Imaging and radiation enhancements from metamaterials...mula for extended charged particle embedded...

MOHAMED-RABIGH KHODJA

IMAGING AND RADIATION ENHANCEMENTS FROM

METAMATERIALS

PHD DISSERTATION

NORTHEASTERN UNIVERSITY BOSTON, MASSACHUSETTS

2008

IMAGING AND RADIATION ENHANCEMENTS FROM METAMATERIALS

A Dissertation Presented

by

Mohamed-Rabigh Khodja

to

The Department of Electrical and Computer Engineering

in partial fulfillment of the requirements

for the degree of

Doctor of Philosophy

in

Electrical Engineering

Northeastern University Boston, Massachusetts

September 2008

c° Copyright 2008 by Mohamed-Rabigh Khodja

All Rights Reserved

Vita

Mohamed-Rabigh Khodja was born in Biskra, Algeria. He obtained a Diplôme

d’Études Supérieurs (DES) in Theoretical Physics from the University of Constan-

tine (now Mentouri University of Constantine), Constantine, Algeria, in 1994, and

a Master’s degree in Physics from King Fahd University of Petroleum and Minerals,

Dhahran, Saudi Arabia, in 1998. (The title of his Master’s thesis was “Radiative and

Nonleptonic Decays of Bottom Baryons in the Quark Model”). He came to North-

eastern University in 2004 to do a PhD in Physics then moved to the Department

of Electrical and Computer Engineering in 2005. Since then he has been engaged

in research, supervised by Professor Edwin A. Marengo, on electromagnetic field

theory. Prior to coming to Northeastern, he was a Lecturer in Physics at King Fahd

University of Petroleum and Minerals, Dhahran, Saudi Arabia. He is a member

of the American Physical Society (APS), the Institute of Electrical and Electronics

Engineers (IEEE), and the American Association for the Advancement of Science

(AAAS).

iv

Publication Record

To Date

1. M. R. Khodja and E. A. Marengo. Radiation enhancement due to metama-

terial substrates from an inverse source theory, Phys. Rev. E, 77:046605,

2008.

2. E. A. Marengo, M. R. Khodja, and A. Boucherif. Inverse source problem in

non-homogeneous background media. Part II: Vector formulation and antenna

substrate performance characterization, SIAM J. App. Math., 69(1):81-110,

2008.

3. M. R. Khodja and E. A. Marengo. Comparative study of radiation enhance-

ment due to metamaterials, invited paper, Radio Sci., doi:10.1029/2007RS003803,

in press, 2008.

4. E. A. Marengo and M. R. Khodja. Generalized power-spectrum Larmor for-

mula for extended charged particle embedded in a harmonic oscillator, Phys.

Rev. E, 74:036611, 2006.

5. P. Abdel-Jalil, M. R. Khodja, and A. Al-Suwayyan. A Physical model for the

distribution of ions and electrons in laminar premixed hydrocarbon flames,

Arab. J. Sc. Eng., 26(2A):127-135, 2001.

6. M. R. Khodja, Riazuddin, and Fayyazuddin. Two-body nonleptonic decays of

bottom baryons in the quark model, Phys. Rev. D, 60:053005, 1999.

v

7. A. Aksoy and M. R. Khodja. Calibration of natural gamma-ray spectrometer

at the ERL, Arab. J. Sc. Eng., 24(1A):73-79, 1999.

8. E. A. Marengo and M. R. Khodja. Source synthesis in substrate media: Fun-

damental bounds, International Symposium on Electromagnetic Theory, URSI

Commision B, Ottawa, Ontario, Canada, July 26-28, 2007.

9. E. A. Marengo and M. R. Khodja. Source inversion in metamaterial sub-

strate media: Fundamental limits and applications, URSI/CNC/USNC North

American Radio Science Conference, Ottawa, Ontario, Canada, July 22-26,

2007.

Projected

1. M. R. Khodja and E. A. Marengo. The Casimir effect in the presence of

metamaterials, to be submitted to Phys. Rev. Lett., 2008.

2. M. R. Khodja and E. A. Marengo. The Casimir effect and the dispersive

nature of metamaterials, to be submitted to Phys. Rev. E, 2008.

3. M. R. Khodja and E. A. Marengo. Inverse scattering and imaging resolution

enhancements in metamaterial backgrounds, to be submitted to J. Opt. Soc.

Am., 2008.

4. R. Hernandez, H. Lev-Ari, A. M. Stankovic, M. R. Khodja, and E. A. Marengo.

Fundamental performance limits in lossy polyphase systems: Apparent power

and optimal compensation, to be submitted to IEEE Trans. on Circuits and

Systems, 2008.

5. E. A. Marengo, M. R. Khodja, and A. Widom. Possibilities and limitations of

employing the Aharonov-Bohm effect for remote sensing and communications,

in preparation.

vi

Dedication

I dedicate this dissertation to the spirit of my late father, Abdelhamid, and to mybeloved mother, Saida, for all their sacrifices to advance my education and to getme to this point in my life.

I also dedicate this dissertation to my wife, Fatima-Zohra, for sharing withme the ups and downs of my journey, and to my parents-in-law Abdelkader andNadjia for all they have done to see me succeed, and they have done so much.

Last but not least, I dedicate this dissertation to the lights of my world, my sons

Abdelhamid (Didu) and Belkacem-Eltaher (Weewer), for enriching my lifein so many ways.

vii

Acknowledgments

I wish to express my deepest gratitude to my mentor and friend Dr. Edwin A.

Marengo for his guidance and support over the past three years. Throughout these

years, I have greatly enjoyed his dedication, creativity, and untiring enthusiasm. His

trust and unflagging pursuit of excellence have made this dissertation what it is. I

could not have hoped for a better dissertation supervisor.

I am privileged to have had Drs. Anthony J. Devaney and Philip E. Serafim

as teachers and members in my dissertation committee. I thank them for their

continuous encouragement and for all the wonderful things I have learned from

them. My gratitude goes to my father-in-law Dr. Abdelkader Boucherif for his

wholehearted support and for introducing me to functional analysis. I am also

grateful to Dr. Stephen W. McKnight for his kind help in the transfer process

from the Department of Physics to the Department of Electrical and Computer

Engineering when he was Chairman of the latter.

All of my family and my wife’s family supported me throughout the years, despite

the great distances between us. I especially thank my brothers, Noureddine-Omar,

Karim-Abdelaziz, and Tarek-Abdelhafidh, my sisters, Nadia, Linda, and Merzaka,

my brothers-in-law, Ammar and Bachir, and my sisters-in-law, Nabila and Dalal. I

also thank my wife’s grand parents, Youcef and Halima, for always remembering us

in their prayers.

I thank my friends and my family’s friends for being such an important part of

our lives. They are too many to mention by name. My family and I are especially

indebted to Mr. Belkacem Naidjate and his family, Dr. Jamal Lebeche and his

family, Dr. Amine Nehari-Talet and his family, Dr. Ghassen Benbrahim, and Mr.

viii

Abubakr Naidjate, for their unconditional kindness.

I cannot thank my wife, Fatima-Zohra, enough for her emotional support, pa-

tience, and unwavering faith during the hard times.

Finally, I gratefully acknowledge financial support provided by Northeastern

University in the form of a Dissertation Writing Fellowship during the summer of

2008. The work embodied in this dissertation was also supported in part by the

United States Air Force Office of Scientific Research under Grant No. FA9550-06-

01-0013, and by the National Science Foundation under Grant No. 0746310.

ix

Abstract

This dissertation is divided into two parts. The first part solves the full-vector, elec-

tromagnetic inverse source problem of synthesizing an unknown source embedded in

a given substrate medium of volume V and radiating a prescribed exterior field. Im-

portantly, the derived formulation is non-antenna-specific. It comprises forward, or

radiation, characterization as well as inverse-theoretic characterization. The forward

characterization is focused on the singular value spectrum while the inverse-theoretic

characterization is performed via the “minimum-energy” sources. Particular atten-

tion is given to the case of two nested spheres made up of materials with oppositely

signed constitutive parameters. We find that, for a given antenna radiating at a

prescribed frequency, the singular values spectra exhibit resonances that correspond

to maximum radiation enhancements. These resonances are primarily due to the

presence of polaritons which, in turn, correspond to sets of constitutive parameters

that maximize the radiated fields. We also find that for electrically small antenna

systems made up of materials with oppositely signed constitutive parameters the

emergence of resonances depends on the ratio of the two radii rather than on the

overall size of the system.

The second part of this dissertation explores the effects that the presence of

electromagnetic metamaterials would have on the Casimir forces. Along with their

companion van der Waals forces, the Casimir forces, are identified as the primary

cause for the collapse of microelectromechanical systems (MEMS) and nanoelectro-

mechanical systems (NEMS). Hence, it is crucial for the future development of nan-

otechnology to investigate the possibility of shielding these potentially destructive

forces. One such avenue is explored by means of electromagnetic metamaterials. A

x

surface modes sum is performed to derive an analytic expression for the Casimir force

in a dispersive four-region system with planar geometry. In the case of two perfectly

conducting plates separated by two media, the numerical simulations demonstrate

that the Casimir force is characterized by a new singular behavior in the presence of

DPS-DNG combinations. The possible utilization of such systems for the shielding

of MEMS and NEMS and for the realization of quantum levitation is discussed.

The theoretical studies are accompanied by numerical illustrations.

xi

Contents

Vita iv

Publication Record v

Dedication vii

Acknowledgments viii

Abstract x

1 Introduction 1

1.1 Electromagnetic Metamaterials . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Physical Constraints on the Response Functions . . . . . . . 2

1.1.2 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Inverse Source Problem in Substrate Media . . . . . . . . . . . . . . 3

1.3 The Casimir Effect and Electromagnetic Metamaterials . . . . . . . 8

2 Inverse Source Problem in Non-homogeneous Background Media 12

2.1 The Forward Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.1.1 Electromagnetic Generalities . . . . . . . . . . . . . . . . . . 14

2.1.2 Source-to-Multipole-Moment Mapping . . . . . . . . . . . . . 16

2.2 Inverse Source Theory Based on Constrained Optimization . . . . . 19

2.2.1 Minimum Energy Solution by Constrained Optimization . . . 19

2.2.2 Minimum Energy Source Having Zero Reactive Power . . . . 22

2.3 Computer Simulation Study . . . . . . . . . . . . . . . . . . . . . . . 32

xii

2.3.1 Minimum Energy Sources . . . . . . . . . . . . . . . . . . . . 33

2.3.2 Tuned Minimum Energy Sources: Additional Zero Reactive

Power Constraint . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3 Radiation Enhancement due to Metamaterial Substrates: Core-Shell System 543.1 The Radiation Problem . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.2 Inverse Source Theory Based on Constrained Optimization . . . . . 58

3.3 Numerical Results and Case Studies . . . . . . . . . . . . . . . . . . 59

3.3.1 Lossless Substrates . . . . . . . . . . . . . . . . . . . . . . . . 62

3.3.2 Lossy Substrates . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4 Comparative Study of Radiation Enhancement due to Metamate-rials 754.1 Reference Antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.2 Numerical Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5 The Casimir Effect in the Presence of Metamaterials 825.1 General Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.1.1 Eigenfrequencies . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.1.2 The Casimir Force . . . . . . . . . . . . . . . . . . . . . . . . 86

5.2 Particular Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.2.1 One Slab of Material Separating Two Half-Spaces . . . . . . 88

5.2.2 Two Slabs of Material Separating Two Perfectly Conducting

Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.2.3 The One-Slab Case and Casimir’s Classical Result . . . . . . 91

5.3 Numerical Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.3.1 Cavities with DPS-DPS and DNG-DNG Slab Combinations . 93

5.3.2 Cavities with DPS-DNG Slab Combinations . . . . . . . . . . 93

5.3.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

xiii

6 Future Directions 1006.1 Extensions to the Inverse Source Problem in Antenna Substrate Media100

6.2 Extensions to the Inverse Scattering Problem in Substrate Media . . 101

6.3 Electrodynamics of Metamaterials . . . . . . . . . . . . . . . . . . . 101

A Definition and Properties of Vector Spherical Harmonics 103

B WavefunctionsB(j)l,m for Piecewise-Constant Radially-Symmetric Back-

grounds 106

C Wavefunctions B(j)l,m for a System of Two Nested Spheres 113

D Calculations of the Fréchet Derivatives of the Objective Functionaland the Constraints 115

E Wavefunctions B(j)l,m for Spherically Symmetric Backgrounds 117

F Establishment of The Dispersion Relations (5.2) and (5.3) 119

G Application of the Argument Principle to the Derivation of Eq.(5.20) 123

Bibliography 127

xiv

List of Figures

2.1 Schematic of a general antenna whose driving points and material

structure are confined within a spherical volume V of radius a. . . . 13

2.2 Free-space singular valueshσ(1)l (x = x0, r = +1 = µr)

i2versus l for

a few representative values of x0 ≡ k0a/π. (The unit of the singular

values is V 2m/A2.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.3 Normalized singular valuesh(1)l

i2versus l for x0 = 1/4 (quarter-wave

case), r = +1 and a few representative values of x. . . . . . . . . . 35


i2versus l for x0 = 10 (resonant or

electrically-large antenna), r = +1 and a few representative values

of x. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36



case), r = +1 and a few representative values of x. . . . . . . . . . . 37



electrically-large antenna), r = +1 and a few representative values

of x. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.7 Logarithmic mesh plot of the source energy E(j=1=l)ME versus x0 and x

for a double-positive material with r = +1. . . . . . . . . . . . . . . 41

2.8 Logarithmic plot of the source energy E(j=1=l)ME versus x for r = +1

and some representative values of x0 for a double-positive medium. . 42

2.9 Logarithmic plot of the source energy E(j=1=l)ME versus x for r = −1and some representative values of x0 for a double-negative medium. . 43

2.10 Logarithmic plot of the source energy E(j=1=l)ME versus x for µr = 1

and some representative values of x0. . . . . . . . . . . . . . . . . . . 44

xv

2.11 Logarithmic plot of the source energy E(j=1=l)ME (x = x0) versus x = x0

for r = +1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.12 Logarithmic plot of the gainG, versus x for r = +1 and some selected

values of x0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.13 Plot of the normalized reactive power g(1)1 versus χ for x0 = 1/4 = x

and r = +1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.14 Plot of the source energy E(j=1=l)EP versus χ for for x0 = 1/4 = x and

r = +1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.15 Plot of the normalized reactive power g(1)1 versus χ for x0 = 1/4 = −x

and r = −1 (“anti-vacuum.”) . . . . . . . . . . . . . . . . . . . . . . 48

2.16 Plot of the source energy E(j=1=l)EP versus χ for x0 = 1/4 = −x and

r = −1 (“anti-vacuum.”). . . . . . . . . . . . . . . . . . . . . . . . 49

3.1 Geometry of the three-region system under consideration. The driving

points and material structure of the antenna are confined within a

spherical volume V of radius a. The inner sphere of radius a has rel-

ative permittivity a and relative permeability µa. This inner sphere

is surrounded by a spherical shell of inner radius a and outer radius

b and has relative permittivity b and relative permeability µb. The

core-shell system is immersed in the vacuum. . . . . . . . . . . . . . 55

3.2 Logarithmic plot of the normalized electric singular valuesh(1)l

i2for a λ/4 antenna versus the radii ratio d ≡ b/a. The inner sphere

is assumed not to contain any material (i.e., a = 1 = µa) and the

surrounding shell is assumed to be a lossless DPS material with µb = 1

and xb = 50 m−1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62





and xb = 150 m−1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

xvi





and xb = 500 m−1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.5 Logarithmic plot of the normalized singular valuesh(1)l

i2for a λ/400

antenna versus the radii ratio d ≡ b/a. The inner sphere is assumed

not to contain any material (i.e., a = 1 = µa) and the surrounding

shell is assumed to be a lossless DPS material with µb = 1 and xb =

150 m−1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65


i2for a λ/4



shell is assumed to be a lossless DNG material with µb = −1 andxb = −50 m−1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66


i2for

a λ/40 antenna versus the radii ratio d ≡ b/a. The inner sphere


surrounding shell is assumed to be a lossless DNG material with µb =

−1 and xb = −50 m−1. . . . . . . . . . . . . . . . . . . . . . . . . . . 67


i2for



surrounding shell is assumed to be a lossless DNG material with µb =

−1 and xb = −50 m−1. . . . . . . . . . . . . . . . . . . . . . . . . . . 68


i2for a λ/400



shell is assumed to be an ENG material with µb = 1 and xb = i50 m−1. 69

xvii


i2for



surrounding shell is assumed to be a lossless DNG material (black

curves), a lossy DNG shell with a loss tangent set to 1/20 (blue

curves), and a lossy DNG material with a loss tangent set to 1/60

(red curves). In all three cases µb = −1, Re[xb] = −150 m−1. . . . . 70

3.11 Logarithmic plot of the gainG for a λ/4-electric-dipole antenna versus

the radii ratio d ≡ b/a. The surrounding shell is assumed to be a DNG

material with µb = −1. . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.12 Logarithmic plot of the gain G for a λ/40-electric-dipole antenna

versus the radii ratio d ≡ b/a. The surrounding shell is assumed to

be a DNG material with µb = −1. . . . . . . . . . . . . . . . . . . . 72

3.13 Logarithmic plot of the gain G for a λ/400-electric-dipole antenna

versus the radii ratio d ≡ b/a. The surrounding shell is assumed to

be a DNG material with µb = −1. . . . . . . . . . . . . . . . . . . . 73

4.1 Logarithmic plot of the enhancementh(1)l

i2for a quarter-wavelength

antenna with radius a = 3.75 mm versus the radii ratio b/a. The shell

is a lossless DNG material with b = −4 and µb = −1. The referenceantenna is assumed to be RA1 antenna. . . . . . . . . . . . . . . . . 77




is a lossless DNG material with b = −4 and µb = −1. The referenceantenna is assumed to be an RA2 antenna. . . . . . . . . . . . . . . 78






i2for a full—wavelength

antenna with radius a = 1.5 cm versus the radii ratio b/a. The shell


xviii


i2for a full-wavelength




i2for a full-wavelength



5.1 Planar geometry under consideration. Two slabs of thicknesses a and

b and relative constitutive parameters a, µa and b, µb, respec-tively, are sandwiched between two half-spaces with relative consti-

tutive parameters 1, µ1 and 2, µ2 . . . . . . . . . . . . . . . . . 83

5.2 A rectangular cavity of dimensions L×L×d is constructed by puttingtwo perfectly conducting plates with dimensions L×L× τ (LÀ dÀτ) in the vacuum parallel to one another and separated by a distance

d. Two slabs of isotropic, homogeneous, lossless material are inserted

between the two plates. The first slab has dimensions L × L × a

and relative constitutive parameters a, µa; the second slab hasdimensions L × L × b and relative constitutive parameters b, µb .(a+ b = d). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.3 Plot of the normalized Casimir force f for DPS-DPS and DNG-

DNG slab combinations as a function of the ratio of thicknesses

r ≡ a/b. The constitutive parameters of the DPS-DPS combina-

tion are a, µa = 2, 1 and b, µb = 2.5, 1.5 . The constitutiveparameters of the DNG—DNG combination are a, µa = −2,−1and b, µb = −2.5,−1.5 . For both combinations the thickness ofthe second slab is set to b = 0.001 mm. . . . . . . . . . . . . . . . . . 94

xix

5.4 Plot of the normalized Casimir force f for a DPS-DNG slab com-

bination as a function of the ratio of thicknesses r ≡ a/b. For this

combination we have b/ a = −a/b = −r = µb/µa.The constitutive

parameters of the DPS slab are a, µa = 2, 1 and those of theDNG slab are b, µb = −2r,−r . The thicknesses of the DNGslabs were set to the arbitrary values 0.001 mm, 0.1 mm and 10 mm.

Because the curves coincide exactly with one another, a purely arti-

ficial shift of magnitude −10 has been introduced in the plot of thenormalized force that corresponds to b = 0.1 mm and a similar arti-

ficial shift of magnitude −20 has been introduced in the plot of thenormalized force that corresponds to b = 10 mm. . . . . . . . . . . . 95

5.5 Plot of the normalized Casimir force f for a DPS-DNG slab com-

bination as a function of the ratio of thicknesses 1/r ≡ b/a. The

constitutive parameters of the DPS slab are a, µa = 2, 1 andthose of the DNG slab are b, µb = −2.5,−1.5 . The thickness ofthe DPS slab is set to a = 0.001 mm. . . . . . . . . . . . . . . . . . . 96

5.6 Plot of the normalized Casimir force f for a DPS-DNG slab combi-

nation as a function of the ratio of thicknesses r ≡ a/b. The consti-

tutive parameters of the DPS slab are a, µa = 2, 1 and those ofthe DNG slab are b, µb = −2.5,−1.5 . The thickness of the DNGslab is set to a = 0.001 mm. . . . . . . . . . . . . . . . . . . . . . . . 97

5.7 Plot of the normalized Casimir force f described in Fig.( 5.6) for

different values of r. . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

G.1 The contour, C, used in Eqs.(G.4,G.5); it is defined by the imaginary

axis and the semicircle of infinite radius directed to the right. The

points ω1, ..., ωN represent the N solutions of the dispersion relations

g (ω) = 0 or gµ (ω) = 0. . . . . . . . . . . . . . . . . . . . . . . . . . 125

xx

List of Tables

2.1 Results of the numerical study for the constrained quarter-wave an-

tenna embedded in a double-positive material with r = +1. (The

unit of the source energies is A2/m.) . . . . . . . . . . . . . . . . . . 51

2.2 Results of the numerical study for the constrained half-wave antenna

embedded in a double-positive material with r = +1. (The unit of

the source energies is A2/m.) . . . . . . . . . . . . . . . . . . . . . . 51

2.3 Results of the numerical study for the constrained quarter-wave an-

tenna embedded in a double-negative metamaterial with r = −1.(The unit of the source energies is A2/m). . . . . . . . . . . . . . . . 51

2.4 Results of the numerical study for the constrained half-wave antenna

embedded in a double-negative metamaterial with r = −1. (Theunit of the source energies is A2/m.) . . . . . . . . . . . . . . . . . . 52

xxi

Chapter 1

Introduction

1.1 Electromagnetic Metamaterials

An electromagnetic wave with wavelength λ much larger than a certain object does

not really “see” that object. Consequently, a wave propagating in a homogeneous

substrate with small inclusions that are, let us say, ∼ λ/10 will interact effectively

not with the individual inclusions but with the system substrate,inclusions as a

whole. The wave, thus, sees a new effective medium having electromagnetic proper-

ties different from those of its components. This suggests that effective media with

properly chosen inclusions may exhibit electromagnetic responses not found in na-

ture. Metamaterials are such media. They are engineered materials whose effective

constitutive parameters can, in principle, have any value, even negative.

The onset of the “metamaterials era” is frequently attributed to the landmark

theoretical article by Veselago [1], though the study of complex media has a long

history that dates back, at least, to the late part of the nineteenth century [2, 3, 4].

Yet, until just a few years ago the study of these materials was largely ignored by

scientists and engineers alike. Then, at the turn of the century, Smith et al. [5]

announced the fabrication of a metamaterial with negative index of refraction in

the microwave regime. This discovery instigated several studies that resulted in

the identification of several novel and exciting applications of metamaterials. They

include perfect lenses [6], subwavelength cavities [7], highly efficient electrically-

small antennas [8], subwavelength waveguides [9], cloaking devices [10], ultrathin

1

laser cavities [11], etc.

1.1.1 Physical Constraints on the Response Functions

Despite the fact that electromagnetic field theory in metamaterial media relaxes the

constraint on the response functions range for these media, it still requires them to

satisfy certain conditions for them to correspond to physically acceptable materials.

Below we merely list these physical constraints. For more detail see, for instance,

[12].

Dissipation and Passive Media

Let f be the response function of a certain isotropic medium. f could either be the

electric permittivity or the magnetic permeability µ. We have, in the most general

case,

f = Re [f ] + i Im [f ] , (1.1)

where i2 = −1, Re [·] stands for the real part, and Im [·] stands for the imaginarypart. The appearance of an imaginary part for f is an indication of the lossy nature

of the medium in question.

For metamaterials, the real part of the response function f is not necessarily

positive, it is a number that could take on any real value, be it positive or negative,

that is,

Re [f ] ∈ R. (1.2)

However, for passive media, i.e., media within which electromagnetic energy is es-

sentially dissipated not created, we must have

Im [f ] > 0. (1.3)

Positive-Definiteness of Electric and Magnetic Energies

The positive-definiteness of the electric and magnetic energies also imposes the fol-

lowing constraints on the response functions

∂ (ωf)

∂ω≥ 0 (1.4)

2

Causality and the Kramers-Kronig Relations

If the medium is to be causal, i.e., if the cause is to precede the effect in such

a medium, then the real and imaginary parts of the response functions must be

Hilbert transforms of each other. It can be shown that this is equivalent to the

following Kramers-Kronig dispersion relations

Re [f (ω)] =2

πP

Z ∞

0

Im [f (ω0)]ω02 − ω2

ω0dω0 (1.5)

and

Im [f (ω)] = −2ωπP

Z ∞

0

Re [f (ω0)]ω02 − ω2

dω0, (1.6)

wherein P stands for the Cauchy principal value. Clearly, according to our current

understanding, metamaterials cannot be nondispersive nor nondissipative. Never-

theless, such assumptions are routinely made in the literature for convenience.

1.1.2 Terminology

There exist different terminologies to describe the different types of metamaterials,

here we adopt the following one (due to Richard Ziolkowski): When Re [ ] > 0 and

Re [µ] > 0 the material is said to be double-positive (DPS); when Re [ ] < 0 and

Re [µ] < 0 the material is said to be double-negative (DNG); and when Re [ ] Re [µ] <

0 the material is said to be single-negative (SNG).1 These particular choices for the

signs of the constitutive parameters are required if wave propagation in the medium

is to be causal [13].

1.2 Inverse Source Problem in Substrate Media

In the first part of this dissertation (Chapters 2-4) the full-vector, electromagnetic

inverse source problem is used to reconstruct an unknown source (antenna) that is

embedded, within a source region V , in a given material or metamaterial substrate,

and that radiates a given exterior field outside V . The derived formulation and

1A more detailed terminology labels materials for which Re [ α] < 0 as ENG or -negativemedia, and materials for which Re [µα] < 0 as MNG or µ-negative media.

3

results on this inverse source problem in substrate media generalize, within the full-

vector formulation, previous work on the inverse source problem in free space (cf.

[14] and the references therein for review and applications), as well as previous work

on the scalar version of the problem for non-homogeneous backgrounds [15, 16],

particularly [17]. Here, as in [17], the inverse problem is addressed in the context

of the Helmholtz operator with emphasis on piecewise-constant radially symmetric

backgrounds. The formulation is based on constrained optimization and, unlike the

vast majority of previous presentations, can be used to implement different kinds

of constraints. Two such possibilities are emphasized in the sequel, in particular,

the minimizing of the source L2-norm or functional energy characterizing the “cur-

rent level”, with and without tuning to resonance, the former case corresponding

to zero source reactive power. Fundamental radiation limits related to the realiz-

ability of given fields or radiation performance with given source resources (antenna

size, current level as measured by the source energy, reactive power, and so on)

or, alternatively, of the minimal resources needed for a given performance, are also

elucidated as a by-product of the derived inverse source theory.

Motivation for this research is provided by the possibility of embedding an an-

tenna in a substrate of a given size, where the original antenna plus the substrate

are treated as the total antenna, so as to generate a given field or performance level

which could not be achieved under the same physical constraints by another antenna

in free space (i.e., without the substrate medium). This possibility has attracted in-

terest from time to time in the antenna community; of particular interest have been

a variety of antenna-embedding materials, including plasmas [18], non-magnetic di-

electrics [19, 20, 21, 22, 23, 24], magneto-dielectrics [25, 26, 27, 28, 29, 30], and,

more recently, double-negative and single-negative metamaterials which are receiv-

ing much attention as antenna performance-enhancing substrates by a number of

groups [8, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49]. The

envisaged property is miniaturization of antennas by controlling electric size, via

larger wavenumber, but other effects are involved, particularly when metamaterials

are used. (A review of the pertinent state-of-the-art can be found in [50]).

For instance, it is well known [51, 52] that in the free-space case the source

energy increases exponentially, for a given radiation pattern, with decreasing k0a,

where k0 is the free-space wavenumber of the field. This increase occurs below a

4

critical point determined by the fine detail that is desired in the radiation pattern,

specifically, the antenna directivity. The question then is whether the critical source

size in question can be made smaller by embedding the source in a properly se-

lected substrate which becomes integral to the antenna. For small antennas (whose

dimensions are, for instance, smaller than about 1/3 of the wavelength [53, 54])2

one is particularly interested in achieving radiation of an elemental dipolar mode,

using minimal resources. Can antenna substrates help toward this goal? Alterna-

tively, in certain applications using larger, resonant antennas whose dimensions are

comparable to or larger than the wavelength one can dispose of some “extra space”

to accommodate a substrate, and the question is: Does antenna embedding yield

enhancement of antenna directivity? Which values of the constitutive parameters

give better performance?

Contrary to previous presentations, which focused on particular devices, we em-

phasize in the present work a non-device-specific analysis that aims at understanding

the practical possibilities opened by antenna-embedding substrates. The source in-

version is approached by including solution constraints yielding minimum energy

sources generating a given radiation pattern. Formal tractability as well as engi-

neering applications both dictate the particular choices made. All the results are

derived for time-harmonic fields, and thus the values of the constitutive parameters,

which generally vary with frequency, are considered in this work for a given central

frequency only. We also employ effective media theory considerations and ignore

material dissipation as well as the general bi-anisotropic nature of metamaterials

[55, 56] whose explicit consideration within the inverse theory is left for the future.

Thus in this dissertation we treat all materials and metamaterials as having zero

conductivity, and generally real-valued scalar permittivity and permeability, (r)

and µ(r), respectively, at the relevant frequency.

As far as we know, there are only three treatments of the inverse source problem

for the non-free-space case [15, 16, 17]. They consider only the scalar version of the

problem. Reference [15] emphasizes the minimum energy solution within an integral

equation framework pertinent to holography (as is also shown in [57]). Reference

[16] generalizes [15] to lossy media. Reference [17] uses an optimization-theoretic

framework which is a simpler, scalar counterpart of the full-vector treatment derived

2Cf. definition given in Section 3.3.

5

in the present sequel.

Electromagnetically the ultimate sources of radiation are: (1) an impressed cur-

rent distribution which under a suppressed time dependence e−iωt is described byits space-dependent part J(r) and which can be controlled at antenna terminals;

and (2) an induced current distribution Jind(r) appearing over the antenna ma-

terial structure upon the presence of the former primary source (as schematically

illustrated in the same figure). The superposition of the two, i.e., the total current

distribution Jtot(r) = J(r) + Jind(r), generates, via the linear total source-to-field

mapping dictated by the free-space outgoing wave dyadic Green function [58, 59],

the respective total electric field E(r). This point of view is relevant to the formula-

tion of an inverse source problem in free space, consisting of deducing an unknown

total source Jtot(r), of known support, say, the entire volume V , that is consistent

with a measured exterior electric field E(r). It is this formulation in free space that

has been the subject of the vast majority of investigations on the inverse source

problem (cf. [51, 52, 14] and the references therein.) Its solutions are electric cur-

rent distributions that are equivalent to the true total antenna currents in that they

generate the same field outside V .

However, for antennas embedded in material substrates or if a number of can-

didate background media are known a priori as part of the antenna analysis and

design, then such a characterization in terms of total currents in free space is very

inadequate. In particular, the true currents are of a certain form dictated by the

wave propagation in the relevant substrate medium that is not taken into account in

the inversion of the total source in free space. In so doing, the thus-found “equiva-

lent sources in free space” can even be non-physically realizable with the particular

driving and material components of the antenna. In other words, source inversion

in free space is unsuited in the presence of antenna-embedding media. It may even

give false leads when addressing the fundamental physical limits of the far-fields, as

determined, e.g., from the Picard conditions [60].

Consequently, a better approach is to formulate an inverse source problem in

substrate media, whose objective is to deduce an unknown primary current density

J(r) that is contained, along with the substrate, in the spherical volume V , and that

generates a prescribed exterior field. After all, from a practical point of view, it is

only this primary source J(r) that one has control of at antenna terminals. Thus in

6

this approach the background medium is treated not as an equivalent source which

could even be non-realizable with the allowable driving excitations and antenna

material, but instead as a propagating medium that is integral to the antenna and

within which the impressed source radiates.

The sought-after source is generally non-unique. This is due to the presence,

within the source region, of nonradiating sources [61, 52, 62, 63] whose generated

fields vanish identically outside the source region. But as constraints are imposed,

one manages to arrive at a unique source generating a given exterior field, such as

the familiar minimum energy solution to the inverse source problem [51, 52, 14].

The most commonly adopted constraint is that of minimizing the square of the L2

norm of the source (as defined in Eq.(2.9)), usually termed “the source energy” in

the inverse problems literature. It is at the heart of the Picard conditions defining

the range of the source-to-field linear mapping from L2 sources to L2 far fields. It

has also been used, recently, in addressing the realizability of electromagnetic pulsed

beams or wavelet fields launchable from finite-size sources [64, 65]. The solution to

the inverse source problem that minimizes the source energy is usually termed “the

minimum (source) energy solution”. It is related to the real image field generated

by a point-reference hologram of the field recorded on a closed surface completely

surrounding the source volume [57, 66, 67]. The ability of an antenna to radiate

a prescribed power with reduced current levels as characterized by this norm is

an indication of efficiency which has been adopted as constraint in the antenna

synthesis problem [68, 69, 70, 71, 72]. Reduction of the source energy also accounts

for reduction of ohmic losses in driving metallic elements, thus this constraint is of

both mathematical convenience as well as physical importance.

A minimized source energy would indicate that the resources of the antenna

generating the given field pattern are optimally used within the prescribed volume

of the source. Furthermore, comparison of the required minimum source energies

for different substrate configurations enables quantification of the enhancement due

to such structures. Upon solving the inverse source problem in these media for the

prescribed exterior fields, one can proceed to tackle the comparison of the required

resources, embodied in the source functional energy in the present case, that are

needed for the launching of the given fields. Substrate configurations for which the

required source energy is lower are then more optimal than alternative configura-

7

tions which require higher source energy for the launching of the same fields. As

mentioned earlier this mathematical framework to characterize substrate enhance-

ment is non-device-specific. In particular, one is then comparing the “best” source,

which minimizes the required source energy for the launching of the given field via a

given substrate, versus the “best” source, which minimizes the source energy for the

launching of the same field but at a different substrate (including the “no-substrate”

or free space-case).

Also, of importance are constraints related to the reactive near fields. High

reactive energy as characterized by the quality factor Q [73, 74, 75, 76, 77] reduces

system bandwidth, and thus it is important to require that the designed antenna

have tolerable reactive near fields. (Some treatments using this constraint can be

found in [78, 79, 80].) Another consideration of practical importance is the tuning

of the antenna to resonance so that its reactive power is zero. In this dissertation

we consider the source energy constraint with and without this tuning to resonance

by means of a generalization to sources in substrates of a constrained optimization

approach to the electromagnetic inverse problem introduced for free space in [14].

1.3 The Casimir Effect and Electromagnetic Metama-terials

The second part of this dissertation (Chapter 5) explores the effects that the pres-

ence of electromagnetic metamaterials would have on the Casimir forces. Quantum

field theory predicts the existence of fluctuations in the ground, or vacuum, state of

all quantum fields [81]. These fluctuations yield an infinite energy for the vacuum

state which ultimately leads to disastrous consequences for the theory [82]. Yet, the

existence of these so-called zero-point fluctuations is necessary for the consistency

of quantum field theory [83]. In spite of the inherently infinite nature of the ener-

gies attributed by quantum field theory to the vacuum, the presence of boundary

conditions alters these energies in a way that makes it possible to extract finite

quantities from the otherwise nonsensical infinities. In 1948, Casimir did just that:

he calculated the change in the electromagnetic vacuum energy due to the pres-

ence of two perfectly conducting parallel plates [84, 85]. He found out that the two

8

plates would attract each other by a force per unit area that is proportional to d−4,where d is the distance between them. This force has now been measured for this

geometry [86] as well as for other geometries [87, 88, 89, 90, 91] with an excellent

agreement between theory and experiment. Nevertheless, the interpretation of the

Casimir effect as proof for the reality of the vacuum quantum fluctuations is still

a debatable issue since Casimir’s result could also be obtained without recurring

to zero-point fluctuations [92, 83, 93]. The debate also extends to the question of

what constitutes a correct theoretical formulation for a given physical problem as

well (see, for instance, [94, 95]).

This being said, the technological impact of the Casimir effect is certainly real.

Besides the experimental vindications mentioned earlier, the Casimir forces, along

with their companion van de Waals forces, are now identified as the primary cause for

the collapse of microelectromechanical systems (MEMS) and nanoelectromechanical

systems (NEMS) [96, 97]. Hence, it is very important for the future development of

the emerging field of nanotechnology to investigate the possibility of shielding these

potentially destructive forces. The nature of the phenomenon allows for several

approaches. The Casimir forces depend strongly on the geometry of the system

and they are known to change sign and become repulsive for certain geometries

[85], although the possibility of geometry-based repulsive forces has been excluded

for a large class of geometries. The Casimir forces also depend on the constitutive

parameters of the media involved. In the context of the Dzyaloshinskii-Lifshitz-

Pitaevskii theory [98, 99] it has been shown that repulsive Casimir-Lifshitz forces

are achievable when the electric permittivity of a nonmagnetic slab sandwiched

between two nonmagnetic half-spaces is itself sandwiched between the permittivities

of the two half-spaces [100], i.e., ε1 (iξ) < εslab (iξ) < ε2 (iξ) , where ξ ≡ −iω is theimaginary frequency.

Another situation in which repulsive Casimir forces may arise involves meta-

materials [101, 102]. Metamaterials are electromagnetically engineered materials

whose constitutive parameters could in principle have any real value [103, 12]. In

particular, it has been suggested [102] that a slab of metamaterial with an index of

refraction n = −1 (also known as “perfect lens”) sandwiched between two perfectlyconducting plates should cause the Casimir force between the two plates to reverse

its sign.

9

Intrigued by the possibility of using metamaterials to produce repulsive Casimir

forces independently of the topology and effectively controlling the magnitude of

these forces we investigate in this dissertation the Casimir effect in the presence of

metamaterials. In particular we focus our attention on the case of two metamaterial

slabs having oppositely signed constitutive parameters sandwiched between two per-

fectly conducting plates. Several authors have considered the multilayered planar

geometry with and without metamaterials [104, 105, 106, 107, 94, 101, 108, 109, 102,

110, 111]. However, to our knowledge, no study has ever investigated the proposed

system. The relevance of this system stems from the realization that the most dra-

matic improvements attributed to metamaterials are often generated in situations

where two or more oppositely signed constitutive parameters are involved [50]. For

instance, the first slab would be made up of a double-positive, or DPS, medium,

i.e., a medium for which both the permittivity and the permeability are positive,

and the second slab would be made up of a double-negative, or DNG, medium, i.e.,

a medium for which both the permittivity and the permeability are negative.

In choosing this system we were particularly inspired by the subwavelength cavity

resonators described in [7]. The idea is to examine the possibility that for such

a cavity the peculiar properties of metamaterials would permit the existence of

electromagnetic vacuum fluctuation modes with wavelengths much larger than the

transverse dimension of the cavity. This would effectively cause an increase in the

outward pressure produced by these modes within the cavity, thus, reversing the

sign of the Casimir force between the two perfectly conducting plates and turning

it into a repulsive force.

The remainder of this dissertation is organized as follows. Chapter 2 develops

the essential theoretical tools needed for the quantitative study of piecewise-constant

radially symmetric backgrounds with particular focus on the lossless homogeneous

sphere. Chapter 3 generalizes the mathematical formulation of Chapter 2 to the

lossy core-shell system. Chapter 4 presents a comparative numerical study whose

aim is to understand the effect of reference antennas on enhancement level estimates.

Chapter 5 explores the effects that the presence of electromagnetic metamaterials

would have on the Casimir forces. Chapters 2, 3, and 5 are, each, supplemented by a

computer simulation study that illustrates the relevant theoretical results. Chapter

6 outlines some future research directions of our interest. Appendix A is a glossary

10

of important results related to spherical multipole theory. Appendices B, C, D, F, G

provide technical details needed in Chapters 2—5. Appendix E presents an important

result that shows how the current study of antenna radiation for antennas embedded

in piecewise-constant backgrounds may be extended to the more general case of a

radially symmetric background. All equations are in the SI system.

11

Chapter 2

Inverse Source Problem inNon-homogeneous BackgroundMedia

In this chapter we solve analytically and illustrate numerically the full-vector, elec-

tromagnetic inverse source problem of synthesizing an unknown source embedded

in a given substrate medium of volume V and radiating a prescribed exterior field.

The derived formulation and results generalize previous work on the scalar version

of the problem, especially [17]. Emphasis is put on substrates having constant con-

stitutive properties within the source volume V , which, for formal tractability, is

taken to be of spherical shape. The adopted approach is one of constrained opti-

mization which also relies on spherical wavefunction theory. The derived theory and

associated implications for antenna substrates are illustrated numerically.

We pay particular attention to lossless piecewise-constant radially symmetric

backgrounds having electric permittivity s and permeability µs, in particular, the

total permittivity distribution is of the form

(r) = sΘ(a− r) + 0Θ(r − a) (2.1)

where Θ denotes Heaviside’s unit step function (Θ(x) = 1, for x > 1, otherwise

12

Figure 2.1: Schematic of a general antenna whose driving points and material struc-ture are confined within a spherical volume V of radius a.

Θ(x) = 0), and the total permeability distribution is of the form

µ(r) = µsΘ(a− r) + µ0Θ(r − a). (2.2)

Generalization of the analysis for distributions of the constitutive parameters which

are spherically symmetric within the source support V and take the free-space values

outside V is also outlined1. Our results, published in [112], reveal the performance

improvements due to antenna-embedding substrates from an inverse antenna theory

point of view which is different than and complementary to efforts by other groups

in this fruitful area (e.g., [8, 25, 26, 27, 28, 31, 32, 33, 34, 35, 36, 37, 38]; an expanded

bibliography can be found in [50]).

Referring to the schematic given in Fig. 2.1, we consider a general antenna which

is composed of a driving structure and a material structure. The driving structure

could be constituted by information-carrying currents, voltages (e.g., as in a dipole

antenna) and/or equivalent primary fields (e.g., as in a driving horn). The material

structure, on the other hand, could be a reflecting body (e.g., a parabolic reflector),

1See Appendix E.

13

an antenna substrate (e.g., as in ceramic-embedded antennas for mobile telephone

handsets [23]), etc. The antenna radiates at angular oscillation frequency ω. The

smallest spherical volume V within which the entire antenna resides is assumed to

have radius a, that is, V ≡ ©r ∈ R3 : r ≡ |r| ≤ aª.

2.1 The Forward Problem

2.1.1 Electromagnetic Generalities

Our starting point is provided by the frequency-domain Maxwell equations for a

generally lossless, non-homogeneous medium, in particular [58, 59],

∇×E(r) = iωµ(r)H(r) (2.3)

∇×H(r) = J(r)− iω (r)E(r), (2.4)

where J(r) represents an impressed current density (i.e., the source) confined within

the spherical volume V , and E(r) and H(r) are, respectively, the electric and mag-

netic fields it generates. (These fields are subject to Sommerfeld’s radiation condi-

tion [113].) Substituting H(r), from Eq. (2.3), into Eq. (2.4) yields the vector wave

equation

∇×µ∇×E(r)

µ(r)

¶− ω2 (r)E(r) = iωJ(r). (2.5)

The partial differential operator in Eq. (2.5) admits an outgoing-wave dyadic

Green’s function G(r, r0) which, along with Sommerfeld’s radiation condition, obeys

∇×µ∇× G(r, r0)

µ(r)

¶− ω2 (r)G(r, r0) = iωδ(r− r0)I, (2.6)

where I denotes the identity dyadic and δ the Dirac delta.

For future convenience we define the weighted inner product

(f , f 0) =Z

drM(r)f∗(r) · f 0(r), (2.7)

where f and f 0 are any two functions of position and the asterisk ∗ denotes thecomplex conjugate; M(r) is a characteristic (indicator or masking) function defined

14

as

M(r) =

(1 ; r ∈ V

0 ; r /∈ V.(2.8)

Using this inner product, we express the source energy E as

E ≡ (J,J), (2.9)

and the complex interaction power P (cf. [58]) as

P = −12(J, eGJ) = −1

2(J ,E) , (2.10)

where we have introduced the linear mapping eG defined by

[ eGJ](r) ≡ Z dr0G(r, r0) · J(r0). (2.11)

The real part of P, i.e., Re [P], represents the radiated power. This radiatedpower is determined by the radiated field or, equivalently, by the multipole moments

a(j)l,m, in the form of an incoherent sum of the multipole contributions [114, 115, 116],

namely

Re [P] = 1

2η0

2Xj=1

∞Xl=1

lXm=−l

l(l + 1)|a(j)l,m|2, (2.12)

where η0 =pµ0/ 0 is the free-space wave impedance.

On the other hand, the imaginary part of P, i.e., Im [P], corresponds to theenergy-storage reactive power [58]. It can have a prescribed value, say, zero (as was

shown in [14, 52]), which corresponds to a tuned antenna, and is one of the solution

constraints to be employed in the formulation to follow. We note that the reactive

power can be expressed as

Im [P] = −12

ZVdrJ∗(r) ·

ZVdr0GS(r, r

0) · J(r0)

≡ −12(J, eGSJ), (2.13)

15

where

GS(r, r0) ≡ Im £G(r, r0)¤ = 1

2i

£G(r, r0)− G∗(r, r0)¤ , (2.14)

and where we have introduced the linear mapping eGS defined by Eq. (2.11) after

the substitutions eG→ eGS and G→ GS .

2.1.2 Source-to-Multipole-Moment Mapping

To formulate the inverse problem for the cases described in Eqs. (2.1) and (2.2) (as

well as cases described by Eqs. (3.1,3.2) and Eqs. (E.1,E.2)) it is necessary to first

have at our disposal the solution of the associated forward or radiation problem. To

accomplish this, we note that, for these cases, the electric field E(r) generated by the

most general source of support V can be represented, outside V , by the multipole

expansion [114]

E(r) =2X

j=1

∞Xl=1

lXm=−l

a(j)l,mΛ

(j)l,m(r), r /∈ V, (2.15)

where the complex-valued expansion coefficients a(j)l,m are the multipole moments of

the field, and where the multipole fields are

Λ(j)l,m(r) =

⎧⎪⎨⎪⎩∇× [h(+)l (k0r)Yl,m(r)] ; j = 1

ik0h(+)l (k0r)Yl,m(r) ; j = 2,

(2.16)

where r ≡ r/r , h(+)l denotes the spherical Hankel function of the first kind and

order l (as defined in [117]), corresponding to outgoing spherical waves in the far

zone, Yl,m is the vector spherical harmonic of degree l and order m (as defined in

[114], Eqs. (4.7) and (4.8))2, and j = 1 and j = 2 correspond to electric (TMr) and

magnetic (TEr) multipole fields, respectively. For the convenience of the reader we

are outlining the key properties of the vector spherical harmonics in Appendix A.

Note that in (2.15) the summation over l starts from l = 1 because there are no

vector spherical harmonics of zero degree [114]. On the other hand, the magnetic

2Physically, the index l characterizes the so-called multipolarity or modal order of the field;thus l = 1 corresponds to 21-pole (dipole) radiation, l = 2 corresponds to 22-pole (quadrupole)radiation, l = 3 corresponds to 23-pole (octupole) radiation, and so on.

16

field outside the source region is defined by substituting from these results into Eq.

(2.3) with the free-space substitution µ(r)→ µ0.

The electric and magnetic multipole moments, a(1)l,m and a(2)l,m, respectively, are

related to the current distribution J by

a(j)l,m = (B

(j)l,m,J), j = 1, 2, (2.17)

i.e., they are the projections of the current distribution J onto the set of source-free

vector fields B(j)l,m which need to be determined for the particular antenna back-

ground medium. For the special free-space case where µ(r)/µ0 = 1 = (r)/ 0 the

latter fields are the familiar source-free multipole fields, in particular (cf. [74] and

[59]),

B(j)l,m(r) ≡

⎧⎪⎪⎨⎪⎪⎩− η0

l(l+1)∇× [jl(k0r)Yl,m(r)] ; j = 1

−i k0η0l(l+1)jl(k0r)Yl,m(r) ; j = 2,

(2.18)

where jl is the spherical Bessel function of the first kind and order l (as defined in

[117], for instance). On the other hand, it is shown in Appendix B that for piecewise-

constant radially symmetric backgrounds whose permittivity and permeability are

given by Eqs. (2.1) and (2.2)

B(j)l,m(r) ≡

⎧⎪⎪⎨⎪⎪⎩−η0l(l+1)F

∗(1)l ∇× [jl(k∗r)Yl,m(r)] ; j = 1

−ik0η0l(l+1) F

∗(2)l jl(k

∗r)Yl,m(r) ; j = 2,

(2.19)

where the substrate wavenumber k = ω√µs s, the relative permittivity r ≡ s/ 0,

the relative permeability µr ≡ µs/µ0, and where we have defined the complex am-

plitudes

F(j)l ≡

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩i/(k0ka2)

( r/µr)1/2jl(ka)Vl(k0a)−h(+)l (k0a)Ul(ka)

; j = 1

iµr/(k0ka2)(µr/ r)

1/2jl(ka)Vl(k0a)−h(+)l (k0a)Ul(ka); j = 2,

(2.20)

where

Ul (λa) ≡ Ul (λr)|r=a ≡∙djl(λr)

d (λr)+

jl(λr)

λr

¸¯r=a

, (2.21)

17

and

Vl (λa) ≡ Vl (λr)|r=a ≡"dh(+)l (λr)

d (λr)+

h(+)l (λr)

λr

#¯¯r=a

. (2.22)

Note that F (1)l and F(2)l are functions of k0a, ka, r, and µr.

Because of the self-imposed restriction to the study of lossless substrates in

this chapter, the relative constitutive parameters µr and r admit only real values.

Consequently, the wavenumber k can assume only real values (positive for DPS

materials and negative for DNG metamaterials) or purely imaginary values (for

single-negative metamaterials). When k is purely imaginary, i.e., k = iα, α ∈ R,the arguments of the spherical Bessel functions involving k in Eqs. (2.20)-(2.22)

are, accordingly, purely imaginary. In this case one notes that the regular spherical

Bessel functions jl and h(+)l are replaced, respectively, with the modified spherical

Bessel functions il and kl such that (see, for instance, [117])

jl (ka) ≡ ilil(αa), (2.23)

and

h(+)l (ka) ≡ −i−lkl(αa). (2.24)

(There shall be no confusion between the modified spherical Bessel functions il and

kl and the imaginary unit i and the wavenumber k since these latter ones do not

carry a subscript.)

We draw the attention of reader to the fact that F (j)l , j = 1, 2, represent the

Mie amplitudes due to the scattering of a plane electromagnetic wave off a sphere

of radius a and wavenumber k embedded in an infinite homogeneous medium of

wavenumber k0; F(1)l being the amplitudes of the electric oscillations and F (2)l those

of the magnetic oscillations. This should not come as a surprise in view of the

physics as well as the formulation itself of the problem.3

It is not hard to show that the amplitudes F (j)l −→r,µr→1

1, as expected, since, in

that case, Eqs. (2.19) and (2.20) reduce to the free-space result (2.18). The corre-

sponding results for more general spherically-symmetric backgrounds are outlined

in Appendix E. Substitution of the associated results into Eq. (2.17) completes

3See Section 2.3 and Appendix B.

18

the description of the forward problem. Armed with these developments, we are in

position to formulate next the corresponding inverse source problem.

2.2 Inverse Source Theory Based on Constrained Opti-mization

The inverse source problem of deducing the source J(r), confined within V from

knowledge of the exterior field E(r) is seen from Eqs. (2.15) and (B.4)4 to be equiv-

alent to that of determining the source from knowledge of the multipole moments,

i.e., to that of inverting Eq. (2.17). The respective inversion is addressed next via

a generalization of the free-space optimization theory in [14] to non-homogeneous

backgrounds. Emphasis is given to the particular case of piecewise-constant radially

symmetric backgrounds, but the derived expressions apply to more general cases as

long as one uses the appropriate projective wavefunctions B(j)l,m which vary from a

medium type to another.5

2.2.1 Minimum Energy Solution by Constrained Optimization

We start by addressing the problem of determining the minimum energy source JME

embedded in a substrate of volume V with fixed constitutive parameters r, µr and

generating a given exterior field. The problem can be cast as

minJ∈S

E (J) , (2.25)

where

S ≡nJ ∈ L2

¡V ;C3

¢: a(j)l,m − (B(j)

l,m ,J) = 0o. (2.26)

Note that the constraint set S is convex; also, the objective functional E is coerciveand strictly convex. The convexity of E along with its continuity at some pointguarantee its continuity on the whole space L2

¡V ;C3

¢[118].

If a minimizer JME exists, then its uniqueness and global minimality are insured

4This can also be derived form Eqs.(B.16)-(B.19)). See Appendix B for more detail.5An important result related to the spherically symmetric substrate case is derived in Appendix

E.

19

by the strict convexity of E and the convexity of S [119]. But what guarantees theexistence of at least one such minimizer? We address this question next. First, we

note that since E is a continuous and convex functional on a Hilbert space then itis also weakly sequentially lower semi-continuous [119]. Given this property of E ,the fact that it is coercive, and the fact that S is a closed and convex subset of a

reflexive Banach space (e.g., a Hilbert space), we can, now, assert that there exists

at least one point JME that minimizes E over S [120]. Hence, problem (2.25,2.26)

admits only one global solution.

Due to the Fréchet differentiability of the objective functional E and the con-straints

ha(j)l,m − (B(j)

l,m, ·)i, the continuity of their Fréchet derivatives6, and the fact

that ∇Jha(j)l,m − (B(j)

l,m ,J)i¯J=JME

maps L2¡V ;C3

¢onto C, there exist [120] La-

grange multipliers c(j)l,m ∈ C such that the generalized Lagrangian

L³J, c

(j)l,m

´≡ E + 2Re

⎡⎣ 2Xj=1

∞Xl=1

lXm=−l

c(j)l,m

³a(j)l,m − (B(j)

l,m , J)´⎤⎦ (2.27)

is stationary at JME.

To compute the solution we require that

δL = 2Re⎡⎣⎛⎝δJ , J−

2Xj=1

∞Xl=1

lXm=−l

c(j)∗l,mB

(j)l,m

⎞⎠⎤⎦ = 0. (2.28)

From the Dubois-Raymond lemma, Eq. (2.28), and the forward mapping relations

(2.17,2.19,2.20,2.21,2.22), one finds that the minimum-energy source is given by

JME(r) =2X

j=1

∞Xl=1

lXm=−l

a(j)l,mh

σ(j)l

i2B(j)l,m(r), (2.29)

where we have introduced the positive-definite “singular values”

[σ(j)l ]

2 ≡ (B(j)l,m,B

(j)l,m), j = 1, 2, (2.30)

6See Appendix D.

20

specifically, hσ(j)l

i2= |F (j)l |2

hκ(j)l

i2, j = 1, 2, (2.31)

where hκ(j)l

i2 ≡⎧⎪⎪⎨⎪⎪⎩

η20R a0 dr

h|jl(kr)|2 + |kr|2

l(l+1) |Ul(kr)|2i

; j = 1

η20k02

l(l+1)

R a0 dr r2 |jl(kr)|2 ; j = 2.

(2.32)

For real k2 the integral associated with the j = 2 case is calculable through the use of

the second Lommel integral (see , for instance, [117]) and the recurrence relations of

the Bessel functions along lines similar to those employed in [51] to evaluate similar

inner products. Afterwards, the recurrence relations are also used to express the

integral associated with the j = 1 case in terms of the calculated integral associated

with the j = 2 case. Consequently, Eqs. (2.32) reduce to,

hκ(j)l

i2=

⎧⎪⎪⎨⎪⎪⎩η20a|ka|2

l(l+1)(2l+1)

£(l + 1)γ2l−1(ka) + lγ2l+1(ka)

¤; j = 1

η20a(k0a)2

l(l+1) γ2l (ka) ; j = 2,

(2.33)

where we have introduced the unitless quantity (cf. [51], Eq. (17))

γ2l (ka) ≡1

a3

Z a

0dr r2j2l (kr)

=1

2

£j2l (ka)− jl−1(ka)jl+1(ka)

¤. (2.34)

For k = iα, α ∈ R as is the case for single-negative metamaterials, one uses definition(2.23) to express Eqs. (2.33) and (2.34) in terms of il.7

Furthermore, the minimum source energy

EME ≡ (JME,JME) =2X

j=1

∞Xl=1

lXm=−l

|a(j)l,m|2

[σ(j)l ]

2. (2.35)

7Note that the lone appearance of the size parameter a in (2.33), i.e., its appearance decoupledfrom the wavenumbers, is a direct consequence of the fact that the multipole moments a(j)l,m aredimensionful quantities. It is, as well, a reminder of the boundedness of the enclosing volume V ,i.e., of the embedding sphere of substrate material.

21

As expected, these developments reduce, for r = 1 = µr, to the free-space result

(Eqs. (13), and (14) in [14]) since F (j)l (k0a, ka, r, µr) = 1; that is, the free-space

minimum-energy solution is given by Eq. (2.29) with B(j)l,m given by Eq. (2.18) and

[σ(j)l ]

2 substituted byhκ(j)l (k0a = ka)

i2.

2.2.2 Minimum Energy Source Having Zero Reactive Power

We consider next the constrained optimization problem of minimizing the functional

energy of the source subject to the additional constraint that the reactive power of

the source has a prescribed value. The results for this problem will be elaborated

next for the particular and important case of zero reactive power, i.e., Im [P] = 0.This corresponds to the minimizing of the antenna currents (the physical resources)

while simultaneously enforcing perfect antenna reactance tuning inside the antenna.

The problem can be cast as

minJ∈X

E (J) , (2.36)

where

X ≡nJ ∈ L2

¡V ;C3

¢: a(j)l,m − (B(j)

l,m ,J) = 0, (J, eGSJ) = 0o. (2.37)

The constraint set X is closed, unbounded, and nonconvex. Its nonconvexity stems

from that of the newly introduced constraint (J, eGSJ) = 0. The set X is assumed to

be nonempty. (If it turns out to be empty this would mean that it is not possible for

an antenna having a substrate medium of constitutive parameters r, µr to produce

the prescribed external field and at the same time have a vanishing reactive power.)

It is clear that problem (2.36,2.37) is an inherently difficult nonconvex program-

ming problem. Not only do we seek to minimize an objective functional under

nonconvex functional constraints but, also, we have to do that on an unbounded

set. Proving, for instance, the existence of a solution to problem (2.36,2.37) would

have been easier if X were convex, but it is straightforward to show that the only

way for X to become convex is to have Reh(J1, eGSJ2)

i≤ 0, ∀J1,J2 ∈ L2

¡V ;C3

¢.

This would amount to imposing a new constraint which appears not to correspond

to anything meaningful, physically speaking.

Now let us address the issue of the existence of a solution to problem (2.36,2.37)

22

in the absence of the convexity and boundedness of the constraint set X. Since X is

a closed subset of a normed vector space and since E is a coercive functional, thenthere exist [120] J0 ∈ X and Γ > 0 such that

infJ∈X

E (J) = infnE (J) : J ∈ X ∩BΓ (J0)

o, (2.38)

where BΓ (J0) is the closed (and bounded) ball of radius Γ and center J0. This is

a powerful result. What this tells us is that minimizing E over the unbounded setX can be reduced to the minimizing of E over a bounded subset in X that could

be much smaller than X. All that remains to complete the proof of existence of a

solution to problem (2.36,2.37) is to demonstrate the existence of a solution to the

auxiliary problem

minJ∈X∩BΓ(J0)

E (J) . (2.39)

A useful variant of the generalized Weierstrass theorem stipulates that for a weakly

sequentially lower semi-continuous functional defined on a weakly sequentially com-

pact subset of a Hilbert space there exists, at least, one solution to the minimization

problem [119]. But we have already shown that E is a weakly sequentially lower semi-continuous functional (see the discussion of problem (2.25,2.26)). Consequently, the

existence of a solution to problem (2.39,2.37) depends entirely on the demonstra-

tion that X ∩BΓ (J0) is a weakly sequentially compact subset. But this, too, is true

because any bounded subset of a reflexive Banach space (e.g., a Hilbert space) is

also weakly sequentially compact [121]. Hence, assuming that X ∩ BΓ (J0) is non-

empty, we are, from the preceding discussion, in position to affirm the existence of at

least one global minimizer JE,P ∈ X ∩BΓ (J0) for the auxiliary problem (2.39,2.37).

However, by virtue of (2.38), this point JE,P is also the sought solution of problem(2.36,2.37), which completes our proof.

Unfortunately, though, we have yet to guarantee the uniqueness of this solution

or even write down a minimality condition that would yield this solution. We shall

now focus on writing down a necessary minimality condition whose solution would

yield the minimizer JE,P .

Let JE,P be a minimizer. In view of the Fréchet differentiability of the objective

23

functional and the constraints, the continuity of their Fréchet derivatives,8 and the

fact that ∇Jha(j)l,m − (B(j)

l,m ,J)i¯J=JE,P

is surjective and the range of ∇J³J, eGSJ

´¯J=JE,P

is closed, there exist Lagrange multipliers χ ∈ R, and c(j)l,m ∈ C such that [120]

Reh³∇JL

³JE,P , χ, c

(j)l,m

´, J− JE,P

´i≥ 0, ∀J ∈ L2

¡V ;C3

¢, (2.40)

where the generalized Lagrangian functional is given by

L³J, χ, c

(j)l,m

´≡ E (J) + χ

³J, eGSJ

´+ 2Re

⎧⎨⎩2X

j=1

∞Xl=1

lXm=−l

c(j)l,m

ha(j)l,m − (B(j)

l,m,J)i⎫⎬⎭ . (2.41)

Condition (2.40,2.41) reduces to

Re

⎡⎣⎛⎝JE,P + χ eGSJE,P −2X

j=1

∞Xl=1

lXm=−l

c(j)l,mB

(j)l,m , J− JE,P

⎞⎠⎤⎦ ≥ 0, ∀J ∈ L2¡V ;C3

¢.

(2.42)

According to Eq. (2.42), to determine JE,P one needs to solve an infinite numberof equations with infinite number of unknowns. That, of course, is not the case in

practical situations. For any real problem the radiation emitted by the source has a

maximum multipolarity lmax ∼ ka (<∞). Thus, for real problems one would needto solve 2lmax (lmax + 2) + 4 integral equations with 2lmax (lmax + 2) + 4 unknowns.

By all standards this is a tedious task, even for small values of lmax. One should

try to find a more clever way of determining what the solution is. For instance, one

could resort to numerical techniques and algorithms available in the literature (see,

e.g., [122] and the references therein). In the sequel we plan on adopting a similar

approach that combines analytical and numerical methods.

We shall assume that X 6= ∅ and adopt partly-analytical partly-numerical

strategies to find a minimizer JE,P , which we proved that it existed, without havingto solve a large number of complicated equations. The “hybrid” approach below is

very much in line with the spirit of those adopted for this kind of problems. We

8See Appendix D.

24

shall also explore some of the properties of the solution. Once a feasible point JE,Pis found by means of the technique below one would substitute it in the derived

minimality conditions to check if it satisfies these conditions.

Let, now, L be the generalized Lagrangian defined as

L³J, χ, c

(j)l,m

´≡ E (J) + χ

n(J, eGSJ) + 2 Im[P]

o+ 2Re

⎧⎨⎩2X

j=1

∞Xl=1

lXm=−l

c(j)l,m

ha(j)l,m − (B(j)

l,m,J)i⎫⎬⎭ , (2.43)

wherein the constraint on the reactive power is now written in such a way that it

permits the latter to have an arbitrary value Im[P ] that is not necessarily zero.

The first variation of the last term in Eq. (2.43) is found from Eqs. (2.13) and

(2.14) to be

χδ(J, eGSJ) = 2Rehχ(δJ, eGSJ)

i. (2.44)

It follows from Eqs. (2.27), (2.28), and (2.44) that the first variation of the La-

grangian in Eq. (2.43) is

δL = 2Re⎡⎣(δJ,J) + χ(δJ, eGSJ)−

2Xj=1

∞Xl=1

lXm=−l

c(j)∗l,m (δJ,B

(j)l,m)

⎤⎦ . (2.45)

By equating the variation in Eq. (2.45) to zero one deduces that the sought

solution, to be denoted as JE,P(r), must obey, within its support V , the relation

JE,P(r) + χ eGSJE,P(r) =2X

j=1

∞Xl=1

lXm=−l

c(j)∗l,mB

(j)l,m(r). (2.46)

If χ = 0 then this approach coincides with the one given earlier, leading to the

minimum energy source in Eq. (2.29) (in such a situation, that source generates

zero reactive power) while for the more general case χ 6= 0 the two formulations

(and their solutions) differ. However, we note that for certain peculiar constitutive-

parameter values the constraint is not active and therefore χ = 0. In that peculiar

case the minimum energy sources are intrinsically resonant.

By letting the vector wave equation operator (∇×∇×− (k∗)2) = (∇×∇×−k2)

25

(the equality stems from the requirement that the substrate be lossless) act on both

sides of Eq. (2.46) and with the aid of the fact that the fields B(j)l,m are solutions

of the homogeneous wave equation associated to the same operator, one concludes

that the source JE,P(r) obeys the homogeneous wave equation

∇×∇× JE,P(r)−K2JE,P(r) = 0, (2.47)

in the interior of the source region V ; the quantity K which appears in Eq. (2.47)

is a modified wavenumber defined by

K2 ≡ k2 − χµsω. (2.48)

(Note that K quickly becomes purely imaginary as χ becomes large and positive.)

Now, the most general source that is confined within the spherical source volume

V and is a solution of Eq. (2.47) in the interior of V must admit the representation

JE,P(r) =2X

j=1

∞Xl=1

lXm=−l

v(j)l,mD

(j)l,m(r), (2.49)

where v(j)l,m are expansion coefficients that need to be determined (for the constraints

of the problem) and where

D(j)l,m(r) =

⎧⎪⎪⎨⎪⎪⎩− η0

l(l+1)∇× [jl(Kr)Yl,m(r)] ; j = 1

− iη0Kl(l+1)jl(Kr)Yl,m(r) ; j = 2.

(2.50)

From the formal similarity of B(j)l,m and D

(j)l,m (cf. Eq. (2.19)) it follows at once from

Eqs. (2.30), (2.31), (2.33), and (2.34) that the inner product³D(j)l,m,D

(j)l,m

´= p(j)

hκ(j)l (k0a,Ka)

i2, (2.51)

26

where

p(j) =

⎧⎪⎨⎪⎩1 ; j = 1

|K|2/k20 ; j = 2.

(2.52)

By substituting from Eqs. (2.49) and (2.50) into Eq. (2.17) while using well-

known orthogonality properties9 of the vector spherical harmonics Yl,m(r) and the

associated vector functions r×Yl,m(r) one obtains the form of the desired solution,

in particular,

JE,P(r) =2X

j=1

∞Xl=1

lXm=−l

a(j)l,m

(B(j)l,m,D

(j)l,m)

D(j)l,m(r), (2.53)

where

(B(j)l,m,D

(j)l,m) =

⎧⎪⎪⎨⎪⎪⎩η20F

(1)l

R a0 dr

hjl(kr)jl(Kr) + kKr2

l(l+1)Ul (kr)Ul (Kr)i

; j = 1

η20F(2)l

k0Kl(l+1)

R a0 drr2jl(kr)jl(Kr) ; j = 2.

.

(2.54)

Similarly to the integrals in (2.32), the integral associated with the j = 2 case

in (2.54) is calculable through the use of the first Lommel integral (cf., for instance,

[117].) The above inner product takes on the form

(B(j)l,m,D

(j)l,m) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩η20kKa3F

(1)l

l(l+1)(2l+1)

£(l + 1)ψl−1 (ka,Ka) + lψl+1 (ka,Ka)

¤; j = 1

η20k0Ka3F(2)l

l(l+1) ψl (ka,Ka) ; j = 2,

(2.55)

where we have introduced the unitless quantity

ψl (ka,Ka) ≡ 1

a3

Z a

0drr2jl(kr)jl(Kr)

=1

a (k2 −K2)[Kjl(ka)jl−1(Ka)− kjl−1(ka)jl(Ka)] . (2.56)

(Note that (2.56) is valid only for k 6= K, i.e., for χ 6= 0. The case k = K, i.e., for

χ = 0, has already been discussed.)9See Appendix A.

27

The source energy corresponding to (2.53) is of the form10

EE,P ≡ (JE,P ,JE,P) =2X

j=1

∞Xl=1

lXm=−l

(D(j)l,m,D

(j)l,m)

|(B(j)l,m,D

(j)l,m)|2

|a(j)l,m|2. (2.57)

We need to incorporate the reactive power constraint, i.e., Eq. (2.13), which

defines the value of the remaining Lagrange multiplier χ. Since the desired reactive

power is specified to be zero, the problem now is to find an expression for the reactive

power in terms of χ from which one can deduce the value of χ which minimizes the

source energy under the constraint Im [P] = 0. This value of χ will be called χ0.

A number of partly analytical, partly numerical strategies can be implemented to

accomplish this step.

One such approach, which generalizes the development for the free-space case

in [14], consists of determining the field E(r) generated by the source JE,P(r) inthe interior of the source region V . In particular, after evaluating the field, one

can compute the interaction power via Eqs. (2.10) and (2.11) and require that its

imaginary part vanish. In particular, plotting Im [P] and EE,P versus χ one can

finally select the value of χ which yields minimum EE,P out of all values of χ forwhich Im [P] = 0. We adopt this approach next.

By rewriting Eq. (2.47) as

¡∇×∇×−k2¢ [JE,P(r)− iχE(r)] = 0, (2.58)

where we have borrowed from Eq. (2.5), one concludes that the field E(r) must

admit in the interior of the source region V an expansion of the form

E(r) =1

iχ

⎡⎣JE,P(r) + 2Xj=1

∞Xl=1

lXm=−l

u(j)l,mB

(j)l,m(r)

⎤⎦ , r ∈ V, (2.59)

where the expansion coefficients u(j)l,m need to be determined taking into account

continuity of the tangential components of the field on the boundary ∂V ≡ r ∈R3 : r = a of V . Continuing on this idea, it is not hard to show from these

10Note that Eqs.(2.53) and (2.57) do not assume any particular value for Im [P].

28

developments, and by straightforward generalization of the discussion of the free-

space version of the problem in [14], Eqs. (30)-(42), that the complex interaction

power can be expressed as

P =2X

j=1

∞Xl=1

lXm=−l

q(j)l |a(j)l,m|2, (2.60)

where

q(j)l =

i

2χ

⎡⎢⎣³D(j)l,m,D

(j)l,m

´¯³B(j)l,m,D

(j)l,m

´¯2 + u(j)l,m

a(j)l,m

⎤⎥⎦ , (2.61)

where the quantity u(j)l,m/a(j)l,m is given by

u(j)l,m

a(j)l,m

=

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

1

F∗(1)l k∗Ul(k∗a)

"−ik0η0χl(l + 1)Vl(k0a)−

KUl(Ka)³B(1)l,m,D

(1)l,m

´#

; j = 1

1

F∗(2)l k0jl(k∗a)

"−ik0η0χl(l + 1)h

(+)l (k0a)− Kjl(Ka)³

B(2)l,m,D

(2)l,m

´#

; j = 2,

(2.62)

where the radial functions Ul and Vl have already been defined in Eq. (2.21) and

Eq. (2.22), respectively.

Thus the reactive power of the source JE,P , is given by

Im [P] =2X

j=1

∞Xl=1

lXm=−l

g(j)l |a(j)l,m|2, (2.63)

where

g(j)l =

1

2χ

⎧⎪⎨⎪⎩³D(j)l,m,D

(j)l,m

´¯³B(j)l,m,D

(j)l,m

´¯2 +Re⎡⎣u(j)l,ma(j)l,m

⎤⎦⎫⎪⎬⎪⎭ . (2.64)

By taking the real part of the complex interaction power, as given by Eqs. (2.60)-

(2.62), one also recovers Eq. (2.12) which is the well-known expression for the

radiated power in terms of the multipole moments.

Eqs. (2.62)-(2.64) relate χ directly to Im [P], as desired. For a certain problem,

29

where a(j)l,m and Im [P] are given, one can compute the values of χ for which Im [P] = 0by using these expressions, and pick, out of those values, the one which minimizes

the functional energy in Eq. (2.57). By substituting that value of χ (i.e., χ0) into

Eqs. (2.48), (2.50), and (2.53) one arrives at the desired solution.

Let

Ξ ≡ χ ∈ R : Im [P (χ)] = 0 . (2.65)

It is found11, numerically, that the minimum source energy is achieved for the value

of χ that is closest to χ = 0, i.e.,

|χ0| = infχ∈Ξ

|χ| . (2.66)

It appears only natural to assume that an increase in the source energy from EME

should correspond to χ0 (and any other value of χ ∈ Ξ for that matter). Thiswould be understood, intuitively, as a cost that one would have to pay to realize

a tuned antenna. The numerical simulations suggest that this, in fact, is the case:

substituting any nonzero value χ ∈ Ξ in the expression for EE,P yields a value thatis larger than EE,P |χ=0 = EME. Does this mean that EME is a lower bound of EE,Pand that EE,P (χ0) is a global minimum? Before we examine this question we notethat the above observations remind us of Eq. (2.38). Indeed, a convenient way of

viewing these observations is to think of the origin of the sphere BΓ (J0) 3 JE,P asthe point J0 = JME and to think of its radius as Γ ≥ |χ0|.

Supposing that EE,P (χ0) corresponds to a feasible point, let us derive a conditionfor it to be a global minimum. By definition EE,P (χ0) is said to be a global minimumwhen

EE,P (χ) ≥ EE,P (χ0) , ∀χ ∈ Ξ. (2.67)

If the inequality is strict then the global minimum is also unique.

11See Section 2.3.

30

It follows from Eqs. (2.17), (2.13), and (2.46) that

EE,P − 2χ Im[P] =2X

j=1

∞Xl=1

lXm=−l

c(j)∗l,m (JE,P ,B

(j)l,m)

=2X

j=1

∞Xl=1

lXm=−l

c(j)∗l,m a

(j)∗l,m . (2.68)

Thus if we require Im [P (χ)] = 0, Eq. (2.68) yields

EE,P =2X

j=1

∞Xl=1

lXm=−l

c(j)∗l,m a

(j)∗l,m . (2.69)

Furthermore, by projecting both sides of Eq. (2.46) onto the functions B(j)l,m while

recalling Eqs. (2.17) and (2.30) one obtains

a(j)l,m + χ(B

(j)l,m,

eGSJE,P) = c(j)∗l,m [σ

(j)l ]

2. (2.70)

By solving for c(j)∗l,m then substituting into Eq. (2.69) one obtains

EE,P = EME + χ2X

j=1

∞Xl=1

lXm=−l

(B(j)l,m,

eGSJE,P)a(j)∗l,m

[σ(j)l ]

2, (2.71)

where χ ∈ Ξ. Upon substituting JE,P(r), from Eq. (2.53), into Eq. (2.71) and usingstandard orthogonality properties of the spherical harmonics, one obtains

EE,P = EME + χ2X

j=1

∞Xl=1

lXm=−l

(B(j)l,m,

eGSD(j)l,m (χ))

(B(j)l,m,D

(j)l,m (χ))

|a(j)l,m|2

[σ(j)l ]

2. (2.72)

Expression (2.72) for the source energy directly assumes that Im [P] = 0, while

expression (2.57) holds for any value of the reactive power Im [P].

It follows from (2.67) and (2.72) that the condition for EE,P (χ0) to be a global

31

minimum is given by

2Xj=1

∞Xl=1

lXm=−l

⎧⎨⎩χ(B

(j)l,m,

eGSD(j)l,m (χ))

(B(j)l,m,D

(j)l,m (χ))

− χ0(B

(j)l,m,

eGSD(j)l,m (χ0))

(B(j)l,m,D

(j)l,m (χ0))

⎫⎬⎭ |a(j)l,m|2

[σ(j)l ]

2≥ 0 (2.73)

for any value of χ ∈ Ξ. Condition (2.73) is a necessary and sufficient condition forEE,P (χ0) to be a global minimum. The way it should be used is as follows. Fora given substrate, solve Im [P (χ)] = 0 for χ (where Im [P (χ)] is given by (2.62)-(2.64)). If condition (2.73) is satisfied for all values of χ ∈ Ξ then EE,P (χ0) is aglobal minimum. If it is not satisfied for at least one value of χ ∈ Ξ then EE,P (χ0)is not a global minimum (but it may still be a local minimum).

Condition (2.73) was written down based on the presumption that EE,P (χ0 6= 0)was a global minimum. For EE,P (χ0 = 0) = EME, condition (2.73) reduces to

χ2X

j=1

∞Xl=1

lXm=−l

(B(j)l,m,

eGSD(j)l,m)

(B(j)l,m,D

(j)l,m)

|a(j)l,m|2

[σ(j)l ]

2≥ 0, ∀χ ∈ Ξ. (2.74)

The preceding developments have emphasized the case of piecewise-constant ra-

dially symmetric backgrounds. Generalization to more general spherically-symmetric

backgrounds follows the same overall approach but then the formulation is based

on the respective forward solver sketched in Appendix E. We shall not dwell on this

here, and instead address numerical illustration of the preceding theory next.

2.3 Computer Simulation Study

The previous theory and algorithms are applied next to elucidate the effect of the

antenna-embedding medium on radiation performance for two classes of antennas:

electrically small, and larger (resonant) antennas. Within each class, we consider

both the minimum energy solution without tuning constraint, and the minimum

energy solution subject to the additional zero reactive power or tuning constraint.

The goal is to gain an understanding of the effect of the antenna substrate on the

minimum source energy for a given radiation pattern. Other related considerations

are also discussed. To present the results with focus, more attention is given in the

32

following to DPS and DNG materials.12

2.3.1 Minimum Energy Sources

It follows from Eq. (2.35) that, generally, the larger the singular valueshσ(j)l

i2, the

smaller the minimum source energy EME required for the launching of a given radi-

ation pattern with a source of a given size. The singular valueshσ(j)l ( r = 1 = µr)

i2correspond to the source in free space, that is, without the substrate. Thus, the

larger the singular valueshσ(j)l

i2for a given substrate wavenumber ka relative to

the corresponding free-space values, the greater the associated enhancement, due to

the substrate, of radiation of the lth multipole order field with given resources. It

is thus important to understand the dependence of the singular valueshσ(j)l

i2on

k0a, ka, r, µr and l, for both the electric (j = 1) and the magnetic (j = 2) cases.

Large singular values, such as resonances or peaks in the plots of the singular values

versus these variables, will indicate enhanced radiation for such operational modes

or conditions, with the given resources. This aspect is investigated numerically next.

Before engaging in the numerical illustrations we make some remarks: (1) the

multipolarity l is handled in the plots as a continuous variable to facilitate under-

standing of the curves, yet the meaningful results correspond solely to the discrete

values of l; (2) in the simulations the size parameter (radius) a of the antenna in-

cluding the substrate has been set to unity, i.e., a = 1 meter; and (3) in the plots

and associated discussion we consider the normalized wavenumbers defined by

x ≡ ka/π (2.75)

and

x0 ≡ k0a/π. (2.76)

The normalized wavenumber x represents the wavenumber of the field in the mater-

ial, hence, the effective electric size in the material, while the normalized wavenum-

ber x0 measures the respective size in free space.

12Having r > 0 and µr > 0 does not necessarily imply that the material is a conventionalone. Indeed, naturally occurring materials do not usually exhibit positive permeabilities and per-mittivities smaller that those of the vacuum. This makes manufacturing a material whose relativepermittivity is equal to 0 < r < 1 still require the use of metamaterials.

33

Behavior of the Singular Valueshσ(j)l

i2Fig. 2.2 shows, for different antenna sizes, the free-space singular values

hσ(1)l ( r = 1 = µr)

i2and

hσ(2)l ( r = 1 = µr)

i2, respectively. No local maxima or resonances are seen for

the free-space cases, in particular, in those cases the singular value spectrum decays

exponentially. Fig. 2.3 shows, for an antenna whose size corresponds to that of a

quarter-wave antenna in free space (x0 = 1/4), plots of the normalized electric singu-

lar valuesh(1)l (x0, x, r, µr)

i2 ≡ hσ(1)l (x0 = 1/4, x, r = 1)i2/hσ(1)l (x0 = 1/4 = x, r = 1 = µr)

i2versus l, parameterized by the normalized wavenumber in the substrate, x. From

now on, the normalized singular valuesh(j)l (x0, x, r, µr)

i2defined as

h(j)l (x0, x, r, µr)

i2 ≡hσ(j)l (x0, x, r, µr)

i2hσ(j)l (free-space case)

i2=

hσ(j)l (x0, x, r, µr)

i2hσ(2)l ( r = 1 = µr)

i2 (2.77)

will be referred to, simply, as singular values, unless otherwise specified. The singular

value spectrum plots for the larger x values considered (x = 5 and 10) reveal well-

defined resonances (local peaks).

The dominant resonances for these larger x values occur around l ∼ π. In fact,

the resonances in question appear to arise only when x>∼ 1. Overall, it is seen that

as the material becomes electromagnetically denser, i.e., as the substrate normalized

wavenumber x increases, the magnitudes of the singular values become accordingly

larger. Since electrically small antennas such as the one considered here can effec-

tively radiate only the lowest multipole orders (such as the dipolar mode), then of

particular interest for small antenna applications is the antenna substrate-induced

enhancement for low multipolarity l. The plots reveal that the dipolar-mode (l = 1)

singular values can be significantly higher for the embedding substrate case than for

the free-space case. The improvement for x = 5 and 10 relative to the free-space case

is of more than 3 orders of magnitude (decades). This means that the magnitude of

34

1 3 5 7 9 11 13 15

l10−15

10−11

10−7

10−3

10

1051 3 5 7 9 11 13 15

10−15

10−11

10−7

10−3

10

105

x=10

x=5

x=1

x=0.5

x=0.25

Figure 2.2: Free-space singular valueshσ(1)l (x = x0, r = +1 = µr)

i2versus l for

a few representative values of x0 ≡ k0a/π. (The unit of the singular values isV 2m/A2.)

1 3 5 7 9 11 13 15

l

1

102

104

106

108

1 3 5 7 9 11 13 15

1

102

104

106

108

x=10

x=5

x=1

x=0.5

x=0.25

Figure 2.3: Normalized singular valuesh(1)l


case), r = +1 and a few representative values of x.

35

1 5 10 15 20 25 30 35 40 45

l

10−2

1

102

104

106

1 5 10 15 20 25 30 35 40 45

10−2

1

102

104

106

x=20

x=10

x=5

x=1

x=0.5

x=0.25



electrically-large antenna), r = +1 and a few representative values of x.

the exciting current or source required for launching of the given dipolar field can

be made correspondingly smaller than in free space by embedding the antenna in

a high wavenumber or electromagnetically dense substrate. Alternatively, for fixed

source energy, the antenna size parameter a can be reduced relative to its value

without the embedding substrate. The improvement for l = 2 and 3 associated to

the larger wavenumber cases (x = 5 and 10) is also noticeable.

The respective plot for the case of a resonant or electrically-large x0 = 10 antenna

is shown in Fig. 2.4. The respective magnetic singular value spectra are shown in

Figs. 2.5 and 2.6. Many of the key features outlined above while explaining the

particular electric quarter-wave antenna case also arise for these other cases. Yet

other aspects become salient. A summary of the main results is given next, along

with some of the former observations, as general conclusions learned from these

simulations as a whole.

It is seen that, for sufficiently large multipolarity l (i.e., for l >∼ 6), and for thevalues of x0 considered which comprise both small and large or resonant antennas,

the singular values are consistently higher for the denser substrates (larger x) than

for the less dense substrates including the free-space (x = x0) case. This is true

36

1 3 5 7 9 11 13 15

l

1

102

104

106

108

1 3 5 7 9 11 13 15

1

102

104

106

108

x=10

x=5

x=1

x=0.5

x=0.25



case), r = +1 and a few representative values of x.

1 5 10 15 20 25 30 35 40 45

l

10−8

10−6

10−4

10−2

1

102

104

106

108

1 5 10 15 20 25 30 35 40 45

10−8

10−6

10−4

10−2

1

102

104

106

108

x=20

x=10

x=5

x=1

x=0.5

x=0.25



electrically-large antenna), r = +1 and a few representative values of x.

37

for both electric (j = 1) and magnetic (j = 2) modes. As we had indicated for the

particular electric quarter-wave antenna case, generally for x = x0 (no embedding

medium or free-space case), the singular value spectrum decays exponentially with

l, i.e., without resonances. This decay is more or less exponential for the smaller

antenna cases. For larger antennas the singular values remain more or less within a

given order of magnitude until about the cutoff l ∼ k0a,13 but this cutoff is clearly

higher, i.e., includes higher order multipoles, for the large wavenumber cases.14 This

further shows performance enhancement via larger wavenumber or electromagneti-

cally denser substrates since higher multipoles represent higher antenna directivity,

i.e., higher level of details or narrower width in the radiation pattern. It is also im-

portant to note that the enhancement in the singular values due to larger substrate

wavenumber k holds for both small and large multipolarities l.

Having shown some of the radiation enhancing possibilities offered by electro-

magnetically denser substrates, we discuss next the question of local optimal se-

lection of the wavenumber x. Consider, for example, a half-wave antenna (so

that x0 = 1/2) embedded in a substrate with r = 1 and launching purely mag-

netic modes (j = 2). Local maxima of the respective normalized singular valuesh(2)l (x0 = 1/2, x, r = 1, µr)

i2for l = 1, 2, and 3 were found to occur as follows: For

the emission of dipole radiation (l = 1) at x ' 1.430, with an enhancement or gainh(2)l (x0 = 1/2, x = 1.430, r = 1, µr)

i2 ' 3.110× 105, relative to free space; for theemission of quadrupole radiation (l = 2) at x ' 1.833, with a gain relative to freespace of 1.925×107; and for the emission of octupole radiation (l = 3) at x ' 2.224,with a gain of 1010. For antennas embedded in denser substrates the numerical

study indicates, however, that the improvement attained is comparatively marginal.

For example, the gain associated to going from the aforementioned values of x to the

local maxima at x ∼ 10 is only 44.03, 23.86, and 5.55 for the dipole, quadrupole andoctupole radiation cases, respectively. Conversely, a half-wave antenna radiating

purely electric modes instead displays a significantly different behavior in this re-

13This value, l ∼ k0a, approximately corresponds to the inflection point in the singular valuespectrum curve for the free-space case.

14Note that in order to see this visually one would need to plot not the normalized singular

valuesh

(j)l

i2but the singular values

hσ(j)l

i2themselves. Yet a careful comparison of normalized

values in Figs. 2.3-2.6 and the illustrative free-space values in Fig. 2.2 leads to the same conclusion.

38

gard, and the overall improvements of the substrate are also more significant. Thus

for modest values of x, a locally maximum improvement in the radiation ability of

the half-wave antenna can be attained for the following values: For electric dipole

radiation at x ' 0.946 with a gainh(1)1 (x0 = 1/2, x = 0.946, r, µr)

i2 ' 5.36; for

quadrupole radiation at x ' 1.362 with a gain relative to free space of 163.6; for

octupole radiation at x ' 1.800 with a gain of 2.952 × 104. For denser materialsthe enhancement relative to free space can be significantly larger. Thus numerical

maximization ofh(1)l (x0, x, r, µr)

i2yields the following gains associated to going

from the aforementioned values of x to the local maxima at x ∼ 10 : 126, 70.81, and33.56 for the dipole, quadrupole and octupole radiation cases, respectively. The first

two of those numbers are relatively significant enhancements, yet for much denser

materials the enhancements are less dramatic, though still meaningful. These con-

siderations are reinforced from an alternative point of view in the next subsection

which revisits the topic of locally optimal antenna substrate wavenumber for a broad

range of antenna sizes under electric dipole radiation.

A legitimate question arises as to the physical reason behind the appearance

of these resonances in the spectra of the non-free-space singular values. As noted

earlier a careful examination of the quantities F (j)l defined in (2.20) shows that

these quantities are essentially the amplitudes of the internal electromagnetic Mie

fields due to the scattering of a plane wave by a sphere of radius a and propagation

constant k embedded in an infinite homogeneous medium of propagation constant

k0; F(1)l being the amplitudes of the electric modes and F

(2)l those of the magnetic

modes [123, 113]. The question that arises now is: how does the presence of these

amplitudes affect the behavior of the singular valuesh(j)l (x0, x, r, µr)

i2, or, more

precisely, are those resonant peaks, which correspond to local maximum enhance-

ment, related to Mie resonances? Before answering this question we review very

briefly the features of Mie resonance that are most relevant to our results. Mie

resonances of the abovementioned sphere are characterized by the vanishing of the

denominators of the amplitudes F (j)l , or, more realistically, by the requirement that

those denominators be minimum [113]. Thus the resonance conditions can be cast

39

in the form of approximate transcendental equations, viz.,

√rVl(x0)

h(+)l (x0)

' √µrUl(x)

jl(x), (2.78)

for the electric modes, and

√µr

Vl(x0)

h(+)l (x0)

' √ rUl(x)

jl(x), (2.79)

for the magnetic modes, where Ul and Vl are the functions defined in (2.21) and

(2.22). (These rather loosely stated conditions could certainly be made more rigor-

ous but this is enough for our purposes.) Because of the presence of Bessel functions,

Eqs. (3.8) and (3.9) admit a discrete, albeit infinite, set of solutions. These solutions

correspond to the so-called Mie resonances.

Now, we can go back to the question of how the observed resonant peaks

which correspond to local maximum enhancement relate to Mie resonances. Sin-

gular valuesh(j)l

i2, defined by Eqs. (2.77), (2.31), and (2.32), are composed

not only of the quantities¯F(j)l

¯2, defined in (2.20), but also of another term,

viz.,hκ(j)l (x0, x)

i2, defined in (2.32), and unless these latter quantities are suf-

ficiently well-behaved one cannot conclude anything as to the relationship of the

resonant values ofh(j)l (x0, x, r, µr)

i2to Mie resonances. Incidentally, the quan-

titieshκ(j)l (x0, x)

i2where x ∈ R are essentially non-pathological combinations of

the spherical Bessel functions jl (λa) which are well-behaved for all integer values

of l and λ ∈ R [124] (which represent the most general cases considered in this

work). Hence, one can confidently claim that the observed peaks in the spectrum

of the singular values are primarily due to the phenomenon of Mie resonance and

maximum enhancement conditions are effectively summarized by the two conditions

(3.8) and (3.9). Therefore, for a given antenna radiating at a prescribed frequency,

the discrete set of solutions x corresponds to a set of constitutive parameters s and

µs that maximize the radiated electromagnetic fields. As their amplitudes increase

these radiated fields draw energy from the embedding medium. But because this

medium is of finite extent the energy extraction process saturates, ultimately, and

40

02

46

810 x

01

23

45

x0

1. × 10−13

1. × 10−11

1. × 10−9

1. × 10−7

02

46

8 x

12

34

5

Figure 2.7: Logarithmic mesh plot of the source energy E(j=1=l)ME versus x0 and x fora double-positive material with r = +1.

as a result of this saturation the fields fall short of effectively “blowing up.”

Further Details: Electric Dipole Radiation

This part examines in greater detail the fundamental electric dipole radiation case, in

particular, the multipole moment a(j)l,m = 1 if j = 1 = l and m = 0, and a(j)l,m = 0 oth-

erwise. The minimum source energy reduces in this case to E(j=1=l)ME (x0, x, r, µr) =hσ(1)1 (x0, x, r, µr)

i−2. Fig. 2.7 shows a mesh plot of the minimum source energy

E(j=1=l)ME versus the normalized wavenumbers x0 and x for a DPS substrate material

with r = 1. For a DNG substrate material having r = −1, the numerical studyshows that the minimum source energy displays a very similar, though not com-

pletely symmetrical, behavior when x changes sign, for a given x0. Consequently,

source energy E(j=1=l)ME is not an even function of x, and hence distinguishes between

DPS and DNG embedding substrates. Fig. 2.8 shows slices or cross-sections of the

41

0 1 2 3 4 5 6 7 8 9 10 11x

10−8

10−6

10−4

10−2

Source

Energy

0 1 2 3 4 5 6 7 8 9 10 11

10−8

10−6

10−4

10−2

x0=10

x0=5

x0=1

x0=0.5

x0=0.25

Figure 2.8: Logarithmic plot of the source energy E(j=1=l)ME versus x for r = +1 andsome representative values of x0 for a double-positive medium.

mesh plot in Fig. 2.7 for particular values of the free-space normalized wavenumber

x0, and Fig. 2.9 shows a slice similar to 2.8 but for negative x. Fig. 2.7 also shows

that, in general terms, source energy tends to decrease as the size of the antenna

increases, this is also true when x is negative. Thus as the antenna size increases

it tends to be easier to distribute the source currents in a more efficient way. As

shown in Figs. 2.8 and 2.9, for small antennas the source energy exhibits its first

local minima at |x| ∼ 1. In particular, for x0 = 1/4 (quarter-wave antenna case)

and x0 = 1/2 (half-wave case) the first local minimum of E(j=1=l)ME appears for pos-

itive x at x ' 0.960 and x ' 0.946, respectively, and for negative x at x = −0.760and x = −0.860, respectively. For x0 = 1 (full-wave antenna case) the first lo-

cal minimum of E(j=1=l)ME appears for positive x at x ' 1.155 and for negative x atx ' −1.200. However, for large antennas a slightly more subtle behavior is observed.If |x| < x0 the local minima of E(j=1=l)ME appear at |x| ∼ (2n+ 1) /2, n = 1, 2, 3, ...,while if |x| > x0 the minima appear at |x| ∼ n, n = 1, 2, 3, ..., with the least minimum

still belonging to the smallest antenna (cf. Figs. 2.8 and 2.9). These rules-of-thumb

depend on the particular combination of constitutive parameters r and µr under in-

vestigation. To illustrate this we display in Fig. 2.10 the logarithmic plot of E(j=1=l)ME

42

0−1−2−3−4−5−6−7−8−9−10−11x

10−8

10−6

10−4

10−2

Source

Energy

0−1−2−3−4−5−6−7−8−9−10−11

10−8

10−6

10−4

10−2

x0=10

x0=5

x0=1

x0=0.5

x0=0.25

Figure 2.9: Logarithmic plot of the source energy E(j=1=l)ME versus x for r = −1 andsome representative values of x0 for a double-negative medium.

versus x for µr = 1. In this figure one clearly sees that the rules are interchanged,

i.e., now, if x < x0 the local minima of E(j=1=l)ME appear at x ∼ n, n = 1, 2, 3, ..., while

if x > x0 the minima appear at x ∼ (2n+ 1) /2, n = 1, 2, 3, ...(This, of course, is

rather expected as the positions of the minima depend on the particular combination

of constitutive parameters r and µr.) Another feature which is worth noting is that

the local minima of the functional energy keep decreasing for increasing values of |x|at an even slower rate, though the numerical study does not seem to conclusively

indicate whether for |x| À 10 there is a limiting value.

A fundamental aspect is presented in Fig. 2.11 which shows a plot of the free-

space case source energy E(j=1=l)ME (x = x0, r = 1 = µr) versus x0. This plot illus-

trates the well-known behavior that the source functional energy increases extremely

fast as x0 decreases below x0 ∼ 1/2: apparently it increases exponentially. The en-ergy reaches a mildly oscillating valley at x0 ∼ 1/2, but the oscillations die out veryrapidly, while the energy approaches a certain limiting value as ka increases. On the

one hand, these observations imply that launching the dipolar mode costs signifi-

cantly more energy if the size of the antenna is smaller than, say, half a wavelength,

and, on the other hand, that increasing the size of the antenna beyond this threshold

43

0 1 2 3 4 5 6 7 8 9 10 11x

10−8

10−6

10−4

10−2Source

Energy

0 1 2 3 4 5 6 7 8 9 10 11

10−8

10−6

10−4

10−2

x0=10

x0=5

x0=1

x0=0.5

x0=0.25

Figure 2.10: Logarithmic plot of the source energy E(j=1=l)ME versus x for µr = 1 andsome representative values of x0.

0 1 2 3 4 5 6x0=x

10−5

10−4

10−3

10−2

10−1

Source

Energy

0 1 2 3 4 5 6

10−5

10−4

10−3

10−2

10−1

Figure 2.11: Logarithmic plot of the source energy E(j=1=l)ME (x = x0) versus x = x0for r = +1.

44

0 2 4 6 8 10 12 14 16 18 20x

10−2

1

102

104

Gain

0 2 4 6 8 10 12 14 16 18 20

10−2

1

102

104

x0=10

x0=0.5

x0=0.25

Figure 2.12: Logarithmic plot of the gain G, versus x for r = +1 and some selectedvalues of x0.

has little effect on the minimization of the source functional energy associated to

this particular mode. In Fig. 2.12 the gain, defined as

G ≡ E(j=1=l)ME (x = x0, r = 1 = µr)

E(j=1=l)ME

=h(1)1

i2, (2.80)

has been plotted as a function of the normalized substrate wavenumber x. The

information depicted in this plot is equivalent to that in Fig. 2.8, but the latter figure

highlights more clearly the reduction in required source energy for the radiation of

the dipolar mode for small and large antennas. For the small antenna cases, the

enhancement is seen to be of more than 4 orders of magnitude for the resonant x

values. Similar results (not shown) were found pertinent to radiation enhancement

for higher order multipoles in the large antenna cases.

Finally, to further illustrate the possibility of reducing radiator size while achiev-

ing a given radiation pattern with prescribed source resources, specifically, source

energy, we considered the free-space wavenumber k0 = π/4, and sought for values

45

of the size parameter a for which the minimum source energy of a source embedded

in a medium having k = 10π renders the same source energy as a unit-valued a

embedded in free space, for which k = k0 = π/4. For an embedding substrate with

r = 1 the first such values of a are 0.098, 0.101, 0.196, 0.204, ..., (units of meters)

which are seen to occur in pairs around 0.1, 0.2, 0.3, etc. This is not surprising in

light of the formula introduced earlier, in particular, the locally optimal values of ka

are ka ∼ nπ, n = 1, 2, 3, ..., that is, a ∼ 1/10, 2/10, 3/10, .... The values of the sizeparameter a for which the source energy in question coincides with the free-space

case source energy for a larger source having unit-valued radius then occur in pairs

around these optimal values, which completes the picture.

2.3.2 Tuned Minimum Energy Sources: Additional Zero ReactivePower Constraint

Next we consider minimum energy sources subjected to the additional zero reactive

power constraint. In particular, we require the reactive power to vanish, that is,

Im [P] = 0. As in the preceding subsection, the focus is the fundamental case

of an electric dipole radiator (specifically, a(j)l,m = 1 if j = 1 = l and m = 0,

and a(j)l,m = 0 otherwise). Particular attention is given to the quarter-wave and the

half-wave antenna cases, though some results related to larger antennas are also

presented.

As in [14], we define the normalized reactive power

g(1)1 ≡

g(1)1

Re [P]= η0g

(1)1 , (2.81)

where Re [P] = 1/2η0 is the radiated power and where the free-space wave

impedance η0 =pµ0/ 0 ' 120π Ω. We define χ0 as the Lagrange multiplier value

χ which annuls the normalized reactive power g(1)1 , i.e., g(1)1 (χ)

¯χ=χ0

= 0 and for

which the resulting source energy is minimal among all such zero reactive power

Lagrange multiplier values. The value in question was consistently found to occur

in the vicinity of χ = 0. This is not surprising since the absolute or unconstrained

minimum energy source and its energy E(j=1=l)ME correspond to χ = 0, that is, the

46

-4 -2 0 2 4Lagrange Multiplier

-4

-2

0

2

4

ReactivePower

Figure 2.13: Plot of the normalized reactive power g(1)1 versus χ for x0 = 1/4 = xand r = +1.

minimum energy source is min E(j=1=l)EP ≡ limχ→0E

(j=1=l)EP (x0, x, r, µr, χ) = E(j=1=l)ME

(see section 3).

Fig. 2.13 is a plot of the normalized reactive power g(1)1 versus the Lagrange

multiplier χ for a quarter-wavelength antenna, embedded in substrates with r = 1

and x = 1/4. The Lagrange multiplier value χ0 is sought for which the respective

source energy (showed in Fig. 2.14) is minimized among all χ values rendering zero

reactive power. Figs. 2.15 and 2.16 show the respective plots for r = −1, andTables 2.1, 2.2, 2.3, and 2.4 summarize the values of χ0, source energy for χ = χ0,

i.e., E(j=1=l)E,P , and absolute minimum energy E(j=1=l)ME , for the cases addressed in

these plots, as well as for other cases (other values of x).

One notes from these results that the minimum energy solution J(j=1=l)ME yields

minimum source energy E(j=1=l)ME or current level but its reactive power is comparable

to the maximum, saturated value corresponding to χ À 1 for the DPS materials,

and χ ¿ −1 for DNG materials (cf. Figs. 2.13 and 2.15). On the other hand,

the new solution J(j=1=l)E,P corresponding to χ0 yields zero reactive power at the

expense of a raised source energy or current level (cf. Tables 2.1, 2.2, 2.3, and 2.4).

The difference between the source energies E(j=1=l)EP and E(j=1=l)ME of the two sources

47


10−3

10−1

10

103

105SourceEnergy

10−3

10−1

10

103

105

Figure 2.14: Plot of the source energy E(j=1=l)EP versus χ for for x0 = 1/4 = x andr = +1.

-4 -2 0 2 4

Lagrange Multiplier

-10

-5

0

5

10

Reactive

Power

Figure 2.15: Plot of the normalized reactive power g(1)1 versus χ for x0 = 1/4 = −xand r = −1 (“anti-vacuum.”)

48


10−4

10−2

1

102

104

106

SourceEnergy

10−4

10−2

1

102

104

106

Figure 2.16: Plot of the source energy E(j=1=l)EP versus χ for x0 = 1/4 = −x andr = −1 (“anti-vacuum.”).

JE,P and JME , respectively, is the source energy of the additional nonradiating part

contained in JE,P whose role in the new source is to counteract the reactive power ofthe minimum energy source alone. It decreases as the electromagnetic density of the

substrate increases, this being true for both DPS and DNG substrates. We found

that, for x0 = 1/4 and 1/2, performances better than those of the free-space cases

(i.e., for which k = k0 and r = 1 = µr) can be achieved.15 Superior performance

can also be obtained by means of a judicious choice of the substrate constitutive

properties, as we explain below. In addition to this, we note that the minimum of the

energy decreases as the electromagnetic density of the substrate increases, whether

the substrate is DPS or DNG. In Figs. 2.14 and 2.16 it is clear that as χ → 0 the

source energy E(j=1=l)E,P reaches an absolute minimum min E(j=1=l)EP , as expected. This

minimum is not the same for DPS materials and DNG metamaterials (cf. Tables

2.1, 2.2, 2.3, and 2.4), as we discussed earlier. Interestingly, the cancellation of the

reactive power is not always possible. For instance, for a quarter-wave antenna, and

for r = 1, the Eq. g(1)1 (χ)

¯χ=χ0

= 0 admits no solutions if x = −1/4 or −1/2, as

15Cf. Tables 2.1, 2.2, 2.3, and 2.4, though for the sake of space the energy differenceE(j=1=l)EP (χ0)−min E(j=1=l)EP is not explicitly displayed in the tables.

49

illustrated in Fig. 2.15 for the x = −1/4 case.16Furthermore, it also follows that if one allows the electromagnetic properties of

the embedding substrate (i.e., r and µr) to vary then one could make the reactive

power vanish for χ0 = 0, this being a matching condition under which the minimum

energy sources are not only of local minimum energy (see below) but also self-

matched to resonance. Let us illustrate this for a quarter-wavelength antenna. For

a given positive relative electric permittivity, for instance r = 1, we find that the

matching condition mentioned above is satisfied for x ' 0.511, i.e., in this case

r = 1 and µr ' 4.18. Now, for a given negative relative electric permittivity, for

instance for r = −1, we find that the matching condition is satisfied for x ' −1.338,i.e., in this case r = −1 and µr = −28.64. A word of caution is necessary at thispoint. From the definition itself of the above-mentioned matching one obtains for

χ0 = 0 : E(j=1=l)E,P = min E(j=1=l)E,P (cf. Tables 2.1 and 2.3.) Yet one must not be

lured into thinking that the substrate constitutive properties r and µr associated

to the matching cases must correspond to global minima for E(j=1=l)E,P , i.e., that they

represent the best substrate values. This is very clearly illustrated in Tables 2.1 and

2.3 where for x = 1, 5 and 10 in Table 2.1 and for x = −1,−5 and −10 in Table2.3 one has E(j=1=l)E,P

¯χ0 6=0

< E(j=1=l)E,P¯χ0=0

. In other words a quarter-wavelength

antenna embedded in substrates having those values of x as their electromagnetic

densities exhibit source energies smaller than those exhibited by the antenna when

it is embedded in a substrate whose constitutive parameters satisfy the matching

condition.

16This is also true for x0 = 1/2, and 1 though the figures are not presented here.

50

Table 2.1: Results of the numerical study for the constrained quarter-wave antennaembedded in a double-positive material with r = +1. (The unit of the sourceenergies is A2/m.)

x χ0[10−4] E(j=1=l)EP (χ0) min E(j=1=l)EP

1/4 −421.2 4.637× 10−4 1.161× 10−41/2 −5.803 8.031× 10−5 8.014× 10−50.511 0 7.809× 10−5 7.809× 10−51 −44.52 2.090× 10−5 2.206× 10−65 −8.039 4.782× 10−6 7.094× 10−810 −4.008 2.486× 10−6 1.763× 10−8

Table 2.2: Results of the numerical study for the constrained half-wave antennaembedded in a double-positive material with r = +1. (The unit of the sourceenergies is A2/m.)


1/4 −899.5 3.087× 10−4 5.854× 10−51/2 −80.37 4.857× 10−5 4.191× 10−51 −35.08 1.513× 10−5 8.824× 10−65 −13.51 3.690× 10−6 2.838× 10−710 −7.234 2.081× 10−6 7.051× 10−8

Table 2.3: Results of the numerical study for the constrained quarter-wave antennaembedded in a double-negative metamaterial with r = −1. (The unit of the sourceenergies is A2/m).


-1/4 · · 3.071× 10−5-1/2 · · 1.462× 10−5-1.338 0 7.980× 10−5 7.980× 10−5-1 24.27 6.288× 10−5 2.540× 10−5-5 7.218 1.708× 10−5 8.167× 10−7-10 3.757 9.392× 10−6 2.029× 10−7

51

Table 2.4: Results of the numerical study for the constrained half-wave antennaembedded in a double-negative metamaterial with r = −1. (The unit of the sourceenergies is A2/m.)


-1/4 · · 3.718× 10−5-1/2 · · 2.554× 10−5-1 37.88 2.696× 10−5 1.462× 10−5-5 13.74 6.821× 10−6 4.702× 10−7-10 7.304 3.825× 10−6 1.168× 10−7

2.4 Conclusion

In this chapter we presented a mathematical theory of the full-vector, electromag-

netic inverse source problem which is applicable to sources embedded in substrates,

with applications to the analysis, source-synthesis and characterization of anten-

nas embedded in substrates. The present work completes the research program on

the inverse source problem in non-homogeneous background media initiated in [17].

The developments considered sources in a spherical volume of radius a, so that the

results and overall conclusions fundamentally hold for rather general sources of a

given maximal dimension (2a), and this also conveniently enabled us to treat the

relevant source-to-field mappings in the spherical coordinate system or multipole

wavefunction domain.

A key objective was to gain understanding from first principles of potential radi-

ation enhancements (reduction of required antenna resources (physical size, current

levels, level of tuning, and so on) for a given far field) due to such substrates. This

problem was treated in the present work within a general and non-device-specific

framework whose predictions (such as performance bounds) under normalized re-

sources are fundamental. The derived theory and the associated numerical illustra-

tions also yielded fundamental insight about the interplay of the variables involved,

as well as an idea of good values for antenna design parameters.

The results were discussed addressing separately the cases of small versus large

or resonant antennas, with the overall conclusion that for small antennas one can

significantly enhance the radiated power or compress source size via the substrates

52

under normalized antenna resources, while for larger antennas the use of substrates

can significantly enhance both radiated power and directivity (related to the num-

ber of essentially independent field modes that can be radiated effectively) under

the given resources. Our analysis thus formally explains, from a first principles,

non-device-specific source-inversion point of view, similar findings by other groups

working in the general area of substrate-enhanced antennas.

53

Chapter 3

Radiation Enhancement due toMetamaterial Substrates:Core-Shell System

To present the non-antenna-specific theory of substrate enhancement developed in

Chapter 2 in a context that relates to important work done by other groups in

this area, we generalize, in this chapter,1 the formalism developed in Chapter 2

to sources that are embedded within two nested spheres2 made up of arbitrary

lossy substrates. This particular configuration is of great practical importance for

both antenna radiation and scattering and hence imaging resolution enhancement.

It enables enhancements that are more dramatic than those due to homogeneous

substrates. Particular emphasis is given to the special case when the two nested

spheres are made up of materials with oppositely signed constitutive parameters.

The radiative as well as the scattering properties of a system of two nested

spheres of ordinary materials associated with a dipole have been considered by

several authors [126, 127, 128, 129]. These studies have now been extended to

cases where metamaterials are present. For instance, Gao and Huang [130] have

calculated the extinction efficiency of the core-shell system. Following the steps of

Aden and Kerker [131], Alù and Engheta [132, 133] have looked at the resonant

1This study has been published in [125].2Other terminologies used for this system are: Core-shell system or three-region system.

54

a b

, aa

b , b

Vacuum

Figure 3.1: Geometry of the three-region system under consideration. The drivingpoints and material structure of the antenna are confined within a spherical volumeV of radius a. The inner sphere of radius a has relative permittivity a and relativepermeability µa. This inner sphere is surrounded by a spherical shell of inner radiusa and outer radius b and has relative permittivity b and relative permeability µb.The core-shell system is immersed in the vacuum.

scattering that arises when the two spheres are constructed by combining a pair

of materials with oppositely signed constitutive parameters. Ziolkowski and Kipple

[40] have established the reciprocity of the peculiar scattering properties described

by Alù and Engheta [132] and the enhanced radiation power they realized would

occur when an electrically small dipole antenna is surrounded by a metamaterial

shell [8].

The geometry of the system to be investigated in the following is depicted in

Fig.(3.1). The inner sphere, of radius a, has relative electric permittivity a ≡sphere/ 0 and relative magnetic permeability µa ≡ µsphere/µ0. This inner sphere

constitutes the core of the system and is the smallest spherical volume V that cir-

cumscribes the largest physical dimension of the original antenna which is treated

next, under a suppressed time dependence e−iωt, as a primary, or impressed, currentdensity J(r). The core is surrounded by a spherical shell, of inner radius a and outer

radius b. The relative constitutive parameters of the shell are relative electric per-

mittivity b ≡ shell/ 0 and relative magnetic permeability µb ≡ µshell/µ0. Thus the

resulting three-region system may be characterized by a total electric permittivity

55

distribution of the form

(r)

0= aΘ(a− r) + b [Θ(r − a)Θ(b− r)] +Θ(r − b) (3.1)

and a total magnetic permeability distribution of the form

µ(r)

µ0= µaΘ(a− r) + µb [Θ(r − a)Θ(b− r)] +Θ(r − b), (3.2)

where Θ denotes Heaviside’s unit step function (Θ(x) = 1, for x > 1, otherwise

Θ(x) = 0), and 0 and µ0 are, respectively, the electric permittivity and magnetic

permeability of the vacuum.

The core and the surrounding shell, being assumed to be generally lossy, are as-

signed relative constitutive parameters that are generally complex. These constitu-

tive parameters are, thus, assumed to have the generic forms: α = Re [ α]+i Im [ α]

and µα = Re [µα] + i Im [µα], α = a, b (where a is for the inner sphere and b is for

the outer shell.) Note that the losses are indicated by the presence of non-negative

imaginary parts of the permittivity and the permeability.

The approach adopted next is to follow the same steps as in Chapter 2 to formu-

late an inverse source problem in substrate media, whose objective is to deduce an

unknown primary current density J(r) that is contained, along with the substrate,

in the spherical volume V , and that generates a prescribed exterior field for |r| > b.

It is important to point out that the present formulation of the inverse source

problem in the two-nested-spheres configuration is also relevant to that of the com-

panion inverse scattering problem. In fact, it is mathematically equivalent to that

of an inverse scattering problem in which a single incident field is used as excita-

tion. Hence, the results on the inverse source problem and on the possibility of

extracting higher spatial frequency information about the unknown object thanks

to the presence of the embedding medium [17] also point out the possibility of

similarly enhancing imaging resolution in the associated inverse scattering problem

with helper substrate media. In addressing the inverse source problem in these me-

dia one is automatically paving the way for inverse scattering formulations in such

media. This is particularly pertinent in the modern context of so-called qualita-

tive imaging methods based on support estimation of induced sources and which

56

are non-iterative (e.g., linear sampling, factorization method, time-reversal MUSIC,

and so on) [134, 135, 136]. Those inverse scattering methods are of the so-called

“inverse-source” type, in other words, they are based on a companion inverse source

problem.

3.1 The Radiation Problem

The electric and magnetic multipole moments, a(1)l,m and a(2)l,m, respectively, are related

to the current distribution J by Eq.(2.17). It is shown in Appendix C that for

backgrounds whose permittivity and permeability are given by Eqs.(3.1,3.2) one

has3

B(j)l,m ≡

⎧⎪⎪⎨⎪⎪⎩−η0l(l+1)F

∗(1)l ∇× [jl(k∗ar)Yl,m(r)] ; j = 1

−ik0η0l(l+1) F

∗(2)l jl(k

∗ar)Yl,m(r) ; j = 2,

(3.3)

where ka = ω√

a 0µaµ0 is the inner sphere substrate wavenumber, and where we

have defined4

F(j)l ≡

⎧⎪⎨⎪⎩− b

∆1k0kba2b2; j = 1

−µaµb∆2k0kba2b2

; j = 2,

(3.4)

where (cf. Eqs. (26,27) in [131], and Eqs.(8,9) in [132, 133])

∆1 =

¯¯¯

0 kbUl(kbb) kbVl(kbb) −k0Vl(k0b)0 bjl(kbb) bh

(+)l (kbb) −h(+)l (k0b)

kaUl(kaa) −kbUl(kba) −kbVl(kba) 0

ajl(kaa) − bjl(kba) − bh(+)(kba) 0

¯¯¯ (3.5)

and

∆2 =

¯¯¯

0 jl(kbb) h(+)l (kbb) −h(+)l (k0b)

0 kbµbUl(kbb)

kbµbVl(kbb) −k0Vl(k0b)

jl(kaa) −jl(kba) −h(+)l (kba) 0kaµaUl(kaa) − kb

µbUl(kba) − kb

µbVl(kba) 0

¯¯¯ . (3.6)

3Cf. Eq.(2.19).4Cf. Eq.(3.4).

57

In (3.5) and (3.6) the quantities Ul and Vl are defined such that by Eqs.(2.21) and

(2.22).

Note that in Eqs.(3.4,3.5,3.6) k0 = ω√

0µ0 and kb = ω√

b 0µbµ0 are the prop-

agation constants in the vacuum and in the shell, respectively. Also it is clear from

Eqs.(3.4,3.5,3.6,2.21,2.22) that the parameters F(j)l , j = 1, 2, are not constants but

in fact functions of several parameters.

It is not hard to show that the terms F(j)l reduce to the Mie amplitudes F (j)l ,

as defined in (2.20) , when a = b, as expected.5 Consequently, they also reduce to

unity in the free-space case, i.e., when a = b, a = 1 = b, and µa = 1 = µb, causing

Eqs.(3.3) to reduce to the free-space case equations [59, 74].

3.2 Inverse Source Theory Based on Constrained Opti-mization

Following the same procedure as in Chapter 2 one finds that the minimum-energy

source is given by (2.29) and that the corresponding minimum source energy is (2.35)

where the positive-definite singular values are nowhσ(j)l

i2 ≡ (B(j)l,m,B

(j)l,m) ≡ |F(j)l |2

hκ(j)l

i2, (3.7)

where are given by Eqs.(3.4,3.5,3.6,2.21,2.22) andhκ(j)l

i2are given by (2.32). In

Chapter 2, analytical expressions where derived for the integrals appearing in (2.32),

for k2a ∈ R, by means of Lommel’s second integral. In the general case, however, thecomplex argument of the Bessel functions makes it necessary to recur to numerical

methods of integration.

The dimensionless positive-definite normalized singular valuesh(j)l

i2have been

defined in Chapter 2 (cf. Eq.(2.77)). In that definition the quantitieshσ(j)l ( r = 1 = µr)

i2referred to what we may call “the free-space case.” In the case of a homogeneous

sphere this was easy to define: Setting r = 1 = µr was equivalent to defining a

reference antenna of length 2a and radiating in the vacuum, with respect to which

radiation performance was evaluated. For the core-shell system investigated in this

5See also [17, 137, 112].

58

chapter, however, the free-space case is not as simple to define. Here, it is defined

as the case in which the original antenna, as defined in Section I, radiates in the

vacuum. Quantitatively this corresponds to the case: a = b, a = b = µa = µb = 1.

In other words, the reference antenna with respect to which the comparisons are

carried out is the original antenna without the shell. This may sound as an unfair

comparison, after all the new antenna, i.e., the core-shell system, represents a totally

new antenna with its own new dimensions and induced currents due to the addition

of the shell to the original antenna. What is more, we allowed, in the numerical

simulations, the dimensions of the outer shell to be of comparable size to the core.

Consequently, this definition may not sound as the best definition for a reference or

standard antenna to measure the enhancement with respect to. Nevertheless, this

definition is underlain by a simple, if not naive, answer to the question of how the

addition of a metamaterial shell would affect the performance of an existing antenna,

or, similarly, how the embedding of an existing antenna in a given core-shell system

with oppositely signed constitutive parameters would affect the performance of the

antenna. A more detailed investigation of the effect of reference antennas on en-

hancement level estimates and the issue of fairness in antenna radiation performance

is presented in Chapter 4.6

3.3 Numerical Results and Case Studies

In this section we turn to the application of the theory exposed above to the elu-

cidation of the effect of embedding media on antenna radiation performance. The

goal is to gain an understanding of the effect of the antenna substrate on the mini-

mum source energy for a given radiation pattern. Because of the dependence of the

problem on so many parameters we limit ourselves to a few illustrative cases. Three

classes of antennas are investigated: a quarter-wavelength antenna (i.e., 2a = λ/4), a

λ/40 antenna (i.e., 2a = λ/40) and, a λ/400 antenna (i.e., 2a = λ/400). The driving

frequency of the antenna is set to f = 3.75 GHz. This corresponds to a = 1 cm, 0.1

cm, 0.01 cm, for the λ/4 antenna, λ/40 antenna, and λ/400 antenna, respectively.

Needless to say that these particular choices of the numerical values of f and a are

arbitrary. However, they lie well within the range of values used in the scientific and

6See also [138].

59

engineering literatures [8, 40, 41, 132, 133]. Particular attention is paid to electri-

cally small antennas. This is in view of the exciting properties that these antennas

exhibit in the subwavelength limit when embedded in a pair of oppositely signed

materials. Here we adopt the definition [139] according to which an electrically

small antenna in the vacuum is defined as an antenna for which k0a = 2πa/λ ≤ 0.5,where a is the radius of the sphere that encompasses the entire original antenna.

Hence, a more detailed investigation is carried out for the λ/400 multipolar and

dipolar antennas. At this point it is appropriate to give one more definition. In the

plots and associated discussion we consider the normalized wavenumber defined by

xb ≡ kb/π. The normalized wavenumber xb represents the wavenumber of the field

in the shell.

It follows from Eqs.(2.35,2.77) that, generally, the larger the singular valuesh(j)l

i2the smaller the minimum source energy EME required for the launching

of a given radiation pattern with a source of a given size. Therefore the larger the

singular valuesh(j)l

i2the greater the associated enhancement, due to the associated

substrates, of radiation of the lth multipole order field with given resources. It is

thus important to understand the dependence of the singular valuesh(j)l

i2on the

several parameters it depends on for both the electric (j = 1) and the magnetic

(j = 2) cases. Large singular values, such as resonances or peaks in the plots

of the singular values versus these variables, will indicate enhanced radiation for

such operational modes or conditions, with the given resources. This aspect is

investigated numerically next. For the sake of conciseness, however, and because of

the noted similarity (see next section) between the behavior of the electric singular

values and the magnetic singular values we concentrate our attention in what follows

on the study of the electric singular values.

As noted above there is a tight relationship between the local behavior of the

singular valuesh(j)l

i2and the launching ability of the antenna: any resonant peaks

in the spectra ofh(j)l

i2would indicate the presence of local enhancements in the

launching ability of the antenna. Nevertheless, it is well known [132, 133, 40] that the

core-shell system does possess a resonant behavior (resonant scattering and resonant

radiation) that can be traced back to the presence of the natural modes (polaritons)

in the system. Thus we should anticipate the occurrence of such resonant behavior

60

in our case too. The question that arises then is: should they appear, can we ascribe

the resonant peaks in the spectra of the singular valuesh(j)l

i2to the presence of

polaritons? The conditions for the existence of polaritons in the core-shell system

are summarized by their dispersion relations [131, 132, 133]

∆1 = 0 (3.8)

for the electric modes, where ∆1 has been defined in (3.5), and

∆2 = 0 (3.9)

for the magnetic modes, where ∆2 has been defined in (3.6). But the singular valuesh(j)l

i2are given by Eqs.(2.77,2.32,3.7), i.e., they are composed of two quantities:¯

F(j)l

¯2, defined in (3.4), and

hκ(j)l

i2, defined in (2.32). It is true that (3.8) and (3.9)

when substituted in the definition of amplitudes¯F(j)l

¯2(i.e., in Eq.(3.4)) would

provide the resonance conditions (3.8) and (3.9) with a very strong effect on the

behavior of the singular valuesh(j)l

i2. Yet, in order for us to be able to confidently

attribute the resonant peaks to the presence of polaritons we have to show that the

quantitieshκ(j)l

i2do not exhibit a resonant behavior similar to that of

¯F(j)l

¯2that

would potentially shift or even kill the peaks created by the resonance conditions

of the polaritons. Actually the quantitieshκ(j)l

i2are essentially non-pathological

combinations of the spherical Bessel functions of the first kind jl and their derivatives

which are sufficiently well-behaved for all integer values of l and complex values of the

argument [124]. Hence, we can confidently claim that the spectrum of the singular

valuesh(j)l

i2will indeed exhibit resonant peaks, and thus maximum enhancements,

and that these peaks will be primarily due to the presence of polaritons as stipulated

by the resonance conditions (3.8) and (3.9).

Finally, we point out that the MATHEMATICA code used for the numerical

simulations has been validated against some well-known cases such as the free-space

case [14] and the single spherical substrate case.7

7See Section 2.3.

61

1.5 2 2.5 3 3.5 4

d10−2

1

102

104

Singular

Values

10−2

1

102

104

l=5

l=4

l=3

l=2

l=1

Figure 3.2: Logarithmic plot of the normalized electric singular valuesh(1)l

i2for

a λ/4 antenna versus the radii ratio d ≡ b/a. The inner sphere is assumed not tocontain any material (i.e., a = 1 = µa) and the surrounding shell is assumed to bea lossless DPS material with µb = 1 and xb = 50 m−1.

3.3.1 Lossless Substrates

Vacuum—Core—DPS—Shell System

We focus on the case of a spherical shell of non-magnetic lossless DPS material

( b > 1, µb = 1) surrounding an inner sphere with no substrate material in it ( a =

1 = µa), a system that we will refer to as a vacuum-core-DPS-shell system.

In Figs.3.2-3.4 we plot the electric singular valuesh(1)l

i2versus the radii ratio

d ≡ b/a, for a quarter-wavelength antenna of maximum length 2a = 2 cm operating

at a the frequency f = 3.75 GHz. The plots show that the singular valuesh(1)l

i2exhibit a cyclic behavior with resonant peaks, i.e., local enhancements, appearing at

specified values of the radii ratio d. (The same behavior is exhibited by the magnetic

singular valuesh(2)l

i2, though the plots are not shown here.) As mentioned above

these resonant peaks correspond to an enhancement in the launching ability of the

antenna for such operational modes with the given resources.

The plots also show that for the smaller values of the radii ratio d (by “smaller

values” we mean d ≤ r (xb), where r (xb) is a value that decreases as xb increases)

62

1.5 2 2.5 3 3.5 4

d10−2

1

102

104

106

108

Singular

Values

10−2

1

102

104

106

108

l=5

l=4

l=3

l=2

l=1


i2for


the best local enhancements, though not always resonant, are observed for the lower

multipole modes starting with the dipolar modes. As the radii ratio increases the

best enhancements shift to the modes with higher multipolarities: quadrupole, then

octupole, and so on. The explanation of this observation is that the higher the

multipolarity of the mode the more “intricate” is its structure such that exciting

higher multipolarity modes in an efficient way requires thicker shells i.e., shells that

possess a “richer” charge structure. The peaks become sharper and more packed as

the electromagnetic density (i.e., the wavenumber) of the shell material increases.

This occurs because as the electromagnetic density of the material increases, i.e., as

b increases (since µb = 1, in this case), the ability of a given thickness of the material

to support more natural modes of oscillation (i.e., polaritons) also increases. For

very electromagnetically-dense materials, however, the heights of the peaks saturate

indicating a saturation in the launching enhancement levels and the peaks pile up at

almost the same values of the radii ratio d, which are now closely packed (cf. Fig.3.4.)

These closely-packed peaks indicate, on the one hand, that an enhancement in the

launching ability of the antenna occurs, for high values of the wavenumber xb, at

63

1 1.2 1.4 1.6 1.8 2

d10−4

10−2

1

102

104

106

Singular

Values

10−4

10−2

1

102

104

106

l=5

l=4

l=3

l=2

l=1


i2for


almost the same radii ratios for the electromagnetic multipolar modes with a pile up

of the resonant peaks at particular values of the radii ratio that is less pronounced for

the magnetic modes. On the other hand, this also indicates that the enhancement

in the launching ability of the antenna occurs for the two types of fields, i.e., electric

and magnetic, at roughly the same values of the radii ratio. Moreover, a closer

examination of the plots in Fig.3.4 and its magnetic counterpart (figure not shown)

reveals that for the electric dipolar mode and its magnetic counterpart, i.e., the

magnetic dipolar mode, in particular the local enhancement peaks appear now at

almost the same values of the radii ratio d ' 1.2, 1.4, 1.6, etc.In Fig.3.5 we plot the normalized electric singular values

h(1)l

i2for a λ/400

antenna versus the radii ratio d. The surrounding shell is assumed to be a lossless

non-magnetic DPS medium with xb = 150 m−1 (i.e., µb = 1 and b = 36.) Fig.3.5

clearly shows that for an electrically small antenna the resonant peaks disappear

over the same range of d values that had been considered for a quarter-wavelength

antenna and that had in several enhancement peaks present in it in that case.

Furthermore, the numerical simulations show that increasing the wavenumber of

64

1 1.2 1.4 1.6 1.8 2

d10−4

10−2

1

102

Singular

Values

10−4

10−2

1

102

l=5

l=4

l=3

l=2

l=1

Figure 3.5: Logarithmic plot of the normalized singular valuesh(1)l

i2for a λ/400

antenna versus the radii ratio d ≡ b/a. The inner sphere is assumed not to containany material (i.e., a = 1 = µa) and the surrounding shell is assumed to be a losslessDPS material with µb = 1 and xb = 150 m−1.

the shell medium not only does not restore the peaks but may make things even

worse in terms of the launching ability of the antenna with respect to the free-

space case. This is in perfect agreement with the fact that the actual total physical

dimensions of a resonating cavity made up of ordinary material, in particular a+ b

in this case, is the determining factor when it comes to which modes are supportable

by the cavity, and not just the radii ratio.

65

1 1.1 1.2 1.3 1.4

d10−2

1

102

104

106

108

1010

Singular

Values

10−2

1

102

104

106

108

1010

l=5

l=4

l=3

l=2

l=1


i2for a λ/4

antenna versus the radii ratio d ≡ b/a. The inner sphere is assumed not to containany material (i.e., a = 1 = µa) and the surrounding shell is assumed to be a losslessDNG material with µb = −1 and xb = −50 m−1.

Vacuum—Core—DNG—Shell System

The next lossless system we wish to investigate is a vacuum-core-DNG-shell system,

i.e., a = 1 = µa, and b < 0, µb < 0 (in fact in all what follows the relative electric

permittivity of the surrounding DNG shells is b = −4 and its relative magneticpermeability is µb = −1.) The driving frequency is as before set to f = 3.75 GHz.

In Fig.3.6 we plot the electric singular valuesh(1)l

i2versus the radii ratio d, for

a quarter-wavelength antenna of maximum length 2a = 2 cm; in Fig.3.7 we ploth(1)l

i2versus d, for a λ/40 antenna of maximum length 2a = 0.2 cm; and in Fig.3.8

we ploth(1)l

i2versus d, for a λ/400 antenna of maximum length 2a = 0.02 cm.

The simulations show that (cf. Figs.3.6-3.8) as the ratio of the length of the

antenna to the wavelength of the radiation in the vacuum, viz., 2a/λ, decreases the

resonant peaks, which correspond to a local enhancement in the launching ability

of the antenna for different modes, appear at certain fixed values of the radii ratio

d. This indicates that for a small enough k0a the enhancement for all the modes

appears at certain specified values of d regardless of the total physical dimensions

66

1 1.1 1.2 1.3 1.4

d10−2

1

102

104

106

108

1010

Singular

Values

10−2

1

102

104

106

108

1010

l=5

l=4

l=3

l=2

l=1


i2for a

λ/40 antenna versus the radii ratio d ≡ b/a. The inner sphere is assumed not tocontain any material (i.e., a = 1 = µa) and the surrounding shell is assumed to bea lossless DNG material with µb = −1 and xb = −50 m−1.

of the antenna. This is in total agreement with the reported subwavelength res-

onator concept [132, 133, 40] where the determining parameter for the existence of

a natural mode (polariton), and thus the occurrence of a local enhancement in the

launching ability of the antenna in this case, turns out to be the ratio of the two

radii rather than the total physical size of the antenna itself as would be the case

in the presence of only ordinary media. This clearly shows that encompassing a

subwavelength antenna in a judiciously chosen DNG metamaterial shell makes it

possible to distribute the resources of the antenna in a fashion that is as efficient as

that made possible only through the use of a much larger volume in free space.

A natural continuation to our previous investigation of the radiation efficiency

of an electrically small antenna embedded in a metamaterial substrate is displayed

in Fig.3.9. In this figure the normalized singular valuesh(1)l

i2for a λ/400 antenna

have been plotted versus the radii ratio d. The shell circumscribing the antenna

is assumed to be an ENG material with µb = 1 and xb = i50 m−1 (i.e., b =

−4). A comparison of Fig.3.9 and Fig.3.8 shows that these figures are in fact the

same though they describe two totally different systems. This clearly demonstrates

67

1 1.1 1.2 1.3 1.4

d10−2

1

102

104

106

108

1010

Singular

Values

10−2

1

102

104

106

108

1010

l=5

l=4

l=3

l=2

l=1


i2for a

λ/400 antenna versus the radii ratio d ≡ b/a. The inner sphere is assumed not tocontain any material (i.e., a = 1 = µa) and the surrounding shell is assumed to bea lossless DNG material with µb = −1 and xb = −50 m−1.

the fact that one can attain the same level of performance achieved through the

utilization of a DNG shell by using an SNG (in this case an ENG) shell. This, too,

is in total agreement with the results reported in the literature [8, 40, 132, 133, 41]

which stipulate that the use of DNG media is not really necessary in order to achieve

high performance levels and that similar performance levels could be achieved by

pairing two materials that possess oppositely signed values of at least one of the

constitutive parameter (In our case we had on one hand a DPS medium, i.e., the

vacuum core, and on the other hand the ENG shell such that µb = 1 = µa while

b = −4 = −4 a). The problem of pairing other types of substrates, such as an

MNG core and an ENG shell, has also been considered and the obtained results are

consistent with the published literature. Attaining high performance levels, such

as high radiation enhancement, through the utilization of an ENG medium is an

interesting possibility since such media exist in nature (plasmonic materials such as

silver, etc.)

68

1 1.1 1.2 1.3 1.4

d10−2

1

102

104

106

108

1010

Singular

Values

10−2

1

102

104

106

108

1010

l=5

l=4

l=3

l=2

l=1


i2for a λ/400

antenna versus the radii ratio d ≡ b/a. The inner sphere is assumed not to containany material (i.e., a = 1 = µa) and the surrounding shell is assumed to be an ENGmaterial with µb = 1 and xb = i50 m−1.

3.3.2 Lossy Substrates

The case of lossy substrates is illustrated in Fig.3.10. In this figure the normalized

electric singular valuesh(1)l

i2have been plotted versus the radii ratio d for a λ/400

antenna embedded in a vacuum-core-DNG-shell system. The surrounding DNG

shell is assumed to have a magnetic permeability µb = −1 and Re[xb] = −150m−1 (i.e., Re [ b] = −36.) The investigated cases are: 1) lossless case (loss tangentIm [ b] /Re [ b] = 0), 2) DNG shell with loss tangent Im [ b] /Re [ b] ' 1/60, and 3)DNG shell with loss tangent Im [ b] /Re [ b] ' 1/20. Fig.3.10 clearly shows that theinclusion of losses simply reduces the heights of the peaks but does not make the

peaks disappear. Also the decrease in the height of the resonant peaks becomes

larger relative to the lossless cases as the loss tangent of the shell increases. These

findings are not surprising and are in agreement with the results reported in the

literature [40, 132, 133, 41].

69

1 1.02 1.04 1.06 1.08 1.1

d10−2

1

102

104

106

108

Singular

Values

10−2

1

102

104

106

108

l=5

l=4

l=3

l=2

l=1


i2for

a λ/400 antenna versus the radii ratio d ≡ b/a. The inner sphere is assumed not tocontain any material (i.e., a = 1 = µa) and the surrounding shell is assumed to bea lossless DNG material (black curves), a lossy DNG shell with a loss tangent set to1/20 (blue curves), and a lossy DNG material with a loss tangent set to 1/60 (redcurves). In all three cases µb = −1, Re[xb] = −150 m−1.

Further Look at the Electric Dipole Case

We now initiate an investigation focused on the electric dipole case, i.e., in this case

j = 1 = l . We define the electric dipole antenna gain as8

G ≡ E(j=1=l)ME (free-space case)

E(j=1=l)ME

=h(1)1

i2. (3.10)

This quantity is plotted next versus the radii ratio d for some representative systems

and some selected values of the shell wavenumber xb. The systems considered here

are electric dipoles of different physical sizes embedded in vacuum—core-DNG-shell

systems. As explained above this means that in all the cases the inner sphere is

assumed not to contain any material, i.e., a = 1 = µa, while the outer shell is made

8Cf. Eq.(2.80).

70

1 1.2 1.4 1.6 1.8 2

d

10−2

1

102

Gain

10−2

1

102

xb=−500

xb=−150

xb=−100

xb=−50

Figure 3.11: Logarithmic plot of the gain G for a λ/4-electric-dipole antenna versusthe radii ratio d ≡ b/a. The surrounding shell is assumed to be a DNG materialwith µb = −1.

up of a DNG material with µb = −1. The driving frequency f is still set to 3.75

GHz.

In Figs.3.11-3.13 the gain G has been plotted versus the radii ratio d for a λ/4-

electric-dipole antenna (a = 1 cm), a λ/40-electric-dipole antenna (a = 0.1 cm), and

a λ/400-electric-dipole antenna (a = 0.01 cm). The aim is to study the effect of the

physical size a on the performance of the antenna. From Figs.3.11-3.13 we notice

that there is something that is counterintuitive here. It appears that as the physical

size of the dipole antenna becomes smaller the antenna’s ability to optimize the

utilization of its resources to radiate the dipolar field efficiently increases. This is

counterintuitive because what one would expect is that as the volume encompassing

the antenna decreases it becomes more difficult to distribute the resources of the

antenna so as to allow the antenna to radiate efficiently [51, 52, 17, 137, 112]. The

explanation of this seemingly counterintuitive situation lies in the physical interpre-

tation of the resonant peaks. As established above the resonant peaks correspond

to the presence of polaritons. These polaritons have a certain dispersion relation,

viz., Eq.(3.8) for the electric modes and Eq.(3.9) for the magnetic modes. These dis-

persion relations, or resonance conditions, establish a certain relationship between

71

1 1.2 1.4 1.6 1.8 2

d10−6

10−4

10−2

1

102

104

106

108

1010

Gain

10−6

10−4

10−2

1

102

104

106

108

1010

xb=−500

xb=−150

xb=−100

xb=−50

Figure 3.12: Logarithmic plot of the gainG for a λ/40-electric-dipole antenna versusthe radii ratio d ≡ b/a. The surrounding shell is assumed to be a DNG materialwith µb = −1.

the different parameters relevant to the problem. When all the parameters are fixed

except for the physical size of the antenna, as is the situation in this case, one should

be able, at least numerically, to solve for the optimum value of the physical size that

would satisfy the resonance condition. This optimal value of the physical size of the

antenna is what we are dealing with in this case. But if this is true then values on

both sides of this optimal size should cause a reduction in the ability of the antenna

to radiate the dipolar field which is no the case. The simulations show that as the

physical size of the antenna is reduced further the resonant peaks remain at the

same location. This objection may be explained away by invoking the concept of

subwavelength resonator [132, 133, 40]. Indeed, if the optimal value of the physical

size turns out to satisfy the subwavelength resonator conditions [132, 133, 40], that

is, if the size of the core-shell system turns out to be smaller than the wavelength

in all three regions then further reducing the physical size of the antenna will not

affect the radiation performance of the antenna, as discussed above.

72

1 1.2 1.4 1.6 1.8 2

d10−6

10−4

10−2

1

102

104

106

108

1010

Gain

10−6

10−4

10−2

1

102

104

106

108

1010

xb=−500

xb=−150

xb=−100

xb=−50

Figure 3.13: Logarithmic plot of the gain G for a λ/400-electric-dipole antennaversus the radii ratio d ≡ b/a. The surrounding shell is assumed to be a DNGmaterial with µb = −1.

3.4 Conclusion

To conclude, we have investigated, both analytically and numerically, the effects

that the presence of metamaterials would have on the performance of a general

antenna embedded in a generally lossy system of two nested spheres (core-shell

system) in terms of the efficiency with which the available resources of the antenna

could be distributed within a prescribed volume so as to generate a given radiated

field. The derived developments constitute a fundamental inverse-source-theoretic

framework for analysis and design of different substrate structures. This framework

also complements in analytical and computational tools and insight the pioneering

work by some of the leading authors in this area.

The adoption of the inverse-source-theoretic approach is aimed at enabling in-

trinsic, i.e., non-antenna-specific, and fair characterization of different substrate con-

figurations by comparing optimal radiation in either configuration (i.e., the “best”

in each one). This characterization is governed by a formally tractable source-energy

cost function that is physically motivated by ohmic loss control. Via analytical and

numerical examples we have explained and illustrated important enhancements due

73

to the presence of metamaterials in the context of the two-nested-spheres configu-

ration, in particular for media with oppositely signed constitutive parameters.

74

Chapter 4

Comparative Study ofRadiation Enhancement due toMetamaterials

In this chapter we present a comparative (numerical) study of radiation enhancement

due to metamaterial substrates. This study, published in [138], complements our

investigations of the homogeneous substrate case presented in Chapter 21 and the

core-shell system case presented in Chapter 3.2 As in Chapter 3 we consider the

core-shell system but contrary to the study carried out in that chapter where we

studied only the effects that adding a shell would have on the performance of an

existing antenna, we shall explore in this chapter the effects that different definitions

of the reference antenna would have on enhancement level estimates.

We shall limit ourselves to a few illustrative examples. We focus on the case

when the core does not contain any material, i.e., a = 1 = µa, and the shell is

a lossless double-negative medium (DNG). In particular, we shall set b = −4 andµb = −1. Two types of antennas are investigated: A quarter-wavelength antenna

(i.e., 2a/λ = 1/4) and a full-wavelength antenna (i.e., 2a/λ = 1). The driving

frequency of the antennas is set to f = 10 GHz. This corresponds to a = 3.75

mm for the quarter-wavelength antenna and a = 1.5 cm for the full-wavelength

1See also [112] and [137].2See also [125].

75

antenna. These particular choices of the numerical values of f and a are, of course,

arbitrary. However, they lie well within the range of values used in the scientific

and engineering literatures [8, 40, 41, 132, 133].

4.1 Reference Antennas

In Chapter 2 we defined the normalized singular valuesh(j)l

i2(cf. Eq.(2.77)). In

this chapter we shall refer to them as enhancements. Three “reasonable” definitions

of the reference antenna are considered: (1) it is an antenna which resides within

a sphere of radius a, (2) it is an antenna which resides within a sphere of radius b

, or, (3) it is an antenna which resides within a sphere of radius bmax, where bmax

is the outer radius of the shell that maximizes the singular valuehσ(1)l

i2. These

three classes of reference antennas will subsequently be referred to as RA1, RA2,

and RA3, respectively. The idea behind the first definition is a simple answer to the

question of how the addition of a metamaterial shell would affect the performance

of an existing antenna radiating in free space. This definition proved to be adequate

when the antenna substrate was merely a homogeneous sphere of given radius.3 The

second definition is motivated by the realization that adding a shell to an existing

antenna creates, in fact, a new antenna with new dimensions. As for the third

definition the idea behind it is to compare optimal radiation in either configuration,

i.e., to compare different antennas when they operate at their best [8, 40, 41].

4.2 Numerical Study

Because of the noted similarity between the behavior of the electric enhancements

and the magnetic enhancements we concentrate our attention in what follows on the

study of the electric enhancements.4

In Figs.4.1-4.3 we plot the enhancementsh(1)l

i2for a quarter-wavelength an-

tenna with radius a = 3.75 mm versus the radii ratio b/a. In Fig.4.1 the reference

antenna is assumed to be an antenna radiating in free space and circumscribed by

a sphere of radius a; in Fig.4.2 the reference antenna is assumed to be an antenna

3See Section 2.3.4See Sections 2.3 and 3.3.

76

1 1.2 1.4 1.6 1.8 2

bêa10−2

1

102

104

106

108

1010

Enhancement

10−2

1

102

104

106

108

1010

l=5

l=4

l=3

l=2

l=1

Figure 4.1: Logarithmic plot of the enhancementh(1)l


antenna with radius a = 3.75 mm versus the radii ratio b/a. The shell is a losslessDNG material with b = −4 and µb = −1. The reference antenna is assumed to beRA1 antenna.

radiating in free space and circumscribed by a sphere of radius b; in Fig.4.3 the refer-

ence antenna is assumed to be an antenna radiating in free-space and circumscribed

by a sphere of radius bmax, where bmax is the outer radius of the shell that maximizes

the singular valueshσ(1)l

i2. The maxima occur at bmax/a = 1.40, 1.29, 1.17, 1.13, 1.1,

for l = 1, 2, 3, 4, 5, respectively. In Figs.4.4-4.6 we show plots similar to those of the

quarter-wavelength antenna case. The outer radii ratios that maximize the singular

valueshσ(1)l

i2are in this case bmax/a = 1.35, 1.16, 1.16, 1.18, 1.13, for l = 1, 2, 3, 4, 5.

Reference antennas that belong to RA2, i.e., reference antennas for the radius of

the circumscribing volume is equal to b, tend to shift the positions of the resonant

peaks. For instance, the first electric dipole peak appears for the singular valuehσ(1)1

i2at bmax/a = 1.40, as mentioned earlier, but when an RA2 antenna is adopted

the position of the peak shifts slightly: it appears at b/a = 1.32 forh(1)1

i2. However,

no relative shifts in the positions of the peaks are produced by the adoption of RA1

or RA3 antennas. This is due to the fact that with these antennas the enhancements

77

1 1.2 1.4 1.6 1.8 2

bêa10−2

1

102

104

106

108

1010

Enhancement

10−2

1

102

104

106

108

1010

l=5

l=4

l=3

l=2

l=1



antenna with radius a = 3.75 mm versus the radii ratio b/a. The shell is a losslessDNG material with b = −4 and µb = −1. The reference antenna is assumed to bean RA2 antenna.

are nothing but the singular valueshσ(1)l

i2scaled by a constant.

Adopting an RA1 reference antenna, i.e., a reference antenna for which the cir-

cumscribing sphere has radius a, yields relatively higher estimates of the performance

levels with respect to the performance levels obtained by means of the remaining

two reference antenna classes, cf. Figs.4.1-4.6. This is quite understandable because

the size of an RA1 reference antenna is by definition smaller that the sizes of the

corresponding RA2 and RA3 antennas.

The simulations show also that the discrepancies between the estimates yielded

by the three reference antenna classes are not significant in the immediate vicinity of

the resonant peak which corresponds to bmax: the ratios between any two estimates,

in particular those obtained by means of RA1 and RA3 antennas on one hand and

RA2 antennas on the other hand, are less than one order of magnitude in favor of

the estimates obtained by means of RA1 and RA3.5 It is only when one attempts

to compare estimates yielded by the three reference antenna classes far away from

5Note that RA2 and RA3 antennas yield identical enhancements at b/a = bmax/a.

78

1 1.2 1.4 1.6 1.8 2

bêa10−2

1

102

104

106

108

1010

Enhancement

10−2

1

102

104

106

108

1010

l=5

l=4

l=3

l=2

l=1



antenna with radius a = 3.75 mm versus the radii ratio b/a. The shell is a losslessDNG material with b = −4 and µb = −1. The reference antenna is assumed to bean RA3 antenna.

this resonant peak that the discrepancies become significant (the ratios between

the estimates obtained by means of the RA2 and RA3 antennas, for instance, are

about one to two orders of magnitude larger, in favor of RA3 estimates, for radii

ratios b/a > 1.8). Note also that the discrepancies are more pronounced for higher

multipole modes (cf. Figs.4.1-4.6). Thus for RA3 reference antennas, i.e., for those

which have a circumscribing volume of radius bmax , to yield a fair comparison they

have to be used locally. In other words it is not suitable to use a value of bmax which

corresponds to a maximum enhancement that appears for a specific range of shell

thickness values as an absolute reference for enhancement level characterization.

The discrepancies would be even larger if bmax corresponded to an infimum of the

set of shell thickness values at which resonant peaks appear in the singular values

spectrum.

79

1 1.2 1.4 1.6 1.8 2

bêa10−2

1

102

104

Enhancement

10−2

1

102

104

l=5

l=4

l=3

l=2

l=1


i2for a full—wavelength an-

tenna with radius a = 1.5 cm versus the radii ratio b/a. The shell is a lossless DNGmaterial with b = −4 and µb = −1. The reference antenna is assumed to be anRA1 antenna.

1 1.2 1.4 1.6 1.8 2

bêa10−2

1

102

104

Enhancement

10−2

1

102

104

l=5

l=4

l=3

l=2

l=1


i2for a full-wavelength an-


80

1 1.2 1.4 1.6 1.8 2

bêa10−2

1

102

104

Enhancement

10−2

1

102

104

l=5

l=4

l=3

l=2

l=1


i2for a full-wavelength an-


4.3 Conclusion

These observations are true for all multipolar modes and for small (quarter-wavelength)

and large (full-wavelength) antennas alike. (In the simulations we also investigated

5λ-antennas but the results are not shown here.) They lead us to the conclusion that

RA2 antennas, i.e., the antennas radiating in free space and having a circumscribing

volume of radius b, are more convenient as reference antennas, especially when one

is interested in performance characterization over a wide range of shell thicknesses,

though the adoption of an RA2 reference antenna may result in a slight shift of the

resonant peaks position.

81

Chapter 5

The Casimir Effect in thePresence of Metamaterials

In this chapter we investigate the Casimir effect in the presence of electromagnetic

metamaterials. We consider in particular the planar geometry sketched in Fig.5.1:

we consider two slabs of thicknesses a and b and constitutive parameters a, µa and b, µb, respectively, sandwiched between two half-spaces of constitutive parame-ters 1, µ1 and 2, µ2 . The four media are all assumed to be isotropic, dispersivemedia (although dissipation shall be assumed to be negligible in this case)1. In par-

ticular, this means that all the constitutive parameters are in general real functions

of the frequency.

It is true that several authors have considered the multilayered planar geometry

with and without metamaterials [104, 105, 106, 107, 94, 101, 108, 109, 102, 110, 111].

However, to our knowledge, all the published studies concentrate on the case of two

stacks of materials separated by a single medium. The concentration on such systems

is, of course, justified by the need for a better understanding of the multilayered

system which is ubiquitous in experimental research on the Casimir effect. In effect,

the mirrors used in the measurements of the Casimir force are practically always

made up of substrate media coated with one, two, or possibly even more layers of

other materials.

After deriving the formula that gives the Casimir force between the two half-

1See Appendix G for more detail.

82

2,µ2

a,µaa

b,µbb

1,µ1

z

x

Figure 5.1: Planar geometry under consideration. Two slabs of thicknesses a andb and relative constitutive parameters a, µa and b, µb, respectively, are sand-wiched between two half-spaces with relative constitutive parameters 1, µ1 and 2, µ2 .

spaces (see Fig.5.1), we validate this formula against some well-known results then

use it to derive the force between two perfectly conducting plates separated by two

slabs of material. Subsequently, a numerical study is performed on the Casimir

force in this case. As we mentioned in the Introduction, in choosing to focus on this

system we are particularly inspired by the subwavelength cavity resonators described

in [7]. The idea is to examine the possibility that, for similar cavities, the peculiar

properties of metamaterials would give rise to some interesting phenomena.

5.1 General Formulation

5.1.1 Eigenfrequencies

By definition, the vacuum energy is given by [83]

U ≡Xψ

1

2~ωψ, (5.1)

83

where ~ is Planck’s constant and ψ stands for all the parameters upon which the

modes depend. To calculate this energy we first have to determine the allowed

electromagnetic vacuum modes for the system under consideration. To do this we

shall adopt the semiclassical surface modes approach initiated in [140]. These modes

correspond to waves propagating parallel to the interfaces and form a complete set

of solutions [141]. The calculation given below is a straightforward generalization of

[140]. Here, we shall follow closely some well-known presentations given, in partic-

ular, in [83, 85].

After imposing the relevant boundary conditions at the interfaces between the

slabs and requiring that the solutions to Maxwell’s equations be nontrivial, one can

show that the allowed wave frequencies in the four-region system sketched in Fig.5.1

are the ω solutions to the dispersion relations2

g (ω) =(K1 a +Ka 1) (K2 b +Kb 2)

(K1 a −Ka 1) (K2 b −Kb 2)e2(Kaa+Kbb)

+(Kb a −Ka b) (K1 a +Ka 1)

(Kb a +Ka b) (K1 a −Ka 1)e2Kaa

+(Kb a −Ka b) (Kb 2 +K2 b)

(Kb a +Ka b) (Kb 2 −K2 b)e2Kbb − 1

= 0 (5.2)

gµ (ω) =(K1µa +Kaµ1) (K2µb +Kbµ2)

(K1µa −Kaµ1) (K2µb −Kbµ2)e2(Kaa+Kbb)

+(Kbµa −Kaµb) (K1µa +Kaµ1)

(Kbµa +Kaµb) (K1µa −Kaµ1)e2Kaa

+(Kbµa −Kaµb) (Kbµ2 +K2µb)

(Kbµa +Kaµb) (Kbµ2 −K2µb)e2Kbb − 1

= 0, (5.3)

2See Appendix F for more detail.

84

wherein

K2j ≡ k2x + k2y − j (ω)µj (ω)

ω2

c2

≡ κ2 − j (ω)µj (ω)ω2

c2, j = 1, 2, a, b. (5.4)

These dispersion relations, (5.2) and (5.3), correspond to two different polarizations

of the electric field. On the one hand, we have the waves that obey the dispersion

relation (5.2). These are the waves for which the polarization lies in the plane formed

by κ ≡ k⊥ and the z-axis (i.e., they have ey = 0). We shall call these waves “ -type”modes.3 On the other hand, we have the waves that obey the dispersion relation

(5.3). These are the waves for which the polarization is perpendicular to the plane

formed by κ ≡ k⊥ and the z-axis (i.e., they have ez = 0). We shall call these waves“µ-type” modes.4 We also note that κ and ω are independent variables and unless

ω/κ ≤ c the waves are evanescent.

To simplify these expressions we define the following quantities

A ≡ (K1 a +Ka 1) (K2 b +Kb 2)

(K1 a −Ka 1) (K2 b −Kb 2), (5.5)

B ≡ (Kb a −Ka b) (K1 a +Ka 1)

(Kb a +Ka b) (K1 a −Ka 1), (5.6)

C ≡ (Kb a −Ka b) (Kb 2 +K2 b)

(Kb a +Ka b) (Kb 2 −K2 b), (5.7)

and

Aµ ≡ (K1µa +Kaµ1) (K2µb +Kbµ2)

(K1µa −Kaµ1) (K2µb −Kbµ2), (5.8)

Bµ ≡ (Kbµa −Kaµb) (K1µa +Kaµ1)

(Kbµa +Kaµb) (K1µa −Kaµ1), (5.9)

Cµ ≡ (Kbµa −Kaµb) (Kbµ2 +K2µb)

(Kbµa +Kaµb) (Kbµ2 −K2µb). (5.10)

3They are similar to TMz (or E) waveguide modes.4They are similar to TEz (or H) waveguide modes.

85

With these quantities (5.2) takes on the simple form

g (ω) = A e2(Kaa+Kbb) +B e2Kaa + C e2Kbb − 1 = 0 (5.11)

and similarly (5.3) takes on the form

gµ (ω) = Aµe2(Kaa+Kbb) +Bµe

2Kaa +Cµe2Kbb − 1 = 0. (5.12)

5.1.2 The Casimir Force

In view of the fact that two types of modes—“ -type” modes and “µ-type” modes—

exist for the system under consideration, Eq.(5.1) transforms into

U =Xψ

1

2~ω(ψ) +

Xψ

1

2~ω(ψ)µ (5.13)

where, now, ψ labels all the parameters on which the -type and µ-type modes de-

pend. After substituting the standard cavity quantum electrodynamics prescription

Xψ

→Z

L

2πdkx

ZL

2πdky

XN

=

µL

2π

¶2 Z ∞

0κdκ

Z π

−πdφXN

, (5.14)

in (5.13), the energy takes on the form

U =~L2

4π

Z ∞

0κdκ

XN

h~ω(N) (κ) + ~ω(N)µ (κ)

i, (5.15)

where N stands for the solutions of (5.2) or (5.3).

From (5.15) we see that the evaluation of the vacuum energy has reduced to a

sum over the zeros of the two functions g (ω) and gµ (ω). If these functions are

viewed as complex functions, it is clear from (5.2) and (5.3) that g (ω) and gµ (ω)

are analytic except for poles (i.e., they are meromorphic functions [142]). Thus,

to evaluate the right-hand side of (5.15) we recur to the theory of meromorphic

functions. We introduce a new variable ξ such that

ξ ≡ −iω. (5.16)

86

With this definition, Eq.(5.4) becomes

K2j = κ2 + j (iξ)µj (iξ)

ξ2

c2, j = 1, 2, a, b. (5.17)

We also define the two new functions

G (ξ) ≡ g (iξ) (5.18)

and

Gµ (ξ) ≡ gµ (iξ) . (5.19)

Consequently, the argument principle for meromorphic functions allows us to write5

U =~L2

8π2

Z ∞

0κdκ

Z ∞

−∞[lnG (ξ) + lnGµ (ξ)] dξ. (5.20)

It can be shown that writing the vacuum energy in this form is equivalent to renor-

malizing it [85].

The Casimir force per unit area between the two half-spaces separated by the

distance d ≡ a+ b is defined as

F ≡ − ∂

∂d

U

L2

= −12

µ∂

∂a+

∂

∂b

¶U

L2, (5.21)

where U is given by Eq.(5.20). Substituting U from (5.20) into (5.21) yields

F = − ~8π2

Z ∞

−∞

∙Ka +Kb +Kb

1−B e2Kaa

G (ξ)+Ka

1− C e2Kbb

G (ξ)

¸dξ

− ~8π2

Z ∞

−∞

∙Ka +Kb +Kb

1−Bµe2Kaa

Gµ (ξ)+Ka

1− Cµe2Kbb

Gµ (ξ)

¸dξ.

(5.22)

Note that the terms that do not contain G (ξ) or Gµ (ξ) are independent of the two

5See Appendix G.

87

half-spaces. They do not correspond to any force between them. Hence

F = − ~4π2

Z ∞

0κdκ

Z ∞

0

"Kb

¡1−B e2Kaa

¢+Ka

¡1− C e2Kbb

¢G (ξ)

#dξ

− ~4π2

Z ∞

0κdκ

Z ∞

0

"Kb

¡1−Bµe

2Kaa¢+Ka

¡1− Cµe

2Kbb¢

Gµ (ξ)

#dξ.

(5.23)

This is the general formula for the Casimir force between two half-spaces separated

by two slabs of material.

5.2 Particular Cases

We shall now show that (5.23) reduces in some special cases to results published in

the literature.

5.2.1 One Slab of Material Separating Two Half-Spaces

Our aim here is to compare our general formula (5.23) to the results published by

several authors on the case of two half-spaces separated by only one slab [98, 99, 92].

If only one slab is assumed to separate the two half-spaces one has B = 0 = C

and Bµ = 0 = Cµ. Consequently

F = − ~2π2

Z ∞

0κdκ

Z ∞

0

£G−1 (ξ) +G−1µ (ξ)

¤Kdξ. (5.24)

where K = Ka = Kb. Now let

κ2 ≡ aµaξ2

c2¡p2 − 1¢ , p ∈ [1,∞). (5.25)

This allows us to rewrite K2j as

K2j ≡ a (iξ)µa (iξ)

ξ2

c2s2j j = 1, 2, a, b, (5.26)

88

where

s2j ≡ p2 − 1 + jµj

aµa. (5.27)

It is straightforward to show that

F = − ~2π2c3

Z ∞

1p2dp

Z ∞

0

∙(s1 + p 1) (s2 + p 2)

(s1 − p 1) (s2 − p 2)e2ξ

√µpd/c − 1

¸−1( µ)3/2 ξ3dξ

− ~2π2c3

Z ∞

1p2dp

Z ∞

0

∙(s1µ+ pµ1) (s2µ+ pµ2)

(s1µ− pµ1) (s2µ− pµ2)e2ξ

√µpd/c − 1

¸−1( µ)3/2 ξ3dξ

(5.28)

where d ≡ 2a = 2b, ≡ a = b, µ ≡ µa = µb. If the slab separating the two media

is nonmagnetic, i.e., if µ = 1, this expression reduces to the result obtained within

the Dzyaloshinskii-Lifshitz-Pitaevskii theory [98, 99] regarded by many authors as

the standard theory of the Casimir effect in dielectric media.

5.2.2 Two Slabs of Material Separating Two Perfectly ConductingPlates

The result (5.23) can be used to derive a formula for the Casimir force between two

perfectly conducting plates separated by two media (see Fig.5.2). In the remainder

of this chapter we shall focus on the abovementioned system.

The case of two perfectly conducting plates is recovered by taking the limits

1 →∞ and 2 →∞ in (5.23). We obtain

F = − ~4π2c3

Z ∞

1p2dp

Z ∞

0

1− γ (p) eβ(p)ξ

e[α(p)+β(p)]ξ − γ (p)£eα(p)ξ − eβ(p)ξ

¤− 1ξ3 ( aµa)3/2 dξ

− ~4π2c3

Z ∞

1p sb (p) dp

Z ∞

0

1 + γ (p) eα(p)ξ

e[α(p)+β(p)]ξ − γ (p)£eα(p)ξ − eβ(p)ξ

¤− 1ξ3 ( aµa)3/2 dξ

− ~4π2c3

Z ∞

1p2dp

Z ∞

0

1− γµ (p) eβ(p)ξ

e[α(p)+β(p)]ξ + γµ (p)£eα(p)ξ − eβ(p)ξ

¤− 1ξ3 ( aµa)3/2 dξ

− ~4π2c3

Z ∞

1p sb (p) dp

Z ∞

0

1 + γµeα(p)ξ

e[α(p)+β(p)]ξ + γµ (p)£eα(p)ξ − eβ(p)ξ

¤− 1ξ3 ( aµa)3/2 dξ,

(5.29)

89

z

a,µaa

b,µbb

x

Figure 5.2: A rectangular cavity of dimensions L×L×d is constructed by putting twoperfectly conducting plates with dimensions L×L× τ (LÀ dÀ τ) in the vacuumparallel to one another and separated by a distance d. Two slabs of isotropic,homogeneous, lossless material are inserted between the two plates. The first slabhas dimensions L× L× a and relative constitutive parameters a, µa; the secondslab has dimensions L×L×b and relative constitutive parameters b, µb . (a+b =d).

wherein

α (p) ≡ 2ac

√aµap ≡ 2

a

cnap, (5.30)

β (p) ≡ 2b

c

√aµasb (p)

≡ 2b

c

√aµa

rp2 − 1 + bµb

aµa

≡ 2b

cna

sp2 − 1 +

µnbna

¶2, (5.31)

γ (p) ≡ sb (p) a − p b

sb (p) a + p b, (5.32)

and

γµ (p) ≡sb (p) µa − pµbsb (p) µa + pµb

. (5.33)

Thus, the Casimir force between two perfectly conducting plates separated by two

material slabs is given by (5.29).

90

5.2.3 The One-Slab Case and Casimir’s Classical Result

Two simple cases that can be studied analytically emerge from (5.29). They are the

cases for which either

γ (p) = 0 = γµ (p) (5.34)

or

α (p) = β (p) . (5.35)

In the first case, i.e., when γ (p) = 0 = γµ (p) is satisfied, it is straightforward to

show that (5.34) is equivalent to ⎧⎪⎨⎪⎩a = b

and

µa = µb,

(5.36)

that is, condition (5.34) corresponds to one slab of total thickness d ≡ a+ b sand-

wiched between two perfectly conducting plates. If n is the index of refraction of

the slab, one obtains in this case

F = − ~4π2c3

Z ∞

1p2dp

Z ∞

0ξ3 ( aµa)

3/2 4

e2√

aµaξd/c − 1dξ (5.37)

= − ~c16π2d4

Z ∞

0

1

n

x3

ex − 1dx. (5.38)

If the slab is nondispersive the formula reduces further to

F = − ~cπ2

240d41

n, (5.39)

which is the same result that one obtains by means of zeta-functional regularization,

for instance. If now a = 1 = b and µa = 1 = µb, that is, when n = 1 and vacuum

is what separates the two plates, this result, (5.39), reduces to Casimir’s classical

result. Note also that when n = −1 in (5.39), one obtains an expression for theforce identical to the one derived by Leonhardt and Philbin [102].

It is worth noting, at this point, that the case of a single slab separating the two

conducting plates cannot be recovered by setting the thickness of one of the slabs

91

to zero. This is due to the fact that the very formulation of the problem assumes

that ab 6= 0. It is also worth noting that the semiclassical nature of the formulationimposes a lower limit on the values of the slabs’ thicknesses: in order for the classical

response functions to be physically meaningful the slabs’ thicknesses must be much

larger than the Bohr radius, i.e., a (or b)À a0 ∼ 10−10 meter.The second special case, i.e., when α (p) = β (p), leads to an interesting new

result. Eq.(5.35) is equivalent to the two new conditions⎧⎪⎨⎪⎩√

a µa = −p | b| |µb|

and

a = b.

(5.40)

where we have taken a and µa to be positive, for definiteness. In other words, (5.40)

corresponds to the situation when the plates are separated by two slabs having the

same thickness and oppositely signed refraction indices√

a µa = −p | b| |µb| ≡ |n|.

Note that (5.40) necessarily implies that a = − b and µa = −µb. In this case too

one arrives at a very simple formula reminiscent of (5.39), namely

F = − ~cπ2

240d41

|n| (5.41)

In other words a DPS-DNG combination used such that nDPS/nDNG = −1 anda = b produces the same attractive Casimir force caused by the presence of one

slab of material having thickness d = 2a and index of refraction nDPS . This makes

sense as in the case of a DPS-DNG combination the ratio a/b is expected to play

an important role.

92

5.3 Numerical Study

Here we wish to investigate further the behavior of the Casimir force in the case

of two slabs sandwiched between two perfectly conducting plates. In view of our

inability to integrate analytically the right hand side of (5.29), we recur to numerical

simulations. The MATHEMATICA code used for this simulation was validated

against Casimir’s classical result and (5.39).6

We define the normalized (unitless) Casimir force

f ≡ F

(−~cπ2/240d4) ; (5.42)

it is the force normalized to that associated with Casimir’s original cavity (i.e., two

plates separated by vacuum), for a total separation d = a+ b. The normalized force

f , thus defined, is an estimate of the effect that the use of materials between the

two plates would have on the Casimir force. Note that, because of (5.42), negative

values in the plots of f indicate the presence of a repulsive force.

5.3.1 Cavities with DPS-DPS and DNG-DNG Slab Combinations

The first question that we would like to address is whether there is any advantage

in putting two slabs instead of one between the plates. In Fig.5.3 we plot the

normalized Casimir force f for a DPS-DPS slab combination and a DNG-DNG slab

combination as a function of the ratio of thicknesses r ≡ a/b. Fig.5.3 shows that

even in those cases the Casimir force changes its sign, clearly demonstrating that

there is a fundamental difference between one-media systems and two-media systems

even when the two media are both DPS or DNG.

5.3.2 Cavities with DPS-DNG Slab Combinations

Now, we turn to systems with DPS-DNG combinations. In Fig.5.4 we plot the

normalized Casimir force for a particular type of DPS-DNG slab combinations as a

6 In the code the usual procedure of converting an improper integral into a proper one byrescaling the variable of integration has been used.

93

0 1 2 3 4 5 6r

-20

-10

0

10

20

NormalizedCasimirForce

DNG−DNG

DPS−DPS

Figure 5.3: Plot of the normalized Casimir force f for DPS-DPS and DNG-DNG slabcombinations as a function of the ratio of thicknesses r ≡ a/b. The constitutive para-meters of the DPS-DPS combination are a, µa = 2, 1 and b, µb = 2.5, 1.5 .The constitutive parameters of the DNG—DNG combination are a, µa = −2,−1and b, µb = −2.5,−1.5 . For both combinations the thickness of the second slabis set to b = 0.001 mm.

function of the ratio of thicknesses r. For this type of combination we require that

b

a= −a

b= −r = µb

µa. (5.43)

This particular choice is considered because the corresponding normalized Casimir

force exhibits a simple profile in the plots. Fig.5.4 clearly demonstrates three impor-

tant features of the force: (1) the use of metamaterials in a DPS-DNG combination

does change the sign of the Casimir force and turns it from an attractive force to

a repulsive force, (2) the magnitude of the repulsive force could be made orders of

magnitude larger than the magnitude of the attractive Casimir force in the vacuum,

and (3) the Casimir force depends on the ratio of thicknesses r in an essential way.

In fact, the numerical simulations suggest that the Casimir force F has, in this case,

the generic profile

f ∼ R³ab

´⇒ F ∼ 1

(a+ b)4R³ab

´(5.44)

94

0 0.5 1 1.5 2 2.5 3r

-200

-180

-160

-140

-120

-100

-80

-60


b=10 mm

b=0.1 mm

b=0.001 mm

Figure 5.4: Plot of the normalized Casimir force f for a DPS-DNG slab combi-nation as a function of the ratio of thicknesses r ≡ a/b. For this combination wehave b/ a = −a/b = −r = µb/µa.The constitutive parameters of the DPS slab are a, µa = 2, 1 and those of the DNG slab are b, µb = −2r,−r . The thick-nesses of the DNG slabs were set to the arbitrary values 0.001 mm, 0.1 mm and10 mm. Because the curves coincide exactly with one another, a purely artificialshift of magnitude −10 has been introduced in the plot of the normalized force thatcorresponds to b = 0.1 mm and a similar artificial shift of magnitude −20 has beenintroduced in the plot of the normalized force that corresponds to b = 10 mm.

where R (·) is a function of the ratio of thicknesses only. These features have beenconfirmed for different DPS-DNG combinations. They appear to be characteristic

of DPS-DNG-based systems not just some peculiar behavior due to the particular

values of the constitutive parameters.

The numerical study also shows that the DPS-DNG combination exhibits some

other interesting features. Fixing the constitutive parameters for a certain combi-

nation and varying the individual thicknesses yields wildly oscillating profiles of the

normalized force with many separation values at which the force assumes a doublet

structure, as shown in Fig.5.5 and Fig.5.6. Then, when the total distance between

the two plates reaches a certain threshold, an “oscillation threshold,” the magnitude

of the normalized Casimir force starts decaying. However, depending on which slab

thickness is being varied the decay is either to f = 0+ or f = 0−. If the thicknessof the DPS slab is held fixed and the thickness of the DNG slab is varied the force

95

40 42 44 46 48 501êr

-1×108

-5×107

0

5×107

1×108


Figure 5.5: Plot of the normalized Casimir force f for a DPS-DNG slab combinationas a function of the ratio of thicknesses 1/r ≡ b/a. The constitutive parametersof the DPS slab are a, µa = 2, 1 and those of the DNG slab are b, µb =−2.5,−1.5 . The thickness of the DPS slab is set to a = 0.001 mm.

is repulsive, beyond the “oscillation threshold,” and its magnitude decays to 0−, ascan be seen in Fig.5.5. While if the thickness of the DNG slab is held fixed and the

thickness of the DPS slab is varied, instead, the force is attractive and its magnitude

decays to 0+, as can be seen in Fig.5.6.

The wildly oscillating profiles exhibited in Figs.5.5 and 5.6 are an indication

of instability in the corresponding systems. Visibly, the slightest variation in the

ratio of thicknesses r around an inflection point causes the systems to collapse. An

inspection of Figs.5.5 and 5.6 beyond the “oscillation threshold” also shows that

the smoother behavior of the force manifested there does not actually prevent the

collapse of the system.

5.3.3 Applications

The behavior of the Casimir force noted above is not the kind of behavior that would

permit the shielding of MEMS and NEMS components or quantum levitation. We

need situations characterized by the opposite behavior. Ideally, we would like to

have a system for which the Casimir force would vanish for a certain separation d0,

96

40 50 60 70 80r

-1×108

-5×107

0

5×107

1×108


Figure 5.6: Plot of the normalized Casimir force f for a DPS-DNG slab combinationas a function of the ratio of thicknesses r ≡ a/b. The constitutive parametersof the DPS slab are a, µa = 2, 1 and those of the DNG slab are b, µb =−2.5,−1.5 . The thickness of the DNG slab is set to a = 0.001 mm.

is attractive and increases for increasing separations, and is repulsive but increasing

for decreasing separations. Intuitively, we seek a situation in which the Casimir force

behaves somewhat like the elastic force of a loosely wound spring. Such a situation

is achievable, as illustrated in Fig.5.7. In this figure we focus our attention on the

profile of the force in the interval 37 < r . 40.6, for instance. For the most part,

the Casimir force is repulsive in this interval. But it exhibits an inflection point at

r ' 38.8, vanishes at r ' 40.3 then becomes an attractive force afterwards. This

exactly is the sought behavior. In effect, inside this interval of thicknesses ratio,

37 < r . 40.6, the force exhibits a behavior reminiscent of that of a loosely woundspring. A direct consequence of this is that the upper plate 7 would not only levitate

but also remain confined to a certain height.

It has been suggested that quantum levitation is achievable even in the presence

of only one DNG metamaterial slab. In particular, it has been speculated [102]

that a “perfect lens” of fixed thickness sandwiched between two conducting plates

would produce a repulsive Casimir force (whose magnitude is obtained by setting

n = −1 in (5.39)) and that this force would be able to levitate a metallic foil. Our7Cf. Fig.5.2.

97

30 35 40 45 50r

-1×108

-5×107

0

5×107

1×108


Figure 5.7: Plot of the normalized Casimir force f described in Fig.( 5.6) for differentvalues of r.

results suggest that this is an intricate issue. Because the perfect lens has a fixed

thickness, levitating the metallic foil would necessarily introduce a second slab of

DPS medium, say, vacuum, over the perfect lens which would then bring in regions

of instability similar to those discussed above. Quantum levitation, as described in

[102], entails a hidden assumption. It assumes that, as the foil levitates, no gap is

created between the foil and the perfect lens. But that, of course, is not the case.8

We also note that (5.39) suggests another situation in which one could obtain

large repulsive Casimir forces. Actually, the inverse dependence of the Casimir

force on the index of refraction in this case suggests the possibility of dramatically

increasing the magnitude of the repulsive Casimir force by sandwiching a metama-

terial with n ∼ 0− between two conducting plates. To our knowledge, this proposalhas not been suggested in the literature.

8See Fig.2 in [102]. There exist, however, one situation in which it would be theoreticallypossible to realize quantum levitation in the manner described in [102]. The space separating thefoil and the lower mirror would have to be continuously occupied by some kind of a "metamaterialfluid" having a refraction index n = −1. That, obviously, is not the case considered by the authors.

98

5.4 Conclusion

The numerical simulations described above establish the fact that, in the context of

the Casimir force between two perfectly conducting plates, major differences exist

between the situations that involve two-slab combinations separating the plates,

and the situations that involve only one slab. We found that (1) the presence of

two slabs greatly affects the behavior of the Casimir force causing it to turn into

a repulsive force, even when the response functions of the two slabs have the same

sign, (2) the force exhibits a profile determined in an essential way by the ratio of

the slabs’ thicknesses, (3) the force could be made orders of magnitude larger by

means of carefully selected response functions and slab separations, and (4) the force

exhibits a singular behavior at particular plate separations when the two slabs have

oppositely signed response functions. Evidently, clear advantages may result from

this behavior of the force depending on the application. Most notably, based on

these results, it appears that quantum levitation is indeed possible. It also appears

that, surprisingly, the energy of the vacuum could be used as a source for both the

actuation and shielding of MEMS and NEMS.

These are very interesting results. They could open up a whole new horizon

in technology. Yet, for all intents and purposes the system illustrated in Fig.5.2

is a highly idealized one. The force-decreasing skin-depth effects due to finite con-

ductivity are ignored; the force-increasing roughness effects of the metallic surfaces

are ignored; nonzero-temperature effects are ignored; the anisotropic nature of cur-

rent metamaterials is ignored; and, most importantly, the dispersive and dissipative

nature of metamaterials are also ignored.9 This, certainly, is the most question-

able assumption we have made as nondispersive or nondissipative metamaterials are

acausal (causality being epitomized by the Kramers-Kronig relations [115]). More-

over, the Casimir effect depends in a nontrivial way on the essentially dispersive

nature of metamaterials [110]. However, we believe that the simplifying assump-

tions made here do not diminish the insight that this study provides, as our primary

intention is to gain a rudimentary understanding of the class of systems proposed.

The effects of dispersion and dissipation shall be addressed elsewhere.

9As emphasized at the onset, the theory is valid for dispersive media (see Eq.(5.23)) and justi-fications exist for its extension to dissipative media as well [143].

99

Chapter 6

Future Directions

6.1 Extensions to the Inverse Source Problem in An-tenna Substrate Media

The theory developed in the first part of this dissertation holds for a given fre-

quency. I am interested in the development of a broadband inverse theory, which

can be derived as a stepped-frequency approach or as a theory directly in the time

domain. Extension of the published scalar formulation to the full-vector context and

dispersive embedding media is an important open problem of my interest. Another

future research avenue is the extension of the theory to prolate and oblate spheroidal

support regions so as to more tightly characterize dipole and planar antennas. Po-

tential future directions also include the development of a general multiport antenna

theory for small antennas formed by a number of independently addressable feeding

ports which may benefit, e.g., from nanotechnology. This approach is expected to

facilitate physical synthesis of near-optimal, wave-like current distributions in given

source volumes.

100

6.2 Extensions to the Inverse Scattering Problem inSubstrate Media

At the moment, I am also working on the scalar inverse scattering counterpart of the

abovementioned inverse source problem in metamaterial backgrounds. In particu-

lar, my goal is to investigate possible enhancements in the imaging resolution of an

object embedded in a metamaterial background. Imaging resolution enhancement

is expected to be possible in this case due to multiple scattering interactions of the

object with a helper substrate that acts as a near-field agent, i.e., a “re-transmitting

station,” that facilitates communication to the far field of evanescent field informa-

tion about the object. In fact, the achieving of super-resolution thanks to multiple

scattering, and the re-evaluation of the so-called “diffraction limit” in imaging, is an

area that has been receiving much attention in recent years, and is closely connected

to the developments in radiation and scattering enhancements due to metamaterials.

Once the current study of metamaterial-engendered imaging super-resolution is

finished, I am interested in expanding it to the full-vector inverse scattering problem

in metamaterial substrates including multiple scattering. In addition to this, I would

like to quantitatively characterize the “enhancement” in imaging in the presence

of noise via the fundamental Cramer-Rao bound. Another interesting open area

for further exploration is the utilization of metamaterials in optical imaging with

phaseless data, particularly in the imaging with far-field data.

6.3 Electrodynamics of Metamaterials

On account of the identified exciting potential applications of metamaterials, a bet-

ter understanding of their electrodynamics is in order. For instance, one major

problem that plagues the manufacturing of metamaterials is the ubiquitous lossy

nature of the meta-cells. I am very motivated to explore the effects that replacing

ordinary conductors as a base material for the meta-cells with superconducting ones

would have on the overall response of these materials. Another area that is still al-

most completely open and where I believe I can make a difference is the development

of a quantum electrodynamical theory in metamaterial backgrounds.

Because of the promising interplay between metamaterial research and nanotech-

101

nology, there is another important topic related to the study of the electrodynamics

of metamaterials that I find very interesting. That is nanoelectromagnetics. As a

matter of fact, new theoretical models supplemented by numerical studies are essen-

tial for the understanding of electromagnetic-wave propagation in media embedded

with nanostructures. At a more fundamental level, but no less important and rele-

vant, one may need to recur to quantum electrodynamics not just electromagnetics

to elucidate some of the physical properties of these media. This already is an ac-

tive line of research pursued in connection with the study of carbon nanotubes and

graphene sheets and I am interested in exploring it in the context of metamaterials.

I am planning to further investigate the Casimir effect in the presence of meta-

materials. The next obvious step would be to take into account the dispersive and

dissipative nature of metamaterials. The practical possibility of quantum levitation

suggested by the results of the second part of this dissertation is an exciting avenue

that I wish to pursue. I also would like to extend my study to other Casimir-type

effects such as the dynamical Casimir effect, where the boundary conditions are no

longer static but dynamic.

102

Appendix A

Definition and Properties ofVector Spherical Harmonics

The scalar spherical harmonics are defined as (cf., e.g., [117])

Yl,m(r) ≡ Yl,m(θ, ϕ)

≡s(2l + 1) (l −m)!

4π (l +m)!Pml (cos θ)e

imϕ, (A.1)

where the associated Legendre polynomials are given by the Rodriguez formula

Pml (x) =

(−1)m2ll!

¡1− x2

¢m/2µ

d

dx

¶l+m ¡x2 − 1¢l , (A.2)

l is a positive integer and |m| 6 l. Note that the Condon-Shortley phase factor

(−1)m has been absorbed in the definition of the Pml ’s.

The vector spherical harmonics are defined as [114]

Yl,m(r) ≡ Yl,m(θ, ϕ)

≡ LYl,m(θ, ϕ), (A.3)

103

L being the orbital angular momentum operator defined as

L ≡ −ir×∇

≡ −iÃ−bθsin θ

∂

∂ϕ+ bϕ ∂

∂θ

!. (A.4)

From the above definitions it can be shown that the scalar spherical harmonics

Yl,m(r) and the vector spherical harmonics Yl,m(r) satisfy the analytic continuation

properties

Y ∗l,m(r) = (−1)m Yl,−m(r), (A.5)

and

Y∗l,m(r) = (−1)m+1Yl,−m(r), (A.6)

along with the orthogonality conditions [114]ZYl,m(r)Y

∗l0,m0(r) dr = δll0δmm0 , (A.7)

ZYl,m(r) ·Y∗l0,m0(r) dr = l (l + 1) δll0δmm0 , (A.8)Z

r×Yl,m(r) · r×Y∗l0,m0(r) dr = l (l + 1) δll0δmm0 , (A.9)ZY∗l,m(r) · r×Yl0,m0(r) dr = 0, (A.10)

where the δxx0 ’s are the Kronecker deltas, and dr ≡ sin θdϕdθ.The following formulae [144] were useful in the derivation of some of the results

∇× [rφl(r)Yl,m(r)] =−irφl(r)Yl,m(r), (A.11)

∇× [φl(r)Yl,m(r)] = ril(l + 1)

rφl(r)Yl,m(r) +

1

r

d

dr[rφl(r)]r×Yl,m(r), (A.12)

∇× [φl(r)r×Yl,m(r)] = −1r

d

dr[rφl(r)]Yl,m(r), (A.13)

∇×∇× [φl(r)Yl,m(r)] = Yl,m(r)

∙l (l + 1)

r2−∇2

¸φl(r). (A.14)

104

A concise source on vector spherical harmonics and their properties is [145].

Note, though, that the notation and normalization adopted in this reference are

slightly different from the ones above.

105

Appendix B

Wavefunctions B(j)l,m forPiecewise-ConstantRadially-SymmetricBackgrounds

The aim of this appendix is to show that the multipole moments a(j)l,m are given by

Eq.(2.17) with the source-free wavefunctions B(j)l,m (r) given by Eqs.(2.19,2.20). A

straightforward way of arriving at these results would be to use the dyadic Green’s

function that governs the propagation of electromagnetic radiation between the

source-enclosing inner sphere and the surrounding vacuum. The spectral-domain

electromagnetic Green’s function linking the different layers of a spherically multi-

layered medium has been calculated by Li et al. [146].1 Afterwards, one calculates

the external electric field by means of Eq.(2) in [146]. Finally one uses Eqs.(2.15)

and (2.16), in this paper, to arrive at the desired results, i.e., Eqs.(2.17,2.19,2.20).

Another way of arriving at Eqs.(2.17,2.19,2.20) is to invoke Lorentz’s reciprocity

theorem and the concept of reaction (also called coupling). Here we adopt this

approach. One reason behind this choice is that this latter approach attests to the

1For the homogeneous sphere the Green function is given by Eqs.(25a) and (26b) along with therelevant definitions while for the three-region geometry of the core-shell system the Green functionis given in [146] by Eqs.(14), (29a)-(29d), and the relevant definitions.

106

visible similarity between the mathematics of the radiation problem at hand and its

scattering counterpart.

The reaction RE→J0 of a field E(r) produced by a source J(r) on another sourceJ0(r), is defined as the integral

RdrE(r) · J0(r). The reciprocity theorem can be

stated as follows (see, for instance, [58]): The reaction of the field E(r) produced

by a source J(r) on another source J0(r) is equal to the reaction of the field E0(r)

produced by the source J0(r) on the source J(r), i.e.,

RE→J0 = RE0→J. (B.1)

We focus, first, on the two-region (homogeneous sphere) case. The calculation

below outlines also the general approach that will be used in the case of the three-

region (core-shell) system. The details of that calculation are presented in Appendix

C.

To evaluate the field due to a current distribution J(r) that is embedded in the

piecewise-constant radially symmetric background of interest, we consider, without

loss of generality, the following two classes of canonical sources:hJ(1)l,m

i0(r) = δ(r −R)r×Yl,m(r) (B.2)

and hJ(2)l,m

i0(r) = δ(r −R)Yl,m(r), (B.3)

where in both expressions R > a represents the radius of the helper source cen-

tered around the origin. (Ultimately, the multipole moments a(j)l,m are expected to,

and in fact will, turn out to be independent of R.) The justification for the above

considerations relies on two results:

• The transverse component of an arbitrary vector field on the spherical sur-face of radius R > a centered about the origin is uniquely characterized by

its expansion in terms of the vector spherical harmonics Yl,m(r) and their

associated vector functions r×Yl,m(r), and

• the multipole moments characterizing any electric field outside the support ofthe emitting sources is uniquely determined by the tangential component of

107

the field on any such spherical surface. Indeed, it can easily be shown that if

R > a is the radius of a sphere centered about the origin, then

a(j)l,m =

⎧⎪⎪⎨⎪⎪⎩−i

l(l+1)k0h(+)l (k0R)

RY∗l,m(r) ·E(Rr)dr ; j = 1

1l(l+1)k0Vl(k0R)

Rr×Y∗l,m(r) ·E(Rr)dr ; j = 2.

(B.4)

The fieldhE(j)l,m

iincthat would be produced by the source

hJ(j)l,m

i0in Eq. (B.3)

if it were in free space (this will be the incident field in the following) is given byhE(j)l,m

iinc=

Zdr0G0(r, r

0) ·hJ(j)l,m

i0

¡r0¢, (B.5)

where G0(r, r0) is the multipole representation of the free-space electric dyadic Green

function [147], viz.,

G0(r, r0) =

∞Xl=1

lXm=−l

−ωµ0k0l (l + 1)

nk20 [jl(k0r<)Yl,m(r<)]

hh(+)l (k0r>)Y

∗l,m(r>)

i+ ∇× [jl(k0r<)Yl,m(r<)]∇×

hh(+)l (k0r>)Y

∗l,m(r>)

io+

i

ω 0rrδ

¡r− r0¢ . (B.6)

The < (>) subscript designates the smaller (larger) of r and r0.

For r < R the fieldhE(1)l,m

iincis found to be given by

hE(1)l,m

iinc(r) = τ l(k0, R)∇× [jl(k0r)Yl,m(r)], (B.7)

where we have defined

τ l(k0, R) ≡ −η0k0R2Vl (k0R) . (B.8)

Along analogous lines, the (incident) fieldhE(2)l,m

iincproduced by the source

hJ(1)l,m

i0

108

in Eq. (B.2) in free space is found, for r < R, to be given byhE(2)l,m

iinc(r) = ζ l(k0R)jl(k0r)Yl,m(r) (B.9)

where we have introduced

ζ l(k0R) ≡ −η0 (k0R)2 h(+)l (k0R). (B.10)

The obtainment of the above results requires the use of orthogonality properties

of the vector spherical harmonics Yl,m(r) and the associated vector functions r ×Yl,m(r).2

Introducing the index of refraction n =√µs s/

√µ0 0 =

√µr r, the evaluation of

the total fieldshE(j)l,m

i0, j = 1, 2, associated with these sources for r < R is seen from

Eqs. (2.1), (2.2), (2.3)-(2.5) to correspond to the solution of the forward scattering

problem associated for r < R with the equation

£∇2 + k20Θ(r − a) + n2k20Θ(a− r)¤ hE(j)l,m

i0(r) = 0 (B.11)

upon excitation by the incident fieldshE(j)l,m

iinc.

The total (incident plus scattered) fieldhE(1)l,m

i0must be, due to considerations

of causality (in the scattered field) and well-behavedness of the interior field for

r < a, of the form

hE(1)l,m

i0(r) =

⎧⎪⎪⎨⎪⎪⎩∇×

hτ l(k0, R)jl(k0r)Yl,m(r) +D1h

(+)l (k0r)Yl,m(r)

i; r > a

A1∇× [jl(kr)Yl,m(r)]; r ≤ a,

(B.12)

where k = nk0 is the wavenumber of the field in the background material confined

within the source volume V and A1 and D1 are coefficients that are to be determined

by imposing continuity of the tangential components of the electric and magnetic

fields on the boundary ∂V ≡ r ∈ R3 : r = a. Analogously, the total field

2See Appendix A.

109

hE(2)l,m

i0must be of the form

hE(2)l,m

i0(r) =

⎧⎪⎪⎨⎪⎪⎩hζ l(k0R)jl(k0r) +D2h

(+)l (k0r)

i×Yl,m(r); r > a

A2jl(kr)Yl,m(r); r ≤ a,

(B.13)

where A2 and D2 are coefficients that need to be determined from the boundary

conditions on ∂V.

By imposing the continuity requirements on the boundary ∂V and using the

Wronskian relation for spherical Bessel functions [117], one obtains (for j = 1)

A1τ l(k0, R)

=i/ (k0a) (ka)

( r/µr)1/2 jl(ka)Vl(k0a)− h

(+)l (k0a)Ul(ka)

≡ F(1)l (k0a, ka, r, µr) , (B.14)

where Ul and Vl have already been defined in Eqs. (2.21) and (2.22). The coefficient

A2 associated with the fieldhE(2)l,m

i0can be obtained by an analogous procedure

which yields

A2ζ l(k0R)

=iµr/ (k0a) (ka)

(µr/ r)1/2 jl(ka)Vl(k0a)− h

(+)l (k0a)Ul(ka)

≡ F(2)l (k0a, ka, r, µr) . (B.15)

Along with Eqs. (B.12) and (B.13) the above results define the fieldshE(1)l,m

i0andh

E(2)l,m

i0.

By applying the reciprocity relation Eq. (B.1) to the preceding results (in par-

ticular, Eqs. (B.2), (B.3), (B.8), (B.10), (B.12), (B.13), (B.14), (B.15)), one finds

that the multipole moments a(j)l,m are indeed independent of R and given by Eq.

(2.17) with the source-free wavefunctions B(j)l,m(r) given by Eqs. (2.19)-(2.22). In

obtaining these results we have also recalled the multipole expansion for the electric

field E(r), i.e., (2.15) and (2.16), along with the orthogonality and analytic con-

tinuation properties of the vector spherical harmonics, cf. Appendix A., and the

analytic continuation property of the spherical Bessel functions of the first kind,

110

viz., j∗l (ka) = jl(k∗a). The derived results (2.19,2.20) reduce to the free-space result

Eq. (2.18) as r → 1 and µr → 1, as expected.

An alternative way of arriving at Eq. (2.17) and Eqs. (2.19)-(2.22) would be

to follow the same procedure as above but to use the far-field expressions for E(r)

and H(r) instead of the multipole expansion given by Eqs. (2.15) and (2.16). The

far-field expressions are obtained from Eqs. (2.15) and (2.16) by using the large-

argument approximation for the spherical Hankel function h(+)l . This yields the

far-field approximations (cf. also [51], Eqs. (4)-(6))

E(r r) ∼k0rÀ1

eik0r

rfe(r) =

η0eik0r

rr× fm(r) (B.16)

H(r r) ∼k0rÀ1

eik0r

η0rr× fe(r) = eik0r

rfm(r), (B.17)

where η0 =pµ0/ 0 is the free-space wave impedance. The two vector quantities

fe(r) and fm(r) are, respectively, the far electric field radiation pattern and the far

magnetic field radiation pattern. They are given as a function of the observation

direction r by

fe(r) =∞Xl=1

lXm=−l

(−i)lha(1)l,mr×Yl,m(r) + a

(2)l,mYl,m(r)

i(B.18)

and

fm(r) =1

η0

∞Xl=1

lXm=−l

(−i)lh−a(1)l,mYl,m(r) + a

(2)l,mr×Yl,m(r)

i. (B.19)

It follows from Eqs.(B.18,B.19) and the orthogonality of the vector spherical har-

monics Yl,m(r) and the associated vector functions r×Yl,m(r) that the multipole

moments a(j)l,m are uniquely defined by projections of the far-field radiation patterns

onto the orthogonal set of functions Yl,m(r) and r×Yl,m(r) (see, e.g., [51], Eq.(7)

and the associated discussion). Thus either the far fields or the multipole moments

uniquely define each other as well as the exterior field everywhere outside the source

volume V (via Eq.(2.15)).

In order to arrive at Eq.(2.17) and Eqs.(2.19,2.20) we follow the same procedure

as above but instead of the multipole expansion (2.15,2.16) we use Eqs.(B.16,B.17,B.18,B.19).

111

We then require that k0R À 1 and use the large-argument approximation for the

spherical Hankel function, in particular, h(+)l (k0R) ∼ (−i)l+1eik0R/(k0R) (see, forinstance, [117]). Afterwards one lets k0R→∞. Whether one uses this approach orthe previous one, the final results are the same.

112

Appendix C

Wavefunctions B(j)l,m for aSystem of Two Nested Spheres

To determine the source-free fields B(j)l,m for the spherical core-shell system we follow

the same procedure used in appendix B. However, we note that the radius R of the

helper source centered around the origin is, now, R > b.

The total fieldhE(1)l,m

i0must be, due to considerations of causality in the scattered

field and well-behavedness of the interior field for r < b, of the form

hE(1)l,m

i0(r) =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

∇×nh

τ l(k0, R)jl(k0r) +D1h(+)l (k0r)

iYl,m(r)

o; r > b

∇×nh

B1jl(kbr) + C1h(+)l (kbr)

iYl,m(r)

o; a < r ≤ b

A1∇× [jl(kar)Yl,m(r)]; r ≤ a,

(C.1)

where A1, B1, C1 and D1 are coefficients that are to be determined by imposing con-

tinuity of the tangential components of the electric and magnetic fields on the inner

and outer boundaries of the spherical shell. Analogously, the total fieldhE(2)l,m

i0must

113

be of the form

hE(2)l,m

i0(r) =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

hζ l(k0R)jl(k0r) +D2h

(+)l (k0r)

iYl,m(r); r > b

hB2jl(kbr) + C2h

(+)l (kbr)

iYl,m(r); a < r ≤ b

A2jl(kar)Yl,m(r); r ≤ a,

(C.2)

where A2, B2, C2 and D2 are coefficients that need to be determined from the bound-

ary conditions.

Imposing the abovementioned continuity requirements on the inner and outer

boundaries of the spherical shell, i.e., on the spheres of radii a and b, respectively,

yields two systems (one for j = 1 and one for j = 2) of four equations each linear in

the unknown coefficients.

Upon solving the two linear systems of equations and using the Wronskian re-

lation for spherical Bessel functions, specifically jl(x)h0l(+)(x) − j0l(x)h

(+)l (x) = i

x2

[117], we find that for j = 1

A1τ l(k0, R)

=b

∆1k0kba2b2

≡ F(1)l , (C.3)

where ∆1 is the determinant given by Eq.(3.5). Similarly for j = 2 one obtains

A2ζ l(k0R)

=1

µb∆2k0kba2b2

≡ F(2)l , (C.4)

where∆2 is the determinant given by (3.6). The remaining constantsB1, C1,D1, B2, C2,

and D2 are straightforwardly obtained; here they are omitted because of their irrel-

evance to the rest of the problem. Along with Eqs.(C.1,C.2), Eqs. (C.3,C.4) define

the fieldshE(1)l,m

i0and

hE(2)l,m

i0for the spherical core-shell system.

114

Appendix D

Calculations of the FréchetDerivatives of the ObjectiveFunctional and the Constraints

Let f : L2¡V ;C3

¢ → C be a functional. The Fréchet derivative of f at J0 ∈L2¡V ;C3

¢is defined as the gradient ∇Jf (J0) : L2

¡V ;C3

¢→ C such that [119]

|f (J0 + d)− f (J0)− (∇Jf (J0) ,d)|CkdkL2(V ;C3)

−→kdkL2(V ;C3)→0

0, ∀d ∈L2 ¡V ;C3¢ . (D.1)

However, it is well-known [119] that difficulties related to the definition of linearity

of the Fréchet differentiation operator would be encountered when f maps a complex

Banach space (in our case L2¡V ;C3

¢) into a real Banach space (in our case R). This

is in particular the case for E and³J, eGSJ

´. In such cases one considers L2

¡V ;C3

¢as a Hilbert space over R instead of C [119]. Consequently, the definition of the

Fréchet derivative given by (D.1) takes on the slightly modified form [119]

[f (J0 + d)− f (J0)−< (∇Jf (J0) ,d)]kdkL2(V ;C3)

−→kdkL2(V ;C3)→0

0, ∀d ∈L2 ¡V ;C3¢ . (D.2)115

Given these definitions, one can show that

< (∇JE (J0) ,d) = < (2J0 , d) , (D.3)

whereby one identifies ∇JE (J0) with 2J0. Similarly, one obtains for the first groupof (convex) constraints³

∇Jha(j)l,m − (B(j)

l,m,J0)i,d´=³−B∗(j)l,m ,d

´, (D.4)

whereby one identifies ∇J³a(j)l,m − (B(j)

l,m,J0)´with −B(j)

l,m. Finally, for the sec-

ond group of (nonconvex) constraints, one also obtains (after using the symmetry

property GS(r, r0) = GS(r

0, r))

<³∇J(J0, eGSJ0),d

´= <

³2 eGSJ0, d

´. (D.5)

In this case, one identifies ∇J(J0, eGSJ0) with 2eGSJ0.

116

Appendix E

Wavefunctions B(j)l,m forSpherically SymmetricBackgrounds

Another important and formally tractable case is that of a spherically symmetric

background for which

(r) = s(r)Θ(a− r) + 0Θ(r − a) (E.1)

µ(r) = µs(r)Θ(a− r) + µ0Θ(r − a), (E.2)

In this case the key step is the determination of the eigenfunctions of the homoge-

neous equation

∇×∇×∙E(r)

µs(r)

¸− ω2 s(r)E(r) = 0 (E.3)

which follows from equations (E.1), (E.2) and (2.5).

Fundamental for the dealing of this case, is the verification that equation (E.3)

admits solutions of the separable form E(2)l,m(r) = φl(r)Yl,m(r). Using well-known

properties of the vector spherical harmonics Yl,m(r) and the associated vector func-

tions r ×Yl,m(r) as well as the useful results (A.12)-(A.14) one readily finds that

the postulated solution E(2)l,m(r) = φl(r)Yl,m(r) obeys equation (E.3) as long as the

117

radial eigenfunctions φl(r) obey

µ−1s (r)∙l(l + 1)

r2−ω2 s(r)− 1

r2d

dr

µr2

d

dr

¶¸φl(r)=

1

r

d

dr

∙rφl(r)

dµ−1s (r)dr

¸. (E.4)

This is a key result, thanks to which one can readily carry out the forward mapping

and subsequent source inversion in this kind of spherically symmetric background.

In particular, following a procedure similar to that employed in appendix B for

piecewise-constant radially-symmetric backgrounds, but with the new radial eigen-

functions φl(r) playing for r < a the role of the spherical Bessel functions jl(kr) of

that appendix, one finds that

B(2)l,m(r) =

−η0Yl,m(r)

l(l + 1)a2

⎧⎨⎩ µr (r)φl(r)

µr (r) k0φl(a)Vl(k0a)− h(+)l (k0a)

hφ0l(a) +

φl(a)a

i⎫⎬⎭∗

.

(E.5)

The relations governing the associated electric source-free wavefunctions B(1)l,m(r)

can be derived similarly.

118

Appendix F

Establishment of The DispersionRelations (5.2) and (5.3)

The aim of this appendix is to establish the dispersion relations (5.2) and (5.3). As

mentioned in Chapter 5, we shall follow closely the simple presentations given in

[83, 85].

The proper modes for the system depicted in Fig.(5.1) are obtained by solving

Maxwell’s equations. For isotropic sourceless media, as is our case, the problem is

equivalent to solving the sourceless Helmholtz equations

£∇2 + k2j (ω)¤Ej (r) = 0, (F.1)

£∇2 + k2j (ω)¤Bj (r) = 0, (F.2)

where j = 1, 2, a, b and

k2j (ω) =ω2 j (ω)µj (ω)

c2, (F.3)

subject to the following static Dirichlet boundary conditions

z ·Dj ≡ D⊥j is continuous, (F.4)

z×Ej ≡ Ekj is continuous, (F.5)

z ·Bj ≡ Bkj is continuous, (F.6)

119

and

z×Hj ≡ Hkj is continuous. (F.7)

which must be satisfied at the interfaces between the media, i.e., at z = 0, z = a,

and z = a+ b.

We consider solutions of the form

Ej (r) = ej (z) eiκ·r⊥

= [ejx (z) bx+ ejy (z) by + ejz (z)bz] ei(kxx+kyy) (F.8)

and

Bj (r) = bj (z) eiκ·r⊥

= [bjx (z) bx+ bjy (z) by + bjz (z)bz] ei(kxx+kyy). (F.9)

It follows thatdejνdz−K2

j ejν = 0 (F.10)

anddbjν (z)

dz−K2

j bjν = 0, (F.11)

where the Kj ’s are defined as1

K2j ≡ k2x + k2y − j (ω)µj (ω)

ω2

c2

≡ κ2 − j (ω)µj (ω)ω2

c2, (F.12)

with ν = x, y, z. The solutions to Maxwell’s equations that satisfy (F.10) and (F.11)

form a complete set of solutions [141]. They are called surface modes because they

correspond to waves propagating parallel to the interfaces and decaying exponen-

tially for z < 0 and z > a+ b.

1Cf. with (5.4).

120

It is easy to show that all the boundary conditions (F.4-F.7) are satisfied if2⎧⎪⎨⎪⎩j (ω) ejz (z) is continuous

anddejz(z)dz is continuous,

(F.13)

and ⎧⎪⎪⎨⎪⎪⎩ejy (z) is continuous

and1

µj(ω)dejz(z)dz is continuous.

(F.14)

where j = 1, 2, a, b and ν = x, y, z.

Ignoring exponentially growing solutions to Eqs.(F.10) leaves us only with the

following physically acceptable ones

ejν (z) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩Aνe

K1z , z < 0

BνeKaz + Cνe

−Kaz , 0 < z < a

DνeKbz +Eνe

−Kbz , a < z < a+ b

Fνe−K2z , z > a+ b,

(F.15)

where j = 1, 2, a, b as before but where the values of ν are now restricted to y and

z. These solutions along with the boundary conditions (F.13) and (F.14) lead to

two systems of linear algebraic equations for the coefficient Aν , Bν , Cν ,Dν , Eν , and

Fν . Imposing nontriviality conditions on the solutions of these systems yields the

following dispersion relations

g (ω) = 0 (F.16)

and

gµ (ω) = 0, (F.17)

2One can arrive at (F.13), for instance, in the following way. Demanding that (F.4) be satisfiednecessitates the continuity of j (ω) ejz (z) . Also, requiring that ∇ ·Dj = 0 and taking into accountthat ∇ ×Ej = (iω/c)Bj , automatically guarantees that ∇ ·Bj = 0 . Now, (F.5) is guaranteed ifejx is continuous. But this in turn is guaranteed if dejx/dz is continuous. Hence, (F.13). Conditions(F.14) are established in a similar fashion (See [83] for more detail.)

121

where

g (ω) ≡

¯¯¯¯

1 − a − a 0 0 0

0 aeKaa

ae−Kaa − be

Kba − be−Kba 0

0 0 0 beKb(a+b)

be−Kb(a+b) − 2e

−K2(a+b)

K1 −Ka Ka 0 0 0

0 KaeKaa −Kae

−Kaa −KbeKba Kbe

−Kba 0

0 0 0 KbeKb(a+b) −Kbe

−Kb(a+b) K2e−K2(a+b)

¯¯¯¯

(F.18)

and

gµ (ω) ≡

¯¯¯¯

1 −1 −1 0 0 0

0 eKaa e−Kaa −eKba −e−Kba 0

0 0 0 eKb(a+b) e−Kb(a+b) −e−K2(a+b)

K1µ1

−Kaµ1

Kaµa

0 0 0

0 KaµaeKaa −Ka

µae−Kaa −Kb

µbeKba Kb

µbe−Kba 0

0 0 0 KbµbeKb(a+b) −Kb

µbe−Kb(a+b) K2

µ2e−K2(a+b)

¯¯¯¯.

(F.19)

After some straightforward algebraic maniplulations one can show that (F.16) and

(F.17) do in fact reduce to (5.2) and (5.3), respectively.

122

Appendix G

Application of the ArgumentPrinciple to the Derivation ofEq. (5.20)

For the convenience of the reader we present here some technical details skipped

in Chapter 5 in the establishment of the Casimir energy formula (5.20). The idea

is to convert the sums on the right-hand side of (5.15) into contour integrals of

complex functions. This is a well-known technique [148]. We follow closely the

simple presentation given in [83].

A useful variant of the argument principle for meromorphic functions [142] stip-

ulates that

1

2πi

IC

f 0 (z)f (z)

zdz =

⎛⎝Xj

zj

⎞⎠f(zj)=0

−⎛⎝X

j

zj

⎞⎠f(zj)→∞

, (G.1)

i.e., the difference between the sum of the zeros of a meromorphic function f (z)

inside a contour C and the sum of its poles inside the same contour is given by the

integral on the left-hand side of (G.1).

Now, we want to apply (G.1) to the evaluation of the right hand side of (5.15).

123

Treating ω as a complex variable we haveXN

ω(N) (κ) = sum of zeros of g (ωj) (G.2)

and XN

ω(N)µ (κ) = sum of zeros of gµ (ωj) . (G.3)

The poles of gα (ωj) and gµ (ωj) are independent of d because d appears only in the

exponentials. Therefore,

1

2πi

IC

g0(ω)

g (ω)ωdω =

XN

ω(N) (κ)− a quantity independent of d (G.4)

and similarly

1

2πi

IC

g0µ (ω)

gµ (ω)ωdω =

XN

ω(N)µ (κ)− a quantity independent of d, (G.5)

where C is the contour depicted in Fig.(G.1). Implicit in this definition of C is

the fact that all physically acceptable solutions ω to the equations g (ω) = 0 and

gµ (ω) = 0 are positive real numbers. This is guaranteed if the constitutive parame-

ters are real functions of ω. Consequently, we shall restrict our selves to dissipation-

less backgrounds.1

Obviously the quantities that are independent of d do not contribute to the force.

1However, one must mention that when the technique is applied “blindly” to the case of complex-valued constitutive parameters it does reproduce the exact same results obtained by means of moresophisticated methods. An intuitive justification for the unanticipated success of the technique inthe presence of the dissipation is given in [143].

124

Im[ ]

C

1… N Re[ ]

Figure G.1: The contour, C, used in Eqs.(G.4,G.5); it is defined by the imaginaryaxis and the semicircle of infinite radius directed to the right. The points ω1, ..., ωNrepresent the N solutions of the dispersion relations g (ω) = 0 or gµ (ω) = 0.

125

Hence, we can ignore this term and write that

U =~L2

4π

1

2πi

Z ∞

0κdκ

⎡⎣IC

g0(ω)

g (ω)ωdω +

IC

g0µ (ω)

gµ (ω)ωdω

⎤⎦=

~L2

8π2i

Z ∞

0κdκ

Z −∞

∞

∙1

g (iξ)

∂g (iξ)

∂ (iξ)(iξ) +

1

gµ (iξ)

∂gµ (iξ)

∂ (iξ)(iξ)

¸d (iξ)

+~L2

8π2i

Z ∞

0κdκ

ZSemicircle

∙1

g (iξ)

∂g (iξ)

∂ (iξ)(iξ) +

1

gµ (iξ)

∂gµ (iξ)

∂ (iξ)(iξ)

¸d (iξ)

=~L2

8π2i

Z ∞

0κdκ

Z −∞

∞

∙1

g (iξ)

∂g (iξ)

∂ (iξ)(iξ) +

1

gµ (iξ)

∂gµ (iξ)

∂ (iξ)(iξ)

¸d (iξ) +Υ,

(G.6)

where Υ is a quantity that is independent of d in the limit of infinite semicircle

radius. Ignoring the d-independent term and substituting G (ξ) and Gµ (ξ) for

gµ (iξ) and g (iξ), respectively, yields

U =~L2

8π2

Z ∞

0κdκ

Z ∞

−∞

∙1

G (ξ)

∂G (ξ)

∂ξξ +

1

Gµ (ξ)

∂Gµ (ξ)

∂ξξ

¸dξ

=~L2

8π2

Z ∞

0κdκ

Z ∞

−∞[lnG (ξ) + lnGµ (ξ)] dξ, (G.7)

which establishes Eq.(5.20).

126

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