Polarization in Electromagnetic Systems [2 ed.] 1630811076, 9781630811075

Table of contents :
Polarization in Electromagnetic Systems,
Second Edition
Contents
Preface
1
Introduction
1.1 Polarization Basics and a Brief History of Polarization
1.2 Overview of the Book
References
2
Wave Polarization Principles
2.1 Introduction
2.2 Plane Waves
2.3 Concept and Visualization of Polarized Waves
2.4 Quantifying Polarization States
2.5 Decompostion of Waves
2.6 Problems
References
3
Polarization State Representations
3.1 Introduction
3.2 The Polarization Ellipse
3.3 The Poincaré Sphere
3.4 The Polarization Vector
3.5 Stokes Parameters
3.6 Polarization Ratio
3.6.1 Polarization Ratio for Linear Polarization
3.6.2 Polarization Ratio for Circular Polarization
3.7 Examples
3.8 Determination of Orthogonal Polarization States
3.8.1 Orthogonal State for the Polarization Ellipse Using ε, τ
3.8.2 Orthogonal State for the Polarization Ellipse Using γ, δ
3.8.3 Orthogonal States on the Poincaré Sphere
3.8.4 Orthogonal Polarization Vector
3.8.5 Stokes Parameters for an Orthogonal State
3.8.6 Polarization Ratio for an Orthogonal State
3.9 Problems
References
4
Partially Polarized Waves
4.1 Unpolarized Waves
4.2 Partially Polarized Waves and Degree of Polarization
4.3 Stokes Parameters Representation for Partially Polarized Waves
4.4 Other Representations for Partially Polarized Waves
4.5 Problems
References
5
Antenna Polarization
5.1 Antenna Basics
5.2 Antenna Polarization Principles
5.2.1 Antenna Pattern Types
5.2.2 Antenna Copolarization and Cross Polarization
5.3 Omnidirectional Antennas
5.3.1 Linearly Polarized Omnidirectional Antennas
5.3.2 Circularly Polarized Omnidirectional Antennas
5.4 Directional Antennas
5.4.1 Linearly Polarized Directional Antennas
5.4.2 Circularly Polarized Directional Antennas
5.5 Broadband Antennas
5.5.1 Linearly Polarized Broadband Antennas
5.5.2 Circularly Polarized Broadband Antennas
5.6 Polarization Purity of Circularly Polarized Antennas
5.7 Problems
References
6
Antenna-Wave Interaction
6.1 Polarization Efficiency
6.2 Calculation of Polarization Efficiency
6.2.1 Polarization Efficiency Evaluation Using the Poincaré Sphere
6.2.2 Polarization Efficiency Evaluation Using Stokes Parameters
6.2.3 Polarization Efficiency Evaluation Using Polarization Ellipse Quantities
6.2.4 Polarization Efficiency Using Axial Ratios
6.2.5 Polarization Efficiency Expressed Using Polarization Ratios
6.2.6 Polarization Efficiency Expressed Using Polarization Vectors
6.2.7 Decomposition of Polarization Efficiency into Unpolarized and Completely Polarized Parts
6.2.8 Decomposition of Polarization Efficiency into Copolarized and Cross-Polarized Parts
6.3 Vector Effective Length of an Antenna
6.4 Normalized Complex Antenna Output Voltage
6.5 Problems
References
7
Dual-Polarized Systems
7.1 Introduction to Dual-Polarized Systems
7.2 Cross-Polarization Ratio
7.3 Cross-Polarization Discrimination and Cross-Polarization Isolation
7.3.1 Definitions
7.3.2 Dual Decomposition
7.3.3 Calculating XPD
7.3.3.1 Ideal Dual Linearly Polarized Receiving Antenna
7.3.3.2 General Dual Polarization
7.3.3.3 Near-Dual Circular Polarization
7.4 Performance Evaluation of Dual-Polarized Systems
7.4.1 Isolation Degradation Caused by Imperfect Antennas
7.4.2 Calculation of Isolation in Systems
7.5 Polarization Control Devices
7.5.1 Polarizers
7.5.2 Orthomode Transducers
7.5.3 Polarization Grids
7.6 Problems
References
8
Depolarizing Media and System Applications
8.1 Introduction
8.2 Principles of Depolarizing Media
8.3 Depolarization at Interfaces
8.3.1 General Formulation of Interface Polarization Effects
8.3.2 Reflection from a Plane, Perfect Conductor
8.3.3 Reflection from the Ground
8.4 Dual-Polarized Communication Systems with a Depolarizing Medium in the Path
8.4.1 General Formulation for a Depolarizing Medium
8.4.2 Rain on a Radio Path
8.4.3 Inclusion of Antenna Effects in System Calculations
8.4.4 Depolarization Caused by Faraday Rotation
8.5 Depolarization Compensation and Adaptive Systems
8.6 Polarization in Radar
8.6.1 Radar Basics
8.6.2 Polarimetric Radar
8.6.2.1 Point Radar Targets
8.6.2.2 Distributed Radar Targets
8.7 Polariztion in Radiometry
8.7.1 Radiometer Basics
8.7.2 Radiometer Applications
8.8 Problems
References
9
Polarization in Wireless Communication Systems Including Polarization Diversity
9.1 Introduction
9.2 Polarization in Wireless Communications: System Principles
9.2.1 Overview
9.2.2 Single-Polarized Systems
9.2.3 Advantages of Using Circular Polarization
9.2.4 Dual-Polarized Systems
9.3 Diversity in Wireless Communications
9.3.1 Diversity Principles
9.3.2 Diversity Types
9.2.3 Diversity Combining
9.4 Polarization Diversity at Base Stations
9.5 Polarization Diversity at Terminals
9.6 Performance Comparison of Polarization Diversity to Other Diversity Types
9.6.1 Comparison of Base Station Diversity Techniques
9.6.2 Comparison of Terminal Diversity Techniques
9.7 Future Directions in Polarization Applications to Wireless
9.8 Chapter Summary
9.9 Problem
References
10
Polarization Measurements
10.1 Introduction to Polarization Measurements
10.2 Antenna Pattern Measurement Principles Including Polarization
10.2.1 Pattern Measurement Techniques
10.2.2 Copolarized and Cross-Polarized Radiation Patterns
10.2.3 Polarization Pattern Measurement
10.2.4 The Spinning Linear and Dual-Linear Pattern Methods
10.3 Complete Polarization State Measurement
10.3.1 Phase-Amplitude Method
10.3.2 Multiple Amplitude-Component Methods
10.3.3 Measurement of the Polarization of Large Antennas
10.4 Measurement of Partially Polarized Waves
10.5 Gain Measurement
10.5.1 Gain Measurement of Linearly Polarized Antennas
10.5.2 Gain Measurement of Circularly Polarized Antennas
10.5.3 Absolute Gain Measurement
10.6 Measurements on Handsets and Other Small Devices
10.7 Problems
References
Appendix A Frequency Bands
A.1 Radio Bands
A.2 Microwave Bands
Appendix B
Useful Mathematical Relations
B.1 Unit Vector Representations
B.2 Trigonometric Relations
List of Symbols
About the Author
Index

Citation preview

Polarization in Electromagnetic Systems Second Edition

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For a listing of recent titles in the Artech House Antennas and Electromagnetics Analysis Library, turn to the back of this book.

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Polarization in Electromagnetic Systems Second Edition

Warren L. Stutzman

artechhouse.com

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Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the U.S. Library of Congress British Library Cataloguing in Publication Data A catalog record for this book is available from the British Library. ISBN-13: 978-1-63081-107-5 Cover design by John Gomes © 2018 Artech House 685 Canton St. Norwood, MA All rights reserved. Printed and bound in the United States of America. No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without permission in writing from the publisher. All terms mentioned in this book that are known to be trademarks or service marks have been appropriately capitalized. Artech House cannot attest to the accuracy of this information. Use of a term in this book should not be regarded as affecting the validity of any trademark or service mark. 10 9 8 7 6 5 4 3 2 1

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I cannot thank my wife, Claudia, enough for providing endless emotional support. She supported me even when I was generally missing in action at home because of being immersed in writing the manuscript. She also shared some of the pain when I was stressed out searching for a missing square root of two.

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Contents Preface

xv

Acknowledgments Part I

xix

Polarization Fundamentals

1

Introduction

1.1

Polarization Basics and a Brief History of Polarization Overview of the Book References

3 6 10

Wave Polarization Principles

11

1.2

2

3

2.1 Introduction 2.2 Plane Waves Concept and Visualization of Polarized Waves 2.3 2.4 Quantifying Polarization States 2.5 Decompostion of Waves

11 11 17 27 32

vii

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viii

Polarization in Electromagnetic Systems

2.6 Problems References 3

Polarization State Representations

38 38 41

3.1 Introduction 41 3.2 The Polarization Ellipse 43 The Poincaré Sphere 48 3.3 The Polarization Vector 50 3.4 3.5 Stokes Parameters 55 3.6 Polarization Ratio 59 3.6.1 Polarization Ratio for Linear Polarization 59 3.6.2 Polarization Ratio for Circular Polarization 64 3.7 Polarization State Representation Examples 66 Determination of Orthogonal Polarization States 69 3.8 3.8.1 Orthogonal State for the Polarization Ellipse Using ε , τ 71 3.8.2 Orthogonal State for the Polarization Ellipse Using γ , δ 73 3.8.3 Orthogonal States on the Poincaré Sphere 75 3.8.4 Orthogonal Polarization Vector 75 3.8.5 Stokes Parameters for an Orthogonal State 78 3.8.6 Polarization Ratio for an Orthogonal State 79 3.9 Problems 80 References 82 4

Partially Polarized Waves

83

4.1 4.2

Unpolarized Waves Partially Polarized Waves and Degree of Polarization Stokes Parameters Representation for Partially Polarized Waves Other Representations for Partially Polarized Waves

83

4.3 4.4

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86 87 91

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Contentsix



4.5 Problems References Part II

System Applications

5

Antenna Polarization

93 93

97

Antenna Basics 97 Antenna Polarization Principles 101 5.2.1 Antenna Pattern Types 102 5.2.2 Antenna Co-polarization and CrossPolarization 104 Omnidirectional Antennas 109 5.3 5.3.1 Linearly Polarized Omnidirectional Antennas 109 5.3.2 Circularly Polarized Omnidirectional Antennas 112 5.4 Directional Antennas 112 5.4.1 Linearly Polarized Directional Antennas 112 5.4.2 Circularly Polarized Directional Antennas 118 5.5 Broadband Antennas 123 5.5.1 Linearly Polarized Broadband Antennas 123 5.5.2 Circularly Polarized Broadband Antennas 125 5.6 Polarization Purity of Circularly Polarized Antennas 125 5.7 Problems 128 References 128

5.1 5.2

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6

Antenna-Wave Interaction

131

6.1 6.2

Polarization Efficiency Calculation of Polarization Efficiency 6.2.1 Polarization Efficiency Evaluation Using the Poincaré Sphere 6.2.2 Polarization Efficiency Evaluation Using Stokes Parameters

131 136 137 140

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x

Polarization in Electromagnetic Systems

6.2.3

Polarization Efficiency Evaluation Using Polarization Ellipse Quantities 140 6.2.4 Polarization Efficiency Using Axial Ratios 143 6.2.5 Polarization Efficiency Expressed Using Polarization Ratios 149 6.2.6 Polarization Efficiency Expressed Using Polarization Vectors 150 6.2.7 Decomposition of Polarization Efficiency into Unpolarized and Completely 152 Polarized Parts 6.2.8 Decomposition of Polarization Efficiency into Co-polarized and Cross-Polarized Parts 154 6.3 Vector Effective Length of an Antenna 155 6.4 Normalized Complex Antenna Output Voltage 161 6.5 Problems 163 References 166 7

Dual-Polarized Systems

7.1 7.2 7.3

Introduction to Dual-Polarized Systems 167 Cross-Polarization Ratio 170 Cross-Polarization Discrimination and CrossPolarization Isolation 176 7.3.1 Definitions 176 7.3.2 Dual Decomposition 180 7.3.3 Calculating XPD 182 7.3.3.1 Ideal Dual-Linearly Polarized Receiving Antenna 183 184 7.3.3.2 General Dual Polarization 7.3.3.3 Near Dual-Circular Polarization 186 Performance Evaluation of Dual-Polarized Systems 192 7.4.1 Isolation Degradation Caused by Imperfect Antennas 193 193 7.4.2 Calculation of Isolation in Systems

7.4

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167

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Contentsxi



7.5

7.6

8

Polarization Control Devices 7.5.1 Polarizers 7.5.2 Orthomode Transducers 7.5.3 Polarization Grids Problems References

196 196 199 199 200 201

Depolarizing Media and System Applications 203

8.1 Introduction 203 8.2 Principles of Depolarizing Media 203 Depolarization at Interfaces 211 8.3 8.3.1 General Formulation of Interface 211 Polarization Effects 8.3.2 Reflection from a Plane, Perfect Conductor 219 8.3.3 Reflection from the Ground 220 Dual-Polarized Communication Systems with a 8.4 Depolarizing Medium in the Path 226 8.4.1 General Formulation for a Depolarizing Medium 226 8.4.2 Rain on a Radio Path 230 8.4.3 Inclusion of Antenna Effects in System Calculations 237 8.4.4 Depolarization Caused by Faraday Rotation 242 8.5 Depolarization Compensation and Adaptive Systems 245 8.6 Polarization in Radar 249 8.6.1 Radar Basics 249 251 8.6.2 Polarimetric Radar 8.6.2.1 Point Radar Targets 252 256 8.6.2.2 Distributed Radar Targets 8.7 Polariztion in Radiometry 257 8.7.1 Radiometer Basics 257 259 8.7.2 Radiometer Applications 8.8 Problems 259 References 260

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xii

Polarization in Electromagnetic Systems

9

Polarization in Wireless Communication Systems Including Polarization Diversity

265

9.1 Introduction 265 9.2 Polarization in Wireless Communications: System 266 Principles 9.2.1 Overview 266 9.2.2 Single-Polarized Systems 267 9.2.3 Advantages of Using Circular Polarization 268 270 9.2.4 Dual-Polarized Systems 9.3 Diversity in Wireless Communications 270 270 9.3.1 Diversity Principles 9.3.2 Diversity Types 272 274 9.3.3 Diversity Combining 9.4 Polarization Diversity at Base Stations 274 9.5 Polarization Diversity at Terminals 276 9.6 Performance Comparison of Polarization Diversity to Other Diversity Types 277 9.6.1 Comparison of Base Station Diversity Techniques 278 9.6.2 Comparison of Terminal Diversity Techniques 281 Future Directions in Polarization Applications to 9.7 Wireless 282 Chapter Summary 284 9.8 9.9 Problem 284 References 284

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10

Polarization Measurements

287

10.1 10.2

Introduction to Polarization Measurements Antenna Pattern Measurement Principles Including Polarization 10.2.1 Pattern Measurement Techniques 10.2.2 Co-polarized and Cross-Polarized Radiation Patterns 10.2.3 Polarization Pattern Measurement 10.2.4 The Spinning Linear and Dual-Linear Pattern Methods

287 289 289 293 295 297

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10.3

Complete Polarization State Measurement 299 10.3.1 Amplitude-Phase Method 300 10.3.2 Multiple Amplitude Component Methods 304 10.3.3 Measurement of the Polarization of Large Antennas 307 10.4 Measurement of Partially Polarized Waves 309 10.5 Antenna Gain Measurement 312 10.5.1 Gain Measurement of Linearly Polarized Antennas 315 10.5.2 Gain Measurement of Circularly Polarized Antennas 315 317 10.5.3 Absolute Gain Measurement 10.6 Measurements on Handsets and Other Small Devices 318 10.7 Problems 320 References 322 Appendix A: Frequency Bands

325

A.1 A.2

325 325

Radio Bands Microwave Bands

Appendix B: Useful Mathematical Relations

327

B.1 B.2

Unit Vector Representations Trigonometric Relations

327 327

List of Symbols

331

About the Author

337

Index 339

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Preface Polarization is the fourth dimension of electromagnetic waves, with the other three being frequency, direction of propagation, and intensity. Of the four dimensions in electromagnetic systems, polarization is often the most misunderstood and neglected. If polarization is not properly included in the design phase of systems, performance can be seriously compromised or the system can even fail. On the other hand, polarization features can be exploited in communication systems to improve reliability and increase capacity. In sensing applications, multiple polarizations are used to increase information about targets and scenes. This book had its beginnings with a technical report published by the author in 1977 and submitted to the research sponsor, NASA. The associated research was on Earth-space propagation in the 10 to 30 GHz frequency range. At these frequencies rain along the propagation path can cause severe attenuation and depolarization and thus the frequencies were not used at the time. Virginia Tech was among the first few organizations in the world to investigate the effects with the goal of understanding propagation impairments and of learning how to configure systems to use dual polarization to double satellite communications capacity. The depolarization effects of rain on communication links were investigated through extensive measurement programs, including a terrestrial link and satellite links using the ATS-6, CTS, xv

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xvi

Polarization in Electromagnetic Systems

COMSTAR, SIRIO, INTELSAT, and OLYMPUS satellites. The mathematical representations of polarization and the models to predict weather induced impairments were developed and verified with measured data. The models were used by government and industry to design systems that implement dual polarization on satellite links, which are now common in today’s operational systems. Colleagues and engineers in industry who used the report encouraged me to turn the report into this book. This second edition is an expanded and reorganized version of the first edition, which was published in 1993. It now has two parts. Part I covers the fundamental theory and mathematical formulations. Part II applies the fundamentals to application areas such as antenna polarization, antenna-wave interaction, dual-polarized systems, and depolarizing media. New to the second edition is specific information on applications to: wireless communications, adaptive systems, radar, and radiometry. Chapter 9 on wireless communication systems is entirely new. Chapter 10 is a greatly expanded treatment of measurement considerations for polarization. The book should be useful to the practicing engineers involved with antennas, propagation, communications, radar, or radiometry. It is organized for rapid understanding of the principles and for easily locating material needed in computations. At the same time, the book is suitable for use in the classroom, especially as a part of radio systems-oriented courseware that is intended to study the complete propagation channel. Several examples with full computations are included to reinforce the important quantative concepts. Many end-of-chapter problems have been included for self-study enrichment and for classroom use. It is assumed that the reader has some knowledge of electromagnetics, but he/she need not have a complete understanding of electromagnetic theory. Maxwell’s equations will not be presented or solved. Instead, Chapter 2 presents the needed formulas for wave polarization calculations. Emphasis is always on understanding the concepts and mathematics needed for system calculations. Chapter 3 presents the polarization state representations. The techniques for treating partially polarized waves are presented in Chapter 4. Chapter 5 starts Part II. It treats antenna polarization (greatly expanded from the first edition) with many examples of specific antennas that can be used for generating linear, circular, and dual polarization. The important problem of computing the power received by an arbitrarily polarized wave incident on a receiving antenna is presented in Chapter 6. Dual-polarized radio systems are presented Chapter 7, including system evaluation methods and hardware components needed to implement dual polarization. Chapter 8

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Contentsxvii

covers all aspects of medium effects on polarized waves, including propagation through a medium and reflections from media. Applications to communications, radar, and radiometry are included. Chapter 9 treats many topics in wireless communications related to polarization, such as frequency reuse with dual polarization and polarization diversity. Chapter 10 presents principles and techniques for measuring the polarization of waves and antennas.

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Acknowledgments I would like to thank the many people with whom I collaborated at Virginia Tech during research programs in propagation through rain on millimeter-wave radio links and in reflector antenna design. Graduate students and faculty colleagues contributed much to this book. Fellow faculty (Charles Bostian, Tim Pratt, and Gary Brown) provided many valuable suggestions. I also extend gratitude to Hal Schrank for encouraging me to write this book and for generously sharing his notes on polarization with me. Many graduate students performed research in areas related to this book and material from their labor is evident herein. In particular, I give special mention to Randy Persinger, Bill Overstreet, Steve Lane, Keith Dishman, Don Runyon, Kerry Yon, and Koichiro Takamizawa. For the second edition, Neill Kefauver provided valuable input on near field ranges.

xix

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Part I: Polarization Fundamentals

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1 Introduction 1.1  Polarization Basics and a Brief History of Polarization Electromagnetic waves have the following characteristics: 1. Frequency of the oscillation of the wave; 2. Direction of propagation; 3. Intensity (i.e., strength) of the wave; 4. Polarization. The first three parameters are common to any type of wave. But the fourth characteristic, polarization, is unique to electromagnetic waves. For example, acoustic waves are not polarized. Polarization in electromagnetic systems is the topic of this book. Far from being just a curiosity of physics, polarization is of significant practical importance. However, polarization is often not well understood by engineers, and this misunderstanding can lead to less than optimum system performance, and in some cases even complete system failure. In addition, polarization can be exploited. For example, communication capacity can be doubled by using orthogonal polarizations. The goal of this book is to understand polarization concepts and to provide useful tools to improve the design of electromagnetic systems. 3

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4

Polarization in Electromagnetic Systems

Including polarization consideration into system design has several benefits, such as improved system performance or increased capacity in a communication system. An example of performance improvement in a communication system is the use of polarization diversity to combat multipath fading on a non-line-of-sight link. The performance of a line-of-sight communication link is increased by using orthogonally polarized channels on the same frequency and path. Information carrying capacity in theory can be doubled, and is realized in practice as well. A second application area is remote sensing. The use of multiple polarizations in a remote sensing system increases the amount of information that can be collected about the target/image compared to a single-polarized system. Polarization, sometimes spelled polarisation in the literature, was defined by Etienne-Louis Malus in 1808 for explaining optical refraction in crystals [1]. Malus used optical experiments to investigate polarization effects associated with refraction and reflection. A mechanical analogy is a wave on a rope. Consider a rope passing through a picket fence. Suppose vertical motion waves are generated on the rope (i.e., vertically polarized waves). The wave can pass through a vertical slit in fence pickets, but if the rope oscillations are horizontal the wave will not pass through the pickets. Augustine Jean Fresnel in 1821 proposed that polarization can be decomposed into two mutually orthogonal components, each perpendicular to the direction of propagation [2, p. 15]. James Clerk Maxwell laid the theoretical foundation for electromagnetics in 1864. Maxwell’s equations unified the areas of electricity, magnetism, and optics, previously thought to be separate physical phenomena. The study of optics (i.e., light) and many of its properties was centuries old before the work of Maxwell, who showed that visible light is a part of what we now call the electromagnetic spectrum. Both light and radio waves obey Maxwell’s equations that describe the wave motion associated with electric and magnetic fields. Because the electric field (and magnetic field) of an electromagnetic wave is perpendicular to the wave propagation direction, it is referred to as a transverse wave. In contrast to the transverse nature of electromagnetic waves, sound waves are longitudinal waves and are not polarized. The mechanical analogy of a wave on a rope is again helpful in understanding polarization and wave motion. The actual motion of the rope is vertical (and transverse) whereas the wave motion propagates in the direction of the rope axis. The orientation of the electric field of an electromagnetic wave carries the polarization information. Specifically, polarization is the behavior of the electric field with time. For example, if the electric field of a wave (i.e., the electric field vector) viewed at one point along the wave propagation path oscillates back and forth along a line, the wave is said to be linearly polarized. Mathematically, electromagnetic

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Introduction5

waves are vector waves, whereas acoustic waves are scalar waves (i.e., carry no field orientation information). Polarization implies orientation sensitivity. A simple demonstration can be performed with polarized sunglasses. Using two pairs of sunglasses, hold one lens from each pair back-to-back. As one lens is rotated the transmitted light will diminish, reducing to zero (appearing black) when the polarization states of the lenses are orthogonal. Another simple demonstration is the dim appearance of a digital device display, such as found in an LCD display, when viewed with polarized sunglasses. The dim appearance is caused by the optical display, which is polarized, being cross-polarized to the sunglasses lenses polarization. Similarly, with communication systems, when the receive antenna on a link is polarized identically to that of the transmit antenna, maximum signal will be received (neglecting any degrading propagation effects). But no signal will be received when the receive antenna is orthogonally polarized to the transmit antenna. In 1887–88, in a series of experiments Heinrich Hertz verified Maxwell’s equations in the laboratory by generating, propagating, and detecting electromagnetic waves. In addition, Hertz demonstrated reflection and refraction of electromagnetic waves as well as polarization [3]. Of course, in the twentieth century many applications for electromagnetics evolved and matured. These include electric power transmission, radio, and television. Radar came to prominence in World War II. In the field of communications, satellite communications emerged in the 1960s and optical communications (via lasers and fibers) became common in the 1980s, followed by the current wireless communications revolution. Radiometers are used to passively detect natural electromagnetic emissions; applications for radiometers include radio astronomy and imaging. Electromagnetic waves are also used to process materials. Industrial drying and cooking using a microwave oven are good examples. Natural emissions, like those from the sun, tend to be randomly polarized; that is, the electric field orientation is completely random with time. Polarized sunglasses take advantage of this by using lenses that pass only vertical polarization, thereby reducing glare. Glare is mostly caused by horizontally polarized waves that have significant reflection from horizontal surfaces such as a calm lake. When interacting with media, electromagnetic phenomena are sometimes best understood by postulating particles (or quanta). This conceptualization is key to explaining the photoelectric effect. For our purposes, however, we use only the wave nature of electromagnetics. There is no way to directly observe waves; instead wave effects are observed indirectly. For example, the wavelength of a wave is found experimentally by creating a standing wave,

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6

Polarization in Electromagnetic Systems

moving a probe along the propagation direction, and measuring the distance between successive nulls (which is a half wavelength). Maxwell presented his equations as laws. A physical law cannot be proved, but instead is found to be always consistent with physical observations. The importance of polarization in electromagnetic systems continues to grow. With analog systems, there are often several implementation difficulties when building an electromagnetic system to utilize polarization features, mainly the necessity for expensive and bulky hardware components. However, as digital implementations become more pervasive in electromagnetic systems, the fourth dimension of electromagnetic waves (polarization) will be exploited more fully by using software processing to replace hardware functionality.

1.2  Overview of the Book This book presents principles and techniques that are needed for applications such as communications, radar, and radiometry. Communications (e.g., wireless) receives the most attention due to its importance. Although the material in this book applies to the whole electromagnetic spectrum, applications to optics are not presented in any depth. Part I of the book (Chapters 1 to 4) treats the fundamentals of wave polarization to facilitate the understanding of polarization and to develop analytic formulas that permit easy calculation. Included is a discussion of the physical principles, the mathematical representations of polarization states of completely polarized waves, and the topic of partially polarized waves. Chapters 2 and 3 present the fundamental physical and mathematical descriptions of polarization in electromagnetic systems. The material is not a rigorous physics treatment, but instead provides a basic understanding of polarization. Mathematics is introduced for two purposes. First, derivations are presented for understanding the principles involved. Second, mathematics is presented to the extent necessary for use in calculations to evaluate realworld systems quantitatively. Electromagnetics can be addressed in either the time domain or the frequency domain. We begin with time domain representations of wave polarization in Chapter 2 because it facilitates understanding of polarization via visualization of the space-time behavior of waves. Frequency domain representations, introduced in Chapter 3 through the use of phasors (complex-valued vectors), are usually employed in practice because the resulting mathematics are simplified. Frequency domain formulations are exact for a monochromatic

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Introduction7

wave, which is a single frequency signal, but yield accurate approximate results for narrow bandwidth signals. Several common mathematical representations for polarization states are detailed in Chapter 3. The representations each have appropriate uses either for displaying/visualizing polarization states or for performing calculations in various system applications. A wave that contains a randomly polarized component in addition to a completely polarized portion is referred to as a partially polarized wave. Most light sources are randomly polarized. Antennas generate only completely polarized waves. Natural sources of radio waves, such as radio stars, are partially polarized, containing both completely polarized and randomly polarized parts. Partially polarized waves are treated in Chapter 4. Part II (Chapters 5 to 10) applies the fundamentals of Part I to the following topics: antenna polarization, interaction of a wave with an antenna, dualpolarized systems, depolarization of waves in media, polarization in wireless systems, and measurement of polarization. Figure 1.1 is a simplified diagram of a general electromagnetic system that highlights the topics in Part II. This typical generic system could apply to communications, radar, radiometry, or other sensing and industrial applications. The general electromagnetic system can be divided into three major blocks: 1. Generating subsystem. If the generating subsystem is active, there is a signal transmitter as the source of electromagnetic waves. In a passive system (e.g., a radiometer), generation of electromagnetic waves is by a natural radiation process such as noise. 2. Propagation medium. In general, the medium through which an electromagnetic wave propagates through can alter the polarization state and direction of the wave. An example is a communication signal

Figure 1.1  Overview of a general electromagnetic system.

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8

Polarization in Electromagnetic Systems

passing through rain. In radar, the medium (or target) backscatters a signal that differs in polarization from the transmitted signal. 3. Receiving subsystem. In a communication system, the receiver is distant from the transmitter. In radar the receiver is usually colocated with the signal transmitter. In a radiometric system, there is only a receiver. The polarization of an antenna is simply the polarization of the wave radiated by the antenna when transmitting. In nearly all cases, the polarization of a particular antenna is the same when used either to transmit or to receive. Antenna polarization is presented in Chapter 5, including discussion of many antenna types available to generate various polarizations and radiation pattern shapes. In the most general case, the transmitting and receiving subsystems can be multipolarized (typically dual-polarized). A common design situation is to determine the transmit antenna polarizations that reduce deleterious propagation medium effects and to use them in operational systems. An example is a communication link between an Earth terminal and a satellite at ultrahigh frequency (UHF) frequencies and below, where the Faraday rotation effect rotates a linearly polarized wave as it propagates along the portion of the path through the ionosphere. A second example is a communication link (terrestrial or Earth-space) operating at several gigahertz or higher frequencies when rain along the path depolarizes the wave. In both examples, the wave generated by the transmitting subsystem is altered by the medium along the propagation path. In the absence of a depolarizing medium, the system is designed to match the polarization of the receiving antenna to that of the generated wave to assure maximum received power (assuming all system parameters other than polarization are properly configured). If the medium depolarizes the wave along the propagation path, the fraction of received power is reduced and can go to zero in the extreme case. It is possible in some cases to employ an adaptive system that responds to dynamic changes in polarization and maintain a low level of polarization mismatch. The antenna-wave interaction process is examined in detail in Chapter 6. Two simultaneous signal channels can be operated over a communication link on the same frequency and over the same propagation path using a dual-polarized system (similar to the block diagram in Figure 1.1). There are two design issues. First, the antennas must be of sufficient quality that the dual polarizations are close to purely orthogonal, thus minimizing self-interference between the channels. In other words, the system should have very little cross talk due to cross-polarization between channels. Second, the propagation

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Introduction9

medium should not depolarize to the extent that the medium-generated cross talk becomes unacceptable. Dual-polarized systems are presented in Chapter 7. The propagation medium effects on a polarized wave passing through it are treated in Chapter 8. Maxwell’s equations can be used to describe medium effects, including the anisotropic (direction dependent) properties of matter, but we use the simpler approach of employing system-level formulations for calculations. The following common depolarizing medium situations are discussed: ground plane effects, Faraday rotation, and rain along a path. Propagation path depolarization effects in communications, radar, and radiometry are included. The effects of imperfect antennas are treated. Methods for compensation of propagation medium generated depolarization are also discussed. Finally, how to use polarization information associated with propagation through (or scattering back from) a medium to infer characteristics of the medium is presented. Chapter 9 treats polarization in wireless communications. Dual-polarized systems are employed at base stations and at user terminals to either improve communication capacity or improve performance. Capacity is doubled using a dual-polarized link. Normally clear line-of-sight between the terminals is needed and thus many wireless communication systems such as cellular telephones cannot employ dual polarization for doubling capacity. However, dual polarization is commonly employed in non-line-of-sight links for improving performance (i.e., combating multipath fading); the technique is referred to as polarization diversity. Chapter 9 includes discussion of diversity. Chapter 10 on polarization measurement provides the reader with a greater understanding of: antenna polarization properties, the quantities used to evaluate antenna polarization, and techniques for measuring polarization. It is important to reemphasize that the goal of this book is to provide a thorough understanding of the basic principles of wave polarization, antenna polarization, and the effects of the propagation medium on polarization in systems. Many examples are included to show how to exploit polarization in system design and to illustrate how to perform computations for quantifying system performance. There are only a few books on the topic of electromagnetic wave polarization other than this work. Beckmann [2] is the first such book and remains a classic; it treats the physics and mathematics of polarization with focus on depolarization of waves traveling through media. A book by Mott [3] includes material on polarization in antennas and radars. A second book by Mott [4] has some topics on polarization principles and applications that concentrate on scattering, targets, and radar. References [5, 6] cover polarization of light.

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Polarization in Electromagnetic Systems

In closing this chapter, it is appropriate to comment on notation and terminology. Notation is a very important part of any engineering topic. Judicious symbol choices simplify calculations and understanding. If not done carefully, confusion and ambiguities can arise. The symbols used in this book are found in the List of Symbols at the end of the book for easy reference. The terms used are guided by IEEE standard definitions of terms for antennas [7]. Not all literature sources use the same definitions and certainly not the same symbols, so it is important to be sure how terms and symbols are defined when reading the literature.

References [1]

Pelosi, G., “Ethienne-Louis Malus: The Polarization of Light by Refraction and Reflection is Discovered,” IEEE Ant. and Prop. Magazine, Vol. 51, Aug. 2009, pp. 226–228.

[2]

Beckmann, P., The Depolarization of Electromagnetic Waves, Boulder, CO: Golem Press, 1968.

[3]

Sarkar, T., R. Mailloux, A. Oliner, M. Salazar-Palma, and D. Sengupta, History of Wireless, Hoboken, NJ: Wiley, 2006, pp. 18–19.

[3]

Mott, H., Polarization in Antennas and Radar, John Wiley and Sons, New York, 1986.

[4]

Mott, H., Remote Sensing with Polarimetric Radar, John Wiley and Sons, New York, 2007.

[5]

Born, M., and E. Wolf, Principles of Optics, Pergamon Press (Elsevier), Oxford, 1959; and Cambridge University Press, Cambridge, UK, 1997.

[6]

Collett, E., Field Guide to Polarization, SPIE Press, Bellingham, WA, 2005.

[7]

IEEE Standard Definitions of Terms for Antennas: IEEE Standard 145-2013, IEEE, 38 pp., 2013.

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2 Wave Polarization Principles

2.1  Introduction In this chapter, we present the basic principles of electromagnetic waves with emphasis on polarization. We begin with a review of plane wave theory that is formulated to easily understand the polarized nature of electromagnetic waves. The mathematics used is not difficult and the formulas are easily applied to the electromagnetic systems introduced later in the book. In this and the subsequent chapter, we confine our attention to completely polarized waves. Antennas emit only completely polarized waves. Unpolarized (or randomly polarized) electromagnetic waves also exist. Natural sources, such as a celestial object, emit randomly polarized waves that are essentially noise. In general, an electromagnetic wave consists of both completely and randomly polarized parts and is referred to as a partially polarized wave. Partially polarized waves are discussed in Chapter 4.

2.2 Plane Waves Electromagnetic waves are composed of electric fields and magnetic fields that exist together and are related through Maxwell’s equations. Maxwell’s equations address the dynamic behavior of electric and magnetic fields. Specifically, 11

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Polarization in Electromagnetic Systems

if a time-varying electric field exists, it will be accompanied by a time-varying magnetic field. If a wave is not confined, such as in free-space propagation, traveling-wave motion results and the wave spreads outward from the source. As a free-space wave travels away from its source, it transitions into a plane wave in which the electric and magnetic fields are perpendicular to each other and to the wave direction, with phase of the fields equal over a plane perpendicular to the propagation direction. If the wave is confined, such as in a metallic waveguide, higher-order modes exist and the electric and magnetic fields will have components in the direction of propagation. For an unbounded free-space wave with no materials nearby, the wave is called a transverse wave because the associated electric and magnetic fields are transverse to the direction of propagation of the wave. The wave type is determined by the form of its source. If the source is an infinite line current, the waves are cylindrical waves in which the phase of the wave is constant over an imaginary cylindrical surface whose axis coincides with the line source. If the radiator is a point source, the resulting waves are spherical waves, in which the phase of the wave is constant over any sphere with the source at the origin. In practice, of course, all sources must be finite in extent. Therefore, at large distances from a finite source of any size the radiation is a spherical wave. At sufficiently large distances from a source, the phase front (surface of constant phase) becomes so large that over small regions the phase front is approximately planar independent of the source type. This is referred to as a plane wave. In applications such as communications and sensing, the distance from the source is typically very large and the waves are treated as plane waves. For example, in a communication system, the wave arriving at the receiver from a distant transmitter is a close approximation to a plane wave. A good rule of thumb for knowing whether the wave from a transmitter can be considered to be a plane wave is to use the far-field distance of 2D2/λ where D is transmit antenna diameter and λ is the wavelength of the wave [1, p. 42]. We assume plane wave behavior unless otherwise noted. Polarization is described by time variations of the electric field vector. We begin our discussion by introducing the definitions and principles of plane waves along with their associated electric fields. The mathematical formulations for a plane wave are obtained by solving Maxwell’s equations. The details of Maxwell’s equations and derivations of plane waves are not given here and can be found in any basic electromagnetics textbook under the topic of plane waves. Here we present the results along with explanations of the equations and their physical significance. The direction of propagation of a plane wave will be taken as the +z-direction. Then the electric field vector (or electric field) variation in time and space is

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Wave Polarization Principles13



! E(t,z) = Eo cos(wt − bz)xˆ



[V/m]

+z-propagating wave (2.1)

where

ω = 2π f = radian frequency of the wave [rad/s] f = frequency of the wave [Hz] β = phase constant [rad/m] λ = wavelength [m] The direction of the electric field must be transverse to the propagation direction, which is the z-direction. In this case, the electric field is chosen to be in the x-direction to illustrate wave principles. Later we will see that this is a linearly polarized wave. The associated magnetic field vector (or, magnetic field) is ! E H (t,z) = o cos(wt − bz)yˆ h



[A/m]

+z-propagating wave (2.2)

where

η = intrinsic impedance of the medium = m/e [Ω] ε = medium permittivity [F/m] μ = medium permeability [H/m] For a free space medium, the intrinsic impedance is



ho =

mo 4p ×10−7 = = 377Ω (2.3) eo 10−7 /36p

Notice that the magnetic field expression looks exactly like the electric field expression except that it is y-directed and is reduced by the intrinsic impedance because the electric and magnetic fields of a plane wave are related as

H y (t,z) =

E x (t,z) (2.4) h

This is Ohm’s law for waves, because current per meter equals voltage per meter divided by resistance. This result can be generalized to be independent of coordinates:

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! ! 1 H (t,z) = nˆ × E(t,z) (2.5) h

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14

Polarization in Electromagnetic Systems

where nˆ = unit vector in the direction of propagation. For our example geometry nˆ = ˆz , so the magnetic field direction is yˆ = ˆz × xˆ . The wave motion contained in the expression of (2.1) is revealed through graphic representation in Figure 2.1, which shows the electric field as a function of position for three instants at time in terms of the period of the wave T which is related to frequency as f = 1/T. Shown in the figure is an imaginary electromagnetic surfboard that rides the crest of a wave, which aids in visualizing the motion of invisible electromagnetic waves. In Figure 2.1(a), the crest of the wave is at the origin where the surfboarder starts his or her ride. Figure 2.1(b) shows the electric field a quarter-period later when t = T/4, ω t = 2π f (T/4) = π /2. Figure 2.1(c) shows the electric field at time t = T/2. From this sequence of advancing times we see that the surfboard riding the wave crest is traveling in the +z-direction and thus so is the wave. The analogy to surfing on an ocean wave serves to visualize electromagnetic wave motion, which is similar to other forms of wave motion that we are familiar with. The important point is that the wave crest is a point of constant phase. The velocity (actually its magnitude, which is speed) of the constant phase point is found by setting the phase expression of (2.1) equal to a constant wt − bz = constant (2.6)



Taking the time derivative gives ω − β (dz/dt) = 0, or

v=

w dz = = phase velocity (2.7) b dt

If the wave is progressing in the −z-direction, (2.6) becomes ω t + β z = constant and dz/dt = −ν . Therefore, the electric field associated with the wave traveling in the −z-direction is ! E ( t,z ) = Eo cos(wt + bz)xˆ − z-propagating wave (2.8) We now examine the wave motion in more detail. First, the frequency, f, is the number of oscillations that the electric field makes per second at a fixed point in space. Wavelength, λ , is the distance between successive constant phase points on the waveform. Figure 2.1 shows waves as a function of position for times t = 0, T/4, and T/4. In Figure 2.1(a), wavelength is indicated as the distance between successive positive peaks. The phase shifts 2π radians in one wavelength, so the phase constant (which is the phase shift per unit distance), β , is

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Wave Polarization Principles15

Figure 2.1  Illustration of wave motion for a plane wave traveling in the +z-direction: (a) t = 0, (b) t = T/4, ω t = π /2, (c) t = T/2, ω t = π . Note the electromagnetic surfboard is riding the wave in the +z-direction.



b=

2p l

phase constant

(2.9)

Using this in (2.7) gives a very fundamental relationship for all types of waves: w 2pf = b 2p/l (2.10) n= fl n=

This indicates the well-known result that for a fixed wave velocity, the wavelength and frequency are inversely related; that is, as frequency increases wavelength decreases.

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Polarization in Electromagnetic Systems

For a free-space medium (and dry air), the velocity is

n = c = 3 × 108 m/s

velocity in free space (2.11)

Appendix A gives the frequency band designations for radio and microwave bands. For example, the VHF band begins at 30 MHz where the wavelength is 3 × 108/30 × 106 = 10m. As a second example, the low cellular telephone bands are near 900 MHz where the wavelength is 3 × 108/900 × 106 = 33 cm (about 1 foot). This band has excellent propagation properties and antennas are of convenient size, hence its popularity for many applications such as cellular telephony. Figure 2.2 shows the electric field of a plane wave in three-dimensional space for a fixed instant of time. The wave nature is evident in the plot. The phase front is highlighted with the dashed area labeled phase front. The phase is constant over the phase front and it is planar in shape. The intensity and direction of propagation of a wave can be expressed with a single vector, the Poynting vector: ! ! ! S(t,z) = E(t,z) × H (t,z) (2.12)

Figure 2.2  The electric field of a generalized plane wave shown in three-dimensional space for a fixed instant of time.

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Wave Polarization Principles17

with units of W/m2. The cross product of the electric and magnetic field vectors gives the direction of wave propagation. Substituting the x-directed electric field of (2.1) and the y-directed magnetic field of (2.2) into (2.12) gives the direction of propagation as the z-direction:



! E2 S(t,z) = o cos(wt − bz)ˆz (2.13) h

We have now provided mathematical representations for all four characteristics of an electromagnetic wave mentioned in Section 1.1. The wave frequency f is a specified quantity. The direction and intensity of the wave are contained in the Poynting vector. Finally, polarization information is found in the electric field vector behavior as time progresses, which is the topic of the next section.

2.3  Concept and Visualization of Polarized Waves In the previous section we discussed the principles and mathematics of plane waves. Plane waves are polarized, as are most manmade electromagnetic waves. In its simplest terms the polarization of a wave is a description of the motion of the tip of the instantaneous electric field vector with time (one period is sufficient) at a fixed point in space. In a plane wave, the magnetic field behaves just like the electric field, but it is conventional to base all polarization definitions on the electric field. In this section, we explore the types of polarization. Electromagnetic waves are, of course, invisible, so we will explain how to visualize them and their polarization. Subsequent sections will develop mathematical relations for each polarization type that are useful in system calculations. Figure 2.3 shows a linearly polarized wave. It is linearly polarized because the electric field is parallel to the x-axis. The x-directed electric field and y-directed magnetic field are in-phase, are perpendicular to each other, and are perpendicular to the direction of propagation ( nˆ = ˆz in this case); these properties are expressed in mathematical form in (2.5). Figure 2.4 shows the electric field vector for the linearly polarized wave of (2.1) and Figure 2.3 for z = 0 at progressive instants of time for one half of a period (wave cycle). The electric field vector in Figure 2.4 starts at time t = 0 with maximum extent on the +x-axis. A half cycle later it reaches maximum extent in the −x-direction. After a full cycle (t = T) the vector has returned to its original strength in the +x-direction. This sinusoidal oscillation process continues at a rate (frequency) of f cycles per second. Since the tip of the

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Polarization in Electromagnetic Systems

Figure 2.3  The spatial behavior of the electric and magnetic fields of a plane wave for a fixed instant of time and wave propagation in the z-direction ( nˆ = zˆ ).

electric field vector moves only in one dimension (along the x-axis), the wave is said to be linearly polarized. The orientation of the electric field of a linearly polarized wave can be referenced to the local ground plane, usually Earth’s surface. Thus, we define a horizontally polarized wave as one whose electric field vector oscillates along a line parallel to Earth (i.e., horizontal) as shown in Figure 2.5. Similarly, a vertically polarized wave is one whose electric field oscillates along a line that is vertical (perpendicular to Earth), also shown in Figure 2.5. In general, the polarization of a wave is determined by observing the motion of the electric field vector in a fixed plane perpendicular to the direction

Figure 2.4  The linearly polarized wave of (2.1) and Figure 2.3 at z = 0 for several instants of time.

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Wave Polarization Principles19

Figure 2.5  Horizontally polarized and vertically polarized cases of linear polarization.

of propagation, called the stationary plane of observation or plane of polarization. The plane labeled “phase front” in Figure 2.2 is a plane of polarization. The direction of wave propagation (which is perpendicular to the observation plane) is also called the ray direction. A ray is commonly used in both radio and optics to show the wave propagation direction. The motion of the electric field vector within the plane of polarization as a function of time determines the polarization of the wave. So far our illustrations have only been for a linearly polarized wave. In general, the tip of the electric field vector can trace out an ellipse and the vector can be decomposed into orthogonal linear components (in the plane of polarization), often taken to be horizontal and vertical as shown in Figure 2.5. The relative amplitudes and phases of these components are used to mathematically describe the polarization state. In the remainder of this section, we define and provide visualizations of polarization states. In the next section, we develop the mathematical representations for polarization states. But first we digress slightly to provide a basic discussion of antenna polarization so that the engineering applications of polarization can be appreciated ahead of the detailed treatment in Part II of the book. Figure 2.6 shows a vertical dipole antenna. It is a straight wire antenna fed from a parallel wire transmission line. For discussion purposes it is taken to be much shorter than a wavelength (i.e., electrically small), the so-called short dipole antenna; see Sections 5.4 and 6.3. The current from the transmission line flows out onto the wire, and the electric field created by the current is parallel to the wire. This can be seen most easily by visualizing the current on the upper half of the dipole to supply positive charges and the return current

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Polarization in Electromagnetic Systems

Figure 2.6  A vertical short dipole antenna that produces a linearly polarized wave: (a) coordinate system and field components, (b) the E-plane radiation pattern, and (c) the H-plane radiation pattern. (d) Three-dimensional view of the radiation pattern with a slice removed. (Source for (d): W. Stutzman and G. Thiele, Antenna Theory and Design, Third Ed., John Wiley & Sons, Inc., 2013. Reprinted with permission.)

to be negative charges appearing in the bottom half of the dipole. This condition is responsible for an electric field directed from the upper half to the lower half of the dipole. The sinusoidal oscillating nature of the source current creates an oscillating electric field that leads to a spherical wave propagating away from the dipole (the r-direction in Figure 2.6). See [1, Sec. 2.3] for a

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Wave Polarization Principles21

complete development of the fields of a short dipole. At long distances from the dipole the electric field becomes entirely perpendicular to the direction of propagation, resulting in an electric field as shown in Figure 2.6(a), which is θ -directed. This is a linearly polarized wave because the electric field is generated by the dipole oscillating along a line and the! resulting radiated field also oscillates along a line. The magnetic field vector ! H , as shown in Figure 2.6(a), is perpendicular to the electric field vector E and is ϕ -directed. In the far-field region of an antenna only radiated fields exist and decay with distance as 1/r. The magnitude of the fields (electric or magnetic) in the far-field region as a function of angle around the antenna is called the radiation pattern [1, Sec. 2.4]. The radiation pattern (or simply, the pattern) is threedimensional, but the patterns in principal planes through the antenna provide a good representation. The principal planes are the E-plane and the H-plane. The patterns in these planes are called the E-plane pattern and the H-plane pattern. The E-plane pattern is a plot of the radiation strength as a function of angle in the plane containing the electric field. The E-plane pattern for the short dipole is shown in Figure 2.6(b). This pattern shape is often referred to as a dumbbell pattern. It is maximum in directions perpendicular to the dipole (that is, in the xy-plane). The radiation goes to zero in directions off the end of the wire, forming pattern nulls along the z-axis. The electric field in this plane is parallel to the z-axis, or vertically polarized. The magnetic field vector lies in the xy-plane; therefore, the H-plane for the short dipole is the xy-plane. The H-plane pattern of the short dipole in Figure 2.6(c) is constant with angle. This is called an omnidirectional pattern because of its uniform radiation in one plane. The electric field in this plane is perpendicular to the xy-plane and is parallel to the z-axis, or vertically polarized. Returning to polarization state basics, we first examine the general linear polarization case; that is, the line of polarization can be arbitrarily oriented (but always perpendicular to the direction of propagation). The geometry is shown in Figure 2.7 where the propagation direction is in the + z-direction and the line of polarization is at an angle τ with respect to the x-axis; τ is called the tilt angle. The amplitudes of the components E1 and E2 determine the tilt angle of the linear polarization. For horizontal polarization (electric field along the x-axis) τ = 0 and the vertical component amplitude is zero. Similarly, for vertical polarization (electric field along the y-axis) τ = 90° and the horizontal component amplitude is zero. As another, slightly more complicated example, consider again two orthogonal linear electric field components with amplitudes E1 and E2, but now they are in phase quadrature; that is, Ey leads Ex by 90°. The components and resulting total electric field are shown in Figure 2.8(a) at various points in

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Polarization in Electromagnetic Systems

Figure 2.7  Linear polarization at an arbitrary tilt angle τ . It has in-phase horizontal and vertical components along the x- and y-directions. (a) The electric field at various points in space for a fixed time (t = 0). (b) The electric field behavior with time at the origin (z = 0).

space for a fixed instant of time (t = 0). The time behavior in Figure 2.8(b) is noted by imagining the diagram in Figure 2.8(a) to move along the z-axis (the direction of wave propagation). At t = 0 the electric field is along the x-axis. A quarter period later (ω t = π /2), the electric field is in the −y-direction. The resultant instantaneous vector in the z = 0 plane of Figure 2.8(b) appears to be rotating clockwise as time advances, yielding a left-hand sensed, elliptically polarized wave. This example serves to emphasize the understanding the

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Wave Polarization Principles23

Figure 2.8  Elliptical polarization resulting from orthogonal electric field components that are in phase quadrature (δ = 90°). The sense is left-handed. (a) Orthogonal electric field components at t = 0 for various points in space. (b) Resultant electric field at z = 0 for two instants of time; the wave is approaching.

definition of polarization: it is the motion of the electric field vector over time at a fixed point in space. In the most general case, the horizontal and vertical components can have any amplitude values E1 and E2 and any relative phase δ . δ is the phase by which the y-component leads the x-component. The resultant locus of points of the electric field vector tip is an ellipse; we will derive this in the next section.

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Polarization in Electromagnetic Systems

Figure 2.9 shows how the shape and orientation of the polarization ellipse is controlled by the relative amplitude and phase of the components. In Figure 2.9(a), the x- and y-components are in phase (δ = 0). In Figure 2.9(b), the y-component lags the x-component by 90° (δ = −90°). Figure 2.9(c) shows a

Figure 2.9  Examples illustrating how the relative amplitude and phase of the xand y-components of the electric field control the polarization ellipse shape and orientation. The wave is approaching (in the +z-direction). (a) A linearly polarized wave; δ = 0. (b) A right-hand elliptically polarized wave with major axis horizontal; δ = −90°. (c) A general left-hand elliptically polarized wave; δ > 90°.

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Wave Polarization Principles25

general polarization ellipse. The electric field vector rotates around the ellipse f times per second. The shape of the ellipse is quantified by magnitude of the axial ratio, R, whose magnitude is given by

R =

major axis length minor axis length (2.14)

The sign of R is positive for right-hand sense and negative for left-hand sense, as is discussed below. The orientation of the ellipse is given by the tilt angle τ of the major axis of the ellipse relative to the x-axis. Linear polarization has an axial ratio of infinity because the minor axis is of zero length. The special case when the polarization ellipse is circular for |R| = 1 is called circular polarization (CP); see Figure 2.10. One remaining piece of information is required to completely specify the polarization state of a wave. It is the sense of the polarization, which is determined by how the electric field vector rotates with time in the plane of polarization (the xy-plane in Figure 2.9). In Figure 2.9 the direction of travel of the wave is toward the observer (i.e., out of the paper); some presentations use direction of travel into the paper. If the electric field vector rotates counterclockwise with the wave approaching, the wave is right-hand sensed as in Figure 2.9(b). Similarly, a wave is left-hand sensed if the electric field vector rotates clockwise as in Figure 2.9(c). In Figure 2.9 if the wave is receding (into the paper), right-hand sense will be clockwise and left-hand sense will be counterclockwise. Conventions of graphical presentation differ, so the reader needs to note the convention used. In any case, the handedness of the sense follows the right-hand rule: with the thumb of one’s hand being pointed in the direction of propagation the fingers will curl in the direction of electric field rotation. That is, with the thumb in the direction of propagation, if the fingers of the right, (left) hand curl in the direction of the electric field vector rotation, the sense is right- (left) handed. Figure 2.9(b) is an example of a righthand (RH) sensed wave. The fingers of the right-hand curl in the direction of the electric field vector rotation with the thumb in the z-direction. Figure 2.9(c) is an example of a left-hand (LH) sensed wave. Linear polarization is a special case of elliptical polarization with the ellipse collapsed into a line as in Figure 2.9(a), rendering sense of rotation meaningless. Thus, sense is not stated for linear polarization. The definition we use for polarization sense is the official IEEE definition [2]. However, some classic references that precede the IEEE definition define sense oppositely; one such reference is [3]. When using a literature source, the

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26

Polarization in Electromagnetic Systems

reader should check the reference for the definition of sense used. Confusion over sense has found its way into real operational systems. One prominent historical case was the first live transatlantic transmission of television program material in 1962 using the AT&T Telstar-1 low-Earth orbit satellite [4]. The system was configured for the United States to transmit and to have reception in England and France. On the first pass of the satellite, reception was successful in France but failed in England. The sense of CP transmitted from the United States was opposite of that of the receiving antenna in England. This mismatch of the incoming wave sense to that of the receive antenna CP sense resulted in a weak signal and a blank picture. By the second pass of the satellite, the receiver in England was reconfigured for the correct sense and TV picture came through (Timothy Pratt, private communication). The difficulty on the first satellite pass was confusion over the definition of sense of CP. The United States and France used the IEEE definition and the British followed a definition yielding the opposite implementation of the intended sense. In Figure 2.9, the wave is traveling in the +z-direction and is approaching (i.e., out of the paper). In Figure 2.9(b), the electric field vector rotates counterclockwise and the wave is right-handed. Thus, the electric field crosses the x-axis before it crosses the y-axis as it rotates. Therefore, the phase of the y-component lags the phase of the x-component, or δ < 0 (in this case by 90°, so δ = −90°). In Figure 2.9(c), the electric field vector rotates clockwise and the wave is left-hand sensed. Therefore, Ey leads E x in phase and δ > 0°. Figure 2.10 shows a three-dimensional perspective of a spatial sequence of electric field vectors for a circularly polarized wave at a fixed instant of time (similar to the left-hand elliptical polarization [LHEP] wave in Figure 2.8). Since the vectors are of equal length, the wave is CP. The sense is determined by examining the time progression of the vector pattern in a fixed plane parallel to the xy-plane as the wave travels in the direction of propagation (+z-direction). The electric field vector rotates clockwise, as is also shown in time sequence in Figure 2.10 of t 1, t 2, t 3, t4. This meets the definition of left-hand sense. Thus, the wave is left-hand circularly polarized. It may seem confusing that the helix shown in Figure 2.10 is RH sensed, but the sense definition is based on the rotation of the electric field with time in a plane as shown and this is LH. To summarize the characteristics of a circularly polarized wave, the linear components are spatially orthogonal, equal in amplitude, and in time quadrature. Stated differently, any two orthogonal linear components of a CP wave are equal in amplitude and 90° out of phase. The sense is right-hand for δ = −90° and left-hand for δ = 90°.

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Wave Polarization Principles27

Figure 2.10  Perspective view of a left-hand circularly polarized wave at a fixed instant of time. Also shown in a fixed plane perpendicular to the direction of propagation (+z-direction) is the time sequence of the rotating electric field vector.

2.4  Quantifying Polarization States The basic definitions associated with wave polarization were introduced in the previous section. In this section, we develop the mathematics needed for quantifying polarization and for use in system calculations. The mathematics only involve vectors and complex numbers. There is no standard set of notation (e.g., symbols) used in polarization. In 1949, Kraus [5] provided the first comprehensive engineering treatment of polarization. Kraus’ notation has been widely adopted in the literature and we use it here. It is the time variation of the electric field of a wave that contains the polarization information, so we begin to quantify wave polarization by examining the time variations of the electric field. The instantaneous electric field associated with a general plane wave traveling in the +z-direction can be decomposed into x- and y-components. Following (2.1), these components are written as

6765_Book.indb 27

Ex (t,z) = E1 cos(wt − bz) (2.15a)

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28

Polarization in Electromagnetic Systems

E y (t,z) = E2 cos(wt − bz + d) (2.15b)

where

E1, E2 = amplitudes of the instantaneous electric fields in the x-, y-directions [V/m] ω = 2π f = radian frequency of the wave [rad/s] β = phase constant [rad/m] δ = phase by which the y-component leads the x-component of the electric field [rad] Each component varies in time and space as described in Sections 2.1 and 2.2. For example, spatial variation of Ex is plotted in Figure 2.1(a) for t = 0. The Ey plot is similar except it is shifted in space. Each component essentially is a linearly polarized wave. The resultant electric field is the vector sum of the components at each instant of time and at each point in space: ! E(t,z) = E x (t,z)xˆ + E y (t,z)yˆ (2.16) To examine the motion of the electric field vector with time we take z = 0 for simplicity, but the results are general. Using (2.15) in (2.16) the resultant vector is ! ! E(t) = E(t,z = 0) = E1 coswt xˆ + E2 cos(wt + d)yˆ (2.17) We will prove that the length of this vector as a function of time traces out an ellipse as shown in Figure 2.11, making one revolution each period of oscillation (T = 1/f ). We now reexamine examples in the previous section in light of the mathematical description for the electric field in (2.17). When the phase angle δ is 0: ! E(t) = ( E1xˆ + E2 yˆ ) coswt (2.18) The factor in parentheses forms a fixed straight-line vector. Therefore, this is linear polarization; see Figure 2.7. The cos ω t factor is responsible for the sinusoidal oscillations. The tilt angle, τ , follows from Figure 2.7(b) as



6765_Book.indb 28

t = tan−1

E2 E1

for linear polarization (2.19)

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Wave Polarization Principles29

Figure 2.11  The polarization ellipse, which is the locus of points formed by the tip of ! the instantaneous electric field vector E (t) as it changes with time.

If the phase is not zero, the polarization becomes elliptical. In the case of phase quadrature where δ = 90° the electric vector expression from (2.17) is ! E(t) = E1 coswt xˆ − E2 sinwt yˆ (2.20) At times corresponding to ω t = 0 and π /2: ! ! E(t = 0) = E1xˆ and E(t = T /4) = − E2 yˆ (2.21) These vectors are shown in Figure 2.8(b). This is a left-hand sensed elliptical polarization state because the electric field vector rotates in the direction of the curl of fingers of the left hand (clockwise) with the thumb pointing in the direction of propagation (out of the paper). That is, the vector is along the x-axis at t = 0 and at t = T/4 aligns with the −y-axis. Thus, as time progresses, there is a clockwise rotation, which is left-hand sense. Next we verify that the time behavior of the electric field actually describes an ellipse. Substituting the trigonometric identity of (B.9) in Appendix B into (2.17) gives

E y (t) = E2 (coswt cosd − sinwt sind) (2.22)

Working from the x-component in (2.17) we find



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coswt =

Ex (2.23) E1

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30



Polarization in Electromagnetic Systems

(

sinwt = 1 − cos2 wt = 1 − Ex /E1

)

2

(2.24)

Substituting (2.23) and (2.24) into (2.22) leads to the following result that eliminates time variation: 2



2 ⎛ Ey ⎞ ⎛ Ex ⎞ ⎛ Ex ⎞ ⎛ E y ⎞ 2 (2.25) − 2 cosd + ⎜⎝ E ⎟⎠ ⎜⎝ E ⎟⎠ ⎜⎝ E ⎟⎠ ⎜⎝ E ⎟⎠ = sin d 1 1 2 2

This is the equation of an ellipse when Ex and Ey are treated as x- and y-coordinates in a rectangular graph as in Figure 2.11. Hence, we conclude that the general expression of (2.17) for the electric field associated with a plane wave is a vector whose tip traces out an ellipse with time. The polarization ellipse can have any shape (i.e., axial ratio) and orientation (i.e., tilt angle). The general polarization ellipse is shown in Figure 2.12 along with quantifying parameters. Using the general definition of axial ratio in (2.14) with the geometry of Figure 2.12, axial ratio is expressed as



R =

major axis length Emax OA = = ≥ 1 (2.26a) minor axis length Emin OB R(dB) = 20 log R (2.26b)

The factor of 20 is used in the decibel definition because ⎪R⎪ is a ratio of field quantities, not a ratio of power quantities. For ⎪R⎪ = ∞ the ellipse degenerates into a line (distance OB is zero), representing the special case of linear polarization. The sign of R carries information on the sense of rotation of the electric field around the polarization ellipse and follows the IEEE definition [2]. A plus (minus) on R represents right- (left-) hand sense rotation; more will be said about this topic in Section 3.3. The ellipticity angle ε shown in Figure 2.12 is related to axial ratio through the following relations: e = cot −1(−R) or R = −cot e

− 45° ≤ e ≤ 45°

sign(R) = + for RH and − for LH

(2.27)

As the axial ratio magnitude varies from ∞ (linear polarization) to 1 (circular polarization), ⎪ε ⎪ = cot–1 ⎪R⎪ varies from 0° to 45°. The orientation of the ellipse is given by the tilt angle, τ :

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t = tilt angle

0 ≤ t ≤ 180° (2.28)

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Wave Polarization Principles31

where τ is the angle of the major axis relative to the x-axis as shown in Figure 2.12. The pair of angles (ε , τ ) completely describes the polarization ellipse. As time advances, the tip of the electric field in (2.17) traces out the polarization ellipse in Figure 2.12. The quantities E1, E2, and δ in (2.17) are shown in the ellipse geometry of Figure 2.12. Note from Figure 2.12 that



g = tan−1

E2 E1

0 ≤ g ≤ 90° (2.29)

Here γ is the angle from the x-axis to the diagonal of a box that just encloses the ellipse and with sides parallel to the x- and y-axes. The angle γ gives the relationship of the amplitude components and δ is the phase by which Ey leads Ex. This is sufficient to describe the polarization ellipse. Therefore, (γ , δ ) is another angle pair that completely describes the polarization ellipse. Here we are referring to the shape of the ellipse. The size of the ellipse is determined by the strength of the electric field of the wave, ⎪E⎪. And information about the sense of rotation of the electric field around the ellipse is carried in the following ellipse parameters: R (− for LH; + for RH); δ (+ for LH; − for RH); ε (>0 for LH; EL0), we see from (2.40) that R > 0, indicating right-hand sense. If EL0 = 0, (2.40) yields R = +1, which is a pure RHCP. If ER0 < EL0, R < 0 and the sense is left-hand. Finally, if ER0 = 0, (2.40) yields R = −1, which is LHCP. Table 2.2 Decomposition of Polarization States into Circularly Polarized Components

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Polarization

EL0

ER0

δ′

LHCP

1

0

NA

RHCP

0

1

NA

Linear horizontal (HP)

1

1

  0°

Linear, slant-45°

1

1

 90°

Linear vertical (VP)

1

1

180°

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38

Polarization in Electromagnetic Systems

The IEEE definition for axial ratio is embodied in (2.40), wherein R is positive for the right-hand sense [2]. Caution must be exercised when using results from the literature due to differences in this definition. To maintain consistency with the IEEE definition the negative sign shown in (2.27) for the definition of ε must be included.

2.6  Problems







1. A spherical wave has an electric field expression given by ! E = Eo e − jbr θˆ Use (2.5) to find the corresponding magnetic field expression. 2. Write an expression for the time shift Δt corresponding to the phase shift of δ in the y-component of the electric field of (2.15b). 3. Draw the polarization ellipse for a plane wave with the following electric field parameters: E1, E2 = E1/2, δ = π /2. Use the instantaneous rectangular component expressions for various instants of time at z = 0. What is the axial ratio magnitude? 4. In Figure 2.10, the tip of the electric field vector in the spatial sequence shown for an instant of time has the shape of a helix. What is the handedness of the helix (right or left) shown? Comment on the relationship to the sense of the wave. 5. Verify (2.25). 6. Show that opposite-sense, equal-amplitude, out-of-phase (δ ′ = 180°) circularly polarized wave components yield vertical linear polarization as indicated in Table 2.2. 7. Derive (2.39) for the case of a general linearly polarized wave. (Hint: form Ey(t)/Ex(t) by using both linear and circular polarization decompositions and equate.)

References [1]

Stutzman, W., and G. Thiele, Antenna Theory and Design, Third Edition, Hoboken, NJ: Wiley & Sons, 2013.

[2]

IEEE Standard 145-2013, “IEEE Standard Definitions of Terms for Radio Antennas,” IEEE, 2013.

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Wave Polarization Principles39

[3]

Silver, S., ed., Microwave Antenna Theory and Design, M.I.T. Radiation Laboratory Series, Vol. 12, McGraw-Hill, New York, 1949. p. 91 (available through Institution of Engineering and Technology).

[4]

Pierce, J. R., Almost Everything About Waves, MIT Press, Cambridge, MA, 1974, pp. 130–131.

[5]

Kraus, J. D., Antennas, McGraw-Hill, New York, 1949, pp. 464–475.

[6]

IEEE Standard 149-1979, “IEEE Standard Test Procedures for Antennas,” IEEE, 1979, Sec. 11.1.

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3 Polarization State Representations

3.1  Introduction There are many ways to represent the polarization state, from ones that are graphic in nature and easy to visualize, to ones that are quantitatively based and well suited to calculations. Table 3.1 lists the approaches we present in this chapter together with their advantages. Only those with complete polarization are included; the topic of partial polarization is treated in the next chapter. Since intensity is a wave characteristic and not a polarization parameter, it is not found in the representations in Table 3.1. The usefulness of a representation depends on the specific application. If one is merely trying to visualize various polarization states, either the polarization ellipse representation or the Poincaré sphere representation is a good choice. For calculations associated with a wave interacting with a depolarizing medium, polarization ratio is an excellent choice. The Stokes parameters and polarization vector representations allow easy calculation of the power delivered by a receiving antenna from an incident wave, each of an arbitrary polarization state. The topic of antennawave interaction is presented in Chapter 6. The first two polarization state representations discussed in this chapter make use of the polarization ellipse of Figure 2.12. Both ellipse representations provide clear geometrical display of the polarization parameters. One of two sets of angle pairs is used for a polarization ellipse representation and gives the 41

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42

Polarization in Electromagnetic Systems Table 3.1 Polarization State Representations Representation

Quantities

Advantages

1

Polarization ellipse (Sections 2.4, 3.2)

ε, τ

Angles directly related to ellipse geometry

2

Polarization ellipse (Sections 2.4, 3.2)

γ, δ

Angle γ directly related to ellipse geometry

3

Poincaré sphere (Section 3.3)

Two spherical angles

Points on the surface of a sphere correspond to all possible polarization states

4

Polarization vector (Section 3.4)

cosg xˆ + sing ej d yˆ

Easy to evaluate with a program; Preserves phase in antenna-wave interaction calculations

5

Stokes parameters (Section 3.5)

s1, s2 , s 3 (real values)

Antenna-wave interaction easy to evaluate

6

Polarization ratio (Section 3.6)

ρ L (complex number)

Useful in calculations involving depolarizing media

ellipse shape and orientation relative to the x-axis. The (ε , τ ) representation has direct geometric significance with ε being the ellipticity angle of (2.27) and τ being the tilt angle of the major axis relative to the x-axis. The (γ , δ ) representation uses the geometric angle γ shown in Figure 2.12 and angle δ , the phase by which the y-component of the electric field leads the x-component. The Poincaré sphere displays the full range of all possible polarization states on the surface of a sphere. This is particularly useful for displaying multiple polarizations. The Poincaré sphere, however, is somewhat awkward to use in quantitative work. The polarization vector representation offers mathematical simplicity and lends itself to being easily evaluated quantitatively. The Stokes parameter representation can be used to evaluate antenna-wave interaction in a straight forward fashion. The polarization ratio representation is the most compact of the representations because it uses only a single complex number. It is very useful in problems involving depolarizing media. On the other hand, it suffers from using infinity to represent vertical linear polarization, which could lead to computational issues when used in a computer code. A minimum of two quantities are required to completely specify a polarization state. All of the representations in Table 3.1 require only two

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Polarization State Representations43

parameters except for the normalized Stokes parameters. The normalized Stokes parameters representation requires three parameters and therefore is less compact than the others. If wave intensity is included in polarization state representation, three independent quantities are required to fully characterize a wave state. So far we have only considered completely polarized waves. To represent a partially polarized wave (including its intensity), four independent quantities are required. Partially polarized waves will be discussed in Chapter 4. The polarization ellipse representations were discussed in the previous chapter, but we begin this chapter with more details on the polarization ellipse before proceeding to a discussion of the remaining representations in Table 3.1. This is followed by a treatment of how to determine the polarization state orthogonal to a given polarization state.

3.2  The Polarization Ellipse The fact that the polarization of a plane wave is elliptical was derived mathematically in Section 2.4. This is an extremely general result in spite of the plane wave assumption. Over local regions of interest many waves behave as plane waves. For example, consider a source of radiation such as a radio transmitter. Waves travel away from the source as spherical waves. That is, the electric field amplitude falls off as l/r, where r is the distance from the source and the phase is constant over a sphere of radius r. At long distances, where r is very large compared to a local region of interest (such as the extent of a receiving antenna), the curvature of the phase front is very flat and is well approximated by a plane. And, of course, for many applications, such as communications, long distances are the objective. Then the plane wave assumption is valid and all the polarization principles for elliptical polarization apply. The full range of polarization ellipse shapes, orientations (i.e., tilt angles), and senses are displayed as a rectangular chart in Figure 3.1. The angle τ ranges from 0° to 180°, taking the tilt of the major axis from horizontal to vertical at 90° and back to horizontal at 180°. The ellipticity angle ε varies from RHCP at −45° through linear at 0° to LHCP at +45°. Left-hand sensed elliptical polarization states are on the upper half of the chart, and right-hand sensed states are in the lower half. A similar chart is shown for γ and δ in Figure 3.2. All possible polarization states are bounded by the range of phase angles δ from −180° to +180° and of γ from 0 to 180°. The relationship of angle γ to the rectangular components of the electric field is shown geometrically in Figure 3.3.

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44

Polarization in Electromagnetic Systems

Figure 3.1  The range of possible elliptical polarization states displayed as a rectangular chart with ε and τ as variables.

We showed in Section 2.4 that either one of the pairs of angles (ε , τ ) or (γ , δ ) are sufficient to completely describe the polarization ellipse. The angles (ε , τ ) can be found directly from the angles (γ , δ ) and vice versa using trigonometric relations as we now show. To go from (γ , δ ) to (ε , τ ) we start with the following relations [1, p. 113; 2]:

sin2e = sin2g sind (3.1)



tan2t = tan2g cosd (3.2)

to solve for ε and δ :

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1 e = sin−1(sin2g sind) (3.3) 2

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Polarization State Representations45

Figure 3.2  The range of possible elliptical polarization states displayed as a rectangular chart with γ and δ as variables.

Figure 3.3  The box containing the polarization ellipse; see Figure 2.12. E1 and E2 are the peak electric fields along the x- and y-axes.

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46



Polarization in Electromagnetic Systems

1 t = tan−1 2

⎡ sin2g cosd ⎤ (3.4) ⎢⎣ cos2g ⎥⎦

There are some special cases of note. When γ = 0° (3.3) yields ε = 0°, which indicates linear polarization. Also, when γ = 0° (3.4) yields τ = 0° which is linear polarization with zero tilt angle, or horizontal linear polarization; this is seen in Figure 3.1 for ε = 0° and τ = 0°. Figure 3.2 also indicates horizontal linear polarization for γ = 0°. Now consider the case for δ = 0°. Then (3.3) and (3.4) yield ε = 0° and τ = γ , which is a linear polarization with a tilt angle equal to γ ; this is also seen using Figure 2.12 for ε = 0°. To find δ and γ from ε and τ , we start with the relations [1, p. 113; 2]:

cos2g = cos2ecos2t (3.5) tand =

tan2e (3.6) sin2t

from which we find γ and δ as

1 g = cos−1(cos2ecos2t) (3.7) 2



tan2e ⎤ (3.8) d = tan−1 ⎡⎢ ⎣ sin2t ⎦⎥

Care must be exercised when τ = 0° or 90°; then the argument on the righthand side of (3.8) is infinite. In this case δ = 90°. Also, the quadrant of the arguments in (3.4) and (3.8) must be noted when there are negative signs in the numerator or denominator. That is, when γ > 45° or τ > 90°, the quadrant must be determined. The following examples address the quadrant issue. Example 3.1  Given ε = 30°, τ = 135°. Find (γ , δ ).

The given parameters correspond to a LHEP wave. It can to be represented in (γ , δ ) form using (3.8): 1.73 ⎤ ⎡ tan(60°) ⎤ = 120° = tan−1 ⎡⎢ d = tan−1 ⎢ ⎥ ⎣ sin(270°) −1 ⎦⎥ ⎣ ⎦ One must be sure that the arctangent computing routine being used preserves the proper quadrant; many calculators do not and give δ = tan–1(−1.73) = −60°,

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Polarization State Representations47



which is incorrect. The remaining angle, γ , is calculated using (3.7) which gives γ = 45°. The results of γ = 45° and δ = 120° can be checked using (3.3) and (3.4), which yield ε = 30° and τ = 135°; these are the values we started with, verifying the procedure used. Example 3.2  Given ε = −30°, τ = 110°. Find (γ , δ ).

The given parameters correspond to a RHEP wave. It can be represented in (γ , δ ) form using (3.7): 1 1 g = cos−1 [ cos(−60°)cos(220°)] = cos−1(−0.383) = 56.2° 2 2 From (3.8) −1.73 ⎤ ⎡ tan(−60°) ⎤ = tan−1 ⎡⎢ d = tan−1 ⎢ = 249.6° = −110.4° ⎥ ⎣ −0.643 ⎥⎦ ⎣ sin(220°) ⎦ Note that in the preceding equation the argument is in the third quadrant, and therefore the angle must be between 180° and 270°. By convention δ is between −180° and +180°, so we use −110.4° for δ instead of 249.6°. We can check this result by substituting γ = 56.2° and δ = −110.4° into (3.3) and (3.4), giving 1 e = sin−1(sin2g sind) 2 1 = sin−1 [ sin(2 ⋅ 56.2°)sin(−110.4°)] = −30.0° 2 1 ⎡ sin2g cosd ⎤ t = tan−1 ⎢ 2 ⎣ cos2g ⎥⎦ 1 ⎡ sin(2 ⋅ 56.2°) cos(−110.4°) ⎤ = tan−1 ⎢ ⎥⎦ cos(2 ⋅ 56.2°) 2 ⎣ 1 −0.322 ⎤ 1 = tan−1 ⎡⎢ = (220°) = 110° 2 ⎣ −0.381 ⎥⎦ 2 which are the given starting values. An interesting, as well as useful, result is that orthogonal linear components that align with the major and minor axes of the polarization ellipse are 90° out of phase. This is easily proved by using τ = 0° in (3.8), which yields δ = ±90°.

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48

Polarization in Electromagnetic Systems

3.3  The Poincaré Sphere Every possible polarization state for a completely polarized wave can be assigned to one point on the surface of a sphere, which is the basis for the Poincaré sphere in Figure 3.4. Conversely, all points on the surface of the sphere correspond to a possible polarization state. Deschamps in [2] applied the early work of Poincaré [3] on optics to antenna polarization. The equator contains all linear polarization states. Elliptical polarization states are anywhere off the equator, right-hand sensed in the southern hemisphere and left-hand sensed in the northern hemisphere. Circular polarizations are at the poles. It is obvious from Figure 3.4 that opposite points on the sphere represent orthogonal polarization states. Points can be located using angle pairs (ε , τ ) or (γ , δ ) as illustrated in Figure 3.5, which expands one octant of the Poincaré sphere. The angle

Figure 3.4  The Poincaré sphere.

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Polarization State Representations49

pairs (2τ , 2ε ) or (δ , 2γ ) are the spherical angles from the origin (which corresponds to horizontal polarization) to the arbitrary polarization state point on the sphere, P. The angles used on the Poincaré sphere and their ranges are



2e = latitude 2t = longitude 2g = great-circle distance from HP point

− 90° ≤ 2e ≤ 90° 0° ≤ 2t ≤ 360° 0° ≤ 2g ≤ 180° (3.9)

2e = angle of great-circle with respect to equator

−180° ≤ d ≤ 180°

The ellipticity angle is related to the wave axial ratio through (2.27) and is repeated here:

e = cot −1(−R) (3.10)

The minus sign in the argument is introduced to make two conventions associated with sense consistent. The IEEE definition of the sign of axial ratio is positive for right-hand sense; see (2.27) [4]. For the Poincaré sphere

Figure 3.5  Location of polarization states on the Poincaré sphere with pairs (2 τ , 2 ε ) or (δ , 2 γ ).

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50

Polarization in Electromagnetic Systems

representation, the negative sign in (3.10) is necessary for the angle 2ε to be in the proper hemisphere. That is, if R is positive (RH sense), ε is negative from (3.10), corresponding to a point in the lower hemisphere of the Poincaré sphere, which is RH sense; see Figure 3.4. When converting between two representations it is also very helpful to visualize the polarization state on the Poincaré sphere. But the allowed ranges on the angles must be observed, as shown in Figure 3.1 or 3.2 and summarized in (3.9). Care must be exercised when consulting literature on the Poincaré sphere. Deschamps in [5] and Born and Wolf [6] use the upper hemisphere for righthand sense. We take the upper hemisphere to correspond to left-hand sensed polarization states to be consistent with the IEEE definition [4] and most of the modern literature.

3.4  The Polarization Vector It is convenient to use phasor !fields in the treatment of electromagnetics. The phasor electric field intensity E is related to the instantaneous field as follows: ! ! E(t) = Re[ Ee jwt ] (3.11) ! ! ! Here E is complex valued and has only spatial variation; that is, E = E(x, y,z). The instantaneous ! ! field is real valued and has both time and spatial dependence; that is, E(t) = E(x, y,z,t). In practice many signals are narrow band and can be treated as harmonic (i.e., monochromatic), as implied by (3.11) in which the electric field is associated with a sinusoidal wave of radian frequency ω . For an example consider the 2.4-GHz Wi-Fi band (2.4–2.4835 GHz). One 20-MHz channel in the band is a bandwidth of 0.8% of the center frequency which in most situations would be considered to be narrowband. In this case polarization analysis at any single frequency across the band (usually the center frequency) is adequate. The polarization vector representation for a polarization state could also be called the rectangular component representation because phasor components of the electric field along the x- and y-axes are used: ! E = Ex xˆ + E y yˆ (3.12) where Ex and Ey are complex valued. This complex vector representation was originally studied by Kales in [2]. The complex valued form facilitates inclusion

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Polarization State Representations51

of the vector component amplitudes (E1 and E2) and the relative phase (δ ) directly as

Ex = E1 and E y = E2 e jd (3.13)

The corresponding instantaneous forms with time variation included are given in (2.15). The magnitudes and phases for the phasors associated with the vector components in (3.13) are

Ex = E1 and E y = E2 (3.14)

( )

phase ( Ex ) = 0 and phase E y = d (3.15)

The phase of Ex is arbitrary because it is only the relative phase of the two vector components that matter, so we choose it to be zero for simplification. We can recover the instantaneous form of the electric field vector by using (3.13) in (3.11) to give ! E(t) = E1 coswt xˆ + E2 cos(wt + d)yˆ (3.16) which is (2.17). The ratio of amplitudes E2/E1 and the relative phase δ can be used to find ε and τ , which completely characterize a polarization state. This can be seen from the box with sides parallel to the x- and y-axes that just encloses the polarization ellipse shown in Figure 3.3, which follows from Figure 2.12. From Figure 3.3 it is apparent that γ = tan–1 (E2/E1), which is (2.29). The intensity of a wave is also contained in the polarization vector representation, as we now show. For intensity we use the time-average Poynting vector with units of W/m2. In a plane wave, the phasor forms of the electric and magnetic fields are related as they were in (2.5) for instantaneous fields:



! 1 ! H = nˆ × E (3.17) h

where nˆ is a unit vector in the direction of propagation. The complex-valued Poynting vector is

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! 1! ! S = E × H ∗ (3.18) 2

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52

Polarization in Electromagnetic Systems

Substituting (3.17) into (3.18) and using a vector identity for vector triple products gives

! 1 ! ! ! ! 1 ! !∗ ⎡⎣( E ⋅ E ) nˆ − ( E ⋅ nˆ ) E ∗ ⎤⎦ (3.19) S= E × ( nˆ × E ∗ ) = 2h 2h



! 1 ! !∗ S= E ⋅ E nˆ (3.20) 2h

! ! But E ⋅ nˆ = 0 because E is perpendicular to the direction of propagation for a plane wave and then (3.19) becomes

and the magnitude of the Poynting vector is S=



1 ! !∗ E ⋅ E (3.21) 2h

The time-average Poynting vector in the direction of propagation is the real part of S, or Sav = Re[S] =



1 ! !∗ E ⋅ E (3.22) 2h

which is the same as (3.21) because S is a real-valued quantity of power density in W/m2. Substituting (3.12) into (3.22) and assuming z-directed propagation (i.e., nˆ = ˆz ) gives 1 ! !∗ 1 E⋅E = E xˆ + E y yˆ ⋅ Ex∗ xˆ + E ∗y yˆ 2h 2h x 1 2 2 (3.23) = E + Ey 2h x 1 = ( E 2 + E22 ) 2h 1

(

S=

(



)(

)

)

Therefore, the intensity of the wave is contained in the polarization vector representation through E1 and E2. Intensity is not a necessary quantity for describing the polarization state of a wave. By dividing the polarization vector by its magnitude the intensity is eliminated and we obtain a normalized unit vector, ê, defined as

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Polarization State Representations53



! E eˆ = ! = e x xˆ + e y yˆ (3.24) E

such that the following normalization condition is satisfied: 2 2 eˆ ⋅ eˆ∗ = e x + e y = 1 (3.25)



It is common to refer to ê as the normalized polarization vector or simply the polarization vector. The term used by the IEEE [4] is polarization vector, but other terms are used such as normalized complex vector. As a check on the normalization, we substitute (3.24) into (3.22): ! ! 2 2 E ⋅ E ∗ = Eeˆ ⋅ E ∗eˆ∗ = E eˆ ⋅ eˆ∗ = E (1) = E12 + E22 = 2hS (3.26) which agrees with (3.23). The polarization vector ê carries all polarization information, but no intensity information. We now relate the polarization vector to polarization ellipse (γ , δ ) parameters. From (3.24): eˆ = e x xˆ + e y yˆ (3.27)

Then ˆ∗

(

)(

eˆ ⋅ e = e x xˆ + e y yˆ ⋅

e∗x xˆ

+

e∗y yˆ

)= e

2 x

2

+ e y = ex

2

2 ⎛ ey ⎞ ⎜ 1 + 2 ⎟ (3.28) ex ⎠ ⎝

But Ey

Ex

=

ey E ey = (3.29) E ex ex

and from (2.29) Ey

Ex

=

E2 = tang (3.30) E1

So (3.28) becomes

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54



Polarization in Electromagnetic Systems

ˆ∗

eˆ ⋅ e = e x

2

2

e (1 + tan g ) = cosx2 g (3.31) 2

However, (3.25) requires that ê · ê∗ = 1, so 2



e x = cos2 g (3.32)

Further, if we let ex be real valued as we did Ex, then

e x = e1 = cosg (3.33)

To satisfy (3.25) we must have then have

e y = sing (3.34)

Note that the phase of ey can be anything and still satisfy (3.25). It is logical to choose it to be the same as we did for Ey. Then

e y = e2 e jd = sing e jd (3.35)

Using (3.33) and (3.35) in (3.27), we have the polarization vector in its most useful form based on the (γ , δ ) representation:

eˆ = cosg xˆ + sing e jd yˆ (3.36)

It is instructive to present a second and more direct derivation of the result in (3.36). First, combining (3.12) and (3.13) we have ! E = Ex xˆ + E y yˆ = E1xˆ + E2 e jd yˆ (3.37) The magnitude of the electric field vector from (3.26) is ! E = E = E12 + E22 (3.38) Dividing this electric field magnitude into the full electric field vector expression of (3.36) gives polarization vector that is normalized to unity magnitude as in (3.24): ! E E E eˆ = = e x xˆ + e y yˆ = 1 xˆ + 2 e jd yˆ (3.39) E E E

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Polarization State Representations55

From the triangle in Figure 3.3, or using (3.34) and (3.35):

E E1 = cosg and 2 = sing (3.40) E E

Using these in (3.39) we again obtain (3.36):

eˆ = e x xˆ + e y yˆ = cosg xˆ + sing e jd yˆ (3.41)

Note that for δ = 0° ê is real valued and (3.41) is an expression for a linear vector at angle γ . This is linear polarization with tilt angle τ = γ . We will present some additional important properties of the polarization vector in Section 3.8.4.

3.5 Stokes Parameters A quantitative approach to describe the polarization of light was introduced by Sir George Stokes in 1852 and as such is probably the oldest polarization representation [7]. The Stokes parameters representation is not as suitable as others discussed in this chapter for many applications. But the Stokes parameters representation is useful for partial polarization (to be discussed in Chapter 4) and is encountered in several treatments of polarization in the literature. It is also easy to use in antenna-wave interaction calculations to be discussed in Chapter 6. Although Stokes parameters are more involved than many other representations, they do not require complex numbers. Here we present Stokes parameters for a completely polarized wave. The Stokes parameters written as a matrix and using the conventional notation for them are [1, Sec. 4.4] ⎡ S0 ⎤ ⎢S ⎥ ⎢ 1 ⎥ (3.42a) ⎢ S2 ⎥ ⎢ ⎥ ⎢⎣ S3 ⎥⎦

where

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S0 = S =

1 ( E 2 + E22 ) = Sx + S y (3.42b) 2h 1

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56



Polarization in Electromagnetic Systems

S1 = Sx − S y = S cos2ecos2t (3.42c)

(

)

(

)

S2 = S x − S y tan2t = S cos2esin2t (3.42d) S3 = S x − S y tan2esec2t = S sin2e (3.42e)

Here Sx and Sy are the intensities associated with the electric field components in the x- and y-directions; as usual the direction of propagation of the wave is +z-direction. In the literature, the symbols I, Q, U, and V are often used instead of S 0, S1, S2, and S3. The Stokes parameters in (3.42) include the wave intensity S. In fact, all of the parameters are positive, real and have units of W/m2. S 0 is not an independent parameter but is obtainable from the other three Stokes parameters as

S02 = S12 + S22 + S32 (3.43)

Normalized Stokes parameters are introduced to eliminate intensity and reduce the number of parameters from four to three. Dividing all matrix entries of (3.42) by the wave intensity S gives the normalized Stokes matrix:



⎡ ⎢ = s [ i ] ⎢⎢ ⎢⎣

1 s1 s2 s3

⎤ ⎥ ⎥ (3.44a) ⎥ ⎥⎦

where

s1 = cos2ecos2t (3.44b)



s2 = cos2esin2t (3.44c)



s3 = sin2e (3.44d)

Note that the Stokes parameters are real valued. Since only ε and τ are needed to find the normalized Stokes parameters, the three normalized Stokes

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Polarization State Representations57

parameters have excess information. Therefore, the three parameters must depend on each other. It is easy to prove the following using (3.44):

1 = s12 + s22 + s32 (3.45)

This shows that having two of the three Stokes parameters the third can be determined. The Stokes parameters can be derived rather simply from the Poincaré sphere. Consider a spherical coordinate system with its z-axis upward through the North pole (see Figure 3.5). Then the conventional spherical coordinate system angles θ and ϕ relate to the angles on the Poincaré sphere 2ε and 2τ as follows:

2e = 90° − q (3.46a)



2t = f (3.46b)

Using a unit radius for the sphere and (B.4) for spherical-to-rectangular coordinate system translations yields

x = 1 sinqcosf = cos2ecos2t = s1 (3.47a)



y = 1 sinqsinf = cos2esin2t = s2 (3.47b)



z = 1 cosq = sin2e = s3 (3.47c)

Used in the foregoing were the following transformations from spherical angles: θ = 90° − 2ε and ϕ = 2τ . Thus, we see that the Stokes parameters s1, s2, s3 are projections onto the rectangular coordinate system axes of the Poincaré sphere as shown in Figure 3.5. The equations of (3.42) do not have any apparent physical interpretation. However, with the connection just established with the Poincaré sphere we can draw some inferences. The s1, s 2, s3 parameters from (3.47) are the projections along the x-,y-, and z-axes of the Poincaré sphere that correspond to: horizontal/vertical, +45° LP/−45°, and LHCP/RHCP, respectively. Therefore, we can summarize the interpretation from Stokes parameters as shown in Table 3.2.

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58

Polarization in Electromagnetic Systems Table 3.2 Physical Interpretation of the Stokes Parameters

Unnormalized Parameter

Interpretation

Normalized Parameter

S0 = S

Total power density of the wave

s0 = 1

S1

Portion of wave that is HP or VP

s1

S2

Portion of the wave that is ±45° LP

s2

S3

Portion of wave that is RHCP or LHCP

s3

For a general elliptically polarized case (3.45) must be satisfied. For linearly polarized cases, there is no CP portion, so (3.45) reduces to

1 = s12 + s22

LP (3.48)

Similarly, for circular polarization then

1 = s32

CP (3.49)

The examples in Table 3.5 satisfy the two foregoing relations. Stokes parameters can also be represented in terms of parameters used to describe circular polarization. [1, p. 122] We start by using (2.27) and (2.40) to express axial ratio as follows:

R=

ER0 + EL0 = −cot e (3.50) ER0 − EL0

From this the following can be derived: cos2e = sin2e =

2 EL0 ER0 R2 − 1 (3.51a) 2 2 = EL0 + ER0 R2 + 1 2 2 − ER0 EL0 −2R (3.51b) 2 2 = EL0 R2 + 1 + ER0

These two relations can be used in (3.44) to find the Stokes parameters in terms of axial ratio and tilt angle.

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Polarization State Representations59

Another matrix formulation that is used as a polarization state representation occasionally is the coherency matrix, which is defined as follows [1, p. 124]: ⎡ s11 s12 ⎤ ⎡⎣ sij ⎤⎦ = ⎢ s s ⎥ (3.52a) ⎣ 21 22 ⎦



where the matrix entries are related to Stokes parameters through



s11 =

1 (1 + s1 ) 2

s21 =

s12 =

1 s − js3 ) 2( 2

1 s + js3 ) (3.52b) 2( 2

s22 =

1 (1 − s1 ) (3.52c) 2

3.6 Polarization Ratio 3.6.1  Polarization Ratio for Linear Polarization The polarization ratio is a single complex number that represents the polarization state of a wave. The definition of polarization ratio for linear polarization is



rL =

Ey

Ex

=

ey (3.53) ex

Substituting (3.13), (3.33), and (3.35) into the above gives



rL =

E2 e2 jd = e (3.54) E1 e1

The special cases for ρ L in Table 3.3 help to illustrate its relationship to the polarization states of the polarization ellipse. Note that the convention of Im(ρ L) > 0 for left-hand sense used here is opposite that of the classic book by Beckmann [8, p. 26]; this is done to maintain consistency with other commonly accepted definitions. Polarization ratio is the most compact of all the representations. It is a single complex number and not a vector. However, to achieve this compact form the ratio is infinite valued for the vertical linear polarization case and caution must be used when using numerical routines for computations using the polarization ratio.

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60

Polarization in Electromagnetic Systems Table 3.3 Example Polarization States in the Polarization Ratio

ρL

State

Comment

0

Horizontal linear (HP)

E2 = 0

∞

Vertical linear (VP)

E1 = 0

j

LHCP

E1 = E2 ; δ = 90°

−j

RHCP

E1 = E2 ; δ = −90°

Im(ρ L ) = 0

Linear

δ =0

Im(ρ L ) > 0

Left-hand elliptical

0 < δ < 180°

Im(ρ L ) < 0

Right-hand elliptical

−180° < δ < 0

1

LP, τ = 45°

E1 = E2 ; δ = 0°

−1

LP, τ = 135°

E1 = E2 ; δ = 180°

Rumsey in [2] introduced polarization ratio in 1951. Beckmann [8, p. 24] developed its theory and applications, using the term complex polarization factor instead of polarization ratio. In this discussion we have followed Beckmann’s results modified to our notation. There is a one-to-one correspondence between all possible polarization states and all points on a plane of complex numbers. Figure 3.6 shows the complex plane with polarization ratio values displayed. Linear polarizations are along the horizontal axis with horizontal linear polarization at the origin

Figure 3.6  Polarization ratio plane (complex ρ L -plane).

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Polarization State Representations61

and vertical LP at infinity. The vertical axis corresponds to imaginary values. Left-hand sensed states are in the upper half plane and right-hand sense states are below the horizontal axis. The left-hand circular polarization state is the point +j and −j corresponds to RHCP. The correspondence between the Poincaré sphere and the complex-ρ L plane is established using a stereographic projection as shown in Figure 3.7. [8, p. 33] The point corresponding to horizontal linear polarization of the Poincaré sphere touches the complex ρ L-plane at the origin of the plane labeled “O”. Corresponding points are found using a line starting at the vertical linear polarization point of the Poincaré sphere (that is, ρ L = ∞ at the top point in Figure 3.7) and passing through the sphere to the complex plane. The two points which pierce the Poincaré sphere and complex ρ L-plane are the same polarization state. The equatorial plane of the Poincaré sphere that contains the linear polarization states becomes the real ρ L-axis when projected onto the complex plane. The North and South poles of the Poincaré sphere (corresponding to LHCP and RHCP) map to the imaginary ρ L-axis on the complex plane as shown in Figures 3.6 and 3.7 The normalized Stokes parameters can be derived from the polarization ratio (following the work of Beckmann [8, p. 33] and correcting for errors):





s1 =

s2 =

1 − rL 1 + rL

2 2

2Re ( rL ) 1 + rL

2

(3.55a)

(3.55b)

Figure 3.7  Relationship between points on the Poincaré sphere and the complex ρ L -plane.

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62

Polarization in Electromagnetic Systems

s3 =

2Im ( rL ) 1 + rL

2

(3.55c)

The polarization ratio is obtained from the normalized Stokes parameters using rL =



1 − s1 (3.56a) 1 + s1

⎛s ⎞ d = tan−1 ⎜ 3 ⎟ (3.56b) ⎝ s2 ⎠



The polarization ratio is related very simply to the (γ , δ ) representation: rL =



E2 = tang (3.57a) E1

phase ( rL ) = d (3.57b)



The polarization ratio is also conveniently related to the polarization vector representation using (3.12):



Ey ⎞ ! ⎛ yˆ = Ex ( xˆ + rL yˆ ) (3.58) E = Ex xˆ + E y yˆ = Ex ⎜ xˆ + Ex ⎟⎠ ⎝

From (3.36) we have



xˆ + rL yˆ = xˆ + tang e jd yˆ =

1 eˆ cosg xˆ + sing e jd yˆ ) = (3.59) ( cosg cosg

Or

eˆ = cosg ( xˆ + rL yˆ ) (3.60)

The polarization ellipse relates to polarization ratio through (3.57). This conversion is displayed graphically in Figure 3.8 [9]. Axial ratio magnitude, ⎪R⎪, is plotted versus phase angle magnitude, ⎪δ ⎪, with polarization ratio magnitude, ⎪ρ L⎪, as a parameter. The plot allows determination of axial ratio magnitude from the relative amplitude and relative phase of the rectangular components of the electric field. Horizontal linear polarization is anywhere on

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Polarization State Representations63

the horizontal axis where E2 = 0,⎪ρ L⎪= 0, and ⎪R⎪= ∞. Circular polarizations are at the point where ⎪δ ⎪= 90° and R⎪ = 1 = 0 dB, which is the point on the top of the vertical axis. The plot shows only positive δ values corresponding to left-hand sense. But the plot does not determine sense (i.e., the sign on R), so it applies to both right- and left-hand sense cases. This plot is based on (3.1) as is shown in the following example. Example 3.3  Using Figure 3.8 to Find Axial Ratio

We illustrate one way to use Figure 3.8 by starting with the polarization ellipse parameters of γ and δ to find the axial ratio. First, we find the polarization ratio magnitude using the following example values of γ and δ for a specific left-hand elliptical polarization state:

γ = 21.8° and δ = 60°

Figure 3.8  Axial ratio magnitude determined from polarization ratio magnitude and phase angle δ for possible values of δ corresponding to left-hand sense. The plot applies to right-hand sense cases where δ is negative by taking the magnitude of δ . (Source: W. D.Yarnall, “Antenna Design Supplement,” Microwaves, Vol. 4, May 1965, p. 60. Reprinted with permission. ©1965, Penton Media, 255513:1117SH.)

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64

Polarization in Electromagnetic Systems

Then, using (3.57a) rL = tang = tan(21.8°) = 0.4 From (3.1) sin2e = sin2g sind = sin(43.6°)sin(60°) = 0.5972 Solving this we find ε = 18.33°. Using (3.50) gives⎪R⎪= cot ⎪ε ⎪= 3.02. So R(dB) = 20 log 3.02 = 9.60 dB This value of 9.60 dB for axial ratio can be located on the plot in Figure 3.8 using the values of δ = 60° and ⎪ρ L⎪ = 0.4. For completeness, we can also find the tilt angle of polarization ellipse using (3.4): 1 ⎡ sin2g cosd ⎤ t = tan−1 ⎢ 2 ⎣ cos2g ⎥⎦ 1 ⎡ sin(43.6°)cos(60°) ⎤ = tan−1 ⎢ ⎥⎦ = 12.7° cos(43.6°) 2 ⎣

3.6.2  Polarization Ratio for Circular Polarization In a fashion similar to (3.53) for linear polarization ratio, we can define polarization ratio for circular polarization as



rC =

ER (3.61) EL

where ER and EL are the phasor CP components. The time-varying forms of the CP electric field components were presented in (2.30) and (2.31). Rewriting these in phasor-vector form gives



! 1 [ xˆ + j yˆ ] E (3.62a) EL0 xˆ + jEL0 yˆ ] = EL = [ L0 2 2



! 1 [ xˆ − j yˆ ] E e jd ′ (3.62b) ⎡⎣ ER0 e jd ′ xˆ − jER0 e jd ′ yˆ ⎤⎦ = ER = R0 2 2

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Polarization State Representations65



As a check on these results, using them in (3.11) yields the instantaneous forms in (2.30) and (2.31). The phasor electric field CP field components in (3.62) are

EL = EL0 (3.63a)



ER = ER0 e jd ′ (3.63b)

Then (3.61) becomes rC =



ER0 jd ′ e (3.64) EL0

The complete vector electric field expressed in terms of circularly polarized component phasors from (3.62) is then ! ! ! E = E L + E R (3.65a)

! 1 ⎡ E + ER0 e jd ′ xˆ + j EL0 − ER0 e jd ′ yˆ ⎤ (3.65b) E= ⎦ 2 ⎣ L0

(

)

(

)

The circular component phasors can be expressed in terms of x- and y-linearly polarized electric field components by starting with the following expression based on (3.12) and (3.13) ! E = Ex xˆ + E y yˆ = E1xˆ + E2 e jd yˆ (3.66) and comparing this to (3.65b) gives

EL = EL0 =

1 E − jE2 e jd (3.67a) 2 1

ER = ER0 e jd ′ =

1 E + jE2 e jd (3.67b) 2 1

(

)

(

)

(Problem 19 at the end of this chapter considers the measurement of the polarization state of wave that is a realization of these equations.) To illustrate (3.67) consider a RHCP wave. From Table 2.1, E1 = E2 = 1/ 2 and δ = −90°. The LHCP component should be zero. This is verified by substituting into (3.67a): EL = (E1 − jE2ejδ)/ 2 = (1/ 2 − je−j90/ 2 )/ 2 = [1 − j(−j)]/2 =0. Evaluating

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66

Polarization in Electromagnetic Systems

(3.67b) gives the RHCP ER = (1/ 2 )[1/ 2 + j(1/ 2 (−j))] = 1. ! component: ! Unit vector forms of E L and E R are developed in Section 3.8.4. Polarization ratio for CP can be related to polarization ratio for LP using (3.67) in (3.61):

rC =



rC =

ER = EL

1 E + jE2 e jd 2 1 1 E − jE2 e jd 2 1

( (

) )

E2 e jd E1 = E e jd 1− j 2 E1 (3.68) 1+ j

1 + jrL 1 − jrL

Another very useful relationship converts axial ratio to polarization ratio for CP. From (2.40)

R=

ER0 + EL0 ER0 − EL0

ER0 +1 r +1 EL0 = = C (3.69) ER0 rC − 1 −1 EL0

Solving this to isolate the polarization ratio magnitude for CP:

rC =

R +1 (3.70) R −1

For RHCP (R = +1) this yields the correct answer that ⎪ρ C⎪ = ∞. For LHCP (R = −1), the correct result of ⎪ρ C⎪ = 0 is obtained. For LP (R = ∞), ⎪ρ C⎪ = 1, which is correct. From (2.39) the phase of ρ C is related to tilt angle from

d ′ = 2t (3.71)

Polarization ratio is usually assumed to be for linear polarization unless circular polarization is indicated.

3.7  Polarizing State Representation Examples In this section we examine the parameter values for two common polarization states, horizontal linear and right-hand circular polarization. The values

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Polarization State Representations67

are given for all six polarization representations in Table 3.1. The parameter values for the examples are also given in Table 3.5. Example 3.4  Representations for Horizontal Linear Polarization

1. Polarization ellipse: (ε , τ )

ε = 0°, τ = 0°; see Figure 3.1 2. Polarization ellipse: (γ , δ ) tan2e ⎤ 0 d = tan−1 ⎡⎢ = tan−1 ⎡⎢ ⎤⎥ = 0°, or any value ⎣ sin2e ⎦⎥ ⎣0 ⎦ 1 1 g = cos−1(cos2ecos2t) = cos−1(1⋅1) = 0° 2 2 3. Poincaré sphere The HP point is on the equator at the x-axis in Figure 3.4. 4. Polarization vector e x = cosg = cos0 = 1 e y = sing e jd = sin0e j0 = 0 eˆ = e x xˆ + e y yˆ = 1xˆ 5. Stokes parameters s1 = cos2e cos2t = cos0cos2t = 1 s2 = cos2esin2t = cos0sin2t = 0 s3 = sin2e = sin0 = 0 ⎡ ⎢ ⎡⎣ si ⎤⎦ = ⎢ ⎢⎣

1 1 0 0

⎤ ⎥ ⎥ ⎥⎦

6. Polarization ratio rL =

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ey 0 = =0 ex 1

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68

Polarization in Electromagnetic Systems

or rL =

1 − s1 = 1 + s1

1−1 =0 1+1

()

0 ⎛s ⎞ d = tan−1 ⎜ 3 ⎟ = tan−1 = 0°, or any value 0 ⎝ s2 ⎠ or rL = tang = tan0 = 0 Example 3.5  Representations for Right-Hand Circular Polarization

1. Polarization ellipse: (ε , τ )

ε = −45°, τ = any value; see Figure 3.1 2. Polarization ellipse: (γ , δ ) tan2e ⎤ ⎡ tan(−90°) ⎤ −1 ⎡ −∞ ⎤ d = tan−1 ⎡⎢ = tan−1 ⎢ = −90° tan ⎢ ⎥ ⎣ sin2t ⎦ ⎣ 0 ⎥⎦ ⎣ sin 0 ⎥⎦ 1 1 g = cos−1(cos2e cos2t) = cos−1 [ cos(−90°)cos0 ] 2 2 1 −1 = cos (0) = 45° 2 3. Poincaré sphere The RHCP point is at the South pole; see Figure 3.4. 4. Polarization vector e x = cosg = cos45° =

1 2

1 2 1 1 xˆ − jyˆ xˆ − j yˆ = eˆ = e x xˆ + e y yˆ = 2 2 2

e y = sing e jd = sin45°e− j90° = −j

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Polarization State Representations69



5. Stokes parameters s1 = cos2ecos2t = cos(−90°)cos2t = 0 s2 = cos2esin2t = cos(−90°)sin2t = 0 s3 = sin2e = sin(−90°) = −1 ⎡ 1⎤ [ si ] = ⎢⎢ 00 ⎥⎥ ⎢⎣ −1 ⎥⎦ 6. Polarization ratio rL =

e y − j/ 2 = = −j ex 1/ 2

or rL =

1 − s1 1− 0 = =1 1+ 0 1 + s1

( )

−1 ⎛s ⎞ d = tan−1 ⎜ 3 ⎟ = tan−1 = tan−1(−∞) = −90° 0 ⎝ s2 ⎠ or rL = tang = tan(45°) = 1

3.8  Determination of Orthogonal Polarization States In system studies involving polarization we frequently want to know the wave state that is orthogonal to a given polarization state. The principles of orthogonal polarization states in this Section apply to both waves and antennas; also see Section 2.5. Dual-polarized systems, to be discussed in Chapter 7, make use of orthogonally polarized antennas. Dual polarization (and multi-polarization) are used in sensing systems such as radar and radiometry to collect information about a target or medium; see Sections 8.5 and 8.6. Communication systems use orthogonally polarized antennas on one or

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70

Polarization in Electromagnetic Systems

both ends of a link to either improve performance or to increase information carrying capacity. In this section, we show how to obtain the polarization state that is orthogonal to an arbitrary state for each of the representations discussed earlier in this chapter. The more common states along with their orthogonal states are examined in detail. The orthogonal states for the six representations of Table 3.1 are summarized in Table 3.4. These orthogonal relationships apply to antenna polarization states as well as to wave states. Table 3.5 gives the values of the parameters for common pairs of orthogonal states for the six polarization state representations of Table 3.4. Subsequent discussion in this section and several problems at the end of the chapter relate to these example cases. Table 3.4 Orthogonal State Representations

1.

Representation

Parameters

Orthogonal State Parameters

Polarization ellipse

ε w, τ w

ε wo , τ wo ε wo = −ε w τ wo = τ w ± 90° 0° ≤ τ wo ≤ 180°

2.

Polarization ellipse

γ w, δ w

γ wo , δ wo γ wo = 90° − γ w 0° ≤ γ wo ≤ 90°

δ wo = δ w ± 180° −180° ≤ δ wo ≤ 180° 3.

Poincaré sphere

Point on surface of the sphere

Point directly opposite and on the surface of the sphere

4.

Polarization vector

êw

êwo êw ·ê∗wo = 0

5.

Stokes parameters

1, s1w , s2w , s 3w

1, s1wo , s2wo , s 3wo s1wo = −s1w s2wo = −s2w s 3wo = −s 3w

6.

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Polarization ratio

ρ Lw

rLwo = −

1 ∗ rLw

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Polarization State Representations71



Table 3.5 Polarization State Representation Parameter Values for Common Pairs of Orthogonal States Linear Linear

(Vertical and Horizontal)

2.

VP

LP τ = 45°

LP τ = 135°

LHCP

RHCP

ε

0°

0°

0°

0°

45°

−45°

τ

0°

90°

45°

135°

any

any

0°

90°

45°

45°

45°

45°

any

any

0°

180°

90°

−90°

Equator– front

Equator– back

Equator– right side

Equator– left side

North pole

South pole

ex

1

0

1/ 2

– 1/ 2

ey

0

1

1/ 2

1/ 2

1/ 2 j/ 2

1/ 2 – j/ 2

s1

1

−1

0

0

0

0

s2

0

0

1

−1

0

0

s3

0

0

0

0

1

−1

0

∞

1

−1

j

−j

Polarization ellipse

Polarization ellipse

γ δ 3.

Poincaré sphere point location

4.

5.

6.

Circular

HP

Representation 1.

(Slant 45°)

Polarization vector, ê

Stokes parameters

Polarization ratio

ρL

3.8.1  Orthogonal State for the Polarization Ellipse Using ε , τ In the ε , τ polarization ellipse representation (see Section 3.2), if ε w and τ w are the wave state parameters, then the parameters for the orthogonal state are

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ewo = −ew (3.72a)



two = tw ± 90°

such that 0° ≤ two ≤ 180° (3.72b)

The angle ε wo is of opposite sign from ε w because the sense of the polarization must be opposite (e.g., left-hand instead of right-hand). This follows from (3.50), which shows that a sign change of ε causes a sign change of R. The tilt angle for the orthogonal state differs from the original state by 90° because the respective major axes are perpendicular. These conditions are illustrated in Figure 3.9, which shows the orthogonal polarization ellipses of general orthogonal polarization states. The wave state w has a polarization ellipse with tilt angle τ w and is shown as right-hand sensed in Figure 3.9. The orthogonal state wo has a polarization ellipse (dashed line) of the same axial ratio magnitude and opposite sense (left-hand). The tilt angle of the wo ellipse is τ wo = τ w + 90°, and, thus, the major axes of the two ellipses are perpendicular. Since axial ratio information is contained in ε , orthogonality can also be described in terms of the axial ratio of the polarization ellipse instead of ε ; see (3.50). The polarization state wo is orthogonal to state w if the polarization ellipse is identical in shape and opposite in sense; that is, ⎪Rwo⎪ = ⎪Rw⎪ and sign(Rwo) = −sign(Rw). Also, the major axes of the orthogonal states must be perpendicular to each other as given by (3.72b). Therefore,

Figure 3.9  Orthogonal polarization states illustrated with generalized polarization ellipses. The polarization state w, shown as the solid ellipse, has tilt angle τ w . The orthogonal state wo, shown as the dashed ellipse, is of the same axial ratio magnitude and has its major axis perpendicular to that of the state w and, hence, tilt angle τ wo = τ w + 90°. The senses are opposite; the sense of state w is right-hand and state wo is left-hand.

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Polarization State Representations73



Rwo = −Rw (3.73a)



two = tw ± 90° (3.73b)

3.8.2  Orthogonal State for the Polarization Ellipse Using γ , δ The state orthogonal to a wave represented by a polarization ellipse using γ w and δ w is found as follows:

g wo = 90° − g w dwo = dw ± 180°

such that 0° ≤ g wo ≤ 90° (3.74a) such that − 180° ≤ dwo ≤ 180° (3.74b)

These conditions on orthogonality are best understood by examining the Poincaré sphere in Figures 3.4 and 3.5. A point opposite the point (2γ w, δ w) is the point (180° − 2γ w, δ w ± 180°) consistent with (3.74). The orthogonal state representation using ε w and τ w of (3.72) is consistent with the γ w and δ w representation. We show this for γ wo using (3.5): cos2g wo = cos2ewo cos2two

cos2( 90° − g w ) = cos2( −ew ) cos2( tw ± 90° ) −cos2g w = cos2ew ( −cos2tw )

(3.75)

cos2g w = cos2ew cos2tw

which is, of course, the relationship of (3.5) that ε w and τ w must satisfy. Next, we show that δ wo satisfies (3.6): tandwo =

tan2ewo sin2two

tan ( dw ± 180° ) =



tandw =

tan2( −ew ) (3.76) sin2( tw ± 90° )

−tan2ew tan2ew = −sin2tw sin2tw

All of the polarization ellipse representations and their orthogonal states are included in the following example.

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Example 3.6  Polarization Ellipse Representations Orthogonal to a Wave with ε w = 20° and τ w = 45°

The wave state orthogonal to (ε w = 20°, τ w = 45°) in terms of ε and τ from (3.72) is

ε wo = −ε w = −20° τ wo = τ w ± 90° = 45° ± 90° = 135° (to keep τ wo between 0° and 180°) The corresponding (γ , δ ) values for this wave follow from (3.7) and (3.8) as 1 1 g w = cos−1 ( cos2ew cos2tw ) = cos−1 [ cos(40°)cos(90°)] = 45° 2 2 ⎡ tan2ew ⎤ −1 ⎡ tan(40°) ⎤ dw = tan−1 ⎢ ⎥ = tan ⎢⎣ sin(90°) ⎥⎦ = 40° sin2t w ⎦ ⎣ The orthogonal wave state in terms of γ and δ from (3.74) is

γ wo = 90° − γ w = 90° − 45° = 45° δ wo = δ w ± 180° = 40° ± 180° = −140° (to keep δ wo between −180° and 180°) Note that (3.75) is satisfied: cos2g wo = cos2ewo cos2two

cos2(45°) = cos2(−20°)cos2(135°) cos(90°) = cos(40°)cos(270°) 0=0 as well as (3.76): tandwo =

tan2ewo sin2two

tan2(−20°) sin2(135°) −0.839 0.839 = = 0.839 −1 tan(−140°) =

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Polarization State Representations75

The axial ratio for this example follows from (2.27) as Rw = −cot ε w = −cot(20°) = −2.75 Then from (3.73a) Rwo = −Rw = 2.75 So, the wave state w is left-hand elliptical with an axial ratio magnitude of 2.75 and a tilt angle of 45°. The state orthogonal to it, wo, is right-hand elliptical with an axial ratio magnitude of 2.75 and a tilt angle of 135°. The orthogonal states shown in Figure 3.9 are similar to those in this example. 3.8.3  Orthogonal States on the Poincaré Sphere The simplest representation for visualizing orthogonal wave states is the Poincaré sphere, which was discussed in detail in Section 3.3. Points that are directly opposite on the surface of the sphere correspond to orthogonal wave states. For example, the polarization state on the x-axis (ε = 0, τ = 0°) is horizontal LP, and 180° around on the equator is the vertical LP state (ε = 0, τ = 90°); see Figure 3.4. As a second example, the point opposite of the North pole (LHCP) is the South pole (RHCP), which are orthogonal states. 3.8.4  Orthogonal Polarization Vector The polarization vector representation was introduced in Section 3.4. A wave with polarization vector êw is orthogonal to a wave state with polarization vector êwo if the following is satisfied:

∗ = 0 (3.77) eˆw ⋅ eˆwo

This is an extension of the concept of spatial perpendicularity. From this condition for orthogonality, we can find an explicit form for êwo in terms of êw. To do this we begin with vector component expressions using (3.27):

eˆw = ewx xˆ + ewy yˆ (3.78a)



eˆwo = ewox xˆ + ewoy yˆ (3.78b)

Using these in (3.77) gives

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∗ ∗ (3.79) + ewy ewoy 0 = ewx ewox



This relation is satisfied if the following are true:

∗ ∗ ewox = ewy and ewoy = −ewx (3.80)

Performing the complex conjugate on these two equations yields

∗ ∗ (3.81) ewox = ewy and ewoy = −ewx

We now have the vector components of êwo in terms of êw and can write the êwo as

∗ ∗ xˆ − ewx yˆ (3.82) eˆwo = ewox xˆ + ewoy yˆ = ewy

Next we examine the special cases of polarization vectors for orthogonal linear and orthogonal circular polarization states. The orthogonal linear states of horizontal and vertical polarizations are commonly used in electromagnetic systems (see Figure 2.5), so we use them to illustrate the orthogonal linear case. The horizontal and vertical linear polarization vectors are simple to write: eˆH = xˆ and eˆV = yˆ (3.83)



The orthogonality condition of (3.77) is satisfied: eˆH ⋅ eˆV∗ = xˆ ⋅ yˆ = 0 (3.84)



The normalization requirement of (3.25) is also satisfied:

eˆH ⋅ eˆ∗H = xˆ ⋅ xˆ = 1 (3.85a)



eˆV ⋅ eˆV∗ = yˆ ⋅ yˆ = 1 (3.85b)

The orthogonal state pair of left-hand and right-hand circular polarizations have unit vectors that follow from (3.62) as

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eˆL =

xˆ + jyˆ 2

eˆR =

xˆ − jyˆ (3.86) 2

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Polarization State Representations77



These polarization vectors for circular polarization satisfy the orthogonality requirement of (3.77): eˆL ⋅ eˆ∗R =



1 ∗ xˆ + jyˆ ) ⋅ ( xˆ − jyˆ ) = 0 (3.87) ( 2

and the normalization requirement of (3.25):

eˆR ⋅ eˆ∗R =

1 ( xˆ − jyˆ ) ⋅ ( xˆ − jyˆ )∗ = 1 (3.88) 2

eˆL ⋅ eˆ∗L =

1 ( xˆ + jyˆ ) ⋅ ( xˆ + jyˆ )∗ = 1 (3.89) 2

and

The linear component polarization vectors are obtainable from circular component polarization vectors as follows:

eˆH = xˆ =

eˆL + eˆR 2

eˆV = yˆ = − j

eˆL − eˆR (3.90) 2

To tie the circular polarization vector back into the time-domain expression for the full electric field, we illustrate with a general right-hand circularly polarized wave. It has amplitude ER0 and polarization vector êR . Then the electric field follows from the (3.86) along with (3.11):



r r xˆ − jyˆ jwt ⎤ ER (t) = Re ⎡⎣ ER e jwt ⎤⎦ = Re ⎡⎣ ER0 ˆe R e jwt ⎤⎦ = Re ⎡⎢ ER0 e ⎥ 2 ⎣ ⎦ (3.91) ER0 coswtxˆ + sinwtyˆ ) = ( 2

which is (2.36b). Our discussion in this section so far has been for plane waves traveling in the + z-direction. A similar approach can be applied to spherical waves where the wave is traveling in the +r-direction; the circularly components of the polarization vectors are

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eˆL =

1 ⎡ (sinf − jcosf)qˆ + (cosf + jsinf)fˆ ⎤⎦ (3.92a) 2⎣

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eˆR =

1 ⎡ (sinf + jcosf)qˆ + (cosf − jsinf)fˆ ⎤⎦ (3.92b) 2⎣

There is further discussion of representations of an electric field in terms of co-polar and cross-polar unit vectors in Section 7.2. 3.8.5  Stokes Parameters for an Orthogonal State The orthogonal representation in the Stokes parameter formulation can be derived from the (ε , τ ) parameters using the Stokes parameters relationships in (3.44). The Stokes matrix for the wave w is



⎡ ⎢ = s [ iw ] ⎢⎢ ⎢⎣

1 s1w s2w s3w

⎤ ⎡ 1 ⎥ ⎢ cos2ew cos2tw ⎥ = ⎢ cos2e sin2t w w ⎥ ⎢ sin2ew ⎥⎦ ⎢⎣

⎤ ⎥ ⎥ (3.93) ⎥ ⎥⎦

The Stokes matrix for the orthogonal state wo is



⎡ ⎢ [ siwo ] = ⎢⎢ ⎢⎣

1 s1wo s2wo s3wo

⎤ ⎡ 1 ⎥ ⎢ cos2ewo cos2two ⎥ = ⎢ cos2e sin2t wo wo ⎥ ⎢ sin2ewo ⎥⎦ ⎢⎣

⎤ ⎥ ⎥ (3.94) ⎥ ⎥⎦

where the values for (ε wo, τ wo) are given in (3.72). To find the relationship between the Stokes parameters for waves w and wo, we substitute (3.72) into (3.94): s1wo = cos2ewo cos2two = cos2( −ew ) cos 2( tw ± 90° ) = −cos2ew cos2tw = −s1w

s2wo = cos2ewo sin2two = cos2( −ew ) sin2( tw ± 90° ) (3.95) = −cos2ew sin2tw = −s2w



s3wo = sin2ewo = sin 2( −ew ) = −sin2ew = −s3w

Thus, the Stokes parameters for the wave state orthogonal to wave state w are

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Polarization State Representations79



s0wo = 1, s1wo = −s1w , s2wo = −s2w , s3wo = −s3w (3.96)

As an example, the Stokes parameters for the LHCP wave and its orthogonal state RHCP are (see [3.49] and Table 3.5): ⎡1 ⎤ ⎢0 ⎥ LHCP: ⎢ ⎥ ⎢0 ⎥ ⎢1 ⎥ ⎣ ⎦



⎡1 ⎤ ⎢0 ⎥ RHCP: ⎢ ⎥ (3.97) ⎢0 ⎥ ⎢ −1⎥ ⎣ ⎦

More examples are found in Table 3.5. 3.8.6  Polarization Ratio for an Orthogonal State The polarization ratio representation expressed in (γ , δ ) from (3.57) is

rLw = tang w (3.98a)



phase ( rLw ) = dw (3.98b)

The polarization ratio for the orthogonal state to ρ Lw is found using (3.73) in the above to give

rLwo = tang wo = tan ( 90° − g w ) = cotg w (3.99a)



phase ( rLwo ) = dwo = dw ± 180° (3.99b)

These two relations lead to the final desired expression for the orthogonal state: rLwo = rLwo e jdwo = cotg w e (

j dw ±180° )

  

rLwo

1 =− ∗ rLw

=−

1 tang w e

− jdw

=−

1 ∗ rLw

(3.100)

Beckmann [8, p. 37] presented a similar proof. As an example, consider orthogonal linear cases of τ w = 45° and τ wo = 135°. Then from Figures 3.1 and 3.2, γ w = 45° and δ w = 0°, and γ wo = 45° and δ wo = 180°. Then from (3.98):

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rLw = tang w e jdw = tan45°e j0° = 1 (3.101)



and the orthogonal state from (3.100) is rLwo = −



1 1 ∗ = − ∗ = −1 (3.102) rLw 1

This verifies the entry in Table 3.5. The relation for orthogonal circular polarization states similar to (3.100) is rCwo = −



1 ∗ (3.103) rCw

The polarization ratio for perfect left-hand circular polarization from Table 3.3 is rCw = j



LHCP (3.104)

Using (3.103) gives the correct orthogonal polarization ratio value:



rCwo = −

1 1 1 = −j ∗ = − ∗ = − −j rCw j

RHCP (3.105)

which verifies the entry in Table 3.5.

3.9  Problems

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1. Use the conversion formulas of (3.3) and (3.4) to find (ε , τ ) from (γ , δ ) for the entries in Table 2.1. 2. For the four elliptical polarization states shown in Figure 3.2 with δ = ±90° and γ = 22.5°, 67.5°, determine (ε , τ ) and axial ratio, R, including sign. 3. Determine the tilt angles of the linear polarizations shown in Figure 3.2 for δ = 0° and 180°. 4. Given a polarization state with ε = −20° and τ = 180°, find the corresponding values of (γ , δ ). 5. Given a polarization state with γ = 90° and δ = 180°, find the corresponding values of (ε , τ ). What is this state?

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Polarization State Representations81

6. Given ε = −45°, find the values of (γ , δ ) for this polarization state. What is the state? 7. For linear polarization, δ and ε are both zero. How are γ and τ related? 8. Derive (3.50). 9. Show that (3.55) satisfies the normalization property of Stokes parameters in (3.45). 10. (a) Use (3.62) to derive (2.30) and (2.31). (b) Use (3.65) to derive (2.33). 11. Show that (3.90) follows from (3.86). 12. Write the polarization vectors for the pair of orthogonal slant linear polarizations with tilt angles 45° and 135°. Prove that they are orthogonal. 13. Derive the Stokes parameters for orthogonal slant linear polarizations with tilt angles 45° and 135°. Prove that they are orthogonal. 14. Verify that (3.82) satisfies (3.77). 15. Find the (ε wo, τ wo) values for a wave orthogonal to a wave with the following polarization state expressed in Stokes parameters:

⎡ 1 ⎤ ⎢0.296 ⎥ [ sw ] = ⎢⎢0.171 ⎥⎥ ⎢0.940 ⎥ ⎦ ⎣



6765_Book.indb 81

16. A right-hand sensed transmitting antenna has a polarization state with an axial ratio of 1 dB at a tilt angle of 10°. The receiver in the communication link is dual-polarized with a co-polarized channel matched to the transmit polarization and with a cross-polarized channel orthogonally polarized to the transmit polarization. What are the co-polarized and cross-polarized receiving antenna polarization states in (ε , τ )? 17. Use (3.57) to find the linear polarization ratios for the following orthogonal linear polarization pairs: (a) HP and VP, and (b) slant LP with τ = 45° and slant VP with τ = 135°. 18. For the orthogonal linear polarization states of slant LP with τ = 45° (state w) and slant LP with τ = 135° (state wo), find the circular polarization ratios of ρ Cw and ρ Cwo. 19. (a) For the polarimeter configuration shown demonstrate which output port corresponds to phasors ER and EL by tracking phasors through the network.

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(b) Show that Channel A gives peak response for the incident wave polarization identified in (a).



20. Sketch the components of the electric field phasor on the surface of a sphere indicating their phase relationship for the RHCP complex vector of (3.92). Do this for the following three directions: (a) θ = 0°, (b) θ = 0° and ϕ = 0°, and (c) θ = 90° and ϕ = 90°.

References [1]

Kraus, J. D., Radio Astronomy, New York: McGraw-Hill, 1966.

[2]

Rumsey, V. H., G. A. Deschamps, M. I. Kales, and J. I. Bohnert, “Techniques for Handling Elliptically Polarized Waves with Special Reference to Antennas,” Proceedings of the IRE, Vol. 39, May 1951, pp. 533–552.

[3]

Poincare, H., Theorie Mathematique de La Lumier, Paris: G. Carre, 1892.

[4]

IEEE Standard Definition of Terms for Antennas: IEEE Standard 145-2013, IEEE, 38 pp., 2013.

[5]

Deschamps, G., and P. E. Mast, “Poincare Sphere Representation of Partially Polarized Fields,” IEEE Trans. on Ant. and Prop., Vol. AP-21, July 1973, pp. 474–478.

[6]

Born, M., and E. Wolf, Principles of Optics, Oxford, UK: Pergamon Press, 1959.

[7]

Stokes, G., “On the Compositional Resolution of Streams of Polarized Light from Different Sources,” Trans. Cambridge Phil. Soc., Vol. 9, Part 3, 1852, pp. 399–416.

[8]

Beckmann, P., The Depolarization of Electromagnetic Waves, Boulder, CO: Golem Press, 1968.

[9]

Yarnall, W. M., “Antenna Design Supplement,” Microwaves, Vol. 4, May 1965, p. 60.

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4 Partially Polarized Waves

4.1 Unpolarized Waves So far we have confined our treatment of polarization to complete polarization. Complete polarization is appropriate for most system applications. Waves can also be unpolarized. An unpolarized wave, also called a randomly polarized wave, is an electromagnetic wave whose time variation is not repetitive. The tip of the electric field vector does not trace out a repeating pattern with time as does a completely polarized wave, but instead is random. Said differently, the polarization parameters (such as axial ratio and tilt angle) are not constant but vary with time. Unpolarized waves arise from the superposition of a large number of statistically independent waves with various polarizations [1, p. 116]. Man-made signals are completely polarized. Unpolarized waves are generated by natural sources of radiation with noise coming from a celestial radio source as a good example. In general, natural emissions consist of both complete polarization and random polarization and are referred to as partially polarized. In this section we present the principles of unpolarized (randomly polarized) waves. The remainder of the chapter treats partially polarized waves. An important theoretical example of a source of unpolarized waves is a blackbody radiator. It is said to be black because at optical frequencies the object will appear to be black in color; it absorbs all incident radiation and reflects 83

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none. A white body, on the other hand, reflects all energy incident upon it. A gray body reflects part of the incident energy. A blackbody reemits the same energy incident upon it, but it does so according to a law of physics referred to as Planck’s law, or the blackbody law. Any object with a temperature above absolute zero temperature (0° Kelvin, or −273°C) emits electromagnetic waves that depend on its temperature as governed by Planck’s law. If an object is heated to a sufficiently high temperture, visible light will be emitted. The emissions will increase in frequency as the body's temperature rises. A campfire is a good example. The portion of the flame farthest from the burning wood is orange and the hottest part near the most intense part of the fire is blue. This is true because the lower end of the visible spectrum is red/orange and the high frequency end is blue, thus demonstrating that radiation increases in frequency with the increasing temperature of the source. A blackbody is an idealization and does not exist. However, some objects, such as those made of carbon, can come close to blackbody behavior. The physics of blackbody radiation was explained in 1900 by Max Planck through the law he developed, Planck’s law. Planck based his work on the assumption that matter emits only small units of energy, or quanta. This revolutionary concept formed the foundation for modern quantum mechanics. Other physicists used Planck’s theory to make further major advances. Albert Einstein in 1905 explained the photoelectric effect using quanta. Niels Bohr in 1913 presented his model for the atom in which the electrons around the nucleus occupy discrete orbits and when transitioning from one orbit to another emit quanta (or packets) of energy. Planck used classical mechanics principles together with quanta of energy in multiples of hf [1, p. 76]. The blackbody law is where

PN ( f ) =

f3 2h (4.1) c2 e hf /kTN − 1

PN = power density of the emitted radiation [W/m2/Hz/rad2] TN = physical temperature of the body [K] h = Planck’s constant = 6.63 × 10 –34 Joule-second k = Boltzmann’s constant = 1.38 × 10 –23 Joule/K This formula shows that as frequency increases the radiated power density increases at first, then peaks and reduces with further increase in frequency.

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Partially Polarized Waves85

The frequency of the radiation peak increases with body temperature. Due to the temperature dependence of blackbody noise, it is often referred to as thermal noise. An approximation to Planck’s law, known as the Rayleigh-Jeans law is given by [1, p. 85]

PN =

2k T f 2 (4.2) c2 N

This formula has radiation increasing linearly with temperature and with frequency squared. It is a valid approximation as long as hf 90°, but the pattern above the ground plane (θ < 90°) is the same as the dipole counterpart antenna. The directivity of a half-wave dipole is 2.15 dB and the quarter-wave monopole directivity is 5.15 dB; the increase of 3 dB is due to the ground plane shorting out all radiation below the ground plane. Low-profile omnidirectional antennas are also popular for compact devices such as handheld cellular telephones. These antennas are not purely omnidirectional, but have a very broad single main beam with little directionality in the horizontal plane. The normal mode helix antenna (NMHA) was one of the first low-profile antennas used on cell phones. It is similar to the axial mode helix of Figure 5.11(a) except that the NMHA is electrically small with a diameter and length that are much smaller than a wavelength. The NMHA is very simple to design and inexpensive to produce. However, lower-profile (smaller height) antennas have become more popular than the NMHA in compact devices such as cell phones, which we treat next. The low-profile antennas of Figure 5.4(b), (c), and (d) have a ground plane, which is often an existing portion of the device the antenna is mounted in. The pattern is broad above the ground plane, quasi-omnidirectional in the horizontal plane, and with reduced radiation below the ground plane. The inverted-L antenna (ILA) of Figure 5.4(b) is essentially a bent monopole antenna. The short vertical segment is responsible for the radiation. It is relatively narrowband with only a few percent bandwidth. Bandwidth can be increased in several ways, including adding a small conducting loop near the feed, adding a parasitic wire parallel to it, and using wide metal strips instead of wire (forming a planar inverted-F antenna [PIFA]) [1, pp. 502–506]. The PIFA of Figure 5.4(c) has about 8% bandwidth, which is adequate for many single-band wireless applications. Multiband versions of the PIFA for use in applications such as cell phones are created by adding arms of various lengths that resonate in desired additional operating bands. The microstrip antenna (MSA) of Figure 5.4(d) is perhaps the lowest profile quasi-omnidirectional antenna. However, special measures must be taken to achieve more than a few percent bandwidth. The antennas mentioned so far are vertical linearly polarized because in the horizontal plane (xy-plane) the radiation is vertically polarized. It is sometimes desirable to have a horizontal linearly polarized omindirectional antenna. The big wheel antenna is a full wavelength wire formed into a circle in the horizontal plane and with three feed points equally spaced around the loop to create a smooth current distribution. Transmission lines from the feed points are joined at the loop center. Measured results with a big wheel antenna

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Polarization in Electromagnetic Systems

constructed for operation at 2.45 GHz had a co-polarized pattern in the horizontal plane that was very close to omnidirectional and with cross-polarization that averaged 13 dB down from the co-polarized level [9]. A horizontally polarized omindirectional antenna that is simpler to implement than the big wheel is the halo antenna. It is popular in the amateur radio community, especially for use on vehicles. The halo antenna is a half-wave dipole shaped into a circle, but the ends do not quite meet. An impedancematching section at the feed point, such as a gamma match, is needed. 5.3.2  Circularly Polarized Omnidirectional Antennas The circularly polarized microstrip antenna is very low profile, just like its linearly polarized counterpart. There are several ways to make a MSA circularly polarized [1, pp. 475–476]. But as with any microstrip antenna, special measures are needed to achieve more than narrow bandwidth. However, a narrowband MSA is well-suited to GPS (more generally, called satellite-based global navigation) applications because the GPS signal is very narrowband and is circularly polarized. Because the circularly polarized MSA can be made inexpensively, it is widely used in GPS receivers.

5.4 Directional Antennas Directional antennas have a prominent main beam and reduced radiation in the side directions. In terms of Figure 5.3 with the antenna main beam in the z-direction, the pattern is several decibels down in the horizontal plane (xyplane). An alternate term for a directional antenna is a narrowbeam antenna. Directional antennas have a many applications, including point-to-point communications, direction finding, and sensing. 5.4.1  Linearly Polarized Directional Antennas A dipole antenna can be used as the primary radiator (the driver) and with one wire nearby and parallel to the driver (the reflector) and several parallel wires on the opposite side of the driver (the directors) to form the Yagi-Uda array (Yagi) of Figure 5.5(a). The Yagi has a narrow beam in the direction of the directors. The beamwidth in the E-plane (i.e., vertical plane of Figure 5.5[a]) is wider than the beamwidth in the H-plane. The basic three-element Yagi (one reflector, one driver, and one director) has a gain of about 9 dB [1,

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Antenna Polarization113

(a)

(b)

(c)

Figure 5.5  Linearly polarized directional antennas: (a) Yagi-Uda array, (b) slot antenna, and (c) horn antenna.

p. 168]. Adding directors increases the gain. The polarization is, of course, linear oriented parallel to the driven element. Very high gains can be achieved using an aperture antenna, which is an antenna with electrically large physical aperture (i.e., an opening through which electromagnetic waves travel outward). Thus, aperture antennas are usually operated at UHF frequencies and above. The simplest aperture antenna is

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the slot antenna of Figure 5.5(b). While the slot antenna is about a half wavelength long and does not fit the definition of an aperture antenna requiring an electrically large aperture of multiple wavelengths, it is usually included with aperture antenna discussions because of its physical aperture. The ground plane around the slot antenna reduces radiation in the side and back directions to low levels. The main beam peak is perpendicular to the aperture. The polarization is linear with tilt orientation aligned with the electric field direction across the narrow dimension of the aperture as shown in Figure 5.5(b). Large directivities (gains) are achieved with the horn antenna of Figure 5.5(c), which resembles a funnel that acts to guide emerging waves in the forward direction. The polarization of a horn antenna is determined by the excitation probe in the connecting waveguide. The probe in Figure 5.5(c) is vertical, so the radiation will be vertical linear and parallel to the aperture field shown. The polarization properties of the horn antenna, and most aperture antennas, tend to remain relatively stable over the main beam. Also, crosspolarization levels can be made quite low, more than 40 dB below the co-polar main beam peak. In the side lobe region of directive antennas, polarization is dominated by construction details and fields diffracted from antenna and support hardware. Therefore, the polarization state of a given side lobe can be quite different from that of the main beam. A reflector antenna offers the potential of extremely high gain and has some interesting polarization properties. The antenna system of Figure 5.6(a) consists of a parabolic reflector of diameter D = 100 wavelengths and a focallength-to-diameter ratio, F/D, of 0.5. The feed antenna is a half-wave dipole at the focal point. In general, this axisymmetric parabolic reflector is fed at the focal point by a feed antenna that determines the polarization of the reflectorfeed combination. A linearly polarized feed, such as a dipole, will have crosspolarized components in the aperture plane of the reflector that are generated by the curvature of the reflector. The polarization and radiation properties of an axisymmetric parabolic reflector are presented in Figure 5.6. The radiation patterns of Figure 5.6(b) and 5.6(e) are rectangular-decibel plots that show details of main beam and first several side lobes. The far-field (secondary pattern) of the reflector-feed combination will have no cross polarization in the principal planes due to cancellation of cross-polar components in these symmetry planes. However, there will be cross-polarization lobes off the principal axes with peaks in the 45° planes, as seen in Figure 5.6(d) and (e). In contrast to the parabolic reflector, a cylindrical reflector does not depolarize [10]. The radiation properties of the reflector antenna of Figure 5.6 were evaluated using GRASP, a sophisticated physical optics reflector antenna computer code [11]. The co-polar radiation patterns in the principal planes

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are shown in Figure 5.6(b). The main beam peak of the patterns at θ = 0 is 43.8 dB, corresponding to the maximum gain value. The ϕ = 0° plane pattern has lower side lobes and a wider main beam than the ϕ = 90° plane pattern because the illumination in the x-direction in the aperture is tapered in amplitude due to a decrease in the feed dipole radiation with off-axis angles. In the ϕ = 90° plane, the dipole feed has an omnidirectional pattern, leading to a stronger illumination of the reflector in the yz-plane, and thus, narrower

Figure 5.6  An axisymmetric parabolic reflector antenna of diameter D = 100 λ and F/D = 0.5 with a half-wave dipole feed at the focus. (a) Reflector geometry, (b) principal plane patterns, (c) co-polar normalized contours, and (d) cross-polar normalized contours. (The intensity in the contours is proportional to shading and the dotted area is between −30 and −40 dB below the peak.) (e) The ϕ = 45° plane co-polarized and cross-polarized patterns. The peak cross-polarization level in the 45° plane is −26.3 dB below the co-polar peak.

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Figure 5.6  (Continued)

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Figure 5.6  (Continued)

beamwidth for the reflector pattern. The cross-polarization is zero in both of these planes. Figure 5.6(c) and (d) show contour maps of the normalized co-polarized and cross-polarized radiation. Note that the patterns of Figure 5.6(b) are cuts through the contour pattern of Figure 5.6(c) along the u-axis and v-axis, respectively. The nonexistent cross-polarization in the principal planes is evident in Figure 5.6(d). However, off the principal planes there will be cross-polarization. The co-polarized and cross-polarized patterns in the ϕ = 45° plane are plotted in Figure 5.6(e). Cross-polarization is strongest in this ϕ = 45° plane. The large cross-polarization lobes that peak near the edges of the co-polar main beam are typical for reflector antennas. The ideal reflector feed antenna would have an axially symmetric pattern (i.e., the same beamwidth in all planes) and a unique phase center. This Huygens source produces an illumination that cancels the reflector-induced cross-polarization, yielding pure linear polarization in the aperture [12]. When the classical Huygens source feeding an axisymmetric parabolic reflector is analyzed using physical optics techniques a small reflector-induced crosspolarization is found [7]. However, this contribution is very small for narrowbeam patterns as obtained from reflectors many wavelengths in diameter.

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The primary depolarization effect of symmetric and offset reflector antennas has been found to manifest itself as degradation of the axial ratio [13]. The feed antenna is usually responsible for the dominant contribution to the cross-polarization in the secondary pattern. A conical corrugated feed horn antenna has very good pattern symmetry and phase center characteristics and it is a popular feed antenna [14]. 5.4.2  Circularly Polarized Directional Antennas In common with omnidirectional CP antennas, special measures are required to produce circular polarization with directive antennas. Two general methods are used to generate circular polarization. Type 1 CP antennas are those antennas that produce CP by virtue of their unique physical structure. Examples are helix and spiral antennas, shown in Figure 5.11, which will be discussed in the next section. The sense of polarization is determined by the sense of the winding of the helix and the spiral. Type 2 CP antennas contain hardware to explicitly generate spatially orthogonal components that are in-phase quadrature. Type 1 antennas do this too, but naturally due to the antenna’s internal structure. An example of a Type 2 CP antenna is the turnstile antenna shown in Figure 5.7. The turnstile is made of vertical and horizontal dipoles (usually half-wave dipoles) that are perpendicular to meet the requirements of equal amplitude and spatially orthogonal linear components. The quadrature phase condition is met in this case with feed lines such that the vertical dipole leads the horizontal dipole by 90° in phase. In general, identical orthogonal linear antennas excited 90° out of phase produce circular polarization, with the sign of the 90° phase determining the sense. Because the dipoles are also equal in length, the x- and y-components of the electric field are equal in amplitude as well as being in phase quadrature; this yields circular polarization normal to the plane of the crossed dipoles. Its operation is understood by imagining the turnstile antenna to be transmitting. Using the left- and right-hand rules we see that there is left-and right-hand CP radiation along the +z- and −z-axes of Figure 5.7, respectively. The electric field vector rotates clockwise (counterclockwise) when viewed from the +z (−z) axis. Thus, along the +z-axis where θ = 0° the rotating electric field starts along the y-axis due to its advanced phase and a quarter period later it is along the x-axis, so the sense of the radiation is LHCP. In the −z-direction where θ = 180° the electric field rotation sense is opposite, making the radiation RHCP. In the y-direction (θ = 90°, ϕ = 90°) the radiated electric field is parallel to the x-directed dipole, which is x-polarized

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Figure 5.7  Turnstile antenna.

so the polarization is horizontal linear. Similarly, in the x-direction (θ = 90°, ϕ = 0°) the radiated electric field is parallel to the y-directed dipole, which is y-polarized so the polarization is vertical linear. It may seem surprising that the turnstile radiates RHCP in one direction and LHCP in the opposite direction, but the simple analogy of the hands of a clock explain the situation. When viewed from the front, the hands of a clock rotate clockwise. However, when viewed from the rear the clock hands rotate in a counterclockwise sense. Phase quadrature excitation for the turnstile antenna can be accomplished using an extra quarter-wavelength section of transmission line from a common transmission line before the dipole that lags in phase; see Figure 6.6. This, of course, is a narrow bandwidth method. Improved bandwidth is obtained using a quadrature hybrid coupler in the feed network. The turnstile antenna demonstrates a general trend for circularly polarized antennas: circular polarization purity tends to deteriorate (axial ratio increases) for increasing angles off the main beam peak. In Figure 5.7 there is pure LHCP in +z-direction. For increasing angles off the z-axis in the xz-plane at long distances from the transmitting turnstile, the axial ratio of radiation increases from 0 dB (CP) to ∞ dB (LP) when the x-axis is reached at 90° off the z-axis. This is easy to understand because the horizontal dipole presents an end view of zero profile and has no contribution to radiation in the x-direction. The radiation is linear polarization parallel to the y-axis in the x-direction. Similarly, along the y-axis, the radiation is LP parallel to the x-axis (and has infinite axial ratio).

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The normalized radiation pattern for the turnstile antenna is [3, p. 158]



F (q) =

1 + cos2 q (5.16) 2

This pattern expression has peaks in the θ = 0° and 180° directions. The lowest pattern points are in the xy-plane (θ = 90°) where the level is down by 3 dB. The pattern is null-free and the directivity is 3 dB. So we see that the turnstile is a null-free antenna with maxima in the +z (−z) directions that are LHCP (RHCP) polarization. The half-power beamwidth is 180° because F(θ = 90°) = 0.707. The pattern expression of (5.16) is for the total radiation and is not relative to a specific polarization state. The pattern relative to LHCP in the forward direction (+z-axis) is [3, p. 158]:

1 FCP (q) = (1 + cosq) (5.17) 2

The half-power beamwidth for this LHCP pattern is 131°. The turnstile antenna has a bidirectional pattern with a wide lobe in both the +z and −z-directions (of opposite CP senses). A unidirectional pattern can be created using the standard technique of backing the crossed dipoles with a ground plane (in practice, a relatively large good conductor) that is parallel to the plane of the dipoles and a quarter wavelength behind them. The wave from the dipoles encounters 180° phase change upon reflection from the ground plane and another 180° phase change due to the round-trip distance of a half wavelength from the source dipole and back again. Thus, the total phase shift of 360° brings the reflected wave inphase with the direct radiation from the dipoles, giving a maximum in the +z-direction. There is no radiation in the back hemisphere, so there is only one major lobe to the pattern. Array theory can be used to find the contribution of the dipoles and their images in the ground plane, resulting in a total pattern relative to circular polarization for the ground-plane backed turnstile antenna [3, p. 160]:



1 p FCP (q) = (1 + cosq) cos ⎡⎢ (1 + cosq)⎤⎥ (5.18) ⎣2 ⎦ 2

The half-power beamwidth is 98.4°, which is much narrower than the corresponding beamwidth for the turnstile without a ground plane of 131°. See [15] for a review of turnstile principles and applications.

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In Type 2 CP antennas, space quadrature is usually accomplished using spatially orthogonal linear elements such as with the turnstile antenna. In contrast, within Type 1 CP antennas the spatial and temporal quadrature conditions occur naturally; see Figure 5.11 for examples. Examples of Type 2 CP antennas are illustrated in Figure 5.8. The crossed Yagi antenna of Figure

(a)

(b)

(c)

Figure 5.8  Examples of Type 2 circularly polarized antennas. The two input terminals are combined ±90° out of phase and equal in amplitude to produce circular polarization. (a) Crossed Yagi-Uda antennas, (b) dual-polarized horn antenna, and (c) reflector antenna with crossed dipole feed.

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5.8(a) is two perpendicular Yagi antennas like the one in Figure 5.5(a). The two driven dipoles are fed with equal amplitude and 90° out of phase. The circularly polarized horn antenna of Figure 5.8(b) is similar to the linearly polarized horn of Figure 5.5(c) but has a two-probe feed with the probes fed with equal amplitude and 90° out of phase. A reflector antenna can be made to be circularly polarized by simply changing the feed antenna to a circularly polarized one, as shown in Figure 5.8(c). The feed is illustrated as a turnstile antenna, but a variety of CP feeds can be used. It is important to note that the sense of a CP feed antenna should be opposite of the desired sense of CP radiated by the reflector. This is because sense of feed radiation is reversed after reflection from the parabolic reflector antenna. Offset reflectors have interesting polarization properties. A linearly polarized feed antenna with an axially symmetric pattern feeding an offset parabolic reflector, as shown in Figure 5.9, results in no cross-polarization in the plane of symmetry (xz-plane), but does have a cross-polarization component in the plane of asymmetry (yz-plane) that increases with offset angle, θ o [10]. If the feed is circularly polarized, there is no cross-polarization, but the main

Figure 5.9  Offset reflector antenna.

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beam peak is displaced off the z-axis (in the yz-plane of Figure 5.9) [10, 16, 17]. This so-called beam squint tilts the main beam off the axis of the reflector (z-axis) in opposing directions for opposite senses of CP. Note that the sense of polarization is opposite that of the feed because of the sense reversal created by the reflection process. An interesting way to create a directive pattern that is circularly polarized is to employ an array of linearly polarized elements that are disposed in subarrays of 2 × 2 elements with the four elements progressively rotated in space (for example, 0°, 90°, 180°, 270°) and each element fed with the same magnitude and the same feed phase progression as the spatial orientation [18]. This is similar to the turnstile antenna with two elements with both 0° and 90° spatial and phase progression, but the four-element subarray has more symmetry, leading to better CP stability at off-axis angles and wider bandwidth. The advantage of this approach over using an array of circularly polarized elements is that linearly polarized elements are easier to realize in practice and require only one feed at each element, leading to reduced feed complexity. Huang [18] has built a number of arrays using 2 × 2 subarrays of linearly polarized microstrip elements and obtained wide axial ratio bandwidth (10% or more), a very symmetrical pattern, and high CP quality when the beam is phase scanned.

5.5 Broadband Antennas The bandwidths of the antennas in Figure 5.4 have rather narrow bandwidth. The half-wave dipole has a bandwidth of about 10% depending on the wire diameter; the inverted-L has 1% bandwidth; and the planar inverted F antenna has 8% bandwidth. The microstrip antenna has the lowest profile, but at a price. Its bandwidth can be as low as 1%, which is not adequate for many modern applications which, in fact, often require a broadband antenna. A broadband antenna (or wideband antenna) is defined to be an antenna with at 2:1 bandwidth (i.e., Br = 2). In this section we present a few prominent examples of broadband antennas that are linearly or circularly polarized. Wide bandwidth is achieved by emphasizing angles instead of fixed physical lengths in the physical structure of the antenna. See [1, Ch. 7] for a complete treatment of broadband antennas. 5.5.1  Linearly Polarized Broadband Antennas One of the earliest broadband antennas was the log-periodic dipole array (LPDA) antenna of Figure 5.10(a). It is made up of progressively smaller dipoles, each

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(a)

(b)

(c) (c)

Figure 5.10  Examples of linearly polarized broadband antennas: (a) log-periodic dipole array antenna, (b) log-periodic toothed trapezoid wedge antenna, and (c) the foursquare antenna array.

connected to the same feed line. In spite of having many fixed physical length segments the LPDA has an angular profile, leading to bandwidths much larger than 10:1. A sheet metal version of the LPDA formed into a wedge shape is the log-periodic toothed trapezoidal wedge antenna of Figure 5.10(b). Its emphasis on angles leads to wide bandwidth with a narrower beamwidth than the LPDA. The polarization of these two log-periodic antennas is linear with orientation parallel to the wires of LPDA and parallel to the teeth edges of the log-periodic toothed trapezoidal wedge antenna. The foursquare antenna array of Figure 5.10(c) is a very compact, lowprofile planar antenna capable of 3.5:1 bandwidth [1, p. 335]. Each foursquare

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element of the array consists of opposing square elements that are driven with a balanced feed, forming linear polarizations on the diagonal. Thus, dual-linear orthogonal polarizations are possible with the foursquare antenna. A variation of the foursquare element that is specifically designed to be used alone or in an array that is not closely spaced (like the foursquare antenna array) is the fourpoint antenna. The fourpoint antenna is a dual-linearly polarized antenna capable of 2.7:1 bandwidth [1, p. 544]. 5.5.2  Circularly Polarized Broadband Antennas Examples of Type 1 circularly polarized broadband antennas are shown in Figure 5.11. The axial mode helix antenna of Figure 5.11(a) has a circumference of about one wavelength and a bandwidth of about 2:1. It has a nearly circularly symmetric endfire beam off the end of the helix. The size of the helix antenna can be reduced by introducing internal stubs in the windings, forming the stub-loaded helix antenna. The stub-loaded helix antenna has properties similar to the axial mode helix, but with reduced length and diameter [19]. Spiral antennas are circularly polarized and have broader beamwidth and wider bandwidth than the helix antenna. The operation of the Archimedean spiral antenna of Figure 5.11(b) and the equiangular spiral antenna of Figure 5.11(c) is explained in [1, Sec. 7.7]. The sense of polarization for both the helix and spiral antennas is determined by the sense of the windings, so the left-hand wound helix of Figure 5.11(a) produces LHCP radiation. The spiral antennas of Figure 5.11(b) and (c) are in free space and create two beams, one out of the page and one into the page. They are right-hand spirals, so they produce RHCP out of the page and LHCP into the page; this behavior is similar to that of the turnstile antenna. Spirals are usually backed with the ground plane to produce a unidirectional beam. The foursquare antenna array of Figure 5.10(c) can be used to generate circular polarization if each pair of opposing squares is fed in the normal fashion for Type 2 circular polarization.

5.6  Polarization Purity of Circularly Polarized Antennas There are several sources of error that will reduce the polarization purity of an antenna. Polarization purity is most commonly quantified using axial ratio. An axial ratio of 1 (0 dB) is pure circular polarization; see Section 2.3 for circular polarization definitions and principles. Therefore, as axial ratio increases above 0 dB the polarization purity degrades. Polarization purity will be a function of

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Figure 5.11  Examples of Type 1 circularly polarized broadband antennas: (a) axial mode helix antenna, (b) Archimedean spiral antenna, and (c) equiangular spiral antenna.

frequency, and thus will impact an antenna’s polarization bandwidth. Circular polarization purity decreases as phase and amplitude errors in the orthogonal linear components increase. For example, the turnstile antenna of Figure 5.7 is perfectly circularly polarized (in the on-axis directions of radiation) when the perpendicular linear components are equal in amplitude and 90° out of phase.

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If the components are not exactly perpendicular or are elliptically polarized rather than perfectly linear, perfect circular polarization will not be produced. Further, if the components are not equal in amplitude and/or not 90° out of phase, the radiated waves will not be perfectly circularly polarized. An important question that arises when designing a turnstile type antenna is how accurately do the excitations need to be realized? Convenient results for the axial ratio values in the presence of errors in the perpendicular linear components are available in the literature [20]. Figure 5.12 shows the axial ratio of the wave generated by orthogonal linear components with amplitude and phase errors [21]. The values are symmetric with respect to amplitude and phase imbalance; that is, whether the x-component is larger than the y-component or vice versa, or whether the phase error is positive or negative [22]. As an example of how to use Figure 5.12, consider two orthogonal linear components that have an amplitude difference of 1.5 dB and a phase difference of 15°. The graph shows the resulting wave will have an axial ratio of 2.5 dB. Phase errors can be caused by differences in feed line lengths in Type 2 CP antennas. Any method of Type 2 circular polarization will have sources of errors, reducing the CP purity. For example, amplitude and phase errors will be introduced in devices such as a quadrature hybrid. Type 1 CP antennas avoid feed issues and CP purity is not as frequency sensitive as Type 2 CP antennas, but of course, all CP antennas are not purely CP polarized.

Figure 5.12  The axial ratio of the wave generated by orthogonal linear components for errors in amplitude and phase of those components. (Source: D. Pozar, “Antenna Designer’s Notebook: Axial Ratio of Circularly Polarized Antennas with Amplitude and Phase Errors,” IEEE Ant. and Prop. Magazine, Vol. 32, Oct. 1990, pp. 45–46. ©1990 IEEE. Reprinted with permission.)

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5.7  Problems



1. Calculate percentage bandwidth values for ratio bandwidths of: 1.105, 1.67, 2, 3, and 10. 2. For a y-polarized test antenna sketch the co-polarization and crosspolarization unit vectors, based on (5.7), in the principal planes of a quadrant of a spherical surface. 3. Verify that the relations in (5.14) are satisfied. 4. Verify that (5.1) is satisfied. 5. For the turnstile antenna in Figure 5.7, sketch the full two-wire transmission lines connecting to a single feeding transmission line. Show the extra quarter-wavelength of line required to produce the phase quadrature condition. 6. Write a program to plot the pattern of a turnstile antenna using (5.12). Plot the pattern in polar-linear form. 7. What is the sense of CP radiated by the following antennas: (a) The helix of Figure 5.11(a) in the upward direction, and (b) the Archimedean spiral of Figure 5.11(b) for radiation out of the page. 8. Two orthogonal dipoles are used to generate a CP wave. The amplitudes of excitation are 1.1 and 0.825 and they are excited 80° out of phase. Evaluate the axial ratio in decibels of the generated wave. Compare to the value you find to that from Figure 5.12.

References [1]

Stutzman, W. L., and G. A. Thiele, Antenna Theory and Design, Third Edition, New York: John Wiley and Sons, 2013.

[2]

Mathis, H. F., “A Short Proof that an Isotropic Antenna is Impossible,” Proc. IRE, Vol. 39, Aug. 1951, p. 970.

[3]

Mott, H., Polarization in Antennas and Radar, New York: Wiley, 1986.

[4]

Fulton, F., “The Combined Radiation Pattern of Three Orthogonal Dipoles,” IEEE Trans. Ant. and Prop., Vol. AP-13, March 1965, pp. 323–324.

[5]

Saunders, W. K., “On the Unity Gain Antenna,” in Electromagnetic Theory and Antennas, Part 2, E.C. Jordan (ed.), Oxford, UK: Pergamon Press, 1963, pp. 1125–1130.

[6]

Scott, W. G., and K. M. Soo Hoo, “A Theorem on Polarization of Null Free Antennas,” IEEE Trans. on Ant. and Prop., Vol. AP-14, Sept. 1966, pp. 587–590.

[7]

Collin, R. E., Antennas and Radiowave Propagation, New York: McGraw-Hill, 1985, pp. 216–224.

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[8]

Ludwig, A. C., “The Definition of Cross Polarization,” IEEE Trans. on Ant. and Prop., Vol. AP-21, Jan. 1973, pp. 116–119.

[9]

Dietrich,C., K. Dietze, J. R. Nealy, and W. Stutzman, “Spatial, Polarization, and Pattern Diversity for Wireless Handheld Terminals, IEEE Trans. on Ant. and Prop., Vol. 49, Sept. 2001, pp. 1271–1281.

[10] Chu, T. -S., and R. H. Turrin, “Depolarization Properties of Offset Reflector Antennas,” IEEE Trans. on Ant. and Prop., Vol. AP-21, May 1973, pp. 339–345. [11] GR ASP-Single and Dual Ref lector Antenna Program Package, TICR A Eng., Copenhagen, Denmark. [12] Koffman, I., “Feed Polarization for Parallel Currents in Reflectors Generated by Conic Sections,” IEEE Trans. on Ant. and Prop., Vol. AP-14, Jan. 1966, pp. 37–40. [13] DiFonzo, D. F., “The Measurement of Earth Station Depolarization Using Satellite Signal Sources,” COMSAT Laboratories Tech. Memo. CL-42-75, 1975. [14] Clarricoats, P. J. B., and A. D. Olver, Corrugated Horns for Microwave Antennas, Stevenage Herts: IET, 1983. [15] Ta, S., Park, I., and R. Ziolkowski, “Crossed Dipole Antennas: A Review,” IEEE Ant. and Prop. Mag., Vol. 57, Oct. 2015, pp. 107–122. [16] Terada, M., and W. Stutzman, “Cross Polarization and Beam Squint in Single and Dual Offset Reflector Antennas,” Electromagnetics, Vol. 16, Nov/Dec 1996, pp. 633–650. [17] Duan, D. W., and Y. Rahmat-Samii, “Beam Squint Determination in Conic-Section Reflector Antennas with Circularly Polarized Feeds,” IEEE Trans. on Ant. and Prop., Vol. 39, May 1991, pp. 612–619. [18] Huang, J., “A Technique for an Array to Generate Circular Polarization with Linearly Polarized Elements,” IEEE Trans. on Ant. and Prop., Vol. AP-34, Sept. 1980, pp. 1113–1124. [19] Barts, R., and W. Stutzman, “Stub Loaded Helix Antenna,” U.S. Patent No. 5,986,621, Nov. 16, 1999. [20] Parekh, S., “Antenna Designer’s Notebook: Simple Formulas for Circular-Polarization Axial Ratio Calculations,” IEEE Ant. and Prop. Magazine, Vol. 33, Feb. 1991, pp. 30–32. [21] Pozar, D. M., “Antenna Designer’s Notebook: Axial Ratio of Circularly Polarized Antennas with Amplitude and Phase Errors,” IEEE Ant. and Prop. Magazine, Vol. 32, Oct. 1990, pp. 45–46. [22] Keen, K. M., “Feeder Errors Cause Antenna Circular Polarization Deterioration,” Microwave Systems News, Vol. 14, May 1984, pp. 102–108.

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6 Antenna-Wave Interaction 6.1 Polarization Efficiency One of the most important uses for polarization theory is in evaluating the interaction of an electromagnetic wave and a receiving antenna. A typical situation in communication system design is matching the polarization of an incoming wave from a distant transmitting antenna to the polarization of a receiving antenna. This matching of wave and antenna polarization provides maximum transfer of the power available to the receiver. Polarization is also important in sensing systems where the polarization state of the incoming wave must be determined accurately. Discussions of antenna-wave interaction in this chapter are in terms of the receiving antenna properties and the parameters of the incoming wave. If the wave is generated by a remote transmitting antenna rather than by a natural process and the intervening propagation path is not a depolarizing medium (which is discussed in Chapter 8), then the wave polarization arriving at the receiver is that of the transmit antenna. The antenna-wave interaction process can be treated by evaluating the conversion of incident wave power density to antenna output power or by evaluating the conversion of incident complex field intensity to complex antenna output voltage. The former approach is the most common and is usually sufficient. The latter approach is necessary if phase information is required. In this chapter we 131

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discuss both. Several antenna-wave interaction power equations are presented for easy evaluation of received signal calculations. The general situation of a plane wave incident on a receiving antenna is shown in Figure 6.1. The receiving antenna converts wave power density S in W/m2 incident on it to the power PD delivered to a load. The power available in watts at the output terminals of the antenna is P and assumes the antenna is matched to the polarization of the wave and is matched to its load impedance. The available output power depends on the polarization of both the incoming wave and of the receiving antenna. A measure of the conversion of wave power density to power available from the receiving antenna is the polarization efficiency, p [1]. It is also frequently called the polarization mismatch factor [1] and occasionally the polarization match factor [2, p. 187]. The quantitative definition of polarization efficiency is developed in this section using a derivation that is based on the receiving antenna quantities and on the physics of the incoming wave. This discussion also serves to explain how a wave interacts with a receiving antenna. The flux density in a plane wave incident on an antenna from (3.23) is

S=

1 ( E 2 + E22 ) (6.1) 2h 1

where E1 and E2 are the x-and y-component electric field amplitudes for a z-directed wave. Assume for the moment that the receive antenna peak pattern response is aligned toward the direction of arrival of the plane wave, and

Figure 6.1  The general case of a plane wave incident on a receiving antenna.

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further, that the receive antenna is matched in polarization to that of the wave. Then the power available from the receiving antenna is

P = SAe (6.2)

This equation serves as a definition of the effective aperture (effective area) of a receiving antenna: Ae = P/S. It assumes there is a perfect polarization match between the wave and antenna. Effective aperture is the ratio of the power received (in watts) to the incident wave power density (in watts/m2). Thus, effective aperture has units of m2 as expected from an area quantity, and is a measure of the collecting area of the antenna. In general, Ae is directiondependent with its angle dependence being that of the power pattern, as expressed by

2

Ae (q,f) = Ae F (q,f) (6.3)

As an aside, we introduce a fundamental relationship in antennas. Effective aperture is an effective collecting area for an antenna, and it is directly related to gain through [3, p. 108]

Ae =

l2 G (6.4) 4p

Comparing this to (5.3) gives a fundamental relationship connecting antenna quantities:

er l2 = Ω A Ae (6.5)

This equation represents the trade-off between beam solid angle and effective aperture; that is, the larger the effective aperture, the smaller is the beam solid angle at a given frequency (wavelength). If the receive antenna is lossless so that er = 1, the effective aperture is replaced by the maximum effective aperture, Aem, so

l2 = Ω A Aem (6.6)

The received power relationship of (6.2) is modified to include an imperfect match of antenna polarization to incoming wave polarization by including polarization efficiency p as follows:

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Pr = pSAe (6.7)

where Pr is the actual power received by the receive antenna (assuming there is an impedance match to the receiving circuitry). Rearranging this equation yields the formal definition of polarization efficiency:



p=

Pr SAe (6.8)

This is the ratio of the power received by an antenna from a plane wave of arbitrary polarization to the power that would be received by the same antenna from a plane wave of the same flux density and same direction of propagation whose polarization state has been adjusted for maximum received power [1]. Polarization efficiency varies from a total mismatch condition (p = 0) to the perfect polarization match case (p = 1) where the wave polarization state is identical to that of the antenna. Therefore

0 ≤ p ≤ 1 (6.9)

Figure 6.2 gives the polarization efficiency values for several common combinations of wave and antenna polarization. Note that the values are symmetric about the main diagonal, indicating that wave and polarization states can be interchanged. For example, a slant-45 LP wave and a RHCP antenna lead to an efficiency of 0.5, and a RHCP wave and a slant-45 LP antenna also lead to an efficiency of 0.5. The polarization efficiency values for more general wave and antenna polarization cases are given in Table 6.1. Again that the wave and antenna polarizations can be interchanged with the exception that partial polarization applies only to the wave, such as Case e with an unpolarized wave incident on any antenna. If the wave is completely polarized and the antenna is matched to it with an identical state (Case a), all power is available and p = 1. If, on the other hand, the antenna polarization state is orthogonal to that of the wave (Case b), no power is received and p = 0. In practice, perfect orthogonality of the antenna to the wave is not possible, but the output level in the cross-polarized state often is 40 dB or more below that in the co-polarized state. Example Cases c and d in Table 6.1 have polarization efficiencies of one half. In Case c, the incident wave is circularly polarized and receive antenna is linearly polarized with any tilt angle. In Case d, a vertical linearly polarized wave is received by an antenna linearly polarized with a 45° tilt angle. These general results will be verified later after we develop techniques for quantitative evaluation

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Antenna-Wave Interaction135

Figure 6.2  Polarization efficiency values for several combinations of several common wave and receiving antenna polarizations.

of polarization efficiency in the next section. Half the available power is also lost when the wave is randomly polarized (Case e). This is because on average the incoming wave is identically polarized to the antenna half the time and orthogonally polarized to the antenna the other half of time. In addition to polarization mismatch, the power delivered to a load attached to the receiving antenna is reduced by the mismatch of the antenna impedance to load impedance. Impedance mismatch is accounted for similar to polarization mismatch by defining an impedance mismatch factor, q. Impedance mismatch factor is a power efficiency that varies from 0 for a complete mismatch to unity for a perfect impedance match (i.e., 0 ≤ q ≤ 1). See [3, p. 112] for details on how to calculate impedance mismatch. Including both

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Table 6.1 Polarization Efficiency Examples Wave Polarization State

Antenna Polarization State

Polarization Efficiency, p

a

Any complete polarization state

Identical to wave state

1

b

Any complete polarization state

Orthogonal to wave state

0

c

Circular

Linear, any tilt angle

0.5

d

Vertical linear

Linear, 45° tilt angle

0.5

e

Unpolarized

Any state

0.5

Case

polarization and impedance mismatches give the power delivered to the load attached to the receive antenna as

PD = pqP = pqSAe (6.10)

(See Figure 6.1.) It is common to convert efficiency factors into losses expressed in decibels. Because polarization efficiency is a power ratio, the dB form is found using 10 log of the power ratio:

Lp = −10log p [dB]

polarization loss (6.11)

This is often called polarization loss. For matched cases, Lp = 0 dB. For orthogonally polarized situations, Lp = ∞ dB. When the polarization efficiency value is one half (Cases c, d, and e), Lp = 3 dB. Impedance mismatch factor of (6.10) can be expressed as a loss in dB in a similar fashion. Losses due to polarization and impedance mismatch are commonly used in communication link power budget calculations.

6.2  Calculation of Polarization Efficiency Any of the several ways to represent polarization state introduced in Chapter 3 can be used to calculate polarization efficiency. That is, both the wave and antenna polarization states are cast in form of the same representation. This

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Antenna-Wave Interaction137



section presents the formulas used to calculate polarization efficiency for each representation, along with their advantages and disadvantages, as well as numerous worked examples. 6.2.1  Polarization Efficiency Evaluation Using the Poincaré Sphere The Poincaré sphere offers the most convenient way to visualize all polarization states; see Section 3.3 for a discussion of the Poincaré sphere. It also offers a very conceptual approach to polarization efficiency evaluation. Figure 6.3 illustrates the general case. The wave polarization state is denoted as w and is a distance d from the center of the sphere, which is equal to the degree of polarization. The completely polarized portion of the wave is located on the surface of the sphere of unity radius. The antenna polarization state, which is completely polarized, is represented by the point a on the surface of the sphere. The angular separation between w and a, denoted ∠wa, along with degree of polarization completely determine polarization efficiency through the following simple equation [4, Part II]: p=



1 [1 + cos(∠wa)] (6.12) 2

For identical polarizations ∠wa = 0 and then 1 p = (1 + d) 2



identical polarizations (6.13)

Further, if the wave is completely polarized, then d = 1 and p = 1 (Case a). If the wave is randomly polarized, then d = 0 and p = 0.5 (Case e). The polarization efficiency is 0.5 because a receiving antenna can capture only one half of the available power in a randomly polarized wave. Equation (6.12) can be recast to clearly show the contributions of the unpolarized and polarized portions of the wave to polarization efficiency: p=

∠wa 1 1 [1 − d + d(1 + cos ∠wa)] = 2 (1 − d) + d cos2 2 (6.14) 2 !" # # $ !#"#$ unpolarized

polarized

Repeating the case of the antenna polarization state being identical to the polarization state of the completely polarized portion of the wave (∠wa = 0): p = (1 − d)/2 + d = (1 + d)/2 as in (6.13).

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Figure 6.3  The Poincaré sphere used to evaluate polarization efficiency.

Orthogonal polarizations are opposite points on the Poincaré sphere (see Section 3.8.3), then ∠wa = 180° and (6.12) and (6.14) yield

1 p = (1 − d) 2

for ∠wa = 180°

(6.15)

Further, if the wave is completely polarized (d = 1), then p = 0 (Case b). If, instead, the wave is randomly polarized (d = 0), then (6.15) gives p = 0.5. For a completely polarized wave (d = 1) and angular separation of ∠wa = 90°, (6.12) and (6.14) yield

p=

1 2

d = 1, ∠wa = 90° (6.16)

For example, if the wave is circularly polarized (either RHCP or LHCP), w is at a pole on the Poincaré sphere. If the antenna is any linear polarization,

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Antenna-Wave Interaction139

a is on the equator; see Figure 3.4. Then ∠wa = 90° and p = 0.5. This is Case c. As a second example of application of (6.16), let the wave be vertical linear and the antenna be linear at a 45° tilt angle. Then ∠wa = 90°; see Figure 3.4. Again p = 0.5 for this example, which is Case d of Table 6.1. The polarization efficiency values as a function of degree of polarization with angular separation as a parameter are plotted in Figure 6.4; the values are found using (6.12). The plot applies for all possible combinations of wave and antenna polarizations. It clearly shows that p = 0.5 for all receiving antennas if the wave is unpolarized (d = 0) and that p ranges from 0 to 1 if the wave is completely polarized (d = 1).

Figure 6.4  Polarization efficiency of a receiving antenna with angle of separation of ∠wa on the Poincaré sphere between the completely polarized portion of the incident wave and the antenna polarization state, and with the wave having degree of polarization d. The plot is based on (6.12).

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6.2.2  Polarization Efficiency Evaluation Using Stokes Parameters The Poincaré sphere offers an intuitive approach to polarization efficiency evaluation, but for arbitrary polarization states, the angle ∠wa used to compute p from (6.12) is not easily determined. On the other hand, Stokes parameters, although less intuitive, offer a simple calculational method for any polarization situation; see Section 3.5 for details on Stokes parameters. The polarization efficiency expression in Stokes parameters is [5] p=



1 T 1 ai ] [ si ] = (1 + a1s1 + a2 s2 + a3 s3 ) (6.17) [ 2 2

where [ai]T = antenna Stokes parameters (d = 1) = [1  a1  a2  a3] [si]T = wave Stokes parameters = [1  s1  s2  s3] and T indicates matrix transpose. For an antenna matched to the wave (Case a), {ai = si}, and (6.17) becomes p=



1 1 1 + s12 + s22 + s32 = (1 + 1) = 1 2 2

(

)

matched (6.18)

where (3.45) was used. This is the correct result of 100% polarization efficiency. For an antenna orthogonally polarized to the wave (Case b) we have from (3.96) that: a1 = −s1, a2 = −s2, a3 = −s3. Then (6.17) yields

p=

1 1 1 − s12 − s22 − s32 = (1 − 1) = 0 2 2

(

)

orthogonal (6.19)

where (3.45) was used. This is the correct result of no power received. If the wave is unpolarized (Case e), [si]T = [1  0  0  0] and (6.17) gives

p=

1 1 1 + a1 ⋅ 0 + a2 ⋅ 0 + a3 ⋅ 0 ) = ( 2 2

unpolarized wave (6.20)

which is the correct result of half the power being received. 6.2.3  Polarization Efficiency Evaluation Using Polarization Ellipse Quantities Polarization efficiency expressed in polarization ellipse quantities follows directly from the Stokes parameters result of the previous subsection; see

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Antenna-Wave Interaction141



Section 3.2 for details on the polarization ellipse. We first write the Stokes parameters for the antenna based on (3.44) and for the wave using (4.13): ⎤ ⎡ 1 ⎢ cos2ea cos2t a ⎥ [ ai ] = ⎢⎢ cos2ea sin2ta ⎥⎥ sin2ea ⎥⎦ ⎢⎣



⎡ 1 ⎢ d cos2ew cos2tw [ si ] = ⎢⎢ d cos2ew sin2tw d sin2ew ⎢⎣

⎤ ⎥ ⎥ (6.21) ⎥ ⎥⎦

These in (6.17) give



⎞ 1 ⎛ 1 + d cos2ea cos2ew cos2t a cos2tw (6.22) p= ⎜ 2 ⎝ + d cos2ea cos2ew sin2t a sin2tw + d sin2ea sin2ew ⎟⎠

Only the relative tilt angle (angle between major axes of the polarization ellipses of the wave and antenna) is important. Relative tilt angle Δτ is defined as Δt = t a − tw (6.23)



Using this and (B.9) in (6.22) simplifies it to p=



1 (1 + d cos2ea cos2ew cos2Δt + d sin2ea sin2ew ) (6.24) 2

If the (ε , τ ) parameter values for the antenna and wave are known (equivalently: ε a, ε w and Δτ ), the polarization efficiency is easily evaluated with (6.22) or (6.24), as illustrated with the following examples. Example 6.1  Evaluation of Polarization Efficiency Values for the Cases in Table 6.1 Using Polarization Ellipse Parameters

Case a: Antenna matched to the wave: d = 1, ε a = ε w, Δτ = 0 p=



1 (1 + cos2 2ew + sin2 2ew ) = 1 2

using (6.24)

Case b: Antenna orthogonal to the wave: d = 1, ε a = − ε w, Δτ = 90° using (3.72)   

p=

1 (1 + cos2 2ew cos(180°) + sin2 2ew ) = 21 (1 − 1) = 0 2

using (6.24)

Case c: CP wave, LP antenna

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Wave: d = 1, ε w = 45°, τ w = arbitrary Antenna: ε a = 0°, τ a = arbitrary

from Table 3.5 for a general LP antenna

1 p = (1 + 0 + 0) = 0.5 2



using (6.24)

Case d: VP wave, LP antenna with τ a = 45° Wave: d = 1, ε w = 90°, τ w = 90° Antenna: ε a = 0°, τ a = 45° Relative tilt angle: Δτ = τ a − τ w = −45°

from Table 3.5 for a 45°-tilted LP antenna

1 p = (1 + 0 + 0) = 0.5 2



using (6.24)

Case e: Unpolarized wave: d = 0, (ε a, τ a) = arbitrary 1 p = (1 + 0 + 0) = 0.5 2



using (6.24)

Equation (6.24) is easy to use for any polarization. It reduces for special cases. Suppose either the completely polarized part of the wave or the antenna is purely linearly polarized. Then ε a or ε w = 0° and (6.24) becomes

1 p = (1 + d cos2ecos2Δt) 2

wave or antenna LP (6.25)

where ε is for the non-LP polarization. (The reader can apply this to Case c as a simple exercise). If both the wave and the antenna are linearly polarized (ε w = ε a = 0°), (6.25) reduces to

1 p = (1 + d cos2Δt) 2 p = cos2 Δt

wave and antenna LP (6.26a)

wave and antenna LP, and d = 1 (6.26b)

Equation (6.26b) varies between unity for alignment of the axes (Δτ = 0°) down to zero for orthogonal major axes (Δτ = 90°). Also, this result follows directly from the Poincaré sphere polarization efficiency formula of (6.14) because ∠wa = 2Δτ for the linear case.

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6.2.4  Polarization Efficiency Using Axial Ratios Axial ratio is a directly measurable quantity (see Sections 10.2 and 10.3). It is, therefore, useful in measurement situations to express polarization efficiency explicitly in terms of axial ratio. The interaction of waves and antennas of arbitrary axial ratios is an important topic in polarization, including calculation of polarization efficiency with arbitrary axial ratios which is presented in this section. We start with the polarization ellipse parameters. The relative tilt angle between the wave and the antenna polarization states from (6.23) is Δτ = τ a − τ w. The ε parameters are related to the axial ratios from (2.27) as ew = cot −1 ( −Rw )



ea = cot −1 ( −Ra ) (6.27)

Remember that axial ratio R carries a sign, being positive for RH sense and negative for LH sense. Substituting these in (6.24) and performing several manipulations leads to (see Problem 4 at the end of this chapter) p=

4R R + ( Ra2 − 1)( Rw2 − 1) cos2Δt 1 +d a w (6.28) 2 2( Ra2 + 1)( Rw2 + 1)

For a completely polarized wave d = 1 and then p=

2 2 1 4Ra Rw + ( Ra − 1)( Rw − 1) cos2Δt + 2 2( Ra2 + 1)( Rw2 + 1)

completely polarized wave

(6.29) Example 6.2  Evaluation of Polarization Efficiency Values for the Cases in Table 6.1 Using Axial Ratios

Case a: Antenna matched to the wave: d = 1, Ra = Rw, Δτ = 0 2 2 1 4R + ( Rw − 1) cos0 1 1 p= + w = + = 1 2 2 2 2 2( Rw2 + 1) 2



using (6.29)

Case b: Antenna orthogonal to the wave: d = 1, Ra = −Rw, Δτ = 90° using (3.73) 2 2 1 −4Rw + ( Rw − 1) cos 180° 1 1 p= + = − = 0 using (6.29) 2 2 2 2 2( Rw2 + 1) 2



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Case c: CP wave, LP antenna Wave: d = 1, Rw = 1, τ w = arbitrary Antenna: Ra = ∞, τ a = arbitrary

p=

1 + 2

4

RHCP wave for a general LP antenna

Rw ⎛ 1 + ⎜ 1 − 2 ⎞⎟ ( Rw2 − 1) cos2Δt Ra ⎝ Ra ⎠ 1 = using (6.29) 1 2 2 ⎛⎜ 1 + 2 ⎞⎟ ( Rw2 + 1) R ⎝ a ⎠

Case d: VP wave, LP antenna with τ a = 45° vertical linear Wave: d = 1, Rw = ∞, τ w = 90° for a 45°-tilted LP antenna Antenna: Ra = ∞, τ a = 45° Relative tilt angle: Δτ = τ a − τ w = −45° 1 4 1 + ⎛⎜ 1 − 2 ⎞⎟ ⎛⎜ 1 − 2 ⎞⎟ cos2Δt Ra ⎠ ⎝ Rw ⎠ ⎝ 1 R R 1 p= + a w = 1 1 2 2 2 ⎛⎜ 1 + 2 ⎞⎟ ⎛⎜ 1 + 2 ⎞⎟ R R ⎝ a ⎠⎝ w⎠    

using (6.29)

Case e: Unpolarized wave: d = 0, (Ra, τ a) = arbitrary 1 p = (1 + 0 + 0) = 0.5 2



using (6.28)

The following example is illustrative of the common situation where both the wave and the antenna are elliptically polarized. Example 6.3  Evaluation of Polarization Efficiency for an Elliptically Polarized Wave Incident on an Elliptically Polarized Antenna Using Axial Ratios

Consider a wave that is nearly LHCP with 1 dB axial ratio that is incident upon an antenna that is LHCP with a 3-dB axial ratio. We wish to find the polarization efficiency. The parameter values for the wave and antenna are Antenna: Ra,dB = 3, ⎪Ra⎪ = 103/20 = 1.4125, Ra = −1.4125 Wave: Rw,dB = 1, ⎪Rw⎪ = 101/20 = 1.122, Rw = −1.122 From (6.29) p = 0.9685 + 0.01904 cos 2Δτ

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Antenna-Wave Interaction145



The efficiency values from this formula are very close to unity because the antenna and wave are fairly well matched in polarization; that is, both are of same sense and have relatively low axial ratio values. The second term is the variation of efficiency with relative tilt angle between the wave and antenna ellipse major axes, and is small in this case. Polarization efficiency is maximum when the major axes are aligned (Δτ = 0°), giving p = 0.9685 or 0.139 dB loss, and minimum when they are perpendicular (Δτ = 90°), giving p = 0.9494, or 0.225 dB loss. The difference in the maximum and minimum loss is 0.086 dB. To summarize, the worst-case loss is 0.225 dB when the ellipses are orthogonal and the variation with tilt angle of 0.086 dB. Both of these are low, making careful alignment of the major axes unnecessary in many applications. This is not true with linear polarization where large fluctuations in polarization mismatch can occur with tilt angle variations. This highlights an advantage of circularly polarized systems: alignment of the major axes is often not critical. A special-case formula is easily developed for when either the completely polarized portion of the wave or the antenna is linearly polarized by using infinity for the wave or antenna axial ratio in (6.28), which yields



p=

( R2 − 1)cos2Δt 1 +d 2 2( R2 + 1)

wave or antenna LP (6.30)

where R is the axial ratio of the non-LP polarization ellipse. In the further special case of both the antenna and the wave being linearly polarized, R = ∞ and (6.30) reduces (6.26). Another important special case result is when either the completely polarized portion of the wave or the antenna is circularly polarized. Then from (6.28)



p=

1 R ±d 2 2 R +1

wave or antenna CP (6.31)

where ⎪R⎪ is the axial ratio magnitude of the antenna polarization if the wave is pure CP or the axial ratio magnitude of the wave if the antenna is pure CP; the sign is positive when the polarizations are of the same sense and negative if opposite. If the wave is completely polarized and of the same sense as the antenna, (6.31) reduces to



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We can check this simple equation at a few points. For pure CP, ⎪R⎪ = 1 giving p = 1, indicating a perfect match. For LP, ⎪R⎪ = ∞ and (6.32) yields p = 0.5, for which Case c in the foregoing examples is one situation. A special case of (6.29) that illustrates some important points is that of an elliptically polarized wave (completely polarized) and an elliptically polarized receiving antenna with identical axial ratio magnitudes: 1 ±4R2 + ( R2 − 1) cos2Δt + 2 2 2( R2 + 1) 2



p=

identical R , d = 1 (6.33)

where the plus (minus) sign is used when the wave and antenna elliptical polarizations are of the same (opposite) sense. If the sense is the same (plus sign), it is obvious that (6.33) is maximum for the major axes aligned (Δτ = 0°), and then p = 1, which is 100% efficiency. There is no loss in this case because the wave and antenna have identical polarizations. If again both the wave and antenna have the same axial ratio and same sense but have orthogonal major axes (Δτ = 90°), (6.33) reduces to



p=

4R2

( R2 + 1)2

identical R , same sense, d = 1, Δt = 90° (6.34)

For the CP case (⎪R⎪ = 1), this gives p = 1, or a perfect match condition. For the LP case (⎪R⎪ = ∞), (6.34) gives p = 0, or a complete mismatch, which is the expected perfect mismatch result for orthogonal linear polarizations. If the senses are opposite (the minus sign in [6.33] holds) and the major axes are aligned (Δτ = 0°), (6.33) reduces to



( R2 − 1)2 p= ( R2 + 1)2

identical R , opposite sense, d = 1, Δt = 0° (6.35)

For the CP case (⎪R⎪ = 1; tilt angles are irrelevant), this gives p = 0, indicating a complete mismatch. For the LP case (⎪R⎪ = ∞; senses are irrelevant), then p = 1, indicating a polarization match. Polarization loss can be found from the convenient nomograph shown in Figure 6.5 that is referred to as the Ludwig chart [6, 7]. It applies to completely polarized waves; if the wave is partially polarized, the techniques of Section 6.2.7 are used. Each axis covers the possible axial ratio values in decibels, including sense. Circular polarizations are at the ends of the axes and linear polarization is at the center of each axis. The chart is used by locating

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Antenna-Wave Interaction147

Figure 6.5  The Ludwig chart. Polarization loss in decibels is found for a completely polarized wave with axial ratio value on the ordinate and an antenna of axial ratio value on the abscissa. The loss ranges from a minimum (solid lines) to a maximum (dashed lines) loss depending on the relative tilt angle between the major axes. The point shown is the case in Example 6.4. (Source: Ludwig, A.C., “A Simple Graph for Determining Polarization Loss,” Microwave J., Vol. 19, Sept. 1976, p. 63. Reprinted with permission.)

the axial ratio for the receiving antenna polarization state on the abscissa and locating the axial ratio for the wave polarization state on the ordinate. The pair of axial ratio values determines a point on the chart. There is a range of polarization loss values that depend on the relative tilt angle between the major axes of the antenna and wave polarization ellipses. The extremes of polarization loss lie on the solid lines (minimum) and dashed lines (maximum). The

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0-dB minimum loss contour is the 45° solid line, which represents equal axial ratios and aligned major axes. All cases of identical axial ratios and senses for the wave and antenna polarizations lie on this line. For example, the case of a right-hand sensed wave incident on a right-hand sensed antenna both with 4-dB axial ratio has a minimum loss (for Δτ = 0°) of 0 dB on the 45° solid line in the lower left portion of the chart and has a maximum loss (for Δτ = 90°) of 1 dB. Pure circular polarization, same-sense cases are at the ends of the 45° line and correspond to 0-dB loss. The maximum loss depends on relative tilt angle as well as the axial ratio, reaching a peak of ∞ dB in the center of the chart corresponding to the case of linear polarizations with orthogonal major axes; the minimum loss is, of course, 0 dB corresponding to aligned linear polarizations. In fact, the negative-slope 45° dashed line corresponds to ∞ dB maximum loss and represents equal axial ratios and orthogonal major axes. It is very easy to use the Ludwig chart of Figure 6.4. First, the receiving antenna axial ratio is located on the abscissa and a vertical line is erected. Next, the axial ratio of the wave is located on the ordinate and a horizontal line is drawn. The intersection of the lines determines a point. Finally, the minimum and maximum losses are read by interpolating the appropriate contours at the intersection point. See if you can locate a point on the chart corresponding to the situation in Example 6.3. The following example also illustrates use of the Ludwig chart. Other graphical techniques in addition to the Ludwig chart are also available [8, 9]. Example 6.4  Polarization Loss Using the Ludwig Chart

Consider the following specific antenna and wave states: ⎪Ra⎪ = 10 dB, Ra = −1010/20 = −3.16 Left-hand elliptical ⎪Rw⎪ = 5 dB, Rw = −105/20 = −1.78 Left-hand elliptical Lines corresponding to these values are shown in Figure 6.5. At the intersection point, the solid contours give a minimum loss value of about 0.2 dB and the dashed contours give a maximum loss value of about 3 dB. We can verify this result using (6.29). For minimum loss case (Δτ = 0°) 1 4(−3.16)(−1.78) + ( 3.162 − 1)(1.782 − 1)(1) + 2 2( 3.162 + 1)(1.782 + 1) = 0.958 = −0.18 dB

p=

For maximum loss case (Δτ = 90°)

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1 4(−3.16)(−1.78) + ( 3.162 − 1)(1.782 − 1)(−1) + 2 2( 3.162 + 1)(1.782 + 1) = 0.533 = −2.73 dB

p=

Thus, the losses associated with polarization efficiency range from 0.18 to 2.7 dB. This agrees with the approximate values read from the Ludwig chart of 0.2 to 3 dB. 6.2.5  Polarization Efficiency Expressed Using Polarization Ratios The polarization ratio representation was introduced in Section 3.6. Let ρ Lw and ρ La be the polarization ratios for the wave and antenna. The wave state here is taken as completely polarized; for cases where the wave is partially polarized, the polarization efficiency is modified as shown in Section 6.2.7. The polarization efficiency expression using polarization ratios is [10, p. 186] p=

∗ 1 + rLw rLa

(1 + r )(1 + r ) 2

Lw



2 2

(6.36)

La

Example 6.5  Evaluation of Polarization Efficiency Values for the Cases in Table 6.1 Using Polarization Ratios

Case a: Antenna matched to the wave: ρ La = ρ Lw p=

1 + rLw ρ ∗La

2

=

1 + rLw

22

(1 + r )(1 + r ) (1 + r )(1 + r ) 2

Lw

2

La

2

2

Lw

=1

Lw

Case b: Antenna orthogonal to the wave: ρ La = −1/ρ ∗Lw from (3.100)

p=

1+

2 rLw ρ ∗La 2

=

1 + rLw

1 −rLw

2

(1 + r )(1 + r ) (1 + r )⎛⎜ 1 + 1 r ⎝ Lw

2

La

2

Lw

⎞ 2⎟ Lw ⎠

=0

Case c: CP wave, LP antenna: ρ Lw = −j (RHCP); ρ La = 0 (HP); see Table 3.3

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p=

1 + rLw ρ ∗La

2

2

(1 + r )(1 + r ) 2

Lw

2

La

1+ 0 = = 0.5 (1 + 1)(1 + 0)

Case d: VP wave, LP antenna (Δτ = 45°): ρ Lw = ∞; ρ La = 1; see Table 3.3

p=

1 ∗ + rLa rLw

(

2

⎛ 1 + 1⎞ 1 + r 2 La ⎜⎝ r 2 ⎟⎠ Lw

)

=

2

0 +1 = 0.5 (0 + 1)(1 + 1)

Case e: Unpolarized wave: p = 0.5 when d = 0; from (6.44) 6.2.6  Polarization Efficiency Expressed Using Polarization Vectors Polarization vectors offer the most compact formula polarization efficiency; see Section 3.4 for details on polarization vectors. The derivation presented here is for a completely polarized wave but can be extended to include partially polarized waves as discussed in Section 6.2.7. The efficiency is calculated by projecting the normalized complex vector of the wave êw onto that of the receiving antenna êa, which has intuitive appeal. The projection process involves a vector dot product [4, Part III]: 2



p = eˆw i eˆ∗a (6.37)

The squaring in this expression is because the polarization vectors are fieldbased quantities whereas efficiency is a power related quantity. The complex conjugate is required to correct for the fact that êa is defined in a transmitting coordinate system but used here for reception; this is discussed further in Section 6.3. We can use this polarization vector approach to derive the polarization efficiency expression in terms of polarization ratios presented in (6.36) without proof. From (3.60)

eˆw = cosg w ( xˆ + rLw yˆ ) (6.38a)



eˆa = cosg a ( xˆ + rLa yˆ ) (6.38b)

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Then (6.37) yields ∗ yˆ ) p = eˆw i eˆa∗ = cos2 g w cos2 g a ( xˆ + rLw yˆ ) i ( xˆ + rLa 2

2



(6.39)

1 + r r∗ = 2 Lw La2 sec g w sec g a

But from (3.57) ⎪ρ L⎪ = tanγ , so 1 + ⎪ρ L⎪2 = 1 + tan2γ ≡ sec2γ , where the identity (B.28) was used. Using this in (6.39) gives p=

∗ 1 + rLw rLa

(1 + r )(1 + r ) (6.40) 2

Lw



2 2

La

which proves (6.36). Example 6.6  Evaluation Polarization Efficiency Values for the Cases in Table 6.1 Using Polarization Vectors

Case a: Antenna matched to the wave: eˆa = eˆw = ewx xˆ + ewy yˆ from (3.24) 2



)(

(

∗ ∗ xˆ + ewy yˆ p = eˆw i eˆ∗a = ewx xˆ + ewy yˆ i ewx

)

2

2

2 2 = ⎡ ewx + ewy ⎤ = 1 ⎣ ⎦

using (3.25)

∗ ∗ xˆ − ewx yˆ from (3.82) Case b: Antenna orthogonal to the wave: eˆa = ewy 2

(

)(

p = eˆw i eˆ∗a = ewx xˆ + ewy yˆ i ewy xˆ + ewx yˆ

)

2

2

= ewx ewy − ewy ewx = 0 Case c: CP wave, LP antenna:

eˆw =

( xˆ − yˆ ) 2 eˆa = xˆ ; 2

p = eˆw i eˆ∗a =

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(RHCP) (HP)

from (3.86) from (3.83)

2

1 ( xˆ + yˆ ) i xˆ = 0.5 2

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Polarization in Electromagnetic Systems

Case d: VP wave, LP antenna: eˆw = yˆ



eˆa =

(xˆ + yˆ ) 2 2

p = eˆw i eˆ∗a = yˆ i

(VP)

from (3.83)

(slant-45 LP)

from Table 3.5

2

1 ( xˆ + yˆ ) = 0.5 2

Case e: Unpolarized wave: p = 0.5 when d = 0; see Section 6.2.7. Of the polarization efficiency formulas presented in this section, the ones that are well suited to mathematical evaluation for arbitrary polarization states are Stokes parameters of Section 6.2.2, polarization ellipse based formulas (either ε and tilt angle of Section 6.2.3 or axial ratio and tilt angle of Section 6.2.4), polarization ratios of Section 6.2.5, and polarization vectors of Section 6.2.6. This is in contrast to the Poincaré sphere method of Section 6.2.1, which requires some spherical geometry interpretation. 6.2.7  Decomposition of Polarization Efficiency into Unpolarized and Completely Polarized Parts Some polarization efficiency evaluation techniques are not well suited to inclusion of partially polarized waves. Examples include the polarization ratio of Section 6.2.5 and the polarization vector form of Section 6.2.6, as well as the complex voltage form to be presented in Section 6.4. However, any polarization efficiency representation can be evaluated for an incident wave that is completely polarized by setting the degree of polarization to unity. The resulting polarization efficiency for a completely polarized wave is denoted pc. pc is found using any of the techniques we have introduced for the interaction of a completely polarized wave with an antenna. For a general partially polarized wave state, the polarization efficiency can be found knowing the degree of polarization of the wave d and the polarization efficiency for the completely polarized part of the wave. The general formula is p = 0.5 + d(pc − 0.5), which we now derive. First, consider a partially polarized wave. It can be decomposed into unpolarized and completely polarized parts by separating into proportions of 1 − d for the unpolarized portion and d for the completely polarized part:

6765_Book.indb 152

Sav = Su + S p = (1 − d)Sav + dSav (6.41)

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Antenna-Wave Interaction153

The available power received by an antenna of effective aperture Ae by extending (6.7) is

1 P = Su Ae + pc S p Ae (6.42) 2

where the one-half factor represents the polarization efficiency for an unpolarized wave incident on any antenna and the polarization efficiency for the polarized part of the wave, pc, ranges from zero to unity. Substituting the flux density partitions of (6.41) into (6.42) gives

1 P = Su Ae ⎡⎢ (1 − d) + pc d ⎤⎥ (6.43) ⎣2 ⎦

Comparing this with (6.7) we have the desired result for the total polarization efficiency in terms of pc and d:

(

)

1 1 1 p = (1 − d) + pc d = + d pc − (6.44) 2 2 2

Equation (6.44) permits evaluation of polarization efficiency of a partially polarized wave, p, from the efficiency for the completely polarized part of the wave and the antenna, pc, and the degree of polarization of the wave, d. To check this result for a completely polarized wave (d = 1), we see that it reduces to p = pc, as it should. Equation (6.44) also gives the correct result of p = 0.5 for an unpolarized wave (d = 0). The general decomposition of (6.44) is readily apparent in the Poincaré sphere polarization efficiency formula of (6.14) from which we find

pc = cos2

∠wa

2

(6.45)

The following example illustrates application of the polarization efficiency calculation approach of (6.44) for a partially polarized wave. Example 6.7  Evaluation of Polarization Efficiency for a Partially Polarized Wave Using the Polarization Efficiency for the Completely Polarized Part

Consider a partially polarized wave with a degree of polarization d = 0.3. The polarization states for the completely polarized portion of the wave as well as the antenna are identical to those of Example 6.4; that is, the wave has a 5-dB wave axial ratio and the antenna an axial ratio of 10 dB, and both are

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Polarization in Electromagnetic Systems

right-hand sensed. From Example 6.4, the completely polarized portion of the wave interacting with the antenna give pc(Δτ = 0°) = 0.958 and pc(Δτ = 90°) = 0.533. Then the polarization efficiencies for this partially polarized incident wave follow from (6.44):

(

)

(

1 1 1 1 + d pc − = + 0.3 0.958 − 2 2 2 2 = 0.637 = −1.96 dB

pc (Δt = 0°) =

(

)

(

)

1 1 1 1 + d pc − = + 0.3 0.533 − 2 2 2 2 = 0.510 = −2.92 dB

pc (Δt = 90°) =

)

The minimum and maximum polarization losses as a function of relative tilt angle are then 1.96 and 2.92 dB. 6.2.8  Decomposition of Polarization Efficiency into Co-polarized and Cross-Polarized Parts In the previous section, we addressed decomposition of a wave into unpolarized and polarized (i.e, completely polarized) parts. It is also useful for system applications to decompose the power in a wave (and polarization efficiency) into two orthogonally polarized parts. As noted in Section 3.8, any polarization state can be decomposed mathematically into co-polarized and cross-polarized components. The choice of the co-polarization and cross-polarization states depends on the application. As an example, consider a terrestrial point-topoint communication system that uses vertical linear polarization on both the transmit end and the receive terminal. The co-polarized state is vertical linear and the cross-polarized state is horizontal linear. In a dual-polarized communication system, there are two channels that have orthogonal copolarized states. Thus, such a communication link can have noninterfering signals on the two co-polarized states. In such a dual-polarized communication system the cross-polarization power levels must be low enough so as not to cause self-interference. The total power available to a receiving system can be expressed in orthogonally polarized parts as

P = Pco + Pcr (6.46)

This can be visualized for two receiving antennas polarized in the copolarized and cross-polarized states, co and cr, and with the same effective

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Antenna-Wave Interaction155

aperture Ae. The antennas would capture the orthogonal power components, so

SAe = pco SAe + pcr SAe (6.47)

where (6.7) was used. The polarization efficiencies pco and pcr are the efficiencies associated with the incident wave polarization state and the receiving polarization states, co and cr. Dividing this equation by the total power, P = SAe, gives

1 = pco + pcr (6.48)

This general result applies to both partially and completely polarized waves. For example, if a wave is unpolarized, pco = pcr = 0.5 and (6.48) is satisfied. For a completely polarized wave that exactly matches the co-polarized state (and thus is orthogonal to the state cr), pco = 1 and pcr = 0; again (6.48) is satisfied.

6.3  Vector Effective Length of an Antenna Polarization, radiation pattern, and effective aperture information can be rep-! resented through a single antenna quantity, called the vector effective length, h [1]; the term vector effective height is also used. It is a complex-valued vector that describes both the phase (i.e., temporal) and polarization (i.e., spatial) properties of the antenna. It is very useful in analysis of receiving antennas. To introduce vector effective length we use the simplest of all antennas, the short dipole. The short dipole antenna is a center-fed straight wire antenna of length L that is much less than a wavelength. The current falls from I at the terminals to 0 at the wire ends, creating a triangular shaped current distribution. Its radiation properties are similar to that of the ideal dipole antenna that has a uniform current over its length L, which is also much less than a wavelength. Figure 2.6 shows the pattern and fields of a short dipole and these also apply to the ideal dipole. The expression for the radiated electric field from an ideal dipole oriented along the z-axis is [3, p. 138]

! jwm e − jbr h sinq θˆ E(q) = 4p r

ideal dipole (6.49)

where h is the effective length of the ideal dipole, which is equal to its length L. The expression for the radiated field from a short dipole (which was discussed

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156

Polarization in Electromagnetic Systems

in Section 2.3) of terminal current I and length L is identical to (6.49) with h = L/2. This is because the triangular current distribution of the short dipole has a current moment (area under the current versus position plot) which is half that for the ideal dipole that has a uniform current distribution. Retaining the dipole length and the radiation pattern parts of (6.49) gives the vector effective length of an electrically small dipole (either an ideal dipole or a short dipole) for the geometry used in Figure 2.6 as [11; 3, p. 102]: ! h(q) = hsinq θˆ electrically small dipole (6.50) This form includes the radiation pattern of the dipole, sin θ , its effective ˆ The effective length has a simple (maximum) length, h, and its polarization, θ. interpretation as the projection of the physical length, L = h, viewed from the angle θ . So, when viewed from broadside (θ = 90°), the magnitude of (6.50) is h. The short dipole is the practical version the ideal dipole realized by a center-fed straight wire. The maximum vector ! effective length of the small dipole occurs for θ = 90° and from (6.50) is h = hẑ for the geometry of Figure 2.6. In general, maximum vector effective length of an antenna is given by ! h = heˆa (6.51) where êa is the polarization vector for the antenna. Unless otherwise noted, the vector effective length of an antenna is usually assumed to be the maximum ! value. Note that h has dimensions of length. The ideal dipole has h = L and the short dipole has h = L/2, where L is the physical length, as noted above. A receiving antenna converts the incoming electromagnetic wave to a voltage at its output terminals. Knowledge of this voltage is useful in system calculations. Inclusion of phase information is also important. This leads us to introduce open circuit complex voltage V, which is defined as the vector projection of the incident electric field onto the vector effective length [10, p. 184]: ! ! V = E i ⋅ h ∗ (6.52) This is an intuitive relation. The incident field in volts per meter appears at the antenna terminals as an output voltage in volts. Polarization information for the wave and antenna are contained in the vectors. For example, if the incident wave electric field is perpendicular to the antenna, the dot product renders a zero voltage which is the true because the wave is cross-polarized to

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Antenna-Wave Interaction157

the antenna. The result in (6.52) is quite general. We make use of it and discuss properties of effective length in the remainder of this section. The receiving antenna relationship of (6.52) applies to any antenna. The complex conjugate is not important for linear polarized antennas such as a short dipole. However, the conjugate is necessary in general polarization cases. This is because the definition of polarization is based on the polarization state of the wave radiated when the antenna is transmitting. For consistency we retain this convention in all situations. Then the polarization representation of a receiving antenna in (6.52) must have a conjugate operator, which acts to reverse the direction of the wave from transmit to receive for the polarization. This principle is illustrated with a receiving antenna consisting of crossed dipoles as shown in Figure 6.6; this is the turnstile antenna of Figure 5.7 except for a different sense of CP. An incident RHCP plane wave travels in the +z-direction in the wave coordinate system and illuminates the crossed dipoles as shown. The receiving antenna coordinate system is oriented with its z-axis parallel to that of the incoming wave and the x- and y-axes parallel to each dipole. This antenna polarization is for the antenna transmitting, which is also RHCP. The conjugate operation accounts for the fact that the antenna is receiving not transmitting. Recall our discussions in Section 5.1 where we

Figure 6.6  Reception of a RHCP wave with a RHCP turnstile antenna. The electric field components of the incoming wave are shown. The response V at the antenna terminals is maximum (output from two dipoles add in phase). If the antenna were used to transmit it would radiate RHCP in +z-direction.

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158

Polarization in Electromagnetic Systems

pointed out that a reciprocal antenna, like this one, has the same polarization properties on transmitting and receiving. The wave and antenna polarizations are to be expressed in their own coordinate systems as shown in Figure 6.6. That is, the incoming wave and the wave radiated by the antenna when transmitting both travel in the +z-direction of their own coordinate systems. The antenna when transmitting has a vector effective length of

! xˆ − jyˆ h = heˆa = h 2

RHCP (6.53)

where the polarization vector for RHCP from Example 3.5 was used in (6.51). The incident electric field is

! xˆ − jyˆ E i = Eo 2

RHCP (6.54)

The voltage output from the antenna is from (6.52)

∗ ! i !∗ xˆ − jyˆ ⎡ xˆ − jyˆ ⎤ V = E ⋅ h = Eo ⋅ h⎢ = Eo h (6.55) 2 ⎣ 2 ⎥⎦

This is the maximum output, which is the expected result for a receive antenna matched to an incoming wave striking the antenna from its pattern maximum direction. The mathematical approach of (6.52), of course, works in all cases. But we would also like to understand how the wave interacts with the antenna to produce the output voltage without using vectors; the result should be consistent with that obtained using (6.52). To do this we revisit the RHCP wave striking a RHCP receive antenna as in Figure 6.6. The incoming wave produces a voltage contribution in the x-oriented dipole of the receiving antenna; that is, 1∠180°—the 180° is present because of the opposite reference directions of the x-axes of the wave and antenna. The y-oriented dipole is excited with 1∠−90° and then is delayed by the quarter-wave transmission, introducing a −90° phase shift and giving a net voltage of 1∠−180° = 1∠180°. This contribution adds in-phase with the x-dipole. Therefore, the output for this receiving antenna is the maximum for a RHCP incident wave given in (6.55). To! summarize, the output voltage ! ifrom a receiving antenna of effective field incident on it is given by (6.52). To E length h with a wave of electric ! ! evaluate (6.52) the vectors E i and h are expressed in terms of their own xand y-coordinates with the wave propagation direction toward the antenna

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Antenna-Wave Interaction159

and the receiving antenna polarization as if it were transmitting; see Figure 6.6. The dot product performs the wave-antenna interaction function, and the complex conjugate accounts for the antenna operating as a receive antenna. The reader is cautioned that some literature sources do not use the conjugate, which then requires one to adjust for the change in transmit/receive functions. Our approach performs the wave-antenna interaction function automatically. Polarization efficiency can be expressed in terms of effective length. To do this we assume the receiving antenna is loaded in a matched impedance of ZL = Ra − jXa, where the antenna impedance is Z A = Ra + jXa. The received power is [3, p. 102] 2



1V P= 2 4Ra (6.56)

Substituting (6.52) into this expression gives

P=

1 ! i !∗ 2 E ⋅ h (6.57) 8Ra

The maximum of this function is

P=

1 !i 2 ! 2 E h (6.58) 8Ra

Polarization efficiency is the ratio of power received for a wave of a specific polarization state to that power received when the wave is polarization matched to the antenna [4, Part III]:



! ! 2 E i ⋅ h∗ P p= = ! 2 !2 = Pmax Ei h

! ! 2 2 E i h∗ ! i ⋅ ! = eˆw ⋅ hˆ∗ (6.59) E h

where (6.58) was used and êw is the complex unit vector for the incoming wave. (Note that the same terminating impedances must be used; here we assume a matched load.) If the wave is generated by a distant antenna of polarization state êa and the transmitted wave does not depolarize before it arrives at the receiving antenna, then the incident wave polarization êw equals êa. The output voltage from the receiving antenna can now be expressed from (6.52) and (6.59) as ! ! ! ! V = E i ⋅ h ∗ = p E i h = Eo h p = Eo h eˆw ⋅ hˆ∗ (6.60)

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Polarization in Electromagnetic Systems

As an example consider the RHCP wave and antenna of Figure 6.6. Using (6.53) and (6.54) in (6.60) gives 2



∗ 2 xˆ − jyˆ ⎛ xˆ − jyˆ ⎞ ∗ ˆ ˆ p = ew ⋅ h = ⋅ = 1 (6.61) 2 ⎝ 2 ⎠

If, instead, the wave were linearly polarized at 45°: 2



∗ 2 2 xˆ + yˆ ⎛ xˆ − jyˆ ⎞ 1 1 ∗ ˆ ˆ p = ew ⋅ h = ⋅ = (1 + j) = (6.62) 2 ⎝ 2 ⎠ 2 2

which is correct for a CP wave incident on an LP antenna; see Figure 6.2. Effective length can be related to conventional antenna parameters. The received power from (6.56) and (6.60) is



P=

2

1V 1 2 2 = E h p (6.63) 2 4Ra 8Ra o

But the received power can be expressed in terms of effective aperture from (6.7) as

P = pSAe (6.64)

Equating these two expressions and solving for h gives



1 2 E SA Ra 2h o h2 = 8Ra 2e = 8Ra A (6.65) 2 Ae = 4 h e Eo Eo

So



h=2

Ra A (6.66) h e

This result has the correct units of length and shows how effective aperture relates to effective length. Equation (6.66) can be generalized to include the radiation pattern and polarization of an antenna as follows:



6765_Book.indb 160

! R h(q,f) = 2 a Ae F (q,f)hˆ (6.67) h

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Antenna-Wave Interaction161

As a final comment, we note that complex-valued electric field components of an arbitrary wave can be found using an approach similar to that used to find (6.55): ! E(w,a) = Ew ⋅ eˆ∗a (6.68) Here êa is the unit vector of a component of the polarization state w. E(w,a) can be thought of as the response to an imaginary probe antenna of polarization a to a wave of state w.

6.4  Normalized Complex Antenna Output Voltage Polarization efficiency, p, is a power ratio and therefore cannot include phase information. The output voltage of an antenna with an incident completely polarized wave, however, can include phase. Knowledge of phase is important in many situations. For example, some polarization measurement systems require phase information; see Section 10.3.1. Also, if the output of an antenna is to be combined with that of another antenna (as in an array antenna) phase information must be preserved. Calculations for these types of applications can be performed using the methods introduced in this section to evaluate the complex-valued output voltage (i.e., including phase). The antenna output voltage expression of (6.52) for incoming wave of state w and a receiving antenna of state a includes amplitude and phase. Intro! ducing the incident wave polarization of êw, the ! electric field is Ew = Eo eˆw . The effective length expression from (6.51) is h = heˆa . Using these in (6.52) gives V = Eo eˆw ⋅ heˆ∗a . Normalizing this by dividing by the amplitude of Eoh gives the normalized complex voltage:

v(w, a) = eˆw ⋅ eˆ∗a (6.69)

Remember that the unit vectors in this equation are the complex vectors representing the incoming wave and the receive antenna when transmitting in their own coordinate systems. Thus, the wave and the antenna are treated separately and (6.69) performs the antenna-wave interaction without the need for separate accounting for phase references or coordinate systems. This simple relation yields the normalized complex voltage from a receive antenna of polarization state a for an incident wave of polarization state w. Most importantly it includes phase. The voltage (or power level) can be reintroduced by a separate calculation involving, for example, a communication link; see [3,

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Polarization in Electromagnetic Systems

p. 119]. Using the properties of polarization vectors in Section 3.4, it is easy to see that v(w,a) = 1 for a perfect match of the wave and antenna polarizations (i.e., êw = êa). Also in general, if the wave and antenna are orthogonally polarized, v(w,a) = 0. The received normalized complex voltage expression can also be given in terms of the polarization ellipse parameters of (γ , δ ) of the wave w and receiving antenna a. Using (3.41) in (6.69) gives

(

)(

v(w, a) = eˆw ⋅ eˆ∗a = cosg w xˆ + sing w e jdw yˆ ⋅ cosg a xˆ + sing a e jda yˆ = cosg w cosg a + sing w sing a e j(dw −da )

) (6.70) ∗

We can check this formula as usual by examining matched and orthogonal cases. For matched polarizations (γ w = γ a, δ w = δ a), (6.70) yields unity. For orthogonal polarizations, we have from Table 3.4 that γ w = 90° − γ a and δ w = δ a ± 180°. Then (6.70) along with (B.11) and (B.10) give v(w, a) = cosg w cos( 90° − g a ) + sing w sin ( 90° ± g w ) e [

j dw −(da ±180°)]

= cosg w sing w + sing w cosg w e j±180° = 0

(6.71)

which is the correct result. Polarization efficiency can be found very easily from normalized complex voltage: 2

p = v(w, a) (6.72)



This follows from comparing (6.69) to (6.37). Example 6.8  Normalized Complex Voltage Calculation for a RHCP Wave Incident on a RHCP Antenna

Consider a RHCP wave incident on a RHCP receive antenna. Figure 6.6 is an illustration of one such case. The polarization vectors for the wave and antenna from (3.86) are eˆw =

xˆ − jyˆ 2

eˆa =

xˆ − jyˆ 2

The received complex voltage from (6.69) is v(w, a) = eˆw ⋅ eˆ∗a =

6765_Book.indb 162

xˆ − jyˆ xˆ + jyˆ ⋅ =1 2 2

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Antenna-Wave Interaction163



This indicates a complete polarization match of the wave to the antenna, as it should be. Using this in (6.72) gives the polarization efficiency value of p = 1, again showing there is no polarization mismatch as we found in (6.61) using the vector effective length approach.

6.5  Problems

1. Show that the normalized Stokes parameters can be written as 2

s1 = e x − e y

2

s2 = 2 e x e y cosd s3 = 2 e x e y sind





6765_Book.indb 163

2. Use Stokes parameters to evaluate the polarization efficiency for a vertically polarized antenna receiving a RHCP wave. 3. Three different waves have the following characteristics: w1: Unpolarized w2: LHEP (left-hand elliptical polarization) with d =1/2, ⎪AR⎪=3, τ = 135° w3: LHCP Three different antennas are polarized as follows: a1: HP (horizontal linear polarization) a2: RHEP with ⎪AR⎪=3, τ = 45° a3: RHCP If all antennas have an effective aperture of 1 m2 and all waves have a flux density of 1 W/m2, calculate the output powers in W for all possible wave-antenna combinations. 4. Derive (6.28) from (6.24). Hint: make use of identities (B.22) and (B.24). 5. A completely polarized, right-hand elliptically polarized wave of 8-dB axial ratio is incident on an antenna with an axial ratio of 4 dB (RHEP). Use the appropriate formula to evaluate the maximum polarization loss in dB. Compare to values from the Ludwig chart of Figure 6.5. 6. An incoming wave with an axial ratio of 4 dB is incident on an antenna also with an axial ratio of 4 dB. Calculate the maximum polarization

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Polarization in Electromagnetic Systems

loss for the situations given below. (You should also be able to find an approximate value from Figure 6.5.) (a) Antenna and wave of the same sense with orthogonal major axes (b) Antenna and wave of the same sense with aligned major axes (c) Antenna and wave of opposite sense with orthogonal major axes (d) Antenna and wave of opposite sense with aligned major axes 7. A completely polarized wave illuminates a receiving antenna and both are elliptically polarized with an axial ratio of 6 dB. For the following cases, find the polarization loss in decibels in two ways: using appropriate formulas and using the Ludwig chart of Figure 6.5. (a) Antenna and wave of the same sense and major axes perpendicular (b) Antenna and wave of opposite sense and major axes aligned 8. A transmitting antenna in a communication link is right-hand sensed with an axial ratio of 1 dB and a tilt angle of 10°. The receiving antenna is also right-hand sensed and has an axial ratio of 0.7 dB and a tilt angle of 35°. Compute the polarization loss in dB. 9. Show that the Stokes parameter form of polarization efficiency of (6.17) is consistent with the form in terms of polarization ratios in (6.36). 10. Derive the following polarization efficiency formula, which is expressed in polarization ratios for circular polarizations [12]. (Hint: use [3.69].) 1 + rCa rCw + 2 rCa rCw cos2Δt 2

p=

2

(r

Ca





6765_Book.indb 164

2

)(

)

+ 1 rCw + 1 2

11. Three waves have the following polarization states: w1: unpolarized w2: RHCP w3: LHEP with d = 0.5, ⎪R⎪ = 1 dB, τ = 90° Three receive antennas have the following polarization states: a1: VP a2: LHCP w3: RHEP with ⎪R⎪= 2 dB, τ = 135° Determine the polarization efficiency associated with all nine combinations of incident waves and receiving antennas. Tabulate your answers. (a) Use Stokes parameters in the solutions (b) Repeat using the axial-ratio formulation

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Antenna-Wave Interaction165



(c) Repeat using polarization ratios 12. Work Example 6.4 for the maximum loss case using polarization ratios to find polarization efficiency; for example, use (6.36). 13. Use the appropriate axial-ratio based formulas to evaluate the polarization efficiency for the following cases: (a) Wave RHCP; antenna RHCP (b) Wave RHCP; antenna LHC (c) A partially polarized wave that has half its power polarized (RHCP); a receiving antenna of 3.5-dB axial ratio and of left-hand sense (d) A LHEP wave of 8-dB axial ratio with major axis vertical; a RHEP receiving antenna of 3-dB axial ratio with major axis 30° from horizontal 14. Repeat Problem 13 using Stokes parameters. 15. Repeat Problem 13 using polarization ratios. 16. Repeat Problem 13 using polarization vectors. 17. Polarization efficiency can be evaluated using the coherency matrices of (3.52) as follows [5]:



⎧⎪ ⎡ a a ⎤ ⎡ s s ⎤ ⎫⎪ p = Tr ⎨ ⎢ 11 12 ⎥ ⎢ 11 12 ⎥ ⎬ ⎪⎩ ⎣ a21 a22 ⎦ ⎣ s21 s22 ⎦ ⎪⎭







6765_Book.indb 165

where Tr is the trace (sum of diagonal entries) of a square matrix. Evaluate polarization efficiency using this approach for parts (a) and (b) of Problem 13. 18. Calculate the loss in decibels due to polarization mismatch for a partially polarized wave with 0.4 degree of polarization and a totally polarized portion that is right-hand elliptically polarized with a 1.5-dB axial ratio and 10° tilt angle incident on an antenna that is left-hand elliptically polarized with a 0.5-dB axial ratio and 0° tilt angle. Make use of (6.43). 19. (a) Prove that the completely polarized form of (6.48), pc,co + pc,cr = 1, using the polarization vector form of polarization efficiency in (6.37). (b) Prove that (6.48) holds for partially polarized waves given that it does for completely polarized waves. 20. Verify (6.48) using the axial ratio form of polarization efficiency in (6.29). 21. Show that the polarization efficiency expressed in terms of antenna vector effective length is equivalent to polarization efficiency expressed in terms of polarization ratios; that is, derive (6.39) from (6.59).

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Polarization in Electromagnetic Systems

References [1] IEEE, IEEE Standard for Definitions of Terms for Antennas, Standard 145-2013, 2013. [2]

Mott, H., Polarization in Antennas and Radar, New York: Wiley, 1986.

[3]

Stutzman, W. L., and G. A. Thiele, Antenna Theory and Design, Third Edition, New York: Wiley, 2013.

[4]

Rumsey, V. H., G. A. Deschamps, M. I. Kales, and J. I. Bohnert, “Techniques for Handling Elliptically Polarized Waves with Special Reference to Antennas,” Proceedings of the IRE, Vol. 39, May 1951, pp. 533–552.

[5]

Kraus, J. D., Radio Astronomy, Second Edition, Cygnus-Quasar Books, 1986, p. 124. (Originally published by McGraw-Hill in 1966.)

[6]

Ludwig, A. C., “A Simple Graph for Determining Polarization Loss,” Microwave J., Vol. 19, Sept. 1976, p. 63.

[7]

Schrank, H., “Antenna Designer’s Notebook,” IEEE Antennas and Propagation Soc. Newsletter, Aug. 1983, pp. 28–29.

[8]

Kramer, E., “Determine Polarization Loss the Easy Way,” Microwaves, Vol. 14, July 1975, pp. 54–55.

[9]

Milligan, T., Modern Antenna Design, Second Edition, New York: Wiley, 2005, p. 25.

[10] Beckmann, P., The Depolarization of Electromagnetic Waves, Boulder, CO: Golem Press, 1968. [11] Sinclair, G. “The Transmission and Reception of Elliptically Polarized Waves,” Proc. of IRE, Vol. 38, Feb. 1950, pp. 148–151. [12] Hollis, J. S., T. J. Lyon, and L. Clayton, Microwave Antenna Measurements, ScientificAtlanta Inc., Atlanta, 1970, pp. 3–40 (available from MI Technologies).

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7 Dual-Polarized Systems 7.1  Introduction to Dual-Polarized Systems Electromagnetic systems employing two polarizations that are orthogonal are referred to as dual-polarized. Dual-polarized systems have many applications. Usually they fall into one of two categories: communications or sensing. In communications, a dual-polarized communication link can be used to increase capacity by as much as a factor of two. This is possible because orthogonally polarized channels, each with its own transmitting and receiving hardware, can operate over the same link on the same frequency at the same time. This is referred to as frequency reuse because the same frequencies are reused on different polarizations. As the electromagnetic spectrum becomes more crowded, demand for frequency reuse will increase. The requirement on dual-polarized communication systems is that there be adequate isolation between channels operating on the same frequency (ideally, they should be orthogonally polarized) to avoid excessive cross-channel interference, or cross talk. Such systems are usually only deployed in situations where propagation effects do not depolarize the signals; Chapter 8 addresses depolarization. Clear line-ofsight links are good candidates for dual-polarized operation. An example is a terrestrial microwave communication link, such as in Figure 7.6(a), which sends information between elevated sites. 167

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Frequency reuse is widely used in satellite communications. In C-band satellite communications, the uplink (i.e., Earth-to-space) operates at 6 GHz and the downlink (i.e., space-to-Earth) operates at 4 GHz. Figure 7.1 shows the frequency plan for the downlink. Note that adjacent channels are offset by half of the frequency interval for the transponder and guard band (20 MHz), and the adjacent channels are of orthogonal polarization. A similar frequency reuse approach is used for Ku-band satellite communications on frequencies 14/11 GHz. In practice, the odd/even channels are assigned either VP/ HP or HP/VP on U.S. domestic satellites. Also in use is a similar frequency plan with dual-circular polarization. The downlink frequency band of 12.2 to 12.7 GHz is set aside for direct broadcast satellite (DBS) service, which has 32 TV channels each of 24-MHz interleaved with alternate RHCP and LHCP polarizations. The second main use for dual-polarized systems in addition to frequency reuse is polarization diversity. Polarization diversity employs two orthogonally polarized antennas to receive (or transmit) a signal. Consider two orthogonally polarized receive antennas. The corresponding polarization components of the incoming wave interact with the propagation medium differently and fade differently with time. In the simplest diversity processing scheme, the larger of the two outputs at the receiver is selected for use as a function of time. Polarization diversity is very effective in improving system performance and will be discussed in detail in Section 9.3. Polarization diversity is popular in non-line-of-sight communication links to reduce fading. A simple example of polarization diversity is found in FM broadcast radio and broadcast television (TV), and both have an interesting history related to polarization. After it was found that man-made RF electrical noise was

Figure 7.1  The frequency and polarization plan illustrating frequency reuse used in C-band satellite communication downlinks from the satellite to Earth stations.

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Dual-Polarized Systems169

predominantly vertically polarized, the FCC selected horizontal polarization for the use in an FM broadcast transmitting antenna. Also considered was the fact that horizontal polarization has better over-the-horizon propagation than vertical polarization. However, vehicles used vertical antennas, such as fender-mounted whip antennas, because of their simplicity. This led to a mismatch between transmit and receive polarizations, giving a loss of several decibels in the received signal compared to if the polarizations were matched [1]. In the 1960s, the FCC permitted FM broadcast stations to transmit both horizontal and vertical polarization, often implemented with a circularly polarized antenna, which greatly improved vehicular reception because it provides a diversity of polarizations for the user terminal to receive. In 1977, the FCC allowed TV stations to also broadcast circular polarization. This improved indoor reception with antennas such as rabbit ears and also reduced ghosting due to reflections from objects such as buildings. Ghost signals are reduced because the reflections tend to be of a sense opposite that of the transmitted CP signal; this principle is discussed in Section 8.3 [2]. The use of vertical and horizontal polarization (often implemented using CP) for transmission affords a polarization diversity dimension to the communication channel. It allows a receiver to use either vertical or horizontal polarization components with no major difference in performance. Cellular communication systems also make extensive use of diversity, including polarization diversity, to improve performance. The primary use of diversity in a macrocellular system is for diversity reception on the base station of the uplink signal from the user terminal. Typical cells are divided into three 120° sectors; more sectors are used on the high frequency bands. Diversity for cellular is traditionally implemented with spatial diversity that uses two antennas spaced about 10 wavelengths apart. The signals received on the antennas experience different propagation paths and thus will have different multipath fading characteristics, leading to a high probability that one signal level will be acceptable when the other is in a deep fade; see Section 9.3 for more details. The base station antenna of Figure 7.2(a) shows a three-sector triangular mount with three antennas on each face that serve one sector. The two antennas at the ends provide for diversity reception, while the center antenna is used for transmitting the downlink signal to the user. There are many variations on this basic configuration. Spatial diversity equipment uses a large structure with associated mechanical and esthetic issues. Polarization diversity offers a more compact configuration for a base station as illustrated in Figure 7.2(b). Polarization diversity provides similar performance to spatial diversity; more quantitative treatments of diversity are in Section 9.6.

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Figure 7.2  A typical three sector base station antenna configuration utilizing (a) spatial diversity, and (b) polarization diversity.

Dual polarization is also used in sensing systems. We discuss the use of a dual-polarized receiver to measure the polarization state of the incoming wave in Section 10.3, where the application is to infer the polarization of the antenna transmitting the wave. There are other such sensing applications. For example, additional information is available about a target or scene with a dual-polarized radar (see Section 8.6) or a radiometer (see Section 8.7). Systems with more than two polarizations are also possible and are called multipolarized systems. The polarizations cannot be orthogonal, but instead are usually as widely separated as possible such as in the case of a quad-polarized system using linear polarizations tilted at angles of 0°, 45°, 90°, and 135°. In this chapter we define quantities used to evaluate dual-polarized systems and discuss common calculations with such systems. Proper design of dual-polarized systems requires a thorough knowledge of these principles as well as hardware limitations and realizations.

7.2 Cross-Polarization Ratio Any wave or antenna polarization state can be decomposed into component orthogonal polarization states. In Section 3.8 we saw how to determine the

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state orthogonal to any given polarization state for several representations. The two orthogonal decomposition states are usually called the co-polarized and cross-polarized states. The co-polarized state is usually taken to be closest in polarization state to the desired or nominal state. This is illustrated with a terrestrial microwave relay system that employs polarizations that are vertical and horizontal linear, where vertical and horizontal usually taken relative to Earth (that is, horizontal is parallel to Earth’s surface). Suppose one channel is nominally vertically polarized. We then take perfect VP and HP as co-­ polarized and cross-polarized states, respectively. The transmitting and receiving antennas are not perfect and can be decomposed into co-polarized and cross-polarized components, or desired and undesired components. Another, and often more meaningful, way to describe this terrestrial link is to choose the co-polarized state to be the exact polarization state of the transmitting antenna, which is near VP but has a noninfinite axial ratio. The cross-polarized state is then orthogonal to this. Ideally the receiving antenna polarization would be matched to that of the transmitting antenna. That is, the co-polarized polarization for the receiving antenna would be the same as the transmitted wave. Then no polarization loss is experienced. More important, however, is the cross-channel interference. If dual-polarized operation is used, the receiving antenna that is nominally HP will respond to the cross-polarized part of the adjacent channel that is nominally VP. In this case, it is best to decompose the states into perfect VP and HP. We now show how to use the antenna-wave interaction principles developed in Chapter 6 to evaluate dual-polarized systems. Let w be the state we wish to decompose into co-polarized state co and cross-polarized state cr. States co and cr are taken to be orthogonal to each other. However, in the steps that follow this assumption is not necessary. The decomposition could also be used to describe waves in space, antennas, or devices but the discussions that follow wave terminology is used. In other words, we are assuming that the polarization of a wave equals the polarization of the transmitting antenna and no depolarization in the medium occurs; depolarization is the topic of Chapter 8. First we decompose the wave polarization state w into co-polarized and cross-polarized states in terms of complex vectors: ! Ew = Eco eˆco + Ecr eˆcr (7.1)

where

! Ew = electric field vector for the wave Eco = electric field for the co-polarized component of the wave

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Ecr = electric field of the cross-polarized component of the wave êco = co-polarized complex (unit) vector êcr = cross-polarized complex (unit) vector From (3.25) and (3.77) we have the following properties of the orthogonal decomposition:

1 = eˆco ⋅ eˆco∗ (7.2a)



0 = eˆco ⋅ eˆcr∗ (7.2b)

Cross-polarization ratio (CPR) is defined as the ratio of the power density in the cross-polarized component of the wave to that in the co-polarized component: 2

CPR =

Ecr Scr 21 Ecr ⋅ Ecr∗ (7.3) = 1 = 2 Sco 2 Eco ⋅ Eco∗ Eco

where (3.22) was used. CPR is expressed in decibels as

CPR(dB) = 10log CPR (7.4)

If no cross-polarization is present (Ecr = 0), then CPR = 0 and CPR(dB) = −∞. If the cross-polarization level is as high as the co-polarization level (Ecr = Eco), then CPR = 1 and CPR(dB) = 0. CPR can be calculated using polarization efficiencies. The polarization efficiencies are found from the co-polarized and cross-polarized power densities as follows:



pco =

Sco Sw

pcr =

Scr (7.5) Sw

where Sw is the power density in the wave. Using these in (7.3) gives CPR in terms of the efficiencies:



CPR =

pcr (7.6) pco

which is very convenient for calculations.

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The linear polarization case of CPR is the easiest one to understand. Figure 7.3 illustrates an LP wave with the co-polarized state VP and the cross-polarized state HP. The wave is linearly polarized with the electric field oriented at an angle Δτ to the co-polarized state (VP). CPR follows directly from (7.6) with (6.26b):



CPR L =

pcr cos2 (90° − Δt) = = tan2 Δt (7.7) pco cos2 Δt

As expected, CPR only depends on the angle Δτ . When the wave electric field is aligned with the vertical component, Δτ = 90° and CPR L = 0, indicating that there is no cross-polarization. When the wave is horizontally polarized Δτ = 0°, CPR L = ∞, indicating that all wave power is in the cross-polarized state and none is in the co-polarized state. CPR L is plotted in Figure 7.4 as a function of angle offset. Note that at 45° of offset CPR L is unity (0 dB), indicating equal power density in the co-polarized and cross-polarized components. In the foregoing case, the wave polarization is a linear polarization state rotated from the desired tilt angle. More generally the wave is elliptically polarized. Then the CPR for an elliptically polarized wave expressed in terms of linear components along the principal axes of the wave polarization ellipse is CPR L =

Eminor Emajor

2 2

=

1 2 (7.8) Rw

where (2.26a) was used. The desired linear polarization here is assumed to be along the major axis of the ellipse. If there is no wave component along the minor axis, Eminor = 0, Rw = ∞ and (7.8) yields CPR L = 0; thus, there is

! Figure 7.3  Decomposition of a linearly polarized wave with electric field Ew decomposed into orthogonal linear components, Eco and Ecr . The cross-polarization ratio is CPR = ⎪Ecr⎪/⎪Eco⎪.

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Figure 7.4  Cross-polarization ratio of a linearly polarized wave as a function of electric field angle relative to a reference co-polarized linear polarization state, Δ τ . The plot is based on (7.7).

no cross-polarization. This form for CPR is commonly used for a wave with low axial ratio. If the wave is close to circularly polarized, the following formulation is used. An elliptically polarized wave can also be decomposed into orthogonal CP components. The component axial ratios are

Rco = ±1 and Rcr = m1 (7.9)

The signs for the axial ratios are opposite ( i.e., Rcr = −Rco) and Rco has the same sign as the wave. That is, if the wave is LHEP (RHEP), then Rco is +1 (−1) and Rcr is −1 (+1). Using these in (6.29) gives





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pco =

( R + 1)2 (7.10a) 2( R2 + 1)

pcr =

( R − 1)2 (7.10b) 2( R2 + 1)

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Dual-Polarized Systems175



where R is the wave axial ratio. Then the cross-polarization ratio for circular polarization components using (7.6) with (7.10) is pcr ( R − 1) (7.11) = pco ( R + 1)2 2

CPR C =



A more direct derivation of this result follows from the axial ratio definition in (2.40): R =

1 + CPR C 1 + EL0 /ER0 (7.12) = 1 − EL0 /ER0 1 − CPR C

where CPR C is EL0/ER0 (or ER0/EL0) if right-hand (or left-hand) circular polarization is used as the co-polarized state. Solving (7.12) for CPRC gives (7.11). CPRC is plotted as a function of axial ratio in Figure 7.5.

Figure 7.5  Cross-polarization ratio of a wave relative to a pure circular co-polarized state of the same sense as a function of its axial ratio. The plot is based on (7.12).

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Example 7.1  CPR in Terms of CP Components for a Wave of 0.3 dB Axial Ratio

The wave axial ratio of 0.3 dB is ⎪R⎪ = 100.3/20 = 1.0351. The cross-polarization ratio from (7.11) expressed in dB is CPR C (dB) = 20log

R −1 = 20log(0.01727) = −35.26 R +1

This example can be located on Figure 7.5. It is often useful to express CPR in terms of normalized complex voltage (see Section 6.4). Using (6.72) in (7.6), we have CPR =

2

pcr v(w,cr) (7.13) = pco v(w,co) 2

7.3  Cross-Polarization Discrimination and CrossPolarization Isolation The performance of general two-channel devices is characterized by coupling between the channels, quantified with isolation. The same is true of dualpolarized radio systems. If two nominally orthogonally polarized channels are operated over the same radio channel, coupling (e.g., cross talk) between channels must be avoided. Such coupling can arise due to the following: high cross-polarization in a transmitting antenna, high cross-polarization in the receiving antenna, or depolarization along the transmission path. Antenna polarization properties were discussed in Section 5.2. Depolarization by a medium is to be discussed in Chapter 8. In this section, we show how to model dual-polarized systems and evaluate coupling effects. 7.3.1  Definitions A general dual-polarized radio link is shown in Figure 7.6. It consists of dualpolarized transmitting and receiving antennas and an intervening propagation medium. The antennas have nominally orthogonal polarizations and the transmit and receive antennas are nominally aligned in polarization; that is, the polarization of channel 1 of the transmit antenna is approximately the same as the polarization on channel 1 of the receive antenna. It is designed for frequency reuse (see Section 7.1). Channels 1 and 2 are transmitting on the same frequency at the same.

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Dual-Polarized Systems177

Figure 7.6  A general dual-polarized radio link: (a) antenna configuration, (b) illustration of XPI coupling, (c) illustration of XPD coupling.

Cross-polarization isolation, XPI, or isolation, I, is defined as the ratio of the wanted power level to the unwanted power level in the same channel when the transmitting antenna is radiating nominally orthogonally polarized signals of the same frequency and power level [3]. This is illustrated in Figure 7.6(b). For definition illustration purposes, we consider channel 1 to be the wanted channel. Channel 1 at the receiving antenna, which is nominally matched to the polarization of channel 1 of the transmitting antenna, has two responses in its channels 1 and 2. The wanted response expressed as a signal

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voltage is V11, where the first subscript indicates the channel of the receiver and the second subscript indicates the channel of the transmitter. Thus, V11 is the voltage out of channel 1 of the receiver due to the signal from channel 1 of the transmitter. The second, and unwanted, response is V12; this is the contribution to receiver channel l arising from of the signal from transmitter channel 2. For perfectly orthogonally polarized and aligned transmit and receive systems the only mechanism to generate V12 is a depolarizing medium along the path. Cross-polarization isolation for channel 1 is 2



V wanted power level in ch. 1 XPI1 = = 11 2 (7.14) unwanted power level in ch. 1 V12

If no signal from channel 2 is coupled into channel 1 at the receiver (V12 = 0), the isolation is infinite (XPI1= ∞). If just as much signal from transmitter channel 2 appears in channel 1 as that coming from transmitter channel 1 (V12 = V11), then XPI1 = 1. The isolation in channel 2 is defined in a similar fashion:



XPI2 =

V22 V21

2 2

(7.15)

XPI is expressed in decibels using

XPI(dB) = 10log XPI (7.16)

So, if the XPI is 40 dB, this means that the wanted signal power level is 40 dB above the unwanted power level in the same receive channel. In communication systems an XPI of 25 dB or more is a common specification for good-quality performance. The received voltages Vij can be found using the effective length of an antenna as in (6.52). The definition of XPI requires that the two transmitting channels carry identical power. That is, isolation includes the coupling properties of the system hardware, not conditions of the input signals. In practice, a system with good isolation can still experience interference if the companion channel carries much higher power. Any power imbalance needs to be taken into account when finding the interference power level in a receiver channel. A related definition for dual-polarized systems is cross-polarization discrimination, XPD, which is the ratio of the power level at the output of a receiving antenna that is nominally co-polarized to the transmitting antenna to the output of a receiving antenna of the same gain but nominally orthogonally

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Dual-Polarized Systems179

polarized to the transmitting antenna [3]. Figure 7.6(c) illustrates XPD coupling. Channel 1 of the transmitter is excited, but not channel 2. The copolarized output of the receiver is V11 and the output from the cross-polarized receiver channel is V21. Thus, XPD1 is the voltage out of receiver channel 2 due to excitation on transmitter channel 1:



XPD1 =

V11

2

V21

2

V22

2

(7.17)

Similarly,



XPD2 =

V12

2

(7.18)

The gains of the two receiving channels are assumed to be identical, which is usually the case in practice. Isolation is representative of the real operational situation. That is, the system designer needs to know the interference level on a given channel due to nonideal polarization effects. However, XPI is difficult to measure. On the other hand, XPD is directly measurable, as suggested in Figure 7.6(c). Channel 1 of the transmitter is excited and the levels of the two receiving antenna ports are measured. This fully characterizes the antennas and the medium. It would be very useful to be able to measure XPD and infer XPI from it. This, in fact, can be done in most cases. If the channel hardware is balanced (the transmitter channels are excited equally and the receiver channels are of the same gain), then

XPI = XPI1 = XPI2 (7.19a)



XPD = XPD1 = XPD2 (7.19b)

Furthermore, if the propagation medium has orthogonal axes of symmetry, then isolation and discrimination values are identical [4]:

XPI = XPD (7.20)

Many hardware systems are designed to be balanced and most propagation media have orthogonal axes of symmetry, so (7.19) and (7.20) hold. Regardless of whether these assumptions can be verified, it is customary to assume that (7.20) holds for operational systems of many kinds.

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In the ideal case, where the antennas have no cross-polarization response (and are aligned in polarization) and the medium does not depolarize, there is infinite isolation (and discrimination) between channels. In this ideal case, assumptions about equal input powers and equal receive gains are not necessary. The foregoing definitions can be written in terms of polarization efficiency and normalized complex voltages:



v (w,aco ) p(w,aco ) XPI = = (7.21) p(wx ,aco ) v (w ,a ) 2 x co



p(w,aco ) v (w,aco ) XPD = = (7.22) p(w,acr ) v (w,a ) 2 cr

2

2

where (6.72) was used. Here the polarization state w is that of wave incident on the receiving antenna generated by the co-polarized channel of the transmitter and aco is the co-polarized state of the receiving antenna. Similarly, the polarization state wx is that of wave incident on the receiving antenna generated by the cross-polarized channel of the transmitter and acr is the cross-polarized state of the receiving antenna. For subsequent developments we will use XPD because a system typically is characterized by measurement of XPD and it is equated to XPI. We use isolation, I, to denote XPI. Isolation is used in evaluating system performance. 7.3.2 Dual Decomposition It is sometimes convenient to decompose a wave into orthogonal components in a reference frame that is not the normal horizontal/vertical linear components before calculating the interaction with a receiving antenna. This dual decomposition technique is performed by treating the components separately and then combining their contributions. We denote the orthogonal decomposition components as co for co-polarized and cr for cross-polarized. The wave polarization state, w, has a normalized complex vector êw, which can be written in terms of co and cr components as

eˆw = v(w, co)eˆco + v(w, cr)eˆcr (7.23)

The receive antenna has two ports that are “co” and “cross,” denoted as states aco and acr. However, these are only nominally orthogonal to each other

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Dual-Polarized Systems181

to allow for imperfect realization. The output from the co-polarized antenna port from (6.69) is v (w, aco ) = eˆw ⋅ eˆ∗aco = v(w, co)eˆco ⋅ eˆ∗aco + v(w, cr)eˆcr ⋅ eˆ∗aco

= v(w, co)v ( co,aco ) + v(w, cr)v ( cr,aco )

(7.24)

where (7.23) was used. The two terms in this expression represent the projections of the co and cr components of the wave onto the antenna co-polarized state aco. A similar line of reasoning for the cross-polarized component leads to

v (w, acr ) = v(w, co)v ( co, acr ) + v(w, cr)v ( cr, acr ) (7.25)

The polarization efficiencies for the antenna ports follow directly from the normalized complex voltages using (6.72) as

p(w, aco ) = v (w, aco ) (7.26a)



p(w, acr ) = v (w, acr ) (7.26b)

2

2

These are general results and the intermediate dual decomposition forms of (7.24) and (7.25) need not be used when finding polarization efficiencies. If the antennas happen to be orthogonal, then these efficiencies sum to unity; see (6.48). Example 7.2  Dual Decomposition of a VP Wave Incident on a VP Antenna

As a simple example to validate the dual decomposition results we consider a case with a known answer: a vertical linearly polarized wave incident on a vertical linearly polarized antenna. Let vertical be in the y-direction, then

eˆw = yˆ

eˆaco = yˆ

eˆacr = xˆ (7.27)

Let the co and cr states for the dual decomposition be 45° and 135° slant linear. The polarization vectors from Table 3.5 are

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( xˆ + yˆ ) 2

45° LP (7.28a)

(− xˆ + yˆ ) 2

135° LP (7.28b)

eˆco = eˆcr =

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Obviously, the output from the antenna VP port should be unity and the output from the antenna HP port should be zero. Using (7.26) we find this to be true: 2 p(w, aco ) = v (w, aco ) = eˆw ⋅ eˆ∗aco = yˆ ⋅ yˆ = 1 (7.29a) 2

2



2 p(w, acr ) = v (w, acr ) = eˆw ⋅ eˆ∗acr = yˆ ⋅ xˆ = 0 (7.29b) 2

2



We can find the normalized complex voltages at the antenna ports using dual decomposition of the wave. First,

v(w, co) = eˆw ⋅ eˆco∗ =

yˆ ⋅( xˆ + yˆ ) 1 = (7.30a) 2 2

v(w, cr) = eˆw ⋅ eˆcr∗ =

yˆ ⋅(− xˆ + yˆ ) 1 = (7.30b) 2 2

v ( co, aco ) = eˆco ⋅ eˆ∗aco =

( xˆ + yˆ ) 1 ⋅ yˆ = (7.30c) 2 2

v ( cr, acr ) = eˆcr ⋅ eˆ∗acr =

(− xˆ + yˆ ) −1 ⋅ xˆ = (7.30d) 2 2

Using these in (7.24) and (7.25) give v (w, aco ) = v(w, co)v ( co, aco ) + v(w, cr)v ( cr, aco )

=

1 1 1 1 + =1 2 2 2 2

(7.31a)

v (w, acr ) = v(w, co)v ( co, acr ) + v(w, cr)v ( cr, acr )

=

1 1 1 ⎛ 1 ⎞ =0 + − 2 2 2⎝ 2⎠

(7.31b)

which are the correct results. 7.3.3 Calculating XPD The polarization state of a wave arriving at the receiving antenna of dualpolarized radio system is the polarization of the transmitting antenna altered

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by any depolarizing effects of the intervening propagation medium. Interaction between the arriving wave with the receiving antenna determines the link isolation. It is most convenient to evaluate systems using XPD because it directly relates to measurements; see (7.22). The XPD expression of (7.22) is similar to the CPR expression of (7.13). However the states aco and acr are the receiving antenna polarization states that are only nominally orthogonal, whereas the co and cr states used in CPR are exactly orthogonal. XPD includes system effects. If the receiving antenna has perfectly orthogonal states, then

CPR =

1 XPD

for orthogonality: acr (=cr) ⊥ aco (=co) (7.32)

As already noted, in most cases isolation is equal to cross-polarization discrimination. So, combining (7.20) with (7.21) gives

p(w,aco ) XPD = I = (7.33) p(wx ,aco ) If the states w and wx are orthogonal, then there is no cross-polarized power contribution to the receiving co-polarized channel. This would be the case if transmitting antenna ports are perfectly orthogonally polarized, the propagation medium is nondepolarizing, and the receiving antenna is matched to the wave (aco = w). Then, isolation is infinite. We will use XPD in the following treatments because the resulting expressions relate more directly to measurements. 7.3.3.1  Ideal Dual-Linearly Polarized Receiving Antenna

First we consider a receiving antenna with perfectly orthogonal, linearly polarized ports. Let the co-polarized state be HP (aco: ε co = 0, τ co = 0). The cross-polarized port polarization state is VP (acr: ε cr = 0, τ cr = 90°). The most convenient form for polarization efficiency calculations uses axial ratios and tilt angles. From (6.29) with Ra = ∞





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p(w, aco ) =

2 1 ( Rw − 1) cos2Δtco + (7.34a) 2 2( Rw2 + 1)

2 1 ( Rw − 1) cos2Δtcr p(w, acr ) = + (7.34b) 2 2( Rw2 + 1)

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where

Δtco = tco − tw (7.35a)



Δtcr = tcr − tw = tco ± 90° − tw = Δtco ± 90° (7.35b)

Using (7.34) and (7.35) in (7.22): XPD =

Rw2 + 1 + ( Rw2 − 1) cos2Δtco Rw2 + 1 − ( Rw2 − 1) cos2Δtco

(7.36)

If the major axis of the wave polarization ellipse is aligned with the copolarized LP orientation, then Δτ co = 0 and the following simple results are obtained from (7.36):

XPD = Rw2

for Δtco = 0 (7.37)

The major and minor axes of the wave polarization ellipse are aligned with the co-polarized and cross-polarized LP receive state major and minor ellipse axes. This result is the inverse of CPR for an elliptical wave given in (7.8). For a circularly polarized wave (⎪Rw⎪= 1) and the XPD is unity, indicating there is as much power in the cross polarized part of the wave as in the co-polarized part (for this situation of a dual-LP system). If the wave axial ratio is infinite (i.e., an LP wave) and is aligned with the co-polarization major axis (Δτ co = 0), then (7.37) gives XPD = ∞. For a general linearly polarized wave (⎪Rw⎪= ∞) with tilt angle τ w (7.36) reduces to



1 + cos2Δtco 2cos2 Δtco XPD = = = cot2 Δtco 1 − cos2Δtco 2sin2 Δtco

LP wave (7.38)

where (B.23) and (B.24) were used. If the wave polarization is aligned with the co-polarized antenna state (Δτ co = 0), (7.41) gives XPD = ∞. If, on the other hand, the wave is aligned with the cross-polarized receiving antenna state (Δτ co = 90°), then XPD = 0 and the wave is completely cross-polarized to the co-polarized antenna state. 7.3.3.2  General Dual Polarization

In the practical case, neither of the receiving antenna polarization states (aco or acr) is ideal (pure linear or circular) nor are the two orthogonal to each other.

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Therefore we start with the general polarization efficiency expression of (6.29) to find XPD using (7.22):

( Rco2 + 1)( Rw2 + 1) + 4Rco Rw + ( Rco2 − 1)( Rw2 − 1)cos2Δtco 2( Rco2 + 1)( Rw2 + 1) p(w,aco ) XPD = = 2 p(w,acr ) ( Rcr + 1)( Rw2 + 1) + 4Rcr Rw + ( Rcr2 − 1)( Rw2 − 1) cos2Δtcr 2( Rcr2 + 1)( Rw2 + 1) ( R2 + 1) ( Rco2 + 1)( Rw2 + 1) + 4Rco Rw + ( Rco2 − 1)( Rw2 − 1)cos2Δtco = cr2 ( Rco + 1) ( Rcr2 + 1)( Rw2 + 1) + 4Rcr Rw + ( Rcr2 − 1)( Rw2 − 1)cos2Δtcr (7.39) Although rather involved, this general expression is easily evaluated; however, caution should be exercised to avoid numerical difficulties in LP cases where axial ratios go to infinity. For orthogonal LP receiving antenna states (Rco = Rcr = ∞; Δτ cr = Δτ co ± 90°), (7.39) reduces to (7.36). If the wave and antennas are perfect CP, then ⎪Rw⎪, ⎪Rco⎪, and ⎪R cr⎪ are all unity, and (7.39) becomes XPD =



1 + Rco Rw =∞ 1 + Rcr Rw

CP case (7.40)

This infinity value arises because w and co are identical so Rco and Rw are either both +1 or both −1. If Rco = +1 (RHCP), then Rcr = −1 (LHCP), or if Rco = −1, then Rcr = +1 because the receiving ports are of opposite sense. Then, RcrRw = −1, making the denominator in (7.40) zero. The performance of a dual- (nearly) linearly polarized receiving system will deteriorate as the axial ratios of the receive antenna decrease. A realistic model uses a balanced receiving antenna (⎪Rco⎪ = ⎪Rcr⎪ = ⎪Ra⎪, Rcr = −Rco, Δτ cr = Δτ co ± 90° ) that is aligned with the incoming wave polarization ellipse major axis (Δτ co = 0). Then (7.39) reduces to XPD =

( Ra Rw + 1)2 ( Ra + Rw )2

balanced receiving antenna (7.41)

If the antenna is perfect (Ra = ∞) this again reduces to (7.37), and the antenna does not limit the system in any way. This result is plotted in Figure 7.7 for various antenna axial ratios. The abscissa is wave cross-polarization ratio, CPR w = 1/Rw2.

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Figure 7.7  XPD of a balanced dual- (nominally) linearly polarized receiving antenna (⎪Rco⎪ = ⎪Rcr⎪ = ⎪Ra⎪, Rcr = −Rco , Δ τ cr = Δ τ co ± 90° ) aligned with the incoming wave polarization ellipse major axis (Δ τ co = 0) as a function of the wave cross-polarization ratio with antenna axial ratio as a parameter. This is a plot of (7.41).

Figure 7.7 shows the effect of antenna polarization purity on the resulting cross-polarization discrimination. If the antenna is perfect (CPR a = −∞ dB), then XPD = 1/CPR w. As antenna performance deteriorates (CPR a reduces), XPD also deteriorates. As an example, for a wave with CPR w = −20 dB striking an antenna with CPR w = −20 dB, the cross-polarization discrimination is XPD = 14 dB. 7.3.3.3  Near Dual-Circular Polarization

Dual- (nominally) circularly polarized radio systems have axial ratios that are nearly unity, and XPD varies only slightly with rotation. The numerator in (7.39) represents the co-polarized power and does not fluctuate much with rotation; this permits introduction of approximations. Because the axial ratio magnitudes ⎪Rco⎪ and ⎪Rcr⎪ are near unity, the terms containing (R 2co − 1) and (R2cr − 1) factors will be small compared to the other terms and can be neglected. Note that the middle term in the numerator is positive because the wave and

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co-polarized antenna are of the same sense. Then (7.39) is approximated for antenna polarization states that are near CP as XPD ≈

( Rcr2 + 1) ( Rco2 + 1)( Rw2 + 1) + 4Rco Rw ( Rco2 + 1) ( Rcr2 + 1)( Rw2 + 1) + 4Rcr Rw + ( Rcr2 − 1)( Rw2 − 1)cos2Δtcr

nearly CP

(7.42) It is helpful to bracket the XPD values (i.e, maximum and minimum values) over the range of possible orientation angles. These are easily found from (7.42). Maximum XPD occurs when the wave polarization ellipse major axis is perpendicular to the cross-polarized antenna polarization ellipse major axis; that is, when Δτ cr = 90°. Similarly, minimum XPD occurs when Δτ cr = 0°. The extrema of (7.42) then are XPDmax = XPD( Δtcr = 90° ) ≈

XPDmin

Rcr2 + 1 ( Rco2 + 1)( Rw2 + 1) + 4Rco Rw (7.43a) 2 Rco2 + 1 2( Rw + Rcr )

Rcr2 + 1 ( Rco2 + 1)( Rw2 + 1) + 4Rco Rw (7.43b) = XPD( Δtcr = 0° ) ≈ 2 2 Rco + 1 2( Rw Rcr + 1)

If the antenna polarizations are of the same axial ratio magnitude as well as being nearly CP, then Rcr = −Rco and (7.43) reduces to





XPDmax

Rco2 + 1)( Rw2 + 1) + 4 Rco ( ≈ 2( Rw − Rco )

2

Rco2 + 1)( Rw2 + 1) + 4 Rco ( XPDmin ≈ 2 2( Rw Rco − 1)

Rw

Rcr = −Rco (7.44a)

Rw

Rcr = −Rco (7.44b)

In the case of a perfect circularly polarized wave, XPD is not dependent on the tilt angle of the incoming wave and (7.44) yields



⎡ R +1 ⎤ XPD ≈ ⎢ co ⎥ ⎣ Rco − 1 ⎦

2

(

)

CP wave Rw = 1 and Rcr = −Rco (7.45)

Note the similarity of this result to that in (7.11). We can establish the correspondence as follows. If instead of the wave being perfect CP, the antenna

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polarizations are perfect circular (Rco = ±1, Rcr = ∓1) and the wave has axial ratio Rw, the XPD formula is of the same form as (7.45) but with ⎪Rco⎪ replaced by ⎪Rw⎪:



⎡ R +1 ⎤ XPD ≈ ⎢ w ⎥ ⎣ Rw − 1 ⎦

2

(

)

perfect CP antennas Rco = Rcr = 1 (7.46)

This is a realization of the decomposition into perfect CP components embodied in the CPR definition. Comparing (7.46) to (7.11) gives

XPD =

1 CPR C

(

)

perfect CP antennas Rco = Rcr = 1 (7.47)

The XPD values found from (7.45) for a perfect CP wave are shown in Figure 7.8. XPD is plotted as a function of antenna axial ratio assuming the co-polarized and cross-polarized antennas are of equal axial ratio and opposite in sense. As the antenna axial ratio (⎪Rco⎪ = ⎪Rcr⎪) approaches unity (0 dB), XPD goes to infinity, indicating perfect channel cross-polarization discrimination, as it should for all polarizations being perfect CP. As another example, if the antenna axial ratio is 1 dB (⎪Rco⎪ = ⎪Rcr⎪= 1.122), then (7.45) gives XPD ≈ 24.8 dB; this case can be located on the figure. This is a commonly used good performance threshold; for perfect CP wave and an antenna with axial ratios lower than 1 dB, XPD will be greater than 25 dB. Figure 7.8 also applies to the case of perfect CP antennas and wave of axial ratio value given on the abscissa; this is the case of (7.46). In the perfect CP receiving antenna case, the axial ratio of a wave is uniquely determined from an XPD measurement. This is shown by solving (7.46):

Rw ≈

XPD + 1 XPD − 1

perfect dual-CP receiving antenna (7.48)

Repeating the above example of a measured XPD value of 24.8 dB (302.0), (7.48) gives the axial ratio of the incoming wave as 1.122 (1.00 dB). If the XPD is instead 15 dB (31.6), then (7.48) gives the axial ratio of the incoming wave as 1.433 (3.1 dB). Figure 7.8 can be used to find these two points. The general practical case is for both the wave and the antenna near CP but not perfect CP. The XPD value is determined by measuring the signal levels in the co-polarized and cross-polarized antenna output ports and using (7.22). The range of XPD values for this general situation are a useful guide on what the possible values are and are best represented graphically with maximum

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Dual-Polarized Systems189

Figure 7.8  XPD for a perfect circularly polarized wave incident on an antenna of the axial ratio value on the abscissa (with Rcr = −Rco ). The plot is based on (7.45). The plot can also be used for a perfect CP antenna (with ⎪Rcr⎪ = ⎪Rco⎪ = 1) and with the wave axial ratio value on the abscissa; see (7.46).

and minimum curves with wave axial ratio as a parameter. Figure 7.9 shows XPD obtained using (7.44) as a function of antenna axial ratio for wave axial ratios of 0.2, 0.4, 0.5, and 1.0 dB, and for a balanced receiving antenna. The maximum (Δτ cr = 90°) and minimum (Δτ cr = 0°) curves bound the actual XPD value. The actual value depends on the relative tilt angle Δτ cr, which often is not known. Note that the maximum curve goes to infinity when the antenna axial ratio (abscissa) value equals that of the wave. This point corresponds to the case of the wave being cross-polarized to the antenna. This occurs because the wave and antenna have the same axial ratios and the major axes are orthogonal, rendering a perfect mismatch. The curves in Figure 7.9 can also be used for a wave of varying axial ratio (abscissa) and an antenna of (co-polarized and cross-polarized) axial ratio magnitude given by the fixed parameter axial ratio values.

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Figure 7.9(a)  XPD values from (7.44) for a wave incident of known axial ratio on a dual-polarized antenna of known axial ratios. Curves are shown for antenna axial ratios of 0.2 and 0.4 dB with the upper curve in each case giving the maximum XPD and the lower cure the minimum. It is assumed that the receiving antenna is balanced: Rcr = −Rco .

The curves of Figure 7.9 show that there are multiple XPD values for a given antenna axial ratio value. The perfect CP antenna case of (7.46) and plotted in Figure 7.8 is an exception; there is a unique XPD for each CPR value. For instance, a perfect CP wave (⎪Rw⎪ = 0 dB) incident on the antenna of 1-dB axial ratio in both co-polarized and cross-polarized ports will produce an XPD of 24.8 dB. But if the incoming wave is not perfect CP and has an axial ratio of 0.5 dB (the antenna again with 1-dB axial ratio on each port), the XPD values will range from 30.8 to 21.3 dB. If the maximum and minimum XPD are measured, antenna axial ratio is uniquely determined if the antenna axial ratio is known. For example, for an antenna of 2-dB axial ratio and the maximum

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Figure 7.9(b)  XPD values from (7.44) for a wave of known axial ratio incident on a dual-polarized antenna of known axial ratios. Curves are shown for antenna axial ratios of 0.5 and 1.0 dB with the upper curve in each case giving the maximum XPD and the lower curve the minimum. It is assumed that Rcr = −Rco .

and minimum XPD values are measured to be 24.8 and 15.3 dB, respectively, the wave axial ratio must be 1 dB. Such a measurement is performed by rotating the co-polarized and cross-polarized antennas (about their beam maxima axes) while aimed in the direction of the incoming wave. If the antenna system consists of a reflector antenna with a dual-polarized feed assembly, it is usually not possible to rotate the antenna about its own (roll) axis. Also, it is often not sufficient to rotate the feed assembly to create this effect because of asymmetric reflector hardware (the reflector itself and support struts). Similarly, if the wave axial ratio is known and the maximum and minimum XPD values are measured, the antenna axial ratio can be found (still

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assuming the co-polarization and cross-polarization of the antenna have the same axial ratio). Repeating the example of a known wave axial ratio of 1 dB, if the measured maximum and minimum values of XPD are 24.8 and 15.3 dB, the antenna axial ratio must equal 2 dB; see Figure 7.9(b). Example 7.3  Dual-CP Earth Station Antenna

A dual-circularly polarized Earth station antenna at Virginia Tech has a 12-foot (3.66m) diameter prime-focus reflector antenna with a mechanically rotatable feed assembly consisting of a feed horn followed by an orthomode transducer (see Section 7.5) that forms RH and LH, nearly CP, outputs. At 11.7 GHz the axial ratios were measured to be 0.30 dB and 0.27 dB. These values are very close, offering some justification for the assumption of equal axial ratio magnitudes, allowing use of Figures 7.8 and 7.9 in practical situations. This is especially good cross-polarization performance in a moderately large antenna. This is demonstrated by the XPD values of Table 7.1, where the co-polarized and cross-polarized antenna states are 0.3 (RH) and 0.27 dB (LH), respectively. For an incoming wave axial ratio of 0.3 dB or lower the XPD will be about 30 dB or greater, which is good discrimination between channel outputs. As the wave axial ratio increases the XPD degrades.

7.4  Performance Evaluation of Dual-Polarized Systems Dual-polarized systems are characterized by isolation, I, which is a measure of the cross talk introduced into one channel from a second channel operating on the same frequency and propagation path. Path effects on isolation are Table 7.1 XPD Variation for the Antenna of Example 7.3 with Various Incident Wave Axial Ratios* XPD Limits (dB) Wave Axial Ratio, ⎪Rw⎪

Maximum

Minimum

0  dB

 1.000

36.2

36.2

0.3 dB

1.0351

55.3

29.7

0.5 dB

1.0593

37.6

27.1

0.7 dB

1.0839

32.1

25.1

1.0 dB

1.1220

27.5

22.7

*Antenna polarizations are 0.3 dB RH and 0.27 dB LH.

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considered in Chapter 8. Here we consider antenna effects on system isolation and how isolation is included in system performance evaluation. 7.4.1  Isolation Degradation Caused by Imperfect Antennas Antenna dual polarization performance is characterized by XPD because it is directly measureable, as discussed in Section 7.3.1 where it was noted that isolation is identical to XPD for a balanced receiving system. This also assumes there are no asymmetric path effects, but we exclude path effects altogether in this chapter; they are treated in the next chapter. For a dual orthogonal linearly polarized receiving system the isolation is found from (7.36) in terms of the wave CPR of (7.8): 1 − CPR L cos2Δtco 1 + CPR L I= 1 − CPR L 1− cos2Δtco 1 + CPR L 1+



LP antennas (7.49)

This reduces to (7.38) for a pure linearly polarized wave (CPR L = 0). Next we consider a dual nearly CP receiving system. The incoming wave CPR equals that of the wave radiated by the transmit antenna (because no propagation path degradation is present here) and is expressed in circular polarization terms by (7.11) using the wave axial ratio. We chose the orientation of the incoming wave that gives the lowest isolation to find the lower bound on performance (i.e., worst-case analysis). For a balanced receiving system, the minimum isolation follows from (7.44b) as



Rco2 + 1)( Rw2 + 1) + 4 Rco ( I min ≈ 2 2( Rw Rco − 1)

Rw

CP antennas (7.50)

7.4.2  Calculation of Isolation in Systems Radio systems are contaminated by noise and interference. Dual-polarized systems have an additional contaminant, cross talk that is introduced through imperfect polarization isolation. Cross talk is essentially a self-interference. It introduces independent information on the same frequency as the desired information. Both cross talk and interference are random disturbances and they are both treated as noise.

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For system calculations the XPDs associated with the various subsystem blocks must be combined. The simplest way to view the situation is to consider the cross-polarized components that each subsystem generates and then add them as phasors. So here we examine electric field phasors rather than output voltage components. Typically a worst-case analysis is performed for total system evaluation. The electric field phasor analysis is illustrated in Figure 7.10 using perpendicular co-polar and cross-polar components. The transmitting antenna XPD and depolarization of the intervening medium between the transmitting and receiving antennas together produce a cross-polarized electric field amplitude Ecrt . Because we are ignoring path effects until the next chapter, the associated XPD arriving at the receiver from the transmitter is XPDt. The receiving antenna XPD is denoted XPDr and internally generates an electric field amplitude Ecrr that is orthogonal to the incident co-polarized field Eco. The cross-polarized phasor components arising from the various sources add at the receiver. The dashed circle in Figure 7.10 indicates how Ecrr varies with its phase. However, phase information is usually not available. Lacking phase information, it is customary to perform a worst-case analysis by assuming the cross-polarized fields all add in-phase as illustrated in Figure 7.10. Then

Ecr ,max = Ecrt + Ecrr (7.51)

where amplitudes Ecrt and Ecrr are nonnegative and real valued. The worst-case XPD (that equals the system isolation) at the receiving antenna is I min = XPDmin

⎡E ⎤ = ⎢ cr ,max ⎥ ⎣ Ecr ⎦ 2



2

⎡ Et Er ⎤ 1 = ⎢ cr + cr ⎥ = 2 E E 1 ⎤ ⎡ 1 co ⎦ ⎣ co + ⎢ XPDt XPDr ⎥⎦ ⎣

(7.52)

Equation (7.52) is similar to parallel impedances in circuit theory. Just as the lowest impedance element dominates in a parallel network, the worst XPD device dominates in a dual-polarized system. For example, if the receiving antenna is perfect XPDr = ∞, then XPDmin = XPDt and the system isolation is determined only by the arriving wave polarization. Stated differently, the system isolation will be limited by the worst (lowest) XPD and will be equal to or lower than the lowest XPD in the system. Second-order effects have been ignored in the foregoing analysis. That is, in cascaded subsystems some

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Figure 7.10  The cross-polarized electric field component Ecrt arising from the XPD of both the transmitting antenna and path depolarization combine with the electric field r generated in the receiving antenna Ecr to produce the total electric field Ecr . Shown is the worst-case combination.

of the cross-polar signal will reappear in the co-polar channel. This, however, is rarely a significant contribution. Example 7.4  System Isolation for Imperfect Transmit and Receive Antennas

The transmitting and receiving antennas of a link have XPDs of 30 dB = 1000 and 36 dB = 3981. The propagation path introduces no depolarization. The worst-case system XPD from (7.52) is I min =

1

1 ⎤ ⎡ 1 ⎢ XPDt + XPDr ⎥ ⎣ ⎦ = 443.7 = 26.5 dB

2

=

1 ⎡ 1 + 1 ⎤ ⎢⎣ 1000 3981 ⎥⎦

2

So, the system isolation of 26.5 dB is slightly lower that the worst antenna XPD of 30 dB. Radio systems are characterized by their carrier-to-noise ratio, (C/N or CNR). In a general treatment there are three contributions to the noise term. First is conventional thermal noise power, Nt, consisting of noise accompanying the desired incoming signal, noise collected by the antenna arising from the sky, ground, and atmosphere, and noise generated in the receiver. Second is interference from signals in other systems operating on the same frequency and polarization and entering the receiving system, usually via antenna side lobes. All such signals are uncorrelated with the desired signal and appear as noiselike disturbances. Therefore, interference power, INT, is treated as

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additive noise. Third is power arriving in the desired channel due to imperfect isolation I of a dual-polarized system; this undesired power behaves in exactly the same way as interference. The net system carrier-to-noise ratio is found by combining the three noise sources in a parallel network fashion [5]. The development is similar to that for XPDs that we used in (7.52) and leads to



( ) ()

−1 C ⎡⎛ C ⎞ C = + N ⎢⎣⎜⎝ N t ⎟⎠ INT

−1

C + I

−1 −1

⎤ ⎥ ⎦

(7.53)

Similar to (7.52), the system carrier-to-noise ratio is limited by the worst of the carrier-to-noise contributions.

7.5  Polarization Control Devices There are hardware devices useful in processing signals to control polarization properties. Often these are realized most easily at microwave frequencies and higher. In addition to controlling polarization, systems that employ frequency reuse require polarization separation hardware. This is usually performed with an orthomode transducer (OMT). An OMT separates HP and VP, or LHCP and RHCP, signals. A related function is frequency separation. A diplexer separates two different frequencies. The most common application for a diplexer is in the separation of transmitting and receiving frequencies with the common port connected to the antenna and the remaining two ports passing frequencies appropriate to the transmitting and receiving bands while rejecting undesired frequencies. This enables use of a single, often expensive, antenna system for both transmitting and receiving. In this section we describe the operation of polarization control devices (polarizers) and OMTs. 7.5.1  Polarizers A polarizer is a two-port device that alters the polarization state of the wave in a controlled fashion. The most common forms are 90° polarizers and 180° polarizers. The 90° polarizer is used to convert an LP wave to a CP wave or vice versa. A 180° polarizer rotates the plane of polarization of an LP wave. Consider the 90° polarizer operation shown in Figure 7.11. The incident wave is CP, usually coming from an antenna. The polarizer converts the TE11 circular waveguide mode to the TE10 rectangular waveguide mode. To understand the CP to LP conversion process in the device the CP wave is visualized as decomposed into linear components parallel and perpendicular to the thin

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Dual-Polarized Systems197

dielectric plate. The plate delays the traveling wave, which is polarized along the plate by 90° relative to the perpendicularly polarized component, as illustrated in Figure 7.11. In effect, the dielectric material slows the wave relative to the same wave in air. The dielectric plate is often mechanically rotatable for polarization alignment. Of course, the plate will introduce some loss and an associated increase in noise the system. The incident LHCP shown in Figure 7.11 has a parallel component that leads the perpendicular component by 90°: E‖i = 1∠90° and E⟘i = 1∠0°. The 90° phase delay of the plate brings the two components in phase at the output: Eo‖ = 1∠0° and E⟘o = 1∠0°. The total electric field of the output wave, Eo, is then linearly polarized 45° to the plate. If a LP wave (E‖i = 1∠0° and E⟘i = 1∠0°) is incident on the 90° polarizer and the plate is oriented 45° to the electric field, the output components are Eo‖ = 1∠−90° and E⟘o = 1∠0°, producing a RHCP output. The 90° polarizer can also be used to linearize (or circularize) an elliptically polarized wave by rotating the polarizer to bring the orthogonal components in phase (or out of phase).

Figure 7.11  Use of a 90° polarizer to convert a left-hand circularly polarized input wave to a linearly polarized output wave at 45° to the dielectric plate.

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The 180° polarizer rotates the orientation angle of an incoming LP electric field, as shown in Figure 7.12. The angle of the incident field Ei relative to the dielectric plate is α . The action of the polarizer is explained by decomposing the incident electric field into parallel and perpendicular components as shown. The parallel component is delayed 180° so that Eo‖ = −E‖i . The perpendicular components are (ideally) not changed in phase by the dielectric so that E⟘o = E⟘i . Reconstructing the electric field at the output yields Eo, as illustrated in Figure 7.12. This orientation is rotated 2α relative to the input polarization. Another way to view the action of the 180° polarizer is that the polarizer plate bisects the input and output LP polarization orientations. This is seen in Figure 7.12 by the angle β = 90° − α relative to either side of the plate along which the input and output electric field planes of polarization are oriented. The 90° and 180° polarizers are often used in cascade; see Figure 8.11(c). The 90° polarizer linearizes the input wave and the 180° polarizer is mechanically rotated to change the LP orientation. This configuration has the following two applications: (1) to produce a match condition on one port and a null condition on the other for an arbitrary wave (this use is discussed in Section 10.3.3), and (2) to completely separate two orthogonal waves, which is to be explained in Section 7.5.2. Polarizer design and performance data are available in the literature [6].

Figure 7.12  The 180° polarizer rotates the plane of polarization of an incident LP wave, Ei, by 2 α , where α is the angle of Ei relative to the normal to the dielectric plate. The wave is propagating out of the page. The dielectric plate shown is placed in a circular waveguide similar to that shown in Figure 7. 11.

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7.5.2 Orthomode Transducers The orthomode transducer separates orthogonal TE11 modes in a circular waveguide. Figure 7.13 shows a form of an OMT that has good isolation between output ports [7]. The two modes are denoted as horizontal and vertical with corresponding electric fields EH and EV. The output ports and their linear polarizations and are displayed in Figure 7.13. The OMT can also be used to separate dual-CP waves. This is accomplished by preceding the OMT by cascaded 90° and 180° polarizers, which produce orthogonal TE11 modes that can be aligned with the OMT ports by mechanical adjustments of the polarizers [8]. See Figure 8.11(c). 7.5.3 Polarization Grids A grid of parallel conductors, such as wires, strips, or plates, used to alter the polarization of a wave is called a polarization grid. The grid acts to short out the wave component parallel to the grid long dimension. The electric field component perpendicular to the grid propagates through the grid with low attenuation. Figure 8.2 shows a conceptual diagram of a polarization grid using parallel wires that are oriented to pass horizontal polarization and eliminate vertical polarization. An elliptically polarized wave incident on the grid would have its component parallel to the grid shorted out, leaving only the component perpendicular to the grid. Thus, elliptical polarization is converted to linear

Figure 7.13  A waveguide orthomode transducer (OMT).

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polarization. The spacing between the grid elements needs to be a fraction of a wavelength apart; see [9]. A common application for a polarization grid is to reduce the crosspolarization level of a linearly polarized antenna. For example, an offset reflector antenna often does not produce an LP wave with low CPR. To improve performance, a polarization grid is placed over the aperture of the reflector antenna perpendicular to the desired LP orientation. The grid greatly reduces the cross-polarization level. Commercial polarizing grids are available that use grid element spacings of 0.1λ or less. Polarization grids can also be used as a linear-to-circular polarizer. This is accomplished using layers made of meander-line grating sheets [10, 11]. The layers are essentially polarization grids that convert the incident LP wave into LP components that are oriented ±45° relative to the incident wave orientation. The structure also introduces a 90° phase shift between the two components, creating a transmitted wave that is CP. The polarizer is capable of producing CP with an axial ratio as low as 1 dB over an octave of bandwidth [10].

7.6  Problems



1. A dual-polarized radio system has a transmitting antenna with 23-dB isolation and operates with 10W of power on channel 1 and 100W of power on channel 2. Channel 1 of the receiving antenna has 20 dB of isolation from channel 2. Find the interference level in channel 1 relative to the desired signal. 2. It is tempting to use polarization efficiency quantities in the dual decomposition technique in the following way: p(w, aco ) = p(w, co) p( co, aco ) + p(w, cr) p( cr, aco )



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(a) Show that this relation is incorrect and not equivalent to the similar expression for voltages (7.24). (b) Evaluate this expression for the situation in Example 7.2 to show the result differs from that in the example. 3. Show that the dual decomposition techniques for linear polarization gives p(w, aco) = cos2Δτ . The incident LP wave has tilt angle Δτ relative to the LP antenna co-polarized state aco. Take co and cr to be any orthogonal linear components.

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4. Use the dual decomposition technique for co and cr states that are LHCP and RHCP, respectively. Find the polarization efficiency in the antenna co-polarized port for the following cases. (a) Wave RHCP, antenna co-polarized state LHCP. (b) Wave VP, antenna co-polarized state LHCP. 5. Verify the values in Table 7.1 for the cases of wave axial ratios of 0 dB and 0.5 dB. 6. A wave of 0.5-dB axial ratio is incident on a dual-polarized antenna with an axial ratio of 0.3 dB on both the co-polarized and crosspolarized ports. Evaluate the XPD extremes using the general form in (7.39) and using the approximate form in (7.44). Tabulate these values in decibels along with those read from Figure 7.8. 7. Show that (7.49) reduces to (7.38) for an LP wave. 8. A satellite transmitting antenna has an XPD of 28 dB. The signal passes through rain and is degraded to an effective XPD of 21 dB. The receive antenna XPD is 31 dB. Compute the worst-case downlink XPD. 9. Find the best-case XPD for the situation described in Problem 8. 10. When evaluating total XPD for a system, field quantities are added as in (7.51); however, for evaluation of system carrier-to-noise ratio, noise powers are added. Explain. 11. Derive the C/N formula of (7.53) using additive noise ideas analogous to additive current components in a parallel network. 12. Show how path depolarization can be explicitly included in (7.52). 13. Sketch the input and output electric fields for cases of LHEP and RHEP waves incident on a 90° polarizer for cases of the dielectric plate along the major axis, and then along the minor axis of the polarization ellipse.







References [1]

Volakis, J. L., (ed.), Antenna Engineering Handbook, Fourth Edition, New York: McGraw-Hill, 2007, Section 39.3.

[2]

Johnson, R. C. (ed.), Antenna Engineering Handbook, Third Edition, New York: McGraw-Hill, 1993, Section 28-4.

[3]

IEEE Standard Definitions of Terms for Radio Wave Propagation, IEEE Standard 211-1997, 1997.

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[4]

Watson, P. A., and M. Arbabi, “Cross-Polarization Isolation and Discrimination,” Electronics Letters, Vol. 9, 1973, pp. 516–517.

[5]

Ha, Tri T., Digital Satellite Communications, Second Edition, New York: McGraw-Hill, 1990, Sec. 4.2.

[6]

Kitsuregawa, T., Satellite Communications Antennas: Electrical and Mechanical Design, Norwood, MA: Artech House, 1990, Sec. 1.3.3.

[7]

Rizzi, P. A., Microwave Engineering: Passive Circuits, Englewood Cliffs, NJ: PrenticeHall, 1988, Sec. 8-4.

[8]

Allnutt, J. E., Satellite-to-Ground Radiowave Propagation, London: Peter Peregrinus, 1989, p. 263.

[9]

Silver, S. (ed.), Microwave Antenna Theory and Design, MIT Radiation Laboratory Series, Vol. 12, McGraw-Hill, 1949, pp. 449–450. Available from IET at www.theiet. org.

[10] Young, L., L. A. Robinson, and C. A. Hacking, “Meander-Line Polarizer,” IEEE Trans. on Ant. & Prop., Vol. AP-21, May 1973, pp. 376–378. [11] Chu, R.- S., and K.- M. Lee, “Analytical Model of a Multilayered Meander-Line Polarizer Plate with Normal and Oblique Plane-Wave Incidence,” IEEE Trans. on Ant. & Prop., Vol. AP-35, June 1987, pp. 652–661.

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8 Depolarizing Media and System Applications 8.1  Introduction The medium through which a radio wave propagates influences its amplitude, phase, and polarization. For wideband signals, frequency-dependent effects (i.e., dispersive properties) are sometimes important as well. In this chapter we consider the effects that various propagation media have on the polarization of a wave as well as some other wave effects such as attenuation and direction change. We are primarily interested in how polarization is altered as it interacts with a medium in one of the following ways: reflection from, scattering by, or transmission through the medium. After the basic principles are covered several applications are considered, including communication systems, radar, and radiometry. Also included in this chapter are discussions of methods of compensating for depolarization effects.

8.2  Principles of Depolarizing Media A medium that alters the polarization state of a wave interacting with the medium is referred to as a depolarizing medium. In general, a depolarizing medium can cause a change in the direction of propagation of a wave as well 203

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as a change in its polarization. The coordinates and reference polarizations used for all depolarizing media situations are shown in Figure 8.1. For generality, a scatterer is shown as a plane, but it could be any medium, interface, or object. A scatterer is an object that redistributes power from an incident wave in multiple directions. A reflector might be considered to be a scatterer that scatters in only one direction; a planar reflector is an example and is discussed in Section 8.3.1. The plane of incidence is formed by the plane normal to the scatterer, nˆ , and the direction of propagation of the incident wave, ûi. The plane of scatter is the plane formed by surface normal and the direction of propagation of the exiting (i.e., output or scattered) wave, ûs. The incident wave is a plane wave consisting of electric field components E‖i and E⟘i that are parallel and perpendicular, respectively, to the plane of incidence. These phasor components can have any magnitude and phase, and thus when combined are capable of representing any polarization state. The exiting wave has definitions similar to those in Figure 8.1. For most types of scatterers, though, the scattered wave will not become a local plane wave until it is a long distance from the scatterer. The simplest general scattering model is the depolarization model developed by Beckmann [1, pp. 39–47]. It is based on representing the polarization states of the input and output waves by polarization ratio ρ L , which was discussed in Section 3.6. The depolarization introduced by the scatterer is found through four coefficients. Although the output wave depends on the input wave polarization state, the scattering coefficients do not. Also, the output wave

Figure 8.1  Coordinates and reference polarization orientations for the general depolarizing scattering problem.

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Depolarizing Media and System Applications205

depends on the electrical properties of the scatterer as well as the incident and exit angles. As indicated in Figure 8.1, θ i is the angle of incidence and is the angle between the direction of propagation of the incident wave and the surface normal. θ s is the angle is the angle between the direction of propagation of the scattered wave and the surface normal. ϕ s is the angle of the scattered wave relative to horizontal component of the incident wave direction. If no scatterer is present, the output polarization (with polarization ratio ρ Ls) equals the input state polarization (with polarization ratio ρ Li) and is not altered in direction; that is, ϕ s = 0°, θ i + θ s = 180°. If the scatterer is a reflecting plane, the angle of the scattered wave equals that of incident wave (θ s = θ i) and ϕ s = 0; this follows from Snell’s law of reflection. But the reflected wave can be altered in polarization from the incident wave state; this topic is covered in the next section. The depolarization introduced in the scattering process is handled by decomposing the incident wave into parallel and perpendicular electric components, interacting the components with the scatterer, and reconstructing the exiting wave electric field. We now present the necessary details for these steps. First, the incident and scattered electric fields are decomposed into orthogonal components as follows: ! E i = E⊥i uˆ i⊥ + E"i uˆ "i (8.1) ! E s = E⊥s uˆ ⊥s + E"s uˆ "s (8.2) At the scatterer each component can, in general, be altered and power from one component can be scattered into the other component. To model this, the scattered electric field components are decomposed as follows: E⊥s = D⊥⊥ E⊥i + D⊥! E!i (8.3a)



EPs = DP⊥ E⊥i + DP P EPi (8.3b)





⎡ Es ⎢ ⊥s ⎢⎣ EP

⎤ ⎡ D⊥ ⊥ D⊥P ⎥ = ⎢ ⎥⎦ ⎢⎣ DP⊥ DP P

⎤ ⎡ E⊥i ⎤ ⎥ ⎢ i ⎥ (8.3c) ⎥⎦ ⎢⎣ EP ⎥⎦

where [D] is the depolarization matrix and the entries are polarization coefficients. The coefficient D⟘‖, for example, represents the scattering from the perpendicular component into the parallel component. If the parallel and

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perpendicular directions align with the axes of symmetry of the scatterer, the cross terms D⟘‖ and D‖⟘ are zero. However, this does not eliminate the possibility of depolarization, as we will see. The general solution for depolarization by a scatterer is easily cast in terms of polarization ratio that was introduced in Section 3.6. The derivation follows from (8.3b) divided by (8.3a):

rLs =

EPs E⊥s

EPi + DP⊥ DP P rLi + DP⊥ E⊥i (8.4) = = EPi D⊥ ⊥ + D⊥P rLi D⊥ ⊥ + D⊥P i E⊥ DP P

Here we have used the parallel and perpendicular orthogonal directions in the polarization ratio definition. This very general relationship allows the polarization state of a scattered wave given by ρ Ls to be found from the incident wave polarization state ρ Li and the polarization coefficients of the medium. This shows clearly that the scattered wave polarization depends on the scatterer characteristics and the incident wave polarization. The general result in (8.4) gives the scattered polarization state for any incident polarization state. All that is required is the calculation or measurement of the four polarization coefficients. The polarization coefficients can be explicitly expressed in terms of the linear polarizations (parallel and perpendicular) based on (8.3) for incident components alternately set to zero: D⊥ ⊥ = D!⊥ =

E⊥s E⊥i E!s E⊥i

E!i =0

E!i =0

D⊥! =

D!! =

E⊥s E!i

(8.5a) E⊥i =0

E!s E!i

(8.5b) E⊥i =0

In a medium that passes all polarizations unaltered in polarization state, such as air, the off-diagonal entrees in the depolarization matrix are zero and the diagonal entrees are equal: D⊥ ⊥ = DP P , D⊥ P = 0, DP⊥ = 0 for a nondepolarizing medium (8.6) Then (8.4) yields

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rLs = rLi



for a nondepolarizing medium (8.7)

as expected. If the medium does not alter the wave in any way (i.e., no depolarization or attenuation), then D⟘⟘ = D‖‖ = 1, D⟘‖ = D‖⟘ = 0 and the depolarization matrix is the identity matrix; air is a good example of such a medium. The polarization coefficient formulation permits some interesting observations on depolarizing media. Beckmann [1, pp. 39–47] has a thorough discussion, but here we present the results most relevant to applications. Other than the nondepolarizing medium, which does not depolarize for any incident polarization, usually only two characteristic polarizations pass undepolarized. In some cases only one polarization passes undepolarized. These properties are formally stated as follows. Theorem 8.1: For every scattering medium there must be for each direction of scattering at least one (and usually two) incident polarizations that pass through the scattering medium unchanged. This excludes a medium that does not depolarize at all. Corollary 8.1: For every scattering medium one, two, or an infinite number of incident polarization states pass undepolarized. The nondepolarizing medium case of (8.6) corresponds to the infinite number of polarization states being passed undepolarized. The situation of two characteristic polarizations that pass undepolarized are found by solving (8.4) subject to (8.6). The resulting condition on the incident wave polarization ratio for no depolarization is [1, p. 43] rLi = 

1 ⎡ D − D⊥ ⊥ ± 2D⊥! ⎣ ! !

( D!! − D⊥ ⊥ )

2

+ 4D⊥! D!⊥ ⎤ ⎦

no depolarization  (8.8)

For a medium that passes all polarizations unchanged in state, (8.6) applies and (8.8) is indeterminate, meaning all incident states are undepolarized. An example of a medium that passes only one polarization state unchanged is the polarization grid consisting of parallel closely spaced conducting wires or strips; see Section 7.5.3. The polarization grid shown in Figure 8.2 has conducting wires that are vertically oriented. The coordinate reference system is chosen so that the parallel polarization state is vertical linear and the perpendicular polarization state is horizontal linear. For an incident

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wave that is VP, as in Figure 8.2(a), the wire grid shorts out the electric field and there is no transmission through the grid. (There is, however, a reflected field, but we are interested in the transmission properties here; i.e., forward scatter.) For horizontal polarization input as in Figure 8.2(b), the wave is passed unchanged. This is the one and only characteristic polarization for this medium, so Corollary 8.1 is satisfied. For the general case of inclined linear polarization incident on the polarization grid as in Figure 8.2(c), the vertical component is extinguished and the horizontal component is passed. In all

Figure 8.2  A polarization grid. The electric fields for the incident and scattered (transmitted) waves are shown. The wire grid shorts out vertical polarization, as shown in (a). The grid passes only one polarization state undepolarized: horizontal polarization, as shown in (b). The general case of inclined linear polarization in (c) has its vertical component removed and passes only horizontal polarization.

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cases we see that there is only HP output regardless of the input state. This is the purpose of the polarization grid discussed in Section 7.5.3; that is, the goal is to remove the unwanted vertical component. From these observations we can write the polarization coefficients as D⊥ ⊥ = 1

D! ! = 0

D⊥! = D!⊥ = 0

only one polarization is undepolarized  (8.9)

 Then (8.4) reduces to

rLs = 0 (8.10)



which corresponds to HP; see Table 3.3. This shows mathematically that only one polarization, HP, is passed through the grid undepolarized. Another very general observation for depolarizing media can be stated as follows. Corollary 8.2: If a medium does not depolarize for three different polarizations, it does not depolarize at all. This is proved by returning to the nondepolarizing medium that obeys (8.7) for an infinite number of polarizations (i.e., characteristic polarizations). Referring to Corollary 8.1, it is the only case with more than two characteristic polarizations. Therefore, if three characteristic polarizations exist, there must be an infinite number, which in turn means the medium does not depolarize at all. This result implies that only three calculations or measurements are needed that yield no depolarization to conclude the medium does not depolarize at all. The most common depolarizing medium has two characteristic polarizations that are orthogonal linear states. If the parallel and perpendicular axes are chosen along characteristic directions that are, say, vertical and horizontal, no cross-polarization terms will be present. Then D⊥! = D!⊥ = 0   medium with two characteristic polarizations

(8.11)

and (8.4) reduces to rLs =

D! ! D⊥ ⊥

rLi = qrLi  medium with two characteristic polarizations  (8.12)

where we introduced depolarization factor:

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q=

D! ! D⊥ ⊥

  medium with two characteristic polarizations  (8.13)

This formula also applies to a medium that passes only one polarization undepolarized. For example, the polarization grid in Figure 8.2 shorts out the ‖ polarization (D‖‖ = 0). Therefore, q = 0 and ρ Ls = 0, indicating that the scattered wave is horizontally polarized. Table 8.1 summarizes the properties of a medium with two LP characteristic polarizations. The two characteristic polarizations that leave the incident polarization state ρ Li unchanged are, of course, vertical and horizontal linear (ρ Li = ∞ and 0, respectively). Rain is a good example of such a propagation medium; rain is discussed further in Section 8.4.2. One more type of medium is the symmetric depolarizing medium, which is one that satisfies D⊥! = D!⊥ (8.14)



This a common situation that applies to backscattering processes as well as transmission cases [1, p. 47]. We conclude the treatment of the general theory of depolarization by discussing reciprocity. For linear, isotropic media it is true that a wave with polarization state ρ 1 when traveling from point A to point B in a medium is depolarized to state ρ 2, and it will also be depolarized from the input state ρ 1 to state ρ 2 when traveling in the reverse direction. This is location reciprocity. Polarization reciprocity occurs if input state is ρ 1 is depolarized to output state ρ 2, and input state is ρ 2 is depolarized to output state ρ 1 for travel in either direction. But this is not true for all media [1, p. 45]. Table 8.1 Depolarization Properties of a Medium with Two Linear Characteristic Polarizations that are HP and VP and q ≠ 0 Incident Polarization

Scattered Polarization

State

ρ Li

ρ Ls

State

Comment

HP

0

0

HP

Not depolarized

VP

VP

∞

∞

Arbitrary

ρ Li

ρ Ls = q ρ Li

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Depolarizing Media and System Applications211

8.3  Depolarization at Interfaces One common electromagnetic wave propagation situation is the presence of an interface between different media. It is desired to understand the effect on the polarization of a wave traveling across the interface. Applications include the problem of identification of a medium through backscatter using remote sensing techniques with radar or radiometry. Transmission applications also often involve interface problems. One example is long-distance communication at HF via ionospheric skip (i.e., reflection). In this section we investigate the formulation of the interface problem emphasizing polarization effects. 8.3.1  General Formulation of Interface Polarization Effects Many interface situations can be modeled using the canonical problem of a plane wave obliquely incident on an infinite planar interface as shown in Figure 8.3. The scattered waves of the general situation in Figure 8.1 for this problem are the reflected wave (r) and the transmitted wave (t), or refracted wave. Sometimes the term specular reflection is used to emphasize the assumption of the interface being large and flat in terms of a wavelength. For the electric field phasors shown in Figure 8.3, the phase progression (−β s) must be included, where s is the distance away from the interface along the exit ray direction. The reference point for phase is the interface (z = 0). Note in Figure 8.3 that the directions of the parallel component of reflected electric field vector is selected to match that of the incident electric field vector. So at normal incidence (θ i = 0) E‖i and E‖r align in the same direction. The angles of reflection and transmission are determined by the constituitive parameters of the two media (denoted 1 and 2): permittivities ε 1 and ε 2, permeabilities μ 1 and μ 2, and conductivities σ 1 and σ 2. We know from elementary electromagnetic theory that the angle of reflection equals the angle of incidence. As well, Snell’s law of refraction requires the phase shift per meter of the incident wave to match that of the reflected and transmitted waves. Thus

qr = qt

law of reflection (8.15a)



b1 sinqi = b2 sinqt

law of reflection (8.15b)

where b = w me . Combining these gives the angle of the transmitted wave as



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Figure 8.3  A plane wave in medium 1 incident on an infinite planar interface at angle θ i experiencing reflection back into medium 1 at angle θ r and transmission into medium 2 at angle θ t .

The polarization of the exiting reflected and transmitted waves are found by first decomposing the incident wave electric field vector into parallel and perpendicular components using the reference orientations shown in Figure 8.3. Next, the interaction of each component with the interface is evaluated. Finally, the components of the exiting waves are recombined to determine the polarization states of the reflected and transmitted waves. In the following we present the solution for each of these components and then apply those results to a general polarization case. We begin by defining reflection coefficient Γ and transmission coefficient T for each orthogonal component as





Γ! = T! =

E!r E!i

E!t E!i

Γ⊥ = T⊥ =

E⊥r E⊥i

E⊥t E⊥i

reflection coefficients (8.17a) transmission coefficients (8.17b)

where all electric fields are evaluated at the interface (z = 0). These are complex valued quantities that contain the magnitude and phase change of the

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Depolarizing Media and System Applications213

components at the interface. These changes are responsible for depolarization. The reflection and transmission coefficients are functions of incident angle (θ i) and exit angles (θ r and θ t) as well as the constituitive parameters of the media. Applying boundary conditions on the electric and magnetic fields at the interface yields



Γ! =

h2 cosqt − h1 cosqi (8.18a) h2 cosqt + h1 cosqi

T! =

2h2 cosqi (8.18b) h2 cosqt + h1 cosqi

Γ⊥ =

h2 cosqi − h1 cosqt (8.18c) h2 cosqi + h1 cosqt

T⊥ =

2h2 cosqi (8.18d) h2 cosqi + h1 cosqt

where the intrinsic impedance η for a general medium is h=



jwm (8.19) s + jwe

If the media are identical (η 1 = η 2), then the obvious results are obtained from (8.18):

Γ! = Γ ⊥ = 0

T! = T⊥ = 1

identical media (8.20)

These indicate that there is no reflection and the incident wave is transmitted through the interface unaltered. Thus the interface has no effect. An important special case is that of a plane, perfect conductor (called a perfect electric conductor [PEC]); also see Section 8.3.2. The conductor is assumed to be perfectly conducting with σ 2 = ∞, and thus η 2 = 0 from (8.19). A good conductor, such as most metals, is well approximated as being perfectly conducting. The conductor presents a short circuit and (8.18) gives

Γ = Γ! = Γ ⊥ = −1

T = T! = T⊥ = 0

PEC plane (8.21)

Note that these values hold for all angles of incidence. The inference from the reflection coefficient being −1 is that the electric field, regardless of its linear

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polarization orientation, flips over on reflection. And, of course, there is no transmitted field into the PEC (T = 0). The reflection coefficient result follows directly from the boundary condition on the electric field at the PEC. This boundary condition is that the total electric field component tangent to the r i r i + Etan = 0, or Etan = −Etan ), giving PEC at z = 0 is zero (i.e., Etan



r i Etan − Etan Γ = i = i = −1 Etan Etan

PEC plane (8.22)

Returning to the general media case for normal incidence (θ i = 0°) we have from (8.15) and (8.16) that θ r = θ t = 0°. Then (8.18) reduces to



Γ = Γ! = Γ ⊥ =

h2 − h1 h2 + h1

normal incidence (8.23a)

T = T! = T⊥ =

2h2 h2 + h1

normal incidence (8.23b)

The reference directions in Figure 8.3 for oblique incidence were chosen to be consistent with this simple result for normal incidence. That is, for these reference directions the reflection and transmission coefficients for normal incidence are identical for both polarizations. Note that the special cases of identical media and for a perfect conductor in (8.20) and (8.21), respectively, also follow from (8.23). Another common situation is that of dielectric media on both sides of the interface. We assume that the materials are nonmagnetic (μ 1 = μ 2 = μ o) and lossless (σ 1 = σ 2 = 0), which is usually the case in practice. Then (8.18) together with (8.15) yields e1 e2 Γ! = − e cosqi + 1 e2

e2 − sin2 qi e1 e2 − sin2 qi e1

dielectric media (8.24a)

e1 cosqi e2 T! = e e cosqi + 1 2 − sin2 qi e2 e1

dielectric media (8.24b)

cosqi −



2



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Depolarizing Media and System Applications215

e2 − sin2 qi e1 Γ⊥ = e cosqi + 2 − sin2 qi e1 cosqi −



T⊥ =

2cosqi e cosqi + 2 − sin2 qi e1

dielectric media (8.24c)

dielectric media (8.24d)

These results also can be applied to lossy dielectrics by replacing the permittivities with complex-valued effective quantities given by ec = e − j



s (8.25) w

The loss is included via the conductivity σ . A perfect dielectric has zero conductivity and then (8.25) reduces to ε c = ε . If medium 2 is a perfect conductor σ 2 = ∞ and ε c = ∞, and then the coefficients in (8.24) reduce to (8.21). An incident wave that is either parallel or perpendicularly polarized to the interface will not generate a cross-polarized component in the reflected or refracted waves. Therefore, (8.12) to (8.14) apply. For the reflection case

Γ! = D! !

Γ ⊥ = D⊥ ⊥ (8.26)

which follows by comparing (8.5) and (8.17). For a nondepolarizing medium, from (8.6) D‖‖ = D⟘⟘ and then (8.26) indicates that



ΓP = Γ ⊥

nondepolarizing interface (8.27)

An example situation is normal incidence on an interface; see (8.23). Using (8.26) in (8.13) gives the depolarization factor for an interface:



q=

Γ! Γ ⊥ (8.28)

For identical media, (8.20) in (8.28) yields q = 0, indicating no depolarization. For medium 2 being a perfect conductor, (8.21) in (8.28) yields q = 1 because the reflection coefficients for parallel and perpendicular polarizations are the same (and both equal −1). For normal incidence and general media, (8.23a)

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used in (8.28) yields q = 1 and there is no depolarization associated with the reflected or transmitted waves. As mentioned, the procedure for analyzing an arbitrarily polarized wave interacting with an interface is to decompose the incident electric field into parallel and perpendicular components, apply the known reflection and transmission behavior for the components, and then reconstruct the electric field after reflection. The incident electric field vector is given in (8.1). The vector forms for the scattered (reflected and transmitted) electric fields then follow from (8.2) and (8.17) as r E r = E⊥r uˆ r⊥ + EPr uˆ Pr = Γ ⊥ E⊥i uˆ r⊥ + ΓP EPi uˆ Pr (8.29a) r E t = E⊥t uˆ t⊥ + EPt uˆ Pt = T⊥ E⊥i uˆ t⊥ + TP EPi uˆ Pt (8.29b) These vector expressions can be put into compact matrix forms:



⎡ Er ⎤ ⎡ Γ 0 ⎤ ⎡ E⊥i ⎤ ⎢ ⊥r ⎥ = ⎢ ⊥ ⎥ ⎢ i ⎥ (8.30a) ⎢⎣ E! ⎥⎦ ⎢⎣ 0 Γ! ⎦⎥ ⎢⎣ E! ⎥⎦



⎡ Et ⎤ ⎡ T 0 ⎤⎡ Ei ⎤ ⊥ ⎢ ⊥t ⎥ = ⎢ ⊥ ⎥ ⎢ i ⎥ (8.30b) 0 T E E ! ⎥ ⎢⎣ ! ⎥⎦ ⎢⎣ ⎦ ⎢⎣ ! ⎥⎦

The complete expressions for the electric fields traveling away from the interface are constructed from the components in (8.29) and the directions of propagation as follows: r E r ( sr ) = ⎡⎣ E⊥r uˆ r⊥ + EPr uˆ Pr ⎤⎦ e − jbr sr (8.31a)

r E t ( sr ) = ⎡⎣ E⊥t uˆ t⊥ + EPt uˆ Pt ⎤⎦ e − jbt st (8.31b)

These are the field expressions for the reflected and transmitted waves for an arbitrary incident wave. They are valid at distances of sr and st away from the interface for the reflected and transmitted waves, respectively. Of course, all wave polarization information is contained in the perpendicular and parallel components. The interface problem can also be formulated in terms of circularly polarized components. The derivation is straightforward using matrix

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Depolarizing Media and System Applications217

representations. We begin with conversions between LP and CP phasors. First, the electric field phasors for left and right circular polarizations expressed in orthogonal linear components based on (3.67) in matrix form are



⎡ EL ⎤ 1 ⎡ 1 − j ⎤ ⎡ E⊥ ⎤ ⎢ E ⎥ = 2 ⎢ 1 j ⎥ ⎢ E ⎥ (8.32) ⎣ ⎦ ⎣⎢ ! ⎥⎦ ⎣ R ⎦

where the general orthogonal linear components electric fields E⟘ and E‖ replace the more specific x- and y-components, E1 and E2. Inverting this expression gives the linear components in terms of circular components:



⎡ E⊥ ⎤ 1 ⎡ 1 1 ⎤ ⎡ EL ⎤ (8.33) ⎢ E ⎥= 2 ⎢⎣ j − j ⎥⎦ ⎢⎣ ER ⎥⎦ ⎢⎣ ! ⎥⎦

For a wave reflected from an interface, the reference direction of the parallel electric field component is reversed from the incident wave (see Figure 8.3), thus the conversion from CP components to LP components in (8.33) becomes



⎡ Er ⎢ ⊥r ⎢⎣ E!

r ⎤ 1 ⎡ 1 1 ⎤ ⎡ EL ⎤ ⎥= ⎥ (8.34) ⎢ 2 ⎢⎣ − j j ⎥⎦ ⎢ ERr ⎥ ⎥⎦ ⎦ ⎣

Inverting this expression gives the conversion from LP to CP:



r ⎡ Er ⎤ 1 ⎡ 1 j ⎤ ⎡ E⊥ L ⎢ ⎢ r ⎥= 2 ⎢⎣ 1 − j ⎥⎦ ⎢ E!r ⎢⎣ ER ⎥⎦ ⎣

⎤ ⎥ (8.35) ⎥⎦

This interface reflection matrix electric field formulation can be recast in terms of reflection coefficients using (8.29a) in (8.35) to give



⎡ Er ⎤ ⎡ Γ Γ ⎤⎡ Ei ⎤ ⎢ Lr ⎥ = ⎢ C X ⎥ ⎢ Li ⎥ (8.36a) ⎢⎣ ER ⎥⎦ ⎣ Γ X Γ C ⎦ ⎢⎣ ER ⎦⎥

where

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Γ C = Γ LL = Γ RR =

1 Γ − Γ! ) (8.36b) 2( ⊥

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Γ X = Γ LR = Γ RL =

1 Γ + Γ! ) (8.36c) 2( ⊥

If the interface does not depolarize (i.e., ΓX = 0), Γ‖ = −Γ⟘ from (8.36c) and then ΓC = Γ⟘ from (8.36b). For a depolarizing interface, consider the example of a pure right-hand circularly polarized wave incident on an interface (i.e., ELi = 0); also take the amplitude of the incident right-hand polarized wave electric field to be unity (ERi = 1). Then (8.36a) yields the co-polarized reflected wave field to be ERr = ΓC and the cross-polarized field generated due to reflection is ELr = ΓX. For transmission through an interface for circular polarization a formulation similar to (8.36) for the reflection case yields



⎡ Et ⎤ ⎡ T T ⎤⎡ Ei ⎤ ⎢ Lt ⎥ = ⎢ C X ⎥ ⎢ Li ⎥ (8.37a) ⎢⎣ ER ⎥⎦ ⎣ TX TC ⎦ ⎢⎣ ER ⎥⎦

where

TC = TLL = TRR =

1 T + T! ) (8.37b) 2( ⊥

TX = TLR = TRL =

1 T + T! ) (8.37c) 2( ⊥

As a check, a nondepolarizing interface has T‖ = T⟘ from (8.20). Then (8.37) gives TC = T‖ = T⟘ and TX = 0. Thus, the wave polarization state remains unchanged in passing through the interface, as expected. To summarize, (8.36) and (8.37) show that interface reflections and transmission involving CP wave components are easily evaluated using the linearly polarized reflection and transmission coefficients. The CP reflection and transmission matrices in (8.36a) and (8.37a) have cross terms present, denoted by the subscript X. That is, there is coupling between one sense of circular polarization to the other at the interface. The cross-polarized reflection coefficient, ΓX, will go to zero when (8.27) holds, which is the general relation for nondepolarizing interfaces. In general, if a pure CP wave is incident on an interface, both CP sense components will be present in the reflected and transmitted waves; that is, the exiting wave polarization will be elliptically polarized.

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Depolarizing Media and System Applications219

8.3.2  Reflection from a Plane, Perfect Conductor The reflection process is very important in communications, remote sensing, and many other applications. This subsection and the next treat reflections for the two most important cases, a perfect conductor and real earth ground (referred to here as ground). It is assumed that the air-to-material interface is planar and infinite. This model is valid if the surface irregularities are small compared to a wavelength and the surface extent is large compared to a wavelength. The perfectly conducting assumption, as well as the planar and infinite extent assumptions, are often easily satisfied. Common metals such as copper, aluminum, and steel have large conductivities (greater than 107 S/m) and the reflection coefficient over the radio spectrum is well approximated by a perfect conductor (PEC); wave interaction with a plane, perfect conductor was treated initially in (8.21). The perfect conductor assumption remains true for all incidence angles and is illustrated in Figure 8.4(a). The angle of reflection equals the angle of incidence; see (8.15a). The reflection coefficients are often presented in terms of grazing angle, which is the angle the incident ray makes with the interface plane and is the complement of the incidence angle:

ψ = 90° − qi

grazing angle (8.38)

Because medium 2 is a perfect conductor, medium 2 conductivity is σ 2 = ∞ and its intrinsic impedance from (8.19) is η 2 = 0. Then (8.18a) and (8.18c) yield Γ‖ = Γ⟘ = −1 as noted in (8.21). The reflected electric field directions shown in Figure 8.4(a) include the reflection coefficient value of −1. The field flips over on reflection for both normal and oblique incidence cases. For normal incidence (θ i = 0°) the parallel and perpendicular polarization cases are identical. The incident and reflected electric field vectors for normal incidence oppose each other as shown in Figure 8.4(a) and are of the same magnitude. Thus the total field (sum of both polarizations) is zero as is required for the boundary condition of the tangent component of the total electric field to a perfect conductor being zero. A very interesting phenomenon occurs when a circularly polarized wave reflects from a plane, perfect conductor. The reflected wave has its circularity preserved, but the sense changes. Therefore, a RHCP incident wave is reflected from a planar conductor as a LHCP wave, as illustrated in Figure 8.4(b). This sense change occurs because both parallel and perpendicular components flip over while maintaining their equal amplitude, phase-quadrature relationship.

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Figure 8.4  Reflection from a plane, perfect conductor (well approximated by most metals): (a) linear polarization, and (b) circular polarization.

However, the direction of propagation is reversed leading to a sense reversal (see Problem 3 at the end of this chapter). 8.3.3  Reflection from the Ground The important problem of ground (i.e., real earth) reflections and the associated depolarizing effects is rather involved for several reasons. It follows from (8.24a) and (8.24c) that the reflection coefficients depend on incidence angle. Also, the ground properties (i.e., electrical characteristics) vary with frequency as indicated in (8.25) and also can change greatly from site to site. The permittivity and conductivity of various soil types, water, and ice are available as a function of frequency [2]. Generalizing, conductivity (and thus loss) increases rapidly with frequency above about 10 MHz for most soil types.

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Figure 8.5 represents typical reflection coefficient behavior as a function of grazing angle at UHF frequencies. The ground permittivity is ε r = 15 + j0. At grazing incidence (ψ = 0° or θ i = 90°), the perpendicular and parallel reflection coefficient values are Γ = Γ! = Γ ⊥ = −1 grazing incidence (ψ = 0°) on a planar dielectric (8.39) This is a general result not dependent on the dielectric constant value; it is found from (8.24a) and (8.24c) with θ i = 90°. These values can be located in Figure 8.5. The transmission coefficient values from (8.24b) and (8.24d) are zero, indicating that the wave does not penetrate into medium 2. As grazing angle increases from zero the reflection coefficient decreases from unity. At normal incidence ψ = 90° and (8.23a) applies, from which it follows that



Γ = Γ! = Γ ⊥ = −

er − 1 er + 1

normal incidence,

h2 1 = (8.40) er h1

If ε r = 1, then Γ = 0 and there is no reflection because the two media have the same permittivity values.

Figure 8.5  Parallel and perpendicular polarization reflection coefficients as a function of grazing angle for typical ground at UHF frequencies: ε r = 15 + j0.

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The parallel reflection coefficient passes through zero at what is called the Brewster angle, also called the polarizing angle. It is obtained by solving for the incidence angle that renders the numerator of (8.24a) zero; this process yields



er ⎞ ⎛ y B = cos−1 ⎜ ⎝ 1 + er ⎟⎠

Brewster angle ( Γ! = 0 ) (8.41)

where ε r is the relative permittivity. The Brewster angle is shown in Figure 8.5 for ε r = 15 and is ψ B = 14.5°. Essentially the parallel polarization component of a wave incident at the Brewster angle is extinguished upon reflection from the interface. Also note from Figure 8.5 that the phase of Γ‖ changes by 180° as the grazing angle passes through the Brewster angle. That is, the electric field vector flips over. For a relative dielectric constant of 1 (8.41) gives ψ B = 45°, but this is an unrealistic limiting case because the media are identical and there will be no reflection anyway. As the relative dielectric constant value increases from 1 the Brewster angle decreases, approaching grazing incidence for large ε r values. An example of a Brewster angle effect is experienced while near a calm lake on a sunny day. When the Sun is low in the sky, light is reflected causing glare, preventing any visibility below the water surface. Thus, the grazing angle for Sun rays is below the Brewster angle. For high Sun angles light can penetrate the surface, illuminating underwater objects. Also for this situation is the observation that polarized sunglasses filter out the glare of horizontal (perpendicular) polarization that causes most of the glare (i.e., has higher reflection coefficient). Thus, sunglasses provide better visibility of underwater objects. Figure 8.5 gives the reflection coefficients for linearly polarized components of an incident wave. Figure 8.6 presents the reflection coefficients for circular polarization as a function of grazing angle. The plots are for the reflection coefficient values for states co-polarized and cross-polarized to the incident circular polarized wave. For example, if the incident wave is RHCP, ΓC is the reflection coefficient for RHCP and ΓX is the relative amount of reflection that is LHCP. Note that the co-polarization reflection and cross-polarization reflection coefficient curves in Figure 8.6 intersect at the Brewster angle. At the Brewster angle Γ‖(ψ B) = 0. Using this with (8.36) gives 1 ΓC ( y B ) = Γ X ( y B ) = Γ ⊥ ( y B ) (8.42) 2

showing that the circular polarization reflection coefficient values are equal at the Brewster angle. However, the case of no reflection at the Brewster angle only occurs for a parallel polarized LP wave.

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Depolarizing Media and System Applications223

Figure 8.6  Reflection coefficients for circularly polarized waves co-polarized (ΓC) and cross-polarized (ΓX ) to the incident circularly polarized wave. The values are for typical ground at UHF frequencies: ε r = 15 + j0.

Example 8.1  Reflection Coefficients for Soil at UHF

Wet ground has a relative permittivity of ε r = 15 at UHF frequencies. The conductivity is about 10 –1 S/m [2]. Then from (8.25) the complex permittivity at 1000 MHz is ec = e − j

s s ⎞ = eo ⎛⎜ er − j = eo (15 − j1.8) ≈ 15eo w 2pf eo ⎟⎠ ⎝

The imaginary part is small compared to the real part, so we ignore it and simply use ε r = 15 in calculations. We now verify some points in Figures 8.5 and 8.6. For the linear polarization results in Figure 8.5, the reflection coefficients with ε r = 15 are found at normal incidence using (8.40): Γ (y = 90°) = Γ! (y = 90°) = Γ ⊥ (y = 90°) =−

er − 1 15 − 1 =− = −0.59 15 + 1 er + 1

This point can be found on the curves in Figure 8.5. This situation is similar to the perfectly conducting ground plane case that has Γ = −1 at normal

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incidence (ψ = 90°); see (8.21). As the grazing angle decreases from normal incidence, Γ⟘ becomes more negative, reaching −1 at grazing incidence. Γ‖ equals is 0 at the Brewster angle, which is found from (8.41): er ⎞ ⎛ ⎛ 15 ⎞ = 14.5° y B = cos−1 ⎜ = cos−1 ⎜ ⎟ ⎝ 1 + 15 ⎟⎠ ⎝ 1 + er ⎠ The null for the ⎪Γ‖⎪ curve in Figure 8.5 occurs at this value. The CP reflection coefficients for this example are plotted in Figure 8.6. Note that curves cross at the Brewster angle, where the reflection coefficients from (8.42) with (8.24c) are 1 0.875 Γ C (14.5°) = Γ X (14.5°) = Γ ⊥ (14.5°) = − = −0.44 2 2 This value for reflection coefficient at the Brewster angle can be found on both curves in Figure 8.6. At grazing incidence (ψ = 0°, θ i = 90°), the reflection coefficients for linear polarization from Figure 8.5 or from (8.24) are

Γ ⊥ (y = 0°) = −1

Γ! (y = 0°) = +1 (8.43)

These coefficients are based on the reference direction conventions in Figure 8.3. Therefore, we see that both electric field vectors flip over at grazing incidence, as shown in Figure 8.4(a), which applies to a perfectly conducting plane. The circular polarization reflection coefficient for a planar dielectric interface can be found using (8.43) in (8.36):

Γ C (y = 0°) =

1 1 Γ − Γ! ) = [ −1 − (1)] = −1 (8.44) 2( ⊥ 2

Γ X (y = 0°) =

1 1 Γ ⊥ + Γ! ) = (−1 + 1) = 0 (8.45) ( 2 2

These points can be found on Figure 8.6. The polarization states of a wave reflected from an earth ground are shown in Figure 8.7 at several grazing angles for an incident wave that is circularly polarized (RHCP). For normal incidence the reflected wave is also circularly polarized, but of opposite sense (LHCP) as explained earlier. Note

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Depolarizing Media and System Applications225

that the amplitude of the wave is diminished (indicated by a smaller circle) because the reflection coefficient magnitude is not unity for a typical ground; it may be close to unity, however, for a calm water surface. The sense reversal is explained by noting that the orthogonal linear components of the incident wave both encounter approximately 180° phase shift. In fact, from Figure 8.5 Γ ⊥ (y = 90°) = Γ! (y = 90°) ≈ Γ ∠180°

normal incidence (8.46)

The phase shift of approximately 180° for each linear component leads to a reversal of the sense for the incident CP upon reflection just as with a perfect conductor; see Section 8.3.2 and Problem 3. As the grazing angle is decreased from normal incidence, ΓX increases as shown in Figure 8.6. This increased level of cross-polarization causes the reflected wave to be more elliptical; this is illustrated in Figure 8.7. When the grazing angle equals the Brewster angle the reflection coefficient for the parallel component goes to zero; see Figure 8.5. With the parallel component extinguished after reflection, the reflected wave is linearly polarized perpendicular to the plane of incidence, (i.e., parallel to the ground plane as shown in Figure 8.7). Thus, a CP wave incident at the Brewster angle becomes an LP wave after reflection. The parallel reflection coefficient magnitude in general lossy media will not go all the way to zero, but instead reaches a minimum at the Brewster angle. Then the reflected wave is LP but not horizontally oriented. For grazing angles less than the Brewster angle, the reflected wave is not purely CP but is of the same sense as the incident pure CP wave.

Figure 8.7  Polarization of waves reflected from real earth ground for an incident circularly polarized wave (RHCP) for several angles of incidence. The polarization ellipses are shown for each ray.

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8.4  Dual-Polarized Communication Systems with a Depolarizing Medium in the Path Dual polarization is frequently used in communications, radar, and radiometer systems. For this situation, there are two parts to the analysis: (1) wave interaction with the medium along the propagation path, and (2) wave-antenna interaction effects at the transmitter and receiver. We have discussed these two parts in Section 8.2 and Chapter 6. In this section we present a unified, general formulation for depolarization along the path in a communication link. More generally, this is referred to as the transmission problem and the application need not only be for communications. Methods of compensating for unwanted depolarization effects will be treated in Section 8.5. Radar and radiometry applications are discussed in Sections 8.6 and 8.7, respectively. 8.4.1  General Formulation for a Depolarizing Medium Communication systems consist of a transmitter (t) and a remote receiver (r). In many systems the straight-line path between the transmitter and receiver is clear of large obstructions such as hills and buildings. For purposes of discussion here we consider this to be the case. The problem we are examining is the change in received signal due to introduction of a depolarizing medium along the path, such as rain. The electric field intensity arriving at! the receiver under initial (clear path) conditions we call the incident field, E i . When medium characteristics change along the path, the electric field arriving at the receiver is altered. In general, it is changed in amplitude, phase, and polarization. It can also be changed in frequency, but that can be accounted for separately. ! The altered field we refer to as the depolarized field, E d . For communication applications wave field. It is important to remem! ! this is the forward-directed ber that E d differs from E i only through the effects of the newly introduced medium changes, just as !we did ! in Sections 8.2 and 8.3. Therefore, with no variations along the path E d = E i . The field propagating along the path always encounters spreading loss and phase shift. That is, the electric field exiting the ! transmitter E t is changed as follows



! e − jbr ! t Ei = E (8.47) 4pr

where r is the path length along the link. The free-space path spherical spreading factor (1/r) in the above expression is not included in the following development.

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The propagation medium effects are represented by the depolarization matrix [D] that was introduced in Section 8.2. The electric field components (i.e., scattered fields) that exit the depolarizing medium are found using a generalized form of (8.3c):



⎡ Ed ⎤ ⎡ D D ⎤⎡ Ei ⎤ ⎢ 1d ⎥ = ⎢ 11 12 ⎥ ⎢ 1i ⎥ (8.48) ⎢⎣ E2 ⎥⎦ ⎣ D21 D22 ⎦ ⎢⎣ E2 ⎦⎥

The subscripts 1 and 2 represent any orthogonal polarizations but are usually either of the two classical orthogonal pairs: horizontal and vertical LP, and left- and right-hand sense CP. Then (1, 2) represent (H, V) or (L, R) and H = HP, V = VP, R = RHCP, and L = LHCP. For a symmetric depolarizing medium, D21 = D12 as in (8.14). For the common case of horizontal and vertical polarization the axes (1, 2) are (H, V), and in general can be in any rotational orientation. Then the depolarization matrix for a symmetric medium is ⎤ ⎡ D D ⎤ ⎡ D D [D] = ⎢ 11 12 ⎥ = ⎢ HH HV ⎥ Linear polarizations (H,V) (8.49) D12 D22 DHV DVV ⎣ ⎦ ⎣ ⎦ The most common depolarizing medium has two characteristic polarizations that pass undepolarized, which we denote as ⟘ and ‖. Then the (1, 2) axes are (⟘, ‖), and (8.11) applies and when used in (8.3c) yields

⎡ E d ⎤ ⎡ D⊥⊥ 0 ⎤ ⎡ E i ⎤ ⎡ Ei ⎢ ⊥d ⎥ = ⎢ ⎢ ⊥i ⎥ = [ Dc ] ⎢ ⊥i ⎥ ⎢⎣ E! ⎥⎦ ⎢⎣ 0 D! ! ⎥⎦ ⎢⎣ E! ⎥⎦ ⎢⎣ E!

⎤ ⎥ (8.50) ⎥⎦

where [Dc] is the depolarization matrix for a medium with two orthogonal linear characteristic polarizations. Figure 8.8(a) shows an example of such a medium (rain in this case). The ⟘ and ‖ axes align with the characteristic axes of the medium, which are the major and minor axes of a raindrop; for simplicity, the raindrops all have their major axes aligned. The coordinate transformations from (H, V) to (⟘, ‖) and from (⟘, ‖) to (H, V) are

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⎡ E⊥ ⎤ ⎡ cosq sinq ⎤ ⎡ EH ⎤ ⎡ EH ⎤ ⎢ E ⎥ = ⎢ −sinq cosq ⎥ ⎢ E ⎥ = [R]⎢ E ⎥ (8.51a) ⎦⎣ V ⎦ ⎢⎣ ! ⎥⎦ ⎣ ⎣ V ⎦

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⎡ E⊥ ⎤ ⎡ EH ⎤ ⎡ cosq −sinq ⎤ ⎡ E⊥ ⎤ T ⎢ E ⎥ = ⎢ sinq cosq ⎥ ⎢ E ⎥ = [R] ⎢ E ⎥ (8.51b) ⎦ ⎢⎣ ! ⎦⎥ ⎢⎣ ! ⎥⎦ ⎣ V ⎦ ⎣

These follow from Figure 8.8(a) which shows the two sets of axes rotated relative to each other by the angle θ , the canting angle of the raindrops relative to horizontal. [R] is the rotation matrix of (H, V) axes into the (⟘, ‖) axes. Now we can write the general depolarization matrix for arbitrary axis (H, V) based on the depolarization matrix that uses characteristic axes (⟘, ‖). Substituting (8.51) into (8.50) gives ⎡ Ed ⎤ ⎡ Ei ⎢ ⊥d ⎥ = [ Dc ] ⎢ ⊥i ⎢⎣ E! ⎥⎦ ⎢⎣ E!

⎤ ⎡ Ei ⎥ = [ Dc ][R]⎢ Hi ⎥⎦ ⎢⎣ EV

⎡ Ed ⎤ ⎡ Ed ⎤ ⊥ ⎥ ⎢ = [R]⎢ Hd ⎥ and d ⎢⎣ E! ⎥⎦ ⎢⎣ EV ⎥⎦

⎤ ⎥ ⎥⎦

so



⎡ Ed ⎢ Hd ⎢⎣ EV

⎤ ⎡ Ei ⎥ = [R]T [ Dc ][R]⎢ Hi ⎥⎦ ⎢⎣ EV

⎤ ⎡ Ei ⎥ = [D]⎢ Hi ⎥⎦ ⎢⎣ EV

⎤ ⎥ (8.52) ⎥⎦

because the inverse of the matrix [R] equals its transpose. So the conversion of the depolarization matrix from the (⟘, ‖) characteristic axes to the rotated (H, V) axes is [D] = [R]T [ Dc ][R] (8.53)



A medium that does not depolarize the wave obeys (8.6). Further, if the medium does not attenuate the wave, [Dc] is the identity matrix (for any axis orientation). This follows from (8.51) and (8.53) as [D] = [R]T [ Dc ][R] (8.54) = ⎡⎢ cosq −sinq ⎤⎥ ⎡⎢ 1 0 ⎤⎥ ⎡⎢ cosq sinq ⎤⎥ = ⎡⎢ 1 0 ⎤⎥ ⎣ sinq cosq ⎦ ⎣ 0 1 ⎦ ⎣ −sinq cosq ⎦ ⎣ 0 1 ⎦ The general two-characteristic polarization medium will have unequal magnitude and phase changes along the characteristic axes. We represent this as



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⎡ D⊥⊥ 0 [D] = ⎢ 0 D! ! ⎢⎣

⎤ ⎡ 1 0 ⎤ ⎥ = D⊥⊥ ⎢ 0 q ⎥ (8.55) ⎣ ⎦ ⎥⎦

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where q is the depolarization factor that was defined in (8.13). If q = 1, the second matrix in (8.55) becomes the identity matrix and the medium does not depolarize. Further, if D⟘⟘ is unity, the medium has no effect. If D⟘⟘ is not unity and q = 1, it simply means that the electric fields are altered in an identical manner along both the ⟘ and ‖ axes, which implies no depolarization. There is an (identical) magnitude and/or phase change in the characteristic polarizations. The depolarization matrix can be expressed in terms of the depolarization factor by substituting (8.55) into (8.53): ⎡ cos2 q + qsin2 q (1 − q)cosqsinq ⎤ ⎡ D D ⎤ [D] = ⎢ 11 12 ⎥ = D⊥⊥ ⎢ ⎥ (8.56) 2 2 D D ⎢⎣ (1 − q)cosqsinq qcos q + sin q ⎥⎦ ⎣ 21 22 ⎦ For a nondepolarizing medium (q = 1), the matrix in (8.56) reduces to the identity matrix, as it should. Also, if the rotation angle θ is zero, (8.56) reduces to (8.55), as it should. A wave propagating through a depolarizing medium can also be analyzed by decomposed it into circular polarization components. The circular polarization states (L, R) used for states (1, 2) in (8.48) give the exiting depolarized electric field components: ⎡ Ed ⎤ ⎡ D DLR ⎤ ⎡ ELi ⎤ LL L ⎢ d ⎥=⎢ ⎥⎢ i ⎥ ⎢⎣ ER ⎥⎦ ⎣ DRL DRR ⎦ ⎢⎣ ER ⎥⎦ ⎡ Ei ⎤ = [ DCP ] ⎢ Li ⎥ ⎢⎣ ER ⎥⎦

  circular polarizations (L, R)

(8.57)

The entries of the depolarization matrix [D CP ] can be expressed in terms of linear polarizations using the conversion formulas for general transmission in (8.37) and changing notation:

DLL = DRR =

1 D + D! ! (8.58a) 2 ⊥⊥

DLR = DRL =

1 D − D! ! (8.58b) 2 ⊥⊥

( (

) )

Using these entries in [D CP ]with (8.13) gives

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⎡ DLL DLR ⎤ 1 ⎡ 1+ q 1− q ⎤ = D⊥⊥ ⎢ ⎥ ⎥ (8.59) D D ⎣ 1− q 1+ q ⎦ ⎣ RL RR ⎦ 2

[ DCP ] = ⎢

The matrix reduces to the identity matrix (corresponding to no depolarization) for q = 1, as it should. 8.4.2  Rain on a Radio Path A radio wave is changed in magnitude, phase, and polarization as it propagates through rain. Rain effects can be significant above frequencies of a few gigahertz on both terrestrial and satellite links. Earth-space communication links are particularly sensitive to rain because of the low link power margins used. Rain not only reduces the signal level on each of the polarizations on a dual-polarized link, but creates a cross-polarized component. The depolarization caused by rain is especially troubling on dual-polarized links that will be discussed further in the next chapter. Although rain is the most common weather impairment to radio waves, all hydrometeors (including snow and ice crystals) can cause problems. Clear air is lossy but is not a depolarizer. Here we concentrate on rain since it is the most limiting of the weather effects. Rain is a lossy dielectric that absorbs as well as scatters incident power. Rain-induced signal fading, called attenuation, increases with frequency at microwave frequencies. On terrestrial links, the attenuation caused by rain depends on the portion of the propagation path filled with rain and the intensity of the rainfall. Earth-space satellite links have a propagation path that is a slanted and forming an angle ε , the elevation angle, relative to horizontal, and the rain-filled portion of the path is usually shorter than on a terrestrial link. Satellite link attenuation in decibels increases rapidly with frequency, approximately as a power of 1.9 with frequency in the 10- to 30-GHz range [3]. For example, for a satellite link operating at 12 GHz and experiencing 4-dB attenuation would have attenuation go up by a factor of (30/12)1.9 = 5.7 when operated at 30 GHz, leading to an attenuation of 22.8 dB at 30 GHz. Generally speaking satellite links are attenuation limited above about 20 GHz and are depolarization limited below about 20 GHz. That is, at low frequencies the signals are not attenuated enough to cause loss of signal, but rain can cause sufficient depolarization (low XPD) that cross talk between orthogonally polarized channels is significant enough to cause unacceptable performance. In the stated example, 4-dB attenuation at 12 GHz would not affect link performance, whereas the 22.8-dB attenuation at 30 GHz would probably lead to a signal outage. In this case, depolarization is not a factor in

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performance. But for rain rates that increase the 12-GHz attenuation much more than 4 dB, depolarization could become a problem. Rain depolarization is rather complicated, but the processes can be explained using the principles of polarization we have developed. Raindrops occur in a distribution of sizes and shapes. Light rain is made up of small spherical drops, which cause significant attenuation only at high frequencies (many gigahertz) and cause no depolarization. Spherical raindrops present the same shape profile (circular) to an incident wave of any polarization, and therefore, affect all polarizations (and polarization components) the same. This is an excellent example of a medium that passes an infinite number of polarizations undepolarized; see Corollary 8.1 in Section 8.2. Moderate- to high-intensity rains have large raindrops (as large as about 4 mm in diameter) that are distorted from spherical shape with their long axis oriented close to horizontal. Then the rain medium is anisotropic due to the nonspherical shape of raindrops and will depolarize. The majority of the raindrops are oblate spheroidal in shape (doorknob-shaped), usually modeled as all aligned as shown in Figure 8.8(a). In reality, the drops have a distribution of tilt angles about the mean tilt angle θ , but the depolarization process is the same as that for the aligned-drop model. The aligned oblate spheroid shaped raindrop model of Figure 8.8(a) allows explanation of the entire rain-filled path through examination of a single drop. The single drop shown in Figure 8.8(b) represents a rain medium in which oblate spheroidal drops are aligned and have major and minor axes (⟘, ‖). The entries in the depolarization matrix of (8.59) are found from the following [4]: D! ! = e



−a! L − jb! L

e

(8.60a)

D⊥⊥ = e −a⊥ L e − jb⊥ L (8.60b)

and

q=



D! ! D⊥⊥

= e − ΔaL e − jΔbL (8.61)

where

α ⟘, α ‖ = attenuation constant for the ⟘ and ‖ linear polarizations [Np/m] β ⟘, β ‖ = phase constant for ⟘ and ‖ linear polarizations [rad/m]

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L = path length through rain [m] Δα = α ‖ − α ⟘ = differential attenuation constant [Np/m] Δβ = β ‖ − β ⟘ = differential phase constant [rad/m] q = differential propagation factor = depolarization factor The exponentials involving α are attenuation factors and the exponentials involving β are phase shift factors. Because the raindrops are thicker along the major axis (⟘) than along the minor axis (‖), there will be more attenuation for a linear polarization oriented along the major axis; thus

Figure 8.8  Rain depolarization: (a) raindrops canted at angle θ , and (b) incident and depolarized electric field components showing the change of tilt angle caused by a raindrop.

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Depolarizing Media and System Applications233

q =



D! ! D⊥⊥

=

−a L

e ! > 1 (8.62) e −a⊥ L

It is easy to demonstrate and display the depolarization process using only differential attenuation as shown in Figure 8.8(b). The electric field incident ! on the drop, E i , is LP at tilt angle τ i. The scattered field follows from (8.50) and (8.55) as



i ⎡ Ed ⎤ ⎡ D 0 ⎤ ⎡ E⊥i ⎤ ⎡ 1 0 ⎤ ⎡ E⊥ ⎤ ⊥ ⊥⊥ ⎢ d ⎥=⎢ ⎥ ⎢ i ⎥ = D⊥⊥ ⎢ 0 q ⎥ ⎢ i ⎥ (8.63) ⎣ ⎦ ⎢⎣ E! ⎥⎦ ⎢⎣ E! ⎥⎦ ⎢⎣ 0 D! ! ⎦⎥ ⎢⎣ E! ⎥⎦

Therefore

E⊥d = D⊥⊥ E⊥i and E!d = D!! E!i = qD⊥⊥ E!i (8.64)

Using ⎪q⎪ > 1 from (8.62) with (8.64) gives E!d E⊥d



= q

E!i E⊥i

>

E!i E⊥i

(8.65)

This shows more directly that the electric field along the major axis direction (⟘) experiences more attenuation than that along the minor axis direction (‖). This result is illustrated in Figure! 8.8(b) with proportional length vectors. When the depolarized electric field E d is reconstructed from its components, it is rotated toward the minor axis. This follows from (8.65) directly: E!d

E⊥d

= tant d > tanti =

E!i E⊥i

⇒ t d > ti (8.66)

A rotation of the linear polarization orientation is, of course, depolarization. Including both differential phase shift as well as differential attenuation, (8.64) leads to E!d



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E⊥d

=q

E!i

E⊥i

or

rLd = qrLi (8.67)

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Polarization in Electromagnetic Systems

as in (8.12). Differential phase shift effects will change a linear polarization (if not along a characteristic axis) into an elliptical polarization because the phase δ of the polarization state has changed in traveling through the medium, which changes the polarization. Differential phase effects dominate below about 10 GHz and differential attenuation effects dominate above about 20 GHz [5, p. 254]. However, above 20 GHz rain fading can be severe enough to cause an outage, making the amount of depolarization irrelevant [6]. A very interesting result appears as a direct consequence of the preceding mathematical developments. A linear polarization tilted at τ i = 45° has the same depolarization properties as circular polarization. This follows from (8.56) with the canting angle value of θ = 45°; then (8.56) reduces to the depolarization matrix for circular polarization of (8.59). Example 8.2  Rain-Induced Depolarization on a 20-GHz Terrestrial Link

A 20-GHz terrestrial microwave link experiences a 2-km rain cell of 20-mm/h rain rate (a moderate rain intensity) along its line-of-sight path. There are two channels, one linearly polarized at 45° and the orthogonal one at 135° to horizontal. The raindrops all have their major axis horizontal (i.e., no tilt). The attenuations for vertical and horizontal polarization at 20 GHz in a 20-mm/ hr rain are 1.7 and 2.0 dB/km, respectively; so α V and α H are 0.1957 and 0.2394 km–1, respectively. It is desired to calculate the cross-polarization generated by the rain cell. First, with no tilt the ⟘ and ‖ directions align with the H and V directions, respectively. Next, the depolarization factor from (8.61) is q = e − ΔaL = e −(aV −aH )L = e −(0.1957− 0.2303)2 = 1.0717 The tilt angle of the incident wave is given to be τ i = 45°. The tilt angle of the exiting wave from Figure 8.8(b) and (8.66) td =

E!d E⊥d

= tan−1 q = tan−1(1.0717) = 47.0°

The change in tilt angle of the linearly polarized input electric field vector is Δτ = τ d − τ i = 47 − 45 = 2°. Then the cross-polarization level of the exiting wave from (7.7) is CPR L = tan2 Δt = tan2 2° = 0.00122 = −29.1 dB

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Therefore, the isolation between channels is 29.1 dB; see (7.32) and (7.33). For most links this is quite acceptable. However, if the 20-mm/h rain extends over 10 km along the path instead of 2 km the isolation reduces to 15.3 dB, which is unacceptable for an operational dual-polarized system. Also, the attenuation for 10 km of rain is about 19 dB, which also could render the link unusable due to low signal power. Attenuation increases with the intensity of the rain R (rain rate in mm/hr) and with the extent of the rain Leff (in km), the effective path length, as follows:

A = a s Leff = aR b Leff [dB] (8.68)

where α s is specific attenuation, which is the attenuation per kilometer of rain of constant rain rate R. Specific attenuation increases with rain rate. It also increases with frequency until about 100 GHz, beyond which it is relatively constant [5, p. 162]. It is much less than 1 dB/km below a few gigahertz and rises to 1 dB/km or more at 10 GHz and as much as 30 dB/ km at 100 GHz. The frequency dependence is contained in the empirically derived constants a and b. The a and b constants are also a function of polarization because attenuation is sensitive to the polarization state of the propagating wave [7]. The effective path length equals the actual extent of rain along the path for a uniform rain. For general situations we account for spatial and time variations of rain along a path using an effective path length Leff. For terrestrial links, a model for effective path length is found in [8]. For satellite links, a simple model is found in [9] and a more comprehensive model in [10]. Cross-polarization is related to the attenuation through rain in a predictable fashion. Thus, the XPD on a dual-polarized link can be found easily for a given attenuation value. For terrestrial links, XPD is found from attenuation on the link, A, as [8]

XPD = 15 + 30 log f − V log A terrestrial link (8.69)

where V is 12.8 f 0.19 for 8 ≤ f ≤ 20 GHz and 22.6 for 20 < f ≤ 35 GHz. Empirical modeling based on extensive measurements for a satellite-to-Earth radio links operating in the range of 10 to 30 GHz has produced a simple model to predict XPD as a function of attenuation and on link parameters [11]: XPD = 14 + 17.3 log( f ) − 42 log(cos e) − 19 log A satellite link (8.70)

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where f is frequency in gigahertz, ε is elevation angle of the slant path in degrees, and A is attenuation in decibels. The leading constant of 14 is an average value and actually has mild dependence on the following parameters: mean and standard deviation of the raindrop canting angles, tilt angle of the polarization, and the fraction of oblate spheroidal raindrops. See [11] for more details. The ITU also has a model [10]. Figure 8.9 shows a plot of XPD versus A using (8.70) for the SIRIO satellite link operating at 11.6 GHz. Three years of measurements were performed with the satellite and the data agreed well with Figure 8.9. [12] Note that XPD deteriorates rapidly at this frequency for increasing attenuation. Only 5 dB of attenuation reduces XPD to 20 dB, which is unacceptable for most applications. The link margin on satellite links is often about 10 dB; that is, if the rain attenuation exceeds 10 dB the receiver will lose the signal. Evaluating (8.70) for 10-dB attenuation for an elevation angle of 40° at frequencies of 10, 20,

Figure 8.9  XPD versus attenuation for a satellite link operating at 11.6 GHz with an elevation angle of 10.7° from Blacksburg, Virginia, with a latitude of 37° and a elevation of 600 m. Values are computed using (8.70). Three years of measured data using the SIRIO satellite agree closely with this model [12].

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Depolarizing Media and System Applications237

and 30 GHz gives XPD values of 17.2, 22.4, and 25.5 dB, respectively. The XPD value of 25.5 dB at 30 GHz might be acceptable on most dual-polarized links, but the values at 10 and 20 GHz probably would not be. It would appear at first from this example that XPD is better as frequency increases, but for a given rain rate attenuation increases greatly with frequency and would often be the limiting factor. A and XPD are usually quantified statistically and presented as the percent of time during a year that given values of A and XPD are equaled or exceeded. Reliability on communication links is commonly specified to be better that 99.99% (no more than 0.01% outage, or 53 minutes per year). Rain rate statistics are available on a worldwide basis [13]. To summarize, the steps involved in predicting rain effects on communication links are 1. Specify a reliability level (i.e., tolerable total outage percent for a year); 2. Find the attenuation A for the outage level on the link for rain rate associated with the the specified reliability at the location of the link; see (8.68) and [13]; 3. Find the XPD value for the attenuation found in step 2 using (8.69) or (8.70). The computed values of A and XPD are then compared to the specifications for acceptable performance to see if the reliability will be achieved. 8.4.3  Inclusion of Antenna Effects in System Calculations The depolarization properties of the propagation medium along a communication link can be combined with the antenna characteristics to determine the performance of the complete system. The main performance quantities are the variations of signal intensity and isolation between channels in a dual-polarized system. These could be treated on a power formulation basis; however, in some cases phase information is important. Therefore, we use a voltage representation that preserves phase information. The voltages received at a two-channel receiver (usually being nominally orthogonally polarized) with a depolarizing medium along the path are obtained by calculating the electric field interaction with the antenna using (6.52):

! ! ! ∗ (8.71a) V1r = E r i h1∗ = E r i hr1eˆr1



! ! ! V2r = E r i h2∗ = E r i hr 2 eˆr∗2 (8.71b)

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where

! E r = the electric field arriving at the receiver

êr1, êr2 = receive antenna normalized polarization vectors for channels 1 and 2 hr1, hr2 = effective lengths associated with the receive antenna channels 1 and 2 = hr for identical gain antennas on channels 1 and 2 (which we assume) The received electric field is decomposed into rectangular (horizontal and vertical) components and is interacted with the receive polarization states using (3.41) in (8.71) to give the received voltages in the two channels:

(

V1r = ( EHr xˆ + EVr yˆ ) i hr cosg 1r xˆ + sing 1r e jd1 yˆ r

)

∗

= EHr hr cosg 1r + EVr hr sing 1r e − jd1

r



(

V2r = ( EHr xˆ + EVr yˆ ) i hr cosg 2r xˆ + sing 2r e jd2 yˆ r

= EHr hr cosg 2r + EVr hr sing 2r e − jd2 r



)

(8.72a)

∗

(8.72b)

Channels 1 and 2 of the receive antenna need not be orthogonal. This approach permits complete characterization of actual system hardware. Casting (8.72) in matrix form gives



⎡ Vr ⎢ 1r ⎢⎣ V2

⎡ r ⎤ ∗ E ⎥ = hr [ Ar ] ⎢ Hr ⎢⎣ EV ⎥⎦

⎤ ⎥ (8.73) ⎥⎦

where [Ar] is the dual-channel receiving antenna matrix:



⎡ cosg r sing r e jd1r [ Ar ] = ⎢⎢ cosg 1r sing 1r e jd2r 2 2 ⎣

⎤ ⎥ ⎥ ⎦

receiving antenna matrix (8.74)

As a reminder, free-space loss effects in (8.47) are not included in the foregoing. The medium effects are included by introducing the depolarization matrix of (8.3c) into (8.73):

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Depolarizing Media and System Applications239

⎡ Vr ⎢ 1r ⎢⎣ V2

⎡ Et ⎤ ∗ ⎥ = hr [ Ar ] [D]⎢ Ht ⎢⎣ EV ⎥⎦

⎤ ⎥ (8.75) ⎥⎦

This expression relates the received voltage components to the transmitted electric field components. The final step in finding the received voltages from the voltages at the transmitter is to include the electric field components from the dual-channel transmitter. A system model for the transmitter is similar to the receiver model in (8.73). Inverting that expression to find the electric field components gives



⎡ Et ⎤ ⎡ t ⎤ T V ⎢ Ht ⎥ = hr−1 [ At ] ⎢ 1t ⎥ (8.76) ⎢⎣ EV ⎥⎦ ⎢⎣ V2 ⎥⎦

where [At ] is the dual-channel transmitting antenna matrix, and

[ At ]

T



⎡ cosg t cosg 2t 1 =⎢ t jd r t jd t ⎢⎣ sing 1 e 1 sing 2 e 2

⎤ ⎥ (8.77) ⎥⎦

Combining this with (8.75) gives the complete path coupling representation [14]:



⎡ Vr ⎢ 1r ⎢⎣ V2

⎤ ⎡ Vt ⎤ ⎥ = hr ht −1[C ]⎢ 1t ⎥ (8.78) ⎥⎦ ⎢⎣ V2 ⎥⎦

where we have introduced the path coupling matrix: [C ] = [ Ar ] [D][ At ] (8.79) ∗



T

The path coupling matrix of (8.79) includes all polarization effects of the transmitting and receiving antennas. Also included are effects of the medium along the path; that is, attenuation, phase shift, and polarization effects introduced by the medium are accounted for. The antennas are characterized by their (γ , δ ) polarization state parameters. No assumptions are made other than equal gains in the channels of the transmitter and receiver. The polarizations of the channels can be of any states. The path coupling matrix includes all channel coupling mechanisms due to both imperfect antennas and a depolarizing medium.

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Example 8.3  A Communication Link with Dual Orthogonal Antennas and a Clear Air Path

To gain some experience with using this compact approach to full antennapropagation characterization we consider some known results. The path is clear air and there are orthogonal linear antennas on each end that are aligned. The depolarization matrix for free space applies and is the identity matrix, as in (8.54). Then (8.79) reduces to [C ] = [C ° ] = [ Ar ] [ At ] ∗

T

clear air

For the transmitting and receiving antennas being orthogonal linearly polarized and oriented along horizontal and vertical: g 1t = 0°, g 2t = 90°, and choose d2t = 0°

transmit antenna

g 1r = 0°, g 2r = 90°, and choose d2r = 0°

receive antenna

Using these in (8.74) and (8.77) make [At]T and [Ar]∗ both identity matrices, and then [C] is also an identity matrix. Thus (8.78) gives V1r = V1t

V2r = V2t

where we have assumed the transmit and receive antennas are such that ht = hr. As expected, perfect antennas and a clear path give ideal performance. Now suppose the antennas are perfect CP with channels (1, 2) being (L, R); then g 1t = 45°, d1t = 90°

g 2t = 45°, d2t = −90°

transmit antennas

g 1r

g 2r

receive antennas

= 45°,

d1r

= 90°

= 45°,

d2r

= −90°

Using these with (8.74) and (8.77) gives [C ] = [C °] = [ Ar ] [ At ] ∗

=

T

1 ⎡ 1 −j ⎤ 1 ⎡ 1 1 ⎤ ⎡ 1 0 ⎤ = 2 ⎢⎣ 1 j ⎥⎦ 2 ⎢⎣ j − j ⎥⎦ ⎣⎢ 0 1 ⎦⎥

Again we find no coupling between channels when using ideal antennas (CP in this case) and propagation in clear air.

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Depolarizing Media and System Applications241

Now we develop expressions for finding cross-polarization and attenuation values from the coupling matrix for general antennas and path conditions. Expanding (8.78):

V1r = hr ht −1 (C11V1t + C12V2t ) = V11r + V12r (8.80a)



V2r = hr ht −1 (C21V1t + C22V2t ) = V21r + V22r (8.80b)

The cross-polarization isolation in channel 1 when both transmit channels are excited equally (V1t = V2t = 1) follows from (7.14): XPI1 =

V11r V12r

2 2

=

hr ht −1C11V1t hr ht −1C12V2t

2 2

=

C11 C12

2 2

(8.81)

Similarly, for channel 2 XPI2 =



C22 C21

2 2

(8.82)

Signal intensity often is reduced in the presence of medium disturbances and is called attenuation or fading. Attenuation is the ratio of the received power in channel 1 with a clear air path to that with a disturbed path (V 1t = 1, V 2t = 0): A1 =

V1r (clear path)

2

V1r (disturbed path)

2

=

o C11

C11

2 2

(8.83)

For example, if the signal voltage received for the disturbed path case is half that for the clear path case (⎪V1r [clear path]⎪2/⎪V1t [disturbed path]​⎪2 = 22 = 4), the attenuation is 4, or 6 dB. Similarly for channel 2



A2 =

o C22

C22

2 2

(8.84)

For perfect antennas ⎪C11o⎪ = ⎪C22o⎪ = 1 and then (8.83) and (8.84) become

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A1 =

1 2 C11

A2 =

1 C22

2

perfect antennas (8.85)

So we have formulas for determining attenuation and isolation from the coupling matrix entries. It is helpful to summarize the steps in system evaluation. First, the system coupling matrix of (8.79) is found from the transmitting antenna, the medium, and the receiving antenna properties. Next, the receiver voltages in each channel are found using (8.80). Then isolation and attenuation values are found using (8.81) to (8.85). We can also find phase information from (8.80). Usually, the relative phase between the two channels is measured. Its fluctuations as a function of the medium disturbance are sometimes useful, as in probing ice layers in the atmosphere with a dualpolarized receiver observing a satellite beacon [15]. Example 8.4  Channel Isolation with Imperfect Antennas on a Radio Link

The antenna on the transmit end of a communication link is dual-linearly polarized but the LP states are not exactly orthogonal: e1t = 0°

e2t = 0°

t1t = 2°

t2t = 89°

The receiving antenna has perfectly orthogonal LP polarizations: e1r = 0°

e2r = 0°

t1r = 0°

t2r = 90°

All four antennas have the same effective length, h. It is desired to find the isolation between the two channels. First, the polarization states for the antennas, from (3.7) and (3.8) are γ 1t = 2°, δ 1t = 0°; γ 2t = 89°, δ 2t = 0°; γ 1r = 0°, δ 1r = 0°; γ 2r = 90°, δ 2r = 0°. Using these values with (8.79), (8.81), and (8.82) give XPI1 = 15.1 dB and XPI2 = 29.1 dB. 8.4.4  Depolarization Caused by Faraday Rotation Propagation through the ionosphere, such as encountered on satellite links, can alter the polarization of radio waves in certain frequency bands. This effect must be considered in designing and using such links. The ionosphere extends from about 50 to 20,00 km above Earth’s surface. The name comes from the fact that electromagnetic radiation from the sun ionizes atmospheric particles in this layer. The presence of free electrons together with Earth’s magnetic field in the ionosphere creates conditions for

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the propagation phenomena called Faraday rotation. We treat Faraday rotation from a propagation behavior stand point rather than giving a detailed physical explanation, which can be found elsewhere [16]. Faraday rotation also occurs at optical frequencies and can be used to infer chemical properties using a polarimeter [1, p. 141]. If a linearly polarized wave propagates through the ionosphere, such as with an Earth-to-satellite path, the orientation angle of the linearly polarized electric field will rotate by an angular amount for an elevation angle of 30° that is given by [5, p. 66] Δt = 108

TEC [deg] Faraday rotation angle in the ionosphere (8.86) f2

where TEC is the total electron content, which is the integral of the electron density along the propagation path and equals the number of electrons per square meter perpendicular to the path. TEC varies from 1016 to 1019 electrons/ m2 with the minimum (maximum) occurring at midnight (midafternoon). The Faraday rotation angle decreases as inverse frequency squared and is significant at UHF. For example, a satellite terminal (assuming an elevation angle of 30° and TEC = 1018) can experience as much as 1188°, 108°, and 12° of rotation at 300, 1,000, and 3,000 MHz, respectively [5, p. 66]. Rotations of this magnitude can lead to significant polarization mismatch loss on a linearly polarized satellite link and significant reduction in isolation on a dual-polarized LP link. In fact, most UHF Earth-to-space communication links use CP to avoid the problem. The ionosphere is a medium with two characteristic polarizations, RHCP and LHCP. There is negligible differential attenuation. In fact, the ionosphere introduces very little attenuation at UHF frequencies. Therefore, the depolarization model for CP in (8.57) reduces to



⎡ E d ⎤ ⎡ e jdL 0 ⎤ ⎡ E i ⎤ L (8.87) ⎢ Ld ⎥ = ⎢ i ⎥ jdR ⎥ ⎢ E E 0 e ⎢⎣ H ⎥⎦ ⎣ ⎦ ⎢⎣ H ⎥⎦

Therefore

ELd = e jdL ELi

EHd = e jdR ERi (8.88)

The phase angles δ L and δ R are not equal, leading to depolarization of an LP wave as illustrated in Figure 8.10. The incident wave is horizontal linearly

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polarized. In Figure 8.10(a) the incident electric field vector is decomposed into CP components. These LHCP and RHCP components undergo the phase shifts of δ L and δ R, respectively; these angles are shown in Figure 8.10(b). Reconstructing the electric field components of the exiting wave gives an LP state rotated by a net angle of Δτ relative to the input LP orientation angle. The rotation angle is found as follows:

Δt =

dL − dR bL − bR = z (8.89) 2 2

for a distance of propagation z through the ionosphere. The Faraday rotation angle formula in (8.89) is derived graphically in Figure 8.10. It can also be proved by examining the electric field vector expressions. We begin with the incident wave electric field vector as shown in Figure 8.10(a): ! ! E i = E d (z = 0) = 2 Eo xˆ (8.90) This is decomposed into CP components, each with its own phase constant, as follows:

! ! ! E E E d = ELd (z) + ERd (z) = o ( xˆ + jyˆ )e − jbL z + o ( xˆ − jyˆ )e − jbR z (8.91) 2 2

where (3.65) was used. This reduces to (8.90) for z = 0. A few mathematical manipulations allow interpretation and reconstruction of the electric field vector after the wave travels a distance z through the medium:

Figure 8.10  A linearly polarized wave undergoing Faraday rotation: (a) electric field of the incident wave decomposed into CP components, and (b) electric field after a Faraday rotation angle of Δ τ .

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Depolarizing Media and System Applications245



r r r E E d = ELd (z) + ERd (z) = o ⎡⎣ xˆ ( e − jbL z + e − jbR z ) ⎤⎦ + jyˆ ⎡⎣( e − jbL z − e − jbR z ) ⎤⎦ 2 E = o e − j( bL + bR )z/2 ⎡⎣ xˆ ( e − j( bL − bR )z/2 + e j( bL − bR )z/2 ) + jyˆ ( e − j( bL − bR )z/2 − e j( bL − bR )z/2 ) ⎤⎦ 2 = 2 Eo e − j( bL + bR )z/2 [ cosΔtxˆ + sinΔtyˆ ]

(8.92) where (B.7) was used. The expression in brackets is identified as a linear vector at angle Δτ with respect to the x-axis; this vector is shown in Figure 8.10(b). Therefore, Δτ in (8.89) gives the Faraday rotation angle. This exercise also demonstrates the usefulness of decomposition in polarization analysis.

8.5  Depolarization Compensation and Adaptive Systems Operational radio systems do not have pure polarization states, which is often acceptable. For example, for a typical single polarized system the transmitter and receiver will have slightly different polarizations. The only consequence of this is a small loss due to polarization mismatch; see Section 6.1 for a complete discussion of polarization efficiency. However, in dual-polarized applications polarization impurities lead to isolation degradation. Applications for dual polarization include communication systems with frequency reuse by using two orthogonally polarized channels operating over the same radio links. Remote sensing systems also often use a dual orthogonally polarized receiving system. In either case, slight polarization imperfections cause coupling into the second channel from the first. Imperfect antennas are often responsible for reduced performance. In this section we discuss depolarization compensation techniques. They are used to reduce the undesired system polarization characteristics to compensate, at least partially, for the decrease in polarization isolation. In addition to imperfect hardware causing reduction in isolation between channels, the propagation medium can also lower isolation, as discussed in Section 8.4. In this case the isolation variation is time-varying. There are two implementations for depolarization compensation: static and dynamic. In static depolarization compensation systems the polarization properties are constant with time. The polarization characteristics in dynamic depolarization compensation systems vary with time to track the changing polarization at the receiver, usually caused by depolarization in the propagation medium. An example is compensation for rain depolarization to maintain high polarization isolation. The techniques used to compensate for depolarization

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in static and dynamic systems are similar, except in dynamic systems the goal is to track the time variations. Compensation can be performed with software in remote sensing applications by measuring the amplitudes and phases and subtracting the residual XPD, which is usually antenna induced [5, p. 268]. The simplest polarization compensation scheme is a pure linearly polarized system in which either the transmitting or receiving antenna is rotated to align the orientation of the transmitting and receiving antenna LP states. In most radio systems this is done during installation and requires no further attention. However, in Earth-space systems the wave LP orientation is rotated on passing through Earth’s atmosphere due to Faraday rotation; see Section 8.4.4. This effect is time-varying and may require polarization tracking (e.g., rotating the receive antenna). Tracking, however, can be avoided by using circular polarization. Antenna depolarization effects with proper system design or through the use of static compensation can be made small, with isolation on the order of 30 dB or more. On the other hand, medium depolarization effects can produce substantial (and maybe time-varying) effects. Rain along a radio path is a prominent example and will be used in this discussion as a specific illustration. Consider a dual-LP radio link with perfect orthogonally polarized channels operating in clear weather. Two effects lead to depolarization: differential phase shift (DPS) and differential attenuation (DA) between the orthogonal states introduced by the medium. Differential phase shift depolarizes but maintains orthogonality, whereas differential attenuation destroys the orthogonality of the states [17]. These results are now verified; this brief proof also aids in understanding the principles. Consider two general orthogonal polarization states represented in complex vector form:

eˆ1i = cosg xˆ + sing e jd yˆ (8.93a)



eˆ2i = −sing e − jd xˆ + cosg yˆ (8.93b)

where (3.78), (3.82), and (3.41) were used. These two vectors are orthogonal since they satisfy (3.77). If a medium introduces a differential phase shift Δϕ between the x- and y-axes, the output states are

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o eˆ1f = cosg e jΔf xˆ + sing e jd yˆ (8.94a) o eˆ2f = −sing e − jd e jΔf xˆ + cosg yˆ (8.94b)

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These remain orthogonal:

o o∗ = −cosg sing e jd x ˆ + sing cosg e jd yˆ = 0 (8.95) eˆ1f ⋅ eˆ2f

For a medium introducing differential attenuation the output wave states are

o eˆ1k = kcosg xˆ + sing e jd yˆ (8.96a) o eˆ2k = −ksing e jd xˆ + cosg yˆ (8.96b)

where the x-component is altered in amplitude by a factor k relative to the y-component. The orthogonality check gives

o ˆo∗ eˆ1k ⋅ e2k = ( −k2 + 1) cosg sing e jd ≠ 0 (8.97)

showing that orthogonality is lost, unless of course if k = 1 (no differential attenuation). This situation is illustrated in Figure 8.8(b) for linear polarization where we now take the ⟘-axis to be the x-axis and the ‖-axis to the y-axis. First, the input wave electric field is decomposed into x- and y-components. The x-component experiences more attenuation than the y-component because the raindrops are thicker in the horizontal plane. Reconstruction of the output wave yields a linear polarization state rotated counterclockwise from the input state. This is a form of depolarization. In addition to this differential attenuation, rain can cause differential phase shift as discussed in Section 8.4.2. The orthomode transducer (see Section 7.5.2) is used to separate a wave into orthogonal components such as dual-linear polarization as illustrated in Figure 7.13. It can be used in a receiving system to recover the two channels of information in a dual-polarized communication system. The OMT can also be used to generate two outputs for orthogonal states in a sensing system. If the intervening medium introduces depolarization, the OMT can sometimes be rotated to improve polarization isolation between output channels. But often a compensation network is required, as discussed next. The principle of depolarization compensation in the general case is a simple concept: the same amount of differential phase shift and differential attenuation is introduced in the direction perpendicular to that encountered in the propagation medium [6]. A depolarization compensation network is used for this purpose. Means for introducing variable phase shift and a variable attenuation are required. The adjustments must follow the time variations of the system and propagation medium using a controller and control algorithm.

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In general, four independent parameters must be sensed and controlled, two for each orthogonal polarization. Often the amplitude and phase of a pilot tone on each channel are detected [17, 18]. There are two basic forms of compensation networks and algorithms; the restoration method and the cancellation method [17]. The restoration method uses a variable phase shifter and a variable attenuator, both of which can be rotated as shown in Figure 8.11(a) [17, 19]. It is a direct application of compensation for the medium-induced differential attenuation and phase shift. The cancellation method shown in Figure 8.11(b)

Figure 8.11  Depolarization compensation networks. The network in (c) is useful only to compensate for medium-induced differential phase shift.

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Depolarizing Media and System Applications249

avoids any mechanical alignments by introducing a small amount of signal cross coupling between channels using two coupling paths, each with variable amplitude and phase. The four required parameters are apparent in Figure 8.11(b). Information is available in the literature on realizing compensation networks [5, pp. 266–270; 20]. It is instructive as a final illustration of compensation networks to explain how a DPS network can be realized; this also further explains the operation of polarizers discussed in Section 7.5. Figure 8.11(c) shows such a network consisting of a 90° polarizer followed by a 180° polarizer, both mechanically rotatable. This two-parameter (θ 1, θ 2) control network can be used to match to a wave of arbitrary polarization or to separate orthogonally polarized waves propagating in a medium introducing (at most) only differential phase. It offers the advantage of not having variable phase shifters (i.e., the insertion phases are fixed). The 90° polarizer linearizes the input wave, which can be circular or elliptical in polarization. If the input state is elliptical, the 90° polarizer plate angle θ 1 must be aligned along a major axis. The 180° polarizer plate angle θ 2 is then mechanically rotated to orient the output LP orientation.

8.6  Polarization in Radar Remote sensing using electromagnetic waves consists of receiving the waves emanating from a target. Both passive and active techniques are used in remote sensing. Passive remote sensing, referred to as radiometry, relies on the fact that any object above absolute zero temperature radiates by virtue of its thermal energy. A radiometer is a receiver used to detect such radiation and is discussed in the next section. Active remote sensing employs a transmitter to illuminate the target and a receiver to measure the waves scattered off of the target. A radar is the instrument commonly used and is the topic of this section. A scatterometer is similar to a radar specialized to the sensing of targets like ocean surfaces and near-surface winds. Unlike a conventional radar, a scatterometer provides no range information (i.e., distance). All of these techniques have the advantage over optical sensing that they are not restricted to day time and good weather operation. 8.6.1 Radar Basics A radar consists of a transmitter, a transmit antenna, a receiving antenna, and a receiver. If the transmitter and receiver are colocated it is referred to as monostatic radar. Most often the same antenna is used for transmission and

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reception as shown in Figure 8.12. A switch is required to switch the transmitter and receiver to the antenna. If the transmitter and receiver are separated, it is referred to as a bistatic radar. The antenna pattern main beam peak is aimed at the target or is scanned across a target area. A radar measures the distance between the antenna and the target based on the time T taken for transmitted pulse to travel from the transmit antenna and back to the receive antenna. For a monostatic configuration then the distance is simply r = 2Tc. Distance determining radars are common in vehicles for warning of nearby objects. In the following we assume the radar is monostatic, but the principles apply to all radars. The treatment also applies to imaging systems such as synthetic aperture radar (SAR). If the target is moving, its speed can be determined using a Doppler radar. A continuous wave (CW) signal is transmitted and target motion changes the frequency of the returned signal. The frequency of the return signal is proportional to the speed component in the line from the transmitter to the target. In addition, if the object is receding, the frequency relative to the transmitted signal will decrease. The frequency increases for approaching objects. Police radars that detect the speed of vehicles use Doppler systems. The shape and location of objects below the surface of Earth can be determined using ground-penetrating radar. A single pulse is transmitted that creates a wideband spectrum. The power received with a radar is related to the transmitter power (Pt), the gains of the transmit and receive antennas (Gt and Gr), the distance to the target (r), and the radar cross section (σ ), and is calculated using the radar equation [21]:



l2GrGt Pr = Pt s (8.98) (4p)3 r 4

Figure 8.12  Block diagram of a monostatic radar.

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The radar cross section is an area quantity and is the equivalent area of the target based on the target reradiating the incident power isotropically. We are only concerned with the power scattered in the direction of the receiver and can assume the target scatters isotropically. The received power is, of course, directly proportional to the radar cross section. The received power decreases with the distance to the target as 1/r4. This is because the electric field of the incident wave falls off as 1/r as in (6.49); so the incident power density falls off as 1/r 2. The power density of the returned power density also experiences another 1/r 2. 8.6.2 Polarimetric Radar The waves reflected from a target depend on the polarization of the incident wave as well as the shape of the target. Thus, the radar return signals in each polarization provide additional information about the target. The polarization of the wave reflected from a target can be significantly different from the incident wave; that is, significant cross-polarization levels can arise. For example, we saw in Section 8.3.2 that a CP wave incident on a flat plate changes its sense on reflection, giving a reflected wave totally cross-polarized to the incidental wave. The wave scattered from a target depends on the following: (1) the illuminating wave, including its incidence angle and polarization, (2) the characteristics of the target, including its size, shape, and composition, and (3) the receiving antenna location and polarization. The operating frequency of the radar is, of course, an important parameter. Higher frequencies with smaller wavelengths are capable of providing more target shape detail. The full vector nature of the waves scattered from the target can be measured using a polarimetric radar which yields the polarization characteristics of the target. The radar cross section for a basic radar is a single number; it is replaced by a matrix for polarimetric radar. The radar cross section of the target is measured for combinations of transmit and receive antenna polarizations. See [22, 23, 24] for further reading on polarimetric radar. In conventional polarimetric radar, orthogonal polarizations are transmitted (usually linear polarization states) and the co-polarized states are received. More information is available using a multipolarized radar that collects the cross-polarized information and perhaps also the associated phase information. For example, in the case of horizontal and vertical polarizations, data is collected for the combinations of HH, HV, VH, and VV where the first character is the transmit polarization and the second is the receive polarization. A radar capable of changing its polarization over time is called a polarization agile radar.

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The general radar problem is modeled with the geometry of Figure 8.1. For monostatic radar the direction of scatter (referred to as backscatter) is back toward the transmitter; that is, in the −ûi direction (ϕ s = 180°, θ s = θ i). A fixed coordinate system must be selected to describe polarizations. We use the backscatter alignment convention in which the polarization vectors are referenced to the radar antenna coordinates [22, p. 20]. This allows use of all our previous developments. Radar problems are treated differently depending on whether the observed scene is a point target or a distributed target, which is often made up of randomly distributed targets such as soil surfaces, ocean waves, or vegetation canopies. The principles for analyzing these target types follow. 8.6.2.1  Point Radar Targets

A point radar target is one that is of an angular extent that is smaller than the radar antenna beamwidth. A point target is said to be deterministic if the temporal fluctuations of the target are much longer than the radar sampling interval [25]; the term coherent is also used. If a point target is deterministic, its scattering characteristics can be represented by a (polarimetric) time-independent scattering matrix. That is, the components of the scattered electric field arriving back at the radar receiver are given by [26, p. 228; 27, p. 19] ⎡ Es ⎤ 1 = s ⎥ E ⎢⎣ 2 ⎥⎦

[ Es ] = ⎢

i 1 ⎡ S11 S12 ⎤ ⎡ E1 ⎤ ⎢ ⎥= 4pr ⎢⎣ S21 S22 ⎥⎦ ⎢ E2i ⎥ ⎣ ⎦

1 [S][ E i ] (8.99) 4pr

where [Ei] contains the electric field components incident on the target. Subscripts (1, 2) represent two polarizations, usually taken to be (H, V) or (R, L). For linear polarization (1 → H, 2 → V) the scattering matrix is



⎡ S ⎤ S [S] = ⎢ HH HV ⎥ (8.100) S S ⎣ VH VV ⎦

For monostatic radar, as we are discussing here, SVH = SHV. If the target is rotationally symmetric, then SVH = SHV = 0. The measurements of the parameters in (8.100) are made by transmitting H and V in rapid succession and measuring the received field intensity in both polarizations. The scattering matrix entries, Sij, are complex valued. The first subscript i on the scattering matrix entries represents the scattered polarization and the second subscript j corresponds to the incident polarization. The radar must measure both the magnitude and phase of the received field components relative

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Depolarizing Media and System Applications253

to those of the transmitted field. The factor 1/r in (8.99) is the spherical wave behavior of the scattered field leaving the finite extent target and arriving at the radar receiver. The incident field Ei in (8.99) is due to the transmitted field Et as indicated in (8.47). The properties of the scattering matrix are similar to those of the depolarization matrix of (8.49). The difference lies in the fact that the depolarization matrix is a transmission matrix that represents only the changes in the wave introduced along the propagation path. In contrast, the scattering matrix includes scattered fields because it includes the influence of the target from which the radar return originates. To illustrate further, if there are no obstacles along the transmission path, the depolarization matrix reduces to the identity matrix as in (8.54) and the received signal is unaltered. In contrast, if no radar target is present, the scattering matrix becomes the null matrix and no signal is returned to the radar. Once the scattering matrix is found, the target is completely characterized and its polarimetric response can be calculated for any incident polarization using (8.100). The scattering matrix has eight unknown quantities: four magnitudes and four phases. Enforcing symmetry (SVH = SHV ) reduces this number to six. It is practical to make relative phase measurements, so only two relative phases are required. Thus, five measurements (three magnitudes and two relative phases) will yield the necessary scattering matrix for a target. See [28, 29; 22, Ch. 5] for examples of hardware implementation of a polarimetric radar. The primary quantity in radar is the radar cross section, which is the effective scattering area of the target. RCS is obtained for any combination of incident and scattered polarization states from the scattering matrix elements as [27, p. 18]

s ij = Sij

2

[ m2 ]

i,j = H or V (8.101)

Conventional radars usually operate in a single polarization and RCS is a single scalar number representing the echo area of the target in that polarization. Note that (8.101) is independent of range (distance r). The scattering matrix [S] as we have presented it represents the linearly polarized electric field scattering process. It is a full-phasor and full-vector representation; that is, both the polarization and the phase changes introduced by the target are included. The vector nature of the scattering process is accounted for through the scattering matrix entries, which yield co-polarized and crosspolarized linear polarization responses. The scattering matrix contains all the information to characterize the target response for any polarization.

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The scattering matrix derivation for CP antennas is similar to that for the transmission case of (8.79) where 1 → L 2 → R:

[ Sc ] = [ Ar ][S][ At ]T =

1 ⎡ 1 j ⎤ ⎡ SHH SHV ⎤ ⎡ 1 1 ⎤ 2 ⎢⎣ 1 − j ⎥⎦ ⎢⎣ SVH SVV ⎥⎦ ⎢⎣ j − j ⎥⎦

1 ⎡ 1 j ⎤ ⎡ SHH + jSHV SHH − jSHV ⎤ = 2 ⎢⎣ 1 − j ⎥⎦ ⎢⎣ SVH + jSVV SVH − jSVV ⎥⎦

(8.102)

The complex conjugate is not present in the receiving antenna matrix for this monostatic radar case as there was for transmission. This is due to the reversed direction propagation for the field scattered from the target. Similar differences were encountered in reflection and transmission at an interface; see (8.36) and (8.37). Performing the final matrix multiplication in (8.102) and using SVH = SHV gives

SLL =

1 ( S − SVV ) + jSHV (8.103a) 2 HH

SLR =

1 ( S + SVV ) = SRL (8.103b) 2 HH

SRR =

1 ( S − SVV ) − jSHV (8.103c) 2 HH

Example 8.5 illustrates the polarimetric radar principles for a spherical conducting target. It demonstrates the use of a scattering matrix for different polarizations. Polarimetric scattering matrices for other scatterers are available in the literature [26, Ch. 6]. Example 8.5  Polarimetric Scattering from a Metallic Sphere

A metallic sphere with radius a that is greater than several wavelengths in extent has a scalar RCS that is simply the cross sectional area:

s = pa2

a >> l (8.104)

This is valid for any orientation (tilt angle) of the incident linear polarization because of the symmetry of the target. Therefore

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s HH = pa2

s VV = pa2 (8.105)

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The target symmetry also dictates that no cross-polarization is generated at the target. Therefore

s HV = 0

s VH = 0 (8.106)

The complete RCS matrix for a conducting sphere is then



[S] = pa2 ⎡⎢ −1 0 ⎤⎥ (8.107) ⎣ 0 −1 ⎦

where minus signs were inserted into the scattering matrix to account for the 180° phase change on reflection from the conductor. The scattering matrix in (8.107) fully characterizes the vector scattering properties of the sphere. It can be used for other polarizations. We demonstrate the results for a dual-CP radar. Channel 1 of both the transmitter and receiver are taken to be LHCP and channel 2 is RHCP. Using the LP scattering matrix values of (8.107) in (8.103) gives the CP scattering matrix



[ Sc ] =

pa2 ⎡⎢ 0 −1 ⎤⎥ (8.108) ⎣ −1 0 ⎦

This CP scattering matrix indicates that the co-polarized responses SLL and SRR are zero. This, of course, is because a LHCP wave reflecting off a metal surface changes to RHCP. The cross-polarized responses are ⎪SLR⎪ = ⎪SRL⎪ = pa2 . The polarization response (also called the polarization signature) of a large conducting sphere to arbitrary polarizations is shown in Figure 8.13. The co-polar responses for identical transmitting and receiving dual-polarized antennas are shown in Figure 8.13(a). Polarization state representations use tilt angle τ and ellipticity angle ε . The polarization response does not depend on tilt angle, of course, because of the symmetry of the spherical target. The copolar response does strongly depend on ε . It is maximum for LP (ε = 0°) and null for CP (ε = ±45°). The cross-polar response is shown in Figure 8.13(b). For LP (ε = 0°) there is no cross-polarization. The cross-polarization response is maximum for CP (ε = ±45°). The scattering properties of a sphere are used to advantage in operational radars. Rain along the radar path introduces clutter, or unwanted return. However, if a radar uses transmitting and receiving antennas that are same-sense CP and the rain is not very intense (thus the raindrops are approximately spherical),

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Figure 8.13  Polarization response of a polarimetric radar for a large conducting sphere showing backscatter RCS normalized to unity maximum for (a) the receiving polarization co-polarized to the transmitting state, and (b) cross-polarized to the transmitting state. (Source: Ulaby, F. T., M. W. Whitt, and K. Sarabandi, “AVNA-Based Polarimetric Scattermeters,” IEEE Ant. and Prop. Magazine, Vol. 32, Oct. 1990, pp. 6–17. ©1990 IEEE. Reprinted with permission.)

the echo signal will be of opposite-sense CP, just as in Example 8.5, and it is rejected by the receiving antenna. But the intended target embedded in the rain could have an observable echo due to double bounces (thereby changing the sense back to the original incident CP sense) from the facets of the target. 8.6.2.2  Distributed Radar Targets

A target that extends over several radar antenna beamwidths is considered to be a distributed radar target. An example situation is a down-looking radar on an airplane that is observing ground features. Distributed targets are typically composed of randomly distributed, small target components. Instead of using RCS to quantify the scene, the backscatter coefficient σ ° is used. This is the ensemble average of RCS per unit area [30]:

so =

s (8.109) A

where A is the illuminated area of the scene. One measurement approach to implementing this is to collect N independent samples and average over (8.101):

σo =

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1 N 2 Sij (8.110) A∑ i=1

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A computationally superior approach is to employ partial polarization techniques such as Stokes parameters. A 4 × 4 Mueller matrix is used to relate the 1 × 4 Stokes parameter matrices of the transmitting and receiving polarization states [23, Ch. 1; 30]. Tragl [25] uses a 3 × 3 covariance matrix, which is related to the Mueller matrix. A technique has been reported that uses noncoherent measurements with six different polarization states (linear polarizations rotated by 45°) to form the Mueller matrix [31]. There are several remote sensing applications for radar. One application is to examine rain for communications system design [32]. A radar beam (2.8 GHz in the study) is scanned along a potential communication link path during precipitation events. The reflectivity in horizontal and vertical polarization is used to estimate the raindrop size distribution along the path, which in turn is used to calculate the attenuation that would be experienced on a communication link on that path at any frequency of interest. Multipolarized radars provide co-polarized as well as cross-polarized backscatter data that is used to greatly enhance image formation of scenes [33]. Radars are finding wide use for many security applications such as weapon detection. Some airport scanners use rotating antenna illuminators to construct three-dimensional images to locate weapons. Multipolarized radars enhance such images [34]. Other applications for radar sensors include observations of ground terrain [25, 28, 30] and estimation of soil moisture [35]. Perhaps the most widely known multipolarized radar is next-generation radar (NEXRAD) that is used to generate the weather maps shown on TV weather forecasting and on smartphone weather apps. The images are created with an elaborate network of 159 S-band radars distributed around the United States and operated by the National Weather Service. Each NEXRAD high-resolution Doppler radar installation is equipped with an 8.5m diameter reflector antenna that is scanned 360° in azimuth and stepped in elevation angle increments. Precipitation and winds are observed. These polarimetric radars transmit simultaneous horizontal and vertical polarization signals. Comparison of returned relative powers and phases from the H and V channels allows distinguishing of rain, hail, snow, and ice in the atmosphere [36].

8.7  Polariztion in Radiometry 8.7.1 Radiometer Basics A radiometer senses natural emissions from a target or scene. This process of passive remote sensing is based on the physics principle that all objects above absolute zero in temperature emit electromagnetic waves that extend potentially

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all across the spectrum. If the source is a so-called blackbody, its emissions follow the blackbody law of (4.1); see Section 4.1. The emissions intensity increases with the temperature of the body. Intensity also increases with frequency (as frequency squared), peaks, and then reduces. The sun for example has a temperature of 6,000K, peaking in the visible region of the spectrum. It is interesting that our eyes also respond best at the same frequencies. Instrumentation for detecting emissions from celestial objects was pioneered by the radio astronomers who searched the heavens for emissions from radio sources. Radio astronomy began in the 1930s and was the first passive remote sensing application. The output of a radiometer is antenna temperature TA . It is an equivalent temperature. The noise emitted by the actual source is equal to that of a perfect blackbody at the same temperature. TA is calculated by integrating the noise temperature of the scene T(θ , ϕ ) in all angular space around the antenna weighted by the antenna power pattern |F(θ , ϕ )|2 [21, p. 105]:



TA =

p 2p

1 2 T (q,f) F (q,f) sinqdqdf (8.111) Ω A ∫0 ∫0

where ΩA is the beam solid angle of the antenna pattern; see (5.2). For an isotropic scene of uniform temperature To, the antenna temperature is TA = To. For a small discrete source subtending a solid angle Ωs that is less than that of the main beam (i.e., Ωs M). Each additional transmit-receive antenna pair increases the volume of data that can be handled. Ideally capacity increases linearly with the number of additional antenna pairs. Conventional MIMO uses two, four, or eight antennas on each end. Future systems will use hundreds of antennas or more in the millimeter-wave bands where the elements in the array are physically small. This massive MIMO technology will further increase the data rate and improve performance. Massive MIMO usually deploys 64 or more antennas. Also needed to support future high data-rate wireless system is increased sectorization with beamforming. Orthogonal polarizations can be used to implement MIMO

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9.8 Chapter Summary We conclude this chapter with at summary of key facts relating to polarization in wireless systems. •

• • •

A variety of diversity schemes are available for use at a base station or terminal with spatial, polarization and angle diversity being the most popular. See Section 9.3. All three diversities offer similar improvement as measured by diversity gain for outdoor and indoor environments. See Section 9.6. Polarization diversity offers the most compact realization. Future wireless systems will employ: –– Smart antennas that adapt to varying terminal conditions using beamforming and polarization adjustment –– High-order MIMO (including massive MIMO) will be deployed

9.9  Problem

1. What is the diversity gain at 0.1% probability for the experiment in Figure 9.4?

References [1]

Doble, J., Introduction to Radio Propagation for Fixed and Mobile Communications, Norwood, MA: Artech House, 1996, p. 49.

[2]

Stapor, D., “Optimal Receive Antenna Polarization in the Presence of Interference and Noise,” IEEE Trans. on Ant. and Prop., Vol. 43, May 1995, pp. 473–477.

[3]

Nagy, L. L., “Automobile Antennas,” Chapter 39 in Antenna Engineering Handbook, Fourth Edition, J. L. Volakis (ed.), New York: McGraw-Hill, 2007.

[4]

Pathak, V., S. Thornwall, M. Krier, S. Rowson, G. Poilasane, and L. Desclos, “Mobile Handset System Performance Comparison of a Linearly Polarized GPS Internal Antenna with a Circularly Polarized Antenna,” Proc. of IEEE Ant. and Prop. Soc. Symp., June 2003, pp. 666–669.

[5]

Licul, S., J. Marks, and W. L. Stutzman, “Method and Apparatus for Quadrifilar Antenna with Open Circuit Element Terminations,” U. S. Patent 7,999,755, Aug. 16, 2011.

[6]

Kajiwara, A., “Line-of-Sight Indoor Radio Communication Using Circular Polarized Waves,” IEEE Trans. Vehicular Tech., Vol. 44, Aug. 1995, pp. 497–493.

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[7]

Rappaport, T. S., J. C. Liberti, K. L. Blackard, and B. Tuch, “The Effects of Antenna Gains and Polarization on Multipath Delay Spread and Path Loss at 918 MHz on Cross-Campus Radio Links,” IEEE 42nd Trans. Vehicular Tech., Vol. 1, May 1992, pp. 550–553.

[8]

Rappaport, T. S., and D. Hawbaker, “Wide-Band Microwave Propagation Parameters Using Circular and Linear Polarized Antennas for Indoor Wireless Channels,” IEEE Trans. on Comm., Vol. 40, Feb. 1992, pp. 240–245.

[9]

Manabe, T., Y. Miura, and T. Ihara, “Effects of Antenna Diversity and Polarization on Indoor Multipath Propagation Characteristics,” IEEE J. on Selected Areas in Comm., Vol. 14, April 1996, pp. 441–448.

[10] Kajiwara, A., “Effects of Polarization, Antenna Directivity, and Room Size on Delay Spread in LOC Indoor Radio Channel,” IEEE 46th Trans. Vehicular Tech., Feb. 1997, pp. 169–175. [11] Barts, R. M. and W. L. Stutzman, “A Reduced Size Helical Antenna,” Proc. of IEEE Ant. and Prop. Soc. Symp., July 1997, pp. 1588–1591. [12] Barts, R. M. and W. L. Stutzman, “Stub Loaded Helix Antenna,” U. S. Patent 5,986,621, Nov. 16, 1997. [13] Chen, Z., and K.- M. Luk, Antennas for Base Stations in Wireless Communications, New York: McGraw-Hill, 2009. [14] Lee, W. C. Y., Mobile Communications Design Fundamentals, Second edition, New York: Wiley, 1993. [15] Rappaport, T. S., Wireless Communications: Principles and Practice, Piscataway, NJ: IEEE Press, 1996, pp. 167–176. [16] Dietrich, C. B., W. L. Stutzman, B.- K. Kim, and K. Dietze, “Smart Antennas in Wireless Communications: Base-Station Diversity and Handset Beamforming,” IEEE Antennas and Propagation Magazine, Vol. 42, Oct. 2000, pp. 142–151. [17] Beverage, H. H., and H. O, Peterson, “Diversity Receiving System of RCA Communications, Inc., for Radio Telegraphy,” Proc. of IRE, Vol. 19, April 1931, pp. 531–561. [18] Fujimoto, K. (ed.), Mobile Antenna Systems Handbook, Third Edition, Norwood, MA: Artech House, 2008, p. 167. [19] Stutzman, W. L., and G. A. Thiele, Antenna Theory and Design, Third Edition, New York: Wiley, 2013. [20] Beckman, C., and U. Wahlberg, “Antenna Systems for Polarization Diversity,” Microwave J., Vol. 40, May 1997, pp. 330–334. [21] Bergada, P., R. Alsin-Pages, and M. Hervas, “Polarization Diversity in a Long-Haul Transequatorial HF Link From Antarctica to Spain,” Radio Science, Vol. 52, Jan. 2017, pp. 105–117.

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[22] Preradovic, S., “Printed 3D Stacked Chipless RFID Tag with Spectral and Polarization Encoding,” Microwave Journal, Vol. 59, April 2016, pp. 122–132. [23] Lemieux, J.- F., and M. El-Tanny, “Experimental Evaluation of Space/Frequency/ Polarization Diversity in the Indoor Wireless Channel,” IEEE Trans. Vehicular Tech., Vol. 40, Aug. 1993, pp. 569–574. [24] Narayanan, R., K. Atanassov, V. Stoilikovic, and G. Kadambi, “Polarization Diversity Measurements and Analysis for Antenna Configurations at 1800 MHz,” IEEE Trans. Ant. and Prop., Vol. 52, July 2004, pp. 1795–1810. [25] Neelakanta, P., W. Preedalumpabut, and S. Morgera, “Making a Robust Indoor Microwave Wireless Kink: A Novel Scheme of Polarization-Sense Diversity,” Microwave J., Vol. 47, Aug. 2004, pp.84–98. [26] Dietrich, C., K. Dietze, J. R. Nealy, and W. Stutzman, “Spatial, Polarization, and Pattern Diversity for Wireless Handheld Terminals,” IEEE Trans. Ant. and Prop., Vol. 49, Sept. 2001, pp. 1271–1281. [27] Dietrich, C., R. Barts, W. Stutzman, and W. A. Davis, “Trends in Antennas for Wireless Communications,” Microwave J., Vol. 46, Jan. 2003, pp. 22–44. [28] Litva, J., and T. Lo, Digital Beamforming in Wireless Communications,” Norwood, MA: Artech House, 1996. [29] Rappaport, T., S. Sun, R. Mayzus, H. Zhao, et al., “Millimeter Wave Mobile Communications for 5G Cellular: It Will Work!,” IEEE Access, Vol. 1, 2013, pp. 335–349.

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10 Polarization Measurements 10.1  Introduction to Polarization Measurements Modern simulation software packages can predict the polarization properties of a device quite accurately in most cases. However, experimental verification is required in many applications for at least a few representative designs. In cases with demanding specifications, a complete measurement set is needed. This chapter discusses the principles and techniques for measurement of the polarization state of a wave or device such as an antenna. In Chapters 2 and 3 we saw that a minimum of two parameters are required to completely specify the polarization state of a wave or antenna. For a wave two additional quantities are frequently of interest as well: the wave intensity and the degree of polarization if the wave is partially polarized. For wave polarization evaluation the wave is measured with a receiver specialized to determine polarization. For antenna polarization evaluation the antenna under test can be operated in transmission or reception. In a traditional measurement scenario the test antenna is rotated and its response is measured as a function of rotation angle, frequency, and source polarization. In this chapter we discuss the measurement of the polarization of waves and antennas. Explanations will be described in terms of an antenna as a test article, but most of the techniques are directly applicable to characterizing the polarization of a general device or of an incoming wave arising from any 287

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source. Section 10.4 addresses the special case of measurement of partially polarized waves. Accurate antenna measurement requires specialized facilities and instrumentation. The facility can be as simple as an outdoor measurement range that uses towers or buildings to elevate the test article and/or the source of electromagnetic waves to reduce the effects of the ground. This is a far-field range, in which the test article is in the far field of the source. The range should have a clear path to the test article and have few structures nearby that can scatter waves from the source toward to the test article. The wave arriving at the test article should be as close to a plane wave as possible; that is, nearly uniform phase and amplitude across the extent of the test article. The source is typically an antenna with a beamwidth just wide enough to accomplish the near planewave conditions and narrow enough to reduce illumination of surroundings. Also the source antenna should be of high-quality polarization, usually linear polarized with at least 30-dB XPD. A generator creates the transmitted power and a receiver/data collection system receives and processes data from the test antenna. Network analyzers can perform both of these functions. The measurement facility can also be indoors. This offers the advantages of all-weather operation and security. The principles and instrumentation are the same as for the outdoor far-field range. However, the walls and floor of the indoor range cause reflections that must be controlled or avoided. The traditional method is to use an anechoic chamber, which is a room with walls covered with absorbing material. In order to achieve the required far-field distance the room can be quite large (especially at low frequencies), greatly increasing costs. This can be avoided by using alternate range types. A reduced size range is the compact range, which uses the near field of a large reflector antenna as the source to produce a nearly uniform illumination of the test article. A near-field range is very small. It uses a probe system to sample the near fields of the test article and then the data are processed to yield far-field patterns, polarization, and so forth. There are many variations on these basic range types used in practice and several are commercially available. The reader is referred to the literature for details on antenna ranges: [1; 2, Ch. 13; 3–7] and on absorbing material: [8]. In this chapter we discuss instrumentation and facilities used in polarization measurements as the antenna characteristics are treated, including pattern, polarization, and gain. The other important parameters for an antenna are impedance and bandwidth. It is worth mentioning that a test antenna can be measured in either the transmitting mode or the receiving mode due to antennas obeying reciprocity in most situations. But we discuss antenna

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measurements in terms of it operating in the receiving mode, which is the common situation.

10.2  Antenna Pattern Measurement Principles Including Polarization 10.2.1  Pattern Measurement Techniques As explained in Chapter 5, an antenna pattern is the graphical representation of the angular distribution of radiation around an antenna when transmitting or the angular response to an antenna when receiving. For full characterization, radiation data should be available for all directions around the antenna. In addition the polarization state in each direction should be known. Usually only partial information on the pattern and its polarization state is needed. This is normally in the form of the response as a function of test antenna rotation angle to source antennas in orthogonal polarization states that are nominally co-polarized and cross-polarized to the test antenna. This is done for at least the two principal planes. These are referred to as the co-polarized and cross-polarized radiation patterns; see Section 5.2.2. The measurement of an antenna pattern is conceptualized by keeping the test antenna stationary and operating in transmission while moving a receiving probe around at a fixed distance in the far field of the antenna as shown in Figure 10.1. Polarization

Figure 10.1  Illustration of pattern measurement by moving a receiving probe over a spherical surface in the far field of the test antenna operating in transmission. (Source: Stutzman, W., and Thiele, G., Antenna Theory and Design, Third Edition. ©2013 John Wiley and Sons. Reprinted with permission.)

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information is obtained by either having a dual orthogonally polarized probe or measuring the pattern twice for orthogonal probe polarizations. The conceptual arrangement of Figure 10.1 is not space-efficient. Instead, traditional far-field pattern measurements are made by rotating the test antenna using a turntable while being illuminated by a distant source antenna as indicated in Figure 10.5. The figure shows the test antenna being operating in reception, but the role of transmission and reception can be reversed; the pattern being measured is that of the antenna being rotated. The pattern cuts determine the polarization component that is measured as is discussed in Section 10.2.2. Some modern antenna measurements have returned to the moving probe concept of Figure 10.1 but with the distance to the probe greatly reduced, forming a near-field range. Here the probe is in the near field of the test antenna, and amplitude and phase samples are collected and processed to determine the properties of the antenna such as far-field pattern and polarization [9–11]. The basic configuration is the planar near-field range as shown in Figure 10.2. The planar scanner provides accurate data only in the forward hemisphere and thus is most suited for narrowbeam antennas with low back directed radiation. Figure 10.2 illustrates how a planar scanner operates. Scanners covering cylindrical or spherical surfaces are used for broadbeam antennas. For faster sampling of the near fields a system with multiple fixed probes is used. A popular configuration has probes placed along a circular arc with the

Figure 10.2  The planar near-field range. (Source: Stutzman, W., and Thiele, G., Antenna Theory and Design, Third Edition. ©2013 John Wiley and Sons. Reprinted with permission.)

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test antenna at the origin. The test antenna is rotated around its axis for each pattern cut. In near field ranges, dual-polarized probes are used to collect orthogonal polarization data simultaneously. Near-field ranges are available commercially; see [12, 13]. The compact range shown in Figure 10.3 is smaller than a far-field range but larger than a near-field range. The reflector forms a collimated field that creates a uniform phase and partially uniform amplitude distribution across the test antenna as long as the test antenna is much smaller than the reflector. The amplitude distribution can be improved by introducing a subreflector and shaping both reflectors, forming a dual-compact range. Indoor far-field ranges have walls that are absorber lined to reduce reflections. Near-field and compact ranges also use absorber, but room reflections are often not as important as with a far-field range. Figure 10.4 shows the anechoic chamber at Virginia Tech (conceived by the author). It is a combination far-field/near-field range. The view in the figure is from the location where the source antenna is placed when operated in the far-field mode. The chamber is tapered, which creates shorter path lengths from reflections off the absorber, leading to a smooth total (direct plus reflected) field over the quiet zone where the test antenna is placed (on the turntable assembly in the center). The box portion is 3.4 × 3.4m in cross section and 3.5m long. The length of tapered section is 4.4m long, giving a total range length of 7.9m. All interior surfaces are absorber lined. For operation in the near-field mode, the probe positioning system on the back wall is used. When operating in the far-field mode the probe is moved to one side. Polarization information is often needed in addition to pattern data, or sometimes just the polarization state of a wave is required. The several methods

Figure 10.3  The compact range. (Source: Stutzman, W., and Thiele, G., Antenna Theory and Design, Third Edition. ©2013 John Wiley and Sons. Reprinted with permission.)

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Figure 10.4  A combination far-field/near-field range viewed from the apex of the tapered section of the chamber where the source is placed during far-field measurements. The test antenna is shown on the positioner in the foreground. On the back wall is the probe assembly used for near-field measurements.

to measure polarization are summarized in Table 10.1. Method 1, the radiation pattern method, was discussed in Section 5.2.2 and will be expanded on in this section. Method 2, the polarization pattern method, also will be discussed in detail in this section. Method 3, the amplitude-phase method, and method 4, the multiple-component method, both provide complete polarization state information and are discussed in Section 10.3. Method 1 is restricted to antenna measurement. The remaining methods apply to an incoming wave, but can also be used for an antenna. In fact, methods 2, 3, and 4 can be performed as a function of test antenna rotation angle. The remainder of this section provides details on methods 1 and 2.

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Table 10.1 Polarization Measurement Methods Method

Parameters Measured

1.  Radiation Pattern Methods   a. Co-polarized and cross-polarized

Radiation patterns for sources nominally co-polarized and cross-polarized to the test antenna

  b. Spinning linear

The radiation pattern is measured while a linearly polarized source spins rapidly

2.  Polarization pattern method

The response from the stationary test antenna is measured while a linearly polarized source is rotated

3.  Amplitude-phase methods   a. Orthogonal linear components and their relative phase

E2 /E1 and δ

  b. Opposite sense circular components and their relative phase

EL0 /ER0 and δ ′

4.  Multiple-component methods   a. Polarization pattern plus sense

Polarization pattern information (⎪R⎪, τ ) plus sense (sign[R]) via an auxiliary sense determination

  b. Six-component method

E1, E2 , E3 , E4 , EL0, ER0

The next section explains methods 3 and 4, which provide complete polarization information. 10.2.2  Co-polarized and Cross-Polarized Radiation Patterns The polarization properties of an antenna are commonly determined by a series of radiation patterns, which is method 1 in Table 10.1. Patterns are taken in at least the principal planes and sometimes in a few intermediate planes. This is done for two source antennas that are nominally co-polarized and crosspolarized to the test antenna, as mentioned in Section 5.2.2. Co-polarization is defined as the intended (or reference) polarization [14]. Cross-polarization is the state orthogonal to the co-polarization state; see Section 3.8. Of course, for all but circular polarization co-polarization and cross-polarization depend on specifying a reference polarization orientation (i.e., co-polarized state). For observation angles other than the co- and cross-reference directions, more

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specific definitions are needed. Ludwig [15] pointed this out and introduced three definitions [16]. Ludwig’s third definition is in common use; it relates directly to measurement and is adopted here. It is given mathematically using (5.9) with (5.10) along with Figure 5.3. We illustrate basic co-polarized and cross-polarized pattern measurements using linearly polarized dipole antennas as shown in Figure 10.5. The linearly polarized dipole source antenna is excited by a transmitter and illuminates the test antenna which is connected to a receiver. It is important to remember that the pattern being measured is that of the antenna undergoing rotation. For a far-field range, the test antenna should be in the far field of the source, which is a distance of at least

r ff =

2D2 (10.1) l

where largest dimension of the antenna and λ is the wavelength of the illuminating wave. This formula gives the far-field distance for general test antennas where D is the maximum dimension of the antenna. The polarization properties of the dipoles used in Figure 10.5 are shown in Figure 2.6. The transmitting dipole has current I as shown. The resulting radiated field intensity vectors over a far-field sphere are displayed. The dipole antenna essentially produces an electric field parallel to the dipole wire projected onto the surrounding sphere. Any plane containing the z-axis ! in Figure 2.6 contains the dipole and also contains the electric field vector E; this is the E-plane. The magnetic field ! vector H must be perpendicular to the electric field, which means the Poynting vector of (3.18) points radially away from the origin of the sphere. Then the H-plane magnetic field is parallel to the xy-plane and tangent to the sphere. A detailed discussion of antenna polarization was presented in Chapter 5. Here we review a few important points. The E-and H-plane co-polarized radiation patterns for the dipole antenna are shown in Figure 2.6(b) and (c), respectively. The cross-polarized patterns depend on the details of the dipole construction, but will usually be 20 or more decibels down from the copolarization pattern peak at all points. Aperture antennas can produce crosspolarization levels that are 40 dB down from the co-polarization pattern peak. Figure 5.6(e) represents an example of a typical cross-polarization pattern for a good quality reflector antenna. The complete polarization state measured over many planar cuts through the antenna axis is referred to as the polarization distribution. However, often the polarization state is quantified by measuring the co-polarized and crosspolarized patterns only over the principal planes.

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Figure 10.5  Illustration of measuring co-polarized and cross-polarized radiation patterns. The test antenna, operating in the receiving mode in this case, is rotated about its axis.

10.2.3  Polarization Pattern Measurement As noted under method 2 in Table 10.1, the polarization pattern of an antenna is the amplitude response of an antenna as it is rotated about its axis when illuminated by a linearly polarized plane wave [14]. The polarization pattern is a polar plot of the response as a function of the relative angle between the illuminating LP wave orientation and a reference orientation of the antenna; see Figure 10.6. Actually, it is easier to explain the polarization pattern with the test antenna operated as an elliptically polarized transmitting antenna and the receiving antenna as a linearly polarized probe. Reciprocity permits us to do this. The tip of the instantaneous electric field vector of the incoming wave

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from the test antenna lies on the polarization ellipse and rotates at the wave frequency; that is, the electric vector completes f rotations around the ellipse per second. The LP probe response (rms output voltage) is proportional to the peak projection of the electric field onto the LP orientation line at angle α . This is the distance OP in Figure 10.6 projected from the tangent point T on the ellipse. The locus of points P as the LP probe is rotated is fatter than the ellipse. The exact shape of the curve can be derived using techniques in Chapter 6; see Problem 10 at the end of this chapter. Of course, for a CP antenna both curves in Figure 10.6 are circular. Note that the maximum and minimum of the polarization pattern are identical to the corresponding maximum and minimum of the polarization ellipse (when scaled to the same size). Thus, while the measured polarization pattern does not give the polarization ellipse, it does reveal the axial ratio magnitude of the antenna polarization. It is also obvious from Figure 10.6 that the tilt angle of the ellipse is determined as well. Therefore The axial ratio magnitude, ⎪R⎪, and tilt angle, τ , of the polarization ellipse can be found from the polarization pattern, but not the sense.

Figure 10.6  Polarization pattern (solid curve) of a typical elliptically polarized test antenna. It is the response to an incoming LP wave with orientation angle α . The polarization ellipse for the test antenna polarization state is the dashed curve.

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The sense can be determined by additional measurements. For example, two nominally CP antennas that are identical except for sense can be used as receiving antennas with the test antenna transmitting. The sense of the antenna with the greatest output is the sense of the test antenna. The polarization pattern method in many cases is a practical way to measure antenna polarization. If the test antenna is nearly circularly polarized, the axial ratio is near unity and measured results are insensitive to the purity of the LP probe. If the test antenna is exactly circularly polarized, tilt angle is irrelevant. In the case of a test antenna that is nearly linearly polarized, axial ratio measurement accuracy depends on the quality of the LP probe, which must have an axial ratio much greater than that of the test antenna. A second use of the term polarization pattern (of an antenna) mentioned by the IEEE [14] is the spatial distribution of the electric field vector over the radiation sphere of an antenna when transmitting; Figure 2.6 is an example. But the foregoing discussed definition is the usual meaning and is used here. 10.2.4  The Spinning Linear and Dual-Linear Pattern Methods If the test antenna is not close to linearly polarized, the spinning linear (or rotating source) method provides a rapid measurement technique for determining the axial ratio magnitude as a function of pattern angle. The test antenna is rotated as in a conventional pattern measurement while an LP probe antenna (usually transmitting) is spun. The spin rate of the LP antenna should be such that the test antenna main pattern does not change appreciably during one-half revolution of the LP antenna while the test antenna rotates slowly. Figure 10.7(a) shows an example spinning linear pattern for the case of a helix antenna. Superimposed on the normal antenna pattern are rapid variations representing twice the rotation rate of the probe antenna. For linear valued patterns the axial ratio of the test antenna at the pattern angle is the ratio of the adjacent maxima and minima (or ratio of the average of two maxima and the included minimum). For logarithmic (dB) patterns, as in Figure 10.7, the difference between adjacent minima and maxima gives the axial ratio at that angle. Sense cannot be obtained with this method. Tilt angle could, in theory, be obtained if probe orientation information was known accurately corresponding to pattern points, but this is usually not done in practice [6, pp. 10–13]. A related method is the dual-linear pattern method. In this method two patterns are run for each planar cut (at a fixed angle ϕ ). The patterns are for orthogonal orientations of the LP probe source antenna so that they align with the major and minor axes of the test antenna polarization ellipse. Thus, it is presumed these axes do not change during the patterns. Figure 10.7(b)

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illustrates the resulting patterns for the same example antenna as used in Figure 10.7(a) for the spinning linear method. Of course, the gains and other equipment settings must remain constant during the entire measurement period. Having to locate the major and minor axis orientations and the longer measurement time makes the dual-linear pattern method somewhat inferior to the spinning linear method.

Figure 10.7  Axial ratio measurement as a function of pattern angle for a helix antenna operating at X-band. (a) The spinning linear method; the axial ratio ⎪R⎪ is the difference in decibels between adjacent maxima and minima at each angle. (b) The dual-linear pattern method; axial ratio ⎪R⎪ is the difference in dB between the two patterns at each angle, which are taken in the planes containing the major and minor axes.

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Figure 10.8  Error bounds in measured axial ratio as a function of test antenna axial ratio ⎪Ra⎪ for a few source antenna (wave) axial ratio values ⎪Rw⎪.

Measurement accuracy of axial ratio by these methods is degraded when the LP source antenna is not perfect. Figure 10.8 shows the bounds on the error in axial ratio of the test antenna ⎪Ra⎪ as a function test antenna axial ratio for a few source antenna axial ratios ⎪Rw⎪; see Problem 12 at the end of this chapter. Also, for small test antenna axial ratios accuracy will degrade because of the reliance on resolving small differences in measured powers.

10.3  Complete Polarization State Measurement For measurement of the polarization state of an antenna or wave, the parameters for a polarization state representation must be determined. Chapter 3 presented the parameters for the various polarization state representations. Table 3.1 summarizes the quantities required in each representation. From this table we see that at least two independent parameters are required to completely determine a polarization state. Some representations are directly applicable to a measurement configuration. For example the polarization pattern method yields the tilt angle τ directly and ε from the measured axial ratio; see (3.10). These results match the polarization ellipse representation with parameters ε

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and τ . Three independent parameters are necessary if wave intensity is also measured. The Stokes parameters representation of (3.42) is a case with three independent parameters that contain wave intensity information. Four independent parameters are needed if, in addition, the wave is partially polarized; see Chapter 4. Methods 3 and 4 in Table 10.1 provide the information to completely determine the polarization state. These methods are discussed in this section. 10.3.1 Amplitude-Phase Method The requirement of two independent parameters to completely determine polarization state is met by a receiving system that measures both the amplitude and phase data from two orthogonal linear antennas. This is method 3a in Table 10.1. A measurement that provides both amplitude and phase data makes instantaneous polarization state determination possible. This permits rapid measurement of the complete polarization state as a function of angle during pattern cuts. Amplitude and phase are measured with a polarimeter, which is a receiving system that processes the measured data and supplies the polarization state. Many applications require instantaneous polarization measurements; for example, in radar the target echo must be sampled in the time interval of the arriving radar pulse. Allen and Tompkins [17] describe a hardware configuration for maintaining amplitude and phase accuracy to the intermediate frequency (IF) point in the receiver chain. Figure 10.9 shows a simple model for an amplitude-phase polarimeter. The system yields the relative amplitude and phase of an incoming plane wave normal to the plane of the dipole antennas. The dipoles are identical (usually half-wave dipoles) so they do not introduce an amplitude bias. The antenna pattern of the probes is not a factor. Other antennas can be used, such as horns, as long as they both are aimed at the source. A better approach is to use a single dual-polarized horn such as a quad-ridged horn antenna [18]. The measurement system in Figure 10.9 produces the relative amplitude and relative phase (phase of the y-component relative to that of the x-component) that yield the (γ , δ ) representation parameters directly as follows:



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g = tan−1

E2 (10.2a) E1

( )

d = phase E y − phase ( Ex ) (10.2b)

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Figure 10.9  Model for a polarimeter that measures the relative amplitude and phase of orthogonal linear components of an incoming wave.

The polarization ratio of Section 3.6 also follows directly from this measurement system as



rL = rL ∠d =

E2 ∠d (10.3) E1

It is fortuitous that only relative phase is required because absolute phase can be difficult to measure accurately.

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The receiver outputs can be processed for immediate understanding of the polarization measured. A good way to do this is to show the polarization state on a video display. Figure 3.2 is an example polarimeter display format in which the actual polarization state ellipse is highlighted. The amplitude-phase method can be used with circularly polarized antennas also. In this case EL0 and ER0 (or ER0/EL0 = ⎪ρ C⎪) and relative phase, δ ′, are measured; see (3.64). The receiver is a circular polarization polarimeter. Polarization state parameters based on linear components can be determined from these CP based quantities. We illustrate this by showing how the (ε , τ ) representation is obtained. The axial ratio value follows from (2.40) as



R=

ER0 /EL0 + 1 rC + 1 = (10.4) ER0 /EL0 − 1 rC − 1

The sign on this axial ratio quantity carries the sense information with + corresponding to RH and − to LH. When and ER0 = EL0, (10.4) indicates that R goes to infinity, corresponding to linear polarization as should be the case. Also, when EL0 > ER0 (i.e., the LHCP component is stronger than the RHCP component) R is negative. This, of course, corresponds to a left-hand sensed elliptical polarization. Similarly, when ER0 > EL0 R is positive, indicating RHEP. The tilt angle follows from (2.39) as

t=

d′ (10.5) 2

And the ellipticity angle follows from (2.27) as

e = cot −1(−R) (10.6)

Thus, with relative amplitude and phase measurements using circularly polarized antennas, the complete polarization state of the wave is determined by finding (ε , τ ). A polarimeter for determining linearly polarized components of a wave using CP antennas is also possible and is illustrated in Figure 10.10. The phasor representations of the linear components follow from (3.67) as



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Ex = E1 = EH =

E L + ER (10.7a) 2

E y = E2 e jδ = EV =

j( EL − ER ) (10.7b) 2

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The divide operation shown in Figure 10.10 is performed with a perfect power divider that divides power equally giving a 1/ 2 voltage change from input to output. In practice, the dividers and 90° phase shifters are realized in hardware as a single device called a quadrature hybrid, which is indicated in Figure 10.10 by dashed boxes. Processing of the orthogonal linear state information then proceeds as in Figure 10.9. An interesting feature of this arrangement is that rotation of one of the CP antennas rotates the response tilt angle of each of the resulting LP states by half that rotation angle; this follows from (10.5). The polarimeter of Figure 10.9 that is based on linear polarization component measurements can also be configured to produce circular polarization components. This often a better approach than the polarimeter in Figure 10.10 that uses CP antennas because polarization purity is easier to achieve with LP antennas than with CP antennas. Problem 19 at the end of this chapter illustrates the configuration. The circular polarization phasors are found from the measured linear polarization phasors as

EL =

EH − jEV (10.8a) 2

ER =

EH + jEV (10.8b) 2

which follow from (3.67). Also the polarization ellipse quantities can be found by processing the measured circular polarization phasors EL and ER to find (⎪ρ C⎪, δ ′) or (ε , τ ) as in (10.4) to (10.6).

Figure 10.10  Block diagram of a polarimeter for measurement of linear phasor components using CP antennas. Equal amplitude and phase responses of the antennas are assumed. The dashed box is realized using a quadrature hybrid component.

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Of course, the two antennas in a polarimeter are not perfectly polarized (that is, not very close to perfect dual-linear or dual-circular). This is particularly true for wideband measurement systems because polarization purity may not remain stable over a large bandwidth, especially with CP antennas. Imperfect polarization state responses can be compensated for in hardware by using a polarization adjustment network [5, pp. 90–91]. One realization of a polarization network is to insert an attenuator and a phase shifter in each channel (at RF or IF) of a dual-channel amplitude-phase system that is either nominally HP/VP or LHCP/RHCP. Correction can also be performed with postprocessing software based on the multiple component method. Another method using the amplitude and phase of polarization components is the variable polarization source antenna method [19]. In this technique there is a nulling network between the input co-polarized and cross-polarized ports (connected to a source that can be switched between the ports) and the dual linear terminals of the source antenna. Amplitude and phase adjustments are made to the nulling network to produce a null in the received signal from the test antenna when the cross port of the source is excited, thereby producing a cross-polarized condition. Co-polarization is then produced by exciting the co-port of the source. Finally, we mention that in theory a polarimeter can be made using the complex output voltages from two arbitrary antennas as long as they are not of identical polarization state [20]. Of course, the best accuracy is obtained for orthogonal state antennas. 10.3.2  Multiple Amplitude-Component Methods The required phase information needed in the techniques of the previous section can add greatly to the measurement complexity. A phase and amplitude receiver is more expensive and more complicated than an amplitudeonly receiver. As well, phase can be difficult to measure accurately, leading to accuracy questions in the resulting polarization values. For this reason multiple amplitude-component methods are attractive. Consider an incoming wave from a transmit antenna of unknown polarization. In general, four amplitude measurements using four receive antennas (of the same gain) are needed to determine wave polarization. Comparing the four measurements gives three relative power values [21–23]. Graphical techniques with the Poincaré sphere can be used to find the unknown state [21, 23]. In this section we present simple mathematical formulas for finding an unknown polarization state based on amplitude measurements. These approaches are summarized in Table 10.1 as method 4.

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Method 4a uses the polarization pattern method plus an auxiliary measurement set. In Section 10.2.3 we showed that the polarization pattern method yields the axial ratio, ⎪R⎪, and tilt angle, τ , of the polarization ellipse. The only remaining quantity required is the sense. Sense is found by using two circularly polarized antennas of opposite sense. Suppose these are nominally identical except for sense. Then the sense of the incoming wave will be that associated with the greatest output: ⎧⎪ + sign(R) = ⎨ − ⎩⎪



ER > E L

E L < ER

(10.9)

where ⎪EL⎪ and ⎪ER⎪ are the measured amplitudes with LHCP and RHCP antennas. This is an intuitive result; the sense of the measured wave will be that of the receiving antenna with the greatest received signal. The resulting sense information along with ⎪R⎪, and τ completely determine the polarization. The approach can be visualized with the Poincaré sphere. Ellipticity angle ε is found from (10.6) using ⎪R⎪, except for its sign. Thus, angles (2τ , 2ε ) place the polarization state on the Poincaré sphere of Figures 3.4 and 3.5 in one of two places corresponding to the two possible signs of ε , one in the upper hemisphere (+) and one in the lower hemisphere (−). The sign determination of (10.9) resolves this ambiguity and then (10.6) yields a unique value. Note that (10.4) is consistent with (10.9). For example, when ⎪ER⎪ = ER0 > ⎪EL⎪ = EL0 (10.4) gives a positive R value, which is consistent with (10.9). The polarization state of a wave can also be determined from the six amplitude measurements listed in method 4b in Table 10.1 and illustrated in Figure 10.11. The electric field values are the amplitude responses to the six antenna polarizations shown. We assume that all antennas have the same gains and the wave in each case is normally incident. From the three amplitude pairs the following ratios are formed: rL =

E2 E1

ratio of vertical to horizontal LP components (10.10a)

rD =

E4 E3

ratio of 135° to 45° LP components (10.10b)

rC =

ER0 EL0

ratio of right-hand to left-hand CP components (10.10c)

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Figure 10.11  The six polarization components measured with the multiple-component method. The components are 1: horizontal, 2: vertical, 3: 45° linear, 4: 135° linear, 5: left circular, and 6: right circular.

Each pair of measurements gives the total power in the wave:

2 2 + ER0 (10.11) E12 + E22 = E32 + E42 = EL0

Intrinsic impedance in (3.22) was omitted here for simplicity. We define the following intermediate quantities from the measured powers:



E22 2 E12 1 − rL (10.12a) YL = = E22 1 + rL 2 1+ 2 E1



E42 2 E32 1 − rD YD = = (10.12b) E42 1 + rD 2 1+ 2 E3

1−

1−

It can be shown that (see Problem 10.11)

YL = cosg (10.13a)



YD = sin2g cosd (10.13b)

Forming the ratio of these



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YD sin2g cosd = = tan2g cosd = tan2t (10.14) YL cos2g

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Polarization Measurements307

where the last of these follows from (3.2). Thus,



Y 1 t = tan−1 D (10.15) 2 YL

The axial ratio R is found from (10.4) using ⎪ρ C⎪ and then we can find ε from (10.6) as ε = cot–1(−R). Therefore, the polarization state using (ε , τ ) has been determined using the multiple components in Figure 10.11. Other complete polarization state representations of the incoming wave can be found from the multiple component method. First, the polarization ratio for the CP representation requires ρ C and δ ′. The quantity ⎪ρ C⎪ is found from measured circular polarization components using (10.10c) and δ ′ follows from δ ′ = 2τ using (2.39) and (10.15). Thus the circular polarization ratio representation ρ C = ⎪ρ C⎪/δ ′ is determined. The polarization ellipse representation (γ , δ ) follows from (10.13a) using the following:





g =

1 cos−1 YL (10.16a) 2

⎛ Y ⎞ d = cos−1 ⎜ D ⎟ (10.16b) ⎝ sin2g ⎠ ⎪⎧ + E L0 > ER0 (10.16c) sign(d) = ⎨ − EL0 < ER0 ⎩⎪

The sense of the polarization is given by sign (δ ) and is the sense of the larger of the circular component amplitude. The linear complex polarization ratio representation follows from (10.10a) and (10.16) that give ⎪ρ L⎪ and δ ; see (3.54). Summarizing, we have shown how to form several complete polarization state representations based on the six measured amplitude quantities in Figure 10.11. These methods rely on the availability of high-quality polarized antennas from which three relative quantities are formed; see (10.10). 10.3.3  Measurement of the Polarization of Large Antennas Antennas that are physically large can be difficult to measure because they might not fit into the available measurement facility. Antennas that are electrically

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large (many wavelengths in diameter) also present difficulty due to the need to illuminate the test antenna with a wave as close to planar as possible. In a conventional far-field antenna range the required measurement distance of 2D2/λ in (10.1) will be physically large, which makes the range difficult to build. Large indoor ranges are expensive because of the volume required and amount of absorber needed to eliminate multipath waves arriving at the test antenna. Outdoor ranges suffer from multipath interference due to ground reflections and have issues with frequency use permissions and weather. Some specialized indoor facilities can accommodate large antennas. The near-field range at Lockheed Martin Space System Company in Denver is an example. The 26m × 26m planar scanner is the largest scanner in the world by scan area. The facility was developed jointly in 1978 with Allen Newell of NIST (then NBS) to demonstrate that planar near field scanning is the most efficient method for evaluation of electrically large aperture antennas precisely (Neill Kefauver, Lockheed Martin Space System Company, private communication). It is mounted in the ceiling of a large building to allow for measuring mesh reflectors in a cup-up configuration to give symmetric gravity conditions. A test at 60 GHz showed that axial ratio measurements can be made to better than 0.2 dB accuracy. The range has also characterized antennas from 400 MHz to 183 GHz and gains exceeding 60 dB. The necessary long path lengths can be obtained using satellites or cosmic radio sources. Cosmic radio sources present two difficulties. First, a radiometric receiver is required. Second, the signal from cosmic radio sources can be randomly polarized or partially polarized and thus is unsuitable for highaccuracy polarization measurements. The advantage of the radiometer method over a satellite-based source is that it can be difficult to find a satellite source at the needed frequency. Satellites are excellent sources for direct polarization measurements if an appropriate signal is transmitted from the satellite such as a narrowband beacon signal on an acceptable frequency. Ideally, the wave should arrive at the Earth terminal site with the test antenna in a pure LP or CP polarization state. However, often the satellite signal itself is not purely polarized and the intervening medium can introduce depolarization. Methods are available to correct for an impure illuminating wave for both circularly polarized [24] and linearly polarized [25] systems. These methods rely on measuring the polarization of the wave arriving from the satellite with an auxiliary antenna that is much smaller and of higher polarization purity than the test antenna; this restriction can be relaxed for LP antennas [26]. The wave polarization state can be measured using an adaptive polarization network similar to the one in Figure 8.11(c). This network is adjusted by rotating the two polarizer angles

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Polarization Measurements309

θ 1 and θ 2 to produce a null on one port of the OMT. Then the other port is matched to the wave and the wave state can be uniquely solved for [25]. This can be done for both the auxiliary and test antennas. The wave polarization is determined from the auxiliary antenna and used to solve for the test antenna characteristics. Then a radiation pattern can be measured from the satellite corrected to give an effectively pure polarization illumination. For dual-circularly polarized systems, such as satellite ground stations, the axial ratios of the dual-polarized ground station(s) and the source (satellite) wave axial ratio can be determined [27]. The maximum and minimum isolations between the co-polarized and cross-polarized channels of the two receive antennas are measured (found by rotating each antenna). These four measured values are used to calculate the axial ratios of the wave and both receive antennas. Alternatively, one dual-polarized ground station can be used if phase is measured.

10.4  Measurement of Partially Polarized Waves Measurement of completely polarized waves was discussed in the previous sections in this chapter where we saw that a minimum of two independent parameters are required to fully characterize a completely polarized wave state. If wave intensity is included, three parameters are required. The measurement of partially polarized waves requires one additional independent parameter, the degree of polarization. These principles guide our discussion of the measurement of partially polarized waves in this section. Characterization of a partially polarized wave through measurement is most easily performed by rotating a linearly polarized receiving antenna about an axis in the direction of the incoming wave. The response will be maximum when the antenna (i.e., its linear polarization angle) is parallel to the major axis of the polarization ellipse of the completely polarized part of the wave; this power output is P‖. The output power for the orthogonal orientation of the receive antenna, P⟘, is the minimum response. The degree of linear polarization is then [28, p. 126]



dL =

P! − P⊥ PLP = (10.17) PT P! + P⊥

PT is the total power in the partially polarized wave; that is, it includes both the power in the completely polarized and unpolarized portions of the wave and is given by PT = S av Ae from (6.2). As a note of caution with terminology,

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degree of polarization is sometimes used to mean the degree of linear polarization in a completely polarized wave [6, pp. 3–28; 29]. The numerator in (10.17) represents the power in the linearly polarized portion of the partially polarized wave. If the wave is unpolarized, P ‖ = P⟘ and then (10.17) gives dL = 0, as it should. If the wave is completely polarized and is CP, then P ‖ = P⟘ and dL = 0, as it should. If the wave is partially polarized with the polarized part being CP, again P ‖ = P⟘ and dL = 0, wherein now P ‖ and P⟘ both contain the same amount of unpolarized wave power as well as equal polarized power levels. There is a similar relation using measured the power outputs from two identical, opposite-sense CP antennas, PL and PR . They determine the degree of circular polarization as [28, p. 126]



dC =

P − PR PCP = L (10.18) PT PL + PR

Again the denominator is the total power in the wave, S av = I. The numerator gives the circularly polarized power. If the wave is unpolarized, then PL = PR and (10.18) gives dC = 0, as it should. If the wave is completely polarized and LP (or partially polarized with the polarized part linearly polarized), then PL = PR and dC = 0. Measurements of the degree of linear and degree of circular polarization give the total information necessary for determining the degree of polarization. We can develop this beginning with examining the following relation to give the degree of linear polarization, which is based on (4.13) and (4.14):

d L = s12 + s22 = d cos2e (10.19)

This equals unity for LP (ε = 0); see (3.48). It is zero for CP (ε = 45°), proving that it does indeed represent only the linear portion of the polarized part of the wave. The degree of circular polarization also follows from (4.13) and is

dC = s3 = d sin2e (10.20)

For a wave with only CP content (ε = 45°) in its polarized part, this reduces to dC = d; see (3.49). For a wave with the polarized part LP (ε = 0), dC = 0 as it should. Combining (10.19) and (10.20), we see that

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d = d L2 + dC2 (10.21)

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Polarization Measurements311



This gives the degree of polarization of an arbitrary wave in terms of its degrees of linear and circular polarization. For waves whose completely polarized parts are purely linear (or purely circular), d = dL (or dC). For a completely polarized wave d =1 and (10.21) reduces to

1 = d L2 + dC2

completely polarized wave (10.22)

The Stokes parameters of an arbitrary wave can be determined directly from power measurements. To do this we use three pairs of antennas, all of the same effective aperture (or account is taken for differences) and with the following polarizations: horizontal and vertical (x and y); +45° and −45° LP (x′ and y′); and left-and right-hand CP (L and R). First consider HP and VP antennas (or equivalently one linearly polarized antenna rotated to the two positions). The received measured powers: Px = P! = Ae Sx (t) (10.23a)



Py = P⊥ = Ae S y (t) (10.23b)

From (4.5b) and (10.23)



s1 =

Sx (t) − S y (t) Px − Py S1 (10.24) = = Sav Px + Py Sx (t) + S y (t)

Next, as the s1 parameter gives the fraction of power in the HP and VP polarization, s2 and s3 give the power fractions in ±45° LP and in CP; see Table 3.2. Thus [28, p. 128]





s2 = s3 =

Px ′ − Py ′ Px + Py

(10.25)

PL − PR (10.26) Px + Py

The total wave power can be found in three ways:

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PT = Px + Py = Px ′ + Py ′ = PL + PR (10.27)

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Polarization in Electromagnetic Systems

Six measurements are used with this technique and therefore some redundancy is present. This provides a means to verify results through satisfaction of (10.27). However, the measurements can be reduced to only four; for example the x, x′, L, and R polarizations. In this case, the Stokes parameters are [28, p. 128]





s1 =

2Px − PL − PR (10.28a) PL + PR

s2 =

2Px ′ − PL − PR (10.28b) PL + PR

s3 =



PL − PR (10.28c) PL + PR

Another measurement method is to measure the amplitude and phase of Ex and Ey and perform the operations of (4.5) [30, p. 510]. This, however, requires the measurement of phase.

10.5  Antenna Gain Measurement The gain of an antenna is the ratio of the radiation intensity in a given direction to the radiation intensity that would be obtained if the power accepted by the antenna was radiated isotropically [14]: G(q,f) =



U (q,f) (10.29) U iso

where [2, p. 51] U(θ , ϕ ) = radiation efficiency representing power lost on the antenna structure 1 2 = power density (at distance r) i r 2 = E(q,f) r 2 2h U iso =

P 4p

P = power delivered to the antenna from the transmitter

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Polarization Measurements313



1 = er

2p p

∫ ∫U (q,f)sinqdqdf

0 0 er = radiation efficiency (0≤ er ≤ 1), representing loss on the antenna structure

A more useful form of the gain expression is G(q,f) = er



2

4p F (q,f) (10.30) ΩA

where ΩA = beam solid angle; also see (5.2) =

2p p

∫ ∫ F (q,f)

2

sinqdqdf

F(θ , ϕ ) = radiation pattern normalized to unity maximum in which 0 0



E(q,f) =

Eo F (q,f) r

Gain is a power gain that quantifies how much larger is the power density in W/m2 (or equivalently, radiation intensity in watts per solid angle) relative to the power obtained by distributing the power uniformly over the radiation sphere. Gain G(θ , ϕ ) is a function of angle around the antenna. The peak value G is often simply referred as gain. In the specifications of an antenna if a single value is given for gain, it is safe to assume this is maximum gain. Thus, when G is used without angle agruments it is customarily taken to be the maximum gain (that is for ⎪F⎪ = 1). Hence,



G = er

4p ΩA

maximum gain (10.31)

Beam solid angle, ΩA, is a measure of how narrowly the radiated power density is restricted in angular extent. As (10.31) indicates, gain varies inversely with beam solid angle. That is, the more directive an antenna is, the narrower is its radiation pattern (ΩA smaller) and the greater the gain. Gain could be obtained by making many radiation pattern cuts with time-stable instrumentation, say θ -cuts that give F(θ , ϕ n) for several ϕ n values.

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Then beam solid angle could be obtained by numerical integration, and the gain evaluated using (10.30). Many software packages that accompany antenna range equipment perform this calculation to give an approximate gain value. The most common gain measurement method is the gain comparison (or gain transfer) method, which is a relative measurement relying on a standard gain antenna of known gain Gs over its operating frequency range. The gain of the test antenna (often referred to as the antenna under test, or AUT) is given by where

Gt (dB) = Pt (dBm) − Ps (dBm) + Gs (dB) (10.32)

Gt(dB) = (maximum) gain of the test antenna in dB Gs(dB) = known (maximum) gain of the standard gain antenna in dB Pt(dBm) = power received with the test antenna in dBm Ps(dBm) = power received with the standard gain antenna in dBm. This result is derived in [2, p. 572], but is a very intuitive result not requiring derivation. The term Pt(dBm) − Ps(dBm) in (10.32) is the increase in received power of the test antenna over that of the standard gain antenna. This equals gain in decibels over that of the standard gain antenna. For example, if Pt(dBm) = Ps(dBm), then Gt(dB) = Gs(dB) and the gain of the test antenna equals that of the standard gain antenna. Figure 10.12 illustrates the measurement setup. A transmitter of fixed power output is connected to a suitable source antenna that is aimed so that its pattern is peaked in the direction of the receiving antenna location. Received power is measured with the test antenna in the receiving location and then

Figure 10.12  Measurement of the gain of a test antenna Gt using the gain comparison method based on the known gain Gs of the standard gain antenna.

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Polarization Measurements315

replaced with the standard gain antenna; in both cases the receiving antenna is aimed at the transmitter. The distance between the source and receiving antennas is not changed, nor are the losses or gains in the instrumentation. In practice the inaccuracies of power level measurements can be eliminated using a calibrated variable attenuator. The attenuator is set to produce the same power level for the two receiving antennas. The difference in the attenuation settings is the gain value. This is the RF substitution method. Often the role of transmitter and receiver are reversed with a valuable attenuator on the transmitting side adjusted to keep the receiving power constant. So far we have not mentioned polarization in our gain discussion. For the idealized situation in Figure 10.12 we assume that all antennas are matched in polarization, as well as matched in impedance. Mismatch losses are not included in the definition of antenna gain and are accounted for separately. However, a gain value is sometimes given relative to a perfect polarization state, such as pure linear or pure circular. Gain measurement of a CP antenna is less straight-forward that for an LP antenna and we give it special attention below. 10.5.1  Gain Measurement of Linearly Polarized Antennas High-quality polarization (i.e., close to perfect) is not difficult to achieve with several types of linearly polarized antennas; LP antennas are discussed in Chapter 5. Therefore, accurate measurements can be made as shown in Figure 10.12 as long as the antennas all aligned in polarization. An example is all three antennas shown of linear polarization with their major axes in the horizontal plane. 10.5.2  Gain Measurement of Circularly Polarized Antennas If a good-quality CP source antenna and standard gain antenna are available, the gain comparison method of Figure 10.12 can be used. But achieving good polarization purity is more difficult with CP antennas than LP antennas. That is, circularly polarized standard gain antennas are not common and therefore high-quality LP antennas are used to measure the gain of near-CP antennas (or more generally, elliptically polarized antennas). This is done using two orthogonal LP antennas or a single LP antenna used in two orthogonal orientations. Suppose the gains are measured for vertical and horizontal LP cases; that is, two measurements are made with the setup in Figure 10.12 for the source antenna in vertical and horizontal orientations. All antennas should be of good LP purity, except of course for the test antenna. The measured

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gains are the partial gains Gtv and Gth. These partial gains combine to give the total gain [5, p. 100]:

Gt (dB) = 10 log (Gtv + Gth )

[dBic] (10.33)

The partial gains in this equation are power ratios not decibels. This is referred to as the partial gain method. Any perpendicular orientations can be used because the power in an elliptically polarized wave is contained in the sum of any two orthogonal components. A CP antenna actually performs this sum instantaneously. Therefore, the gain in (10.33) is relative to an ideal CP antenna. The unit dBic is used for gain relative to a perfect CP isotropic antenna. Example 10.1  Calculation of Gain Using the Partial Gain Method

Consider the dual-linear patterns in Figure 10.13. The test antenna is a cavitybacked spiral antenna, as in Figure 5.11(b), operating at 1,054 MHz. Also shown is the pattern of an LP standard gain horn. The gain from the manufacturer’s gain versus frequency curve is 14.15 dB at 1054 MHz. The power levels associated with the peaks of the patterns taken in vertical and horizontal polarizations are 16.1 and 13.25 dB, respectively, below the standard gain pattern at θ = 0°. Thus, the partial gains are Gtv(dB) = 14.15 − 16.1 = −1.95 dB Gth(dB) = 14.15 − 13.25 = 0.9 dB Converting the dB values to power ratios: Gtv = 10 –1.95/10 = 0.64 Gth = 100.9/10 = 1.23 The final gain value for the test antenna using (10.33) is Gt(dB) = 10 log(0.64 + 1.23) = 2.71 dBic A low-gain CP test antenna was used to illustrate that negative decibel partial gains are possible. If the LP standard gain antenna has an axial ratio of 40 dB or better, it will not contribute to errors in the test antenna gain value. However, if the source antenna is not of high polarization purity the accuracy of a gain measurement will be reduced. The error in gain contributed by a source antenna

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Polarization Measurements317

Figure 10.13  Measured patterns of a cavity-backed spiral antenna at 1,054 MHz using vertically and horizontally polarized source antennas. The pattern of standard gain horn in place of the test antenna is also shown and is used in Example 10.1 to calculate gain.

increases as its axial ratio decreases. The error in gain is plotted in Figure 10.14 as a function of the source antenna axial ratio; see Problem 13 at the end of this chapter. Here the test antenna is taken to be perfect CP and the gain standard is assumed to be perfect LP. These are worst-case error bounds; a test antenna with imperfect axial ratio has slightly smaller error bounds. Errors for a standard gain antenna with low axial ratio (i.e., not close to pure LP) are available in [31]. 10.5.3  Absolute Gain Measurement The gain measurements discussed so far are relative and depend on an antenna of known gain. Absolute gain measurements can also be made that do not depend on knowing the gain of any antennas used. The three-antenna method uses measurements of received power for all three combinations of three antennas to solve for the gain of each antenna [5, p. 96].

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Figure 10.14  Worst-case error in the gain of a perfect CP test antenna using the partial gain method and a nominally LP source antenna with the axial ratio value given on the abscissa.

10.6  Measurements on Handsets and Other Small Devices Handheld devices are unique in that high directivity and high polarization purity are not goals, and in fact are not wanted. This is because a handheld device in normal use can be positioned in almost any orientation and is still expected to perform adequately. Thus, the pattern should be very broad. As well, the polarization should not be pure such as vertical linear because it could become cross-polarized to the base station polarization when oriented horizontally. Another factor is that the device is often located in a high-clutter environment that will depolarize its radiation anyway. These facts ease the antenna design problem because a compact device often will have an electrically small antenna, which naturally has a broad pattern and low polarization purity. As wireless devices become smaller and more integrated, the antenna is no longer a separate discrete device but is instead integral to the whole device. This necessitates making in situ measurements; that is, measurements are made on the device with the antenna in place. During the development phase, traditional antenna measurements (impedance, gain, pattern, and polarization) are made on the antenna when it is out of the device. These are often referred to as passive measurements. Measurements are also made on the antenna alone while the antenna is in the device, but this requires attaching a cable,

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Polarization Measurements319



which can affect the results. Thus, there is heavy reliance on measurements of the performance of the device as a whole (i.e., in situ), rather than just the antenna. These measurements are often referred to as active measurements and will be the focus of this section. Active measurements for wireless devices are called over the air (OTA) performance testing. Certification measurement procedures for OTA testing are available from the CTIA [32]. Here we define and introduce the key figureof-merit quantities in OTA testing; see [33] for details. The first parameter is effective isotropic radiated power (EIRP), which is the amount of power (in units of watts or dBm) emitted from an isotropic antenna needed to obtain the same power density in the direction of the antenna pattern peak [2, p. 110]. EIRP is the product of the antenna gain and the power accepted by the antenna from the transmitter, PG. EIRP can also be a function of angle around the antenna, as we will be using here. More commonly used for compact wireless devices is the quantity total radiated power (TRP). TRP is the total power radiated in all directions around the device when it is transmitting, relative to an isotropic antenna of the same input power. It is found by summing (using simulation or measured data) the EIRP all around the device in both orthogonal polarizations: TRP =

1 4p

2p p

∫ ∫ ⎡⎣ EIRPq (q,f) + EIRPf (q,f)⎤⎦ sinqdqdf (10.34) 0 0

where EIRPθ and EIRPϕ are θ - and ϕ -polarizations of EIRP. Of course, measurements are usually made at discrete angles and the integrals become summations. A second measurement quantity used to characterize wireless devices is total isotropic sensitivity (TIS). TIS is a sensitivity, which means it is the lowest power level transmitted by the base station required to give acceptable receive device performance averaged over all space around the device. Common performance parameters are frame error rate or bit error rate. Of course, a lower sensitivity value is desired, indicating the receiver can detect a weaker signal. It is found by averaging sensitivity of the receiving device to a single incident wave when the angles of arrival are uniformly distributed [34]. Equivalent isotropic sensitivity (EIS) with units of watts is the TIS reduced by the antenna gain in order to reference the measurement to an ideal isotropic antenna and is used to find TIS: TIS =

6765_Book.indb 319

2p p

4p

1 1 ⎡ ⎤ ∫ ∫ ⎢ EISq (q,f) + EISf (q,f) ⎥ sinqdqdf ⎦ 0 0⎣

(10.35)

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Polarization in Electromagnetic Systems

Again, both polarizations are included. The terms inside the integral have inverse EIS values to give low-sensitivity component values weight in the integration, which in turn leads to lower TIS or better performance. TRP and TIS together provide a measure of the device effectiveness at the maximum range from the base station while meeting the performance requirements. The OTA quantities are measured in either a reverberation chamber or an anechoic chamber. Measurements are repeated for many angles of arrival at the test device. While the anechoic chamber reduces reflections to low levels the reverberation chamber creates multipath components with many angles of arrival similar to that encountered with the terminal device in a room of a building. The reverberation chamber is a room with metallic walls and a mode stirrer with moving metallic blades to introduce reflections.

10.7  Problems



1. Plot in polar form the (voltage) response of a rotating LP probe antenna as a function of angle α (see Figure 10.6) to an incoming LP wave along the x-axis. 2. Prove that the phase of a linear component of a CP wave varies in a one-to-one fashion with angular rotation. Do this by assuming a LHCP wave illuminates an LP receiving antenna with orientation angle α relative to horizontal. 3. Demonstrate that rotation of one of the CP antennas in the polarimeter of Figure 10.10 by angle ϕ rotates the response LP states by half the same amount. 4. Prove that the following relation is satisfied, showing that the total power in a wave is found by summing any orthogonal component intensities; see (10.11). 2 2 + ER0 = E12 + E22 EL0



5. Find ε and τ for each of the parameter sets below and name the polarization state of the wave:

E1 = 1, E2 = 0, E3 = (a) E1 = (b)

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1 1 1 1 , E4 = , EL0 = , ER0 = 2 2 2 2

1 1 1 1 ,E = , E = 1, E4 = 0, EL0 = ,E = 2 2 2 3 2 R0 2

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Polarization Measurements321



E1 = 0, E2 = 1, E3 = (c)

1 1 1 1 , E4 = , EL0 = , ER0 = 2 2 2 2

(d) ⎪ρ L⎪ = 0.668, ⎪ρ D⎪ = 0.450, ⎪ρ C⎪ = 0.466 E1 = (e)





1 1 1 1 , E2 = , E3 = , E4 = , E = 0, ER0 = 1 2 2 2 2 L0

6. Repeat Problem 5 giving the γ and δ angles for each state. 7. The gain of an elliptically polarized test antenna is measured with a linearly polarized standard gain antenna first for vertical polarization and then for horizontal polarization, giving partial gains of 9 and 8 dB, respectively. (a) Evaluate the gain of the test antenna relative to CP. (b) Make a statement about the possible values of the test antenna axial ratio magnitude. 8. Prove (10.21) using (10.17) and (10.18). 9. The Stokes parameter entries given in (10.24) to (10.26) are in terms of measured powers. Reexpress them in terms of polarization efficiencies. 10. An LP probe antenna, such as a dipole, is rotated about an axis parallel to the direction of propagation of an incoming elliptically polarized wave. (a) Write an expression for the dipole output voltage magnitude as a function of the orientation angle of the dipole, α ; see Figure 10.6. Assume τ w = 0°. (b) Polar plot one quadrant of the antenna response for a CP wave. (c) Repeat (b) for an elliptically polarized wave of 3-dB axial ratio. 11. Derive (10.13) by finding the appropriate electric field components using (6.68) and the wave polarization expressed in the form of (3.41). 12. Axial ratio error obtained when measuring a test antenna of axial ratio magnitude ⎪Ra⎪ using an imperfect rotating LP antenna of axial ratio magnitude ⎪Rw⎪ was discussed in Section 10.2 and plotted in Figure 10.8. (a) Derive the following formula for the measured axial ratio: Rm =

Ra Rw ± 1 Ra ± Rw

where plus (minus) is used for same (opposite) sense.

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Polarization in Electromagnetic Systems

(b) Evaluate the axial ratio error limits (measured axial ratio divided by the test antenna axial ratio) in decibels for a source antenna of 20-dB axial ratio and a test antenna of 2-dB axial ratio. 13. The gain error encountered in the partial gain method of Section 10.5.2 for measuring the gain of a test antenna with axial ratio ⎪Ra⎪ can be derived for a source antenna which is not perfect LP and with axial ratio ⎪Rw⎪. (a) Derive the fractional gain error expression (ratio of measured to actual gain):

( Ra Error =

(R

a



)2 (

Rw ± 1 + Ra ± Rw 2

)(

2

)

)2

+ 1 Rw + 1

where the plus (minus) signs are used for same (opposite) senses. (b) For a pure CP test antenna evaluate the measured gain error limits for source antenna axial ratios of 20, 30, and 40 dB. Compare to Figure 10.14.

References [1]

Rodriguez, V., “Basic Rules for Indoor Anechoic Chamber Design,” IEEE Ant. and Prop. Mag., Vol. 58, Dec. 2016, pp. 82–93.

[2]

Stutzman, W. L., and G. A. Thiele, Antenna Theory and Design, Third Edition, Hoboken, NJ: Wiley, 2013.

[3]

Evans, G. E., Antenna Measurement Techniques, Norwood, MA: Artech House, 1990.

[4]

Parini, C., S. Gregson, J. McCormick, and D. van Rensburg, Theory and Practice of Modern Antenna Range Measurements, Herts, UK: IET, 2014.

[5]

IEEE Standard Test Procedures for Antennas, IEEE Standard 149-1979, IEEE, 1979.

[6]

Hollis, J. S., T. Lyon, and L. Clayton (eds.), Microwave Antenna Measurements, ScientificAtlanta, 1970. Available from MI-Technologies.

[7]

Arai, H., Measurement of Mobile Antenna Systems, Norwood, MA: Artech House, 2001.

[8]

Hemming, L. H., Electromagnetic Anechoic Chambers: A Fundamental Design and Specification Guide, New York: IEEE Press, 2002.

[9]

Slater, D., Near-Field Antenna Measurements, Norwood, MA: Artech House, 1991.

[10] Hansen, J. E. (ed.), Spherical Near-Field Antenna Measurements, London: IET and Peter Peregrinus Ltd., 1988.

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[11] Gregson, S., J. McCormick, and C. Parini, Principles of Planar Near-Field Antenna Measurements, Herts, UK: IET, 2007. [12] Microwave Vision Group, www.mvg-world.com. [13] NSI-MI Technologies, www.nsi-mi.com. [14] IEEE Standard Definitions of Terms for Antennas, IEEE Standard145-2013, 2013. [15] Ludwig, A. C., “The Definition of Cross Polarization,” IEEE Trans. on Ant. and Prop., Vol. AP-21, Jan. 1973, pp. 116–119. [16] Jacobson, J., “On the Cross Polarization of Asymmetric Reflector Antennas for Satellite Applications,” IEEE Trans. on Ant. and Prop., Vol. AP-25, March 1977, pp. 276–283. [17] Allen, P. J., and R. D. Tompkins, “An Instantaneous Microwave Polarimeter,” Proc. IRE, Vol. 47, July 1979, pp. 1231–1237. [18] Volakis, J. (ed.), Antenna Engineering Handbook, Fourth Edition, New York: McGrawHill, 2007, Ch. 14, p. 62. [19] Bodnar, D. G., “Polarization Characteristics of a Monopulse Tracking Feed, Microwaves, Vol. 27, Dec., 1984, pp. 123–136. [20] Lee, S-W, E. Okubo, and H. Ling, “Polarization Determination Utilizing Two Arbitrarily Polarized Antennas,” IEEE Trans. on Ant. and Prop., Vol. 36, May 1988, pp. 720–723. [21] Deschamps, G. A., “Geometrical Representation of the Polarization of a Plane Wave, Proc. IRE, Vol. 39, May 1951, pp. 540–544. [22] Clayton, L., and S. Hollis, “Antenna Polarization Analysis by Amplitude Measurement of Multiple Components,” Microwave Journal, Vol. 8, Jan. 1965, pp. 35–41. [23] Knittel, G. H., “The Polarization Sphere as a Graphical Aid in Determining the Polarization of an Antenna by Amplitude Measurements Only,” IEEE Trans. on Ant. and Prop., Vol. AP-15, March 1967, pp. 217–221. [24] DiFonzo, D. F., “The Measurement of Earth Station Depolarization Using Satellite Signal Sources,” COMSAT Laboratories Tech. Memo CL-42-75, 1975. [25] Keen, K. M., and A. K. Brown, “Techniques for Measurement of the Cross-Polarization Radiation Patterns of Linearly Polarized, Polarization-Diversity Satellite GroundStation Antennas,” Proc. of IEE, Part H, Vol. 129, June 1982, pp. 103–108. [26] Keen, K. M., “Ground Station Antenna Crosspolarisation Measurements with an Imperfectly Polarized Ancillary Antenna,” Electronics Letters, Vol. 18, 14 Oct. 1982, pp. 924–926. [27] Stutzman, W. L., and W. P. Overstreet, “Axial Ratio Measurements of Dual Circularly Polarized Antennas,” Microwave J., Vol. 24, Oct. 1981, pp. 75–78. [28] Kraus, J. D., Radio Astronomy, New York: McGraw-Hill, 1966. [29] Beckmann, P., The Depolarization of Electromagnetic Waves, Boulder, CO: Golem Press, 1968, p. 30.

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[30] Ishimaru, A., Electromagnetic Wave Propagation, Radiation, and Scattering, Engelwood, NJ: Prentice-Hall, 1991, p. 510. [31] Parekh, S., “The Measurement Column–Uncertainties in Gain Measurements of a Circularly Polarized Test Antenna,” IEEE Ant. and Prop. Magazine, Vol. 32, April 1990, pp. 41–44. [32] www.ctia.org/initiatives/certification. [33] Zhang, Z., Antenna Design for Mobile Devices, Singapore: Wiley, 2011, pp. 215–226. [34] Hussain, A., P.- S. Kildal, and A. Glazunov, “Interpreting the Total Isotropic Sensitivity and Diversity Gain of LTE-enabled Wireless Devices from Over the Air Throughput Measurements in Reverberation Chambers,” IEEE Access, Vol. 3, 2015, pp. 131–145.

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Appendix A Frequency Bands A.1 Radio Bands Band

Frequencies

ELF

30 to 300 Hz

VF

300 to 3,000 Hz

VLF

3 to 30 kHz

LF

30 to 300 kHz

MF

300 to 3,000 kHz

HF

3 to 30 MHz

VHF

30 to 300 MHz

UHF

300 to 3,000 MHz

SHF

3 to 30 GHz

EHF

30 to 300 GHz

A.2 Microwave Bands Band

Frequencies (GHz)

L

1 to 2

S

2 to 4

C

4 to 8

X

8 to 12

Ku

12 to 18

K

18 to 27

Ka

27 to 40

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Appendix B Useful Mathematical Relations B.1  Unit Vector Representations

ˆ ˆ (B.1) xˆ = ˆr sinqcosf + θcosqcosf − φsinf



ˆ ˆ (B.2) yˆ = ˆr sinqsinf + θcosqsinf + φcosf



ˆ (B.3) ˆz = ˆr cos q − θsinq



ˆr = xˆ sinqcosf + yˆ sinqsinf + ˆz cosq (B.4)



θˆ = xˆ cosqcosf + yˆ cosqsinf − ˆz sinq (B.5)



φˆ = − xˆ sinf + yˆ cosf (B.6)

B.2 Trigonometric Relations



e ± ja = cosa ± jsina (B.7a) cosa =

e ja + e − ja (B.7b) 2

sina =

e ja − e − ja (B.7c) 2j 327

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sin(a ± b) = sina cos b ∓ cosa sin b (B.8)



cos(a ± b) = cosa cos b ∓ sina sin b (B.9)



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( ) ( )

sin

p ± a = cosa (B.10) 2

cos

p ± a = ∓sina (B.11) 2

sin(p ± a) = ∓sina cos(p ± a) = −cosa (B.12) sina cos b =

1 [ sin(a + b) + sin(a − b) ] (B.13) 2

cosa sin b =

1 [ sin(a + b) − sin(a − b) ] (B.14) 2

cosa cos b =

1 [ cos(a + b) + cos(a − b) ] (B.15) 2

sina sin b = −

1 [ cos(a + b) − cos(a − b) ] (B.16) 2

sina + sin b = 2sin

a+b a−b cos (B.17) 2 2

sina − sin b = 2cos

a+b a−b sin (B.18) 2 2

cosa + cos b = 2cos

a+b a−b cos (B.19) 2 2

cosa − cos b = −2sin

a+b a−b sin (B.20) 2 2

a a sina = 2sin cos (B.21) 2 2

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Useful Mathematical Relations329



sin2a = 2sina cosa (B.22)



cosa = 2cos2

a a − 1 = 1 − 2sin2 (B.23) 2 2



cos2a = 2cos2 a − 1 = cos2 a − sin2 a = 1 − 2sin2 a (B.24)



cos2 a + sin2 a = 1 (B.25)



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cota =

1 cosa (B.26) = tana sina

cot −1 x =

p − tan−1 x (B.27) 2

1 + tan2 a = sec2 a (B.28)

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List of Symbols aco

co-polarized port of an antenna, nominally orthogonal to acr acr cross-polarized port of an antenna, nominally orthogonal to aco A attenuation B fractional bandwidth Bp percent bandwidth Br ratio bandwidth BW bandwidth [Hz] co co-polarized wave state (orthogonal to cr) CPR cross-polarization ratio CPR for linear polarization CPR L CPR for circular polarization CPRC CPR (dB) cross-polarization ratio in dB CP or C circularly polarized cr cross-polarized wave state (orthogonal to co) [C] path coupling matrix degree of polarization d degree of circular polarization, for a partially dC polarized wave degree of linear polarization, for a partially dL polarized wave 331

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Polarization in Electromagnetic Systems

D! ! , D!⊥ , D⊥ ! , D⊥ ⊥ ! E ! E(t) ! Ed ! Ei ! Er ! Er ! Ew ! ! Eco , Ecr

polarization coefficients phasor electric field intensity vector [V/m] instantaneous electric field intensity vector [V/m] electric field exiting a depolarizing medium [V/m] incident electric field [V/m] reflected electric field [V/m] received electric field [V/m] vector-phasor electric field intensity of wave w [V/m]



co-polarized and cross-polarized electric field components [V/m]

Ex, Ey E1, E2

complex valued x- and y-linear components [V/m] x- and y-directed linear component amplitudes (real valued) [V/m] 45° slant linear component amplitudes (real valued) E 3 , E 4 [V/m] circular components (complex valued) [V/m] EL , ER circular component amplitudes (real valued) [V/m] EL0, ER0 electric field components parallel/perpendicular to the E‖, E⟘ plane of incidence [V/m] EP elliptical polarization normalized polarization vector, also called ê polarization vector radiation efficiency (0 ≤ er ≤ 1) er f frequency [Hz] upper, center, and lower frequency of an operating band f U, fC , f L [Hz] ! phasor magnetic field vector [A/m] H ! instantaneous magnetic field vector [A/m] H (t) HP or H horizontally polarized HPBW half-power beamwidth of an antenna pattern [deg]

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List of Symbols333

! vector effective length of an antenna [m] h I isolation (= XPI) INT interference power [W] L path length [m] L subscript indicating left-hand sense effective path length through rain [m] Leff Lp polarization loss LP or L linearly polarized left-hand sensed LH or L left-hand sensed circular polarization LHCP left-hand elliptical polarization LHEP polarization efficiency, also called polarization p mismatch factor polarization efficiency for a completely polarized wave pc polarization efficiency of the co-polarized component of pco a wave polarization efficiency of the cross-polarized component pcr of a wave P power output of an antenna [W] power received by an antenna [W] Pr power received by a test antenna [W] Pt power received by a standard gain antenna [W] Ps perfect electrical conductor PEC depolarization factor, also called differential q propagation factor impedance mismatch factor q right-hand sense RH or R right-hand circular polarization RHCP RHEP right-hand elliptical polarization R axial ratio axial ratio in dB R(dB) axial ratio of the co-polarized component of a wave Rco axial ratio of the cross-polarized component of a wave Rcr

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RCS radar cross section [R] rotation matrix S time-average Poynting vector power density (flux density) for a monochromatic wave S 0, S1, S2, S3 Stokes parameters [W/m2] time-average Poynting vector power density (flux Sav density) for a quasi-monochromatic, partially polarized wave; time averaging is performed over the coherency time [W/m2] maximum radiated power density [W/m2] Smax Δt coherency time [S] T period of an electromagnetic wave = 1/f [s] TEC total electron content [electrons/m2] V complex voltage [V/m] complex voltage out of receiver channel i due to Vij excitation of transmitter channel j [V/m] normalized complex voltage resulting from a wave v(w, a) of polarization state w incident on an antenna of polarization state a vertically polarized VP or V w wave polarization state wave polarization state exactly orthogonal to wave wo state w wave polarization state from a channel nominally cross w x polarized to wave state w XPD cross-polarization discrimination XPD (dB) cross-polarization discrimination in dB XPI cross-polarization isolation XPI (dB) cross-polarization isolation in dB Greek symbols

α β γ

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attenuation constant [nepers/m] phase constant [deg/m] polarization angle [deg]

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List of Symbols335



Γ! ! , Γ!⊥ , Γ⊥ ! , Γ⊥ ⊥

reflection coefficients

δ ε ε ε η θ θ θ ρ L ρ C σ σ τ Δτ co, Δτ cr Δτ ϕ ψ ΩA

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phase angle of an electric field [deg] ellipticity angle [deg] elevation angle [deg] permittivity [deg] intrinsic impedance [Ohms] spherical polar angle [deg] incidence angle [deg] tilt angle of raindrops [deg] polarization ratio for linear polarization polarization ratio for circular polarization conductivity [S/m] radar cross section [m2] tilt angle [deg] relative tilt angle [deg]; Δτ co = τ co − τ w, Δτ cr = τ cr − τ w Faraday rotation angle [deg] plane polar angle [deg] grazing angle [deg] beam solid angle [steradians]

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About the Author Warren L. Stutzman received a B.S. degree in electrical engineering and a B.A. degree in mathematics from the University of Illinois in 1964, and M.S. and Ph.D. degrees in electrical engineering from Ohio State University in 1965 and 1969. Dr. Stutzman has been in the Electrical and Computer Engineering Department at Virginia Tech since 1969 and is currently Emeritus Professor. He founded the Virginia Tech Antenna Group and was Interim Head of the ECE Department twice. His research activities include antennas for wireless systems, polarization theory, reflector antennas, array design, and atmospheric effects on Earth-space communication links. In addition to authoring Polarization in Electromagnetic Systems, he is coauthor with Gary A. Thiele of the textbook Antenna Theory and Design, John Wiley, 1981, 1998, 2013. He is a Life Fellow of the Institute of Electrical and Electronics Engineers (IEEE) and served as president of the IEEE Antennas and Propagation Society in 1992. Dr. Stutzman has won the Wheeler best applications paper in the IEEE Transactions on Antennas and Propagation twice, and was awarded the Third Millennium Medal from the IEEE in 2000. He is a Distinguished Alumnus of the University of Illinois ECE Department.

337

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Index

Antenna polarization, 97–128 antenna pattern types and, 102–4 co-polarization, 104–8 cross-polarization, 104–8 defined, 101 as key operating characteristic, 100 principles, 101–8 receiving case, 101 See also Polarization Antennas aperture, 101, 113–14 array, 101 bandwidth, 100 big wheel, 111–12 block diagram, 97 broadband, 101, 123–25 directional, 112–23 electrically small, 101 halo, 112 horn, 114 ideal dipole, 155 inverted-L (ILA), 111 isotropic, 104 log-periodic toothed trapezoidal wedge, 124 microstrip (MSA), 111, 112 monopole, 109–11 null-free, 103–4

Absolute gain measurement, 317–18 Access points, 275 Active remote sensing, 249 Adaptive antennas, 282 Adaptive systems, 245–49 Amplitude-phase method, 300–304 defined, 301 ellipticity angle, 302 illustrated, 301 polarimeter block diagram, 303 receiver outputs, 302 See also Polarization state measurement Angle diversity, 278 Angles of incidence, 213–14 Antenna impedance, 100 Antenna pattern measurements compact range, 291 co-polarized, 293–95 cross-polarized, 293–95 dual-compact range, 291 illustrated, 289 near-field range and, 290 polarization pattern, 295–97 principles, 289–99 spinning linear and dual-linear methods, 297–99 techniques, 289–93 See also Polarization measurements 339

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340

Polarization in Electromagnetic Systems

Antennas (Continued) omnidirectional, 109–12 performance parameters, 98 planar inverted-F (PIFA), 111 quadrifilar helix, 269 reflector, 114–18, 122 resonant, 101 short dipole, 19, 20, 103, 155 slot, 114 smart, 282 stub-loaded helix, 125 turnstile, 118–20, 157 types of, 101 use of, 97 vector effective length of, 155–61 wideband, 101 Antenna temperature, 258 Antenna-wave interaction, 131–65 normalized complex output voltage, 161–63 polarization efficiency, 131–36 polarization efficiency calculation, 136–55 power equations, 132 vector effective length, 155–61 Aperture antennas, 101, 113–14 Arbitrary polarization, 34–35 Arrays colinear, 109 defined, 101 log-periodic dipole (LPDA), 123–24 phased, 101 Yagi-Uda, 112 Atmospheric noise, 86 Attenuation from coupling matrix entries, 242 differential, 233, 247 through rain, 235 XPD versus, 236 Axial ratio circularly polarized waves and, 37 cross-polarization discrimination (XPD) and, 188

6765_Book.indb 340

defined, 25 as directly measurable quantity, 143 finding, 63–64 magnitude, 30 magnitude, from polarization ratio, 63 measurement accuracy of, 299 measurement illustration, 298 orthogonal linear components, 127 polarization efficiency using, 143–49 polarization ratio and, 66 for receiving antenna, 147 Stokes parameters and, 58 Backscatter, 252 Bandwidth array, 101 defined, 100 functional, 100 gain, 100 impedance, 100 percent, 100 polarization, 100 ratio, 100 Base stations comparison of diversity techniques, 278–81 defined, 274 diversity gain measured using receiving antennas, 281 experimental, diagram of, 279 polarization diversity at, 274–76 Beam solid angle, 99 Beam squint, 123 Big wheel antennas, 111–12 Bistatic radar, 250 Blackbody law, 84 Blackbody radiation defined, 84–85 noise, 85 physics of, 84 Brewster angle, 222 Broadband antennas circularly polarized, 125

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Index341



defined, 101, 123 linearly polarized, 123–25 Canting angle, 228 Carrier-to-noise (C/N) ratio, 195 Characteristic polarizations, 207 Circularly polarized antennas broadband, 125 gain measurement of, 315–16 omnidirectional, 112 Circularly polarized directional antennas, 112–18 methods, 118 Type 1, 118, 121 Type 2, 118–21 Circularly polarized waves axial ratio and, 37 components, 35 contra-rotating components, 35 decomposition into components, 37 equal-amplitude, 37 reflection coefficients for, 223 Circular polarization in clear line-of-sight applications, 268–69 defined, 25 degree of, 310 in low-/medium-Earth orbit satellite systems, 269 near-dual, 186–92 polarization ratio for, 64–66 purity, 119, 125–27 right-handed, 68–69 Stokes parameters and, 58 in wireless communication, 268–70 Clutter, 255 Coherence time, 271 Coherency matrix, 59 Coherency time, 88 Coherent waves, 87 Colinear arrays, 109 Compact range, 291 Complex conjugate, 254

6765_Book.indb 341

Complex polarization factor, 60 Complex voltage, 156 Co-polarization decomposition of polarization efficiency and, 154–55 defined, 105 reference polarization orientation and, 108 unit vectors, 108 Co-polarized patterns, 105, 293–95 Co-polarized state, 171 Cross-polarization attenuation through ran and, 235 decomposition of polarization efficiency and, 154–55 defined, 105 isolation, 241 levels, 106 reference polarization orientation and, 108 unit vectors, 108 Cross-polarization discrimination (XPD) attenuation versus, 236 axial ratio and, 188 of balanced dual-linearly polarized receiving antenna, 186 balanced receiving antenna, 185 calculation of, 182–92 CP case, 185 defined, 178–79 as directly measurable, 179 expression, 183 general dual polarization, 184–86 ideal dual-linearly polarized antenna, 183–84 illustrated, 177 maximum and minimum values, 191–92 near-dual-circular polarization, 186–92 for perfect circularly polarized wave, 189

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342

Polarization in Electromagnetic Systems

Cross-polarization discrimination (XPD) (Continued) range of values for general situation, 188 tilt angle and, 187 transmitting antenna, 194 values, bracketing, 187 values for wave incident of axial ratio, 190 values for wave known axial ratio, 191 with various incident wave axial ratios, 192 worst-case, 194 Cross-polarization isolation (XPI) defined, 177 denoting, 180 expression in decibels, 178 illustrated, 177 Cross-polarization ratio (CPR), 170–76 calculation, 172 defined, 172 inverse for elliptical wave, 184 of linearly polarized wave, 174 linear polarization for, 173 in terms of CP components, 176 of wave relative to circular co-polarized state, 175 Cross-polarized patterns, 105, 106, 293–95 Cross talk, 194 Cumulative distribution function (CDF), 278, 280 Decomposition of linearly polarized wave, 173 polarization efficiency (co-polarized and cross-polarized), 154–55 polarization efficiency (unpolarized and completely polarized), 152–54 of waves, 32–38 Degree of circular polarization, 310 Degree of polarization, 86–87, 90 Delay spread, 269

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Depolarization by Faraday rotation, 242–45 general theory of, 203–10 at interfaces, 211–25 introduced in scattering process, 205 properties, 210 rain, 230–31 by scatterer, general solution, 206 Depolarization compensation defined, 245 DPS network realization, 249 dynamic, 245 networks, 247–48 principle of, 247 restoration method, 248 simple scheme, 246 static, 245 Depolarization factor, 209–10, 229 Depolarized field, 226 Depolarizing matrix, 205, 227, 228, 229 Depolarizing media change in direction and, 203–4 characteristic polarizations, 209 defined, 203 depolarization at interfaces, 211–25 dual-polarized system with, 226 principles of, 203 properties, 210 reciprocity and, 210 symmetric, 210 three different polarizations and, 209 wave propagation through, 229 Dielectric media, 214–15 Differential attenuation, 233, 247 Differential phase shift, 233 Diplexers, 196 Dipole antennas ideal, 155 short, 19–20, 103, 155 vector effective length, 156 Directional antennas circularly polarized, 112–18 defined, 112

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Index343

linearly polarized, 112–18 Directivity, 99 Distributed radar targets composition of, 256 defined, 256 Stokes parameters and, 257 Diversity angle, 278 combining, 274 frequency, 272–73 pattern, 273 polarization, 168, 170, 273, 274–82 principles, 270–72 spatial, 169–70, 272, 273 time, 273 types of, 272–73 in wireless communications, 270–74 Diversity gain, 278–79 Doppler radar, 250 Dual-compact range, 291 Dual composition technique, 180–82 Dual CP Earth station antenna, 192 Dual-linear pattern method, 297–99 Dual polarization, 170, 184–86 Dual-polarized systems, 8–9, 167–201 cross-polarization discrimination (XPD) and, 178–80, 182–92 cross-polarization isolation (XPI) and, 177–79 cross-polarization ratio (CPR) and, 170–76 cross talk, 193 defined, 167 dual decomposition and, 180–92 frequency reuse and, 167–68 introduction to, 167–70 isolation, 192–96 orthogonally polarized antennas and, 69 performance evaluation of, 192–96 polarization (wireless communication), 270 polarization control devices, 196–200

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Dual-polarized systems (depolarizing medium), 226–45 antenna effects in calculations, 237–42 depolarization caused by Faraday rotation and, 242–45 general formulation, 226 rain on radio path, 230–37 Dynamic depolarization compensation, 245 Effective aperture, 133 Effective isotropic radiated power (EIRP), 319 Effective path length, 235 Electrically small antennas, 101 Electric fields complex-valued components, 161 cross-polarized component, 195 decomposed into rectangular components, 238 direction of, 13 for incident and scattered waves, 208 perpendicular, 12 phasor intensity, 50 of plane waves, 16, 18 time-varying, 12, 27 vector magnitude, 54 vector motion, 28 Electromagnetics in frequency domain, 6–7 in time domain, 6 Electromagnetic system generating subsystem, 7 overview illustration, 7 propagation medium, 7–8 receiving subsystem, 8 Electromagnetic waves characteristics, 3 composition of, 11–12 principles of, 11–38 Elevation angle, 230 Elliptical polarization, 23 Ellipticity angle, 30, 302

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Polarization in Electromagnetic Systems

E-plane, 21, 294 E-plane pattern, 21 Equal-gain combining, 274 Equivalent isotropic sensitivity (EIS), 319 Fading causes of, 270–71 fast, 271 multipath, 271–72 slow, 271 Faraday rotation angle in ionosphere, 243, 245 defined, 243 depolarization caused by, 242–45 linearly polarized wave undergoing, 244 Far-field distance, 12 Far-field/near-field range combination, 292 Far-field range, 288 Fast fading, 271 Flux density, 132 Frequency, wavelength and, 15 Frequency bands, 325 Frequency diversity, 272–73 Frequency reuse defined, 167 illustrated, 168 use of, 168 Fresnel, Augustine Jean, 4 Functional bandwidth, 100 Gain defined, 99 diversity, 278–79 expression form, 313 as power gain, 313 ratio, 312 standard antenna, 314 Gain bandwidth, 100 Gain comparison (gain transfer) method, 314 Gain measurement

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absolute, 317–18 of circularly polarized antennas, 315–16 comparison method, 314 illustrated, 314 of linearly polarized antennas, 315 obtaining, 313–14 overview, 312–13 partial gain method, 316–17, 318 three-antenna method, 317 Generating subsystem, 7 Global Navigation Satellite Systems (GNSSs), 269 Global Positioning System (GPS), 269 Gray body, 84 Grazing angle, 224 Grazing incidence, 221, 224 Ground permittivity, 221 polarization of waves reflected from, 225 reflection from, 220–25 relative permittivity, 223 Half-power beamwidth (HPBW), 99 Halo antennas, 112 Handsets, measurements on, 318–20 Harmonic waves, 87 Hertz, Heinrich, 5 Horizontal polarization (HP) defined, 32 linear, representations for, 67–68 Horn antennas, 114 H-plane, 21, 294 H-plane pattern, 21 Huygens source, 117 Identity matrix, 230 Impedance bandwidth, 100 Impedance mismatch factor, 135 Interface circularly polarized components and, 216–17

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Index345

dielectric media on both sides of, 214–15 polarization effects, 211–18 reflection matrix electric field formulation, 217 Intrinsic impedance, 213 Inverted-L antennas (ILA), 111 Isolation calculation of, 193–96 channel, with imperfect antennas, 242 from coupling matrix entries, 242 cross-polarization, 241 defined, 180, 192 degradation cause by imperfect antennas, 193 for imperfect transmit and receive antennas, 195–96 path effects on, 192–93 Isotropic antennas, 104 Isotropic pattern, 103 Large antennas difficulty, 308 for dual circularly polarized systems, 309 polarization measurement of, 307–9 Law of reflection, 211 Left-hand circularly polarized (LHCP) wave, 32, 33–34 arbitrary vector, 34 incident wave, 224 superposition of, 35 Linearly polarized antennas broadband, 123–25 directional, 112–18 gain measurement of, 315 omnidirectional, 109–12 Linear polarization for cross-polarization ratio (CPR), 173 defined, 28, 32 depolarizing matrix for, 227 horizontal, 67–68 illustrated, 22

6765_Book.indb 345

polarization ratio for, 59–64 Line-of-sight (LOS) systems, 266 Log-periodic dipole array (LPDA), 123–24 Log-periodic toothed trapezoidal wedge antenna, 124 Lossy dielectrics, 215, 230 Low-profile omnidirectional antennas, 111 Ludwig chart, 146, 147, 148–49 Ludwig’s third definition, 108, 294 Magnetic fields perpendicular, 12 of plane waves, 18 y-directed, 13 Man-made noise, 86 Mathematical relations, 327–29 Maximum effective aperture, 133 Maximum-ratio combining, 274 Maxwell, James Clerk, 4 Maxwell’s equations, 4, 5, 11 Microstrip antennas (MSA), 111, 112 Microwave bands, 325 Monochromatic waves, 87 Monopole antennas, 109–11 Monostatic radar, 249–50 Multipath fading, 271–72 Multiple amplitude-components methods defined, 304 polarization components measured with, 306 ratios from amplitude pairs, 305 summary, 307 See also Polarization state measurement Multiple-input/multiple-output (MIMO), 283 Multipolarized radar, 251, 257 Multipolarized systems, 170 Near-dual-circular polarization, 186–92 Near-field range, 290 Next-generation radar (NEXRAD), 257

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Polarization in Electromagnetic Systems

90° polarizers, 196–97 Noise temperature, 258 Non-depolarizing medium, 206–7, 215 Non-line-of-sight (NLOS) systems, 267 Normal incidence, 221 Normalized complex voltage calculation for RHCP wave, 162–63 defined, 161 polarization efficiency from, 162 received, 162 Normalized polarization vector. See Polarization vectors Normalized Stokes parameters, 56, 61–62 Normal mode helix antenna (NMHA), 111 Null-free antennas, 103–4 Ohm’s law, 13 Omnidirectional antennas, 109–12 circularly polarized, 112 defined, 103 illustrated, 110 linearly polarized, 109–12 low-profile, 111 Omnidirectional patterns, 21 180° polarizers, 198 Organization, this book, 6–10 Orthogonality, 72 Orthogonal polarization states defined, 32 determination of, 69–80 illustrated, 72 parameters for common pairs, 71 on Poincaré sphere, 75 polarization ellipse, 71–75 polarization ratio for, 79–80 representations for, 70 Stokes parameters for, 78–79 Orthogonal polarization vector, 75–78 Orthomode transducers (OMTs), 196, 199 Output voltage

6765_Book.indb 346

expression for incoming wave, 161 normalized complex antenna, 161–63 receiving antenna, 158 Over the air (OTA) performance testing, 319 Partial gain method, 316–17, 318 Partially polarized waves, 83–93 characterization through measurement, 309 defined, 83 degree of polarization and, 86–87 occurrence of, 86 Poincaré sphere representation, 91–92 polarization efficiency for, 152–54 polarization measurements, 309–12 Stokes parameters representation for, 87–91 total power density and, 88 Passive remote sensing, 257–58 Path coupling matrix, 239 Pattern diversity, 273 Percent bandwidth, 100 Perfect electric conductor (PEC), 213, 219–20 Phase angles, 28, 243–44 Phased arrays, 101 Phase shifts, 14, 225, 233 Phasors circular component, 65 electric field intensity, 50 Planar inverted-F antennas (PIFA), 111 Planck’s law, 84, 85 Plane of incidence, 204 Plane of polarization, 19 Plane of scatterer, 204 Plane waves, 11–17 defined, 12 direction of polarization, 12–13 electric field of, 16 electric fields of, 18 incident on infinite planar interface, 212

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magnetic fields of, 18 motion, 14, 15 Poincaré sphere, 42, 48–50 angles, 49 defined, 48 graphical techniques with, 304 horizontal linear polarization, 67 illustrated, 48 location of polarization states on, 49 orthogonal states on, 75, 138 of partially polarized waves, 91–92 polarization efficiency evaluation using, 137–39 relationship of points on, 61 right-hand circular polarization, 68 Point radar targets defined, 252 polarimetric scattering from metallic sphere, 254–56 scattering matrix, 252–54 Polarimeter, for linear phase component measurement, 303 Polarimetric radar, 251–57 defined, 251 distributed radar targets, 256–57 point radar targets, 252–54 polarization response from, 256 scattering from metallic sphere, 254–56 Polarization antenna, 9, 97–128 arbitrary, 34–35 basics, 3–6 brief history of, 4–5 characteristic, 207 circular, 25, 58, 64–66 defined, 4 degree of, 86–87, 90 dual, 170, 184–86 elliptical, 23 horizontal, 32 importance of, 6 linear, 22, 28, 32, 59–64

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Index347 orientation sensitivity and, 5 plane of, 19 in radar, 249–57 in radiometry, 257–59 in system design, 4 vertical, 32 wave, principles, 11–38 Polarization (wireless communication), 9, 265–84 circular polarization advantages and, 268–70 dual-polarized systems, 270 future directions in, 282–83 introduction to, 265–66 line-of-sight (LOS) systems, 266 millimeter-wave licensed bands and, 282–83 non-line-of-sight (NLOS) systems, 267 overview, 266–67 polarization diversity and, 274–82 single-polarized systems, 267–68 system principles, 266–70 Polarization adjustment network, 304 Polarization agile radar, 251 Polarization bandwidth, 100 Polarization coefficients, 205, 206, 207, 209 Polarization control devices diplexers, 196 orthomode transducers (OMTs), 196, 199 polarization grids, 199–200 polarizers, 196–98 Polarization diversity antenna illustration, 276 at base stations, 274–76 comparison of base station diversity techniques, 278–81 comparison of terminal diversity techniques, 281–82 defined, 168, 273 illustrated, 170

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348

Polarization in Electromagnetic Systems

Polarization diversity (Continued) performance comparison to other diversity types, 277–82 at terminals, 276–77 Polarization efficiency, 131–36 for antenna ports, 181 with axial ratios, 143–49 calculation of, 136–55 decomposition of (co-polarized and cross-polarized), 154–55 decomposition of (unpolarized and completely polarized), 152–54 defined, 132 effective length and, 159 evaluating using Poincaré sphere, 137–39 examples, 136 from normalized complex voltage, 162 for partially polarized waves, 152–54 with polarization ellipse quantities, 140–42 with polarization ratios, 149–50 with polarization vectors, 150–52 as power ratio, 161 of receiving antenna, 139 with Stokes parameters, 140 values, 134, 135 values as function of degree of polarization, 139 Polarization ellipse, 24, 43–47 amplitudes, 31 box containing, 45 derivation, 43 horizontal linear polarization, 67 illustrated, 29 orthogonal state for, 71–75 in polarization efficiency evaluation, 140–42 polarization ratio and, 62 polarization vector and, 53–54 range of possible states, 44, 45 representations, 43 right-hand circular polarization, 68

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shape and orientation, 30 Polarization grids common application for, 200 defined, 199 electric fields for incident and scattered waves, 208 as linear-to-circular polarizer, 200 Polarization loss defined, 136 finding, 146 Ludwig chart and, 146, 147, 148–49 values, 147 Polarization measurements, 9, 287–322 antenna gain, 312–18 antenna pattern, 289–99 facility, 288 far-field range, 288 handsets and small devices, 318–20 introduction to, 287–89 methods, 293 partially polarized waves, 309–12 polarization state, 299–309 requirements, 288 Polarization mismatch factor. See Polarization efficiency Polarization pattern defined, 295 illustrated, 296 maximum and minimum, 296 measurement, 295–97 Polarization purity of circularly polarized antennas, 125–27 in dual-polarized base station antenna, 276 as function of frequency, 125 phase errors, 127 turnstile antenna and, 127 Polarization ratio, 59–66 assumption for linear polarization, 66 axial ratio and, 66 for circular polarization, 64–66 as compact representation, 59

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defined, 59 example polarization states in, 60 horizontal linear polarization, 67–68 left-hand circular polarization, 80 for linear polarization, 59–64 magnitude, 62 normalized Stokes parameters and, 62 for orthogonal polarization states, 79–80 plane, 60 polarization efficiency and, 149–50 polarization ellipse and, 62 polarization vector and, 62 right-hand circular polarization, 69 Polarization response, 255 Polarization signature, 255 Polarization state measurement amplitude-phase method, 300–304 complete, 299–309 large antennas, 307–9 multiple amplitude-components methods, 304–7 parameter determination, 299–300 Polarization states decomposition of waves, 37 defined, 25 horizontal, 32–33 intensity, 52 location on Poincaré sphere, 49 orthogonal, 32, 69–80 in polarization ratio, 60 quantifying, 27 representation examples, 66–69 representations, 33, 41–80 scattered, 206 vertical, 32–33 of wave, determining, 305 Polarization vectors, 50–55 circular polarization, 77 defined, 53 horizontal linear polarization, 67 linear component, 77 orthogonal, 75–78

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Index349 polarization efficiency using, 150–52 polarization ellipse parameters and, 53–54 polarization ratio and, 62 representation, 50 right-hand circular polarization, 68 Polarized waves circularly polarized, 26, 27 concept and visualization of, 17–27 horizontally, 18, 19 linearly, 17, 18 partially, 83–93 vertically polarized, 18, 19 Polarizers defined, 196 90°, 196–97 180°, 198 Polarizing angle, 222 Polychromatic waves, 87 Power pattern, 98 Poynting vector, 16–17 complex-valued, 51 time-average, 52 Propagation medium, 7–8 depolarization properties of, 237 effects on polarized wave, 9, 227 Quadrifilar helix antennas, 269 Quasi-monochromatic waves, 87 Radar basics, 249–51 bistatic, 250 defined, 249 Doppler, 250 monostatic, 249–50 multipolarized, 251, 257 next-generation (NEXRAD), 257 polarimetric, 251–57 polarization agile, 251 polarization in, 249–57 Radar cross section (RCS), 250, 253, 255 Radar equation, 250

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350

Polarization in Electromagnetic Systems

Radiation efficiency, 99 Radiation pattern, 21, 97–98 Radio astronomy, 258 Radio bands, 325 Radiometers applications, 259 basics, 257–59 defined, 86, 257 passive remote sensing, 257–58 Radiometry, polarization in, 257–59 Rain attenuation through, 235 as lossy dielectric, 230 on radio path, 230–37 Rain depolarization defined, 230–31 illustrated, 232 processes, 231 on terrestrial link, 234–37 Raindrops, 232, 234 Randomly polarized waves. See Unpolarized waves Ratio bandwidth, 100 Rayleigh-Jeans law, 85 Rays, 19 Receiving subsystem, 8 Reflected wave, 210 Reflection from the ground, 220–25 from a plane, perfect conductor, 219–20 process, 219 Reflection coefficients, 212, 213, 221, 224 for circularly polarized waves, 223 for soil at UHF, 223–25 Reflector antennas defined, 114 Huygens source and, 117 ideal, 117 offset, 118, 122 parabolic, 115–17 radiation properties, 114–17

6765_Book.indb 350

Refracted wave, 210 Relative tilt angle, 141 Resonant antennas, 101 Restoration method, 248 Reverberation chamber, 320 Right-hand circularly polarized (RHCP) wave, 32, 33–34 arbitrary vector, 34 incident wave, 224 superposition of, 35 Right-hand circular polarization representations, 68–69 Satellite Digital Audio Radio System (SDARS), 269 Scattered polarization state, 206 Scattered waves, 211 Scatterers defined, 204 depolarization by, 206 of reflecting plane, 205 Scattering matrix, 252–54 Scattering medium, 207 Scattering model, 204–5 Scatterometer, 249 Selection combining, 274 Short dipole antennas as center-fed straight wire antenna, 155 defined, 19 illustrated, 20 as linearly polarized, 103 Side lobes, 98 Single-polarized systems with clear-line-of-sight, 267 polarization in wireless communication, 267–68 two-way terrestrial communications, 268 Slot antennas, 114 Slow fading, 271 Smart antennas, 282 Snell’s law, 211

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Index351

Spatial diversity base station technique comparison, 278 defined, 169, 273 equipment, 169 illustrated, 170 measured data, 272 Specular reflection, 211 Spherical coordinate system, 107 Spinning linear method, 297 Static depolarization compensation, 245 Stokes, George, 55 Stokes matrix, 78 Stokes parameters, 55–59 of arbitrary waves, 311 axial ratio and, 58 circular polarization and, 58 defined, 55 determination of, 57 distributed radar targets and, 257 horizontal linear polarization, 67 interpretation for, 57, 58, 90 normalized, 56, 61–62 for orthogonal state, 78–79 for partially polarized waves, 87–91 polarization efficiency using, 140 as projections, 57 as real valued, 56 right-hand circular polarization, 69 tilt angle and, 58 written as matrix, 55–56 Stub-loaded helix antenna, 125 Switched combining, 274 Symbols, list of, 331–35 Symmetric depolarizing media, 210 Synthetic aperture radar (SAR), 250 Terminals comparison of diversity techniques, 281–82 polarization diversity at, 276–77 Thermal noise, 85 Three-antenna method, 317

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Tilt angle cross-polarization discrimination (XPD) and, 187 defined, 21 of exiting wave, 234 of linearly polarized input electric field vector, 234 Stokes parameters and, 58 Time diversity, 273 Time-harmonic wave, 87 Total electron content (TEC), 243 Total isotropic sensitivity (TIS), 319 Total power, 310, 311 Total power density, 88 Total radiated power (TRP), 319 Transmission coefficients, 212, 213 Transmitted waves, 211 Transverse waves, 4, 12 Trigonometric relations, 327–29 Turnstile antenna bidirectional pattern, 120 defined, 118 illustrated, 119 normalized radiation pattern, 120 polarization purity and, 127 RHCP, 157 trend, 119 Type 1 circularly polarized antennas, 118, 121 Type 2 circularly polarized antennas, 118–21 Ultraviolet catastrophe, 85 Unit vector representations, 327 Unpolarized waves blackbody radiator, 83–84 defined, 83 generation of, 89 Vector effective length, 155–61 defined, 155 maximum, of small dipole, 156 polarization efficiency and, 159

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352

Polarization in Electromagnetic Systems

received power and, 160 Vertical polarization (VP), 32 Wavelength defined, 14 frequency and, 15 Wave motion, 14, 15 Wave velocity, 15 White body, 84

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Wideband antennas, 101 Wireless communications diversity in, 270–74 polarization in, 9 See also Polarization (wireless communication) Yagi-Uda array, 112

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