Optimization methods in electromagnetic radiation

The subject of antenna design, primarily a discipline within electrical en- neering, is devoted to the manipulation of structural elements of and/or the electrical currents present on a physical object capable of supporting such a current. Almost as soon as one begins to look at the subject, it becomes clear that there are interesting mathematical problems which need to be addressed, in the ?rst instance, simply for the accurate modelling of the electromagnetic ?elds produced by an antenna. The description of the electromagnetic ?elds depends on the physical structure and the background environment in which thedeviceistooperate. It is the coincidence of a class of practical engineering applications and theapplicationofsomeinterestingmathematicaloptimizationtechniquesthat is the motivation for the present book. For this reason, we have thought it worthwhile to collect some of the problems that have inspired our research in appliedmathematics,andtopresenttheminsuchawaythattheymayappeal to two di?erent audiences: mathematicians who are experts in the theory of mathematical optimization and who are interested in a less familiar and importantareaofapplication,andengineerswho,confrontedwithproblemsof increasing sophistication, are interested in seeing a systematic mathematical approach to problems of interest to them. We hope that we have found the right balance to be of interest to both audiences. It is a di?cult task. Our ability to produce these devices at all, most designed for a part- ular purpose, leads quite soon to a desire to optimize the design in various ways. The mathematical problems associated with attempts to optimize p- formance can become quite sophisticated even for simple physical structures.

Preface	p. IX
1 Arrays of Point and Line Sources, and Optimization	p. 1
1.1 The Problem of Antenna Optimization	p. 1
1.2 Arrays of Point Sources	p. 2
1.2.1 The Linear Array	p. 3
1.2.2 Circular Arrays	p. 10
1.3 Maximization of Directivity and Super-gain	p. 15
1.3.1 Directivity and Other Measures of Performance	p. 15
1.3.2 Maximization of Directivity	p. 19
1.4 Dolph-Tschebysheff Arrays	p. 21
1.4.1 Tschebysheff Polynomials	p. 22
1.4.2 The Dolph Problem	p. 24
1.5 Line Sources	p. 26
1.5.1 The Linear Line Source	p. 30
1.5.2 The Circular Line Source	p. 36
1.5.3 Numerical Quadrature	p. 43
1.6 Conclusion	p. 47
2 Discussion of Maxwell's Equations	p. 49
2.1 Introduction	p. 49
2.2 Geometry of the Radiating Structure	p. 49
2.3 Maxwell's Equations in Integral Form	p. 50
2.4 The Constitutive Relations	p. 51
2.5 Maxwell's Equations in Differential Form	p. 52
2.6 Energy Flow and the Poynting Vector	p. 55
2.7 Time Harmonic Fields	p. 56
2.8 Vector Potentials	p. 58
2.9 Radiation Condition, Far Field Pattern	p. 60
2.10 Radiating Dipoles and Line Sources	p. 63
2.11 Boundary Conditions on Interfaces	p. 68
2.12 Hertz Potentials and Classes of Solutions	p. 70
2.13 Radiation Problems in Two Dimensions	p. 73
3 Optimization Theory for Antennas	p. 77
3.1 Introductory Remarks	p. 77
3.2 The General Optimization Problem	p. 80
3.2.1 Existence and Uniqueness	p. 81
3.2.2 The Modeling of Constraints	p. 84
3.2.3 Extreme Points and Optimal Solutions	p. 88
3.2.4 The Lagrange Multiplier Rule	p. 93
3.2.5 Methods of Finite Dimensional Approximation	p. 96
3.3 Far Field Patterns and Far Field Operators	p. 101
3.4 Measures of Antenna Performance	p. 103
4 The Synthesis Problem	p. 113
4.1 Introductory Remarks	p. 113
4.2 Remarks on Ill-Posed Problems	p. 115
4.3 Regularization by Constraints	p. 121
4.4 The Tikhonov Regularization	p. 127
4.5 The Synthesis Problem for the Finite Linear Line Source	p. 133
4.5.1 Basic Equations	p. 134
4.5.2 The Nyström Method	p. 135
4.5.3 Numerical Solution of the Normal Equations	p. 137
4.5.4 Applications of the Regularization Techniques	p. 138
5 Boundary Value Problems for the Two-Dimensional Helmholtz Equation	p. 145
5.1 Introduction and Formulation of the Problems	p. 145
5.2 Rellich's Lemma and Uniqueness	p. 148
5.3 Existence by the Boundary Integral Equation Method	p. 151
5.4 L 2 - Boundary Data	p. 157
5.5 Numerical Methods	p. 163
5.5.1 Nyström's Method for Periodic Weakly Singular Kernels	p. 164
5.5.2 Complete Families of Solutions	p. 168
5.5.3 Finite Element Methods for Absorbing Boundary Conditions	p. 174
5.5.4 Hybrid Methods	p. 181
6 Boundary Value Problems for Maxwell's Equations	p. 185
6.1 Introduction and Formulation of the Problem	p. 185
6.2 Uniqueness and Existence	p. 188
6.3 L 2 - Boundary Data	p. 193
7 Some Particular Optimization Problems	p. 195
7.1 General Assumptions	p. 195
7.2 Maximization of Power	p. 197
7.2.1 Input Power Constraints	p. 198
7.2.2 Pointwise Constraints on Inputs	p. 202
7.2.3 Numerical Simulations	p. 204
7.3 The Null-Placement Problem	p. 211
7.3.1 Maximization of Power with Prescribed Nulls	p. 213
7.3.2 A Particular Example - The Line Source	p. 216
7.3.3 Pointwise Constraints	p. 219
7.3.4 Minimization of Pattern Perturbation	p. 221
7.4 The Optimization of Signal-to-Noise Ratio and Directivity	p. 226
7.4.1 The Existence of Optimal Solutions	p. 227
7.4.2 Necessary Conditions	p. 228
7.4.3 The Finite Dimensional Problems	p. 231
8 Conflicting Objectives: The Vector Optimization Problem	p. 239
8.1 Introduction	p. 239
8.2 General Multi-criteria Optimization Problems	p. 240
8.2.1 Minimal Elements and Pareto Points	p. 241
8.2.2 The Lagrange Multiplier Rule	p. 247
8.2.3 Scalarization	p. 249
8.3 The Multi-criteria Dolph Problem for Arrays	p. 250
8.3.1 The Weak Dolph Problem	p. 251
8.3.2 Two Multi-criteria Versions	p. 253
8.4 Null Placement Problems and Super-gain	p. 262
8.4.1 Minimal Pattern Deviation	p. 264
8.4.2 Power and Super-gain	p. 270
8.5 The Signal-to-noise Ratio Problem	p. 278
8.5.1 Formulation of the Problem and Existence of Pareto Points	p. 278
8.5.2 The Lagrange Multiplier Rule	p. 280
8.5.3 An Example	p. 282
A Appendix	p. 285
A.1 Introduction	p. 285
A.2 Basic Notions and Examples	p. 286
A.3 The Lebesgue Integral and Function Spaces	p. 292
A.3.1 The Lebesgue Integral	p. 292
A.3.2 Sobolev Spaces	p. 295
A.4 Orthonormal Systems	p. 298
A.5 Linear Bounded and Compact Operators	p. 300
A.6 The Hahn-Banach Theorem	p. 307
A.7 The Fréchet Derivative	p. 310
A.8 Weak Convergence	p. 312
A.9 Partial Orderings	p. 315
References	p. 319
Index	p. 327

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