Optimization Methods in Electromagnetic Radiation

Contents Preface 1 Arrays of Point and Line Sources, and Optimization 1.1 The Problem of Antenna Optimization 1.2 Arrays of Point Sources 1.2.1 The Linear Array 1.2.2 Circular Arrays 1.3 Maximization of Directivity and Super-gain 1.3.1 Directivity and Other Measures of Performance 1.3.2 Maximization of Directivity 1.4 Dolph-Tschebyshe. Arrays 1.4.1 Tschebyshe. Polynomials 1.4.2 The Dolph Problem 1.5 Line Sources 1.5.1 The Linear Line Source 1.5.2 The Circular Line Source 1.5.3 Numerical Quadrature 1.6 Conclusion 2 Discussion of Maxwell's Equations 2.1 Introduction 2.2 Geometry of the Radiating Structure 2.3 Maxwell's Equations in Integral Form 2.4 The Constitutive Relations 2.5 Maxwell's Equations in Differential Form 2.6 Energy Flow and the Poynting Vector 2.7 Time Harmonic Fields 2.8 Vector Potentials 2.9 Radiation Condition, Far Field Pattern 2.10 Radiating Dipoles and Line Sources 2.11 Boundary Conditions on Interfaces 2.12 Hertz Potentials and Classes of Solutions 2.13 Radiation Problems in Two Dimensions 3 Optimization Theory for Antennas 3.1 Introductory Remarks 3.2 The General Optimization Problem 3.2.1 Existence and Uniqueness 3.2.2 The Modeling of Constraints 3.2.3 Extreme Points and Optimal Solutions 3.2.4 The Lagrange Multiplier Rule 3.2.5 Methods of Finite Dimensional Approximation 3.3 Far Field Patterns and Far Field Operators 3.4 Measures of Antenna Performance 4 The Synthesis Problem 4.1 Introductory Remarks 4.2 Remarks on Ill-Posed Problems 4.3 Regularization by Constraints 4.4 The Tikhonov Regularization 4.5 The Synthesis Problem for the Finite Linear Line Source 4.5.1 Basic Equations 4.5.2 The Nystrom Method 4.5.3 Numerical Solution of the Normal Equations 4.5.4 Applications of the Regularization Techniques 5 Boundary Value Problems for the Two-Dimensional Helmholtz Equation 5.1 Introduction and Formulation of the Problems 5.2 Rellich's Lemma and Uniqueness 5.3 Existence by the Boundary Integral Equation Method 5.4 L2-Boundary Data 5.5 Numerical Methods 5.5.1 Nystrom's Method for Periodic Weakly Singular Kernels 5.5.2 Complete Families of Solutions 5.5.3 Finite Element Methods for Absorbing Boundary Conditions 5.5.4 Hybrid Methods 6 Boundary Value Problems for Maxwell's Equations 6.1 Introduction and Formulation of the Problem 6.2 Uniqueness and Existence 6.3 L2-Boundary Data 7 Some Particular Optimization Problems 7.1 General Assumptions 7.2 Maximization of Power 7.2.1 Input Power Constraints 7.2.2 Pointwise Constraints on Inputs 7.2.3 Numerical Simulations 7.3 The Null-Placement Problem 7.3.1 Maximization of Power with Prescribed Nulls 7.3.2 A Particular Example -- The Line Source 7.3.3 Pointwise Constraints 7.3.4 Minimization of Pattern Perturbation 7.4 The Optimization of Signal-to-Noise Ratio and Directivity 7.4.1 The Existence of Optimal Solutions 7.4.2 Necessary Conditions 7.4.3 The Finite Dimensional Problems 8 Conflicting Objectives: The Vector Optimization Problem . 8.1 Introduction 8.2 General Multi-criteria Optimization Problems 8.2.1 Minimal Elements and Pareto Points 8.2.2 The Lagrange Multiplier Rule 8.2.3 Scalarization 8.3 The Multi-criteria Dolph Problem for Arrays 8.3.1 The Weak Dolph Problem 8.3.2 Two Multi-criteria Versions 8.4 Null Placement Problems and Super-gain 8.4.1 Minimal Pattern Deviation 8.4.2 Power and Super-gain 8.5 The Signal-to-noise Ratio Problem 8.5.1 Formulation of the Problem and Existence of Pareto Points 8.5.2 The Lagrange Multiplier Rule 8.5.3 An Example A Appendix A.1 Introduction A.2 Basic Notions and Examples A.3 The Lebesgue Integral and Function Spaces A.3.1 The Lebesgue Integral A.3.2 Sobolev Spaces A.4 Orthonormal Systems A.5 Linear Bounded and Compact Operators A.6 The Hahn-Banach Theorem A.7 The Frechet Derivative A.8 Weak Convergence A.9 Partial Orderings References Index