯ V¯ , W = E t −∇  ψ¯ + Ht ∇  



55

 

 ε ||2 + |ψ|2 d x.



It follows from Theorem 2.2.2, that V¯ is the transversal , - component of eigenwave corresponding to γ. ¯ Using (2.2.86), we obtain that V¯ , W = 0 for γ = γ. ¯ Thus, for eigenwaves of multiplicity 1 corresponding to γn = γ¯ n we have          2       (n) 2  (0) 2  (0) 2  ψ + ε  E n,t ε (n) d x.  +  Hn,t  d x =   0 0  

(2.2.90)



Application of Lemmas 2.1.3 and 2.1.4 and formulas (2.2.89) and (2.2.90) allows us to prove the following statement. Theorem 2.2.3 Let ε1 = ε2 , the spectrum of pencil L (γ) contain an infinite set of ˜ and γn → ∞ for n → ∞. Then the 1, n ∈ A, isolated eigenvalues γn of multiplicity  T $ ( p) ( p) system of vector-functions E n,t , Hn,t related with the system of eigenvectors

(n)  and associated vectors ϕ p ( p = 0, 1, . . . , m n ) of pencil L (γ) corresponding to eigenvalues γn , n ∈ A, A˜ ⊂ A, is not a basis in L 42 ().   ( p) Proof Assume that the system Vn , p = 0, 1, . . . , m n , n ∈ A, is a basis  ( p) $ ! !  ( p) ( p) ! ( p) ! 4 in L 2 (). Define the normalized system V n , V n := Vn / !Vn !. From Lemma 2.2.4, it follows that this system is also basis in L 42 ().   ( p) Using linear combinations of the elements of the system Wn , p = 0, 1, . . . , m n , n ∈ A, we define, similarly  ( p) $to the proof of Theorem 2.2.2, a unique biorthogonal    ( p) ˜ to basis V n . Set system Wn  ( p)

Wn

⎧ ˜ ⎨ W˜ n( p) , n ∈ / A, = ! Wn(0) ! , n ∈ A˜ ( p = 0) . ⎩ ! (0) ! !Wn !

For n, m ∈ A˜ we have 0 / (0) !2   (0) - ! , V n , W m = δnm Nn , Nn := Vn(0) , Wn(0) / !Vn(0) ! , ! ! ! ! ! ( p) ! ! ( p) ! because !Wn ! = !Vn !. From (2.2.89) and (2.2.90) for sufficiently large n and γn = γ¯ n (see Theorems 2.1.3 and 2.1.7) we have

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2 Shielded Waveguide

 |Nn | ≤ εmax |γn |−1



              (0) 2  (0) 2  (n) 2  (n) 2 ε E n,t  + Hn,t  d x + ε 0  + ψ0  d x          (0) 2  (0) 2 ε E n,t  + Hn,t  d x 

= 2εmax |γn |−1 → 0 ˜$ for n → ∞, n ∈ A.  ( p) $ ( p)  and W n satisfy the conditions of Lemma 2.1.3; conseSystems V n  ( p) $  quently, V n in not a basis in L 42 (). This contradiction proves the theorem.



Note that conditions of Theorem 2.2.3 are typical for partially filled rectangular and waveguides. Theorem 2.2.3 shows that the basis property of system  circular  ( p) Vn is not valid for these problems. The property of normal waves to form a basis is important for solving the problems  ( p) is applied in [16]. of the waveguide excitation. The basis property of system Vn   ( p) It is interesting to note that the basis property of system Vn is not valid in the case ε1 = ε2 . However, using decomposition into E- and H-waves one can show that a“basis of subspaces” exists and the expansions in [16] can be applied. When ε1 = ε2 , the problem is not reducable to scalar problems for E- and Hwaves.

References 1. V.I. Gvozdev, S.S. Hitrov, Transmission lines for integral devices. For. Radio Electron. (5), 86–107 (1982). (in Russian) 2. R.Z. Dautov, E.M. Karchevskii, Existence and properties of solutions to the spectral problem of dielectric waveguide theory. Comp. Maths. Math. Phys 40(8), 1200–1213 (2000) 3. Yu.V. Egorov, Partially Filled Rectangular Waveguides (Sovetskoe Radio, Moscow, 1967). (in Russian) 4. G.F. Zargano, A.M. Lerer, V.P. Lyapin, G.P. Sinyavskii, Transmission Lines with Complex Cross-Sections (Rostov University Press, Rostov-on-Don, 1983). (in Russian) 5. A.S. Ilinski, Yu.V. Shestopalov, Applications of the Methods of Spectral Theory in the Problems of Wave Propagation (Moscow State University, Moscow, 1989) 6. L. Levin, Theory of Waveguides (Newnes-Butterworths, London, 1975) 7. V.P. Shestopalov, Summation Equations in the Modern Theory of Diffraction (Naukova Dumka, Kiev, 1983). (in Russian) 8. A.A. Samarskii, A.N. Tikhonov, On Excitation of Radio Waveguides. I. Zhurn. Teoretich. Fiziki 17(11), 1283–1296 (1947). (in Russian) 9. A.A. Samarskii, A.N. Tikhonov, On Excitation of Radio Waveguides. II. Zhurn. Teoretich. Fiziki 17(12), 1431–1440 (1947). (in Russian)

References

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10. A.A. Samarskii, A.N. Tikhonov, The representation of the field in waveguide in the form of the sum of TE and TM modes. Zhurn. Teoretich. Fiziki 18(7), 971–985 (1948). (in Russian) 11. A.S. Ilyinsky, Yu.G. Smirnov, Electromagnetic Wave Diffraction by Conducting Screens (VSP, Utrecht, 1998) 12. A.M. Belyancev, A.V. Gaponov, On the waves with complex propagation constants in coupled transmission lines without dissipation. Radiotekhnika i Electronika 9(7), 1188–1197 (1964). (in Russian) 13. G.I. Veselov, P.E. Krasnushkin, On the dispersion properties of shielded circular waveguides and their complex waves. Dokl. AN SSSR 260(3), 576–579 (1981). (in Russian) 14. A.S. Ilinski, G.Ja. Slepyan, Oscilations and Waves in Electromagnetic Systems with Losses (Moscow University Press, Moscow, 1983). (in Russian) 15. P.E. Krasnushkin, E.N. Fedorov, On the multiplicity of wavenumbers of normal waves in layered media. Radiotekhnika i Electronika 17(6), 1129 (1972). (in Russian) 16. A.S. Ilinski, Yu.V. Shestopalov, Mathematical models for problem of propagation of waves in micro-strip transmission lines. Vych. Method and Programming, vol. 32 (Moscow University Press, Moscow, 1980), pp. 85–103. (in Russian) 17. A.S. Ilinski, Yu.V. Shestopalov, On the spectrum of normal waves of slot transmission line. Radiotekhnika i Electronika 26(10), 2064–2073 (1981). (in Russian) 18. Yu.V. Shestopalov, Normal waves of open and shielded slot transmission lines formed by arbitrary cross-sections. Dokl. AN SSSR 289(4), 840–845 (1986) 19. Yu.V. Shestopalov, Existence of discrete spectrum of normal waves of microstrip transmission lines with layered dielectric filling. Dokl. AN SSSR 273(3), 594 (1983) 20. A.S. Ilinski, E.V. Chernokozhin, Yu.V. Shestopalov, Method of operator equations for solving problem of normal waves of coupled microstrip transmission lines with layered dielectric substrate. Mathematical Models of Applied Electrodynamics (Moscow State University Press, Moscow, 1984), pp. 116–136. (in Russian) 21. A.S. Ilinski, Yu.G. Smirnov, Methods of mathematical modeling and automation of processing observation data and their application. Analysis of Mathematical Models of Microstrip Transmission Lines (Moscow State University Press, Moscow, 1986), pp. 175–198. (in Russian) 22. A.S. Ilinski, Yu.G. Smirnov, Mathematical modelling of wave propagation in the slot transmission line. Zh. Vych. Matem. i Matem. Fis. 27(2), 252–261 (1987). (in Russian) 23. A.S. Ilinski, Yu.G. Smirnov, Numerical solution of the slot lines formed by rectangular waveguides with various cross-section. Radioteknika i Electronika 34(5), 908–916 (1989). (in Russian) 24. M.V. Keldysh, On Eigenvalues and Eigenfunctions of certain classes of non-self-adjoint equations. Dokl. AN SSSR 77(1), 11–14 (1951). (in Russian) 25. T.Ja. Azizov, I.S. Iohvidov, Linear Operators in Hilbert Spaces with G-metric (Nauka, Moscow, 1986). (in Russian) 26. I.Tz. Gokhberg, M.G. Krein, Introduction in the Theory of Linear Nonselfadjoint Operators in Hilbert Space (American Mathematical Society, Providence, 1969) 27. I.Tz. Gokhberg, M.G. Krein, Basic Results on Defect Numbers, Kernel Numbers and Indexes of Linear Operators. Uspekhi Mat. Nauk 12(2), 43–118 (1957). (in Russian) 28. I.Tz. Gokhberg, E.I. Sigal, An operator generalization of the logarithmic residue theorem and the theorem of Rouche. Math. USSR-Sbornik 13(4), 603–625 (1971) 29. A.G. Kostyuchenko, M.B. Orazov, Certain properties of the roots of a self-adjoint quadratic pencil. Functi. Anal. Appl. 9(4), 295–305 (1975) 30. V.B. Lidskii, Conditions of completeness of system of eigenvectors and associated vectors for non-self-adjoint operators with discrete spectrum. Trydu MMO 8, 84–220 (1958). (in Russian) 31. A.S. Markus, Conditions of completeness of system of eigenvectors and associated vectors of linear operator in banach space. Matemat. Sbornik 70, 526–561 (1966). (in Russian) 32. A.S. Markus, On holomorphic operator-functions. Dokl. AN SSSR 119(6), 1099–1102 (1958). (in Russian) 33. A.S. Markus, On the completeness of part of eigenvectors and associated vectors of analytical operator-function. Matematich-eskie Issledovaniya. Kishinev 9(3(33)), 105–126 (1974). (in Russian)

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34. A.S. Markus, V.I. Matsaev, Spectral theory of holomorphic operator-functions in Hilbert space. Funct. Anal. Appl. 9(1), 73–74 (1975) 35. A.L. Delitsyn, An approach to the completeness of normal waves in a waveguide with magnitodielectric filling. Diff. Equ. 36(5), 695–700 (2000) 36. A.S. Zilbergleit, Yu.I. Kopilevich, Spectral Theory of Regular Waveguides (Izd-vo Fiz.-tech. in-ta, Leningrad, 1983). (in Russian) 37. P.E. Krasnushkin, E.I. Moiseev, On the excitation of oscillations in layered radiowaveguide. Dokl. AN SSSR 264(5), 1123–1127 (1982). (in Russian) 38. G.V. Radzievskii, Multiple completeness of eigen and associates vectors of some classes of operator-functions analytic in a disk. Funct. Anal. Appl. 7(1), 76–77 (1973) 39. R. Adams, Sobolev Spaces (Academic, New York, 1975) 40. V.G. Maz’ya, Sobolev Spaces. Springer Ser. Soviet Math. (Springer, Berlin, 1985) 41. L.A. Weinschtien, Electromagnetic Waves (Radio i Svyaz’, Moscow, 1988). (in Russian) 42. J.L. Lions, E. Magenes, Non-homogeneous Boundary Value Problems and Applications (Springer, Berlin, 1972) 43. M. Costabel, Boundary, integral operators, on Lipschitz domains: elementary results. SIAM J. Math. Anal. No. (3), 613–626 (1988) 44. O.A. Ladyzhenskaja, N.N. Uralceva, Linear and Quasilinear Equations of Parabolic Type (Nauka, Moscow, 1973). (in Russian) 45. V.P. Mihailov, Partial Differential Equations? (Nauka, Moscow, 1983). (in Russian) 46. V.A. Kondrat’ev, Boundary value problems for elliptic equations in domains with conical and corner points. Trudy MMO 16, 209–292 (1967). (in Russian) 47. T. Kato, Perturbation Theory for Linear Operators (Springer, Berlin, 1966) 48. R. Courant, D. Hilbert, Methods of Mathematical Physics, vol. 1 (Interscience Publishers, New York, 1953) 49. G.I. Veselov, S.B. Raevskii, Metal-Dielectric Waveguides Formed by Layers? (Radio i Svyaz’, Moscow, 1988). (in Russian) 50. A.I. Markushevich, Theory of Functions of a Complex Variable, 2nd edn. (Chelsea Publishing Company, New York, 1977) 51. G.V. Radzievskii, Multiple completeness of root vectors of a Keldys pencil perturbed by an operator-valued function analytic in a disc. Math. USSR-Sbornik 20(3), 323–347 (1973) 52. A.N. Kolmogorov, S.V. Fomin, Elements of the Theory of Functions and Functional Analysis (Dover Publications, New York, 1999) 53. M.V. Keldysh, On the completeness of the eigenfunctions of some classes of non-selfadjoint linear operators. Russ. Math. Surv. 26(4), 15–44 (1971) 54. A.S. Ilinski, Yu.G. Smirnov, Variation method in the eigenvalue problem for partially filled waveguide with nonregular boundary. Computational Methods on the Inverse Problems in Mathematical Physics (Moscow State University, Moscow, 1988), pp. 127–137 55. Yu.G. Smirnov, On the completeness of the system of eigen-waves and joined waves of partially filled waveguide with non-regular boundary. Dokl. AN SSSR 297(4), 829–832 (1987). (in Russian) 56. Yu.G. Smirnov, Investigation of solvability of vector electrodynamic problems on unclosed surfaces. Thesis of Doctor of Science (in physics and mathematics) (Moscow State University, Moscow, 1995). (in Russian) 57. Yu.G. Smirnov, The method of operator pencils in the boundary transmission problems for elliptic system of equations. Differentsialnie Uravnenia 27(1), 140–147 (1991). (in Russian) 58. A.N. Kolmogorov, S.V. Fomin, Elements of the Theory of Functions and Functional Analysis (Dover Publications, New York, 1999) 59. H.F. Baker, Abelian functions. Abels Theorem and the Allied Theory of Theta Functions (Cambridge University Press, Cambridge, 1897). (reprinted in 1995) 60. E.B. Bykhovsky, N.V. Smirnov, On orthogonal decomposition of vector space of square integrable functions on a certain domain and operators of vector analysis. Trudy MIAN SSSR 59, 536 (1960). (in Russian) 61. V.A. Sadovnichii, Theory of Operators (Moscow University Press, Moscow, 1989). (in Russian) 62. I.Tz. Gokhberg, M.G. Krein, Introduction in the Theory of Linear Nonselfadjoint Operators in Hilbert Space (American Mathematical Society, Providence, 1969)

Chapter 3

Planar Waveguide

Analysis of the spectra, real or complex, of the waves or oscillations in open guiding structures is much less developed as compared with the theory of shielded waveguides which enters many textbooks and monographs in electromagnetics, e.g., [1–4]. In fact, the former is possible only within the frames of the spectral theory of open structures [5, 6] involving mathematically correct statements of non-self-adjoint boundary eigenvalue problems with generalized conditions at infinity that enable one to consider complex modes of all types. The dielectric layer (DL) is one of the most well-studied waveguide structures in electromagnetics [1–4]. In fact, DL is the simplest plane-parallel waveguide (from the geometrical point of view) and its dispersion equation (DE) can be written explicitly. On the other hand, such a structure is widely used in practice (planar optical waveguides). However, still, there is no rigorous proof of the presence (or absence) of infinitely many real or complex eigenwaves propagating in DL, to the best of our knowledge. The main attention of this study is paid to the analysis of the surface and leaky waves in DL. Note that the closed-form DE of DL can easily be solved numerically (for real propagation constants). A large number of numerical results and calculations of surface waves in DL were obtained [7–9], however without completing the justification of the method including rigorous proofs of the existence of the DE roots. The existence of a finite number of surface propagating waves is well known and has been proved graphically [10] (note that here a mathematically rigorous proof has never been completed). For leaky waves (including complex ones), the solution of DE on a two-sheet Riemann surface was considered in [11]. In this paper, we focus on theoretical analysis rather than on numerical results and numerical methods ending up with mathematically correct proofs. The general results about the existence of propagating waves in nonhomogeneous waveguides and localization of propagation constants on the complex plane have been recently obtained in [12–15]. Some theoretical findings concerning the existence of propagating waves in nonhomogeneous planar waveguides can be found in [16]. This paper addresses a DL shielded by a perfectly conducting screen from one side, a partially shielded DL (PSDL). Wave propagation in such a structure is a classical © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 Y. Shestopalov et al., Optical Waveguide Theory, Springer Series in Optical Sciences 237, https://doi.org/10.1007/978-981-19-0584-1_3

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3 Planar Waveguide

problem [1–4]. The main purpose of this article is not only the correct verification of the occurrence of complex waves in a DL but also a rigorous proof of the existence of an infinite number of complex leaky waves. We will look for the odd TE modes of DL [2, 3]. For even TE waves, as well as for odd and even TM waves, the same results hold. Before proceeding to the mathematical consideration of complex and leaky waves one may adopt the following physical model that leads to the appearance of an infinite wave spectrum. If one considers a PSDL as a limiting case of a DL shielded from both sides shifting one boundary to infinity, one may assume that the infinite number of decreasing waves in a closed structure will pass to the improper sheet of the Riemann surface and form an infinite number of leaky waves. But it must be proved rigorously. It is preferable to handle surface and leaky waves separately which will be performed in this study. Note that in doing so we will not consider the solution of DE on the Riemann surface limiting our analysis to its one particular sheet. The paper [17] proves the existence of an infinite number of waves at the frequency ω = 0. It is natural to assume therefore (due to continuity with respect to the problem parameters) that at a frequency ω = 0, again, there will be an infinite number of waves. However, this reasonable conjecture requires a special rigorous proof. The knowledge (and rigorous mathematical proof) of the occurrence of infinitely many eigenwaves in an open waveguide plays crucial role in the construction of the eigenwave expansion of an arbitrary electromagnetic field excited in the structure. This problem was solved for shielded guides in classical works [18–20] (TE- and TM-mode expansions in hollow homogeneous waveguides) and then extended in [21, 22] to arbitrary inhomogeneous waveguides. However, for open structures (where the basic types are DL, dielectric waveguides, Goubau line, open transmission lines) the proof of the occurrence of infinitely many eigenwaves and the corresponding complex eigenvalues remained unfinished. This study fills this gap, at least for PSDL, and opens the way to generalization of the results to more complicated families of open waveguides. Note that for open structures (possessing circular symmetry and finite in the cross section) the first findings confirming the existence of infinitely many complex waves have been obtained recently [23]. However, planar layer-type dielectric and metal-dielectric waveguides (like DL and PSDL) are infinite in the cross section so that for them these results cannot be applied and the development of a specific approach is required which is an objective of this study. Highlight that rigorous proofs of the existence of infinitely many complex leaky waves and an absence of complex surface waves in PSDL are obtained for the first time, to the best of our knowledge. This result is of fundamental character for the theory of wave propagation in electromagnetics and can be extended to more general families of open waveguides. The analysis of all the considered types of waveguides and running waves is performed employing a universal scheme repeated for every waveguide family: description of the waveguide geometry and material parameters → spectral problem for the Maxwell equations → reduction to a boundary eigenvalue problem for a system of ordinary differential equations → re-formulation in terms of a variational relation in Sobolev spaces → reduction to a problem of finding characteristic numbers of an OVF → proofs of the discreteness and existence of the spectrum for the OVF

3 Planar Waveguide

61

of the problem. Such an organization of the book makes it possible for a reader to consider every particular type of a waveguide independently staying within one specific chapter equipped with all necessary information and notations (that may be, for clarity, repeated throughout the text).

3.1 TE-Polarized Waves in a Partially Shielded Dielectric Layer 3.1.1 Statement of the Problem Consider the three-dimensional half space R3 equipped with the Cartesian coordinate system O x yz. The half space is filled with an isotropic source-free medium with permittivity ε0 ε2 ≡ const and permeability μ0 ≡ const, where ε0 and μ0 are permittivity and permeability of vacuum. We consider electromagnetic waves propagating through a dielectric layer located between two half spaces x < 0 and x > h:  := {(x, y, z) : 0  x  h} . The boundaries x = 0 and x = h are the projections of, respectively, the surfaces of the perfectly conducting shield and dielectric. The geometry of the problem is shown in Fig. 3.1. Determination of normal TE-polarized waves in the considered waveguide structure is formulated as a problem of finding nontrivial running-wave solutions of the homogeneous system of Maxwell equations, i.e., the solutions with the dependence eiγz on the coordinate z along which the structure is regular, 

Fig. 3.1 Geometry of the problem

curl H = −i εE, ˜ curl E = iH,

(3.1.1)

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3 Planar Waveguide

    E = 0, E y (x)eiγz , 0 , H = Hx (x)eiγz , 0, Hz (x)eiγz ,

(3.1.2)

with the boundary conditions for the tangential electric field component on the perfectly conducting surface (x = 0) E y (0) = 0;

(3.1.3)

the transmission conditions for the tangential electric and magnetic field components on the surfaces of discontinuity (“jump”) of the permittivity (x = h)  [E y ]x=h = 0, [Hz ]|x=h = 0, where [ f ]|x0 = lim

x→x0 −0

f (x) − lim

x→x0 +0

(3.1.4)

f (x); and the radiation condition at infinity

which will be formulated and discussed later. The Maxwell system (3.1.1) is written in the normalized form. The passage to dimensionless variables has been carried out, namely, k0 x → x, γ → γ/k0 , √ μ0 /ε0 H → H, E → E , where k02 = ω 2 ε0 μ0 , ω is a circular frequency (the time factor e−iωt is omitted everywhere). We assume that the relative permittivity in the entire space has the form   ε=

ε1 , 0 ≤ x ≤ h, x > h, ε2 ,

(3.1.5)

where ε1 > ε2 . The problem on normal waves is an eigenvalue problem for the Maxwell equations with spectral parameter γ, which is the normalized propagation constant of the normal wave. Rewrite system (3.1.1) in the expanded form: ⎧  εEy, ⎪ ⎨ iγ Hx − Hz = −i −iγ E y = i Hx , ⎪ ⎩ E y = i Hz ,

(3.1.6)

and express Hx and Hz via function E y from system (3.1.1) Hx = −γ E y , Hz = −i E y .

(3.1.7)

It follows from (3.1.7) that the normal wave field in the waveguide can be represented with the use of one scalar function u := E y (x).

(3.1.8)

Thus, the problem has been reduced to finding tangential component u of the electric field. In the text which follows, ( · ) stands for the differentiation with respect to x.

3.1 TE-Polarized Waves in a Partially Shielded Dielectric Layer

63

We have the following eigenvalue problem for the tangential electric field component u: find γ ∈ C such that there exist nontrivial solutions of the differential equation   (3.1.9) ε − γ 2 u = 0, x > 0, u  +  satisfying the boundary condition for x = 0 u(0) = 0,

(3.1.10)

the transmission conditions for x = h  [u]|h = 0, u  h = 0

(3.1.11)

and the condition at infinity. Thus field (E, H) will satisfy all conditions (3.1.1)–(3.1.4). Definition 3.1.1 The propagating wave is characterized by real parameter γ. Definition 3.1.2 The decreasing wave is characterized by pure imaginary parameter γ. Definition 3.1.3 The complex wave is characterized by complex parameter γ such that Re γ Im γ = 0. Definition 3.1.4 The surface wave is a normal wave such that u(x) → 0, x → ∞. Definition 3.1.5 The leaky wave is a normal wave such that u(x) → ∞, x → ∞. For 0 < x < h, we have  ε = ε1 ; then from (3.1.9) we obtain the equation u  + λ2 u = 0,

(3.1.12)

λ 2 = ε1 − γ 2 ,

(3.1.13)

where and λ is a new complex (spectral) parameter. In view of the boundary condition for the tangential electric field component on the perfectly conducting surface (3.1.10), we obtain a solution of this equation in the form (3.1.14) u(x; λ) = C1 sin λx, 0 < x < h, where C1 is a constant. For x > h, we have  ε = ε2 ; then from (3.1.9) we obtain the equation   u  − 2 − λ2 u = 0, where 2 = ε1 − ε2 > 0.

(3.1.15)

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3 Planar Waveguide

We choose a solution of this equation in the form √

u(x; λ) = C2 e−(x−h)

2 −λ2

, x > h,

(3.1.16)

where C2 is a constant. R

In the general case we have the solution of equation (3.1.12) in the form u = C2 e−(x−h)

√ 2 −λ2

+ C3 e(x−h)

√ 2 −λ2

, x > h,

(3.1.17)

where C2 and C3 are arbitrary constants. Below we explain our choice in the form (3.1.16). Note that the study of the waves determined by solution (3.1.16) was carried out earlier (see, for example, [4]). However, a rigorous proof of the existence of complex and leaky waves was not performed, as well as the classification of waves.

From transmission conditions (3.1.11), solutions (3.1.14), and (3.1.16) we obtain the system of equations  C1 sin λh = C2 , √ λC1 cos λh = −C2 2 − λ2 . Solving this system we obtain the dispersion equation (DE) tan λh + √

λ 2 − λ2

= 0.

(3.1.18)

In (3.1.18) the square root is two-valued complex function. We will choose a specific branch of the square root below. Solutions λ of equation (3.1.18) define different types of waves which are considered below.

3.1.2 Surface Waves In this section, we will consider the surface waves such that the electromagnetic field decays for x → ∞. We assume that Re



2 − λ2 > 0

(3.1.19)

in order to specify the radiation condition at infinity. This condition determines the surface wave, i.e., the wave decaying at infinity according to formula (3.1.16). Following (3.1.19) we obtain that C3 = 0 in (3.1.17) for surface waves. Consider (3.1.18) subject to condition (3.1.19) for three different cases. Case 3.1. Let λ = α, α ∈ R. Then (3.1.18) takes the form

3.1 TE-Polarized Waves in a Partially Shielded Dielectric Layer

65

α tan αh + √ = 0. 2  − α2 +

(3.1.20)

√ Here and below sign + denotes the arithmetic root. The following statements hold. The proofs of the statements are straightforward. Lemma 3.1.1 Equation (3.1.20) has no solution for α2 > 2 . Lemma 3.1.2 If α is a root of (3.1.20) then −α is a root of this equation as well. Lemma 3.1.3 If m − 1/2 ≤ h/π ≤ m + 1/2 then there exist a nonzero positive number m ≥ 1 (and m nonzero negative roots) of equation (3.1.20). The jth positive root α j is located so that π π πj (2 j − 1) < α j < ( j = 1, . . . , m − 1), (2m − 1) < αm < . 2h h 2h If 0 < h/π ≤ 1/2 then there are no roots of equation (3.1.20). Case 3.2. Let λ = iβ, β ∈ R. Then equation (3.1.18) takes the form tanh βh +  +

β 2

+ β2

= 0.

(3.1.21)

Lemma 3.1.4 Equation (3.1.21) has no nontrivial solution. Case 3.3. Let λ = α + iβ, α = 0, β = 0, α, β ∈ R. Theorem 3.1.1 Equation (3.1.18) under condition (3.1.19) has no (complex) solutions. Proof Let us rewrite (3.1.18) as follows: λ tan λh = − √ ; 2  − λ2 +

(3.1.22)

taking the squares of the right- and left-hand sides we get sin2 λh =

λ2 , 2

hence, ( sin λh − λ) = 0 or ( sin λh + λ) = 0.

(3.1.23)

Let the first equation of (3.1.23) have a solution λm = αm + iβm . Then we have λm and from (3.1.22) we obtain sin λm h = 

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3 Planar Waveguide

 Re

2 − λ2m = − Re λm cot λm h = − Re cos λm h = − cos αm cosh βm > 0,

and therefore cos αm h < 0. On the other hand we have Im sin λm = Im λm /, hence βm cos αm h = > 0 which is a contradiction. Then the first equation in (3.1.23)  sinh βm  has no solution λm satisfying the condition Re 2 − λ2m > 0. Repeating the above reasoning for the second equation in (3.1.23) we get the same conclusion. Thus (3.1.18) has no solution under condition (3.1.19).  We have shown that there are only a finite  number of propagating surface waves sur f

with the propagation constants γ j = ±+ ε1 − α2j , j = 1, . . . , m. Thus, there are no complex surface waves satisfying condition (3.1.19). As far as we know, this result is obtained for the first time.

3.1.3 Leaky Waves In this section we consider leaky waves. The study of leaky waves has been started in electromagnetics and microwave engineering in the early 1950s (see, e.g., [24–26]). We assume that  (3.1.24) Re 2 − λ2 < 0 in order to specify the radiation condition at infinity. This condition determines the leaky waves increasing at infinity. We assume that C3 = 0 in (3.1.17) and solution u has the form (3.1.16). We will consider (3.1.18) under condition (3.1.24) for three different cases. Case 4.1. Let λ = α, α ∈ R. Taking into account (3.1.24) write equation (3.1.18) in the form α = 0. (3.1.25) tan αh − √ 2  − α2 + The following statements hold. Lemma 3.1.5 Equation (3.1.25) has no solution for α2 > 2 . Lemma 3.1.6 If α is a root of (3.1.25) then −α is a root of this equation as well. ( j)

Lemma 3.1.7 Let z 0 be the jth positive root of the equation tan z = z such that ( j) ( j+1) z0 < z0 , j = 1, 2, . . . . There are no roots of equation (3.1.25) for 0 < h ≤ 1 and π/2 ≤ h ≤ z 0(1) . If h = z 0(n) then there exist n ≥ 1 nonzero positive roots (and n nonzero negative roots) of equation (3.1.25); the jth root α j is located so that πj π < αj < (2 j + 1), j = 1, . . . , n. h 2h

3.1 TE-Polarized Waves in a Partially Shielded Dielectric Layer

67

If z 0(n) < h < π (2n + 1) /2 then there exist n + 1, n ≥ 1 nonzero positive roots (and n + 1 nonzero negative roots) of equation (3.1.25); the jth root α j is located so that π πj < αj < (2 j + 1), j = 1, . . . , n − 1, h 2h and

π π π π < αn < (2n + 1), < αn+1 < (2n + 1). h 2h h 2h

Finally, if π (2n + 1) /2 ≤ h < z 0(n+1) then there exist n ≥ 1 nonzero positive roots (and n nonzero negative roots) of equation (3.1.25); the jth root α j is located so that πj π < αj < (2 j + 1), j = 1, . . . , n. h 2h Proofs of Lemma 7 are based on the analysis of the number of solutions to the equations sin αh = ±α/ for different h and . Case 4.2. Let λ = iβ, β ∈ R. Then equation (3.1.18) takes the form tanh βh −  +

β 2 + β 2

= 0.

(3.1.26)

Lemma 3.1.8 Equation (3.1.26) has two nontrivial solutions ±β0 if h < 1. If h > 1 then there are no nontrivial solutions of (3.1.26). Case 4.3. Let λ = α + iβ, α = 0, β = 0, α, β ∈ R. Theorem 3.1.2 Equation (3.1.18) under condition (3.1.24) has infinitely many roots which form a sequence that tends to infinity.   1 1 Proof Let us consider the function f (z) := z sin , where z = . Function z λh f (z) has an isolated essential singularity at z = 0. Let a := h > 0. Consider two equations f (z) = a −1 and f (z) = −a −1 . From Picard’s theorem (see [27]) it follows that one of these equations has infinitely many roots z k (z k = 0), z k → ∞, k → ∞ in a neighborhood of the point z = 0. Indeed, according to Picard’s theorem there is only one exceptionable value A such that the equation f (z) = A has no infinitely many roots. Let us consider the first equality         1 1 1 1 2 1 = . Then cos = ± 1 − sin = ± 1 − 2 2 . (3.1.27) sin z az z z a z

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3 Planar Waveguide

          z 1 z 1 cosh ,z= , we get Re cos = cos 2 z z |z|2 |z|   1 1 z  + i z  (z  = 0 because there are no real solutions of the equation z sin = z a    z 1 > 0, we obtain the following equality: for |z| < ). Since cosh a |z|2

Taking a real part of cos

     1 z sign Re cos . = sign cos z |z|2 On the other hand, taking imaginary part of sin Im sin

  1 we have z

           z  z −z 1 z = cos sinh = − = ⇒ cos z |z|2 |z|2 a|z|2 |z|2

z  a|z|2 sinh



z  |z|2

 > 0.

  1 > 0. According to condition (3.1.24) we have Consequently, sign Re cos z Re



 2

−

λ2

= h Re

1−

1 < 0, a2 z2

i.e., we should choose the sign of the root in (3.1.27) as follows: cos

   1 1 =− 1− 2 2 z a z

  1 > 0. in order to satisfy the condition sign Re cos z −1 Let z k be a root of the equation f (z) = a ; then 

1 sin zk



   1 1 1 = = − 1− 2 2. , cos az k zk a zk

Next  tan

1 zk



Thus, we obtain that the equation

+

1 az k 1 1− 2 2 a zk

= 0.

(3.1.28)

3.1 TE-Polarized Waves in a Partially Shielded Dielectric Layer

tan λk h + 

λk h 2 − λ2k h 2 h2

69

=0

1 . zk h Repeating the above consideration for the equation f (z) = −a −1 we get the same conclusion. Since one of the equations f (z) = a −1 or f (z) = −a −1 has infinite number of roots in the neighborhood of z = 0 we obtain that (3.1.18) has infinitely many roots at infinity under condition (3.1.24).  has a solution λk =

Lemma 3.1.9 If λ is a root of equation (3.1.18) under condition (3.1.24) then −λ is a root of (3.1.18) under condition (3.1.24). Thus, we have shown that there  are a finite number of real leaky waves with the leaky

propagation constants γ j

= ±+ ε1 − α2j , j = 1, . . . , n, and two leaky waves  leaky with the propagation constants γˆ 1,2 = ±+ ε1 + β02 . Theorem 3.1.2 establishes the existence of infinite number of complex leaky waves (waves growing at infinity and satisfying  condition (3.1.24)). The propagation leaky

constants of such leaky waves are  γk is obtained for the first time.

=

ε1 − λ2k . As far as we know, this result

3.2 TE-Polarized Waves in an Inhomogeneous Partially Shielded Dielectric Layer 3.2.1 Statement of the Problem Consider electromagnetic waves propagating along a partially shielded dielectric layer:  := {(x, y, z) : 0  x  h} . The boundaries x = h and x = 0 are, respectively, the projections of the dielectric surface and of the perfectly conducting screen. Determination of the surface TE-polarized waves reduces to finding nontrivial running-wave solutions of the homogeneous system of Maxwell equations depending on the coordinate z along which the structure is regular in the form eiγz . The relative permittivity in the entire space has the form   ε=

ε, 0 ≤ x ≤ h, x > h. εc ,

(3.2.29)

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3 Planar Waveguide

We assume that ε(x) > εc is a continuous function on the segment [0, h], i.e., ε(x) ∈ C[0, h] and Im ε(x) = 0. The problem on surface waves is an eigenvalue problem for the Maxwell equations with spectral parameter γ which is the propagation constant of the normal wave. The normal wave field in the waveguide can be represented using one scalar function (3.2.30) u := E y (x). Thus, the problem is reduced to finding tangential component u of the electric field. Throughout the text below, ( · ) stands for the differentiation with respect to x. We have the following eigenvalue problem for the tangential electric field component u: find γ ∈ C such that there exist nontrivial solutions of the differential equation   (3.2.31) ε − γ 2 u = 0, u  + ω 2 μ0 ε0 satisfying the boundary conditions u|x=0 = 0,

(3.2.32)

 [u]|x=h = 0, u  x=h = 0.

(3.2.33)

and the transmission conditions

We do not specify the radiation condition at infinity because we want to consider the problem for arbitrary γ. For x > h, we have  ε = εc ; then from (3.2.31) we obtain the equation u  − λ2 u = 0.

(3.2.34)

We choose a solution of this equation in the form u(x; λ) = e(h−x)λ , x > h,

(3.2.35)

where λ2 = γ 2 − ω 2 ε0 μ0 εc and λ is a new (complex) spectral parameter. If Re λ > 0 then we have a surface wave and if Re λ < 0, a leaky wave. For 0 < x < h, we have  ε = ε(x); thus from (3.2.31) we obtain the equation   Lu := u  + 2 − λ2 u = 0,

(3.2.36)

2 (x; ω) = ω 2 ε0 μ0 (ε(x) − εc ) .

(3.2.37)

where Definition 3.2.1 λ ∈ C is called a characteristic number of the problem if there exists nontrivial solution u of equation (3.2.36) for 0 < x < h satisfying (3.2.35) for x > h, boundary condition (3.2.32), and transmission conditions (3.2.33).

3.2 TE-Polarized Waves in an Inhomogeneous Partially Shielded Dielectric Layer

71

3.2.2 Sobolev Spaces and Variational Relation We will look for solutions u in Sobolev space H01 (0, h) with the inner product and the norm h h   2 f g dρ, f 1 = ( f, f )1 = | f  |2 dρ, ( f, g)1 = 0

0

where H01 (0, h) :=

⎧ ⎨

h f :

⎩

⎫ ⎬

| f  |2 dρ < ∞; f (0) = 0 . ⎭

0

Note that we use nonstandard notation for this Sobolev space because f (0) = 0 but, in general, f (h) = 0. Let us give a variational formulation of the problem under consideration. Multiplying (3.2.36) by an arbitrary test function v ∈ H01 (0, h) (preliminarily we can assume that this function is continuously differentiable on [0, h]) and applying Green’s formula, we obtain h

h v Ludρ =

0



h

u vdρ + 0

h  uvdρ − λ 2

2

0

h = u  v 0 −

h

uvdρ 0

u  v  dρ +

0

= u  (h)v(h) −

h

h 2 uvdρ − λ2

0

h

u  v  dρ +

0

uvdρ 0

h

h 2 uvdρ − λ2

0

uvdρ = 0.

(3.2.38)

0

Using (3.2.35), we express the values of the derivatives at x = h from relations (3.2.33) as follows: (3.2.39) u  (h) = −λu(h). Then, in view of (3.2.38) and (3.2.39), we obtain h λ

h uvdρ + λu(h)v(h) +

2 0

 

h

u v dρ − 0

2 uvdρ = 0.

(3.2.40)

0

Variational relation (3.2.40) has been obtained for a smooth function v. Consider the following sesquilinear forms and the corresponding operators:

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3 Planar Waveguide

h k(u, v) :=

uvdρ = (Ku, v), ∀v ∈ H01 , 0

 k(u, v) :=

h

u, v), ∀v ∈ H01 , 2 uvdρ = (K

0

h a(u, v) :=

u  v  dρ = (Iu, v), ∀v ∈ H01 ,

0

s(u, v) = u(h)v(h) = (Su, v), ∀v ∈ H01 . The boundedness of k(u, v) and  k(u, v) follows from the Poincarè inequality [28]. The boundedness of the form k(u, v) will be shown in the next subsection. Now variational problem (3.2.40) can be written in the operator form (N(λ)u, v) = 0, ∀v ∈ H01 , or, equivalently,

   u = 0. N(λ)u := λ2 K + λS + I − K

(3.2.41)

Equation (3.2.41) is the operator form of variational relation (3.2.40). The characteristic numbers and eigenvectors of operator N by definition coincide with the eigenvalues and eigenvectors of the problem under consideration.

3.2.3 Properties of the Operator Pencil We have reduced the problem on surface TE-polarized waves to the study of spectral properties of operator pencil N. In this connection, we first consider the properties of the operators introduced in the preceding section.  : H01 → H01 are compact and K > Lemma 3.2.1 The bounded operators K, K  0, K > 0. Eigenvalues and eigenfunctions of K are λn (K) = (ß2 (n + 1/2)2 /h2 )−1 , (n ≥ 0). Asymptotics of eigenvalues of operator K is λn (K) ∼ u n (x) = sin π(n+1/2)x h h2 , n → ∞. ß2 n2 Proof Since the embedding H01 (0, h) ⊂ L 2 (0, h) is compact, it follows that the  : H01 → H01 are compact. Further, for u = 0 (as an element of operators K and K 1 H0 (0, h))

3.2 TE-Polarized Waves in an Inhomogeneous Partially Shielded Dielectric Layer

73

h (Ku, u) =

|u|2 dρ > 0 0

and u, u) = (K

h 2 |u|2 dρ > 0; 0

 > 0. consequently, K > 0, K The eigenvalue problem for operator K has the form ⎧ ⎨ u  (x) = − 1 u(x), λ(K ) ⎩ u(0) = 0, u  (h) = 0.

(3.2.42)

Eigenvalues and eigenfunctions (eigenvectors) of this problem are λn (K) = (n ≥ 0).  (ß2 (n + 1/2)2 /h2 )−1 , u n (x) = sin π(n+1/2)x h Lemma 3.2.2 The operator S : H01 → H01 is compact and dim Im S = 1. The operator S : H01 → H01 is nonnegative. There exists only one nonzero eigenvalue λ∗1 (S) = h with the corresponding eigenfunction (eigenvector) u ∗1 (x) = x. Proof Consider the form h s(u, v) := u(h)v(h) =

s  (x)v  (x)d x,

(3.2.43)

0

where s(x) = Su. Let v = 0 in a neighborhood of point h; then h 0 = s (x)v(x)0 − 

h

s  (x)v(x)d x,

0

or

h

s  (x)v(x)d x = 0.

(3.2.44)

0

Since function v(x) is arbitrary, it follows from (3.2.44) that s  (x) = 0, 0 < x < h. Let v = 0 at point h. From (3.2.43) we get

(3.2.45)

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3 Planar Waveguide

s(0) = 0, s  (h) = u(h).

(3.2.46)

Thus we obtain the boundary value problem 

s  (x) = 0, s(0) = 0, s  (h) = u(h).

(3.2.47)

Then s(x) = u(h)x, dim Im S = 1. Similarly we get that a nonzero eigenvalue of operator S is λ∗1 (S) = h with the corresponding eigenfunction (eigenvector) u ∗1 (x) = x. The proof of the lemma is complete.  Lemma 3.2.3 There exists a  λ ∈ R such that operator N( λ) is continuously invertλ) is ible, i.e., the resolvent set ρ(N) := {λ : ∃ N−1 (λ) : H01 → H01 } of OVF N( nonempty, ρ(N) = ∅. Proof Let λ ∈ R. Since function 2 belongs to the space C 2 [−h, h], it follows from  and S that for  the properties of operators K, K λ > max |(x)| 0≤x≤h

u, u) ≥ u 2 λ(Su, u) + u 2 − (K (N( λ)u, u) =  λ2 (Ku, u) +  for each u. Hence  λ ∈ ρ(N), where ρ(N) is the resolvent set of operator N.



Theorem 3.2.1 The operator pencil N(λ) : H01 → H01 is bounded, Fredholm, and holomorphic on C. Proof We obtain the desired result by the application of the above lemmas.



Theorem 3.2.2 The spectrum of operator pencil N(λ) : H01 → H01 is discrete on C, i.e., this OVF has a finite set of characteristic numbers with finite algebraic multiplicity on any compact set K 0 ⊂ C. Proof The assertion of the theorem is a corollary of Theorem 3.2.1 and a theorem on a holomorphic operator function (see [29], Theorem 5.1). 

3.2.4 Completeness of System of Eigenvectors and Associated Vectors of Operator Pencil First, let us analyze the quadratic form of the operator pencil  u). (N(λ)u, u) = λ2 (Ku, u) + λ(Su, u) + (u, u) − (Ku,

(3.2.48)

3.2 TE-Polarized Waves in an Inhomogeneous Partially Shielded Dielectric Layer

75

Set λ = λ + iλ and (u, u) = 1. Then, the real and imaginary parts of quadratic form (3.2.48) are u, u), Re(N(λ)u, u) = ((λ )2 − (λ )2 )(Ku, u) + λ (Su, u) + 1 − (K Im(N(λ)u, u) = λ (2λ (Ku, u) + (Su, u)).

(3.2.49) (3.2.50)

Denote λ0 := max |(x)|, 1 := {λ : λ = 0, λ > 0}, 2 := {λ : λ = 0, λ > λ0 }. 0≤x≤h

Theorem 3.2.3 1 ∪ 2 ⊂ ρ(N ). Proof From (3.2.49) and (3.2.50) we have that (N(λ)u, u) = 0. Taking into account Theorem 3.2.1 we conclude that bounded operator N−1 (λ) exists for λ ∈ 1 ∪ 2 .  Thus, we have the following propositions: Corollary 3.2.4 There are no complex surface TE waves. Corollary 3.2.5 There are not more than a finite set of propagating surface TE waves. It was recently shown in [30, 31] that there may be finitely many surface waves in an open dielectric layer. Since characteristic numbers of the pencil have finite algebraic multiplicity, we should unite the sets of surface and leaky waves to obtain an infinite set of normal waves and provide completeness of the system. Introduce operators K, P, and R.   Denote μk := π(k + 1/2)/ h. We have u k (x)= sin μk x, u k 2 =μ2k h2 + sin4μ2μk k h . Taking into account Lemma 3.2.1 we obtain that the system of vectors {ek }∞ k=0 , ek (x) = u k (x)/ u k forms an orthogonal normalized basis in H01 (0, h): (ek , el ) = δkl , where δkl is the Kronecker delta. Set ∞  1/2 K1/2 ϕ = λk (ϕ, ek )ek k=0

for ϕ =

∞ 

(ϕ, ek )ek .

k=0

A one-dimensional projector P is determined by the formula Pϕ = (ϕ, e0 )e0 for ϕ =

∞  k=0

(ϕ, ek )ek .

Let V1 := {u ∈ H01 (0, h) : u(x) = Cu ∗1 (x), C ∈ C}, V = V1 ⊕ V1⊥ . Since u ∗1 2 = h, we have that ψ1 = u ∗1 (x)/ u ∗1 is a normalized eigenvector of operator S.

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3 Planar Waveguide

Vector e0 can be decomposed into an orthogonal sum of vectors e0 = (e0 , ψ1 )ψ1 + ψ1⊥ , where ψ1⊥ := e0 − (e0 , ψ1 )ψ1 and ψ1 ∈ V1 , ψ1⊥ ∈ V1⊥ . Taking into account that (ψ1 , e0 ) = ( u 0 u ∗1 μ20 )−1 = 0, we determine a onedimensional projector −1/2

Rϕ = (ϕ, e0 ) for ϕ =

∞ 

−1/2

λ∗1 λ0 λ∗1 λ ψ1 − (ϕ, ψ1⊥ ) 0 ψ1 (ψ1 , e0 ) (ψ1 , e0 )

(ϕ, ek )ek .

k=0

Let us define the operator T = RP. It is easy to check that S = TK1/2 . Indeed, TK1/2 ψ1 = λ∗1 ψ1 and TK1/2 ϕ = 0 for ϕ ∈ V1⊥ . Thus, Sϕ = λ∗1 (ϕ, ψ1 )ψ1 = ϕ(h)x. Hence, we can rewrite pencil (3.2.41) in the form    u = 0, N(λ)u := λ2 (K1/2 )2 + λTK1/2 + I − K

(3.2.51)

where K1/2 is a positive compact operator of order 2 (with eigenvalues λn (K1/2 ) ∼ h  are compact operators. Thus, all conditions of the Keldysh , n → ∞) and T, K πn theorem (see, [29], Theorem 9.1) are fulfilled and we obtain the following statement. Theorem 3.2.6 System of eigen- and associated vectors of pencil N(λ) is double complete in H01 (0, h) and all characteristic numbers are located in the angles πm/2 − δ < argλ < πm/2 + δ for arbitrary δ > 0, except maybe for a finite set of characteristic numbers; m = 0, 1, 2, 3.

R

One can find definitions of associated vectors of the pencil in [29]. The corresponding definitions of the eigen- and associated waves of the waveguiding structure can be found in [32].

Corollary 3.2.7 There are infinitely many leaky complex TE waves. Proof Consider the case Re λ = 0 and Im λ = 0. It follows from (3.2.50) that (Su, u) = 0 and u(h) = 0. Taking into account (3.2.39) we obtain the Cauchy problem for the linear homogeneous differential equation (3.2.36) with zero initial conditions u(h) = u  (h) = 0. Hence, we have u(x) = 0, 0 ≤ x ≤ h. Thus {λ : Re λ = 0, Im λ = 0} ⊂ ρ(L). Finally, we conclude that the statement follows from Theorems 3.2.3 and 3.2.6. 

3.3 TE-Polarized Waves in an Inhomogeneous Metamaterial Partially Shielded Layer

77

3.3 TE-Polarized Waves in an Inhomogeneous Metamaterial Partially Shielded Layer 3.3.1 Statement of the Problem Consider electromagnetic waves propagating through a partially shielded metamaterial layer:  := {(x, y, z) : 0  x  h} . The boundaries x = h and x = 0 are the projections of, respectively, the metamaterial surface and the perfectly conducting screen. Determination of surface TE-polarized waves reduces to finding nontrivial running-wave solutions of the homogeneous system of Maxwell equations depending on the coordinate z along which the structure is regular in the form eiγz . Assume that the relative permittivity in the entire space has the form   ε=

−ε2 , 0 ≤ x ≤ h, x > h, εc ,

(3.3.52)

where ε2 (x) > εc are continuous functions on the segment [0, h], i.e., ε(x) ∈ C[0, h] and Im ε(x) = 0. The problem on surface waves is an eigenvalue problem for the Maxwell equations with spectral parameter γ, which is the propagation constant of the running wave. The normal wave field in the waveguide can be represented using one scalar function (3.3.53) u := E y (x). Thus, the problem is reduced to finding tangential component u of the electric field. Below, symbol ( · ) stands for the differentiation with respect to x. We have the following eigenvalue problem for the tangential electric field component u: find γ ∈ C such that there exist nontrivial solutions of the differential equation   (3.3.54) ε − γ 2 u = 0, u  + ω 2 μ0 ε0 satisfying the boundary conditions u|x=0 = 0,

(3.3.55)

 [u]|x=h = 0, u  x=h = 0.

(3.3.56)

and the transmission conditions

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3 Planar Waveguide

We will not formulate the radiation condition at infinity because we want to consider the problem for arbitrary γ. For x > h, we have  ε = εc ; then from (3.3.54) we obtain the equation u  − λ2 u = 0.

(3.3.57)

We choose a solution of this equation in the form u(x; λ) = e(h−x)λ , x > h,

(3.3.58)

where λ2 = γ 2 − ω 2 ε0 μ0 εc and λ is a new (complex) spectral parameter. If Re λ > 0 then we have a surface wave and if Re λ < 0 a leaky wave. For 0 < x < h, we have  ε = ε(x); thus from (3.3.54) we obtain the equation

where

  Lu := u  − 2 + λ2 u = 0,

(3.3.59)

  2 (x; ω) = ω 2 ε0 μ0 ε(x)2 + εc .

(3.3.60)

Definition 3.3.1 λ ∈ C is called a characteristic number of the problem if there exists nontrivial solution u of equation (3.3.59) for 0 < x < h satisfying (3.3.58) for x > h, boundary condition (3.3.55), and transmission conditions (3.3.56).

3.3.2 Sobolev Spaces and Variational Relation We will look for solutions u in Sobolev space H01 (0, h) with the inner product and the norm h h   2 = f g dρ, f = f, f = | f  |2 dρ, f, g) ( )1 ( 1 1 0

where

0

h H01

(0, h) := { f :

| f  |2 dρ < ∞; f (0) = 0}.

0

Note that we use nonstandard notation for this Sobolev space because f (0) = 0 but, in general, f (h) = 0. Let us give a variational formulation of the problem under consideration. Multiplying (3.3.59) by arbitrary test function v ∈ H01 (0, h) (preliminarily we can assume that this function is continuously differentiable on [0, h]) and applying Green’s formula, we obtain

3.3 TE-Polarized Waves in an Inhomogeneous Metamaterial Partially Shielded Layer

h

h vLudρ =

0

h



u vdρ − 0

h  uvdρ − λ 2

2

0

h = u  v 0 −

h

79

uvdρ 0

u  v  dρ −

0

h

h 2 uvdρ − λ2

0

h



= u (h)v(h) −

 

uvdρ 0

h

u v dρ − 0

h  uvdρ − λ 2

0

uvdρ = 0. (3.3.61)

2 0

Using (3.3.58), we express the values of the derivatives at x = h from relations (3.3.56) as follows: u  (h) = −λu(h). (3.3.62) Then, in view of (3.3.61) and (3.3.62), we obtain h λ

h uvdρ + λu(h)v(h) +

2 0

 

h

u v dρ + 0

2 uvdρ = 0.

(3.3.63)

0

Variational relation (3.3.63) has been obtained for smooth function v. The integrals occurring in (3.3.63) can be viewed as sesquilinear forms over the field C defined on H01 and depending on arguments u and v. These forms t define some bounded linear operators T : H01 → H01 by formula [33] t(u, v) = (Tu, v), ∀v ∈ H01 ,

(3.3.64)

provided that the forms themselves are bounded, |t(u, v)| ≤ C u v . The linearity follows from the linearity of the form in the first argument, and the continuity follows from the estimates Tu 2 = t(u, Tu) ≤ C u Tu . Consider the following sesquilinear forms and the corresponding operators: h uvdρ = (Ku, v), ∀v ∈ H01 ,

k(u, v) := 0

80

3 Planar Waveguide

 k(u, v) :=

h

 v), ∀v ∈ H01 , 2 uvdρ = (Ku,

0

h a(u, v) :=

u  v  dρ = (Iu, v), ∀v ∈ H01 ,

0

s(u, v) = u(h)v(h) = (Su, v), ∀v ∈ H01 . The boundedness of k(u, v) and  k(u, v) follows from the Poincarè inequality [28]. The boundedness of form k(u, v) will be shown in the next section. Now variational problem (3.3.63) can be written in the operator form (N(λ)u, v) = 0, ∀v ∈ H01 or, equivalently,

   u = 0. N(λ)u := λ2 K + λS + I + K

(3.3.65)

Equation (3.3.65) is the operator form of variational relation (3.3.63). The characteristic numbers and eigenvectors of operator N by definition coincide with the eigenvalues and eigenvectors of the problem under consideration.

3.3.3 Properties of the Operator Pencil We have reduced the problem on surface TE-polarized waves to the study of spectral properties of operator pencil N. In this connection, we first consider the properties of the operators introduced in the preceding section.  : H01 → H01 are compact, and K > Lemma 3.3.1 The bounded operators K, K  0, K > 0. Eigenvalues and eigenfunctions of operator K are λn (K) = (ß2 (n + (n ≥ 0). Asymptotics of eigenvalues of opera1/2)2 /h2 )−1 , u n (x) = sin π(n+1/2)x h 2 tor K is λn (K) ∼ ßh2 n2 , n → ∞. Lemma 3.3.2 The operator S : H01 → H01 is compact and dim Im S = 1. The operator S : H01 → H01 is nonnegative. There exists only one nonzero eigenvalue λ∗1 (S) = h with the corresponding eigenfunction (eigenvector) u ∗1 (x) = x. Lemma 3.3.3 There exists a  λ ∈ R such that operator N( λ) is continuously invertible, i.e., the resolvent set ρ(N) := {λ : ∃ N−1 (λ) : H01 → H01 } of OVF N( λ) is nonempty, ρ(N) = ∅. , and S that for Proof Let λ ∈ R. It follows from the properties of operators K, K  λ>0

3.3 TE-Polarized Waves in an Inhomogeneous Metamaterial Partially Shielded Layer

81

u, u) ≥ u 2 (N( λ)u, u) =  λ2 (Ku, u) +  λ(Su, u) + u 2 + (K for each u. Hence  λ ∈ ρ(N), where ρ(N) is the resolvent set of operator N.



Theorem 3.3.1 The operator pencil N(λ) : H01 → H01 is bounded, Fredholm, and holomorphic on C. Proof We obtain the desired result by the application of the above lemmas.



Theorem 3.3.2 The spectrum of operator pencil N(λ) : H01 → H01 is discrete on C, i.e., this OVF has a finite set of characteristic numbers with finite algebraic multiplicity on any compact set K 0 ⊂ C. Proof The assertion of the theorem is a corollary of Theorem 3.3.1 and a theorem on a holomorphic operator function (see [29], Theorem 5.1). 

3.3.4 Completeness of the System of Eigenvectors and Associated Vectors of Operator Pencil First, let us analyze the quadratic form of the operator pencil u, u). (N(λ)u, u) = λ2 (Ku, u) + λ(Su, u) + (u, u) + (K

(3.3.66)

Set λ = λ + iλ and (u, u) = 1. Then, the real and imaginary parts of quadratic form (3.3.66) are u, u), Re(N(λ)u, u) = ((λ )2 − (λ )2 )(Ku, u) + λ (Su, u) + 1 + (K Im(N(λ)u, u) = λ (2λ (Ku, u) + (Su, u)).

(3.3.67) (3.3.68)

Denote 1 := {λ : λ = 0, λ > 0}, 2 := {λ : λ = 0, λ > 0}. Theorem 3.3.3 1 ∪ 2 ⊂ ρ(N ). Proof From (3.3.67) and (3.3.68) we have that (N(λ)u, u) = 0. Taking into account Theorem 3.3.1 we conclude that bounded operator N−1 (λ) exists for λ ∈ 1 ∪ 2 .  Thus we have the following propositions: Corollary 3.3.4 There are no complex surface TE waves. Corollary 3.3.5 There is not more than a finite number of propagating surface TE waves.

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It was recently shown in [30, 31] that there are finitely many surface waves in an open metamaterial layer. Since characteristic numbers of the pencil have finite algebraic multiplicity, we should unite the surface and leaky waves to obtain an infinite set of normal waves providing thus completeness of the system. Let us introduce operators K, P, and R.   Denote μk := π(k + 1/2)/ h. We have u k (x)= sin μk x, u k 2 =μ2k h2 + sin4μ2μk k h . Taking into account Lemma 3.3.1 we obtain that the system of vectors {ek }∞ k=0 , ek (x) = u k (x)/ u k , forms an orthogonal normalized basis in H01 (0, h): (ek , el ) = δkl , where δkl is the Kronecker delta. Set ∞  1/2 K1/2 ϕ = λk (ϕ, ek )ek k=0

for ϕ =

∞ 

(ϕ, ek )ek .

k=0

A one-dimensional projector P is determined by the formula Pϕ = (ϕ, e0 )e0 for ϕ =

∞ 

(ϕ, ek )ek .

k=0

Let V1 := {u ∈ H01 (0, h) : u(x) = Cu ∗1 (x), C ∈ C}, V = V1 ⊕ V1⊥ . Since u ∗1 2 = h, we have that ψ1 = u ∗1 (x)/ u ∗1 is a normalized eigenvector of operator S. Vector e0 can be decomposed into an orthogonal sum of vectors e0 = (e0 , ψ1 )ψ1 + ψ1⊥ , where ψ1⊥ := e0 − (e0 , ψ1 )ψ1 ; ψ1 ∈ V1 , ψ1⊥ ∈ V1⊥ . Taking into account that (ψ1 , e0 ) = ( u 0 u ∗1 μ20 )−1 = 0, we determine the onedimensional projector −1/2

Rϕ = (ϕ, e0 ) for ϕ =

∞ 

−1/2

λ∗1 λ0 λ∗1 λ ψ1 − (ϕ, ψ1⊥ ) 0 ψ1 (ψ1 , e0 ) (ψ1 , e0 )

(ϕ, ek )ek .

k=0

Define the operator T = RP. It is easy to check that S = TK1/2 . Indeed, TK1/2 ψ1 = and TK1/2 ϕ = 0 for ϕ ∈ V1⊥ . Thus, Sϕ = λ∗1 (ϕ, ψ1 )ψ1 = ϕ(h)x. Hence, we can rewrite pencil (3.3.65) in the form

λ∗1 ψ1

   u = 0, N(λ)u := λ2 (K1/2 )2 + λTK1/2 + I + K

(3.3.69)

h where K1/2 is positive compact operator of order 2 (with eigenvalues λn (K1/2 ) ∼ πn ,  n → ∞) and T, K are compact operators. Thus, all conditions of the Keldysh theorem (see, [29], Theorem 9.1) are fulfilled and we obtain the following proposition.

3.3 TE-Polarized Waves in an Inhomogeneous Metamaterial Partially Shielded Layer

83

Theorem 3.3.6 System of eigen- and associated vectors of pencil N(λ) is double complete in H01 (0, h) and all characteristic numbers are located in the angles πm/2 − δ < argλ < πm/2 + δ for arbitrary δ > 0, except maybe for a finite set of characteristic numbers; m = 0, 1, 2, 3.

R

Definitions of the associated vectors of the pencil can be found in [29] and of the corresponding eigenwaves and associated waves in [32].

Corollary 3.3.7 There are infinitely many complex leaky TE waves. Proof Consider the case Re λ = 0 and Im λ = 0. It follows from (3.3.68) that (Su, u) = 0 and u(h) = 0. Taking into account (3.3.62) we obtain the Cauchy problem for the linear homogeneous differential equation (3.3.59) with zero initial conditions u(h) = u  (h) = 0. Hence, we have u(x) = 0, 0 ≤ x ≤ h. Thus {λ : Re λ = 0, Im λ = 0} ⊂ ρ(L). Then the statement follows from Theorems 3.3.3 and 3.3.6. 

3.4 TE-Polarized Waves in an Inhomogeneous Partially Shielded Lossy Dielectric Layer 3.4.1 Statement of the Problem Consider electromagnetic waves propagating through a partially shielded layer with lossy dielectric:  := {(x, y, z) : 0  x  h} . The boundaries x = h and x = 0 are the projections of, respectively, the dielectric surface and the perfectly conducting screen. Determination of surface TE-polarized waves reduces to finding nontrivial running-wave solutions of the homogeneous system of Maxwell equations depending on the coordinate z along which the structure is regular in the form eiγz . Assume that the relative permittivity in the entire space has the form   ε=

ε + iεlos , 0 ≤ x ≤ h, x > h, εc ,

(3.4.1)

where ε(x) > εc are continuous functions on the segment [0, h], i.e., ε(x) ∈ C[0, h] and Im ε(x) = 0 and εlos > is a real positive constant.

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The problem on surface waves is an eigenvalue problem for the Maxwell equations with spectral parameter γ which is the propagation constant of the running wave. The normal wave field in the waveguide can be represented using one scalar function (3.4.2) u := E y (x). Thus, the problem is reduced to finding tangential component u of the electric field. Below, symbol ( · ) stands for the differentiation with respect to x. We have the following eigenvalue problem for the tangential electric field component u: find γ ∈ C such that there exist nontrivial solutions of the differential equation   (3.4.3) ε − γ 2 u = 0, u  + ω 2 μ0 ε0 satisfying the boundary conditions u|x=0 = 0,

(3.4.4)

 [u]|x=h = 0, u  x=h = 0.

(3.4.5)

and the transmission conditions

We do not formulate the radiation condition at infinity because we want to consider the problem for arbitrary γ. For x > h, we have  ε = εc ; then from (3.4.3) we obtain the equation u  − λ2 u = 0.

(3.4.6)

We choose a solution of this equation in the form u(x; λ) = e(h−x)λ , x > h,

(3.4.7)

where λ2 = γ 2 − ω 2 ε0 μ0 εc and λ is a new (complex) spectral parameter. If Re λ > 0 then we have a surface wave and if Re λ < 0 a leaky wave. For 0 < x < h, we have  ε = ε(x); thus from (3.4.3) we obtain the equation   2 − λ2 u = 0, Lu := u  + 2 + ilos

(3.4.8)

where 2 (x; ω) = ω 2 ε0 μ0 εlos . 2 (x; ω) = ω 2 ε0 μ0 (ε(x) − εc ) and los

(3.4.9)

Definition 3.4.1 λ ∈ C is called a characteristic number of the problem if there exists nontrivial solution u of equation (3.4.8) for 0 < x < h satisfying (3.4.7) for x > h, boundary condition (3.4.4), and transmission conditions (3.4.5).

3.4 TE-Polarized Waves in an Inhomogeneous Partially Shielded Lossy Dielectric Layer

85

3.4.2 Sobolev Spaces and Variational Relation Let us give a variational formulation of the problem under consideration. Multiplying (3.4.8) by an arbitrary test function v ∈ H01 (0, h) (preliminarily we can assume that this function is continuously differentiable on [0, h]) and applying Green’s formula, we obtain h

h v Ludρ =

0

u  vdρ +

0

h

2 2 uvdρ + ilos

0

h = u  v 0 −

h

h

uvdρ − λ2

h

0

u  v  dρ +

0

= u  (h)v(h) −

h

2 2 uvdρ + ilos

h

0

h

uvdρ 0

uvdρ − λ2

0

u  v  dρ +

0

h

2 2 uvdρ + ilos

0

h uvdρ 0

h

uvdρ − λ2

0

h uvdρ = 0.

(3.4.10)

0

Using (3.4.7), we express the values of the derivatives at x = h from relations (3.4.5) as follows: u  (h) = −λu(h). (3.4.11) Then, in view of (3.4.10) and (3.4.11), we obtain h λ

h

 

uvdρ + λu(h)v(h) +

2 0

h

u v dρ − 0

h  uvdρ − 2

0

uvdρ = 0. (3.4.12)

2 ilos 0

Variational relation (3.4.12) is obtained for smooth function v. Consider the following sesquilinear forms and the corresponding operators: h k(u, v) :=

uvdρ = (Ku, v), ∀v ∈ H01 , 0

 k(u, v) :=

h

u, v), ∀v ∈ H01 , 2 uvdρ = (K

0

h a(u, v) :=

u  v  dρ = (Iu, v), ∀v ∈ H01 ,

0

s(u, v) = u(h)v(h) = (Su, v), ∀v ∈ H01 .

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The boundedness of k(u, v) and  k(u, v) follows from the Poincarè inequality [28]. The boundedness of form k(u, v) will be shown in the next section. Now variational problem (3.4.12) can be written in the operator form (N(λ)u, v) = 0, ∀v ∈ H01 or, equivalently,   2  − ilos K u = 0. N(λ)u := λ2 K + λS + I − K

(3.4.13)

Equation (3.4.13) is the operator form of variational relation (3.4.12). The characteristic numbers and eigenvectors of operator N by definition coincide with the eigenvalues and eigenvectors of the problem under consideration.

3.4.3 Properties of the Operator Pencil We have reduced the problem on surface TE-polarized waves to the study of spectral properties of operator pencil N. In this connection, we first consider the properties of the operators introduced in the preceding section.  : H01 → H01 are compact, and K > Lemma 3.4.1 The bounded operators K, K  > 0. Eigenvalues and eigenfunctions of operator K are λn (K) = (ß2 (n + 0, K (n ≥ 0). Asymptotics of eigenvalues of opera1/2)2 /h2 )−1 , u n (x) = sin π(n+1/2)x h h2 tor K is λn (K) ∼ ß2 n2 , n → ∞. Proof Since the embedding H01 (0, h) ⊂ L 2 (0, h) is compact, it follows that the  : H01 → H01 are compact. Further, for u = 0 (as element of operators K and K 1 H0 (0, h)) h (Ku, u) = |u|2 dρ > 0 0

and u, u) = (K

h 2 |u|2 dρ > 0; 0

 > 0. consequently, K > 0, K The eigenvalue problem for operator K is as follows: ⎧ ⎨ u  (x) = − 1 u(x), λ(K ) ⎩ u(0) = 0, u  (h) = 0.

(3.4.14)

3.4 TE-Polarized Waves in an Inhomogeneous Partially Shielded Lossy Dielectric Layer

87

Eigenvalues and eigenfunctions (eigenvectors) of this problem are λn (K) = (n ≥ 0).  (ß2 (n + 1/2)2 /h2 )−1 , u n (x) = sin π(n+1/2)x h Lemma 3.4.2 The operator S : H01 → H01 is compact and dim Im S = 1. The operator S : H01 → H01 is nonnegative. There exists only one nonzero eigenvalue λ∗1 (S) = h with the corresponding eigenfunction (eigenvector) u ∗1 (x) = x. Lemma 3.4.3 There exists a  λ ∈ R such that operator N( λ) is continuously invertλ) is ible, i.e., the resolvent set ρ(N) := {λ : ∃ N−1 (λ) : H01 → H01 } of OVF N( nonempty, ρ(N) = ∅.  and S that for Proof Let λ ∈ R. It follows from the properties of operators K, K,  λ>0  u) ≥ u 2 λ(Su, u) + u 2 − (Ku, Re(N( λ)u, u) =  λ2 (Ku, u) +  for each u. Hence  λ ∈ ρ(N), where ρ(N) is the resolvent set of operator N. Theorem 3.4.1 The operator pencil N(λ) : holomorphic on C.

H01

→

H01



is bounded, Fredholm, and

Proof We obtain the desired result by the application of the above lemmas.



→ is discrete on Theorem 3.4.2 The spectrum of operator pencil N(λ) : C, i.e., this OVF has a finite set of characteristic numbers with finite algebraic multiplicity on any compact set K 0 ⊂ C. H01

H01

Proof The assertion of the theorem is a corollary of Theorem 3.4.1 and a theorem on a holomorphic operator function (see [29], Theorem 5.1). 

3.4.4 Completeness of System of Eigenvectors and Associated Vectors of Operator Pencil First, let us analyze the quadratic form of the operator pencil 2 u, u) − los (Ku, u). (3.4.15) (N(λ)u, u) = λ2 (Ku, u) + λ(Su, u) + (u, u) − (K

Let λ = λ + iλ and (u, u) = 1. Then, the real and imaginary parts of quadratic form (3.4.15) are u, u), Re(N(λ)u, u) = ((λ )2 − (λ )2 )(Ku, u) + λ (Su, u) + 1 − (K

(3.4.16)

2 (Ku, u). Im(N(λ)u, u) = λ (2λ (Ku, u) + (Su, u)) − los

(3.4.17)

  2 Denote λ0 := max0≤x≤h |(x)|, 1 := λ : λ > 0, λ λ > los , 2 := {λ : λ = 2 0, λ > λ0 }.

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Theorem 3.4.3 1 ∪ 2 ⊂ ρ(N ). Proof From (3.4.16) and (3.4.17) we have that (N(λ)u, u) = 0. Taking into account Theorem 3.4.1 we conclude that bounded operator N −1 (λ) exists for λ ∈ 1 ∪ 2 .  Thus we have the following propositions: Corollary 3.4.4 There are no complex surface TE waves. Corollary 3.4.5 There is not more than a finite number of propagating surface TE waves. It was recently shown in [30, 31] that there are finitely many surface waves in an open dielectric layer. Since characteristic numbers of the pencil have finite algebraic multiplicity we should unite the surface and leaky waves to obtain an infinite set of normal waves providing completeness of the system. Introduce operators K, P, and R.   Denote μk := π(k + 1/2)/ h. We have u k (x)= sin μk x, u k 2 =μ2k h2 + sin4μ2μk k h . Taking into account Lemma 3.4.1 we obtain that the system of vectors {ek }∞ k=0 , ek (x) = u k (x)/ u k , is an orthogonal normalized basis in H01 (0, h): (ek , el ) = δkl , where δkl is the Kronecker delta. Let K1/2 ϕ =

∞ 

1/2

λk (ϕ, ek )ek

k=0

for ϕ =

∞ 

(ϕ, ek )ek .

k=0

A one-dimensional projector P is determined by the formula Pϕ = (ϕ, e0 )e0 for ϕ =

∞ 

(ϕ, ek )ek .

k=0

Let V1 := {u ∈ H01 (0, h) : u(x) = Cu ∗1 (x), C ∈ C}, V = V1 ⊕ V1⊥ . Since u ∗1 2 = h, we have that ψ1 = u ∗1 (x)/ u ∗1 is a normalized eigenvector of operator S. Vector e0 can be decomposed into the orthogonal sum of vectors e0 = (e0 , ψ1 )ψ1 + ψ1⊥ , where ψ1⊥ := e0 − (e0 , ψ1 )ψ1 and ψ1 ∈ V1 , ψ1⊥ ∈ V1⊥ . Taking into account that (ψ1 , e0 ) = ( u 0 u ∗1 μ20 )−1 = 0, we determine the onedimensional projector −1/2

Rϕ = (ϕ, e0 ) for ϕ =

∞  k=0

(ϕ, ek )ek .

−1/2

λ∗1 λ0 λ∗1 λ ψ1 − (ϕ, ψ1⊥ ) 0 ψ1 (ψ1 , e0 ) (ψ1 , e0 )

3.4 TE-Polarized Waves in an Inhomogeneous Partially Shielded Lossy Dielectric Layer

89

Define the operator T = RP. It is easy to check that S = TK1/2 . Indeed, TK1/2 ψ1 = and TK1/2 ϕ = 0 for ϕ ∈ V1⊥ . Thus, Sϕ = λ∗1 (ϕ, ψ1 )ψ1 = ϕ(h)x. Hence, we can rewrite pencil (3.4.13) in the form

λ∗1 ψ1

  2  − los K u = 0, N(λ)u := λ2 (K1/2 )2 + λTK1/2 + I − K

(3.4.18)

where K1/2 is a positive compact operator of order 2 (with eigenvalues λn (K1/2 ) ∼ h  are compact operators. Thus, all conditions of the Keldysh , n → ∞) and T, K πn theorem (see, [29], Theorem 9.1) are fulfilled and we obtain the following statement. Theorem 3.4.6 The system of eigen- and associated vectors of pencil N(λ) is double complete in H01 (0, h) and all characteristic numbers are located in the angles πm/2 − δ < argλ < πm/2 + δ for arbitrary δ > 0, except maybe for finite set of characteristic numbers; m = 0, 1, 2, 3.

R

Definitions of the associated vectors of the pencil can be found in [29] and of the corresponding eigenwaves and associated waves in [32].

Corollary 3.4.7 There are infinitely many complex leaky TE waves. Proof Consider the case Re λ = 0 and Im λ = 0. It follows from (3.4.17) that (Su, u) = 0 and u(h) = 0. Taking into account (3.4.11) we obtain the Cauchy problem for the linear homogeneous differential equation (3.4.8) with zero initial conditions u(h) = u  (h) = 0. Hence, we have u(x) = 0, 0 ≤ x ≤ h. Thus {λ : Re λ = 0, Im λ = 0} ⊂ ρ(L). Then the statement follows from Theorems 3.4.3 and 3.4.6. 

3.5 TM-Polarized Waves in an Inhomogeneous Partially Shielded Dielectric Layer 3.5.1 Statement of the Problem Consider the three-dimensional space R3 equipped with the Cartesian coordinate system O x yz and filled with an isotropic source-free medium having permittivity εc ε0 ≡ const and permeability μ0 ≡ const, where ε0 and μ0 are permittivity and permeability of vacuum. We consider electromagnetic waves propagating through a partially shielded dielectric layer:  := {(x, y, z) : 0  x  h} .

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3 Planar Waveguide

The boundaries x = h and x = 0 are the projections of, respectively, the dielectric surface and the perfectly conducting screen. Determination of surface TM-polarized waves reduces to finding nontrivial running-wave solutions of the homogeneous system of Maxwell equations depending on the coordinate z along which the structure is regular in the form eiγz , 

curl H = −iωε0 εE, ˜ curl E = iωμ0 H,

(3.5.70)

    E = E x (x)eiγz , 0, E z (x)eiγz , H = 0, Hy (x)eiγz , 0 , with the boundary condition for the tangential electric field component on the perfectly conducting screen (3.5.71) E z |x=0 = 0, and the transmission conditions for the tangential electric and magnetic field components on the permittivity discontinuity surface (x = h)  [E z ]|x=h = 0, [Hy ]x=h = 0, where [ f ]|x0 = lim

x→x0 −0

f (x) − lim

x→x0 +0

(3.5.72)

f (x). We will not formulate the radiation con-

dition at infinity because we want to consider the problem for arbitrary γ. Assume that the relative permittivity in the entire space has the form   ε=

ε(x), 0 ≤ x ≤ h, x > h, εc ,

(3.5.73)

where ε(x) > εc is continuous functions on the segment [0, h], i.e., ε(x) ∈ C[0, h] and Im ε(x) = 0. The problem on surface waves is an eigenvalue problem for the Maxwell equations with spectral parameter γ, which is the propagation constant of the running wave. Rewrite system (3.5.70) in the expanded form: ⎧  ⎪ ⎨ iγ E x − E z = iωμ0 Hy , −iγ Hy = iωε0 εE ˜ x, ⎪ ⎩  Hy = iωε0 εE ˜ z,

(3.5.74)

and express functions E x , E z via Hy from system (3.5.74) Ex = −

γ 1 Hy , E z = −i H. ωε0 ε˜ ωε0 ε˜ y

(3.5.75)

3.5 TM-Polarized Waves in an Inhomogeneous Partially Shielded Dielectric Layer

91

It follows from (3.5.75) that the normal wave field in the waveguide can be represented with the use of one scalar function u := Hy (x).

(3.5.76)

Thus, the problem is reduced to finding tangential component u of the magnetic field. Throughout the text below, ( · ) stands for differentiation with respect to x. We have the following eigenvalue problem for the tangential magnetic field component u: find γ ∈ C such that there exist nontrivial solutions of the differential equation   ε − γ 2 u = 0, (3.5.77) u  + ω 2 μ0 ε0 satisfying the boundary conditions  u  x=0 = 0,

(3.5.78)

and the transmission conditions  [u]|x=h = 0,

 u   = 0.  ε x=h

(3.5.79)

Here, conditions (3.5.78) and (3.5.79) are obtained from representations (3.5.75). We will not formulate the radiation condition at infinity because we want to consider the problem for arbitrary γ. For x > h, we have  ε = εc ; then from (3.5.77) we obtain the equation u  − λ2 u = 0.

(3.5.80)

We choose a solution of this equation in the form u(x; λ) = Ce(h−x)λ , x > h,

(3.5.81)

where C is an unknown constant, λ2 = γ 2 − ω 2 ε0 μ0 εc , and λ is a new (complex) spectral parameter. If Re λ > 0 then we have a surface wave and if Re λ < 0 a leaky wave. For 0 < x < h, we have  ε = ε(x); thus from (3.5.77) we obtain the equation   Lu := u  + 2 − λ2 u = 0,

(3.5.82)

2 (x; ω) = ω 2 ε0 μ0 (ε(x) − εc ) .

(3.5.83)

where Definition 3.5.1 λ ∈ C is called a characteristic number of the problem if there exists nontrivial solution u of equation (3.5.82) for 0 < x < h satisfying (3.5.81) for x > h, boundary condition (3.5.78), and transmission conditions (3.5.79).

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3 Planar Waveguide

3.5.2

Sobolev Spaces and Variational Relation

Let us give a variational formulation of the problem under consideration. Multiplying (3.5.82) by an arbitrary test function v ∈ H 1 (0, h) (preliminarily we can assume that this function is continuously differentiable on [0, h]) and applying Green’s formula (where boundary condition (3.5.78) is taking into account), we obtain h

h vLudρ =

0

h



u vdρ + 0

h  uvdρ − λ 2

0

h = u v 0 − 

2

uvdρ 0

h

h

 

u v dρ + 0

h  uvdρ − λ 2

2

0

h



m = u (h)v(h) −

 

h

u v dρ + 0

uvdρ 0

h  uvdρ − λ 2

0

uvdρ = 0.

2 0

(3.5.84) Using (3.5.81), we express the values of the derivatives at x = h from relations (3.5.79) as follows: (3.5.85) u  (h) = −λχ2 u(h), where χ2 =

ε(h) . Then, in view of (3.5.84) and (3.5.85), we obtain εc

h λ

h uvdρ + λχ u(h)v(h) +

2

2

0

0

    u v + uv dρ −

h

 2   + 1 uvdρ = 0.

0

(3.5.86) Variational relation (3.5.86) has been obtained for smooth function v. Consider the following sesquilinear forms and the corresponding operators: h k(u, v) :=

uvdρ = (Ku, v), ∀v ∈ H 1 , 0

 k(u, v) :=

h

 2  u, v), ∀v ∈ H 1 ,  + 1 uvdρ = (K

0

h a(u, v) := 0

    u v + uv dρ = (Iu, v), ∀v ∈ H 1 ,

3.5 TM-Polarized Waves in an Inhomogeneous Partially Shielded Dielectric Layer

93

s(u, v) = u(h)v(h) = (Su, v), ∀v ∈ H 1 . The boundedness of k(u, v) and  k(u, v) follows from the Poincarè inequality [28]. The boundedness of form k(u, v) will be shown in the next section. Now variational problem (3.5.86) can be written in the operator form (N(λ)u, v) = 0, ∀v ∈ H 1 or, equivalently,

   u = 0. N(λ)u := λ2 K + λχ2 S + I − K

(3.5.87)

Equation (3.5.87) is the operator form of variational relation (3.5.86). The characteristic numbers and eigenvectors of operator N by definition coincide with the eigenvalues and eigenvectors of the problem under consideration.

3.5.3 Properties of the Operator Pencil We have reduced the problem on surface TM-polarized waves to the study of spectral properties of operator pencil N. In this connection, we first consider the properties of the operators introduced in the preceding section.  : H 1 → H 1 are compact, and K > Lemma 3.5.1 The bounded operators K, K  > 0. Eigenvalues and eigenfunctions of operator K are λn (K) = 0, K (n ≥ 0). Asymptotics of eigenvalues of oper((h 2 + π 2 n 2 )/ h 2 )−1 , u n (x) = cos πnx h h2 ator K is λn (K) ∼ h 2 +π2 n 2 , n → ∞. Proof Since the embedding H 1 (0, h) ⊂ L 2 (0, h) is compact, it follows that the  : H 1 → H 1 are compact. Further, for u = 0 (as element of operators K and K 1 H (0, h)) h (Ku, u) = |u|2 dρ > 0 0

and  u) = (Ku,

h

 2   + 1 |u|2 dρ > 0;

0

 > 0. consequently, K > 0, K The eigenvalue problem for operator K is as follows:

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3 Planar Waveguide

⎧ ⎨ u  (x) = − 1 − λ(K) u(x), λ(K) ⎩  u (0) = 0, u  (h) = 0.

(3.5.88)

Eigenvalues and eigenfunctions (eigenvectors) of this problem are λn (K) = (n ≥ 0).  ((h 2 + π 2 n 2 )/ h 2 )−1 , u n (x) = cos πnx h Lemma 3.5.2 The operator S : H 1 → H 1 is compact and dim Im S = 1. The operator S : H 1 → H 1 is nonnegative. There exists only one nonzero eigenvalue λ∗1 (S) = h with the corresponding eigenfunction (eigenvector) u ∗1 (x) = x. Proof Consider the form h s(u, v) := u(h)v(h) =





h

s (x)v (x)d x + 0

s(x)v(x)d x,

(3.5.89)

0

where s(x) = Su. Let v = 0 in a neighborhood of the point h; then h 0 = s (x)v(x)0 − 

h

h



s (x)v(x)d x + 0

or

h

s(x)v(x)d x, 0

(s  (x) − s(x))v(x)d x = 0.

(3.5.90)

0

Since function v(x) is arbitrary, it follows from (3.5.90) that s  (x) − s(x) = 0, 0 < x < h.

(3.5.91)

Let v = 0 at the point h. From (3.5.89) we get s  (0) = 0, s  (h) = u(h).

(3.5.92)

Thus we obtain the boundary value problem 

s  (x) − s(x) = 0, s  (0) = 0, s  (h) = u(h).

Then s(x) =

u(h) cosh x , sinh h

(3.5.93)

3.5 TM-Polarized Waves in an Inhomogeneous Partially Shielded Dielectric Layer

95

dim Im S = 1. Similarly we get that nonzero eigenvalue of operator S is λ∗1 (S) = coth h with the corresponding eigenfunction (eigenvector) u ∗1 (x) = cosh x. The proof of the lemma is complete.  Lemma 3.5.3 There exists a  λ ∈ R such that the operator N( λ) is continuously invertible, i.e., the resolvent set ρ(N) := {λ : ∃ N−1 (λ) : H 1 → H 1 } of the operator function N( λ) is nonempty, ρ(N) = ∅. Proof Let λ ∈ R. Since the function 2 belongs to the space C 2 [−h, h], it follows  and S that for  from the properties of operators K, K, λ > max0≤x≤h |(x)| u, u) ≥ u 2 λχ2 (Su, u) + u 2 − (K (N( λ)u, u) =  λ2 (Ku, u) +  for each u. Hence  λ ∈ ρ(N), where ρ(N) is the resolvent set of operator N.



Theorem 3.5.1 The operator pencil N(λ) : H → H is bounded, Fredholm, and holomorphic on C. 1

1



Proof We obtain the desired result by application of above lemmas.

Theorem 3.5.2 The spectrum of the operator pencil N(λ) : H → H is discrete on C, i.e., this function has a finite set of characteristic numbers with finite algebraic multiplicity on any compact set K 0 ⊂ C. 1

1

Proof The assertion of the theorem is a corollary of Theorem 3.5.1 and a theorem on a holomorphic operator function (see [29], Theorem 5.1). 

3.5.4 Completeness of System of Eigenvectors and Associated Vectors of Operator Pencil First, let us analyze the quadratic form of the operator pencil u, u). (N(λ)u, u) = λ2 (Ku, u) + λχ2 (Su, u) + (u, u) − (K

(3.5.94)

Let λ = λ + iλ and (u, u) = 1. Then, the real and imaginary parts of quadratic form (3.5.94) are  u), (3.5.95) Re(N(λ)u, u) = ((λ )2 − (λ )2 )(Ku, u) + λ χ2 (Su, u) + 1 − (Ku, Im(N(λ)u, u) = λ (2λ (Ku, u) + χ2 (Su, u)).

(3.5.96)

Denote max := max0≤x≤h |(x)|, 1 := {λ : λ = 0, λ > 0}, 2 := {λ : λ = 0, λ > max }.

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3 Planar Waveguide

Theorem 3.5.3 1 ∪ 2 ⊂ ρ(N ). Proof From (3.5.95) and (3.5.96) we have that (N(λ)u, u) = 0. Taking into account Theorem 3.5.1 we conclude that bounded operator N−1 (λ) exists for λ ∈ 1 ∪ 2 .  Thus we have the following propositions: Corollary 3.5.4 There are no complex surface TM waves. Corollary 3.5.5 There is not more than a finite number of propagating surface TM waves. It was recently shown in [30, 31] that there are finitely many surface waves in an open dielectric layer. Since characteristic numbers of the pencil have finite algebraic multiplicity we should unite the sets of surface and leaky waves to obtain an infinite set of normal waves providing completeness of the system. Introduce operators K, P, and R.  h 1 + μ2k . Taking Denote μk := πk/ h. We have u k (x) = cos μk x, u k 2 = 2 into account Lemma 3.5.1 we obtain that the system of vectors {ek }∞ k=0 , ek (x) = u k (x)/ u k forms an orthogonal normalized basis in H 1 (0, h): (ek , el ) = δkl , where δkl is the Kronecker delta. Let ∞  1/2 1/2 K ϕ= λk (ϕ, ek )ek k=0

for ϕ =

∞ 

(ϕ, ek )ek .

k=0

A one-dimensional projector P is determined by the formula Pϕ = (ϕ, e0 )e0 for ϕ =

∞ 

(ϕ, ek )ek .

k=0

V1 := {u ∈ H 1 (0, h) : u(x) = Cu ∗1 (x), C ∈ C}, V = V1 ⊕ V1⊥ . Since sinh 2h u ∗1 2 = , we have that ψ1 = u ∗1 (x)/ u ∗1 is a normalized eigenvector of 2 operator S. Vector e0 can be decomposed into the orthogonal sum of vectors e0 = (e0 , ψ1 )ψ1 + ψ1⊥ , where ψ1⊥ := e0 − (e0 , ψ1 )ψ1 ; ψ1 ∈ V1 , ψ1⊥ ∈ V1⊥ . sinh h Taking into account that (ψ1 , e0 ) = = 0, we determine the one u 0 u ∗1 -dimensional projector Set

−1/2

Rϕ = (ϕ, e0 )

−1/2

max λ∗1 max λ∗1 ψ1 − (ϕ, ψ1⊥ ) ψ1 (ψ1 , e0 ) (ψ1 , e0 )

3.5 TM-Polarized Waves in an Inhomogeneous Partially Shielded Dielectric Layer

for ϕ =

∞ 

97

(ϕ, ek )ek .

k=0

Define the operator T = RP. It is easy to check that S = TK1/2 . Indeed, TK1/2 ψ1 = and TK1/2 ϕ = 0 for ϕ ∈ V1⊥ . Thus, Sϕ = λ∗1 (ϕ, ψ1 )ψ1 = ϕ(h)x. Hence, we can rewrite pencil (3.5.87) in the form

λ∗1 ψ1

   u = 0, N(λ)u := λ2 (K1/2 )2 + λχ2 TK1/2 + I − K

(3.5.97)

where K1/2 is a positive compact operator of order 2 (with eigenvalues λn (K1/2 ) ∼  are compact operators. Thus, all conditions of the √ h , n → ∞) and T and K π 2 n 2 +h 2 Keldysh theorem (see, [29] Theorem 9.1) are fulfilled and we obtain the following proposition. Theorem 3.5.6 System of eigen- and associated vectors of pencil N(λ) is double complete in H 1 (0, h) and all characteristic numbers are located in the angles πm/2 − δ < argλ < πm/2 + δ for arbitrary δ > 0, except maybe for a finite set of characteristic numbers; m = 0, 1, 2, 3.

R

Definitions of the associated vectors of the pencil can be found in [29] and of the corresponding eigenwaves and associated waves in [32].

Corollary 3.5.7 There are infinitely many complex leaky TM waves. Proof Consider the case Re λ = 0 and Im λ = 0. It follows from (3.5.96) that (Su, u) = 0 and u(h) = 0. Taking into account (3.5.85) we obtain the Cauchy problem for the linear homogeneous differential equation (3.5.82) with zero initial conditions u(h) = u  (h) = 0. Hence, we have u(x) = 0, 0 ≤ x ≤ h. Thus {λ : Re λ = 0, Im λ = 0} ⊂ ρ(L). Then, the statement follows from Theorems 3.5.3 and 3.5.6. 

3.6 TM-Polarized Waves in an Inhomogeneous Metamaterial Partially Shielded Layer 3.6.1 Statement of the Problem Consider electromagnetic waves propagating through a partially shielded dielectric layer:  := {(x, y, z) : 0  x  h} .

98

3 Planar Waveguide

The boundaries x = h and x = 0 are the projections of, respectively, the dielectric surface and the perfectly conducting screen. Determination of surface TM-polarized waves reduces to finding nontrivial running-wave solutions of the homogeneous system of Maxwell equations depending on the coordinate z along which the structure is regular in the form eiγz . Assume that the relative permittivity in the entire space has the form   ε=

−ε2 , 0 ≤ x ≤ h, x > h, εc ,

(3.6.116)

where ε2 (x) > εc are continuous functions on the segment [0, h], i.e., ε(x) ∈ C[0, h] and Im ε(x) = 0. The problem on surface waves is an eigenvalue problem for the Maxwell equations with spectral parameter γ, which is the propagation constant of the running wave. The normal wave field in the waveguide can be represented using one scalar function (3.6.117) u := Hy (x). Thus, the problem is reduced to finding tangential component u of the electric field. Throughout the text below, ( · ) stands for the differentiation with respect to x. We have the following eigenvalue problem for the tangential electric field component u: find γ ∈ C such that there exist nontrivial solutions of the differential equation   ε − γ 2 u = 0, (3.6.118) u  + ω 2 μ0 ε0 satisfying the boundary conditions  u  x=0 = 0,

(3.6.119)

and the transmission conditions  [u]|x=h = 0,

 u   = 0.  ε x=h

(3.6.120)

We will not formulate the radiation condition at infinity because we want to consider the problem for arbitrary γ. For x > h, we have  ε = εc ; then from (3.6.118) we obtain the equation u  − λ2 u = 0.

(3.6.121)

We choose a solution of this equation in the form u(x; λ) = e(h−x)λ , x > h,

(3.6.122)

3.6 TM-Polarized Waves in an Inhomogeneous Metamaterial Partially Shielded Layer

99

where λ2 = γ 2 − ω 2 ε0 μ0 εc and λ is a new (complex) spectral parameter. If Re λ > 0 then we have a surface wave and if Re λ < 0 a leaky wave. For 0 < x < h, we have  ε = ε(x); thus from (3.6.118) we obtain the equation

where

  Lu := u  − 2 + λ2 u = 0,

(3.6.123)

  2 (x; ω) = ω 2 ε0 μ0 ε2 (x) + εc .

(3.6.124)

Definition 3.6.1 λ ∈ C is called a characteristic number of the problem if there exists nontrivial solution u of equation (3.6.123) for 0 < x < h satisfying (3.6.122) for x > h, boundary condition (3.6.119), and transmission conditions (3.6.120).

3.6.2 Sobolev Spaces and Variational Relation Let us give a variational formulation of the problem under consideration. Multiplying (3.6.123) by arbitrary test function v ∈ H 1 (0, h) (preliminarily we can assume that this function is continuously differentiable on [0, h]) and applying Green’s formula, we obtain h

h vLudρ =

0

h



u vdρ − 0

h  uvdρ − λ 2

2

0

h = u v 0 − 

h

h

 

u v dρ − 0



uvdρ 0

h  uvdρ − λ 2

0

h

= u (h)v(h) −

 

uvdρ 0

h

u v dρ − 0

2

h  uvdρ − λ 2

uvdρ = 0.

2

0

0

(3.6.125) Using (3.6.122), we express the values of the derivatives at x = h from relations (3.6.120) as follows: u  (h) = λχ2 u(h), (3.6.126) where χ2 =

ε2 (h) > 1. Then, in view of (3.6.125) and (3.6.126), we obtain εc

h λ

h uvdρ − λχ u(h)v(h) +

2 0

 

h

u v dρ +

2

0

2 uvdρ = 0. 0

(3.6.127)

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3 Planar Waveguide

Variational relation (3.6.127) has been obtained for smooth function v. Consider the following sesquilinear forms and the corresponding operators: h k(u, v) :=

uvdρ = (Ku, v), ∀v ∈ H 1 , 0

 k(u, v) :=

h

 v), ∀v ∈ H 1 , 2 uvdρ = (Ku,

0

h a(u, v) :=

u  v  dρ = (Iu, v), ∀v ∈ H 1 ,

0

s(u, v) = u(h)v(h) = (Su, v), ∀v ∈ H 1 . The boundedness of k(u, v) and  k(u, v) follows from the Poincarè inequality. The boundedness of form k(u, v) will be shown in the next section. Now variational problem (3.6.127) can be written in the operator form (N(λ)u, v) = 0, ∀v ∈ H 1 or, equivalently,

   u = 0. N(λ)u := λ2 K − λχ2 S + I + K

(3.6.128)

Equation (3.6.128) is the operator form of variational relation (3.6.127). The characteristic numbers and eigenvectors of operator N by definition coincide with the eigenvalues and eigenvectors of the problem under consideration.

3.6.3 Properties of the Operator Pencil We have reduced the problem on surface TM-polarized waves to the study of spectral properties of operator pencil N. In this connection, we first consider the properties of the operators introduced in the preceding section.  : H 1 → H 1 are compact, and K > Lemma 3.6.1 The bounded operators K, K  0, K > 0. Eigenvalues and eigenfunctions of operator K are λn (K) = (n ≥ 0). Asymptotics of eigenvalues of oper((h 2 + π 2 n 2 )/ h 2 )−1 , u n (x) = cos πnx h h2 ator K is λn (K) ∼ h 2 +π2 n 2 , n → ∞.

3.6 TM-Polarized Waves in an Inhomogeneous Metamaterial Partially Shielded Layer

101

Lemma 3.6.2 The operator S : H 1 → H 1 is compact and dim Im S = 1. The operator S : H 1 → H 1 is nonnegative. There exists only one nonzero eigenvalue λ∗1 (S) = h with the corresponding eigenfunction (eigenvector) u ∗1 (x) = x. Lemma 3.6.3 There exists a  λ ∈ R such that operator N( λ) is continuously invertible, i.e., the resolvent set ρ(N) := {λ : ∃ N−1 (λ) : H 1 → H 1 } of OVF N( λ) is nonempty, ρ(N) = ∅.  and S that for Proof Let λ ∈ R. It follows from the properties of operators K, K,  λ 0, except maybe for a finite set of characteristic numbers; m = 0, 1, 2, 3.

R

Definitions of the associated vectors of the pencil can be found in [29] and of the eigenwaves and associated waves in [32].

Corollary 3.6.7 There are infinitely many complex leaky TM waves. Proof Consider the case Re λ = 0 and Im λ = 0. It follows from (3.6.131) that (Su, u) = 0 and u(h) = 0. Taking into account (3.6.126) we obtain the Cauchy problem for the linear homogeneous differential equation (3.6.123) with zero initial conditions u(h) = u  (h) = 0. Hence, we have u(x) = 0, 0 ≤ x ≤ h. Thus {λ : Re λ = 0, Im λ = 0} ⊂ ρ(L). Then the statement follows from Theorems 3.6.3 and 3.6.6. 

3.7 TM-Polarized Waves in an Inhomogeneous Partially Shielded Lossy Dielectric Layer 3.7.1 Statement of the Problem Consider electromagnetic waves propagating through a partially shielded dielectric layer:  := {(x, y, z) : 0  x  h} .

104

3 Planar Waveguide

The boundaries x = h and x = 0 are the projections of, respectively, the dielectric surface and the perfectly conducting screen. Determination of surface TM-polarized waves reduces to finding nontrivial running-wave solutions of the homogeneous system of Maxwell equations depending on the coordinate z along which the structure is regular in the form eiγz . We assume that the relative permittivity in the entire space has the form   ε=

ε + iεlos , 0 ≤ x ≤ h, x > h, εc ,

(3.7.133)

where ε(x) > εc are continuous functions on the segment [0, h], i.e., ε(x) ∈ C[0, h], Im ε(x) = 0, and εlos > is a real positive constant. The problem on surface waves is an eigenvalue problem for the Maxwell equations with spectral parameter γ, which is the propagation constant of the running wave. The normal wave field in the waveguide can be represented using one scalar function (3.7.134) u := Hy (x). Thus, the problem is reduced to finding tangential component u of the electric field. Throughout the text below, ( · ) stands for the differentiation with respect to x. We have the following eigenvalue problem for the tangential electric field component u: find γ ∈ C such that there exist nontrivial solutions of the differential equation   (3.7.135) ε − γ 2 u = 0, u  + ω 2 μ0 ε0 satisfying the boundary conditions  u  x=0 = 0,

(3.7.136)

and the transmission conditions  [u]|x=h = 0,

 u   = 0.  ε x=h

(3.7.137)

For x > h, we have  ε = εc ; then from (3.7.135) we obtain the equation u  − λ2 u = 0.

(3.7.138)

We choose a solution of this equation in the form u(x; λ) = e(h−x)λ , x > h,

(3.7.139)

where λ2 = γ 2 − ω 2 ε0 μ0 εc and λ is a new (complex) spectral parameter. If Re λ > 0 then we have a surface wave and if Re λ < 0 a leaky wave. For 0 < x < h, we have  ε = ε(x); thus from (3.7.135) we obtain the equation

3.7 TM-Polarized Waves in an Inhomogeneous Partially Shielded Lossy Dielectric Layer

  2 Lu := u  − 2 + ilos − λ2 u = 0,

105

(3.7.140)

where 2 (x; ω) = ω 2 ε0 μ0 εlos . 2 (x; ω) = ω 2 ε0 μ0 (ε(x) − εc ) and los

(3.7.141)

Definition 3.7.1 λ ∈ C is called a characteristic number of the problem if there exists nontrivial solution u of equation (3.7.140) for 0 < x < h satisfying (3.7.139) for x > h, boundary condition (3.7.136), and transmission conditions (3.7.137).

3.7.2 Sobolev Spaces and Variational Relation We will look for solutions u in Sobolev space H 1 (0, h) , with the inner product and the norm h h   2 f g dρ, f 1 = ( f, f )1 = | f  |2 dρ, ( f, g)1 = 0

0

where

h H (0, h) := { f : 1

| f  |2 dρ < ∞}.

0

Let us give a variational formulation of the problem under consideration. Multiplying (3.7.140) by arbitrary test function v ∈ H 1 (0, h) (preliminarily we can assume that this function is continuously differentiable on [0, h]) and applying Green’s formula, we obtain h

h v Ludρ =

0

h



u vdρ + 0

h 

2

uvdρ − λ

0

h = u v 0 − 

2

0

h

u v dρ +

h 

2

2 uvdρ + ilos

0

h

= u (h)v(h) −

 

h uvdρ − λ

0

h

u v dρ + 0

uvdρ 0

h

 

0 

h

2 uvdρ + ilos

uvdρ 0

h 

2

0

2

2 uvdρ + ilos

h uvdρ − λ

0

uvdρ = 0.

2 0

(3.7.142) Using (3.7.139), we express the values of the derivatives at x = h from relations (3.7.137) as follows: u  (h) = −λχ2 u(h), (3.7.143)

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3 Planar Waveguide

ε(h) + iεlos ε(h) , Re χ = > 0. εc εc Then, in view of (3.7.142) and (3.7.143), we obtain

where χ =

h λ

h uvdρ − λχu(h)v(h) +

2

 

h

u v dρ −

0

0

h  uvdρ − 2

uvdρ = 0.

2 ilos

0

0

(3.7.144) Variational relation (3.7.144) has been obtained for smooth function v. The integrals occurring in (3.7.144) can be viewed as sesquilinear forms over the field C defined H 1 and depending on arguments u and v. These forms t define some bounded linear operators T : H 1 → H 1 by formula [33] t(u, v) = (Tu, v), ∀v ∈ H 1 ,

(3.7.145)

provided that the forms themselves are bounded, |t(u, v)| ≤ C u v . The linearity follows from the linearity of the form in the first argument, and the continuity follows from the estimates Tu 2 = t(u, Tu) ≤ C u Tu . Consider the following sesquilinear forms and the corresponding operators: h k(u, v) :=

uvdρ = (Ku, v), ∀v ∈ H 1 , 0

 k(u, v) :=

h

 v), ∀v ∈ H 1 , 2 uvdρ = (Ku,

0

h a(u, v) :=

u  v  dρ = (Iu, v), ∀v ∈ H 1 ,

0

s(u, v) = u(h)v(h) = (Su, v), ∀v ∈ H 1 . The boundedness of k(u, v) and  k(u, v) follows from the Poincarè inequality. The boundedness of form k(u, v) will be shown in the next section. Now variational problem (3.7.144) can be written in the operator form

3.7 TM-Polarized Waves in an Inhomogeneous Partially Shielded Lossy Dielectric Layer

107

(N(λ)u, v) = 0, ∀v ∈ H 1 or, equivalently,   2  − ilos K u = 0. N(λ)u := λ2 K + λχS + I − K

(3.7.146)

Equation (3.7.146) is the operator form of variational relation (3.7.144). The characteristic numbers and eigenvectors of operator N by definition coincide with the eigenvalues and eigenvectors of the problem under consideration.

3.7.3 Properties of the Operator Pencil We have reduced the problem on surface TM-polarized waves to the study of spectral properties of operator pencil N. In this connection, we first consider the properties of the operators introduced in the preceding section.  : H 1 → H 1 are compact, and K > Lemma 3.7.1 The bounded operators K, K  > 0. Eigenvalues and eigenfunctions of operator K are λn (K) = 0, K (n ≥ 0). Asymptotics of eigenvalues of oper((h 2 + π 2 n 2 )/ h 2 )−1 , u n (x) = cos πnx h h2 ator K is λn (K) ∼ h 2 +π2 n 2 , n → ∞. Lemma 3.7.2 The operator S : H 1 → H 1 is compact and dim Im S = 1. The operator S : H 1 → H 1 is nonnegative. There exists only one nonzero eigenvalue λ∗1 (S) = h with corresponding eigenfunction (eigenvector) u ∗1 (x) = x. Lemma 3.7.3 There exists a  λ ∈ R such that operator N( λ) is continuously invertible, i.e., the resolvent set ρ(N) := {λ : ∃ N−1 (λ) : H 1 → H 1 } of OVF N( λ) is nonempty, ρ(N) = ∅. , and S that for Proof Let λ ∈ R. It follows from the properties of operators K, K  λ 0, except maybe for a finite set of characteristic numbers; m = 0, 1, 2, 3.

R

Definitions of the associated vectors of the pencil can be found in [29] and of the eigenwaves and associated waves in [32].

Corollary 3.7.7 There are infinitely many complex leaky TM waves.

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3 Planar Waveguide

Proof Consider the case Re λ = 0 and Im λ = 0. It follows from (3.7.149) that (Su, u) = 0 and u(h) = 0. Taking into account (3.7.143) we obtain the Cauchy problem for the linear homogeneous differential equation (3.7.140) with zero initial conditions u(h) = u  (h) = 0. Hence, we have u(x) = 0, 0 ≤ x ≤ h. Thus {λ : Re λ = 0, Im λ = 0} ⊂ ρ(L). Then the statement follows from Theorems 3.7.3 and 3.7.6. 

3.8 Numerical Simulation 3.8.1 Statement of the Problem We consider electromagnetic waves propagating along the layer  := {(x, y, z) : 0  x  h} . We assume that the fields depend harmonically on time as exp(−iωt), where ω > 0 is the circular frequency and consider electromagnetic fields of the form     E = 0, E y (x)eiγz , 0 , H = Hx (x)eiγz , 0, Hz (x)eiγz , where γ is an unknown complex (spectral) parameter. Determination of surface TE-polarized waves reduces to finding nontrivial running-wave solutions of the homogeneous system of Maxwell equations depending on the coordinate z along which the structure is regular in the form eiγz , 

curl H = −iωε0 εE, ˜ curl E = iωμ0 H,

(3.8.151)

with the transmission conditions for the tangential electric and magnetic field components on the permittivity discontinuity surface (x = 0 and x = h)

and where [ f ]|x0 = lim

x→x0 −0

 [E y ]x=0 = 0, [Hz ]|x=0 = 0,

(3.8.152)

 [E y ]x=h = 0, [Hz ]|x=h = 0,

(3.8.153)

f (x) − lim

x→x0 +0

f (x). We will not formulate the radiation con-

dition at infinity because we want to consider the problem for arbitrary γ. We assume that the relative permittivity in the entire space has the form

3.8 Numerical Simulation

111

⎧ x > h, ⎨ εc ,  ε = ε, 0 ≤ x ≤ h, ⎩ x > h, εc ,

(3.8.154)

where ε is a continuous function on [0, h] and ε(x) > εc , Im ε(x) = 0. The problem on TE waves is an eigenvalue problem for the Maxwell equations with spectral parameter γ which is the propagation constant of the normal wave. The normal wave field in the waveguide can be represented using one scalar function (3.8.155) u := E y (x). Thus, the problem is reduced to finding tangential component u of the electric field. We have the following eigenvalue problem for the tangential electric field component u: find γ ∈ C such that there exist nontrivial solutions of the differential equation   (3.8.156) ε − γ 2 u = 0, u  + ω 2 μ0 ε0 satisfying the transmission conditions

and

 [u]|x=0 = 0, u  x=0 = 0

(3.8.157)

 [u]|x=h = 0, u  x=h = 0.

(3.8.158)

For x > h, we have  ε = εc ; then from (3.8.156) we obtain the equation u  − λ2 u = 0.

(3.8.159)

We choose a solution to this equation in the form  (h−x)λ , x > h, u(x; λ) = Ae

(3.8.160)

 is a where λ2 = γ 2 − ω 2 ε0 μ0 εc and λ is a new (complex) spectral parameter, A constant. If Re λ > 0 we have a surface wave and if Re λ < 0 a leaky wave. For x < 0, we have  ε = εc ; then from (3.8.156) we obtain the equation u  − λ2 u = 0.

(3.8.161)

We choose a solution to this equation in the form u(x; λ) = Ae xλ , x < 0,

(3.8.162)

where A is a constant. For 0 < x < h, we have  ε = ε(x); then from (3.8.156) we obtain the equation

112

3 Planar Waveguide

  u  +  − λ2 u = 0,

(3.8.163)

(x; ω) = ω 2 ε0 μ0 (ε(x) − εc ) .

(3.8.164)

where Definition 3.8.1 γ ∈ C is called a propagation constants of the problem if there exists nontrivial solution u of equation (3.8.164) for 0 < x < h satisfying for x < 0 and x > h (3.8.162) and (3.8.160), respectively, and transmission conditions (3.8.157) and (3.8.158).

3.8.2 Numerical Method Consider the Cauchy problem for the equation   u  +  − λ2 u = 0

(3.8.165)

u (0) := A, u  (0) := Aλ.

(3.8.166)

with the initial conditions

We assume that the Cauchy problem given by (3.8.165) and (3.8.166) is globally and uniquely solvable on the segment [0, h] for a given h and its solution continuously depends on parameter γ. Applying the transmission condition on the boundary x = h, one obtains the dispersion equation (3.8.167)  (λ) ≡ λu(h) + u  (h) = 0. Let λ = α + iβ, where α, β ∈ R. Then equating to zero the real and imaginary parts of  (γ), one obtains a system of equations involving real-valued functions for determining the real and imaginary parts of complex parameter γ: 

1 (α, β) := Re  (λ) = 0, 2 (α, β) := Im  (λ) = 0.

(3.8.168)

We solve numerically system of equations (3.8.168) to determine a number pair (α, β). The solution to each equation of system (3.8.168) is a curve in the Oαβplane. Next, we determine the points of intersection of the curves; these points are approximate eigenvalues of the problem. Introduce a grid

3.8 Numerical Simulation

113

Fig. 3.2 Numerical solution of system (3.8.168): the blue and red curves are, respectively, solution of the first and second equations of (3.8.168); the yellow point is a solution of system (3.8.168)

  α(i) , β ( j) : α(i) = a1 + iτ1 , β ( j) = b1 + iτ2 , i = 0, n, τ1 =

a2 − a1 b2 − b1 , j = 0, m, τ2 = n m

with the steps τ1 > 0 in α and τ2 > 0 in β, where a1 , a2 , b1 , and b2 are real fixed constants. The grid points are used in the implementation of the shooting method below. Solving the Cauchy problem for each grid point, one obtains u(i, j)(r ) and u  (i, j)(r ), i = 0, n, j = 0, m. Since solution u(r ; α, β) is continuously  dependent  !( j) in the plane on parameters α and β, it follows that there exists a point α(i) , β     !( j) ∈ β ( j) , β ( j+1) such that 1 α(i) , β !( j) = 0. The smaller τ1 and Oαβ where β τ2 , the more accurate is the solution. Proceeding in the same manner, one determines a set of pairs (α( p) , β (q) ) that form a curve in the plane Oαβ (the blue curve in Fig. 3.2). Applying the same approach to the second equation of (3.8.168), one obtains another curve in the plane Oαβ (the red curve in Fig. 3.2). This curve is an approximate solution of the equation 2 (α, β) = 0. It is clear that the intersection point of the curves (the yellow point in Fig. 3.2) is an approximate solution of the problem. Decreasing the steps τ1 and τ2 , we can obtain the solutions with the prescribed (arbitrary) accuracy.

114

3 Planar Waveguide

ω = 0.25

ω = 0.5

ω = 0.75

ω = 1.0

Fig. 3.3 Homogeneous dielectric with ε = 4. Numerical solution of system (3.8.168): blue and red curves are solutions of, respectively, the first and the second equations of (3.8.168); green and purple intersection points are, respectively, propagation constants of the propagating surface and leaky TE waves; yellow intersection points are propagation constants of the complex leaky TE waves

3.8.3 Numerical Results Figures 3.3, 3.4, 3.5, 3.6, 3.7 and 3.8 display the calculated propagation constants of the TE-polarized waves in a layer filled with homogeneous and inhomogeneous dielectric, dielectric with losses, and metamaterial. We have carried out numerical experiments for four frequency values. Propagating, evanescent, and complex surface and leaky TE waves are determined.

3.8 Numerical Simulation

115

ω = 0.25

ω = 0.5

ω = 0.75

ω = 1.0

Fig. 3.4 Inhomogeneous dielectric with ε = 4 + hx . Numerical solution of system (3.8.168): blue and red curves are solutions of, respectively, the first and the second equations of (3.8.168); green and purple intersection points are, respectively, propagation constants of the propagating surface and leaky TE waves; yellow intersection points are propagation constants of the complex leaky TE waves

The following values of parameters are used in calculations: εc = 1, ε0 = 1, μ0 = 1, A = 1, h = 4, a1 = −5, a2 = 5, τ1 = 0.05, b1 = −5, b2 = 5, and τ2 = 0.05. Figures 3.3, 3.4, 3.5, 3.6, 3.7, and 3.8 show that there are no complex surface TE waves (no intersection points of blue and red curves in the domain Re γ > 0, Im γ = 0). In Figs. 3.3, 3.4, 3.5, and 3.6 we present the results of calculations for a dielectric and dielectric with losses.

116

3 Planar Waveguide

ω = 0.25

ω = 0.5

ω = 0.75

ω = 1.0

Fig. 3.5 Lossy homogeneous dielectric with ε = 4 + 0.1ωi. Numerical solution of system (3.8.168): blue and red curves are solutions of, respectively, the first and the second equations of (3.8.168); green and purple intersection points are, respectively, propagation constants of the propagating surface and leaky TE waves; yellow intersection points are propagation constants of the complex leaky TE waves

Note that in the case of a metamaterial (see Figs. 3.7 and 3.8), as the frequency increases, the absolute values of the propagation constants corresponding to the complex leaky TE waves increase.

3.8 Numerical Simulation

117

ω = 0.25

ω = 0.5

ω = 0.75

ω = 1.0

Fig. 3.6 Lossy inhomogeneous dielectric with ε = 4 + 0.1ωi. Numerical solution of system (3.8.168): blue and red curves are solutions of, respectively, the first and the second equations of (3.8.168); green and purple intersection points are, respectively, propagation constants of the propagating surface and leaky TE waves; yellow intersection points are propagation constants of the complex leaky TE waves

118

3 Planar Waveguide

ω = 0.25

ω = 0.5

ω = 0.75

ω = 1.0

Fig. 3.7 Homogeneous metamaterial with ε = −4. Numerical solution of system (3.8.168): blue and red curves are solutions of, respectively, the first and the second equations of (3.8.168); yellow intersection points are propagation constants of the complex leaky TE waves

3.8 Numerical Simulation

119

ω = 0.25

ω = 0.5

ω = 0.75

ω = 1.0

√ Fig. 3.8 Inhomogeneous metamaterial with ε = −3 x + 1 − 1. Numerical solution of system (3.8.168): blue and red curves are solutions of, respectively, the first and the second equations of (3.8.168); yellow intersection points are propagation constants of the complex leaky TE waves

120

3 Planar Waveguide

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

14. 15.

16.

17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27.

M.J. Adams, An Introduction to Optical Waveguide (Wiley, New York, 1981) A. Snyder, J. Love, Optical Waveguide Theory (Springer, New York, 1983), p. 735 L.A. Vainshtein Electromagnetic Waves (Sovetskoe Radio, 1957) (in Russian) D. Marcuse Light Transmission Optics (Krieger Pub Co, 1989) V. Shestopalov, Yu. Shestopalov, Spectral Theory and Excitation of Open Structures (IET, London, 1996) G. Hanson, A. Yakovlev, Operator Theory for Electromagnetics: An Introduction (Springer, New York, 2002) W.-P. Yuen, A simple numerical analysis of planar optical waveguides using wave impedance transformation. IEEE Photon. Technol. Lett. 5(8) (1993) R.E. Smith, S.N. Houde-Walter, G.W. Forbes, Mode determination for planar waveguides using the four-sheeted dispersion relation. IEEE J. Quantum Electron. 28, 1520–1526 (1992) A.A. Shishegar, A. Safavi-Naeini, A hybrid analysis method for planar lens-like structures, in IEEE Antennas and Propagation Society International Symposium (1996) R. Collin, Field Theory of Guided Waves (IEEE Press, Piscataway, 1991), p. 852 L.B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (Prentice Hall, Englewood Cliffs, 1973) Y.G. Smirnov, E.Y. Smolkin, Discreteness of the spectrum in the problem on normal waves in an open inhomogeneous waveguide. Differ. Equ. 53(10), 1168–1179 (2018) Y.G. Smirnov, E. Smolkin, M.O. Snegur, Analysis of the spectrum of azimuthally symmetric waves of an open inhomogeneous anisotropic waveguide with longitudinal magnetization. Comput. Math. Math. Phys. 58(11), 1887–1901 (2018) Y.G. Smirnov, E. Smolkin, Operator function method in the problem of normal waves in an inhomogeneous waveguide. Differ. Equ. 54(9), 1262–1273 (2017) Y.G. Smirnov, E. Smolkin, Investigation of the spectrum of the problem of normal waves in a closed regular inhomogeneous dielectric waveguide of arbitrary cross section. Dokl. Math. 97(1), 86–89 (2017) Y.G. Smirnov, Eigenvalue transmission problems describing the propagation of TE and TM waves in two-layered inhomogeneous anisotropic cylindrical and planar waveguides. Comput. Math. Math. Phys. 55(3), 461–469 (2015) G.W. Hanson, A.B. Yakovlev, Investigation of mode interaction on planar dielectric waveguides with loss and gain. Radio Sci. 34, 1349–1359 (1999) A.N. Tikhonov, A.A. Samarskii, On the excitation of radio waves. I. Zh. Tekh. Fiz. 17, 1283– 1296 (1947) A.N. Tikhonov, A.A. Samarskii, On the excitation of radio waves. II. Zh. Tekh. Fiz. 17, 1431– 1440 (1947) A.N. Tikhonov, A.A. Samarskii, On the representation of the field in a waveguide as a sum of the fields TE and TM. Zh. Tekh. Fiz. 18, 959–970 (1948) Y. Shestopalov, Y. Smirnov, Eigenwaves in waveguides with dielectric inclusions: spectrum. App. Anal. 93(2), 408–427 (2014) Y.V. Shestopalov, Y.G. Smirnov, Eigenwaves in waveguides with dielectric inclusions: completeness. Appl. Anal. 93(9), 1824–1845 (2014) Y.V. Shestopalov, E. Kuzmina, On a rigorous proof of the existence of complex waves in a dielectric waveguide of circular cross section. PIER B 82, 137–164 (2018) N. Marcuvitz, On field representations in terms of leaky modes or eigenmodes. IRE Trans. Antennas Propag. 4(3), 192–194 (1956) A.A. Oliner, Leaky waves: basic properties and applications. Proc. Asia-Pacific Microw. Conf. 1, 397–400 (1997) F. Monticone, A. Alu, Leaky-wave theory, techniques, and applications: from microwaves to visible frequencies. Proc. IEEE 103(5), 793–821 (2015) G.A. Korn, T.M. Korn, Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review (General Publishing Company, 2000)

References

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28. R. Adams, Sobolev Spaces (Academic Press, New York, 1975) 29. I. Gohberg, M. Krein, Introduction to the Theory of Linear Nonself-adjoint Operators in Hilbert Space, vol. 18 (American Mathematical Society, 1969) 30. Y.G. Smirnov, E.Y. Smolkin, On the existence of an infinite number of leaky complex waves in a dielectric layer. Dokl. Math. 101(1), 53–56 (2020) 31. Y.G. Smirnov, E.Y. Smolkin, Complex waves in dielectric layer. Lobachevskii J. Math. 41(7), 1396–1403 (2020) 32. Y.G. Smirnov, Mathematical Methods for Electromagnetic Problems (Penza State University Press, Penza, 2009) 33. T. Kato, Perturbation Theory for Linear Operators (Springer, New York, 1980)

Chapter 4

Waveguides of Circular Cross Section

The chapter is devoted to the study of the electromagnetic wave propagation in cylindrical waveguides of circular cross section. A homogeneous dielectric waveguide of circular cross section, a dielectric rod (DR), and a perfectly conducting cylinder covered with a homogeneous dielectric material layer, the Goubau line (GL), are taken as model waveguides. The following types of dielectric materials are considered: inhomogeneous dielectric, metamaterial, anisotropic dielectric, absorption dielectric, and chiral medium. As far as benchmark structures are concerned, a DR is a basic structure in the family of open waveguides of circular cross section. However, rigorous mathematical proof of the existence of complex waves in a DR was until recently one of fundamental unsolved problems in the theory of electromagnetic wave propagation (the proof has been accomplished in [1]). Another basic open metal-dielectric structure for which the occurrence of waves has been rigorously verified [2, 3] is GL with linear and nonlinear dielectric cover [4–8]. In fact, the analysis that leads to a correct proof of the existence of complex waves must necessarily contain investigations of the properties of the functions of several complex variables involved in dispersion equations (DEs), direct proofs of the existence of complex roots of DEs, analysis of various types of singularities in the dependence of roots on the problem parameters, and the like. Such comprehensive studies have never been finalized, and correspondingly rigorous proof of the existence and correct determination of complex waves in a DR. This circumstance has become a driving force to complete the proofs and make the corresponding mathematical methods available and open for scientific community. In view of this, a goal of this chapter is to set up in a detailed manner elements of a rigorous theory of the electromagnetic wave propagation, specifically for structures possessing circular symmetry, using the most advanced to date mathematical tools. For a waveguide having circular cross section, it is reasonable to consider azimuthally symmetric electromagnetic waves, i.e., solutions with dependence of the from eimϕ+iγz on the coordinates ϕ and z, along which the structure is regular. Two types of waves are mainly considered in the study: surface waves decaying at © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 Y. Shestopalov et al., Optical Waveguide Theory, Springer Series in Optical Sciences 237, https://doi.org/10.1007/978-981-19-0584-1_4

123

124

4 Waveguides of Circular Cross Section

infinity and leaky waves growing at infinity. The results of calculating the fields in such structures for the indicated types of waves are presented. Note once again that the analysis of all the considered types of waveguides and running waves is performed, as in the previous chapters, within the frames of a universal scheme repeated for every waveguide family: description of the waveguide geometry and material parameters → spectral problem for the Maxwell equations → reduction to a boundary eigenvalue problem for a system of ordinary differential equations → re-formulation in terms of a variational relation in Sobolev spaces → reduction to a problem of finding characteristic numbers of an operator-valued function → proofs of the discreteness and existence of the spectrum for the operatorvalued function of the problem. Such an organization of the book makes it possible for a reader to consider every particular type of a waveguide independently staying within one specific chapter equipped with all necessary information and notations (that may be, for clarity, repeated throughout the text).

4.1 Surface Waves in Inhomogeneous Waveguides 4.1.1 Statement of the Problem Consider the three-dimensional space R3 equipped with the cylindrical coordinate system Oρϕz. The space is filled with an isotropic source-free medium with the permittivity ε = ε0 ≡ const and the permeability μ = μ0 ≡ const, where ε0 and μ0 are permittivity and permeability of vacuum. A GL with the cross section  := {(ρ, ϕ, z) : r0  ρ  r, 0  ϕ < 2π} and the generatrix parallel to the Oz-axis is placed in R3 . The cross section of the waveguide, which is perpendicular to its axis, consists of two concentric circles of radii r0 and r (see Fig. 4.1): r is the radii of the internal (perfectly conducting) cylinder and r − r0 is the thickness of the external (dielectric) cylindrical shell. The geometry of the problem is shown in Fig. 4.1. Determination of normal waves in the waveguide structure reduces to the problem of finding nontrivial running wave solutions of the homogeneous system of the Maxwell equations, i.e., solutions with the dependence eimϕ+iγz on coordinates ϕ and z along which the structure is regular, 

curl H = −i εE, ˜ curl E = i μH, ˜

  E = E ρ (ρ) eρ + E ϕ (ρ) eϕ + E z (ρ) ez eimϕ+iγz ,   H = Hρ (ρ) eρ + Hϕ (ρ) eϕ + Hz (ρ) ez eimϕ+iγz ,

(4.1.1)

(4.1.2)

4.1 Surface Waves in Inhomogeneous Waveguides

125

Fig. 4.1 Geometry of the problem

with the boundary conditions for the tangential electric field components on the perfectly conducting surfaces (ρ = r0 ) E ϕ (r0 ) = 0, E z (r0 ) = 0,

(4.1.3)

the transmission conditions for the tangential electric and magnetic field components on the surfaces of discontinuity (“jump”) of the permittivity and permeability (ρ = r )   [E ϕ ]ρ=r = 0, [E z ]|ρ=r = 0, [Hϕ ]ρ=r = 0, [Hz ]|ρ=r = 0,

(4.1.4)

the finite energy condition ∞ ( ε|E|2 +  μ|H|2 )dρ < ∞,

(4.1.5)

r0

and the radiation condition at infinity: the electromagnetic field decays as o(ρ-1/2 ) for ρ → ∞. The Maxwell system (4.1.1) is written in the normalized form, where the passageto dimensionless variables has been carried out [9]; namely, k0 ρ → ρ, γ → μ0 γ , H → H, E → E, where k02 = ωε0 μ0 (the time factor e−iωt is omitted k0 ε0 everywhere). We assume that the permittivity and permeability in the entire space have the form  ε=

ε (ρ) , r0 ≤ ρ ≤ r, and  μ= 1, ρ > r,



μ (ρ) , r0 ≤ ρ ≤ r, 1, ρ > r,

(4.1.6)

where ε(ρ) > 1 and μ(ρ) > 1 are twice continuously differentiable function on the segment [r0 , r ], i.e., ε(ρ) ∈ C 2 [r0 , r ] and μ(ρ) ∈ C 2 [r0 , r ], Imε(ρ) = 0, Imμ(ρ)= 0.

126

4 Waveguides of Circular Cross Section

The problem on normal waves is an eigenvalue problem for the Maxwell equations with spectral parameter γ, which is the normalized propagation constant of a running (normal) wave of GL. Rewrite system (4.1.1) in the expanded form: ⎧ m ⎪ i Hz − iγ Hϕ ⎪ ⎪ ρ ⎪ ⎪ ⎪ ⎪ ⎪ iγ Hρ − Hz ⎪ ⎪ ⎪ ⎪ im ⎪1  ⎪ ⎪ Hρ ⎨ (ρHϕ ) − ρ ρ m ⎪ ⎪ i E z − iγ E ϕ ⎪ ⎪ ρ ⎪ ⎪ ⎪ ⎪ ⎪ iγ E ρ − E z ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1 (ρE ϕ ) − im E ρ ρ ρ

= −i ε Eρ, = −i ε Eϕ, = −i ε Ez , = i μ Hρ ,

(4.1.7)

= i μ Hϕ , = i μ Hz ,

and express E ρ , Hρ , E ϕ , Hϕ via the functions E z and Hz from the first, second, fourth, and fifth equations in system (4.1.1) m μ Hz − iγρE z , ρ(γ 2 −  ε μ) γm E z + iρ μ Hz , Eϕ = ρ(γ 2 −  ε μ)

Eρ =

iγρHz + m εEz , 2 ρ(γ −  ε μ) γm Hz − iρ ε E z Hϕ = . ρ(γ 2 −  ε μ)

Hρ = −

(4.1.8)

It follows from (4.1.8) that the normal wave field in the waveguide can be represented with the use of two scalar functions u e := i E z (ρ), u m := Hz (ρ).

(4.1.9)

Thus, the problem has been reduced to finding longitudinal components u e and u m of the electric and magnetic fields. Throughout the text, ( · ) stands for differentiation with respect to ρ. We have the following eigenvalue problem for the longitudinal field components u e and u m : find γ ∈ C such that, for given m ∈ Z, there exist nontrivial solutions to the system of differential equations ⎧      ερ  ε m2 ε μ) ( ⎪  2 ⎪ u ρ u e = γm  − + ⎪ 2 u m , e ⎪ 2 2 ⎨ γ − ε μ ρ γ − ε μ γ2 −  ε μ     ⎪  μρ  μ 2 m2 ε μ) ( ⎪  ⎪ u ρ u − + = γm ⎪ 2 u e , m  m ⎩ γ2 −  ε μ ρ γ2 −  ε μ ε μ γ2 −  satisfying the boundary conditions for ρ = r0

(4.1.10)

4.1 Surface Waves in Inhomogeneous Waveguides

127

u e (r0 ) = 0, u m (r0 ) = 0,

(4.1.11)

the transmission conditions for ρ = r [u e ]|r = 0, [u m ]|r = 0,           ρ μu m  um ρ εu e  ue  −  − = 0, γm = 0, γm  ε μ − γ 2 r  ε μ − γ 2 r  ε μ − γ 2 r  ε μ − γ 2 r (4.1.12) where [ f ]|ρ0 = lim f (ρ) − lim f (ρ) and the condition of boundedness of the 

ρ→ρ0 −0

ρ→ρ0 +0

field in any finite domain and the decay condition at infinity. Once we determine the longitudinal field components u e and u m by solving problem (4.1.10)–(4.1.12), we can find the transverse components by formulas (4.1.8). The field (E, H) thus obtained satisfies all conditions (4.1.1)–(4.1.5). The equivaε μ; lence of the reduction to problem (4.1.10)–(4.1.12) is not valid only for γ 2 =  in this case, it is necessary to study system (4.1.1) directly. For ρ > r , we have  ε = 1,  μ = 1; then from (4.1.10), we obtain the system     ρ(ρu e ) − ρ2 κ2 + m 2 u e = 0, ρ(ρu m ) − ρ2 κ2 + m 2 u m = 0, where κ2 = γ 2 − 1. In view of the condition at infinity, we obtain a solution of this system in the form u e (ρ; γ, m) = C1 K m (κρ), (4.1.13) u m (ρ; γ, m) = C2 K m (κρ), where K m is the modified Bessel function (the Macdonald function) [10]; C1 and C2 are constants. R

We suppose that constants C1 and C2 are such that C12 + C22 = 0;

(4.1.14)

i.e., the field (E, H) outside the waveguide is not zero.

R

In view of the condition at infinity, we select the following branch of the square root κ=

R

        1  |γ 2 − 1| + Re γ 2 − 1 + isignIm γ 2 − 1 |γ 2 − 1| − Re γ 2 − 1 . γ2 − 1 = √ 2

κ(γ) is an analytical function in the domain C\ K , where  K := {γ : Imγ 2 = 0, γ 2 ≤ 1}.

128

4 Waveguides of Circular Cross Section

For r0 ≤ ρ ≤ r , we have  ε = ε (ρ) and  μ = μ (ρ), and from (4.1.10), we obtain the system of differential equations       L e u e : = με − γ 2 u e + pe γ 2 + qe u e + γ 4 + r1 γ 2 + r2 u e = γ f e u m ,       L m u m : = με − γ 2 u m + pm γ 2 + qm u m + γ 4 + r1 γ 2 + r2 u m = γ f m u e (4.1.15) where 1 1 ε μ pm = − − , pe = − − , ε ρ μ ρ qe = −μ ε + r1 =

με , ρ

με , ρ

  m2 r2 = με με − 2 , ρ

m2 − 2με, ρ2

fe = −

qm = −με +

m (με) , ρ ε

fm = −

m (με) . ρ μ

Definition 4.1.1 If for given m there exist nontrivial functions u e and u m corresponding to some γ ∈ C such that they are given by (4.1.13) for ρ > r , form a solution to system (4.1.15) for r0 ≤ ρ ≤ r , and satisfy transmission conditions (4.1.12), then γ is called a characteristic number of problem Pinh . Definition 4.1.2 A pair u e and u m , |u e |2 + |u m |2 ≡ 0, will be called an eigenvector of problem Pinh corresponding to the characteristic number γ ∈ C.

4.1.2 Sobolev Spaces and Variational Relation We will look for solutions u e and u m of problem Pinh in the Sobolev spaces   H01 (r0 , r ) = f : f ∈ H 1 (r0 , r ) , f (r0 ) = 0 and H 1 (r0 , r ) , with the inner product and the norm r ( f, g)1 = r0

R



 



f g + f g dρ,

r f 21

= ( f, f )1 =

 2  | f | + | f |2 dρ.

r0

Here, we use the notation for the Sobolev space H01 (r0 , r ), which does not coincide with the standard one: in our case f (r0 ) = 0; however, generally f (r ) = 0.

4.1 Surface Waves in Inhomogeneous Waveguides

129

Let us give a variational formulation of problem Pinh . Multiplying equations (4.1.15) by arbitrary test functions ve ∈ H01 (r0 , r ) and vm ∈ H 1 (r0 , r ) (we can assume that these functions are continuously differentiable in (r0 , r )) and applying Green’s formula, we obtain r r0

vLudρ =

r r0

    r  r  v με − γ 2 u  dρ + v pγ 2 + q u  dρ + v γ 4 + r1 γ 2 + r2 udρ r

r

0 0  r    r     = v με − γ 2 u   − v με − γ 2 + v (με) u  dρ

r0

r0

 r  v pγ 2 + q u  dρ + v γ 4 + r1 γ 2 + r2 udρ r0  r0  r r r r r    r 4 2      =γ uvdρ + γ u v dρ + pu vdρ + r1 uvdρ − u v r + μεu  v r +

r0

−



r



r0

r

μεu  v  dρ −

r0

r0

r

(με) u  vdρ +

r0

0

r0

r r0

qu  vdρ +

r r0

0

r2 uvdρ,

(4.1.16) where u = u j , v = v j , p = p j , q = q j , j = e or m. Taking into account the boundary condition for ρ = r0 , we get r

r vLudρ = γ 4

r0

⎛ uvdρ + γ 2 ⎝

r0

r

u  v  dρ +

r0

r −

r

pu  vdρ +

r0

μεu  v  dρ −

r0

r

r

⎞   r1 uvdρ − u  v r ⎠ + μεu  v r

r0

(με) u  vdρ +

r0

r

qu  vdρ +

r0

r r2 uvdρ.

(4.1.17)

r0

We separately apply formula (4.1.17) to the first and second equations in system (4.1.15) on the interval [r0 , r ], add the results, and find that the sum of the left-hand sides of (4.1.15) satisfies the relation r

r (v e L e u e + v m L m u m )dρ = γ 4

r0

(u e v e + u m v m )dρ r0

 r +γ 2

(u e v e + u m v m )dρ +

r0

r

( pe u e v e + pm u m v m )dρ +

r0

r − r0

r + r0

με(u e v e + u m v m )dρ −

r

 r1 (u e v e + u m v m )dρ

r0

r

(με) (u e v e + u m v m )dρ

r0

(qe u e v e + qm u m v m )dρ +

r r2 (u e v e + u m v m )dρ r0

  + μ(r )ε(r ) − γ 2 (u e (r )v e (r ) + u m (r )v m (r )).

(4.1.18)

130

4 Waveguides of Circular Cross Section

On the other hand, for the right-hand sides of the equations in system (4.1.15), we have r r (v e L e u e + v m L m u m )dρ = γ ( f e u m v e + f m u e v m )dρ. (4.1.19) r0

r0

Given the solutions (4.1.13), we express the values of the normal derivatives at the ρ = r from relations (4.1.12) as follows: γm μ(r )ε(r ) − 1 μ(r )ε(r ) − γ 2 K m (κr ) u e (r ), u (r ) − m r ε(r ) κ2 κε(r ) K m (κr ) γm μ(r )ε(r ) − 1 μ(r )ε(r ) − γ 2 K m (κr ) u m (r ) = u (r ) − u m (r ). e r μ(r ) κ2 κμ(r ) K m (κr ) u e (r ) =

(4.1.20)

Then, in view of (4.1.18), from (4.1.19) and (4.1.20), we obtain r γ4

(u e v e + u m v m )dρ r0

 r +γ 2

(u e v e + u m v m )dρ +

r0

( pe u e v e + pm u m v m )dρ +

r0

r −

με(u e v e + u m v m )dρ−

r0

r + r0

+

r



r r1 (u e v e + u m v m )dρ r0

r

(με) (u e v e + u m v m )dρ

r0

(qe u e v e

+ qm u m v m )dρ +

r r2 (u e v e + u m v m )dρ r0

 μ(r )ε(r ) − γ 2  γm μ(r )ε(r ) − 1 μ(r )ε(r ) − γ 2 K m (κr ) u (r )− (r ) v e (r ) u m e ε(r ) r κ2 κ K m (κr )  μ(r )ε(r ) − γ 2 K m (κr ) μ(r )ε(r ) − γ 2  γm μ(r )ε(r ) − 1 u m (r ) v m (r ) u e (r ) − + 2 μ(r ) r κ κ K m (κr ) r −γ ( f e u m v e + f m u e v m )dρ = 0. (4.1.21) r0

Note that variational relation (4.1.21) has been obtained for smooth functions v e and vm .

4.1 Surface Waves in Inhomogeneous Waveguides

131

4.1.3 Spectrum of the OVF Let H = H01 (r0 , r ) × H 1 (r0 , r ) be the Cartesian product of Hilbert spaces with the inner product and the norm (u, v) = (u 1 , v1 )1 + (u 2 , v2 )1 , u 2 = u 1 21 + u 2 21 ; u, v ∈ H, u = (u 1 , u 2 )T , v = (v1 , v2 )T , u 1 , v1 ∈ H01 (r0 , r ) , u 2 , v2 ∈ H 1 (r0 , r ) . Then the integrals occurring in (4.1.21) can be viewed as sesquilinear forms over the field C defined on H and depending on the arguments u = (u e , u m )T , v = (v e , v m )T . Consider the following sesquilinear forms and the corresponding operators: r k(u, v) :=

(u e v e + u m v m )dρ = (Ku, v), ∀v ∈ H, r0

r k1 (u, v) :=

(r1 − 1)(u e v e + u m v m )dρ = (K1 u, v), ∀v ∈ H, r0

r k2 (u, v) :=

(r2 − με)(u e v e + u m v m )dρ = (K2 u, v), ∀v ∈ H, r0

 k(u, v) :=

r

u, v), ∀v ∈ H, ( f e u m v e + f m u e v m )dρ = (K

r0

r a1 (u, v) :=

(u e v e + u m v m + u e v e + u m v m )dρ = (Iu, v), ∀v ∈ H,

r0

r a2 (u, v) :=

με(u e v e + u m v m + u e v e + u m v m )dρ = (Au, v), ∀v ∈ H,

r0

r b1 (u, v) := r0

( pe u e v e + pm u m v m )dρ = (B1 u, v), ∀v ∈ H,

132

4 Waveguides of Circular Cross Section

r b2 (u, v) :=

(με) (u e v e + u m v m )dρ = (B2 u, v), ∀v ∈ H,

r0

r b3 (u, v) :=

(qe u e v e + qm u m v m )dρ = (B3 u, v), ∀v ∈ H,

r0

  μ(r )ε(r ) − γ 2 γm μ(r )ε(r ) − 1 μ(r )ε(r ) − γ 2 K m (κr ) u (r ) − (r ) v e (r ) u m e ε(r ) r κ2 κ K m (κr )   μ(r )ε(r ) − γ 2 K m (κr ) μ(r )ε(r ) − γ 2 γm μ(r )ε(r ) − 1 u e (r ) − + u m (r ) v m (r ) 2 μ(r ) r κ κ K m (κr )

s(u, v) =

= (Su, v), ∀v ∈ H.

(4.1.22) The boundedness of the form a2 (u, v) is obvious. The boundedness of the forms k(u, v) follows from the Poincar’e inequality [11]. k(u, v), k1 (u, v), k2 (u, v) and  Now variational problem (4.1.21) can be written in the operator form (N(γ)u, v) = 0, ∀u ∈ H or, equivalently,    + K2 − A − B2 + B3 + S(γ) u = 0. N(γ)u := γ 4 K + γ 2 (K1 + B1 + I) − γ K (4.1.23) Equation (4.1.23) is the operator form of variational relation (4.1.21). The characteristic numbers and eigenvectors of the operator N by definition coincide with the eigenvalues and eigenvectors of problem Pinh for γ 2 = μ(ρ)ε(ρ).

4.1.4 Properties of the Spectrum of OVF We have reduced the problem on normal waves to the study of spectral properties of OVF N. In this connection, we first consider the properties of the operators introduced in the preceding section. Lemma 4.1.1 The bounded operator A : H → H is positive definite A ≥ γ∗2 I, √ where 0 < γ∗ = min μ(ρ)ε(ρ). r0 ≤ρ≤r

Proof The assertion of the lemma is obvious.



 : H → H are compact, Lemma 4.1.2 The bounded operators K, K1 , K2 and K and K > 0.

4.1 Surface Waves in Inhomogeneous Waveguides

133

Proof Since the embedding H 1 (r0 , r ) ⊂ L 2 (r0 , r ) is compact, it follows that the  are compact. Further, operators K, K1 , K2 and K r

  |u e |2 + |u m |2 dρ > 0;

(Ku, u) = r0

consequently, K > 0. This completes the proof of the lemma.



Lemma 4.1.3 The operators B1 , B2 and B3 : H → H are compact. Proof Consider the form r b(u, v) :=

(u e v e + u m v m )dρ = (Bu, v), ∀v ∈ H.

r0

We have the estimates  r    r r         |u e ||ve | + |u m ||vm | dρ |b(u, v)| =  (u e v e + u m v m )dρ ≤ u e v e + u m v m  dρ ≤≤   r0

⎛

≤⎝ ⎛ ≤⎝

r0

r 

|u e |2

+ |u m |2



⎞1 ⎛ 2

dρ⎠ ⎝

r0

r 

r0

r 

|ve | + |vm | 2

2



⎞1 2

dρ⎠

r0



⎞1 ⎛ 2

|u e |2 + |u m |2 + |u e |2 + |u m |2 dρ⎠ ⎝

r0

r 



⎞1 2

|ve |2 + |vm |2 dρ⎠

r0

= u

v L 2

(4.1.24)

Setting v = Bu, we obtain the estimate

Bu 2 = |b(u, Bu)| ≤ u

Bu L 2 .

(4.1.25)

 and Bun → 0 weakly in H . Let un → 0, n → ∞ weakly in H . Then un ≤ C Since the embedding H 1 (r0 , r ) ⊂ L 2 (r0 , r ) is compact, it follows that Bu L 2 → 0, n → ∞ strongly in L 2 . Hence, Bu L 2 → 0, n → ∞, and consequently, the operator B : H → H is compact.  In a similar way, one proves the compactness of operators B1 , B2 , B3 . Lemma 4.1.4 The operator S : H → H is compact. Proof Consider the form r s1 (u, v) := u(r )v(r ) =





r

s (ρ)v (ρ)dρ + r0

s(ρ)v(ρ)dρ, r0

(4.1.26)

134

4 Waveguides of Circular Cross Section

where s(ρ) = S1 u. Let v = 0 in a neighborhood of the points r0 and r ; then r 0 = s (ρ)v(ρ)r0 − 

r



s (ρ)v(ρ)dρ + r0

or

r

r s(ρ)v(ρ)dρ, r0

   −s (ρ) + s(ρ) v(ρ)dρ = 0.

(4.1.27)

r0

Since function v(ρ) is arbitrary, it follows from (4.1.27) that s  (ρ) − s(ρ) = 0, r0 < ρ < r.

(4.1.28)

We take v = 0 at the points r0 and r ; then the form (4.1.26) u(r )v(r ) = s  (r )v(r ) − s  (r0 )v(r0 ) ⇒ s  (r0 ) = 0, s  (r ) = u(r ).

(4.1.29)

Now consider and solve the boundary value problem 

s  (ρ) − s(ρ) = 0, s  (r0 ) = 0, s  (r ) = u(r ).

(4.1.30)

The general solution of the equation of the problem has the form s(ρ) = C1∗ cosh (ρ − r0 ) + C2∗ sinh (ρ − r0 ), hence

s  (ρ) = C1∗ sinh (ρ − r0 ) + C2∗ cosh (ρ − r0 ).

Since s  (r0 ) = 0, it follows that C2∗ = 0. Let us determine constant C1∗ : s  (r ) = C1∗ sinh (r − r0 ) = u(r ), C1∗ = Then s(ρ) =

u(r ) . sinh(r − r0 )

u(r ) cosh (ρ), sinh(r − r0 )

and S1 u is a rank-one operator, dim ImS1 = 1; consequently, this operator is compact.

4.1 Surface Waves in Inhomogeneous Waveguides

135

The form s(u, v) contains two summands s1 (u, v) and each of them defines a compact operator. The proof of the lemma is complete.  Lemma 4.1.5 The operator γ 2 I − A : H → H is bounded and continuously invertible in the domain C\ E and  E := {γ : Imγ = 0, γ∗ ≤ |Reγ| ≤ γ ∗ }, where 0 < γ ∗ = max

r0 ≤ρ≤r

√

μ(ρ)ε(ρ).

Proof The proof follows from the analysis of the real and imaginary parts of the expression μ(ρ)ε(ρ) − γ 2 and the form corresponding to operator A.  Lemma 4.1.6 There exists a  γ ∈ R such that operator N( γ ) is continuously invertible; i.e., the resolvent set (N) := {γ : ∃ N−1 (γ) : H → H } of OVF N( γ ) is nonempty, (N) = ∅. Proof Assume that γ ∈ R, γ > 0 and γ → +∞. Consider the OVF 1 N(γ) = N1 (γ) + N2 (γ), γ2 where N1 (γ) = γ 2 K + K1 + B1 + I + γS0 , and

 1 1 + K − A − B + B + S(γ) − γS0 , N2 (γ) = − K 2 2 3 γ γ2

and operator S0 : H → H is defined by the sesquilinear form   u e (r )v e (r ) u m (r )v m (r ) S0 u, v := + . ε(r ) μ(r )

(4.1.31)

In view of the asymptotics of functions K m as γ → +∞ [12], we have K m (κr ) ∼ −1, K m (κr ) and

Consequently,

1 μ(r )ε(r ) − γ 2 ∼ −γ + O . κ γ     1   S(γ 2 ) − γS0  = O 1 , as γ → +∞.  γ2  γ

(4.1.32)

136

4 Waveguides of Circular Cross Section

Then (OVFs) γ −2 N(γ) can be viewed as a perturbation of operator pencil N1 by N2 for large γ, r Reb1 (u, u) := Re

( pe u e u e + pm u m u m )dρ

r0

=

1 2

r 

     pe |u e |2 + pm |u m |2 dρ

r0

  1        1 pe (r ) |u e (r )|2 + pm (r ) |u m (r )|2 − pe (r0 ) |u e (r0 )|2 + pm (r0 ) |u m (r0 )|2 = 2 2 r     1    |u m |2 dρ − pe |u e |2 + pm 2 r0

    1 pe (r ) |u e (r )|2 + pm (r ) |u m (r )|2 − = 2 r 1 − 2

  1 pm (r0 ) |u m (r0 )|2 2       pe |u e |2 + pm |u m |2 dρ.

r0

Since functions ε and μ belong to space C 2 [r0 , r ], it follows from the last relation, definition (4.1.31), the asymptotic relation (4.1.32), and the properties of the γ such that operators K and S0 that there exists a large number  Re(N1 ( γ )u, u) =  γ 2 (Ku, u) + (K1 u, u) + Re(B1 u, u) + (u, u) +  γ (S0 u, u) ≥ u 2

for each u. Hence,  γ ∈ (N1 ), where (N1 ) is the resolvent set of OVF N1 . Moreover, Theorem 4.1 in [13] gives the estimate γ ) ≤ 1.

N1−1 ( γ ) < 1 is satisfied and show then that there We take  γ such that the inequality N2 ( exists a bounded operator −1    γ ) + N2 ( γ) = I + N1−1 ( γ )N2 ( γ ) N1−1 ( γ ). N1 (  Theorem 4.1.1 OVF N(γ) : H → H is bounded, holomorphic, and Fredholm in the domain  = C\ ( K ∪  E ). Proof The functions κ

K m+1 (κr ) m K m (κr ) = −κ + . K m (κr ) K m (κr ) r

(4.1.33)

4.1 Surface Waves in Inhomogeneous Waveguides

137

are analytical in the domain C\ K with respect to variable γ because functions K m (κr ) have no zeros for Reκ > 0 [12]. Finally, we obtain the desired result applying Lemmas 4.1.5 and 4.1.6. The proof of the theorem is complete.  Theorem 4.1.2 The spectrum of OVF N(γ) : H → H is discrete in the domain ; i.e., this function has finitely many characteristic points of finite algebraic multiplicity on any compact set K 0 ⊂ . Proof The assertion of the theorem is a corollary of Theorem 4.1.1 and a theorem on a holomorphic OVF [13].  Lemma 4.1.7 The spectrum of OVF N(γ) : H → H is symmetric with respect to the real and imaginary axes. If γ0 is eigenvalue of N(γ) corresponding to eigenvector u1 = (u e , u m )T , then −γ0 , γ 0 , −γ 0 are eigenvalues of OVF N(γ) corresponding to eigenvectors u2 = (−u e , u m )T , u3 = (u e , u m )T , u4 = (−u e , u m )T with the same multiplicity. Proof The assertion of the lemma can readily be proved by a straightforward verification with the use of (4.1.10) and conditions (4.1.11) and (4.1.12). Note that the condition at infinity is satisfied for all four characteristic numbers. 

4.2 Surface Waves in a Waveguide Filled with Metamaterial Media 4.2.1 Statement of the Problem The statement of the problem of determining normal surface waves in a GL described in Sect. 4.1.1 having the dielectric cover filled with metamaterial media which is reduced to an eigenvalue problem for the Maxwell equations with spectral parameter γ, the normalized propagation constant of the running wave, is given by (4.1.1)– (4.1.5). We assume that the permittivity and permeability in the entire space have the form 2 −μ2 , r0 ≤ ρ ≤ r, −ε , r0 ≤ ρ ≤ r, and  μ= (4.2.34)  ε= 1, ρ > r, 1, ρ > r, where ε2 > 1 and μ2 > 1 are real positive constants. Rewrite Maxwell’s equation in the expanded form:

138

4 Waveguides of Circular Cross Section

⎧ m ⎪ i Hz − iγ Hϕ ⎪ ⎪ ρ ⎪ ⎪ ⎪ ⎪ ⎪ iγ Hρ − Hz ⎪ ⎪ ⎪ ⎪ 1 im ⎪  ⎪ ⎪ Hρ ⎨ (ρHϕ ) − ρ ρ m ⎪ ⎪ i E z − iγ E ϕ ⎪ ⎪ ρ ⎪ ⎪ ⎪ ⎪ ⎪ iγ E ρ − E z ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1 (ρE ϕ ) − im E ρ ρ ρ

= −i ε Eρ, = −i ε Eϕ, = −i ε Ez , = i μ Hρ ,

(4.2.35)

= i μ Hϕ , = i μ Hz ,

and express functions E ρ , Hρ , E ϕ , Hϕ via E z and Hz from the first, second, fourth, and fifth equations of system (4.2.35) m μ Hz − iγρE z , ρ(γ 2 −  ε μ) γm E z + iρ μ Hz , Eϕ = ρ(γ 2 −  ε μ)

Eρ =

iγρHz + m εEz , ρ(γ 2 −  ε μ) γm Hz − iρ ε E z Hϕ = . ρ(γ 2 −  ε μ)

Hρ = −

(4.2.36)

It follows from (4.2.36) that the normal wave field in the waveguide can be represented using two scalar functions u e := i E z (ρ), u m := Hz (ρ).

(4.2.37)

Thus, the problem has been reduced to finding longitudinal components u e and u m of the electric and magnetic fields. In what follows, ( · ) stands for differentiation with respect to ρ. We have the following eigenvalue problem for the longitudinal field components u e and u m : find γ ∈ C such that, for given m ∈ Z, there exist nontrivial solutions of the system of differential equations ⎧     1  2 2   ⎪ ⎪ ρ γ − ε μ + m 2 u e = 0, ⎨ ρu e − ρ   ⎪    1  2  2 ⎪ ⎩ ρu m − ρ γ − ε μ + m 2 u m = 0, ρ

(4.2.38)

satisfying the boundary conditions for ρ = r0 u e (r0 ) = 0, u m (r0 ) = 0, the transmission conditions for ρ = r

(4.2.39)

4.2 Surface Waves in a Waveguide Filled with Metamaterial Media

139

[u e ]|r = 0, [u m ]|r = 0,           ρ μu m  um ρ εu e  ue  −  − = 0, γm = 0, γm  ε μ − γ 2 r  ε μ − γ 2 r  ε μ − γ 2 r  ε μ − γ 2 r (4.2.40) where [ f ]|ρ0 = lim f (ρ) − lim f (ρ), and the condition of boundedness of the 

ρ→ρ0 −0

ρ→ρ0 +0

field in any finite domain and the decay condition at infinity. Once we determine longitudinal field components u e and u m by solving problem (4.2.38)–(4.2.40), we can find the transverse components by formulas (4.2.36). The equivalence of the reduction to problem (4.2.38)–(4.2.40) is not valid only ε μ. for γ 2 =  For ρ > r , we have  ε = 1,  μ = 1; then from (4.2.38), we obtain the system     ρ(ρu e ) − ρ2 κ2 + m 2 u e = 0, ρ(ρu m ) − ρ2 κ2 + m 2 u m = 0, where κ2 = γ 2 − 1. In view of the condition at infinity, we obtain a solution of this system in the form u e (ρ; γ, m) = C1 K m (κρ), (4.2.41) u m (ρ; γ, m) = C2 K m (κρ), where K m is the modified Bessel function (the Macdonald function) [10] and C1 and C2 are constants. ε = −ε2 and  μ = −μ2 , and from (4.2.38) we obtain the For r0 ≤ ρ ≤ r , we have  system of differential equations   L e u e : = u e + pu e − γ 2 + q u e = 0,   L m u m : = u m + pu m − γ 2 + q u m = 0 where p=

1 , ρ

q = −ε2 μ2 +

(4.2.42)

m2 . ρ2

Definition 4.2.1 If for given m there exist nontrivial functions u e and u m corresponding to γ ∈ C such that these functions are given by (4.2.41) for ρ > r , form a solution of system (4.2.42) for r0 ≤ ρ ≤ r , and satisfy transmission conditions (4.2.40), then γ is called a characteristic number of problem Pmet . Definition 4.2.2 The pair u e and u m , |u e |2 + |u m |2 ≡ 0, will be called an eigenvector of problem Pmet corresponding to the characteristic number γ ∈ C.

4.2.2 Sobolev Spaces and Variational Relation Let us give a variational formulation of problem Pmet . Multiplying equations (4.2.42) by arbitrary test functions ve ∈ H01 (r0 , r ) and vm ∈ H 1 (r0 , r ) (we can assume that

140

4 Waveguides of Circular Cross Section

these functions are continuously differentiable in (r0 , r )) and applying Green’s formula, we obtain r

r vLudρ =

r0

r



vu dρ + r0

=

r vu  r0

r



pvu dρ − r0 r

−

r0 r 

 

u v dρ + r0

= −γ

r



pu vdρ − r0

r r0

  v γ 2 + q udρ

r0

r

r

uvdρ −

2

  v γ 2 + q udρ

quvdρ + r0

r pu vdρ + u v r0 − 



r0

r

u  v  dρ.

r0

(4.2.43) Taking into account the boundary condition for ρ = r0 , we get r

r vLudρ = −γ

r0

r uvdρ −

2 r0

r quvdρ +

r0

 pu vdρ + u v r − 



r0

r

u  v  dρ

r0

(4.2.44) We separately apply formula (4.2.44) to the first and second equations in system (4.2.42) on the interval [r0 , r ], add the results, and find that the sum of the left-hand sides of (4.2.42) satisfies the relation r

r (v e L e u e + v m L m u m )dρ = −γ

r0

(u e v e + u m v m )dρ

2 r0

r −

r q(u e v e + u m v m )dρ +

r0

p(u e v e

+

u m v m )dρ−

r0

r

(u e v e + u m v m )dρ

r0

+u e (r )v e (r ) + u m (r )v m (r ). (4.2.45) Given the solutions (4.2.41), we express the values of the normal derivatives at the ρ = r from relations (4.2.40) as follows: γ 2 − ε2 μ2 κε2 2 γ − ε2 μ2 u m (r ) = κμ2 u e (r ) =

K m (κr ) γm ε2 μ2 − 1 u e (r ) + 2 u m (r ), K m (κr ) rε κ2 K m (κr ) γm ε2 μ2 − 1 u m (r ) + 2 u e (r ). K m (κr ) rμ κ2

Then, in view of (4.2.45), from (4.2.46), we obtain

(4.2.46)

4.2 Surface Waves in a Waveguide Filled with Metamaterial Media

r γ

r (u e v e + u m v m )dρ +

2

141

r0

q(u e v e + u m v m )dρ r0

r −

p(u e v e

+

u m v m )dρ

r0

r +

(u e v e + u m v m )dρ

r0

 γ −ε μ γm ε2 μ2 − 1 u m (r ) v e (r ) − u e (r ) + 2 κε2 K m (κr ) rε κ2  2  γ − ε2 μ2 K m (κr ) γm ε2 μ2 − 1 − u m (r ) + 2 u e (r ) v m (r ) = 0. κμ2 K m (κr ) rμ κ2 (4.2.47) 

2

2 2

K m (κr )

Note that variational relation (4.2.47) has been obtained for smooth functions v e and vm .

4.2.3 Spectrum of the OVF Let H = H01 (r0 , r ) × H 1 (r0 , r ) be the Cartesian product of the Hilbert spaces with the inner product and the norm (u, v) = (u 1 , v1 )1 + (u 2 , v2 )1 , u 2 = u 1 21 + u 2 21 ; u, v ∈ H, u = (u 1 , u 2 )T , v = (v1 , v2 )T , u 1 , v1 ∈ H01 (r0 , r ) , u 2 , v2 ∈ H 1 (r0 , r ) . Then the integrals occurring in (4.2.47) can be viewed as sesquilinear forms over the field C defined on H and depending on the arguments u = (u e , u m )T , v = (v e , v m )T . Consider the following sesquilinear forms and the corresponding operators: r k(u, v) :=

(u e v e + u m v m )dρ = (Ku, v), ∀v ∈ H, r0

 k(u, v) :=

r r0

 v), ∀v ∈ H, q(u e v e + u v v m )dρ = (Ku,

142

4 Waveguides of Circular Cross Section

r a(u, v) :=

(u e v e + u m v m + u e v e + u m v m )dρ = (Iu, v), ∀v ∈ H,

r0

r b(u, v) :=

p(u e v e + u m v m )dρ = (Bu, v), ∀v ∈ H,

r0



γ 2 − ε2 μ2 κε2  2 γ − ε2 μ2 + κμ2

s(u, v) =

 K m (κr ) γm ε2 μ2 − 1 u e (r ) + 2 u (r ) v e (r ) m K m (κr ) rε κ2  K m (κr ) γm ε2 μ2 − 1 u m (r ) + 2 u (r ) v m (r ) e K m (κr ) rμ κ2 = (Su, v), ∀v ∈ H. (4.2.48)

The boundedness of the form a(u, v) is obvious. The boundedness of k(u, v) and  k(u, v) follows from the Poincar’e inequality [11]. Now variational problem (4.2.47) can be written in the operator form (N(γ)u, v) = 0, ∀u ∈ H or, equivalently,    − K + I − B − S(γ) u = 0. N(γ)u := γ 2 K + K

(4.2.49)

Equation (4.2.49) is the operator form of variational relation (4.2.47). The characteristic numbers and eigenvectors of the operator N by definition coincide with the eigenvalues and eigenvectors of problem Pmet for γ 2 = μ2 ε2 .

4.2.4 Properties of the Spectrum of OVF We have reduced the problem on normal waves to the study of spectral properties of OVF N. In this connection, we first consider the properties of the operators introduced in the preceding section. The validity of Lemmas 4.2.1–4.2.3 is shown in the previous paragraph.  : H → H are compact and K > 0. Lemma 4.2.1 Bounded operators K and K Lemma 4.2.2 The operators B : H → H are compact. Lemma 4.2.3 The operator S : H → H is compact.

4.2 Surface Waves in a Waveguide Filled with Metamaterial Media

143

Lemma 4.2.4 There exists a  γ ∈ R such that the operator N( γ ) is continuously invertible; i.e., the resolvent set (N) := {γ : ∃ N−1 (γ) : H → H } of operatorvalued function N( γ ) is nonempty, (N) = ∅. Proof Let γ ∈ R, γ > 0 and γ → +∞. Consider the OVF N(γ) = N1 (γ) + N2 (γ), where

 − K + B + I + γS0 , N1 (γ) = γ 2 K + K

and N2 (γ) = −S(γ) − γS0 , and the operator S0 : H → H is defined by the sesquilinear form   u e (r )v e (r ) u m (r )v m (r ) S0 u, v := + . ε2 μ2

(4.2.50)

In view of the asymptotics of functions K m as γ → +∞ [12], we have K m (κr ) ∼ −1, K m (κr ) and

1 γ 2 − μ2 ε2 ∼γ+O . κ γ

Consequently,

N2 (γ) = O

1 γ

, as γ → +∞.

(4.2.51)

Then for large γ (OVFs) N(γ) can be viewed as a perturbation of operator pencil N1 by OVF N2 . r Reb(u, u) := Re

p(u e u e + u m u m )dρ =

r0

1 2

r p

     |u e |2 + |u m |2 dρ

r0

 p(r0 )   p(r )  = |u e (r )|2 + |u m (r )|2 − |u e (r0 )|2 + |u m (r0 )|2 2 2 r   1 − p  |u e |2 + |u m |2 dρ 2 r0

144

4 Waveguides of Circular Cross Section

=

 p(r0 ) p(r )  |u e (r )|2 + |u m (r )|2 − |u m (r0 )|2 2 2 r   1 − p  |u e |2 + |u m |2 dρ. 2 r0

It follows from the last relation, definition (4.2.50), asymptotic relation (4.2.51), and γ such that the properties of operators K and S0 that there exists a large number  u, u) − (Ku, u) + Re(Bu, u) + (u, u) +  Re(N1 ( γ )u, u) =  γ 2 (Ku, u) + (K γ (S0 u, u) ≥ u 2

for each u. Hence,  γ ∈ (N1 ), where (N1 ) is the resolvent set of OVF N1 . Moreover, Theorem 4.1 in [13] gives the estimate γ ) ≤ 1.

N1−1 ( γ ) < 1 is satisfied and find that there exists We take  γ such that the inequality N2 ( a bounded operator −1    γ ) + N2 ( γ) = I + N1−1 ( γ )N2 ( γ ) N1−1 ( γ ). N1 (  Theorem 4.2.1 The OVF N(γ) : H → H is bounded, holomorphic, and Fredholm in the domain  = C\ K . Proof The functions κ

K m+1 (κr ) m K m (κr ) = −κ + . K m (κr ) K m (κr ) r

(4.2.52)

are analytical in the domain C\ K with respect to variable γ because functions K m (κr ) have no zeros for Reκ > 0 [12]. Finally, we obtain the desired result by Lemmas 4.2.4. The proof of the theorem is complete.  Theorem 4.2.2 The spectrum of the OVF N(γ) : H → H is discrete in the domain ; i.e., this function has finitely many characteristic numbers of finite algebraic multiplicity on any compact set K 0 ⊂ . Proof The assertion of the theorem is a corollary of Theorem 4.2.1 and a theorem on a holomorphic OVF [13].  Lemma 4.2.5 The spectrum of the OVF N(γ) : H → H is symmetric with respect to the real and imaginary axes. If γ0 is an eigenvalue of N(γ) corresponding to the eigenvector u1 = (u e , u m )T , then −γ0 , γ 0 , −γ 0 are also eigenvalues of OVF

4.2 Surface Waves in a Waveguide Filled with Metamaterial Media

145

N(γ) corresponding to the eigenvectors u2 = (−u e , u m )T , u3 = (u e , u m )T , u4 = (−u e , u m )T with the same multiplicity. Proof The assertion of the lemma can readily be proved by a straightforward verification with the use of (4.2.38) and conditions (4.2.39) and (4.2.40). Note that the condition at infinity is satisfied for all four characteristic numbers. 

4.3 Surface Waves in a Waveguide Filled with Lossy Medium 4.3.1 Statement of the Problem Consider GL described in Sect. 4.1.1. We assume that the permittivity and permeability in the entire space have the form  ε=

ε + iεlos , r0 ≤ ρ ≤ r, and  μ= 1, ρ > r,



μ, r0 ≤ ρ ≤ r, 1, ρ > r,

(4.3.53)

where ε > 1, εlos > 0 and μ2 > 1 are real positive constants; ε > ε >los . The problem on normal waves is an eigenvalue problem for the Maxwell equations with spectral parameter γ, which is the normalized propagation constant of a normal wave. Rewrite of Maxwell’s equation in the expanded form: ⎧ m ⎪ i Hz − iγ Hϕ ⎪ ⎪ ρ ⎪ ⎪ ⎪ ⎪ ⎪ iγ Hρ − Hz ⎪ ⎪ ⎪ ⎪ 1 im ⎪  ⎪ ⎪ Hρ ⎨ (ρHϕ ) − ρ ρ m ⎪ ⎪ i E z − iγ E ϕ ⎪ ⎪ ρ ⎪ ⎪ ⎪ ⎪ ⎪ iγ E ρ − E z ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1 (ρE ϕ ) − im E ρ ρ ρ

= −i ε Eρ, = −i ε Eϕ, = −i ε Ez , = i μ Hρ ,

(4.3.54)

= i μ Hϕ , = i μ Hz ,

and express functions E ρ , Hρ , E ϕ , Hϕ via E z and Hz from the first, second, fourth, and fifth equations in system (4.3.54) m μ Hz − iγρE z , ρ(γ 2 −  ε μ) γm E z + iρ μ Hz Eϕ = , ρ(γ 2 −  ε μ)

Eρ =

iγρHz + m εEz , 2 ρ(γ −  ε μ) γm Hz − iρ ε E z Hϕ = . ρ(γ 2 −  ε μ)

Hρ = −

(4.3.55)

146

4 Waveguides of Circular Cross Section

It follows from (4.3.55) that the normal wave field in the waveguide can be represented using two scalar functions u e := i E z (ρ), u m := Hz (ρ).

(4.3.56)

Thus, the problem has been reduced to finding longitudinal components u e and u m of the electric and magnetic fields. In what follows, ( · ) stands for differentiation with respect to ρ. We have the following eigenvalue problem for the longitudinal field components u e and u m : find γ ∈ C such that, for a given m ∈ Z, there exist nontrivial solutions of the system of differential equations ⎧     1  2 2   ⎪ ⎪ ρ γ − ε μ + m 2 u e = 0, ⎨ ρu e − ρ  2 2     1 ⎪  ⎪ ⎩ ρu m − ρ γ − ε μ + m 2 u m = 0, ρ

(4.3.57)

satisfying the boundary conditions for ρ = r0 u e (r0 ) = 0, u m (r0 ) = 0,

(4.3.58)

the transmission conditions for ρ = r [u e ]|r = 0, [u m ]|r = 0,           ρ μu m  um ρ εu e  ue  −  − = 0, γm = 0, γm  ε μ − γ 2 r  ε μ − γ 2 r  ε μ − γ 2 r  ε μ − γ 2 r (4.3.59) where [ f ]|ρ0 = lim f (ρ) − lim f (ρ) and also with the condition of bounded

ρ→ρ0 −0

ρ→ρ0 +0

ness of the field in any finite domain and the decay condition at infinity. Once we determine the longitudinal field components u e and u m by solving problem (4.3.57)–(4.3.59), we can find the transverse field components by formulas (4.3.55). The equivalence of the reduction to problem (4.3.57)–(4.3.59) is not valid only ε μ. for γ 2 =  For ρ > r , we have  ε = 1,  μ = 1; then from (4.3.57), we obtain the system     ρ(ρu e ) − ρ2 κ2 + m 2 u e = 0, ρ(ρu m ) − ρ2 κ2 + m 2 u m = 0, where κ2 = γ 2 − 1. In view of the condition at infinity, we obtain a solution of this system in the form u e (ρ; γ, m) = C1 K m (κρ), (4.3.60) u m (ρ; γ, m) = C2 K m (κρ),

4.3 Surface Waves in a Waveguide Filled with Lossy Medium

147

where K m is the modified Bessel function (the Macdonald function) [10]; C1 and C2 are constants. R

Function κ(γ) is analytical in the domain C\ K , where  K := {γ : Imγ 2 = 0, γ 2 ≤ 1}.

For r0 ≤ ρ ≤ r , we have  ε = ε + iεlos and  μ = μ, and from (4.3.57), we obtain the system of differential equations   L e u e : = u e + pu e − γ 2 + q u e = 0,   L m u m : = u m + pu m − γ 2 + q u m = 0 where 1 , ρ

p=

q = −(ε + iεlos )μ +

(4.3.61)

m2 . ρ2

Definition 4.3.1 If for a given m there exist nontrivial functions u e and u m corresponding to some γ ∈ C such that they are given by (4.3.60) for ρ > r , form a solution of system (4.3.61) for r0 ≤ ρ ≤ r , and satisfy transmission conditions (4.3.59), then γ is called a characteristic number of problem Plos . Definition 4.3.2 The pair u e and u m , |u e |2 + |u m |2 ≡ 0, will be called an eigenvector of problem Plos corresponding to the characteristic number γ ∈ C.

4.3.2 Sobolev Spaces and Variational Relation Let us give a variational formulation of problem Plos . Multiplying equations (4.3.61) by arbitrary test functions ve ∈ H01 (r0 , r ) and vm ∈ H 1 (r0 , r ) (we can assume that these functions are continuously differentiable in (r0 , r )) and applying Green’s formula, we obtain r

r vLudρ =

r0

r



vu dρ + r0

=

r



pvu dρ − r0

r vu  r0

r −

r0

r

 

u v dρ + r0

r = −γ

r0

r



pu vdρ − r0

r uvdρ −

2

  v γ 2 + q udρ

r0

r quvdρ +

r0

  v γ 2 + q udρ

r0

r pu vdρ + u v r0 − 



r

u  v  dρ.

r0

(4.3.62)

148

4 Waveguides of Circular Cross Section

Taking into account the boundary condition for ρ = r0 , we get r

r vLudρ = −γ

r0

r

r

uvdρ −

2 r0

quvdρ + r0

 pu vdρ + u v r − 



r0

r

u  v  dρ

r0

(4.3.63) We separately apply formula (4.3.63) to the first and second equations in system (4.3.61) on the interval [r0 , r ], add the results, and find that the sum of the left-hand sides of (4.3.61) satisfies the relation r

r (v e L e u e + v m L m u m )dρ = −γ

(u e v e + u m v m )dρ

2

r0

r0

r −

r q(u e v e + u m v m )dρ +

r0

p(u e v e

+

u m v m )dρ−

r0

r

(u e v e + u m v m )dρ

r0  +u e (r )v e (r )

+ u m (r )v m (r ). (4.3.64)

Given solutions (4.3.60), we express the values of the normal derivatives at ρ = r from relations (4.3.59) as follows: (ε + iεlos )μ − 1 γm (ε + iεlos )μ − γ 2 K m (κr ) u e (r ), u m (r ) − 2 r (ε + iεlos ) κ κ(ε + iεlos ) K m (κr ) γm (ε + iεlos )μ − 1 (ε + iεlos )μ − γ 2 K m (κr ) u m (r ). u m (r ) = u e (r ) − 2 rμ κ κμ K m (κr ) (4.3.65) Then, in view of (4.3.64), from (4.3.65), we obtain u e (r ) =

r γ2

r (u e v e + u m v m )dρ +

r0

r0

r − 

q(u e v e + u m v m )dρ

r0

p(u e v e

+ u m v m )dρ +

r

(u e v e + u m v m )dρ

r0

 γm (ε + iεlos )μ − 1 (ε + iεlos )μ − γ 2 K m (κr ) − u u (r ) − (r ) v e (r ) m e r (ε + iεlos ) κ2 κ(ε + iεlos ) K m (κr )   γm (ε + iεlos )μ − 1 (ε + iεlos )μ − γ 2 K m (κr ) u m (r ) v m (r ) = 0. − u e (r ) − 2 rμ κ κμ K m (κr )

(4.3.66) Note that variational relation (4.3.66) has been obtained for smooth functions v e and vm .

4.3 Surface Waves in a Waveguide Filled with Lossy Medium

149

4.3.3 Spectrum of the OVF of the Problem Let H = H01 (r0 , r ) × H 1 (r0 , r ) be the Cartesian product of Hilbert spaces with the inner product and the norm (u, v) = (u 1 , v1 )1 + (u 2 , v2 )1 , u 2 = u 1 21 + u 2 21 ; u, v ∈ H, u = (u 1 , u 2 )T , v = (v1 , v2 )T , u 1 , v1 ∈ H01 (r0 , r ) , u 2 , v2 ∈ H 1 (r0 , r ) . Then the integrals occurring in (4.3.66) can be viewed as sesquilinear forms over the field C defined on H and depending on the arguments u = (u e , u m )T , v = (v e , v m )T . Consider the following sesquilinear forms and the corresponding operators: r k(u, v) :=

(u e v e + u m v m )dρ = (Ku, v), ∀v ∈ H, r0

 k(u, v) :=

r

u, v), ∀v ∈ H, q(u e v e + u v v m )dρ = (K

r0

r a(u, v) :=

(u e v e + u m v m + u e v e + u m v m )dρ = (Iu, v), ∀v ∈ H,

r0

r b(u, v) :=

p(u e v e + u m v m )dρ = (Bu, v), ∀v ∈ H,

r0

 s(u, v) =

  (κr ) γm (ε + iεlos )μ − γ 2 K m (ε + iεlos )μ − 1 u m (r ) − u e (r ) v e (r ) r (ε + iεlos ) κ(ε + iεlos ) K m (κr ) κ2    (κr ) γm (ε + iεlos )μ − 1 (ε + iεlos )μ − γ 2 K m + u m (r ) v m (r ) u e (r ) − rμ κμ K m (κr ) κ2 = (Su, v), ∀v ∈H.

(4.3.67) The boundedness of form a(u, v) is obvious. The boundedness of forms k(u, v) and  k(u, v) follows from the Poincar’e inequality [11].

150

4 Waveguides of Circular Cross Section

Now variational problem (4.3.66) can be written in the operator form (N(γ)u, v) = 0, ∀u ∈ H or, equivalently,    − K + I − B − S(γ) u = 0. N(γ)u := γ 2 K + K

(4.3.68)

Equation (4.3.68) is the operator form of variational relation (4.3.66). The characteristic numbers and eigenvectors of operator N by definition coincide with the eigenvalues and eigenvectors of problem Plos for γ 2 = (ε + iεlos )μ.

4.3.4 Properties of the Spectrum of the OVF We have reduced the problem on normal waves to the study of spectral properties of OVF N. In this connection, we first consider the properties of the operators introduced in the preceding section. The validity of Lemmas 4.3.1–4.3.3 is shown in Sect. 4.2.4.  : H → H are compact, and K > 0. Lemma 4.3.1 Bounded operators K and K Lemma 4.3.2 Operators B : H → H are compact. Lemma 4.3.3 Operator S : H → H is compact. Lemma 4.3.4 There exists a  γ ∈ R such that the operator N( γ ) is continuously invertible; i.e., the resolvent set (N) := {γ : ∃ N−1 (γ) : H → H } of operatorvalued function N( γ ) is nonempty, (N) = ∅. Proof Let γ ∈ R, γ > 0 and γ → +∞. Consider the OVF N(γ) = N1 (γ) + N2 (γ), where

 − K + B + I + γS0 , N1 (γ) = γ 2 K + K

and N2 (γ) = −S(γ) − γS0 , and the operator S0 : H → H is defined by the sesquilinear form   u e (r )v e (r ) u m (r )v m (r ) S0 u, v := . + ε + iεlos μ

(4.3.69)

4.3 Surface Waves in a Waveguide Filled with Lossy Medium

151

In view of the asymptotics of functions K m as γ → +∞ [12], we have K m (κr ) ∼ −1, K m (κr ) and

1 μ(ε + iεlos ) − γ 2 ∼ −γ + O . κ γ

Consequently,

N2 (γ) = O

1 γ

, as γ → +∞.

(4.3.70)

Then (OVFs) N(γ) can be considered as a perturbation of operator pencil N1 by OVF N2 for large γ. r Reb(u, u) := Re

p(u e u e

+

u m u m )dρ

r0

1 = 2

r p

     |u e |2 + |u m |2 dρ

r0

 p(r0 )   p(r )  = |u e (r )|2 + |u m (r )|2 − |u e (r0 )|2 + |u m (r0 )|2 2 2 r   1 − p  |u e |2 + |u m |2 dρ 2 r0

 p(r0 ) p(r )  = |u e (r )|2 + |u m (r )|2 − |u m (r0 )|2 2 2 r   1 − p  |u e |2 + |u m |2 dρ. 2 r0

It follows from the last relation, definition (4.3.69), asymptotic relation (4.3.70), and the properties of operators K and S0 that there exists a large number  γ , such that u, u) − (Ku, u) + Re(Bu, u) + u 2 +  Re(N1 ( γ )u, u) =  γ 2 (Ku, u) + Re(K γ Re(S0 u, u) ≥ u 2

for each u. Hence,  γ ∈ (N1 ), where (N1 ) is the resolvent set of OVF N1 . Moreover, Theorem 4.1 in [13] gives the estimate γ ) ≤ 1.

N1−1 ( γ ) < 1 is satisfied and find that there exists We take  γ such that the inequality N2 ( a bounded operator −1    γ ) + N2 ( γ) = I + N1−1 ( γ )N2 ( γ ) N1−1 ( γ ). N1 (

152

4 Waveguides of Circular Cross Section

 Theorem 4.3.1 The OVF N(γ) : H → H is bounded, holomorphic, and Fredholm in the domain  = C\ ( K ∪  E ) and  E := {γ : γ 2 = (ε + iεlos )μ}. Proof The functions κ

K m (κr ) K m+1 (κr ) m = −κ + . K m (κr ) K m (κr ) r

(4.3.71)

are analytical in the domain C\ K as functions of variable γ because functions K m (κr ) have no zeros for Reκ > 0 [12]. Then we obtain the desired result by Lemma 4.3.4. The proof of the theorem is complete.  Theorem 4.3.2 The spectrum of OVF N(γ) : H → H is discrete in ; i.e., this function has finitely many characteristic numbers of finite algebraic multiplicity on any compact set K 0 ⊂ . Proof The assertion of the theorem is a corollary of Theorem 4.3.1 and a theorem on a holomorphic OVF [13].  Lemma 4.3.5 The spectrum of OVF N(γ) : H → H is symmetric with respect to the real and imaginary axes. If γ0 is an eigenvalue of N(γ) corresponding to the eigenvector u1 = (u e , u m )T , then −γ0 , γ 0 , −γ 0 are also eigenvalues of OVF N(γ) corresponding to the eigenvectors u2 = (−u e , u m )T , u3 = (u e , u m )T , u4 = (−u e , u m )T with the same multiplicity. Proof The assertion of the lemma can readily be proved by a straightforward verification with the use of (4.3.57) and conditions (4.3.58) and (4.3.59). Note that the condition at infinity is also satisfied for all four characteristic numbers. 

4.4 Surface Waves in a Waveguide Filled with Anisotropic Media 4.4.1 Statement of the Problem Consider the problem on normal waves in GL described in Sect. 4.1.1. We assume that the permittivity and permeability in the entire space have the form  ε= and

ε, ˆ r0 ≤ ρ ≤ r, and  μ= 1, ρ > r,



μ, ˆ r0 ≤ ρ ≤ r. 1, ρ > r,

(4.4.72)

4.4 Surface Waves in a Waveguide Filled with Anisotropic Media

153

⎡

⎤ ⎡ ⎤ ερ 0 0 μρ 0 0 εˆ = ⎣ 0 ερ 0 ⎦, μˆ = ⎣ 0 μρ 0 ⎦, 0 0 εz (ρ) 0 0 μz (ρ)

(4.4.73)

where ερ > 1 and μρ > 1 are constants; εz (ρ) > 1 and μz (ρ) > 1 are twice continuously differentiable function on the segment [r0 , r ], i.e., εz (ρ) ∈ C 2 [r0 , r ] and μz (ρ) ∈ C 2 [r0 , r ], and Imεz (ρ) = Imμz (ρ) = 0. The problem on normal waves is an eigenvalue problem for the Maxwell equations with spectral parameter γ, which is the normalized propagation constant of the normal wave. Rewrite Maxwell’s equation in the expanded form: ⎧ m ⎪ i Hz − iγ Hϕ ⎪ ⎪ ρ ⎪ ⎪ ⎪ ⎪ ⎪ iγ Hρ − Hz ⎪ ⎪ ⎪ ⎪ 1 im ⎪ ⎪ ⎪ (ρHϕ ) − Hρ ⎨ ρ ρ m ⎪ ⎪ i E z − iγ E ϕ ⎪ ⎪ ρ ⎪ ⎪ ⎪ ⎪ ⎪ iγ E ρ − E z ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎩ (ρE ϕ ) − im E ρ ρ ρ

= −i ερ E ρ , = −i ερ E ϕ , = −i εz E z , = i μρ Hρ ,

(4.4.74)

= i μρ Hϕ , = i μz Hz ,

where  ερ =

ερ , r0 ≤ ρ ≤ r, and  μρ = 1, ρ > r,



μρ , r0 ≤ ρ ≤ r. 1, ρ > r,

(4.4.75)

μz (ρ), r0 ≤ ρ ≤ r. 1, ρ > r.

(4.4.76)

and  εz =

εz (ρ), r0 ≤ ρ ≤ r, and  μz = 1, ρ > r,



Express functions E ρ , Hρ , E ϕ , Hϕ via E z and Hz from the first, second, fourth, and fifth equations in system (4.4.74) m μρ Hz − iγρE z , ρ(γ 2 −  ερ  μρ ) γm E z + iρ μρ Hz Eϕ = , ρ(γ 2 −  ερ  μρ )

Eρ =

iγρHz + m ερ E z , ρ(γ 2 −  ερ  μρ ) γm Hz − iρ ερ E z Hϕ = . ρ(γ 2 −  ερ  μρ )

Hρ = −

(4.4.77)

It follows from (4.4.77) that the normal wave field in the waveguide can be represented with the use of two scalar functions

154

4 Waveguides of Circular Cross Section

u e := i E z (ρ), u m := Hz (ρ).

(4.4.78)

Thus, the problem has been reduced to finding longitudinal components u e and u m of the electric and magnetic fields. In what follows, ( · ) stands for differentiation with respect to ρ. We have the following eigenvalue problem for the longitudinal field components u e and u m : find γ ∈ C such that, for a given m ∈ Z, there exist nontrivial solutions of the system of differential equations ⎧      1 2  2  εz m2 ⎪ ⎪ ⎪ u e = 0, ερ  μρ + ⎨ ρu e − ρ ρ γ −   ερ  ερ   ⎪  μz m2 ⎪    1 2  2 ⎪ u m = 0, ρ γ − ερ  μρ + ⎩ ρu m − ρ  μρ  μρ

(4.4.79)

satisfying the boundary conditions for ρ = r0 u e (r0 ) = 0, u m (r0 ) = 0,

(4.4.80)

the transmission conditions for ρ = r [u e ]|r = 0, [u m ]|r = 0,            ρ μρ u m ρ ερ u e um ue  −  = 0, γm  −  = 0, γm  ερ μρ − γ 2 r  ερ μρ − γ 2 r  ερ μρ − γ 2 r  ερ μρ − γ 2 r 

where [ f ]|ρ0 = lim

ρ→ρ0 −0

f (ρ) − lim

ρ→ρ0 +0

(4.4.81) f (ρ) and the condition of boundedness of the

field in any finite domain and the decay condition at infinity. Once we determine the longitudinal field components u e and u m by solving problem (4.4.79)–(4.4.81), we can find the transverse components by formulas (4.4.77). The equivalence of the reduction to problem (4.4.79)–(4.4.81) is not valid only for ερ  μρ . γ2 =  For ρ > r , we have ε = 1,  μ = 1; then from system (4.4.79), we obtain the system     ρ(ρu e ) − ρ2 κ2 + m 2 u e = 0, ρ(ρu m ) − ρ2 κ2 + m 2 u m = 0, where κ2 = γ 2 − 1. In view of the condition at infinity, we obtain a solution of this system in the form u e (ρ; γ, m) = C1 K m (κρ), (4.4.82) u m (ρ; γ, m) = C2 K m (κρ), where K m is the modified Bessel function (the Macdonald function) [10]; C1 and C2 are constants.

4.4 Surface Waves in a Waveguide Filled with Anisotropic Media R

155

Function κ(γ) is analytical in the domain C\ K , where  K := {γ : Imγ 2 = 0, γ 2 ≤ 1}.

For r0 ≤ ρ ≤ r , we have  ε = εˆ and  μ = μ, ˆ and from (4.4.79) we obtain the system of differential equations   L e u e : = u e + pu e − γ 2 f e + qe u e = 0,   L m u m : = u m + pu m − γ 2 f m + qm u m = 0 where p=

1 , ρ

qe = −εz μρ +

fe =

εz , ερ

(4.4.83)

μz , μρ

fm =

m2 m2 , qm = −ερ μz + 2 . 2 ρ ερ ρ μρ

Definition 4.4.1 If for a given m there exist nontrivial functions u e and u m corresponding to some γ ∈ C such that these functions are given by (4.4.82) for ρ > r , form a solution of system (4.4.83) for r0 ≤ ρ ≤ r , and satisfy transmission conditions (4.4.81), then γ is called a characteristic number of problem Pani . Definition 4.4.2 The pair u e and u m , |u e |2 + |u m |2 ≡ 0, will be called an eigenvector of problem Pani corresponding to the characteristic number γ ∈ C.

4.4.2 Sobolev Spaces and Variational Relation Let us give a variational formulation of problem Pani . Multiplying equations (4.4.83) by arbitrary test functions ve ∈ H01 (r0 , r ) and vm ∈ H 1 (r0 , r ) (we can assume that these functions are continuously differentiable in (r0 , r )) and applying Green’s formula, we obtain r

r vLudρ =

r0

r



vu dρ + r0

=

pvu dρ − r0

r vu  r0

r −

u v dρ + r0

r0

r



pu vdρ − r0

r f uvdρ −

2

  v γ 2 f + q udρ

r0

r

 

r = −γ

r



r0

r quvdρ +

r0

  v γ 2 f + q udρ

r0

r pu vdρ + u v r0 − 



r

u  v  dρ,

r0

(4.4.84)

156

4 Waveguides of Circular Cross Section

where u = u j , v = v j , f = f j , q = q j , j = e or m. Taking into account the boundary condition for ρ = r0 , we get r

r vLudρ = −γ

r0

r f uvdρ −

2 r0

r quvdρ +

r0

 pu vdρ + u v r − 



r0

r

u  v  dρ

r0

(4.4.85) We separately apply formula (4.4.85) to the first and second equations in system (4.4.83) on the interval [r0 , r ], add the results, and find that the sum of the left-hand sides of (4.4.83) satisfies the relation r

r (v e L e u e + v m L m u m )dρ = −γ

( f e u e v e + f m u m v m )dρ

2

r0

r0

r −

r qe u e v e + qm u m v m )dρ +

r0

p(u e v e

+

u m v m )dρ

r0

r −

(u e v e + u m v m )dρ

r0

+u e (r )v e (r ) + u m (r )v m (r ). (4.4.86) Given the solutions (4.4.82), we express the values of the normal derivatives at ρ = r from relations (4.4.81) as follows: u e (r ) = u m (r )

γm μρ ερ − 1 μρ ερ − γ 2 K m (κr ) u e (r ), u m (r ) − 2 r ερ κ κερ K m (κr )

γm μρ ερ − 1 μρ ερ − γ 2 K m (κr ) u m (r ). = u (r ) − e r μρ κ2 κμρ K m (κr )

(4.4.87)

Then, in view of (4.4.86), from (4.4.87), we obtain r γ

r ( f e u e v e + f m u m v m )dρ +

2 r0

(qe u e v e + qm u m v m )dρ r0

r − r0

p(u e v e

+

u m v m )dρ

r +

(u e v e + u m v m )dρ

r0

 μρ ερ − γ 2 K m (κr ) γm μρ ερ − 1 u e (r ) v e (r ) u m (r ) − − r ερ κ2 κερ K m (κr )   γm μρ ερ − 1 μρ ερ − γ 2 K m (κr ) u − u (r ) − (r ) v m (r ) = 0. e m r μρ κ2 κμρ K m (κr ) 

(4.4.88)

Variational relation (4.4.88) has been obtained for smooth functions v e and v m .

4.4 Surface Waves in a Waveguide Filled with Anisotropic Media

157

4.4.3 Spectrum of the OVF Let H = H01 (r0 , r ) × H 1 (r0 , r ) be the Cartesian product of Hilbert spaces with the inner product and the norm (u, v) = (u 1 , v1 )1 + (u 2 , v2 )1 , u 2 = u 1 21 + u 2 21 ; u, v ∈ H, u = (u 1 , u 2 )T , v = (v1 , v2 )T , u 1 , v1 ∈ H01 (r0 , r ) , u 2 , v2 ∈ H 1 (r0 , r ) . Then the integrals occurring in (4.4.88) can be viewed as sesquilinear forms over the field C defined on H and depending on the arguments u = (u e , u m )T , v = (v e , v m )T . Consider the following sesquilinear forms and the corresponding operators: r k(u, v) :=

( f e u e v e + f m u m v m )dρ = (Ku, v), ∀v ∈ H, r0

r

 k(u, v) :=

u, v), ∀v ∈ H, ((qe − 1)u e v e + (qm − 1)u v v m )dρ = (K

r0

r a(u, v) :=

(u e v e + u m v m + u e v e + u m v m )dρ = (Iu, v), ∀v ∈ H,

r0

r b(u, v) :=

p(u e v e + u m v m )dρ = (Bu, v), ∀v ∈ H,

r0



 γm μρ ερ − 1 μρ ερ − γ 2 K m (κr ) u u (r ) − (r ) v e (r ) m e r ερ κ2 κερ K m (κr )   γm μρ ερ − 1 μρ ερ − γ 2 K m (κr ) u + u (r ) − (r ) v m (r ) e m r μρ κ2 κμρ K m (κr ) s(u, v) =

= (Su, v), ∀v ∈ H.

(4.4.89)

The boundedness of a(u, v) is obvious. The boundedness of forms k(u, v) and  k(u, v) follows from the Poincar’e inequality [11]. Now variational problem (4.4.88) can be written in the operator form

158

4 Waveguides of Circular Cross Section

(N(γ)u, v) = 0, ∀u ∈ H or, equivalently,    + I − B − S(γ) u = 0. N(γ)u := γ 2 K + K

(4.4.90)

Equation (4.4.90) is the operator form of variational relation (4.4.88). The characteristic numbers and eigenvectors of operator N by definition coincide with the eigenvalues and eigenvectors of problem Pani for γ 2 = μρ ερ .

4.4.4 Properties of the Spectrum of OVF We have reduced the problem on normal waves to the study of spectral properties of OVF N. In this connection, we first consider the properties of the operators introduced in the preceding section. The validity of Lemmas 4.4.1–4.4.3 is shown in Sect. 4.2.3.  : H → H are compact, and K > 0. Lemma 4.4.1 Bounded operators K and K Lemma 4.4.2 Operators B : H → H are compact. Lemma 4.4.3 Operator S : H → H is compact. Lemma 4.4.4 There exists a  γ ∈ R such that operator N( γ ) is continuously invertγ ) is ible; i.e., the resolvent set (N) := {γ : ∃ N−1 (γ) : H → H } of OVF N( nonempty, (N) = ∅. Proof Let γ ∈ R, γ > 0 and γ → +∞. Consider the OVF N(γ) = N1 (γ) + N2 (γ), where

 + B + I + γS0 , N1 (γ) = γ 2 K + K

and N2 (γ) = −S(γ) − γS0 , and the operator S0 : H → H is defined by the sesquilinear form   u e (r )v e (r ) u m (r )v m (r ) S0 u, v := + . ερ μρ In view of the asymptotics of functions K m as γ → +∞ [12], we have K m (κr ) ∼ −1, K m (κr )

(4.4.91)

4.4 Surface Waves in a Waveguide Filled with Anisotropic Media

and

159

1 ερ μρ − γ 2 . ∼ −γ + O κ γ

Consequently,

N2 (γ) = O

1 γ

, as γ → +∞.

(4.4.92)

Then (OVFs) N(γ) can be considered as a perturbation of operator pencil N1 by OVF N2 for large γ. We have r Reb(u, u) := Re

p(u e u e

+

u m u m )dρ

r0

1 = 2

r p

     |u e |2 + |u m |2 dρ

r0

 p(r0 )   p(r )  = |u e (r )|2 + |u m (r )|2 − |u e (r0 )|2 + |u m (r0 )|2 2 2 r   1 − p  |u e |2 + |u m |2 dρ 2 r0

 p(r0 ) p(r )  = |u e (r )|2 + |u m (r )|2 − |u m (r0 )|2 2 2 r   1 − p  |u e |2 + |u m |2 dρ. 2 r0

It follows from the last relation, definition (4.4.91), asymptotic relation (4.4.92), and the properties of operators K and S0 that there exists a large number  γ , such that u, u) + Re(Bu, u) + (u, u) +  γ )u, u) =  γ 2 (Ku, u) + (K γ (S0 u, u) ≥ u 2 Re(N1 ( for each u. Hence,  γ ∈ (N1 ), where (N1 ) is the resolvent set of OVF N1 . Moreover, Theorem 4.1 in [13] gives the estimate γ ) ≤ 1.

N1−1 ( γ ) < 1 is satisfied and conclude that there We take  γ such that the inequality N2 ( exists a bounded operator −1    γ ) + N2 ( γ) = I + N1−1 ( γ )N2 ( γ ) N1−1 ( γ ). N1 ( 

160

4 Waveguides of Circular Cross Section

Theorem 4.4.1 The OVF N(γ) : H → H is bounded, holomorphic, and Fredholm √ in the domain  = C\( K ∪  E ), where  E := {γ : Imγ = 0, γ = ± ερ μρ }. Proof The functions κ

K m (κr ) K m+1 (κr ) m = −κ + . K m (κr ) K m (κr ) r

(4.4.93)

are analytical in the domain C\ K as functions of γ because functions K m (κr ) have no zeros for Reκ > 0 [12]. Finally, we obtain the desired result by Lemma 4.4.4. The proof of the theorem is complete.  Theorem 4.4.2 The spectrum of OVF N(γ) : H → H is discrete in ; i.e., this function has finitely many characteristic numbers of finite algebraic multiplicity on any compact set K 0 ⊂ . Proof The assertion of the theorem is a corollary of Theorem 4.4.1 and a theorem on a holomorphic OVF [13].  Lemma 4.4.5 The spectrum of OVF N(γ) : H → H is symmetric with respect to the real and imaginary axes. If γ0 is an eigenvalue of N(γ) corresponding to the eigenvector u1 = (u e , u m )T , then −γ0 , γ 0 , −γ 0 are also eigenvalues of OVF N(γ) corresponding to the eigenvectors u2 = (−u e , u m )T , u3 = (u e , u m )T , u4 = (−u e , u m )T with the same multiplicity. Proof The assertion of the lemma can readily be proved by a straightforward verification with the use of (4.4.79) and conditions (4.4.80) and (4.4.81). Note that the condition at infinity is satisfied for all four characteristic numbers. 

4.5 Surface Waves in a Waveguide Filled with Inhomogeneous Chiral Media 4.5.1 Statement of the Problem Consider the three-dimensional space R3 with the cylindrical coordinate system Oρϕz. The space is filled with an isotropic source-free medium having permittivity ε = ε0 ≡ const and permeability μ = μ0 ≡ const, where ε0 and μ0 are permittivity and permeability of vacuum. A waveguide with a cross section  := {(ρ, ϕ, z) : r0  ρ  r, 0  ϕ < 2π} and a generating line parallel to the axis Oz is placed in R3 .

4.5 Surface Waves in a Waveguide Filled with Inhomogeneous Chiral Media

161

Fig. 4.2 Geometry of the problem

The cross section of the waveguide by a plane perpendicular to its axis consists of two concentric circles of radii r0 and r (see Fig. 4.2): r is the radius of the internal (perfectly conducting) cylinder and r − r0 is the thickness of the external (dielectric) cylindrical shell. The geometry of the problem is shown in Fig. 4.2. The problem on normal waves in a waveguide structure reduces to finding nontrivial running-wave solutions to the homogeneous system of Maxwell equations, i.e., solutions with the dependence of the from eiγz on the coordinate z along which the structure is regular,  curl H = −iω εE − ω χH, (4.5.94) curl E = iω μH − ω χE,   E = E ρ (ρ) eρ + E ϕ (ρ) eϕ + E z (ρ) ez eiγz ,   H = Hρ (ρ) eρ + Hϕ (ρ) eϕ + Hz (ρ) ez eiγz ,

(4.5.95)

with the boundary conditions for the tangential electric components on perfectly conducting surfaces (ρ = r0 )  E ϕ ρ=r0 = 0, E z |ρ=r0 = 0, (4.5.96) transmission conditions for tangential electric and magnetic components on the surfaces of “discontinuity” of permittivity and permeability (ρ = r )  [E ϕ ]ρ=r = 0, [E z ]|ρ=r = 0,  [Hϕ ]ρ=r = 0, [Hz ]|ρ=r = 0, the condition of the finiteness of energy

(4.5.97)

162

4 Waveguides of Circular Cross Section

∞ ( ε|E|2 +  μ|H|2 )dρ < ∞,

(4.5.98)

r0

and the radiation condition at infinity: the electromagnetic field decays as o(ρ-1/2 ) for ρ → ∞. Permittivity, permeability, and chirality in the whole space have the form



μ(ρ), r0 ≤ ρ ≤ r, , χ= μ0 , ρ > r.



χ, r0 ≤ ρ ≤ r, , 0, ρ > r. (4.5.99) where ε(ρ) ∈ C 1 [r0 , r ], min ε(ρ) > ε0 , μ(ρ) ∈ C 1 [r0 , r ], min μ(ρ) > μ0 and χ  ε=

ε(ρ), r0 ≤ ρ ≤ r, , μ= ε0 , ρ > r. [r0 , r ]

[r0 , r ]

is a constant. The problem on normal waves is an eigenvalue problem for the Maxwell equations with spectral parameter γ which is the normalized propagation constant. Rewrite system (4.5.94) in the expanded form: ⎧ −iγ Hϕ = −iω ε E ρ − ω χ Hρ , ⎪ ⎪ ⎪  ⎪ ⎪ iγ H − H = −iω ε E − ω χ Hϕ , ρ ϕ ⎪ z ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ (ρHϕ ) = −iω ε E z − ω χ Hz , ⎨ ρ ⎪ −iγ E ϕ = iω μ Hρ − ω χ Eρ, ⎪ ⎪ ⎪ ⎪  ⎪ iγ E ρ − E z = iω μ Hϕ − ω χ Eϕ, ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎩ (ρE ϕ ) − = iω μ Hz − ω χEz , ρ

(4.5.100)

and express functions E ρ , Hρ , E z , Hz via functions E ϕ and Hϕ from the 1st, 3rd, 4th, and 6th equations in system (4.5.94) χ Eϕ −  μ Hϕ γ i , ω  χ2 −  ε μ χ Hϕ +  ε Eϕ γ i Hρ = , ω  χ2 −  ε μ

Eρ =

χ(ρE ϕ ) + i μ(ρHϕ ) 1  , ωρ  χ2 −  ε μ χ(ρHϕ ) − i ε(ρE ϕ ) 1  Hz = − . ωρ  χ2 −  ε μ

Ez = −

(4.5.101)

It follows from (4.5.101) that the normal wave field in the waveguide can be represented with the use of two scalar functions u e := i E ϕ (ρ), u m := Hϕ (ρ).

(4.5.102)

Thus, the problem has been reduced to finding the tangential components u e and u m of the electric and magnetic fields. Throughout the text, ( · ) stands for differentiation with respect to ρ.

4.5 Surface Waves in a Waveguide Filled with Inhomogeneous Chiral Media

163

For the tangential field components u e and u m , we have the following eigenvalue problem: find such γ ∈ C for which there exist nontrivial solutions of the following system of differential equations:   u e +  h e u e +  ge − γ 2 u e =  f m (ρu m ) +  km u m ,     2  um +  hm um +  ge − γ u m =  f e (ρu e ) +  ke u e , where  he =  ge = ω 2 ( χ2 + ε μ) −

(4.5.103)

1 1 μ  ε μ  ε   + , and h + , = m 2 2  χ − ε μ ρ  χ − ε μ ρ μ ε μ ε 1 1  1 1  and  gm = ω 2 ( , + χ2 + ε μ) − 2 + 2 2 2 ρ ρ χ − ε μ ρ ρ χ − ε μ

 fe =

 χ ε  χ μ   and   , = f m ρ  χ2 −  ε μ ρ  χ2 −  ε μ

 χ ε and  km = −2ω 2  χ μ, ke = −2ω 2  satisfying the boundary and transmission conditions  u e |r0 = 0, u m r0 = 0, [u e ]|r = 0, [u m ]|r = 0,      χ(ρu m ) −   χ(ρu e ) −  μ(ρu m )  ε(ρu e )  = 0,   = 0,  χ2 −  ε μ  χ2 −  ε μ r

(4.5.104)

r

the condition of boundedness of the field in any finite domain, and the condition of decay at infinity. μ = μ0 and  χ = 0; then from (4.5.103), we obtain For ρ > r , we have  ε = ε0 ,  the system   1   2 (ρu e ) − ρ κ + 2 u e = 0, ρ   1   2 (ρu m ) − ρ κ + 2 u m = 0, ρ where κ2 = γ 2 − ω 2 ε0 μ0 . In view of the condition at infinity, a solution of this system is sought in the form

u e (ρ; γ, m) = C1 K 1 (κρ), u m (ρ; γ, m) = C2 K 1 (κρ),

(4.5.105)

164

4 Waveguides of Circular Cross Section

where K 1 is the modified Bessel function (the Macdonald function) [12] and C1 and C2 are constants. R

Due to the condition at infinity, we choose the following branch of the square root:     1 κ = γ 2 − εc μ0 = √ |γ 2 − ε0 μ0 | + Re γ 2 − ε0 μ0 2      2 |γ 2 − ε0 μ0 | − Re γ 2 − ε0 μ0 . (4.5.106) + isignIm γ − ε0 μ0

ε = ε (ρ) ,  μ = μ (ρ) and  χ = χ, and from system For r0 ≤ ρ ≤ r , we have  (4.5.103), we obtain the system of differential equations   Lu e : = u e + h e u e + ge − γ 2 u e = f m (ρu m ) + km u m ,   Lu m : = u m + h m u m + ge − γ 2 u m = f e (ρu e ) + ke u e , where he =

(4.5.107)

1 1 εμ ε μ + , and h + , = m 2 2 χ − εμ ρ χ − εμ ρ

ε μ 1 1 εμ 2 (χ2 + εμ) − 1 + 1 ge = ω 2 (χ2 + εμ) − 2 + = ω and g , m ρ χ2 − εμ ρ χ2 − εμ ρ ρ2

fe =

χε χμ  and f m =  , ρ χ2 − εμ ρ χ2 − εμ 

ke = −2ω 2 χε and km = −2ω 2 χμ, Knowing the solution in free space, the problem (4.5.103)–(4.5.104) can be reduced to an eigenvalue problem on the segment [r0 , r ]. Introduce the following. Definition 4.5.1 If there exist nontrivial functions u e and u m corresponding to some γ ∈ C such that these functions are the solutions (4.5.105) for ρ > r , solve system (4.5.107) for r0 ≤ ρ ≤ r , and satisfy conditions (4.5.104), then γ is called a characteristic number of the problem. Definition 4.5.2 The pair u e and u m , |u e |2 + |u m |2 ≡ 0 is called an eigenvector of the problem corresponding to the characteristic number γ ∈ C.

4.5 Surface Waves in a Waveguide Filled with Inhomogeneous Chiral Media

165

4.5.2 Sobolev Spaces and Variational Relation We will look for solutions u e and u m of problem Pm in the Sobolev spaces   H01 (r0 , r ) = f : f ∈ H 1 (r0 , r ) , f (r0 ) = 0 and H 1 (r0 , r ) , with the inner product and the norm r ( f, g)1 =



r



 

f g + f g dρ,

f 21

= ( f, f )1 =

r0

 2  | f | + | f |2 dρ.

r0

Let us give a variational formulation of problem Pm . Multiply equations (4.5.107) by arbitrary test functions ve ∈ H01 (r0 , r ) and vm ∈ H 1 (r0 , r ) (one can assume that these functions are continuously differentiable in (r0 , r )) and apply Green’s formula to obtain r

r vLudρ =

r0

r



u vdρ + r0

r



hu vdρ − r0

 2  γ − g uvdρ

r0

r = u v r0 − 

r

r

 

u v dρ + r0



r

hu vdρ− r0

 2  γ − g uvdρ,

r0

(4.5.108) where u = u j , v = v j , h = h j , g = g j , j = e or m. We separately apply formula (4.5.108) to the first and second equations in system (4.5.107) on the interval [r0 , r ] and add the results to verify that the sum of left-hand sides of (4.5.107) satisfies the relation r

r (v e Lu e + v m Lu m )dρ = −γ

r0

(u e v e + u m v m )dρ

2 r0

r − r0

r +

(u e v e + u m v m )dρ +

r

(h e u e v e + h m u m v m )dρ

r0

(ge u e v e + gm u m v m )dρ + (u e (r )v e (r ) + u m (r )v m (r )).

r0

(4.5.109) On the other hand, for the right-hand sides of the equations in system (4.5.107), we have

166

4 Waveguides of Circular Cross Section

r

r (v e Lu e + v m Lu m )dρ ==

r0

( f e (ρu e ) v m + f m (ρu m ) v e )dρ +

r0

r (ke u e v m + km u m v e )dρ. r0

(4.5.110) Given the solutions (4.5.105), we express the values of the normal derivatives at ρ = r from relations (4.5.104) as follows:  μ 1 χ u e (r ) + κ F(γ)u m (r ), F(γ) − ε0 r ε0   ε 1 χ u m (r ) + κ F(γ)u e (r ), F(γ) − u m (r ) = κ μ0 r μ0 u e (r ) = κ



where F(γ) = −

(4.5.111)

K 0 (κr ) . K 1 (κr )

From (4.5.109), taking into account (4.5.110) and (4.5.111), we obtain the sought variational relation r γ

r (u e v e + u m v m )dρ +

2 r0

(u e v e + u m v m + u e v e + u m v m )dρ

r0

r −

r ((ge + 1)u e v e + (gm + 1)u m v m )dρ −

r0

(h e u e v e + h m u m v m )dρ

r0

r + r0

( f e (ρu e ) v m + f m (ρu m ) v e )dρ +

r (ke u e v m + km u m v e )dρ r0

    1 χ μ u e (r ) + κ F(γ)u m (r ) v e (r ) F(γ) − + κ ε0 r ε0     1 χ ε u m (r ) + κ F(γ)u e (r ) v m (r ) = 0. F(γ) − + κ μ0 r μ0 (4.5.112)

We note that variational relation (4.5.112) has been obtained for smooth functions v e and v m . Let H = H01 (r0 , r ) × H 1 (r0 , r ) be the Cartesian product of the Hilbert spaces with the inner product and the norm (u, v) = (u 1 , v1 )1 + (u 2 , v2 )1 , u 2 = u 1 21 + u 2 21 ; u, v ∈ H, u = (u 1 , u 2 )T , v = (v1 , v2 )T , u 1 , v1 ∈ H01 (r0 , r ) , u 2 , v2 ∈ H 1 (r0 , r ) .

4.5 Surface Waves in a Waveguide Filled with Inhomogeneous Chiral Media

167

Consider the following sesquilinear forms and the corresponding operators: r k(u, v) :=

(u e v e + u m v m )dρ = (Ku, v), ∀v ∈ H, r0

r k1 (u, v) :=

((ge + 1)u e v e + (gm + 1)u m v m )dρ = (K1 u, v), ∀v ∈ H, r0

 k(u, v) :=

r

u, v), ∀v ∈ H, (ke u e v m + km u m v e )dρ = (K

r0

r a(u, v) :=

(u e v e + u m v m + u e v e + u m v m )dρ = (Iu, v), ∀v ∈ H,

r0

r b1 (u, v) :=

(h e u e v e + h m u m v m )dρ = (B1 u, v), ∀v ∈ H,

r0

r b2 (u, v) :=

( f e (ρu e ) v m + f m (ρu m ) v e )dρ = (B2 u, v), ∀v ∈ H,

r0

    μ 1 χ u e (r ) + κ F(γ)u m (r ) v e (r ) s(u, v) = κ F(γ) − ε0 r ε0     1 χ ε u m (r ) + κ F(γ)u e (r ) v m (r ) = (S(γ)u, v), ∀v ∈ H. F(γ) − + κ μ0 r μ0

(4.5.113) Now variational problem (4.5.112) can be written in the operator form (N(γ)u, v) = 0, ∀u ∈ H or, equivalently,    + S(γ) u = 0. N(γ)u := γ 2 K + I − K1 − B1 + B2 + K

(4.5.114)

Equation (4.5.114) is the operator form of variational relation (4.5.112). The characteristic numbers and eigenvectors of operator N by definition coincide with the eigenvalues and eigenvectors of the original problem.

168

4 Waveguides of Circular Cross Section

4.5.3 Properties of the OVF The problem on surface electromagnetic waves in an inhomogeneous open waveguide has been reduced to the study of the spectral properties of OVF N. We present the following statements about the properties of operators entering N(γ) (see the proof in [14]):  : H → H are compact and Lemma 4.5.1 The bounded operators K, K1 and K K > 0. Lemma 4.5.2 The operators B1 and B2 : H → H are compact. Lemma 4.5.3 The operator S : H → H is compact. Lemma 4.5.4 There exists a  γ ∈ R such that the operator N( γ ) is continuously invertible; i.e., the resolvent set (N) := {γ : ∃ N−1 (γ) : H → H } of the operator function N( γ ) is nonempty, (N) = ∅. √ Proof Let γ0 = ω ε0 μ0 , then κ=

 γ 2 − ω 2 ε0 μ0 = 0.

(4.5.115)

Consider OVF N(γ) on the set [γ0 , γ0 + t) , for some t > 0. This function is continuous on the indicated set due to (4.5.113), (4.5.114) and (4.5.115) and the asymptotics of functions as z → +0, (z ∈ R) K 0 (z) ∼ − ln z, K 1 (z) ∼

1 , F(γ0 ) = 0. z

Then if the operator N−1 (γ0 ) : H → H, then there is a 0 < t0 < t, such that the operator N−1 (γ0 + t0 ) will be bounded and γ0 + t0 ∈ (N ). Thus, since N(γ) is a Fredholm operator, it suffices to prove that the equation N(γ0 )u = 0 has only a trivial solution. Varying the functions ve and vm in (4.5.112), we see that (4.5.111) holds. Then for γ = γ0 (respectively, κ = 0) we have u e (r ) = u m (r ) = 0. Substituting these expressions into conditions (4.5.104), it is easy check that for χ2 = (ε(r ) − ε0 )(μ(r ) − μ0 ) it is necessary that u e (r ) = u m (r ) = 0. We obtain a Cauchy problem for a system of second-order differential equations (4.5.103) with homogeneous (zero) initial conditions. Whence, due to the smoothness of the coefficients, we find that u e (ρ) ≡ u m (ρ) ≡ 0. Note that the point γ0 + t0 lies in the domain where operatorvalued function N(γ) is holomorphic. Thus, we have proved that the resolvent set is not empty.  Theorem 4.5.1 OVF N(γ) : H → H is bounded, holomorphic, and Fredholm in ,   = {γ : Imγ 2 = 0, γ 2 ≤ ω 2 ε0 μ0 }. the domain  = C\

4.5 Surface Waves in a Waveguide Filled with Inhomogeneous Chiral Media

169

Proof In the domain , N(γ) : H → H is bounded and holomorphic. Operatorvalued function  N(γ) is Fredholm as the sum of invertible I and compact K, K1 , B1 ,  and S operators.  B2 , K Theorem 4.5.2 The spectrum of OVF N(γ) : H → H is discrete in the domain ; i.e., this function has finitely many characteristic numbers of finite algebraic multiplicity on any compact set K 0 ⊂ . Proof The assertion of the theorem is a corollary of Theorem 4.5.1 and properties of a holomorphic OVF [13]. 

4.5.4 Numerical Results Using the projection method [15, 16], we reduce variational equation (4.5.112) to a system of algebraic equations. First, split an interval [r0 , r ] into n subintervals with the length h=

r0 − r . n

Define a set of n subintervals i = [r0 + (i − 1)h, r0 + (i + 1)h], i = 1, ..., n − 1 and n = [r0 + (n − 1)h, r ], and set of n + 1 subintervals 1 = [r0 , r0 + h],  j = [r0 + (i − 2)h, r0 + i h], j = 2, ..., n and n+1 = [r0 + (n − 1)h, r ]. These subintervals are called base finite elements. The basis functions φi defined on i are ⎧ ⎪ ⎨

ρ − r0 − (i − 1)h , h φi = ⎪ ⎩ − ρ − r0 − (i + 1)h , h and

ρ < r0 + i h, ρ > r0 + i h,

, i = 1, n − 1

(4.5.116)

170

4 Waveguides of Circular Cross Section

φn =

ρ−r +h ; l

(4.5.117)

The basis functions ψi defined on i are ψ1 = −

ρ2 − 2r0 ρ + r02 − h 2 , h2

⎧ ⎪ ⎪ ⎨

ρ2 − 2r0 ρ + r02 , h2 ψ2 = ⎪ ρ − r0 − 2h ⎪ ⎩− , h ⎧ ⎪ ⎨

ρ − r0 − (i − 2)h , h ψj = ⎪ ⎩ − ρ − r0 − i h , h

(4.5.118)

ρ < r0 + h, (4.5.119) ρ > r0 + h,

ρ < r0 + (i − 1)h,

, j = 3, n

(4.5.120)

ρ > r0 + (i − 1)h,

and ψn+1 =

ρ−r +h . h

(4.5.121)

Such basis functions take into account the physical nature of the problem under consideration. We look for an approximate solution with real coefficients αi and β j such that ue =

n $

αi φi ,

i=1

um =

n+1 $

βjψj.

(4.5.122)

j=1

Substituting functions u e and u m with representations (4.5.122) into the variational equation (4.5.112), we obtain a system of linear equations with respect to αi and β j (for a fixed value of γ) A(γ)x = 0, (4.5.123) where matrices A(γ) and x have the form ⎛

A1,1 · · · A1,n ee ee ⎜ .. .. .. ⎜ . . . ⎜ n,1 n,n ⎜ A · · · A ee ⎜ ee A=⎜ ⎜ 1,1 1,n ⎜ A ⎜ me · · · Ame ⎜ .. .. .. ⎝ . . . n+1,1 · · · An+1,n Ame me

A1,1 mm .. .

An+1,1 mm

⎞

⎛

⎞ α1 ⎜ .. ⎟ ⎟ ⎜ . ⎟ ⎟ ⎜ ⎟ ⎟ ⎜ αn ⎟ ⎟ ⎜ ⎟ ⎟ ⎟, x = ⎜ ⎟, ⎜ ⎟ ⎟ 1,n+1 ⎟ ⎜ ⎟ · · · Amm ⎟ ⎜ β1 ⎟ ⎜ ⎟ .. . ⎟ .. ⎝ .. ⎠ ⎠ . . · · · An+1,n+1 βn+1 mm

1,n+1 A1,1 em · · · Aem .. .. .. . . . n,n+1 An,1 · · · A em em

4.5 Surface Waves in a Waveguide Filled with Inhomogeneous Chiral Media

171

and 

i, j

Aee = γ 2 i

i, j Aem =





i i, j



Ame = i

i, j

 φi φ j dρ +



i

φi φj dρ −



 ge φi φ j dρ −

i



h e φi φ j dρ

i

 μ(r ) 1 +κ φi (r )φ j (r ), i, j = 1, n; F(γ) − ε0 r

 χ f e (ρφi ) + ke φi ψ j dρ + κ F(γ)φi (r )ψ j (r ), i = 1, n, j = 1, n + 1, μ0

 χ f m (ρψi ) + km ψi φ j dρ + κ F(γ)ψi (r )φ j (r ), i = 1, n + 1, j = 1, n, ε0





Amm = γ 2

ψi ψ j dρ + i

i

ψi ψ j dρ −



 gm ψi ψ j dρ −

i

h m ψi ψ j dρ

i

 ε(r ) 1 ψi (r )ψ j (r ), i, j = 1, n + 1. +κ F(γ) − μ0 r 

As a model problem, consider the following configuration of parameters: r0 = ρ 1, r = 3, ε = 4 + , μ = 1, ε0 = μ0 = 1. Dispersion curves (graph of the depenr dence of the normalized propagation constant γ/ω on the frequency ω) are shown in the following figures. Figures 4.3 and 4.4 show that the spectrum of surface waves propagating in a waveguide filled with a chiral medium (χ = 0.00125), at low frequencies coincide with the spectrum of a dielectric waveguide. However, with an increase of the chirality coefficient, the wave spectrum undergoes a noticeable deviation from the corresponding spectrum of an inhomogeneous dielectric waveguide. Note that when the parameter χ is equal to zero, the problem splits into two independent problems of TE- and TM- polarized surface waves. The dispersion curves for such problem are shown in Figs. 4.3 and 4.4 in blue and green color. Next, we plot solutions to the problem inside the waveguide for the selected frequency ω = 1.0. These solutions are presented in Figs. 4.5 and 4.6.

172

4 Waveguides of Circular Cross Section

Fig. 4.3 Dispersion curves. The red dot lines correspond to the chiral filling of the waveguide χ = 0.0125. Blue and green curves correspond to a waveguide filled with an inhomogeneous dielectric χ=0

Fig. 4.4 Dispersion curves. The red dot lines correspond to the chiral filling of the waveguide χ = 1.25. Blue and green curves correspond to a waveguide filled with an inhomogeneous dielectric χ=0

4.6 Leaky Waves in an Inhomogeneous Waveguide 4.6.1 Statement of the Problem The problem on normal waves in GL described in Sect. 4.1.1 is an eigenvalue problem for the Maxwell equations with spectral parameter γ, which is the normalized propagation constant of the normal wave. Rewrite Maxwell’s equation in the expanded form:

4.6 Leaky Waves in an Inhomogeneous Waveguide

173

Fig. 4.5 Fields u e and u m . The red dots correspond to the chiral filling of the waveguide χ = 0.0125. Blue lines corresponds to a waveguide filled with an inhomogeneous dielectric χ = 0

Fig. 4.6 Fields u e and u m . The red dots correspond to the chiral filling of the waveguide χ = 1.25. Blue lines corresponds to a waveguide filled with an inhomogeneous dielectric χ = 0

174

4 Waveguides of Circular Cross Section

⎧ m ⎪ i Hz − iγ Hϕ ⎪ ⎪ ρ ⎪ ⎪ ⎪ ⎪ ⎪ iγ Hρ − Hz ⎪ ⎪ ⎪ ⎪ 1 im ⎪  ⎪ ⎪ Hρ ⎨ (ρHϕ ) − ρ ρ m ⎪ ⎪ i E z − iγ E ϕ ⎪ ⎪ ρ ⎪ ⎪ ⎪ ⎪ ⎪ iγ E ρ − E z ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1 (ρE ϕ ) − im E ρ ρ ρ

= −i ε Eρ, = −i ε Eϕ, = −i ε Ez , = i μ Hρ ,

(4.6.124)

= i μ Hϕ , = i μ Hz ,

and express functions E ρ , Hρ , E ϕ , Hϕ via E z and Hz from the first, second, fourth, and fifth equations in system (4.6.124) m μ Hz − iγρE z , ρ(γ 2 −  ε μ) γm E z + iρ μ Hz , Eϕ = ρ(γ 2 −  ε μ)

Eρ =

iγρHz + m εEz , ρ(γ 2 −  ε μ) γm Hz − iρ ε E z Hϕ = . ρ(γ 2 −  ε μ)

Hρ = −

(4.6.125)

It follows from (4.6.125) that the normal wave field in the waveguide can be represented with the use of two scalar functions u e := i E z (ρ), u m := Hz (ρ).

(4.6.126)

Thus, the problem has been reduced to finding longitudinal components u e and u m of the electric and magnetic fields. We have the following eigenvalue problem for the longitudinal field components u e and u m : find γ ∈ C such that, for a given m ∈ Z, there exist nontrivial solutions of the system of differential equations ⎧      ερ  ε m2 ε μ) ( ⎪  2 ⎪ ρ u − + = γm u ⎪ e  2 u m , ⎪ ⎨ γ2 −  ε μ e ρ γ2 −  ε μ γ2 −  ε μ     ⎪  μρ  μ 2 m2 ε μ) ( ⎪  ⎪ u ρ u m = γm  − + ⎪ 2 u e , m ⎩ γ2 −  2 ε μ ρ γ − ε μ γ2 −  ε μ

(4.6.127)

satisfying the boundary conditions for ρ = r0 u e (r0 ) = 0, u m (r0 ) = 0, the transmission conditions for ρ = r

(4.6.128)

4.6 Leaky Waves in an Inhomogeneous Waveguide

175

[u e ]|r = 0, [u m ]|r = 0,           ρ μu m  um ρ εu e  ue  −  − = 0, γm = 0, γm  ε μ − γ 2 r  ε μ − γ 2 r  ε μ − γ 2 r  ε μ − γ 2 r (4.6.129) where [ f ]|ρ0 = lim f (ρ) − lim f (ρ) and the conditions of the boundedness 

ρ→ρ0 −0

ρ→ρ0 +0

of the field in any finite domain and increasing at infinity. Once we determine the longitudinal field components u e and u m by solving problem (4.6.127)–(4.6.129), we can find the transverse components by formulas (4.6.125). The equivalence of the reduction to problem (4.6.127)–(4.6.129) is not ε μ. valid only for γ 2 =  For ρ > r , we have  ε = 1,  μ = 1; then from (4.6.127), we obtain the system     ρ(ρu e ) − ρ2 κ2 + m 2 u e = 0, ρ(ρu m ) − ρ2 κ2 + m 2 u m = 0, where κ2 = γ 2 − 1. In view of the condition at infinity, we obtain a solution of this system in the form u e (ρ; γ, m) = C1 Im (κρ), (4.6.130) u m (ρ; γ, m) = C2 Im (κρ), where Im is the modified Bessel function (the Infield function) [10] and C1 and C2 are constants. R

Function κ(γ) is analytical in the domain C\ K , where  K := {γ : Imγ 2 = 0, γ 2 ≤ 1}.

For r0 ≤ ρ ≤ r , we have  ε = ε (ρ) and  μ = μ (ρ), and from system (4.6.127), we obtain the system of differential equations       L e u e : = με − γ 2 u e + pe γ 2 + qe u e + γ 4 + r1 γ 2 + r2 u e = γ f e u m ,       L m u m : = με − γ 2 u m + pm γ 2 + qm u m + γ 4 + r1 γ 2 + r2 u m = γ f m u e , (4.6.131) where 1 1 ε μ pm = − − , pe = − − , ε ρ μ ρ qe = −μ ε + r1 =

με , ρ

m2 − 2με, ρ2

fe = −

m (με) , ρ ε

qm = −με +

με , ρ

  m2 r2 = με με − 2 , ρ fm = −

m (με) . ρ μ

176

4 Waveguides of Circular Cross Section

Definition 4.6.1 If for a given m there exist nontrivial functions u e and u m corresponding to γ ∈ C such that these functions are given by (4.6.130) for ρ > r , form a solution of system (4.6.131) for r0 ≤ ρ ≤ r , and satisfy transmission conditions (4.6.129), then γ is called a characteristic number of problem Pinh . Definition 4.6.2 The pair u e and u m , |u e |2 + |u m |2 ≡ 0, will be called an eigenvector of problem Pinh corresponding to the characteristic number γ ∈ C.

4.6.2 Sobolev Spaces and Variational Relation Let us give a variational formulation of problem Pinh . Multiplying equations (4.6.131) by arbitrary test functions ve ∈ H01 (r0 , r ) and vm ∈ H 1 (r0 , r ) (we can assume that these functions are continuously differentiable in (r0 , r )) and applying Green’s formula, we obtain r

r

  v με − γ 2 u  dρ +

vLudρ = r0

r0

r





r



v pγ + q u dρ + 2

r0

  r = v με − γ 2 u  r −

  v γ 4 + r1 γ 2 + r2 udρ

r0

r

    v με − γ 2 + v (με) u  dρ

0

r0

r r0

= γ4



r



  v γ 4 + r1 γ 2 + r2 udρ

v pγ + q u dρ +

+ r



⎛

uvdρ + γ 2 ⎝

r0

2

r0

r

u  v  dρ +

r0

r −

r

pu  vdρ +

r0

r

 

μεu v dρ − r0

r

⎞  r r r1 uvdρ − u  v r ⎠ + μεu  v r 0

0

r0 

r



(με) u vdρ + r0

r



qu vdρ + r0

r2 uvdρ,

(4.6.132)

r0

where u = u j , v = v j , p = p j , q = q j , j = e or m. Taking into account the boundary condition at ρ = r0 , we get r

r vLudρ = γ 4

r0

⎛ r ⎞  r r   uvdρ + γ 2 ⎝ u  v  dρ + pu  vdρ + r1 uvdρ − u  v r ⎠ + μεu  v r

r0

r0

r −

 

r0

r

μεu v dρ − r0





r0

r

(με) u vdρ + r0



r

qu vdρ + r0

r2 uvdρ. r0

(4.6.133)

4.6 Leaky Waves in an Inhomogeneous Waveguide

177

We separately apply formula (4.6.133) to the first and second equations in system (4.6.131) on the interval [r0 , r ], add the results, and find that the sum of the left-hand sides of (4.6.131) satisfies the relation r

r (v e L e u e + v m L m u m )dρ = γ 4

r0

r0

 r +γ

(u e v e + u m v m )dρ

2

(u e v e

+ u m v m )dρ +

r0

r

( pe u e v e

pm u m v m )dρ +

+

r0

r −

με(u e v e

+

r1 (u e v e + u m v m )dρ r0

+ u m v m )dρ −

r0

r



r

r

(με) (u e v e + u m v m )dρ

r0

(qe u e v e

r0

r

+ qm u m v m )dρ +

r2 (u e v e + u m v m )dρ r0

  + μ(r )ε(r ) − γ 2 (u e (r )v e (r ) + u m (r )v m (r )).

(4.6.134)

On the other hand, for the right-hand sides of the equations in system (4.6.131), we have r

r (v e L e u e + v m L m u m )dρ = γ

r0

( f e u m v e + f m u e v m )dρ.

(4.6.135)

r0

Given the solutions (4.6.130), we express the values of the normal derivatives at ρ = r from relations (4.6.129) as follows: γm μ(r )ε(r ) − 1 μ(r )ε(r ) − γ 2 Im (κr ) u e (r ), u (r ) − m r ε(r ) κ2 κε(r ) Im (κr ) γm μ(r )ε(r ) − 1 μ(r )ε(r ) − γ 2 Im (κr ) u m (r ). u m (r ) = u (r ) − e r μ(r ) κ2 κμ(r ) Im (κr ) u e (r ) =

(4.6.136)

Then, in view of (4.6.134), from (4.6.135) and (4.6.136), we obtain r γ4

(u e v e +u m v m )dρ r0

 r +γ 2

(u e v e + u m v m )dρ +

r0

r

( pe u e v e + pm u m v m )dρ +

r0

r − r0

με(u e v e + u m v m )dρ −



r r1 (u e v e + u m v m )dρ r0

r r0

(με) (u e v e + u m v m )dρ

178

4 Waveguides of Circular Cross Section

r +

(qe u e v e + qm u m v m )dρ +

r0

r r2 (u e v e + u m v m )dρ r0

 μ(r )ε(r ) − γ 2 Im (κr ) γm μ(r )ε(r ) − 1 u (r ) − (r ) v e (r ) u m e r κ2 κ Im (κr )   μ(r )ε(r ) − γ 2 Im (κr ) μ(r )ε(r ) − γ 2 γm μ(r )ε(r ) − 1 u (r ) − (r ) v m (r ) + u e m μ(r ) r κ2 κ Im (κr ) r (4.6.137) − γ ( f e u m v e + f m u e v m )dρ = 0. +

μ(r )ε(r ) − γ 2 ε(r )



r0

Variational relation (4.6.137) has been obtained for smooth functions v e and v m .

4.6.3 Spectrum of the OVF Let H = H01 (r0 , r ) × H 1 (r0 , r ) be the Cartesian product of Hilbert spaces with the inner product and the norm (u, v) = (u 1 , v1 )1 + (u 2 , v2 )1 , u 2 = u 1 21 + u 2 21 ; u, v ∈ H, u = (u 1 , u 2 )T , v = (v1 , v2 )T , u 1 , v1 ∈ H01 (r0 , r ) , u 2 , v2 ∈ H 1 (r0 , r ) . Then the integrals occurring in (4.6.137) can be viewed as sesquilinear forms over the field C defined on H and depending on the arguments u = (u e , u m )T , v = (v e , v m )T . Consider the following sesquilinear forms and the corresponding operators: r k(u, v) :=

(u e v e + u m v m )dρ = (Ku, v), ∀v ∈ H, r0

r k1 (u, v) :=

(r1 − 1)(u e v e + u m v m )dρ = (K1 u, v), ∀v ∈ H, r0

r k2 (u, v) :=

(r2 − με)(u e v e + u m v m )dρ = (K2 u, v), ∀v ∈ H, r0

4.6 Leaky Waves in an Inhomogeneous Waveguide

 k(u, v) :=

r

179

u, v), ∀v ∈ H, ( f e u m v e + f m u e v m )dρ = (K

r0

r a1 (u, v) :=

(u e v e + u m v m + u e v e + u m v m )dρ = (Iu, v), ∀v ∈ H,

r0

r a2 (u, v) :=

με(u e v e + u m v m + u e v e + u m v m )dρ = (Au, v), ∀v ∈ H,

r0

r b1 (u, v) :=

( pe u e v e + pm u m v m )dρ = (B1 u, v), ∀v ∈ H,

r0

r b2 (u, v) :=

(με) (u e v e + u m v m )dρ = (B2 u, v), ∀v ∈ H,

r0

r b3 (u, v) :=

(qe u e v e + qm u m v m )dρ = (B3 u, v), ∀v ∈ H,

r0

  μ(r )ε(r ) − γ 2 γm μ(r )ε(r ) − 1 μ(r )ε(r ) − γ 2 Im (κr ) u (r ) − (r ) v e (r ) u m e ε(r ) r κ2 κ Im (κr )   μ(r )ε(r ) − γ 2 γm μ(r )ε(r ) − 1 μ(r )ε(r ) − γ 2 Im (κr ) + u (r ) − (r ) v m (r ) u e m μ(r ) r κ2 κ Im (κr )

s(u, v) =

= (Su, v), ∀v ∈ H.

(4.6.138) The boundedness of a2 (u, v) is obvious. The boundedness of forms k(u, v), k(u, v) follows from the Poincar’e inequality [11]. k1 (u, v), k2 (u, v) and  Now variational problem (4.6.137) can be written in the operator form (N(γ)u, v) = 0, ∀u ∈ H or, equivalently,    + K2 − A − B2 + B3 + S(γ) u = 0. N(γ)u := γ 4 K + γ 2 (K1 + B1 + I) − γ K (4.6.139)

180

4 Waveguides of Circular Cross Section

Equation (4.6.139) is the operator form of variational relation (4.6.137). The characteristic numbers and eigenvectors of operator N by definition coincide with the eigenvalues and eigenvectors of problem Pinh for γ 2 = μ(ρ)ε(ρ).

4.6.4 Properties of the Spectrum of the OVF We have reduced the problem on normal waves to the study of spectral properties of OVF N. In this connection, we first consider the properties of the operators introduced in the preceding section. The validity of Lemmas 4.6.1–4.6.5 is shown in Sect. 4.1.4. Lemma 4.6.1 The bounded operator A : H → H is positive definite A ≥ γ∗2 I, √ where 0 < γ∗ = min μ(ρ)ε(ρ). r0 ≤ρ≤r

 : H → H are compact, and Lemma 4.6.2 Bounded operators K, K1 , K2 and K K > 0. Lemma 4.6.3 Operators B1 , B2 and B3 : H → H are compact. Lemma 4.6.4 Operator S : H → H is compact. Lemma 4.6.5 The operator γ 2 I − A : H → H is bounded and continuously invertible in the domain C\ E and  E := {γ : Imγ = 0, γ∗ ≤ |Reγ| ≤ γ ∗ }, where 0 < γ ∗ = max

r0 ≤ρ≤r

√

μ(ρ)ε(ρ).

Lemma 4.6.6 There exists a  γ ∈ R such that operator N( γ ) is continuously invertγ ) is ible; i.e., the resolvent set (N) := {γ : ∃ N−1 (γ) : H → H } of OVF N( nonempty, (N) = ∅. Proof Let γ ∈ R, γ > 0 and γ → +∞. Consider the OVF 1 N(γ) = N1 (γ) + N2 (γ), γ2 where N1 (γ) = γ 2 K + K1 + B1 + I − γS0 , and

 1 1 + K − A − B + B + S(γ) + γS0 , N2 (γ) = − K 2 2 3 γ γ2

4.6 Leaky Waves in an Inhomogeneous Waveguide

181

and the operator S0 : H → H is defined by the sesquilinear form   u e (r )v e (r ) u m (r )v m (r ) S0 u, v := + . ε(r ) μ(r )

(4.6.140)

In view of the asymptotics of functions Im as γ → +∞ [12], we obtain Im (κr ) ∼ 1, Im (κr ) and

1 μ(r )ε(r ) − γ 2 ∼ −γ + O . κ γ

Consequently,

    1   S(γ 2 ) + γS0  = O 1 , as γ → +∞.  γ2  γ

(4.6.141)

Then OVF γ −2 N(γ) can be considered as a perturbation of operator pencil N1 by OVF N2 for large γ. We have r Reb1 (u, u) := Re

( pe u e u e + pm u m u m )dρ

r0

1 = 2

r 

     pe |u e |2 + pm |u m |2 dρ

r0

      1    1 = pe (r ) |u e (r )|2 + pm (r ) |u m (r )|2 − pe (r0 ) |u e (r0 )|2 + pm (r0 ) |u m (r0 )|2 2 2 r    1    pe |u e |2 + pm |u m |2 dρ − 2 r0

    1 = pe (r ) |u e (r )|2 + pm (r ) |u m (r )|2 − 2 r 1 − 2

  1 pm (r0 ) |u m (r0 )|2 2 

     pe |u e |2 + pm |u m |2 dρ.

r0

Since functions ε and μ belong to the space C 2 [r0 , r ], it follows from the last relation, definition (4.6.140), the asymptotic relation (4.6.141), and the properties of operators γ such that K and S0 that there exists a large number  Re(N1 ( γ )u, u) =  γ 2 (Ku, u) + (K1 u, u) + Re(B1 u, u) + (u, u) −  γ (S0 u, u) ≥ u 2

182

4 Waveguides of Circular Cross Section

for each u. Hence,  γ ∈ (N1 ), where (N1 ) is the resolvent set of OVF N1 . Moreover, Theorem 4.1 in [13] gives the estimate γ ) ≤ 1.

N1−1 ( γ ) < 1 is satisfied and conclude that there We take  γ such that the inequality N2 ( exists a bounded operator −1    γ ) + N2 ( γ) = I + N1−1 ( γ )N2 ( γ ) N1−1 ( γ ). N1 (  Theorem 4.6.1 OVF N(γ) : H → H is bounded, holomorphic, and Fredholm in the domain  = C\ ( I ∪  E ). Proof The functions κ

Im (κr ) Im+1 (κr ) m =κ + . Im (κr ) Im (κr ) r

(4.6.142)

are analytical in C\ I as functions of γ because Im (κr ) have no zeros for Reκ > 0 [12]. Finally, we obtain the desired result by Lemmas 4.6.5 and 4.6.6. The proof of the theorem is complete.  Theorem 4.6.2 The spectrum of OVF N(γ) : H → H is discrete in the domain ; i.e., this function has finitely many characteristic numbers of finite algebraic multiplicity on any compact set K 0 ⊂ . Proof The assertion of the theorem is a corollary of Theorem 4.6.1 and a theorem on a holomorphic operator function [13].  Lemma 4.6.7 The spectrum of OVF N(γ) : H → H is symmetric with respect to the real and imaginary axes. If γ0 is an eigenvalue of N(γ) corresponding to the eigenvector u1 = (u e , u m )T , then −γ0 , γ 0 , −γ 0 are eigenvalues of OVF N(γ) corresponding to the eigenvectors u2 = (−u e , u m )T , u3 = (u e , u m )T , u4 = (−u e , u m )T with the same multiplicity. Proof The assertion of the lemma can readily be proved by a straightforward verification with the use of (4.6.127) and conditions (4.6.128) and (4.6.129). Note that the condition at infinity is also satisfied for all four characteristic numbers. 

4.7 Leaky Waves in a Waveguide Filled with Metamaterial Media

183

4.7 Leaky Waves in a Waveguide Filled with Metamaterial Media 4.7.1 Statement of the Problem We assume that the permittivity and permeability in the entire space have the form  ε=

−ε2 , r0 ≤ ρ ≤ r, and  μ= 1, ρ > r,



μ2 , r0 ≤ ρ ≤ r, 1, ρ > r,

(4.7.143)

where ε2 > 1 and μ2 > 1 are real positive constants. The problem on normal waves in GL described in Sect. 4.1.1 is an eigenvalue problem for the Maxwell equations with spectral parameter γ, which is the normalized propagation constant of the normal wave. The Maxwell system is written in the normalized form. The passage  to dimensionless variables has been carried μ0 γ H → H, E → E, where k02 = ωε0 μ0 (the out [9]; namely, k0 ρ → ρ, γ → k0 , ε0 time factor e−iωt is omitted everywhere). Rewrite Maxwell’s equation in the expanded form: ⎧ m ⎪ i Hz − iγ Hϕ ⎪ ⎪ ρ ⎪ ⎪ ⎪ ⎪ ⎪ iγ Hρ − Hz ⎪ ⎪ ⎪ ⎪ 1 im ⎪  ⎪ ⎪ Hρ ⎨ (ρHϕ ) − ρ ρ m ⎪ ⎪ i E z − iγ E ϕ ⎪ ⎪ ρ ⎪ ⎪ ⎪ ⎪ ⎪ iγ E ρ − E z ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1 (ρE ϕ ) − im E ρ ρ ρ

= −i ε Eρ, = −i ε Eϕ, = −i ε Ez , = i μ Hρ ,

(4.7.144)

= i μ Hϕ , = i μ Hz ,

and express functions E ρ , Hρ , E ϕ , Hϕ via E z and Hz from the first, second, fourth, and fifth equations in system (4.7.144) m μ Hz − iγρE z , ρ(γ 2 −  ε μ) γm E z + iρ μ Hz , Eϕ = ρ(γ 2 −  ε μ)

Eρ =

iγρHz + m εEz , 2 ρ(γ −  ε μ) γm Hz − iρ ε E z Hϕ = . ρ(γ 2 −  ε μ)

Hρ = −

(4.7.145)

It follows from (4.7.145) that the normal wave field in the waveguide can be represented with the use of two scalar functions

184

4 Waveguides of Circular Cross Section

u e := i E z (ρ), u m := Hz (ρ).

(4.7.146)

Thus, the problem has been reduced to finding longitudinal components u e and u m of the electric and magnetic fields. We have the following eigenvalue problem for the longitudinal field components u e and u m : find γ ∈ C such that, for a given m ∈ Z, there exist nontrivial solutions of the system of differential equations ⎧     1  2 2   ⎪ ⎪ ρ γ − ε μ + m 2 u e = 0, ⎨ ρu e − ρ  2 2     1 ⎪  ⎪ ⎩ ρu m − ρ γ − ε μ + m 2 u m = 0, ρ

(4.7.147)

satisfying the boundary conditions for ρ = r0 u e (r0 ) = 0, u m (r0 ) = 0,

(4.7.148)

the transmission conditions for ρ = r [u e ]|r = 0, [u m ]|r = 0,           ρ μu m  um ρ εu e  ue  −  − = 0, γm = 0, γm  ε μ − γ 2 r  ε μ − γ 2 r  ε μ − γ 2 r  ε μ − γ 2 r (4.7.149) where [ f ]|ρ0 = lim f (ρ) − lim f (ρ) and the conditions of the boundedness 

ρ→ρ0 −0

ρ→ρ0 +0

of the field in any finite domain and increasing at infinity. Once we determine the longitudinal field components u e and u m by solving problem (4.7.147)–(4.7.149), we can find the transverse components by formulas (4.7.145). The equivalence of the reduction to problem (4.7.147)–(4.7.149) is not ε μ. valid only for γ 2 =  For ρ > r , we have  ε = 1,  μ = 1; then from (4.7.147), we obtain the system     ρ(ρu e ) − ρ2 κ2 + m 2 u e = 0, ρ(ρu m ) − ρ2 κ2 + m 2 u m = 0, where κ2 = γ 2 − 1. In view of the condition at infinity, we choose a solution of this system in the form u e (ρ; γ, m) = C1 Im (κρ), (4.7.150) u m (ρ; γ, m) = C2 Im (κρ), where Im is the modified Bessel function (the Infeld function) [10] and C1 and C2 are constants. R

Function κ(γ) is analytical in the domain C\ K , where  K := {γ : Imγ 2 = 0, γ 2 ≤ 1}.

4.7 Leaky Waves in a Waveguide Filled with Metamaterial Media

185

For r0 ≤ ρ ≤ r , we have  ε = −ε2 and  μ = μ2 , and from (4.7.147) we obtain the system of differential equations   L e u e : = u e + pu e − γ 2 + q u e = 0,   L m u m : = u m + pu m − γ 2 + q u m = 0, where p=

1 , ρ

q = ε2 μ2 +

(4.7.151)

m2 . ρ2

Definition 4.7.1 If for a given m there exist nontrivial functions u e and u m corresponding to γ ∈ C such that these functions are given by (4.7.150) for ρ > r , form a solution of system (4.7.151) for r0 ≤ ρ ≤ r , and satisfy transmission conditions (4.7.149), then γ is called a characteristic number of problem Pmet . Definition 4.7.2 The pair u e and u m , |u e |2 + |u m |2 ≡ 0, will be called an eigenvector of problem Pmet corresponding to the characteristic number γ ∈ C.

4.7.2 Sobolev Spaces and Variational Relation Let us give variational formulation of problem Pmet . Multiplying equations (4.7.151) by arbitrary test functions ve ∈ H01 (r0 , r ) and vm ∈ H 1 (r0 , r ) (we can assume that these functions are continuously differentiable in (r0 , r )) and applying Green’s formula, we obtain r

r vLudρ =

r0

r



vu dρ + r0

=

r



pvu dρ − r0



r vu  r0

r −

r0

r

 

u v dρ + r0

= −γ

r



pu vdρ − r0

r r0

r quvdρ +

r0

  v γ 2 + q udρ

r0

r uvdρ −

2

  v γ 2 + q udρ

r pu vdρ + u v r0 − 



r0

r

u  v  dρ.

r0

(4.7.152) Taking into account the boundary condition for ρ = r0 , we get r

r vLudρ = −γ

r0

r uvdρ −

2 r0

r quvdρ +

r0

r0

 pu vdρ + u v r − 



r

u  v  dρ

r0

(4.7.153)

186

4 Waveguides of Circular Cross Section

We separately apply formula (4.7.153) to the first and second equations in system (4.7.151) on the interval [r0 , r ], add the results, and find that the sum of the left-hand sides of (4.7.151) satisfies the relation r

r (v e L e u e + v m L m u m )dρ = −γ

r0

(u e v e + u m v m )dρ

2 r0

r −

r q(u e v e + u m v m )dρ +

r0

p(u e v e

+

u m v m )dρ−

r0

r

(u e v e + u m v m )dρ

r0

+u e (r )v e (r ) + u m (r )v m (r ). (4.7.154) Given solutions (4.7.150), we express the values of the normal derivatives at ρ = r from relations (4.7.149) as follows: γm ε2 μ2 + 1 ε2 μ2 + γ 2 Im (κr ) u e (r ), u (r ) − m r ε2 κ2 κε2 Im (κr ) γm ε2 μ2 + 1 ε2 μ2 + γ 2 Im (κr ) u m (r ). u m (r ) = − 2 u (r ) + e rμ κ2 κμ2 Im (κr ) u e (r ) =

(4.7.155)

Then, in view of (4.7.154), from (4.7.155), we obtain r γ2

r (u e v e + u m v m )dρ +

r0

q(u e v e + u m v m )dρ r0

r −

p(u e v e

+

u m v m )dρ

r0

r +

(u e v e + u m v m )dρ

r0

 ε μ + γ 2 Im (κr ) γm ε μ + 1 u e (r ) v e (r ) u m (r ) − − r ε2 κ2 κε2 Im (κr )   ε2 μ2 + γ 2 Im (κr ) γm ε2 μ2 + 1 u m (r ) v m (r ) = 0. (4.7.156) u e (r ) + − − 2 rμ κ2 κμ2 Im (κr ) 

2 2

2 2

Variational relation (4.7.156) has been obtained for smooth functions v e and v m .

4.7 Leaky Waves in a Waveguide Filled with Metamaterial Media

187

4.7.3 Spectrum of the OVF of the Problem Let H = H01 (r0 , r ) × H 1 (r0 , r ) be the Cartesian product of Hilbert spaces with the inner product and the norm (u, v) = (u 1 , v1 )1 + (u 2 , v2 )1 , u 2 = u 1 21 + u 2 21 ; u, v ∈ H, u = (u 1 , u 2 )T , v = (v1 , v2 )T , u 1 , v1 ∈ H01 (r0 , r ) , u 2 , v2 ∈ H 1 (r0 , r ) . Then the integrals occurring in (4.7.156) can be viewed as sesquilinear forms over the field C defined on H and depending on the arguments u = (u e , u m )T , v = (v e , v m )T . Consider the following sesquilinear forms and the corresponding operators: r k(u, v) :=

(u e v e + u m v m )dρ = (Ku, v), ∀v ∈ H, r0

 k(u, v) :=

r

u, v), ∀v ∈ H, q(u e v e + u v v m )dρ = (K

r0

r a(u, v) :=

(u e v e + u m v m + u e v e + u m v m )dρ = (Iu, v), ∀v ∈ H,

r0

r b(u, v) :=

p(u e v e + u m v m )dρ = (Bu, v), ∀v ∈ H,

r0



 γm ε2 μ2 + 1 ε2 μ2 + γ 2 Im (κr ) u u (r ) − (r ) v e (r ) m e r ε2 κ2 κε2 Im (κr )   γm ε2 μ2 + 1 ε2 μ2 + γ 2 Im (κr ) + − 2 u u (r ) + (r ) v m (r ) e m rμ κ2 κμ2 Im (κr ) = (Su, v), ∀v ∈ H. (4.7.157)

s(u, v) =

The boundedness of a(u, v) is obvious. The boundedness of forms k(u, v) and  k(u, v) follows from the Poincarè inequality [11].

188

4 Waveguides of Circular Cross Section

Now variational problem (4.7.156) can be written in the operator form (N(γ)u, v) = 0, ∀u ∈ H or, equivalently,    − K + I − B − S(γ) u = 0. N(γ)u := γ 2 K + K

(4.7.158)

Equation (4.7.158) is the operator form of variational relation (4.7.156). The characteristic numbers and eigenvectors of operator N by definition coincide with the eigenvalues and eigenvectors of problem Pmet for γ 2 = −μ2 ε2 .

4.7.4 Properties of the Spectrum of the OVF We have reduced the problem on normal waves to the study of spectral properties of OVF N. In this connection, we first consider the properties of the operators introduced in the preceding section. The validity of Lemmas 4.7.1–4.7.3 is shown in Sect. 4.1.4.  : H → H are compact, and K > 0. Lemma 4.7.1 The bounded operators K and K Lemma 4.7.2 Operators B : H → H are compact. Lemma 4.7.3 Operator S : H → H is compact. Lemma 4.7.4 There exists a  γ ∈ R such that operator N( γ ) is continuously invertible; i.e., the resolvent set (N) := {γ : ∃ N−1 (γ) : H → H } of OVF N( γ ) is nonempty, (N) = ∅. Proof Let γ ∈ R, γ > 0 and γ → +∞. Consider the OVF N(γ) = N1 (γ) + N2 (γ), where

 − K + B + I + γS0 , N1 (γ) = γ 2 K + K

and N2 (γ) = −S(γ) − γS0 , and the operator S0 : H → H is defined by the sesquilinear form 

 u e (r )v e (r ) u m (r )v m (r ) S0 u, v := − + . ε2 μ2

(4.7.159)

4.7 Leaky Waves in a Waveguide Filled with Metamaterial Media

189

In view of the asymptotics of functions Im as γ → +∞ [12], we have Im (κr ) ∼ 1, Im (κr ) and

1 μ2 ε2 + γ 2 ∼γ+O . κ γ

Consequently,

N2 (γ) = O

1 γ

, as γ → +∞.

(4.7.160)

Then (OVFs) N(γ) can be viewed as a perturbation of operator pencil N1 by operatorvalued function N2 for large γ. We have r Reb(u, u) := Re

p(u e u e

+ u m u m )dρ

r0

1 = 2

r p



    |u e |2 + |u m |2 dρ

r0

 p(r0 )   p(r )  = |u e (r )|2 + |u m (r )|2 − |u e (r0 )|2 + |u m (r0 )|2 2 2 r   1 p  |u e |2 + |u m |2 dρ − 2 r0

 p(r0 ) p(r )  |u m (r0 )|2 |u e (r )|2 + |u m (r )|2 − = 2 2 r   1 − p  |u e |2 + |u m |2 dρ. 2 r0

It follows from the last relation, definition (4.7.159), asymptotic relation (4.7.160), γ such that and the properties of operators K and S0 that there exists a large number   u) − (Ku, u) + Re(Bu, u) + (u, u) +  Re(N1 ( γ )u, u) =  γ 2 (Ku, u) + (Ku, γ (S0 u, u) ≥ u 2

for each u. Hence,  γ ∈ (N1 ), where (N1 ) is the resolvent set of OVF N1 . Moreover, Theorem 4.1 in [13] gives the estimate γ ) ≤ 1.

N1−1 ( γ ) < 1 is satisfied and conclude that there We take  γ such that the inequality N2 ( exists a bounded operator −1    γ ) + N2 ( γ) = I + N1−1 ( γ )N2 ( γ ) N1−1 ( γ ). N1 (



190

4 Waveguides of Circular Cross Section

Theorem 4.7.1 OVF N(γ) : H → H is bounded, holomorphic, and Fredholm in the domain  = C\ K . Proof The functions κ

Im (κr ) Im+1 (κr ) m = −κ + . Im (κr ) Im (κr ) r

(4.7.161)

are analytical in the domain C\ K as functions of variable γ because Im (κr ) have no zeros for Reκ > 0 [12]. Finally, we obtain the desired result by Lemma 4.7.4. The proof of the theorem is complete.  Theorem 4.7.2 The spectrum of OVF N(γ) : H → H is discrete in the domain ; i.e., this function has finitely many characteristic numbers of finite algebraic multiplicity on any compact set I0 ⊂ . Proof The assertion of the theorem is a corollary of Theorem 4.7.1 and a theorem on a holomorphic OVF [13].  Lemma 4.7.5 The spectrum of OVF N(γ) : H → H is symmetric with respect to the real and imaginary axes. If γ0 is an eigenvalue of N(γ) corresponding to the eigenvector u1 = (u e , u m )T , then −γ0 , γ 0 , −γ 0 are eigenvalues of OVF N(γ) corresponding to the eigenvectors u2 = (−u e , u m )T , u3 = (u e , u m )T , u4 = (−u e , u m )T with the same multiplicity. Proof The assertion of the lemma can readily be proved by a straightforward verification with the use of (4.7.147) and conditions (4.7.148) and (4.7.149). Note that the condition at infinity is satisfied for all four characteristic numbers. 

4.8 Leaky Waves in a Waveguide Filled with Lossy Medium 4.8.1 Statement of the Problem We assume that the permittivity and permeability in the entire space have the form  ε=

ε + iεlos , r0 ≤ ρ ≤ r, and  μ= 1, ρ > r,



μ, r0 ≤ ρ ≤ r, 1, ρ > r,

(4.8.162)

where ε > 1, εlos > 0 and μ2 > 1 are real positive constants and ε > ε >los . The problem on normal waves in GL described in Sect. 4.1.1 is an eigenvalue problem for the Maxwell equations with spectral parameter γ, which is the normalized propagation constant of the normal wave. Rewrite Maxwell’s equation in the expanded form:

4.8 Leaky Waves in a Waveguide Filled with Lossy Medium

⎧ m ⎪ i Hz − iγ Hϕ ⎪ ⎪ ρ ⎪ ⎪ ⎪ ⎪ ⎪ iγ Hρ − Hz ⎪ ⎪ ⎪ ⎪ 1 im ⎪  ⎪ ⎪ Hρ ⎨ (ρHϕ ) − ρ ρ m ⎪ ⎪ i E z − iγ E ϕ ⎪ ⎪ ρ ⎪ ⎪ ⎪ ⎪ ⎪ iγ E ρ − E z ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1 (ρE ϕ ) − im E ρ ρ ρ

191

= −i ε Eρ, = −i ε Eϕ, = −i ε Ez , = i μ Hρ ,

(4.8.163)

= i μ Hϕ , = i μ Hz ,

and express functions E ρ , Hρ , E ϕ , Hϕ via E z and Hz from the first, second, fourth, and fifth equations in system (4.8.163) m μ Hz − iγρE z , ρ(γ 2 −  ε μ) γm E z + iρ μ Hz , Eϕ = ρ(γ 2 −  ε μ)

Eρ =

iγρHz + m εEz , ρ(γ 2 −  ε μ) γm Hz − iρ ε E z Hϕ = . ρ(γ 2 −  ε μ)

Hρ = −

(4.8.164)

It follows from (4.8.164) that the normal wave field in the waveguide can be represented with the use of two scalar functions u e := i E z (ρ), u m := Hz (ρ).

(4.8.165)

Thus, the problem has been reduced to finding longitudinal components u e and u m of the electric and magnetic fields. Throughout the following, ( · ) stands for differentiation with respect to ρ. We have the following eigenvalue problem for longitudinal field components u e and u m : find γ ∈ C such that, for a given m ∈ Z, there exist nontrivial solutions of the system of differential equations ⎧     1  2 2   ⎪ ⎪ ρ γ − ε μ + m 2 u e = 0, ⎨ ρu e − ρ   ⎪    1  2  2 ⎪ ⎩ ρu m − ρ γ − ε μ + m 2 u m = 0, ρ

(4.8.166)

satisfying the boundary conditions for ρ = r0 u e (r0 ) = 0, u m (r0 ) = 0, the transmission conditions for ρ = r

(4.8.167)

192

4 Waveguides of Circular Cross Section

[u e ]|r = 0, [u m ]|r = 0,           ρ μu m  um ρ εu e  ue  −  − = 0, γm = 0, γm  ε μ − γ 2 r  ε μ − γ 2 r  ε μ − γ 2 r  ε μ − γ 2 r (4.8.168) where [ f ]|ρ0 = lim f (ρ) − lim f (ρ) and the conditions of the boundedness 

ρ→ρ0 −0

ρ→ρ0 +0

of the field in any finite domain and increasing at infinity. Once we determine longitudinal field components u e and u m by solving problem (4.8.166)–(4.8.168), we can find the transverse components by formulas (4.8.164). The equivalence of the reduction to problem (4.8.166)–(4.8.168) is not valid only ε μ. for γ 2 =  For ρ > r , we have  ε = 1,  μ = 1; then from (4.8.166), we obtain the system     ρ(ρu e ) − ρ2 κ2 + m 2 u e = 0, ρ(ρu m ) − ρ2 κ2 + m 2 u m = 0, where κ2 = γ 2 − 1. In view of the condition at infinity, we obtain a solution of this system in the form u e (ρ; γ, m) = C1 Im (κρ), (4.8.169) u m (ρ; γ, m) = C2 Im (κρ), where Im is the modified Bessel function (the Infeld function) [10] and C1 and C2 are constants. R

Function κ(γ) is analytical in the domain C\ K , where  K := {γ : Imγ 2 = 0, γ 2 ≤ 1}.

For r0 ≤ ρ ≤ r , we have  ε = ε + iεlos and  μ = μ, and from (4.8.166) we obtain the system of differential equations   L e u e : = u e + pu e − γ 2 + q u e = 0,   L m u m : = u m + pu m − γ 2 + q u m = 0 where p=

1 , ρ

q = −(ε + iεlos )μ +

(4.8.170)

m2 . ρ2

Definition 4.8.1 If for given m there exist nontrivial functions u e and u m corresponding to γ ∈ C such that these functions are given by (4.8.169) for ρ > r , form a solution of system (4.8.170) for r0 ≤ ρ ≤ r , and satisfy transmission conditions (4.8.168), then γ is called a characteristic number of problem Plos . Definition 4.8.2 The pair u e and u m , |u e |2 + |u m |2 ≡ 0, will be called an eigenvector of problem Plos corresponding to the characteristic number γ ∈ C.

4.8 Leaky Waves in a Waveguide Filled with Lossy Medium

193

4.8.2 Sobolev Spaces and Variational Relation Let us give variational formulation of problem Plos . Multiplying equations (4.8.170) by arbitrary test functions ve ∈ H01 (r0 , r ) and vm ∈ H 1 (r0 , r ) (we can assume that these functions are continuously differentiable in (r0 , r )) and applying Green’s formula, we obtain r

r vLudρ =

r0

r



vu dρ + r0

=

r



pvu dρ − r0

r vu  r0

r −

r0

r

 

u v dρ + r0

r =−γ

r0

r



pu vdρ − r0

r

r pu vdρ + u v r0 − 

quvdρ + r0

  v γ 2 + q udρ

r0

r uvdρ −

2

  v γ 2 + q udρ



r0

r

u  v  dρ.

r0

(4.8.171) Taking into account the boundary condition for ρ = r0 , we get r

r vLudρ = −γ

r0

r uvdρ −

2 r0

r quvdρ +

r0

 pu vdρ + u v r − 



r0

r

u  v  dρ

r0

(4.8.172) We separately apply formula (4.8.172) to the first and second equations in system (4.8.170) on the interval [r0 , r ], add the results, and find that the sum of the left-hand sides of (4.8.170) satisfies the relation r

r (v e L e u e + v m L m u m )dρ = −γ

r0

(u e v e + u m v m )dρ

2 r0

r −

r q(u e v e + u m v m )dρ +

r0

r0

p(u e v e

+

u m v m )dρ−

r

(u e v e + u m v m )dρ

r0

+u e (r )v e (r ) + u m (r )v m (r ). (4.8.173) Given solutions (4.8.169), we express the values of the normal derivatives at ρ = r from relations (4.8.168) as follows:

194

4 Waveguides of Circular Cross Section

(ε + iεlos )μ − 1 γm (ε + iεlos )μ − γ 2 Im (κr ) u e (r ), u (r ) − m r (ε + iεlos ) κ2 κ(ε + iεlos ) Im (κr ) γm (ε + iεlos )μ − 1 (ε + iεlos )μ − γ 2 Im (κr ) u m (r ). u m (r ) = u (r ) − e rμ κ2 κμ Im (κr ) (4.8.174) Then, in view of (4.8.173), from (4.8.174), we obtain u e (r ) =

r γ2

r (u e v e + u m v m )dρ +

r0

q(u e v e + u m v m )dρ r0

r −

p(u e v e + u m v m )dρ +

r0

r

(u e v e + u m v m )dρ

r0

 γm (ε + iεlos )μ − 1 (ε + iεlos )μ − γ 2 Im (κr ) u e (r ) v e (r ) u m (r ) − − r (ε + iεlos ) κ(ε + iεlos ) Im (κr ) κ2   γm (ε + iεlos )μ − 1 (ε + iεlos )μ − γ 2 Im (κr ) u m (r ) v m (r ) = 0. u e (r ) − − rμ κμ Im (κr ) κ2 

(4.8.175) Variational relation (4.8.175) has been obtained for smooth functions v e and v m .

4.8.3 Spectrum of the OVF Let H = H01 (r0 , r ) × H 1 (r0 , r ) be the Cartesian product of Hilbert spaces with the inner product and the norm (u, v) = (u 1 , v1 )1 + (u 2 , v2 )1 , u 2 = u 1 21 + u 2 21 ; u, v ∈ H, u = (u 1 , u 2 )T , v = (v1 , v2 )T , u 1 , v1 ∈ H01 (r0 , r ) , u 2 , v2 ∈ H 1 (r0 , r ) . Then the integrals occurring in (4.8.175) can be viewed as sesquilinear forms over the field C defined on H and depending on the arguments u = (u e , u m )T , v = (v e , v m )T . Consider the following sesquilinear forms and the corresponding operators: r k(u, v) :=

(u e v e + u m v m )dρ = (Ku, v), ∀v ∈ H, r0

4.8 Leaky Waves in a Waveguide Filled with Lossy Medium

 k(u, v) :=

r

195

u, v), ∀v ∈ H, q(u e v e + u v v m )dρ = (K

r0

r a(u, v) :=

(u e v e + u m v m + u e v e + u m v m )dρ = (Iu, v), ∀v ∈ H,

r0

r b(u, v) :=

p(u e v e + u m v m )dρ = (Bu, v), ∀v ∈ H,

r0



 γm (ε + iεlos )μ − γ 2 Im (κr ) (ε + iεlos )μ − 1 u (r ) − (r ) v e (r ) u m e r (ε + iεlos ) κ2 κ(ε + iεlos ) Im (κr )   (ε + iεlos )μ − γ 2 Im (κr ) γm (ε + iεlos )μ − 1 u (r ) − (r ) v m (r ) + u e m rμ κ2 κμ Im (κr )

s(u, v) =

= (Su, v), ∀v ∈ H.

(4.8.176) The boundedness of a(u, v) is obvious. The boundedness of forms k(u, v) and  k(u, v) follows from the Poincarè inequality [11]. Now variational problem (4.8.175) can be written in the operator form (N(γ)u, v) = 0, ∀u ∈ H or, equivalently,    − K + I − B − S(γ) u = 0. N(γ)u := γ 2 K + K

(4.8.177)

Equation (4.8.177) is the operator form of variational relation (4.8.175). The characteristic numbers and eigenvectors of operator N by definition coincide with the eigenvalues and eigenvectors of problem Plos for γ 2 = (ε + iεlos )μ.

4.8.4 Properties of the Spectrum of the OVF We have reduced the problem on normal waves to the study of spectral properties of OVF N. In this connection, we first consider the properties of the operators introduced in the preceding section. The validity of Lemmas 4.8.1–4.8.3 is shown in Sect. 4.1.4.  : H → H are compact, and K > 0. Lemma 4.8.1 The bounded operators K and K

196

4 Waveguides of Circular Cross Section

Lemma 4.8.2 Operators B : H → H are compact. Lemma 4.8.3 Operator S : H → H is compact. Lemma 4.8.4 There exists a  γ ∈ R such that operator N( γ ) is continuously invertible; i.e., the resolvent set (N) := {γ : ∃ N−1 (γ) : H → H } of OVF N( γ ) is nonempty, (N) = ∅. Proof Let γ ∈ R, γ > 0 and γ → +∞. Consider the OVF N(γ) = N1 (γ) + N2 (γ), where

 − K + B + I + γS0 , N1 (γ) = γ 2 K + K

and N2 (γ) = −S(γ) − γS0 , and the operator S0 : H → H defined by the sesquilinear form   u e (r )v e (r ) u m (r )v m (r ) S0 u, v := . + ε + iεlos μ

(4.8.178)

In view of the asymptotics of functions Im as γ → +∞ [12], we have Im (κr ) ∼ 1, Im (κr ) and

1 μ(ε + iεlos ) − γ 2 ∼ −γ + O . κ γ

Consequently,

N2 (γ) = O

1 γ

, as γ → +∞.

(4.8.179)

Then (OVFs) N(γ) can be viewed as a perturbation of operator pencil N1 by operatorvalued function N2 for large γ. r Reb(u, u) := Re

p(u e u e

+

u m u m )dρ

r0

1 = 2

r p

     |u e |2 + |u m |2 dρ

r0

 p(r0 )   p(r )  = |u e (r )|2 + |u m (r )|2 − |u e (r0 )|2 + |u m (r0 )|2 2 2 r   1 − p  |u e |2 + |u m |2 dρ 2 r0

4.8 Leaky Waves in a Waveguide Filled with Lossy Medium

=

197

 p(r0 ) p(r )  |u e (r )|2 + |u m (r )|2 − |u m (r0 )|2 2 2 r   1 − p  |u e |2 + |u m |2 dρ. 2 r0

It follows from the last relation, definition (4.8.178), asymptotic relation (4.8.179), and the properties of operators K and S0 that there exists a large number  γ such that u, u) − (Ku, u) + Re(Bu, u) + u 2 +  Re(N1 ( γ )u, u) =  γ 2 (Ku, u) + Re(K γ Re(S0 u, u) ≥ u 2

for each u. Hence,  γ ∈ (N1 ), where (N1 ) is the resolvent set of OVF N1 . Moreover, Theorem 4.1 in [13] gives the estimate γ ) ≤ 1.

N1−1 ( γ ) < 1 is satisfied and conclude that there We take  γ such that the inequality N2 ( exists a bounded operator −1    γ ) + N2 ( γ) = I + N1−1 ( γ )N2 ( γ ) N1−1 ( γ ). N1 (  Theorem 4.8.1 OVF N(γ) : H → H is bounded, holomorphic, and Fredholm in the domain  = C\ ( K ∪  E ) and  E := {γ : γ 2 = (ε + iεlos )μ}. Proof The functions κ

Im+1 (κr ) m Im (κr ) = −κ + . Im (κr ) Im (κr ) r

(4.8.180)

are analytical in the domain C\ K as functions of γ because Im (κr ) have no zeros for Reκ > 0 [12]. Finally, we obtain the desired result by Lemmas 4.8.4. The proof of the theorem is complete.  Theorem 4.8.2 The spectrum of OVF N(γ) : H → H is discrete in the domain ; i.e., this function has finitely many characteristic numbers of finite algebraic multiplicity on any compact set I0 ⊂ . Proof The assertion of the theorem is a corollary of Theorem 4.8.1 and a theorem on a holomorphic OVF [13].  Lemma 4.8.5 The spectrum of OVF N(γ) : H → H is symmetric with respect to the real and imaginary axes. If γ0 is an eigenvalue of N(γ) corresponding to the eigenvector u1 = (u e , u m )T , then −γ0 , γ 0 , −γ 0 are eigenvalues of OVF N(γ) corresponding to the eigenvectors u2 = (−u e , u m )T , u3 = (u e , u m )T , u4 = (−u e , u m )T with the same multiplicity.

198

4 Waveguides of Circular Cross Section

Proof The assertion of the lemma can readily be proved by a straightforward verification with the use of (4.8.166) and conditions (4.8.167) and (4.8.168). Note that the condition at infinity is satisfied for all four characteristic numbers. 

4.9 Leaky Waves in a Waveguide Filled with Anisotropic Media 4.9.1 Statement of the Problem We assume that the permittivity and permeability in the entire space have the form  ε=

ε, ˆ r0 ≤ ρ ≤ r, and  μ= 1, ρ > r,



μ, ˆ r0 ≤ ρ ≤ r. 1, ρ > r,

⎡ ⎤ ⎡ ⎤ ερ 0 0 μρ 0 0 εˆ = ⎣ 0 ερ 0 ⎦, μˆ = ⎣ 0 μρ 0 ⎦, 0 0 εz (ρ) 0 0 μz (ρ)

and

(4.9.181)

(4.9.182)

where ερ > 1 and μρ > 1 are constants; εz (ρ) > 1 and μz (ρ) > 1 are twice continuously differentiable function on the segment [r0 , r ], i.e., εz (ρ) ∈ C 2 [r0 , r ] and μz (ρ) ∈ C 2 [r0 , r ], and Imεz (ρ) = Imμz (ρ) = 0. The problem on normal waves in GL described in Sect. 4.1.1 is an eigenvalue problem for the Maxwell equations with spectral parameter γ, which is the normalized propagation constant of the normal wave. Taking into account the GL anisotropic filling, write Maxwell’s equation in the expanded form: ⎧ m ⎪ i Hz − iγ Hϕ = −i ερ E ρ , ⎪ ⎪ ρ ⎪ ⎪ ⎪ ⎪ ⎪ iγ Hρ − Hz = −i ερ E ϕ , ⎪ ⎪ ⎪ ⎪ 1 im ⎪  ⎪ ⎪ Hρ = −i εz E z , ⎨ (ρHϕ ) − ρ ρ (4.9.183) m ⎪ ⎪ E i − iγ E = i μ H , ⎪ z ϕ ρ ρ ⎪ ρ ⎪ ⎪ ⎪ ⎪ ⎪ iγ E ρ − E z = i μρ Hϕ , ⎪ ⎪ ⎪ ⎪ ⎪ 1 im ⎪ ⎩ (ρE ϕ ) − E ρ = i μz Hz , ρ ρ

where  ερ =

ερ , r0 ≤ ρ ≤ r, and  μρ = 1, ρ > r,



μρ , r0 ≤ ρ ≤ r. 1, ρ > r,

(4.9.184)

4.9 Leaky Waves in a Waveguide Filled with Anisotropic Media

199

and  εz =

εz (ρ), r0 ≤ ρ ≤ r, and  μz = 1, ρ > r,



μz (ρ), r0 ≤ ρ ≤ r. 1, ρ > r.

(4.9.185)

Express functions E ρ , Hρ , E ϕ , Hϕ via E z and Hz from the first, second, fourth, and fifth equations in system (4.9.183) m μρ Hz − iγρE z , ρ(γ 2 −  ερ  μρ ) γm E z + iρ μρ Hz Eϕ = , ρ(γ 2 −  ερ  μρ )

Eρ =

iγρHz + m ερ E z , 2 ρ(γ −  ερ  μρ ) γm Hz − iρ ερ E z Hϕ = . ρ(γ 2 −  ερ  μρ )

Hρ = −

(4.9.186)

It follows from (4.9.186) that the normal wave field in the waveguide can be represented with the use of two scalar functions u e := i E z (ρ), u m := Hz (ρ).

(4.9.187)

Thus, the problem has been reduced to finding longitudinal components u e and u m of the electric and magnetic fields. We have the following eigenvalue problem for the longitudinal field components u e and u m : find γ ∈ C such that, for a given m ∈ Z, there exist nontrivial solutions of the system of differential equations ⎧      1 2  2  εz m2 ⎪ ⎪ ⎪ u e = 0, ερ  μρ + ⎨ ρu e − ρ ρ γ −   ερ  ερ   ⎪  μz m2 ⎪    1 2  2 ⎪ u m = 0, ρ γ − ερ  μρ + ⎩ ρu m − ρ  μρ  μρ

(4.9.188)

satisfying the boundary conditions for ρ = r0 u e (r0 ) = 0, u m (r0 ) = 0,

(4.9.189)

the transmission conditions for ρ = r [u e ]|r = 0, [u m ]|r = 0,             ρ μ ρ ερ u e u um ue ρ m  −  = 0, γm  −  = 0, γm  ερ μρ − γ 2 r  ερ μρ − γ 2 r  ερ μρ − γ 2 r  ερ μρ − γ 2 r 

where [ f ]|ρ0 = lim

ρ→ρ0 −0

f (ρ) − lim

ρ→ρ0 +0

(4.9.190) f (ρ) and the conditions of the boundedness

of the field in any finite domain and increasing at infinity. Once we determine longitudinal field components u e and u m by solving problem (4.9.188)–(4.9.190), we can find the transverse components by formulas (4.9.186).

200

4 Waveguides of Circular Cross Section

The equivalence of reduction to problem (4.9.188)–(4.9.190) is not valid only for ερ  μρ . γ2 =  For ρ > r , we have  ε = 1,  μ = 1; then from (4.9.188), we obtain the system     ρ(ρu e ) − ρ2 κ2 + m 2 u e = 0, ρ(ρu m ) − ρ2 κ2 + m 2 u m = 0, where κ2 = γ 2 − 1. In view of the condition at infinity, we obtain a solution of this system in the form u e (ρ; γ, m) = C1 Im (κρ), (4.9.191) u m (ρ; γ, m) = C2 Im (κρ), where Im is the modified Bessel function (the Infeld function) [10] and C1 and C2 are constants. R

Function κ(γ) is analytical in the domain C\ K , where  K := {γ : Imγ 2 = 0, γ 2 ≤ 1}.

For r0 ≤ ρ ≤ r , we have  ε = εˆ and  μ = μ, ˆ and from (4.9.188) we obtain the system of differential equations   L e u e : = u e + pu e − γ 2 f e + qe u e = 0,   L m u m : = u m + pu m − γ 2 f m + qm u m = 0 where p=

1 , ρ

qe = −εz μρ +

fe =

εz , ερ

fm =

(4.9.192)

μz , μρ

m2 m2 , q = −ε μ + . m ρ z ρ2 ερ ρ2 μρ

Definition 4.9.1 If for a given m there exist nontrivial functions u e and u m corresponding to γ ∈ C such that these functions are given by (4.9.191) for ρ > r , are a solution of system (4.9.192) for r0 ≤ ρ ≤ r , and satisfy the transmission conditions (4.9.190), then γ is called a characteristic number of problem Pani . Definition 4.9.2 The pair u e and u m , |u e |2 + |u m |2 ≡ 0, will be called an eigenvector of problem Pani corresponding to the characteristic number γ ∈ C.

4.9.2 Sobolev Spaces and Variational Relation Let us give variational formulation of problem Pani . Multiplying equations (4.9.192) by arbitrary test functions ve ∈ H01 (r0 , r ) and vm ∈ H 1 (r0 , r ) (we can assume that

4.9 Leaky Waves in a Waveguide Filled with Anisotropic Media

201

these functions are continuously differentiable in (r0 , r )) and applying Green’s formula, we obtain r

r vLudρ =

r0

r



vu dρ + r0

=

r vu  r0

r



pvu dρ − r0 r

−

r0 r 

 

u v dρ + r0

r =−γ

r0

r



pu vdρ − r0

  v γ 2 f + q udρ

r0

r

r

f uvdρ −

2

  v γ 2 f + q udρ

quvdρ + r0

r pu vdρ + u v r0 − 



r0

r

u  v  dρ,

r0

(4.9.193) where u = u j , v = v j , f = f j , q = q j , j = e or m. Taking into account the boundary condition for ρ = r0 , we get r

r vLudρ = −γ

r0

r f uvdρ −

2 r0

r quvdρ +

r0

 pu vdρ + u v r − 



r0

r

u  v  dρ

r0

(4.9.194) We separately apply formula (4.9.194) to the first and second equations in system (4.9.192) on the interval [r0 , r ], add the results, and find that the sum of the left-hand sides of (4.9.192) satisfies the relation r

r (v e L e u e + v m L m u m )dρ = −γ

( f e u e v e + f m u m v m )dρ

2

r0

r0

r −

r qe u e v e + qm u m v m )dρ +

r0

p(u e v e

+

u m v m )dρ

r0

r −

(u e v e + u m v m )dρ

r0

+u e (r )v e (r )+u m (r )v m (r ). (4.9.195) Given solutions (4.9.191), we express the values of the normal derivatives at ρ = r from relations (4.9.190) as follows: u e (r ) = u m (r )

γm μρ ερ − 1 μρ ερ − γ 2 Im (κr ) u e (r ), u (r ) − m r ερ κ2 κερ Im (κr )

γm μρ ερ − 1 μρ ερ − γ 2 Im (κr ) u m (r ). = u (r ) − e r μρ κ2 κμρ Im (κr )

Then, in view of (4.9.195), from (4.9.196), we obtain

(4.9.196)

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4 Waveguides of Circular Cross Section

r γ

r ( f e u e v e + f m u m v m )dρ +

2 r0

(qe u e v e + qm u m v m )dρ r0

r −

p(u e v e

+

r0

u m v m )dρ

r +

(u e v e + u m v m )dρ

r0

 γm μρ ερ − 1 μρ ερ − γ 2 Im (κr ) u m (r ) − − u e (r ) v e (r ) r ερ κ2 κερ Im (κr )   γm μρ ερ − 1 μρ ερ − γ 2 Im (κr ) u − u (r ) − (r ) v m (r ) = 0. e m r μρ κ2 κμρ Im (κr ) (4.9.197) 

Variational relation (4.9.197) has been obtained for smooth functions v e and v m .

4.9.3 Spectrum of the OVF Let H = H01 (r0 , r ) × H 1 (r0 , r ) be the Cartesian product of Hilbert spaces with the inner product and the norm (u, v) = (u 1 , v1 )1 + (u 2 , v2 )1 , u 2 = u 1 21 + u 2 21 ; u, v ∈ H, u = (u 1 , u 2 )T , v = (v1 , v2 )T , u 1 , v1 ∈ H01 (r0 , r ) , u 2 , v2 ∈ H 1 (r0 , r ) . Then the integrals occurring in (4.9.197) can be viewed as sesquilinear forms over the field C defined on H and depending on the arguments u = (u e , u m )T , v = (v e , v m )T . Consider the following sesquilinear forms and the corresponding operators: r k(u, v) :=

( f e u e v e + f m u m v m )dρ = (Ku, v), ∀v ∈ H, r0

r

 k(u, v) :=

 v), ∀v ∈ H, ((qe − 1)u e v e + (qm − 1)u v v m )dρ = (Ku,

r0

r a(u, v) := r0

(u e v e + u m v m + u e v e + u m v m )dρ = (Iu, v), ∀v ∈ H,

4.9 Leaky Waves in a Waveguide Filled with Anisotropic Media

r b(u, v) :=

203

p(u e v e + u m v m )dρ = (Bu, v), ∀v ∈ H,

r0



 γm μρ ερ − 1 μρ ερ − γ 2 Im (κr ) u e (r ) v e (r ) s(u, v) = u m (r ) − r ερ κ2 κερ Im (κr )   γm μρ ερ − 1 μρ ερ − γ 2 Im (κr ) u + u (r ) − (r ) v m (r ) e m r μρ κ2 κμρ Im (κr ) = (Su, v), ∀v ∈H. (4.9.198) The boundedness of a(u, v) is obvious. The boundedness of forms k(u, v) and  k(u, v) follows from the Poincarè inequality [11]. Now variational problem (4.9.197) can be written in the operator form (N(γ)u, v) = 0, ∀u ∈ H or, equivalently,    + I − B − S(γ) u = 0. N(γ)u := γ 2 K + K

(4.9.199)

Equation (4.9.199) is the operator form of variational relation (4.9.197). The characteristic numbers and eigenvectors of operator N by definition coincide with the eigenvalues and eigenvectors of problem Pani for γ 2 = −με2 .

4.9.4 Properties of the Spectrum of the OVF We have reduced the problem on normal waves to the study of spectral properties of OVF N. In this connection, we first consider the properties of the operators introduced in the preceding section. The validity of Lemmas 4.9.1–4.9.3 is shown in the Sect. 4.1.4.  : H → H are compact, and K > 0. Lemma 4.9.1 The bounded operators K and K Lemma 4.9.2 Operators B : H → H are compact. Lemma 4.9.3 Operator S : H → H is compact. Lemma 4.9.4 There exists a  γ ∈ R such that operator N( γ ) is continuously invertible; i.e., the resolvent set (N) := {γ : ∃ N−1 (γ) : H → H } of OVF N( γ ) is nonempty, (N) = ∅. Proof Let γ ∈ R, γ > 0 and γ → +∞. Consider the OVF

204

4 Waveguides of Circular Cross Section

N(γ) = N1 (γ) + N2 (γ), where

 + B + I + γS0 , N1 (γ) = γ 2 K + K

and N2 (γ) = −S(γ) − γS0 , and the operator S0 : H → H is defined by the sesquilinear form   u e (r )v e (r ) u m (r )v m (r ) S0 u, v := + . ερ μρ

(4.9.200)

In view of the asymptotics of functions Im as γ → +∞ [12], we have Im (κr ) ∼ 1, Im (κr ) and

1 ερ μρ − γ 2 ∼ −γ + O . κ γ

Consequently,

N2 (γ) = O

1 γ

, as γ → +∞.

(4.9.201)

Then (OVFs) N(γ) can be viewed as a perturbation of operator pencil N1 by operatorvalued function N2 for large γ. We have r Reb(u, u) := Re

p(u e u e

+

u m u m )dρ

r0

1 = 2

r p

     |u e |2 + |u m |2 dρ

r0

 p(r0 )   p(r )  = |u e (r )|2 + |u m (r )|2 − |u e (r0 )|2 + |u m (r0 )|2 2 2 r   1 − p  |u e |2 + |u m |2 dρ 2 r0

 p(r0 ) p(r )  = |u e (r )|2 + |u m (r )|2 − |u m (r0 )|2 2 2 r   1 − p  |u e |2 + |u m |2 dρ. 2 r0

4.9 Leaky Waves in a Waveguide Filled with Anisotropic Media

205

It follows from the last relation, definition (4.9.200), asymptotic relation (4.9.201), γ , such that and the properties of operators K and S0 that there exists a large number  u, u) + Re(Bu, u) + (u, u) +  γ )u, u) =  γ 2 (Ku, u) + (K γ (S0 u, u) ≥ u 2 Re(N1 ( for each u. Hence,  γ ∈ (N1 ), where (N1 ) is the resolvent set of OVF N1 . Moreover, Theorem 4.1 in [13] gives the estimate γ ) ≤ 1.

N1−1 ( γ ) < 1 is satisfied and conclude that there We take  γ such that the inequality N2 ( exists a bounded operator −1    γ ) + N2 ( γ) = I + N1−1 ( γ )N2 ( γ ) N1−1 ( γ ). N1 (  Theorem 4.9.1 OVF N(γ) : H → H is bounded, holomorphic, and Fredholm in √ the domain  = C\( K ∪  E ), where  E := {γ : Imγ = 0, γ = ± ερ μρ }. Proof The functions κ

Im (κr ) Im+1 (κr ) m = −κ + . Im (κr ) Im (κr ) r

(4.9.202)

are analytical in the domain C\ K as functions of variable γ because Im (κr ) have no zeros for Reκ > 0 [12]. Finally, we obtain the desired result by Lemmas 4.9.4. The proof of the theorem is complete.  Theorem 4.9.2 The spectrum of the operator function N(γ) : H → H is discrete in the domain ; i.e., this function has finitely many characteristic points of finite algebraic multiplicity on any compact set I0 ⊂ . Proof The assertion of the theorem is a corollary of Theorem 4.9.1 and a theorem on a holomorphic operator function [13].  Lemma 4.9.5 The spectrum of OVF N(γ) : H → H is symmetric with respect to the real and imaginary axes. If γ0 is an eigenvalue of N(γ) corresponding to the eigenvector u1 = (u e , u m )T , then −γ0 , γ 0 , −γ 0 are eigenvalues of OVF N(γ) corresponding to the eigenvectors u2 = (−u e , u m )T , u3 = (u e , u m )T , u4 = (−u e , u m )T with the same multiplicity. Proof The assertion of the lemma can readily be proved by a straightforward verification with the use of (4.9.188) and conditions (4.9.189) and (4.9.190). Note that the condition at infinity is satisfied for all four characteristic numbers. 

206

4 Waveguides of Circular Cross Section

4.10 Numerical Simulation 4.10.1 Statement of the Problem We consider electromagnetic waves propagating in GL described in Sect. 4.1.1. The fields depend harmonically on time as exp(−iωt), where ω > 0 is the circular frequency. Determination of surface TE-polarized waves reduces to finding nontrivial running-wave solutions of the homogeneous system of Maxwell equations depending on the coordinate z along which the structure is regular in the form eiγz , 

curl H = −iωε0 εE, ˜ curl E = iωμ0 H,

(4.10.203)

    E = 0, E ϕ (ρ)eiγz , 0 , H = Hρ (ρ)eiγz , 0, Hz (ρ)eiγz , with the boundary condition for the tangential electric field component on the perfectly conducting cylindrical surface  E ϕ ρ=r0 = 0,

(4.10.204)

and the transmission conditions for the tangential electric and magnetic field components on the permittivity discontinuity surface (ρ = r )  [E ϕ ]ρ=r = 0, [Hz ]|ρ=r = 0, where [ f ]|x0 = lim

x→x0 −0

f (x) − lim

x→x0 +0

(4.10.205)

f (x). We will not fix the radiation condition

at infinity because we want to consider the problem for arbitrary γ. We assume that the relative permittivity in the entire space have the form   ε=

ε(ρ), r0  ρ  r, ρ > r, εc ,

where ε(ρ) is a continuous function on [r0 , r ], i.e., ε(ρ) ∈ C[r0 , r ]. The problem on surface waves is an eigenvalue problem for the Maxwell equations with spectral parameter γ which is the propagation constant of the normal wave. The normal wave field in the waveguide can be represented using one scalar function u := E ϕ (ρ). Thus, the problem is reduced to finding tangential component u of the electric field. Throughout the text below, ( · ) stands for differentiation with respect to ρ.

4.10 Numerical Simulation

207

Let us formulate definitions of the normal waves considered in the study. Definition 4.10.1 The propagating wave is characterized by a real parameter γ. Definition 4.10.2 The evanescent wave is characterized by a purely imaginary parameter γ. Definition 4.10.3 The complex wave is characterized by complex parameter γ such that ReγImγ = 0. Definition 4.10.4 The surface wave satisfies the condition u(ρ) → 0, ρ → ∞. Definition 4.10.5 The leaky wave satisfies the condition u(ρ) → ∞, ρ → ∞.

R

Propagation constant γ characterizes the behavior of a wave (propagating, evanescent, or complex) in the z-direction. Classification of waves as surface or leaky depends on the behavior in the ρ-direction in the cross-sectional domain.

We have the following eigenvalue problem for the tangential electric field component u: find γ ∈ C such that there exist nontrivial solutions of the differential equation   (4.10.206) ε − γ 2 u = 0, u  + ρ−1 u  − ρ−2 u + ω 2 μ0 ε0 satisfying the boundary conditions u|ρ=r0 = 0,

(4.10.207)

' ( [u]|ρ=r = 0, u  ρ=r = 0.

(4.10.208)

and the transmission conditions

Thus, the resulting field (E, H) will satisfy all conditions (4.10.203)–(4.10.205). For ρ > r , we have  ε = εc ; then from (4.10.206), we obtain the equation u  + ρ−1 u  − ρ−2 u − κ2 u = 0.

(4.10.209)

In accordance with the condition at infinity (see Definition 2.1.4 and 2.2.1), we choose a solution of the last equation for surface waves in the form u = C1 K 1 (κρ), ρ > r,

(4.10.210)

u = C2 I1 (κρ), ρ > r,

(4.10.211)

and for leaky waves  where κ = γ 2 − ω 2 εc ε0 μ0 and Reκ > 0, C1 , C2 are constants and K m and Im are the modified Bessel functions (Macdonald and Infeld functions) [12].

208

4 Waveguides of Circular Cross Section

For r0 < ρ < r , we have  ε = ε(ρ); thus from (4.10.206), we obtain the equation   u  + ρ−1 u  − ρ−2 u +  − γ 2 u = 0,

(4.10.212)

(ρ; ω) = ω 2 ε0 μ0 ε(ρ).

(4.10.213)

where Definition 4.10.6 γ ∈ C is called a propagation constants of the problem if there exists nontrivial solution u of equation (4.10.212) for r0 < ρ < r given by (4.10.210), ρ > r , for surface waves and (4.10.211) for leaky waves, respectively, boundary condition (4.10.207), and transmission conditions (4.10.208).

4.10.2 Numerical Method Consider the Cauchy problem for equation (4.10.212) with the initial conditions u (r0 ) := 0, u  (r0 ) := A,

(4.10.214)

where A is a known constant. We assume that the Cauchy problem (4.10.212), (4.10.214) is globally and uniquely solvable on the segment [r0 , r ] for given values r0 and r and its solution continuously depends on parameter γ. Applying the transmission condition on the boundary ρ = r (4.10.208), one obtains DE for surface waves    (γ) ≡ K 1 (κr )u  (r ) + κK 0 (κr ) + r −1 K 1 (κr ) u(r ) = 0,

(4.10.215)

and DE for leaky waves    (γ) ≡ I1 (κr )u  (r ) − κI0 (κr ) − r −1 I1 (κr ) u(r ) = 0,

(4.10.216)

where quantities u(r ) and u  (r ) are obtained from the solution to the Cauchy problem (4.10.212), (4.10.214). Let γ = α + iβ, where α, β ∈ R. Then equating to zero the real and imaginary parts of  (γ), one obtains a system of equations involving real-valued functions for determining the real and imaginary parts of complex parameter γ: 

1 (α, β) := Re (γ) = 0, 2 (α, β) := Im (γ) = 0.

(4.10.217)

We solve numerically system of equations (4.10.217) to determine a number pair (α, β). The solution to each equation of system (4.10.217) is a curve in the Oαβ-

4.10 Numerical Simulation

209

Fig. 4.7 Numerical solution of system (4.10.217): blue and red curves are solutions of, respectively, the first and the second equations of (4.10.217); yellow point marks a solution of (4.10.217)

plane. Next, we determine the points of intersection of the curves; these points are approximate eigenvalues of the problem. Introduce a grid )  α(i) , β ( j) : α(i) = a1 + iτ1 , β ( j) = b1 + iτ2 , i = 0, n, τ1 =

a2 − a1 b2 − b1 * , j = 0, m, τ2 = n m

with the steps τ1 > 0 in α and τ2 > 0 in β, where a1 , a2 , b1 , and b2 are real fixed constants. The grid points are used in the implementation of the shooting method below. Solving the Cauchy problem (4.10.212), (4.10.214) for each grid point, one obtains u(i, j)(r ) and u  (i, j)(r ), i = 0, n, j = 0, m. Since the solution u(r ; α, β) is dependent on parameters αand β, it follows that there exists a point     continuously +( j) ∈ β ( j) , β ( j+1) , such that 1 α(i) , β +( j) in the plane Oαβ where β +( j) = α(i) , β 0. The smaller τ1 and τ2 , the more accurate is the solution. Proceeding in the same manner, one determines a set of pairs (α( p) , β (q) ) that form a curve in the plane Oαβ (the blue curve in Fig. 4.7). Applying the same approach to the second equation of (4.10.217), one obtains another curve in the plane Oαβ (the red curve in Fig. 4.7). This curve is an approximate solution of the equation 2 (α, β) = 0. It is clear that the intersection point of the curves (the yellow point in Fig. 4.7) is an approximate solution of the problem. Decreasing the steps τ1 and τ2 , we can obtain the solutions with the prescribed (arbitrary) accuracy.

210

4 Waveguides of Circular Cross Section

4.10.3 Numerical Results Figures 4.8, 4.9, 4.10, 4.11, 4.12, 4.13, 4.14 and 4.15 display the calculated propagation constants of the TE-polarized waves in GL filled with homogeneous and inhomogeneous dielectric, dielectric with losses, and metamaterial. We have carried out numerical experiments for four frequency values. Propagating, evanescent, and complex surface and leaky TE-waves are determined.

ω = 0.25

ω = 0.5

ω = 0.75

ω = 1.0

Fig. 4.8 Surface waves. Homogeneous dielectric with ε = 4. Numerical solution of system (4.10.217): blue and red curves are solutions of, respectively, the first and the second equations of (4.10.217); green and purple intersection points are, respectively, propagation constants of the propagating and evanescent surface TE-waves

4.10 Numerical Simulation

211

ω = 0.25

ω = 0.5

ω = 0.75

ω = 1.0

ρ Fig. 4.9 Surface waves. Inhomogeneous dielectric with ε(ρ) = 4 + r −r . Numerical solution 0 of system (4.10.217): blue and red curves are solutions of, respectively, the first and the second equations of (4.10.217); green and purple intersection points are, respectively, propagation constants of the propagating and evanescent surface TE-waves

The following values of parameters are used in calculations: εc = 1, ε0 = 1, μ0 = 1, A = 1, r0 = 2, r = 4, a1 = −3, a2 = 3, τ1 = 0.025, b1 = −3, b2 = 3, and τ2 = 0.025.

212

4 Waveguides of Circular Cross Section

ω = 0.25

ω = 0.5

ω = 0.75

ω = 1.0

Fig. 4.10 Surface waves. Lossy dielectric with ε = 4 + 0.1ωi. Numerical solution of system (4.10.217): blue and red curves are solutions of, respectively, the first and the second equations of (4.10.217); green and purple intersection points are, respectively, propagation constants of the propagating and evanescent surface TEwaves

4.10 Numerical Simulation

213

ω = 0.25

ω = 0.5

ω = 0.75

ω = 1.0

Fig. 4.11 Surface waves. Metamaterial with ε = −4. Numerical solution of system (4.10.217): blue and red curves are solutions of, respectively, the first and the second equations of (4.10.217); green and purple intersection points are, respectively, propagation constants of the propagating and evanescent surface TE-waves

214

4 Waveguides of Circular Cross Section

ω = 0.25

ω = 0.5

ω = 0.75

ω = 1.0

Fig. 4.12 Leaky waves. Homogeneous dielectric with ε = 4. Numerical solution of system (4.10.217): blue and red curves are solutions of, respectively, the first and the second equations of (4.10.217); green and purple intersection points are, respectively, propagation constants of the propagating and evanescent surface TE-waves; yellow intersection points are propagation constants of the complex leaky TE-waves

4.10 Numerical Simulation

215

ω = 0.25

ω = 0.5

ω = 0.75

ω = 1.0

ρ Fig. 4.13 Leaky waves. Inhomogeneous dielectric with ε(ρ) = 4 + r −r . Numerical solution 0 of system (4.10.217): blue and red curves are solutions of, respectively, the first and the second equations of (4.10.217); green and purple intersection points are, respectively, propagation constants of the propagating and evanescent surface TE-waves; yellow intersection points are propagation constants of the complex leaky TE-waves

216

4 Waveguides of Circular Cross Section

ω = 0.25

ω = 0.5

ω = 0.75

ω = 1.0

Fig. 4.14 Leaky waves. Lossy dielectric with ε = 4 + 0.1ωi. Numerical solution of system (4.10.217): blue and red curves are solutions of, respectively, the first and the second equations of (4.10.217); green and purple intersection points are, respectively, propagation constants of the propagating and evanescent surface TE-waves; yellow intersection points are propagation constants of the complex leaky TE-waves

4.10 Numerical Simulation

217

ω = 0.25

ω = 0.5

ω = 0.75

ω = 1.0

Fig. 4.15 Leaky waves. Metamaterial with ε = −4. Numerical solution of system (4.10.217): blue and red curves are solutions of, respectively, the first and the second equations of (4.10.217); green and purple intersection points are, respectively, propagation constants of the propagating and evanescent surface TE-waves; yellow intersection points are propagation constants of the complex leaky TE-waves

Figures 4.8, 4.9, 4.10 and 4.11 show the solutions to the problem of surface TEwaves in GL. There are no complex surface TE-waves (yellow points are absent). In Figs. 4.8 and 4.11 we present the results of calculations for a dielectric with constant permittivity and with losses. In these cases, the propagating surface TEwaves (green points) do not exist at all frequencies. If the waveguide is filled with inhomogeneous dielectric (as in Fig. 4.9), then with increasing frequency, the number of the propagating surface TE-waves increases together with the absolute value of the real propagation constants.

218

4 Waveguides of Circular Cross Section

In the case of a metamaterial (see Fig. 4.11), as the frequency increases, the absolute values of the propagation constants corresponding to the evanescent surface TE-waves decrease, while the absolute values of the propagation constants of the propagating surface TE-waves increase. Figures 4.12, 4.13, 4.14 and 4.15 demonstrate the calculated solutions to the problem of the leaky TE-waves in GL. In the case of a homogeneous dielectric with constant permittivity all types of leaky TE-waves exist at all chosen frequencies (see Fig. 4.12). It can be seen, that with the increasing frequency, the absolute values of the real propagation constants increase and of the complex propagation constants decrease; the number of the waves having imaginary propagation constants increases. This property is preserved in the cases of homogeneous (Fig. 4.13) and lossy (Fig. 4.14) dielectrics. For a metamaterial filling (see Fig. 4.15), the propagating leaky TE-waves (green points) do not exist at all frequencies. When the frequency increases, the absolute values of the imaginary part of the propagation constants corresponding to the complex surface TE-waves (yellow points) increase, while the absolute values of the real part of the propagation constants tend to zero.

References 1. Y. Shestopalov, Complex waves in a dielectric waveguide. Wave Motion 82, 16–19 (2018) 2. Y. Shestopalov, E. Kuzmina, A. Samokhin, On a Mathematical Theory of Open MetalDielectric Waveguides FERMAT, vol. 5 (2014) 3. E. Kuzmina, Waves in a Lossy Goubau line, in Proceedings of the 2016 European Conference on Antennas and Propagation EuCAP (2016) 4. E.Y. Smolkin, Goubau line filled with nonlinear medium: Numerical study of TM-polarized waves, in Proceedings of the ICEAA, Torino, Italy, September 7–11 (2015), pp. 1572–1575 5. E.Y. Smolkin, On the problem of propagation of nonlinear coupled TE-TM waves in a doublelayer nonlinear inhomogeneous cylindrical waveguide, in Proceedings of the Internatonal Conference Days on Diffraction (2015), pp. 318–322 6. E.Y. Smolkiin, D.V. Valovik, Guided electromagnetic waves propagating in a two-layer cylindrical dielectric waveguide with inhomogeneous nonlinear permittivity. Adv. Math. Phys. 1–11 (2015) 7. E.Y. Smolkin, Y.U. Shestopalov, Nonlinear Goubau line: numerical study of TE-polarized waves, in Proceedings of the PIER Symposium (2015), pp. 1513–1517 8. Y. Shestopalov, Y. Smirnov, E. Kuzmina, Mathematical aspects of the theory of wave propagation in metal-dielectric waveguides, in Proceedings of the XXXI URSI General Assembly (2014) 9. Y.G. Smirnov, Mathematical Methods for Electromagnetic Problems (Izdatelstvo PSU, Penza, 2009) 10. A.F. Nikiforov, V.B. Uvarov, Spetsial’nye funktsii matematicheskoi fiziki (Special Functions of Mathematical Physics) (Nauka, Moscow, 1978) 11. R. Adams, Sobolev Spaces (Academic, New York, 1975) 12. M. Abramovic, I. Stigan, Handbook on Special Functions (Nauka, Moscow, 1979). ([in Russian]) 13. I. Gohberg, M. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators in Hilbert Space, vol. 18 (American Mathematical Society, 1969)

References

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14. Y.G. Smirnov, E.Y. Smolkin, Discreteness of the spectrum in the problem on normal waves in an open inhomogeneous waveguide. Differ. Equ. 53(10), 1168–1179 (2018) 15. R. Kress, Linear Integral Equations (Springer, New York, 1999) 16. E.Y. Smolkin, Numerical method for electromagnetic wave propagation problem in a cylindrical inhomogeneous metal dielectric waveguiding structures. Math. Model. Anal. 22(3), 271–282 (2017)

Chapter 5

Open Waveguides of Arbitrary Cross Section

Analysis of the wave propagation in open metal-dielectric waveguides constitutes an important class of vector electromagnetic problems. In the case of hollow shielded waveguides (filled with homogeneous dielectric) the spectral parameter enters the equations and not the transmission conditions, ending up with an eigenvalue problem for a self-adjoint operator. However, a general setting for a metal-dielectric waveguide yields non-self-adjoint boundary eigenvalue problems for the systems of Helmholtz equations with piecewise constant coefficients, the transmission conditions and the conditions at infinity containing the spectral parameter; the transmission conditions are stated on the discontinuity lines (surfaces) of the permittivity and the resulting problem becomes non-self-adjoint. An approach based on the reduction within the frames of this setting to eigenvalue problems for OP considered in Sobolev spaces was proposed by Smirnov in [1–3] (see also [4, 5]) based on the fundamental work [6]. The general theory of polynomial OVF called OPs is sufficiently well elaborated. Here, a fundamental work by Keldysh [7] pioneered investigation of non-self-adjoint polynomial pencils. The OP method is known to be a natural and efficient approach for the investigation of the wave propagation in regular waveguides. OPs were applied to the analysis of electromagnetic problems in [8–10]. Open waveguides were investigated by a great number of authors [8, 11, 12]. Significant contributions to the theory were made in the monographs [13–16]. However, for open (unshielded) structures of arbitrary cross section, a complete theory of the wave propagation has not been constructed, mainly because the problem becomes much more complicated (due to the non-compactness of the corresponding operators). This fact serves as a driving force to perform the present study. Analysis of running (normal) waves constitutes one of central topics of electromagnetics [6, 14], and acoustics [17], and is a core subject of the spectral theory of open structures [6]. The verification of the existence and analysis of the distribution

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 Y. Shestopalov et al., Optical Waveguide Theory, Springer Series in Optical Sciences 237, https://doi.org/10.1007/978-981-19-0584-1_5

221

222

5 Open Waveguides of Arbitrary Cross Section

of the wave spectra on the complex plane are key issues for open structures: dielectric and metal-dielectric waveguides and transmission lines. Apart from the knowledge of eigenvalues of the corresponding non-self-adjoint boundary eigenvalue problems in open domains, which is important by itself, the information about the location of the wave spectra enables researchers to clarify many phenomena of the wave propagation and facilitate the use of different numerical approaches and solvers. For canonical coordinate domains forming the cross section of a cylindrical scatterer like a circle and an ellipse, determination of the propagation constants of the running waves can be reduced, using separation of variables in cylindrical coordinates [18–20], to the determination of complex zeros of functions of one complex variable and several real or complex parameters [21]. However, even for a dielectric cylinder of circular shape, a dielectric rod (DR), a rigorous proof of the existence of the wave spectra has been an open question. Only recently, this problem has been solved in [22] where it has been shown that the problem can be reduced to finding zeros of a family of weighted cross-products of cylindrical functions. The latter have been referred to as generalized cylindrical polynomials (GCPs) in [23] where the elements of the GCP theory have been elaborated that enable one to determine the occurrence and location of zeros. In [6, 24–27], the existence of the real and complex running wave and resonance frequency spectra were demonstrated for certain families of shielded and open metal-dielectric scatterers formed in the cross section by (bounded) domains (particularly, in [25] for the union or rectangular domains with slots on the common part of the boundary) and ’slotted’ open domains (a cavity-backed slotted screen in [24]). However, these results have never been extended to the case when the boundary curve of a cylindrical dielectric scatterer is a single arbitrary smooth (or piecewise smooth) simply connected contour. For such structures, the methods of boundary integral equations (BIEs) developed in [28, 29] (see comprehensive lists of references therein reflecting the state of the art) and summation equations [6, 30] have been a subject of intense studies. A characteristic feature (and a drawback) of the BIE method as applied to the calculation of complex eigenvalues for dielectric waveguides in open domains is the absence of preliminary information about the existence and location of the sought eigenvalue spectrum. This is one of the main reasons to accomplish the present study; an objective is, apart from the theoretical findings and novelty, to complement the BIE method with a tool to perform justified and efficient computations of eigenvalues (characteristic numbers of OVFs and OPs). Spectral theory of OVFs and OPs with a nonlinear dependence on the spectral parameter relevant to our studies was developed mainly by Gohberg and his coworkers in [31, 32]. The beginning of application of this theory to the analysis of oscillations, waves, and resonance scattering in electromagnetics and acoustics dates back to the late 1970s when the pioneering works [33, 34] were published. A characteristic feature of the approach is introduction in [35] of a correct statement of a boundary eigenvalue problem with the generalized Sveshnikov-Reichardt conditions at infinity that enable one to determine complex eigenvalues of open structures. The findings were summarized then in monographs [6, 26, 36] and addressed [37] the wave propagation in transmission lines of various types. Substantiation of the method

5 Open Waveguides of Arbitrary Cross Section

223

of generalized potentials [38, 39] (included then to [6, 26]) gave rise to elaborating an approach that employs layer-potential boundary integral operators and BIEs and further development of the spectral theory for electromagnetics [39, 40]. The latter involves integral OVFs with a logarithmic singularity of the kernel [26], in particular finite-meromorphic integral OPs (IOPs) associated with the determination of resonance frequencies and running waves of open waveguides. For the first time, finite-meromorphic OVFs and IOPs were applied for the analysis of resonance scattering in [34, 37]; a consistent presentation of the method was performed later in [6] and [26] where, among all, explicit formulas for the IOP characteristic numbers (CNs) were obtained. Comprehensive asymptotic analysis and explicit formulas for narrow-slot scatterers (as segments of small-parameter series) are presented in [6, 26]. The study was extended in [24, 25] where particularly multi-slot and multi-strip scatterers were considered, both shielded and open. Methods of the OVF spectral theory in combination with the layer BIEs and IOPs were extensively applied in a series of papers among which we note [41] and were summarized then in the monograph [42]. It should be noted however that a priority here is connected with the results obtained (a decade earlier) in our studies which led to the construction in the 1990s of a unified approach called the spectral theory of open structures [6]. This study may be considered as part of a broader framework of the BIE applications [28, 29] to the analysis of open dielectric structures and beyond. The efforts are aimed to fill the existing gap in the BIE theory, particularly to validate the eigenvalue determination using numerical methods. Note that for open structures of arbitrary cross section a breakthrough has been recently made in [43] where the existence of complex resonance frequencies has been rigorously verified by the reduction to eigenvalue problems for OPs and OVFs using BIEs. For coordinate open structures (of circular symmetry; the simplest are DR and GL), the determination of the wave spectra can be reduced to the solution of DEs; i.e., to finding (the parameter dependence) of real or complex zeros of certain families of functions. This task may be a severe mathematical problem by itself. For structures of more general cross sections, a typical approach [28, 30, 44–46] employs summation equations (infinite systems involving the unknown Fourier coefficients of the solution series expansions) ending up, for eigenvalue problems, with the determination of zeros of infinite determinants. The most essential drawback of this approach is again the absence of rigorous information and proofs about the existence and localization of the sought zeros (eigenvalues of the considered problems) on the complex plane. The chapter deals with open structures where the exterior domains (filled with a homogeneous medium) are unbounded. The first results on the investigation of such problems using OPs where the normal wave spectra are described have been recently obtained in [47–49] for a circular waveguide. For layered metal-dielectric waveguides of circular cross section (cylindrical slot and microstrip lines), the findings obtained using the spectral theory of OVFs in combination with the BIE method (applied for dielectric waveguides of arbitrary cross section) are summarized, respectively, in [50, 51] (the first work [52] dates back to 1985) and [45] (see the references therein)

224

5 Open Waveguides of Arbitrary Cross Section

The difficulties connected with the study of the wave propagation in open metaldielectric waveguides can be overcome by introducing a fictitious outer region (the exterior of a circle) and representing the solution in this region in terms of the Green function. This leads to the occurrence of a trace operator in the variational relation (on the boundary of the fictitious region) which depends in a nonlinear way on the spectral parameter. In the end, one has to analyze an OVF rather than OP (as in [3]). Using the recently obtained results, it is possible to study the properties of OVF in sufficient detail and prove the existence and discreteness of the spectrum of the problem on normal waves (which is the set of the OVF characteristic numbers) and describe its distribution on the complex plane. Note that we consider the normal waves that decrease in the cross-sectional plane at a distance from the waveguide (we impose the corresponding conditions at infinity). Other types of waves are not considered. This approach was used to study the shielded waveguide structures as well [53, 54].

5.1 Normal Waves in an Open Metal-Dielectric Waveguide 5.1.1 Statement of the Problem Consider the three-dimensional space R3 with the cylindrical coordinate system Oρϕz. The space is filled with an isotropic source-free medium with permittivity ε2 ε0 ≡ const, ε2 > 1 and permeability μ0 ≡ const, where ε0 and μ0 are permittivity and permeability of vacuum. We will consider a mathematical model of a regular (along the Oz-axis), open waveguide structure whose transverse section by the plane z = const is formed by the bounded domain 1 . The boundary 0 is the cross section of the surface of the infinitely thin and perfectly conducting screens and 1 is the cross section of the dielectric surfaces. The waveguide is filled with a homogeneous isotropic dielectric with a relative permittivity ε > ε0 , Im ε = 0, μ = μ0 . We choose r > 0 so that Br := {x = (ρ, ϕ) : ρ < r } ⊃ 1 and introduce the domain 2 := Br \ Q; 2 := {x : ρ = r } (see Fig. 5.1). ε where The dielectric permittivity in the whole space is given by the expression ε0 ⎧ ⎪ ⎨ ε1 ,  ε = ε2 , ⎪ ⎩ ε2 ,

x ∈ 1 , x ∈ 2 , x ∈ R \ 1 . 2

We will consider monochromatic waves  T  T Ee−iωt = e−iωt Eρ , Eϕ , Ez , He−iωt = e−iωt Hρ , Hϕ , Hz ,

(5.1.1)

5.1 Normal Waves in an Open Metal-Dielectric Waveguide

225

Fig. 5.1 Geometry of the problem

where ( · )T denotes the transpose operation. Each component of the field E, H is a function of three spatial variables. Determination of normal waves in a waveguide structure is the problem of finding nontrivial propagating running-wave solutions of the homogeneous system of the Maxwell equations, i.e., solutions with dependence of the from eiγz on coordinate z [55], curl H = −i εE, ˜ (5.1.2) curl E = iH,   E = E ρ (x) eρ + E ϕ (x) eϕ + E z (x) ez eiγz ,   H = Hρ (x) eρ + Hϕ (x) eϕ + Hz (x) ez eiγz ,

(5.1.3)

with the boundary conditions for the tangential electric field components on the perfectly conducting surfaces (5.1.4) E τ |0 = 0, the transmission conditions for the tangential electric and magnetic field components on the surfaces of the discontinuity (“jump”) of the permittivity [E τ ]|1 = 0, [Hτ ]|1 = 0, [E τ ]|2 = 0, [Hτ ]|2 = 0,

(5.1.5)

the finite energy condition

( ε|E|2 + |H|2 ) d X < ∞, V = {X : x ∈ 1 ∪ 2 }, X = (ρ, ϕ, z) , V

(5.1.6)

226

5 Open Waveguides of Arbitrary Cross Section

and the radiation condition at infinity: the electromagnetic field decays as O (1/ρ) for ρ → ∞. Here, γ is the normalized propagation constant of the waveguide (unknown spectral parameter of the problem). (5.1.2)–(5.1.6) is an eigenvalue problem for the Maxwell equations with spectral parameter γ. In what follows we often omit the arguments of functions when it does not lead to misunderstanding. The Maxwell system (5.1.2) is written in the normalized form. The passage to dimensionless variables has been carried out [3]; namely, k0 ρ → ρ, γ → kγ0 ,  μ0 H → H, E → E, where k02 = ω 2 ε0 μ0 (the time factor e−iωt is omitted everyε0 where.). Substituting E and H with components (5.1.3) into (5.1.2), we obtain ⎧ 1 ∂ Hz ⎪ ⎪ − iγ Hϕ ⎪ ⎪ ρ ∂ϕ ⎪ ⎪ ⎪ ⎪ ⎪ ∂ Hz ⎪ ⎪ iγ Hρ − ⎪ ⎪ ∂ρ ⎪ ⎪ ⎪   ⎪ ⎪ 1 ∂ Hρ 1 ∂ ρHϕ ⎪ ⎪ ⎪ − ⎨ ρ ∂ρ ρ ∂ϕ 1 ∂ Ez ⎪ ⎪ ⎪ − iγ E ϕ ⎪ ⎪ ρ ∂ϕ ⎪ ⎪ ⎪ ⎪ ∂ Ez ⎪ ⎪ ⎪ iγ E ρ − ⎪ ⎪ ∂ρ ⎪ ⎪   ⎪ ⎪ ⎪ 1 ∂ Eρ 1 ∂ ρE ϕ ⎪ ⎪ ⎩ − ρ ∂ρ ρ ∂ϕ

= −i ε Eρ, = −i ε Eϕ, = −i εEz , (5.1.7) = i Hρ , = i Hϕ , = i Hz ,

Expressing functions E ρ , Hρ , E ϕ , Hϕ in terms of E z and Hz from the first, second, fourth, and fifth equations of system (5.1.7), we find

∂ Ez ∂ Hz 1 i + , E ρ = − 2 γρ ρ κ ∂ρ ∂ϕ

∂ Ez ∂ Hz 1 i −ρ , Eϕ = − 2 γ ρ κ ∂ϕ ∂ρ



∂ Hz ∂ Ez 1 i Hρ = − 2 γρ − ε , ρ κ ∂ρ ∂ϕ

(5.1.8) ∂ Hz ∂ Ez 1 i Hϕ = − 2 γ + ρ ε , ρ κ ∂ϕ ∂ρ

where  κ2 = γ 2 −  ε. It follows from (5.1.8) that the field of the normal wave can be represented via two scalar functions  := E z (x),  := Hz (x). For functions  and  from (5.1.2)–(5.1.6), we have the following eigenvalue problem: to find γ ∈ C (called eigenvalues) such that there are nontrivial solutions of the system

5.1 Normal Waves in an Open Metal-Dielectric Waveguide

 −  κ2  = 0,  −  κ2  = 0,

227

(5.1.9)

satisfying the boundary conditions on 0 , |0 = 0,

 ∂  = 0, ∂n 0

(5.1.10)

the transmission conditions on 1 , []|1 = 0, []|1 = 0,    1 ∂  1 ∂  − = 0, γ  κ2 ∂τ 1  κ2 ∂n 1     1 ∂   ε ∂  γ + = 0,  2  κ ∂τ 1  κ2 ∂n 1 

(5.1.11)

the transmission conditions on 2 , 

[]|2 = 0, []|2 = 0,    ∂  ∂  = 0, = 0, ∂n 2 ∂n 2

(5.1.12)

where n denotes the normal unit vector such that ρ × ϕ = τ × n, the energy condition

  |∇|2 + |∇|2 + ||2 + ||2 d x < ∞,

(5.1.13)



where  = 1 ∪ 2 ∪ 1 , and the radiation condition at infinity (ρ, ϕ) = O

1 1 , (ρ, ϕ) = O , ρ → ∞, uniformly w.r.tϕ. (5.1.14) ρ ρ

Once we determine the longitudinal field components  and , we can find the transverse components by formulas (5.1.6). The equivalence of the reduction to problem (5.1.9)–(5.1.14) is not valid only for γ 2 =  ε; in this case, it is necessary to study system (5.1.2) directly.

5.1.2 Variational Formulation We will find solutions  and  of problem (5.1.9)–(5.1.14) in Sobolev spaces: H01 () and H 1 () , respectively, with the inner product and the norm

228

5 Open Waveguides of Arbitrary Cross Section

( f, g)1 =



 ∇ f ∇g + f g d x, f 21 = ( f, f )1 .



R

Here, we use the notation for Sobolev space H01 () which does not coincide with the standard one: in our case, f |0 = 0 but generally f |2 = 0.

Let us give a variational formulation of problem (5.1.9)–(5.1.14). Multiplying the equations of system (5.1.9) by arbitrary test functions u ∈ H01 () , v ∈ H 1 () (we can assume that these functions are continuously differentiable in ) and applying Green’s formula [56], we obtain



∂ 2 ud x + κ1 ud x = udτ ∂n 1 1 1



∂ − udτ − ∇∇ud x + κ21 ud x = 0, ∂n 0

1

1

and





∂ vdτ ∂n 1 1



∂ − vdτ − ∇∇vd x + κ21 vd x = 0, ∂n

vd x + 1

κ21 vd x =

0

1

1

where κ21 = ε1 − γ 2 . Taking into account boundary conditions (5.1.10), we get



ud x + 1

κ21 ud x

=

1

1

∂ udτ − ∂n



∇∇ud x +

1

κ21 ud x = 0, 1

(5.1.15) and



vd x +

1

κ21 vd x

1

= 1

∂ vdτ − ∂n



∇∇vd x +

1

κ21 vd x = 0. 1

(5.1.16) ε1 1 Multiplying (5.1.15) and (5.1.16) by 2 and 2 , respectively, adding and subκ1 κ1 γ  ∂ γ  ∂ tracting 2 udτ and 2 vdτ , we get κ1 1 ∂τ κ1 1 ∂τ

5.1 Normal Waves in an Open Metal-Dielectric Waveguide

γ κ21



∂ ε1 udτ + 2 ∂τ κ1

1

1

229

∂ udτ ∂n



γ ∂ 1 = 2 udτ + 2 ε1 ∇∇ud x − ε1 ud x, ∂τ κ1 κ1 1

1

1

(5.1.17) and γ κ21

1

∂ ∂ 1 vdτ − 2 vdτ ∂τ ∂n κ1 1



γ ∂ 1 = 2 vdτ − 2 ∇∇vd x + vd x. ∂τ κ1 κ1 1

1

(5.1.18)

1

For domain 2 , we obtain in a similar way −

γ κ22



∂ ∂ ε2 udτ − 2 udτ ∂τ ∂n κ2 1 1





1 ∂ ∂ γ 1 =− 2 udτ − 2 udτ + 2 ε2 ∇∇ud x + ε2 ud x, ∂n ∂τ κ2 κ2 κ2 2

1

2

2

(5.1.19) and −

γ κ22

1

∂ ∂ 1 vdτ + 2 vdτ ∂τ ∂n κ2 1





1 ∂ ∂ γ 1 = 2 vdτ − 2 vdτ − 2 ∇∇vd x − vd x. ∂n ∂τ κ2 κ2 κ2 2

1

2

2

(5.1.20) where κ22 = γ 2 − ε2 . Applying transmission conditions (5.1.11) in the following form γ κ21 γ κ21

1

1

∂ ε1 udτ + 2 ∂τ κ1 ∂ 1 vdτ − 2 ∂τ κ1

1

1

∂ γ udτ = − 2 ∂n κ2 ∂ γ vdτ = − 2 ∂n κ2

1

1

∂ ε2 udτ − 2 ∂τ κ2 ∂ 1 vdτ + 2 ∂τ κ2

1

1

∂ udτ , ∂n ∂ vdτ , ∂n

230

5 Open Waveguides of Arbitrary Cross Section

we have

2



κ2 ∂ ∂ ∂  ε 2 2 udτ = −γ udτ − κ ∇∇ud x + κ  εud x, + 2 2 ∂n ∂τ κ21 ∂τ k˜ 2 2

1





and



2



κ2 ∂ ∂ ∂ 1 2 2 − + vdτ = −γ vdτ + κ ∇∇vd x − κ vd x. 2 2 ∂n ∂τ κ21 ∂τ k˜ 2 2

1





Since boundary 2 is a circle with radius r , we can rewrite the last formulas as follows:

2



κ2 ∂ ∂ ∂  ε 2 2 udl = −γ udτ − κ2 ∇∇ud x + κ2  εud x, + ∂ρ ∂τ κ21 ∂τ k˜ 2 2

1





(5.1.21) and

2

∂ vdl = γ ∂ρ

1



κ22 ∂ ∂ 1 2 2 + vdτ − κ ∇∇vd x + κ vd x. 2 2 ∂τ κ21 ∂τ k˜ 2 



(5.1.22)

5.1.2.1

The Green Function

Outside domain , we have  ε = ε2 . Then system (5.1.9) takes the form L :=  − κ22  = 0,

L :=  − κ22  = 0,

(5.1.23)

where κ22 = γ 2 − ε2 . The Green function G of the exterior Dirichlet problem for the Helmholtz equation (5.1.23) in the exterior of circle 2 is defined as a solution of the following boundary value problem: ⎧ δ (ρ − ρ0 ) δ (ϕ − ϕ0 ) ⎪ ⎪ LG = − , ρ0 > r ; ⎪ ⎪ ρ ⎪ ⎨ G(x, x0 )|2 = 0, ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎩ G(x, x0 ) = O , ρ → ∞. ρ We have G(x, x0 ) = G(x0 , x), x = (ρ, ϕ) and x0 = (ρ0 , ϕ0 ). We have [57] in the sense of distributions

5.1 Normal Waves in an Open Metal-Dielectric Waveguide

δ (ϕ − ϕ0 ) =

231

∞ 1  cos [m (ϕ − ϕ0 )]. 2π m=−∞

Then the Green function G(x, x0 ) takes the form G(x, x0 ) =

∞ 1  G m (ρ, ρ0 ) cos [m (ϕ − ϕ0 )], 2π m=−∞

where G m (ρ, ρ0 ) is a Green function of the following boundary value problem:



2 ⎧ dG m m 1 d δ (ρ − ρ0 ) ⎪ 2 ⎪ L m G m := ρ − , ρ0 ≥ r ; + κ2 G m = − ⎪ ⎪ 2 ρ dρ dρ ρ 2πρ ⎪ ⎨ G m (r, ρ0 ) = 0, ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎩ G m (ρ, ρ0 ) = O , ρ → ∞, uniformly w.r.t m. ρ It is easy to verify that the Green function G m has the form

G m (ρ, ρ0 ) =

⎧ K m (κ2 ρ0 ) ⎪ ⎪ (I (κ r )K m (κ2 ρ) − Im (κ2 ρ)K m (κ2 r )) , r ≤ ρ < ρ0 , ⎪ ⎨ K m (κ2 r ) m 2 ⎪ ⎪ K (κ ρ) ⎪ ⎩ m 2 (Im (κ2 ρ0 )K m (κ2 r ) − Im (κ2 r )K m (κ2 ρ0 )) , r ≤ ρ0 < ρ, K m (κ2 r )

where Im and K m are the modified Bessel functions [58]. Finally, we obtain ⎧ ∞  K m (κ2 ρ0 ) ⎪ ⎪ ⎪ ξ1 cos [m (ϕ − ϕ0 )], r ≤ ρ < ρ0 ⎪ ⎪ K m (κ2 r ) ⎪ ⎨ m=−∞ 1 G(x, x0 ) = 2π ⎪ ∞ ⎪  ⎪ K m (κ2 ρ) ⎪ ⎪ ξ2 cos [m (ϕ − ϕ0 )], r ≤ ρ0 < ρ, ⎪ ⎩ K m (κ2 r ) m=−∞ where ξ1 := Im (κ2 r )K m (κ2 ρ) − Im (κ2 ρ)K m (κ2 r ) and ξ2 := Im (κ2 ρ0 )K m (κ2 r ) − Im (κ2 r )K m (κ2 ρ0 ). Using the second Green formula, we have

232

5 Open Waveguides of Arbitrary Cross Section



2π ∂v ∂u  u (u Lv − vLu)d x = r dϕ. −v ∂ρ ∂ρ ρ=r

R2 \Br

0

Next, setting v = G, we get



2π (u LG − G Lu)d x = r

R2 \Br

0

 ∂G  u dϕ. ∂ρ ρ=r

Moreover,

(u LG − G Lu)d x = −u(x0 ), x0 ∈ R2 \ B r . R2 \Br

Finally, we obtain

2π u(x0 ) = −r 0

 ∂G(x, x0 )  2 u(x)  dϕ, x0 ∈ R \ B r , ∂ρ ρ=r

(5.1.24)

where  ∞ 1  K m (κ2 ρ0 ) ∂G(x, x0 )  cos [m (ϕ − ϕ0 )]. = −  ∂ρ 2πr m=−∞ K m (κ2 r ) ρ=r

(5.1.25)

Taking into account the continuity of the normal derivative of the double layer potential on the boundary [59] and using (5.1.24), we define the normal derivatives on boundary 2  

2π ∂u  ∂ 2 G(x, x0 )  = −r u(x) dϕ ∂ρ0 ρ0 =r ∂ρ0 ∂ρ  ρρ=r =r 0

(5.1.26)

0

and  ∞ κ2  K m (κ2 r ) ∂ 2 G(x, x0 )  cos [m (ϕ − ϕ0 )]. =− ∂ρ0 ∂ρ  ρρ=r 2πr m=−∞ K m (κ2 r ) =r

(5.1.27)

0

The convergence of series (5.1.25) and (5.1.27) is understood in the sense of distributions [57].

5.1 Normal Waves in an Open Metal-Dielectric Waveguide

233

Further, rearranging the variables x ↔ x0 , multiplying by test functions u and v,  ∂u vdl, we get and taking the integrals 2 ∂ρ

2

∂ udl = −r 2 ∂ρ

2π 2π 0

  ∂ 2 G(x, x0 ) (x0 )u(x) dϕdϕ0 ρ=r ∂ρ∂ρ0 ρ =r

(5.1.28)

  ∂ 2 G(x, x0 ) (x0 )v(x) dϕdϕ0 . ρ=r ∂ρ∂ρ0 ρ =r

(5.1.29)

0

0

and

2

∂ vdl = −r 2 ∂ρ

2π 2π 0

0

0

Applying transmission conditions (5.1.13), we obtain −r 2



2π 2π 2  ∂ G(x, x0 )  (x0 )u(x) dϕdϕ0  ρ=r ∂ρ∂ρ0 ρ0 =r 0 0 

 2



κ2 ∂ ∂  ε 2 2  + = −γ udτ − κ ∇∇ud x + κ εud x, 2 2 ∂τ κ21 ∂τ k˜ 2 1





and −r 2



2π 2π 2  ∂ G(x, x0 )  (x0 )v(x) dϕdϕ0  ρ=r ∂ρ∂ρ0 ρ0 =r 0 0 

 2



κ2 ∂ ∂ 1 2 2 + =γ vdτ − κ ∇∇vd x + κ vd x. 2 2 ∂τ κ21 ∂τ k˜ 2 1





Summing up the last expressions, we get

κ22 

1 ε∇∇u + ∇∇v) d x − κ22 ( k˜ 2 +γ

εu + v) d x ( 



∂ κ22 ∂ +1 v− u dτ ∂τ ∂τ κ21 1

2π 2π − r2 0

0

 ∂ 2 G(x, x0 )  (x0 )u(x) + (x0 )v(x)  dϕdϕ0 = 0. ρ=r ∂ρ∂ρ0 ρ =r 0

234

5 Open Waveguides of Arbitrary Cross Section

Multiplying the last relation by γ 2 , we obtain the variational relation

εu + v) d x (

γ4 

⎞ ⎛



+ γ 2 ⎝ ( ε∇∇u + ∇∇v) d x − (ε1 + ε2 ) ( εu + v) d x ⎠ 



+ γ (ε1 − ε2 ) 1



∂ ∂ Phi v− u dτ ∂τ ∂τ

⎛ ⎞



1 ∇∇u + ∇∇v d x ⎠ + ε1 ε2 ⎝ ( εu + v) d x −  ε 



2π 2π − r 2 κ21 0

0

 ∂ 2 G(x, x0 )  (x0 )u(x) + (x0 )v(x)  dϕdϕ0 = 0. ρ=r ∂ρ∂ρ0 ρ =r 0

(5.1.30) Definition 5.1.1 The pair of functions  ∈ H01 () ,  ∈ H 1 () , (  1 +  1 = 0) is called an eigenvector of problem (5.1.9)–(5.1.14) corresponding to the eigenvalue γ0 ∈ C if variational relation (5.1.30) holds for u ∈ H01 () , v ∈ H 1 ().

5.1.3 Spectrum of the OVF Let us consider the following forms and corresponding operators:

εu + v) d x = (Ku, v), ∀v ∈ H, (

k(u, v) := 

 k(u, v) :=



 ε+1 u, v), ∀v ∈ H, εu + v) d x = (K (  ε

εu + v) d x = (Au, v), ∀v ∈ H, ε∇∇u + ∇∇v +  (

a(u, v) := 

5.1 Normal Waves in an Open Metal-Dielectric Waveguide

 a(u, v) :=

235

1 1 ∇∇u + ∇∇v + u + v d x = ( Au, v), ∀v ∈ H,  ε  ε



s(u, v) := 1

2π 2π p(u, v) := 0

∂ ∂ v− u dτ = (Su, v), ∀v ∈ H, ∂τ ∂τ

   F(ϕ, ϕ0 ) (x0 )u(x) + (x0 )v(x) 

0

F(ϕ, ϕ0 ) := −r 2 κ21

ρ=r ρ0 =r

dϕdϕ0 = (Pu, v), ∀v ∈ H,

 ∂ 2 G(x, x0 )  . ∂ρ∂ρ0  ρρ=r =r

(5.1.31)

0

It is known that the Green function G(x, x0 ) has a logarithmic singularity that can be isolated. Then, as in [60], we find  ∞ κ2  K m (κ2 r ) ∂ 2 G(x, x0 )  cos [m (ϕ − ϕ0 )] = − ∂ρ0 ∂ρ  ρρ=r 2πr m=−∞ K m (κ2 r ) =r 0

∞ κ2  K m (κ2 r ) κ2 K 1 (κ2 r ) − cos [m (ϕ − ϕ0 )] = πr K 0 (κ2 r ) πr m=1 K m (κ2 r )

∞ κ2 K 1 (κ2 r ) 1  K (κ2 r ) κ2 r m = − 2 + m cos [m (ϕ − ϕ0 )] πr K 0 (κ2 r ) πr m=1 K m (κ2 r ) ∞ 1  m cos [m (ϕ − ϕ0 )] πr 2 m=1

∞ 1  κ2 K 1 (κ2 r ) K (κ2 r ) κ2 r m − 2 + m cos [m (ϕ − ϕ0 )] = πr K 0 (κ2 r ) πr m=1 K m (κ2 r )

+

−

1 1 . 2 πr 1 − cos (ϕ − ϕ0 )

1 K m (κ2 r ) +m = O for m → ∞, and the series converges K m (κ2 r ) m conditionally for ϕ = ϕ0 . We represent operator P(γ) as a sum of a compact and a hypersingular operator We have κ2 r

  P, P(γ) = KP (γ) + γ 2 − ε1  where

(5.1.32)

236

5 Open Waveguides of Arbitrary Cross Section

1  p(u, v) = − 2π

2π 2π 0

0

 (x0 )u(x) + (x0 )v(x)    ρ=r dϕdϕ0 = (Pu, v), ∀v ∈ H, 1 − cos (ϕ − ϕ0 ) ρ =r 0

and κ2 κ2 r K 1 (κ2 r ) ˇp (u, v) = − 1 π K 0 (κ2 r ) κ2 + 1 π

2π 2π ((x0 )u(x) + (x0 )v(x))| 0

ρ=r ρ0 =r

dϕdϕ0

0

2π 2π  ∞ K m (κ2 r ) κ2 r +m × K m (κ2 r ) m=1 0

0

× cos [m (ϕ − ϕ0 )] ((x0 )u(x) + (x0 )v(x))|

ρ=r ρ0 =r

dϕdϕ0

= (KP (γ)u, v), ∀v ∈ H. Since , u, , v ∈ H 1 (), then the traces of these functions on 2 belong H (2 ) [61]. The hypersingular operator  P : H 1/2 (2 ) → H −1/2 (2 ) is bounded [62]. Taking into account the antidual coupling of the spaces H 1/2 (2 ) and H −1/2 p(u, v) is bounded in H . Further, taking (2 ) [63], we get that sesquilinear form  into account the rate of decrease of the coefficients of the series for  p(u, v), we find [60] that this series will define an integral operator with a logarithmic singularity of P : H 1/2 (2 ) → H −1/2 (2 ) will be compact. Thus, the kernel, and the operator K P(γ) : H → H is bounded. The boundedness of the first five sesquilinear forms is proved in [3]. Now variational problem (5.1.30) can be written in the operator form 1/2

(N(γ)u, v) = 0, ∀u ∈ H or, equivalently, N(γ)u = 0, N(γ) : H → H,    + P(γ). A−K N(γ) := γ 4 K+γ 2 (A − (ε1 + ε2 + 1) K) + γ (ε1 − ε2 ) S − ε1 , ε2  (5.1.33) Equation (5.1.33) is the operator form of variational relation (5.1.30). The characteristic numbers and eigenvectors of OVF N(γ) by definition coincide with the ε. eigenvalues and eigenvectors of problem (5.1.9)–(5.1.14) for γ 2 =  Thus, the problem on normal waves is reduced to the eigenvalue problem for OVF N(γ).

5.1 Normal Waves in an Open Metal-Dielectric Waveguide

237

5.1.4 Properties of OVF N(γ) We formulate the following statements about the properties of operators occurring in N(γ) (the proof is in [3]): Lemma 5.1.1 Operator K is positive, K > 0, and compact. The following estimate holds for its eigenvalues: λn (K) = O(n −1 ), n → ∞. Lemma 5.1.2 Operators A,  A are uniformly positive: I ≤ A ≤ ε2 I,

1 I≤ A ≤ I, ε1

(5.1.34)

where I is the unit operator in H . Lemma 5.1.3 Operator S is self-adjoint, S = S∗ , and the following inequalities hold −

1 1 I ≤ S ≤ I. 2 2

(5.1.35)

A special attention is required to consider the properties of the trace operator P. In [62] this operator is studied in detail as a pseudo-differential operator in Sobolev spaces. Lemma 5.1.4 P is a self-adjoint bounded operator. Definition 5.1.2 We will denote by ρ(N) the resolvent set of N(γ) (consisting of all values of γ ∈ C, where there exists the bounded inverse operator N−1 (γ)) and by σ(N) = C \ ρ(N) the spectrum of N(γ). Properties of the spectrum of OVF N(γ) are summarized in the following theorems. Theorem 5.1.1 There exists  γ ∈ R such that operator N( γ ) is continuously invertible; i.e., the resolvent set (N) := {γ : ∃ N−1 (γ) : H → H } of OVF N( γ ) is not empty. Proof Let γ ∈ R and γ → +∞. Consider the following OVF: 1 N(γ) = N1 (γ) + N2 (γ), γ2 where N1 (γ) = γ 2 K + A − (ε1 + ε2 + 1) K + γP0 ,

238

5 Open Waveguides of Arbitrary Cross Section

and N2 (γ) =

 ε 1 − ε2 ε 1 ε2  1  + P(γ) − γP0 , S− 2  A−K γ γ γ2

and the operator P0 : H → H is defined by a sesquilinear form   p0 u, v =

2π  ∞ 0

2π    (x0 )u(x) + (x0 )v(x)  cos [m (ϕ − ϕ0 )]dϕ

m=−∞

0

ρ=r ρ0 =r

dϕ0

= (P0 u, v), ∀v ∈ H. It is known [57] that (in the sense of distributions) ∞ 

cos mφ = 2π

m=−∞

∞ 

δ (φ − 2πl) = 2πδ(φ),

l=−∞

where φ = |ϕ − ϕ0 |, 0 ≤ |φ| ≤ π. We get that ∞ 

cos [m (ϕ − ϕ0 )] = 2πδ(ϕ − ϕ0 ).

m=−∞

Then operator P0 takes the form

2π      (x)u(x) + (x)v(x)  P0 u, v := 2π

ρ=r

dϕ, ∀v ∈ H.

(5.1.36)

0

It is obvious that

2π      |(x)|2 + |(x)|2  P0 u, u = 2π

ρ=r

dϕ > 0 for u = 0.

0

Since the traces of functions , u, , v on 2 belong to the space H 1/2 (2 ), we conclude that P0 : H → H is bounded. Taking into account the asymptotic property of function K m for γ → +∞ [64], we have: 1 1 , F(ϕ, ϕ ) ∼ γ + O 0 γ2 γ and hence

5.1 Normal Waves in an Open Metal-Dielectric Waveguide

239

    1   P(γ) − γP0  = O 1 , for γ → +∞.  γ2  γ

(5.1.37)

Then OVF γ −2 N(γ) can be considered as a perturbation of operator pencil N1 by OVF N2 for large γ. Taking into account the properties of operators K and A1 , we conclude that there is a sufficiently large  γ > 0 such that Re(N1 ( γ )u, u) =  γ 2 (Ku, u) + (Au, u) − (ε1 + ε2 + 1) (Ku, u) +  γ (P0 u, u) ≥ u 2

for any u. Hence,  γ ∈ ρ(N1 ), where (N1 ) is the resolvent set of OVF N1 . Moreover, Theorem 4.1 in [65] yields the estimate γ ) ≤ 1. N1−1 ( γ ) < 1 is satisfied and find that there exists We take  γ such that the inequality N2 ( a bounded operator −1    γ ) + N2 ( γ) = I + N1−1 ( γ )N2 ( γ ) N1−1 ( γ ). N1 ( 

The proof of the theorem is complete. Lemma 5.1.5 Operator  P is positive,  P > 0.

Proof Consider the inner product ( Pu, u). We represent functions  and  in the form of Fourier series (ϕ) =

∞  n=−∞

and (ϕ) =

∞  p=−∞

∞ 

X n einϕ , (ϕ0 ) =

X m e−imϕ0

m=−∞

Ype

i pϕ

∞ 

, (ϕ0 ) =

Y q e−iqϕ0 ,

q=−∞

and using the technique of distribution [66], rewrite the quadratic form as ( u = 0) 1 ( Pu, u) = − 2π

2π 2π 0

0

 (x0 )(x) + (x0 )(x)    1 − cos (ϕ − ϕ0 ) =

ρ=r ρ0 =r ∞ 

dϕdϕ0

  2π j |X j |2 + |Y j |2 > 0.

j=1



240

5 Open Waveguides of Arbitrary Cross Section

We can represent OVF N(γ) as follows: N(γ) := NK (γ) + N0 (γ), where

and

(5.1.38)

 + KP (γ) NK (γ) = γ 4 K − γ 2 (ε1 + ε2 + 1) K + ε1 ε2 K     +  P + γ (ε1 − ε2 ) S − ε1 ε2 A P . N0 (γ) = γ 2 A + 

Theorem 5.1.2 Operator N(γ) : H → H is bounded, holomorphic, and Fredholm in the domain  and   := {γ : Im γ 2 = 0, γ 2 ≤ ε2 } ∪ {γ : Im γ = 0, γ∗ ≤ |γ| ≤ γ ∗ },  = C\

where   P )(ε2 +  P ) − + 2 + 4ε2  + 2 + 4ε1 (ε1 +  ∗ , , γ := γ∗ :=  2 2(ε1 + P ) =

ε1 − ε 2  √ √  , γ∗ < ε2 , γ ∗ > ε1 . . 2

Proof In the domain {γ : Im γ 2 = 0, γ 2 ≤ ε2 }, the functions κ2

K m (κ2 r ) K m+1 (κ2 r ) m = −κ2 + K m (κ2 r ) K m (κ2 r ) r

(5.1.39)

are analytical (as a function of γ). Since Re κ2 > 0, functions K m (κ2 r ) have no zeros [64]. Let γ = γ + iγ

, γ

= 0. Then    1 ε1  N0 (γ) = A + P + 2 ε2  A + P ≥ I. Im

γ γ |γ| Next, assume that Im γ = 0. Consider the following quadratic form: (N0 (γ)u, u) =

     2  A + P u, u = 0. P + γ (ε1 − ε2 ) S − ε1 ε2  γ A +

Expressing (Su, u) from the last equation and taking into account estimate (5.1.35), we obtain the following double inequality: −≤

   ε1  ε2  A +  P − γ A +  P ≤ . γ

(5.1.40)

5.1 Normal Waves in an Open Metal-Dielectric Waveguide

241

Solving inequalities (5.1.40) with respect to γ and taking into account estimates (5.1.34), we find that outside the set γ∗ ≤ |γ| ≤ γ ∗ operator N0 (γ) is positive or negative definite and therefore continuously invertible [65]. Hence, N(γ) is a Fredholm OVF as the sum of an invertible and compact operators, and ind N(γ) = 0. The second assertion of the theorem follows from the fact that variational relation (5.1.30) for γ = γ2 or γ = γ1 is identically equal to zero for functions  and  such that   ,  ∈ C0∞ ∗ , for ∗ ⊂ 1 or ∗ ⊂ 2 , respectively.



Theorem 5.1.3 The spectrum of OVF N(γ) : H → H is discrete in , i.e., it has a finite number of eigenvalues √ of finite algebraic multiplicity in any compact set K0 ⊂ . The points γ j = ±  ε denote the values of degeneration of OVF N(γ), where dim ker N(γ j ) = ∞. Proof The assertion of the theorem is a corollary of Theorem 5.1.2 and the theorem on the holomorphic OVF [65]. The second assertion follows from the fact that variational relation (5.1.30) for γ = γ2 or γ = γ1 is identically equal to zero for functions  and  such that   ,  ∈ C0∞ ∗ , for ∗ ⊂ 1 or ∗ ⊂ 2 , respectively.



Lemma 5.1.6 The spectrum of OVF N(γ) is symmetric with respect to the origin, σ(N) = σ(−N). If γ0 is an eigenvalue of N(γ) corresponding to the eigenvector u = (, )T , then −γ0 is an eigenvalue of N(γ) corresponding to the eigenvector u = (−, )T with the same multiplicity. Proof The proof is a simple verification of variational relation (5.1.30).



5.2 Normal Waves in an Open Inhomogeneous Metal-Dielectric Waveguide 5.2.1 Statement of the Problem Consider the three-dimensional space R3 with the cylindrical coordinate system Oρϕz. The space is filled with an isotropic source-free medium with permittivity ε2 ε0 ≡ const, ε2 ≥ 1 and permeability μ0 ≡ const, where ε0 and μ0 are permittivity and permeability of vacuum.

242

5 Open Waveguides of Arbitrary Cross Section

Permittivity in the whole space is ε0 ε, where ⎧ ⎪ ⎨ ε1 (x),  ε = ε2 , ⎪ ⎩ ε2 ,

x ∈ 1 , x ∈ 2 ,

(5.2.41)

x ∈ R \ 1 . 2

The field of the normal wave can be represented via two scalar functions  := E z (x),  := Hz (x). For functions  and , we have the following eigenvalue problem: find γ ∈ C called eigenvalues such that there are nontrivial solutions of the system ⎧ γ2 γ ⎪ ⎪ ⎨ L :=  +  κ2  = 2 ∇ ε∇ + 2 J ( ε, ) , x ∈ 1 ∪ 2 ,  ε κ  ε κ ⎪ 1 γ ⎪ ⎩ L :=  +  κ2  = 2 ∇ ε∇ − 2 J ( ε, ) , x ∈ 1 ∪ 2 ,  κ  κ and J (u, v) :=

(5.2.42)

∂u ∂v ∂u ∂v − ; ∂x ∂ y ∂ y ∂x

satisfying the boundary conditions on 0 , |0

 ∂  = 0, = 0, ∂n 0

(5.2.43)

the transmission conditions on 1 , []|1 = 0, []|1 = 0,    1 ∂  1 ∂  − = 0, γ  κ2 ∂τ 1  κ2 ∂n 1     1 ∂   ε ∂  γ + = 0;  κ2 ∂τ 1  κ2 ∂n 1 

(5.2.44)

the transmission conditions on 2 , []|2 = 0, []|2 = 0, 

   ∂  ∂  = 0, = 0, ∂n 2 ∂n 2

(5.2.45)

where n denotes the normal unit vector such that ρ × ϕ = τ × n; the energy condition

5.2 Normal Waves in an Open Inhomogeneous Metal-Dielectric Waveguide

243

  |∇|2 + |∇|2 + ||2 + ||2 d x < ∞,

(5.2.46)



where  = 1 ∪ 2 ∪ 1 ; and radiation condition at infinity 1 1 , (ρ, ϕ) = O , ρ → ∞, uniformly w.r.t ϕ. ρ ρ (5.2.47) The equivalence of the reduction to problem (5.2.42)–(5.2.47) is not valid only for γ 2 =  ε. (ρ, ϕ) = O

5.2.2 Variational Formulation Let us give a variational formulation of problem (5.2.42)–(5.2.47). Multiplying the equations of system (5.2.42) by arbitrary test functions u ∈ H01 () , v ∈ H 1 () (one may assume that these functions are continuously differentiable in ), and applying Green’s formula, we obtain

1

ε1 uLd x = κ21

+ 1

1

ε1 ∂ udτ − κ21 ∂n

γ2 u∇ε1 ∇d x + κ41



0

ε1 ∂ udτ − κ21 ∂n

1

ε1 ∇∇ud x κ21

ε1 ud x, 1

and

1



1 1 ∂ 1 ∂ vLd x = vdτ − vdτ 2 2 ∂n κ1 κ1 κ21 ∂n 1 0



1 1 − ∇∇vd x + v∇ε ∇d x + vd x, 1 κ21 κ41 1

1

1

where κ21 (x) = ε1 (x) − γ 2 . Taking into account boundary conditions (5.2.43), we get

ε1



uLd x = 2

1

and

κ1

1

ε1 ∂ udτ − κ2 ∂n 1



1

ε1



∇∇ud x + 2

κ1

1

γ2



u∇ε1 ∇d x + 4

κ1

1

ε1 ud x,

244

5 Open Waveguides of Arbitrary Cross Section





1

vLd x = 2

κ1

1

1

1 ∂ vdτ − κ2 ∂n 1



1



1

κ1



1

∇∇vd x + 2

v∇ε1 ∇d x + 4

κ1

1

vd x.

1

Taking into account the right-hand sides of equations of system (5.2.42), we obtain

ε1 ∂ udτ − κ21 ∂n

1

1

ε1 ∇∇ud x + κ21



ε1 ud x −

1

1

γ u J (ε1 , )d x = 0, κ41

and

1 ∂ vdτ − κ21 ∂n

1

1

1 ∇∇vd x + κ21

Adding and subtracting





vd x +

1



γ v J (ε1 , )d x = 0. κ41

 γ ∂  γ ∂ udτ and vdτ , we get 2 ∂τ 2 1 κ1 1 κ1 ∂τ

ε1 ∂ γ ∂ udτ + udτ 2 ∂n κ1 κ21 ∂τ 1





γ ∂ ε1 γ = udτ + ∇∇ud x − ε1 ud x + u J (ε1 , )d x, κ21 ∂τ κ21 κ41

1

1

1

1

1

(5.2.48) and



γ ∂ 1 ∂ vdτ − vdτ 2 ∂τ κ1 κ21 ∂n 1





γ ∂ 1 γ = vdτ − ∇∇vd x + vd x + v J (ε1 , )d x. κ21 ∂τ κ21 κ41

1

1

1

1



(5.2.49) For domain 2 , we obtain in a similar way −

γ κ22



∂ ∂ ε2 udτ − 2 udτ ∂τ ∂n κ2 1 1





1 ∂ ∂ γ 1 =− 2 udτ − 2 udτ + 2 ε2 ∇∇ud x + ε2 ud x, ∂n ∂τ κ2 κ2 κ2 2

1

2

2

(5.2.50)

5.2 Normal Waves in an Open Inhomogeneous Metal-Dielectric Waveguide

245

and −



γ κ22

1

∂ ∂ 1 vdτ + 2 vdτ ∂τ ∂n κ2 1





1 ∂ ∂ γ 1 = 2 vdτ − 2 vdτ − 2 ∇∇vd x − vd x. ∂n ∂τ κ2 κ2 κ2 2

1

2

2

(5.2.51) where κ22 = γ 2 − ε2 . Applying transmission conditions (5.2.44) in the following form

1

1

γ ∂ udτ + κ21 ∂τ γ ∂ vdτ − κ21 ∂τ



ε1 ∂ γ udτ = − 2 2 ∂n κ1 κ2

1



1 ∂ γ vdτ = − 2 2 ∂n κ1 κ2

1

1

1



∂ ε2 udτ − 2 ∂τ κ2

∂ udτ , ∂n

1

∂ 1 vdτ + 2 ∂τ κ2



∂ vdτ , ∂n

1

we get

∂ udτ = −γ ∂n

2

 1

κ22 κ21

 +1

∂ udτ − κ22 ∂τ



 ε ∇∇ud x + κ22  κ2



 εud x − 

γ 

κ22 u J ( ε, )d x,  κ4

and

2

∂ vdτ = γ ∂n

 1

κ22 κ21

 +1

∂ vdτ − κ22 ∂τ



1 ∇∇vd x + κ22  κ2



vd x + 

γ 

κ22 v J ( ε, )d x.  κ4

Since boundary 2 is a circle with radius r , we can rewrite the last formulas as follows

2

κ2 ∂ ∂ udl = −γ + 1 udτ 2 ∂ρ ∂τ κ1 2

1



− κ22 

 ε ∇∇ud x + κ22  κ2



 εud x − 

γ 

κ22 u J ( ε, )d x,  κ4 (5.2.52)

and

246

5 Open Waveguides of Arbitrary Cross Section

2

∂ vdl = γ ∂ρ

1

κ22 ∂ +1 vdτ ∂τ κ21

−

κ22 

1 ∇∇vd x + κ22  κ2



vd x + 

γ 

κ22 v J ( ε, )d x.  κ4 (5.2.53)

Outside domain , we have  ε = ε2 . Then system (5.2.42) takes the form L :=  − κ22  = 0, L :=  − κ22  = 0.

(5.2.54)

Green’s function G of the exterior Dirichlet problem for the Helmholtz equation (5.2.54) in the exterior of circle 2 is defined as the solution of the following boundary value problem: ⎧ δ − ρ0 ) δ (ϕ − ϕ0 ) ⎪ ⎪ LG = − (ρ , ρ0 > r ; ⎪ ⎨ ρ G(x, x0 )|2 = 0, ⎪ ⎪ ⎪ ⎩ G(x, x0 ) = O (1/ρ) , ρ → ∞. We have G(x, x0 ) = G(x0 , x), x = (ρ, ϕ) and x0 = (ρ0 , ϕ0 ). Green’s function G m has the form ⎧ K (κ ρ ) ⎪ ⎪ m 2 0 (Im (κ2 r )K m (κ2 ρ) − Im (κ2 ρ)K m (κ2 r )) , r ≤ ρ < ρ0 , ⎨ K m (κ2 r ) G m (ρ, ρ0 ) = K ⎪ m (κ2 ρ) ⎪ ⎩ (Im (κ2 ρ0 )K m (κ2 r ) − Im (κ2 r )K m (κ2 ρ0 )) , r ≤ ρ0 < ρ, K m (κ2 r ) where Im and K m are the modified Bessel functions [58]. Applying transmission conditions (5.2.46), we obtain

2π 2π −r

2 0

0

 

 2  κ2 ∂ 2 G(x, x0 ) ∂  (x0 )u(x) dϕdϕ0 = −γ +1 udτ 2 ρ=r ∂ρ∂ρ0 ∂τ κ1 ρ =r −κ22 

2π 2π −r

2 0

0

1

0



 ε ∇∇ud x + κ22  κ2





 εud x − 

γ 

κ22 u J ( ε, )d x,  κ4

 

 2  κ ∂ ∂ 2 G(x, x0 ) 2 (x0 )v(x) dϕdϕ0 = γ +1 vdτ 2 ρ=r ∂ρ∂ρ0 ∂τ κ 1 ρ =r 0

−κ22 

1

1 ∇∇vd x + κ22  κ2



vd x +



γ 

κ22 v J ( ε, )d x.  κ4

5.2 Normal Waves in an Open Inhomogeneous Metal-Dielectric Waveguide

247

Summing up the last expressions and simplifying them, we get



1 ε∇∇u + ∇∇v) d x (  κ2 



ε1 − ε2 ∂ γ ∂ +γ v − u dτ − ε, ) − v J ( ε, )) d x (u J ( ∂τ ∂τ  κ4 κ21 κ22

εu + v) d x − ( 

1



+

r2 κ22

2π 2π 0

0

 ∂ 2 G(x, x0 )  dϕdϕ0 = 0. (x0 )u(x) + (x0 )v(x)  ρ=r ∂ρ∂ρ0 ρ =r 0

Multiplying the last relation by γ 2 , we obtain the variational relation



εu + v) d x + (

γ2 

+ γ3 1

+

r 2γ2 κ22

ε1 − ε 2 κ21 κ22

2π 2π 0

0



ε∇∇u + ∇∇v) d x − (





 ε ε∇∇u + ∇∇v) d x (  κ2

3 γ ∂ ∂ v− u dτ − ε, ) − v J ( ε, )) d x (u J ( ∂τ ∂τ  κ4 

 ∂ 2 G(x, x0 )  dϕdϕ0 = 0. (x0 )u(x) + (x0 )v(x)  ρ=r ∂ρ∂ρ0 ρ =r

(5.2.55)

0

Definition 5.2.1 The pair of functions  ∈ H01 () ,  ∈ H 1 () , (  1 +  1 = 0) is called the eigenvector of problem (5.2.42)–(5.2.47) corresponding to the eigenvalue γ0 ∈ C if variational relation (5.2.55) holds for u ∈ H01 () , v ∈ H 1 ().

5.2.3 Spectrum of the OVF Let us consider the following forms and corresponding operators:

εu + v) d x = (Ku, v), ∀v ∈ H, (

k(u, v) := 

k1 (u, v) := 

γ2 εu + v) d x = (K1 (γ)u, v), ∀v ∈ H, (  κ2

248

5 Open Waveguides of Arbitrary Cross Section

a(u, v) :=

εu + v) d x = (Au, v), ∀v ∈ H, ε∇∇u + ∇∇v +  ( 

a1 (u, v) := 

 ε εu + v) d x = (A1 (γ)u, v), ∀v ∈ H, ε∇∇u + ∇∇v +  ( κ˜ 2

s(u, v) := γ

3 1

b(u, v) := 

2π 2π p(u, v) := 0

ε 1 − ε2 κ21 κ22



∂ ∂ v− u dτ = (S(γ)u, v), ∀v ∈ H, ∂τ ∂τ

γ3 ε, ) − v J ( ε, )) d x = (B(γ)u, v), ∀v ∈ H, (u J (  κ4

   F(γ, ϕ, ϕ0 ) (x0 )u(x) + (x0 )v(x) 

0

ρ=r ρ0 =r

dϕdϕ0 = (Pu, v), ∀v ∈ H,

 r 2 γ 2 ∂ 2 G(x, x0 )  . F(γ, ϕ, ϕ0 ) := 2 ∂ρ∂ρ0  ρρ=r κ2 =r

and

(5.2.56)

0

The boundedness of forms a(u, v), a(u, v), k(u, v), and  k(u, v) is obvious. In what follows, we need the continuation of function ε1 (x) into domain 2 preserving its smoothness. Since ε1 ∈ C 1 (1 ), it follows that this can readily be done [67]. The continued function preserves its notation and belongs to space C 1 (). R

Obviously, for any δ > 0, the continuation can be chosen so that to satisfy the estimate ε1 − ε2 C() ≤ (1 + δ) ε1 − ε2 C(1 ) . (5.2.57)

It is known that Green’s function G(x, x0 ) has a logarithmic singularity that can be isolated. Then, as in [60], we find  ∞ ∂ 2 G(x, x0 )  κ2  K m (κ2 r ) cos [m (ϕ − ϕ0 )] = − ∂ρ0 ∂ρ  ρρ=r 2πr m=−∞ K m (κ2 r ) =r 0

∞

κ2  K m (κ2 r ) κ2 K 1 (κ2 r ) − cos [m (ϕ − ϕ0 )] = πr K 0 (κ2 r ) πr m=1 K m (κ2 r )

∞ κ2 K 1 (κ2 r ) K (κ2 r ) 1  κ2 r m = − 2 + m cos [m (ϕ − ϕ0 )] πr K 0 (κ2 r ) πr m=1 K m (κ2 r )

5.2 Normal Waves in an Open Inhomogeneous Metal-Dielectric Waveguide

249

∞ 1  m cos [m (ϕ − ϕ0 )] πr 2 m=1

∞ κ2 K 1 (κ2 r ) K (κ2 r ) 1  κ2 r m = − 2 + m cos [m (ϕ − ϕ0 )] πr K 0 (κ2 r ) πr m=1 K m (κ2 r )

+

−

1 1 . 2 πr 1 − cos (ϕ − ϕ0 )

1 K m (κ2 r ) +m = O for m → ∞, and the series converges We have κ2 r K m (κ2 r ) m conditionally for ϕ = ϕ0 . We represent P(γ) as the sum of a compact operator and a hypersingular operator P(γ) =

 γ2  PK (γ) +  P , 2 κ2

(5.2.58)

where 1  p(u, v) = − 2π

2π 2π 0

0

 (x0 )u(x) + (x0 )v(x)    ρ=r dϕdϕ0 = (Pu, v), ∀v ∈ H, 1 − cos (ϕ − ϕ0 ) ρ =r 0

and pk (u, v) = −

+

1 π

κ2 r K 1 (κ2 r ) π K 0 (κ2 r )

2π 2π ((x0 )u(x) + (x0 )v(x))| 0

ρ=r ρ0 =r

dϕdϕ0

0



2π 2π  ∞  K (κ2 r )  κ2 r m + m cos [m (ϕ − ϕ0 )] ((x0 )u(x) + (x0 )v(x))  K m (κ2 r ) 0

0

m=1

ρ=r ρ0 =r

dϕdϕ0

= (PK (γ)u, v), ∀v ∈ H.

Since , u, , v ∈ H 1 (), the traces of these functions on  belong to P : H 1/2 () → H −1/2 () is the space H 1/2 () [61]. The hypersingular operator  bounded [62]. Taking into account the antidual coupling of spaces H 1/2 () and H −1/2 () [63], we get that sesquilinear form ph (u, v) is bounded in H . Further, taking into account the rate of decrease of the coefficients of the series for kp (u, v), we find [60] that this series will define an integral operator with a logarithmic singularity of the kernel and the operator PK : H 1/2 () → H −1/2 () will be compact. Thus, P(γ) : H → H is bounded. The boundedness of forms s(u, v) and b(u, v) is proved in [68]. Now variational problem (5.2.55) can be written in the operator form

250

5 Open Waveguides of Arbitrary Cross Section

(N(γ)u, v) = 0, ∀u ∈ H or, equivalently, N(γ)u = 0, N(γ) : H → H, (5.2.59) N(γ) := γ K+A + K1 (γ) + S(γ) − B(γ) − A1 (γ) + P(γ). 2

Equation (5.2.59) is the operator form of variational relation (5.2.55). The characteristic numbers and eigenvectors of OVF N(γ) by definition coincide with the ε. eigenvalues and eigenvectors of problem (5.2.42)–(5.2.47) for γ 2 =  Thus, the problem on surface waves is reduced to an eigenvalue problem for OVF N(γ).

5.2.4 The Properties of OVF We give the following statements about the properties of operators forming N(γ) (the proof is in [3]): Lemma 5.2.1 Operator A is positive definite I ≤ A ≤ max ε1 I, where I is the unit x∈

operator in H . Lemma 5.2.2 Operators K and K1 (γ) are compact. Operator K is positive definite and its characteristic numbers have the asymptotics λn (K) = O(n −1 ), n → ∞. Lemma 5.2.3 Operators B(γ) are compact and OVF B(γ) is holomorphic in the domain   √ C\0 and 0 := {γ : Im γ = 0, min ε1 (x) ≤ |γ| ≤ max ε1 (x), |γ| = ε2 }. x∈

x∈

A special attention is required to consider the properties of the trace operator P. In [62] this operator is studied in detail as a pseudodifferential operator in Sobolev spaces. The following result is known. Lemma 5.2.4 P is a self-adjoint bounded operator. Lemma 5.2.5 Operator  P is positive,  P > 0. Proof Consider the inner product ( Pu, u). We represent functions  and  in the form of Fourier series

5.2 Normal Waves in an Open Inhomogeneous Metal-Dielectric Waveguide

(ϕ) =

∞  n=−∞

and (ϕ) =

∞ 

∞ 

X n einϕ , (ϕ0 ) =

251

X m e−imϕ0

m=−∞

Ype

i pϕ

∞ 

, (ϕ0 ) =

p=−∞

Y q e−iqϕ0 ,

q=−∞

and using the technique of distributions [66], write the quadratic form as ( u = 0): 1 ( Pu, u) = − 2π

2π 2π 0

0

 (x0 )(x) + (x0 )(x)    1 − cos (ϕ − ϕ0 ) =

ρ=r ρ0 =r ∞ 

dϕdϕ0

  2π j |X j |2 + |Y j |2 > 0. 

j=1

Definition 5.2.2 We will denote by (N) the resolvent set of N(γ) (consisting of all values of γ ∈ C where there exists bounded inverse operator N−1 (γ)) and by σ(N) = C \ ρ(N) the spectrum of N(γ). Properties of the spectrum of OVF N(γ) are given in the following theorems. Theorem 5.2.1 There exists  γ ∈ R such that operator N( γ ) is continuously invertible, i.e., the resolvent set (N) := {γ : ∃ N−1 (γ) : H → H } of OVF N( γ ) is not empty. Proof Let γ ∈ R and γ → +∞. Consider the following OVF: N(γ) = N1 (γ) + N2 (γ), where N1 (γ) = γ 2 K + A + K1 (γ) + γP0 , and N2 (γ) = S(γ) − B(γ) − A1 (γ) + P(γ) − P0 , and the operator P0 : H → H is defined by a sesquilinear form   p0 u, v =

2π  ∞ 0

m=−∞

2π    (x0 )u(x) + (x0 )v(x)  cos [m (ϕ − ϕ0 )]dϕ 0

ρ=r ρ0 =r

dϕ0

= (P0 u, v), ∀v ∈ H.

252

5 Open Waveguides of Arbitrary Cross Section

It is known [57] that (in the sense of distributions) ∞ 

cos mφ = 2π

m=−∞

∞ 

δ (φ − 2πl) = 2πδ(φ),

l=−∞

where φ = |ϕ − ϕ0 |, 0 ≤ |φ| ≤ π. We get that ∞ 

cos [m (ϕ − ϕ0 )] = 2πδ(ϕ − ϕ0 ).

m=−∞

Then operator P0 takes the form 



P0 u, v := 2π

2π 

  (x)u(x) + (x)v(x) 

ρ=r

dϕ, ∀v ∈ H.

(5.2.60)

0

It is obvious that

2π      |(x)|2 + |(x)|2  P0 u, u = 2π

ρ=r

dϕ > 0 for u = 0.

0

Since the traces of functions , u, , v on 2 belong to space H 1/2 (2 ), we conclude that P0 : H → H is bounded. Taking into account the asymptotic properties of function K m for γ → +∞ [64], we have: 1 1 , F(ϕ, ϕ ) ∼ γ + O 0 γ2 γ and hence

    1   P(γ) − γP0  = O 1/γ , for γ → +∞.  γ2 

(5.2.61)

Then OVF γ −2 N(γ) can be considered as a perturbation of operator pencil N1 by OVF N2 for large γ. Taking into account the properties of operators K and A, we find that there is a large  γ > 0 such that γ )u, u) =  γ 2 (Ku, u) + (Au, u) + (K1 ( γ )u, u) +  γ (P0 u, u) ≥ u 2 Re(N1 ( for any u. Hence,  γ ∈ ρ(N1 ), where (N1 ) is the resolvent set of the OVF N1 . Moreover, Theorem 4.1 in [65] gives the estimate

5.2 Normal Waves in an Open Inhomogeneous Metal-Dielectric Waveguide

253

N1−1 ( γ ) ≤ 1. γ ) < 1 is satisfied and find that there exists We take  γ such that the inequality N2 ( a bounded operator −1    γ ) + N2 ( γ) = I + N1−1 ( γ )N2 ( γ ) N1−1 ( γ ). N1 ( 

The proof of the theorem is complete.

Since relation (5.2.55) holds for any u ∈ H01 () , v ∈ H 1 (), we introduce test functions u and v such that u=

ε 1 − ε2 ε 1 − ε2 u = f −1 u, v = 2 2 v = f −1 v. κ21 κ22 κ1 κ2

(5.2.62)

Obviously, functions u and v thus defined belong to the same spaces as initial functions u and v. Relation (5.2.55) becomes





∂ ∂ + +γ v− u dτ ∂τ ∂τ  1

1 u ε ∇∇ ε + v∇∇ ε u J ( ε , ) − v J ( ε, ) 4 2 − − γ κ∗ κ2 dx  κ2 ε 1 − ε2 ε 1 − ε2

 εu + v κ21 κ22 dx ε 1 − ε2



 ε∇∇u + ∇∇v  κ2∗ dx ε 1 − ε2



 ∂ 2 G(x, x0 ) (x0 )u(x) + (x0 )v(x)   ρ=r dϕdϕ0 = 0, ∂ρ∂ρ0 ε 1 − ε2 ρ =r

2π 2π + r2

κ21 0

(5.2.63)

0

0

κ2 κ2 where κ2∗ = 1 2 2 .  κ Then the integrals in (5.2.63) can be considered as sesquilinear forms over field C. These forms determine the bounded linear operators:  k(u, v) :=



 a1 (u, v) :=

 ε∗ 

 a2 (u, v) := 

 2 2  εu + v (γ)u, v), ∀v ∈ H, κ1 κ2 − κ2∗ d x = (K ε 1 − ε2  ε∇∇u + ∇∇v +  εu + v d x = ( A1 u, v), ∀v ∈ H, ε 1 − ε2

 ε∇∇u + ∇∇v +  εu + v d x = ( A2 u, v), ∀v ∈ H, ε 1 − ε2

254

5 Open Waveguides of Arbitrary Cross Section

 s(u, v) := 1

 b(u, v)

= 

1  κ2

∂ ∂ v− u dτ = ( Su, v), ∀v ∈ H, ∂τ ∂τ



u ε∇∇ ε + v∇∇ ε u J ( ε, ) − v J ( ε, ) κ42 dx − γ κ2∗ ε1 − ε2 ε1 − ε2 = ( B(γ)u, v), ∀v ∈ H,

 p1 (u, v) = −

1 2π

2π 2π 0

1  p2 (u, v) = − 2π

0

2π 2π 0

0

 ε1 (x0 )u(x) + (x0 )v(x)    ρ=r dϕdϕ0 = (P1 u, v), ∀v ∈ H, ε1 − ε2 1 − cos (ϕ − ϕ0 ) ρ =r 0

 1 (x0 )u(x) + (x0 )v(x)    ρ=r dϕdϕ0 = (P2 u, v), ∀v ∈ H, ε1 − ε2 1 − cos (ϕ − ϕ0 ) ρ =r 0

and κ2 r K 1 (κ2 r )  pk (u, v) = − π K 0 (κ2 r ) 1 + π

2π 2π κ21 0

0

2π 2π κ21 0

0

 (x0 )u(x) + (x0 )v(x)   ρ=r dϕdϕ0 ε1 − ε2 ρ =r 0



∞  (x0 )u(x) + (x0 )v(x)  K m (κ2 r ) κ2 r + m cos [m (ϕ − ϕ0 )]   K m (κ2 r ) ε1 − ε2

m=1

ρ=r ρ0 =r

dϕdϕ0

= ( PK (γ)u, v), ∀v ∈ H,

ε1 ε2 .  ε   Operators K(γ), B(γ),  A1 and  A2 preserve all properties of operators A + K1 (γ) and B(γ) listed in Lemmas 5.2.1–5.2.3.

where  ε∗ =

Lemma 5.2.6 Operator  S is self-adjoint,  S = S∗ , and the following inequalities hold −

1 1 I ≤ S ≤ I. 2 2

(5.2.64)

In accordance with Lemma 5.2.5, the following corollary holds: Corollary 5.2.2 Operators  P1 and  P2 are positive,  P1 > 0 and  P2 > 0. We can rewrite OVF N(γ) as follows: N(γ) := NK (γ) + N0 (γ),

(5.2.65)

5.2 Normal Waves in an Open Inhomogeneous Metal-Dielectric Waveguide

where

255

 P (γ) + B(γ) − K NK (γ) = −K(γ)

and

    A2 +  S−  A1 +  P2 − γ P1 . N0 (γ) = γ 2 

Theorem 5.2.3 Operator N(γ) : H → H is bounded, holomorphic, and Fredholm in the domain    and   := 0 ∪ {γ : Im γ 2 = 0, γ 2 ≤ ε2 } ∪ γ : Im γ = 0, γ∗ ≤ |γ| ≤ γ ∗ ,  = C\

where γ∗ =

√ 

   1 1 + 16−2 − 1  and γ ∗ = 1 + 16  A1 +  P1  A2 +  P2 + 1 , 2  2 A2 +  P2

where  = max ε1 (x) − ε2 . x∈

Proof In the domain {γ : Im γ 2 = 0, γ 2 ≤ ε2 }, the functions κ2

K m (κ2 r ) K m+1 (κ2 r ) m = −κ2 + K m (κ2 r ) K m (κ2 r ) r

(5.2.66)

are analytical (with respect to γ). Since Re κ2 > 0, functions K m (κ2 r ) have no zeros [64]. Let γ = γ + iγ

, γ

= 0. Then    1 N0 (γ) 1  = A2 +  A1 +  Im P2 + 2  P1 ≥ I.

γ γ |γ| Further, let Im γ = 0. Consider the following quadratic form (N0 (γ)u, u) =

     2  A2 +  S−  A1 +  γ  P2 − γ P1 u, u = 0.

Expressing (Su, u) from the last equation and taking into account estimate (5.2.64), we obtain the following double inequality:   1  1 1  A2 +  A1 +  P2 − P1 ≤ I. − I≤γ  2 γ 2

(5.2.67)

Solving inequalities (5.2.67) with respect to γ, we find that outside the set γ∗ ≤ |γ| ≤ γ ∗

256

5 Open Waveguides of Arbitrary Cross Section

operator N0 (γ) is positive or negative definite and therefore continuously invertible [65]. Hence, N(γ) is a Fredholm OVF as the sum of invertible and compact operators, and ind N(γ) = 0.  Theorem 5.2.4 The spectrum of OVF N(γ) : H → H is discrete in , i.e., it has a finite number of eigenvalues of finite algebraic multiplicity in any compact set K0 ⊂ . Proof The assertion of the theorem is a corollary of Theorem 5.2.3 and properties of the holomorphic OVF [65]. 

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Chapter 6

Conclusion

The book deals with the problems of the electromagnetic wave propagation in a wide class of waveguide structures: from planar waveguides to waveguides of arbitrary cross section (including the shielded ones). The settings for the Maxwell are reduced to boundary eigenvalue problems for the longitudinal components of the electromagnetic field in Sobolev spaces using a universal approach applicable to all the considered waveguide families. To determine and analyze the solution, variational formulations are used and the problems are reduced to the study of the spectra of OVF; many of them are polynomial operator pencils. The discreteness of the eigenwave spectra is proved and the distribution of characteristic numbers on the complex plane is described. Numerical study is performed on the spectrum of surface, leaky, and complex waves using variational formulations of the problems. The determination of running waves is reduced to eigenvalue problems for a system of ordinary differential equations which are solved numerically using a version of the “shooting method”. The proposed numerical technique is proved to be an effective tool to calculate the wave propagation constants. The results of a number of numerical experiments are reported. List the most important results concerning the properties of the wave spectra for all the types of waveguides and waves obtained in the study. For shielded metal-dielectric waveguides of arbitrary cross section with inclusions, the running-wave spectrum is discrete (forms a set of isolated points on the complex plane without finite accumulation points), is located in a band, and the system of eigenwaves and associated waves is complete (in the sense and in the function spaces specified in Chap. 2 of the book). For open planar metal-dielectric waveguides: There are no complex surface TE waves, not more than a finite set of propagating surface TE waves, and infinitely many leaky complex TE waves in an inhomogeneous partially shielded dielectric layer with the spectrum located in a certain cone on the complex plane; the system © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 Y. Shestopalov et al., Optical Waveguide Theory, Springer Series in Optical Sciences 237, https://doi.org/10.1007/978-981-19-0584-1_6

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of eigenwaves and associated waves is complete (in the sense and in the function spaces specified in Chap. 3 of the book). We have proved fundamental properties of the spectrum of normal waves propagating in waveguide of circular cross section including the discreteness and a statement describing localization of eigenvalues of the operator function on the complex plane (in the sense and in the function spaces specified in Chap. 4 of the book). We have shown that the difficulties connected with the investigation of the wave spectra in open waveguides of arbitrary cross section can be overcome by introducing a fictitious outer region (the exterior of the circle) and representing the solution in this region in terms of Green’s function. This leads to the appearance of a trace operator (on the boundary of the fictitious region) in the variational relation, which depends in a nonlinear way on the spectral parameter. In the end, we have to analyze not the OP but an OVF and it becomes possible to study the properties of OVF in sufficient detail and obtain the results on its spectrum using the techniques developed in the book. The discreteness of the spectrum of the problem of normal waves is proved in Chap. 5 and the distribution of characteristic numbers on the complex plane is presented.