Basic Electromagnetism and Materials

Basic Electromagnetism and Materials Basic Electromagnetism and Materials André Moliton Université de Limoges Limoges,...

Author: André Moliton

106 downloads 1660 Views 3MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

Basic Electromagnetism and Materials

Basic Electromagnetism and Materials André Moliton Université de Limoges Limoges, France

André Moliton Laboratory Xlim-MINACOM Faculte des Sciences et Techniques 123 Avenue Albert Thomas 87060 Limoges Cedex France [email protected]

Library of Congress Control Number: 2005939183 ISBN-10: 0-387-30284-0 ISBN-13: 978-0387-30284-3 ©2007 Springer Science+Business Media, LLC All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. 9 8 7 6 5 4 3 2 1 springer.com

Foreword

Electromagnetism has been studied and applied to science and technology for the past century. It is mature field with a well developed theory and a myriad of applications. Understanding of electromagnetism at a deep level is important for core understanding of physics and engineering fields and is an asset in related fields of chemistry and biology. The passing of the knowledge of electromagnetism and the skills in application of electromagnetic theory is usually done in university classrooms. Despite the well developed literature and existing texts there is a need for a volume that can introduce electromagnetism to students in the early mid portion of their university training, typically in their second or third year. This requires bringing together many pieces of mathematics and physics knowledge and having the students understand how to integrate this information and apply the information and concepts to problems. André Moliton has written a very clear account of electromagnetic radiation generation, and its propagation in free space and various dielectric and conducting media of limited and also infinite dimensions. Absorption and reflection of radiation is also described. The book begins by reminding the reader in an approachable way of the mathematics necessary to understand and apply electromagnetic theory. Thus chapter one refreshes the reader’s knowledge of operators and gradients in an understandable and concise manner. The figures, summaries of important formulas, and schematic illustrations throughout the text are very useful aids to the reader. Coulomb’s law describing the force between two charges separated by a distance r and the concept of electric field produced at a distance from a charge are introduced. The scalar potential, Gauss’s theorem, and Poisson’s equation are introduced using figures that provide clarity to the concepts. The application of these concepts for a number of different geometries, dimensionalities and conditions is particularly useful in cementing the reader’s understanding. Similarly in chapter one, Ohm’s law, Drude model, and drift velocity of charges are clearly introduced. Ohm’s law and its limits at high frequency are described. The author’s comments provide a useful perspective for the reader.

vi

Basic electromagnetism and materials

The introduction of magnetostatics and the relationships between current flow, magnetic fields, and vector potential, and Ampère’s theorem are also introduced in chapter one. Again the figures, summaries of important formulas, and author’s comments are particularly helpful. The questions and detailed answers are useful for retaining and deepening understanding of the knowledge gained. Chapter two provides a useful and practical introduction to dielectrics. The roles of dipolar charges, discontinuities, space charges, and free charges as well as homoand hetero-charges are illustrated in Figure 2.2. The applications of the formulas introduced in chapter one and the corresponding problems and solutions complete this chapter. Similarly magnetic properties of materials are introduced in chapter three. Dielectric and magnetic materials are introduced in Figure 3. The properties of magnetic dielectric materials are described in Figure 4 together with a number of useful figures, summaries, problems, and solutions. Maxwell’s equations together with oscillating electromagnetic fields propagating in materials of limited dimensions are introduced and described in chapters five through seven. Propagation of oscillating electromagnetic waves in plasmas, and dielectric, magnetic, and metallic materials is described in chapter eight. The problems and solutions at the end of each chapter will be particularly helpful. The generation of electromagnetic radiation by dipole antennae is described in chapter nine, with emphasis on electric dipole emission. Absorption and emission of radiation from materials follow in chapters ten and eleven. Chapter twelve concludes with propagation of electromagnetic radiation in confined dimensions such as coaxial cables and rectangular waveguides. In sum, I recommend this book for those interested in the field of electromagnetic radiation and its interaction with matter. The presentation of mathematical derivations combined with comments, figures, descriptions, problems, and solutions results in a refreshing approach to a difficult subject. Both students and researchers will find this book useful and enlightening. Arthur J. Epstein Distinguished University Professor The Ohio State University Columbus, Ohio October 2006

Preface

This volume deals with the course work and problems that are common to basic electromagnetism teaching at the second- and third-year university level. The subjects covered will be of use to students who will go on to study the physical sciences, including materials science, chemistry, electronics and applied electronics, automated technologies, and engineering. Throughout the book full use has been made of constructive exercises and problems, designed to reassure the student of the reliability of the results. Above all, we have tried to demystify the physical origins of electromagnetism such as polarization charges and displacement and Amperian currents (“equivalent” to magnetization). In concrete terms, the volume starts with a chapter recalling the basics of electromagnetism in a vacuum, so as to give all students the same high level at the start of the course. The formalism of the operators used in vectorial analysis is immediately broached and applied so as to help all students be well familiarized with this tool. The definitions and basic theories of electrostatics and magnetostatics then are established. Gauss’s and Ampère’s theories permit the calculation—by a simple route—of the electric and magnetic fields in a material. The calculations for charges due to polarization and Amperian currents caused by magnetization are detailed, with attention paid to their physical origins, and the polarization and magnetization intensity vectors, respectively. A chapter is dedicated to the description of dielectric and magnetic media such as insulators, electrets, piezoelectrets, ferroelectrics, diamagnets, paramagnets, ferromagnets, antiferromagnets, and ferrimagnets. Oscillating environments are then described. As is the tradition, slowly oscillating systems—which approximate to quasistationary states—are distinguished from higher-frequency systems. The physical origin of displacement currents are detailed and the Maxwell equations for media are established. The general properties of electromagnetic waves are presented following a study of their propagation in a vacuum. Particular attention has been paid to two different types of notation—used by dielectricians and opticians—to describe what in effect is the same wave. The properties of waves propagating in infinitely large materials are then described along with the description of a general method allowing determination of dynamic polarization in a material that disperses and absorbs the waves. The Poynting vector and its use in determining the energy of an electromagnetic wave is then detailed, followed by the behavior of waves in the more widely encountered materials such as dielectrics, plasmas, metals, and uncharged magnetic materials.

viii


The following two chapters are dedicated to the analysis of electromagnetic field sources. An initial development of the equations used to describe dipole radiation in a vacuum is made. The interaction of radiation with electrons in a material is detailed in terms of the processes of diffusion, notably Rayleigh diffusion, and absorption. From this the diffusion of radiation by charged particles is used to explain the different colors of the sky at midday and sunset. Rutherford diffusion along with the various origins of radiation also are presented. The theory for absorption is derived using a semiquantic approximation based on the quantification of a material but not the applied electromagnetic field. This part, which is rather outside a normal first-degree course, can be left out on a first reading by undergraduate students (even given its importance in materials science). It ends with an introduction to spectroscopy based on absorption phenomena of electromagnetic waves, which also is presented in a more classic format in a forthcoming volume entitled Applied Electromagnetism and Materials. The last two chapters look at the propagation of waves in media of limited dimensions. The study of reflection and refraction of waves at interfaces between materials is dominated by the optical point of view. Fresnel’s relations are established in detail along with classic applications such as frustrated total internal reflection and the Malus law. Reflection by an absorbing medium, in particular metallic reflections, is treated along with studies of reflection at magnetic layers and in antiradar structures. Guided propagation is introduced with an example of a coaxial structure; then, along with total reflection, both metallic wave guides and optical guides are studied. The use of limiting conditions allows the equations of propagation of electromagnetic waves to be elaborated. Modal solutions are presented for a symmetrical guide. The problem of signal attenuation, i.e., signal losses, is related finally to a material’s infrared and optical spectral characteristics. I would like to offer my special thanks to the translator of this text, Dr. Roger C. Hiorns. Dr. Hiorns is following post-doctorial studies into the synthesis of polymers for electroluminescent and photovoltaic applications at the Laboratoire de PhysicoChimie des Polymères (Université de Pau et des Pays de l’Adour, France).

Contents

Chapter 1. Introduction to the fundamental equations of electrostatics and magnetostatics in vacuums and conductors……….. 1 1.1. Vectorial analysis……………………………………………………. 1.1.1. Operators………………………………………………………. 1.1.2. Important formulae……………………………………………. 1.1.3. Vectorial integrations…………………………………………. 1.1.4. Terminology……………………………………………………. 1.2. Electrostatics and vacuums………………………………………… 1.2.1. Coulomb's law………………………………………………….. 1.2.2. The electric field: local properties and its integral……………. 1.2.3. Gauss's theorem………………………………………………… 1.2.4. The Laplace and Poisson equations…………………………… 1.3. The current density vector: conducting media and electric currents……………………………………………………………… 1.3.1. The current density vector …………………………………….. 1.3.2. Local charge conservation equation…………………………... 1.3.3. Stationary regimes …………………………………………….. 1.3.4. Ohm's law and its limits………………………………………… 1.3.5. Relaxation of a conductor……………………………………… 1.3.6. Comment: definition of surface current (current sheet)……… 1.4. Magnetostatics……………………………………………………….. 1.4.1. Magnetic field formed by a current…………………………….. 1.4.2. Vector potential …………………………………………………

1 1 4 4 7 8 8 9 10 14 16 16 17 17 20 23 24 24 25 26

x


G

1.4.3. Local properties and the integral of B……………………….… 1.4.4. Poisson's equation for the vector potential ……………………. G 1.4.5. Properties of the vector potential A ………………………….… 1.4.6. The Maxwell-Ampere relation………………………………….. 1.5 Problems…………….……………………………………………….. 1.5.1. Calculations……………………………………………………… 1.5.2. Field and potential generated inside and outside of a charged sphere……………….……………………………….

26 26 27 28 31 31

Chapter 2. Electrostatics of dielectric materials…………………….

39

2.1. Introduction: dielectrics and their polarization………………..…… 2.1.1. Definition of a dielectric and the nature of the charges………… 2.1.2. Characteristics of dipoles…………………………………………. 2.1.3. Dielectric in a condenser………………………………………… 2.1.4. The polarization vector ……………………………………….… 2.2. Polarization equivalent charges…………………………………….… 2.2.1. Calculation for charges equivalent to the polarization…………. 2.2.2. Physical characteristics of polarization and polarization charge distribution……………………………………………… 2.2.3. Important comment: under dynamic regimes the polarization charges are the origin of polarization currents………………… G G 2.3. Vectors ( E ) and an electric displacement ( D ) : characteristics at interfaces…………………………………………. G 2.3.1. Vectors for an electric field ( E ) and an electric G displacement ( D ): electric potentials…………………………. 2.3.2. Gauss's theorem…………………………..……………………... G G 2.3.3. Conditions under which E and D move between two dielectrics ………………………………………………………. 2.3.4. Refraction of field or induction lines…………………….…….. 2.4. Relations between displacement and polarization vectors……… … 2.4.1. Coulomb's theorem…………………………..…………………... 2.4.2. Representation of the dielectric-armature system…………….… 2.4.3. Linear, homogeneous and isotropic dielectrics………………….. 2.4.4. Comments………………………………………………………… 2.4.5. Linear, inhomogeneous and non-isotropic dielectrics…………..

33

39 39 40 42 43 44 44 46 50 51 51 52 53 56 56 56 57 58 59 60

Contents

xi

2.5 Problems: Lorentz field………………………………………………. 2.5.1. Dielectric sphere ………………………………………………… 2.5.2. Empty spherical cavity………………………………………….. 2.6. The mechanism of dielectric polarization: response to a static field…………………………………………………………………… 2.6.1. Induced polarization and orientation…………………………… 2.6.2. Study of the polarization induced in a molecule………………… 2.6.3. Study of polarization by orientation……………………………. 2.7. Problems……………………………………………………………… 2.7.1. Electric field in a small cubic cavity found within a dielectric….. 2.7.2. Polarization of a dielectric strip………………………………... 2.7.3. Dielectric planes and charge distribution (electric images)…… 2.7.4. Atomic polarizability using J.J. Thomson's model……………. 2.7.5. The field in a molecular sized cavity…………………………….

Chapter 3: Magnetic properties of materials………………………. 3.1. Magnetic moment……………………………………….…………… 3.1.1. Preliminary remarks on how a magnetic field cannot be derived from a uniform scalar potential………………………. 3.1.2. The vector potential and magnetic field at a long distance from a closed circuit…………………………………… 3.1.3. The analogy of the magnetic moment to the electric moment..... JJG G i ³³ dS …………… 3.1.4. Characteristics of magnetic moments M

61 61 62 63 63 66 67 72 72 75 77 81 82 89 89 89 89 93 94

S

3.1.5. Magnetic moments in materials………………………………… 97 3.1.6. Precession and magnetic moments……………………………. 98 3.2. Magnetic fields in materials………………………………………… 100 3.2.1. Magnetization intensity………………………………………... 100 3.2.2. Potential vector due to a piece of magnetic material…………. 101 3.2.3. The Amperian currents…………………………………………. 103 G G 3.2.4. Definition of vectors B and H in materials…………………… 106 3.2.5. Conditions imposed on moving between two magnetic media…. 107 3.2.6. Linear, homogeneous and isotropic (l.h.i) magnetic media….... 109 3.2.7. Comment on the analogy between dielectric and magnetic media. …………………………………………………………… 109

xii


3.3. Problems……………………………………………………………… 3.3.1. Magnetic moment associated with a surface charged sphere turning around its own axis…………………………………………… 3.3.2. Magnetic field in a cavity deposited in a magnetic medium…… 3.3.3. A cylinder carrying surface currents…………………………... 3.3.4 Virtual current…………..……………………………………….

111 111 112 114 116

Chapter 4: Dielectric and magnetic materials………………………. 119 4.1. Dielectrics……………………………………………………………. 4.1.1. Definitions……………………………………………………… 4.1.2. Origins and types of breakdowns…………...…………………. 4.1.3 Insulators……………………………………………………….. 4.1.4 Electrets ………………………………………………………… 4.1.5. Ferroelectrics…………………………………………………... 4.2. Magnetic materials………………………………………………….. 4.2.1. Introduction…………………………………………………….. 4.2.2. Diamagnetism and Langevin's theory………………………… 4.2.3. Paramagnetism………………………………………………… 4.2.4. Ferromagnetism………………………………………………… 4.2.5. Antiferromagnetism and ferrimagnetism……………………... 4.6. Problem………………………………………………………………. Dielectrics, electrets, magnets, and the gap in spherical armatures…

119 119 120 121 123 126 129 129 131 132 137 150 151 151

Chapter 5. Time-Varying Electromagnetic Fields and Maxwell’s equations…………………………………………………. 157 5.1. Variable slow rates and the rate approximation of quasi-static states (RAQSS)………………………………………………………… 157 5.1.1. Definition……………………..………………………………… 157 5.1.2. Propagation…………………………………………………….. 157 5.1.3. Basics of electromagnetic induction…………………………… 158 5.1.4. Electric circuit subject to a slowly varying rate………………… 158 5.1.5. The Maxwell-Faraday relation………………………………….. 159

Contents

xiii

G 5.2. Systems under frequencies ( div j z 0 ) and the Maxwell – Ampere

relation……………………………………………………………….. 160 JJG G G 5.2.1. The shortfall of rot H jA (first form of Ampere's theorem for static regimes)……………………………………………… 5.2.2. The Maxwell-Ampere relation……………………………….. 5.2.3. Physical interpretation of the displacement currents…..…… 5.2.4. Conclusion……………………………………………………... 5.3. Maxwell's equations………………………………………………… G G 5.3.1. Forms of div E and div B under varying regimes…………... 5.3.2. Summary of Maxwell's equations…………………………….. 5.3.3. The Maxwell equations and conditions at the interface of two media………………………………………………………….. 5.4. Problem ……………...………………………………………………. Values for conduction and displacement currents in various media…

160 161 162 165 167 167 167 168 170 170

Chapter 6. General properties of electromagnetic waves and their propagation through vacuums………………………… 173 6.1. Introduction: equations for wave propagation in vacuums……… 173 6.1.1. Maxwell's equations for vacuums: U A = 0 and j A = 0……... 173 6.1.2. Equations of wave propagation………………………………… 173 6.1.3. Solutions for wave propagation equations…………………….. 174 6.2. Different wave types………………………………………………… 176 6.2.1. Transverse and longitudinal waves……………………………. 176 6.2.2. Planar waves…………………………………………………… 176 6.2.3. Spherical waves………………..………………………………… 177 6.2.4. Progressive waves ……………………………………………… 178 6.2.5. Stationary waves………………………………………………… 179 6.3. General properties of progressive planar electromagnetic waves (PPEMW) in vacuums with U A = 0 and j A = 0…………….. 180 G G 6.3.1. E and B perpendicular to the propagation: TEM waves……… 181 G G 6.3.2. The relation between E and B ………………………………… 181 6.3.3. Breakdown of a planar progressive electromagnetic wave (PPEMW) to a superposition of two planar progressive EM waves polarized rectilinearly………………………………… 183 6.3.4. Representation and spectral breakdown of rectilinearly polarized PPEMWs………………………………………………. 184

xiv


6.4. Properties of monochromatic planar progressive electromagnetic waves (MPPEMW)…………………………………………………. 186 6.4.1. The polarization………………………………………………… 186 6.4.2. Mathematical expression for a monochromatic planar wave…. 191 6.4.3. The speed of wave propagation and spatial periodicity………... 194 6.5. Jones's representation……..…………………………………………. 195 6.5.1. Complex expression for a monochromatic planar wave………. 195 6.5.2. Representation by way of Jones's matrix……………………… 196 6.6. Problems……………………………………………………………… 199 6.6.1. Breakdown in real notation of a rectilinear wave into 2 opposing circular waves……………………………………….. 199 6.6.2. The particular case of an anisotropic medium and the example of a phase retarding strip…………………………..... 200 6.6.3. Jones's matrix based representation of polarization…….…….. 202

Chapter 7. Electromagnetic waves in absorbent and dispersing infinite materials and the Poynting vector……………………….. 205 7.1. Propagation of electromagnetic waves in an unlimited and uncharged material for which U A = 0 and j A = 0. Expression for the dispersion of electromagnetic waves………………………… 205 7.1.1. Aide memoir: the Maxwell equation for a material where UA 0 and jA 0 ……………………………………… 205 7.1.2. General equations for propagation……………………………. 205 7.1.3. A monochromatic electromagnetic wave in a linear, homogeneous and isotropic material………………………. 206 7.1.4. A case specific to monochromatic planar progressive electromagnetic waves (or MPPEM wave for short)………….. 208 7.2. The different types of media……………………………………….. 210 7.2.1. Non-absorbing media and indices……………………………... 210 7.2.2. Absorbent media, and complex indices………………………… 212 7.3. The energy of an electromagnetic plane wave and the Poynting vector …………………………………………………….. 215 7.3.1. Definition and physical significance for media of absolute permittivity (H), magnetic permeability (µ) and subject to a conduction current (j A )…………………………………….. 215 7.3.2. Propagation velocity of energy in a vacuum …………………. 217

Contents

7.3.3. Complex notation……………………………………………… 7.3.4. The Poynting vector and the average power for a MPPEM wave in a non-absorbent (k and n are real) and non-magnetic (µr = 1, so that µ = µ0) medium………………………………… 7.3.5. Poynting vector for a MPPEM wave in an absorbent dielectric such that µ is real………………………………. 7.4 Problem……………………………………………………………….. Poynting vector………………………………………………………...

xv

218

219 219 220 220

Chapter 8. Waves in plasmas and dielectric, metallic, and magnetic materials ……………………………………………….. 227 8.1. Interactions between electromagnetic wave and materials…………227 8.1.1. Parameters under consideration………………………………….227 8.1.2. The various forces involved in conventionally studied materials……………………………………………………… 228 8.2. Interactions of EM waves with linear, homogeneous and isotropic (lhi) dielectric materials: electronic polarization, dispersion and absorption…………………………………………………………….. 229 8.2.1. The Drude Lorentz model……………………………………… 231 8.2.2. The form of the polarization and the dielectric permittivity….. 232 8.2.3. Study of the curves Fe’(Z) and Fe’’(Z) for Z | Z0 …………….. 234 8.2.4. Study of the curves of Fe’(Z) and Fe’’(Z) when Z z Z0 …….. 238 8.2.5. The (zero) pole of the dielectric function……………………… 240 8.2.6. Behavior of a transverse plane progressive EM wave which has a pulsation between Z0 and Z A sufficiently far from

Z0 so that H’’ | 0……………………………………………… 241 8.2.7. Study of an MPPEM wave both outside the absorption zone and the range [Z0,Z A ]………………………………………… 241 8.2.8. Equation for dispersion n = f(O0) when Z R ) is in the form G ȡ.R 3 G E ext = u OP . 3İ 0 r 2 G Ur G u OP . 2. Show that for a point P inside the sphere ( OP < R ) we have Eint = 3H 0 G 3. Plot the curve E = f(r) .

4. Show that when r > R , the equation for the electric potential is Vext = Similarly, show that when r < R , we have Vint =

U.R 3

UR² ª r² º «1 » 2H0 ¬ 3R ² ¼

3H0 r

.

and

plot V = f(r) .

5. Show how the calculation for the potential can also be made using Poisson's equation.

34


Answers 1. Given that the charge is spread symmetrically, the resulting electric field also will have the same symmetry, as at any point P in a given sphere with radius r = OP (point Pe or Pi shown in Figure 1) and in space there is the same charge distribution. Thus, for a sphere with a radius denoted by r, the modulus of the field (E) is independent of P on the sphere; that is to say the angles T and M given to define P in terms of spherical coordinates (Figure 2). The E is therefore only dependent on r, so that it is possible to write in terms of modulus that E = E(r) . In terms of vectors, G G and again due to symmetry, E is parallel to er , a unit vector of the radial direction G under consideration, which corresponds to the normal ( n ) outside of the sphere (for U > 0 ).

M 2S

dS =

G Eext Pe

R O

³

d²S

2Sr² sin TdT

G er

M 0

G n

P

G E inint

d²S = r² sin T dT d

Pi

T r

O M

p

Figure 2

Figure 1

G Gaussian surface. For the calculation of the field E at P, which is at a distance from O given by r = OP , the sphere with radius r around the center O is called the Gaussian surface, for which E is independent of P or in other terms, E is a constant for a given value of r. The electric field flux ()) through a Gaussian surface is represented by a sphere with a radius given by r, and thus is G JJJG )= w ³³ E.d²S w ³³ E.d²S T,M

T,M

JJJG G G as E and d²S are with respect to the external normal, in that n

G er .

Chapter 1. Introduction to fundamental equations of electrostatics and magnetostatics

35

With d²S = r² sinT dT dM , we have: T S

2S

) = r² E w ³³ sin T dT dM= r² E ³ sin T dT ³ dM= 4 S r² E T,M

T 0

M=0

The calculation of the field denoted Eext at Pe, which is such that OPe = r > R may be performed using Gauss's theorem, which gives here r R T S M 2S

) = 4Sr²E ext = ¦

³

Qint H0

i

³

r 0 T 0

³ UdW

M 0

,

H0

so that with U = Cte and dW = r² sinT dr dT dM , then 4

) = 4Sr²E ext

3

SR 3

U H0

R3

G . From this can be deduced that E ext

3r²H0

G U er .

4 U SR 3 , we also can write that 3 G Q G E ext er 4SH0 r² and the electric field corresponds to that of a charge (Q) placed at O at a distance r = OPe from Pe . In other words, the electric field generated at Pe can be considered due to a point charge at O. With Q

2. For a point denoted Pi inside a charged sphere, which is such that OPi = r < R , Gauss's theorem gives: r OPi T S M 2 S

) = 4Sr²Eint = ¦

Qint

³ r 0

³

T 0

³ UdW

M 0

H0 H0 from which can be determined that G Eint

4 3

i

Ur G er . 3H0

3. The plot of E = f(r) is given in Figure 3.

S OPi3

U

4

H0

3

S r3

U H0

,

36


V UR² 2H 0 UR² 3H0

E UR 3H0

r

R

R Figure 4

Figure 3

4. With the electric field being radial, it is possible to state that: G JJG dV ³ E.dr ³ E.dr , and that outside the sphere, we have: Vext

³ E ext dr

UR 3 dr

UR 3

3H0

3H0 r

³

r²

C . The integral constant (C) can be

determined using the limiting condition Vext (r o f) o 0 , from which C = 0 and UR 3

Vext U

Vext (R) = Vint (R) and

UR 3 3H0 R

Vint

³ r dr

UR² 6H0

. Ur²

D . The constant denoted D is 3H0 6H0 determined using the continuous potential condition for r = R, so that Inside, Vint

³ Eint dr

3H0 r

D , from which D

UR² 2H0

. Finally,

UR² ª r² º «1 ». 2H0 ¬ 3R ² ¼

The plot of V = g(r) is shown in Figure 4.

5. Poisson's equation, written in the form 'V

U H0

0 , is written in spherical

coordinates for a function, which is independent of the variables T and M; thus 1 w ²(rV) U , so that: r wr² H0

r

Chapter 1. Introduction to fundamental equations of electrostatics and magnetostatics

x when r t R, with U = 0 : Vext = A1 + Vext =

A2 r

A2

1 w ²(rVext ) r

; the condition

r

;

wr² rVint

Ar B, so that

Vext (r o f) o 0 gives A1 = 0 , from which

1 w ²(rVint )

x when r d R, with "U = U z 0" , w (rVnt )

0 rVext

wr²

37

Ur² 2H 0

Ur 3

wr²

r

U H0

w ²(rVint ) wr²

Ur H0

so that:

B1 . The result is that:

B1r B2 Vint

Ur 2

B1

B2

. 6H 0 6H0 r In physical terms, the former potential retains a finite value and cannot diverge from r = 0 , from which B2 = 0 . The determination of the two constants denoted A2 and B1 necessitates two equations with two unknowns, which can be contained by writing the continuity, for r = R, in terms of the potential (V), as in (Vext =Vint )r=R , and in terms of the electric field (E), as in E = - dV/dr . From these can be determined that A2

UR 3

and B1

UR 2

6H0 2H 0 from which the same forms as those of Vext and Vint.

,

Chapter 2

Electrostatics of Dielectric Materials

2.1. Introduction: Dielectrics and Their Polarization 2.1.1. Definition of a dielectric and the nature of the charges A dielectric is effectively an insulator, meaning that a priori it does not have free and mobile charges. Nevertheless, different types of charges can be found, as shown in Figure 2.1. They are: 1. Dipolar charges, which are attached to a molecule that makes up the dielectric; examples are the dipoles attached to HCl molecules where the center of negative electronic charge is displaced towards the Cl atom, while the H has an excess positive charge, thus giving rise to the dipole H+Cl-. These charges are inseparable, tied to the bonded H and Cl, and are called bound charges. 2. Charges due to discontinuities such as interfaces between aggregates. Where the solid dielectrics exhibit defaults, charges can accumulate giving rise to particular electronic phenomena, such as the Maxwell-Wagner-Sillars effect at low frequencies. 3. Homocharges, which have the same sign as the electrodes to which they are adjacent. 4. Heterocharges, which have the opposite sign to the electrode at which they are near. 5. Space charges, which are charges localised within a region of space. 6. Free charges, in principle, are little or not present in dielectrics. They can appear, however, when there is a breakdown caused the application of an electric field and a sudden loss in the ability of the material to insulate. When the current is relatively weak it is called a leak current and it can be due to a wide range of causes, such as impurities in the dielectric and so forth.

40


electrode charges, which polarise the dielectric

homocharges (possibly free) heterocharges (possibly free)

space charges dipolar bound charges

+ + + + + + + + + + + + + + ++ + +++++++++++++++++

compensation charges (opposite facing bound charges)

charges at discontinuities (interface of aggregate shown by dotted line)

Figure 2.1. Various types of charges in a dielectric.

2.1.2. Characteristics of dipoles 2.1.2.1. Electric potential produced by an electric dipole If two charges bound to an electric dipole, detailed in Figure 2.2a, are placed at JJJG JG A (-q) and at B (+q) so that AB dl (in the sense going from the negative to the positive charge), the dipolar moment is by definition JJG dµ

JG qdl

To calculate the potential (dV) for a point (P) produced by the dipole that has a center at M, the following equation can be directly obtained:

dV

1 § q q ¨¨ 4SH0 © rB rA

Accordingly: dV

· ¸¸ ¹

q (rA rB ) 4SH0 rA rB

G JJG r dµ 4SH0 r 3 1

.

|

q dl cos T 4SH0 r²

.

Chapter 2. Electrostatics of dielectric materials

G E

P rA rA - rB A(- q)

G fB

r

G qE

G E B +q

A-q

rB T JJG M B(+q) dµ JG dl

41

JG qdl G fA

(a)

G -q E

(b)

Figure 2.2. (a) Electric dipole; and (b) force couple.

G 2.1.2.2. Energy of an electric dipole in a uniform electric field ( E )

It is worth recalling here that an electric charge placed at a point N at an electric potential (VN) takes on a potential energy (Ep) E p = W(q) = q VN . The energy associated with a dipole placed as detailed in Figure 2.2, with –q at A and +q at B, is therefore E p = - q V(A) + q V(B) = q (VB - VA ). In addition, it is possible to write A

that: VB - VA = ³ dV B

A

G JG

G

JG

³ E.dl , so that with E being uniform (over a space dl ), B

JG A JG VB - VA =E ³ dl

JG JJJG JJJG JG JJJG E ª¬OA - OB º¼ = -E.AB . B JJG JG JJJG Then with dµ qdl= q AB we obtain: JJJG G JJG G G E p = q (VB - VA ) = - qAB.E - dµ.E . For a dipole with a moment µ

G ql,

then G G E P = - µ. E

2.1.2.3. Electric couple This section refers to Figure 2.2(b). The moment of the couple of forces JJJG G JJJG JJJG G G G G G G ( f A , f B f A ) is by definition: C AB u f B AB u qE= q AB u E , so that JJJG G with P q AB , G G G C µuE

42


2.1.3. Dielectric in a condenser 2.1.3.1. Study of a condenser with a vacuum between the armatures Figure 2.3 shows a condenser that has armatures carrying the charges +Q and –Q: The potential gradient is therefore only between the armatures that have a known surface area (S), outside of which the electric field can be considered zero. Gauss's theorem, applied across the surface 6, gives rise to )

G JJG

G JJG

JJG

G

w ³³ E.dS ³³ E.dS= E S (assuming that E and dS are collinear) 6

S

Q

VS

H0

H0

where V is the superficial charge density at the electrodes,

from which can be determined : E =

V H0

.

d

S

--- -Q

--H0

+ + + +Q (6) Figure 2.3. Vacuum-based condenser.

2.1.3.2. The presence of a depolarizing field in the dielectric G E

(a)

+ + + + + + +

0

G Ez0

+ +

+ +

+ +

+

+

+

+

+ +

+ (b)

Dielectric surface charges

Mutually cancelled internal charges

Figure 2.4. Dipole orientation: (a) without field and (b) with an applied field.


43

If a dielectric is inserted into an unpolarized condenser (zero field between the armatures), as schematized in Figure 2.4a, then the dipoles associated with the molecules of the dielectric are randomly distributed. Statistically, the opposite charges cancel each other out and the material is electrically neutral overall. If the dielectric is inserted into an electric field, as shown in Figure 2.4b, then the positive charges are pulled in on direction and the negative charges in the G other. The dipoles dµ are aligned in parallel in such a way that their potential GG energy (Ep) is minimized, as in E P = - dP.E . As the bonded charges cannot move, the result is a chain of dipoles in the volume; however, at the surface of the dielectric, charges opposite to that of the armatures appear. G If an applied external field ( E ext ) is in the sense indicated in Figure 2.4c, given the presence of surface dielectric charges, then an opposing field appears G called the depolarizing field ( E d ). The result is an effective field, which can more precisely be termed an external effective field, and is such that: G Ea

G G E ext E d . G G G As E d is antiparallel to E ext , the field E a is less than the external applied field G E ext . G + E + + + + + + E d : E depolarising + + + + Dielectric surface charges caused by its polarization

Figure 2.4(c). The action of a depolarizing field.

2.1.4. The polarization vector In a dielectric sits a small element with volume dW, length dl, and cross section dS, such that it is parallel to the surface of the armatures, as shown in Figure 2.5. If the volume of this element is equivalent to that of a dipole, with respect to the armatures, then it is possible to consider that the elementary charges (q) of the dipoles are such that q = VP dS , where VP represents the superficial charge density compared with the armature and therefore the superficial polarization charge density.

44


The moment of dipoles compared to the armatures can be written as JG JG dµ G dµ q dl VP dS dl , and as dµ = VP dW , we can state that VP . dW The magnitude of the dielectric polarization is used to define the JG G dµ , and has as apparent polarization vector ( P ), which is such that P VP dW modulus the dipolar moment of the unit volume. In terms of vectors JG P

JJG dµ dW

.

JG G G G G G As dµ // E ext or E a , we also have P // E ext or E a . Accordingly, when we speak of G G the electric field ( E ) without any further precision, it is the effective field E a which is the subject of discussion. G E ext G + Ea + + dS JJG dµ + + + + dl + +

Figure 2.5. Dipolar polarization and orientation.

2.2. Polarization Equivalent Charges 2.2.1. Calculation for charges equivalent to the polarization 2.2.1.1. The potential generated at P by a finite volume of polarized dielectric The above figures indicate that a polarized dielectric can be represented by a vacuum, of permitivity H0, in which is placed dipoles orientated by an applied field. So, to calculate the potential generated at a point (P) by a volume (V) of a dielectric, as shown in Figure 2.6, it suffices to calculate the potential generated in a vacuum G by the orientated dipoles ( dµ ) occupying a fraction (dW) of V and then to integrate over the total volume V.


45

P

(V)

G

M x dµ dW

(S)

Figure 2.6. Calculation of charges equivalent to a polarization.

Thus, the element of volume dW taken from around a point M forms at P a potential G 1 r JJG dV dµ (result obtained in Section 2.1.2.1) which also can be written as: 4SH0 r 3 1 JJJJG § 1 · JJG 1 JJJJG § 1 · G dV grad M ¨ ¸ dµ grad M ¨ ¸ PdW . 4SH0 4SH0 ©r¹ ©r¹ The potential V(P) formed at P by the total volume V is therefore: V(P) =

1 4SH0

JJJJG

§1· G

³³³ grad M ¨ ¸ .PdW . V

©r¹

JG JG JG JJJJG Now using the notable equation div (a A) = a div A + A . grad a , for which here G G a = 1/r and A P , we arrive at: JG JJJJG § 1 · G JG P 1 grad M ¨ ¸ .P = div M div M P . r r ©r¹ Substituting this into V(P), we have

G G º ª P 1 V(P) = « ³³³ div M dW - ³³³ div M P dW » , and by using the Ostrogradsky 4SH0 «¬ V r »¼ V r equation on the first triple integral, we obtain: G JJG G º 1 ª P.dS 1 V(P) = - ³³³ div M P dW » . «w ³³ 4SH0 ¬« S r V r ¼»

1

GG If we make VP = P.n

JG PN and UP = - div M P , we finally arrive at:

46


V(P) =

ª VP º UP dW » . «w ³³ dS + ³³³ 4SH0 «¬ S r »¼ V r

1

2.2.1.2. Polarization equivalent charges We can conclude by stating that a volume V of polarized dielectric is equal to the JG sum in V of a volume distribution of charge density UP = - div M P along with a G G surface charge density VP P. n PN over the total surface (S) of the dielectric, with all charges distributed in a vacuum. The volume (UP) and surface (VP) densities thus appear as charges equivalent to the polarization, with the condition that they are in a vacuum! There is nothing imaginary about them and are in complete contrast to the term “imaginary charges” which has been used to describe them. In other words, a polarized material which takes up a space (E) can be represented as taking up the equivalent vacuum space (E) in which are distributed volume and surface charge densities UP and VP, respectively, which represent the polarization effect of the dipoles making up the material, evidently from an electrical point of view. If additional (real) charges with a volume density UA and surface density VA are added to the dielectric, the potential at a point P and that generated by the polarized dielectric can be termed in the generality:

V(P) =

ª VA V P º U UP dS + ³³³ A dW » «w ³³ 4SH0 ¬« S r r »¼ V

1

.

2.2.2.

Physical characteristics of polarization and polarization charge distribution Up to now, the equivalence of polarization and polarization charge distributions has been essentially mathematical because the determination of UP and VP requires particularly long and abstract calculations.

2.2.2.1. Preliminary comments: charge displacement and the corresponding charge polarizations G Comment 1: How moving a charge qi over Gi is the same as applying G G (superimposing) a dipole moment Gµi q i Gi To verify the proposition, a schematic verification can be used to show how the two transformations are equivalent in that going from the same initial state (qi at O) both JJJG G arrive at the same final state (qi at O’, which is such that OO ' Gi ).


47

G First transformation displacement of qi by Gi : G

G

O (qi ) G i O’ initial state: qi at O

st

1 transformation: movement of

G

O Gi O’(qi ) final state: qi at O’ such that

JJJG OO '

qi by G i

G Gi

Second transformation superposition on starting system of a dipole of moment G G G Gµi qi Gi : GPi G - qi G li + qi G G 2nd transformation: superposition O (qi ) G i O’ initial state: qi at O

O Gi O’(qi ) final state: qi at O’ such that

of starting system of a dipole of

G

G

moment G µi

qi G i

JJJG OO '

G

Gi

Comment 2. Calculation of the polarization associated with a charge displacement To arrive at a solution to this problem, it suffices to calculate the dipole moment per G unit volume which is apparent following the displacement of qi through Gi . If the number of qi per unit volume is ni, the polarization that would come about following G G the displacement by Gi of only the qi charges would be ni qi Gi , which is the dipolar moment per unit volume due to the charge displacement. If in the dielectric there are several types of charge (qi) such as q1, q2, qQ and so on, then the dipolar moment per unit volume that would appear following a G G movement of all the charges, i.e. q1 being moved by G1 , qQ by GQ and so on, is G G ¦ n i qi Gi P , which is the system polarization and is the dipole moment per unit i

volume. 2.2.2.2. The corollary of charge movement: Number of charges entering and leaving the dielectric volume following polarization and the volume charge density equal to the polarization D q ({ q) (a)

+q ({ q+)

JJG dS G Gi

G Gi

(b)

Figure 2.7. (a) Characteristics of the electric charges and (b) their displacement.

48


A neutral zone (D) in a dielectric contains various type qi charges, where in reality each charge (q- =-q, and q+ = + q, with identical densities n- and n+) meets at each dipole, as shown in Figure 2.7, with n- and n+ so that D is definitely neutral. In the absence of an electric field and also therefore a polarization, qi are JG G G not moved (Gi 0) and the polarization P ¦ n i qi Gi is zero. i

the

G Under the effect of an applied electric field, qi are moved (by Gi z 0), and JG G G polarization vector becomes P ¦ n i qi Gi ( ¦ n i qi Gi ) as the i

i

-,

summation is performed over all charges q ({ q) and q+ ({ + q). JJG The algebraic flux for qi traversing a part ( dS ) of the surface following the G algebraic displacements ( Gi ) indicated in Figure 2.7(b) is equal to the number of JJG G carriers contained in a cylinder with base dS and length Gi , and therefore a volume G JJG G JJG = Gi . dS . The number of carriers therefore is equal to dNi = n i Gi dS , and the G JJG corresponding amount of charge is dQi =qi n i Gi dS . With respect to the various charges qi, which include both q- and q+, the JJG total algebraic charge traversing the surface element dS thus is G JJG G JJG dQ = ¦ n i qi Gi dS P dS . The algebraic charge Q crossing the closed i

-,

surface S therefore is Q

w ³³ dQ

G JJG

w ³³ P.dS and the zone D, which initially was S

neutral, now contains after the polarization a total charge (QP) opposite to Q. In order to calculate the number of charges gained by entering into D, the reentrant JJJG surface -dS can be used. G G JJG The final result is that Q P = - w ³³ P.dS , so that QP = - ³³³ div P dW . S

D

We thus have shown that on a macroscopic scale, polarization is equal to a G volume density of charge UP = - div P , which is indeed the same result as that obtained from the calculation for a potential generated by a polarized dielectric. 2.2.2.3. Surface charges Following a polarization there is an accumulation of charges at a surface, as detailed in Section 2.1.4 and described in Figure 2.8.


G G On calculating P n

G dP G n dW

G q dl

G with dP

49

G VP dS d l , we find

G G UP P n . Once again, this time for surface polarization charges, the result is equivalent to that for the charge generated by a polarized dielectric. G Ez0

+ + + + + + +

+ +

+ +

+

+

+

+

internal charges which cancel each other out

+ +

dS

+ +

G n

surface charges with density VP

Figure 2.8. Surface polarization charges.

2.2.2.4. Conclusion The physical evidence for polarization charges shows that the densities VP and UP are not associated with imaginary charges or simple mathematical equivalent, but are the result of localized excesses in bound charges, real excesses caused by polarization.

G P

G P

G P G P

+Q

UP

G divP 0

G (divP ! 0)

G P Figure 2.9. Schematization of the divergence operator.

As shown in Figure 2.9, when a dielectric is not deliberately charged in its volume (with real charges), the resultant for bound charges over the volume is zero and UP = 0. So that UP becomes nonzero, the divergence polarization vector also is

50


G nonzero (as UP = - div P ). In the presence of real charges in the volume, the charge +Q indicated in Figure 2.9, which gives a physical representation of the divergence G operator, localized polarization charges appear, so that UP = - div P z 0 . If the material is neutral prior to polarization it must remain neutral following polarization as it is only the existing, bound charges which are displaced. This property can be verified by considering a volume (V) of dielectric delimited by a surface (S). Here, the total of polarized charges is given by: G

G JJG

GG

G JJG

³³³ UP dV w ³³ VP dS ³³³ div P dV w ³³ P.n dS w ³³ P. dS w ³³ P. dS 0 V

S

V

S

S

S

The equation shows how the algebraic sum of the charges equals zero. We also can see in Figure 2.8, where UP 0 and the surface polarization negative charges of density VP facing the positive electrode are exactly compensated for by the surface polarization, positive charges facing the negative electrode. 2.2.3.

Important comment: Under dynamic regimes the polarization charges are the origin of polarization currents Even though this section deals with static states, it is worth making the occasional sortie into dynamic systems. In effect, for the results obtained from static systems to be acceptable for their dynamic counterparts, with polarization values in particular, the variations in magnitude should be negligible with respect to a given macroscopic domain. A reasonable estimate for this is to assume, for example, that a polarization is constant over a distance (d) of around 10 nm (the “atomic” dimensions being of an order of less than 1 nm). So if the length of a polarization signal wave (Ȝ) is such that O > 10d, we can assume that for a “macroscopic” domain of size d the signal remains more or less constant and therefore definable. Figure 2.10 gives an example c of this where the frequency (v) under consideration is Q < | 3 x 1015 Hz . 100 nm Polarization

P | cte

O

x d

O

Figure 2.10. Distance d over which P remains more or less constant.

Above a frequency of this order, it is necessary to take into account the variation with time for a polarization in the macroscopic domain. If the bound


51

d G (Gi ) , there is a dt corresponding current termed a “polarization current”, which has a density K G ji n i qi vi . The average resultant current density is given by G JG G G dGi G , so that with P ¦ n i qi Gi , JP ¦ n i q i vi ¦ n i q i dt i i i G charge carriers (qi) are vibrated with an average velocity vi

G JP

G wP wT

This current density is of a macroscopic scale and represents an average over the microscopic currents associated with the slight displacements of bound charges. The current therefore is not imaginary and gives rise to the same magnetic effects as conduction currents. Its specificity is that it cannot leave the dielectric material since it is tied to bound charges and therefore cannot be measured or used in an external circuit. G G Vectors for an Electric field ( E ) and an Electric Displacement ( D ) : Characteristics at Interfaces Section 2.2.1.1 showed that the potential formed at a point (P) in space by a polarized dielectric is given by an equation that brings in charge densities. The JJJJG G electric field at P can be determined by the often used equation E - grad P V .

2.3.

The inconvenience though is that it is assumed that the polarization vector is known. In a more classic method from the Anglo-Saxon school of thought, it is interesting to reason in terms of an electric field in a vacuum on which is imposed a limiting law in order to take into account the characteristics of bound charges associated with a dielectric medium. 2.3.1

G G Vectors for an electric field ( E ) and an electric displacement ( D ):

electric potentials 2.3.1.1 Coulomb's law applied to dielectrics In general terms, if q and q' are two electric charges placed in a linear, homogeneous JJJG G and isotropic (l.h.i.) dielectric at two points M and P, which are such that r MP , the force that should appear between the two charges is given by Coulomb's law, P i.e.: G q’ G q q' r G F . r M 4 S H0 H r r 3 q

52


where Hr is the relative dielectric permittivity characteristic to each dielectric material. The quantity given by H H0 H r is the absolute dielectric permittivity and in the electrostatic system unity (e.s.u.), H0 = 1, so that H = H r and is the magnitude generally called the dielectric constant. 2.3.1.2 Electric field In simple terms an electric field can be defined over all points P, each defined in JJJG G MP , just as if there were a force respect to M, where there is a charge q by r exerted on the charge unit (unit here is q' = 1), which is assumed here to be very small, such that its dimensions tend toward zero in order to limit the singularity at P. This gives: G G q r E(P) . 4 S H0 H r r 3 G For a group of charges (qi), we have E(P)

1 4 S H0 Hr

¦ qi i

G ri ri3

.

2.3.1.3. Electrical potential Just as for the system in a vacuum, detailed in Section 1.2.2.1, the potential (V) is JJJJG G always such that E gradV . Therefore, for a dielectric, V=

q 4 ʌ İ0İ r r

.

2.3.1.4. Electric displacement vector By definition, the vector, which is also termed electrical induction, is given by: G D

G HE

G H0 H r E

and for a linear homogeneous and isotropic (lhi) is dielectric, as detailed in Section 2.4.3.1. 2.3.2. Gauss's theorem The calculation is identical as that carried out for a vacuum, with the exception that here the absolute permittivity of the medium (H = H0 H r ) is substituted in place of that of a vacuum (H0). We therefore have: 1. for a discrete distribution of charges, where qi represents the internal real charges at the surface S: qi G JJG ¦ i ) w ; ³³ E . dS H0 Hr S


53

2. for a continuous distribution of charges with a volume density ( UA ) of deliberately added free charges

)

G JJG

1

w ³³ E . dS

³³³ UA dW

H0 H r

S

V

1 H

³³³ UA dW

G JJG

Use of Ostrogradsky's theory, )

.

V

G

w ³³ E . dS ³³³ div E dW , gives rise to a S

localized Gauss's theorem: JG U divE = A H0 H r G With D

UA H

.

G H E , we also have: G divD

UA .

G For its part, the integral form of Gauss's theorem with respect to the flux from D is written: G JJG 1. for a discrete distribution of charges, ) D w ³³ D.dS ¦ qi and S

2. for a continuous distribution

i

G JJG

)D

w ³³ D.dS ³³³ UA dW . S

2.3.3.

G G Conditions under which E and D move between two dielectric materials

2.3.3.1. The continuity of potential medium 1

A1

x

x

A2

medium 2 Figure 2.11. Potential at an interface.

54


Figure 2.11 details two points (A1 and A2) each situated in, respectively, medium 1 and medium 2 and both close to the interface. If these two points are sufficiently G close to one another so that E is a constant between them, then A 2 G JG G JJJJJJG VA1 VA2 ³ E.dl | E. A1A 2 A1

At the immediate neighborhood of the interface, A1 tends toward A2 so that A1A 2 | 0 , and therefore VA1 VA 2 so that the potential can be known at the interface. G 2.3.3.2. Continuity of the tangential component of E For two trajectories, A1B1 (in medium 1) and A2B2 (in medium 2), which are equal neighbors around the interface surface (S) described in Figure 2.12a, given the continuity of the potential at the interface, we have: VA1 VB1 VA 2 VB2 so that E1t A1B1 that:

E 2t A 2 B2 . From also knowing that A1B1

E1t

A1

H1

B1

A2

H2

B2

(a)

medium 1 medium 2

A 2 B2 , we can determine

E 2t . G -n

G G n = - n1 G 2 medium 1: D1 G medium 2: D 2 G G (b) n1 = n

G n12

Slateral

Figure 2.12. Setup to study continuity of (a) Et and (b) Dn.

G 2.3.3.3. Continuity of the normal component of D 2.3.3.3.1. No real charges at the interface The construction of a small parallelepipedal element around the interface with infinitely small lateral sides makes it possible to study the properties of the G displacement vector D in the neighborhood of the interface where Slateral o 0. For this system shown in Figure 2.12, we have G JJG )D w ³³ D . dS ³³³ UA . dW . Sparallelepiped

As there are no real charges inside the parallelepiped, we have:


)D

w ³³

G JJG D . dS

Sparallelepiped

³³

0

G JJG D . dS

Ssup erior

55

G JJG D . dS

³³ Sinf erior

JJG

G

³³ D . dS . S lateral

With the upper and lower faces being opposite one another and identical, we have Ssuperior = Sinferior = Ssi . In addition, Slateral § 0. G G G n1 - n 2 , and by stating that the displacement vectors are By making n G G such that D1 is in medium 1, and D2 is in medium 2, we then have: G JJG G JJG ) D = ³³ D . dS ³³ D . dS Ssup erior

Sinf erior

G G G G ³³ (D2 .ndS D1.( n)dS)

³³ (D2 .n D1.n)dS 0

G G

G G

Ssi

Ssi

JJG JJG where n1 is normal to the exterior of medium 1 and n 2 is normal exterior to G G G G medium 2. From this can be determined that D2 .n D1.n 0 , and thus D1n

D2n .

2.3.3.3.2. Polarization charges and no real charges at the interface There are not always real charges in the Gaussian volume (the parallelepiped), which means that ) D 0 and that the preceding result, D1n D2n , again is valid. 2.3.3.3.3. Distribution ( VA ) of real charges at the interface (surface layer of real charges) For Gauss's theorem applied in Figure 2.12, there now is a parallelepiped that contains a total interior charge ³³ VA dS , which can be written as: Sinterface

)D

G G G G ³³ (D2 .n1dS D1.n 2 dS) Ssi

³³ VA dS . Si nterface

Where Sinterface = Ssi , and with the same notations as above, G G G G ³³ (D2 .n D1.n )dS ³³ VA dS , which is to say Ssi

Ssi

D2n D1n

VA .

56


2.3.4. Refraction of field or induction lines G D1

D1n = D2n

G n M 1 M2

G E1 G E2

G D2

medium H 2

E1t = E2t

medium H1

Figure 2.13. Field and induction lines refracted.

It is assumed here that there are no real charges at the interface, so that we can go on to write the two equations of continuity (Figure 2.13): E1t

E 2t , so that E1 sin M1

E 2 sin M2 (1)

D2n , so that D1 cos M1 E Dividing Eq. (1) by Eq. (2) we obtain 1 tan M1 D1

D2 cos M2 (2)

D1n

tan M1

H1

tan M2

H2

E2 D2

tan M2 , which also gives:

,

and with H1 < H 2 , we find also that M1 < M2 . 2.4. Relations between Displacement and Polarization Vectors 2.4.1. Coulomb's theorem The theorem concerns the conductor-dielectric interface. A condenser with an armature of a known surface (S) carries a total charge (QT) so that QT VT S (Figure 2.14). medium , QT = VT S ++++++++++++ medium 2, H (dielectric) --------------medium 1, -QT = -VT S

VT

- VT

Figure 2.14. Configuration used to establish Coulomb's theorem.


57

With the condenser in equilibrium, on the inside of its metallic electrodes, which as such are equipotential, we have: JJG G G E1 0 and D1 Hmetal E1 0 . At the interface between the two materials, we have D2n D1n VT , so that here, G VT with D1 0 , D2n VT which also can be written as E 2n . H The continuity of the tangential component of the electric field allows us to see that E 2t 0 (as E1t 0 ), so that in the dielectric the field is normal to the armature of the condenser. The field that acts between the armatures (effective field) therefore is equal to VT Ea . (3) H G G 2.4.2. Representation of the dielectric-armature system and P (H H0 )E a As previously mentioned in this text, a material with a dielectric permittivity (H) can also be seen (on a microscopic scale) as a vacuum in which dipolar charges sit (attached to atoms making up the material). Therefore, regarding the armatures, the surface density of dipolar charges ( B VP), detailed in Section 2.1.4, annihilate an equivalent number of charges (with an opposing sign r VP) carried by the armature surfaces, as shown in Figure 2.15a. Charges belonging to the upper armature not canceled out by the dipolar charges thus have a density V0 such that VT

VP V0 (4).

The resulting problem with respect to the charges shown in Figure 2.15b is that of a condenser with arms carrying charges of density r V0, while there is a vacuum with a permittivity H0 separating the armatures. The effective electric field between the armatures therefore is: V0 Ea (5). H0 From Eqs. (3) and (4), we have V0 H0 E a , we also find that G V P P VT V0

VT

V P V0

HE a H 0 E a

HE a , as from Eq. (5), (H H 0 )E a .

G G In terms of vectors and for P // E a (detailed in Section 2.1.4) we arrive at G G P (H H0 )E a .

58


V0

VP

V0 self-cancelling charges

-VP vacuum H0

vacuum H0

Ea

V0 . H0

VP

- VT

- V0

Figure 2.15. Microscopic analysis of charges in (a) a polarized dielectric and (b) a vacuum.

2.4.3. Linear, homogeneous, and isotropic dielectrics and the relation G G G D H0 E a P 2.4.3.1. Lhi dielectrics G A lhi dielectric is one that gives a response to either a displacement ( D ) or G G polarization ( P ) vector in which the excitation of an effective electric field ( E a ) can be described by a relation: G G 1. D HE a where H is a real number in a linear system and the dielectric can be called perfect, which is to say it has no dielectric losses associated with leak currents through the material (as detailed later on, when there are losses, H becomes a complex number); 2. which is true at any point in the dielectric, i.e. it is homogeneous; and 3. which is verifiable in all directions throughout the dielectric (isotropic). G This restates the working hypothesis set out in Section 2.3.1.4 with the notation E { G Ea . G G G G The relation between P and E a is P (H H0 )E a , and this can be rewritten as: G G G P (H H0 )E a H0 (H r 1)E a , which makes: Fd

(Hr 1)


59

where Fd is the dielectric susceptibility, and leads to: G P

G H0 Fd E a

G G G 2.4.3.2. The relation D = H0 E a P G G G Starting from P (H H0 )E a , we can write P G D

.

G G HE a H0 E a

G G D H0 E a , so that:

G G H0 E a P .

2.4.4. Comments Comment 1. As indicated in Section 2.1.4 and recalled in Section 2.4.3.1, when an G electric field ( E ) is denoted without any further precision, it is the effective field which is under discussion. G G G G G G G In addition, we have D HE , P H0 Fd E , and D H0 E P . G Comment 2. The latter relationship ( D

G G H0 E P ) shows how the electric G induction comes from the polarization ( P ) of the dielectric, on which is G superimposed an effect ( H0 E ) of the corresponding space vacuum, which would mean that a vacuum can be seen as a dielectric with zero polarization.

Comment 3. Poisson's equation and dielectrics JG JJJJG JG UA U , we directly have 'V A On substituting E grad V into divE H H replacing H0 by H, we have the same relation for a vacuum.

0 . By

Comment 4. Gauss's equation and dielectric media without real volume charges where UA 0 Gauss's theorem, which was written for a dielectric of permittivity H containing a distribution of real charges ( UA ) in the form G JJG G UA ) w ³³ E . dS ³³³ div E dW ³³³ dW . This can be rewritten to take into account H S the equivalence of a dielectric to a vacuum in which the volume densities of real charges ( UA ) with a polarization UP are found distributed so that G G JJG G UA U P UA div P ) w dW ³³³ dW . ³³ E . dS ³³³ div E dW ³³³ H0 H0 S The localized form can be written as

60


G JG UA div P divE = H0

.

However, if the dielectric does not contain real polarization charges, then: G UA 0 (if UA 0 ) 1. div E H G G G UA div P div P 2. and div E (if UA 0 ) H0 H0 G div P

0 if UA 0 . G As UP divP , therefore UP 0 when UA 0 , and we return to the result already given in Section 2.2.2.4. The volume charges equivalent to the polarization are zero when the dielectric contains no real volume charges, and so polarization charges exist only at the surface of the dielectric (Vn Pn ) .

2.4.5. Linear, inhomogeneous, and nonisotropic dielectrics For this particular case, within a Cartesian framework, the Dx, Dy, and Dz G G components of D , and the Ex, Ey, and Ez components of E are all related by the relation: H xz · ª E x º ª D x º § H xx H xy ¸« » « » ¨ H yz ¸ « E y » « D y » ¨ H yx H yy ¸ « D » ¨¨ H H zy H zz ¸¹ «¬ E z »¼ ¬ z ¼ © zx The matrix for permittivities is symmetric in that Hij

H ji , so the matrix is

diagonal for a system with axes OX, OY and OZ, which are the electric axes for the media, making it possible to state that: DX H XX E X , DY H YY E Y and D Z H ZZ E Z .


61

2.5. Problems: Lorentz field 2.5.1. Dielectric sphere z dielectric (H) d²S’’ G E ext

VP > 0

G P

T

G n

+ d²S’ + + + ++++ dS + + +PG O T + + d²Ed’ d²E d d²E+d’’ T + G n VP < 0

y

vacuum: H0

+ G A uniform external field ( E ext ) is applied on a dielectric sphere with absolute permittivity H. The sphere is assumed to be of small dimensions with respect to the G distance between the armatures producing E ext , so it is worth noting that the above scheme is not to scale. 1. Determine the charges equivalent to the polarization. G G P 2. Show that at the center (O) of the sphere, the depolarizing field is E d . 3H0

3. Calculate the resultant field at O. Give the expression for the polarization vector. G G 1. The polarization vector ( P ) is parallel to E ext , and the surface polarization charge densities on the upper half of the sphere are Vp P cos T > 0 as in this zone S . On the lower half, < T < S and Vp < 0 . 2 2 These signs are evidently in accordance with the orientation of the dipoles due to the electric field. 0 H0. (b) The polarization charges can be of two types: volume ( UP ) or surface ( VP ). As G G G UP divP , so that UP (İ-İ 0 )div E , with E and the localized form of Gauss's G UA (İ-İ 0 ) theorem, divE , we have UP UA , so that UP 0 , as the volume H İ density of real charges in the dielectric ( UA ) is zero.

74 Basic electromagnetism and materials

GG The surface charges, VP P.n ext , use external normals to the dielectric. Given the directions they follow, which are Oy at the two lateral interfaces of the cavity (parallel to the Oxz plane) and Ox at the two sides behind and in front (parallel to the plane Oyz), only surface charge densities at the upper and lower sides (parallel to the plane Oxy with normals along Oz, in the direction of the polarization) are zero. Therefore: GG G G • on the upper face, VP P.n ext upper face P.( ez ) - P < 0 ; and GG G G • on the lower side VP P.n ext lower face P.(ez ) P>0.

2 (a) Any point M on either the upper or lower side can be associated with a symmetrically equivalent point M' through Oz. The components following Oy' for JJJJG an electric field ( dE P ) generated by the polarization charges at these two points are opposed so that the resultant component of the electric field is directed along Oz. In addition, given the sign of the polarization charges, the sense of the electric field is JJG the same as Oz whether the points are respectively on upper or lower sides. As a JJJG JJG result, we have E P & Oz . (b) We have dE p dE z

1

VP dSP

4SH0

r²

, for which the projection along Oz is:

dE P cos T . With |VP | = P , we also find: dE z

M’

1

P cosT dSP

4SH0

r²

.

M dSP

JJJJG dE 'P

JJJJG dE P dS T

T

O

As the solid angle through which a surface element dS of the sphere is dS d: where dS and dSP are such that dS dSP cos T , from which r² 1 P dS P dS dSP cos T dE z d: and d: represents the solid angle 4SH0 r² 4SH0 r² r² through which the surface dSp is seen from O.


75

P

³ d: where the integral must be taken over 4SH0 the solid angles that cover the charge-carrying upper and lower sides of the cavity seen from O. (c) The solid angle through which the whole of the cube can be seen from the point O, and therefore all of its six sides, and is a solid angle through all space is 4ʌ steradians. With the six sides all being equivalent, 1 side can be seen through the 4S 4S 4S u2 solid angle , and two sides thus are seen through steradians. We 6 6 3 P P 4S P therefore find that E Z . ³ d: 4SH0 4SH0 3 3H0 Finally, we have E Z

(d) In terms of vectors, and given the answers in 2 (a) and (b), we can state that: G G P G P Ez ez . 3H0 3H0

2.7.2. Polarization of a dielectric strip z

vacuum H0 G dielectric strip H E ext O

y

A thin dielectric strip with an absolute dielectric permittivity equal to H is placed parallel to and in between condenser armatures which generate an uniform JG “external” field denoted E ext , as shown in the figure above. It is assumed that the dielectric, like the armatures, has infinite dimensions and a center of gravity called O. Its thickness is such that the field external to the strip is not modified by polarization charges, and that the field inside the strip is uniform. 1. Algebraically indicate the value of the polarization charges and indicate their position on a figure. The value of the charges should be expressed as a function of the dielectric polarization (P). 2. With the help of an equation for the continuity for the dielectric strip-vacuum JG interface and as a function of E ext and Hr, directly find the expression for the


G resultant field in the strip ( E L ) and then that of the depolarising field and polarization vector.

Answers 1. The polarization vector is in the same direction and sense as the electric field and follows Oz. The external normal at the upper face of the strip goes along Oz, but at the lower face goes along –Oz. The result is that the superficial charge density GG equivalent to the polarization VP = P.n ext is positive on the upper side and negative on the bottom. These polarities are realized simply by considering the orientation of the permanent dipoles in the strip when subject to an external field. With the strip not being charged in its volume, then UP

0.

z

---vacuum H0 G P

+dielectric + + + + +strip + + H+ O G E ext

G n ext

G G P & n ext VP ! 0 + ++++ y G G G P & n ext VP 0 n ext

++++ 2. By taking the upper interface as an example, the continuity of the component normal to the electric (displacement) induction can be used. As there is no real charge surface density ( VA 0 ), we have Dn - D n0 = 0 where Dn HE L (induction in the dielectric for a field denoted E L ) and Dn0 H0 E ext . The result is HE L H0 E ext 0 , and given the same sense of the two fields, G G H0 G E ext EL E ext . H Hr G G G G The depolarizing field ( E d ) is such that E L E ext E d , where G Ed

§ H 1· H 1 G E ext . When H r > 1 , we have ¨ r r ¸ < 0 and it is confirmed that ¨ Hr ¸¹ Hr © G G E d is antiparallel to E ext . In the dielectric strip, the polarization vector given by G G G G G G E ext P H H0 E a is such that the effective field E a E L , so that E a and Hr

G P

H0 H r 1 G E ext . Hr


77

2.7.3. Dielectric planes and charge distribution (electric images) In a trihedral about Oxyz, the plane xOz separates a vacuum (permittivity H0) in the region y > 0 from a dielectric (permittivity H) which extends through the lower half of the space where y < 0. At S, which has the coordinates (0, a, 0), there is a charge (q). An additional point of reference is S' which is symmetrical to S about O and therefore has the coordinates (0, -a, 0); we also have r = SP and r’ = S’P. This problem concerns the Oxy plane and a point P with coordinates (x, y, 0) from an electrical point of view. y S H0

P (x, y,0) z O

x

H S’

1. In order to deal with this problem, it is worth showing that at the point P (x, y, 0), the scalar potential (V) is such that: 1 ª q q1 º • when y >0 : V1 (P) « » 4SH0 ¬ r r' ¼ 1 q2 • when y < 0 : V2 (P) 4SH r where q1 and q2 are constants. 1 q 1 q1 1 q2 (a) What do the potentials , , represent? What actual 4SH0 r 4SH0 r ' 4SH r physical origin might they have? (b) Calculate as a function of x and y the Cartesian components for the field G G vectors E1 (P) and E 2 (P) which are derived from V1(P) and V2(P), respectively. (c) Determine the two constant q1 and q2. 2. (a) Show that the charges equivalent to the polarization are only at the surface. (b) Determine the density of the surface charges and the total charge through the plane xOz, noting that the distribution is around the axis Oy.


Answers 1. (a) V (1)

1

q

represents the potential generated at P by the charge q at S. The 4SH0 r dielectric in the region y < 0 is polarized by q. The overall polarization of the dielectric can be taken into account by using an intermediate charge (q1), which 1 q1 remains to be determined, at S'. The potential V'(1) = is a representation of 4SH0 r ' the problem using a reflected electric image for which q1 must be determined. This representation is more acceptable when seen as an analogy of an optical system, where it is possible to imagine a mirror, for example, reflecting an image of an object of a similar size and shape to q to give q1, much as a fisherman might see his float (F) dangling in the air taking on the form F1 as y an image in the water. The total potential in the vacuum is therefore: G JG S q uJJJJ S 'P 1 ª q q1 º (1) (1) V1 (P) V V ' « ». P H0 G 4SH0 ¬ r r ' ¼ uJJG SP This equation also can be interpreted by imagining x the fisherman sitting on the bank in the air H (permittivity equal to H0), and him being able to see B + B1, both his float and its reflection. S’ q1 1 q2 represents the The potential V (2) 4SH r potential at point P (which is in the dielectric), generated by a charge (q2) at S. Carrying on with the analogy, here it is as if the charge q at S is being observed in the shape of q2 by the fisherman who has dived into the water to be at P (y < 0). Once again though, this representative is only valid if q2 can be determined. G JJG G JJJJG (b) Bringing in the unit vectors uSP and uS' in the directions SP and S’P gives: P

G JJG uSP

JJG SP JJG SP

x ° °r ° (y-a) ® ° r ° ° ¯0

and

G JJJJG uS' P

JJJG S' P JJJG S' P

x ° ' °r ° (y a) ® ' ° r ° ° °¯0


79

G G The result is that the fields E1 and E 2 are from the same charges generating V1(P) and V2(P) and are such that: x ° ° 4SH0 G °° 1 E1 ® ° 4SH0 ° ° °¯0

q1 º ªq « 3 3» r' ¼ ¬r ª q y a q1 y a º « » 3 r '3 ¬« r ¼»

1 q2 x ° 3 ° 4SH r G °° 1 q 2 y a E2 ® r3 ° 4SH ° ° °¯0

and

(c) There remain two constants to determine, namely q1 and q2. To do this we need two equations with two unknowns, and these can be derived from the limiting conditions at the interface y = 0. At this plane there is the continuity of the tangential component of the electric field, and in the absence of a real superficial charge a continuity of the component normal to D. At the interface where r = r’, y = 0, and 'x = x', we thus have: E1x = E 2x and H0 E1y = HE 2y , which give, respectively: ª q q1 º « » 4SH0 ¬ r 3 r 3 ¼ x

1 q2 x 4SH r

3

and

1 ª q a q1 a º « » 4S ¬ r 3 r3 ¼

1 q 2 a 4S

r3

.

From which we also find: q q1

q2

H0

H

and q q1

q2 .

From this can be deduced that q1

H0 H H H0

q , and

2 (a) There are no volume charges in a dielectric, so UA 0 with the result that UP 0 (see course work). Only surface charges equivalent to the polarization can be present. GG G (b) We have VP P.n where n is the normal outside of the dielectric as shown in the figure. The result is that VP Pcos (S - T) = - P cosT .

2H

that q 2

H H0

q.

y S

H0 H

a

T r

G n

T x

x G P


G To calculate the polarization, P

G

H H0 E 2

G wherein the term E 2 intervenes as

the field in the dielectric. Deriving the potential, V2 (P) G E2

1

q2

4SH

r G r

1 4SH

q2

1

r3

from which G P

G , we have for the field E 2 ,

1

1

q

G r

2S H H 0 r 3 G H H0 r

. 2S H H 0 r 3 q H H0 1 cos T , and with cos T From this can be derived VP =- P cosT 2S H H 0 r 2 H H0 1 cos ² T from which . By making A q , we find that r² a² H H0 VP

q

A 2Sa²

cos3 T .

y T

dS = 2S R dR R

plane Oxz

In the Oxz plane, the resultant of charges equivalent to the polarization is: a dT . qp ³ VP dS , where dS = 2S R dR , and R= a tan T and dR cos ²T planOxz S/2

This gives q p

A

³ sin T dT A

T 0

q

H H0 H H0

.

The angle T in the figure varies from 0 to ʌ/2 so as to cover the whole surface when integrating over the whole surface of the Oxz plane.

a r


81

2.7.4. Atomic polarizability using J.J. Thomson's model To represent an atom of hydrogen, J.J. Thomson used the following model. A fixed sphere with a radius R, a center O contains a charge (+q) uniformly spread throughout its volume. The electron charge (-q) is thought of as a point charge and moves in the sphere, assuring the overall neutrality of the atom. All charges are considered to be moving in a vacuum. G 1. Calculate the internal electric field ( Eint ) generated by +q for a point M inside G G G the sphere and located by the vector OM = r = r er . 2. Determine the attractive force between +q and the electron (-q) at M. Detail the equilibrium position of minimum energy for the electron. G G 3. An external field ( E ext ) is now applied along er to the system. Determine the

G

G

value of d which r must take so that the system is in equilibrium. 4. For the equilibrium position, derive the equation for the dipole moment induced G by E ext and then from this the polarizability of the atom.

Answers 1. Gauss's theorem applied to sphere with radius r (Gaussian surface) gives: G JJG 1 ) ³³ Eint .dS ³³³Volume U dW , from which H0 int érieur 4S r² Eint =

U 4 H0 3

derive that Eint

q

Sr 3 . Then with U q

4 / 3 SR 3

R O

G r

, we can

r

, and as q > 0, in terms of vectors gives: 4SH0 R 3 G G q r G r G Eint er where er . 3 4SH0 R r G G G 2. The force exerted on an electron at M, such that OM = r = r er , is given by G G q² r G F qEint er . The energy of the electron ( W(r) ) is such that 4SH0 R 3 r r G JJG JJJJG G F gradW , so that ³ dW ³ F.dr , from which can be deduced that 0

q²r²

0

+ constant. Thus W(r) is at a minimum when r = 0, and therefore 8SH0 R 3 G also the force F is canceled out at this position, which would make it an equilibrium position for the electron. W(r)


G G 3. In the presence of E ext applied along er , the resultant force on the electron is G G G G G FT q Eint E ext . The equilibrium position, the value of r equal to d , again G G G corresponds to when FT 0 in that Eint G G E ext . From this can be

G determined that d

r

d

3

G G 4SH0 R G E ext and that d is antiparallel to E ext . q

G G 4. E ext moves –q by d . This in return induces a dipole moment, which can be G G G defined by Pi q d 4SH0 R 3 E ext . The polarizability (D), as given by the relation G G Pi D E ext , therefore has a value of D 4SH0 R 3 .

2.7.5. The field in a molecular- sized cavity Preliminary comments Debye's theory The calculation for the effective localized field in a spherical cavity above was carried out with the assumption that the field was the result of a superposition of the applied external field and Lorentz's field, the latter being generated by polarization charges at the surface of a dielectric. Assuming that inside the cavity the molecules are distributed with a symmetry such that the local field they generate is zero, then the resultant field in the cavity experienced by the molecules is the Debye field G G G K P ( ED ), given by E D E ext . 3H0 Onsager's theory Onsager's theory was introduced to account for localised interactions between molecules. The method used is to superimpose two steps to give the result, respectively, given below in 1, 2, and 3. G G 1. Internal field ( G ) formed by external field in a small cavity ( E z 0 and P 0 ) An external field is applied to a small, molecule-sized cavity that does not contain molecules. First the internal field (G) in the cavity is calculated assuming that it is of the same form as the external field except modified by refraction at the dielectric interface. A molecular-sized spherical cavity, of radius a and a center that is taken as the origin, filled with a dielectric of known absolute permittivity (H1), is placed in a G dielectric medium, with an absolute permittivity denoted by H2. If a field ( E ) is applied such that it is parallel to the Oz axis and far from the cavity, as shown in the


83

figure, the sphere distorts the field lines in its neighboring volume. As there is a symmetry around the axis Oz, the problem is not reliant on an azimuthal angle M. Given this symmetry, for a point P with spherical coordinates [r,T,M], we can use the Oxz plane where M = 0 so that the potential can be written as: z • in the volume outside of the cavity ((r > a), A cosT P ; and V2 = - E r cosT + r² T G E • in the cavity (r < a), V1 = B r cosT , r where A and B are constants. x O H1 (a) Give a physical justification for the form of the two potentials. H2 (b) Determine the constants A and B. G (c) Give an expression for the internal electric field ( G ), G H2 which can be expressed as a function of E and H r . H1 N.B. It also is given that (gradV)r

dV dr

and (gradV)T

1 dV r dT

G 2. Reaction field ( R ) in the absence of an external field, due to polarization charges G induced by dipole in the cavity ( E=0 , dipole moment (µ) is non zero) A pinpoint dipole with an axis in the direction Oz is now at the point O. This dipole causes a polarization of the spherical surface. It is supposed that the potentials are similar to those found in Section 1 just above.

(a) Show how the potentials now can be described by: C cosT • outside the cavity (r > a), U 2 ; and r² D 'cos T • in the cavity (r < a), U1 = D r cosT + r² where C, D and D' are constants to be determined. (b) By using the limiting conditions of the interface, determine C and D as a function of D'. Considering a limiting case, give D' as a function of the total moment (µ). Give the final form of U1 and U2. G (c) Given that the reaction field ( R ) is that which is induced by a dipole at the G surface of the cavity, find R in terms of the corresponding component of U1 .


3. Onsager's field (a) Show how the internal field acting on the molecule (Onsager's field) takes on G G r µG the form F gE o . Detail the constants g and ro as a function of Hr and H1 (often a3 assumed equal to H0). G (b) Formulate the expression of the resulting dipole moment ( µ ) for the molecule G located at the center of the cavity that exhibits a permanent moment ( m ) and a G polarizability (D) in the presence of the electric field E . z Answers G 1. (a) Given the direction of E , it can be supposed that the polarization charges at the interface of the two media P will give rise to dipoles with a resultant directed along G T Oz. As the sphere is of molecular dimensions, the E r µr distance between dipoles is around the same as that as the x µ1 dimension of a molecule—which well generates dipoles. µ 2 In effect, any given dipole (µ1) will have a symmetrical +++ equivalent (µ2) so that combined their resultant (µr) will H2 follow Oz, as shown in the adjacent figure. The sense of direction of the dipoles will depend in the polarity of the polarization charges, and hence also the respective values of H1 and H2. By consequence, for a point P outside of the P G JG G G JJJG cavity, the potential due to E is such that VP VO ³ E.dl E.OP= - E r cosT , O

so that VP = - E r cosT (at potentials with respect to the origin O), to which must be added the potential due to the resultant dipole µr which is of the form: G G 1 G JJJJJJG 1 1 P r .r 1 P r r cos T VdP P r .grad P , with 3 4SH2 r 4SH 2 r 4SH 2 r3 Pr A cos T A VdP . 4SH 2 r2 The resultant potential in the medium outside the cavity with permittivity H2 is therefore: A cos T V2 = VP + VdP = - E r cosT ( Q.E.D.). r2 In the cavity the field E is modified by the refraction of field lines so that form of the first term for V2 is modified to become B r cos T where B is unknown for the present moment. The second term will disappear inside the cavity: If it were


85

kept, when r o 0 the term would diverge and so that this does not happen A has to go to zero, thus removing the component. In the cavity (permittivity H1), the potential is thus in the form V1 = Br cosT . (b) The two equations of continuity for the electric field and induction can be used to find the unknown constant A and B. For r = a, the equations are E1t E 2t and G D1n = D 2n . The “tangential” and “normal” directions following, respectively, eT G and er , can be described by (E1T E 2T ) r a and (D1r D 2r ) r a , so that 1 § wV1 · ¨ ¸ a © wT ¹r a

1 § wV2 · § wV · and H1 ¨ 1 ¸ ¨ ¸ a © wT ¹r a © wr ¹r a

§ wV · H2 ¨ 2 ¸ . © wr ¹r a

From these can be determined that H 2 H2 A B E and B 2 E A so that 3 H a a 3 H1 1 A

H1 H 2 H1 2H 2

a 3 E and B

3H 2 H1 2H 2

E.

Therefore, the potentials V1 and V2 are: 3H 2 H H1 E cos T 3 V1 Er cos T , and V2 Er cos T 2 a . H1 2H 2 H1 2H 2 r 2 JJJJG gradV1 , where V1 = Br cosT , so G that with z = r cosT we have V1 = B z . The result is that the only component of G is: wV 3H 2 G z 1 B E , which in terms of vectors gives wz H1 2H 2 G G (c) The field G in the cavity is such that G

G G

3H r 2H r 1

G E.

2. (a) In general terms, the potentials can be written as: C cos T U 2 = -Er cosT , r2 C cos T . so that with E = 0, we have U 2 r2


And U1 = D r cosT

D 'cos T

where O is a singularity (a pinpoint dipole with zero r2 moment) so that r can no longer go towards zero. (b) The limiting conditions 1 § wU1 · ¨ ¸ a © wT ¹r a C

3H1 2H 2 H1

1 § wU 2 ¨ a © wT

· § wU · and H1 ¨ 1 ¸ ¸ ¹r a © wr ¹r a

D ' and D

2(H2 H1 ) D ' 2H 2 H1 a 3

§ wU H2 ¨ 2 © wr

· give ¸ ¹r a

.

The potentials are therefore: U1

2(H 2 H1 ) D ' r cos T D 'cos T and U 2 2H2 H1 a3 r2

3H1

D 'cos T

2H 2 H1

r2

.

If it is proposed that the surface of the spherical surface can become quite large, the potential (U1) will be that generated by an isolated dipole of total moment µ in a medium of permittivity H1, with a value at a coordinate point (r,T) given by µ cos T Ud so that when the surface increases, and 1/a² o0, we have 4SH1r 2 µ cos T 4SH1r

2

U d | (U1 ) 1 a

Finally, U1

2

D 'cos T o0

r

2

, from which by identification D '

µ 4SH1

.

2(H 2 H1 ) r cos T µ cos T , 4SH1 2H 2 H1 a 3 4SH1r 2 µ

and U2

3

µ cos T

2H2 H1 4Sr 2

.

(c) In the equation for U1, the potential inside the sphere, the second term is simply due to the dipole itself and the first term is the potential caused by charges at the surface of the sphere induced by the dipole. The resulting field is the reaction field G ( R ). G With z = r cosT , R can be derived from the potential


2(H 2 H1 ) z , so that R 4SH1 2H 2 H1 a 3 G 2(H 2 H1 ) µG terms of vectors, R 2H2 H1 4SH1a 3 U

µ

2(H 2 H1 ) µ and in wz 2H 2 H1 4SH1a 3 G 2(H r 1) µ . 2H r 1 4SH1a 3

wU

3. G (a) The internal field acting on the molecule is given by F G F

87

G G G R , and thus

G 2(H r 1) µG E . 2H r 1 2H r 1 4SH1a 3 3H r

G r µG gE o , where a3 1 2(H r 1) . 4SH1 2H r 1

G This can be written then as F g

3Hr 2H r 1

and ro

G G (b) The total moment ( µ ) of the molecule is the sum of its permanent moment ( m ) G G G G G r Dµ G G and its induced moment ( DF ), so that µ m DF m DgE o , from which a3 G G G m DgE we have µ . ro D 1 3 a

This equation shows how the reaction field modifies the molecule's moment.

Chapter 3

Magnetic Properties of Materials

3.1. Magnetic Moment 3.1.1. Preliminary remarks on how a magnetic field cannot be derived from a uniform scalar potential JJG G JJG G G µ0 j , shows how rot P B is Ampere's theorem written for a vacuum, rot P B only zero at points (P) which are without current, and therefore is nonzero G elsewhere. The B therefore cannot be derived from a uniform scalar potential (V), G JJJJG and in effect, if we could write for all P that B grad P V , we will end up with JJG G rot P B = 0 , which is not true as we have just seen. Nevertheless, we can define a JJJJG G pseudoscalar potential (V*) such that B grad D V where D represents points JJG G without current, as at these points rot D B 0 (a pseudoscalar is a scalar that is defined by its being limited to certain points). 3.1.2. The vector potential and magnetic field at a long distance from a closed circuit 3.1.2.1. Form of the vector potential G G G Here we will calculate the vector potential A and the induction B at a distance rM far from a current in a closed coil that is considerably smaller than the distance JJJG G G shown in Figure 3.1. For a point P in a vacuum, for which rM MP , the vector A given by the coil (C) in which a current of intensity i moves is given by: JG G P0i dl A v³ . 4S rM This also is referred to in Section 1.4.2.

90


(S)

N’ N

G uN

JJJG G rM = MP

G uM

(C) (i)

P

M JJG dl

Figure 3.1. Coiled current.

G G G If for M we consider a unit vector ( u M ) such that u M constant u whatever JJJG P i u JG G G the position used, the scalar u M .A v³ 0 M dl can correspond to the circulation 4S rM JJJG G P0i u M . around C of the vector G M 4S rM G Stokes theory written for the vector G M gives: JG JJG G JJG G G G u M .A v³ G M . dl ³³ rot N G N .dS , C

S

where N is on the surface (S) through C. JJJG JJJG G P0i u N G So now with G N and rN NP , we find 4S rN JJG G rot N G N

JJJJG § µ i µ0i JJG G rot N u N grad N ¨ 0 ¨ 4Sr 4SrN N ©

· G ¸¸ u u N . ¹ G Yet with the vector u N (detailed in Figure 3.1), the components JJG G G (xN’ – xN), (yN’ – yN), (zN’ – zN), which allow u N 1 , are such that rot N u N (as for example

w (z N ' - z N ) wy N

We therefore have: JJG G JJG G G u M .A ³³ rot N G N .dS S

0 ).

µ0i § JJJJG 1 G · JJG u u N ¸ .dS ¨¨ grad N ¸ rN S 4S © ¹

³³

0

Chapter 3. Magnetic properties of materials

G G u M .A

µ0 i 4S

§ JJG

JJJJG

³³ ¨¨ dS u grad N ©

91

1 ·G ¸.u N rN ¸¹

JJJJG 1 · º G ª µ i § JJG u N . « 0 ³³ ¨ dS u grad N ¸» . ¨ rN ¹¸ »¼ «¬ 4S © G G From u M .A

G G u N .A

G G G A cos (u, A) it can be inferred that A can be written as: G A

µ0 i 4S

§ JJG

JJJJG

³³ ¨¨ dS u grad N S©

1 · ¸. rN ¸¹

(1)

3.1.2.2. The approximation r >> C and the magnetic moment

3.1.2.2.1. The nature of the approximation We can justify the approximation by recognizing that in real applications the coils through which the current passes are extremely small with respect to the distances over which the magnetic effects can be dealt with. More specifically, the JJJJG 1 is constant for each and every point (N) over approximation states that grad N rN the surface S, which can be whatever size, although it depends directly on the size of JJJJG G C, and that rN NP | constant, with respect to the position of point N. In other words, from a position P outside of the coil, we would see that any part of the coil would be a distance r away, and indeed, the coil would seem miniscule from P. We also can see that there is an analogy with the electric doublet considered in the section on electrostatics, in which it is also supposed that r § constant. 3.1.2.2.2. Magnetic moment G iS

G M

(C) i Figure 3.2. The magnetic moment from a circular current.

The magnetic moment is introduced by definition as a vector: G M

i

JJG

³³ dS S

(2)

92


thus giving it its name – spin angular magnetic moment – as schematized in Figure 3.2. Given that only C has been determined and that S is dependent on C, the magnitude of Eq. (2) results from Eq. (1). Taking into account Stokes relation, the magnetic moment therefore depends only on the size of C. If C is a flat coil, then S can be simply that defined by the coil G C, and in which case M is normal to the plan of the circuit so that its modulus is iS. G The direction of M is defined by the corkscrew rule, which gives the rotational G direction of i through the coil. (Stokes theory states S that the direction is given by JG that of the current i dl .) JJJJG 1 | constant Carrying Eq. (2) into Eq. (1), we can write (when grad N rN JJJJG 1 grad N ) that r JJJJG 1 G µ0 G A M u grad N . 4S r This expression can be turned around:

G A

JJJJG 1 G M u grad P . 4S r

µ0

(3)

JJG G G Given that the circuit is fixed in space, and that rot P M = 0 as M is independent of P: JJG § 1 P0 G · rot P ¨ M¸ © r 4S ¹

G 1 P0 JJG G § P0 JJJJG 1 · rot P M ¨ grad P ¸ u M r 4S r¹ © 4S

we therefore arrive at: G A

G JJG § P M · rot P ¨ 0 ¸ . ¨ 4Sr ¸ © ¹

3.1.2.2.3. Magnetic field G G We know that B , for a point P, is given by B JJG JJG so (with rot rot

JJJJG G G grad div - ' ) we have B

P0 G JJJJG 1 Mu grad P . 4S r

(4)

G JJG JJG § P0 M · rot P rot P ¨ , ¨ 4Sr ¸¸ © ¹ G G M G § M ·º P0 ª JJJJG - 'P ¨ ¸» . « grad P div P ¨ r ¸ 4S ¬« r © ¹ ¼»

JJG G rot P A


93

G However, with M fixed and independent from P, we find that: G G § M · G ª § M x ·º G ª § M y ·º G ª § Mz ·º 'P ¨ ¸ i «'P ¨ ¸¸ » k « ' P ¨ ¸ » j « ' P ¨¨ ¸» ¨ ¸ «¬ © r ¹ »¼ ¬ © r ¹¼ ¬ © r ¹¼ © r ¹ G G G §1· i Mx j M y k Mz 'P ¨ ¸ , ©r¹ G G §M· §1· so, with ' P ¨ ¸ 0 , then ' P ¨ ¸ 0 . ¨ r ¸ ©r¹ © ¹ This finally gives us: G G P0 JJJJG M . (5) B grad P div P 4S r

G Therefore, at a large distance from a coil through which moves i, the vector B is derived from a (pseudo) scalar potential ( V* ) and that G P0 M * V div P . (6) 4S r G The vector B is such that JJJJG G B - grad P V* . (7)

3.1.3. The analogy of the magnetic moment to the (dipolar) electric moment and JJG G I ³³ dS the justification of the term magnetic doublet for M S

We can recall briefly that the scalar potential generated by an electric dipole of G G moment µ q l is such that 1 G JJJJG 1 V=P . grad P . (8) 4SH0 r On comparing Eqs. (3) and (8), we rapidly can see that the potentials take on the same form, the scalar product (in the expression for the scalar potential) can be G substituted by the vectorial product (in the expression for the potential vector A ), JJG G G and the vector M i ³³ dS can take the place of the electric dipolar moment ( µ ). S

G Therefore, by analogy, the term M is called the magnetic dipole moment. At the level of the pseudoscalar potential given by Eq. (6), it also is possible to write:


94

G M

G P G JJJJJG 1 P G JJJJJG 1 (6’), div P M 0 M. grad P 0 M. grad P 4S r 4S r 4S r 4S r G G because div P M 0 ( M is independent of P). We can see in the expression for the magnetic pseudoscalar a resemblance to the relationship for electric potential when G 1 P G exchanging M for µ . There also is the coefficient, 0 to change to for the 4S 4SH0 *

V

P0

div P

P0 1

electrostatic potential, and µ0 which takes on the role of

1

(as detailed in Chapter 1 H0 in the formulation of Poisson's equation). This equivalence again justifies calling G M the magnetic dipole moment. To conclude this section, we have seen that for a very small coil with respect to G the distance r to where the magnetic effects are observed, the vector potential, the pseudoscalar potential, and the magnetic field are given by expressions such as Eqs. (3), (6) or (6’), and (7). These relationships are analogous to those found in dielectrics—which are supposed as vacuums in which “sit” electrostatic dipoles (Section 2.2.1.1). Similarly, a small closed circuit can be associated with a magnetic dipole, and outside of the volume defined by the circuit the magnetic effects can be described by potentials expressed with the help of the magnetic moment JJG G M i ³³ dS . This study is carried over to Section 3.2.2. S

Interestingly enough, in material, the same closed currents can be tied to the movement of electrons in their orbitals. G 3.1.4. Characteristics of magnetic moments M

JJG i ³³ dS S

3.1.4.1. “The right hand rule”: magnetic moments are positive when the rotational sense of the current corresponds to the north pole G K G For electrons in their orbitals, U = - ne, and as i Uv where U < 0, we find i G JJG K antiparallel to v , so that from the “north face”, as defined in Figure 3.3(a), M // Oz , G G G while the kinetic moment l r u mv is such that l is antiparallel to Oz , and G G therefore M is antiparallel to l .


JJG i ³³ dS

z G M JJG dS

i plan north

S

O

G v G r

i G l

G G r u mv

Figure 3.3(a). Northern plan and the direction of the magnetic moment.

With respect to the “south face” shown in Figure 3.3(b), G G Oz , although M remains antiparallel to l . G zl

south face

G G r u mv

O

i

i G M

G M is antiparallel to

JJG dS

JJG i ³³ dS

G v G r

S

Figure 3.3(b). South face and the direction of the magnetic moment. 3.1.4.2. The energy of a magnetic dipole in a magnetic field G B G G M iS

i Figure 3.4. Magnetic dipole in a magnetic field.

95

96


G If the dipole is very small, then it can be assumed that B is constant with G respect to the rest of the dipole, a supposition that indeed is correct if B is uniform. G G If S is the surface of the dipole, the flux from B going through the dipole is: JJ G G G JJG ) = ³³ B.dS B³³ dS . S

S

JJG JJG i ³³ dS , we can carry forward ³³ dS

G From M

S

S

G M i

into ) to obtain

G GM ) B . i In addition, the potential energy of the closed loop circuit (magnetic dipole) traversed by i and placed within the flux ) is E p = - i ) , so that with the preceding expression for ), we have G G E p = - M.B . G G 3.1.4.3. Forces and couples active on M placed in B 3.1.4.3.1. Forces JJJJG G In general terms, we have F - grad E p , so that more specifically JJJJG G G G F grad M.B .

3.1.4.3.2. Couples JG JJJG G G While moving through dl, d: of a dipole, if F and * are the elements of reduction around O for the torsor of forces acting upon the dipole, we have G G G G dW dE p M dB Bd M . G G G G G G § wB · wB wB M dB M ¨ dx dy dy ¸ , Performing the calculation ¨ wx ¸ wy wy © ¹ G recognizing that M is independent of the position of the dipole, means that G G M dB

w

G G

M.B dx wx

w

G G

M.B dy wy

w

G G

M .B dz wz

G G It is also possible to derive the expression B.d M JJJG JJJG G G d: u M . given that d: is defined by d M

G G JG ªgrad M.B º .dl . ¬ ¼

G JJJG G B. d: u M

JJJG G G d:.(Mu B) ,


97

G G G G G G JG G G JJJG dE p M dB Bd M ªgrad M.B º .dl (Mu B).d: ¬ ¼ G JG G JJJG F.dl *d: , which by identity gives G G G G G G F grad M.B and * Mu B .

Finally, from dW

G G G G G Comment: If B is uniform, then F 0 and * Mu B . In effect, the dipole is G G G subject to a single couple with moment * Mu B .

3.1.5. Magnetic moments in materials 3.1.5.1. Introduction to the elementary terms G The kinetic moment ( V ) of a particle with a known mass (m) is the moment of a G G G G G known movement such that V r u mv r u p . G The moment of a force is defined by: * The theory for kinetic moment states that G K therefore if * , V is constant.

G G r u F. G dV dt

G dr

G G G dp up ru dt dt

G G ruF

G * , and

3.1.5.2. Atomic magnetic moments

For every kinetic moment there is an associated magnetic moment. Similarly, for each orbital kinetic moment there is an orbital magnetic moment, and for a spin kinetic moment a spin magnetic moment. Concerning the orbital kinetic and magnetic moments, the orbital kinetic G G G G moment is given by l r u mv and the related magnetic moment is µl iS , as schematized in Figure 3.4. G l

i G µl

G v G r

Figure 3.5. Kinetic moment and the atomic orbital.

98


The current intensity is given by i = -ne where n is the number or orbital rotations (about a radius r) made by an electron of charge –e and speed v. This then v v rv G , and µl = e ʌ r 2 =-e , so that in terms of vectors, gives n = 2ʌr 2ʌr 2 G G r×mv e G G µl = - e =l. 2m 2m e µB , where µB is the Bohr magneton, we have: If we make E 2m G G µl -E l . Concerning the spin kinetic and magnetic moments, the same reasoning can be repeated; however, the speed of the system is so much greater as the distance involved is so considerably smaller that the calculation must be carried out using relativity. So, in relativistic quantum mechanics, the resulting spin magnetic moment G G G K ( µs ) is µs = - 2 ȕ s , where s is the spin kinetic moment. For the atomic magnetic moment then, the resulting moment is G G G µT - E(l 2 s) . For an atom with more than one peripheral electron and within the spin-orbit coupling approximation we have: G G G µT - E(L 2 S) .

3.1.6. Precession and magnetic moments 3.1.6.1. A magnetic field and the gyroscopic effect and the Larmor precession G B G G l dl G

ZL

G l

G v G µl

G E l

electron rotating in its orbital

Figure 3.6. Magnetic field effect in a magnetic moment.


99

G A field B acting on the movement of an electron in its orbital can be schematized as G G in Figure 3.6 as a moment couple due to B on µl . G G G G dl G So * µl u B G G G G dl dt EB u l ZL u l , G G G G dt E l u B E B u l G e G G G and then placing ZL E B B where ZL is the rotational vector, here 2m G collinear to B . G G So the vector l undergoes a rotation around B with an angular frequency, or eB rather a Larmor frequency given by ZL EB . 2m G G The movement of an electron in the plane of its orbit is not altered by B (as l remains constant during the rotation), but the plane of the orbit goes through a G G rotation, with a rotational vector ZL , around B in what is called the Larmor precession. The movement of precession resembles that of a gyroscope. G G dl G ZL u l by E, the direct result is Comment If we multiply the equation dt G G dµl G G G ZL u µl . In effect, the moment µl also goes through a rotation about B , dt G e G G B. characterized by the same rotational vector ZL E B 2m G G 3.1.6.2. Precession and the coupling L,S : internal precession

JJG µL

2

G L

G µS

G µJ

G µS G J

µA G µT

2 G S

G µS 2

Figure 3.7. Precession of the couple L,S.

100


G G When there is a couple such as L,S , first the orbital kinetic moments are coupled, G G G then the spin moments; this gives rise to a resultant kinetic moment of J L S . In G G G G G dJ the absence of an external field, * µ u B 0 and J is a constant of dt movement, so that a fixed direction is retained in space, as shown in Figure 3.7. The figure also shows the kinetic and magnetic moments as, by design, E { -1 , so that we also can see P L L . JJG JJG GG G G However, µL and µS display an interaction of energy 'E v L.S cos(L,S) . G G G G At a given energy, cos(L,S) must be constant and L and S can only conduct one G precession movement (their direction may change) about J , which retains a fixed direction in the absence of an external field. The result of this is that G e G G µT (L 2S) , which is invariably tied to the triangle (L,S,J), which also goes 2m G through a precession about J . G G G In addition, µT can be broken down into a component µJ along J , and a G G perpendicular component µA that turns around J at the precession rate of motion. If G we consider the average value of µ A over an interval of time much greater than the G time required for one rotation, it is actually zero and the effect of µT is the same as G that of µJ (the apparent magnetic moment). G e G G It is possible then to write µJ gJ E gJ where g is the Landé 2m G e G g , where the factor J, such that µJ JJ is factor. We also can state that J 2m called, for its part, the gyromagnetic ratio. G G The factor g can be calculated easily. For example, by calculating µJ and J , we obtain: g

1

J(J 1) S(S 1) L(L 1) 2J(J 1)

.

3.2. Magnetic Fields in Materials 3.2.1. Magnetization intensity Here we suppose that there is a bar of material of a volume dW that is equivalent to having a volume of atoms and/or molecules sitting in a vacuum each having an elementary magnetic moment associated with the movement of electrons in their orbits. This setup is comparable to that used for dielectrics.


101

A priori, the moments are randomly orientated by thermal agitation when there is no external applied field. The dipoles, however, will tend to orientate themselves G along the lines of an applied field ( B ) so that they assume a minimum potential energy. This will confer on each element in the volume of the material a nonzero G dipole moment equal to d M . In addition, this corresponds to the hypothesis given by Ampere for molecular currents. By analogy with dielectrics, where the polarization vector is defined G dµG , we define the vector for the magnetization intensity at a point in the by P dW magnetic medium such that: G G dM I . dW This is quite often termed a vector of “magnetic polarization”.

3.2.2. Potential vector due to a piece of magnetic material (magnetized and G characterized by I ) and Amperian currents P

dW

M

V Figure 3.8. Calculation for a potential vector through P.

In physical terms, the potential vector is the sum of all the potential vectors generated by the electronic atomic or molecular currents in a vacuum that give rise G to a magnetization intensity ( I ) from the points (M) in the volume (M). In other words, the potential vector at a point P is the sum over all the elementary potentials G G G ( dA ) due to the elementary magnetic moments ( d M IdW ) carried by the elements in the volume dW . Using Eq. (3) in Section 3.1.2. we can state that: G dA

G JJJJG § 1 · d Mu grad M ¨ ¸ 4S ©r¹

P0

G P0 d M

JJJJG § 1 · u grad M ¨ ¸ dW 4S dW ©r¹

P0 G JJJJG § 1 · I u grad M ¨ ¸ dW , 4S ©r¹

G from which can be determined that expression for the resultant potential vector ( A ):

102


G A

P0

G

JJJJG

§1·

³ I u grad M ¨ ¸ dW .

4S V ©r¹ JJG G JJG G JJJJG G Again using the relationship rot (aT) a rot T (grad a) u T where now a = 1/r G JJG G G G JJJJJJJG 1 I 1 JJJJJG G and T I , we can say that rot M ( ) rot M I I u (grad M ) , from which r r r can be derived: G JJG I G P0 1 JJG G P0 W A rot I d rot dW . M M ³ ³ 4S V r 4S V r Taking into account the equation for a rotational in Section 1.1, JJG G JJG G G ³ rot T dW ³ dS u T , the second integral of the equation giving A can be V

S

G transformed with T

G I r G A

JJG G 1 JJG G P0 dS u I , giving ³ rot M I dW ³ 4S V r 4S S r G G P0 1 JJG G P0 Iun dS . ³ rot M I dW ³ 4S V r 4S S r

G , so that A

Magnetized material characterized G by I

P0

{

Vacuum in which is a volume JJG G G Ja rot M I with surface G G G j a = I×n

Figure 3.9. Equivalence of magnetized material to Amperian current densities sitting in a vacuum.

To conclude, the preceding equation shows how the potential vector due to a volume V of magnetized material, characterized by a magnetization intensity vector G I , is equal to the potential vector formed in a vacuum by so-called Amperian (or molecular) currents distributed—as also shown in Figure 3.9—so that:

x there is one part characterized by the volume current density vector, such that JJG G G Ja rot M I ;


103

x there is another paper at the surface (limited of course by V) characterized by the superficial (or surface) current density vector, such that G G G ja Iun. These calculations can be used to determine the internal or external vector potential due to the magnetized material.

Example: Determine the Amperian currents equivalent to a uniformly magnetized cylinder and then the magnetization intensity through its axis, as shown in Figure 3.10. G I G ja G n

Figure 3.10. Cylinder magnetized along its axis.

JJG G G rot I = 0 (as I is uniform throughout the volume). G G G G G x at the ends, I and n are parallel, and ja I u n 0 ; G G G G G G x on the sides, I and n are perpendicular, and ja I u n IeT , where G eT is at a tangent to the surface of the cylinder and normal to its axis.

G We have Ja JJG ja

3.2.3. Physical representation of the magnetization of material and the Amperian currents The Amperian currents, defined above, are sometimes simply called magnetization currents. However, as they are the indirect result of a calculation, and therefore seem to have little bearing in reality, they are also often called imaginary currents. In effect, they cannot really be termed macroscopic and in addition are not associated with the movement of charge through a material, as is the case for a current density vector. Amperian currents are not then what one would call “your normal type of current”. Yet they are not fictional, as for example in orbital magnetism (spin magnetism is even more complex) they are the charge carriers tied to their nuclei, which give rise to localized, individual currents that generate the magnetic moments and result G G dM . in I dW

104


We can use two simple examples with geometrical representations to show how Amperian currents can be seen as averages over space of individual current densities, which also qualify as microscopic currents. G 3.2.3.1. A uniformly magnetized material perpendicular to the plane I

Iz

x

currents cancel each other out

G I (a)

z

y K uniform I

(b)

G I G Ja

G j az 0

G n

0

(S)

Figure 3.11. Equivalence of (a) a uniformly magnetized material and (b) a material crossed by surface Amperian currents.

In a parallelepipedal volume uniformly magnetized along Oz and in a plane perpendicular to the Oz axis, there are ring currents associated with the movement of electron orbitals. In order to detail these currents in geometrical terms, the orbitals are assimilated into rectangular trajectories as indicated in Figure 3.11 a. It can be observed in the figure that, given the direction that the currents take, they cancel one another out in the volume of the material but not on the surface, and hence the representation of the overall current in Figure 3.11b. This result is in perfect agreement with the calculation of the density of Amperian volume currents JJG G G G rot M I = 0 as I is uniform and independent of the points M in the material). ( Ja G G G For its part, the calculation ja I u n indicates the correct direction and sense of the resultant current, already seen using the geometrical argument detailed above.


105

G 3.2.3.2. A nonuniformly magnetized material with I along Oz and such that wI ! 0. wy

(a)

x G I z y K non-uniform I : wI !0 wy

G I

(b)

G ja

G n

(S)

Figure 3.12. Equivalence of (a) a nonuniformly magnetized material and (b) a material crossed by surface and volume Amperian currents.

For a nonuniform I crossing toward positive values of y as shown in Figure 3.12(a), the geometric resultant of the volume currents is along Ox as detailed in Figure 3.12 b. The calculations for the Amperian volume currents, however, give the same result, as:

G Ja

JJGG rotI

w

w

w

wx

wy

wz =

0

0

I

wwII !0 Ox wy ! 0along : selon Ox wy w - IwI!0 along Oy wy 0 : selon Oy wx 0 along Oz 0 selon Oz

The geometric result of the Amperian surface currents also is reported in G Figure 3.12 b, where the nonuniform density, the direction, and the sense of ja are G G G all in agreement with the calculation ja I u n .

106


G G 3.2.4. Definition of vectors B and H in materials G 3.2.4.1. Definition of B G JJG G G The vector B is defined by the relation B rot A , as detailed in Section 3.1.2.2.3. From this definition, the result is: G divB 0 , and therefore in a vacuum, as in the medium of a magnetic material, the flux vector G B retains the same value (through conservation of flux). G For its part, the vector A can be calculated as if in a vacuum and subject to magnetic forces, with the proviso that all currents be brought into play, notably the JJG G G G G G volume ( Ja rot I ) and surface ( ja I u n ) density molecular currents. Assuming that a certain volume of magnetic material is identical to the same volume of Amperian currents (and real currents when present) distributed in a vacuum, Ampere's theory can be written for the magnetic material as JJG G rot B

G µ0 JT ,

G under which classic form only the volume currents ( J ) appear, so that here G G G JT =JA + Ja ,

G where JA is the volume density of real currents (that is to say free currents, deliberately applied in the material). G 3.2.4.2. Definition of the excitation vector H JJG G G G G µ0 JT µ0 (JA + Ja ) equally can be written: The relation rot B G G JJG B JJG § B G · G G G G G JJG G - I ¸ JA . rot JT JA + Ja =JA + rot I , so also, rot ¨ ¨µ ¸ µ0 © 0 ¹ G G § B G· - I¸ Finally, the introduction of the vector H ¨ ¨µ ¸ © 0 ¹ gives JJG G G rot H = JA .

G This last equation shows us that the real current density ( JA ) (deliberately G applied) is the source current for the vector H , generally called the magnetic excitation vector. For this vector, Ampere's theory is identical whether for a a JJG G G vacuum or a material, which is rot H = JA .


In the material then, we have G B G G (while in a vacuum, B µ0 H ).

107

G G µ0 H I ,

3.2.5. Conditions imposed on moving between two magnetic media G 3.2.5.1. Continuity of the normal component of B G n1 medium 2 G G G n n12 n1

G n2

G n2

medium 1

Figure 3.13. Continuity of Bn at the interface between two magnetic media.

In order to simplify the calculation, a cylindrical form is used of which the volume (V) is made up of two halves called medium 1 and medium 2. The height of the cylinder is infinitely small so that we can concentrate on the characteristics of the region about the interface—the sides are negligible in size with respect to the base of the volume. G JJG

G With n

G

G n12

G n1

G G n 2 , B can be written as:

w ³³ B.dS ³³³ div B dW 0 S

V

G JJG B.dS

³³ Sbase sup erior

JJG JJG B1.n

³³

G JJG B.dS

S lower base

³³

G JJG B.dS

Sside o 0

³³ Sbase

JJJG G JJG JJJJG B2 .n B1.( n) dS

JJJG JJG B2 .n , which is stated as B1n

B2n .

This result indicates that there is a continuity of the normal component of B in the region of separating surfaces between the two media.

108


G 3.2.5.2. Relation between the tangential components of the vector H crossing a layer G of current traversed by a superficial current of surface density js z B A G G medium 2 n12 n G F E j y s x medium 1 D C G

Figure 3.14. Study of the continuities across a layer of current ( js ).

Figure 3.14 describes the outline of the form ABCDA drawn at the vicinity of an G interface that contains a superficial current js , which is perpendicular to the plane of the figure (intersection between ABCDA and the layer of current gives EF then G traversed by the current js ). G G G G Ampere's theory may be written as v³ H.d l I js .ey EF js EF , ABCDA

G G because by construction js & e y . Therefore, with AD = BC | 0 and by designating G the component of H tangential to the current layer as H t , then JJJJG JJJG JJJG JJJG H t2 .AB H t1.CD js EF . G G If H t A is the component of H perpendicular to js and collinear with the JJJG JJJG JJJG G vector AB (itself perpendicular to js ), then with AB CD we have

H t2 A H t1A AB H t2 A H t1A js .

js EF .

As

AB

EF , we can immediately see that

In effect, this relation can be written in the very general vectorial form, JJJJG JJJG JG G H t2 - H t1 js u n12 . If js

0 , the formula simplifies to H t2

H t1 .

G Comment: In the same way in which the current JA , which appears in Ampere's JJG G G theory ( rot H=JA ), is a real current (deliberately applied), the superficial current G ( js ), which is such that I js EF and introduced above, also is a real current (deliberately applied) and in no circumstances contains the Amperian current component due to the Amperian surface current density ( ja ).


109

3.2.6. Linear, homogeneous, and isotropic (l.h.i) magnetic media 3.2.6.1. Definition These materials are such that at any point in the magnetic media (is therefore homogeneous) and in whatever direction (therefore isotropic) there is a linear G relationship between the excitation ( H ) and the response, which takes on the form

G I

G Fm H

where F m is the magnetic susceptibility of the medium. The upshot is that

G B

G G µ0 H I

with µ=µ0 1 F m and P r =

G G µ0 1 F m H =µH

µ

1 F m , where P r is relative permittivity and µ0 µ is the (magnetic) absolute permittivity.

3.2.6.2. Result 1 JJG G From Ampere's theorem, rotH G so that PJA

JJGJJG G rotrotA

G G G JA , we can determine that with B µH : JJGG G rotB PJA , JJJJG G JJJGG G graddivA 'A and, finally, as divA 0 : JJJGG G ǻA+ µJA

0.

3.2.6.3. Result 2 G If the deliberately applied current density JA 0 , the result is that the volume JJG G G G G density of the Amperian volumes JA = 0 . In effect, as rot H = JA , if JA 0 , JJG G JJG G G G G 1 JJG G rot H = 0 , so that with I F m H we have rot I = 0 JA = rot I = 0 . Ȥm

3.2.7. Comment on the analogy between dielectric and magnetic media, terms for K K H and B , and magnetic masses defined as calculable equivalents G G G 3.2.7.1. The well-used analogy based on the equations D = İ0 E + P and G G G G G B µ0 (H I)= µ0 H µ0 I

110


G The direct transposition from the above equations would seem to correspond D with G G G G G G G B , E with H , and P with µ0 I . As D is called the electric induction vector, B is G generally called the magnetic induction vector. With E being the electric field G vector, in the same way, H is called the electric field vector. 3.2.7.2. Analogy based on the “source” equations of Gauss's and Ampere's theories These equations are: G divD ȡA G ȡA +ȡ P divE İ0

and and

JJG G G rotH JA ; and JJG G JJG JJG rot B µ0 (J A J a ) .

G In this case, it is the vector H that generally is termed the magnetic excitation G (or induction) vector by analogy with the electric excitation (or induction) vector D , as the two cases share the same type of real source (deliberately applied). G The vector B really should be called the magnetic field vector through an G analogy with the electric field vector ( E ), as the sources in both cases are both real and equivalent to the polarization/magnetization, all placed in a vacuum (of permittivity H0 or permeability µ0, which intervene in the Gaussian or Amperian equations).

3.2.7.3. Conclusion These two arguments driving toward two different sets of names cannot be anything but admissible and it seems reasonable to accept both terms. In order to make the problem easier, it seems nevertheless more simple to call the two magnitudes G G G “vector H” for H and “vector B” for B . In any case, in mathematical terms, H just G like B corresponds to a field of vectors each determined by their three components in the trihedral reference grid. 3.2.7.4. Comment. Magnetic masses as intermediates in calculations for permanent magnets

G JJG G G G G G G As B rotA , divB 0 P0 div(H I) , where divH divI . By analogy with electrostatics, where the charge volume density for G polarization is Up divP , in magnetism the volume density of magnetic masses G equivalent to the magnetization can expressed as U* divI . Similarly, the surface density of magnetic mass can be characterized through V* with Vp Pn ).

I n (by analogy


G So we have divH

111

G divI )H

U* . The result is that G JJG G w ³³ H.dS ³³³ div H dW ³³³ ȡ*dĲ ,

which is Gauss's theory for magnetism, and is a relation that is equivalent to Gauss's theory for electrostatics: G JJG 1 )E w ³³ E.dS ³³³ ȡA dĲ . İ G JJG * Placing m* ³³³ ȡ*dĲ , we also find ) H w ³³ H.dS m (the flux calculated over all space, that is : = 4S), and for a solid angle d:, the elementary flux is: JJG G G m* m * dS.u G JJG 1 m*G u and is H.dS , where by identification H d) H d: 4S r² 4S 4S r² the magnetic field due to a magnetic mass m*. It is worth noting that the magnetic masses introduced here are only equivalents used in calculations and as such are not physically real, in contrast to the Amperian currents. Experience tells us that the negative and positive masses cannot be separated; indeed, the two poles are “inseparable” and can be physically attached only to two faces, north and south, of a closed current, as in Amperian currents.

3.3. Problems 3.3.1. Magnetic moment associated with a surface charged sphere turning around its own axis A sphere of radius R turns around its axis Oz at an angular velocity Z. The sphere is uniformly charged at its surface with a charge density VA . 1. Directly obtain the expression for the charge carried by an elementary part of the surface defined by dS 2SR² sin TdT . Calculate the intensity equivalent to this amount of charge rotating at the angular velocity Ȧ. 2. Given that the closed loop of current of radius R sin T is traversed by the above current dI , determine the magnetic moment of the rotating sphere. Answers 1. The charge carried by the surface element dS 2SR² sin TdT is dq 2SVA R² sin TdT . The intensity dI can be thought of as the ratio of the quantity of charge that goes through the section dS with each complete turn of the sphere (i.e., dq) , over the time (dt) for the charge to pass through this section. This time is 2S that which the sphere takes to make one turn, which is dt T . The result is Z

112


therefore dI =

dq dt

2ʌı A R²sinșdș

=

2ʌ/Ȧ

= ıA R²Ȧsinșdș .

2. The magnetic moment (dµ) that corresponds current ring crossed by dI is by definition such that dP dI.S , where S is the surface limited by the closed circuit. Given that the ring of current has a radius Rsinș , its surface S is therefore: S = S (R sinT)² , where dP

dI.S

ı A R²Ȧsinșdș S (R sinT)² = ʌı A ȦR 4 sin 3șdș .

RsinT dS T

R

O

The resultant magnetic moment over the whole sphere is obtained by summing over all the elementary magnetic moments associated with the circuits of radius RsinT, where T varies from 0 to S:

µ=

ʌ

ʌ

ʌ

ș=0

ș=0

ș=0 ʌ

4 3 4 3 ³ dµ = ³ ʌıA ȦR sin șdș = ʌıA ȦR ³ sin șdș

= ʌı A ȦR

4

, where

³ (1-cos²ș)d -cosș

ș=0

§

µ = ʌı A ȦR

4¨

¨> ¨ ©

ʌ 0

-cosș @

ʌ ª cos3ș º ·¸ 4 +« » = ʌı A ȦR 4 . «¬ 3 »¼ 0 ¸¸ 3 ¹

3.3.2 Magnetic field in a cavity deposited in a magnetic medium A magnetic medium with permeability µ and G uniform magnetic intensity I parallel to Oz contains an empty spherical cavity of radius R. G 1. Determine the Amperian currents JA and G jA equivalent to the magnetization.

z

G I

RsinT T

R

O µ0 µ


113

2. Calculate the elementary induction dB at the center O of the cavity created by a coil of size RdT carrying a current dI associated with a layer of surface current of G density jA . 3. From this, determine the total induction formed at O only by the magnetized material. N.B. It is worth remembering that the field created by a coil of radius r carrying an intensity i at a point on the axis that observes the coil from an angle T is i sin 3 T

given by H

2r

.

Answers JJGG G G 1. We have JA rotI 0 as I is uniform. In addition, we have G G G G jA I u n ext , where n ext is the normal external to the magnetic medium, as shown G in the figure on the right. Therefore jA is perpendicular to the plane of the figure and corresponds to a surface current tangential to the sphere's surface, which can be seen as a layer of current. Its modulus is jA I sin(S - T) = I sin T .

O

G I G T G jA n ext µ0

µ

2. As noted in the question, for a coil with a radius r, the field H formed at a point M at a distance z from the coil is directed along the axis of the coil and has a modulus of H=

i r² 3/2

2 r²+z²

=

i sin 3ș 2r

z M z

.

A coil carrying a layer of current with current density equal to jA is traversed by the intensity di jA u dl jA u RdT R u I sinTdT . The result is the formation at O of an G JJG I elementary induction dB of sense opposite to G µ0 that of I and such that dB dI sin 3 T . 2r With r = R sin T , we have µ0 µ I dB = R I sin 4 ș dș = 0 sin 3ș dș . 2Rsinș 2 µ

N

G H T

h r

z dl =RdT di T

R

O µ0


114

The total induction (B) is therefore: µ I ʌ B = 0 ³ sin 3șdș 2 ș=0 = B

µ0I ʌ 2 2 3

³ (1-cos²ș)d(-cosș) =

0

µ0I 4 2 3

, that is to say:

µ0 I .

3.3.3. A cylinder carrying surface currents A cylinder of radius a has a surface revolving around an axis Oz, a direction through which the cylinder is infinitely long. The surface is carrying superficial currents that have a value for each current G G point M(a,T,z) of the form k k 0 sin T ez .

z

O T 1. Show that the distribution of the current is equivalent to a uniformly magnetized cylinder with a G x magnetization intensity vector I to be determined. 2. Take the problem in the inverse direction and show that its resolution is actually faster. N.B. For a cylinder, JJGG ª 1 wI z wIT º G 1 wI º G ª wIr wI z º G ª1 w rotI « rIT r » ez . » er « » eT « r wT ¼ wz ¼ wr ¼ ¬ r wT ¬ wz ¬ r wr

M

a

y

Answers 1. Given the equivalence of “magnetized material with a magnetization intensity G G I ”, and “material traversed by currents of surface density k and volume density JJGG G G G G G G G G G K ”, we can write that k ja I u n and K Ja rotI . As here, k k 0 sin T ez G G and K 0 (as no volume current is indicated in the question), the vector I should JJGG G G G verify on one hand k 0 sin T ez I u n , and on the other rotI 0 . Given the symmetry of the problem, which would tend to invite the use of §I · §0 · §1 · § 0 · G¨ G¨ r ¸ G¨ ¸ ¨ ¸ ¸ cylindrical coordinates, we have I ¨ IT ¸ u n ¨ 0 ¸ ¨ Iz ¸ = k ¨ 0 ¸ , where, by ¨ 0 ¸ ¨ I ¸ ¨ k sin T ¸ Ï ¸ © ¹ © T¹ © 0 ¹ © z¹


§I G ¨ r identification, I { ¨ IT Ï © z

115

· ¸ k 0 sin T ¸ . ¸ 0 ¹

?

JJGG In order to determine Ir , we are obliged to use the second equation, rotI 0 , for which the three components give three equations (taking into account that Iz 0 ) so

wIT

0,

wz

0,

wz

1 w r wr

rIT

1 wI r r wT

0.

k 0 sin T (= constant with respect to r), the last equation gives IT

As IT where

wI r

wIr wT

IT Ir

0,

wT

k 0 sin T , or rather Ir =k 0 cosT .

I 2r IT2

Finally, I As

IT

wIr

k 02 cos ²T sin ²T

k0 .

T

tan T , we have the geometrical

G G representation which details how I // ex , where G G I k 0 ex .

G n

Ir

IT G ex

G T I

2. Turning question 1 on its head implies finding the Amperian currents equivalent G G to a magnetic cylinder with a magnetization intensity of I k 0 ex . As this magnetization intensity is uniform and has k0 = constant and an orientation at all points following Ox, the volume Amperian currents are zero and all that remains to G G G G determine are the surface Amperian currents ja I u n . Taking into account that I is orientated following Ox, we are more than obliged to use Cartesian coordinates, and thus directly find: cos T · § 0 · §k · §n G G G ¨ 0¸ ¨ x ¸ ¨ ¸ ja I u n ¨ 0 ¸ u ¨ n y sin T ¸ ¨ 0 ¸ , ¨ 0¸ ¨ ¸ ¨ k sin T ¸ 0 © ¹ © ¹ ¹ © 0 so that G G ja k 0 sin T ez . The resolution of the first question is greatly facilitated by using Cartesian coordinates, rather than the cylindrical coordinates that the symmetry of the problem G otherwise would suggest when I is unknown!

116


3.3.4 Virtual current y S µ0 P(x, y,0) z

O

x

µ S’

In the trihedral indicated by Oxyz, the plane xOz separates a vacuum (of permeability µ0) corresponding to the region y > 0, from a magnetic medium (of permeability µ) that extends through the inferior part of the defined space where y < 0. An infinitely long conducting wire, both rectilinear and parallel to Oz, is placed at S, which has coordinates (0, a, 0). A current of intensity I goes through the G wire in sense u , a unit vector along the Oz axis. An additional point, S' which is used in the problem, has coordinates (0, -a, 0). It is possible to show that for any point P with coordinates (x, y, 0), the vector G B is in the form G G G µ0 I G r r' u×( +K ); and x when y > 0, B0 = 2ʌ r² r'² G JG µI G Lr x when y < 0, B= u× , 2ʌ r² G JJG G JJJG noting that in both cases, r=SP and r'=S'P . Also, K and L are two constant that are P P0 2P0 , respectively. and L given by K P P0 P P0

1. Recall the method by which the constants K and L can be determined. JJJJJJJG 2. Directly calculate the expression for the magnetizing field H e (P) formed by the current I. JJJJJJJG 3. As a function of H e (P) , give the expressions for: JJJJJG (a) the field H(P) in the medium (y < 0);


117

JJJJJJJG (b) the demagnetizing field H d (P) ; and JJJJJG (c) the magnetization intensity vector J(P).

4. Determine the Amperian currents equivalent to the magnetization. JG 5. Calculate for the point S the expression for B via two different routes: JG (a) directly from the above derived equations for B expressed as a function of predetermined constants; and (b) by bringing in to bear on the calculation all the currents, both real and Amperian. Answers 1. Ampere's theory states that H 2Sr I , so at point P, G G G I G r I H= u× , and in the medium with permeability µ : H S x G 2ʌ r² G r B G µ I G Gr B= u× . P 2ʌ r² At a point P in the medium with permeability µ0, the field results from the current of intensity I (in the medium where the permeability is µ0) and the current KI (K remains to be determined) at S' (where the permeability is µ). S' is symmetric to S about the origin and KI is the “reflected” (virtual) current. When P is in the medium with permeability µ, the field to which it is subject is of a “perturbed” current—by the presence of the two different media—in the form LI where L has yet to be determined. The problem therefore is to find two unknowns, K and L, which can be determined by introducing two equations, namely for the continuity at the interface (plane xOz) of the tangential component of H which gives [1 - K] = L , and for the normal component of B which gives K[P+P0 ] = P-P0 . G G I G r G JJG u× , with r = SP . 2. The calculation was performed in (1), where H e (P) = 2ʌ r² G G G 3. H = H e + H d for a point inside the material and given that the fields are: G G B L I G Gr G (a) in the material H = u× L He µ 2ʌ r² G G G G G (b) H d = H - H e (L-1) H e = - K H e G G G G G (c) B= µ0 H + Ia = µ H where Ia is the magnetization intensity, we find

118


G Ia

G G B - µ0 H

µ - µ0 2µ0 G µ - µ0 G He 2 He µ0 µ0 µ + µ0 µ + µ0 G G G µ 1 G (where Ia = Ȥ H= µr - 1 L H e = 2 r H e ). µr 1

G 2K H e

JJG G JJG G JJG G G G 4. JA = rot Ia = (µr -1) rot H , while rot H = jA = 0 (no current source in the G material) JA = 0 . G G G G G 2KI G G G KI G jA = Ia × n=2K H e u n = u×r×n= au 2ʌr² ʌr²

5. We have G G µIG Lr 2µ0 I 1 G u× = ex ; (a) B(r = 2a) = 2ʌ r² µ+µ0 2ʌ 2a G G G G (b) B = B(produced by I in vacuum) + B(produced by jA in vacuum) . B(produced in S' by I located in S in vacuum) = dBampérien

µ0 jA dx 2ʌ r

4ʌa

.

; the relevant component, given its symmetry, is:

(dBAmperian ) useful = dBAmperian cosT With x

µ0 I

a tan T and cos T

a r

µ0 jA dx 2ʌ r

cos ș (see Figure below).

, we have:

µ KI ș=ʌ/2 µ KI BAmperian= ³ș=-ʌ/2 0 cos²ș dș = 0 , 2ʌa² 4Sa G from which comes the same result for B as found in part (a). S T a

dx’

x

dx a T S’

dB’Amperian (dBAmperian)relevant dBAmperian

Chapter 4

Dielectric and Magnetic Materials

4.1. Dielectrics 4.1.1. Definitions Dielectrics are in effect electrical insulators. A scale of conductivity can be divided into somewhat arbitrary but well recognized characteristics for each group of materials. 10 + 2

10 + 6

V (:-1cm-1)

Metals (copper)

10 –2

Doped semiconductors

10 – 6

Intrinsic semiconductors

10 – 10

Insulators (dielectrics)

10 – 18

There are certain characteristics that are specific to dielectrics. These include: x Dielectric withstand strength (Ec) (usually given in the units kV mm-1). When the electric field E > Ec, the dielectric is no longer an insulator and an electrical discharge is generated. x Breakdown potential (Uc). When the potential applied is such that U > Uc, the dielectric is no longer an insulator. x Electrical discharge, which is the current passing through the dielectric when it breaks down. The discharge is due to the formation of a highly conductive passage between two electrodes.

120


4.1.2. Origins and types of breakdowns 4.1.2.1. Thermal breakdown When a dielectric exhibits losses through dipolar absorptions or leak currents, a Joule effect ensues that generates heat. If the heat produced is greater than the heat given out by the insulator, the temperature rises and this can come about rapidly as dielectrics are often both good electrical and thermal insulators. As the conductivity (

EG

)

(V) is related to temperature by V V0 e kT , where EG is the band gap of the insulator or semiconductor, the conductivity also increases with temperature to the point where the material can no longer be termed an insulator. 4.1.2.2. Intrinsic breakdown Such a breakdown is caused by a snowball effect rather than the Joule effect, which no longer plays a role. Once the electric field is sufficiently high, a significant number of electrons impact with and ionize the dielectric. Electron-hole pairs are then separated by the electric field, and holes (poorly mobile) tend to accumulate near the cathode. The resulting space-charge reinforces the local electric field and contributes to an increase in the number of ionizations. Field effects also can give rise to additional emissions. 4.1.2.3. Ageing and changes in Ec with time If a dielectric contains inhomogeneities such as cavities or imperfections due to foreign particles, partial discharges can develop around these defaults and an erosion or even a localized melting of the dielectric can result. A network of more or less conducting channels may then develop resembling so closely the branches on a tree that the effect is indeed called treeing. An example is shown in Figure 4.1. Just as mechanical strains can generate cracks, humidity and ionizing radiation, present in our everyday environment, can provoke similar disruptions in certain polymers. In addition, the shape of the electrodes (or contacts) can play a central role, so that to limit localized breakdowns, bumps, and deformities are avoided. An important example of a way in which such effects are limited is the use of ʌ-conjugated polymers in the insulation of high tension cables, where they are used as an inner sheath around the copper core. insulating sheath

conducting wire

cracks initiating treeing Figure 4.1. Electric cable covered by an insulating sheath showing treeing.

Chapter 4. Dielectric and magnetic materials

121

4.1.3 Insulators 4.1.3.1. Natural and inorganic insulators Examples include: x composites made from natural materials such as paper or cotton impregnated with oil (paraffin) and wax; x electronegative gases, in particular those with halogen atoms, for example, SF6, which have high electron affinities, and thus reduce discharges by reducing the density of free charges; and x inorganic materials such as ceramics prepared at high temperatures and pressures (eliminating the need for binders) and engineering ceramics (containing titanium resulting in a high permittivities) that are used in specific applications, for example high value capacitors. Ceramics containing high amounts of aluminum facilitate metal plating, a useful property when used as substrates for electrical circuits. 4.1.3.2. Synthetic organic insulators For the most part, these materials are based on polymers, which consist of a chain of a high number of monomers (M). The principal types of polymers are: linear homopolymer branched polymer

–M – M – M – M – M–M–M–M–M– M–M–M M–M–M–M–M– –M–M–M–M–M– M–M–M–M–

reticulated polymer

M

M

M–M–M–M–M-M–M–M–M–

alternating copolymer

– M1 – M2 – M1 – M2 –

The degree of crystallinity is defined by Ĳ

volume of crystalline part . sample volume

The most common examples are: 1. Polyethylene (PE), which is based on the structure:

122


H H H C

H H

C

C C

H H

H H

H

and is prepared from ethylene, which has the chemical structure H2C = CH2. Highdensity polyethylene (often indicated as HDPE) has a high level of crystallinity. 2. Poly(vinyl chloride) (PVC), is based on the structure:

H Cl H Cl H C

C C

C

H H H H and is prepared from vinyl chloride. 3. Polystyrene (PS)

H H H H H C

C C

H

H

C

comes from the polymerization of styrene, which unlike the above-noted systems, carries aromatic phenyl rings. 4. Polypropylene (PP)

H H H H H C

C C

C

H C H C H H HH H H as its name indicates results from the polymerization of propylene. 5. Polytetrafluoroethylene (PTFE) is also known under its commercial name of Teflon, owned by DuPont. This is an amorphous polymer that, on carrying a “thermal history”, and having undergone mechanical treatment, does not tend toward a crystalline state.

123


F

F

F

F

C

C C

C

F

F

F

F

Dielectric characteristics of the polymers

Hr (50 Hz) TanG (1 MHz) U=1/V (: m) Ec (kV mm-1)

PE 2.3 < 5x10-4 > 1014 17-28

PVC 4 10-1 109-104 11-32

PS 2.4 10-4 > 1014 16-28

PP 2.3 1014 20-26

PTFE 2 < 5.10-4 > 1013 17-24

4.1.4 Electrets 4.1.4.1. Definition and properties Electrets are dielectric materials that carry a quasipermanent charge and are analogous to magnets. They have a permanent polarization; however, the charges involved are relatively small. Under normal ambient conditions, ions in the atmosphere, resulting from natural ionizing radiation, can neutralize the deposited charges. In Figure 4.2, an example of the evolution of surface charge on an electret is given. Typically, after around 2 or 3 months, there is only 20 to 30 % of the initial charge.

1

V

charge density at time t

V0

charge density at time 0

0.2 t

t=0

ca 2 months Figure 4.2. Diminution of charge on an electret with time.

4.1.4.2. Operational details of the voltammeter (devices which measure charge densities) These devices operate on the principle of compensation. That is to say, that for a charged dielectric placed on the lower electrode, the following equation can be

124


written: B

V0 = VA VB =

G JG

³ E.dl E s + Ec. sc

(1)

A

where ı is the surface charge density of the dielectric, s is its thickness, and s' is the distance from the upper electrode. E and E' are the electric fields at the dielectric and between the electret and the upper electrode, respectively.

s

s’

B

electrodes V0 = VA - VB

A

charged dielectric Figure 4.3. Schematization of the operating principles of an electrostatic voltammeter.

Imposing the condition of continuity on the component, normal to the displacing vector gives: Dn D'n = V , which in turn yields HE H0 E c

V.

(2)

V0 is adjusted until E' = 0 (by using the compensation method), so that (1) gives (1’): V0 = E s , (2) gives (2’): V = HE HV0 Determining the ratio (2’)/(1’) directly gives V . s

4.1.4.3. Piezoelectrets These dielectrics become polarized when subject to an applied force. Inversely, when subject to a polarization and in the absence of any mechanical constraints, they change their dimensions. The more common piezoelectrets are quartz (SiO2), barium titanate (BaTiO3), and aluminum phosphate (AlPO4). There is a linear relationship between the causal applied force and the resulting polarization, and elasticity theory can be used to describe the phenomenon. Piezoelectricity is a property tied closely to the structure of a material. For it to exist, the centers of gravity of positive and negative charges, which coincide in the absence of any strain as shown in Figures 4.4a and 4.5a, are separated by deformation. In Figure 4.4b, the dipolar moment remains at 0 because of symmetry,


125

so the material is not piezoelectric; however, in Figure 4.5b, the deformation in Oy leads to a nonsymmetry and a dipolar moment in the direction Ox.

O +

-

+

-

-

+

-

+

y

x

-

+

-

+

+

-

+

-

+

-

+

-

-

+

-

+

-

+

-

+

+

-

+

-

(a)

(b)

G before deformation, ¦ pi

after deformation (through Oy), G ¦ pi 0 not piezoelectric

0

i

i

Figure 4.4. Deformation yielding no piezoelectric effect.

O

(a)

(b)

y

+

+

-

-

+

+

x -

-

+

+

+

-

i

-

+

+

+

-

0

-

-

px

-

G Before deformation, ¦ pi

-

-

distance conservée

-

Décalage sous l’effet de la déformation

O

+

+

-

After deformation, (through Oy) : G ¦ pi p x z 0 piezoelectric (SiO2) i

Figure 4.5. Deformation resulting in a piezoelectric effect (SiO2).

126


(a) Random orientation of the spontaneous polarisation due to ferroelectric domains

P Ps

G E (b) Orientation of the spontaneous polarisations by an external applied field (E). When E=Em, all domains are aligned, as shown here.

1st polarisation plot

4.1.5. Ferroelectrics 4.1.5.1. Definition Ferroelectric materials possess domains, called ferroelectric, inside which dipolar moments are coupled with each other, thus giving rise to spontaneous polarizations. 4.1.5.2. Properties The dielectric permittivity (Hr) of ferroelectric materials is very high and can reach values of around 103 to 104. It is for this reason that they are used in high-strength capacitors. They are very sensitive to temperature: above the so-called Curie ferroelectric temperature, the ferroelectricity disappears.

Pr -Em

O

Eco

Em

E (c) (c)

Ferroelectric hysteresis cycle

Figure 4.6. Ferroelectric domains and their (a) random orientation; (b) organization under an external field; and (c) hysteresis plot.

A plot of polarization against the applied electric field resembles a normal cycle of hysteresis: x At the initial zero field strength, as in Figure 4.6a, the overall polarization of the sample is zero even though each individual domain gives rise to a polarization. This polarization is due to the gradual coupling of dipolar moments up to the limits of the domain, which are structural dislocations of various origin. As the orientation of all the domains is completely random, the initial overall polarization is zero. x When an electric field is applied, a coupling energy tends to orientate the ferroelectric domains in the directional sense of the field. This coupling energy (W) follows the equation W = - p.E cos T and is directly proportional to E. The disorientated polarized domains orientate themselves to the field bit by bit with the increasing field, and this first polarization gives the plot shown in Figure 4.6b. Once the value Em is attained, saturation occurs, i.e., P = Ps.


127

The completed cycle is shown in Figure 4.6c, where E is varied from –Em to + Em. At E = 0, the permanent polarization (Pr) remains, so to return to P = 0, the coercive field (Eco) needs to be applied. 4.1.5.3. Polarization with respect to temperature 4.1.5.3.1 Conditions for a spontaneous polarization (Ps z0) At a temperature T < Tc , the ferroelectric material can exhibit a nonzero spontaneous polarization (i.e., Ps z 0 without an external field). To understand how this can be brought about, we shall look at a highly polar ferroelectric material in which the orientated polarization (Por) dominates the other polarized components; that is to say that Por | P . For this system, and from Sections 2.5.3.2-3, we have: P = n µ n P p = n P p L(E) (1), G G G µp E al P and Eal is such that E al E a where E . kT 3H0 In the absence of an applied field, the presence of P is such that Ea = 0 so that µp P P 3H0 kT , and accordingly P Eal = E (2). , E 3H0 kT 3H0 µP

Given that Eq. (1) = Eq. (2), we find that L(E)=

3H0 kT n µ2p

E . This equation,

which contains the condition P z 0 when E a = 0 , is in fact the straight slope due to 3H0 kT n µ2p

. To find the solution, which corresponds to P z 0 when E a = 0 , the line of

the equation

3H0 kT n µ2p

E must intercept the Langevin function ( L(E) ), which is a

tangent with a slope of 1/3 and has certain aspects detailed in Chapter 2. This means 3H kT that the slope of the equation 0 E must be less that the slope of the tangent to n µ2p the origin, i.e.,

3H0 kT n

µ2p

0 , where µr > 1 , we have H1 =

H0

G < H 0 . On writing H1 in the form

µr G G G G G G H1 H 0 h , the vector h needs to be antiparallel to H 0 . Here, h is said to be a demagnetizing field. G H x If F m < 0 , where µr < 1 , we have H1 = 0 > H 0 . Once again, writing H1 in the µr G G G G G G form H1 H 0 h , the vector h is now parallel to H 0 , and h is now a magnetizing field.

4.2.1.2. General properties As we shall see, there are two main classes of magnetic materials.

4.2.1.2.1. Linear materials G These materials have an diamagnetic intensity ( I ) that is proportional to the G G G magnetic field ( H ), so that I F m H . When F m < 0 , the material is diamagnetic, and when F m > 0 , it is paramagnetic. In fact, diamagnetism is a quite general phenomenon and can be found in all materials, as it results from orbital magnetic moments, while paramagnetism can be observed only in materials with a total, resulting magnetic moment PT z 0 . Apart from the sign of F m , its constancy or variation with temperature also can be used to indicate the type of magnetism one is dealing with: in the case of diamagnetism, F m is independent of temperature, whereas in paramagnetism, F m (T) v

1 T

. This relationship is observed for dilute

systems, which is detailed below. 4.2.1.2.2. Nonlinear materials (essentially the ferromagnets) G G In this class, the relationship I F m H is still appropriate; however, Fm is now a G G G G G G function of H such that F m F m (H) . Similarly, P P(H) , so that B P(H) H .


131

In addition, P and F m are now not only dependent on the strength of H at any particular time t, but also on its anterior values: the system is now subject to hysteresis.

4.2.2. Diamagnetism and Langevin's theory The Larmor precession, associated with the orbital moment, can be found even in atoms where the resultant magnetic movement is zero (such as noble gases). To understand the effect of a resultant magnetic field on an intraatomic orbital, we suppose that the field is applied as indicated by the Larmor precession (Section 3.1.6), shown in Figure 4.8.

G l

G B

Rotational direction of an electron. The current flows in the opposite direction.

iL=-QLe G

ZL

G µl

e G l 2m

G µL

JG iL S

Figure 4.8. Effect of magnetic field on the orbital magnetic moment.

ZL

The frequency of the Larmor precession ( Q L

) forms a current (iL) 2S which is such that i L = - Q L e , where Q L is the number of rotations per second. We also can write i L If U²

ZL 2S

e . With ZL

EB , where E

e 2m

, we find i L

e²B 4Sm

.

is the average value of the square of the distance between the

electron and the axis Oz through which the magnetic field is applied, then

132


PL

r² = x² +y² + z² U2 = x² +y²

i L S U²

e²B

U² . 4m Note that µL is in the opposite direction to B, just as µl is opposite to l. In addition, i LS

= <x²> + + = 3<x²>

= <x²> + = 2<x²> 2 = . 3

For a number n of atoms per volume, each containing z electrons, the magnetic moment per unit volume for the precession is nzµL . The magnetic G intensity ( I ) therefore is such that G nze² G G I nzµL B r² . 6m Given that the magnetic material is represented as a vacuum through which currents associated with orbiting electrons progress, and that the magnetic moments moving in this vacuum are such that B = µ0 H , we can write that Fm

I

µ0 I

H

B

µ0 nze² r² 6m

,

where F m therefore is negative and temperature independent, and

r²

can be

calculated for atoms or ions using quantum mechanics. Even when the orbital magnetic moments and the spin give a resultant equal to zero, this susceptibility related only to the orbital magnetic moment is still apparent. This is because it is tied to the single Larmor precession. When the resultant is not equal to zero, then there is diamagnetism, however, its contribution to the magnetic susceptibility is less intense than that of paramagnetism. Indeed, the latter masks the former.

4.2.3. Paramagnetism Paramagnetism appears for atoms that carry a permanent magnetic moment, such that PT z 0 . The effect due to diamagnetism is less than that caused by paramagnetism, and in atoms where PT z 0 paramagnetic effects dominate. 4.2.3.1. Langevin's theory Langevin's theory can be thought of as an analogy of the theory developed for dielectrics under an orientating polarization (which gives rise to Clausius Mossotti's general formula). Here we consider in terms of magnetism the distribution of


133

magnetic dipoles, rather than the dielectric effect due to a distribution of dielectric G dipoles. In the presence of an external field ( B ) and at a certain temperature (T), the dipoles are subject to: G x on one hand, an orientation due to B , for which the coupling energy in the stable JJJG G state is in the form (E p )min = - PT B - µT .B cos T - µT .B , so that T = 0 JJJG G (2S) gives PT //B ; and x on the other hand, a disorientation with respect to the direction OO' of the applied G field B due to a thermal agitation of energy kT, where k is Boltzmann's constant.

d:

G B

x

T O

G

Figure 4.9. Spatial distribution of magnetic moments subject to B .

The number of atoms (dN) with a moment within the solid angle ( d: ) shown in Figure 4.9 is given by dN = A' d: . Given Boltzmann's distribution, we can write that

dN = A exp( +

µT B cos T kT

)d: .

Taking into account the symmetry of the calculation around the axis Ox, the resultant of the magnetic moments is with respect to the axis. The contribution of dN atoms is therefore dM = µT cos TdN . The resultant of the dipolar moment from all the atoms together with respect to Ox is thus T S

M

³dM

³ µT cosT dN .

T 0

As each “average” magnetic dipole makes a contribution, it is possible to state: µT B 1 M . When E is µT = µT cosT= µT L(E) where E and L(ȕ) = coth ȕ ȕ² N kT

134


small (or B is not too intense) we find that L(E) |

ȕ 3

. If we suppose that

B = 1 Wb m-2, T = 300 K, and µT | 10-22 MKS , we find E | 1/ 400 . We thus arrive at the definitive equation: µT ȕ

µT = µT L(ȕ) |

3

µT2 B

=

3kT

.

For a given number (n) of atoms per unit volume, the magnetic moment per unit JJJG JJG G volume is n µT = n µT L(ȕ) ex , where e x is the unit vector in the direction Ox G G through which the field B is applied. By definition, the magnetic intensity ( I ), JJG G which is the magnetic moment per unit volume, is precisely I nµT , where G G nµT2 B . As we have represented our magnetic material as a vacuum in which I| 3kT bath magnetic atoms with magnetic moments equal to µT, we can write that G G B µ0 H , and therefore Fm

I

µ0 I

nµ0 µT2

H

B

3kT

.

The relationship Fm C

is in the form F m defined by C

T

3kT

, which is Curie's law, wherein C is Curie's constant that is

nµ0 µT2 3k

nµ0 µT2

. This law shows F m to be positive and to vary inversely

with temperature. 4.2.3.2. Correction required by quantum theory

G Quantum theory gives the magnetic moment as µT µT2

e² 4m²

e 2m

G gJ , so that

g²J² . Given the particular values for J², <J²> = =²J(J + 1) , and the

definition of Bohr magneton (µB), P B µT2

e² 4m²

e= 2m

, the expression

g² <J²> = µ2B g²J(J + 1) gives


Fm

nµ0

µT2

3kT

n µ0 J(J 1) g² µ2B

, or rather F m

3kT

135

.

4.2.3.3. Paramagnetism and molecular fields: the Curie-Weiss theory In solid materials, molecules are not independent of one another and consequently Boltzmann's law, so well adapted to gases, is no longer directly applicable. Particularly when the applied magnetic field is weak and the temperature is low enough to make thermal vibrations weak, the magnetic interactions of electrons and thus neighboring atoms in condensed systems become nonnegligible. Then, atoms G subject to the action of an external field B and also subject to an additional effect due to a so-called molecular field that results from neighboring molecules or atoms. Weiss hypothesized that this additional effect can be expressed in the G G G K I and as such must be added to the external field ( H ). This form H m hypothesis, that the molecular field is proportional to each magnetic material, seems G reasonable as I depends on the magnetic moment (µT) of the very molecules or atoms that determine the intensity of the magnetic interactions between neighbors. G G G G G Thus with a resultant field of the form H K I , we have B= µ0 (H K I) . G G nµT2 B Taking the equation given in Section 4.2.3.1, namely, I , we now find that 3kT G G G nµ0µT2 (H K I) C G G I (H P I) , 3kT T G G§ K C · CH from which it can be deduced that I ¨1 , so that ¸ T ¹ T © C G C I T , hence F m Fm . G H 1 (KC) T KC T Given that the Curie temperature ( 4 ) is such that 4 KC , we find that F m is

Fm

C T 4

.

This last equation, or law, accords well with experimental results and is a notable characteristic of paramagnetism for condensed materials. It is worth noting that in order that F m > 0 , the temperature must be greater than 4 ; so that the temperature 4 is real, it must be greater than zero, so that

136


G with 4 KC , K is also greater than zero. As H m G as I , and therefore the molecular field is positive.

G G K I , H m has the same sign

4.2.3.4. Comments While the Curie-Weiss law is verifiable for most situations, there is a point at low temperatures, notably for T d 4 , where a spontaneous ferromagnetism appears. On this and other related points three remarks can be made apparent: 1 E x First, at low temperatures, the approximation L(E) = coth E | is no longer E 3 valid, as E is no longer small. As detailed in Chapter 2, for higher values of E, L(E) | 1 , and I = n PT = nµT L(E) | nµT = Is , where Is is termed a saturated

magnetization and is independent of the applied field. Qualitatively, this means that at low temperatures, thermal agitation no longer limits dipole orientation or the magnetism. More quantitatively, the approximation L(E) |

E is no longer 3

acceptable when µT BA > kT ; that is to say at a temperature equal to or less than Tc, where Tc is defined by the relationship kTc = µT BA and is approximately equal to 4 . BA is the field local to a molecule or atom. However, when T > Tc , L(E) |

E

, then the paramagnetism described above returns due to the creation of 3 structural disorder by thermal agitation. G G K I appeared only in x Second, we have assumed that the molecular field H m G the presence of the magnetism I originating itself from the effect of orientation G by an applied field B . Therefore, the above-established Curie law will no longer apply to materials that already have a molecular field in the absence of an external field. This molecular field also can orientate the magnetic moments parallel to one another. There is in effect a premagnetism, or spontaneous magnetism, that corresponds to ferromagnetism. The Curie law also is not observed by antiferromagnets, such as MnO and Cr2O3. While there is still spontaneous magnetism, it is such that particles compensate one another (compensated premagnetization). x Third, paramagnetism also may be caused by electron spin and is in this case called spin paramagnetism, or Pauli's paramagnetism. Free unpaired electrons, by way of their spin and the resulting spin magnetic moment, can couple with a e G G magnetic field of intensity B. For a spin µs s the coupling energy is m


137

G G E p = - µs .B . The orbital movement of the electron is not taken into account, otherwise a factor of ½ would have to be introduced into the general theory developed by Thomas who used a frame of reference appropriately tied to the composite spin and orbital movements. If Oz is the direction along which B is applied, then e e 1 |Ep| = B <s z > , which gives |Ep| = B = ms . With ms = r , two different m m 2 e= . Two values for energy are obtained, namely, Ep = ±µB B , where P B 2m calculations then can be carried out: x One with a Boltzmann distribution of the different energy electrons, so that with µB B x , we find I = (n - - n + )µB = nµB thx where n- and n+ are the number of kT electrons per unit volume with spins parallel or antiparallel to the field B, respectively. With thx | x , we arrive at F

µ0 I

|

nµ0µ2B

, and this law is of the B kT same type as Curie's law. It is worth noting though that this law is poorly verified for the Pauli paramagnetism or nonferromagnetic metals due to the small susceptibilities and temperature independence of such systems. x Two, with a Fermi-Dirac distribution, which is better adapted to electron distributions. The calculation results in finding F

3 µ0µ2B 2 kTF

, where TF is the Fermi

temperature defined by E F = kTF . This result was established by Pauli in 1927 and is applicable to free electrons in metals.

4.2.4. Ferromagnetism 4.2.4.1. The orientation of a ferromagnetic bar in a magnetic field G As we saw in Section 4.2.1.1, when a bar is placed in a magnetic field ( H 0 ) and is: G G x In a longitudinal position, parallel to the field, so that H 0 H L , we have G G G G G G G H1 H 0 H L . The result of this is that I IL F m H L (µr - 1) H L , and if G the susceptibility ( F m ) is large, then IL is also large. G IL G H1

G Fm H L G G H0 H L

G H0

G HL

G H1

138


G G x In a transverse position perpendicular to the field so that H 0 H T , we have B1 = BT = B0 . From this G G G G G G G G B1 G B0 G H1 I I , so H1 H 0 I = H T I . µ0 µ0 G G G G G µr 1 G IT | H T G G G In addition, I IT HT | HT , H1 G G G H H 0 T µr H1 = H 0 I G if µr , and consequently F m , are large. G In general terms, a bar of any shape placed in an external field H 0 , within the G limiting values of H1 detailed above, then G H0 G G G H1 H 0 f I

where f is a form factor that should take on in the above limiting situations the values: G G x f = 0 (where H 0 H L ); G G x f = 1 (where H 0 H T ). The value of f decreases as the ellipsoid flattens out. G G G G G G In addition, if H1 is considered in the form H1 H 0 h d , then h d f I ; in other words the field is demagnetizing. With respect to the resultant magnetization intensity, we have: G I

G G IL IT

G G Fm H L HT .

IL = FHL G H0

G I

HL G H0

H T = IT

G

G

Figure 4.10. Orientation of I with respect to the excitation H 0 .


139

From this we can go on to draw Figure 4.10, where the resultant magnetization G dµG G , where dµ is the magnetic moment of a bar of any shape such as intensity ( I dW G an iron filing) does not have the same direction as the external field ( H 0 ). A G G coupling appears that tends to orientate the filing in such a way that I // H 0 . The GG GG potential energy of the system, E P v dµ.B0 I.B0 dW , is at a minimum when G G G JG I // B0 and I // H 0 .

Fluxmeter

4.2.4.2. Ferromagnets and magnetization plots 4.2.4.2.1. Plot of the primary magnetization

Figure 4.11. Set up used to plot the first magnetization.

The device shown in Figure 4.11 shows how the first magnetization can be plotted. There is a large torus that is cut at a cross section so that there is a small gap into which a coil can be placed, which is in turn connected to a fluxmeter. The large iron torus is covered with N turns per unit length, through which flows a current I. For G G this, v³ H.d l ¦ I , which gives H = NI , where H is the field in the torus iron. Given that the component normal to B does not change throughout, the field inside the torus is the same as that in the air gap, which can be measured using the fluxmeter. Thus knowing both H and B, we can find out the magnetization intensity B in the torus from I = - H and then plot I = f(H) . µ0 For a sample that has been demagnetized, H is increased from zero and the first magnetization plot shows three main zones, which can be divided as shown in Figure 4.12 a: x zone 1, which exhibits an essentially linear increase;


140

x zone 2, in which there is a considerable increase in I with H; and x zone 3, where the magnetization reaches its saturation point (Is). The latter is specific to the material under study and is dependent on purity and temperature. I

zone 3

B

Is

Fma

slope = µmax

B o = µ 0I s zone 2

Fi

(a)

H

G G G B µ 0 (H Is )

P

µi

(b) H

O

zone 1 µ µmax µi (c) µ0

H

Figure 4.12. The plots for (a) I(H); (b) B(H); and (c) µ(H).

G As B

G G µ0 (H I) , the plot showing B(H) in Figure 4.12(b) is the result of

plotting P0 I = f(H) [from the plot in (a) with the homothetic ratio µ0] following an insertion of the linear variation P0 H . At higher values of H, the plot tends toward G G G an oblique asymptote such that B µ0 (H Is ) and for which the coordinate of the origin at H = 0 is Bo = µ0 Is . Given the plot B(H) such that B = µ(H) .H , we can plot the line in Figure 4.12(c) which shows µ = g(H) . Geometrically speaking, µ is the slope of the line OP in Figure 4.12(b). For hardened iron, (µr )max | 100 , and this value can reach around 80000 for certain alloys such as Mumetal™.

141


4.2.4.2.2. Magnetization at the point of saturation paramagnetism, where F = C/(T-4)

ferromagnetism at

Is low temperature

1

F

Is(0)

T Tf

T Tf

4

Figure 4.13. Plots of (a) Is = f(T) and (b) 1/F = f(T)

The Is is temperature (T) dependent. As shown in Figure 4.13(a), as T rises from absolute zero, Is diminishes quite regularly and then quite quickly before reaching zero at a temperature Tf , which is called Curie's ferromagnetic temperature. Above Tf the material is not ferromagnetic but paramagnetic, as indicated in Figure 4.13(b). At temperatures considerably greater than Tf , the value of F follows the Curie-Weiss law where F

C

, in which 4 is slightly above Tf . In the case of T4 iron, Tf = 1043 K and 4 1101 K . 4.2.4.2.3. Hysteresis loop and magnetic state A hysteresis loop can appear on having increased H from 0 to a maximum value (Hmax) at saturation and then on decreasing H, finding that the current is below that G G B G H, described by the first magnetization plot, shown in Figure 4.14(a). As I µ0 this phenomenon results in a delay in B, which is the effect called hysteresis. By varying H between Hm and –Hm, the current follows a closed loop, otherwise known as the hysteresis loop. There are two notable points: x the remanence magnetization (Ir) remains when H = 0; and x the coercive field (Hc) is the value of the opposing field H which needs to be applied to remove the magnetization.

142


I

(a)

(b)

Ir - Hmax

Hc

Hmax H

I

E E’ E’’ I1 D hd D’ H0 D’’

H H0 H0 = H

H0 = 0 Figure 4.14. (a) Hysteresis loop and (b) operation.

For a material with a well-defined hysteresis and a form factor f, its magnetic state can be determined from both: x its participation in the plot I = f(H) , which is characteristic of the hysteresis of a material; and H H x its part in the slope I 0 (equation I = D(H) directly deduced from f f G G G H H 0 f I ). The line has a slope of

1

and is such that I 0 if H = H 0 [so that H 0 is at the f intersection of the line I = D(H) with the abscissas]. The hysteresis intervenes at two points in Figure 4.14(b) where at the intersection, I = f(H) for a hysteresis along the line I = D(H) : x at D when H is increasing along with I; x at E when H is decreasing, as is I. 1 When the external field H 0 changes, the line moves but retains its slope . f H If H 0 = 0 , the plot is simplified to I and goes through the origin. The points f G G G of intervention are now D’’ and E’’. Given that H H 0 h d and that here H 0 = 0 ,


143

G h d . The intensity of the magnetization, I1 = I(H = hd) , when H 0 = 0 , is the actual remanence magnetization.

G we now find that H

If H = H 0 , I 0 and the point at which the hysteresis functions is D’ (the equation G G G G H H 0 h d demonstrates that for this scenario, h d 0 ). 4.2.4.2.4. Energy loss by hysteresis For the experimental setup shown in Figure 4.15—which also may be used to study hysteresis loops—Ohm's law also can be written as: dI Ri . u + e = Ri , or rather, u dt R

i + u

Figure 4.15. Experimental setup to study hysteresis.

Multiplying both sides by idt, we have: u i dt - i dI= Ri² dt . On integrating this differential equation between the points t = 0 and t = T , i = 0 and i = i , and I I0 and I ) , we find T

T

T

T

T

T

³ uidt ³ idI

³ Ri²dt , which also gives ³ uidt ³ Ri²dt

³ idI .

0

0

0

0

0

0

(1)

(2)

(3)

T

Term (1) represents the energy WG = ³ uidt supplied by the generator of fem u, so 0

that with i

dq dt

T

, we have WG = ³ udq

Qu , if q = 0 when t = 0 and q= Q

0

when t = T . Term (2) represents the energy lost through the Joule effect between t = 0 and t = T which is associated with the resistance of the solenoid.

144

Basic electromagnetism and materials )

Term (3), which should be written in the form

³ idI , represents the energy lost

I0

through hysteresis and it is this term that is detailed below. If we suppose that the solenoid is infinitely long and has N turns each with a surface area S over the solenoid length l, then H = N i and I B N l S . In turn, )

excusing the pun, we have dI= N l S dB , so that

)

³ idI

I0

³ NlSidB .

I0

With V = l S , where V is the volume of the solenoid, and H = N i , we have: )

B

³ idI V ³ HdB ,

I0

B'

where B' is the field at the initial instant t = 0 and while B is the field at the last instant t = T. In terms of the coordinates (B,H), the hysteresis takes on the form described in Figure 4.16; H dB is indicated by the hatched surface dS where H is increasing in the first quadrant. When H decreases, H dB is given by an area dS', associated with a value H' < H and such that dS' < dS. B dS’ dS

H’ H

Figure 4.16. B(H) coordinates of the hysteresis loop.

Schematized in Figures 4.17(a) and 4.17(b), respectively, are the hatched areas swept in the first quadrant when H increases and decrease. The difference between ³ dS and ³ dS' corresponds to the area (A) of the hysteresis loop. As H increases, the system gains energy (if H n , dB ! 0 and HdB ! 0 ) , and when H decreases, the system releases energy (if H p , dB 0 and HdB 0 ) .


As

145

³ dS' ³ dS , the released energy is less than that received, and the energy

absorbed through one complete cycle can be written as: )

B

³ idI V ³ HdB V [S - S'] = V.A .

I0

B'

The higher the value of A, the greater the energy absorbed.

B

B

(a) H

(b) H

Figure 4.17. Areas covered in the first quadrant when (a) H increases; and (b) H decreases.

4.2.4.3. Soft and hard ferromagnets 4.2.4.3.1. Soft ferromagnets Soft ferromagnets are characterized by their weak coercive field, where H c < 100 A.m-1 . They magnetization therefore is relatively easy to change. Given that with a low value of Hc hysteresis is small, and energy losses are also small, these materials often are used in transformers, electromagnets, relays, and telephone loud speakers. Examples include Permalloy™ (Fe = 21.5 %, Ni = 78.5 %) and Mumetal™ (Fe = 16%, Ni = 77%, Cu = 5%, Cr = 2%). 4.2.4.3.2. Hard ferromagnets These magnets exhibit values for H c > 103 A.m-1 and their remanence is relatively difficult to remove. They are generally used as permanent magnets. Examples include steels with around 1% carbon, or even with Co, Mn, or W. More recently, alloys have been prepared such as the Alnico™ series, which includes Alnico 5 based on Fe 51.5%, Al = 8%, Ni = 13.5%, Co = 24%, and Cu = 3%, or alloys with titanium such as Ticonal™.

4.2.4.4. Aspects of the theory of ferromagnetism 4.2.4.4.1. Theoretical conditions required for spontaneous magnetisation to appear: influence of temperature

146


It is worth trying to find that conditions required at which a paramagnetic substance remains magnetic (I z 0 ) simply by its molecular field (Hm) and without any external field. G G G G Hm , can be The relationship that defines Hm : H m K I , or rather I K G G T Hm multiplied by T above and below the line to give I . In the absence of an K T

§H· applied field, H is due only to the molecular field. In the representation I = f ¨ ¸ , ©T¹ T . As the magnetization state therefore is shown as a straight line (D) with slope K detailed in Section 4.2.3.1, the magnetization in the presence of magnetic moments (µT) can be given as: µT B µ0µT H I = n µT n µT L(E) where E . Therefore kT kT §µ µ H · I = n µT L ¨ 0 T ¸ . © kT ¹ For a material to be spontaneously magnetized, simply by its own molecular field, its magnetization state should be represented by the point A that is both on the line D and on the curve I = n µT L(E) , as presented in Figure 4.18.

line D (slope T/K) Is IA

slope nµ0µT2/3k

A curve nµTL(E)

H/T

Figure 4.18. Plots of I = f(H/T) (labeled line D) and I = nµTL(E). Note: plots are for a given temperature.


147

Given that at the origin [E o 0] L(E) tends toward a straight line with an equation I = nµT nµ0µT2

E 3

E 3

, the function I tends [also when E o 0] toward a straight line written by nµ0µT2

H

3kT

nµ0µT2 3k

§H· , for which I = f ¨ ¸ has a slope equal to T ©T¹

H

; for a point A to exist, the slope of D must be less than the slope of the

3k

tangent to the origin of I, which is represented by a line of slope condition is more concisely given by

T

K

T4

K

nµ0µT2 3k

nµ0µT2 3k

. This

, or rather

nµ0µT2

. 3k In order to have T positive, the Curie temperature ( 4 ) must also be positive. This in turn requires that the molecular field must be positive ( K ! 0 ). The conclusion therefore is that a paramagnetic material with a positive molecular field is susceptible to being spontaneously magnetized—in the absence of an external field—at a temperature below 4 .

Comment 1. It should be noted that low temperatures enable spontaneous order, and µT B therefore E must be relatively high, even if the approximation made by kT E L(E) | is not accurate under such conditions. In reality, it is sufficient that the 3 slope (

T

K

) of the line D is less than P

T < KP = Tf < K

nµ0µT2 3k

, that is to say that

nµ0µT2

4 , where Tf is the ferromagnetic temperature. As 3k noted elsewhere in this chapter, representative values given for iron are Tf and 4 equal, respectively, 1043 and 1101 K. Comment 2. Tf separates the two temperature domains above which a disordered phase reigns resulting in paramagnetism and below which the temperature is sufficiently low for an ordered regime to result in ferromagnetism (see also Comment 1 of Section 4.2.3.4).

148


4.2.4.4.2. Weiss domains and the Barkhausen effect Associated with the point A, which represented the ferromagnetic state, is a magnetization intensity (IA). In the absence of an external field, the intensity is no more than that which is found in small domains, called Weiss domains, which are around a micron cubed in size (Figure 4.19). With respect to the larger macroscopic volumes, the spontaneous magnetization is zero in the absence of an applied field and because of the random domain orientation. Iaverage z 0

Iaverage = 0

Hexternal=0

Hexternal weak and increasing

Figure 4.19. Evolution of Weiss domains with a growing magnetic field.

Once an external field (H) is applied, and while H is relatively weak, the walls between the Weiss domains deform to the point where domains facing in the same or nearest direction as H become greater in number than their neighbors. While the displacements (distortions) are relatively small in the weak field, they are also reversible and give rise to the smooth change observed for I(H) in Figure 4.20.

zone 1

I IA

Figure 4.20. The Barkhausen effect.

H


149

Under a strong field the changes in domain orientation become abrupt so that the magnetization curve becomes discontinuous and resembles a stairway, each step corresponding to the orientation of a single domain in what is known as the Barkhausen effect (Figure 4.20). The origin of hysteresis can be identified in the movement of these domains. As the displacements required are nonnegligible, they are susceptible to being stopped by obstacles such as impurities and defaults, and the phenomenon is therefore irreversible and nonlinear. The effect of the field becomes delayed and the domain orientation continues, for example, even when the field has disappeared, hence hysteresis. When all the domains are orientated in the direction of the field H, then the average magnetization for all domains tends toward the individual domain value of spontaneous magnetization (IA). What now remains in is an explanation of the physical origin of IA for each domain in the absence of an external magnetic field, that is to say the reason for the existence of an internal field in the absence of an external field. This is opposed to the supposition expressed earlier on paramagnetism where the molecular field appeared due to the generation of order by an external field. 4.2.4.4.3. Origin of the spontaneous magnetization of domains Ferromagnetism only occurs with elements that have their internal electronic layers incomplete, as is the case with iron which has an incomplete 3d orbital. The unpaired electrons from these inner layers are coupled through spin, with interaction or so-called “exchange” energies being of the form

G G We = - 2 J e s1.s2 . In this equation, Heisenberg's theory, Je is the exchange integral and varies with the overlapping wavefunctions of the electrons. A positive Je favors an alignment of same-sense spins and spin magnetic moments and little by little we can see that an order can be imposed upon the material through the spin magnetic moments. While for individual atoms Je is determined by the way in which the different orbitals are filled which follow Hund's rules whereby spins are organized so that S is at a maximum, in a metal Hund's rule is no longer applicable as the atoms contribute to valence bonds. Indeed, the interactions of the 3d orbitals from the atoms placed together result in the formation of the half bands 3d+ and 3d-, which correspond, respectively, to the parallel or antiparallel alignment of spins. A small shift in the energies of these two bands results in a considerable separation of the two populations, in turn resulting in a large spontaneous magnetization. If the bands are very different, for example, if 3d+ is more populous than 3d-, then there can be an intense magnetization caused by a high proportion of parallel spins ordering the magnetic moments.

150


individual free atom Fe 3d 4s

metal 3d

4s

3d+ band 3d- band Mn 3d+ band 3d- band

Figure 4.21. Electronic structure of the transition elements iron (ferromagnetic) and manganese (antiferromagnetic) in both free and metallic states.

4.2.5. Antiferromagnetism and ferrimagnetism 4.2.5.1. Antiferromagnetism In the case of chromium or manganese, the 3d+ and 3d- bands are pretty well equally populated, so that the average spin magnetic moments are antiparallel and there is no longer any spontaneous magnetization as the spin magnetic moments cancel each other out. Above a certain temperature, the Néel temperature (TN), this ordered state C disappears and F follows a law of the type F . When T increases, F TT decreases so that F goes through a maximum at T = TN , in a behavior characteristic of antiferromagnets. For chromium, TN = 475 K .

(a)

(b)

Figure 4.22. (a) Organization of spin magnetic moments due to antiferromagnetism; and (b) distribution of spin magnetic moments, of alternating value, in ferrimagnetism. 4.2.5.2. Ferrimagnetism For materials based on a mixture of two types of atoms, exchange interactions can orientate all similar atoms in one sense and all the other different atoms in another sense. The overall effect is a nonzero spontaneous magnetization, which can be


151

strong, but not as strong as ferromagnetism. Ferrites make up the main class of ferrimagnets and are based on iron(III) oxides mixed with metals such as Ni, Al, Zn, Mn, and Co in their secondary oxidation states (II). An example is that of Fe2O3, MO. There are soft ferrites, based on mixtures of manganese and zinc, which are of considerable commercial interest. They are insulators (U < 1 : m) and exhibit limited losses through Foucault currents, hence their use in high-frequency transformers. Hard ferrites containing barium, are normally prepared with high temperatures and pressures, and are used as permanent magnets.

4.3. Problems Dielectrics, electrets, magnets, and the gap in spherical armatures 1. A lhi dielectric of absolute permittivity H is placed between the electrodes of a spherical capacitor which is defined by spheres of radius a and b centered about O. If +Q and –Q are the charges on the electrodes, where a < b, then: G G G (a) Determine the vectors E (M), D(M), P(M) for a point M situated at a distance r from O ( a < r < b ) as well as the potential difference Va - Vb between the two G JJJJG electrodes, where r OM . (b) Calculate the surface and volume densities equivalent to the polarizations Va, Vb, and UP. (c) Calculate the total quantity of these charges due to the polarizations in the dielectric. What remarks can be made on the results? 2. After having discharged the capacitor used in the pervious problem, the dielectric is changed for an electret, which has a permanent polarization expressed as G G Dr P (where a < r < b and D is a constant). The spheres, of radius a and b, do 4Sr 3 not carry real charges. (a) Calculate the charge densities equivalent to the surface polarizations V’a, and V’b, and the volume polarization U’P. G G (b) Use the preceding result to find the vectors E and D for when r > a . 3. Now the same capacitor has, in place of an electret, a magnet with a permanent G G G Er magnetization intensity I which is such that I ( where a < r < b , and E is a 4Sr 3 constant). (a) Calculate the Ampere surface and volume current densities identical to the magnetization.

152


(b) (i) Calculate the imaginary surface (V*a and V*b) and volume (U*) magnetic mass densities. (ii) From the preceding result, calculate the total magnetic mass carried by the material with respect to an armature of radius a, and then of radius b. Conclude. (c) The notion of magnetic mass is used as an intermediate in calculations of the G magnetic field at point M located by the vector r (where a < r < b ). (i) Recalling that at a distance r from a magnetic mass m*, the field is given by: G G G 1 r H(r) m * . Give the form of Gauss's theory for H . 4S r3 G G (ii) Determine H(r) when a < r < b , and then find B .

Answers 1. (a) When a < r < b , Gauss's theory states that: Q ) = 4Sr²E = , so that as the field is radial, H G G 1 r E Q . 4SH r 3 Therefore, G G G 1 r D HE Q 4S r 3 and G G (H H0 ) Gr P (H H0 )E Q . 4SH r3 b G JJG Va - Vb = ³ E.dr a

Q b dr

³

4SH a r²

Q

A’ G na ' O G

r

G aA

-+ -+

+Q

G b

G nb

M

Q ª1 1 º « ». 4SH ¬ a b ¼

(b) As UA = 0, the localized form of Gauss's theory states that UP = 0 . Equally valid, a direct calculation can be made when the polarization vector is known:

G UP (M) = - div M P

(H H 0 )

Q div M

4SH (H H 0 ) § JJJJG 1 · Qdiv M ¨ grad M ¸ 4SH r¹ ©

G r r3 (H H 0 ) 4SH

§1· Q' M ¨ ¸ ©r¹

0.


x

153

G G at a point A such that r a , we have G (H H0 ) Q G G G K K a.n a . With a antiparallel at na so that Va P(A).n a 3 4SH a (H H 0 ) Q G G a. n a = - a , we find Va ; 4SH a 2 Va is indeed negative as can be discerned by inspection of the orientation of the dipoles influenced by the polarization (due to the polarization charge +Q on the electrode with radius a) and shown in the figure above. When

M

is

G G x In the same way, when M is at the point B, so that r b , we have: G GG G (H H 0 ) Q G G G G b.n b with b & n b , so b.n b b , and we find that Vb P(B).n b 3 4SH b (H H 0 ) Q Vb , where Vb ! 0 (again see the figure above). 4SH b 2 (c) If Qa and Qb designate the total polarization charges with respect to the armatures with radii a and b, we have Qa = 4Sa²Va et Q b = 4Sb²Vb , so that Q T = Qa + Q b =

(H H 0 ) H

Q

(H H 0 ) H

Q

0.

To conclude, the dielectric material is overall electrically neutral and the polarization has the effect of simply displacing the charges to give localized charge surpluses.

2. G (a) As P

G Dr

: 4Sr 3 G G and x With A such that r a G D GG D G Vca P(A).n a a.n a . 4Sa 3 4Sa 2 G G x With B such that r b, G G D D G G Vcb P(B).n b b.n b . 3 4Sb 4Sb 2 G G r In addition, U'P (M) = - div M P v div M r3 and as elsewhere we can also state that UA = Gauss's theory, U’P =0.

K a

we

antiparallel

have

at

K na ,

G G b & nb ,

then

then

0 where v means proportional to, 0, so that in the localized form of

154


(b) Writing Gauss's theory for charges sitting in a vacuum thus gives: ¦ Qint G JJG Q 'a all charges x when a < r < b , ) (M) ³³ E.dS 4Sr²E . H0 H0 G G D r With Q'a = 4Sa²V'a = -D , we have in terms of vectors E . 4SH0 r 3

x when r > b , we have ¦ Qint = 4Sa²V'a + 4Sb²V'b = -D D With respect to induction, G G G x when a < r < b: D H0 E P

G D r

G D r

4S r 3 4S r 3 G G G x for r > a , as E and P are zero, D is also zero.

3. G (a) We have I

G Er 4Sr 3

G 0 , in which E

0.

0 ; and

when a < r < b so we can state that the Ampere currents are

such that:

G E JJG r E JJG JJJJG 1 rot M rot M grad M 0 ; and 4S 4S r r3 x surface current: on an electrode of radius a we have: G G G G Ea G G G ja ) r a I u n a u n a 0 as sin(a, n a ) sin S 0 . 3 4Sa G G G G Eb G G G Similarly, ja ) r b I u n b u n b 0 as sin(a, n b ) sin 2S 0 . 3 4Sa (b)(i) For the imaginary magnetic masses equivalent to the magnetization, we have: JJJJG 1 G E E §1· x for the volume densities: U*(M) = - divI(M) div M grad M '¨ ¸ 0 4S r 4S © r ¹ x for the surface densities: G G Ea G E G K K

- for A: V (A) I(A).n a .n as a antiparallel at na , 3 a 4Sa² 4Sa G G G Eb G E G G - for B : V (B) I(B).n b .n b as b & n a . 3 S 4 b² 4Sb The upshot is that the total magnetic masses with respect to the armatures of radii a and b are, respectively, m*a = 4ʌa2ı*a = -ȕ and m*b = 4ʌa2ı*b = ȕ. G x volume current: Ja (M)

JJGG rotI

We can conclude by verifying that the resulting magnetic mass from the two faces of the magnet is m*R = m*a +m*b = 0 . This result is analogous to that found


155

in answer 1(b), in that localized excesses in the positive (m*a) and negative (m*b) magnetic masses appear in the calculation so as to take into account the magnetization. (b)(ii) G With H(r)

1 4S

m*

G r

1

r3

4S

m*

)H so that with d: that m

G JJG u.dS r²

G u r2

, we can write that

G JJG w ³³ H.dS

and w ³³ d:

m* 4S

w ³³

G JJG u.dS r²

,

4S we arrive at ) H

m , where m* is such

³³³ U * dW and represents the sum of magnetic masses inside the Gaussian x

surface in the form of Gauss's magnetic theory. If we then consider a point M such that a < r < b, with a Gaussian surface being a sphere of radius a, then G G JJG G 1 r

)H w H.dS 4 S r²H m E , where H E . ³³ 4S r 3 G G G G G E r · § E r From this can be deduced B P0 (H I) µ0 ¨ ¸ 0. © 4S r 3 4S r 3 ¹ K K Comment: In nonlinear magnetic materials, the relationship between B and H is G G G G G nonlinear and B =µ(H) H so that the only usable relationship is B P0 (H I) . To G G be more explicit, it is worth remembering to not write B = µ H , from which can K K arise B = 0 and H = 0, which would be completely wrong, as we can see in this example!

Chapter 5

Time-Varying Electromagnetic Fields and Maxwell’s Equations

5.1. Variable Slow Rates and the Rate Approximation of Quasistatic States (RAQSS) 5.1.1. Definition In this region of frequencies, the applied field varies sufficiently slowly with time so that it is possible to state that at a given instant the current intensity is the same throughout all parts of a closed circuit. Given then that for any value of S, G JJG G I = ³³ j.dS = constant implies that the current density flux ( j ) across a current S

G JJG “tube” (which delimits a closed surface) is zero, as in w ³³ j.dS

0 (see also Chapter G 1). Following on from Ostrogradsky's theory, it would indicate that div j 0 , and

the conservation of charge therefore would give

wU

0 . This hypothesis indeed wt could be used as a starting point in defining quasistatic states.

5.1.2. Propagation Concentrating on systems where the intensity is the same in all parts at a given time means neglecting propagation phenomena that would appear if the intensity were to vary rapidly with time. Once the intensity of the electrical configuration starts to vary, its effects felt at a distance will be delayed related to the velocity of signal propagation (which is the speed of light in a vacuum). When the phenomenon varies periodically, at a frequency Q, a pair of closest neighboring points undergoing the same vibrational state are at a distance of one spatial period, that is to say one c wavelength, which is defined by O = . If the length (L) of the circuit is very small Q with respect to O, i.e., L 0 for a c velocity c. The phenomenon can be examined by studying the position in space of the signal G at times t1 and t2, as shown in Figure 6.1.

Chapter 6. General properties of electromagnetic waves

G (t -

z ) c

G (t1 – z’1/c)

t1

t2 (= t1 + 't = t1 +'z/c) P’

P’

G (t1 – z1/c)

175

P

P

z z’1

z1

z2=z1 +'z = z1 + c't

z’2

Figure 6.1. Signal displacement between the times t1 and t2.

The signal represented by the function G (t1 – z1/c) at a point (P) on the abscissa (z1) and at time t1, will be such that at a following moment in time (t2) z where t 2 = t1 +'t , it can be represented by the function G (t 2 - 2 ) where z2 is c the place on the abscissa where P has reached by the time t2. The signal is assumed to be the same at t1 and t2, and indeed for all the points P, so we have z z z z G (t1 - 1 ) = G (t 2 - 2 ) , and if t1 - 1 = t 2 - 2 we find that c c c c z 2 - z1 = c (t 2 - t1 ) , which can also be written as 'z c't . As the result above is valid for all the points denoted P, it also can be supposed that it is valid for all the points moving in the signal following the same law, that is: 'z

c't .

This indicates that the signal itself, given by the function G (t -

z c

) , undergoes a

propagation along z > 0 at a velocity c. z

) represents the propagation of the c signal following a law such that 'z = - c 't, derived from the condition z z t1 + 1 = t 2 + 2 , which indicates that the signal is still moving at a velocity c, c c but along a point where z < 0: the F function represents a retrograde wave.

For its part, the function

F(t +

176


6.2. Different wave types 6.2.1. Transverse and longitudinal waves 6.2.1.1. Transverse signals A transverse wave is one that has a movement (of its signal in the most general terms) that is perpendicular to the direction of its propagation. An example is a signal that moves perpendicularly along a stretched rope.

displacement

propagation

z

Figure 6.2. Transverse signal.

6.2.1.2. Longitudinal signal A longitudinal wave is one that has a displacement parallel to the direction of the propagation. An example of this is a compression signal applied parallel to and along the length of a spring. propagation

displacement Figure 6.3. Longitudinal signal.

6.2.2. Planar waves 6.2.2.1. Definition G G A vector ( A ) is said to be propagated by a planar wave if at a given moment A is the same at all points in a plane ʌ—called the wave plane—parallel to a given plane (S0) that is perpendicular to the propagation direction (Oz) of the signal. This is schematized in Figure 6.4.


x G A( z0 , t ) z0

G A( z1 , t )

z1

z

G A( z1 , t )

G A( z0 , t ) S0

y

177

S1 t

Figure 6.4. Planar wave at a given point in time (t).

6.2.2.2. The implication of a planar wave G The signal represented by the vector A depends only on z and t, so that G G G G A A(z, t) . The solutions looked for in Section 6.1.3 for E and B also correspond G to planar waves, as their solution components were in the form ) (z, t) . In effect, E G and B propagate as planar waves, so it can be said that we have an electromagnetic planar wave (EMPW). 6.2.2.3. Planar waves in practice In strict terms, in order to have a system of planar waves, all points should be within G an infinite plane represented as an example by the vectors A(z1 ) in an infinite plane denoted S1. In practical terms that is not possible; nevertheless, at a sufficient G distance from a source, identical A vectors can be determined within a large enough dimension to be considered a planar wave. 6.2.3. Spherical waves While for a planar wave it is supposed that only one direction of propagation is used (Oz in the above example), for a spherical wave the propagation is isotropic in space. The field components ([) depend only on the distance (r) from the observation point of the wave. Thus [ [(r, t) , and the Laplacian is now reduced to '[

1 w² r wr²

turn gives r[= g + (t-

r

(r[) and the propagation equation becomes '[ w² wr²

(r[)

1 w ²(r[) c² wt²

r ) + f - (t+ ) , so that c c

1 w ²[ c² wt²

0 , which in

0 . The solutions for this are in the form

178


[= 1

g + (t-

1 r

g + (t-

r c

)+

1

r f - (t+ ) . r c

r

) represents the propagation of a wave of speed c for when r c r is positive, with an attenuation of the wave as r increases.

The solution

6.2.4. Progressive waves 6.2.4.1. Progressive planar waves (PPW) z As an example, if the function G (t - ) represents a signal that progresses toward c z > 0 at a speed c, then the sense and direction of the wave are perfectly defined and we can say that it is a progressive wave (and is also called a “direct” wave given its z propagation toward z > 0). Similarly, F(t + ) is a progressive wave that goes c toward the points z < 0 also at a speed c. When in addition these progressive waves are one dimensional, and are only dependent on one component in space, such as the waves given by G(z,t) or F(z,t) , then they are progressive planar waves that have wave planes defined by z = constant. 6.2.4.2. Monochromatic progressive planar waves (MPPW) Additionally, if the waves have sinusoidal signals, then the following can be stated: z z º ª G and F waves have been G (t - ) = a cos «Z (t - ) » attributed with the same c c ¼ ¬ amplitude and that F is out of z z ª º phase with G by M. F(t + ) = a cos « Z (t + ) +M» c c ¬ ¼ 2ʌ By introducing the period T , it is possible to say that: Ȧ z z · §t G (t - ) = G(z,t) = a cos 2S ¨ ¸. c © T cT ¹ And then by introducing the wavelength O = cT , we have: § t z· G(z,t) = a cos 2S ¨ ¸ . ©T O¹ Z With the introduction of the wavenumber, defined by k , the wave can be c written in the form: G(z,t) = a cos (Zt - kz) . G For its part, the wave vector k is defined by the relation:


179

G G k =ku

ZG u , c G G G where u is the unit vector of the direction of the propagation (here u ez ). As shown graphically in Figure 6.5, it is possible to see that this function presents both temporal and spatial periodicities.

G(z,t)

G(z,t)

O

O

t

z

(a)

T

(b)

O

Figure 6.5. (a) Temporal and (b) spatial periodicity.

6.2.5. Stationary waves z

z

) . Following a c c simple, direct calculation, we have for a monochromatic wave: M· M· § z M· § § S(z,t)= 2 a cos ¨ Z ¸ .cos ¨ Zt ¸ =A(z).cos ¨ Zt ¸ =A(z)C(t) . 2¹ 2¹ © c 2¹ © © in which the term C(t) is the same over all points at any given instant, so that the signal [S(z,t)] is in the same phase at any point. Only the amplitude [A(z)] changes with respect to the abscissa of the point in question, and there is no propagation term. Signals with the form S(z,t)=A(z)C(t) are stationary waves and give rise to nodes at the abscissa where z is such that t : A(z)=0 (and also S(z,t)=0 ).

A stationary wave would require that S(z,t) = G (t -

) + F(t +

Standing modes S(z,t)

O

t = T/4... t = 0, T/2

L t = 3T/4...

Figure 6.6. A stationary wave exhibiting nodes at extremities O and L.

z

180


In order to simplify the calculations, additionally we can assume that M =S z z so that: S(z,t) = 2a sin(Z ).sin (Zt) = 2a (sin 2S ) (sinZt) . O c It can be seen immediately that when z = 0, S(z,t) = 0 and that S(z,t) represents a node at the origin that is repeated at points on the abscissa where O z = m . For the moment the nodes (and likewise the antinodes) can occur at any 2 value of Ȧ, and it can be stated that the number of normal vibrational modes is infinite, a characteristic of continuous media. If a second node is now imposed on the other end of the signal at z = L, where the signal can be imagined as, for example, a cord now fixed at either extremity or an electric field crossing between two conductors which are such that at their interface Etangential = 0, then as shown in Figure 6.6 we have two conditions: condition (1), S(z = 0, t) = 0; and condition (2), S(z = L, t) = 0. z Given the form of S(z,t) = 2a (sin 2S ) (sinZt) , condition (1) is always O fulfilled; however, in order for condition (2) to be fulfilled, it is required that 2L L L sin 2S 0 , so that 2S nS , and that O must be such that O . The n O O c (n) and that of the angular frequency frequency (vn ) is restricted now to Q n 2L (Ȧn) is: Sc (n) . Zn L These conditions define the standing (stationary) modes and limit their number, as they give rise to discrete values for each of n whole number. 6.3. General Properties of Progressive Planar Electromagnetic Waves (PPEMW) in Vacuums with U A = 0 and j A = 0 G G With the electromagnetic waves being planar, E and B only depend on a single G G G G spatial coordinate (z) and time (t) in that E E(z,t) , B B(z,t) . In this section we consider a direct progressive wave that follows a propagation along z > 0 and as G G z solutions for its components has E and B in the form G (t - ). c


181

G G 6.3.1. E and B perpendicular to the propagation: transverse electromagnetic waves G JJGG wB , taken along Oz, yields Equation (3), rotE wt wB (z, t) wE y (z, t) wE x (z, t) z 0. wt wx wy The result is that Bz must be independent from t. G Equation (2), divB 0 , makes it possible to state that: wBx (z, t) wB y (z, t) wBz (z, t) wBz (z, t) 0 , wx wy wz wz =0 =0

with the result that Bz should be independent of z. Therefore Bz must be independent of z and t and can be only constant. However, this solution does not represent a propagation and cannot be acceptable, and we therefore must simply consider that Bz = 0. The same is true for Ez (Ez = 0), as Eqs. (4) and (1) have the same structure as Eqs. (3) and (1). G G The E and B therefore have the nonzero components Ex, Ey, Bx, and By. The G G fields E and B are definitely perpendicular to the direction of the propagation and the electromagnetic wave, by consequence, is termed a transverse electromagnetic (TEM) wave. G G 6.3.2. The relation between E and B

G For example, E

E x (z, t)

z G1 (t ) c

E y (z, t)

z G 2 (t ) c

E z (z, t)

0

(9)

Bx (z,t)

G while the following remain as yet unknown B

By (z,t)

Bz (z,t)

JJGG Equation (3), rotE

G wB wt

, gives:

0

182


x with respect to Ox,

wE y

wBx

(10); and wt wB y wE x x with respect to Oy, . (11) wz wt G JJG G 1 wE Similarly, Eq. (4), rot B , gives: c² wt wBy 1 wE x x in part . (12) wz c² wt wBx 1 wE y x along with . (13) wz c² wt derived with respect to (t – z/c) So from : wE y wBx 1 z 1 z Eq. (10), G '2 (t ) Bx G 2 (t ) [C te wrt t] c c c wt wz c wBx 1 wE y 1 ' z 1 z G 2 (t ) Bx G 2 (t ) [C te wrt z] Eq. (13), 2 c c wz c² wt c c 1 z Bx G 2 (t ) . c c (Note: wrt means “with respect to”.) Similarly, from Eqs. (11) and (12) we can 1 z G1 (t ) . determine that By c c wz

Bx

G G Finally we have B such that B

1 z G 2 (t ) c c

Ey c

(14)

1 z Ex G1 (t ) . c c c The following minor sections give the conclusions to this part. By

G G 6.3.2.1. E and B are perpendicular to one another G G G G If the calculation E B is made, we find that E B

E x

Ey

Ey

Ex

0 , and as c c G G both E and B are perpendicular to the direction of propagation and in the plane Oxy, the configuration shown in Figure 6.7 arises.


183

x

G E

G G u { ez

z

G B

y

Figure 6.7. Structure of a progressive planar electromagnetic wave.

6.3.2.2. Equation for E and B In fact we have: E

c

B

where E

E 2x

E 2y

E 2y

and B

c²

E 2x

, and hence the result.

c²

6.3.2.3. Involving the unit vector G JJG Bringing in the unit vector u ez for the Oz axis gives rise to two properties that can be condensed into the same equation: G E

G G c Bu u

.

(15)

6.3.3. Breakdown of a planar progressive electromagnetic wave (PPEMW) to a superposition of two planar progressive EM waves polarised rectilinearly 6.3.3.1. Preliminary definition: rectilinear polarized wave G A wave can be termed rectilinearly polarized when the vector E stays over all points and instants parallel to a given direction in the plane of the wave. This direction is the direction of polarization. G G 6.3.3.2 . Breakdown of E and B G G Equations (9) and (14) show that E and B can be considered as vectors in the form G G G G G G G G G G E E1 E 2 and B B1 B2 in which the vectors E1 , E 2 , B1 , and B2 are necessarily such that:

184


G E1

G E2

E1x

G1

E1y

0

E 2x

0

E 2y

G2

y

G B2

1 c

G1

B2x

1 G2 c

B2y

0

(16)

x

G E1 G G u { ez

z

G B

0

B1y

x

G E

B1x

G B1

y

G B1

x G G u { ez

G E2

z

wave 1 polarised // Ox

y

G G u { ez

G B2

z

wave 2 polarised // Oy

Figure 6.8. Breakdown of a direct PPEMW in to two rectilinearly polarized PPEMWs.

G G As schematized in Figure 6.8, the direct PPEMW ( E , B ) is such that G G G E(z, t) and B B(z, t) , can appear thus as a superposition of: G z G x “wave 1” polarized along Ox, as E1 G1 (t )ex ; and c G z G x “wave 2” polarized along Oy, as E 2 G 2 (t )e y . c G E

6.3.4. Representation and spectral breakdown of rectilinearly polarized PPEMWs The first wave polarized along Ox can be studied in three-dimensional space and as a function of time. It is defined by the function z E1x G1 (t ) . c


185

6.3.4.1. Spatial representation To have a spatial representation, a given instant, t = 0, is chosen. The signal then z progresses at a speed c, so that E1x (z, t) E1x (z, 0) G1 ( ) . We then have the c G G spatial representation shown in Figure 6.9, where throughout E and B are normal G to each other and u . G G E1 =E1x eGx = G1(–z/c) ex

x

G u

z G B1

y

Figure 6.9. Spatial presence of a rectilinearly polarized PPEMW.

6.3.4.2. Representation with respect to time At a given point, z = 0, E1x (z, t) E1x (0, t)

G1 (t) . The same signal can be

observed at any point on the abscissa (z1) shifted in time by 't

'z

. The observed c amplitude observed at z = 0 and t = 0 then will be seen after an interval of time at z1 'z z1 and t1 such that t1 't (see Figure 6.10 a). c c G1(t)

G1(t)

t

t (a)

t=0 signal at z = 0

h(Z)

t1 = z1/c signal at z = z1

Z0 (b)

Figure 6.10. Representations of a rectilinearly polarized PPEMW (a) v.s. time ; and (b) a component centered about Z0 .

Z

186


6.3.4.3. Spectral breakdown If the wave under study is not monochromatic, then it corresponds to a collection of waves that can be seen as the superposition of an infinite number of waves with a distribution of frequencies spread about an average value (Z0) as shown in Figure 6.10 b. It thus is possible to express the function G1(t) as a Fourier integral: f

G1 (t)

iZt ³ h(Z) e dZ ,

f

where h(Z) represents a distribution of the amplitude as a function of Ȧ and is such 1 f iZt that the reciprocal Fourier transformation gives h(Z) ³ G1 (t)e dt . 2S f In effect, it is possible that all progressive planar waves can be broken down into an infinite number of monochromatic and rectilinearly polarized progressive planar waves. 6.4. Properties of Monochromatic Planar Progressive Electromagnetic Waves (MPPEMW) 6.4.1. The polarization Given the above results, a MPPEMW can be seen as a superposition of two monochromatic, rectilinearly polarized, planar progressive waves detailed by the collection of equations denoted (16). The G1 and G2 functions can be written in a more general form that takes into account any possible dephasing between G1 and G2 in that: z z º ª G1 (t ) a1 cos «Z(t ) » , and c c ¼ ¬ z z ª º G 2 (t ) a 2 cos «Z(t ) M » . c c ¬ ¼ Here the amplitudes can be denoted as a1 = E1x { E mx and again, by denotation, a 2 = E 2y { E my . Additionally, the signals are reproduced at each point in space identical to themselves, with a delay of

'z

, so that the study can be limited to the c G point z = 0, the origin in space, where the resultant electric field ( E ) is such that G G G G G G E E1 E 2 E1x E 2 y . The evolution of the vector E as a function of time is JJJJG G given for a point M(x,y,z) at the extremity of the vector OM E such that:

OM

G E

x

G1 (t)

E mx cos Zt

y

G 2 (t)

E my cos(Zt M) .

z

0

(17)


187

6.4.1.1. When M = 0 (waves “1” and “2” are in phase) y E my tgD and this ratio, equal to that of the amplitudes, is constant We have x E mx like the angle D. The M denotes a line in the 1st or 3rd quadrants (tgD

(

E my E mx

) > 0)

G and the resultant wave ( E ) is rectilinearly polarized along a diagonal, schematized in Figure 6.11 a.

Emy

y

y Emy

M

E

D (a)

M=0

Emx

x Emx

(b)

x

M=S

Figure 6. 11. Rectilinearly polarized wave when (a) M = 0; and (b) when M = S .

6.4.1.2. When M = S (waves “1” and “2” are out of phase) E my y tgE and this ratio is again constant just like the introduced We have x E mx angle E. The M this time denotes a line through 2nd or 4th quadrants G E my (tgE ( ) < 0) , and E thus is rectilinearly polarized along the other E mx diagonal, as shown in Figure 6.11 b. 6.4.1.3. When M

If M JJJJG OM

r

S

(waves “1” and “2” in quadrature): circular polarization 2 or elliptic lines (straight ellipse) S S r , M B and the components x, y, z are: 2 2 x G1 (t) E mx cos Zt G S y G 2 (t) E my cos(Zt B ) r E my sin Zt , E 2 z 0

the calculation directly gives:

188


x² E 2mx

y²

1.

E 2my

This is the equation for a straight ellipse (elliptical polarization) with a larger axis denoted Emx and a smaller axis denoted Emy. The ellipse becomes a circle when the amplitudes are equal, i.e., E mx E my (see Figure 6.12 a for circular polarization). We also have when M

S

y

r

x

E my E mx

tan Zt

r r tan Zt

and the – sign is for when M

2 amplitudes) is constant.

tan G(t) . The + in front of r is for S 2

E my

n, while r

E mx

> 0 (ratio of real

We thus have: d dt

tan G

1

dG

cos ²G dt

rr

1

d(Zt)

cos ²Zt

dt

rr

Z cos ² Zt

.

(a) M = + S/2 , Emx = Emy anticlockwise circular polarisation

M = - S/2 , Emx = Emy clockwise circular polarisation

y

y

Emy

Emy M G(t) Emx

(b) M = + S/2: elliptical anticlockwise polarisation

x

M G(t) Emx

x

(c) M = - S/2: elliptical clockwise polarisation

Figure 6.12. (a) Circular; (b) anticlockwise elliptical; and (c) clockwise elliptical polarizations.


189

For the first quadrant (Figure 6.12 b and c), G(t) changes in the same sense as G(t), it can be determined that when: S Z d dG > 0 , and also > 0 . The point M x M , we have tan G r dt cos ²Zt dt 2 describes an ellipse (or circle) in the trigonometric sense and we have an anticlockwise polarization, as in Figure 6.12 b. S Z d dG cos Zt cos M sin Zt sin M@ , so that: y E my

cos Zt cos M

sin Zt sin M .

With cos Zt

x E mx 1/ 2

sin 2 Zt

ª x² º 1 cos 2 Zt, from which sinZt = «1» 2 ¬« E mx ¼»

,

and then by substitution of the values for cosZt and sinZt into the previous relation : y E my

x E mx

1/ 2

cos M

§ x² · sin M ¨1 ¸ ¨ E 2mx ¸¹ ©

.

Squaring all round, we have: x² E 2mx

y² E 2my

2xy E mx E my

cos M

sin 2 M ,

which is the equation for an ellipse. The component of the wave denoted by x G1 (t) E mx cos Zt is at a maximum when t = 0, and is such that x E mx . This is detailed by point A in Figure 6.13 a. At the same time, and according to Eq. (17), the point A has coordinates along Oy that are y E my cos M .

190


For this instant when t = 0, we can state that: dx > ZE mx sin Zt @t 0 0 dt dy [ZE my sin(Zt I)]t 0 ZE my sin M dt G ª dE º G « » = ZE my sin M e y . dt ¬« ¼» t 0 The sense of direction in which the movement A { M is given by the sign of sin M. G dE y G //e y , then the sense in which M moves is x If M @0, S> , sin M > 0 and dt the trigonometric sense and an anticlockwise elliptical polarization, as in Figure 6.13 a, b, and c; and G dE y G is antiparallel to e y , then M moves in a x if M @S, 2S> , sin M < 0 and dt direction opposite to the trigonometric sense and undergoes a clockwise elliptical polarization, as in Figure 6.13 d, e, and f. It is worth noting that in Figure 6.13 (a), the point A' is such that y

E my

is at a maximum. This gives Zt = M , while the coordinates for A' in terms of Ox are x E mx cos Zt E mx cos M .

Anticlockwise elliptical polarization

EmycosM

y

Emy

A’

(a)

Clockwise elliptical polarization

0 < M < S/2

(d)

- S < M < - S/2

A Emx

x

EmxcosM

(b)

(c)

M = S/2

S/2 < M < S

(e) M = - S/2

(f) - S/2 < M < 0

Figure 6.13. The various polarizations as a function of M.


191

6.4.2. Mathematical expression for a monochromatic planar wave propagating in a direction OH 6.4.2.1. Rectilinearly polarized wave 6.4.2.1.1. Classic system and derivation If we denote the amplitude of a rectilinearly polarized wave as E0 and that JJJG G E 0 A OH so the planar monochromatic wave propagates through OH in the direct sense as indicated in Figure 6.14, then it is possible to write that JJJG G G h E(h, t) E 0 cos[Z(t )] , where OH h. c

z

M H

G r

h G k

G E G u

y

O x

G B JJJG

Figure 6.14. Waves propagating in any direction OH .

G JJJJG The field at a point M located in space by the vector r OM can be found GG G using the following method. In Figure 6.14 we see that h r.u where u is the unit vector in the direction of propagation. Given that M is in the same wave plane as H, we thus have: GG G G G (notation) G G r.u E(h, t) E(r, t) E E 0 cos[Z(t )] . c On introducing the wave vector: G ZG k u , c G G for the fields E and B we have: GG G G E = E 0 cos ª¬Ȧ t - k r º¼ GG G G B = B0 cos ª¬Ȧ t - k r º¼ .

(18)

192


6.4.2.1.2. Complex system and derivation x Using the normal notation for electrokinetics and dielectrics, where Z is the unit of reference for pulsation G The goal of this method is to arrive at an expression for E that contains the classic electrokinetic term exp(+iZt) . As is the tradition, we will omit the symbol R for the “real part” of the waves. A complex system can be given to the waves under consideration, so that for example the direct wave can be written as GG G G E E 0 cos[Zt k.r] , and in turn: GG GG G G G G E 0 exp i[Zt k.r] E 0 exp(ik.r) exp(iZt) E 0 (r) exp(iZt) , (19) GG G G G where E 0 (r) E 0 exp(ik.r) for the complex amplitude of the direct wave. G E

G Really it should be written that E

GG G G G E 0 exp(i[Zt - k.r]) =E 0 (r) exp(iZt) , so that

a complex number is in both the left- and right-hand sides of the equation; however, G G as only the real part ( E ) of E has a physical presence, by notational simplification the waves are often written as presented. GG G G For the retrograde wave, E E 0 cos[Zt k.r] and the corresponding complex GG GG G G G G G form is: E E 0 exp i[Zt k.r] E 0 exp(ik.r) exp(iZt) E 0 (r) exp(iZt) where GG G G G E 0 (r) E 0 exp(ik.r) is the complex amplitude of a retrograde wave.

x Using the normal notation for optics, where the unit of reference is the wave G vector ( k ) GG The goal now in the equations is to bring out the term exp(+i k.r) in the direct wave GG and the term exp(- i k.r) in the retrograde wave. As cos (D) = cos (-D), we can write GG GG G G G E E 0 cos[Zt k.r] E 0 cos[k.r Zt] for the direct wave, so that the complex form also can be written: GG GG G G G G E 0 exp i[k.r Zt] E 0 exp(ik.r) exp(iZt) E 0 (r) exp(iZt) , GG G G G where E 0 (r) E 0 exp(ik.r) is the complex amplitude of the direct wave. G E

(19’)

For the wave in retrograde, we can write that: GG GG G G G E E 0 cos[Zt k.r] E 0 cos( [Zt k.r]) , from which: GG GG G G G G G E E 0 exp(i[Zt k.r]) E 0 exp(ik.r) exp(iZt) E 0 (r) exp(iZt) , where GG G G G E 0 (r) E 0 exp(ik.r) is the complex amplitude of the retrograde wave.


193

It is worth noting that for whatever notation used, in the two exponentials in the G equations for E , the sign that appears must be different for the direct wave, but is the same sign that intervenes in the two exponentials for the retrograde wave. 6.4.2.2. Wave polarized in any sense z G e1

G u

y

x

G e2

Figure 6.15. A polarized wave seen as the superposition of two rectilinear waves G G polarized in e1 and e2 .

As we have already seen for a more general case, a wave can be considered to be the G G resultant of two waves rectilinearly polarized in the directions e1 and e2 G G G perpendicular to the plan of the wave. This gives E E1 E 2 with (for a direct wave): GG G G E1 E m1e1 cos(Zt k.r) ; and GG G G E 2 E m2 e2 cos(Zt k.r M) . The complex form thus can be written: GG GG G G G G G E E1 E 2 E m exp[i(Zt k.r)] E m exp[ik.r)]exp[iZt] G G G where E m ( E m1 e1 E m2 exp[-iM]e2 ) is a complex vector for an elliptical G G G G polarization. By again writing E in the form E E 0 (r) exp(iZt) , we have: GG G G G E 0 (r) E m exp[ik.r)] (20), G G where E m E 0 (0). G G In this case, E m z E 0 , while for a rectilinearly polarized wave G G G E m E 0 E m (actual magnitude). In this general case, we therefore have in complex notation for the direct wave GG G G E E m exp[-ik.r)]exp(iȦt) , so that also: GG G G in electrokinetic notation, E E m exp[i(Zt - k.r)] , or GG G G in optical notation, E E m exp[i(k.r - Zt)] .

194


6.4.3. The speed of wave propagation and spatial periodicity

z

H’ M G r

O

G h k

x

G E

H

G C

G C

G E

G u

y

O

Figure 6.16. Propagation speed for the wave plane.

For a monochromatic rectilinear plane polarized electromagnetic wave GG G G (MRPPEW) of the form E E 0 exp i[Zt kr ] , the phase of the wave is given by GG I Zt k.r . As time varies, the plane of the wave defined GG G G G G by E E 0 exp i[Zt k.r] C , where C is a constant vector, moves in such a way GG that I Zt k.r Zt- kh = K where K is a constant and as shown in Figure 6.16. For this, dI must equal zero so that (Zdt - k dh) = 0 , from which we find: vM

dh

Z

dt

k

G The point H, and therefore the signal E

.

G C , moves at a speed vM

Z

, which is k the speed of the wave phase, and also therefore the speed of the propagation of the wave plane. Z In a vacuum, where k , we have vM c . c G G At a given instant t, E C is repeated at intervals in space, equivalent to Ȝ, GG GG along the plane of the wave such that cos(Zt - k.r) = cos(Zt - k.r - kO ) , and so that kO

2S , from which we rediscover the physical significance of O

spatial period.

2S k

as the


195

6.5. Jones's Representation 6.5.1. Complex expression for a monochromatic planar wave propagating in the direction Oz A given monochromatic planar wave propagating in a given direction Oz can be seen in the most general of cases, and as detailed in Section 6.3.3.2, as the resultant K K of two waves rectilinearly polarized in two directions ex and ey (classically taken as Ox and Oy) perpendicular to the plane of the wave, as shown in Figure 6.17.

G ex

x

G ey

G u

G ez

z

y Figure 6.17. Planar monochromatic EM wave propagating along Oz.

In general terms, and without considering the origin of the phases on the G wave E x otherwise Mx = 0 as supposed in Section 6.4.1, G G G E E x E y , where G G E x E mx ex cos(Zt k.z Mx ) G G E y E my e y cos(Zt k.z M y ) . The wave can be written under the form: G G E(z) E 0 (z) exp(iZt) , G G G with E 0 (z) E m exp[ik.z)] , where E m is a complex vector such that (relation also noted in Section 6.4.2.2): In addition: G G E(0) = E 0 (0) exp(iZt)

G E 0 (0)

JG Em

Eq. (1).

G G E x (0) E y (0) G G = (E mx exp > -jMx @ ex E my exp ¬ª-jM y ¼º e y )exp(iZt) ,

so that:

G G G E 0 (0) = (E mx exp > -jMx @ ex E my exp ª¬-jM y º¼ e y )

Eq. (2).

By comparing Eqs. (1) and (2), we can directly determine that: G G G G G E m = (E mx exp > -jMx @ ex E my exp ª¬-jM y º¼ e y ) = E mx ex + E my e y .

196


By making E mx = E mx exp > -jMx @ and E my =E my exp ª¬-jM y º¼ , the complex vector G E m appears as the vectors for the (complex) components E mx and E my . To sum up then, the wave can be represented in the plane of the wave z = 0 by the vector G G E(0) { E (by notation) such that G E G where E m

G Em

G E m exp(iZt)

G E 0 (0) is a complex vector with components:

E mx =E mx exp > -jMx @ E my =E my exp ª¬-jM y º¼ .

0

r

The ratio defined by the complex number, E my E my , and its exp i[M y Mx ] , is such that its modulus is r r E mx E mx E mx E my

argument is: M

Arg r

(M x M y ) .

It is worth noting that once again a common abuse of notation has G G intervened, for example, E E m exp(iZt) would be, in a more rigorous notation, G G E E m exp(iZt) . The lapse in notation though is because we are simply looking for G G the physical solution E R (E) . 6.5.2. Representation by way of Jones's matrix G G The state of a wave at z = 0, given by E E m exp(iZt) , can be simply written using a column matrix (Jones's notation): § E mx · ¨¨ ¸¸ © E my ¹

§ E mx exp(iMx ) · ¨¨ ¸¸ © E my exp(iM y ) ¹

1 § · E mx exp(iMx ) ¨ ¸ r exp(i ) M © ¹

By choosing the origin of the phases so that Mx = 0, one can also write that: § E mx · ¨¨ ¸¸ © E my ¹

E mx § · ¨¨ ¸ E exp( i ) M y ¸¹ © my

1 § · E mx ¨ ¸ © r exp(iM) ¹


197

y E0

Emy D

Emx

x

Figure 6.18. Definition of Emx and Emy.

By making E mx

E 0 cos D and E my

E 0 sin D , as in Figure 6.18, and still with

Mx = 0, we have: G Em

E mx

E mx

E 0 cos D

E my =E my exp ª¬-jM y º¼

E 0 sin D exp ª¬ -jM y º¼

so that: § E mx · ¨¨ ¸¸ © E my ¹

E mx § · ¨¨ ¸¸ M E exp( i ) y ¹ © my

E 0 cos D § ¨ ¨ E sin D e iMy © 0

· ¸ ¸ ¹

cos D § E0 ¨ ¨ sin D e iMy ©

· ¸ . ¸ ¹

For a rectilinear polarization, we can take My = 0, and the Jones's vector representing the wave is, by notation G

Em

§ E 0 cos D · ¨ ¸ © E 0 sin D ¹

§ cos D · E0 ¨ ¸ , and the normalized Jones's vector is such that: © sin D ¹ G § Em ¨¨ © E0

· ¸¸ ¹

§ cos D · ¨ ¸. © sin D ¹

A rectilinear polarization through Ox corresponds to D and sin D 0 , from which G §1· G Em E 0 ¨ ¸ E 0 (eh ) x ©0¹

0 , so that cos D

1

§1· ¨ ¸. ©0¹ Similarly, a rectilinear polarization along Oy corresponds to D = S/2 so that cos D = 0 and sin D = 1, from which G G where eh is represented by the column matrix (eh )


198

G

§0· E0 ¨ ¸ ©1¹

Em y

G E 0 (ev ) §0· ¨ ¸. ©1¹

G G where ev is represented by the column matrix (ev )

For a clockwise polarized wave, S

D

4

My

E 0 cos D

, so that E mx

S 2

, from which, with e

G

E m d

2

E0

iM y

2 i

G

E0

2 2

S

e2

i , we have:

§ 2 · ¨ ¸ 2 ¸ E0 ¨ ¨ 2¸ ¨¨ i ¸¸ © 2 ¹

For an anticlockwise polarized wave, D

E m g

E 0 sin D

and E my

§ 2 · ¨ ¸ 2 ¸ ¨ E0 ¨ ¸ ¨¨ i 2 ¸¸ 2 ¹ ©

E 0 2 §1· ¨ ¸ 2 ©i¹ S 4

and M y

S 2

, from which:

E0 2 § 1 · ¨ ¸. 2 © i ¹

G G These waves can be represented by a linear combination of ( eh ) and ( ev ), so for example, G

E m d

E 0 2 § 1· ¨ ¸ 2 ©i¹

E0 2 § 1 · E0 2 § 0 · ¨ ¸ ¨ ¸ 2 © 0¹ 2 ©i¹

E0 2 G E 2 G ( eh ) i 0 (ev ) . 2 2

Inversely, the wave obtained by a superposition of the clockwise (negative) and anticlockwise (positive) waves is a rectilinear polarized wave because we have G

G

E m d E m g

E 0 2 § 1 · E 0 2 § 1· ¨ ¸ ¨ ¸ 2 © i ¹ 2 ©i¹

§1· E0 2 ¨ ¸ © 0¹

where the wave is polarized rectilinearly with respect to Ox.

G E 0 2 ( eh ) ,


199

6.6. Problems 6.6.1. Breakdown in real notation of a rectilinear wave into two opposing circular waves A rectilinearly polarized, monochromatic plane electromagnetic wave propagates G G G along Oz in the form E E 0 cos Zt kz and is such that E & Oxy makes an angle D with the axis Ox.

y G B

G E

D x Show that this wave can be seen as a superposition of two planar waves circularly polarized in the opposite sense.

Answers G The components Ex and Ey of E on Ox and Oy are, {with 1 1 cos a cos b = [cos(a + b) + cos(a - b)] and cos a sin b = [sin(a + b) - sin(a - b)]} 2 2 Ex

E 0 cos Zt kz cos D

Ey

E 0 cos Zt kz sin D

E0 2 E0 2

cos(Zt kz D) sin(Zt kz D )

E0 2 E0 2

cos(Zt kz D ) sin(Zt kz D ).

This wave can be seen as the superposition of two following waves with the components E cG x

E

cG y

E cD x

E cD y

E0 2 E0 2 E0

cos(Zt kz D)

anticlockwise circularly polarized wave; and sin(Zt kz D)

cos(Zt kz D) 2 clockwise polarized wave. E0 sin(Zt kz D) 2


200

Comment: It is worth remembering that a wave with the components Ex Ey

E m cos Zt E m cos Zt M

is one that has an anticlockwise polarization when M

S 2

, which means that the

components of this anticlockwise polarized wave are: Ex Ey

E m cos Zt M

E m cos Zt S· § E m cos ¨ Zt ¸ 2¹ ©

E m sin Zt

Similarly, a wave which is polarized clockwise corresponds to M

S 2

, and has

components Ex Ey

E m cos Zt M

E m cos Zt S· § E m cos ¨ Zt ¸ 2¹ ©

E m sin Zt

6.6.2. The particular case of an anisotropic medium and the example of a phaseretarding strip Neutral lines in an anisotropic strip are such that rectilinear polarized waves along the two directions of these neutral lines, which are perpendicular to another, retain G G their polarization during their propagation. It is supposed that ex and e y are the vectors locating the directions of the neutral lines. G G 1. In a phase-retarding strip, the rectilinearly polarized waves along ex and e y

each propagate as if in an isotropic media but with different indices, which are nx for G G ex and ny for e y , and we will suppose that ny > nx. The components of the incident wave in the plane z = 0 are:

G E

E mx e jZt E my e j(Zt M) 0

Give the form of the wave emerging from the side plane where z = e, and determine the additional difference in phase caused by the wave's propagation


201

between the two components with respect to Ox and Oy through the strip of thickness e. Express the result as a function of the additional differences in optical pathways (GL) between the two components of the wave under consideration created by the strip. O 2. Under consideration is a half-wave strip such that GL . Study the effect of 2 such a strip on an incident rectilinear polarized wave (M 0) . Answers 1. Phase delay. For the waves polarized rectilinearly with respect to the directions c c G G ex and e y , if we suppose that n y > n x , we have v x . The axis > vy = nx ny G G with respect to ex is termed the fast axis, while the axis along e y is the slow axis. The two components Emx and Emy of the emerging wave from the strip do not have the same dephasing to that which exists between the two same components of the incident wave. So the components of the incident wave in the side plane z = 0 are: G E

E mx e jZt E my e j(Zt M) ,

0 and the emerging wave, from the side plane z = e, has the components: G E

E mx e j(Zt k x e) E my e

j( Zt k y e M)

.

0

The additional phase difference, due to the propagation of the wave through a strip of thickness e, between two components with respect to Ox and Oy, is therefore: 2S 2S ) = (k y e - k x e) = G L where GL represents the difference in the (n y n x )e O O additional optical pathway generated by the strip between the two components of the wave under consideration. S O When ) , we have GL = and the strip is called a quarter wave strip. 2 4 O When ) = S, we have GL and the strip is called a half wave strip. In fact, the 2 strip is not a true half wave strip unless O 2 GL = 2 (n y - n x ) e .

202


2. Effect of a half wave strip on a rectilinearly polarized wave (M incident rectilinearly polarized wave, we can take (at z = 0):

G E

0) . For the

E 0 cos D e jZt E 0 sin D e jZt 0

G The emergent wave is such that (with z = e) : E

E 0 cos D e j(Zt k x e) E 0 sin D e 0

j(Zt k y e)

By changing the origin of times by taking for the temporal origin a value that ensures a zero dephasing of kx e with respect to the x components, the emerging wave can be written as:

G E

E 0 cos D e jZt E 0 sin D e

j( Zt k y e k x e)

0

With )

G E

G , so that E

E 0 cos D e jZt E 0 sin D e j(Zt ) ) 0

S (for the half wave strip) and e jS

1 , we reach:

E 0 cos D e jZt

E 0 sin D e jZt . 0

The emergent wave remains rectilinearly polarized and is symmetric to the incident wave with respect to the x axis. Note: Effect of the quarter wave strip. Similarly, we can show that a quarter wave strip can transform an elliptical vibration into a rectilinear vibration if the axes of the ellipse coincide with the neutral axes of the strip. All circularly polarized waves are transformed into rectilinearly polarized waves by a quarter wave strip.

6.6.3. Jones's matrix based representation of polarization In the representation demonstrated by Jones, the effect of a polarizer can be given by a matrix ( P ) such that:


G

Em

§ E mx · ¨¨ ¸¸ © E my ¹exit

203

§ E mx · ( P )¨ ¨ E my ¸¸ © ¹entrance

The representation directed along Ox is that of a projection type operator: §1 0· ( PX )= ¨ ¸. © 0 0¹ With respect to the action of a strip that introduces a delay equal to My between the components Emx and Emy of the field, it can be represented by a matrix such as: 0 §1 (L)= ¨ ¨ 0 e iM y ©

· ¸. ¸ ¹

1. Indicate the form of the matrix ( Py ) that can be associated with a polarizer directed along Oy so that the matrix is associated with a rotation \. 2. Use Jones's formalism to study the effect of a half wave strip on a rectilinearly polarized wave. 3. Give the product matrix that can be associated with an effect of two crossed polarizers between which sits at 45 ° a quartz strip on an incident light. Answers §0 0· 1. The action of a polarizer directed along Oy is represented by ( Py ) = ¨ ¸. ©0 1¹ For its part, the rotation (\) of an element (strip, polarizer) is represented by the § cos \ sin \ · rotational matrix: R(\ ) ¨ ¸. © sin \ cos \ ¹

Finally, the state of polarization on leaving the system can be obtained by bringing together the successive orientational effects subjected on the incident wave. The resultant effect thus is written as a product of the matrices associated with the transformations. 2 The effect of a half wave on a rectilinearly polarized wave. Using Jones's notation and two dimensions, the incident wave is: G

1 0 Em inc E0 cos D §¨ 0 ·¸ E0 sin D §¨ 1 ·¸ .

© ¹

© ¹

The transformation associated with a half wave strip is given by the matrix:

204


( L )My=S G

Em exit

0 §1 ¨ iM y ¨ ©0 e

· ¸ ¸ ¹

0 · §1 ¨¨ iS ¸¸ ©0 e ¹

§1 0 · ¨ ¸ , and © 0 1¹

G ( L )My=S E m

1 inc E0 cos D §¨ 0

©

0 ·§1· §1 0 ·§ 0· ¸ ¨ ¸ E 0 sin D ¨ ¸¨ ¸ , 1¹ © 0 ¹ © 0 1¹ © 1 ¹

so that: G

1 0 1 0 Em exit E0 cos D §¨ 0 ·¸ E0 sin D §¨ 1·¸ E0 cos D §¨ 0 ·¸ E0 sin D §¨ 1 ·¸ .

© ¹

©

¹

© ¹

© ¹

We rediscover of course (see preceding problem) the conclusion that the emerging wave is symmetrical to the incident wave with respect to the x axis. 3. The effect on a incident wave by two crossed polarizes between which is inserted a strip of quartz at 45 ° is obtained directly from the product of the following matrices: § E mx · ¨¨ ¸¸ © E my ¹exit

( Py ) R(45D ) ( L )

M=

S ( Px

2

§ E mx · . )¨ ¨ E my ¸¸ © ¹entrance

Chapter 7

Electromagnetic Waves in Absorbent and Dispersing Infinite Materials and the Poynting Vector

7.1. Propagation of Electromagnetic Waves in an Unlimited and Uncharged Material for Which U A = 0 and j A = 0. Expression for the Dispersion of Electromagnetic Waves 7.1.1. Aide mémoire: the Maxwell equation for a material where UA 0 and jA 0 These equations are:

G divE

G divB

JJGG rotE

0 (1)

0

G wB wt

JJG G rot B

(2)

(3)

Hµ

G wE wt

(4)

7.1.2. General equations for propagation G Just as in a vacuum, we can eliminate B between Eqs. (3) and (4) by calculating the rotational of Eq. (3): G w JJG G (4) w ²E (using Eq. (4)) (rotB) = -Hµ wt wt ² JJJGG (1) JJJGG JJJJG G grad(divE) 'E 'E (using Eq. (1))

JJJJG JJG G rot(rotE)

G JJJGG w ²E 'E = İµ wt²

H0 H r P0P r

G w ²E

wt²

(5’)

206


With H0µ0 =

1 c2

And similarly:

, we can then go on to write: G JJJGG H P w ²E r r 'E c² wt² G JJJGG H P w ²B r r 'B c² wt²

0

0

(5)

(6)

7.1.3. A monochromatic electromagnetic wave in a linear, homogeneous, and isotropic material For a monochromatic electromagnetic (EM) wave in a linear, homogeneous, and G G G G isotropic (lhi) material, E and D are directly related by D H(Z) E where H(Z), dependent on Z, has for most materials (i.e., imperfect ones), a complex magnitude given by H(Z) H . V0eiZt

C = HrC0

if

Figure 7.1. A leak current ( if) in a real dielectric.

The electrical behavior of an imperfect dielectric, otherwise called a real dielectric, does not resemble that of a single capacitor as the presence of leak currents (if), caused for example by residual charges, gives rise to a component due to resistance (Figure 7.1). Therefore, H(Z) needs to be written in a complex form in order to give a correct Fresnel diagram for the real dielectric. This means that the current intensity (I) should simultaneously present a component due to a dephasing S by with respect to the applied tension (capacitive component) and a component 2 in phase with the tension due to the resistance associated—in this example—with leak currents.

Chapter 7. Electromagnetic waves in materials and the Poynting vector

207

With Q = CV = Hr C0 V0 e jZt where H r is the complex relative dielectric permittivity

(given by H r = Hr ' - jH r '' and H r =

H

=

C

, where C0 is the capacitance of a H 0 C0 capacitor in a vacuum), it is possible to directly obtain I in terms of two these two components (capacitive and resistive components), as dQ I= =ZH r '' C0 V + j ZH'r C0 V = IR + jIC . dt It is just worth noting that, as detailed in Chapter 2 of the second volume in this series, C is complex, and to be strictly correct we should write in the preceding equations for Hr that Hr = C/C0. G G G Given that the waves are monochromatic, the fields E , D , and B can be given by: G G G G E(r,t) = E 0 (r)exp(+jZt) JG G JG G D(r, t) = D0 (r)exp(+jZt) JG G JG G B(r, t) = B0 (r)exp(+jZt)

Following simplification of the two members by exp(+jZt), and given that derivation with respect to time is the same as multiplying by jZ, i.e., w multiplication by jZ, the Maxwell Eqs. (1) to (4) take on the form: wt JJG G G G G G G divE 0 (r) 0 (1’) rotE 0 (r)= - jZB0 (r) (3’) JJG G G G G G G divB0 (r) 0 (2’) rotB0 (r)= jZHµE 0 (r) (4’) Similarly, Eq. (5') for propagation yields: JJJJ G JG G G G ' E 0 (r) Z2 HµE 0 (r)

0

(7)

If the medium is simply a nonmagnetic one ( µr =1 ), then H Hµ = H0 H rµ0 = r and Eq. (7) then becomes: c² JJJJ G JG G Z² G G ' E 0 (r) H r E 0 (r) c²

0

(7’)

208


Alternatively, multiplying by exp[jZt] the equation can be given as: JJJGG G Z² G G (7’’) ' E(r) H r E(r) 0 c² The relationship H

H(Z) (or H r

H r (Z) ) accounts for the fact that in a

vacuum waves of different frequencies do not propagate at the same velocity due to a dispersion phenomenon. Just as was justified above for a real dielectric, H(Z) is normally complex, resulting in absorption phenomena associated with its imaginary component.

7.1.4. A case specific to monochromatic planar progressive electromagnetic waves (or MPPEM wave for short) The complex amplitude of a MPPEM wave is in the form [see Eq. (20) in Chapter 6], that is: GG G G G E 0 (r) E m exp(- jk.r) (8) In this case the material can absorb, so in the same way as we made H = H , k is used in its complex form: k = k .

7.1.4.1. Structure of a MPPEM wave G G First, Eq. (1') gives us divE 0 (r) 0 3

¦

d

i 1 dxi

GG 3 GG G E mxi exp[ j(k x1x1 k x2 x 2 k x3 x 3 = - j ¦ k xi E mxi e jk.r =-j k.E 0 (r)

The result,

i 1

GG G G G divE 0 (r)= - j k.E 0 (r)

(9)

is of a general form and concerns the action of the divergence operator on a wave GG G determined by Eq. (8). By using Eq. (1), we can determine that k.E0 (r)= 0 , so that GG simplification with exp(- jk.r) yields: GG k.E m = 0

(10)

GG G G G Second, Eq. (2') similarly gains divB0 (r)= - j k.B0 (r) = 0 , from which we find: GG k.Bm = 0

(11)


209

JJG G G JJG G G G G Third, Eq. (3’), rot E 0 (r)= - jZ B0 (r) , demands a calculation around rot E 0 (r) .

Therefore, GG GG GG GG JJG G G d d [rot E0 (r)]x = (E mz e jk.r ) (E my e jk.r )= - jk y E mz e jk.r + jk z E my e jk.r dy dz from which yields: JJG G G G G G [rot E 0 (r)]x = -j k u E 0 (r) .

x

The general result of the action of the rotational of Eq. (8) finally ends up with JJG G

G

G

rot E0 (r)G = -j k u E0 (r)G

.

(12)

G G G G G With the help of Eq. (3’), we find that the relation k u E 0 (r) =Z B0 (r) , which is GG simplified on using exp(- jk.r) , gives

G G k u Em

G ZBm .

(13)

Fourth, Eq. (4’) is such that we also find: JJG G G G G G rotB0 (r) j k u B0 (r) G = jZ H µ E 0 (r)

Therefore, as µr = 1 , H = H0 H r GG with exp( jk.r) gives us G G k u Bm

and simplification of the above equation

G G Ȧ - Ȧ İ µ0 E m = İr Em . c²

(14)

Fifth, the MPPEM in a material has a planar progressive structure that is in direct G G G relationship with the trihedral (E, B, k) . 7.1.4.2. General equation for dispersion The calculation of the Laplacian vector for the wave's complex amplitude given in Eq. (8) directly gives: JJJJ G G G G JG G ' E 0 (r) = - k².E 0 (r) .

(15)

210


JJJJ G JG G By identification with the value given by ' E 0 (r) from Eq. (7'), as in JJJJ GG G JG G Z² G G ' E 0 (r)= H r E 0 (r) and the additional division by exp(- jk.r) , the result is: c² Z² ª ºG Hr (Z) » E m 0 . « k² c² ¬ ¼ G With E m z 0 , the general form of the so-called dispersion equation is: k²

Z² c²

Hr (Z)

.

(16)

7.2. The Different Types of Media 7.2.1. No absorbing media and indices For nonabsorbent materials with real values for Hr(Z), Eq. (16) indicates that k is also real, within the constraints of Hr(Z) > 0, as otherwise Hr(Z) would correspond to an evanescent wave for which k = k = - ik'' (see Chapter 9). According Eq. (16) can be written more simply as: Z k= H r (Z) . (16’) c The velocity of the phase of a MPPEM wave with frequency Z is now given by: Z c . (17) vM k H r (Z) For its part, the group velocity of the wave is now: dZ vg = . (18) dk

7.2.1.1. In a vacuum Now H r =1 and therefore from Eq. (17) gives vM = c (the velocity of the phase is constant with respect to Z and equal to c). As Z = c k , Eq. (18) shows us that vg = c , so that in a vacuum there is vM = vg = c .

7.2.1.2. In a nondispersing medium Now Hr is independent of the frequency and accordingly from Eq. (17): vM A , where A is a constant. In addition, as indicated in the plot shown in

211


Figure 7.2, Z = A k .

Indeed, vg =

dZ dk

A , so that for nondispersing media,

vM = vg . Z slope A =

c

Hr

= vM =vg

k Figure 7.2. Plot of Z = f(k) for nondispersing media.

7.2.1.3. Normal dispersing media Now we find that if Z increases, then Hr(Z) also increases. However, Eq. (17) dZ indicates that vM decreases, so that vM2 < vM1 and the tangent ( ) to the dk dispersion curve is above the curve, and therefore vg < vM , as shown in Figure 7.3a. Z

Z

vg1 = dZ/dk

Z2

Z2

Z1

vM 2 vM 1

k1

vg1 = dZ/dk vM 2

Z1

vM 1

(a) k2

Normal dispersing medium

(b) k

k1

k2

k

Abnormally dispersing medium

Figure 7.3. Dispersion curves showing Z=f(k) for (a) normal dispersion medium; and (b) abnormal dispersion medium.

8.2.1.4. Abnormally dispersing media Now as Z increases, then Hr(Z) decreases, and Eq. (17) indicates that vM also dZ increases so that vM2 > vM1 and the tangent ( ) is below the dispersion curve and dk vg > vM , as shown in Figure 7.3b.

212


7.2.1.5. Indices

An index (n) is defined by:

c

n

vM

(19)

vM (Z) , it is also possible to state that n = n(Z) .

and as Eq. (17) points out vM

Placing Eq. (17) into Eq. (19) gives: n(Z)= H r (Z)

(19’)

from which through Eq. (16') we gain Z (20) k n k0n . c Z where k 0 and is the wavenumber in a vacuum. As the wavelength in a c c 2S 2S 2Sc medium can be determined by O vM T , if we introduce O 0 n Z k Z (wavelength in vacuo) we arrive at: O

O0 n(Z)

.

(21)

7.2.1.6. Comment It is possible also to use Eq. (13') by multiplying the two members by GG G G G exp i[Zt k.r] so that k u E ZB which gives in terms of moduli kE ZB . With

k

Z c

n , the equation yields: E

c

B

n

vM .

(19’’)

7.2.2. Absorbent media and complex indices 7.2.2.1. Absorbent media (dielectric permittivity and wavevector both complex) 7.2.2.1.1. Equation for dielectric permittivity As above detailed, absorbing materials means that H(Z) is complex, and by notation, H(Z) { H . Dielectric losses caused by Joule effects from the resistive component are absorbed by the dielectric—hence the name absorbent. More precisely, and depending on notational convention, complex permittivity can be defined in terms of:


electrokinetics, where H(Z) H H ' jH '' wave optics, where H(Z) H H ' jH ''

213

(22) or (22’)

Whether a minus or positive sign is chosen in the above equations (and as detailed below in equations for k ) is relative to the final expression used for the wave, wherein its application to dielectric media signifies an absorption and not a (spontaneous) amplification of the wave, the latter of which would be unrealistic. dQ was Staying with dielectrics, in Section 7.1.3 the calculation of I = dt performed using an electrokinetic notation, i.e., V = V0 exp(+jZt) from which with H H ' jH '' from Eq. (22') it can be determined that I = ZH''r C0 V + j ZH'r C0 V . The term due to resistance (IR) which is such that IR = +ZH''r C0 V is in phase with the tension; however, if the term H had been taken into Eq. (22'), then the result would have been IR = - ZH''r C0 V , indicating an unphysical dephasing of the intensity and the tension by S at the resistance terminals. Inversely, if the optical notation were used so that V = V0 exp(-jZt), then Eq. (22') must be used to obtain a physically acceptable result, as in IR = +ZH''r C0 V . 7.2.2.1.2. Equation for the wavenumber and the MPPEM wave In optical terms, according to Eq. (16) for dispersion, the complex form of H indicates that k also must be complex (k o k). As detailed below, the optical wave can be absorbed during its propagation, and therefore k should be written in the form: k = k' - j k'' (electrokinetic notation) (23) and k = k' + j k'' (optical notation) (23’) Again following from Eq. (16), with a r sign given for both possibilities, Z² k² k ' ² k '' ² r 2 jk 'k '' (H 'r r jH ''r ) , we have: c² so that by identification of the real and imaginary parts, Z² k'² - k''² = ( ) Hr ' c² Z² 2 k' k'' = ( ) H r '' . c² With H r '' > 0 (dielectric absorption) and k' > 0 (propagation in the sense z > 0), the second relation indicates that k'' > 0 and is an optical absorption.

214

G E

G E


The MPPEM wave along Oz therefore is such that: G E m exp( j[Zt kz]) (using electrokinetic notation) G E m exp( k ''z) exp( j[Zt k ' z]) G E m exp( j[kz Zt])

G E m exp( k '' z) exp( j[k ' z Zt])

(by optical notation).

The term exp(k '' z) represents the exponential absorption—also called attenuation in optics—of the wave during its propagation. It is worth noting that the wrong choice of sign in Eqs. (23) and (23') would mean instead of representing an attenuation of the signal as it propagates that the signal would be progressively amplified, a physical impossibility. 7.2.2.2. Complex index In turn, the index also must be complex given that the defining equation, as an extension to Eqs. (19') and (20) where k is complex, is: n² = H r (Z) where k =

Z c

n

(24)

Wherein: n = n' - jn''

(electrokinetic notation)

(25)

n = n' + jn''

(optical notation)

(25’)

where the negative sign is correctly used to represent the absorption in electrokinetic notation and the positive sign is correct in optical terms. Identification of the real and imaginary parts gives: Z

n'² - n''² = H r '

k' = n'

2 n' n'' = H r ''

k'' = n''

c Z c

The progressive wave along Oz therefore can be represented by:


G E

G E m exp( j[Zt kz])

G E

G E m exp( j[kz Zt])

215

G Z E m exp( j[Zt nz]) c (electrokinetic notation) G Z Z E m exp( n '' z) exp( j[Zt n ' z]) c c (26) G Z E m exp( j[ nz Zt]) c (optical notation) G Z Z E m exp( n '' z) exp( j[n ' z Zt]) c c

The term for absorption and attenuation is thus in the form exp(- n''

Z

z) where the c component n'' for the index is called the extinction index while n' is the refraction index associated with the passage of the wave through an additional medium.

7.3. The Energy of an Electromagnetic Plane Wave and the Poynting Vector 7.3.1. Definition and physical significance for media of absolute permittivity (H), magnetic permeability (µ), and subject to a conduction current (j A ) 7.3.1.1. Definition The following two Maxwell equations are used for the titled material: JJGG rotE JJG rot

G B

G wB

(3)

wt

G G w (HE) jA wt

µ

(4’’)

[ Eq. (3) multiplied by

G B µ

]

G [ Eq. (4’’) multiplied by [-E] ]

Following multiplication of the two members of each equation, as above, the addition of respective members yields:

G G G G G B JJG G G JJG B B wB G w (HE) G G jA .E . rot E - E rot = -E wt µ µ µ wt G G G JJG G G JJG G On using div a u b =b rot a - a rot b , the equation is changed to:

G §G B· w § E² B² · G G div ¨ E u ¸ = - ¨¨ H ¸ jA .E . ¨ µ ¸¹ wt © 2 2µ ¸¹ ©

(27’)

216


By definition then, the Poynting vector is: G G G B S Eu . µ

(27)

7.3.1.2. Physical significance By introducing the Poynting vector into Eq. (27'), it is found that:

G w § E² B² · G G div S= - ¨¨ H ¸ jA .E . 2µ ¸¹ wt © 2 For a volume (V) with a surface limited to 6 , we can state that: G G G w § E² B² · - ³³³ div S dW= ³³³ ¨¨ H ¸¸ dW ³³³ jA .E dW , which also yields wt © 2 2µ ¹ G G G G w § E² B² · -w ³³ S.d6 ³³³ ¨¨ H ¸¸ dW ³³³ jA .E dW . wt © 2 2µ ¹ 6 (1)

(2)

7.3.1.2.1. Physical significance of term (1) It is well known (at least in vacuum, where we have H0 and µ0 instead of H and µ) that the volume density of electrical and magnetic energies are, respectively, equal dw m B² dw e H E² dw e dw m + to and . . In all, the energy is equal to w = dW dW dW 2µ dW 2 dw

dW and represents the power (for a total dt volume of the material) in electrical and magnetic energies created by the electromagnetic field in the Term (1) can be written in the form

³³³

7.3.1.2.2. Physical significance of term (2) G G Simplification using dW = d6.dr permits the equation G G G G G G ³³³ jA .E dW = ³³ jA .d6 ³ E.dr = I.V = PJ , which represents the electrical power given by the charges in the system, which typically correspond to the power dissipated by the Joule effect caused by jA . G To conclude, the Poynting vector flux ( S ) through a surface ( 6 ) is equal to the power which gives rise to the electromagnetic field through 6 of the material under consideration.


217

7.3.2. Propagation velocity of energy in a vacuum For this calculation we suppose that jA = 0 and that the energy is being carried by a plane progressive wave, for which the Poynting vector is directed in the same sense G G G G as the propagation given by the vector denoted u (with the direct trihedral E, B, u ). K K In a vacuum, where the components for the vectors E and B are given by Eqs. (13) and (14) of Chapter 6, we have: G §G B · 1 S Sz ¨ E u (E x B y E y Bx ) ¸¸ ¨ µ0 ¹ z µ0 © (28) 1 2 2 2 2 (G1 G 2 ) H0 c (G1 G 2 ). µ0 c With respect to a unit surface normal to the direction of the EM wave propagation, it therefore can be written with the power (P) transmitted by the EM field through a surface unit ( 6 = 1 ) that: P = ³³ Sz d6 = Sz = H0 c (G12 G 22 ).

(29)

6 1

Additionally, with jA = 0, the volume density of the energy associated with the EM wave (w) is equal to: H E² B² 1 1 w= 0 = H0 (G12 G 22 )+ (G12 G12 )= H0 (G12 G 22 ). (30). 2 2µ0 2 2µ0 c² From Eqs. (28) and (30), we can determine that S

cw , which in terms of vectors

gives G S

G cwu

.

(31)

Given P through 6 =1 and w, with the help of Eqs. (29) and (30), we find the relationship: P = c w . (32)

l=1 6=1 L =ve Figure 7.4. Power transmitted through 6.

218


As shown in Figure 7.4., P transmitted as energy per unit time through 6 = 1 corresponds to the energy contained within the volume (V) given by V = 6.L ¦ =1 where L = ve ('t) 't=1 = ve and V = ve in which ve is the energy propagation velocity. As the energy density (w) represents the energy in the unit volume (surface 6 =1 and length l =1 ), we therefore can state that P = ve w . As above noted, P = c w , we can directly find: ve = c which indicates that the energy propagates at the velocity of light in a vacuum.

7.3.3. Complex notation Calculations for the Poynting vector or the energy cannot be directly performed G G using the complex values of the fields E or B as the real part of the resulting product is not equal to the product of the real parts. Such a problem has been found G G G for S or w given by Eqs. (27) and (30) which are not linear with respect to E or B . G However, it is possible to obtain the average values of S or w. G G Given the following expressions for the fields E or B G G E E 0 exp(iZt) , where a rigorous treatment would mean writing: G G E= R(E 0 exp(iZt)) , G G G G where E= R(E) and E=E 0 exp(iZt) . As z denotes a complex number, as in z = a + ib , we find that R(z) = a =

zz* 2

,

where z* is the conjugated complex of z . Given the above, the Poynting vector can be calculated with µ being real by: G G G B 1G G 1 G G S E u = E u B = R(E) u R(B) µ µ µ G G G G 1 = { [E 0 exp(iZt) E 0 exp(iZt) ] u [B0 exp(iZt) B 0 exp(iZt) ]} 4µ G G G G G G G 1 G = {[E 0 u B0 ] + [E 0 u B0 ]+[E 0 u B0 exp(2iZt)]+[E 0 u B 0 exp(2iZt)]} . 4µ The first two terms are conjugated complexes and their total value is twice their real parts. The third and fourth terms vary as functions of 2Z so that their average value over a period is equal to zero. This leaves us with


219

G G G 1 S = R(E 0 u B0 ) . (33) 2µ GG G G G For a planar progressive wave along u , we have E 0 =E m exp[ik.r)] , which yields: JG G G 1 S = R(E m u Bm ) . 2µ

(33’)

In a vacuum, where µ = µ0 , H0µ0 c² = 1 , and Bm

Em c

:

JG G G H cG S = 0 Em u Em u . 2 G In the more specific case of a rectilinear plane wave, where E m

G E 0 , we have:

G H c G S = 0 E 02 u . 2 G G G B This result can be obtained directly from S E u [Eq. (27) written for a vacuum µ0 where µ = µ0].

7.3.4. The Poynting vector and the average power for a MPPEM wave in a nonabsorbent (k and n are real) and nonmagnetic (µr = 1, so that µ = µ0) medium G Z G G 1G G From Eq. (13), B n u , we have k u E and k c Z G G G B n G G G S Eu E u (u u E) . µ0 cµ0 G G G GG G GG G By using the equation for a paired vector, a u (b u c) a.c b a.b c , we obtain G GG G GG K G S= ncH0 [(E.E).u (E.u)E]= ncH0 E 2 u .

By taking an average value over a given period, for the power transmitted (P) by the EM wave, 1
= <Sz > = ncH0 E 02 . 2

7.3.5. Poynting vector for a MPPEM wave in an absorbent dielectric such that µ is real Using the optical notation given in Section 7.2.2.1,

220


G E G and B

G E m exp(k '' z) exp( j[k ' z Zt]) k G G n G G uuE uuE . Z c

According to Eq. (33), established for a real value of µ , we find that G G G G 1 E 0 E m exp(k '' z) exp( jk ' z) <Sz > = R(E 0 u B0 ) , where 2µ G n G G B0 u u E0 c Therefore

ª n G G º n G G G G G ªn G G º G G E 0 u B0 E 0 u « u u E 0 » E0 u « u u E0 » E0 u u u E0 c «¬ c »¼ ¬c ¼ n G G G n G E 0 .E 0 u E m ² exp(2k '' z) , from which c c G G 1 1 n' G <Sz > = R(E 0 u B0 ) E m ² exp(2k '' z) , 2µ 2µ c so that for a nonmagnetic medium (µ = µ0) cn ' H0 G <Sz > = E m ² exp(2k '' z) . 2 The power transmitted by the electromagnetic field exponentially decreases with the Z distance traversed. The extinction coefficient for this is given by 2k'' = 2n'' . c

7.4. Problem Poynting vector It is worth recalling that the physical magnitude of flux represents an amount associated with a physical magnitude which traverses a unit surface per unit time. 1.

The analogy between the charge current density and the energy current density vectors G (a) Here j denotes the charge current density vector and U the volume charge G G G G density. The flux of the vector j through a surface dȈ is dI = j dȈ . What does j alone represent? Give the equation for the conservation of charge for a rapidly varying regime in an isolated system.


221

(b) By analogy, the energy flux (which thus exhibits a power per unit surface) G G G through the surface dȈ is defined by a relationship of the type dP = S .dȈ in which G S represents the energy current density, otherwise termed the Poynting vector. What does S alone represent? By introducing the magnitude w T , which represents the volume density of the total energy, give an equation for the conservation of G G energy analogous to that for the conservation of charge ( S plays the role of j and w T that of U ).

2. The total volume density of energy w T can be considered as the sum of two terms, one being due to the density of kinetic energy ( w c ) and the other due to the energy density ( w em ), which will be detailed below. (a) The theorem for kinetic energy makes it possible to state that the variation in kinetic energy is equal to the work of the applied forces. A variation in the kinetic G G energy density (for a unit volume containing a charge U) is given by dw c F. v dt G G G G G with F q ª¬ E v u Bº¼ when charges with volume density U and velocity v are G G placed in an electromagnetic field characterized by the fields E and B . Determine from the equation for the conservation of energy the value of the expression G G G dw em . divS + E.j + dt JJG G (b) Give the Maxwell-Faraday (M-F) relationship (M-F which gives rot E ) and the G JJG B ) for a nonMaxwell-Ampere (M-A) relationship (M-A which gives rot µ0 G G B ] and magnetic medium traversed by a real current ( j ). By multiplying M-F by [ µ0 GG G M-A by [ -E ], calculate E. j . G (c) Determine the expressions for S and w em by identification between the results of 2a and 2b. Conclude. 3. (a) It is worth recalling here that the real part of a product of two complex numbers is different from the product of the real parts of two complex numbers. Show how, in a similar way, the complex amplitude of the product of two complexes is different from the complex amplitude of these two complexes. What is under consideration when dealing with the vectorial products in place of the simple products? K K (b) If E 0 and B0 are the complex amplitudes of a complex field, then the complex electric field and the complex magnetic field:

222


notation G G G G G E 0 (r) exp(iZt) = E0 exp(iZt) and B B0 exp(iZt) , are such that: G G G G E Re(E) and B Re(B). G G G G 1 From the calculation for S [Re(E) u Re(B)] , show that the average value of S µ0 is of the form: G · §G G 1 ¨ E0 u B 0 ¸ ¢S² , Re ¸¸ 2 ¨¨ µ0 © ¹ G G where B 0 is the complex conjugated with B0 . G (c) Calculate ¢S² for a monochromatic plane progressive wave propagating in the G G direction of the unit vector u z in a medium where the wave vector k is real. G E

Answers 1. l = (v 't)'t = 1 = v (a) By definition, j represents the flux of electric charges, also termed the charge current density. This flux therefore represents the amount of electrical 6=1 charges that traverse a unit surface per unit time. Quantitatively, we can say that this quantity of charges is localized within a cylinder (or parallelepiped) with a cross-sectional unit area ¦ = 1 and length (l) given by l = (v 't) 't = 1 = v , where v is the velocity of the charges. If U = n q , is the volume charge density where n is the charge density, i.e., the number of charges per unit volume, then the quantity of charge found in the volume (V) given by V = ( ¦ ) ¦ =1 (v 't) 't = 1 = v is such that j = n q ( ¦ )¦ =1 (v 't) 't = 1 = U v . G G G The vector j is defined by j Uv . For its part, the equation for the conservation of charge is written G wU divj 0. (1) wt G (b) Similarly, if S represents the energy current density vector, S represents the quantity of energy that traverses a unit surface in a unit time. S therefore can be seen as the electrical power through a unit surface. If w T represents the total energy, the equation for the conservation of energy can be written by analogy to the equation for the conservation of charge as:


G ww T divS wt

0.

223

(2)

2. (a) Therefore, w T = w c + w em , and

dw T

dw c

dt

dt

dw em dt

. (3)

G G In addition, the theorem for the kinetic energy allows the equation dw c = F. v dt so G that by taking into account the form of F , we have: G G G G GG G G dw c G G = F. v = U[ E + v u B] v = U E v = j . E . (4) dt dw T G G dw em j. E By moving this expression into Eq. (3), we find . By dt dt introducing this equation into Eq. (2) for the conservation of energy, we arrive at G G G dw em divS j. E 0 . (5) dt (b) We have: G G JJG G wB B rot E = (equation multiplied by ) (M-F) µ wt

(M-A)

JJG rot

G B µ0

G G w (HE) j wt

G (equation multiplied by [ -E ]).

Following multiplication of the two members of each equation as indicated, we find by addition that: G G G G G B JJG G G JJG B B wB G w (HE) G G j.E . rotE - E rot =-E wt µ0 µ0 µ0 wt G G G JJG G G JJG G Using the relation div a u b =b rot a - a rot b , the above equation becomes G §G B · w § E² B² · G G H div ¨ E u =¸ ¨ ¸ j.E which can also be written as ¨ wt ¨© 2 2µ0 ¸¹ µ0 ¸¹ © G §G B · GG w § E² B² · (6) div ¨ E u ¸¸ +j.E + ¨¨ H ¸ 0. ¨ wt © 2 2µ0 ¸¹ µ0 ¹ ©

G (c) By identification with Eq. (5), it can be determined that S the Poynting vector, and that

G G B Eu which is µ0

224

w em


H

E² 2

B² 2µ0

is the density of electromagnetic energy in the form:

w e w m , with w e

H

E²

and w m

B²

. 2 2µ0 These two expressions are for the electrical and magnetic energy densities under a stationary regime and they prove that they are still valid for variable regimes. w em

3. (a) The use of complex numbers needs to take into account (with z n = a n + i b n ): x if the real part of the sum of two complex numbers is actually equal to the sum of the real parts of the two complex numbers, as in R(z1 + z 2 ) = R(z1 ) + R(z 2 ) = a1 +a 2 ; and x or if the real part of the two complex numbers is not equal to the product of the two real parts of the two complex numbers, as in R(z1z 2 ) = R([a1 + i b1 ] [a 2 + i b 2 ]) = R (a1b1 - a 2 b 2 +i[a1b2 + a 2 b1 ]) = a1b1 - a 2 b 2 z a1b1 = R(z1 )R(z 2 ). Similarly, if the product (P) of the complexes is P = A.B , then its complex amplitude is such that (with A = A 0 eiZt and B = B0 eiZt ): P = A 0 eiZt B0 eiZt = A 0 .B0 ei2Zt = P 0 eiZt , from which P0 = A 0 .B0 eiwt z A 0 .B0 . The complex amplitude of the product of two complexes appears different from the product of the complex amplitude of two complexes. Similarly, the vectorial product, which brings in the components of the products, differs from the simple products, for example: JG G G G G G G 1 R(E u B) . R(E u B) z R(E) u R(B) , so that here R(S) z µ0 G G G G (b) However, if E E0 exp(iZt) and B = B0 exp(iȦt) , we have: G · §G G 1 ¨ E0 u B 0 ¸ ¢S² . Re ¸¸ 2 ¨¨ µ0 © ¹ G In effect, S G With E

G G G G · 1 G G 1 § 1 E u B= R(E u B) ¸ . ¨¨ ¬ª R(E) u R(B) ¼º z ¸ µ0 µ0 © µ0 ¹

JG E 0 exp(iZt)= A + iB , we have:


225

G G* G G* G E 0 eiZt E eiZt EE , from which R(E)= A = 2 2 G G G G G 1 S [E 0 exp(iZt) E 0 exp(iZt)] u [B0 exp(iZt) B 0 exp(iZt)] 4µ0 G G G G G G G 1 G [E 0 u B0 ] + [E 0 u B0 ]+[E 0 u B0 exp(2iZt)]+[E 0 u B 0 exp(2iZt)] . 4µ0

^

`

^

`

The first two terms are conjugated complexes and their sum is equal to twice their real parts [as z1 + z1* = (a1 + i b1 ) + (a1 - i b1 ) = 2 a1 ) ]. The third and fourth terms each vary as a function of 2Z so that they have average values over a half period, and therefore also a full period, equal to zero. Therefore: G G G 1 S R(E 0 u B0 ) . 2µ0

(c) If the plane wave progresses along Oz, then the electrical field can be supposed, for example, to be moving with respect to Ox. Therefore, GG GG G notation G G G E0 E 0 (r) E m eik.r E m eik.r u x . Similarly, for a magnetic field moving through Oy, we have GG GG G notation G G G B0 B0 (r) Bm eik.r Bm e ik.r u y . Therefore: G G G 1 S R(E 0 u B0 ) 2µ0 Given that E m G S

1

1

1 G G G R(E m .Bm* u x u u y ) = R(E m .Bm* u z ) . 2µ0 2µ0

vM Bm in a material (or E m

n G 2G R(E m .E m* u z ) = Em uz 2µ0 vI 2µ0 c

c Bm if in a vacuum), we have: H0 2

2G nc E m u z .

If the MPPEM wave is polarized rectilinearly, so that E m = E 0 is real, then G H0 G S nc E 02 u z . 2

Chapter 8

Waves in Plasmas and Dielectric, Metallic, and Magnetic Materials

8.1. Interactions between Electromagnetic Wave and Materials 8.1.1. Parameters under consideration So that geometrical considerations do not become a problem, this chapter will look at the interactions of electromagnetic (EM) waves with materials that have infinite dimensions. Materials with limited forms will be looked at, most notably, in Chapters 11 and 12. The propagation of EM waves can be studied by considering: x a description of the EM wave in the material with Maxwell's equations; and x a representation of the material as a collection of electronic and ionic charges that interact with the electromagnetic field through the Lorentzian force, which can be written as: G G dv G G m qE qv u B . dt In terms of moduli, the ratio of magnetic and electronic contributions are given by Fm qvB vB . Fe qE E For a plane EM wave, the E = vM B and the ratio

Fm

v

Fe

vM

. As v 0) , from which Z0 - Z- = H > 0 and Fe '(Z- ) > 0 ; and x negative on the right hand side of Z0 so that Z+ = Z0 +H , from which Z0 -Z+ =-H < 0 , and Fe '(Z+ ) 0 . 8.2.3.2. Expression for Fe’’ Similarly to that in the previous section, from Eq. (7), Fe’’ can be found in: * 2 Z Z * p 0 2 . (13) = Fe ''(Z|Z0 ) | Z2p 2 2 2 2 2 Z 4Z0 Z0 Z Z0 * ² 0 2 §*· Z0 Z ¨ ¸ ©2¹ For its part, Fe’’ is at a maximum when the dominator is at a minimum, that is to say when Z - Z0 = 0 , so that Z = Z0 . Thus: Fe ''(Z0 ) = Fe ''max =

Z2p 1 Z0 *

.

(14)

By substituting this expression in Eq. (13):

Fe ''(Z|Z0 ) | Fe ''(Z0 )

§*· ¨ ¸ ©2¹

Z

0

Z

2

2

§*· ¨ ¸ ©2¹

2

1

=Fe ''(Z0 ) 1

4 *2

Z0 Z

2

.

(13’)

236 Basic electromagnetism and materials Fe’’(Z

| Z0)

is a function paired with (Z0 -Z) , and therefore is symmetrical with

respect to Z = Z0 . This type of curve is termed a Lorentzian curve, and as Z is separated from Z0, (Z0 -Z)2 increases and Fe’’(Z) o 0. 8.2.3.3. Evolution of Fe’(Z) and Fe’’(Z) when Z | Z0 The functions Fe’(Z) and Fe’’(Z) when Z | Z0 evolve as shown in Figures 8.1 and 8.2. The halfpeak width (hpw) of the curve representing Fe’’(Z) gives the pulsations Z1 and Z2 such that Fe ''(Z1 ) = Fe ''(Z2 ) = Fe ''(Z0 )/2 . From Eq. (13’), pulsations Zi (where i = 1 or 2) should accord to: 1

Fe ''(Zi ) = Fe ''(Z0 ) 1

4

Z0 Zi

1

2

2

Fe ''(Z0 ) .

*2 From which the following equation can de determined: * *² (Z0 -Zi )2 , so that Zi = Z0 ± . 2 4 These equations finally give: * Z1 = Z0 2 * Z2 = Z0 + 2 The hpw thus is equal to 'Z = Z2 - Z1 = * = 1/W (15). In addition, it is possible to see that the angular frequencies Zi (Z1 and Z2) give rise to the extreme limiting values of Fe’(Z). To verify this, it suffices that ª dFe ' º « » = 0 , a relatively simple calculation. ¬ dZ ¼ Zi According to Eq. (12), the corresponding values for Fe’(Zi) are: * r Z0 Fe '(0) Z F '(0) Z W 2 Fe '(Zi ) = =r 0 e = ± 0 Fe '(0) . 2 2 2 2* 2 §*· §*· ¨ ¸ ¨ ¸ ©2¹ ©2¹ Therefore: Z Z W Fe '(Z1 ) = 0 Fe '(0)= 0 Fe '(0) 2* 2 (16) Z0 Z W Fe '(Z2 ) = Fe '(0)= - 0 Fe '(0). 2* 2

Chapter 8. Waves in plasmas and dielectric, metallic, and magnetic materials

Fe’= Hr’-1

Z0W 2

F e, (0)

'Z =

1

W

Z0 Z1

0

-

Z0W 2

F e, (0)

Z2

Z

Z0

Figure 8.1. Plot of Fe’= f(Z) when Z | Z0 .

Fe’’

Fe’’(Z0) =

Z 2p Z0 *

Z0WF e, (0)

F e'' (Z0 ) 2

Z1

Z0

Z2

Z 'Z =1/W

Figure 8.2. Plot of Fe’’= f(Z) when Z | Z0 .

237


8.2.4. Study of the curves of Fe’(Z) and Fe’’(Z) when Z z Z0 (well outside of an absorption zone) Far from an absorption zone, the oscillation can be assumed to undergo only a very weak dragging force. This small force is such that it can be assumed that * ²Z² 0 (see Figure 8.4, noting that when Z < Z0 , H'r > 1 and when Z > ZA , 0 < H'r < 1 ). Thus, the dispersion equation Z2

Hr =

Z2

H'r > 0 , from which k = k' = k . c2 c2 With H’r(Z) given when Z z Z0 by Eq. (17’), then

takes on the form k² =

k² =

Z2p º Z2 ª «1 ». c2 «¬ Z02 Z2 »¼

(20)

242 Basic electromagnetism and materials By developing this further, it is possible to obtain: Z4 - Z²(k²c² + Z0 ² + Zp ²) + k²c²Z02

0 .

8.2.7.2. The relation Z = f(k) The preceding equation is a “bi-squared” equation of the type x 4 + b x² + c = 0 , which has solutions in the form x² = -

Zr2

k²c² Z02 Z2p 2

b 2

r

b² 4c 2

ª§ 2 2 « k²c² Z0 Zp r «¨ ¨ 2 «¬©

, so that it can be stated that: 1/ 2

2 º · 2 ¸ k²c²Z0 » » ¸ »¼ ¹

.

(21)

8.2.7.3. Shape of the dispersion curve When Z ZA , then k² = where H'r (Z) = 1 +

Z2 c2

H'r ,

Z2p

Z02 Z2 It is worth considering the two limiting values of k: x k | 0 according to the equation for dispersion, there are two possible solutions. These are either: o Z | 0 , which corresponds to the solution Z = Z- , [which also can be directly checked by introducing k = 0 into Eq. (21), giving a solution for Z- as Z- = 0 ]; or

o H'r (Z) | 0 , in which case when Z | ZA , such that ZA2

Z2p Z02 , the

solution for Z+ is Z+ = ZA , [which again can be found by direct substitution of k = 0 into Eq. (21)]. x

k o f where once again two solutions are possible: Z Z o f and k | o for a solution to Z+. The introduction into Eq. (21) of c c k o f so that k²c² >> Z0 ² or k²c² >> Zp ² , gives Z+ = ck ; or

o H'r (Z) o f , from which (1 +

Z2p Z02 Z2

that Z o Z0 , hence the solution to Z-

) o f , and in other words means


243

A representation of Z = f(k) therefore shows two branched curves associated with the transversal EM waves (T branches) corresponding to the two solutions Z+ and Z- The curves are separated by a zone, or band, into which propagation of the EM waves cannot occur. The value at which Z = ZA shown in Figure 8.5 corresponds for its part to the longitudinal electric wave (L).

Z T

Z = ck

Z+ L

ZA

zone unavailable to propagation: reflection zone

Z0 T

ZO

k Figure 8.5. Dispersion curves indicated by T.

The scheme presented below gives a résumé of the different behaviors of a dielectric undergoing electric polarizations at different frequencies (and pulsations). Absorption

Hr’ > 1

0 < Hr’ < 1

Hr’ < 0 Z0

propagation with dispersion

ZA

no propagation, evanescent wave

Z

propagation, and when Z o f, Hr’ o 1, Behavior analogous to that in a vacuum, the e- cannot follow these high frequencies

8.2.8. Equation for dispersion n = f(O0) when Z 1 , and the relation for dispersion k² = for a real Hr, k = k =

Z

Z² c²

Hr gives,

Hr . This region is where Z < Z0 is in the visible range and c therefore is nonabsorbing and transparent. More precisely, the conditions on Z can

244 Basic electromagnetism and materials be written Zi 1 (region Z < Z0e ), n = the result is that, with k 0 =

Z c

2Sc

Z

c

Hr

1 , and vM =

2S

2S

O0

k

k0n

n

< O0 ,

and is the wavelength associated with Z in a vacuum.

8.2.8.2. Dispersion equation for Z < Z0e and Cauchy's formula In the region Z < Z0e , H r (Z) = Hr '(Z) is given by Eq. (17’), so that with Z0 = Z0e the resulting equation is: H r (Z) = 1 +

Z2p 2 Z0e

2

Z

=1+

Z2p 2 Z0e

1

1 = 1 +Fe (0) Z² 2 Z0e

1

1 Z² 2 Z0e

ª º Z² = 1 +Fe (0) «1 ...» 2 «¬ Z0e »¼ With n² = Hr (Z) and Z

2Sc

O0

n² | 1 +Fe (0)+

it is possible to say that:

Fe (0) 2 Z0e

Z2 = 1 +Fe (0)+

Fe (0) 2 Z0e

By making A = 1 +Fe (0) = H r (0) = n 02 , and B = 2Sc

2

2Sc 2

Fe (0) 2 Z0e

1

O 02

.

, then n² = A +

B

O 02

.

Chapter 8. Waves in plasmas and dielectric, metallic, and magnetic materials 1/ 2

From this can be determined that n so that by making C =

B 2A1/ 2

12

A

§ B · ¨1 ¸ ¨ AO 02 ¸¹ ©

245

· § B | A1 2 ¨1 ... ¸ , 2 ¸ ¨ 2AO 0 ¹ ©

, it can be stated that

n | n0 +

C

O 02

.

(22)

Equation (22), called Cauchy's equation, shows that the index varies with the wavelength.

8.2.8.3. The Rayleigh relation and group and phase velocities dZ dQ dQ = - O² The group velocity is given by vg = . In addition, dk d(1/ O ) dO O0 =

2Sc

c

Z

Q

and O =

O0

c

n

nQ

, so therefore Q =

c nO

. The result is:

ª § 1 ·º d¨ ¸» « d(1/ nO ) 1 § 1 · 1 © n ¹» c § 1 dn · vg = - cO ² = - cO ² « ¨ ¸ = cO ¨ ¸ . « » dO n © O ² ¹ O dO n © n ² dO ¹ « » «¬ »¼ c With vM = , it is possible to obtain: n O dn · § (23) vg = vM ¨1 ¸ . n dO ¹ ©

8.2.8.4. Comment: normal and abnormal dispersions In Sections 7.2.1.3 and 7.2.1.4 normal and abnormal dispersions were defined. In dielectrics, as shown in Figure 8.4 and as a general rule, Hr’(Z) increases with Z. However, as indicated in Figure 8.1, when Z | Z0 , the function Hr’(Z) decreases as Z increases, describing a behavior in the abnormal dispersion zone. 8.3. Propagation of a MPPEM Wave in a Plasma (or the Dielectric Response of an Electronic Gas) 8.3.1. Plasma oscillations and pulsations 8.3.1.1. Definition of a plasma Overall a plasma is neutral and is made up of ions, assumed to be in fixed positions (due to their high mass which accords them considerable inertia), and by electrons,

246 Basic electromagnetism and materials assumed to be highly mobile. All of this is in a vacuum. At equilibrium (rest state), the ion volume densities (n0) and the electron volume densities (ne) are identical. However, following a Coulombic interaction, if the perturbation obliges an electron to move from its equilibrium position, then it will have a tendency to return to original position due to a returning electrical force. Below, electron and ion charges are denoted by –e and +e, respectively.

8.3.1.2. Study of the displacement of electrons in a plasma subject to a mechanical perturbation

S

u(x,t)

x

u(x+dx,t)

x

x + dx

Figure 8.6. Moving a “slice” of plasma.

Supposing that under the influence of a mechanical perturbation, for example, an acoustic effect, the electrons shown in Figure 8.6 are moved together along the Ox axis by a small distance u(x,t). The thermal agitation and weights involved are negligible, much as the frictional or mechanical recall forces, as the electrons are not tied to any specific atom as is the case for dielectrics. To start there is a “slice” of electron “fluid” which has a cross-sectional area S and is placed between x and x + dx at its rest state. The initial corresponding volume of the slice is V0 = S dx . Under the effect of a perturbation, the deformed fluid moves in such a way that its limits move to x + u(x,t) and x + dx + u(x + dx, t) . Therefore its volume also changes to : Vloss = S [dx + u(x +dx, t) - u(x, t)] . With

u(x +dx, t) = u(x) + dx

wu

, then

wx variation in volume can be considered as :

Vloss = S [dx + dx

GV = Vpert - V0 = S dx

wu

wu wx

] , and the final

. wx By denoting the concentration of the charge following perturbation by ne + Gne, the conservation of charge in the occupied volume before and after the perturbation can be described by: n e V0 = (n e + Gn e ) Vpert , or in other words, n e S dx = (n e +Gn e ) S [dx + dx

wu wx

] , from which comes


247

wu · wu § + Gn e ¨ 1 Zp , then 1 > , and H r (Z) > 0 and k² > 0 . k is therefore real, so Z2 G G G k = k' , and the wave has the form E=E m ei(Ȧt-kz) E m ei(Ȧt-k'z) . The wave thus is

(b)

When Z < Zp , then in turn

progressive and does not exhibit a term for absorption ( k'' = 0 ). The dispersion equation thus can be written, with the more simple k' = k , as Z =

Z2p c²k² .

The curve due to this equation is shown in Figure 8.7. The slope of the dispersion dZ curve, vg = , gives vg which is the group velocity. Note that the slope is less dk than the slope of the plot Z = ck and therefore less than the speed of light.


251

Z Z=

Z=ck

Z2p c²k²

Hr > 0 propagation

Z > Zp

Zp

Hr < 0 : evanescent wave

Z < Zp

ck

Figure 8.7. Dispersion curve for a plasma.

(d) To conclude, the plasma (or electron gas) acts much as a high band filter toward incident waves. As indicated in Figure 8.8, the gas is only transparent when H r (Z) > 0 ; that is to say when Z > Zp . However, when Z < Zp , a component due to attenuation by e-k’’z without a term for propagation, appears.

5. Hr(Z)

1

0 O

0.5

'drag' region Z < Zp vacuum

plasma Z>Z O < Op

1

1.5

region for propagation Z > Zp vacuum

Op Z

Z/Zp

plasma

O(µm)

Zp Z > Zp

Figure 8.8. High band filtering characteristics of a plasma.

252 Basic electromagnetism and materials Given that Z ! Zp corresponds to when O < O p , waves with lengths less than Op can propagate with k'' = 0 , that is without attenuation or in other terms absorption, through the medium under consideration. Thus the alkali metals should be transparent to radiation with wavelengths below 300 nm, and this includes UV radiation.

6. The condition H(Z = ZL ) = 0 makes it possible to determine ZL, which in turn should be such that H(ZL ) = 1 -

Z2p

= 0 and that ZL = Zp . The (zeros) poles of the Z2L dielectric function therefore are equal to the plasma frequency, that is to say the frequency at which the electron gas undergoes a longitudinal oscillation. The cutoff frequency (Z = Zp) of the transversal EM waves corresponds to the longitudinal oscillation mode of the electron gas. The electrons are subject to displacement pulsations, in a direction along that of the associated field ( E = E 0 eiZt ), that is to say in the longitudinal direction of the field and along the line of electron displacement.

8.4. Propagation of an EM Wave in a Metallic Material (Frictional Forces) A study of the complex conductivity and the dispersion of waves within a metal can be carried out by resolving the following question, which considers the velocity (v) of electrons subject to a Coulombic force generated by an applied alternating field G G given by E E 0 exp(iZt) . The question given in Sections 8.2 and 8.3 is revived here except of course the electrons are now considered to be in a metallic environment so that the volume density for electron charge is given by UA = - n e e and the electrons are subject to a m

v , where Ȟ and W are the conduction W velocity and relaxation time, respectively. It is assumed that these conduction electrons are not subject to returning forces and that their movement can be studied in one dimension.

frictional force which is in the form f t = -

1. Give the equation for the movement of each electron and from this determine that the velocity when v = v0 eiZt .

2. The material is thought of as a vacuum in which the characteristic conduction electrons are spread. In the following order, calculate:


253

(a) the conduction current density jA = UA v = VA E and express VA as a function of Zp ² =

n e e²

, and also as a function of the conductivity (V0) for a stationary H0 m regime, i.e., when Z = 0 ; (b) the current density associated with movement through vacuum (jD0); and (c) from (b) derive the total current density (jT) and the conductivity (V), which appears in its complex form.

3.

This

question

concerns

ZW 1 , give the expression for Hr comparable to that obtained in question 3 above. From the result, draw a conclusion about the nature of the MPPEM wave. (c) Here, ZW 1 . Calculate Hr and indicate the form of the associated wave.

Answers 1. The fundamental dynamic equation along the direction of the velocity is: dv m F=m ¦ f eE v , from which the complex terms derived are: dt W dv m m v eE dt W

254 Basic electromagnetism and materials With E in the form E = E 0 eiZt , a solution for v in the form v = v0 eiZt is required. dv By substitution into the differential equation, and with iZv , it is possible to dt m v eE , so that following division by eiZt , the equation obtain: imZv + W m v0 eE 0 . becomes imZv0 + W From this can be derived: eE 0 W v0 = . m(1 iZW)

2. (a) Ohm's law gives the conduction current associated with the volume density of electrons (ne), as in: n e e² W n e e² W jA = UA v = - n e e v = E 0 eiZt E= VA E , so that m(1 iZW) m(1 iZW) VA = By introducing Zp ² =

n e e² H0 m

n e e² W m(1 iZW)

into VA , then VA =

. H0 Z2p W 1 iZW

.

When Z = 0 (i.e., under a stationary regime), VA (Z = 0) = V0 = H0 Z2p W , so that it is possible to also state that: 1 V A =V 0 . 1 iZW

(b) The displacement current associated with a vacuum, given that in a vacuum D = H0 E 0 eiZt , is jD0 =

wD wt

= i ZH0 E .

(c) The total current density is therefore: jT = jA + jD0 = (

H0 Z2p W

§ Z2 W · p + i ZH0 )E = V E , from which V = H0 ¨ iZ ¸ . ¨ 1 iZW ¸ 1 iZW © ¹


255

3. (a) With VA = V0

1 1 iZW

j

(b) We have

D0

j

A

|

and with ZW Zp they are progressive and when Z Zp they are evanescent.

256 Basic electromagnetism and materials When Z W 0

–e


and also that tanT =

u

v

261

= tan(-M) = - tan M , so that:

M = Arc tan

u v

= Arc tan

b a

= Arc tan

Z*

Z02

Z²

.

To conclude, the movement of an oscillator is cosinusoidal and undergoes the same pulsation (Z) as the excitation force, given by FC = - e E cos Zt , the same amplitude (x0) and the same dephasing (M) with respect to E. The resonance pulsation is such that x0 is at a maximum. If * = 0 , then the oscillator is unrestricted and x0 is at a maximum when Z = Z0 .

(d) Moving a charge (q) by a distance (x) is the same as applying to the system a dipole moment given by µ = qx (as detailed in Section 2.2.2.1.). Similarly, moving an electronic charge (-e) by x is the same as applying a dipole moment µ = - e x, so that: e² µ = - e x 0 cos (Zt - M) = E 0 cos (Zt - M) . 2 m[ (Z0 Z2 )² Z²* ² ] The polarization P =

¦ n i qi Gi can be written as P = N µ = - e N x , and hence i

P=

Ne²

1

m [ (Z2 Z2 )² Z²* ² ] 0

E 0 cos (Zt - M) .

(e) With the applied field being given by cos Zt, there is no proportionality between the instantaneous polarization, given by P(t) = P0 cos (Zt-M) and the instantaneous field, given by E(t) = E 0 cos Zt . The polarization is established with a phase delay (M) with respect to the applied electric field. 2. (a) The complex polarization is: P = - N e x = - N e x 0 exp(-jM) exp(jZt) = - N e x 0 exp(jZt) = P 0 exp(jZt) , which P0 = - N e x 0 exp(-jM)= - N e x 0 P0 =

Ne²

Ne² m[(Z02

1

m [ (Z2 Z2 )² Z²* ² ] 0

Z2 ) jZ*]

E 0 = P0 e-jM , with

E 0 and M= Arc tan

Z*

Z02

Z²

.

from


(b) The complex susceptibility is defined by P = H0 F E , which means that P0 exp(-jZt) = H0 F E 0 exp(jZt) = H0 F E0 exp(jZt) . From this can be immediately derived that: P0 = H0 F E 0 Ne² = E0 2 m[(Z0 Z2 ) jZ*]

(c) Here D = H0 E + P =HE

H = H0 +

So H r = 1 + F = 1 + that: H r = 1 +

P E

2

Z ) jZ*]

[(Z02 Z ) jZ*]

Ne² mH0 [(Z02

Z2 ) jZ*]

= H0 + H0 F = H0 (1 + F) =H 0 H r .

Ne² mH0 [(Z02 Z2p 2

F=

Ne²

, and by making Zp ² =

mH0

, we find

.

3. (a) With H = H' - j H'' , D = H E = (H' - jH'') E and E = E 0 exp(jZt) = E 0 cosZt + j E 0 sinZt D = H'(E 0 cosZt + j E 0 sinZt) - j H'' E 0 cosZt + H'' E 0 sinZt , and D = R(D) = (H'cos Zt + H''sin Zt) E 0 , or: D = Acos Zt + Bsin Zt , with A = İ'E0 and B = İ"E0 .

(b) D therefore is given by D = A cos Zt + B sin Zt , which can be changed to D = C cos (Zt - M) . In effect, if we make: A = C cos M B = C sin M

C = A² B² and tg M =

B A

, then

D = A cos Zt + B sin Zt = C cos M cos Zt + C sin M sin Zt = C cos (Zt - M) . Consequently, the real induction is D = C cos (Zt - M) , with C = A² B² = E 0 H ' ² H '' ² , tg M =

B

H ''

A

H'

Finally, this can be written as: D = C cos (Zt - M) = E 0 H ' ² H '' ² cos (Zt - M) , with M = Arc tan

H '' H'

.

.


263

8.6.2. A study of the electrical properties of a metal using the displacement law of electrons by x and the form of the induced electronic polarization (based on Section 8.4) This exercise concerns the polarization, conductivity, and optical properties of a metal assumed to consist of a collection of N fixed ions (per unit volume) with N (N = ne) free electrons, there being one free electron for each atom, of mass and charge denoted, respectively, by m and –e. All are assumed to be in a vacuum with a permittivity denoted by H0. The electronic gas is subjected to the following forces: x a Coulombic force induced by an alternating field (applied along Ox) that is given G G in its complex form as E E 0 exp(iZt) ; and G G m dx x a frictional force given by Ft . W dt The metal of choice for this problem is copper, which has a relaxation time (W) of 10-14 sec. Its conductivity is denoted by V(0) and is approximately equal to 6 x 107 ȍ-1 m-1. The plasma angular frequency (Zp) is defined by the relationship Nq² Zp 2 and typically is of the order of 1016 rad sec-1. mH0 1. Under a forcing regime, we are looking for the expression for displacement in the G G G form x x 0 exp(iZt) (complex expression). Determine x 0 . G 1 2. Give the expression for the complex polarization ( P ) as a function of . 1 iZW G G G wD 3. The complex conductivity (ı) is defined in the general equation: j VE . wt G (a) By giving D as a function notably of polarization, give the equation for V that will then be used to give the static conductivity [V(0)] which describes the conductivity when the frequency is zero and can be given as a function of N, q, W, and m. (b) Give the physical significance of the two terms that appear in the equation for ı. (c) Express V(0) and then İ r as a function of Zp².

Answers G 1. The dynamic fundamental solution given for ¦ Fapplied gives

m dx

W dt

eE

m

d²x dt²

, so that

G mJ moving along Ox

264 Basic electromagnetism and materials d²x

1 dx

e

E. dt² W dt m The solution for this differential equation, under a forcing regime where E = E 0 exp(iZt) , is required in the form x = x 0 exp(iZt) . With x iZx and

x

Z² x substituted into the differential equation, it is possible to obtain Z e - Z²x 0 + i x 0 = E 0 , following division of the two terms by exp(iZt), and m W hence : e eW x0 = E0 = E0 . Zº ª imZ[1 iZW] m « Z² i » W¼ ¬

2. The electronic polarization is such that for each individual electron (of charge –e) is displaced by x by the electric field and generates a dipole moment given by µ = - ex. Denoting the electron density by ne, the corresponding polarization is given n e² W 1 E 0 e jZW . It is interesting to note that by by P = - n e e x = e imZ [1 iZW] n e e² , it is possible to write that: introducing Zp ² = mH0 P=

Z2p H0 W iZ[1 iZW]

E 0 e jZW .

3. (a) With D = H0 E + P , it is possible to state that n e² W 1 ª º = iZ «H0 E e E » . It is possible to imZ 1 iZW ¼ wt wt wt ¬ immediately derive the relationship n e² W 1 V = iZH0 + e = V(Z) . m [1 iZW] n e² W , and finally, From this, when Z = 0 , V = V(0) = e m V(0) V = iZH0 + . [1 iZW] wE . This term can be considered (b) The first term given as iZH0 corresponds to H0 wt as being tied to the displacement current through a vacuum, as in V vide = iZH0 . The j = VE =

wD

= H0

wE

+

wP


V(0)

wP

, meaning the current due to the volume [1 iZW] wt polarization of the material (that is to say inside the material). This conductivity is an internal conductivity V(0) Vinternal = . [1 iZW]

second term,

, corresponds to

265

(c) With D = H0 Hr E , we have j = VE =

wD wt

= H0 H r

wE wt

= iZH0 H r E , from which

V = iZH0 H r . By equalizing the preceding expression for conductivity, V = iZH0 + possible to obtain H r = 1 +

(d) With V(0) = Zp ² =

V(0)

1

iZH0 [1 iZW]

=1-

i V(0) ZH0 [1 iZW]

.

n e e² W

n e e²

m

mH0

V(0) = Zp ² H0 W , and H r = 1 - i

Z2p W Z[1 iZW]

(= 1 -

Z2p W[ZW i] Z[1 Z² W²]

).

V(0) [1 iZW]

, it is

Chapter 9

Electromagnetic Field Sources, Dipolar Radiation, and Antennae

9.1. Introduction Until now the properties of electromagnetic (EM) waves have been covered without considering the mechanisms for their production or destruction. In fact, the laws of electromagnetism and classic mechanics applied to quasipoint charges such as electrons gives rise to a theory for wave emission, the principal result of which is that when a particle undergoes an acceleration, an EM wave may be emitted. In this chapter, it will be shown how the sinusoidal oscillation of an electric dipole can yield an EM wave. Electric dipolar radiation can be obtained by either sinusoidally oscillating the distance between dipolar charges or the actual dipolar charges themselves, as long as the dipolar moment is in the form p = p0 e jZt . This configuration is used in antennae, the principal types of which are detailed below with particular attention being paid to the half-wave antenna. G A preliminary determination of V and A potentials, associated with the dipoles, is required to calculate the EM radiation field. In addition, this calculation requires the solutions to Poisson's equations, which guide the propagation of potentials, generally called “retarded potentials”. In this chapter we will limit ourselves to the study of radiation in a vacuum, while in next chapters (Chapter 10 among others) we will look at interactions between EM waves and materials. Complex notation: It is worth noting that if a signal is in the form g = G m cos(Zt - M) , the complex notation is g = R(G m e jZt e-jM ) , and that by

convention we can write more simply g = G m e jZt e-jM . In this case, the complex amplitude is G = G m e- jM , which makes it possible to state that g = G e jZt . Strictly speaking, and as mentioned in Chapter 6, it is best to write g with an underline so that the equation becomes g = G e jZt ; however, the simplified if criticized notation G G also is used. If g and Gm are the vectors g and G m , G becomes a complex vector

268


G denoted by G , which generally has the components Gx, Gy, and Gz. We thus have for the differentiation or integration operations of g = G ejZt with respect to time, the following: dg = jȦ G e jȦt = jȦg , dt dg dG jȦt dG R = e , so by identification = jȦG (= jȦ G m e-jM ); and dt dt dt

³ gdt =

1 jȦ

1

G e jȦt

jȦ

g,

R ³ gdt=( ³ Gdt)e jȦt , and ³ Gdt =

1 jȦ

G.

9.2. The Lorentz Gauge and Retarded Potentials 9.2.1. Lorentz's gauge 9.2.1.1. Poisson's equations for potentials within an approximation of quasistationary states In a homogeneous dielectric media, and under a regime of an approximation of quasistationary states (AQSS), we have: G JJJJG G JJG G G wA (1) B rot A (2) E grad V wt G The V and A potentials are expressed as a function of the deliberately applied and acting charge ( UA ) and current ( jA ) densities, as in: G G P jA dW 1 UA dW (3) (4) V= A ³³³ ³³³ 4SH r 4S r Under an AQSS regime,

wUA wt

0 , and the equation for the conservation of charge

G G gives div jA = 0 , from which can be deduced that div A = 0 (see Section 1.4.5). In addition, the potentials follow the Poisson equations, so that: JJJGG G U (5) 'A µjA 0 (6) 'V + A 0 H

269

Chapter 9. Sources, dipolar radiation, and antennae

9.2.1.2. Under rapidly changing regimes For systems subject to rapid variations, the displacement current must be considered G G G G wD by adding in jD to the conduction current jA VE , so that he current density wt which now intervenes is: G G G j jA jD . G The V and A potentials are instantaneous potentials calculated in Eqs. (3) and (4) for a time t at a certain point (r) from charges and currents at the same t. Propagation does not intervene in these equations for the potentials. The G instantaneous potentials V and A thus appear as intermediates in the calculation of G G E and B , having brought in a term for the displacement currents rather than the G G G G G propagation. In effect, in Eq. (4), where j jA jD now takes the place of jA , A G G G wD is bound to the derivative of D (in terms of jD ) and is not an intermediate in wt G G G the simple calculation of E in Eq. (1) and D , because knowledge of A supposes G G that the magnitude of D , or rather the derivative of D , is known. In order to resolve the problem more simply, the displacement currents are ignored and other potentials are used, such as those defined and denoted below G as Vr and A r , to represent the propagation. 9.2.1.3. Lorentz's gauge

G In Maxwell's equations, div B G ( A r ), such that:

JJGG The equation rotE

G wB

wt

G 0 so that B is still derived from a potential vector G B

JJG G rotA r .

(7)

G JJG § G wA r becomes rot ¨ E ¨ wt ©

· ¸¸ ¹

0 , which indicates that

G G wA r E is derived from a scalar potential (Vr) such that: wt G JJJJG G wA r E gradVr . (8) wt G In a homogeneous media, possibly charged, the Poisson equation, div E

be written as:

UA H

, can

270


- 'Vr -

w

G (divA r )

UA

. (9) H G JJG G §G wE · In addition, Ampere's theorem given by rotB µ ¨ jA H ¸ , brings us to: ¨ wt ¹¸ © G JJG G JJGJJG G § wE · (10) rotB rotrotA r µ ¨ jA H ¸. ¨ wt ¹¸ © wt

JJG JJG With rot rot

JJJJG G grad div ' , and by bringing Eq. (8) into Eq. (10), we have: G JJJJG JJJJG wVr G GG §G w ²A r · (11) grad divA r 'A r = µ ¨ jA H grad H ¸. ¨ wt wt² ¸¹ ©

Equations (9) and (10) can be rewritten in the following form to give Poisson's equations for a rapidly varying regime: G w ²Vr UA w wV (9’) 'Vr - HP (divA r Hµ r ) wt² H wt wt G G JJJJG G GG w ²A r wV 'A r - Hµ + µ jA =grad(divA r Hµ r ) wt² wt

(11’)

G G JJG G In addition, Eq. (7), B rotA r , can define only A r to the closest gradient, as JJG G JJG G JJJJG G G rotA r rotA 'r when A 'r A r gradf . JJJJG G G wf For its part, Vr can be determined only to within , as A r = A 'r gradf wr substituted into Eq. (8) gives: G JJJJG G wA 'r JJJJG wf E grad Vr grad wt wt G JJJJG § wf · wA 'r grad ¨ Vr ¸ wt ¹ wt © G' JJJJG wA r wf = gradVr' , with Vr' Vr . wt wt G G The change of ( Vr , A r ) to ( Vr' , A 'r ) is called a gauge transformation, as in: Vr o Vr'

Vr

wf wt


271

G G G JJJJG A r o A 'r = A r +gradf .

The determination of the potentials is carried out by imposing a measuring G condition so as to find Vr and A r ; f is an arbitrary function called a gauge G G function or simply a gauge. The invariance of E and B is termed the EM field gauge invariance. The most convenient method to determine the potentials therefore is to G impose on A r and Vr a condition that simplifies Eqs. (9) and (11) by removing in both cases the terms in parentheses. This condition, G wV divA r Hµ r 0 , (12) wt wV 0 it takes the place of the is called Lorentz's gauge, noting when in addition wt G condition div A 0 for stationary regimes. Equations (9’) and (10’) therefore yield: U A 0 (13) wt² H G JJJGG G w ²A r 'A r - H µ + µ jA = 0 (14) wt² 'Vr - HP

w ²Vr

G In these equations, Vr and A r are decoupled, in contrast to Eqs. (9’) and (11’), and follow Eqs. (13) and (14) if f satisfies the wave equations discussed in Comment 3 below. Equations (13) and (14) generalize Eqs. (5) and (6) to rapidly varying regimes.

Comment

1:

We

have

H µ = H 0 µ0 H r µ r =

1 c²

H r µr ,

where

H r µr = n²

(with µr = 1 , we again find Maxwell's equation, H r = n² ), so that H µ = Equations (13) and (14) can also be written as: 1 w ²Vr UA 'Vr H v² wt² G JJJGG 1 w ²A G r 'A r + µ jA = 0 v² wt²

0

n²

1

c²

v²

.

272


Comment 2: In a vacuum without charge or current, Eqs. (13) and (14) reduce to: G JJJGG 1 w ²A 1 w ²Vr r = 0. 'Vr 0 and 'A r c² wt² c² wt² By convention, the following operators are noted thus: 1 w² '{ , called the d'Alembertian, Vr = 0 c² wt² G 1 w² '{ c² wt²

, called the d'Alembertian vector

G Ar = 0 .

Comment 3: A Lorentz condition is imposed on the gauge function, in which case, G wV we still find that divA r Hµ r = 0 . wt G ' G JJJJG wf With A r = A r +gradf and Vr' Vr we also have: wt JJJJG G G div A 'r = div A r + divgradf (a); and Hµ

wVr' wt

Hµ

wVr wt

Hµ

w ²f wt²

(b).

By adding each successive member in (a) and (b), we obtain JJJJG G G wV ' wV w ²f 0 div A 'r +Hµ r = div A r +Hµ r + div gradf Hµ wt wt wt² = 0 as the Lorentzian gauge JJJJG w ²f =0 divgradf Hµ wt² 1 w ²f R 'f =0 v² wt²

Finally, Lorentz's gauge condition requires that f equation.

satisfies a wave

Comment 4: This additional remark concerns Coulomb's gauge. If there are no charges or currents, then Eq. (9) becomes: G w 'Vr + (divA r ) 0 wt


If Vr is chosen so that 'Vr G div A r = 0 as in the stationary regime.

0

273

(Coulomb's gauge), we than have

9.2.2. Equation for the propagation of potentials and retarded potentials

r M x dW q(t)

Figure 9.1. Potential generated by a charge [q(t)] throughout dW.

Equations (13) and (14) are the equations for propagation and they can be integrated by hypothesizing that at a given instant (t), a charge [q(t)] is in an elementary volume (dW), which is assumed to be spherical given the symmetry caused by the isotopic propagation and is situated at a point (M) as described in Figure 9.1. The action of the charge depends only on the distance (r) from which its effect is studied. Within the spherical symmetry of the problem, the Laplacian of Vr is in the form: 1 w ² rVr 'Vr = , r wr² And Eq. (13) gives: 1 w ² rVr 1 w ² Vr UA 0 (13’) r wr² v² wt² H Outside of the element dW, the singularity of which can be considered to be a quasipoint, we have U = 0, and therefore can write for the outside that: w ² rVr 1 w ² rVr 1 w ² rVr 1 w ² Vr 0 =0. (15) r wr² v² wt² wr² v² wt² Equation (15) is for propagation, and the function rVr at t and r has a solution in the form: r r rVr (r, t) G(t - ) F (t ) . v v As the source is at M and it is only waves emitted toward r > 0 that have physical r solutions which are in the form G( t - ) , we therefore find: v r G(t ) v . (16) Vr (r, t) r

274


In order to determine G, a limiting scenario might be used where r o 0 and in 1 1 w ² rVr , which is such that: Eq. (13) we find that becomes very large, just as r r wr² 1 w ² rVr 1 w ² Vr >> . r wr² v² wt² 1 w ² rVr UA + 0 , so that : Equation (13') thus tends toward r wr² H U 'Vr + A = 0 . (13’’) H Equation (13') therefore tends toward the more usual Poisson equation, which has a solution that is for r o0 (point charge at M): q(t) q(t) Vr (r,t) = , so that also, [r Vr (r,t)]r o 0 = . 4SHr 4SH From this can be determined that ª § r ·º q(t) ½ [r Vr (r, t)]r o 0 = q(t) ° «G ¨ t ¸ » . v ¹¼ r o0 4SH ¾ G(t) = ¬ © 4SH ° = G(t) ¿ And therefore, from Eq. (16), we find: r q(t ) v . Vr (r,t) = 4SHr

(17)

If in a volume (V) there is a distribution of charges q(t), then there is an accumulation of potentials from the elementary charges such that q(t) = U(t) dW and the general expression for the potential at t is: r· § U¨ t ¸ 1 v © ¹ dW . (18) Vr (r,t) = ³³³ 4SH r In addition, Eq. (14) yields potential vector components similar to those of Eq. (13), with the condition that 1/H is replaced by µ. The components of the potential vector thus have solutions of the type given by Eq. (18) in which 1/H must be replaced by µ. These solutions can be brought together in a single equation: G § r· j ¨t ¸ G µ v¹ © (19) A r (r,t) = dW . ³³³ 4S r


275

For a distribution of rectilinear currents, we have: r· § I¨ t ¸ G µ v © ¹ dlG . (19’) A r (r,t) = ³ 4S r To conclude, we thus obtain as a function of the active charges and currents at an G r r instant (t - ) the potentials Vr and A r for t and r with a delay of , which takes v v G into account the duration of the propagation. The potentials Vr and A r are delayed G G potentials and substituted into Eqs. (7) and (8) yield E and B . 9.3. Dipole Field at a Great Distance G 9.3.1. Expression for the potential vector A generated by a domain D 9.3.1.1. General formula dWi

ri

Mi

qi O

P r

G u

domain D JJJG

Figure 9.2. Distribution of charges (qi) placed around O by OM i

G Gi .

Given an element, with a volume (dWi) and an associated charge (qi) such that qi = U dWi , which is within a domain (D) that has a distribution of qi charges placed JJJJG around O by OMi , we can associate with D a nonzero dipole moment given by JJJJG G G G p ¦ qi Gi z 0 where Gi OMi . i

G Of interest is the form of the potential vector A at a point (P) located with respect to the origin (O) at the heart of D and the distribution of charges, by JJJG G G OP r ru . Given Eq. (19) and that the system is in a vacuum, with ri = Mi P , its possible to write that: G § r · ji ¨ t i ¸ G G µ0 c¹ © A r (r,t) =A(P,t) = dWi . ³³³ 4S ri

G It is possible to state that if vi (t) is the velocity of qi associated with dWi in D, G G G G G G with ji Uvi , we have ji dWi Uvi dWi vi UdWi vi qi , and can then write:

276


MP G qi vi (t i ) G µ0 c , A(P,t) = ¦ 4S i Mi P

(20)

where the summation is overall the charges within D. In order to calculate the electromagnetic field that comes from this potential vector, the space can be divided into three: x a zone defined by r > O and where the radiation is the concern of this study. G 9.3.1.2. The form of A in the radiation zone In the outer zone, it can be assumed that ri = MiP | r, as all the points in D are, in practical terms, a distance r from P. From this, Eq. (20) can be rewritten so that G µ r G A(P,t) = 0 ¦ qi vi (t ) . (21) 4Sr i c JJJJJG dOMi G G G G With vi , we have ¦ qi vi (t) p(t) , where p(t) is the derivative with dt i respect to time of the resultant dipole moment due to all charges in D. This finally gives: r G ) p(t G µ0 c A(P,t) = . (21’) 4S r G It is interesting to note that the form of A therefore corresponds to a spherical wave, which at a great distance from D would be considered to be a source generating a plane wave (see Section 6.2.2.3). Indeed, at a great distance from the source, the surface of such a sphere can be associated with a tangential plane wave. With r dW W = t - , and as 1 , it can also be noted that: c dt G dp(W) d G dW d G = p(W) p(W) . dt dW dt dW

The derivative therefore can be taken indifferently in respect of t or W.


277

9.3.2. Expression for the electromagnetic field in the radiation zone (r >>O) G JJG G r G G G ) = p( W) = p W (which in effect With B rotA , and placing by notation, p(t c means placing into lower magnitudes with respect to the index W): r º ª G p(t ) » JJG G JJG pG W G µ0 JJG « c » µ0 rot B(P, t) rot A(P, t)= rot « . 4S r r « » 4S «¬ »¼ JJG G JJG G JJJJG G By using rot(aA) arotA grada u A , it is possible to write: G µ0 JJGG µ JJJJG 1 G B rotp W + 0 grad u p W , the second term of which brings in 4Sr 4S r G G JJJJG 1 1 r u grad , a term that varies with respect to and at a great distance 3 r r² r² r 1 becomes negligible with respect to the first term which only varies as . Therefore: r G µ0 JJGG B| rotp W . (22) 4Sr G JJGG u G w² K G p W , where A quite involved calculation then gives rotp W =- u p p(W) . In c wW² order to get this result, one can verify that: x on the one hand, we have JJG d wp (W) d ª wp y (W) º ª rotpG W º = ª« z º» ¬ ¼ x dy ¬ wW ¼ dz «« wW »» ¬ ¼ G JJJJG ²p w w ²p z wW w ²p(W) º ª y wW «gradW u » wW² wy wW² wz ¬ wW² ¼ x so that by reproducing the calculation in three dimensions, we find: G JJGG JJJJG w ²p( W) rotp W gradW u ; and wW² G G JJJJG r u x on the other hand, gradW . This result can be obtained by calculating, cr c JJJJG wW 1 w x for example, that [gradW]x = (x² y² z²)1/ 2 , from which we wx c wx cr have: G JJJJG G G 1 G r gradW [xi yj zk] . cr cr

278


JJGG On taking the result rotp W

G B

u c

G p W into Eq. (22), we finally reach: u

G G µ0 p(W) u u 4Sc

r

.

(23)

G In the plane wave approximation, for an electric field, E G E

µ0 1 G G G ª p(W) u u º u u . ¬ ¼ 4S r

G G cB u u , so that :

(24)

G To conclude, if p(W)= 0 , the Eqs. (23) and (24) show that the radiated magnetic field is zero. By consequence, only accelerated charges radiate, as G dv G G p ¦ qi i and the wave corresponding to p z 0 is termed the “acceleration dt i wave”.

9.3.3. Power radiated by a dipole G In vacuum, the Poynting vector is defined as : S

G G EuB µ0

(Chapter 7, Section 7.3.). G G G cB u u , S can be written

G Staying with the planar wave approximation, so that E as : G cB² G S u. µ0

By taking Eq. (23) and substituting it into this equation, for an angle (T) between G G p(W) and u as shown in Figure 9.3(a), we find: G S

µ0

G p 2 (W)

16S²c

r²

sin ²T

G u.

(25)

If T = 0 , then in other words if the radiation intensity is observed in the direction G of p it would be found to be zero. However, the radiation maximum is found when T =

S

2

.


G p

T=0

P r

T

G u

T=-

(a)

279

G S

S

T=+

2

(b)

S 2

Figure 9.3. Definition of T (a) and the variation of S with T (b).

The variation in S as a function of T is represented in Figure 9.3.b, and it G S can be seen that |S| is at a maximum when T = ± , and S = 0 when T = 0 . 2 G S G d6 n O

G

Figure 9.4. Calculation of the flux ( S ) through a sphere (6) with center O.

G The power radiated through a surface can be calculated from the flux of S through the total surface (see Section 7.3.1.2). For the spherical surface under G consideration, the total radiation is given by a calculation of the flux S across the sphere (6) around a center O of radius r, as detailed in Figure 9.4. G G As S // n.d²6 ), we have P = w ³³ S.d²6 . With d²6 = r²sinT dT dM , so that also 6

d6 = 2S r² sinT dT , we find that T S

P = ³ S 2Sr² sin T dT= T 0

T S

With: ³ sin 3 T dT T 0

cos T 1

³

sin 2 T d(cos T)

cos T 1

P=

µ0 G 2 p (W) 8Sc

µ0 G 2 p (W) 6Sc

4 3

T S T 0

, we obtain the Larmor equation:

2 G 2 p (W) . 4SH0 3c3 1

3 ³ sin T dT .

(26)

280


9.4. Antennas 9.4.1. Principle: a short antenna where A ez u er @ u er

§ 0 · §1 · ¨ ¸ ¨ ¸ ¨ 0 ¸ u ¨ 0¸ ¨ sin T ¸ ¨ 0 ¸ © ¹ © ¹

G r

r

§0 · ¨ ¸ ¨ sin T ¸ ¨0 ¸ © ¹

(from Figure

G sin T eT

From this can be deduced that: G E

µ0 1 G G G ª p(W) u u º u u ¬ ¼ 4S r

so that in addition, with Z² G E

k² 4SH0 r

k²c²

k²

H0µ0

G p(W) sin T eT

µ0 Z² 4Sr

G p(W)sin T eT ,

, and k

S H0 O ²r

2S O

(31)

, we find:

G p(W) sin T eT .

(32)

It is possible to see that a field radiated by a Hertzian dipole has only one G component with respect to eș , is inversely proportional to r, but also depends on r and t through the term p(W). 9.4.1.4. Radiation power of a Hertzian dipole when A = sm O. Larmor's formula given in Eq. (26), obtained by removing p from the integral, therefore can be applied. Thus we find that with Q = Qm cosZt , and that I =

dQ dt

= - Z Qm sinZt = Im sinZt, then

p = Qm A cosȦt , and p= - Q m AȦ2 cos Zt , from which p²= Q 2m A 2 Ȧ4 cos 2 Zt and we now have: P =

µ0 6ʌc

p 2 (Ĳ)

µ0 Q2m A 2 Ȧ4

cos 2 Zt .

6ʌc

By taking I²m = Z²Q²m and given that = ½, we finally have: P =

µ0 A 2 Ȧ 2 12Sc

1

I 2m

2

R ray I2m .

(33)

This relationship gives a definition of Rray, the so-called radiation resistance, which has resistance as a dimension and is directly obtained from: µ Ȧ2 R ray = 0 A 2 6Sc

With µ0 c = µ0

1 İ 0µ0

=

R ray =

µ0 İ0

2

2ʌµ0 c § A · ¨ ¸ . 3 ©O¹

= Z0 = 378 : , we can also write that:

2

2

§A· §A· Z0 ¨ ¸ | 780 ¨ ¸ . 3 ©O¹ ©O¹

2ʌ

(34)

(35)

9.4.2. General remarks on various antennae: half-wave and “whip” antennae Antennae can be seen everywhere in modern day life. They can be found in portable telephones, televisions, cars, and automatic doors, just to give a few examples. According to Eq. (35), short antennae radiate poorly, and it is for this reason that antennae with sizes of the order of the wavelength are preferred. The most basic versions are still made up of a simple dipole, as shown in Figure 9.8a. When the antenna is emitting, it is powered by an oscillator and parallel wire cables as detailed above (Section 9.4.1.2). If the antenna has a length O/2 and establishes a stationary current so that there are nodes at the two ends and an antiO node at the origin in the middle ( z r ), then the current has the form: 4 I(t) = Im cos kz sin Zt.

284


Each part along z with a length dz thus radiates as if it were a small dipole. This concept is used in a problem at the end of the chapter. When the antenna is in the receptive mode, the electric field around it induces a current (see more precisely Section 12.3.3.3), which propagates through the cable up to the receiver.

(a)

O O

(b)

4

2

Figure 9.8. (a) Half-wave antenna; and (b) quarter wave or “whip” antenna.

One of the more common types of antennae are the “whip” antennae often seen on cars. They are electrical monopoles, as shown in Figure 7.8a, and form an electrical image in the plane of the conductor so that a positive charge that moves along the antenna toward the top produces a “reflected” image negative charge that moves symmetrically toward the bottom. Typically, the length of an antenna is O/4, but its actual length needs to be determined quite precisely before determining its exact radiation diagram. When its length is equal to O/4, the quarter-wave length antenna acts in effect as a half-wave antenna as it can generate its symmetrically opposite image in the plane of the conductor. There are many other types of antennae. To cite just a few: x “disk” antennae formed from microstrips that are applied for example onto airplane wings; x frame antennae, also called a magnetic dipole antenna, based on a ferrite iron core, which concentrates the magnetic field; x horn antennae (used with wave guides); and x parabolic antennae that concentrate waves at a focal point where a horn antenna is placed.


285

9.5. Problem Radiation from a half-wave antenna For a half-wave antenna, that is with a length ( A ) such that A = O/2, and ignoring any energy losses such as the radiation, it can be assumed that it carries a current (I) which can be written as: 2S I Im cos kz exp jZt Im cos z exp jZt . O Assuming that the radiation is isotropic, it is independent of the angular coordinate (M). This problem concerns the radiation zone where the radius or distance from the antenna (r) is such that r >> O. 1. Determine the expression for the electric field at a point (P), which is such that P { P(r,ș) , i.e., has spherical coordinates (r,T) and M = 0 in the plane of the paper. K K 2. Assuming that when r >> O, the vectors E and H are orthogonal as in a plane wave and are related in a vacuum by

E

c . Given that

B

G calculate H . 3. Calculate the average Poynting vector.

E

µ0

H

H0

Z0 | 378 : ,

Answers 1. z A

P(r,T) r’

O/4 z

M

T’ dz T

I(z)

O

Im cos

I A

O/4

r 2S

O

z

O 2

B It can be assumed that T’ | T at P in the radiation zone where r >> O | A . The actual dipole can be considered to be a resultant of many small dipoles of a length dz centered on a point M such that MP = r’ = r – z cos T, as schematized in the diagram above.


286

An elementary Hertzian dipole with a moment denoted by dp radiates, as G detailed in Section 9.4.1.3, an elementary electric field dE that follows the form: G µ Ȧ² G dE = - 0 dp(W) sinș eș , where r' = MP, [see Eq. (31)]. 4ʌ r' From Eq. (29), we can also write that: G µ0 jȦ I(Ĳ) G dE = dz sinș eș , where sm = dz, so that in this problem, 4ʌ r' § ª r 'º· § r' · I(W) = I ¨ t- ¸ Im cos kz exp ¨ jZ « t » ¸ . c ¼¹ © c¹ © ¬ As r >> O, and r >> A , the denominator r' can be replaced by r. This cannot be said to be true for the numerator as the exponential phase varies rapidly with r'. So, integrating over the length of the antenna and using r’ = r – z cos T to clean up, we end up with: O / 4 G § ª µ0 r º· jZz cos T G E jZIm exp ¨ jZ « t » ¸ sin T ³ cos kz exp dz eT . 4Sr c c ¼¹ © ¬ O / 4 By using cos kz

G E

µ 0 jZI m 8Sr

eikz eikz 2

and

Z

2S

c

O

k , we obtain:

§ ª r º· exp ¨ jZ « t » ¸ sin T c¼¹ © ¬ O / 4

u ³

O / 4

êxp jkz >cos T 1@ exp jkz >cos T 1@ ` dz eGT

The integral gives: O / 4

³

O / 4

êxp jkz >cos T 1@ exp jkz >cos T 1@ ` dz

2S O 2S O 2S O j 2 S O > cos T1@ j j j >cos T1@ >cos T1@ >cos T1@ ½° 1 °° e O 4 e O 4 e O 4 e O 4 ° = ® ¾ jk ° cos T 1 cos T 1 cos T 1 cos T 1 ° °¯ °¿ S ½ S ½½ ° 2 jsin ® > cos T 1@¾ 2 jsin ® > cos T 1@¾ ° 1 ° ¯2 ¿ ¯2 ¿° ® ¾ jk ° cos T 1 cos T 1 ° °¯ °¿


With

µ0 jZIm 2 8ʌr

k

=

µ0 jZI m c Ȧ

4ʌr

=

jIm 4ʌİ 0 cr

287

, we obtain:

S ½ S ½½ ° sin ® > cos T 1@¾ sin ® > cos T 1@¾ ° ° ¯2 2 ¿ ¯ ¿ ° eG sin T exp jZW ® ¾ T 4ʌİ 0 cr cos T 1 cos T 1 ° ° °¯ ¿°

G E

jI m

As: S ½ sin ® > cos T 1@¾ ¯2 ¿ S ½ sin ® > cos T 1@¾ 2 ¯ ¿ G we have E

§S · cos ¨ cos T ¸ ©2 ¹ S § · cos ¨ cos T ¸ 2 © ¹

° 2 §S · ½° G sin T exp jZW ® cos ¨ cos T ¸ ¾ eT so that finally: 4ʌİ 0 cr °¯ sin ²T ©2 ¹ ¿° jI m

§S · cos ¨ cos T ¸ j ©2 ¹ I exp jZW eG . T m 2ʌİ 0 cr sin T

G E

It is worth noting that this expression is indeterminate when sin T = 0, i.e., when T = 0 or T = S. 2. Assuming that at great distance the wave still assumes a plane wave structure, the G G G G G G vector H will be orthogonal to E . So with the trihedral E, H, er where er indicates G G the direction and sense of propagation, H is collinear with eM . Therefore with H

E H0 / µ0 we obtain

G H

§S · cos ¨ cos T ¸ ©2 ¹ I exp jZW eG . M m 2ʌr sin T j

3. By applying Eq. (33) of Section 7.3.3, we can write that

288


G S

1 2

G G Re E u H* where B = µ0 H,

from which can be obtained

G S

§S · cos 2 ¨ cos T ¸ 2 2 © ¹ Im eG u eG T M 2 2 2 2Sr H0 c sin T §S · cos 2 ¨ cos T ¸ 2 1 1 ©2 ¹ Ieff eG = r 2 4SH0 Sc sin T r2 1

§S · cos 2 ¨ cos T ¸ 2 © ¹ 9.5 sin 2 T

2 Ieff G e (S.I.), that is in W/m². 2 r r

Integration over the sphere of radius r gives the total radiated power, much as the same method used to obtain the Larmor relation given in Section 9.3.3. The 2 result that can be obtained from the ensuing numerical calculation is P | 73 Ieff and gives a final radiation resistance for the half-wave antenna of the order of 73 :.

Chapter 10

Interactions between Materials and Electromagnetic Waves, and Diffusion and Absorption Processes

10.1. Introduction Having looked at the way in which electromagnetic (EM) waves are emitted by a particle accelerating in a vacuum in Chapter 9, this chapter is concerned with how EM waves interact with materials, and in particular: x the form of wave emitted by a material subject to an incident EM wave (Rayleigh diffusion); x how EM waves are formed by charged particles interacting with a material as the so-called Rutherford or Bremsstrahlung (German for “braking radiation”) radiation; and x the absorption and emission phenomena of EM waves interacting with excited (more exactly “perturbed” as detailed below by the intervention of Hamiltonian perturbation) materials. The description used will be semiclassical, or possibly semiquantic, depending on whether the glass is half full or half empty with regards to a knowledge of quantum mechanics! That is to say that EM radiation can be written in a classic form while the material can be understood in quantic terms generally known by readers. The quantic terms used however, generally are for students following their first university degree course, and although Section 10.4 details in the simplest terms this semiquantic theory, it can be skipped over during an initial read. The essential results have been set out in Section 10.5 and these can be compared in a second read with the results given in chapter 3 of the volume, “Applied Electromagnetism and Materials” (notably the last Figure of that chapter). 10 .2. Diffusion Mechanisms 10.2.1. Rayleigh diffusion: radiation diffused by charged particles An electron with a charge denoted by q is subject to a monochromatic planar polarized electromagnetic (MPPEM) wave which is polarized along Ox (complex

290


JJJG G amplitude of the incident wave given by Einc //Ox ) and propagates along Oz, as G G shown in Figure 10.1. The electric wave can be described by E Einc exp(jZt) .

x G E inc

P

T

z

q

G Binc

M y

Figure 10.1. MPPEM wave incident on an electron with charge denoted by q.

If the electron is within a dense material, following its displacement along Ox by the electric field of the incident wave, it can be assumed that it will be subject to a returning force in the direction of its equilibrium position and in the form f r = - k x while simultaneously being subject to frictional forces of the form f t = - f

dx

. dt Rigorously speaking, the interactive force between the EM wave and the velocity G G G G G ( v ) of the electron is given by Fem q(E v u B) E

E

< E assuming that v c , so that the c c electromagnetic force can be reduced simply to Coulomb's force, as in :

With B

, we have vB

v

G G Fem | FC G The fundamental dynamic equation, ¦ F qEinc e jZt - k x - f

G qE . G mJ , given in terms of Ox thus gives: dx dt

=m

d²x dt²

The solution for a steady state can be looked for in the form x = x 0 e jZt , and the placing of this into the preceding fundamental equation gives: qEinc x e jZt x 0 e jZt . k mZ² jZf

The result is that q displaced by x generates a dipole moment:

Chapter 10. Diffusion and absorption processes

291

p(t) = q x(t) = q x 0 e jZt and radiates according to the result of Section 9.3.2 concerning an EM field such G G G G G p(W)= q x(W)= q J , so that that E and B are expressed as a function of

p(W)= -Z²q x = - Z²q x 0 e jZW . The power radiated and given by Eq. (26) of Section 1.3.3 is proportional to G2 p (W) , and thus p(W)= - Z²q x 0 e jZW . Two simple examples can now be considered.

10.2.1.1.

Diffusion by bound electrons (valence electrons of the atmospheric molecules O2 and N2) The electrons are well bound to the molecules, and at the level of the forces involved, the constants introduced are such that k >> mZ² and k >> Zf . The result is that the dynamic fundamental equation, as in qEinc x= e jZt , k mZ² jZf qEinc jZt qE e , so that x 0 = inc , where x 0 thus appears is reduced to x | k k p(W)= - Z²q x 0 e jZW , we have: independent of Z . With G p 2 (W) v Z4 . G p 2 (W) and is in the form P v Z4 , that is The radiated power is proportional to the power radiates to the fourth power of the angular frequency (Z).

sun at zenith average yellow radiation

(a) maximum diffusion is blue sky (b) radiation with reduced blue appears more orange Earth

Figure 10.2. Observations made when the sun is at its zenith.

292


Given that it is radiation with a high angular frequency that dominates: x When we look at the sky, but not at the sun, the sky appears blue as shown in Figure 10.2 a. The power radiated, or rather diffused, by electrons in the molecules that make up the atmosphere is at a maximum for the highest angular frequencies, i.e., those that correspond to the color blue in the incident optical radiation from the sun. x When we look at the sun when at its zenith (an observation worth avoiding due to possible eye damage!) as shown in Figure 10.2 b, the sun appears on earth more orange than the yellow observed by a satellite outside of earth's atmosphere. This is because the transmitted light is equal to the incident light minus light lost to diffusion during its passage through the atmosphere. In effect, the light we see has an “impoverished” blue region, especially if the sky is cloudy, resulting in an apparent color shift toward red. x When we see the setting sun, as shown in Figure 10.3, the observed waves have had to travel through a long distance in the atmosphere, much longer than that when the sun is at its zenith, with the result that there is an greater impoverishment of blue light diffused all along the light's pathway. This makes the light look red as all other light has been diffused.

setting sun long pathway resulting in considerable diffusion of blue light sky

radiation in red due to loss of blue earth

Figure 10.3. Observation of the setting sun.

10.2.1.2. Diffusion by free electrons This section looks at the study of free electrons using, for example, Laue type diffusions in crystalline solids. Free electrons, which are little or even not at all bound to atoms, are not subject to returning forces toward a given atomic nucleus. Thus, k = 0 and the frictional forces are also negligible, so that f = 0 . qEinc e jZt thus is The solution to the differential equation, x = k mZ² jZf


reduced to x =

qEinc mZ²

e jZt ; p(W)= - Z²q x 0 e jZW then gives q²

p(W)

m

Einc e jZW .

In addition, according to Eq. (24) of Section 9.3.2, we have:

G E

µ0 1 G G G ª p(W) u u º u u ¬ ¼ 4S r

G G and as in Figure 10.4, we have p // ex .

x

M G G G G G puu puu uu G eT T G p

G u

z y Figure 10.4. Orientation of the vectors used in the text.

G E

G For a point (M) located by the direction of a vector ( u ), it can be written that G EeT , so that in terms of moduli, E = E T .

S G G G G ª º ª º ¬ p(W) u u ¼ sin 2 ¬ p(W) u u ¼ µ µ0 q² we have E T = 0 p sin T Einc sin T e jZW . 4Sr 4Sr m r With W = t - , and F = m J = q Einc , so that J c As

G G G ª º ¬ p(W) u u ¼ u u

ET =

qJ

sin T

4SH0 c²

r

jZ

e

p sin T

qEinc m

, finally we have

r c e jZt

= E 0 T e j Zt .

293


294

The wave appears to be well proportional to the acceleration denoted by J ; hence the term “accelerating wave”.

10.2.2. Radiation due to Rutherford diffusion 10.2.2.1. A general description of Rutherford diffusion Here the incident particle is an electron denoted –e with a mass given by m that is a great distance from a proton that is assumed to be fixed and at a reference origin (O). The trajectory of the electron is considered rectilinear with a velocity vf . If there are no forces acting on the incident electron, then it will have a trajectory given by the straight line (D), as shown in Figure 10.5, which passes at a minimum distance (b) from the target (b is called the impact parameter).

D vf

b

R

T

U

O

Figure 10.5. Trajectory and the Rutherford diffusion.

In reality, when an electron nears a proton, the Coulombic interactive force is no G longer negligible. The electron is subject to a Coulombic force ( f ) that is directed G along the line between the proton and electron distance (R). If V denotes the G G electron's kinetic moment, the theory of kinetic moment is written (with f and R being collinear) as G dV G G Ruf 0. dt G G G The result is that V R u mv is a first integral of the movement where: G x the direction of V is fixed and the movement is thus plane; and G x the modulus (ı) of the vector V is a constant.


295

With the electron being located in the plane of its movement by the polar angle T, it is possible to state that: V

mRv

mR²

dT

, dt from which the surface law can be deduced, as in: dT dS V R² 2 C . dt dt µ By denoting the potential energy of an electron by Ep(R) and its total energy by E, the application of the law of energy conservation results in the equation: 2 V² ª E E p (R) º mR² ¬ ¼ dT dR V m m ²R² r r dR d T V 2 V² ª E E p (R) º mr² ¼ m²R² m¬ dR

changes sign and R abruptly changes from a decreasing to an dT increasing region, or the inverse, when the numerator of the last equation cancels out so that for a value U for R is such that: V² E p (U) E . 2mU² The derivative

In addition, by taking the origin of the potential energies at infinity, it also is possible to write for the two constants for the movement that the kinetic moment (ı) and the energy (E) that 1 2 V mbvf and E , mvf 2 from which can be deduced for E p (U) that 2 § mvf b² · ¨1 ¸¸ . ¨ 2 © U² ¹ Here the interaction potential [ E p (U) ] is negative and the equation shows that U < b.

E p (U)

It is equally feasible to introduce a parameter (G) that corresponds to a distance between the two particles at which the Coulombic energy of the electron, given by e² e² , is equal to its mass energy as in mc2, so that G . 4SH0 R 4SH0 mc

296


10.2.2.2. Rutherford diffusion and radiation During the deviation caused by electrostatic interactions, the electron – proton pair G G constitutes a dipole that has a dipole moment given by p e( R) , which varies G continuously, as R varies just as well in terms of direction as in terms of modulus. G G The sign changes [in (R) ] due to the fact that R is orientated with respect to the origin (the proton) toward the electron, while the dipole moment goes from the negative (the electron) to the positive (the proton) charge. The consequence of this is that the dipole radiates, which is assumed, to a first approximation, to not modify the movement. It thus is possible to calculate the electromagnetic field radiated at a point (P), which is in the radiation zone. Also, OP >> O , where the origin of the dipole also is at O (by supposing that R/2 Z0 Z@t

*

e

1

4

j> Z0 Z@t

1

304


a m (t)

2

2

: mn

¬ª1 cos(Z0 Z)t ¼º .

(22)

Z0 Z 2

2= ²

The term shown in Eq. (22) represents the probability of finding a particle in the state m at an instant t, and the probability of the transition Pn o m is defined by the 2

variation as a function of time of the probability a m (t) of finding the particle in the final state m, as in: sin Z0 Z t 1 d 2 2 Pn o m a n (t) m|:|n . dt Z0 Z 2=² If we suppose that the time t is large with respect to the period of the electromagnetic field, which is f the order of 10-15 sec in the optical domain, and if we make x = Z0 - Z, we obtain: Pn o m

1 2= ²

m|:|n

2

lim

sin xt

t of

x

.

Dirac's function is defined by Gx

1

lim

S t of

sin xt x

and the probability of the transition thus is written as Pn o m

S 2= ²

m|:|n

2

G Z0 Z .

(23)

This indicates that for the transition probability from one state n to another state m is zero, then G Z0 Z z 0 Z = Z0 . Therefore, the angular frequency (Z) of the incident electromagnetic wave must be such that:

=Z

=Z0

E 0n E 0m .

(24)

This expression confirms in some way the principle of the conservation of energy and can be considered as a “resonance condition”.


305

With Z Z0 , we have E 0n E 0m + =Z . There is an absorption of energy ( =Z ) by the atom. If we had taken as the more important term in Eq. (21) that which corresponded to Z Z0 , we would have obtained E 0n

E 0m - =Z . Therefore, there is an

emission of an energy =Z induced by the perturbation.

10.4.3.3. Determination of the transition rule, and dipole transition moment The probability of transition between the state n and the state m is given by Eq. (23), in which the resonance condition intervenes through the intermediary of Dirac's function ( G Z0 Z ). With the resonance condition being fulfilled ( Z Z0 ), the transition probability thus brings in the term P0

S 2= ²

m|:|n

2

, which

evaluated with the help of Eq. (17) is of the form:

P0

S 2= ²

m|:|n

2

2 JJJG G 2 S § =e · 2 ¨ ¸ A0 m | | n . 2= ² © m ¹

In order to evaluate the term

(25)

G m | | n the following mathematical term

(described in the problems at the end of this Chapter) can be used: G =² G \m | | \n \m | r | \n En Em . (26) m Under these condition, Eq. (25) which can be written as:

P0

G G º ª§ = ²e · S G 2 ª§ =²e · A m | | n » «¨ ¸ m||n 4 0 «¨ m ¸ 2= ¹ ¬© ¼ ¬© m ¹

G brings in the terms in the form (where µ G § = ²e · ¨ ¸ m||n © m ¹

G \ m | er | \ n

º » ¼

*

G er , the dipole moment)

En Em

G µmn E n E m .

To conclude, the probability of a transition is proportional to the transition dipole moment defined by: µmn

\m | µ | \n

* ³ \ m µ \ n dW

(27)

306


Once the resonance condition is established, it is the result of the calculation for this integral that finally indicates the permitted transitions between the various possible m and n atomic states.

10.5. Conclusion: Introduction to Atomic and Molecular Spectroscopy In this last part, which gives the most important results from Section 10.4 on the semiclassic calculations for dipolar approximations, we will underline the importance of dipolar radiation and apply it to the description of the possible different internal transitions (for absorption or emission) in atoms and molecules. 10.5.1. Result concerning the dipole approximation In very general terms, the dipolar approximation, which states that the electromagnetic field is practically constant at the level of an atom or a molecule, means assuming that there is only a weak coupling between an electromagnetic field and an atom or a molecule. In effect, it is inefficient because its length is way below that of the length of the electromagnetic waves. We thus can treat the effect of an electromagnetic field as that of a perturbation. By carrying out a development of the interaction energy of the electromagnetic field in terms of various contributions in an order of decreasing importance (analogous to that of a distribution of charges at multipoles, with successive terms due to total charge, dipole moment, quadripolar moment, etc.) we find that the term for the electric dipole interaction is greater than the term for the magnetic dipole interaction, which in turn is greater than the term for the quadripolar electric interaction and so on. In physical terms, the luminous perturbation, characterized by its electric field G GG ( E ) moves electrons through a coupling energy in the form W µ.E , where G G G wA E ZA 0 sin Zt in the Coulombic gauge, where the scalar potential wt associated with the luminous wave field is zero (we can see that this coupling energy intervenes in Eq. (25) taking Eq. (26) into account).

If the following electronic redistribution is symmetric, there is no overall variation in the electric dipole and the transition is forbidden. Inversely, if the redistribution is asymmetric, then the transition is possible even if the atom (or molecule) showed no initial permanent dipole moment.

10.5.2. Different transitions possible in an electromagnetic spectrum Various types of transitions (for emissions or absorptions) can be written, depending on the wavelength of the electromagnetic field.


307

10.5.2.1. Rotational transitions A molecule with a permanent magnetic dipole can appear as a carrier of a variable dipole moment when touring on its own axis (figure 10.6). Inversely, if the molecule is symmetric (without a permanent moment), there is no apparent variation in the dipole moment with rotation. Only molecules that possess a permanent electric dipole can emit or absorb radiation (in the far infrared or microwave region) by making a transition between rotational states.

+

+

Figure 10.6. Variation of the dipole moment of a rotating polar molecule.

Figure 10.7a shows a rotating dipolar molecule, such as HCl, being influenced by an alternating electric field (E) along Ox at an initial time t = 0. Figure 10.7b represents in the plane Oxy the rotation of the dipole at the various instant when t = 0, t1, t2, t3 etc., and Figure 10.7c shows the variation of a component of the dipole in the direction Ox with respect to time. As shown in Figure 10.7b, the dipole periodically changes position with the electric field due to the variation of the orientation of E and the variation in the GG coupling energy, which is of the form W µE and when t = 0, Wmin µ x E as E is directed along Ox at the initial time. As a consequence, the component of the dipole in a given direction (shown for Ox in Figure 10.7c) fluctuates regularly, and thus the component µx exhibits a fluctuation with a form resembling that of the electric field from the electromagnetic wave. (In emission, the radiation would be of the same frequency as long as the rotation is not slowed by the interaction of the dipole with neighboring molecules in the material.)


308

Ex

(a) 0 z

t4

t3

t

y

(b)

t2

t1

+

µx

x

+

+

+

+

+

+ y

x

µx

(c) 0

t1

t2

t3

t4

t

Figure 10.7. Variation of (a) a component (Ex) of a varying electric field; (b) the position of a permanent dipole moment (µ) in the plane Oxy; and (c) the value of µx with time.

10.2.2. Vibrational transitions R

(a)

(b)

+

R

+

Figure 10.8. A vibration results in a stretching of the bond, and for (a) the nonpolar molecule shows no variation in dipole moment, but (b) a polar molecule exhibits a change in dipole moment due to the oscillation.


309

These transitions appear when there is a vibration associated with, for example, a stretching or a torsion of the bonds between the atoms of a molecule, resulting in a variation in the dipole moment of the molecule. Only those vibrations that are associated with a modification of the dipole moment of a molecule are accompanied with an emission of absorption of electromagnetic radiation. Given the energies brought into play, the mechanism is apparent in the infrared.

10.5.2.3. Electronic transitions Electronic transitions are apparent if, following an electromagnetic excitation or emission, the resulting redistribution of electrons changes the dipole moment of the molecule. For this to happen, the redistribution must be asymmetric; if the redistribution is symmetric, then the corresponding transition is forbidden. The energies brought into play are in the optical region. 10.5.3. Conclusion This chapter has described different phenomena of absorption and emission of EM waves by materials. Absorption phenomena will be detailed further in Chapter 3 of the second volume called “Applied Electromagnetism and Materials” using a more phenomenological approach based simply on the interaction of electromagnetic waves with molecules. The deformation that the molecules undergo will be described as a function of the region in which the frequency of the EM wave falls and as part of a more classic mechanical treatment, as detailed in the last figure of Chapter 3 of Volume 2. It is worth adding that if the energies are of the order of radio frequencies, then it is the magnetic spin moment that interacts with the electromagnetic wave. The reversing magnetic spin dipole thus is at the origin of electronic paramagnetic resonance (EPR) that follows the reversing electron spin. If the effect involves the proton spin, then phenomena associated with nuclear magnetic resonance (NMR) are then observed. 10.5. Problems 10.5.1. Diffusion due to bound electrons This problem concerns a bound electron with a charge denoted by –q, which is initially situated at the origin of a trihedral defined by Oxyz of an atom or molecule subject to: x a monochromatic plane polarized (along Ox) electromagnetic wave incident along G G Oz and with an electric field in the form E E 0 exp(iZt) u x ; and G G x and a returning force toward its equilibrium position of the form f -mZ0 ² x u x . It is assumed that the angular frequency (Z) of the incident wave is well below that of Z0 . The atom or molecule thus is an oscillating dipole.

310


1. It is also assumed that the velocity acquired by the electron is negligible with respect to that of the speed of light. (a) Give the fundamental dynamic equation. (b) From this deduce the solution for the permanent steady state along the x abscissa of the electron. (c) What is the value of the dipole moment (p(t)) of the system? 2. The equation for an electromagnetic field radiated at a great distance by a dipole JJJJG G at a point (M) such that OM r is: G G G G µ0 1 G µ0 p(W) u r W) uG ) uG . B and E (p( r r 4S c r 4S r

G ur G uMG uT

x M

G p T M

m

G uM

z

G G G What does W correspond to? Write the equation for p(W) . Thus calculate B and E . G G 3. If E W and BW are the complex amplitudes of the electric field and the magnetic G G G G field, respectively, given by E E w exp(iZt) and B Bw exp(iZt) , in general G terms, the average value of the Poynting vector ( S ) is of the form G ¢S²

G G § E B w w Re ¨ 2 ¨© µ0

1

· G G ¸ , where B w is the conjugate complex of Bw . ¸ ¹

Determine the average Poynting vector for the problem.

4. Determine from this the total power radiated by the electron and from the result makes a conclusion.


311

Answers 1. (a) The displacement with respect to Ox can be found by considering that for this axis: G G G Fem Frappel mJ , so that G G G q E v u B Freturn

With B =

E c

, we have v B =

G mJ vE c

> Z , we find

, and finally

x

qE 0 iZt e mZ02

In terms of vectors, G x

G xu x

G x 0 eiZt u x

G where u x is the unit vector along Ox.

(c) Moving the charge –q is equivalent to applying a dipole moment of p = - q x, for q²E 0 iZt which p(t) qx e , so that in terms of vectors, mZ02

312


q²E 0 iZt G e ux . mZ02

G p(t)

G 2. We have B(M)

G G µ0 p(W) u r

4S c

r

DD G JJJJ with the propagation) . With p(W)

, where W

t

r

(change in variable associated

c

r

§ · q²E 0 iZ¨© t c ¸¹ G e u x , we find mZ02

G Z²p(W) and p(W)

G that for B(M) G B(M)

r

G G Similarly, with u x u u r results in: G E(M)

µ0 q² Z²E 0 4ScrmZ02

§ · G iZ¨© t c ¸¹ sin T u M e

G Bw eiZt . G G sin T u T , the equation for E

G G G sin T u M and sin T u M u u r

r

µ0 q²Z²E 0 4SrmZ02

§ · G iZ¨© t c ¸¹ sin T u T e

G G 3. The average Poynting vector thus is (where u T u u M G ¢S²

G G § E B w w Re ¨ 2 ¨© µ0

1

· ¸ ¸ ¹

G E w eiZt . G u r ) given by

µ0 q 4 Z2 E 02 32S²r²m²Z04 c

G sin 2 T u r .

4. The total power radiated is given by: P=

G JJJG

w ³³ S d²6 . 6

JJJG G With d²6 = r²sinT dT dM u r , we have:

P

µ0 q 4 Z2 E 02 32S²r²m²Z04 c

r² ³³ sin 3TdTdM T,M

µ0 q 4 Z2 E 02 32S²m²Z04 c

S

2S

3 ³ sin TdT .

T 0


With

S

4

T 0

3

3 ³ sin TdT

, we finally obtain:

µ0 q 4 E 02 § Z4 ¨ 12Sm²c ¨© Z04

P

2Sc

Comment: With Z

313

O

2Sc

and Z0

O0

, we also can write that:

µ0 q 4 E 02 § O 04 ¨ 12Sm²c ¨© O 4

P

· ¸ ¸ ¹

· ¸. ¸ ¹

As detailed in the course, the smaller Ȝ (0.4 µm in the visible spectrum, that is blue light), the greater the power radiated (hence the blue color diffused by electrons in molecules present in the sky).

10.5.2. Demonstration of the relationship between matrix elements [see Eq. (26) of Section 10.4.3.3 as in G =² G \m | | \n \m | r | \n En Em ] m For this, we will establish the following equation: d A

1

dt

i=

> A, H @

which is a relative equation for the physical magnitude A, which is not explicitly l ) is Hermitian (proper real values) time dependent, so that the associated operator ( A

l|\ . \|A

and is such that A

Answers d A

d\ l l | d\ . |A|\ \|A dt dt dt dt d\ d \ l \ j= , for which However, according to Schrödinger, H dt dt Calculate

d

l|\ \|A

1 l H\ . j=


314

We thus obtain:

d A

1 l l|\ \|A l| 1 H l\ H\ | A j= j=

dt =-

1 l l|\ 1 \|A l|H l\ . H\ | A j= j=

H and A are Hermitian (being equal to their associated terms and are such that, for example, \ | H | M H\ | M ), so we also find that: d A dt 1

=

1 ª l l | \ \ | AH ll | \ º \ | HA »¼ j= «¬

l H lº | \ \ | ª A, ¬ ¼ j=

1 j=

> A, H @

j =

> H, A @

.

l { mrG , in terms of operators we also can write that (see also, for With A example, E. Durand, Quantum Mechanics, Masson, p.69, 1970):

m As m

G dr dt

G dr dt

jm ª l G º H, r . ¼ = ¬

G corresponds to an operator of a quantity of movement, j= , we also

have: G j=

jm ª l G º =² G H, r , from which ¼ m = ¬

H lº . ª Gr, ¬ ¼

Multiplication of the left side of the last expression by \*m and the right side by \ n , and then carrying out with a scalar product gives: =² m

G \m | | \n

Gl l G | \ . \ m | r H | \ n \ m | Hr n

l is Hermitian (just as is Gr ), so that: H l G | \ \ m | Hr n

l | Gr\ \m | H n

G l|\ * r\ n | H m

G

l\ dr . ³ r\ n H m *


Similarly, we obtain: Gl \ m | r H | \n

G l \ m | r | H \n

l\ | Gr\ * H n m

G *

l\ dr . ³ r\ m H n

The result is that: =² m

l\* As H m

G \m | | \n

l\ E m \*m and H n =² m

G \m | | \n

G

\* H l l * ³ r[ m \ n \ n H\ m ]dr .

E n \ n , we have: G G E m ³ \*m r\ n dr E n ³ \*m r\ n dr ,

and hence, the looked-for equation: =² m

G \m | | \n

G \ m | r | \ n

En Em .

315

Chapter 11

Reflection and Refraction of Electromagnetic Waves in Absorbent Materials of Finite Dimensions

11.1. Introduction In practical terms, electromagnetic (EM) waves in materials of limited dimensions correspond to those in systems such as coaxial cables, optical fibers, and wave guides. It is by this process that signals, and therefore information, are transmitted with as low a degree of attenuation and parasitic phenomena as possible. This chapter will look at the influence of discontinuity, at the interface between two materials, on the propagation of EM waves. Typically, the two materials are: x linear, homogeneous, and isotropic (lhi) dielectrics and not magnetic so that their G G G G G magnetization intensity can be written as I = 0 , so that B=µ0 H+I=µH , µ = µ0 G with I = 0 ; G x uncharged, as in UA 0 , and not traversed by real currents, as in jA 0 ; and x described by their complex dielectric permittivity as H = H' - jH'' (electrokinetic notation), or as H = H' + jH'' (optical notation). In this chapter, and then in Chapter 12, which mostly looks at applications using the optical region, the notation used is that classically used elsewhere in optics as in sinusoidal planar progressive EM waves are written in the form (see also Section 6.4.2): GG G G E=E m exp j[kr-Ȧt] . (1)

318


G ki

medium (1)

11.2. Law of Reflection and Refraction at an Interface between Two Materials 11.2.1. Representation of the system

x

G G n12 = n z incident plane

z

medium (2)

y

O

Figure 11.1. Interface between media (1) and (2).

As a general system, the volume under consideration is made up of two media, denoted (1) and (2) which have permittivities H1 and H 2 , respectively, and are separated by an interface on the plane Oxy. The incident wave has an angular GG G G frequency denoted by Zi, is written in the form Ei =Eim exp j[k i r-Ȧi t] , and thus by G G notation for the incident wave E m = Eim . In addition, by choosing the origin of G G G phases on this wave such that Eim = Ei0 exp(jMi ) = Ei0 when Mi 0 , then it is possible to state: GG G G Ei =Ei0 exp j[k i r-Ȧi t] . (2) G The incident plan is defined by the plane that contains the incident wavevector ( k i ) G G as well as the normal to the interface ( n12 = n z ). In order to simplify the representations, the incident plane can be taken as Oxy in which consequently, is the G wavevector k i . Also, we will assume that the first material, medium (1), is G G nonabsorbent so that H r1 = n12 so that the modulus of k i is k i = k1 = k 0 n1 .

At the reflection on the plane of the interface Oxy, the incident wave will observe a modification in one part of its amplitude (which now will be denoted G E r0 ), while the other part will be subject to a dephasing by a quantity denoted as Mr. The reflected wave then can be written in a general form where Zr designates the angular frequency following reflection:

Chapter 11. Reflection and refraction in absorbent finite materials

G G G G E r =E 0r exp(jMr )exp j[k r r-Ȧr t]

G G G0 E r exp j[k r r-Ȧr t] .

319

(3)

Similarly, for the transmitted wave, self-evident notations are used: G G G G G G G0 E t =E 0t exp(jMt )exp j[k t r-Ȧ t t] E t exp j[k t r-Ȧt t] . (4) G G In these equations, k r and k t are the wavevector relating to the reflected and G transmitted wave, respectively. As detailed below, k t can be a complex magnitude G (thus denoted k t ).

11.2.2. Conservation of angular frequency on reflection or transmission in linear media Two different types of reasoning can be used in this case: the first is by a physical analysis of the problem, and the second via an equation using the conditions of continuity at an interface. 11.2.2.1. Physical reasoning As shown in Chapter 10 dipoles associated with a displacement of charges inside a G dielectric that are subject to an incident wave ( Ei ) with a given angular frequency ( Zi = Z ), and within an approximation of a linear oscillator (returning force proportional to the induced displacement), emit in turn a wave with the same angular frequency as the incident wave. The result is that little by little, in media (1) and (2), the incident wave with angular frequency is reemitted with the same angular frequency, so that Zi = Zr = Zt = Z . (5) 11.2.2.2. Reasoning with the help of equations of continuity G G0 G0 By denoting ai, ar and at as the proper projections of Ei0 , E r , and E t at the plane of the interface Oxy, the continuity condition of the tangential component of the electric field gives rise to: GG G G G G a i exp j[k i r-Ȧi t] a r exp j[k r r-Ȧr t] a t exp j[k t r-Ȧ t t] . By multiplying the two GG members by exp - j[k i r-Ȧi t] , we obtain:

ai + a r e

G G G j[(k r - k i )r-(Ȧr - Ȧi )t]

G

G G

a t e j[(k t - k i )r-(Ȧt - Ȧi )t] .

(6)

This equation must be true for all instants (t), and each term of the preceding equation should have the same temporal dependence, so that 0 Zr - Zi = Zt - Zi . The same result is found as that given by Eq. (5), that is Zi = Zr = Zt = Z .

320


11.2.3. Form of the wavevectors with respect to the symmetry of the media Once again, the same two types of reasoning may be applied. 11.2.3.1. Reasoning based on physics of the symmetry of a system The EM field of the incident wavevector is given by: k 0 G ° ix k i ®k iy z 0 ° ¯k iz z 0 Taking into account the disposition of the interface, which is such that the discontinuity only occurs in the direction Oz, the propagation in the directions x and y is not perturbed at the interface. This results in perturbation being only in the direction Oz, where the wavevector thus is modified on reflection and transmission. Physically then we should have: k ix = k rx = k tx = 0 k iy = k ry = k ty

(7)

k iz z k rz z k tz

11.2.3.2. Mathematically based reasoning and the use of the continuity equation Again using the result given by Eq. (6) and also from the imposition of the condition of continuity of the tangential component of the electric field, the equation is valid G only at the level of the interface where the vector ( r ) has the components given by x z 0 G° r ®y z 0 °z = 0 ¯ G ki G

G kr Ti

Tr

G

[i

[r G

[t Tt

z

y

G kt

Figure 11.2. Representation of the projected vectors ( [ ) at the interface.


321

G G G By denoting ȟ i , ȟ r , and ȟ t , the projected vectors at the interface Oxy of the G G G vectors k i , k r , k t , respectively, we can state that they have components given by: k G ° ix ȟ i ®k iy ° ¯0

k G ° rx ȟ r ®k ry ° ¯0

k G ° tx ȟ t ®k ty ° ¯0

G Eq. (6) for the vectors r located at the interface thus is in the form: G

G

G G

G G

a t e j[([ t - [i )r-(Ȧt - Ȧi )t] .

a i +a r e j[([r - [i )r-(Ȧr - Ȧi )t]

(6’)

G This Eq. (6') should be true for all r vectors at the interface, and we should find for each term in Eq. (6') the same spatial dependence, which can be written: G G G G G G G G G 0 = ȟ r - ȟ i r = ȟ t - ȟ i r, so that ȟi = ȟ r = ȟ t . G With k i in the plane of the wave k ix 0 , it can be determined that:

k ix = k rx = k tx = 0 and k iy = k ry = k ty .

(7’)

Following reflection and transmission of the EM wave at the level of the interface, there remains only k rz and k tz to be determined. This can be done only through a use of the equations for the propagation in each of the media.

11.2.4. Symmetry and linear properties of the media and the form of the related field 11.2.4.1. In medium (1)

A medium (1) medium (2) Figure 11.3. Representation of a wave at point A in medium 1.

A wave at any point (A) in medium (1) is in fact a superposition of the incident and reflected wave. Taking the results of Sections 11.2.2 and 11.2.3 into account, which made it possible to state that Zr = Z , that k rx = k ix = 0 and that k ry = k iy , along with that of the form of Eq. (2) for the incident wave, the resultant wave in medium (1) therefore must be of the form: G G G0 E1 (r,t) = E1 (z) exp ª j k iy y- Ȧt º . (8) ¬ ¼

322


The exponential term contains the well-determined (and unchanged) part of the wave, where the temporal variation is Zr = Z and the spatial variation with respect to x is given by k rx = k ix = 0 and with respect to y by k ry = k iy . G0 The term E1 (z) contains the behavior with respect to Oz, which remains for the moment unknown; the resolution of the propagation equation in medium (1) will G0 allow a determination of E1 (z) and krz (as a function of kiz).

11.2.4.2. In the medium (2) Once again, Zt = Z and k ty = k iy , while only ktz remains unknown. In medium (2) the wave thus is in the form: G G G0 E 2 (r,t) = E 2 (z) exp ª j k iy y- Ȧt º . (9) ¬ ¼ G0 The exponential contains the known terms, while E 2 (z) encloses that which is unknown for the moment (behaviour with respect to Oz) and it is the resolution of the propagation equation for medium (2) which will permit a determination of G0 E 2 (z) and k tz { k 2z .

11.2.5. Snell-Descartes law for the simple scenario where media (1) and (2) are nonabsorbent and k1 and k2 are real 11.2.5.1. The wavevectors associated with incident, reflected, and transmitted waves are all in the same plane (incidence plane) The result given in the subtitle can be determined immediately from the fact that we have k ix = k rx = k tx = 0 from the first expressions of Eq. (7). The result is that the G G G wavevectors k i , k r , and k t are all in the plane Oyx, which is their hypothetical plane of incidence. 11.2.5.2. Law of reflection: Ti Tr The first equivalence of the second expression of the equations given in (7) is that Ȧ Ȧ k ry = k iy . With k i = n1 and also k r = n1 , the relation k ry = k iy , which c c G G represents the equality of the projection of vectors k i and k r with respect to Oy, Ȧ n1 sin și = n1 sin ș r , so that Ti c c G G we orientate the angles with respect to the normal, then n12 = n z , and

gives in moduli (see Figure 11.2)

Ti

Ȧ

Tr .

(10)

Tr , and if


323

11.2.5.3. Law of refraction: n1 sin și = n2 sin șt Still with respect to the second of the expressions in Eq. (7), we also have k ty = k iy . When the materials are nonabsorbing, the wavevectors are real just as the optical indices. We thus have k i =

Ȧ c

n1 and k t =

Ȧ c

n 2 and the relation

G G k ty = k iy , which brings in a projection of the vectors k i and k t with respect to Oy, Ȧ

n1 sin și =

Ȧ

n 2 sin ș t , so that : c c n1 sin și = n 2 sin ș t . (11)

gives rise to (Figure 11.2)

11.2.6. Equation for the electric field in medium (1): the law of reflection As above indicated, medium (1) is not magnetic or absorbent, n1 and G Ȧ k i = n1 = k 0 n1 are real, and the components of k i are (0, k iy z 0 and k iz z 0 ). c In medium (1), the monochromatic wave is in accordance with the wave equation [Eq. (7") in Chapter 7]: JJJGG Ȧ² 2 G ǻE1 + n1 E1 = 0 (12) c² G where E1 is the form given by Eq. (8), as in G G G0 E1 (r,t) = E1 (z) exp ª j k iy y- Ȧt º . ¬ ¼ G G G0 By making f(y)=exp ª j k iy y- Ȧt º , we have E1 (r,t)= E1 (z) f(y) , with the result ¬ ¼ that

JJJGG G G G ǻE1 = i ' E1x + j ' E1y + k ' E1z G ªG0 G G G0 G0 i ' E1x (z) f(y) º j ' ª E1y (z) f(y) º k ' ª E1z (z) f(y) º »¼ ¬« ¼» ¬« ¼» ¬« G G G G ª w² 0 º ª 2 0 0 0 = f(y) « E1 (z) » + i² k iy i E1x +j E1y +k E1z f(y)º « ¬ ¼» ¬ wz² ¼

ª w² G 0 º 2 G0 E1 (z)-k iy E1 (z) » exp ªi k iy y-Ȧt º . « ¬ ¼ w z² ¬ ¼

Following simplification of the two f(y)=exp ª j k iy y- Ȧt º , Eq. (12) becomes: ¬ ¼

latter

terms

with

the

term

324


G0 d²E1 (z) dz² 2 By making k1z =

Ȧ² c²

§ Ȧ² 2 · G0 + ¨ n12 - k iy ¸ E1 (z) = 0 . © c² ¹

(13)

2 , then Eq. (13) can be written as n12 - k iy

G0 d²E1 (z) dz²

2 G0 + k1z E1 (z) = 0 .

(13’)

It is notable that: 2 2 2 2 k i2 = k ix + k iy + k iz = k iy + k i2z

k i2

k 0 n1

2

Ȧ² c²

2 = k iz

n12

Ȧ² c²

2 2 n12 - k iy = k1z .

Where k iz = k1z > 0 so that the incident propagation can occur for a positive value G0 of Oz, the general solution for E1 (z) is given by G0 G0 G0 E1 (z)= Ei exp(i k1z z)+E r exp(- i k1z z) . (14) incident propagation

reflected propagation

G0 By substituting E1 (z) into Eq. (8), we obtain: G G G0 E1 (r,t) = E1 (z) exp ª j k iy y- Ȧt º ¬ ¼ G0 G0 Ei exp ªi k iy y + k1z z - Zt º +E r exp ªi k iy y - k1z z - Zt º ¬ ¼ ¬ ¼ G G Ei + E r . (15)

The first term details the incident wave for which the wavevector is given by: k = 0 G ° ix k i ®kiy °k = k > 0. 1z ¯ iz The second term represents the reflected wave at the level of the interface and is the G monochromatic plane progressive wave for which the vector k r has the component:


325

k = k ix = 0 G ° rx k r ®k ry = k iy z 0 ° ¯k rz = - k1z = - k iz < 0 From the relations k rx = k ix = 0 and k ry = k iy , we find once again the laws of reflection obtained in Section 11.2.5. The incident and reflected rays are in the incident plane Oyz and Ti Tr . It is worth noting that these laws do not depend on the nature of medium (2), as no hypothesis has been formulated concerning it, and it could be nonabsorbing or absorbent and even a perfect insulator or conductor.

11.2.7. The Snell-Descartes law for reflection a system where medium (2) can be absorbent: n2 and k2 are complex Here medium (1) is nonabsorbent and n1 and k1 are real. 11.2.7.1. Form of the field in medium (2) Here medium (2) can be absorbent and is characterized by a complex dielectric permittivity, so that İ r2

n 22 .

JJJGG Ȧ² 2 G The wave equation in medium (2) is ǻE 2 + n 2 E 2 =0 . c² G Into this the substitution of the form of E 2 given by Eq. (9), G G G0 E 2 (r,t) = E 2 (z) exp ª j k iy y- Ȧt º , gives ¬ ¼ G0 d²E 2 § Ȧ² 2 2 · G0 (16) + ¨ n 2 - k iy ¸ E2 = 0 . dz² © c² ¹

By making k 22z

i.e.,

§ Ȧ² 2 2 · ¨ n 2 - k iy ¸ , so that with k iy = k 0 n1 sin și , then c² © ¹ k 22z = k 02 (n 22 - n12sin²și ) .

(17)

Then the equation for the propagation of the wave takes on its more normal form [see for example Eq. (15) in Chapter 7]: G0 G0 d²E 2 (16’) + k 22z E 2 = 0 . dz² The general solution is thus of the form: G0 G0 G E 2 (z) = E t exp(ik 2z z) + E '0t exp(- ik 2z z) .

(18)

326


Physically, in medium (2) there is no source of light and the only wave that actually can be found is a wave transmitted and distancing itself from the interface. In this case, the second component of Eq. (18) must be equal to zero, as it represents G the effect due to a wave that nears the interface (the E '0t also must be taken as equal to zero in order that the condition is held). Only the first component of Eq. (18), which clearly represents the effect of a wave leaving from the interface, can be retained on a physical basis, so that finally we have G0 G0 E 2 (z) = E t exp(ik 2z z) . (18’) For its part, k 2z defined by Eq. (17), generally has a complex form and can be written using a notation in the form: k 2z = k tz = k 'tz + ik 'tz' .

(19)

On substituting Eq. (18’) into Eq. (9), we obtain: G G G0 G0 E 2 (r,t) = E 2 (z) exp ªi k iy y- Ȧt º E t exp ªi k iy y + k 2z - Ȧt º , ¬ ¼ ¬ ¼ so that with Eq. (19): G G G0 (21) E 2 (r,t) = E t exp( - k ''tz z) exp ªi k iy y + k 'tz z - Ȧt º . «¬ »¼

(20)

propagation term

attenuation term

11.2.7.2. Form of the wavevector in the medium (2) Equation (21) for the wave in medium (2) can be rewritten in the form: G G G G G G G0 (21’) E 2 (r,t) = E t exp( - k ''t r) exp ª i k 't r - Ȧt º . «¬ »¼ G G This relation defines the vectors k 't and k ''t for which the components are given from Eq. (21):

G k 't

k 'tx = 0 (= k ix ) k 'ty = k iy =

(22)

k 0 n1 sin și

(19) (17) k 'tz = R(k 2z ) = k 0 R ® n 22 - n12 sin²și ¯

1/2 ½

¾ ¿


G k ''t

k ''tx = 0

327

(23)

k ''ty = 0

19 (17) k ''tz = Im(k 2z ) = k 0 Im ® n 22 - n12sin²și ¯

1/2 ½

¾ ¿

Finally, the wave can be written in the form: G G G0 ª G G º E 2 (r,t) = E t (z) exp «i §¨ k t r - Ȧt ·¸ » , ¹¼ ¬ © G G G G where k t is defined by k t = k 't + ik 't' , such that: 1/2 ½ G G' 2 G 2 ¾ ez °k t =k iy e y + k 0 R ® n 2 - n1 sin²și G ° ¯ ¿ kt ® ° kG '' = k Im n 2 - n 2 sin²ș 1/2 ½ eG . ® 2 1 ¾ z 0 i ° t ¯ ¿ ¯

G G k 'ty e y + k 'tz ez

(24)

Consequences: inhomogeneous and homogeneous waves Generally speaking, the wavevector for the transmitted wave can be written in accord with Eq. (24) in the form: G G G G k t =k 'ty e y + k 'tz ez + i (k ''tz ez ) . (25)

According to Eq. (21), the propagation in medium (2), which is determined by G the real component of k t , is along Oy and Oz, while the attenuation associated with G the imaginary component of k ''t is only in Oz. The directions of propagation and attenuation thus are different and the wave is termed inhomogeneous. In the specific case where k 'ty = 0, so that when k 'ty = k1 sin ș1 we have T1 (normal incidence), we have:

0

G G G k t = k 'tz ez + i (k ''tz ez )

and the propagation and attenuation are in the same direction (Oz), and thus the wave is termed homogeneous.

328


11.2.7.3. First law of refraction

G G Here we have k 'tx = 0 and k ''tx = 0 , so that ª k t º = 0 . The k t has no component «¬ »¼ x along Ox and the transmitted wave is propagated in the plane of incidence, i.e., Oyz. 11.2.7.4. Second law of refraction

When k 'tz z 0 , the transmitted wave effectively propagates in medium (2); G G supposing T2 =T t =(n12 ,k 't ) , so that k 'ty = k 't sin ș 2 (see Figure 11.2). In addition, it also is possible to state, from Eq. (22), that k 'ty = k iy = k 0 n1 sin și which can written with k1 = k 0 n1 and by making T1

Ti by notation, that k 'ty = k1 sin ș1 . As

a consequence, we have the equation: k1 sin ș1

k 't sin ș 2 ,

(26)

which can be rewritten using the notations introduced, as k i sin și

k 't sin ș t .

(26’)

G G 11.2.7.4.1. Case where k t is real R k ''t = 0

Here, according to Eq. (23), the term n 22 - n12sin²și

must have a positive and real

magnitude: if it were negative, its square root would be purely imaginary and k ''t would be nonzero. Looking at the following conditions:

x first condition, n 22 - n12sin²și that n 22

n 22

has a real magnitude, imposes that n

2 2

is real, so

H r 2 where Hr2 has a real magnitude. Medium (2) thus is intrinsically

[that is to say by itself without medium (1)] nonabsorbent as k 22 =

Ȧ² c²

is real if H r2 = H r2 is real (and positive)

n 22 - n12sin²ș1 = n 22 - n12sin²ș1 ; x second condition, the term n 22 - n12sin²ș1 has a positive magnitude if

n 22 > n12sin²ș1 .

(27)

n 22 =

Ȧ² c²

İ r2


329

This is the supplementary condition to n 22 being real [medium (2) nonabsorbent and G G Hr2 real] so that k t can be real (R k ''t = 0) when medium (2) is in the presence of medium (1). 0 , we have k t = k 't , which is such that according to Eq. (22):

When k ''t

k 2t = k '2t = k '2ty + k '2tz = (k 0 n1 sin ș1 ) 2 +(k 02 [n 22 - n12 sin² ș1 ]) = k 02 n 22 = Also, when

k ''t

0 , Eq. (26) is written as k1 sin ș1 n1 sin ș1

n 2 sin ș 2 .

Ȧ² c²

n 22 = k 22 .

k 2 sin ș 2 , so that in turn: (28)

G The conditions required so that n 22 > n12sin²ș1 (R k ''t = 0) are: x by simple evidence, that n 2 > n1 (see also Figure 11.4); x or n 2 < n1 and T1 TA where TA is defined by n1 sin TA = n 2 .

(29)

In effect, so that n 2 n1 we have T2 ! T1 (Figure 11.5). The T2 cannot exceed T2A

S

2

and TA also cannot exceed a limiting value given by T1

such that n1 sinTA = n 2 sin

TA and

S

. The limiting angle ( TA ) above which there can be 2 no refracted radiation is associated with the total reflection and thus is defined by the n equation sinTA = 2 . n1 The condition T1 TA yields sinT1 < sinTA =

n2 n1

, which otherwise can be

G stated as n1 sinT1 < n 2 and corresponds well to Eq. (27), for which k t is real G (R k ''t = 0) .

330


G ki

n2 > n1, T2 < T1. if T1= T’1 = S/2, T2 = T’2 such that sin T’2 = n1/n2

T1

n1

n2

T1 TA

n1

T1 = S/2

n2 T2

n2 < n1, T2 > T1. if T2 = T2A = S/2, T1= TA such that sin TA = n2/n1

G kt

T2l= S/2 G kt T2

T’ 2

n2 > n1 Figure 11.4. Refraction when n2 > n1 .

n2 < n1 Figure 11.5. Refraction when n2 < n1 .

G G 11.2.7.4.2. When k t is complex: R k ''t z 0 G When k t is complex, then k ''tz z 0 must be true. The attenuation occurs in the Oz

direction. According to the third relation given in Eq. (23), the following must be true: x either n 22 = İ r2 is imaginary and medium (2) is absorbent; or x n 22 = İ r2 = İ r2 is real [medium (2) is not absorbent] and n 2 n1 and T1 ! TA

(which correspond to n1 sinT1 ! n1 sinTA = n 2 ). We thus have n 22 - n12 sin²ș1 0 , so that n 22 - n12sin²ș1 i 2 ( n12sin²ș1 - n 22 ) and, according to Eqs. (22) and (23), we can deduce that, respectively: ' °k tz ° ® ° '' °k tz ¯

1/ 2 ½ k 0 R ® ª i²(n12sin²ș1 - n 22 ) º ¾ 0 ¬ ¼ ¯ ¿ 1/ 2 ½ k 0 Im ® ª i²(n12sin²ș1 - n 22 ) º ¾ = r k 0 n12 sin²ș1 - n 22 ¬ ¼ ¯ ¿

1/ 2

(30) z 0.

So that the wave undergoes an exponential “braking”, Eq. (21) must have k ''tz z

! 0 . As z ! 0 in medium (2), the positive solution from Eq. (30) for k ''tz is retained, and in terms of vectors, we have [from Eq. (24)]:


G k ' =k eG =k n sin ș eG iy y 0 1 1 y ° t ®G 1/2 G ° k ''t = k 0 n12 sin²ș1 - n 22 ez ¯

331

(30’)

the wave is inhomogeneous. It is as shown in Figure 11.6. T1 > TA

G ki

TA

n1

k 't = k 0 n1 sinș1

T1

y n2

G ki

k ''t

z n 2 < n1 Figure 11.6. Inhomogeneous wave where k ''t z 0 , n2 < n1 and T1 > TA .

11.3. Coefficients for Reflection and Transmission of a Monochromatic Plane Progressive EM Wave at the Interface between Two Nonabsorbent lhi Dielectrics (n1 and n2 are real), and the Fresnel Equations 11.3.1. Hypothesis and aim of the study We thus can suppose that n12 = İ r1 = İ r1 are real, just as n 22 = İ r2 = İ r2 . Once again it is assumed that the wave source imposes an angular frequency (Z) and that the state of the polarization of the incident wave, which is always a combination of two orthogonal polarization states, is either such that: G G x Ei0 is perpendicular to the plane of incidence (Ei0A ) in which case it is termed a transverse electric (TE) polarization, as the electric field is orthogonal to the plane of incidence (it can also be stated that the wave is in an orthogonal polarization as in Figure 11.7 a); or G G x Ei0 is parallel to the plane of incidence ( Ei0& ) and in which case the term is one of transverse magnetic (TM) polarization as the magnetic field thus is orthogonal to the plane of incidence. Another term used is that of parallel polarization, as shown in Figure 11.7b.

332


G Ei0

G0 G 0 Bi E i Ti ki

Ti

ki

n1 n2

y

G x n12

Tt

n1 y

G x n12

n2

kt

Tt

kt

z

z Fig. 11.7(a). TE polarization.

Fig. 11.7(b) TM polarization.

G G Given the incident wave, as in Ei = Ei0 exp(j[k i sinși y + k i cosși z - Ȧt]) , it should G0 G0 be possible to determine either the vectors E r and E t for the reflected and

transmitted waves, respectively, or the coefficients for the amplitudes of reflection (r) and transmission (t) defined by: r=

E 0r Ei0

and t=

E 0t Ei0

.

(31)

The r and t are a priori complex magnitudes that can take into account any possible dephasing between the reflection and the transmission. We thus have two unknowns to determine, which can be done with the help of two equations established from the limiting conditions. With the media assumed to be nonmagnetic, the following two relations for continuity at the interface are used: E1t = E 2t and B1t = B2t (as B1t = µ0 H1t = µ0 H 2t = B2t in nonmagnetic media where µ = µ0 ). The indices “1” and “2” denote the media with refractive indices n1 and n2, respectively. Given that the study concerns monochromatic planar G G progressive electromagnetic (MPPEM) waves, between E and B there now is the G G G kuE . well-used equation B = Z


333

11.3.2. Fresnel equations for perpendicular polarizations (TE) 11.3.2.1. When kt is real (n2 ! n1 or n2 > n1 and T1 TA )

G Ei0

n1

G Bi0 Ti G Ti ki

Tr Tr

G B0r

G kr G0 E r

y

G x n12

Tt G0 Et

n2

Tt

G B0t

G kt

z Figure 11.8 In TE mode where n2 > n1 or n2 < n1 and ș1 < șA .

By symmetry, the reflection and transmission fields conserve the same polarization directions as the incident wave. It is supposed that the fields thus are placed as indicated in Figure 11.8. The eventual dephasing with respect to the reflection or the transmission will be determined through an argument around the coefficients for reflection or transmission. For the configuration then it is possible to write equations for the continuity at the interface: x with respect to Ox: Ei0 + E 0r = E 0t , so that on dividing the two members by Ei0 ,

we have: 1 + rA = tA

(32);

x with respect to Oy: Bi0 cosși - B0r cosși = B0t cosș t , so that also

ki Ȧ

Ei0 cosși -

kr Ȧ

E 0r cosși =

kt Ȧ

E 0t cosș t from which with

k i = k r = k 0 n1 and k t = k 0 n 2 and division of two members by Ei0 : (33) (1 - r1 ) n1 cos T1 = t A n 2 cos T2 .

334


By substituting

rA = tA - 1

from Eq. (32) into Eq. (33), and with

și = ș r = ș1 and ș t = ș 2 , it can be immediately deduced that: tA =

2n1cosș1

and r A =

n1cosș1 +n 2 cosș 2

n1 cosș1 - n 2 cosș 2 n1 cosș1 + n 2 cosș 2

.

(34)

With the help of Eq. (28) ( n1 sin ș1 n 2 sin ș 2 ), we can eliminate n1 and n2 from Eq. (34), and then by multiplying the top and bottom of the preceding sin ș 2 yields: equations by n1 tA =

2 sinș 2 cosș1 sin(ș1 +ș 2 )

and r A =

sin(ș 2 -ș1 ) sin(ș1 +ș 2 )

The angles ș1 and ș 2 vary at most between 0 and

.

(34’)

S

, and sin T2 and cos ș1 2 are always positive just as is sin(ș1 +ș 2 ) as (ș1 +ș 2 ) also has a variation limited to between 0 and S. The tA is in fact always real and positive here, and the transmitted wave does not exhibit a dephasing with respect to the incident wave. 10.3.2.1.1. When n 2 > n1 With kt being real, in agreement with the relation n1 sin ș1 n 2 sin ș 2 , we have ș 2 < ș1 , and rA is always real and negative as the maximum variation in (ș 2 - ș1 ) is given by -

ʌ

< (ș 2 -ș1 ) < 0 . We can conclude therefore that a TE wave reflected by 2 the more refractive material undergoes at the reflection a dephasing ( MA ) such that exp(jMA ) = -1 , so that: MA = S (as exp(jS) = cos S + j sin S = - 1). In the limiting case, Eq. (34) shows that: n -n x for a normal incidence, where ș1 = 0, rA = 1 2 0 , so that in terms of moduli: n1 +n 2 rA = x for a glancing incidence, where T1

n 2 -n1 n1 +n 2 S

(35)

then rA 1 . 2 As T1 increases, the modulus of rA continuously increases from a value given by Eq. (35) up to unity. Finally, we obtain the representation given in Figure 4.9 for when n 2 > n1 .


1

335

S

n2 > n1

n2 > n1

|rA|

MA

0

90°

T1

0

90°

T1

Figure 11.9. Modulus and phase angle of the reflection coefficient as a function of the angle of incidence (T1) for a TE wave with n2 > n1 .

11.3.2.1.2. When n 2 n1 and T1 TA (kt real) According to Eq. (28), T2 ! T1 and 0 (T2 T1 )

S

so that sin(T2 T1 ) ! 0 , 2 and according to Eq. (34'), rA is real and positive, so that when medium (2) is less refractive than medium (1) and when T1 TA , the reflected wave rests in phase with the incident wave. According to Eq. (34), we can see that as T1 increases, the

modulus of rA steadily increases from a value given by rA =

n1 -n 2 n1 +n 2

! 0 , obtained

for a normal incident wave (ș1 = 0) , up to the unit value given for the limit of incidence ( ș1 = șA ), as at this point ș 2 takes on the value representation of this zone. 1

S

2

. Figure 11.10 gives a

0

n2 < n1

n2 < n1

|rA|

MA

0

Tl

T1

90°

- S0

Tl

T1

90°

Figure 11.10. Modulus and phase angle of the reflection coefficient as a function of the angle of incidence ( T1 ) for a TE wave and with n2 n1 .

336


11.3.2.2. When kt is complex (n2 n1 and T1 ! TA ) , the wave is inhomogeneous and is as shown in Figure 11.6 The relations for the continuity at the interface (Figure 11.8) give: x with respect to Ox, we again find Eq. (32); and x with respect to Oy, we have: G G0 ·G ki 0 k 1 G G0 G 1§ G Ei cosși - r E 0r cosși = k t ×E t e y ¨ ¬ª k t ¼º y ¬ª k t ¼º z ×E t ¸ e y . Ȧ Ȧ Ȧ Ȧ© ¹ G0 As E t is directed along Ox (TE wave), the term on the right-hand side is reduced to

^

`

G G G G Ei0 '' G ª k t º ×E 0t e y . With ª k t º ik e and by simplifying with the two tz z ¬ ¼z Z Ȧ ¬ ¼z members of the continuity equation for Oy we have:

1

(1 - r A ) k1 cosT1 = i k ''tz t A ,

with according Eq. (30’) : k ''tz = k 0 n12 sin²ș1 - n 22

1/ 2

. We thus obtain

k cosș1 - ik ''tz and r A = 1 . (36) k1cosș1 +ik ''tz k1 cosș1 + ik ''tz For r A , the numerator is the conjugate complex of the denominator, for which r A =1 . The angle for dephasing at the reflection thus is given by: tA =

2k1cosș1

MA = Arg (r A ) tel que r A = |r A | exp(iMA ) = exp(iMA ). We

therefore

z = k1 cosș1 + ik ''tz r A = exp(iMA ) =

§ M · tan ¨ - A ¸ © 2 ¹

have

z

exp(iMA )

rA

z

,

and

then

by

making

U exp(i D ) , we also find that

z

ȡ exp(- iĮ)

z

ȡ exp(iĮ)

tan D

k ''tz k1 cosș1

= exp(- i 2Į) , so that

.

By using n 2 = n1 sinT1 (which makes it possible to replace n 22 in Eq. (30’) by n12 sin 2 TA ), we find that (in the shaded zone of Figure 11.10):

§M · tan ¨ A ¸ © 2 ¹ If T1

TA , MA

0 , and if T1

shaded part of Figure 11.10).

=-

S 2

sin²ș1 - sin²șA 1/2 cosș1 ,

MA 2

S 2

.

, thus MA

(37)

S (see the plot on the


337

10.3.3. Fresnel's equations for parallel magnetic field polarizations 10.3.3.1. When kt is real (n 2 ! n1 or n 2 n1 and T1 TA ) . G G0 G0 kr Bi Br G0 Ti Tr G0 G T T r i Ei Er ki y

G x n12 Tt G0 Et

G Tt B0t G kt

z Figure 11.11. TM wave with n2 > n1 or n2 < n1 , and ș1 < șA .

As detailed in Section 10.3.2, the fields retain the same polarization directions at reflection or transmission through symmetry. We thus can assume that the fields, once reflected and transmitted, are laid out as described in Figure 11.11. For this configuration, the equations for continuity at the interface (still with |Ti | = |Tr | = T1 and Tt T2 ) give:

x with respect to Ox: Bi0 + B0r = B0t , so in addition, k 0 n1 Ei0 Ȧ

k 0 n1 E 0r Ȧ

k 0 n 2 E 0t from which can be deduced that: Ȧ (38); and n1 (1 + r|| ) = n 2 t||

- with respect to Oy: - Ei0 cosș1 + E 0r cosș1 = - E 0t cosș 2 , from which: cosT1 (1 - r|| ) = t|| cosT2 . (39) From this can be deduced that: n 2 cosș1 - n1 cosș 2 r& and t& n 2 cosș1 + n1 cosș 2

2n1 cosș1 n 2 cosș1 + n1 cosș 2

.

(40)

By again using Eq. (28), it is possible to eliminate n1 and n2 from the above equations, so that they become:

338


r&

sinș1 cosș1 - sinș 2 cosș 2 sinș1 cosș1 + sinș 2 cosș 2

from which r&

tan(ș1 - ș 2 ) tan(ș1 + ș 2 )

Similarly, we obtain t &

.

sin2ș1 - sin2ș 2

2 cos(ș1 +ș 2 ) sin(ș1 -ș 2 )

sin2ș1 + sin2ș 2

2 sin(ș1 +ș 2 ) cos(ș1 -ș 2 )

,

(40’)

2 cosș1 sinș 2 . sin(ș1 + ș 2 ) cos(ș 2 - ș1 )

(40’’)

Using Eq. (40), it is immediately evident that t & is always real and positive as cosș1 and cosș 2 are both always positive ( ș1 and ș 2 being between 0 and

S

). 2 Hence the dephasing between the incident wave and the transmitted wave is always zero. With regard to the reflected wave, according to Eq. (40), it is possible to state that r& r& is a real magnitude and the dephasing of the reflected wave is dependent on the sign of r|| , which in turn is such that: sgn r|| = {sgn > tan(ș1 - ș 2 ) @}x{sgn > tan(ș1 + ș 2 ) @} . 11.3.3.1.1. When n 2 ! n1 : kt is real and ș 2 ș1 For a normal incidence (ș1 r&

0D ) , according to Eq. (40) r&

n 2 - n1 ! 0 so that n 2 + n1

r& is a magnitude that is both real and positive, and the dephasing M|| thus is

zero. In general terms though, ș 2 0, S / 2@ , (ș1 - ș 2 ) > 0, S / 2@ , and sgn > tan(ș1 - ș 2 ) @

always

is

positive. The result is such that {sgn r|| } changes sign with {sgn > tan(ș1 + ș 2 ) @} and r& goes from being positive to negative when ș1 ș B so that: (ș B + ș 2 ) =

S

. (41) 2 ș B , is called Brewster's angle and is such that

The angle of incidence, ș1 S tan(ș B + ș 2 ) = tan f , with r|| = 0 . 2 transmitted.

The

wave

therefore

is

entirely


When ș1

S

ș B , so that T2 =

2

339

- TB , we have

S n1 sin TB = n 2 sin T2 = n 2 sin ( - TB ) = n 2 cosTB , from which: 2 n tan TB = 2 . (42) n1

At an air/water interface, we have TB = Arc tan 1.5 | 57 ° . With ș1 = 90°, we have r& =-1 and |r|| | = 1 and M& =S. So, to display the results, the plots of |r|| | = f(T1 ) and M|| = g(T1 ) are shown in Figure 11.12.

S

1

n2 > n1

n2 > n1 M||

|r|||

0

TB

T1

90°

0

TB T1

90°

Figure 11.12. Modulus and phase angle for the reflection coefficient as a function of ș1 for a TM wave with n2 > n1 .

11.3.3.1.2. When n 2 n1 and T1 TA (kt is real), thus here with ș 2 ! ș1 For r&

a

normal

incident

n 2 - n1 0 and r& n 2 + n1

wave

(ș1

0D )

according

to

Eq.

(40)

r& is a real and negative magnitude. The dephasing (M||)

thus is equal to S. When ș 2 ! ș1 , (ș1 - ș 2 ) > 0, -S / 2@ and sgn > tan(ș1 - ș 2 ) @ is always negative; the result is that r|| goes from being negative to positive when > tan(ș1 + ș 2 ) @


340

changes

sign.

This

also

happens

ș1

when

ș B such that (ș B + ș 2 ) =

S 2

.

Simultaneously, M|| goes from S to 0. n We still have tan TB = 2 though, and also can state that ș B șA . In effect, n1 n2

sin TB

> sinTB , so that șA > ș B . n1 cos TB A representation of this is given in Figure 11.13. sin TA =

= tan TB

It is worth noting that for the example of air and water, n1 = 1.5 and n 2 = 1 . § 1 · § 1 · We therefore have TA = Arc sin ¨ ¸ | 42 ° and TB = Arc tan ¨ ¸ | 34 ° . © 1.5 ¹ © 1.5 ¹

S 1

n2 < n1

n2 < n1 M||

|r|||

0

TB

Tl

T1

TB Tl

0

90°

T1

90°

Figure 11.13. Modulus and phase angle for the reflection coefficient as a function of T1 for a TM wave, with n2 < n1 .

11.3.3.2. When kt is complex ( n 2 < n1 and T1 > TA ) In this situation it is not an easy task to reuse the reasoning given in Section 11.3.2.2; the projections made demand a symmetry around the angle Tt { T2 , which has no real physical significance as the wave does not propagate only along the Oy axis. Mathematically, we can check that T2 is not a real angle, as in effect, n sin T1 sin T2 = 1 sin T1 = , so that with T1 >TA we have sin T2 > 1 . n2 sin TA In cos T2

Fresnel's

equations,

in

the

place

of

the

usual

equation,

1 sin ²T2 , now with sin²T2 > 1 we must bring in for the term cos T2 a

purely imaginary value given by cos T2

i sin ²T2 1 . The negative sign is


341

required by the need to obtain at a later point an attenuation of the wave in medium (2). We also can state that cos T2

i

sin ² T1 sin ²TA

1 , and with

n2 n1

sin TA , Eq. (40)

gives: r&

sin²TA cosș1 + j sin²T1 sin ²TA sin²TA cosș1 - j sin²T1 sin ²TA

.

(43)

Once again the numerator is the conjugate complex of the denominator, for which z | r || |² = 1 . By writing r|| = exp(iM|| ) = , and by making z z = sin²TA cosș1 + j sin²T1 sin ² TA = ȡ exp(i ȕ) , we obtain: r|| = exp(iM|| ) =

z z

=

§ M& · = exp( i 2E) , so that with tan ¨ ¸ = tan E : ȡ exp(- iE) © 2 ¹ ȡ exp( iE)

§ M& tan ¨ © 2

· ¸= ¹

sin²T1 sin ²TA cosș1 sin²TA

.

(44)

§ M& · S TA , then tan ¨ ¸ 0 and M|| = 0 . If T1 , then 2 © 2 ¹ § M& · cos T1 = 0 , tan ¨ ¸ = f , and M|| = S (see also the plot on the shaded area of © 2 ¹ Figure 11.13).

If T1

11.3.3.3. Comment

TB

Figure 11.14. A Brewster angle incidence for a TM wave at an air/glass window.

342


To ensure that a ray passes through a glass/air interface, as shown in Figure 11.14, without loss by reflection, the wave should be TM polarized and undergo a Brewster type incidence as ª¬ r& º¼ 0 (such as is used in Brewster lasers with a Brewster T1 TB

window).

11.3.4. Reflection coefficients and energy transmission 11.3.4.1. Aide mémoire: energy flux through a surface The energy flux, associated with the propagation of an electromagnetic field in a G material, is equal to the flux traversing the surface given by the Poynting vector ( S ). Equation (33’) of Chapter 7, Section 7.3.3, shows that for a MPPEM wave in nonmagnetic materials the average value for the Poynting vector is given by G G G G G G 1 S R(E m u Bm ) , so that with ȦBm = k×E m [Eq. (13) of Chapter 7] we 2µ0 have

G S =

1

G G 2 k E m , and with

İ c = 0 , we can write that µ0Ȧ k0 1

2µ0 Ȧ G G İ0c k G 2 S E m . With our notations, and for incident, reflected, and transmitted 2 k0 wave, the associated average value of the Poynting vector is, respectively: G G G G G İ0c ki G 0 2 İ0c k r G 0 2 G İ0c k t G 0 2 Si Ei , Sr Er , S = Et . 2 k0 2 k0 2 k0

The energy flux across the surface 6 is the energy transmitted per unit time through the surface. In effect, it also represents the power transmitted by the wave under consideration.

T1 x

T1 n1

6

T2

y

h

G n12

n2

z

Figure 11.15. Reflection and transmission of an incident flux traversing 6 .


343

The radiated power of the wave traversing an interface with surface (6) from a medium with an index denoted by n1 toward a medium with an index n2 (Figure 11.15) thus is given by:

G G G G 2 İ0c ki G 0 2 G İ c < Pi > = Si n12 ¦ = Ei n12 ¦ = 0 n1 Ei0 Ȉ cosș1 2 k0 2 G 2 G G G0 2 İ c k G0 G İ c < Pr > = Sr n12 ¦ 0 r E r n12 ¦ 0 n1 E r Ȉ cosș1 2 k0 2 G G G İ c k G0 2 G İ c G 0 2 k' < Pt > = St n12 ¦ 0 t E t n12 ¦ 0 E t 6 tz . 2 k0 2 k0

(45)

11.3.4.2. Equation for the reflection(R) and transmission (T) coefficients: total energy The titled coefficients are defined so as to give positive values, so that we have: Pt Pr R=and T = . (46) Pi Pi From this can be determined with the equations in Eq. (45) that: G0 2 G0 2 Er Et k 'tz k 'tz 2 2 R= = r and T = = t . G0 2 G 0 2 k n cosș k n cosș 0 1 1 0 1 1 Ei Ei

(47)

It is notable that in the case of the transmitted power, T z | t |² . This is due to the fact that the way in which the power is transported depends on the different cross sections, so that ¦ i = ¦ cos T1 for an incident wave and

¦ t = ¦ cos T2 for a transmitted wave (while for incident and transmitted powers, ¦ i = ¦ r = ¦ cos T1 ). With respect to the sum of the energy, we can calculate this by using a closed surface, such as a parallelepiped of a given height (h) such that h o 0, and then drawn around the level of the surface (Figure 11.15). In physical terms, for an isolated system, the incoming radiation power must be equal to that leaving, so: x the incoming radiation power is that of the incident wave, such that G G G n12 : = Si n12 ¦ .

x the outgoing power radiated has two components:


344

G x power radiated by the reflected wave with respect to - n12 G G Sr n12 ¦ = - < Pr >

G x the power radiated by the transmitted wave with respect to n12 G G = St n12 ¦ We thus should have = - < Pr > + < Pt > . Dividing through term by term by we obtain: R+T=1

.

(48)

Comment: The transmission coefficient for a homogeneous transmitted wave (n1 n 2 or n1 ! n 2 and T1 TA ) . G G' In this case, k t is real and equal to k t . With k 'tz k 0 n 2 cos T2 , we have T

2

t

n 2 cosș 2 . n1 cosș1

(49)

11.3.4.3. Representation of the reflection coefficient as a function of the angle of incidence 1 1 n2 > n1 R R

§ n1 - n 2 ¨¨ © n1 + n 2

· ¸¸ ¹

2

RA

§ n1 - n 2 ¨¨ © n1 + n 2

R|| 0

TB

T1

90°

Figure 11. 16 a. Factors in the reflection of energy: R A = |rA |² and R || = | r|| |² as a function of

T1, for n2 ! n1 ( with rA and r|| from Figures 11.9 and 11.12)

T B = 57 ° for the air/glass interface.

0

· ¸¸ ¹

2

RA R||

TB Tl

T1

90°

Figure 11.16 b. Factors in the reflection of energy:

R A = |rA |² and R || = | r|| |² as a function of T1, for n2 n1 ( with rA and r|| from Figures 11.10 and 11.13). TB = 34 ° for an air/glass interface, and TA = 42 ° .

Taking the equation R = r

2

into account, we can use the representations given in

Figure 11.15 for when n 2 ! n1 , or n1 ! n 2 .


For normal incidence, ș1 = ș 2 = 0, and R

At the same time, T

t

2

n2 n1

§ 2n1 ¨¨ © n1 +n 2

2

· n2 ¸¸ ¹ n1

§ n1 - n 2 ¨¨ © n1 + n 2

4n1n 2

n1 +n 2 2

345

2

· ¸¸ . ¹

1- R .

It can be seen in the plots of Figure 11.16 that when n 2 ! n1 the reflection is always partial and relatively weak, except in the neighborhood of the glancing angle ʌ (ș1 | ) . This is the reason why the air/water interface is always transparent 2 ( n 2 | 1.33) except when there is a glancing angle, or under conditions of reflection. Under a Brewster angle, only a wave polarized perpendicularly to the plane of incidence (TE waves) is partially reflected. At this incidence, the TM wave is entirely transmitted (R || = 0 when ș1 = ș B ).

polarized TM wave for 2nd mirror

1st mirror

Wave loss after 2nd reflection at Brewster angle TB

TB incident natural light

2nd mirror with incidence plane normal to that of the 1st

Figure 11.17. Malus's experiment.

Under natural light, the incident wave exhibits all polarization states, and for an incidence with the Brewster angle it is only waves polarized perpendicularly to the plane of incidence which is reflected (TE wave). It is this result that explains physically Malus's law, which demonstrates the polarization of light with the help of two mirrors set at a Brewster angle, so that the planes of incidence are orthogonal. The two successive reflections are at Brewster angles so that after the first reflection

346


of natural light there remains only a component that is polarized perpendicularly to the plane of incidence of the first mirror (thus a TE wave for the first mirror). The incidence plane of the second mirror is perpendicular to that of the first, and the incoming wave is seen as a TM wave. Following a reflection at the Brewster angle on the second mirror, there is a complete extinction of the wave.

11.3.5. Total and frustrated total reflection

G G When n1 ! n 2 and ș1 > șA , we have k 'tz = 0 and k 't = k 'ty ey ; there no longer is a propagation along Oz and only propagation along Oy persists. As R = | r |² = 1 , we have T = 1 - R = 0 , while at the same time with t z 0 (see Eq. (36), for example). T1 > TA

G ki

TA

n1

k 't = k 0 n1 sinș1

T1

y

n2

G n 2 < n1

k ''tz

evanescent wave: - propagation along Oy - attenuation for Oz

z Figure 11.18. The evanescent wave and frustrated total reflection.

Therefore, even though there is a transmitted wave, as t z 0 , the transmission coefficient for energy is zero. This is because the propagation through G medium (2) can occur only at the interface (along e y ). The wave itself is evanescent (Figure 11.18 being the “completed” Figure 11.6) with wavevector G components ( k t ) given by Eq. (30’). By substituting the value for the components into Eq. (21), we find that the transmitted wave has the form: G G G0 E t (r,t) = E t exp( - k ''tz z) exp ªi kiy y - Ȧt º , ¬ ¼

with k ''tz = k 0 (n12sin²ș1 -n 22 )1/2

k0n2 (

n12 n 22

sin²ș1 - 1)1/2 .

G G 2 The intensity of the transmitted wave, proportional to E t (r,t) , is thus of the form:


347

G 2 2z 1 1 n2 I t (z)= E0t exp(-2k ''tz )= I t (0) exp(- ) , with į = = ( 1 sin²ș1 - 1)- 1/2 į k 0 n 2 n 22 k ''tz where G represents the degree of attenuation (when z =

G 2

, the wave intensity is

divided by e). The wavelength in medium (2) is given by O 2 = į=

Ȝ0 n2

, so that also we have:

O 2 n12 ( sin²ș1 - 1)- 1/2 . 2 2S n 2

If T1 increases, sin T1 also increases, as (

n12 n 22

(50)

sin²ș1 - 1) , while (

n12 n 22

sin²ș1 - 1)-1/2

decreases with G and the evanescent wave penetrates less medium (2). Numerically, for the glass/air interface, and at what is practically a glancing ʌ angle, (ș1 | ), we find that G | 0.14 O 2 . 2

glass (1) (n1)

TA interface I

air (n2)

h|G

interface II glass (2) (n1)

Figure 11.19. Experimental demonstration of frustrated total reflection.

The presence of an electromagnetic wave in a thickness of the order of G of the second medium (such that T = 0) can be demonstrated by using the setup shown in Figure 11.19. At the level of the first interface (interface I) the wave penetrates to a depth of the order of G and is then transmitted to the level of the second interface II that is such that its index [glass (2) with index n1] is higher than that of the first medium (air with index n2). In other terms, following its passage from interface I, the wave is attenuated over a depth of h | G and what “remains”' at this depth is then transmitted to the level of the interface II, which in turn does not follow the conditions of total reflection. The result is that of frustrated total reflection.

348


In order to carry out such an experiment, a total reflection prism can be used with angles equal to 45°, which is greater than the 42° required for the glass/air interface ( ș1 = 45o > ș A ). By placing a second prism at a distance h | G from the first, as described in Figure 11.20, a frustrated total reflection occurs exhibited by the presence of part of the incident wave in the right-hand side. However, as mentioned above, the two prisms must be placed a distance apart given by h | G | 0.14 O 2 , which means that h | 0.1 µm for waves in the optical domain. This requires extremely delicate handling and can be made easier by the use of waves with wavelengths of the order of centimeters thus requiring h | several millimeters (which is easy with paraffin prisms). The results obtained for electromagnetic waves can be extended to waves associated with material particles. Frustrated total reflection can be considered that of a tunneling effect, in this case applied to photons. h

air

glass incident wave

T1 > TA

wave after frustrated total reflection

Figure 11.20. A total reflection prism used to demonstrate frustrated total reflection.

11.4. Reflection and Absorption by an Absorbing Medium 11.4.1. Reflection coefficient for a wave at a normal incidence to an interface between a nonabsorbent medium (1) (index of n1) and an absorbent medium (2) (index of n 2 ) The index of medium (2) is complex and given by n 2 = n '2 + i n '2' . For its part, the incident wave is assumed to be normal to the interface so that T1 = 0° and sin T1 = 0 , and the wavevector of the transmitted wave thus can be written, with k 'ty = k 0 n1 sinș1

0 , as:

Chapter 11. Reflection and refraction in absorbent finite materials 1/2 ' k tz = k 0 R n 22 - n12 sin²ș1 = k 0 n '2 ° G ° ' '' G k t = k t + ik t ez , where ® 1/2 °k '' = k Im n 2 - n 2sin²ș = k 0 n ''2 tz 0 1 1 2 °¯ G G G so that k t = k 0 n '2 + n ''2 ez k 0 n 2 ez . G G The EM field associated with the transmitted wave E t , Bt

349

G G0 E =E exp -k 0 n ''2 z exp i[k 0 n '2 z - Ȧt] °° t t G ® G kG ×E G °B = t t = n 2 eGz ×E . t t Ȧ c ¯°

thus is given by:

G G With the index n 2 being complex, Bt is no longer in phase with E t . Taking into account the form of the fields, we can equally state that this MPPEM wave Ȧ c propagates along Oz with a phase velocity given by vM = and is = ' k 0 n 2 n '2 attenuated exponentially along Oz in the absorbent medium (2). The condition of continuity at the interface of the tangential components for the G G fields E and B results in the same values as those cited previously for reflection and amplitude transmission coefficients, while nevertheless noting that n2 is in a complex form and that: r=

n1 - n 2

=

n1 - n '2 - in ''2 n1 +

n1 + n 2

n '2 +

in ''2

and t =

2n1 n1 + n 2

=

2n1 n1 + n '2 + in '2'

.

The arguments with respect to r and t, respectively, give the dephasing of the reflected and the transmitted wave with respect to the incident wave. For the reflection and transmission coefficients in terms of energy, we also obtain: R= r

2

2

n - n + n = n + n + n 1

' 2

1

' 2

2

''2 2 ''2 2

and

T=1-R=

4 n1 n '2

n + 1

n '2

2

+

. n ''2 2

11.4.2. Optical properties of a metal: reflection and absorption at low and high frequencies by a conductor See Problem 1 of the present chapter.

350


11.5. The Antiecho Condition: Reflection from a Magnetic Layer; a Study of an Antiradar Structure; and a Dallenbach Layer 11.5.1. The antiecho condition: reflection from a nonconducting magnetic layer 11.5.1.1. Reflection from a magnetic layer This follows on from the study in Section 11.3.2 on TE polarization by reflection at an interface between media (1) and (2) with the difference that here the media have a permeability different to that of a vacuum such that media (1) and (2), respectively, exhibit permeabilities µ1 and µ2. It is assumed that the media are sufficiently insulating so that the density of free charges is negligible so that VA = 0 and jA = 0. Under such conditions, the continuity conditions for the interface mean that G G G G G G B1t B2t E1t E 2t and H1t H 2t and in turn with in addition B1t = B2t as µ1 µ2

specifically to this case µ1 z µ2 (see also the end of Section 11.3.1). Equation (32) then becomes: G x along Ox: Ei0 + E 0r = E 0t in an unchanged relation due to the conservation of the continuity equation. By dividing the two terms by Ei0 , we also find 1 + rA = tA , (51) and x along Oy:

ki

Ei0 cosși -

kr

E 0r cosși =

kt

E 0t cosș t .

(52) Ȧµ1 Ȧµ1 Ȧµ2 With the media being nonconducting, i.e., V | 0, we can state that in each medium, according to Eq. (5) of Section 8.5.1, k i = k r = Z İ1 µ1 and Oy becomes:

H1 µ1

k t = Z İ 2 µ 2 . The preceding equation with respect to

Ei0 cosși -

H1 µ1

E 0r cosși =

With Eq. (6) from Section 8.5.2, i.e.,

H2 µ2

1

H1

Z1

µ1

E 0t cosș t . , and

1

H2

Z2

µ2

, we have:

0 Z1-1Ei0 cosși - Z1-1 E 0r cosși = Z-1 2 E t cosș t .

By dividing the two terms by Ei0 , we obtain: Z1-1cosși - Z1-1 r A cosși = Z-1 2 t A cosș t , so that with the same notations as those used in Section 11.3.1, (1 - r A ) Z1-1cosș1 = t A Z-1 2 cosș 2 ,

(52’)


351

From Eqs (51) and (52’), we can determine using a method analogous to that in Section 11.3.2 (where Z1-1 was substituted for n1 and Z-1 2 for n2): cosș1 cosș 2 Z1 Z2 rA = , so that following multiplication of top and bottom with cosș1 cosș 2 + Z1 Z2 Z1Z2 we have: rA

Z2 cosș1 - Z1 cosș 2 Z2 cosș1 + Z1 cosș 2

, from which according to Eq. (47) we have:

RA =

Z2 cosș1 - Z1 cosș 2 Z2 cosș1 + Z1 cosș 2

2

.

(53)

G G As the incidence is normal (with directions k i and ez being merged) the notion of incidence plane loses its significance (otherwise defined by the vector G G G directions k i and n12 { ez ), and we can state for the energy reflection factor for whatever polarization direction of the incident wave, that: R =

Z1 - Z2 Z1 + Z2

2

.

(54)

1.5.1.2. The antiecho condition Here the reflected wave is annulled so that R = 0. For this to occur, taking Eq. (54) into account being applicable for a great distance where the wave exhibits a normal İ İ2 , and hence: incidence on the target, it suffices that Z1 = Z2 , that is, 1 µ1 µ2

İ1r

İ 2r

. (55) µ1r µ2r So that this condition is true, a specific sort of coating must be applied to medium (2). If medium (1) is air, then Eq. (55) means that between the dielectric permittivity and magnetic permeability of the second medium, there is a simple equation given by H r = µr . (55’) Comment: the antiecho condition for EM waves is just one part of the conditions under research in optics in order to find a material that is completely anti-reflective.

352


In such a case, the condition R = 0 means that R =

n1 - n 2

2

n1 + n 2

0 , so that

n1 = n 2 . This condition has no sense in itself, as it would mean that the media (1) and (2) would be identical and that therefore there would be no interface. One method though is to insert between the two media a layer of a given thickness and index n such that n1 < n < n 2 and then find the conditions for which a destructive interference is formed between the two reflected waves, one being at the interface of the materials with indices n1 and n and the other being at the interface of the materials with interfaces n and n 2 .

11.5.2. The Dallenbach layer: an anti-radar structure 11.5.2.1. The stealth concept

G Ei G ki O

G Et G kt

d

G Er

air (H0, µ0)

G kr

y dielectric (H) and/or magnetic (µ) medium which is slightly conducting, although not equal to zero

metal

z Figure 11.21. Antiradar structure.

Assuming that a wave with a frequency denoted by Z has a normal incidence to a metallic surface, the reflection is total. In order to make the metallic layer stealthlike, it is covered with a dielectric and/or magnetic material adapted so as to satisfy the antiecho condition for an air/coating interface. In addition, the wave transmitted in the layer also must be totally attenuated in order to suppress the total reflection that would be produced by the metallic layer. The attenuation is obtained with a slightly conducting layer of sufficient thickness (d) as indicated in Figure 11.21.


353

11.5.2.2. Condition (1): antiecho at the air/dielectric-magnetic interface The classically used absorbent materials are generally composites based on carbon, iron carbonyls, or ferrites (see also Chapter 4, Section 4.2.5.2). These materials do, however, suffer from certain inconveniences such as their weight or their mechanical rigidity due to the high number of charges necessary to reach the required absorption. These composite materials are presently being replaced by conducting polymers, which can be acquired through doping a conductivity appropriate to the system.

11.5.2.2.1. Equation for the reflection coefficient In the system under study, the conductivity (ı) is such that V z 0 and in the layer k = k t so that from Eq. (5) of Chapter 8, Section 8.5.1, where “electrokinetic” notation was used to establish the equation we have: 1/ 2

ı · § k t = Ȧ µİ ¨ 1 - i (56) ¸ . Ȧİ © ¹ Under normal incidences, the preceding Eq. (52) can be written for the air/layer Ȧ interface (where for air k i = k r = k 0 = ): c k0 µ0 k0 µ0

Ei0 -

k0

E 0r =

µ0

kt

1 - r =

µ

k0 r

µ0 k0 µ0

+

kt µ

E 0t . By dividing the two terms by Ei0 it is determined that:

t , from which with the help of Eq. (51) we determine that: kt µ , so that also r kt

µk 0 - µ0 k t µk 0 + µ0 k t

µ

µr µr

Ȧ c Ȧ c

- kt .

(57)

+ kt

11.5.2.2.2. Approximate calculation using Ȟ = 5 GHz, H r = 15, V = 5 x 10-1 :-1m-1 = 5 x 10-3 cm-1 ) Figures 11.22a and b give a qualitative indication of the evolution of the conductivity and dielectric permittivity of polyaniline films with varying levels of doping as a function of frequency. On going from A to D the plots are for increasing levels of doping. It can be seen that for sufficiently high levels of doping, quite high values can be attained.


354

V(:-1cm-1)

Hr

D

Vdc

105

10-3

D C

Vdc

(b)

103

10-6 (a)

B

A

10-9 1 kHz

1 MHz

C

B 10

frequency 1 GHz

A

frequency

1 kHz

1 MHz

1 GHz

Figure 11.22. Evolution of V and Hr’ as a function of frequency for doped polyanilines.

Using ı Ȧİ

=

the

values

given

5 x 10-1 2ʌ x 5 x 109 x 15 x 8.85 x 10-12

in

Eq.

(56)

for

kt ,

the

| 1.2 x 101 is very much less than 1. With a

ı · § limited development, from Eq. (56) for k t we obtain k t Ȧ µİ ¨ 1 - i ¸. 2Ȧİ ¹ © ı By making E , we thus can write that: 2Ȧİ Ȧ (56’) k t Ȧ µİ 1 - iȕ µ r İ r 1 - iȕ . c The reflection coefficient given by Eq. (57) then takes on the form:

r

µr -

µr Hr (1 iȕ)

µr +

µr H r (1 iȕ)

term

1, so that finally r

1+

İr µr İr µr

(1 iȕ) . (1 iȕ)

Numerically, as E | 10-1 > 1 >> 1 and n' >> 1 just as n'' >>1 . (d) We have « » ZH0 «¬ ZH0 »¼ i min


The result is that r =

n1 - n n1 + n

=

1 - n' + in''

|-

1 + n'- in''

n' - in''

eiS and on reflection

1

n' - in''

359

there is a dephasing by S while the reflecting power is in the neighborhood of 1, i.e., R = |r²| | 1 . The ability to reflect energy is given by R= r

2

1- n + n = 1+ n + n '

2

'

''2

2

n '2 +n ''2

n '2 +n ''2

''2

=1

where n' >>1 . (e) With k =

Z c

Z

n = k' - i k'' , we have k''

c

n''

Z

V(0)

1

Ȧ ı(0)

c

2ZH0

c

2İ 0

E 0t eiZt

A wave that propagates in the form E t

.

E 0t eikz eiZt can be written for

this frequency range as E = E 0t ei(Zt k ' z) e k '' z . The equation carries a term for the propagation as in ei(Zt k ' z) , and a term for attenuation as in e k ''z that also can be

rewritten as e

z G

where G

1 k ''

c

2İ 0 Ȧ ı(0)

. With G representing the depth at

which the electric wave is attenuated by the ratio

1 e

, for distances greater than

several G , the wave is practically zero. As G

c

2İ 0 Ȧ ı(0)

, we can see that as Z increases, G decreases. For the

highest frequencies in this region ( Z d

1

| 1014 Hz ) the EM waves are localized at

W the surface of the conductors, in an effect called the skin effect.

(f) The equation for T is deduced from that of R, as in T = 1 - R = With n'' = n' = n >> n1 = 1 , the result is that T

2 n

2Ȧį

2ʌ

2ʌ

c Ȧ which is known as the Hagen-Rubens equation.

k0

2 k 0 į , so that with Ȝ 0

c

(n1 + n')2 + n ''2

.

, so in addition T 0 (as n >>

1). In more precise terms, when n' = n' = n , and with G T

4n1n'

1

c 1

k ''

Ȧ n''

, we also have

, we have T

4ʌ į Ȝ0

,

360


The energy received by the metal is transmitted by carriers to the lattice that dissipates the energy through the Joule effect. So, the factor T is often called the absorption power of the metal. In numerical terms, for copper, V(0) = 6 x 107 :-1m -1 , Z = 2 S 109 rad/s, H0 = 8.85 10-12 MKS so that G As O 0

c Ȟ

c

2İ 0

2.06 µm .

Ȧ ı(0)

0.3 m (region of centimeter wavelengths), the Hagen-Rubens

formula gives T | 8 x 10- 5 o 0 , to which R | 0.99990 o 1 . Hence the use of microwave frequency radars for police use in determining car speeds!

2. (a) This part uses for H r the general formula (2) obtained from question 3(d) of Section 8.6.2, for metals, as in: Z2p W Hr 1 i . Z[1 iZW] When ZW>> 1 , we have H r | 1 i (b) When Z t1 z

(a) progressive wave

signal t1 t2 > t1

z

(b) stationary wave

Figure 12.5. (a) Progressive; and (b) stationary waves.

The physical solution for the resultant wave in a vacuum thus is finally given by G G G E1 = R(E1 )= 2 Ei0 sink 0 z sinȦt ex where the spatial and temporal dependences are now separated. The phase velocity thus is zero, and the wave is stationary, as shown in Figure 12.5 b (see also Section 6.2.5).

12.3.3. Study of the form of the surface charge densities and the current at the metal 12.3.3.1. The surface charge density of zero

G G The normal components of the electric field are zero in medium (1) ( E1 // ex ) as in G medium (2) (where E C = 0 ), and the normal component of the electric field does G G not give rise to any discontinuity, so E 2n = E1n (= 0) , where Vs = 0 .

378


12.3.3.2. Form of the surface current density G 12.3.3.2.1. Equation for js As can be seen in Figure 12.4, which schematizes the results obtained for a reflected G G wave, following reflection Bi and Br remain in phase. In medium (1) the resultant magnetic field thus is given by: G G G G G 1 G G ª k i ×Ei º + ª k r ×E r º , B1 = Bi + Br = ¼ ¬ ¼ Ȧ ¬ G G G G G so that with - k r = k i = k 0 ez and -E 0r ex = Ei0 e x we have G G G e ×e B1 = z x k 0 Ei0 exp ª¬ik 0 z º¼ +exp ª¬-ik 0 z º¼ exp > -iȦt @ . Ȧ Z With k 0 = , we obtain for the physical solution for a vacuum c G G 2Ei0 G B1 = R(B1 ) = e y cos k 0 z cos Ȧt . c

The magnetic field thus is directed parallel to the interface. G G As in a metal BC = 0 (and B2t = 0 ), the tangential component of the magnetic field undergoes a discontinuity at the interface. Therefore, there must appear a G surface current density ( js ) at the interface so that the equation for continuity (end of preceding Section 11.3.1) G B1t G G G G G G G z=0 ez × js . B1t = µ0 js ×n 21 µ0 js × ez , so that z=0 P0 G G G By multiplying by vectors the right-hand side by ez and on noting that ez A js , we obtain: G B1t G 2Ei0 G z 0 G js = ×ez = ex cos Ȧt . µ0 P0 c

G G G Of note then is that js // Ei ( //ex ) and the current density is collinear to the incident electric field. Physically, the current density is the result of the polarization of the metal by the electric field.

Chapter 12. Total reflection and guided propagation

379

12.3.3.2.2. Comment As Figure 12.4 illustrates, the magnetic field vectors of the incident and reflected waves are parallel to the interface. Their resultant normal component in medium (1) G thus is zero, thus assuring the continuity of the normal component of B , as in G G medium (2) the magnetic field is zero ( BC = 0 implies that B2n = 0 ). 12.3.3.3. Applications

This property is the basis of the use of metallic antennae for detecting EM waves. With the antenna lines parallel to the direction of polarization of the incident wave, the current density detected is at a maximum and directly proportional to the intensity of the incident electric field. If the lines of the antenna are not parallel to the incident wave, then the detected current density is no longer proportional to the intensity of the component of the electric field at a parallel incidence to the antenna lines, as shown in Figure 12.6.

0 Eix

Ei0

detected current density 0 proportional to Eix

0 Eiy

Figure 12.6. Relation between the disposition of the antenna lines and the detected current.

With a metal never being a perfect conductor, in effect the current circulates over a thickness several times greater than the “skin” thickness ( G ). If the thickness (d) of the metal is such that d < G , a part of the wave in fact can be transmitted. However, if d >> G , the metal becomes a real obstacle to the propagation of the EM wave and will create an electromagnetic shield.

380


12.4. Study of Propagation Guided between Two-Plane Conductors: Extension to Propagation through a Guide with a Rectangular Cross Section 12.4.1. Wave form and equation for propagation between two conducting planes

z a y O

y

x

Figure 12.7. Propagation between two-plane conductors.

This section considers a wave propagated along Oy in a vacuum without attenuation between two-plane and perfect conductors separated by a given distance (a) and assumed to be of great dimensions (with respect to a) in directions Oy and Ox. G With the system being invariant with respect to Ox, the field ( E ) does not G depend on x and it can be assumed that E is a monochromatic wave in the form

G G E = E m (z) exp[i(k g y - Zt)] ,

(1)

where kg is a positive constant. Equation (1) is valid uniquely between the two-plane conductors. Above and G below these planes, the electric field is zero (metallic planes inside which E= 0 ). Additionally, in a plane normal to the direction of propagation, the wave is not uniform and cannot be assumed to be planar. Between the two plane conductors, dominated by a vacuum, the equation for G JJJGG 1 w ²E propagation is ǻE = 0 [from Eq. (5) of Chapter 6]. Taking Eq. (1) into c² wt² account and by making a calculation resembling that in Section 11.2.6, we have

G d²E m (z) § Ȧ² ·G +¨ - k g ² ¸ E m (z) = 0 . dz² © c² ¹

(2)


381

The tangential component of the electric field, situated in a plane parallel to Oxy, exhibits two components denoted Ex and Ey which are such that the limiting G G conditions are [with E m (0) = 0 and E m (a) = 0 ] E mx (0) = E my (0) = 0 , and E mx (a) = E my (a) = 0 . G For its part, the continuity of the normal component of B is Bmz (0) = Bmz (a) = 0 . (4)

(3)

12.4.2. Study of transverse EM waves Here the possibility of the transverse wave propagating at a velocity (c) along Oy with, on the one side vM = c (nondispersion solution as vM = constant), and on the other vM =

Ȧ kg

[equation for vM in agreement with Eq. (1)], we have k g =

Ȧ c

.

We shall now determine the corresponding exact form of the electric and magnetic fields.

G 12.4.2.1. Form of the electric field (given by E m (z) ) Equation (2) for propagation thus is reduced to

G d²E m (z) dz²

G G G integrations with respect to z give E m (z)= C1 z + C2 , (5) G G where C1 and C2 are two constant vectors.

= 0 . Two successive

The

conditions

E mx (0) = 0 and E my (0) = 0 , respectively, give: C2x = 0 and C2y = 0 and similarly, from E mx (a) = E my (a) = 0 we can deduce that C1x = 0 and C1y = 0 . G G Finally the vectors C1 and C2 only have components in the z direction, and are of G G G G the form C1 = C1z ez and C2 = C2z ez . Eq. (5) can therefore be rewritten as G G G E m (z)= C1z z + C 2z ez = E m (z) ez . (6)

G dE m (z) Gauss's equation gives then for its part div E= 0 , where =0, dz dE m (z) = C1z . From this can be deduced that C1z = 0 , such that while Eq. (6) gives dz G G G E m (z) = C2z ez = E m ez

382


G where E m is a constant. For this type of wave, E m (z) is a constant vector parallel G G to ez . The electric field, also parallel to ez , thus is in the form G G E = E m {exp[i(k g y - Zt)]} ez .

(7)

12.4.2.2. Form of the magnetic field With the wave assumed to be monochromatic, the Maxwell-Faraday equation makes G G wB G it possible to state that rotE= =- (- iȦB) , from which can be deduced directly wt JJG G G i that B = - rotE . Ȧ G From Eq. (7) for E , we can deduce that: G w G G rotE = E m exp[i(k g y-Ȧt)] ex ik g E m exp[i(k g y-Ȧt)] ex , with the result that wy

G i²k g G B=E m exp[i(k g y-Ȧt)] ex , so with k g = Ȧ G E B = m exp[i(k g y-Ȧt)] c

^

`

Ȧ c G ex .

we have (8)

We can note that this result is identical to that which would be obtained had we G G G k g ×E used the equation for plane waves as in B = , which a priori cannot be used Ȧ here directly as the wave cannot be assumed to be completely planar. Nevertheless, G G Eqs. (7) and (8) demonstrate that here again E and B are transverse, and hence the term “transverse electromagnetic wave” or TEM wave.

12.4.2.3. Comment: The rectangular cross-sectional wave guide If we envisage that in a rectangular guide, as shown in Figure 12.1 b with an axis as in Figure 12.7, there is a wave that propagates at a given speed (c), then the limiting conditions imposed on that wave are: x in one part by the parallel metallic planes with sides z = 0 and z = a , which result in the electric field taking on a form given by the equation (see also results from Section 12.4.2.1): G G E = E m {exp[i(k g y - Zt)]} ez G G where E m is a constant and E also is parallel to ez (at whatever value of x);


383

x and in another part, fixed by the second system of metallic planes parallel at the sides to x = 0 and x = b with respect to Ox (0 being on the side of the plane Oyz and b on the side of the plane to which it is parallel), impose that for x = 0 and G x = b , E= 0 so that E m = 0 . This trivial solution is not a physical one, as the possibility of propagation at the speed c along Oy is not available to a guide with a rectangular cross section.

12.4.3. Study of transverse electric waves (TE waves) Here we study the possibility of a wave propagating in the guide plane Oxy. The wave is thus polarized in accord with the breakdown given by: G G G E = E x ex + E y e y ,

(9)

G G G and has a form given by Eq. (1), as in E = E m (z) exp[i(k g y - Zt)] = E(y,z) .

12.4.3.1. Polarization direction of the electric field Under these conditions, Gauss's equation gives: G div E= 0 dE x dE y dE z = + + = ik g E y Ey = 0 dx dy dz =0 (as E x = E x (y,z) )

=0 (as E z = 0 )

G G Equation (9) thus is reduced to E = E x ex and the electric field is polarized along Ox perpendicularly to the direction of propagation Oy. The wave is termed “transverse electric” and denoted TE. The electric field can be written: G G G E = E mx (z) ex exp[i(k g y - Zt)] = E mx (z)exp[i(k g y - Zt)] . 12.4.3.2. Solutions for the propagation equation Equation (2) for propagation becomes: d²E mx (z) § Ȧ² · +¨ - k g ² ¸ E mx (z) = 0 . dz² © c² ¹

(11)

(10)

384


12.4.3.2.1. First case Ȧ § Ȧ² · If ¨ - k g ² ¸ = - D ² < 0 (where D is real), so that k g > , the solution to c © c² ¹ Eq. (11) is: E mx (z)= C1 exp(Dz) + C2 exp(-Dz) . With E mx (0) = 0 and E mx (a) = 0 , we can deduce that, respectively,

½° ¾ 2C1 shDa = 0 C1 exp(Da) + C2 exp(-Da) = 0 °¿ C1 = - C2

With shDa z 0 , we deduce that C1 = - C2 = 0 , so that E mx (z)= 0 , and there is no possible physical solution in this case. 12.4.3.2.2. Second case Ȧ § Ȧ² · If ¨ - k g ² ¸ = K² > 0 (with K being real), then k g < and the solution to Eq. c © c² ¹ (11) is given by E mx (z)= C1 exp(iKz) + C2 exp(-iKz) .

(12)

The limiting conditions, E mx (0)= 0 and E mx (a)= 0 , respectively, give:

½° ¾ 2iC1 sin ka = 0 , C1 exp(ika) + C2 exp(-ika) = 0 °¿ mʌ so that K = , with m being a whole nonzero number. a Substituting these results into Eq. (12), we gain: C1 = - C2

E mx (z)= 2 i C1 sin

mʌ a

z.

(13)

G G The electric field, according to Eq. (10), is of the form E = E mx ex exp[i(k g y - Zt)] , G mʌ G so that with Eq. (13), we find that E= 2 i C1 sin z e y exp[i(k g y - Zt)] . a By choosing the origin of the phases as being at y = 0 at a time t = 0 , then it is


385

imposed that 2 i C1 is real, as in 2 i C1 = E° , and we have

G mʌ · G § E= E° ¨ sin z ¸ ex exp[i(k g y - Zt)] . a ¹ ©

(14)

12.4.3.3. Proper vibrational modes With K =

mʌ a

, we thus have: Ȧ² c²

from which Z = c k g2 + m²

= k g2 +

m²ʌ² a²

,

(15)

ʌ²

. The equation Z = f(k g ) is not a linear relation and a² the system undergoes dispersion. ʌc When m = 1 and k g o 0 , Z o Zc = (Figure 12.8). a Sc For any value of m and and k g o 0 , Z o Zm = mZc = m . a

Z m=4

4Zc

Z

Z = ck

m=3 3Zc m=2 2Zc

Ȧc =

ʌc

m=1

a k3

k2

k1

kg

Figure 12.8. Dispersion plots for Z = f(k).

For a given value of Z, there are many possible values of k g : k1 , k 2 , k 3 . They are associated with whole values of m which are of a limited number as in the

386


second case,

Ȧ² c²

-

m²ʌ² a²

= k g ² > 0 , so that m
c , then vg < c while noting that vg vM = c² also is verified.

388


v vM

TE wave

c

TEM wave

vg

Z

Zc

Figure 12.9. Plots of vg = f(Z) and vM = g(Z).

Variations in vg and in vM as a function of Z are indicated in Figure 12.9 for m = 1 . When Z >> Zc , so that also O 0 Zc ), we obtain 2ʌ 2a for when m = 1 that T1 | 66 ° and when m = 2 , we have T2 | 37 ° . Finally, it is the angle of incidence (i') at the entrance of the guide that determines the mode of propagation (detailed further on Section 12.5.2 and Figure 12.17). 12.4.4.3. Physical representations

G ks

z a P O

x x

T G - ks

G G k 0 = ki G kG g kg

G kr

y G

Figure 12.13. Geometrical representation of the k vectors.


394

12.4.4.3.1. Wave breakdown The superposition zones of the incident wave with the reflected wave on the metallic planes give (see Figure 12.13): x with respect to the Oy axis, a progressive wave with a doubled amplitude due to the superposition of two progressive waves with the same amplitudes and both G with the same wavevectors given by k g , such that for the incident wave G G G G k g = |k i | sinT e y and for the reflected wave k g = |k r | sinT e y ; and x with respect to the Oz axis, a stationary wave that by superposition of the two progressive waves with the same amplitude but propagating in opposing senses G such that the wavevector of the incident wave is k s and for the reflected wave G is - k s . 12.4.4.3.2. Different speeds z N G a ki P

M T H

O xx

Q G kr y

N’

Figure 12.14. Definition of the points P, M, N, and H used to illustrate the various speeds.

Figure 12.14 shows an incident ray along PM which is a progressive wave such that PM = c W . We thus have: kg PM §S · = cos ¨ - ș ¸ = sinș = x , so that with (see Section 12.4.3.6) k0 NM ©2 ¹ kg = PM NM

Ȧ vM

, and k 0 =

= sinș =

c vM

Ȧ c

then:

. From this can be deduced that NM =

which in turn means that NM = vM W . x

PH PM

= sinș , from which PH = PM sinT = NM sin²T , so that

§ c PH = (vM W) ¨ ¨ vM ©

2

· c² ¸ = Ĳ = vg Ĳ . ¸ v M ¹

PM sinș

=

cĲ c/vM

,


395

The three speeds, c, vM, and vg, respectively, correspond to the following: x c is the speed of point P on the zigzag trajectory of incident and reflected waves Ȧ (k i = k r = k 0 = ) ; c x vg is the displacement speed from P to H, in other words the displacement of P along a trajectory globally parallel to Oy, such that vg < c (the resultant speed given by vg of the point P on the resultant direction thus is inferior to the speed along the instantaneous zigzag trajectory); and x vM is the speed of a point N with respect to the direction Oy. The N has no material reality and as it physically represents none of the characteristics of a wave it is not subject to relativistic physical laws. To give an idea of what it represents, it can be thought of as a “running shadow” on the line z = a of the point P following its illumination from N'. This means that vM > c simply states that the immaterial shadow is faster than the material point that moves at the speed c. 12.4.4.3.3. Representation of the resultant amplitude for the resultant progressive wave According to Eq. (14), the resultant wave is given by: G mʌ G G E= E° sin( z ) ex exp[i(k g y - Zt)] = E0x ex exp[i(kg y - Zt)] , a mʌ where E 0x = E° sin( z) . a

a

(2nd case)

z

(1st case)

z

+a/2

y

O E 0x -a/2

O m=1

m=2

m=3

m=4

Figure 12.15. Representation of the amplitudes E 0x

f (z) for two axial systems where the

origin O can be taken on the lower metallic plane or as between the two metallic planes.

396


Whatever the y point of the trajectory, the amplitude ( E 0x ) of the wave only depends on z and the excited m mode (which is fixed by the choice of the incidence angle). So, Figure 12.15 represents the form of the electric field amplitudes associated with the various possible modes, and E 0x = f(z) is represented in the plane Oxz on the y = 0 side. It is notable that the wave is polarized with respect to Ox, so for the representation that has to be in the 0xz plane, the Oy axis is directed toward the back of the figure. Comment: Role of the choice of the origins for the guide planes with respect to stationary solutions In general terms, the propagation is given by Eq. (11) so that with d²E m (z) Ȧ² - k g ²= K² and E mx = E m (as E // Ox ), then +K² E m (z) = 0 . The c² dz² general solutions, which also can be written as E m (z) = A cos(Kz) + B sin(Kz) , can be placed into one of the two following forms, depending on the how the guide planes parallel to Oxy are assumed to be placed:

z

z a

+ a/2

0 x x

O x x

scenario 1

O

y

scenario 2

y

- a/2

(a)

(b)

Figure 12.15. bis. The origin (O) placed (a) on a plane; or (b) midway between planes.

1. At z = 0 and z = a as shown in Figure 12.15bis, scenario 1, which is the same as the representation used up till now, so that the limiting conditions are thus: x E m (0) = 0 , for which A = 0 , so that E m (z) = B sin(Kz) = E 0 sin(Kz) , with B = E 0 (constant); and x E m (a) = 0 ,

for

which

so

that mʌ

, and finally E m = E 0 sin z a a = 0 , which is a nonphysical solution).

m = 1, 2, 3...., so K = that E m

mʌ

E 0 sin(Ka) = 0 ,

Ka = m S with ( m = 0 means


397

The form of the wave is thus: § mʌ · E = E m exp[i(k g y - Zt)]= E0 sin ¨ z ¸ exp[i(k g y - Zt)] ), © a ¹ which is what has been used until now.

2. At z =

-a

and z =

+a

(scenario 2 in Figure 12.15bis). The limiting conditions 2 2 thus are now: a) a a a a a E m ( ) = A cos(K ) + B sin(K ) = 0 = E m (- ) = A cos(K ) - B sin(K ; 2 2 2 2 2 2 Two solutions are possible for A and B, either: a x B = 0 and A z 0 ( A = constant denoted E 0 ) with cos(K ) = 0 , so that 2 a ʌ ʌ Mʌ K = + nS , for which K = [1 + 2n] = with odd M, and 2 2 a a Mʌ E m = E 0 cos z with odd whole M; or a a x A = 0 and B z 0 ( B = constant denoted E0 ) with cos(K ) = 0 , so that 2 a ʌ Mʌ with an even value of M. K = nS , for which K = 2n = a a 2

The form of the wave E = E m exp[i(k g y - Zt]) thus is given by either: § Mʌ · z ¸ exp[i(k g y - Zt]) with a whole odd value for M; or x E = E 0 cos ¨ © a ¹ § Mʌ · x E = E 0 sin ¨ z ¸ exp[i(k g y - Zt]) with M being whole and even. © a ¹ We can verify that the plots of Em(y) in scenario (1) where m = 1, 2, 3, 4... or for scenario (2) with the two forms of solutions for even or odd values of M, where M = 1, 2, 3, 4... ( { m ) , are identical, which is reassuring as the problem is the same! (This remark also can be made in terms of the quantum mechanics of wells with symmetrical potentials, where the origin can be taken either as at an extremity of the well where there is a potential wall, or at the center of the well.)


398

12.4.4.4. Multimodal fields

It should be noted that it is not only the field corresponding to a given mode that can be guided, meaning in other words those fields that have a transversal intensity (with respect to Oz) independent of the axial position (with respect to Oy) as shown in Figures 12.16 a and b corresponding, respectively, to the given modes (using notation of scenario 1): x complex amplitude, with m = 1 E10 = 2 Ei0 sin([k 0 cos T1 ]z) exp(i[k 0 sinT1 ]y) , so that its intensity is independent of y; and complex amplitude, with m = 2

x

E 02 = 2 Ei0 sin([k 0 cos T2 ]z) exp(i[k 0 sinT2 ]y) , so that its intensity is independent of y. The fields simply have to verify the limiting conditions (E = 0 on metallic planes) and thus can correspond to the superposition of several modes. Thus in the case given in Figure 12.16 c, taking into account the analytical forms of the fields, it is possible to see that the transversal intensity (with respect to Oz) of the wave is dependent on the axial position (with respect to Oy) in which it is placed. G In effect, the complex amplitude of the wave E1,2 , as a superposition of the modes m = 1 and m = 2 , is in the form 2 Ei0 ^ sin([k 0 cos T1 ]z) exp(i[k 0 sinT1 ]y) + sin([k 0 cos T2 ]z) exp(i[k 0 sinT2 ]y)` , and its intensity gives rise to a variation in its distribution with respect to Oy as k 0 sinT1 z k 0 sinT2 . z

z y

y (a) m = 1

z

(b) m = 2

y (c) mode m = 1 and mode m = 2

Figure 12.16. (a and b) Guided waves corresponding to a mode; and (c) without a mode.

12.4.4.5. Comment: TM modes

TM modes occur when it is the magnetic field that is applied with respect to Ox. They can be studied in the same way as TE modes and as such can be guided. The electric thus has components with respect to Oy and Ox.


399

12.5. Optical Guiding: General Principles and How Fibers Work 12.5.1. Principle Optical fibers are used, by way of a support, to guide an incident luminous wave from its point of injection at an entrance face to its exit point at the other. Typically, there are two main types of fiber: fibers with a jump in indices and fibers with a gradient in indices. This text will limit itself to detailing the first type, which are constituted of a core made from a material with a circular cross section and a given index (n) surrounded by a gain with an index denoted n1 such that n > n1 . Generally, the whole structure sits in air, which has an index given by n' | 1 . The luminous wave injected at the entrance face of the fiber is under an angle i', which is such that n' sin i' = n sin i . This wave meets the core/gain interface at an ʌ angle of incidence given by Ti = T , such that i +T = . When T >TA where TA is 2 a limiting angle above which the phenomenon of total reflection occurs when n > n1 , the wave makes no penetration of the gain material and is guided along the fiber in a zigzag trajectory associated with successive total reflections at the core/gain interface, and as illustrated in Figure 12.17. The calculations carried out below in the plane of the longitudinal section of the fiber are equally valid for the longitudinal plane section of a symmetrical rectangular guide where the Or axis is simply replaced by a Oz axis.

r

lost ray

n1 n’

O

T’ i

guided ray

T

i’

n (> n1) n1 L Figure 12.17. Trajectories when T > TA and T’ < TA .

y

400


12.5.2. Guiding conditions 12.5.2.1. Condition on i

The limiting angle TA is defined by n sinTA = n1 , wherein at the entry face an angle i has a value given by iA =

ʌ 2

- TA . A condition for the total reflection T t TA is

therefore i d iA . In addition, by moving TA = obtain cos iA =

ʌ 2

- iA into n sinTA = n1 , we directly

n1

. n This condition for wave guiding thus brings us to a condition for the incidence at the n entry face, which must be such that i d iA , so that cos i t cos iA = 1 , which n n1 . means that i d iA = Arc cos n All rays with i ! iA have at the core/gain interface an incidence such that T' < TA . A fraction of the wave is transmitted toward the gain, and following several successive reflections, the remaining fraction of the wave in the core becomes smaller and smaller so finally it is extinguished along its journey. In numerical terms, with n = 1.50 and n1 = 1.49 , we obtain iA = Arc cos

n1 n

= 6.62 ° .

When n = 1.60 and n1 = 1.50 , we have iA = Arc cos

n1 n

= 20.36 ° .

12.5.2.2. Condition on i’ (acceptance angle): numerical aperture

The condition T t TA makes it possible to state that: cosT d cosTA = ʌ

1 sin ²șA =

1

n12 n2

.

-T , from which with n' sin i' = n sin i 2 ʌ n n' sin i' = n sin( - T) = n cosT , so that sin i' = cosT and the preceding equation 2 n' therefore leads to

We also have i =


sin i' =

n n'

cosT d

n n'

cosTA

n n'

1

n12

n § n² - n12 ¨ n' ¨© n 2

n2

1/2

n - n 2

2 1

n'

, so that:

1/2

n - n sin i' d 2

1/ 2

· ¸ ¸ ¹

401

2 1

.

n'

To conclude, all rays introduced under an angle of incidence i' verify the preceding inequality and therefore can propagate. The quantity given by

NA = n 2 - n12

1/2

is the numerical aperture of the system, which although here is an optical fiber, also can be applied to a guide structure. The acceptance angle is by definition: 2

i'A

1/2

n - n = Arc sin 2 1

n'

.

In numerical terms n = 1.50 - n1 = 1.49 - n' = 1, so the NA = 0.173, so that i'A = 9.96 ° . When n = 1.6 - n1 = 1.5 - n' = 1, and the NA = 0.557, so that i'A = 33.83 ° .

12.5.3. Increasing the signals Once a luminous impulsion is injected into a fiber, various pathways can be followed, as in: x the shortest trajectory which is along the pathway Oy and corresponds exactly to c the length (L) of the fiber. With the speed of propagation being given by vM = , n nL ; the pathway time (ty) is t y = c x zigzag trajectories which each have a given value for the angle i and have a L > L . With the pathway speed still being pathway length (Li) given by Li = cos i c L nL vM = , the pathway times (ti) are now given by t i = i = > t y (ti is n vM c cos i great when cos i is small, so that i is great).

402


The difference in propagation times thus is given by: i² · § 11 ¸ nL§ 1 · nL¨ 2 ¸ | n L i² . ¨ 't i = t i - t y = 1¸ | ¨ i² c © cos i c ¨ 1 ¸ 2c ¹ ¨ ¸ 2 ¹ © In numerical terms, when L = 1 km, n = 1.5 and i = 6 ° < iA , so that 't i = 28 ns. Therefore, an impulsion emitted at y = 0 at a time t 0 = 0 over an interval of time assumed to be Gt | 0 will be emitted at y = L starting from the moment t 0 + t y = t y (for the wave that takes the shortest pathway) up to the moment t 0 + (t i )max = (t i )max where (t i ) max is the pathway time for the longest possible trajectory. This latter value comes from the highest possible value of i, i.e., i = iA . The corresponding interval in time therefore is given by 't iA =

nL 2 iA where 't iA 2c

is the duration time of the emission at the exit. The interval in time (T) between two impulsions at the entrance which is required so that they do not mix at the exit therefore must be such that T t 't iA , which is the same as stating that the frequency of the impulsions must be lower than 1 1 Q iA = (we can state that we should have T = t 't iA so that 't iA Q

Q

1

't i A

=QiA . and in terms of actual numbers, this means that for the

aforementioned values, QiA | 30 MHz ).

12.6. Electromagnetic Characteristics of a Symmetrical Monomodal Guide Here we look at a buried guide, which for technological reasons is simplified to the symmetrical structure shown in Figure 12.18. - a/2

y

+ a/2

z

n

G E x

Figure 12.18. Buried guide.

n1


403

The propagation of an electromagnetic wave with an angular frequency given by Z is along Oy and is polarized with respect to Ox. The field is reflected at the interface, between a medium with an index given by n and another medium with an index denoted by n1, along planes parallel to Oxy with the EM wave alternating its -a +a and z = . This configuration has the reflection at the planes defined by z = 2 2 same geometry as that given in Section 12.4, and as that presented in Figure 12.19a, making direct comparisons possible between systems that only differ by virtue of their planes of interface. The metallic planes, which are imposed as a limiting G condition where E= 0 , are now replaced by dielectric diopters with the aforementioned index n1 where n1 < n (and as a consequence, with different limiting conditions). In addition, the two dielectrics are assumed to be nonabsorbent. In the zones (1), (2), and (3) the electric field can be written by notation as G 1 G 2 G 3 E , E , and E , respectively, and in a manner analogous to that of Section 12.4, taking into account the symmetry of the problem, the solutions for the TE wave polarized with respect to Ox are looked for in each of the three zones in the form: G p p G E =E mx (z) ex exp[j(k yp y - Ȧt)] where p = 1, 2, 3 characterizes the zone under consideration. G 3 zone (3), index n1 < n, and field E

z +a/2 O x x

T i

G 2 zone (2), index > n1, and field E

y

- a/2 G 1 zone (1), index n1 < n, and field E Figure 12.19(a). Propagation in a symmetrical dielectric guide.

12.6.1. General form of the solutions The equations for continuity of the tangential component for the different field G 1 G 2 G 3 E , E , and E directed exactly along Ox, true for all moments t and whatever values of y, lead to k y1 = k y2 = k y3 = k y [see also Section 11.2.3, concerning Eq. (7) or (7’) with respect to Oy].

404


With the media being nonabsorbent, H rp = n p ² , we then have: G p G JJG G p n p2 w E p wE rot B = µ0 İ 0 İ rp = . By taking the rotational of the two terms c² wt wt G p JJG G p wB , and by using the preceding equation, we obtain of the equation rot E = wt an equation for propagation in a medium with an index denoted np [see Chapter 7, Eq. (5) this volume, and with µr = 1 ] G p JJJGG p n 2p w ²E ǻE =0. c² wt² With the wave being monochromatic, we also can state, using Eq. (7") again from Chapter 7, G p JJJGG p Ȧ2 2 w ²E ǻE + np = 0. wt² c² Taking into account the form of the required solution, this equation gives [using a calculation analogous to that yielding Eq. (11) of Section 12.4.3.2]:

p

d²E mx (z) § Ȧ² 2 · p + ¨ n p - k y ² ¸ E mx (z) = 0 . dz² © c² ¹

§ Ȧ² · n 2p - k y ² , and as ¨ n 2p - k y ² ¸ can be, a priori, positive or c² c² © ¹ negative, and Į p can be real or purely imaginary, under these conditions the Making Į 2p =

Ȧ²

solutions are in the general form:

p

E mx (z)= A exp(jD p z) + B exp(-jD p z) .

12.6.2. Solutions for zone (2) with an index denoted by n kz { D G k i

T

ky { E G

Figure 12.19 (b) Geometrical component of k .


In this zone, the wavenumber is given by k =

nȦ c

405

and its projection along the axis

of propagation Oy is k y = k sinT = k cos i { E (by notation) which is a constant bound to the propagation phase. Ȧ² 2 n - k y ² = k² - k² cos²i = k²(1 - cos²i) = k²sin²i , so The result is that Į 2 ² = c² that D 2 = ± k sin i . The sign is not important with respect to the general solution given by E (2) mx (z) , which contains both signs as exponentials, so by notation we use

D 2 = k sin i = k cosT = k z { D . This magnitude is finally tied to the amplitude of a stationary wave due to reflections at the interfaces (see the metallic guide in Sections 12.4.2 and 12.4.3). Therefore, in the zone (2) we have, with D = k sin i ,

2

E mx (z)= D1 exp(jD z) + D2 exp(-jD z) where D1 and D2 are the constants to be determined for the limiting conditions. In fact, given the symmetry of the problem, we should have D1 = ± D2 , so that D1 = D and D2 = HD with H = ±1 . In effect, given this symmetry of the guide, with respect to the z axis and for two points symmetrical placed about O located by z = z 2 and z = - z 2 , the intensity of the wave should be the same (in quantum mechanics, the probabilities of presence would be stated to be equal). Thus, | D1 exp(jD z 2 ) + D 2 exp(-jD z 2 )|² = | D1 exp(-jD z 2 ) + D2 exp(jD z2 )|² , so that | D1 + D2 exp(-2jD z 2 )|² = | D1 exp(-2jD z 2 ) + D2 |², for which |D1|² = |D2 |² , and D1 = ± D2 . Finally, G 2 G E = > D exp(j Į z) + İ D exp(-j Į z) @ ex exp[j(k y y-Ȧt)] .

12.6.3. Solutions for the zones (1) and (3) In the titled zones [zones (1) and (3)], we have D1 ² = D3 ² = n12 Ȧ²

n12 n 2 Ȧ²

n12

c²

n²c²

n²

n12 Ȧ² c²

- k y ² , where

k² , so that k y = k cos i is always true. The result is that


406

§ n2 · D1 ² = D3 ² =k² ¨ 1 cos ²i ¸ , ¨ n² ¸ © ¹

and as: x x

n12

< 1, n² and on the other, in order to guide the incident ray (i) to the entry face shown in Figure 12.19a, we should have i d iA as a result of Section 12.5.2.1 which uses the same geometry, so that n cos i t cos iA = 1 n 2 §n · We have ¨ 1 cos ²i ¸ < 0 , so that we can make: ¨ n² ¸ © ¹ on the one hand, n > n1 ,

§ n2 · D1 ² = D3 ² =k² ¨ 1 cos ²i ¸ = j² J ² < 0 , for which D1 = D3 = jJ, where: ¨ n² ¸ © ¹

n2 J = k cos²i - 1 > 0 . n²

1

In zone (1) we thus find that E mx (z)=C1 exp(-j D1 z) + C3 exp(j D1 z) , and in this zone where z < 0 , so that the wave is evanescent (corresponding to the guiding condition as in T > TA or i < iA ), we should find C3 = 0 [If not, the component G 1 C3 exp(j D1 z) yields for E a contribution of the form C3 exp(j² J z) = C3 exp(-J z) , which will diverge when z o -f to give the amplification term rather than the term for attenuation as required]. In zone (1), the solution thus is in the form E1mx (z)= C1 exp(-j D1 z) , so that:

1

E mx (z)= C1 exp(J z) . In the zone (1), the final expression for the wave is: G 1 G E = C1 exp(Ȗ z) ex exp[j(k y y-Ȧt)] . This wave is evanescent with respect to Oz and progressive with respect to Oy. It is


407

in effect an inhomogeneous MPP wave that has an amplitude that is dissimilar at different points along the plane of the wave defined by y = constant. Similarly, in zone (3), we have:

3

E mx (z)= C1 exp(-j D3 z) + C3 exp(j D3 z) and in this zone where z > 0, the physically acceptable solution (yielding an attenuation and not an amplification) is E3mx (z)= C3 exp(j D3 z) , so that:

3

E mx (z)= C3 exp(- J z) . In zone (3) then, the final equation for the wave is given as G 3 G E = C3 exp(-Ȗ z) ex exp[j(k y y-Ȧt)] which also is an evanescent wave with respect to Oz and a progressive wave with respect to Oy. It is worth now looking at the relationship between C1 and C3 due to the symmetry of the guide. With the problem being one of symmetry around O with respect to the z axis, we should have the same wave intensity at two symmetrical points, one located with z = z1 and in zone (1), and the other located with z = z3 = - z1 and in zone (3). So, |C1 exp(J z1 )|²

½° C1 = ± C3 = {|C3 exp(-J z3 )|²}z3 = -z1 = |C3 exp(J z1 )|² ¾°¿

By making C1 = C and C3 = HC with H = ± 1 , then we will have as solutions for zone (1): G 1 G 1 E mx (z)= C exp(J z) , and E = C exp(Ȗ z) ex exp[j(k y y-Ȧt)] .

On zone (3), the solutions will be the same: G 3 G 3 E mx (z)= H C exp(- J z), and E = İ C exp(- Ȗ z) ex exp[j(k y y-Ȧt)] .


408

12.6.4. Equations for the magnetic field Zone (1) G JJG G (1) wB(1) , we have : From Maxwell-Faraday law, which states rot E = wt JJG G (1) rot E =

0 G (1) °° G (1) wB (1) (1) and = jZB Ex / z = E w w J ® wt ° (1) (1) °¯- wEx / wy=- jk y E

0 G (1) 1 °° B = ®-j J E (1) Z° (1) °¯- k y E

Zone (3) 0 JJG G (3) °° rot E = ®wEx (3) / wz =- J E (1) ° (1) (3) °¯- wEx / wy=- jk y E

and -

G (3) wB wt

G (3) = jZB

0 G (3) 1 °° B = ® j J E (3) Z° (3) °¯- k y E

Zone (2) 0 ° ° (2) ®wEx / wz =- jD > D exp(j Į z) - İ D exp(-j Į z) @ exp[j(k y y-Ȧt)] and ° (2) (2) °¯- wEx / wy=- jk y E 0 G (2) G (2) 1 °° = jZB B = ®D > D exp(j Į z) - İ D exp(-j Į z) @ exp[j(k y y-Ȧt)] Z° (2) °¯- k y E

JJG G (2) rot E =

-

G (2) wB wt

12.6.5. Use of the limiting conditions: determination of constants §Ȗa· § Įa· Here we make G = exp ¨ ¸. ¸ and µ = exp ¨ j © 2 ¹ © 2 ¹ The continuity of the tangential components of E at z =

a 2

are written:


409

a a E (3) ( ) = E (2) ( ) , 2 2 from which a a a H C exp(-J ) = D exp(jD ) + H D exp(-jD ) , 2 2 2 so that HC H = D( µ + ). (1) G µ With the media being nonmagnetic, we can equally write that the continuity of a are such that we have the tangential components for B at z = 2 a a By (3) ( ) = B y(2) ( ) , from which 2 2 jJHC exp(-J a/2) = D > D exp(j Į a/2) - İ D exp(-j Į a/2)@ , so that: j J H C/G = D [µ - H/µ] D .

(2)

We thus obtain two linearly independent equations that allow a determination of the unknowns C and D. The other equations of continuity give for their part equations leading to the same relations as Eqs. (1) or (2). If the reasoning permitted by the effect of symmetry had not been possible, then there would have been four constants, namely C1,C2, D1, and D2, which these additional equations would have permitted to determine. a For example, at z =- , we have: 2 a a E (1) (- ) = E (2) (- ) , 2 2 for which a a a C exp(-J ) = D exp(-jD ) + H D exp(jD ) , 2 2 2 so that C D = + HµD . G µ By multiplying by H we find Eq. (1).

410


12.6.6. Modal equation 12.6.6.1. Placing into equations First, we write the Eqs. (1) and (2) for H = +1 , as in: C/G = ( µ + 1/µ) D

(1’)

j J C/G = D [µ - 1/µ] D

(2’)

the ratio (2’)/(1’) gives

jĮa

jJ=D

µ² - 1 µ² + 1

, so that

Ȗ

2

1e

Į

j

jĮa

e 2

jĮa

tan

jĮa

Įa 2

, and therefore

e 2 e 2 tan

Įa

=

2

Ȗ Į

.

Similarly, when H = -1 , we obtain: - C/G = ( µ - 1/µ) D (1’’) - j J C/G = D [µ + 1/µ] D

(3)

the ratio (2’’)/(1’’) gives

(2’’) jĮa

jJ=D

µ² + 1 µ² - 1

, for which

Ȗ

j

Į

e

2

jĮa

e 2

jĮa

jĮa

e 2 e 2 Įa

tan

Equation (3) gives Į a 2

1 x

=

ʌ 2

Į Ȗ

.

(4)

Į a

§Ȗ· = Arctan ¨ ¸ + rS , where r is a whole number t 0 as 2 ©Į¹

> 0 . So, we have

Arctan x + Arctan

2

=-

-1 , so that Įa t an 2

,

Da 2

=

§Į· - Arctan ¨¨ ¸¸ + r S , by using the equation 2 ©Ȗ¹

ʌ

§ 1 · ʌ 1 ʌ ( tan( - y) = gives - y = Arctan ¨¨ ¸¸ by 2 tan y 2 © tan y ¹

making x = tan y ). We deduce that: §Į· ʌ + Arctan ¨¨ ¸¸ = (2r + 1) 2 2 ©Ȗ¹

Į a

[ r is an integer t 0 ] .

(5)


411

From Eq. (4) we pull out: §Į· Į a Į a > 0 , we thus have = - Arctan ¨¨ ¸¸ + s S , and with s integer > 0 as 2 2 ©Ȗ¹ § Į · 2 s ʌ . + Arctan ¨¨ ¸¸ = 2 2 ©Ȗ¹

Į a

(6)

The solutions from Eqs. (5) and (6) can be regrouped into a single expression, as in: §Į· qʌ + Arctan ¨¨ ¸¸ = , 2 2 ©Ȗ¹

Į a

(7)

where q is an integer and q t 1 as r t 0 and s > 0.

12.6.6.2. General equation and solutions n2 n In addition, we made D = k sin i , and J = k cos²i - 1 , with 1 = cos iA . The n² n left-hand side of Eq. (7) is by consequence a function of the angle i, and Eq. (7) can be rewritten: a f(i) = k sin i + Arctan 2

sin i cos²i - cos²iA

q

ʌ 2

.

(8)

Equation (8) is a modal equation that permits a determination of q when i [0, iA ] . For this interval, sin i and

1

are strictly increasing functions of i, just

cos²i - cos²i A like the arc tangent function. Indeed, f(i) is also a strictly incremental function, so that for given values of a and q, there is a single value of i as a solution to f(i). The maximum value of the function f(i) is obtained for the maximum acceptable value of i, that is i = iA . For this value, we therefore have: a a ʌ ʌ [f(i)]max = k sin iA + arc tanf = k sin iA + = q . 2 2 2 2 The solution q = 1 therefore is always obtained, even when a o 0 , as it suffices to take i o iA as the arc tangent function tends toward S/2.

412


12.6.6.3. Monomodal solution To remain in a monomodal regime where q = 1, the solution q = 2 must remain unattainable. For this to be true [f(i)]max < S must be imposed, so that a ʌ ʌ nȦ 2ʌ . With k = k sin iA < , which means that a < a1 = = n k0 = n 2 2 k sin iA c Ȝ0 and sin iA =

n2 1 cos ²iA = 1 1 , we obtain: n2 a1 =

Ȝ0 2 n 2 - n12

.

In numerical terms, with O 0 = 1.3 µm (a typical telecommunications wavelength), n = 1.50 and n1 = 1.49 . For the monomodal condition, where q = 1 uniquely, the guide must have a width less than a1 = 3.76 µm. When O 0 = 1.55 µm, n = 1.50 and n1 = 1, we obtain a1 = 0.7 µm .

12.6.6.4. Multimodal solutions When a > a1 , the modal Eq. (8) in effect permits several different values of q as solutions. In order to find the general solution to Eq. (8), we can write that with 2ʌ sin²iA - sin²i , we have k= and cos²i - cos²iA Ȝ a sin i ʌ ʌ sin i + Arctan q , Ȝ 2 sin²iA - sin²i so that on applying Arctan x + Arctan

1

=

ʌ

, we find that x 2 sin²iA - sin²i a ʌ ʌ sin i - q-1 = Arctan . Ȝ 2 sin i By setting m = (q - 1) and as q t 1 , m therefore is an integer such that m = 0, 1, 2, 3.... , and so on using the fact that x = Arctan y , we have y = tan x , and 1/2

ʌ · § sin²iA § ʌa · tan ¨ sin i - m ¸ = ¨ - 1¸ 2 ¹ © sin²i © Ȝ ¹

.

(9)

The two terms in this equation are functions of the variable sin i, and the solutions can be found at the intersection of the plots representing:


413

ʌ· § ʌa x the function g(sin i) = tan ¨ sin i - m ¸ ; and 2¹ © Ȝ 1/2

§ sin²iA · - 1¸ x the function h(sin i) = ¨ © sin²i ¹

.

In reality, the function g(sin i) can be more simply seen as the function given by: § ʌa · 1. g1 (sin i) = - cotan ¨ sin i ¸ when m is odd (as © Ȝ ¹ ʌº ª ªʌ º tan « x - » = - tan « - x » = - cotan x ); and 2¼ ¬ ¬2 ¼ ʌa § · 2. g 2 (sin i) = tan ¨ sin i ¸ when m is even as tan > x ± ʌ @ = tanx . Ȝ © ¹ The resolution of Eq. (9) thus leads to plotting, as a function of sin i, the two systems given by: ʌa ʌ x the plots for 2 defined for when 0 < sin i< when m = 0 , Ȝ 2 ʌa 3ʌ S < sin i< when m = 2 , etc..., so that Ȝ 2 Ȝ Ȝ 3Ȝ 0 < sin i < ( m = 0 ), < sin i < ( m = 2 ), and so on; and 2a a 2a ʌ ʌa x the plots for 1 defined for when < sin i< S ( m = 1 ), 2 Ȝ 3ʌ ʌa < sin i< 2S ( m = 3 ), etc..., so that 2 Ȝ Ȝ Ȝ 3Ȝ 2Ȝ < sin i < ( m = 1 ), < sin i < ( m = 3 ), and so on. 2a a 2a a The solutions for sin i in each interval precisely defined by the value of m Ȝ for the solution), thus can be found as shown in (hence the notation sin i m = m 2a 1/2

§ sin²iA · - 1¸ , which is a Figure 12.20 at the intersection with the plot of h(sin i) = ¨ © sin²i ¹ function that decreases monotonically with sin i. When i = iA , this function cancels out so that h(sin iA ) = 0 .

414


Figure 12.20. Graphical solutions for the modal equations obtained by the intersection of the plots h(sin i) and g(sin i).

For given values of a, O, n, and n1, for each value of m there is a corresponding value of I denoted Im and between 0 and iA . Associated with Im is the wavevector with a component nk0 cos Im with respect to the axis of propagation along Oy. Em = k y = nk 0 cos i m is the propagation constant. With respect to the axis Oz, the component of the wavevector associated with Im is for its part given by k z = nk 0 sin i m { D m . kz nk0 n 1k 0 n k0 sin il

Q m = 3 (q = 4)

n k0 sin i3

m = 2 (q = 3) m = 1 (q = 2) i3

m = 0 (q = 1)

il nk0 n k0 cos il = n1k0 n k0 cos i3

E = ky = n k0 cos i

Figure 12.21. Determination of Im angles corresponding to m modes.


415

n1

, Em varies between nk 0 and n1k 0 . n The im angles corresponding to the m modes, as well as the corresponding components given by k y { Em and k z { D m , are shown in Figure 12.21. As cos i m varies between 1 and cos iA =

12.6.6.5. Number of nodes From Figure 12.20, each interval associated with a mode has a width given by O/2a, which means that for a width defined by L as a function of sin i such that L = sin iA , the number of possible modes (M) is equal to the first integer immediately above the sin iA sin iA . By convention, we can write that M . number given by the ratio Ȝ/2a Ȝ/2a 12.6.6.6. Cutoff frequency Returning to Eq. (8) and the reasoning used in Sections 12.6.6.2 and 12.6.6.3, for a ʌ ʌ mode given by q (= m + 1) to not be affected, then [f(i)]max < q = (m + 1) 2 2 a ʌ 2ʌ must be true, so that k sin iA < (m + 1) , and from which with k = n and 2 2 Ȝ0 n1 n

= cos iA =

Ȝ 0m > Ȝ 0mc =

1 - sin²iA included we can deduce that sin iA = c (m + 1)

2a n 2 - n12

n 2 n12 n

and

.

12.6.7. Comments: alternative methodologies 12.6.7.1. Comment 1: by analogy to solutions for potential wells The type of mode, whether even or odd, can be apprehended directly through Eqs. (3) and (4) which are identical to those found in a resolution of potential wells with the given width as a. Į a Ȗ a and Y = ; Considering Eq. (3), and by making X = 2 2 Eq. (3) can be written as Y = X tan X Similarly, Eq. (4) gives Y = - X cotan X . This then yields:

416


X²+Y² =

a² 4

a² ª n2 º « k² sin²i + k² cos²i - k² 1 » 4 «¬ n² »¼ § n 2 · a² a² = k² ¨ 1 - 1 ¸ = k 02 (n² - n12 ) . ¨ ¸ n² ¹ 4 4 ©

>Į² + Ȗ² @ =

For a given guide, with indices n and n1 and of dimension a, it is possible to state that X²+Y²= R² is a constant characteristic of the guide. The solutions for X and Y can be found at the intersection of the plots of the preceding Y = f(X) with a circle of radius R. From these solutions D can be determined, thus i, and then, as will be detailed in Section 12.6.8, the distribution of the field that varies with cos Dz (for even modes corresponding to the solutions to the equation Y = X.tan X ) or with sin Dz (for odd modes corresponding to solutions to the equation Y = - X cotan X ). 12.6.7.2. Comment 2: methodology from principles of optics

We have assumed up till now that for a wave to be guided that (1) it has to undergo a reflection at an incidence such that T > TA at an interface between two dielectrics that make up the guide; and (2) the equation for the propagation should be verified using the limiting conditions of the problem. It is by this route that we have selected solutions corresponding to the modes. For a given mode, the solutions are such that the transversal intensity of the wave is independent of the position in Oy of the resultant direction of propagation. The second condition can be directly replaced due to the fact that for the wave to propagate it must be part of the same system of plane waves, meaning that after several reflections the planes of the waves orthogonal to the direction of propagation are conserved. This is the same as saying that the dephasing between the wave that propagates along AB (subject to dephasing of MA and MB on reflection at A and B) and the wave that propagates directly from A to C is a multiple of 2S, as shown in Figure 12.22. Thus: 2ʌ k.AB - k.AC + MA + MB = [AB - AC]+ 2 MA = 2S m . (10) Ȝ In effect, the two interfaces are identical (the same dielectrics with indices n and n1) and the wave is polarized perpendicularly to the plane of the figure so that MA = MB = MA , and therefore we either have MA + MB =2MA , or MA as determined in Section 11.3.2.2 with Eq. (37). ʌ This with (i + T) = , gives: 2

§M tan ¨ A © 2

1/2

sin²ș - sin²șA · ¸= cosș ¹

1/ 2

§ sin ²iA · 1¸ = -¨ sin ²i © ¹

.

(11)


C

dielectric with index = n1

A

417

A’

i i

T

a

O

dielectric

dielectric with index = n1

with

index = n

B

Figure 12.22. Conservation of wave planes in guided propagation.

With [AB - AC] = 2 a sin i

(classic dephasing calculated, for example, in

establishing Braggs law, so that here AB = AB - AC = 2ʌ Ȝ

a sin i

a sin i

and AC = AB cos 2i , to give

[1 -cos 2i] = 2a sin i ), then Eq. (10) gives:

2 a sin i + 2 MA = 2S m , which in turn yields: ʌ · MA § ʌa . ¨ sin i - m ¸ = 2¹ 2 © Ȝ

By taking the tangent of this equation into which is also plugged in Eq. (11), we obtain the preceding Eq. (9), as in: 1/2

ʌ · § sin²iA · § ʌa tan ¨ sin i - m ¸ = ¨ - 1¸ 2 ¹ © sin²i ¹ © Ȝ

.

12.6.8. Field distribution and solution parity It is worth looking at the distribution of the electric field in each of the zones (1), (2), and (3) for the possible values of q.

418


q=1 m=0

z + a/2

q=2 m=1

q=3 m=2

q=4 m=3

y

- a/2

Figure 12.23. Distribution of the electric field for various m modes.

12.6.8.1. In zone (2)

The form of the field established from Section 12.6.2 is such that E 2mx (z)= ª¬ D exp(jĮ q z) + İ D exp(-jĮq z) º¼ . 12.6.8.1.1. Even solutions (as cosines) When H = +1 , we have:

2 E mx (z)= ª¬ D exp(jĮq z) + D exp(-jĮq z) º¼ v cos Įq z = cos [(k sin iq )z] .

In addition, H = + 1 corresponds to Eq. (3) which leads to q = 2r+1 [Eq. (5)], with r an integer t 0. The r = 0 gives q = 1 so that m = 0; r = 1 gives q = 3 so that m = 2, while r = 2 gives q = 5 so that m = 4, and so on (see also Figure 12.21 and 12.23). 12.6.8.1.2. Odd solutions (as sins) When H = -1, we have:

2

E mx (z)

ª D exp(jĮ q z) - D exp(-jĮq z) º v sin Į q z = sin [(k sin iq )z] . ¬ ¼

Here, H = - 1 corresponds to Eq. (4) which leads to q = 2s (Eq. (6)), with s being an integer > 0, so that for s = 1, we have q = 2 and m = 1, and when s = 2 we have q = 4 and m = 3, and so on (see also Figures 12.21 and 12.23).


419

12.6.8.2. In zone (1)

The form of the field established in Section 12.6.3 is such that when z < - a/2 ,

1

E mx (z)= C exp(J z) , and with J as the extinction coefficient, we can state that: n2 J = k cos²i - 1 = k 0 n cos²i - cos²iA n² = k 0 n cos iA

cos²i cos²iA

- 1= k 0 n1

cos²i cos²iA

-1

12.6.8.3. In zone (3)

The form of the field established in Section 12.6.3 is such that when z > a/2 ,

3

E mx (z)= H C exp(-J z) and J takes on the same form as above. 12.6.8.4. The Goos–Hänchen effect

We can note that in the presence of a dielectric (with an index = n1) in place of the plane conductors, the electric field no longer gives rise to nodes (E = 0) at z = ± a/2 (see Figure 12.23) and the penetration to a depth of the order of G | 1/J of the ray undergoing a total reflection is called the Goos–Hänchen effect.

12.6.9. Guide characteristics 12.6.9.1. Form of the signal leaving the guide If a >> O , numerous values are possible for q, which results in discreet but close values for the injection angle (iq). The group of modes allows the guide to function in multimode. To each value of iq there is a different path length in the guide along with an associated time (tq), and under multimodal regimes, there is a temporal increase of an impulsion leaving a wave with respect to that at the entrance, in what is an effect termed “intermodal dispersion” (Figure 12.24).

420


I

I all modes

all modes

t mode 1 mode 2

t

mode 1

mode 2

mode 3

t

at entrance to guide

mode 3

t at a given time (t) the exiting signal is weaker than the starting signal

Figure 12.24. Comparing guide entrance and exit signals.

In effect, for a given value of q, i is dependent on Z [as k =

nZ c

intervenes in

the modal Eq. (8)], and E = k cos i thus is dependent on Z just as is vg =

dZ

. dE The pathway time also is dependent on Z (and hence also the wavelength), and hence the slight increase in the impulsions with each mode. This is termed the “intramodal effect”, which is superimposed on that of dispersion due to a variation of the index with wavelength. The result is a chromatic dispersion. 12.6.9.2. Nature of losses in a guide

Losses in a guide have various origins and can be organized into two main classes. They are, first, characterized by a attenuation coefficient, or second, an attenuation factor. 12.6.9.2.1. Attenuation coefficient and the linear attenuation factor The transmitted power (P) for a penetrated length (L) varies in accordance with the equation P(L) = P(0) exp(- 2J L) . P(0) is the power introduced at the entrance point, and J here represents the linear attenuation coefficient for the amplitude.


421

For its part, the linear attenuation factor (D) is defined with the help of the expression: P(L) = P(0)10-ĮL/10 , so that D =

10 L

log10

P0 P

where D is expressed in decibels per meter (dB m-1). As an example, a fiber with 0.22 dB km-1 transmits around 95 % of the energy over 1 km. 12.6.9.2.2. Losses due to the physical configuration of the guide These losses can include: x losses due to a defective guiding (in principle, these losses do not count for guided modes); x losses due to curves (unnatural losses due to deformations of the fiber) for example, when the radius of curvature is less than a centimeter, the higher modes can be refracted within the gain; x losses at joints, which in turn can be divided into two groups: 1. losses due to poor alignment of adjacent components; 2. Fresnel losses, associated with the reflection of the injection signal at the entrance face (as in Figure 12.25). Under normal incidence, using classic 2

experimental

conditions,

we

have

ª n - n1 º R= « » , ¬« n + n1 ¼»

so

that

with

n = 1.5 and n1 = 1, we have R | 0.04 = 4 % . Thus the losses by Fresnel reflections are given by KFresnel = 10 log10 (1-R) = - 0.18 dB .

injection guide with index = n medium with index = n1 reflection Figure 12.25. Fresnel losses.

12.6.9.2.3. Losses due to the material making up the guide Losses due to the properties of the guide material can be divided into two physical phenomena, absorption and diffusion. First, the losses due to absorption (which also are further detailed in Chapter 10 of this volume, or Chapter 3 of the second volume entitled Applied Electromagnetism and Materials) which can come about:

422


x in the visible domain (zone 3) due to electronic absorption; x in the infrared domain (zone 4) due to network vibrations (vibrations of atoms or ions depending on the nature of the guide and/or fiber). In polymer based materials, the wavelength for the fundamental absorption by the C-H group is approximately 3.3 µm and over distances of any consequence, the window of transparency is in fact limited to 0.8 µm due to absorptions associated with harmonics. This phenomenon resembles absorptions due to hydroxyl bonds (-O-H) that exhibit a transparency window up to 2 µm, which includes in particular two transmission windows at 1.3 and 1.55 µm used in optical telecommunications. The use of fluorinated polymers permits an extension of this region of transparency; x because of a wide number of impurities that present absorption characteristics, and hence the necessity of using highly purified materials. For silicon-based systems, absorptions due to impurities including hydroxyl groups are situated around the above-mentioned windows at 1.3 and 1.55 µm. The improvement of manufacturing techniques has meant that the presence of water has been reduced to less than 1 part per 107 and accordingly the performances of silicon-based fibers have increased considerably to less than 0.2 dB km-1 at 1.55 µm. The use of fluorinated glasses can extend the transparency window to 5 µm, and although the absorption bands are always present, they are well shifted to longer wavelengths. D (dB km-1) peaks due to OH- ions

10

infrared absorption (network vibrations)

1 Rayleigh diffusion increases at shorter O

0.1

0.5

1

1.5

2

O(µm)

Figure 12.26. Plot of D = f(O) showing the origin of losses in a silicon based fiber.


423

Second, losses due to diffusion that can be further separated into two groups: x extrinsic diffusion which itself can be due to two possible causes, the first being O tied to the possible inclusion of dust (of a size greater than ) and the second 20 being associated with the presence of microcrystallites. It is evident therefore that there is an interest in using extremely clean and amorphous materials. x intrinsic diffusion due to Rayleigh diffusion (see also Chapter 10) which is caused 1 by interactions of light with materials. It varies with respect to so that the O4 effect becomes more important at shorter wavelengths. Thus, at very short wavelengths, there is a deviation in the plot of D = f(O ) due to an attenuation of the system (see Figure 12.26 for silicon fiber optics).

12.7. Problems Monomodal conditions By way of recall, the total reflection at the interface between two dielectric media brings into consideration two different and real indices denoted here as n and n1 and such that n > n1 . We thus have n1 = n - 'n . x

+a

i0 n0

n1 = n - 'n

n guided wave O i T -a n1 evanescent wave z

y

Gc

The figure shows the configuration of injection and propagation of light in a medium with an index denoted as n. When T > TA , where TA represents the limiting angle (and iA is the corresponding angle on the inside of the injection surface), G electromagnetic theory tells us that the transmitted wave ( E t ) in the medium with index n1 and for the zone defined by z > a/2 , is in the form:


424

G JJG E t =E 0t exp(- Jz) exp(i[k y y - Zt]), avec J = k 0 n1 (cos²i/cos²iA ) - 1]1/2 .

1. Show that the intensity of the wave in the medium with index = n1 can be written § 2z · in the form I t (z) = I t (0) exp ¨ ¸ . Give an equation for G, which then can be used © G ¹ to give an approximate value, assuming that T is sufficiently close to S/2 to state that sin²T | 1 . 2. Determine the condition for 'n that permits a weak penetration of a wave (evanescent) into the medium with an index = n1. The condition for weak penetration is fixed using the condition G d Gc | 3 O1 (wavelength in a medium with an index = n1). In numerical terms, determine the condition on 'n when n | n1 | 1.5 . 3. Monomodal condition. In this chapter it is shown that: a < a1 =

Ȝ0 2 n 2 -n12

. With

'n being very small, which can be verified later on, and with the same numerical characteristics as given before in that O 0 = 1.3 µm and a = 5 µm , give the numerical condition that 'n must verify so that the guide is monomodal. Conclude.

Answers JG 1. We have I t (z) = E t

2

JJJJG 2 § 2z · = E 0t exp(- 2 Jz) = I t (0) exp ¨ ¸ with © G ¹ 1/ 2

§ cos²i · G= = 1¸ . ¨¨ ¸ Ȗ k 0 n1 © cos²iA ¹ cos²i as a function of T, we have in one Expressing cos²iA 1

1

part, cos i = cos (S/2 - T) = sin T , and in the other cos iA = cos (S/2 - TA ) = sin TA =

§ n2 · ¨ G= sin²ș 1¸ ¸ k 0 n1 ¨© n12 ¹ 1

1/ 2

n1 n

, with the result that

§ n2 · ¨ | 1¸ ¸ k 0 n1 ¨© n12 ¹ 1

1/ 2

.


2. With k 0 n1 =

Ȧ c

· Ȝ § n2 G | 1¨ 1¸ 2 ¸ 2ʌ ¨© n1 ¹ 1/ 2

§ n2 · ¨ 1¸ 2 ¸ 6S ¨© n1 ¹ 1

425

n1 , G < 3 O1 gives rise to the condition:

1/ 2

< 3 O1 , and the condition for weak penetration therefore is 2

2 n2 2 ǻn n 'n § 1 · , so that ¨ ¸ < 1= 1 1 | . 2 2 n1 n1 n1 © 6S ¹

We therefore have as a condition, finally, 'n >

n1 72 ʌ 2

.

In numerical terms with n1 = 1.5, we have 'n > 0.0021 .

3. The monomodal condition can be written as Ȝ0 Ȝ0 Ȝ0 , so that: a < a1 = = | 2 2 2 2 ' 2 n -n1 2 2n n 2 n1 +'n - n1 1

'n

Basic Electromagnetism and Materials

Applied Electromagnetism and Materials

Applied Electromagnetism and Materials

Basic Laws of Electromagnetism

Basic laws of electromagnetism

Basic Laws of Electromagnetism

Characterization and Measurement of Magnetic Materials (Electromagnetism)

Basic Photographic Materials and Processes

Electromagnetism

Electromagnetism

Electromagnetism

Electromagnetism

Transmission Lines and Lumped Circuits (Electromagnetism) (Electromagnetism)

Permanent Magnet & Electromechanical Devices: Materials, Analysis, and Applications (Electromagnetism)

Electromagnetism and Life

Computational electromagnetism

Classical Electromagnetism

Classical electromagnetism

MAS207: electromagnetism

Classical Electromagnetism

Classical Electromagnetism

Classical electromagnetism

Computational Electromagnetism

Computational Electromagnetism

Classical Electromagnetism

Computational electromagnetism

Problems and Solutions on Electromagnetism

Topological foundations of electromagnetism

Physics of Classical Electromagnetism

Electromagnetism for Electronic Engineers

Numerical Methods in Electromagnetism

Basic Electromagnetism and Materials

Applied Electromagnetism and Materials

Applied Electromagnetism and Materials

Basic Laws of Electromagnetism

Basic laws of electromagnetism

Basic Laws of Electromagnetism

Characterization and Measurement of Magnetic Materials (Electromagnetism)

Basic Photographic Materials and Processes

Electromagnetism

Electromagnetism

Electromagnetism

Electromagnetism

Transmission Lines and Lumped Circuits (Electromagnetism) (Electromagnetism)

Permanent Magnet & Electromechanical Devices: Materials, Analysis, and Applications (Electromagnetism)

Electromagnetism and Life

Computational electromagnetism

Classical Electromagnetism

Classical electromagnetism

MAS207: electromagnetism

Classical Electromagnetism

Classical Electromagnetism

Classical electromagnetism

Computational Electromagnetism

Computational Electromagnetism

Classical Electromagnetism

Computational electromagnetism

Problems and Solutions on Electromagnetism

Topological foundations of electromagnetism

Physics of Classical Electromagnetism

Electromagnetism for Electronic Engineers

Numerical Methods in Electromagnetism

Recommend Documents