Infinite-Space Dyadic Green Functions in Electromagnetism · Takao Sakurai and Raj Kishor Gartia Chemical bonds well outside metal surfaces J Mahanty and N H March The massive scalar

This content has been downloaded from IOPscience. Please scroll down to see the full text.

Download details:

IP Address: 54.39.106.173

This content was downloaded on 08/02/2021 at 12:17

Please note that terms and conditions apply.

You may also be interested in:

Dyadic Green Function for an Electromagnetic Medium Inspiredby General Relativity

Akhlesh Lakhtakia and Tom G. Mackay

Erratum: Dyadic Green Function for an Electromagnetic Medium Inspired by General Relativity [Chin.

Phys. Lett. 23 (2006) 832]

Akhlesh Lakhtakia and Tom G. Mackay

Multipole radiation in a chiral medium

Silvestre Ragusa

The Schrodinger equation with an anharmonic oscillator potential

P M Radmore

The determination of intrinsic trapping parameters of a thermoluminescence peak of BeO

Takao Sakurai and Raj Kishor Gartia

Chemical bonds well outside metal surfaces

J Mahanty and N H March

The massive scalar meson field in a Schwarzschild background space

D J Rowan and G Stephenson

A problem in two-dimensional flow

H H Macey

Electronic control of a synchronous motor

L U Hibbard

IOP Concise Physics

Infinite-Space Dyadic Green Functions in Electromagnetism

Muhammad Faryad and Akhlesh Lakhtakia

Chapter 1

Introduction

The Green function is the solution of an inhomogeneous linear differential equationin which the source function is localized both in space and time. Such a source isreferred to as a point source (space) and impulse (time). The Green function isutilized to find the solutions for a source that occupies finite spatial and temporaldomains by representing the source as a dense distribution of impulsive pointsources.

Green functions can be scalars, dyadics, or tensors. Dyadics are commonly usedin electromagnetism. A dyadic can be considered as equivalent to a square matrix—most commonly, a 3 × 3 matrix. Characterizing the physical process represented bya linear differential equation, the dyadic Green function maps a general source (avector) into the corresponding solution (a vector) of a differential equation.

Mathematics textbooks focusing on Green functions usually deal with scalarGreen functions as tools to solve linear differential equations that often describephysical processes [1–3]. The scalar Green function is useful when both the sourceand the field (solution of the differential equation) are scalar functions. Examplesinclude the electric scalar potential due to an unchanging distribution of charges,sound waves produced by objects moving in water, and the voltage or current in anelectrical circuit in response to either abruptly opening or abruptly closing a switch.

The dyadic Green function is useful when both the source and the field are vectorfunctions. The physical processes that require a dyadic Green function include theradiation of electromagnetic waves by a current source in a medium and the generationof elastodynamic waves by a mechanical source in a material. A point source can havethree linearly independent orientations, and the field (e.g. electric field and mechanicaldisplacement) can have components along three mutually orthogonal directions. Thus,nine scalar functions are required to map the source to the field. Thereby the need arisesfor dyadic Green functions in electromagnetism and elastodynamics. This monographdeals with dyadic Green functions in electromagnetism, but most of the techniquesshould be directly applicable in elastodynamics as well. Since the stress tensor due to a

doi:10.1088/978-1-6817-4557-2ch1 1-1 ª Morgan & Claypool Publishers 2018

source is often a quantity of interest in elastodynamics, triadic Green functions also areused in that field of research [4, 5].

Numerous research papers and several books, as well as book chapters, havebeen written on dyadic Green functions in electromagnetism, especially when boththe source and the field are time harmonic. A book by Chen [6] deals with dyadicGreen functions for (i) a homogeneous isotropic dielectric medium occupying allspace and (ii) a homogeneous uniaxial dielectric medium occupying all space.Although the book also derives the infinite-space dyadic Green function for theelectric field phasor in a homogeneous isotropic dielectric-magnetic mediumtranslating with uniform velocity, we caution the reader that the underlyingMinkowski formulation is noncausal and, therefore, unphysical. The crowningglory of Chen’s book is an excellent introductory chapter on dyadics. A book byTai [7] presents dyadic Green functions applicable for situations in which ahomogeneous isotropic dielectric-magnetic medium occupies finite space withspecific boundary conditions imposed on the boundary. Other notable booksdealing with dyadic Green functions are those of Felsen and Marcuvitz [8], VanBladel [9, 10], and Lindell [11].

This monograph deals with infinite-space dyadic Green functions in electro-magnetism. The basic concept is presented in section 1.1. Several examples arepresented in section 1.2 to illustrate the relevant types of physical problems. Section 1.3contains the formulation of the dyadic Green functions in electromagnetism whenall space is occupied by a linear, homogeneous, bianisotropic medium. Themost commonly used approaches to find dyadic Green functions are described insection 1.4. The organization and contents of the remainder of this monograph arediscussed in section 1.5.

In this monograph, the ω−i texp( ) time-dependence has been assumed wherevertime-harmonic quantities are considered, with ω being the angular frequency, t thetime, and = −i 1 . Furthermore, ε0, μ0, and ω ε μ=k0 0 0 represent, respectively, thefree-space permittivity, permeability, and wavenumber. Boldface letters representvectors anddyadics are underlined twice.All linear operators are representedbyupper-case calligraphic letters, e.g.,A and B. Dyadic operators are additionally underlinedtwice, e.g., A and B. Dyadic Green functions are represented by upper-case lettersthat are underlined twice, for example G, whereas scalar Green functions arerepresented by lower-case letters such as g. Time-dependent quantities are decoratedabove with a tilde like G. Spatial-Fourier-transformed (q-dependent) quantities aredecorated by a breve above, for example, g. Finally, · *( ) represents the complex

conjugate, · −( ) 1 the inverse, ·adj( ) the adjoint, ∣ · ∣( ) the determinant, and ·( )T thetranspose.

1.1 Concept of infinite-space dyadic Green functionsTo understand the concept of infinite-space Green functions, let us consider a lineardifferential operator L ∇ ∂( , )t such that

L ∇ ∂ · ˜ = ˜t tF r S r( , ) ( , ) ( , ), (1.1)t

Infinite-Space Dyadic Green Functions in Electromagnetism

1-2

where ∇ is the gradient operator with respect to the spatial coordinates, ∂t ≡ ∂/∂t isthe time derivative, ˜ tS r( , ) is the source function, ˜ tF r( , ) is the field due to thatsource, and the dot product (·) or the contraction between a vector and a dyadic isdefined in equation (A.4) in appendix A. Several physical phenomenons are modeledby differential equations of the type (1.1), as shown in section 1.2.

The dyadic Green function is a tool employed to solve equation (1.1) to determineF for an arbitrary source S. By definition, the dyadic Green function ˜ ′ ′G t tr r( , , , ) isthe solution of

L δ δ˜ ∇ ∂ · ˜ ′ ′ = − ′ − ′G t t t t Ir r r r( , ) ( , , , ) ( ) ( ) , (1.2)t

where I is the identity dyadic [6]. The delta functions are defined as follows:

∭δπ

− ′ = · − ′−∞

∞

i dr r q r r q( )1

(2 )exp [ ( )] , (1.3)3

3

∫δπ

ω ω− ′ = − − ′−∞

∞

t t i t t d( )1

2exp [ ( )] . (1.4)

The different signs of the exponential functions in equations (1.3) and (1.4) arenecessary to maintain causality.

Taking the dot product of both sides of equation (1.2) from the right with ˜ ′ ′tS r( , ),we have

L δ δ˜ ∇ ∂ · ˜ ′ ′ · ˜ ′ ′ = − ′ − ′ ˜ ′ ′G t t t t t tr r S r r r S r( , ) ( , , , ) ( , ) ( ) ( ) ( , ). (1.5)t

Next, let us integrate over both ′r and ′t to get

L ∫ ∭

∫ ∭ δ δ

˜ ∇ ∂ · ˜ ′ ′ · ˜ ′ ′ ′ ′

= − ′ − ′ ˜ ′ ′ ′ ′

= ˜

−∞

∞

−∞

∞−∞

∞

−∞

∞

G t t t d dt

t t t d dt

t

r r S r r

r r S r r

S r

( , ) ( , , , ) ( , )

( ) ( ) ( , )

( , ), (1.6)

t3

3

by virtue of a defining property of the delta function [12]. Now, a comparison ofequations (1.1) and (1.6) leads to

∫ ∭˜ = ˜ + ˜ ′ ′ · ˜ ′ ′ ′ ′−∞

∞

−∞

∞

t t G t t t d dtF r F r r r S r r( , ) ( , ) ( , , , ) ( , ) , (1.7)cf 3

where ˜ tF r( , )cf is the solution of the homogeneous differential equation

L ∇ ∂ · ˜ =tF r 0( , ) ( , ) . (1.8)tcf

Infinite-Space Dyadic Green Functions in Electromagnetism

1-3

Equation (1.7) encapsulates the usefulness of the dyadic Green function, i.e., thesolution of equation (1.1) can be found for an arbitrary source function ˜ tS r( , ) interms of the dyadic Green function ˜ ′ ′G t tr r( , , , ).

1.2 Examples of linear operatorsBefore we move on to the solution methods in section 1.4, let us present a fewexamples of physical processes that can be modeled as linear differential equations.

1.2.1 RL circuit

Consider an electrical circuit comprising lumped elements. A simple one contains aninductor with inductance L and a resistor with resistance R, connected in series anddriven by a voltage source υ t( ), as shown schematically in figure 1.1. Let the currenti t( ) through the circuit be the variable of interest. In that case, the differentialequation relating υ t( ) and i t( ) is

⎛⎝⎜

⎞⎠⎟ υ+ ˜ = ˜L

ddt

R i t t( ) ( ). (1.9)

The linear operator L ≡ +L d dt R( / ) is then scalar. When υ δ˜ =t t( ) ( ), the currenti t( ) equals the scalar Green function h t( ), which is usually referred to as the impulseresponse [13].

v(t)

R

L

i(t)

Figure 1.1. A series RL circuit driven by a voltage source υ t( ).

1.2.2 Sound wave

Consider an inviscid, uncharged, and stagnant fluid with uniform mass density ρm.The propagation of a sound wave with small amplitude obeys the Euler fieldequations [8]

⎫⎬⎪⎪

⎭⎪⎪

γ

ρ

∂∂

˜ + ∇ · ˜ = − ˜

∇ ˜ + ∂∂

˜ = ˜

p tp t t s t

p tt

t t

r v r r

r v r f r

1( , ) ( , ) ( , )

( , ) ( , ) ( , )

, (1.10)

m

0

Infinite-Space Dyadic Green Functions in Electromagnetism

1-4

where p tr( , ) is the pressure differential from the background pressure p0, ˜ tv r( , ) isthe velocity of fluid particles, and γ is the ratio of the specific heat of the fluid atconstant pressure to the specific heat of the fluid at constant volume. The sourceterms s tr( , ) and ˜ tf r( , ) represent, respectively, the scalar particle source density andthe vector force density.

Elimination of ˜ tv r( , ) from equations (1.10) leads to the wave equation

⎛⎝⎜

⎞⎠⎟ ρ∇ − ∂

∂˜ = −∇ · ˜ + ∂

∂˜

c tp t t

ts tr f r r

1( , ) ( , ) ( , ), (1.11)

am

22

2

2

where γ ρ=c p /a m0 is the speed of acoustic waves. For this equation, the operator

L ≡ ∇ − ∂∂c t

1(1.12)

a

22

2

2

is scalar and is often called the scalar wave operator.Similarly, the elimination of p tr( , ) leads to

⎛⎝⎜

⎞⎠⎟ γ

∇∇− ∂∂

· ˜ = −∇˜ + ∂∂

˜Ic t

t s tp t

tv r r f r1

( , ) ( , )1

( , ). (1.13)a2

2

20

In this equation, the linear operator

L ≡ ∇∇ − ∂∂

Ic t1

(1.14)a2

2

2

is dyadic.

1.2.3 Plate vibration

Consider the vibration of a uniform plate of area A and thickness h. It is made ofan isotropic material of mass density ρm, Young modulus E, and Poisson ratio ν.The stiffness of the plate is ν= −D Eh /[12(1 )]3 2 . If the plate lies in the xy plane,the displacement w x y t( , , ) along the z axis satisfies the biharmonic equation [3]

⎛⎝⎜

⎞⎠⎟ρ∇ + ∂

∂˜ = ˜D h

tw x y t p x y t( , , ) ( , , ), (1.15)m

42

2

where p x y t( , , ) is the pressure on the plate and the operator ∇4 ≡ ∇2∇2 is called thebi-Laplacian. For this example, the linear operator

L ρ˜ ≡ ∇ + ∂∂

D ht

(1.16)m4

2

2

is scalar.

Infinite-Space Dyadic Green Functions in Electromagnetism

1-5

1.2.4 Helmholtz operator

The scalar operator

L ≡ ∇ + k (1.17)2 2

is known as the Helmholtz operator and is ubiquitous in physics and engineering. Itcan also be considered to arise from the temporal Fourier transform of the waveoperator (1.12). The function g kr( , )H that satisfies the equation

δ∇ + = −k g kr r( ) ( , ) ( ) (1.18)H2 2

is referred to as the scalar Green function in electromagnetism.

1.3 Linear electromagnetismThis monograph deals with the infinite-space dyadic Green functions in electro-magnetism with the assumption that all space (except for some bounded region) isoccupied by a linear homogeneous medium. Let us assume that this medium isbianisotropic [14, 15].

Consider the time-dependent macroscopic Maxwell equations

∇ × ˜ − ∂∂

˜ = ˜tt

t tH r D r J r( , ) ( , ) ( , ), (1.19)e

∇ × ˜ + ∂∂

˜ = − ˜tt

t tE r B r J r( , ) ( , ) ( , ), (1.20)m

ρ∇ · ˜ = ˜t tD r r( , ) ( , ), (1.21)e

and

ρ∇ · ˜ = ˜t tB r r( , ) ( , ), (1.22)m

where ˜ tJ r( , )e and ˜ tJ r( , )m are electric and magnetic source current densities,respectively, and ρ tr( , )e and ρ tr( , )m are the electric and magnetic source chargedensities, respectively. The magnetic source current and charge densities arefictitious and have been kept not only for generality but also because severalradiation and scattering problems can be solved more easily if certain equivalent(but fictitious) magnetic current and charge densities are used. The source currentand source charge densities are related by the continuity conditions

⎫⎬⎪⎪

⎭⎪⎪

ρ

ρ

∇ · ˜ + ∂∂

˜ =

∇ · ˜ + ∂∂

˜ =

tt

t

tt

t

J r r

J r r

( , ) ( , ) 0

( , ) ( , ) 0. (1.23)

e e

m m

Infinite-Space Dyadic Green Functions in Electromagnetism

1-6

To proceed, let us define the temporal Fourier transform from time t to theangular frequency ω as

∫ω ω= ˜−∞

∞

F F t i t dt( ) ( ) exp ( ) , (1.24)

and its inverse transform as

∫πω ω ω˜ = −

−∞

∞

F t F i t d( )1

2( ) exp ( ) . (1.25)

Whereas F t( ) is a real-valued function of t, F(ω) is a complex-valued function of ω.When F t( ) is a field, F(ω) is classified as a phasor.

The temporal Fourier transforms of the Maxwell equations (1.19)–(1.22) are asfollows:

ω ω ω ω∇ × + =iH r D r J r( , ) ( , ) ( , ), (1.26)e

ω ω ω ω∇ × − = −iE r B r J r( , ) ( , ) ( , ), (1.27)m

ω ρ ω∇ · =D r r( , ) ( , ), (1.28)e

ω ρ ω∇ · =B r r( , ) ( , ). (1.29)m

Equations (1.26)–(1.29) are the same as equations (1.19)–(1.22) for the time-harmonic electromagnetic fields with ω−i texp( ) time dependence.

The frequency-domain constitutive relations for a linear, homogeneous, bianiso-tropic medium can be written as [14, 15]

ω ε ω ω ξ ω ω= · + ·D r E r H r( , ) ( ) ( , ) ( ) ( , ) (1.30)

and

ω ζ ω ω μ ω ω= · + ·B r E r H r( , ) ( ) ( , ) ( ) ( , ), (1.31)

where ε ω( ) is the permittivity dyadic, μ ω( ) is the permeability dyadic, and ξ ω( ) as

well as ζ ω( ) are called magnetoelectric dyadics. As E and B are considered to be the

primitive fields, whereas D and H are the induction fields in modern electro-magnetism, the constitutive dyadics must satisfy the Post constraint [14–17]

⎡⎣ ⎤⎦μ ω ζ ω ξ ω· + =−{ }Tr ( ) ( ) ( ) 0 (1.32)1

when the whole space is occupied by a linear homogeneous medium, with Tr a( )denoting the trace of the dyadic a. The Post constraint may seem to break downwhen the medium occupies a finite region, but it can be preserved by assuming

Infinite-Space Dyadic Green Functions in Electromagnetism

1-7

surface states consistent with the surface Hall effect [18, 19] in the boundaryconditions [20, 21].

1.3.1 Dyadic Green functions for field phasors

Since the frequency-domain Maxwell curl equations (1.26) and (1.27) are linear inωE r( , ) and ωH r( , ), the electric and magnetic field phasors can be written in terms

of four dyadic Green functions as

∭ω ω ω ω ω= ′ · ′ + ′ · ′ ′−∞

∞

G G dE r r r J r r r J r r( , ) [ ( , , ) ( , ) ( , , ) ( , )] (1.33)eee

emm

3

and

∭ω ω ω ω ω= ′ · ′ + ′ · ′ ′−∞

∞

G G dH r r r J r r r J r r( , ) [ ( , , ) ( , ) ( , , ) ( , )] . (1.34)mee

mmm

3

Here and hereafter, ω′G r r( , , )ee is the electric dyadic Green function, ω′G r r( , , )mm

is the magnetic dyadic Green function, and ω′G r r( , , )em as well as ω′G r r( , , )me arethe magnetoelectric dyadic Green functions.

From now onwards, the ω-dependences are implicit and therefore are omitted.Furthermore, the sources are taken to occupy a finite region Vs. Therefore, theforegoing equations can be written as

V

∭= ′ · ′ + ′ · ′ ′G G dE r r r J r r r J r r( ) [ ( , ) ( ) ( , ) ( )] (1.35)ee

eem

m3

s

and

V

∭= ′ · ′ + ′ · ′ ′G G dH r r r J r r r J r r( ) [ ( , ) ( ) ( , ) ( )] . (1.36)me

emm

m3

s

The delta function allows the electric current density phasor to be represented as

∭ δ= ′ − ′ ′−∞

∞

dJ r J r r r r( ) ( ) ( ) (1.37)e e3

and the magnetic current density phasor as

∭ δ= ′ − ′ ′−∞

∞

dJ r J r r r r( ) ( ) ( ) . (1.38)m m3

Infinite-Space Dyadic Green Functions in Electromagnetism

1-8

Substitution of equations (1.30), (1.31), and (1.35)–(1.38) in the frequency-domainMaxwell curl equations (1.26) and (1.27) results in four coupled linear differentialequations as follows:

ωξ ωε δ∇ × + · ′ + · ′ = − ′I i G i G Ir r r r r r( ) ( , ) ( , ) ( ), (1.39)me ee

ωζ ωμ∇ × − · ′ − · ′ =I i G i Gr r r r( ) ( , ) ( , ) 0, (1.40)ee me

ωξ ωε∇ × + · ′ + · ′ =I i G i Gr r r r( ) ( , ) ( , ) 0, (1.41)mm em

ωζ ωμ δ∇ × − · ′ − · ′ = − − ′I i G i G Ir r r r r r( ) ( , ) ( , ) ( ). (1.42)em mm

These can be manipulated to yield the following four second-order linear differentialequations [14, 15]:

L ω δ∇ · ′ = − ′G i Ir r r r( ) ( , ) ( ), (1.43)eee

L ωζ ε δ∇ · ′ = ∇ × − · − ′−G I ir r r r( ) ( , ) ( ) ( ), (1.44)mme 1

L ωξ μ δ∇ · ′ = − ∇ × + · − ′−G I ir r r r( ) ( , ) ( ) ( ), (1.45)eem 1

L ω δ∇ · ′ = − ′G i Ir r r r( ) ( , ) ( ). (1.46)mmm

In these equations, the linear electric dyadic operator

L ωξ μ ωζ ω ε∇ ≡ ∇ × + · · ∇ × − −−I i I i( ) ( ) ( ) (1.47)e1 2

and the linear magnetic dyadic operator

L ωζ ε ωξ ω μ∇ ≡ ∇ × − · · ∇ × + −−I i I i( ) ( ) ( ) . (1.48)m1 2

Finding the dyadic Green functions in electromagnetism is essentially solvingequations (1.43)–(1.46) for specific constitutive dyadics. However, not all fourequations (1.43)–(1.46) need to be solved, because ′G r r( , )me and ′G r r( , )em can bedetermined from equations (1.40) and (1.41) if ′G r r( , )ee and ′G r r( , )mm are foundfirst. All four Green functions are dyadics, as each maps a vector to a vector. Thevector located at ′r is a source current density phasor, whereas the vector located at ris a field phasor radiated by that source. Accordingly, one can think of a Greenfunction as a transfer function in a linear system [22].

Infinite-Space Dyadic Green Functions in Electromagnetism

1-9

1.3.2 Dyadic Green functions for vector potential phasors

Electromagnetic field phasors can be written in terms of vector potential phasorsA r( ) and F r( ) as [23]

⎫⎬⎪⎭⎪

ω ε ω ξ

ω μ ω ζ

= − __ · ∇ × __ + __ ·

= + __ · ∇ × __ − __ ·

−

−

i I i

i I i

E r A r F r

H r F r A r

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ). (1.49)

1

1

Substitution of equations (1.30), (1.31), and (1.49) in equations (1.26) and (1.27)results in the differential equations

⎪

⎪⎫⎬⎭

L

L

∇ · =

∇ · =

A r J r

F r J r

( ) ( ) ( )

( ) ( ) ( ). (1.50)

e e

m m

The dyadic Green functions for the two vector potential phasors can be defined as

V

∭= ′ · ′ ′G dA r r r J r r( ) ( , ) ( ) (1.51)A

e3

s

and

V

∭= ′ · ′ ′G dF r r r J r r( ) ( , ) ( ) . (1.52)F

m3

s

Substitution of equations (1.37), (1.38), (1.51), and (1.52) in equations (1.50) gives

⎪⎪⎫⎬⎭

L

L

δ

δ

∇ · ′ = − ′

∇ · ′ = − ′

G I

G I

r r r r

r r r r

( ) ( , ) ( )

( ) ( , ) ( ). (1.53)

eA

mF

Equations (1.35), (1.36), (1.49), (1.51), and (1.52) can be used to relate the dyadicGreen functions for the field phasors and the vector potential phasors as follows:

ω′ = ′G i Gr r r r( , ) ( , ), (1.54)ee A

μ ωζ′ = · ∇ × − · ′−G I i Gr r r r( , ) ( ) ( , ), (1.55)me A1

ε ωξ′ = − · ∇ × + · ′−G I i Gr r r r( , ) ( ) ( , ), (1.56)em F1

ω′ = ′G i Gr r r r( , ) ( , ). (1.57)mm F

These equations are in agreement with equations (1.40) and (1.41). In light of equations(1.54)–(1.57), there is insignificant, if any, advantage in casting boundary-value

Infinite-Space Dyadic Green Functions in Electromagnetism

1-10

problems in terms of vector potential phasors than in terms of the field phasors.Furthermore, boundary conditions still have to be specified on the components of thefield phasors.

1.4 Solution approachesThe temporal Fourier transform of equation (1.2) gives

L ω ω δ ω∇ − · ′ ′ = − ′ ′i G t i t Ir r r r( , ) ( , , , ) ( ) exp ( ) . (1.58)

As t′ can be arbitrarily chosen if we impose the condition of translation invariancewith respect to time, and because we are restricted to frequency-domain electro-magnetic fields in linear materials, equation (1.58) is recast simply as

L δ∇ · ′ = − ′G Ir r r r( ) ( , ) ( ) (1.59)

for further use. Equations (1.43), (1.46), and (1.53) exemplify equation (1.59).

1.4.1 Spatial-Fourier-transform approach

An analytical approach based on the spatial Fourier transform is, perhaps, theeasiest of all approaches, in principle, though the eventual application of the inversespatial Fourier transform often proves to be challenging and even impossible. Thespatial Fourier transform of a function F r( ) is defined as

∭˘ = − ·−∞

∞

F F i dq r q r r( ) ( ) exp ( ) , (1.60)3

where q is the wave vector and q is its magnitude. The inverse spatial Fouriertransform is defined as

∭π= ˘ ·

−∞

∞

F F i dr q q r q( )1

(2 )( ) exp ( ) . (1.61)3

3

Application of the spatial Fourier transform reduces equation (1.59) to

L · ˘ ′ = − · ′i G i Iq q r q r( ) ( , ) exp ( ) . (1.62)

Thus, the main feature of this approach is the conversion of the differential dyadicoperator L ∇( ) to the dyadic L iq( ). As this dyadic is a 3 × 3 matrix, it can be invertedusing linear algebra [6].

Use of the spatial Fourier transform thus converts the linear differential equation(1.2) into the algebraic equation (1.62). The solution of equation (1.62) can beformally found by taking the dot products of both of its sides from the left with theinverse L− iq( )1 of L iq( ), i.e.

L˘ ′ = − · ′−G i iq r q q r( , ) ( ) exp ( ), (1.63)1

since L L· =− i i Iq q( ) ( )1 . With M iq( ) denoting the adjoint of L iq( ), we get

Infinite-Space Dyadic Green Functions in Electromagnetism

1-11

LML

=− ii

iq

q

q( )

( )

( ), (1.64)1

provided that L iq( ) has the non-zero determinant L iq( ) [6].Now, the dyadic Green function ′G r r( , ) can be found using the inverse spatial

Fourier transform as

∭π′ = ˘ ′ ·

−∞

∞

G G i dr r q r q r q( , )1

(2 )( , ) exp ( ) (1.65)3

3

ML∭π

= · − ′−∞

∞i

ii d

q

qq r r q

1(2 )

( )

( )exp [ ( )] . (1.66)3

3

Analytic evaluation of the integral in equation (1.65) is possible only for very specificmaterials [6, 15], though approximate expressions can sometimes be obtained [8].

An important, though simple, result concerning the spatial Fourier transform iswidely used in electromagnetism. Taking the gradient of both sides of equation(1.61) results in

∭π∇ = ˘ ∇ ·

−∞

∞

F F i dr q q r q( )1

(2 )( ) [ exp ( )] , (1.67)3

3

∭π= ˘ ·

−∞

∞

i F i dq q q r q1

(2 )[ ( )] exp ( ) . (1.68)3

3

Therefore, the spatial Fourier transform of ∇F r( ) is ˘i Fq q( ). Similarly, the spatialFourier transforms of ∇ · F r( ) and ∇ × F r( ) are · ˘iq F r( ) and × ˘iq F r( ), respectively.Hence, while taking the spatial Fourier transform, one can simply replace (i) allfunctions with their transforms and (ii) ∇ by iq. This property can be used whiletaking the inverse Fourier transform as well. In that case, Fq q( ) can be replaced by− ∇i F r( ), and so on.

As an example, let us use the spatial Fourier transforms of equations (1.43) and(1.46) to get

⎡⎣ ⎤⎦ωξ μ ωζ ω ε

ω

× + · · × − + · ˘ ′

= − − · ′

−I I G

i I i

q q q r

q r

( ) ( ) ( , )

exp ( )(1.69)

ee1 2

and

⎡⎣ ⎤⎦ωζ ε ωξ ω μ

ω

× − · · × + + · ˘ ′

= − − · ′

−I I G

i I i

q q q r

q r

( ) ( ) ( , )

exp ( ),(1.70)

mm1 2

Infinite-Space Dyadic Green Functions in Electromagnetism

1-12

respectively. Dot-multiplication of equation (1.69) from the left with the inverse ofthe dyadic in the square brackets yields ˘ ′G q r( , )ee ; likewise, dot-multiplication ofequation (1.70) from the left with the inverse of the dyadic in the square bracketsyields ˘ ′G q r( , )mm . Thereafter, implementation of the inverse Fourier transform yieldsthe electric and magnetic dyadic Green functions in spectral form as

∭ω ωξ μ ωζ

ω ε

′ = −π

× + · · × −

+ · − ′−∞

∞−

−

Gi

I I

i d

r r q q

q r r q

( , )(2 )

[( ) ( )

] exp [ ( )]

(1.71)ee

31

2 1 3

and

∭ω ωζ ε ωξ

ω μ

′ = −π

× − · · × +

+ · − ′−∞

∞−

−

Gi

I I

i d

r r q q

q r r q

( , )(2 )

[( ) ( )

] exp [ ( )] .

(1.72)mm

31

2 1 3

Equations (1.71) and (1.72) can be used to find ′G r r( , )ee and ′G r r( , )mm if all theinverses on the right sides of these equations exist and the integrations can beperformed either analytically or numerically. But, ε−1 does not exist if ε is purelyanti-symmetric, and μ−1 does not exist if μ is purely anti-symmetric. Therefore,

′G r r( , )ee does not exist when ε is purely anti-symmetric and ′G r r( , )mm does not existwhen μ is purely anti-symmetric.

The spatial-Fourier-transform approach is used for finding the dyadic Greenfunctions for isotropic dielectric-magnetic mediums and isotropic chiral mediums inchapter 2, and for uniaxial dielectric mediums in section 3.3.

1.4.2 Direct approach

When L ∇( ) is simple, its adjoint operator M ∇( ) can sometimes be found usingstandard dyadic manipulations [24, 25]. The adjoint operator of L ∇( ) is definedthrough the relationship

L M∇ · = ∇ ·V r W r V r W r( ), ( ) ( ) ( ) ( ), ( ) , (1.73)

where the inner product of two complex vectors is defined as

∭= ·−∞

∞* dV r W r V r W r r( ), ( ) ( ) ( ) . (1.74)3

Accordingly,

M L L M N∇ · ∇ ≡ ∇ · ∇ ≡ ∇ I( ) ( ) ( ) ( ) ( ) , (1.75)

where N ∇( ) is a scalar operator.

Infinite-Space Dyadic Green Functions in Electromagnetism

1-13

On taking first the dot product from the left with M ∇( ) of both sides of equation(1.59) and then the spatial Fourier transforms of both sides of the result, theexpression

MM L

˘ ′ = − · ′·

G ii

i iq r q

q rq q

( , ) ( )exp ( )

( ) ( )(1.76)

emerges, where M L·i iq q( ) ( ) is a polynomial in q. The inverse spatial Fouriertransform yields

M′ = − Δ ′G gr r r r( , ) ( ) ( , ), (1.77)n

where the scalar Green function

N∭π′ = − · − ′

−∞

∞

gi

idr r

q r rq

q( , )1

(2 )exp [ ( )]

( )(1.78)n 3

3

satisfies the scalar differential equation

N δ∇ ′ = − − ′g r r r r( ) ( , ) ( ). (1.79)n

Since M ∇( ) contains second-order derivatives because L ∇( ) also does, N ∇( )contains fourth-order derivatives in general. A successful attempt to factor N ∇( )into operators containing derivatives of order not exceeding two is considered verydesirable [25, 26]. The direct approach is used extensively in chapter 3 for severalmediums including axially uniaxial bianisotropic mediums (sections 3.6 and 3.7) andself-dual bianisotropic mediums (section 3.10).

1.4.3 Eigenfunction-expansion approach

An approach that results in a series solution for the dyadic Green function employsthe eigenfunction spectrum of the linear operator L ∇( ), provided that the operator isself adjoint, i.e.

M L∇ ≡ ∇( ) ( ). (1.80)

In order to solve equation (1.59) to determine the dyadic Green function for theoperator L ∇( ), let us consider the eigenvalue equation [27]

L σ∇ · = ∈ …mV r V r( ) ( ) ( ), {1, 2, 3, }, (1.81)m m m

where V r( )m is the mth eigenvector corresponding to the mth eigenvalue σm. SinceL ∇( ) is a self-adjoint operator, its eigenvectors can be normalized to form anorthonormal set. Let us therefore assume that

δ=V r V r( ), ( ) , (1.82)m n mn

where

Infinite-Space Dyadic Green Functions in Electromagnetism

1-14

⎧⎨⎩δ = =≠

m nm n

1, ,0, ,

(1.83)mn

is the Kronecker delta. The eigenvectors also form a complete set, i.e.

∑ δ′ = − ′* IV r V r r r[ ( ) ( )] ( ) . (1.84)m

m m

As the eigenvectors therefore form a basis, the dyadic Green function can beexpanded in terms of these eigenvectors as

∑′ = ′G r r V r A r( , ) [ ( ) ( )] (1.85)m

m m

with unknown coefficient vectors ′A r( )m . Let us substitute this representation on theleft side of equation (1.59) to get

L L∑ ∑

∑ σ

∇ · ′ = ∇ · ′

= ′

{ }V r A r V r A r

V r A r

( ) [ ( ) ( )] [ ( ) ( )] ( )

[ ( ) ( )],(1.86)

m m

m

m m m m

m m m

by virtue of equation (1.81). The use of equations (1.84) and (1.86) in equation (1.59)then yields

∑ ∑σ ′ = ′*V r A r V r V r[ ( ) ( )] [ ( ) ( )]. (1.87)m m

m m m m m

In consequence of the orthonormality relation (1.82), we get

σ′ = ′*

A rV r

( )( )

. (1.88)mm

m

The end result is the bilinear expansion

⎡⎣⎢

⎤⎦⎥∑

σ′ = ′*

G r rV r V r

( , )( ) ( )

(1.89)m

m m

m

of the dyadic Green function. This bilinear expansion holds for ≠ ′r r .The foregoing approach has been used for scalar Green functions [2] as well as

dyadic Green functions [7, 27]. Provided that the eigenanalyses of the operatorsL ∇( )e and L ∇( )m , defined in equations (1.47) and (1.48), can be performed, theapproach can be used to determine the four infinite-space dyadic Green functions fora general bianisotropic medium as follows [28].

Let the vector wavefunctions Φν r( )j( ) be the eigensolutions of the differentialequation

L Φ∇ · =ν r 0( ) ( ) , (1.90)ej( )

Infinite-Space Dyadic Green Functions in Electromagnetism

1-15

where the subscripted index ν ∈ …{1, 2, 3, } and the superscripted index ∈j {1, 3}.Similar to Φν r( )j( ) , let the vector wavefunctions Ψν r( )j( ) be the eigensolutions of thedifferential equation

L Ψ∇ · =ν r 0( ) ( ) . (1.91)mj( )

Whereas Φν r( )(1) and Ψν r( )(1) are regular at the origin, Φν r( )(3) and Ψν r( )(3) are regular atinfinity. These two sets of vector wavefunctions can then be used to formulate thefollowing two additional sets of wavefunctions:

⎫⎬⎪⎪

⎭⎪⎪

ωμ ω ζ

ωε ω ξ

Θ

Ξ Ψ

= __ · ∇ × __ − __ · Φ

= − __ · ∇ × __ + __ ·

ν ν

ν ν

−

−

iI i

iI i

r r

r r

( )1

( ) ( )

( )1

( ) ( ). (1.92)

j j

j j

( ) 1 ( )

( ) 1 ( )

In fully three-dimensional scattering problems, the functions identified by j = 1 areused to represent the incident field phasors, whereas the functions identified by j = 3are used to represent the scattered field phasors.

The bilinear expansions of the four dyadic Green functions must then be [28]

⎧

⎨⎪⎪

⎩⎪⎪

∑

∑

α β

α β

Φ Φ Ξ Ξ

Φ Φ Ξ Ξ′ =

′ + ′ > ′

′ + ′ > ′

ν

ν

=

∞

=

∞

ν ν ν ν ν ν

ν ν ν ν ν ν

G

r r

r r

r r

r r r r

r r r r

( , )

[ ( ) ( ) ( ) ( )], ,

[ ( ) ( ) ( ) ( )], ,

, (1.93)1

1

ee

(3) (1) (3) (1)

(1) (3) (1) (3)

⎧

⎨⎪⎪

⎩⎪⎪

∑

∑

α β

α β

Θ Θ Ψ Ψ

Θ Θ Ψ Ψ′ = −

′ + ′ > ′

′ + ′ > ′

ν

ν

=

∞

=

∞

ν ν ν ν ν ν

ν ν ν ν ν ν

G

r r

r r

r r

r r r r

r r r r

( , )

[ ( ) ( ) ( ) ( )], ,

[ ( ) ( ) ( ) ( )], ,

, (1.94)1

1

mm

(3) (1) (3) (1)

(1) (3) (1) (3)

⎧

⎨⎪⎪

⎩⎪⎪

∑

∑

α β

α β

Θ Φ Ψ Ξ

Θ Φ Ψ Ξ′ =

′ + ′ > ′

′ + ′ > ′

ν

ν

=

∞

=

∞

ν ν ν ν ν ν

ν ν ν ν ν ν

G

r r

r r

r r

r r r r

r r r r

( , )

[ ( ) ( ) ( ) ( )], ,

[ ( ) ( ) ( ) ( )], ,

, (1.95)1

1

me

(3) (1) (3) (1)

(1) (3) (1) (3)

⎧

⎨⎪⎪

⎩⎪⎪

∑

∑

α β

α β

Φ Θ Ξ Ψ

Φ Θ Ξ Ψ′ = −

′ + ′ > ′

′ + ′ > ′

ν

ν

=

∞

=

∞

ν ν ν ν ν ν

ν ν ν ν ν ν

G

r r

r r

r r

r r r r

r r r r

( , )

[ ( ) ( ) ( ) ( )], ,

[ ( ) ( ) ( ) ( )], ,

, (1.96)1

1

em

(3) (1) (3) (1)

(1) (3) (1) (3)

Infinite-Space Dyadic Green Functions in Electromagnetism

1-16

with the coefficients αν and βν to be determined using a standard technique [27,chapter 13] if equations (1.90) and (1.91) can be solved to find the eigenvectors andeigenvalues. These bilinear expansions satisfy equations (1.43)–(1.46) when ≠ ′r r .The sum over ν is converted to an integral when ν is a continuous variable. Specificexamples are provided in chapter 4.

1.4.4 Scalarization approach

Closely related to the direct approach of section 1.4.2, the scalarization approachaims to reduce the problem of finding the dyadic Green function to the problem offinding a scalar Green function. In this approach, the dyadic equation (1.59) ismanipulated using vector and dyadic operations until it can be written as [24, 25, 29]

H W δ∇ ′ = ∇ − ′G r r r r( ) ( , ) ( ) ( ), (1.97)

where the scalar operator H ∇( ) is such that

H W W H∇ ∇ ≡ ∇ ∇( ) ( ) ( ) ( ). (1.98)

Equation (1.97) can be used to find the dyadic Green function as

W′ = − ∇ ′G gr r r r( , ) ( ) ( , ), (1.99)h

where the scalar Green function ′g r r( , )h satisfies the equation

H δ∇ ′ = − − ′g r r r r( ) ( , ) ( ). (1.100)h

This approach is used in section 3.5 to derive dyadic Green functions for uniaxialdielectric-magnetic mediums.

1.5 Organization of the monographThis monograph is focused mainly on those anisotropic and bianisotropic mediums,all linear and homogeneous, for which the closed-form dyadic Green functions areknown when the chosen medium occupies all space. Many derivations are providedto train the reader in the art and science of finding the infinite-space dyadic Greenfunctions. The usefulness of these dyadic Green functions for finding the electric andmagnetic field phasors due to simple sources is illustrated for simpler cases. Theusefulness of the same functions in setting up scattering problems for generalbianisotropic mediums is also presented. A brief introduction to dyadics and someimportant identities used in this monograph are provided in appendix A. Thedetailed plan of the following chapters is as follows.

The infinite-space dyadic Green functions for the isotropic dielectric-magneticand isotropic chiral mediums are derived in chapter 2 using the spatial-Fourier-transform approach. The electromagnetic fields radiated by point-electric and point-magnetic dipoles are presented with emphasis on fields in the near zone and the farzone. Far-zone radiation from an electrically small current loop is compared withfar-zone radiation from a point-magnetic dipole. A construct called the non-reciprocal biisotropic medium is also briefly discussed in chapter 2.

Infinite-Space Dyadic Green Functions in Electromagnetism

1-17

Chapter 3 is the keystone of this monograph. It presents the derivations of theclosed-form dyadic Green functions for the uniaxial dielectric medium; the uniaxialmagnetic medium; the uniaxial dielectric-magnetic medium; the Lorentz-reciprocal,axially uniaxial, bianisotropic medium; the Lorentz-nonreciprocal, axially uniaxial,bianisotropic medium; the Lorentz-reciprocal, anisotropic dielectric-magneticmedium with cross-handed gyrotropy; the general self-dual bianisotropic medium;and a special gyrotropic bianisotropic medium. For each of these derivations, eithera single approach or a combination of more than one approaches described insection 1.4 is used. The electromagnetic fields radiated by point-electric and point-magnetic dipoles in a few uniaxial mediums are also presented. Furthermore, a set oftransformations is also presented that can be used to determine the dyadic Greenfunctions for mediums obtained by transformations of other mediums.

Series representations of the dyadic Green functions for the isotropic dielectric-magnetic and isotropic chiral mediums in the Cartesian, cylindrical, and sphericalcoordinate systems are presented in chapter 4 using the eigenfunction-expansionapproach. Furthermore, series representation of the dyadic Green functions for aspecial orthorhombic dielectric-magnetic medium are presented in the sphericalcoordinate system.

Chapter 5 is dedicated to the application of the infinite-space dyadic Greenfunctions in scattering problems. The chapter begins with the formulations of theEwald–Oseen extinction theorem and the Huygens principle for quite a generalbianisotropic medium, though not the most general. Application to scatteringproblems follows. Since dyadic Green functions are singular when the source pointand the field point coincide, depolarization dyadics are introduced next in thischapter to determine the electric and magnetic field phasors in the source region.The method of moments and the coupled-dipole method are formulated using thedepolarization dyadics and the dyadic Green functions. The chapter and themonograph conclude with a brief introduction to homogenization formalisms forbianisotropic mediums.

References[1] Stakgold I 1967 Boundary Value Problems of Mathematical Physics (New York: Macmillan)[2] Dudley D G 1994 Mathematical Foundations for Electromagnetic Theory (Piscataway, NJ:

IEEE Press)[3] Hayek S I 2011 Advanced Mathematical Methods in Science and Engineering 2nd edn (Boca

Raton, FL: CRC Press)[4] Varatharajulu V and Pao Y-H 1976 Scattering matrix for elastic waves. I. Theory J. Acoust.

Soc. Am. 60 556–66[5] Lakhtakia A, Varadan V K, Varadan V V and Wall D J N 1984 The T-matrix approach for

scattering by a traction-free periodic rough surface J. Acoust. Soc. Am. 76 1839–46[6] Chen H C 1983 Theory of Electromagnetic Waves: A Coordinate-Free Approach (New York:

McGraw–Hill)[7] Tai C-T 1994 Dyadic Green Functions in Electromagnetic Theory (Piscataway, NJ: IEEE

Press)

Infinite-Space Dyadic Green Functions in Electromagnetism

1-18

[8] Felsen L B and Marcuvitz N 1994 Radiation and Scattering of Waves (Piscataway, NJ: IEEEPress)

[9] Van Bladel J 1991 Singular Electromagnetic Fields and Sources (Oxford: Clarendon)[10] Van Bladel J 2007 Electromagnetic Fields 2nd edn (Hoboken, NJ: Wiley)[11] Lindell I V 1992 Methods for Electromagnetic Field Analysis (Oxford: Clarendon)[12] Kanwal R P 2004 Generalized Functions: Theory and Applications 3rd edn (Boston, MA:

Birkhäuser)[13] Thomas R E, Rosa A J and Toussaint G J 2016 The Analysis and Design of Linear Circuits

8th edn (Hoboken, NJ: Wiley)[14] Mackay T G and Lakhtakia A 2008 Electromagnetic fields in linear bianisotropic mediums

Prog. Opt. 51 121–209[15] Mackay T G and Lakhtakia A 2010 Electromagnetic Anisotropy and Bianisotropy

(Singapore: World Scientific)[16] Post E J 1997 Formal Structure of Electromagnetics (New York: Dover)[17] Lakhtakia A 2004 On the genesis of Post constraint in modern electromagnetism Optik

115 151–8[18] Widom A, Friedman M H and Srivastava Y 1986 Surface Hall effect in magneto-electric

media J. Phys. A: Math. Gen 19 L175–6[19] Kim D J, Thomas S, Grant T, Botimer J, Fisk Z and Xia J 2013 Surface Hall effect and

nonlocal transport in SmB6: evidence for surface conduction Sci. Rep. 3 3150[20] Lakhtakia A and Mackay T G 2016 Classical electromagnetic model of surface states in

topological insulators J. Nanophoton. 10 033004[21] Lakhtakia A and Mackay T G 2016 Electromagnetic scattering by homogenous, isotropic,

dielectric-magnetic sphere with topologically insulating surface states J. Opt. Soc. Am. B 33603–10

[22] Callier F M and Desoer C A 1990 Linear System Theory (New York: Springer)[23] Lindell I V and Olyslager F 2001 Potentials in bi-anisotropic media J. Electromagn. Waves

Appl. 15 3–18[24] Weiglhofer W S 1993 Analytic methods and free-space dyadic Green’s functions Radio Sci.

28 847–57[25] Weiglhofer W S 1995 Frequency-dependent dyadic Green functions for bianisotropic media

Advanced Electromagnetism: Foundations, Theory and Applications ed T W Barrett and D WGrimes (Singapore: World Scientific), 376–89

[26] Olyslager F and Lindell I V 2002 Electromagnetics and exotic media: a quest for the holygrail IEEE Antennas Propagat. Mag. 44(2) 48–58

[27] Morse P M and Feshbach H 1953 Methods of Theoretical Physics, Vol. II (New York:McGraw–Hill)

[28] Lakhtakia A 2018 The Ewald–Oseen extinction theorem and the extended boundarycondition method The World of Applied Electromagnetism: In Appreciation of MagdyFahmy Iskander ed A Lakhtakia and C M Furse (Cham, Switzerland: Springer), 481–513

[29] Weiglhofer W 1987 Scalarisation of Maxwell’s equations in general inhomogeneousbianisotropic media IEE Proc. H 134 357–60

Infinite-Space Dyadic Green Functions in Electromagnetism

1-19