The wiener-hopf method in electromagnetics

This advanced research monograph is devoted to the Wiener-Hopf technique, a function-theoretic method that has found applications in a variety of fields, most notably in analytical studies of diffraction and scattering of waves. It provides a comprehensive treatment of the subject and covers the latest developments, illustrates the wide range of possible applications for this method, and includes an extensive outline of the most powerful analytical tool for the solution of diffraction problems.

This will be an invaluable compendium for scientists, engineers and applied mathematicians, and will serve as a benchmark reference in the field of theoretical electromagnetism for the foreseeable future.

Author Notes

Vito G. Daniele received his degree in Electronic Engineering from the Polytechnic of Turin, Italy, 1966. From 1981 to 2012 he was a Full Professor of Electrical Engineering at the Polytechnic of Turin, and is presently a Senior Professor at the University and a corresponding member of the Academy of Sciences of Turin. His research interests are mainly in analytical and approximation methods for the evaluation of electromagnetic fields. He has published more than 150 papers in refereed journals and conference proceedings, as well as several book chapters. Prof. Daniele has been Chairman and invited speaker at several international symposia, a reviewer for many international journals and a Guest Editor of special issues. He has also been a consultant to various industries in Italy.
Rodolto Zich received his degree in Electronic Engineering in 1962. From 1976 to 2010 he was a Full Professor of Electromagnetic Fields and Circuits at the Polytechnic of Turin, Italy where he served as Rector from 1987 to 2001. He is presently a 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, 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 Istituto Superiore Mario Boella as a bridge between academic and industrial research.

Preface	p. xiii
Foreword	p. xvii
Part 1 Mathematical Aspects	p. 1
1 Forms of Wiener-Hopf equations	p. 3
1.1 The basic Wiener-Hopf equation	p. 3
1.1.1 An electromagnetic example: The half-plane problem	p. 5
1.2 Modified W-H equations (MWHE)	p. 7
1.2.1 Longitudinally modified W-H equations	p. 7
1.2.2 Transversely modified W-H equations	p. 9
1.2.3 The incomplete Wiener-Hopf equations	p. 10
1.3 Generalized W-H equations	p. 12
1.3.1 An electromagnetic example: The PEC wedge problem	p. 12
1.3.2 An electromagnetic example: The dielectric wedge problem	p. 13
1.4 The Hilbert-Riemann problem	p. 14
1.5 Reduction of W-H equations to the classical form	p. 14
1.5.1 Reduction of the transversely modified W-H equations to CWHE	p. 14
1.5.2 Reduction of the longitudinally modified W-H equations to CWHE	p. 15
1.5.3 The Hilbert-Riemann equations	p. 16
1.5.4 Generalized Wiener-Hopf equations	p. 16
1.6 From Wiener-Hopf equations to Fredholm integral equations in the spectral domain	p. 17
1.7 Fundamental literature	p. 19
2 The exact solution of Wiener-Hopf equations	p. 21
2.1 Introduction	p. 21
2.2 Additive decomposition	p. 22
2.3 Multiplicative decomposition or factorization	p. 23
2.4 Solution of the W-H equation	p. 24
2.4.1 Solution of the nonhomogeneous equation	p. 24
2.4.2 Remote source	p. 27
2.5 Unbounded plus and minus unknowns	p. 29
2.6 Factorized matrices as solutions of the homogeneous Wiener-Hopf problem	p. 29
2.7 Nonstandard factorizations	p. 31
2.8 Extension of the W-H technique to the GWHE	p. 34
2.9 Important mappings for dealing with W-H equations	p. 35
2.9.1 The ¿ = $$$ mapping	p. 35
2.9.2 The ¿ = -¿ o cos w mapping	p. 36
3 Functions decomposition and factorization	p. 45
3.1 Decomposition	p. 45
3.1.1 Example 1	p. 47
3.1.2 Decomposition of an even function	p. 51
3.1.3 Numerical decomposition	p. 51
3.1.4 Example 1 revisited	p. 53
3.1.5 The case of meromorphic functions	p. 54
3.1.6 Decomposition using rational approximants of the function	p. 55
3.2 Factorization	p. 57
3.2.1 General formula for the scalar case	p. 57
3.2.2 Example 2	p. 57
3.2.3 Example 3	p. 58
3.2.4 Factorization of meromorphic functions	p. 58
3.2.5 Example 4	p. 60
3.2.6 Factorization of kernels involving continuous and discrete spectrum	p. 63
3.3 Decomposition equations in the w - plane	p. 66
3.3.1 Evaluation of the plus functions	p. 66
3.3.2 Evaluation of the minus functions	p. 69
3.3.4 Use of difference equation for function decomposition	p. 73
3.3.5 The W-H equation as difference equation	p. 73
4 Exact matrix factorization	p. 75
4.1 Introduction	p. 75
4.2 Some possibilities to reduce the order of the kernel matrices	p. 76
4.3 Factorization of triangular matrices	p. 78
4.4 Factorization of rational matrices	p. 80
4.4.1 Introduction	p. 80
4.4.2 Matching of the singularities	p. 81
4.4.3 The factorization in the framework of the Fredholm equations	p. 85
4.5 Techniques for solving the factorization problem	p. 86
4.5.1 The logarithmic decomposition	p. 86
4.6 The factorization problem and the functional analysis	p. 92
4.6.1 The iterative method	p. 92
4.6.2 The Fredholm determinant method	p. 93
4.6.3 Factorization of meromorphic matrix kernels with an infinite number of poles	p. 94
4.7 A class of matrices amenable to explicit factorization: matrices having rational eigenvectors	p. 95
4.8 Factorization of a 2 × 2 matrix	p. 96
4.8.1 The Hurd method	p. 96
4.8.2 The off-diagonal form	p. 98
4.8.3 Reduction of matrices commuting with polynomial matrices to the Daniele matrices	p. 99
4.8.4 Explicit factorization of Daniele matrices	p. 101
4.8.5 The elimination of the offensive behavior for matrices having the Daniele form	p. 104
4.8.6 A relatively simple case	p. 106
4.8.7 The $$$ rational function of ¿ case	p. 108
4.9 The factorization of matrices commuting with rational matrices	p. 110
4.9.1 Introduction	p. 110
4.9.2 Matrix of order two commuting with polynomial matrices	p. 111
4.9.3 Explicit expression of ¿ i (¿) in the general case	p. 113
4.9.4 Asymptotic behavior of the logarithmic representation of-l(¿)P -1 (¿) + 1	p. 117
4.9.5 Asymptotic behavior of the decomposed ¿ i+ (¿)	p. 118
4.9.6 A procedure to eliminate the exponential behavior	p. 120
4.9.7 On the reduction of the order of the system	p. 124
4.9.8 The nonlinear equations as a Jacobi inversion problem	p. 125
4.9.9 Weakly factorization of a matrix commuting with a polynomial matrix	p. 127
5 Approximate solution: The Fredholm factorization	p. 129
5.1 The integral equations in the ¿ - plane	p. 129
5.1.1 Introduction	p. 129
5.1.2 Source pole ¿ 0 with positive imaginary part	p. 130
5.1.3 Analytical validation of a particular W-H equation	p. 131
5.1.4 A property of the integral in the Fredholm equation	p. 132
5.1.5 Numerical solution of the Fredholm equations	p. 134
5.1.6 Analytic continuation outside the integration line	p. 141
5.2 The integral equations in the w - plane	p. 143
5.3 Additional considerations on the Fredholm equations	p. 146
5.3.1 Presence of poles of the kernel in the warped region	p. 146
5.3.2 The Fredholm factorization for particular matrices	p. 147
5.3.3 The Fredholm equation relevant to a modified kernel	p. 147
6 Approximate solutions: Some particular techniques	p. 149
6.1 The Jones method for solving modified W-H equations	p. 149
6.1.1 Introduction	p. 149
6.1.2 Longitudinal modified W-H equation	p. 149
6.1.3 Transversal modified W-H equation	p. 152
6.2 The Fredholm factorization for particular matrices	p. 153
6.3 Rational approximation of the kernel	p. 161
6.3.1 Pade approximants	p. 161
6.3.2 An interpolation approximant method	p. 163
6.4 Moment method	p. 167
6.4.1 Introduction	p. 167
6.4.2 Stationary properties of the solutions with the moment method	p. 169
6.4.3 An electromagnetic example: the impedance of a wire antenna in free space	p. 173
6.5 Comments on the approximate methods for solving W-H equations	p. 175
Part 2 Applications	p. 177
7 The half-plane problem	p. 179
7.1 Wiener-Hopf solution of discontinuity problems in plane-stratified regions	p. 179
7.2 Spectral transmission line in homogeneous isotropic regions	p. 180
7.2.1 Circuital considerations	p. 181
7.2.2 Jump of voltage or current in a section where it is present a discontinuity	p. 182
7.2.3 Jump of voltage or current in a section where a concentrated source is present	p. 182
7.3 Wiener-Hopf equations in the Laplace domain	p. 183
7.4 The PEC half-plane problem	p. 185
7.4.1 E-polarization case	p. 185
7.4.2 Far-field contribution	p. 188
7.5 Skew incidence	p. 191
7.6 Diffraction by an impedance half plane	p. 197
7.6.1 Deduction of W-H equations in diffraction problems by impenetrable half-planes	p. 197
7.6.2 Presence of isotropic impedances Z a and Z b	p. 200
7.7 The general problem of factorization	p. 203
7.7.1 The case of symmetric half-plane	p. 205
7.7.2 The case of opposite diagonal impedances Z b = -Z a	p. 206
7.8 The jump or penetrable half-plane problem	p. 206
7.9 Full-plane junction at skew incidence	p. 207
7.10 Diffraction by an half plane immersed in arbitrary linear medium	p. 208
7.10.1 Transverse equation in an indefinite medium	p. 208
7.10.2 Field equations in the Fourier domain	p. 210
7.10.3 The W-H equation for a PEC or a PMC half-plane immersed in a homogeneous linear arbitrary medium	p. 216
7.11 The half-plane immersed in an arbitrary planar stratified medium	p. 220
8 Planar discontinuities in stratified media	p. 223
8.1 The planar waveguide problem	p. 223
8.1.1 The E-polarization case	p. 223
8.1.2 Source constituted by plane wave	p. 225
8.1.3 Source constituted by an incident mode	p. 227
8.1.4 The skew plane wave case	p. 228
8.2 The reversed half-planes problem	p. 230
8.2.1 The E-polarization case	p. 230
8.2.2 Qualitative characteristics of the solution	p. 231
8.2.3 Numerical evaluation of the electromagnetic field	p. 232
8.2.4 Numerical solution of the W-H equations	p. 233
8.2.5 Source constituted by a skew plane wave	p. 237
8.3 The three half-planes problem	p. 244
8.3.1 The E-polarization case (normal incidence case)	p. 244
8.3.2 The skew incidence case	p. 247
8.4 Arrays of parallel wire antennas in stratified media	p. 248
8.4.1 The single antenna case	p. 248
8.4.2 The W-H equations of an array of wire antennas	p. 250
8.4.3 Spectral theory of transmission lines constituted by bundles of wires	p. 254
8.5 Spectral theory of microstrip and coplanar transmission lines	p. 254
8.5.1 Coplanar line with two strips	p. 254
8.5.2 The shielded microstrip transmission line	p. 260
8.6 General W-H formulation of planar discontinuity problems in arbitrary stratified media	p. 261
8.6.1 Formal solution with the factorization method	p. 263
8.6.2 The method of stationary phase for multiple integrals	p. 267
8.0.3 The circular aperture	p. 268
8.6.4 The quarter plane problem	p. 272
9 Wiener-Hopf analysis of waveguide discontinuities	p. 279
9.1 Marcuvitz-Schwinger formalism	p. 279
9.1.1 Example 1	p. 280
9.1.2 Example 2	p. 283
9.2 Bifurcation in a rectangular waveguide	p. 285
9.3 The junction of two waveguides	p. 287
9.4 A general discontinuity problem in a rectangular waveguide	p. 289
9.5 Radiation from truncated circular waveguides	p. 292
9.6 Discontinuities in circular waveguides	p. 297
10 Further applications of the W-H technique	p. 301
10.1 The step problem	p. 301
10.1.1 Deduction of the transverse modified W-FI equations (E-polarization case)	p. 301
10.1.2 Solution of the equations	p. 303
10.2 The strip problem	p. 303
10.2.1 Some longitudinally modified W-H geometries	p. 304
10.3 The hole problem	p. 304
10.4 The wall problem	p. 305
10.5 The semi-infinite duct with a flange	p. 307
10.6 Presence of dielectrics	p. 308
10.7 A problem involving a dielectric slab	p. 310
10.8 Some problems involving dielectric slabs	p. 313
10.8.1 Semi-infinite dielectric guides	p. 314
10.8.2 The junction of two semi-infinite dielectric slab guides	p. 314
10.8.3 Some problems solved in the literature	p. 314
10.9 Some problems involving periodic structures	p. 315
10.9.1 Diffraction by an infinite array of equally spaced half-planes immersed in free space	p. 315
10.9.2 Other problems solved in the literature	p. 317
10.10 Diffraction by infinite strips	p. 318
10.10.1 Solution of the key problem	p. 319
10.10.2 Boundary conditions	p. 321
10.10.3 Solution of the W-H equation	p. 321
10.11 Presence of an inductive iris in rectangular waveguides	p. 323
10.12 Presence of a capacitive iris in rectangular waveguides	p. 324
10.13 Problems involving semi-infinite periodic structures	p. 324
10.14 Problems involving impedance surfaces	p. 325
10.15 Some problems involving cones	p. 326
10.16 Diffraction by a PEC wedge by an incident plane wave at skew incidence	p. 330
10.17 Diffraction by a right PEC wedge immersed in a stratified medium	p. 334
10.18 Diffraction by a right isorefractive wedge	p. 337
10.18.1 Solution of the W-H equations	p. 342
10.18.2 Matrix factorization of g e (¿)	p. 345
10.18.3 Near field behavior	p. 347
10.19 Diffraction by an arbitrary dielectric wedge	p. 349
References	p. 351
Index	p. 361

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