INTRODUCTION TO The Uniform Geometrical Theory of Diffraction

Transcription

1 INTRODUCTION TO The Uniform Geometrical Theory of Diffraction D.A. McNamara, C.W.I. Pistorius J.A.G. Malherbe University of Pretoria Artech House Boston London

2 CONTENTS Preface xiii Chapter 1 The Nature of High-Frequency Methods Introduction A Brief Historical Overview High-Frequency Phenomena 5 References 6 Chapter 2 Geometrical Optics Fields Introduction Ray Optical Construction of the High-Frequency Field Preliminary Remarks Some Conventional Electromagnetic Theory The Luneberg-Kline Anticipated Solution (Ansatz) The Efkonal Equation The Transport Equations The Geometrical Optics Terms and Their Interpretation Ray Paths, Amplitude Functions, and Phase Functions Sign Conventions and Caustics of the Geometrical Optics Fields The Geometrical Optics Field and Fermat's Principle Summary of the Properties of a High-Frequency Field and Some Special Cases Specific Examples of Geometrical Optics Fields Initial Comments and Some Definitions Uniform Plane Wave Fields The Fields of Electric and Magnetic Line Sources The Fields of a Hertzian Dipole The Far-Zone Fields of Horn Antennas The Fields of a Piecewise-Sinusoidal Dipole 46 vii

3 2.4.7 Sources with Fields That Are Not Geometrical Optics or Ray-Optic Fields Further Comment Reduction of Results to Two-Dimensional Ray Tubes Rays in Lossy Media Concluding Remarks A Taste of Things to Come 53 Problems 57 References 58 Chapter 3 Geometrical Optics Reflected Fields Introduction Initial Remarks A Stroll in the Sun A Strategy for This Chapter The Law of Reflection, Polarization Propertjes, and Phase Functions The Definition of Certain Geometrical Terms and Coordinate Systems The Law of Reflection Trajectories of Reflected Rays Polarization of Reflected Rays Phase Continuation along Reflected Rays Invocation of the Locality Principle More about Shadowing Geometrical Optics Surface Currents An Alternative Interpretation of the Form of R and the Law of Reflection What More Do We Need? The Expressions for the Geometrical Optics Field Reflected from Smooth Conducting Surfaces: Two-Dimensional Problems When Is a Problem of a Two-Dimensional Nature? Description of the Two-Dimensional Reflecting Surface Geometry Simplifications for Two-Dimensional Problems Simplification of the Polarization Description of Reflected GO Fields for Two-Dimensional Problems Amplitude Continuation along Two-Dimensional Reflected Ray Tubes The Classical Geometrical Optics Interpretation Summary of Reflected Field Expressions for Two- Dimensional Problems 93

4 3.3.8 On the Specular Point Q r and Its Location Initial Two-Dimensional Problem Examples Interpretation in Terms of Fundamental Electromagnetic Theory Relationship to Physical Optics Comments on GO Reflected Fields about Shadow Boundaries Further Examples of Two-Dimensional Reflected Field Problems General Expressions for the Reflected Fields from Three- Dimensional Smooth Conducting Surfaces Introduction Principal Radii of Curvature of Reflected Ray Tube at Qr-First Format Principal Radii of Curvature of Reflected Ray Tube at Qf Second Format Important Special Cases Principal Directions of the Reflected Wavefront Alternative Form for the Reflected GO Field at the Specular Point Q r Comments on the Expressions for the Reflected GO Field Alternative Determination of Principal Radii of Curvature of the Reflected Wavefront Examples of Three-Dimensional Reflected Field Problems Concluding Remarks 154 Problems 156 References ' 157 Chapter 4 Two-Dimensional Wedge Diffraction Introduction Diffraction by Huygens' Principle Keller's Original GTD The Uniform Theory of Diffraction Shadow Boundaries Two-Dimensional UTD Diffraction Coefficients Enforcing Continuity across the Shadow Boundaries Transition Regions Grazing Incidence Half-Plane and Curved Screen Continuity across the Shadow Boundary: Grazing Incidence Full-Plane 218 ix

5 4.5 Slope Diffraction General Two-Dimensional Edge Diffracted Fields Dielectric and Impedance Wedges 227 Problems 228 References 231 Chapter 5 Applications of Two-Dimensional Wedge Diffraction Radiation from a Parallel Plate Waveguide with TEM Mode Propagation, Terminated in an Infinite Ground Plane Antenna Gain Radiation from an -Plane Horn Antenna Radiation from an //-Plane Horn Antenna Radar Width of a Two-Dimensional Structure 248 Problems 257 References 260 Chapter 6 Three-Dimensional Wedge Diffraction and Corner Diffraction Introduction Edge-Fixed Coordinate System Three-Dimensional UTD Diffraction Coefficients Examples of Three-Dimensional Wedge Diffraction Corner Diffraction Corner Diffraction from a Flat Plate Corner Diffraction from a Vertex in Which Wedges with Arbitrary Wedge Angles Are Terminated Alternative Forms of the Diffraction Coefficients 300 Problems 301 References 304 Chapter 7 Equivalent Currents Introduction Equivalent Currents for Edge Diffraction Radiation From Equivalent Currents Reflected Fields Using Equivalent Currents 322 Problems 327 References 328 Chapter 8 Diffraction at a Smooth Convex Conducting Surface The Phenomenon of Creeping Waves, or Curved Surface Diffraction Introduction Asymptotic Evaluation of Eigenfunction Solutions for Line Source Illumination of a Conducting Circular Cylinder 332

6 xi Interpretation of the Asymptotic Solution in Terms of Surface Rays Invocation of Locality and the Generalized Fermat Principle The Significance of the UTD Results for Diffraction by Smooth Convex Surfaces Problem Classes for Curved-Surface Diffraction Differential Geometry for 2D Curved-Surface Diffraction The Two-Dimensional Scattering Formulation The Scattering Problem Geometry UTD Scattering Solution in the Lit Region UTD Scattering Solution in the Shadow Region Field Continuity at the SSB UTD Scattering Solution in the Surface-Based Ray Coordinate System The Radiation Problem for a Source Mounted on a Smooth Convex Conducting Surface The Radiation Problem Geometry Sources of the Radiated Fields UTD Solution for the Radiation Problem: Observation Point in the Lit Zone UTD Solution for the Radiation Problem: Observation Point in the Shadow Zone Noninfinitesimal Sources Deep Shadow Zone Field Expressions and Their Interpretation The Two-Dimensional Convex Conducting Surface Coupling Problem Detailed Geometry for the Coupling Problem Preliminaries UTD Coupling Solution for Magnetic Current Sources UTD Coupling Solution for Electric Current Sources Special Geometries A Form of the Coupling Solution in the Deep Shadow Region and Its Interpretation Bibliographic Remarks 408 Problems 409 References 410 Appendix A Unit Vectors 413 A.I Cartesian Coordinate System 413 A. 2 Spherical Coordinate System 413 A.3 Cylindrical Coordinate System 414

7 xii Appendix B Special Functions for the Uniform Geometrical Theory of Diffraction 417 B.I Introduction 417 B.2 The Fresnel Integrals and Transition Function 418 B.3 Bessel and Hankel Functions 420 B.4 The Airy Functions 422 B.5 The Fock Scattering Functions 424 B.6 The Fock Radiation Functions 428 B.7 The Fock Coupling Functions 430 B.8 Concluding Remarks 432 References 432 Appendix C Differential Geometry 435 C.I Curves 435 C.2 Surfaces ' 440 C.2.1 Unit Vector Normal to a Surface 440 C.2.2 Radius of Curvature of a Surface ; 442 References 449 Appendix D The Method of Stationary Phase 451 D.I Introduction 451 D.2 The Method of Stationary Phase 452 D.3 Bibliographical Remarks 456 References 457 Appendix E Additional References 459 Appendix F Computer Subroutine Listings 465 F.I Fresnel Integrals 465 F.2 Transition Function 465 F.3 Wedge-Diffraction Coefficient 466 F.4 Wedge-Slope-Diffraction Coefficient 466 F.5 Fock Scattering Functions 467 F.6 Universal Fock Radiation Functions 467 F.7 Fock Coupling Functions 467 References 467 Index 469