Advanced Theoretical and Numerical Electromagnetics: Vol.2 1839535709, 9781839535703

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Advanced Theoretical and Numerical Electromagnetics

The ACES Series on Computational and Numerical Modelling in Electrical Engineering Andrew F. Peterson, PhD – Series Editor The volumes in this series will encompass the development and application of numerical techniques to electrical and electronic systems, including the modelling of electromagnetic phenomena over all frequency ranges and closely related techniques for acoustic and optical analysis. The scope includes the use of computation for engineering design and optimization, as well as the application of commercial modelling tools to practical problems. The series will include titles for senior undergraduate and postgraduate education, research monographs for reference, and practitioner guides and handbooks. Titles in the Series K. Warnick, “Numerical Methods for Engineering,” 2010. W. Yu, X. Yang and W. Li, “VALU, AVX and GPU Acceleration Techniques for Parallel FDTD Methods,” 2014. A.Z. Elsherbeni, P. Nayeri and C.J. Reddy, “Antenna Analysis and Design Using FEKO Electromagnetic Simulation Software,” 2014. A.Z. Elsherbeni and V. Demir, “The Finite-Difference Time-Domain Method in Electromagnetics with MATLAB® Simulations, 2nd Edition,” 2015. M. Bakr, A.Z. Elsherbeni and V. Demir, “Adjoint Sensitivity Analysis of High Frequency Structures with MATLAB® ,” 2017. O. Ergul, “New Trends in Computational Electromagnetics,” 2019. D. Werner, “Nanoantennas and Plasmonics: Modelling, design and fabrication,” 2020. K. Kobayashi and P.D. Smith, “Advances in Mathematical Methods for Electromagnetics,” 2020. S. Roy, “Uncertainty Quantification of Electromagnetic Devices, Circuits, and Systems,” 2021.

Advanced Theoretical and Numerical Electromagnetics Volume 2: Field representations and the Method of Moments Vito Lancellotti

The Institution of Engineering and Technology

Published by SciTech Publishing, an imprint of The Institution of Engineering and Technology, London, United Kingdom The Institution of Engineering and Technology is registered as a Charity in England & Wales (no. 211014) and Scotland (no. SC038698). © The Institution of Engineering and Technology 2022 First published 2021 This publication is copyright under the Berne Convention and the Universal Copyright Convention. All rights reserved. Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may be reproduced, stored or transmitted, in any form or by any means, only with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publisher at the undermentioned address: The Institution of Engineering and Technology Michael Faraday House Six Hills Way, Stevenage Herts, SG1 2AY, United Kingdom www.theiet.org While the author and publisher believe that the information and guidance given in this work are correct, all parties must rely upon their own skill and judgement when making use of them. Neither the author nor publisher assumes any liability to anyone for any loss or damage caused by any error or omission in the work, whether such an error or omission is the result of negligence or any other cause. Any and all such liability is disclaimed. The moral rights of the author to be identified as author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.

British Library Cataloguing in Publication Data A catalogue record for this product is available from the British Library

ISBN 978-1-83953-564-2 (Volume 1 hardback) ISBN 978-1-83953-565-9 (Volume 1 PDF) ISBN 978-1-83953-568-0 (Volume 2 hardback) ISBN 978-1-83953-569-7 (Volume 2 PDF) ISBN 978-1-83953-570-3 (2 Volume set hardback)

Typeset in India by MPS Limited Printed in the UK by CPI Group (UK) Ltd, Croydon

To the memory of my parents, Loreto and Angelina, with deep gratitude

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Contents

List of figures List of tables List of examples About the author Foreword Preface Acknowledgements 1

Fundamental notions and theorems 1.1 The electromagnetic field 1.2 The Maxwell equations 1.2.1 Integral or global form 1.2.2 Differential or local or point form 1.3 The Faraday law for slowly moving conductors 1.4 Displacement current 1.5 Time-harmonic fields and sources 1.6 Constitutive relationships 1.7 Boundary conditions for fields and currents 1.8 Wave equations 1.8.1 Time domain 1.8.2 Frequency domain 1.9 Electromagnetic radiation 1.10 Conservation of electromagnetic energy 1.10.1 Poynting theorem in the time domain 1.10.2 Poynting theorem in the frequency domain 1.11 Conservation of electromagnetic momentum 1.12 Conservation of electromagnetic angular momentum References

2

Static electric fields I 2.1 Laws of electrostatics 2.2 Scalar potential and the Poisson equation 2.3 Physical meaning of the scalar potential 2.4 Boundary conditions for the scalar potential 2.5 Uniqueness of the static solutions 2.5.1 Scalar potential 2.5.2 Electrostatic field 2.6 The three-dimensional static Green function 2.6.1 Unbounded homogeneous isotropic medium 2.6.2 Unbounded homogeneous anisotropic medium

xv xxv xxvii xxix xxxi xxxiii xxxvii 1 1 4 4 6 14 20 23 26 30 41 43 46 48 51 51 58 64 72 75 79 79 83 89 90 91 91 102 105 106 107

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Advanced Theoretical and Numerical Electromagnetics 2.7 Integral representation of the scalar potential 2.8 Volume potential 2.9 Double-layer potential 2.10 Single-layer potential References

109 120 131 135 144

3 Static electric fields II 3.1 Scalar potential due to surface charges 3.2 Integral representation of the electrostatic field 3.3 Other Green functions for static problems 3.3.1 The Dirichlet Green function 3.3.2 The Neumann Green function 3.4 Properties of the static Green functions 3.5 Laplace equation and boundary value problems 3.5.1 Polar spherical coordinates 3.5.2 Circular cylindrical coordinates 3.6 Multipole expansion of the scalar potential 3.6.1 Taylor series of the Green function 3.6.2 Spherical harmonics 3.7 Polarization vector 3.8 The Kelvin and Earnshaw theorems 3.9 Image principle in electrostatics 3.10 Singular electric fields References

147 147 151 154 154 158 159 163 163 175 182 182 187 191 197 206 211 215

4 Stationary magnetic fields I 4.1 Stationary limit of Maxwell’s equations 4.2 Vector potential and the vector Poisson equation 4.3 Boundary conditions for the vector potential 4.4 Magnetic scalar potential 4.5 Magnetic dipoles 4.6 Energy and momentum in the stationary limit 4.7 Uniqueness of the stationary solutions 4.7.1 Vector potential 4.7.2 Magnetic entities in the presence of magnetic media 4.7.3 Magnetic entities in the presence of conductors References

219 219 229 233 236 238 242 242 242 247 252 260

5 Stationary magnetic fields II 5.1 Integral representations 5.1.1 Vector potential in an isotropic medium 5.1.2 Magnetic induction and magnetic field 5.1.3 Vector potential and magnetic entities in an anisotropic medium 5.2 Vector potential due to surface currents 5.3 Physical meaning of the vector potential 5.4 Geometrical meaning of the scalar potential 5.5 Multipole expansion of the vector potential 5.6 Magnetization vector

263 263 263 275 279 284 289 293 296 300

Contents

ix

5.7 Magnetic forces between steady currents References

311 316

6

Properties of electromagnetic fields 6.1 Principle of superposition 6.2 Well-posedness of the Maxwell equations 6.3 Uniqueness in the time domain 6.3.1 Bounded regions 6.3.2 Unbounded regions 6.4 Uniqueness in the frequency domain 6.4.1 Bounded regions 6.4.2 Unbounded regions 6.5 Magnetic charges and currents 6.6 Boundary conditions with magnetic sources 6.7 Duality transformations 6.8 Reciprocity theorems 6.8.1 Frequency domain 6.8.2 Non-reciprocal media 6.8.3 Time domain 6.9 Other symmetry relationships 6.9.1 Electrostatic fields 6.9.2 Stationary fields and steady currents References

319 319 325 326 327 331 337 338 353 359 365 368 371 371 388 389 391 391 398 402

7

Electromagnetic waves 7.1 Time-domain uniform plane waves 7.2 Time-harmonic plane waves 7.2.1 Lossless isotropic medium 7.2.2 Lossy isotropic medium 7.3 Polarization of plane waves 7.4 Plane-wave propagation in layered isotropic media 7.4.1 Reflection and transmission at a planar interface 7.4.2 Network equivalent of a multi-layered medium 7.5 Time-domain uniform cylindrical waves 7.6 The two-dimensional time-domain Green function 7.7 Time-domain transverse electric-magnetic spherical waves 7.8 Non-radiating sources References

407 407 409 410 421 430 433 434 445 460 470 479 485 490

8

Time-varying electromagnetic fields I 8.1 The Helmholtz decomposition 8.1.1 Unbounded regions 8.1.2 Bounded regions 8.2 Electrodynamic potentials and gauge transformations 8.3 Boundary conditions for the electrodynamic potentials 8.4 Hertzian potentials 8.5 The scalar Helmholtz equation 8.5.1 Polar spherical coordinates

493 493 493 502 513 519 523 527 527

x

Advanced Theoretical and Numerical Electromagnetics 8.5.2 The Rellich theorem 8.5.3 Conservation of ‘energy’ 8.5.4 Uniqueness in unbounded regions 8.5.5 Uniqueness in bounded regions 8.6 Uniqueness of solutions to the D’Alembert equation 8.6.1 Bounded regions 8.6.2 Unbounded regions 8.7 The three-dimensional time-dependent Green function 8.7.1 Frequency domain 8.7.2 Time domain References

529 531 532 536 542 543 545 548 550 556 565

9 Time-varying electromagnetic fields II 9.1 Integral representations of the potentials 9.1.1 Frequency domain 9.1.2 Time domain 9.2 Potentials and fields of a point charge in uniform motion 9.2.1 Velocity smaller than c 9.2.2 Velocity equal to c 9.2.3 Velocity larger than c 9.3 Electrodynamic potentials due to surface sources 9.4 Time-harmonic dyadic Green functions 9.4.1 Observation points away from the sources 9.4.2 Observation points in the source region 9.4.3 Governing equation of GEJ (r, r ) 9.4.4 Governing equation of GH J (r, r ) 9.5 Regions of a localized time-harmonic source 9.6 Fields in the Fraunhofer region of a time-harmonic source 9.7 Symmetry properties of dyadic Green functions 9.8 Quasi-static electromagnetic fields 9.8.1 Electro-quasi-static regime 9.8.2 Magneto-quasi-static regime References

569 569 569 577 585 585 591 592 595 601 602 604 614 619 622 633 646 653 656 657 670

Index

675

10 Integral formulas and equivalence principles 10.1 Integral representations with dyadic Green functions 10.2 The integral formulas of Stratton and Chu 10.3 Integral formulas with Kottler’s line charges 10.4 Surface equivalence principles 10.4.1 The Huygens and Love equivalence principles 10.4.2 The Schelkunoff equivalence principle 10.5 Volume equivalence principle 10.6 The equivalent circuit of an antenna 10.6.1 Antenna port connected to a coaxial cable 10.6.2 Antenna port modelled with the delta-gap approximation References

687 687 693 702 705 708 715 718 725 726 731 733

Contents

xi

11 Spectral representations of electromagnetic fields 11.1 Modal expansion in cavities 11.1.1 Vector eigenvalue problems in cavities 11.1.2 Solenoidal modes 11.1.3 Lamellar modes 11.1.4 Orthogonality properties of the cavity eigenfunctions 11.1.5 Stationarity of the Rayleigh quotient 11.1.6 Completeness of the cavity eigenfunctions 11.1.7 Equivalent sources on a cavity boundary 11.2 Modal expansion in uniform cylindrical waveguides 11.2.1 The Marcuvitz-Schwinger equations 11.2.2 Transverse-magnetic modes 11.2.3 Transverse-electric modes 11.2.4 Transverse-electric-magnetic modes 11.2.5 Orthogonality properties of the transverse eigenfunctions 11.2.6 Sources in waveguides 11.3 Wave propagation in periodic structures 11.3.1 Periodic boundary conditions 11.3.2 Bloch modes in a periodic layered medium 11.4 Sources and fields invariant in one spatial dimension 11.4.1 Two-dimensional TM and TE decomposition 11.4.2 The two-dimensional Helmholtz equation 11.4.3 Reflection and transmission at a planar material interface References

735 735 736 738 739 743 745 747 750 756 758 764 770 773 780 782 790 792 795 802 803 804 812 820

12 Wave propagation in dispersive media 12.1 Constitutive relations in frequency and time domain 12.2 The Kramers-Krönig relations 12.3 Simple models of dispersive media 12.3.1 Conducting medium 12.3.2 Dielectric medium 12.3.3 Polar substances 12.4 Narrow-band signals in the presence of dispersion 12.5 Intra-modal dispersion in waveguides References

823 823 828 833 833 842 852 855 861 866

13 Integral equations in electromagnetics 13.1 General considerations 13.2 Surface integral equations for perfect conductors 13.2.1 Electric-field integral equation (EFIE) 13.2.2 EFIE with delta-gap excitation 13.2.3 Magnetic-field integral equation (MFIE) 13.2.4 Interior-resonance problem 13.2.5 Combined-field integral equation (CFIE) 13.2.6 A modified EFIE for good conductors 13.3 Surface integral equations for homogeneous scatterers 13.3.1 The integral equations of Poggio and Miller (PMCHWT) 13.3.2 The Müller integral equations

869 869 878 878 884 892 898 902 904 907 913 915

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Advanced Theoretical and Numerical Electromagnetics 13.4 13.5

Volume integral equations for inhomogeneous scatterers Hybrid formulations 13.5.1 Electric-field and volume integral equations 13.5.2 Integral and wave equations References

916 922 922 929 935

14 The Method of Moments I 14.1 General considerations 14.2 Discretization of the EFIE 14.3 Discretization of the MFIE 14.4 Discretization of the CFIE 14.5 Discretization of the PMCHWT equations 14.6 Discretization of the Müller equations 14.7 The basis functions of Rao, Wilton and Glisson 14.8 Area coordinates 14.9 Singular integrals over triangles 14.9.1 Integrals involving 1/R 14.9.2 Integrals involving R/R 14.9.3 Integrals involving ∇(1/R) 14.10 Discretization of the EFIE with delta-gap excitation 14.11 Scaling of solutions References

941 941 947 953 955 955 960 964 971 978 980 988 989 994 1002 1009

15 The Method of Moments II 15.1 Discretization of volume integral equations 15.2 The basis functions of Schaubert, Wilton and Glisson 15.3 Volume coordinates 15.4 Singular integrals over tetrahedra 15.4.1 Integrals involving 1/R 15.4.2 Integrals involving R/R 15.4.3 Integrals involving ∇(1/R) 15.4.4 Integrals involving ∇(1/R), a constant dyadic and R 15.5 Discretization of EFIE and volume integral equations 15.6 Discretization of integral and wave equations 15.7 Edge elements for the vector wave equation References

1013 1013 1018 1026 1033 1035 1040 1048 1049 1052 1059 1067 1079

A Vector calculus A.1 Systems of coordinates A.1.1 Circular cylindrical coordinates A.1.2 Polar spherical coordinates A.2 Differential operators A.3 The Gauss theorem A.4 The Stokes theorem A.5 The surface Gauss theorem A.6 The Helmholtz transport theorem A.7 Estimates for vector-valued functions References

1081 1081 1081 1083 1085 1087 1088 1088 1090 1096 1098

Contents

xiii

B Complex analysis B.1 Derivatives and integrals B.2 Poles and residues B.3 Branch points and Riemann surfaces References

1101 1101 1106 1114 1117

C Dirac delta distributions C.1 Definitions and properties C.2 Derivatives and weak operators References

1119 1119 1125 1129

D Functional analysis D.1 Vector and function spaces D.2 The Bessel inequality D.3 Linear operators D.4 The Cauchy-Schwarz inequality D.5 The Riesz representation theorem D.6 Adjoint operators D.7 The spectrum of a linear operator D.8 The Fredholm alternative References

1131 1131 1143 1144 1148 1150 1153 1157 1161 1162

E Dyads and dyadics E.1 Scalars, vectors, and beyond E.2 Dyadic calculus E.2.1 Sum of dyadics and product with a scalar E.2.2 Scalar and vector product E.2.3 Neutral elements E.2.4 Transpose and Hermitian transpose E.2.5 Double scalar product and double vector product E.2.6 Determinant, trace and eigenvalues E.3 Differential operators References

1165 1165 1167 1168 1169 1169 1171 1173 1175 1176 1177

F Properties of smooth surfaces ˆ  ) · (r − r) F.1 An estimate for n(r F.2 Solid angle subtended at a point F.3 Points in an open neighbourhood F.4 Criterion for the Hölder continuity of scalar fields References

1179 1179 1180 1182 1185 1186

G A surface integral involving the time-harmonic scalar Green function G.1 Two estimates for ∇ G(r, r ) G.2 Finiteness and Hölder continuity References

1187 1187 1189 1192

H Formulas H.1 Vector identities and inequalities

1193 1193

Advanced Theoretical and Numerical Electromagnetics

xiv

H.2 H.3 H.4 H.5

Dyadic identities Differential identities Integral identities Legendre polynomials and functions H.5.1 Nomenclature H.5.2 Differential equation H.5.3 Explicit expressions for the lowest orders H.5.4 Orthogonality relationships H.5.5 Functional relationships H.6 Bessel functions H.6.1 Nomenclature H.6.2 Differential equation H.6.3 Functional relationships H.6.4 Asymptotic behavior for small argument H.6.5 Asymptotic behavior for large argument H.6.6 Recursion relationships H.6.7 Wronskians and cross products H.6.8 Integral relationships H.6.9 Series References Index

1194 1195 1197 1199 1199 1199 1199 1200 1200 1200 1200 1201 1201 1201 1202 1203 1203 1203 1203 1203 1205

List of figures

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11

A region of space containing sources  and J, and a test point charge. Surfaces for stating the Maxwell equations in integral form. For the derivation of the Ampère-Maxwell law in local form. For the derivation of the electric Gauss law in local form. The role played by the Maxwell equations and the continuity equation. Sliding metallic strip. Faraday disk. The Ampère law (1.78) fails for time-varying currents. The Ampère law (1.78) fails for time-varying currents. The Ampère-Maxwell law (1.13) applied to the capacitor-wire system of Figure 1.8b. Temporal evolution of time-harmonic electric field and magnetic induction in a point in space. A material body confined in the region V2 ⊂ R3 and the geometrical quantities for determining the jump conditions of H and E across ∂V. A material body confined in the region V2 ⊂ R3 and the geometrical quantities for determining the jump conditions of D, B and J across ∂V. A piecewise-smooth material interface ∂V := ∂V1 ∪ ∂V2 and the geometrical quantities for determining the jump conditions of JS across the line γ ⊂ ∂V. Accelerated charge (◦) causing a ripple in the fabric of the electric field (→). Relative positions of a moving point charge (◦) and an observer at rest. The ball B(w(tr ), R) has been drawn (−−) only partially. A bounded domain V in which sources and a conducting medium reside. Lines of constant Φ(r) (−−) and streamlines of E(r) (−) in the xOz plane for an elementary electric dipole with moment p = pˆz. A test point charge immersed in an electrostatic field. For proving uniqueness of solutions to the Poisson equation. For proving uniqueness of solutions to the Poisson equation in the presence of N conducting bodies. Meridian cross-section of an ideal spherical capacitor. For proving uniqueness of solutions to the electrostatic equations. For the derivation of the integral representation of the scalar potential. A ball B(r, a) for isolating the location of the charge where the Green function is singular. Two concentric balls B(r, a) and B(r, b) for showing that the volume integral in (2.160) is bounded for observation points r within the source region. Auxiliary problems for the derivation of the integral representation of the scalar potential. For proving the mean value theorem of electrostatics.

2 4 8 10 12 16 18 20 21 22 26 31 33 39 42 49 52

88 89 92 96 100 104 110 111 112 113 116

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Advanced Theoretical and Numerical Electromagnetics

2.12 2.13 2.14 2.15

Geometrical construction for proving the continuity of the static volume potential. The graph of an instance of the radial three-dimensional step function (2.201). Geometrical construction for computing the gradient of the volume potential V(r). Geometrical construction for proving that the volume potential V(r) solves the Poisson equation for points r in the source region V . Physical meaning of electrostatic double-layer and single-layer potentials. For computing the limiting values of W(r) as r → S + . For computing the limiting values of W(r) as r → S − . For defining the differential solid angle dΩ with respect to r. Geometrical construction for showing that the static single-layer potential exists for points r ∈ S .

2.16 2.17 2.18 2.19 2.20 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17 3.18 3.19 3.20 4.1 4.2 4.3 4.4 4.5 4.6

For the derivation of the electrostatic potential generated by a layer of charges with density S (r) in a homogeneous isotropic dielectric medium. For the derivation of the electrostatic potential generated by a layer of dipoles with density τS (r) in a homogeneous isotropic dielectric medium. Electrostatic shielding achieved with a grounded conducting shell. Geometry for the Dirichlet Green function in a dielectric half space. Conducting sphere in an impressed uniform electrostatic field. Dielectric sphere immersed in an impressed uniform electrostatic field. Graphical representation of Bessel functions for real values of the argument. The asymptotic expansion (3.187) holds true for points outside a ball B(0, a) ⊃ V . Geometrical setup for the calculation of the electric quadrupole moment. Normalized potential of a quadrupole. Three systems of spherical coordinates with the same origin for the proof of the addition theorem for spherical harmonics. For defining the polarization vector. For determining the local Gauss law in the presence of a dielectric body. A system of N charged conductors for proving the Kelvin theorem. A system of N charged conductors for deriving the Earnshaw theorem. The image principle applied to the calculation of the electrostatic field generated by a charge (◦) in the presence of a conducting half space. Charges and dielectric bodies in the presence of a conducting half space. Equivalent problem for the potential obtained with the image principle. Geometry for studying potential, electric field and induced charges near the corner of a grounded conducting wedge. Singular behavior near the corner of a conducting wedge. Possible ways of realizing a stationary current J(r): (a) a finite-sized closed tube and (b) an infinitely extended tube. Rectangular path Γ in the xOz plane for the calculation of Hz (τ) produced by the current density (4.12). The magnetic field component Hϕ (τ) generated by an infinite straight uniform current density of circular cross section. For the calculation of the magnetic field generated by a solenoid. The boundary of an annular sector in a plane perpendicular to the solenoid of Figure 4.4b for showing that Hϕ (τ) = 0. Longitudinal cut of the solenoid of Figure 4.4b

121 123 125 128 131 133 133 135 136

148 150 155 156 171 174 181 183 186 187 188 192 195 198 203 207 208 211 212 215 220 223 225 226 227 228

List of figures 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 5.1 5.2

5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11

6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9

An open smooth surface S that intersects a region V J occupied by the current density J(r) for showing that the magnetic scalar potential Ψ(r) is many-valued. The line integral of a stationary magnetic field between two points A and B is path-dependent in general. Elementary dipoles: (a) electric dipole in an electric field and (b) magnetic dipole in a magnetic induction field. For proving uniqueness of the solutions to the vector Poisson equation. For proving uniqueness of solutions to the stationary magnetic equations. Ring-shaped isotropic conductor ‘excited’ by external sources. Ring-shaped homogeneous isotropic conductor excited by a localized impressed electric field (source of electromotive force). Secondary field and associated potential in a ring-shaped conductor.

xvii

237 239 240 244 249 253 255 257

For the derivation of the integral representation of the vector potential. Geometrical construction for showing that the volume integral in (5.18) is bounded for observation points within the current region. The ball B(r, b) has been drawn (−−) only partially. Auxiliary problems for the derivation of the integral representation of the vector potential. For the derivation of the magnetic vector potential generated by a stationary electric current sheet with density JS (r) in a homogeneous isotropic magnetic medium. A point charge (◦) at rest in a stationary field. The ball B(rq , a) has been drawn (−−) only in part. For illustrating the geometrical meaning of the magnetic scalar potential Ψ(r) generated by a current loop γ. The ball B(r, a) has been outlined (−−) only in part. Geometrical setup for computing the equivalent magnetic dipole moment of a small circular loop of line current I0 . For defining the magnetization vector. For determining the local Ampère law in the presence of a magnetized body. Permanently magnetized sphere with uniform magnetization M0 zˆ . Streamlines (−) of the magnetic field H(r) produced in the xOz plane by a permanently magnetized sphere (shaded region) with uniform magnetization M0 zˆ .

264

For illustrating the superposition principle. Superposition principle applied to an electrostatic problem. Lines of constant Φ(r) (−−) and streamlines of E(r) (−) in the xOz plane for the semi-infinite line charge density of Figure 6.2a. Example of well-posed problem: direct electromagnetic scattering. Example of ill-posed problem: inverse electromagnetic scattering. For proving uniqueness of the solutions to the Maxwell equations in the time domain in a bounded region of space. A shrinking ball (−−) for proving uniqueness of the solutions to the Maxwell equations in the time domain in an unbounded region. A shrinking cylinder (−−) for proving uniqueness of the solutions to the Maxwell equations in the time domain in an unbounded cylindrical region. For proving uniqueness of the solutions to the Maxwell equations in the presence of a sharp edge.

320 322

267 268 284 290 294 299 301 304 308 310

324 326 326 327 331 334 337

xviii 6.10 6.11 6.12 6.13 6.14 6.15 6.16 6.17 6.18 6.19 6.20 6.21 6.22 6.23 6.24 6.25 6.26 6.27 6.28 6.29 6.30 6.31 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9

Advanced Theoretical and Numerical Electromagnetics For proving uniqueness of the solutions to the time-harmonic Maxwell equations in a finite region of space in the presence of losses. For studying the uniqueness of the solutions to the time-harmonic Maxwell equations in a finite lossless region of space with PEC boundary conditions. Circular cylindrical cavity with PEC walls. Transmission-line model of a source-free lossless cylindrical cavity. Transmission-line model of source-driven cylindrical cavity. For proving uniqueness of the solutions to the time-harmonic Maxwell equations in a lossy unbounded region. For proving uniqueness of solutions to the time-harmonic Maxwell equations in a lossless unbounded region. Geometry for the calculation of the linear and angular momentum due to a pair of electric and magnetic charges in free space. Equivalent magnetic surface current density J MS (r) defined on the aperture S C of a truncated coaxial cable immersed in free space. Application of the duality principle. For illustrating reciprocity in the presence of a material body. For deriving the reciprocity theorem: sources and matter for states (a) and (b). Special case of the reciprocity theorem: sources for states (a) and (b) located outside V. For deriving the reaction theorem: sources and matter for state (a) and state (b). For applying the reciprocity theorem to a two-port device connected to two waveguides. Applying reciprocity to show that an impressed surface electric current flush with a PEC boundary does not radiate: state (a) and state (b). Application of reciprocity to two antennas in free space: states (a) and (b). Close-up of the antenna gaps WGl (see Figure 6.26) and related geometrical quantities. A circulator used as a duplexer to separate transmitted and received signals in a radar system. For deriving a symmetry relation for static fields: charges and matter for state (a) and state (b). For proving the symmetry of the capacitance matrix [C]: a system of two conductors in state (a) and state (b). For deriving a symmetry relation for static fields produced by point charges. Snapshots of a plane wave propagating along sˆ in a homogeneous isotropic medium. Two planes of constant phase for determining the spatial period of a time-harmonic plane wave in an isotropic lossless medium. Plane waves in a homogeneous isotropic lossless unbounded medium. Snapshot for t = 0 of the normalized electric field of an x-propagating inhomogeneous plane wave in a lossless isotropic medium. Uniform plane waves generated by an infinite electric current sheet JS (r). Plane waves in a homogeneous isotropic lossy unbounded medium. Snapshot for t = 0 of the normalized electric field of a homogeneous plane wave in a lossy isotropic medium with ωε = 4σ. Snapshot for t = 0 of the normalized electric field of an inhomogeneous plane wave in a lossy isotropic medium with ωε = 10σ and θ = π/3. Damping of a uniform plane wave in a lossy half space.

338 342 348 349 351 354 358 362 365 370 372 373 376 377 379 381 382 383 388 392 395 396 409 412 416 417 419 422 423 423 426

List of figures 7.10 7.11 7.12 7.13 7.14 7.15 7.17 7.19 7.20 7.21 7.22 7.23 7.24 7.25 7.26 7.27 7.28 7.29 7.30 7.31 8.1 8.2 8.3 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8

Uniform plane waves in a homogeneous isotropic lossy unbounded medium. Polarization of a time-harmonic wave in a lossless medium. Geometry for determining the principal axes C0 and C0 of the polarization ellipsis. Graphical representation of the circular clockwise polarization expressed in (7.141) and (7.142). Reflection and transmission of uniform plane waves at a planar interface between two isotropic media: transverse-electric (TE) polarization. Reflection and transmission of uniform plane waves at a planar interface between two isotropic media: transverse-magnetic (TM) polarization. Reflection coefficients for the plane-wave problems of Figures 7.14 and 7.15. Reflection and transmission of uniform plane waves in the presence of a slab: transverse-electric (TE) polarization. Reflection and transmission of uniform plane waves in the presence of a slab: transverse-magnetic (TM) polarization. Lumped-element circuit equivalents of plane-wave reflection and transmission in the presence of a slab. Example of transmitted power density for the plane-wave problem of Figure 7.19. For choosing the right solution to (7.310). Electric current and associated function E0 (t) for the generation of cylindrical TM waves. Snapshots of the electric field Ez (ρ, t) of a TM cylindrical wave. Inverse Fourier transformation of the two-dimensional spectral Green function. Inverse Fourier transformation of the two-dimensional spectral Green function. Three snapshots of the two-dimensional Green function (7.382). Practical source of spherical TEMr waves. Four snapshots of the electric field Eϑ (r, ϑ, t) of a TEMr spherical wave as a function of the normalized coordinates x/c0 and z/c0 . An ideal non-radiating source made of two concentric charged spheres. For studying the uniqueness of the solutions to the Helmholtz equation (8.194) in a finite region of space. Inverse Fourier transformation of the three-dimensional spectral Green function. Properties of potentials (8.390) and (8.394). For illustrating which part of the sources contributes to the retarded potentials in the observation point r for t > 0. Geometry for the direct solution of the D’Alembert equation in the time domain. For showing that the surface integrals over ∂V in the representation (9.53) vanish when all sources are located inside the region V. Lines of constant ΦE (r, t) (− ·) and streamlines of E(r, t) (−) in the ZOx plane for a charge in uniform motion (v = vˆz). Time variation of electric and induction fields of a charge in uniform motion along the z-axis for a few values of the speed parameter ς in (9.77). Relative positions of a point charge in uniform motion and an observer at rest. Conical shock wave produced by a point charge in uniform motion with velocity larger than the speed of light in the background medium. For the derivation of the electrodynamic potentials generated by time-harmonic electric surface sources in a homogeneous isotropic medium.

xix 428 431 432 433 434 435 442 446 446 453 458 465 468 469 473 476 478 482 484 485

537 554 564

578 580 583 588 589 590 594 596

xx 9.9 9.10 9.11 9.12 9.13 9.14 9.15 9.16 9.17 9.18 9.19 9.20 9.21 9.22 9.23 9.24 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 10.10 10.11 10.12 10.13 10.14 10.15 10.16 10.17

Advanced Theoretical and Numerical Electromagnetics Close-up of the source region VS and an excluded domain VE ⊂ VS for determining the electric dyadic Green function in a point r ∈ VS . Classification of the spatial regions around a localized time-harmonic electromagnetic source according to the distance therefrom. Geometrical meaning of the far-field approximation. Alternative ways of defining an electric Hertzian dipole. For the calculation of the radiation field of a dipole antenna. For illustrating the genesis of a dipole antenna: open-ended transmission line. For illustrating the genesis of a dipole antenna: open and bent transmission line. Radiation solid of the dipole antenna of Figure 9.13 for d = λ/2. For the calculation of the radiation field of a loop antenna. Radiation solid of the loop antenna of Figure 9.17 for a = λ/10. For deriving a special instance of the reciprocity theorem in the presence of Hertzian dipoles. Rotating dielectric sphere in an impressed uniform electrostatic field. Normalized magnetic potential of rotating dielectric sphere. Skin effect in a wire of circular cross section: (a) normalized current density versus normalized radial coordinate for various frequencies; (b) skin depth versus frequency. Skin effect in a wire of circular cross section: normalized current density versus normalized radial coordinate for various conductivities at f = 10 KHz. Generation of eddy currents in a conducting body exposed to a time-varying magnetic induction field. For the derivation of the integral representations with dyadic Green functions: electromagnetic problem to be solved. For the derivation of the integral representations with dyadic Green functions: auxiliary problem. For the derivation of the integral representation with dyadic Green functions: electric Hertzian dipole outside the region of concern. For the derivation of the Stratton-Chu integral representation: electric Hertzian dipole on the boundary of the region of interest. For the derivation of the Stratton-Chu formulas with field discontinuities across the closed line γ on the boundary S : electromagnetic problem to be solved. For illustrating the equivalence principle in network theory: (a) original circuit; (b) equivalent circuit with voltage and current generators in the section AA’. For illustrating the equivalence principle in network theory: (a) equivalent circuit with voltage generator only; (b) equivalent circuit with current generator only. For the alternative derivation of the Love equivalence principle: original problem. For the alternative derivation of the Love equivalence principle: equivalent problem. For the alternative derivation of the Love equivalence principle: equivalent problem. For the alternative derivation of the Love equivalence principle: equivalent problem. For deriving the Schelkunoff equivalence principle. For deriving the Schelkunoff equivalence principle. For deriving the volume equivalence principle. For deriving the volume equivalence principle. For deriving the volume equivalence principle. Application of reciprocity to the region B(0, d) \ (VA ∪ B(rb, b)) for deriving the equivalent circuit of a receiving antenna.

605 623 625 627 639 639 640 642 643 646 647 658 662 666 666 668 688 689 692 697 703 706 708 711 714 715 715 716 717 718 720 721 726

List of figures 10.18 Close-up of the port region of the antenna in Figure 10.17. 10.19 Thevenin equivalent circuit of the antenna in receiving mode. 10.20 Application of reciprocity for deriving the equivalent circuit of a receiving antenna when the port is modelled with the delta-gap approximation. 10.21 Close-up of the port region of the antenna in Figure 10.20. 11.1 11.2 11.3 11.4 11.5

11.6 11.7 11.8 11.9 11.10 11.11 11.12 11.13 11.14 11.15 11.16 11.17 11.18 11.19 11.20 11.21

12.1 12.2 12.3

Transverse cross-section S of a uniform hollow-pipe waveguide V := S × R. Transverse view of a hollow-pipe waveguide V := S × R with contour-wise multiply-connected cross-section. Transverse view of a two-conductor hollow-pipe waveguide V := S × R for the normalization of the TEM eigenfunctions. Transverse view of a coaxial cable for the calculation and normalization of the TEM eigenfunctions. Normalized magnitude and streamlines of the TEM electric (−) and magnetic (— •) eigenfunctions (11.280) in a coaxial cable with radii a = 2 cm and b = a/2 (Figure 11.4). For solving the telegraph equations: infinitely long transmission line excited by a lumped current generator inserted in parallel. For solving the telegraph equations: infinitely long transmission line excited by a lumped voltage generator connected in series. Longitudinal cut of a periodic multi-layered penetrable medium. Longitudinal cut of a hollow-pipe metallic waveguide periodically loaded with thin metallic irises. Front view of a planar two-dimensional periodic structure obtained through the repetition of metallic crosses in a square lattice. Plot of the dispersion relations (11.370) and (11.371) for k x = 0. Dispersion curves of TE and TM Bloch modes for the structure of Figure 11.8 for k x = 0. Plot of the dispersion relations (11.370) and (11.371) for k x = 2/h (see Figure 11.11 for data). Dispersion curves of TE Bloch modes for the structure of Figure 11.8 for k x = 2/h (see Figure 11.11 for data); (−) real kB , (−−) imaginary kB . Dispersion curves of TM Bloch modes for the structure of Figure 11.8 for k x = 2/h (see Figure 11.11 for data); (−) real kB , (−−) imaginary kB . Dispersion curves of Bloch modes for the structure of Figure 11.8. ˜ x , y). Inverse Fourier transformation of the spectral function Ψ(k Physical interpretation of the integral representation (11.425) as superposition of elementary plane waves emerging from current sheets in y = y ∈ [y1 , y2 ]. Reflection and transmission of z-invariant electromagnetic waves at a planar interface between two homogeneous isotropic media. ˜ l (k x , y). Inverse Fourier transformation of the spectral functions Ψ ˜ Inverse Fourier transformation of the spectral functions Ψl (k x , y) in case medium 2 is a good conductor. Analytic properties of F(ω) ∈ {χ˜ e (r; ω), σ(r; ˜ ω), ε˜ c (r; ω) − ε0 }. Poles (×) and contour Γ (−−) in the complex plane Ω for the derivation of the Kramers-Krönig relations. Dispersion in a conducting medium.

xxi 727 730 731 732 757 766 774 778

779 786 787 791 792 793 798 799 800 800 801 802 809 811 812 817 820 829 830 837

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Advanced Theoretical and Numerical Electromagnetics

12.4

Poles (×) and contour Γ (−−) in the complex plane Ω for the calculation of the integral (12.70) with the Cauchy theorem of residues. Geometrical and physical setup for the calculation of the local field experienced by a polarized atom in a dielectric medium. Dispersion in a dielectric medium. Poles (×) and contour Γ (−−) in the complex plane Ω for the calculation of the integral (12.122) with the Cauchy theorem of residues. Dispersion in a polar material: arc plots of permittivity obtained with Debye (—) and Cole-Cole (−−) models. Dispersion in a polar material: permittivity obtained with Debye and Cole-Cole models as a function of the normalized angular frequency. Qualitative spectrum of a narrow-band waveform. Propagation of a sinusoidal carrier (—) modulated by a narrow-band function (−−) in a dispersive environment without attenuation. Interpretation of TE and TM modes in a parallel-plate waveguide as the linear superposition of two plane waves (→ and ). Dispersion curves (ω − kz diagrams) for the TEM mode and the first three TE and TM modes in the parallel-plate waveguide of Figure 12.12.

12.5 12.6 12.7 12.8 12.9 12.10 12.11 12.12 12.13

13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8 13.9 13.10 13.11 13.12 13.13 13.14 13.15 13.16 13.17 13.18 13.19 13.20 13.21 13.22 13.23

Typical applications of the EFIE: (a) electromagnetic scattering from PEC bodies and (b) radiation from antennas comprised of PEC parts. Derivation of the EFIE for a scattering problem. Derivation of the EFIE for a scattering problem. Application of the EFIE to infinitely thin PEC bodies. Modelling an antenna port with the delta-gap source. Close-up of the antenna gap of Figure 13.5b and related geometrical and physical quantities. Network equivalent of an antenna port. Calculation of the average power radiated by an antenna with the aid of the complex Poynting theorem (1.314). Typical application and inappropriate usage of the MFIE. Derivation of the MFIE for a scattering problem. Derivation of the MFIE for a scattering problem. For illustrating why the MFIE is unsuitable for infinitely thin PEC bodies. For illustrating the interior-resonance problem of the EFIE. For illustrating the interior-resonance problem of the MFIE. Wave scattering from a conducting object modelled by means of a surface impedance ZS and the Leontovich boundary condition (13.142). Scattering problem involving a homogeneous penetrable body endowed with isotropic constitutive parameters. Derivation of surface integral equations for the scattering from penetrable objects. Derivation of surface integral equations for the scattering from penetrable objects. Derivation of surface integral equations for the scattering from penetrable objects. Application and derivation of a volume integral equation. Artist’s impression of a plasma thruster. Derivation of coupled EFIE and VIE for a plasma thruster. Practical gaseous plasma antennas.

838 843 848 850 855 856 858 860 862 865

879 880 881 883 885 886 889 892 893 894 894 897 899 901 905 908 910 910 912 917 923 924 930

List of figures

xxiii

13.24 Derivation of coupled surface integral equations and Maxwell equations for a plasma antenna.

931

14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8 14.9 14.10 14.11 14.12 14.13 14.14 14.15 14.16 14.17 14.18 14.19 14.20 14.21 14.22 15.1 15.2 15.3 15.4 15.5

Geometrical interpretation of condition (14.15). Triangular tessellation S M of a circular horn antenna. The support of a subsectional basis function defined over a part Ξm of the tessellation S M . Geometrical setup for the definition of the RWG basis function associated with an inner edge and defined over a pair of adjacent triangles T n+ and T n− . Geometrical setup for the definition of the normalization constant of an RWG basis function. The mapping of the two-dimensional simplex S 2 onto the triangle T . For defining the area coordinates ξ1 , ξ2 and ξ3 on a triangle. Geometrical setup for the calculation of the Gram matrix of a set of RWG functions. Geometrical quantities associated with a triangle T for the evaluation of singular surface integrals: projection P outside T . Geometrical quantities associated with a triangle T for the evaluation of singular surface integrals: projection P inside T . Geometrical quantities associated with a triangle T for the evaluation of singular surface integrals: projection P on the edge γi . Geometrical quantities associated with a triangle T for the evaluation of singular surface integrals: projection P on the vertex Vl . For illustrating the ambiguity of the unit normal on edges and vertices of a triangular tessellation. Modelling the antenna gap of Figure 13.5b with patches: support Ξm of the mth subsectional test function fm (r) for arbitrary-shaped patches and gap of finite size. Modelling the antenna gap of Figure 13.5b with patches: triangular-faceted tessellation for RWG basis functions and gap reduced to a closed curve γ˜ G . Triangular-faceted model of an inverted-F antenna. Input impedance ZA of the inverted-F antenna of Figure 14.16 as a function of the electric size. Average input or radiated power of the inverted-F antenna of Figure 14.16 as a function of the electric size. Magnitude of the surface current density JS (r) of the inverted-F antenna of Figure 14.16 for VG = 1 V and f = 1.2 GHz. Radiation solid of the inverted-F antenna of Figure 14.16 for f = 1.2 GHz. For studying the scaling of solutions to a scattering problem. For studying the scaling of solutions to a radiation problem. Geometrical setup for the definition of the SWG basis function associated with the facet T n and defined over a pair of adjacent tetrahedra Wn+ and Wn− . Geometrical setup for the definition of the normalization constant of an SWG basis function. The mapping of the three-dimensional simplex S 3 onto the tetrahedron W. For the definition of the volume coordinates ξ1 , ξ2 , ξ3 and ξ4 in a tetrahedron. Geometrical quantities associated with the lth facet of a tetrahedron W for the evaluation of singular surface integrals: projection Pl inside ∂Wl .

945 948 950 965 967 971 972 974 981 983 984 985 993 996 997 999 1000 1000 1001 1001 1002 1006

1019 1021 1027 1028 1037

xxiv 15.6 15.7 15.8 15.9

A.1 A.2 A.3 A.4 B.1

Advanced Theoretical and Numerical Electromagnetics Geometrical quantities associated with the pth facet of a tetrahedron W (not shown) for the evaluation of singular volume integrals. Tetrahedron W and geometrical quantities for the definition of the shape functions hil (r), i = 1, . . . , 3, l = i + 1, . . . , 4, and curl-conforming edge elements. For studying the edge element h12 (r) when the point P (identified by the position vector r) belongs to one of the six edges γil , i = 1, . . . , 3, l = i + 1, . . . , 4. Close-up of the surface mesh S MP for visualizing the interaction of four RWG functions (➞) associated with the sides of Ξn ⊂ S MP and the edge element en (r) associated with the edge γn ⊂ S MP . Geometrical setup for the definition of the circular cylindrical coordinates. Geometrical setup for the definition of the polar spherical coordinates. A rigid, slowly moving surface S and two reference frames for proving the Helmholtz transport theorem. A deforming, slowly moving surface S 1 := S (t) and a local system of curvilinear coordinates for proving the Helmholtz transport theorem.

1046 1069 1070

1078 1082 1084 1091 1093

B.6 B.7

For reviewing the properties of analytic functions: an open set U in the complex plane. For reviewing the properties of analytic functions: (a) a contour (a closed path) contained in U; (b) two lines γ1 and γ2 contained in U and joining the same two points. Line integrals around the pole z0 for the application of (B.31) and the calculation of the residue (B.40). Geometrical setup for the derivation of the Cauchy integral formula for points z on the contour γ. For stating and proving the Cauchy theorem: (a) a contour γ encircling N poles (•) of the function f (z); (b) modified path for computing the line integral along γ. Contours (−−) for the calculation of an integral along the real axis. √ Mapping properties of the many-valued function w = z.

C.1 C.2

A sequence of functions which tend to the one-dimensional delta distribution δ(x). A sequence of functions which tend to the two-dimensional Dirac delta distribution.

1120 1122

E.1

For illustrating the genesis of a dyadic field.

1166

F.1 F.2

1181

F.3 F.4 F.5

Calculation of the solid angle subtended at a point on a smooth surface. The open neighborhood Ha (shown in grey) that surrounds a C2 -smooth closed surface S . Geometrical construction for estimate (F.26). Geometrical construction for estimate (F.31). Close-up of part of a smooth surface S and three strips Ub , U2b and U3b .

1182 1184 1184 1186

H.1

Right-handed triple of unit vectors and two vectors forming an angle α.

1194

B.2

B.3 B.4 B.5

1103

1106 1107 1109 1110 1112 1115

List of tables

1.1

Correspondence between time-harmonic fields and phasors

24

6.1

Duality transformations

369

7.1 7.2

Summary of properties of time-harmonic plane waves in a homogeneous isotropic medium Conductivity, permeability and skin depth of some good conductors

424 425

8.1

Duality transformations for electrodynamic potentials

519

9.1

Duality transformations for dipole moments

652

14.1 Quadrature formulas over a triangle 14.2 Scaling of electromagnetic quantities with the size of an object 14.3 Scaling of electromagnetic quantities with the size of an antenna

974 1005 1008

15.1 Cubature formulas over a tetrahedron

1029

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List of examples

1.1 1.2

Electromotive force in a planar circuit with variable shape Electromotive force generated by a Faraday disk

15 17

2.1 2.2 2.3 2.4 2.5 2.6 2.7

Electrostatic field of a spherical uniform distribution of charge Electrostatic field of a point charge Electrostatic scalar potential of a point charge Electrostatic scalar potential of a spherical uniform distribution of charge Electrostatic scalar potential and electric field of an electrostatic dipole Potential, field and capacitance matrix of a spherical capacitor Scalar potential of a spherical uniform distribution of charge (reprise)

80 82 85 85 86 100 117

3.1 3.2 3.3 3.4 3.5 3.6

Electrostatic potential and field of a uniform spherical layer of charges The Dirichlet Green function of a dielectric half space Conducting sphere in a uniform electrostatic field Dielectric sphere in a uniform electrostatic field Multipole expansion of a quadrupole distribution of charges Dielectric sphere in a uniform electrostatic field (reprise)

149 155 170 173 185 197

4.1 4.2 4.3 4.4 4.5

Magnetic field of a circular cylindrical uniform stationary current density Magnetic field of an infinite solenoid Vector potential of an infinite solenoid The Neumann vector field inside a spherical-conical cavity Steady conduction current inside a ring-shaped conductor

222 225 232 250 255

5.1 5.2 5.3 5.4 5.5

Vector potential and magnetic field of a rotating spherical layer of charge Potential momentum of a charge outside an infinite solenoid Magnetic dipole moment of a small circular loop of uniform current Magnetic field and induction produced by a permanently magnetized sphere Magnetic force between cylindrical uniform stationary currents

286 291 299 308 314

6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9

Electrostatic field of a semi-infinite straight line density of charge Uniqueness in a circular cylindrical cavity with PEC walls Momenta of a magnetic charge in the presence of an electric charge Field of a truncated coaxial cable Electric field generated by a steady magnetic current density Symmetry of impedance and admittance matrices Impressed electric currents on PEC surfaces do not radiate Equivalence of mutual admittances of two antennas Symmetry of the capacitance matrix

321 347 361 364 370 378 380 382 394

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Advanced Theoretical and Numerical Electromagnetics

7.1 7.2 7.3 7.4 7.5 7.6

Uniform plane waves generated by an infinite planar current sheet Uniform plane waves in a good conductor Propagation of a uniform cylindrical TM wave in free space Propagation of a TEMr spherical wave in free space Field generated by a pulsating spherical charge density Field generated by the sudden appearance of a point charge

419 425 467 483 488 489

8.1 8.2 8.3

The Helmholtz decomposition of static and stationary fields in the whole space The Helmholtz decomposition of static electric fields in a bounded region The Helmholtz decomposition of stationary magnetic fields in a bounded region

501 507 508

9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9

Depolarization dyadic for a sphere Depolarization dyadic for a regular tetrahedron Field generated by an electric Hertzian dipole Radiation field of a dipole antenna Radiation field of a loop antenna Dielectric sphere slowly rotating in a uniform electrostatic field Skin effect and skin depth in a conducting half space Skin effect and skin depth in a circular cylindrical conductor Induction heating of a circular cylindrical metallic bar

611 611 626 638 643 657 664 665 668

10.1

Equivalence principle in network theory

706

11.1

Transverse-electric-magnetic eigenfunctions in a coaxial cable

777

12.1 12.2

The Kramers-Krönig relations for a conducting medium The Kramers-Krönig relations for a dielectric medium

837 849

13.1

Solving a degenerate-kernel Fredholm equation of the second kind

876

14.1 14.2 14.3

The Gram matrix and other projection integrals involving RWG functions The Fourier transform of an RWG function Analysis of an inverted-F antenna with EFIE and MoM

973 976 999

15.1 15.2

Projection integrals involving SWG functions The Fourier transform of an SWG function

1029 1031

B.1 B.2

Applying the Cauchy-Riemann condition Calculation of inverse Fourier transforms with the Cauchy theorem

1104 1111

C.1

Derivatives of the three-dimensional step function

1127

About the author

Vito Lancellotti received the laurea degree (M.Sc.) with honors in Electrical Engineering and the Ph.D. degree in Electronics and Communications from Politecnico di Torino (Italy) in 1995 and 1999. In early 1999, he joined Telecom Italia Lab in Torino as a Senior Researcher and was involved in projects concerning TCP/IP and ATM networks. In June 2000, as a Senior Researcher, he joined the Milan-based subsidiary of Corning (now Avanex), where he worked on the design of broadband electro-optic lithium-niobate modulators and optical waveguides. From 2002 to 2008, he served as a Research Fellow and lecturer in the Department of Electrical Engineering of the Politecnico di Torino, where he contributed to developing the TOPICA code, a tool devised for the analysis and design of plasma-facing antennas utilized in magnetically controlled nuclear fusion. In 2005, he was appointed Visiting Scientist at the Massachusetts Institute of Technology (Cambridge, MA) and, in 2007, conducted research at the Max-Planck-Institut für Plasmaphysik (Garching, Germany). From April 2008 to August 2017, he was with the Faculty of Electrical Engineering of the Technical University of Eindhoven (Netherlands), where he served as a Senior Researcher and as an Assistant Professor. His specialties and interests include theoretical, applied and computational electromagnetics; antennas; plasma sources and antennas; plasma-facing antennas for nuclear fusion; biological effects of electromagnetic fields; metallic and dielectric waveguides; spectral methods and the Wiener-Hopf technique.

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Foreword

The ACES/IET book series on Computational Electromagnetics and Engineering fosters the development and application of numerical techniques to electrical systems, including the modeling of electromagnetic phenomena over all frequency ranges and closely-related techniques for acoustic and optical analysis. While focusing on research monographs promoting computational practice for engineering design and optimization, the series also includes titles for undergraduate and graduate education. Modern curricula in electromagnetic (EM) fields, especially at the graduate level, require a frontend course that efficiently reviews undergraduate materials and fills in gaps, since most students in undergraduate electrical engineering programs today are only required to take one course on fundamentals. The course must then progress to cover the advanced concepts needed to introduce the students to classical methods of solving boundary value problems, approximate asymptotic techniques, and modern numerical techniques. Along the way, students need to learn about material interactions and metamaterials, and a wide range of applications including microwave devices, antennas, EM propagation and scattering problems, modern communications, and bio-electromagnetics. The present textbook, designed to be the principal source for a one- or two-course sequence on graduate-level EM, does an excellent job of meeting the needs of such a curriculum. This book begins by reviewing the fundamentals of static fields and EM waves, then progresses through more advanced concepts for time-varying fields, including scalar and dyadic Green’s functions and sourcefield relations, surface and volume equivalence principles, and the spectral representation of EM fields. Dispersive materials are discussed at length. The text concludes by addressing integral equations and their numerical solution using the method of moments. Of particular interest is the inclusion of mathematical details seldom found in engineering texts. Appendices review topics such as complex variables, functional analysis, and dyads, all of which are needed for a solid foundation in EM theory. This book should prove useful for both students and professionals with interest in a wide variety of EM applications. Dr. Lancellotti’s approach of combining fundamental EM theory and computational techniques make this an efficient, self-contained resource. Series Editor Andrew F. Peterson School of ECE, Georgia Institute of Technology Atlanta, GA USA August 3, 2021

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Preface

It appears to me, therefore, that the study of electromagnetism in all its extent has now become of the first importance as means of promoting the progress of science. — James C. Maxwell In the early spring of 2008, I gladly found myself joining the Electromagnetics Group of the Faculty of Electrical Engineering of the Technical University of Eindhoven, The Netherlands. From the very beginning, in addition to conducting research on integral equations and domain decomposition methods, I was involved in the lectures of an elective graduate course that, at the time, was known as ‘Mathematics for Electromagnetism’ and had been initiated, a few years earlier, by Prof. Antonius G. Tijhuis, head of the Electromagnetics Group. Around 2011 I had the pleasure and the honor to take over the course and was asked to redesign it in the context of a general overhauling that the electrical-engineering study program was concurrently undergoing. By and large, this book is an outgrowth of the lectures I delivered from 2011 till 2017 for the said course which, at some point along the way, was renamed ‘Advanced Electromagnetics and Moments Methods’ for two reasons. First and foremost, in the meantime, integral equations and the Method of Moments had found their place in the course program, and this feature had to be properly advertised to prospective students. Secondly, experience indicated that hinting at mathematics inexplicably seemed to scare away the less-bold engineering students who, as a consequence, would not register for the course. That might have been so, though one thing is certain, i.e., the organization of the book, for the most part, closely mirrors the logical order with which the topics were presented to the class because I built the material starting from the slides I had prepared and used for the lectures over the years. Nevertheless, being reasonably unconstrained by criteria of time or space, the material collected in the book exceeds by far the contents of the original set of lectures. As was expected of the brave students who dared to attend the course and ultimately enjoyed it — this I can say on the grounds of the positive anonymous feedback I received — in order to benefit from this book the Reader should have an undergraduate knowledge of classical electromagnetism and should know calculus reasonably well. Starting with the Lorentz force, the first part of the book covers classical electromagnetic theory, namely, Maxwell’s equations, the limits of electrostatic and magneto-stationary fields, electromagnetic theorems, wave propagation, time-varying fields and potentials, integral representations and equivalence principles, fields in cavities and waveguides, and dispersion in material media. The second part, computational electromagnetics, deals with surface and volume integral equations and hybrid formulations in the frequency domain, as well as the solution of those equations through the Method of Moments. I shall be the first to admit that I got carried away and perhaps included more mathematical details than would befit an electrical engineering book, advanced as it may be. And yet, it seems to me that the final result is absolutely in keeping with the original name and intent of the course which inspired me. Indeed, in an effort to make the book as self-contained as possible, detailed proof of most formulas and theorems is given, either in the scope of the main text or in dedicated appendices.

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Advanced Theoretical and Numerical Electromagnetics

This applies, e.g., to the convergence of singular integrals that involve Green functions as well as to the asymptotic behavior of integrals at infinity. The smoothness properties of the electrostatic volume, double- and single-layer potentials (a topic which, owing to time limitations was not part of the set of lectures for the course) are examined extensively in Sections 2.8-2.10, with due regard to the delicate matter of interchanging integrals and derivatives, and the check a posteriori that the integral representation does solve the Poisson equation. These results are also relevant to the smoothness of the integral representation of the stationary magnetic potential (Section 5.1) and the electrodynamic potentials (Section 9.1) since all of them exhibit the same singular character as the electrostatic potential. Still, I reckon that the Reader may safely skip Sections 2.8-2.10 and the related Appendices F and G without compromising the full understanding of the remaining topics. Since Dirac delta distributions have the unquestionable merit of making it amazingly simple to include singular sources in the equations (which is why I used them a lot during the lectures), I covered the topic in Appendix C. However, in the text, I favored the classical approach — based on the isolation of the singular point with a sphere — and employed Dirac distributions sparingly, and only when avoiding them would have made the exposition too cumbersome. For instance, the classical approach is used for the derivation of integral formulas for the electrostatic potential in Section 2.7, the magnetic vector potential in Section 5.1, and the electrodynamic potentials in Section 9.1. By contrast, a not-so-trivial problem, in which surface delta distributions ultimately make their appearance, consists of computing the electromagnetic field produced by a point charge that moves with a uniform velocity larger than the speed of light in the underlying medium (Section 9.2). The free-space time-dependent three-dimensional Green function provides another typical example because it is a singular outward-going spherical wave and, as such, can only be expressed with the aid of a surface delta (Section 8.7). On a related score, the potentials generated by surface charges and currents are often obtained by expressing such sources with the aid of suitable surface delta distributions and by invoking the properties thereof to transform volume integrals into surface ones. Contrariwise, in the text, I described the direct derivation based on homogeneous Poisson or Helmholtz equations supplemented with the relevant matching conditions, which naturally allow bringing the surface sources into the picture (Sections 3.1, 5.2, and 9.3). The students of ‘Advanced Electromagnetics and Moments Methods’ would see dyads and dyadics for the first time at the beginning of the course, when I introduced the linear constitutive relationships in anisotropic media. On that occasion, I usually managed to deflect questions and put off the topic by discounting dyadic permittivity, permeability, and conductivity as nothing more than matrices. At a later stage, I explained the need for entities more general than vectors before I discussed the multipole expansion of the electrostatic scalar potential since the quadrupole moment is a dyadic. The short intermezzo also laid the groundwork for the dyadic Green functions, which are singular dyadic fields. Yet, the interested Reader might want to browse Appendix E upfront because I expanded the early discussion of the energy balance in Section 1.10 by deriving, in Sections 1.11 and 1.12, the conservation laws for linear and angular momentum, both of which involve dyadic fields. At some time or the other, the course program also covered the topics of wave propagation in one-dimensional periodic structures and the spectral representation of fields and sources in situations where the geometry is invariant in one spatial direction. These subjects are dealt with in Sections 11.3 and 11.4, where proofs of the Floquet theorem and of the image principle for time-harmonic fields are given. Then again, since the modal expansion in cavities and hollow-pipe cylindrical waveguides is a spectral representation as well, I decided to include these topics in Sections 11.1 and 11.2 for the sake of completeness. In particular, the waveguide problem is reduced to that of finding

Preface

xxxv

waves along fictitious transmission lines by means of the approach introduced by N. Marcuvitz and J. Schwinger. The specific choice of subjects for Chapters 13-15 (namely, integral equations and Method of Moments) was motivated, for the most part, by the research I conducted at the Technical University of Eindhoven. In particular, since I worked on domain decomposition methods for surface integral equations, it seemed almost natural to expose the students to that topic, which enabled them to undertake and carry out graduation projects in computational electromagnetics. Thus, integral equations for radiation and scattering problems are discussed at length, whereas the classical line integral equations for wire antennas are left out. Volume integral equations and hybrid techniques are also presented, as these subjects are in keeping with the idea of mathematically separating a problem into parts. A popular tenet has it that explaining a topic to someone is a good, if not the best, way of learning and understanding it. Not only is this statement true, in my opinion, but it is also, in essence, the very reason why I decided to write this book. I was also spurred by the moderately selfish desire of producing a self-contained reference I can turn to for help whenever my memory fails me. In the process, I learned a great deal, because I was forced to ask myself questions and to come up with satisfactory answers, but more importantly, I had a lot of fun. I wonder what would be of electromagnetics in one hundred years from now since, as I put these thoughts in writing, Maxwell’s theory is already about one hundred and fifty years old and one of the most complete, at that. While it is indisputable that one need not master electromagnetics — or quantum field theory, for that matter — to use a mobile phone or a tv set, still, someone must have the knowledge to build those devices in the first place, right? Vito Lancellotti Torino, Italy November 25, 2021

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Acknowledgements

A curious fact about this work is that, back in the spring of 2017, when I started putting the material together, I did not dream it would develop into being a ‘real’ book. However, now that Advanced Theoretical and Numerical Electromagnetics has indeed been published (and in two volumes, at that) it is my distinct pleasure to give well-deserved credit to all the people who, in different moments, either deliberately or unwittingly, played a role in making this book see the light of day. Dr. Nicki Dennis, Commissioning Editor for Radar, Electromagnetics, and Signal Processing with IET Press, reacted enthusiastically to my book proposal, which I submitted for consideration in February 2021. She boldly believed in the potential of this project and patiently assisted me in the early stages of the publication process. Mrs. Olivia Wilkins, Assistant Editor with IET Press, handled the manuscript submission and helped me stay on schedule by checking on my progress regularly, though unobtrusively. She addressed all of my concerns, especially about the nifty typesetting of the long and complicated equations found in this book. Mrs. Nikki Tarplett, Senior Production Controller with IET Press, was my production contact within the IET. In particular, she made sure that the typesetting process ran smoothly. Mr. Srinivasan N., Project Manager with MPS Ltd., India, was in charge of the final typesetting of the book. He handled all of my requests for amendments and changes in the proofs with considerable patience and professionalism. The Reviewers of my book proposal devoted precious time to assessing this project. They all expressed favorable opinions on the usefulness of the subject, the organization of the book, and its potential audience. What is more, they were kind enough to vouch for my expertise in electromagnetics. One of the Reviewers even quipped he would look forward to reading the completed book, so I humbly hope this scientific work will live up to his optimistic expectations. Prof. Andrew F. Peterson of Georgia Institute of Technology in Atlanta, GA, author of the book Computational Methods for Electromagnetics (IEEE Press, 1998) and Series Editor with IET Press, did me the honor of reviewing the manuscript and wrote an invaluable foreword which makes a convincing case for this work. His support, appreciation, and endorsement mean a lot to me and are gratefully acknowledged. Prof. Antonius G. Tijhuis, Emeritus, literally provided the opportunity for this book when, in 2011, he entrusted me with the responsibility for the lectures of the graduate course ‘Mathematics for Electromagnetism’ at the Technical University of Eindhoven, NL, while he good-naturedly offered to serve as my teaching assistant. The slides of the 2016-2017 edition of said course are the skeleton on which this resource got fleshed out, as it were, in a three-year-long effort. Prof. Renato Orta, Emeritus, was an excellent teacher of mine and also served as thesis adviser for both my graduation project and my doctoral dissertation. Under his authoritative guidance and thanks to his well-prepared lectures for the course ‘Optical Circuits and Components’ (which I attended in 1992), I developed a thorough and deep knowledge of waveguides, periodic structures, and spectral methods, all of which are described in Chapter 11.

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Prof. Vito Daniele, Emeritus, an expert in the Wiener-Hopf method, was another great teacher of mine who, during his graduate course ‘Advances in Electromagnetic Waves’, used to joke that: ‘Electromagnetics is a hard subject.’ And yet, his inspiring lessons, which were jam-packed with mathematical derivations, were crystal clear (I still have and treasure the notes I took) and taught me how to apply the theory of complex functions to electromagnetics. I utilized these invaluable concepts, for example, when discussing the 2-D Green function (Section 7.6) and the KramersKrönig relations (Section 12.2). Dr. Theodore Anderson, Haleakala Research and Development Inc., Brookfield, MA, author of the book Plasma Antennas (Artech House, 2011), kindly granted me permission to include pictures of his gaseous plasma antennas in Figure 13.23. Dr. Davide Melazzi conducted part of his doctoral research on plasma engines for spacecraft propulsion as a guest Ph. D. student at the Technical University of Eindhoven, NL, in 2011. Our pleasant and productive cooperation led to the development of ADAMANT, a FORTRAN code for the numerical analysis of coupled plasma and antennas with integral equations and the Method of Moments. The mathematical and computational aspects of that electromagnetic problem inspired the contents of Sections 13.5.1 and 15.5. In particular, Dr. Melazzi contributed the colorful artist’s impression of the plasma thruster reproduced, with permission, in Figure 13.21. Mr. Anuar D. J. Fernandez-Olvera was one of the most brilliant students of my course ‘Advanced Electromagnetics and Moments Methods’. In 2015, he conducted his graduation project, which entailed, for his part, the coding of a hybrid surface-integral-equation and wave-equation approach for the analysis of plasma antennas. The theoretical investigations stimulated by that activity constitute the subject of Sections 13.5.2, 15.6, and 15.7. Last but not least, I would like to pay tribute to my beloved late parents, Loreto and Angelina. They helped me become the person and the scientist I am today by never failing to sustain me with unconditional love, constant encouragement, and unshakable practical and emotional support. Unfortunately, though, my father, an honest and hard-working man, never had a chance to know of my moving to Eindhoven and of my writing this book because he passed away in 2002. My elderly mother, strong as a lioness, passed away suddenly and unexpectedly, due to the complications from a severe COVID-19 infection, in early April of 2021, just a few weeks after I had excitedly broken the news to her that IET Press would publish my work. Therefore, this book is for my father and my mother: I wish they could see it, but, most of all, I hope that, wherever they are now, they may be proud. Vito Lancellotti Torino, Italy November 25, 2021

Chapter 10

Integral formulas and equivalence principles

10.1 Integral representations with dyadic Green functions So far we have determined integral solutions to the time-harmonic Maxwell equations in a homogeneous unbounded isotropic medium with the aid of the retarded potentials (Section 9.4). In theory, an integral representation of time-harmonic fields in a bounded region of space may be obtained by combining decompositions (9.152) and (9.153) with the integral representations of the potentials (9.15), (9.16) and the dual ones for the potentials due to magnetic sources. Such procedure, though feasible, is quite involved as it entails the non-trivial calculation of the derivatives of a few surface integrals. One more downside is that the resulting formula will initially involve the values of the potentials, rather than the fields, on the boundary of the region of interest, and some more algebra will be needed to arrive at representations in terms of fields only. For all these reasons in this section we tackle the problem by working directly with the Maxwell equations in the frequency domain without the intervention of the electrodynamic potentials. The result we seek extends the integrals representations of static electric fields (Section 3.2) and stationary magnetic fields (Section 5.1.2). The method followed in those situations consisted of the ingenious combination of the original problem and an auxiliary one amenable to a closed-form solution. Here we exploit the same idea, but instead of starting from the wave equations (Section 1.8) we resort to employing the reciprocity theorem (6.253) since the latter establishes a global relationship between two different problems which, as we know, are referred to as state (a) and state (b). Since we wish to determine a vector field rather than a scalar one, we need to employ a vector source, such as a Hertzian dipole, pretty much as we did for the stationary magnetic fields examined in Section 5.1.2. This choice should work out nicely because we managed to determine the fields produced by electric or magnetic time-harmonic Hertzian dipoles in Example 9.3 and Section 9.7. There is a catch, though, because strictly speaking we cannot compute the reactions in (6.256) inasmuch as we derived the theorem by assuming that the sources were not singular. Of course we could interpret the result in a distributional sense and proceed unfazed. Alternatively, we carefully define the two states by excluding the location of the Hertzian dipole, as we did in Section 9.7. We consider a connected volume V1 ⊂ R3 filled with unknown matter (e.g., an anisotropic reciprocal medium) and immersed in an unbounded isotropic medium endowed with constitutive parameters ε and μ. We call V the finite region of space bounded by ∂V1 and a larger smooth closed surface S , whereby the boundary of V is ∂V := S ∪ ∂V1 . Electric and magnetic sources exists in a smaller volume V J ⊂ V. This configuration of sources and matter, exemplified in Figure 10.1, is the actual problem we wish to ‘solve’. To gain more generality and emphasize that the result is formally not affected by the properties of the medium which fills the complementary region R3 \ V we allow for other sources and other

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Figure 10.1 For the derivation of the integral representations with dyadic Green functions: electromagnetic problem to be solved. matter, possibly non-reciprocal, existing outside V. This provision is possible inasmuch as the reciprocity theorem is valid under such broader assumptions. The singular point-source is located in rb ∈ V \ V J , but it is isolated by means of a ball B(rb, b) where the radius b is small enough for the ball to be contained in V \ V J . Last but not least, to apply the reciprocity theorem (6.253) to the region V \ B[rb, b] we indicate the dummy integration variable with r because the latter corresponds, in effect, to the source points. The two states are defined as follows. State (a)

State (b)

The electric current densities J(r ; ω) and J M (r ; ω), r ∈ V J , are switched on and radiate in the whole space in the presence of the media in V1 as well as the non-reciprocal matter and the other sources in R3 \ (V ∪ V 1 ). Conversely, the singular source located in rb is turned off. The fields produced by J(r ; ω) and J M (r ; ω) are regular in rb . The electric current densities J(r ; ω) and J M (r ; ω) and any other sources in R3 \ (V ∪ V 1 ) are turned off, whereas the electric Hertzian dipole, located in rb and having moment j ωp, is switched on. The dipole exists in an unbounded homogeneous isotropic medium with parameters ε and μ. This setup (Figure 10.2) may be pictured as the result of ‘removing’ all other matter, inside and outside V, and ‘replacing’ it with the same underlying isotropic medium.

A few more remarks are in order before we write down the special instance of (6.256) for the states just defined. Contrary to the geometrical setup of Figure 6.21 we choose the unit normal to S ∪ ∂V1 ∪ ∂B directed inwards V \ B[rb, b]. This choice — which amounts to a change of sign of the flux integrals — will facilitate the interpretation of the final integral formula. The reaction fa , gb  is null because the Hertzian dipole — the source of state (b) — has been placed outside V \ B[rb , b]. We may use (6.256) in V \ B[rb , b] even in the presence of the sources in V J because the tangential components of the electric and magnetic fields are continuous across ∂V J (see discussion on page 374). All this been said, the reciprocity theorem applied to V \ B[rb, b] yields  − S ∪∂V1

ˆ  ) · [Ea (r ) × Hb (r ) − Eb (r ) × Ha (r )] dS  n(r

Integral formulas and equivalence principles

689

Figure 10.2 For the derivation of the integral representations with dyadic Green functions: auxiliary problem.  −

ˆ  ) · [Ea (r ) × Hb (r ) − Eb (r ) × Ha (r )] dS  n(r

∂B

 =

dV  [Eb (r ) · J(r ) − Hb (r ) · J M (r )] (10.1)

VJ

ˆ  ). where the minus signs before the flux integrals are a consequence of the adopted orientation for n(r This is our starting point for the derivation of an integral formula for the electric field E(r; ω) = Ea (r; ω) produced by J(r ; ω) and J M (r ; ω) and possibly the sources existing in R3 \ V. First of all, we compute the flux integral over ∂B in the limit as the radius b vanishes. This is the very same singular integral we handled in great details in Section 9.7. If we take the leading minus sign into account, the result may be written down straightaway with the help of (9.381)  ˆ  ) · [Ea (r ) × Hb (r ) − Eb (r ) × Ha (r )] = j ωp · Ea (rb ) (10.2) − lim+ dS  n(r b→0

∂B

which also clarifies the rationale for choosing an electric Hertzian dipole as an auxiliary problem. Indeed, the dipole plays the role of an ideal electric probe of sorts that, having infinitesimal spatial extension, measures or samples the electric field in the location rb . Inserting this result into (10.1) and rearranging the integrands yields  j ωp · Ea (rb ) =

dV  [J(r ) · Eb (r ) − J M (r ) · Hb (r )]

VJ

 contribution from sources inside V  ˆ  ) × Ha (r )] · Eb (r ) − [Ea (r ) × n(r ˆ  )] · Hb (r )} (10.3) + dS  {[n(r S ∪∂V1

 contribution from sources and matter outside V

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which constitutes an ‘explicit’ formula for the projection of Ea (rb ) onto the electric dipole moment j ωp. Therefore, to finalize the result we need to substitute for the fields produced by the Hertzian dipole and rid ourselves of the utilitarian p. Meanwhile, we notice that the right-hand side of (10.3) is comprised of two terms, i.e., a volume integral and a surface integral. Since the known sources of the original problem enter the volume integral over V J , the remaining surface integral must account for sources other than J(r ; ω) and J M (r ; ω) and necessarily located in the complementary domain R3 \ V. We shall show that this contribution is null so long as all the sources are confined within V J ⊂ V and the medium filling the complementary region R3 \ V has the same constitutive parameters as the medium in V. Nonetheless, ˆ  ) and n(r ˆ  )×Ha (r ) are zero for r ∈ S . this does not mean that the tangential components Ea (r )× n(r In fact, we already reached similar conclusions for the integral representations of ΦE (r, t), AE (r, t), ΦE (r; ω), AE (r; ω) as well as static and stationary fields. To obtain the fields produced by the electric Hertzian dipole in an unbounded isotropic medium we resort to (9.370) and (9.371), namely, 

  ∇ ∇ e− j k|r −rb | ·p Eb (r ; ω) = ω μGEJ (r , rb ; ω) · p = ω μ I + 2 k 4π|r − rb |  e− j k|r −rb | Hb (r ; ω) = j ωGH J (r , rb ; ω) · p = j ω∇ ×p 4π|r − rb | 



2

2

(10.4) (10.5)

having noticed that the independent variable (r) in the present setup has been called r and that the position of the dipole is rb . Next, we perform a little algebra on the integrands of (10.3), viz.,

T J(r ) · Eb (r ) = ω2 μJ(r ) · GEJ (r , rb ; ω) · p = ω2 μp · GEJ (r , rb ; ω) ·J(r) = − j ωμ(j ωp) · GEJ (rb , r ; ω) · J(r )



T

(10.6)

−J M (r ) · Hb (r ) = − j ωJ M (r ) · GH J (r , rb ; ω) · p = − j ωp · GH J (r , rb ; ω) ·J M (r ) = j ωp · GEM (rb , r ; ω) · J M (r )

(10.7)

and similarly, ˆ  ) × Ha (r )] · Eb (r ) = − j ωμ(j ωp) · GEJ (rb , r ; ω) · [n(r ˆ  ) × Ha (r )] [n(r 











ˆ )] · Hb (r ) = j ωp · GEM (rb , r ; ω) · [Ea (r ) × n(r ˆ )] −[Ea (r ) × n(r

(10.8) (10.9)

where we have employed the symmetry relationships (9.384) and (9.395). By substituting these expressions back into (10.3) and dividing through by j ω we find  p · Ea (rb ) = − j ωμp ·  − j ωμp ·









dV GEJ (rb , r ; ω) · J(r ) + p · VJ

dV  GEM (rb , r ; ω) · J M (r )

VJ

ˆ  ) × Ha (r )] dS  GEJ (rb , r ; ω) · [n(r 

S

+p·

ˆ  )] dS  GEM (rb , r ; ω) · [Ea (r ) × n(r

(10.10)

S

and, since the dipole moment p and the position thereof are arbitrary, (10.10) can provide the Cartesian components of Ea (rb ) if we take p ∈ {ˆx, yˆ , zˆ } at any point rb ∈ V \ V J . Hence, we collect

Integral formulas and equivalence principles

691

the resulting three scalar expressions, rename rb to r and drop the subscript ‘a’ everywhere to get  E(r) = − j ωμ  − j ωμ S ∪∂V1

VJ

    − j k|r−r | ∇ ∇ e− j k|r−r |    e · J(r × J M (r ) dV  I + 2 ) + dV ∇ k 4π|r − r | 4π|r − r | VJ

   ∇ ∇ e− j k|r−r | ˆ  ) × H(r )] · [n(r dS  I + 2 4π|r − r | k  +



dS  ∇

S ∪∂V1

e− j k|r−r | ˆ  )] (10.11) × [E(r ) × n(r 4π|r − r |

which is the desired integral representation of E(r) for observation points r ∈ V \ V J . Before we proceed with the discussion we write down the analogous integral formula for the magnetic field under the same assumptions of state (a) above. It should be clear that the detailed calculations call for a magnetic Hertzian dipole of moment j ωμm placed at rb . However, we may simply invoke the duality principle and transform (10.11) into its dual counterpart with the aid of Table 6.1, i.e.,  H(r) = − j ωε  − j ωε S ∪∂V1

dV VJ

dS 







   − j k|r−r | ∇ ∇ e− j k|r−r |    e I+ 2 (r ) − dV ∇ · J × J(r ) M 4π|r − r | 4π|r − r | k VJ

  ∇ ∇ e− j k|r−r | ˆ  )] I+ 2 · [E(r ) × n(r 4π|r − r | k  − S ∪∂V1



dS  ∇

e− j k|r−r | ˆ  ) × H(r )] (10.12) × [n(r 4π|r − r |

still for observation points r ∈ V \ V J [1, 2]. In words, (10.11) and (10.12) state that electric and magnetic fields are due to the actual sources confined in V J but also to additional terms that take into account whatever sources and matter may exist in the complementary region. In fact, the flux integrals over S ∪ ∂V1 have been rephrased as contributions which involve the tangential components of E(r ) and H(r ) on S ∪ ∂V1 as well as the very same dyadic Green functions as in the volume integral over V J . Therefore, we argue that ˆ  )×H(r) and E(r )× nˆ (r ) play the role of equivalent surface current densities flowing on S ∪∂V1 , n(r namely, ˆ  ) × H(r ; ω), JS (r ; ω) := n(r ˆ  ), J MS (r ; ω) := E(r ; ω) × n(r

r ∈ S ∪ ∂V1 r ∈ S ∪ ∂V1

(10.13) (10.14)

with physical dimensions of A/m and V/m, respectively. To be fastidious, the surface currents (10.13) and (10.14) are set on the positive side of S ∪ ∂V1 and hence within V. Besides, JS (r ; ω) and J MS (r ; ω) vanish in the presence of PEC or PMC materials, respectively, flush with the boundary S ∪ ∂V1 . We should not refer to (10.11) and (10.12) as the solution to the Maxwell equations (1.98)(1.101) because, in general, the surface currents JS (r ; ω) and J MS (r ; ω) are unknown, and assigning them arbitrarily and independently may cause the formulas to return a non-physical electromagnetic field that does not solve the Maxwell equations for r ∈ V. Thus, as is the case with all the integral representations obtained so far, (10.11) and (10.12) constitute just a formal result which comes in

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Figure 10.3 For the derivation of the integral representation with dyadic Green functions: electric Hertzian dipole outside the region of concern. handy, e.g., in formulating scattering problems with integral equations (Chapter 13). Still, depending on the problem under investigation, one may sometimes come up with a reasonable approximation for JS (r ; ω) and J MS (r ; ω), a classic example being the radiation from apertures cut in an infinite PEC plane [3, Chapter 17], [4, Section 9.6], in which case (10.11) and (10.12) can indeed provide an estimate of the true electromagnetic field. What happens if we try and use (10.11) and (10.12) for observation points in the complementary domain R3 \ V? The lessons learned in Sections 2.7, 5.1 etc. should alert us to expect a null field as a result. To ascertain whether this is the case, we apply the reciprocity theorem again with essentially the same states (a) and (b) considered at the beginning. The only modification concerns the position of the Hertzian electric dipole, which we place in rb ∈ R3 \ V, as is pictorially suggested in Figure 10.3. While we may still isolate the dipole with a ball B(rb , b), this step is not necessary inasmuch as the field produced by the dipole is regular within V, the region of interest. Consequently, we may apply (6.256) to V as a whole to get  ˆ  ) · [Ea (r ) × Hb (r ) − Eb (r ) × Ha (r )] − dS  n(r S ∪∂V1

 =

dV  [Eb (r ) · J(r ) − Hb (r ) · J M (r )]

(10.15)

VJ

where it is important to notice that both the flux integral and the integral over V J are formally the same as those in (10.1). The actual value may different, of course, for it depends on the position of the dipole. Proceeding as before we arrive at the identity      − j k|r−r | ∇ ∇ e− j k|r−r |     e · J(r × J M (r ) ) + dV ∇ 0 = − j ωμ dV I + 2 4π|r − r | 4π|r − r | k 

VJ

− j ωμ S ∪∂V1

dS 



VJ

  ∇ ∇ e− j k|r−r | ˆ  ) × H(r )] · [n(r I+ 2 4π|r − r | k

Integral formulas and equivalence principles  +

693



dS  ∇

S ∪∂V1

e− j k|r−r | ˆ  )] (10.16) × [E(r ) × n(r 4π|r − r |

precisely because the direct contribution of the Hertzian dipole is lacking in the left-hand side. But then, we realize that the terms in the right member have the same form as those in (10.11). Hence, simple comparison of the equations side by side allows us to conclude that (10.11) returns a null electric field when evaluated for points r  V. The same observation applies to (10.12) which is the dual of (10.11). This remarkable result — sometimes referred to as an extinction theorem [3] — implies that the four integrals in the right-hand sides of the representations combine in such a way that they cancel each other out. The fields generated by the true sources J(r; ω) and J M (r; ω) are neutralized, as it were, by the fields produced by the equivalent currents (10.13) and (10.14) on the boundary S ∪ ∂V1 . We emphasize, though, that the ‘true’ electromagnetic field produced by J(r ) and J M (r ) for points outside V is not necessarily null! When we say ‘fields generated’ by true or equivalent currents we mean in the absence of the matter in R3 \ V. Indeed, the integral representations derived above involve the dyadic Green functions (9.159), (9.161), (9.162) and (9.163), which are fundamental solutions in a homogeneous isotropic unbounded medium (cf. Figure 10.2). We may also look at (10.11) and (10.16) from the viewpoint of the matching conditions derived in Sections 1.7 and 6.6, though applied to time-harmonic fields [5, 6]. In this regard, we have to interpret the boundary S ∪ ∂V1 as the interface between two media, say, medium 1 filling V, and medium 2, filling the complementary domain. According to the representation formulas, both electric and magnetic fields in medium 2 are identically zero. However, the fields in V and in particular on the positive side of S ∪ ∂V1 do not vanish. Thus, the jump condition (1.196) and the time-harmonic analogue of (6.215) reduce to (10.13) and (10.14). Further, a quick glance at (1.168) and (6.227) suggests that medium 2 has the properties of a fictitious material which is simultaneously both PEC and PMC. The equivalent surface current densities ‘appear’ on S ∪ ∂V1 because the true electric and magnetic fields do not suffer any jump at all across the boundary of V.

10.2 The integral formulas of Stratton and Chu In deriving (10.11) and (10.12) we have favored the role of the dyadic Green functions — which has a few advantages. For one thing the approach allows obtaining quite compact expressions with very little effort thanks to the symmetry properties discussed in Section 9.7. Secondly, (10.11) and (10.12) lend themselves to generalizations based on the usage of other dyadic Green functions. Finally, the surface integrals in (10.11) and (10.12) involve just the tangential components of E(r ) and H(r ) on the boundary S ∪ ∂V1 , a feature which is somehow consistent with the conditions for uniqueness derived in Section 6.4.1. Nevertheless, (10.11) and (10.12) as they stand cannot be extended for observation points in the source region V J or on S ∪ ∂V1 , essentially because the dyadic Green functions GEJ (r, r ) and GH M (r, r ) exhibit a singularity of the type 1/|r − r |3 , which cannot be integrated over a volume [cf. (9.169) and (9.190)], let alone a surface. It is possible, however, to repeat the procedure of Section 10.1 by starting with a dipole placed in rb ∈ V J and isolated with a ball B(rb , b) ⊂ V J . In the limit as the radius b approaches zero the volume integral over V J \ B(rb, b) passes over into its Cauchy principal value, whereas with a little algebra the surface integral over ∂B can be shown to reduce to the field Ea (rb ) and a term which is the direct contribution of J(rb ) through the depolarizing dyadic (9.197) for a sphere. The final formula generalizes (9.204) with a boundary term over S ∪∂V1 . Alternatively, here we go on to manipulate and transform the integrands so that only less singular

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terms remain. We outline the details for the electric field given by (10.3), as we can retrieve the magnetic field through the duality principle. To lighten the notation a bit we indicate the threedimensional scalar Green function with G(r , rb ) as in (8.356). For the volume integral over V J we have   ∇ ∇   2  J(r ) · Eb (r ) = ω μJ(r ) · I + 2 G(r , rb ) · p k 1   2   = J(r ) · k I + ∇ ∇ G(r , rb ) · p ε 1 k2 = p · J(r )G(rb , r ) + J(r ) · ∇ [p · ∇G(rb , r )]  ε ε scalar field

k2 1 = p · J(r )G(rb , r ) + ∇ · [J(r )p · ∇G(rb , r )] ε ε 1  − ∇ · J(r ) p · ∇G(rb , r ) ε 2 k jω  ρ(r ) p · ∇ G(rb , r ) = p · J(r )G(rb , r ) + ε ε 1 + ∇ · [J(r )p · ∇G(rb , r )] ε −J M (r ) · Hb (r ) = − j ωJ M (r ) · ∇ G(r , rb ) × p = j ωp · ∇G(rb , r ) × J M (r )

(10.17) (10.18)

where we have exploited the fact that the static dipole moment p is a constant vector and used (H.51). Besides, G(r , rb ) is obviously symmetric in its two arguments, and the continuity equation (1.46) holds for r ∈ V J . The last term in the rightmost-hand side of (10.17) is in the form of the divergence of a vector field for r ∈ V J , and we may apply the Gauss theorem so long as rb ∈ V \ V J and J(r ) ∈ C1 (V J )3 ∩ C(V J )3 . If the divergence of J(r )p·∇ G is only piecewise continuous, one invokes the Gauss theorem separately in each region where continuity holds and requires that nˆ · J be continuous across the interfaces. Regardless, we arrive at a flux integral over ∂V J that vanishes because J(r ) is confined ˆ  ) · J(r ) = 0 for r ∈ ∂V J . Moreover, we already know that the singularities of to V J and hence n(r  G(r , rb ) and of the gradient thereof are integrable over a volume. We proceed with the integrand of the flux integral over S ∪ ∂V1 , viz., ˆ  ) × Ha (r )] · Eb (r ) = [n(r  1 ˆ  ) × Ha (r )] · k2 I + ∇ ∇ G(r , rb ) · p = [n(r ε 1  k2 ˆ  ) × Ha (r )]G(rb , r ) + n(r ˆ ) · Ha (r ) × ∇ [∇G(rb , r ) · p] = p · [n(r  ε ε scalar field

k2 1  ˆ  ) × Ha (r )]G(rb , r ) − n(r ˆ ) · ∇ × [Ha (r )∇G(rb , r ) · p] = p · [n(r ε ε 1 ˆ  ) · ∇ × Ha (r )]∇G(rb , r ) · p + [n(r ε k2 ˆ  ) × Ha (r )]G(rb , r ) + j ωp · ∇ G(rb , r )[n(r ˆ  ) · Ea (r )] = p · [n(r ε 1  ˆ ) · ∇ × [Ha (r )∇G(rb , r ) · p] − n(r ε

(10.19)

Integral formulas and equivalence principles ˆ  )] · Hb (r ) = − j ω[Ea (r ) × n(r ˆ  )] · ∇G(r , rb ) × p −[Ea (r ) × n(r  ˆ  )] × ∇G(rb , r ) = − j ωp · [Ea (r ) × n(r ˆ  )] = j ωp · ∇G(rb , r ) × [Ea (r ) × n(r

695

(10.20)

having used (H.50) and the source-free local Ampère-Maxwell law (1.98) because the surface S ∪∂V1 intercepts no sources by hypothesis. When the last term in the rightmost member of (10.19) is integrated, it gives rise to the flux of the curl of a vector field through S ∪ ∂V1 , and hence we would like to apply the Stokes theorem (A.55). This is not an issue provided Ha (r ) is continuous over S ∪ ∂V1 and rb is placed away from the boundary. As a result, since S and ∂V1 are closed surfaces, this term contributes naught. By inserting (10.17)-(10.20) into (10.3) we obtain   k2 jω    p · dV  ρ(r )∇G(rb , r ) j ωp · Ea (rb ) = p · dV J(r )G(rb , r ) + ε ε VJ VJ   2 k ˆ  ) × Ha (r )G(rb , r ) + j ωp · dV  ∇G(rb , r ) × J M (r ) + p · dS  n(r ε VJ S ∪∂V1  ˆ  ) · Ea (r )∇G(rb , r ) dS  n(r + j ωp · S ∪∂V1

 + j ωp ·

ˆ  )] (10.21) dS  ∇ G(rb , r ) × [Ea (r ) × n(r

S ∪∂V1

which we finalize by dividing through by j ω and removing the dependence on p. Dropping the inessential subscript a and renaming rb to r yields   1    E(r) = − j ωμ dV J(r )G(r, r ) + dV  ρ(r )∇G(r, r ) ε VJ V  J ˆ  ) × H(r )G(r, r ) + dV  ∇G(r, r ) × J M (r ) − j ωμ dS  n(r VJ

S ∪∂V1





ˆ ) · E(r )∇G(r, r ) + dS n(r

+



S ∪∂V1



ˆ  )] (10.22) dS  ∇ G(r, r ) × [E(r ) × n(r

S ∪∂V1

which represents the classic integral formula derived first by J. A. Stratton and L. J. Chu [7], [8, Section IV.10], [9, Section 6.2], [1–3,10] for r ∈ V \ V J . The analogous formula for the magnetic field reads   1    H(r) = − j ωε dV J M (r )G(r, r ) + dV  ρ M (r )∇G(r, r ) μ VJ VJ   ˆ  )G(r, r ) dS  E(r ) × n(r − dV  ∇G(r, r ) × J(r ) − j ωε VJ

 +

S ∪∂V1 



ˆ ) · H(r )∇G(r, r ) − dS n(r

S ∪∂V1

by virtue of the duality transformations of Table 6.1.



ˆ  ) × H(r )] (10.23) dS  ∇ G(r, r ) × [n(r

S ∪∂V1

Advanced Theoretical and Numerical Electromagnetics

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In words, the Stratton-Chu integral representations (10.22) and (10.23) yield the electromagnetic field at the interior points of a bounded region V in terms of: 1) the true known electric and magnetic sources located in V and 2) the value of E(r ) and H(r ) over the boundary ∂V. Unlike (10.11) and (10.12), (10.22) and (10.23) involve both current and charge densities. The surface integrals must account for whatever matter and sources exists outside V. In particular, the normal components of the fields over ∂V may be interpreted as equivalent surface charge densities, namely, ˆ  ) · E(r ), ρS (r ) := εn(r ˆ  ) · H(r ), ρ MS (r ) := μn(r

r ∈ S ∪ ∂V1 r ∈ S ∪ ∂V1

(10.24) (10.25)

which we may show to be related to JS (r ) and J MS (r ) defined in (10.13) and (10.14). For instance, we have ˆ  ) × H(r )] = −n(r ˆ  ) · ∇ × H(r ) ∇s · JS (r ) := ∇s · [n(r ˆ  ) · j ωεE(r ) := − j ωρS (r ) = −n(r

(10.26)

on account of (A.60) and the source-free Ampère-Maxwell law (1.98). The relation for the magnetic densities follows in like manner or by invoking duality. These connections make the usage of (10.22) and (10.23) all the more difficult because, once the tangential components of the fields have been assigned on the boundary, the remaining normal components must be determined consistently so ˆ  ) × H(r ) and that the Maxwell equations are satisfied. We have already mentioned that even n(r   ˆ ) are not independent of one another, although we know that in lossless bounded regions E(r ) × n(r ˆ  ) × H(r ) or E(r ) × n(r ˆ  ) on the boundary is not sufficient to ensure uniqueness specifying either n(r of the solutions to the time-harmonic Maxwell equations (Section 6.4.1). Since (10.22) and (10.23) are just an alternative form of (10.11) and (10.12) we take for granted that the Stratton-Chu formulas return zero when evaluated for observation points in the complementary domain R3 \ V. Therefore, we go on to extend the result for points r ∈ ∂V and r ∈ V J . We apply the reciprocity theorem again with the state (a) graphically shown in Figure 10.1, whereas in order to ‘sample’ the electric field Ea (r ) in a point on the boundary ∂V we re-define the state (b) by placing the electric Hertzian dipole in rb ∈ S , as is suggested in Figure 10.4. Then, we isolate the dipole with a ball B(rb , b) which is only partly contained in V. In this configuration the boundary of the finite region V \ B[rb, b] is comprised of three parts: the closed surface ∂V1 , the open surface S  := {r ∈ S : |r − rb |  b}, and the open surface S  := ∂B ∩ V. Having carefully excluded the singular source, we may employ (6.256) on V \ B[rb, b], viz.,  ˆ  ) · [Ea (r ) × Hb (r ) − Eb (r ) × Ha (r )] − dS  n(r ∂V1



−

ˆ  ) · [Ea (r ) × Hb (r ) − Eb (r ) × Ha (r )] dS  n(r

S



−

ˆ  ) · [Ea (r ) × Hb (r ) − Eb (r ) × Ha (r )] dS  n(r

S 

 =

dV  [Eb (r ) · J(r ) − Hb (r ) · J M (r )]

(10.27)

VJ

and we wish to take the limit for vanishing b. Since we have supposed V J ⊂ V, the volume integral is not affected by the process. Likewise, the contribution of the integral over ∂V1 does not depend on b.

Integral formulas and equivalence principles

697

Figure 10.4 For the derivation of the Stratton-Chu integral representation: electric Hertzian dipole on the boundary of the region of interest. For the integral over S  we resort to (10.19) and (10.20) to obtain  −

ˆ  ) · [Ea (r ) × Hb (r ) − Eb (r ) × Ha (r )] dS  n(r

S



ˆ ) dS  G(rb , r )Ea (r ) × n(r

= − j ωp · ∇rb × 2

k − p· ε

S













ˆ ) × Ha (r )G(rb , r ) + j ωp · ∇rb dS n(r S

1 + ε



ˆ  ) · Ea (r )G(rb , r ) dS  n(r

S

ˆ  ) · ∇ × [Ha (r )∇G(rb , r ) · p] (10.28) dS  n(r

S

where we have used ∇ G(rb , r ) = −∇rb G(rb , r ) and interchanged the order of integration and differentiation. This step — which is crucial to ensure the finiteness of the relevant contributions in the limit as b → 0+ — is permitted because G(rb , r ) is not singular at all for r ∈ S  . For the integral over S  ⊂ V we have  −

ˆ  ) · [Ea (r ) × Hb (r ) − Eb (r ) × Ha (r )] dS  n(r

S 



= − j ωp · S 

 1 ˆ  ) × Ha (r )G(rb , r ) ˆ  ) · ∇ G(rb , r ) − p · dS  n(r dS  Ea (r )n(r ε S   1 ˆ  ) · ∇ × [Ha (r )∇G(rb , r ) · p] (10.29) dS  n(r + ε S 

Advanced Theoretical and Numerical Electromagnetics

698

thanks to manipulations analogous to those performed to obtain (9.374). Here it is not necessary to move the derivatives outside the integral in that ˆ  ) · ∇ G(rb , r )dS  = (1 + j kb) −n(r

e− j kb  dΩ , 4π

r ∈ S 

(10.30)

which remains finite as b → 0+ . Then, with the aid of the mean value theorem we compute  Ω(b) ˆ  ) · ∇G(rb , r ) = j ωp · Ea (r0 )(1 + j kb)e− j kb (10.31) − j ωp · dS  Ea (r )n(r 4π S 

where r0 ∈ S  is a suitable point and Ω(b) is the solid angle subtended by S  with respect to rb . Since S is smooth by hypothesis, Ω(b) approaches 2π in the limit as b → 0+ (Appendix F.2). Besides, ˆ  ) × Ha (r ) as follows we estimate the integral of n(r





 c1   

p · dS  n(r ˆ ) × Ha (r )G(rb , r )   c1 b dS  (10.32)



4πb S 

S 

where c1 is a suitable positive constant. Finally, we notice that the last contributions in the right members of (10.28) and (10.29) combine to give the flux of the curl of a vector through the closed surface S  ∪ S  and hence the final result is zero on account of either the Stokes theorem or the Gauss theorem. It is paramount we carry out this step prior to taking the limit for vanishing b. Now, plugging (10.28) and (10.29) into (10.27), using (10.17) and (10.18) for the reaction fb , ga  and letting b → 0+ yields   jω k2 jω    p · Ea (rb ) = p · dV J(r )G(rb , r ) + p · dV  ρ(r )∇G(rb , r ) 2 ε ε VJ VJ   2 k ˆ  ) × Ha (r )G(rb , r ) + j ωp · dV  ∇G(rb , r ) × J M (r ) + p · dS  n(r ε VJ S ∪∂V1  ˆ  ) · Ea (r ) dS  G(rb , r )n(r − j ωp · ∇rb S ∪∂V1



ˆ ) dS  G(rb , r )Ea (r ) × n(r

− j ωp · ∇rb ×

(10.33)

S ∪∂V1

whence the desired formula follows by dividing by j ω, eliminating p, renaming rb as r and dropping the inessential subscript a, viz.,   1 1 E(r) = − j ωμ dV  J(r )G(r, r) + dV  ρ(r )∇G(r, r ) 2 ε VJ V   J ˆ  ) × H(r )G(r, r ) + dV  ∇G(r, r ) × J M (r ) − j ωμ dS  n(r VJ

 −∇

S ∪∂V1

ˆ  ) · E(r ) − ∇ × dS  G(r, r )n(r

S ∪∂V1



ˆ ) dS  G(r, r )E(r ) × n(r

S ∪∂V1

(10.34)

Integral formulas and equivalence principles

699

which is valid for r ∈ ∂V. The magnetic field representation follows through the duality principle. This proves that the Stratton-Chu formulas (10.22) and (10.23) return half the value of the field if the observation point is set on the smooth boundary of the region of interest. All of the surface integrals involve just the Green function — as the derivatives have been left out — and exist. We can show that (10.22) and (10.23) remain valid for observation points within the source region V J . Intuitively, this is to be expected inasmuch the time-harmonic Green function (8.356) and the static counterpart (2.131) exhibit the same singular behavior for r = r. We just need to prove that the volume integrals over V J remains bounded. To this purpose we assume that an electric Hertzian dipole is positioned in rb = r ∈ V J and excluded from the domain of concern by means of a ball B(r, b) (cf. Figures 2.9 and 5.2 for a similar setup) where the radius b is chosen so that B(r, b) ⊂ V J . An application of (6.256) together with a careful revision of (10.19) for points r ∈ ∂B ⊂ V J leads essentially to (10.22) save for the domain of the volume integrals, which now extend over Vb := V J \ B[r, b]. We consider a larger ball B(r, a) whose radius is such that V J ⊆ B(r, a) and by virtue of (9.167) make the following three estimates







 J ∞

dV  G(r, r )J(r )

  J ∞  dV dV 



4πR 4πR

Vb

Vb B(r,a)\B(r,b) a = J ∞

dR R =

1 1 J ∞ (a2 − b2 )  J ∞ a2 2 2

(10.35)

b







 c2

dV  ∇ G(r, r )ρ(r )

  c2 dV ρ  dV  ρ ∞ ∞



2 2 4πR 4πR

Vb

Vb B(r,a)\B(r,b) a = c2 ρ ∞

dR = c2 ρ ∞ (a − b)  c2 ρ ∞ a

(10.36)

b







 c2

dV  ∇ G(r, r ) × J (r )

  c2 dV J M ∞  dV  J M ∞ M



2 2 4πR 4πR

Vb

Vb B(r,a)\B(r,b) a = c2 J M ∞

dR = c2 J M ∞ (a − b)  c2 J M ∞ a

(10.37)

b

and these prove that the integrals are dominated by constants for any value of b and in particular for b → 0+ . The integrals involved in (10.23) are dual to those treated above and thus are likewise bounded. We are now ready to prove that the surface integrals over S ∪ ∂V1 do represent the effect of sources and media existing outside the region of interest V, as was mentioned in conjunction with (10.3). With reference to Figure 10.1, we expect that the surface integrals should vanish if J(r ) and J M (r ) are the only sources of the problem and they exist in an unbounded homogeneous medium. Put another way, the latter condition means that no other matter with different constitutive parameters is allowed for in the complementary domain R3 \ V.

Advanced Theoretical and Numerical Electromagnetics

700

As regards the contribution of ∂V1 we observe that (10.22) may be applied to the volume V1 and, since the latter contains no currents by hypothesis, the Stratton-Chu formula for the electric field yields       ˆ ) × H(r )G(r, r ) − dS  n(r ˆ  ) · E(r )∇G(r, r ) E(r) = j ωμ dS n(r ∂V1

∂V1

 −

ˆ  )] dS  ∇G(r, r ) × [E(r ) × n(r

(10.38)

∂V1

where we have changed sign to the surface integrals because the unit normal points outwards V1 . We already know that the Stratton-Chu formulas return a null value for the fields when evaluated for observation points located outside the region of interest. As a consequence, the right-hand side of (10.38) vanishes if we set r ∈ V. But then, under the same circumstances the right-hand side of (10.38) is identical with the negative of the combination of surface integrals appearing in (10.22) also in light of the continuity conditions (1.196), (1.197) and (1.198). Therefore, by comparison the surface integrals over ∂V1 in (10.22) are null, as anticipated. Notice that this conclusion holds because the materials in V and V1 have been assumed to have the same properties. Were this not case, the Green functions in V and V1 would evidently be different — as they depend on the constitutive parameters through the wavenumber k — and hence we might not deduce that the surface integrals over ∂V1 in (10.22) vanish. In practice, the presence of a material body in V1 is tantamount to having actual sources within V1 . We shall elaborate this point of view in the context of the volume equivalence principle in Section 10.5. To reach the same conclusion for the integral over S we place the observation point r somewhere within the region V and introduce the ball B(r, a) whose radius is large enough for the ball to enclose V ∪ V1 . When we apply the Stratton-Chu integral representation of the electric field to the surfacewise multiply-connected domain B(r, a) \ (V ∪ V 1 ), which is devoid of sources, we get   ˆ  ) × H(r )G(r, r) − ˆ  ) · E(r )∇G(r, r ) dS  n(r dS  n(r 0 = j ωμ S ∪∂B

S ∪∂B

 −

ˆ  )] dS  ∇G(r, r ) × [E(r ) × n(r

(10.39)

S ∪∂B

ˆ  ) points outwards B(r, a) for r ∈ ∂B and inwards V for where we have taken into account that n(r  r ∈ S . The sum of integrals in the right-hand side returns zero because the observation point is outside B(r, a) \ (V ∪ V 1 ) by hypothesis. From (10.39) we derive the following identity   ˆ  ) × H(r )G(r, r ) + dS  n(r ˆ  ) · E(r )∇G(r, r ) − j ωμ dS  n(r  +

S



S 









ˆ )] = j ωμ dS ∇ G(r, r ) × [E(r ) × n(r 

S

−

ˆ  ) × H(r )G(r, r ) dS  n(r

∂B 











ˆ ) · E(r )∇ G(r, r ) − dS n(r ∂B

ˆ  )] dS  ∇ G(r, r ) × [E(r ) × n(r

(10.40)

∂B

and observe that the left member is identical with the combination of integrals over S in (10.22) for r ∈ V. Thus, we just need to prove, if possible, that the right-hand side of (10.40) vanishes in the

Integral formulas and equivalence principles limit as a → +∞. To this purpose we observe that for r ∈ ∂B 

ˆ ˆ ) = −R, n(r

e− j ka , G(r, r ) = 4πa 



 1 e− j ka ˆ ∇ G(r, r ) = j k + R a 4πa 



701

(10.41)

whereby we have ˆ  )G(r, r ) − n(r ˆ  ) · E(r )∇G(r, r ) + ∇G(r, r ) × [n(r ˆ  ) × E(r )] = − j ωμH(r ) × n(r ˆ · E(r )∇ G(r, r ) ˆ × H(r )G(r, r ) + R = − j ωμR ˆ G(r, r ) · E(r ) + E(r )∇G(r, r ) · R ˆ − R∇   − j ka − j ka ˆ · E(r ) j k + 1 e ˆ ˆ × H(r ) e +R R = − j kZ R 4πa a 4πa     1 e− j ka ˆ 1 e− j ka ˆ ˆ   ˆ − R jk + R · E(r ) + E(r ) j k + R·R a 4πa a 4πa − j ka

e− j ka  ˆ × H(r ) + e E(r ) = jk E(r ) − Z R 4πa 4πa2 − j ka e− j ka e− j ka ˆ +e [E(r ) − ZH(r ) × rˆ  ] + j k ZH(r ) × (ˆr + R) = jk E(r ) (10.42) 4πa 4πa 4πa2 which we may use to estimate the right-hand side of (10.40). Indeed, if we demand that the first term in the rightmost member fall off with a rapidly enough for the relevant surface integral over ∂B to vanish, we arrive at the Silver-Müller condition (6.171). Before we proceed, though, we also derive ˆ namely, an estimate for the vector rˆ  + R, 2 



r2 r2 ˆ

2 = 2 1 + rˆ  · R ˆ  r 

ˆr + R  , Rr a(a − r) (a − r)2

a>r

(10.43)

whence

ˆ



ˆr + R

r , a>r (10.44) a−r having employed (9.18). By integrating the rightmost-hand side of (10.42) over ∂B we obtain three integrals which we examine separately. From the first term we find





 − j ka

k



j k dS  e   

E(r ) − ZH(r ) × rˆ 

[E(r ) − ZH(r ) × rˆ ]  dS 

4πa 4πa

∂B ∂B   M k M k aM  dS   dS  =k −−−−−→ 0 (10.45) 4πa r2 4πa (a − r)2 (a − r)2 a→+∞ ∂B

∂B

where M > 0 is a suitable constant. This result provides the mathematical basis for the Silver-Müller condition (6.171). From the second term we have





 − j ka 

kZ

j k dS  e   ˆ

 ˆ

ˆ r ZH(r |H(r )|

rˆ  + R ) × + R dS 

4πa 4πa

∂B ∂B   kZ M r kZ Mr aZMr   dS  dS  =k −−−−−→ 0 (10.46)  2 4πa r a − r 4πa (a − r) (a − r)2 a→+∞ ∂B

∂B

702

Advanced Theoretical and Numerical Electromagnetics

thanks to (10.44) and (6.156). Finally, from the third term we get







 − j ka 1 1 M

dS  e 

  E(r )  dS |E(r )|  dS 



2 2 2 r 4πa 4πa 4πa

∂B

∂B ∂B  M 1 M = −−−−−→ 0  dS  2 4πa a − r a − r a→+∞

(10.47)

∂B

by virtue of (6.155). In view of identity (10.40) these findings tell us that the combination of integrals over S in (10.22) vanishes when r ∈ V under the stated hypotheses. The result for the magnetic field and the other Silver-Müller condition (6.172) follow through similar steps or straightaway by invoking duality. In summary, for sources in a homogeneous medium the Stratton-Chu formulas are comprised only of the contributions from J(r ) and J M (r ), r ∈ V J . Besides, since (10.40) implies that the shape of the boundary S is inconsequential for the integral representations, we may as well choose S ≡ ∂B and let the radius a approach infinity in order to extend (10.22) and (10.23) to the whole space, viz.,  E(r) = − j ωμ dV  J(r )G(r, r ) VJ

+

1 ε



dV  ρ(r )∇G(r, r ) +

VJ

 H(r) = − j ωε



dV  ∇G(r, r ) × J M (r )

(10.48)

VJ

dV  J M (r )G(r, r )

VJ

1 + μ













dV ρ M (r )∇ G(r, r ) − VJ

dV  ∇ G(r, r ) × J(r )

(10.49)

VJ

for points r ∈ R3 . It is important to realize that (10.48) and (10.49) contain only known sources and thus they constitute the solution to the time-harmonic Maxwell equations in a homogeneous unbounded medium. As a matter of fact, it is possible to show that (10.48) and (10.49) are equivalent to (9.156) and (9.157) which were obtained with the aid of the auxiliary electrodynamic potentials.

10.3 Integral formulas with Kottler’s line charges While deriving the Stratton-Chu formulas (10.22), (10.23) and also (10.34) we repeatedly invoked the Stokes theorem on a closed surface in order to simplify the integrals that arose from the application of the reciprocity theorem. The strategy worked out nicely thanks to the assumption that the magnetic field and, by duality, the electric one are continuous over the boundary ∂V := S ∪ ∂V1 of the region of interest (Figure 10.1). As a matter of fact, there should be no need to state such condition again, because the continuity of E and H over ∂V is a requirement for the validity of the reciprocity theorem (6.256), which in turn is based on the Gauss theorem applied to V. However, to model a certain class of scattering problems, e.g., the propagation of waves through perforated metallic screens, it is expedient to allow for discontinuities in the fields over S ∪ ∂V1 . In which instance, the reciprocity and the Stokes

Integral formulas and equivalence principles

703

Figure 10.5 For the derivation of the Stratton-Chu formulas with field discontinuities across the closed line γ on the boundary S : electromagnetic problem to be solved. theorems may not be used directly in V and on S ∪ ∂V1 , respectively, and hence the procedure that led us to (10.22) and (10.23) must be revised. For the sake of argument we suppose that Ha (r ) suffers a jump across a smooth closed line γ ⊂ S , as is suggested in Figure 10.5. Further, we pick up a smooth surface S γ ⊂ V with boundary the line γ and assume for simplicity that S γ does not cross the source region V J . The line γ effectively divides S into two adjoining open surfaces, say, S  and S  , with boundaries ∂S  ≡ ∂S  ≡ γ, whereas the surface S γ divides V into two adjacent regions V  and V  bounded by S  ∪ S γ and S  ∪ S γ . We take the unit vector sˆ(r ) tangent to γ positively oriented according to the right-handed screw rule with respect to S  (cf. Figure 1.2a) whereby −ˆs(r ) is the tangent vector which obeys the same convention for S  . Finally, for state (b) we place the electric Hertzian dipole j ωp at the center of the ball B(rb, b) ⊂ V  . In this setup we may invoke the reciprocity theorem (6.256) separately in V  and V  \ B[rb, b], and when we sum the resulting equations side by side the two flux integrals over S γ cancel out, so long as the tangential components of the fields are continuous across S γ . Thus, for the problem of Figure 10.5 we arrive at the formula  ˆ  ) · [Ea (r ) × Hb (r ) − Eb (r ) × Ha (r )] − dS  n(r ∂V1



−

ˆ  ) · [Ea (r ) × Hb (r ) − Eb (r ) × Ha (r )] dS  n(r

S



−

ˆ  ) · [Ea (r ) × Hb (r ) − Eb (r ) × Ha (r )] dS  n(r

S 



−

ˆ  ) · [Ea (r ) × Hb (r ) − Eb (r ) × Ha (r )] dS  n(r

∂B

 = VJ

where the unit normals point inwards V \ B[rb , b].

dV  [Eb (r ) · J(r ) − Hb (r ) · J M (r )] (10.50)

Advanced Theoretical and Numerical Electromagnetics

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In the limit as b → 0+ the contribution from the fields on ∂B is still given by (10.2). The integrands of the reaction and of the remaining flux integrals are transformed as in (10.17)-(10.20). ˆ  ) · ∇ × [Ha (r )∇G(rb , r ) · p] in (10.19) will contribute By virtue of the Stokes theorem the term n(r naught on ∂V1 , since the latter is closed. Contrariwise, the very same term integrated over S  and S  produces two line integrals along γ that do not cancel each other in view of the surmised discontinuity of Ha (r ) across γ. In symbols, we have  1 ˆ  ) · ∇ × [Ha (r )∇G(rb , r ) · p] dS  n(r − ε S  1 ˆ  ) · ∇ × [Ha (r )∇G(rb , r ) · p] − dS  n(r ε S   1 ds sˆ (r ) · [Ha (r ) − Ha (r )]∇G(rb , r ) · p (10.51) =− ε γ

where Ha (r ) and Ha (r ) denote the magnetic field on either side of γ but in S  and S  , respectively. We now insert (10.17)-(10.20) into (10.50) and make use of (10.51). Upon dividing through by j ω, discarding the common and arbitrary constant vector p, renaming rb to r and dropping the remaining subscript ‘a’ we arrive at   1    E(r) = − j ωμ dV G(r, r )J(r ) + dV  ρ(r )∇G(r, r ) ε VJ VJ   ˆ  ) × H(r )] dS  G(r, r )[n(r + dV  ∇G(r, r ) × J M (r ) − j ωμ S ∪∂V1

VJ















ˆ  )] dS  ∇ G(r, r ) × [E(r ) × n(r

ˆ ) · E(r )∇ G(r, r ) + dS n(r

+ S ∪∂V1

S ∪∂V1

−

1 j ωε



ds sˆ(r ) · [H (r ) − H (r )]∇G(r, r )

(10.52)

γ

which is the Stratton-Chu formula for E(r) and observation points r ∈ V \ V J with H(r ) discontinuous across γ ⊂ S [1, Section 6.1.1], [2, p. 468], [11, pp. 53–56]. The analogous formula for the magnetic field with E(r ) discontinuous across γ reads   1 dV  ρ M (r )∇G(r, r ) H(r) = − j ωε dV  G(r, r )J M (r ) + μ VJ VJ   ˆ  )] dS  G(r, r )[E(r ) × n(r − dV  ∇G(r, r ) × J(r ) − j ωε VJ



+ S ∪∂V1

S ∪∂V1

ˆ  ) · H(r )∇G(r, r ) − dS  n(r



ˆ  ) × H(r )] dS  ∇G(r, r ) × [n(r

S ∪∂V1

+

1 j ωμ

 γ

ds sˆ(r ) · [E (r ) − E (r )]∇G(r, r )

(10.53)

Integral formulas and equivalence principles

705

thanks to the duality transformations of Table 6.1. The extension of (10.52) and (10.53) for points on the boundary but not on γ requires moving the derivatives of the Green function outside the surface integrals as in (10.34) or alternatively casting the surface integrals that involve ∇G as in (9.34). Further, (10.52) and (10.53) remain valid for r ∈ V J , and this can be proved by placing the dipole in the source region as we did in Section 10.2. The line integrals over γ in (10.52) and (10.53) are known as the Kottler terms [11, 12] and by comparison with the integrals of ρ(r ) and ρ M (r ) they may be interpreted as the fields produced by electric and magnetic line densities of charges given by 1 sˆ(r ) · [H (r ) − H (r )], jω 1 sˆ(r ) · [E (r ) − E (r )], ρ ML (r ) := jω ρL (r ) := −

r ∈ γ ⊂ S

(10.54)

r ∈ γ ⊂ S

(10.55)

which accumulate along γ in view of the field discontinuity. More specifically, if we indicate the ˆ  ) with νˆ (r ) = sˆ(r ) × n(r ˆ  ) (Figure 10.5) on account of unit vector perpendicular to sˆ(r ) and n(r definition (10.13) we can write ˆ  ) × νˆ (r ) = νˆ (r ) · [JS (r ) − JS (r )] j ωρL (r ) = [H (r ) − H (r )] · n(r

(10.56)

where JS (r ) and JS (r ) denote the equivalent surface current densities on S  and S  . Relation (10.56) is in agreement with the jump condition for time-harmonic surface currents (1.201) provided we take into account the different orientation of the unit normal on the surface S (cf. Figure 1.14). We may understand the build up of charge along γ as the result of the unbalance between the currents flowing into and away from γ. The same conclusion applies to the magnetic densities of charge and current. Of course, in the absence of electric or magnetic conductors flush with S  or S  all equivalent charges and currents of concern here are fictitious. Indeed, historically (10.52) was used by Stratton and Chu to study the propagation of electromagnetic waves through metallic screens with slits or apertures [2, 7]. For example, with reference to the geometry of Figure 10.5 we could assume the PEC boundary conditions (1.169) on S  and think of S  as being the aperture through which the fields produced by external sources can penetrate into the open cavity V. Since the electric field is not zero on S  whereas the magnetic field may be taken as continuous over S  ∪ S  , the Kottler charge densities become ρL (r ) := 0, ρ ML (r ) := −

1 sˆ(r ) · E (r ), jω

r ∈ γ ⊂ S

(10.57)

r ∈ γ ⊂ S

(10.58)

although the actual value of E (r ) is yet unknown. The singularities of G(r, r ) and ∇G(r, r ) [see (9.167)] cannot be integrated over a line, and hence the Kottler terms diverge if we try to evaluate them for observation points on γ (cf. Example 6.1). Therefore, the extended formulas (10.52) and (10.53) may only be employed to compute the fields everywhere in V \ γ, including the source region [7].

10.4 Surface equivalence principles By and large, an equivalence principle is a ‘recipe’ for devising an electromagnetic problem which is identical with another one at least in the region of space where a solution is desired. In this

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(a)

(b)

Figure 10.6 For illustrating the equivalence principle in network theory: (a) original circuit; (b) equivalent circuit with voltage and current generators in the section AA’. respect, the image principle discussed for electrostatic fields in Section 3.9 may also be regarded as an equivalence principle. The problem comprised of the charges and matter in the physical space (Figure 3.17) is extended with suitable charges and matter in the image space (Figure 3.18) because in doing so the resulting problem is easier to solve. We remarked, though, that the solution thus found represents the desired one only in the physical space. As useful as the image principle may be, the application thereof is predicated on the ability to identify the distribution of sources and matter in the image space, and such procedure is by no means trivial for general geometries. By contrast equivalence principles in electromagnetics are quite general and not restricted to special configurations. Before we delve into the derivation, we turn our attention to an example from network theory in the time-harmonic regime. Example 10.1 (Equivalence principle in network theory) With reference to Figure 10.6a, suppose that a lumped component described by means of its impedance ZL is connected to a voltage generator of intensity VG and internal impedance ZG . We may assume that VG and ZG are known or, alternatively, that they represent the Thévenin model [13] of some complicated network we know little about. We indicate with IL = I0 and VL = V0 the current flowing towards the load and the voltage drop across ZL . Solving the circuit with elementary means yields ZL VG ZG + ZL VG I L = I0 = ZG + ZL

V L = V0 =

(10.59) (10.60)

and, although it may seem trivially obvious, we emphasize that V0 = Z L I 0

(10.61)

which is to say, V0 and I0 in the section AA’ of the circuit are not independent! Next, we consider a modification of the original circuit in which the voltage generator is switched off (VG = 0) whereas two other generators are inserted between ZG and the load ZL in the section AA’. More precisely, we connect a voltage generator of intensity V0 in series and a current generator of intensity I0 in parallel, as is sketched in Figure 10.6b. The graphical symbol chosen for the combination of the two generators serves as a visual reminder that the order in which we connect these active components with respect to the load is utterly inconsequential, as we will show in a

Integral formulas and equivalence principles

707

moment. A key point, though, is that the intensities V0 and I0 must be given by (10.59) and (10.60), that is, they coincide with the current entering the load and the voltage drop relevant to the circuit of Figure 10.6a. So, how does the circuit of Figure 10.6b behave? To answer this question we compute, again, the current IL flowing into the load and the voltage drop VL across ZL , but also the current IA leaving the impedance ZG and the voltage drop VA across ZG . For the calculation we may invoke the linearity of the network as well as the network analogue of the principle of superposition (cf. Section 6.1). In practice, the quantities we seek arise from the combination of currents and voltages due to V0 with the current generator turned off (I0 = 0) plus currents and voltages due to I0 with the voltage generator switched off (V0 = 0). In symbols, we have VL =

ZL ZL ZG V0 + I 0 = V0 ZG + ZL ZG + ZL   with I0 =0

(10.62)

with V0 =0

V0 ZG IL = + I0 = I0 ZG + ZL ZG + ZL ZG ZL ZG VA = − V0 + I0 = 0 ZG + ZL ZG + ZL V0 ZL IA = − I0 = 0 ZG + ZL ZG + ZL

(10.63) (10.64) (10.65)

where the rightmost-hand sides are a consequence of relationship (10.61). We have just proved that, from the viewpoint of the load, the circuit of Figure 10.6b behaves exactly as the original network in Figure 10.6a. In fact, the current IL and the voltage VL are the same for the two circuits. The load cannot ‘tell’, as it were, whether the excitation is due to the original voltage generator VG or to the combination of the generators V0 and I0 ! On the contrary, the current IA and the voltage VA are null, and it is a simple matter to check that they differ from the corresponding values in the circuit of Figure 10.6a. Hence, we may state that the circuits of Figures 10.6a and 10.6b are equivalent but only with regard to the voltage and the current experienced by the load. The whole point of building an equivalent problem is to simplify the task of solving the network, especially in configurations far more complex than the one under investigation. We now see from (10.64) and (10.65) that the internal impedance ZG of the original generator is not excited, because the combined effects of V0 and I0 produce neither a current through ZG nor a voltage drop across ZG . It seems logical that the actual value of ZG should have no relevance whatsoever for the excitation of ZL . We can put this hunch to test by considering two extreme situations, namely, ZG = 0 (a short circuit) and ZG → ∞ (an open circuit). Letting ZG = 0 causes the current generator I0 to direct all the current towards the left part of the circuit where the impedance is null. Thus, I0 does not contribute to the excitation of ZL and the resulting circuit, shown in Figure 10.7a, contains only V0 . Calculations indicate that VL and IL are given by (10.59) and (10.60). Finally, letting ZG → ∞ renders the voltage generator V0 ineffective because it either remains ‘dangling’ (if connected to the left of the section AA’) or becomes inconsequential for the load (if connected to the right of the section AA’). Therefore, V0 does not contribute to the excitation of the load and the resulting circuit (Figure 10.7b) contains only I0 . Again, simple calculations confirm that VL and IL are still given by (10.59) and (10.60). These observations suggest that we are permitted to modify the left part of the circuit of Figure 10.6a provided we introduce two equivalent generators with intensities (10.59) and (10.60)

708

(a)

Advanced Theoretical and Numerical Electromagnetics

(b)

Figure 10.7 For illustrating the equivalence principle in network theory: (a) equivalent circuit with voltage generator only; (b) equivalent circuit with current generator only. as described in Figure 10.6b, though some choices may turn out to be more convenient than others, as remarked by Figures 10.7a and 10.7b. (End of Example 10.1)

We wish to extend these considerations to electromagnetic fields and apply them to the Maxwell equations. Many practical electromagnetic problems can be treated in the frequency domain by invoking the time-harmonic regime. Nevertheless, the direct solution of the Maxwell equations (1.98)-(1.101) with suitable boundary conditions (Section 1.7) may not be feasible especially when the problem of concern involves an unbounded region of space. It is often fruitful to rephrase the original problem into an equivalent one which is more amenable to an efficient numerical solution. Although this strategy is the subject of Chapters 13-15 here we develop the essential tools — known as the surface equivalence principles —- that enable us to devise and formulate the equivalent problem. We remark that equivalent is synonymous with exact, namely, we do not make any approximation in turning the original problem into another one possibly simpler to solve. Rather, we shift focus, as it were, from one set of unknowns (usually the fields in each point of the unbounded domain) to an alternative, more convenient one (often the tangential fields over the finite boundaries of the domain). The basis for at least one equivalence principle is precisely provided by the Stratton-Chu integral representations derived in Section 10.2, though it is possible to arrive at the same result working with the Maxwell equations. We discuss the topic in the following sections.

10.4.1 The Huygens and Love equivalence principles We obtained the Stratton-Chu formulas (10.22) and (10.23) for a bounded region of space V under the assumptions that the ‘true’ sources J(r ), J M (r ), r ∈ V J ⊂ V, were known and that unspecified matter and sources were present in the complementary domain R3 \ V (Figure 10.1). We argued that (10.22) and (10.23) do not constitute a solution to the Maxwell equations (1.98)-(1.101) for the problem of Figure 10.1 and r ∈ V, unless we can specify the values of E(r ) and H(r ) on the boundary ∂V consistently. However, the possibility of representing the fields in a region of space by means of the fields on the boundary of that region — even if they are not known at first — is what we call a surface equivalence principle. We observe that not knowing the fields on the boundary is a minor glitch because we may regard E(r ) and H(r ), r ∈ ∂V, as unknowns and try and determine them. This means that instead of solving (1.98)-(1.101) for the fields in the whole space, we content ourselves with determining

Integral formulas and equivalence principles

709

the fields on ∂V only, which is quite a reduction in complexity. Once we are done we turn back to (10.22) and (10.23) to compute the fields everywhere in V. More often than not, in radiation and scattering problems (e.g., Figure 6.4) we are interested in finding the fields produced by true sources in the whole space in the presence of material bodies of finite extent. Thus, we need to revise (10.22) and (10.23) in order to apply them to unbounded regions. Suppose that a material body — possibly non-reciprocal — exists in a finite volume V1 and that the underlying unbounded medium is isotropic and endowed with parameters ε and μ. We consider the surface-wise multiply-connected region V := B(0, a) \ V 1 where the radius a is large enough for V1 to be contained in the ball B(0, a). We also assume that the true sources are confined in V J ⊂ V. To begin with, we apply the Stratton-Chu formulas to V thus obtaining two integral representations which involve, among other contributions, surface integrals over the sphere ∂B. By manipulating the relevant integrands as we did in (10.42) for the electric field and by using estimates (10.45)-(10.47) we may conclude that the surface integral over ∂B vanishes in the limit as a → +∞. In the end we are left with the formulas   1    E(r) = − j ωμ dV J(r )G(r, r ) + dV  ρ(r )∇G(r, r ) ε VJ VJ   ˆ  ) × H(r )G(r, r ) + dV  ∇G(r, r ) × J M (r ) − j ωμ dS  n(r VJ

 −∇

∂V1

ˆ  ) · E(r )G(r, r ) − ∇ × dS  n(r

∂V1

and



H(r) = − j ωε  −

VJ

dV  ∇G(r, r ) × J(r ) − j ωε  −∇ ∂V1

ˆ  ) (10.66) dS  G(r, r )E(r ) × n(r

∂V1

dV  J M (r )G(r, r ) +

VJ





1 μ



dV  ρ M (r )∇G(r, r )

VJ

ˆ  )G(r, r ) dS  E(r ) × n(r

∂V1

ˆ  ) · H(r )G(r, r ) + ∇ × dS  n(r



ˆ  ) × H(r ) (10.67) dS  G(r, r )n(r

∂V1

which represent the mathematical statement of the surface equivalence principle for the unbounded region R3 \ V1 [1–3,14]. There is no approximation involved, because the effect of the body enclosed in V1 is accounted for by the values of E(r ) and H(r ) over ∂V1 . These ideas were initially propounded by Larmor and Love [15–17], and later developed fully by Schelkunoff [18, 19]. We already had the opportunity to mention that the tangential and normal fields over ∂V1 may be construed as equivalent surface densities of currents and charges, given by (10.13), (10.14), (10.24) and (10.25). The role of the equivalent surface densities of currents and charges on ∂V1 is the same as that of the equivalent generators V0 and I0 in the modified circuit of Figure 10.6b. In some applications it may be known a priori that either the electric densities or the magnetic ones are null. For instance, if the medium that fills V1 is a PEC, then the matching conditions (1.169) and (1.171) state that the equivalent magnetic current and charge over ∂V1 vanish. This situation is the electromagnetic analogue of the modified circuit of Figure 10.7a. Likewise, if the medium in V1 is a PMC, by duality we infer that the equivalent electric current and charge over ∂V1 are null (see Section 6.6). This configuration corresponds to the modified circuit of Figure 10.7b.

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Advanced Theoretical and Numerical Electromagnetics

An interesting situation occurs when we assume that 1) the medium in V1 has the same constitutive parameters of the background isotropic medium existing in the complementary domain and 2) the true sources are located inside V1 . Then, a straightforward application of (10.66) and (10.67) yields       ˆ ) × H(r )G(r, r ) − ∇ dS  n(r ˆ  ) · E(r )G(r, r ) E(r) = − j ωμ dS n(r ∂V1

∂V1

 −∇×

ˆ ) dS  G(r, r )E(r ) × n(r

(10.68)

∂V1

 H(r) = − j ωε ∂V1

ˆ  )G(r, r) − ∇ dS  E(r ) × n(r



ˆ  ) · H(r )G(r, r ) dS  n(r

∂V1

 +∇×

ˆ  ) × H(r ) dS  G(r, r )n(r

(10.69)

∂V1

which are the mathematical formulation of the Love equivalence principle [15] or the vector Huygens principle [14,16,20], [21, Chapter XIII]. In 1690 the Dutch physicist C. Huygens introduced the concept of secondary sources to explain the propagation of light, centuries before the work of Maxwell and the realization that light is, after all, an electromagnetic wave. According to Huygens’s idea, the points over a surface which encloses a source of light (e.g., a candle) become in turn secondary sources of spherical waves. Evidently, this construction can be repeated by considering a larger surface which surrounds the previous one, now to be regarded as a source, and so on. This is precisely what (10.68) and (10.69) state for electric and magnetic fields, especially on the grounds of the asymptotic expansions (9.325) and (9.326) in the Fraunhofer region of ∂V1 . It is important to recall that (10.66)-(10.69) return zero when evaluated for points r ∈ V1 , although the actual fields are not necessarily null. More importantly, while the properties of the material within V1 do not enter (10.66) and (10.67) explicitly, this does not mean that the effect of the body in V1 is irrelevant. In like manner, while the true source within V1 does not appear in (10.68) and (10.69) this does not imply that such source has no effect on the value of the field in the complementary domain R3 \ V 1 . We must keep in mind that the fields E(r ) and H(r ) over ∂V1 are precisely those pertinent to the original problem, that is, in the presence of the body or the sources in V1 ! We now discuss an alternative viewpoint for obtaining the surface equivalence principle that is based on the direct manipulation of the Maxwell equations in the time-harmonic regime. The procedure more closely resembles the construction of an equivalent circuit described in Example 10.1. We suppose that in a homogeneous isotropic unbounded medium endowed with constitutive parameters ε and μ two sets of time-harmonic electric and magnetic sources radiate in the presence of material anisotropic bodies, possibly inhomogeneous and even non-reciprocal, as is exemplified in Figure 10.8. Up to now we have found integral solutions to the Maxwell equations in homogeneous unbounded media essentially because we managed to compute the relevant dyadic Green functions (Section 9.4). Unfortunately, closed-form Green functions cannot be determined that account for anisotropic inhomogeneous media of general shape. Therefore, it is convenient to isolate the troublesome media with a smooth surface S which defines an internal region Vin . We assume that in the

Integral formulas and equivalence principles

711

Figure 10.8 For the alternative derivation of the Love equivalence principle: original problem comprised of sources and matter in a homogeneous isotropic unbounded medium. process also some sources — which we call Jin (r) and J Min (r) — end up enclosed within Vin . The complementary domain Vex := R3 \ V in represents the external region and contains known sources J(r) and J M (r). By contruction S is just a mathematical surface rather than the material interface between two media. For later usage we introduce a ball B(0, d) whose radius d is large enough for all sources and bodies to be contained in B(0, d) (Figure 10.8). Our goal is to devise an equivalent problem for the region Vex while obviating the difficulties posed by the contents of Vin . This strategy should pay off since Vex is filled with a homogeneous isotropic medium, and hence we hope we can use the dyadic Green functions of Section 9.4. Thus, we need to cast the Maxwell equations (1.98)-(1.101) into a form for which the solution is given by, e.g., (9.156) and (9.157). More precisely, guided by the procedure of Example 10.1, we wish to write the Maxwell equations governing the evolution of the electromagnetic field that 1) is null within Vin and 2) for r ∈ Vex coincides with the one relevant to the configuration of sources and matter of Figure 10.8. To this purpose we define the three-dimensional step function [cf. (C.49)] ⎧ ⎪ ⎨0, r ∈ Vin U(r) := ⎪ ⎩1, r ∈ V ex

(10.70)

and observe that the vector fields U(r)E(r), U(r)H(r), U(r)D(r) and U(r)B(r) vanish for r ∈ Vin . We would like to take curl and divergence in accordance with the formal structure of (1.98)-(1.101), but since U(r)E(r) etc. are discontinuous across S ≡ ∂Vin , we need to interpret U(r)E(r) and the other fields as distributions and compute the derivatives in a weak sense as prescribed by (C.44) and (C.45). In the subsequent derivations we adopt the same symbols to denote scalar and vector fields and the distributions therewith associated (Appendix C).

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Advanced Theoretical and Numerical Electromagnetics

We begin by examining U(r)D(r) to obtain the electric Gauss law corresponding to (1.100) and consider   ∇ · [U(r)D(r)] := − dV ∇φ(r) · D(r)U(r) = − dV ∇φ(r) · D(r) R3

 =

Vex



dV φ(r)∇ · D(r) − Vex

dV ∇ · [φ(r)D(r)] Vex



dV φ(r)ρ(r) − lim

=

dV ∇ · [φ(r)D(r)]

d→+∞ B(0,d)\Vin

Vρ



=

ˆ · D(r)φ(r) + dS n(r)

d→+∞

Vρ



∂B

 dV φ(r)ρ(r) +

Vρ

ˆ · D(r)φ(r) dS n(r) S

ˆ · D(r)φ(r) dS n(r) S





dV φ(r)U(r)ρ(r) +

=





dV φ(r)ρ(r) − lim

=



R3

ˆ · D(r)φ(r)δS (r − rS ) dV n(r) R3

:= U(r)ρ(r) + n(r) ˆ · D(r)δS (r − rS )

(10.71)

3 C∞ 0 (R )

is a test function (Appendix C), we have used (1.100) for r ∈ Vex , and we have where φ(r) ∈ applied the Gauss theorem for D(r)φ(r) is differentiable in B(0, d) \ Vin . Besides, the integral over the sphere ∂B vanishes in the limit as d → +∞, because φ(r) has a bounded support by hypothesis. The notation δS (r − rS ) indicates a surface delta distribution localized on the surface S and serves to ˆ · D(r), which plays the role of a surface charge density. Notice that, strictly include the effect of n(r) ˆ speaking, n(r)·D(r) is the value of the normal component of D(r) on the positive side of S (i.e., inside ˆ · D(r) is continuous across S by assumption. Vex ), though this distinction is immaterial here, as n(r) The magnetic Gauss law pertinent to the magnetic induction U(r)B(r) may be derived by carrying out the same steps above but also, more simply, by invoking the duality principle on (10.71). In the end we obtain ˆ · B(r)δS (r − rS ) ∇ · [U(r)B(r)] = U(r)ρ M (r) + n(r)

(10.72)

ˆ ·B(r) constitutes a surface density where ρ M (r) is the magnetic charge density located in Vex and n(r) of magnetic charge on S . Next, we turn our attention to the Ampère-Maxwell law for U(r)H(r) and consider   ∇ × [U(r)H(r)] := − dV ∇φ(r) × H(r)U(r) = − dV ∇φ(r) × H(r) 

R3

dV φ(r)∇ × H(r) −

= Vex



Vex



dV ∇ × [φ(r)H(r)] Vex



dV φ(r)[j ωD(r) + J(r)] − lim

= Vex



dV ∇ × [φ(r)H(r)]

d→+∞ B(0,d)\Vin



dV φ(r)[j ωD(r) + J(r)] − lim

=

ˆ × H(r)φ(r) dS n(r)

d→+∞

Vex

∂B

Integral formulas and equivalence principles

713

 + S

 =

ˆ × H(r)φ(r) dS n(r) 

dV φ(r)[j ωD(r) + J(r)] + Vex

ˆ × H(r)φ(r) dS n(r) S





dV φ(r)[j ωD(r) + J(r)]U(r) +

= R3

ˆ × H(r)δS (r − rS ) dS φ(r)n(r) R3

:= j ωU(r)D(r) + U(r)J(r) + n(r) ˆ × H(r)δS (r − rS )

(10.73)

where we have used (1.98) for r ∈ Vex , and we have applied the integral identity (H.91) for H(r)φ(r) ˆ × H(r) represents an electric surface current is differentiable in B(0, d) \ Vin . We remark that n(r) density confined to the positive side of S . Finally, we invoke the duality principle on (10.73) to obtain the relevant Faraday law, viz., ˆ ∇ × [U(r)E(r)] = − j ωU(r)B(r) − U(r)J M (r) − E(r) × n(r)δ S (r − rS )

(10.74)

ˆ where J M (r) is the magnetic current density outside Vin and E(r) × n(r) plays the role of a surface density of magnetic current on S . Although (10.71)-(10.74) are distributional equations, still they are the desired set of timeharmonic Maxwell equations for the vector fields U(r)E(r), U(r)H(r), U(r)D(r) and U(r)B(r). In particular, since said fields clearly vanish in Vin we have reached our goal — namely, writing the equations of an equivalent problem which ‘conceals’ the complexity within Vin — so long as these fields also coincide with E(r), H(r), D(r) and B(r) for r ∈ Vex . To convince ourselves that this is indeed true we may write (10.71)-(10.74) as equations between ordinary vector fields by restricting them to points r ∈ Vex , viz., ∇ · D(r) = ρ(r)

(10.75)

∇ · B(r) = ρ M (r) ∇ × H(r) = j ωD(r) + J(r)

(10.76) (10.77)

∇ × E(r) = − j ωB(r) − J M (r)

(10.78)

which must be supplemented with the Silver-Müller conditions (6.171)-(6.172) and the boundary ˆ × H(r) or E(r) × n(r) ˆ on S , conditions for r ∈ S ≡ ∂Vin . The latter consist of assigning either n(r) while recalling that these surface fields are not independent of each other and that they represent the actual values of the electromagnetic field of the problem of Figure 10.8 for points on S . Under these hypotheses (10.75)-(10.78) admit a unique solution, as was proved in Section 6.4.2. What is more, (10.75)-(10.78) constitute the Maxwell equations for the problem of Figure 10.8 though restricted to points r ∈ Vex and complemented with the same matching conditions just reviewed. As a result, the unique solution to (10.75)-(10.78) coincides with the electromagnetic field of the problem of Figure 10.8 for r ∈ Vex and vanishes for points r ∈ Vin . The close analogy with the behavior of the network of Figure 10.6b should not go unnoticed. ˆ · D(r), n(r) ˆ · B(r), n(r) ˆ × H(r) and E(r) × nˆ (r) play a role analogous to Indeed, the surface fields n(r) the lumped generators V0 and I0 of Figure 10.6b. As is suggested in Figure 10.9, the fields in Vex are due to two set of sources, namely, the true sources J(r) and J M (r) in Vex and fictitious surface current densities JS eq (r) and J MS eq (r) localized on the positive side of S . More importantly, the combination of true and equivalent sources in Vex produces no fields at all within Vin . Since the true sources Jin (r) and J Min (r) do not enter (10.71)-(10.74) explicitly, for all practical purposes we may regard them as

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Figure 10.9 For the alternative derivation of the Love equivalence principle: equivalent problem comprised of fictitious surface sources and original matter in a homogeneous isotropic unbounded medium. having been switched off. From the viewpoint of an observer located in Vex the effect of sources and matter within Vin is accounted for by the equivalent sources on S . We can state the surface equivalence principle in words by saying that an electromagnetic problem may be defined that is exactly equivalent to the original one in Vex := R3 \ V in , as long as suitable equivalent sources ˆ × H(r), JS eq (r) = n(r) ˆ J MS eq (r) = E(r) × n(r),

r ∈ S ≡ ∂Vin r ∈ S ≡ ∂Vin

(10.79) (10.80)

are placed on the positive side of S . The problem thus defined is not equivalent to the original one in Vin because the fields are zero there. The special case J(r) = 0 = J M (r) for r ∈ Vex is the Love equivalence principle [1, 15]. Having found the ‘recipe’ for constructing an equivalent problem is only helpful if we are able to solve the resulting equations (10.71)-(10.74) or (10.75)-(10.78). Indeed, we have just remarked that, thanks to the fictitious sources JS eq (r) and J MS eq (r), we can rid ourselves of the true sources within Vin . However, the procedure outlined so far has no effect on the material bodies which are still present in Vin , and the solution entails finding the fields radiated by all sources (the true ones in Vex and the fictitious ones on S ) in the presence of the media within Vin . This situation corresponds to having the internal impedance ZG of the generator still in place in the circuit of Figure 10.6b and, unlike the example from network theory, the electromagnetic problem may not be amenable to an integral representation. But here is the way out: since the fields of the equivalent problem are zero in Vin , the material bodies therein are not ‘excited’, so to speak, and the fields in Vex are not altered if we modify the content (i.e., the constitutive parameters) of Vin . Clearly, this procedure corresponds to replacing ZG with other impedances, such as in Figures 10.7a and 10.7b. The electromagnetic analogues of the short and open circuits are realized by filling Vin with a PEC or a PMC, respectively, as is sketched in Figure 10.10. The advantage of either choice lies in the fact that only one type of sources radiates. If the medium in Vin is PEC, then the electric current JS eq is ‘shorted out’, as we proved in Example 6.7. Conversely, if the medium in Vin is PMC, the magnetic current J MS eq amounts to a voltage generator of sorts that, being left ‘dangling’, does not radiate. This conclusion can be reached by using the reciprocity theorem as we did in Example 6.7 or by just invoking the duality principle. On the downside, now we have to solve (10.75)-(10.78) in the presence of a PEC or PMC body, and the relevant Green functions are not available for general

Integral formulas and equivalence principles

715

Figure 10.10 For the alternative derivation of the Love equivalence principle: equivalent problem comprised of fictitious surface sources and a PEC or PMC medium within Vin .

Figure 10.11 For the alternative derivation of the Love equivalence principle: equivalent problem comprised of fictitious surface sources and the same medium in Vin ∪ Vex . shapes of Vin . A notable exception occurs when S ≡ ∂Vin is a plane which divides the space into two half-spaces. One final choice consists of filling Vin with a material whose constitutive parameters are the same as those relevant to the homogeneous isotropic medium in Vex , as is shown in Figure 10.11. The corresponding choice in the network analogue of Example 10.1 consists of replacing ZG with the impedance ZL of the load. Although with this setup we do not achieve any reduction of the number of equivalent sources over S , yet all of them as well as the true ones in Vex radiate in a homogeneous isotropic unbounded medium. Since for this occurrence we do have the closed-form dyadic Green functions, we can write the integral representation of the fields for r ∈ Vex . It is a simple matter to show that, after a few manipulations, one obtains final expressions that are nothing but the Stratton-Chu formulas for the unbounded region Vex .

10.4.2 The Schelkunoff equivalence principle The integral representations of Stratton and Chu and, more generally, the Love equivalence principle provide a means to determine the electromagnetic field in a given domain V filled with a homogeneous isotropic material so long as 1) the true sources (if any are present within V) are known and 2) the values of the fields are assigned consistently on the boundary ∂V. Leaving aside the fact that,

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Advanced Theoretical and Numerical Electromagnetics

(b)

Figure 10.12 For deriving the Schelkunoff equivalence principle: (a) original problem comprising sources in an unbounded homogeneous medium; (b) auxiliary problem comprising sources in a domain bounded by a PEC medium.

after all, the values of E(r) and H(r) on ∂V are unknown in most practical circumstances, one might also object that integral formulas such as (10.66) and (10.67) look like an overkill. Indeed, we recall from Section 6.4.2 that uniqueness of time-harmonic solutions is achieved in unbounded regions of ˆ space if we specify either nˆ × H(r) or E(r) × n(r) on the boundary ∂V1 (see Figure 6.16) and the Silver-Müller conditions (6.171), (6.172) to pick up the outward-going wave solution. Therefore, it should be possible to obtain integral representations that make use only of one type of equivalent sources, namely, either (10.13) and (10.24) or (10.14) and (10.25). The result goes by the name of Schelkunoff’s equivalence principle [18, 22]. For the derivation we consider a configuration of sources which exist in a homogeneous isotropic unbounded medium. We isolate the sources J(r) and J M (r) within a domain Vin (Figure 10.12a) and attempt to devise an equivalent problem for the exterior unbounded region Vex := R3 \ V in . For the subsequent discussion we indicate the field produced by J(r) and J M (r) with Ei (r) and Hi (r) where the superscript ‘i’ stands for incident or impressed. The first step consists of examining an auxiliary problem, related to the original one, in which Vex is conceptually filled with a PEC medium, as is suggested in Figure 10.12b. What can we expect of the electromagnetic field in this configuration? The true sources still produce the incident field but this cannot be the right solution within the PEC where, as we learned in Section 1.6 on page 29, the electromagnetic field must be null. Hence, some other source must generate a suitable field which compensates and cancels Ei (r) and Hi (r) for r ∈ Vex . Since no conduction currents are possible inside a PEC, the source in question is an electric surface current density Jind (r) flowing on the negative side of ∂Vin . The subscript ‘ind’ stands for induced and serves to remind us that Jind (r) is not independent but rather is caused by J(r) and J M (r) within Vin . We also notice that Jind (r) radiates towards Vin and Vex . More specifically, as represented in Figure 10.13a, for points r ∈ Vex the current Jind (r) must produce a field which is the negative of Ei (r) and Hi (r) so as to achieve the expected cancellation. The latter is a consequence of the linearity of the problem and the principle of superposition (Section 6.1). The field produced by Jind (r) towards Vin is indicated with Es (r) and Hs (r) where the superscript ‘s’ signifies scattered or secondary.

Integral formulas and equivalence principles

(a)

717

(b)

Figure 10.13 For deriving the Schelkunoff equivalence principle: (a) auxiliary problem comprising sources in a domain bounded by a PEC medium; (b) equivalent problem comprising equivalent electric sources on the boundary of the excluded region. According to the time-harmonic counterpart of the boundary conditions (1.168)-(1.171) which hold at the interface between a dielectric medium and a PEC, we have ˆ × H(r), Jind (r) = −n(r)

r ∈ ∂Vin

(10.81)

where H(r) = Hi (r) + Hs (r) is the total field on the negative side of ∂Vin and the minus sign is a consequence of the orientation of the unit normal which points towards Vex . The scattered electric field Es (r) produced by Jind (r) combines with Ei (r) in order to ensure that ˆ := [Ei (r) + Es (r)] × n(r) ˆ = 0, E(r) × n(r)

r ∈ ∂Vin

(10.82)

whence, as a result, the average power flux (1.303) into Vex is zero. This finding is consistent with the lossless character of the medium which fills Vex in Figure 10.13a. The key point is that Jind (r) produces −Ei (r) and −Hi (r) for r ∈ Vex . Thus, in the second and final step we consider an electromagnetic problem which only comprises the fictitious current density ˆ × H(r), JS eq (r) := −Jind (r) = n(r)

r ∈ ∂Vin

(10.83)

existing in the original homogeneous isotropic unbounded medium (Figure 10.13b). In light of the previous discussion, we know that JS eq (r) radiates the fields Ei (r) and Hi (r) towards Vex and the latter are, by construction, the very same fields generated by J(r) and J M (r) in the original problem of Figure 10.12a. In words, an electromagnetic problem can be defined that is exactly equivalent to a given one in the external region Vex so long as a suitable equivalent electric surface current density JS eq (r) is placed on the positive side of the boundary ∂Vin . However, not only is the new problem different than the original one for points r ∈ Vin , but the fields are therein not zero because JS eq (r) radiates also towards Vin . Therefore, we are not permitted to alter the content of Vin — as is the case for the Love equivalence principle — because any such modification would destroy the equivalence in the region

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Advanced Theoretical and Numerical Electromagnetics

Figure 10.14 For deriving the volume equivalence principle: original problem comprising sources and inhomogeneous anisotropic media. of interest Vex . Besides, since JS eq (r) depends on the unknown total magnetic field H(r) relevant to the auxiliary problem, the Schelkunoff equivalence principle is fraught with the same difficulties as those presented by the Love equivalence principle. We conclude by observing that, if we construct the auxiliary problem of Figure 10.12b by placing a PMC medium in Vex and repeat the reasoning that has led us to (10.83), we arrive at an equivalent configuration with the fictitious magnetic surface current ˆ J MS eq (r) := E(r) × n(r),

r ∈ ∂Vin

(10.84)

where E(r) is the total electric field on the negative side of ∂Vin and in the presence of the PMC.

10.5 Volume equivalence principle In the previous section we have described rules for devising equivalent problems in a region of space. In particular, equivalence is achieved by introducing suitable surface densities of electric and magnetic sources on the boundary of the excluded region. However, when the latter is ‘filled’ with a medium whose constitutive parameters are inhomogeneous or dyadic or both, an application of the surface equivalence principle may not help formulate an integral equation, as will become clear in Chapter 13. The main reason is that the relevant dyadic Green function of the problem may not available in closed form. To get around this hurdle we aim at accounting for the effect of penetrable bodies by means of equivalent currents which extend throughout the excluded region rather than just on the boundary thereof. A typical problem of electromagnetic scattering is shown in Figure 10.14 and consists of known sources of finite extent that radiate in an isotropic homogeneous background medium in the presence of two penetrable bodies occupying two domains V1 and V2 . For the sake of argument, we suppose that the body within V1 is a dielectric endowed with constitutive parameters ε1 (r), μ, whereas the other one, within V2 , is a magnetic medium characterized by permeability μ2 (r) and permittivity ε. We wish to cast the time-harmonic Maxwell equations into a form which lends itself to a formal integral solution. To this end, we define the permittivity and the permeability of the problem as dyadic fields for r ∈ R3 , viz., ⎧ ⎪ ⎪ ⎨ε1 (r), r ∈ V1 ε(r) = ⎪ (10.85) ⎪ ⎩εI, r ∈ R3 \ V 1

Integral formulas and equivalence principles ⎧ ⎪ ⎪ ⎨μ2 (r), r ∈ V2 μ(r) = ⎪ ⎪ ⎩μI, r ∈ R3 \ V 2

719 (10.86)

where we notice that the components of ε(r) and μ(r) may be discontinuous across the smooth boundaries ∂V1 and ∂V2 . We begin by examining the Ampère-Maxwell law (1.98) which for the problem under study reads ∇ × H(r) = j ωε(r) · E(r) + J(r),

r ∈ R3

(10.87)

in light of (1.126). Since the troublesome term is the displacement current, which contains the dyadic field ε(r), we use the trick of adding and subtracting the displacement current relevant to the background medium, namely,

r ∈ R3 (10.88) ∇ × H(r) = j ωεE(r) + J(r) + j ω ε(r) − εI · E(r), and regard the resulting equation as an equivalent Ampère-Maxwell law for the problem. What are the meaning and the role of the last contribution in the right-hand side of (10.88)? First of all, we observe that the dyadic field ε(r) − εI vanishes for points r ∈ R3 \ V 1 in view of (10.85). Secondly, on account of (3.257) we recognize the vector field

r ∈ V1 (10.89) P(r) := D(r) − εE(r) = ε1 (r) − εI · E(r), as the time-harmonic counterpart of the polarization vector introduced in Section 3.7. We recall that P(r) represents a volume density of electric dipoles induced by the interaction between the external field and the atoms or molecules which form the dielectric body. Finally, we define the entity ⎧ ⎪

⎪ ⎨j ωP(r), r ∈ V1 Jeq (r) := j ω ε(r) − εI · E(r) = ⎪ (10.90) ⎪ ⎩0, r ∈ R3 \ V 1 which we call equivalent polarization current density and interpret, by comparison with (9.279), as a volume density of electric current due to a distribution of electric Hertzian dipoles within V1 . Indeed, Jeq (r) is non-zero only for r ∈ V1 , i.e., in the domain occupied by the dielectric body. Besides, to emphasize the effect of the body in determining the polarization current we introduce the dimensionless dyadic field

−1 κe (r) = ε1 (r) − εI · ε−1 r ∈ V1 (10.91) 1 (r) = I − εε1 (r), which is called the dielectric contrast factor. For all practical purposes, (10.88) constitutes the Ampère-Maxwell law relevant to the electromagnetic problem of Figure 10.15, in which the dielectric body has been replaced by Jeq (r) defined in (10.90). We notice that the tangential component of H(r) remains continuous through ∂V1 since the transformation carried out to obtain (10.88) does not involve the magnetic field. We continue with the Faraday law (1.99) which reads ∇ × E(r) = − j ωμ(r) · H(r) − J M (r),

r ∈ R3

(10.92)

in accordance with (1.127). To deal with the troublesome term that involves the dyadic permeability we add and subtract j ωμH(r), viz.,

∇ × E(r) = − j ωμH(r) − J M (r) − j ω μ(r) − μI · H(r), r ∈ R3 (10.93)

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Advanced Theoretical and Numerical Electromagnetics

Figure 10.15 For deriving the volume equivalence principle: equivalent problem in which the dielectric body is replaced by equivalent electric sources. and consider the final equation as the equivalent Faraday law for the problem of concern. The dyadic field μ(r) − μI vanishes for points r ∈ R3 \ V 2 on the grounds of (10.86). Thanks to (5.183) we identify the vector field M(r) :=

1 1 B(r) − H(r) = μ1 (r) − μI · H(r), μ μ

r ∈ V2

(10.94)

as the time-harmonic counterpart of the magnetization vector introduced in Section 5.6. We recall that M(r) represents a volume density of magnetic dipoles induced by the interaction between the external induction field and the atoms or molecules which form the magnetic body. Lastly, we introduce the entity ⎧ ⎪

⎪ ⎨j ωμM(r), r ∈ V2 (10.95) J Meq (r) := j ω μ(r) − μI · H(r) = ⎪ ⎪ ⎩0, r ∈ R3 \ V 2 which we refer to as an equivalent magnetization current density and interpret, by comparison with (9.389), as a volume density of magnetic current due to a distribution of magnetic Hertzian dipoles within V2 . Also, to highlight the effect of the body in determining the magnetization current we introduce the dimensionless dyadic field

−1 κm (r) = μ2 (r) − μI · μ−1 r ∈ V2 (10.96) 2 (r) = I − μμ2 (r), which is termed the magnetic contrast factor. In practice, (10.93) is the Faraday law relevant to the electromagnetic problem of Figure 10.16, in which also the magnetic body has been replaced by J Meq (r) defined in (10.95). The tangential component of E(r) remain continuous through ∂V2 since the transformation does not involve the electric field. A similar transformation of the Gauss laws in differential form requires a bit more care since D(r) and B(r) are not continuous across the boundaries ∂V1 and ∂V2 in view of (10.85) and (10.86). Therefore, in line with the discussion of Section 1.2.2, if we wish to state the local Gauss laws without resorting to distributions and surface δ’s, we have to write down (1.100) and (1.101) separately in each domain where D(r) and B(r) can be differentiated and enforce the continuity of the normal components thereof. In symbols, this means ⎧ ⎪ ⎪ r ∈ R3 \ V 1 ⎨ε∇ · E(r) = ρ(r), (10.97) ∇ · D(r) = ⎪   ⎪ ⎩∇ · ε1 (r) · E(r) = 0, r ∈ V1

Integral formulas and equivalence principles

721

Figure 10.16 For deriving the volume equivalence principle: equivalent problem in which also the magnetic body is replaced by equivalent magnetic sources. ⎧ ⎪ ⎪ r ∈ R3 \ V 2 ⎨μ∇ · H(r) = ρ M (r), ∇ · B(r) = ⎪   ⎪ ⎩∇ · μ2 (r) · H(r) = 0, r ∈ V2

(10.98)

ˆ · D(r) continuous through ∂V1 and n(r) ˆ · B(r) continuous across ∂V2 . The parts pertinent with n(r) to the domains V1 and V2 occupied by the bodies may then be transformed as follows (cf. Sections 3.7 and 5.6)     ∇ · ε1 (r) · E(r) = ε∇ · E(r) + ∇ · [ε1 (r) − εI] · E(r) = ε∇ · E(r) + ∇ · P(r) = 0,

r ∈ V1

(10.99)

r ∈ V2

(10.100)

    ∇ · μ2 (r) · H(r) = μ∇ · H(r) + ∇ · [μ2 (r) − μI] · H(r) = μ∇ · H(r) + μ∇ · M(r) = 0,

in light of (10.89) and (10.94). By defining equivalent polarization and magnetization densities of charge ⎧ ⎪ ⎪ ⎨−∇ · P(r), r ∈ V1 ρeq (r) := ⎪ (10.101) ⎪ ⎩0, r ∈ R3 \ V 1 ⎧ ⎪ ⎪ ⎨−μ∇ · M(r), r ∈ V2 (10.102) ρ Meq (r) := ⎪ ⎪ ⎩0, r ∈ R3 \ V 2 for the equivalent problem of Figure 10.16 we write the local Gauss laws in the form ε∇ · E(r) = ρ(r) + ρeq (r),

r ∈ R3

(10.103)

μ∇ · H(r) = ρ M (r) + ρ Meq (r),

r∈R

(10.104)

3

ˆ ·D(r) and n(r) ˆ ·B(r) implied.1 However, since we aim at a set of equations with the continuity of n(r) for E(r) and H(r) in an unbounded homogeneous medium with constitutive parameters ε and μ, we 1 The fact that in (10.95) and (10.102) μ appears explicitly in combination with M(r) whereas in (10.90) and (10.101) ε is ‘concealed’ in P(r) is essentially due to H(r) being a secondary entity (or an entity of quantity) and to E(r) being a fundamental entity (or an entity of intensity).

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Advanced Theoretical and Numerical Electromagnetics

have to rephrase the jump conditions (1.155) through ∂V1 and (1.157) across ∂V2 in terms of εE(r) and μH(r), respectively. By adding superscripts ‘+’ or ‘-’ to indicate fields on the positive or negative side of ∂Vl , l = 1, 2, and using (10.89) and (10.94) we have ˆ · ε1 (r) · E− (r) = n(r) ˆ · [εE− (r) + P(r)] = n(r) ˆ · εE+ (r), n(r) ˆ · μ1 (r) · H− (r) = n(r) ˆ · [μH− (r) + μM(r)] = n(r) ˆ · μH+ (r), n(r)

r ∈ ∂V1 r ∈ ∂V2

(10.105) (10.106)

r ∈ ∂V1 r ∈ ∂V2

(10.107) (10.108)

whence ˆ · εE+ (r) − n(r) ˆ · εE− (r) = n(r) ˆ · P(r) = ρS eq , n(r) + − ˆ · μH (r) − n(r) ˆ · μH (r) = n(r) ˆ · μM(r) = ρ MS eq , n(r)

that is, the normal components of εE and μH suffer a jump across the boundaries ∂V1 and ∂V2 due to equivalent surface densities of polarization and magnetization charges. Furthermore, we observe that the polarization and magnetization charge densities are related to Jeq (r) and J Meq (r) by continuity equations. For points r ∈ V1 we have ∇ · Jeq (r) := j ω∇ · P(r) = − j ωρeq (r)

(10.109)

and the dual results holds for r ∈ V2 and J Meq . The continuity on the boundaries ∂V1 and ∂V2 requires the usage of the general jump condition (1.200) and its dual where all the currents and charges are of either the polarization type or the magnetization type. Application of (1.200) with reference to Figures 1.13 and 10.15 thus gives ˆ · P(r) = n(r) ˆ · j ωP(r) − 0, ∇s · JS (r) + j ωn(r)

r ∈ ∂V1

(10.110)

since in particular Jeq (r) = 0 for r  V1 . Condition (10.110) is satisfied if the surface current JS (r), r ∈ ∂V1 , is solenoidal. Since there are no free charges allowed to roam over ∂V1 it is reasonable to assume JS (r) = 0. The corresponding relationship over r ∈ ∂V2 for the magnetization quantities follows through duality. In summary, (10.103) and (10.104) are the equivalent Gauss laws for the problem of Figure 10.14 and together with (10.88) and (10.93) describe the equivalent problem of Figure 10.16. Since all sources (true and equivalent) exist in a homogeneous unbounded medium we may write the integral representation with the help of either the electrodynamic potentials (Section 8.2) or the dyadic Green functions derived in Section 9.4 or also the Stratton-Chu formulas (Section 10.2). In accordance with (9.156) and (9.157) and thanks to the principle of superposition — by which we deal with true, polarization and magnetization sources independently — we simply write     ∇∇ e− j kR e− j kR − ∇ × dV  J M (r ) E(r) = − j ωμ I + 2 · dV  J(r ) 4πR 4πR k 

VS

VS

   ∇∇ e− j kR e− j kR − ∇ × dV  J Meq (r ) − j ωμ I + 2 · dV  Jeq (r ) 4πR 4πR k V1 V2     e− j kR ∇∇ e− j kR H(r) = ∇ × dV  J(r ) − j ωε I + 2 · dV  J M (r ) 4πR 4πR k VS VS     e− j kR ∇∇ e− j kR − j ωε I + 2 · dV  J Meq (r ) + ∇ × dV  Jeq (r ) 4πR 4πR k V1

V2

(10.111)

(10.112)

Integral formulas and equivalence principles

723

for points r ∈ R3 . In order to highlight the role of the equivalent charges ρeq (r) and ρS eq (r) we transform the integral of Jeq (r) over V1 by moving one derivative onto the scalar Green function (cf. Section 2.8), viz.,    ∇∇ e− j kR = − j ωμ I + 2 · dV  Jeq (r ) 4πR k  = − j ωμ

V1

∇ e− j kR − dV Jeq (r ) 4πR j ωε 



V1



= − j ωμ +

∇ j ωε

dV  Jeq (r ) · ∇

V1 − j kR

∇ e − dV Jeq (r ) 4πR j ωε 



V1



 



e− j kR 4πR

e− j kR dV ∇ · Jeq (r ) 4πR 







V1

dV 

− j kR

e ∇ · Jeq (r ) 4πR

(10.113)

V1

on account of (H.51). Application of the Gauss theorem for points r ∈ V1 requires the usual limiting procedure by which we exclude r with a ball B(r, a) ⊂ V1 of vanishing radius a, as we did in (5.32) for the stationary magnetic vector potential. All in all, we arrive at    ∇∇ e− j kR = − j ωμ I + 2 · dV  Jeq (r ) 4πR k V1



dV  Jeq (r )

= − j ωμ

∇ e− j kR − 4πR j ωε

V1

∇ + j ωε



 ∂V1

ˆ  ) · Jeq (r ) dS  n(r

e− j kR 4πR

e− j kR  ∇ · Jeq (r ) dV  4πR

V1

   − j kR − j kR ∇ ∇ e− j kR   e   e − − ρeq (r ) = − j ωμ dV Jeq (r ) dS ρS eq (r ) dV  4πR ε 4πR ε 4πR ∂V1

V1

(10.114)

V1

where we have used (10.90), (10.107) and (10.109). The normal component of Jeq (r) does not vanish on ∂V1 , and this occurrence is modelled as the build up of surface polarization charges described by ρS eq . The dual relationship holds true for the magnetic field produced by J Meq (r), ρ Meq (r) and ρ MS eq (r). It is instructive to obtain the integral ‘solution’ for the problem of Figure 10.16 by means of the Stratton-Chu formulas. We may not employ (10.48) directly to express the electric field inasmuch as it does not tell us how to handle the surface polarization charges ρS eq (r) over ∂V1 . Therefore, we follow a two-step approach in which we apply (10.66) to the unbounded region R3 \ V 1 and to the domain V1 separately. The procedure is similar to the one adopted in Section 9.3 for the electrodynamic potentials generated by surface sources. In computing E(r) we do not concern ourselves with the boundary of V2 , because nˆ × H, E × nˆ and nˆ · E are continuous across ∂V2 . For points outside V 1 we get  E(r) = − j ωμ VS

dV  J(r )G(r, r ) +

1 ε

 VS

dV  ρ(r )∇G(r, r )

Advanced Theoretical and Numerical Electromagnetics

724  +

dV  ∇G(r, r ) × J M (r ) +

VS



dV  ∇ G(r, r ) × J Meq (r )

V2







+







ˆ ) × H (r )G(r, r ) − ∇ dS n(r

− j ωμ ∂V1

ˆ  ) · E+ (r )G(r, r ) dS  n(r

∂V1

 −∇×

ˆ  )G(r, r) dS  E+ (r ) × n(r

(10.115)

∂V1

and inside V 1 we have   1 dV  ρeq (r )∇G(r, r ) E(r) = − j ωμ dV  Jeq (r )G(r, r ) + ε V1 V1   ˆ  ) × H− (r )G(r, r ) + ∇ dS  n(r ˆ  ) · E− (r )G(r, r ) + j ωμ dS  n(r ∂V1

∂V1

 +∇×

ˆ  )G(r, r) dS  E− (r ) × n(r

(10.116)

∂V1

ˆ  ) on ∂V1 points outwards V1 . We recall where we have taken into account that the unit normal n(r that the right member of (10.115) vanishes for r ∈ V1 whereas the right member of (10.116) is null for r ∈ R3 \ V 1 . Besides, both (10.115) and (10.116) return half the value of E(r) for r ∈ ∂V1 . As a consequence, we may add (10.115) and (10.116) side by side to find a formula valid for points r ∈ R3   1 E(r) = − j ωμ dV  J(r )G(r, r ) + dV  ρ(r )∇G(r, r ) ε VS VS       + dV ∇ G(r, r ) × J M (r ) + dV  ∇G(r, r ) × J Meq (r ) VS



− j ωμ 

V1

∂V1



1 ε

VS





VS



− j ωμ 









V2

1 dV Jeq (r )G(r, r ) + ε 





V1

∂V1

dV  ρ(r )∇G(r, r )

VS

dV ∇ G(r, r ) × J M (r ) +

∇ − ε

dV  ρeq (r )∇G(r, r )

V1

dV  J(r )G(r, r ) +

= − j ωμ +



ˆ  ) · [E+ (r ) − E− (r)]G(r, r) dS  n(r

−∇



V2

1 dV  Jeq (r )G(r, r ) + ε

dS  ρS eq (r )G(r, r )

dV  ∇G(r, r ) × J Meq (r ) 

dV  ρeq (r )∇G(r, r )

V1

(10.117)

Integral formulas and equivalence principles

725

in light of (10.107) and the continuity of the tangential components of E(r ) and H(r ) across ∂V1 . Still, since the normal component of E(r ) suffers a jump through ∂V1 , for r ∈ ∂V1 formula (10.117) returns  1 + ˆ n(r) ˆ + n(r) ˆ × [E(r) × n(r)] ˆ E (r) + E− (r) · n(r) 2

(10.118)

by virtue of (10.107) and (2.276) for the gradient of the single-layer potential. It is straightforward to conclude that (10.117) is identical with (10.111) by virtue of (10.114). Analogous steps allow obtaining the expression for the magnetic field by starting from (10.67). We emphasize that (10.111), (10.117) and the analogous expressions for H(r) are formal integral representations in that they involve the polarization and magnetization equivalent sources, which ordinarily are unknown.

10.6 The equivalent circuit of an antenna In Example 9.4 we mentioned in passing that the dipole antenna constitutes a ‘load’ from the viewpoint of the generator. More generally, any single-port antenna can be characterized in the frequency domain by a complex impedance ZA (ω). When the antenna is employed as a transmitter to produce electromagnetic waves, ZA (ω) plays the role of a load connected to the generator, possibly through a piece of transmission line. Conversely, when used as a receiving device — i.e., to ‘collect’ electromagnetic waves and ‘absorb’ energy — the antenna can be modelled as an equivalent voltage generator with internal impedance ZA (ω). Intuition suggests that the strength of the equivalent generator is related to both the antenna shape and the wave impinging thereon. While the proof of this statement is an important application of the reciprocity theorem of Section 6.8, we present the subject at this point because it is based on the approximation of the fields in the Fraunhofer region of a source (Section 9.6) as well as the notion of Hertzian dipole (Example 9.3). We cannot help remarking that a rigorous solution of the problem of a receiving antenna — even when the latter is modelled as a PEC body — requires the calculation of the surface current density JS induced on the conducting parts and the port region as well as an accurate modelling of the electromagnetic field within the coaxial cable or waveguide that connects the antenna to the receiving network. For instance, a solution strategy could be based on formulating one or more integral equations (Chapter 13) and applying the Method of Moments (Chapter 14). In fact, what we seek here for an antenna in receiving mode is an approximate equivalent circuit which is valid so long as the following hypotheses can be made: (1)

(2)

(3)

the antenna is a PEC body except in the port region (cf. Figures 6.26 and 13.5a further on) where a coaxial cable or another transmission line provides the necessary connection with the generator (in transmitting mode) or the intended receiving circuit; the electromagnetic field in the line — whatever the cross section thereof — is described by the fundamental eigenfunction, typically a transverse-electric-magnetic (TEM) mode for classical transmission lines (cf. Example 6.4 and Section 11.2.4) [3, 23–25], [26, Chapter 2]; the antenna and the external radiating source — which generates the field to be received by the antenna under investigation — are located in the Fraunhofer region of one another (Section 9.5) so that the field produced by the antenna is not affected by the presence of the source and vice-versa [6, 27];

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Figure 10.17 Application of reciprocity to the region B(0, d) \ (VA ∪ B(rb, b)) for deriving the equivalent circuit of a receiving antenna; the boundary of the ball B(0, d) has been drawn (−−) only partially. (4)

the size of the antenna is such that the field generated by the external source can be locally approximated as a uniform plane wave (Section 7.2) [2, Chapter IV], [28, Chapter 11], [1].

The way we model the antenna port is bound to play an important role in the actual derivation of the result, even though the final form of the equivalent circuit is unaffected by the particular choice we make. To this purpose, we shall examine two configurations, namely, 1) a horn-like antenna mounted at the open end of a waveguide which in turn is connected to the generator or load by means of a coaxial cable, and 2) an arbitrary-shaped antenna whose port is modelled with the delta-gap approximation (cf. Figure 6.27). In either case, we assume that the underlying background medium is free space.

10.6.1 Antenna port connected to a coaxial cable We start off with the analysis of the horn-like antenna, which we assume hooked to a ‘black box’ via a coaxial cable whose inner conductor protrudes within the waveguide attached to the horn. The black box — whose walls are also made of PEC — is a mathematical expedient for enclosing either the generator in transmission or the intended antenna load in receiving mode. For reasons that will become clearer in the course of the proof it is convenient to set the origin of the main coordinates system at the open end of the coaxial cable. We call VA the finite region of space occupied by the antenna, the cable and the black box, we surround the external radiating source with a ball B(rb , b), and lastly we consider a ball B(0, d) whose radius d is large enough for B(0, d) to enclose both the antenna and the source. In a moment, we shall let d grow infinitely large and b infinitely small. This configuration is pictorially represented in Figure 10.17.

Integral formulas and equivalence principles

727

Figure 10.18 Close-up of the port region of the antenna in Figure 10.17.

In order to apply the Lorentz reciprocity theorem in global form to the surface-wise multiplyconnected region B(0, d) \ (VA ∪ B(rb , b)) we define the required two states as follows. State (a) State (b)

The generator (placed within the black box) is on, and thus the antenna behaves as a transmitter, whereas the source within B(rb, b) is inactive. The external source (surrounded by B(rb, b)) is on, and enclosed in the black box is now the receiving circuit attached to the other end of the cable.

Under the choices just stated the general formula (6.256) becomes  ˆ · [Ea (r) × Hb (r) − Eb (r) × Ha (r)] dS n(r) SA



+

ˆ · [Ea (r) × Hb (r) − Eb (r) × Ha (r)] dS n(r) ∂Bb

 ˆ · [Ea (r) × Hb (r) − Eb (r) × Ha (r)] = 0 dS n(r)

+

(10.119)

∂B

where S A := ∂VA is the surface that tightly wraps the antenna region, ∂Bb := {r ∈ R3 : |r − rb | = b}, ∂B := {r ∈ R3 : |r| = d}, and we have taken the unit normals oriented inwards B(0, d) \ (VA ∪ B(rb , b)). The reactions are null because the true sources are located either in VA or in B(rb, b) by assumption. Besides, in the limit as d → +∞ the flux integral over ∂B tends to zero, inasmuch as the fields produced by the antenna and the finite-sized source in B(rb, b) obey the Silver-Müller conditions (6.171) and (6.172). Indeed, we exploited the same argument in (6.260) to derive the reaction theorem (6.257) in the whole space. Next, we conceptually separate the closed surface S A into two adjoining open surfaces, viz., • •

S P (‘P’ standing for ‘port’) which coincides with the annular section of the truncated coaxial cable; S C (‘C’ a mnemonic for ‘conducting’) which is flush with the PEC bodies contained in VA .

Actually, owing to the presence of the protruding inner conductor of the cable, S C is made of at least two disjoint parts, as is exemplified in the enlarged view of Figure 10.18.

Advanced Theoretical and Numerical Electromagnetics

728

Then, in accordance with the time-harmonic counterpart of (1.168) and (1.169) the matching conditions satisfied by the electromagnetic fields on S A := S C ∪ S P read ⎧ ⎪ r ∈ SC ⎨−JS ν (r), ˆ =⎪ ν ∈ {a, b} (10.120) Hν (r) × n(r) ⎩I (ω)h(r) × n(r), ˆ r ∈ SP ν ⎧ ⎪ r ∈ SC ⎨0, ˆ × [Eν (r) × n(r)] ˆ n(r) =⎪ ν ∈ {a, b} (10.121) ⎩V (ω)e(r), r ∈ S ν P where • • •

JS ν (r), r ∈ S C , denotes the surface electric current density that is induced on the conducting parts of the antenna in either state; e(r) and h(r), r ∈ S P , indicate the electric and magnetic transverse eigenfunctions pertinent to the TEM mode in the coaxial cable (see Example 6.4); Vν (ω) and Iν (ω) are the voltages and currents associated with the TEM mode precisely for r ∈ S P , and they are related to either the generator’s strength in state (a) or the field radiated by the external source in state (b).

For convenience we choose the arbitrary amplitudes of e(r) and h(r) so that the transverse ˆ for r ∈ S P ) are real-valued vector fields [29–31] and also the eigenfunctions (perpendicular to n(r) normalization condition holds  ˆ · e(r) × h(r) = 1 dS n(r) (10.122) SP

in which case the average power carried by the TEM mode through S P reads ⎧1  1 ⎪ ⎪ ⎪ ˆ · Ea (r) × H∗a (r) = Re{Va Ia∗ } Re dS n(r) in state (a) ⎪ ⎪ ⎪ 2 2 ⎪ ⎪ ⎪ S ⎨ P  PF  = ⎪ ⎪ ⎪ 1 1 ⎪ ⎪ ⎪ ˆ · Eb (r) × H∗b (r) = − Re{Vb Ib∗ } in state (b) − Re dS n(r) ⎪ ⎪ ⎪ 2 2 ⎩

(10.123)

SP

ˆ points outwards VA (Figure 10.18), in state (b) the leading minus on account of (1.304). Since n(r) sign signifies that the average power supplied by the external source and captured by the horn enters the cable and flows towards the receiving circuit. We proceed by observing that the flux integral over S A can be computed as  ˆ · [Ea (r) × Hb (r) − Eb (r) × Ha (r)] = dS n(r) SA

 =

ˆ − Eb (r) · Ha (r) × n(r)] ˆ dS [Ea (r) · Hb (r) × n(r) SP

= Va I b − Vb I a

(10.124)

by virtue of the jump conditions (10.120), (10.121) and the normalization integral (10.122). To deal with the remaining flux integral we have to make assumptions on the source enclosed within B(rb, b). In real-life scenarios the field received by the antenna within VA is most likely produced by yet another radiating device. Nonetheless, since we have speculated that the antenna

Integral formulas and equivalence principles

729

and the external source sit in each other’s Fraunhofer regions, the fields Ea (r), Ha (r) detected by an observer within B(rb, b) are nearly spherical waves given by (9.317), (9.318) or possibly (9.325), (9.326). In like manner, also the fields Eb (r), Hb (r) in the neighborhood of the receiving antenna form a spherical wave which, however, we may locally approximate as a uniform plane wave on the grounds that the characteristic size of VA is ‘small’ compared to the distance (rb ) from the external source. In which instance −ˆrb gives the direction of propagation of said plane wave. Furthermore, since we seek a formula which involves the field impinging on VA but we are less interested in the true nature of the radiating device, we make the provisional hypothesis that Eb (r), Hb (r) are produced by an equivalent electric Hertzian dipole of moment j ωpb located at rb and oriented so that pb · rb = 0, as is shown in Figure 10.17. We remark that the Hertzian dipole is one more mathematical expedient for helping us express the field received by the real antenna within VA . Then, by following the same procedure detailed in Section 9.7 for the derivation of the Rayleigh-Carson theorem, we may conclude that in the limit as b → 0+ the integral over ∂Bb yields  ˆ · [Ea (r) × Hb (r) − Eb (r) × Ha (r)] = − j ωpb · Ea (rb ) (10.125) lim+ dS n(r) b→0

∂Bb

where the order of the terms in the integrand is reversed with respect to the one in (9.380) in view of ˆ the different orientation of the unit normal n(r), which here points outwards B(rb , b). By inserting the intermediate results (10.124) and (10.125) into (10.119) and letting d → +∞ we find the relationship Va Ib − Vb Ia − j ωpb · Ea (rb ) = 0

(10.126)

where we should be able to rid ourselves of pb if the latter really is inconsequential for the result, as we have surmised. Worse still, as it stands, (10.126) involves the electric field produced by the antenna in state (a) rather than the received field. Further progress we can make by writing Ea (rb ) ≈ − j ωμ0

e− j k 0 r b e− j k 0 r b PEJ (ϑb , ϕb ; ω) = − j ωμ0 heff (ϑb , ϕb )Ia 4πrb 4πrb

(10.127)

which is based on the combination of the electric Schelkunoff radiation vector (9.328) with either one of the far-field approximations (9.317), (9.325). We recall from (9.331) that the vector quantity heff (ϑ, ϕ) represents the effective length or height of the antenna within VA and depends only on the shape thereof [3, 27, 32]. The polar angles ϑb and ϕb identify the direction of the unit vector rˆ b , i.e., they point to the position of the auxiliary Hertzian dipole j ωpb . To write down the electric field produced by the Hertzian dipole in the origin of the main coordinates system (i.e., the antenna port) it is convenient to introduce local systems of Cartesian and polar spherical coordinates, viz., (ξb , ηb , ζb ) and (τb , αb , βb ), both centered on the point rb . In particular, we choose the ζb -axis and the polar axis parallel to one another and to the moment pb (Figure 10.17). Then, with the aid of (9.297) we obtain Eib (0, ϑb, ϕb ) ≈ − j ωμ0

e− j k 0 r b e− j k 0 r b (αˆ b αˆ b + βˆ b βˆ b ) · j ωpb = − j ωμ0 j ωpb 4πrb 4πrb

(10.128)

because by construction we have τb = rb , αb = π/2, pb = −pb αˆ b , and thus pb · αˆ b = −pb and pb · βˆ b = 0. The notation Eib (0, ϑb , ϕb ) — where ‘i’ stands for ‘incident’ — reminds us that this is the electric field radiated by the dipole in free space in the absence of the antenna within VA in the direction indicated by the polar angles ϑb and ϕb .

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Advanced Theoretical and Numerical Electromagnetics

Figure 10.19 Thevenin equivalent circuit of the antenna in receiving mode. Now, by inserting (10.127) into (10.126) we get Va Ib − Vb Ia = − j ωμ0

e− j k 0 r b j ωpb · heff (ϑb , ϕb )Ia = Eib (0, ϑb, ϕb ) · heff (ϑb , ϕb )Ia 4πrb

(10.129)

where the last step follows by comparison with the rightmost member of (10.128). Remarkably, the dipole moment j ωpb has dropped out of the picture after dutifully serving its purpose, and we have managed to turn the right-hand side of (10.126) into an expression which depends explicitly on (1) (2)

the incident electric field which illuminates the receiving antenna, the radiation properties (concealed in heff ) of the same antenna when the latter is used as a transmitting device.

We see from (10.128) that the incident electric field is aligned with pb and hence is perpendicular to rb by assumption (Figure 10.18). Therefore, we can equivalently interpret Ei (0, ϑb, ϕb ) as the electric field of a uniform plane wave which travels towards the antenna in the direction −ˆrb and is evaluated at the antenna port. This was, after all, one of the hypotheses we made at the very beginning. The final formula ought not depend on Va and Ia either, since these quantities belong to the antenna in transmitting mode. Thus, we divide (10.129) through by Ia to arrive at ZA Ib − Vb = Eib (0, ϑb, ϕb ) · heff (ϑb , ϕb )

(10.130)

where ZA := Va /Ia is the antenna input impedance ‘seen’ at the truncated section of the coaxial cable while looking into the horn. Lastly, we may forgo the inessential subscript ‘b’ and rearrange the terms to find [4, Section 8.7], [27, Section 2.15], [33, 34] V = ZA I − Ei (ϑ, ϕ) · heff (ϑ, ϕ)

(10.131)

with the proviso that E (ϑ, ϕ) signifies the electric field of the plane wave coming in from the direction given by (ϑ, ϕ) and evaluated at the antenna port. Evidently, (10.131) provides the desired Thevenin equivalent circuit [13, 35] of the antenna in receiving mode (Figure 10.19). The term Ei (ϑ, ϕ) · heff (ϑ, ϕ) plays the role of an ideal voltage generator whose strength changes with the direction of arrival of the incident plane wave and is the negative of the open-circuit voltage. The fact that ZA is the same impedance ‘seen’ by the generator when the antenna is in transmission mode is also a consequence of reciprocity. Indeed, we have been able to invoke (6.256) inasmuch as we have precluded the presence of non-reciprocal media anywhere in the region B(0, d) \ (VA ∪ B(rb , b)) in the first place. The specific values of V and I in (10.131) can be computed by closing the circuit with the impedance ZL ‘seen’ at truncated section of the cable while looking into the cable itself. To understand i

Integral formulas and equivalence principles

731

Figure 10.20 Application of reciprocity to the region B(0, d) \ (VA ∪ B(rb, b)) for deriving the equivalent circuit of a receiving antenna when the port is modelled with the delta-gap approximation; the boundary of the ball B(0, d) has been drawn (−−) only partially. why the current I appears to enter the antenna, we recall from (10.120) that in our model I(ω)h(r), r ∈ S P , represents the magnetic field of the TEM mode excited in the cable by the incident plane ˆ wave. Since we have chosen the axis of the cable aligned with n(r) in the port region (Figure 10.18), as a result the current I is positively oriented towards the antenna and thus in the circuit of Figure 10.19 I enters the equivalent generator. Of course, this is just a matter of convention, and we can define a current Ir = −I (‘r’ for ‘received’) so as to conform with the usual setup in the Thevenin equivalent circuit of an electrical network. Indeed, we can convince ourselves that the electromagnetic power captured by the antenna actually flows into the load ZL with the aid of the formula for state (b) in (10.123). In symbols, we have 1 1 (10.132) PF  = − Re{VI ∗ } = Re{ZL }|I|2  0 2 2 since V = −ZL I.

10.6.2 Antenna port modelled with the delta-gap approximation We continue with the construction of the equivalent circuit of an arbitrary-shaped antenna with the port modelled in the delta-gap approximation [27, 32], [4, Section 7.13], [3, Section 20.2]. Much of the work has already been done and, as a matter of fact, we need only revise the way we compute the flux integral over S A in (10.119). As is suggested in Figure 10.20, we assume that the antenna occupies the region VA := V1 ∪ V2 ∪ WG , where V1 ∩ V2 = ∅. The volumes V1 and V2 are filled with PEC, whereas WG — a small cylindrical region of height hG — constitutes the antenna gap (see Example 6.8 for the hypotheses behind this concept). Concerning the details of the reciprocity theorem, in state (a) an ideal voltage generator is conceptually placed within the otherwise empty antenna gap, and consequently a potential difference

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Advanced Theoretical and Numerical Electromagnetics

Figure 10.21 Close-up of the port region of the antenna in Figure 10.20. Va := Φ1a − Φ2a is maintained between any two points r1 ∈ ∂WG ∩ ∂V1 and r2 ∈ ∂WG ∩ ∂V2 . Indeed, the surfaces ∂WG ∩ ∂V1 and ∂WG ∩ ∂V2 are equipotential, because the electromagnetic field is quasistatic in the gap by assumption (Section 9.8). On the contrary, in state (b) the receiving network (i.e., an equivalent impedance ZL ) is inserted in the gap and connected to r1 and r2 . Correspondingly, a current −Ib flows from the contact point on V1 down to the contact point on V2 , and this in turn causes a voltage drop Vb := Φ1b − Φ2b to appear across the gap. The boundary S A of the antenna is the closed surface comprised of the non-overlapping parts of ∂V1 , ∂WG and ∂V2 . To proceed we separate S A into two parts, namely, S P := ∂WG ∩ S A and S C := S A \ S P . More precisely, S P is the lateral surface of the cylinder WG , whereas S C consists of two disjoint parts. Besides, on S P we introduce a local system of curvilinear orthogonal coordinates ˆ and vˆ . The latter satisfy (ξ, η), ξ ∈ [0, 2π], η ∈ [0, hG ], that are associated with the unit vectors u(r) (Figure 10.21) ˆ := u(r) ˆ × vˆ , n(r)

r ∈ SP

(10.133)

ˆ with vˆ aligned to the axis of the cylinder WG and tangent to the coordinate lines γη . The vector u(r) is tangent to the other coordinate lines γξ . Since by assumption the electric fields Ea (r) and Eb (r) obey the time-harmonic counterpart of the jump condition (1.169) for r ∈ S C , only the fields over S P contribute to the integral over S A . In symbols, we have  ˆ · [Ea (r) × Hb (r) − Eb (r) × Ha (r)] = dS n(r) SA

 ˆ × vˆ · [Ea (r) × Hb (r) − Eb (r) × Ha (r)] dS u(r)

= SP



=

 ˆ · Ea (r) vˆ · Hb (r) − dS u(r)

SP

 −

ˆ · Hb (r) vˆ · Ea (r) dS u(r) SP



ˆ · Eb (r) vˆ · Ha (r) + dS u(r) SP

ˆ · Ha (r) vˆ · Eb (r) dS u(r)

(10.134)

SP

by virtue of (10.133) and (H.16). To a good approximation we may assume that Ea (r) and Eb (r) ˆ are perpendicular to u(r) for r ∈ S P and depend only on η, because the electromagnetic field is

Integral formulas and equivalence principles

733

quasi-static in the gap. For the same reason, we may take Ha (r) and Hb (r), r ∈ S P , essentially perpendicular to vˆ and depending only on ξ. Under these conditions the surface integrals in the rightmost-hand side of (10.134) either vanish or turn into the product of two line integrals, namely,  ˆ · [Ea (r) × Hb (r) − Eb (r) × Ha (r)] = dS n(r) SA

 =−

 ˆ · Hb (r) vˆ · Ea (r) + dS u(r)

SP

ds vˆ · Ea (r)

=−

SP





γη

ˆ · Ha (r) vˆ · Eb (r) dS u(r) 

ˆ · Hb (r) + ds u(r) γξ

 ds vˆ · Eb (r)

γη

ˆ · Ha (r) ds u(r) γξ

    =Va

=Ib

=−Vb

Ib

= Va Ib − Vb Ia

(10.135)

where the last step is possible because both Ea (r) and Eb (r) can be derived from a quasi-static electric potential as in (2.15) and both Ha (r) and Hb (r) obey the Ampère law (4.5). After inserting (10.135) back into (10.119), the derivation proceeds from (10.126) as before and once again we arrive at (10.131). If we compute the flux of the complex Poynting vector through S A in state (b), viz.,   1 1 ˆ · Eb (r) × H∗b (r) = − Re dS u(r) ˆ × vˆ · Eb (r) × H∗b (r) PF  = − Re dS n(r) 2 2 S S A  P  1 1 ˆ · H∗b (r) = Re dS vˆ · Eb (r) uˆ (r) · H∗b (r) = Re ds vˆ · Eb (r) ds u(r) 2 2 SP

γη

γξ

1 1 = − Re{VI ∗ } = Re{ZL }|I|2  0 (10.136) 2 2 we conclude that also the delta-gap model correctly predicts a positive power flow into the load ZL when the antenna is in receiving mode.

References [1] [2] [3] [4] [5] [6] [7] [8] [9]

Rothwell EJ, Cloud MJ. Electromagnetics. London, UK: CRC Press; 2001. Stratton JA. Electromagnetic theory. London, UK: McGraw-Hill; 1941. Orfanidis SJ. Electromagnetic Waves and Antennas. www.ece.rutgers.edu/~orfanidi/ewa; 2004. Van Bladel JG. Electromagnetic Fields. Piscataway, NJ: IEEE Press; 2007. Harrington RF. Time-harmonic Electromagnetic Fields. London, UK: McGraw-Hill; 1961. Balanis CA. Advanced Engineering Electromagnetics. New York, NY: John Wiley & Sons, Inc.; 2005. Stratton JA, Chu LJ. Diffraction Theory of Electromagnetic Waves. Physical Review. 1939;56:99–107. Müller C. Foundations of the Mathematical Theory of Electromagnetic Waves. Berlin Heidelberg: Springer-Verlag; 1969. Colton DL, Kress R. Inverse Acoustic and Electromagnetic Scattering Theory. 3rd ed. New York, NY: Springer; 2013.

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Fradin AZ. Microwave Antennas. Oxford, UK: Pergamon Press; 1961. Jones DS. The Theory of Electromagnetism. Oxford, UK: Pergamon; 1964. Kottler F. Electromagnetische Theorie der Beugung an schwarzen Schirmen. Ann der Physik. 1923;71:457. Brenner E, Javid M. Analysis of electric circuits. 2nd ed. McGraw-Hill electrical and electronic engineering series. London, UK: McGraw-Hill; 1967. Lindell IV. Huygens’ principle in electromagnetics. Science, Measurement and Technology, IEE Proceedings. 1996 Mar;143(2):103–105. Love AEH. The Integration of the Equations of Propagation of Electric Waves. Philos Trans R Soc London, Ser A. 1901;197:1–45. Baker BB, Copson ET. The Mathematical Theory of Huygens’ Principle. 2nd ed. Oxford, UK: Clarendon Press; 1950. Bouwkamp CJ. Diffraction Theory. Repts Progr Phys. 1954;17:35. Schelkunoff SA. Some Equivalent Theorems of Electromagnetics and their Application to Radiation Problems. Bell Sys Tech J. 1936;15:92. Schelkunoff SA. On Diffraction and Radiation of Electromagnetic Waves. Phys Rev. 1939;56:308. Huygens C. Traité de la lumière. 2nd ed. Idem. Dunod; 2015. Slater JC, Frank NH. Electromagnetism. New York, NY: McGraw-Hill; 1947. Collin RE. Field Theory of Guided Waves. Piscataway, NJ: IEEE press; 1991. Adler RB, Chu LJ, Fano RM. Electromagnetic energy transmission and radiation. New York, NY: John Wiley & Sons, Inc.; 1960. Popovi´c Z, Popovi´c BD. Introductory Electromagnetics. Upper Saddle River, NJ: Prentice Hall; 2000. Shen LC, Kong JA. Applied electromagnetism. Monterey, CA: Brooks/Cole; 1983. Kong JA. Electromagnetic Wave Theory. 2nd ed. New York, NY: Wiley; 1990. Balanis CA. Antenna Theory: Analysis and Design. 2nd ed. New York, NY: John Wiley & Sons, Inc.; 1997. Helrich CS. The classical theory of fields - Electromagnetism. Berlin Heidelberg: SpringerVerlag; 2012. Pozar D. Microwave Engineering. 4th ed. New York, NY: John Wiley & Sons, Inc.; 2012. Marcuvitz N. Waveguide Handbook. 2nd ed. Electromagnetic Waves Series. London, UK: The Institution of Engineering and Technology; 1985. Felsen LB, Marcuvitz N. Radiation and scattering of waves. Piscataway, NJ: IEEE Press; 2001. King RWP. The Theory of Linear Antennas. Cambridge, MA: Harvard University Press; 1956. Sinclair G. The Transmission and Reception of Elliptically Polarized Waves. Proceedings of IRE. 1950 February;38:148–151. Park PK, Tai C. Receiving Antennas. In: Lo YT, Lee SW, editors. Antenna Handbook: Theory, Applications, and Design. Boston, MA: Springer US; 1988. p. 347–378. Available from: https://doi.org/10.1007/978-1-4615-6459-1_6. Millmann J, Grabel A. Microelectronics. 2nd ed. New York, NY: McGraw-Hill; 1988.

[13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34]

[35]

Chapter 11

Spectral representations of electromagnetic fields

A possible way of solving the set of Maxwell differential equations in a region of space V ⊆ R3 — especially when an analytic approach is unfeasible — consists of writing the unknown electromagnetic entities as linear combination of suitable vector fields. Then, the coefficients of the expansions are ‘adjusted’ in order to achieve, in some sense, the ‘best’ approximation to the desired solution: this is the essence of the Finite Element Method [1–6], [7, Chapter 6], [8, Chapter 9]. In applying this technique particularly convenient is the set of functions which solve the source-free instance of the Maxwell equations in local form or, more generally, some closely related problem [9–13]. The relevant source-free solutions — usually called modes or eigenfunctions — constitute the spectrum of the operator of concern. Which is why a linear superposition of modes is called a spectral or modal representation of the field. As it turns out, if the region V is bounded or unbounded in one spatial dimension the modes are infinite but denumerable. In otherwise unbounded regions, the modes form a continuum, and the linear superposition thereof necessarily becomes an integral. Even though the eigenfunctions are infinite, they usually have interesting properties, such as being orthogonal to one another with respect to a suitable inner product, and this facilitates the calculation of the expansion coefficients. In this chapter we discuss the spectral representation of the electromagnetic field in a cavity and in a cylindrical waveguide, both with metallic walls. Then, we investigate the important special case of layered structures in which the constitutive parameters depend periodically on one coordinate. Lastly, we examine the case of a continuum spectrum in unbounded layered structures, which are essentially two-dimensional problems and can be regarded as the limit of a waveguide whose crosssection grows infinitely large.

11.1 Modal expansion in cavities With the expression electromagnetic cavity we refer to a finite region of space V ⊂ R3 bounded by metallic walls and possibly filled with a penetrable medium other than air (e.g., a dielectric material). Similarly to lumped low-frequency electric circuits made up of capacitors and inductors, the main purpose of a cavity is to store electromagnetic energy in the field within V hopefully with small losses. In fact, such device would be of little or no use at all if it were perfectly closed and thus isolated from the outer space, i.e., the complementary region R3 \ V. For this reason, in practical configurations one or more apertures or holes S h are excised in the cavity wall ∂V in order to enable a two-way energy flow into and out of the region V [12, Section 7.4]. Cavities are employed especially at frequencies above 1 GHz and coupled to metallic waveguides (see Section 11.2) to realize, e.g., resonators, oscillators, and filters. Cavities of cuboidal or circular cylindrical shape are most common in that they are easily manufactured and integrated with waveguides of rectangular or circular cross-section.

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11.1.1 Vector eigenvalue problems in cavities When designing a microwave system which comprises a cavity it is important to know beforehand the electromagnetic field that the intended sources will generate in V. This goal can be conveniently achieved by first determining the modes of an ideal cavity with PEC walls and no apertures, i.e., the solutions of suitable vector eigenvalue problems specially set up for r ∈ V. In a subsequent step, said solutions can be used to represent the field in an actual cavity with the same shape as the ideal one, even if apertures are present and the metallic walls are lossy owing to high, though finite conductivity (cf. Table 7.2). In this section we focus on the mathematical aspects of the determination of the modes in a region V bounded by PEC walls and filled, in particular, with a lossless homogeneous isotropic penetrable medium [10, 12, 14–16]. We recall that, in accordance with the Helmholtz theorem in the form (8.57) or (8.60) for a finite domain V, a complex-valued vector field F(r) ∈ C2 (V)3 may be separated into lamellar and solenoidal parts. This observation suggests that in order to represent the electromagnetic entities within a cavity we employ sets of lamellar and solenoidal vector fields defined for r ∈ V. The latter may be found by solving the homogeneous Helmholtz equation ∇2 F(r) + ς2 F(r) = ∇∇ · F(r) − ∇ × ∇ × F(r) + ς2 F(r) = 0,

r∈V

(11.1)

with ς2 being a constant as yet unspecified. If we fail to enforce some constraint for F(r) on ∂V, then (11.1) ends up having infinitely many solutions — including, e.g., the well-known exp(− j ς|r|)/|r| — which depend continuously on ς and do not serve our purposes. If we plan on employing the solutions to (11.1) to expand the electric field in V, it seems logical to require that on ∂V F(r) obey the jump condition (1.169) at a PEC interface. Likewise, demanding that F(r) meet the matching condition (1.171) makes the solutions suitable to represent the magnetic field in a cavity with PEC walls. Still, either one constraint is not sufficient by itself to ensure that (11.1) admits a ‘unique’ eigenfunction for an admissible value of ς2 . To elaborate, since an eigenfunction is determined up to a multiplicative constant,1 here ‘uniqueness’ refers to the absence of multiple linearly independent eigenfunctions associated with the same ς2 . Conversely, the solution to (11.1) is not unique in the sense that infinitely many eigenfunctions characterized by distinct values of ς2 are indeed possible. To gain insight into the question, following Collin’s approach [12, Section 7.7], we suppose two solutions F1 (r) and F2 (r) have been found that are associated with the same constant ς2 = ς02 . We define the difference field F0 (r) := F1 (r) − F2 (r) which, thanks to the linearity of the Helmholtz operator, solves an instance of (11.1), too. We dot-multiply such equation with F∗0 (r) and integrate over V to obtain   ς02 dV |F0 (r)|2 = − dV F∗0 (r) · ∇2 F0 (r) V

 =

V

dV

F∗0 (r)

 · ∇ × ∇ × F0 (r) −

V



 dV |∇ × F0 (r)|2 −

= V

 −

dV ∇ · V

1 This



V

F∗0 (r)∇ ·

dV F∗0 (r) · ∇∇ · F0 (r)

V

  dV ∇ · F∗0 (r) × ∇ × F0 (r)   F0 (r) + dV |∇ · F0 (r)|2 V

follows by observing that if F(r) solves (11.1), then AF(r), with A an arbitrary non-null constant, is also a solution, by virtue of the linearity of the Helmholtz operator (∇2 + ς2 ){•} (Section D.3).

Spectral representations of electromagnetic fields  =

737

  dV |∇ × F0 (r)|2 + |∇ · F0 (r)|2

V

 −

  ˆ · F∗0 (r) × ∇ × F0 (r) + F∗0 (r)∇ · F0 (r) dS n(r)

(11.2)

∂V

where we have invoked the differential identities (H.59), (H.49) and (H.51), and eventually applied ˆ positively oriented towards the complementary the Gauss theorem (A.53) with the unit normal n(r) domain R3 \ V. Apparently, the expression in the rightmost member of (11.2) vanishes if we require ∇ · F0 (r) = 0,

∇ × F0 (r) = 0,

r∈V

(11.3)

r ∈ ∂V

(11.4)

and either ˆ × F0 (r) = 0, n(r)

∇ · F0 (r) = 0,

or ˆ · F0 (r) = 0, n(r)

ˆ × [∇ × F0 (r)] = 0, n(r)

r ∈ ∂V

(11.5)

whereby, provided ς02  0, we infer that F0 (r) = 0 in V, and ‘uniqueness’ is established. In summary, the solution to the Helmholtz equation (11.1) for a permissible ς2 is ‘unique’ provided (i) (ii)

the curl and the divergence of F(r) are assigned in V and two boundary conditions are enforced on ∂V.

Nevertheless, even when the above requirements are met, if the shape of V exhibits some degree of symmetry, then two or more linearly independent eigenfunctions may still be associated with the same eigenvalue, in which case we say they are degenerate. While two pairs of constraints alternative to (11.4) and (11.5) can be constructed from (11.2), the previous analysis shows that just the vanishing of the tangential or the normal component of F(r) on ∂V is not sufficient to determine ‘unique’ eigenfunctions. The second parts of (11.4) and (11.5) are consistent with representation (8.60), but we could not have guessed them merely on the grounds of the known jump conditions (1.168)-(1.171) on a PEC surface. As a result, we are led to formulate two eigenvalue problems [10, 12, 14–16] ⎧ 2 ⎪ ∇ e(r) + ς2 e(r) = 0, ⎪ ⎪ ⎪ ⎨ ˆ × e(r) = 0, n(r) ⎪ ⎪ ⎪ ⎪ ⎩∇ · e(r) = 0, ⎧ 2 ⎪ ∇ h(r) + ς2 h(r) = 0, ⎪ ⎪ ⎪ ⎨ ˆ · h(r) = 0, n(r) ⎪ ⎪ ⎪ ⎪ ⎩n(r) ˆ × [∇ × h(r)] = 0,

r∈V r ∈ ∂V

(electric modes)

(11.6)

(magnetic modes)

(11.7)

r ∈ ∂V r∈V r ∈ ∂V r ∈ ∂V

where •

the non-trivial solutions to (11.6) are referred to as electric modes because the jump condition ˆ × e(r) = 0 may be construed as the vanishing of the tangential electric field on the cavity n(r) walls and amounts to a short circuit in network terms;

738 •

Advanced Theoretical and Numerical Electromagnetics ˆ · the non-trivial solutions to (11.7) are said magnetic modes in that the matching condition n(r) h(r) = 0 may be interpreted as the vanishing of the normal magnetic field on ∂V and amounts to an open circuit in network terms.

Problems (11.6) and (11.7) allow obtaining two complete sets of eigenfunctions which are wellsuited to represent the electric and magnetic fields inside a cavity with PEC walls, but (11.6) and (11.7) do not follow from the time-harmonic Maxwell equations. As we shall show in Section 11.1.6, completeness means that any time-varying electromagnetic field in V can be expressed as a linear combinations of the infinite denumerable solutions to (11.6) and (11.7) (cf. [17, Chapter 6]). Further, by making use of the differential identity (H.59) we notice that (11.6) and (11.7) provide only a relation between the curl and the divergence of e and h, respectively. Thus, we are still free to enforce an additional condition in keeping with (11.3). To simplify the solution process we may require that either the divergence or the curl vanish within V. Modes for which ∇ · e(r) = 0,

∇ · h(r) = 0,

r∈V

(11.8)

r∈V

(11.9)

are said solenoidal (divergence-free), and modes which obey ∇ × e(r) = 0,

∇ × h(r) = 0,

are called lamellar (curl-free). Moreover, if the region V is multiply connected, it is possible to enforce the vanishing in V of both the curl and the divergence, in which instance (11.2) says that ς02 must be zero in order for a non-trivial solution to exist. Vector fields which are both lamellar and solenoidal are said harmonic. Hereinbelow we examine all possible cases.

11.1.2 Solenoidal modes In light of (11.1) and condition (11.8), (11.6) and (11.7) become ⎧ ⎪ −∇ × ∇ × e(r) + κ2 e(r) = 0, ⎪ ⎪ ⎪ ⎨ ∇ · e(r) = 0, ⎪ ⎪ ⎪ ⎪ ⎩n(r) ˆ × e(r) = 0, ⎧ ⎪ −∇ × ∇ × h(r) + κ2 h(r) = 0, ⎪ ⎪ ⎪ ⎨ ∇ · h(r) = 0, ⎪ ⎪ ⎪ ⎪ ⎩n(r) ˆ × [∇ × h(r)] = 0,

r∈V r∈V

(solenoidal electric modes)

(11.10)

(solenoidal magnetic modes)

(11.11)

r ∈ ∂V r∈V r∈V r ∈ ∂V

upon letting ς = κ. For solenoidal electric modes the vanishing of the divergence everywhere in V implies ∇ · e = 0 on the boundary by continuity. For solenoidal magnetic modes the condition ˆ · h(r) = 0 on ∂V is implied by the other one because n(r) ˆ · h(r) = n(r) ˆ · ∇ × ∇ × h(r) = −∇s · {n(r) ˆ × [∇ × h(r)]}, κ2 n(r)

r ∈ ∂V

(11.12)

on account of (A.60). Solenoidal eigenfunctions of either type are related to one another. Indeed, by taking the curl one more time in (11.11) we obtain −∇ × ∇ × [∇ × h(r)] + κ2 ∇ × h(r) = 0,

r∈V

(11.13)

Spectral representations of electromagnetic fields

739

and if we let κe(r) = ∇ × h(r),

r∈V

(11.14)

then (11.13) reduces to the eigenvalue problem (11.10) also because the vector field e(r) is evidently solenoidal. If we now take the curl of (11.14) we get κ∇ × e(r) = ∇ × ∇ × h(r) = κ2 h(r),

r∈V

(11.15)

whence κh(r) = ∇ × e(r),

r∈V

(11.16)

by virtue of (11.11). Thus, the solutions e(r) and h(r) are degenerate, being associated with the same eigenvalue, and we shall prove in Section 11.1.4 that κ2 is a real non-negative number. What is more, the volume integrals of |e(r)|2 and |h(r)|2 coincide. Dot-multiplying (11.14) with κe∗ (r) where κ ∈ R yields   2 2 dV |e(r)| = κ dV e∗ (r) · ∇ × h κ V

 =κ

V

dV h(r) · ∇ × e∗ − κ

V





=κ

dV |h(r)| − κ

2

2

V

∂V



dV ∇ · [e∗ × h]

V

ˆ · e∗ (r) × h(r) = κ2 dS n(r)   =0

 dV |h(r)|2

(11.17)

V

on account of (H.49), the Gauss theorem, (11.16) and the boundary condition for electric modes. Since (11.10) is formally identical with the source-free instance of the time-harmonic wave equation (1.234), non-trivial solutions to (11.10) physically represent the electric field in a loss-free cavity with PEC walls with the wavenumber k ∈ R given by κ, which indeed has the physical dimension of the inverse of a length. In like fashion, since (11.11) has the same structure as the homogeneous instance of the time-harmonic wave equation (1.236), the eigenfunctions h(r) may be interpreted as the magnetic field in a loss-free cavity with PEC walls. Moreover, (11.17) is compatible with (1.327), namely, the fact that average electric (We ) and magnetic (Wh ) energies in V are equal at a resonance. All in all, the existence of eigenfunctions which solve (11.10) and (11.11) is consistent with the result proved in Section 6.4.1, i.e., that the solution to the time-harmonic source-free Maxwell equations is not unique in lossless bounded regions of space. On the other hand, if we restricted the search for modes in V to just the solution of (11.10) and (11.11), we would not be able to represent all possible fields in the cavity, in particular, those which are not solenoidal!

11.1.3 Lamellar modes Enforcing (11.9) explicitly in (11.6) and (11.7) yields ⎧ ⎪ ∇∇ · e(r) + α2 e(r) = 0, r ∈ V ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ r∈V ⎨∇ × e(r) = 0, ⎪ ⎪ ⎪ ∇ · e(r) = 0, r ∈ ∂V ⎪ ⎪ ⎪ ⎪ ⎪ ⎩n(r) ˆ × e(r) = 0, r ∈ ∂V

(lamellar electric modes)

(11.18)

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Advanced Theoretical and Numerical Electromagnetics

⎧ ⎪ ∇∇ · h(r) + β2 h(r) = 0, r ∈ V ⎪ ⎪ ⎪ ⎨ ∇ × h(r) = 0, r∈V ⎪ ⎪ ⎪ ⎪ ⎩n(r) ˆ · h(r) = 0, r ∈ ∂V

(lamellar magnetic modes)

(11.19)

where we have renamed ς as α and β, since e(r) and h(r) will be in general associated with different eigenvalues because they obey different boundary conditions. The lamellar electric modes are further divided into two categories, namely, for α2  0 and 2 α = 0, and in both cases we may derive the field e(r) from an auxiliary scalar potential Υ(r). If the eigenvalue α2 is not zero, we may let e(r) =

∇Υ , α

r∈V

(11.20)

whereby the first equation in (11.18) passes over into   ∇ ∇2 Υ(r) + α2 Υ(r) , = 0

r∈V

(11.21)

which is certainly satisfied if we require ∇2 Υ(r) + α2 Υ(r) = Ce ,

r∈V

(11.22)

where Ce is an arbitrary constant. However, it is always possible to choose Ce = 0 and demand ∇2 Υ(r) + α2 Υ(r) = 0,

r∈V

(11.23)

since Υ(r) is determined up to yet another additive constant, which on account of (11.20) bears no consequence on the eigenfunction e(r). The jump conditions in (11.18) must be rephrased in terms of the auxiliary potential, viz., ∇2 Υ(r) = 0,

ˆ × ∇Υ(r) = 0, n(r)

r ∈ ∂V

(11.24)

and the first constraint along with (11.23) requires Υ(r) = 0,

r ∈ ∂V

(11.25)

because we have assumed α2  0 for the time being. The other constraint is superfluous in that it entails the derivative of Υ(r) along the boundary ∂V. Indeed, on account of (H.14) we have ˆ × [n(r) ˆ × ∇Υ(r)] = ∇Υ(r) − n(r) ˆ 0 = −n(r)

∂Υ = ∇s Υ(r), ∂nˆ

r ∈ ∂V

(11.26)

where ∇s {•} indicates the surface gradient, and this equation is satisfied as Υ(r) vanishes on ∂V. Being curl-free, e(r) can be construed as the static electric field produced by fixed electric charges in the region V bounded by a conducting medium, and Υ(r) is, in fact, an electrostatic potential (cf. Section 2.2). Accordingly, in (11.18) and (11.23) α does not represent a wavenumber, inasmuch as no oscillations are involved. We also notice that thanks to the division by α in (11.20) e(r) and Υ(r) carry the same physical dimension. Conversely, if the eigenvalue happens to be null, (11.18) becomes ⎧ ∇ · e(r) = 0, r∈V ⎪ ⎪ ⎪ ⎪ ⎨ ∇ × e(r) = 0, r∈V (11.27) ⎪ ⎪ ⎪ ⎪ ⎩n(r) ˆ × e(r) = 0, r ∈ ∂V

Spectral representations of electromagnetic fields

741

and we let e(r) = ∇Υ

(11.28)

whereby (11.23) passes over into the Laplace equation ∇2 Υ(r) = 0,

r∈V

(11.29)

with the only boundary condition nˆ × ∇Υ(r) = 0,

r ∈ ∂V

(11.30)

because the first requirement in (11.24) is automatically fulfilled on account of (11.29). Since in light of (11.26) condition (11.30) demands that the potential be constant on the walls of the cavity, the only possible solution is a potential Υ(r) which is everywhere constant in V, if the latter is bounded by a single closed surface. Indeed, from (11.28) and the Gauss theorem we have   2 ˆ · e(r) 0= dV ∇ Υ = dS n(r) (11.31) ∂V

V

 0=

dV Υ∗ (r)∇2 Υ =



V

V

 = Υ∗0





dV |∇Υ|

∂V



dV |∇Υ| = −

dV |e(r)|2

2

V

 

dV |∇Υ|2

2

V

ˆ · e(r) − dS n(r)

 V



∗

ˆ · e(r)Υ (r) − dS n(r) ∂V

=

dV ∇ · [Υ∗ (r)∇Υ] −

(11.32)

V

=0

where Υ0 is the constant value of the potential on the wall ∂V. Since the quantity in the rightmosthand side is always non-positive, it follows that e(r) = 0 in V. By contrast, if the region V is surface-wise multiply connected, i.e., the boundary ∂V is comprised of two or more non-intersecting closed surfaces (see Figure 2.6a) then non-trivial solutions to (11.29) are possible. For the sake of argument, we suppose that ∂V := S 1 ∪ S 2 with S 1 ∩ S 2 = ∅, and apply the Gauss theorem as before, namely,    2 0= dV ∇ Υ = dS nˆ 1 (r) · e(r) + dS nˆ 2 (r) · e(r) (11.33) V

S1

 0=

∗

dV Υ (r)∇ Υ = V

∗

∗

= Υ∗01

 S1

dV |∇Υ|2 V



dS nˆ 1 (r) · e(r)Υ (r) + S1



dV ∇ · [Υ (r)∇Υ] − V

 =

S2



2



∗

dS nˆ 2 (r) · e(r)Υ (r) − S2

dS nˆ 1 (r) · e(r) + Υ∗02



 dS nˆ 2 (r) · e(r) − S2

dV |∇Υ|2 V

dV |e(r)|2

(11.34)

V

where Υ01 and Υ02 are the constant values of Υ(r) on S 1 and S 2 . When Υ01 and Υ02 are not coincident, then we may not invoke (11.33) to conclude that the last integral in (11.34) must vanish. As a result, harmonic electric eigenfunctions associated with α = 0 can be found.

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Advanced Theoretical and Numerical Electromagnetics

Since e(r) is both lamellar and solenoidal, from a physical viewpoint it represents the electrostatic field that exists in V when the various metallic surfaces which make up ∂V are raised to different, constant potentials. A practical geometry is represented by a spherical capacitor where V is the spatial region between two concentric metallic spheres of which the larger is hollow so as to accommodate the smaller one (Example 2.6). The lamellar magnetic modes, too, are further divided into two sub-sets, namely, for β2  0 and β2 = 0. If the eigenvalue β2 is not zero, we may let h(r) =

∇Υ , β

r∈V

whereby (11.19) passes over into   ∇ ∇2 Υ(r) + β2 Υ(r) = 0,

(11.35)

r∈V

(11.36)

which we can satisfy by demanding ∇2 Υ(r) + β2 Υ(r) = 0,

r∈V

(11.37)

thanks to the arbitrary additive constant implied in definition (11.35). The matching condition in (11.7) becomes ˆ · ∇Υ(r) = n(r)

∂Υ = 0, ∂nˆ

r ∈ ∂V.

(11.38)

Since h(r) is lamellar but not solenoidal, we may interpret it as the static magnetic field generated by fixed magnetic charges (Section 6.5) present in the region V bounded by a PEC surface. Thus, the scalar potential Υ(r) is single-valued because it is generated by charges, albeit magnetic, rather than steady electric currents (cf. Section 4.4). Also in this case, in (11.19) and (11.23) β does not represent a wavenumber, since the field configuration is static. In view of definition (11.35) the physical dimensions of h(r) and Υ(r) coincide. If the eigenvalue β2 does vanish, (11.19) becomes ⎧ ∇ · h(r) = 0, ⎪ ⎪ ⎪ ⎪ ⎨ ∇ × h(r) = 0, ⎪ ⎪ ⎪ ⎪ ⎩n(r) ˆ · h(r) = 0,

r∈V r∈V

(11.39)

r ∈ ∂V

and non-trivial solutions are possible only if the region V is contour-wise multiply connected. The prototypical geometry is represented by the inside of a torus (see Figure 4.11). We may derive h(r) from either a scalar potential Υ(r) — which must be many-valued (cf. Section 4.4) — or a vector potential V(r). Indeed, if we let h(r) = ∇Υ,

r∈V

(11.40)

r∈V

(11.41)

we obtain the Laplace equation ∇2 Υ(r) = 0,

Spectral representations of electromagnetic fields

743

subject to the boundary condition (11.38). From (11.41) we derive    0= dV Υ∗ (r)∇2 Υ = dV ∇ · [Υ∗ (r)∇Υ] − dV |∇Υ|2 V

V

 =

∗

dS Υ (r) nˆ (r) · h(r) −   =0

S

V



 dV |∇Υ| = −

dV |h(r)|2

2

V

(11.42)

V

where we have applied the Gauss theorem (A.53) under the provisional hypothesis that Υ(r) is singlevalued in V. But then, since the normal component of h(r) vanishes on ∂V, (11.42) necessarily requires that Υ(r) be constant in V, whence we conclude that h(r) is everywhere null in the cavity. Thus, in order for h(r) to be a non-trivial eigenfunction of (11.39) the generating scalar potential must be many-valued so that (11.42) does not apply as it stands. On the contrary, if we let h(r) = ∇ × V(r),

r∈V

(11.43)

r∈V

(11.44)

(11.39) passes over into ∇ × ∇ × V(r) = 0, with the only jump condition ˆ · ∇ × V(r) = −∇s · [n(r) ˆ × V(r)] = 0, n(r)

r ∈ ∂V

(11.45)

on account of (A.60). For (11.45) to be true the normal component of ∇ × V must vanish on the ˆ × V(r) has to be solenoidal, though not necessarily null for cavity boundary or, equivalently, n(r) r ∈ ∂V. If we dot-multiply (11.44) with V∗ (r) and integrate over the region V we obtain  dV V∗ (r) · ∇ × ∇ × V(r) 0= V

 =

 dV |∇ × V(r)|2 +

V



 dV |h(r)| +

=

2

dV ∇ · {[∇ × V(r)] × V∗ (r)}

V

ˆ · h(r) × V∗ (r) dS n(r)

(11.46)

∂V

V

and since the tangential components of h(r) and V(r) do not vanish on ∂V, we cannot infer that h(r) must vanish in V. The eigenfunctions which solve (11.39) are called Neumann vector fields, and physically represent the stationary magnetic field produced by a steady electric current that flows on ∂V (see Section 4.7.2 and Example 4.4).

11.1.4 Orthogonality properties of the cavity eigenfunctions We begin by showing that with the chosen matching conditions for r ∈ ∂V the Laplace operator ∇2 {•} is self-adjoint [18, Section 10.4] (also see the discussion on page 385 and Appendix D.6). A suitable three-dimensional inner product is (Appendix D.1) 

 (f, g)V := dV f ∗ (r) · g(r) = (g, f)V ∗ (11.47) V

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Advanced Theoretical and Numerical Electromagnetics

where f(r) and g(r) denote twice-differentiable real- or complex-valued vector fields defined for r ∈ V. Then, on account of the differential identities (H.59), (H.51) and (H.49) and the Gauss theorem (A.53) we have    

 dV f ∗ (r) · ∇2 g(r) = dV f ∗ (r) · ∇∇ · g(r) − ∇ × ∇ × g(r) f, ∇2 g = V

V

 =

V ∗

dV {∇ · [f (r)∇ · g(r)] − ∇ · f ∗ (r) ∇ · g(r)}

V

 +

 =

dV {∇ · [f ∗ (r) × ∇ × g(r)] − ∇ × f ∗ (r) · ∇ × g(r)}

V

ˆ · {f ∗ (r)∇ · g(r) + f ∗ (r) × [∇ × g(r)]} dS n(r)

∂V

 −

dV [∇ · f ∗ (r) ∇ · g(r) + ∇ × f ∗ (r) · ∇ × g(r)]

(11.48)

V

where the flux integral vanishes invariably if f(r) and g(r) are two electric modes [see (11.6)] or two magnetic modes [see (11.7)]. In like manner and for the same reasons it is found that    ∇2 f, g = − dV [∇ · f ∗ (r) ∇ · g(r) + ∇ × f ∗ (r) · ∇ × g(r)] (11.49) V

V

whence if follows     f, ∇2 g = ∇2 f, g V

(11.50)

V

by comparison with (11.48). An immediate consequence of (11.50) is that the eigenvalue ς2 is always real and non-negative. Indeed, from (11.6) we have     (11.51) 0 = e, ∇2 e + ς2 e = e, ∇2 e + ς2 (e, e)V V V      ∗ 0 = ∇2 e + ς2 e, e = ∇2 e, e + ς2 (e, e)V (11.52) V

V

and subtracting these identities side by side yields   ∗    ∗    0 = ς2 − ς2 (e, e)V = ς2 − ς2 e 2V = 2 j Im ς2 e 2V

(11.53)

whence the real nature of ς2 follows because the norm of e(r) is non-null. Besides, by solving (11.51) for ς2 and invoking (11.48) we obtain an expression for the Rayleigh quotient    e, ∇2 e   1 V ς2 = − = dV |∇ · e(r)|2 + |∇ × e(r)|2  0 (11.54) 2 (e, e)V e V V

and since the quantity in the second part is non-negative, so is the eigenvalue. Similar proofs hold for the magnetic modes (11.7). All in all, this means in practice that we may conveniently choose real-valued eigenfunctions e(r) and h(r), and we shall make this assumption from now on.

Spectral representations of electromagnetic fields

745

Two solenoidal or lamellar electric eigenfunctions e1 (r) and e2 (r) associated with different eigenvalues ς12 and ς22 are orthogonal with respect to the inner product (11.48). From (11.6) we have     (11.55) 0 = e1 , ∇2 e2 + ς22 e2 = e1 , ∇2 e2 + ς22 (e1 , e2 )V V    V 0 = e2 , ∇2 e1 + ς12 e1 = e2 , ∇2 e1 + ς12 (e2 , e1 )V (11.56) V

V

and by subtracting these relations side by side we get   ς22 − ς12 (e2 , e1 )V = 0

(11.57)

by virtue of (11.50) and the fact that the inner product (11.47) becomes symmetric when the operands are real-valued. Since the eigenvalues are different by hypothesis, the orthogonality of e1 (r) and e2 (r) follows immediately. Mutatis mutandis, this derivation shows that two solenoidal or lamellar magnetic eigenfunctions are likewise orthogonal. Lastly, solenoidal and lamellar eigenfunctions are orthogonal to one another by construction (cf. Section 8.1). Assuming that e1 (r), h1 (r) are solenoidal and e2 (r), h2 (r) are lamellar we have    1 1 1 (e1 , e2 )V = dV e1 (r) · ∇Υ2 = dV ∇ · (e1 Υ2 ) − dV Υ2 (r)∇ · e1 α2 α2 α2 V V V  1 ˆ · e1 (r) Υ2 (r) = 0 = dS n(r) (11.58)  α2 =0

∂V

on account of (11.25), and    1 1 1 (h1 , h2 )V = dV h1 (r) · ∇Υ2 = dV ∇ · (h1 Υ2 ) − dV Υ2 (r)∇ · h1 β2 β2 β2 V V V  1 ˆ · h1 (r) Υ2 (r) = 0 = dS n(r)   β2 ∂V

(11.59)

=0

by virtue of the boundary condition for solenoidal magnetic modes. The norm of the lamellar eigenfunctions associated with non-null eigenvalues may be rephrased in terms of the auxiliary potential Υ(r). For an electric mode associated with α2 > 0 this means    |∇Υ|2 1 1 e 2V = dV = dV ∇ · [Υ(r)∇Υ] − dV Υ(r)∇2 Υ α2 α2 α2 V V V   1 ˆ · ∇Υ(r) + dV |Υ(r)|2 = Υ 2V = 2 dS Υ(r)n(r) (11.60) α ∂V

V

where the flux integral is null on account of (11.38). A similar formula holds for lamellar magnetic modes provided β2 > 0.

11.1.5 Stationarity of the Rayleigh quotient We may regard the Rayleigh quotient (11.54) as a linear operator (Appendix D.3) or functional, say, ⎧ + ⎪ ⎪ ⎨ VL −→ R  (11.61) L {•} : ⎪ ⎪ ⎩ g(r) −→ g, −∇2 g / g 2V V

746

Advanced Theoretical and Numerical Electromagnetics

that takes vector fields g(r) defined in the cavity V [see (D.64)] and returns a real non-negative number. For the existence of ∇2 g as an ordinary function the domain VL is comprised of fields g(r) that are at least continuously differentiable for r ∈ V. If, in addition, we agree to restrict VL to realvalued vector fields which on ∂V obey the same boundary conditions as the electric modes (11.6) or the magnetic modes (11.7), then the Rayleigh quotient is stationary in the ‘neighborhood’ of an eigenfunction. In words, stationarity means that the quotient is ‘insensitive’ to reasonably small variations of e(r) and h(r) [11, Section 7.5]. To elucidate, we consider the vector field e˜ ν (r) := eν (r) + hu(r),

h ∈ R+

r ∈ V,

(11.62)

where ν := (m, n, p) ∈ N3 indicates a triple of whole indices, eν (r) is the electric eigenfunction associated with ςν2 , h  1 is a dimensionless parameter, and u(r) ∈ C2 (V)3 is a vector field in V that obeys the same boundary conditions as eν (r) but is not required to solve (11.6). For relatively small values of h, we may regard e˜ ν (r) as a ‘perturbation’ of eν (r) if the latter is known or, more generally, an approximation to eν (r). When we insert e˜ ν (r) in the right-hand side of (11.54) and carry out the prescribed integrals, we do not obtain the eigenvalue ςν2 but rather we end up with a positive number or, more precisely, a positive function of h, namely,       e˜ ν , ∇2 e˜ ν eν + hu, ∇2 eν + h∇2 u eν + hu, ςν2 eν − h∇2 u V V V f (h) = − =− = (˜eν , e˜ ν )V (eν + hu, eν + hu)V (eν + hu, eν + hu)V     ςν2 eν 2V + hςν2 (u, eν )V − h eν , ∇2 u − h2 u, ∇2 u V V = eν 2V + 2h (eν , u)V + h2 u 2V   ςν2 eν 2V + 2hςν2 (eν , u)V + h2 u, −∇2 u V = 0 (11.63) eν 2V + 2h (eν , u)V + h2 u 2V where we have recalled that eν (r) does solve (11.6), applied relation (11.50), and exploited the symmetry of the inner product (11.47) for real-valued fields. Direct calculation shows that the first derivative of f (h) vanishes for h = 0, viz.,  2ς2 (eν , u)V eν 2V − 2ςν2 (eν , u)V eν 2V d f   = ν f  (0) = =0 (11.64) dh h=0 eν 4V whereby we can write the Taylor power expansion of f (h) around h = 0 as  d2 f  h2 h2    2 = ς f h , f (h) = f (0) + + h1  ν 2 dh2 h=h 2

(11.65)

where h ∈ [0, h] is a suitable value [19, Formula 38.1]. Since we may consider f (h) an approximation of the true eigenvalue ςν2 , the formula above tells us that the error | f (h) − ςν2 | decreases quadratically with h. Further, from (11.62) the error on the eigenfunction eν (r) reads ˜eν − eν V = h u V

(11.66)

and by solving this algebraic equation for h we may write      ˜eν − eν 2V  f (h) − ςν2  =  f  h  , 2 u 2V

h1

(11.67)

Spectral representations of electromagnetic fields

747

whence it becomes apparent that the error on ςν2 diminishes with the square of the error on eν (r). Therefore, if a ‘good’ approximation of the true eigenfunction eν (r) is available, then (11.54) returns an even better approximation of the associated eigenvalue. This property is, in essence, the stationary character of the Rayleigh quotient. Whether (11.54) and the analogous functional for magnetic modes reach a minimum or a maximum when e(r) or h(r) are any of the true eigenfunctions largely depends on the difference field hu(r) in (11.62). In principle, we can write u(r) as a linear superposition of electrical eigenfunctions, say,  (eυ , u)V uυ eυ (r), uυ = , r∈V (11.68) u(r) = eυ 2V υ where we may either compute the coefficients uυ as indicated or, quite more simply, assign them to come up with a suitable vector field which meets the electric boundary conditions on ∂V. Of course, this step heavily relies on our knowing the electric eigenfunctions beforehand, and if all of them were available we would not need (11.54) to find approximations to the eigenvalues! What we are trying to do here is investigate the nature of the extrema of the Rayleigh quotient by exploiting the theoretical knowledge of the eigenfunctions in the process. This point been clarified, from (11.63) and (11.68) we have   u, −∇2 u − ςν2 u 2V V 2 2 f (h) = ςν + h ˜eν 2V   h2    = ςν2 + uυ u, −∇2 eυ − ςν2 uυ (u, eυ )V 2 V ˜eν V υ  h2   2 2 − ς (11.69) = ςν2 + ς u2υ eυ 2V υ ν ˜eν 2V υ having made use of (11.6) with e(r) = eυ (r) and ς2 = ςυ2 . If ςν2 is the smallest electric eigenvalue, then the summation in the rightmost-hand side of (11.69) surely is a positive number, since ςυ2  ςν2 for all υ. In which instance we conclude that f (h) is invariably larger than ςν2 , and the latter is a minimum of (11.54). When ςν2 is any other eigenvalue, the nature of the extremum is not clear-cut, since the result of the summation can be positive or negative. Nonetheless, we may take u(r) orthogonal to all the eigenfunctions eυ (r) such that ςυ2  ςν2 . This means that the terms of the summation are either null, because uυ = 0, or positive, in that ςυ2 > ςν2 , whereby ςν2 is again a minimum of (11.54).

11.1.6 Completeness of the cavity eigenfunctions We now prove an assertion made at the very beginning of this section, namely, that the infinite sets {eν (r)}ν and {hν (r)}ν of electric and magnetic eigenfunctions obtained by solving (11.6) and (11.7) are complete. This means that in a cavity with PEC walls any time-varying electromagnetic field which solves the Maxwell equations can be expanded as  (eυ , E)V cυ (t)eυ (r), cυ (t) = , r∈V (11.70) E(r, t) = eυ 2V υ  (hυ , H)V dυ (t)hυ (r), dυ (t) = , r∈V (11.71) H(r, t) = hυ 2V υ

748

Advanced Theoretical and Numerical Electromagnetics

where the sums extend over all the modes in the cavity. While in principle the expansion coefficients may be functions of time, in the following we lighten the notation by omitting the explicit dependence on t, inasmuch as this feature is inconsequential for the result. Since our proof will make use of the Rayleigh quotient as a functional given by (11.61), we need to assume that E(r) and H(r) are at least continuously differentiable in V, whereby ∇2 E and ∇2 H are piecewise continuous. Focusing on the electric quantities for the sake of argument, we define the difference field (cf. [17, pp. 771 and ff.])   ˜ := E(r) − Eν (r) = E(r) − E(r) cυ eυ (r) = cυ eυ (r), r∈V (11.72) υ∈Iν

υIν

where Iν ⊂ N3 denotes the subset of triples υ := (m, n, p) which identify the eigenvalues ςυ2 smaller ˜ than ςν2 . We may regard Eν (r) as an approximation to E(r) and E(r) as the approximation error. In the limit as we broaden Iν in order to encompass the triples associated with all the eigenvalues, the ˜ norm of E(r) should approach zero. If this is the case, then the set {e(r)}υ is complete. First of all, we show that the expansion coefficients cυ , υ ∈ Iν , in the approximation Eν (r) can ˜ be chosen so as to minimize the norm of the difference field E(r). From definition (11.47) of inner product in the cavity we have ⎛ ⎞   ⎟⎟⎟  2 ⎜⎜⎜ ⎜ ˜  = ⎜⎜E(r) − E cυ eυ (r), E(r) − cυ eυ (r)⎟⎟⎠⎟ ⎝ V υ∈Iν υ∈Iν V   2 2 2 = E V + cυ eυ V − 2 cυ (eυ , E)V (11.73) υ∈Iν

υ∈Iν

on account of the orthogonality of the electric eigenfunctions eυ (r) (Section 11.1.4). Since the expression in the right-hand side is a real and positive function of cυ , υ ∈ Iν , we obtain a minimum if weadjust  the coefficients so as to nullify each and every component of the multidimensional gradient ˜ 2 with respect to cυ , viz., of E V ∂  ˜ 2 EV = 2cυ eυ 2V − 2 (eυ , E)V = 0, ∂cυ

υ ∈ Iν

(11.74)

which, when solved explicitly for cυ , yields the expression anticipated in (11.70). Further, inserting the result back into (11.73) gives   2 ˜  = E 2 − E c2υ eυ 2V  0 (11.75) V V υ∈Iν

and, since the norm is a non-negative real number, from the definition of Eν (r) in (11.72) we have  Eν 2V = c2υ eυ 2V  E 2V (11.76) υ∈Iν

which shows that, with the chosen coefficients, the summation introduced in (11.70) converges because on the one hand the set Iν can be broadened to encompass an arbitrary large number of eigenvalues whereas, on the other, the norm of E(r) is finite since the latter field is at least continuously differentiable by assumption. We are left with the task of proving that Eν (r) actually converges to E(r). To proceed, we observe that by using the expansion coefficients provided by (11.74) in the first ˜ part of (11.72) it becomes immediately evident that E(r) is orthogonal to all eigenfunctions eυ (r),

Spectral representations of electromagnetic fields

749

˜ as a linear υ ∈ Iν , with respect to the inner product (11.47). Therefore, it is possible to write E(r) 2 2 superposition of the eigenfunctions associated with all the eigenvalues ςυ  ςν , υ  Iν , as is stated in ˜ the last part of (11.72). Then, we evaluate the Rayleigh quotient (11.61) in E(r) by making use of both representations for the difference field in (11.72), viz.,   ˜ ∇2 E ˜ E, 1 −  2 V =  2 E E ˜  ˜  V V 1 =  2 E ˜  V   ˜ ˜ ∇2 E E, 1 −  2 V =  2  E  ˜ ˜  E V

⎛ ⎞   ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜E(r) − cυ eυ (r), −∇2 E(r) + cυ ∇2 eυ (r)⎟⎟⎟⎠ ⎝ υ∈Iν

υ∈Iν

V

 1  2 2 cυ ςυ eυ 2V E, −∇2 E −  2 V E ˜  υ∈Iν V ⎞ ⎛   ⎟⎟⎟ ⎜⎜⎜ 2 ⎟⎟⎟ =  1 ⎜⎜⎜ cυ eυ (r), − c ∇ e (r) c2υ ςυ2 eυ 2V υ υ ⎠ ⎝ 2 E ˜ υIν V V υIν V υIν ⎛ ⎞  ⎜ ⎟  2 ⎟⎟ 1 ⎜⎜ ˜  ⎟⎟⎟ = ςν2 +  2 ⎜⎜⎜⎝ c2υ ςυ2 eυ 2V − ςν2 E V⎠  E ˜  υIν V  1  2 2 cυ ςυ − ςν2 eυ 2V  ςν2 = ςν2 +  2 E ˜  

(11.77)

(11.78)

V υIν

having taken into account that the Laplace operator is self-adjoint with respect to the inner product (11.47) [see (11.50)]. Besides, we have used definition (11.70), the defining equation (11.6) for electric modes and the orthogonality thereof. In (11.78) we have swapped the order of inner product and summations on the grounds that the latter converge in light of the discussion above. The purpose of the second result is to show that the Rayleigh quotient of concern can never be smaller than ςν2 , i.e., the eigenvalue associated with the first eigenfunction excluded from the truncated expansion Eν (r). On the other hand, since on account of (11.54) the first term in the last part of (11.77) is a positive quantity and the sum over υ ∈ Iν is evidently non-negative, we have   1  1  2 2 1  cυ ςυ eυ 2V   2 E, −∇2 E ςν2   2 E, −∇2 E −  2 V V E E E ˜  ˜  υ∈Iν ˜  V V V

(11.79)

 2 ˜  we arrive at the estimate thanks to (11.78). By solving for E V    2 ˜   1 E, −∇2 E E V V ςν2

(11.80)

which in words states that the norm of the difference field can become arbitrarily small as ςν2 grows infinitely large, since the numerator in the right member is a constant independent of ν. This is possible in turn, because the set of eigenvalues of (11.6) has no upper bound. Hence, Eν (r) tends to E(r) in the norm • V , and the set {eν (r)}ν is complete. A perfectly similar proof holds for magnetic fields and the set of magnetic modes. Incidentally, proving that (11.6) and (11.7) do have infinite denumerable solutions (cf. the proof for the Helmholtz equation in Section 8.5.5) requires notions from functional analysis, such as Sobolev spaces of vector functions (Appendix D.1) [20], the Riesz representation theorem of linear operators (Appendix D.5) [21, Chapter 1], [18, Chapter 10], and the Fredholm theory for integral equations (Appendix D.8) [22].

750

Advanced Theoretical and Numerical Electromagnetics

We conclude by observing that if in (11.71) we insert the explicit representation of the coefficients dυ into the expansion of the magnetic field, we obtain an identity of sorts. More specifically, starting with the approximation Hν (r) we have   hυ (r)hυ (r )  (hυ , H)V  Hν (r) := hυ (r) = dV · H(r ), r∈V (11.81) 2 2 h h υ υ V V υ∈Iν υ∈Iν V

where the interchange of integration and summation is possible since the latter is just a combination of a finite number of terms. In the limit as Iν → N3 we obtain   hυ (r)hυ (r ) H(r) = lim dV  · H(r ) 2 Iν →N3 h υ V υ∈Iν V   hυ (r)hυ (r ) dV  · H(r ), r∈V (11.82) = 2 h υ V υ V

where the last integral is formal in that the series does not converge uniformly. The result of the integration is finite in keeping with (11.76) and (11.80) applied to the magnetic field, but the components of the dyadic field within the integrand are no ordinary functions. Indeed, in accordance with (C.19) we may interpret (11.82) as an alternative definition of the dyadic three-dimensional Dirac distribution, namely,  hυ (r)hυ (r )   = δ(3) r − r I, r, r ∈ V (11.83) 2 h υ V υ and this provides the expansion of δ(3) (r − r ) I in terms of magnetic eigenfunctions of the cavity V with PEC walls. The expansion in terms of electric eigenfunctions is analogous.

11.1.7 Equivalent sources on a cavity boundary We introduce the notation for the cavity eigenfunctions [10, 12] ⎧ ⎪ ⎨Eν (r) solenoidal electric modes e(r) = ⎪ ⎩F (r) lamellar and harmonic electric modes ν ⎧ ⎪ H ⎨ ν (r) solenoidal magnetic modes h(r) = ⎪ ⎩G (r) lamellar and harmonic magnetic modes ν

(11.84) (11.85)

where ν := (m, n, p) ∈ N3 , and indicate the relevant eigenvalues with κν2 , α2ν and β2ν . As already noticed, the functions e(r) and h(r) are suitable to expand electric and magnetic fields, respectively, in a cavity with PEC walls. Nonetheless, since e(r) and h(r) obey jump conditions on ∂V which are more general than those expected of electric and magnetic fields on a PEC surface, care must be exercised when computing the spatial derivatives of E(r, t) and H(r, t). More specifically, it is not permitted to interchange the order of differentiation and summation in general. For instance we may express the time-varying electromagnetic entities in V as   eν (t)Eν (r) + fν (t)Fν (r), r∈V (11.86) E(r, t) = ν

H(r, t) =

 ν

ν

hν (t)Hν (r) +

 ν

gν (t)Gν (r),

r∈V

(11.87)

Spectral representations of electromagnetic fields

751

where harmonic eigenfunctions of both types are needed if the region V is surface-wise and contourwise multiply connected. If in order to compute ∇ × E we naïvely swapped the curl operator with the series, we would find    ∇ × E(r, t) = eν ∇ × Eν (r) + fν ∇ × Fν (r) = κν eν Hν (r) (11.88) ν

ν

ν

given that solenoidal electric and magnetic eigenfunctions are related by (11.16) and Fν (r) is lamellar. Alternatively, we may treat the curl of E(r, t) as yet another vector field in V and let   ∇ × E(r, t) = aν (t)Hν (r) + bν (t)Gν (r), r∈V (11.89) ν

ν

because the curl of an electric field has the physical character of a magnetic field, according to the Faraday law (1.20). To determine the auxiliary expansion coefficients aν and bν we exploit the orthogonality of the modes with respect to the inner product (11.47). From (11.89) and (11.86) we have  aν Hν 2V = (Hν , ∇ × E)V = dV Hν (r) · ∇ × E(r, t) V





dV ∇ · [E(r, t) × Hν (r)] +

= V

V

 =



ˆ · [E(r, t) × Hν (r)] + κν dS n(r) ∂V

ˆ dS Hν (r) · [E(r, t) × n(r)] + κν eν Eν 2V ∂V

dV Gν (r) · ∇ × E(r, t) V





dV ∇ · [E(r, t) × Gν (r)] + V

dV E(r, t) · ∇ × Gν (r) V

 =

(11.90)



= (Gν , ∇ × E)V = =

dV E(r, t) · Eν (r) V



=− bν Gν 2V

dV E(r, t) · ∇ × Hν (r)



ˆ · [E(r, t) × Gν (r)] = − dS n(r) ∂V

ˆ dS Gν (r) · [E(r, t) × n(r)]

(11.91)

∂V

having used the Gauss theorem (the eigenfunctions possess the required derivatives in V) with the ˆ unit normal n(r) pointing outward V, relationship (11.14) and the fact that Gν (r) is curl-free by construction. Substituting the expressions of aν and bν back into (11.89) yields ∇ × E(r, t) =

 ν

κν eν Hν (r) −

 Hν (r)  ˆ  )] dS  Hν (r ) · [E(r , t) × n(r 2 H ν V ν ∂V  Gν (r)  ˆ  )] (11.92) − dS  Gν (r ) · [E(r , t) × n(r 2 ν Gν V ∂V

where we have invoked (11.17) and renamed the dummy integration variable to avoid confusion.

Advanced Theoretical and Numerical Electromagnetics

752

By comparing (11.88) with (11.92) we see that the two results coincide only if the surface integrals on the wall of the cavity vanish identically. This is indeed the case so long as we assume PEC boundary conditions (1.169) for r ∈ ∂V. By contrast, if the cavity is coupled to the complementary domain by means of a small opening S h ⊂ ∂V then the surface integrals are not necessarily null ˆ = 0 for r ∈ S h . As a matter of fact, the field inasmuch as we may not assume E(r, t) × n(r) ˆ J MS eq (r, t) := −E(r, t) × n(r),

r ∈ Sh

(11.93)

plays the role of an equivalent surface magnetic current density on S h (see Section 10.4), and if we obtained the curl through (11.88) we would erroneously overlook this source contribution. Proceeding in the same way with the magnetic field we write   aν (t)Eν (r) + bν (t)Fν (r), r∈V (11.94) ∇ × H(r, t) = ν

ν

because the curl of a magnetic field has the physical nature of an electric field, according to the Ampère-Maxwell law (1.34). After a few manipulations from (11.94) we have  aν Eν 2V = κν dV Hν (r) · H(r, t) = κν hν Hν 2V (11.95)  bν Fν 2V

=

V



dV H(r, t) · ∇ × Fν (r) −

ˆ × Fν (r)] · H(r, t) = 0 dS [n(r)

(11.96)

∂V

V

on account of (11.87) and the properties of Eν (r), Fν (r). Substituting into (11.94) yields ∇ × H(r, t) =



κν hν

ν

Hν 2V Eν 2V

Eν (r) =



κν hν Eν (r)

(11.97)

ν

in light of (11.17). In this case, the contribution of the electric surface current ˆ × H(r, t), JS eq (r, t) := −n(r)

r∈S

(11.98)

ˆ × Fν (r). In particular, we obtain the same result by differentiis filtered out by the vanishing of n(r) ating (11.87) term by term. We can explain the presence of the source term J MS eq on S h with the Love equivalence principle of Section 10.4.1. An equivalent problem for r ∈ V is obtained by placing suitable surface sources ˆ is positively directed outwards V) and ‘switching off’ the true sources on ∂V − (the unit normal n(r) outside V. Then, since we are allowed to conceptually ‘fill’ the complementary region R3 \ V with any medium, we choose to leave the cavity wall in place and additionally to ‘close’ the hole S h with a PEC patch. These steps effectively turn the source-free open-cavity problem into the solution of a closed cavity excited by equivalent sources on the wall. However, JS eq (r, t) does not radiate as it is flush with a PEC (see Example 6.7) and hence it can be ignored. Further, the equivalent magnetic current is null everywhere but on S h in view of the original boundary conditions on ∂V − \ S h . To determine the field excited in the cavity by JMS eq (r, t) we insert expansions (11.86), (11.87), (11.92) and (11.97) into the Maxwell equations (1.20) and (1.34). In symbols, we have  ν

κν eν Hν (r) = −μ

 dhν ν

dt

Hν (r) − μ

 dgν ν

dt

Gν (r)

Spectral representations of electromagnetic fields  Hν (r)  dS  Hν (r ) · J MS eq (r , t) 2 H ν V ν Sh  Gν (r)  dS  Gν (r ) · J MS eq (r , t) − 2 ν Gν V

753

−



κν hν Eν (r) = ε

 deν

ν

dt

ν

Sh

Eν (r) + ε

 d fν ν

dt

Fν (r)

(11.99)

(11.100)

which by virtue of the orthogonality properties of the eigenfunctions amount to infinitely many separate systems of four ordinary differential equations for each ν, viz.,  1 dhν κν eν (t) + μ =− dS  Hν (r ) · J MS eq (r , t) (11.101) dt Hν 2V Sh

deν κν hν (t) − ε =0 dt  1 dgν =− μ dS  Gν (r ) · J MS eq (r , t) dt Gν 2V

(11.102) (11.103)

Sh

d fν =0 ε dt

(11.104)

where the first two equations are still coupled, but the remaining ones are trivially solved, once the initial condition for the field in V has been prescribed. Obviously, one also needs to specify the form of J MS eq (r, t). If the aperture S h is relatively small, as a first approximation we may employ the incident electric field produced by the ‘true’ sources — located outside the cavity — at the location of the opening (cf. [10, Section 9.6]). Or else, J MS eq (r, t) must be treated as an unknown, whereby the system (11.101)-(11.104) couples to the equations enforced in the complementary region R3 \ V [10, Section 10.6]. By assuming time-harmonic dependence for the sake or argument and letting (Section 1.5) J MS eq (r, t) = Re{J MS eq (r)ej ωt } and so forth, (11.101)-(11.104) pass over into  1 dS  Hν (r ) · J MS eq (r ) κν e˜ ν + j ωμh˜ ν = − Hν 2V

eν (t) = Re{˜eν ej ωt }

(11.105)

(11.106)

Sh

j ωε˜eν − κν h˜ ν = 0 j ωμ˜gν = −

1 Gν 2V

j ωε f˜ν = 0



(11.107) dS  Gν (r ) · J MS eq (r )

(11.108)

Sh

(11.109)

which in principle we may solve to get the coefficients e˜ ν , h˜ ν , g˜ ν and f˜ν . While the latter two are invariably found to be  1 1 dS  Gν (r ) · J MS eq (r ) f˜ν = 0 (11.110) g˜ ν = − j ωμ Gν 2V Sh

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Advanced Theoretical and Numerical Electromagnetics

for the calculation of e˜ ν and h˜ ν we have three possibilities (cf. Example 6.2) depending on the determinant of the system matrix, namely,    κν j ωμ 2 2 ν ∈ N3 (11.111)  = k − κν ,  j ωε −κν  where k is the real wavenumber defined in (1.248). (a)

If the driving angular frequency ω is such that k2  κν2 for the eigenvalues of all solenoidal modes, the determinant (11.111) never vanishes, and (11.106) and (11.107) can be inverted to find  1 κν dS  Hν (r ) · J MS eq (r ) (11.112) e˜ν = 2 k − κν2 Hν 2V Sh  1 j ωε dS  Hν (r ) · J MS eq (r ) (11.113) h˜ ν = 2 k − κν2 Hν 2V Sh

(b)

whereby the solution to the aperture-coupled cavity problem is unique. This happens because the homogeneous equations associated with (1.98)-(1.101) in V and supplemented with condition (1.169) on ∂V admit only the trivial solution E(r) = 0 = H(r) when k2  κν2 . By contrast, if ω is such that for some ν = η the wavenumber k2 coincides with the eigenvalue κη2 , then the determinant (11.111) is null, the system is rank-deficient and, by means of straightforward linear combinations of the equations, (11.106) and (11.107) can be cast into the equivalent reduced form  1 dS  Hη (r ) · J MS eq (r ) (11.114) κη e˜ η + j ωμh˜ η = −  2 H  η V S h  κη dS  Hη (r ) · J MS eq (r ) (11.115) 0 = −  2 H  η V S h

whence we see the system is solvable if (and only if) [23, Theorem 3.17]  dS  Hη (r ) · J MS eq (r ) = 0

(11.116)

Sh

i.e., the driving surface current density J MS eq (r) is orthogonal to the solenoidal magnetic eigenfunction Hη (r) [cf. (6.90)] that is also the solution to the homogeneous equations (6.89) with ν = η. Of course, for ν  η the coefficients e˜ ν and h˜ ν still follow from (11.112) and (11.113) with k2 = κη2 . However, since from (11.114) we get the only constraint κη e˜ η = − j ωμh˜ η

(c)

(11.117)

the coefficients e˜ η and h˜ η remain undetermined, and hence the solution to the cavity problem is not unique. Lastly, if under the hypotheses of case (b) condition (11.116) is not fulfilled, the set formed by (11.106) and (11.107) is inconsistent and cannot be solved. Then, the aperture-coupled cavity problem has no solution at all.

Spectral representations of electromagnetic fields

755

We now substitute the expansion coefficients given by (11.110), (11.112) and (11.113) into the time-harmonic instance of (11.86) and (11.87). By exchanging summation and integration over S h with the aid of a limiting process as in (11.82) we arrive at the formal integral representations for the phasors of the fields in V   κν Eν (r)Hν (r ) E(r) = dS  · J MS eq (r ) (11.118) 2 − κ2 ) H 2 (k ν ν V ν Sh ⎤   ⎡⎢⎢ Hν (r)Hν (r ) Gν (r)Gν (r ) ⎥⎥⎥   ⎢ + (11.119) H(r) = j ωε dS ⎣⎢ 2 ⎦⎥ · J MS eq (r ) 2 ) H 2 2 G 2 (k − κ k ν ν ν V V ν Sh

where the dyadic parts of the integrands play the role of electric and magnetic dyadic Green functions for magnetic sources [cf. (9.162) and (9.163)]. Moreover, since the lamellar magnetic modes Gν (r) are derived from a scalar potential when β2ν > 0, we may cast the coefficients g˜ ν and hence (11.119) into an alternative format. From (11.35) and (11.108) we get   ∇ Υν 2     ˆ ) · ˆ j ωμ˜gν Gν V = dS Gν (r ) · [E(r ) × n(r )] = dS  n(r × E(r ) βν ∂V ∂V   1 1 ˆ  ) · ∇ × (EΥν ) − ˆ  ) · ∇ × E = dS  n(r dS  Υν (r )n(r βν βν ∂V ∂V   =0  j ωμ ˆ  ) · H(r ) = dS  Υν (r )n(r (11.120) βν ∂V

whence by virtue of an identity analogous to (11.60) we find   1 1     ˆ g˜ ν = dS Υ (r ) n(r ) · H(r ) = dS  Υν (r )ρ MS eq (r ) ν βν Υν 2V μβν Υν 2V ∂V

(11.121)

Sh

where ˆ · H(r), ρ MS eq (r) := μn(r)

r ∈ S h ⊂ ∂V

(11.122)

is the phasor of a surface density of equivalent magnetic charge that is related to J MS eq (r) by a time-harmonic surface continuity equation. This shows that lamellar magnetic modes are excited whenever the normal component of the magnetic field does not vanish everywhere on the cavity walls. Now, provided the domain V is contour-wise simply connected — so that harmonic magnetic eigenfunctions are absent — the magnetic field may be written as    Hν (r)Hν (r )  Gν (r)Υν (r ) 1    · J (r ) + dS ρ MS eq (r ) H(r) = j ωε dS MS eq 2 2 − κ2 ) H 2 μ (k β Υ ν ν ν ν V V ν ν Sh Sh   Hν (r)Hν (r ) = j ωε dS  · J MS eq (r ) 2 − κ2 ) H 2 (k ν V ν ν Sh   Υν (r)Υν (r ) ∇ dS  ρ MS eq (r ) (11.123) + 2 Υ 2 μ β ν ν V ν Sh

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Advanced Theoretical and Numerical Electromagnetics

where we have formally swapped the order of summation, integration and differentiation. The result of the series in the second contribution to (11.123) is a static scalar Green function. Dyadic and scalar Green functions in the cavity are evidently symmetric under the exchange of source (r ) and observation (r) point in the sense of (9.384) and (3.46), respectively, and this property is a consequence of the reciprocal nature of the medium that fills V. Still, the dependence is on r and r separately, rather than just the distance |r − r |. This can be explained by noticing that, unlike an observer in a homogeneous unbounded medium, an ideal observer in the cavity detects different fields depending on her position relative to the boundary ∂V even if the distance from the singular point source in V is kept fixed in the process. When k2 = κη2 and the orthogonality condition (11.116) holds true, all the expansion coefficients except e˜ η and h˜ η are univocally determined. The cavity resonates, and the electromagnetic field inside V takes on the form   κν Eν (r)Hν (r ) dS  · J MS eq (r ) + e˜ η Eη (r) (11.124) E(r) = 2 2 2 νη (kη − κν ) Hν V Sh   Hν (r)Hν (r ) κη e˜ η Hη (r) H(r) = j ωε dS  · J MS eq (r ) − 2 2 2 j ωμ νη (kη − κν ) Hν V Sh   Gν (r)Gν (r ) · J MS eq (r ) (11.125) − dS  2 ν j ωμ Gν V Sh

on account of (11.117). While formulas (11.118), (11.119) and (11.123) break down at exactly k2 = κη2 , still these integral representations tell us that for k ≈ κη the field in the cavity is essentially dominated by the solenoidal electric and magnetic modes associated with κη2 , whereas the remaining ones contribute far less. Then again, if we account for possible losses in the medium that fills V, k becomes the complex wavenumber (1.249), and for no angular frequency ω ∈ R can the term k2 − κη2 vanish. Thus, formulas (11.118), (11.119) and (11.123) are always well defined in the presence of losses.

11.2 Modal expansion in uniform cylindrical waveguides Broadly speaking, a waveguide is a passive device specially devised for transferring electromagnetic radiation (Section 1.9) from one point in space to another one while hopefully losing as little energy as possible in the process [24, Chapter 44], [25, Chapters 6, 7, 8], [26, Chapter 22], [27, Chapter 5], [28, Chapter 13], [29, Chapter 9], [30, Chapter 8], [31, Chapter XI], [32, Chapter 5], [9–11, 33, 34]. Unlike electromagnetic waves in a three-dimensional homogeneous unbounded medium, waves generated by a source in a waveguide are forced (i.e., guided, hence the name) to travel along the path defined by a material boundary. Notable examples include hollow-pipe waveguides with walls made of a good conductor (Table 7.2), coaxial cables, but also dielectric waveguides and optical fibers, which are made of a low-loss penetrable medium. Waves excited in a metallic waveguide cannot escape the intended path of propagation because the electromagnetic field gets rapidly attenuated inside a conductor due to the skin effect (see Examples 7.2, 9.7 and 9.8). Metallic waveguides are thus referred to as closed, in that the field confinement is theoretically absolute. By contrast, dielectric waveguides exploit the fact that the relevant guided waves become evanescent (i.e., exponentially damped) in the medium which surrounds the intended region of propagation owing to the phenomenon of total internal reflection at the material boundary (see Section 7.4). Consequently, dielectric waveguides are termed open, because the field

Spectral representations of electromagnetic fields

757

Figure 11.1 Transverse cross-section S of a uniform hollow-pipe waveguide V := S × R.

extends outside the guiding region [32, Chapter 6], [35,36]. What is more, bents and other unwanted deviations from the ideal intended path can cause an open waveguide to ‘leak power’ towards the external medium and to act effectively as an antenna. Still, metallic waveguides become inefficient at relatively high frequencies because much of the useful power ends up being dissipated in the walls (cf. Figure 12.3) which is why at optical frequencies (say, from 200 to 800 THz) it is more convenient to employ dielectric structures to guide electromagnetic radiation. In this section we examine in some detail the propagation of waves in uniform hollow-pipe waveguides whose boundary is made of PEC (e.g. [9, 10, 33, 37]). The geometry of interest is essentially an infinitely long right cylinder V := S × R, where S is a planar surface with smooth or piecewise-smooth boundary (Figure 11.1). We assume the waveguide is filled with a homogeneous lossless medium endowed with constitutive parameters ε and μ. The propagation of time-harmonic electromagnetic waves in V is governed by the set of Maxwell’s equation ∇ × E(r) + j ωμH(r) = −J M (r), ∇ × H(r) − j ωεE(r) = J(r), ε∇ · E(r) = ρ(r), μ∇ · H(r) = ρ M (r), E(r) × νˆ (r) = 0,

r∈V r∈V

(11.126) (11.127)

r∈V r∈V

(11.128) (11.129)

r ∈ ∂V

(11.130)

plus suitable asymptotic conditions for r − zˆz = ρ ∈ S and |z| → +∞ (Section 11.2.6). The sources are confined to a finite region V J ⊂ V, and νˆ (r) is the unit vector normal to the PEC wall and positively oriented outwards the waveguide. The solution of the source-free counterpart of (11.126)(11.130) provides the modes of the waveguide, and these in turn can be employed to solve the general source-driven problem. We observed in Section 1.5 that in the time-harmonic regime the Gauss laws (11.128) and (11.129) follow from the curl equations in combination with the conservation laws given by (1.102) and the time-harmonic analogue of (6.187). As a result, we can make do with just the Faraday law and the Ampère-Maxwell law. More importantly, since (11.128) provides a link between the three components of the electric field, it should be possible to express any component of E(r) as a function of the remaining two. The same goes for (11.129) and the magnetic field H(r). Therefore, the solution strategy will consist of the following main steps:

758 (i)

(ii) (iii)

Advanced Theoretical and Numerical Electromagnetics from (11.126) and (11.127) a reduced set of differential equations named after N. Marcuvitz and J. Schwinger (Section 11.2.1) is derived that involves only four out of six field components, viz., two electric and two magnetic; we solve the homogeneous (i.e., source-free) instance of the Marcuvitz-Schwinger equations to find the modes of the waveguide (Sections 11.2.2-11.2.5); we expand the electromagnetic field in V as a linear superposition of modes and solve the source-driven Marcuvitz-Schwinger equations (Section 11.2.6).

11.2.1 The Marcuvitz-Schwinger equations In order to obtain a reduced set of equations from (11.126) and (11.127), we choose to align the axis of the cylinder V with the z-coordinate of a suitable system of cylindrical coordinates and write the position vector as r := ρ + zˆz ∈ V,

ρ ∈ S,

z∈R

(11.131)

where ρ is the part of r perpendicular to zˆ . Since observation points on the waveguide boundary ∂V are given by rγ := ργ + zˆz ∈ V,

ργ ∈ γ := ∂S ,

z∈R

(11.132)

it is convenient to choose the remaining coordinates, say, u and v, which are possibly curvilinear and not orthogonal (e.g., [14, Chapter 1], [17, Chapter 1], [38]) so that the line γ can be described by a simple equation like u = u0 or v = v0 , with u0 and v0 being two real constants. Such arrangement facilitates the statement and fulfillment of the jump condition (11.130) in the solution process. For example, if S is a circle of radius a, a system of circular cylindrical coordinates (Appendix A.1) is best suited, in that γ is just specified by letting ρ = a, i.e., in a way that is independent of the azimuthal coordinate ϕ. Then again, for the subsequent derivations we need not specify a basis to expand ρ. In line with representation (11.131) it is also fruitful to separate fields and current densities into transverse parts (perpendicular to zˆ ) and longitudinal parts (parallel to zˆ ), viz., E(ρ, z) = Et (ρ, z) + Ez (ρ, z)ˆz,

H(ρ, z) = Ht (ρ, z) + Hz (ρ, z)ˆz

(11.133)

J(ρ, z) = Jt (ρ, z) + Jz (ρ, z)ˆz,

J M (ρ, z) = J Mt (ρ, z) + J Mz (ρ, z)ˆz

(11.134)

since the waveguide is invariant along the z-direction, that is, the cross-section S does not change with z. Actually, we may even extend this decomposition, at least symbolically, to the del operator, i.e., ∇ := ∇t + zˆ

∂ ∂z

(11.135)

where ∇t is the transverse del operator. Although the explicit form of ∇t can only be given after choosing the coordinates u and v in any plane perpendicular to zˆ , we shall resort to Cartesian coordinates and let ∇t := xˆ

∂ ∂ + yˆ ∂x ∂y

whenever this assumption is expedient for intermediate calculations.

(11.136)

Spectral representations of electromagnetic fields

759

If we insert (11.133) and (11.134) into (11.126) and (11.127) and separate the derivatives with respect to the components of ρ from those with respect to z we get ∂Et + j ωμHt (r) + j ωμHz (r)ˆz = −J Mt (r) − J Mz (r)ˆz ∂z ∂Ht − j ωεEt (r) − j ωεEz (r)ˆz = Jt (r) + Jz (r)ˆz ∇t × Ht (r) + ∇t Hz (r) × zˆ + zˆ × ∂z ∇t × Et (r) + ∇t Ez (r) × zˆ + zˆ ×

(11.137) (11.138)

which can be proved with the aid of (A.35) and (11.136). Unlike the initial equations, (11.137) and (11.138) explicitly show the vectorial contributions that are either perpendicular or parallel to the z-direction. Therefore, we extract the transverse parts of (11.137) and (11.138) by cross-multiplying with zˆ from the left, namely, ∂Et + ∇t Ez (r) + j ωμˆz × Ht (r) = −ˆz × J Mt (r) ∂z ∂Ht − + ∇t Hz (r) − j ωεˆz × Et (r) = zˆ × Jt (r) ∂z −

(11.139) (11.140)

having made use of the algebraic identity (H.14). In like manner, the longitudinal parts of (11.137) and (11.138) follow by dot-multiplying with zˆ , viz., j ωμHz (r) + zˆ · ∇t × Et (r) = j ωμHz (r) − ∇t · [ˆz × Et (r)] = −J Mz (r) − j ωεEz (r) + zˆ · ∇t × Ht (r) = − j ωεEz (r) + ∇t · [Ht (r) × zˆ ] = Jz (r)

(11.141) (11.142)

where the second steps can be regarded as a special instance of the differential identity (A.60) with the surface divergence ∇s · {•} becoming in particular ∇t · {•}, the divergence transverse to zˆ . Finally, since (11.141) and (11.142) involve the longitudinal components of the fields algebraically we use them to eliminate Ez and Hz from (11.139) and (11.140) and arrive at the Marcuvitz-Schwinger equations [9] $ % ∂Et ∇t ∇t − = j ωμ Itz + 2 · [Ht (r) × zˆ ] − zˆ × J˜ Mt (r), r∈V (11.143) ∂z k $ % ∂Ht ∇t ∇t − = j ωε Itz + 2 · [ˆz × Et (r)] + zˆ × J˜ t (r), r∈V (11.144) ∂z k where Itz = I − zˆ zˆ

(11.145)

is the identity dyadic transverse to z, k is the wavenumber given by (1.248), and ∇t J Mz J˜ t (r) := Jt (r) + × zˆ , j ωμ ∇t Jz J˜ Mt (r) := J Mt (r) + zˆ × , j ωε

r ∈ VJ

(11.146)

r ∈ VJ

(11.147)

are total transverse electric and magnetic current densities. Even in the absence of a ‘true’ magnetic current density J M (r), J˜ Mt (r) may not be null provided the electric current density has a z-component that depends on ρ. The Marcuvitz-Schwinger equations involve only the transverse components of the fields, are first-order in z and second-order with respect to ρ. Once Et (r) and Ht (r) have been found from (11.143) and (11.144), the longitudinal components follows from (11.141) and (11.142).

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Advanced Theoretical and Numerical Electromagnetics

In order for (11.130) to be compatible with (11.143) and (11.144) we must express the boundary condition at a PEC surface by means of transverse field components. To this end, on account of (11.133) we write 0 = [Et (r) + Ez (r)ˆz] × νˆ (r) = Et (r) × νˆ (r) + Ez (r)τˆ (r) τˆ (r) ∇t · [Ht (r) × zˆ ] = Et (r) × νˆ (r) + j ωε

(11.148)

where τ(r) ˆ is the unit vector tangential to γ (Figure 11.1), and we have made use of (11.142) to eliminate the longitudinal electric field. There is no contribution from Jz (r), since (11.148) holds for r ∈ ∂V, whereas V J ⊂ V by hypothesis.2 Lastly, we can split (11.148) into two independent conditions, viz., Et (r) × νˆ (r) = 0,

∇t · [Ht (r) × zˆ ] = 0,

r ∈ ∂V

(11.149)

because the vector field Et (r) × νˆ (r) is perpendicular to τˆ (r). Before we set about to calculate the waveguide eigenfunctions by solving the source-free instance of the Marcuvitz-Schwinger equations, we obtain a general result which can be regarded as the two-dimensional version of the Helmholtz theorem we investigated at length in Section 8.1. Given an arbitrary non-null vector u ∈ R3 perpendicular to zˆ , we may expand the identity dyadic as I=

uu zˆ × u zˆ × u + + zˆ zˆ = Itz + zˆ zˆ |u|2 |u|2

(11.150)

in light of (E.32) with the choices a = u, b = zˆ × u and c = zˆ . Further, in accordance with (E.27) any transverse vector field Vt (r) can be written as Vt (r) = I · Vt (r) =

u[u · Vt (r)] zˆ × u[ˆz × u · Vt (r)] + |u|2 |u|2    

(11.151)

⊥u

u

and this identity provides the decomposition of Vt (r) in vectors parallel and perpendicular to u. In other words, we have the representation of Vt (r) on a basis of vectors without resorting to a particular system of coordinates in a plane perpendicular to zˆ . Next, by regarding ∇t as an ordinary transverse vector [see (11.136)] and mirroring the structure of (11.151) we may formally write Vt (r) ∈ C2 (R3 )2 as [9, Chapter 1] Vt (r) =

∇t [∇t · Vt (r)] zˆ × ∇t [ˆz × ∇t · Vt (r)] + = VtL (r) + VtS (r), ∇2t ∇2t     transverse lamellar

r ∈ R3

(11.152)

transverse solenoidal

where ∇2t {•} is the Laplace operator transverse to z defined in (7.341). The formal division by ∇2t merely indicates the inversion of a two-dimensional Poisson equation supplemented with suitable boundary conditions. Similarly to the three-dimensional counterpart (8.2), (11.152) tells us that any transverse vector field endowed with the necessary partial derivatives can be separated into parts ‘parallel’ and ‘perpendicular’ to ∇t . For the sake of brevity we shall content ourselves with putting (11.152) on solid ground, though we will not attempt to derive explicit general integral formulas as 2 In fact, even if J(r) did extend up to the waveguide boundary and J (r) were indeed non-zero for r ∈ ∂V, we could z neglect that contribution in (11.148), inasmuch as an impressed electric current flush with a PEC surface produces no field (cf. Example 6.7).

Spectral representations of electromagnetic fields

761

in Section 8.1, partly because, in order to compute the modes of the PEC waveguide under study, all we need to know is that decomposition (11.152) is feasible. By starting with the transverse Laplace operator (7.341) and invoking (H.59) backwards we find ∂2 Vt ∂2 V t = ∇∇ · Vt − ∇ × ∇ × Vt − 2 ∂z ∂z2 $ % ∂Vt ∂2 Vt = ∇∇t · Vt − ∇ × ∇t × Vt + zˆ × − ∂z ∂z2 ∂Vt ∂2 Vt ∂ ∂Vt + zˆ · ∇ − = ∇t ∇t · Vt + zˆ ∇t · Vt − ∇ × ∇t × Vt − zˆ ∇t · ∂z ∂z ∂z ∂z2

∇2t Vt = ∇2 Vt −

(11.153)

also on account of (H.54) with A = zˆ and B = ∂Vt /∂z. Since Vt (r) ∈ C2 (R3 )2 we can invoke the Schwarz theorem [39, pp. 235–236] to swap the order of derivatives in the second term in the rightmost member. As a result, this term becomes the negative of the fourth contribution, and the two of them cancel out. The last two terms, too, cancel each other, since from (A.26) we infer that zˆ · ∇ = ∂/∂z. Next, we observe that the vector field ∇t × Vt is aligned with zˆ , and thus we can write −∇ × ∇t × Vt = −∇ × [ˆz (ˆz · ∇t × Vt )] = −∇(ˆz · ∇t × Vt ) × zˆ   scalar field

= zˆ × ∇t (ˆz · ∇t × Vt ) = zˆ × ∇t [(ˆz × ∇t ) · Vt ]

(11.154)

in light of (H.50) with A = zˆ . In practice, the last step amounts to ‘swapping the cross with the dot’ and can be proved by using Cartesian coordinates to show that zˆ × ∇t = yˆ

∂ ∂ − xˆ ∂x ∂y

(11.155)

on account of (11.136). Plugging (11.154) back into (11.153) finally yields ∇2t Vt (r) = ∇t ∇t · Vt (r) + zˆ × ∇t [ˆz × ∇t · Vt (r)]

(11.156)

which is the analogue of (H.59) for transverse vector fields. The first term in the right member of (11.156) is evidently lamellar. To show that the second term is solenoidal we notice that the outer operator zˆ × ∇t acts on a scalar field, say, Υ(r), and hence we have ∇t · [ˆz × ∇t Υ(r)] = −ˆz · ∇t × ∇t Υ(r) = 0

(11.157)

by means of a special instance of (A.60). To determine the lamellar part of Vt (r) we observe that the condition ∇t × VtL (r) = 0,

r ∈ R3

(11.158)

is certainly satisfied if we let VtL (r) = ∇t ΥL (r),

r ∈ R3

(11.159)

where ΥL (r) is a scalar field twice differentiable with respect to ρ. Taking the transverse divergence of (11.152) yields ∇t · Vt (r) = ∇t · VtL (r) + ∇t · VtS (r) = ∇2t ΥL (r),

r ∈ R3

(11.160)

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Advanced Theoretical and Numerical Electromagnetics

which — complemented with suitable boundary conditions — provides ΥL (r). As for the solenoidal part of Vt (r), the constraint ∇t · VtS (r) = 0,

r ∈ R3

(11.161)

holds true if we assume VtS (r) = −ˆz × ∇t ΥS (r),

r ∈ R3

(11.162)

where ΥS (r) is yet another scalar field twice differentiable with respect to ρ. Cross-multiplying (11.152) with zˆ and then taking the transverse divergence yields ∇t · [ˆz × Vt (r)] = ∇t · [ˆz × VtL (r)] + ∇t · [ˆz × VtS (r)] = −∇t · {ˆz × [ˆz × ∇t ΥS (r)]} = ∇2t ΥS (r),

r ∈ R3

(11.163)

by virtue of (11.158) and (H.14). Since (11.163) differs from (11.160) only for the ‘source term’, so to speak, we need only know how to solve the two-dimensional Poisson equation. In order to find all the non-trivial solutions to $ % ∇t ∇t ∂Et = j ωμ Itz + 2 · [Ht (r) × zˆ ], r∈V (11.164) − ∂z k $ % ∂Ht ∇t ∇t − = j ωε Itz + 2 · [ˆz × Et (r)], r∈V (11.165) ∂z k we would like to separate Et and Ht into their respective lamellar and solenoidal parts thanks to the two-dimensional Helmholtz decomposition (11.152) discussed above. However, since Et and Ht are related to one another through (11.164) and (11.165), the decompositions of electric and magnetic fields are not independent of each other. To elaborate, suppose we start off by assuming Et (r) = Et (r) + Et (r),

r∈V

(11.166)

where Et is lamellar and Et is solenoidal. Invoking the linearity of the operators we first insert the lamellar part of Et into (11.165) to obtain a relationship for Ht , viz., $ % ∂H ∇t ∇t − t = j ωε Itz + 2 · [ˆz × Et (r)] = j ωεˆz × Et (r), r∈V (11.167) ∂z k because Et satisfies an equation like (11.158). By taking the transverse divergence of both sides and applying the Schwarz theorem we get −

∂ ∇t · Ht (r) = j ωε∇t · [ˆz × Et (r)] = 0, ∂z

r∈V

(11.168)

whence we conclude ∇t · Ht (r) = fH (ρ),

r∈V

(11.169)

with fH (ρ) denoting a suitable scalar field independent of z. If we insert Et into the source-free instance of the longitudinal equation (11.141) we obtain Hz (r) = −

1 zˆ · ∇t × Et (r) = 0, j ωμ

r∈V

(11.170)

Spectral representations of electromagnetic fields

763

which in combination with the homogeneous magnetic Gauss law (11.129) leads to ∇t · Ht (r) +

∂Hz = ∇t · Ht (r) = 0, ∂z

r∈V

(11.171)

and this requirement sets fH (ρ) to zero. In summary, we have shown that • •

the lamellar part of Et is associated with the solenoidal part of Ht , the longitudinal component of the solenoidal transverse magnetic field vanishes.

The non-trivial solutions to (11.164) and (11.165) that are endowed with these features are called transverse-magnetic (TM) modes, inasmuch as Hz is null, or also E-modes, in that Ez is the only component along z. At this stage it seems logical that the solenoidal part of Et be related to the lamellar part of Ht . To confirm this expectation, we let Ht (r) = Ht (r) + Ht (r),

r∈V

(11.172)

where Ht is solenoidal and Ht is lamellar. Invoking the linearity of the operators we insert the lamellar part of Ht into (11.164) to obtain a relationship for Et , viz., $ % ∂E ∇t ∇t − t = j ωμ Itz + 2 · [Ht (r) × zˆ ] = j ωμHt (r) × zˆ , r∈V (11.173) ∂z k because this time Ht satisfies an equation like (11.158). By taking the transverse divergence of both sides and applying the Schwarz theorem we get −

∂ ∇t · Et (r) = j ωμ∇t · [Ht (r) × zˆ ] = 0, ∂z

r∈V

(11.174)

whence we conclude ∇t · Et (r) = fE (ρ),

r∈V

(11.175)

with fE (ρ) being a suitable scalar field independent of z. If we insert Ht into the source-free instance of the longitudinal equation (11.142) we obtain Ez (r) =

1 zˆ · ∇t × Ht (r) = 0, j ωε

r∈V

(11.176)

which in combination with the homogeneous electric Gauss law (11.128) leads to ∇t · Et (r) +

∂Ez = ∇t · Et (r) = 0, ∂z

r∈V

(11.177)

and this constraint sets fE (ρ) to zero. In conclusion, we have shown that • •

the lamellar part of Ht is associated with the solenoidal part of Et , the longitudinal component of the solenoidal transverse electric field vanishes.

The non-trivial solutions to (11.164) and (11.165) that possess these characteristics are called transverse-electric (TE) modes, in that Ez is null, or also H-modes, inasmuch as Hz is the only component along z.

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Advanced Theoretical and Numerical Electromagnetics

Further, since the dyadic differential operator in the right-hand sides of (11.164) and (11.165) acts only on ρ we apply the separation argument [40, Chapter 3], [41, Appendix A.4] and seek solutions in the form Et (r) = V  (z)e (ρ) + V  (z)e (ρ), 







Ht (r) = I (z)h (ρ) + I (z)h (ρ),

ρ ∈ S,

z∈R

(11.178)

ρ ∈ S,

z∈R

(11.179)

where V  , V  , I  and I  are symbolic or modal voltages and currents for both types of modes. The vector fields e , h , etc. are called transverse eigenfunctions and all carry the physical dimension of the inverse of a length (1/m), so long as we agree to measure V  and V  in volts and I  and I  in amperes. We are now ready to address the calculation of the TM and TE eigenfunctions separately.

11.2.2 Transverse-magnetic modes If we insert the TM parts of (11.178) and (11.179) into (11.164) and (11.165) we arrive at $ % dV  ∇t ∇t = j ωμ Itz + 2 · [h (ρ) × zˆ ]I  (z), −e (ρ) r∈V (11.180) dz k dI  = j ωεˆz × e (ρ)V  (z), r∈V (11.181) −h (ρ) dz because e (ρ) is lamellar by assumption. For the second equation to be true the magnetic eigenfunction h (ρ) must be proportional to zˆ × e (ρ), say, h (ρ) = zˆ × e (ρ),

ρ∈S

(11.182)

although we are most certainly free to choose other proportionality factors without altering (11.181). Parenthetically, this condition together with (11.178) and (11.179) says that electric and magnetic fields of the TM modes are orthogonal to each other at any point within the waveguide. Using (11.182) in (11.180) yields & ' dV  ∇t ∇t = j ωμ e (ρ) + 2 · e (ρ) I  (z), r∈V (11.183) −e (ρ) dz k whence we gather that by operating on e (ρ) with ∇t ∇t · {•} we must obtain a transverse vector field proportional to e (ρ) for the solvability of the equation. In other words, ∇t ∇t ·{•} must relate elements of the same vector space, and e (ρ) must be an eigenfunction of ∇t ∇t · {•} (Appendix D.7). We may enforce this requirement by demanding ⎧ ⎪ ∇t ∇t · e (ρ) + kt2 e (ρ) = 0, ρ ∈ S ⎪ ⎪ ⎪ ⎨  ρ∈γ e (ρ) × νˆ (ρ) = 0, ⎪ ⎪ ⎪ ⎪ ⎩∇ · e (ρ) = 0, ρ∈γ t

(11.184)

where the boundary conditions follow from (11.149) and (11.182). The constant kt (physical dimension: 1/m) is called the transverse wavenumber or transverse propagation constant. We defer the proof that kt2 is a non-negative real number until Section 11.2.5. The solution to (11.184) associated with the smallest permissible value of kt is called the fundamental or principal mode. On a side note, we may regard the eigenvalue problem (11.184) as the two-dimensional instance of (11.18). Thus, TM waveguide modes are analogous to the lamellar electric modes in a cavity with PEC walls.

Spectral representations of electromagnetic fields

765

It should be apparent that e (ρ) and h (ρ) are determined only by the shape of the waveguide cross-section and the chosen boundary conditions on γ. Indeed, two identical waveguides filled with different materials have the same transverse eigenfunctions, since ε and μ do not enter the eigenvalue problem (11.184)! Therefore, we may carry out the calculation of the transverse eigenfunctions once and for all independently of V  and I  . More specifically, since the eigenfunctions e (ρ) are lamellar, we may derive them from an auxiliary scalar potential Υ (ρ) (sometimes called a Dirichlet eigenfunction), viz., e (ρ) = −

∇t Υ , kt

ρ∈S

(11.185)

provided the transverse wavenumber is not null (we discuss this possibility further below). This strategy is the two-dimensional analogue of the approach we followed for the solution of the electrostatic equations in Section 2.2. Substituting (11.185) into (11.184) yields   ρ∈S (11.186) ∇t ∇2t Υ (ρ) + kt2 Υ (ρ) = 0, which is certainly satisfied if we require ∇2t Υ (ρ) + kt2 Υ (ρ) = C  ,

ρ∈S

(11.187)

with C  denoting an arbitrary constant. Still, we may always choose C  = 0 and solve ∇2t Υ (ρ) + kt2 Υ (ρ) = 0,

ρ∈S

(11.188)

because in accordance with (11.185) Υ (ρ) is determined within an additive constant. The jump conditions in (11.184) must be expressed in terms of the auxiliary potential, namely, ∇t Υ (ρ) × νˆ (ρ) = 0,

∇2t Υ (ρ) = 0,

ρ∈γ

(11.189)

and the second constraint together with (11.188) leads to Υ (ρ) = 0,

ρ∈γ

(11.190)

since we have provisionally assumed that kt does not vanish. As regards the other constraint we write νˆ = τˆ × zˆ (Figure 11.1) and obtain νˆ (ρ) × ∇t Υ (ρ) = [τˆ (ρ) × zˆ ] × ∇t Υ (ρ) = zˆ τˆ (ρ) · ∇t Υ (ρ) = zˆ

∂Υ =0 ∂τˆ

(11.191)

in light of (H.14). The last equation requires that the derivative of the potential along the boundary vanish, but this is surely true on account of (11.190), which is the only condition we really need to enforce. The special case of null eigenvalue (kt = 0) deserves a treatment of its own. For starters, since (11.185) is meaningless we assume instead e (ρ) = −∇t Υ (ρ),

ρ∈S

(11.192)

whereby (11.184) passes over into ∇2t Υ (ρ) = 0,

ρ∈S

(11.193)

Advanced Theoretical and Numerical Electromagnetics

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Figure 11.2 Transverse view of a hollow-pipe waveguide V := S × R with contour-wise multiply-connected cross-section. thanks to the arbitrariness of the auxiliary potential. On account of this equation, the second jump condition in (11.189) is automatically satisfied. From (11.191) we see that the remaining jump condition becomes ∂Υ = 0, ∂τˆ

ρ∈γ

(11.194)

which in turn requires that Υ (ρ) be constant on the waveguide wall. Unfortunately, this constraint is too strong for the geometry of Figure 11.1, and the only admissible, useless solution is a potential Υ (ρ) constant everywhere on S , which gives e (ρ) = 0, thus forbidding the occurrence of a null eigenvalue for (11.184). To prove this result we apply the two-dimensional Gauss theorem, which is a special case of (H.101) for a flat surface. In symbols, we have  (  dS ∇t · e = ds νˆ (ρ) · e (ρ) (11.195) 0 = − dS ∇2t Υ = S

S

 dS Υ

0=−  =

∗

(ρ)∇2t Υ

S

dS |∇t Υ |2 + 2

dS |∇t Υ | +

= S

=

2



dS |∇t Υ | − S

dS ∇t · [Υ∗ (ρ)∇t Υ ]

S

ds νˆ (ρ) · e (ρ)Υ∗ (ρ)

γ

S



(

γ



Υ∗ 0

(





ds νˆ (ρ) · e (ρ) = γ

 

dS |e (ρ)|2

(11.196)

S

=0

Υ0

where is the constant value of Υ on the boundary γ. The last two surface integrals may be interpreted as the squared norms of ∇t Υ and e (ρ) [see (11.283) further on]. Since by definition the norm vanishes only if the vector field in question is null, we conclude that Υ (ρ) = Υ0 for ρ ∈ S and e (ρ) = 0 everywhere. As an exception, this conclusion is faulty — and one or more null eigenvalues do occur — if the cross-section of the waveguide is contour-wise multiply connected, that is to say, the boundary γ is comprised of two or more non-intersecting closed lines, as is graphically shown in Figure 11.2. In all respects, this situation is the two-dimensional analogue of a surface-wise multiply-connected cavity

Spectral representations of electromagnetic fields

767

with PEC walls [cf. (11.18) and (11.27)]. A practical configuration is constituted by a coaxial cable, sketched for instance in Figure 6.18. To keep the discussion simple, we suppose that γ := γ1 ∪ γ2 , γ1 ∩ γ2 = ∅, and apply the two-dimensional Gauss theorem as before, viz.,  ( (  dS ∇t · e = ds νˆ 1 (ρ) · e (ρ) + ds νˆ 2 (ρ) · e (ρ) (11.197) 0 = − dS ∇2t Υ = S

 0=− 

S

dS Υ∗ (ρ)∇2t Υ =

S

=

2

=

dS |e (ρ)|2 + Υ∗ 01

γ2

dS ∇t · [Υ∗ (ρ)∇t Υ ]

S 

∗

(

( γ2

ds νˆ 1 (ρ) · e (ρ) + Υ∗ 02

γ1

S

Υ01

dS |∇t Υ |2 −



ds νˆ 1 (ρ) · e (ρ)Υ (ρ) + γ1

S



γ1

S

(

dS |∇t Υ | +



ds νˆ 2 (ρ) · e (ρ)Υ∗ (ρ) (

ds νˆ 2 (ρ) · e (ρ)

(11.198)

γ2

where and are the constant values of Υ (ρ) on γ1 and γ2 . We see that when Υ01 and Υ02 are different, we may not rely on (11.197) to conclude that the norm of e (ρ) is null in S . As a result, an eigenfunction associated with kt = 0 can be found, in which case, in addition to being lamellar, e (ρ) is also solenoidal in S , on account of (11.192) and (11.193). From (11.182) it then follows that h (ρ) is solenoidal and lamellar as well, and the source-free instance of (11.142) gives Ez (r) =

Υ02



1  I (z)∇t · [h (ρ) × zˆ ] = 0, j ωε

r∈V

(11.199)

that is, the longitudinal component of the electric field vanishes as well. For this reason the nontrivial solutions to (11.180) and (11.181) with kt = 0 are called transverse-electric-magnetic (TEM) modes (Section 11.2.4). A TEM mode, if it exists, is always the fundamental one. Having found the eigenvalue problem for the transverse eigenfunctions, we go on to determine the equations that govern the behavior of modal voltages and currents along z. By using (11.184) to eliminate the field ∇t ∇t · e from (11.183) and inserting (11.182) in (11.181) we obtain $ % k2 dV  = j ωμ 1 − t2 e (ρ)I  (z), −e (ρ) r∈V (11.200) dz k dI  = j ωεh (ρ)V  (z), r∈V (11.201) −h (ρ) dz which, being a pair of vector relationships, can be satisfied by equating the coefficients of e (ρ) and h (ρ) in the two sides of the equations. This step yields a first-order homogeneous differential system for TM modal voltages and currents, namely, $ % kt2  k2 − kt2  dV    = j ωμ 1 − 2 I (z) = j I (z) = j kz Z∞ I (z), z∈R (11.202) − dz ωε k kz dI  = j ωεV  (z) = j  V  (z), z∈R (11.203) − dz Z∞ where

⎧) ⎪ 2 2 ⎪ ⎪ ⎪ ⎨ k − kt  kz (ω) := ⎪ ) ⎪ ⎪ ⎪ ⎩− j k2 − k2 t

k

kt

k < kt

⎧ k ⎪ z ⎪ ⎪ ⎪ ⎪ ⎨ ωε  Z∞ (ω) := ⎪ ⎪ ⎪ |k | ⎪ ⎪ ⎩− j z ωε

k  kt (11.204) k < kt

Advanced Theoretical and Numerical Electromagnetics

768

are the longitudinal wavenumber (or propagation constant) and the characteristic impedance for TM modes. Thus, we have shown that V  and I  satisfy a differential system formally identical with the telegraph equations for classical uniform transmission lines [11, 26–29, 34, 42–47], [48, Chapter 2]. Absent any other constraint the general solution to (11.202) and (11.203) reads 



V  (z) = V0+ e− j kz z + V0− ej kz z = V + (z) + V − (z), 

I (z) =

 I0+ e− j kz z

+

 I0− ej kz z

+

−

= I (z) + I (z),

z∈R

(11.205)

z∈R

(11.206)

where  + I0 V0+ = Z∞

 − V0− = −Z∞ I0

(11.207)

and I0+ and I0− are two arbitrary constants. When the frequency is such that k = kt , then the longitudinal wavenumber vanishes and the solution takes on the special form V  (z) = V0 ,

I  (z) = − j ωεV0 z + I0 ,

z∈R

(11.208)

though rarely is a waveguide operated in these conditions. The full TM modal solution for a given kt is obtained by inserting (11.205) and (11.206) into the first parts of (11.178) and (11.179) and by retrieving the z-component of the electric field from the source-free instance of (11.142). In symbols, this reads ' ' & & ∇t · e ∇t · e       zˆ I0+ e− j kz z + −Z∞ zˆ I0− ej kz z e (ρ) + e (ρ) + (11.209) E (r) = Z∞ j ωε j ωε 



H (r) = h (ρ)I0+ e− j kz z + h (ρ)I0− ej kz z

(11.210)

for r ∈ V. The superscript ‘+’ reminds us that, when kz is real and positive, the first term in the right-hand sides of (11.209) and (11.210) represents a progressive wave, i.e., ‘propagating’ in the positive z-direction. Similarly, the superscript ‘−’ indicates that, under the same condition, the second contribution in the right-hand sides of (11.209) and (11.210) constitutes a regressive wave, that is, ‘travelling’ in the negative z-direction. These assertions are proved by first turning the modal solutions back in the time domain with the aid of the transformation rule (1.88) for phasors and then by examining ‘snapshots’ of the field at a few successive instants of time. For instance, the modal magnetic field reads  −  −  H (r, t) = |I0+ | cos(ωt − kz z + φ+ 0 )h (ρ) + |I0 | cos(ωt + kz z + φ0 )h (ρ)

(11.211)

where we have taken h (ρ) real (we shall show in Section 11.2.5 that this choice is always possible) − + − and φ+ 0 , φ0 indicate the phases of the possibly complex constants I0 , I0 . From (11.211) we notice that the magnetic field varies periodically in time and with the position along the waveguide axis. To find the shortest spatial period λg > 0 we set a time t and consider two points r1 and r2 within the waveguide such that r2 = r1 + λg zˆ ,

z2 = z1 + λg

(11.212)

and demand that the fields in r1 and r2 be related as H (r1 , t) = H (r2 , t) = H (r1 + λg zˆ , t)

(11.213)

which is satisfied if (ωt − kz z1 ) − (ωt − kz z2 ) = kz λg = 2π

(11.214)

Spectral representations of electromagnetic fields (ωt + kz z1 ) − (ωt + kz z2 ) = −kz λg = −2π

769

(11.215)

since the dependence on ρ is factored out through the transverse eigenfunction. From either condition above we obtain λg (ω) =

2π 2π λ = ) = 1/2 , kz 1 − kt2 /k2 k2 − kt2

ω  ckt

(11.216)

where λ := 2πc/ω is the wavelength (7.31) of a uniform plane wave propagating in an unbounded medium endowed with the same constitutive parameters as the material that fills the region V. The spatial period λg (ω) in the waveguide is called the guide wavelength and it is always larger than λ, since the denominator of the ratio in the rightmost-hand side of (11.216) is smaller than unity (when kz is real, that is). The phase velocity v p = v p zˆ of the progressive wave may be defined as the velocity of an ideal observer who, ‘travelling’ alongside the wave, manages to invariably detect the same field. The latter appears to be constant in time if the argument of the cosine function remains constant, that is, 0=

∂  dz (ωt − kz z + φ+ = ω − kz v p 0 ) = ω − kz ∂t dt

whence we find ω ω c = v p (ω) =  = ) 1/2 , kz 2 1 − kt /k2 k2 − kt2

(11.217)

ω  ckt

(11.218)

on account of (1.248). Evidently, the phase velocity of the regressive wave is the negative of v p . More importantly, the phase velocity is always larger than the speed of light c in the medium that fills the waveguide. Actually, v p (ω) is infinite for k = kt and tends asymptotically to c as the angular frequency grows infinitely large (cf. Section 12.5). Finally, by eliminating the propagation constant from (11.216) and (11.218) and recalling (1.87) we obtain λg (ω) = v p (ω)

ω = v p (ω)T 2π

(11.219)

that is, λg (ω) and v p (ω) obey the same relation (7.31) as λ and c for uniform plane waves in an unbounded isotropic lossless medium. When kz is imaginary the guide wavelength (11.216) and the phase velocity (11.218) are meaningless, and both progressive and regressive waves are evanescent and damped along the positive and negative z-direction, respectively. Such attenuation is not due to losses in the medium within V or in the waveguide walls (both of which we have not contemplated) but rather to a geometrical effect having to do with the finite cross-section of the waveguide. Since kz becomes imaginary when the wavenumber k is smaller than the transverse propagation constant kt , the latter is also called cut-off or critical constant. Put another way, for a chosen value of angular frequency ω only the modes for which kt < k can propagate in the waveguide, whereas all the infinitely many remaining ones are damped. As a matter of fact, in practical applications the modal cut-off phenomenon is a blessing rather than an issue, and waveguides are usually operated at frequencies for which only the fundamental lowest-order mode is above cut-off and charged with the task of carrying radiation. Finally, since kt = 0 for a TEM mode, when the latter exists, it can propagate in the waveguide for values of ω all the way down to zero. In which case, the modal solution reduces to the static electric field and the stationary magnetic field produced by a steady current flow over the surface of the conductors (two or more) which form the waveguide [34] (also see Section 12.5 and Figure 12.13).

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Advanced Theoretical and Numerical Electromagnetics

11.2.3 Transverse-electric modes If we insert the TE parts of (11.178) and (11.179) into (11.164) and (11.165) we obtain dV  = j ωμh (ρ) × zˆ I  (z), dz $ % dI  ∇t ∇t  = j ωε Itz + 2 · zˆ × e (ρ)V  (z), −h (ρ) dz k

−e (ρ)

r∈V

(11.220)

r∈V

(11.221)

since h (ρ) is lamellar by hypothesis. For the first equation to hold true the electric eigenfunction e (ρ) must be proportional to h (ρ) × zˆ , say, e (ρ) = h (ρ) × zˆ ,

ρ∈S

(11.222)

though other proportionality constants can be adopted without affecting (11.220). Also condition (11.222) in tandem with (11.178) and (11.179) says that electric and magnetic fields of the TE modes are orthogonal for r ∈ V. Plugging (11.222) in (11.221) yields & ' dI  ∇t ∇t = j ωε h (ρ) + 2 · h (ρ) V  (z), −h (ρ) r∈V (11.223) dz k whence we conclude that for the solvability of the equation h (ρ) must be an eigenfunction of ∇t ∇t · {•}. To fulfill this requirement we demand ⎧  2  ⎪ ⎪ ⎨∇t ∇t · h (ρ) + kt h (ρ) = 0, ρ ∈ S (11.224) ⎪ ⎪ ⎩νˆ (ρ) · h (ρ) = 0, ρ∈γ which constitutes an eigenvalue problem for h (ρ) with eigenvalue kt2 . The jump condition follows from (11.222) and the first part of (11.149). The second part of (11.149) is automatically satisfied by TE modes, because Ez (r) = 0 everywhere in V and hence, in particular on the waveguide boundary. The constant kt (physical dimension: 1/m) is the transverse wavenumber or transverse propagation constant for TE modes. The proof that kt2 is a non-negative real number is outlined in Section 11.2.5. Besides, since (11.224) is the two-dimensional analogue of (11.19), TE waveguide modes correspond to the lamellar magnetic modes existing in a cavity with PEC walls. Also the TE eigenfunctions can be computed once and for all independently of ω, ε, μ, V  and I  . Since the eigenfunctions h (ρ) are lamellar by construction, we may derive them from an auxiliary scalar potential Υ (ρ) (sometimes called a Neumann eigenfunction). In symbols, we let h (ρ) = −

∇t Υ , kt

and substitute into (11.224) to get   ∇t ∇2t Υ (ρ) + kt2 Υ (ρ) = 0,

ρ∈S

(11.225)

ρ∈S

(11.226)

whence we find ∇2t Υ (ρ) + kt2 Υ (ρ) = C  ,

ρ∈S

(11.227)

with C  and arbitrary constant. However, we may let C  = 0 and demand that ∇2t Υ (ρ) + kt2 Υ (ρ) = 0,

ρ∈S

(11.228)

Spectral representations of electromagnetic fields

771

since Υ (ρ) is determined up to an additive constant — which in light of (11.225) has no effect on the eigenfunction h (ρ). The boundary condition on h (ρ) becomes νˆ (ρ) · ∇t Υ (ρ) =

∂Υ =0 ∂ˆν

ρ ∈ γ.

(11.229)

In definition (11.225) we have tacitly assumed that kt is non-zero because, in fact, under no circumstance can the problem (11.224) admit a null eigenvalue. Indeed, if we suppose that kt vanishes and let h (ρ) = −∇t Υ ,

ρ∈S

(11.230)

then (11.228) becomes ∇2t Υ (ρ) = 0,

ρ∈S

(11.231)

in view of the arbitrariness of Υ (ρ). Next, we multiply (11.231) by Υ∗ (ρ), integrate over S , and apply the two-dimensional Gauss theorem, viz.,    ∗ 2   2 dS |∇t Υ | − dS ∇t · [Υ∗ (ρ)∇t Υ ] 0 = − dS Υ (ρ)∇t Υ = 

S  2

(

dS |∇t Υ | +

= S

γ

S

S

∗





ds Υ (ρ) νˆ (ρ) · h (ρ) =   =0

dS |h (ρ)|2

(11.232)

S

and the conclusion follows because the last integral is always positive, unless h (ρ) vanishes everywhere in S . Unlike the case of TM eigenfunctions, the result for TE modes remains true even if the waveguide cross-section is contour-wise multiply connected (Figure 11.2) because such assumption has no effect on the boundary condition νˆ · h (ρ) = 0. This should be contrasted with the threedimensional case and the existence of Neumann fields in a torus-like cavity [cf. (11.19) and (11.39)]. On the other hand, we notice that (11.231) subject to the matching condition (11.229) is indeed solved by a constant, non-zero potential, say,  1 AS := dS , ρ∈S (11.233) Υ (ρ) = Υ0 = √ , AS S

though the latter necessarily yields null transverse eigenfunctions by virtue of (11.230) and (11.222). Nonetheless the mode associated with Υ0 can be construed as the z-directed constant magnetic field which for r ∈ V := S × R solves the source-free stationary magnetic equations (4.9) and (4.10) with νˆ (ρ) · H(r) = 0 automatically satisfied for r ∈ γ × R. To shed further light on the meaning of Υ0 and Υ (ρ) we observe that a consistent set of scalar functions hz (ρ) for the representation of the longitudinal magnetic field in V can be built by starting from h (ρ) and the source-free instance of the longitudinal equation (11.141), namely, hz (ρ) :=

∇2 Υ (ρ) ∇t · [ˆz × e (ρ)] ∇t · h (ρ) = = − t 2 = Υ (ρ),   kt kt kt

kt2  0,

ρ∈S

(11.234)

in view of (11.222) and (11.228). However, this approach does not give Υ0 straightaway, unless we agree to take it as the limit of the second-to-last member for kt2 → 0. Yet, a detailed analysis, similar to the one we carried out in Sections 11.1.1 and 11.1.6 for PEC cavities, indicates that such constant

Advanced Theoretical and Numerical Electromagnetics

772

potential is indeed necessary to correctly represent the longitudinal component of the magnetic field in the source region V J when magnetic currents are present [10, Section 15.1]. Calculations show that in the end one recovers the direct contribution of J Mz (r) which, with the Marcuvitz-Schwinger approach, is already present in (11.141). By employing (11.224) to eliminate the transverse field ∇t ∇t · h from (11.223) and inserting (11.222) in (11.220) we obtain dV  = j ωμe (ρ)I  (z), dz $ % kt2  dI   = j ωε 1 − 2 h (ρ)V  (z), −h (ρ) dz k

−e (ρ)

r∈V

(11.235)

r∈V

(11.236)

which can be satisfied by equating the coefficients of e (ρ) and h (ρ) in the two sides of the equations. With this step we arrive at a first-order homogeneous differential system for TE modal voltages and currents, viz., kz dV  = j ωμI  (z) = j  I  (z), dz Y∞ $ 2 %  k2 − kt2  k dI   V (z), − = j ωε 1 − t 2 V  (z) = j I (z) = j kz Y∞ dz ωμ k

−

z∈R

(11.237)

z∈R

(11.238)

where ⎧) ⎪ 2 2 ⎪ ⎪ ⎪ ⎨ k − kt kz (ω) := ⎪ ) ⎪ ⎪ ⎪ ⎩− j k2 − k2 t

k

kt

k
y1 . To obtain physical wave-like solutions to (11.396) that are valid for y ∈ R we require that ˜ dΨ ˜ x , y) = 0, + j ky Ψ(k dy ˜ dΨ ˜ x , y) = 0, − j ky Ψ(k dy

y > y2

(11.398)

y < y1

(11.399)

in order to choose waves which emanate from the source and travel towards plus or minus infinity, respectively. Such requirements are physically motivated by the absence of other sources anywhere

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along the y-direction, and play the role of radiation conditions of sorts [cf. (11.313) and (11.321) for uniform transmission lines and the Sommerfeld radiation condition (8.239)]. The source S˜ (k x , y) is distributed in [y1 , y2 ], and hence in order to solve (11.396) we first obtain ˜ x , y, y ) where y and y denote the coordithe relevant one-dimensional spectral Green function G(k nate of observation and source point. To this purpose, we seek the solution to the problem ⎧ 2˜ ⎪ dG ⎪ ⎪ ˜ x , y) = 0, ⎪ + ky2 G(k y ∈ R \ {y0 } ⎪ 2 ⎪ ⎪ dy ⎪ ⎪ ⎪ ⎪ ⎪ ˜ x , y+0 ) = G(k ˜ x , y−0 ), ⎪ G(k y = y0 ⎪ ⎪ ⎪   ⎪ ⎪ ⎪ ˜ ˜ ⎪   dG  dG  ⎪ ⎪ ⎪ − = −1, y = y0   ⎨ dy dy y=y− (11.400) + ⎪ y=y0 ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ dG˜ ⎪ ⎪ ˜ x , y) = 0, ⎪ + j kyG(k y > y+0 ⎪ ⎪ ⎪ dy ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ dG˜ ⎪ ⎪ ˜ x , y) = 0, ⎪ − j kyG(k y < y−0 ⎩ dy where y0 ∈ R specifies the arbitrary location of the singular ‘point source’. To elaborate, we recall that in the physical space S (ρ) is a function of x and y, whereas in the spectral domain the dependence of S˜ (k x , y) on x disappears, as it were. Thus, when looked at back in the physical space, a ‘point source’ in y0 may actually be a planar current sheet with finite extension along x. According to (11.386), if Ψ(ρ) stands for Ez (ρ), then the current sheet is a total surface electric current or a jump of the tangential magnetic field across the plane y = y0 [see (1.196)]. If Ψ(ρ) represents Hz (ρ), then the current sheet is a total surface magnetic current or a jump of the tangential electric field across the plane y = y0 [cf. (6.215)]. ˜ y , y) is continuous and that the derivaThe first two matching conditions in (11.400) state that G(k tive thereof suffers a jump across the location of the source. The remaining constraints are analogous to (11.398) and (11.399) and serve to select wave-like solutions which propagate away from the source point y = y0 in the y-direction towards plus or minus infinity. ˜ x , y) is continuous across y0 , it has the general form Since G(k ˜ x , y0 )e− j ky |y−y0 | , ˜ x , y) = G(k G(k

y∈R

(11.401)

˜ x , y0 ) is the unknown amplitude in y0 . Enforcing the jump condition on the first derivative where G(k yields − j ky |y−y0 | ˜ x , y) = e G(k , 2 j ky

y∈R

(11.402)

where we may finally let y0 = y and write 

− j ky |y−y | ˜ x , y, y ) = e , G(k 2 j ky

(y, y ) ∈ R × R

(11.403)

as the coordinate y0 is arbitrary. The spectral Green function is regular for y = y , the first derivative is discontinuous but finite, whereby the second derivative has a meaning in a distributional sense (Appendix C.2). Besides, since the underlying medium is reciprocal (cf. Section 6.8.1) then the Green function is symmetric under the exchange of source and observation point. Further, since the ˜ x , y, y ) depends on the distance |y − y | rather than on y and y region of concern in unbounded, G(k individually.

Spectral representations of electromagnetic fields

807

˜ x , y) generated by an arbitrary source S˜ (k x , y) is the The procedure to find the scalar field Ψ(k one-dimensional analogue of the approach we followed to solve the three-dimensional Helmholtz equation in Section 9.1.1. In light of (11.396) and (11.400) we consider the harmonic equations d2 ˜ ˜ x , y ) = −S˜ (k x , y ), Ψ(k x , y ) + ky2 Ψ(k dy2 d2G˜ ˜ x , y, y ) = 0, + ky2 G(k dy2

y ∈ R

(11.404)

y ∈ R \ {y}

(11.405)

where using the symbol y to indicate the independent variable will come in handy in the solution ˜ x , y ) and subtract the resulting ˜ x , y, y ) and (11.405) by Ψ(k process. We multiply (11.404) by G(k expressions to arrive at & ' ˜ ˜ d ˜  dΨ ˜ x , y ) dG = −S˜ (k x , y )G(k ˜ x , y, y ), G(k , y, y ) − Ψ(k y ∈ R \ {y} (11.406) x dy dy dy an equation which may not be integrated with respect to y over an interval that includes the point y because the function in brackets is discontinuous for y = y and hence the left-hand side is not defined in y = y. Thus, to handle the troublesome point we choose a length a > 0 large enough so that [y1 , y2 ] ⊂ [y − a, y + a], and we integrate separately from y − a to y− and from y+ to y + a. In symbols, this step gives & '− ˜ ˜ y  ˜   dΨ  dG ˜ ˜ ˜ dy S (k x , y )G(k x , y, y ) = G(k x , y, y )  − Ψ(k x , y )  − dy dy y−a y−a   ˜  1 dΨ dG˜   ˜ =  − Ψ(k x , y)   2 j ky dy y− dy y− y

−



˜ − a) + Ψ(y ˜ − a) j kyG(k ˜ x , y, y − a) j ky Ψ(y ˜ x , y, y − a) − G(k & ' y+a ˜ ˜ y+a  ˜  ˜   dΨ  dG ˜ ˜ dy S (k x , y )G(k x , y, y ) = G(k x , y, y )  − Ψ(k x , y )  − dy dy y+

(11.407)

y+

˜ + a) + Ψ(y ˜ + a) j kyG(k ˜ x , y, y + a) j ky Ψ(y ˜ x , y, y + a) = −G(k   ˜ ˜ 1 dΨ  + Ψ(k ˜ x , y) dG  −   2 j ky dy y+ dy y+

(11.408)

˜ x , y ) for y = y ± a. If the ˜ x , y, y ) and Ψ(k on account of the ‘radiation conditions’ obeyed by G(k  singular point y = y is located at the left (resp. at the right) of the source interval [y1 , y2 ], then the integral in (11.407) [resp. in (11.408)] vanishes. Regardless, by summing (11.407) and (11.408) side by side and simplifying we obtain ⎛   ⎞ y+a ˜  ⎟⎟⎟ ⎜⎜⎜ dG˜  d G  ˜  ˜    ⎟⎟ = −Ψ(k ˜ x , y) ⎜⎜⎝ ˜ x , y) − dy S (k x , y )G(k x , y, y ) = Ψ(k  − dy y+ dy y− ⎠

(11.409)

y−a

˜ x , y) and the derivative thereof are continuous, and also used the where we have recalled that Ψ(k ˜ x , y, y ) included in (11.400). In summary, the solution to jump condition of the derivative of G(k

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Advanced Theoretical and Numerical Electromagnetics

(11.396) valid in a homogeneous unbounded medium reads √ y2 − j k2 −k2x |y−y | e  ˜ x , y) = ˜ y , y) ∗ S˜ (ky , y), Ψ(k dy S˜ (k x , y ) = G(k 2 2 2 j k − kx

y∈R

(11.410)

y1

on account of (11.397) and (11.403) and the fact that S˜ (k x , y ) vanishes outside [y1 , y2 ] ⊂ [y − a, y + a]. The integral representation (11.410) has the typical form of a convolution product between the relevant spectral Green function and the Fourier transform of the source [cf. (2.187), (7.327), (7.347) and (8.341)]. The next step towards the determination of Ψ(ρ; ω) consists of inserting the expression of ˜ x , y) into (11.393) and computing the integral with respect to k x . In this regard, we observe Ψ(k ˜ x , y) may be singular, because the that, as a function of the complex variable k x = kx + j kx , Ψ(k integrand in (11.410) diverges in k x = ±k, where it also has algebraic branch points of order one, since the complex function ) f (k x ) := ky = k2 − k2x , kx ∈ C (11.411) is many-valued (Section B.3). Incidentally, the spectral Green function (11.403) has the same singularities as f (k x ). If the wavenumber k is real, the branch points are located on the real axis, and this occurrence makes it difficult to evaluate and interpret the inverse Fourier transformation (11.393), inasmuch as the integrand is ambiguous for real values of the spectral variable. Thus, to give a meaning to the integral in k x we suppose temporarily that the underlying medium has a small non-null conductivity σ > 0, in which instance the wavenumber k becomes complex and can be approximated as in (7.98). As a result, the branch points ‘migrate’ away from the real axis into the second and the fourth ˜ x , y) is regular for quadrant of the complex plane, as is illustrated in Figure 11.17a. Therefore, Ψ(k k x ∈ R, and the inverse Fourier transformation can be carried out by integrating along the real axis, provided in so doing we do not cross the branch lines of f (k x ), which we have not specified yet. We recall that branch cuts may have arbitrary shapes but they have to connect two branch points, possibly by reaching and including the point at infinity. A particularly convenient choice for the function f (k x ) is constituted by the locus of points k x ∈ C such that ) Im{ f (k x )} := Im{ky } = Im k2 − k2x = 0 (11.412) and the reason will become clear in a moment [9, Section 4.3a]. If k is complex, (11.412) is better solved with the help of the dispersion relationship (11.397) where we enforce ky = ky and let k = k + j k with k > 0 and k < 0. Separating real and imaginary parts provides us with 2 ky2 = k2 − k2 − k2 x + kx

(11.413)

kx kx

(11.414)

 

=kk

where the second relation represents a rectangular hyperbola [62, Section 8.4] that has center in the origin, lies in the second and fourth quadrants (since k k < 0) and tends asymptotically to the straight lines kx = 0, kx = 0. The branch points k x = ±k belong to said hyperbola because relationship (11.414) is evidently satisfied by kx = ±k and kx = ±k . Besides, since ky2 must be a positive real number by definition, from (11.413) we deduce the constraint 2 2 2 k2 x − kx  k − k

(11.415)

Spectral representations of electromagnetic fields

(a) k ∈ C

809

(b) k ∈ R

˜ x , y): branch points (×), Figure 11.17 Inverse Fourier transformation of the spectral function Ψ(k branch cuts (••) and paths (−) for integration in the complex plane k x . which restricts the range of admissible values of kx and kx in (11.414). Indeed, (11.415) specifies the unbounded connected region of the complex plane enclosed by the two branches of the hyperbola 2 2 2 k2 x − kx = k − k

(11.416)

which has asymptotes kx = ±kx and — for |k | < k — vertices in kx = ±(k2 − k2 )1/2 and kx = 0. For example, it can be verified that the points on the imaginary axis (kx = 0) satisfy (11.415) because the right-hand side is positive. The branch points k x = ±k belong also to the hyperbola (11.416). In conclusion, the solution to (11.412) is given by (11.414) subject to (11.415), whereby we may define the branch cuts of f (k x ) as γb− := {k x ∈ C : kx kx = k k , −k  kx < 0, kx > 0} γb+ := {k x ∈ C : kx kx = k k , 0 < kx  k , kx < 0}

(11.417) (11.418)

and they are graphically shown in Figure 11.17a by means of dotted lines. One usually speaks of branch cuts but as a matter of fact γb− and γb+ form a single line which runs from k x = −k to infinity and back to k x = k. With this choice the integration path (the real axis) in (11.393) does not intercept the branch lines, as desired. If k is real, then (11.412) is manifestly satisfied when the argument of the square root is a nonnegative real number. Such condition can be realized in two ways, viz., (a) (b)

the spectral variable k x is real and |kx |  k or k x is purely imaginary, because then k2 − k2x = k2 + k2 x .

whereby we take as branch cuts the piecewise-straight lines γb− := {k x ∈ C : k x = ξ ∪ k x = j η, ξ ∈ [−k, 0− ], η  0} γb+ := {k x ∈ C : k x = k − ξ ∪ k x = − j η, ξ ∈ [0, k− ], η  0}

(11.419) (11.420)

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Advanced Theoretical and Numerical Electromagnetics

which are graphically shown in Figure 11.17b. It is understood that γb− lies in the second quadrant and γb+ in the fourth quadrant. As we make the conductivity σ conceptually approach zero, the branch lines (11.417) and (11.418) move towards the coordinate axes, while deforming, and ultimately come to coincide with the lines (11.419) and (11.420). ˜ x , y) are located on the real axis, This analysis also suggests that, when the singularities of Ψ(k we trade the integral (11.393) for an alternative, perfectly equivalent one along a path γ, say, 1 Ψ(ρ; ω) = 2π



dk x e− j kx x

γ

y2 y1

dy

e− j

√

k2 −k2x |y−y |

S˜ (k x , y ), 2 j k2 − k2x

ρ ∈ R2

(11.421)

where k ∈ R, and γ is the jagged line defined as (Figure 11.17b) γ := {k x ∈ C : k x = ξ, |ξ| > k + b, |ξ| < k − b ∪ k x = −k − b exp(j α) ∪ k x = k − b exp(− j α), α ∈ [0, π], 0 < b < k} (11.422) so that we steer clear of poles and do not cross any branch cut. This alteration is possible on account of (B.32) in that the integrand is analytic everywhere except at infinity and for points on γb± , and the endpoints of γ and the real axis coincide. Lastly, as f (k x ) is many-valued we still have to decide which value to use in the calculation of ˜ x , y). Since ky vanishes on γ− and γ+ by construction [see (11.412)], then Im{ f (k x )} changes Ψ(k b b sign upon crossing either branch line once. Consequently, Im{ f (k x )} will be positive or negative everywhere on either sheet of the Riemann surface associated with f (k x ) (Appendix B.3) and this feature amply motivates our choice of branch lines. Now, from (11.421) we gather that the imaginary part of f (k x ) ought to be non-positive in order to guarantee the convergence of the integral along γ, in that the integrand contains the term e− j

√

k2 −k2x |y−y |

√ 2 √ 2   2 2 = e− j |y−y |Re k −kx e|y−y |Im k −kx

(11.423)

and hence, we may arbitrarily decide that the complex plane in Figures 11.17a and 11.17b is the sheet of the Riemann surface such that ) Im{ f (k x )} = Im k2 − k2x < 0.

(11.424)

The points of the real axis and the line γ meet this condition precisely because the integration paths in both cases do not intercept any branch line. Having taken due care of the singularities of the spectral Green function (11.403) and also ensured the convergence of the line integral, we may swap the order of integration in (11.421) and obtain y2 Ψ(ρ; ω) = y1

dy

 γ

√ 2  2 e− j kx x e− j k −kx |y−y | ˜ dk x S (k x , y ), 4π j k2 − k2x

ρ ∈ R2

(11.425)

an expression that lends itself to the following physical interpretation. Since Ψ(ρ; ω) represents either Ez (ρ; ω) or Hz (ρ; ω) on account of (11.386) and (11.387), we may regard the inner integral as

Spectral representations of electromagnetic fields

(a) uniform plane waves

811

(b) inhomogeneous plane waves

Figure 11.18 Physical interpretation of the integral representation (11.425) as superposition of elementary plane waves emerging from current sheets in y = y ∈ [y1 , y2 ]. a linear superposition of elementary plane waves (Section 7.2). More specifically, the time-harmonic scalar field √ 2 ⎧  2 ⎪ ⎪ S˜ (k x , y )ej y k −kx − j kx x − j y √k2 −k2x ⎪ ⎪ ⎪ e e y > y ⎪ ⎪ ⎪ 2 − k2 ⎪ 4π j k x ˜ y ) = ⎨ Υ(ρ, y ∈ [y1 , y2 ] (11.426) √ 2 ⎪ ⎪  ⎪ 2 −k √  − j y k ⎪ x ˜ ⎪ , y )e S (k 2 2 ⎪ x ⎪ ⎪ e− j kx x ej y k −kx y < y ⎪ ⎩ 4π j k2 − k2x represents the z-component of either the electric or the magnetic field of plane waves which have wavevector ) ⎧ ⎪ ⎪ ⎪ ˆ x + k2 − k2x yˆ , y > y k ⎪ x ⎨ k=⎪ (11.427) ) ⎪ ⎪ ⎪ ⎩k x xˆ − k2 − k2x yˆ , y < y and conceptually emerge from a planar current sheet located on the plane y = y . When the underlying medium is lossless, if k > k x ∈ R the plane waves are uniform and propagate in the direction set by k ∈ R2 (Figure 11.18a). Under the same hypothesis, if k < k x ∈ R we compute the wavevector as ) ⎧ ⎪ ⎪ ⎪ ˆ x − j k2x − k2 yˆ , y > y k ⎪ x ⎨   (11.428) k = k − jk = ⎪ ) ⎪ ⎪ ⎪ ⎩k x xˆ + j k2x − k2 yˆ , y < y in accordance with (11.424). The resulting plane waves are inhomogeneous, travel along the xdirection and are damped along ±ˆy away from the plane y = y (Figure 11.18b). Regardless, the amplitude of these plane waves is determined by the Fourier transform of the actual source S (ρ; ω) in the physical space. Finally, we notice that (11.425) can be turned into the spatial convolution of the two-dimensional time-harmonic Green function (7.316) with the source S (ρ, ω) as in (7.347), by taking advantage of

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Advanced Theoretical and Numerical Electromagnetics

Figure 11.19 Reflection and transmission of z-invariant electromagnetic waves at a planar interface between two homogeneous isotropic media. the integral representation (7.362) for the Hankel function (with obvious changes to account for the different notations) and the convolution theorem for the Fourier transform [17, Section 4.8].

11.4.3 Reflection and transmission at a planar material interface Having found the formal solution to the two-dimensional Helmholtz equation (11.392) in an unbounded homogeneous medium, we go on to study the fields generated by a z-invariant source S (ρ; ω) that radiates in the presence of a planar material interface between two half-spaces filled with different penetrable media. Without loss of generality we can assume that the interface coincides with the plane y = 0, medium 1 and the source occupy the half-space y > 0, and medium 2 fills the half space y < 0. This problem is sketched in Figure 11.19. Besides, for the sake of argument we focus on TM electromagnetic fields generated by a total electric source J˜z (ρ; ω). We shall retrieve the TE case at the end by invoking the duality principle (Section 6.7). Based on the observations made at the beginning of Section 11.4.2, on the one hand we need to state two distinct Helmholtz equations of the type (11.386), on the other we may introduce a spectral representation only along x, in that the medium of concern is piecewise-homogeneous along the y-direction. To lighten the notation a bit we indicate the z-component of the electric field in medium l with Ψl (ρ; ω), l = 1, 2, and formulate the electromagnetic problem mathematically as ∇2t Ψ1 (ρ; ω) + k12 Ψ1 (ρ; ω) = −S (ρ; ω),

x ∈ R,

y>0

(11.429)

∇2t Ψ2 (ρ; ω)

x ∈ R,

y0

(11.433)

y y2

(11.438)

y 0 ascribes the electromagnetic field to the true sources S (ρ; ω) and to infinitely extended equivalent sources in the half-space y < 0. All sources are invariant along z and reside in a homogeneous unbounded space that has the properties of medium 1. Under these circumstances the field ˜ i (k x , y). produced by S˜ (k x , y) is essentially the solution (11.410), and represents the incident wave Ψ 1 The equivalent sources are responsible for the generation of a secondary or scattered or reflected ˜ r (k x , y), which the observer perceives as emerging from the interface y = 0. Conversely, wave, say, Ψ 1 from the viewpoint of another observer sitting in the half-space y < 0 the electromagnetic field is produced by infinitely extended equivalent sources that are located in the half-space y > 0 and exist in a homogeneous unbounded space with the constitutive parameters of medium 2. Accordingly, the ˜ t (k x , y) which the observer sees as only contribution to the field in medium 2 is a transmitted wave Ψ 2 emerging from the plane y = 0.

Advanced Theoretical and Numerical Electromagnetics

814

All in all, we are led to put the solution to (11.433) and (11.434) in the form ˜ 1 (k x , y) := Ψ ˜ i1 (k x , y) + Ψ ˜ r1 (k x , y) Ψ y2  e− j ky1 |y−y | ˜ = dy S (k x , y ) + A1 (k x )e− j ky1 y , 2 j ky1

y>0

(11.440)

y 0 — and obeys the radiation condition (11.438). The reflected wave satisfies (11.438) too, and solves the source˜ r (k x , y) are non-zero only free instance of (11.433) because, as we argued above, the ‘sources’ of Ψ 1 for y < 0. Lastly, the transmitted wave solves (11.434) and fulfills (11.439). Thus, we only have ˜ 1 (k x , y) and Ψ ˜ 2 (k x , y) satisfy the matching to ‘adjust’ the amplitudes A1 (k x ) and A2 (k x ) so that Ψ conditions (11.436) and (11.437). By inserting (11.440) and (11.441) into (11.436) and (11.437) we obtain the algebraic linear system y2 y1

ky1 − ωμ1

y2



dy

e− j ky1 y ˜ S (k x , y ) + A1 (k x ) = A2 (k x ) 2 j ky1

(11.442)



dy

y1

ky1 ky2 e− j ky1 y ˜ A1 (k x ) = − A2 (k x ) S (k x , y ) + 2 j ky1 ωμ1 ωμ2

(11.443)

which is readily solved to give  Z  − Z∞1 A1 (k x ) = ∞2  + Z Z∞2 ∞1

A2 (k x ) =

 2Z∞2  + Z Z∞2 ∞1

y2 y1 y2

y1



dy

e− j ky1 y ˜ S (k x , y ) 2 j ky1

dy

e− j ky1 y ˜ S (k x , y ) 2 j ky1

(11.444)



(11.445)

where  := Z∞1

ωμ1 , ky1

 := Z∞2

ωμ2 ky2

(TM waves)

(11.446)

are the characteristic impedances for the TM polarization. For ease of manipulation we also define reflection and transmission coefficients as RT M :=

  Z∞2 − Z∞1   , Z∞2 + Z∞1

T T M :=

 2Z∞2  + Z∞1

 Z∞2

(TM waves)

(11.447)

which should be compared with the quantities introduced in (7.190). It is important to realize that, aside from apparent minor notational differences (which are due to the alternative orientation of the system of Cartesian coordinates) the coefficients in (11.447) coincide with those in (7.190). Indeed, in Section 7.4.1 plane waves having no component of the electric field perpendicular to the material

Spectral representations of electromagnetic fields

815

interface (see Figure 7.14) were termed transverse-electric. Here, the very same plane waves are called transverse-magnetic because the magnetic field is perpendicular to the direction (z) of spatial invariance. In fact, this definition is consistent with the nomenclature adopted in Section 7.5 for cylindrical waves generated by straight line currents parallel to the z-axis. With these findings and positions reflected and transmitted waves read y2 ˜ r (k x , y) = RT M Ψ 1 y1 y2

˜ t (k x , y) = T T M Ψ 2



e− j ky1 (y+y ) ˜ dy S (k x , y ), 2 j ky1 

y>0

(11.448)

y0

(11.450)

image source in y 0. To finalize the solution of (11.429) and (11.430) one needs to carry out the inverse Fourier ˜ l (ky , y) according to (11.393). To this purpose we have to study nature and transformation of Ψ position of the singularities in the complex plane k x . We notice that reflected and transmitted waves involve the complex functions ) fl (k x ) := kyl = kl2 − k2x , k x ∈ C, l = 1, 2 (11.457) which are many-valued and exhibit first-order algebraic branch points in k x = ±kl . The latter are located on the real axis if medium 1 and 2 are lossless, whereas they ‘migrate’ into the second and fourth quadrant if the materials possess even a small non-zero conductivity σl . As a consequence, the branch lines may be taken as in (11.419), (11.420) or (11.417), (11.418), and these choices are illustrated in Figures 11.20a and 11.20b. The Riemann surface (Appendix B.3) associated with ˜ t (k x , y) is comprised of four sheets ‘glued’ together along the branch lines γ± . We ˜ r (k x , y) and Ψ Ψ 1 bl 2 suppose that the complex plane in Figures 11.20a and 11.20b is the sheet whereon Im{ fl (k x )} < 0, in order to ensure the convergence of the inverse Fourier transformations. Finally, the incident wave depends only upon f1 (k x ), whereby the considerations of Section 11.4.2 apply. ˜ l (ky , y) may be the poles of RT M and T T M , i.e., the roots of the Additional singularities of Ψ irrational equation   Z∞1 (k x ) + Z∞2 (k x ) = 0

(11.458)

which we can write as μ1 f2 (k x ) + μ2 f1 (k x ) = 0

(11.459)

by virtue of (11.446) and (11.457). If we exclude magnetic losses, though, (11.459) as it stands has no solutions, and hence there exists no poles of RT M and T T M . Indeed, if the electric conductivity in media 1 and 2 is null and k2x < min{k12 , k22 }, then (11.459) is impossible, since the left-hand side is the sum of two positive quantities. If either f1 (k x ) or f2 (k x ) is complex (for whatever reason) the equation is impossible. Finally, if both f1 (k x ) and f2 (k x ) are complex, the equations is still impossible because, irrespective of the sign of Re{ fl (k x )}, we must compute the square roots so that Im{ fl (k x )} < 0, and thus the requirement μ1 Im{ f2 (k x )} + μ2 Im{ f1 (k x )} = 0 cannot be satisfied, since μl > 0.

(11.460)

Spectral representations of electromagnetic fields

(a) kl ∈ C

817

(b) kl ∈ R

˜ l (k x , y): branch points (×), Figure 11.20 Inverse Fourier transformation of the spectral functions Ψ branch cuts (••) and paths for integration (−) in the complex plane k x . We write the solution in the spatial domain formally as 1 Ψ1 (ρ) = 2π

y2



 − j kx x e

−j

√

k12 −k2x |y−y |

S˜ (k x , y ) ) 2 2 2 j k1 − k x γ y1 √  y2 − j k12 −k2x (y+y ) 1 e + dk x dy e− j kx x RT M S˜ (k x , y ), ) 2π 2 2 2 j k1 − k x γ y1 √ √ y 2  j k22 −k2x y − j k12 −k2x y e 1  − j kx x e Ψ2 (ρ) = dk x dy e T T M S˜ (k x , y ), ) 2π 2 2 j k1 − k2x γ y1 dk x

dy e

y>0

(11.461)

y 0 (Figure 11.20a). By contrast, when the media are lossless and, e.g., k1 < k2 we take γ as the jagged line γ := {k x ∈ C : k x = ξ, |ξ| < k1 − b1 , k1 + b1 < |ξ| < k2 − b2 , |ξ| > k2 + b2 , ∪ k x = −k − b1 exp(j α) ∪ k x = k − b1 exp(− j α), ∪ k x = −k − b2 exp(j α) ∪ k x = k − b2 exp(− j α), α ∈ [0, π], 0 < b1 < k1 , k1 + b1 < k2 − b2 } (11.463) which runs in the third and first quadrants while avoiding branch points and associated branch lines (Figure 11.20b). The solution of the problem of reflection and transmission of TE waves can be accomplished, for the most part, by systematically applying the duality transformations listed in the right part of Table 6.1. The scalar field Ψl (ρ) denotes the z-component of H(ρ) in medium l, and S (ρ) accounts for the total magnetic current density J˜Mz (ρ). A ‘true’ magnetic current is not really necessary because,

Advanced Theoretical and Numerical Electromagnetics

818

as we gather from (11.389), TE waves can be excited by a transverse electric current density Jt (ρ) that is not lamellar [see (11.152)]. The matching condition (11.437) entails the permittivities εl , whereas the ‘radiation conditions’ (11.438) and (11.439) are formally unaffected. With the help of (11.448) and (11.449) we write reflected and transmitted TE waves in the spectral domain as y2 ˜ r1 (k x , y) Ψ



dy

e− j ky1 (y+y ) ˜ S (k x , y ), 2 j ky1

dy

ej ky2 y e− j ky1 y ˜ S (k x , y ), 2 j ky1

= RT E y1

y2 ˜ t (k x , y) = T T E Ψ 2

y>0

(11.464)

y k1 + b, |ξ| < k1 − b ∪ k x = −k1 − b exp(j α) ∪ k x = k1 − b exp(− j α), α ∈ [0, π], 0 < b < k1 } (11.480) which runs in the third and first quadrants while avoiding the branch points ±k1 (Figure 11.21).

References [1] [2] [3] [4] [5] [6] [7] [8]

Zienkiewicz OC. The Finite Element Method in Engineering Science. London, UK: McGrawHill; 1971. Jin JM. The Finite Element Method in Electromagnetics. New York, NY: John Wiley & Sons, Inc.; 1993. Crandall SH. Engineering Analysis. New York, NY: McGraw-Hill; 1956. Becker EB, Carey GF, Oden JT. Finite Elements. Englewood Cliffs, NJ: Prentice-Hall; 1981. Binns KJ, Lawrenson PJ, Trowbridge CW. The analytical and numerical solution of electric and magnetic fields. Chichester: John Wiley & Sons, Inc.; 1992. Bastos JPA, Sadowski N. Electromagnetic modeling by finite element methods. New York, NY: Marcel Dekker, Inc.; 2003. Sadiku MNO. Numerical techniques in electromagnetics. 2nd ed. Boca Raton, FL: CRC Press; 2001. Jin JM. Theory and Computation of Electromagnetic Fields. 2nd ed. Hoboken, NJ: IEEE Press; 2015.

Spectral representations of electromagnetic fields [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39]

821

Felsen LB, Marcuvitz N. Radiation and scattering of waves. Piscataway, NJ: IEEE Press; 2001. Van Bladel JG. Electromagnetic Fields. Piscataway, NJ: IEEE Press; 2007. Harrington RF. Time-harmonic Electromagnetic Fields. London, UK: McGraw-Hill; 1961. Collin RE. Foundations for Microwave Engineering. New York, NY: McGraw-Hill; 1992. Hill DA. Electromagnetic Fields in Cavities. Piscataway, NJ: IEEE Press; 2009. Stratton JA. Electromagnetic theory. London, UK: McGraw-Hill; 1941. Kurokawa K. The expansions of electromagnetic fields in cavities. IRE Trans Microwave Theory Tech. 1958;p. 178–187. Schelkunoff SA. On representation of electromagnetic fields in cavities in terms of natural modes of oscillation. Journal of Applied Physics. 1955;26:1231–1234. Morse P, Feshbach H. Methods of Theoretical Physics. New York, NY: McGraw-Hill; 1953. Muscat J. Functional Analysis. London, UK: Springer; 2014. Dwight HB. Tables of Integrals and Other Mathematical Data. 4th ed. New York, NY: MacMillan Publishing Co., Inc.; 1961. Adams RA, Fournier JJF. Sobolev spaces. 2nd ed. New York, NY: Academic Press; 2003. Conway JB. A Course in Functional Analysis. 2nd ed. Graduate Texts in Mathematics. New York, NY: Springer-Verlag; 1990. Colton DL, Kress R. Integral Equation Methods in Scattering Theory. New York, NY: John Wiley & Sons, Inc.; 1983. Blyth TS, Robertson EF. Basic Linear Algebra. 2nd ed. Springer Undergraduate Mathematics Series. London, UK: Springer-Verlag; 2002. Schwinger J, De Raad LL, Milton KA, et al. Classical electrodynamics. Perseus Books; 1998. Milton KA, Schwinger J. Electromagnetic Radiation: Variational Methods, Waveguides and Accelerators. Berlin Heidelberg: Springer-Verlag; 2006. Holt CA. Introduction to Electromagnetic Fields and Waves. New York, NY: John Wiley & Sons, Inc.; 1963. Shen LC, Kong JA. Applied electromagnetism. Monterey, CA: Brooks/Cole; 1983. Kraus JD, Carver KR. Electromagnetics. 2nd ed. Tokyo, Japan: McGraw-Hill; 1973. Orfanidis SJ. Electromagnetic Waves and Antennas. www.ece.rutgers.edu/~orfanidi/ewa; 2004. Dobbs R. Electromagnetic Waves. London, UK: Routledge & Kegan Paul; 1985. Slater JC, Frank NH. Electromagnetism. New York, NY: McGraw-Hill; 1947. Zhang K, Li D. Electromagnetic Theory for Microwaves and Optoelectronics. Berlin Heidelberg: Springer-Verlag; 1998. Collin RE. Field Theory of Guided Waves. Piscataway, NJ: IEEE press; 1991. Marcuvitz N. Waveguide Handbook. 2nd ed. Electromagnetic Waves Series. London, UK: The Institution of Engineering and Technology; 1985. Marcuse D. Light Transission Optics. New York, NY: Van Nostrand Reinhold Company; 1972. Vassallo C. Optical Waveguide Concepts. Amsterdam, NL: Elsevier Science Publishers B.V.; 1991. Lorrain P, Corson DR, Lorrain F. Electromagnetic Fields and Waves. 3rd ed. New York, NY: W. H. Freeman and Company; 1988. Moon P, Spencer DE. Field Theory Handbook: Including Coordinate Systems, Differential Equations and Their Solutions. 2nd ed. Berlin Heidelberg: Springer-Verlag; 1971. Rudin W. Principles of mathematical analysis. 3rd ed. New York, NY: McGraw-Hill; 1976.

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[40] [41] [42] [43]

Jackson JD. Classical Electrodynamics. 3rd ed. Chichester, UK: Wiley; 1999. Rothwell EJ, Cloud MJ. Electromagnetics. London, UK: CRC Press; 2001. Pozar D. Microwave Engineering. 4th ed. New York, NY: John Wiley & Sons, Inc.; 2012. Hayt WH, Buck JA. Engineering Electromagnetics. 8th ed. New York, NY: McGraw-Hill; 2012. International edition. Guru BS, Hiziroglu HR. Electromagnetic field theory fundamentals. 2nd ed. New York, NY: Cambridge University Press; 2004. Adler RB, Chu LJ, Fano RM. Electromagnetic energy transmission and radiation. New York, NY: John Wiley & Sons, Inc.; 1960. Popovi´c Z, Popovi´c BD. Introductory Electromagnetics. Upper Saddle River, NJ: Prentice Hall; 2000. Schwab AJ. Field Theory Concepts. Berlin Heidelberg: Springer-Verlag; 1988. Kong JA. Electromagnetic Wave Theory. 2nd ed. New York, NY: Wiley; 1990. Sommerfeld A. Partial Differential Equations in Physics. vol. 1 of Lectures on theoretical physics. New York, NY: Academic Press; 1949. Meise R, Vogt D. Introduction to Functional Analysis. Oxford, UK: Clarendon Press; 1997. Limaye BV. Linear Functional Analysis for Scientists and Engineers. Singapore: Springer Science+Business Media; 2016. Abramowitz M, Stegun IA. Handbook of mathematical functions. New York, NY: Dover Publications, Inc.; 1965. Flanders H. Differentiation under the integral sign. American Mathematical Monthly. 1973 June-July;80(6):615–627. Brillouin L. Wave Propagation in Periodic Structures. New York, NY: Dover Publications, Inc.; 1953. Elachi C. Waves in active an passive periodic structure: a review. Proceedings of the IEEE. 1976 Dec;64(12):1666–1693. Zangwill A. Modern electrodynamics. Cambridge, UK: Cambrigde University Press; 2013. Navarro MS, Rozzi TE, Lo YT. Propagation in a rectangular waveguide periodically loaded with resonant irises. IEEE Transactions On Microvawe Theory And Techniques. 1980 August;28(8):857–865. Munk BA. Frequency Selective Surfaces: Theory and Design. New York, NY: John Wiley & Sons, Inc.; 2000. Golub GH, van Loan CF. Matrix Computations. Baltimore, MD: Johns Hopkins University Press; 1996. Bau III D, Trefethen LN. Numerical linear algebra. Philadelphia, PA: Soci. Indus. Ap. Math.; 1997. Stein EM, Weiss G. Introduction to Fourier Analysis on Euclidean Spaces. Princeton, NJ: Princeton University Press; 1971. Coxeter HSM. Introduction to Geometry. 2nd ed. New York, NY: John Wiley & Sons; 1989.

[44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57]

[58] [59] [60] [61] [62]

Chapter 12

Wave propagation in dispersive media

The subject of this chapter is a detailed analysis of the wave propagation in linear penetrable media, that is, materials whose constitutive parameters depend only on the properties of the underlying medium but not on the applied electromagnetic field. Thus non-linear phenomena such as the Kerr effect [1, Section V.4] or magnetic hysteresis in ferromagnetic materials [2] are excluded from the following discussion. After introducing the constitutive relationship in the frequency domain (Section 12.1) we go on to show that the principle of causality poses limits on the form of the constitutive parameters when considered as complex functions of the complex angular frequency ω (Section 12.2). Next, we develop models of the constitutive parameters on the basis of a classical description of the atomic or molecular structure of penetrable media and conductors (Section 12.3). Finally, we examine how the frequency dependence of the constitutive parameters affects the propagation of waves in a medium (Section 12.4), and we show that similar effects exist also in waveguides (Section 12.5) even when the underlying medium is free-space, whose constitutive parameters are constant.

12.1 Constitutive relations in frequency and time domain We have seen that in many a situation it is convenient to rephrase the Maxwell equations in the frequency domain by either making the hypothesis of time-harmonic dependence of fields and sources (Section 1.5) or by introducing spectral entities by means of a temporal Fourier transformation (e.g., Section 7.5). There are no substantial differences between the two approaches except for, perhaps, the added difficulty of computing the inverse Fourier transformation of the solution found, should this last step be really necessary. The analysis in the frequency domain comes in handy also to establish the linear constitutive relations we outlined in Section 1.6. Linearity of the equations is an essential requisite, or else we may neither apply the principle of superposition (Section 6.1) nor exchange the order of the operators as we did in, e.g., (1.95) and (7.309). The simplest and, admittedly, most trivial constitutive relationships are those relevant for vacuum or free space, and they are given by (1.112) and (1.113) for a general time-dependence. In accordance with either (1.88) or the following pair of forward and inverse temporal Fourier transformations [3, Section 4.8], [4] E(r; ω) :=

+∞ dt e− j ωt E(r, t) −∞

1 E(r, t) := 2π

+∞ dω ej ωt E(r, ω) −∞

(12.1)

(12.2)

824

Advanced Theoretical and Numerical Electromagnetics

the constitutive relations in the frequency domain read D(r; ω) = ε0 E(r; ω),

r ∈ R3

(12.3)

B(r; ω) = μ0 H(r; ω),

r∈R

(12.4)

3

because the permittivity and the permeability of vacuum are fundamental constants in a given set of physical units. Whether the field is time-harmonic or has a general time-dependence we would like to extend (12.3) and (12.4) in order to describe the electromagnetic response of a material medium. Obviously, the constitutive parameters should carry information about the interaction between an external field and the atoms or molecules at a microscopic level. Therefore, we expect that the permittivity and the permeability may depend on the frequency of the driving field and also change from point to point within the finite region V occupied by the medium. Furthermore, if we allow for the possibility of different directions between entities of intensity (primary) and entities of quantity (secondary) we are led to speculate the following equations [5, Chapter 16]   ˜ ω) · E(r; ω), r∈V (12.5) D(r; ω) = ε0 I + χ˜ e (r; ω) · E(r; ω) = ε(r;   ˜ ω) · H(r; ω), B(r; ω) = μ0 I + χ˜ m (r; ω) · H(r; ω) = μ(r; r∈V (12.6) ˜ ω) · E(r; ω), Jc (r; ω) = σ(r;

r∈V

(12.7)

where we have added the diacritical symbol ˜ to indicate dyadic susceptibility, permittivity, permeability, and conductivity in the frequency domain. Expressions (12.5) and (12.6) are supported by the approximate relations (3.262) and (5.192). In fact, it is expected that permittivity, permeability and conductivity reduce to their static or stationary counterparts in the limit as ω → 0. By contrast, we say that a material is dispersive or that it exhibits dispersion if the constitutive parameters thereof are a function of the angular frequency ω [6, Section 18.2]. Dispersion occurs in all material media to some degree [7]. At this stage (12.5)-(12.7) are no more than a working assumption, though we shall find explicit expressions for isotropic media further on in Section 12.3. For the sake of consistency with the constitutive relationships for free space, we assign the frequency-domain parameters of a material medium the usual physical dimensions, viz.,       ˜ ω) = H/m, ˜ ω) = 1/(Ωm) ˜ ω) = F/m, ε(r; μ(r; σ(r; (12.8) whereas we note that, so long as we assume time-harmonic fields, the physical dimensions of electric and magnetic entities do not change in light of (1.88) and the like. Conversely, if we define the entities in the frequency domain through (12.1) and analogous formulas for the other entities and the conduction current, we must tweak the relevant units as follows [E(r; ω)] = Vs/m,

[D(r; ω)] = Cs/m2 ,

[H(r; ω)] = As/m,

[B(r; ω)] = Wb s/m = Ts

[Jc (r; ω)] = As/m2 2

(12.9) (12.10)

in that the differential of time dt carries physical dimension as well! What do the constitutive relations look like in the time domain? To answer this question we examine the Ohm law (12.7) for the sake simplicity. If the driving field is time-harmonic, we multiply through exp(j ωt) and take the real part, viz.,   ˜ ω) · E(r; ω)ej ωt Jc (r, t) = Re{Jc (r; ω)ej ωt } = Re σ(r;     ˜ ω) · E(r; ω) cos(ωt) − Im σ(r; ˜ ω) · E(r; ω) sin(ωt), = Re σ(r; r∈V (12.11)

Wave propagation in dispersive media

825

˜ ω) is available. More although not much more can be said unless an explicit expression of σ(r; generally, if we interpret (12.7) as a relation between the temporal Fourier transforms of Jc (r, t) and E(r, t) we have 1 Jc (r, t) := 2π

+∞ +∞ j ωt ˜ dω e σ(r; ω) · E(r; ω) = dt σ(r, t − t ) · E(r, t ), −∞

r∈V

(12.12)

−∞

by virtue of the convolution theorem [3, Section 4.8] applied backwards to the terms σ ˜ xx E x , σ˜ xy Ey ˜ and so forth. The symbol σ indicates the inverse Fourier transform of σ, and, in effect, implicit ˜ ω) as a function of ω has the properties expected of a Fourier in (12.7) is the assumption that σ(r; transform. Comparing (12.12) with (1.128) we notice that as a result of dispersion the conduction current does not mirror the variations of the electric field instantaneously. Rather, (12.12) says that the conduction current at time t depends — through the dyadic field σ(r, t −t ) — on the electric field at earlier times t  t. All in all, the medium behaves as if it had a memory of sorts [8, Section 8.1]. Do we really need to carry out the integration with respect to t all the way up to +∞? In fact, this procedure would amount to admitting that the conduction current were influenced, quite strangely, also by the values taken on by E(r, t ) at later times t > t, i.e., by the future driving electric field! However, this bizarre occurrence is not physically plausible inasmuch as it violates the postulate of causality. Accordingly, a physical phenomenon (the effect) may not take place prior to the event (the cause) which determines the former. It follows that if our surmised linear constitutive relationship (12.7) must be physically sound, then the dyadic kernel σ(r, t − t) must obey σ(r, t − t ) = 0,

t > t

(12.13)

a constraint referred to as a causality requirement [9, Section 4.3], [10, Chapter 1], [11, Section 6.1], [6, Section 18.7]. Consequently, we may stop the integration at t = t and write instead t Jc (r, t) =

+∞ dt σ(r, t − t ) · E(r, t ) = dτ σ(r, τ) · E(r, t − τ), 



−∞



r∈V

(12.14)

0

having performed the change of dummy variable t = t − τ. The expression in the rightmost-hand side better highlights the dependence of the current (the effect) at t on the values of the electric field at past values of time t − τ. Similar conclusions hold for the other constitutive relationships, and we may write (Volterra, 1912) [8, Section 8.1], [11, Chapter 6], [12, Chapter 51], [9, Section 4.2], [13, 14] +∞ D(r, t) = ε0 E(r, t) + dτ ε0 χe (r, τ) · E(r, t − τ), r∈V (12.15) 0

+∞ B(r, t) = μ0 H(r, t) + dτ μ0 χm (r, τ) · H(r, t − τ),

r∈V

(12.16)

0

where the dyadic fields χe (r, τ) and χm (r, τ) — which are the inverse Fourier transforms of χ˜ e (r; ω) and χ˜ m (r; ω) — vanish for τ < 0 so as to meet the requirement imposed by causality. In view of the role they play, the kernels of the integral relationships (12.14)-(12.16) are also called memory functions [8, Section 8.1]. This distinction is quite helpful to remember that the physical dimensions of the memory functions, viz.,       χe (r, t) = 1/s, χm (r, t) = 1/s, σ(r, t) = 1/(Ωm s) (12.17)

826

Advanced Theoretical and Numerical Electromagnetics

are different than those of the frequency-domain counterparts. If the dyadics χ˜ e (r; ω) and χ˜ m (r; ω) are to represent Fourier transforms, then their nine compo˜ ω) and μ(r; ˜ ω) defined through nents decay at least as fast as 1/ω for ω → ±∞. As a result, ε(r; (12.5) and (12.6) approach the permittivity and permeability of free space in the limit as ω → ±∞. ˜ ω) are not the Fourier transforms of ordinary dyadic fields. On the ˜ ω) and μ(r; This means that ε(r; other hand, if we extend the constitutive relationships (12.15) and (12.16) into the realm of distributions (Appendix C) and recall that the Dirac δ(•) is the neutral or unitary element of the convolution product we have +∞   D(r, t) = dτ ε0 δ(τ)I + χe (r, τ) · E(r, t − τ) 0

+∞ = dτ ε(r, τ) · E(r, t − τ) = ε(r, t) ∗· E(r, t),

r∈V

(12.18)

r∈V

(12.19)

0 +∞

  dτ μ0 δ(τ)I + χm (r, τ) · H(r, t − τ)

B(r, t) = 0

+∞ = dτ μ(r, τ) · H(r, t − τ) = μ(r, t) ∗· H(r, t), 0

˜ ω) and where the memory ‘functions’ ε(r, τ) and μ(r, τ) are the inverse Fourier transforms of ε(r; ˜μ(r; ω) in the sense of distributions, and the star-dot symbol ‘ ∗· ’ indicates three-dimensional convolution and dot product between the operands. We observe that ε(r, τ) and μ(r, τ) have physical dimensions     ε(r, t) = F/(m s), μ(r, t) = H/(m s) (12.20) and obey the causality requirement as well, in that the delta distribution is null for τ  0 (Appendix C). Although the constitutive parameters in (12.5)-(12.7) are complex functions of the angular frequency ω, the nine components of χe (r, t) and so forth ought to be real functions of t in order for the relations (12.14)-(12.16) to make sense in the time domain. Assuming an isotropic homogeneous dielectric with susceptibility χ˜ e (ω) for the sake of illustration, we may determine the conditions for χe (t) to be real. We start with the definition of inverse Fourier transformation and separate out real and imaginary parts 1 χe (t) := 2π

+∞ +∞   1 j ωt dω e χ˜ e (ω) = dω ej ωt Re{χ˜ e (ω)} + j Im{χ˜ e (ω)} 2π

−∞

−∞

+∞   1 = dω Re{χ˜ e (ω)} cos(ωt) − Im{χ˜ e (ω)} sin(ωt) 2π −∞

j + 2π

+∞   dω Im{χ˜ e (ω)} cos(ωt) + Re{χ˜ e (ω)} sin(ωt) −∞

(12.21)

Wave propagation in dispersive media

827

and we need the last contribution (the imaginary part) to disappear. Evidently, the latter occurs if, on the whole, the relevant integrand is an odd function of ω because the integration is carried out over an interval which is symmetric around ω = 0. This observation in turn implies Re{χ˜ e (−ω)} = Re{χ˜ e (ω)},

Im{χ˜ e (−ω)} = −Im{χ˜ e (ω)},

ω∈R

(12.22)

that is, the real (resp. imaginary) part of χ˜ e must be an even (resp. odd) function of the angular frequency. Correspondingly, the memory function χe (t) follows from 1 χe (t) = π

+∞   dω Re{χ˜ e (ω)} cos(ωt) − Im{χ˜ e (ω)} sin(ωt)

(12.23)

0

which means, in essence, that just the values of χ˜ e (ω) for ω > 0 are sufficient to determine χe (t). More generally, even-odd symmetry conditions such as (12.22) hold true for the nine components of the dyadic fields involved in (12.5)-(12.7). To shed light on the behavior of χ˜ e (ω) for large real values of the angular frequency we consider the asymptotic expansion +∞ dt e− j ωt χe (t) χ˜ e (ω) = 0

=



 1 1 dχe 1 d2 χe χe (0+ ) − 2 + 3 + O , jω ω dt t=0+ j ω dt2 t=0+ ω4

ω → +∞

(12.24)

which is obtained by repeatedly integrating by parts and noticing that χe (t) and its derivatives must vanish at infinity for the improper integrals to exist in the first place. It is evident that the real part of the expansion contains even powers of 1/ω only, whereas the imaginary part involves odd powers of 1/ω only — which is in agreement with our previous finding (12.22). The susceptibility and its derivatives are all evaluated at t = 0+ . Besides, since causality dictates that χe (t) = 0 for times t  0− we are led to assume χe (0+ ) = 0, too. Indeed, a memory function discontinuous for t = 0 would imply that the dielectric medium responded instantaneously to the incoming electromagnetic wave, and this occurrence is nonphysical [7, 15]. Therefore, we conclude



 1 1 Re{χ˜ e (ω)} = O 2 , Im{χ˜ e (ω)} = O 3 , ω → +∞ (12.25) ω ω that is, the susceptibility decays even faster than 1/ω as initially surmised. This conclusion holds in general for the nine components of χ˜ e (r; ω) and χ˜ m (r; ω). Before we delve into the analysis of the most important property of the constitutive parameters (Section 12.2) we anticipate that the imaginary parts are associated with either dissipation and losses or gain [15]. In particular, we distinguish (a)

passive media if Im{ε(ω)} ˜ < 0,

(b)

Im{μ(ω)} ˜ < 0,

ω>0

(12.26)

Im{μ(ω)} ˜ > 0,

ω > 0.

(12.27)

active media if Im{ε(ω)} ˜ > 0,

828

Advanced Theoretical and Numerical Electromagnetics

Sometimes, to emphasize that the lossy character of a material is predominantly due to free charges (e.g., electrons in a conductor) it is convenient to introduce the notion of complex permittivity ε˜ c (r; ω). To elucidate we write the local form of the Ampère-Maxwell law (1.98) by including the conduction current Jc (r; ω) explicitly [also see (1.119)], viz., ˜ ω) · E(r; ω) + σ(r; ˜ ω) · E(r; ω) + J(r; ω) ∇ × H(r; ω) = j ωε(r;       displacement

conduction

impressed

  1 ˜ ˜ = j ω ε(r; ω) + σ(r; ω) · E(r; ω) + J(r; ω), jω

r∈V

(12.28)

where we have grouped together displacement and conduction currents. The dyadic field ˜ ω), ˜ ω) − j σ(r; ε˜ c (r; ω) := ε(r; ω

r∈V

(12.29)

˜ ω) may be complex per se is called the complex permittivity of the medium [14]. Notice that ε(r; owing to loss mechanisms other than the drift of free charges.

12.2 The Kramers-Krönig relations In the previous section we have pointed out that the memory functions (or constitutive parameters in the time domain) must vanish prior to a given time — which we choose to be t = 0 as a matter of convenience — in order to ensure that the postulate of causality is not violated. Since we began our discussion by noticing that it is more natural to obtain the constitutive parameters in the frequency domain, we may wonder what consequences the causality condition bears for the dyadic ˜ ω). fields ε˜ c (r; ω) and μ(r; As it turns out, in addition to the even-odd properties such as (12.22) and the asymptotic behavior (12.25), the real and the imaginary parts of the constitutive parameters are not independent of each other but, on the contrary, are intimately linked by a pair of integral relations which we wish to derive hereinbelow [9, 14–19], [20, Section 8.1], [6, Section 18.7]. From a mathematical viewpoint, this remarkable connection stems, in essence, from the Cauchy integral formula (B.48), which may ˜ ω) are holomorphic be invoked because the nine components of the dyadic fields ε˜ c (r; ω) and μ(r; := functions (Appendix B) in the open region U {ω ∈ C : Im{ω} < 0}, i.e., the lower complex half-plane excluding the real axis. To keep the exposition lucid we deal with an isotropic lossy dielectric medium endowed with scalar memory functions χe (r, t) and σ(r, t), r ∈ V. This is not a limitation in that for general anisotropic, possibly non-reciprocal media the following discussion holds separately for each of the ˜ ω). On account of the causality requirements nine components χ˜ e (r; ω) and σ(r; χe (r, t) = 0,

t  0−

σ(r, t) = 0,

(12.30)

the susceptibility and conductivity in the frequency domain may be written formally as +∞ dt e− j ωt χe (r, t), χ˜ e (r; ω) :=

r∈V

(12.31)

r∈V

(12.32)

0

+∞ σ(r; ˜ ω) := dt e− j ωt σ(r, t), 0

Wave propagation in dispersive media

829

Figure 12.1 Analytic properties of the function F(ω) ∈ {χ˜ e (r; ω), σ(r; ˜ ω), ε˜ c (r; ω) − ε0 } in the complex plane ω. and, finally, the complex permittivity reads  σ(r;  ˜ ω) ε˜ c (r; ω) := ε0 1 + χ˜ e (r; ω) + jω +∞ +∞ 1 dt e− j ωt σ(r, t), = ε0 + ε0 dt e− j ωt χe (r, t) + jω 0

r∈V

(12.33)

0

in accordance with (12.5) and definition (12.29). Irrespective of the form of χe (r, t) and σ(r, t), the Fourier integrals in (12.31) and (12.32) exist for complex values of the angular frequency only if Im{ω}  0. Indeed, since exp(− j ωt) = exp(− j Re{ω}t) exp(Im{ω}t)

(12.34)

we see that only when Im{ω}  0 is the real exponential bounded for t → +∞. It is worth remarking, though, that the complex functions χ˜ e (r; ω) and σ(r; ˜ ω) may exist for values ω ∈ C but only for Im{ω}  0 can they be represented by means of Fourier integrals as in (12.31) and (12.32). As a consequence, χ˜ e (r; ω) and σ(r; ˜ ω) are certainly analytic in the lower complex half-plane U. To support this claim we consider a contour γ ⊂ U (Figure 12.1) and examine the integral of χ˜ e (r; ω) along γ, namely, 

 dω χ˜ e (r; ω) = γ

γ

 +∞ +∞ − j ωt dω dt e χe (r, t) = dt χe (r, t) dω e− j ωt = 0 0

0

(12.35)

γ

where, in accordance with the Fubini theorem for iterated integrals, we have interchanged the order of integration because the integrand is continuous for (t, ω) ∈ R+ × C. The contour integral of exp(j ωt) vanishes inasmuch as the complex exponential is analytic in particular for ω ∈ U (see Appendix B.1). Since γ is absolutely arbitrary, we conclude that any contour integral of χ˜ e (r; ω) vanishes for ω ∈ U, and hence by virtue of Morera’s theorem [21] χ˜ e (r; ω) is analytic for ω ∈ U. By contrast, we may not repeat the same reasoning for values of ω in the upper complex halfplane, because the integral representations (12.31) and (12.32) do not apply. As a matter of fact, we ˜ ω) have singularities for shall see in Section 12.3 that the explicit expressions of χ˜ e (r; ω) and σ(r;

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Advanced Theoretical and Numerical Electromagnetics

Figure 12.2 Poles (×) and contour Γ (−−) in the complex plane Ω for the derivation of the Kramers-Krönig relations. Im{ω} > 0. Furthermore, χ˜ e (r; ω) and σ(r; ˜ ω) vanish for ω → ∞ with Im{ω}  0. This property was proved in (12.24) and (12.25) for the susceptibility. We continue by observing that ε˜ c (r; ω) − ε0 , regarded as a function of the complex variable ω, satisfies the following two properties: (1)

(2)

ε˜ c (r; ω) − ε0 is holomorphic in the lower complex half-plane U except in ω = 0 where it exhibits a first-order pole [see (B.34)]; this is a consequence of the analyticity of χ˜ e (r; ω) and σ(r; ˜ ω) and the very definition of complex permittivity (12.33). ε˜ c (r; ω) − ε0 behaves asymptotically at least as

 1 ε˜ c (r; ω) − ε0 = O , |ω| → +∞, Im{ω}  0 (12.36) |ω| once again in light of definition (12.33).

Armed with these findings we start the derivation proper by considering the contour Γ in the complex plane Ω (Figure 12.2) and defined as Γ := γab ∪ Ca ∪ C0 ∪ Cω with γab Ca C0 Cω

:= := := :=

{Ω ∈ C : Re{Ω} ∈ [−a, −b] ∪ [b, ω − b] ∪ [ω + b, a], Im{Ω} = 0} {Ω ∈ C : Ω = ae− j α , α ∈ [0, π]} {Ω ∈ C : Ω = bej α , α ∈ [−π, 0]} {Ω ∈ C : Ω = ω + bej α , α ∈ [−π, 0]}

(12.37)

(12.38)

where a > ω + b > 0, and ω ∈ R \ {0} is the angular frequency. The complex function [ε˜ c (r; Ω) − ε0 ]/(Ω − ω) exhibits a first-order pole in Ω = ω and possibly another one in Ω = 0 on account of (12.33), but it surely is analytic in the region U and in particular for values of Ω in the inside of the graph of Γ. Therefore, we conclude  ε˜ c (r; Ω) − ε0 = 0, r∈V (12.39) dΩ Ω−ω Γ

Wave propagation in dispersive media

831

which can also be interpreted as an application of the Cauchy integral formula (B.48) with ω outside the contour Γ. We are interested in evaluating the contour integral formally though in the limit as a → +∞ and b → 0+ , under which circumstances the half circle Ca recedes to infinity and the half circles C0 and Cω reduce to the points Ω = 0 and Ω = ω. We split the integration into four line integrals along the curves that comprise Γ, viz.,  0=

dΩ C0

ε˜ c (r; Ω) − ε0 + Ω−ω

 dΩ Cω

ε˜ c (r; Ω) − ε0 Ω−ω

 +

dΩ Ca

ε˜ c (r; Ω) − ε0 + Ω−ω

 dΩ γab

ε˜ c (r; Ω) − ε0 , Ω−ω

r∈V

(12.40)

and observe that for a → +∞ and b → 0+ the last contribution reduces to the Cauchy principal value of an integral along the whole real axis Im{Ω} = 0. For the integral along Ca we let Ω = a exp(− j α) and, for a > ω > 0, estimate  π π a|ε˜ c (r; Ω) − ε0 | dΩ ε˜ c (r; Ω) − ε0 = dα (− j ae− j α ) ε˜ c (r; Ω) − ε0  dα − j α − ω Ω − ω ae |ae− j α − ω| Ca

0

0

π 

dα 0

πCε a Cε = −−−−−→ 0 a−ω a a − ω a→+∞

(12.41)

where Cε > 0 is a suitable constant related to the asymptotic behavior (12.36). Thus, this integral tends to zero as it is dominated by a constant which vanishes for a → +∞. As for the integral around the pole in Ω = ω we choose the parameterization Ω = ω + b exp(j α) and write  Cω

ε˜ c (r; Ω) − ε0 = dΩ Ω−ω

0

  dα j ε˜ c (r; ω + bej α ) − ε0

−π

  = j π ε˜ c (r; ω + bej α ) − ε0 −−−−→ j π [˜εc (r; ω) − ε0 ] + b→0

(12.42)

having invoked the mean value theorem [22] with α ∈ [−π, 0] a suitable angle. The result follows because ε˜ c (r; Ω) is analytic for Ω ∈ U \ {0}, and thus also a continuous function of b. Finally, for the integral along C0 we need to exhibit the pole Ω = 0 before we let Ω = b exp(j α), viz.,     1 ε˜ c (r; Ω) − ε0 σ(r; ˜ Ω) = dΩ dΩ ε0 χ˜ e (r; Ω) + Ω−ω jΩ Ω−ω C0

C0

0 =

dα −π

=π

˜ bej α ) j ε0 bej α χ˜ e (r; bej α ) + σ(r; bej α − ω

j ε0 bej α χ˜ e (r; bej α ) + σ(r; ˜ bej α ) σ(r; ˜ 0) −π −−−−→ j α b→0+ ω be − ω

(12.43)

832

Advanced Theoretical and Numerical Electromagnetics

again by virtue of the mean value theorem and the continuity of χ˜ e (r; Ω) and σ ˜ e (r; Ω). In the end, this means that the integral along C0 contributes the term π σ(r; ˜ 0) =− −π ω ω

+∞ σ0 (r) , dt σ(r, t) = −π ω

r∈V

(12.44)

0

where σ0 (r) is the conductivity in stationary or steady-state conditions. By making use of (12.41)-(12.43) in (12.40) we arrive at the identity +∞ ε˜ c (r; Ω) − ε0 σ0 (r) + j π[ε˜ c (r; ω) − ε0 ] + 0 + PV dΩ , 0 = −π ω Ω−ω

r∈V

(12.45)

−∞

which we write as 1 σ0 (r) ε˜ c (r; ω) − ε0 − = − PV jω jπ

+∞ ε˜ c (r; Ω) − ε0 , dΩ Ω−ω

r∈V

(12.46)

−∞

by solving formally with respect to the complex permittivity. We may construe (12.46) as an instance of the Cauchy integral formula (B.55) though modified due to the presence of the extra pole Ω = 0 on the boundary of the region U. Moreover, since (12.46) is a relation between complex functions of ω, it amounts to two separate equations for real and imaginary parts of both sides, namely, 1 Re{ε˜ c (r; ω)} − ε0 = − PV π Im{ε˜ c (r; ω)} +

σ0 (r) 1 = PV ω π

+∞ Im{ε˜ c (r; Ω)} , dΩ Ω−ω

−∞ +∞

dΩ

−∞

Re{ε˜ c (r; Ω)} − ε0 , Ω−ω

r∈V

(12.47)

r∈V

(12.48)

which constitute a pair of Hilbert transformations [21] and are also known as Plemelj formulas. As anticipated, real and imaginary part of the complex permittivity are not independent if the corresponding memory function in time domain must obey the postulate of causality. More importantly, (12.47) and (12.48) suggest that if either the real part or the imaginary part of ε˜ c (r; Ω) − ε0 is known by means of a suitable model or through experimental data (e.g., [23]), then the other part may be determined consistently by carrying out an integral along the real axis in the complex plane Ω. ˜ ω) and By recalling the symmetry properties (12.22) for χ˜ e (r; ω), the analogous ones for σ(r; the defining equation (12.33) we see that the complex permittivity satisfies the symmetry properties Re{ε˜ c (r; −Ω)} = Re{ε˜ c (r; Ω)},

Im{ε˜ c (r; −Ω)} = −Im{ε˜ c (r; Ω)},

Ω∈R

(12.49)

which we may use to cast (12.47) and (12.48) into yet another form. It is straightforward to show that (12.47) and (12.48) are equivalent to 2 Re{ε˜ c (r; ω)} − ε0 = − PV π

+∞ Ω Im{ε˜ c (r; Ω)} dΩ , Ω2 − ω2

r∈V

(12.50)

r∈V

(12.51)

0

+∞ σ0 (r) 2ω Re{ε˜ c (r; Ω)} − ε0 Im{ε˜ c (r; ω)} + = PV dΩ , ω π Ω2 − ω2 0

Wave propagation in dispersive media

833

which are the classical Kramers-Krönig relations [8, 14–18] named after Hendrik A. Kramers [24] and Ralph de L. Krönig [25] who derived them independently in 1927 and 1926. Since (12.50) requires knowing the imaginary part of the complex permittivity only for positive values of the real angular frequency Ω, it is even simpler to derive the corresponding real part. Perfectly analogous steps may be taken to arrive at the Kramers-Krönig relations for the permeability of an isotropic magnetic medium. We quote the relevant result 2 Re{μ(r; ˜ ω)} − μ0 = − PV π

+∞ Ω Im{μ(r; ˜ Ω)} dΩ , Ω2 − ω2

r∈V

(12.52)

r∈V

(12.53)

0

+∞ 2ω Re{μ(r; ˜ Ω)} − μ0 Im{μ(r; ˜ ω)} = PV dΩ , π Ω2 − ω2 0

which also follow directly from (12.50) and (12.51) by invoking the duality transformations of Table 6.1 and setting to zero the magnetic conductivity so long as we do not account for the occurrence of free magnetic charges.

12.3 Simple models of dispersive media We are now ready to develop explicit expressions for the constitutive parameters of three common types of materials, namely, conducting media, dielectrics and polar substances [18, Chapter 19], [6, Section 18.5]. Arguably, the resulting formulas must meet all the requirements discussed in Sections 12.1 and 12.2 and in particular must obey the Kramers-Krönig relations (12.50) and (12.51).

12.3.1 Conducting medium A rigorous microscopic analysis of a conductor calls for a treatment in the framework of quantum mechanics in order to describe the interactions (scattering) between the free electrons and the positive ions. In what follows we content ourselves with the classical picture of electrons and ions as charged particles and derive the scalar conductivity σ(ω) ˜ of an isotropic homogeneous conducting medium so as to state the Ampère-Maxwell law in the form (12.28). The result — also known as the Drude model — is well-suited to characterize metals as well as gasses of charged particles such as nonmagnetized, cold and weakly ionized plasma [14,15,26,27]. The mathematics involved is admittedly simple, but the model requires several subtle assumptions which are far from trivial. We envision the conducting medium as a collection of free electrons moving around in a background lattice of positive ions in an otherwise empty region of space V [28, Chapter 1]. We make the reasonable assumption of neutrality, namely, that electrons and ions are present in equal numbers and, more importantly, arrange themselves within V in such a way that the total field due to each and every charged particle remains null on average at any given moment both in V and outside V. This hypothesis is tantamount to saying that the electrons and the ions in the conductor produce no field at all, as if they were test charges (see Section 1.1). Ordinarily, the free electrons in the conductor do not move along any special path and their motion is chaotic. However, when the body is immersed in an electromagnetic field or wave E(r, t), B(r, t) generated by impressed sources external to V, the electrons experience the Lorentz force (1.4) which imparts them a preferred direction of motion aligned with E(r, t) and perpendicular to B(r, t). Nonetheless, while driven by the electromagnetic field E(r, t), B(r, t), the electrons do not proceed unimpeded but rather happen to ‘collide’ with the ions. We indicate with νe the effective

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Advanced Theoretical and Numerical Electromagnetics

constant collision frequency or rate (physical dimension: 1/s), that is, the number of collisions per unit of time that occur between electrons and ions, and hence 1/νe is the mean time between any two such collisions. As the latter tend to slow down an electron, we model this phenomenon by means of a damping force proportional to the velocity v(r, t) and the collision frequency. Furthermore, since the electron mass me = 9.11 · 10−31 kg is arguably far smaller than the mass mi of the relevant ions and the latter in turn are tightly bound together by intense local electric forces, we may stipulate that the ions are essentially fixed, i.e., 1) we overlook the transfer of kinetic energy from electrons to ions due to collisions and 2) we neglect the direct interaction of the electromagnetic wave with the ions. These assumptions may not be entirely true for a gas of charged particles wherein electrons and ions are equally free to move, and hence one needs to consider the velocity of each type of particles. Finally, since the velocity v(r, t) that an electron can attain in the medium is far smaller than the speed of light c0 , we may resort to the Newton second law of dynamics [28, Chapter 9] to describe the motion of an electron in V. In symbols, we have me

dv = −qe [E(r, t) + v(r, t) × B(r, t)] −νe me v(r, t),   dt

r∈V

(12.54)

damping force

where −qe (qe = 1.6021·10−19 C) is the electron charge.1 The presence of the ion lattice is accounted for by the damping force which opposes the electron motion. Notice that, strictly speaking, the electromagnetic entities E(r, t) and B(r, t) in (12.54) are unknown at this stage, as they must be determined through the solution of the Maxwell equations with the conduction current Jc (r, t) included. Even though we have carefully avoided considering the intricate local fields due to the charged particles in V (since neutrality holds) still the electromagnetic field in V and outside the conductor is altered by the very presence thereof. After all, this is precisely the purpose of the conductivity we are seeking, namely, to provide an equivalent macroscopic description of the medium while ‘sweeping electrons and ions under the carpet’, as it were. As magnetism is a relativistic effect (cf. Sections 1.3 and 9.2) the magnetic induction B(r, t) is generally far smaller than the associated electric field E(r, t). In order to justify this statement, suppose that E(r, t) and B(r, t) are the field entities of a plane wave which propagates in free space along the direction specified by the constant unit vector sˆ (Section 7.1), then 1 μ0 sˆ × E(r, t) = sˆ × E(r, t) Z0 c0 |v(r, t)| |v(r, t) × B(r, t)|  |v(r, t)||B(r, t)| = |E(r, t)| c0 B(r, t) = μ0 H(r, t) =

(12.55) (12.56)

by virtue of definition (1.358) stated in free space. If we identify v(r, t) with the electron velocity of concern, we see that the magnetic part of the Lorentz force in (12.54) is substantially smaller than the contribution of the electric field, since |v(r, t)| c0 by hypothesis. Consequently, in our analysis we may safely forgo the term v(r, t) × B(r, t) in (12.54) [29, Section 3.1.E]. Our argument, albeit plausible, is only approximate because, as already mentioned, the field E(r, t), B(r, t) is not known a priori within or outside V and it may not be a plane wave at all. More importantly, this approximation is not permitted if E(r, t) = Ei (r, t) is the electric field of the impinging wave but B(r, t) = Bi (r, t) + B0 (r), where B0 (r) represents the magnetic induction due to an external permanent magnet or a solenoid (Example 4.2) placed around the conducting medium. In this case Ei (r, t) and B0 (r) are generated by two different sources, so |Bi (r, t)| may be neglected but |B0 (r)| may be quite intense 1

In plasma physics (12.54) is called the (linearized) Langevin equation for electrons.

Wave propagation in dispersive media

835

as is required in specific applications such as nuclear magnetic-resonance imaging (MRI) diagnostic devices [30] and nuclear fusion reactors [31]. We proceed by turning (12.54) into a relation in the frequency domain by representing the electric field as in (12.2) and the velocity as the temporal inverse Fourier transform 1 v(r, t) := 2π

+∞ dω ej ωt V(r; ω)

(12.57)

−∞

which when substituted into (12.54) yield 1 2π

+∞   dω ej ωt j ωme V(r; ω) + qe E(r; ω) + νe me V(r; ω) = 0,

r∈V

(12.58)

−∞

having swapped the order of integration and temporal derivative on account of the continuity of V(r; ω) exp(j ωt) and its derivative for (t, ω) ∈ R+ × R. Since the exponential function never vanishes, (12.58) is satisfied for any value of t only if j ωme V(r; ω) = −qe E(r; ω) − νe me V(r; ω),

r∈V

(12.59)

whence we find −qe E(r; ω), me (νe + j ω)

V(r; ω) =

r∈V

(12.60)

for the Fourier transform of the velocity of a single electron. We suppose now that within V there exist Ne electrons per unit of volume, where Ne is the number density (physical dimension: 1/m3 ). Notice that in view of the neutrality of the medium Ne must needs equal the number density of ions in the lattice. The hypothesis of homogeneous conductor translates into the requirement that the number density be uniform, i.e., independent of position. Moreover, we do not consider any possible variation of Ne with ω caused by the very presence of the electric field. With the aid of Ne we may write the charge density as −qe Ne and define the conduction current in the frequency domain as [see discussion on page 2 and (1.3)] Jc (r; ω) := ρe V(r; ω) = −qe Ne V(r; ω),

r∈V

(12.61)

where V(r; ω) is the velocity of an electron as given by (12.60). Actually, the notion of charge density makes sense so long as the wavelength λ0 = c0 / f = 2πc0 /ω is far larger than the typical ‘dimension’ of electrons and ions in the medium. Then, by substituting for V(r; ω) and rearranging we have Jc (r; ω) =

ω2pe ε0 Ne q2e ε0 E(r; ω) = E(r; ω), ε0 me νe + j ω νe + j ω

where the quantity  Ne q2e ω pe := ε0 me

r∈V

(12.62)

(12.63)

is called the plasma frequency of the electrons [26,27]. By comparing (12.62) with (12.7) and (1.120) we are prompted to identify the factor := σ(ω) ˜

ω2pe ε0 νe + j ω

=

ω2pe ε0 ω2 + ν2e

(νe − j ω)

(12.64)

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Advanced Theoretical and Numerical Electromagnetics

as the conductivity of the medium. The collective effect of the lattice of positive ions is captured by the collision frequency νe , and this results in the appearance of a pole for ω = j νe on the positive imaginary axis in the complex plane ω ∈ C. It is interesting to notice that σ(ω) ˜ is analytic in the lower complex half-plane as is demanded by the postulate of causality (Section 12.2 and Figure 12.1). It takes one more step to arrive at the complex permittivity of the material. In (12.33) we set χ˜ e (ω) = 0 because the medium is modelled by just a conductivity, and we find ⎛ ⎞ ⎜⎜⎜ ω2pe ε0 ω2pe ⎟⎟⎟ j ε0 νe ω2pe σ(ω) ˜ ⎜ ⎟⎟ − = ε0 − = ε0 ⎜⎝1 − 2 ε˜ c (ω) := ε0 − j (12.65) ω ω(ω − j νe ) ω + ν2e ⎠ ω ω2 + ν2e having separated real and imaginary parts [32, Section 1.4]. The latter satisfy the even-odd symmetry properties (12.49) as well as the typical asymptotic conditions (12.25). The imaginary part of the permittivity is evidently negative for ω > 0, and hence it is associated with losses due to the collisions between electrons and ions in the lattice. Indeed, if we neglect the collisions — which is possible if νe is very small — the permittivity becomes real ⎛ ⎞ ⎜⎜⎜ ω2pe ⎟⎟⎟ ⎜ ε˜ c (ω) = ε0 ⎜⎝1 − 2 ⎟⎟⎠ (12.66) ω and this formula is useful to characterize a cold collision-free non-magnetized electron plasma [26]. It also sheds some light on the physical meaning of the plasma frequency ω pe . Apparently, for |ω| < ω pe the permittivity is negative, whereby the wavenumber (1.248) becomes imaginary, viz.,  ω2pe j  2 √ ω pe − ω2 , |ω| < ω pe (12.67) k := ω ε0 μ0 1 − 2 = − c0 ω and, as a result, electromagnetic waves are damped and cannot propagate within the medium. Therefore, ω pe plays the role of cut-off frequency, similarly to the cut-off frequencies of guided modes in a classic hollow-pipe waveguide (Section 11.2) [8, 33–39], although in the latter case the basis for the effect is purely geometrical if the waveguide is empty. Figure 12.3 shows a normalized plot of the conductivity σ(ω) ˜ versus the angular frequency. As ˜ is nearly zero and can be seen, at relatively low frequencies, i.e., ω νe , the imaginary part of σ(ω) the conductivity is essentially real. This behavior is typical of most good conductors for which the approximation σ(ω) ˜ ≈

ω2pe ε0 νe

=

Ne q2e , νe me

ω νe

(12.68)

is valid for frequencies well into the microwave part of the electromagnetic spectrum (ω the order of 102 GHz) because the collision frequency νe ranges from 10 to 102 THz (also see Table 7.2). At higher frequencies the imaginary part of σ(ω) ˜ becomes dominant while the real part tends more rapidly to zero. Thereby, in the optical regime conductors behave as penetrable opaque lossy dielectric media. The same is true for electron plasma driven at frequencies in the microwave regime, because in that case ω > νe with the collision frequency being the order of 10 MHz. At very high frequencies (ω νe ) the Drude model (12.64) becomes inaccurate because other effects come into play such as the oscillations of the electrons bound to the nuclei, and a resonance model (Section 12.3.2) is better suited. When the motion of the ions is important, such as in a cold gaseous plasma, one needs to consider conduction currents of the form (12.62) for each species of ions as well. Since the various

Wave propagation in dispersive media

837

Figure 12.3 Dispersion in a conducting medium: real and imaginary parts of the normalized conductivity as a function of the normalized angular frequency. contributions combine additively, conductivity and complex permittivity are found to be [40, Chapter 6], [26, 27] := σ(ω) ˜

 ω2pα ε0 α

ε˜ c (ω) := ε0 −

να + j ω



ω2pα ε0

α

ω(ω − j να )

(12.69)

 where α signifies summation over all types of charged particles (electrons and ions alike) and να denotes the phenomenological rate of collision between the particles of species α and the particles of the remaining species. The plasma frequencies ω pα are obtained through formulas analogous to (12.63) with the appropriate charges, masses and number densities. Example 12.1 (The Kramers-Krönig relations for a conducting medium) The most important property expected of the complex permittivity provided by the Drude model (12.65) is that ε˜ c (ω) satisfies the postulate of causality. Indeed, we may ascertain with direct calculations that the expression in the rightmost-hand side of (12.65) obeys the Kramers-Krönig relations, despite the result being based on a classical picture of the atomic nature of the medium as well as a number of simplifying assumptions. According to (12.50) we should recover the real part of ε˜ c (ω) from the knowledge of the associated imaginary part. The latter follows from (12.65), and substitution into (12.50) yields +∞ +∞ 2 ε0 νe ω2pe 2 Ω 1 1 ε0 νe ω pe Re{ε˜ c (r; ω)} − ε0 = PV dΩ 2 = PV dΩ π π Ω − ω2 Ω Ω2 + ν2e (Ω2 − ω2 )(Ω2 + ν2e ) 0

(12.70)

−∞

where in the last passage we have exploited the even symmetry of the integrand to arrive at an improper integral along the entire real axis in the complex plane of the variable Ω ∈ C. In this way

Advanced Theoretical and Numerical Electromagnetics

838

Figure 12.4 Poles (×) and contour Γ (−−) in the complex plane Ω for the calculation of the integral (12.70) with the Cauchy theorem of residues. we may try and evaluate the integral with the aid of the Cauchy theorem of residues (B.57) if we choose a suitable contour. Besides, we observe that the integrand behaves asymptotically as

 ε0 νe ω2pe 1 =O , |Ω| → +∞ (12.71) (Ω2 − ω2 )(Ω2 + ν2e ) |Ω|4 and it possesses four simple poles, i.e., two in Ω = ± j νe and two on the real axis in Ω = ±ω. These latter two must be excluded through a limiting process in accordance with the definition of principal value. To proceed, in the complex plane Ω we consider the contour Γ shown in Figure 12.4 and defined as Γ := γab ∪ Ca ∪ C−ω ∪ Cω

(12.72)

with γab Ca C−ω Cω

:= := := :=

{Ω ∈ C : |Re{Ω}| ∈ [0, ω − b] ∪ [ω + b, a], Im{Ω} = 0} {Ω ∈ C : Ω = aej α , α ∈ [0, π]} {Ω ∈ C : Ω = −ω − be− j α , α ∈ [0, π]} {Ω ∈ C : Ω = ω − be− j α , α ∈ [0, π]}

(12.73)

where a > ω + b > 0. Since the contour Γ encircles only the pole in Ω = + j νe in a counterclockwise fashion, we have  ε0 νe ω2pe 1 dΩ 2 = π (Ω − ω2 )(Ω2 + ν2e ) Γ

=

1 π

 dΩ γab

ε0 νe ω2pe (Ω2 − ω2 )(Ω2 + ν2e )

+

1 π

 dΩ C−ω

ε0 νe ω2pe (Ω2 − ω2 )(Ω2 + ν2e )

Wave propagation in dispersive media 1 + π

 Cω

ε0 νe ω2pe

1 + dΩ 2 2 2 2 (Ω − ω )(Ω + νe ) π

 dΩ Ca

(Ω − j νe )ε0 νe ω2pe

= 2 j lim

Ω→+ j νe

(Ω2 − ω2 )(Ω − j νe )(Ω + j νe )

839

ε0 νe ω2pe (Ω2 − ω2 )(Ω2 + ν2e )

=−

ε0 ω2pe ω2 + ν2e

(12.74)

on account of (B.57) and the residue of the integrand in Ω = + j νe . We are interested in evaluating the contour integral in the limit as a → +∞ and b → 0+ , whereby the half circle Ca recedes to infinity, the half circles C−ω and Cω reduce to the points Ω = −ω and Ω = ω, and finally γab becomes the real axis Im{Ω} = 0. In particular, the integral along γab passes over into the integral in (12.70) that we wish to compute. For the integral along Ca we let Ω = a exp(j α) and estimate  π 2 2 ν ω ε ν ω ε 0 e 0 e 1 1 pe pe = jα dΩ dα a j e π 2 − ω2 )(Ω2 + ν2 ) 2 e2 j α − ω2 )(a2 e2 j α + ν2 ) π (Ω (a e e C 0

a

1  π

π 0

aε0 νe ω2pe

1  dα 2 2 j α 2 2 2 j α 2 |a e − ω ||a e + νe | π

π dα a

M M = 3 −−−−−→ 0 4 a a a→+∞

(12.75)

0

where M > 0 is a suitable constant involved in the asymptotic behavior (12.71). Therefore, this integral tends to zero because it is dominated by a constant which vanishes as a → +∞. As regards the contributions of the half-circles Cω and C−ω we let Ω = ω − b exp(− j α) and Ω = −ω − b exp(− j α), respectively, and carry out the integration formally with the help of the mean value theorem, viz., 1 π

 dΩ Cω

=

1 π



ε0 νe ω2pe (Ω2 − ω2 )(Ω2 + ν2e )

(2ω −

1 π

1 dΩ 2 = (Ω − ω2 )(Ω2 + ν2e ) π =

(2ω +

π dα 0

− j ε0 νe ω2pe  be− j α ) (ω − be− j α )2 ε0 νe ω2pe

C−ω

=

j ε0 νe ω2pe − − − − → −  + ν2e b→0+ 2ω(ω2 + ν2e )

π dα 0

j ε0 νe ω2pe  be− j α ) (ω + be− j α )2

− j ε0 νe ω2pe   (2ω − be− j α ) (ω − be− j α )2 + ν2e (12.76)

j ε0 νe ω2pe   (2ω + be− j α ) (ω + be− j α )2 + ν2e

j ε0 νe ω2pe − − − − →  + ν2e b→0+ 2ω(ω2 + ν2e )

(12.77)

where α ∈ [0, π] indicates a suitable angle in both cases, and the results follow because the integrands are continuous functions of Ω and hence of the radius b. Clearly, these terms cancel out, and all in all only the integral along γab survives. Thus, (12.74) yields +∞ ε0 νe ω2pe ε0 ω2pe 1 PV dΩ = − π (Ω2 − ω2 )(Ω2 + ν2e ) ω2 + ν2e

(12.78)

−∞

which by comparison with (12.70) and (12.65) proves that the Drude model does satisfy (12.50), the first of the Kramers-Krönig relations. The check of (12.51) proceeds along the same lines.

Advanced Theoretical and Numerical Electromagnetics

840

We observe that the conclusion is necessarily independent of the contour Γ which, after all, is just a mathematical expedient for the calculation of the integral. For instance, the contour obtained by reflecting Γ in Figure 12.4 around the real axis entails the same calculations listed above. Alternatively, we could also have opted for a contour which did encircle the poles on the real axis. (End of Example 12.1)

The memory function σ(t) of a conducting medium follows by taking the inverse Fourier transformation of (12.64), viz., σ(t) :=

1 2π

+∞ ε0 ω2pe dω ej ωt νe + j ω

(12.79)

−∞

where the improper integral exists finite for t  0. Indeed, integrating by parts over the finite interval [−a, a] and taking the absolute values allows obtaining the estimate +a +a  j ωt dω e 2 a 2 2  dω + + arctan (12.80) = νe + j ω |t|(ν2e + a2 )1/2 νe |t|(ν2e + ω2 ) |t|(ν2e + a2 )1/2 |t|νe −a

−a

and the quantity in the rightmost member remains finite for a → +∞. For t = 0 the integral exists in the Cauchy-principal-value sense and yields +∞ 2 ε0 ω2pe ε0 ω2pe ε0 ω2pe νe + j a ε0 ω pe 1 = lim log = log(−1) = PV dω 2π νe + j ω 2π j a→+∞ νe − j a 2π j 2

(12.81)

−∞

where we have taken the fundamental value π j of the many-valued complex logarithm [41]. For the evaluation of (12.79) when t  0 we resort to the Cauchy theorem as in Example B.2 by considering two contours in the complex plane ω ∈ C, namely, Γ+a := {ω ∈ C : ω = ξ, ξ ∈ [−a, a] ∪ ω = aej α , α ∈ [0, π]}

(12.82)

Γ−a

(12.83)

:= {ω ∈ C : ω = ξ, ξ ∈ [−a, a] ∪ ω = ae

−jα

, α ∈ [0, π]}

with a > 0. With these definitions, the orientation of Γ+a is counterclockwise and that of Γ−a is clockwise. Both Γ+a and Γ−a are comprised of a part of the real axis and a half-circle of radius a, and hence we expect to recover the improper integral along the real axis in the limit as a → +∞, provided the contour integrals along Γ+a and Γ−a converge. To begin with, we observe 1 2π 1 2π



dω ej ωt

Γ+a



Γ−a

dω ej ωt

ε0 ω2pe νe + j ω ε0 ω2pe νe + j ω

= j lim ej ωt ε0 ω2pe ω→j νe

=0

ω − j νe = ε0 ω2pe e−νe t νe + j ω

(12.84)

(12.85)

because the integrand exhibits a first-order pole which is encircled by Γ+a and is analytic within Γ−a .

Wave propagation in dispersive media

841

Since our goal is the calculation of (12.79) we split the contour integrals into line integrals along part of the real axis and the half-circles, namely,  ε0 ω2pe 1 dω ej ωt ε0 ω2pe e−νe t = 2π νe + j ω Γ+a +a

1 = 2π and 1 0= 2π 1 = 2π



dξ e −a

dξ e

j ξt

−a

ε0 ω2pe

1 + νe + j ξ 2π

π

dα a j ej α ej at cos α e−at sin α

0

ε0 ω2pe νe + j aej α

(12.86)

ε0 ω2pe

dω ej ωt

Γ−a +a

j ξt

νe + j ω ε0 ω2pe

1 − νe + j ξ 2π

π

dα a j e− j α ej at cos α eat sin α

0

ε0 ω2pe νe + j ae− j α

(12.87)

where the left-hand sides do not vary with the radius a. The last terms in the rightmost members may remain finite in the limit as a → +∞ so long as t > 0 and t < 0, respectively. The other integrals pass over into the desired inverse Fourier transformation in (12.79). Therefore, we may employ (12.86) to determine σ(t) for positive values of time and (12.87) for t < 0. We just have to show that the integrals along the half-circles vanish as the radius is increased to infinity. For positive times and a > νe we have π π  ae−at sin α ε0 ω2pe ε0 ω2pe j α j at cos α −at sin α dα a j e e  e dα νe + j aej α |νe + j aej α | 0

0





aε0 ω2pe a − νe

π 0

2aε0 ω2pe a − νe

dα e−at sin α =

2aε0 ω2pe a − νe

π/2 dα e−at sin α 0

π/2 ε0 ω2pe π dα e−2atα/π = (1 − e−at ) −−−−−→ 0 a→+∞ a − νe t

(12.88)

0

where we have employed the Jordan inequality (B.59). Hence, in the limit as a → +∞ the contour integral along Γ+a reduces to the improper integral in (12.79) if t > 0. By means of similar steps we prove that the integral along the half-circle in the lower complex half-plane vanishes for a → +∞. Putting these intermediate results together yields [36, pp. 17–18] ⎧ 2 −νe t ⎪ ⎪ ⎨ε0 ω pe e , t  0 σ(t) = ⎪ (12.89) ⎪ ⎩0, t 0. The pair of charges −Zqe and Zqe separated by the distance |u| constitutes an electric dipole of moment given by Zme

p(r, t) := −Zqe u(r, t),

r∈V

(12.96)

844

Advanced Theoretical and Numerical Electromagnetics

according to the discussion of Example 2.5 and Section 3.6. Therefore, we obtain an equation for p(r, t) by multiplying (12.95) through with −qe , namely, me

d2 p dp + me ω2r p(r, t) = Zq2e Elocal (r, t), + 2νe me dt dt2

r∈V

(12.97)

and one more equation for the polarization vector P(r, t) = Nd p(r, t) (see Section 3.7) me

d2 P dP + me ω2r P(r, t) = Nd Zq2e Elocal (r, t), + 2νe me 2 dt dt

r∈V

(12.98)

assuming that Nd is the uniform number density of dipoles induced within V by the electric field. We now derive an expression for the local electric field. We have anticipated above that an atom experiences both the field produced by external sources and the field generated by the remaining polarized atoms induced within V. Since the atoms are either regularly arranged (as in crystals) or randomly distributed (as in gasses) it is reasonable to suppose that the atoms very close to the one in question produce a zero net field on average. This hypothesis was suggested by O. F. Mosotti in 1850 [49] and actually proved by Lorentz [50] for cubic crystals. Following the Mosotti model, we isolate the atom with a ball B(r0, a), where r0 is the position vector of the nucleus, as is sketched in Figure 12.5, and write the local electric field as Elocal (r, t) := E(r, t) + Eatoms (r, t)

(12.99)

that is, as being due partly to the external driving wave-field E(r, t) and partly to the presence of the surrounding atoms existing outside B(r0, a). Since these atoms, too, are modelled by means of equivalent dipoles characterized by a polarization vector P(r, t) we assume that an unbalanced ˆ · P(r, t) (see Section 3.7) exists on the sphere ∂B, with the unit normal surface charge density n(r) ˆ n(r) pointing inwards B(r0, a) towards the nucleus. Any other atom which happens to be enclosed by B(r0 , a) is then neglected in the calculation of Eatoms (r, t). From the discussion in Example 9.3 we know that the electromagnetic field is predominantly static very close to a time-varying source. As a result, we may capture the main contribution to Eatoms (r, t) by considering it to be quasi-static and deriving it from an electrostatic potential (Sections 2.2 and 9.8), viz.,  ˆ  ) · P(r , t) n(r Eatoms (r, t) = −∇Φatoms (r, t) = −∇ dS  , r ∈ B(r0 , a) (12.100) 4πε0 |r − r | ∂B

on the grounds of (2.15), (3.5) and (3.245). Notice that in writing the integral representation of the scalar potential we have used the permittivity of free space because at a microscopic level the atoms which constitute the medium are indeed immersed in free space. But then, since P(r, t) is precisely the entity we wish to determine by solving (12.98), how do we compute the integral above if we do not know P(r, t) yet? ˆ  ) · P(r , t) over the boundary ∂B. We do know that the polarizaWell, luckily all we need is n(r tion vector depends on the applied electric field E(r, t) and — having assumed that the medium is homogeneous and isotropic — we may stipulate that P(r, t) is parallel to E(r, t). We introduce two local systems of Cartesian coordinates (ξ, η, ζ) and (ξ , η , ζ  ) centered in r0 . Then, we have r = r˜ + r0 = ξξˆ + ηηˆ + ζ ζˆ + r0 r = r˜  + r0 = ξ ξˆ + η ηˆ + ζ  ζˆ + r0

(12.101) (12.102)

Wave propagation in dispersive media

845

for the relevant changes of variables. We may always take the ζ-axis parallel to the external electric field in the sphere, and hence we have ˆ P(r , t) = P(˜r , t) = P0 (t)ζ,

r , r˜  ∈ ∂B

(12.103)

where P0 (t) changes with time but remains essentially constant over the ball so long as the spatial variation of the electric field is large as compared to the typical size of the lattice of atoms in the body. This hypothesis is fundamental for the very introduction of a macroscopic parameter such as the susceptibility. Lastly, we also define two local spherical coordinates (˜r, α, β) and (˜r , α , β ) centered in r0 with α and α the polar angles measured from the ζ-axis (Figure 12.5). With these positions we have ˆ r ) · P(˜r , t) = −P0 (t) cos α n(˜

(12.104)

and the relevant potential becomes Φatoms (r, t) = Φatoms (˜r, t) = −



P0 (t) 4πε0

dS 

∂B

cos α , |˜r − r˜  |

|˜r | = a

(12.105)

on account of (12.101) and (12.102). To evaluate the integral in (12.105) we resort to the addition theorem for spherical harmonics (3.235) and observe that cos α = P1 (cos α ), i.e., the Legendre polynomial of order one. By substituting (3.235) into (12.105) with r> = a and r< = r˜, we find Φatoms (˜r, t) = −

+n +∞ P0 (t)  r˜n  (n − m)! m P (cos α)e− j mβ 4πε0 n=0 an+1 m=−n (n + m)! n

2π ×

 j mβ

dβ e 0

=−

π

P0 (t) r˜ P1 (cos α)2π 4πε0

 dα a2 sin α P1 (cos α )Pm n (cos α )

0

π

dα sin α [P1 (cos α )]2

0

P0 (t) P0 (t) 2 P0 (t) =− r˜ P1 (cos α) = − r˜ cos α = − ζ 2ε0 3 3ε0 3ε0

(12.106)

because the associated Legendre functions and the complex exponentials are orthogonal over the intervals [0, π] and [0, 2π], respectively, and only the term with indices n = 1, m = 0 survives in the double summation. Usage of the normalization integral (H.127) yields the result. The inversion of the order of summation and integration is permitted since the series converges provided r˜ < a, as is the case, since we are interested in the field at the location of the atom. We observe that the radius a of the ball B(r0, a) does not enter in the expression of Φatoms (r, t). Finally, to take the gradient of the potential with respect to r = xˆx + yˆy + zˆz we notice that ζ = ζˆ · r˜ = ζˆ · (r − r0 ) in view of (12.101), and then 

∂ζ ∂ζ P0 (t) ∂ζ Eatoms (r, t) = −∇Φatoms (ζ, t) = xˆ + yˆ + zˆ 3ε0 ∂x ∂y ∂z " P0 (t) P0 (t) ! ˆ 1 = P(r, t) (12.107) ζ · xˆ xˆ + ζˆ · yˆ yˆ + ζˆ · zˆ zˆ = ζˆ = 3ε0 3ε0 3ε0

846

Advanced Theoretical and Numerical Electromagnetics

whence Elocal (r, t) = E(r, t) +

1 P(r, t), 3ε0

r∈V

(12.108)

is the sought form of the local electric field. Although we have obtained this expression by taking the observation point inside the ball B(r0 , a), the result is valid in the region occupied by the dielectric medium, as the procedure can, in principle, be repeated for each and every atom present in V. Not surprisingly Elocal (r, t) still depends on the unknown polarization vector, thus (12.108) is only useful insofar as it enables us to insert the true external field in the governing equation (12.98) which passes over into

 Nd Zq2e Nd Zq2e dP d2 P 2 + ω + 2ν − E(r, t), r∈V (12.109) P(r, t) = ε0 e r 2 dt 3ε0 me ε0 me dt or more succinctly dP d2 P + ω20 P(r, t) = ε0 ω2pe E(r, t), + 2νe dt dt2 where

r∈V

(12.110)



ω pe :=

Nd Zq2e ε0 me

(12.111)

is the plasma frequency of the electron cloud [26]. We see that the resonant (binding) frequency ωr of an atom in isolation is reduced according to  # 2 N Zq 1 d e ω0 = ω2r − = ω2r − ω2pe < ωr (12.112) 3ε0 me 3 since each atom in the dielectric is surrounded by polarized atoms. To solve (12.110) we represent the time-varying electric field as in (12.2) and the polarization vector as the temporal inverse Fourier transform 1 P(r, t) := 2π

+∞ dω ej ωt P(r; ω)

(12.113)

−∞

which by substitution into (12.110) yields 1 2π

+∞  dω ej ωt − ω2 P(r; ω) + j ω2νe P(r; ω) −∞

 + ω20 P(r; ω) − ε0 ω2pe E(r; ω) = 0,

r∈V

(12.114)

having swapped the order of integration and time derivative on account of the continuity of P(r; ω)ej ωt and its first- and second-order derivatives for (t, ω) ∈ R+ × R. The exponential function never vanishes, and thus (12.114) is satisfied only if " ! r∈V (12.115) ω20 − ω2 + 2 j ωνe P(r; ω) = ε0 ω2pe E(r; ω),

Wave propagation in dispersive media

847

whence we find P(r; ω) =

ε0 ω2pe ω20 − ω2 + 2 j ωνe

E(r; ω),

r∈V

(12.116)

for the Fourier transform of the polarization vector within V. By comparing (12.116) with its static counterpart (3.260) we are led to identify the dimensionless factor χ˜ e (ω) :=

ω2pe ω20

− ω2 + 2 j ωνe

= ω2pe

ω20 − ω2 − j ω2νe (ω20 − ω2 )2 + 4ν2e ω2

(12.117)

as the dielectric susceptibility of the medium [14, 15, 36, 47]. The real part of χ˜ e (ω) is even, whereas the imaginary part is odd. Besides, (12.117) obeys the expected asymptotic behavior (12.25). Finally, χ˜ e (ω) is analytic in the lower complex half-plane ω ∈ C and exhibits two simple poles in ω = j νe ± (ω20 − ν2e )1/2 when ω0 > νe (see Example 12.2 further on). To obtain the permittivity of the dielectric we set σ(ω) ˜ = 0 in (12.33) since there are no free electrons by hypothesis, and we find ⎛ ⎞ ⎜⎜⎜ ⎟⎟⎟ ω2pe := ε0 [1 + χ˜ e (ω)] = ε0 ⎜⎜⎝1 + 2 ε(ω) ˜ ⎟⎟⎠ 2 ω0 − ω + 2 j ωνe = ε0 +

ε0 ω2pe (ω20 − ω2 ) (ω20 − ω2 )2 + 4ν2e ω2

−j

2νe ωε0 ω2pe (ω20 − ω2 )2 + 4ν2e ω2

(12.118)

having outlined real and imaginary parts. In a dielectric medium the physical mechanism responsible for the losses is constituted by the ‘collisions’ that occur between the bound electron clouds and the neighboring atoms. Indeed, in formula (12.118) the permittivity becomes real if the collision frequency is set to zero. Figure 12.6 shows a plot of the susceptibility χ˜ e (ω) versus the normalized angular frequency ω/ω0 . We consider only positive frequencies in view of the even-odd symmetry properties (12.22). At relatively low frequencies, i.e., for ω ω0 , the susceptibility is essentially real and equal to the static limit ω2pe /ω20 . For ω in the neighborhood of the binding frequency ω0 the real part of χ˜ e (ω) has a negative slope, exhibits a steep variation and changes sign for ω = ω0 . For relatively large values of ω the real part gently tends to zero. We call regions of normal dispersion the intervals of the angular frequency for which the variation of Re{χ˜ e (ω)} is slow. Conversely, we call regions of anomalous dispersion the intervals of ω for which Re{χ˜ e (ω)} changes very rapidly. Some Authors (e.g., [36, 47]) define regions of normal or anomalous dispersion the ranges of frequency where the real part of the susceptibility (or the permittivity) increases or decreases, respectively. We observe that the imaginary part of χ˜ e (ω) remains mostly negligible except for ω ≈ ω0 . In particular, −Im{χ˜ e (ω)} reaches its maximum values for ω = ω0 , i.e., −Im{χ˜ e (ω0 )} =

ω2pe 2νe ω0

(12.119)

and we say that ω = ω0 is a resonance frequency. From a physical viewpoint, only the spectral components of the driving field (12.2) that oscillate at frequencies close to ω0 interact strongly with the bound electron clouds in the medium. Since such components are in sync with the natural resonance frequency of the atoms, an efficient transfer of electromagnetic energy takes place from the field or wave to the atoms when ω ≈ ω0 , and this phenomenon (absorption) is appropriately

848

Advanced Theoretical and Numerical Electromagnetics

Figure 12.6 Dispersion in a dielectric medium: real and imaginary parts of the susceptibility χ˜ e (ω) as a function of the angular frequency normalized to the resonance frequency. perceived as polarization losses. It is worthwhile noticing, though, that the imaginary part of χ˜ e (ω), as given by (12.117), is always negative for ω > 0. This feature is a substantial shortcoming of the classical electron oscillator model of an atom, whereby the possible emission of energy and amplification of the external driving field for some frequency ω > 0 — in which instance Im{χ˜ e (ω)} should be positive — is actually overlooked (cf. [44, Chapter 5]). √ The real part of χ˜ e√(ω) reaches its extrema for ω = 0, ω √= ω0 (ω0 + 2νe ) and, so long as ω0  2νe , also for ω = ω0 (ω0 − 2νe ). The peak value for ω = ω0 (ω0 − 2νe ) is half the maximum (12.119) of −Im{χ˜ e (ω)} and occurs at the intersection of the graphs of Re{χ˜ e (ω)} and −Im{χ˜ e (ω)}. Finally, we notice that owing to the absence of free charges in a dielectric the permittivity (12.118) remains finite for ω → 0, whereas the permittivity of a conductor according to the Drude model (12.65) exhibits a pole in the origin. For substances comprised of more than one type of atoms or molecules, the resonance model (12.117) and (12.118) is easily generalized by stating (12.110) for each species. Since the total polarization vector is the sum of contributions such as (12.116), we obtain χ˜ e (ω) :=



ω2peυ

υ

ω2υ − ω2 + 2 j ωνυ

ε˜ (ω) := ε0 +



ε0 ω2peυ

υ

ω2υ − ω2 + 2 j ωνυ

(12.120)

 where υ means summation over all the types of atoms or molecules, 2νυ and ωυ denote the collision rate and the resonance frequency for the species υ. The special collision-free case := ε0 + ε(ω) ˜

 ε0 ω2peυ υ

ω2υ − ω2

(12.121)

is referred to as the Sellmeier equation and is used for modelling the material dispersion in optical fibers [36, 47].

Wave propagation in dispersive media

849

Example 12.2 (The Kramers-Krönig relations for a dielectric medium) It is instructive to show that the permittivity in (12.118) obtained with the classical microscopic model of a dielectric body does obey the postulate of causality. We check that the Kramers-Krönig relations hold true and, since the steps are quite similar to those described in Example 12.1, we just outline the proof. On account of (12.118), the first Kramers-Krönig relation (12.50) reads +∞ ε0 ω2pe 2νe Ω 2 Ω Re{ε(r; ˜ ω)} − ε0 = PV dΩ 2 π Ω − ω2 (ω20 − Ω2 )2 + 4ν2e Ω2 0

+∞ 2νe ε0 ω2pe Ω2 1  = PV dΩ  π (Ω2 − ω2 ) (Ω2 − ω2 )2 + 4ν2e Ω2

(12.122)

0

−∞

having exploited the even symmetry of the integrand so as to extend the integration along the entire real axis in the complex plane of the variable Ω ∈ C. The integrand behaves asymptotically as  1 f (Ω) := 2 , =O |Ω|4 (Ω − ω2 )[(Ω2 − ω20 )2 + 4ν2e Ω2 ]

2νe ε0 ω2pe Ω2

|Ω| → +∞

(12.123)

and exhibits six simple poles, namely, Ω = Ω1 = j νe +



 ω20 − ν2e  Ω = Ω4 = − j νe + ω20 − ν2e

ω20 − ν2e  Ω = Ω3 = − j νe − ω20 − ν2e

Ω = Ω2 = j νe −

(12.124) (12.125)

and, finally, two poles on the real axis in Ω = ±ω. The latter must be avoided in accordance with the definition of Cauchy principal value. In order to invoke the theorem of residues (B.57) we choose the contour Γ shown in Figure 12.7 and defined again by (12.72) and (12.73). Since Γ encircles only the poles Ω1 and Ω2 in a counterclockwise fashion, we compute the relevant residues with the aid of (B.46), viz., Res f (Ω1 ) = lim (Ω − Ω1 ) f (Ω) = Ω→Ω1

ε0 ω2pe Ω1

Res f (Ω2 ) = lim (Ω − Ω2 ) f (Ω) = − Ω→Ω2

(12.126)

2 j(Ω21 − ω2 )(Ω1 − Ω2 ) ε0 ω2pe Ω2

(12.127)

2 j(Ω22 − ω2 )(Ω1 − Ω2 )

whence we get 2π j

2  n=1

Res f (Ωn ) =

π(ω20 − ω2 )ε0 ω2pe

(12.128)

(ω2 − ω20 )2 + 4ν2e ω2

with a little algebra. Then, application of (B.57) provides      1 1 1 1 1 dΩ f (Ω) = dΩ f (Ω) + dΩ f (Ω) + dΩ f (Ω) + dΩ f (Ω) π π π π π Γ

γab

Cω

C−ω

Ca

850

Advanced Theoretical and Numerical Electromagnetics

Figure 12.7 Poles (×) and contour Γ (−−) in the complex plane Ω for the calculation of the integral (12.122) with the Cauchy theorem of residues.

=

ε0 ω2pe (ω20 − ω2 )

(12.129)

(ω2 − ω20 )2 + 4ν2e ω2

on account of (12.128). We are interested in taking the limits of both sides for a → +∞ and b → 0+ , whereby the integral along γab yields the desired integral in (12.122). The integral along the half-circle Ca vanishes in the limit as the radius a approaches infinity thanks to the asymptotic decay (12.123) [cf. (12.41)]. The integrals along the half-circles Cω and C−ω can be formally evaluated with the mean value theorem for a finite value of b after letting Ω = ω − b exp(− j α) and Ω = −ω − b exp(− j α), respectively. In the limit for a vanishing b we get 1 lim b→0+ π 1 π

lim+

b→0

 dΩ Cω



dΩ C−ω

2νe ε0 ω2pe Ω2 (Ω2 − ω2 )[(Ω2 − ω20 )2 + 4ν2e Ω2 ] 2νe ε0 ω2pe Ω2 (Ω2 − ω2 )[(Ω2 − ω20 )2 + 4ν2e Ω2 ]

=− =

j ωνe ε0 ω2pe (ω2 − ω20 )2 + 4ν2e ω2 j ωνe ε0 ω2pe

(ω2 − ω20 )2 + 4ν2e ω2

(12.130)

(12.131)

and thus these contributions cancel out. In the end we are left with 1 PV π

+∞ dΩ

−∞

2νe ε0 ω2pe Ω2 (Ω2 − ω2 )[(Ω2 − ω20 )2 + 4ν2e Ω2 ]

=

ε0 ω2pe (ω20 − ω2 ) (ω2 − ω20 )2 + 4ν2e ω2

(12.132)

which on account of (12.122) and (12.118) proves that the resonance model obeys the KramersKrönig relation (12.50). Also in this case other choices for the contour Γ are possible and lead to the same conclusion. (End of Example 12.2)

Wave propagation in dispersive media

851

We obtain the memory function χe (t) of a homogeneous isotropic dielectric medium by subjecting χ˜ e (ω) to the temporal inverse Fourier transformation, viz., 1 χe (t) := 2π

+∞ dω ej ωt −∞

ω2pe

(12.133)

ω20 − ω2 + j 2νe ω

where the improper integral exists finite for t ∈ R because, integrating over the finite interval [−a, a], we can find a constant M > 0 for 0 < ω M < ω < a such that a ωM a  j ωt dω e M dω  2 + 2 dω 2 (12.134)  ω ω20 − ω2 + j 2νe ω 2 − ω2 )2 + 4ν2 ω2 (ω −a ωM e 0 0 and the second integral in the right member remains finite as a grows infinitely large. The integral is readily evaluated by means of the Cauchy theorem as in Example B.2. We use the same two contours Γ+a and Γ−a defined by (12.82) and (12.83). The poles of the integrand are in ω = Ω1 and ω = Ω2 given by (12.124). The relevant residues follow from Resexp(j ωt)χ˜ e (Ω1 ) = lim ej ωt ω→Ω1

Resexp(j ωt)χ˜ e (Ω2 ) = lim ej ωt ω→Ω2

(ω − Ω1 )ω2pe ω20 − ω2 + j 2νe ω (ω − Ω2 )ω2pe ω20 − ω2 + j 2νe ω

= =

ω2pe Ω2 − Ω1 ω2pe Ω1 − Ω2

ej Ω1 t

(12.135)

ej Ω2 t

(12.136)

whence we obtain 2π j

$  % 2πω2pe −ν t Resexp(j ωt)χ˜ e (Ωn ) =  e e sin t ω20 − ν2e n=1 ω20 − ν2e

2 

(12.137)

with a little algebra. Then, we observe 1 2π 1 2π



dω ej ωt

Γ+a



dω ej ωt

Γ−a

ω2pe ω20 − ω2 + j 2νe ω ω2pe ω20 − ω2 + j 2νe ω

= 

ω2pe ω20 − ν2e

$  % e−νe t sin t ω20 − ν2e

=0

(12.138)

(12.139)

because Γ+a encircles the poles whereas the integrand is analytic in the lower complex half-plane. Since we aim at computing (12.133) we split the contour integrals into contributions along part of the real axis and the half-circles, viz.,  $  % ej ωt ω2pe ω2pe 1 −νe t 2 2 e sin t ω0 − νe = dω 2  2π ω0 − ω2 + j 2νe ω ω20 − ν2e Γ+a 1 = 2π

+a −a

ej ξt ω2pe

1 + dξ 2 ω0 − ξ2 + j 2νe ξ 2π

π 0

dα a j ej α

ej at cos α e−at sin α ω2pe ω20 − a2 e2 j α + j 2νe aej α

(12.140)

Advanced Theoretical and Numerical Electromagnetics

852 and

1 0= 2π

 Γ+a

ej ωt ω2pe

1 dω 2 = ω0 − ω2 + j 2νe ω 2π

1 − 2π

π

dα a j e− j α

0

+a dξ −a

ej ξt ω2pe ω20 − ξ2 + j 2νe ξ

ej at cos α eat sin α ω2pe ω20 − a2 e−2 j α + j 2νe ae− j α

(12.141)

where the left-hand sides are independent of the radius a. The last terms in the rightmost members may remain finite for a → +∞ provided t > 0 and t < 0, respectively. The other two integrals reduce to the desired inverse Fourier transformation in (12.133). Therefore, we use (12.140) to determine χe (t) for t > 0 and (12.141) for negative values of time. In order to show that the integrals along the half-circles vanish as the radius is increased to infinity, from (12.124) we notice that |Ω1 | = |Ω2 | = ω0 , and for t > 0, a > ω0 we consider π j at cos α −at sin α 2  π e e ω ae−at sin α ω2pe pe dα a j ej α  dα |aej α − Ω1 ||aej α − Ω2 | ω2 − a2 e2 j α + j 2νe aej α 0

0



aω2pe (a − ω0 )2

0

π

dα e−at sin α 

0

πaω2pe (a − ω0 )2

−−−−−→ 0 a→+∞

(12.142)

and thus in the limit as a → +∞ the contour integral along Γ+a reduces to the improper integral in (12.133) if t > 0. Analogous steps allow concluding that the integral along the half-circle in the lower complex half-plane vanishes as well. Thanks to these intermediate results we have found (for ω0 > νe ) ⎧ $  % ⎪ ω2pe ⎪ −νe t ⎪ 2 − ν2 , t  0 ⎪ e sin t ω  ⎪ e ⎪ 0 ⎨ χe (t) = ⎪ (12.143) ω20 − ν2e ⎪ ⎪ ⎪ ⎪ ⎪ ⎩0, t 0 with no oscillations [cf. (12.89) and (12.143)]. Experimental evidence suggests that the second formula in (12.147) be generalized by defining the permittivity as := ε∞ + ε(ω) ˜

ε s − ε∞ ε s − ε∞ ε s − ε∞ − j ωτ = ε∞ + 2 2 1 + j ωτ 1+ω τ 1 + ω2 τ2

(12.148)

which is known as the Debye equation [49]. The parameters ε s and ε∞ represent the static permittivity and the high-frequency (or optical) limit thereof. Since the substance is passive, then ε s > ε∞ to ensure that Im{ε(ω)} ˜ < 0. While the dispersion of water follows almost perfectly the Debye model (12.148) with ε s = 80.1ε0 , ε∞ ≈ 5.3ε0 , and τ ≈ 9.37 s for frequencies up to 100 GHz [51, Section 3.5], some other polar materials are best modelled with := ε∞ + ε(ω) ˜ = ε∞ +

ε s − ε∞ 1 + (j ωτ)1−α   (ε s − ε∞ ) 1 − (ωτ)1−α sin(απ/2) 1 + (ωτ)2(1−α) + 2(ωτ)1−α sin(απ/2) (ε s − ε∞ )(ωτ)1−α cos(απ/2) −j 1 + (ωτ)2(1−α) + 2(ωτ)1−α sin(απ/2)

(12.149)

which is referred to as the Cole-Cole equation (1941) with the exponent α ∈ [0, 1[ [51]. For instance, transformer oil obeys (12.149) with α = 0.23.

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Advanced Theoretical and Numerical Electromagnetics

In order to separate real and imaginary parts of ε(ω) ˜ in (12.149) we have noticed that the complex function z1−α with z ∈ C is many-valued and exhibits branch points for z = 0 and z = ∞ (Appendix B.3). If 1 − α is a rational number, the branch points are algebraic, otherwise they are transcendental. So, to compute (j ωτ)1−α we observe (j ωτ)1−α = (ωτ)1−α j e− j πα/2 e2πnα ,

n∈Z

(12.150)

and we let n = 0 whereby choosing j ωτ on the fundamental sheet of the Riemann surface associated with z1−α . When substances, such as biological tissues, are characterized by both polarization and conduction mechanisms, the Debye and Cole-Cole models are extended by introducing a conduction term [51], namely, σs ε s − ε∞ −j 1 + j ωτ ω ε s − ε∞ σs ε˜ c (ω) := ε∞ + −j ω 1 + (j ωτ)1−α

ε˜ c (ω) := ε∞ +

(12.151) (12.152)

where σ s represents the residual static conductivity of the material [cf. (12.29)]. Therefore, losses are due to friction experienced by the rotating molecules as well as currents of free ions. The Debye and the Cole-Cole relaxation models satisfy the Kramers-Krönig relations and hence provide causal memory functions. To investigate the effect of the parameter α and the differences between the Debye and the ColeCole models, it is fruitful to interpret (12.148) and (12.149) as the parametric definition of lines in the complex plane ε˜ ∈ C. In fact, when ωτ ∈ R is varied from zero all the way up to infinity, the complex number ε(ω) ˜ describes a line in the complex plane that joins the points ε(0) ˜ = ε s and ε˜ (+∞) = ε∞ . Thus, to find said line we eliminate the ‘parameter’ ωτ ∈ R and obtain an implicit equation for the real and imaginary parts of the permittivity. We carry out the procedure for the Cole-Cole model whence the result for the Debye model is derived as a special case by setting α = 0. Solving the first part of (12.149) formally for (ωτ)1−α yields $ πα %' ε s − ε˜ & $ πα % (ωτ)1−α = sin − j cos , ωτ  0 (12.153) ε˜ − ε∞ 2 2 on account of (12.150) with n = 0. Now, since the left-hand side is real whereas ε˜ is complex, the last relationship can be true only if the imaginary part of the right member vanishes as well, that is, ( $ πα %') ε s − ε˜ & $ πα % sin − j cos =0 (12.154) Im ε˜ − ε∞ 2 2 which, being independent of ωτ, provides us with the desired implicit representation. With a little more algebra we arrive at $

Re{ε} ˜ −

$ πα %'2 $ ε − ε %2 & $ πα %' ε s − ε∞ ε s + ε∞ % 2 & s ∞ tan + Im{ε} ˜ − = 1 + tan2 2 2 2 2 2

(12.155)

which constitutes the equation of a bundle of circles whose centers and radii change with α. In a system of Cartesian coordinates (Re{ε}, ˜ −Im{ε}) ˜ ∈ R2 the circles (12.155) intersect the horizontal axis (−Im{ε} ˜ = 0) in two fixed points located at Re{ε} ˜ ∈ {ε∞ , ε s }. Notice that relation (12.155) represents a larger set of values (Re{ε}, ˜ −Im{ε}) ˜ than it is possible to obtain with either (12.148) or (12.149), because in (12.155) the variable −Im{ε} ˜ is not limited to

Wave propagation in dispersive media

855

Figure 12.8 Dispersion in a polar material: arc plots of permittivity obtained with Debye (—) and Cole-Cole (−−) models.

positive values. Only the parts of the circles for which −Im{ε} ˜  0 correspond to the ‘parametric’ representations (12.148) and (12.149). Figure 12.8 shows two such arcs for α = 0 (Debye model) and α = 0.4. Graphical representations of this type are said arc plots of the permittivity or Argand diagrams [52, 53]. Finally, Cartesian plots of the permittivity predicted by the Debye and Cole-Cole relaxation models are reported in Figure 12.9 as a function of the normalized angular frequency ωτ. The maximum of −Im{ε(ω)} ˜ occurs for ωτ = 1 for any value of α ∈ [0, 1[, although the peak grows ever broader as α is increased. A general remark is that the peak of the imaginary part is broader than the similar peak exhibited by the resonance model (12.118).

12.4 Narrow-band signals in the presence of dispersion Thanks to the macroscopic description of material media by means of constitutive parameters (Sections 1.6 and 12.1) the Maxwell equations in the frequency domain take on the very same functional form they have for fields and waves in free space. This feature facilitates the formulation and the solution of electromagnetic problems, and especially so when the time-harmonic regime (Section 1.5) can be invoked. Then again, truly time-harmonic fields are an abstraction and, more often than not, one deals with general time-varying electromagnetic fields which can be represented as inverse Fourier transforms. In fact, we can construe the right member of (12.2) and like expressions as the linear superposition of a continuous set of infinitely many time-harmonic constituents. If the underlying medium is dispersive, e.g., in accordance with (12.5)-(12.7), then any two time-harmonic waves characterized by different frequencies, say, ω1 and ω2 , effectively propagate in two different materials. Consequently, the field we reconstruct through (12.2) may actually vary from point to point as a result of the dispersive nature of the medium. This effect combines with the spatial variation of the field due to the source being localized and finite (e.g., Chapter 8). We wish to investigate how dispersion affects the propagation of electromagnetic waves [5, Chapter 16 ].

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Advanced Theoretical and Numerical Electromagnetics

Figure 12.9 Dispersion in a polar material: permittivity obtained with Debye and Cole-Cole models as a function of the normalized angular frequency. To gain more generality while keeping the exposition lucid we consider a wave-like solution to the time-domain Maxwell equations that depends only on time and one spatial coordinate, say, z. Suppose that for z = 0 the solution reads Ψ(0, t) = f (t) cos(ω0 t)

(12.156)

where f (t) = 0 for t < 0 and ω0 > 0 is a fixed angular frequency. The function f (t) is called the envelope, whereas the periodic part of the solution cos(ω0 t) is called the carrier. This combination is quite common in digital communications where f (t) is a simple-shaped waveform which contains information and varies slowly as compared to the carrier. For example, Ψ(z, t) may represent the voltage or the current along a classical transmission line (Example 6.2), or the modal voltage or current in an optical fiber or in a waveguide (Section 11.2), or even the electric or magnetic field of a uniform plane wave (Sections 7.1 and 7.2). In order to determine Ψ(z, t) we turn to the frequency domain because the dispersive properties of the medium are more easily characterized through algebraic constitutive relations such as (12.5)(12.7). The spectral counterpart of Ψ(0, t) is formally defined as ˜ ω) := Ψ(0;

+∞ dt e− j ωt f (t) cos(ω0 t) −∞

1 = 2

+∞ +∞ 1 − j(ω−ω0 )t dt e f (t) + dt e− j(ω+ω0 )t f (t) 2

−∞

−∞

1 1 = F(ω − ω0 ) + F(ω + ω0 ) 2 2

(12.157)

Wave propagation in dispersive media

857

where F(ω) denotes the Fourier transform of the envelope f (t). We recall that F(−ω) = F ∗ (ω),

ω∈R

(12.158)

because f (t) is a real-valued function of time. For the sake of simplicity we make the assumption that the medium of concern is infinitely extended along z, so that we need not bother with boundary conditions at the ends of the region of interest, and we can write the wave in z as ˜ ω) = e− j kz (ω)z Ψ(0; ˜ ω) = 1 F(ω − ω0 )e− j kz (ω)z + 1 F(ω + ω0 )e− j kz (ω)z Ψ(z; 2 2

(12.159)

˜ ω) is a modal voltage where kz (ω) is the propagation constant of the problem. For instance, if Ψ(0; or current in the section z = 0 of a waveguide, (12.159) is supported by the telegraph equations (11.202) and (11.203) or (11.237) and (11.238) with kz (ω) given by (11.204) or (11.239). If, on the ˜ ω) is the electric field of uniform plane wave propagating along z in a homogeneous other hand, Ψ(0; isotropic medium, (12.159) is motivated by the general solution (7.20), and kz (ω) coincides with the wavenumber (1.249). In both cases, the constitutive parameters are replaced by ε˜ (ω), μ(ω) ˜ and σ(ω) ˜ so as to include the dispersion of the material in our discussion. In general the propagation constant may be a complex function of ω ∈ R, hence we write it by separating real and imaginary parts kz (ω) := β(ω) − j α(ω)

(12.160)

where β(ω) ∈ R and α(ω) ∈ R are called the phase constant and the attenuation constant, respectively. If kz (ω) happens to be real, β(ω) is oft-times referred to as the propagation constant as well. Besides, it is not difficult to show that β(ω) ∈ R and α(ω) ∈ R must obey the symmetry relations β(−ω) = −β(ω),

α(−ω) = α(ω),

ω∈R

(12.161)

in order that the inverse Fourier transform of (12.159) be a real-valued function of time and position. ˜ ω), The wave-like solution Ψ(z, t) is formally given by the inverse Fourier transformation of Ψ(z; viz., 1 Ψ(z, t) = 2π

+∞ 1 dω ej ωt F(ω − ω0 )e− j β(ω)z e−α(ω)z 2

−∞

1 + 2π

+∞ 1 dω ej ωt F(ω + ω0 )e− j β(ω)z e−α(ω)z 2

(12.162)

−∞

which we would like to cast into an alternative form that is more convenient to investigate the effect of dispersion. We observe that, by performing the change of the dummy variable ω = −Ω, the second integral in the right-hand side can be written as 1 2π

+∞ 1 dω ej ωt e− j β(ω)z e−α(ω)z F(ω + ω0 ) = 2

−∞

1 = 2π

+∞ 1 dΩ e− j Ωt e− j β(−Ω)z e−α(−Ω)z F(ω0 − Ω) 2

−∞

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Advanced Theoretical and Numerical Electromagnetics

Figure 12.10 Qualitative spectrum of a narrow-band waveform centered around ω0 and phase (β(ω)) and attenuation (α(ω)) constants of a hypothetical dispersive environment. +∞ 1 1 dΩ e− j Ωt ej β(Ω)z e−α(Ω)z F ∗ (Ω − ω0 ) = 2π 2 −∞ ⎡ ⎤∗ ⎢⎢⎢ +∞ ⎥⎥⎥ 1 ⎢⎢⎢ 1 ⎥ = ⎢⎢ dΩ ej Ωt e− j β(Ω)z e−α(Ω)z F(Ω − ω0 )⎥⎥⎥⎥ ⎣ 2π ⎦ 2

(12.163)

−∞

where we have made use of (12.158) and (12.161). Apparently, the second term in (12.162) is the complex conjugate of the first one, and, as a result, we obtain ⎫ ⎧ +∞ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ 1 j ωt − j β(ω)z −α(ω)z dω e e e F(ω − ω ) Ψ(z, t) = Re ⎪ (12.164) 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ 2π −∞

on account of (B.6). Finding Ψ(z, t) is now just a matter of computing an inverse Fourier transformation, but clearly this step is feasible — at least numerically — provided we specify the functional form of kz (ω) and, ultimately, the constitutive parameters of the medium. Still, we can draw general conclusions if we assume that the envelope f (t) is a narrow-band function of time. With this hypothesis we mean that the Fourier transform F(ω) is substantially different than zero only in a narrow range of angular frequencies around ω = 0. Therefore, we speculate that the main contribution to the integral in (12.164) comes from a small interval of frequencies centered around ω0 , as is graphically, though qualitatively, suggested in Figure 12.10. As a result, we may approximate phase and attenuation constants with asymptotic expansions in the neighborhood of ω0 , namely, dβ 1 d2 β β(ω) ≈ β(ω0 ) + (ω − ω0 ) + (ω − ω0 )2 dω ω=ω 2 dω2 0

ω=ω0

1 = β0 + β0 (ω − ω0 ) + β0 (ω − ω0 )2 , 2 dα 1 d2 α α(ω) ≈ α(ω0 ) + (ω − ω0 ) + (ω − ω0 )2 dω ω=ω0 2 dω2 ω=ω

ω → ω0

(12.165)

1 − ω0 ) + α0 (ω − ω0 )2 , 2

ω → ω0

(12.166)

0

= α0 +

α0 (ω

Wave propagation in dispersive media

859

because the contribution for values away from ω0 — albeit non-zero — is filtered out by the rapidly decaying F(ω − ω0 ). We suppose for the time being that the asymptotic expansion of β(ω) may be stopped at the first-order term and α(ω) may be taken as constant. This works pretty well if ω0 falls in a region of normal dispersion (see Figures 12.3 and 12.6) where the variation of the constitutive parameters is relatively slow and losses are almost negligible. By inserting the chosen approximations for β(ω) and α(ω) into (12.164) and letting Ω = ω − ω0 we obtain [14] ⎧ ⎫ +∞ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  1 ⎨ ⎬ j ωt−j β0 z − j β0 z(ω−ω0 ) Ψ(z, t) ≈ e−α0 z Re ⎪ dω e e F(ω − ω ) ⎪ 0 ⎪ ⎪ ⎪ ⎪ 2π ⎩ ⎭ −∞ ⎧ ⎫ +∞ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ j ω0 t−j β0 z 1 ⎬ −α0 z j Ω(t−β0 z) dΩ e F(Ω)⎪ = e Re ⎪ e ⎪ ⎪ ⎪ ⎪ 2π ⎩ ⎭ −∞  

! " β0 z (12.167) = e−α0 z f t − β0 z cos ω0 t − ω0 having recognized the remaining integral as the inverse Fourier transform of F(Ω) though evaluated at the time t − β0 z. By comparing (12.167) with (12.156) we see that a narrow-band waveform travels in a dispersive environment essentially undisturbed except for the attenuation of the amplitude due to, e.g., losses. Moreover, since envelope and carrier exhibit the dependence on time and position that is typical of progressive travelling waves [cf. (7.11)], it is instructive to determine the relevant velocities of propagation. The velocity of the carrier equals the velocity of an ideal observer who rides alongside the waveform and ‘sees’ the same shape of the carrier. This situation is achieved if the argument of the cosine in (12.167) remains constant with time, viz., 0=

dz ∂ (ω0 t − β0 z) = ω0 − β0 = ω0 − β0 v p ∂t dt

(12.168)

whereby we find v p (ω0 ) =

ω0 β(ω0 )

(12.169)

which is referred to as the phase velocity. The latter is, as the name suggests, the rate of variation of the phase (the argument of the cosine) and as such can be larger than the speed of light (1.209) without violating the postulate that information or matter cannot move faster than light [cf. (11.218)]. In like manner, the velocity of the envelope is the velocity of an ideal observer who, while speeding along the waveform, ‘detects’ the same shape of the envelope. Again, this occurrence requires the argument of the narrow-band function f (t − β0 z) to be independent of time, namely, 0=

" dz ∂ ! t − β0 z = 1 − β0 = 1 − β0 vg ∂t dt

(12.170)

whence we obtain vg (ω0 ) =

1 1 = dβ β0 dω ω0

(12.171)

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Advanced Theoretical and Numerical Electromagnetics

Figure 12.11 Propagation of a sinusoidal carrier (—) modulated by a narrow-band function (−−) in a dispersive environment without attenuation. which is called the group velocity. Then again, the envelope is not a physical entity but rather the set of maximum values attained by Ψ(z, t) at any given position as the time goes by. It is the apparent collective motion of a group of crests and troughs that creates the illusion of an envelope moving with velocity vg . The latter is indeed associated with the transport of energy or information and, as such, is always smaller than the speed of light (1.209) in the medium. Last but not least, we can construe the envelope of the carrier in (12.167) as a shifted replica of f (t) that lags behind by a time dβ  (12.172) τg (ω0 , z) := β0 z = z dω ω=ω0 a quantity referred to as the group delay. The longer is the distance travelled by the wave (12.156), the larger grows the group delay. The ultimate explanation for the existence of group velocity and delay lies in the fact that f (t) results from the superposition of infinitely many elementary time-harmonic waves (a group or packet, as it were) that, while oscillating at different frequencies around ω = 0, propagate at different phase velocities (12.169). As an example, Figure 12.11 shows plots of a sample waveform of the type (12.167) with α0 = 0 for z = 0 and a point z > 0. The situation gets more involved if we include higher-order terms in the asymptotic expansion of the phase constant. This step may be necessary in order to better approximate β(ω) in the range of frequency where F(ω − ω0 ) happens to be substantially non-null (Figure 12.10). For instance, by stopping at the second-order term in (12.165) we arrive at Ψ(z, t) ≈ e

−α0 z

(

1 Re 2π

) +∞   2 dω ej ωt−j β0 z e− j β0 z(ω−ω0 ) e− j β0 z(ω−ω0 ) /2 F(ω − ω0 ) −∞

Wave propagation in dispersive media =e

−α0 z

=e

−α0 z

( Re e 

j ω0 t−j β0 z

1 2π

j ω0 t−j β0 z

) +∞ − j Ω(t−β0 z) − j β0 zΩ2 /2 dΩ e e F(Ω)  

−∞

β0 z)

=F1 (Ω)



Re e f1 (t −   = e−α0 z Re ej ω0 t−j β0 z+j θ1 (z,t) f1 (t − β0 z) = e−α0 z f (t − β z) cos[ω t − β z + θ (z, t)] 1

0

861

0

0

(12.173)

1

where f1 (z, t) is the possibly complex, inverse Fourier transform of the auxiliary function F1 (Ω), and θ1 (z, t) indicates the phase of f1 (z, t). In this case not only is the waveform attenuated as it travels along, owing to losses, it also undergoes distortion, i.e., it changes shape, which can be problematic in digital communications if information is associated with the very shape of the envelope that is being received. Since at this level of approximation β0 is responsible for distorting the envelope, if possible, it is better to choose the frequency ω0 of the carrier so that β0 = 0 because, as a result, the spectrum of the narrow-band envelope spans a frequency range in which the phase constant β(ω) is practically linear. If β0 = 0, distortion may be kept to a minimum though not completely eliminated, inasmuch as then the effect of the third-order derivative of β(ω) comes into play.

12.5 Intra-modal dispersion in waveguides We conclude this chapter by showing with an example that dispersion routinely occurs when electromagnetic waves are made to propagate in spatial regions with boundaries even in the absence of truly dispersive materials. The typical situation is represented by guided waves (also known as modes) in classical hollow-pipe metallic waveguides (Section 11.2) [2, 8, 33, 35–39, 54]. Although the calculation of the modes and the analysis of the dispersion properties thereof may be tackled systematically with the network formalism developed by N. Marcuvitz and J. Schwinger [34] (Section 11.2.1), here we resort to an ad hoc approach applied to the electromagnetic propagation in the unbounded region of space between two parallel PEC planes (Figure 12.12) [2], [55, Section 5.5], [29, Sections 1.7.E, 4.2.A], [56, Section XI.1], [57, Chapter 8]. Admittedly, this configuration — known as a parallel-plate waveguide — is unpractical but serves our purposes quite well. Suppose that in a system of Cartesian coordinates the metallic plates are perpendicular to the y-axis and described by the equations y = 0 and y = h. We begin by seeking solutions to the timeharmonic source-free Maxwell equations that are characterized by an electric field aligned with the x-axis and dependent only on y and z. This assumption makes sense for reasons of geometrical symmetry as the plates are, after all, invariant along the x-direction. Accordingly, the Helmholtz equation (1.238) becomes ∂2 E x ∂2 E x + + ω2 ε0 μ0 E x (y, z) = 0 ∂y2 ∂z2

(12.174)

supplemented with the boundary conditions (1.169) which here read E x (y, z)ˆx × yˆ = E x (y, z)ˆz = 0,

y ∈ {0+ , h− },

z ∈ R.

(12.175)

The general solutions to (12.174) are uniform or inhomogeneous plane waves (Section 7.2) in the form E x (y, z) = e− j ky y e− j kz z

(12.176)

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Advanced Theoretical and Numerical Electromagnetics

Figure 12.12 Interpretation of TE and TM modes in a parallel-plate waveguide as the linear superposition of two plane waves (→ and ). with the propagation constants ky , kz related by ky2 + kz2 = ω2 ε0 μ0

(12.177)

which is a special case of the dispersion relation (7.50). A solitary plane wave propagates in the direction specified by the wavevector k = ky yˆ +kz zˆ and clearly does not satisfy the required boundary conditions. However, in view of the linearity of (12.174) and the principle of superposition (Section 6.1) a linear combination of two or more solutions still solves (12.174). Thus, we try (Figure 12.12) E x (y, z) = A1 e− j ky1 y e− j kz1 z + A2 e− j ky2 y e− j kz2 z

(12.178)

and determine A1 , A2 , ky1 , ky2 , kz1 and kz2 in order to meet (12.175) and the pertinent two instances of the dispersion relation (12.177). In fact, since (12.175) must be true at any point along the plates, we also choose kz1 = kz2 = kz , whereby in (12.178) the dependence on z can be factored out. At a closer look, we notice that all in all, we have five conditions and six parameters, so we can only fix five of them. Now, in view of kz1 = kz2 = kz the dispersion relations relevant to the two waves in question demand 2 2 = ω2 ε0 μ0 − kz2 = ky2 ky1

(12.179)

which can only be satisfied if either ky1 = ky2 or ky1 = −ky2 . The former condition is unsuitable because then the two waves in (12.178) would coincide, and we would be back to square one. So, we are prompted to choose ky1 = −ky2 = ky , whereby our trial solution (12.178) takes on the form ! " (12.180) E x (y, z) = A1 e− j ky y + A2 ej ky y e− j kz z . In order to satisfy the boundary conditions at the location of the plates we enforce E x (0+ , z) = (A1 + A2 )e− j kz z = 0, ! " E x (h− , z) = A1 e− j ky h + A2 ej ky h e− j kz z = 0,

z∈R

(12.181)

z∈R

(12.182)

and since the complex exponential exp(− j kz z) never vanishes, the equations above amount to a homogeneous algebraic linear system for the coefficients A1 and A2 . Non-trivial solutions exist if

Wave propagation in dispersive media the equations are linearly dependent or, equivalently, the system matrix

 1 1 [M] = − j ky h j ky h e e

863

(12.183)

is singular or rank-deficient, in which case the determinant of [M] vanishes [58–60]. By computing the determinant and equating it to zero we get det[M] = ej ky h − e− j ky h = 2 j sin(ky h) = 0

(12.184)

which represents a transcendental equation for the propagation constant ky . The infinite but denumerable solutions are π n∈Z (12.185) kyn = n, h and can be interpreted as the eigenvalues of the non-linear eigenvalue problem represented by (12.181) and (12.182). We need to exclude n = 0 and kyn = 0 because the resulting field (12.178) would then be constant along y and, in order to meet (12.175), necessarily null everywhere between the plates. For all other values of n we may set A2 = −A1 = E0 /(2 j) in light of (12.181) and define # $ nπ %2 , n ∈ N \ {0} (12.186) kzn = ω2 ε0 μ0 − h on the grounds of (12.179). In the end, we obtain $ nπ % y , E x (y, z) = E0 e− j kzn z sin h

n ∈ N \ {0}

(12.187)

as the infinite, though discrete set of eigensolutions to (12.174) subject to (12.175). Since negative values of n produce a set of eigensolutions which are just the negative of those given by (12.187), we need not consider n < 0. Each one of (12.187) is called a transverse-electric (TE) mode of the parallel-plate waveguide in that the electric field is perpendicular (transverse) to z. Notice that the amplitude E0 remains undetermined, as we have used up the five conditions at our disposal, whereas kzn depends on the angular frequency ω. We shall elaborate this point further on. We continue the analysis of the parallel-plate waveguide by looking for solutions to the timeharmonic source-free Maxwell equations that are characterized by a magnetic field aligned with the x-axis and dependent only upon y and z. The relevant Helmholtz equation is (1.239), which becomes ∂2 H x ∂2 H x + + ω2 ε0 μ0 H x (y, z) = 0 (12.188) ∂y2 ∂z2 still supplemented with the boundary conditions (1.169). However, since the latter involve the electric field whereas our primary unknown is H x , we need to express E(y, z) in terms of H x , namely,

 ∂H x ∂H x 1 yˆ − zˆ E(y, z) × yˆ = × yˆ j ωε0 ∂z ∂y xˆ ∂H x = 0, y ∈ {0+ , h− }, z∈R (12.189) = j ωε0 ∂y on account of the Ampère-Maxwell law (7.17). Building on the arguments we have invoked to find the TE modes, we are prompted to try solutions in the form ! " (12.190) H x (y, z) = A1 e− j ky y + A2 ej ky y e− j kz z .

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In order to satisfy the boundary conditions at the location of the plates we enforce ∂H x = − j ky (A1 − A2 ) e− j kz z = 0 z∈R ∂y y=0+ ! " ∂H x = − j ky A 1 e− j k y h − A 2 ej k y h e− j k z z = 0 z∈R ∂y −

(12.191) (12.192)

y=h

and since the complex exponential exp(− j kz z) never vanishes, the above equations constitute a homogeneous algebraic linear system for A1 and A2 . To find the non-trivial solutions we consider the system matrix

 1 −1 [M] = − j ky h (12.193) e −ej ky h and require the determinant of [M] to vanish [58–60]. In symbols, we have det[M] = −ej ky h + e− j ky h = −2 j sin(ky h) = 0

(12.194)

a transcendental equation for ky that is solved by the same eigenvalues as in (12.185). This time, however, we may also retain n = 0 and kyn = 0 inasmuch as this choice yields a non-zero field which does meet (12.191) and (12.192). We proceed by setting A1 = A2 = H0 /2 on account of (12.191) whereby the infinite discrete set of eigensolutions to (12.188) subject to (12.189) read $ nπ % y , n ∈ N \ {0} (12.195) H x (y, z) = H0 e− j kzn z cos h with the propagation constants kzn still given by (12.186). Evidently, we need not consider negative values of n, since the corresponding eigensolutions coincide with those given by (12.195) for n > 0. Each one of (12.195) is referred to as a transverse-magnetic (TM) mode of the parallel-plate waveguide because the magnetic field is perpendicular (transverse) to the z-direction. The case n = 0, kyn = 0 is special because the propagation constant kzn reduces to k0 = ω(ε0 μ0 )1/2 , that is, the wavenumber of the medium which occupies the region of space between the plates. Furthermore, by computing H x from (12.195) and the electric field with the aid of (7.17) we arrive at H x (y, z) = H0 e− j k0 z ,

Ey (y, z) = −

H0 k 0 − j k 0 z e = −Z0 H0 e− j k0 z ωε0

(12.196)

where Z0 = (μ0 /ε0 )1/2 is the intrinsic impedance of free space [cf. (1.358)]. Since electric and magnetic fields are both perpendicular to z, this eigensolution is called a transverse-electric-magnetic (TEM) mode. The latter is constant in any section z = z0 of the parallel-plate waveguide, and this configuration is feasible because the electric field is everywhere perpendicular to the plates. From a different perspective, the TEM mode is constituted by a single plane wave which by itself is capable of satisfying the necessary boundary conditions (12.189). As already noted, the propagation constants kzn (12.186) depend on the angular frequency ω and in particular become imaginary for k0 < kyn , namely, #$ % nπ 2 − k02 = − j |kzn |, n ∈ N \ {0} (12.197) kzn (ω) = − j h whereby we see that the eigenvalues kyn have the meaning of critical or cut-off constants. For any given value of ω only a limited number of TE and TM modes are characterized by a real propagation

Wave propagation in dispersive media

865

Figure 12.13 Dispersion curves (ω − kz diagrams) for the TEM mode and the first three TE and TM modes in the parallel-plate waveguide of Figure 12.12.

constant kzn and hence are permitted to travel between the plates. All the remaining modes — for which it holds k0 < kyn — exhibit an imaginary propagation constant and thus are exponentially damped and do not propagate. It is worthwhile emphasizing that the attenuation here is not related to losses in the medium, but rather to the very geometry of the region where the propagation takes place. On the other hand, the TEM mode can always propagate, since the propagation constant thereof is real for any value of ω. It is customary to present (12.186) graphically as plots of ω versus kz if kz ∈ R and versus − j |kz | if kz is imaginary. The resulting diagrams are called dispersion curves of the modes or simply ω − kz plots [14, 33, 34, 39]. The dispersion curves for the first few modes of the parallel-plate waveguide are shown in Figure 12.13 for ω  0. For propagating modes and kzn ∈ R, the dispersion curves (in the right half of the diagram) are hyperbolas, whereas for cut-off modes and kzn = − j |kz | the curves (in the left half of the diagram) are circular arcs of radii nπ/h. The hyperbolas tend asymptotically to the straight line ω = c0 k0 , which also constitutes the dispersion curve of the TEM mode. Tracing back our steps we may construe TE and TM modes as the combination of two uniform plane waves (with kzn > 0) which endlessly carom back and forth between the plates (Figure 12.12). These multiple reflections give rise to waves of the form (12.187) and (12.195) (with n > 0) that effectively travel in the positive z-direction with propagation constants kzn . According to definition (12.169), the phase velocities of the propagating modes read v pn (ω) =

ω = # kzn

ω ω2 ε

0 μ0

−

$ nπ %2 = # h

1−

c0 $ nπc %2 ,

n ∈ N \ {0}

(12.198)

0

hω

by virtue of (1.53) and (12.169). The key point is that, although the medium that ‘fills’ the waveguide is free space — which is not dispersive at all — the phase velocity of TE and TM modes evidently depends on the frequency. We talk of intra-modal dispersion to distinguish the phenomenon from the material dispersion due to the atomic or molecular structure of the medium in which the propagation takes place.

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Advanced Theoretical and Numerical Electromagnetics

For a narrow-band waveform of the type (12.156) that propagates in a parallel-plate waveguide as predicted by (12.162) we may also define the group velocity for propagating TE and TM modes in accordance with (12.171), viz., # $ nπc %2 1 0 2 kzn = c0 1 − vgn (ω) = , n ∈ N \ {0} (12.199) = c0 dkzn ω hω dω whereby we see that vgn (ω) never exceeds the speed of light. In fact, phase and group velocities in the waveguide are evidently related to one another by vgn (ω)v pn (ω) = c20 ,

n ∈ N \ {0}.

(12.200)

The TEM mode represents a notable exception in that, having a propagation constant kz equal to the wavenumber k0 , it travels with constant phase and group velocities equal to c0 , and no intramodal dispersion occurs.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

Coelho R. Physics of Dielectrics for the Engineer. Amsterdam, NL: Elsevier; 1979. Shen LC, Kong JA. Applied electromagnetism. Monterey, CA: Brooks/Cole; 1983. Morse P, Feshbach H. Methods of Theoretical Physics. New York, NY: McGraw-Hill; 1953. Stein EM, Weiss G. Introduction to Fourier Analysis on Euclidean Spaces. Princeton, NJ: Princeton University Press; 1971. Helrich CS. The classical theory of fields - Electromagnetism. Berlin Heidelberg: SpringerVerlag; 2012. Zangwill A. Modern electrodynamics. Cambridge, UK: Cambrigde University Press; 2013. Brillouin L. Wave propagation and group velocity. New York, NY and London, UK: Academic Press; 1960. Van Bladel JG. Electromagnetic Fields. Piscataway, NJ: IEEE Press; 2007. Oughstun KE. Electromagnetic and Optical Pulse Propagation. vol. 1. New York, NY: Springer Science+Business Media; 2006. Nussenzveig HM. Causality and Dispersion Relations. New York, NY: Academic; 1972. Post EJ. Formal structure of electromagnetics. Amsterdam, NL: North-Holland Publishing Company; 1962. Schwinger J, De Raad LL, Milton KA, et al. Classical electrodynamics. Perseus Books; 1998. Jones DS. The Theory of Electromagnetism. Oxford, UK: Pergamon; 1964. Rothwell EJ, Cloud MJ. Electromagnetics. London, UK: CRC Press; 2001. Jackson JD. Classical Electrodynamics. 3rd ed. Chichester, UK: Wiley; 1999. Böttcher CJF. Theory of electric polarisation. vol. 1. 2nd ed. Amsterdam, NL: Elsevier Publishing Company; 1973. Böttcher CJF. Theory of electric polarisation. vol. 2. 2nd ed. Amsterdam, NL: Elsevier Publishing Company; 1978. de Hoop AT. Handbook of radiation and scattering of waves. London, UK: Academic Press; 1995. Hu BY. Kramers-Kronig in two lines. Americal Journal of Physics. 1989;57(9):821. Zhang K, Li D. Electromagnetic Theory for Microwaves and Optoelectronics. Berlin Heidelberg: Springer-Verlag; 1998.

Wave propagation in dispersive media [21] [22] [23]

[24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50]

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LePage. Complex variables and the Laplace transform for engineers. New York, NY: Dover Publications, Inc.; 1961. Makarov B, Podkorytov A. Real Analysis: Measures, Integrals and Applications. Universitext. London, UK: Springer; 2013. Tofighi M, Daryoush A. Measurement Techniques for the Electromagnetic Characterization of Biological Materials. In: Bansal R, editor. Engineering Electromagnetics Applications. Boca Raton, FL: CRC press; 2006. p. 235–275. Kramers HA. La diffusion de la lumière par les atomes. In: Estratto dagli Atti del Congresso Internazional de Fisici Como, Bologna. Nicolo Zonichelli; 1927. p. 545–557. de L Krönig R. On the theory of dispersion of X-rays. J Opt Soc Am & Rev Sci Instrum. 1926;12(6):547–557. Stix TH. Plasma Waves. New York, NY: Springer-Verlag; 1992. Fitzpatrick R. Plasma Physics: An Introduction. Boca Raton, FL: CRC Press; 2014. Feynmann RP, Leighton R, Sands M. The Feynmann lectures on physics. vol. 1. Reading, MA: Addison-Wesley; 1964. Kong JA. Electromagnetic Wave Theory. 2nd ed. New York, NY: Wiley; 1990. Brown RW, Cheng YN, Haacke EM, et al. Magnetic Resonance Imaging: Physical Principles and Sequence Design. 2nd ed. Hoboken, NJ: John Wiley & Sons, Inc.; 2014. Wesson J. Tokamaks. 3rd ed. Oxford, UK: Clarendon Press; 2004. Ashcroft NW, Mermin ND. Solid State Physics. 1st ed. Orlando, FL: Harcourt College Publishers; 1976. Collin RE. Field Theory of Guided Waves. Piscataway, NJ: IEEE press; 1991. Felsen LB, Marcuvitz N. Radiation and scattering of waves. Piscataway, NJ: IEEE Press; 2001. Harrington RF. Time-harmonic Electromagnetic Fields. London, UK: McGraw-Hill; 1961. Orfanidis SJ. Electromagnetic Waves and Antennas. www.ece.rutgers.edu/~orfanidi/ewa; 2004. Weeks WL. Electromagnetic Theory for Engineering Applications. New York, NY: John Wiley & Sons, Inc.; 1964. Lorrain P, Corson DR, Lorrain F. Electromagnetic Fields and Waves. 3rd ed. New York, NY: W. H. Freeman and Company; 1988. Marcuvitz N. Waveguide Handbook. 2nd ed. Electromagnetic Waves Series. London, UK: The Institution of Engineering and Technology; 1985. Boyd TJM, Sanderson JJ. The physics of plasmas. Cambridge, UK: Cambridge University Press; 2003. Marsden JE. Basic Complex Analysis. San Francisco, CA: W. H. Freeman and Company; 1973. Pantell RH, Puthoff HE. Fundamentals of quantum electronics. New York, NY: Wiley; 1969. Ditchburn RW. Light. New York, NY: Interscience; 1963. Yariv A. Optical Electronics. 4th ed. Fort Worth, TX: Saunders College Publishing; 1991. Biggs HF. The Electromagnetic Field. London, UK: Oxford University Press; 1934. Hayt WH, Buck JA. Engineering Electromagnetics. 8th ed. New York, NY: McGraw-Hill; 2012. International edition. Born M, Wolf E. Principles of Optics. Oxford, UK: Pergamon Press; 1980. Feynmann RP, Leighton R, Sands M. The Feynmann lectures on physics. vol. 2. Reading, MA: Addison-Wesley; 1964. Debye P. Polar molecules. New York, NY: Dover Publications, Inc.; 1945. Lorentz HA. Theory of electrons. New York, NY: Dover Publications, Inc.; 1952.

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[51]

Gabriel C. Dielectric properties of biological materials. In: Greenebaum B, Barnes FS, editors. Bioengineering and Biophysical Aspects of Electromagnetic Fields. Boca Raton, FL: CRC press; 2007. p. 51–100. Daniel VV. Dielectric Relaxation. London, UK: Academic Press; 1967. Frohlich H. Theory of Dielectrics, Dielectric Constants, and Dielectric Loss. Amen House, London, UK: Oxford University Press; 1958. Milton KA, Schwinger J. Electromagnetic Radiation: Variational Methods, Waveguides and Accelerators. Berlin Heidelberg: Springer-Verlag; 2006. Dudley DG. Mathematical Foundations for Electromagnetic Theory. New York, NY: WileyInterscience; 1994. Slater JC, Frank NH. Electromagnetism. New York, NY: McGraw-Hill; 1947. Sevgi L. Electromagnetic Modeling and Simulation. Hoboken, NJ: IEEE Press; 2014. Golub GH, van Loan CF. Matrix Computations. Baltimore, MD: Johns Hopkins University Press; 1996. Bau III D, Trefethen LN. Numerical linear algebra. Philadelphia, PA: Soci. Indus. Ap. Math.; 1997. Blyth TS, Robertson EF. Basic Linear Algebra. 2nd ed. Springer Undergraduate Mathematics Series. London, UK: Springer-Verlag; 2002.

[52] [53] [54] [55] [56] [57] [58] [59] [60]

Chapter 13

Integral equations in electromagnetics

Since charge and current densities are related to the electromagnetic field they generate through the Maxwell equations, solving the latter seems quite natural and especially so if the source distribution is known. This strategy is sometimes referred to as a field or domain approach, in that it aims at determining the electromagnetic field in the spatial region of interest. So far in this book we have tacitly used the field approach while seeking closed-form solutions for static, stationary, and timeharmonic fields in the whole space but also for the study of electromagnetic waves in spatial regions with planar, cylindrical, and spherical symmetry. Still, either the integral representations of potentials and fields obtained in Sections 2.7, 3.2, 5.1, 9.1 and 10.2 or the equivalence principles proved in Sections 10.4 and 10.5 provide the basis for the alternative formulation of electromagnetic problems in terms of integral equations. This strategy is called a source approach because oft-times the unknowns are densities of equivalent sources localized on lines or surfaces or distributed within volumes. A classic example is the direct scattering problem of Figure 6.4 in which one wishes to determine the field reflected back by an obstacle in response to an impinging wave generated by some remote source. The scattered field Es (r, t), Hs (r, t) can be ascribed to equivalent currents that flow on the surface of the body (if the latter is a PEC or a PMC) or in the bulk thereof (if the obstacle is penetrable). Therefore, it may be convenient to compute the unknown sources first, inasmuch as they are confined just to the finite surface or bulk of the body. Conversely, the field approach would require to compute Es (r, t), Hs (r, t) straightaway for all points of space. In this chapter, after reviewing general concepts related to integral equations, we focus mostly on the derivation of surface (Sections 13.2 and 13.3) and volume (Section 13.4) integral equations for equivalent sources in the time-harmonic regime. The last part (Section 13.5) is devoted to two mixed formulations, i.e., one that uses coupled surface and volume integral equations and another one that combines the local Maxwell equations with a suitable set of surface integral equations.

13.1 General considerations As the name suggests, an integral equation is essentially a mathematical expression in which the unknown quantity appears as part of the integrand of one or more integrals. We can justify an integral equation intuitively by starting with a typical problem in linear algebra, that is, a system of N linear equations in as many unknowns [1, 2], say, ⎛ ⎜⎜⎜ a11 ⎜⎜⎜ . ⎜⎜⎜ .. ⎜⎜⎝ aN1

⎞⎛ ⎞ ⎛ ⎞ · · · a1N ⎟⎟⎟ ⎜⎜⎜ x1 ⎟⎟⎟ ⎜⎜⎜ b1 ⎟⎟⎟ .. ⎟⎟⎟⎟ ⎜⎜⎜⎜ .. ⎟⎟⎟⎟ ⎜⎜⎜⎜ .. ⎟⎟⎟⎟ .. . . ⎟⎟⎟⎟ ⎜⎜⎜⎜ . ⎟⎟⎟⎟ = ⎜⎜⎜⎜ . ⎟⎟⎟⎟ ⎠⎝ ⎠ ⎝ ⎠ · · · aNN xN bN

or

[A][x] = [b]

(13.1)

Advanced Theoretical and Numerical Electromagnetics

870 where • • •

[A] is a square matrix with N rows and N columns comprised of complex entries amn , m, n ∈ {1, . . . , N}; [x] is a column vector with N unknown entries xn , n = 1, . . . , N; [b] represents a column vector with N known entries bm , m = 1, . . . , N.

We may regard the matrix [A] as a finite-rank continuous linear operator (Appendix D.3) which takes the vector [x] ∈ CN and produces the vector [b] ∈ CN as a result. Since [b] is known and [x] is desired, then solving the system (13.1) translates into the problem of finding the inverse operator corresponding to [A]. In the language of matrices, though, the inverse operator is called the inverse matrix and — when it exists — is indicated with the symbol [A]−1 . In particular, we write the solution to (13.1) as [x] = [A]−1 [b]

(13.2)

with the understanding that [A]−1 [A] = [I] = [A][A]−1

(13.3)

where [I] is the identity matrix. Next, we cast (13.1) into yet another format which emphasizes the effect of the operator [A] on the unknown vector [x], viz., N 

amn xn = bm ,

m = 1, . . . , N

(13.4)

n=1

where the indices n and m may be regarded as two discrete variables. Suppose now that the dimension of the space of the vectors [x] and [b] is, in effect, infinite. This means that [x] and [b] have infinitely many entries and so does [A], whereby the explicit notation of (13.1) becomes clumsy and unpractical. On the contrary, (13.4) is perfectly suited for the job and reads +∞ 

amn xn = bm ,

m∈Z

(13.5)

n=−∞

with amn denoting the entries of an infinite-dimensional matrix. A simple circumstance in which (13.5) can actually be solved analytically and exactly arises when the matrix [A] is diagonal, that is, [A] = diag {amm } ,

m∈Z

(13.6)

because the infinitely many equations represented by (13.5) decouple and, as a consequence, can be inverted independently of one another. Finally, we may think of the indices n and m as being continuously distributed between −∞ and +∞. Heuristically, we understand that the summation in (13.5) passes over into an improper integral and thus we write (but this is no proof) +∞ L {x} = dν a(μ, ν)x(ν) = b(μ), −∞

μ∈R

(13.7)

Integral equations in electromagnetics

871

which is termed an integral equation for x(ν) [3, Chapter 8]. Broadly speaking, the linear operator +∞ L {•} := dν a(μ, ν){•},

μ∈R

(13.8)

−∞

extends the notion of square matrix. The function of two variables a(μ, ν) is variously called the kernel or the nucleus or the Green function of the integral operator and must satisfy certain requirements in order for (13.7) to have solutions [4, Chapter 6]. We shall devote the following sections to showing how integral equations arise in electromagnetics where quite often the variables μ and ν represent space coordinates, x(ν) is (the component of) a vector current density, and b(μ) is (the component of) a known electromagnetic field generated by some distant sources. For the time being we stick to general one-dimensional problems, and hence we trade the symbols μ and ν for x and x to indicate the relevant space coordinate. A typical linear one-dimensional integral equation may take on the following three forms b h(x) =

dx g(x, x ) f (x ),

x ∈ [a, b]

(13.9)

x ∈ [a, b]

(13.10)

x ∈ [a, b]

(13.11)

a

b h(x) = f (x) − α

dx g(x, x ) f (x ),

a

b h(x) = a(x) f (x) − α

dx g(x, x ) f (x ),

a

which are referred to as Fredholm’s integral equations of the first, the second and the third kind, respectively. The number α  0 is a parameter, and the end points a and b may also be infinite, in which instance the equations are said singular. When one of the end points is a function of the independent variable x, we obtain a slightly different set of integral equations x h(x) =

dx g(x, x ) f (x ),

x ∈ [a, b]

(13.12)

x ∈ [a, b]

(13.13)

x ∈ [a, b]

(13.14)

a

x h(x) = f (x) − α

dx g(x, x ) f (x ),

a

x h(x) = a(x) f (x) − α

dx g(x, x ) f (x ),

a

which are named Volterra’s integral equations of the first, the second and the third kind. The notation g(x, x ) for the kernel serves as a reminder that more often than not g(x, x ) is the relevant Green function of the problem. If g(x, x ) can be written in the form g(x, x ) =

N  i=1

ai (x)bi (x ),

x, x ∈ [a, b]

(13.15)

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Advanced Theoretical and Numerical Electromagnetics

where ai (x) and bi (x ) are functions, the kernel is said to be separable or degenerate, in which case (13.10) can, in principle, be turned into an algebraic linear system of order N (cf. Example 13.1 further on) [5]. If g(x, x ) depends on the difference x− x rather than on x and x separately, we write g(x, x ) = g(x − x ),

x, x ∈ [a, b]

(13.16)

and say that the kernel is convolutional inasmuch as the integrals in (13.9)-(13.14) become, in fact, convolution products. From a physical standpoint the positive number |x − x | represents the distance between the two points x and x . A kernel is said to be weakly singular if it takes on the forms g(x, x ) = b(x, x ) log

1 , |x − x |

g(x, x ) =

b(x, x ) , (x − x )ν

x, x ∈ [a, b]

(13.17)

where b(x, x ) is a bounded function and ν ∈]0, 1[ [6,7]. The case ν = 1 is referred to as a singular or Cauchy kernel, and the integrals in (13.9)-(13.14) are interpreted as Cauchy principal values. Finally, the case ν = 2 constitutes a strongly singular kernel. If the kernel g(x, x ) is square-integrable over the square [a, b] × [a, b], viz., ⎡ b ⎤1/2 b ⎢⎢⎢ ⎥⎥⎥



g(x, x )

2 := ⎢⎢⎢⎢⎢ dx dx |g(x, x )|2 ⎥⎥⎥⎥⎥ < +∞ ⎣ ⎦ a

(13.18)

a

then the integral operators in (13.9)-(13.11) are called Hilbert-Schmidt operators and can be shown to be compact (Appendix D.3) [4, Section 7.3], [8, pp. 368 and ff.], [9, Example 9.23]. As can be seen, the main difference between the first type of equations and the other two is that, in the latter case, the unknown function f (x) appears inside the integral but also directly in the equation. This occurrence is no minor detail, because it substantially changes the properties of the equation or, equivalently, of the operator one needs to invert to find the solution f (x). More specifically, the equation of the first kind (13.9) constitutes an ill-posed problem (see Section 6.2), whereas the equation of the second kind (13.10) can always be solved with the aid of a method developed in 1903 by E. I. Fredholm, under the hypotheses that the desired solution and the source term are continuous and so is the kernel on the square [a, b] × [a, b]. Said approach [10, Chapter 3] generalizes the method of matrix determinants for linear matrices [1, 2], though it is far too involved to be useful in practical situations. Fredholm also proved that under suitable conditions (13.10) either has a unique solution or the associated homogeneous equation (i.e., with h(x) = 0) has a finite number of linearly independent solutions [11]. This fundamental result goes by the name of Fredholm alternative (Appendix D.8) [12, Chapter 8], [4, Chapter 8], [5, Chapter 1], [13–15]. The Fredholm equation of the second kind is also amenable to an iterative solution procedure known as the method of successive approximations. To elucidate, we cast (13.10) into the alternative form b f (x) = h(x) + α

dx g(x, x ) f (x ) = h(x) + L { f (x)}

(13.19)

a

and argue that the right-hand side can be used to generate ever-better approximations to f (x) starting with an initial guess f0 (x). For instance, if we let f0 (x) = 0, the repeated application of (13.19) yields f1 (x) = h(x) + L { f0 (x)} = h(x)

(13.20)

f2 (x) = h(x) + L { f1 (x)} = I {h(x)} + L {h(x)}

(13.21)

Integral equations in electromagnetics

873

f3 (x) = h(x) + L { f2 (x)} = h(x) + L {h(x) + L {h(x)}} = I {h(x)} + L {h(x)} + L2 {h(x)}

(13.22)

where the symbol I {•} denotes a suitable identity operator, and L2 {•} is a shorthand notation for L {L {•}}. At the step n + 1 we have fn+1 (x) = h(x) + L { fn (x)} = I {h(x)} + L {h(x)} + . . . + Ln {h(x)}

(13.23)

which represents a partial sum. Do these partial sums converge as n → +∞ and, if so, is the limit the desired unknown f (x)? To answer the first question we turn to definition (D.124) for the norm of an operator L {•} : VL → VL , where the function space VL plays the role of domain and contains the range of L {•} [see (D.118)]. Then, we observe that L {h(x)}VL  L {•} h(x)VL



2

L {h(x)}

V  L {•} L {h(x)}VL  L {•}2 h(x)VL L

(13.24) (13.25)

and so forth, with •VL being a suitable norm in VL . We go on to take the norm of the partial sum fn+1 (x) which is still an element of VL by construction, i.e.,  fn+1 (x)VL = h(x) + L {h(x)} + · · · + Ln {h(x)}VL  h(x)VL + L {h(x)}VL + · · · + Ln {h(x)}VL  h(x)VL + L {•} h(x)VL + · · · + L {•}n h(x)VL =

1 − L {•}n+1 h(x)VL 1 − L {•}

(13.26)

having used the triangular inequality (D.11) and the sum of a geometrical progression [16, Formula 26], [17, Formula 3.1.10]. We see that, provided L {•} < 1

(13.27)

the rightmost-hand side remains finite as n → +∞ [12, p. 396], [18, p. 166]. If L {•} satisfies (13.27) then, for any two functions f1 (x), f2 (x) ∈ VL we have L { f1 (x)} − L { f2 (x)}VL  L {•}  f1 (x) − f2 (x)VL <  f1 (x) − f2 (x)VL

(13.28)

whereby the operator is Lipschitz continuous [8, Definition 4.14] and, in particular, referred to as a contraction. Condition (13.27) is usually difficult to check but it is likely to hold for integral equations which stem from physical problems. An alternative criterion for the convergence is obtained by interpreting the integral in (13.10) as an inner product for x fixed and considering first the estimate  2 b b  b  2 2      2   2 dx |g(x, x )| dx |h(x )|2 |L {h} | = |α|  dx g(x, x )h(x )  |α|   a a a b = |α|

2

h22 a

dx |g(x, x )|2

(13.29)

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Advanced Theoretical and Numerical Electromagnetics

which is based on the Cauchy-Schwarz inequality (D.145) specialized to scalar functions and the definition of 2-norm (D.69). Since both sides in the above expression are still functions of x, integrating over [a, b] and taking the square roots yields L {h}2  |α| g2 h2

(13.30)

where the 2-norm of g(x, x ) over the square [a, b] × [a, b] is defined in (13.18). These steps can be iterated, viz., 2  b b b    2 2     2   2   2 L {h} = |α|  dx g(x, x )L h(x )   |α| dx |g(x, x )| dx |L {h} |2   a a a b = |α|

2

L {h}22



 2

dx |g(x, x )|  |α|

b 4

g22

h22

a

dx |g(x, x )|2

(13.31)

a

whence

2

L {h} 2  |α|2 g22 h2

(13.32)

and so forth. Thus, taking the 2-norm of fn+1 (x) gives  fn+1 (x)2 = h(x) + L {h(x)} + . . . + Ln {h(x)}2  h(x)2 + L {h(x)}2 + . . . + Ln {h(x)}2  h(x)2 (1 + |α| g2 + . . . + |α|n gn2 ) =

1 − |α|n+1 gn+1 2 h(x)2 1 − |α| g2

(13.33)

whereby we see that the right-hand side remains finite as n → +∞ if the parameter α meets the constraint |α| <  b a

dx

b a

1 dx |g(x, x )|

1 1/2 = g(x, x ) 2 2

(13.34)

and of course g(x, x ) obeys (13.18). At any rate, by virtue of either (13.27) or (13.34) the partial sums tend to the Neumann series, i.e., lim fn+1 (x) =

n→+∞

+∞ 

Ln {h(x)}

(13.35)

n=0

with L0 {•} = I {•} by definition. To answer the second question, namely, whether fn+1 (x) actually tends to f (x), at step n + 1 we examine the difference between the sought solution and the partial sum, namely, f (x) − fn+1 (x) = f (x) − h(x) − L {h(x)} − L2 {h(x)} − · · · − Ln {h(x)} = L { f (x) − h(x)} − L2 {h(x)} − · · · − Ln {h(x)}

Integral equations in electromagnetics = L2 { f (x) − h(x)} − · · · − Ln {h(x)} = · · · = Ln { f (x) − h(x)}

875 (13.36)

having repeatedly invoked (13.19) and the linearity of L {•}. Now, taking a generic norm in VL or the 2-norm •2 yields  f (x) − fn+1 (x)VL = Ln { f (x) − h(x)}VL  L {•}n  f (x) − h(x)VL −−−−−→ 0 n→+∞

n n n





 f (x) − fn+1 (x)2 = L { f (x) − h(x)}2  |α| g(x, x ) 2  f (x) − h(x)2 −−−−−→ 0 n→+∞

(13.37) (13.38)

n ∞ n where the last steps follow because the numerical sequences {L {•}n }∞ n=0 and {|α| g2 }n=0 tend to zero in view of conditions (13.27) and (13.34), respectively [19–21]. Alternatively, we notice that (13.35) provides us with the formal series expansion of the inverse operator corresponding to I {•} − L {•}, namely,

(I − L)−1 {•} = I {•} + L {•} + L2 {•} + . . .

(13.39)

in light of (13.10) and (13.19). To show that this is indeed the case we consider ⎧ +∞ ⎫ +∞ ⎧ +∞ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ n ⎬  n ⎨ n ⎬ (I − L) ⎪ L {h(x)}⎪ L {h(x)} − L ⎪ L {h(x)}⎪ = ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ n=0

=

+∞ 

Ln {h(x)} −

n=0

n=0

+∞ 

Ln+1 {h(x)} =

n=0

n=0

+∞ 

Ln {h(x)} −

n=0

+∞ 

Lm {h(x)}

m=1

= L0 {h(x)} = h(x)

(13.40)

that is, ⎫ ⎧ +∞ ⎪ ⎪ ⎪ ⎬ ⎨ n ⎪ L {•}⎪ (I − L) ⎪ = I {•} ⎪ ⎪ ⎭ ⎩

(13.41)

n=0

whereby (13.39) follows [12, p. 396], [8, p. 295], [18, p. 166]. Thus, so long as either (13.27) or (13.34) is satisfied, the following equation +∞ 

Ln {h(x)} = (I − L)−1 {h(x)} = f (x)

(13.42)

n=0

gives the formal inverse of (13.10) and (13.19) and together with (13.35) shows again that fn+1 (x) tends to the solution. If a solution to (13.19) can be found by means of the method of successive approximations, then such solution is unique. Indeed, if two functions f1 (x) and f2 (x) solved (13.19), we would have  f1 − f2 VL = L { f1 } − L { f2 }VL = L { f1 − f2 }VL  L {•}  f1 − f2 VL  f1 − f2 2 = L { f1 } − L { f2 }2 = L { f1 − f2 }2  |α| g2  f1 − f2 2

(13.43) (13.44)

which, by dividing through by  f1 − f2 VL and  f1 − f2 2 , respectively, would lead to the conditions L {•}  1,

|α| 

1 g(x, x )2

(13.45)

which in turn contradict assumptions (13.27) and (13.34). Therefore, necessarily f1 (x) = f2 (x).

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Advanced Theoretical and Numerical Electromagnetics

According to (13.39) a sufficiently small ‘perturbation’ L {•} of the identity I {•} surely produces an invertible operator. However, it is important to realize that, even though L {•} is not ‘small’ enough — in the sense of (13.27) or (13.34) — for the Neumann series to converge, (13.10) may still be solvable and the inverse operator (I − L)−1 {•} may exist. In practice, this only means that we cannot write (I − L)−1 {•} as in (13.35) and the method of successive approximations does not yield the solution to (13.10). We explore this possibility in Example 13.1. Example 13.1 (Solving a degenerate-kernel Fredholm equation of the second kind) We consider the following special instance of (13.10) with g(x, x ) = xx , viz., b f (x) = h(x) + α





b



dx xx f (x ) = h(x) + αx a

dx x f (x ),

x ∈ [a, b]

(13.46)

a

where h(x) is assigned and integrable over [a, b]. Clearly, the kernel of the integral operator has the separable form (13.15) with N = 1, a1 (x) = x, and b1 (x ) = x . What is more, since we have ⎞1/2 ⎛ b ⎞1/2 ⎛ b b b ⎟⎟⎟ ⎟⎟⎟ ⎜⎜⎜ ⎜⎜⎜



⎟ ⎜ ⎜

g(x, x )

2 = ⎜⎜⎜⎜ dx dx |xx |2 ⎟⎟⎟⎟ = ⎜⎜⎜⎜ dx x2 dx x2 ⎟⎟⎟⎟⎟ = x22 < +∞ ⎠ ⎠ ⎝ ⎝ a

a

a

(13.47)

a

the relevant operator in the right member of (13.46) is of the Hilbert-Schmidt type and hence compact (Appendix D.3) [4, Section 7.3], [8, pp. 368 and ff.], [9, Example 9.23]. In which case, I {•}−L {•} is a perturbation of the identity under the action of a compact operator, and we can invoke the Fredholm theory (Appendix D.8) [4, Chapter 8], [12, Chapter 8], [5, Chapter 1], [22, Section VIII.2.4], [23, pp. 227–243], [24, Section 1.9], [13–15]. Still, in order to determine the function f (x) we introduce the auxiliary unknown [5] b







b

dx x f (x ) =

C := a

dx x f (x)

(13.48)

a

and, after multiplying (13.46) by x, we integrate both sides over x from a to b to arrive at a first-order algebraic equation b C=

  1 − α x22 C = dx x h(x ) b

dx xh(x) +

α x22 C

=⇒

a

(13.49)

a

which presents us with three possibilities [cf. part (ii) of Example 6.2]. (a)

If the parameter α is such that the condition



α

g(x, x )

2 = α x22  1

(13.50)

is fulfilled, then C can be computed from (13.49) as 1 C= 1 − α x22

b a

dx x h(x )

(13.51)

Integral equations in electromagnetics

877

and by inserting this value back into (13.46) we obtain the unique solution in closed form — if the remaining integral can be computed analytically, that is — as b αx dx x h(x ) 1 − α x22 a b   α dx xx h(x ) L {h(x)} = h(x) + a = h(x) + , 2 1 − α xx 2 1 − α x2

f (x) = h(x) +

x ∈ [a, b]

(13.52)

whence we gather (I − L)−1 {•} = I {•} +

(b)

L {•} 1 − α xx 2

(13.53)

and these formulas confirm that (13.46) can be solved even though the Neumann series (13.35) does not converge, because (13.50) is far more general than constraint (13.34). If (13.50) is violated because α x22 = 1, but the source term h(x) obeys the compatibility condition b 0=

dx x h(x )

(13.54)

a

then (13.46) admits infinitely many solutions, viz., f (x) = h(x) + αCx

(13.55)

inasmuch as (13.49) is identically satisfied by any value of C ∈ C. Further, we notice that b L {x} = α





b



dx g(x, x )x = α a

(c)

dx x|x |2 = xα x22 = x,

x ∈ [a, b]

(13.56)

a

i.e., the function x is an eigenvector of the operator L {•} with eigenvalue ν = 1 (Appendix D.7). If α x22 = 1 and h(x) does not satisfy (13.54), then (13.49) is incompatible, and (13.46) does not have a solution.

And yet, if the more stringent requirement (13.34) holds true, then we can show that (13.52) and (13.53) are perfectly equivalent to (13.35) and (13.39). To this purpose we notice that b L2 {h(x)} = α

  dx xx L h(x ) = α

a

b =α a

b a

dx |x |2 α

b

dx xx α

b

dx x x h(x )

a



dx xx h(x ) = α x22 L {h(x)} = α

xx

2 L {h(x)}

(13.57)

a

2 L3 {h(x)} = α2

xx

2 L {h(x)}

(13.58)

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Advanced Theoretical and Numerical Electromagnetics

and so forth. Therefore, by writing down the Taylor expansion of the second term in the rightmost member of (13.52) we find f (x) = h(x) +

+∞ 

+∞ +∞  

m αm

xx

2 L {h(x)} = L0 {h(x)} + Lm+1 {h(x)} = Ln {h(x)}

m=0

m=0

(13.59)

n=0

which is nothing but the Neumann series (13.35), as anticipated. (End of Example 13.1)

An integral operator L {•} : VL → VL is said to be positive definite if for all f (x) ∈ VL \ {0} it holds ( f (x), L { f (x)})VL > 0

(13.60)

where (•, •)VL is an inner product in the function space VL . It can be shown that integral equations involving positive definite operators admit a unique solution. Indeed, suppose that (13.9) is solved by f1 (x) and f2 (x) and define the difference f0 (x) := f1 (x) − f2 (x). Since the operator is linear, the function f0 (x) must satisfy the homogeneous equation b L { f0 (x)} :=

dx g(x, x ) f0 (x ) = 0

(13.61)

a

and by taking the inner product with f0 (x) we arrive at ( f0 (x), L { f0 (x)})VL

⎛ ⎞ b ⎜⎜⎜ ⎟⎟ ⎜⎜⎜ ⎟    ⎟ = ⎜⎜ f0 (x), dx g(x, x ) f0 (x )⎟⎟⎟⎟ ⎝ ⎠ a

=0

(13.62)

VL

which contradicts condition (13.60). Thus, we are led to conclude that f0 (x) = 0 necessarily, whence the uniqueness of solutions follows.

13.2 Surface integral equations for perfect conductors In this section we address the problem of determining the fields produced by time-harmonic sources in the presence of perfectly electric conducting (PEC) bodies immersed in a homogeneous unbounded medium. Although many integral equations have been proposed for this task, we shall consider in details only the three most commonly employed, namely, the electric-field integral equation (EFIE), the magnetic-field integral equation (MFIE), and the combined-field integral equation (CFIE) [25– 28], [29, Chapter 6]. The latter are all termed surface integral equations because, as we will show in the following, the unknown is distributed only over the surface of the objects. Lastly, we shall discuss a modified EFIE which can be used when the conducting bodies are characterized by a very high, though finite conductivity.

13.2.1 Electric-field integral equation (EFIE) The electric-field integral equation (EFIE) is so called because it is a relationship which involves the electric fields produced by primary and secondary sources. The primary sources generate the impressed or incident field Ei (r) which, by definition, constitutes the solution to the Maxwell

Integral equations in electromagnetics

(a)

879

(b)

Figure 13.1 Typical applications of the EFIE: (a) electromagnetic scattering from PEC bodies and (b) radiation from antennas comprised of PEC parts.

equations in the absence of the conducting bodies. The secondary sources, in this case, are the surface electric current densities JS (r) induced by the incident field on the surface of the PEC bodies. The physical motivation for the onset of such currents is that the impressed field by itself cannot satisfy the required boundary conditions (1.168)-(1.171). More importantly, the incident field is certainly not null within the region occupied by the bodies (see Section 1.6). Therefore, the induced current JS (r) takes care of generating a secondary or scattered field Es (r) charged with the tasks of (1) (2)

ensuring that the appropriate boundary conditions are met on the surfaces of the bodies and cancelling the incident field inside the PEC bodies.

Notice that JS (r) is, in effect, the main unknown of the problem. Parenthetically, the very separation of the fields in incident and scattered parts is afforded by the principle of superposition, which in turn is a consequence of the linearity of the Maxwell equations. The EFIE is well-suited for both scattering problems (Figure 13.1a) and radiation problems (Figure 13.1b). In the former, one is interested in determining the field reflected back by a PEC object when illuminated, as it were, by the incident field due to a distant source. Sometimes the sources are supposed to be located infinitely far away from the object, and the incident field is then assumed to be a uniform plane wave (Section 7.2). The applications are numerous, e.g., radar systems, remote sensing and imaging, to name a few. By contrast, in radiation problems the source is placed in close proximity of or even on the body whose shape, in turn, is designed so as to enhance or control the scattered (radiated) field, as is typical for antenna systems [30–34]. In order to derive the EFIE we focus on the scattering problem of Figure 13.1a so as to avoid, at least for the time being, the additional difficulty that is posed by the modelling of the source (generator) in the case of antenna problems, a topic which we postpone to Section 13.2.2. We have already argued that responsible for the scattered field Es (r) is the current JS (r), and hence we need to find a way of relating Es (r) to JS (r). This goal may be accomplished, for a PEC body immersed in a homogeneous unbounded medium, by invoking either the integral representation of Stratton and Chu (10.34) or the surface equivalence principle of Section 10.4.1. We follow the

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Advanced Theoretical and Numerical Electromagnetics

(a)

(b)

Figure 13.2 Derivation of the EFIE for a scattering problem: (a) the equivalence principle is invoked to establish an equivalent problem in the external region Vex ; (b) the internal region Vin is taken to be coincident with the region V occupied by the body. second strategy as it is more general and, to this purpose, we enclose the body within a larger region of space Vin with smooth boundary ∂Vin , as is suggested in Figure 13.2a. First of all, we introduce equivalent electric and magnetic surface current densities JS eq (r) and J MS eq (r), r ∈ ∂Vin , given by (10.79) and (10.80). We recall that in accordance with the equivalence principle, the combined effect of JS eq (r), J MS eq (r) and the true sources — located somewhere in the external region Vex := R3 \ V in — produces no fields at all within Vin and the ‘right’ field in Vex , that is, the one due to the true sources in the presence of the body. We should also keep in mind that both JS eq (r) and J MS eq (r) are not known at this stage since they are related to the very total field we wish to compute. Secondly, we let the surface ∂Vin conceptually shrink all the way down to the very boundary ∂V of the region V occupied by the PEC body, as is shown in Figure 13.2b. While this modification does not alter the characteristics of the fields generated by true and equivalent sources in Vin — which now coincides with V — we now find ourselves in a better position to make statements about JS eq (r) and J MS eq (r) for r ∈ ∂V. More specifically, since ∂V is the boundary of a PEC body, the time-harmonic counterpart of the jump conditions (1.168) and (1.169) apply, and on account of (10.79) and (10.80) we have ˆ = 0, J MS eq (r) := E(r) × n(r) ˆ × H(r) = JS (r), JS eq (r) := n(r)

r ∈ ∂V + r ∈ ∂V

+

(13.63) (13.64)

ˆ where E(r) and H(r) denote the total electric and magnetic field on ∂V, and the unit normal n(r) is directed positively outward V. Therefore, we see that choosing Vin ≡ V is quite convenient in that this geometrical setup allows us to get rid of the equivalent magnetic current density J MS eq (r) altogether. In the third step we take advantage of the freedom afforded by the equivalence principle in order to conceptually ‘remove’ the PEC body and ‘fill’ the region V with the same medium as the one existing in Vex , as was illustrated in Figure 10.11 for a general situation. The outcome for the problem of concern is shown in Figure 13.3 and it amounts to two sets of currents, namely, JS (r) and the true sources, which both exist in the same homogeneous unbounded medium. We know how to relate the fields to the currents at least formally with the aid of the electrodynamic potentials ΦE (r) and AE (r). The result is provided by (9.152) without the magnetic vector

Integral equations in electromagnetics

881

Figure 13.3 Derivation of the EFIE for a scattering problem: the body in V is ‘removed’ and replaced with the same medium existing in Vex . potential, whereas ΦE (r) and AE (r) are essentially given by (9.23) and (9.24). However, on the one hand our current density is a surface one, whereas on the other, the scalar potential entails the integration of a charge density. Then again, we observe that JS (r), as formal as it is, is related to the surface charge density ρS (r) by the surface continuity equation (10.26). Thus, the relevant potentials read e− j kR  1 ΦE (r) = − ∇ · JS (r ), dS  r ∈ R3 (13.65) j ωε 4πR s ∂V e− j kR JS (r ), r ∈ R3 (13.66) AE (r) = μ dS  4πR ∂V

according to the formulas that were proved in Section 9.3. By definition the total electric field reads E(r) := Ei (r) + Es (r) − j kR e− j kR  1 i  e  JS (r ) + ∇ dS  ∇ · JS (r ) = E (r) − j ωμ dS 4πR j ωε 4πR s ∂V

(13.67)

∂V

on account of (9.152), (13.65) and (13.66). Although this representation for E(r) is valid for r ∈ R3 — and returns E(r) = 0 for r ∈ V — (13.67) cannot be used to compute the field explicitly outside the region occupied by the body unless we find JS (r). There is, however, a convenient subset of points r for which we do have information on the electric field beforehand and irrespective of the value of JS (r). More specifically, since the matching condition (13.63) must be true for r ∈ ∂V + , from (13.67) we may extract the part of E(r) that is tangential to ∂V + and equate it to zero. Since the tangential component — which we denote with the symbol Et (r) — is perpendicular ˆ on ∂V, thanks to the algebraic identity (H.14) we have to the unit normal n(r) ˆ × [E(r) × n(r)] ˆ ˆ n(r) ˆ · E(r) = E(r) − En (r)n(r) ˆ = Et (r) n(r) = E(r) − n(r)

(13.68)

where En (r) is the component of E(r) normal to the boundary of the PEC object. The same decomposition applies to incident and scattered fields. In particular, to write Est (r) in terms of JS (r) we

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Advanced Theoretical and Numerical Electromagnetics

observe that since the surface current density is tangential to ∂V so is the vector potential (13.66). On the contrary, we need to extract the tangential part of ∇ΦE (r) for r ∈ ∂V + , namely, ˆ × [∇ΦE (r) × n(r)] ˆ ˆ n(r) ˆ · ∇ΦE (r) n(r) = ∇ΦE (r) − n(r) ∂ΦE ˆ = ∇s ΦE (r) n(r) = ∇ΦE (r) − ∂nˆ

(13.69)

where ∇s {•} indicates the tangential or surface gradient. The proof that ∇s ΦE (r) is continuous across ∂V is reduced to that for the single-layer static potential (Section 2.10, page 140 and ff.) because | exp(− j kR)/R| = 1/R. All in all, these observations lead to the condition Eit (r) − j ωμ ∂V

dS 

e− j kR ∇s JS (r ) + 4πR j ωε

∂V

dS 

e− j kR  ∇ · JS (r ) = 0, 4πR s

r ∈ ∂V +

(13.70)

which represents the sought EFIE [24, Section 12.1], [29, Chapter 6], [28, Section 2.7.1], [35, Section 6.4.4], [36], [25, 26]. According to the classification laid out in Section 13.1, (13.70) is in the form of a Fredholm equation of the first kind. Nonetheless, (13.70) is sometimes referred to as an integrodifferential equation in that it also involves ∇s · JS (r ) and the surface gradient of the scalar potential ΦE (r). Implied in the very definition of the scattered field is the requirement that JS (r) be at least of class C1 (∂V)3 for we need to compute the surface divergence and, more importantly, there is no physical ground for a discontinuous surface charge density ρS (r) ∝ ∇s ·JS (r) so long as the boundary of the PEC body is smooth. Actually, the latter specification may be relaxed a bit, and the EFIE can also be applied to bodies with piecewise-smooth boundaries inasmuch as the Gauss theorem — upon which the reciprocity theorem, the equivalence principle, and the integral representations are based — holds true under such more general circumstances. We shall come back on the topic of the regularity of JS (r) in Section 14.2 while discussing the numerical solution of (13.70) with the Method of Moments. The EFIE is quite popular because it may be formulated and solved even in the special case where the PEC body (or just a part thereof) reduces to an open conducting surface in the sense of Figure 1.2a. To elaborate, we start with a PEC object with smooth boundary ∂V = S which we think of as being divided into two adjoining and non-overlapping parts S + and S − . This idea is illustrated in Figure 13.4a for an object with specular symmetry around a plane whose trace is shown with a dotdashed line. We denote with J+S (r) and J−S (r) the parts of JS (r) that flow on S + and S − , respectively. We remark that in this configuration J+S (r) = J−S (r) for points r ∈ ∂S + ≡ ∂S − . Next, we examine what happens to the currents as the body is conceptually made to shrink around the symmetry plane. As is suggested in Figure 13.4b, the shape of the object becomes ever more elongated with the two surfaces S + and S − being deformed and brought closer to one another. Nothing much happens to the currents away from the separation line ∂S + ≡ ∂S − . By contrast, we expect that the transition of J+S (r) into J−S (r) and vice-versa through ∂S + ≡ ∂S − may grow more and more hampered owing to the sharp curvature of S at ∂S + ≡ ∂S − . In the limiting situation where the body is flattened out and S ≡ S + ≡ S − we end up with two separate currents J+S (r) and J−S (r) which flow on either side of a conducting surface (Figure 13.4c). Since the latter is immersed in a lossless dielectric medium by hypothesis, there cannot be any conduction current for r ∈ R3 \ S . Therefore, the appropriate boundary condition for the currents for points r ∈ ∂S is (1.195) [37] νˆ (r) · J+S (r) = 0 = νˆ (r) · J−S (r)

(13.71)

Integral equations in electromagnetics

(a)

(b)

883

(c)

Figure 13.4 Application of the EFIE to infinitely thin PEC bodies: (a) the induced current JS on a PEC body is separated into two parts; (b) the body is conceptually made to shrink; (c) in the limit the supports of the two currents coincide. ˆ ˆ with νˆ (r) := sˆ(r) × n(r) being the unit normal to ∂S perpendicular to both n(r) and the unit vector sˆ(r) tangent to ∂S . Having ascertained that for the infinitely thin PEC surface of Figure 13.4c the equivalent current JS (r) splits up into J+S (r) and J−S (r), we also realize that the EFIE (13.70) separates into two equations stated on either side of S , namely, 0=

Eit (r)

− j ωμ S+

∇s e− j kR +  e− j kR  +  JS (r ) + ∇ · J (r ) dS dS  4πR j ωε 4πR s S S+ e− j kR −  e− j kR  −  ∇s JS (r ) + ∇ · J (r ), − j ωμ dS  dS  4πR j ωε 4πR s S 

S−

r ∈ S+

(13.72)

r ∈ S−

(13.73)

S−

and 0=

Eit (r)

− j ωμ S+

e− j kR +  e− j kR  +  ∇s JS (r ) + ∇ · J (r ) dS dS  4πR j ωε 4πR s S + S − j kR e− j kR  −  ∇s  e −  JS (r ) + ∇ · J (r ), − j ωμ dS dS  4πR j ωε 4πR s S 

S−

S−

where we have highlighted the contributions of J+S (r) and J−S (r). The structure of (13.72) embodies the physical fact that the secondary field produced by J−S is non-zero on the opposite side S + , whereas the dual observation holds true for the other equation. Nevertheless, can we actually solve both (13.72) and (13.73)? A moment’s thought allows concluding that in reality (13.72) and (13.73) are one and the same equation because S + ≡ S − . As a result, (13.72) and (13.73) do not constitute a system of integral equations and cannot be solved for J+S (r) and J−S (r) individually. In other words there is too little information to obtain the currents on the two sides of S . Then again, since integrating over S + or S −

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is equivalent, we can write (13.72) in a more compact fashion as Eit (r)

− j ωμ S

dS 

! e− j kR +  J (r ) + J−S (r ) 4πR S ! ∇s e− j kR  + ∇ · J+S (r ) + J−S (r ) = 0, dS  j ωε 4πR s

r∈S

(13.74)

S

where now S := S + ≡ S − is an orientable open surface. In summary, the EFIE for infinitely thin conductors may be stated and solved for an induced current density defined as JS (r) := J+S (r) + J−S (r),

r∈S

(13.75)

which physically represents the sum of the two currents that flow on either side of S (Figure 13.4c). This approach is possible essentially because the boundary condition (1.169) takes on the same form on S + and S − , and the tangential electric field is thus trivially continuous across S . Once the electric current density JS (r) has been computed by solving either (13.70) or (13.74), the total field E(r) may be obtained in any point of interest by means of (13.67). In particular, the scattered electric field in the Fraunhofer region of the object is given by (9.325).

13.2.2 EFIE with delta-gap excitation We now turn our attention to the radiation problem sketched in Figure 13.1b and, more precisely, to the derivation of the relevant electric-field integral equation in the time-harmonic regime. While the body of the antenna is made of PEC for the most part, so that the matching conditions (1.168)-(1.171) apply, the real difficulty here is to come up with a physically sound model of the source region — where the ‘true’ generator resides — and consequently with an appropriate mathematical description of the primary field Ei (r) produced by such source. As a matter of fact, we can reasonably expect that the EFIE we seek will look like (13.70) — even though that one was specifically developed for the electromagnetic scattering of waves from PEC bodies — but the very form of Ei (r) is far from obvious and depends on the hypotheses we make concerning the generator. First off, we recall that in antenna problems the ultimate electromagnetic source may be a highfrequency oscillator [38] connected to the device by means of a classical transmission line, e.g., a coaxial cable [39]. The typical setup can be appreciated in the left part of Figure 6.20 where we get a glimpse of the cable which taps into the mid-section of a folded-dipole antenna. Conversely, the cartoon of Figure 9.13 shows the idealized link between a dipole antenna and its generator by means of twin-lead transmission line. The comparatively small region where the cable or the line is joined to the antenna is called the port. In fact, we already made use of this concept in Example 6.8 where we proved the equivalence of the input admittances of two antennas and, in particular, introduced the concept of antenna gap or delta gap as a model of the port [24, Section 7.13], [32, Section 20.2], [30, 31]. But how, exactly, do we go from the realistic configuration of Figure 9.13 to the antenna gap drawn in Figure 6.27. And, more importantly, how does this help us write the EFIE for the radiation problem of Figure 13.1b? We wish to address all of these questions in the following. We suppose that the antenna — immersed in free space — is connected to the oscillator with a twin-lead transmission line as in Figure 13.5a. Since the purpose of the line is to deliver electromagnetic energy from the oscillator to the antenna rather than to radiate power towards infinity, we stipulate that the electromagnetic field is substantially non-null only between the two wires and very close to them, whereas it rapidly decays away from the wires. Moreover, the electromagnetic wave

Integral equations in electromagnetics

(a)

885

(b)

Figure 13.5 Modelling an antenna port with the delta-gap source: (a) port connected to a twin-lead line; (b) equivalent model of antenna and port thereof.

guided along the line is typically a transverse-electric-magnetic (TEM) mode, which means that the electric field is perpendicular to the wires, and the magnetic field streamlines describe closed loops around the wires [39] (cf. Examples 6.4 and 11.1). As a result, in the neighborhood of the antenna port, the electromagnetic field guided along the line is still essentially non-zero only in the region of space between the wires and, by construction, just between the two PEC bodies of which the antenna is comprised. Furthermore, the electric field in the port is perpendicular to the two disjoint PEC bodies which form the antenna. For all these reasons, we are prompted to replace the configuration of Figure 13.5a with the one of Figure 13.5b where we assume that the electric field Ei (r) is known in the port and somehow related to the potential difference existing between the wires of the line. The port model of Figure 13.5b is suggestively called the delta gap of the antenna because it really is a small empty region or discontinuity between two PEC bodies. What can we say about the effect of the twin-lead transmission line on the field radiated by the antenna? Well, as already mentioned above, the line does not radiate in principle, and hence for the sole purpose of computing the radiated field we may safely overlook the actual presence of the wires (cf. Figure 9.14). By contrast, radiation can indeed occur precisely at the antenna port, where the wires connect to the antenna. This happens because, even though the transition between the line and the antenna were smooth, still the electromagnetic field of the line should change and adapt itself so as to meet the mutated boundary conditions in the port. Simply put, the TEM mode alone does not obey (1.169) in the port region, and the final, correct field may be described by including, on the line side, higher-order guided modes and, on the antenna side, an induced current JS (r) (see Figure 13.5) responsible for producing a secondary electric field Es (r). Both the higher-order modes and Es (r) help fulfill (1.169) on whatever PEC surface that condition must hold true. Nonetheless, in Example 6.8 we already had the opportunity to mention that if the gap is ‘small’, then the electromagnetic field is quasi-static in the port region (Section 9.8). For this hypothesis to hold it is necessary that the size of the gap be small compared to the wavelength in the background medium and the characteristic size of the antenna. As a matter of fact, especially in combination with the Method of Moments (Section 14.10) it will be possible and convenient to think of the gap as

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Figure 13.6 Close-up of the antenna gap of Figure 13.5b and related geometrical and physical quantities. being infinitely short and hence degenerated into a line (e.g., see Figure 13.22a further on). The main advantages of the quasi-static assumption are that we can derive the electric field in the gap from a scalar potential according to (2.15) and the associated magnetic field obeys the Ampère law (4.5). With reference to Figure 13.5b we indicate with V1 and V2 the two disjoint domains filled with PEC, and with WG the antenna gap which, for the sake of argument, we suppose to be a small right cylinder, though the cross section need not be circular. We call V := V1 ∪ V1 ∪ WG the region comprised of V1 , V2 and the gap, and let h be the height of WG . By construction the intersections ∂WG ∩∂Vl , l ∈ {1, 2}, are non-empty and represent the bases of the cylinder WG . We consider the line integral of Ei (r) along a path γ21 which connects any two points P1 ∈ ∂WG ∩∂V1 and P2 ∈ ∂WG ∩∂V2 , as is shown in Figure 13.6. Since we have argued above that Ei (r) is curl-free in the gap and each one the surfaces ∂WG ∩∂Vl is equipotential, we may conveniently choose γ21 as a straight segment perpendicular to the bases of the cylinder WG . Then, we have

h ds sˆ (r) · Ei (r) = −

γ21

dη vˆ · ∇Φgap (η) = Φ(r2 ) − Φ(r1 )

(13.76)

0

where η ∈ [0, h] is a local coordinate along γ21 , vˆ is the unit vector tangent to γ21 and positively oriented toward P1 , rl is the position vector of Pl , and finally Φ(rl ) is the quasi-static potential on ∂WG ∩ ∂Vl . We may let VG := Φ(r1 ) − Φ(r2 )

(13.77)

and interpret the potential difference between P1 and P2 as if it were due to an ideal voltage generator which has strength VG and the positive terminal connected to ∂WG ∩ ∂V1 . Consequently, (13.76) and (13.77) imply that the electric field throughout the gap is uniform and can be expressed as Ei (r) = −

VG vˆ , h

r ∈ WG

(13.78)

since edge effects are negligible, so long as WG is small. This also means that the boundary condition (1.169) developed for PEC bodies applies only on ∂Vl ∩ ∂V but not on ∂WG ∩ ∂V, i.e., the lateral surface of the cylindrical gap, because Ei (r) is tangential to ∂WG ∩ ∂V according to (13.78). All in

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887

all, the delta-gap model demands we revise the time-harmonic counterpart of (1.169) for the device of Figure 13.5b as follows ⎧ ⎪ 0, r ∈ ∂Vl ∩ ∂V ⎪ ⎪ ⎨ ˆ × [E(r) × n(r)] ˆ n(r) = Et (r) = ⎪ (13.79) V ⎪ ⎪ ⎩− G vˆ , r ∈ ∂WG ∩ ∂V h which represents the matching condition on ∂V. Thus, in order to obtain the EFIE for an antenna problem we essentially have to follow the steps outlined in Section 13.2.1 for scattering problems while enforcing the jump condition (13.79) for r ∈ ∂V + (cf. Figure 13.3). In this regard, we must realize that the electric field for points r ∈ R3 \ V is generated only by the equivalent surface currents JS eq (r) and J MS eq (r) which are set against ∂V + , because no other sources are contemplated in addition to the voltage generator in the gap. While JS eq (r), r ∈ ∂V + , is still given by (13.64), the magnetic counterpart J MS eq (r) does not vanish everywhere on ∂V + , as is stated by (13.63). More precisely, on account of (13.79) we have ⎧ ⎪ 0, r ∈ ∂Vl ∩ ∂V ⎪ ⎪ ⎨ ˆ =⎪ J MS eq (r) := E(r) × n(r) (13.80) V ⎪ ⎪ ⎩− G u(r), ˆ r ∈ ∂WG ∩ ∂V h where ˆ := vˆ × n(r) ˆ u(r)

r ∈ ∂WG ∩ ∂V

(13.81)

denotes the unit vector tangential to the lateral surface of the gap and perpendicular to vˆ (Figure 13.6). Still, we have to conclude that J MS eq (r) does not contribute to the field for r ∈ R3 \ V. Indeed, on the one hand such contribution should be negligible owing to the supposedly small size of the support ∂WG ∩ ∂V. On the other hand, J MS eq (r), if present at all, is quasi-static and does not radiate. Evidently, this assumption is in line with the delta-gap model of the port inasmuch as we have traded the field produced by a truncated transmission line (Figure 13.5a) for the static field Ei (r) which is non-null only within WG (Figure 13.5b). By keeping all of these observations into account we see that the electric field produced by JS (r) := JS eq (r) reads e− j k 0 R e− j k 0 R  ∇ JS (r ) + ∇ · JS (r ) dS  (13.82) E(r) := Es (r) = − j ωμ0 dS  4πR j ωε0 4πR s ∂V

∂V

thanks to (9.152), (13.65) and (13.66). When inserted into (13.79), (13.82) leads to − j ωμ0 ∂V

dS 

e− j k 0 R JS (r ) 4πR ∇s + j ωε0

∂V

⎧ ⎪ 0, − j k0 R ⎪ ⎪ e ⎨ ∇s · JS (r ) = ⎪ dS  V ⎪ ⎪ 4πR ⎩− G vˆ , h

r ∈ ∂Vl ∩ ∂V r ∈ ∂WG ∩ ∂V

(13.83)

which is the sought EFIE for the problem of Figure 13.5. The minus sign in the right member results from the way we decided to connect the poles of the voltage generator to the PEC bodies in V1 and V2 . Had we made the opposite choice, the minus sign would have been absent. This extra degree of freedom is not critical at all and ultimately arises from the linearity of the Maxwell equations.

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Advanced Theoretical and Numerical Electromagnetics

The EFIE (13.83) also works when the antenna is comprised (at least partly) of infinitely thin PEC conductors inasmuch as the matching condition (13.79) remains valid. In particular, the threedimensional region WG , which constitutes the antenna gap, degenerates into a small open surface, e.g., a rectangle. The limiting case as h → 0+ , which describes a vanishingly short antenna gap, is best treated in the realm of distributions, because we expect to have

h ds sˆ(r) · E (r) = − lim+ i

lim

h→0+

dη

h→0

γ21

VG = −VG h

(13.84)

0

which is the telltale of the one-dimensional Dirac delta distribution VG δ(η) (see Appendix C). We do not attempt to write (13.83) in distributional form but rather we observe that it is easier to take the limit as h → 0+ after the Method of Moments has been applied for the numerical calculation of the unknown JS (r) (Section 14.10). In antenna problems we are interested in knowing the field produced by the device (usually in the Fraunhofer region) but also the input impedance ZA (ω) which is ‘seen’ at the end of the transmission line (Figure 13.5a). Such quantity is essential for the proper design of the feeding network, which has to make sure the generator is suitably ‘matched’ to the antenna [38, 40]. To compute the antenna impedance we observe that the circuit equivalent to the configuration shown in Figures 13.5b and 13.6 is comprised of an ideal voltage generator of strength VG connected to a single-port lumped element whose impedance is precisely ZA (ω). This arrangement is sketched in Figure 13.7a. We indicate with IA the time-harmonic current that conceptually leaves the generator and enters the lumped element, that is to say, the antenna. By applying the very definition of impedance we have ZA :=

VG IA

(13.85)

but, more precisely, the EFIE (13.83) allows us to first determine the inverse of ZA , namely, the input admittance given by YA :=

IA VG

(13.86)

in that VG is the known forcing term in accordance with (13.78) and (13.79) whereas IA constitutes the response of the antenna to the stimulus VG . Therefore, now the problem of finding YA (ω) has been turned into the calculation of the current IA . To this purpose we recall that the fields in the gap region WG and in the neighborhood thereof are quasi-static. Since the current IA is actually related to the part of JS (r) which flows on the lateral surface of WG (Figure 13.6), we may invoke the global Ampère law (4.5) to obtain " ds sˆ(r) · H(r) (13.87) IA = ∂S

where ∂S is the boundary of an arbitrary open surface S which cuts through the gap, i.e., WG ∩ S must be non-empty. Then again, the EFIE (13.83) does not provide us with the magnetic field around the gap, let alone in any other point r ∈ R3 \ V. As a consequence, we are better off if we choose S coinciding with a cross section of WG so that ∂S becomes the line γG ⊂ (∂WG ∩∂V), as is suggested in

Integral equations in electromagnetics

(a)

889

(b)

Figure 13.7 Network equivalent of an antenna port: (a) ideal voltage generator and (b) voltage generator with non-zero internal impedance ZG . Figure 13.6. Indeed, thanks to the boundary condition (13.64) we do know the tangential component of the magnetic field on γG so long as we solve the EFIE. Hence, we may cast (13.87) as "

L

IA =

ds sˆ(r) · H(r) = γG

ˆ · H(r) = dξ u(r) 0

ˆ × H(r) dξ vˆ · n(r) 0

"

L dξ vˆ · JS (r) =

=

L

ds vˆ · JS (r)

(13.88)

γG

0

ˆ given by (13.81) is the unit vector tangent where ξ ∈ [0, L] is a local coordinate along γG , and u(r) to γG . The integral in the rightmost-hand side may be interpreted as the flux of the surface vector field JS (r) through the line γG . Besides, since JS (r) is nearly stationary for r ∈ ∂WG ∩ ∂V (as the quasi-static regime holds in the gap) IA does not depend on η ∈ [0, h], the local coordinate along γ12 . This statement can be verified by assuming ∇s · JS (r) = 0 and applying the surface Gauss theorem (A.59) over ∂WG ∩ ∂V. As a result, we can write (13.88) further as 1 IA = h

h

L dξ vˆ · JS (r) =

dη 0

0

1 VG

dS JS (r) · ∂WG ∩∂V

VG vˆ h

which in light of the jump condition (13.79) and the EFIE (13.83) becomes ⎡ ⎤ ⎢⎢⎢ ⎥⎥⎥ − j k0 R − j k0 R e e 1 ∇ ⎢⎢⎢ ⎥ s IA = JS (r ) − ∇s · JS (r )⎥⎥⎥⎥ dS JS (r)· ⎢⎢j ωμ0 dS  dS  ⎣ ⎦ VG 4πR j ωε0 4πR ∂V ∂V ∂V e− j k 0 R j ωμ0 JS (r ) dS JS (r) · dS  = VG 4πR ∂V ∂V e− j k 0 R  1 ∇ · JS (r ) dS ∇s · JS (r) dS  + j ωε0 VG 4πR s ∂V

∂V

(13.89)

(13.90)

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Advanced Theoretical and Numerical Electromagnetics

having used formula (H.77) and applied the surface Gauss theorem (A.59) over ∂V. If the latter is a closed surface, then no contribution comes from the boundary of ∂V. If the antenna surface is open (e.g., it comprises infinitely thin perfectly conducting parts as in Figure 13.4c) then the timeharmonic counterpart of the jump condition (1.195) holds, and the line integral along the boundary of ∂V is null once again. All in all, by inserting the expressions found for IA back into (13.86) we can write the antenna admittance as " 1 YA = ds vˆ · JS (r) (13.91) VG γG

or j ωμ0 YA = VG2



dS JS (r) · ∂V

∂V

dS 

e− j k 0 R JS (r ) 4πR 1 + j ωε0 VG2



dS ∇s · JS (r) ∂V

dS 

∂V

e− j k 0 R  ∇ · JS (r ) 4πR s

(13.92)

formulas which make sense so long as we have solved the EFIE and obtained the current JS (r) on ∂V (see Section 14.10). As expected of a linear system, YA does not depend on the particular value assigned to VG , since the formal inversion of (13.83) shows that JS (r) is, in fact, proportional to VG . The approach described and the related formulas should be compared to Harrington’s procedure [41, Section 7.9], where the antenna is excited by a current filament in the gap and hence the impedance is computed first. The delta-gap model of an antenna port also allows contemplating the case of a voltage generator with non-zero internal impedance ZG , as is suggested in Figure 13.7b. The combination of VG and ZG may be the Thevenin equivalent lumped circuit of the true generator plus the transmission line (Figure 13.5a) plus other loads connected to the antenna port. Since the potential difference between the two PEC bodies filling V1 and V2 now reads Φ(r1 ) − Φ(r2 ) = VG − ZG IA = VA

(13.93)

the matching condition (13.79) based on (13.78) must be modified accordingly, viz., ⎧ ⎪ 0, r ∈ ∂Vl ∩ ∂V ⎪ ⎪ ⎨ ˆ × [E(r) × n(r)] ˆ n(r) = Et (r) = ⎪ I − V Z G ⎪ ⎪ ⎩ G A vˆ , r ∈ ∂WG ∩ ∂V h

(13.94)

where IA is actually unknown and related to JS (r) through (13.88). Therefore, the EFIE (13.83) passes over into − j ωμ0 ∂V

∇s e− j k 0 R dS JS (r ) + 4πR j ωε0 

∂V

dS 

e− j k 0 R  ∇ · JS (r ) 4πR s

⎧ 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ZG " VG =⎪ ⎪ vˆ vˆ , ds vˆ · JS (r) − ⎪ ⎪ ⎪ h ⎩ h γG

r ∈ ∂Vl ∩ ∂V r ∈ ∂WG ∩ ∂V

(13.95)

Integral equations in electromagnetics

891

which is useful when one is concerned with the effect of ZG on the radiated field, because the presence of ZG is bound to modify the current distribution JS (r) a bit. However, for the sole purpose of computing the input admittance, (13.83) may be used. What is more, since the problem is linear, the effect of ZG on the average power absorbed by the antenna is conveniently investigated by turning to the network equivalent of Figure 13.7b after ZA has been found. In this regard, intuition strongly suggests that, if the equivalent circuit of Figure 13.7a is to be trusted, we can compute the average power (1.303) radiated by the antenna alternatively as PF  =

1 Re{VG IA∗ } 2

(13.96)

under the additional hypothesis that the background medium is lossless. In order to confirm this idea we resort to the complex Poynting theorem applied to the region of space B(0, a) \ V, where the radius a is large enough for the ball B(0, a) to enclose V completely (Figure 13.8). By noticing that there are no sources in B(0, a) \ V and that the conductivity of the background material is null by assumption, (1.314) becomes ˆ · S(r) = Re dS rˆ · S(r) − Re dS n(r) ˆ · S(r) dS n(r) (13.97) 0 = Re ∂B∪∂V

∂B

∂V

where we have taken the unit normal on ∂B positively oriented towards infinity and the unit normal on ∂V pointing outwards V. In particular, the flux of the Poynting vector through ∂V may be transformed as follows 1 1 ˆ · S(r) = ˆ · E(r) × H∗ (r) = − dS n(r) dS n(r) dS Et (r) · J∗S (r) 2 2 ∂V ∂V ∂V 1 VG 1 = dS vˆ · J∗S (r) = VG IA∗ (13.98) 2 h 2 ∂WG ∩∂V

on account of (13.64), (13.79) and (13.89). Since the integral over ∂B represents the average radiated power that leaves the region B(0, a), (13.97) and (13.98) lead us to 1 (13.99) PF  = Re dS rˆ · S(r) = Re{VG IA∗ } 2 ∂B

which is in agreement with our presumption (13.96). When the antenna is excited by a generator with internal impedance ZG , the voltage drop across ZA and the current IA read (Figure 13.7b) VA =

ZA VG , ZA + ZG

IA =

VG ZA + ZG

(13.100)

whereby (13.96) becomes PF  =

1 1 Re{ZA } Re{VA IA∗ } = |VG |2 2 2 |ZA + ZG |2

(13.101)

and this calculation does not require we solve the EFIE all over again, in that VA and IA are computed through the network analogue of the antenna.

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Figure 13.8 Calculation of the average power radiated by an antenna with the aid of the complex Poynting theorem (1.314).

13.2.3 Magnetic-field integral equation (MFIE) As the name suggests, the magnetic-field integral equation (MFIE) or Maue equation (1949) [42] is a relationship which involves the magnetic fields produced by primary and secondary sources. The primary sources generate the incident field Hi (r) which, by definition, constitutes the solution to the Maxwell equations in the absence of the conducting bodies. Also in this case the secondary sources are the electric current densities JS (r) induced by the incident field on the boundary of the PEC objects. The secondary currents take care of producing a scattered magnetic field Hs (r) which in combination with the incident field Hi (r) (1) (2)

yields a zero total field within the region V occupied by the bodies and makes sure that the matching conditions (1.168), (1.169), (1.170) and (1.171) are satisfied.

Perhaps it is not superfluous to mention that the MFIE is not, as one might erroneously guess, the dual to the EFIE obtained by applying the transformation rules of Table 6.1 to (13.70). Similarly to EFIE (13.70) the MFIE is well-suited for scattering problems (Figure 13.9a) in which one determines the magnetic field scattered by a PEC body exposed to the incident magnetic field generated by a distant source. In this regard, a uniform plane wave may be employed as well (Section 7.2). The analysis of radiation problems with the MFIE is not as widespread as it is with the EFIE perhaps owing to the difficulty of characterizing the antenna generator by means of its magnetic field (see Figure 13.1b). Said difficulty can be traced back to the fact that the electromagnetic field in the neighborhood of a source is mainly electrostatic in nature rather than magnetostatic (cf. Example 9.3) whereas the MFIE actually requires assigning the incident magnetic field. Furthermore, unlike the EFIE, the MFIE is not appropriate for infinitely thin PEC bodies (Figure 13.9b), that is, open surfaces characterized by the PEC boundary condition (1.168). We shall elaborate on the reasons for this limitation after we have derived the MFIE with the aid of the equivalence principle of Section 10.4.1. We examine a scattering problem involving a PEC body with smooth boundary ∂V. Since responsible for the secondary magnetic field Hs (r) is the surface current JS (r) induced on the boundary of the object, we need to find a way of expressing Hs (r) in terms of JS (r).

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(b)

Figure 13.9 Typical application and inappropriate usage of the MFIE: (a) electromagnetic scattering from closed surfaces with PEC boundary conditions and (b) electromagnetic scattering from infinitely thin PEC bodies. To begin with, we enclose the body within a larger region of space Vin ⊃ V with smooth boundary ∂Vin (Figure 13.10a) and we define equivalent electric and magnetic surface current densities JS eq (r) and J MS eq (r), r ∈ ∂Vin . The latter produce an electromagnetic field which combines with Hi (r) to give the right solution to the Maxwell equations in the presence of the body in Vex := R3 \Vin and a null field in Vin . Secondly, we let the surface ∂Vin conceptually shrink and deform until it tightly wraps the object, in which instance Vin ≡ V, as is suggested in Figure 13.10b. Since the body is a PEC by hypothesis, the matching conditions (1.169) and (1.168) must hold, and by virtue of (10.79) and (10.80) we have r ∈ ∂V +

ˆ = 0, J MS eq (r) := E(r) × n(r) ˆ × H(r) = JS (r), JS eq (r) := n(r)

r ∈ ∂V

+

(13.102) (13.103)

ˆ is where E(r) and H(r) denote the total electric and magnetic field on ∂V, and the unit normal n(r) directed positively outward V. As was the case for EFIE, thanks to the judicious choice of Vin we are left with just the surface electric current JS (r) on ∂V. Next, we recall that the ‘right’ field in the external region Vex is not affected by whatever modification we make on the properties of the medium within the excluded region V. As often is the case, the most convenient setup consists of ‘removing’ the PEC object and replacing it with a medium endowed with the same constitutive parameters as the medium existing in Vex . This arrangement is illustrated in Figure 13.11 and amounts to two sets of sources, i.e., the original ones together with JS (r), which radiate in a unbounded homogeneous isotropic medium. From Section 9.4 we know how to relate the magnetic field to the sources through the electrodynamic potentials. In particular, the representation of interest is (9.153) without the contribution of A M (r) and Φ M (r) because no magnetic sources are present in this problem. The magnetic vector potential due to JS (r) is still given by (13.66). Hence, by definition the total magnetic field reads H(r) := Hi (r) + Hs (r) = Hi (r) + ∇ × ∂V

dS 

e− j kR JS (r ) 4πR

(13.104)

Advanced Theoretical and Numerical Electromagnetics

894

(a)

(b)

Figure 13.10 Derivation of the MFIE for a scattering problem: (a) the equivalence principle is invoked to establish an equivalent problem in the external region Vex ; (b) the internal region Vin is taken to be coincident with the region V occupied by the body.

Figure 13.11 Derivation of the MFIE for a scattering problem: the body in V is ‘removed’ and replaced with the same medium existing in Vex . on account of (9.153) and (13.66). This expression is valid for r ∈ R3 — and returns H(r) = 0 for r ∈ V — but cannot be employed to find the magnetic field explicitly outside the region occupied by the object until we determine JS (r). Then again, we do have information on the current density and the magnetic field for points r ∈ ∂V + inasmuch as the matching condition (1.168) must be satisfied. ˆ × H(r), where n(r) ˆ Therefore, we need to examine the behavior of n(r) is the unit normal on ∂V positively oriented towards Vex and H(r) is given by (13.104), in the limit as the observation point r approaches the boundary ∂V. For ease of manipulation and to highlight the dependence on source and observation points we indicate the time-harmonic scalar Green function in (13.104) with G(r, r ) and also notice that G(r, r ) obeys the symmetry condition ∇G(r, r ) = −∇ G(r, r ), as can be proved by direct calculation in Cartesian coordinates. Then, we pick a point ˆ 0 ) ∈ Ha \ ∂V, r(w) := r0 + wn(r

0 < |w| < a

(13.105)

where r0 ∈ ∂V, and Ha is an open neighbourhood of ∂V [see (F.15) and Figure F.2]. With this definition the vector r(w) − r0 is perpendicular to ∂V at r0 , and r(w) tends to r0 as w → 0. Since the

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incident magnetic field is regular on ∂V by assumption, we focus on the scattered field at r(w) and write ⎡ ⎤ ⎢⎢⎢ ⎥⎥⎥ ⎢⎢⎢  s     ⎥ ˆ 0 ) × H (r) = n(r ˆ 0 ) × ⎢⎢∇ × dS G(r, r )JS (r )⎥⎥⎥⎥ n(r ⎣ ⎦ ∂V ˆ 0 ) × [∇G(r, r ) × JS (r )] dS  n(r = ∂V



= ∂V



= ∂V

ˆ 0 ) · ∇G(r, r ) − dS  JS (r )n(r



ˆ 0 ) · JS (r )∇G(r, r ) dS  n(r

∂V

# $ ∂G ˆ  ) − n(r ˆ 0 ) · ∇ G(r, r ) dS JS (r )  − dS  JS (r ) n(r ∂nˆ ∂V $ # ˆ  ) − n(r ˆ 0 ) · JS (r )∇G(r, r ) + dS  n(r 



(13.106)

∂V

where in the first step we have interchanged the order of integration and differentiation because the integrand is regular for r(w) ∈ Ha \ ∂V. Next, we have used the vector identity (H.14) and noticed ˆ  ) · JS (r ) = 0. that n(r The Cartesian components of the first integral in the rightmost-hand side of (13.106) have the form of the double-layer potential (9.14). Moreover, the singularity of the time-harmonic Green function is the same as that of the static counterpart (2.131). Therefore, it is relatively straightforward to extend the proof of Section 2.9 and in particular result (2.244), namely, % & ∂G ∂ 1 lim± dS  JS (r )  = lim± dS  JS (r )  G(r, r ) − w→0 w→0 ∂nˆ ∂nˆ 4π|r − r | ∂V ∂V ∂ 1 + lim± dS  JS (r )  w→0 ∂nˆ 4π|r − r | ∂V % & ∂ 1 = dS  JS (r )  G(r0 , r ) − ∂nˆ 4π|r0 − r | '()* ∂V regular

1 1 ∂ ± JS (r0 ) + PV dS  JS (r )   2 ∂nˆ 4π|r0 − r |

(13.107)

∂V

where the function within square brackets is regular because the singularity of G(r0 , r ) is cancelled by one of the same kind exhibited by the static Green function. The surface integral of the last singular contribution remains finite in the limit as r → r0 by virtue of estimate (F.1) which applies since ∂V is smooth [cf. (2.256) and (2.257)]. With some tedious but otherwise simple algebraic manipulations the remaining two integrals in the rightmost member of (13.106) can be transformed into combinations of potential integrals of the type defined in (G.1). Such terms are examined in Appendix G where, by means of lengthy derivations, it is shown that they are Hölder continuous in a neighbourhood of ∂V and remain finite ˆ  ) − n(r ˆ 0 ) for for r(w) → r0 . In a nutshell, this happens because the vanishing of the surface field n(r      r → r0 mitigates the strong singularity of ∇ G(r, r ) for r(w) → r0 .

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ˆ 0 ) × Hs (r) has a continuous extension to either side of ∂V. All in all this analysis proves that n(r  In light of the arbitrariness of r0 we may rename that point to r ∈ ∂V and finally write 1 ˆ × Hs (r) = ± JS (r) + PV dS  n(r) ˆ × [∇G(r, r ) × JS (r )], n(r) r ∈ ∂V ± (13.108) 2 ∂V

having used the manipulations outlined in (13.106) and (13.107) backwards. From (13.108) we gather that the rotated tangential component of Hs (r) suffers a jump across ∂V, namely, ˆ × Hs (r) − lim− n(r) ˆ × Hs (r) = JS (r) lim n(r)

w→0+

(13.109)

w→0

and this is in agreement with (13.103) inasmuch as the incident magnetic field is continuous through the boundary ∂V by hypothesis. Thanks to (13.108) we may now write down the matching condition (13.103) explicitly as % − j kR & e 1 ˆ × ∇ ˆ × Hi (r) + JS (r) + PV dS  n(r) × JS (r ) , r ∈ ∂V + (13.110) JS (r) = n(r) 2 4πR ∂V

and eventually as % − j kR & 1 e ˆ × ∇ ˆ × Hi (r) + PV dS  n(r) JS (r) = n(r) × JS (r ) , 2 4πR

r ∈ ∂V +

(13.111)

∂V

which is the sought MFIE [29, Chapter 6], [24, Section 12.2], [28, Section 2.7.2], [35, Section 6.4.4], [36]. We have to choose the plus sign in (13.108) in that (13.103) holds on the positive side of the surface ∂V. Interestingly, if on the negative side of ∂V we enforce the boundary condition ˆ × [Hi (r) + Hs (r)] = 0, n(r)

r ∈ ∂V −

(13.112)

which is motivated by the vanishing of H(r) in r ∈ V (Figure 13.11) we arrive again at (13.111) on account of (13.108) used with the negative sign of JS (r)/2. According to the classification of Section 13.1, (13.111) is a Fredholm equation of the second kind. Since the MFIE involves JS (r) but not the derivatives thereof, the surface current density need not be differentiable, a feature which makes the numerical solution with the Method of Moments relatively easier (Section 14.3). We now have the tools to understand why the MFIE runs into trouble if applied to infinitely thin PEC conductors (Figure 13.9b). We begin with a PEC object with smooth boundary ∂V = S which we think of as being divided into two adjoining and non-overlapping parts S + and S − , as is shown in Figure 13.12a for an object with specular symmetry around a plane whose trace is shown with a dash-dotted line. We denote with J+S and J−S the parts of JS that flow on S + and S − . At this stage J+S (r) = J−S (r) for points r ∈ ∂S + ≡ ∂S − . As we make the body conceptually flatten around the symmetry plane (Figure 13.12b), the shape of the object becomes increasingly elongated with the two surfaces S + and S − being brought ever closer to one another. While nothing much happens to the currents away from the separation line ∂S + ≡ ∂S − , the flow of J+S (r) into J−S (r) and vice-versa through ∂S + ≡ ∂S − gets more and more difficult in view of the steep curvature of S at ∂S + ≡ ∂S − . In the limit as the body is completely flattened out and S ≡ S + ≡ S − we end up with two separate currents J+S (r) and J−S (r) which flow on either side of a conducting surface (Figure 13.12c). Since the latter is immersed in a lossless dielectric medium by assumption, there cannot exist any conduction current for r ∈ R3 \ S and the edge condition (13.71) applies.

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Figure 13.12 For illustrating why the MFIE is unsuitable for infinitely thin PEC bodies: (a) the induced current JS on a PEC body is separated into two parts; (b) the body is conceptually made to shrink; (c) in the limit the supports of the two currents coincide. Since JS (r) actually splits up into two distinct but overlapped current densities J+S (r) and J−S (r), we may enforce the jump condition (1.168) separately on both sides of S , but in view of the limiting behavior (13.108) we must be careful in extending (13.111) to the geometry of Figure 13.12c. In symbols, we have 1 + + i ˆ × [∇G(r, r ) × J+S (r )] ˆ × H (r) + JS (r) + PV dS  n(r) JS (r) = n(r) 2 S+ 1 − ˆ × [∇G(r, r ) × J−S (r )], + JS (r) + PV dS  n(r) r ∈ S+ (13.113) 2 − S 1 − i ˆ × [∇G(r, r ) × J+S (r )] ˆ × H (r) + J+S (r) − PV dS  n(r) JS (r) = −n(r) 2 S+ 1 − ˆ × [∇G(r, r ) × J−S (r )], + JS (r) − PV dS  n(r) r ∈ S− (13.114) 2 S−

ˆ where we have taken into account that the unit normal on S − is −n(r). Consequently, to write the right member of (13.114) we choose the minus sign in (13.108) since the observation point r conceptually ˆ  ). As was the case for approaches the surface S − from the negative side with respect to the normal n(r the EFIE, (13.113) and (13.114) actually represent the selfsame relationship which we may cast as 1 + ˆ × Hi (r) [J (r) − J−S (r)] = n(r) 2 S + PV

  ˆ × ∇G(r, r ) × [J+S (r ) + J−S (r )] , dS  n(r)

r∈S

(13.115)

S +

−

where now S := S ≡ S is an orientable open surface in the sense of Figures 1.2a and (13.9b). However, unlike the EFIE (13.74), (13.115) cannot be solved because it involves two unknowns or, equivalently, both the sum and the difference of the currents J+S (r) and J−S (r). From a physical viewpoint, (13.115) contains too little information on the current density induced on S because

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the tangential magnetic field is discontinuous across S in accordance with the boundary condition (1.168). That is why the MFIE by itself is unsuitable for infinitely thin conductors. In spite of this limitation (13.115) is not entirely useless, though. We recall from Section 13.2.1 that in the situation of Figure 13.4c the EFIE (13.74) can be solved for the sum J+S (r )+J−S (r ). Therefore, if one is really interested in finding J+S (r) and J−S (r) separately, one may employ (13.115) to compute the difference in the left-hand side after the sum has been determined through (13.74). Once sum and difference of J+S (r) and J−S (r) are known, it is possible to find each contribution individually. After the electric current density JS (r) has been obtained by solving (13.111), the total field H(r) may be computed in any point of interest by means of (13.104). Moreover, the scattered magnetic field in the Fraunhofer region of the object is given by (9.326).

13.2.4 Interior-resonance problem In Sections 13.2.1 and 13.2.3 we mentioned that both the EFIE and the MFIE are particularly wellsuited for formulating and solving problems concerned with the scattering of time-harmonic electromagnetic waves from conducting bodies whose boundaries are comprised of closed surfaces in the sense of Figure 1.2b. Nevertheless, owing to the so-called interior-resonance problem [26,43] which plagues both (13.70) and (13.111), the numerical solution of the EFIE and the MFIE may become unstable or outright impossible for particular values of the angular frequency ω. Essentially, the interior-resonance problem is due to the fact that neither the EFIE nor the MFIE alone contains sufficient information to ensure the uniqueness of the sought induced current JS (r). As a result, (13.70) and (13.111) may lead to currents that generate fields which are incorrectly nonzero within the PEC body, contrary to the expectation. This is not to say by all means that in the actual scattering problem the electromagnetic field does not vanish within the body! Rather, it is just the mathematical formulation of the problem through either (13.70) and (13.111) that fails to yield the correct current density. To elaborate on this topic we refer to the scattering problem of Figure 13.13a. The relevant EFIE (13.70) may be written as ˆ ˆ = −Ei (r) × n(r), Es (r; JS ) × n(r)

r ∈ ∂V +

(13.116)

where the shorthand notation Es (r; JS ) serves as a reminder that the scattered field is produced by JS (r) flowing on the positive side of ∂V. Compared to (13.70), (13.116) involves the rotated tangential electric fields on ∂V + , but this modification does not affect the solution JS (r). Secondly, we consider the PEC cavity that has the very same shape (i.e., same boundary ∂V) of the object involved in the previous problem, as is pictorially suggested in Figure 13.13b. We already mentioned, while discussing the uniqueness theorem for the Maxwell equations in the frequency domain (Section 6.4.1), that in a source-free bounded region, for discrete values of the angular frequency ων there exist non-trivial solutions Eν (r), Hν (r) called eigenfunctions or modes [24, 41] (see Section 11.1 for the full derivation). Since we exclude the presence of primary sources in the region V by hypothesis, a non-zero electromagnetic field can be regarded as being produced by induced secondary currents flowing on the PEC wall of the cavity. More specifically, given that induced magnetic currents cannot exist on a PEC surface, the current density in question must needs be electric in nature. We indicate such current with J˜ eS (r) with the understanding that it flows on ∂V − . In order to come up with an EFIE for J˜ eS (r) we follow the procedure outlined in Section 13.2.1. We start off by applying the surface equivalence principle with Vex := V. Accordingly, the current J˜ eS (r) does not radiate in the ‘interior’ region Vin , which is, in fact, the unbounded region outside the cavity. Then we ‘remove’ the PEC medium that surrounds (more precisely, defines) the cavity and ‘fill’ Vin := R3 \ V with the same lossless isotropic medium existing in V. In the end we enforce the

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(b)

Figure 13.13 For illustrating the interior-resonance problem of the EFIE: (a) electromagnetic scattering from a PEC body and (b) lossless PEC cavity associated with the object. jump condition (1.169) and arrive at a homogeneous EFIE for the unknown J˜ eS (r) associated with the eigenfunctions of the cavity. In symbols, we have r ∈ ∂V −

ˆ = 0, Es (r; J˜ eS ) × n(r)

(13.117)

where the right-hand side is null because there is no incident field in the cavity by definition. We observe that (13.117) represents the condition for solenoidal electric modes (11.10) in a cavity with PEC walls and the relevant eigenfunction is an electric cavity mode [24, 44]. Besides, (13.117) is a homogeneous Fredholm equation of the first kind and as such it may admit non-trivial solutions J˜ eS ν (r) for ω = ων . So, is this an issue for the seemingly unrelated scattering problem of Figure 13.13a? To answer this question we have to look carefully into the explicit form of the scattered field appearing in (13.116) and (13.117). While there is no denying that the equations are different, still the way we compute the tangential component of the secondary electric field Est (r; •) in (13.70) and (13.117) is the same, viz., − j kR e− j kR  ∇s s  e  JS (r ) + ∇ · JS (r ), dS dS  r ∈ ∂V + (13.118) Et (r; JS ) := − j ωμ 4πR j ωε 4πR s + + ∂V ∂V − j kR e e− j kR  ˜ e  ∇ s s e  e  ˜ ˜ JS (r ) + ∇ · J (r ), dS dS  r ∈ ∂V − (13.119) Et (r; JS ) := − j ωμ 4πR j ωε 4πR s S ∂V −

∂V −

because the integration over ∂V + is equivalent to the one over ∂V − and the Green function for the scattering problem coincides with the one for the cavity problem. In other words, both (13.70) and (13.117) involve the same integral operator, and as a result the EFIE ‘cannot distinguish’ the body from the associated cavity, so to speak. Thus, here is the catch. With an eye to the scattering problem of Figure 13.13a we state (13.70) or (13.116) in an attempt to compute the current JS (r). However, if the angular frequency ω of the incident field Ei (r) happens to coincide with one of the infinitely many resonance frequencies

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ων pertinent to the cavity associated with the object, (13.117) in turn is satisfied by the current J˜ eS ν (r) related to the corresponding eigenfunction. In which case, since (13.117) is nothing but the homogeneous instance of (13.116), any linear combination of the form JS (r) = JS (r) + Cνe J˜ eS ν (r),

r ∈ ∂V

(13.120)

with Cνe being an arbitrary unknown constant, is a solution to (13.116). This means that the solution of the EFIE is not unique at the frequencies which represent the resonances of the cavity having the same shape of the body under investigation. This phenomenon motivates the name of interiorresonance problem anticipated at the beginning. For frequencies close to a resonance of the cavity the EFIE becomes unstable and the numerical solution thereof is inaccurate, even though JS (r) is unique in principle. We continue our discussion by examining the MFIE applied to the scattering problem sketched in Figure 13.14a. We write (13.111) succinctly as r ∈ ∂V +

ˆ × Hs+ (r; JS ) − JS (r) = −n(r) ˆ × Hi (r), n(r)

(13.121)

where the subscript ‘+’ appended to the scattered magnetic field signifies that we choose the plus sign in (13.108). Prompted by the analysis carried out previously for the EFIE, we might think that the MFIE runs into trouble for angular frequencies ω which correspond to the resonances ων of the associated cavity with PEC walls shown in Figure 13.13b. As a matter fact, if we apply the equivalence principle to construct an equivalent problem for the inside Vex := V and enforce the boundary condition (1.168) we get ˆ × Hs− (r; J˜ hS ) + J˜ hS (r) = 0, n(r)

r ∈ ∂V −

(13.122)

having taken into account the orientation of the normal, which points towards the outside of the cavity. In this regard, the subscript ‘−’ appended to the scattered field means that we select the minus sign in (13.108). Finally, there is no excitation term in the right-hand side as the incident magnetic field is absent in V by assumption. Although (13.122) is a homogeneous Fredholm equation of the second kind and it may admit non-trival solutions, the combination of current and scattered magnetic field in the left member is clearly different than the one occurring in the MFIE (13.121). Hence, the solutions to (13.122) have nothing to do with the stability or uniqueness of (13.121). Apparently, the resonances of the PEC cavity associated with the body in Figure 13.14a do not pose any problem. Therefore, we proceed backwards and start by considering the homogeneous instance of (13.121), namely, ˆ × Hs+ (r; J˜ hS ) − J˜ hS (r) = 0, n(r)

r ∈ ∂V −

(13.123)

with the current J˜ hS (r) flowing on the negative side of ∂V. Evidently, provided (13.123) has nontrivial solutions J˜ hS ν (r) for ω = ων , then any linear combination of the type JS (r) = JS (r) + Cνh J˜ hS ν (r),

r ∈ ∂V

(13.124)

with Cνh an arbitrary unknown constant solves (13.121) as well. This implies that the MFIE does not have a unique solution for those frequencies which correspond to the eigenfunctions of (13.123). It remains to ascertain what physical problem is mathematically described by (13.123). To this purpose we cast (13.123) into the alternative format ˆ × Hs− (r; J˜ hS ) = 0, n(r)

r ∈ ∂V −

(13.125)

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Figure 13.14 For illustrating the interior-resonance problem of the MFIE: (a) electromagnetic scattering from a PEC body and (b) lossless PMC cavity associated with the object.

by virtue of (13.108). Since we recognize the vanishing of the tangential magnetic field on a surface as the hallmark of a PMC medium, thanks to the matching condition (6.226), we may interpret (13.125) as the governing equation of the eigenfunctions of a PMC cavity with the same shape as that of the original PEC body (Figure 13.14b). In fact, (13.125) is the condition for solenoidal magnetic modes in a PMC cavity, that is, the dual situation of the solenoidal electric modes in a PEC cavity that are provided by (11.10). The relevant eigenfunctions are called magnetic cavity modes [44]. Still, the way we have arrived at (13.125) may sound a bit contrived. What is more, one might question the very appearance of an electric current density on the surface of a PMC, on the grounds that (6.226) says indeed that this occurrence is forbidden. To clarify the nature of J˜ hS (r) we need to derive an integral representation pertinent to the PMC cavity problem of Figure 13.14b. From the Stratton-Chu formula (10.23) applied from the viewpoint of an observer located at r ∈ V we deduce the relationship Hs (r) = j ωε

ˆ  )G(r, r ) + ∇ dS  E− (r ) × n(r

∂V −





ˆ  ) · H− (r ) dS  G(r, r )n(r

∂V −

−∇×

ˆ  ) × H− (r ), dS  G(r, r )n(r

r∈V

(13.126)

∂V −

with due regard to the outwardly oriented normal and the absence of true sources within V. The subscript ‘−’ is a reminder that the fields in the surface integrals are evaluated on the negative side of ∂V. In particular, since the surface ∂V − is flush with a PMC medium, the last contribution in the right member is actually zero. Next, we consider an auxiliary problem with sources located in V and the medium in the complementary domain R3 \ V endowed with the same constitutive parameters as the medium in the cavity of Figure 13.14b. We apply (10.23) once again but this time from the viewpoint of an observer at

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Advanced Theoretical and Numerical Electromagnetics

r ∈ R3 \ V, viz., HA (r) = − j ωε

ˆ  )G(r, r ) − ∇ dS  E+ (r ) × n(r

∂V +

+∇×



ˆ  ) · H+ (r ) dS  G(r, r )n(r

∂V −

ˆ  ) × H+ (r ), dS  G(r, r )n(r

r ∈ R3 \ V

(13.127)

∂V −

where the subscript ‘+’ means that the fields in the integrands are taken on the positive side of ∂V. A fundamental remark here is that the combination of integrals in the right-hand side of (13.127) returns zero when evaluated for points r ∈ V. More importantly, since we are at liberty of choosing the auxiliary problem arbitrarily we make the assumption that the speculated sources in V produce an electromagnetic field such that ˆ  ) = E− (r ) × n(r ˆ  ), E+ (r ) × n(r    ˆ ) · H+ (r ) = n(r ˆ ) · H− (r ), n(r

r ∈ ∂V r ∈ ∂V

(13.128) (13.129)

although the second relation is a consequence of the first one, the differential identity (A.60), and the Faraday law (1.99). As a result, when we sum (13.126) and (13.127) side by side we get ˆ  ) × H+ (r ), Hs (r) = ∇ × dS  G(r, r )n(r r∈V (13.130) ∂V

that is, the scattered field in the PMC cavity can be derived from an equivalent surface density of electric current given by ˆ × H+ (r), J˜ hS (r) := n(r)

r ∈ ∂V

(13.131)

and to this purpose it is immaterial whether we think of J˜ hS (r) on the plus or the negative side of ∂V. The equivalent electric current density J˜ hS (r) is not the current induced on the boundary of the PMC — which is zero — but rather the tangential magnetic field of a suitable auxiliary problem so ˆ  ) which is indeed ‘designed’ as to cancel out the effect of the magnetic current density E− (r ) × n(r − induced on ∂V . To finalize the derivation we extract the tangential part of (13.130) and, with the help of (13.108), take the limit as r approaches ∂V − from within V. This procedure yields 1 ˆ ˆ × Hs− (r; J˜ hS ) = 0, r)× J˜ hS (r )] − J˜ hS (r) = n(r) PV dS  n(r)×[∇G(r, r ∈ ∂V − (13.132) 2 ∂V

which is exactly (13.125). We notice that J˜ hS (r), unlike J˜ eS (r) in (13.117), generates a non-zero field also in the complementary domain R3 \ V, as is required by the integral representation (13.127) for the auxiliary problem.

13.2.5 Combined-field integral equation (CFIE) Many remedies have been devised in order to counteract the interior-resonance problem of the EFIE and the MFIE [43, 45–54]. Since, in principle, the EFIE and the MFIE applied to the same geometry yield the same current JS (r), a simple and relatively old approach consists of ‘mixing’ (13.70) and (13.111) for scattering problems into a single equation in an attempt to prevent the eigenfunctions of

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the associated PEC and PMC cavities from sneaking into the solution. The resulting relationship — referred to as the combined-field integral equation (CFIE) [28, Section 2.7.3], [26, 44] — reads ˆ × [E(r) × n(r)] ˆ ˆ × H+ (r) − JS (r)] = 0, αn(r) + Z(1 − α)[n(r)

r ∈ ∂V +

(13.133)

where α ∈]0, 1[ is the mixing parameter, and Z is the intrinsic impedance (1.358) of the homogeneous medium in which the body and the source are immersed (Figures 13.13a and 13.14a). The subscript ‘+’ reminds us to pick up the plus sign in (13.108) for the calculation of the scattered magnetic field. The insertion of Z in the linear combination serves to render the contributions of EFIE and MFIE dimensionally homogeneous and, more importantly, of the same order of magnitude. In fact, without the scaling effect of Z the magnetic field would be substantially smaller than the electric field. Evidently, (13.133) reduces to the EFIE for α = 1 and to the MFIE for α = 0. The rationale behind the combination is that we draw information from both (13.70) and (13.111) simultaneously, given that the EFIE or the MFIE alone are not able to distinguish the scattering problem from the respective PEC or PMC resonant cavities. Consequently, α = 0.5 is the typical value used for the mixing parameter. We wish to show that the CFIE is actually immune to the interiorresonance problem. By separating incident and scattered fields we write (13.133) as ˆ ˆ × Hs+ (r; JS ) − JS (r)] ˆ × [Es (r; JS ) × n(r)] + Z(1 − α)[n(r) αn(r) ˆ ˆ × Hi (r), ˆ × [Ei (r) × n(r)] − Z(1 − α)n(r) = −αn(r)

r ∈ ∂V +

(13.134)

where we have highlighted the dependence of the scattered fields on the induced current JS (r). The homogeneous counterpart of (13.134) reads ˆ × [Es (r; J˜ S ) × n(r)] ˆ ˆ × Hs− (r; J˜ S ) = 0, αn(r) + Z(1 − α)n(r)

r ∈ ∂V −

(13.135)

on account of (13.108) as in (13.125). We may interpret (13.135) as an impedance matching condition of sorts [cf. (6.78)] that the source-free field must obey on the boundary of a cavity which has the same shape as that of the object. In this case the surface impedance of the cavity wall is ZS := Z

1−α > 0, α

α ∈]0, 1[

(13.136)

and since a real impedance is synonymous with losses, we may expect that resonances are precluded. This was, after all, the argument used at the end of Section 6.4.1 to prove uniqueness of time-harmonic solutions in a closed region with losses in the boundary. In fact, the scattering problem governed by (13.134) has a unique solution for all frequencies so long as (13.135) is solved only by J˜ S (r) = 0 for r ∈ ∂V − . To check whether this is true, we examine the squared magnitude of the left-hand side of (13.135) [26, 44], namely, 2 2    + Z 2 (1 − α)2 n(r) ˆ × [Es (r; J˜ S ) × n(r)] ˆ × Hs− (r; J˜ S ) ˆ α2 n(r) + , ˜ ˆ · Re Es (r; J˜ S ) × Hs∗ − 2Zα(1 − α)n(r) − (r; JS ) = 0,

r ∈ ∂V −

(13.137)

having made use of (H.13) and (B.6). Next, we integrate over the wall ∂V − and arrange the three contributions to get − α2 ∂V −

 2  2  − Z 2 (1 − α)2 ˆ × [Es (r; J˜ S ) × n(r)] ˆ × Hs− (r; J˜ S ) ˆ dS n(r) dS n(r) ∂V −

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Advanced Theoretical and Numerical Electromagnetics = 2Zα(1 − α) Re

˜ ˆ dS Es (r; J˜ S ) × Hs∗ (13.138) − (r; JS ) · [−n(r)]

∂V −

where the quantity in the right member is proportional to the average radiated power (1.303) flowing ˆ from the wall into the cavity, because n(r) points towards the complementary region R3 \ V, as is illustrated in Figures 13.13b and 13.14b. However, since the cavity is lossless by hypothesis, the current J˜ S (r), if it is non-null, does not deliver power to the medium within V. As a result, (13.138) demands  2  2  + Z 2 (1 − α)2 ˆ × [Es (r; J˜ S ) × n(r)] ˆ × Hs− (r; J˜ S ) = 0 ˆ dS n(r) dS n(r) (13.139) α2 ∂V −

∂V −

where the integrands are always positive quantities. Hence, for α ∈]0, 1[ condition (13.139) can only be satisfied if r ∈ ∂V −

ˆ × [Es (r; J˜ S ) × n(r)] ˆ ˆ × Hs− (r; J˜ S ), n(r) = 0 = n(r)

(13.140)

that is, the tangential electric and magnetic fields vanish on the wall of the cavity. The application of the Stratton-Chu representation formulas (10.22) and (10.23) allows us to determine that Es (r; J˜ S ) = 0 = Hs (r; J˜ S ) everywhere in the cavity, as there are no other sources in V. In particular, from the viewpoint of the associated PEC cavity, ˆ × Hs− (r; J˜ S ), J˜ S (r) = −n(r)

r ∈ ∂V −

(13.141)

whence we conclude that J˜ S (r) = 0 in light of (13.140). Therefore, (13.135) is only verified by the trivial solution J˜ S (r) = 0, and the CFIE (13.133) does not suffer from the interior-resonance problem.

13.2.6 A modified EFIE for good conductors The propagation of electromagnetic waves in good conductors (i.e., media for which σ  ωε) is forbidden for all practical purposes, as was shown for the special case of plane waves in Examples 7.2, 9.7, and 9.8. In principle, the matching conditions (1.142), (1.144), (1.155) and (1.157) are still valid at the interface between a dielectric medium and a conducting body. However, the electromagnetic field inside the conductor is rapidly damped in the bulk thereof and is substantially non-null only in a very small layer whose thickness is given by the skin depth (7.102). Thus, motivated by the impedance relationship (7.112) we argue that it is more convenient to characterize a good conductor of arbitrary shape by means of an approximate boundary condition of the Leontovich type [55–58], viz., ˆ × [E(r) × n(r)] ˆ n(r) = ZS (r) nˆ (r) × H(r),

r ∈ ∂V +

(13.142)

ˆ is the outward unit normal on the surface ∂V which bounds the conducting region V, and where n(r) ZS (r) denotes the surface impedance, which is possibly changing from point to point. An expression for ZS (r) is provided by (7.111), though the latter was derived for a planar interface. Hence, (13.142) and (7.111) may be utilized for non-planar surfaces so long as ∂V is smooth and the local curvature of ∂V is large as compared to the wavelength in the background medium. In other words, we expect (13.142) and (7.111) to become less accurate in the neighborhood of sharp corners. Nonetheless, we wish to rely on (13.142) to devise an integral equation for solving scattering problems such as the one sketched in Figure 13.15 in which the conductivity of the object is very large but not infinite.

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Figure 13.15 Wave scattering from a conducting object modelled by means of a surface impedance ZS and the Leontovich boundary condition (13.142). The time-harmonic incident electromagnetic field Ei (r), Hi (r) is generated by distant sources which are not affected by the presence of the body for r ∈ V. Then, Ei (r) and Hi (r) by themselves cannot, in general, obey the impedance relationship (13.142) on the surface of the body. As a consequence, within V an electric conduction current (1.120) is induced that has the task of producing a secondary electromagnetic field Es (r) and Hs (r). However, since the medium in V is a good conductor, the current is confined to a tiny layer just beneath the surface ∂V. In fact, we may retain retain the jump condition (1.168) which is pertinent to a truly PEC body and assume the onset of a surface electric current density given by ˆ × [Hi (r) + Hs (r)], JS (r) := n(r)

r ∈ ∂V +

(13.143)

which when inserted into (13.142) yields r ∈ ∂V +

ˆ × {[Ei (r) + Es (r)] × n(r)} ˆ n(r) = ZS (r)JS (r),

(13.144)

as the alternative form of the impedance relationship. In order to relate the secondary electric field to JS (r) we follow the strategy based on the surface equivalence principle and extensively described in Section 13.2.1. We choose the internal region Vin coincident with the volume V occupied by the body and introduce electric (JS eq ) and magnetic (J MS eq ) equivalent surface currents on ∂V + . Finally, we replace the conducting body with a penetrable medium endowed with the same scalar constitutive parameters as those of the medium existing in the exterior volume Vex := R3 \ V. In view of definition (10.79) and the jump condition (13.143) we are led to set JS eq (r) ≡ JS (r),

r ∈ ∂V +

(13.145)

but what can be said of J MS eq (r)? Well, this current is not exactly null, as was the case for the PEC body examined in Sections 13.2.1 and 13.2.3, because we have traded the matching condition (1.169) for (13.142). In theory we should regard J MS eq (r) as a second unknown of the problem and employ both JS (r) and J MS eq (r) to compute the secondary electric field. In practice we hardly need be so fastidious inasmuch as the actual medium within V is a good conductor for which we may suppose that ˆ ≈ 0, J MS eq (r) := E(r) × n(r)

r ∈ ∂V +

(13.146)

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Advanced Theoretical and Numerical Electromagnetics

which confessedly is in contrast to (13.142). As a matter of fact, we should keep in mind that (13.142) is approximated anyway, though it constitutes a fairly better approximation than (13.146). In summary, we neglect J MS eq (r) when it comes to the contribution thereof to the radiation of the secondary field, whereas we suppose that J MS eq (r) is non-null precisely to model the lossy character of the boundary ∂V + . Now, the sum of incident and secondary electric fields can be written explicitly as in (13.67) for r ∈ R3 . The triple cross product in the left-hand side of (13.142) serves to single out the components of Ei (r) and Es (r) that are tangential to the boundary ∂V, as in (13.70). All in all, we arrive at Eit (r) − j ωμ ∂V

dS 

e− j kR JS (r ) 4πR ∇s + j ωε



dS 

∂V

e− j kR  ∇ · JS (r ) = ZS (r)JS (r), 4πR s

r ∈ ∂V +

(13.147)

which is the desired modified electric-field integral equation [35, Section 6.4.5], [57]. In light of the nomenclature of Section 13.1, (13.147) is a Fredholm integral equation of the third kind. For good conductors the surface impedance computed through (7.111) is small, and hence (13.147) may be regarded as a perturbation of (13.70). It is possible to apply (13.147) even when the scattering object is a thin conducting sheet with finite thickness, in which case an expression for ZS (r) — more general than (7.111) — can be found in [59]. Moreover, unlike the MFIE, (13.147) also works for infinitely thin conducting bodies in spite of the presence of the surface current JS (r) in the right member. The ultimate reason for ˆ this behavior is that the relevant integral operator does not depend on the unit normal n(r). With reference to Figures 13.4a-13.4c and the related discussion, we recall that, in the limit as a body can be modelled as an orientable open conducting surface S := S + ≡ S − , the current JS (r) splits up into two disjoint overlapped parts J+S (r) and J−S (r) that flow on S + and S − . In this situation (13.147) degenerates into two coupled equations valid on either side of S , namely, e− j kR +  e− j kR  +  ∇s JS (r ) + ∇ · J (r ) dS  4πR j ωε 4πR s S + + S S − j kR − j kR e e ∇ s J− (r ) + ∇ · J− (r ), r ∈ S + − j ωμ dS  dS  4πR S j ωε 4πR s S S− S− − j kR ∇s e− j kR  +  − − i  e +  JS (r ) + ∇ · J (r ) ZS (r)JS (r) = Et (r) − j ωμ dS dS  4πR j ωε 4πR s S S+ S+ e− j kR −  e− j kR  −  ∇s JS (r ) + ∇ · J (r ), r ∈ S − dS  − j ωμ dS  4πR j ωε 4πR s S ZS+ (r)J+S (r) = Eit (r) − j ωμ

S−



dS 

(13.148)

(13.149)

S−

where ZS+ (r) and ZS− (r) indicate the surface impedance on S + and S − . We observe that the integrodifferential operators in the right members are the same, whereas the left-hand side remain different even when ZS+ (r) and ZS− (r) coincide. Hence, at least in principle (13.148) and (13.149) can be solved simultaneously to obtain J+S (r) and J+S (r). In practice, though, the system formed by (13.148) and (13.149) as it stands is nearly singular because ZS+ (r) and ZS− (r) are small for a good conductor. Indeed, (13.148) and (13.149) reduce to (13.72) and (13.73) in the limit as the conductivity grows

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infinitely large and the medium approaches a PEC. Then again, since (13.148) and (13.149) evidently imply ZS+ (r)J+S (r) = ZS− (r)J−S (r),

r∈S

(13.150)

we may eliminate either current density from one of the equations so long as the surface impedances on the two sides of S are such that ZS+ /ZS− = O (1). For instance, replacing J−S (r) with J+S (r) in (13.148) yields ZS+ (r)J+S (r)

% & ZS+ (r ) +  e− j kR − j ωμ dS 1 + −  JS (r ) 4πR ZS (r ) + S -% . & − j kR ZS+ (r ) +  ∇s  e  ∇ · 1 + −  JS (r ) , + dS j ωε 4πR s ZS (r )

=

Eit (r)



r ∈ S+

(13.151)

S+

and once J+S (r) is known, J−S (r) can be determined by means of (13.150). This procedure is as stable as solving the EFIE can be under the same circumstances in that the factor 1+ZS+ /ZS− in the integrands is regular. Owing to the lossy character of the medium in V, a fraction of the power delivered by the remote sources is absorbed by the conductor and dissipated therein. In this regard, the complex Poynting theorem (1.314) applied to V states 1 1 ˆ dV σ(r)|E(r)|2 = −Re dS E(r) × H∗ (r) · n(r) (13.152) PC  := 2 2 V

∂V

whereby we see that the average absorbed power PC  can be computed by integrating the normal component of the complex Poynting vector (1.304). The minus sign in the left-hand side is consistent ˆ with the fact that n(r) points outwards the lossy region V (see Figure 13.15) and there exists a net average efflux of power directed from the complementary region into V. In light of the boundary condition (13.142) and our assumption (13.143), we have 1 1 ˆ × H∗ (r)] = Re ˆ × H(r)|2 PC  = Re dS Et (r) · [n(r) dS ZS (r) |n(r) 2 2 ∂V ∂V 1 2 = dS Re {ZS (r)} |JS (r)| (13.153) 2 ∂V

which is real and positive as anticipated.

13.3 Surface integral equations for homogeneous scatterers We continue with the task of determining the fields produced by time-harmonic sources in the presence of homogeneous penetrable bodies and immersed in a homogeneous unbounded medium, as is suggested in Figure 13.16 [24, Section 12.5], [26]. Admittedly, the scattering problem is more complicated than it is for PEC objects (Section 13.2) because the electromagnetic field is non-null within the bodies of concern. However, so long as the constitutive parameters of the object in question are scalar quantities, we can write a set of coupled surface integral equations that involve the unknown tangential components of the total electromagnetic field over the boundary of the body. Said approach

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Figure 13.16 Scattering problem involving a homogeneous penetrable body endowed with isotropic constitutive parameters. is possible essentially because the three-dimensional Green function (8.356) is available. The integral equations are based on a linear combination of the matching conditions (1.196) and (1.197). Evidently, this goal can be achieved in a few different ways. We shall discuss the two most commonly used strategies, namely, the formulation of Poggio and Miller [60] and the Müller equations [26, 61]. We suppose that the background homogeneous medium — which contains the true sources — is endowed with constitutive parameters ε1 and μ1 , whereas we indicate with ε2 and μ2 the permittivity and permeability of the medium which comprises the penetrable object occupying a smooth region of space V ⊂ R3 . Linearity and the principle of superposition (Section 6.1) allow us to conceptually separate the field into two parts, viz., E1 (r) := Ei1 (r) + Es1 (r), E2 (r) := Es2 (r),

H1 (r) := Hi1 (r) + Hs1 (r), H2 (r) := Hs2 (r),

r ∈ R3 \ V r∈V

(13.154) (13.155)

where Ei1 (r), Hi1 (r), r ∈ R3 , denote the incident or primary field which is radiated by the known sources in the absence of the object, and Esl (r), Hsl (r), l = 1, 2, indicate the scattered or secondary field which is caused by the very presence of the penetrable body. As a result, the equations we seek will be relationships between incident field produced by the true sources and scattered fields existing both in the background medium and within the body, as pictorially shown in Figure 13.16. So, a legitimate question is this: What are the sources of the secondary field? Time-harmonic electromagnetic fields must be generated by charges performing a periodic motion. Still, in a dielectric medium in equilibrium there are no free charges which can roam under the action of an external field, but we know from Section 3.7 that polarization charges are induced within a dielectric body and on the surface thereof. The associated polarization vector — which in Section 3.7 was derived for electrostatic fields — passes over into the time-harmonic counterpart P(r; ω). Since by hypothesis the medium is homogeneous, P(r; ω) is constant within the body and hence net polarization charges appear only on the boundary ∂V of the object and are given ˆ ˆ · P(r; ω), with n(r) representing the unit normal oriented positively towards by ρS (r; ω) := n(r) the background (Figure 13.16). In like fashion, in a magnetic medium exposed to an induction field magnetic dipoles are induced, as we argued in Section 5.6 for the stationary or steady-state regime. The associated magnetization vector — which we obtained for stationary magnetic fields — must be traded for the time-harmonic counterpart M(r; ω). Once again, the medium being homogeneous, M(r; ω) is constant within the object and net magnetization magnetic charges given by ˆ · M(r; ω) accumulate only on the boundary of the body. ρ MS (r; ω) := μ1 n(r)

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Responsible for generating the secondary electromagnetic field are the surface polarization and magnetization currents associated with ρS (r; ω) and ρ MS (r; ω) through the surface continuity equations. In the macroscopic form of the Maxwell equations, though, we describe the effect of matter by means of constitutive parameters, such as permittivity and permeability, which account for the polarization and magnetization charges, so that we do not deal with ρS (r; ω) and ρ MS (r; ω) directly. The secondary electromagnetic field is charged with the task of making the matching conditions (1.196)-(1.199) obeyed by the total field at the interface between the object and the background medium. This becomes especially evident if we write (1.198) and (1.199) explicitly as ˆ · [Ei1 (r) + Es1 (r)] = ε2 n(r) ˆ · Es2 (r), ε1 n(r)

r ∈ ∂V

(13.156)

ˆ · μ1 n(r)

r ∈ ∂V

(13.157)

[Hi1 (r)

+

Hs1 (r)]

ˆ · = μ2 n(r)

Hs2 (r),

whence we notice that • •

failing to account for a scattered field in the body would cause the normal component of the total electromagnetic field to suffer a non-physical abrupt jump across ∂V; omitting a scattered field in the background medium would make it hard to ensure the continuity of the normal component of E(r) and H(r) for arbitrary combinations of the constitutive parameters.

To proceed we need to find a way of linking the scattered fields to their sources. In this regard, the integral representations of Stratton and Chu (10.22) and (10.23) provide an almost natural answer to the problem. In fact, we just need to apply (10.22) and (10.23) twice, i.e., from the viewpoint of the sources in the unbounded region R3 \ V and then within the body from the standpoint of an observer located in V. This strategy has the merit of showing that the scattered field may be thought of as being generated by equivalent surface currents and charges (both electric and magnetic) flowing on either side of ∂V. The other way of writing the scattered fields requires the application of the surface equivalence principle (Section 10.4) so as to turn the original problem into equivalent ones for which the solution — albeit formal — is available in closed form. As is well known, equivalence with the original problem is preserved in a selected region of space so long as suitable equivalent surface sources are defined. Since the true sources — in the region where they are located — generate the incident field, we are led to conclude that the equivalent sources cannot but produce the scattered field! We follow the second approach and separate the workload into two parts, namely, the exterior problem and the interior problem. In the former we focus on the secondary field in the background medium, whereas in the latter we deal with the scattered field within the body. For the exterior problem we start off by enclosing the body within a larger region of space Vin with smooth boundary ∂Vin . We apply the equivalence principle to the unbounded region Vex := R3 \ V in and accordingly define equivalent current densities J1S eq (r) and J1MS eq (r) (Figure 13.17a). At this stage the subscript ‘1’ serves only as a reminder that J1S eq (r) and J1MS eq (r) exists in the medium 1, the background. The combined effect of the true sources — placed somewhere in Vex — and of J1S eq (r) and J1MS eq (r) produces the ‘right’ field in Vex and no field at all within the excluded region Vin . We let the surface ∂Vin conceptually shrink all the way down to the boundary ∂V of the penetrable body within V ⊂ Vin . With this choice for Vex we are able to identify J1S eq (r) and J1MS eq (r) with the tangential electric and magnetic fields just at the boundary of the body though within the background (Figure 13.17b). In symbols, this reads ˆ = J MS (r), J1MS eq (r) := E1 (r) × n(r) ˆ × H1 (r) = JS (r), J1S eq (r) := n(r)

r ∈ ∂V +

(13.158)

r ∈ ∂V +

(13.159)

910

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Advanced Theoretical and Numerical Electromagnetics

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Figure 13.17 Derivation of surface integral equations for the scattering from penetrable objects: (a) the equivalence principle is invoked to establish an equivalent problem in the external region Vex ; (b) the internal region Vin is taken to be coincident with the region V occupied by the body.

Figure 13.18 Derivation of surface integral equations for the scattering from penetrable objects: the body is ‘removed’ and replaced with the same medium existing in Vex .

where E1 (r) and H1 (r) denote the total electric and magnetic field on the positive side of ∂V. The trouble with the original configuration of Figure 13.16 is that we do not have the relevant Green function for a piecewise-homogeneous medium. Therefore, we exploit the freedom afforded by the equivalence principle to alter the material properties of the medium within Vin while preserving full equivalence with the original problem for r ∈ Vex . As already done in the derivation of the EFIE and the MFIE, we conceptually ‘replace’ the body with another one whose constitutive parameters are ε1 and μ1 (Figure 13.18). In this way, we end up with a set of true and equivalent sources which exist and radiate in a homogeneous unbounded medium. We know how to relate currents to fields in this situation, at least formally, by resorting to the auxiliary electromagnetic potentials and the representations (9.152) and (9.153). Since our sources are surface densities over ∂V + the relevant integral formulas for the potentials are (9.136), (9.151)

Integral equations in electromagnetics and the dual thereof, viz., e− j k 1 R  1 ΦE1 (r) = − ∇ · JS (r ), dS  j ωε1 4πR s ∂V e− j k 1 R AE1 (r) = μ1 dS  JS (r ), 4πR ∂V e− j k 1 R  1 Φ M1 (r) = − ∇ · J MS (r ), dS  j ωμ1 4πR s ∂V e− j k 1 R J MS (r ), A M1 (r) = ε1 dS  4πR

911

r ∈ R3

(13.160)

r ∈ R3

(13.161)

r ∈ R3

(13.162)

r ∈ R3

(13.163)

∂V

where k1 := (ε1 μ1 )1/2 denotes the wavenumber in the background medium. We have exploited the fact that JS (r) and J MS (r) are related to the respective charge densities ρS (r) and ρ MS (r) by surface continuity equations which are special instances of (1.200) and its dual counterpart. The total field in the background medium reads

E1 (r) = Ei1 (r) − j ωμ1

e− j k 1 R JS (r ) 4πR ∂V e− j k 1 R  e− j k 1 R 1 ∇s · JS (r ) − ∇ × dS  J MS (r ) ∇ dS  + j ωε1 4πR 4πR dS 

∂V



H1 (r) = Hi1 (r) − j ωε1

(13.164)

∂V

e− j k 1 R J MS (r ) 4πR ∂V − j k1 R 1 e− j k 1 R  e   ∇ dS + ∇s · J MS (r ) + ∇ × dS  JS (r ) j ωμ1 4πR 4πR dS 

∂V

(13.165)

∂V

on account of (9.152), (9.153) and (13.160)-(13.163). These representations are valid for r ∈ R3 , although they produce a null field within V. More importantly, we do not know yet the actual values of JS (r ) and J MS (r ). Therefore, we turn to the interior problem in order to gather additional information. We apply the surface equivalence principle this time from the viewpoint of an observer located within Vin who wishes to establish an equivalent simpler problem in Vin . To this purpose, we choose the separation surface coincident with the inner, negative side of ∂V, whereon we place the required equivalent current densities (Figure 13.19a) ˆ = −J MS (r), J2MS eq (r) := −E2 (r) × n(r) ˆ × H2 (r) = −JS (r), J2S eq (r) := −n(r)

r ∈ ∂V − r ∈ ∂V

−

(13.166) (13.167)

where E2 (r) and H2 (r) denote the total electric and magnetic field on ∂V − . Since there are no true sources in the body by assumption, the total field is the same as the scattered field. The minus sign in the definitions above is a consequence of the opposite orientation of the normal on ∂V − with

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Advanced Theoretical and Numerical Electromagnetics

(a)

(b)

Figure 13.19 Derivation of surface integral equations for the scattering from penetrable objects: (a) the equivalence principle is invoked to establish an equivalent problem in the interior region Vin ; (b) the external region Vex is ‘filled’ with the same material of the body. respect to the exterior problem. Since the tangential components of magnetic and electric field are continuous through the interface between two material media, according to (1.196) and (1.197), the equivalent currents J2S eq (r) and J2MS eq (r) coincide with the negative of JS (r) and J MS (r) introduced in (13.159) and (13.158). The currents J2S eq (r) and J2MS eq (r) reproduce the correct field within V due to the sources outside and in the presence of the body as well as null fields in Vex . However, J2S eq (r) and J2MS eq (r) are not independent of J1S eq (r) and J1MS eq (r)! As already done before we simplify the task of relating fields and sources by ‘replacing’ the medium in Vex with one characterized by ε2 and μ2 (Figure 13.19b). With this choice the equivalent sources −JS (r) and −J MS (r) (the only ones in our setup) radiate in a homogeneous medium for which the Green function is available. Indeed, based on the derivations of Section 9.3 the electrodynamic potentials take on the form e− j k 2 R  1 ∇ · JS (r ), dS  r ∈ R3 (13.168) ΦE2 (r) = j ωε2 4πR s ∂V e− j k 2 R JS (r ), r ∈ R3 (13.169) AE2 (r) = −μ2 dS  4πR ∂V e− j k 2 R  1 ∇ · J MS (r ), dS  r ∈ R3 (13.170) Φ M2 (r) = j ωμ2 4πR s ∂V e− j k 2 R J MS (r ), r ∈ R3 (13.171) A M2 (r) = −ε2 dS  4πR ∂V

where k2 := (ε2 μ2 )1/2 is the wavenumber relevant to medium 2. In the end, the scattered fields within the body read Es2 (r) = j ωμ2 ∂V

dS 

e− j k 2 R JS (r ) 4πR

Integral equations in electromagnetics − Hs2 (r) = j ωε2 ∂V

dS 

1 ∇ j ωε2



dS 

∂V

e− j k 2 R  ∇ · JS (r ) + ∇ × 4πR s



dS 

∂V

e− j k 2 R J MS (r ) 4πR

e− j k 2 R J MS (r ) 4πR e− j k 2 R  e− j k 2 R 1 ∇s · J MS (r ) − ∇ × dS  JS (r ) ∇ dS  − j ωμ2 4πR 4πR ∂V

913

(13.172)

(13.173)

∂V

where we have used (13.166) and (13.167). These representations are valid for r ∈ R3 , though they vanish for points in the exterior region. We are now left with the task of setting up two equations to determine JS (r) and J MS (r).

13.3.1 The integral equations of Poggio and Miller (PMCHWT) The strategy proposed by A. J. Poggio and E. K. Miller in 1973 [60] to arrive at a set of integral equations consists of using (13.164), (13.165), (13.172) and (13.173) to enforce the matching conditions (1.196) and (1.197) for r ∈ ∂V. Since a few other Authors investigated the same approach building on the original work by Poggio and Miller, the system of equations has come to be named after Poggio, Miller, Chang, Harrington, Wu and Tao, and the initialism PMCHWT is used for brevity. We emphasize that the surface density JS (r) in (1.196) is a conduction current — which is null at the interface between two penetrable lossless media — and should not be confused with our unknown in (13.159) and (13.167), i.e., the equivalent surface density JS (r) which arises from the onset of polarization charges. In symbols, we have ˆ × {[Ei1 (r) + Es1 (r)] × n(r)} ˆ ˆ × [Es2 (r) × n(r)], ˆ n(r) = n(r) ˆ × n(r)

{[Hi1 (r) +

Hs1 (r)]

ˆ ˆ × × n(r)} = n(r)

[Hs2 (r) ×

ˆ n(r)],

r ∈ ∂V

(13.174)

r ∈ ∂V

(13.175)

and to write the field explicitly we need to take the limit of (13.164), (13.165), (13.172) and (13.173) as the observation point r approaches the boundary ∂V of the object. We focus on the electric field and notice that we may retrieve the formulas for the magnetic field by invoking the principle of duality (Section 6.7). By taking a look at (13.164) and (13.172) we observe that the contribution of JS (r) is of the same type considered while deriving the EFIE. In particular, the integrals are in the form of singlelayer potentials and hence they exists also for r ∈ ∂V, whereas the triple cross product with the unit ˆ extracts the surface gradient of ΦE1 (r) and ΦE2 (r), as in (13.69). By contrast, handling normal n(r) the contribution of the magnetic current J MS (r) calls for the same careful treatment we devised for the MFIE in Section 13.2.3. However, a straightforward application of the duality transformations of Tables 6.1 and 8.1 to (13.108) yields the result immediately for J MS (r) in medium 1, viz., −

1 1 ˆ × [∇ × A M1 (r)] = ∓ J MS (r) n(r) ε1 2 % − j k1 R & e ˆ × ∇ × J MS (r ) , − PV dS  n(r) 4πR ∂V

whence we deduce the formula for −J MS (r) in medium 2, namely, −

1 1 ˆ × [∇ × A M2 (r)] = ± J MS (r) n(r) ε2 2

r ∈ ∂V ±

(13.176)

914

Advanced Theoretical and Numerical Electromagnetics + PV ∂V

% − j k2 R & e ˆ × ∇ × J MS (r ) , dS  n(r) 4πR

r ∈ ∂V ±

(13.177)

ˆ once by simply trading ε1 for ε2 and k1 for k2 . By cross-multiplying (13.176) and (13.177) with n(r) more from the right we get & % − j k1 R 1 e 1 ˆ − PV dS  ∇ × J MS (r ) , r ∈ ∂V ± (13.178) − [∇ × A M1 (r)]t = ∓ J MS (r) × n(r) ε1 2 4πR t ∂V & % − j k2 R 1 1 e ˆ + PV dS  ∇ × J MS (r ) , r ∈ ∂V ± (13.179) − [∇ × A M2 (r)]t = ± J MS (r) × n(r) ε2 2 4πR t ∂V

where the notation

  ˆ = {nˆ × [•]} × nˆ = I − nˆ nˆ · [•] [•]t := nˆ × {[•] × n}

(13.180)

indicates the extraction of the tangential-to-∂V component of the vector field within brackets. In ˆ in writing the contribution of J MS (r) to Es1 (r) for r ∈ ∂V + we pick up the term −J MS × n(r)/2 (13.178) because in (13.164) the observation point approaches the boundary from Vex . Likewise, in writing the contribution of J MS (r) to Es2 (r) for r ∈ ∂V − we choose the minus sign in (13.179) because in (13.172) the observation point is brought towards ∂V from Vin . By inserting the representations of the secondary electric fields into the continuity condition (13.174) we get

e− j k 1 R e− j k 1 R  ∇s JS (r ) + ∇ · JS dS  4πR j ωε1 4πR s ∂V ∂V & % − j k1 R 1 e   ˆ − PV dS ∇ × J MS (r ) − J MS (r) × n(r) 2 4πR t ∂V e− j k 2 R e− j k 2 R  ∇s JS (r ) − ∇ · JS = j ωμ2 dS  dS  4πR j ωε2 4πR s ∂V ∂V & % − j k2 R 1 e ˆ + PV dS  ∇ × J MS (r ) , − J MS (r) × n(r) 2 4πR t

Eit (r) − j ωμ1

dS 

r ∈ ∂V

(13.181)

∂V

where we notice that the direct contributions of J MS (r) appearing in both sides have the same sign and hence cancel each other out. To finalize the derivation we group together the pairs of similar integrals and invoke duality on the result to obtain − jω ∂V

/ − j k1 R / − j k1 R 0 0 e− j k 2 R  e e− j k 2 R ∇s   e + μ2 + dS μ1 dS JS (r ) + ∇ · JS 4πR 4πR jω 4πε1 R 4πε2 R s ∂V & % / − j k1 R 0 e e− j k 2 R + − PV dS  ∇ r ∈ ∂V × J MS (r ) = −Eit (r), 4πR 4πR t 

∂V

− jω ∂V

/ − j k1 R / − j k1 R 0 0 e e− j k 2 R  e e− j k 2 R ∇s + ε2 + dS  ε1 dS  J MS (r ) + ∇ · J MS 4πR 4πR jω 4πμ1 R 4πμ2 R s ∂V

(13.182)

Integral equations in electromagnetics + PV ∂V

& % / − j k1 R 0 e e− j k 2 R + dS  ∇ × JS (r ) = −Hit (r), 4πR 4πR t

r ∈ ∂V

915

(13.183)

which are the desired PMCHWT equations. According to the nomenclature of Section 13.1 they represent a set of Fredholm integral equations of the first kind.

13.3.2 The Müller integral equations The strategy devised by Müller [23, 26, 61] begins with the very definition of equivalent electric and magnetic currents on either side of ∂V. We rewrite (13.158), and (13.166) by highlighting the contributions of incident and scattered fields, viz., r ∈ ∂V + r ∈ ∂V −

ˆ = J MS (r), [Ei1 (r) + Es1 (r)] × n(r) ˆ = J MS (r), Es2 (r) × n(r)

(13.184) (13.185)

where in computing Es1 (r) and Es2 (r) we need to employ (13.176) and (13.177) with the minus sign. Then, we make a linear combination of (13.184) and (13.185) by multiplying the former with ε1 and the latter with ε2 . Summing the resulting equations side by side yields ˆ + ε2 Es2 (r) × n(r) ˆ = (ε1 + ε2 )J MS (r), ε1 [Ei1 (r) + Es1 (r)] × n(r)

r ∈ ∂V

(13.186)

r ∈ ∂V

(13.187)

whence we obtain the corresponding condition for the magnetic field ˆ × [Hi1 (r) + Hs1 (r)] + μ2 n(r) ˆ × Hs2 (r) = (μ1 + μ2 )JS (r), μ1 n(r)

by invoking the duality principle. By substituting the explicit expressions of the secondary fields we arrive at / 0 e− j k 1 R e− j k 2 R ˆ × dS  μ1 ε1 − μ2 ε2 j ωn(r) JS (r ) 4πR 4πR ∂V / − j k1 R 0 e ∇s e− j k 2 R  ˆ × dS  − n(r) − ∇s · JS jω 4πR 4πR ∂V % / − j k1 R & 0 e e− j k 2 R ˆ × ∇ ε1 − ε2 + PV dS  n(r) × J MS (r ) 4πR 4πR ∂V

1 ˆ × Ei (r), − (ε1 + ε2 )J MS (r) = ε1 n(r) 2 / 0 e− j k 1 R e− j k 2 R ˆ × dS μ1 ε1 j ωn(r) − μ2 ε2 J MS (r ) 4πR 4πR ∂V / − j k1 R 0 e e− j k 2 R  ∇s ˆ × − dS  − n(r) ∇s · J MS jω 4πR 4πR ∂V % / − j k1 R & 0 e e− j k 2 R ˆ × ∇ μ1 − μ2 − PV dS  n(r) × JS (r ) 4πR 4πR

∂V



r ∈ ∂V

(13.188)

Advanced Theoretical and Numerical Electromagnetics

916

1 ˆ × Hi (r), + (μ1 + μ2 )JS (r) = μ1 n(r) r ∈ ∂V (13.189) 2 which constitute the system of the Müller integral equations. According to the definitions of Section 13.1 (13.188) and (13.189) are Fredholm integral equations of the second kind and are amenable to an iterative solution [62].

13.4 Volume integral equations for inhomogeneous scatterers The surface integral equations of Sections 13.2 and 13.3 are quite convenient inasmuch as they turn the solution of the Maxwell equations plus boundary conditions into the task of finding one or more equivalent current densities flowing on just the boundary of the region of interest or the material interfaces between regions filled different media. Nonetheless, it is clear from the discussion that such formulation is feasible so long as the bodies in question are either PEC or good conductors or homogeneous penetrable isotropic media. As already noted, the reason for this limitation is that the integral representations of the secondary fields due to the objects are hinged on the Green function of a suitable unbounded isotropic medium. Since the relevant time-harmonic Green functions for penetrable media which are anisotropic or inhomogeneous or possibly both may not be known in closed form [35, Chapter 5], we are not able to obtain integral formulas analogous to (10.22) and (10.23). Consequently, it is not easy to turn a scattering problem involving an anisotropic or inhomogeneous object into a set of surface integral equations. Still, as an alternative to the direct solution of the Maxwell equations in local form we can formulate the problem as one or more volume integral equations [26]. The fundamental tool for the derivation is the volume equivalence principle of Section 10.5, and the resulting equations must be solved for unknown vector fields which extend in the region of interest. To keep the exposition lucid we consider the time-harmonic electromagnetic scattering from a penetrable object comprised of an isotropic inhomogeneous dielectric medium and immersed in free space, as is suggested in Figure 13.20a. Therefore, the permittivity of the object is ε(r), r ∈ V, and the permeability is μ0 .1 An application of the volume equivalence principle allows us to transform the Ampère-Maxwell law as in (10.88) and the electric Gauss law as in (10.103). This mathematical expedient amounts to replacing the effect of the body with equivalent electric current and charge densities given by Jeq (r) := j ω[ε(r) − ε0 ]E(r) = j ωP(r), ρeq (r) := −∇ · {[ε(r) − ε0 ]E(r)} = −∇ · P(r), ˆ · {[ε(r) − ε0 ]E(r)} = n(r) ˆ · P(r), ρS eq (r) := n(r)

r∈V

(13.190)

r∈V r ∈ ∂V −

(13.191) (13.192)

where the electric field is the total one, i.e., the sum of • •

the incident field Ei (r) radiated by the true sources located in R3 \ V and the secondary field Es (r) generated by the equivalent sources in V.

Since both true sources and equivalent ones radiate in an unbounded homogeneous medium with the constitutive parameters of free space (Figure 13.20b), we may formally write the electric field that solves the Maxwell equations as E(r) := Ei (r) + Es (r) 1 Unlike

the symbols used in Chapter 12, to lighten the notation we dispense with the diacritical ˜ in the frequency-domain constitutive parameters.

Integral equations in electromagnetics

(a)

917

(b)

Figure 13.20 Application and derivation of a volume integral equation: (a) electromagnetic scattering from an inhomogeneous isotropic dielectric body; (b) the volume equivalence principle is invoked to replace the object with equivalent sources in free space. ⎛ ⎞ ⎜⎜ e− j k 0 R ∇∇ ⎟⎟ Jeq (r ), = Ei (r) − j ωμ0 ⎜⎜⎝I + 2 ⎟⎟⎠ · dV  4πR k0

r ∈ R3

(13.193)

V

by virtue of (10.111) minus the contribution of the equivalent magnetic currents, which are absent in the problem under study. Although this equation yields the electric field in the whole space, we cannot use it because the equivalent electric current is yet unknown. However, there is a subset of points, namely, r ∈ V, where (13.190) must hold true and as such provides us with additional information on Jeq (r). Stated another way, the ‘right’ electric current Jeq (r) must produce a secondary electric field Es (r) which combines with Ei (r) in such a way that (13.190) is satisfied for r ∈ V. Indeed, by inserting the Ansatz (13.193) into (13.190) we get ⎛ ⎞ ⎜⎜⎜ ∇∇ ⎟⎟⎟ e− j k 0 R Jeq (r ) Jeq (r) = ω μ0 [ε(r) − ε0 ] ⎝⎜I + 2 ⎟⎠ · dV  4πR k0 2

V

+ j ω[ε(r) − ε0 ]Ei (r),

r∈V

(13.194)

which constitutes a Fredholm integral equation of the second kind for the unknown polarization current Jeq (r). For the numerical solution of (13.194) with the Method of Moments (Section 15.1) it is convenient to handle derivatives of order lower than the second. Thus, since the integral is in the form of a time-harmonic volume potential (9.12) we may move the divergence operator past the integral sign and onto the current as described in (10.114). This step along with trivial algebraic manipulations yields Jeq (r) =

⎤ ⎡ ⎥⎥⎥ ⎢ − ε0 ⎢⎢⎢⎢ e− j k 0 R e− j k 0 R  ⎥     ⎢⎢⎢ dV Jeq (r ) + ∇ dV ∇ · Jeq (r )⎥⎥⎥⎥ ⎦ ⎣ ε0 4πR 4πR V V e− j k 0 R ε(r) − ε0 ˆ  ) · Jeq (r ) + j ω[ε(r) − ε0 ]Ei (r), n(r ∇ dS  − ε0 4πR

ε(r) k02

∂V

r∈V

(13.195)

918

Advanced Theoretical and Numerical Electromagnetics

where we notice that the normal component of Jeq (r ) on the negative side of the boundary ∂V is ˆ  ) · Jeq (r )/(j ω) coincides non-null. This happens because Jeq (r ) is a polarization current, and n(r with the layer of unbalanced polarization charges (13.192) which are induced by the incident electric field at the interface between the object and free space. Incidentally, we could have written (13.195) directly with the aid of the electrodynamic potentials (9.23), (9.24) and (9.136) generated by the sources (13.190)-(13.192) in free space. An alternative volume integral equation for the problem of Figure 13.20a may be obtained by introducing the displacement vector D(r) as follows ε(r) − ε0 ε(r)E(r) = j ωκe (r)D(r), ε(r) & % ε(r) − ε0 ε(r)E(r) = −∇ · [κe (r)D(r)], ρeq (r) := −∇ · ε(r) & % ε(r) − ε0 ˆ · κe (r)D(r), ˆ · ρS eq (r) := n(r) ε(r)E(r) = n(r) ε(r) Jeq (r) := j ω

r∈V

(13.196)

r∈V

(13.197)

r ∈ ∂V −

(13.198)

where κe (r) is the scalar instance of the dielectric contrast factor (10.91). Secondly, we write the total electric field as E(r) =

1 D(r) = Ei (r) + Es (r), ε(r)

r∈V

(13.199)

and since incident and secondary fields are still given by (10.111) we may enforce the condition ⎛ ⎞ ⎜⎜ ε0 e− j k 0 R ∇∇ ⎟⎟⎟ i 2⎜ D(r) = ε0 E (r) + k0 ⎝⎜I + 2 ⎠⎟ · dV  κe (r )D(r ), r∈V (13.200) '()* ε(r) 4πR k0 Di (r)

V

on the grounds of (13.196). This is a Fredholm integral equation of the third kind for the unknown vector field D(r). The known term in the right member may be construed as the incident displacement vector in the background medium. Since the formulation is based on the division by the permittivity field ε(r), it is required that ε(r) not vanish for r ∈ V. Also in this case it is possible to interchange the volume integral and the divergence operator (cf. Section 2.8), viz., ε0 D(r) = ε0 Ei (r) + k02 ε(r)

V

dV 

e− j k 0 R κe (r )D(r ) 4πR e− j k 0 R , + ∇ dV  κe (r )D(r ) · ∇ 4πR

r∈V

(13.201)

r∈V

(13.202)

V

which passes over into ε0 D(r) = k02 ε(r)

V

dV 

e− j k 0 R e− j k 0 R  κe (r )D(r ) + ∇ dV  ∇ · [κe (r )D(r )] 4πR 4πR V − j k0 R e ˆ  ) · κe (r )D(r ) + ε0 Ei (r), n(r − ∇ dS  4πR ∂V

Integral equations in electromagnetics

919

by means of the usual limiting procedure which entails excluding the observation point r with a ball B(r, a) of vanishing radius in order to invoke the Gauss theorem (A.53). This last format requires that ε(r) be differentiable for r ∈ V. On the other hand, if ε(r) is only piecewise continuous, then the surface integral in (13.202) is augmented with contributions from the interfaces across which ε(r) ˆ  ) · D(r ) is formally evaluated on the negative side of the boundary ∂V, suffers jumps. Notice that n(r ˆ  ) · D(r ) is continuous across ∂V in light of the though in practice this detail is irrelevant because n(r matching condition (1.155). Once the equivalent electric current Jeq (r) has been found either from (13.195) or (13.202) together with (13.196), the scattered electric field may be computed everywhere by means of (13.193). An important feature of (13.201) and (13.202) is that they remain valid and numerically stable for arbitrarily small frequencies ω. As a matter of fact, in the limit as ω → 0, (13.201) and (13.202) become ε0 1 i D(r) = ε0 E (r) + ∇ dV  κe (r )D(r ) · ∇ , r∈V (13.203) ε(r) 4πR V

ε0 D(r) = ∇ ε(r)



dV 

V

1  ∇ · [κe (r )D(r )] 4πR 1 ˆ  ) · D(r )κe (r ) + ε0 Ei (r), n(r − ∇ dS  4πR

r∈V

(13.204)

∂V

which can be used to solve an electrostatic boundary value problem where Ei (r) represents the impressed static field (Section 3.5 and [63, Chapter 3]). The contrast factor κe (r) must be evaluated using the static limit of ε(r) if the medium is dispersive, as is reasonable (Chapter 12). Furthermore, since κe (r)D(r) = [ε(r) − ε0 ]E(r) = P(r),

r∈V

with a little algebra we may cast (13.203) and (13.204) as P(r) = [ε(r) − ε0 ]Ei (r) − [ε(r) − ε0 ]∇ dV  P(r ) · ∇ V



1 , 4πε0 R

(13.205)

r∈V

−∇ · P(r ) 4πε0 R V ˆ  ) · P(r ) n(r − [ε(r) − ε0 ]∇ dS  + [ε(r) − ε0 ]Ei (r), 4πε0 R

P(r) = −[ε(r) − ε0 ]∇

(13.206)

dV 

r∈V

(13.207)

∂V

and solve for the static polarization vector P(r). The volume integral in (13.206) is the electrostatic scalar potential Φ(r) due to a volume distribution of electrostatic dipoles (Section 3.7). Likewise, the integrals in (13.207) may be interpreted as the electrostatic scalar potential produced by volume and surface densities of polarization charges in V and on ∂V − , respectively. Two volume integral equations for determining the total time-harmonic magnetic field produced by distant sources in the presence of a magnetic medium endowed with permeability μ(r)

920

Advanced Theoretical and Numerical Electromagnetics

and permittivity ε0 — i.e., the dual configuration of the setup of Figure 13.20a — can be formulated immediately by invoking the principle of duality on (13.194) or (13.202). In symbols, we have ⎡ ⎤ ⎢⎢⎢ ⎥⎥⎥ − j k0 R − j k0 R e e ⎢⎢⎢ ⎥  J Meq (r ) + ∇ dV  ∇ · J Meq (r )⎥⎥⎥⎥ ⎢⎢⎣ dV ⎦ 4πR 4πR V V e− j k 0 R μ(r) − μ0 ˆ  ) · J Meq (r ) + j ω[μ(r) − μ0 ]Hi (r), r ∈ V n(r ∇ dS  − μ0 4πR

μ(r) − μ0 J Meq (r) = k02 μ0

(13.208)

∂V

μ0 B(r) = k02 μ(r)



dV 

V

e− j k 0 R e− j k 0 R  κm (r )B(r ) + ∇ dV  ∇ · [κm (r )B(r )] 4πR 4πR V e− j k 0 R ˆ  ) · B(r )κm (r ) − ∇ dS  n(r + μ0 Hi (r), '()* 4πR ∂V

r∈V

(13.209)

Bi (r)

where J Meq (r) is the equivalent magnetic current which obtains from (13.190) with the duality transformations or Table 6.1, and κm (r) is the scalar instance of the magnetic contrast factor (10.96). The magnetic permeability field μ(r) in (13.209) must not vanish for r ∈ V. Also (13.209) works well for frequency values all the way down to zero, in which case it yields the formulation of a magnetostatic boundary value problem [63, Chapter 5], viz., μ0 B(r) = μ0 Hi (r) +∇ '()* μ(r) Bi (r)



dV 

1  ∇ · [κm (r )B(r )] 4πR

V

−∇ ∂V

dS 

1 ˆ  ) · B(r )κm (r ), n(r 4πR

r∈V

(13.210)

where now μ(r) and κm (r) signify the static limit of the permeability and the contrast factor, if the medium is dispersive (Chapter 12). The analysis of the time-harmonic electromagnetic scattering from objects comprised of media which are both dielectric and magnetic calls for a system of coupled equations of the types considered so far because the volume equivalence principle leads to the introduction of both Jeq (r) and J Meq (r) in the region occupied by the body. The total field appearing in (13.190) and the dual thereof is now due to the external source but also to Jeq (r) and J Meq (r). Hence, the pertinent equations read ⎞ ⎛ ⎜⎜ e− j k 0 R ∇∇ ⎟⎟ Jeq (r ) Jeq (r) = j ω[ε(r) − ε0 ]Ei (r) + ω2 μ0 [ε(r) − ε0 ] ⎜⎜⎝I + 2 ⎟⎟⎠ · dV  4πR k0 V e− j k 0 R J Meq (r ), − j ω[ε(r) − ε0 ]∇ × dV  r∈V 4πR V

⎞ ⎛ ⎜⎜ e− j k 0 R ∇∇ ⎟⎟ J Meq (r ) J Meq (r) = j ω[μ(r) − μ0 ]Hi (r) + ω2 ε0 [μ(r) − μ0 ] ⎜⎜⎝I + 2 ⎟⎟⎠ · dV  4πR k0 V

(13.211)

Integral equations in electromagnetics + j ω[μ(r) − μ0 ]∇ ×

dV 

e− j k 0 R Jeq (r ), 4πR

r∈V

921

(13.212)

V

by virtue of (10.111) and (10.112). In like manner the formulation in terms of flux densities D(r) and B(r) reads e− j k 0 R ε0 D(r) = ε0 Ei (r) + k02 dV  κe (r )D(r ) ε(r) 4πR V − j k0 R e e− j k 0 R     ˆ  ) · κe (r )D(r ) ∇ · [κe (r )D(r )] − ∇ dS  n(r + ∇ dV 4πR 4πR V ∂V e− j k 0 R κm (r )B(r ), − j ωε0 ∇ × dV  4πR

r∈V

(13.213)

r∈V

(13.214)

V

e− j k 0 R μ0 i 2 B(r) = μ0 H (r) + k0 dV  κm (r )B(r ) μ(r) 4πR V − j k0 R e e− j k 0 R ˆ  ) · κm (r )B(r ) ∇ · [κm (r )B(r )] − ∇ dS  n(r + ∇ dV  4πR 4πR V ∂V e− j k 0 R κe (r )D(r ), + j ωμ0 ∇ × dV  4πR V

where we have used (13.202), (13.209) and simply added the contribution of J Meq (r) and Jeq (r) to Es (r) and Hs (r), respectively, on account of the principle of superposition. Finally, we write down the volume integral equations which are suited for inhomogeneous and anisotropic penetrable bodies. The steps needed for the derivation are the same, and the result simply follows by judiciously replacing the scalar constitutive parameters with dyadic ones and paying attention to the order of multiplication of dyadic and vector fields. For instance, (13.202) and (13.209) become [64] ε0 [ε(r)]

−1

· D(r) =

k02 V

e− j k 0 R e− j k 0 R    ∇ · [κe (r ) · D(r )] dV κe (r ) · D(r ) + ∇ dV  4πR 4πR V − j k0 R e ˆ  ) · κe (r ) · D(r ) + ε0 Ei (r), n(r − ∇ dS  r ∈ V (13.215) 4πR 

∂V

μ0 [μ(r)]−1 · B(r) = k02

V

e− j k 0 R e− j k 0 R  ∇ · [κm (r ) · B(r )] κm (r ) · B(r ) + ∇ dV  4πR 4πR V − j k0 R e ˆ  ) · κm (r ) · B(r ) + μ0 Hi (r), n(r − ∇ dS  r ∈ V (13.216) 4πR

dV 

∂V

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Advanced Theoretical and Numerical Electromagnetics

where the dyadic contrast factors κe and κm were defined in (10.91) and (10.96). The extension of the other equations is achieved in a similar fashion. As a final remark we mention that the equivalent charge densities (13.191), (13.192), (13.197) and (13.198) were introduced under the hypothesis that the constitutive parameters are differentiable throughout the volume V occupied by the object of concern. When this is not the case, e.g., because the constitutive parameters are just piecewise differentiable in V, then in (13.195), (13.202), (13.204) and (13.207) the volume integrals must be interpreted as a sum of integrals over the regions where the permittivity is differentiable, and the surface integrals pass over into a sum of integrals over the boundaries of the regions where the permittivity is differentiable. Similar considerations apply to the integral equations for magnetic media as well as to the scalar potential contributions in (13.213)(13.216).

13.5 Hybrid formulations Real-life scattering or radiation problems quite often involve both good conductors and penetrable media. As a consequence, none of the integral-equation-based approaches described so far is sufficient by itself to compute the relevant scattered or radiated fields. It is possible, though, to employ two or more formulations simultaneously, as the nature of the problem requires (e.g., [65]). For example, the electromagnetic scattering from a composite structure comprised of PEC bodies and homogeneous isotropic dielectric media may be formulated by combining the EFIE (13.70) with the PMCHWT equations (13.182), (13.183). It should be apparent that the three equations of concern are all coupled. Indeed, the secondary electric and magnetic field generated by the equivalent electric current density JS (r) on the surface of the conducting parts (Figure 13.3) also illuminates the dielectric parts. Conversely, the equivalent electric and magnetic current densities at the interface of the dielectric parts (Figure 13.18) produce a secondary electric field which must be considered while setting up the EFIE. To elaborate on mixed or hybrid formulations of electromagnetic problems in this section we address two of the numerous possibilities, namely, the combination of EFIE and volume integral equations and, even more generally, the mixing of surface integral equations with the local form of the Maxwell equations.

13.5.1 Electric-field and volume integral equations Computing electromagnetic fields in the presence of PEC objects and anisotropic, possibly inhomogeneous media calls for a mix of surface and volume integral equations (e.g., [66]). The EFIE is the preferred formulation for the conducting bodies because, as compared to the MFIE, it is more flexible, inasmuch as it works equally well for scattering and radiation problems and, perhaps more importantly, the range of applicability is not limited to closed surfaces (Figure 13.9a). To be more specific, we focus on solving the radiation from a PEC antenna in the presence of a cold magnetized plasma [67,68], [69, Section 6.5]. The problem has practical interest for the analysis and design of plasma sources [70, 71], plasma thrusters [72–75], and even plasma antennas [76]. In these devices a simple-shaped low-profile metallic antenna operated at relatively low frequencies is employed as a means to generate plasma or to heat it up or, sometimes, for both tasks simultaneously. In a plasma thruster, for instance, the same radio-frequency antenna may be used to produce the plasma by starting from a weakly naturally ionized gas (argon or nitrogen) and then to accelerate the charged particles, as is suggested in the artist’s impression of Figure 13.21.

Integral equations in electromagnetics

923

Figure 13.21 Artist’s impression of a plasma thruster: a longitudinal cut along a symmetry plane is shown in order to expose the radio-frequency antenna and the gas tube (© 2014 Elsevier B.V. Reprinted, with permission, from [74]). At a microscopic level this mechanism is essentially governed by the Lorentz force (1.4) which causes the ions to rotate around the streamlines of a static background magnetic field B0 , if the latter is present [77, Chapter 5], [68]. When a macroscopic description is adopted, though, the plasma enters the equations through a complex dyadic permittivity ε(r; ω) which is not symmetric in case B0 is non-zero.2 As a result, the acceleration of the particles is perceived as average power being absorbed by or lost into the plasma (Section 1.10.2). This phenomenon in turn is accounted for quantitatively by the anti-Hermitian part of the permittivity [see (E.55) and further below]. To get a feeling of how the system of Figure 13.21 can work as an engine, first of all we need to mention that, as the ions drift before and past the antenna placed in the mid-section of the device, the transverse component of the velocity of the ions / 0 B0 B0 B0 B0 × v× (13.217) =v−v· v⊥ := |B0 | |B0 | |B0 |2 increases thanks to the interaction with the electromagnetic field radiated by the antenna. Notice that here ‘transverse’ means perpendicular to the axial magnetic field B0 produced by the coil that surrounds the antenna. Secondly, as the ions enter the magnetic nozzle — a transition region where the background magnetic field becomes non-uniform, spreads, and ultimately dwindles — the transverse velocity diminishes as well, because the ratio m|v⊥ (r)|2 (13.218) μB := 2|B0 (r)| which is called the first adiabatic invariant (physical dimension: joule per tesla, J/T), is nearly constant [68], [78, Section 2.6]. Last but not least, within the nozzle the longitudinal or axial component 2

Unlike the symbols used in Chapter 12, to lighten the notation we do not insert the diacritical ˜.

924

Advanced Theoretical and Numerical Electromagnetics

(a)

(b)

Figure 13.22 Derivation of coupled EFIE and VIE for a plasma thruster [74]: (a) basic model of PEC antenna and magnetized plasma column; (b) result of the application of surface and volume equivalence principles.

of the velocity v (r) := v(r) ·

B0 (r) |B0 (r)|

(13.219)

must increase accordingly, since the total kinetic energy of a particle WK :=

 1 1  mv · v = m |v⊥ (r)|2 + |v (r)|2 2 2

(13.220)

is conserved, too. Therefore, the ions are expelled through the nozzle at high axial velocities, and this effect imparts the thruster a net velocity in the opposite direction because the linear momentum of the system is conserved as well [79]. In an engine of this type the exhaust flow through the nozzle can be modulated by simply controlling the injection of propellant from the reservoir into the plasma generation/acceleration stage. As a result, a plasma thruster has the remarkable ability to ‘shift gear’ during normal operations. That been said, we turn our attention to the task of writing down an EFIE and a VIE of the type (13.215). Since the transfer of electromagnetic energy from the actual generator to the plasma occurs in the neighborhood of the radio-frequency antenna, we consider a simplified, though satisfactory model which comprises the antenna and a cylindrical region of space occupied by plasma. The antenna and the plasma are immersed in free space. A typical antenna well-suited for the job is nothing but a conducting loop with the port modelled by means of a delta-gap source (Section 13.2.2). While this geometry, sketched in Figure 13.22a, shall serve as a basis for the development of the equations, the formulation we will obtain works also for arbitrary-shaped antennas and regions as well as any penetrable medium that can be described through a non-singular dyadic permittivity ε(r; ω). Besides, it is worthwhile mentioning that the static magnetic field B0 (included in Figure 13.22a for completeness) does not explicitly enter the equations to be derived but rather it is accounted for in the nine components of ε(r; ω) [67, 68], [74, Section 2.1]. Our first step consists of invoking both surface and volume equivalence principles in order to set up an equivalent problem which involves unknown electric current densities in free space. As regards

Integral equations in electromagnetics

925

the PEC antenna, by following the procedure detailed in Section 13.2.1 we end up with an electric surface current density (Figure 13.22b) r ∈ S +A

ˆ × H(r), JA (r) := n(r)

(13.221)

ˆ where n(r) is the outward unit normal on the antenna boundary S A , and H(r) is the magnetic field generated by all the sources, true and equivalent; we shall come back to this topic in a moment. The application of the surface equivalent principle is finalized by ‘replacing’ the PEC medium in VA with free space. In like manner, applying the volume equivalence principle to the plasma region, as is outlined in Section 13.4, leads us to the introduction of electric volume densities of current and charge, viz., JP (r) := j ωκe (r) · D(r), ! ρP (r) := −∇ · κe (r) · D(r) , ˆ · κe (r) · D(r), ρPS (r) := n(r)

r ∈ VP r ∈ VP

(13.222) (13.223)

r ∈ ∂VP−

(13.224)

where the dyadic contrast factor κe (r) is defined in (10.91). In these equations the displacement vector D(r) is generated by all the sources. Since the plasma is described macroscopically as a penetrable dielectric medium with magnetic permeability μ0 , then the equivalent magnetic currents and charges are null (Figure 13.22b). We now hark back to the question of sources. Clearly, in the system of Figure 13.22a truly responsible for the build-up of a time-harmonic electromagnetic field is the generator connected to the antenna port. If we adopt the delta-gap approximation of the port [30, 31], [24, Section 7.13], the latter can be modelled as a line γA ⊂ S A along which the known quasi-static electric field EiAA is non-null (actually, infinite). For this reason, we provisionally suppose that the gap is a small cylinder WG ⊂ VA with finite height h (cf. Figure 13.6) in order to avoid working with distributions. We will let h approach zero as we apply the Method of Moments in Section 15.5. As we learnt in Section 13.2.2, the equivalent current JA (r) produces an electric field which satisfies the boundary condition (13.79). But this is not the end of the story, given that in the equivalent configuration of Figure 13.22b in addition to JA (r) also JP (r) generates a secondary non-zero electric field on S A . Including this contribution is crucial for modelling the effect the plasma has on the antenna current JA (r). Thus, all in all the matching condition (13.79) for the electric field reads ⎧ ⎪ 0, r ∈ S +A \ (∂WG ∩ S +A ) ⎪ ⎪  !  ⎨ s s ˆ × ˆ n(r) EAA (r) + EAP (r) (13.225) × n(r) =⎪ V ⎪ ⎪ '()* '()* ⎩− G νˆ , r ∈ ∂WG ∩ S +A antenna antenna-plasma h self-interaction

interaction

where with evident notation • • •

EsAA (r) denotes the secondary electric field produced onto S A by sources located on S A , and this term represents the interaction of the antenna with itself; EsAP (r) indicates the secondary electric field produced onto S A by sources located in VP , and this term constitutes the action of the plasma onto the antenna; VG is the voltage drop across the gap (also see Figure 13.6).

Likewise, we need to carefully revise condition (13.199) which led us to the volume integral equation (13.215) for a dielectric body. While the role of the secondary field due to JP (r) is easily understood and identified, we see that the incident electric field cannot be EiAA (r). Indeed, in accordance with the delta-gap model of the antenna port, EiAA (r) vanishes everywhere but for r ∈ ∂WG ∩S +A ,

Advanced Theoretical and Numerical Electromagnetics

926

and hence it has no direct effect on the plasma. Then again, the electric current JA (r) flowing on S A radiates a secondary field which happens to be non-zero also for r ∈ VP . It is precisely this field that plays the role of ‘incident’ contribution from the viewpoint of the plasma. Therefore, for the equivalent problem of Figure 13.22b we state (13.199) in the form ! ε(r) −1 · D(r) = EsPA (r) + EsPP (r) , r ∈ VP (13.226) '()* '()* plasma-antenna interaction

plasma self-interaction

where • • •

D(r) is the displacement vector within VP generated by all sources; EsPA (r) denotes the secondary electric field produced within VP by sources located on S A , and this term constitutes the action of the antenna onto the plasma; EsPP (r) indicates the secondary electric field produced within VP by sources located in VP , and this term represents the interaction of the plasma with itself. More explicitly, on account of (13.65), (13.66), and (10.111), we have e− j k 0 R e− j k 0 R  ∇ EsAA (r) = − j ωμ0 dS  JA (r ) + ∇ · JA (r ), dS  4πR j ωε0 4πR s SA SA ⎛ ⎞ − j k0 R ⎜ ⎟ ∇∇ e ⎜ ⎟ κe (r ) · D(r ), EsAP (r) = ω2 μ0 ⎜⎝⎜I + 2 ⎟⎠⎟ · dV  4πR k0 V − j k0 R e e− j k 0 R  ∇ s   EPA (r) = − j ωμ0 dS JA (r ) + ∇ · JA (r ), dS  4πR j ωε0 4πR s SA SA ⎛ ⎞ − j k R 0 ⎜⎜ e ∇∇ ⎟⎟ κe (r ) · D(r ), EsPP (r) = ω2 μ0 ⎜⎝⎜I + 2 ⎟⎠⎟ · dV  4πR k0

r ∈ S +A

(13.227)

r ∈ S +A

(13.228)

r ∈ VP

(13.229)

r ∈ VP

(13.230)

VP

where ∇s · {•} denotes the surface divergence on S A . With these positions, (13.225) and (13.226) constitute a system of coupled electric-field and volume integral equations to be solved for the unknowns JA (r), r ∈ S +A , and D(r), r ∈ VP [74]. Inserting (13.227)-(13.230) into (13.225) and (13.226) yields − j ωμ0 SA



+ ω2 μ0 VP

dS 

e− j k 0 R 1 JA (r ) + ∇s 4πR j ωε0

!−1

dS 

SA

e− j k 0 R 1 dV  κe (r ) · D(r ) + ∇s 4πR ε0

1 − ∇s ε0

ε0 ε(r)



SP



e− j k 0 R  ∇ · JA (r ) 4πR s dV 

e− j k 0 R  ∇ · [κe (r ) · D(r )] 4πR

VP

⎧ ⎪ 0, r ∈ S +A \ (∂WG ∩ S +A ) ⎪ ⎪ e− j k 0 R ⎨    ˆ ) · κe (r ) · D(r ) = ⎪ n(r dS V ⎪ ⎪ 4πR ⎩− G νˆ , r ∈ ∂WG ∩ S +A h 

· D(r) = − j ωε0 μ0 SA

dS 

1 e− j k 0 R JA (r ) + ∇ 4πR jω

SA

dS 

e− j k 0 R  ∇ · JA (r ) 4πR s

(13.231)

Integral equations in electromagnetics + k02

dV 

e− j k 0 R κe (r ) · D(r ) + ∇ 4πR

VP

VP

dV 

−∇

927

e− j k 0 R  ∇ · [κe (r ) · D(r )] 4πR

dS 

e− j k 0 R ˆ  ) · κe (r ) · D(r ), n(r 4πR

r ∈ VP

(13.232)

SP

where we have multiplied (13.226) by ε0 and moved the divergence inside the integral over VP by following the procedure described in (10.114). In so doing we have tacitly assumed that the plasma permittivity tensor is differentiable throughout VP . Or else, if κe (r) is only piecewise differentiable, in (13.231) and (13.232) the volume integrals contributed by the scalar potential in the plasma must be interpreted as a sum of integrals over the regions where the contrast factor is differentiable, and the associated surface integrals pass over into a sum of integrals over the boundaries of the regions where κe (r) is differentiable. While in most applications conduction losses are unwanted and regarded as problematic, in the case of a plasma thruster one actually seeks to maximize the average power ‘dissipated’ within the plasma, because the energy that is drawn from the generator through the antenna serves to increase the transverse velocity v⊥ of the ions and is ultimately converted into linear momentum of the device. Therefore, it is necessary to derive a suitable formula for the calculation of the average power PP  absorbed by the plasma. To this purpose we consider a ball B(0, a) large enough to contain the system of Figure 13.22a and we apply the complex Poynting theorem (1.314) to the region of space B(0, a) \ VA . We exclude the region occupied by the PEC antenna because therein the electromagnetic field is zero, but we retain VP where the polarization or plasma current JP (r) flows. In symbols, we have 1 Re 2



ˆ · E(r) × H∗ (r) dV n(r)

B(0,a)

1 − Re 2



1 ˆ · E(r) × H∗ (r) = − Re dV n(r) 2

SA



dV E(r) · J∗P (r) (13.233)

VP

where the negative sign before the flux integral over S A is due to the previous choice for the local unit normal which is oriented outward VA and inward B(0, a) \ VA . From Section 1.10.2 we recall that the first term in (13.233) is PF , i.e., the average power radiated by the sources towards infinity. It is worthwhile mentioning that the electromagnetic field over B(0, a) is constituted solely by the secondary contributions produced by JA (r) and JP (r), in that the quasi-static field EiAA (r) in the gap WG does not radiate. The second term in (13.233) should be cast into a form that highlights the voltage VG of the ideal generator connected to the antenna port. We observe that (Section 13.2.2 and also Example 6.8) 1 VG 1 1 ˆ · E(r) × H∗(r) = dS n(r) dS Et (r) · J∗A (r) = − dS νˆ · J∗A (r) = − VG IA∗ (13.234) − 2 2 2h 2 SA

SA

∂WG ∩S A

having made use of (13.64), (13.225) and (13.89). The negative sign tells us that the corresponding integral represents average power delivered by the generator. The time rate of energy transfer from electric sources to the field is given by (1.309). The negative sign is precisely an indication of the energy leaving the sources and getting either stored in the field or radiated towards infinity never to come back. Thus, the third term in (13.233) would

928

Advanced Theoretical and Numerical Electromagnetics

represent the power delivered by the sources if the current JP (r) were actually an impressed one, but this is not the case. On the contrary, the polarization current depends on the applied field, and hence this term is the negative of the time rate at which the electric field does work on the charges within VP [41, 63, 80, 81]. In fact, we should rewrite (13.233) as 1 1 1 ∗ ∗ ˆ · E(r) × H (r) + Re dV E(r) · J∗P (r) Re{VG IA } = Re dV n(r) (13.235) 2 2 2 VP

B(0,a)

and succinctly as PS  = PF  + PP 

(13.236)

which clarifies how the power delivered by the generator is spent. Since the task of the radiofrequency antenna in the thruster is not to radiate toward the background environment, ordinarily the term PF  is negligible, and the approximate relationship PP  ≈

1 Re{VG IA∗ } 2

(13.237)

may be employed to get a very good estimate of the power transferred to the plasma. It takes a little more algebra to arrive at a formula for PP  that explicitly involves D(r) and the anti-Hermitian part of [ε(r)]−1 [cf. (E.55)], namely, 1 PP  = Re dV E(r) · (− j ω)κ∗e (r) · D∗ (r) 2 VP # $−1 ω = Im dV D(r) · εT (r) · κ∗e (r) · D∗ (r) 2 VP ! ω (13.238) = Im dV D∗ (r) · κeH (r) · ε(r) −1 · D(r) 2 VP

and the integrand can be manipulated as follows + , $ 1 2−1 1 # ∗ H −1 Im D∗ · κeH · ε−1 · D = D · κe · ε · D − D · κTe · ε∗ · D∗ 2j   −1 1  ∗ H −1 · κe · D = D · κ e · ε · D − D∗ · ε H 2j    −1 1 = D∗ · κeH · ε−1 − εH · κe · D 2j   −1 −1  H −1  −1 −1  1 ·ε − ε + ε0 ε H ·ε ·D = D∗ · ε−1 − ε0 εH 2j   −1  + , 1 · D = Im D∗ · ε−1 · D = D∗ · ε−1 − εH 2j '()*

(13.239)

anti-Hermitian

on account of (10.91) and (B.7). Finally, the average absorbed power is conveniently computed as ω ! PP  = Im dV D∗ (r) · ε(r) −1 · D(r) (13.240) 2 VP

Integral equations in electromagnetics

929

essentially because ε(r) is assigned, and D(r) is one of the unknowns of the integral formulation. Nonetheless, we can retrieve the more familiar expression that involves ε(r) and the total electric field [81]

PP  =

ω Im 2



dV E∗ (r) · εH (r) · ε(r)

!−1

· ε(r) · E(r)

VP

ω = Im 2



dV E∗ (r) · εH (r) · E(r)

VP

=−

ω 2



dV E∗ (r) ·

ε(r) − εH (r) · E(r) 2j

(13.241)

VP

but this form is less useful for the present problem inasmuch as E(r) is not a primary unknown of (13.231) and (13.232).

13.5.2 Integral and wave equations The second hybrid formulation we discuss in details consists of combining surface integral equations with the wave equation for the solution of wave scattering and radiation problems which involve PEC antennas and inhomogeneous isotropic penetrable media (e.g., [82, 83]). This approach — specialized to time-harmonic sources and fields — allows us to exploit the robustness of integral equations along with the versatility of the local Maxwell equations. More precisely, we shall use the former to handle the PEC bodies and the unknown fields on the boundary of the penetrable objects, whereas the wave equations will describe the evolution of the fields within the penetrable objects without resorting to a Green function. For this reason, as we shall see in Section 15.6, the numerical solution of the wave equation gives rise to extremely sparse matrices. While the applications are numerous, for the sake of argument here we focus on the analysis and design of a gaseous plasma antenna (GPA) which at the very least is comprised of a metallic feeding system and a spatial region occupied by plasma. Two practical configuration are shown in Figure 13.23. A GPA is a radiating device in which a plasma discharge is employed to transmit and receive electromagnetic waves [76, 84, 85]. Among the potential advantages of GPAs over conventional metallic antennas we mention that GPAs can be electrically reconfigurable [86–88] in terms of, e.g., gain and frequency on timescales the order of milliseconds or less, GPAs are virtually invisible when de-energized, and are transparent to electromagnetic waves above the plasma frequency [68] (cf. Section 12.3.1). In fact, GPAs designed to operate at different frequencies do not interfere with one another. As a result, it is possible to stack GPAs and metallic elements together to form a plasma antenna array (PAA) [89]. PAAs allow steering the main beam and improving the antenna directivity [30, Section 2.5] by adding nulls to the radiation pattern. In practice, an array of plasma discharges surrounding a radiating element (which can be either metallic or another plasma antenna) effectively realizes a ‘plasma blanket’ that can shield unwanted incoming waves and let outward waves through ‘windows’ opened in selected positions. With reference to Figure 13.24a we suppose that the GPA exists in free space. Besides, we model the metallic parts as PEC and the antenna port with a delta-gap WG of finite height h which we will let approach zero in the end (Section 13.2.2). The volume occupied by the metallic bodies is VA . The plasma occupies the domain VP and constitutes an inhomogeneous lossy medium characterized by a

Advanced Theoretical and Numerical Electromagnetics

930

(a) antenna array

(b) reflector

Figure 13.23 Practical gaseous plasma antennas (pictures courtesy of T. Anderson, Haleakala Research and Development Inc., Brookfield, MA, USA) scalar complex permittivity ω2pe ε0

ε˜ c (r; ω) := ε0 −

ω(ω − j νe )

,

r ∈ VP

(13.242)

where • •

ω pe is the plasma electron frequency defined in (12.63); νe is the phenomenological collision frequency which accounts for the rate of transfer of kinetic energy from the lighter electrons to the heavier ions in the gas.

Notice that (13.242) has precisely the functional form we already derived in (12.65) for a conducting medium [90, Section 1.4]. The plasma is inhomogeneous if the electron number density Ne — which affects both ω pe and νe — is a function of position. To distinguish the fields on either side of S P := ∂VP we call region ➀ the unbounded domain R3 \ (V P ∪ V A ) and region ➁ the volume VP . Region ➂ is the domain occupied by the metallic parts, though we will not need to reference the fields within VA , as they are zero by hypothesis. Finally, we indicate the boundary of VA with S A . To derive a set of coupled integral and differential equations we apply the Love equivalence principle (Section 10.4.1) in order to obtain suitable integral formulas for the electromagnetic field in region ➀, i.e., E1 (r) and H1 (r) with r ∈ R3 \ (V A ∪ V P ). According to the procedure outlined in Section 10.4.1 we construct an equivalent problem from the viewpoint of an observer who sits in R3 \ (V A ∪ V P ) by ‘replacing’ the PEC bodies and the plasma with free space and by setting surface current and charge densities against S +A and S +P (Figure 13.24b). Then, by invoking the principle of superposition to combine the effect of sources on S +A and S +P we can write E1 (r) = − j ωμ0

dS 

S A ∪S P

e− j k 0 R  ˆ ) × H1 (r ) − ∇ n(r 4πR



dS 

S A ∪S P

e− j k 0 R  ˆ ) · E1 (r ) n(r 4πR

Integral equations in electromagnetics

(a)

931

(b)

Figure 13.24 Derivation of coupled surface integral equations and Maxwell equations for a plasma antenna [88]: (a) basic model of antenna comprising PEC feeder and plasma region; (b) result of the application of the surface equivalence principle. −∇×

dS 

e− j k 0 R ˆ  ), E1 (r ) × n(r 4πR

SP



H1 (r) = − j ωε0

dS 

e− j k 0 R ˆ ) − ∇ E1 (r ) × n(r 4πR

SP



+∇×

r ∈ R3 \ (V A ∪ V P )

dS 

(13.243)

e− j k 0 R ˆ  ) · H1 (r ) n(r 4πR

SP

dS 

S A ∪S P

− j k0 R

e ˆ  ) × H1 (r ), n(r 4πR

r ∈ R3 \ (V A ∪ V P )

(13.244)

where we have used (9.152) and (9.153) in combination with (9.136), (9.151) and the dual formulas ˆ  ) for r ∈ S A thanks to the jump thereof. Furthermore, there is no contribution from E1 (r ) × n(r condition (13.79) on a PEC surface with a delta-gap region and the fact that the field is quasi-static in ˆ  ) · H1 (r ) are not independent unknown quantities ˆ  ) · E1 (r ) and n(r the gap. Lastly, we recall that n(r because for r ∈ S A ∪ S P we have ˆ  ) · E1 (r ) ˆ  ) × H1 (r )] = −n(r ˆ  ) · ∇ × H1 (r ) = − j ωε0 n(r ∇s · [n(r ˆ  ) · H1 (r ) ˆ  )] = n(r ˆ  ) · ∇ × E1 (r ) = − j ωμ0 n(r ∇s · [E1 (r ) × n(r

(13.245) (13.246)

on account of (A.60), (1.98) and (1.99). Next, we observe that, so long as the plasma is inhomogeneous, we cannot write down similar formulas from the standpoint of an observer within VP , as we did to derive the equations of Poggio and Miller in Section 13.3.1 or those of Müller in Section 13.3.2, the reason being that a Green function is not available for an arbitrary inhomogeneous unbounded isotropic medium! Nor do we wish to invoke the volume equivalence principle (Section 10.5) which ultimately would lead us to an integral equation for the fields within VP . Thus, we proceed by introducing an auxiliary electromagnetic problem comprised of sources that are located somewhere outside V P ∪ V A and radiate in the absence of the plasma and the conductors. While said sources are quite arbitrary, we demand that the electromagnetic field Eaux (r), Haux (r) they produce obey the condition ˆ = E1 (r) × n(r), ˆ Eaux (r) × n(r)

r ∈ SP

(13.247)

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Advanced Theoretical and Numerical Electromagnetics

i.e., the tangential electric field on the boundary of VP coincides with the true and yet unknown tangential component of E1 (r) on S P in the original problem. Additionally, (13.247) implies ˆ · μ0 Haux (r) = n(r) ˆ · μ0 H1 (r), n(r)

r ∈ SP

(13.248)

by virtue of a relationship analogous to (13.246). The actual form of the auxiliary sources is unessential for the discussion. If we further apply the Love equivalence principle to the auxiliary problem for observation points r ∈ VP we get e− j k 0 R e− j k 0 R  ˆ  ) × Haux (r ) + ∇ dS  ˆ ) · Eaux (r ) Eaux (r) = j ωμ0 dS  n(r n(r 4πR 4πR SP



+∇×

SP

e− j k 0 R ˆ  ), Eaux (r ) × n(r dS  4πR

r ∈ VP

SP

Haux (r) = j ωε0

dS 

e− j k 0 R ˆ ) + ∇ Eaux (r ) × n(r 4πR

SP



−∇×



dS 

(13.249)

e− j k 0 R ˆ  ) · Haux (r ) n(r 4πR

SP

e− j k 0 R ˆ  ) × Haux (r ), dS  n(r 4πR

r ∈ VP

(13.250)

SP

where we have taken into account that the unit normal on S P is positively oriented towards the outside of VP . As we learned in Section 10.4.1, these integral representations return zero when they are evaluated for points r  V P . For this reason, we can sum (13.243) and (13.249) side by side without altering the values of the electric field for r ∈ R3 \ (V A ∪ V P ); the same goes for (13.244) and (13.250). These manipulations are advantageous inasmuch as they allow us to eliminate the ˆ  ) on S P in light of (13.247). All in all, for the electric field we have contribution of E1 (r ) × n(r E1 (r) = − j ωμ0 ∇ + j ωε0



dS  JA (r )

e− j k 0 R 4πR

SA

dS 

e− j k 0 R  ∇ · JA (r ) − j ωμ0 4πR s

SA

∇ + j ωε0





dS 

e− j k 0 R JP (r ) 4πR

SP

dS 

e− j k 0 R  ∇ · JP (r ), 4πR s

(13.251)

SP

and similarly, for the magnetic field we get e− j k 0 R e− j k 0 R JA (r ) + ∇ × dS  JP (r ), H1 (r) = ∇ × dS  4πR 4πR SA

r ∈ R3 \ (V A ∪ V P )

r ∈ R3 \ (V A ∪ V P )

(13.252)

SP

where ˆ × H1 (r), JA (r) := n(r) ˆ × [H1 (r) − Haux (r)], JP (r) := n(r) are two unknown electric surface current densities.

r ∈ S +A

(13.253)

S +P

(13.254)

r∈

Integral equations in electromagnetics

933

Apart from serving our purposes well, (13.251) and (13.252) are theoretically important because they show that an electromagnetic field in an unbounded region of space can be determined by assigning only suitable electric surface current densities on the boundary of the domain. This result is not surprising and quite in line with the statement of the uniqueness theorem derived in Section 6.4.2. On the other hand, the integral formulas (13.251) and (13.252), unlike the representations of Stratton and Chu (Section 10.2), do not return a null field for observation points in the complementary region. More importantly, the unknown current JP (r) is not just the tangential magnetic field of the problem ˆ under investigation, but rather the combination of n(r) × H1 (r) and the tangential magnetic field of the auxiliary problem. This setup should also be contrasted with the Schelkunoff equivalence principle of Section 10.4.2. Parenthetically, had we chosen the hypothetical auxiliary sources so as to enforce the dual counterparts of (13.247) and (13.248) in accordance with the substitutions of Table 6.1, we would have obtained integral representations in terms of a magnetic surface current density J MP (r) over S +P . Since we plan on setting up three surface integral equations by enforcing the matching conditions (13.79) on S A as well as (1.197) and (1.196) across S P , we need to examine the limiting values of (13.251) for r → S +A and of (13.251) and (13.252) for observation points on S +P . In particular, we are interested in the parts of the fields that are tangential to S A and S P . As we let r approach S +A the contributions of the currents flowing on S P remain regular, whereas the integrals over S A becomes singular. Nonetheless, we recall from Section 13.2.1 that only the normal component of the gradient of the single-layer potential suffers a jump, whereby we conclude e− j k 0 R ˆ × [E1 (r) × n(r)] ˆ n(r) JA (r ) = Et1 (r) = − j ωμ0 dS  4πR +

∇s j ωε0



SA

dS 

− j k0 R

e ∇ · JA (r ) − j ωμ0 Is (r) · 4πR s

SA



dS 

e− j k 0 R JP (r ) 4πR

SP

+

∇s j ωε0



dS 

e− j k 0 R  ∇ · JP (r ), 4πR s

r ∈ S +A

(13.255)

SP

where ˆ n(r), ˆ Is (r) := I − n(r)

r ∈ SA

(13.256)

ˆ is the identity dyadic transverse to n(r) on S A . Dot-multiplying a vector field by Is (r) yields the ˆ component perpendicular to n(r), and this is equivalent to carrying out the triple cross product in the left-hand side of (13.255). We notice that if JA (r) and JP (r) were the ‘right’ current densities, then the right member of (13.255) would satisfy (13.79). Stated another way, we seek JA (r) and JP (r) so that ∇s e− j k 0 R  e− j k 0 R + ∇ · JA (r ) − j ωμ0 dS  JA (r ) dS  4πR j ωε0 4πR s SA



− j ωμ0 Is (r) · SP

SA

e− j k 0 R ∇s JP (r ) + dS  4πR j ωε0

SP

dS 

e− j k 0 R  ∇ · JP (r ) 4πR s ⎧ V G ⎪ ⎪ ⎪ ⎨− vˆ , h =⎪ ⎪ ⎪ ⎩0,

r ∈ ∂WG ∩ S +A r ∈ S +A \ (∂WG ∩ S +A )

(13.257)

934

Advanced Theoretical and Numerical Electromagnetics

where vˆ is the unit vector parallel to the impressed quasi-static electric field in the gap (see Figure 13.6), and VG is the voltage drop through the gap. Clearly, (13.257) is a special instance of the EFIE (13.83), where the PEC surfaces are also illuminated by the secondary electric field scattered by the plasma. Next, we let r approach S +P and observe that only the integrals over S P now become singular. Concerning the electric field, the tangential component of the gradient of the single-layer potential is continuous across S P . As a result, we obtain e− j k 0 R e− j k 0 R  ∇s JA (r ) + ∇ · JA (r ) Et1 (r) = − j ωμ0 Is (r) · dS  dS  4πR j ωε0 4πR s − j ωμ0 SP

SA

dS 

− j k0 R

e ∇s JP (r ) + 4πR j ωε0



SA

dS 

e− j k 0 R  ∇ · JP (r ), 4πR s

r ∈ S +P

(13.258)

SP

where Is (r) is defined as in (13.256) though for points on S P . As regards the magnetic field, we must be a little more careful, inasmuch as H1 (r) consists of contributions which are the curl of vector potential terms, as was the case for the MFIE examined in Section 13.2.3. We recall that the tangential components of said contributions suffer a jump across the relevant current-carrying surface. This observation applies to S P but not to S A for r → S ±P . Then again, owing to the way it was derived, the integral representation (13.252) holds only for points outside V P (and V A ). By following the limiting procedure detailed in Section 13.2.3 we get % − j k0 R & e ˆ × ∇ ˆ × H1 (r) = dS  n(r) n(r) × JA (r ) 4πR SA

% − j k0 R & e ˆ × ∇ × JP (r ) , dS  n(r) 4πR

r ∈ S +P

(13.259)

which in principle can be further written as % − j k0 R & 1 e ˆ × ∇ ˆ × [H1 (r) + Haux (r)] = n(r) × JA (r ) dS  n(r) 2 4πR SA % − j k0 R & e ˆ × ∇ × JP (r ) , + PV dS  n(r) 4πR

r ∈ S +P

(13.260)

1 + JP (r) + PV 2



SP

SP

by virtue of (13.254). We must seek JA (r) and JP (r) so that (1.197) and (1.196) hold true for r ∈ S P . In symbols, we find e− j k 0 R JA (r ) − j ωμ0 Is (r) · dS  4πR ∇s + j ωε0



SA

dS 



e− j k 0 R  ∇ · JA (r ) − j ωμ0 4πR s

SA

dS 

e− j k 0 R JP (r ) 4πR

SP

∇s + j ωε0



SP

dS 

e− j k 0 R  ∇ · JP (r ) = Et2 (r), 4πR s

r ∈ SP

(13.261)

Integral equations in electromagnetics

935

% − j k0 R & e 1 ˆ × ∇ × JA (r ) + JP (r) dS  n(r) 4πR 2

SA

+ PV

% − j k0 R & e ˆ × ∇ ˆ × H2 (r), × JP (r ) = n(r) dS  n(r) 4πR

r ∈ SP

(13.262)

SP

ˆ × H2 (r) constitute the electromagnetic field on the negative side of S P , i.e., where Et2 (r) and n(r) within the plasma. The combination of (13.257), (13.261) and (13.262) represents a system of three coupled inteˆ × H2 (r). Since we are short gral equations in four unknowns, namely, JA (r), JP (r), Et2 (r) and n(r) of one relationship, we supplement (13.257), (13.261) and (13.262) with the homogeneous wave equation for the electric field E2 (r) within the plasma, viz., ∇ × ∇ × E2 (r) − ω2 μ0 ε˜ c (r)E2 (r) = 0,

r ∈ VP

(13.263)

because we have not contemplated other sources within VP . From the viewpoint of (13.263), it ˆ and n(r) ˆ × H2 (r) on would appear that (13.261) and (13.262) assign the values of both E2 (r) × n(r) the boundary of the solution domain VP . In accordance with the uniqueness theorem in bounded lossy regions (Section 6.4.1) we know that not only is this double specification unnecessary, it may ˆ and n(r) ˆ × H2 (r) are not independent of also lead to erroneous results in general, since E2 (r) × n(r) each other. However, in our setup we also have the current densities JP (r) and JA (r) at our disposal. Therefore, we can think of (13.261) as a boundary condition for Et2 (r) and of (13.262) as a means ˆ × H2 (r) once JP (r) and JA (r) have been found. to determine n(r) The numerical solution of (13.257), (13.261), (13.262) and (13.263) is discussed in Section 15.6, where the coupling between the wave equation and (13.261), (13.262) will be made explicit. As a concluding remark, we observe that the hybrid formulation works equally well for any penetrable inhomogeneous anisotropic medium endowed with dyadic constitutive parameters εc (r) and μ(r), in which case (13.263) is replaced by the source-free instance of (1.244).

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Chapter 14

The Method of Moments I

Since the time-harmonic integral equations as well as the hybrid formulations developed in Chapter 13 can rarely be solved analytically, one resorts to numerical approaches in order to construct approximate, though accurate, solutions. Although a few strategies are available, in this and the following chapter we examine the Method of Moments in great detail. The technique, which is also known as the method of weighted residuals, originated in the Russian scientific community [1], but the application to integral equations for electromagnetic problems is attributed to J. H. Richmond (1965) [2] and even more so to R. F. Harrington (1967) [3, 4]. After describing the main features of the strategy in general terms (Section 14.1) we go on to specialize the MoM to surface integral equations for PEC and homogenous isotropic scatterers in Sections 14.2-14.7, where it will become clear that, as part of the solution process, one has to compute nested surface integrals. This topic is dealt with in Sections 14.8 and 14.9. The solution of the EFIE for antenna problem is addressed separately in Section 14.10 because of the additional difficulty posed by the delta-gap model of the antenna port. Finally, in Section 14.11 we show how the solution to a given scattering or antenna problem is affected when the size of the object in question is scaled and the frequency is modified accordingly.

14.1 General considerations To cover a larger number of cases we state a generic integral equation symbolically and succinctly as L {J(r; ω)} = −F(r; ω),

r∈D

(14.1)

where • • • •

D ⊂ R3 is the region of space (a line, a surface or a volume) where condition (14.1) must be fulfilled; L {•} : VL −→ WL denotes a linear integral operator (see Appendix D.3); J(r; ω), r ∈ D, indicates the unknown vector field, quite often a current density, which is an element of the space VL , the domain of L {•}; F(r; ω), r ∈ D, stands for the known forcing term or source, usually the incident or impressed electromagnetic field, which is an element of the range of L {•}, i.e., the subspace ran L ⊆ WL [see (D.118)].

The numerical solution of (14.1) with the Method of Moments [1, 3, 5, 6], [7, Chapter 3], [8, Chapter 5], [9, Chapter 10], [10, Section 8.4], [11, Section 1.8], [12, Chapter 10], [13, Chapter 13] consists essentially of three steps: (1)

the integral equation is first transformed into a discrete (matrix) equation through a process referred to as discretization;

942 (2) (3)

Advanced Theoretical and Numerical Electromagnetics the entries of the system matrix and of the excitation vector are computed; the resulting algebraic linear system is inverted.

The discretization process begins with the introduction of a suitable set of basis functions or expansion functions B := {fn (r)}+∞ n=1 ⊆ VL ,

r∈D

(14.2)

and the subsequent formal expansion of the unknown, namely, J(r; ω) =

+∞ 

r∈D

In (ω)fn (r),

(14.3)

n=1

where the series is assumed to be convergent and, more importantly, to converge to the sought solution to (14.1). Thanks to (14.3) the expansion coefficients In (ω) become the new unknown of the problem, whereby the search for J(r) translates into finding In (ω). It is customary to call B the basis of expansion functions, even though most times the elements fn (r) are not orthogonal to one another with respect to the relevant inner product (see Sections 14.7 and 15.2). On the other hand, it is important that the fn (r) be linearly independent [14, Section 9.1], that is, any element of the set B cannot be represented by means of a linear combination of the remaining basis functions, as is done for J(r) in (14.3). In symbols, this means that the equation +∞ 

cn fn (r) = 0,

r∈D

(14.4)

n=1

is satisfied only for cn = 0, n ∈ N. If VL is an infinite-dimensional function space, using an infinite set of basis functions, as is hinted at by (14.2), is necessary in principle, but this choice is unpractical nonetheless, inasmuch as the discretization process transforms the original integral equation (14.1) into an algebraic system of infinitely many coupled equations. An exception worth mentioning is the special case where L {•} is self-adjoint (Appendix D.6) and B is the set of orthogonal eigenfunctions of L {•} (Appendix D.7), that is, L {fn (r)} = νn fn (r),

r ∈ D,

n∈N

(14.5)

in which instance the algebraic equations of concern are all uncoupled and the system can be solved trivially. Unfortunately, though, if the eigenfunctions of L {•} are not known analytically, the task of computing them numerically — i.e., solving (14.5) with the Method of Moments — may be even more demanding than solving (14.1) directly. As a consequence, one is usually compelled to limit the number of basis functions to a certain N < +∞ and assume J(r; ω) =

N 

In (ω)fn (r) −  N (r),

r∈D

(14.6)

n=1

where  N (r) is the approximation error caused by the truncation of the series in (14.3). To continue the discretization procedure we insert the Ansatz (14.6) into the integral equation (14.1) to get L {J(r; ω)} =

N  n=1

In (ω)L {fn (r)} − L { n (r)} = −F(r; ω),

r∈D

(14.7)

The Method of Moments I

943

where we have exploited the linearity of the operator to transfer the action of L {•} from J(r; ω) onto the basis functions. The quantity RN (r) := L { N (r)}

(14.8)

is called the residual of the approximation and should not be confused with  N (r). Ideally, we would like to choose the coefficients In (ω) so that RN (r) = F(r; ω) +

N 

In (ω)L {fn (r)} = 0,

r∈D

(14.9)

n=1

but this condition cannot be achieved in practice, if the integral equation (14.1) is actually solvable. To understand why, we observe that the truncation error is non-null in general. Consequently, the requirement L { N (r)} = 0,

r∈D

(14.10)

implies that  N (r) is a non-trivial solution to the homogeneous integral equation associated with (14.1), i.e., it belongs to the null space of L {•}. But if the null-space is non-empty, then the operator L {•} is singular and cannot be inverted. Still, if (14.1) is the mathematical statement of a ‘well-posed’ physical problem, chances are that a solution exists and a unique one at that, whereby the operator L {•} must be invertible and the coefficients In (r) cannot be adjusted in order for  N (r) to solve (14.9) or (14.10). In conclusion, the residual RN (r) cannot be made identically null no matter how we choose the finite set of N basis functions. Then, we cannot help but relax the requirements on RN (r) somewhat. More specifically, we aim at obtaining the smallest possible residual. This goal is accomplished by minimizing the norm (or the square thereof) of the residual,  2 N    2 (RN , RN )D := RN D = F(r; ω) + In (ω)L {fn (r)} (14.11)   n=1 D

where (•, •)D denotes a suitable inner product of the type (D.68) in the range of the operator L {•}. The quantity in the rightmost member is a real and positive function of the N expansion coefficients In (ω). Thus, we obtain the minimum of the norm if we choose In (ω) in such a way that the Ndimensional gradient of RN 2D with respect to In (ω) vanishes, viz.,   ∂ ∂ 2 2 2 RN D · · · RN D = 0 (14.12) gradIn RN D = ∂I1 ∂IN which amounts to N coupled algebraic equations for the N expansion coefficients. To find the mth equation we observe     ∂RN ∂ ∂ ∂RN 2 (RN , RN )D = RN D = , RN + RN , ∂Im ∂Im ∂Im ∂Im D D      ∗  ∂RN ∂RN ∂RN = , RN + , RN = 2Re , RN =0 (14.13) ∂Im ∂Im ∂Im D D D on account of (D.68) and (B.6). We have gained more generality by assuming a complex residual and a non-symmetric inner product. Now, the derivative of the residual reads ⎡ ⎤ N  ⎥⎥ ∂RN ∂ ⎢⎢⎢⎢ = In (ω)L {fn (r)}⎥⎥⎥⎦ ⎢⎣F(r; ω) + ∂Im ∂Im n=1

944

Advanced Theoretical and Numerical Electromagnetics =

N  ∂ In (ω)L {fn (r)} = L {fm (r)} , ∂Im n=1

r∈D

(14.14)

in light of (14.9) and the fact that the expansion coefficients are independent of one another. Besides, since a complex number or function is null if both real and imaginary parts thereof vanish, (14.13) is equivalent to 

∂RN , RN ∂Im

 D

⎛ ⎞ N  ⎜⎜⎜ ⎟⎟ ⎜ = ⎜⎝L {fm (r)}, F(r; ω) + In (ω)L {fn (r)}⎟⎟⎟⎠ = 0 n=1

(14.15)

D

for m = 1, . . . , N. These are the N algebraic equations implied by (14.12), and the N vector fields wm (r) := L {fm (r)} ,

r∈D

(14.16)

are called weighting or test functions and formally belong to WL . Condition (14.15) lends itself to an interesting geometrical interpretation. Since two functions are ‘orthogonal’ in the sense that the inner product thereof vanishes, then (14.15) says that we certainly minimize the norm of the residual if we select the coefficients In (ω) in such a way that the N . Alternatively, we may say that residual is orthogonal to the set of weighting functions {wm (r)}m=1 the residual must have no component in the function space WN spanned by the N weighting functions. If on account of (14.1), (14.11) and (14.16) we write the norm of the residual as RN 2D

 2 N   =  In (ω)wn (r) − L {J(r; ω)}  n=1 

(14.17)

D

we realize that minimizing RN D is equivalent to minimizing the ‘distance’ between the image of the exact solution and the image of the approximation afforded by the combination of N basis  functions. Arbitrary choices of the coefficients In (ω) make the approximation n In wn worse in the  sense that n In wn moves farther away from L {J(r; ω)}, which in general lies outside the function space WN . These situations are pictorially represented in Figure 14.1 for N = 2 and supported in general Hilbert spaces by properties (D.50) and (D.58) proved in Appendix D.1. In practice, instead of enforcing the point-wise relation (14.9) — which is too strong — we require the residual to be zero in a weak or average sense over D. The calculation of the inner products and the transformation of (14.1) into its weak or algebraic counterpart, namely, (wm , RN )D = 0,

m = 1, . . . , N

(14.18)

is referred to as the testing procedure. The adoption of suitable test functions wm (r) for the problem is paramount to ensure the accuracy of the approximate representation of J(r) in (14.6). The strategy followed thus far points naturally to (14.16). However, computing L {fm } is generally complicated and not possible in closed form, so this choice is rarely used. At the opposite end of the complexity scale we find wm (r) := νˆ m δ(3) (r − rm )

(14.19)

which, more precisely, are distributions (Appendix C) and require we interpret (14.15) in a distributional sense. This choice has the merit of making the calculation of the inner products particularly

The Method of Moments I

945

Figure 14.1 Geometrical interpretation of condition (14.15): since L {J(r)} = −F(r) lies outside the space WN (the shadowed area) spanned by the weighting functions, the best approximation obtains when the residual RN (r) is ‘orthogonal’ to WN . simple thanks to the sifting property (C.19) of the delta distribution. In fact, the procedure is equivalent to forcing the residual to vanish in a set of N points rm ∈ D, viz., RN (rm ) = F(rm ; ω) +

N 

L {fn (r)} In (ω) = 0,

m = 1, . . . , N

(14.20)

n=1

and this strategy is called the collocation method or point-matching method [3]. On the downside we mention that the delta distributions may not be in the range of the operator and that (14.20) may yield inaccurate results for r  rm . A widespread selection for wm (r) consists of requiring the test functions to be the same as the basis functions wm (r) := fm (r),

r∈D

(14.21)

in which instance the approach is called the Galerkin testing or the Method of Moments in the form of Galerkin [1]. If the function spaces VL and WL coincide, (14.21) is equivalent to (14.16). Indeed, we can always represent L {fm } as a linear combination of N functions fn (r), now taken conceptually in the range of L {•}. Alternatively, we observe that the orthogonality condition (14.15) can be fulfilled by replacing L {fm } with any set of N linearly independent functions belonging to the range of L {•}. The ultimate goal of the testing procedure is the transformation of (14.1) into a linear algebraic system. In this regard, we may cast (14.15) as N 

(wm , L {fn })D In (ω) = − (wm , F)D ,

m = 1, . . . , N

(14.22)

n=1

and then in compact matrix form [L][J] = −[F]

(14.23)

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Advanced Theoretical and Numerical Electromagnetics

where

⎛ ⎜⎜⎜ (w1 , L {f1 })D ⎜⎜ .. [L] := ⎜⎜⎜⎜⎜ . ⎜⎝ (wN , L {f1 })D ⎞ ⎛ ⎜⎜⎜ I1 (ω) ⎟⎟⎟ ⎜⎜⎜ . ⎟⎟⎟ [J] := ⎜⎜⎜⎜ .. ⎟⎟⎟⎟ , ⎟⎠ ⎜⎝ IN (ω)

⎞ · · · (w1 , L {fN })D ⎟⎟⎟ ⎟⎟⎟ .. .. ⎟⎟⎟ . . ⎟⎟⎠ · · · (wN , L {fN })D ⎞ ⎛ ⎜⎜⎜ (w1 , F)D ⎟⎟⎟ ⎟⎟⎟ ⎜⎜⎜ .. ⎟⎟⎟ . [F] := ⎜⎜⎜⎜ . ⎟⎟⎠ ⎜⎝ (wN , F)D

(14.24)

(14.25)

The square matrix [L] has rank N and is complex, possibly asymmetric, and full. The latter is a consequence of the non-locality of the integral operators. This is in contrast to the behavior of differential operators, which are local (cf. Section 15.7). In other words, the value of the field L {J} in a point r depends on the value of the current not only in the neighborhood of r but virtually in any other point where the current is defined [e.g., see (10.22)]. This property is in turn reflected in the matrix [L] which is meant to be, after all, the algebraic approximation of the original integral operator L {•}. Thus, from a numerical viewpoint it is important to adopt basis functions which can afford good accuracy while keeping the size of the matrix [L] to a minimum in order to reduce both the amount of computer memory required to store [L] and the time needed to invert the system (14.23). The solution of (14.23) yields the vector [J] of expansion coefficients. In view of the truncation of the series (14.3) in general the Method of Moments does not yield J(r; ω) but rather an approximation thereof, viz., JN (r; ω) =

N 

In (ω)fn (r),

r∈D

(14.26)

n=1

though in the following we shall often use the same symbol to denote both the exact solution and the approximation in order to lighten the notation. On the whole, the physical properties of the unknown J(r) in (14.1) can guide us in the selection of the basis functions, which in any case ought to belong to the domain of L {•}. If the geometry of the region D is relatively simple, it may be possible to come up with functions fn (r) defined everywhere in D, in which instance one talks of entire-domain basis functions. However, in most real-life situations entire-domain basis and test functions are difficult to come by and especially so when the region D has a complicated shape. To circumvent this hurdle and still apply the Method of Moments with profit, one resorts to the expedient of approximating D with a cluster of M adjoining smaller, simple-shaped domains. This process is referred to as the meshing of D, and the resulting aggregate of elementary domains is called the mesh of the original region. Then, basis functions are defined that are non-zero on just one or few of the elementary sub-domains used to model D. When this approach is taken, the elements fn (r) are called sub-domain or subsectional basis functions. The choice of the sub-domains depends on the number of spatial dimensions of the region D. More precisely, (a)

if D := γ is a piecewise-smooth line, we replace γ with the set γ M :=

M 

γp

p=1

where γ p are straight segments;

(14.27)

The Method of Moments I (b)

947

if D := S is a piecewise-smooth surface, we replace S with the tessellation S M :=

M 

Sp

(14.28)

p=1

(c)

where S p are elementary flat surfaces (patches); if D := V is a smooth volume, we replace V with a set of adjoining cells V M :=

M 

Vp

(14.29)

p=1

where V p are ‘small’ regions with canonical shapes. It is important to notice that the meshing brings in a modelling error — which adds to the truncation error — because the relevant integral equation is not formulated exactly on D but rather on the representation D M thereof. The problem becomes particularly evident in the construction of meshes for modelling canonical curved surfaces such as a sphere. However, if the original surface is already piecewise-smooth and comprised of open flat parts, the meshing error may be minimal or outright zero. The flat facets most commonly used for tessellations are rectangles, triangles and quadrilaterals. Rectangles are easily defined but their usage is limited to the cases where the domain D := S is already comprised of a few larger rectangular surfaces. Triangles are quite versatile and popular in that they allow meshing curved boundaries with great accuracy. Quadrilaterals afford flexibility, too, and may be combined with triangles to achieve an even better model of S . The resulting faceted ˆ is not univocally defined at the corners surface is only piecewise-smooth because the unit normal n(r) where the edges of two or more facets are joined. Typical cells employed to approximate threedimensional regions are tetrahedra and hexahedra. Like triangles, tetrahedra are most versatile and can afford great accuracy in the modelling of penetrable objects with curved boundaries. By way of example, the triangular-faceted model of a circular horn antenna is shown in Figure 14.2; the mesh is comprised of M = 3944 triangles. Replacing S with flat surfaces such as triangles makes it easier to evaluate the integrals arising from the discretization of integral equations with the Method of Moments, essentially because we do not need local curvilinear coordinates on the facets. In like manner, special local coordinates may be devised to specify points inside a tetrahedron, which also facilitates the numerical calculation of volume integrals. All in all, we see that the meshing procedure helps implement the Method of Moments. In Sections 14.7 and 15.2 we shall discuss concrete examples of subsectional basis functions which are suited for solving the integral equations presented in Chapter 13.

14.2 Discretization of the EFIE We wish to specialize the general discretization procedure outlined in Section 14.1 to the numerical solution of the time-harmonic EFIE (13.70) [6, Section 7.3], [15]. For the Method of Moments in the N . form of Galerkin we choose the same set of N real-valued vector basis and test functions {fn (r)}n=1 We make the Ansatz JS (r) =

N  n=1

In fn (r),

r ∈ ∂V

(14.30)

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Advanced Theoretical and Numerical Electromagnetics

Figure 14.2 Triangular tessellation S M of a circular horn antenna.

with the understanding that JS (r) signifies the approximation of the exact solution, and observe that the fn (r), regarded as the set of basis functions, must satisfy a few basic properties: (a)

(b)

(c)

Since the fn (r) are employed to expand a surface current, they must be vector fields defined on the boundary ∂V of the object of concern but also be tangential to ∂V because JS (r) has no component normal to ∂V. Since the integral operator of the EFIE involves the surface divergence of JS (r), the fn (r) must admit a finite surface divergence for r ∈ ∂V; basis functions equipped with this feature are said to be divergence-conforming. In case the object is infinitely thin (Figure 13.4c), the boundary S ≡ ∂V is an open surface and hence we demand fn (r) · νˆ (r) = 0,

r ∈ γ ≡ ∂S

(14.31)

in order to account for the boundary condition νˆ (r) · JS (r) = 0, r ∈ γ ≡ ∂S [cf. (1.195)], which is also implied in (13.74). By contrast, the tangential component of fn (r) along the sharp edge of the PEC should obey the asymptotic condition 

 1 fn (r) · sˆ(r0 ) = O , |r − r0 |1/2

r → r0 ∈ γ ≡ ∂S

(14.32)

in that sˆ(r0 ) · JS (r0 ) is an infinite of the same order along the edge [16]. We define the inner product in the range of L {•} as  dS f(r) · g(r)

(f, g)∂V := ∂V

(14.33)

The Method of Moments I

949

and go on to write the formal expression of the entries Lmn , m, n ∈ {1, . . . , N}, of the matrix [L] in the algebraic system (14.23). The integral operator of the EFIE formally reads   ∇s



dS G(r, r )∇ s · {•}, r ∈ ∂V + (14.34) L {•} := − j ωμ dS G(r, r ){•} + j ωε ∂V ∂V   vector potential part

scalar potential part

where G(r, r ) is the Green function given in (8.356). We have emphasized that L {•} is determined by two contributions, namely, the magnetic vector potential and the electric scalar potential (see Section 8.2). In view of the linearity of the inner product such separation may be evidenced in the matrix entries as well, viz., A + LΦ Lmn := Lmn mn ,

where



A Lmn = − j ωμ

LΦ mn =

m, n ∈ {1, . . . , N}

 ∂V



dS G(r, r )fn (r )

(14.36)

dS G(r, r )∇ s · fn (r )

(14.37)

dS fm (r) · ∂V

(14.35)

∂V

∇s dS fm (r) · j ωε



∂V

and if the basis and test functions are normalized so as to be dimensionless, the entries of [L] carry the physical dimension of Ωm2 . We notice that, after the inner integration over ∂V has been carried out, the contribution of the scalar potential requires the numerical evaluation of the surface gradient and the subsequent combination of the result with fm (r). This procedure is cumbersome and can be avoided altogether if we cast LΦ mn into an alternative format beforehand. Since the test function fm (r) can stand a derivative as well we let  1 Υn (r) = dS G(r, r )∇ s · fn (r ) (14.38) j ωε ∂V

and observe    dS fm (r) · ∇s Υn (r) = dS ∇s · [fm (r)Υn (r)] − dS Υn (r)∇s · fm (r) ∂V

∂V

(14.39)

∂V

having used the differential identity (H.77). We are now faced with two possibilities, viz., the boundary S ≡ ∂V is either an open surface or a closed one. Regardless, we would like to apply the surface Gauss theorem (A.59) in order to get rid of the first contribution in the right-hand side. To this purpose, we notice that Υn (r) is a time-harmonic single-layer potential as in (9.13) and hence it is certainly differentiable over ∂V, whereas we have supposed that fn (r) is just divergence-conforming. Thus, only if fn (r) is also of class C1 (∂V)3 may we use (A.59) and get    dS fm (r) · ∇s Υn (r) = ds νˆ (r) · fm (r)Υn (r) − dS Υn (r)∇s · fm (r) ∂V

∂S

∂V



=−

dS Υn (r)∇s · fm (r) ∂V

(14.40)

950

Advanced Theoretical and Numerical Electromagnetics

Figure 14.3 The support of a subsectional basis function defined over a part Ξm of the tessellation S M . where, if ∂V is open, the line integral vanishes because the test functions obey condition (14.31) by hypothesis. If, on the other hand, ∂V is closed, the line ∂S degenerates into a point and the associated contribution vanishes as well. For arbitrary-shaped obstacles, though, we may have a hard time finding entire-domain functions equipped with the necessary derivatives over ∂V. The typical way out consists of modelling ∂V with a tessellation S M made up of M simple-shaped parts (i.e., sub-domains) and then of defining the functions fn (r) on one or more sub-domains S p ∈ S M , p = 1, . . . , M. We indicate with Ξn ⊂ S M the support of fn (r), as is suggested in Figure 14.3. Thus, we may succeed in ensuring the differentiability of basis and test functions, so that we have    dS fm (r) · ∇s Υn (r) = ds νˆ (r) · fm (r)Υn (r) − dS Υn (r)∇s · fm (r) (14.41) Ξm

Ξm

∂Ξm

where the line integral contributes only if the property νˆ (r) · fm (r) = 0,

r ∈ ∂Ξm ,

m = 1, . . . , N

is not satisfied. Accordingly, we may write (14.37) as   1 Φ Lmn = − dS ∇s · fm (r) dS G(r, r )∇ s · fn (r ) j ωε ∂V

or as LΦ mn = −

1 j ωε

(14.43)

∂V



 dS ∇s · fm (r) Ξm

(14.42)

dS G(r, r )∇ s · fn (r )

(14.44)

Ξn

which are more convenient inasmuch as the divergence of the basis functions can usually be computed analytically (cf. Section 14.7). Furthermore, if in (14.36), (14.43) and (14.44) we swap the order of the two surface integrals and make the substitutions r → r and r → r, it remains proved that A A Lmn = Lnm ,

Φ LΦ mn = Lnm ,

m, n ∈ {1, . . . , N}

(14.45)

whence we conclude that the matrix [L] is symmetric. This result is a consequence of the chosen inner product, the Galerking testing, conditions (14.31) and (14.42), and ultimately the reciprocity

The Method of Moments I

951

of the background medium, which makes the Green function (8.356) invariant with respect to the interchange of source and observation points. Since a symmetric square matrix of order N has only N(N + 1)/2 independent entries (the diagonal and, e.g., the upper triangular part) we see that the symmetry of [L] is quite a desirable feature in that it helps reduce (1) (2) (3)

the time needed for setting up the matrix roughly by 50 per cent; the computer memory necessary to store [L] roughly by half; the time required to solve the system (14.23).

However, notice that the symmetry of [L] is lost if the sub-domain basis functions fn (r) do not obey (14.42) because the line integrals along ∂Ξm are non-null. The calculation of the right-hand side of (14.23) requires evaluating surface integrals of the type    := Fm = fm , Ei dS fm (r) · Ei (r), m = 1, . . . , N (14.46) ∂V

∂V

where the integration is restricted to r ∈ Ξm if the fm (r) are subsectional test functions. In problems of electromagnetic scattering the incident field is quite often taken to be a uniform plane wave (Section 7.2) in which case we find  i (14.47) Fm = E0 · dS e− j k·r fm (r), ∂V

and we may construe the integral as a two-dimensional Fourier transform of the test function evaluated in −k, the negative of the wavevector. A formula for the scattered electric field in the Fraunhofer region of the PEC body can be obtained by inserting the expansion of JS (r) (14.30) into (9.325). In symbols, we have Es (r) ≈ − j ωμ = − j ωμ

e− j kr ˆ ˆ (ϑϑ + ϕˆ ϕ) ˆ · 4πr e− j kr ˆ ˆ (ϑϑ + ϕˆ ϕ) ˆ · 4πr



dS ej kˆr·r

n=1

N 

In fn (r )

n=1

∂V N 



 In



dS ej kˆr·r fn (r )

(14.48)

∂V

which essentially requires the calculation of the Fourier transforms of the basis functions in the spectral points kˆr (cf. Example ! 14.2). The average power PsF reflected or scattered back by the object is due to the radiation of electromagnetic waves by the equivalent surface current density JS (r) that flows over ∂V. To com! pute PsF we may obviously employ formula (9.334) where the integral must be evaluated over a sphere which contains the object and has a radius large enough for (14.48) to hold true (see Section 9.5). However, if the underlying medium is lossless, a suitable application of the Poynting theorem ! (1.314) to just the scattered fields Es (r) and Hs (r) allows us to conclude that we may get PsF by considering the flux of the complex Poynting vector (1.304) across any surface enclosing the body in question (cf. Figure 13.2a). The one surface which is particularly well-suited for the task is precisely ∂V + , the positive side of the body’s boundary (Figure 13.2b). And the reason is that — thanks to the very solution of (13.70) — on ∂V + we know both JS (r) and the scattered electric field. Starting with definition

952

Advanced Theoretical and Numerical Electromagnetics

(1.303) we express the Poynting vector in terms of JS (r) and known electromagnetic entities, viz.,  " s# 1 ˆ · Es (r) × Hs∗ (r) PF = Re dS n(r) 2 ∂V +  $ %∗ 1 ˆ × H(r) − Hi (r) = − Re dS Es (r) · n(r) 2 ∂V +  $ % 1 ˆ × Hi∗ (r) · Est (r) = − Re dS J∗S (r) − n(r) 2 ∂V +   1 1 ˆ dS J∗S (r) · Eit (r) + Re dS Eit (r) × Hi∗ (14.49) = Re t (r) · n(r) 2 2 ∂V +

∂V +

where we have made use of the properties of the triple scalar product (H.13), the jump condition (13.64), and lastly the matching condition (13.63) to replace the tangential scattered electric field with the negative of the tangential incident electric field. Now, since the unit normal on ∂V is positively oriented outwards V (Figure 13.3), the second integral in the right member equals the negative of the average power radiated by the true sources — located somewhere in R3 \ V — and flowing into the region V when occupied by a material with the same parameters as the background medium. But then, since the underlying medium is lossless by hypothesis, no dissipation occurs within V either, which means that under the stated conditions no power due to the external sources enters V on average. As a consequence, we have  ! 1 ˆ dS Eit (r) × Hi∗ =0 (14.50) PiF := − Re t (r) · n(r) 2 ∂V +

and the average scattered power follows from "

# PsF

1 = Re 2



∂V +

dS

J∗S (r) · Eit (r)

1 = Re 2



∂V +

dS

N  m=1

Im∗ fm (r) · Eit (r) =

N  1  ∗ Re Im fm , Eit (14.51) ∂V 2 m=1

on account of (14.30) and (14.33). By virtue of (14.46) the inner product of a test function fm (r) with the tangential incident field Eit (r) is the mth element of the N-by-1 column vector [F] [cf. (14.25)]. Therefore, we can cast the average power further as "

N N ' ' # 1 & 1 & 1  ∗ I Lmn In PsF = Re [J]H [F] = − Re [J]H [L][J] = − Re 2 2 2 m=1 n=1 m

(14.52)

where the 1-by-N vector [J]H is the Hermitian transpose of [J]. The second step is a consequence of (14.23) applied to the EFIE. Clearly, evaluating (14.52) is trivial as compared to calculating (9.334), though (14.52) may be employed only if the background medium is lossless. ! The average power PsF must be positive, as the equivalent source JS (r) radiates towards the outside of the PEC body. This feature poses some constraints on the entries of [L], viz., & ' [J]H [L][J] + [J]H [L]∗ [J] [J]H [L][J] + [J]T [L]∗ [J]∗ =− −Re [J]H [L][J] = − 2 2 ∗ H [L] + [L] H [J] = −[J] Re {[L]} [J] > 0 = −[J] 2

(14.53)

The Method of Moments I

953

having invoked (B.6) and exploited the fact that [L] is symmetric. The last condition implies that the real part of [L] is a negative definite matrix [17, 18]. While the matrix [L] embodies information on the geometry of the PEC object, the current coefficients [J] depend also on the incident field. Since the latter may be arbitrary, (14.53) must be true for any vector [J] ∈ CN . In particular, we may take [J] as an eigenvector of [L] associated with the eigenvalue ν [18, Chapter 9], in which case we observe [L][J] = ν [J]

=⇒

[J]H [L][J] = ν [J]H [J]

and by forming the Rayleigh quotient associated with ν we get  H [J] [L][J] [J]H Re {[L]} [J] Re{ν} = Re 0 dS (r − r ) · (r − r ) = + l − l (14.162) 3 3 ⎝ 16A 1 2 3 4A2 T ⎛ ⎞  ⎟⎟ l1 l2 ⎜⎜⎜ l21 + l22 l1 l2 2⎟ ⎜ ⎟ − l dS (r − r ) · (r − r ) = (14.163) 1 2 ⎝ 3⎠ . 2 16A 3 4A T

Another matrix which sometimes arises in the discretization of integral equations (e.g., [28]) is comprised of integrals involving f(r) · nˆ ×g(r), where f(r) and g(r) denote two RWG functions which share a triangle T and nˆ is the unit vector normal to T . When f(r) and g(r) coincide, the result is evidently null. The other two occurrences may be reduced to the calculation of integrals of the type (see Figure 14.8b)  l1 l2 dS (r − r1 ) · nˆ × (r − r2 ) (14.164) I3 := 4A2 T

which we can carry out with the aid of the area coordinates (14.141). In symbols, we have  l1 l2 I3 = dS nˆ · (r − r2 ) × (r − r1 ) 4A2 T  l1 l2 dξ1 dξ2 nˆ · [ξ1 (r1 − r3 ) + (ξ2 − 1)(r2 − r3 )] × [(ξ1 − 1)(r1 − r3 ) + ξ2 (r2 − r3 )] = 2A S 2

Advanced Theoretical and Numerical Electromagnetics

976 =

l1 l2 2A

 dξ1 dξ2 [2Aξ1 ξ2 − 2A(ξ1 − 1)(ξ2 − 1)] 

S2

dξ1 dξ2 (ξ1 + ξ2 − 1) = −

= l1 l2 S2

l1 l2 6

(14.165)

where we have also made use of (14.142). (End of Example 14.1)

Example 14.2 (The Fourier transform of an RWG function) In the numerical solution of surface integral equations with the Method of Moments and RWG basis functions we often come across integrals of the type [e.g., see (14.47)]    l l j k·r + j k·r F(k) := dS f(r)e = dS (r − r3 )e − − dS (r − r−3 )ej k·r (14.166) 2A+ 2A T + ∪T − T+ T−   F+ (k)

F− (k)

where f(r) is any one of the RWG functions defined in (14.116) and associated with the pair of adjacent triangles T + and T − . The operation implied in (14.166) may be interpreted as a Fourier transformation of sorts which maps the surface vector field f(r) defined over a triangular tessellation (see Section 14.1) to the vector field F(k) in the domain of the spectral variable k ∈ C3 . To compute the integral we split it into two parts as indicated and we resort to area coordinates (ξ1 , ξ2 ) for both of them. We begin with the integration over T + which, in accordance with (14.150), becomes  + + + + + j k·r+3 F (k) = le dξ1 dξ2 [(r+1 − r+3 )ξ1 + (r+2 − r+3 )ξ2 ]ej k·[(r1 −r3 )ξ1 +(r2 −r3 )ξ2 ] S2  ˜+ ˜+ j k·r+3 + = le (r1 − r+3 ) dξ1 dξ2 ξ1 ej k1 ξ1 +j k2 ξ2 S2  ˜+ ˜+ j k·r+3 + + + le (r2 − r3 ) dξ1 dξ2 ξ2 ej k1 ξ1 +j k2 ξ2 (14.167) S2

where S 2 is the two-dimensional simplex drawn in Figure 14.6 and k˜ 1+ := k · (r+1 − r+3 ),

k˜ 2+ := k · (r+2 − r+3 )

(14.168)

are the components of a normalized dimensionless spectral variable. We have reduced the calculation of F+ (k) to a linear combination of two simpler integrals. To further simplify the integration we can write   ∂ ˜+ ˜+ j k˜ 1+ ξ1 +j k˜ 2+ ξ2 dξ1 dξ2 ξ1 e = −j + dξ1 dξ2 ej k1 ξ1 +j k2 ξ2 (14.169) ˜ ∂ k1 S2 S2   ∂ ˜+ ˜+ ˜+ ˜+ dξ1 dξ2 ξ2 ej k1 ξ1 +j k2 ξ2 = − j + dξ1 dξ2 ej k1 ξ1 +j k2 ξ2 (14.170) ˜ ∂ k2 S2 S2 where the interchange of derivatives and integrals is permitted because the integrands and the derivatives thereof are regular functions of (ξ1 , ξ2 , k˜ 1+ , k˜ 2+ ). The remaining integral is formally the twodimensional Fourier transformation of 1 − U(ξ1 , ξ2 ) where ⎧ ⎪ ⎪ ⎨0, (ξ1 , ξ2 ) ∈ S 2 (14.171) U(ξ1 , ξ2 ) := ⎪ ⎪ ⎩1, (ξ1 , ξ2 ) ∈ R2 \ S 2

The Method of Moments I

977

is a two-dimensional step function. After some lengthy, though straightforward manipulations the result can be written succinctly as  Υ(k˜ 1+ ) − Υ(k˜ 2+ ) ˜+ ˜+ (14.172) dξ1 dξ2 ej k1 ξ1 +j k2 ξ2 = j(k˜ 1+ − k˜ 2+ ) S2 where the auxiliary function Υ(α), α ∈ C, and the derivatives thereof are given by ⎧ jα ⎪ e −1 ⎪ ⎪ ⎪ , α0 ⎨ jα Υ(α) := ⎪ ⎪ ⎪ ⎪ ⎩1, α=0 ⎧ jα ⎪ e (j α − 1) + 1 ⎪ ⎪ ⎪ , α0 ⎪ ⎪ dΥ ⎨ j α2 := ⎪ ⎪ ⎪ dα ⎪ j ⎪ ⎪ ⎩ , α=0 2 ⎧ jα ⎪ e (2 − 2 j α − α2 ) − 2 ⎪ ⎪ ⎪ , α0 2 ⎪ ⎪ d Υ ⎨ j α3 := ⎪ ⎪ ⎪ dα2 ⎪ 1 ⎪ ⎪ α = 0. ⎩− , 3 With these positions we have ∂ Υ(k˜ 1+ ) − Υ(k˜ 2+ ) −j + = ∂k˜ k˜ + − k˜ + 1

−j

1

2

∂ Υ(k˜ 1+ ) − Υ(k˜ 2+ ) = ∂k˜ + k˜ + − k˜ + 2

1

Υ(k˜ 1+ ) + (k˜ 2+ − k˜ 1+ )

dΥ − Υ(k˜ 2+ ) dk˜ 1+

(14.174)

(14.175)

(14.176)

(k˜ 2+ − k˜ 1+ )2 dΥ Υ(k˜ 2+ ) + (k˜ 1+ − k˜ 2+ ) + − Υ(k˜ 1+ ) dk˜

2

(14.173)

2

(14.177)

(k˜ 1+ − k˜ 2+ )2

where the singularity of the complex functions in the right-hand sides must be only apparent, since the Fourier transform of a bounded function null everywhere except over a finite domain is analytic over the whole complex space C2 (see Appendix B). Indeed, in the limit as k˜ 2+ approaches k˜ 1+ we find dΥ Υ(k˜ 1+ ) + (k˜ 2+ − k˜ 1+ ) + − Υ(k˜ 2+ ) dk˜ 1 1 d2 Υ lim = − 2 d(k˜ + )2 k˜ 2+ →k˜ 1+ (k˜ + − k˜ + )2 2

1

(14.178)

1

which is finite on account of (14.175). Finally, we can write the result symbolically as Υ(k˜ 1+ ) + (k˜ 2+ − k˜ 1+ ) +

F+ (k) = lej k·r3 (r+1 − r+3 )

dΥ − Υ(k˜ 2+ ) dk˜ + 1

(k˜ 2+ − k˜ 1+ )2 Υ(k˜ 2+ ) + (k˜ 1+ − k˜ 2+ ) +

+ lej k·r3 (r+2 − r+3 ) with k˜ 1+ and k˜ 2+ given by (14.168).

dΥ − Υ(k˜ 1+ ) dk˜ 2+

(k˜ 1+ − k˜ 2+ )2

(14.179)

978

Advanced Theoretical and Numerical Electromagnetics

Concerning F− (k) we observe that it follows from (14.179) by replacing the superscript ‘+’ with ‘−’ throughout, and this completes the calculation of (14.166). (End of Example 14.2)

14.9 Singular integrals over triangles At the end of the previous section we have alluded at the usage of quadrature formulas for the numerical calculation of surface integrals over triangles. The procedure works fine provided the integrand is sufficiently smooth. On the whole, the higher the number of continuous derivatives of the integrand, the higher the order of the formula which may be applied to evaluate an integral, as is suggested in (14.150). Stated another way, there is no point in choosing a quadrature formula of order NO unless the integrand can be differentiated at least NO times. Compounding the problem for the implementation of the Method of Moments on a given mesh S M is the fact that the surface integrals encountered in Sections 13.2 and 13.3 involve the threedimensional Green function (8.356). So long as the observation point r and the source point r

belong to two different facets of the tessellation, the integrand is regular, and a quadrature formula may be profitably used. However, when r and r range on the same triangle, G(r, r ) is singular for r ≡ r. Under such circumstances a quadrature formula can hardly guarantee the accuracy for which it was designed, since the integrand is definitely not smooth on the triangle in question. Worse still, if one of the nodes (ξi , ηi ) is mapped onto r ≡ r, the integrand diverges because the denominator R = |r − r | of G(r, r ) vanishes. When this happens, a computer program most likely incurs into overflow during runtime while attempting to carry out a division by zero. For these reasons we need to devise strategies for the stable numerical calculation of the entries of the MoM matrix [L] in (14.23). We suppose that the material interface ∂V has been approximated by means of a triangularfaceted tessellation S M comprised of M patches and that, in accordance with the nature of the problem, one or more sets of RWG functions have been associated with the interior edges of the mesh. For the sake of clarity we indicate with S n := T n+ ∪ T n− , n = 1, . . . , N, the pair of triangles on the which the RWG function fn (r) is non-zero (Figure 14.4). A closer look at the discretization schemes developed in Sections 14.2-14.6 shows that the relevant surface integrals amount to suitable linear combinations of few basic types that involve G(r, r ), the gradient thereof, and two RWG functions fn (r ), fm (r), namely, 

 dS fm (r) ·

L1 := Sm

Sn



dS G(r, r )fn (r ) 

dS ∇s · fm (r)

L2 := Sm



 dS fm (r) ·

L3 := Sm



dS fm (r) · Sm



Sm

(EFIE, PMCHWT)

(14.181)

(MFIE, Müller)

(14.182)

(PMCHWT)

(14.183)

(Müller)

(14.184)

Sn

ˆ × [∇G(r, r ) × fn (r )] dS n(r) dS ∇G(r, r ) × fn (r )

Sn



ˆ × fm (r) · dS n(r)

L5 :=

(14.180)

Sn



L4 :=

dS G(r, r )∇ s · fn (r )

(EFIE, PMCHWT)

Sn

dS G(r, r )fn (r )

The Method of Moments I 

 ds sˆ(r) · fm (r)

L6 := ∂S m

dS G(r, r )∇ s · fn (r )

(Müller)

979

(14.185)

Sn

where L2 and L6 are further simplified by recognizing that the divergence of an RWG function is piecewise-constant. One more type of nested integrals, namely,   ˆ × fm (r)] dS G(r, r )∇ s · fn (r ) = 0 L7 := dS ∇s · [n(r) (Müller) (14.186) Sm

Sn

also arises, in general, from the Müller equations [see (14.105) and (14.106)]. When fm (r) is an RWG function, though, the result is zero, as indicated, by virtue of identity (A.60) and (14.132). The contribution of the line δ-distributions that appear in (14.134) is accounted for by integrals of type (14.185). When S n and S m coincide or are just partially overlapped — which happens when the domains of fn (r ) and fm (r) share a triangle — the Green function is singular. An old-fashioned, though efficacious countermeasure consists of decomposing G(r, r ) into the sum of two parts, one regular and one still singular, i.e.,

G(r, r ) :=



e− j k|r−r | − 1 e− j k|r−r | 1 = +



4π|r − r | 4π|r − r | 4π|r − r |   regular

(14.187)

singular

where we recognize the singular term as the limit of G(r, r ) for k → 0 or the static Green function (2.131). That the regular part remains finite for r → r is proved by letting n = 0 in formula (9.175). To illustrate how representation (14.187) helps us handle the singularity of the Green function we write (14.180) as    

˜ r )fn (r ) + dS fm (r) · dS fn (r ) (14.188) dS fm (r) · dS G(r, L1 := 4π|r − r | Sm

Sn



Sm

Sn



˜ r ) denotes the regular part of G(r, r ) in (14.187). The first integral now exhibits a regular where G(r, integrand and can be evaluated numerically with a quadrature formula. Indeed, overflow is prevented ˜ r ) numerically when |r − r|  , with  > 0 being a from occurring, if instead of computing G(r,

˜ r ) with its known limiting value k/(4π j). The remaining integral suitable threshold, we replace G(r, in (14.188) is simpler than the original one in (14.180), and we can perform at least the innermost integration analytically. Since the resulting function of r is regular, the outermost integration can be carried out numerically. The overall strategy is referred to as the singularity extraction in that the singular character of G(r, r ) is isolated and treated separately [5]. The same remarks apply to (14.181), (14.184) and (14.185). By virtue of (9.175) with n = 1 and (3.188) with ∇ R = −∇R it is easy to check that the gradient ˜ r ) is finite, i.e., of G(r,  

k2 ˆ d e− j kR − 1 e− j k|r−r | − 1 lim = lim ∇R = − R (14.189) ∇ r →r R→0+ dR 4π|r − r | 4πR 8π ˜ r ) as r → r is not well defined, because the result though, strictly speaking, the limit of ∇G(r,

depends on the direction taken to make r approach r. What is more, the higher-order derivatives of ˜ r ) are even divergent, e.g., G(r, ⎞ ⎛

ˆR ˆ dG˜ ⎟⎟ ⎜⎜⎜ e− j k|r−r | − 1 d2G˜ I − R ⎟⎟⎠ = ∞ ˆ ˆ ⎜ = lim+ ⎝RR 2 + lim ∇∇ (14.190) r →r R→0 4π|r − r | R dR dR

980

Advanced Theoretical and Numerical Electromagnetics

on account of (9.166) and the fact that 1/R diverges. Consequently, we cannot achieve high precision in the calculation of the integrals in (14.182) and (14.183) by just applying higher-order quadrature formulas, as observed at the beginning. The application of the singularity extraction to (14.180)-(14.185) leads us to the calculation of three basic types of integrals on a triangle T , namely,  1 (14.191) dS

I1 := R T  r − r (14.192) I2 := dS

R T  1 (14.193) I3 := dS ∇ R T

where we have omitted immaterial constant multiplication factors. The second type follows from, e.g., (14.188) by noticing that r − r3 r − r r − r3 = + (14.194) R R R where r3 is the position vector of the vertex V3 of the triangle (see Figure 14.4), and r is the observation point. Since the vector r−r3 is constant with respect to the integration variable r , the integration of the second term above boils down to an integral of the type (14.191). The integral of the third type is not well-defined per se as r → r (cf. Sections 2.9 and 13.2.3), but it yields a bounded result ˆ and r − r3 in the construction of the algebraic MFIE as well nonetheless when combined with n(r) as the discretized PMCHWT and Müller equations.

14.9.1 Integrals involving 1/R While the analytic calculation of the integral in (14.191) can be effected in a few ways [29–31], it is preferable to pursue a strategy that leads to more compact, symmetric formulas in that the latter are easier to automate in a dedicated computer program [32]. To gain more generality we suppose that the observation point r is not located on the plane which contains the triangle T . Besides, we indicate with P the projection of r onto such plane and let r0 be the position vector of P (Figure 14.9). It pays to introduce a local system of circular cylindrical coordinates (τ , α , ζ ) with the origin in P and the ζ-axis aligned with the unit vector nˆ orthogonal to T . Therefore, the distance R may be written as  1/2 R = |r − r| = |τ − w 0 nˆ | = τ 2 + w 2 (14.195) 0 where w 0 := nˆ · (r − r )

(14.196)

is the component of r − r perpendicular to T , and |w 0 | is the distance of r from the plane of the triangle T . It is possible to express 1/R as the surface divergence of an auxiliary vector field F(τ ) which — since 1/R depends on τ only — must be in the form Fτ τˆ , viz., 1 1 d 1 = = (τ Fτ ) 1/2 R (τ 2 + w 2 τ dτ ) 0

(14.197)

The Method of Moments I

981

Figure 14.9 Geometrical quantities associated with a triangle T for the evaluation of singular surface integrals: projection P outside T (cf. [32]). and integrating this differential equation yields 1/2 R 1  =

=⇒ Fτ (τ ) = τ 2 + w 2 0 τ τ

F(τ ) =

R

τ τ 2

(14.198)

whence we obtain

 

1 |r − r|

1

= = ∇s · τ , R |r − r| τ 2

r ∈ T,

r ∈ R3 \ {r }

(14.199)

where the prime means that the derivatives are taken with respect to the polar coordinates (τ , α ). This formula is a special case of the following more general result   ⎧ 1

⎪

m+2 τ ⎪ ⎪ ∇ · R , m ∈ Z \ {−2} ⎪ s ⎪ ⎪ τ 2 ⎨m + 2 m R =⎪ 

 (14.200) ⎪ ⎪ τ ⎪

⎪ ⎪ ∇ · log R , m = −2 ⎩ s τ 2 with the exponent m possibly extended to complex numbers. Then, we distinguish four cases: (1) (2) (3) (4)

the projection P falls outside T ; the projection P belongs to T ; the projection P falls on the boundary ∂T ; the projection P coincides with one of the vertices of T .

If the point r0 falls outside T then (14.199) may be employed in (14.191) to transform the surface integral into a line one along the boundary ∂T with the aid of the surface Gauss theorem (A.59). The latter may be invoked because τ > 0 and the vector field Rτ /τ 2 is sufficiently smooth for r ∈ T . In symbols, we have   4R 5  1 R dS = dS ∇ s · 2 τ = ds νˆ (τ ) · τ 2 (14.201) R τ τ T

T

∂T

Advanced Theoretical and Numerical Electromagnetics

982

where we have applied (A.59) with the unit normal νˆ (τ ) positively oriented outward T . To evaluate the line integral we split the calculations along the three edges of T , say, γi , i = 1, 2, 3. To this purpose we introduce the geometrical quantities (Figure 14.9): r−i and r+i , the position vectors of the initial and final endpoints of γi ; sˆ i , the unit vector tangent to the edge γi ; νˆ i := sˆ i × nˆ , the unit vector perpendicular to the edge γi ; τ i0 := νˆ i · τ , the component of τ orthogonal to γi , thus |τ i0 | is the distance of r0 from the edge γi ; s := sˆ i · τ , a local coordinate along γi .

• • • • •

Accordingly, the position vector τ ∈ γi may be expressed as τ = τ i0 νˆ i + s sˆ i ,

+ s ∈ [s − i , si ]

(14.202)

+ − ˆ i and s − ˆ i . With these positions we have where s + i = (r − ri ) · s i = (r − ri ) · s

 ∂T

 R ds νˆ (τ ) · τ 2 = τ i=1 3

+

si

 R ds τ i0 2 = τ i=1 3

s − i

+

si s − i

⎛

⎞ ⎜⎜⎜ τi0 w 2 ⎟⎟⎟ 0 τi0 ⎟ + ds ⎝⎜ ⎠ R Rτ 2

(14.203)

so we are left with the calculation of one-dimensional integrals of two types. If in light of (14.195) we let

2 1/2 = u − s

R = (s 2 + w 2 0 + τi0 )

(14.204)

and solve for s with respect to u, we can make a change of dummy variable, i.e., s =

2 u2 − w 2 0 − τi0 , 2u

ds =

2 u2 + w 2 0 + τi0 du, 2u2

R=

2 u2 + w 2 0 + τi0 2u

and then can compute the following indefinite integrals (see [33, Formula 200.1])   τ

τ

ds i0 = du i0 = τ i0 log |u| = τ i0 log(s + R) R u  

2

2 4uτ i0 w 2 w 0 (u2 − w 2 w τ 0 0 − τi0 )

du 2 2 = w arctan ds 0 2i0 = 0

2 1/2

2 Rτ 4u τi0 + (u2 − w 2 τ i0 (u2 + w 2 0 − τi0 ) 0 + τi0 ) = w 0 arctan

w 0 s

|w 0 |s

= |w | arctan 0 Rτ i0 Rτ i0

(14.205)

(14.206)

(14.207)

where in the last step we have exploited the fact that the arctangent is an odd function of its argument. Using these formulas in (14.201) and (14.203) yields  T

dS

3 3 3   |w 0 |s + |w |s − s + + R+i 1 

i



= τi0 log i − + |w | arctan − |w | arctan 0− i 0 0 − +

R i=1 si + Ri Ri τi0 Ri τi0 i=1 i=1

(14.208)

where $ % 2

2

2 1/2 R+i := (s + , i ) + w0 + τi0

$ % 2

2

2 1/2 R−i := (s − i ) + w0 + τi0

are the distances of the point r from the endpoints of γi .

(14.209)

The Method of Moments I

983

Figure 14.10 Geometrical quantities associated with a triangle T for the evaluation of singular surface integrals: projection P inside T (cf. [32]).

If, on the other hand, the point r0 belongs to T , the vector field Rτ /τ 2 is singular for τ = 0 2 and w 0  0. Consequently, by choosing a two-dimensional radial test function φ(τ ) ∈ C∞ 0 (R ) and computing the surface divergence in a distributional sense [cf. (C.44)] we find

∇ s



  2π +∞ Rτ

∂φ Rτ



· 2 := − dS ∇s φ · 2 = − dα dτ R ∂τ τ τ R2

0

0

 +∞

( ) φ(τ )

τ

= 2π|w = −2π φ(τ )R +∞ + 2π dτ φ(τ ) |φ(0) + dS

0 0 R R

(14.210)

R2

0

where we have integrated by parts with respect to the radial coordinate τ . In shorthand notation this means   Rτ 1 6 7 1 =

= ∇ s · 2 − 2π|w 0 |δ(2) τ

R |r − r| τ

(14.211)

and this formula extends (14.199) for all points r ∈ T and r ∈ R3 . Since for r0 ∈ T we may not apply the Gauss theorem (A.59) right away, we proceed by excluding the offending point P with a circle B2 (r0 , a) centered in r0 (Figure 14.10). The radius a is chosen small enough for B2 (r0 , a) to be contained in T but it is otherwise arbitrary. Then, we have  T

dS



 4R 5 1

τ dS

+ R τ 2 B2 (r0 ,a) T \B2 (r0 ,a)    R R 1 = ds νˆ (τ ) · τ 2 + ds νˆ (τ ) · τ 2 + dS 2 1/2 τ τ (τ + w 2 0 )

1 = R

∂T

dS ∇ s ·

∂B2 (r0 ,a)

B2 (r0 ,a)

(14.212)

984

Advanced Theoretical and Numerical Electromagnetics

Figure 14.11 Geometrical quantities associated with a triangle T for the evaluation of singular surface integrals: projection P on the edge γi (cf. [32]). where we have applied (A.59) with the unit normal νˆ (τ ) positively oriented outward T \ B2 (r0 , a). The first integral is given by (14.203) and (14.208). The remaining ones are evaluated in polar coordinates (τ , α ) by observing that νˆ (τ ) · τ = −a and τ = a on ∂B2, thus   R 1 ds νˆ (τ ) · τ 2 + dS 2 = 1/2 τ (τ + w 2 0 ) B2 (r0 ,a)

∂B2 (r0 ,a)

2π =−

  1/2 

dα a2 + w 2 + dα dτ

0 a

2π



0

0

0

τ

1/2 (τ 2 + w 2 0 )

  1/2 1/2 = −2π a2 + w 2 + 2π a2 + w 2 − 2π|w 0 | = −2π|w 0 | 0 0

(14.213)

where we see that the result does not depend on the radius a. Lastly, by inserting everything back into (14.212) we get  T

dS

3 s + + R+i 1 

= τi0 log i − R i=1 si + R−i

+ |w 0 |

3 

arctan

i=1

3  |w 0 |s + |w |s − i

− |w | arctan −0 i − 2π|w 0 | 0 +

Ri τi0 Ri τi0 i=1

(14.214)

a result which is valid strictly for P ∈ T . It should be noticed that the last term in the right-hand side of (14.214) accounts precisely for the contribution of the term proportional with the Dirac delta distribution in (14.211). If the point P falls on one of the edges of T (Figure 14.11) we exclude the offending point with a circle B2 (r0 , a) and split the integration over T \ B2 (r0 , a) and over the half-circle B2 (r0 , a) ∩ T . We also define two lines, namely, γ := {r ∈ ∂T : |r − r0 |  a} and γ

:= ∂B2 ∩ T , and write    4R 5 1 1 dS = dS ∇ s · 2 τ + dS

R R τ T

T \B2 (r0 ,a)

B2 (r0 ,a)∩T

The Method of Moments I

985

Figure 14.12 Geometrical quantities associated with a triangle T for the evaluation of singular surface integrals: projection P on the vertex Vl (cf. [32]).  = γ

ds νˆ (τ ) · τ

R + τ 2



ds νˆ (τ ) · τ

γ

R + τ 2



dS

B2 (r0 ,a)∩T

(τ 2

1 1/2 + w 2 0 )

(14.215)



where we have applied (A.59) with the unit normal νˆ (τ ) positively oriented outward T \ B2 (r0 , a). The integrals along γ

and over B2 (r0 , a)∩T are easily evaluated in local polar coordinates (τ , α ), i.e.,   R 1 ds νˆ (τ ) · τ 2 + dS 2 = 1/2 τ (τ + w 2 0 )

γ

B2 (r0 ,a)∩T

π =−

  1/2 

dα a2 + w 2 + dα dτ

0 π

0

a

0

0

τ

1/2 (τ 2 + w 2 0 )

  1/2 1/2 = −π a2 + w 2 + π a2 + w 2 − π|w 0 | = −π|w 0 | 0 0

(14.216)

so the combined contribution of these two terms is independent of a. The integral along γ is further split into three parts, namely, along γ ∩ γi , γi+1 and γi+2 . Since P belongs to γi , the contribution from γ ∩ γi vanishes, because νˆ (τ ) · τ = νˆ i · τ = 0 on γ ∩ γi (Figure 14.11). The integrals along the other two edges can be computed as in (14.203) with the aid of (14.206) and (14.207). All in all, the result is  T

dS

3 s + + R+i 1 

= τi0 log i − R i=1 si + R−i il

+ |w 0 |

3  i=1 il

arctan

3  |w 0 |s + |w |s − i − |w 0 | arctan 0− i − π|w 0 | +

Ri τi0 Ri τi0 i=1

(14.217)

il

which is valid for P ∈ γl , l = 1, 2, 3. Finally, if P coincides with the vertex Vl , l = 1, 2, 3, of T (Figure 14.12), by proceeding as in the previous case we arrive formally again at (14.215). We compute the integrals along γ

and over B2 (r0 , a) ∩ T in local polar coordinates (τ , α ), i.e.,   R 1 ds νˆ (τ ) · τ 2 + dS 2 = 1/2 τ (τ + w 2 0 )

γ

B2 (r0 ,a)∩T

Advanced Theoretical and Numerical Electromagnetics

986

αl =−

    2

2 1/2

dα a + w0 + dα dτ

αl

a



0

0

0

τ

1/2 (τ 2 + w 2 0 )

 1/2  1/2 = −αl a2 + w 2 + αl a2 + w 2 − αl |w 0 | = −αl |w 0 | 0 0

(14.218)

where αl denotes the angle determined by the edges of T joined at Vl . As for the integral along γ , we divide it into two parts, namely, along γl — the edge opposite Vl — and along γ \ γl . The latter contribution vanish because νˆ (τ ) · τ = 0 for r ∈ γ \ γl , whereas the integral along γl is evaluated as in (14.203) with the aid of (14.206) and (14.207). In conclusion, the result reads  s + + R+l |w |s + |w |s − 1 dS = τ l0 log l − + |w 0 | arctan +0 l − |w 0 | arctan 0− l − αl |w 0 | (14.219) − R sl + Rl Rl τl0 Rl τl0 T

valid strictly for P ≡ Vl . Although we have covered all bases, having to choose among four different formulas for the same integral is cumbersome and especially so in a computer program. Therefore, we wish to collect the results into a single compact expression. We begin by observing that ∇ s ·

τ

= 0, τ 2

τ ∈ R2 \ {0}

(14.220)

and we distinguish again four cases: •

if the point P falls outside T , then τ > 0 and  0=

τ

dS ∇ s · 2 = τ

T

=

3 

arctan

i=1

•

 ∂T

νˆ (τ ) · τ  ds

= τ 2 i=1 3

T \B2(r0 ,a)

=

3   i=1

ds

s − i

s − i

τ i0 s 2 + τ 2 i0 (14.221)

∂T



ds

3  s + s − i − arctan i

τi0 i=1 τi0

if the point r0 ∈ T , then  

νˆ (τ ) · τ

τ dS ∇s · 2 = ds

+ 0= τ τ 2 s + i

+

si

τ i0 s 2 + τ 2 i0

2π −

dα =

3 

0



ds

∂B2 (r0 ,a)

arctan

i=1

νˆ (τ ) · τ

τ 2

3  s + s − i i − arctan − 2π τ i0 i=1 τ i0

(14.222)

whence 3  i=1

•

arctan

3  s + s − i i − arctan = 2π τ i0 i=1 τ i0

(14.223)

if the point P belongs to the edge γl , then with γ := {r ∈ ∂T : |r − r0 |  a} and γ

:= ∂B2 ∩ T we have    ˆ (τ ) · τ

νˆ (τ ) · τ

τ

ν dS ∇ s · 2 = ds

+ ds 0= τ τ 2 τ 2 T \B2(r0 ,a)

γ

γ

The Method of Moments I =

3 

arctan

i=1 il

3  s + s + i i − arctan −π τ i0 i=1 τ i0

987

(14.224)

il

whence 3 

arctan

i=1 il

•

3  s + s + i i − arctan =π τ i0 i=1 τ i0

(14.225)

il

if the point P coincides with the vertex Vl , then  s + s − τ

0= dS ∇ s · 2 = arctan l − arctan l − αl τl0 τl0 τ

(14.226)

T \B2 (r0 ,a)

whence arctan

s + s − l l − arctan = αl . τ l0 τ l0

(14.227)

Thanks to identities (14.223), (14.225) and (14.227) we can express 2π|w 0 |, π|w 0 | and αl |w 0 | in (14.214), (14.217) and (14.219) in terms of arctangent functions which we may combine with like contributions. Moreover, we may subtract the null quantity in the rightmost-hand side of (14.221) from the result of (14.208) without modifying the value of the integral. More specifically, by using the trigonometric identity [27, Formula 4.4.34], [34, Formula 1.625.9] arctan z1 − arctan z2 = arctan

z1 − z2 , 1 + z1 z2

z1 z2 > −1

(14.228)

we obtain arctan

|w 0 |s + s + s + i i τi0 i − arctan = arctan

2

+ τ i0 R+i τ i0 τ 2 i0 + w0 + |w0 |Ri

(14.229)

as well as a similar expression for quantities evaluated at the initial endpoint of γi . Thus, the final compact expression for the surface integral of type (14.191) reads [32]  T

dS

3

 s + 1 i τi0 = −|w 0 | arctan 2

+ R τi0 + w 2 0 + |w0 |Ri i=1

+ |w 0 |

3  i=1

arctan

s − i τi0

2

− τ 2 i0 + w0 + |w0 |Ri

+

3  i=1

τ i0 log

+ s + i + Ri − s − i + Ri

(14.230)

and it is valid for any mutual position of the projection P and the triangle T . In this formula the same geometrical quantities pertinent to each edge of T are handled in succession, and this feature makes it easy to evaluate (14.230) in a computer program. What is more, (14.230) can be immediately extended to the case of a polygon with an arbitrary number of edges. We observe that I1 has the form a static single-layer potential of the type (2.189). We know from the discussion of Section 2.10 that such potential is a continuous scalar field for all observation points r ∈ R3 . This property ought to be reflected in the right-hand side of (14.230). Since the

988

Advanced Theoretical and Numerical Electromagnetics

integrand is singular for r ∈ T , we need to check out what happens for w 0 = 0 and τ i0 = 0. The former condition implies that r coincide with its projection r0 , whereas the latter (τ i0 = 0) may be true if P falls on γi or one of the vertices of T (Figures 14.11 and 14.12). We notice that in (14.230) even though the arguments of the arctangent functions tend to infinity, the arctangent remains bounded. Regardless, these terms are proportional to |w 0 | and for w 0 = 0 do not contribute to the result. On the contrary the logarithmic terms are weakly singular and may diverge if the observation point lies on a vertex of T , on an edge or the extension thereof, in that the argument lim lim



w0 →0 τi0 →0

+

+ s + s + i + Ri i + |si | = −

− s − s − i + Ri i + |si |

(14.231)

is either null or infinite or undetermined. Nonetheless, the singular behavior of the logarithm is dominated by the vanishing of the multiplicative factor τ i0 , whereby the ultimate value of this contribution is null, and the single-layer potential remains finite as expected.

14.9.2 Integrals involving R/R To speed up the calculation of (14.192) we draw heavily on the results of Section 14.9.1 and observe that the integrand may be separated into a tangential part and a normal component with respect to T , viz., r − r r − r0 r − r0 τ w 0

= − = − nˆ R R R R R

(14.232)

whence we see that the second term gives rise to an integral of type (14.191) because w 0 nˆ is a constant vector. Given that R depends on τ only, the other contribution may be derived from the gradient of an auxiliary scalar field F(τ ), namely, τ

τ

dF = 2 τˆ = τˆ

2 1/2 R dτ (τ + w0 )

=⇒

F(τ ) = R

(14.233)

whereby w

r − r = ∇ s R − 0 nˆ

R R

(14.234)

with ∇ s denoting the surface gradient over T with respect to (τ , α ). The first summand in the right-hand side is a special instance of the more general formula ⎧ 1 ⎪ ⎪ ⎪ ∇ Rm+2 , m ∈ Z \ {−2} ⎨ m

m+2 s (14.235) R τ =⎪ ⎪ ⎪ ⎩∇ log R, m = −2 s where the exponent m can even take on complex values. For the integral of ∇ s R we may invoke theorem (H.102) with first curvature J(r ) = 0, because

R(τ ) ∈ C1 (T ) ∩ C(T ) if w 0  0, and T is a flat surface, viz.,  T

dS ∇ s R =

 ∂T



ds νˆ (τ )R =

3  i=1

+

νˆ i

si s − i

 

2 1/2 ds s 2 + τ 2 i0 + w0

The Method of Moments I =

3 3 + s + 1  + + 1  2 i + Ri

2 ˆ νˆ i (Ri si − R−i s − ν ) + (τ + w ) log i 0 − 2 i=1 2 i=1 i i0 s − i + Ri

989

(14.236)

having made use of the indefinite integral given in [33, Formula 230.01], which can be proved starting with the change of variable (14.205). Combining (14.236) and (14.230) yields  T

+

dS

r − r 1  + + νˆ (R s − R−i s − = i ) R 2 i=1 i i i

w 0 |w 0 |nˆ

3

3 ⎛  ⎜⎜⎜ ⎜⎝arctan i=1

s + i τi0

2

+ τ 2 i0 + w0 + |w0 |Ri

− arctan +

s − i τi0

⎞ ⎟⎟⎟ ⎟ −⎠

2

τ 2 i0 + w0 + |w0 |Ri

⎞ 3 ⎛ 2  s + + R+i ⎜⎜⎜ τi0 + w 2 ⎟⎟ 0

⎜⎝ νˆ i − w 0 τ i0 nˆ ⎟⎟⎠ log i − 2 si + R−i i=1

(14.237)

which is valid for any mutual position of the projection P and the triangle T . Also I2 is in the form of a static single-layer potential with linear vector density which vanishes for r = r, thus (14.237) ought to be well-defined for any choice of r ∈ R3 . The singular character of ˆ is the integrand manifests itself as the non-existence of the limit as r → r. Indeed, (r − r)/R = −R

ˆ finite but the orientation of the unit radial vector R depends on the direction taken by r to approach r. On the other hand, the right-hand side of (14.237) remains bounded even when w 0 = 0 and τ i0 = 0. For the finiteness of the logarithmic terms we refer to the discussion at the end of Section 14.9.1.

14.9.3 Integrals involving ∇(1/R) These integrals occur in the discretization of the MFIE, the PMCHWT and the Müller equations. Since the integrand exhibits a singularity of the type 1/R2 , (14.193) cannot be integrated when r and r belong to the same triangle T . Thus, we follow a limiting procedure by starting with the observation point (Figure 14.9) r(w 0 ) = r0 + w 0 nˆ

(14.238) w 0

w 0

 0. Taking the limit as → 0 will enable us to give that is, away from the triangle T so long as a meaning to the integral for r ∈ T . We begin by separating the integrand into two parts, i.e., tangential to T and perpendicular to T ∇

w

1

1



1

1

r−r d 1

1

0 d 1 ˆ ˆ ˆ ˆ ˆ n n = −∇ − n = −∇ + n = −∇ + n ·∇ · s s s |r − r | R R R R dR R R R dR R

(14.239)

where ∇ s is the surface gradient with respect to the local polar coordinates (τ , α ). Looking forward to the careful application of the surface Gauss theorem (A.59) we transform the normal component into the surface divergence of a suitable vector field F(τ ) = Fτ (τ )τˆ on T , since the distance R is a function of τ . We require w 0 d 1 w 0 d 1 1 d = = (τ Fτ ) R dR R τ dτ R τ dτ whereby we finally get   wτ 1

1



∇ = −∇s + nˆ ∇s · 20 ,

|r − r | R τ R

=⇒

r ∈ T,

Fτ (τ ) =

w 0 τ R

r ∈ R3 \ {r }.

(14.240)

(14.241)

Advanced Theoretical and Numerical Electromagnetics

990

The tangential part of the integrand is regular for w 0  0 and hence we may invoke the integral theorem (H.102) with first curvature J(r ) = 0  −

dS ∇ s

1 =− R

T



ds

∂T





νˆ (τ ) =− R

s +

3 i  i=1

s − i

ds

3 +  s + νˆ i i + Ri

ˆ ν = − log i − 1/2 s − (τ 2 + w 2 i + Ri 0 ) i=1

(14.242)

on account of (14.206) with due regard to the appropriate multiplicative constant. In spite of having assumed w 0  0, the normal component of the integrand may be singular not because of R but rather owing to the presence of τ in the denominator of the function Fτ (τ ). In 2 particular, by introducing a two-dimensional radial test function φ(τ ) ∈ C∞ 0 (R ) and evaluating the surface divergence in a distributional sense [cf. (C.44)] we have ∇ s

  2π +∞ w 0 τ

w 0 τ

∂φ w



· 2 := − dS ∇s φ · 2 = − dα dτ 0 τ R τ R ∂τ R 

R2

0

0

  +∞ w +∞ w

= −2π φ(τ ) 0 − 2π dτ φ(τ )τ 03 R 0 R 0  w d 1 = 2πφ(0) sign(w 0 ) + dS φ(τ ) 0 R dR R

(14.243)

R2

where we have integrated by parts with respect to τ , and the symbol ⎧

⎪ w

⎨+1 w0 > 0 sign(w 0 ) := 0 = ⎪ |w0 | ⎩−1 w 0 < 0

(14.244)

denotes the signum function of argument w 0 . In shorthand notation this means   w τ w 0 d 1 6 7 = ∇ s · 20 − 2π sign(w 0 )δ(2) τ

R dR R τ R

(14.245)

which is valid for r ∈ T and r ∈ R3 . Then, we examine four cases as we did for (14.191). If the projection P falls outside T (Figure 14.9) the integrand is smooth, and we have 

w d 1 = dS 0 R dR R

T

 T

=

+



s   3 i  w 0 τ

w 0 τ w





dS ∇s · 2 ds νˆ (τ ) · τ 2 = ds i0 2 0 = τ R τ R i=1 τ R

3 

∂T



τ i0 arctan

i=1





s − s + i |w0 | i |w0 | − arctan sign(w 0 ) R+i τ i0 R−i τ i0

s − i

(14.246)

having used (14.207) after dividing by w 0 . If the projection P belongs to T we exclude the offending point τ = 0 with a circle B2 (r0 , a) as in Figure 14.10 and split the integration into two parts      wτ w

w d 1

0 d 1



= dS dS ∇s · 20 dS 0 + R dR R R dR R τ R T

T \B2 (r0 ,a)

B2 (r0 ,a)

The Method of Moments I  =

ds νˆ (τ ) · τ

∂T

w 0 + τ 2 R



ds νˆ (τ ) · τ

∂B2 (r0 ,a)

w 0 + τ 2 R



dS

B2 (r0 ,a)

w 0 d 1 R dR R

991

(14.247)

where we have applied (A.59) with the unit normal νˆ (τ ) positively oriented outward T . The integral along ∂T is the same as in (14.246), whereas the remaining ones are evaluated in polar coordinates (τ , α ) by observing that νˆ (τ ) · τ = −a and τ = a on ∂B2 (r0 , a), viz., 

w

ds νˆ (τ ) · τ 20 + τ R





∂B2 (r0 ,a)

=−





2πw 0 1/2 (a2 + w 2 0 )

+

w d 1 =− dS 0 R dR R

B2 (r0 ,a)

2π

w

dα 0 + R

0

2πw 0 1/2 (a2 + w 2 0 )

2π

a



dα 0

dτ

d w 0 dτ R

0

− 2π sign(w 0 ) = −2π sign(w 0 )

(14.248)

so these terms combine is such a way that the result is independent of a, and hence it is not necessary to take the limit for vanishing a. All in all, we have found  T

 3

 s − w 0 d 1  s + i |w0 | i |w0 |

= dS τ arctan + − arctan − sign(w 0 ) − 2π sign(w 0 ) R dR R i=1 i0 Ri τi0 Ri τi0 

(14.249)

residue

a formula which is valid strictly for P ∈ T . Since the integrals along ∂B2 and B2 (r0 , a) remain bounded even as w 0 → 0 (i.e., r → r0 ) when a > 0, if we subsequently let a → 0+ in (14.247) we are led to identify the first two terms in the right member of (14.247) as the principal value of the integral — which is null — and the remaining term as a residue of sorts due to the singularity. We recall that this separation was carried out for the general formulation of the MFIE and the Müller equations. Further, the residue in (14.249) precisely accounts for the contribution of the term proportional to the Dirac delta distribution in formula (14.245). If the projection point falls on one of the edges of T as in Figure 14.11, we define two paths γ := {r ∈ ∂T : |r − r0 |  a} and γ

:= ∂B2 ∩ T and proceed as in (14.215), viz., 

w d 1 = dS 0 R dR R





T

 =

dS

∇ s

T \B2 (r0 ,a)

ds νˆ (τ ) ·

γ



w

τ 20 τ R



 w 0 τ

· 2 + τ R



dS

B2 (r0 ,a)∩T w

ds νˆ (τ ) · τ 20 τ R

 + γ

w 0 d 1 R dR R 

+

dS

B2 (r0 ,a)∩T

w 0 d 1 R dR R

(14.250)

where the integrals along γ

and over B2 (r0 , a) ∩ T are computed in polar coordinates (τ , α ). In symbols, we have 



ds νˆ (τ ) · τ

γ

=−

w 0 + τ 2 R

πw 0 1/2 (a2 + w 2 0 )



w d 1 =− dS 0 R dR R

B2 (r0 ,a)∩T

+

πw 0 1/2 (a2 + w 2 0 )

π 0

w

dα 0 + R

π



a

dα 0

− π sign(w 0 ) = −π sign(w 0 )

dτ

d w 0 dτ R

0

(14.251)

992

Advanced Theoretical and Numerical Electromagnetics

which is independent of a. For the integral along γ we notice that the contribution from γ ∩ γi vanishes because νˆ i · τ = 0 on γ ∩ γi . The contributions of the other two edges follow through (14.246). Thus, we have found  T

dS

  3 s + |w | s − |w | w 0 d 1  = τ i0 arctan i + 0 − arctan i − 0 sign(w 0 ) − π sign(w 0 ) R dR R i=1 Ri τi0 Ri τi0 

(14.252)

residue

il

for P ∈ γl . Also in this case the principal value of the integral (for w 0 = 0 and a → 0+ ) vanishes, while the residue is only half the value appearing in (14.249). Finally, if P coincides with the vertex Vl , l = 1, 2, 3, of T we introduce the circle B2 (r0 , a) and consider the two paths γ := {r ∈ ∂T : |r − r0 |  a} and γ

:= ∂B2 ∩ T . We indicate with αl the angle determined by the edges of T joined at Vl (Figure 14.12). By proceeding as in (14.250) we arrive at    s − |w | w d 1 s + |w | = τ l0 arctan l + 0 − arctan l − 0 sign(w 0 ) − αl sign(w 0 ) dS 0 (14.253) R dR R Rl τl0 Rl τl0  T

residue

for P ≡ Vl . The principal part (for w 0 = 0 and a → 0+ ) vanishes, and the residue is proportional to the angle with vertex Vl . The result of the integration of the normal component is given by four different formulas which differ for the value of the residue. As already noted in Section 14.9.1 it is desirable to have a single compact expression which collects all possible cases. To this end we just have to resort to identities (14.221), (14.223), (14.225) and (14.227) together with formula (14.229). Including also the tangential part yields  T

dS ∇

3 3   τ i0 s + s + + R+i 1 i



ˆ νˆ i log i − n =− − sign(w ) arctan 0 −

2 + w 2 + |w |R+ R s + R τ i i i0 0 0 i i=1 i=1

+ sign(w 0 )

3  i=1

nˆ arctan

τ i0 s − i

2

− τ 2 i0 + w0 + |w0 |Ri

(14.254)

which is valid for w 0  0 and any mutual position of P and T . The formula is readily extended to polygons with an arbitrary number of edges. We observe that the normal component of the result depends on the sign of w 0 , and this behavior is consistent with the discontinuity of a static doublelayer potential [see (2.244) and (2.276)]. As already noted at the end of Section 14.9.1 the logarithmic part in formula (14.254) is weakly singular for r ∈ ∂T . However, we have to keep in mind that (14.193) appears in combination with other vectors which arise from the discretization of surface integral equations with RWG basis functions. For instance, if r = r0 + w 0 nˆ with r0 ∈ T , for the MFIE and the Müller equations we need to consider the integrand [see (14.182)]       1 1 1 ˆ 0 ) × ∇ × (r − r3 ) = nˆ × ∇ × (r − r3 ) = nˆ × ∇ × (r0 + w 0 nˆ − r3 ) n(r (14.255) R R R ˆ 0 ) ≡ nˆ and r0 − r3 is a vector tangential to T . Accordingly, in combination with (14.254) where n(r we have nˆ × [ˆν i × (r0 + w 0 nˆ − r3 )] = νˆ i nˆ ·(r0 + w 0 nˆ − r3 ) − (r0 + w 0 nˆ − r3 )nˆ · νˆ i = w 0 νˆ i

(14.256)

The Method of Moments I

993

Figure 14.13 For illustrating the ambiguity of the unit normal on edges and vertices of a triangular tessellation. and this factor, being predominant over the singularity of the logarithms, drives those contributions to zero as w 0 → 0. Thus, we can give a meaning to the result for w 0 = 0 and r ∈ T for, e.g., the MFIE, namely, ⎧ 2π, r ∈ T ⎪    ⎪ ⎪ ⎪ 1

⎨ π, r ∈ γl dS nˆ × ∇ × (r − r3 ) = (r − r3 ) sign(w0 ) ⎪ l = 1, 2, 3 (14.257) ⎪ ⎪ R ⎪ ⎩ T αl , r ∈ Vl whereby we even obviate the calculation of six arctangent functions. When the factor 2π is divided by the normalization constant 4π — which is brought in by the static Green function — the residue equals one-half of the RWG function defined over T , and this result should be compared with the general finding (13.108). The other two cases are not contemplated in (13.108) because there the surface ∂V was assumed smooth, whereas the mesh S M of concern here has corners and tips. In this regard, taking the observation point r on the edges or the vertices of the tessellation should be avoided in that the unit normal is not uniquely defined in those situations. Indeed, if r is made to approach r0 ∈ ∂T while being constrained on a neighboring triangle, as is suggested in Figure 14.13, in the limit the unit normal nˆ will not coincide with nˆ , an occurrence which invalidates (14.256) and our conclusion (14.257). In particular, the logarithmic terms would diverge, which poses an issue in the numerical implementation of (14.254). This problem can be circumvented by computing the outer integrals in (14.182) and (14.183) with quadrature formulas on triangles that employ nodes placed away from the edges and the corners (Table 14.1 and [26]). If integral (14.193) is encountered in the construction of the algebraic counterpart of the PMCHWT equations, for r = r0 + w 0 nˆ with r0 ∈ T we have to handle scalar functions of the type 1 1 × (r − r3 ) = (r0 + w 0 nˆ − rl ) · ∇ × (r0 + w 0 nˆ − r3 ), l = 1, 2, 3 (14.258) R R where r0 − rl is a vector tangential to T . For l = 3 the resulting integral is clearly null, since the integrand is a triple scalar product which involves two identical multiplicands. For l = 1, 2 in tandem with (14.254) we have (r − rl ) · ∇

(r − rl ) · νˆ i × (r − r3 ) = (r0 + w 0 nˆ − rl ) · νˆ i × (r0 + w 0 nˆ − r3 ) = (r0 − rl ) · νˆ i × (r0 − r3 ) +w 0 nˆ · νˆ i × (r0 − r3 )  =0

994

Advanced Theoretical and Numerical Electromagnetics + (r0 − rl ) · νˆ i × w 0 nˆ + w 0 nˆ · νˆ i × w 0 nˆ

 = w 0 nˆ · νˆ i × (rl − r3 )

=0

(14.259)

whereby the logarithmic contribution in (14.254) is driven to zero as w 0 → 0. Thanks to this result, we can give a meaning to the integral for w 0 = 0 and r ∈ T for the PMCHWT equations, viz, 

dS (r − rl ) · ∇

T

1 × (r − r3 ) R ⎧ ⎧ ⎪ 2π, r ∈ T ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪

⎨ ⎪ π, r ∈ γl ⎪ ⎨−(r − rl ) · nˆ × (r − r3 ) sign(w0 ) ⎪ ⎪ ⎪ =⎪ ⎪ ⎩α , r ∈ V ⎪ ⎪ ⎪ l l ⎪ ⎪ ⎪ ⎪ ⎩0

l = 1, 2

(14.260)

l = 3.

14.10 Discretization of the EFIE with delta-gap excitation Solving the EFIE (13.83) developed for antenna problems in combination with the delta-gap model of the port requires we revise only the calculation of the excitation vector [F] in the algebraic system (14.23). Indeed, from the comparison of (13.70) and (13.83) it becomes apparent that the relevant integral operator L {•} takes the same form (14.34) already considered in Section 14.2. In particular, N , the Ansatz (14.30) for the current density the properties expected of the basis functions {fn (r)}n=1 JS (r), and the inner product (14.33) remain valid in the present case. As a result, the matrix [L] is still obtained by combining (14.36) for the vector potential contribution with either (14.43) or (14.44) for the scalar potential contribution. By contrast, we need to exercise some care in the evaluation of the entries Fm , m = 1, . . . , N, in view of the discontinuous character of the right-hand side of the EFIE (13.83). For the sake of argument we begin by supposing that the basis functions are defined over the whole surface ∂V. Application of (14.33) with reference to Figure 13.6 yields  5 4 V VG G := U∂V vˆ vˆ , m = 1, . . . , N (14.261) F m = fm , dS fm (r) · h h ∂V ∂WG ∩∂V

where U∂V (r) is a suitable surface step function on ∂V so conceived as to vanish outside ∂WG ∩ ∂V and to equal unity for r ∈ ∂WG ∩ ∂V. We wish to show that the integral over ∂WG ∩ ∂V makes sense even in the limit as the antenna gap reduces to a line γG ⊂ ∂V. From (13.76) and definition (13.77) of the voltage drop across the gap we have   Φ(r1 ) − Φ(r2 ) VG vˆ = ∇Φgap (r) = ∇ η + Φ(r2 ) , η ∈ [0, h] (14.262) h h where Φgap (r) is the electric potential in the gap, and η is a local coordinate on ∂WG ∩ ∂V. Since Φgap (r) is continuous and differentiable for r ∈ ∂WG ∩ ∂V, and the test functions are divergenceconforming by hypothesis, we can transform the surface integral in (14.261) as follows   VG vˆ = dS fm (r) · dS fm (r) · ∇s Φgap (r) h ∂WG ∩∂V

∂WG ∩∂V

The Method of Moments I  =

 dS ∇s · [fm (r)Φgap (r)] −

∂WG ∩∂V



=

995

∂WG ∩∂V



ds vˆ · fm (r)Φ(r1) − γG1

dS Φgap (r)∇s · fm (r) 

ds vˆ · fm (r)Φ(r2) − γG2

dS Φgap (r)∇s · fm (r)

(14.263)

∂WG ∩∂V

where we have invoked the surface Gauss theorem (A.59), and γG1 ∪ γG2 represents the boundary of the open surface ∂WG ∩ ∂V (see Figure 13.6). Besides, the outward unit normal is vˆ along γG1 and −ˆv along γG2 , and this arrangement motivates the minus sign before the relevant curvilinear integral. Evidently, in the limit as h → 0+ the line γG1 approaches γG2 and the surface ∂WG ∩ ∂V degenerates into a line, say, γG ≡ γG1 ≡ γG2 . Nonetheless, we must consider the two surfaces ∂WG ∩ ∂Vl , l = 1, 2, conceptually distinct and held at two different potentials Φ(rl ). In other words, the potential Φgap (r) suffers a jump across the line γG (cf. the stationary case examined in Example 4.5). Finally, the last integral in (14.263) can be shown to vanish in that the integrand remains finite, namely, 33 33  33 33  33  33 dS Φ (r)∇ · f (r) dS |Φgap (r)||∇s · fm (r)| gap s m 33 33 3 ∂WG ∩∂V 3∂WG ∩∂V  max{|Φ(r1 )|, |Φ(r2)|} ∇s · fm ∞ hL −−−−→ 0 + h→0

where L is the length of γG . By putting everything together we find   VG Fm = lim+ dS fm (r) · m = 1, . . . , N vˆ = VG ds vˆ · fm (r), h→0 h

(14.264)

(14.265)

γG

∂WG ∩∂V

on account of (13.77). Unless the integral of vˆ · fm (r) vanishes for some m, all the entries of [F] are non-zero in general. We continue with the calculation of Fm when the functions fm (r) are subsectional and associated with one or more patches S p , p = 1, . . . , M, of the tessellation S M that models the smooth surface ∂V [see (14.28)]. We indicate the support of fm (r) with Ξm and notice that the latter is comprised of one or more patches S p . Further, we call S G ⊂ S M the part of the tessellation that approximates the surface ∂WG ∩ ∂V. In light of the boundary condition (13.79) — which must be rephrased for r ∈ S M — we expect Fm to vanish unless Ξm ∩ S G  ∅, that is to say, unless the support of fm (r) extends somehow over the antenna gap. When this condition is verified, by taking the same steps as in (14.263) we have   4 V 5 VG G := ˆ ˆ v v= dS fm (r) · dS fm (r) · ∇s Φgap (r) F m = fm , h Ξm ∩S G h Ξm ∩S G Ξm ∩S G    = ds vˆ · fm (r)Φ(r1 ) − ds vˆ · fm (r)Φ(r2 ) + ds νˆ (r) · fm (r)Φgap (r) Ξm ∩γ˜ G1

−

Ξm ∩γ˜ G2



dS Φgap (r)∇s · fm (r)

∂Ξm ∩S G

(14.266)

Ξm ∩S G

where γ˜ G1 ∪ γ˜ G2 is the boundary of the open surface S G , and νˆ (r) is the outward unit normal on ∂Ξm ∩ S G which, in turn, is the part of the boundary of Ξm on the gap. An instance of the geometry is depicted in Figure 14.14.

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Advanced Theoretical and Numerical Electromagnetics

Figure 14.14 Modelling the antenna gap of Figure 13.5b with patches: support Ξm of the mth subsectional test function fm (r) for arbitrary-shaped patches and gap of finite size. In the limit as h → 0+ , the line ∂Ξm ∩ S G degenerates into a discrete set of points, and hence we have 33 33 33 33  33  f  max{|Φ(r )|, |Φ(r )|}d(h) −−−−→ 0 33 ˆ ds ν (r) · f (r)Φ (r) (14.267) m gap m ∞ 1 2 33 33 h→0+ 3 3 ∂Ξm ∩S G

where d(h) is the initial length of ∂Ξm ∩ S G . All in all, as S G reduces to a line, say, γ˜ G , we obtain   VG Fm = lim+ dS fm (r) · ds vˆ · fm (r) (14.268) vˆ = VG h→0 h Ξm ∩S G

Ξm ∩γ˜ G

because the last integral in (14.266) vanishes by virtue of estimates similar to (14.264). Notice that if Ξm ∩ γ˜ G = ∅, the pertinent integral is absent, and Fm = 0. For later derivations it is convenient to write all the entries of [F] as  Fm = Um VG ds vˆ · fm (r), m = 1, . . . , N (14.269) Ξm ∩γ˜ G

where ⎧ ⎪ ⎨0, Um := ⎩ ⎪ 1,

if Ξm ∩ γ˜ G = ∅ if Ξm ∩ γ˜ G  ∅

m = 1, . . . , N

(14.270)

is a discrete step function. As a special case we consider a triangular-faceted tessellation S M made of M triangles and the expansion of JS (r) by means of the basis functions of Rao, Wilton and Glisson [25] described in Section 14.7. In order to use (14.269) we choose γ˜ G so that it is comprised of a number of internal edges of the triangular mesh (Figure 14.15). In view of property (14.119) only the test functions associated with two triangles which share an edge being part of γ˜ G contribute a non-zero entry to [F]. Moreover, since S M is piecewise-smooth, the definition of vˆ along γm := Ξm ∩ γ˜ G is not unique,

The Method of Moments I

997

Figure 14.15 Modelling the antenna gap of Figure 13.5b with patches: triangular-faceted tessellation for RWG basis functions and gap reduced to a piecewise-straight closed curve γ˜ G . unless the triangles T m+ and T m− are coplanar. In consequence, we may interpret fm (r) · vˆ for r ∈ γm as the limit obtained by letting r approach the edge from either T m+ or T m− (see Figures 14.5a and 14.5b). Regardless, the result is unique because fm (r) · vˆ = fm (r) · νˆ +m = fm (r) · (−ˆν−m ) = 1,

r ∈ γm

(14.271)

in accordance with (14.117) and (14.118). Keeping this into account, from (14.269) we find  Fm = Um VG ds fm (r) · vˆ = lm VG Um , m = 1, . . . , N (14.272)  γm

=1

where lm denotes the length of γm , the mth internal edge of the mesh. We see that the usage of RWG functions renders the calculation of Fm trivial. In writing (14.271) we have tacitly assumed that T m+ belongs to the part of the mesh that models ∂V1 ∩ ∂V, i.e., the ‘upper’ conducting body of the antenna in Figure 13.5b. Likewise, T m− belongs to the part of the mesh that models ∂V2 ∩ ∂V, the ‘lower’ conducting body. This arrangement is shown in Figure 14.15 along with the opposite choice, namely, T m+ belonging to the part of the mesh that models ∂V2 ∩ ∂V and T m− belonging to the part of the mesh that models ∂V1 ∩ ∂V. In which case we have fm (r) · vˆ = fm (r) · (−ˆν+m ) = fm (r) · νˆ −m = −1, whereby the coefficient Fm reads  Fm = Um VG ds fm (r) · vˆ = −lm VG Um ,  γm

r ∈ γm

m = 1, . . . , N

(14.273)

(14.274)

=−1

that is, the negative of what we have found in (14.272). A numerical code which implements the solution of (13.83) with the Method of Moments and RWG functions must be able to distinguish the two possibilities discussed above in order to ensure the correct evaluation of the entries of Fm . Next, we seek expressions for the current IA flowing into the antenna port and for the antenna admittance YA . In case entire-domain basis functions are employed, (13.88) becomes  N  In ds vˆ · fn (r) (14.275) IA = n=1

γG

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Advanced Theoretical and Numerical Electromagnetics

on account of expansion (14.30). Conversely, if subsectional basis functions are adopted on a tessellation S M , then we can write IA =

N 

 U n In

n=1

ds vˆ · fn (r)

(14.276)

Ξn ∩γ˜ G

with Un given by (14.270). Interestingly, in both cases the relevant integral is the same as the one arising from the calculation of Fm . Thus, invoking (14.265) or (14.269) and carrying out a simple substitution yields IA =

N 1  1 F m Im = [J]T [F] VG m=1 VG

(14.277)

where we have changed name to the dummy summation variable and used (14.25). Further, in light of (13.86) we obtain three equivalent formulas for the admittance YA =

1 1 1 [J]T [F] = − 2 [J]T [L][J] = − 2 [F]T [L]−1 [F] VG2 VG VG

(14.278)

thanks to (14.23) and the formal solution thereto. The first and the second expressions above are of little use, though, until the expansion coefficients In have been found. If the basis and test functions are dimensionless, the vector [J] and the matrix [L] have the physical dimensions A/m and Ωm2 , respectively, and thus the expressions for the admittance are physically correct. If the current density is expanded by means of RWG basis functions and the delta-gap happens to be modelled by just one inner edge, say, γm , the current IA becomes  IA = Im ds vˆ · fn (r) = lm Im (14.279) γm

and hence the admittance reads YA =

Im Im lm lm VG = VG VG2

Im2 1 I L I = − Lmm m mm m VG2 VG2     1 = − 2 lm VG [L]−1 lm VG = −l2m [L]−1 mm mm VG =−

(14.280) (14.281) (14.282)

where ([L]−1 )mm indicates the entry corresponding to the mth row and mth column of [L]−1 . In particular, (14.282) confirms that YA , which plays the role of a transfer function of a linear system, is independent of the specific value of VG . Lastly, we notice that if Fm is determined through (14.274), the current IA changes sign too, so that expressions (14.278) for the admittance maintain their form. However, (14.280) becomes YA =

Im Im lm (−lm VG ) = − 2 VG VG

whereas (14.281) and (14.282), being quadratic expressions in Im and Fm , are unaffected.

(14.283)

The Method of Moments I

999

Figure 14.16 Triangular-faceted model of an inverted-F antenna.

So long as the antenna port is modelled with the delta-gap approximation, the impressed field Ei (r) is non-zero only in the gap and the total electric field in the Fraunhofer region of the antenna coincides with the secondary field Es (r) produced by JS (r). Therefore, once the expansion coefficients In have been determined, formula (14.48) can be used to compute the radiation fields as well as the radiation pattern defined in (9.336). Example 14.3 (Analysis of an inverted-F antenna with EFIE and MoM) An inverted-F antenna is so called because its shape is vaguely reminiscent of a toppled letter F lying on a ground plane. Figure 14.16 shows the triangular-faceted model of an instance of such antenna on a finite-sized square ground plane. The tessellation S M is comprised of M = 358 triangles, and the line γ˜ G which models the delta-gap of the antenna is made up of just one inner edge. The set N contains N = 480 RWG basis functions. The length of the longer arm (which runs parallel {fn (r)}n=1 to the ground) is d1 = 65 mm, whereas the distance between the vertical arms and the distance to ground are d2 = d3 = 20 mm. Solving the EFIE (13.83) with VG = 1 V for frequencies in the range f = ω/(2π) ∈ [0.1, 2] GHz and using (14.282) allows us to compute the input admittance. Plotted in Figure 14.17 is ZA = 1/YA as a function of the electric size (d1 + d3 )/λ0 , with λ0 indicating the wavelength in free space. At the frequencies for which the imaginary part of the impedance changes sign the antenna resonates, and specifically this happens approximately for f ∈ {0.76, 1.2, 2} GHz. However, the first and the third resonances are not quite useful, because the real part of ZA is very large, which makes it difficult to match the antenna to the characteristic impedance of the feeding transmission line (see Figure 13.5a). On the other hand, at the second resonance we have ZA ≈ 15 Ω, and this corresponds with the largest average input or radiated power PF , as can be seen with the aid of Figure 14.18. The reason for the relatively small value of Re{ZA } at around f = 1.2 GHz is that the current IA flowing into the antenna port has a very large magnitude. This information may be gathered from Figure 14.19 which plots

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Advanced Theoretical and Numerical Electromagnetics

Figure 14.17 Input impedance ZA of the inverted-F antenna of Figure 14.16 as a function of the electric size.

Figure 14.18 Average input or radiated power of the inverted-F antenna of Figure 14.16 as a function of the electric size.

The Method of Moments I

1001

Figure 14.19 Magnitude of the surface current density JS (r) of the inverted-F antenna of Figure 14.16 for VG = 1 V and f = 1.2 GHz.

Figure 14.20 Radiation solid of the inverted-F antenna of Figure 14.16 for f = 1.2 GHz; a slice of the graph has been cut away for the sake of visualization.

|JS (r)| for f = 1.2 GHz when the voltage drop VG in the gap is set to 1 V [see (13.83)]. Contrariwise, |IA | is nearly zero at the first and the third resonance. Finally, the radiation pattern (9.336) of the inverted-F antenna at f = 1.2 GHz is plotted in Figure 14.20. The pattern is nearly omnidirectional in the xOy plane (ϑ = π/2), but the antenna radiates essentially in the positive z-direction due the presence of the ground plane. (End of Example 14.3)

Advanced Theoretical and Numerical Electromagnetics

1002

(a)

(b)

(c)

Figure 14.21 For studying the scaling of solutions to a scattering problem: (a) original PEC body, (b) body scaled through a similarity transformation; (c) scaled body and scaled incident field.

14.11 Scaling of solutions Under certain hypotheses (to be discussed in the following) the solution to a given electromagnetic problem may be ‘recycled’, in a manner of speaking, to determine the solution to a scaled instance of the original problem. To be more specific, suppose that we have solved the electromagnetic scattering from a PEC body whose characteristic size is d (Figure 14.21a). What can be said of the solution to the scattering from another PEC body with the same shape as the first one but of characteristic size d/α (Figure 14.21b) where α is a real positive number? Besides, what happens if we further multiply the original incident field Ei (r) by the same factor α, as is hinted at in Figure 14.21c? If the object of concern is immersed in an unbounded homogeneous non-dispersive medium, we can work out useful relations between the solutions to the two problems outlined above. To this purpose we turn to the EFIE developed in Section 13.2.1 together with the algebraic counterpart obtained in Section 14.2 with the Method of Moments. We append the subscript a (resp. b) to physical and geometrical quantities relevant to the original body (resp. the scaled one). The spatial region occupied by object b is obtained by subjecting the domain occupied by body a to a similarity transformation centered in the origin of a common system of coordinates. Therefore, source and observation points in problems a and b are related as r b =

r a , α

rb =

ra , α

Rb = |rb − r b | =

Ra , α

α>0

(14.284)

whereby it is not difficult to prove the formulas  Sb

∇ sb = α∇ sa ,  1 dS b {•} = 2 dS a {•}, α Sa

 γb

∇ sb = α∇ sa  1 dsb {•} = dsa {•} α

(14.285) (14.286)

γa

where γa and γb are two loops contained in S a and S b and related by the same transformation. Formulas (14.285) and (14.286) conform to the intuitive idea that distances and areas must scale linearly or quadratically with α. On the other hand, since the time-harmonic Green function (8.356)

The Method of Moments I

1003

contains the exponential function, we need to make sure that the scaling defined by (14.284) leaves the argument of exp{•} unaltered. This goal is achieved by requiring kb R b = kb

Ra = ka R a α

(14.287)

whence we obtain the transformation rules for wavenumbers, angular frequencies, and wavelengths kb = αka ,

ωb = αωa ,

λb = λa /α,

(14.288)

although the third relationship could have been guessed from (14.284), since the wavelength is, after all, a distance. The scaling of frequency and wavelength as stated above holds true if the speed of light in the background medium is constant, i.e., there is no dispersion (Chapter 12). As a consequence of (14.288) the solution found for body a at a frequency ωa may at best be connected to the solution for body b at another frequency. Now, the EFIE (13.70) stated on the surface S a reads Eit0 e− j ka ·ra

 − j ωa μ

dS a

e− j k a R a JS a (r a ; ωa ) 4πRa

Sa

∇ sa + j ωa ε



dS a

e− j k a R a

∇ · JS a (r a ; ωa ) = 0, 4πRa sa

ra ∈ S a+

(14.289)

Sa

where we have also assumed that the incident field is a uniform plane wave with wavevector ka = ka kˆ (Section 7.2). Next, we notice that the EFIE stated on the surface S b , namely, Eit0 e− j kb ·rb − j ωb μ



dS b

e− j k b R b JS b (r b ; ωb ) 4πRb

Sb

+

∇ sb j ωb ε



dS b

e− j k b R b

∇ · JS b (r b ; ωb ) = 0, 4πRb sb

rb ∈ S b+

(14.290)

ra ∈ S a+

(14.291)

Sb

can be turned into Eit0 e− j ka ·ra

 − j ωa μ

dS a



 ra e− j k a R a ; αωa JS b 4πRa α

Sa

∇ sa + j ωa ε



dS a



 ra e− j k a R a

; αωa = 0, ∇ · JS b 4πRa sa α

Sa

by virtue of the transformation rules (14.284)-(14.288) and the fact that the unit vector kˆ is unaffected by the scaling. Comparison with (14.289) indicates that (14.291) coincides with the EFIE formulated for the scattering from object a, save for the very unknown current density. However, the solution to the EFIE (14.289) is unique for frequencies other than those possibly associated with interior resonances of the PEC cavity with boundary S a (Section 13.2.4) whence we conclude that 

 ra

; αωa = JS b (r b ; ωb ) (14.292) JS a (ra ; ωa ) = JS b α

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Advanced Theoretical and Numerical Electromagnetics

that is, the surface current induced on the scaled body at ω = ωb coincides with the one induced on the original body at ω = ωa . With similar derivations applied to the scattered field in (13.67) it is immediately proved that Esa (ra ; ωa ) = Esb (rb ; ωb )

(14.293)

on account of (14.292). The radar cross section σ3D (physical dimension: m2 ) is a far-field quantity that provides an indication of the average power reflected back by an object that is illuminated by a plane wave [35]. Since σ3D is proportional to the squared magnitude of the scattered electric field in the far region of the source, we have σ3Db = lim 4πrb2

|Esb (rb ; ωb )|2

rb →+∞

σ3Db σ3Da σ3Da = 2 2 = 2 λa λ2b α λb

|Ei0 |2

= lim 4πra2 ra →+∞

|Esa (ra ; ωa )|2 σ3Da = 2 α α2 |Ei0 |2

(14.294) (14.295)

in light of (14.284), (14.293) and (14.288). As regards the numerical solution of (13.70) with the Method of Moments we recall that the RWG functions (14.116) are normalized shifted position vectors, and hence scaling the size of the triangular-faceted mesh has no effect on basis and test functions. For instance, with evident notation we have lb la la α2 ra − r3a = (rb − r3b ) = (ra − r3a ) 2Ab α 2Aa α 2Aa

(14.296)

thanks to (14.284). The relation between the entries of the system matrices [La ] and [Lb ] follows from (14.36) and (14.44), viz., A (ωb ) + LΦ Lmn,b (ωb ) = Lmn,b mn,b (ωb )   e− j k b R b = − j ωb μ dS b fmb (rb ) · dS b

fnb (r b ) 4πRb Ξmb

− =−

1 j ωb ε

j ωa μ α2 −



Ξnb





dS b

dS b ∇ sb · fmb · Ξmb



dS a fma (ra ) · Ξma

1 j ωa εα2



Ξnb

dS a

Ξna

e− j k a R a fna (r a ) 4πRa



dS a ∇ sa · fma · Ξma

e− j k b R b

∇ · fnb 4πRb sb

Ξna

dS a

e− j k a R a

∇ · fna 4πRa sa

1 = 2 Lmn,a (ωa ) α

(14.297)

and for the entries of the excitation vector we have   1 Fma (ωa ) dS b fmb (rb ) · Ei0 e− j kb ·rb = 2 dS a fma (ra ) · Ei0 e− j ka ·ra = . Fmb (ωb ) = α α2 Ξmb

Ξma

(14.298)

The Method of Moments I

1005

Table 14.2 Scaling of electromagnetic quantities with the size of an object Scattering problem

Figure 14.21a

Figure 14.21b

Figure 14.21c

Characteristic size† Incident field Surface current density Scattered field Radar cross section (RCS) RCS normalized to λ2 Entries of the MoM matrix‡ Entries of the source vector§

d Ei (ra ; ωa ) JS (ra ; ωa ) Es (ra ; ωa ) σ3D (ωa ) σ3D (ωa )/λ2a Lmn (ωa ) Fn (ωa )

d/α Ei (rb ; ωb ) JS (rb ; ωb ) Es (rb ; ωb ) σ3D (ωb )/α2 σ3D (ωb )/λ2b Lmn (ωb )/α2 Fn (ωb )/α2

d/α αEi (rb ; ωb ) αJS (rb ; ωb ) αEs (rb ; ωb ) σ3D (ωb )/α2 σ3D (ωb )/λ2b Lmn (ωb )/α2 Fn (ωb )/α

(†) The dimensionless scaling factor α is a real positive number. (‡) Lmn is the sum of (14.36) and (14.37) for the algebraic EFIE. (§) Given by (14.47) for the algebraic EFIE.

As a result, the algebraic system (14.23) relevant to the EFIE for problem b can be transformed into the corresponding system for problem a, viz., [Lb ][Ib ] = −[Fb ]

=⇒

[La ][Ib ] = −[Fa ]

(14.299)

and since the solution is unique for frequencies different than those associated with possible interior resonances, we have [Ib ] = [Ia ]

(14.300)

that is, another proof that the current density induced on the scaled object at ω = ωb coincides with the current density on the original body at ω = ωa . Concerning the simultaneous scaling of object a and multiplication of the incident field by α we observe that (14.291) passes over into αEit0 e− j ka ·ra

 − j ωa μ

dS a



 ra e− j k a R a ; αωa JS b 4πRa α

Sa

∇ sa + j ωa ε



dS a



 ra e− j k a R a

; αωa = 0, ∇ · JS b 4πRa sa α

ra ∈ S a+

(14.301)

Sa

whence by comparison with (14.289) and thanks to the linearity of the operators we find 

 ra 1

; αωa JS a (ra ; ωa ) = JS b =⇒ JS b (r b ; ωb ) = αJS a (r a ; ωa ) α α

(14.302)

that is, the current JS b is α times the original, and the same conclusion applies to the scattered field Esb . The radar cross section, being a quantity independent of the magnitude of the incident plane wave by construction, is evidently not affected further. Similarly, the entries of the system matrix [Lb ] are still given by (14.297), whereas for the entries of the excitation vector [Fb ] we have  Fma (ωa ) (14.303) dS b fmb (rb ) · αEi0 e− j kb ·rb = Fmb (ωb ) = α Ξmb

Advanced Theoretical and Numerical Electromagnetics

1006

(a)

(b)

(c)

Figure 14.22 For studying the scaling of solutions to a radiation problem: (a) original PEC antenna, (b) antenna scaled through a similarity transformation; (c) scaled antenna and scaled generator strength. which, when used to scale the relevant algebraic system, allows proving [Ib ] = α[Ia ]

(14.304)

in agreement with (14.302). All the relationships obtained thus far are summarized in Table 14.2 for ease of reference. We continue the discussion with the solutions to radiation problems by supposing that the characteristic size d of a PEC antenna immersed in a homogeneous unbounded space (Figure 14.22a) is likewise scaled by a factor α (Figure 14.22b). We first consider a scaled antenna excited by the same input voltage VG at the port as the original device, and then we factor in the effect of an excitation VG /α (Figure 14.22c). The transformation rules (14.284)-(14.288) still apply. The starting point is the EFIE (13.83) which we write for antenna a, viz.,  − j ωa μ0

dS a

e− j k a R a JS a (r a ; ωa ) 4πRa

Sa

+

∇ sa j ωa ε0

 Sa

⎧ ⎪ r ∈ ∂Vla ∩ S a ⎪ ⎪ ⎪0, ⎨

e



dS a ∇ · JS a (ra ; ωa ) = ⎪ VG ⎪ ⎪ 4πRa sa ⎪ ⎩− h vˆ , r ∈ ∂WGa ∩ S a a − j ka R a

(14.305)

and for antenna b  − j ωb μ0

dS b

e− j k b R b JS b (r b ; ωb ) 4πRb

Sb

∇ sb + j ωb ε0

 Sb

e− j k b R b

dS b

∇ · JS b (r b ; ωb ) = 4πRb sb

⎧ ⎪ 0, r ∈ ∂Vlb ∩ S b ⎪ ⎪ ⎪ ⎨ V ⎪ G ⎪ ⎪ ⎪ ⎩− h vˆ , r ∈ ∂WGa ∩ S b b

(14.306)

The Method of Moments I

1007

which with a few manipulations passes over into 

dS a

− j ωa μ0



 r e− j k a R a 1 JS b a ; αωa 4πRa α α

Sa

+

∇ sa j ωa ε0



dS a



− j ka R a



r a

1 e ; αωa ∇ · JS b 4πRa sa α α

Sa



⎧ ⎪ ⎪ ⎪ ⎪0, ⎨ =⎪ VG ⎪ ⎪ ⎪ ⎩− h vˆ , a

r ∈ ∂Vla ∩ S a r ∈ ∂WGa ∩ S a

(14.307)

on account of (14.284)-(14.288). We observe that (14.307) has the same structure as (14.305). Thus, if ωa is not a frequency of interior resonance of the PEC cavity bounded by S a , the solution is unique and we conclude that 

 r 1 =⇒ JS b (r b ; ωb ) = αJS a (r a ; ωa ) (14.308) JS a (r a ; ωa ) = JS b a ; αωa α α that is, the current flowing on the scaled antenna is α times the current in the original problem, even though the strength of the generator has not been changed. This can be understood by noticing that in the scaling process the size of the delta-gap (Figures 14.14 and 14.22b) is modified as well, and this geometrical effect can be alternatively construed as the multiplication of VG times α, i.e., as the excitation of the original antenna with a voltage αVG . The current flowing into the antenna port is computed through (13.88), namely,  IAa (ωa ) = dsa JS a (ra ; ωa ) · vˆ (14.309) γGa



IAb (ωb ) =

 dsb JS b (rb ; ωb ) · vˆ =

γGb

dsa JS a (ra ; ωa ) · vˆ = IAa (ωa )

(14.310)

γGa

in light of (14.286), (14.308) and the fact that the unit vector vˆ is not affected by the similarity transformation. If the gap is made to shrink and the same amount of charge has to flow through a shorter contour, as is stated by (14.310), then the surface current density increases in agreement with (14.308). The antenna admittance is provided, e.g., by (13.86), whereby we conclude immediately YAb (ωb ) = YAa (ωa )

(14.311)

since the generator strength and the current at the port coincide for the two problems. In view of definition (13.82) for the radiated field it can be shown that Esb (rb ; ωb ) = αEsa (ra ; ωa )

(14.312)

on the grounds of (14.308). The average radiated power is not affected by the scaling. This can be ascertained by means of formulas (13.96) and (14.310) so long as the background medium is lossless. Alternatively, one chooses two balls B(0, ra) and B(0, rb) with radii ra and rb = ra /α that are large enough for the spheres ∂Ba and ∂Bb to be located in the respective Fraunhofer regions of antennas a and b, and then uses the far-field approximation (9.325) to compute the complex Poynting vector (1.304). Thanks to the impedance relationship (9.327) we have   32 3 1 dS a 33rˆ × Esa (ra ; ωa )33 (14.313) PFa (ωa ) = Re 2Z ∂Ba

1008

Advanced Theoretical and Numerical Electromagnetics

Table 14.3 Scaling of electromagnetic quantities with the size of an antenna Antenna problem

Figure 14.22a

Figure 14.22b

Figure 14.22c

Characteristic size† Input voltage at antenna port Surface current density Current at antenna port Antenna admittance Radiated field Average radiated power Entries of the MoM matrix‡ Entries of the source vector§

d VG JS (ra ; ωa ) IA (ωa ) YA (ωa ) Es (ra ; ωa ) PF (ωa ) Lmn (ωa ) Fn (ωa )

d/α VG αJS (rb ; ωb ) IA (ωb ) YA (ωb ) αEs (rb ; ωb ) PF (ωb ) Lmn (ωb )/α2 Fn (ωb )/α

d/α VG /α JS (rb ; ωb ) IA (ωb )/α YA (ωb ) Es (rb ; ωb ) PF (ωb ) /α2 Lmn (ωb )/α2 Fn (ωb )

(†) The dimensionless scaling factor α is a real positive number. (‡) Lmn is the combination of (14.36) and (14.37) for the algebraic EFIE. (§) Given by (14.272) for the algebraic EFIE.

and

 32 3 1 PFb (ωb ) = Re dS b 33rˆ × Esb (rb ; ωb )33 2Z ∂Bb   32 3 1 = Re dS a 33ˆr × Esa (ra ; ωa )33 = PFa (ωa ) 2Z 

(14.314)

∂Ba

on account of (14.286) and (14.312). Although (9.327) is only approximated — because terms proportional to higher-order powers of 1/r are neglected — the average radiated power is computed correctly, since only first-order terms in 1/r contribute to radiation (cf. Example 9.3 for the Hertzian dipole). Further, while it is true that, in principle, any surface in the Fraunhofer region can be employed for the calculation of the average power, we are compelled to consider ∂Ba and the scaled version thereof (∂Bb) because (14.312) provides us with a link between the field on two different spheres precisely related through (14.284). The relation between the entries of the MoM matrices [La ] and [Lb ] is the same as that proved in (14.297). By contrast, from (14.272) for the excitation vector we find Fmb (ωb ) = lmb VG (ωb )Um =

lma Fma (ωa ) VG (ωa )Um = , α α

m = 1, . . . , N

(14.315)

1 1 [La ][Ib ] = − [Fa ] 2 α α

(14.316)

whereby the algebraic system (14.23) becomes [Lb ][Ib ] = −[Fb ]

=⇒

and formal solution proves (14.308) again. Lastly, when the simultaneous scaling of antenna a and division of the generator strength by α is contemplated (14.307) is modified as  − j ωa μ0 Sa

dS a



 r e− j k a R a JS b a ; αωa 4πRa α

The Method of Moments I ∇ sa + j ωa ε0

 Sa

⎧ 

 ⎪ 0, ⎪ − j ka R a ⎪ ⎪ r e ⎨ dS a

∇ sa · JS b a ; αωa = ⎪ VG ⎪ ⎪ 4πRa α ⎪ ⎩− h vˆ , a

r ∈ ∂Vla ∩ S a r ∈ ∂WGa ∩ S a

whence we conclude that 

 r JS b (r b ; ωb ) = JS b a ; αωa = JS a (r a ; ωa ) α

(14.317)

(14.318)

by comparison with (14.305). Then, from (13.88) it follows that   1 IAa (ωa ) dsb JS b (rb ; ωb ) · vˆ = dsa JS a (ra ; ωa ) · vˆ = IAb (ωb ) = α α γGb

1009

(14.319)

γGa

by virtue of (14.286) and comparison with (14.309). The antenna admittance is unaffected by the scaling of the generator, because YA (ω) represents, after all, the response of a linear system. In like manner, computing the radiated field through (13.82) shows that Esb (rb ; ωb ) and Esa (ra ; ωa ) coincide. The radiated power for the scaled object reads   32 3 1 dS b 33rˆ × Esb (rb ; ωb )33 PFb (ωb ) = Re 2Z ∂Bb   32 PFa (ωa ) 3 1 1 = Re dS a 33rˆ × Esa (ra ; ωa )33 = (14.320) 2 2Z α α2 ∂Ba

by comparison with (14.313). Scaling the generator strength does not alter the entries of the MoM matrix either, whereas for the excitation vector we have Fmb (ωb ) = lmb VG (ωb )Um =

lma VG (ωa ) Fma (ωa ) Um = , α α α2

m = 1, . . . , N

(14.321)

which in combination with (14.23) indicates that the vectors of current coefficients [Ia ] and [Ib ] coincide, in agreement with (14.318). We have collected all these findings in Table 14.3.

References [1] [2] [3] [4] [5]

Kantorovich LV, Krylov VI. Approximate methods of higher analysis. 4th ed. New York, NY: John Wiley & Sons, Inc.; 1959. Translated by C. D. Benster. Richmond JH. Digital computer solutions of the rigourous equations for scattering problems. Proceedings of the IEEE. 1965 Aug;53:796–804. Harrington RF. Field Computation by Moment Methods. New York, NY: MacMillan Publishing Co., Inc.; 1968. Harrington RF. The method of moments in electromagnetics. Journal of Electromagnetic Waves and Applications. 1987;1(3):181–200. Peterson AF, Ray SL, Mittra R. Computational Methods for Electromagnetics. Piscataway, NJ: IEEE Press; 1998.

1010 [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

[16] [17] [18] [19] [20] [21] [22] [23] [24]

[25] [26] [27] [28] [29]

Advanced Theoretical and Numerical Electromagnetics Gibson WC. The Method of Moments in electromagnetics. Boca Raton, FL: Chapman & Hall/CRC; 2008. Atkinson KE. The Numerical Solution of Integral Equations of the Second Kind. Cambridge, UK: Cambridge University Press; 1997. Sadiku MNO. Numerical techniques in electromagnetics. 2nd ed. Boca Raton, FL: CRC Press; 2001. Zhou P. Numerical Analysis of Electromagnetic Fields. Berlin Heidelberg: Springer-Verlag; 1993. Binns KJ, Lawrenson PJ, Trowbridge CW. The analytical and numerical solution of electric and magnetic fields. Chichester: John Wiley & Sons, Inc.; 1992. Dudley DG. Mathematical Foundations for Electromagnetic Theory. New York, NY: WileyInterscience; 1994. Jin JM. Theory and Computation of Electromagnetic Fields. 2nd ed. Hoboken, NJ: IEEE Press; 2015. Sevgi L. Electromagnetic Modeling and Simulation. Hoboken, NJ: IEEE Press; 2014. Rudin W. Principles of mathematical analysis. 3rd ed. New York, NY: McGraw-Hill; 1976. Ylä-Oijala P, Markkanen J, S Järvenpää e. Surface and volume integral equation methods for time-harmonic solutions of Maxwell’s equations. Progress in Electromagnetic Research. 2014;149:15–44. Van Bladel JG. Singular Electromagnetic Fields and Sources. Piscataway, NJ: IEEE Press; 1991. Bau III D, Trefethen LN. Numerical linear algebra. Philadelphia, PA: Soci. Indus. Ap. Math.; 1997. Blyth TS, Robertson EF. Basic Linear Algebra. 2nd ed. Springer Undergraduate Mathematics Series. London, UK: Springer-Verlag; 2002. Ylä-Oijala P, Taskinen M. Calculation of CFIE impedance matrix elements with RWG and n x RWG functions. IEEE Trans Antennas Propag. 2003 Aug;51(8):1837–1846. Golub GH, van Loan CF. Matrix Computations. Baltimore, MD: Johns Hopkins University Press; 1996. van der Vorst H. Iterative Krylov methods for large linear systems. Cambridge, UK: Cambridge University Press; 2003. Dongarra JJ, Duff IS, Sorensen DC, et al. Numerical linear algebra for high-performance computers. Philadelphia, PA: Society for Industrial and Applied Mathematics; 1998. Ylä-Oijala P, Taskinen M. Well-conditioned Müller formulation for electromagnetic scattering by dielectric objects. IEEE Trans Antennas Propag. 2005 Oct;53(10):3316–3323. Sheng XQ, Jin J, Song J, et al. Solution of combined-field integral equation using multilevel fast multipole algorithm for scattering by homogeneous bodies. IEEE Trans Antennas Propag. 1998 Nov;46(11):1718–1726. Rao SM, Wilton DR, Glisson AW. Electromagnetic scattering by surfaces of arbitrary shape. IEEE Trans Antennas Propag. 1982 May;30(3):409–418. Stroud AH. Approximate calculation of multiple integrals. Englewood Cliffs, NJ: Prentice Hall; 1971. Abramowitz M, Stegun IA. Handbook of mathematical functions. New York, NY: Dover Publications, Inc.; 1965. Lancellotti V, Milanesio D, Maggiora R, et al. TOPICA: an accurate and efficient numerical tool for analysis and design of ICRF antennas. Nuclear Fusion. 2006;46(7):S476–S499. Rao SM. Electromagnetic scattering and radiation of arbitrarily-shaped surfaces by triangular patch modeling [Research Output]. University of Mississippi. Oxford, MS (USA); 1980.

The Method of Moments I [30] [31]

[32]

[33] [34] [35]

1011

Okon EE, Harrington RF. The polarizabilities of electrically small apertures of arbitrary shape. IEEE Transactions on Electromagnetic Compatibility. 1981 Nov;23(4):359–366. Birtles AB, Mayo BJ, Bennett AW. Computer technique for solving 3-dimensional electronoptics and capacitance problems. Proceedings of Institute of Electrical Engineering. 1973 Feb;120(2):213–220. Wilton DR, Rao SM, Glisson AW, et al. Potential integrals of uniform and linear source distributions on polygonal and polyhedral domains. IEEE Trans Antennas Propag. 1984 March;32(3):276–281. Dwight HB. Tables of Integrals and Other Mathematical Data. 4th ed. New York, NY: MacMillan Publishing Co., Inc.; 1961. Gradshteyn IS, Ryzhik IM. Table of Integrals, Series, and Products. 5th ed. Amsterdam, NL: Academic Press, Inc.; 1994. Knott EF, Turley MT, Shaeffer JF. Radar Cross Section. Norwood, MA: Artech House, Inc.; 1985.

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Chapter 15

The Method of Moments II

We continue the discussion of the Method of Moments presented in its general aspects in Section 14.1 by describing the application to volume integral equations for inhomogeneous anisotropic objects in Sections 15.1-15.4. The solution of coupled electric-field and volume integral equations is addressed in Section 15.5. Lastly, in Section 15.6 we apply the MoM to the combination of surface integral equation and wave equation. The latter calls for specially devised sub-domain basis functions whose properties are considered in Section 15.7.

15.1 Discretization of volume integral equations For the sake of argument we tailor the general procedure of Section 14.1 to (13.215) in which the unknown quantity is the time-harmonic displacement vector D(r) in an anisotropic and possibly inhomogeneous dielectric medium. Specifically, we observe that the integral operator involved in (13.215) reads  L {•} :=









dV G(r, r )κe (r ) · {•} + ∇

k02 V

  dV  G(r, r )∇ · κe (r ) · {•}

V





vector potential

scalar potential (volume charges)



−∇

  ˆ  ) · κe (r ) · {•} −ε0 ε(r) −1 · {•}, dS  G(r, r )n(r

r∈V

(15.1)

∂V

 scalar potential (surface charges)

where G(r, r ) is the Green function (8.356) and the dyadic field κe (r ) is the dielectric contrast factor (10.91). This operator stems essentially from the integral representation of the electric field produced by polarization currents and charges. The former, flowing within V, enter through the vector potential AE (r) (9.24), whereas the charges — distributed both within V and on the interface ∂V — are accounted for by the gradient of the scalar potentials ΦE (r) (9.23) and (9.136). Further,   since ε0 ε(r) −1 is the inverse of the relative permittivity dyadic defined in (1.129), L {•} transforms displacement vectors into other displacement vectors. Since the basis functions fn (r) used to discretize L {•} must mimic the physical properties expected of D(r), the form of the entries of the matrix [L] in (14.24) strongly depends on whether the basis functions we choose to expand D(r) are entire-domain or sub-domain. This statement will become clearer in the course of the discussion.

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Advanced Theoretical and Numerical Electromagnetics

N We begin with the set of N real-valued entire-domain expansion functions {fn (r)}n=1 defined for r ∈ V. We also introduce the symmetric inner product in the space WL ≡ VL for the operator in (15.1) as  (f, g)V := dV f(r) · g(r) (15.2) V

and make the Ansatz D(r) =

N

r∈V

Dn fn (r),

(15.3)

n=1

with the understanding that D(r) is a shorthand notation for the approximation DN (r) to the exact solution [cf. (14.6)]. The expansion functions fn (r) should have the following characteristics. Since the fn (r) are employed to expand the flux density D(r), they must be vector fields defined within the region V occupied by the object of concern. Since the integral operator in (15.1) involves the divergence of the field κe (r) · D(r) which by virtue of (H.73) [1, Eq. (A4.57)] can be expanded as     ∇ · κe (r) · D(r) = ∇ · κe (r) · D(r) + κe (r) : ∇D(r) (15.4)

(a) (b)

the fn (r) must have finite first-order derivatives for r ∈ V. In the special instance of isotropic inhomogeneous medium, κe (r) = κe (r)I, and from (H.51) we just have ∇ · [κe (r)D(r)] = D(r) · ∇κe (r) + κe (r)∇ · D(r)

(15.5)

whereby we require the divergence of fn (r) to be finite. Vector fields endowed with this feature are said to be divergence-conforming (cf. Section 14.2). Since the fn (r) must correctly describe the occurrence of a layer of polarization charges on the material interface ∂V (but inside V) we require

(c)

ˆ = 1, fn (r) · n(r)

r ∈ ∂V.

(15.6)

Splitting (15.1) of the operator L {•} is mirrored by the entries of [L], viz., A Φ,S Lmn := Lmn + LΦ,V mn + Lmn + G mn ,

where





A = k02 Lmn

LΦ,V mn =



V

V



=−

dV  G(r, r )κe (r ) · fn (r )

(15.8)

  dV  G(r, r )∇ · κe (r ) · fn (r )

(15.9)



dV fm (r) · ∇ V

(15.7)

V



dV fm (r) · ∇ V

LΦ,S mn

dV fm (r) ·

m, n ∈ {1, . . . , N}



Gmn = −ε0 V

ˆ  ) · κe (r ) · fn (r ) dS  G(r, r )n(r

∂V

  dV fm (r) · ε(r) −1 · fn (r) = −

 V

  dV fm (r)fn (r) : ε(r) −1 ε0

(15.10) (15.11)

The Method of Moments II

1015

and, if the basis and test functions are dimensionless, then the coefficients Lmn carry the physical dimension of a volume (m3 ). If the permittivity dyadic ε(r) is symmetric positive definite, so is its inverse [ε(r)]−1 , and the matrix [G] formed with the coefficients (15.11) constitutes a Gram matrix of sorts associated with N . What is more, (15.11) can be interpreted as a weighted inner product — with the set {fn (r)}n=1 weighting function −ε0 [ε(r)]−1 — which clearly reduces to (15.2) when −ε0 [ε(r)]−1 coincides with the identity dyadic. As already noticed for the entries of the algebraic EFIE, also (15.9) and (15.10) are in a form which is ill-suited for the numerical calculation of the outer integrals inasmuch as the gradient of the inner integral has to be computed first. Therefore, we try to cast (15.9) and (15.10) into an alternative format by ‘moving the gradient onto the test function’. However, this step is permissible so long as fm (r) can stand a derivative. To elucidate, we define the auxiliary scalar potential        ˆ  ) · κe (r ) · fn (r ) dV G(r, r )∇ · [κe (r ) · fn (r )] − dS  G(r, r )n(r (15.12) Υn (r) := ∂V

V

and observe    dV fm (r) · ∇Υn (r) = dV ∇ · [fm (r)Υn (r)] − dV Υn (r)∇ · fm (r) V

V

(15.13)

V

thanks to the differential identity (H.51). We would like to apply the Gauss theorem (A.53) to the first integral in the right member. To this purpose, the components of the vector field fm (r)Υn (r) must be differentiable in V and continuous in V. Now, the auxiliary function Υn (r) is certainly smooth, since it is the linear combination of a volume potential [see (9.11) and Section 2.8] and a single-layer potential (see Section 2.10). As regards the test function fm (r), we must require — in light of (H.51) — that the divergence be continuous throughout V. This been said, we have    ˆ · fm (r)Υn (r) − dV Υn (r)∇ · fm (r) dV fm (r) · ∇Υn (r) = dS n(r) (15.14) ∂V

V

V

where the flux integral does not vanish on account of the surmised property (15.6). All in all, for entire-domain basis and test functions we arrive at the alternative format     ˆ LΦ,V = dS n(r) · f (r) dV  G(r, r )∇ · κe (r ) · fn (r ) m mn ∂V

V

 −

 dV ∇ · fm (r)

V

LΦ,S mn = −



 ˆ · fm (r) dS n(r)

∂V

  dV  G(r, r )∇ · κe (r ) · fn (r ) (15.15)

V

ˆ  ) · κe (r ) · fn (r ) dS  G(r, r )n(r

∂V

 +

 dV ∇ · fm (r)

V

ˆ  ) · κe (r ) · fn (r ) (15.16) dS  G(r, r )n(r

∂V

where     ∇ · κe (r ) · fn (r ) = ∇ · κe (r ) · fn (r ) + κe (r ) : ∇ fn (r )

(15.17)

1016

Advanced Theoretical and Numerical Electromagnetics

or ∇ · [κe (r )fn (r )] = fn (r ) · ∇ κe (r ) + κe (r )∇ · fn (r )

(15.18)

on account of (H.73) and (H.51). In practice, though, it is difficult to find entire-domain basis functions for arbitrary-shaped bodies. Thus, one resorts to modelling V by means of a volumetric mesh V M [see (14.29)] made up of M simple-shaped adjoining sub-domains and considers N real-valued subsectional basis functions associated with one or more sub-domains, so the number of sub-domains is not necessarily the same as the number of basis functions. We need to distinguish two cases, namely, (i)

the sub-domain V p is internal to V M , which means that the boundary ∂V p of V p does not intersect the material interface ∂V M , that is, V p ⊂ VM ,

(ii)

∂V p ∩ ∂V M = ∅,

p = 1, . . . , MI

(15.19)

where MI < M is the number of internal sub-domains; the boundary ∂V p is partly internal to V M and partly belongs to the material interface ∂V M , viz., V p ⊂ VM ,

∂V p ∩ ∂V M  ∅,

p = MI + 1, . . . , MI + MB

(15.20)

where MB = M − MI is the number of boundary sub-domains. Accordingly, the support of some of the basis functions will have part of the boundary lying on ∂V M . Thus, we introduce a set of N = NI + NB functions as

N I

NI +NB N := fnI (r) {fn (r)}n=1 ∪ fnB (r) (15.21) n=NI +1

n=1

where • •

fnI (r) is a function whose support Δn is comprised of one or more internal sub-domains, fnB (r) is a function whose support Δn taps into ∂V M , i.e., ∂Δn ∩ ∂V M  ∅.

The operator L {•} in (15.1) is traded for one which maps vector fields defined over the volumetric mesh V M onto fields with support over V M . As a consequence, we define the inner product in the range of the new operator as  (f, g)V M := dV f(r) · g(r) (15.22) VM

and assume D(r) =

NI

Dn fnI (r) +

n=1

N

Dn fnB (r),

r ∈ VM

(15.23)

n=1+NI

where again D(r) signifies DN (r), the approximation of the exact solution that is afforded by the use of N expansion functions associated with the mesh V M . As regards the properties of fnI (r) and fnB (r) we observe that both fnI (r) and fnB (r) must have finite derivatives. Besides, we require ˆ = 0, fnI (r) · n(r)

r ∈ ∂Δn ,

n = 1, . . . , NI

(15.24)

The Method of Moments II fnB (r) ·

⎧ ⎪ ⎨1, ˆ =⎪ n(r) ⎩0,

r ∈ ∂Δn ∩ ∂V M

n = 1 + NI , . . . , NI + N B

r ∈ ∂Δn ∩ V M

1017 (15.25)

since surface polarization charges exist only on the boundary ∂V M . As a result, with subsectional basis and test functions the coefficients Lmn are still given by (15.7) with   A = k02 dV fm (r) · dV  G(r, r ) κe (r ) · fn (r ) (15.26) Lmn Δm

Δn



dV fm (r) · [ε(r)]−1 · fn (r)

Gmn = −ε0

(15.27)

Δm ∩Δn

LΦ,V mn



 = Um

ˆ · fm (r) dS n(r)

∂Δm

  dV  G(r, r )∇ · κe (r ) · fn (r )

Δn

 −

 dV ∇ · fm (r)

Δm

LΦ,S mn



 = −Um ∂Δm

ˆ · fm (r) dS n(r)

  dV  G(r, r )∇ · κe (r ) · fn (r ) (15.28)

Δn

ˆ  ) · κe (r ) · fn (r ) dS  G(r, r ) n(r

∂Δn



 dV ∇ · fm (r)

+ Δn

ˆ  ) · κe (r ) · fn (r ) (15.29) dS  G(r, r ) n(r

∂Δn

where ⎧ ⎪ ⎨0, m = 1, . . . , NI Um := ⎪ ⎩1, m = N + 1, . . . , N + N I I B

(15.30)

is a discrete step function of sorts. With this notation we make it clear that the terms involving surface integrals over ∂Δm in (15.28) and (15.29) are included only when fm (r) ≡ fmB (r). As a matter of fact, when the medium is isotropic and fn (r) ≡ fnI (r), also the integral over ∂Δn in LΦ,S mn vanishes because ˆ  ) · κe (r ) · fn (r ) = κe (r )n(r ˆ  ) · fn (r ) = 0 n(r

(15.31)

in view of (15.24). A Further development of the expressions for Lmn , etc., is only possible once the form of fn (r) is explicitly known (see Section 15.2). Regardless, if the permittivity dyadic ε(r) is symmetric, then by direct inspection of (15.8), (15.11), (15.26) and (15.27) we conclude A A Lmn = Lnm ,

Gmn = Gnm ,

m, n ∈ {1, . . . , N}

(15.32)

Φ,S whereas this property does not hold for LΦ,V mn and Lmn . Hence, the matrix [L] is not symmetric with our choice of test and basis functions and inner product.

1018

Advanced Theoretical and Numerical Electromagnetics

The entries of the excitation vector [F] in the right-hand side of (14.24) read    m = 1, . . . , N Fm = fm , ε0 Ei := ε0 dV fm (r) · Ei (r), V

(15.33)

V

where the integration is restricted to the support Δm if fm (r) is a subsectional basis function. If the incident electric field is a uniform plane wave propagating in the background medium (Section 7.2), we have  := ε0 E0 · dV e− j k·r fm (r), m = 1, . . . , N (15.34) Fm V

where the domain integral may be interpreted as the three-dimensional Fourier transform of fm (r) evaluated in the spectral point −k, the negative of the wavevector. In order to compute the scattered electric field in the Fraunhofer region of the dielectric body we first obtain the polarization current density [cf. (13.196)] Jeq (r) := j ωκe (r) · D(r) = j ωκe (r) ·

N

r ∈ VM

Dn fn (r),

(15.35)

n=1

which we then substitute into (9.317) to get e− j k 0 r ˆ ˆ (ϑϑ + ϕˆ ϕ) ˆ · Dn 4πr n=1 N

Es (r) ≈ ω2 μ0

≈ ω2 μ0

e− j k 0 r ˆ ˆ (ϑϑ + ϕˆ ϕ) ˆ · 4πr

N n=1



dV  κe (r ) · fn (r )ej k0 rˆ ·r

Δn



Dn κe (rn ) ·

dV  fn (r )ej k0 rˆ ·r





(15.36)

Δn

where in the second expression we have also approximated the dyadic contrast factor with the value taken on in the centroid rn of the support Δn . While this step makes sense so long as the variation of κe (r ) is relatively slow within Δn , perhaps the remaining integral can be evaluated in closed form (see Example 15.2). A formula for the calculation of the average power absorbed by or lost into the material medium that fills the region V is derived further on in Section 15.5.

15.2 The basis functions of Schaubert, Wilton and Glisson Although the particular choice of basis functions has a marginal impact on the application of the Method of Moments to volume integral equations, still coming up with entire-domain functions defined in arbitrary-shaped volumetric domains is even more difficult than it is for surfaces also in view of the additional degree of freedom. Therefore, prior to the discretization process one resorts to replacing the volume V with a three-dimensional mesh V M of M adjoining simple-shaped subdomains V p , p = 1, . . . , M. When the latter are tetrahedra it is possible to define the elements of the N set {fn (r)}n=1 as normed and shifted position vectors associated with one or two sub-domains. These basis functions may be rightfully regarded as the three-dimensional extension of the RWGs (Section 14.7) and, since they were proposed by D. H. Schaubert, D. R. Wilton and A. W. Glisson in 1984 [2], they are sometimes referred to as SWG basis functions for short.

The Method of Moments II

1019

Figure 15.1 Geometrical setup for the definition of the SWG basis function associated with the facet T n and defined over a pair of adjacent tetrahedra Wn+ and Wn− . A tetrahedral mesh V M is comprised of vertices, triangular facets and tetrahedra. Most facets are shared by two adjoining sub-domains, but the facets that approximate the original boundary ∂V belong to just one tetrahedron. We call inner or interior facets the NI triangles that belong to two sub-domains, and boundary facets the NB patches which are part of a single tetrahedron; clearly, it holds N = NI + NB . We shall see in a moment that it is necessary to define SWG functions associated with both inner and boundary facets, in contrast to the definition of RWG functions (14.116). Thus, the number N of functions coincides with the total number of facets, not the number of tetrahedra M in the mesh. For the mathematical description of the nth SWG function we introduce the following geometrical quantities with the help of Figure 15.1: • • • • • • • • • • • •

T n , n = 1, . . . , N, all the triangular facets of the mesh V M ; Wn+ and Wn− , the two tetrahedra which share the common interior facet T n , with n = 1, . . . , NI ; Wn+ , the tetrahedron associated with the boundary facet T n , where n = NI + 1, . . . , NI + NB ; + + + + + V1n , V2n , V3n and V4n , the four vertices of the tetrahedron Wn+ ; by convention V4n indicates the vertex that does not belong to the triangle T n ; there is no simple labelling of the remaining three vertices that ensures a numbering in increasing order locally on all facets of Wn+ ; − − − − − V1n , V2n , V3n and V4n , the four vertices of the tetrahedron Wn− ; by convention V4n denotes the − + − + ≡ V2n , and vertex that does not belong to the triangle T n ; we arbitrarily choose V1n ≡ V1n , V2n − + V3n ≡ V3n ; + + + + + T 1n , T 2n , T 3n and T 4n = T n , the four facets forming Wn+ ; the vertex V1n is opposite the triangle + T 1n , and so forth; − − − − − T 1n , T 2n , T 3n and T 4n = T n , the four facets forming Wn− ; the vertex V1n sits across from the − triangle T 1n , and so on; A+1n , A+2n , A+3n and A+4n = An , the areas of the four facets belonging to Wn+ ; A−1n , A−2n , A−3n and A−4n = An , the areas of the four facets belonging to Wn− ; Vn+ and Vn− , the volumes of the tetrahedra Wn+ and Wn− ; + + + + r+1n , r+2n , r+3n and r+4n , the position vectors of the vertices V1n , V2n , V3n and V4n ; − − − − − − − − r1n , r2n , r3n and r4n , the position vectors of the vertices V1n , V2n , V3n and V4n ;

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Advanced Theoretical and Numerical Electromagnetics

+ + + + nˆ +1n , nˆ +2n , nˆ +3n and nˆ +4n ≡ nˆ +n the unit vectors normal to the facets T 1n , T 2n , T 3n and T 4n = T n ; the 3 + normals are positively oriented towards R \ Wn ; − − − − nˆ −1n , nˆ −2n , nˆ −3n and nˆ −4n ≡ nˆ −n the unit vectors normal to the facets T 1n , T 2n , T 3n and T 4n = T n ; the 3 − − + normals are positively oriented towards R \ Wn , and it holds nˆ n ≡ −nˆ n .

Having laid down the necessary nomenclature, we can write the SWG basis function associated with the inner facet T n as the dimensionless vector field ⎧ An ⎪ ⎪ ⎪ (r − r+4n ), r ∈ Wn+ ⎪ ⎪ + ⎪ 3V ⎪ n ⎪ ⎪ ⎪ ⎨ An n = 1, . . . , NI (15.37) fn (r) := ⎪ ⎪ − − (r − r−4n ), r ∈ Wn− ⎪ ⎪ ⎪ 3V ⎪ n ⎪ ⎪ ⎪ + − ⎪ ⎩0, r ∈ V M \ (W ∪ W ) n

n

whence we see that the support of fn (r) is Δn := Wn+ ∪ Wn− and also • •

for points r ∈ Wn+ , fn (r) is proportional to the position vector of a point in Wn+ with respect to + the vertex V4n ; for points r ∈ Wn− , fn (r) is proportional to the position vector of a point in Wn− with respect to − the vertex V4n .

If T n is a boundary facet, the definition of the (half) SWG function reads ⎧ A ⎪ n + + ⎪ ⎪ ⎪ + (r − r4n ), r ∈ Wn ⎨ 3V n fn (r) := ⎪ n = NI + 1, . . . , NI + NB ⎪ ⎪ + ⎪ ⎩0, r ∈ VM \ W n

(15.38)

so that the support of fn (r) is just Δn = Wn+ , though we might as well agree to label the relevant tetrahedron as Wn− and include a minus sign in the definition. The reasoning behind the choice of the normalization constant is similar to that followed for the RWG functions. We consider the tetrahedron Wn+ which is sketched in Figure 15.2a from a different perspective. When the point r lies on the facet T n , the component of r − r+4n along the unit normal nˆ +n is precisely the height h+n of the tetrahedron with respect to the base T n . In like fashion, if T n is an interior facet, the component of r − r−4n along the unit normal nˆ −n equals h−n , the height of Wn− with respect to T n (Figure 15.2b). As a result, we have An h+n An (r − r+4n ) · nˆ +n = = 1, + 3Vn 3Vn+ An h−n An fn− (r) · (−nˆ −n ) = − − (r − r−4n ) · (−nˆ −n ) = = 1, 3Vn 3Vn− fn+ (r) · nˆ +n =

r ∈ ∂Wn+ ∩ T n

(15.39)

r ∈ ∂Wn− ∩ T n

(15.40)

which we can summarize by stating that the component of fn (r) orthogonal to the shared facet T n is continuous through T n . In the case where T n is a boundary facet, then fn (r) possesses unitary normal component on T n . This behavior makes the SWG functions well-suited for the representation of the displacement vector D(r) and the magnetic induction field B(r) in a tetrahedral mesh V M , because the normal components of D(r) and B(r) satisfy the matching conditions (1.198) (with no surface ˆ charges) and (1.199). Moreover, the subset of functions (15.38) allows modelling the fact that n·D(r) and nˆ · B(r) do not vanish on the boundary of a dielectric or a magnetic medium, respectively. When the point r lies on the facets T ln+ , l = 1, 2, 3, of Wn+ , the vector r − r+4n becomes parallel to that patch, and a similar remark holds for the facets T ln− of Wn− . Hence, we have fn (r) · nˆ +ln = 0,

r ∈ T ln+ ,

fn (r) · nˆ −ln = 0,

r ∈ T ln−

(15.41)

The Method of Moments II

(a) tetrahedron Wn+

1021

(b) tetrahedron Wn−

Figure 15.2 Geometrical setup for the definition of the normalization constant of an SWG basis function. +

−

and, since fn (r) vanishes for r ∈ V M \ (W n ∪ W n ), we conclude that the normal component of fn (r) is even continuous (i.e., null) across the boundary of Wn+ ∪ Wn− . On the other hand, the vector field fn (r) per se is only piecewise-continuous for r ∈ V M and suffers jumps across the facets of the mesh. Therefore, to compute the divergence we need, in principle, to invoke the distributional form of the derivatives (Appendix C.2). In fact, for all we know at this stage, the divergence of fn (r) may include surface Dirac distributions on the boundary ∂Wn+ ∪ ∂Wn− . To elaborate, we introduce the weak divergence for vector functions defined over the mesh V M , viz., ∇ · F(r) := −

M  p=1 V

dV ∇φ(r) · F(r)

(15.42)

p

3 where F(r) is an arbitrary, possibly complex vector field over V M , and φ(r) ∈ C∞ 0 (R ) is a test function (see remark on page 967). The expression in the right-hand side is a linear and continuous functional (Appendix D.3) which associates φ(r) with the complex number resulting from the integration over V M . The difference between (15.42) and the corresponding definition (14.120) for surface fields over a tessellation S M is only formal and lies in the dimensionality of the gradient and the integration. When we take F(r) as the nth SWG function of type (15.37), only the two domains that coincide with Wn+ and Wn− contribute to the summation in (15.42), i.e.,   ∇ · fn (r) := − dV ∇φ(r) · fn (r) − dV ∇φ(r) · fn (r) (15.43) Wn+

Wn−

which, since fn (r) is separately continuous in Wn+ and Wn− , may be cast as   ∇ · fn (r) = dV φ(r)∇ · fn (r) − dV ∇ · [φ(r)fn(r)] Wn+

 +

Wn+



dV φ(r)∇ · fn (r) − Wn−

dV ∇ · [φ(r)fn(r)] Wn−

(15.44)

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Advanced Theoretical and Numerical Electromagnetics

having used the differential identity (H.51). We may also invoke the Gauss theorem (A.53), because fn (r) is locally smooth in Wn+ and Wn− , and arrive at 



∇ · fn (r) =

dV φ(r)∇ · fn (r) − Wn+

∂Wn+

 +

ˆ · fn (r) dS φ(r)n(r)

Wn−

 = Wn+

ˆ · fn (r) dS φ(r)n(r) ∂Wn−



dV φ(r)∇ · fn (r) +



dV φ(r)∇ · fn (r) − Wn−



=



dV φ(r)∇ · fn (r) −

  dS nˆ +n · fn (r) + nˆ −n · fn (r) φ(r)

Tn

dV φ(r)∇ · fn (r)

(15.45)

Wn+ ∪Wn−

where the integration over T ln+ ⊂ ∂Wn+ and T ln− ⊂ ∂Wn− , l = 1, 2, 3, amounts to naught in light of the orthogonality properties (15.41), and the net contribution of the surface integral over the shared triangle T n vanishes by virtue of the normalization conditions (15.39) and (15.40). This result tells us that the divergence of the SWG function (15.37) exists in the sense of ordinary functions despite fn (r) being generally discontinuous across ∂Wn+ ∪ ∂Wn− . In other words, the vector field defined by (15.37) is divergence-conforming. To finalize the derivation we determine ∇ · fn (r) separately on Wn+ and Wn− by introducing local + − and V4n — the orientation of the polar systems of polar spherical coordinates (τ, α, β) centered in V4n axis is inessential. We let τ = r − r+4n and with the help of (A.31) compute ∇ · fn (r) =

An 1 d 2 3 An τ = +, 3Vn+ τ2 dτ2 Vn

r ∈ Wn+

(15.46)

− because the SWG is a radial vector. A similar procedure with spherical coordinates centered in V4n + − − yields the result for r ∈ Wn . Moreover, the divergence is evidently null for points r ∈ V M \(W n ∪W n ), + − in that fn (r) vanishes outside W n ∪ W n . In summary, we may write [2]

⎧ An ⎪ ⎪ ⎪ , ⎪ ⎪ + ⎪ V ⎪ n ⎪ ⎪ ⎪ ⎨ An ∇ · fn (r) := ⎪ ⎪ − −, ⎪ ⎪ ⎪ Vn ⎪ ⎪ ⎪ ⎪ ⎪ ⎩0,

r ∈ Wn+ n = 1, . . . , NI

r ∈ Wn− +

(15.47)

−

r ∈ V M \ (W n ∪ W n )

whereby we see that the divergence of fn (r) associated with the interior facets of V M is bounded and piecewise constant. Still, the divergence of the SWG associated with a boundary facet is quite another story. Applying definition (15.42) to (15.38) and using (15.39) and (15.41) leads us to 

 dV φ(r)∇ · fn (r) −

∇ · fn (r) := Wn+

dS Tn

φ(r)nˆ +n

An · fn (r) = + Vn



 dV φ(r) − Wn+

dS φ(r) Tn

(15.48)

The Method of Moments II

1023

where the contribution of the surface integral over T n remains. Since the ability of turning a volume integral into a surface one is the hallmark of a surface Dirac delta distribution [cf. (C.32)] we conclude that the divergence of the half SWG function exists as a distribution which comprises a surface delta on T n . With the shorthand notation of Appendix C we may write the result succinctly as ∇ · fn (r) =

 An  1 − Un+ (r) − δT n (r − rT n ), Vn+

n = NI + 1, . . . , NI + NB

(15.49)

where

⎧ + ⎪ ⎪ ⎨0, r ∈ Wn + Un (r) := ⎪ ⎪ ⎩1, r ∈ V M \ W + n

(15.50)

is a three-dimensional unit step function defined over the volumetric mesh, rT n means points on T n , and δT n is a surface delta having support on the patch T n . In truth, by including (15.38) in the set of basis functions we correctly model the onset of polarization charges (see Sections 3.7 and 10.5) on the boundary of a dielectric body illuminated by an electromagnetic field [cf. (13.224)]. By duality (15.38) allows modelling the onset of magnetization charges on the boundary of a magnetic body exposed to an electromagnetic field. The curl of an SWG function — whether it is defined by (15.37) or (15.38) — is not an ordinary function. To elaborate, we specialize the definition of weak curl (C.45) to vector fields defined for r ∈ VM M  := ∇ × F(r) − dV ∇φ(r) × F(r) (15.51) p=1 V

p

3 where φ(r) ∈ C∞ 0 (R ) is a test function. By letting F(r) be an SWG function associated with an inner facet, we find   ∇ × fn (r) := − dV ∇φ(r) × fn (r) − dV ∇φ(r) × fn (r)



Wn+

=



dV φ(r)∇ × fn (r) − Wn+

dV ∇ × [φ(r)fn(r)] Wn+

 +

Wn−



dV φ(r)∇ × fn (r) − Wn−

dV ∇ × [φ(r)fn(r)]

(15.52)

Wn−

having made use of (H.50). The curl of fn (r) for r ∈ Wn+ can be carried out in a local system of polar + spherical coordinates (τ, α, β) with origin in the vertex V4n and polar angle α, namely,   An An 1 αˆ ∂τ ˆ ∂τ −β ∇×τ = (15.53) ∇ × fn (r) = = 0, r ∈ Wn+ + 3Vn 3Vn+ τ sin α ∂β ∂α and the same conclusion holds for the curl of fn (r) inside r ∈ Wn− . Furthermore, fn (r) is smooth separately inside Wn+ and Wn− , and hence we may apply the integral theorem (H.91) to obtain ∇ × fn (r) := −

4  i=1

T in+

dS

nˆ +in

× fn (r)φ(r) −

4  i=1

T in−

dS nˆ −in × fn (r)φ(r),

n = 1, . . . , NI

(15.54)

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Advanced Theoretical and Numerical Electromagnetics

whereby we see that ∇ × fn (r) must comprise surface Dirac delta distributions with support on the facets of Wn+ and Wn− . In the notation of distributions we may write ∇ × fn (r) = −

4

nˆ +in × fn (r) δT in+ (r − rT in+ )

i=1

−

4

nˆ −in × fn (r) δT in− (r − rT in− ),

n = 1, . . . , NI

(15.55)

i=1

where rT in± specifies points on T in± , and δT in± are surface deltas. The formulas for the other definition (15.38) follow from the previous ones by considering only the sum over the facets of Wn+ . In any case, the SWG functions are not curl-conforming. The gradient of an SWG function — irrespective of the definition thereof — is a distributional dyadic field. To be specific, we define the weak gradient of a vector field over the mesh V M as ∇F(r) := −

M  p=1 V

dV [∇φ(r)]F(r)

(15.56)

p

3 where φ(r) ∈ C∞ 0 (R ) is a test function. We take F(r) as an SWG function associated with an inner facet and observe   ∇fn (r) := − dV [∇φ(r)]fn(r) − dV [∇φ(r)]fn(r)

 =

Wn+



dV φ(r)∇fn (r) − Wn+

dV ∇[φ(r)fn (r)] Wn+

 +

Wn−



dV φ(r)∇fn (r) − Wn−

dV ∇[φ(r)fn(r)]

(15.57)

Wn−

having applied (H.52). The gradient of fn (r) for r ∈ Wn+ and r ∈ Wn− can be computed in Cartesian coordinates, viz.,  An An  An ∇(r − r+4n ) = I, ∇(xˆx) + ∇(yˆy) + ∇(zˆz) = 3Vn+ 3Vn+ 3Vn+ An ∇fn (r) = − − I, 3Vn

∇fn (r) =

r ∈ Wn+

(15.58)

r ∈ Wn−

(15.59)

where the result follows because the coordinates are independent of each other. Moreover, since fn (r) is separately smooth within Wn+ and Wn− we may invoke the integral theorem (H.104) [1, Formula (A4.71)] to get ∇fn (r) :=

An 3Vn+

 dV φ(r)I −

4  i=1

Wn+

An − 3Vn−



dS φ(r)nˆ +in (r)fn (r)

T in+

dV φ(r)I − Wn−

4  i=1

T in−

dS φ(r)nˆ −in (r)fn (r),

n = 1, . . . , NI

(15.60)

The Method of Moments II

1025

whence, owing to the presence of non-vanishing surface integrals, we conclude that ∇fn (r) comprises surface delta distributions localized on ∂Wn+ ∪ ∂Wn− . In the shorthand notation of distributions, we have  An  nˆ +in (r)fn (r) δT in+ (r − rT in+ ) 1 − Un+ (r) I − + 3Vn i=1 4

∇fn (r) =

 An  − nˆ −in (r)fn (r) δT in− (r − rT in− ), (r) I − 1 − U n 3Vn− i=1 4

−

n = 1, . . . , NI

(15.61)

where rT in± specifies points on T in± , and δT in± are surface deltas. The unit step function Un+ (r) is the same as in (15.50), and Un− (r) simply follows from (15.50) by replacing the superscript ‘+’ with ‘−’. The formula for the SWG function given by (15.38) reads An   + nˆ +in (r)fn (r) δT in+ (r − rT in+ ), (r) I − 1 − U n 3Vn+ i=1 4

∇fn (r) =

n = NI + 1, . . . , NI + NB

(15.62)

and is obtained from (15.61) by considering only the terms with the superscript ‘+’. Lastly, we observe that the scalar field (15.47) satisfies the property  dV ∇ · fn (r) = An − An = 0,

n = 1, . . . , NI

(15.63)

Wn+ ∪Wn−

similarly to the divergence of the RWG functions. Besides, if we integrate (15.49) in the sense of distributions we have    An dV ∇ · fn (r) = dV + − dS = An − An = 0, n = NI + 1, . . . , NI + NB (15.64) Vn Wn+

R3

Tn

that is, the same result regardless of fn (r) being associated with an interior or a boundary triangle of a tetrahedral mesh V M . When we employ NI + NB SWG functions to expand the displacement vector D(r) in a dielectric body modelled with the mesh V M , say, D(r) =

N I +NB

r ∈ VM

Dn fn (r),

(15.65)

n=1

the distribution ∇ · D(r) represents the electric charge density ρ(r) in any point in space. However, since there are no true charges in the body, the integral of ∇ · D(r) over V M must vanish. Indeed, we have ⎡ ⎤   ⎥⎥ NI  N I +NB ⎢ ⎢⎢⎢  ⎢⎢⎢ dV D ∇ · f (r) − dS D ⎥⎥⎥⎥⎥ = 0 dV ∇ · D(r) = dV Dn ∇ · fn (r) + (15.66) n n n ⎢⎢⎣ ⎥⎥⎦ n=1 n=N +1 VM

Δn

I

Wn+

Tn

by virtue of (15.63) and (15.64). Dual considerations apply to the expansion of B(r) within a magnetic body.

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Advanced Theoretical and Numerical Electromagnetics

15.3 Volume coordinates Solving volume integral equations with the Method of Moments and SWG functions calls for the calculation of domain integrals over tetrahedra. To this purpose it is convenient to define a set of normalized variables to identify points inside a tetrahedron. Since the following considerations apply to a generic tetrahedron of the mesh V M , we omit the index n as well as the superscripts + and − and work with the shifted position vector r − r4 . Since a point in a spatial region is specified by three numbers, we introduce a linear mapping Ψ(ξ1 , ξ2 , ξ3 ) ∈ R3 r = Ψ(ξ1 , ξ2 , ξ3 ) = ξ1 (r1 − r4 ) + ξ2 (r2 − r4 ) + ξ3 (r3 − r4 ) + r4

(15.67)

which associates the local coordinates (ξ1 , ξ2 , ξ3 ) with the point r. In order for r to belong to the tetrahedron V ⊂ R3 we must limit the range of variability of ξ1 ,ξ2 and ξ3 . We observe that • • • • •

ξ1 = 1, ξ2 = 0, ξ3 ξ1 = 0, ξ2 = 1, ξ3 ξ1 = 0, ξ2 = 0, ξ3 ξ1 = 0, ξ2 = 0, ξ3 the set of points

= 0 is mapped onto r1 = 0 is mapped onto r2 = 1 is mapped onto r3 = 0 is mapped onto r4

•

(15.68) τ4 := {(ξ1 , ξ2 , ξ3 ) ∈ R3 : ξ1 + ξ2 + ξ3 = 1, ξ1 ∈ [0, 1], ξ2 ∈ [0, 1], ξ3 ∈ [0, 1]} √ represents an equilateral triangle with edges of length 2 and is mapped onto the triangle T 4 which is opposite V4 ; the right triangles

(the vertex V1 ); (the vertex V2 ); (the vertex V3 ); (the vertex V4 );

τ3 := {(ξ1 , ξ2 , ξ3 ) ∈ R3 : ξ1 + ξ2  1, ξ1  0, ξ2  0, ξ3 = 0}

(15.69)

τ2 := {(ξ1 , ξ2 , ξ3 ) ∈ R : ξ3 + ξ1  1, ξ3  0, ξ1  0, ξ2 = 0}

(15.70)

τ1 := {(ξ1 , ξ2 , ξ3 ) ∈ R : ξ2 + ξ3  1, ξ2  0, ξ3  0, ξ1 = 0}

(15.71)

3 3

are mapped onto the facets T 3 , T 2 and T 1 , respectively. All in all, as is pictorially shown in Figure 15.3, the map Ψ(ξ1 , ξ2 , ξ3 ) transforms the unitary tetrahedron S 3 := {(ξ1 , ξ2 , ξ3 ) ∈ R3 : ξ1 ∈ [0, 1], ξ2 ∈ [0, 1], ξ3 ∈ [0, 1], ξ1 + ξ2 + ξ3  1}

(15.72)

into the tetrahedron W specified by the vertices r1 , r2 , r3 and r4 . The three-dimensional domain S 3 is called a simplex. The geometrical interpretation of the variables may be regarded as the three-dimensional extension of the meaning of the area coordinates (14.141). To mirror the discussion of Section 14.8 we write (15.67) as r = ξ1 r1 + ξ2 r2 + ξ3 r3 + (1 − ξ1 − ξ2 − ξ3 )r4 = ξ1 r1 + ξ2 r2 + ξ3 r3 + ξ4 r4

(15.73)

where the constraint ξ1 + ξ2 + ξ3 + ξ4 = 1

(15.74)

is the equation of a hyperplane in a four-dimensional space. With reference to Figure 15.4 we observe that the point P ∈ W determined by r divides W into four the tetrahedra, i.e.,

The Method of Moments II

1027

Figure 15.3 The mapping of the three-dimensional simplex S 3 onto the tetrahedron W. • • • •

W1 , with vertices P, V2 , V3 and V4 , W2 , with vertices P, V1 , V3 and V4 , W3 , with vertices P, V1 , V2 and V4 , W4 , with vertices P, V1 , V2 and V3 ,

and we denote the respective volumes with VW1 , VW2 , VW3 and VW4 . The total volume V = VW1 + VW2 +VW3 +VW4 of the tetrahedron can be computed by taking any one of the four facets as reference and by considering a triple scalar product formed with the vectors which identify the edges of W, e.g., (r3 − r4 ) × (r2 − r4 ) · (r1 − r4 ) = 6V

(15.75)

(r1 − r4 ) × (r3 − r4 ) · (r2 − r4 ) = 6V (r3 − r4 ) × (r2 − r4 ) · (r1 − r4 ) = 6V

(15.76) (15.77)

(r2 − r1 ) × (r3 − r1 ) · (r4 − r1 ) = 6V

(15.78)

though other combinations are possible. In specific, the result of the triple scalar product is not altered for any even permutation of the vectors [see (H.13)] or if we ‘swap the dot with the cross’. Then, we derive the inverse mapping (ξ1 , ξ2 , ξ3 ) = Ψ−1 (r) by manipulating (15.67) (r3 − r4 ) × (r2 − r4 ) · (r − r4 ) = ξ1 (r3 − r4 ) × (r2 − r4 ) · (r1 − r4 ) = 6Vξ1

(15.79)

(r1 − r4 ) × (r3 − r4 ) · (r − r4 ) = ξ2 (r1 − r4 ) × (r3 − r4 ) · (r2 − r4 ) = 6Vξ2 (r3 − r4 ) × (r2 − r4 ) · (r − r4 ) = ξ3 (r3 − r4 ) × (r2 − r4 ) · (r1 − r4 ) = 6Vξ3

(15.80) (15.81)

whence 1 VW1 (r3 − r4 ) × (r2 − r4 ) · (r − r4 ) = 6V V 1 VW2 ξ2 = (r1 − r4 ) × (r3 − r4 ) · (r − r4 ) = 6V V 1 VW3 ξ3 = (r3 − r4 ) × (r2 − r4 ) · (r − r4 ) = . 6V V The fourth coordinate follows similarly by casting (15.67) as

ξ1 =

r − r1 = ξ2 (r2 − r1 ) + ξ3 (r3 − r1 ) + ξ4 (r4 − r1 )

(15.82) (15.83) (15.84)

(15.85)

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Advanced Theoretical and Numerical Electromagnetics

Figure 15.4 For the definition of the volume coordinates ξ1 , ξ2 , ξ3 and ξ4 in a tetrahedron. with the aid of (15.74) for the elimination of ξ1 . Then, we have ξ4 =

1 VW4 (r2 − r1 ) × (r3 − r1 ) · (r − r1 ) = 6V V

(15.86)

which is consistent with condition (15.74). In conclusion, ξ1 , ξ2 , ξ3 and ξ4 represent the volumes of the four tetrahedra W1 , W2 , W3 and W4 normalized to the volume of W. Accordingly, ξ1 , ξ2 , ξ3 and ξ4 are called volume coordinates or also simplex coordinates [3]. When the ith coordinate ξi , i = 1, . . . , 4, vanishes, the point P belongs to the facet T i , and Wi degenerates into a triangle. When two volume coordinates are null, then P falls on one of the six edges of W, and two tetrahedra are degenerate. Finally, if only one coordinate ξi does not vanish, then P coincides with the vertex Vi , and the only non-degenerate tetrahedron Wi coincides with W. The rationale behind the introduction of the volume coordinates is that they facilitate the analytical and numerical evaluation of domain integrals over tetrahedra. To carry out the change of dummy variable we need the 3-by-3 Jacobi matrix [4, p. 234] of the mapping (15.67), viz., ⎞ ⎛  ⎜⎜ x1 − x4 x2 − x4 x3 − x4 ⎟⎟  ⎟⎟ ⎜⎜⎜ ∂Ψ ∂Ψ ∂Ψ [JΨ ] = = ⎜⎜⎜⎜y1 − y4 y2 − y4 y3 − y4 ⎟⎟⎟⎟⎟ , , (15.87) ∂ξ1 ∂ξ2 ∂ξ3 ⎠ ⎝ z1 − z4 z2 − z4 z3 − z4 whereby we get | det[JΨ ]| = det[JΨ ]T =

∂Ψ ∂Ψ ∂Ψ × · = 6V ∂ξ1 ∂ξ2 ∂ξ3

(15.88)

because the elements of the columns of [JΨ ] are the Cartesian components of the vectors ri − r4 , i = 1, 2, 3, which identify three edges of W, and the determinant of [JΨ ] is identical with the triple scalar product of the three vectors ri − r4 . With these positions an integral over W is turned into one over the simplex S 3 [4, p. 252]    dV F(r) = dξ1 dξ2 dξ3 F(ξ1 , ξ2 , ξ3 )| det[JΨ ]| = 6V dξ1 dξ2 dξ3 F(ξ1 , ξ2 , ξ3 ) (15.89) W

S3

S3

The Method of Moments II

1029

Table 15.1 Cubature formulas over a tetrahedron Order NO

Weights wi

Nodes (ξ1 , ξ2 , ξ3 )

4

1/240 1/240 1/240 1/240

(0, 0, 0) (1, 0, 0) (0, 1, 0) (0, 0, 1)

4

3/80 3/80 3/80 3/80

(1/3, 0, 1/3) (1/3, 1/3, 0) (0, 1/3, 1/3) (1/3, 1/3, 1/3)

on account of (15.88). When F(r) involves SWG functions which are non-null over W, the factor 6V combines with the normalization constants included in the definitions (15.37) and (15.38). Cubature formulas are typically devised over the simplex S 3 , and the calculation of the integral is approximated as  NC dV F(r) ≈ 6V wi F(ξ1i , ξ2i , ξ3i ) (15.90) i=1

W

where • • •

NC is the number of function samples and cubature points; wi , i = 1, . . . , NC , constitute the weights of the formula; the triples (ξ1i , ξ2i , ξ3i ) denote the independent volume coordinates of the sampling points or nodes of the formula.

A given cubature formula may be exact for vector functions whose components are polynomials of ξ1 , ξ2 and ξ3 of order up to NO , and the latter is the degree of the cubature formula. Two examples are provided in Table 15.1 [5, Formula 25.4.70, p. 895]. Example 15.1 (Projection integrals involving SWG functions) The discretization of volume integral equations for penetrable and anisotropic bodies with SWG functions (15.37) and (15.38) leads to the evaluation of, among other quantities, projection integrals of the type  A4 Al dV (r − r4 ) · η · (r − rl ), l = 1, . . . , 4 (15.91) Il := 9V 2 W

where the coefficients Al are the areas of the facets of the tetrahedron W, and η is a dyadic field which is constant within W. The entries of the matrix [G] in (15.11) provide a practical example of (15.91). This is where the volume coordinates come in handy. We observe that the linear mapping (15.67) may be written alternatively in matrix form as ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎜⎜⎜ x − x4 ⎟⎟⎟ ⎜⎜⎜ x1 − x4 x2 − x4 x3 − x4 ⎟⎟⎟ ⎜⎜⎜ξ1 ⎟⎟⎟ ⎟⎟⎟ ⎜⎜⎜ ⎟⎜ ⎟ ⎜⎜⎜ (15.92) ⎜⎜⎜y − y4 ⎟⎟⎟ = ⎜⎜⎜y1 − y4 y2 − y4 y3 − y4 ⎟⎟⎟⎟⎟ ⎜⎜⎜⎜⎜ξ2 ⎟⎟⎟⎟⎟ ⎠ ⎝ ⎠⎝ ⎠ ⎝ z − z4 z1 − z4 z2 − z4 z3 − z4 ξ3

Advanced Theoretical and Numerical Electromagnetics

1030

where we recognize the matrix of the transformation as the Jacobian (15.87). It is also convenient to write the dyadic η in the form of a 3-by-3 matrix [η], whose elements are the nine Cartesian components of η (see Appendix E). Thanks to these preliminaries we can change the integration variables to obtain  2A24 dξ1 dξ2 dξ3 (r − r4 ) · η · (r − r4 ) I4 = 3V S3 ⎛ ⎞  ⎜⎜⎜ξ1 ⎟⎟⎟   2A24 ⎜ ⎟ = dξ1 dξ2 dξ3 ξ1 ξ2 ξ3 [JΨ ]T [η][JΨ ] ⎜⎜⎜⎜⎜ξ2 ⎟⎟⎟⎟⎟ 3V ⎝ ⎠ S3 ξ3  3 3 2A24 η˜ is dξ1 dξ2 dξ3 ξi ξ s (15.93) = 3V i=1 s=1 S3 and for l  4  Al A4 Al Il = I4 − dV (r − r4 ) · η · (rl − r4 ) A4 9V 2 W

2A4 Al Al I4 − = A4 3V





dξ1 dξ2 dξ3 ξ1

⎛ ⎞ ⎜⎜⎜δ1l ⎟⎟⎟ ⎜ ⎟ T ξ3 [JΨ ] [η][JΨ ] ⎜⎜⎜⎜⎜δ2l ⎟⎟⎟⎟⎟ ⎝ ⎠ δ3l 

ξ2

S3

 3 2A4 Al Al I4 − η˜ il dξ1 dξ2 dξ3 ξi = A4 3V i=1 S3

(15.94)

where the coefficients η˜ is are the nine entries of the matrix [η] ˜ := [JΨ ]T [η][JΨ ], and the symbol ⎧ ⎪ ⎨1, i = l δil := ⎪ (15.95) ⎩0, i  l is the Kronecker delta. The expression of rl − r4 follows immediately from (15.67) or (15.92). For reasons of symmetry we just have to carry out three integrals over the simplex, namely, for i = 1, s = 2 1−ξ3 1−ξ 1  1 −ξ3  dξ1 dξ2 dξ3 ξ1 ξ2 = dξ3 dξ1 ξ1 dξ2 ξ2 S3

0

0

1

1−ξ3 

=

dξ3 0

0

dξ1

1 ξ1 (1 − ξ1 − ξ3 )2 = 2 120

(15.96)

0

for i = s = 3  1 1 1 1 1 dξ1 dξ2 dξ3 ξ32 = dξ3 (1 − ξ3 )2 ξ32 = dξ3 (ξ32 − ξ33 + ξ34 ) = 2 2 60 S3 0

(15.97)

0

and lastly  dξ1 dξ2 dξ3 ξ3 = S3

1 2

1 dξ3 ξ3 (1 − ξ3 )2 = 0

for the calculation of Il , l  4.

1 24

(15.98)

The Method of Moments II

1031

Now we can obtain the formula for I4 I4 =

3 3 3 3 2A24 1 A2 wis η˜ is = 4 wis η˜ is 3V 60 i=1 l=1 90V i=1 s=1

(15.99)

with the weights ⎧ ⎪ i=s ⎨1, := wis ⎪ ⎩1/2, i  s

(15.100)

and the one relevant to Il , l  4 Il =

3 3 3 3 3 3 Al A24 2Al A4 1 Al A4 Al A4 wis η˜ is − η˜ il = wis η˜ is − wii η˜ il A4 90V i=1 s=1 3V 24 i=1 90V i=1 s=1 36V i=1

(15.101)

though a single compact formula can be written as Il =

3 3 3 Al A4 Al A4 wis η˜ is − wii η˜ il (1 − δl4 ), 90V i=1 s=1 36V i=1

l = 1, . . . , 4

(15.102)

where δis is defined as in (15.95). Since four SWG functions of either type are associated with a tetrahedron, we need to evaluate 4 × 4 = 16 integrals such as Il . This number reduces to 4 · 5/2 = 10 if the dyadic η and hence the matrices [η] and [η˜ ] are symmetric. In a computer program the results of the application of (15.102) to the M tetrahedra of the volumetric mesh V M may be obtained at the beginning and stored. By letting η = I in (15.91) we obtain, as a special case, the integrals required for the calculation N . The result follows by inserting the entries of the entries Gmn of the Gram matrix of the set {fn (r)}n=1 T of [η˜ ] = [JΨ ] [JΨ ] in (15.102). (End of Example 15.1)

Example 15.2 (The Fourier transform of an SWG function) The volume coordinates may also be used to compute integrals of the type    A A + j k·r dV f(r)ej k·r = dS (r − r )e − dS (r − r−4 )ej k·r F(k) := 4 3V + 3V − W + ∪W − W+ W−   F+ (k)

(15.103)

F− (k)

where f(r) indicates any one of the SWG functions defined in (15.37) and associated with the pair of adjacent tetrahedra W + and W − , and k ∈ C3 is a vector parameter. The right-hand side of (15.103) constitutes the Fourier transform of the vector field f(r) defined over a tetrahedral mesh (see Section 14.1). The contribution of F− (k) is absent if the SWG function is associated with a boundary facet and defined as in (15.38). Therefore, we focus on the calculation of F+ (k) by means of volume coordinates (15.67) as suggested in (15.89). We introduce the column vector ⎛ + ⎞ ⎜⎜⎜F1 (k)⎟⎟⎟ ⎟ ⎜ [F + ] := ⎜⎜⎜⎜⎜F2+ (k)⎟⎟⎟⎟⎟ (15.104) ⎝ + ⎠ F3 (k)

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Advanced Theoretical and Numerical Electromagnetics

and from (15.103) and (15.89) we get [F + ] = 2Ae

j k·r+4

⎛ ⎞ ⎜⎜⎜ξ1 ⎟⎟⎟ ⎜ ⎟ ˜+ ˜+ ˜+ dξ1 dξ2 dξ3 ⎜⎜⎜⎜⎜ξ2 ⎟⎟⎟⎟⎟ ej k1 ξ1 ej k2 ξ2 ej k3 ξ3 ⎝ ⎠ S3 ξ3

 [JΨ ]

(15.105)

where we have invoked (15.92), [JΨ ] is the Jacobi matrix (15.87), S 3 is the three-dimensional simplex drawn in Figure 15.3, and the scalars k˜ l+ = k · (r+l − r+4 ),

l = 1, 2, 3

(15.106)

are dimensionless spectral variables. We are left with the evaluation of three domain integrals  ˜+ ˜+ ˜+ Il := dξ1 dξ2 dξ3 ξl ej k1 ξ1 ej k2 ξ2 ej k3 ξ3 , l = 1, 2, 3 (15.107) S3

the calculations of which are lengthy and tedious but do not present any special difficulties. We list the results ˜+

˜+

I1 =

Υ(k˜ 1+ ) − ej k2 Υ(k˜ 1+ − k˜ 2+ ) Υ(k˜ 1+ ) − ej k3 Υ(k˜ 1+ − k˜ 3+ ) − k˜ + k˜ + 2

3

k˜ 2+ − k˜ 3+ ˜+

I2 =

I3 =

(15.108)

˜+

Υ(k˜ 2+ ) − ej k3 Υ(k˜ 2+ − k˜ 3+ ) Υ(k˜ 2+ ) − ej k1 Υ(k˜ 2+ − k˜ 1+ ) − k˜ + k˜ + 3

1

k˜ 3+ − k˜ 1+ ˜+ ˜+ Υ(k˜ 3+ ) − ej k1 Υ(k˜ 3+ − k˜ 1+ ) Υ(k˜ 3+ ) − ej k2 Υ(k˜ 3+ − k˜ 2+ ) − k˜ + k˜ + 1

2

k˜ 1+ − k˜ 2+

where the auxiliary function Υ(α), α ∈ C, and the derivative thereof are given by ⎧ jα e (1 − j α) − 1 ⎪ ⎪ ⎪ , α0 ⎪ ⎪ ⎨ α2 Υ(α) := ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎩ , α=0 2 ⎧ jα 2 e (α + 2 j α − 2) + 2 ⎪ ⎪ ⎪ , α0 ⎪ ⎪ dΥ ⎨ α3 := ⎪ ⎪ ⎪ dα j ⎪ ⎪ ⎩ , α=0 3 ⎧ jα 3 e (j α − 3α2 − 6 j α + 6) − 6 ⎪ ⎪ ⎪ , α0 ⎪ ⎪ d2 Υ ⎨ α4 := ⎪ ⎪ ⎪ dα2 1 ⎪ ⎪ ⎩− , α = 0. 4 Since Il is the three-dimensional Fourier transform of ξl − ξl U(ξ1 , ξ2 , ξ3 ), where ⎧ ⎪ ⎪ ⎨0, (ξ1 , ξ2 , ξ3 ) ∈ S 3 U(ξ1 , ξ2 , ξ3 ) := ⎪ ⎪ ⎩1, (ξ1 , ξ2 , ξ3 ) ∈ R3 \ S 3

(15.109)

(15.110)

(15.111)

(15.112)

(15.113)

(15.114)

The Method of Moments II

1033

is a three-dimensional step function, the entries of [F + ] must be analytic functions for (k˜ 1+ , k˜ 2+ , k˜ 3+ ) ∈ C3 (see Appendix B). In other words, the singularities of Il must be only apparent. For instance, the first term in the numerator of I3 looks singular for k˜ 1+ = 0, but in actuality by taking the limit we get ˜+

" ˜+ ! Υ(k˜ 3+ ) − ej k1 Υ(k˜ 3+ − k˜ 1+ ) [ 00 ] = lim − j Υ(k˜ 3+ − k˜ 1+ ) + Υ (k˜ 3+ − k˜ 1+ ) ej k1 + k˜ 1+ →0 k˜ 1+ →0 k˜ lim

1

= − j Υ(k˜ 3+ ) +

dΥ dk˜ 3+

(15.115)

where Υ (•) signifies differentiation with respect to the argument. The result is finite for all values of k˜ 3+ on account of (15.111) and (15.112), and the other term behaves similarly. Furthermore, as k˜ 2+ approaches k˜ 1+ we get [ 00 ] lim I3 = lim

k˜ 2+ →k˜ 1+

k˜ 2+ →k˜ 1+

=

e

j k˜ 1+

˜+

d Υ(k˜ 3+ ) − ej k2 Υ(k˜ 3+ − k˜ 2+ ) dk˜ + k˜ + 2

2

[(1 − j k˜ 1+ )Υ(k˜ 3+ − k˜ 1+ ) + k˜ 1+ Υ (k˜ 3+ − k˜ 1+ )] − Υ(k˜ 3+ ) (k˜ + )2

(15.116)

1

which is evidently bounded for k˜ 1+  0. Next, we obtain ˜+ 2 ej k1 [(1 − j k˜ 1+ )Υ(k˜ 3+ − k˜ 1+ ) + k˜ 1+ Υ (k˜ 3+ − k˜ 1+ )] − Υ(k˜ 3+ ) 1 ˜ + ) + j dΥ − 1 d Υ (15.117) lim = Υ( k 3 2 k˜ 1+ →0 (k˜ 1+ )2 dk˜ 3+ 2 d(k˜ 3+ )2

which is finite by virtue of (15.111)-(15.113). Last but not least, the value of I3 for k˜ l+ = 0, l = 1, 2, 3, follows from (15.117) by letting k˜ 3+ → 0, viz., ⎡ ⎤ ⎢⎢⎢ 1 dΥ 1 d2 Υ ⎥⎥⎥ 1 + ˜ ⎢ ⎥⎦ = lim I3 = lim ⎣ Υ(k3 ) + j + − (15.118) + 2 ˜ ˜ ˜k+ →0 ˜k+ →0 2 2 24 d k3 d(k3 ) l 3 though the result can also be obtained directly with the aid of (15.98). Perfectly similar arguments apply to I1 and I2 . Eventually, [F + ] may be concisely written as ⎛ ⎞ ⎜⎜⎜I1 ⎟⎟⎟ ⎜ ⎟ + j k·r+4 [F ] = 2Ae [JΨ ] ⎜⎜⎜⎜⎜I2 ⎟⎟⎟⎟⎟ (15.119) ⎝ ⎠ I3 whereas [F − ] — which is defined analogously to [F + ] — follows from (15.119) by systematically changing the superscript ‘+’ with ‘−’ throughout, and this completes the calculation of F(k). (End of Example 15.2)

15.4 Singular integrals over tetrahedra We suppose that the spatial region occupied by the penetrable body of concern has been modelled by means of a tetrahedral mesh V M comprised of M tetrahedra, and that a set of N = NI + NB SWG

Advanced Theoretical and Numerical Electromagnetics

1034

functions of types (15.37) and (15.38) has been associated with the facets of the mesh. The typical MoM integrals in (15.26), (15.28) and (15.29) can be further broken up into linear combinations of nested integrals over tetrahedra, namely,   L1 := dV fm (r) · dV  G(r, r ) κe (rn ) · fn (r ) (15.120) Wm

Wn





  dV  G(r, r )∇ · κe (rn ) · fn (r )

dV ∇ · fm (r)

L2 := W

W

m =

n

dV  G(r, r ) κe (rn ) : ∇ fn (r )

dV ∇ · fm (r) Wm



Wn



ˆ · fm (r) dS n(r)

L3 := ∂Wm



dV  G(r, r ) κe (rn ) : ∇ fn (r )

(15.122)

ˆ  ) · κe (rn ) · fn (r ) dS  G(r, r ) n(r

(15.123)

ˆ · fm (r) dS n(r) ∂Wm





Wn

dV ∇ · fm (r)

L4 := Wm



∂Wn



ˆ · fm (r) dS n(r)

L5 :=

  dV  G(r, r )∇ · κe (rn ) · fn (r )

Wn



=

(15.121)

∂Wm

ˆ  ) · κe (rn ) · fn (r ) dS  G(r, r ) n(r

(15.124)

∂Wn

where we have assumed that κe (r ) is piecewise-constant and used (H.73) in (15.121) and (15.122). If the medium is not homogeneous but the contrast factor does not vary much over Wn , approximating κe (r ) with the value taken on in rn , the centroid of the tetrahedron, may be justified and, more importantly, facilitate the evaluation of the integrals. For the sake of completeness we mention that in the formulation of some complex electromagnetic problems one may want to combine surface and volume integral equations that arise from the simultaneous application of the surface and the volume equivalence principles (e.g., [6–8]). If both RWG and SWG functions are then introduced on triangular tessellations and volumetric meshes, respectively, it is necessary to deal with integrals of the type   dS gm (r) · dV  ∇G(r, r ) × κe (rn ) · fn (r ) (15.125) L6 := Tm

Wn

where gm (r) denotes an RWG function (14.116) partly associated with the triangle T m . The numerical calculation of (15.120)-(15.125) may be effected by means of cubature and quadrature formulas (Tables 15.1 and 14.1) expressed in terms of simplex coordinates (15.67) and (14.138). However, this strategy cannot be blindly applied to those cases where the supports of basis and test functions are at least partially overlapped because, as already noted in Section 14.9 for triangular tessellations, a computer program will run into trouble unless special care is exercised to handle the singularity of the Green function (8.356). Thus, to make sure that the integrands exhibit a certain degree of smoothness we subject G(r, r ) to the regularization technique described in Section 14.9 for surface integrals over triangles, e.g.,     κe (rn ) · fn (r )  ˜    := (15.126) L1 dV fm (r) · dV G(r, r ) κe (rn ) · fn (r ) + dV fm (r) · dV  4π|r − r | Wm

Wn

Wm

Wn

The Method of Moments II

1035

˜ r ) is defined in (14.187). since the Green function is the same, and the regularized part G(r, The singularity extraction (14.187) and the very form of the SWG functions lead us to the calculation of few basic types of integrals on a tetrahedron W, namely,  1 (15.127) I1 := dV  R W     r −r ˆ := I2 = − dV  R dV (15.128) R W

 I3 := W

 I4 :=

dV  ∇

1 =− R



W

dV 

ˆ R R2

(15.129)

W

1 dV  ∇ × κ · (r − r) R

(15.130)

W

where we have dispensed with inconsequential multiplicative factors, and κ denotes a constant dyadic. Integrals of the type I1 and I2 originate from L1 , L2 and L3 by writing     1 1   r −r  + + (15.131) dV  κ(rn ) · (r − r+ ) = κ(r ) · dV κ (r ) · (r − r ) dV  e n n 4n 4n R R R W Wn Wn  n  # $ 1 1  (15.132) dV  κe (rn ) : ∇ (r − r+ dV  4n ) = Tr κe (rn ) R R  Wn

=I

Wn

#  $  and noticing that κe (rn ), κe (rn )·(r−r+ 4n) and Tr κe (rn ) are constant with respect to the source point r . The innermost surface integrals appearing in L4 and L5 give rise to singular integrals over triangles of the form (14.191) and (14.192), and these were investigated in Sections 14.9.1 and 14.9.2. Lastly, I3 and I4 arise from the innermost integrals in (15.125) thanks to manipulations analogous to those carried out in (15.131). The strategy followed to evaluate (15.127)-(15.130) is a two-step procedure. First we try and write the integrand as the divergence or the gradient of a suitable vector or scalar field with support on W. Secondly, with the help of the Gauss theorem (A.53), the gradient theorem (H.89) or the curl theorem (H.91) we try and transform the volume integral into a combination of surface integrals over the facets of W. Since the facets are triangles, we shall see that most of the remaining integrals can be computed by building on the results of Section 14.9.

15.4.1 Integrals involving 1/R In order to lay the groundwork for the calculation of (15.127) we provisionally introduce a local system of polar spherical coordinates (R, α , β ) with the origin in the source point r and the polar axis aligned with the vector r . Indeed, we shall employ this unusual arrangement for the sole ˆ whose divergence equals 1/R. Based on purpose of seeking a radial vector field F(R) = F(R)R (A.31) we have " 1 d ! 2 1 1 = 2 R>0 (15.133) =⇒ F(R) = , R F(R) R R dR 2 whence    ˆ r −r 1 1 R  = =∇· =∇ · r ∈ R3 \ {r } (15.134) , r ∈ W, R |r − r| 2 2R

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Advanced Theoretical and Numerical Electromagnetics

where the prime means that the derivatives are carried out with respect to the source point r . The validity of (15.134) can be checked a posteriori by invoking (H.51) and computing the remaining divergences in Cartesian coordinates. More generally, one has ⎧ 1   ⎪ ⎪ ˆ , m ∈ Z \ {−3} ⎪ ∇ · Rm+1 R ⎪ ⎪ ⎪ ⎨ m + 3  (15.135) Rm = ⎪ ⎪ ˆ ⎪ R ⎪ ⎪ ⎪ m = −3 ⎩∇ · R2 log R , where m can even be a complex number. To proceed we distinguish five cases which we list in order of increasing complexity: (1) (2) (3) (4) (5)

the observation point r is located outside the tetrahedron W; the observation point r is located inside W; the observation point r lies on one of the four facets of W; the observation point r lies on one of the six edges of W; the observation point r coincides with one of the four vertices of W.

When r  W we insert representation (15.134) into (15.127) and apply the Gauss theorem (A.53) directly to cast the domain integral into a flux one over the four facets of W. Such procedure is permitted in that the radial vector field F(R) and the divergence thereof are evidently regular for r  W and r ∈ W. In symbols, we have    ˆ 1 1 R ˆ ˆ ) · R dV  = − dV  ∇ · = − dS  n(r R 2 2 W

∂W

W

=−

4  1

2

dS  nˆ l ·

l=1 ∂W

l

Rl 1 =− Rl 2 l=1 4



dS 

∂Wl

wl0 Rl

(15.136)

where • • • •

∂Wl , l = 1, . . . , 4, denotes the lth triangular facet of W; nˆ l denotes the unit normal on ∂Wl and positively oriented outwards W; Rl = r − r is the position vector of r with respect to r ∈ ∂Wl , and Rl = |Rl |; w0l = nˆ l · Rl is the component of r − r , r ∈ ∂Wl , perpendicular to ∂Wl , and |w0l | is the distance of r from the plane which contains the facet ∂Wl .

Since w0l is a constant for a given index l, with this first step we have reduced the calculation to the evaluation of four surface integrals of the type encountered in Section 14.9.1. If we indicate with Pl the projection of the observation point r onto the plane whereon ∂Wl lies, in principle we have to examine four more sub-cases, namely, (i) (ii) (iii) (iv)

the point Pl falls outside ∂Wl , the point Pl belongs to ∂Wl (see Figure 15.5 for a pictorial representation of this case), the point Pl falls on the boundary γl := ∂(∂Wl ), the point Pl coincides with one of the vertices of ∂Wl ,

but in practice we can just resort to the final compact formula (14.230) provided we update the notation so as to match the one relevant to the problem of interest here. In symbols, we have  ∂Wl

dS 

3  s+ 1 li τli0 = −|wl0 | arctan 2  + Rl τli0 + w2 l0 + |wl0 |Rli i=1

The Method of Moments II

1037

Figure 15.5 Geometrical quantities associated with the lth facet of a tetrahedron W for the evaluation of singular surface integrals: projection Pl inside ∂Wl (cf. [9]).

+ |wl0 |

3

arctan

i=1

 s− li τli0

τ2 li0

+

w2 l0

+

|wl0 |R−li

+

3

τli0 log

i=1

+ s+ li + Rli − sli + R−li

(15.137)

where the index pair (l, i) identifies geometrical quantities pertinent to the ith edge of the lth facet. In particular, with reference to Figure 15.5 • • • • • • • • •

γli , i = 1, 2, 3, is the ith edge of γl ; τl is the position vector of r with respect to rl0 which in turn specifies the projection Pl of r onto (the plane that contains) ∂Wl ; sˆli is the unit vector tangent to γli ; νˆ li := sˆli × nˆ l is the unit vector perpendicular to γli ; τli0 := νˆ li · τl is the component of τl orthogonal to γli , thus |τli0 | represents the distance of rl0 from γli ; sl := sˆli · τl denotes a local coordinate along γli ; r−li and r+li are the position vectors of the initial and final endpoints of γli ; + − ˆli and s− ˆli are the coordinates of r−li and r+li along the edge γli ; s+ li = (r − rli ) · s li = (r − rli ) · s finally, the quantities ! " 2 2 2 1/2 , R+li := (s+ li ) + wl0 + τli0

! " 2 2 2 1/2 R−li := (s− li ) + wl0 + τli0

(15.138)

are the distances of the point r from the endpoints of γli , that is, the vertices Vl,i+1 and Vl,i+2 . Now, inserting (15.137) into (15.136) yields [9]  W

⎡ 4 3  ⎢⎢⎢ s+ 1 1 li τli0 dV  = wl0 ⎢⎢⎢⎣|wl0 | arctan 2  + R 2 l=1 τli0 + w2 l0 + |wl0 |Rli i=1 −|wl0 |

3 i=1

arctan

 s− li τli0 2  − τ2 li0 + wl0 + |wl0 |Rli

−

3 i=1

τli0

⎤ +⎥ s+ ⎥⎥⎥ li + Rli ⎥ log − ⎥ (15.139) sli + R−li ⎦

Advanced Theoretical and Numerical Electromagnetics

1038

which, because of the way we have derived it, for the moment is only valid for r  W and any mutual position of the projection Pl and the facet ∂Wl . If, on the other hand, the observation point r is strictly contained in the tetrahedron, we may not ˆ is not defined uniquely in the limit as use the Gauss theorem directly because the vector field R/2  ˆ r → r, and the divergence of R/2 is singular anyway. As already noted while discussing (14.237), the field is bounded but the limit depends on the way we let the source point r approach the observation point r. Thus, in order to transform (15.127) into a flux integral we need to exclude the offending point r with a small ball B(r, a) ⊂ W and organize the integration as follows [9] 



1 1 =− dV R 2 

W







=−



ˆ −1 ˆ )·R dS n(r 2 

∂W

4  1

2

1 R

B(r,a)

W\B(r,a)

1 =− 2

dV 

ˆ + dV ∇ · R



dS

l=1 ∂W

w  l0 Rl



a



ˆ ·R ˆ + 4π dS R ∂B

dR R 0

− 2πa2 + 2πa2 = −

4  1

2

l

dS 

l=1 ∂W

wl0 Rl

(15.140)

l

where we have integrated over ∂B and B(r, a) by means of a local system of polar spherical coordinates (R, α , β ) with the origin in the observation point r. In this regard, the orientation of the polar axis is not important. The rightmost member coincides with the previous result (15.136) obtained for r  W. This is not surprising, because the volume integral of 1/R exists finite for r ∈ R3 . Evidently, from this point onward the calculations proceed exactly as before and lead again to (15.139), which we can claim holds for r ∈ R3 \ ∂W. Next, we address the occurrence of the observation point r on the pth facet of W, p = 1, . . . , 4. For the same reasons just recalled, we take care of the singularity of the integrand by excluding r with a small ball B(r, a) which the facet ∂W p divides into two hemispheres. We also define two surfaces, namely, S  := {r ∈ ∂W : |r − r|  a} and S  := ∂B(r, a) ∩ W, so that S  ∪ S  is the boundary of W \ B(r, a). As for the integration we have 



1 1 =− dV R 2 

W



ˆ + dV ∇ · R

1 2



1 R

dV 

B(r,a)∩W

W\B(r,a)

=−





ˆ − ˆ ) · R dS  n(r

S

1 2





ˆ ·R ˆ + dS  R

S 

dV 

1 R

(15.141)

B(r,a)∩W

where we have applied (A.53) with the unit normal positively oriented outwards W \ B(r, a). The integral over S  is further split into four parts, i.e., over S  ∩ ∂W p — this is just ∂W p \ B2 (r, a) — and over the remaining three facets of W. Since r ∈ ∂W p , the contribution from S  ∩ ∂W p vanishes, ˆ = 0. The integrals over S  and the hemisphere B(r, a) ∩ W are evaluated in local polar in that nˆ p · R spherical coordinates, viz., 1 − 2





ˆ ·R ˆ + dS R S 



1 1 =− dV R 2

B(r,a)∩W







a

dS + 2π S 

dR R = −πa2 + πa2 = 0 0

(15.142)

The Method of Moments II

1039

whence we see that the combined contribution of S  and B(r, a) ∩ W is null and, in fact, independent of the radius a. Thanks to these intermediate findings we obtain  4  w 1  1 =− dV dS  l0 , r ∈ ∂W p , p = 1, . . . , 4 (15.143) R 2 l=1 Rl lp ∂Wl

W

and the final formula is given by (15.139) provided l  p. In practice, we obtain the correct result by setting wp0 = 0 in (15.139). The case of r belonging to the edge shared by the facets ∂W p and ∂Wq , p, q ∈ {1, . . . , 4}, p  q, may be treated in like manner by excluding the observation point with a small ball B(r, a). We consider again the two surfaces S  and S  which are formally defined as before. However, now S  consists of the boundary of W minus two semicircles of radius a and centered in r, and S  is a meridian slice of ∂B with area Ω pq a2 , where Ω pq indicates the solid angle determined by ∂W p and ∂Wq . By proceeding as before for the integration we arrive formally at (15.141). The integral over S  is expanded into four contributions, viz., from S  ∩ ∂W p , S  ∩ ∂Wq and from the remaining facets ˆ = nˆ q · R ˆ = 0. The integrals of W. The integrals over S  ∩ ∂W p and S  ∩ ∂Wq vanish because nˆ p · R over S  and the spherical sector B(r, a) ∩ W are evaluated in local polar spherical coordinates, viz., 

1 − 2





ˆ ·R ˆ + dS R S 

1 1 =− dV R 2 

B(r,a)∩W



a



1 1 dR R = − Ω pq a2 + Ω pq a2 = 0 (15.144) 2 2

dS + Ω pq S 

0

whereby we conclude   4 w 1  1 =− dV dS  l0 , R 2 l=1 Rl

r ∈ ∂(∂W p ) ∩ ∂(∂Wq )

(15.145)

l{p,q} ∂Wl

W

and the result follows from (15.139) provided l  {p, q}. Also in this instance we arrive at the correct answer by simply setting wp0 = wq0 = 0 in the general formula (15.136), and hence we can claim that (15.139) applies as well for points on the six edges of W. Lastly, we discuss the situation in which the observation point r coincides with the vertex of W opposite the facet ∂Wq , q = 1, . . . , 4. In light of the previous results and the existence of the integral for r ∈ R3 we expect a contribution from just the triangle ∂Wq . To verify this supposition we isolate the point r with a small ball B(r, a) and define the usual surfaces S  and S  , where S  consists of ∂W minus three semicircles with radius a and center the vertex of concern, and S  is a spherical triangle determined by the intersection of ∂B and the three facets ∂Wl , l  q. We split the integration as in (15.141) and the contribution from S  into four parts, i.e., from S  ∩ ∂Wl , l  q, and from ∂Wq . ˆ = 0 for l  q. The integrals over S  and Only the latter is non-null, as anticipated, because nˆ l · R B(r, a) ∩ W are carried out in local polar spherical coordinates centered in r, i.e., 1 − 2







ˆ ·R ˆ + dS R S 

1 1 =− dV R 2 

B(r,a)∩W





a

dS + Ωq S 

1 1 dR R = − Ωq a2 + Ωq a2 = 0 2 2

(15.146)

0

where Ωq is the solid angle subtended by ∂Wq with respect to r. In conclusion, we find  W

dV 

1 1 =− R 2

 ∂Wq

dS 

wq0 Rq

,

r∈

4 % l=1 lq

∂(∂Wl ),

q = 1, . . . , 4

(15.147)

1040

Advanced Theoretical and Numerical Electromagnetics

and we get the expected result by letting wl0 = 0 for l  q in (15.136). Since (14.230) works fine for any mutual position of the observation point and a triangle T , with this analysis we have showed that (15.139) holds true for r ∈ R3 .

15.4.2 Integrals involving R/R Once again in an attempt to transform the domain integral (15.128) into one over the boundary of W we momentarily consider a local system of polar spherical coordinates (R, α , β ) with the origin in the source point r and the polar axis aligned with the vector r . With these positions and on account of (A.28) the vector field we wish to integrate becomes r − r R ˆ = −∇R = ∇ |r − r|, = − = −R R R

r ∈ W,

r ∈ R3 \ {r }

(15.148)

where as usual ∇ signifies differentiation with respect to the primed coordinates. More generally, one has ⎧ 1 ⎪ ⎪ ⎪ ∇Rm+1 , ˆ =⎨ m + 1 Rm R ⎪ ⎪ ⎪ ⎩∇ log R,

m ∈ Z \ {−1}

(15.149)

m = −1

with the exponent m possibly complex. We continue by examining the same five cases listed at the beginning of Section 15.4.1 on page 1036. The simplest situation occurs when the observation point r is located outside the tetrahedron W, in which instance the scalar field R and the components of ∇ R, when considered as functions of r , are of class C1 (W) ∩ C(W), and we may invoke the gradient theorem (H.89) directly. Specifically, we have  W

r − r = dV R 









dV ∇ R = W





ˆ )R = dS n(r ∂W

4 l=1

nˆ l



dS  Rl

(15.150)

∂Wl

where • • •

∂Wl , l = 1, . . . , 4, denotes the lth triangular facet of W; nˆ l denotes the unit normal on ∂Wl and positively oriented outwards W; Rl = |Rl | with Rl = r − r being the position vector of r with respect to r ∈ ∂Wl .

Since nˆ l is constant on ∂Wl , we have reduced the original task to evaluating the integral of the scalar field Rl over ∂Wl . However, since we did not deal with this type of integral in Section 14.9, we shall develop the calculations in details here. To lighten the notation and to obtain a general result we forego the indices for the time being and focus on the integral of R over an arbitrary triangle T with the aid of the geometrical setup of Figure 14.9. Following the strategy of Section 14.9.1 we suppose that r  T and indicate with P the projection of r onto the plane that contains T . Next, we define a local system of circular cylindrical coordinates (τ , α , ζ  ) with origin in P and the ζ-axis aligned with the unit normal nˆ  on T . The distance R can be written as in (14.195) and a surface radial vector field F(τ ) = Fτ (τ )τˆ  should exist such that  1/2 1 d R(τ ) = τ2 + w2 =   (τ Fτ ) 0 τ dτ

(15.151)

The Method of Moments II

1041

where w0 = (r − r ) · nˆ  is the component of R along the unit normal. Integrating the differential equation above yields Fτ (τ ) =

R3 3τ

F(τ ) =

=⇒

whence finally we obtain  3  R τ   R = |r − r| = ∇s · , 3τ2

r ∈ T,

R3 τ 3τ2

r0  r if w0  0

(15.152)

(15.153)

where r0 is the position vector of the projection P, and ∇s · {•} denotes the surface divergence over T and carried out with respect to the source point r . To proceed further and transform the integral over T into a line integral along the boundary ∂T we distinguish four cases, namely, (1) (2) (3) (4)

the point P falls outside T ; the point P belongs to T ; the point P lies on the boundary ∂T ; the point P coincides with one of the three vertices of T .

If r0  T , we can exploit (15.153) and apply the surface Gauss theorem (A.59) right away because the vector field R3 τ /τ2 is regular and the divergence thereof is bounded so long as τ > 0 if w0  0. Specifically, we have  3  &   3 R τ        R ˆ ν dS R = dS ∇s · ds (τ ) · τ (15.154) = 3τ2 3τ2 T

∂T

T 

with the unit normal νˆ on ∂T positively oriented outward T . To evaluate the contour integral we separate the calculation along the three edges γi , i = 1, 2, 3, of T . With reference to Figure 14.9 and the relevant geometrical quantities already defined on page 982 and ff. we have & ∂T

R3 ds νˆ  (τ ) · τ 2 = 3τ i=1 3

+

si s− i

R3 ds τi0 2 = 3τ i=1 3

+

si s− i

⎞ ⎛ w4 τi0 ⎜⎜⎜ w2 ⎟ 0 0 ⎟ ⎜⎝R + + 2 ⎟⎟⎠ ds 3 R Rτ 

(15.155)

where the last two integrands are of the same type occurring in (14.203) except for different and immaterial multiplicative constants. To handle the first integrand we carry out an integration by parts, viz., ⎞ ⎛    w2 + τ2 ⎜⎜ s2 ⎟⎟⎟ i0 ⎟ = s τi0 R − ds τi0 ⎜⎜⎝R − 0 ds τi0 R = s τi0 R − ds τi0 (15.156) ⎠ R R whence [10, Formula 230.01]   w2 + τ2 1 1   1 i0   2  = s τi0 R + τi0 (w2 ds τi0 R = s τi0 R + ds 0 0 + τi0 ) log(s + R) 2 2R 2 2

(15.157)

having made use of the previous result (14.206). By combining (15.157), (14.206) and (14.207) we obtain the indefinite integral  |w |s τ τ R3 τ  ds i02 = i0 s R + |w0 |3 arctan 0  + i0 (τ2 + 3w2 (15.158) 0 ) log(s + R) 2 Rτi0 2 i0 τ

Advanced Theoretical and Numerical Electromagnetics

1042

which allows us to finalize the calculation along the edges of T . In conclusion, we find 

dS  R =

T

3 ' |w0 |s+ 1 τi0 + + i −  3 (si Ri − s− R ) + |w | arctan i i 0 3 i=1 2 R+i τi0

−|w0 |3 arctan

+ |w0 |s− τi0 2 s+ i i + Ri 2 (τ + + 3w ) log 0 − R−i τi0 2 i0 s− i + Ri

( (15.159)

where the distances R+i and R−i are defined in (14.209). Remarkably, while the scalar field R is certainly well-defined for all points r and r, still the generating surface vector field F(τ ) in (15.153) actually diverges for τ = 0 if w0  0. As a 2 consequence, picking up a two-dimensional radial test function φ(τ ) ∈ C∞ 0 (R ) and computing the surface divergence in a distributional sense [cf. (C.44)] yields ∇s ·



  2π +∞ R3 τ ∂φ R3 R3 τ     := − dS ∇ φ · = − dα dτ  s 2 2 ∂τ 3 3τ 3τ R2

0

0

' (  +∞ 3 +∞ 2  R     3 = −2π φ(τ ) + 2π dτ φ(τ )Rτ = π|w0 | φ(0) + dS  φ(τ )R 3 0 3

(15.160)

R2

0

where we have integrated by parts with respect to the radial coordinate τ . In shorthand notation this means  3  ) * R τ 2   R = |r − r| = ∇s · − π|w0 |3 δ(2) τ (15.161) 3 3τ2 and this formula extends (15.153) for all points r ∈ T and r ∈ R3 . Thus, before we go back to (15.150) we suppose that r0 ∈ T and compute the integral of R again. This time we exclude the offending point P with a small circle B2 (r0 , a) ⊂ T , as is suggested in Figure 14.10. Then, we have  3     R τ    dS R = dS ∇s · dS  R + 3τ2 T

& = ∂T

T \B2 (r0 ,a)

R3 ds νˆ  (τ ) · τ 2 + 3τ

&

B2 (r0 ,a)

ds νˆ  (τ ) · τ

∂B2 (r0 ,a)

R3 + 3τ2



  2 1/2 dS  w2 0 +τ

(15.162)

B2 (r0 ,a)

by virtue of (A.59) with the unit normal pointing outward T . We discuss the three integrals in the rightmost-hand side one by one. As a matter of fact, the first has already been calculated and is given by (15.159). The second and the third one are carried out in local polar coordinates (τ , α ) by noticing that νˆ  (τ ) · τ = −a and τ = a for r ∈ ∂B2, thus &  3       R 2 1/2 ds νˆ (τ ) · τ 2 + dS  w2 = 0 +τ 3τ B2 (r0 ,a)

∂B2 (r0 ,a)

2π =− 0

(w2 + a2 )3/2 dα a 0 2 + 3a 

2π

2



a

dα 0

0

  2 1/2 dτ τ w2 0 +τ

The Method of Moments II

1043

  2  2 2 2  2 3/2 2 3/2 + π w2 − π|w0 |3 = − π|w0 |3 (15.163) = − π w2 0 +a 0 +a 3 3 3 3 where we have used Formula 231.01 in [10]. The result is evidently independent of the radius a. By combining (15.159) with (15.162) and (15.163) we get 

dS  R =

T

3 ' |w0 |s+ 1 τi0 + + i −  3 (si Ri − s− R ) + |w | arctan i i 0 3 i=1 2 R+i τi0

−|w0 |3

+( |w0 |s− τi0 2 s+ 2 i i + Ri 2 (τ + 3w0 ) log − arctan −  + − π|w0 |3 (15.164) Ri τi0 2 i0 si + R−i 3  residue

a formula which is strictly valid for r0 ∈ T . It is not superfluous to notice that the residue precisely accounts for the contribution of the term proportional with the two-dimensional Dirac delta distribution in (15.161). When the projection P belongs to the edge γ p of T , p = 1, 2, 3, (see Figure 14.11) we isolate r0 by means of a small circle B2 (r0 , a) which γ p divides into two halves. We define the lines γ := {r ∈ ∂T : |r −r0 |  a} and γ := ∂B2 ∩T , and organize the calculation of the integral as follows  3     R τ dS  R = dS  ∇s · dS  R + 3τ2 B2 (r0 ,a)∩T

T \B2 (r0 ,a)

T



= γ

R3 ds νˆ  (τ ) · τ 2 + 3τ



ds νˆ  (τ ) · τ

γ

R3 + 3τ2



  2 1/2 dS  w2 0 +τ

(15.165)

B2 (r0 ,a)∩T

thanks to (A.59) with the unit normal pointing outward T . The integral along γ is split into three contributions, i.e., along γ ∩ γ p and the two remaining edges of T . The contribution from γ ∩ γ p vanishes because the position vector τ = r − r0 , with r , r0 ∈ γ p , is perpendicular to the local normal νˆ p . The integrals along γi , i  p, can be written with the aid of the previous result (15.159). The integrals along γ and the semicircle B2 (r0 , a) ∩ T can be evaluated in local polar coordinates (τ , α ) with origin in the point r0 , viz.,     R3 2 1/2 ds νˆ  (τ ) · τ 2 + dS  w2 = 0 +τ 3τ γ

B2 (r0 ,a)∩T

π =−

dα a2

2 3/2 (w2 0 +a ) + 2 3a

0

π

dα

0

a

  2 1/2 dτ τ w2 0 +τ

0

 3/2 π π π  2 π 2 3/2 = − w2 + − |w0 |3 = − |w0 |3 w0 + a 2 0 +a 3 3 3 3 and putting all these intermediate findings together gives  T

dS  R =

(15.166)

3 ' |w0 |s+ 1 τi0 + + i −  3 (si Ri − s− i Ri ) + |w0 | arctan 3 i=1 2 R+i τi0 ip

−|w0 |3 arctan

+( |w0 |s− τi0 2 s+ π i i + Ri 2 (τ + + 3w ) log − |w0 |3 i0 0 −  − − Ri τi0 2 si + Ri 3  residue

(15.167)

1044

Advanced Theoretical and Numerical Electromagnetics

regardless of the value assigned to a. This time the residue amounts to half the contribution of the Dirac delta in (15.161) because conceptually the support of δ(2) (τ ) is split into two identical parts by the edge γ p and only the part inside T contributes. Finally, if the point P coincides with the vertex V p of T , p = 1, 2, 3, we exclude the offending point with a small circle B2 (r0 , a) as in Figure 14.12. We introduce the usual lines γ and γ , where now γ consists of the boundary ∂T external to B2 (r0 , a), and γ is a circular arc. The integral over T looks formally as in (15.165), so we refrain from writing it down again, though the various contributions are actually different. More specifically, we have  γ

+

R3 ds νˆ  (τ ) · τ 2 = 3τ

s p

ds τp0

s− p

R3 , 3τ2

p = 1, 2, 3

(15.168)

where the integrals along γi , i  p, vanish because the position vector τ = r − r0 with r0 ∈ V p and r ∈ γi , i  p, is perpendicular to the local normal νˆ i . The integrals along the circular arc γ and over the circular sector B2 (r0 , a) ∩ T can be evaluated in local polar coordinates (τ , α ) with origin in the point r0 , viz.,  γ

R3 ds νˆ (τ ) · τ 2 + 3τ  





α p =− 0



  2 1/2 dS  w2 = 0 +τ

B2 (r0 ,a)∩T

(w2 + a2 )3/2 dα a 0 2 + 3a  2

α p



a

dα 0

  2 1/2 dτ τ w2 0 +τ

0

3/2 α p   α p  2 αp  3 αp 2 3/2 |w0 | = − |w0 |3 =− + − w0 + a 2 w2 0 +a 3 3 3 3

(15.169)

where α p is the angle subtended by the edge γ p with respect to the vertex V p (see Figure 14.12). Therefore, in the end we find  T

⎡  |w0 |s+ 1 ⎢⎢⎢⎢ τ p0 + + p −  3 dS R = ⎢⎣ (s p R p − s− p R p ) + |w0 | arctan +  3 2 R p τ p0 

−|w0 |3 arctan

|w0 |s− p R−p τp0

+

τp0 2

(τ2 p0

+

3w2 0 ) log

⎤ + s+ ⎥⎥⎥ α p  3 p + Rp ⎥ |w | (15.170) ⎥− −⎦ s− + R 3 0 p p  residue

which applies when P ≡ V p . We already mentioned in Sections 14.9.1 and 14.9.3 that it is more convenient to have a single compact expression which works for all possible relative positions of P and the triangle T , inasmuch as this facilitates the numerical calculations in a computer program. Expressing the residues as the sum of suitable angles as we did in (14.221), (14.223), (14.225) and (14.227) does the trick. Then, we make use of formula (14.229) to simplify the difference of two arctangent functions. All in all, these manipulations applied to (15.159), (15.164), (15.167) and (15.170) yield  T

dS  R =

3 ⎡  s+ 1 ⎢⎢⎢ τi0 + + i τi0 −  3 ⎢⎣ (si Ri − s− R ) − |w | arctan i i 0 2  + 3 i=1 2 τ2 i0 + w0 + |w0 |Ri

The Method of Moments II +|w0 |3 arctan

 s− i τi0 2  − τ2 i0 + w0 + |w0 |Ri

+

1045

+⎤ τi0 2 s+ ⎥⎥⎥ i + Ri ⎥ (τi0 + 3w2 ) log 0 − ⎦ (15.171) 2 s− + R i i

which holds for any reciprocal position of the projection P and the triangle T . This formula is easily automated and, if need be, even extended by inspection to a polygon with an arbitrary number of edges. The weak singularity of the logarithm — which may diverge when τi0 = 0 = w0 — is dominated by the simultaneous vanishing of the multiplicative factor, and hence the formula is well-defined even for observation points on ∂T , vertices included. At long last by specializing (15.171) to the facets of the tetrahedron W we can finalize the calculation of the integral of interest in (15.150) [9], viz.,  W

dV 

+ 3 ' 4 r − r  1 τli0 + + − nˆ l = (sli Rli − s− li Rli ) R 3 2 l=1 i=1

− |wl0 |3 arctan

 s+ li τli0 2  + τ2 li0 + wl0 + |wl0 |Rli

+ |wl0 |3 arctan

 s− li τli0 2  − τ2 li0 + wl0 + |wl0 |Rli +(, τli0 2 s+ li + Rli 2 + (τli0 + 3wl0 ) log − 2 sli + R−li

(15.172)

which for the moment is strictly valid for r  W and any reciprocal position of the projection Pl and the facet ∂Wl . If the observation point r belongs to W, the gradient theorem (H.89) may not be invoked directly ˆ albeit bounded, is singular for r → r. We have as in (15.150) inasmuch as the vector field −R, already had the opportunity to mention that the singular character is due to the fact that the limit of (r − r)/R depends on the direction followed to let r approach r. Thus, in order to reduce (15.128) to a surface integral we exclude r by means of a small ball B(r, a) ⊂ W and observe    r − r r − r = dV  dV  ∇ R + dV  R R W

W\B(r,a)



= ∂W

ˆ  )R + dS  n(r

B(r,a)



ˆ− dS  RR

∂B



ˆ = dV  R

B(r,a)

  =0

4 l=1

nˆ l



dS  Rl

(15.173)

∂Wl

=0

where we have made use of the transformation of unit vectors (A.11) and integrated over ∂B and B(r, a) by means of the local system of polar spherical coordinates (τ , α , β ) with the origin in the observation point r. Since the rightmost member evidently has the same form as was found in (15.150), we just refer to (15.171) to arrive again at (15.172) for any position of the projection Pl and the facet ∂Wl . In this way we have extended (15.172) for points r ∈ R3 \ ∂W. Next, we consider the occurrence of r on the facet ∂W p of W, p = 1, . . . , 4, and proceed by excluding the singular point by means of a small ball B(r, a). We define the surfaces S  := {r ∈ ∂W : |r − r|  a} and S  := ∂B ∩ W. For the integration over W we write    r − r r − r = dV  dV  ∇ R + dV  R R W B(r,a)∩W W\B(r,a)    ˆ − ˆ ˆ  )R + dS  RR = dS  n(r dV  R S

S 

B(r,a)∩W

Advanced Theoretical and Numerical Electromagnetics

1046

Figure 15.6 Geometrical quantities associated with the pth facet of a tetrahedron W (not shown) for the evaluation of singular volume integrals: observation point r on ∂W p and local systems of polar and spherical coordinates.

=

4

nˆ l

l=1 lp

=



∂Wl

4

nˆ l

l=1



dS  Rl +



dS  nˆ p R p +

dS  Rl −

∂Wl

dS  nˆ p R p +

B2 (r,a)





ˆ − dS  RR

S 

∂W p \B2 (r,a)





ˆ − dS  RR

S 

ˆ dV  R

B(r,a)∩W



ˆ dV  R

(15.174)

B(r,a)∩W

where B2 (r, a) := ∂W p ∩ B(r, a), and in the last step we have exploited the linearity of the integral over ∂W p \ B2 (r, a) and the regularity of the integrand nˆ p R p . In this way the contribution of the four facets ∂Wl is readily retrieved from (15.172). For the evaluation of the remaining integrals it pays to introduce local systems of polar and spherical coordinates, viz., (ξ , β ) and (τ , α , β ), both centered in r (Figure 15.6). The polar axis is aligned with the unit normal nˆ p on ∂W p , and the unit radial  ˆ Then, we observe vectors are ξˆ and τˆ  = −R.  −

dS



nˆ p R p

=

−nˆ p

B2 (r,a)

2π



−

2π

ˆ = nˆ p dV  R

B(r,a)∩W

0



π

dβ 0

a

2 dξ ξ2 = − πa3 nˆ p 3

(15.175)

0

ˆ = −nˆ p a3 dS RR

S 



a

dβ 0





dτ τ2

dα sin α cos α = πa3 nˆ p

(15.176)

π/2

2π 0

dβ

π

π/2

π dα sin α cos α = − a3 nˆ p 3

(15.177)

ˆ = −ˆτ parallel to the having noticed that R p = ξ when r , r ∈ ∂W p and that the components of R  facet ∂W p contribute naught once integrated over β ∈ [0, 2π] [cf. (A.11)]. All in all, the combined effect of the last three contributions in (15.174) is null.

The Method of Moments II

1047

ˆ for the This somewhat involved procedure has been necessary to handle the singularity of R sole purpose of applying the gradient theorem in the first place. Still, the final result is the same as the one we obtained in (15.150), an outcome we could have predicted, since the original integrand is bounded anyway. Therefore, we use (15.171) in (15.174) to arrive at (15.172) which is thus extended for all observations points except the edges and the corners of W. When the observation point r falls on the edge shared by the facets ∂W p and ∂Wq , p, q ∈ {1, . . . , 4}, p  q, we isolate r with a ball B(r, a) and consider again the surfaces S  and S  and also the semicircles S l := ∂Wl ∩ B(r, a), l ∈ {p, q}. In this instance, S  consists of the facets of W minus S p and S q . We split the integration as in (15.174)     r − r  r −r   = dV dV ∇ R + dV  R R W W\B(r,a) B(r,a)∩W    ˆ − ˆ ˆ  )R + dS  RR = dS  n(r dV  R S

=

4

nˆ l

l=1 l{p,q}



+

=

S 

ˆ − dS  RR

nˆ l



dS  nˆ p R p +

∂W p \S p

+

ˆ − dS  RR





dS  nˆ q Rq

∂Wq \S q

ˆ dV  R 

dS  Rl − nˆ p

∂Wl

S 



B(r,a)∩W

B(r,a)∩W

l=1





dS  Rl +

∂Wl

S  4



dS  R p − nˆ q

Sp



dS  Rq

Sq

ˆ dV  R

(15.178)

B(r,a)∩W

and it remains to compute the last four integrals. The calculations for an arbitrary radius a are lengthy, tedious and not particularly instructive. Since the original integral over W must not depend on a, we may as well consider the limiting case of a vanishing radius. In fact, it is easy to ascertain that the three integrals are dominated by constants proportional to a3 , and hence they vanish in the limit as a → 0+ . This is also the result for an arbitrary value of a, and invoking (15.171) in (15.178) leads us to the conclusion that (15.172) holds even for r on the six edges of W. Finally, if r coincides with the vertex opposite ∂Wq , q = 1, . . . , 4, we exclude that point with the ball B(r, a) and define the surfaces S  and S  and the three circular sectors S l := ∂Wl ∩ B(r, a), l  q. This time S  consists of the facets of W minus S l , l  q. It takes a few steps to show that the integral over W becomes    r − r r − r = dV  dV  ∇ R + dV  R R W

W\B(r,a)

=

4 l=1

nˆ l



∂Wl

B(r,a)∩W

dS  Rl −

4 l=1 lq

nˆ l

 Sl

dS  Rl +

 S 

ˆ − dS  RR



ˆ dV  R

(15.179)

B(r,a)∩W

where the net contribution of the last three terms may be shown to vanish in particular in the limit as a → 0+ . Thus, for the same reasons mentioned above and the application of (15.171) we conclude that (15.172) is valid for r ∈ R3 .

1048

Advanced Theoretical and Numerical Electromagnetics

15.4.3 Integrals involving ∇(1/R) Integrals of type (15.129) exist and converge for points r ∈ R3 . This statement can be verified by estimating 

1 dV ∇  R 

W

 dV

  a ˆ R  1  dV 2 = dR dΩ = 4πa R2 R



W

B(r,a)

0

(15.180)

4π

where we have used (D.152), and the ball B(r, a) is chosen large enough to contain the tetrahedron W. More importantly, the integrand of I3 is already in a form which conveniently lends itself to the application of the gradient theorem (H.89). To begin with we suppose that r  W so that both 1/R and the gradient thereof are regular for r ∈ W. Invoking (H.89) directly provides     4 ˆ ) n(r 1 1 1 nˆ l =− dV  ∇ = − dV  ∇ = − dS  dS  (15.181) R R R R l l=1 W

∂W

W

∂Wl

∂Wl , nˆ l

where and Rl , l = 1, . . . , 4, are the same quantities defined after (15.136). Now, the four surface integrals in the rightmost-hand side of (15.181) are of type (14.191) which was computed in Section 14.9.1 and reviewed in Section 15.4.1. Consequently, we just need to recall the general formula (15.137) to get  W

dV  ∇

3 ' 4  s+ 1   li τli0 nˆ l = |wl0 | arctan 2  + R l=1 τli0 + w2 l0 + |wl0 |Rli i=1

− |wl0 | arctan

 s− li τli0 2  − τ2 li0 + wl0 + |wl0 |Rli

− τli0 log

+ s+ li + Rli − s− li + Rli

( (15.182)

where the geometrical quantities are shown in Figure 15.5 and described in details after (15.137). The formula just found strictly holds for r  W owing to the way we have arrived at (15.181). If, on the other hand, the observation point belongs to W we follow the usual strategy of isolating the troublesome point r with a small ball B(r, a) ⊂ W. In this way we may write    ˆ 1 1 R dV  ∇ = − dV  ∇ − dV  2 R R R W

B(r,a)

W\B(r,a)

 =− ∂W

ˆ ) n(r − dS R 



a   4 ˆ  R 1  ˆ  nˆ l dS − dR dΩ R = − dS  R R l l=1 0 4π ∂B ∂Wl   

=0

(15.183)

=0

where the integrals over ∂B and B(r, a) can be computed by means of polar spherical coordinates ˆ Since (15.183) looks formally (τ , α , β ) centered in r, in which case the radial vector is τˆ  = −R. as (15.181) then we conclude that (15.182) holds for r ∈ R3 \ ∂W. If the observation point r lies on the facet ∂W p , p = 1, . . . , 4, we introduce the ball B(r, a) which is divided into two hemispheres by ∂W p . We define the open surfaces S  := {r ∈ ∂W : |r − r|  a} and S  := ∂B(r, a) ∩ W, and evaluate the integral as follows    ˆ 1 1 R dV  ∇ = − dV  ∇ − dV  2 R R R W

W\B(r,a)

B(r,a)∩W

The Method of Moments II  =− S

=−

ˆ ) n(r − dS R

4

 dS

nˆ   l Rl

l=1 lp ∂Wl

=−





4 

S 



−

nˆ  dS l + Rl

ˆ R R2

dV 

B(r,a)∩W

dS 

∂W p \B2 (r,a)



l=1 ∂W l



ˆ R − dS R 

 dS



nˆ p Rp

nˆ p Rp

B2 (r,a)



−

dS 

S 

 − S 

1049



ˆ R − R

dV 

ˆ R R2

B(r,a)∩W

ˆ R − dS R 



dV 

ˆ R R2

(15.184)

B(r,a)∩W

where B2 (r, a) := ∂W p ∩ B(r, a). Besides, in the last step we have exploited the fact that the integral of 1/R p over B2 (r, a) exists for any value of a. To compute the last three integrals we resort to local systems of polar and spherical coordinates, viz., (ξ , β ) and (τ , α , β ), both centered in r (Figure  15.6). The polar axis is aligned with the unit normal nˆ p on ∂W p , and the unit radial vectors are ξˆ ˆ Therefore, we have and τˆ  = −R.  dS

nˆ p



Rp

B2 (r,a)



a

dβ

ˆ R = nˆ p a dS R 

− S 

−

2π 0





=

nˆ p

dξ = 2πanˆ p

(15.185)

0

2π

π



dβ

dα sin α cos α = −πanˆ p

   ˆ R dV 2 = nˆ p dτ dβ dα sin α cos α = −πanˆ p R

B(r,a)∩W

(15.186)

π/2

0 a

2π

π



0

0

(15.187)

π/2

ˆ = −ˆτ parallel to the having noticed that R p = ξ when r , r ∈ ∂W p and that the components of R  facet ∂W p contribute naught once integrated over β ∈ [0, 2π] [cf. (A.11)]. As a result, (15.184) reduces to (15.181) and hence formula (15.182) holds true for any point r save the edges and the vertices of W for the moment. Finally, the extension of (15.182) for observation points on the edges and the vertices of the tetrahedron may be effected along the same lines by showing that (15.181) invariably applies in the limit as a → 0+ for the sake of simplicity (see Section 15.4.2). Indeed, (15.182) contains a weakly singular logarithmic term which for τli0 = 0 = wl0 is driven to zero by the multiplicative factor τli0 .

15.4.4 Integrals involving ∇(1/R), a constant dyadic and R Due to the presence of the vector r − r, the integrand of (15.130) is even less singular than was the case for (15.129). The domain integral exists finite for r ∈ R3 , because  W

dV  ∇

 1 1 × κ · (r − r)  dV  ∇ κ · (r − r) R R W W   --κ-- 1 -  = 2πa2 --κ-dV  2 --κ-- R  dV  R R

1 × κ · (r − r)  R



W

dV  ∇

B(r,a)

where we have used (D.152), (E.76), and the ball B(r, a) is large enough to contain W.

(15.188)

Advanced Theoretical and Numerical Electromagnetics

1050

Since the dyadic κ is constant within W by hypothesis, we can transform the integrand as follows ∇

  1 1 ˆ − 1 ∇ × )κ · R* × κ · (r − r) = ∇ × κ · R = ∇ × κ · R R R R

(15.189)

by virtue of the differential identity (H.50) with A := κ · R. We may compute the curl of the vector κ · R in Cartesian coordinates with the aid of the symbolic determinant in (A.32), viz.,

∇ × (κ · R) =

xˆ

yˆ

zˆ

∂ ∂x

∂ ∂y

∂ ∂z

xˆ · κ · (r − r )

yˆ · κ · (r − r ) zˆ · κ · (r − r )

= (κyz − κzy )ˆx + (κzx − κ xz )ˆy + (κ xy − κyx )ˆz (15.190) = −ˆx × κ · xˆ − yˆ × κ · yˆ − zˆ × κ · zˆ ) * whereby we see that ∇ × κ · R is a constant vector which, in particular, vanishes identically when κ is a symmetric dyadic and this occurrence includes, as special cases, κ being diagonal or just proportional to I. Therefore, the second term in the rightmost member of (15.189) is just proportional to 1/R, and hence this contribution gives rise to integrals of type (15.127) already studied in Section 15.4.1. Thus, we turn our attention to the other term which, being in the form of a curl, we would like to transform into a surface integral over ∂W by means of the curl theorem (H.91). Since the integrand is not well-defined for r → r we need to distinguish the five cases listed on page 1036 at the beginning of Section 15.4.1. We begin by considering the simplest situation of the observation point located outside the tetrahedron, in which instance thanks to (H.91) we have 

ˆ = dV  ∇ × (κ · R)



ˆ =− ˆ ) × κ · R dS  n(r

l=1

∂W

W

4

nˆ l × κ ·

 ∂Wl

dS 

r − r Rl

(15.191)

with ∂Wl , nˆ l and Rl , l = 1, . . . , 4, introduced and described after (15.136). The four surface integrals in the rightmost-hand side of (15.191) are of type (14.192) which we examined in Section 14.9.2. Therefore, we just need to specialize the general formula (14.237) to the facets of W to arrive at 





ˆ =− dV ∇ × (κ · R) W

⎛ ⎜   ⎜ + wl0 |wl0 |nˆ l ⎜⎜⎝arctan

4 l=1

τ2 li0 +

nˆ l

×κ·

3 '  νˆ

li

i=1

 s+ li τli0  + w2 l0 + |wl0 |Rli

2

− − (R+li s+ li − Rli sli )

− arctan

 s− li τli0

⎞ ⎟⎟⎟ ⎟ −⎠

2  τ2 li0 + wl0 + |wl0 |Rli ⎞ ⎛ 2 ( 2 s+ + R+li ⎟⎟ ⎜⎜ τ + wl0  νˆ li − wl0 τli0 nˆ l ⎟⎟⎠ log li− (15.192) + ⎜⎜⎝ li0 2 sli + R−li

with the geometrical quantities sketched in Figure 15.5 and detailed after (15.137). The formula thus found holds strictly for r  W in view of the hypothesis which has led us to (15.191). Parenthetically, the right-hand side of (15.192) should vanish identically when κ = κI, because   1 r − r r − r  r  r (15.193) ∇ × κI · = κ∇ × (r − r ) = κ 3 × (r − r ) = 0, R R R

The Method of Moments II

1051

by virtue of (H.50) in that r − r is curl-free. Indeed, this implies that (see Figure 15.5) 4

' 3 νˆ 

+ s+ li + Rli − 2 s− li + Rli i=1 l=1 ' 4 3 +( s+ 1  + + li + Rli − − 2 2 =− sˆ R s − Rli sli + (τli0 + wl0 ) log − 2 l=1 i=1 li li li sli + R−li

0=−

nˆ l ×

li

− − 2 2 R+li s+ li − Rli sli + (τli0 + wl0 ) log

(

(15.194)

which is true inasmuch as each and every edge of W enters this expression twice with two equal and opposite vector contributions that cancel out. We should be able to show that (15.191) is valid for any other choice of the observation point r in that the estimate (15.188) guarantees that the integral is always bounded. If we suppose r ∈ W, ˆ is singular for r = r, so we isolate the offending point with a small ball B(r, a) ⊂ W. In then κ · R symbols, we have    ˆ = ˆ + ˆ dV  ∇ × (κ · R) dV  ∇ × (κ · R) dV  ∇ × (κ · R) W

B(r,a)

W\B(r,a)



=

ˆ + ˆ ) × κ · R dS  n(r

∂W

=−

4

nˆ l × κ ·

l=1

 + ∂B

 ∂Wl



ˆ + ˆ ×κ·R dS  R

∂B

dS 



ˆ dV  ∇ × (κ · R)

B(r,a)



r −r Rl

ˆ + ˆ ×κ·R dS  R



ˆ dV  ∇ × (κ · R)

(15.195)

B(r,a)

where we expect the two integrals over ∂B and B(r, a) to contribute naught. With the aid of the usual ˆ we local system of polar spherical coordinates (τ , α , β ) centered in r and radial vector τˆ  = −R write ˆ ×κ·R ˆ = )sin α cos β xˆ + sin α sin β yˆ + cos α zˆ * × R ) * κ · xˆ sin α cos β + κ · yˆ sin α sin β + κ · zˆ cos α

(15.196)

which, after the cross products have been computed, yields nine vector fields. However, with an eye towards the integration over the angles α and β , we carry along only the three terms which contribute a non-null result, namely, 



2π

ˆ =a ˆ ×κ·R dS R ∂B

2



π

dβ 0

 dα sin α xˆ × κ · xˆ sin2 α cos2 β

0

+ yˆ × κ · yˆ sin2 α sin2 β + zˆ × κ · zˆ cos2 α   4 = πa2 xˆ × κ · xˆ + yˆ × κ · yˆ + zˆ × κ · zˆ 3 4 ) * = − πa2 ∇ × κ · R 3 in light of (15.190).



(15.197)

1052

Advanced Theoretical and Numerical Electromagnetics

As regards the integral over the ball we use (15.189) backwards, as it were, viz., 

    ) * 1 1 ˆ = dV  ∇ × κ · R dV  ∇ × κ · R + ∇ × κ · R dV  R R

B(r,a)

B(r,a)

 = B(r,a) a

=

 



dτ τ 0

1 = 2

B(r,a)

ˆ R ˆ + 2πa2 ∇ × )κ · R* dV  2 × κ · RR R



ˆ ×κ·R ˆ + 2πa2 ∇ × )κ · R* dΩ R

4π

ˆ + 2πa2 ∇ × )κ · R* ˆ ×κ·R dS  R

∂B

2 ) * ) * 4 ) * = − πa2 ∇ × κ · R + 2πa2 ∇ × κ · R = πa2 ∇ × κ · R 3 3

(15.198)

where we have made use of (15.197) to evaluate the integral over ∂B. Evidently, the term in the rightmost-hand side cancels out the contribution found in (15.197). These calculations prove that (15.195) reduces to (15.191), as anticipated, so it follows that (15.192) applies for r ∈ W as well. Finally, the extension for points on the edges and the vertices of W may be effected along the same lines. Tracing back our steps we are ready to write down the complete formula for the singular integral (15.130), namely, 4 3 '  νˆ li + + 1   nˆ l × κ · (Rli sli − R−li s− dV ∇ × κ · (r − r) = − li ) R 2 l=1 i=1 W ⎛ ⎞   s− s+ ⎟⎟⎟ ⎜⎜⎜ li τli0 li τli0   ⎜ ⎟ − arctan 2 + wl0 |wl0 |nˆ l ⎝arctan 2 2  + 2  −⎠ τli0 + wl0 + |wl0 |Rli τli0 + wl0 + |wl0 |Rli ⎞ ⎛ 2 ( s+ + R+li ⎟⎟ ⎜⎜ τ + w2 l0  νˆ li − wl0 τli0 nˆ l ⎟⎟⎠ log li− + ⎜⎜⎝ li0 2 sli + R−li 3 ⎡ 4  s+ ) *  ⎢⎢⎢  1 li τli0 ⎢⎣|wl0 | arctan 2 − ∇ × κ · R wl0  + 2 τli0 + w2 l0 + |wl0 |Rli l=1 i=1





−|wl0 | arctan

 s− li τli0 2  − τ2 li0 + wl0 + |wl0 |Rli

− τli0 log

+⎤ s+ ⎥⎥⎥ li + Rli ⎥ − ⎦ (15.199) s− + R li li

on account of (15.189) and the previous result (15.139).

15.5 Discretization of EFIE and volume integral equations Since (13.231) and (13.232) constitute nothing but special instances of EFIE and VIE, respectively, the application of the Method of Moments is accomplished by following the procedures already

The Method of Moments II

1053

detailed in Sections 14.2 and 15.1. In particular, the integral operator of this hybrid formulation may be written symbolically as a 2 × 2 abstract matrix of four integro-differential operators, viz.,  L {•} :=

antenna

plasma

LAA {•} LPA {•}

LAP {•} LPP {•}

with the definitions





antenna

(15.200)

plasma

 e− j k 0 R 1 e− j k 0 R  {•} +∇s ∇ · {•}, LAA {•} := − j ωμ0 dS dS  4πR j ωε0 4πR s SA SA   

vector potential part

LAP {•} :=

r ∈ S +A

(15.201)

scalar potential part

  k02 e− j k 0 R e− j k 0 R  ∇s ∇ · [κe (r ) · {•}] dV  κe (r ) · {•} + dV  ε0 4πR ε0 4πR VP VP   vector potential part

∇s − ε0



scalar potential (volume charges)

e− j k 0 R ˆ  ) · κe (r ) · {•}, n(r 4πR SP  dS 

r ∈ S +A

(15.202)

scalar potential (surface charges)

LPA {•} :=

  k02 e− j k 0 R e− j k 0 R  ∇ {•} + ∇ · {•}, dS  dS  jω 4πR jω 4πR s SA SA   vector potential part

(15.203)

scalar potential part

 e− j k 0 R e− j k 0 R  ∇ · [κe (r ) · {•}] κe (r ) · {•} +∇ dV  4πR 4πR VP VP   

LPP {•} := k02

r ∈ VP

dV 

vector potential



scalar potential (volume charges) − j k0 R

  e ˆ  ) · κe (r ) · {•} −ε0 ε(r) −1 · {•}, n(r 4πR ∂VP 

−∇

dS 

r ∈ VP

(15.204)

scalar potential (surface charges) 

where R = |r − r |. The operators in the first column of L {•} take a surface current density on S +A as input, whereas those in the second column act upon a displacement vector defined within VP . The operators of the first row return a tangential electric field on S +A , and those in the second row return a field in VP that carries the physical dimension of a displacement vector. The ‘right’ input for the problem of Figure 13.22a is the only admissible pair of JA (r) and D(r) such that all in all L {•} returns the negative of the tangential impressed electric field on ∂WG ∩ S +A [see (13.231)] and a null displacement vector within VP . Indeed, for consistency with (15.200) the excitation term of the symbolic equation (14.1) is defined as an abstract column vector, namely,   EA antenna := F(r) (15.205) 0 plasma

1054

Advanced Theoretical and Numerical Electromagnetics

where EA , given by the negative of the right member of (13.231) for r ∈ S +A , contains the antenna excitation with the delta-gap model of the port. To keep the solution procedure simple we suppose that the surface S A has been modelled by means of a tessellation S MA comprised of MA patches S p , p = 1, . . . , MA [see (14.28)] and that the three-dimensional domain VP has been approximated with a volumetric mesh V MP made of MP subdomains V p , p = 1, . . . , MP [see (14.29)]. In this regard, we also need to distinguish between subdomains internal to V MP and boundary sub-domains, as we did in (15.19) and (15.20), respectively. We notice that these assumptions require the formal modification of the four entries of the operator L {•} in (15.200), namely, the surface integrals are to be carried out over the tessellation S MA and the domain integrals over the volumetric mesh V MP . Then, keeping in mind that the numbers of patches and sub-domains may differ from the numbers of basis and test functions, for the discretization step of the Method of Moments in the form of Galerkin we consider • •

a set of NA subsectional real-valued divergence-conforming surface vector basis and test funcNA [cf. (14.30)]; tions {fn (r)}n=1 a set of NP subsectional real-valued divergence-conforming volumetric vector basis and test NP functions {gn (r)}n=1 [cf. (15.23)].

The support Ξn of the function fn (r) is constituted by one or more adjacent patches S p ∈ S MA and satisfies the required properties outlined in Section 14.2. Similarly, the support Δn of gn (r) is constituted by one or more adjoining sub-domains V p ∈ V MP , and gn (r) is endowed with the features laid out in Section 15.1. Next, we make the Ansatz JA (r) =

NA

In fn (r),

r ∈ S MA

(15.206)

r ∈ V MP

(15.207)

n=1

D(r) =

NP

Dn gn (r) =

n=1

NIP

Dn gnI (r)

n=1

+

NIP +NBP

Dn gnB (r),

n=1+NIP

where it is understood that JA (r) and D(r) signify the approximations JNA (r) and DNP (r) that are afforded by the expansion on a grand total of NA + NP basis functions. Furthermore, we recall that • •

gnI (r), n = 1, . . . , NIP , is a function whose support Δn is comprised of one or more internal sub-domains and satisfies (15.24), and NIP < NP is the total number of functions with this property; gnB (r), n = 1 + NIP , . . . , NIP + NBP , is a function whose support Δn taps into the interface ∂V MP and obeys (15.25), and NBP = NP − NIP is the total number of functions with this characteristic.

In order to obtain the algebraic counterpart of L {•} we use the inner products (14.33) and (15.2) for the EFIE (13.231) and for the VIE (13.232), respectively. The resulting system matrix [L] comes partitioned into four blocks, namely,   [LAA ] [LAP ] [L] = (15.208) [LPA ] [LPP ] which evidently mirrors the structure of L {•}. By referring to the findings of Sections 14.2 and 15.1 we can easily write down the entries of the diagonal sub-matrices, viz.,   e− j k 0 R fn (r ) LAA,mn = − j ωμ0 dS fm (r) · dS  4πR Ξm

Ξn

The Method of Moments II 1 j ωε0

−

dS ∇s · fm (r) Ξm

 + Um 

Δm

ˆ · gm (r) dS n(r)

∂Δm

−



dV ∇ · gm (r) Δm

 +

dV 

e− j k 0 R  ∇ · [κe (r ) · gn (r )] 4πR

Δn

dV 

Δn



− Um

e− j k 0 R κe (r ) · gn (r ) 4πR

Δn





dV ∇ · gm (r) Δn

∂Δn

m, n ∈ {1, . . . , NA } (15.209)

e− j k 0 R  ∇ · [κe (r ) · gn (r )] 4πR



dS 

ˆ · gm (r) dS n(r)

∂Δm

e− j k 0 R  ∇ · fn (r ), 4πR s

dV 

dV gm (r) ·

k02

dS 

Ξn



 LPP,mn =





1055

∂Δn

e− j k 0 R  ˆ ) · κe (r ) · gn (r ) n(r 4πR

e− j k 0 R ˆ  ) · κe (r ) · gn (r ) n(r 4πR    dV gm (r) · ε(r) −1 · gn (r), − ε0

dS 

m, n ∈ {1, . . . , NIP + NIB } (15.210)

Δm ∩Δn

where ⎧ ⎪ ⎨0, m = 1, . . . , NIP Um := ⎪ ⎩1, m = N + 1, . . . , N + N IP IP BP

(15.211)

is a suitable discrete step function which singles out the boundary test functions. The entries of the off-diagonal blocks [LAP ] and [LPA ] are characteristic of this mixed formulation and deserve some attention as they involve nested volume and surface integrals. For the mutual effect of the plasma onto the antenna and vice-versa we write at first LAP,mn = + −

1 ε0 1 ε0



k02 ε0



 dS fm (r) · Ξm



dS fm (r) · ∇s Ξm

dV 

Δn

dV 

e− j k 0 R  ∇ · [κe (r ) · gn (r )] 4πR

dS 

e− j k 0 R ˆ  ) · κe (r ) · gn (r ), n(r 4πR

Δn





dS fm (r) · ∇s Ξm

e− j k 0 R κe (r ) · gn (r ) 4πR

∂Δn

m = 1, . . . , NA ,

LPA,mn =

k02 jω



 dV gm (r) ·

Δm

Ξn

dS 

e− j k 0 R fn (r ) 4πR

n = 1, . . . , NP

(15.212)

Advanced Theoretical and Numerical Electromagnetics

1056 +





1 jω

dV gm (r) · ∇

dS 

Ξn

Δm

e− j k 0 R  ∇ · fn (r ), 4πR s m = 1, . . . , NP ,

n = 1, . . . , NA

(15.213)

but since we are better off without computing derivatives numerically, we prefer to ‘move’ the operators ∇s {•} and ∇{•} in the scalar potential contributions onto the test functions. By following the steps outlined in Sections 14.2 and 15.1 we obtain the alternative expressions LAP,mn 1 − ε0 1 + ε0



−



 dS fm (r) · Ξm



dS ∇s · fm (r) Ξm



dS ∇s · fm (r) Ξm

Um jω 1 jω

Δn



e− j k 0 R  ∇ · [κe (r ) · gn (r )] 4πR

dS 

e− j k 0 R ˆ  ) · κe (r ) · gn (r ), n(r 4πR





dS 

dV gm (r) · Ξn

Δm



ˆ · gm (r) dS n(r) ∂Δm



 dV ∇ · gm (r) Ξn

e− j k 0 R κe (r ) · gn (r ) 4πR

dV 

∂Δn

k2 = 0 jω

Δm

dV 

Δn



LPA,mn +

k2 = 0 ε0

dS 

Ξn

dS 

m = 1, . . . , NA ,

n = 1, . . . , NP

(15.214)

m = 1, . . . , NP ,

n = 1, . . . , NA

(15.215)

e− j k 0 R fn (r ) 4πR

e− j k 0 R  ∇ · fn (r ) 4πR s

e− j k 0 R  ∇ · fn (r ), 4πR s

under the additional hypotheses that fm (r) obeys (14.42) and gm (r) satisfies (15.24) and (15.25). Accordingly, the column vector of the NA + NP unknown expansion coefficients introduced in (15.206) and (15.207) reads ⎞ ⎛ ⎞ ⎛ ⎜⎜⎜ I1 ⎟⎟⎟ ⎜⎜⎜ D1 ⎟⎟⎟   ⎟ ⎜ ⎜ [I] ⎜ . ⎟ ⎜ . ⎟⎟ [X] = with [I] = ⎜⎜⎜⎜⎜ .. ⎟⎟⎟⎟⎟ , [D] = ⎜⎜⎜⎜⎜ .. ⎟⎟⎟⎟⎟ . (15.216) [D] ⎟⎠ ⎜⎝ ⎟⎠ ⎜⎝ INA DNP Finally, on account of (15.205) we write the excitation vector as ⎛ ⎛ ⎞ ⎞ ⎜⎜⎜ F A,1 ⎟⎟⎟ ⎜⎜⎜0⎟⎟⎟   ⎜⎜ . ⎟⎟ ⎜⎜⎜⎜ . ⎟⎟⎟⎟ [F A ] with [F A ] = ⎜⎜⎜ .. ⎟⎟⎟ , [F] = [0NP ] = ⎜⎜⎜⎜⎜ .. ⎟⎟⎟⎟⎟ [0NP ] ⎜⎝ ⎜⎝ ⎟⎠ ⎟⎠ F A,NA 0

(15.217)

where [0NP ] is a column vector with NP null entries. We observe that the only non-null elements of [F A ] are contributed by the test functions fm (r) whose support happens to overlap S G ⊂ S MA

The Method of Moments II

1057

(Figures 14.14, 14.15 and 13.22b) that is, the part of the surface mesh which models the boundary of the delta-gap region. At this stage we take the limit as h → 0+ whereby the gap S G reduces to the line γ˜ A ⊂ S MA . Therefore, on the grounds of (14.268) we have

F A,m

  ⎧ VG ⎪ ⎪ ⎪ νˆ = VG dS fm (r) · ds νˆ · fm (r), Ξm ∩ γ˜ A  ∅ lim ⎪ ⎪ ⎨h→0+ h =⎪ Ξm ∩S G Ξm ∩γ˜ A ⎪ ⎪ ⎪ ⎪ ⎩0, Ξm ∩ γ˜ A = ∅

(15.218)

for m = 1, . . . , NA . We notice that the sub-matrix [LAA ] is essentially the one relevant to the application of the Galerkin testing to the EFIE operator and hence, as we showed in Section 14.2, it is symmetric. Regrettably, a similar beneficial property does not hold for the plasma self-interaction block [LPP ] nor can we conclude that the off-diagonal blocks are the transpose of each other as was the case, e.g., for the algebraic system arising from the PMCWHT equations (Section 14.5). Thus, the matrix [L] is non-symmetric, the ultimate reason being the presence of the contrast dyadic κe (r) within the inner integrals only. Not much can be done at this stage, unless we start over and review our choice of test functions or inner products or both. On a related note we observe that, even though the elements LAP,mn and LPA,mn are patently different, still the following relationship holds [11, Appendix] ε0 LAP,mn = j ωLPA,nm

when

κe (r) ≡ I

(15.219)

when

κe (r) ≡ I

(15.220)

and also ε0 [LAP ] = j ω[LPA ]T

which can be proved, e.g., by inverting the order of integration in (15.214), changing names to the dummy variables according to r → r and r → r, and finally observing that the distance R is unaffected by such substitutions. Thus, from a numerical standpoint the computer program module charged with the task of evaluating LAP,mn — which is more general — may also be run to determine LPA,nm provided one allows for the possibility of letting κe ≡ I. The latter assumption amounts to setting ε(r) → ∞ in (10.91) and, as such, has no physical meaning, it is merely a clever programming expedient. If the basis and test functions are dimensionless, the expansion coefficients In and Dn must perforce carry the same physical dimension as the unknown vector fields for which they were introduced in the first place. Thus, the entries of the unknown vector [X] are dimensionally unbalanced and even more so are the elements of the system matrix [L]. Indeed, a thorough dimensional check reveals that the physical dimensions are as follows • • • •

LAA,mn , area times an impedance (Ωm2 ), LAP,mn , area divided by a permittivity (m3 /F), LPA,mn , area divided by a frequency (m2 /Hz) or volume divided by a velocity (m3 /(m/s)), LPP,mn , a volume (m3 ).

Dimensional balance is desirable in order to reduce the condition number of the system matrix [12] and consequently improve the accuracy of the computed numerical solution [X] (see discussion in Section 14.5). In the present case, though, multiplying [L] from the left and the right side with the same matrix [P]−1 as was done in (14.89) will not do the trick, because the off-diagonal sub-blocks

1058

Advanced Theoretical and Numerical Electromagnetics

of [L] have different physical dimensions. We need a more general approach to the pre-conditioning that involves different left and right splitting matrices [13, Chapter 13], [14, Chapter 9], namely, [P2 ]−1 [L][P1 ]−1 [P1 ][X] = − [P2 ]−1 [F]    [L ]

[X  ]

(15.221)

[F  ]

with

⎛√ ⎞ ⎜⎜⎜ Z0 [INA ] [0NA ,NP ] ⎟⎟⎟ ⎟⎟⎟ ⎜⎜ ⎟⎟ [P2 ] := ⎜⎜⎜⎜⎜ [I ] N P ⎜⎝ [0NP ,NA ] hP √ ⎟⎟⎟⎠ c0 Z0

⎛√ ⎞ ⎜⎜⎜ Z0 [INA ] [0NA ,NP ] ⎟⎟⎟ ⎟⎟⎟ , [P1 ] := ⎜⎜⎜⎜⎝ √ ⎟⎠ [0NP ,NA ] c0 Z0 [INP ]

(15.222)

where Z0 := (μ0 /ε0 )1/2 and c0 := (ε0 μ0 )−1/2 are the intrinsic impedance and the speed of light in free space, and hP represents a suitable characteristic length of the problem. Besides, [Iα ], α ∈ {NA , NP }, is the identity matrix of order α, and [0α,β ], α, β ∈ {NA , NP }, α  β, denotes a matrix of zeros with size α × β. By carrying out the matrix-matrix and matrix-vector multiplications indicated in (15.221) we obtain ⎛ [L ] ⎞ ⎛ [LAP ] ⎞⎟⎟ ⎛ √ [F A ] AA ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎞ ⎟⎟⎟ ⎜ √ ⎜⎜⎜ Z ⎟⎟⎟ ⎜ ⎜⎜⎜ c0 Z0 ⎟⎟⎟ ⎜⎜⎜ Z0 [I] ⎟⎟⎟⎟ 0 Z 0 ⎟⎟⎟ ⎜⎜⎜⎜ ⎜ = − (15.223) ⎟ ⎟ ⎜ √ ⎜ ⎠ ⎟⎟⎟ ⎝ ⎟⎟ ⎜⎜⎜ c0 . ⎜⎜⎜ c0 ] [L c Z [D] PP ⎟ 0 0 ⎝ ⎠ ⎝ [LPA ] Z0 [F P ]⎟⎠ hP hP hP where now all the elements of [L ] have the dimension of an area. The entries of [X  ] and [F  ] have the dimension W1/2 /m and W1/2 m, respectively. The factor hP may be chosen to be the maximum chord of VP or the average characteristic size of the sub-domains V p ∈ V MP or even the wavelength λ0 . For the calculation of the average power absorbed by or lost into the medium which occupies the region VP we start from the exact formula (13.240) and substitute the expansion of D(r), viz., ω PP  = Im 2



V MP

dV

NP

D∗m gm (r)

NP −1 · ε(r) · Dn gn (r)

m=1



n=1

 NP NP   ω ∗ = Im D Dn dV gm (r) · ε0 ε(r) −1 · gn (r) 2ε0 m=1 n=1 m Wm ∩Wn

=−

NP NP

 ω ω Im D∗mGmn Dn = − Im [D]H [G][D] 2ε0 m=1 n=1 2ε0

(15.224)

where the NP × NP matrix [G] has entries Gmn defined in (15.27), and the superscript ‘H’ indicates complex conjugation and transposition. So long as the medium which fills VP is passive, the rightmost-hand side of (15.224) must be positive or null. Therefore with a few extra manipulation we have   

1 H 1 H Im [D]H [G][D] = [D] [G][D] − [D]T [G]∗ [D]∗ = [D] [G][D] − [D]H [G]H [D] 2j 2j H [G] − [G] = [D]H [D]  0 (15.225) 2j which states that the Hermitian matrix ([G] − [G]H )/(2 j) is negative semi-definite [12, 15, 16].

The Method of Moments II

1059

15.6 Discretization of integral and wave equations We wish to extend the Method of Moments described in Section 14.1 to the hybrid system of integral and wave equations developed in Section 13.5.2 for the analysis of the device sketched in Figure 13.24a. Although the treatment of the surface integral equations is a minor generalization of the approach outlined for the EFIE and the MFIE, we need to devise a similar way to turn the wave equation (13.263) into an algebraic system, while at the same time evidencing the expected coupling with (13.258). Since the problem of Figure 13.24a has been reduced to the solution of integral equations plus the time-harmonic source-free wave equation for the electric field E2 (r) inside the plasma, we begin by focusing on the three coupled integral equations (13.257), (13.261), (13.262). In accordance with (14.1) we define L {•} as an abstract 3 × 3 matrix of integro-differential operators, namely, PEC feeder

plasma

⎛ ⎜⎜⎜ LAA {•} ⎜⎜ L {•} := ⎜⎜⎜⎜⎜ EPA {•} ⎜⎝ HPA {•}

plasma

LAP {•}

0

EPP {•}

0

HPP {•}

−IPP {•}

⎞ ⎟⎟⎟ ⎟⎟⎟ ⎟⎟⎟ ⎟⎟⎠

PEC feeder

(15.226)

plasma plasma

with the operator entries given explicitly by 

dS 

LAA {•} := − j ωμ0 SA

∇s e− j k 0 R {•} + 4πR j ωε0 dS 

LAP {•} := − j ωμ0 Is (r) · SP



dS 

EPA {•} := − j ωμ0 Is (r) · SA

dS 

EPP {•} := − j ωμ0  HPA {•} :=

SP



dS 

SA







− j k0 R

∇s e {•} + 4πR j ωε0 − j k0 R

∇s e {•} + 4πR j ωε0

− j k0 R

∇s e {•} + 4πR j ωε0

ˆ × ∇ dS  n(r)

− j k0 R

e 4πR

 × {•} ,



e− j k 0 R  ∇ · {•}, 4πR s



r ∈ S +A

(15.227)

dS 

e− j k 0 R  ∇ · {•}, 4πR s

r ∈ S +A

(15.228)

dS 

e− j k 0 R  ∇ · {•}, 4πR s

r ∈ SP

(15.229)

r ∈ SP

(15.230)

r ∈ SP

(15.231)

r ∈ S +P

(15.232)

SP



SA

dS 

e− j k 0 R  ∇ · {•}, 4πR s

SP

SA

HPP {•} :=

 − j k0 R   e 1 ˆ × ∇ {•} + PV dS  n(r) × {•} , 2 4πR SP

and IPP {•} := Is (r) · {•},

r ∈ SP

(15.233)

being the identity operator that takes a surface vector field on S P and returns the selfsame field for r ∈ S P . The three operators in the first column take the surface current density JA (r) and return the tangential electric field over S A , the tangential electric field over S P , and the rotated tangential magnetic field over S P , respectively. The operators in the second column return the same surface

1060

Advanced Theoretical and Numerical Electromagnetics

fields by acting on the surface current density JP (r). In fact, not only do LAA {•} and EPP {•} have the same form as the typical EFIE operator defined in (14.34), but also the abstract 2 × 2 matrix   LAA {•} LAP {•} (15.234) LEFIE {•} := EPA {•} EPP {•} may be interpreted as a single EFIE operator defined on S A ∪ S P . Finally, the operators in the last ˆ × H2 (r) on the negative side of column of (15.226) act on the twisted tangential magnetic field n(r) S P. For consistency with (15.226) the right-hand side of the symbolic equation (14.1) gets replaced by an abstract column vector, viz., ⎛ ⎞ ⎜⎜⎜EA ⎟⎟⎟ PEC feeder ⎜ ⎟ F(r) := ⎜⎜⎜⎜⎜EP ⎟⎟⎟⎟⎟ plasma (15.235) ⎝ ⎠ 0 plasma where EA , given by the negative of the right member of (13.257) for r ∈ S +A , contains the antenna excitation under the delta-gap model of the port, and r ∈ S −P

EP := −Et2 (r),

(15.236)

where Et2 (r) is the tangential electric field on the negative side of S P , i.e., within the plasma. In prinˆ × H2 (r), ciple, we can invert the operator L {•} defined in (15.226) to compute JA (r), JP (r) and n(r) but this step is of little use inasmuch as also Et2 (r) in the right member (15.235) is an unknown of the problem. Indeed, the missing piece of information is precisely provided by the wave equation (13.263) within VP . As an alternative viewpoint, we could regard (14.1) under the assumptions (15.226) and (15.235) as a means to assign the tangential electric field Et2 (r) on S −P , which is necessary to ensure the uniqueness of solution to (13.263). To keep the description of the discretization procedure simple, for the time being we pretend that suitable entire-domain vector test and basis functions are available to expand surface currents and fields, though we are perfectly aware that this is hardly feasible for arbitrary-shaped surfaces and three-dimensional domains. That been said, we introduce • • •

NA , r ∈ S A (Seca set of NA real-valued divergence-conforming surface vector functions {fn (r)}n=1 tion 13.2.1); NP , r ∈ S P (Seca set of NP real-valued divergence-conforming surface vector functions {gn (r)}n=1 tion 13.2.1); NE , r ∈ VP (see a set of NE real-valued curl-conforming volumetric vector functions {en (r)}n=1 Section 15.7 further on);

whereby we can define the four expansions JA (r) =

NA

An fn (r),

r ∈ S +A

(15.237)

Pn gn (r),

r ∈ S +P

(15.238)

Hn gn (r),

r ∈ S −P

(15.239)

n=1

JP (r) =

NP n=1

ˆ × H2 (r) = n(r)

NP n=1

The Method of Moments II E2 (r) =

NE

r ∈ VP

En en (r),

1061 (15.240)

n=1

with the usual understanding that the vector fields in the left members above signify the approximation of the true unknown currents and fields that is afforded by the usage of a grand total of NA + 2NP + NE expansion functions. In light of (15.234) the requirements expected of fn (r) and gn (r) are the same as those we ˆ × H2 (r) as already discussed for the solution of the EFIE. Moreover, we employ gn (r) to expand n(r) well, because a twisted magnetic field is physically equivalent to an electric surface current density. This statement, in addition to being supported by dimensional analysis, stems from the matching condition (1.168) at a PEC interface but also from the Stratton-Chu integral representations (10.22) ˆ × H2 (r) enters (15.226) only through the identity operator IPP {•}, and (10.23). However, since n(r) basis functions that are not divergence-conforming might be used in lieu of gn (r). In fact, we reached the same conclusion while applying the Method of Moments to the numerical solution of the MFIE in Section 14.3. Last but not least, the volumetric basis functions en (r) must be curl-conforming [3, Section 9.7], meaning that the components of en (r) must admit a finite curl for r ∈ VP . We shall elaborate on this constraint further on, but we notice for the moment that some degree of differentiability of en (r) is clearly motivated by the very form of the wave equation (13.263) which is part of the formulation. Next, guided by the testing procedure devised for the EFIE and the MFIE, we introduce the symmetric inner product  (f, g)S A := dS f(r) · g(r) (15.241) SA

with f and g real surface vector fields in the spaces WLAA and WLAP , and the symmetric inner product  (f, g)S P := dS f(r) · g(r) (15.242) SP

with f and g real surface vector fields in the spaces WEPA , WEPP , WHPA and WHPP . Carrying out the testing procedure with (15.241) and (15.242) lets us transform (14.1) into an algebraic system comprised of NA +2NP equations and NA +2NP + NE unknowns. At this stage such system is undetermined inasmuch as the NE expansion coefficients En are not known yet. Skipping the mathematical details, which are no different than those discussed in Sections 14.2, 14.3 and 14.10, we arrive at the resulting matrix [L] which comes naturally partitioned into nine blocks ⎞ ⎛ ⎜⎜⎜ [LAA ] [LAP ] [0NA ,NP ]⎟⎟⎟ ⎟ ⎜⎜⎜ [L] = ⎜⎜⎜ [E PA ] [E PP ] [0NP ,NP ]⎟⎟⎟⎟⎟ (15.243) ⎠ ⎝ [HPA ] [HPP ] −[G PP ] where with evident notation the entries of the non-zero sub-matrices read   e− j k 0 R fn (r ) LAA,mn = − j ωμ0 dS fm (r) · dS  4πR SA

1 − j ωε0



SA



dS ∇s · fm (r) SA

SA

dS 

e− j k 0 R  ∇ · fn (r ), 4πR s

m, n ∈ {1, . . . , NA } (15.244)

1062

Advanced Theoretical and Numerical Electromagnetics 

 LAP,mn = − j ωμ0 −

1 j ωε0

SA



dS 

dS ∇s · fm (r) SA

 dS gm (r) ·

SP





dS ∇s · gm (r) SP

dS 

dS 

 dS gm (r) ·

SP

n = 1, . . . , NA

(15.246)

e− j k 0 R gn (r ) 4πR 

dS 

e− j k 0 R  ∇ · gn (r ), 4πR s

SP



' − j k0 R ( e  ˆ × ∇ dS n(r) × fn (r ) 4πR 

ˆ · dS gm (r) × n(r) SP

m, n ∈ {1, . . . , NP } (15.247)



SA



dS 

SP

dS gm (r) · SP

dS  ∇

e− j k 0 R × fn (r ) 4πR

SA

m = 1, . . . , NP , 

n = 1, . . . , NA

(15.248)

' − j k0 R (   e ˆ × ∇ × gn (r ) dS gm (r) · gn (r) + PV dS gm (r) · dS  n(r) 4πR

SP

1 2

m = 1, . . . , NP ,

e− j k 0 R  ∇ · fn (r ), 4πR s

dS ∇s · gm (r)



=

(15.245)

e− j k 0 R fn (r ) 4πR

SP



1 − j ωε0

1 2

n = 1, . . . , NP

SA

E PP,mn = − j ωμ0

HPP,mn =

m = 1, . . . , NA ,

SA



=

e− j k 0 R  ∇ · gn (r ), 4πR s

SP



HPA,mn =

e− j k 0 R gn (r ) 4πR

SP



E PA,mn = − j ωμ0 1 − j ωε0

dS 

dS fm (r) ·

SP

SP





dS gm (r) · gn (r) +PV SP





ˆ · dS gm (r) × n(r)

SP

dS  ∇

e− j k 0 R × gn (r ) 4πR

SP

GPP,mn

m, n ∈ {1, . . . , NP }

(15.249)

 G PP,mn = (gm , gn )S P =

dS gm (r) · gn (r),

m, n ∈ {1, . . . , NP }

SP

and [0α,β ], α, β ∈ {NA , NP }, denotes a matrix of zeros with size α × β.

(15.250)

The Method of Moments II

1063

As was the case for the solution of the EFIE with Galerkin testing (Section 14.2), the matrices [LAA ] and [E PP ] are symmetric, and this feature may be exploited to reduce the filling time by roughly 50 per cent. Besides, direct inspection of (15.245) and (15.246) shows that the matrices [LAP ] and [E PA ] are the transpose of each other, and hence only one of the two needs to be filled. The algebraic counterpart of the identity operator IPP {•} is [G PP ], the so-called Gram matrix or projection matrix NP associated with the basis {gn (r)}n=1 . Only if the functions gn (r) formed an orthonormal basis on S P , would [G PP ] coincide with the identity matrix of rank NP . The column vector of the NA + 2NP unknown expansion coefficients introduced in (15.237), (15.238) and (15.239) reads ⎛ ⎞ ⎜⎜⎜ [A] ⎟⎟⎟ ⎜ ⎟ [X] = ⎜⎜⎜⎜⎜ [P] ⎟⎟⎟⎟⎟ , ⎝ ⎠ [H]

with

⎞ ⎛ ⎜⎜⎜ A1 ⎟⎟⎟ ⎜⎜ . ⎟⎟ [A] = ⎜⎜⎜⎜⎜ .. ⎟⎟⎟⎟⎟ , ⎟⎠ ⎜⎝ ANA

⎞ ⎛ ⎜⎜⎜ P1 ⎟⎟⎟ ⎜⎜ . ⎟⎟ [P] = ⎜⎜⎜⎜⎜ .. ⎟⎟⎟⎟⎟ , ⎟⎠ ⎜⎝ PNP

⎞ ⎛ ⎜⎜⎜ H1 ⎟⎟⎟ ⎜⎜ . ⎟⎟ [H] = ⎜⎜⎜⎜⎜ .. ⎟⎟⎟⎟⎟ ⎟⎠ ⎜⎝ HNP

(15.251)

so as to be compatible with the very form of (15.243). In like manner, the abstract ‘excitation’ vector F(r) is turned into a column vector with NA + 2NP elements, i.e., ⎞ ⎛ ⎜⎜⎜ [F A ] ⎟⎟⎟ ⎟ ⎜⎜⎜ [F] = ⎜⎜⎜ [F E ] ⎟⎟⎟⎟⎟ , ⎠ ⎝ [0NP ]

with

⎛ ⎞ ⎜⎜⎜ F A,1 ⎟⎟⎟ ⎜⎜ . ⎟⎟ [F A ] = ⎜⎜⎜⎜⎜ .. ⎟⎟⎟⎟⎟ , ⎜⎝ ⎟⎠ F A,NA

⎛ ⎞ ⎜⎜⎜ F E,1 ⎟⎟⎟ ⎜⎜ . ⎟⎟ [F E ] = ⎜⎜⎜⎜⎜ .. ⎟⎟⎟⎟⎟ , ⎜⎝ ⎟⎠ F E,NP

and non-zero entries given by  & VG vˆ = VG ds vˆ · fm (r), dS fm (r) · F A,m = lim+ h→0 h  F E,m = −

∂WG ∩S A

dS gm (r) · Et2 (r) = − SP

⎛ ⎞ ⎜⎜⎜0⎟⎟⎟ ⎜⎜ . ⎟⎟ [0NP ] = ⎜⎜⎜⎜⎜ .. ⎟⎟⎟⎟⎟ , ⎜⎝ ⎟⎠ 0

(15.252)

m = 1, . . . , NA

(15.253)

m = 1, . . . , NP

(15.254)

γA

NE 

dS gm (r) · en (r) En ,

n=1 S

P 

Cmn

on the grounds of (13.257) and (15.240). By letting h, the height of the gap region WG , approach zero we reduce the delta-gap to the line γA (Figure 13.24b). The local quasi-static electric potential Φgap suffers a jump equal to VG across γA [see (13.76) and (13.77)], and the network equivalent of the device is sketched in Figure 13.7a. Having assumed entire-domain basis functions on S A , we expect that all the integrals in (15.253) are non-null. The dot product gm (r) · en (r), for r ∈ S P , depends only on the part of en (r) that is tangential to the interface S P . The coefficients Cmn — involved in the calculation of the elements F E,m — constitute the entries of an NP × NE real matrix [C] which eventually quantifies the surmised coupling between (13.261) and the wave equation (13.263). Indeed, we can write [F E ] in (15.252) explicitly as [F E ] = −[C][E]

(15.255)

where [E] is the column vector which contains the NE unknown coefficients En . Some of the entries of [C] may be null even though gm (r) and en (r) are entire-domain functions on S P and VP , respectively. The discretization procedure outlined in Section 14.1 was described with an eye towards integral equations in electromagnetics. Nonetheless, since the linear operator L {•} of (14.1) need not be an

Advanced Theoretical and Numerical Electromagnetics

1064

integral one after all, we may also apply the Method of Moments to the homogeneous wave equation (13.263) with the definition LPP {•} := ∇ × ∇ × {•} − ω2 μ0 ε˜ c (r)I · {•},

r ∈ VP

and the inner product  (f, g)VP := dV f(r) · g(r)

(15.256)

(15.257)

VP

where f and g are volumetric vector fields in the space WLPP . For the sake of completeness, though, we must mention that in the specialized literature the Method of Moments applied to purely (partial) differential equations is most commonly known as the Finite Elements Method [17, Chapter 6], [18, Chapter 9], [19–25], but the two techniques are essentially the same. The operator LPP {•} takes an electric field in VP and returns an electric field within the same region, apart from the inconsequential multiplicative factor ω2 μ0 ε˜ c (r); thus, the spaces VLPP and WLPP coincide. In fact, to make this statement even more evident, we might divide (13.263) through by k02 and accordingly adjust the definition of LPP {•} which would become dimensionless. Since the plasma region VP is devoid of sources, (13.263) demands we seek the electric field E2 (r), r ∈ VP , in the null space of LPP {•} subject to the condition that the tangential component Et2 (r) on S −P is non-zero and specified by (13.261). The best approximation to the true electric field E2 (r) that is afforded by the linear combination (15.240) obtains when the (squared) norm of the residual RNE (r), r ∈ VP , 



R∗NE , RNE VP

--2 -NE ---2 -- := -RNE -V = -- LPP {en (r)} En --P - n=1

(15.258)

VP

is minimized [cf. (14.12) and (14.15) and Figure 14.1]. Since (15.257) is a symmetric inner product but the residual RNE (r) is a complex vector field, the complex conjugation of either factor in (15.257) is necessary in order to obtain a real positive number, as expected of a norm.1 With our positions and the choice of em (r) as weighting functions in the range of LPP {•} (Galerkin testing) from (14.18) and (15.257) we have  ! " 0 = (em , LPP {E2 })VP = dV em (r) · ∇ × ∇ × E2 (r) − ω2 μ0 ε˜ c (r)E2 (r) VP

 =



dV ∇ · {[∇ × E2 (r)] × em (r)} + VP

 − ω2 μ0

dV ∇ × em (r) · ∇ × E2 (r) VP

dV ε˜ c (r)em (r) · E2 (r) VP





ˆ · [∇ × E2 (r)] × em (r) + dS n(r)

= SP

 − ω2 μ0

dV ∇ × em (r) · ∇ × E2 (r) VP

dV ε˜ c (r)em (r) · E2 (r) VP

1

By contrast, in (14.11) the complex conjugation was implied in the very definition of inner product (D.68).

The Method of Moments II  =

! " dV ∇ × em (r) · ∇ × E2 (r) − ω2 μ0 ε˜ c (r)em (r) · E2 (r)

VP

 +

 =

1065

ˆ × [∇ × E2 (r)] dS em (r) · n(r) SP

! " dV ∇ × em (r) · ∇ × E2 (r) − ω2 μ0 ε˜ c (r)em (r) · E2 (r)

VP

 − j ωμ0

ˆ × H2 (r), dS em (r) · n(r)

m = 1, . . . , NE

(15.259)

SP

having used the differential identity (H.49) with A = ∇ × E2 and B = em , the Gauss theorem (A.53), the algebraic identity (H.13), and eventually the Faraday law (1.99), which holds in VP . To finalize the transformation of (13.263) into an algebraic system we insert the expansions (15.239) and (15.240), namely, NE 

! " dV ∇ × em (r) · ∇ × en (r) − ω2 μ0 ε˜ c (r)em (r) · en (r) En

n=1 V

P 

Mmn

= j ωμ0

NP 

dS em (r) · gn (r) Hn ,

m = 1, . . . , NE

(15.260)

n=1 S

P 

Cmn

where the explicit appearance of terms such as ∇ × en (r) in (15.260) clarifies why the basis functions en (r) must be curl-conforming. It is worthwhile noticing that the testing procedure (14.18) has also ‘moved’ one of the curl operators onto the weighting function em (r). As a result, the original secondorder differential operator appearing in LPP {•} has been turned into a combination of first-order differential ones, and this welcome side effect allows relaxing the requirements on the smoothness of the basis and test functions en (r). A moment’s thought helps us realize that the surface integrals over S P in (15.260) constitute the entries of a matrix which is the transpose of [C], already encountered in (15.254) and (15.255). Such connection can be made even more explicit by writing (15.260) succinctly in matrix form as [M][E] = j ωμ0 [C]T [H]

(15.261)

where the superscript ‘T ’ signifies matrix transposition. In the context of the Finite Elements Method, [M] is usually referred to as the stiffness matrix [20,25], [17, Chapter 6], [18, Chapter 9]. Since [M] is a square matrix of rank NE , (15.261) provides the additional NE equations which allow determining [A], [P], [H] and [E] consistently as a function of the generator strength VG in turn concealed in the entries of [F A ]. For instance, formally solving (15.261) for [E], plugging the result into (15.255) and (15.252), and lastly performing a little algebra yields ⎛ ⎞⎛ ⎞ ⎞ ⎛ [0NA ,NP ] ⎜⎜⎜ [LAA ] [LAP ] ⎟⎟⎟ ⎜⎜⎜ [A] ⎟⎟⎟ ⎜⎜⎜ [F A ] ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎟ ⎜⎜⎜ ⎜⎜⎜ [E PA ] [E PP ] j ωμ0 [C][M]−1 [C]T ⎟⎟⎟ ⎜⎜⎜ [P] ⎟⎟⎟ = − ⎜⎜⎜[0NP ]⎟⎟⎟⎟⎟ (15.262) ⎜⎜⎝ ⎟⎟⎠ ⎜⎜⎝ ⎟⎟⎠ ⎟⎟⎠ ⎜⎜⎝ [HPA ] [HPP ] −[G PP ] [0NP ] [H]

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Advanced Theoretical and Numerical Electromagnetics

where now the system matrix neatly incorporates the effect of the plasma (or any dielectric medium) present within VP . This rearrangement is particularly advantageous if one is interested in studying the device of Figure 13.24a by changing the shape of the feeder while keeping the physical properties of the material in VP and the shape thereof fixed. Since the permittivity ε˜ c (r) only affects the calculation of the matrix [M], the block [C][M]−1 [C]T must be computed once and for all. Likewise, if the shape of VP is not supposed to vary, the blocks [E PP ], [HPP ] and [G PP ] must be filled the first time only. On the contrary, if the focus is on the performance of the device for different properties of the medium within VP , it is more convenient to favor the calculation of [E] through (15.261) after eliminating [H] with the aid of (15.243), (15.251) and (15.252). If the basis functions fn (r), gn (r) and en (r) are dimensionless, then the coefficients An , Pn , Hn and En have the same physical dimension as the vector field they are meant to represent through the expansions (15.237)-(15.240). Because of the way we wrote down the integral and differential equations for the problem of Figure 13.24a, the entries of the system matrices in (15.243) and (15.262) do not have the same physical dimensions. In particular, we observe that • • • •

LAA,mn , LAP,mn , E PA,mn and E PP,mn represent an impedance times an area (Ωm2 ), HPA,mn and HPP,mn represent an area (m2 ), Cmn is an area (m2 ), Mmn is a length (m).

One way to achieve dimensional balance and reduce the condition number [12] is to employ left ([P2 ]) and right ([P1 ]) splitting matrices [13, Chapter 13], [14, Chapter 9], as we did in (15.221). More specifically, we define ⎛√ ⎞ [0NA ,NP ] ⎟⎟ ⎜⎜⎜ Z0 [INA ] [0NA ,NP ] ⎟⎟⎟ √ ⎜⎜ [P1 ] := ⎜⎜⎜⎜⎜ [0NP ,NA ] (15.263) Z0 [INP ] [0NA ,NP ] ⎟⎟⎟⎟⎟ ⎜⎝ ⎟⎠ √ [0NA ,NP ] [0NA ,NP ] Z0 [INP ] ⎛√ ⎞ [0NA ,NP ] ⎟⎟⎟ ⎜⎜⎜ Z0 [INA ] [0NA ,NP ] ⎜⎜⎜ ⎟⎟ √ Z0 [INP ] [0NA ,NP ] ⎟⎟⎟⎟⎟ [P2 ] := ⎜⎜⎜⎜ [0NP ,NA ] (15.264) ⎜⎝ √ ⎟⎠ [0NA ,NP ] [INP ]/ Z0 [0NA ,NP ] where Z0 := (μ0 /ε0 )1/2 is the intrinsic impedance of free space, and [Iα ], α ∈ {NA , NP }, is the identity matrix of order α. Applying the pre-conditioning procedure embodied in (15.221) to (15.262) yields √ ⎞ ⎛ ⎞ ⎛√ ⎞ ⎛ [0NA ,NP ] ⎜⎜⎜ [LAA ]/Z0 [LAP ]/Z0 ⎟⎟⎟ ⎜⎜⎜ Z0 [A] ⎟⎟⎟ ⎜⎜⎜ [F A ]/ Z0 ⎟⎟⎟ ⎜⎜⎜⎜ ⎟⎟ ⎜⎜ √ ⎟⎟ ⎟⎟ ⎜⎜ (15.265) ⎜⎜⎜[E PA ]/Z0 [E PP ]/Z0 j k0 [C][M]−1 [C]T ⎟⎟⎟⎟⎟ ⎜⎜⎜⎜⎜ Z0 [P] ⎟⎟⎟⎟⎟ = − ⎜⎜⎜⎜⎜ [0NP ] ⎟⎟⎟⎟⎟ ⎜⎝ ⎟⎠ ⎜⎝√ ⎟⎠ ⎟⎠ ⎜⎝ [0NP ] [HPA ] [HPP ] −[G PP ] Z0 [H] where all the elements of the system matrix carry the dimension of an area. Having laid down the groundwork for the numerical solution of (13.257), (13.261), (13.262) and (13.263) with the Method of Moments, we briefly discuss the necessary modifications when the shapes of S A , S P and VP do not lend themselves to the construction of ad hoc entire-domain basis and test functions. As regards the surface integral equations enforced on S A and S P , we resort to modelling the relevant boundaries by means of suitable tessellations made up of simple-shaped patches, e.g., triangles. In which case, NA and NP subsectional basis functions, such as the RWG basis functions (Section 14.7) may be associated with the inner edges of the meshes. The specific form of LAA,mn , LAP,mn , E PA,mn , E PP,mn , HPA,mn , HPP,mn and G PP,mn is easily deduced with the aid of

The Method of Moments II

1067

the analogous derivations presented in Sections 14.2, 14.3 and 14.10. Contrariwise, the construction of subsectional basis functions en (r) for the discretization of (13.263) deserves a detailed discussion of its own, which we lay out in Section 15.7.

15.7 Edge elements for the vector wave equation In the general description of the Method of Moments in Section 14.1 we remarked that the selection of the basis functions for an electromagnetic problem should be inspired somehow by the known physical properties of the field (scalar or vectorial) which is to be expanded in accordance with (14.3). Since so far we have mostly dealt with surface and volume integral equations, we have used RWG (Section 14.7) and SWG (Section 15.2) basis functions, respectively, in order to represent surface current densities (e.g., JS and J MS ) and the flux densities D and B. In this regard, it is expedient to recall that •

•

the RWG basis functions [26] are associated with pairs of adjacent triangles and, in addition to being divergence-conforming, they enforce, by construction, the continuity of the normal component of the surface field across the edge shared by the two triangles in question (Figure 14.4); when the SWG basis functions [2] are associated with two adjacent tetrahedra, then by construction, they enforce the continuity of the normal component of the vector field across the triangle shared by the two tetrahedra in question (Figure 15.1).

It is precisely the aforementioned features that, in view of the jump conditions (1.201), (1.198) and (1.199), make the RWG and the SWG functions well-suited to expand surface current densities and electric or magnetic flux densities. In like manner, we ask ourselves what properties to require of the subsectional basis functions NE in order for them to be good candidates for the expansion (15.240) of the electric field E2 (r) {en (r)}n=1 in the wave equation (13.263). Before delving into the topic, we begin by replacing the region VP of Figure 13.24b with a volumetric mesh V ME comprised of ME adjoining tetrahedra W p , p = 1, . . . , ME . As always, the number of sub-domains and the number of basis functions may be different. The choice of a tetrahedral mesh is quite necessary, given that it makes the piecewise-flat boundary of V ME coincide with the triangular tessellation S MP which is introduced to model S P := ∂VP and define NP of RWG basis functions to be used in (15.238) and (15.239). The functions en (r) the set {gn (r)}n=1 will be non-zero on a certain number of adjacent tetrahedra of the mesh V ME . We already pointed out that en (r) must be a curl-conforming vector field. Evidently, this constraint rules out the usage of SWG basis functions because, as we showed in Section 15.2, they are not curl-conforming. Worse yet, we have just noted above that the SWG functions would enforce ˆ the continuity of n(r) · E2 (r) across the facet shared by two neighboring tetrahedra — an occurrence which is not verified in general. A quick look at the matching condition (1.198) will help us understand where the trouble lies. Even if the medium in the region VP (which is now modelled by the mesh V ME ) is a plasma, the free charges are accounted for by the macroscopic complex permittivity ε˜ c (r) defined in (13.242), so that ρS (r) = 0 at the interface between any two neighboring tetrahedra in V ME . However, it may well be the case that, if the medium is inhomogeneous, we have to assign two different values of ε˜ c (r) to each of the tetrahedra in question, whereby (1.198) becomes ˆ · E2 (r) = ε˜ c (rq )n(r) ˆ · E2 (r), ε˜ c (r p )n(r)

r ∈ ∂W p ∩ ∂Wq

(15.266)

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Advanced Theoretical and Numerical Electromagnetics

ˆ where r p and rq , with p, q ∈ {1, . . . , ME }, are suitable points within W p and Wq , and n(r) is the unit normal to ∂W p ∩ ∂Wq . The intersection of ∂W p and ∂Wq is the common triangular facet across ˆ · D2 (r) is, in fact, expected to be continuous. Regrettably, though, using SWG functions which n(r) ˆ to expand E2 (r) — rather than D2 (r) — would clearly force n(r) · E2 (r) to be continuous across ∂W p ∩ ∂Wq , a condition which, in view of (15.266), is only correct so long as ε˜ c (r p ) ≡ ε˜ c (rq ), i.e., for a homogeneous medium! Having extensively motivated why the SWG functions are unsuited for the task of concern, we seek basis functions en (r) which, by construction, can satisfy the matching condition (1.197) across the edges and facets shared by neighboring tetrahedra in the mesh V ME . For this reason, it is better to think of en (r) as being associated with the edge γn of the mesh, which is why the basis functions en (r) are also called edge elements. Owing to the greater complexity of a three-dimensional mesh, a single edge γn is likely shared by two or more tetrahedra (see, for instance, Figure 15.1 and, just for reference, compare with the simpler, two-dimensional case in Figure 14.4). Therefore, in order to describe en (r) and analyze the properties thereof, we focus on the behavior of six elementary vector fields defined within a tetrahedron, say, W p . The number of such fields is six because each one of them is conceptually associated with one of the edges of W p (Figure 15.7). In the context of the Finite Element Methods, said fields are called shape functions [20–25], [17, Chapter 6]. Now, all the shape functions that belong to adjoining tetrahedra and are associated with the same common edge γn concur to the definition of the desired basis function en (r). To study the shape functions, we dispense with the subscript p for the moment and with the help of Figure 15.7 we introduce the following geometrical quantities related to a tetrahedron W: • • • • • • • • • •

γil , with i = 1, . . . , 3, l = i + 1, . . . , 4, the six edges of W; T i , with i = 1, . . . , 4, the four triangles that form ∂W; Vi , with i = 1, . . . , 4, the four vertices of the tetrahedron; by convention, V4 indicates the vertex that does not belong to the triangle T 4 , and so forth; ri , with i = 1, . . . , 4, the position vectors of the vertices Vi ; wil := |rl − ri |, with i = 1, . . . , 3, l = i + 1, . . . , 4, the lengths of the edges γil ; Ai , with i = 1, . . . , 4, the areas of the four facets T i ; VW , the volume of the tetrahedron; hi := VW /(3Ai ), with i = 1, . . . , 4, the heights of the tetrahedron with respect to the facets T i ; in other words, hi is the distance of the vertex Vi from T i ; nˆ i , with i = 1, . . . , 4, the unit vectors normal to the facets T i ; the normals are positively oriented towards R3 \ W; sˆil , the six unit vectors tangential to the edges γil and given by sˆil :=

rl − ri , wil

i = 1, . . . , 3,

l = i + 1, . . . , 4.

(15.267)

The six shape functions defined for r ∈ W are more easily formulated in terms of the normalized volume coordinates ξi , i = 1, . . . , 4, introduced in Section 15.3 and graphically explained in Figure 15.4. In symbols, we have [3, Section 9.7], [27] ⎧ ⎪ ⎪ i = 1, . . . , 3, ⎨wil (ξi ∇ξl − ξl ∇ξi ), r ∈ W hil (r) := ⎪ (15.268) ⎪ ⎩0, l = i + 1, . . . , 4 rW where the variables ξi are explicitly related to the position vectors r and ri , e.g., by (15.82)-(15.84) and (15.86). The notation hil (r) reminds us that the shape function in question is a vector field associated with the straight segment γil which connects the vertex Vi to the vertex Vl , with l > i. The

The Method of Moments II

1069

Figure 15.7 Tetrahedron W and geometrical quantities for the definition of the shape functions hil (r), i = 1, . . . , 3, l = i + 1, . . . , 4, and curl-conforming edge elements. shape functions (15.268) are dimensionless in that wil is a length, but the gradients of ξi and ξl carry the physical dimension of the inverse of a length. Since ξi is a linear function of the position vector r, the gradient thereof is a constant vector perpendicular to the facet T i . For the proof, we focus on h12 (r) and observe that (Figure 15.7) (r − r2 ) · (r3 − r2 ) × (r4 − r2 ) A1 (r2 − r) · nˆ 1 = (r2 − r) · nˆ 1 = 6VW 3VW h1 (r − r1 ) · (r4 − r1 ) × (r3 − r1 ) A2 (r1 − r) · nˆ 2 = (r1 − r) · nˆ 2 = ξ2 := 6VW 3VW h2

ξ1 :=

whence, by using Cartesian coordinates and (A.26) we obtain ' ( ∂ 1 ∂ ∂ xˆ (r2 − r) · nˆ 1 + yˆ (r2 − r) · nˆ 1 + zˆ (r2 − r) · nˆ 1 ∇ξ1 = h1 ∂x ∂y ∂z 1 1 = − (ˆxxˆ · nˆ 1 + yˆ yˆ · nˆ 1 + zˆ zˆ · nˆ 1 ) = − nˆ 1 h1 h1 ' ( ∂ 1 ∂ ∂ xˆ (r1 − r) · nˆ 2 + yˆ (r1 − r) · nˆ 2 + zˆ (r1 − r) · nˆ 2 ∇ξ2 = h2 ∂x ∂y ∂z 1 1 = − (ˆxxˆ · nˆ 2 + yˆ yˆ · nˆ 2 + zˆ zˆ · nˆ 2 ) = − nˆ 2 h2 h2

(15.269) (15.270)

(15.271)

(15.272)

by virtue of (E.29). (An obvious exchange of indices yields the results for ∇ξ3 and ∇ξ4 .) To shed light on the behavior of the components of hil (r) that are tangential to the edges of W for observation points r on the edges, we stick to h12 (r) for the sake of argument and examine the six possible cases: (1)

if the observation point P — identified by the position vector r — belongs to the edge γ12 , as is suggested in Figure 15.8a, we have ξ3 = ξ4 = 0, ξ1 + ξ2 = 1, and sˆ12 · h12 (r) = −

ξ1 ξ2 (r2 − r1 ) · nˆ 2 + (r2 − r1 ) · nˆ 1 = ξ1 + ξ2 = 1 h2  h1  −h2

h1

(15.273)

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Advanced Theoretical and Numerical Electromagnetics

(a) r ∈ γ12

(b) r ∈ γ13

(c) r ∈ γ14

(d) r ∈ γ23

(e) r ∈ γ24

(f) r ∈ γ34

Figure 15.8 For studying the edge element h12 (r) when the point P (identified by the position vector r) belongs to one of the six edges γil , i = 1, . . . , 3, l = i + 1, . . . , 4.

The Method of Moments II

(2)

on account of (15.267) and (15.268); if the observation point P belongs to γ13 (Figure 15.8b) we have ξ2 = ξ4 = 0, ξ1 + ξ3 = 1, and sˆ13 · h12 (r) = −w12 ξ1 sˆ13 ·

(3)

1 nˆ 2 = 0 h2

(15.275)

1 nˆ 1 = 0 h1

(15.276)

because sˆ23 and nˆ 1 are perpendicular by construction; if the observation point P belongs to γ24 (Figure 15.8e) we have ξ1 = ξ3 = 0, ξ2 + ξ4 = 1, and sˆ24 · h12 (r) = w12 ξ2 sˆ24 ·

(6)

(15.274)

because sˆ14 and nˆ 2 are perpendicular by construction; if the observation point P belongs to γ23 (Figure 15.8d) we have ξ1 = ξ4 = 0, ξ2 + ξ3 = 1, and sˆ23 · h12 (r) = w12 ξ2 sˆ23 ·

(5)

1 nˆ 2 = 0 h2

because sˆ13 and nˆ 2 are perpendicular by construction; similarly, if the observation point P belongs to γ14 (Figure 15.8c) we have ξ2 = ξ3 = 0, ξ1 + ξ4 = 1, and sˆ14 · h12 (r) = −w12 ξ1 sˆ14 ·

(4)

1071

1 nˆ 1 = 0 h1

(15.277)

because sˆ24 and nˆ 1 are perpendicular by construction; finally, if the observation point P belongs to γ34 (Figure 15.8f) we have ξ1 = ξ2 = 0, ξ3 + ξ4 = 1, and sˆ34 · h12 (r) = 0

(15.278)

because, in accordance with (15.268), h12 (r) vanishes identically for r ∈ γ34 . In summary, the tangential component of h12 (r) is unitary on γ12 and vanishes on the five remaining edges of W. These conclusions can be extended in a straightforward manner to the other five shape functions defined in W by means of (15.268). More importantly, since γ12 ≡ γn may actually be shared by a number of adjoining tetrahedra, all the shape functions associated with γn — and hence forming en (r) — satisfy (15.273). As a result, we conclude that the edge element en (r) obeys the condition [28] sˆn · en (r) = 1,

r ∈ γn

(15.279)

where sˆn denotes the unit vector tangential to γn . This property guarantees that the matching condition for the electric field (1.197) is intrinsically enforced along γn . Next, we study the behavior of the tangential component nˆ i ×h12 (r), i = 1, . . . , 4, for observation points on the four facets of W. (1)

For P ∈ T 4 we examine the vectors 1 1 nˆ 4 × nˆ 1 = nˆ 4 × [(r3 − r2 ) × (r4 − r2 )] h1 6VW 1 1 nˆ 4 · (r4 − r2 )(r3 − r2 ) − nˆ 4 · (r3 − r2 )(r4 − r2 ) = 6VW 6VW 

nˆ 4 × ∇ξ1 = −

=0

1072

Advanced Theoretical and Numerical Electromagnetics 1 nˆ 4 · (r4 − r1 )(r3 − r2 ) 6VW 1 1 nˆ 4 × ∇ξ2 = − nˆ 4 × nˆ 2 = nˆ 4 × [(r4 − r1 ) × (r3 − r1 )] h2 6VW 1 1 nˆ 4 · (r3 − r1 )(r4 − r1 ) − nˆ 4 · (r4 − r1 )(r3 − r1 ) =  6VW 6VW =

(15.280)

(15.281)

=0

having used (H.14), (15.271) and (15.272), and observed that nˆ 4 · (r1 − r2 ) = 0. Then, we form the cross product   w12 nˆ 4 · (r4 − r1 ) ξ1 (r3 − r1 ) + ξ2 (r3 − r2 ) 6VW w12 = nˆ 4 · (r4 − r1 )(r − r3 ), r ∈ T4 6VW

nˆ 4 × h12 (r) = −

(15.282)

where the result follows by noticing that, when r ∈ T 4 , the volume coordinate ξ4 vanishes and we have ξ1 (r1 − r3 ) + ξ2 (r2 − r3 ) = r − r3

(15.283)

thanks to constraint (15.74). Lastly, we observe that nˆ 4 =

1 (r2 − r1 ) × (r3 − r1 ) 2A4

(15.284)

and go on to write nˆ 4 × h12 (r) =

w12 (r2 − r1 ) × (r3 − r1 ) · (r4 − r1 )(r − r3 ) 12A4 VW 

w12 (r3 − r), = 2A4 (2)

r ∈ T4

w12 (r − r4 ), 2A3

r ∈ T 3.

(15.286)

For P ∈ T 2 we have ξ2 = 0 by construction and nˆ 2 × h12 (r) = nˆ 2 × w12 ξ1 ∇ξ2 = −

(4)

(15.285)

which is the desired result. For P ∈ T 3 the calculations are similar to those carried out in the previous case, and we find nˆ 3 × h12 (r) =

(3)

−6VW

w12 nˆ 2 × nˆ 2 = 0, h2

r ∈ T2

(15.287)

r ∈ T1

(15.288)

by virtue of (15.272). For P ∈ T 1 we have ξ1 = 0 by construction and nˆ 1 × h12 (r) = −nˆ 1 × w12 ξ2 ∇ξ1 = by virtue of (15.271).

w12 nˆ 1 × nˆ 1 = 0, h1

The Method of Moments II

1073

The tangential component of h12 (r) vanishes on T 1 and T 2 whereas it is non-null on T 3 , T 4 , i.e., the two facets of W that share the common edge γ12 . More importantly, it is apparent from (15.285) that the result depends only on geometrical quantities pertinent to the triangle T 4 . Therefore, an expression of the form (15.285) must hold for the adjacent tetrahedron which shares γ12 and T 4 with the one of concern. The same observation applies to the shape functions which share γ12 and T 3 . All in all, this means that the tangential component of the edge element en (r) associated with γn ≡ γ12 is continuous across the facets of the tetrahedra which share γn . As a result, when the edge elements en (r) are used to represent an electric field, as in (15.240), the matching condition (1.197) is intrinsically satisfied [29]. As regards the component of h12 (r) that is perpendicular to the facets of W, we examine the scalar functions nˆ i · h12 (r) for r ∈ T i , i = 1, . . . , 4. (1)

When P ∈ T 4 , from (15.269) and (15.270) we obtain ξ1 =

1 nˆ 1 · (r3 − r), h1

ξ2 =

1 nˆ 2 · (r3 − r), h2

r ∈ T4

(15.289)

because r2 − r3 and r1 − r3 are perpendicular to nˆ 1 and nˆ 2 , respectively. Using these expressions along with (15.271) and (15.272) into the general definition (15.268) yields   w12 nˆ 1 nˆ 2 nˆ 4 · h12 (r) = w12 nˆ 4 · nˆ 4 · (nˆ 1 nˆ 2 − nˆ 2 nˆ 1 ) · (r3 − r) ξ2 − ξ1 = h1 h2 h1 h2 w12 nˆ 4 × (nˆ 2 × nˆ 1 ) · (r3 − r), = r ∈ T4 (15.290) h1 h2 (2)

on account of (H.14). When P ∈ T 3 , from (15.269) and (15.270) we find ξ1 =

1 nˆ 1 · (r4 − r), h1

ξ2 =

1 nˆ 2 · (r4 − r), h2

r ∈ T3

(15.291)

since r2 − r4 and r1 − r4 are perpendicular to nˆ 1 and nˆ 2 , respectively. By inserting these expressions into the general definition (15.268) we arrive at nˆ 3 · h12 (r) = (3)

w12 nˆ 3 × (nˆ 2 × nˆ 1 ) · (r4 − r), h1 h2

(15.292)

When P ∈ T 2 , ξ2 = 0, and we have nˆ 2 · h12 (r) = w12 nˆ 2 · ξ1 ∇ξ2 =

(4)

r ∈ T3.

w12 (r − r2 ) · nˆ 1 , h1 h2

r ∈ T2

(15.293)

from (15.269) and (15.272). When P ∈ T 1 , ξ1 = 0, and we find nˆ 1 · h12 (r) = −w12 nˆ 1 · ξ2 ∇ξ1 =

w12 (r1 − r) · nˆ 2 , h1 h2

r ∈ T1

(15.294)

from (15.270) and (15.271). The scalar fields nˆ i · h12 (r) do not vanish on the four facets of W. Evidently, the result for r ∈ T i depends also on geometrical quantities other than those pertinent to the facet in question. Therefore, for a given facet shared by two adjacent tetrahedra we obtain different results in general. This implies that the normal components of the edge element en (r) associated with γn ≡ γ12 suffer jumps across the facets of the tetrahedra that have γn in common. Consequently, even if the medium

1074

Advanced Theoretical and Numerical Electromagnetics

filling the mesh V ME is homogeneous, the edge elements en (r) cannot satisfy the matching condition (15.266) [29]. In view of the very definition (15.268) the curl and the divergence of h12 (r) for r ∈ W exist as ordinary fields. In symbols, we have ∇ × h12 (r) = w12 ∇ × (ξ1 ∇ξ2 − ξ2 ∇ξ1 ) = 2w12 ∇ξ1 × ∇ξ2 =

2w12 nˆ 1 × nˆ 2 , h1 h2

∇ · h12 (r) = w12 ∇ · (ξ1 ∇ξ2 − ξ2 ∇ξ1 )   = w12 ∇ξ1 · ∇ξ2 − ξ1 ∇2 ξ2 − ∇ξ2 · ∇ξ1 + ξ2 ∇2 ξ1 = 0,

r∈W

r∈W

(15.295)

(15.296)

by virtue of (15.271), (15.272). We conclude this part by observing that it is possible to express the shape functions (15.268) in terms of position vectors only [28]. The alternative representation may come in handy for the calculation of the surface integrals in (15.254) and (15.260). With reference to the shape function h12 (r) given by (15.268) for i = 1 and l = 2, we write for r ∈ W w12 A1 A2 w12 A2 A1 nˆ 1 · (r − r4 ) nˆ 2 − nˆ 2 · (r − r4 ) nˆ 1 3VW 3VW 3VW 3VW w12 A1 A2 = (nˆ 1 × nˆ 2 ) × (r − r4 ) 2 9VW

h12 (r) =

(15.297)

by virtue of (15.269)-(15.272) and the algebraic identity (H.14). Then, with the aid of Figure 15.7 we observe 2A1 nˆ 1 × 2A2 nˆ 2 = [(r3 − r4 ) × (r2 − r4 )] × [(r1 − r4 ) × (r3 − r4 )] = − (r3 − r4 ) × (r2 − r4 ) · (r1 − r4 )(r3 − r4 ) = 6VW (r3 − r4 ) 

(15.298)

−6VW

on account of (H.14) and (H.13). Substituting back into (15.297) yields ⎧ w12 ⎪ ⎪ ⎪ ⎨ 6VW (r3 − r4 ) × (r − r4 ), r ∈ W h12 (r) = ⎪ ⎪ ⎪ ⎩0, r  W.

(15.299)

In order to write en (r) formally and examine curl and divergence thereof we first introduce the subsets of indices

 n = 1, . . . , NE (15.300) In := p ∈ {1, . . . , ME } : W p ∩ γn  ∅ , that is, In contains the indices of the adjacent tetrahedra which, having the edge γn in common, form the support of en (r). Hence, we may define the nth edge element as ⎧ ⎪ ⎪ ⎨hnp (r), r ∈ W p , p ∈ In en (r) := ⎪ (15.301) ⎪ ⎩0, r ∈ V M E \ Dn where the symbol hnp (r) now indicates the shape function which is associated with the edge γn and is non-zero for r ∈ W p , and / Wp, n = 1, . . . , NE (15.302) Dn := p∈In

The Method of Moments II

1075

is a shorthand for the support of en (r). For the calculation of the curl we start from the weak definition (15.51) for r ∈ V ME . By recalling that en (r) vanishes outside Dn ⊂ V ME we have  ∇ × en (r) := −

dV ∇φ(r) × en (r) = −



dV φ(r)∇ × hnp (r) −

p∈In W p

=

dV ∇φ(r) × hnp (r)

p∈In W p

Dn

=

 

! " dV ∇ × φ(r)hnp (r)

p∈In W p



dV φ(r)∇ × hnp (r) −

p∈In W p

 p∈In

ˆ × hnp (r)φ(r) dS n(r)

(15.303)

∂W p

having applied the curl theorem (H.91) in the last step because, according to (15.295), the shape function hnp (r) is differentiable within W p . The linear combination of surface integrals can be further split into contributions over the facets of W p , for p ∈ In . However, the net result of the sum is zero, ˆ × hnp (r) = 0 on account of formulas such as (15.288), or the two contributions because either n(r) over shared facets cancel out by virtue of formulas such as (15.285). Moreover, each one of the remaining domain integrals represents the curl of hnp (r) within W p , and since ∇ × hnp (r) exists as an ordinary function for r ∈ W p we can write (15.303) alternatively as ⎧ ⎪ ⎪ ⎨∇ × hnp (r), r ∈ W p , p ∈ In ∇ × en (r) := ⎪ ⎪ ⎩0, r ∈ V M E \ Dn

(15.304)

that is, the curl of the edge elements (15.301) exists in the sense of ordinary functions despite en (r) being generally discontinuous across ∂W p , p ∈ In . In other words, the vector field defined by (15.301) is curl-conforming as desired. The explicit expression of ∇ × en (r) — which is piecewise-constant — is obtained with the aid of (15.295). Next, to compute the divergence we employ the weak definition (15.42) for r ∈ V ME . Since en (r) is non-zero only for r ∈ Dn we have  ∇ · en (r) := −

dV ∇φ(r) · en (r) = −

p∈In W p

Dn

=



dV φ(r)∇ · hnp (r) −

p∈In W p

=−

 p∈In

 

dV ∇φ(r) · hnp (r) ! " dV ∇ · φ(r)hnp (r)

p∈In W p

ˆ · hnp (r)φ(r) dS n(r)

(15.305)

∂W p

where we have used (15.296), (H.51) and the Gauss theorem (A.53). The linear combination of flux integrals does not vanish in general inasmuch as the normal components of the shape functions suffer jumps across the facets shared by neighboring tetrahedra. Therefore, we conclude that ∇·en (r) is comprised of surface delta distributions (Appendix C) with supports over the facets of W p with p ∈ In , and en (r) is not divergence-conforming.

1076

Advanced Theoretical and Numerical Electromagnetics

Thanks to (15.301) and (15.304) the expression of the entries of the stiffness matrix [M] in (15.260) is formally modified as  ! " dV ∇ × em (r) · ∇ × en (r) − ω2 μ0 ε˜ c (r)em (r) · en (r) , m, n ∈ {1, . . . , NE } (15.306) Mmn := Dm ∩Dn

which shows that Mmn does not vanish only when the supports of basis and test functions happen to be at least partially overlapped. As a result, unlike the case of entire-domain basis functions, [M] is extremely sparse, and efficient techniques may be used for handling the linear system (15.261) [30, Chapter 1], [31, Chapters 3, 4], [3]. NP are defined over the surface mesh S MP ≡ ∂V ME , the When RWG basis functions {gm (r)}m=1 entries Cmn of the coupling matrix [C] in (15.254) become  Cmn = dS gm (r) · en (r) (15.307) Ξm ∩∂Dn

where Ξm indicates the pair of adjacent triangles which form the support of gm (r) (Figure 14.4). This integral may be non-null only when the support Dn of en (r) consists of at least one tetrahedron with a facet which is also part of Ξm . The calculation of (15.307) may be further broken up into a number of basic integrals which involve the shape functions (15.268) and the shifted position vector r − r3 over the facet T = ∂W4 of a tetrahedron W (Figure 15.7), say,  w12 dS (r − r3 ) · hil (r), i = 1, . . . , 4, l = i + 1, . . . , 4 (15.308) Iil := 2A4 T

where w12 is the length of the edge opposite r3 and A4 is the area of T , the facet opposite the vertex V4 of W. In this way, the leading multiplicative factor expresses the normalization constant in (14.116) in terms of geometrical quantities relevant to W. We begin by showing that out of the six shape functions in (15.268) only h13 (r) and h23 (r), in fact, lead to a non-null value of the integrals Iil . In particular, I13 and I23 are proportional to the integral of r − r3 over T . (1)

For i = 1 and l = 2 we make use of the alternative expression (15.297)  w212 A1 A2 I12 = dS (r − r3 ) × (r4 − r) · (nˆ 1 × nˆ 2 ) 2 18A4 VW T  w212 A1 A2 ˆ ˆ = (n1 × n2 ) × (r3 − r4 ) · dS (r − r3 ) = 0 2 18A4 VW

(15.309)

T

(2)

where the result follows with the aid of (H.14) by noticing that the edge vector r3 − r4 is orthogonal to both nˆ 1 and nˆ 2 (Figure 15.7). For i = 1 and l = 3 we first obtain the representation h13 (r) =

w13 A1 A3 (r4 − r) × (nˆ 1 × nˆ 3 ), 2 9VW

r∈W

(15.310)

from (15.268) and expressions of ξ3 and ∇ξ3 analogous to (15.269)-(15.272). Then, we compute  w12 w13 A1 A3 dS (r − r3 ) × (r4 − r) · (nˆ 1 × nˆ 3 ) I13 = 2 18A4 VW T

The Method of Moments II  w12 w13 A1 A3 ˆ ˆ ( n × n ) × (r − r ) · dS (r − r3 ) 1 3 3 4 2 18A4 VW T  w12 w13 A1 A3 (r4 − r3 ) · nˆ 3 nˆ 1 · dS (r − r3 ) = 2 18A4 VW

1077

=

(15.311)

T

(3)

where the last step follows from (H.14) and (r3 − r4 ) · nˆ 1 = 0. For i = 1 and l = 4 in like manner we obtain the expression h14 (r) =

w14 A1 A4 (r3 − r) × (nˆ 1 × nˆ 4 ), 2 9VW

r∈W

(15.312)

whereby I14 =

(4)

w12 w14 A1 A4 2 18A4 VW

 dS (r − r3 ) × (r3 − r) · (nˆ 1 × nˆ 4 ) = 0

(15.313)

T

the integrand being obviously null. For i = 2 and l = 3 we have h23 (r) =

w23 A2 A3 (r4 − r) × (nˆ 2 × nˆ 3 ), 2 9VW

whence I23 =

w12 w23 A2 A3 2 18A4 VW

r∈W

(15.314)

 dS (r − r3 ) × (r4 − r) · (nˆ 2 × nˆ 3 ) T

 w12 w23 A2 A3 ˆ ˆ ( n × n ) × (r − r ) · dS (r − r3 ) = 2 3 3 4 2 18A4 VW T  w12 w23 A2 A3 = (r4 − r3 ) · nˆ 3 nˆ 2 · dS (r − r3 ) 2 18A4 VW

(15.315)

T

(5)

with the last step following from (H.14) and (r4 − r3 ) · nˆ 2 = 0. For i = 2 and l = 4 we have h24 (r) =

w24 A2 A4 (r3 − r) × (nˆ 2 × nˆ 4 ), 2 9VW

whence I24

(6)

w12 w24 A2 A4 = 2 18A4 VW

r∈W

(15.316)

 dS (r − r3 ) × (r3 − r) · (nˆ 2 × nˆ 4 ) = 0

(15.317)

T

analogously to (15.313). For i = 3 and l = 4 we have h34 (r) =

w34 A3 A4 [(r3 − r) · nˆ 4 nˆ 3 − (r4 − r) · nˆ 3 nˆ 4 ] , 2 9VW

r∈W

(15.318)

Advanced Theoretical and Numerical Electromagnetics

1078

Figure 15.9 Close-up of the surface mesh S MP for visualizing the interaction of four RWG functions (➞) associated with the sides of Ξn ⊂ S MP and the edge element en (r) associated with the edge γn ⊂ S MP . whence I34 =

w12 w34 A3 2 18VW

 dS (r − r3 )·[(r3 − r) · nˆ 4 nˆ 3 − (r4 − r) · nˆ 3 nˆ 4 ] = 0

(15.319)

T

where we have noticed that the position vector r−r3 is perpendicular to nˆ 4 when r ∈ T = ∂W4 (Figure 15.7). To finalize the calculation of I13 and I23 we introduce area coordinates over T , though we prefer to adopt the symbols η1 and η2 to avoid confusion with the volume coordinates in (15.268). On account of definition (14.140) and the general change of variables (14.150) we have    1  r − r3 1 dS = dη1 dη2 η1 (r1 − r3 ) + η2 (r2 − r3 ) = (r1 − r3 ) + (r2 − r3 ) (15.320) 2A4 6 6 S2 T

where S 2 is the two-dimensional simplex drawn in Figure 14.6. By inserting this result into (15.311) and (15.315) and noticing that (r2 − r3 ) · nˆ 1 = 0 = (r1 − r3 ) · nˆ 2 (Figure 15.7) we arrive at I13 =

w12 w13 A1 A3 w12 w13 A1 h1 A3 h3 w12 w13 (r4 − r3 ) · nˆ 3 (r1 − r3 ) · nˆ 1 = − =− 2   4 3VW 3VW 4 36VW h3

I23

(15.321)

−h1

w12 w23 A2 A3 w12 w23 = (r4 − r3 ) · nˆ 3 (r2 − r3 ) · nˆ 2 = − 2  4 36VW

(15.322)

−h2

where hi , i = 1, 2, 3, denote the heights of W with respect to the facets ∂Wi . Remarkably, the integrals of interest depend only on the lengths of the edges of T = ∂W4 . Of course, if the part of the RWG function of concern is proportional to −(r − r3 ) [see (14.116) and (15.308)], we simply have to change sign to the results in the rightmost members of (15.321) and (15.322). In light of (15.321) and (15.322) it turns out that the coefficient Cmn is also null — even though Ξm ∩ ∂Dn  ∅ — when the RWG function gm (r) and the edge element en (r) are associated with the same edge γn ⊂ S MP . By the same token, if Ξm ∩∂Dn  ∅ the nth column of [C] will contain four nonzero entries that arise from the interaction of en (r) associated with γn and the four RWG functions associated with the four edges that form the boundary of the pair of triangles Ξn (Figure 15.9).

The Method of Moments II

1079

References [1] [2]

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[7]

[8]

[9]

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[20]

Van Bladel JG. Electromagnetic Fields. Piscataway, NJ: IEEE Press; 2007. Schaubert DH, Wilton DR, Glisson AW. A tetrahedral modeling method for electromagnetic scattering by arbitrarily shaped inhomogeneous dielectric bodies. IEEE Trans Antennas Propag. 1984;32(1):77–85. Peterson AF, Ray SL, Mittra R. Computational Methods for Electromagnetics. Piscataway, NJ: IEEE Press; 1998. Rudin W. Principles of mathematical analysis. 3rd ed. New York, NY: McGraw-Hill; 1976. Abramowitz M, Stegun IA. Handbook of mathematical functions. New York, NY: Dover Publications, Inc.; 1965. Lancellotti V, Tijhuis AG. Linear embedding via Green’s operators. In: Mittra R, editor. Computational Electromagnetics. New York, NY: Springer Science + Business Media; 2014. p. 227–257. Lancellotti V, Melazzi D. Hybrid LEGO-EFIE method applied to antenna problems comprised of anisotropic media. Forum in Electromagnetic Research Methods and Application Technologies (FERMAT). 2014;6:1–19. www.e-fermat.org. Lancellotti V, Tijhuis AG. Extended linear embedding via Green’s operators for analyzing wave scattering from anisotropic bodies. International Journal of Antennas and Propagation. 2014;11 pages, Article ID 467931. Wilton DR, Rao SM, Glisson AW, et al. Potential integrals of uniform and linear source distributions on polygonal and polyhedral domains. IEEE Trans Antennas Propag. 1984 March;32(3):276–281. Dwight HB. Tables of Integrals and Other Mathematical Data. 4th ed. New York, NY: MacMillan Publishing Co., Inc.; 1961. Melazzi D, Lancellotti V. ADAMANT: A Surface and Volume Integral-Equation Solver for the Analysis and Design of Helicon Plasma Sources. Computer Physics Communications. 2014;185:1914–1925. Golub GH, van Loan CF. Matrix Computations. Baltimore, MD: Johns Hopkins University Press; 1996. van der Vorst H. Iterative Krylov methods for large linear systems. Cambridge, UK: Cambridge University Press; 2003. Dongarra JJ, Duff IS, Sorensen DC, et al. Numerical linear algebra for high-performance computers. Philadelphia, PA: Society for Industrial and Applied Mathematics; 1998. Bau III D, Trefethen LN. Numerical linear algebra. Philadelphia, PA: Soci. Indus. Ap. Math.; 1997. Blyth TS, Robertson EF. Basic Linear Algebra. 2nd ed. Springer Undergraduate Mathematics Series. London, UK: Springer-Verlag; 2002. Sadiku MNO. Numerical techniques in electromagnetics. 2nd ed. Boca Raton, FL: CRC Press; 2001. Jin JM. Theory and Computation of Electromagnetic Fields. 2nd ed. Hoboken, NJ: IEEE Press; 2015. Notarˇos BM, Yan S. New trends in finite element methods. In: Özgür Ergül, editor. New Trends in Compuational Electromagnetics. ACES Series on Computational and Numerical Modelling in Electrical Engineering. London, UK: SciTech Publishing; 2019. p. 259–313. Zienkiewicz OC. The Finite Element Method in Engineering Science. London, UK: McGrawHill; 1971.

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Appendix A

Vector calculus

Starting with the representations of vectors and vector fields in the ordinary three-dimensional space, we recall the definition of the differential operators most commonly used in electromagnetism and, more generally, in physics. The notions of flux and circulation of a vector field are considered, and the fundamental integral theorems on bounded domains and open surfaces are reviewed. Then, a result concerning the derivative of a flux integral depending on a parameter is obtained.

A.1

Systems of coordinates

We specify the position of a point P in the three-dimensional space with the vector r = xˆx + yˆy + zˆz ∈ R3

(A.1)

where the real numbers x, y and z are the Cartesian components of r, and xˆ , yˆ and zˆ are the fundamental unit vectors aligned with the principal axes. The fundamental vectors form an orthogonal right-handed triple in the sense of (H.1), (H.2) and Figure H.1. The same position vector — and more generally any vector field — can be identified by giving the components thereof with respect to other triples of unit vectors, possibly orthogonal [1, Section 1.3], [2, Chapter 1], [3]. The two most-used systems of coordinates (besides the Cartesian one, that is) are the circular cylindrical and the polar spherical ones [4, Chapter 1], [5, Appendix II], [6, Appendix A.3].

A.1.1 Circular cylindrical coordinates In the circular cylindrical system, sketched in Figure A.1, we identify a point P in space by means of the triple of real numbers (ρ, ϕ, z) where • • •

ρ ∈ [0, +∞[ is the distance of the projection Q of P onto the xOy plane from the origin O or, equivalently, the distance of P from the z-axis, ϕ ∈ [0, 2π[ is the angle formed by the x-axis and the straight line passing through O and Q, z ∈ R is the signed distance of the point P from the xOy plane.

We decide that ϕ must not take on the value 2π because, otherwise, the triples of numbers (ρ, 0, z) and (ρ, 2π, z) would constitute a double representation of the same point in space. What is more, we could also choose the open interval [−π, π[ as the range of permitted values for ϕ, in which case we exclude π because (ρ, −π, z) and (ρ, π, z) identify the very same point. The relationships between the unit vectors are as follows ρˆ = xˆ cos ϕ + yˆ sin ϕ

ϕˆ = −ˆx sin ϕ + yˆ cos ϕ

zˆ = zˆ

(A.2)

Advanced Theoretical and Numerical Electromagnetics

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Figure A.1 Geometrical setup for the definition of the circular cylindrical coordinates. xˆ = ρˆ cos ϕ − ϕˆ sin ϕ

yˆ = ρˆ sin ϕ + ϕˆ cos ϕ

zˆ = zˆ .

(A.3)

Since points on the z-axis are identified by ρ = 0 and, of course, the z-coordinate, the angle ϕ becomes irrelevant or ignorable. As a consequence, the unit vectors ρˆ and ϕˆ are not unique on the z-axis. The canonical surfaces are [3, Figure 1.02] • • •

ˆ ρ = ρ0 > 0, a circular cylinder whose axis coincides with the z-axis and is perpendicular to ρ, ϕ = ϕ0 ∈ [0, 2π[, a half-plane with edge coinciding with the z-axis and perpendicular to ϕ, ˆ z = z0 ∈ R, a plane orthogonal to zˆ .

The transformation rules which allow passing from cylindrical to Cartesian coordinates and vice-versa are x = ρ cos ϕ 

ρ = x2 + y 2

1/2

y = ρ sin ϕ ⎧ y ⎪ ⎪ arctan ⎪ ⎪ ⎪ x ⎪ ⎪ ⎪ ⎪ y ⎨ +π arctan ϕ=⎪ ⎪ ⎪ x ⎪ ⎪ ⎪ ⎪ y ⎪ ⎪ ⎩arctan + 2π x

z=z

(A.4)

z=z

(A.5)

x > 0, y  0 x < 0, y ∈ R x > 0, y < 0

where the many-valued function arctan(•) is restricted to the interval [−π/2, π/2]. The Jacobi matrix [7, p. 234] pertinent to transformation (A.4) reads ⎛ ⎞  ⎜⎜cos ϕ −ρ sin ϕ 0⎟⎟ ⎜⎜⎜ ⎟⎟ ∂r ∂r ∂r [J] = , , = ⎜⎜⎜⎜ sin ϕ ρ cos ϕ 0⎟⎟⎟⎟⎟ ∂ρ ∂ϕ ∂z ⎝ ⎠ 0 0 1

(A.6)

whereby the differential elements of area on the canonical surfaces and the elemental volume are found to be   ∂r ∂r  ×  dϕ dz = ρ dϕ dz (A.7) dS ρ =  ∂ϕ ∂z

Vector calculus   ∂r ∂r  dS ϕ =  ×  dρ dz = dρ dz ∂z ∂ρ   ∂r ∂r    dρdϕ = ρ dρ dϕ dS z =  × ∂ρ ∂ϕ  dV = | det[J]| dρ dϕ dz = ρ dρ dϕ dz.

1083 (A.8) (A.9) (A.10)

A.1.2 Polar spherical coordinates In the polar spherical system, shown in Figure A.2, a point P in space is identified by means of the triple (r, ϑ, ϕ) where • • •

r ∈ [0, +∞[ is the distance of P from the origin O, ϑ ∈ [0, π] is the angle formed by the z-axis and the straight line passing through P and O, ϕ ∈ [0, 2π[ is the angle formed by the x-axis and the straight line passing through O and the projection Q of P onto the xOy plane. The relationships between the unit vectors are as follows rˆ = xˆ sin ϑ cos ϕ + yˆ sin ϑ sin ϕ + zˆ cos ϑ ϑˆ = xˆ cos ϑ cos ϕ + yˆ cos ϑ sin ϕ − zˆ sin ϑ

(A.11) (A.12) (A.13)

ϕˆ = −ˆx sin ϕ + yˆ cos ϕ xˆ = rˆ sin ϑ cos ϕ + ϑˆ cos ϑ cos ϕ − ϕˆ sin ϕ yˆ = rˆ sin ϑ sin ϕ + ϑˆ cos ϑ sin ϕ + ϕˆ cos ϕ

(A.14)

zˆ = rˆ cos ϑ − ϑˆ sin ϑ.

(A.16)

(A.15)

Since points on the z-axis are specified by setting ϑ = 0 or ϑ = π and, obviously, by giving the distance r from the origin, the angle ϕ becomes ignorable. As a special case, the origin itself (which also belongs to the z-axis) is identified by setting r = 0, in which instance both ϑ and ϕ are ignorable. Consequently, on the z-axis the unit vectors ϑˆ and ϕˆ are not defined univocally. In addition, all of the unit vectors become multiply defined in the origin O. The canonical surfaces are [3, Figure 1.05] • • •

r = r0 > 0, a sphere with center the origin and thus perpendicular to rˆ , ϑ = ϑ0 ∈ [0, π], a one-sheeted cone with apex in the origin, axis coinciding with the z-axis, and ˆ perpendicular to ϑ, ˆ ϕ = ϕ0 ∈ [0, 2π[, a half-plane with edge coinciding with the z-axis and perpendicular to ϕ.

The transformation rules which allow passing from spherical to Cartesian coordinates and viceversa are x = r sin ϑ cos ϕ  1/2 r = x2 + y2 + z2 ⎧ z ⎪ ⎪ z0 ⎪ ⎪arccos r ⎨ ϑ=⎪ ⎪ z ⎪ ⎪ ⎩arccos + π z < 0 r

y = r sin ϑ sin ϕ

z = r cos ϑ

(A.17) (A.18) (A.19)

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Figure A.2 Geometrical setup for the definition of the polar spherical coordinates. ⎧ y ⎪ ⎪ arctan ⎪ ⎪ ⎪ x ⎪ ⎪ ⎪ ⎪ y ⎨ arctan + π ϕ=⎪ ⎪ ⎪ x ⎪ ⎪ ⎪ y ⎪ ⎪ ⎪ ⎩arctan + 2π x

x > 0, y  0 x < 0, y ∈ R

(A.20)

x > 0, y < 0

where the many-valued functions arccos(•) and arctan(•) range from −π/2 to π/2. The Jacobi matrix [7, p. 234] pertinent to transformation (A.17) reads ⎛ ⎞ ⎜⎜sin ϑ cos ϕ r cos ϑ cos ϕ −r sin ϑ sin ϕ⎟⎟ ⎟⎟ ∂r ∂r ∂r ⎜⎜⎜⎜ [J] = , , = ⎜⎜⎜ sin ϑ sin ϕ r cos ϑ sin ϕ r sin ϑ cos ϕ ⎟⎟⎟⎟⎟ ∂r ∂ϑ ∂ϕ ⎝ ⎠ cos ϑ −r sin ϑ 0 

(A.21)

whereby the differential elements of area on the canonical surfaces and the elemental volume are found to be   ∂r ∂r   dϑdϕ = r2 sin2 ϑ dϑdϕ × dS r =  ∂ϑ ∂ϕ    ∂r ∂r  dS ϑ =  ×  dr dϕ = r sin ϑ dr dϕ ∂ϕ ∂r   ∂r  ∂r  dr dϑ = r dr dϑ dS ϕ =  × ∂r ∂ϑ  dV = | det[J]| dr dϑ dϕ = r2 sin2 ϑ dr dϑ dϕ.

(A.22) (A.23) (A.24) (A.25)

Vector calculus

A.2

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Differential operators

Hereinbelow we list the main differential operators expressed in Cartesian, circular cylindrical and polar spherical coordinates [8], [6, Appendix A.3], [9, Section 2.5], [10, Chapter 10], [11, Appendix 2], [12, Appendix VII.2]. We let F ∈ {R, C} and consider a scalar field Ψ(r) ∈ Cn (F3 ) as well as a vector field F(r) ∈ n 3 3 C (F ) with n ∈ N. The gradient of Ψ(r) is the vector field given by ∂Ψ ∂Ψ ∂Ψ + yˆ + zˆ ∂x ∂y ∂z ∂Ψ 1 ∂Ψ ∂Ψ = ρˆ + ϕˆ + zˆ ∂ρ ρ ∂ϕ ∂z ∂Ψ ˆ 1 ∂Ψ 1 ∂Ψ = rˆ +ϑ + ϕˆ . ∂r r ∂ϑ r sin ϑ ∂ϕ

grad Ψ(r) = ∇Ψ(r) := xˆ

(A.26) (A.27) (A.28)

The divergence of F(r) is the scalar field given by ∂F x ∂Fy ∂Fz + + ∂x ∂y ∂z 1 ∂ 1 ∂Fϕ ∂Fz = ( ρFρ ) + + ρ ∂ρ ρ ∂ϕ ∂z 1 ∂ 2 ∂ 1 1 ∂Fϕ = 2 (r Fr ) + (Fϑ sin ϑ) + . r sin ϑ ∂ϑ r sin ϑ ∂ϕ r ∂r

div F(r) = ∇ · F(r) :=

(A.29) (A.30) (A.31)

The curl of F(r) is the vector field given by curl F(r) = rot F(r) = ∇ × F(r)    ∂Fy ∂Fz ∂Fy ∂F x ∂F x ∂Fz := xˆ − − − + yˆ + zˆ ∂z ∂y ∂z ∂x ∂x ∂y    ∂Fρ ∂Fρ ∂Fz 1 ∂Fz ∂Fϕ zˆ ∂ − − (ρFϕ ) − = ρˆ + ϕˆ + ρ ∂ϕ ∂z ∂z ∂ρ ρ ∂ρ ∂ϕ  ˆ rˆ ∂ ϑ 1 ∂Fr ∂ ∂Fϑ (Fϕ sin ϑ) − − (rFϕ ) = + r sin ϑ ∂ϑ ∂ϕ r sin ϑ ∂ϕ ∂r  ϕˆ ∂ ∂Fr + (rFϑ ) − r ∂r ∂ϑ

(A.32) (A.33)

(A.34)

or, equivalently, in compact form as the determinant of an abstract 3 × 3 matrix of unit vectors, differential operators and components of F(r)   yˆ zˆ   xˆ ∂ ∂ ∂   (A.35) ∇ × F(r) :=   ∂x ∂y ∂z   F x Fy Fz    ρϕˆ zˆ   ρˆ ∂ ∂  1∂  =  (A.36) ρ  ∂ρ ∂ϕ ∂z   Fρ Fϕ Fz 

1086

Advanced Theoretical and Numerical Electromagnetics   rˆ 1  ∂  = 2 r sin ϑ  ∂r  Fr

rϑˆ ∂ ∂ϑ rFϑ

 r sin ϑ ϕˆ   ∂  ∂ϕ  Fϕ r sin ϑ

(A.37)

where swapping the rows is meaningless and thus not permitted. If n  2, the gradient of a scalar field is a curl-free (or also lamellar, irrotational, conservative) vector, and the curl of a vector field is a divergence-free (or solenoidal) vector, viz., [13, pp. 355–357] curl grad Ψ(r) = ∇ × ∇Ψ(r) = 0

(A.38)

div curl F(r) = ∇ · ∇ × F(r) = 0

(A.39)

which can be proved, e.g., by expanding the operators in Cartesian coordinates and invoking the Schwarz theorem [7, pp. 235–236]. Nonetheless, for merely mnemonic purposes we may regard the symbol ∇ as an ordinary vector whose components, though, are differential operators. Then, (A.38) is intuitively true in that it involves the cross product of two parallel vectors. Likewise, (A.39) formally amounts to a triple scalar product as in (H.13) and, since two vectors out of three coincide, the result follows. If n  1, then the Laplacian of Ψ(r) is the scalar field defined by div grad Ψ(r) = ∇ · ∇Ψ(r) = ∇2 Ψ(r) = ΔΨ(r) ∂2 Ψ ∂2 Ψ ∂2 Ψ + 2 + 2 ∂x2 ∂y ∂z  1 ∂ ∂Ψ 1 ∂2 Ψ ∂2 Ψ = ρ + 2 2 + 2 ρ ∂ρ ∂ρ ρ ∂ϕ ∂z   1 ∂Ψ 1 ∂2 Ψ 1 ∂ 2 ∂Ψ ∂ r + 2 sin ϑ + 2 = 2 . r ∂r ∂r r sin ϑ ∂ϑ ∂ϑ r sin ϑ ∂ϕ2 :=

(A.40) (A.41) (A.42)

If n  1, the Laplacian of a vector field is a vector that can be defined through the successive application of (A.26) and (A.29) to the Cartesian components of F(r). As a word of caution, this simple rule does not apply for fields expressed in cylindrical and spherical coordinates. In symbols, we have ∇2 F(r) := xˆ ∇2 F x + yˆ ∇2 Fy + zˆ ∇2 Fz   Fρ Fϕ 2 ∂Fϕ 2 ∂Fρ = ρˆ ∇2 Fρ − 2 − 2 + ϕˆ ∇2 Fϕ − 2 + 2 + zˆ ∇2 Fz ρ ρ ∂ϕ ρ ρ ∂ϕ    2 cos ϑ 1 ∂Fϕ ∂Fϑ = rˆ ∇2 Fr − 2 Fr + Fϑ + + sin ϑ sin ϑ ∂ϕ ∂ϑ r    Fϑ ∂Fr cos ϑ ∂Fϕ 1 − 2 + 2 + ϑˆ ∇2 Fϑ − 2 ∂ϑ r sin2 ϑ sin2 ϑ ∂ϕ    2 ∂Fr cos ϑ ∂Fϑ 1 Fϕ − −2 2 + ϕˆ ∇2 Fϕ − 2 r sin ϑ sin ϑ ∂ϕ sin ϑ ∂ϕ

(A.43) (A.44)

(A.45)

whence we gather, e.g., that ˆ 2 Fϑ + ϕ∇ ∇2 (Fr rˆ + Fϑ ϑˆ + Fϕ ϕ) ˆ  rˆ ∇2 Fr + ϑ∇ ˆ 2 Fϕ

(A.46)

the reason being that rˆ , ϑˆ and ϕˆ themselves depend on ϑ and ϕ in accordance with (A.11)-(A.13).

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1087

More generally, in coordinate-free notation the Laplacian of a vector field can be computed with the differential identity [see (H.59)] ∇2 F(r) = ∇∇ · F(r) − ∇ × [∇ × F(r)]   curl-free

(A.47)

solenoidal

which shows that ∇2 F(r) can be decomposed into a curl-free part and a solenoidal part, in light of (A.38) and (A.39) (see Section 8.1 for more details). The gradient, divergence and curl operators examined above can be specialized so as to act on surface fields, i.e., scalar and vector functions Ψ(r) and F(r) that are defined for points r on a surface S ⊂ R3 (Figures 1.2a and 1.2b). Notable examples in electromagnetics include the surface charge density S (r) in (1.155), the surface current density JS (r) in (1.142), and the surface density of electric dipoles τS (r) (2.238). ˆ perpendicular to S plays a key role, it is important that the unit normal Since the unit vector n(r) be univocally defined, and thus we assume that S is smooth (cf. Appendix F). This is not a serious limitation because, if the surface of interest is only piecewise smooth, it is possible to separate S into parts whereon the normal is unique. The surface gradient of a scalar function Ψ(r) ∈ Cn (S ) is the vector field ˆ ∇s Ψ(r) := ∇Ψ(r) − n(r)

∂Ψ ˆ [n(r) ˆ · ∇Ψ(r)] = ∇Ψ(r) − n(r) ∂nˆ

(A.48)

with ∇Ψ given formally by (A.26). The surface divergence of a vector field F(r) ∈ Cn (S )3 is the scalar function ˆ ∇s · F(r) := ∇s · Ft (r) − J(r)F(r) · n(r) = ∇ · F(r) −

∂ ˆ [F(r) · n(r)] ∂nˆ

(A.49)

where J(r) is the first curvature of S defined in (H.100) [11, Appendix 3], and ˆ × [F(r) × n(r)] ˆ ˆ ˆ · F(r)], Ft (r) := n(r) = F(r) − n(r)[ n(r)

r∈S

(A.50)

is the part of F tangential to the surface or, equivalently, the projection of F onto S at r. The surface curl of a vector field F(r) ∈ Cn (S )3 is the vector field ˆ × ∇s × F(r) := ∇ × F(r) − n(r)

∂F ∂nˆ

(A.51)

which is not necessarily orthogonal to S . Explicit, coordinate-dependent forms of (A.48), (A.49) and (A.51) can be given once a local set of curvilinear coordinates on S is introduced [11, Appendix 3].

A.3

The Gauss theorem

We suppose that V ⊂ R3 is a bounded open region of space with sufficiently regular boundary ∂V (Appendix F). Then, ∂V is a closed surface, as in Figure 1.2b, and the set V := V ∪ ∂V is the closure of V. The flux of a vector field F(r) : R3 ⊃ V → C3 through the surface ∂V is the scalar defined as  ΨF := dS (r) nˆ (r) · F(r) (A.52) ∂V

ˆ where n(r), r ∈ ∂V, is the unit vector perpendicular to ∂V and pointing outwards V.

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Advanced Theoretical and Numerical Electromagnetics

Then, if F(r) ∈ C1 (V)3 ∩C(V)3 is a single-valued vector field, the Gauss (or divergence) theorem states that [14, Chapter 4], [15, Chapter 8], [1, Section 1.4], [8]   dV(r) ∇ · F(r) = dS (r) nˆ (r) · F(r) (A.53) ∂V

V

and in particular the volume integral in the left-hand side exists. The theorem was first proved by J. L. Lagrange in 1764 and then again by C. F. Gauss in 1813. For divergence-free fields (A.53) implies that the flux through a closed surface vanishes. Fields with this property are called incompressible or source free.

A.4 The Stokes theorem We assume that S ⊂ R3 is a sufficiently regular, open and two-sided surface (Appendix F) with unit ˆ normal n(r). The regular closed curve ∂S is the boundary of S (Figure 1.2a), and the set S := S ∪ ∂S is the closure of S . The circulation of a vector field F(r) : R3 ⊃ S → C3 along the closed curve ∂S is the number defined as  := ΥF ds(r) sˆ(r) · F(r) (A.54) ∂S

where sˆ(r), r ∈ ∂S , is the unit vector tangent to ∂S and oriented in accordance with the right-handed ˆ In this setup, an ideal observer who stands upright on ∂S and walks in screw rule with respect to n. the direction indicated by sˆ(r) sees S to his left. Then, if F(r) ∈ C1 (S )3 ∩C(S )3 is a single-valued vector field, the Stokes (or circulation) theorem states that [14, Chapter 4], [1, Section 1.4], [8]   dS (r) nˆ (r) · ∇ × F(r) = ds(r) sˆ(r) · F(r) (A.55) ∂S

S

or in words, that the flux of the curl of F through S equals the circulation of F along the boundary of S , in light of (A.52) applied to S . The theorem — first proved by A. Ampère in 1825 and then published again by G. Stokes in 1854 — does not hold for non-orientable open surfaces, a case in point being the Möbius strip.

A.5 The surface Gauss theorem We suppose that S ⊂ R3 is a sufficiently regular open connected surface (Appendix F) with regular ˆ boundary ∂S . The unit normal n(r) and the unit tangent sˆ(r) are oriented according to the righthanded screw rule (Figure 1.2a). The flux of a surface vector field F(r) : R3 ⊃ S → C3 through the line ∂S is the number defined as  ΨF := ds(r) νˆ (r) · F(r) (A.56) ∂S

where ˆ νˆ (r) := sˆ(r) × n(r),

r ∈ ∂S

(A.57)

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1089

is the unit vector perpendicular to the boundary ∂S . Moreover, by construction νˆ (r) lies in the plane tangential to S at r ∈ ∂S . If F(r) ∈ C1 (S )3 ∩ C(S )3 is a surface vector field tangential to S , i.e., ˆ · F(r) = 0, n(r)

r∈S

(A.58)

the surface Gauss theorem states that   dS (r) ∇s · F(r) = ds(r) νˆ (r) · F(r)

(A.59)

∂S

S

where ∇s · {•} is the surface divergence (A.49) [11, Appendix 3]. Additionally, the result holds for a generic vector field provided S is flat, i.e., its curvature J(r) (H.100) is null. By contrast, if the curvature of S is finite and F(r) does have a component perpendicular to S , then the right-hand side of (A.59) must be augmented with one more surface integral which involves J(r) and, of course, ˆ · F(r) [see (H.101)]. n(r) With the aid of (A.59) we can show that ˆ × F(r)] + n(r) ˆ · ∇ × F(r) = 0, ∇s · [n(r)

r∈S

(A.60)

ˆ where n(r) is the unit vector normal to S , and F(r) is a surface field over S though not necessarily ˆ tangent to S . Indeed, since taking the cross product with n(r) filters out the component of F(r) ˆ × F(r) is a field tangential to S , and (A.59) can be applied under the same perpendicular to S , n(r) hypotheses stated above. We pick up an arbitrary, though regular scalar field f (r) defined in particular for r ∈ S and start with the surface integral  ˆ × F(r)] = dS (r) f (r)∇s · [n(r) S



 ˆ × F(r)] − dS (r) ∇s · [ f (r)n(r)

= S

ˆ × F(r) dS (r) ∇s f (r) · n(r) S





=

ˆ · F(r) f (r) + ds(r) νˆ (r) × n(r) ∂S



 ds(r) sˆ(r) · F(r) f (r) +

=− ∂S



=−

dS (r) nˆ (r) × ∇s f (r) · F(r) S

dS (r) nˆ (r) × ∇ f (r) · F(r) S



dS (r) nˆ (r) · ∇ × [ f (r)F(r)] + S

S

 =−

dS (r) nˆ (r) · ∇ f (r) × F(r)

ˆ · ∇ × F(r) dS (r) f (r)n(r)

(A.61)

S

where we have applied (H.77) in the first step, then (A.59) and (H.13), the Stokes theorem and finally the differential identity (H.50). In the third step we have also noticed that nˆ × ∇s f = nˆ × ∇ f on account of definition (A.48). By walking backwards through the chain of identities in (A.61) we conclude that  ˆ × F(r)] + n(r) ˆ · ∇ × F(r)} = 0 dS (r) f (r) {∇s · [n(r) (A.62) S

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1090

and, since the scalar field f (r) is arbitrary, then the latter condition is satisfied only if (A.60) holds true. Alternatively, by choosing a surface S 0 ⊂ S we have    ˆ × F(r)] = ˆ × F(r) = − ds(r) sˆ(r) · F(r) dS (r) ∇s · [n(r) ds(r) νˆ (r) · n(r) ∂S 0

S0

∂S 0

 dS (r) nˆ (r) · ∇ × F(r)

=−

(A.63)

S0

having applied the Stokes theorem (A.55) in the last passage. Thanks to the mean value theorem [15], then ˆ × F(r)]|r=r0 + n(r ˆ 0 ) · ∇ × F(r)|r=r0 = 0 ∇s · [n(r)

(A.64)

with r0 ∈ S 0 a suitable point. Then, (A.60) follows in view of the arbitrariness of S 0 .

A.6 The Helmholtz transport theorem We consider a sufficiently regular open connected surface S ⊂ R3 with smooth boundary γ := ∂S (Appendix F) and a vector field F(r, t) which is defined throughout space and, in particular, for points r ∈ S . Evidently, the flux of F(r, t), as given by formula (A.52) applied to S , must be a function of time because the vector field of concern is time dependent. Then, if the surface S is fixed — i.e., it changes neither shape nor position as time goes by — the time derivative of the flux can be computed as   d ∂ d ˆ · F(r, t) = ˆ · F(r, t) ΨF (t) := (A.65) dS n(r) dS n(r) dt dt ∂t S

S

provided both F(r, t) and ∂F/∂t are continuous fields for (r, t) ∈ S × R. A clever strategy for the proof of (A.65) consists of integrating the rightmost member with respect to time, say, from t0 up to t > t0 

t dτ t0

S

∂ ˆ · F(r, τ) = dS n(r) ∂τ



t

∂ F(r, τ) ∂τ t0 S   ˆ · F(r, t) − dS n(r) ˆ · F(r, t0 ) dS n(r) = ˆ · dS n(r)

S

dτ

(A.66)

S

 constant

where we have interchanged the order of integration in accordance with the Fubini theorem because the surface integral can be reduced to combinations of double ones over suitable two-dimensional subsets of points of R2 [8, Chapter 2]. Needless to say, this step makes sense inasmuch as S does not change with τ. Now, taking the time derivative of the leftmost- and rightmost-hand sides and invoking the fundamental theorem of calculus yields (A.65) since the flux of F(r, t0 ) through S is a constant. A formula exists that is more general than (A.65) and valid for the time derivative of ΨF (t) when the surface S = S (t) is slowly moving in space with small velocity v(r, t) and is possibly changing shape. The result goes by the name of Helmholtz transport (or integral) theorem (1867) [16,

Vector calculus

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Figure A.3 A rigid, slowly moving surface S and two reference frames for proving the Helmholtz transport theorem. pp. 419 and 456], [17, p. 130], [18, Formula 10.812], [11,19–22], and is especially important in fluid dynamics. The theorem can be regarded as an extension to higher dimensions of the Leibniz rule for the differentiation of integrals depending on a parameter [23, Formula 3.3.7], [24, Formula 69.3], [25]. In electromagnetics the formula is used to derive (1.56), the kinematic form of the Faraday law. We give first a derivation for the case where the surface maintains its shape as it drifts in space. The requirement that the velocity be small means |v| = v c0 , whereby classical mechanics holds and we may invoke the inverse Galilean transformation [26, Section 13.2] r = r + vt t=t

(A.67)



(A.68)

between a suitable reference frame (characterized by primed space and time coordinates) that moves precisely with velocity v and the laboratory reference frame (unprimed coordinates). Thanks to this choice, the surface S  ≡ S (t) appears to be instantaneously at rest to an observer who travels alongside S (t). The surface and the reference frames are sketched in Figure A.3. The vector area element dS nˆ looks the same to an observer standing in the laboratory and to the observer ‘attached’ to S (t), essentially because (A.67) acts as a change of dummy variable that does not distort the shape of S . Indeed, by introducing a local system of curvilinear coordinates (ξ, η) on S we have ∂r ∂r ∂r ∂r ˆ ) ˆ := dξdη × (A.69) = dξdη × = dS  n(r dS n(r) ∂ξ ∂η ∂ξ ∂η inasmuch as the vector vt is independent of ξ and η. Therefore, we can write the flux equivalently as   ˆ · F(r, t) = dS n(r) dS  nˆ  (r ) · F(r + vt , t ) = ΨF (t ) (A.70) ΨF (t) = S

S

in light of the Galilean transformation (A.67) and (A.68). However, in the rest frame of S (t) the variation of F with time appears different than what it looks like to the stationary observer in the laboratory. This phenomenon is embodied in (A.67) and accounts for the underlying motion of S  , unbeknownst to the observer who rides along with the surface. Applying (A.65) in the rest frame of S (t) implies  d d  Ψ (t ) = dS  nˆ  (r ) ·  F(r + vt , t ) F dt dt S

1092

Advanced Theoretical and Numerical Electromagnetics  =

dS  nˆ  (r ) ·

S



∂ ∂x ∂ F(r + vt , t ) +  F(r + vt , t ) ∂t ∂t ∂x

∂y ∂ ∂z ∂ +  F(r + vt , t ) +  F(r + vt , t ) ∂t ∂y ∂t ∂z   ∂ ∂ = dS  nˆ  (r ) · F(r + vt , t ) + v x F(r + vt , t ) ∂t ∂x S  ∂ ∂       + vy F(r + vt , t ) + vz F(r + vt , t ) ∂y ∂z    ∂          = dS nˆ (r ) · F(r + vt , t ) + (v · ∇)F(r + vt , t ) ∂t S    ∂ ˆ · F(r, t) + (v · ∇)F(r, t) = dS n(r) ∂t



(A.71)

S (t)

having used (A.67) and (A.68) in the last step. In essence, rephrasing the calculation in the reference frame where S (t) is at rest has allowed us to move the time derivative inside the flux integral. Furthermore, (A.70) and (A.68) lead us to state    ∂ d d  ˆ · ΨF (t) =  ΨF (t ) = F(r, t) + (v · ∇)F(r, t) (A.72) dS n(r) dt dt ∂t S (t)

which we can write in a more convenient way by invoking the differential identity (H.54). Specifically, we have ∇ × (v × F) = v∇ · F − F∇ · v + (F · ∇)v − (v · ∇)F

(A.73)

and observe that ∇·v =∇·

dr d d = ∇·r= 3 =0 dt dt dt

(A.74)

by virtue of (A.31) and also dr d d ∂r d ∂r d ∂r = F · ∇r = F x + Fy + Fz dt dt dt ∂x dt ∂y dt ∂z d d d = F x xˆ + Fy yˆ + Fz zˆ = 0 dt dt dt

F · ∇v = F · ∇

(A.75)

where in the last step we have noticed that the reference unit vectors are fixed in time. (We could alternatively compute ∇v and ∇r by using the concepts of dyads and dyadics presented in Appendix E.) All in all, these calculations provide us with the identity (v · ∇)F = v∇ · F − ∇ × (v × F) which, when inserted into (A.72), gives    d ∂ ˆ · ΨF (t) = F(r, t) + v∇ · F(r, t) − ∇ × [v × F(r, t)] dS n(r) dt ∂t S (t)

(A.76)

(A.77)

Vector calculus

1093

Figure A.4 A deforming, slowly moving surface S 1 := S (t) and a local system of curvilinear coordinates for proving the Helmholtz transport theorem.

whence by applying the Stokes theorem we get 

d dt

S (t)



 ˆ · F(r, t) = dS n(r)

ˆ · dS n(r)

  ∂ F(r, t) + v∇ · F(r, t) − ds sˆ(r) · v × F(r, t) ∂t

(A.78)

∂S (t)

S (t)

the desired generalization of (A.65). A second proof of (A.78) that is valid for moving and deforming surfaces is hinged on the classic notion of derivative of a scalar function applied to ΨF (t) [20, Appendix A.2], [19, 21, 27]. In this regard we suppose that in the ‘small’ time interval Δt the surface of concern S 1 := S (t) sweeps a region of space V while moving and changing into S 2 := S (t + Δt) with velocity v. By construction, V is a cylinder-like domain bounded by S l , l = 1, 2, and a lateral surface, say, S L , which is formed with the streamlines of the velocity field v (Figure A.4). We begin by generating an integral identity which involves the fluxes of F(r, t + Δt) through S l and S L by applying the Gauss theorem (A.53) to the domain V, viz., 1 Δt

 dV ∇ · F(r, t + Δt) = V

1 Δt

 ˆ · F(r, t + Δt) dS n(r) S2

1 − Δt

 S1

1 ˆ · F(r, t + Δt) + dS n(r) Δt

 ˆ · F(r, t + Δt) dS n(r)

(A.79)

SL

ˆ on S 1 positively oriented inwards where, for later convenience, we have chosen the unit normal n(r) V. Since we are interested in the time rate of variation of the flux of F(r, t) through S 1 we cast (A.79)

Advanced Theoretical and Numerical Electromagnetics

1094

into the alternative format   ˆ · F(r, t + Δt) − dS n(r) ˆ · F(r, t) dS n(r) S2

S1

Δt



F(r, t + Δt) − F(r, t) Δt S1   1 1 ˆ · F(r, t + Δt) + dS n(r) dV ∇ · F(r, t + Δt) − Δt Δt =

ˆ · dS n(r)

SL

(A.80)

V

wherefrom we see that for vanishing Δt • •

the quotient in the left-hand side provides (by definition, [7, Chapter 5]) the desired time derivative of the function ΨF (t) since S 2 tends to S 1 ; the first term in the right member reduces to the flux of the time-derivative of F(r, t) through S (t).

We are left with the task of showing that as Δt → 0+ the integrals over S L and V remain finite and pass over, respectively, into integrals over ∂S (t) and S (t). To this purpose we introduce a local system of curvilinear cylindrical coordinates (ξ, η, ζ) — possibly not orthogonal — conceived in a such a way that (Figure A.4) • • •

the ‘azimuthal’ coordinate η ∈ [0, 2π] becomes ignorable on the ζ-axis, which is thus identified by ξ = 0, S L is part of the canonical surface ξ = ξL , S l is punctured by the ζ-axis and is contained in the canonical surface ζ = ζl .

Further, we assume that the coordinate vectors form a right-handed triple whereby the normal vector on S L , viz.,   ∂r  ∂r    , N(ξL , η, ζ) = × η ∈ [0, 2π], ζ ∈ [ζ1 , ζ2 ] (A.81) ∂η ξL ∂ζ ξL points outwards V and the determinant of the Jacobi matrix (cf. [7, page 252])  ∂r ∂r ∂r ∂r ∂r ∂r , , × · det[J] = det = ∂ξ ∂η ∂ζ ∂ξ ∂η ∂ζ

(A.82)

is positive. We notice that the velocity field in any point r ∈ V is given by v(r) =

dr ∂r dζ = dt ∂ζ dt

(A.83)

because v is directed along the positive ζ-coordinate by construction. Since at time τ ∈ [t, t + Δt] the surface S (τ) coincides with the canonical surface ζ ∈ [ζ1 , ζ2 ] we introduce the change of variable ζ = ζ(τ)

(A.84)

so that ζ(t) = ζ1 and ζ(t + Δt) = ζ2 . With these positions we can rephrase the integrals over S L and V in the local coordinates (ξ, η, ζ), viz., 1 Δt

 SL

1 ˆ · F(r, t + Δt) = dS n(r) Δt

2π

ζ2 dη

0

dζ ζ1

∂r ∂r × · F(r, t + Δt) ∂η ∂ζ

Vector calculus 1 = Δt 2π = 0

1 Δt

 V

1 dV ∇ · F(r, t + Δt) = Δt 1 Δt

=

ξL dξ 0

ξL =

2π

dξ 0

dη 0

2π 0

t+Δt 

dτ

2π

1095

t+Δt  ∂r dζ ∂r · × F(r, t + Δt) dη dτ ∂η dτ ∂ζ t

0

 ∂r   · v(ξL , η, ζ(τ)) × F(ξL , η, ζ(τ), t + Δt) dη ∂η τ=τ

ξL

2π dξ

0

ζ2 dη

dζ ζ1

0

(A.85)

∂r ∂r ∂r × · ∇ · F(r, t + Δt) ∂ξ ∂η ∂ζ

∂r ∂r ∂r dζ × · ∇ · F(r, t + Δt) ∂ξ ∂η ∂ζ dτ

t

  ∂r  ∂r   ×  · v(ξ, η, ζ(τ))∇ · F(ξ, η, ζ(τ), t + Δt) dη ∂ξ τ=τ ∂η τ=τ

(A.86)

where we have used (A.83), (A.84) and applied the mean value theorem [15] to the innermost integrals with τ ∈ [t, t + Δt] a suitable instant of time. Perhaps it is not superfluous to remark that in (A.85) and (A.86) we have integrated a ‘snapshot’ of the field F taken at time t + Δt with respect to an auxiliary time variable which is associated with the successive positions of the surface S (t) along the ζ-axis. The contributions in the rightmost members of (A.85) and (A.86) — which are are well-defined for any value of Δt — amount to a contour integral along ∂S (τ) and a flux integral across S (τ). For vanishing Δt these terms pass over into a contour and a flux integral along ∂S (t) and over S (t) in that τ tends to t for Δt → 0+ , namely, 1 lim Δt→0+ Δt



2π ˆ · F(r, t + Δt) = dS n(r)

SL

0



 ∂r   · v(ξL , η, ζ1 ) × F(ξL , η, ζ1 , t) dη ∂η ζ1

=

ds sˆ(r) · v(r) × F(r, t)

(A.87)

∂S (t)

1 lim Δt→0+ Δt

ξL

 dV ∇ · F(r, t + Δt) = V

2π dξ

0

 =

0

  ∂r  ∂r    · v(ξ, η, ζ1 )∇ · F(ξ, η, ζ1 , t) dη × ∂ξ ζ1 ∂η ζ1

ˆ · v(r)∇ · F(r, t) dS n(r)

(A.88)

S (t)

where v(r) denotes the velocity of S (t) at time t. With these results it remains proved that (A.80) yields (A.78) as Δt tends to zero. Although the surface S (t) may be changing in some weird manner, it is always possible to divide it into N relatively ‘small’ parts which sweep regions with piecewise-smooth boundaries. The procedure outlined above can be applied to each region perhaps with minor conceptual modifications concerning the local

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Advanced Theoretical and Numerical Electromagnetics

systems of coordinates. The result for the original surface S (t) is then recovered by summing N equations of type (A.78) inasmuch as the flux integrals add up and the contour integrals reduce to a single one along ∂S (t) due to cancellations. For the sake of completeness, we mention that another rigorous, quite technical derivation of the transport theorem in a n-dimensional space can be found in [25] and relies on the notions of differential forms, exterior derivatives and interior products.

A.7 Estimates for vector-valued functions The derivative of a function f (x) : [a, b] → R of class C1 (a, b) ∩ C[a, b] satisfies f  (x0 ) =

f (b) − f (a) b−a

(A.89)

where x0 ∈]a, b[ is a suitable point. This result is known as the mean value theorem for derivatives [7, Chapter 5]. The geometrical meaning of (A.89) is that, under the stated conditions, there exists a point x0 where the straight line tangential to the graph of the function (see note on page 1119) is parallel to the straight line drawn from the point (a, f (a)) to the point (b, f (b)). In kinematics, if x represents the time variable t, and f (t) indicates the position of a point along a smooth curve γ ⊂ R3 , then f  (t) constitutes the instantaneous velocity of the point as it travels along γ. In which case, (A.89) states that at some moment t0 between two times a = t1 and b = t2 the average velocity of the point equals the instantaneous velocity. A property analogous to (A.89) does not exist for vector fields or, more generally, vector-valued functions of many variables. Nevertheless, if F(r) is a real- or complex-valued vector field of class C1 (D)3 with D ⊂ R3 an open domain, the following estimate holds true |F(r2 ) − F(r1 )|  M|r2 − r1 |,

r1 , r2 ∈ D

(A.90)

where M is a suitable positive constant (cf. [7, Theorems 5.19 and 9.19]). A vector field which obeys (A.90) with M independent of r1 and r2 is said to be Lipschitz continuous [28, Definition 4.14]. To prove (A.90) in the case where the domain D is convex as prescribed by definition (D.6) we may define three scalar functions of one variable s, namely, s ∈ [0, 1],

gα (s) := Fα (r(s)),

α ∈ {x, y, z}

(A.91)

where r(s) := (1 − s)r1 + sr2 = [(1 − s)x1 + sx2 ]ˆx + [(1 − s)y1 + sy2 ]ˆy + [(1 − s)z1 + sz2 ]ˆz

(A.92)

represents the straight segment that connects r1 and r2 and is contained in D. Next, we consider the quantities 1 Fα (r2 ) − Fα (r1 ) = gα (1) − gα (0) =

dgα = ds ds

0

1 ds ∇Fα · (r2 − r1 )

(A.93)

0

where we have used the fundamental theorem of calculus [7, Theorem 6.21] and the chain rule for derivatives. By taking the absolute values of both sides we find 1 |Fα (r2 ) − Fα (r1 )| 

1 ds |∇Fα · (r2 − r1 )| 

0

ds |∇Fα ||r2 − r1 |  ∇Fα ∞ |r2 − r1 | 0

(A.94)

Vector calculus

1097

by virtue of the Cauchy-Schwarz inequality (D.151). Finally, on account of (H.19) applied to the Cartesian components of F(r2 ) − F(r1 ) we consider   |F(r2 ) − F(r1 )|  |Fα (r2 ) − Fα (r2 )| 

∇Fα ∞ |r2 − r1 | (A.95) α∈{x,y,z}

α∈{x,y,z}

and this ends the proof. Likewise, when F(r) is a real- or complex-valued surface vector field of class C1 (S )3 with S ⊂ 3 R being an open or closed C2 -smooth surface (see Figure 1.2 and Appendix F) it can be shown that |F(r2 ) − F(r1 )|  M|r2 − r1 |,

r1 , r2 ∈ S

(A.96)

for some M > 0. Still, we cannot proceed as we did to arrive at (A.90) inasmuch as the straight segment that connects r1 to r2 lies, in general, outside S — an obvious exception being the case of a flat surface — whereas F(r) is only defined for points on S . Therefore, we indicate the characteristic size of S with d and pick up a distance a > 0 small enough so that the intersection S ∩ B(r1 , a) is non-empty and comprised of only one connected part of S . Then, if r1 and r2 are such that a < |r2 − r1 |  d, we have immediately |F(r2 ) − F(r1 )| =

|F(r2 )| + |F(r1 )| |F(r2 ) − F(r1 )| 2 a |r2 − r1 |  F ∞ |r2 − r1 | a a a

(A.97)

on account of (H.19). On the contrary, if |r2 − r1 |  a, we introduce dimensionless local coordinates ξ := (u, v) on S ∩ B(r1 , a) by means of the bijective mapping (cf. Figure 2.20) Ψ : R2 ⊃ B2 (0, 1) −→ S ∩ B(r1, a) ⊂ R3

(A.98)

where B2 (0, 1) is the unit circle (D.22). Further, we suppose that ξ1 = (u1 , v1 ) and ξ2 = (u2 , v2 ) are the local coordinates which are mapped to the points r1 and r2 on S ∩ B(r1 , a) by Ψ(•). Finally, we define the constant, possibly complex, vector w := F(r2 ) − F(r1 )

(A.99)

for ease of manipulation. If w happens to be null for some reason, then (A.96) is obviously true, and hence we may assume that w  0. Since B2 (0, 1) is a convex two-dimensional domain [see (D.6)] we can define the real scalar function g(s) := Re {w∗ · [F(Ψ(ξ(s))) − F(r1 )]} ,

s ∈ [0, 1]

(A.100)

where ξ(s) := (1 − s)ξ1 + sξ2 = ( u(s), v(s) ) = ( (1 − s)u1 + su2 , (1 − s)v1 + sv2 )

(A.101)

represents the straight segment that joins the local points ξ1 to ξ2 and is mapped by Ψ(•) onto a line γ12 ⊂ S ∩ B(r1 , a) that connects r1 to r2 . Then, we notice that by construction we have g(0) = 0

g(1) = |w|2 = |F(r2 ) − F(r1 )|2

(A.102)

and, more importantly, |w|2 = |F(r2 ) − F(r1 )|2 = g(1) − g(0) = (1 − 0)g (s0 ),

s0 ∈]0, 1[

(A.103)

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Advanced Theoretical and Numerical Electromagnetics

precisely by virtue of the mean value theorem (A.89), as the differentiability of g(s) follows from that of F(Ψ), Ψ(ξ) and ξ(s). Since F(r) depends on u(s) and v(s) through r = Ψ(ξ) = Ψ( u(s), v(s) ), by invoking the chain rule for derivatives [7, Theorem 5.5] on (A.100) and letting ξ0 = ξ(s0 ) in light of (A.101) we obtain      ∗ ∂F  ∗ ∂F  (u2 − u1 ) + w · (v2 − v1 ) g (s0 ) = Re w · ∂u ξ0 ∂v ξ0       ∗ ∂F  ∗ ∂F  = Re w · (u2 − u1 ) + Re w · (v2 − v1 ) ∂u ξ0 ∂v ξ0              ∗ ∂F       ∗ ∂F     ∂F    ∂F      ξ2 − ξ1  + w ·   ξ2 − ξ1   |w|    +    ξ2 − ξ1   w ·    ∂u ξ0   ∂v ξ0  ∂u ξ0  ∂v ξ0   M     −1 −1   = M Ψ (r2 ) − Ψ (r1 ) |w|  M M |r2 − r1 ||w| = M|r2 − r1 ||w| (A.104) where we have used the Cauchy-Schwarz inequality (D.151) and applied estimate (A.90) to the inverse mapping Ψ−1 (•), since the latter exists and is twice-differentiable by definition of C2 -smooth surface. Invoking (A.90) is permitted because Ψ(•) represents the restriction to S ∩ B(r1, a) of a ˜ vector-valued function, say, Ψ(•), which maps B(0, 1) onto B(r1 , a). Lastly, using inequality (A.104) in the leftmost member of (A.103) and dividing through by |w| yields (A.96). ˆ defined on S . In Appendix F.3 we shall apply (A.96) to the unit normal n(r)

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

Morse P, Feshbach H. Methods of Theoretical Physics. New York, NY: McGraw-Hill; 1953. Stratton JA. Electromagnetic theory. London, UK: McGraw-Hill; 1941. Moon P, Spencer DE. Field Theory Handbook: Including Coordinate Systems, Differential Equations and Their Solutions. 2nd ed. Berlin Heidelberg: Springer-Verlag; 1971. Hayt WH, Buck JA. Engineering Electromagnetics. 8th ed. New York, NY: McGraw-Hill; 2012. International edition. Balanis CA. Advanced Engineering Electromagnetics. New York, NY: John Wiley & Sons, Inc.; 2005. Schwab AJ. Field Theory Concepts. Berlin Heidelberg: Springer-Verlag; 1988. Rudin W. Principles of mathematical analysis. 3rd ed. New York, NY: McGraw-Hill; 1976. Schey HM. Div, grad, curl and all that. 4th ed. New York, NY and London, UK: W. W. Norton & Company; 2005. Helrich CS. The classical theory of fields - Electromagnetism. Berlin Heidelberg: SpringerVerlag; 2012. Holt CA. Introduction to Electromagnetic Fields and Waves. New York, NY: John Wiley & Sons, Inc.; 1963. Van Bladel JG. Electromagnetic Fields. Piscataway, NJ: IEEE Press; 2007. Balanis CA. Antenna Theory: Analysis and Design. 2nd ed. New York, NY: John Wiley & Sons, Inc.; 1997. Mason M, Weaver W. The electromagnetic field. New York, NY: Dover Publications, Inc.; 1929. Kellogg OD. Foundations of potential theory. Berlin Heidelberg: Springer-Verlag; 1929. Makarov B, Podkorytov A. Real Analysis: Measures, Integrals and Applications. Universitext. London, UK: Springer; 2013.

Vector calculus [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]

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Loomis LH, Sternberg S. Advanced Calculus. Reading, MA:Addison-Wesley; 1968. Sommerfeld A. Mechanics of Deformable Bodies. vol. 2 of Lectures on theoretical physics. New York, NY: Academic Press; 1950. Gradshteyn IS, Ryzhik IM. Table of Integrals, Series, and Products. 5th ed. Amsterdam, NL: Academic Press, Inc.; 1994. Tai CT. Generalized Vector and Dyadic Analysis. New York, NY: IEEE Press; 1997. Rothwell EJ, Cloud MJ. Electromagnetics. London, UK: CRC Press; 2001. Pauli W. Theory of relativity. New York, NY: Dover Publications, Inc.; 1981. Abraham M, Becker R. The Classical Theory of Electricity and Magnetism. 2nd ed. London, UK: Blackie; 1932. Abramowitz M, Stegun IA. Handbook of mathematical functions. New York, NY: Dover Publications, Inc.; 1965. Dwight HB. Tables of Integrals and Other Mathematical Data. 4th ed. New York, NY: MacMillan Publishing Co., Inc.; 1961. Flanders H. Differentiation under the integral sign. American Mathematical Monthly. 1973 June-July;80(6):615–627. Lorrain P, Corson DR, Lorrain F. Electromagnetic Fields and Waves. 3rd ed. New York, NY: W. H. Freeman and Company; 1988. Whitaker S. Introduction to fluid mechanics. Englewood Cliffs, NJ: Prentice-Hall; 1968. Muscat J. Functional Analysis. London, UK: Springer; 2014.

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Appendix B

Complex analysis

We review the fundamental notions and results concerning complex functions of one complex variable [1, Chapter 1], [2, Chapter 4], [3–6], which come in handy throughout this book and especially so in Sections 7.6, 8.7 and 12.2.

B.1

Derivatives and integrals

We begin by recalling that a complex number z ∈ C and its complex conjugate z∗ ∈ C are defined as ordered pairs of real numbers, viz., z := (x, y) = x + j y z := (x, −y) = x − j y

(B.1)

∗

(B.2)

where x, y ∈ R and √ j := (0, 1) = −1

(B.3)

is the imaginary unit. The latter satisfies the properties j2 = j · j = −1 j4 = j2 · j2 = 1

j3 = j · j2 = − j 1 = −j j

(B.4) (B.5)

thanks to its very definition (B.3). Since (B.1) and (B.2) associate x and y with z and z∗ we may regard these relations as a change of variables. In fact, the inverse transformation yields the real and imaginary parts of z, i.e., z + z∗ = Re{z} 2 ∗ z−z = Im{z} y= 2j x=

(B.6) (B.7)

whereby it should be quite evident that z and z∗ are two independent numbers. Indeed, a complex function of the two real variables x and y may be alternatively interpreted as a function of z and z∗ , viz., f (x, y) = f (z, z∗ )

(B.8)

precisely by virtue of (B.6) and (B.7). Suppose now that the partial derivatives of f (x, y) in a point (x0 , y0 ) ∈ R2 exist. The complex number corresponding to (x0 , y0 ) is z0 , and z∗0 is defined by means

1102

Advanced Theoretical and Numerical Electromagnetics

of (B.2). Then, by invoking the chain rule for the derivatives of f (z(x, y), z∗ (x, y)) [1, Theorem 5.5] we have ∂f ∂f ∂f (x0 , y0 ) = (z0 , z∗0 ) + ∗ (z0 , z∗0 ) ∂x ∂z ∂z ∂f ∂f ∂f ∗ (x0 , y0 ) = j (z0 , z0 ) − j ∗ (z0 , z∗0 ) ∂y ∂z ∂z

(B.9) (B.10)

whence by solving for the derivatives with respect to z and z∗ we find ∂f (z0 , z∗0 ) = ∂z ∂f (z0 , z∗0 ) = ∂z∗

1 ∂f (x0 , y0 ) + 2 ∂x 1 ∂f (x0 , y0 ) − 2 ∂x

1 ∂f (x0 , y0 ) 2 j ∂y 1 ∂f (x0 , y0 ) 2 j ∂y

(B.11) (B.12)

and these expressions may serve as a definition for the partial derivatives of f (z, z∗ ), inasmuch as the right-hand sides exist by hypothesis. We observe that if the function f does not depend on z∗ , then (B.12) demands that ∂f 1 ∂f (x0 , y0 ) = (x0 , y0 ) ∂x j ∂y

(B.13)

since the derivative with respect to z∗ must vanish. Conversely, if (B.13) is verified, then f (x0 , y0 ) = f (z0 ). A complex function for which (B.13) holds true is said to be analytic in z0 . The requirement (B.13) is referred to as the Cauchy-Riemann condition in Cartesian form [7, Section 12.3]. The combination of (B.11) and (B.13) provides two equivalent ways for computing the derivative of f (x, y) = f (z). More importantly, (B.13) also implies that such derivative is ‘independent of the direction’. To elucidate, by extending the classic notion of derivative of a real function [1, Chapter 5] we may write f (z) − f (z0 ) df (z0 ) := lim z→z0 dz z − z0

(B.14)

which apparently makes sense if, and only if, the limit is unique no matter how we let z approach z0 in the complex plane. Let us see how (B.13) helps by letting z = x0 + j y0 + h first df f (x0 + j y0 + h) − f (x0 + j y0 ) ∂ f (z0 ) := lim = (x0 , y0 ) h→0 dz h ∂x

(B.15)

which amounts to reaching z0 along a straight line parallel to the real axis x, because the imaginary part of z is constant and equal to y0 . Secondly, we consider z = x0 + j(y0 + h) df f (x0 + j y0 + j h) − f (x0 + j y0 ) 1 ∂ f (z0 ) := lim = (x0 , y0 ) h→0 dz jh j ∂y

(B.16)

which consists of moving towards z0 along a straight line parallel to the imaginary axis y, since the real part is kept equal to x0 . Now, the results of the two approaches can only coincide if f (z) satisfies the Cauchy-Riemann condition. A function for which (B.13) holds for points z ∈ U where U ⊂ C is an open set (Figure B.1) is said analytic or holomorphic in U. We shall show in Section B.2 that a complex function f (z) possesses infinitely many derivatives for values z in the region U where it is analytic. This feature

Complex analysis

1103

Figure B.1 For reviewing the properties of analytic functions: an open set U in the complex plane. has no counterpart in the context of real functions of one or more real variables where, in general, after taking a certain number of derivatives, one arrives at a distribution (Appendix C). We can express (B.13) in an alternative form by separating the analytic function f (z) into its real and imaginary parts, namely, f (z) = Re{ f (z)} + j Im{ f (z)} = u(x, y) + j v(x, y)

(B.17)

where we interpret u and v as real-valued scalar functions of x and y given by (B.6) and (B.7). By inserting the above into (B.13) and exploiting the linearity of the differential operators we get the identity ∂u ∂u 1 ∂v ∂v (x0 , y0 ) + j (x0 , y0 ) = (x0 , y0 ) + (x0 , y0 ) ∂x ∂y j ∂x ∂y

(B.18)

which, being a relationship between complex quantities, can only be true if real and imaginary parts in either side coincide. This observation leads us to the pair of conditions ∂u ∂v (x0 , y0 ) = (x0 , y0 ) ∂x ∂y

∂u ∂v (x0 , y0 ) = − (x0 , y0 ) ∂y ∂x

(B.19)

on account of (B.5). Further, if we define the two-dimensional real-valued vector field E(x, y) = vˆx + uˆy in the region DU := {(x, y) ∈ R2 : z = x + j y ∈ U} with U ⊂ C being the open set where f (z) satisfies the Cauchy-Riemann conditions, we may clearly rephrase (B.19) as zˆ · ∇ × E(x, y) = 0,

∇t · E(x, y) = 0,

(x, y) ∈ DU

(B.20)

whereby we see that E(x, y) is a harmonic vector field and can be derived from the gradient of a scalar potential Φ(x, y) which in turn is harmonic in DU . Thus, we can alternatively claim that if f (z) is analytic in U, then its imaginary and real parts must be the Cartesian components of a harmonic vector field in DU . What is more, in view of (2.4), (2.3) and (1.117) it is natural to interpret E(x, y) as the two-dimensional electrostatic field in source-free regions of R2 . Indeed, this shows that there is an intimate relationship between analytic functions and two-dimensional static electric fields on the plane R2 [2, Chapter 4]. An important consequence of (B.19) is that the real and imaginary parts of a holomorphic function f (z) obey the two-dimensional Laplace equation in DU [8, Section II.2]. To prove this statement

Advanced Theoretical and Numerical Electromagnetics

1104

we need to take the second derivatives of the two equations in (B.19) for points (x, y) ∈ DU . In symbols, we have ∂2 v ∂2 u ∂2 v ∂2 u = = − = ∂x2 ∂x∂y ∂y∂x ∂y2 2 2 2 ∂u ∂2 v ∂u ∂v = = − = ∂y2 ∂y∂x ∂x∂y ∂x2

∂2 u ∂2 u + =0 ∂x2 ∂y2 ∂2 v ∂2 v ∇2t v = 2 + 2 = 0 ∂x ∂y

=⇒

∇2t u =

=⇒

(B.21) (B.22)

having invoked the Schwarz theorem [1, pp. 235–236]. Thus, the real and imaginary parts of an analytic function are harmonic functions in DU , and as such they cannot have local maxima or minima for (x, u) ∈ DU . It follows that u(x, y) and v(x, y) can only have saddle points in DU and reach their extrema on the boundary ∂DU [3, 4]. Example B.1 (Applying the Cauchy-Riemann condition) The function f (z) = 1

(B.23)

is evidently analytic everywhere in the complex plane, including the point z = ∞. Is the function f (z) = z2 = x2 − y2 + 2 j xy

(B.24)

analytic? Let us compute the derivatives with respect to x and y ∂f = 2x + 2 j y = 2z, ∂x

1 ∂f = 2 j y + 2x = 2z j ∂y

(B.25)

therefore the Cauchy-Riemann condition is fulfilled. As a counterexample we consider the complex function 1 z + 3z∗ f (x, y) = 2x − j y = z + z∗ − (z − z∗ ) = 2 2

(B.26)

and the expression in the rightmost-hand side clearly shows that f (x, y) depends on both z and z∗ , so it should not satisfy (B.13). Let us check this out ∂f = 2, ∂x

1 ∂f = −1 j ∂y

(B.27)

whence the Cauchy-Riemann condition does not hold. Then again, the application of the classic definition of derivative (B.14) reveals that the limit does depend on the direction chosen to approach the point z0 . (End of Example B.1)

A curve or line γ in the complex plane is a map from a real interval [a, b] ⊆ R onto C, namely,  γ:

[a, b] → C s → z(s) = x(s) + j y(s)

(B.28)

Complex analysis

1105

whereas a loop or contour in the complex plane is a curve γ such that z(a) = z(b)

(B.29)

that is, the initial and final points coincide. The graph of γ is the set of points z(s), and for simplicity we will use the same letter to denote both the curve (which is the map, strictly speaking) and the graph thereof. The graph of a contour γ divides the complex plane into two subsets, namely, the inside Uγ of γ, which is bounded, and the complement thereof C \ U γ , which is unbounded. The positive orientation of γ is conventionally chosen so that an ideal observer walking along γ sees Uγ to his or her left side. The derivative of the function z(s) with respect to s is a complex number whose real and imaginary parts represent the Cartesian components of a two-dimensional vector which is tangential to the graph of γ at z(s). Therefore, based on the notion of line integral of a two-dimensional vector field along a path in the plane R2 it is possible to define the line integral of a complex function f (z) as follows  b dz (B.30) dz f (z) = ds f (z(s)) ds γ

a

with γ given by (B.28). Suppose now that the graph of a contour γ is contained in an open set U in the complex plane, as is exemplified in Figure B.2a. If the inside Uγ of (the graph of) γ is entirely contained in U and a function f (z) is analytic for z ∈ U then  dz f (z) = 0 (B.31) γ

that is, the line integral of an analytic function in U along any contour in U vanishes. The result can be proved by using the Cauchy-Riemann conditions (B.19) and the Stokes theorem (A.55). The converse statement (Morera’s theorem [9]) is also true, i.e., if f (z) is continuous in U and (B.31) holds for any γ contained in U, then f (z) is analytic for z ∈ U. Although in this case the result does not depend on the orientation of the contour, the line integral (B.31) is computed by assuming that γ is positively oriented in accordance with the definition mentioned above. An immediate and useful consequence of (B.31) is that the line integral of f (z) along any curve that joins the same two points z1 and z2 in the complex plane (Figure B.2b) is independent of the specific path chosen. In symbols, this reads   dz f (z) = dz f (z) (B.32) γ1

γ2

and it follows from (B.31) applied to the contour Γ = γ1 ∪ (−γ2 ). With −γ2 we mean the line which connects z2 to z1 along γ2 , and this can be visualized as moving along γ2 in the opposite direction. Since from the viewpoint of (B.30) swapping the direction means   dz f (z) = − dz f (z) (B.33) −γ

γ

(B.32) follows easily. It is important to notice the striking similarity between (B.31) and (B.32) on the one hand and (2.2) and (2.45) on the other. This gives further evidence of the relationship between analytic functions and two-dimensional electrostatic fields in source free regions [2, Chapter 4].

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Advanced Theoretical and Numerical Electromagnetics

(a)

(b)

Figure B.2 For reviewing the properties of analytic functions: (a) a contour (a closed path) contained in U; (b) two lines γ1 and γ2 contained in U and joining the same two points.

B.2 Poles and residues We say that a complex function f (z) is singular or has a singularity in those points where it is not analytic. In particular, if f (z) can be put in the form f (z) =

g(z) , (z − z0 )n

n ∈ N \ {0}

(B.34)

where g(z) is analytic in z0 and g(z0 )  0, we say that f (z) exhibits a pole of order n in z0 . For instance, the function f (z) =

1 z−j

(B.35)

is analytic everywhere except for z = j where it has a first-order (n = 1) or simple pole. By contrast, although the function f (z) =

sin(z − 1) z−1

(B.36)

is cast in the form (B.34), the point z = 1 is not a simple pole, because sin(z − 1) vanishes for z = 1. In fact, since the limit of f (z) for z → 1 exists and is finite, viz., lim z→1

sin(z − 1) =1 z−1

(B.37)

we say that z = 1 is an apparent or removable singularity. Suppose that f (z) is analytic in U except at z0 where it has a first-order pole and let γ be a contour that, being contained in U, also encircles z0 (Figure B.3). We consider the line integral of f (z) along γ and observe that (B.31) may not be invoked because f (z) is not analytic everywhere in the inside Uγ of γ. Therefore, we exclude the singularity with the open disk C := {z ∈ C : |z−z0 | < a} where the radius a is chosen small enough for C to be contained in Uγ . The function f (z) is evidently analytic for points z ∈ Uγ \ C, but the boundary of the open set Uγ \ C is comprised of two disjoint contours, namely, γ and the circle ∂C := {z ∈ C : |z − z0 | = a}. So, in order to use (B.31) we connect

Complex analysis

1107

Figure B.3 Line integrals around the pole z0 for the application of (B.31) and the calculation of the residue (B.40). The segments AA and BB — which in actuality are perfectly overlapped — are drawn slightly apart for the sake of visualization. γ and ∂C by means of two straight segments AA and BB , as is graphically suggested in Figure B.3. The set of points γ∪(−∂C)∪AA ∪ BB is now a contour, and the line integral thereon vanishes. More importantly, since AA and BB are, in effect, the same segment oriented in two opposite directions, according to (B.33) the line integrals along AA and BB yield opposite contributions which cancel out. This is an application of the so-called cancellation principle in the complex plane. In the end, we are left with    dz f (z) = dz f (z) − dz f (z) (B.38) 0= γ

γ∪(−∂C)

∂C

whence   dz f (z) = dz f (z) γ

(B.39)

∂C

on account of (B.33). Identity (B.39) states that the line integral along the contour γ can be equivalently computed by integrating f (z) along the circumference ∂C. Since γ is somewhat arbitrary, (B.39) also says that the integrals along any contour which encircles the singularity yield the same result. Therefore, it makes sense to define the unique complex number   1 1 g(z) dz f (z) = dz (B.40) Res f (z0 ) := 2π j 2π j z − z0 γ

γ

which is called the residue of f (z) in z0 . To compute the integral we take advantage of (B.39) and choose γ ≡ ∂C in that this assumption simplifies the calculation by far. The points along ∂C are given by z(s) = z0 + aej s = x0 + a cos s + j(y0 + a sin s),

s ∈ [0, 2π]

whereby (B.40) becomes 1 Res f (z0 ) = 2π j

 ∂C

g(z) 1 dz = z − z0 2π j

2π ds a j ej s 0

g(z0 + a exp(j s)) aej s

(B.41)

1108

Advanced Theoretical and Numerical Electromagnetics 1 = 2π

2π ds g(z0 + a exp(j s))

(B.42)

0

in light of (B.30) and (B.34). The result ought to be independent of the radius a, since (B.40) holds for any contour and, in particular, any disk centered in z0 , so we take the limit of the right-hand side for vanishing a. We observe that the function g(z) is analytic and hence continuous in z0 . This means that for all  > 0 there exists a number δ > 0 such that if |z − z0 | < δ then |g(z) − g(z0 )| < . Thus, we estimate       2π   2π 1   1  ds g(z0 + a exp(j s)) − g(z0 ) =  ds [g(z0 + a exp(j s)) − g(z0 )]  2π 2π     0

0



1 2π

2π ds |g(z0 + a exp(j s)) − g(z0 )| < 

(B.43)

0

by choosing a < δ . Since  and hence a can be made arbitrarily small, we obtain the result 1 Res f (z0 ) = lim+ a→0 2π

2π ds g(z0 + a exp(j s)) = g(z0 )

(B.44)

0

with g(z) = (z − z0 ) f (z)

(B.45)

by virtue of (B.34) for n = 1. Moreover, (B.44) and (B.34) provide an alternative way for computing the residue of f (z) in the pole z0 , namely, Res f (z0 ) = g(z0 ) = lim (z − z0 ) f (z) z→z0

(B.46)

because the function g(z) is analytic and hence continuous in z0 . As a byproduct of the formal calculation of the residue we have also found the expression  1 g(z) dz = g(z0 ) (B.47) 2π j z − z0 γ

under the hypothesis that g(z) is analytic for points z ∈ Uγ . Therefore, we may rephrase the result above for a generic complex function f (z) analytic in U by writing  1 f (ζ) (B.48) f (z) = dζ 2π j ζ−z γ

where z is any point in Uγ ⊂ U. This expression goes by the name of Cauchy integral formula [7, Section 12.6] and essentially states that an analytic function is entirely specified in a region of the complex plane by assigning the values of f on the contour which bounds the region. In fact, we may regard (B.48) as an integral representation of f (z) for z ∈ Uγ . More importantly, since for ζ ∈ γ and

Complex analysis

1109

Figure B.4 Geometrical setup for the derivation of the Cauchy integral formula for points z on the contour γ. z ∈ Uγ the function f (ζ)/(ζ − z) is analytic with respect to ζ and z, we can take take the derivative with respect to z of both sides in (B.48) to obtain  1 f (ζ) df = dζ , z ∈ Uγ (B.49) dz 2π j (ζ − z)2 γ

because interchanging derivative and integral is permitted. Likewise, since now f (ζ)/(ζ − z)2 is also analytic for ζ ∈ γ and z ∈ Uγ , the process can be repeated. At the nth step we find  dn f n! f (ζ) = dζ , z ∈ Uγ , n∈N (B.50) n dz 2π j (ζ − z)n+1 γ

i.e., an integral representation for the nth derivative of f (z) in Uγ . This proves constructively that an analytic function can be differentiated infinitely many times, since n is arbitrary. On the other hand, it is straightforward to check that, if we take z ∈ C \ U γ , (B.48) returns zero, because the integrand is then analytic throughout Uγ and (B.31) applies. Finally, to determine the value predicted by (B.48) in case z ∈ γ we exclude z with the open disk C := {ζ ∈ C : |ζ − z| < a}, where the radius a > 0 is chosen small enough for the intersection γ ∩ C to be non-empty. We may consider the contour Γ := γ ∪ γ

where γ := {ζ ∈ γ : |ζ − z|  a} and γ

:= ∂C ∩ Uγ , as is suggested in Figure B.4. Since the integrand is analytic in UΓ , the inside of γ ∪ γ

, (B.31) holds and we have    1 1 f (ζ) f (ζ) f (ζ) 1 = + (B.51) dζ dζ dζ 0= 2π j ζ − z 2π j ζ − z 2π j ζ −z Γ

γ

γ

and notice that in the limit as a → 0+ the line γ reduces to the contour γ. Therefore, the first integral yields the Cauchy principal value of the right-hand side of (B.48). As for the other integral we denote the start and end points of the line γ with ζ1 and ζ2 (Figure B.4) and choose the following parametric representation of the arc γ

ζ(s) = z + aej{ϑ2 (a)−s[ϑ2 (a)−ϑ1 (a)]} ,

s ∈ [0, 1]

(B.52)

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Advanced Theoretical and Numerical Electromagnetics

(a)

(b)

Figure B.5 For stating and proving the Cauchy theorem: (a) a contour γ encircling N poles (•) of the function f (z); (b) modified path for computing the line integral along γ. where ϑ1 (a) and ϑ2 (a) indicate the phases of the complex numbers ζ1 − z and ζ2 − z, respectively. Then, we write 1 2π j

 γ

ϑ(a) f (ζ) =− dζ ζ−z 2π

1

ϑ(a) ds f (ζ(s)) = − 2π

0

1

  ϑ(a) f (z) ds f (ζ(s)) − f (z) − 2π

(B.53)

0

where ϑ(a) = ϑ2 (a) − ϑ1 (a) > 0 denotes the angle subtended by γ

at z. If the contour γ is smooth at z, then ϑ(a) approaches π as the radius a is made infinitely small.1 The first contribution in the right-hand side of (B.53) vanishes in the limit as a → 0+ by virtue of the estimate    1  1    ds f (ζ(s)) − f (z)   ds | f (ζ(s)) − f (z)| <  (B.54)   0  0 because f (ζ) is analytic and hence continuous in Uγ ⊂ U. By collecting all our findings and taking the limit of (B.51) for a → 0+ we conclude  1 1 f (ζ) f (z) = , z∈γ (B.55) PV dζ 2 2π j ζ−z γ

that is, the Cauchy integral formula returns half the value of the function if z ∈ γ. We may extend the definition of residue (B.40) to the situation where the function f (z) is analytic in U except in a number of points zn , n = 1, 2, . . . , N, in which f (z) exhibits first-order poles (Figure B.5a). We use the expedient of excluding the singularities by means of N disks Cn := {z ∈ C : |z − zn | < a} in order to obtain a contour which bounds a region where f (z) is analytic so that (B.31) may be applied. As exemplified in Figure B.5b, the disks are joined to the original contour γ by means of pairs of straight segments oriented in opposite directions, whereby the corresponding line integrals contribute naught. Thus, we find 0= 1

1 2π j

 dz f (z) − γ

 N  1 dz f (z) 2π j n=1 ∂Cn

This result can be regarded as the complex-plane/two-dimensional analogue of (F.8) for smooth surfaces.

(B.56)

Complex analysis

1111

and observe that in light of (B.39) and (B.40) the nth term of the sum above is, in fact, the residue of f (z) in the pole zn . Hence, we have proved the formula  dz f (z) = 2π j

N 

Res f (zn )

(B.57)

n=1

γ

which is known as the Cauchy theorem of residues (1922). The latter provides a powerful and convenient way for evaluating contour integrals in the complex plane once the position of the poles of the integrand are known, inasmuch as the residues can be computed through (B.46). For the sake of completeness we derive here two inequalities due to C. Jordan [3, pp. 167–168], namely, 1−

2α  cos α  1, π 2α  sin α  α, π

α ∈ [0, π/2]

(B.58)

α ∈ [0, π/2]

(B.59)

which turn out particularly useful in estimating line integrals of complex exponentials when (part of) the integration path is brought to infinity. We observe that for α ∈ [0, π/2] the mean values of sine and cosine are monotonic increasing and decreasing functions of α, respectively. In symbols, we have 1 0 α

α

dα sin α =

1 − cos α 2  α π

(B.60)

dα cos α =

sin α 2  α π

(B.61)

0

1

1 α

α 0

and inversion of the above inequalities with respect to cos α and sin α yields (B.58) and (B.59). Example B.2 (Calculation of inverse Fourier transforms with the Cauchy theorem) As an application of (B.57) we consider the calculation of a one-dimensional inverse spatial Fourier transform. Actually, to show that we indeed arrive at the correct result we proceed backwards, as it were, and with an eye towards electromagnetic waves (Chapter 7) we start with the one-dimensional function ⎧ ⎪ x0 and first obtain the direct spatial Fourier transform. From a physical viewpoint the function f (x) may represent the only non-zero component of the electric field of a uniform plane wave which travels in the positive x-direction in half space x > 0 (see Section 7.2). Then, k is the complex wavenumber given by (1.249), if the region is filled with a homogeneous isotropic lossy medium. By definition of spatial Fourier transformation [10] we have +∞ +∞ +∞



j kx x j k x x − j kx dx e f (x) = dx e e = dx ej(kx −k )x e−(kx +k )x F(k x ) := −∞

0

0

(B.63)

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Advanced Theoretical and Numerical Electromagnetics

(a) upper complex half-plane

(b) lower complex half-plane

Figure B.6 Contours (−−) for the calculation of an integral along the real axis. where k x = k x +j k

x ∈ C is the spectral variable. We notice that the improper integral in the rightmost member converges only if the argument of the real exponential is strictly negative, which occurs only if k

x > −k

. Since by hypothesis k

> 0, the Fourier transform exists for values of k x in the open region U := {k x ∈ C : k

x > −Δ > −k

, Δ > 0}, that is, the upper imaginary half-plane including the real axis k x (Figure B.6a). Proceeding with the calculation we get

F(k x ) =





ej(kx −k )x e−(kx +k j(k x − k)



 )x +∞ 0

=

j , kx − k

k

x > −k

(B.64)

where the function in the rightmost-hand side is analytic everywhere in the complex plane except for k x = k, where it possesses a first-order pole. However, only when k x ∈ U does the same function constitute the Fourier transform of f (x) and hence admit the integral representation (B.63)! Although we already know the answer in this case, we wish to retrieve the function f (x) by employing the definition of inverse Fourier transform, viz., 1 f (x) := 2π

+∞ dk x e− j kx x −∞

j kx − k

(B.65)

and to compute the integral we would like to invoke the Cauchy theorem (B.57). To this purpose we need to consider integrals along two contours in the complex plane k x = k x + j k

x , namely, Γ+a := {k x ∈ C : k x = ξ, ξ ∈ [−a, a] ∪ k x = aej α , α ∈ [0, π]}

(B.66)

Γ−a

(B.67)

:= {k x ∈ C : k x = ξ, ξ ∈ [−a, a] ∪ k x = ae

−jα

, α ∈ [0, π]}

with a > 0. With these definitions, the orientation of Γ+a is counterclockwise and that of Γ−a is clockwise. Both Γ+a and Γ−a are comprised of a part of the real axis and a half-circle of radius a. Thus, in the limit as a → +∞ we expect to recover the improper integral (B.65) along the real axis, provided the contour integrals along Γ+a and Γ−a converge and, in particular, vanish. Starting with Γ+a (see Figure B.6a) we compute  1 j =0 (B.68) dk x e− j kx x 2π j kx − k Γ+a

Complex analysis

1113

thanks to (B.31) because the integrand is analytic for points k x in the inside (of the graph) of Γ+a . We continue with Γ−a and obtain (Figure B.6b)  1 j = −Rese− j kx x j (k) = − j e− j kx dk x e− j kx x (B.69) k x −k 2π j kx − k Γ−a

by virtue of the Cauchy theorem (B.57) and the alternative formula for the residue (B.46). The negative of the residue is required because the orientation of Γ−a is clockwise, and (B.33) applies. Since our goal is the evaluation of (B.65) we split the contour integrals into line integrals along part of the real axis and the half-circles, namely,  1 j dk x e− j kx x 0= 2π j kx − k Γ+a

1 = 2π j

+a dξ e

− j ξx

−a

− j e− j kx =

1 2π j

1 = 2π j



1 j + ξ − k 2π

dk x e− j kx x

Γ−a

+a dξ e −a

− j ξx

π

a j e− j ax cos α eax sin α aej α − k

dα 0

(B.70)

j kx − k

1 j − ξ − k 2π

π dα 0

a j e− j ax cos α e−ax sin α aej α − k

(B.71)

where the left-hand sides are independent of a. The last terms in the rightmost members may remain finite as a → +∞ only if x < 0 and x > 0, respectively. This means, in practice, that we may use (B.70) to determine the inverse Fourier transform for x < 0 and (B.71) for the positive values of the spatial variable. We are left with the task of showing that the integrals along the half-circles vanish in the limit a → +∞. For x > 0 we have the estimate   π π π π/2   − j ax cos α −ax sin α  a j e ae−ax sin α e −ax sin α    dα dα j α = 2M dα e−ax sin α  M dα e   aej α − k |ae − k| 0

0

0

π/2

dα e−2axα/π = M

 2M

0

π (1 − e−ax ) −−−−−→ 0 a→+∞ ax

(B.72)

0

where M > 1 is a suitable constant (see note on page 474) and we have used the Jordan inequality (B.59). Thus, in the limit the contour integral along Γ−a reduces to an improper integral along the real axis, and from (B.71) we arrive at 1 2π j

+∞ dξ e− j ξx −∞

j = − j e− j kx ξ−k

(B.73)

for x > 0. Dividing through by − j yields an explicit expression for the desired inverse Fourier transform (B.65) for x > 0 and, evidently, the result coincides with the original function f (x). Analogous steps applied to (B.70) provide f (x) for the negative values of the spatial variable. (End of Example B.2)

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Advanced Theoretical and Numerical Electromagnetics

B.3 Branch points and Riemann surfaces Complex functions may exhibit singular points other than just poles of order n ∈ N \ {0}, as defined by (B.34). To elaborate, we consider a classic example, namely, the square root of z ∈ C [2, Section 4.4]  C→C √ (B.74) f : z → w = z which is better studied by writing the number z in exponential form, say, z = ρ ej θ+2π j n = ρ ej θ e2π j n ,

n∈Z

(B.75)

where

√ ρ := |z| = + z z∗  0 is the magnitude of z; θ = arctan(y/x) ∈ [−π, π[ is the principal phase of z; θ + 2πn, n ∈ Z, denote all the possible phases of z.

• • •

Confessedly, as far as the complex number z is concerned, we have been too fastidious with notation (B.75), because exp(2π j n) = 1 and hence this factor may be safely and conveniently omitted. However, when we make use of (B.75) in the right-hand side of (B.74) we obtain √ √ n∈Z (B.76) w = z = ρej θ/2 ej πn , and since e

j πn

⎧ ⎪ n = 2m ⎨1, = (−1) = ⎪ ⎩−1, n = 2m + 1 n

m∈Z

in general we end up with two possible values for w, viz., √ θ ∈ [−π, π[ w = ± ρej θ/2 ,

(B.77)

(B.78)

which is the expected well-known result (Figure B.7). Needless to say, had we been so cavalier as to ‘overlook’ the factor exp(2π j n), we would have missed the second value in (B.78). In actuality, this straightforward analysis indicates that the square root is an operation which takes z ∈ C and returns two complex numbers that are the negative of each other. Complex functions which associate z ∈ C with two or even more numbers in the complex plane w ∈ C are said to be many-valued or multivalued. Going back to (B.74) or (B.78) we see by direct inspection that z = 0 and z = ∞ are special points,2 exceptions of sorts, because in these cases ρ = 0 or ρ = +∞, and the square root invariably returns zero or infinity, i.e., one number instead of two. The numbers that a many-valued function f (z) maps to a single value in the plane w ∈ C are termed branch points, and the reason for this name will become clear in a moment. Branch points — which always occur in pairs — constitute singularities of f (z), though this has little to do with the specific value taken on by f (z) therein, e.g., the function may or may not be infinite. Rather, the problem is that something weird happens when ‘moving’, so to speak, on a curve around a branch point, as we are about to discuss. With reference to (B.74) and (B.78), we suppose that an ideal observer walks counterclockwise along the circle ∂C1 := {z ∈ C : |z| = a}, by starting from the point z1 = a exp(− j π) = −a. After 2

Perhaps it is useful to recall that in the set of complex numbers there exists only one point at infinity.

Complex analysis

1115

√ Figure B.7 Mapping properties of the many-valued function w = z. For visualization’s sake the two circles ∂C1 and ∂C2 are given slightly different radii but in reality they are perfectly overlapped.

completing a full lap around the origin, the observer finds himself at the point z2 = a exp(j π) = −a. This process is illustrated in the left part of Figure B.7, where ∂C1 is drawn by means of a continuous line. Meanwhile, according to (B.78) the function f (z) has mapped ∂C1 onto the line γ1 in the plane w ∈ C, i.e.,  γ1 :

[−π/2, π/2] → C √ θ → w(θ) = aej θ/2

(B.79)

√ √ that is, a semi-circle √ of radius a, centered in w = 0 and extending from the point w1 = − j a up to the point w2 = j a, as is sketched in the right half of Figure B.7. In practice, the values assumed √ by z on either side of the negative real axis are different, and this is the reason why a branch point constitutes a singularity. What is more, this also explains why we should treat z1 and z2 as two distinct points as well, even though they seem to coincide. Now, the unwavering observer keeps going and walks a second complete lap around the branch point until he reaches z3 = a exp(3 j π) = −a. The relevant path is the circle ∂C2 := {z ∈ C : |z| = a} which, in the left part of Figure B.7, we have drawn as a dashed line for clarity’s sake. In the plane w, the observer’s stroll produces one more semi-circle which in parametric form reads  γ2 :

[π/2, 3π/2] → C √ θ → w(θ) = aej θ/2

(B.80)

and extends the previous one √ (B.79) from w2 down √ to w3 = w1 , and this gives rise to the loop γ1 ∪ γ2 := {w ∈ C : |w| = a}, a circle of radius a. Since now w3 = w1 , it is also reasonable to assume z3 = z1 in the complex plane z. To recapitulate, it takes the observer two full laps around the branch point z = 0 — the exact shape of the path is not critical, it does not have to be a circle at all — for the line in the plane w to fully encircle w = 0, i.e., the image of the branch point. For this reason, z = 0 is called an algebraic √ branch point of order 2 − 1 = 1 of the function z. More generally, if it takes h laps, the branch point has order h − 1, and if the required number of rounds is infinite, the branch point is termed

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Advanced Theoretical and Numerical Electromagnetics

√4 transcendental. As an example, the function z − 1 has an algebraic branch point of order 3 in z = 1, whereas the function log(z + j) exhibits a transcendental branch point in z = − j. From the observer’s viewpoint, he has just walked twice along one and only circle, say, ∂C ≡ ∂C1 or ∂C ≡ ∂C2 . Nonetheless, to be consistent with (B.78) it is better to think of the total path travelled by the observer as if it was made up of two overlapped, though conceptually distinct circles ∂C1 and ∂C2 which intersect one another at z = −a, precisely as we attempt to show in Figure B.7. Thanks to this expedient, the many-valued function (B.74) maps points of ∂C1 onto γ1 , and a similar √ remark applies to ∂C2 and γ2 . So, if we regard z as a link between ∂C1 ∪ ∂C2 and the circle γ1 ∪ γ2 in the plane w, the function becomes indeed single-valued. In order to make the function (B.74) single-valued for all numbers z we consider two distinct, overlapped complex planes ‘glued’ together along the negative real axis. The set of points ΩR thus √ defined is a mathematical entity called the Riemann surface of the function z, each plane is referred to as a sheet of the surface, and the curve along which the two sheets are joined is termed branch cut or branch line, because it connects two branch points. More generally, if the algebraic branch point is of order h − 1, then the Riemann surface is comprised of h sheets, whereas in the presence of transcendental branch points, the Riemann surface is made of infinitely many sheets. With regard to the function (B.74), the chosen Riemann surface is defined as the set of complex numbers ΩR := {z = ρ exp(j θ) : ρ  0, θ ∈ [−π, 3π]}

(B.81)

and the branch cut which connects z = 0 to z = ∞, i.e., γb := {z ∈ C : Re{z}  0, Im{z} = 0}

(B.82)

consists of the negative real axis. The latter is drawn as a dotted line in the left half of Figure B.7. We may decide quite arbitrarily that the points in the ‘upper’ sheet of the relevant Riemann surface √ are transformed into the right half-plane U1 := {w ∈ C : Re{w} > 0} and consequently, that z associates the points in the ‘lower’ sheet with the left half-plane U2 := {w ∈ C : Re{w} < 0}; U1 and U2 are the shadowed regions in the right part of Figure B.7. On a related score, it is easy to conclude that the branch line γb is transformed into the imaginary axis Re{w} = 0. Armed with these definitions, we can retrace our steps and state that the ideal observer walking √ around z = 0 actually moved first on the upper sheet of the Riemann surface of z, then he crossed the branch line γb to continue his stroll on the lower sheet of ΩR , and finally, by traversing the branch cut once more he found himself exactly at the starting point on the upper sheet of ΩR . It is worthwhile mentioning that the actual shape of a branch cut is arbitrary, the only requirement being that it connect two branch points. In our example (B.74) we could have equally well chosen the positive real axis or any fancy squiggle beginning in z = 0 and reaching infinity, and this of course demands we revise the details of our definition of Riemann surface (B.81). We know from (B.31) that if f (z) is analytic in a region U, then any line integral along a contour γ ⊂ U vanishes identically. When it comes to many-valued functions we need to decide on which sheet of the Riemann surface we are to carry out the integration, because this choice in turn determines the values assumed by f (z). For instance, if we integrate the function (B.74) from z1 to z2 along the circle ∂C1 — which lies on the upper sheet of the pertinent Riemann surface (B.81) — we have to use the branch or part of √ z whose image is the region U1 , viz.,  ∂C1

√ dz z =

π −π

π √ j θ/2 2 3/2 3 j θ/2 4 dθ j ae ae = a e = − j a3/2 3 3 −π jθ

(B.83)

Complex analysis

1117

whereas, by integrating from z2 to z3 = z1 along ∂C2 in the lower sheet we get 

√ dz z =

3π

√ j θ/2 dθ j ae ae = jθ

π

∂C2

π dα j a

3/2 3 j α/2 j π

e

−π

e

π 2 3/2 j α/2 4 = − a e = j a3/2 3 3 −π

(B.84)

that is, clearly two different and non-null numbers. The change of dummy variable θ = α + 2π is not √ essential, but it helps emphasize that for the calculation we must use the branch of z which maps onto U2 in the plane w (Figure B.7). √ The reason for the different results is that z is, in fact, many-valued and comprised of two branches. What is more, the integrals do not vanish, despite being carried out seemingly along loops, because the latter surround a branch point and, as we have extensively motivated, the branch cut γb √ constitutes a line of discontinuity for each of the two branches that make up z. Indeed, since z1 and z2 are conceptually two distinct points on the upper sheet of ΩR , the curve ∂C1 is not exactly  a loop, but rather just a line, and the same goes for ∂C2 . This is why we have not used the symbol dz. √ But then, since z is analytic everywhere on both sheets of ΩR except on the branch cut γb , it is instructive to integrate the function in the upper sheet, e.g., along the contour Γ defined as Γ := {z ∈ C : z = aej θ , θ ∈ [−π, π] ∪ z = ξ − a, ξ ∈ [0, a] ∪ z = −η, η ∈ [0, a]}

(B.85)

that is, the loop formed by the union of ∂C1 and two parallel segments which run on either side of γb , as is also exemplified in the left part of Figure B.7 by means of dash-dotted lines which join around z = 0. By taking into account (B.78) and (B.83) and by using (B.30) we find 

√ dz z =

Γ

 ∂C1

√ dz z +

a



dξ a − ξe 0

j π/2

a −

√ dη ηe− j π/2

0

4 2 2 = − j a3/2 + j a3/2 + j a3/2 = 0 3 3 3

(B.86)

in agreement with (B.31) as Γ does not encircle any singular point. In particular, it is important to √ notice that z takes on opposite values on the two segments set against either side of γb , and that the image of Γ in the plane w lies entirely in the region U1 (right part of Figure B.7).

References [1] [2] [3] [4] [5] [6] [7] [8]

Rudin W. Principles of mathematical analysis. 3rd ed. New York, NY: McGraw-Hill; 1976. Morse P, Feshbach H. Methods of Theoretical Physics. New York, NY: McGraw-Hill; 1953. Colombo S. Holomorphic Functions of One Variable. New York, NY: Gordon and Breach Science Publishers; 1983. Marsden JE. Basic Complex Analysis. San Francisco, CA: W. H. Freeman and Company; 1973. Henrici P. Applied and Computational Complex Analysis. Pure and Applied Mathematics. New York, NY: John Wiley & Sons, Inc.; 1974. Rudin W. Real and Complex Analysis. 3rd ed. London, UK: McGraw-Hill; 1987. Kellogg OD. Foundations of potential theory. Berlin Heidelberg: Springer-Verlag; 1929. Dautray R, Lions JL. Mathematical Analysis and Numerical Methods for Science and Technology - Physical Origins and Classical Methods. vol. 1. Berlin Heidelberg: Springer-Verlag; 1990.

1118 [9] [10]

Advanced Theoretical and Numerical Electromagnetics LePage. Complex variables and the Laplace transform for engineers. New York, NY: Dover Publications, Inc.; 1961. Stein EM, Weiss G. Introduction to Fourier Analysis on Euclidean Spaces. Princeton, NJ: Princeton University Press; 1971.

Appendix C

Dirac delta distributions

Generalized functions or distributions represent the extension of the classic notion of function and can be defined rigorously either as linear and continuous functionals over a set of test functions (Appendix D.3) or, alternatively, as the limit of sequences of functions [1, Chapter 1], [2]. As a result, a distribution can be associated with most well-behaved functions, whereas the converse is generally false. Indeed, there exist distributions that do not have a counterpart as a function, the most relevant case in point being the Dirac delta distribution [3, Appendix B]. In this appendix, we provide a brief review of definitions and basic properties of the delta distribution in one, two, and three dimensions as well as over lines and surfaces.

C.1

Definitions and properties

To begin with, we consider the family of ordinary real-valued functions f : R → R+ defined by   1 a/π 1 Im f (x) = 2 = (C.1) x − ja a + x2 π where a > 0 is a parameter [4, Section 1.4]. By taking a look at the graph1 of f (x) plotted in Figure C.1 for a few values of a, we see that as the latter is made increasingly smaller, the function becomes ever more peaked around the point x = 0. In the limit as a → 0+ f (x) vanishes everywhere except at x = 0, where it grows infinitely large. Nonetheless, the area of the flat unbounded surface limited by the graph of f (x) and the horizontal axis is finite and given by   x +∞ a/π 1  = 1 arctan dx 2 = (C.2) π a −∞ a + x2 R

that is, the result is independent of the parameter a. Therefore, we can formally but correctly write   a/π lim+ dx 2 = dx δ(x) = 1 (C.3) a→0 a + x2 R

R

and assume the rightmost equality as the defining property of the one-dimensional Dirac delta distribution, keeping in mind, though, that the second integral above is just a symbol because δ(x) is no ordinary function. Parenthetically, it is worthwhile remarking that in (C.3) it does not make sense to swap the order of integration with respect to x and limit with respect to a. The reason is that the integrand, regarded 1

The graph is the set of points γ := {(x, y) ∈ R2 : x ∈ R, y = f (x)}; γ is a line in the plane xOy.

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Advanced Theoretical and Numerical Electromagnetics

Figure C.1

A sequence of functions which tend to the one-dimensional delta distribution δ(x).

as a function of two variables (x, a), does not possess a unique limit for x → 0 and a → 0 or, stated another way, the limit depends on the path one takes to approach the troublesome point (0, 0). For instance, if we let x = 0 first, then the function diverges for a → 0, and this is essentially the piece of information we gather from Figure C.1. On the contrary, if we set a = 0 first, the function vanishes everywhere along the x-axis. A consequence of (C.3) is the sifting property of δ(x), namely,  dx φ(x)δ(x) = φ(0) (C.4) R

where the integral is again a symbol and φ(x) ∈ C∞ 0 (R) — called a test function — is an arbitrary infinitely differentiable real or complex function with support on a finite interval I ⊂ R. We recall that the support of a function is the smallest set of points x for which the function does not vanish. To prove (C.4) for the function f (x) defined above we notice that, since φ(x) is differentiable, then it is also Hölder continuous, viz., |φ(x2 ) − φ(x1 )|  M|x2 − x1 |α ,

x1 , x2 ∈ R

(C.5)

where M > 0 is a suitable constant and α ∈ ]0, 1]. Then, we estimate         a/π a/π  dx  =  dx  φ(x) − φ(0) [φ(x) − φ(0)]     a 2 + x2 a 2 + x2 R R   a/π a/π dx 2 |φ(x) − φ(0)|  dx 2 M|x|α  a + x2 a + x2 R



2Ma π

R

+∞ a2 0

xα Maα −−−−→ 0 = 2 cos(πα/2) a→0+ +x

(C.6)

Dirac delta distributions

1121

where we have used the Hölder condition with α ∈]0, 1[, the evenness of the integrand and formula 856.07 in [5] with a change of dummy variable. The support of a distribution is the smallest set of points for which it is non-null. Concerning the support of δ(x) we observe that the right-hand side of (C.4) vanishes if the support I of the test function does not include the origin of the x-axis. It follows that the δ-distribution is null for any point R \ {0}, but it cannot vanish also in the origin, or else the left member of (C.4) would be the null distribution. Thus, we conclude that the support of δ(x) is just the point {0}. It is possible to give a meaning to a Dirac delta that depends on x through a real-valued function g(x) which exhibits simple zeros in, say, xn , n = 1, . . . , N. We require that g(x) be sufficiently regular and in particular that its derivative exist and not vanish for x = xn . Further, we suppose that g(x) is comprised of N parts ξ = gn (x) each defined on a sub-interval In of R so that g−1 n (ξ) exists, xn ∈ In , and gn (xn ) = 0, with xn being the only zero of gn (x). Then, by formally applying the rule for changing a dummy variable of integration we have  dx φ(x)δ(g(x)) =

N  

dx φ(x)δ(gn(x)) =

n=1 I n

R

N  

 φ(xn ) δ(ξ) = |gn (x)| n=1 |gn (xn )| N

dξ φ(g−1 n (ξ))

n=1 Ξ

n

(C.7)

by virtue of (C.4). This result can be stated succinctly by writing δ(g(x)) =

N  δ(x − xn ) n=1

(C.8)

|g (xn )|

again in light of (C.4). As an example, we have δ(ax) =

δ(x) |a|

δ(x2 − a2 ) =

1 [δ(x − a) + δ(x + a)] 2|a|

(C.9)

where a  0. The support of δ(x2 − a2 ) is the set of points {−a, a}. Next, we consider the family of real-valued scalar fields f : R2 → R+ given by f (ρ) =

2π(a2

a + ρ2 )3/2

(C.10)

where a > 0 is a parameter. For ever smaller values of a the function becomes increasingly peaked around the point ρ = 0 (Figure C.2). In the limit as a → 0+ f (ρ) vanishes everywhere save in the origin where it diverges. However, the volume of the unbounded region limited by the graph2 of the function (C.10) and the xOy plane is independent of a and remains finite, namely,  R2

a dS = 2π(a2 + ρ2 )3/2

2π

+∞ dϕ dρ

0

0

+∞  ρa a = −  =1 2π(a2 + ρ2 )3/2 (a2 + ρ2 )1/2 0

and thus similarly to (C.3) we can write   a lim dS = dS δ(2) (ρ) = 1 a→0+ 2π(a2 + ρ2 )3/2 R2

(C.11)

(C.12)

R2

whereby we identify the rightmost-hand side as the definition of the two-dimensional Dirac distribution. Also in this case the integration must be carried out prior to letting a approach zero. Contrariwise, 2

The graph is the set of points Σ := {(x, y, z) ∈ R3 : (x, y) ∈ R2 , z = f (x, y)}; Σ is a surface in the three-dimensional space.

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Advanced Theoretical and Numerical Electromagnetics

Figure C.2

A sequence of functions which tend to the two-dimensional Dirac delta distribution δ(2) (ρ); slices of the graphs have been cut away for visualization’s sake.

it is immediately ascertained that by setting a = 0 in the integrand we obtain a function of ρ which is null everywhere, and the result of the integration would be quite different than what is stated by (C.12). A consequence of (C.12) is the sifting property  (C.13) dS φ(ρ)δ(2) (ρ) = φ(0) R2 2 2 where the integral is formal and φ(ρ) ∈ C∞ 0 (R ) is a scalar field with support S ⊂ R . The support is (2) (ρ) is the point {(0, 0)} ⊂ R2 . the smallest set of points ρ for which φ(ρ)  0, and the support of δ To prove (C.13) for the function (C.10) we notice that since φ(ρ) is differentiable, then it is also Hölder continuous, viz.,

|φ(ρ) − φ(0)|  M|ρ|α ,

ρ ∈ R2

(C.14)

where M > 0 is a suitable constant and α ∈ ]0, 1]. We proceed by estimating           aφ(ρ) a[φ(ρ) − φ(0)]   dS      − φ(0) =  dS  2 + ρ2 )3/2 2 + ρ2 )3/2  2π(a 2π(a     R2   R2   a|φ(ρ) − φ(0)| aM|ρ|α  dS  dS 2 2 3/2 2π(a + ρ ) 2π(a2 + ρ2 )3/2 R2

= aM 0

R2





Γ 1 + α2 Γ 1−α ρα+1 2 α

dρ 2 = Ma −−−−→ 0 3 a→0+ (a + ρ2 )3/2 Γ 2

+∞

(C.15)

Dirac delta distributions

1123

where we have used the Hölder condition with α ∈]0, 1[ and formula 856.11 in [5] after the change of variable ρ = aξ1/2 . The symbol Γ(•) denotes the gamma function or Euler integral of the second kind [6, Section 7.2], [7, Chapter 6], [8, Chapter 8]. Finally, we examine the family of real-valued radial scalar fields f : R3 → R+ given by f (r) =

3a2 4π(a2 + r2 )5/2

(C.16)

where a > 0 is a parameter. As a → 0+ , f (r) becomes increasingly steeper for r = 0 and ever smaller everywhere else. Nonetheless, the integral of f (r) over the whole space is finite and reads  R3

3a2 dV = 2 4π(a + r2 )5/2

+∞ dr 0

+∞  3r2 a2 r3 = 2  =1 2 3/2 2 2 5/2 (a + r ) 0 (a + r )

(C.17)

i.e., the result is independent of a. This quantity is the ‘volume’ of the four-dimensional unbounded region limited by the graph of w = f (r) and the hyperplane w = 0 (i.e., the space xyzO) in the four-dimensional space R4 . Similarly to (C.3) and (C.12) we can write  lim+

dV

a→0

R3

3a2 = 4π(a2 + r2 )5/2

 dV δ(3) (r) = 1

(C.18)

R3

and take the rightmost equality as the defining property of the three-dimensional Dirac delta distribution. A consequence of (C.18) is the sifting property  (C.19) dVφ(r)δ(3) (r) = φ(0) R3 3 3 where the integral is again a symbol and φ(r) ∈ C∞ 0 (R ) is a scalar field with support V ⊂ R , and the (3) (r) 3 is the point {(0, 0, 0)} ⊂ R . To obtain (C.19) from the function defined in (C.17) support of δ we observe that, since φ(r) is differentiable, then

|φ(r) − φ(0)|  M|r|,

r ∈ R3

(C.20)

with M > 0 a suitable constant (see Appendix A.7). Then, we estimate           2 2 3a φ(r) 3a [φ(r) − φ(0)]     dV    − φ(0) =  dV  2 + r2 )5/2 2 + r2 )5/2  4π(a 4π(a     R3   R3   2 2 3a |φ(r) − φ(0)| 3a M|r| dV  dV  4π(a2 + r2 )5/2 4π(a2 + r2 )5/2 R3

+∞ 2 = 3a M dr 0

R3

r3 = 2aM −−−−→ 0 a→0+ (a2 + r2 )5/2

where we have computed the last integral with the aid of formula 203.05 in [5].

(C.21)

Advanced Theoretical and Numerical Electromagnetics

1124

Furthermore, by applying the known properties of integration in more than one variable and invoking (C.3) two or three times, respectively, we can formally write    1= dx δ(x) dy δ(y) = dS δ(x)δ(y) (C.22) R

R

 1=

 dx δ(x)

R

R  dy δ(y) dz δ(z) = dV δ(x)δ(y)δ(z) 2

R

R

(C.23)

R3

which by comparison with (C.12) and (C.18) yield [1, Chapter 1] δ(2) (ρ) = δ(x)δ(y)

(C.24)

δ (r) = δ(x)δ(y)δ(z)

(C.25)

(3)

the representation of δ(2) (r) and δ(3) (r) in Cartesian coordinates. The ‘multiplication’ of one-dimensional delta distributions in (C.24) and (C.25) is called tensor product or direct product of distributions. Likewise, by working with circular cylindrical coordinates (ρ, ϕ, z) and spherical coordinates (r, ϑ, ϕ) (Appendix A.1) we have 2π 1=

 +∞ δ(ρ) δ(ρ) = dϕ dρ ρ dS 2πρ 2πρ

0

0

π 1=

2π dθ sin θ

0

(C.26)

R2

dϕ 0

 +∞ δ(r) δ(r) dr r2 = dV 4πr2 4πr2

(C.27)

R3

0

under the additional assumptions that +∞ dρ δ(ρ) = 1

(C.28)

0

+∞ dr δ(r) = 1

(C.29)

0

which are motivated by (C.3). Again, comparison with (C.12) and (C.18) points to [1, Chapter 1], [9, Chapter 4], [10] δ(ρ) 2πρ δ(r) δ(3) (r) = 4πr2

δ(2) (ρ) =

(C.30) (C.31)

the representations of δ(2) (ρ) and δ(3) (r) in cylindrical and spherical coordinates. A surface delta distribution with support on the surface S is defined formally through   dV φ(r)δS (r − rS ) = dS φ(r) R3

S

(C.32)

Dirac delta distributions

1125

where rS indicates points on S ⊂ R3 . This is a sifting property of sorts in that the surface delta turns a volume integral into a surface one over the support [1, Chapter 1]. A line delta distribution with support the line C is formally given by   dV φ(r)δC (r − rC ) = ds φ(r) (C.33) R3

C

where rC indicates points on C ⊂ R3 . The sifting property in this case turns a volume integral into a line integral along C [1, Chapter 1].

C.2

Derivatives and weak operators

Occasionally, we will need to take the derivative of a distribution and, in particular, of the Dirac delta. How do we compute the derivative of something which is zero everywhere and infinite in just one point? To answer this question we take a step back and consider two complex one-dimensional functions, say, φ(x) ∈ C∞ 0 (R) and f (x) which, for the moment, we suppose to be at least continuous and hence integrable. From the theorem that provides the derivative of a product of functions [8, Chapter 5] and the linearity of the integral operator, we have    dφ df d f (x) + dx φ(x) = dx dx (φ f ) = [φ(x) f (x)]+∞ (C.34) −∞ dx dx dx R

R

R

where the convergence of the improper integrals is guaranteed if φ(x) has bounded support. In which case the finite increment of φ(x) f (x) vanishes and we find   df dφ = − dx f (x) (C.35) dx φ(x) dx dx R

R

where φ(x) constitutes a test function, according to the previous definition. Therefore, we may interpret the left-hand side as the distribution associated with the derivative of f (x) and the right member as an alternative way to compute the same distribution. When f (x) is at least continuous and differentiable, both approaches are permissible, but if f (x) is piecewise-continuous or even more singular, the derivative of f (x) is not defined in the ordinary sense [8, Chapter 5]. Nevertheless, we may compute the derivative of f (x) in a weak sense as the distribution given by  dφ df := − dx f (x) (C.36) dx dx R

because the derivative of the test function is always well defined. Repeated application of these ideas leads to the definition  dm f dm φ m := (−1) dx f (x) (C.37) dxm dxm R

for the mth derivative of a one-dimensional distribution. Apparently, since the test function is infinitely differentiable, a distribution possesses infinitely many derivatives, a property which has

1126

Advanced Theoretical and Numerical Electromagnetics

no counterpart for ordinary functions. It may seem obvious from (C.37), but to define the weak derivative of order m > 2 we do not need the existence of weak derivatives of lower order, contrary to what happens with the classical definition. In particular, when f (x) is the delta distribution, we interpret the integral (C.36) formally as in (C.4) to obtain  dδ dφ := − dx δ(x) = −φ (0) (C.38) δ (x) = dx dx R

by virtue of the sifting property of δ(x). Moreover, when we take f (x) = U(x), the one-dimensional step function — which is discontinuous in the origin — (C.36) yields dU := − dx

+∞ dφ = −[φ(x)]+∞ dx 0 = φ(0) dx

(C.39)

0

and comparison with (C.4) gives  dU = dx φ(x)δ(x) = δ(x) dx

(C.40)

R

which provides both a meaning to the derivative of the step function and an alternative definition of the Dirac δ-distribution. To extend the notion of weak derivative to higher-dimensional spaces we begin with a test 3 function φ(r) ∈ C∞ 0 (R ) and a scalar field f (r) which we take at least continuous at first. Then, we consider a ball B(0, a) and from formulas (H.47) and (H.89) we find    ˆ dV [ f (r)∇φ + φ(r)∇ f ] = dV ∇(φ f ) = dS n(r)φ(r) f (r) (C.41) B(0,a)

B(0,a)

∂B

where the surface integral is null if φ(r) has bounded support V ⊂ B(0, a). Thus, from (C.41) we obtain   dV φ(r)∇ f = − dV f (r)∇φ (C.42) R3

R3

and interpret the left-hand side as the distribution associated with the gradient of f (r). If the scalar field is not continuous, then only the expression in the right-hand side may be computed, for the gradient of the test function has a meaning in the usual sense. Therefore, we define  ∇ f := − dV f (r)∇φ (C.43) R3

as the weak gradient of f (r). By starting with (H.50) and (H.51) and carrying out similar steps with a vector field f(r) we arrive at  ∇ · f := − dV f(r) · ∇φ (C.44) R3

Dirac delta distributions

1127

 ∇ × f := −

dV ∇φ × f(r)

(C.45)

R3

as the definitions of weak divergence and weak curl. Finally, for the extension of the Laplace operator to distributions we consider a test function φ(r) and a scalar field f (r) at least differentiable, and apply the integral identity (H.97) to a ball B(0, a)    ˆ · [ f (r)∇φ − φ(r)∇ f ] dV f (r)∇2 φ − φ(r)∇2 f = dS n(r) (C.46) ∂B

B(0,a)

where the flux integral is null if the bounded support of φ(r) is contained in B(0, a). Hence, we find   dV φ(r)∇2 f = dV f (r)∇2 φ (C.47) R3

R3

and regard the left-hand side as the distribution associated with ∇2 f . However, if the scalar field is not differentiable or even discontinuous, then only the integral in the right-hand side makes sense, and we take  dV f (r)∇2 φ (C.48) ∇2 f := R3

as the definition of the weak laplacian of a scalar field. Extension to vector fields f(r) is straightforward and may be accomplished by applying (C.48) to the Cartesian components of f(r). Example C.1 (Derivatives of the three-dimensional step function) The three-dimensional step function is defined as ⎧ ⎪ ⎪ ⎨0, r ∈ V U(r) := ⎪ ⎪ ⎩1, r ∈ R3 \ V

(C.49)

where V is a finite region of space with smooth or piecewise-smooth boundary ∂V. Since U(r) is discontinuous across ∂V, the gradient of U(r) has a meaning in a distributional sense in accordance with (C.43) and reads     ˆ ˆ ˆ ∇U := − dV ∇φ = dS n(r)φ(r) − dS n(r)φ(r) = dS n(r)φ(r) (C.50) B(0,a)\V

∂V

∂B

∂V

where the support of φ(r) contains V, and the ball B(0, a) is large enough to enclose the support of φ(r). We have applied (H.89) with the unit normals pointing outward V and B(0, a) and also noticed that the integral over ∂B vanishes because the support of φ(r) is contained in B(0, a) \ V. Comparing (C.50) with (C.32) — where we let S ≡ ∂V — yields  ˆ ˆ dV n(r)φ(r)δ (C.51) ∇U = S (r − r∂V ) = n(r)δ S (r − r∂V ) R3

which somehow generalizes (C.40).

Advanced Theoretical and Numerical Electromagnetics

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As a further example we compute the divergence of the radial vector field U(r) rˆ , r ∈ R3 (C.52) 4πr2 where U(r) is the one-dimensional radial step function. For r > 0 the field is regular and the divergence simply vanishes on account of (A.31). However, if we wish to include the point r = 0 as well, then the divergence exists as a distribution. We need to use definition (C.44), viz., f(r) =

  +∞  1 U(r) ∂ U(r) rˆ := − dV rˆ · ∇φ(r) = − dΩ dr φ(r) ∇· 2 2 4π ∂r 4πr 4πr 4π 0 R3   1 1 =− dΩ [φ(r)]+∞ dΩ φ(0) = φ(0) r=0 = 4π 4π 



4π

(C.53)

4π

where the symbol 4π dΩ signifies integration in spherical polar coordinates over the full solid angle. Since the result of the calculation is the same as the formal expression (C.19), we conclude that the divergence of f(r) equals the three-dimensional delta distribution, i.e., in short-hand notation   δ(r) U(r) ˆ r = δ(3) (r) = (C.54) ∇· 4πr2 4πr2 also on account of (C.31). Lastly, we examine the Laplacian of the radial scalar field U(r) , r ∈ R3 (C.55) 4πr which is relevant in electrostatics and for stationary magnetic fields (Chapters 2, 3 and 4). For r > 0 the field is regular, and the Laplacian simply vanishes on the grounds of (A.42). If, on the other hand, we include the point r = 0, the Laplacian is a distribution which is best computed by means of (C.48). To evaluate the improper integral, which converges anyway, we consider two concentric balls B(0, a) and B(0, b) with the radius a < b and the radius b large enough for B(0, b) to contain the support of the test function φ(r). Then, we have    ∇2 φ ∇2 φ ∇2 φ = lim+ lim + lim+ dV dV dV ∇2 f := 4πr a→0 b→+∞ 4πr a→0 4πr B(0,a) B(0,b)\B(0,a) R3  =0    1 ∇φ −∇ · ∇φ = lim+ lim dV ∇ · a→0 b→+∞ 4πr 4πr B(0,b)\B(0,a)      1 ∇φ 1 = lim+ lim − ∇ · φ(r)∇ dV ∇ · + φ(r)∇2 (C.56) a→0 b→+∞ 4πr 4πr 4πr f (r) =

B(0,b)\B(0,a)

on account of (H.51). The integral over the ball B(0, a) vanishes in the limit as the radius tends to zero, in that it is dominated by a constant proportional to a. The last term in the remaining integrand vanishes for r ∈ B(0, b) \ B(0, a), and the Gauss theorem may be applied to the remaining part. In symbols, we find       ∇φ ∇φ 1 1 2 − φ(r)∇ − φ(r)∇ dS rˆ · dS rˆ · − lim+ (C.57) ∇ f = lim b→+∞ a→0 4πr 4πr 4πr 4πr |r|=b

|r|=a

Dirac delta distributions

1129

where the first integral vanishes in that φ(r) = 0 = ∇φ(r) for r = b, by hypothesis. For the remaining one, we have      rˆ · rˆ rˆ · ∇φ rˆ · ∇φ φ(r) ∇2 f = − lim+ + φ(r) − lim (C.58) dS a dΩ dΩ = − lim a→0 a→0+ a→0+ 4πa 4π 4π 4πa2 |r|=a

4π

4π

where the first term vanishes in the limit as a → 0+ because the integral is finite and independent of a. The second one is evaluated with the aid of the mean value theorem [6] ∇2 f = − lim+ φ(a, ϑ0 , ϕ0 ) = −φ(0) a→0

(C.59)

with ϑ0 and ϕ0 two suitable angles. Since the result of the calculation is the negative of the test function evaluated in the origin, we have proved that ∇2

U(r) = −δ(3) (r) 4πr

(C.60)

on the grounds of (C.19). Therefore (C.55) solves the distributional Poisson equation and is the distribution associated with the electrostatic Green function (2.130) derived in Section 2.6. (End of Example C.1)

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]

Van Bladel JG. Singular Electromagnetic Fields and Sources. Piscataway, NJ: IEEE Press; 1991. Jones DS. The Theory of Generalised Functions. Cambridge, UK: Cambridge University Press; 2009. Oughstun KE. Electromagnetic and Optical Pulse Propagation. vol. 1. New York, NY: Springer Science+Business Media; 2006. Milton KA, Schwinger J. Electromagnetic Radiation: Variational Methods, Waveguides and Accelerators. Berlin Heidelberg: Springer-Verlag; 2006. Dwight HB. Tables of Integrals and Other Mathematical Data. 4th ed. New York, NY: MacMillan Publishing Co., Inc.; 1961. Makarov B, Podkorytov A. Real Analysis: Measures, Integrals and Applications. Universitext. London, UK: Springer; 2013. Abramowitz M, Stegun IA. Handbook of mathematical functions. New York, NY: Dover Publications, Inc.; 1965. Rudin W. Principles of mathematical analysis. 3rd ed. New York, NY: McGraw-Hill; 1976. Dudley DG. Mathematical Foundations for Electromagnetic Theory. New York, NY: WileyInterscience; 1994. Stinson DC. Intermediate Mathematics of Electromagnetics. Englewood Cliffs, NJ: PrenticeHall; 1976.

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Appendix D

Functional analysis

D.1

Vector and function spaces

In mathematics a collection V of objects which satisfy the following properties ∀ u, v ∈ V ∀ v ∈ V, ∀ξ ∈ F

∃u+v=w∈V ∃ ξv = w ∈ V

(addition) (multiplication by a scalar)

(D.1) (D.2)

is called a vector space or linear space over the field F ∈ {R, C}, and the members of V are called vectors [1, Chapter 1], [2, Chapter 9], [3, Chapter 7], [4, Chapter 5], [5]. In words, (D.1) and (D.2) say that the operations of sum between two elements and product of an element by a scalar in F are defined and return an element of V as a result (this is the closure property). Important examples of linear spaces include triples of real numbers, ordinary vectors in three dimensions, sequences, matrices, continuous functions, or functionals to name but a few possibilities. A subspace Z of V is a subset of V such that operations (D.1) and (D.2) still hold but are restricted to the elements of Z, namely, ∀ u, v ∈ Z ⊆ V ∀ v ∈ Z ⊆ V,

∃u+v=w∈Z

(addition)

(D.3)

∃ ξv = w ∈ Z

(multiplication by a scalar)

(D.4)

(neutral element of the sum)

(D.5)

∀ξ ∈ F

and additionally 0∈V

=⇒

0∈Z

i.e., the null element of V must also belong to Z. Trivial subspaces which enjoy properties (D.3)(D.5) are V itself and Z := {0 ∈ V}. A subset D ⊂ V is said to be convex when the ‘line segment’ which joins any two ‘points’ v1 and v2 in D is entirely contained in D, viz., ∀ v1 , v2 ∈ D,

τ ∈ [0, 1]

=⇒

v := (1 − τ)v1 + τv2 ∈ D

(D.6)

and hence, linear subspaces are always convex by virtue of (D.3) and (D.4). A norm is a uniformly continuous map which associates an element v of a vector space V with a real non-negative number vV , namely, •V : V −→ R+

(D.7)

and satisfies the following properties: vV  0

(D.8)

Advanced Theoretical and Numerical Electromagnetics

1132

vV = 0 ⇐⇒ v = 0

(D.9)

ξvV = |ξ| vV , ξ∈F u + vV  uV + vV

(D.10) (D.11)

where • • •

(D.8) and (D.9) together form the non-negativity property or positive-definiteness; (D.10) is called the homogeneity property; (D.11) is known as the triangle inequality or subadditivity property and indicates that, in general, the norm is not a linear operation [cf. (D.120) in Appendix D.3].

Uniform continuity [3, Section 4.2] of •V over V follows by observing that for any two elements u, v ∈ V and a positive number η we have u − vV < η

=⇒

| uV − vV |  u − vV < η

(D.12)

thanks to the reverse triangle inequality (H.20). A function which obeys properties (D.8), (D.10) and (D.11) but fails to meet condition (D.9) is referred to as a semi-norm. A vector space equipped with a norm is called a normed space. All elements in a normed space must have a finite norm, that is to say, if an element has infinite norm, then it does not belong to the space. The notion of norm extends the concept of absolute value in R and of magnitude of an ordinary vector. For the same reason, given two vectors u, v ∈ V, the norm of the difference u − vV represents the distance between u and v [6, Chapter 2], [3, Chapgter 2], [7, Chapter 1]. Clearly, different definitions of norm are possible for the same vector space and hence the ‘distance’ between two vectors depends on the chosen norm in V. Given a vector v0 ∈ V and a real positive number a the subset B(v0 , a) := {v ∈ V : v − v0 V < a}

(D.13)

is called the open ball of radius a centered in v0 . In like fashion, the subset B[v0 , a] := {v ∈ V : v − v0 V  a}

(D.14)

is called the closed ball of radius a centered in v0 . It is a simple matter to check that balls are convex subsets of V in accordance with (D.6). We say that a subset D ⊂ V is open if for each vector v ∈ D there exists a ball B(v, a) entirely contained in D. Therefore, V is open because B(v, a) ⊆ V for any v ∈ V and any a > 0, whereas the empty set ∅ is open by default, as it contains no points. Conversely, a set D ⊂ V is said to be closed if its complement Dc := V \ D is open. A vector v0 ∈ D belongs to the frontier or boundary of D if any ball B(v0, a) contains vectors of D and vectors of V \ D. The frontier of D is denoted with ∂D. The closure of D is the set D := D ∪ ∂D, and the closure of a vector space V is the smallest space which contains V. Besides, a set D ⊂ V is bounded or limited if sup v1 − v2 V < +∞

v1 ,v2 ∈D

(D.15)

whereby balls are evidently bounded. If V has finite dimension, a subset D ⊂ V is said compact if it is bounded and closed. To extend the notion of compactness in infinite-dimensional spaces one first considers an open cover of D, i.e., a collection of open balls {B(vn , an )}+∞ n=1 in V, with possibly different radii, such that D⊆

+∞  n=1

B(vn , an )

(D.16)

Functional analysis

1133

and then, if for any given open cover of D ⊂ V it is still possible to cover D with a finite subcollection of balls, namely, D⊆

+∞ 

B(vn, an )

=⇒

∃ n1 , . . . , n N

D⊆

:

n=1

N 

B(vni , ani )

(D.17)

i=1

the subset D ⊂ V is said compact [3, Section 6.3], [2, p. 36]. It follows that any finite set, including the trivial ∅, is compact. Moreover, compact subsets in infinite-dimensional normed spaces are closed [2, Theorem 2.34]. Since throughout this book we are concerned with functions defined over the vector spaces V = Rn for n = 2, 3 (e.g., densities, potentials, fields, constitutive parameters) we specialize part of the previous discussion accordingly. The magnitude of a vector r = xˆx + yˆy + zˆz ∈ R3 is the non-negative number  1/2 |r| = r := x2 + y2 + z2

(D.18)

and it satisfies (D.8)-(D.11) with F = R. The subsets B3 (r0 , a) = B(r0, a) := {r ∈ R3 : |r − r0 | < a}

(D.19)

B3 [r0 , a] = B[r0, a] := {r ∈ R : |r − r0 |  a}

(D.20)

3

represent an open and a closed ball of radius a > 0 centered in r0 ∈ R3 [3, Chapter 2]. Besides, the subset ∂B(r0, a) := {r ∈ R3 : |r − r0 | = a}.

(D.21)

is called the sphere of radius a > 0 centered in r0 ∈ R3 and constitutes the boundary of B3 (r0 , a). In R2 the open ball of radius a > 0 and centered r0 ∈ R2 is the circle defined by B2 (r0 , a) := {r ∈ R2 : |r − r0 | < a}.

(D.22)

A set D ⊂ Rn is bounded (limited) if sup |r| < +∞

(D.23)

r∈D

whereas a set D ⊂ Rn is compact if it is bounded and closed. We recall that a (discrete) sequence of members of V, indicated succinctly with, e.g., {vn }+∞ n=1 , is formally a map from the set N of whole numbers to V. A sequence is said to converge (strongly) to a vector v ∈ V if [7, Definition 1.3], [1, p. 12] ∀η > 0

∃N ∈ N

:

∀n  N

=⇒

vn − vV < η

(D.24)

and this is compactly written as lim vn = v

n→+∞

(D.25)

or equivalently as lim vn − vV = 0

n→+∞

(D.26)

1134

Advanced Theoretical and Numerical Electromagnetics

which explains why strong convergence is also called convergence in the norm. By comparing (D.24) with definition (D.13) we may state that the elements of a convergent sequence are to be found in ever smaller balls centered in v. Among all the possible sequences in V, those which satisfy the property ∀η > 0

∃N ∈ N

:

∀ m, n  N

=⇒

vm − vn V < η

(D.27)

which is written succinctly as lim vm − vn V = 0

(D.28)

m,n→+∞

are termed Cauchy sequences [3, Definition 4.1], [7, Definition 1.3], [1, pp. 12–13]. In light of the geometrical interpretation of the norm we may rephrase (D.27) by saying that the elements of a Cauchy sequence get closer and closer to one another, although, in general normed spaces, such behavior does not imply that {vn }+∞ n=1 converges to some vector v ∈ V. On the contrary, any convergent sequence in the sense of (D.26) is also a Cauchy sequence, inasmuch as for m, n  N we have vm − vn V  vm − vV + vn − vV
0. Indeed, if {vn }n=1 converges to v we can set η = 1 in (D.24), find a suitable N, and choose an index n > N, so that vn V  vn − vV + vV < 1 + vV

(D.30)

whence necessarily we obtain vn   max {v1 V , v2 V , . . . , vN V , 1 + vV } = M

∀n ∈ N

(D.31)

+∞ and this proves that {vn }+∞ n=1 is limited. If {vn }n=1 is Cauchy (not necessarily convergent, though) we set η = 1 in (D.27), find a suitable N, and choose n = N and m > N, so that

vm V  vm − vN V + vN V < 1 + vN V

(D.32)

whence we get vm   max {v1 V , v2 V , . . . , vN V , 1 + vN V } = M

∀m ∈ N

(D.33)

and hence {vn }+∞ n=1 is bounded. The converse of this statement is obviously false, as there may exist bounded sequences in V which are not Cauchy or do not converge. Given two normed vector spaces V1 and V2 we say that V1 is embedded into V2 and write V1 → V2 , if for any u ∈ V1 we also have u ∈ V2 and the condition uV2  M uV1

(D.34)

where the embedding constant M > 0 does not depend on u. The constraint on the norms means that V1 is more than simply contained in V2 . Evidently, (D.34) implies that convergent and Cauchy sequences in V1 remain so in the ‘bigger’ embedding space V2 .

Functional analysis

1135

An inner product is a continuous sesquilinear map (or form) which takes two elements u, v of a vector space V and returns a number (u, v)V ∈ F. In symbols, this reads [3, Section 10.1], [7, Section 7.1], [8, Section 6.1] (•, •)V (ξ1 u1 + ξ2 u2 , v)V (u, η1 v1 + η2 v2 )V (u, v)V

: V × V −→ F = ξ1∗ (u1 , v)V + ξ2∗ (u2 , v)V = η1 (u, v1 )V + η2 (u, v2)V   = (v, u)V ∗

(D.35) (D.36) (D.37) (D.38)

where ξ1 , ξ2 , η2 , η1 , ∈ F, and • • •

(D.36) means that (•, •)V is conjugate linear or antilinear in the first variable; (D.37) means that the inner product is linear in the second variable [see definition (D.120)]; (D.38) means that (•, •)V is sesquilinear.

‘Sesqui’ is a Latin prefix meaning ‘one and a half’, which precisely hints at (•, •)V being only ‘half’ linear when the multiplicative scalar in F ≡ C is in front of the first variable. However, if the result does not depend on the order of the vectors, i.e., (u, v)V = (v, u)V

(D.39)

we say that the inner product is symmetric. It can be shown that the inner product (D.35) is continuous [8, Theorem 6.10] by considering ∞ two sequences {un }∞ n=1 , {vn }n=1 of vectors that converge in V to u and v, respectively. Then, since a convergent sequence is also bounded [see (D.31)], we have | (un , vn )V − (u, v)V | = | (un − u, vn )V + (u, vn − v)V |  | (un − u, vn )V | + | (u, vn − v)V |  un − uV vn V + uV vn − vV −−−−−→ 0 n→+∞

(D.40)

by virtue of (D.36), (D.37), and the Cauchy-Schwarz inequality (D.150). In particular, this means   (D.41) lim (un , vn )V = (u, v)V = lim un , lim vn n→+∞

n→+∞

n→+∞

V

whereby it is permitted to swap the order of inner product and limit [1, p. 12]. Finally, if the inner product is such that (u, u)V  0 (u, u)V = 0 ⇐⇒ u = 0 (ξu, ξu)V = |ξ|2 (u, u)V then we may take uV := (u, u)V

(D.42) (D.43) ξ∈F

(D.44)

(D.45)

and we say that the corresponding norm is induced by the inner product. Notice though that a norm can be defined even if an inner product is not available [7, Chapter 1]. Then again, when an inner product is provided and (D.45) holds true, we may introduce the abstract notion of acute angle formed by any two elements u, v through the formula cos α :=

| (u, v)V | uV vV

(D.46)

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1136

which mimics and extends the familiar result for ordinary vectors in three dimensions. The proof that cos α is no larger than one (and hence the angle α is real) requires the Cauchy-Schwarz inequality (D.150). Taking the absolute value of the inner product in (D.46) — versus just the real part thereof, which in view of (D.45) and u − v2V = u2V − 2Re {(u, v)V } + v2V = u2V − 2 uV vV cos α + v2V

(D.47)

would seem more natural — is necessary to make sure that α = π/2 actually implies (u, v)V = 0. Indeed, two vectors are said to be orthogonal if their inner product vanishes. Furthermore, a pair of subsets W1 and W2 of V are said orthogonal if, and only if, (w1 , w2 )V = 0 for all w1 ∈ W1 and w2 ∈ W2 . Given a subset W of V the set W ⊥ := {v ∈ V : (w, v)V = 0,

∀w ∈ W}

(D.48)

is called the orthogonal complement of W in V. The set W ⊥ is, in fact, a closed subspace of V, since properties (D.3)-(D.5) are evidently satisfied on account of (D.37). As for W ⊥ being closed, it is ⊥ sufficient to consider a sequence {vn }+∞ n=1 ⊂ W which converges to, say, v ∈ V. Since (w, vn )V = 0 for all w ∈ W, we have   0 = lim (w, vn )V = w, lim vn = (w, v)V (D.49) n→+∞

n→+∞

V

by virtue of the continuity of the inner product proved in (D.41). Therefore, the limit v belongs to W ⊥ as well, which in turn implies that W ⊥ is closed [3, Proposition 10.9], [1, pp. 20–21], [8, Theorem 6.12]. We notice that W and W ⊥ can only have the null vector 0 ∈ V in common because, if v ∈ W belonged also to W ⊥ , then in light of (D.48) v would be orthogonal to all members of W, including itself, whereby we should have 0 = (v, v)V = v2V , and this constraint requires v = 0 ∈ V on account of (D.9). A vector space equipped with an inner product is called an inner-product space or sometimes a pre-Hilbert space [9, Chapter 1]. An inner-product space V which is complete with respect to the norm (D.45) induced by the inner product is called a Hilbert space [3, Chapter 10], [10, Chapter 1], [7, Chapter 7]. In many a respect Hilbert spaces extend the finite-dimensional Euclidean spaces Rn . For instance, if V is a Hilbert space and Z is a non-empty closed convex subset (not necessarily a subspace, that is) then for any member v of V there exists a unique element u0 ∈ Z such that v − u0 V = inf v − wV = d

(D.50)

w∈Z

that is, u0 is the member of Z with shortest ‘distance’ from v ∈ V [3, Theorem 10.11], [1, Appendix A.1], [7, Lemma 7.11], [6, Proposition 11.4], [8, Theorem 6.13]. While this statement may sound intuitively true in the familiar three-dimensional space, proving (D.50) requires some abstract reasoning in infinite-dimensional spaces. To begin with, if we pick up a sequence {wn }∞ n=1 ⊂ Z such that lim wn − vV = d

n→+∞

and hence

lim wn − v2V = d 2

n→+∞

(D.51)

we can find a suitably large index n  N ∈ N which yields

η2



wn − v2V − d2

< 4

=⇒

wn − v2V 

η2 + d2 4

(D.52)

Functional analysis with η > 0 being an arbitrary small real number. Then, we use the identity 2  wn + wm  2 2 2  wn − wm V = 2 wn − vV + 2 wm − vV − 4  − v 2 V

1137

(D.53)

which is an application of the general parallelogram law [3, Proposition 10.8], [10, Section 1.2], [7, Theorem 7.10], [1, Example 1.13] and can be verified by inspection keeping in mind that the norm in V is induced by the inner product as in (D.45). Since Z is convex, the vector (wn + wm )/2 belongs to Z as well — this follows by using (D.6) with τ = 1/2 — though it may not be the closest to v, and we obtain 2

2  η w n + wm wn − wm 2V  4 + d2 − 4  − v  η2 + 4d2 − 4d2 = η2 (D.54) 4 2 V on account of (D.50) and (D.52) with m, n  N. This means, by comparison with (D.27), that {wn }∞ n=1 is, in fact, a Cauchy sequence in Z and, as V is complete, it must converge to a vector u0 in V or, more precisely, in Z, because the latter is closed. Therefore, for an arbitrary η > 0 and a suitable n  N, we have | wn − vV − u0 − vV |  wn − u0 V < η

(D.55)

whence we conclude that v − u0 V = lim wn − vV = d = inf v − wV n→+∞

w∈Z

(D.56)

in light of (D.51). This shows that at least a closest element u0 exists. As regards uniqueness, if (D.50) were satisfied by two members u1 and u2 of Z, then from the estimate 2  u1 + u2 − v  2d2 + 2d2 − 4d 2 = 0 u1 − u2 2V = 2 u1 − v2V + 2 u2 − v2V − 4  (D.57) 2 V we would necessarily get u1 − u2 V = 0, whereby it follows u1 = u2 by property (D.9). As a special case, by letting v = 0 ∈ V in (D.50) we also conclude that any non-empty convex closed subset of V contains a unique element with smallest norm. We can say something more about v−u0 if Z is a closed subspace of the Hilbert space V, namely, if u0 ∈ Z obeys (D.50), then for any w ∈ Z it holds (v − u0 , w)V = 0

(D.58)

meaning that the vector v − u0 is also orthogonal to any element of Z (cf. Figure 14.1) [3, Theorem 10.12], [7, Proposition 7.13], [1, Appendix A.1], [8, Theorem 6.13]. Indeed, we can choose a vector w ∈ Z \ {0} (otherwise the assertion is trivial) and a number ν ∈ C so that v − u0 2V  v − u0 − νw2V = v − u0 2V − 2Re {ν (v − u0 , w)V } + |ν|2 w2V

(D.59)

where the first inequality is a consequence of (D.50) since u0 + νw belongs to Z in view of (D.3) and (D.4) —- as Z is a subspace of V — but it is not the closest vector to v. By setting ν = (w, v − u0 )V / w2V in (D.59) we obtain ⎫ ⎧ ⎪ | (v − u0 , w)V |2 ⎬ | (v − u0 , w)V |2 ⎨ | (v − u0 , w)V |2 ⎪ =− (D.60) 0  −2Re ⎪ + ⎪ ⎭ ⎩ 2 2 wV wV w2V

1138

Advanced Theoretical and Numerical Electromagnetics

on account of property (D.38). Then, we have necessarily 0 = (v − u0 , w)V

(D.61)

and this proves (D.58). The vector u0 is called the orthogonal projection of v onto Z, and the combination of (D.50) and (D.58) is known as the projection theorem. Last but not least, we notice that when Z is a proper closed subspace of the Hilbert space V (i.e., Z is strictly contained in V) then we have Z ⊥  {0 ∈ V}

(D.62)

that is, the orthogonal complement Z ⊥ contains non-zero elements. Indeed, since Z is proper, then V \ Z is not empty. Now, for any vector v ∈ V \ Z ⊂ V property (D.50) guarantees the existence of a vector u0 ∈ Z which is the closest to v and property (D.58) says that v − u0 is orthogonal to any member of Z. Thus, v − u0 is non-null and belongs to Z ⊥ by definition (D.48). The vector v − u0 is called the orthogonal projection of v onto Z ⊥ . We shall make good use of (D.62) in the proof of the Riesz representation theorem (D.168) in Appendix D.5. Vector spaces whose elements are scalar or vector fields are also called function spaces [11], and all the notions introduced so far apply as well. Let D be a subset of R3 and F ∈ {R, C}. A function f : D → F for which  dV | f (r)|2 < +∞ (D.63) D

is said to be square-integrable. The set of functions ⎧ ⎫  ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ 2 2 L (D) := ⎪ D → F : dV| f (r)| < +∞ f : ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭

(D.64)

D

is the vector space of square-integrable functions over D. A function F : D → F3 which satisfies    2 dV |F(r)| = dV |Fα (r)|2 < +∞ D

(D.65)

α∈{x,y,z} D

is also square-integrable. The set of vector fields   L2 (D)3 := F : D → F3 : Fα ∈ L2 (D), α ∈ {x, y, z}

(D.66)

is the vector space of square-integrable vector-valued functions over D. Two suitable inner products for the spaces in (D.64) and (D.66) are defined as [3, Chapter 10]  ( f, g)L2 (D) := dV f ∗ (r)g(r) (D.67) D 

(F, G)L2 (D)3 := D

dV F∗ (r) · G(r)

(D.68)

Functional analysis

1139

where the complex conjugation is immaterial if the set F coincide with R. The abbreviated notation in the left-hand sides for the integrals in the right members is due to Courant and Hilbert [12, Chapter 11]. Accordingly, the following definitions  f 2 :=

 ( f, f )L2 (D)

⎞1/2 ⎛ ⎟⎟⎟ ⎜⎜⎜  ⎜⎜⎜ 2⎟ = ⎜⎜ dV | f (r)| ⎟⎟⎟⎟ ⎠ ⎝

(D.69)

D

F2 :=

 (F, F)L2 (D)3

⎞1/2 ⎛ ⎟⎟⎟ ⎜⎜⎜  ⎟ ⎜⎜⎜ = ⎜⎜ dV |F(r)|2⎟⎟⎟⎟ ⎠ ⎝

(D.70)

D

provide the relevant norms induced by the inner products (D.67) and (D.68). If f (r) and F(r) are bounded, other possible norms are defined as  f ∞ := sup | f (r)|

(D.71)

F∞ := sup |F(r)|

(D.72)

r∈D r∈D

without the intervention of an inner product. In particular, we notice that ⎛ ⎞1/2 ⎞1/2 ⎛ ⎞1/2 ⎛ ⎜⎜⎜  ⎜⎜⎜  ⎜⎜⎜  ⎟⎟⎟ ⎟⎟⎟ ⎟⎟⎟ ⎜⎜⎜ ⎜ ⎜ ⎟ ⎟ ⎟  f 2 = ⎜⎜ dV| f (r)|2 ⎟⎟⎟⎟  ⎜⎜⎜⎜ dV  f 2∞ ⎟⎟⎟⎟ = ⎜⎜⎜⎜ dV ⎟⎟⎟⎟  f ∞ = M  f ∞ ⎝ ⎝ ⎝ ⎠ ⎠ ⎠ D

D

(D.73)

D

where the constant M > 0 depends only on the domain D. A similar inequality holds for (D.70) and (D.72) [13, p. 244]. For instance, the time-harmonic Green function (8.356) is a member of L2 (D), D ⊂ R3 , but it is not bounded for r , r ∈ D. Any two functions which satisfy ( f, g)L2 (D) = 0

(D.74)

(F, G)L2 (D)3 = 0

(D.75)

are said to be orthogonal with respect to the inner products (D.67) and (D.68). The set C0 (D) = C(D) of continuous functions f : R3 ⊃ D → F equipped with the norm  f ∞ := max | f (r)|

(D.76)

r∈D

is a normed vector space. In particular, while C(D) is not complete when endowed with the norm (D.69) (e.g., [1, Example 1.17]), it can be completed to become L2 (D). The set of Hölder continuous functions on D is defined as [14], [15, p. 127], [13, p. 41], [16, p. 152]   (D.77) C0,α (D) = Cα (D) := f : R3 ⊃ D → F : | f (r1 ) − f (r2 )|  M|r1 − r2 |α , α ∈]0, 1[ and constitutes a normed space equipped with the norm  f C0,α (D) := sup | f (r)| + sup r∈D

r1 ,r2 ∈D r1 r2

| f (r1 ) − f (r2 )| . |r1 − r2 |α

(D.78)

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Advanced Theoretical and Numerical Electromagnetics

We say that a function f : R3 ⊃ D → F is continuously differentiable in D if the partial derivatives with respect to x, y, z exist and are continuous in D. Similarly, a function f : R3 ⊃ D → F is twice continuously differentiable if the second-order partial derivatives with respect to x, y, z exist and are continuous in D, and so forth. The set of functions   n n (D.79) Cn (D) := f : R3 ⊃ D → F : ∂nxx ∂yy ∂z z f (r) ∈ C(D), n x + ny + nz = n  1 is a vector space. With the aid of this space we may define the function space   Cn (D)3 := F : R3 ⊃ D → F3 : Fν (r) ∈ Cn (D), ν ∈ {x, y, z} .

(D.80)

It is worthwhile noticing that even if the domain D is finite, a function f ∈ Cn (D) or a field F ∈ Cn (D)3 need not be bounded inasmuch as they may grow near the boundary ∂D. The solutions to partial differential equations are better determined in a weak sense in function spaces whose elements possess a norm that is a combination of norms of the derivatives of the function up to a certain order. In this way the norm gives an indication of both the size and the regularity of a function. Vector spaces with these properties are named after the Russian mathematician S. L. Sobolev who studied them in the 1930s in relation to problems of mathematical physics [17], [18, Chapter 3], [19, Chapter IV]. We only review the basic facts which form the groundwork for the discussion of uniqueness of the time-harmonic Maxwell equations (Section 6.4.1) and the Helmholtz equation (Section 8.5.5) in lossless finite spatial regions. The Sobolev space H1 (V), V ⊂ R3 , is defined as   (D.81) H1 (V) = H(grad, V) := u ∈ L2 (V) : ∇u ∈ L2 (V)3 and is endowed with the canonical inner product  (u, v)H1 (V) := dV [u∗ (r)v(r) + ∇u∗ · ∇v] = (u, v)L2 (V) + (∇u, ∇v)L2 (V)3

(D.82)

V

which can be shown to be well-defined by invoking the Cauchy-Schwarz inequality (D.145) separately on the summands in the rightmost member. The following norm  1/2  1/2 = u2L2 (V) + ∇u2L2 (V)3 (D.83) uH1 (V) := (u, u)L2 (V) + (∇u, ∇u)L2 (V) is induced by (D.82). The Sobolev space H01 (V) is defined as H01 (V) := {u ∈ H1 (V) : u(r) = 0, r ∈ ∂V}

(D.84)

i.e., as the space of functions in H1 (V) which additionally vanish on ∂V, and hence H01 (V) is a subspace of H1 (V). We understand H01 (V) as the closure of C∞ 0 (V), the space of infinitely differentiable functions over V that vanish on ∂V. The alternative inner product and norm  := (u, v) dV ∇u∗ · ∇v = (∇u, ∇v)L2 (V)3 (D.85) V

  u := (u, u) 1/2 = ∇uL2 (V)3

(D.86)

are defined in H01 (V), and the functional •2 : H01 (V) −→ R+

(D.87)

Functional analysis

1141

is sometimes called the Dirichlet integral [15, p. 168]. To see that the norm • vanishes if, and only if, u(r) is the null element of H01 (V), we consider  0= dV |∇u|2 =⇒ ∇u = 0, r∈V (D.88) V

which requires that u(r) be constant in V. But then, since u(r) vanishes on ∂V, it vanishes everywhere in V. By contrast, (D.86) cannot be a norm in the larger space H1 (V) — it is, in fact, a semi-norm — because it vanishes for all constant functions in that space. The estimate (e.g., [15, p. 415]) uL2 (V)  M ∇uL2 (V)3 = M u ,

M > 0,

u ∈ H01 (V)

(D.89)

where M is a constant that depends on the domain V, is known as the Poincaré inequality or as an inequality of the Friedrichs type. We notice that (D.89) does not hold for elements of H1 (V). For instance, if u ∈ H1 (V) is a non-null constant, then u = 0, but uL2 (V) does not vanish, and (D.89) is violated. With the help of (D.89) we can show that the norms (D.83) and (D.86) are equivalent in H01 (V). Indeed, we have u2 = ∇u2L2 (V)3  u2L2 (V) + ∇u2L2 (V)3  (M 2 + 1) ∇u2L2 (V)3

(D.90)

whence it follows u  uH01 (V) 

√

M 2 + 1 u

(D.91)

by means of definition (D.83) which applies to H01 (V) ⊂ H1 (V) as well. Furthermore, since  1/2 uL2 (V)  u2L2 (V) + ∇u2L2 (V) = uH01 (V)

(D.92)

the space H01 (V) is embedded in L2 (V), in accordance with (D.34). The spaces H1 (V) and H01 (V) are Hilbert spaces with respect to the inner products (D.82) and (D.85). The Sobolev space H(curl, V), V ⊂ R3 , is defined as   H(curl, V) := f ∈ L2 (V)3 : ∇ × f ∈ L2 (V)3 (D.93) and is endowed with the natural inner product  (f, g)H(curl,V) := dV [f ∗ (r) · g(r) + ∇ × f ∗ · ∇ × g] = (f, g)L2 (V)3 + (∇ × f, ∇ × g)L2 (V)3

(D.94)

V

which in turn induces the following norm  1/2  1/2 = f2L2 (V)3 + ∇ × f2L2 (V)3 . fH(curl,V) := (f, f)L2 (V)3 + (∇ × f, ∇ × f)L2 (V)3

(D.95)

The Sobolev space H0 (curl, V) is defined as [20, Chapter 9] ˆ × [f(r) × n(r)] ˆ = 0, r ∈ ∂V} H0 (curl, V) := {f ∈ H(curl, V) : n(r)

(D.96)

3 where nˆ is the unit normal to ∂V. We understand H0 (curl, V) as the closure of C∞ 0 (V) , the space of infinitely differentiable vector functions over V that vanish on ∂V. Nonetheless, vector fields

Advanced Theoretical and Numerical Electromagnetics

1142

in H0 (curl, V) are not necessarily null on ∂V as compared to the elements of H01 (V). Indeed, if u ∈ H01 (V), then f = ∇u is a lamellar element of H0 (curl, V), inasmuch as ˆ × [∇u × n(r)] ˆ n(r) = ∇s u = 0,

r ∈ ∂V,

u ∈ H01 (V)

(D.97)

but nˆ · ∇u = nˆ · f need not be null on ∂V. We define the subspace H0S (curl, V) of H0 (curl, V) as H0S (curl, V) := {f ∈ H0 (curl, V) : ∇ · f = 0}

(D.98)

as the space of functions whose curl is square-integrable in V, whose tangential component is null on ∂V, and that additionally are solenoidal. The notation in (D.98) may not be standard but surely the subscript S is a good reminder for ‘solenoidal’. The alternative inner product and norm  (f, g) := dV ∇ × f ∗ · ∇ × g = (∇ × f, ∇ × g)L2 (V)3 (D.99) V



f := (f, f)

1/2

= ∇ × fL2 (V)3

(D.100)

are defined in H0S (curl, V). If this norm is null, then f is the gradient of some scalar potential u over ˆ V that, in light of definition (D.98), is even harmonic (cf. Sections 2.1 and 2.2). Since nˆ × (f × n) vanishes on ∂V, then u is equal to a constant u0 on ∂V, and this is sufficient to conclude that f vanishes everywhere in V, because      ∗ 2 ∗ 2 ∗ ˆ · ∇u − dV |∇u| = u0 dS n(r) ˆ · ∇u − dV |∇u|2 dV u (r)∇ u = dS u (r)n(r) 0= V

= u∗0



 dV ∇2 u −

V

∂V



dV |∇u|2 = − V

V

dV |∇u|2 = − V

∂V

 dV |f|2

V

(D.101)

V

whereby f = ∇u = 0, as required of a norm. Nonetheless, it appears that any constant or harmonic ˆ vanishes on ∂V, then non-zero constant fields are field has a null norm. However, since nˆ × (f × n) not elements of the space H0S (curl, V) (because f = f nˆ changes direction on ∂V unless f = 0) and harmonic fields are zero everywhere, as argued above. Contrariwise, notice that (D.100) cannot be a norm in H(curl, V) — it is a semi-norm, though — because it also vanishes for all constant vectors and lamellar vectors, which together form the null space [see (D.119) in Appendix D.3] of the curl operator on V. Also the estimate [21, Lemma 3.4 and Theorem 3.6] fH0 (curl,V)  M ∇ × fL2 (V)3 ,

M>0

(D.102)

is known as the Poincaré inequality, and can be used to prove that • given by (D.100) is an equivalent norm in H0S (curl, V). Indeed, we have f2 = ∇ × f2L2 (V)3  f2L2 (V)3 + ∇ × f2L2 (V)3  M 2 ∇ × f2L2 (V)3 = M 2 f2

(D.103)

whence it follows f  fH0S (curl,V)  M f

(D.104)

Functional analysis

1143

on account of (D.95) which applies to H0S (curl, V) ⊂ H0 (curl, V) ⊂ H(curl, V) as well. Besides, since we have  1/2 = fH0S (curl,V) = fH0 (curl,V) = fH(curl,V) (D.105) fL2 (V)3  fL2 (V)3 + ∇ × fL2 (V)3 the spaces H0S (curl, V), H0 (curl, V) and H(curl, V) are all embedded in L2 (V)3 in accordance with (D.34). The spaces H(curl, V), H0 (curl, V) and H0S (curl, V) are Hilbert spaces with respect to the relevant inner products (D.94) and (D.99). The Sobolev space H(div, V), V ⊂ R3 , is defined as [20, Chapter 9]   H(div, V) := f ∈ L2 (V)3 : ∇ · f ∈ L2 (V) (D.106) and is endowed with the natural inner product  (f, g)H(div,V) := dV [f ∗ (r) · g(r) + ∇ · f ∗ ∇ · g] = (f, g)L2 (V)3 + (∇ · f, ∇ · g)L2 (V)

(D.107)

V

which in turn induces the following norm  1/2  1/2 fH(div,V) := (f, f)L2 (V)3 + (∇ · f, ∇ · f)L2 (V) = f2L2 (V)3 + ∇ · f2L2 (V) .

(D.108)

The space H0 (div, V) is defined as ˆ · f(r) = 0, r ∈ ∂V} H0 (div, V) := {f ∈ H(div, V) : n(r)

(D.109)

where nˆ is the unit normal to ∂V. The spaces H(div, V) and H0 (div, V) are Hilbert spaces with respect to the inner product (D.107). In the context of electromagnetic theory the fields E and H belong to H(curl, V), whereas the flux densities D and B are elements of H(div, V).

D.2

The Bessel inequality

We suppose that f : R3 ⊃ D → C is a complex scalar field in L2 (D) [see (D.64)], and consider a set 2 of orthonormal functions {φn (r)}∞ n=1 also in L (D). The latter hypothesis means that (φn , φm )D = δnm

⎧ ⎪ ⎨0, m  n =⎪ ⎩1, m = n

m, n ∈ N \ {0}

(D.110)

where (•, •)D is the inner product (D.67). We define the scalar field fN (r) :=

N 

cn φn (r),

r∈D

(D.111)

n=1

where cn are complex constant coefficients, and examine the squared norm (D.69) of the difference f (r) − fN (r), namely,  f −

fN 22

:= D



2 N 



dV

f (r) − cn φn (r)





n=1

1144

Advanced Theoretical and Numerical Electromagnetics  =

dV | f (r)|2 +

N  n=1

D

=  f 22 +

N  

⎡ ⎤  ⎥⎥ N ⎢  ⎢⎢⎢  ⎢⎢⎢cn dV f ∗ (r)φn (r) + c∗ dV f (r)φ∗ (r)⎥⎥⎥⎥⎥ |cn |2 − n n ⎥⎦ ⎣⎢ n=1

D

D

& ' cn c∗n − 2 Re cn ( f, φn )D

(D.112)

n=1

where we have used (D.110) and (B.6). It can be shown that the right-hand side of (D.112) — which is a real, non-negative function of cn and c∗n — reaches its minimum if the expansion coefficients cn satisfy c∗n = ( f, φn )D ,

n = 1, . . . , N

(D.113)

a result that follows by setting to zero the partial derivatives with respect to cn and c∗n [cf. (14.12)]. Indeed, as pointed out by Morse and Feshbach [22, Section 4.1, page 349], a complex number and its conjugate are two distinct variables. By substituting (D.113) into (D.112) we have  f − fN 22 =  f 22 −

N 

|cn |2 =  f 22 −

n=1

N



( f, φ )

2 n D

(D.114)

n=1

and, since the left-hand side is a non-negative quantity, we obtain the estimate   f 22 :=

dV | f (r)|2 

N



( f, φ )

2 n D

(D.115)

n=1

D

which goes by the name of Bessel’s inequality [10, p. 15], [7, Theorem 7.23], [23, Proposition 2.19], [8, Theorem 6.24]. If N → +∞ we have +∞

N

 

( f, φ )

2 = lim

( f, φ )

2  lim  f 2 =  f 2 n D n D 2 2 n=1

N→+∞

n=1

N→+∞

(D.116)

on account of (D.115). The result also holds for scalar fields defined on lower-dimensional domains, such as surfaces, and it can be proved in general inner-product spaces (cf. Section 11.1.6).

D.3 Linear operators An operator L {•} is a map (i.e., a rule) which says how to associate elements of a vector space VL over the field F ∈ {R, C} with members of another, possibly different, vector space WL over the same field F. In symbols, we write L {•} : VL −→ WL

(D.117)

where • •

the space VL is called the domain of L {•}; the subspace (see Appendix D.1) of WL defined as ran L := {w ∈ WL : L {v} = w, v ∈ VL } ⊆ WL is termed the range or image of the operator;

(D.118)

Functional analysis •

1145

the subspace of VL defined as ker L := {v ∈ VL : L {v} = 0 ∈ WL } ⊆ VL

(D.119)

is called the zero space or null space or kernel of L {•}. Although the name is the same, ker L is a vector space and should not be confused with the kernel or nucleus of an integral operator, as in, e.g., (13.7), which instead is a function. Still, even an integral operator can have a null space in the sense of (D.119). An important class of operators (which find application in electromagnetics and, more generally, in physics) is constituted by those maps for which it is permitted to interchange the operations of sum of two elements and multiplication by a scalar. To elucidate, let v1 and v2 be any two elements of VL and α1 , α2 ∈ C two numbers. Then, if the following equation L {α1 v1 + α2 v2 } = α1 L {v1 } + α2 L {v2 }

(D.120)

holds true, the operator L {•} is said to be linear or distributive [2, p. 206], [13, Chapter 5], [1, p. 25], and VL and WL are sometimes called the departure space and the arrival space, respectively. The differential operators involved in the statement of Maxwell’s equations in local form (Section 1.2.2) are linear and so are the integral operators discussed in Chapter 13. The inner products (D.67) and (D.68) are linear operators with respect to the second argument, whereas the norms (D.69) and (D.70) are not. Evidently, in light of (D.120) we have α=0∈C

α0 = 0 ∈ VL

=⇒

L {0} = L {α0} = αL {0} = 0 L {0} = 0 ∈ WL

(D.121)

that is to say, a linear operator associates the null element of VL with the null member of WL . In fact, property (D.121) can be used to check whether an operator is linear. A (linear) operator is said to be • •

injective or one-to-one or an injection if L {v1 } = L {v2 } implies that v1 and v2 coincide; surjective or onto or a surjection if ∀w ∈ WL , ∃ v ∈ VL such that L {v} = w.

A linear operator can be injective if, and only if, the kernel contains just the null element of VL . Indeed, if L {•} is injective and v ∈ ker L we have L {v} = 0 = L {0} ∈ WL

=⇒

v ≡ 0 ∈ VL

(D.122)

by invoking (D.119), (D.121) and the very definition of injection. Conversely, if ker L = {0}, we may choose two vectors v1 and v2 in VL such that L {v1 } = L {v2 } and consider 0 = L {v1 } − L {v2 } = L {v1 − v2 }

(D.123)

whereby we conclude that v1 − v2 belongs to the null space of L {•}. However, since ker L contains only the null element, then v1 − v2 = 0 and v1 = v2 , so that L {•} is one-to-one by definition. Further, if an operator is surjective, then the image coincides with the space WL . A linear operator which is both injective and surjective is called bijective or an isomorphism. According to the previous discussion L {•} is bijective if, and only if, ker L = {0} and ran L ≡ WL . The supremum or uniform norm of an operator is defined as [2, p. 208], [15, p. 396], [3, p. 122] L {•} := sup v∈VL

L {v}WL vVL

(D.124)

1146

Advanced Theoretical and Numerical Electromagnetics

for all v except the null element of VL . Notice that this definition requires only that norms (not inner products) be given in VL and WL . What is more, definition (D.124) meets the requirements (D.8)(D.11). Indeed, (D.8) is obvious, whereas (D.10) and (D.11) hold true inasmuch as they are valid for •WL . As for property (D.9) we observe that if L {•} is the null operator — the one which maps each and every v ∈ VL onto 0 ∈ WL — we find L {•} = sup v∈VL

0WL =0 vVL

(D.125)

whereas if the norm of L {•} vanishes, then we have 0  L {v}WL  L {•} vVL = 0

(D.126)

whence it follows L {v}WL = 0 and hence L {v} = 0 for any v ∈ VL , so that L {•} is the null operator. An operator is said to be bounded if its norm is finite, otherwise the operator is unbounded. Moreover, since a linear operator is bounded if, and only if, it is continuous on VL , in practice the two properties are equivalent [3, Proposition 8.2], [1, pp. 27–28], [7, Proposition 1.6], [23, Proposition 3.2]. Indeed, if L {•} is bounded, for a fixed vector u0 ∈ VL and any vector u such that u − u0 VL < η (this condition defines the open ball of radius η > 0 and center u0 in the space VL ) we have L {u} − L {u0 }WL = L {u − u0 }WL  L {•} u − u0 VL < L {•} η =

(D.127)

and continuity at u0 follows by definition and by taking η = / L {•}. Since the vector u0 is arbitrary, we see that boundedness implies continuity for any vector in VL . Conversely, if L {•} is continuous in u0 , then there exists a number η > 0 such that u − u0 VL < η

=⇒

L {u} − L {u0 }WL < 1

(D.128)

and by picking up any element v ∈ VL we can always define a vector u so that the vector u − u0 =

ηv 2 vVL

with

u − u0 VL =

η L {u} − L {u0 }WL = L {u − u0 }WL =  = (D.130)   2 vVL W 2 vVL L whence we get L {v}WL 2 < , vVL η

∀v ∈ VL

(D.131)

and the boundedness of L {•} follows by passing to the supremum and invoking definition (D.124). Since we have just proved that continuity at a vector implies boundedness, then by virtue of (D.127) if a linear operator is continuous at u0 , it is continuous everywhere in VL . An operator is said to have finite rank if it is bounded (or continuous) and the dimensionality of ran L ⊆ WL is finite. An operator is called compact or completely continuous or totally bounded if, and only if, for +∞ every bounded sequence {vn }+∞ n=1 of elements in VL the sequence {L {vn }}n=1 has a subsequence which

Functional analysis

1147

converges in WL [13, Chapter 5], [10, Section 2.4], [6, Chapter 15], [23, Section 3.4]. Equivalently, a compact operator maps the unit ball in VL [see (D.14)] onto a subset of WL that has compact closure (see Appendix D.1). Among the linear operators the compact ones are the simplest in that their behavior mimics and generalizes the more familiar linear algebra of square matrices [4]. It is not difficult to show that, if L {•} is compact and A {•} : WL −→ WA

B {•} : VB −→ VL

(D.132)

L {B {•}} : VB −→ VL −→ WL

(D.133)

are bounded, then A {L {•}} : VL −→ WL −→ WA

are compact operators as well. A compact operator is also bounded, whereas a bounded operator L {•} between Hilbert spaces is compact if, and only if, lim Ln {•} − L {•} = 0

(D.134)

n→+∞

where {Ln {•}}∞ n=1 is a sequence of finite-rank operators [10, Theorem 4.4, p. 41]. This property can often be employed to show that a given operator is compact by devising a suitable sequence {Ln {•}}∞ n=1 which converges to L {•}. We notice that the identity operator is bounded — since I {•} = 1 — but, if the dimensionality of VL is infinite, I {•} is not compact. Indeed, if VL is an inner-product space, it is possible to find a sequence of orthogonal vectors vn ∈ VL , n = 1, 2, . . ., with vn VL = 1, that is evidently bounded. Then, thanks to the very definition of I {•} and for any two whole numbers n, m, we have  1/2 √ = 2, I {vn } − I {vm }VL = vn − vm VL = vn 2VL + vm 2VL

n, m ∈ N \ {0}

(D.135)

whereby we see that the ‘distance’ between any two vectors generated by applying I {•} to a bounded sequence remains constant, and hence a convergent subsequence cannot be extracted, contrary to what is required of a compact operator. If V1 and V2 are two normed spaces, and V1 is embedded in V2 in accordance with condition (D.34), the (linear) imbedding operator is defined as ( V1 −→ V2 (D.136) J {•} : u −→ u and formally transforms an element u ∈ V1 into the same element u though regarded as a member of V2 . Thus, J {•} is an identity of sorts, but not quite inasmuch as it links vectors that belong to two distinct spaces. By virtue of (D.34) the fact that V1 → V2 is equivalent to stating that the imbedding operator (D.136) is bounded (or continuous) in that J {u}V2 = uV2  M uV1 ,

∀u ∈ V1

(D.137)

u0

(D.138)

whence we conclude J {•} = sup u∈V1

J {u}V2  M, uV1

on account of definition (D.124). When the imbedding operator is also compact, we say that the space V1 is compactly embedded in V2 .

Advanced Theoretical and Numerical Electromagnetics

1148

When L {•} is bijective (i.e., one-to-one and onto) we define the inverse operator by means of the properties L−1 {L {v}} = I {v} = v,   L L−1 {w} = I {w} = w,

v ∈ VL

(D.139)

w ∈ WL

(D.140)

where I {•} indicates the identity operators on VL and WL . By taking the vector norms in (D.139) and using definition (D.124) we obtain the estimate       (D.141) v = L−1 {L {v}}  L−1 {•} L {v}  L−1 {•} L {•} v VL

VL

VL

VL

whence we conclude that [3, p. 128]  −1  L {•} L {•}  1

(D.142)

where the quantity in the left-hand side is called the condition number of the operator L {•}. If the condition number is large, then an equation of the type v ∈ VL ,

L {v} = w,

w ∈ WL

(D.143)

is ill-conditioned and the problem is ill-posed (cf. Section 6.2) inasmuch as the relative error on the computed solution v may be far larger than the error on the source term w.

D.4 The Cauchy-Schwarz inequality We assume that F : R3 ⊃ D → C3 and G : R3 ⊃ D → C3 are two complex-valued vector fields in the inner-product space L2 (D)3 [see (D.66)]. To lighten the notation we let (F, G)L2 (D)3 = (F, G)D

(D.144)

thus shifting the focus from the function space to the domain of integration in (D.68). Then, the Cauchy-Schwarz inequality states that [22, p. 82], [3, Proposition 10.3], [10, p. 3], [7, p. 151] | (F, G)D |  F2 G2

(D.145)

where equality clearly holds when G(r) = ηF(r) for some scalar η ∈ C, in light of properties (D.37) and (D.45). The statement is trivial if G(r) = 0, thus for the proof we suppose that G(r) is not the null field over D and let ν ∈ C be a constant. The squared 2-norm (D.70) of the vector field F(r) − ν G(r) reads   2 2 := F − ν G2 dV |F(r) − ν G(r)| = dV [F(r) − ν G(r)]∗ · [F(r) − ν G(r)] D





dV |F(r)| − ν

=

∗

∗



dV F (r) · G(r) − ν

2

D

D

D

= F22 − 2 Re {ν∗ (F, G)D } + |ν|2 G22

∗

dV F(r) · G (r) + |ν| D

 dV |G(r)|2

2 D

(D.146)

on account of (D.68), (D.38) and (B.6). Now, since G2 > 0, we may choose the arbitrary constant ν as ν=

(F, G)D G22

(D.147)

Functional analysis

1149

and substitute the above into (D.146) to get F − ν G22 = F22 − 2

| (F, G)D |2 G22

+

| (F, G)D |2 G22

= F22 −

| (F, G)D |2

(D.148)

G22

and since the leftmost-hand side is a non-negative quantity irrespective of the value taken on by ν, we obtain the estimate F22 

| (F, G)D |2

(D.149)

G22

whence (D.145) follows by taking the square roots. The steps outlined above can be repeated for members of general inner-product spaces (Appendix D.1) and for constant vector fields, thus leading to | (u, v)V |  uV vV

(D.150)

|u · v|  |u||v|

(D.151)

where u, v are two ordinary vectors in the three-dimensional space. Further extension to dyadic fields is addressed in Appendix E.2. The analogous result for complex square-integrable scalar fields, say, f (r) and g(r) can be obtained by choosing a constant vector w and letting F(r) = f (r)w and G(r) = g(r)w in (D.145). Alternatively, the Cauchy-Schwarz inequality for f (r) and g(r) can also be proved by means of the Bessel inequality (D.115) where one sets N = 1 and chooses, e.g., φ1 (r) = g(r)/ g2 to conform with (D.110). As an example, with the aid of (D.151) we can prove the estimate









dV F(r)

 dV |F(r)| (D.152)



D

D

where the complex vector field F(r) is the same as in (D.146). We pick up an arbitrary constant complex unit vector uˆ and observe ⎫ ⎧     ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ ∗ ∗ ∗ ˆ ˆ ˆ · dV F(r) dV Re{ u · F(r)}  dV | u · F(r)|  dV |F(r)| (D.153) Re ⎪ = u ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ D

D

D

D

precisely on account of (D.151) and the fact that |uˆ ∗ | = 1 by definition. Now we make the special choice ) dV F(r)

uˆ :=

)D (D.154)

dV F(r)

D

whereby the leftmost-hand side of (D.153) becomes







⎫ ⎫ ⎧ ⎧ *) +∗   ⎪ ⎪ ⎪ ⎪







 ⎪ ⎪ ⎪ ⎪ dV F(r) ⎪ ⎪ ⎪ ⎪ ⎬ ⎬ ⎨ ∗ ⎨ D

dV F(r)

=

dV F(r)

(D.155)



) ˆ · dV F(r) dV F(r) · Re ⎪ = Re = Re u ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪





⎪ ⎪ ⎪

⎪ ⎭ ⎭ ⎩ ⎩

D dV F(r)

D

D

D

D

and this concludes the proof of (D.152). In particular, (D.152) holds for real-valued vector fields.

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Advanced Theoretical and Numerical Electromagnetics

A more general instance of (D.150) exists that involves a linear positive definite operator L {•} : VL → VL , namely, [23, p. 151] +1/2 * +1/2



*

(u, L {v})VL

 (u, L {u})VL (v, L {v})VL (D.156) under the hypothesis that (u, L {u})VL  0,

∀u ∈ VL

(D.157)

where equality holds if, and only if, u = 0 ∈ VL . Indeed, in the vector space VL we can define the alternative inner product ((u, v))VL := (u, L {v})VL

(D.158)

which with a little algebra can be shown to obey properties (D.36)-(D.38) and (D.42)-(D.44) as L {•} is linear and positive definite. Therefore, since we are allowed to apply (D.150) to the left-hand side of (D.158), we obtain * +1/2 * +1/2 ((v, v))VL | ((u, v))VL |  ((u, u))VL (D.159) whence (D.156) follows by virtue of definition (D.158). We notice that if L {•} is the identity operator, then we recover (D.150) from (D.156).

D.5 The Riesz representation theorem The Riesz representation theorem — proved for the first time by F. Riesz in 1909 [24] — states that any bounded (continuous) linear functional on a Hilbert space V (Appendix D.1), say, ( V −→ C (D.160) A {•} : u −→ ξ can be expressed in a unique way as the inner product (D.35) between u ∈ V and a suitable element v ∈ V that depends on A {•} [3, Chapter 10], [10, Chapter 1], [7, Theorem 7.16], [6, Theorem 11.9], [25, Theorem 4.12], [26, Théorème V.5], [8, Theorem 8.12]. We notice that if A {•} is the null functional — i.e., the one which maps any u ∈ V onto 0 ∈ C — the assertion of the theorem is trivial, because with v = 0 ∈ V we have A {u} = 0 = (0, u)V ,

∀u ∈ V

(D.161)

and hence for the proof we may suppose that A {•} is not the null functional. First, in accordance with (D.119) we define the null space or kernel of A {•} as ker A := {w ∈ V : A {w} = 0 ∈ C} ⊂ V

(D.162)

and notice that ker A is a closed, convex, and proper subspace of V (i.e., strictly contained in V) or else A {•} would be the null functional, an occurrence we have just discarded. Properties (D.3)(D.5) and (D.6) are obviously satisfied. To show that the kernel of A {•} is closed, we may consider a Cauchy sequence {wn }+∞ n=1 ⊂ ker A ⊂ V [see (D.27)]. Since V is complete by hypothesis, the elements converge necessarily to a vector w ∈ V. By choosing an arbitrarily small real number η > 0 of {wn }+∞ n=1 so that w − wn V < η for sufficiently large indices n in accordance with (D.24), we have |A {w} | = |A {w} − A {wn } | = |A {w − wn } |  A {•} w − wn V < A {•} η

(D.163)

Functional analysis

1151

whereby we see that |A {w} | tends to zero, and hence A {w} = 0. Thus, w belongs to ker A as well, and this confirms that ker A is closed. The orthogonal complement (ker A)⊥ [see (D.48)] is also a closed subspace of V in light of (D.49). More importantly, since V is a Hilbert space, we know from (D.62), (D.58) and (D.50) that (ker A)⊥ contains non-zero vectors, though we claim that the dimensionality of (ker A)⊥ is just one. Indeed, if a1 and a2 were two linearly independent members of (ker A)⊥ — so that A {a1 }  A {a2 }  0 — we could choose two complex numbers, say, ξ1 and ξ2 , such that 0 = ξ1 A {a1 } − ξ2 A {a2 } = A {ξ1 a1 − ξ2 a2 }

(D.164)

which means that the vector ξ1 a1 −ξ2 a2 belongs both to (ker A)⊥ by construction and to ker A in view of (D.162). But this can only occur if ξ1 a1 − ξ2 a2 = 0, so we must conclude that, e.g., a1 is a multiple of a2 and, more generally, that the space (ker A)⊥ consists of all the multiples of a single element a. All in all, these observations allow us to choose a vector a ∈ (ker A)⊥ \ {0}, so that A {a}  0 and a is orthogonal to any element in ker A. Secondly, we define the vector w := u −

A {u} a, A {a}

∀u ∈ V

(D.165)

and observe that A {w} = A {u} −

A {u} A {a} = 0 A {a}

(D.166)

by virtue of the linearity of A {•}. Therefore, w ∈ ker A and, since a is orthogonal to w by hypothesis, we have

A {u} A {u} a = (a, u)V − a2V (D.167) 0 = (a, w)V = a, u − A {a} V A {a} where the last step follows from the linearity property (D.37) of the inner product. Finally, by solving the above equation for A {u} we obtain ⎛ ⎞ ⎜⎜⎜ (A {a})∗ ⎟⎟ A {a} (a, u)V = ⎝⎜ a, u⎟⎠⎟ = (v, u)V (D.168) A {u} = 2 2 aV aV V which is the desired result. The vector v, whose existence is predicted by (D.168), is unique inasmuch as (ker A)⊥ is a one-dimensional subspace of V and using any scaled version of a in (D.168) clearly does not affect the definition of v in light of the linearity of A {•} and property (D.10) of a norm. Then again, if the inner product in V obeys properties (D.42) and (D.43), and two options v1 and v2 were possible, we would have A {u} = (v1 , u)V = (v2 , u)V

=⇒

(v1 − v2 , u)V = 0

(D.169)

and, since the last expression must hold for any u ∈ V, by choosing u = v1 − v2 we would arrive at the condition v1 − v2 2V = 0 whence we conclude again that necessarily v1 = v2 by virtue of property (D.9) of a norm.

(D.170)

1152

Advanced Theoretical and Numerical Electromagnetics

The set of all bounded linear functionals on V endowed with operations (D.1) and (D.2) and the uniform norm (D.124) is itself a complete normed vector space that is referred to as the (continuous or topological) dual of V and variously indicated with either V  or V or V . The linear operator (sometimes called the Riesz map or Riesz isomorphism) ( V −→ V (D.171) J {◦} : v −→ A {•} = (v, •)V which formally associates the relevant member v ∈ V with A {•} ∈ V , is bijective and isometric, i.e., it preserves the norm. Indeed, that the Riesz map (D.171) is injective and surjective follows from the uniqueness and existence of v just proved above. To compute the norm of A {•} in (D.160) we exploit (D.168) and notice that |A {u} | = | (v, u)V |  vV uV

(D.172)

thanks to the Cauchy-Schwarz inequality (D.150). Then, by dividing through by uV we obtain the constraint |A {u} | | (v, u)V | =  vV uV uV

(D.173)

whence, by passing to the supremum with respect to u  0, we conclude that A {•}  vV

(D.174)

on account of (D.124). But then, if we combine (D.168) with definition (D.124) we get | (v, u)V | = |A {u} |  A {•} uV

=⇒

| (v, u)V |  A {•} uV

(D.175)

and by making the special choice u = v we arrive at | (v, v)V | = vV  A {•} vV

(D.176)

which in view of the upper bound (D.174) forces us to conclude that vV = A {•} = J {v}V

(D.177)

i.e., the norm of v is preserved by the Riesz map (D.171) which is thus isometric and a non-trivial example of imbedding operator (D.136). The purpose and the usefulness of the Riesz theorem (D.168) and identity (D.177) lie in the fact that one can identify the linear functionals (D.160) on V with the very members of V, according to (D.171). For instance, expression (1.279) which gives the instantaneous power delivered by the source J(r, t) is in the form of an inner product of type (D.68) with the electric field E(r, t). The Riesz theorem allows us to regard E(r, t) alternatively as the linear functional that operates on current densities to yield the power. Furthermore, the Riesz theorem can be applied to rephrase partial differential equations as integral ones in order to study the existence and uniqueness of solutions (we do so in Sections 6.4.1 and 8.5.5). Finally, we remark that representation (D.168) fails to be true in incomplete inner-product spaces (e.g., [23, Remark 4.15]).

Functional analysis

D.6

1153

Adjoint operators

If VL and WL are two Hilbert spaces (Appendix D.1) and L {•} is a bounded linear operator, then the adjoint operator L† {•} : WL −→ VL

(D.178)

is defined so that for any v ∈ VL and w ∈ WL we have   (w, L {v})WL = L† {w}, v

(D.179)

VL

where (•, •)VL and (•, •)WL denote the inner products in VL and WL [13, Section V.17], [3, Section 10.4], [10, Section 2.2], [7, Section 7.2], [23, Section 4.3].1 The existence of the adjoint operator is guaranteed by the Riesz representation theorem (D.168), inasmuch as the mapping ( VL −→ C B {•} : (D.180) v −→ (w, L {v})WL can be interpreted as a bounded linear functional from VL to C that, for a given w, associates a complex number with any v ∈ VL . Indeed, the linearity of B {•} stems from that of L {•} and (•, •)WL , whereas the boundedness of B {•} follows from the estimate |B {v} | = | (w, L {v})WL |  wWL L {•} vVL

(D.181)

by virtue of the Cauchy-Schwarz inequality (D.150), definition (D.124), and the surmised boundedness of L {•}. Then, there exists a unique element in VL , say, gw , which depends on w ∈ WL and such that (w, L {v})WL = B {v} = (gw , v)VL

(D.182)

whence (D.179) follows by letting gw := L† {w}. Further, the adjoint operator is unique because, if there were at least two possibilities, say, L†1 {w} and L†2 {w}, then by definition (D.179) we would have  †     †    L1 {w}, v = L†2 {w}, v =⇒ L1 {w} − L†2 {w}, v = (L†1 − L†2 ){w}, v = 0 (D.183) VL

VL

VL

VL

and since the last expression must hold for any v ∈ VL and, in particular, for v = (L†1 − L†2 ){w}, we conclude that 2  † (L1 − L†2 ){w}V = 0 (D.184) L

which can only be true if (L†1 − L†2 ){w} = 0,

w ∈ WL

(D.185)

owing to property (D.9) of a norm. From this result we see that (L†1 − L†2 ){•} is the null operator, and uniqueness follows. By means of definition (D.179) and a little algebra it is possible to show that also L† {•} is linear. the mathematical literature the adjoint operator is routinely indicated with a superscript asterisk as in L∗ {•} — i.e., the notation we reserve for complex conjugate entities throughout this book — most likely because taking the adjoint of an operator L {•} is analogous to conjugating a complex number, self-adjoint operators correspond with real numbers [see (D.205)], and unitary operators [see (D.208) and (D.212)] are analogous to complex numbers with unitary magnitude [3, Chapter 15].

1 In

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Advanced Theoretical and Numerical Electromagnetics

A bounded operator and the adjoint thereof necessarily have the same supremum norm, viz.,   (D.186) L {•} ≡ L† {•} because first of all we have the estimates       † 2   L {w}V := L† {w}, L† {w} = w, L L† {w}  wWL L {•} L† {w}V L VL WL L    †  2 † L {v}WL := (L {v}, L {v})WL = L {L {v}}, v V  L {•} L {v}WL vVL L

(D.187) (D.188)

in view of definition (D.179), the inequality (D.150), and definition (D.124). Then,   Cauchy-Schwarz by dividing through by wWL L† {w}V and L {v}WL vVL , respectively, we obtain L   † L {w}V  †  L L {•}  L {•}  L {•} =⇒ (D.189) wWL   L {v}WL  †   L {•} =⇒ L {•}  L† {•} (D.190) vVL which can only be consistent if (D.186) holds true. We notice that, if L {•} is compact so is the adjoint operator, since (D.186) implies the boundedness of L† {•}. Indeed, by virtue of (D.133) we know that L L† {•} is compact, since L† {•} is bounded. Therefore, we can find a bounded sequence {wn }+∞ n=1 ⊂ WL such that   and lim L L† {wn } = a ∈ WL (D.191) wn WL  M n→+∞

and, by fixing an arbitrarily small number η > 0 and choosing two sufficiently large indices n and m, on account of (D.179), the Cauchy-Schwarz inequality (D.150), the triangle inequality (H.19), and (D.191) we have       † 2 L {wn } − L† {wm }V = L† {wn − wm }, L† {wn − wm } = wn − wm , L L† {wn − wm } VL WL L      † †    wn − wm WL L L {wn } − L L {wm }  WL            †    wn WL + wm WL L L {wn } − a + L L† {wm } − a WL WL  η η   2M + =η (D.192) 4M 4M which shows that {L† {wn }}+∞ n=1 is a Cauchy sequence in VL [see (D.27)] and hence it converges to a member v of VL in that the latter is a complete inner-product space by hypothesis. Then, we conclude that L† {•} is compact by definition, inasmuch as it transforms any bounded sequence {wn }+∞ n=1 into a convergent one. Similar steps can be taken to support the opposite claim, namely, that L {•} is compact if so is L† {•}. The adjoint of the adjoint operator, say, L†† {•}, can be applied to members of VL and, in fact, coincides with the original operator L {•}. Indeed, by virtue of definition (D.179) and property (D.38) we have ,     -∗ * +∗  = v, L† {w} = L† {w}, v = (w, L {v})WL = (L {v}, w)WL (D.193) L†† {v}, w WL

VL

VL

whence the claim follows by invoking the linearity property (D.37) of the inner product and by choosing w = (L†† − L){v} in order to conclude that (L†† − L){•} is the null operator on VL .

Functional analysis

1155

Furthermore, it is possible to prove the following relationships between null spaces and images of L {•} and L† {•}  ⊥ ker L† ≡ (ran L)⊥ (D.194) ker L ≡ ran L† where the orthogonal complement is defined in (D.48). Indeed, if we take v ∈ ker L in (D.179), then L {v} = 0 by definition (D.119), and we have   L† {w}, v = (w, L {v})WL = 0 =⇒ L† {w} ⊥ v ∀w ∈ WL (D.195) VL

which implies  ⊥ ker L ⊆ ran L†

(D.196)

because v, besides being in the null space of L {•} by hypothesis, is also orthogonal to the image of w under the action of the adjoint operator. On the other hand, if we choose v ∈ (ran L† )⊥ , then (D.179) gives   (w, L {v})WL = L† {w}, v = 0 =⇒ w ⊥ L {v} ∀w ∈ WL (D.197) VL

which can only be true for any w, and in particular for w = L {v}, if L {v} = 0. Therefore, v must also belong to the null space of L {•}, in addition to being in the orthogonal complement of the image of L† {•} by assumption, viz., ⊥  (D.198) ran L† ⊆ ker L and this condition in tandem with (D.196) yields the first part of (D.194). The second part follows immediately by trading L {•} for L† {•} and by recalling that L†† {•} ≡ L {•}. In particular, (D.194) holds when VL ≡ WL . As a special case, when the spaces VL and WL coincide and L† {•} ≡ L {•}, then L {•} is said to be a self-adjoint or Hermitian operator [3, Section 10.4]. Incidentally, the differential operators considered in Sections 6.8.1, 8.7, 11.1.4, and 11.2.5 are self-adjoint, though they are not bounded, let alone compact. In which cases, the existence of the adjoint operators is proved constructively by using integration by parts or suitable integral identities. With the aid of definition (D.179) it is easy to prove that the operators   L L† {•} : WL −→ VL −→ WL (D.199) L† {L {•}} : VL −→ WL −→ VL are self-adjoint [13, p. 244–245]. In order to show that  †  L {L {•}} ≡ L {•}2 we consider the estimates   L† {L {v}}V   L {v}WL  †  L  L† {•}  L {•} L {•} = L {•}2 vVL vVL     L† {L {v}}, v L† {L {v}}V L {v}2WL (L {v}, L {v})WL VL L = =  vVL v2V v2V v2V L

L

L

(D.200)

(D.201)

(D.202)

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Advanced Theoretical and Numerical Electromagnetics

which are based on the Cauchy-Schwarz inequality (D.150) and definitions (D.124), (D.179). Passing to the supremum with respect to v yields the string of inequalities     † L {L {•}}  L {•}2  L† {L {•}} (D.203)   which can only be consistent if (D.200) is true. The same result is proved for L L† {•} by swapping the role of L {•} and L† {•} in (D.200) and invoking (D.186). What is more, from (D.200) we have  2  L {•} ≡ L {•}2 (D.204) when L {•} is self-adjoint. When VL ≡ WL is an inner-product space, a generic linear bounded operator L {•} can always be written as the sum of two self-adjoint operators, namely, L {•} =

L {•} − L† {•} L {•} + L† {•} +j = X{•} + j Y{•} 2 2j

(D.205)

and this expression (sometimes called the Cartesian decomposition of L {•}) mimics and extends the analogous splitting available for square matrices and dyadics [see (E.55)]. Decomposition (D.205) is unique, because if O {•} is the null operator — which we may add to the leftmost-hand side of (D.205) without modifying the action of L {•} — then we have . / O {•} = X0 {•} + j Y0 {•} =⇒ X0 {•} = − j Y0 {•} = j Y0 {•} † = (−X0 {•})† = −X0 {•} (D.206) whence it follows that X0 {•} = O {•} and hence Y0 {•} = O {•}. In view of the remarkable analogy between (D.205) and representation (B.1) of complex numbers, some Authors (e.g., [10, p. 34]) refer to X{•} and Y{•} as the real and imaginary parts of L {•}. An operator is said to be normal if VL ≡ WL and the following identity holds [10, p. 33]   L† {L {v}} = L L† {v} , v ∈ VL (D.207) that is, if L {•} commutes with its adjoint. In particular, an operator L {•} on an inner-product space VL is said unitary if it is normal and additionally   v ∈ VL (D.208) L† {L {v}} = L L† {v} = I {v} = v, in which case by virtue of (D.179) we have   v2VL = (v, v)VL = L† {L {v}}, v = (L {v}, L {v})VL = L {v}2VL VL   † (u, v)VL = L {L {u}}, v = (L {u}, L {v})VL VL

(D.209) (D.210)

that is, the norm of v and the inner product in VL are preserved under the action of L {•}. The geometrical interpretation of (D.209) and (D.210) is that a unitary operator does not alter ‘distances’ and ‘angles’ [see (D.46)] in the space VL . A classic example of unitary operator is provided by the Fourier transformation, in which instance (D.209) is called the Parseval formula [22, Section 4.8]. Further, by comparing (D.208) with (D.139) we gather L−1 {w} ≡ L† {w} ,

w ∈ WL ≡ VL

(D.211)

that is, if L {•} is unitary, the adjoint operator and the inverse one coincide. On account of (D.200) and (D.208) we have   L {•}2 = L† {L {•}} = I {•} = 1 (D.212) whence we conclude that the norm of a unitary operator is equal to one.

Functional analysis

D.7

1157

The spectrum of a linear operator

If VL ≡ WL , we define the spectrum σ(L) of a (bounded) linear operator L {•} as the set of complex numbers ν for which the operator (L − νI){•} : VL −→ VL

(D.213)

is not invertible [3, Chapter 14]. In particular, if the equation L {e} = νe,

ν ∈ C,

e ∈ VL \ {0}

(D.214)

admits non-trivial solutions, we say that e is an eigenvector of L {•} with eigenvalue ν. The set σ(L) may consist of three parts, namely, • • •

the point spectrum σP (L) or set of eigenvalues ν ∈ C for which (L − νI){•} is not injective [cf. (D.122) and (D.123)]; the continuous spectrum σC (L) that contains those ν with (L − νI){•} not surjective, though injective and with image dense in VL (i.e., ran (L − νI) ≡ VL ); the residual spectrum σR (L) that contains the numbers ν for which (L − νI){•} is injective but L {v} − νv is not dense (i.e., ran (L − νI)  VL ).

Although σ(L) provides important information about L {•}, still σ(L) does not characterize L {•} uniquely inasmuch as different operators may have the same spectrum. Eigenvectors associated with different eigenvalues are linearly independent, whereas eigenvalues and eigenvectors of L {•} and L† {•} on Hilbert spaces are related as follows. By assuming that u and v are two vectors such that L {u} = αu,

L† {v} = βv,

α, β ∈ C,

in accordance with (D.214), we consider   α (v, u)VL = (v, L {u})VL = L† {v}, u = β∗ (v, u)VL VL

=⇒

u, v ∈ VL \ {0}

(D.215)

(α − β∗ ) (v, u)VL = 0

(D.216)

whereby we have two possibilities, viz., (i) (ii)

(v, u)VL does not vanish, and hence α = β∗ , so that the eigenvalues of L† {•} are the complex conjugates of those of L {•} and viceversa; α  β∗ , and hence (v, u)VL = 0, so that the eigenvectors of L {•} are always orthogonal to those of the adjoint operator, so long as they are not associated with complex conjugate pairs of eigenvalues.

As a special case, the eigenvalues of a self-adjoint operator are real and the eigenvectors are orthogonal to each other. Indeed, if (D.214) holds true, then ν e2VL = ν (e, e)VL = (e, L {e})VL = (L {e}, e)VL = ν∗ (e, e)VL = ν∗ e2VL

(D.217)

on account of property (D.36) of the inner product. Since e2VL is strictly positive — e cannot be the null vector — then ν = ν∗ . If e1 and e2 are two eigenvectors with associated real eigenvalues ν1 and ν2 , then we have ν1 (e1 , e2 )VL = (L {e1 }, e2 )VL = (e1 , L {e2 })VL = ν2 (e1 , e2 )VL

(D.218)

again by virtue of (D.36) and (D.37). Since ν1  ν2 , it follows that (e1 , e2 )VL = 0, as claimed.

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Advanced Theoretical and Numerical Electromagnetics

A compact operator has at most a countable set of eigenvalues having no limit points with exception of possibly zero [15, p. 398], [23, Section 5.2]. Proving that the eigenvalues tend to zero is easier under the additional hypothesis that L {•} is also self-adjoint, in which case the infinitely many eigenvectors {en }+∞ n=1 are orthogonal and can obviously be chosen to have unitary norm, and the eigenvalues {νn }+∞ n=1 are real. Then, since en VL = 1, the sequence e1 , e2 , . . . is bounded, and the sequence of image vectors L {en } = νn en ,

n ∈ N \ {0}

(D.219)

is convergent (or a subsequence can be extracted that converges) to a vector v ∈ VL , because L {•} is compact. Therefore, for any arbitrarily small number η > 0 it is possible to find two sufficiently large whole numbers n and m so that 1/2  1/2  = νn en 2VL − 2 (νn en , νm em )VL + νm em 2VL ν2n + ν2m = νn en − νm em VL = (νn en − v) − (νm em − v)VL η η  L {en } − vVL + L {em } − vVL  + = η 2 2

(D.220)

where we have exploited the orthogonality of νn en and νm em and the triangle inequality (D.11). Since the leftmost member is never negative, this means that the eigenvalues νn must tend to zero and, as a consequence, the limit vector v = 0 ∈ VL . By contrast, it should be noticed that the sequence of eigenvectors of a compact self-adjoint operator does not converge per se, because  1/2 √ = 2, n, m ∈ N \ {0} (D.221) en − em VL = en 2VL + em 2VL which shows that the ‘distance’ between any two eigenvectors remains constant. When v  0 ∈ VL is not an eigenvector, then L {v} is not ‘parallel’ to v, and we can determine the scaling factor ξ which minimizes the ‘distance’ between ξv and L {v}, i.e., the quantity   L {v} − ξv2VL = L {v}2VL − 2Re ξ∗ (v, L {v})VL + |ξ|2 v2VL



(v, L {v})VL

2 | (v, L {v})VL |2

2 = L {v}VL − + ξ vVL −



vVL

v2V L

 L {v}2VL −

| (v, L {v})VL |2

(D.222)

v2VL

where (•, •)VL is a suitable inner product in VL . Evidently, the distance in question is minimized if equality holds between the leftmost-hand side and the rightmost member, which happens if we choose ξ=

(v, L {v})VL

v ∈ VL \ {0}

v2VL

(D.223)

where the number ξ is called the Rayleigh quotient or numerical range of the operator. When it comes to bounded self-adjoint operators on an inner-product space VL , the Rayleigh quotient is not merely a scaling factor, because it can be shown that [10, Section 2.2], [23, Theorem 4.30], [8, Lemma 8.26] L {•} ≡ L† {•}

=⇒

L {•} = sup v∈VL

| (v, L {v})VL | v2VL

= NL ,

v0

(D.224)

Functional analysis

1159

which provides an alternative way of computing the norm of L {•}. To begin with, we notice that for any vector v ∈ VL the number (v, L {v})VL is real, because * +∗ (v, L {v})VL = (L {v}, v)VL = (v, L {v})VL

(D.225)

since L {•} is self-adjoint and (D.38) applies. From the Cauchy-Schwarz inequality (D.150) and for any element w ∈ VL we have immediately | (w, L {w})VL | w2VL



L {w}VL  L {•} wVL

=⇒

NL  L {•}

(D.226)

by the very definition (D.224) of NL and that of operator norm (D.124). Next, we pick up any two vectors u and v and consider the identity (u + v, L {u + v})VL − (u − v, L {u − v})VL = 2 (v, L {u})VL + 2 (u, L {v})VL   * +∗ = 2 (v, L {u})VL + 2 (v, L {u})VL = 4 Re (v, L {u})VL

(D.227)

where the leftmost member is real by virtue of (D.225), and we have invoked (D.38) and L {•} = L† {•}. By using this equality backwards, applying the triangle inequality (H.19) and the definition of NL with u + v and u − v in lieu of v we find  



4 Re (v, L {u})VL 

(u + v, L {u + v})VL

+

(u − v, L {u − v})VL

    (D.228)  NL u + v2VL + u − v2VL = 2NL u2VL + v2VL also on account of (D.45) and a general instance of the parallelogram law (H.21) valid in innerproduct spaces. Now, in light of the arbitrariness of u and v we may make the special choices * +∗ (v, L {w})VL wVL u=w , v = L {w} , ∀w ∈ VL \ {0}, (D.229) | (v, L {w})VL | L {w}VL to get first   2| (v, L {w})VL |  NL w2VL + v2VL

(D.230)

and then 2| (v, L {w})VL | = 2 wVL L {w}VL  2NL w2VL

(D.231)

because uVL = vVL = wVL by construction. Dividing through by 2 w2VL and passing to the supremum with respect to w gives L {•}  NL

(D.232)

which in combination with (D.226) proves the claim (D.224). What is more, if L {•} is self-adjoint and in addition (v, L {v})VL = 0,

∀ v ∈ VL

(D.233)

then (D.224) says that L {•} = 0, whereby L {•} is the null operator in light of property (D.9).

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Advanced Theoretical and Numerical Electromagnetics

We now have the necessary tools to derive a fundamental result that concerns the point spectrum of a self-adjoint compact operator L {•} on a Hilbert space VL , namely, L {e} = νe

ν = ± L {•}

with

(D.234)

that is to say, either L {•} or − L {•} or possibly both numbers are eigenvalues of L {•} [10, Lemma 5.9, p. 48], [6, Lemma 16.1, p. 148], [8, Theorem 9.16, p. 225]. To begin with, we observe that, if L {•} is the null operator, assertion (D.234) is trivial, because any member of VL \ {0} is evidently an eigenvector associated with the eigenvalue ν = L {•} = 0. Therefore, we may suppose that L {•}  O {•}. In the general case we can formally devise a suitable bounded sequence {un }+∞ n=1 ⊂ VL which satisfies the following conditions un VL = 1 lim | (un , L {un })VL | = L {•} = |ν|

(D.235) (D.236)

n→+∞

lim L {un } = e ∈ VL \ {0}

(D.237)

n→+∞

where (D.236) holds by virtue of property (D.224) which in turn applies since L {•} is self-adjoint, and (D.237) stems from the presumed compactness of L {•}. More importantly, the vector e cannot be the null element of VL , or else (D.237) would imply that for an arbitrarily small number η > 0 we could choose n  N ∈ N so that, thanks to the Cauchy-Schwarz inequality (D.150), we would arrive at | (un , L {un })VL |  L {un }VL < η

(D.238)

a requirement which patently contradicts property (D.236), as we have assumed L {•}  0. Besides, since (D.225) says that (un , L {un })VL is, in fact, a real number, then (D.236) is equivalent to lim (un , L {un })VL = ν = ± L {•}

(D.239)

n→+∞

or, by the very definition of limit [2, Definition 4.1], also to ∀η > 0

∃n  N

:

−η < ν − (un , L {un })VL < η

(D.240)

and we shall put this condition to use in a moment. Now, the initial claim (D.234) is certainly proved if we manage to show that (L − νI) {e} = lim (L − νI) {L {un }} = 0 n→+∞

(D.241)

in light of definition (D.214). The first part of the above string of equalities follows easily from the boundedness of (L − νI){•}, which is granted in that both L {•} and I {•} are bounded operators. In symbols, for an arbitrarily small η > 0 there exists an index n  N so that we have (L − νI){L {un } − e}VL  (L − νI){•} L {un } − eVL  2|ν|η

(D.242)

in view of property (D.11) applied to the supremum norm, the definition of ν in (D.239), and assumption (D.237). As regards the second equality in (D.241) we consider (L − νI){L {un }}2VL = L {L {un }} − νL {un }2VL = L {L {un } − νun }2VL

Functional analysis

1161

* +  L {•}2 L {un } − νun 2VL = ν2 L {un }2VL − 2ν (un , L {un })VL + ν2 * +  ν2 L {•}2 − 2ν (un , L {un })VL + ν2 * + = 2ν3 ν − (un , L {un })VL < 2ν3 η (D.243) on the grounds of normalization (D.235) and the second part of (D.240), and this shows that the leftmost member of (D.243) can be made as small as is desired by taking a sufficiently large index n. In conclusion, (D.242) and (D.243) along with (D.241) provide evidence of the validity of (D.234). Further, from (D.234) and (D.224) we deduce that the magnitude of the eigenvalues of L {•} cannot exceed |ν| = L {•}, because for any eigenvector em ∈ VL associated with the eigenvalue νm we have L {•} 

| (em , L {em })VL | em 2VL

=

| (em , νm em )VL | em 2VL

= |νm |,

m ∈ N \ {0}

(D.244)

whence we conclude that the eigenvalues of a self-adjoint compact operator are to be found in the closed interval [− L {•} , L {•}] ⊂ R.

D.8

The Fredholm alternative

The Fredholm alternative is a famous theorem named after E. I. Fredholm [27] that concerns compact linear operators from a space VL into itself [7, Chapter 8], [15, Chapter 8], [28, Chapter 1], [20, Section VIII.2.4], [13, pp. 227–243], [29, Section 1.9], [30–32] and can be regarded as an extension of the theory of linear mappings in finite dimensional spaces [4, Chapter 3]. In words, if L {•} is a linear compact operator and L† {•} is the adjoint thereof, then (a)

either the following Fredholm equations of the second kind u − L {u} = f,

v − L† {v} = g,

u, v, f, g ∈ VL

(D.245)

have a unique solution for any source terms f and g, and the homogeneous equations associated with (D.245), u0 − L {u0 } = 0, (b)

v0 − L† {v0 } = 0,

u 0 , v 0 ∈ VL

(D.246)

have only the trivial solutions u0 = 0 and v0 = 0 or the homogeneous equations in (D.246) have the same finite number of non-trivial linearly independent solutions u1 , . . . , uN , v1 , . . . , vN , and the Fredholm equations in (D.245) are solvable if, and only if, (vn , f )VL = 0,

(un , g)VL = 0,

n = 1, . . . , N

(D.247)

in which case the solutions to (D.245) are not unique and read u = uP +

N  n=1

an un ,

v = vP +

N 

b n vn

n=1

where uP and vP are particular solutions and an , bn are arbitrary constants.

(D.248)

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Advanced Theoretical and Numerical Electromagnetics

In light of (D.245) the first of conditions (D.247) is equivalent to stating that the members of the image of the operator I {•} − L {•} are orthogonal to the elements of the null space of the adjoint operator I {•} − L† {•}, whereas the converse holds true for the second condition. These features are in agreement with (D.194). Since u = I {u}, where I {•} is the identity operator on VL , the combination u − L {u} in (D.245) is called a perturbation of the identity by a compact operator [7, p. 143].

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

[16] [17]

[18] [19]

[20]

Dudley DG. Mathematical Foundations for Electromagnetic Theory. New York, NY: WileyInterscience; 1994. Rudin W. Principles of mathematical analysis. 3rd ed. New York, NY: McGraw-Hill; 1976. Muscat J. Functional Analysis. London, UK: Springer; 2014. Blyth TS, Robertson EF. Basic Linear Algebra. 2nd ed. Springer Undergraduate Mathematics Series. London, UK: Springer-Verlag; 2002. Dettman JW. Mathematical Methods in Physics and Engineering. New York, NY: McGrawHill; 1962. Meise R, Vogt D. Introduction to Functional Analysis. Oxford, UK: Clarendon Press; 1997. Bowers A, Kalton NJ. An introductory course in functional analysis. Universitext. New York, NY: Springer Science+Business Media; 2014. Hunter JK, Nachtergaele B. Applied Analysis. Singapore: World Scientific; 2001. Available from: https://doi.org/10.1142/4319. Blyth TS, Robertson EF. Further Linear Algebra. 2nd ed. Springer Undergraduate Mathematics Series. London, UK: Springer-Verlag; 2002. Conway JB. A Course in Functional Analysis. 2nd ed. Graduate Texts in Mathematics. New York, NY: Springer-Verlag; 1990. Vulich BZ. Introduction to functional analysis for scientists and technologists. Oxford, UK: Pergamon Press; 1963. Translated by I. N. Sneddon. Courant R, Hilbert D. Methoden der mathematischen Physik. 2nd ed. Berlin: Springer; 1931. Müller C. Foundations of the Mathematical Theory of Electromagnetic Waves. Berlin Heidelberg: Springer-Verlag; 1969. Van Bladel JG. Singular Electromagnetic Fields and Sources. Piscataway, NJ: IEEE Press; 1991. Bassanini P, Elcrat AR. Theory and applications of partial differential equations. Mathematical Concepts And Methods In Science And Engineering. New York, NY: Plenum Press; 1997. Kellogg OD. Foundations of potential theory. Berlin Heidelberg: Springer-Verlag; 1929. Sobolev SL. Applications of functional analysis in mathematical physics. vol. 7 of Translations of Mathematical Monographs. Providence, Rhode Island: American Mathematical Society; 1963. Adams RA, Fournier JJF. Sobolev spaces. 2nd ed. New York, NY: Academic Press; 2003. Dautray R, Lions JL. Mathematical Analysis and Numerical Methods for Science and Technology - Functional and Variational Methods. vol. 2. Berlin Heidelberg: Springer-Verlag; 1990. Dautray R, Lions JL. Mathematical Analysis and Numerical Methods for Science and Technology - Spectral Theory and Applications. vol. 3. Berlin Heidelberg: Springer-Verlag; 1990.

Functional analysis [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]

1163

Girault V, Raviart PA. Finite element methods for Navier-Stokes equations: Theory and algorithms. Berlin, Germany: Springer-Verlag; 1986. Morse P, Feshbach H. Methods of Theoretical Physics. New York, NY: McGraw-Hill; 1953. Limaye BV. Linear Functional Analysis for Scientists and Engineers. Singapore: Springer Science+Business Media; 2016. Riesz F. Sur les opérations fonctionnelles linéaires. C R Acad Sci (Paris). 1909;149:974–977. Rudin W. Real and Complex Analysis. 3rd ed. London, UK: McGraw-Hill; 1987. Brezis H. Analyse fonctionnelle – Théorie et applications. Paris, France: Masson; 1983. Fredholm EI. Sur une classe d’équations fonctionnelles. Acta Mathematica. 1903;27:365– 390. Atkinson KE. The Numerical Solution of Integral Equations of the Second Kind. Cambridge, UK: Cambridge University Press; 1997. Van Bladel JG. Electromagnetic Fields. Piscataway, NJ: IEEE Press; 2007. Colton DL, Kress R. Integral Equation Methods in Scattering Theory. New York, NY: John Wiley & Sons, Inc.; 1983. Kantorovich LV, Akilov GP. Functional analysis in normed spaces. Oxford, UK: Pergamon Press; 1964. Translated by D. E. Brown. Rudin W. Functional Analysis. New York, NY: McGraw-Hill; 1973.

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Appendix E

Dyads and dyadics

E.1

Scalars, vectors, and beyond

The laws of physics are formulated by introducing mathematical entities in turn related by differential or integral equations. More often than not the relevant entities depend on position r and time t, in which case we say that the physical entity in question is a field. Some entities are easily described by specifying the intensity and, correspondingly, a scalar field. Prominent examples in the context of electromagnetism are provided by the charge density (r, t) (Section 1.1) and the electrostatic potential Φ(r) (Section 2.1). More generally, we shall need to complement the intensity with a direction and an orientation, and the corresponding mathematical tool is a vector field. The current density J(r, t) (Section 1.1) and the magnetization M(r, t) (Section 5.6) are typical examples. In a few notable situations, though, we have to express a relationship between two vector fields that is linear in the sense of (D.120). If the fields were just parallel to one another, a simple proportionality factor would do the trick. However, sometimes we may want to signal that the directions and orientations of the two vectors form angles other than 0 or π. To this end, we must introduce a mathematical field entity which is more general than vectors. To illustrate the problem we turn to an example taken from electrostatics or, more specifically, the electric field E(r) produced by an elementary electric dipole with moment p that, for simplicity, we suppose located in the origin of a system of Cartesian coordinates (see Figure E.1 and Example 2.5). Clearly, the vector E(r) — besides being singular at the very location r = 0 of the dipole — is not necessarily parallel to the moment p at any point in space. We begin by supposing that the dipole moment is aligned with the x-axis so that p = p1 = p x xˆ (Figure E.1a). The associated electric field is a vector which, being proportional to p x , may be written as E1 (r) = xˆ G xx (r)p x + yˆ Gyx (r)p x + zˆ Gzx (r)p x = Gx (r)p x

(E.1)

where G xx (r), Gyx (r) and Gzx (r) are suitable scalar fields which represent the Cartesian components of the vector field Gx (r). By definition, we may interpret Gx (r) as the electric field produced by a dipole with unitary moment (p x = 1 Cm) aligned with the x-axis. In like manner, for the dipoles with moments p2 = py yˆ and p3 = pz zˆ (Figures E.1b and E.1c) we may assume E2 (r) = xˆ G xy (r)py + yˆ Gyy (r)py + zˆ Gzy (r)py = Gy (r)py E3 (r) = xˆ G xz (r)pz + yˆ Gyz (r)pz + zˆ Gzz (r)pz =

Gz (r)pz

(E.2) (E.3)

where G xy (r) etc. are scalar fields. Now, in order to express the electric field produced by a generic dipole whose moment is given by p := p1 + p2 + p3 = p x xˆ + py yˆ + pz zˆ

(E.4)

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Advanced Theoretical and Numerical Electromagnetics

(a) p1 = p x xˆ

(b) p2 = py yˆ

(c) p3 = pz zˆ

Figure E.1 For illustrating the genesis of a dyadic field: electrostatic fields produced by three orthogonal electric dipoles located in the origin.

and, as such, is arbitrarily oriented in space, we invoke the principle of superposition (Section 6.1) and combine the fields generated by p1 , p2 and p3 , namely, E(r) = E1 (r) + E2 (r) + E3 (r) = Gx (r)p x + Gy (r)py + Gz (r)pz = xˆ G xx (r)p x + yˆ Gyx (r)p x + zˆ Gzx (r)p x + xˆ G xy (r)py + yˆ Gyy (r)py + zˆ Gzy (r)py + xˆ G xz (r)pz + yˆ Gyz (r)pz + zˆ Gzz (r)pz

(E.5)

which is the most general linear relationship between the components of the moment and those of the electric field in terms of nine scalar fields. Evidently, only when the latter satisfy the conditions G xx (r) = Gyy (r) = Gzz (r) G xy (r) = Gyx (r) = G xz (r) = Gzx (r) = Gyz (r) = Gzy (r) = 0

(E.6) (E.7)

for some point r  0, does (E.5) yield the special case of electric field parallel to the moment. The main shortcoming of (E.5) is that it is lengthy to write and cumbersome to manipulate. Hence, we try to write it in a more compact notation by introducing a symbol for the nine scalar fields G xx (r), G xy (r) etc. E(r) = G xx (r)ˆx(ˆx · p) + Gyx (r)ˆy(ˆx · p) + Gzx (r)ˆz(ˆx · p) + G xy (r)ˆx(ˆy · p) + Gyy (r)ˆy(ˆy · p) + Gzy (r)ˆz(ˆy · p) + G xz (r)ˆx(ˆz · p) + Gyz (r)ˆy(ˆz · p) + Gzz (r)ˆz(ˆz · p)  = G xx (r)ˆxxˆ + G xy (r)ˆxyˆ + G xz (r)ˆxzˆ + Gyx (r)ˆyxˆ + Gyy (r)ˆyyˆ + Gyz (r)ˆyzˆ

 +Gzx (r)ˆzxˆ + Gzy (r)ˆzyˆ + Gzz (r)ˆzzˆ · p = G(r) · p

(E.8)

having recognized that p x = xˆ · p, py = yˆ · p, and pz = zˆ · p. Each one of the nine terms within brackets is a scalar times a pair of vectors and is referred to as a dyad, whereas we call dyadic the linear combination of any number of dyads [1, Appendix A.3], [2, Section 1.6], [3, Chapter 2], [4–6]. Besides, we say that G xx (r) and so on represent the nine Cartesian components of the dyadic field G(r). The rightmost-hand side of (E.8) provides the relationship between the electric field and the dipole moment in dyadic form. More importantly, even though we have worked with Cartesian coordinates for the sake of simplicity, the dyadic form

Dyads and dyadics

1167

is actually a coordinate-free expression. This means we need not bother with the components of either G(r) or p until we really get around to carrying out the dot-product, and even then, we are not bound to using just Cartesian coordinates. By the way, the relevant expression of G(r) in polar spherical coordinates is given in (2.43). If we want to exhibit the components of a dyadic field explicitly we first choose a system of coordinates. In particular, we may express (E.8) alternatively as ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎜⎜⎜E x (r)⎟⎟⎟ ⎜⎜⎜G xx (r) G xy (r) G xz (r)⎟⎟⎟ ⎜⎜⎜ p x ⎟⎟⎟ ⎜⎜⎜⎜E (r)⎟⎟⎟⎟ = ⎜⎜⎜⎜G (r) G (r) G (r)⎟⎟⎟⎟ ⎜⎜⎜⎜ p ⎟⎟⎟⎟ yy yz ⎟⎟⎠ ⎜⎜⎝ y ⎟⎟⎠ ⎜⎜⎝ y ⎟⎟⎠ ⎜⎜⎝ yx Ez (r) Gzx (r) Gzy (r) Gzz (r) pz

 

(E.9)

=[G]

which we call the matrix form of the dyadic relation. The 3 × 3 matrix [G] is uniquely associated with the dyadic G(r). However, had we opted for another system of coordinates, the entries of the resulting matrix [G] would have been different! In specific, though, we observe that the components of the vectors Gx (r), Gy (r) and Gz (r) form the columns of the associated matrix [G]. We may also define the vectors G x (r) := G xx (r)ˆx + G xy (r)ˆy + G xz (r)ˆz

(E.10)

and the like with the entries of the rows of [G]. With these positions and in light of (E.8) we may express the dyadic field in the alternative formats G(r) = Gx (r)ˆx + Gy (r)ˆy + Gz (r)ˆz = xˆ G x (r) + yˆ Gy (r) + zˆ Gz (r)

(E.11)

where the order of the various vectors in each dyad is essential! A dyadic field may be interpreted as a kind of linear operator or mapping (Appendix D.3) which takes a vector and transforms it into another vector. For example, the notation Gyz for the dyadic component yz is a powerful and convenient mnemonic for remembering that it transforms the zcomponent of the ‘input’ vector into (a contribution to) the y-component of the ‘output’ vector, as can be gathered from both (E.8) and (E.9).

E.2

Dyadic calculus

The dyadic formalism and most related formulas were introduced by J. W. Gibbs in 1881 and 1884 [7]. To lighten the notation a bit we work with ordinary vectors in the three-dimensional space V3 , though we should keep in mind that all properties and operations we shall derive apply to dyadics formed with real and complex vector fields as well [3, Chapter 2]. As a matter of fact, dyadic fields may be introduced in abstract vector spaces with more than just three dimensions [2, Section 1.6]. By ‘pairing’ any two vectors u and v we may form two dyads, namely, A = uv,

B = vu

(E.12)

which are different unless u and v are parallel. To determine the pertinent nine Cartesian components of A we need to write u and v explicitly. For instance A = (u x xˆ + uy yˆ + uz zˆ )(v x xˆ + vy yˆ + vz zˆ ) = u x v x xˆ xˆ + u x vy xˆ yˆ + u x vz xˆ zˆ

1168

Advanced Theoretical and Numerical Electromagnetics + uy v x yˆ xˆ + uy vy yˆ yˆ + uy vz yˆ zˆ + uz v x zˆ xˆ + uz vy zˆ yˆ + uz vz zˆ zˆ

(E.13)

and similarly for B. Alternatively, we may try and compute the associated matrix [A] directly by writing ⎛ ⎞ ⎜⎜⎜u x ⎟⎟⎟  ⎜ ⎟ [A] = ⎜⎜⎜⎜⎜uy ⎟⎟⎟⎟⎟ v x ⎝ ⎠ uz

vy

⎛ ⎜⎜⎜u x v x ⎜ vz = ⎜⎜⎜⎜⎜uy v x ⎝ uz v x 

u x vy u y vy u z vy

⎞ u x vz ⎟⎟⎟ ⎟ uy vz ⎟⎟⎟⎟⎟ ⎠ u z vz

(E.14)

where the result is obtained in accordance with the row-by-column multiplication of ordinary matrices [8–10]. Notice that u, the first vector of the pair, is associated with a one-column matrix, whereas v, the second vector of the pair, is associated with a one-row matrix. Moreover, direct inspection allows concluding that the matrix [B] associated with B in (E.12) is the transpose of [A]. We now examine some properties of dyadics and a few operations which involve vectors and dyads or dyadics.

E.2.1 Sum of dyadics and product with a scalar The sum of two dyadics A and B is still a dyadic that is computed by writing A and B down explicitly and summing the coefficients of identical elementary dyads. In Cartesian coordinates this operation reads C = A + B = (A xx + B xx )ˆxxˆ + · · · + (Azz + Bzz )ˆzzˆ

(E.15)

whence it follows that A+B=B+A

(E.16)

i.e., the sum of two dyadics is commutative. Alternatively, one may obtain the matrix associated with the sum by summing the matrices [A] and [B] [8–10]. The product of a dyadic A with a scalar is still a dyadic given by B = αA = αA xx xˆ xˆ + · · · + αAzz zˆ zˆ ,

α∈C

(E.17)

whence we conclude that αA = Aα,

α∈C

(E.18)

so this operation is commutative, too. Alternatively, the resulting associated matrix follows by computing α[A] [8–10]. In light of definitions (E.15) and (E.17) the set of all dyadics formed with pairs of vectors in the ordinary space V3 constitutes a vector space D (Appendix D.1). In particular, the nine elementary dyads xˆ xˆ ,

xˆ yˆ , xˆ zˆ ,

yˆ xˆ , yˆ yˆ ,

yˆ zˆ , zˆ xˆ ,

zˆ yˆ ,

zˆ zˆ

(E.19)

are all different from each other (better yet, linearly independent) and form a basis for the representation of all dyadics in three dimensions.

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1169

E.2.2 Scalar and vector product Given A as in (E.12) we may define two scalar products with a vector w, namely, scalar product from the left A · w := u(v · w)

(E.20)

and scalar product from the right w · A := (w · u)v

(E.21)

which yield different results in general, except when u and v are parallel. Notice that dot-multiplying a dyadic with a vector produces another vector. If A does not come written as a pair of two vectors we need to write it explicitly, e.g., as in (E.8) and apply the rules above to the elementary dyads (E.19). Given four vectors a, b, u and v and the dyads A := ab and B := uv we define the scalar products between A and B as A · B := a(b · u)v = (b · u)av

(E.22)

B · A := u(v · a)b = (v · a)ub

(E.23)

which are clearly different. More generally, we may evaluate the scalar products by computing the resulting associated matrix which is the row-by-column product of the matrices [A] and [B] [8–10] in the same order as that of the dyads. The dot-product of two dyads or dyadics is a dyad or a dyadic. Given A as in (E.12) and a vector w we may introduce two vector products A × w := u(v × w)

(E.24)

w × A := (w × u)v

(E.25)

which are different in general. The result of cross-multiplying a dyadic and a vector is still a dyadic.

E.2.3 Neutral elements The null dyadic O is the neutral element of the sum (E.15) and is defined so that A+O=O+A=A

(E.26)

whence we conclude that the nine components of O are all zero in any system of coordinates. The unit or identity dyadic I is the neutral element of the scalar products (E.20), (E.21), and (E.22) I·u= u·I = u

(E.27)

I·A=A·I=A

(E.28)

and direct inspection shows that I takes on the explicit forms I := xˆ xˆ + yˆ yˆ + zˆ zˆ = ρˆ ρˆ + ϕˆ ϕˆ + zˆ zˆ = rˆ rˆ + ϕˆ ϕˆ + θˆ θˆ

(E.29) (E.30) (E.31)

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Advanced Theoretical and Numerical Electromagnetics

in Cartesian, circular cylindrical, and polar spherical systems of coordinates (Appendix A.1). More generally, if a, b and c are three orthogonal vectors or vector fields, the unit dyadic reads I :=

aa bb cc + 2+ 2 2 |a| |b| |c|

(E.32)

whereby it follows that the matrix associated with I is always the identity matrix [I]. We can also obtain two explicit expressions for I in terms of three arbitrarily oriented vectors a, b and c, so long as they are not co-planar. To prove this statement, we consider a vector u and write u = ξa + ηb + ζc

(E.33)

and try to determine the triple (ξ, η, ζ) of expansion coefficients. Now, dot-multiplying both sides with a, b and c in succession produces a linear system of three coupled equations which, in effect, can be solved for (ξ, η, ζ). However, the system matrix is full because a, b and c are not necessarily orthogonal to one another, and thus the overall procedure is clumsy. In fact, to single out ζ we need a vector which is simultaneously orthogonal to a and b, and such a vector happens to be a × b [11, Section 2.4]. Likewise, b × c is perpendicular to b and c, and c × a is orthogonal to c and a. These are not unit vectors, granted, but we can worry about it later on. Meanwhile, we can write a × b · u = ζ a × b · c = ζJ

(E.34)

c × a · u = η c × a · b = η a × b · c = ηJ b × c · u = ξ b × c · a = ξ a × b · c = ξJ

(E.35) (E.36)

where we have exploited the permutation properties (H.13) of the triple scalar product. Depending on the orientation of a, b and c the quantity J equals plus or minus the volume of the hexahedron with edges a, b and c. Solving the above uncoupled equations for (ξ, η, ζ) and substituting back into (E.33) yields [3, Section 1.6]     c×a a×b b×c c×a a×b b×c +b +c a+ b+ c (E.37) u= a ·u =u· J J J J J J which by comparison with (E.27) provides the desired expressions for I. Since the vectors a, b and c are not perpendicular, we end up with two representations of I that are to be used for dotmultiplication from the left and the right, respectively. The set of three vectors a :=

b×c , J

b :=

c×a , J

c :=

a×b J

(E.38)

is sometimes called the reciprocal or dual basis associated with the set a, b and c. As a byproduct we have proved that ξ, η and ζ are the projections of u along the vectors of the dual basis. Furthermore, in light of (E.28) we can extend (E.37) to a dyadic, namely,     c×a a×b c×a a×b b×c b×c +b +c a+ b+ c (E.39) A= a ·A=A· J J J J J J which provides the starting point to write any dyadic as the sum of three dyads [3, Section 2.1.1]. −1 The inverse dyadic A , when it exists, is the dyadic which satisfies the equation −1

A·A

−1

=I=A

·A

(E.40)

Dyads and dyadics

1171

and it may be determined by inverting the associated matrix [A] [8, 10]. The vector product between the unit dyadic and a vector is independent of the order of multiplication. For instance, in Cartesian coordinates we have I × u = u × I = u x (ˆzyˆ − yˆ zˆ ) + uy (ˆxzˆ − zˆ xˆ ) + uz (ˆyxˆ − xˆ yˆ )

(E.41)

and the associated matrix reads ⎞ ⎛ ⎜⎜⎜ 0 −uz uy ⎟⎟⎟ ⎟ ⎜ I × u −→ ⎜⎜⎜⎜⎜ uz 0 −u x ⎟⎟⎟⎟⎟ ⎠ ⎝ −uy u x 0

(E.42)

but it is also instructive to carry out the proof starting with the very general representations (E.37).

E.2.4 Transpose and Hermitian transpose T

The transpose of a dyadic A is denoted with the symbol A and is the dyadic associated with the matrix [A]T . For the dyads (E.19) computing the transpose amounts to swapping the order of the vectors in the pairs, namely, (ˆxxˆ )T := xˆ xˆ ,

(ˆxyˆ )T := yˆ xˆ ,

(ˆxzˆ )T := zˆ xˆ

(E.43)

and so forth. A dyadic A is said to be symmetric if it coincides with its transpose T

A=A

(E.44)

whence it follows that in a three-dimensional space A is entirely specified by just six independent coefficients, because A xy = Ayx , A xz = Azx and Ayz = Azy . A single dyad A = uv (with u and v real vectors) can be symmetric only if u and v are parallel to one another. Indeed, (E.44) demands vu = uv

(E.45)

which when dot-multiplied from the left by v and from the right by u according to (E.20) and (E.21) yields |v|2 |u|2 = (u · v)2

=⇒

u · v = ±|u||v|

(E.46)

and in view of (H.5) this condition is true precisely if u and v are aligned. Extending the result to a complex dyad A = uv with u and v complex vectors requires dot-multiplying by v∗ and u∗ . A dyadic A is said anti-symmetric or skew-symmetric if it coincides with the negative of its transpose T

A = −A

(E.47)

whence we conclude that in a three-dimensional space A is completely determined by just three independent coefficients, since in particular the diagonal components must be null [cf. (E.42)]. As a result, it is possible to associate an anti-symmetric dyadic with a vector and vice-versa, namely, A=I×u

(E.48)

Advanced Theoretical and Numerical Electromagnetics

1172 where

A xy = −uz ,

A xz = uy ,

Ayz = −u x

(E.49)

thanks to (E.41) and (E.42). A single dyad A = uv (with u and v real vectors) cannot be anti-symmetric. Indeed, (E.47) implies vu + uv = 0

(E.50)

which when dot-multiplied from the left by v and from the right by u yields |v|2 |u|2 + (v · u)2 = 0

(E.51)

and this equation can only be true if u = 0 = v, the left member being a non-negative real number. The extension to a complex dyad A with u and v complex vectors just requires dot-multiplication by v∗ and u∗ . H The Hermitian transpose or adjoint of a complex dyadic A is indicated with the symbol A and is the dyadic associated with the matrix [A]H = ([A]T )∗ [3, Section 2.7]. A complex dyadic A is said to be Hermitian if it coincides with its Hermitian transpose A=A

H

(E.52)

and it is said anti-Hermitian or skew-Hermitian if it coincides with the negative of its Hermitian transpose H

A = −A .

(E.53)

In light of (E.15) and (E.17) an arbitrary dyadic can be decomposed into the sum of symmetric and anti-symmetric dyadics, viz., T

T

A+A A−A A= + = A s + Aa 2 2

    symmetric

(E.54)

anti-symmetric

and a complex dyadic can also be written as the sum of Hermitian and anti-Hermitian dyadics or even two Hermitian dyadics [cf. (D.205)] H

H

H

H

A+A A−A A+A A−A +j . A= + = 2 2 2 2j

    Hermitian

(E.55)

anti-Hermitian

It can be shown that these decompositions are unique. Indeed, suppose that A=B+C

(E.56)

where B is symmetric and C is anti-symmetric. Then, we have T

T

T

A =B +C =B−C

(E.57)

and the system comprised of equations (E.56) and (E.57) can be solved for B and C to find B = A s and C = Aa . A similar proof of uniqueness applies to (E.55).

Dyads and dyadics

1173

E.2.5 Double scalar product and double vector product Given two dyads A := ab and B := uv we define the double scalar product and the double cross product as [3, Chapter 2] A : B := (a · u)(b · v) = u · A · v A × × B := (a × u)(b × v)

(E.58) (E.59)

and the result is a scalar or a dyadic, respectively. For general dyadics we need to write A and B explicitly as combinations of elementary dyads to which we can apply the two rules above. Furthermore, direct inspection shows that T

A:B=B:A=A :B

T

(E.60)

× A × × B=B × A

(E.61)

i.e., in particular the double scalar and double vector product are commutative. For the double scalar product we may write A : B = (Ax xˆ + Ay yˆ + Az zˆ ) : (Bx xˆ + By yˆ + Bz zˆ ) = (Ax · Bx )(ˆx · xˆ ) + (Ay · By )(ˆy · yˆ ) + (Az · Bz )(ˆz · zˆ ) = B : A Ax ,

Ay ,

Az

Bx ,

By ,

(E.62)

Bz

and are the vectors formed with the columns of the associated where matrices [A] and [B], respectively [see (E.9)]. The conclusion follows because the scalar product between ordinary vectors is commutative. The double scalar product of a symmetric dyadic A with an anti-symmetric one B vanishes invariably, because  1  1  T T 1 A:B= A:B+A:B = A:B−A :B = A:B−A:B =0 (E.63) 2 2 2 where in the last step we have used the rightmost part of (E.60). Since the double scalar product associates two dyadics with a number, this operation may be adopted as an inner product in the sense of (D.35) in the vector space D of dyadics, i.e., (•, •)D : D × D → R   A, B := A : B = A xx B xx + · · · + Azz Bzz D

(E.64) (E.65)

which clearly obeys the necessary properties listed in Appendix D.1. We can compute the double scalar product of a dyadic A with itself by using Cartesian coordinates for simplicity, namely, A : A = (Ax xˆ + Ay yˆ + Az zˆ ) : (Ax xˆ + Ay yˆ + Az zˆ ) = (Ax · Ax )(ˆx · xˆ ) + (Ay · Ay )(ˆy · yˆ ) + (Az · Az )(ˆz · zˆ ) = |A xx |2 + |Ayx |2 + |Azx |2 + · · · + |A xz |2 + |Ayz |2 + |Azz |2

(E.66)

and since the rightmost-hand side of (E.66) is non-negative, we can define the norm of a dyadic as [3, Chapter 2]    A := A : A (E.67)

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where a complex conjugation on either factor is in order for a complex dyadic A. It is easy to check that the elementary dyads in (E.19) have unitary norm. Having recognized the double scalar product as a suitable inner product, then it is possible to extend the Cauchy-Schwarz inequality (D.150) to dyadics as well, viz.,          A, B  := A : B  A B (E.68) D by following the steps outlined in Appendix D.4 and invoking the commutativity of A : B. Comparison of (E.65) with (H.5) suggests that we interpret the number A:B cos γ =     A B

(E.69)

as the cosine of the ‘angle formed’ by A and B in the nine-dimensional space D. The angle γ is certainly real because the condition | cos γ|  1 is guaranteed by (E.68). Besides, with the help of (E.68) we can go on to prove       A + B  A + B (E.70) that is, the triangular inequality for dyadics. Assuming that A(r) is a dyadic field which can be integrated together with its norm over a domain D ⊆ R3 we can prove the estimate         dV A(r)  dV A(r) (E.71)   D D by picking up an arbitrary constant dyadic, say, B, with unitary norm and considering                B : dV A(r) =  dV B : A(r)  dV B : A(r)  dV A(r) (E.72)     D D D D   precisely on account of (E.68) and the fact that B = 1 by definition. Now we make the special choice  dV A(r)  B := D (E.73)  dV A(r) D whereby the leftmost-hand side of (E.72) becomes                dV A(r)  B :  : dV A(r) =  D dV A(r) =  dV A(r)     dV A(r)      D

D

D

(E.74)

D

and this concludes the proof of (E.71). An estimate which involves the scalar product (E.20) of a dyadic A and a vector w can be obtained by using the second part of (E.11) and the Cauchy-Schwarz inequality (D.151), namely, 2      A · w = xˆ Ax · w + yˆ Ay · w + zˆ Az · w2 = |Ax · w|2 + Ay · w2 + |Az · w|2

Dyads and dyadics  2    |A x |2 + |Ay |2 + |Az |2 |w|2 = A |w|2 whence we get     A · w  A |w|.

1175 (E.75)

(E.76)

E.2.6 Determinant, trace and eigenvalues The determinant of a dyadic [3, Section 2.4] is the same as the determinant of its associated matrix [8–10], viz.,   A xx A xy A xz  det A := det[A] = Ayx Ayy Ayz    Azx Azy Azz  = A xx (Ayy Azz − Ayz Azy ) + A xy (Ayz Azx − Ayx Azz ) + A xz (Ayx Azy − Ayy Azx ) 1 = A × × A:A 6

(E.77)

where the double cross product has priority over the double scalar product, which, by producing a number, would render the other operation meaningless. The determinant is an invariant, meaning that it is a characteristic number associated with the dyadic and, as such, does not depend on the specific basis chosen. A dyadic whose determinant vanishes does not have an inverse in the sense of (E.40). It is also evident from (E.77) and (E.42) that the determinant of an anti-symmetric dyadic is null. The trace of a dyadic is defined as   (E.78) Tr A := I : A = A xx + Ayy + Azz and coincides with the sum of its diagonal components when expressed in an orthonormal basis, e.g., the Cartesian one. Like the determinant (E.77), the trace is an invariant as well. A dyadic is said to be traceless if its trace vanishes. For instance, anti-symmetric or anti-Hermitian dyadics are perforce traceless. To prove the rightmost part of (E.78) we observe I : A = (ˆxxˆ + yˆ yˆ + zˆ zˆ ) : (Ax xˆ + Ay yˆ + Az zˆ ) = (ˆx · Ax )(ˆx · xˆ ) + (ˆy · Ay )(ˆy · yˆ ) + (ˆz · Az )(ˆz · zˆ ) = A xx + Ayy + Azz

(E.79)

where Ax , Ay and Az are vectors formed with the columns of the associated matrix [A]. As a special case, with the aid of (E.37) we consider     b×c c×a a×b b×c c×a a×b : I:I= a +b +c a+ b+ c J J J J J J        c × a c × a  a×b a×b b×c b×c ·a + b· ·b + c· ·c = a· J J J J J J   2   2   J = 3 2 = 3 = Tr I = I (E.80) J thanks to the permutation properties of the triple scalar product.

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Having defined the left and right scalar product of a dyadic A with a vector w in (E.20) and (E.21) we can regard A as a linear operator (Appendix D.3) which transforms vectors into vectors. In particular, we call eigenvectors of A the special vectors whose direction in the three-dimensional space is not altered by the action of A [cf. (D.214)]. In symbols, this condition means A · u = αu

=⇒

(A − αI) · u = 0

α∈C

(E.81)

v · A = βv

=⇒

v · (A − βI) = 0

β∈C

(E.82)

where u and v denote a right and a left eigenvector, respectively, and α and β are the corresponding eigenvalues of A. Since u and v must satisfy algebraic linear systems of three homogeneous equations, non-trivial solutions are possible only if     det A − αI = 0 = det A − βI (E.83) which, when expanded in accordance with (E.77), give rise to the same algebraic equation of order three [3, Section 2.6]. Therefore, we conclude that left and right eigenvalues coincide and let λn , n = 1, 2, 3, denote the eigenvalues of A. We are now able to show that the trace of A is intimately related to its eigenvalues. We observe that for general real dyadics the eigenvectors may be complex and not necessarily orthogonal to one another. Thus, we consider the three defining equations A · un = λn un ,

n = 1, 2, 3

(E.84)

and dot-multiply each one of them from the left with u2 × u3 , u3 × u1 and u1 × u2 , viz., u2 × u3 · A · u1 = λ1 u2 × u3 · u1 = λ1 J

(E.85)

u3 × u1 · A · u2 = λ2 u3 × u1 · u2 = λ2 J

(E.86)

u1 × u2 · A · u3 = λ3 u1 × u2 · u3 = λ3 J

(E.87)

thanks to the permutation properties (H.13) of the triple scalar product. By dividing through by J — which is non-zero as the eigenvectors are linearly independent by hypothesis — and summing the equations side by side we arrive at 3  n=1

λn = A :

u × u u3 × u1 u1 × u2  2 3 u1 + u2 + u3 J J J

in view of (E.58). Hence, we conclude   λ1 + λ2 + λ3 = A : I = Tr A

(E.88)

(E.89)

by virtue of (E.37).

E.3 Differential operators We continue the discussion by considering a dyadic field A(r) ∈ Cn (R3 )9 , n ∈ N, in the ordinary three-dimensional space. Clearly, if we take the partial derivatives of the nine components of A(r) with respect to x, y and z — provided said components are at least continuous in the region of interest — we end up with a set of 3 × 9 = 27 scalar fields.

Dyads and dyadics

1177

In general, a mathematical object with twenty-seven components is called a triadic [3] or, with modern definition, a tensor of rank three. In the special case where the twenty-seven components are all the possible derivatives that we can obtain from A(r), the triadic represents the gradient of A(r) and is denoted, e.g., with ∇A. However, we shall not pursue this topic any further, as we do not make use of ∇A in this book. There are, in fact, meaningful ways of combining subsets of the twenty-seven derivatives of A(r) in order to construct vector or dyadic fields. In particular, the divergence of A(r) is a vector, whereas the curl of A(r) is still a dyadic [5]. The divergence of A(r) is defined in Cartesian coordinates as ∇ · A(r) := (∇ · Ax )ˆx + (∇ · Ay )ˆy + (∇ · Az )ˆz ∂A x ∂Ay ∂Az = + + ∂x ∂y ∂z

(E.90) (E.91)

and it involves only nine out of all the possible partial derivatives. The curl of A(r) is defined in Cartesian coordinates as ∇ × A(r) = (∇ × Ax )ˆx + (∇ × Ay )ˆy + (∇ × Az )ˆz       ∂Ay ∂Ax ∂Az ∂Ay ∂Ax ∂Az − − − = xˆ + yˆ + zˆ ∂y ∂z ∂z ∂x ∂x ∂y

(E.92) (E.93)

and it combines only eighteen out of all the possible derivatives. Divergence and curl of A(r) in circular cylindrical and polar spherical coordinates can be found in [4, Appendix 4].

References [1] [2] [3] [4] [5] [6] [7]

[8] [9] [10] [11]

Rothwell EJ, Cloud MJ. Electromagnetics. London, UK: CRC Press; 2001. Morse P, Feshbach H. Methods of Theoretical Physics. New York, NY: McGraw-Hill; 1953. Lindell IV. Methods for electromagnetic field analysis. Piscataway, NJ: IEEE Press; 1992. Van Bladel JG. Electromagnetic Fields. Piscataway, NJ: IEEE Press; 2007. Tai CT. Generalized Vector and Dyadic Analysis. New York, NY: IEEE Press; 1997. Van Bladel JG. Singular Electromagnetic Fields and Sources. Piscataway, NJ: IEEE Press; 1991. Gibbs JW. Elements of vector analysis. Privately printed in two parts: New Haven; 1881 and 1884. Reprint in The scientific papers of J. Willard Gibbs, vol. 2, pp. 84-90, Dover, New York, 1961. Bau III D, Trefethen LN. Numerical linear algebra. Philadelphia, PA: Soci. Indus. Ap. Math.; 1997. Blyth TS, Robertson EF. Basic Linear Algebra. 2nd ed. Springer Undergraduate Mathematics Series. London, UK: Springer-Verlag; 2002. Golub GH, van Loan CF. Matrix Computations. Baltimore, MD: Johns Hopkins University Press; 1996. Holt CA. Introduction to Electromagnetic Fields and Waves. New York, NY: John Wiley & Sons, Inc.; 1963.

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Appendix F

Properties of smooth surfaces

The rigorous definition of a C2 -smooth surface S ⊂ R3 is somewhat complicated. For our purposes it ˆ can be defined univocally will suffice to say that, if S is C2 -smooth, then the unit normal vector n(r) in any point r ∈ S or, equivalently, the tangent plane to S exists and is unique for any point r ∈ S . In this sense a sphere is C2 -smooth, whereas the boundary of a cube is not. Then again, the boundary of a cube is made up of six squares which are C2 -smooth, so we may say that such surface is piecewise C2 -smooth. Further, while the notion of smoothness might not seem so relevant to electromagnetic theory in the first place, one should consider that the triangular-faceted tessellations employed in the solution of surface integral equations with the Method of Moments (Chapter 14) are, in fact, piecewise C2 -smooth open or closed surfaces. In this Appendix, we examine a few properties of smooth surfaces that are essential for the study of surface integrals such as those met in the integral representation of electromagnetic potentials and entities.

ˆ  ) · (r − r) An estimate for n(r

F.1

We wish to prove a result that comes in handy when investigating the behavior of flux integrals which involve the derivatives of the three-dimensional scalar Green function (e.g., see Sections 2.7, 5.1, 9.1 and Appendix G). More specifically, provided S is C2 -smooth, then the following estimate holds true    n(r ˆ ) · (r − r)  M|r − r |2 , r → r r, r ∈ S (F.1) ˆ  ) is the unit normal on S at r , and M denotes a suitable positive constant. Intuitively, we where n(r are convinced that (F.1) must be true, since the vector r − r tends to become tangent to S as r → r or the other way around, though it is not obvious that the quantity in the left-hand side falls off as the square of the distance |r − r |. For the proof we pick up a point r ∈ S and consider a ball B(r, a) [see definition (D.13)] with radius a small enough for the intersection S ∩ B(r , a) to be non-empty. We introduce local coordinates ξ := (u, v) on S ∩ B(r , a) by means of the bijective mapping Ψ : R2 ⊃ B2 (0, 1) −→ S ∩ B(r, a) ⊂ R3

(F.2)

where B2 (0, 1) is the unit circle (D.22). Let ξ := (u , v ) ∈ B2 (0, 1) be the local coordinates which are mapped to r ∈ S by Ψ. We expand the vector-valued function Ψ(ξ) in a Taylor series around the point ξ [1, Section 9.13], namely,   ∂Ψ  ∂Ψ   (u − u ) +  (v − v ) + f(ξ, ξ ) (F.3) Ψ(ξ) = Ψ(ξ ) +    ∂u ξ ∂v ξ r

r

second-order terms

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Advanced Theoretical and Numerical Electromagnetics

with  2 |f(ξ, ξ )|  M1 ξ − ξ  for some positive constant M1 . Since the two vectors   ∂Ψ  ∂Ψ   ,  ∂u ξ ∂v ξ are tangential to S in r , the unit vector perpendicular to S in the same point is given by   ∂Ψ  ∂Ψ   ×  ∂u ξ ∂v ξ ˆ  ) := ±  n(r     ∂Ψ  ∂Ψ    ×    ∂u ξ ∂v ξ 

(F.4)

(F.5)

(F.6)

and is well defined because S is C2 -smooth by assumption. Next, with the aid of (F.6), (F.3) and the Cauchy-Schwarz inequality (D.151) we estimate         ∂Ψ  ∂Ψ  ∂Ψ  ∂Ψ          × · (u − u ) + (v − v ) + f(ξ, ξ )    ∂u ξ    ∂v ξ ∂u ξ ∂v ξ  n(r ˆ ) · (r − r ) =     ∂Ψ  ∂Ψ    ×    ∂u ξ ∂v ξ        ∂Ψ  ∂Ψ   ×  · f(ξ, ξ )   2  ∂u ξ ∂v ξ  =  |f(ξ, ξ )|  M1 ξ − ξ      ∂Ψ  ∂Ψ    ×    ∂u ξ ∂v ξ  2  = M Ψ−1 (r) − Ψ−1 (r )  M|r − r |2 (F.7) 1

where the last inequality follows from estimate (A.96), since the inverse mapping Ψ−1 (•) exists and is twice-differentiable by definition.

F.2 Solid angle subtended at a point We consider a finite volume V bounded by a C2 -smooth surface ∂V and assume that the unit normal on ∂V is oriented positively towards the interior of V. We pick up a ball B(r, a) with r ∈ ∂V and observe that, if the radius a is small enough for the intersection ∂V ∩ B(r, a) to be non-empty, then ∂V divides the ball B(r, a) into two regions lying on either side of ∂V. In particular, the surface ∂B+a := ∂B(r, a) ∩ V is contained in V, and its area is a2 Ω(a), where Ω(a) is the solid angle subtended by ∂B+a with respect to the point r (see Figure F.1). We wish to show that lim Ω(a) = 2π

a→0+

(F.8)

under the previous assumptions of smoothness for ∂V. This result was used implicitly, e.g., in the calculation of the values returned by the integral representations (2.167) and (5.25) for observation points on the boundary of the region of interest.

Properties of smooth surfaces

1181

Figure F.1 Calculation of the solid angle subtended at a point on a smooth surface. For the proof we consider the closed curve γ := ∂B ∩ ∂V and the quantity ˆ = a cos α, (r − r) · n(r)

r ∈ γ

(F.9)

ˆ where α ∈ [αm , α M ] ⊂ [0, π] is the angle formed by r − r and n(r) and is measured from the unit normal. As r moves along γ, α = α(r ; a) reaches its extreme values for, say, r = rm and r = rM , as is suggested in Figure F.1. We denote with S m := {r ∈ ∂B : α ∈ [0, αm ]},

S M := {r ∈ ∂B : α ∈ [0, α M ]}

(F.10)

the two spherical caps which bound the surface ∂B+a from below and from above; also notice that γ ⊂ S M ∩ S m . Evidently, the area of ∂B+a is bounded by the areas Am and A M of S m and S M , respectively. In view of the shape of S m and S M it is convenient to compute Am and A M by means of a local system of polar spherical coordinates (a, α, β) centered in r [cf. (A.22)]

αm dS = 2πa

Am := Sm

dα sin α = 2πa2 (1 − cos αm )

2 0

= 2πa[a −

(rm

 ˆ ˆ − r) · n(r)]  2πa[a − |(rm − r) · n(r)|]  2πa a − cm a2

dS = 2πa2

A M :=

(F.11)

αM

SM

= 2πa[a −

dα sin α = 2πa2 (1 − cos α M ) 0

(rM

 ˆ ˆ − r) · n(r)]  2πa[a + |(rM − r) · n(r)|]  2πa a + c M a2

on account of (F.9) and (F.1). Finally, we estimate

 

2πa a − cm a2  Am  a2 Ω(a)  A M  2πa a + c M a2

(F.12)

(F.13)

whence, by dividing through by a2 , we find 2π(1 − cm a)  Ω(a)  2π(1 + c M a)

(F.14)

and, since Ω(a) is manifestly bounded by two continuous functions of a which tend to 2π as a → 0+ , (F.8) follows.

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Advanced Theoretical and Numerical Electromagnetics

Figure F.2 The open neighborhood Ha (shown in grey) that surrounds a C2 -smooth closed surface S .

F.3 Points in an open neighbourhood An open neighbourhood Ha of a C2 -smooth closed surface S ⊂ R3 (see Figure F.2) is the set of points defined by

ˆ  ) ∈ R3 : r ∈ S , |w| < a (F.15) Ha := r = r + wn(r ˆ  ) denotes the outward unit normal to S at r . where n(r We wish to show that there exists a constant a0 > 0 such that for all a < a0 and every point ˆ  ). Since n(r ˆ  ) is orthogonal r ∈ Ha there exists a unique point r ∈ S and |w| < a with r = r + wn(r  to S in r , the point r belongs to the straight line perpendicular to the plane tangential to S in r , and |w| represents the distance of r from the surface or the tangent plane. For the proof by contradiction we suppose that there are indeed two points r1 and r2 on S and show that they cannot but be coincident. Since r ∈ Ha , in accordance with (F.15) we have two possible representations for r, viz., ˆ 1 ) = r2 + w2 n(r ˆ 2 ) r = r1 + w1 n(r

(F.16)

where we also assume that ˆ 1 ) · n(r ˆ 2 )  0 n(r

(F.17)

provided the distance |r1 − r2 |  2a0 . We further take a0 such that a0 M
0

(F.25)

on account of the working hypothesis (F.18). Therefore, the left-hand side of (F.23) is always positive or null, and the condition can only be true if w1 = w2 . As a result, (F.20) implies that r1 = r2 . ˆ  ) ∈ Ha (see Figure Next, we wish to show that given any two points r0 ∈ S and r = r + wn(r F.3) the following estimate holds      r − r0   2 r − r0  (F.26) under hypothesis (F.18). To this purpose we examine 2  2  2  r − r0  = r − r0 + wn(r ˆ  )  r − r0  + 2wn(r ˆ  ) · (r − r0 )      2 2 2  r − r0  − 2|w| ˆn(r ) · (r − r0 )  r − r0  − 2a0 M r − r0   2 = (1 − 2a0 M) r − r0 

(F.27)

by virtue of (F.1). From (F.18) we get a milder requirement a0 M
1 −

3 1 = 4 4

(F.29)

and (F.27) becomes |r − r0 |2 

1  |r − r0 |2 4

(F.30)

whereby (F.26) follows immediately. ˆ 1 ) and Finally, we wish to show that for any two points r1 , r2 ∈ Ha with r1 = r1 + w1 n(r   ˆ 2 ) (see Figure F.4) the following estimate is true r2 = r2 + w1 n(r |r1 − r2 |  2|r1 − r2 | under the same assumption (F.18). We observe  2 ˆ 1 ) − w2 n(r ˆ 2 ) |r1 − r2 |2 = r1 − r2 + w1 n(r 2    ˆ 1 ) − w2 n(r ˆ 2 )  r1 − r2  + 2(r1 − r2 ) · w1 n(r

(F.31)

Properties of smooth surfaces   2   ˆ 1 ) − w2 n(r ˆ 2 )   r1 − r2  − 2 (r1 − r2 ) · w1 n(r     2  ˆ 1 ) − 2|w2 | (r1 − r2 ) · n(r ˆ 2 )  r1 − r2  − 2|w1 | (r1 − r2 ) · n(r    2 2 2  r1 − r2  − 4a0 M r1 − r2  = (1 − 4a0 M) r1 − r2 

1185

(F.32)

where we have invoked (F.1) twice. From (F.18) we derive a less stringent condition a0 M
1 −

3 1 = 4 4

(F.34)

and (F.32) passes over into |r1 − r2 |2 

1  |r − r2 |2 4 1

(F.35)

whereby estimate (F.31) follows.

F.4

Criterion for the Hölder continuity of scalar fields

The set of points        Ub := r ∈ R3 : inf r − r < b 

(F.36)

r ∈S

ˆ ) ∈ is called an open strip around the C2 -smooth surface S . Notice that if r is such that r = r + wn(r Hb , where Hb is the neighborhood of S with thickness b, then       r − r   r − r  = wn(r ˆ  ) < b inf (F.37)  r ∈S

and hence Hb ⊂ Ub . We wish to prove that a scalar field f : R3 → C is Hölder continuous [2,3], that is, for a suitable constant M > 0 | f (r1 ) − f (r2 )|  M|r1 − r2 |ν ,

r1 , r2 ∈ R3 ,

ν ∈ [0, 1[

(F.38)

if the following three conditions are met (Hölder a) (Hölder b)

f (r) is Hölder continuous in some neighborhood Ha of a smooth surface S ; f (r) is Lipschitz continuous in R3 \ Ub for every b > 0, i.e., | f (r1 ) − f (r2 )|  M1 |r1 − r2 |,

(Hölder c)

r1 , r2 ∈ R3 \ Ub

(F.39)

where Ub is the open strip around S with thickness b; f (r) is bounded, i.e.,  f ∞ < +∞ [see (D.71)].

For the proof we set the distance b > 0 in such a way that U3b ⊂ Ha and we distinguish three situations.

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Advanced Theoretical and Numerical Electromagnetics

(a) |r1 − r2 | < b with r1 ∈ U2b

(b) |r1 − r2 | < b with r1  U2b

Figure F.5 Close-up of part of a smooth surface S and three strips Ub , U2b and U3b for proving the Hölder continuity of a scalar field. (i) (ii)

Suppose that |r1 − r2 | < b with r1 ∈ U2b , as is suggested in Figure F.5a. Then r1 , r2 ∈ U3b ⊂ Ha , whereby Hölder continuity follows from condition (Hölder a). Suppose that |r1 − r2 | < b but r1  U2b , as is depicted in Figure F.5b. Then r1 , r2  Ub and hence from condition (Hölder b) we have | f (r1 ) − f (r2 )|  M1 |r1 − r2 |  M1 |r1 − r2 |1−ν |r1 − r2 |ν  M1 b1−ν |r1 − r2 |ν .

(iii)

(F.40)

Suppose that |r1 − r2 | > b, then condition (Hölder c) implies | f (r1 ) − f (r2 )|  | f (r1 )| + | f (r2 )| =

2  f ∞ ν 2  f ∞ b  |r1 − r2 |ν . bν bν

(F.41)

References [1] [2] [3]

Rudin W. Principles of mathematical analysis. 3rd ed. New York, NY: McGraw-Hill; 1976. Van Bladel JG. Singular Electromagnetic Fields and Sources. Piscataway, NJ: IEEE Press; 1991. Bassanini P, Elcrat AR. Theory and applications of partial differential equations. Mathematical Concepts And Methods In Science And Engineering. New York, NY: Plenum Press; 1997.

Appendix G

A surface integral involving the time-harmonic scalar Green function

In this Appendix we examine the properties of a rather general integral over a C2 -smooth surface S that involves the gradient of the time-harmonic scalar Green function (8.356), namely, the complexvalued scalar field     e− j k|r−r | , r ∈ Ha dS  ψ(r ) − ψ(r0 ) V(r ) · ∇ (G.1) Dk (r) := 4π|r − r | S

where ˆ 0 ), |w| < a, is a point in the neighborhood Ha of S [see (F.15) and r0 ∈ S and r = r0 + wn(r Figure F.2]; the scalar field ψ(r ) is Hölder continuous on S [1, 2], say,

• •

|ψ(r1 ) − ψ(r2 )|  Mψ |r1 − r2 |α ,

r1 , r2 ∈ S ,

α ∈]0, 1[

(G.2)

the possibly complex vector field V(r ) is simply continuous for r ∈ S .

•

Under these hypotheses we wish to show that (1) (2)

the right-hand side of (G.1) exists also for r ≡ r0 ∈ S as an improper integral; Dk (r) is a Hölder continuous function of r in Ha for any exponent β < α ∈]0, 1[.

These statements are far from obvious in view of the singular nature of the Green function, though the result may be expected since the difference ψ(r ) − ψ(r) tends to zero as r → r, and this weakens the strong singularity of ∇ G(r, r ) [see (9.167)]. Moreover, by setting the wavenumber k to zero in (G.1) we recover the integral the involves the static Green function (2.131), inasmuch as this step does not affect the order of the singularity for r → r [cf. (3.189)]. Integrals of type (G.1) find application in the study of the smoothness of, e.g., the scalar and the vector potentials discussed in Sections 2.10, 5.1 (with k = 0), 9.1, and 13.2.3.

G.1 Two estimates for ∇G(r, r ) To begin with, we derive the following two estimates    ∇ G(r, r ) 

M1 , |r − r |2    |r − r | ∇ G(r1 , r ) − ∇G(r2 , r )  M2 1 23 , |r1 − r |

r ∈ S , with

r ∈ Ha

(G.3)

|r1 − r |  3|r2 − r |

(G.4)

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Advanced Theoretical and Numerical Electromagnetics

where G(r, r ) indicates the three-dimensional Green function (8.356), rl ∈ Ha , l = 1, 2, and Ml > 0 are suitable constants. For the sake of clarity we introduce the shorthand notation R = |r − r |, Rl = |rl − r |, and G(r, r ) = G(R). The derivatives of G(R) with respect to the distance R, viz., G (R) and G (R), are given explicitly in (9.167) and (9.168). For the proof of (G.3) we use (9.167) and observe that    c c Rc + c M ∇ G(r, r )  1 + 2 2 = 1 2 2  21 (G.5) 4πR 4πR 4πR R where we have noticed that R, as a function of r and r , is bounded from above because r ∈ Ha and r ∈ S (see Figure F.2). For the proof of (G.4) we first obtain the intermediate result         1 1  r − r1 − r − r2  =  r2 − r1 + (r − r ) − 2   |r − r1 | |r − r2 |   |r − r1 | |r − r1 | |r − r2 |    |r − r2 | − |r − r1 | |r2 − r1 | + |r − r2 |   |r − r1 | |r − r1 ||r − r2 | |r2 − r1 | |r2 − r1 | |r1 − r2 | + =2 (G.6)   |r − r1 | |r − r1 | |r1 − r | on account of inequalities (H.19) and (H.20). Secondly, by invoking the fundamental theorem of calculus [3, Theorem 6.21] we observe  ⎧  R ⎛ ⎞ ⎪  R2   2 ⎪ ⎪ ⎪ c3 c3 ⎜⎜⎜ 1 1 ⎟⎟⎟ ⎪    ⎪ ⎜⎝ − ⎟⎠ , R2 > R1  dR G (R)  ⎪ dR = ⎪ ⎪ ⎪ 4πR3 8π R21 R22   ⎪ ⎪    ⎪ R1 ⎨ R1 G (R1 ) − G (R2 ) = ⎪   (G.7) ⎪ R1 ⎪ ⎛ ⎞  R1   ⎪ ⎪   ⎪ c3 c3 ⎜⎜⎜ 1 1⎟ ⎪  dR G (R)  ⎪ ⎪ ⎜⎝ 2 − 2 ⎟⎟⎟⎠ , R1 > R2 dR = ⎪ ⎪   3 ⎪ 8π 4πR R2 R1  ⎪  ⎩ R2

R2

where we have used the dominant term in the estimate (9.167) because r ∈ Ha and r ∈ S . All in all, for any combination of r1 and r2 and r  {r1 , r2 } this implies     ⎛ ⎞    c3  1 R22 − R21  ⎜⎜⎜ 1  c 1 c3 1 ⎟⎟⎟ 3   −  = ⎜ ⎟⎠ G (R1 ) − G (R2 )  |R = − R | + 2 1 ⎝ 8π  R21 R22  8π R21 R22 8π R1 R22 R21 R2   ⎞ ⎛   ⎜⎜⎜ 9 3 ⎟⎟⎟ 3c3 |r2 − r | − |r1 − r | 3c3 |r2 − r1 | c3 |R2 − R1 | ⎜⎝ 3 + 3 ⎟⎠ =  (G.8)  8π 2π 2π |r1 − r |3 |r1 − r |3 R1 R1 since R1 < 3R2 by hypothesis [see the second part of (G.4)]. Finally, we examine      r − r1 r − r2   ∇ G(r1 , r ) − ∇G(r2 , r ) = G (R1 )  − G (R2 )  |r − r1 | |r − r2 |         r − r1 r − r2    G (R1 ) − G (R2 ) + G (R2 )   −  |r − r1 | |r − r2 |  2|r2 − r1 | c2 3c3 |r2 − r1 | +   3  2 2π |r1 − r | 4π|r − r2 | |r − r1 | |r2 − r1 | 3c3 + 9c2 |r2 − r1 |  = M2  2π |r − r1 |3 |r − r1 |3 where we have used (G.8), (9.167), (G.6) and once again R1 < 3R2 ; this proves (G.4).

(G.9)

A surface integral involving the time-harmonic scalar Green function

1189

G.2 Finiteness and Hölder continuity The proof that Dk (r) is finite for observation points on S can be effected by letting r = r0 ∈ S in (G.1), and then by considering a smaller surface S 2 ⊂ S such that r ∈ S 2 . The integral over S \ S 2 is then finite, whereas the integral over S 2 can be treated with a limiting procedure similar to the one followed in Section 2.10 to arrive at (2.256) by invoking (G.3), the boundedness of V(r ), and the Hölder continuity of ψ(r ). ˆ l ) ∈ To prove that Dk (r) in (G.1) is Hölder continuous we pick up any two points rl = rl + wl n(r  Ha , rl ∈ S , l = 1, 2, and try to show that |Dk (r1 ) − Dk (r2 )|  M|r1 − r2 |β ,

r l ∈ Ha

(G.10)

with β < α ∈]0, 1[. To this purpose we divide the surface S into two parts, viz., S r1 ,b := {r ∈ S : |r − r1 | < b} = S ∩ B(r1, b)

with

b = 3|r1 − r2 |

(G.11)

and S \ S r1 ,b . We can think of S r1 ,b as the intersection of S with the ball B(r1, b). Based on definition (G.1) we write the difference Dk (r1 ) − Dk (r2 ) explicitly as 

  dS  ψ(r ) − ψ(r1 ) V(r ) · ∇G(r1 , r )

Dk (r1 ) − Dk (r2 ) =



S

−

  dS  ψ(r ) − ψ(r2 ) V(r ) · ∇G(r2 , r ) (G.12)

S

and split the integration into two parts, namely, over S r1 ,b and over S \ S r1 ,b . For the contribution from the two integrals over S r1 ,b we have        IS   :=  dS  ψ(r ) − ψ(r1 ) V(r ) · ∇ G(r1 , r ) r ,b  1 S r ,b

          − dS ψ(r ) − ψ(r2 ) V(r ) · ∇ G(r2 , r )   S r ,b 1      dS  ψ(r ) − ψ(r1 ) |V(r )| ∇ G(r1 , r )  1



S r ,b



1

+

    dS  ψ(r ) − ψ(r2 ) |V(r )| ∇ G(r2 , r )

S r ,b



1

 Mψ M1 V ∞ S r ,b

dS 



|r − r1 |α |r − r2 |α + |r1 − r |2 |r2 − r |2

 (G.13)

1

because V(r ) is continuous and thus bounded on S r1 ,b , ψ(r ) is Hölder continuous on S r1 ,b ⊂ S , and the gradient of G(rl , r ) is estimated by (G.3). Then, we recall that rl ∈ Ha and apply (F.26) twice to

Advanced Theoretical and Numerical Electromagnetics

1190 get

    |r − r1 |α |r − r2 |α    IS    4Mψ M1 V ∞ dS + 4M M V dS ψ 1 ∞ r ,b 1 |r1 − r |2 |r2 − r |2   S r ,b

S r ,b 1

4Mψ M1 V ∞ dS  + |r1 − r |2−α

1

S r ,b



1

S r ,2b

4Mψ M1 V ∞ dS  |r2 − r |2−α

(G.14)

2

where the last step follows because the integrand is positive and the larger surface S r2 ,2b := {r ∈ S : |r − r2 | < 2b} = S ∩ B(r2, 2b)

b = 3|r1 − r2 |

with

(G.15)

contains S r1 ,b , and this property is proved by means of (2.263). In order to show that the remaining integrals in (G.14) are finite for α > 0 we extend the limiting procedure described for the single-layer potential in Section 2.10. In the interest of conciseness we denote either S r1 ,b or S r2 ,2b with S 2 and rl with just r. Besides, we may use the same bijective mapping Ψ defined in (2.252), and consider the ring-like region B2 (0, 1) \ B2 (ξ, ν) — where ν is a dimensionless radius small enough that B2 (ξ, ν) ⊂ B2 (0, 1) — which is mapped by Ψ onto the surface S 2 \ S ν (cf. Figure 2.20). The smaller surface S ν serves to isolate the singular point r ∈ S ν ⊂ S 2 . Then, we have     1 1  ∂Ψ × ∂Ψ    dS  = du dv |r − r |2−α |Ψ(ξ) − Ψ(ξ )|2−α  ∂u ∂v  S 2 \S ν

B2 (0,1)\ B2 (ξ,ν)



du dv



B2 (0,1)\ B2 (ξ,ν)



du dv



B2 (ξ,2)\ B2 (ξ,ν)

c2  |Ψ(ξ) − Ψ(ξ )|2−α c1 c2 = |ξ − ξ |2−α

2π 0



du dv

B2 (0,1)\ B2 (ξ,ν)

2

dη

dτ τ

ν

c1 c2 α c1 c2 (2 − να )  2α+1 π = 2π α α

c1 c2 |ξ − ξ |2−α

c1 c2 τ2−α (G.16)

on account of estimate (2.253). Since the rightmost member of (G.16) is independent of ν, we conclude that the integral over S 2 \ S ν exists also for ν → 0+ , provided α > 0. Lastly, by formally defining two bijective mappings from the circles B2 (ξ1 , b) and B2 (ξ2 , 2b) onto S r1 ,b and S r2 ,2b , respectively, we have   c1 1 bα     c dS   du dv = (G.17)   α 1 |r1 − r |2−α 2π|ξ1 − ξ |2−α B2 (ξ1 ,b)

S r ,b 1



dS 

S r ,2b 2

1  |r2 − r |2−α



du dv

B2 (ξ2 ,2b)

2π|ξ2

α c2 αb  c  2−α = 2 α 2 −ξ |

(G.18)

where c1 and c2 are other suitable positive constants, and we have integrated in polar coordinates (τ , η ) centered in ξ1 and ξ2 . Inserting (G.17) and (G.18) back into (G.14) and combining the inessential constant factors yield    IS    MI bα = MI bα−β 3β |r1 − r2 |β (G.19) r ,b 1

A surface integral involving the time-harmonic scalar Green function

1191

since b = 3|r1 − r2 | by hypothesis. At this stage we have no compelling reason for choosing β < α, though this condition will turn out to be necessary to ensure the boundedness of the contribution from S \ S r1 ,b . For the difference of the integrals over S \ S r1 ,b we consider        IS \S   :=  dS  ψ(r ) − ψ(r1 ) V(r ) · ∇G(r1 , r ) r ,b  1 S \S r ,b 1             − dS ψ(r ) − ψ(r2 ) V(r ) · ∇ G(r2 , r )   S \S r ,b 1      dS  |V(r )| ∇G(r1 , r )  ψ(r2 ) − ψ(r1 ) S \S r ,b



1

    dS  ψ(r ) − ψ(r2 ) |V(r )| ∇G(r1 , r ) − ∇G(r2 , r )

+

(G.20)

S \S r ,b 1

where we have added and subtracted an integral proportional to ψ(r2 ). We recall that V(r ) is bounded on S , ψ(r ) is Hölder continuous, and we apply estimates (G.3) and (G.4), i.e.,       |r1 − r2 |α    α |r1 − r2 | IS \S    Mψ M3 V ∞ dS + |r − r2 | (G.21) r ,b 1 |r1 − r |2 |r1 − r |3 S \S r ,b 1

where M3 := max{M1 , M2 }. To be precise, the application of (G.4) requires that |r − r1 |  3|r − r2 |. Indeed, this condition is satisfied for points r ∈ S \ S r1 ,b and it follows from (2.269)-(2.271). Furthermore, since r ∈ S \ S r1 ,b , by definition of the distance b and on account of (2.269) we have |r1 − r2 | =

b 1  2  |r − r1 |  |r − r1 |  |r − r1 | 3 3 3

(G.22)

and also |r − r2 |  2|r − r2 |  2|r − r1 | + 2|r1 − r2 |  4|r − r1 |

(G.23)

by virtue of (F.26) and (G.22). By using (F.31), (G.23) and (F.26) in (G.21) and combining the constant factors we get      α |r1 − r2 |  |r1 − r2 | IS \S    M4 dS + r ,b 1 |r1 − r |2 |r1 − r |3−α S \S r ,b 1

= M4 |r1 − r2 |

β

 dS



S \S r ,b

 M5 |r1 − r2 |

β

1

S \S r ,b 1

dS 



|r1 − r2 |α−β |r1 − r2 |1−β +  |r1 − r |2 |r1 − r |3−α

|r1

1 − r |2−α+β



(G.24)

Advanced Theoretical and Numerical Electromagnetics

1192

where β < α, and we have used the fact that |r1 − r2 |  |r − r1 |/3 for r ∈ S \ S r1 ,b by construction [see (G.11)]. Except for a different exponent in the denominator, the last integral in (G.24) is of the type examined above in (G.16), and it is certainly finite for b > 0, i.e., so long as S r1 ,b does not shrink to a point. To show that the integral in question remains finite for any value of b — though only if β < α — we choose a surface S 2 such that S r1 ,b ⊂ S 2 ⊂ S and further split the calculation into two parts, namely, over S 1 := S \ S 2 and over S 2 \ S r1 ,b . The contribution of S 1 is necessarily finite because it is independent of b and the integrand is regular for r  S r1 ,b . For the integral over S 2 \ S r1 ,b we introduce a set of local coordinates ξ = (u , v ) and formally define a bijective mapping Ψ such that Ψ : R2 ⊃ B2 (0, 1) −→ S 2 ⊂ R3

(G.25)

where B2 (0, 1) is the unit circle [see definition (D.22)]. The existence of Ψ is guaranteed inasmuch as the surface S is smooth by hypothesis. In particular, we may require that Ψ map the point ξ1 = (0, 0) onto r1 and the smaller circle B2 (0, ν), ν < 1, onto the surface S r1 ,b . Then, we consider 

1 dS  = |r1 − r |2−α+β





S 2 \S r ,b

1 du dv |Ψ(ξ1 ) − Ψ(ξ )|2−α+β 

   ∂Ψ × ∂Ψ   ∂u ∂v 

B2 (0,1)\B2 (0,ν)



1



du dv



B2 (0,1)\B2 (0,ν)

|ξ1

c1 c2 − ξ |2−α+β

(G.26)

in light of (2.253). The last integral can be computed by using a local system of polar coordinates (τ , η ) centered in ξ1 , viz., 

1 dS   |r1 − r |2−α+β 

S 2 \S r ,b 1

2π



1

dη 0

ν

dτ τ

c1 c2 τ2−α+β

= 2πc1 c2

1 − να−β 2πc1 c2  α−β α−β

(G.27)

where the last term remains finite for any ν = ν(b) provided β < α. Finally, putting (G.19) and (G.24) together yields the Hölder condition (G.10).

References [1] [2] [3]

Van Bladel JG. Singular Electromagnetic Fields and Sources. Piscataway, NJ: IEEE Press; 1991. Bassanini P, Elcrat AR. Theory and applications of partial differential equations. Mathematical Concepts And Methods In Science And Engineering. New York, NY: Plenum Press; 1997. Rudin W. Principles of mathematical analysis. 3rd ed. New York, NY: McGraw-Hill; 1976.

Appendix H

Formulas

H.1 Vector identities and inequalities In the following, a, b, c and d denote either vectors in the ordinary V3 space or vector fields. Besides, eˆ 1 , eˆ 2 and eˆ 3 indicate three mutually orthogonal unit vectors which form a right-handed triple, as is graphically shown in Figure H.1. eˆ 1 · eˆ 2 = 0 eˆ 1 × eˆ 2 = eˆ 3

eˆ 2 · eˆ 3 = 0 eˆ 2 × eˆ 3 = eˆ 1

a = a1 eˆ 1 + a2 eˆ 2 + a3 eˆ 3

eˆ 3 · eˆ 1 = 0 eˆ 3 × eˆ 1 = eˆ 2 b = b1 eˆ 1 + b2 eˆ 2 + b3 eˆ 3

a ± b = (a1 ± b1 )ˆe1 + (a2 ± b2 )ˆe2 + (a3 ± b3 )ˆe3 a · b = a1 b1 + a2 b2 + a3 b3 = |a||b| cos α

 eˆ 1 a × b = (a2 b3 − a3 b2 )ˆe1 + (a3 b1 − a1 b3 )ˆe2 + (a1 b2 − a2 b1 )ˆe3 = a1  b1

eˆ 2 a2 b2

(H.1) (H.2) (H.3)

 eˆ 3  a3   b3 

(H.4) (H.5) (H.6)

|a × b| = |a||b| sin α

(H.7)

a+b =b+a a·b =b·a

(H.8) (H.9)

a × b = −b × a

(H.10)

(a + b) · c = a · c + b · c

(H.11)

(a + b) × c = a × c + b × c

(H.12)

a · (b × c) = c · (a × b) = b · (c × a) a × (b × c) = (a · c)b − (a · b)c

(H.13) (H.14)

(a × b) × c = (a · c)b − (b · c)a

(H.15)

(a × b) · (c × d) = (a · c)(b · d) − (b · c)(a · d) = a · b × (c × d)

(H.16)

1194

Advanced Theoretical and Numerical Electromagnetics

Figure H.1 Right-handed triple of unit vectors and two vectors forming an angle α. (a × b) × (c × d) = (a × b · d)c − (a × b · c)d = (d × a · b)c − (c × a · b)d a × [b × (c × d)] = (a × c)(b · d) − (a × d)(b · c) |a + b|  |a| + |b| ||a| − |b||  |a − b| |a + b| + |a − b| = 2|a| + 2|b| 2

2

2

(H.17) (H.18)

(triangle inequality)

(H.19)

(reverse triangle inequality)

(H.20)

(parallelogram law)

(H.21)

2

H.2 Dyadic identities In the following, a, b and c denote either vectors in the ordinary V3 space or vector fields, A and B are dyadics or dyadic fields [1]. a · A · b = (a · A) · b = a · (A · b)

(H.22)

(a × b) · A = a · (b × A) = −b · (a × A)

(H.23)

(A × a) · b = A · (a × b) = −(A × b) · a

(H.24)

(a × A) · b = a × (A · b)

(H.25)

a · A × b = (a · A) × b = a · (A × b) a × A × b = (a × A) × b = a × (A × b) = −ab × × A

(H.26) (H.27)

a × (b × A) = b(a · A) − A(a · b)

(H.28)

(ab − ba) · c = (b × a) × c

(H.29)

B · (a × b) = (B × a) · b = −(B × b) · a

(H.30)

a · A · B = (a · A) · B = a · (A · B)

(H.31)

A · B · a = (A · B) · a = A · (B · a)

(H.32)

a × A · B = (a × A) · B = a × (A · B)

(H.33)

A · B × a = (A · B) × a = A · (B × a)

(H.34)

(A × a) · B = A · (a × B) T

a·A·b=b·A ·a

(H.35) (H.36)

Formulas a·I=I·a=a a×I =I×a

⎛     ⎜⎜⎜⎜⎜ 0 a × I = I × a = ⎜⎜⎜⎜ az ⎝ −ay

−az 0 ax

a·b×I = a×b·I = a×b

⎞ ay ⎟⎟⎟ ⎟ −a x ⎟⎟⎟⎟⎟ ⎠ 0

1195 (H.37) (H.38)

(H.39)

I × (a × b) = ba − ab

(H.40)

(I × a) · A = a × A = (a × I) · A

(H.41)

 Tr A = I : A = A : I

   T T A : B = A · B : I = Tr A · B

T   T  A : B = A · B : I = Tr A · B

(H.42) (H.43) (H.44)

H.3 Differential identities In the following, Ψ(r, t), Φ(r, t) represent scalar fields and A(r, t), B(r, t) denote vector fields, A(r, t) ˆ is the unit vector normal to a smooth surface. is a dyadic field, and n(r) ∇(Φ + Ψ) = ∇Φ + ∇Ψ ∇(ξΦ) = ξ∇Φ

(H.45) ξ∈C

(H.46)

∇(ΦΨ) = Φ∇Ψ + Ψ∇Φ

(H.47)

∇ · (A + B) = ∇ · A + ∇ · B

(H.48)

∇ · (A × B) = B · ∇ × A − A · ∇ × B

(H.49)

∇ × (ΦA) = Φ∇ × A + ∇Φ × A

(H.50)

∇ · (ΦA) = Φ∇ · A + ∇Φ · A

(H.51)

∇(ΦA) = (∇Φ)A + Φ∇A ∇(A · B) = (A ·∇)B + (B ·∇)A + A × (∇×B) + B × (∇×A)

(H.52) (H.53)

∇ × (A × B) = A∇ · B − B∇ · A + (B · ∇)A − (A · ∇)B

(H.54)

B × (∇ × A) = ∇A · B − B · ∇A

(H.55)

(B × ∇) × A = ∇A · B − B∇ · A

(H.56)

(B × ∇) × A = B × (∇ × A) + B · ∇A − B∇ · A

(H.57)

∇2 (ΦΨ) = Φ∇2 Ψ + 2∇Φ · ∇Ψ + Ψ∇2 Φ

(H.58)

1196

Advanced Theoretical and Numerical Electromagnetics ∇ × ∇ × A = ∇∇ · A − ∇2 A

(H.59)

∇ × ∇ × (ΦA) = ∇Φ × (∇ × A) − A∇ Φ + A · ∇∇Φ + Φ∇ × ∇ × A + ∇Φ∇ · A − ∇Φ · ∇A 2

∇∇ · (ΦA) = (∇Φ)∇ · A + Φ∇∇ · A + ∇Φ × (∇ × A) + A · ∇∇Φ + ∇Φ · ∇A ∇2 (ΦA) = Φ∇2 A + 2∇Φ · ∇A + A∇2 Φ

(H.60) (H.61) (H.62)

∇·∇×A=0

(H.63)

∇ × ∇A = 0

(H.64)

∇ · (ΦI) = ∇Φ

(H.65)

∇ × (ΦI) = ∇Φ × I

(H.66)

∇ · (I × A) = ∇ × A

(H.67)

∇ × ∇ × (ΦI) = ∇∇Φ − I∇ Φ 2

(H.68)

∇(A × B) = (∇A) × B − (∇B) × A

(H.69)

∇(ΦB) = (∇Φ)B + Φ∇B ∇ · (AB) = (∇ · A)B + A · (∇B)

(H.70) (H.71)

∇ · (ΦA) = (∇Φ) · A + Φ∇ · A

(H.72)

∇ · (A · B) = (∇ · A) · B + A : ∇B

(H.73)

∇ · (A × B) = (∇ × A) · B − A · ∇ × B ∇ · (BA − AB) = ∇ × (A × B) ∇ × (AB) = (∇ × A)B − A × ∇B ∇s · (ΦA) = Φ∇s · A + ∇s Φ · A

(H.74) (H.75) (H.76) (H.77)

∇s · (A × B) = B · ∇s × A − A · ∇s × B ∇s × (ΦA) = Φ∇s × A + ∇s Φ × A ˆ = ∇s Φ × nˆ ∇s × (Φn) ∇s × nˆ = 0 ∇s · (nˆ × A) = −nˆ · ∇ × A

(H.78) (H.79) (H.80) (H.81) (H.82)

dA dΦ d (ΦA) = Φ + A dt dt dt d dB dA (A · B) = A · + ·B dt dt dt dB dA d (A × B) = A × + ×B dt dt dt

(H.83) (H.84) (H.85)

∂A dr ∂A d A(r(t), t) = + · ∇A = + v(t) ·∇A  dt ∂t dt ∂t velocity

Formulas =

∂A + v(t)∇ · A + ∇ × [A × v(t)] ∂t

1197 (H.86)

H.4 Integral identities In the following, V denotes a simply connected volume with smooth boundary ∂V and outward unit normal nˆ (Figure 1.2b), S indicates an open smooth surface with smooth boundary ∂S and unit normal nˆ (Figure 1.2a) [2], and γ is a piecewise-smooth line that connects two points r1 and r2 . The unit tangent sˆ to ∂S , the unit normal νˆ , and the unit normal nˆ form a right-handed triple of orthogonal vectors. We suppose that all the scalar, vector and dyadic fields are endowed with the necessary derivatives. eˆ 2 := yˆ eˆ 3 := zˆ eˆ 1 := xˆ   ˆ dV eˆ ν · ∇Φ = dS eˆ ν · n(r)Φ(r)

dV ∇Φ(r) =

ˆ dS n(r)Φ(r)

(gradient theorem)

(H.89)

(Gauss theorem)

(H.90)

(curl theorem)

(H.91)

(Stokes theorem)

(H.92)

∂V

V



dV ∇ · A(r) =

ˆ · A(r) dS n(r) ∂V

V





dV ∇ × A(r) =

ˆ × A(r) dS n(r) ∂V

V



 ˆ · ∇ × A(r) = dS n(r)

S

ds sˆ(r) · A(r) ∂S





ˆ × ∇Φ(r) = dS n(r)

ds sˆ(r)Φ(r)

(H.93)

ds sˆ(r) × A(r)

(H.94)

∂S

S



ˆ × ∇] × A(r) = dS [n(r) ∂S

S



 ˆ · ∇Φ × ∇Ψ = dS n(r)

 ds sˆ(r) · Φ(r)∇Ψ = −

∂S

S

ds sˆ(r) · Ψ(r)∇Φ

(H.95)

∂S



 dV [Ψ(r)∇ Φ + ∇Φ · ∇Ψ] = 2

ˆ · Ψ(r)∇Φ dS n(r)

(H.96)

ˆ · [Ψ(r)∇Φ − Φ(r)∇Ψ] dS n(r)

(H.97)

∂V

V





dV [Ψ(r)∇ Φ − Φ(r)∇ Ψ] = 2

V

(H.88)







ν ∈ {1, 2, 3} (Gauss lemma)

∂V

V



(H.87)

2

∂V

Advanced Theoretical and Numerical Electromagnetics

1198



 dV ∇ · [A(r) × (∇ × B)] =

ˆ × A(r) · ∇ × B dS n(r)

(H.98)

∂V

V

 dV [B(r) · ∇ × ∇ × A − A(r) · ∇ × ∇ × B] V

 =

ˆ × A(r) · ∇ × B − n(r) ˆ × B(r) · ∇ × A] dS [n(r)

(H.99)

∂V

1 1 + , R1 R2

J(r) = 

r∈S 

dS ∇s · A(r) = S

ds νˆ (r) · A(r) − 

dS ∇s Φ(r) = 





(H.104)

ˆ · A(r) dS n(r)

(H.105)

ˆ × A(r) dS n(r)

(H.106)

ds sˆ(r) · A(r)

(H.107)



∂V





dV ∇ × A(r) = V

∂V



ˆ · ∇ × A(r) = dS n(r) 

ˆ dS n(r)A(r) ∂V

V

dV ∇ · A(r) =

S

(H.103)



V



ˆ × A(r) dS J(r)n(r) S

dV ∇A(r) = 

(H.102)



ds νˆ (r) × A(r) − ∂S

S

(H.101)

ˆ dS J(r)n(r)Φ(r) S



dS ∇s × A(r) =

ˆ · A(r) dS J(r)n(r) S

ds νˆ (r)Φ(r) − ∂S

S

(H.100)



∂S



(first curvature of S [3, Appendix 3])

∂S

ds sˆ(r) · ∇A(r) = A(r2 ) − A(r1 )

(H.108)

γ



  dV (∇ × ∇ × B) · A(r) − B(r) · ∇ × ∇ × A

V

 = ∂V

 ˆ × B(r) · ∇ × A + [n(r) ˆ × (∇ × B)] · A(r) (H.109) dS n(r)

Formulas

1199

H.5 Legendre polynomials and functions H.5.1 Nomenclature The independent variable is ξ = cos ϑ ∈ [−1, 1] with ϑ ∈ [0, π], n and m are integer numbers [4, 5]. Pn (ξ) = Pn (cos ϑ) Qn (ξ) = Qn (cos ϑ) m Pm n (ξ) = Pn (cos ϑ) m Qm n (ξ) = Qn (cos ϑ)

Legendre polynomial Legendre function of the second kind associated Legendre function of the first kind associated Legendre function of the second kind

H.5.2 Differential equation     1 d dΘmn m2 Θmn = 0 sin ϑ + n(n + 1) − sin ϑ dϑ dϑ sin2 ϑ   m2 d2 Θmn dΘmn + n(n + 1) − (1 − ξ2 ) Θmn = 0 − 2ξ dξ2 dξ 1 − ξ2

(H.110) (H.111)

H.5.3 Explicit expressions for the lowest orders P0 (ξ) = 1

(H.112)

P1 (ξ) = ξ = cos ϑ 1 P2 (ξ) = (3ξ2 − 1) 2 1 P3 (ξ) = (5ξ3 − 3ξ) 2 1 P4 (ξ) = (35ξ4 − 30ξ2 + 3) 8   1+ξ ϑ 1 = log cot Q0 (ξ) = log 2 1−ξ 2   1+ξ 1 Q1 (ξ) = ξ log −1 2 1−ξ   1+ξ 1 3 Q2 (ξ) = (3ξ2 − 1) log − ξ 4 1−ξ 2   1+ξ 1 55 35 Q4 (ξ) = (35ξ4 − 30ξ2 + 3) log − ξ3 + ξ 16 1−ξ 8 24

(H.113)

P11 (ξ) = −(1 − ξ2 )1/2 = − sin ϑ P12 (ξ) P22 (ξ)

= −3ξ(1 − ξ )

2 1/2

= −3 cos ϑ sin ϑ

= 3(1 − ξ ) = 3 sin ϑ 3 P13 (ξ) = − (5ξ2 − 1)(1 − ξ2 )1/2 2 P23 (ξ) = 15 ξ(1 − ξ2 )

P33 (ξ)

2

2

= −15(1 − ξ )

2 3/2

(H.114) (H.115) (H.116) (H.117) (H.118) (H.119) (H.120) (H.121) (H.122) (H.123) (H.124) (H.125) (H.126)

1200

Advanced Theoretical and Numerical Electromagnetics

H.5.4 Orthogonality relationships π dϑ sin ϑPn (cos ϑ)Pn (cos ϑ) =

2 δn n 2n + 1

(H.127)

m dϑ sin ϑPm n (cos ϑ)Pn (cos ϑ) =

2 (n + m)! δn n 2n + 1 (n − m)!

(H.128)

0

π 0

H.5.5 Functional relationships m P−m n (cos ϑ) = (−1)

(n − m)! m P (cos ϑ) (n + m)! n

⎧ μ μ π Pν (cos ϑ) cos[π(ν + μ)] − Pν (− cos ϑ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ sin[π(ν + μ)] ⎨2 Qνμ (cos ϑ) = ⎪ μ ⎪ ⎪ π Pν (cos ϑ) cos[π(ν + μ)] − Pνμ (− cos ϑ) ⎪ ⎪ ⎪ ⎩ lim ν+μ→n+m 2 sin[π(ν + μ)]

(H.129)

ν+μZ (H.130) ν+μ∈Z

n+m m Pm Pn (cos ϑ) n (− cos ϑ) = (−1) n Pn (− cos ϑ) = (−1) Pn (cos ϑ)

(H.131) (H.132)

n+m+1 m Qn (cos ϑ) Qm n (− cos ϑ) = (−1)

(H.133)

Qn (− cos ϑ) = (−1)

(H.134)

Pm n (1)

n+1

Qn (cos ϑ)

⎧ ⎪ ⎨1, m = 0 =⎪ ⎩0, m > 0

(H.135)

H.6 Bessel functions H.6.1 Nomenclature The independent variable z may be real or complex, the index ν may be a complex number in general [4–7]. Jν (z) Yν (z) Hν(1) (z) Hν(2) (z) Iν (z) Kν (z) jν (z) yν (z) h(1) ν (z) h(2) ν (z)

Bessel function of the first kind Bessel function of the second kind or Weber function or Neumann function Hankel function of the first kind or Bessel function of the third kind Hankel function of the second kind or Bessel function of the third kind Modified Bessel function of the first kind Modified Bessel function of the second kind Spherical Bessel function of the first kind Spherical Bessel function of the second kind Spherical Hankel function of the first kind Spherical Hankel function of the second kind

Formulas

1201

H.6.2 Differential equation

  ν2 d2 Zν 1 dZν + 1 − + Zν (z) = 0 z dz dz2 z2

(H.136)

H.6.3 Functional relationships ⎧ Jν (z) cos(νπ) − J−ν (z) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ sin(νπ) ⎨ Yν (z) = ⎪ ⎪ ⎪ J (z) cos(απ) − J−α (z) α ⎪ ⎪ ⎪ lim ⎩α→ν sin(απ)

νZ , ν∈Z

| arg(z)| < π

(H.137)

Hν(1) (z) = Jν (z) + j Yν (z)

(H.138)

= Jν (z) − j Yν (z)

(H.139)

Hν(2) (z)

⎧ −ν j Jν (j z) = jν J−ν (j z) = jν Jν (− j z), ν ∈ Z ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ π ⎪ ⎨e− j νπ/2 Jν (zej π/2 ), ν  Z, −π < arg(z)  Iν (z) = ⎪ 2 ⎪ ⎪ ⎪ ⎪ π ⎪ j 3νπ/2 − j 3π/2 ⎪ ⎩e < arg(z)  π Jν (ze ), ν  Z, 2 ⎧π π ⎪ ⎪ jν+1 Hν(1) (j z) = (− j)ν+1 Hν(2) (− j z), ν ∈ Z ⎪ ⎪ ⎪ 2 2 ⎪ ⎪ ⎪ ⎪ ⎪ π ⎨ j π j νπ/2 (1) j π/2 e Hν (ze ) ν  Z, −π < arg(z)  Kν (z) = ⎪ ⎪ ⎪ 2 2 ⎪ ⎪ ⎪ ⎪ ⎪ π j π − j νπ/2 (2) − j π/2 ⎪ ⎪ ⎩− e Hν (ze ) ν  Z, − < arg(z)  π 2 2

(H.140)

(H.141)



π J 1 (z) 2z ν+ 2  ν 1 d sin z ν jν (z) = z − z dz z  π yν (z) = Y 1 (z) 2z ν+ 2  ν 1 d cos z yν (z) = −zν − z dz z  π (2) h(2) H (z) ν (z) = 2z ν+ 12 ν  1 d e− j z (2) ν hν (z) = j z − z dz z jν (z) =

(H.142) ν = 0, 1, 2, . . .

(H.143) (H.144)

ν = 0, 1, 2, . . .

(H.145) (H.146)

ν = 0, 1, 2, . . .

(H.147)

H.6.4 Asymptotic behavior for small argument (|z|  1) J0 (z) = 1 + o(z)

(H.148)

1202

Advanced Theoretical and Numerical Electromagnetics

z ! 2 log + γ J0 (z) + o(z) π 2 I0 (z) = 1 + o(z)

z ! K0 (z) = − log + γ I0 (z) + o(z) 2

z ν 1 + o(zν ) Jν (z) = Γ(ν + 1) 2  ν Γ(ν) 2 Yν (z) = − + o(zν ) π z

z ν 1 + o(zν ) Iν (z) = Γ(ν + 1) 2  ν Γ(ν) 2 + o(z) Kν (z) = 2 z

z ! 2 H0(1) (z) = 1 + j log + γ J0 (z) + o(z) π 2

z ! 2 (2) H0 (z) = 1 − j log + γ J0 (z) + o(z) π 2 Y0 (z) =

γ = 0.5772157 . . .

(H.149) (H.150) (H.151) (H.152)

Re{ν} > 0

(H.153) (H.154)

Re{ν} > 0

(H.155) (H.156) (H.157)

H.6.5 Asymptotic behavior for large argument (|z|  1) 

Jν (z) ≈  Yν (z) ≈

νπ π  2 cos z − − πz 2 4

(H.158)

νπ π  2 sin z − − πz 2 4

(H.159)

 Hν(1) (z)

=

 # "

1 νπ π ! 2 4ν2 − 1 exp j z − − +O 2 1− πz 2 4 8jz z

(H.160)

−π < arg z < 2π  Hν(2) (z)

=

 # "

1 νπ π ! 2 4ν2 − 1 exp − j z − − +O 2 1+ πz 2 4 8jz z

(H.161)

−2π < arg z < π h(1) ν (z)

 #

! " 1 π 4ν2 − 1 1 +O 2 = exp j z − (ν + 1) 1 − z 2 8jz z

(H.162)

−π < arg z < 2π h(2) ν (z) =

 #

! " 1 π 4ν2 − 1 1 exp − j z − (ν + 1) 1 + +O 2 z 2 8jz z −2π < arg z < π

(H.163)

Formulas

1203

H.6.6 Recursion relationships The symbol Zν denotes any function that solves the Bessel equation (H.136). zZν−1 (z) + zZν+1 (z) = 2νZν (z)

(H.164)

2Zν (z)

Zν−1 (z) − Zν+1 (z) = zZν (z) + νZν (z) = zZν−1 (z)

(H.165) (H.166)

zZν (z) − νZν (z) = −zZν+1 (z)

(H.167)

H.6.7 Wronskians and cross products Jν (z)Yν+1 (z) − Jν+1 (z)Yν (z) = − Iν (z)Kν+1 (z) + Iν+1 (z)Kν (z) =

2 πz

(H.168)

1 z

(H.169)

2 πz 4 (1) (2) Hν(2) (z)Hν+1 (z) − Hν(1) (z)Hν+1 (z) = j πz Jν (z)Hν(1) (z) − Jν (z)Hν(1) (z) = −

(H.170) (H.171)

H.6.8 Integral relationships   

dz zn+1 Jn (z) = zn+1 Jn+1 (z)

(H.172)

dz z1−n Jn (z) = −z1−n Jn−1 (z)

(H.173)

dz Zν (az)Zν (bz) = z

1 Jn (z) = 2π jn

2π dα e

bZν (az)Zν−1 (bz) − aZν−1 (az)Zν (bz) a2 − b2

j z cos α j nα

0

e

1 = 2π jn

2π

dα ej z cos α cos(nα)

(H.174)

(H.175)

0

H.6.9 Series ej z cos ϕ =

∞ $

jn Jn (z)ej nϕ

(H.176)

n=−∞

References [1] [2] [3] [4]

Lindell IV. Methods for electromagnetic field analysis. Piscataway, NJ: IEEE Press; 1992. Kellogg OD. Foundations of potential theory. Berlin Heidelberg: Springer-Verlag; 1929. Van Bladel JG. Electromagnetic Fields. Piscataway, NJ: IEEE Press; 2007. Abramowitz M, Stegun IA. Handbook of mathematical functions. New York, NY: Dover Publications, Inc.; 1965.

1204 [5] [6] [7]

Advanced Theoretical and Numerical Electromagnetics Jackson JD. Classical Electrodynamics. 3rd ed. Chichester, UK: Wiley; 1999. Watson GN. A Treatise on the Theory of Bessel Functions. 2nd ed. Cambridge Mathematical Library. Cambridge, UK: Cambridge University Press; 1995. Jones DS. The Theory of Electromagnetism. Oxford, UK: Pergamon; 1964.

Index

Abraham density, 64 Adjoint problem frequency domain, 389 time domain, 389 Aharonov-Bohm effect, 292 Ampère-Maxwell law, 5, 7, 25 antenna biconical, 481 coaxial-cable excitation, 726 dipole, 638 effective height, 637, 729 electric-field integral equation, 884, 922, 929 Method of Moments, 994, 1052, 1059 equivalent circuit, 725, 888 gaseous plasma, 929 horn, 726 impedance, 730, 888, 997 inverted-F, 999 loop, 643, 924 plasma thruster, 922 port coaxial-cable, 726 delta-gap, 382, 731, 884, 925, 929, 994 radiation pattern, 638, 642, 645, 1001 reciprocity, 371, 382, 725 anti-ferromagnetic medium, 307 area coordinates, 971, 1078 Argand diagrams, see permittivity, arc plots atomic number, 842 Bessel equation, 179, 464, 528, 665, 669 functions, 180, 213, 645 inequality, 530, 1143 modified functions, 181 spherical functions, 528 Bianchi identities, 12 Biot and Savart law, 278 Bloch propagation constant, 793 boundary conditions asymptotic, 51, 336, 598 Dirichlet, 94, 386

homogeneous conductor, 37 Leontovich, 339, 377, 427, 904 Maxwell’s stress tensor, 71 Neumann, 94, 386 Perfect Electric Conductor, 37 Perfect Magnetic Conductor, 368 Poynting vector, 57 Robin, 386, 534, 537 sharp edge, 337 branch line, 475, 808, 816, 1116 point, 475, 808, 816, 853, 1114 surface, 249 Brewster angle, 442 cancellation principle, 8, 1107 capacitance capacitary potentials, 98, 510 matrix, 98, 394 solitary conductor, 100, 172 TEM modes, 776 Cauchy integral formula, 832, 1108 sequence, 1134 theorem of residues, 1110 Cauchy-Riemann condition, 1101 causality, see principle of causality chain matrix TE polarization, 448, 795 TM polarization, 449, 795 charge density conductor, 29 continuity equation, 6 Hertzian dipole, 627 line, 39, 321, 462 magnetic, 360 magnetization, 525 non-radiating, 488 polarization, 194, 524 surface, 34, 36, 214, 286, 970 volume, 2, 1025 Circulator, 388

1206

Advanced Theoretical and Numerical Electromagnetics

Clebsch-Helmholtz decomposition, 504 Cole-Cole equation, 853 collision frequency conductor, 833 dielectric, 842 plasma, 930 condition number matrix, 959, 1057, 1066 operator, 1147 conduction current anisotropic media, 29 dispersive media, 824, 841 Drude model, 835 eddy, 668 isotropic media, 28 magnetic, 367 magneto-quasi-static regime, 663–670 modified electric-field integral equation, 904 plane waves, 421 Poynting theorem, 53, 62 skin effect, 664–667 stationary fields, 252 wave equations, 44 conservation of charge line, 40, 705 RWG basis functions, 966, 970 surface, 36, 695 volume, 6, 12, 25, 35 continuity equation, 12 convolution product Dirac delta distribution dyadic Green function, 613 Hertzian dipole, 629 reactions, 647 scalar Green function, 557 electrodynamic potentials, 603 electrostatic potential, 120 integral equations, 871 memory functions, 826 Coulomb field, 48, 212, 589 force, 200 gauge, 263, 280, 286, 343, 515 critical angle, 440 constants, 769 curl dyadic field, 1177

Sobolev space, 343, 1141 surface, 969, 1087 three-dimensional, 1085 weak form, 969, 1023, 1126 current density conduction, 28 continuity equation, 6 dipole antenna, 641 eddy, 668 Hertzian dipole, 627 line, 293, 461 loop antenna, 643 magnetic, 154, 360, 364 magnetic dipole moment, 298 magnetization, 303, 525, 720 Neumann vector field, 252 non-radiating, 485 polarization, 524, 719 solenoid, 226 steady, 219, 222, 246, 296 surface, 32, 36, 271, 598 time-harmonic, 419, 598 total, 5, 21 volume, 2 D’Alembert Ansatz, 408, 480 equation, 407, 524, 542, 550, 586, 592 operator, 43, 518 Debye equation, 853 relaxation, 853 delta-gap approximation antenna equivalent circuit, 731 electric-field integral equation, 884, 994 inverted-F antenna, 999 mutual antenna admittances, 382 plasma antenna, 929 plasma thruster, 925 depolarization dyadic, 609 diamagnetic medium, 206, 307 diffusion equation, 663 Dirac delta distribution line, 969, 970, 1125 one-dimensional, 350, 557, 563, 826, 888, 1119 sifting property, 613, 616, 641, 644, 944, 1120, 1122–1125

Index support, 1121–1123 surface, 257, 592, 594, 711, 1021, 1022, 1075, 1124 three-dimensional, 400, 489, 551, 585, 613, 627, 750, 1123 two-dimensional, 315, 462, 1121 Dirichlet boundary condition, 94 Green function, 154 integral, 1140 dispersion relationship parallel-plate waveguide, 861 periodic layered medium, 797 plane waves, 414, 421 divergence dyadic field, 1177 surface, 36, 140, 234, 743, 882, 926, 967, 980, 989, 1087 three-dimensional, 1021, 1085 transverse, 759, 803 weak form, 1021, 1126 Drude model, 833 duality principle, 368 dipoles, 651 electromagnetic potentials, 519 dyadic anti-symmetric, 1171 Cartesian components, 1166 constitutive parameters, 29 curl, 1177 determinant, 1175 divergence, 1177 double cross product, 1173 double scalar product, 1173 eigenvalues, 1175 gradient, 1176 Green function, see Green function Hermitian transpose, 1172 identity, 1169 matrix form, 1167 norm, 1173 scalar product, 1169 surface impedance, 340 symmetric, 1171 trace, 1175 transpose, 1171 vector product, 1169

Earnshaw theorem, 200 eddy currents, 667 eigenvalues associated Legendre equation, 167 Bessel equation, 179 compact operator, 1157 self-adjoint, 1159 discretized EFIE, 953 dyadic, 1176 dyadic field ∇∇Φ(r), 201 dyadic permeability, 279 dyadic permittivity, 108 harmonic equation, 165 Helmholtz equation, 542 Laplace equation, 213 lossless cavity, 63, 341, 349, 738, 740, 744 lossless waveguide hollow-pipe, 764, 770, 780 parallel-plate, 863 Maxwell stress tensor, 69 periodic structures, 795 plane waves, 414 electric charge, 1 conducting sphere, 170–172 Gauss law, 6, 79 line density, 321–324, 462 moving, 48, 585 non-radiating, 489 polarization, 194 potential momentum, 289–292 quantization, 363 radiation, 48 surface density, 149 test, see test charge work, 3 contrast factor, 719, 1013 dipole, 86, 132, 507, 843 displacement, 1 energy, 51 field, 1 potential, see potential quadrupole, 182, 185 electromagnetic torque, 74 electric dipole, 239 magnetic dipole, 240 electromotive force, 15, 254, 667

1207

1208

Advanced Theoretical and Numerical Electromagnetics

electrostatic shielding, 155 energy method electrostatic fields, 102 scalar Helmholtz equation, 537 stationary fields, 250 time-harmonic fields, 338, 341 time-varying fields, 329, 331, 333 equipotential lines, see potential, electrostatic equivalence principle electric circuit, 706 Schelkunoff, 716 surface, 708 volume, 436, 719 ether, 42 Faraday cage electrostatic, 155 general time dependence, 330 disk, see homopolar generator law, 5, 7, 25 ferrimagnetic medium, 307 ferromagnetic medium, 307 Fitzgerald vector, 525 Floquet harmonics, 794 theorem, 793 flux rule, 15, 667 Foucalt currents, see eddy currents Fourier transformations spatial, 471, 552, 634, 805, 951, 1017, 1111 RWG function, 976 SWG function, 1031 temporal, 463, 470, 550, 557, 577, 823, 828, 835, 840, 850, 856 Fraunhofer region, 626 antenna equivalent circuit, 725 dyadic Green functions, 634 inverted-F antenna, 1001 Fredholm alternative, 346, 541, 872, 1161 equations, 871 combined-field, 906 displacement vector, 918 electric-field, 882, 887, 933, 934 Müller, 915 magnetic induction, 919

magnetic-field, 895, 934 magnetization current, 919 Poggio and Miller, 914 polarization current, 917 polarization vector, 919 successive approximations, 872 Fresnel region, 626 frustrated total internal reflection, 457 Galilean transformations, 15, 1091 Gauss law electric, 6, 10, 25, 79 magnetic, 5, 7, 25 solid angle formula, 134 theorem of arithmetic mean, 116 gradient N-dimensional, 748, 943 dyadic field, 1176 Sobolev space, 343, 538, 1140 surface, 90, 250, 295, 520, 611, 740, 882, 913, 926, 988, 989, 1087 three-dimensional, 1085 transverse, 758 weak form, 1024, 1126 Gram matrix, 954, 961, 1063 RWG functions, 973 SWG functions, 1029 Gram-Schmidt orthogonalization, 782 Green function 1-D spectral domain, 806 2-D frequency domain, 466 2-D spectral domain, 472 2-D time domain, 467, 470, 478 3-D dyadic frequency domain, 603, 613, 651, 755 3-D frequency domain, 529, 551, 755 regularized, 605, 979, 1034 3-D spectral domain, 552 3-D time domain, 550, 557, 559 Dirichlet, 154, 156 Neumann, 158 static, 106, 107, 110 distributional, 1128 transmission line, 353, 786, 787 Green reciprocation theorem, 393 group delay, 859

Index velocity dispersive medium, 859 parallel plate waveguide, 865 Hölder continuity, 1139 charge density, 127, 140 criterion, 1185 current density, 604 on a surface, 1187 single-layer potential, 137 test function, 1120, 1122 harmonic equation, 165, 179, 385, 528, 551, 805, 813 function, 204, 241, 245, 264 potential, 156, 163, 250, 502 vector field, 500, 510, 738, 741, 742, 766 Hartree harmonics, 794 Helmholtz decomposition current density, 517 electromagnetic fields, 513 magnetization, 526 PEC cavity, 342, 736 polarization, 526 three-dimensional, 235, 493 two-dimensional, 760 equation, 47, 356, 464, 490 dyadic Green function, 614, 619 energy conservation, 531 Hertzian dipole, 629 parallel-plate waveguide, 861, 863 PEC cavity, 342, 736 plane-wave reflection, 812 radiation conditions, 535 Rellich theorem, 529, 597, 600 scalar Green function, 550 spectral solution, 552 spherical coordinates, 528 surface sources, 596, 598 two-dimensional, 803, 804 weak form, 538 operator, 48, 356 transport theorem, 14, 1091 Hertzian dipole, 687, 728 electric dipole, 626 magnetic dipole, 651, 687 potentials, 523, 577, 585

1209

Hilbert transformations, 832 homopolar generator, 17 Hooke law, 842 identity operator, 961, 1062 perturbation, 875 image principle electrostatic fields, 206 time-harmonic fields, 815, 818 impedance relationship combined-field integral equation, 903 far fields, 636 Leontovich, 329, 339, 340, 377, 904 modified electric-field integral equation, 904 plane waves, 416 radiation vectors, 637 scalar, 534, 537 TEM spherical waves, 481 time domain, 391 inductance TEM modes, 776 induction heating, 668 inner product, 1134 associated Legendre equation, 167 Bessel equation, 179 cavity, 743 EFIE, 948 harmonic equation, 166, 177, 385 Helmholtz decomposition, 499, 510 Method of Moments, 943 Poisson equation, 393 positive definite operator, 878, 1149 reactions, 386 Sobolev spaces, 1140, 1141, 1143 spatial-temporal, 549 surface integral equation, 1061 vector wave equation, 614, 1063 volume integral equation, 1013, 1016 waveguide, 780 weighted, 1015 Jacobi matrix area coordinates, 972 circular cylindrical coordinates, 1082 curvilinear coordinates, 1094 polar spherical coordinates, 1084 volume coordinates, 1028 Jordan inequalities, 1111

1210

Advanced Theoretical and Numerical Electromagnetics

Kelvin theorem, 198 Kirchhoff current law layered media, 443 transmission line, 351, 785, 786 voltage law layered media, 443 stationary fields, 259 transmission line, 785, 786 Kirchooff voltage law, 260 Kottler line charges, 705 Kronecker delta, 97, 169, 510, 1030 Laplace equation, 92, 97, 100, 163, 204, 213, 250, 659, 741, 742 operator inverse, 115, 494 self-adjoint, 743 three-dimensional, 1086 transverse, 471, 591, 760 weak form, 551, 1127 transformation, 477 Larmor formula, 50 Legendre associated equation, 167, 287, 1199 associated functions, 168, 287, 659, 845 polynomials, 170, 188, 309 Leibniz rule, 1090 Lorenz gauge, 523 telegraph equations, 788 time derivative of energy, 332, 545 Lenz rule, 16, 19 eddy currents, 667 lines of force, see streamlines Lipschitz continuity integral operator, 873 scalar field, 1185 single-layer potential, 137 vector field, 1096 Lorentz force, 3 bound electrons in a dielectric, 842 conservation of angular momentum, 72 conservation of momentum, 64 free electrons in a conductor, 833 magnetized plasma, 922

steady current, 311–314 work on a charge, 89 lemma, see reciprocity transformations, 14 Lorenz gauge continuity of potentials, 520 surface sources, 601 time domain, 517, 584 magnetic, 518 time-dependent Green function, 563 time-harmonic regime, 573 magnetic charges, 5, 13, 35, 238, 278, 306, 359, 509, 659, 742 angular momentum, 361 contrast factor, 720 currents, 360 dipole, 240, 288, 509 energy, 52 field, 1 induction, 1 nozzle, 923 potential, see potential quadrupole, 661 magnetization charge density, 721 current density, 720 vector magnetized sphere, 308 rotating dielectric sphere, 658 static, 301 time-dependent, 525 time-harmonic, 720 Marcuvitz-Schwinger equations, 757 Maxwell stress tensor, 66, 72, 172, 242 Maxwell-Boffi equations electrostatic, 196 stationary currents, 305 mean value theorem derivatives, 1096 magnetic field, 278 scalar potential, 116 vector potential, 273 memory functions, 825, 840, 850 Method of Moments algebraic system, 945 basis functions, 942

Index EFIE, 948, 964, 994 Müller equations, 960, 973 MFIE, 953, 973 PMCHWT equation, 956 volume integral equation, 1013 collocation, 945 meshing, 946 point-matching, 945 residual, 942, 1064 test functions, 944 method of weighted residuals, see Method of Moments Mosotti model, 844 Neumann boundary condition, 94 function, 1200 Green function, 158 series, 455, 874 vector fields, 249, 254, 512, 742 norm operator, 1145 vector, 1131 Ohm law anisotropic conductor, 29 dispersive anisotropic conductor, 824 isotropic conductor, 28 isotropic magnetic conductor, 367 Neumann vector field, 254 operator adjoint, 549, 1153 bijective, 1145 bounded, 1146 Cartesian decomposition, 1156 compact, 876, 1146, 1157 condition number, 1147 continuous, 1146 D’Alembert, 43, 542 domain, 1144 energy functional, 198 finite rank, 1146 Green function, 548 Helmholtz, 48, 356, 527 Hilbert-Schmidt, 874, 876 identity, 503, 872, 1059, 1147 image, 1144 imbedding, 1147

1211

injective, 1145 integral, 116, 400, 871 CFIE, 955 EFIE, 948 EFIE and volume, 1053 Müller equation, 960 MFIE, 953 PMCHWT equation, 956 volume, 1013 integral and differential, 1059 inverse, 1147 Laplace, 163 linear, 941, 1145, 1175 normal, 1156 null space, 1145, 1150 positive definite, 878, 1149 Rayleigh quotient, 1158 self-adjoint, 385, 615, 619, 1155 spectrum, 1157 supremum norm, 1145 surjective, 1145 transverse del, 758 transverse Laplace, 471, 591, 760 unitary, 1156 paramagnetic medium, 307 Perfect Electric Conductor antenna equivalent circuit, 725 boundary conditions, 37 combined-field integral equation, 902 definition, 29 electric-field integral equation, 878, 884, 922, 929, 933 magnetic-field integral equation, 892 Schelkunoff equivalence principle, 716 surface equivalence principle, 714 Perfect Magnetic Conductor boundary conditions, 368 definition, 367 magnetic-field integral equation, 900 reciprocity, 381 Schelkunoff equivalence principle, 718 surface equivalence principle, 714 permeability anisotropic medium, 29, 307 causality, 833 dispersive media, 823–828 duality transformation, 368

1212

Advanced Theoretical and Numerical Electromagnetics

isotropic medium, 28, 307 non-reciprocal medium, 388–389 reciprocal medium, 373 relative, 30 vacuum, 27 permittivity anisotropic medium, 29, 196 arc plots, 854 causality, 828–833 conductor, 833–837 dispersive media, 823–828 duality transformation, 368 good conductor, 422 isotropic medium, 28, 196, 842–848 non-reciprocal medium, 388–389 plasma, 929 polar medium, 852–855 reciprocal medium, 373 relative, 30 vacuum, 27 phase velocity dispersive medium, 859 parallel plate waveguide, 865 plane wave, 412 slow wave, 799 waveguide, 769 phasor, 23 Planck constant, 292 plasma absorbed power, 927 antenna, 929 first adiabatic invariant, 923 frequency, 835, 846, 930 permittivity, 836, 837, 929 thruster, 922 Plemelj formulas, see Hilbert transformations Poincaré identity, 504 inequality, 539, 1141, 1142 Poincaré identity, 504 Poisson theorem, 194, 306 polarization charge density, 721 current density, 719 vector dielectric sphere, 197, 658 electric susceptibility, 843–847 integral equation, 919

polar substance, 852 static, 192 time-dependent, 524 time-harmonic, 719 potential electric, 514 equipotential surfaces, 587, 731, 886 electrostatic equipotential lines, 87, 89, 186 scalar, 83, 109, 391, 656 vector, 103 Hertzian, 523, 525 magnetic, 514 magnetostatic, 657 equipotential lines, 661 scalar, 236, 308, 659 vector, 229, 263, 279, 289 polarization, 524 power dissipated, 55 generated, 55 radiated, 54, 1007 accelerated charge, 50 cylindrical wave, 470 delta gap, 891 dipole antenna, 642 Hertzian dipole, 631 inverted-F antenna, 999 loop antenna, 644 plane wave, 417, 429, 445 Schelkunoff vector, 637 scattered EFIE, 951 MFIE, 954 Poynting theorem frequency domain, 61, 62 stationary fields, 242 time domain, 54 vector complex, 60 cylindrical wave, 470 Fraunhofer region, 637 instantaneous, 55 not unique, 58 plane wave, 417, 427 stationary, 242 pre-conditioner

Index electric-field and volume integral equations, 1057 integral and wave equations, 1066 Müller equations, 964 PMCHWT equations, 959 precession, 852 principle of causality Coulomb gauge, 517 Kramers-Krönig relations, 828 memory functions, 825 particles beam, 792 time-dependent Green function, 548 quadrupole electric, 182, 185 magnetic, 661 radar cross section, 1004 radiation accelerated charge, 48 Cherenkov, 595 conditions harmonic equation, 805 Silver-Müller, 357, 376, 701 Sommerfeld, 357, 535 time domain, 547, 559, 582 transmission lines, 785 vector potential, 598 Wilcox, 359, 535 dipole antenna, 638 fields, 633–636 gauge, 263 inverted-F antenna, 1001 loop antenna, 643 pattern, 638 resistance dipole antenna, 642 Hertzian dipole, 633 loop antenna, 644 vectors, see Schelkunoff Rayleigh quotient algebraic EFIE, 953 cavities, 744 geometrical meaning, 1158 harmonic equation, 166, 178 waveguides, 781 region, 626

1213

Rayleigh-Carson theorem, 647, 650, 652 reciprocity antenna equivalent circuit, 726 dyadic Green functions, 647 impedance matrix, 378 Lorentz lemma, 374 mutual antenna admittances, 382 static Green functions, 159 reference frame, 4 relaxation time conductor, 28 polar substance, 852 Rellich theorem, 358, 529 retarded potentials, 577 time, 48, 62, 577, 589 Riemann surface, 475, 810, 816, 853, 1116 Riesz map, 1152 representation theorem, 1150 adjoint operator, 1153 Helmholtz equation, 539 wave equation, 344 right-handed screw rule, 4 triple, 4, 432, 1081, 1094, 1193 Schelkunoff equivalence principle, 716 radiation vectors, 637, 729 self-adjoint operator cavity modes, 743 compact, 346, 541, 1157 Laplace, 159 reciprocation, 393 reciprocity, 385 time-dependent Green function, 549 vector Helmholtz, 615, 619 vector Poisson, 400 waveguide modes, 780 Sellmeier equation, 848 separation argument, 286 Helmholtz equation, 528 Laplace equation, 164, 176, 250 Marcuvitz-Schwinger equations, 763 shape functions, 1068 signum function, 477, 785, 990 similarity transformation, 612, 1002

1214

Advanced Theoretical and Numerical Electromagnetics

simplex coordinates, 971, 1028 singularity extraction, 979 skin depth conducting cylinder, 665, 668 conducting half space, 425, 664, 819 modified EFIE, 904 plane waves, 424 slow waves, 792, 799 smooth surface open neighbourhood, 137, 894, 1182 open strip, 137, 1185 subtended solid angle, 1180 unit normal, 1179 Snell laws, 438, 795 Sobolev spaces scalar functions, 538, 1140 vector functions, 342, 1141 solid angle, 134, 293, 637, 1180 spherical harmonics, 168 addition theorem, 191 stationarity energy functional, 198 Rayleigh quotient, 745 steady current, 20, 219, 296 step function discrete, 996, 1017, 1055 one-dimensional, 321, 436, 467, 478, 486, 628 derivative, 1126 smooth, 122, 141, 606 surface, 969, 994 three-dimensional, 502, 711, 1023, 1032 derivative, 1127 two-dimensional, 976 stiffness matrix, 1065, 1075 streamlines coaxial cable, 779 electric field, 41, 88, 256, 324, 486, 587 magnetic field, 258, 310, 667, 922 twin-lead transmission line, 884 velocity field, 1093 subset ball, 1132 bounded, 1132 closed, 1132 convex, 604, 1131 frontier, 1132

open, 1132 subspace, see vector space susceptibility absorption, 847 anomalous dispersion, 847 electric, 196, 824, 847, 852 magnetic, 307, 824 normal dispersion, 847 telegraph equations sources, 783 TE guided modes, 772 TEM modes, 777 TM guided modes, 767 transmission line, 348 tesseral harmonics, see spherical harmonics test charge, 3, 200, 833 Thomson dipole, 361 theorem, see Kelvin theorem total internal reflection, 439, 457, 756 transmission line, 638 chain matrix, 452 coaxial cable, 364, 777 narrow-band signal, 855 reflection coefficient, 640 telegraph equations, 348 transverse gauge, 263 tunnelling, 459 uniqueness of solutions cavity problem, 753 combined-field integral equation, 903 D’Alembert equation, 542 electric scalar potential, 91 electric-field integral equation, 899 electrodynamic scalar potential, 596 electrodynamic vector potential, 599 electrostatic field, 102 Fredholm equation of the second kind, 875, 1161 Helmholtz equation, 532, 538 magnetic vector potential, 242 magnetic-field integral equation, 900 modes in a PEC cavity, 736 positive definite operators, 878 stationary magnetic entities, 247, 252

Index time domain, 329, 331 cylindrical regions, 333 time-harmonic regime, 339, 342, 353 vector circulation, 250 vector space, 1131 ball, 1132 closure, 1132 compact set, 1132 complete, 1134, 1136 dual, 1151 embedding, 345, 540, 1134 Hilbert, 1136 inner product, 1134

1215

norm, 1131 orthogonal complement, 1136 projection theorem, 1137 semi-norm, 1132 Sobolev, 1140 Volterra equations, 871 volume coordinates, 1026, 1068 wave-function, 292, 459 wavelength plane wave, 411 spherical wave, 625 waveguide, 768 zonal harmonics, see spherical harmonics