The Analytical Foundations of Loop Antennas and Nano-Scaled Rings

Preface

Acknowledgements

Contents

List of Figures

Part I Preliminaries

1 General Introduction

1.1 A Motivation for the Study of Loops and Rings as Radiating Structures

1.2 The Experimental History of Loops

1.3 The Analytical History of Loops

1.4 The Recent History of Nano-Scaled Rings

References

2 Foundations

2.1 The Geometry of the Closed Toroidal Ring

2.1.1 Measurements in Different Coordinate Systems

2.1.2 Coordinate Systems In Detail

2.1.3 A Measure of Thickness of the Toroidal Ring: ?

2.1.4 A Measure of Frequency and Wavelength Related to Loop Geometry: kb

2.1.5 The Distance Between Points Within and on the Toroid

2.2 Useful Expressions of Maxwell's Equations for Toroidal Rings

2.2.1 Materials Linear in Maxwell's Equations

2.2.2 Linear Wave Equations

2.3 Propagation Parameters

2.3.1 Material Impedance and Absorption

2.3.2 Frequency Dependence of Material Parameters

2.3.3 Characteristics of Metals and Dielectrics From RF to Optical Wavelengths

2.4 Vector and Scalar Potentials

2.5 The Governing Equation of the Loop Antenna and Nano-Scaled Ring

2.5.1 The General Equation

2.5.2 The Governing Equation for Perfectly Conducting (PEC) Metals

2.5.3 The Governing Equation for Thin-Wire, PEC Loops and Rings

2.5.4 The Governing Equation for Thick-Wire, PEC Loops and Rings

2.6 Coupling a Driving Source to the Loop or Ring

2.6.1 Direct Coupling

2.6.2 Inductive Coupling

2.6.3 Illuminated Coupling

References

Part II Standalone Loop Antennas and Rings

3 Thin-Wire Perfectly Conducting Loops and Rings

3.1 The Early History

3.2 The Governing Equation and Solutions

3.3 Determining the Coefficients, an and Kn

3.3.1 Storer's Recursive Solution

3.3.2 Storer's Non-recursive Solution

3.3.3 Wu's Solution

3.4 An Elliptical Solution

3.5 Determining Non-solvability and Convergence

3.6 Summary of Solutions to the Thin-Wire PEC Loop

3.6.1 The Symmetry of the Closed Loop Coefficients, In, Around Mode n=0

References

4 The Driving Point Impedance and Admittance of Thin, PEC Loops and Rings

4.1 Formation of the Input Impedance

4.2 The Circuit Element Representation of the Loop

4.2.1 The Closed Loop as an R, L and C Circuit at Any kb

4.2.2 Difficulties with the Series Resonant Model of the Loop

4.3 The Subwavelength Anti-resonance

References

5 Current Distribution and Radiation Characteristics of Thin, PEC Loops and Rings

5.1 Current Characteristics

5.2 Characterizing Radiation of the PEC Loop

5.2.1 Radiating "0245E Field Patterns, Near and Far

5.2.2 Radiated Power, Radiation Intensity, Directivity and Gain

5.3 Characterising the Large Closed Loop

5.4 Characterising the Small Closed Loop

References

6 Lossy Thin Loops and Rings

6.1 The Effect of Surface Impedance on Loop Characteristics

6.1.1 The Functional Dependencies of the Surface Impedance

6.1.2 Modelling the Index of Refraction

6.2 The Driving Point Impedance and Admittance

6.3 The RLC Model for Lossy Metals

6.3.1 The Total R, L and C of the Lossy Metal Loop at Any kb

6.4 Resonance Saturation of Rings in the Optical Region

6.5 Radiation and Radiation Losses in the Thin-Wire Loop

References

7 Lossy Thin Loops and Rings with Multiple Impedance Loads

7.1 The Usefulness of Loop Antennas with Multiple Loads

7.2 Current in Multiply Loaded Loops

7.2.1 A Simplification

7.2.2 Current Coefficients

7.2.3 Asymmetries Due to Loads

7.3 The Input Impedance and Circuit Representation of Multiply Loaded Loops

7.4 Radiation from Multiply Loaded Loops

7.5 The Single Capacitor Loop

7.5.1 Constant Value Capacitor

7.5.2 Tuning by Varying the Capacitance Value, l?

7.6 The Effects of Multiple Capacitors

References

8 Thick PEC Rings

8.1 Thick Rings in Use in the Short-Wavelength Regions

8.2 The Governing Equations

8.2.1 Perfectly Conducting, Thick Ring Equations Applicable at Low Frequency to 150 GHz

8.2.2 Solutions to the Perfectly Conducting, Thick Ring Equations Applicable to 150 GHz

References

Part III Coupled Rings in One, Two and Three Dimensions

9 Meta-atoms & Rings as Large-Scale Atoms

9.1 Introduction

9.2 Plane Wave Illumination of Meta-atom Rings

9.3 Gap Capacitance

9.3.1 Standard Models

9.3.2 Testing the Models

References

10 Coupled Loops and Rings

10.1 Coupled Rings as Meta-atoms

10.2 Near-Field Coupling of Thin Rings

10.3 Optimization Procedure for the Design of Coupled Nano-Loop Antennas

References

A Bessel Functions

A.1 Bessel of the First Kind

A.2 Modified Bessel of the First and Second Kind

A.3 Lommel-Weber Function

A.4 Combinations

References