Microwave NDT

DEVELOPMENTS IN ELECTROMAGNETIC THEORY AND APPLICATIONS

VOLUME 10

Editor

G. F. Roach, Strathclyde University. Department of Mathematics, UK

The titles published in this series are listed at the end of this volume.

Microwave NDT by

N.Ida Department of Electrical Engineering, Vniversity of Akron, Ohio, V.SA.

SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

Library of Congress Cataloging-in-Publication Data

Ida, Nathan. Mtcrowave NOT / by N. Ida.

p. cm. -- (Oevelopments tn electromagnettc theory and appltcattons : 10l

Inc 1 udes index. ISBN 978-94-010-5215-3 ISBN 978-94-011-2739-4 (eBook) DOI 10.1007/978-94-011-2739-4 1. Non-destructtve testtng. 2. Mtcrowave measurements.

1. Title. II. Sertes. TA417.2.133 1992 620.1' 127--dc20 92-33611

ISBN 978-94-010-5215-3

Printed on acid-free paper

AII Rights Reserved © 1992 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1992 Softcover reprint ofthe hardcover Ist edition 1992 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means. electronic or mechanical. inc1uding photocopying. recording or by any information storage and retrieval system. without written permission from the copyright owner.

Contents

Preface ................................................................................... xiii

Introduction ............................................................................ . 1. The Microwave Domain ................................................................ 1 2. Historical ................................................................................. 3 3. Advantages and Disadvantages of Microwaves for Testing ........................ 4 4. Energy Associated with Microwaves . ... . . .. . . ... . .. . . .. . . . . . . .. .. ... . .. . . .. . ... . . ... 6 5. Properties of Fields at High Frequencies ............................................. 7 6. A Note on References and Bibliography ............................................. 8 7. References ................................................................................ 9

Part I. Electromagnetic Field Theory

Chapter 1. The Electromagnetic Field Equations and Theoretical Aspects ..................................................................................... 10

1.1. Introduction:The Electromagnetic Field Equations .... ......... ...... ........ ..... 10 1.2. Maxwell's Equations in Differential Form ......................................... 11

1.2.1. The Time Harmonic Equations .............................................. 13 1.2.2. The Source Free Equations .. ......... ....... ......... .... ....... .... ........ 14

1.3. Maxwell's Equations in Integral Form ....... ...... .... .................... ... ..... 14 1.4. Material Properties and Constitutive Relations .................................... 16

1.4.1. Conductivity ................................................................... 17 1.4.2. Complex Permittivity........ .... ... ................ ........... ...... .... ..... 17 1.4.3. Complex Permeability ........................................................ 18 1.4.4. Anisotropic Materials ......................................................... 20

1.5. The Poynting Theorem and Energy ............................................... 20 1.5.1. The Complex Poynting Vector .............................................. 22

1.6. Potential Functions ....................... '" ................ " . . .. . .. . ... .. .. . . ... ... 24 1.6.1. The Electric Scalar Potential ................................................. 24 1.6.2. The Magnetic Scalar Potential ............................................... 25 1.6.3. The Magnetic Vector Potential ............................................... 25

1. 7. The Field Equations in Terms of Potential Functions ............................ 27

vi

1.7.1. Vector Potentials .............................................................. 27 1.7.2. Scalar Potentials ............................................................... 28 1.7.3. Gage Conditions ............................................................. 29

1.8. The General, Time Dependent Wave Equation .................................... 30 1.8.1. The Time-Harmonic Wave Equation ....................................... 32 1.8.2. The Helmholtz Equations .................................................... 33

1.9. Propagation of Waves: Plane Waves in Lossless Dielectrics ................. 33 1.10. Propagation of Plane Waves in Lossy Media .................................... 38

1.10.1. Losses in Materials .......................................................... 38 1.10.2. Propagation of Waves in Lossy Dielectrics .............................. 39 1.10.3. Propagation of Waves in Low Loss Dielectrics .......................... 42 1.1004. Propagation of Waves in Conductors ..................................... 43

1.11. Electromagnetic Boundary Conditions ............................................ 44 1.11.1. Interface Conditions for the Electric Field ............................. 45 1.11.2. Interface Conditions for The Magnetic Field ............................. 46 1.11.3. Interface Conditions Between Two Lossless Dielectrics ............... 50 LIlA. Interface Between a Dielectric and a Conductor ......................... 50 1.11.5. Other Interface Conditions ................................................. 50

1.12. Bibliography ......................................................................... 52

Chapter 2. Transmission Lines, Waveguides, and Resonant CavItIes ..................................................................................... 54

2.1. Transmission Lines ................................................................... 54 2.1.1. The Lossless Transmission Line ............................................ 59 2.1.2. Reflection on Transmission Lines and the Reflection Coefficient ....... 60 2.1.3. The Transmission Coefficient ............................................... 62 2.1.4. Power Relations in a Lossy Transmission Line ........................... 62 2.1.5. Standing Waves on Transmission Lines .................................. 63 2.1.6. Line Impedance ............................................................... 64 2.1.7. Impedance Matching .......................................................... 66 2.1.8. Stub Matching ................................................................. 67 2.1.9. Quarter Wavelength Transformer Matching .............................. 69

2.2. Waveguides............................................................................ 70 2.2.1. TM and TE Modes in Waveguides ........................................ 73 2.2.2. Rectangular Waveguides ..................................................... 75 2.2.3. TM Modes in Rectangular Waveguides .................................... 76 2.204. TE Modes in Rectangular Waveguides ..................................... 79 2.2.5. Cylindrical Waveguides ...................................................... 81 2.2.6. TM Modes in Cylindrical Waveguides ..................................... 82 2.2.7. TE Modes in Cylindrical Waveguides ...................................... 84

2.3. Cavity Resonators .................................................................... 86 2.3.1. TM and TE Modes in Cavity Resonators .................................. 87 2.3.2. TE Modes in a Rectangular Cavity Resonator ............................. 88 2.3.3. Cylindrical Cavity Resonators............................................... 89

vii

2.3.4. Energy in a Cavity Resonator ............................................... 90 2.3.5. Quality Factor of a Cavity Resonator ....................................... 91 2.3.6. Coupling to Cavities .......................................................... 92

2.4. Perturbation Techniques ............................................................. 93 2.4.1. Volume Perturbation .......................................................... 94 2.4.2. Material Perturbation ......................................................... 98 2.4.3. Perturbation by Material Insertion .......................................... 100

2.5. Bibliography........................................................................... 101

Chapter 3. Reflection, Transmission, and Scattering of Waves 103

3.1. Introduction ............................................................................ 103 3.2. Polarization of Plane Waves ... ............. ... .... .......... ............. ....... .... 103

3 .2. 1. Linear Polarization .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 3.2.2. Elliptical and Circular Polarization .. ......... ....... .......... ....... .... 105

3.3. Reflection and Transmission of Plane Waves .................................... 107 3.3.1. Reflection and Transmission at a General Dielectric Interface:

Normal Incidence.... ... ...... ....... ....... ... ............. .......... ........ 108 3.3.2. Reflection and Transmission at a Lossy Dielectric Interface: ........... 113 3.3.3. Reflection and Transmission at a Lossless Dielectric Interface: ......... 114 3.3.4. Reflection and Transmission at an Air Conductor Interface:

Normal Incidence .... ................... .... ..... ....... ....... .... ........ ... 116 3.3.5. Reflection and Transmission at an Interface: Oblique Incidence ........ 119 3.3.6. Oblique Incidence on a Conducting Interface: Perpendicular

Polarization .................................................................... 120 3.3.7. Oblique Incidence on a Conducting Interface: Parallel Polarization..... 122 3.3.8. Oblique Incidence on a Dielectric Interface: Perpendicular Polarization 123 3.3.9. Oblique Incidence on a Dielectric Interface: Parallel Polarization ....... 126

3.4. Brewster Angle, Total Reflection, and Surface Waves ........................ 126 3.4.1. Total Reflection ............................................................... 129 3.4.2. Surface Waves ................................................................ 131

3.5. Reflection and Transmission for Layered Materials at Normal Incidence...... 132 3.5.1. Reflection and Transmission for a Dielectric Slab at Normal Incidence 135 3.5.2. Reflection and Transmission for a Low Loss Dielectric Slab at Normal

Incidence. ......... .......... .......... ......... ....... ....... ... ... ...... ...... 137 3.5.3. Reflection and Transmission for a High Loss Dielectric Slab at Normal

Incidence....................................................................... 139 3.5.4. Reflection and Transmission for a Lossless Dielectric Slab Backed by a

Perfect Conductor: Normal Incidence. ...... ..... ........ .... ...... ... ..... 140 3.6. Reflection and Transmission for Layered Dielectrics: Oblique Incidence ..... 141

3.6.1. Oblique Incidence on N Dielectric Layers: Perpendicular Polarization . 143 3.6.2. Oblique Incidence on N Dielectric Layers: Parallel Polarization ......... 147

3.7. Scattering .................... ................ ................. ....... .................. 147 3.8. Bibliography........................................................................... 150

viii

Part II. Microwave Techniques and Devices

Chapter 4. Microwave Measurement Techniques ............................... 152

4.1. Introduction ............................................................................ 152 4.2. Power Measurements ................................................................. 152

4.2.1. Definition of Power................ ........................ ................... 153 4.2.2. Methods of Measurement .................................................... 154 4.2.3. Thermo-Resistive Methods .................................................. 154 4.2.4. Thermocouple Measurements ............................................... 155 4.2.5. Measurement of Power Density ............................................. 156

4.3. Frequency Measurements ............................................................ 157 4.3.1. Frequency Counting .......................................................... 157 4.3.2. Heterodyne Frequency Measurement....................................... 158 4.3.3. Wavemeters ..... .......................................... .............. ....... 159

4.4. Q Measurements . .. . . .. . . . . .. . . . . . . . . . . . . . . .. . . .. . .. . . .. .. .. . . . . . .. . . .. . ... . .. . ... ... . 161 4.5. Attenuation Measurements ........................................................... 163 4.6. Standing Wave Ratio and Reflection Coefficient Measurements ...... .......... 165 4.7. Microwave Microscopy .............................................................. 167 4.8. An example of Test Setup: The Balanced Microwave Interferometer........... 168 4.9. Bibliography ........................................................................... 170

Chapter 5. Microwave Sources, Sensors, and Devices ...................... 171

5.1. Introduction ............................................................................ 171 5.2. Generation of Microwave Fields: Microwave Tubes ............................. 172

5.2.1. The Magnetron and M Tubes ................................................ 172 5.3. Microwave Liner-Beam Tubes (O-Tubes) ......................................... 175

5.3.1. The Klystron .................................................................. 175 5.3.2. The Traveling Wave Tube (TWn .......................................... 177

5.4. Solid State Microwave Devices ..................................................... 179 5.4.1. The Tunnel Diode ............................................................. 179 5.4.2. The Gunn Diode .............................................................. 180 5.4.3. The Avalanche Diode......................................................... 181 5.4.4. The BARITT Diode ........................................................... 182 5.4.5. The PIN Diode ................................................................ 183

5.5. Microwave Circuits ................................................................... 184 5.5.1. The Negative Resistance Oscillator ......................................... 184 5.5.2. Transistor Oscillators ...... ..... ...................................... ........ 186 5.5.3. Amplifiers...................................................................... 186

5.6. Coupling of Microwaves Sources to the Test Sample...... ......... ........... 186 5.7. Microwave Probes and Sensors .................................................... 187

5.7.1. The Thermocouple ............................................................ 187 5.7.2. The Thermistor ................................................................ 187

ix

5.7.3. Diode Detectors ....................... .............................. .......... 188 5.8. Waveguide Probes .................................................................... 189 5.9. Antennas ....... ................ .................... ................... ....... ....... ... 189

5.9.1. The Hom Antenna ............................................................ 190 5.9.2. Microstrip or Patch Antennas................................................ 190 5.9.3. Spiral Antennas ............................................................... 190 5.9.4. Slot Antennas .... ... .......... ..... .... .......... ............ .... .............. 191

5.10. Open Waveguides as Sources and Probes .......... .... .......................... 191 5.10.1. The Small Loop .............................................................. 192

5.11. Passive Microwave Devices. ....................................................... 192 5.11.1. Waveguides, Waveguide Sections, and Cavity Resonators ............ 193 5.11.2. The Magic T Hybrid Junction .............................................. 196 5.11. 3. Directional Couplers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 5.11.4. Isolators and Circulators .................................................. 198 5.11.5. Attenuators ................................................................... 199

5.12. Bibliography ......................................................................... 199

Part III. Testing

Chapter 6. Dimensional Testing.............................................................. 201

6.1. Introduction: Testing with Microwaves ............. '" ............................ 201 6.1.1. Reflection Tests ............................................................... 201 6.1.2. Transmission Tests............................................................ 202 6.1.3. Scattering Tests................................................................ 202 6.1.4. Resonant Tests ................................................................ 203 6.1.5. Testing Parameters ............................................................ 204

6.2. Thickness Gaging ..................................................................... 206 6.2.1. Reflectometry at Conducting Interfaces ................................... 206

6.3. Transmission and Attenuation Tests in Dielectrics and Lossy Dielectrics ...... 214 6.4. Standing Wave Measurements ...................................................... 223 6.5. Phase Measurements on the Transmission and Reflection Coefficients .... 225 6.6. Frequency Measurements ............................................................ 226 6.7. Measurement of Coated Conductors ................................................ 228 6.8 Bibliography ............................................................................ 229

Chapter 7. Testing for Discontinuities ................................................... 231

7.1. Introduction ............................................................................ 231 7.2. Scattering Methods of Flaw Detection .............................................. 232 7.3. Location of flaws ..................................................................... 237 7.4. Scanning Measurements .... . .. . . .. .. .. . . ... ... . .. . . .. .... ... ...... .. . . .. . . .. . .. . ... . 242 7.5. Automatic Testing..................................................................... 244

x

7.6. Bibliography ........................................................................... 244

Chapter 8. Testing by Monitoring Material Properties

8.1. Introduction ............................................................................ 247 8.2. Transmission Tests ................................................................... 247 8.3. Reflection and Attenuation Tests .................................................... 250 8.4. Resonant Methods .................................................................... 253

8.4.1. Transmission Line Methods and Transmission Line Resonators ...... 258 8.4.2. Density Tests .................................................................. 264

8.5. Scattering Methods ................................................................... 267 8.6. Bibliography ........................................................................... 270

Part IV. Modeling of the Testing Environment

Chapter 9. Methods of Modeling ............................................................ 274

9.1. Introduction ............................................................................ 274 9.2. Purpose and Scope of Modeling .................................................... 274 9.3. General Approach to Modeling: Numerical ........................................ 275 9.4. The Finite Difference Method ....................................................... 278

9.4.1. The Finite Difference Representation ....................................... 278 9.4.2. Finite Difference Formulation For the 1-D Wave Equation .............. 282

9.5. Finite Element Methods (FEM) ..................................................... 287 9.5.1. The Finite Element Formulation ............................................. 288 9.5.2. Energy Functional for a Quasi-Static Magnetic Problem ............... 288 9.5.3. Finite Element Discretization ................................................ 289 9.5.4. Finite Element Formulation .................................................. 290 9.5.5. Quadrilateral Isoparametric Elements ....................................... 291 9.5.6. Functional Minimization ..................................................... 293 9.5.7. Boundary Conditions ......................................................... 294

9.6. Boundary Integral Methods: The Method of Moments ........................... 297 9.6.1. The Method of Moments for Differential Operators .................... 298 9.6.2. Subsectional Bases ........................................................... 303 9.6.3. The Method of Moments for Integral Operators ........................... 304 9.6.4. Method of Moments for Current Distributions ............................ 309 9.6.5. Formulation .................................................................... 309

9.7. Bibliography ........................................................................... 314

xi

Chapter 10. Modeling of the Time-Dependent Wave Equation 317

10.1. Introduction .......................................................................... 317 10.2. Formulation of the Time Dependent Wave Equation ......................... 317

10.2.1. The Time Dependent Equations............................................ 317 10.2.2. Alternative Formulation: TE and TM Representation ................... 324

10.3. The Axi-Symmetric Formulation .................................................. 327 10.4. Radiation Boundary Conditions ................................................... 329 10.5. Finite Difference Implementation .................................................. 330

10.5.1. Two-Dimensional Applications .................... .... ............. ....... 330 10.5.2. Axi-Symmetric Applications ............................................... 332

10.6. Examples ............................................................................. 334 10.6.1. Scattering by Embedded Cylinders ........................................ 334 10.6.2. Waves Due to a Small Loop................................................ 338

10.7. 3-D Formulation With the Finite Difference Time Domain Method ....... 340 10.8. Bibliography ......................................................................... 344

Chapter 11. Modeling of the Time-Harmonic Wave Equation........... 346

11.1. Introduction .......................................................................... 346 11.2 The Time Hannonic Wave Equations .............................................. 347

11.2.1. The Wave Equation ......................................................... 347 11.2.2. The Generalized Quasi-Static Equation (Eddy Current Equation) .... 348 11.2.3. Formulation of the Two-Dimensional Helmholtz Equation ............ 351 11.2.4. Formulation of the Three-Dimensional Helmholtz Equation ........... 355 11.2.5. Formulation of the Modified Eddy Current Equation: 2-D Case ....... 358 11.2.6. Formulation of the Modified Eddy Current Equation: 3-D Case ....... 359

11.3. The Weak Form Approach......................................................... 361 11.4. Examples ............................................................................. 365

11.4.1. Resonant Frequency of a Cubic Cavity ................................... 365 11.4.2. Modes in Rectangular Waveguides ........................................ 369 11.4.3. Modes in a Loaded Cavity Resonator ..................................... 370

11.5. Bibliography ......................................................................... 371

Part V. Miscellaneous Topics

Chapter 12. Miscellaneous Topics ............................................................ 374

12.1. Tables of Material Properties ....................................................... 374 12.2. Hyperbolic and Exponential Functions ........................................... 377 12.3. Euler's Equation ..................................................................... 378

xii

Appendix A. Vector Relations

A.I. The Gradient, Divergence, and Curl .............................................. .

A.I.l. The V Operator .............................................................. . A.I.2. The Gradient ................................................................. . A.l.3. The Divergence .............................................................. . A.l.4. The Curl ...................................................................... .

A.2. Vector Theorems .................................................................... . A.2.1. The Divergence Theorem ................................................... . A.2.2. Stokes' Theorem ............................................................ . A.2.3. Helmholtz's Theorem ...................................................... ..

A.3. Vector Identities .................................................................... .. A.4. The Laplacian ....................................................................... .. A.S. Expressions in Cartesian, Cylindrical and Spherical Coordinates ............. .

381 381

381 381 382 382 383 383 383 383 384 384 385

Subject Index .......................................................................... 387

Preface

It is perhaps curious that a technique as developed as microwave testing has been given only scant attention as a method for nondestructive testing of materials. This is particularly so since much of the existing work with microwaves is directly applicable to nondestructive testing. In fact, microwave techniques have been used in a large number of applications that can be classified as nondestructive testing applications, ranging from large scale remote sensing to detection of tumors in the body. The instrumentation is also easily available and, unlike for other methods of NDT, require little modification. Perhaps the main reason for this state is the fact that the resolution one can expect of microwave testing is of the order of about half a wavelength which, with standard microwave equipment, is of the order of at most 1 mm (at 100 GHz the wavelength is 3 mm). Other reasons are that microwave measurements are viewed as noisy and "difficult" to perform. While some of these reasons were relevant in the past, they have been largely overcome. Current instrumentation is excellent, computer controlled, and with excellent noise figures. Microwave sources are more stable than have ever been and equipment in the mm-wave region is now available, further reducing the resolution to a fraction of a mm. This is fortunate since microwave testing offers some attractive features, especially for testing of composite and other nonmetallic materials.

Applications of microwaves to nondestructive testing of materials date back to at least the early 1950's. There were a number of attempts at testing of plastics, paper products, and composite materials. This activity has intensified with increased use of microwaves in processing of materials and availability of better, more accurate equipment. However, surprisingly, little has been written on microwave nondestructive testing. A notable exception is a small book written in 1982 by AJ. Bahr ("Microwave Nondestructive Testing Methods", Gordon and Breach Science Publishers, 1982).

The present volume attempts a unified approach to microwave nondestructive testing by presenting the three essential components of testing: theory, practice, and modeling. While recognizing that each of these subjects is wide enough to justify a volume of its own, the presentation of the three topics together is intended to show that these are interrelated and should be practiced together. While few will argue against a good theoretical background, modeling and simulation of the testing environment is seldom part of the NDT training in any method, but particularly so in microwave testing.

The text is divided into four parts. The first part presents the field theory background necessary for understanding the microwave domain. This includes chapters 1, 2, and 3. The second part, treats microwave measurements as well as devices and sources and includes chapters 4 and 5. Chapters 6, 7, and 8 discuss practical tests applicable to a

xiii

xiv Preface

variety of materials and geometries. The fourth part discusses modeling of microwave testing and consists of chapters 9, 10, and 11.

Chapter 1 presents Maxwell's equations in differential and integral forms, as well as the various material properties, potential functions, and derived equations. Aspects of field theory required for work in the microwave domain such as propagation of waves in materials, complex permittivity and energy relations are emphasized.

Aspects of electromagnetic field theory relating to transmission lines, waveguides and electromagnetic cavity resonators are given in chapter 2. Transmission line theory is presented first followed by TE and TM modes in rectangular and cylindrical waveguides. The properties of cavity resonators are discussed mainly from an energy point of view with emphasis on perturbation techniques.

Chapter 3 discusses reflection, transmission and scattering of waves in lossless, low loss, and lossy dielectrics. A variety of topics, including polarization, propagation between materials, reflection and transmission, coefficients, and propagation through layered materials are discussed as used in testing.

Microwave measurement techniques is the subject of chapter 4. While the topic of microwave measurements is extremely wide, the discussion here emphasizes those measurements that are commonly used for testing, from a very basic point of view. Only the techniques are given without attempting to introduce instrumentation. Measurements are described in simple block diagrams which allow understanding of the measuring technique without the need to discuss complicated equipment such as spectrum or network analyzers. While these instruments can, and often are used to accomplish the measurements described here, the diagrams are given in terms of the basic functions of measurement of power, amplitude, phase, frequency and the like.

Chapter 5 is a short chapter in which a number of microwave devices are described. As with previous chapters, only those aspects of microwave power generation, devices and instrumentation that are likely to be used in a nondestructive testing environment are given. Thus, magnetrons and klystrons are discussed because these are so common while solid state devices are given because they are the choice for simple, low cost, low power microwave sources for NDT. A number of passive devices used for measurement setup are described as well.

Chapter 6 is the first of three chapters on testing, discussing methods of dimensional testing. The various methods and test parameters are presented but, while all methods of testing are given, the emphasis in this chapter is on reflectometry and related topics (such as standing wave methods) since this is a universal method for testing of dimensional parameters. Other methods such as transmission, attenuation, scattering and resonant methods are also given but these are emphasized in following chapters where they are more useful. The chapter also contains a general discussion on testing which serves as an introduction to chapters 6, 7 and 8.

Chapter 7 discusses testing for discontinuities and emphasizes scattering methods. Detection of flaws and location of discontinuities are discussed from a general scattering point of view. In doing so, various techniques and notation used in radar work are borrowed. This, in effect, shows testing for discontinuity to be any means of detecting and measuring the scattered field.

In chapter 8, we emphasize resonant methods of testing. Resonant methods rely on detection of variation of material properties and are well adapted for testing of bulk properties such as moisture content or degree of curing in materials. However, the

Preface xv

subject of the chapter is testing for material properties and a variety of other methods can be used for this purpose, including attenuation, transmission and reflection. These methods are also covered.

Chapter 9 introduces modeling for nondestructive testing purposes. The presentation is divided into finite difference, finite element and method of moments to emphasize three different, general methods for modeling. The basic methods are introduced here and some of these, will be then expanded upon in chapters 10 and 11. Some very simple examples of modeling are given to show that modeling need not be complicated.

The time-dependent wave equation and modeling of testing environments related to time-dependent wave propagation are introduced in chapter 10. The finite difference method is emphasized as the method of choice for time-dependent applications. Two­dimensional and axi-symmetric formulations are given first with examples from testing of composite materials. These are then extended to three dimensional modeling.

Chapter 11 gives a number of formulations for time-harmonic applications, using the finite element method. 2-D, 3-D and axisymmetric methods and applications are given including calculation of shift in resonant frequencies, and modes in cavities and waveguides.

In addition, a short chapter (chapter 12) gives information on material properties as well as describing hyperbolic and exponential functions and the Euler equation, in the form of short appendices.

Each chapter contains a bibliography intended to expand on the material given and, in particular, to point to subjects which could not be covered either as not appropriate or for lack of space.