The Wiener-Hopf Method T h e in Electromagnetics W i e n e r - H o p This advanced research monograph is devoted to Vito G. Daniele received his degree in Electronic f the Wiener-Hopf technique, a function-theoretic Engineering from the Polytechnic of Turin, Italy, in M 1966. From 1981 to 2012 he was a Full Professor method that has found applications in a variety e of Electrical Engineering at the Polytechnic of of fields, most notably in analytical studies of t Turin, and is presently a Senior Professor at the h diffraction and scattering of waves. It provides University and a corresponding member of the o a comprehensive treatment of the subject and Academy of Sciences of Turin. His research d interests are mainly in analytical and approximation covers the latest developments, illustrates the wide i methods for the evaluation of electromagnetic n range of possible applications for this method, and fields. He has published more than 150 papers includes an extensive outline of the most powerful in refereed journals and conference proceedings, E l analytical tool for the solution of diffraction as well as several book chapters. Prof. Daniele e has been Chairman and invited speaker at several c problems. t international symposia, a reviewer for many r This will be an invaluable compendium for international journals and a Guest Editor of special o scientists, engineers and applied mathematicians, issues. He has also been a consultant to various m industries in Italy. and will serve as a benchmark reference in the a field of theoretical electromagnetism for the Rodolfo Zich received his degree in Electronic g Engineering in 1962. From 1976 to 2010 he was a n foreseeable future. Full Professor of Electromagnetic Fields and Circuits e t at the Polytechnic of Turin, Italy where he served i c as Rector from 1987 to 2001. He is presently a s Professor Emeritus and a national member of the Academy of Sciences of Turin. His research interests are mainly in analytical and approximation techniques for the evaluation of electromagnetic fields at high frequencies. He has extensive international experience in university management, The Wiener-Hopf Method as a member of the Board of Ecole Polytechnique de Paris (Palaiseau), President of Columbus, and President of CLUSTER. He has also been President of TiLab and INRIM. He founded the in Electromagnetics Istituto Superiore Mario Boella as a bridge between academic and industrial research. D a n i e l e Vito G. Daniele and Rodolfo S. Zich a n The Institution of Engineering and Technology d www.theiet.org Z ISBN 978-1-61353-001-6 ic MARIO BOELLA SERIES ON ELECTROMAGNETISM IN INFORMATION & COMMUNICATION h ISMB-Daniele&Zich-WHME 254x178.indd 3 17/07/2014 11:41 The Wiener-Hopf Method in Electromagnetics Mario Boella Series on Electromagnetism in Information and Communication Piergiorgio L. E. Uslenghi, PhD– Series Editor TheMarioBoellaseriesofferstextbooksandmonographsinallareasofradioscience,witha specialemphasisontheapplicationsofelectromagnetismtoinformationandcommunication technologies. The series is scientifically and financially sponsored by the Istituto Superiore Mario Boella affiliated with the Politecnico di Torino, Italy, and is scientifically co- sponsored by the International Union of Radio Science (URSI). It is named to honor the memory of Professor Mario Boella of the Politecnico di Torino, who was a pioneer in thedevelopmentofelectronicsandtelecommunicationsinItalyforhalfacentury,andavice president of URSIfrom 1966 to 1969. Published Titles in the Series Introduction to Wave Phenomena byAkira Hiroseand Karl Lonngren Scattering of Waves by Wedges and Cones with Impedance Boundary Conditions byMikhail Lyalinov and Ning Yan Zhu Complex Space Source Theory ofSpatially LocalizedElectromagnetic Waves byS.R.Seshadri TheWiener-Hopf Method in Electromagnetics byVito Daniele and Rodolfo Zich Forthcoming Titles Higher OrderNumerical Solution Techniques in Electromagnetics byRoberto Graglia and AndrewPeterson (2015) Slotted Waveguide Array Antennas bySembiam Rengarajan and Lars Josefsson(2015) The Wiener-Hopf Method in Electromagnetics ISMB Series Rodolfo Zich Instituto Superiore de Mario Boella Vito Daniele Polytechnic of Torino theiet.org PublishedbySciTechPublishing,animprintoftheIET. www.scitechpub.com www.theiet.org Copyright(cid:1)2014bySciTechPublishing,Edison,NJ.Allrightsreserved. Nopartofthispublicationmaybereproduced,storedinaretrievalsystemortransmittedinanyformorbyany means,electronic,mechanical,photocopying,recording,scanningorotherwise,exceptaspermittedunderSections 107or108ofthe1976UnitedStatesCopyrightAct,withouteitherthepriorwrittenpermissionofthePublisher,or authorizationthroughpaymentoftheappropriateper-copyfeetotheCopyrightClearanceCenter,222Rosewood Drive,Danvers,MA01923,(978)750-8400,fax(978)646-8600,oronthewebatcopyright.com.Requeststothe PublisherforpermissionshouldbeaddressedtoTheInstitutionofEngineeringandTechnology,MichaelFaraday House,SixHillsWay,Stevenage,Herts,SG12AY,UnitedKingdom. Whiletheauthorandpublisherbelievethattheinformationandguidancegiveninthisworkarecorrect,allparties mustrelyupontheirownskillandjudgementwhenmakinguseofthem.Neithertheauthornorpublisherassumes anyliabilitytoanyoneforanylossordamagecausedbyanyerrororomissioninthework,whethersuchanerroror omissionistheresultofnegligenceoranyothercause.Anyandallsuchliabilityisdisclaimed. 10987654321 ISBN978-1-61353-001-6(hardback) ISBN978-1-61353-031-3(PDF) TypesetinIndiabyMPSLimited PrintedintheUSbyIntegratedBooksInternational PrintedintheUKbyCPIGroup(UK)Ltd,Croydon Contents Preface xiii Foreword xvii PART 1 Mathematical Aspects 1 1 Forms of Wiener-Hopf equations 3 1.1 The basic Wiener-Hopf equation 3 1.1.1 Anelectromagnetic example: The half-plane problem 5 1.2 Modified W-H equations (MWHE) 7 1.2.1 Longitudinallymodified W-H equations 7 1.2.2 Transverselymodified W-H equations 9 1.2.3 The incomplete Wiener-Hopf equations 10 1.3 Generalized W-H equations 12 1.3.1 Anelectromagnetic example: The PEC wedge problem 12 1.3.2 Anelectromagnetic example: The dielectric wedge problem 13 1.4 The Hilbert-Riemann problem 14 1.5 Reduction of W-H equations to the classical form 14 1.5.1 Reduction of the transversely modified W-H equations to CWHE 14 1.5.2 Reduction of the longitudinally modified W-H equations to CWHE 15 1.5.3 The Hilbert-Riemann equations 16 1.5.4 Generalized Wiener-Hopfequations 16 1.6 From Wiener-Hopf equations to Fredholm integral equations in the spectral domain 17 1.7 Fundamental literature 19 v vi Contents 2 The exact solution of Wiener-Hopf equations 21 2.1 Introduction 21 2.2 Additive decomposition 22 2.3 Multiplicative decomposition or factorization 23 2.4 Solution of the W-H equation 24 2.4.1 Solution of the nonhomogeneous equation 24 2.4.2 Remote source 27 2.5 Unbounded plus and minus unknowns 29 2.6 Factorized matrices as solutions of the homogeneous Wiener-Hopf problem 29 2.7 Nonstandard factorizations 31 2.8 Extension of the W-H technique to the GWHE 34 2.9 Important mappings for dealing with W-H equations 35 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2.9.1 The c¼ t2(cid:2)a2 mapping 35 o 2.9.2 The a¼(cid:2)t cosw mapping 36 o 3 Functions decomposition and factorization 45 3.1 Decomposition 45 3.1.1 Example 1 47 3.1.2 Decomposition of an even function 51 3.1.3 Numerical decomposition 51 3.1.4 Example 1 revisited 53 3.1.5 The case of meromorphic functions 54 3.1.6 Decomposition usingrational approximants of the function 55 3.2 Factorization 57 3.2.1 General formula for the scalar case 57 3.2.2 Example 2 57 3.2.3 Example 3 58 3.2.4 Factorization of meromorphic functions 58 3.2.5 Example 4 60 3.2.6 Factorization of kernels involving continuousand discrete spectrum 63 3.3 Decomposition equations in the w – plane 66 3.3.1 Evaluation of the plus functions 66 3.3.2 Evaluation of the minus functions 69 3.3.4 Useof difference equation for function decomposition 73 3.3.5 The W-Hequationas difference equation 73 Contents vii 4 Exact matrix factorization 75 4.1 Introduction 75 4.2 Some possibilities to reduce the order of the kernel matrices 76 4.3 Factorization of triangular matrices 78 4.4 Factorization of rational matrices 80 4.4.1 Introduction 80 4.4.2 Matching of the singularities 81 4.4.3 The factorization in the framework of the Fredholm equations 85 4.5 Techniques for solving the factorization problem 86 4.5.1 The logarithmic decomposition 86 4.6 The factorization problem and the functional analysis 92 4.6.1 The iterative method 92 4.6.2 The Fredholm determinant method 93 4.6.3 Factorizationof meromorphic matrixkernels with an infinite number of poles 94 4.7 A class of matrices amenable to explicit factorization: matrices having rational eigenvectors 95 4.8 Factorization of a 2(cid:3)2 matrix 96 4.8.1 The Hurdmethod 96 4.8.2 The off-diagonal form 98 4.8.3 Reduction of matricescommuting with polynomial matrices to the Daniele matrices 99 4.8.4 Explicit factorization of Daniele matrices 101 4.8.5 The elimination of the offensive behavior for matrices having the Daniele form 104 4.8.6 Arelatively simple case 106 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4.8.7 The aðaÞ=bðaÞ rational function of a case 108 4.9 The factorization of matrices commuting with rational matrices 110 4.9.1 Introduction 110 4.9.2 Matrix of order two commuting with polynomial matrices 111 4.9.3 Explicit expression of yðaÞ in the general case 113 i 4.9.4 Asymptotic behavior of the logarithmic representation of (cid:2)lðaÞP(cid:2)1ðaÞþ1 117 4.9.5 Asymptotic behavior of the decomposedy ðaÞ 118 i(cid:4) 4.9.6 Aprocedure to eliminate the exponential behavior 120 4.9.7 Onthe reduction of the order of the system 124 4.9.8 The nonlinear equations as a Jacobi inversion problem 125 4.9.9 Weakly factorization of a matrix commuting with a polynomial matrix 127 viii Contents 5 Approximate solution: The Fredholm factorization 129 5.1 The integral equations in the a(cid:2)plane 129 5.1.1 Introduction 129 5.1.2 Source pole a with positive imaginary part 130 o 5.1.3 Analytical validation of a particular W-H equation 131 5.1.4 Aproperty of the integral in the Fredholm equation 132 5.1.5 Numerical solution of the Fredholm equations 134 5.1.6 Analytic continuation outside the integration line 141 5.2 The integral equations in the w(cid:2)plane 143 5.3 Additional considerations on the Fredholm equations 146 5.3.1 Presence of poles of the kernel in the warped region 146 5.3.2 The Fredholm factorization forparticular matrices 147 5.3.3 The Fredholm equation relevant toa modified kernel 147 6 Approximate solutions: Some particular techniques 149 6.1 The Jones method for solving modified W-H equations 149 6.1.1 Introduction 149 6.1.2 Longitudinal modified W-Hequation 149 6.1.3 Transversal modified W-H equation 152 6.2 The Fredholm factorization for particular matrices 153 6.3 Rational approximation of the kernel 161 6.3.1 Pade approximants 161 6.3.2 Aninterpolation approximant method 163 6.4 Moment method 167 6.4.1 Introduction 167 6.4.2 Stationary properties of the solutions with the moment method 169 6.4.3 Anelectromagnetic example: the impedance of a wire antenna in free space 173 6.5 Comments on the approximate methods for solving W-H equations 175 PART 2 Applications 177 7 The half-plane problem 179 7.1 Wiener-Hopf solution of discontinuity problems in plane-stratified regions 179 7.2 Spectral transmission line in homogeneous isotropic regions 180 7.2.1 Circuital considerations 181 Contents ix 7.2.2 Jumpof voltage or current in a section where it is present a discontinuity 182 7.2.3 Jumpof voltage or current in a section where a concentrated source ispresent 182 7.3 Wiener-Hopf equations in the Laplace domain 183 7.4 The PEC half-plane problem 185 7.4.1 E-polarizationcase 185 7.4.2 Far-field contribution 188 7.5 Skew incidence 191 7.6 Diffraction by an impedance half plane 197 7.6.1 Deduction of W-Hequationsin diffraction problems by impenetrable half-planes 197 7.6.2 Presence of isotropic impedances Z and Z 200 a b 7.7 The general problem of factorization 203 7.7.1 The case of symmetric half-plane 205 7.7.2 The case of opposite diagonal impedances Z ¼(cid:2)Z 206 b a 7.8 The jump or penetrable half-plane problem 206 7.9 Full-plane junction at skew incidence 207 7.10 Diffraction by an half plane immersed in arbitrary linear medium 208 7.10.1 Transverse equation inan indefinite medium 208 7.10.2 Field equations in the Fourier domain 210 7.10.3 The W-H equation fora PEC or a PMChalf-plane immersed in a homogeneous linear arbitrarymedium 216 7.11 The half-plane immersed in an arbitrary planar stratified medium 220 8 Planar discontinuities in stratified media 223 8.1 The planar waveguide problem 223 8.1.1 The E-polarization case 223 8.1.2 Source constituted byplane wave 225 8.1.3 Source constituted byan incident mode 227 8.1.4 The skewplane wave case 228 8.2 The reversed half-planes problem 230 8.2.1 The E-polarization case 230 8.2.2 Qualitative characteristics of the solution 231 8.2.3 Numerical evaluation of the electromagnetic field 232 8.2.4 Numerical solution of the W-Hequations 233 8.2.5 Source constituted bya skewplane wave 237