green function for maxwells
TRANSCRIPT
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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.
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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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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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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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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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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
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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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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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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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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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.)(),()(
,)(),()(
=
=
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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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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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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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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( ) ( )[ ]
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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=
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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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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=
/
/)
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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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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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 =