Green Function for Maxwells

8/12/2019 Green Function for Maxwells

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Application of Dyadic Greens Function Method in

Electromagnetic Propagation Problems

A Thesis

Presented to the Graduate School

Faculty of Engineering, Alexandria University

In Partial Fulfillment of the

Requirements for the Degree

of

Master of Science

In

Engineering Mathematics

By

Islam Ahmed Abdul Maksoud Ali Soliman

February 2009


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Application of Dyadic Greens Function Method in

Electromagnetic Propagation Problems

Presented by

Islam Ahmed Abdul Maksoud Ali Soliman

For the Degree of

Master of Science

In

Engineering Mathematics

ExaminersCommittee: Approved

Prof. --------------------------- -----------

Prof. --------------------------- -----------

Prof. --------------------------- -----------

Prof. --------------------------- -----------

Prof. Dr./Vice Dean of graduate studies and research

Faculty of Engineering, Alexandria University


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Advisors Committee:

Prof. Dr. Hassan Elkamchouchi -----------

Prof. Dr. Refaat El-Attar -----------


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ABSTRACT

Dyadic Greens functions are widely used in solving electromagnetic problems.

They are used as a mathematical kernel that relates the radiated or propagated

electromagnetic fields with their cause through an integral. Frequency domain models

were commonly used. However, there is a recent tendency in the electromagnetic

literature to use time domain models. This tendency is basically due to the recent

increasing use of short pulses with wide bandwidths in communications and radar

systems. A newly published form for the time domain dyadic Greens function for

Maxwells equations in free-space contains a source region term that seems to be

inconsistent with the extensively studied frequency domain form. One objective of the

thesis, is to clear this apparent inconsistency and to represent a form that is

completely consistent with the frequency domain results. Another objective, is to

show that when the dyadic Greens function is used as a propagator for a certaininitial field, the second derivative term can be completely omitted. This result reduced

greatly the time and effort in computing the propagated field. Verifications and

interpretations of these results are presented.


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TABLE OF CONTENTS

ABSTRACT .................................................................................................................iv

TABLE OF CONTENTS...............................................................................................v

CHAPTER 1 1

INTRODUCTION 1

1.1 Motivation and Contribution................................................................................1

1.2 Organization.........................................................................................................2

CHAPTER 2 3

ELECTROMAGNETIC FUNDUMENTALS 3

2.1 Maxwells Equations ...........................................................................................3

2.1.1 Maxwells Equations in Differential Form...................................................3

2.1.2 Maxwells Equations in Integral Form .........................................................5

2.1.3 Duality of Maxwell's Equations....................................................................52.2 Essence of Electromagnetics................................................................................6

CHAPTER 3 10

FREQUENCY-DOMAIN ANALYSIS 10

3.1 Introduction........................................................................................................10

3.2 Field Equations and Associated Potentials in Frequency-Domain ....................10

3.2.1 Electric and Magnetic Fields in Frequency-Domain ..................................10

3.2.2 Vector Wave and Vector Helmholtz Equations..........................................12

3.2.3 Vector and Scalar Potentials and Associated Helmholtz Equations...........13

3.3 Solution of Field Equations Outside the Source Region ...................................15

3.3.1 Solution of Scalar Helmholtz Equation Using Green's Function Method ..163.3.2 Combined-Source Solution of Maxwells Equations .................................19

3.3.3 Separated-Source Solution of Maxwells Equations ..................................22

3.3.4 Vector Potentials Approach ........................................................................25

3.4 Solution of Field Equations Inside the Source Region ......................................26

3.4.1 Source Region Solution of Scalar Helmholtz Equation..............................27

3.4.2 Source Region Solution of Maxwell's Equations .......................................29

CHAPTER 4 36

TIME-DOMAIN ANALYSIS 36

4.1 Introduction........................................................................................................36

4.2 Field Equations and Associated Potentials in Time Domain.............................37

4.2.1 Wave Equations ..........................................................................................37

4.2.2 Vector and Scalar Potentials .......................................................................37

4.3 Solution of Field Equations Outside the Source Region ...................................38

4.3.1 Solution of Scalar Wave Equation Using Green's Function.......................38

4.3.2 Solution of Maxwells Equations Using Dyadic Green's Function

Felsens Approach ...............................................................................................45

4.4 Field Inside the Source Region and Propagation of Initial Field The Complete

Time-Domain Solution ............................................................................................50

4.4.1 Nevels Approach .......................................................................................50

4.4.2 Time-Domain Vector Potential Approach Proposed Approach ..............60


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CHAPTER 5 78

CONCLUSIONS 78

APPENDIX A..............................................................................................................80

APPENDIX B ..............................................................................................................82

REFERENCES ............................................................................................................84


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CHAPTER 1

INTRODUCTION

1.1 Motivation and Contribution

Integral equations have been widely used to solve electromagnetic scattering and

related problems. A fundamental component of the integral equation model is the

dyadic Greens function. The dyadic Greens function makes it possible for the

integral equation to directly transform the electromagnetic sources to electromagnetic

fields. During the past era, frequency-domain dyadic Greens functions have appeared

regularly in the literature. On the other hand, time-domain forms were much less

common [1]. A principle reason for favoring the frequency-domain over the time-domain is that the frequency-domain approach was generally more tractableanalytically. Furthermore, the experimental hardware available for making

measurements in past years was largely confined to frequency-domain. However, the

recent increasing use of short pulses with wide bandwidths in communication and

radar systems has made time-domain methods more attractive. Some variants of

which has received widespread attention in the literature, mainly owing to their

superiority for solving wide-band problems and studying transient fields, in

comparison with frequency-domain methods.

Recently, [1] have reported a formula for the time-domain dyadic Greens function

of Maxwells equations in an unbounded space. The formulation included bothinfluences of the source currents and propagation of an initial field. The used state-

space approach have raised a new source region term that was not reported before.

However, for a field due to entirely a source current, the new term only contributes a

local nonpropagating field. This shows that the new term is unnecessary when the

field outside the source current region is considered. The new term was not reported

in literature before [1] because consideration had only been on the field due to entirely

a source current and propagating outside the source region. However, it is verified in

[1] that the new term is necessary to obtain the correct results of the propagation of an

initial field. The new term is also needed when the field inside the source current

region is required.

The problem of the field inside the source region was extensively studied in

frequency-domain by Yaghjian and Van Bladel among others. Their work in [2] and

[3] has shown that the strong singularity of the dyadic Greens function inside the

source region must be treated carefully. In order to correctly exclude the source region

singularity to perform the integral in a principle-value sence, a source region term

must be added to the dyadic Greens function. The added term has some properties

that were discussed in detail in the work of Yaghjian in [3]. One main property is that

its value is dependent on the shape of volume excluding the singularity. The principle

value integral shows a similar dependency on the shape of the exclusion volume. Both

contributions add up in just the right way to cancel the dependency on the shape of the

exclusion volume. The combination always results in a unique value for the fieldindependent of the shape of the exclusion volume.


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It is expected that both source region terms; the one reported in [1], and that

deduced in frequency-domain in [3], are two aspects of one thing. In other words, the

two forms for the source region terms are time-frequency transform duals. However,

the form reported [1] seem to be inconsistent with the frequency-domain form in that

it does not show the dependency on the shape of the volume excluding the singularity

as the frequency-domain form does.

The objective of this thesis is to introduce a form of the time-domain dyadic

Greens function that is completely consistent with the frequency-domain form. We

will also explain why such inconsistency occurred for the form in [1]. Another

objective is that we will show that the second derivative term in the form in [1] for the

field propagator is completely unnecessary. This leads to a great simplification in

calculations. Verifications and interpretations are presented afterwards.

1.2 Organization

The thesis consists of five chapters. Chapter 1 is the introduction. Chapter 2

presents some fundamental concepts from classical electromagnetic theory. Chapter

begins with a presentation of the governing Maxwells equations for macroscopic

electromagnetic phenomena, both in differential and integral forms. The property of

duality of Maxwells equations is presented. The chapter concludes with a section on

the essence of electromagnetics scientific. Different models used in solving

electromagnetic problems are discussed.

The third chapter presents the frequency-domain analysis necessary for an integral

equation model. The chapter begins with a representation of the field equations and

the equations governing the vector and scalar potentials in frequency-domain. Themethod of Greens function is presented and applied in finding the solution of the

scalar Helmholtz equation. The free space dyadic Greens function of Maxwells

equations is derived. The chapter ends with a detailed study of the problem of finding

the fields inside the source region.

Chapter 4 introduces time-domain analysis to find the time-domain solution of

Maxwells equations in free space. The chapter starts by a brief overview of the field

equations and the associated vector and scalar potentials in time-domain. The

following section seeks the solution of the time-domain Maxwells equations in free

space. A time-domain Greens function method is used to find the complete solution

of the scalar wave equation including the influences of the initial conditions and thenonhomogeneous boundary conditions. Then, we find the solution of Maxwells

equations as an influence of source currents only. Limitations of the described

solutions are pointed out. The following section describes two approaches that yield

time-domain solutions of Maxwells equations in free space. The described solutions

include both influences of the source currents and initial fields and covers the whole

domain including the source region. The first approach is the one recently described

by Nevels and Jeong in their paper [1]. The second is the proposed approach based on

vector potentials. Verifications and interpretations of the results are presented.

The fifth chapter gives the summary and conclusions.


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CHAPTER 2

ELECTROMAGNETIC FUNDUMENTALS

This chapter gives a brief description of the fundamentals of electromagnetics. The

chapter begins with a presentation of Maxwells equations for macroscopic

electromagnetic phenomena. Maxwells equations are presented in both differential

and integral forms. The duality property of Maxwells equations is also presented.

The chapter concludes with a section on the essence of the electromagnetics discipline

with a presentation of the most common propagator models used in solving

electromagnetics problems and the basic differences between these models.

2.1 Maxwells Equations

2.1.1 Maxwells Equations in Differential Form

Classical macroscopic electromagnetic phenomena are governed by a set of vector

equations known collectively as Maxwell's equations. Maxwell's equations in

differential form are

).,(),(),(

),,(),(),(

),,(),(

),,(),(

ttt

t

ttt

t

tt

tt

e

m

m

e

rJrDrH

rJrBrE

rrB

rrD

+

=

=

=

=

(2.1)

where E is the electric field intensity )m/V( , D is the electric flux density )m/C( 2 ,

B is the magnetic flux density )m/Wb( 2 , H is the magnetic field intensity )m/A( ,

e is the electric charge density )m/C(3 , eJ is the electric current density )m/A(

2 ,

m is the magnetic charge density )m/Wb(

2

, and mJ is the magnetic current density)m/V( 2 , and where V stands for volts, C for coulombs, Wb for webers , A for

amperes, and m for meters.

The equations are known, respectively as, Gauss' law, the magnetic-source lawor

magnetic Gauss' law, Faraday's law and Ampere's law. The magnetic charge and

magnetic current density have not been shown to physically exist, and so often those

terms are set to zero. However, their inclusion provides a nice mathematical

symmetry to Maxwell's equations.

The constitutive equations


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),(),(),(

),(),(),(

00

0

ttt

ttt

rMrHrB

rPrErD

+=

+= (2.2)

provide relations between the four field vectors in a material medium, where P is the

polarization density )m/C(2

, M is the magnetization density )m/A( , 0 is thepermittivity of free space )m/F1085.8( 212 , and 0 is the permeability of free

space )m/H104( 7 , and where Fstands for farads and H for henrys.

The polarization and magnetization densities are associated with electric and

magnetic dipole moments, respectively, in a given material. These dipole moments

include both induced effects and permanent dipole moments. In free space these

quantities vanish.

In the preceding equations r is the "field point" position vector zyx zyxr ++= .

However, r denotes the "source point" position vector zyx ++= zyxr . The vector

that points from the source point to the field point is denoted by

R),(),( rrRrrrrR =

with ),(),( rrrrrr == RR .

An important equation that demonstrates the charge conservation is embedded in

(2.1) is known as the continuity equation. Taking the divergence of Ampere's law we

get

te

+==

DJH0 (2.3)

and, upon interchanging the spatial and temporal derivatives and invoking Gauss' law,

we obtain the continuity equation

0=

+

t

ee

J (2.4)

Similarly, starting with Faraday's law we obtain

0=

+

t

mm

J (2.5)

Conversely, the two divergence equations are not independent equations within the set

(2.1), in the sense that they are embedded in the two curl equations and the continuityequation. Therefore, in macroscopic electromagnetics, one may consider the relevant

set of equations to be written as

),(),(),( ttt

t m rJrBrE

= (2.6)

),(),(),( ttt

t e rJrDrH +

= (2.7)

0),()(

)( =

+

tt

me

me

rJ (2.8)

subject to appropriate boundary conditions.


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2.1.2 Maxwells Equations in Integral Form

Starting with the differential (point) form of Maxwell's equations, an integral

(large-scale) form may be derived. Applying the divergence theorem

== SSV ddSdV SFFnF (2.9)to the divergence and continuity equations, and Stokes' theorem

= lS dd lFSF (2.10)

to the curl equations, leads to the integral form

=

+=

=

=

=

V me

S me

S S e

l

S S m

l

V m

S

V e

S

dVtdt

ddt

dtdtdt

ddt

dtdtdt

ddt

dVtdt

dVtdt

),(),(

),(),(),(

),(),(),(

),(),(

),(),(

)()( rSrJ

SrJSrDlrH

SrJSrBlrE

rSrB

rSrD

(2.11)

assuming that the conditions implied by the divergence and Stokes' theorems are

satisfied and that the differential and integral operators may be interchanged.

2.1.3 Duality of Maxwell's Equations

Maxwell's equations (2.1) are symmetric with respect to electric and magnetic

quantities, except for a sign change. This symmetry can be utilized to simplify some

electromagnetic problems. Considering the set of equations comprising Maxwell's

equations and the continuity equations, the substitutions

,,,,

,,,,

emmeem

me

JJ

JJBDDBEHHE (2.12)

leave the set unchanged. This duality is often used when a solution ( )ee HE , isobtained for the fields caused by electric sources ,, ee J with magnetic sources set to

zero. Then upon the replacements

,,,

,,

me

me JJEHHE

one has the solution for the electric and magnetic fields ( )mm HE , maintained bymagnetic sources.


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2.2 Essence of Electromagnetics

Electromagnetics is the scientific discipline that deals with electric and magnetic

sources and the fields these sources produce in specific environments. Maxwell's

equations provide the starting point for the study of electromagnetic problems,together with certain principles and theorems such as superposition, linearity, duality,

reciprocity, induction, uniqueness, etc., derived therefrom. While a variety of

specialized problems can be identified, a common ingredient of essentially all of them

is that of establishing a quantitative relationship between a cause (forcing function or

input ) and its effect (the response or output), a relationship which is referred to as a

field propagator. This relationship may be viewed as a generalized transfer function as

shown in figure.

In general , we can say that the essence of electromagnetics is the study anddetermination of field propagators to obtain thereby an input-output transfer function

for the problem of interest. This observation, while perhaps appearing transparent, is

an extremely fundamental one as it provides a focus for what elecromagnetics is all

about [4].

It is convenient to classify solution techniques for electromagnetic modeling in

terms of the field propagator that might be used, the anticipated application, and the

problem type. Such classification is outlined in table below.

Field Propagator Description based on:

Integral Operator Green's function for infinite medium

or special boundaries

Differential Operator Maxwell's curl's equations or their

integral counterparts

Transfer Function

derived from

Maxwell's Equations

Input Output

(Excitation) (Near, Far and sources

Fields)

PROBLEM DESCRIPTION

(Electrical, Geometrical)

Fig. The electromagnetic transfer function


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Modal Expansions solutions of Maxwell's equations in

particular coordinate system and

expansion

Optical Description rays and diffraction coefficients

Application Requires:

Radiation determining the originating sources

of a field

Propagation obtaining the fields distant from a

known source

Scattering determining the perturbing effects of

medium inhomogeneities

Problem Type Characterized by:

Solution Domain time or frequency

Solution Space configuration r or wave number k

Dimensionality one, two , or three

Electrical properties of medium

and/or boundary

dielectric; lossy; perfectly

conducting; anisotropic; inhomogeneous;

nonlinear

Boundary Geometry linear; curved; segmented;

compound; arbitrary

Selection of a field propagator is a first step in developing the electromagnetic

model for the problem we are interested in. The two mostly common propagator

models are those which employ Maxwell's curl equations directly or those described

by source integrals which employ a Green's function. The first type is named the

differential equation DE model, and the other is named the integral equation IE

model. Another criteria in constructing the EM model is the selection of the solution

domain. Either IE or DE propagator models can be formulated in time-domain or in

frequency-domain. Hence, basically we have four major models :

1. Time Domain Differential Equation (TDDE) Models: the use of which hasincreased tremendously over the past several years, primarily as a result of

much larger and faster computers.

2. Time Domain Integral Equation (TDIE) Models: although available forwell over 30 years , have gained increased attention in the last decade. The


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recent advances in this area make these methods very attractive for a large of

variety of applications.

3. Frequency Domain Integral Equation (FDIE) Models: which remain themost widely studied and used models, as they were the first to receive detailed

development.

4. Frequency Domain Differential Equation (FDDE) Models: whose use hasalso increased considerably in recent years, although most work to date hasemphasized low frequency applications.

It worth noting that the well-known method of moments (MoM) in general

involves IE modeling, whereas the finite element method (FEM) and finite difference

method (FDM) both use DE formulations.

Basic Differences

We briefly discuss and compare below the characteristics of IE and DE models in

terms of their development and applicability.

1. Integral Equation ModelThe basic starting point for developing an IE model is the selection of a Green's

function appropriate for the problem class of interest. The model is formulated as an

integral from which the fields in a giving contiguous volume of space can be written

in terms of integrals over the surfaces which bound it and volume integrals over those

sources located within it.

2. Differential Equation ModelA DE models requires intrinsically less analytical manipulation than does the

derivation of an IE model. That is because it seeks a direct numerical solution of

Maxwell's equations. It is implemented by discretizing the space of the problem into a

mesh, then repeatedly implement a discretized analog of Maxwell's equations or their

integral counterparts at each lattice cell or element of the mesh. However, in order to

be capable of handling infinite domains, certain absorbing boundary conditions

(ABC) are imposed. ABCs have the advantage of truncating the solution domain and

effectively simulate its extension to infinity.

Some basic differences between DE and IE models are as follows:

DE models include a capability to treat medium inhomogeneities,nonlinearities and the time variations in a more straight forward manner than

does IE models.

For DE models, the solution space includes the object's surroundings, theradiation condition is notenforced in exact sense, thus leading to certain error

in the solution. For the IE solution, the solution integral is confined to the

object and the radiation condition is automatically enforced.

The IE solutions are generally more accurate and efficient. Spurious solutions exist in DE methods, whereas such solutions are absent in

IE methods.

In terms of numerical efficiency, DE methods generate a space matrix, whilethe IE methods generate full dense matrices.


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In IE numerical implementation, discretization is applied only for the volumeof space occupied by the source or the surface of the boundary. Whereas in

DE models, discretization is applied to the whole solution domain. Thats why

DE methods are also called domain methods, while IE methods are called

boundary methods.


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CHAPTER 3

FREQUENCY-DOMAIN ANALYSIS

3.1 Introduction

The study in this thesis is confined to the integral equation model in modeling

electromagnetic problems. Frequency-domain integral equation models are considered

to be the most widely studied and used models. They were also the first to receive

detailed development. Frequency-domain models were favored because they are

generally more tractable analytically.

The chapter starts by a section that represents the field equations and the equations

governing the vector and scalar potentials in the frequency-domain. Expressing thesolution of Maxwells equations as a Greens function integral is considered as the

first step in developing an integral equation model in frequency-domain. Thus, the

second section is concerned in seeking a free space solution for Maxwells equations

using the Greens function method. Also, a vector potentials approach to the solution

is presented. The vector potentials approach yields the same Greens function integral

obtained before.

The represented solution is shown to be limited to find fields that are outside the

source region. That is why the next section is devoted to tackle the problem of finding

the fields inside the source region. Such concern about the fields inside the source

region arises in some applications such as the evaluation of an antenna impedence, the

induced current on a scatterer, and other situations [2][5]. The section reviews the

results of the extensive studies by Yaghjian and Van Bladel, among others.

3.2 Field Equations and Associated Potentials in Frequency-Domain

In this section, we express the electric and magnetic field equations in the

frequency-domain. Next, We represent the equations governing the vector and scalar

potentials in frequency-domain. We also show how can the electric and magnetic

fields be recovered from the vector potentials if the potentials were known.

3.2.1 Electric and Magnetic Fields in Frequency-Domain

When the electromagnetic sources vary arbitrarily with time in a narrow-band, it is

often convenient to work in the frequency-domain. The Fourier transform pair is

given as

{ }

{ }

==

==

det

dtett

tj

tj

),(2

1),(),(

),(),(),(

1 rKrKrK

rKrKrK

(3.1)


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where we have separated the induced effects from the applied source. Repeating

for )()()()()( rJrJrHrJrJ cmi

mm

i

mm +=+= and noting that

0)()( =+ i

me

i

me jJ (3.7)

we have( )

( )

)()()(

)()()(

)()(

)()(

rJrErH

rJrHrE

rrH

rrE

i

e

i

m

i

m

i

e

j

j

+=

=

=

=

(3.8)

where,

= m

j

~ .

For later convenience it is useful to relax our notation in (3.8) and simply work

with,

( )

( )

)()()(

)()()(

)()(

)()(

rJrErH

rJrHrE

rrH

rrE

e

m

m

e

j

j

+=

=

=

=

(3.9)

3.2.2 Vector Wave and Vector Helmholtz Equations

We start with Maxwells curl equations (3.9)

)()()()(

)()()()(

rJrErrH

rJrHrrE

e

m

j

j

+=

=

In order to decouple above equations, we take the curl of )()( 1 rEr and of

)()( 1 rHr which leads to

),()()(

)()()()(

1

21

rJrrJ

rErrEr

mej =

(3.10)

),()()(

)()()()(

1

21

rJrrJ

rHrrHr

emj +=

(3.11)


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where (3.11) could also be obtained from (3.10) using duality. These are the vector

wave equations for the fields. Either (3.10) or (3.11) may be solved, with the

undetermined field quantity found via the curl equations.

Various simplifications to the above can be found. For instance, if the medium is

isotropic and homogeneous, we have

).()()()(

),()()()(

2

2

rJrJrHrH

rJrJrErE

em

me

j

j

+=

=

(3.12)

Of course (3.12) also applies to individual homogeneous subregions within an

isotropic inhomogeneous region.

Noting that ( ) VVV 2= , we also have for isotropic homogeneousmedia

.)()()()(

,)()()()(

22

22

m

em

eme

j

j

+=+

++=+

rJrJrHrH

rJrJrErE

(3.13)

These are known as vector Helmholtz equations. Yet another form can be obtained

using the continuity equations, leading to

)()()()(

),()()()(

2

22

2

22

rJrJIrHrH

rJrJIrErE

em

me

j

j

+=+

+

+=+

(3.14)

where I is the identity dyadic and is the second partial derivatives dyadic. They

can be equivalently presented in the matrix forms

=

100

010

001

I (3.15)

and

=

2

222

2

2

22

22

2

2

zyzxz

zyyxy

zxyxx

(3.16)

.

3.2.3 Vector and Scalar Potentials and Associated Helmholtz Equations

The source terms on the right side of (3.13) and (3.14) are quite complicated.Introducing a potential function can simplify the form of the source term, which in


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turn leads to a reduction of many vector problems to scalar ones. Another benefit of

the potential approach is that the integrals providing the potentials from the sources

are less singular than those relating the electric and magnetic fields to the sources.

For simplicity we proceed assuming homogeneous isotropic media.

Consider first the case of only electric sources in (3.9). By virtue of the identity

0= V , Maxwell's equation 0= B leads to the relationship

AB = , (3.17)

where A is known as the magnetic vector potential )m/Wb( . Substitution of this into

Faraday's law results in ( ) 0AE =+ j . From the vector identity 0= weobtain

ej = AE (3.18)

where e is known as the electric scalar potential )V( .Hence, Ampere's law then

becomes

( )

( ) ),()()(

11)( 2

rJArJrE

AAArH

eee jjj +=+=

==

(3.19)

leading to

( ) )(22 rJAAA eejk +=+ (3.20)

where 22 =k .

So far only the curl of A has been specified. According to the Helmholtz theorem,

a vector field is determined by specifying both its curl and its divergence. We are at

liberty to set A such that the right side of (3.20) is simplified. Accordingly, we

let ej= A , which is known as theLorenz gauge, resulting in

)(22

rJAAek =+ (3.21)

Because we also have /2 eej == AE , then

/22 eee k =+ (3.22)

Now consider only magnetic sources. Maxwell's equation 0= E leads to

FE = (3.23)

where F is known as the electric vector potential ( V ). Substituting into Ampere's

law leads to 0FH =+ )( j , while the vector identity 0= results in

,mj = FH (3.24)

where m is known as the magnetic scalar potential. Faraday's law is then

),()()()(

)()(2

rJFrJrH

FFFrE

mmm jij ==

==

(3.25)

leading to


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)()(22

rJFFFmmjk +=+ (3.26)

Accordingly, let mj= F , resulting in

)(22

rJFF mk =+ (3.27)

Because we also have /2 mmj == FH , then

/22 mmm k =+ (3.28)

In summary, the various potentials in the Lorenz gauge satisfy Helmholtz

equations as

.

/

/)(

22

=

+

m

e

m

e

m

e

kJ

J

F

A

(3.29)

We note that the Helmholtz equations for the potentials have much simpler source

terms than those for the fields, in particular, in a homogeneous space the vectors

A and F will be collinear with the source terms eJ and mJ respectively, often

reducing the vector problem to a simpler scalar one.

Using superposition we obtain the fields from the Lorenz-gauge potentials as

.1

,1

FAAE

FFAB

+=

+=

jj

jj

(3.30)

3.3 Solution of Field Equations Outside the Source Region

This section is devoted to find a free space solution of Maxwells equations. The

presented solution is confined to find the fields outside the source region. The

problem of finding the fields inside the source region is discussed later in sec(3.4). Inthe first subsection we describe the method of Greens function and use the method to

find the solution of the scalar Helmholtz equation. Then, in the second subsection, the

method is generalized to find the solution of the combined-source vector Helmholtz

equation. The generalized method introduces a dyadic Greens function instead of a

scalar one. Since the results of the combined-source solution seem to be inapplicable,

the next subsection presents a compact explicit-source form for the solution. The

solution is obtained by the frequency-domain analog of an approach conducted by

Felsen and Marcuvitz in their book [6]. The last subsection describes a vector

potentials approach that yields the same results of the approach of Felsen and

Marcuvitz.


22/90

16

3.3.1 Solution of Scalar Helmholtz Equation Using Green's Function

Method

The scalar Helmholtz equation is defined as

)()()( 22 rrr =+ k (3.31)

where )(r is the source term, and the solution is assumed to satisfy certain

boundary conditions on a closed surface S.

The solution of (3.31) is expected to include the influence of both the source term

and the boundary conditions of the problem. One way to find an expression of such a

solution is by using the method of Green's function. The method of Green's function

depends, basically, on a simple physical principle; to obtain the field caused by a

distributed source (charge or heat generator or whatever it is that causes the field) we

calculate the effects of each elementary portion of the source and add them all (as

long as the problem is linear). If )( rr, g is the field at the observer's point r caused by

a unit point source at the source point r , then the field at r caused by a source

distribution )(r is the integral of the )( rr, g weighted by the source distribution

over the whole range of r occupied by the source. The function g is called the

Green's function. Boundary conditions can be treated as sources (whether they are

Dirichlet or Neumann conditions) which enables us to include their effect in the

solution in a similar way as we did for .

The Greens function method involves two main steps:

I. Finding the Green's function of the problem.II. Expressing the solution in terms of the Green's function.

The first step is done by solving the partial differential equation of the problem, but

with a point-source )( rr instead of the source distribution )(r . This leads to a

partial differential equation which is homogeneous except at rr = . The obtained

equation is called the Green's function differential equation. It is usually solved

subject to homogeneous boundary conditions to give the Greens function of the

problem.

The second step is based on the application of Green's theorem and the reciprocity

property of Green's functions. These are discussed in detail later in the section.

I. Finding the 3D Green's Function of the Scalar Helmholtz Equation

For the scalar Helmholtz equation defined by (3.31), the Green's function

differential equation is

)(),(),( 22 rrrrrr =+ gkg . (3.32)

Without loss of generality, the source is assumed to be at the origin of the coordinates.

Hence, equation (3.32) becomes

)()()( 22 rrr =+ gkg (3.33)


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17

Since the free space is assumed, )(rg is only a function of r=r due to symmetry.

By using the Laplacian in spherical coordinates, (3.33) is rewritten as

)()()(1 22

2 rrgk

dr

rdgr

dr

d

r=+

(3.34)

The right-hand side of (3.34) is zero except at the origin. Hence, for 0r (3.34) can

be rewritten as

( ) ( ) 0)()( 22

2

=+ rrgkrrgdr

d (3.35)

yielding the solution

r

eMrg

jkr

=)( (3.36)

where Mis an arbitrary constant, and only the traveling wave is assumed (for antje

+time dependence).

The arbitrary constant Mis determined by substituting (3.36) into (3.33), and then

integrating within a small sphere including the origin as follows

( )

=

==

+=

=

=

V

jkrjkr

V

jkrjkr

V S

dV

ek

rejk

MkdVgk

r

ejk

r

eMr

gr

dgdVg

.1)(

111

4

4

4

2

22

2

2

2

2

r

S

By taking the limit 0r , we obtain4

1=M .

When the source is located at an arbitrary position r , the Green's function is

expressed as

rrrr,

rr

=

4)(

jkeg (3.37)

which is recognized as the usual free-space 3D scalar Green's function of Helmholtz

equation.

II. Expressing the Solution in Terms of the Greens Function

To express the solution )(r in (3.31) in terms of the Green's function, Green's

theorem and the reciprocity property of the Green's function are applied. Before we

derive the formula of the solution integral, we give a short description and a

derivation of both the Green's theorem and the reciprocity property.


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18

Green's Theorem

Green's theorem is a variant of Gauss' divergence theorem (2.9). Greens theorem

is stated as a relation between surface and volume integrals given by

=SV

duvvudVuvvu S)()( 22 (3.38)

Derivation

For a closed surface S, Gauss' divergence theorem is

=SV

ddV SFF (3.39)

Consider two scalar fields )(ru and )(rv . By taking uvvu =F and using the

vector identity FFF += fff ,we obtain

.2

2

uvuvvuvu

++= F (3.40)

When substituted in (3.39), it directly yields Greens theorem

Reciprocity of Green's Functions

Reciprocity of Green's functions or sometimes called Maxwell's reciprocityis the

property that )()( r,rrr, = gg . That means that the response at r due to a

concentrated source at r is the same as the response at r due to a concentrated source

at r . It worth noting that this is not physically obvious. It is purely a mathematical

property.

Derivation

Green's theorem is used to prove the reciprocity property. Taking )( 1rr, =gu and

)( 2rr, =gv with both satisfying the same homogeneous boundary conditions , leads to

{ }{ }

).()()()(

)()()(

)()()(

)()()()(

1221

11

2

2

22

2

1

1

2

22

2

1

22

rrrr,rrrr,

rrrr,rr,

rrrr,rr,

rr,rr,rr,rr,

+=

=

=

gg

gkg

gkg

gggguvvu

Substitution in Green's theorem (3.38) leads to

{ }

)conditionsboundaryshomogeneousameesatisfy thbothsince(0

,)()()()()sidehandright(

),()()sidehandleft(

1221

2112

=

=

+=

Sr,rr,rr,rr,rr,rr,r

dgggg

gg

S

By rewriting the variables as rrrr == 21 , ,we finally obtain,

).()( r,rrr, =

gg (3.41)


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Formulation of the Solution

In the following we derive the formula expressing the solution in terms of the

Green's function. This is done by taking )(r=u and )( rr, =gv , then applying

Green's theorem (3.38). So, the integrand of the left-hand side is written as

{ } { }).()()()(

)()()()()()(

)()()()(

22

2222

rr,rrrr

rrrr,rrrr,r

rrr,rr,r

+=

=

=

g

kggk

gguvvu

Therefore, applying Green's theorem (3.38) yields

{ } =+SV

dggdVg Srrr,rr,rrrr,r )()()()()()()(

By interchanging the variables r and r , and using the reciprocity of the Green's

function (3.41),we obtain

[ ] +=SV

dggVdg Srrr,rrr,rrr,r )()()()()()()( (3.42)

Equation (3.42) represents the solution of the scalar Helmholtz equation expressed

in terms of the Greens function. Vis the volume under consideration, S is the

surface of V, and Sd is the outward normal vector of S.

As can be seen, the volume integral in the right-hand side of (3.42) corresponds

to the superposition of the contribution of the source while the surface integral

corresponds to the superposition of the contribution from the equivalent sources on

the boundary.

3.3.2 Combined-Source Solution of Maxwells Equations

As was shown, decoupling of Maxwell's equations in frequency domain have led to

equations (3.14). These can be written as

m

e

k

k

iH(rH(r

iE(rE(r

=+

=+

))

,))

22

22

(3.43)

where

).()(

),()(

2

2

rJrJIi

rJrJIi

emm

mee

j

j

+

+=

+=

(3.44)

Dyadic Greens Function

In the previous subsection, the concept of Green's function was confined to the

scalar case; i.e. , when a scalar field is excited with a scalar field. Clearly, in such a

case, the mediating function, called the Green's function is then a scalar quantity too.For vector problems, however, the idea of a Green's function becomes more involved.


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20

To retain full generality, the propagator or the inverse operator between a vector

source and a vector field must be a dyadic (a second rank tensor). This distinction

provides the main difference in the interpretation of the Green's function in the scalar

and vector cases. For a scalar problem, one essentially has to solve the same scalar

differential equation for the scalar Green's function as for the original fields (with a

delta function term replacing the source term). In the vector problem, however, thevector differential equation for the original vector fields are replaced by a dyadic

differential equation in terms of the dyadic Green's function.

The dyadic Green's function makes the formulation and solution of

electromagnetic problems more compact. Even though many problems may be solved

without using dyadic Green's functions, the symbolic simplicity offered by them

makes its use attractive. This is especially true in multiple scattering problems, in

which complex physics of a vector field is compactly accounted for using the dyadic

Green's function. [7]

I. Finding the Dyadic Greens Function of the ProblemFor the electric and magnetic vector Helmholtz equations (3.43) the dyadic Green's

function is defined to satisfy the dyadic differential equation

)(),(),( 22 rrIrrGrrG =+ k (3.45)

One way to solve the above dyadic equation is by means of the scalar Green's

function which satisfy (3.32). Eliminating the delta functions from both the scalar and

the dyadic equations, leads to

),()(),()( 2222 rrIrrG +=+ gkk

or

( ) 0rrIrrG =+ ),(),()( 22 gk

A particular solution to the above is

),(),( rrIrrG = g (3.46)

That means that, in free space, the dyadic Greens function of the vector Helmholtz

equations (3.43) is

rr

IrrG

rr

=

4

),(

jke

(3.47)

II. Expressingthe Solution in terms of the Dyadic Greens Function

In order to express the solution in terms of the dyadic Green's function, a vector-

dyadic variant of the Greens theorem is used. It is called the vector-dyadic Greens

second theorem [8]. It is given by

( )[ ]

( ) ( ) ( ) [ ]{ }

++=

S

V

dS

dV

ABnBAnBAnBAn

BABA22

(3.48)


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21

Taking )r(r,GB = and E(r)A= or H(r) yields,

( ) ( )

( ) [ ]{ }

( ) ( )

( ) [ ]{ }

++

+=

++

+=

S

SV

m

S

SV

e

dS

dSdV

dS

dSdV

.H(r))r(r,Gn)r(r,GH(r)n)r(r,GH(r)n

)r(r,GH(r)n)r(r,G(r)i)rH(

,E(r))r(r,Gn)r(r,GE(r)n)r(r,GE(r)n

)r(r,GE(r)n)r(r,G(r)i)rE(

(3.49)

By interchanging the roles of r and r , and using the reciprocity property , we

obtain

( ) ( )

( ) [ ]{ }

( ) ( )

( ) [ ]{ }

++

+=

++

+=

S

SV

m

S

SV

e

Sd

SdVd

Sd

SdVd

.)rH()r(r,Gn)r(r,G)rH(n)r(r,G)rH(n

)r(r,G)rH(n)r(r,G)r(iH(r)

,)rE()r(r,Gn)r(r,G)rE(n)r(r,G)rE(n

)r(r,G)rE(n)r(r,G)r(iE(r)

(3.50)

The equations given above for HE and may be further simplified if free space is

considered. This means that we let the surface S recede to infinity, and

GHE and, will satisfy the Sommerfeld radiation condition [8]

( )

( )

( ) .lim

,0lim

,0lim

0GrG

HrH

ErE

=+

=+

=+

ikr

ikr

ikr

r

r

r

(3.51)

In such a case, the surface integrals in (3.50) vanish, leading to the simple intuitive

formulas for HE and ,

.

,

=

=

V

m

V

e

Vd

Vd

)r(r,G)r(iH(r)

)r(r,G)r(iE(r)

(3.52)

Applicability

Unfortunately, the equations given above for HE and are still unsatisfactory due

to the complicated form of the physical source densities )(meJ appearing in )(mei . In


28/90

22

typical situations, the source densities are numerically determined or approximated.

That means that the subsequent differentiation can introduce large errors. That is why

it is better to move the derivative operators onto the known Green's function rather

than the sources. Also, from an analytical standpoint it is more convenient to work

with terms involving a Green's dyadic and an undifferentiated current density [8].

3.3.3 Separated-Source Solution of Maxwells Equations

In this section, we derive the solution for the electric and magnetic fields in a

compact separated-source form. Consider the vector wave equations (3.12) of the

electric and magnetic fields, given by

.))

,))

2

2

me

me

jk

jk

JJH(rH(r

JJE(rE(r

=

= (3.53)

As obvious from these equations, both eJ and mJ has an influence on the value of

the electric field E .The same can be said for H . From the linearity of the problem,

one can separate the effects from eJ and mJ for each equation. Hence, it is expected

that four dyadic Green's functions are needed, namely, mmmeemee GGGG and,, . The

dyadic eeG , for example, accounts for the influence of eJ on E , emG for the

influence of mJ on E , and so on.

In terms of the four dyadic Green's functions, the solution for E and H in free

space is

.)(

,)(

+=

+=

V

mmm

V

eme

V

mem

V

eee

VdVd

VdVd

)r(J)r(r,G)r(J)r(r,GrH

)r(J)r(r,G)r(J)r(r,GrE

(3.54)

where no surface integrals are accounted here because, in free space, surface

integrals vanish due to radiation conditions [8].

A more compact formulation is given by

Vdm

e

V mmme

emee

=

J

J

GG

GG

H

E (3.55)

Let [ ]T

HEF= to be the field vector, [ ]T

me JJJ = to be the source vector, and

let

=

mmme

emee

GG

GGG to be the Maxwell's equations dyadic Green's function. Hence,

equation (3.55) is rewritten as

VdV

= )()()( rJrr,GrF (3.56)

This gives a single compact expression of radiation from both the electric and

magnetic sources in free space. One great advantage of this form over the form (3.52),

is that the source densities are undifferentiated. This allows a way for applying

approximated source densities or those numerically determined without expectinglarge numerical errors.


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23

Finding G

The goal now is to determine the Green's function G for Maxwell's equations with

its four components mmmeemee GGGG and,, . First, we find the dyadic differential

equations governing those four dyadic Green's functions. And second, by solving

those dyadic equations, we obtain the expressions of the four dyadic Green's

functions.

1. Finding the dyadic equations of components of G

Consider the dyadic eeG , for example, which is known in literature as theelectric

field dyadic Green's function.It accounts for the influence of the electric source eJ on

the electric field E . It is known that E , in the case when eJ is the only effective

current source, satisfies

ejk JEE =2

(3.57)

Hence, eeG will satisfy the dyadic wave equation

)(2 rrIGG = jk eeee (3.58)

With a close look in the form of the source terms in (3.53) we can construct the

dyadic equations of the other dyadic Green's functions as

)(2 rrIGG = emem k (3.59)

)(2 rrIGG = meme k (3.60)

)(2 rrIGG = jk mmmm (3.61)

From the above equations, some interrelations between the four dyadic Green's

functions can be deduced. These are

.meem

mmee

GG

GG

=

= (3.62)

It is worth noting that we can find an equivalent set for the equations governing the

four dyadic Green's functions derived directly from Maxwell's curl equations before

decoupling. This is done as follows.

Maxwell's curl equations before decoupling are

m

e

j

j

JHE

JHE

=+

=

(3.63)

Assuming the case of an electric source where 0=mJ , eeG and meG will be the

influence functions of the electric current source eJ on the electric and magnetic

fields respectively. Hence, to find the dyadic equations of eeG and meG , we replace

E with eeG , and H with meG and a unit dyadic delta source instead of eJ in

Maxwell's curl equations, leading to

)( rrIGG = meee

j (3.64)

0GG =+ meee j (3.65)


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24

The same can be done for mmG and emG , in the case of a magnetic source where

0=eJ . This leads to

0GG = mmemj (3.66)

)( rrIGG =+ mmem j (3.67)

Simple mathematical manipulations on the set of equations (3.64)-(3.67) shows a

complete equivalence with the set of equations (3.58)-(3.61).

2. Finding expressions for the four dyadic Green's functions Felsen's Approach

As shown by Felsen and Marcuvitz in [6], expressions for the four dyadic Green's

functions can be derived by a simple set of operations on the scalar Green's function.

The procedure is as follows.

We start with eeG .Applying the dyadic identity CCC2

= in (3.58)

leads to

( ) )(22 rrIGGG = jk eeeeee (3.68)

The divergence of eeG can be found from (3.64) by taking the divergence both sides

and making use of the dyadic identities 0 C and II ff = . This yields

)()( rrIrrIG == eej (3.69)

Substituting in (3.68) leads to

)()(1 22 rrIGGrr =

jkj

eeee

Hence,

)(1

2

22 rrIGG

+=+

kjk eeee (3.70)

which is a dyadic Helmholtz equation. However, since the scalar Green's function

g satisfies

)(22 rr =+ gkg , (3.71)

the delta function can be eliminated between (3.70) and (3.71) to obtain

( ) ( ) gk

kjk ee

++=+

2

2222 1IG

or

( ) 0IG =

+++ g

kjk ee 2

22 1

A particular solution of the above is

gk

jee

+=

2

1IG (3.72)


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25

From equation (3.65) it is easy to find meG in terms of eeG . Applying the dyadic

identity 0 , we obtain

IIG == ggme (3.73)

Also, similar procedures yield

gk

jmm

+=

2

1IG , (3.74)

and

IIG == ggem (3.75)

Hence, the results can be summarized as

gk

jmmee

+==

2

1IGG (3.76)

IIGG === ggmeem (3.77)

whererr

rr,

rr

=

4)(

jke

g .

When expressions (3.76) and(3.77) are applied in (3.54), we obtain the free space

solution of Maxwells equations in frequency domain,

{ } +

+=

V

m

V

e VdgVdgk

j )()(1

)(2

rJIrJIrE (3.78)

{ }

++=

V

m

V

e Vdgk

jVdg )(1

)()(2

rJIrJIrH (3.79)

3.3.4 Vector Potentials Approach

An alternative approach to derive (3.78) and (3.79) is by using the vector electric

and magnetic potentials A and F defined in sec (3.2.3). An attractive property of

vector potentials is that they satisfy vector Helmholtz equations with simple collinearsource terms. Referring to (3.29), the equations of A and F are given by

( )( ) .

,

22

22

m

e

k

k

JF

JA

=+

=+ (3.80)

The dyadic Green's function of the vector Helmholtz equation was shown before to

be gIG= (see sec(3.3.2)). Hence , the solutions of (3.80) for A and F in free space

simply are


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26

.)(),()(

,)(),()(

=

=

V

m

V

e

Vdg

Vdg

rJrrIrF

rJrrIrA

(3.81)

The relations between the fields E and H and the vector potentials A and F ,respectively, were depicted in equations (3.30). Thus solutions for E and H can be

formed by simple substitution yielding

+=

V

m

V

e VdgVdgj

j )()(1

)( rJIrJIrE

(3.82)

++=

V

m

V

e Vdgj

jVdg )(1

)()( rJIrJIrH

(3.83)

Interchanging the order of the differential and integral operators yields,

{ } +

+=

V

m

V

e VdgVdgk

j )()(1

)(2

rJIrJIrE (3.84)

{ }

++=

V

m

V

e Vdgk

jVdg )(1

)()(2

rJIrJIrH (3.85)

which is the same result obtained by using Felsen's approach (3.78) and (3.79).

3.4 Solution of Field Equations Inside the Source Region

It has been shown in the last section that the electric and magnetic fields E and H

outside a current-carrying volume can be given by equations (3.54) or equations

(3.78) and (3.78). One is normally interested in finding the fields in points outside the

source region (i.e., r is outside V). This is the case particularly when computing the

radiation pattern of a current distribution. However, it is not without practical interest

to inquire whether (3.54) are still valid when r is insideV. In other words, when we

are interested in finding the fields inside the source region, can we validly use (3.54)?From the practical point of view, such an interest arises in the evaluation of an

antenna impedance, the power radiation, the induced current on a scatterer, and other

situations[2][5].

Clearly, the dyadic Green's functions become infinite when r approaches r ,

hence, the integrals appearing in (3.54) become improper ones. The singularities of

eeG and mmG are of the order3R , and the singularities of emG and meG are of the

order 2R . That means that for a typical source current distributions, equations (3.54)

would lead to divergent integrals. Such a feature has been extensively studied by

Yaghjian and Van Bladel, among others. Their work in [2] and [3] has shown that the

principle value of the integrals involving the current element and the dyadic Green's


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27

function should be carefully defined. Also, a correction term should be added to the

integrals involving eeG or mmG [9].

In the following, we will start tackling carefully the derivations for the solutions,

with the source region in consideration.

3.4.1 Source Region Solution of Scalar Helmholtz Equation

The problem of the scalar Helmholtz equation was solved in sec (3.3.1). In this

subsection we treat the problem again but with taking the source region into

consideration. Actually the derivation of the solution of scalar Helmholtz equations

depicted in (3.3.1) would not now be rigorously valid. That is because Green's

theorem (3.38) requires the involving functions to be continuous in the region. The

substitution )( rr, =gv violates the conditions of Green's theorem when r

approaches r . We can alleviate this difficulty by following the usual procedure ofexcluding the point rr = from the integration. We exclude the point r from the

volume V by containing it within an arbitrary volume V bounded by the smooth

surface S . The application of Green's theorem to the region VV , with the

substitution )(r=u and )( rr, =gv , is now rigorously valid, leading to

[ ]

[ ]

+

=

SS

VV

dgg

dVgg

.)()()()(

)()()()( 22

Srr,rrrr,

rrr,rr,r

(3.86)

We write this as

[ ]

[ ] .)()()()(

)()()()()()(

=

S

VVS

dgg

dVsgdgg

Srr,rrrr,

rrr,Srr,rrrr,

When taking the limit as 0 , the left side becomes

( )( )

( ) .4

)(lim

4

)(lim)(4

lim

20

00

S

jkR

S

jkR

S

jkR

dSR

e

dSikR

edS

R

e

nRr

nRrnr

The first term vanishes since R ( is the maximum chord of V ) so that the

integrand is ( )/1O , while the surface element is ( )2O . The second term vanishesfor the same reason, while the third term leads to )(r . In evaluating the third term

we assume )(r is well behaved for r near r , so that it can be brought outside the

integral as a constant on S . The solid-angle formula

4

2 =

SdS

R

nR (3.87)


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28

where both unit vectors point outward from S and the point 0=R is contained inside

S then leads to the desired result. We therefore get

{ } =

SVV

dggdVg Srrr,rr,rrrr,r )()()()()()(lim)(0

(3.88)

By interchanging the variables r and r , and using the reciprocity of the Green's

function (3.41), we obtain

{ } +=

SVV

dggVdg Srrr,rrr,rrr,r )()()()()()(lim)(0

(3.89)

which is the solution when r is inside the source region. As obvious, (3.89) has just

the same form as (3.42) except that an infinitesimal volume containing the singularity

is excluded.

The above form raises some questions about exclusion volume V . Does V has acertain shape, or can we arbitrarily chose its shape? If the shape can be arbitrarily

chosen, would that mean that the volume integral in (3.89) does not have a unique

value? The theory of improper integrals gives us the answers.

According to the theory of improper integrals [8], if a function )( rr, f is

piecewise continuous everywhere in a region V , except at rr = where it becomes

unbounded, then the improper integral dVfV

)( rr, is, classically, said to exist

(converge to a unique function of r ) and is equal to

dVfVV

)(lim

0

rr,

if the latter integral exists. In the latter expression V is a small volume containing the

singular point r , and so V is a function of r (i.e., )(r= VV ). The only restrictions

on V are that the point r is interior to V and that the maximum chord of V does

not exceed . As the limit is taken, the shape position, and orientation with respect tor are maintained. The integral is said to exist (converge) if the limiting integral

converges to a finite value independent of the shape of the exclusion region. Such a

case occurs, for instance, for the improper volume integral

[ ]Vn

dVR1

when 20


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The singularity in (3.89) is ( )RO 1 , hence )(r converges to a unique valueindependent of the shape of the exclusion volume.

Now we can state the whole-region (inside and outside the source region) solution

of the scalar Helmholtz equation by

[ ] +=SV

dggVdg Srrr,rrr,rrr,r )()()()()()()(

where it is implicitly known that when r approaches r , the volume integral will have

the form

VV

Vdg )()(lim0

rrr,

where V is a volume excluding the singularity. The shape of V can be arbitrarily

chosen, always leading to a unique result. That is because the singularity here is

removable according to the theory of improper integrals.

3.4.2 Source Region Solution of Maxwell's Equations

In order to account for the fields in the source region using the dyadic Green's

function of Maxwell's equations G , it is useful to use the results of the vector

potentials approach depicted in sec(3.3.4). The solutions for E and H were expressed

in (3.82) and (3.83) as

+=

V

m

V

e VdgVdg

j

j )()(1

)( rJIrJIrE

(3.90)

++=

V

m

V

e Vdgj

jVdg )(1

)()( rJIrJIrH

(3.91)

In those equations, the singularities inside the integrations are ( )RO 1 , which areremovable singularities. Hence, in the same manner as described in sec(3.2.2), if we

are interested in finding the fields inside the source region, the integrals will be

performed as

+=

VV

m

VV

e VdgVdgjj )(lim)(lim1

)( 00 rJIrJIrE (3.92)

++=

VV

m

VV

e Vdgj

jVdg )(lim1

)(lim)(00

rJIrJIrH (3.93)

where V is a volume excluding the singularity of each integral separately, leading to

a unique value for the fields whatever was the shape of V for each integral.

However, these forms will still be impractical as long as the differential operators

are operating on the integrals from outside. That might lead to the use of numerical

differentiation which is expected to give large errors. Alternatively, if the differential


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30

and integral operators are interchanged, we obtain a form with the differential

operator acting on the Greens function directly which has an analytical expression.

In sec(3.3.4) differential and integral operators were validly interchanged since no

singularity occurs in the domain of interest which was the volume outside the source

region. However, when the source region is considered, the interchange of operators

must be treated carefully. To study the validity of such an interchange between

operators it is convenient to find the first and second derivatives of integrals of the

form

==

V

jk

V

Vde

sVdgsrr

rrr,rr

rr

4)()()()( (3.94)

where we assume the source density )(rs is at least piecewise continuous.

First Derivatives of V

Vdgs )()( rr,r

It is known that for the case of Vr (i.e., outside the source region), rr = cannot

occur. Hence, (3.94) represents a proper convergent integral over fixed limits. As such

it can be differentiated arbitrarily often, with derivatives brought under the integral

sign, i.e. ,

VVdgx

sVdgsx

V iVi

=

rrr,rrr,r :)()()()( (3.95)

We now consider the case of Vr (i.e., inside the source region) where the

volume integral is to be interpreted as ( ) VV Vd0lim after excluding thesingularity by the volume V . Because )(r VV = , the validity of passing

ix through the limiting integral needs to be established carefully. As reported in

[8], it can be shown that the volume integral (3.94) uniformly converges to a

continuous function )(r which is differentiable with the derivatives allowed to be

taken under the integral sign, i.e.,

VVdgx

sVdgsx

VV iVVi

=

rrr,rrr,r :.)()(lim)()(lim00

(3.96)

This interchange of operators can also be accomplished through the use ofLeibnitz's theorem which, in the one-dimension case, is stated as

x

xgxgxf

x

xgxgxfdyyxf

xdyyxf

x

xg

xg

xg

xg

+

=

)())(,(

)())(,(),(),( 11

22

)(

)(

)(

)(

2

1

2

1

(3.97)

When (3.96) holds, one can see that the "extra terms" given in the three-dimensional

Leibnitz's theorem, generated by the rigorous interchange of operators, vanish. Thevalidity of this interchange for the curl operator and the type of integrand of interest

here is described in detail in [10].


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Using (3.96), one can see that for first derivatives (usually ix and in the

scalar potential case, and A and A in the case of the vector potential) the final

result is the same as if the derivative was formally passed through the integral without

regard for either the limiting operation or the integration limits depending on the

differentiation variable. Thus we obtain (see [8]).

.)()(lim)()(lim

,)()(lim)()(lim

,)()(lim)()(lim

,)()(lim)()(lim

00

00

00

00

=

=

=

=

VVVV

VVVV

VV iVVi

VVVV

VdgVdg

VdgVdg

Vdgx

Vdgx

VdgsVdgs

rr,rsrr,rs

rr,rsrr,rs

rr,rsrr,rs

rr,rrr,r

(3.98)

Second Derivatives of V

Vdgs )()( rr,r

Second derivatives of (3.94) may not necessarily exist when Vr , but if the

source density is piecewise continuous in V , then at any point in V(the bounding

surface Sis not part of V) where the source density )(rs satisfies a Hlder condition

rrrr kss )()( (3.99)

where 0>k and 10 < , then the second-order partial derivative

=

VVijij

Vdgsxxxx

)()(lim)(0

22

rr,rr (3.100)

exists as well [8]. If the source density satisfies the same Holder condition everywhere

in V, then the second partial derivatives are (Holder) continuous in V, although they

will not, in general, be continuous on the boundary S.

Even if existence of the second- derivative is established, second derivative

operators may not generally be brought under the integral sign without careful

consideration of the source-point singularity. If the integral and second-derivative

operator are formally interchanged, i.e.,

VV ij

Vdgxx

s )()(lim2

0rr,r (3.101)

the integral of the resulting differentiated integrand is often no longer convergent in

the classical sense (the differentiated integrand being singular, )1( 3RO ). However,

according to [8], the concept of convergence can be broadened to say that an integral

is convergent in theprinciple value(P.V.)sensenamed conditionally convergent(i.e.,

exists in a conditional sense) if the limiting integral converges to a finite value that is

dependenton the shape of the exclusion region.

An example of a conditionally convergent improper integral in one dimension is [8]


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32

( ) ( )[ ]

lnlnlim

11lim

1

0

0

0

0=

+==

dxxdx

xdx

xI

a

a

a

a

If the limit variables are related, say = , then ( )ln=I and the integral

converges (conditionally) to a number for a given , but that number is not unique.Note that if , are unrelated then the integral is not even finite. So it is seen that in

this instance the value of the integral depends on the "shape" of the exclusion region.

A similar situation occurs in (3.101), leading to a contradiction between (3.100)

and (3.101). In (3.101) the result is not unique and dependent on the shape of the

exclusion volume unlike the result of (3.100) which has a unique value independent of

the shape of the exclusion volume.

One procedure to correctly evaluate (3.100) is presented and proved in [8]. It states

that

[ ] ,)()()(lim

)()(

)()(lim

2

0

0

2

+

=

VV ji

S

i

j

VVij

Vdgxx

ss

Sdgx

s

Vdgsxx

rr,rr

nxrr,r

rr,r

(3.102)

where Sis the boundary surface of V , n is an outward unit normal vector on S, and

Vrr, . This equation holds for s being Holder continuous. The form (3.102) can be

used to pass various second-order derivative operators ( ,,,2

etc.)through integrals of the form (3.94), scalar or vector case as appropriate.

Alternative Method for Evaluating Second Derivatives of V

Vdgs )()( rr,r

Another method for evaluating the second partial derivatives of (3.94) was

developed in [5] and [11] with regards to the electric dyadic Green's function

singularity. Using concepts from generalized function theory, it is shown that,

operationally,

( )( ).

4

lim)(

4)(lim

4)(

20

2

0

2

=

S

ij

VV

jk

ij

V

jk

ij

Sds

Vde

xxs

Vde

sxx

rr

Rxnxr

rrr

rrr

rr

rr

(3.103)

This can also be written in the operational form


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33

=

V

ji

V

jk

ij

VdgsVde

sxx

)()(4

)(2

rr,rrr

r

rr

(3.104)

where

)()()(P.V.)(2

rrrrr,rr,

jiij

ji Lgxx

g (3.105)

with

( )

=

S

ij

ji SdL 20 4

lim)(

rr

Rxnxr (3.106)

and P.V. indicates the integral for that term should be performed in the principle value

sense. Although the singularity in the volume integral in the right side of (3.103) is

)1( 3RO , which means that the integral is not convergent in the classical sense. The

integral is only conditionally convergentmeaning that the integral will converge to avalue which is dependent on the shape of the exclusion volume. However, under the

conditions specified for (3.100), the integral in the left side of (3.103) will converge to

a unique value independent of the shape of the exclusion volume. That means that, as

described in [3], the volume and surface integrals in the right side of (3.103), which

are both dependent on the shape of the exclusion volume, just add up in a way to

cancel the shape dependency of each other.

In order to give a more compact formulation of (3.106), a dyadic L is defined as

=

SSd

2

4

)(

rr

RnrL (3.107)

Using the this definition, jiL can be written as

ijjiL xrLxr )()( = (3.108)

Also the second derivative of g can be written in the dyadic form

ij

ij

ggxx

xrr,xrr, )()(2

=

(3.109)

Hence, operationally, (3.103) can be rewritten as

.)()(

)()(lim

4)(

0

2

ij

VV

ij

V

jk

ij

s

Vdgs

Vde

sxx

xrLxr

xrr,xr

rrr

rr

=

(3.110)

We now move on to apply (3.98) and (3.110) to correctly interchange the

differential and integral operators in (3.82) and (3.83).


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Interchange of Operators in Field Equations

Consider the electric field. From (3.98), the curl operator in the second term in

(3.90) can be interchanged with the integral operator resulting in

{ }

==

VV

m

VV

m

VV

m

VdgVdg

Vdg

)(lim)(lim

)(lim

00

0

rJIrJ

rJI

(3.111)

To interchange the operators in the first term in (3.90) we use (3.110) to obtain

= =

==

+

=

+

3

1

3

1

3

1

3

10

0

0

,)()(1

)()(lim1

)()(lim

)(lim1

i j

ijeji

VV j

ijej

i

i

e

VV

VV

e

Jj

VdgJj

Vdgj

Vdgj

j

xrLxrx

xrr,xrx

rJrr,

rJI

which can be shown to be equal to the compact dyadic form

.)()(

)()(1

lim20

jVdg

kj ee

VV

rJrLrJrr,I

+

(3.112)

Adding the two terms (3.111) and (3.112) yields

{ } .)(lim

)()()(

1lim)(

0

20

+

+=

VV

m

VV

ee

Vdg

jVdg

kj

rJI

rJrLrJIrE

(3.113)

Similar manipulations, or duality, lead to the magnetic field equation

{ }

.)()(

)(1

lim

)(lim)(

20

0

++

=

VV

em

VV

e

jVdg

kj

Vdg

rJrLrJI

rJIrH

(3.114)

In order to write (3.113) and (3.114) in the form given in (3.54) or (3.55), it is

customary to take

)()()(

)()()(

1P.V.)(

)()()(

1P.V.)(

2

2

rr,Irr,Grr,G

rrrLrr,Irr,G

rrrLrr,Irr,G

==

+=

+=

gj

g

k

j

jg

kj

meem

mm

ee

(3.115)


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35

where P.V. indicates that the associated term is to be integrated in the principle value

sense. This leads to the simpler form

.)(

,)(

+=

+=

V

mmm

V

eme

V

mem

V

eee

VdVd

VdVd

)r(J)r(r,G)r(J)r(r,GrH

)r(J)r(r,G)r(J)r(r,GrE

(3.116)

The Depolarizing Dyadic L

The dyadic L is known as the depolarizing dyadic [3]. It arises mathematically

from the careful consideration of the strong-point singularities in eeG and mmG . As

was previously shown, L arises when the interchange between the differential and

integral operators is done correctly.

Actually, if the electric field was found only by the principle value integral, that

would cause the electric field to have a non-unique value which is dependent on the

shape of the exclusion volume. It is the depolarizing dyadic L that solves the

problem. The depolarizing dyadic term, which is also dependent on the shape of the

exclusion volume, is added in just the right way to cancel the shape dependence of the

principle value integral, resulting in a unique value for E independent of the shape of

the exclusion volume.

Physical Interpretation

Physically [7], the principle value integral corresponds to putting the observation

point r inside a cavity excavated in the current source region. Since the current isdiscontinuous on the surface of this cavity, charges build up on the surface of the

cavity. When the cavity size is very small, the field due to the charges is essentially

electrostatic in nature inside the cavity. Since the electrostatic field satisfies Laplace's

equation which is scale invariant, this field persists even in the limit when the

exclusion volume tends to zero. This electrostatic field is a function of the shape of

the cavity, no matter how small it is. These charges give rise to a field which should

not have been there since the exclusion volume is absent in the actual case. Hence, to

obtain a correct answer, the term of L is added to remove the effect of the surface

charges around the exclusion volume.

The form of the symmetric dyadic L for various exclusion volumes is presented in

a table in [3]. For a sphere, 3IL = is independent of the position of the origin within

the sphere. For a cube with origin at the center of the cube, 3IL = as well. For a

pillbox of arbitrary cross-section iixxL = , where ix is the unit vector in the direction

of the axis of the pillbox. The same dyadic is found for the "slice" exclusion volume,

which is the natural form of the pillbox for laterally infinite layered-media

geometries.


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CHAPTER 4

TIME-DOMAIN ANALYSIS

4.1 Introduction

During the past era, frequency domain dyadic Green's functions in

electromagnetics have appeared regularly in the literature. On the other hand, time

domain forms were much less common [1]. A principle reason for favoring the

frequency domain over the time domain had been that the frequency-domain approach

was generally more tractable analytically. Furthermore, the experimental hardware

available for making measurements in past years was largely confined to frequencydomain.

However, the recent increasing use of short pulses with wide band bandwidths in

communication and radar systems has made time-domain methods more attractive.

Some variants of which has received widespread attention in the literature, mainly

owing to their superiority for solving wide-band problems and studying transient

fields in comparison with frequency domain methods. In the numerical

implementation of time domain methods, the response of the system over a wide

range of frequencies can be obtained with a single simulation.

The chapter starts by a brief overview of the field equations and the associated

vector and scalar potentials in time domain. The following section seeks the solutionof the time-domain Maxwells equations in free space. However, the section is limited

to find solutions only outside the source region. A time-domain Greens function

method is used to find the complete solution of the scalar wave equation including the

influences of the initial conditions and the nonhomogeneous boundary conditions.

Then, using the approach described by Felsen and Marcuvitz in their book [6], we

find the solution of Maxwells equations as an influence of source currents only.

Limitations of the described solutions are pointed out. The following section describes

two approaches that give a complete form of the solution of time-domain Maxwells

equations in free space. By a complete form we mean that the solution includes the

influence of the initial fields, and the solution covers the whole space region including

the source region. The first approach is the one recently described by Nevels andJeong in their paper [1]. Unfortunately, we think that the formula of the solution

conducted by Nevels and Jeong does not present the complete picture since it seems

to be inconsistent with the frequency-domain results known in literature and described

in equations (3.115) and (3.116). Alternatively, we propose another approach based

on vector potentials. Although known in frequency-domain analysis, we think, to our

knowledge, that it is the first time to apply a vector potentials approach in time

domain. The proposed approach yields results that are completely consistent with the

frequency-domain well-known formulations. It also shows that there are some

unnecessary terms in Nevels form, thus, an extreme reduction of the calculation

effort is yielded.


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38

=

/

/)

1(

2

2

2

2

m

e

m

e

m

etc

J

J

F

A

(4.5)

which is the time domain analog of (3.29). And the fields can be obtained from the

Lorenz gauge potentials by

.

,11

2

2

FAAE

FFAB

+

=

+

=

t

c

t

ttc (4.6)

4.3 Solution of Field Equations Outside the Source Region

4.3.1 Solution of Scalar Wave Equation Using Green's Function

In this section we solve the scalar wave equation with a time dependent source,

),(),(1

),(2

2

2

2tt

t

u

ctu rrr =

(4.7)

subject to the two initial conditions,

)()0,( rr fu = (4.8)

)()0,(2

2

rr gt

u=

(4.9)

.

Green's Function for the Scalar Wave Equation

We introduce the Green's function ),,( ttG rr as a solution, due to a concentrated

source atrr =

acting instantaneously only at tt =

, of the differential equation

)()(),,(1

),,(2

2

2

2tttt

t

G

cttG =

rrrrrr (4.10)

where )( rr is the Dirac delta function of the appropriate dimension.

The Green's function is the response at r at time tdue to a source located at r at

time t . Since we desire the Green's function G to be the response only due to thissource acting at tt = ( not due to some nonzero earlier conditions ), we insist that the

response G will be zero before the source acts ( tt < ) :

ttttG


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39

known as the causality principle.

The Green's function ),,( ttG rr only depends on the time after occurrence of the

concentrated source. If we introduce the elapsedtime, tt = ,

,0for0

)()(1

2

2

2

2


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40

The terms on the right side represent contributions from the boundaries: the spatial

boundaries for all time, and the temporal boundaries ( itt= and ftt= ) for all space.

Reciprocity

For the scalar Helmholtz equation, we have shown that the Green's function issymmetric, )()( r,rrr, = gg . We proved this result using Green's theorem for two

different Green's functions )(and)( 21 rr,rr, gg . The result followed because the

boundary terms in Green's theorem vanished.

For the wave equation there is a somewhat analogous property. However, it is not

),,(),,( ttGttG rrrr = . Indeed if tt > the second of these is zero. In order to

obtain a reciprocity relation the following approach is used.

The Green's function ),,( ttG rr satisfies

)()(12

2

2

2 tttG

cG =

rr (4.21)

subject to the causality principle,

ttttG . To utilize the Green's formula (4.20) to prove reciprocity, we

need a second Green's function. If we choose it to be ),,( AA ttG rr , then the

contribution f

i

t

t S

dtduvvu S)( on the spatial boundary vanishes, but the

contribution

V

t

t

dVt

uv

t

vu

f

i

on the time boundary will not vanish at both itt= and ftt= . However, if we let

tti in Green's formula, the "initial" contribution will vanish.

For a second Green's function we are interested in varying the source time t,

),,( 11 ttG rr , what is called the source- varying Green's function [12]. From the

translation property,

),,(),,( 1111 ttGttG = rrrr (4.23)

since the elapsed times are the same [ ]tttt =

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Green Function for Maxwells