Free-Space Metamaterial Superlenses Using Transmission-Line Techniques by Ashwin K. Iyer a Thesis Submitted in Conformity with T

Free-Space Metamaterial Superlenses using Transmission-Line Techniques

Ashwin K. Iyer

A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Electrical Engineering University of Toronto

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Abstract

Free-Space Metamaterial Superlenses using Transmission-Line Techniques

Ashwin K. Iyer

Doctor of Philosophy

Graduate Department of Electrical Engineering

University of Toronto

2009

Free-space imaging with a resolution beyond that dictated by the classical diﬀraction limit may be achieved with a ‘Veselago-Pendry’ superlens made from a metamaterial possessing a number of speciﬁc properties, including a negative refractive index (NRI).

Although a planar NRI transmission-line (NRI-TL) metamaterial based on the periodic lumped loading of a host TL network has successfully verified the phenomenon of superlensing in a 2D microstrip environment, a true Veselago-Pendry superlens capable of interacting with and manipulating fields in free space remained elusive, largely due to the difficulty of meeting its stringent design constraints and also to the problem of realizing a full 3D isotropic, polarization-independent structure. This work presents the first experimental verification of free-space Veselago-Pendry superlensing using a new class of volumetric metamaterials based on 2D NRI-TL layers that, although polarization- specific, may be easily constructed using available lithographic techniques to interact with free-space sources. An equivalent-circuit model is developed to enable accurate design of the metamaterial’s dispersion and transmission characteristics, including those associated with Veselago-Pendry superlensing, and is validated using full-wave simulations. First, a volumetric NRI-TL metamaterial employing fully printed loading elements is fabricated to verify the salient properties of a free-space metamaterial-slab lens. This lens demonstrates diffraction-limited focusing at X-band and, thus, affirms theoretical results that suggest that electrically thick and lossy metamaterials are unable to perform

ii superlensing. Thereafter, a volumetric NRI-TL metamaterial based on discrete lumped elements is designed to meet the conditions of the Veselago-Pendry superlens at 2.40GHz, and experimentally demonstrates a resolution ability over three times better than that afforded by the classical diffraction limit. A microwave superlens designed in this fashion can be particularly useful for illumination and discrimination of closely spaced buried objects over practical distances by way of back-scattering, for example, in tumour or landmine detection, or for targeted irradiation over electrically small regions in tomography or hyperthermia applications. Possible optical implementations of the volumetric topology are also suggested, and finally, a fully isotropic, polarization-independent 3D metamaterial structure related to the volumetric NRI-TL structure is proposed.

iii Acknowledgements

I would like to express my sincere thanks to my advisor, Prof. George Eleftheriades, for his invaluable guidance and encouragement over the years. I am truly grateful for his conﬁdence in me, his interest in my success, and his friendship, and I would be satisﬁed to have imbibed even a small amount of his scholarship, wisdom, and character. I would also like to thank Prof. Keith Balmain for his insightful and helpful suggestions, as well as Profs. Costas Sarris and Mohammad Mojahedi for many stimulating discussions.

I am indebted to Gerald Dubois for his patience and intuition, and for the cheerful manner in which he so often rescued my research from certain disaster. Thanks also to

Peter Kremer for his technical expertise, and Rogers Corporation, Saturn Electronics, and Jamie McIntyre and Malcolm Forge of George Brown College for their help in the experimental portion of this work. To my fellow graduate students - it’s been great - you know all the memories! I’ve learned a lot from all of you, and I wish you the best in your future careers. I also gratefully acknowledge ﬁnancial support that enabled this work, including the NSERC Canada Graduate Scholarship and Edward S. Rogers Sr.

Scholarship.

Most of all, I am deeply grateful to my mother and brother for their unconditional love and unfailing support, without which I could not have fulﬁlled any of my aspirations.

I owe them everything, and I dedicate this work to them.

–Ashwin K. Iyer

Toronto, 2008

iv List of Symbols and Acronyms

β Propagation Constant

χe Electric Susceptibility

χm Magnetic Susceptibility

² Permittivity

²0 Free-Space Permittivity

²r Relative Permittivity

B Magnetic Flux Density Vector

D Electric Flux Density Vector

E Electric Field Intensity Vector

H Magnetic Field Intensity Vector

M Magnetization Vector

P Polarization Vector

µ Permeability

µ0 Free-Space Permeability

µr Relative Permeability

v ω Angular Frequency

σ Electric Conductivity

C Capacitance

L Inductance

R Resistance or Resolution Enhancement Factor (by context)

1D One-dimensional

2D Two-dimensional

3D Three-dimensional

CPS Coplanar-Strip

EN Expanded Node

FEM Finite-Element Method

GHz Gigahertz (109 Hz)

HFSS High-Frequency Structure Simulator (Ansoft)

LH Left-Handed

MHz Megahertz (106 Hz)

MTL Multiconductor Transmission Line

NRI Negative Refractive Index

NRI-TL Negative-Refractive-Index Transmission Line

PCB Printed-Circuit Board

PRI Positive Refractive Index

vi RH Right-Handed

SCN Symmetrical Condensed Node

SRR Split-Ring Resonator

THz Terahertz (1012 Hz)

TL Transmission Line

TLM Transmission-Line Matrix

vii Contents

1 Introduction 1

1.1 Motivation ...... 2

1.2 Objectives ...... 4

1.3 Outline ...... 5

2 Background 7

2.1 Metamaterials: Transcendent Artiﬁcial Dielectrics ...... 7

2.2 The Left-Handed (LH) Medium ...... 11

2.3 Metamaterial Implementations ...... 15

2.4 Metamaterial Applications ...... 21

3 Fundamentals of Transmission-Line Metamaterials 29

3.1 TL Theory of LH Media ...... 29

3.2 Negative Parameters: An Electrodynamical Perspective ...... 35

3.3 TL Metamaterials and the Split-Ring Resonator Connection ...... 49

3.4 Practical Realization of TL-Based LH Metamaterials ...... 58

4 Uniplanar Transmission-Line Metamaterials 60

4.1 Dispersion Characteristics ...... 63

4.2 2D Eﬀective-Medium Properties ...... 71

4.3 Closure of the Stopband: The Impedance-Matched Condition ...... 78

viii 4.4 Practical realization ...... 79

5 Free-Space Volumetric NRI-TL Metamaterials 82

5.1 Development of a Two-Port Equivalent-Circuit Model ...... 84

5.2 Volumetric Eﬀective-Medium Properties ...... 89

5.3 Discrete Lumped-Element Design ...... 90

5.4 Fully Printed Design ...... 98

5.5 Design for Free Space ...... 104

5.6 Implementations from THz to Optical Frequencies ...... 108

6 A Free-Space NRI-TL Lens 111

6.1 Subwavelength versus Diﬀraction-Limited Focusing ...... 111

6.2 Design ...... 113

6.3 Simulation ...... 115

6.4 Experiment ...... 120

7 A Free-Space NRI-TL Superlens 129

7.1 Design ...... 131

7.2 Simulation ...... 134

7.3 Experiment ...... 141

7.4 Subdiﬀraction Imaging by Other Mechanisms ...... 150

7.5 A Comment on Bandwidth ...... 153

8 A 3D NRI-TL Topology for Free-Space Excitation 154

8.1 Metamaterials Based on the Symmetrical Condensed Node (SCN) . . . . 155

8.2 Design ...... 156

8.3 Simulation ...... 158

8.4 Optical Implementation ...... 161

ix 9 Conclusions 163

9.1 Summary ...... 163

9.2 Contributions ...... 165

9.3 Future Work ...... 168

A The Diﬀraction Limit 171

A.1 Imaging: An Interference Phenomenon ...... 172

A.2 Imaging in the Far Field ...... 174

A.3 Subdiﬀraction Imaging ...... 174

A.4 The Veselago-Pendry Superlens ...... 175

B Transmission-Line Networks as Artiﬁcial Materials 177

B.1 Artiﬁcial Materials and the Long-Wavelength Limit ...... 177

B.2 The Lumped-Element Transmission-Line Model ...... 178

B.3 From Modeling to Synthesis: The NRI-TL Metamaterial ...... 183

C Free-Space Measurement Apparatus Design 185

C.1 Standard-Gain Horn Antenna Speciﬁcations ...... 186

C.2 Bulk Rexolite Speciﬁcations ...... 186

C.3 Measurement Setup and Design ...... 187

C.4 Lens Design ...... 190

D The Shielded-Loop Antenna 192

D.1 Unbalanced Currents ...... 193

D.2 Balancing Currents with the Shielded-Loop ...... 194

D.3 Simulations ...... 196

D.4 Radiation Pattern Measurements ...... 197

Bibliography 200

x List of Tables

4.1 Description of the symmetry points of the 2D Brillouin zone for a rectangu-

lar lattice (n is an integer corresponding to a particular Floquet-Bloch spa-

tial harmonic). Reprinted with permission from Ref. 60, copyright °c 2006

Optical Society of America...... 68

4.2 Dispersion relations for isotropic and anisotropic cases (selected propaga-

tion angles). Reprinted with permission from Ref. 60, copyright °c 2006

Optical Society of America, Inc...... 68

5.1 Design parameters employed for a volumetric layered NRI-TL medium

using discrete lumped elements. Reprinted with permission from Ref. 60,

5.2 Design parameters for interdigitated capacitors and strip inductors em-

ployed in the fully printed NRI-TL planar unit cell depicted in Fig. 5.7.

Reprinted with permission from Ref. 60, copyright °c 2006 Optical Society

of America, Inc...... 102

7.1 Surface-mount inductor and capacitor values...... 132

7.2 Element values for the equivalent-circuit model describing on-axis propa-

gation through the NRI-TL free-space metamaterial superlens at 2.40GHz. 133

C.1 Properties and dimensions of ATM standard-gain X-band horn antennas

depicted schematically in Fig. C.1 ...... 187

xi C.2 Dimensions of measurement setup depicted schematically in Fig. C.2 . . 187

xii List of Figures

2.1 (a) Array of thin metallic wires resembling a microwave plasma with ² <

0 for electric ﬁelds polarized as shown; (b) split-ring resonator (SRR)

particle yielding µ < 0 for magnetic ﬁelds polarized as shown. Part (b)

after Ref. 32. Reprinted with permission of John Wiley & Sons, Inc.,

2.2 Orientation of ﬁeld quantities E, H, Poynting vector S, and wavevector k

in RH media and LH media. After Ref. 32. Reprinted with permission of

John Wiley & Sons, Inc., copyright °c 2005...... 12

2.3 Negative refraction at an interface between a RH medium and a LH

medium. The rays indicate the direction of the wavevector, or phase lag. 13

2.4 Focusing of the rays of a cylindrical or spherical excitation by a LH slab

embedded in a RH host medium. The arrows illustrate the focusing eﬀect

of propagating (ray or wavevector) components, and the solid curve depicts

the restoration of the amplitude of evanescent components...... 13

2.5 Depiction of Wire/SRR metamaterial. After Ref. 32. Reprinted with

permission of John Wiley & Sons, Inc., copyright °c 2005...... 16

2.6 Spherical resonances in dielectric particles corresponding to negative per-

meability (TE011) and negative permittivity (TM011). Solid lines: magnetic ﬁelds; dashed lines: electric ﬁelds...... 17

xiii 2.7 (a) NRI-TL unit cell consisting of a host TL network loaded in a dual

conﬁguration using lumped inductors and capacitors. (b) Planar NRI-TL

lens interfaced with a RH grid containing a voltage source located by the

arrow (left) as well as the detecting probe (right). The inset depicts the

realized unit cell in microstrip technology. Reprinted with permission from

Ref. 18, copyright °c 2003 Optical Society of America...... 19

2.8 (a) Depiction of the two-plasmonic-wire system and the ﬁeld and result-

ing currents producing a magnetic response. After Ref. 55. Reprinted

with permission of John Wiley & Sons, Inc., copyright °c 2005; (b) In-

terfaces of plasmonic (² < 0) layers and conventional dielectrics (² > 0)

supporting backward surface waves; (c) interpretation of positive- and

negative-permittivity nanoparticles as nanocapacitors and nanoinductors,

respectively...... 21

2.9 (a) CPW-based NRI-TL phase shifters (top to bottom: one-stage, two-

stage, four-stage, eight-stage), and conventional one-wavelength CPW TL.

Reprinted with permission from Ref. 71, copyright °c 2003 IEEE; (b) fully

printed NRI-TL coupled-line coupler. Reprinted with permission from

Ref. 72, copyright °c 2006 IEEE...... 25

2.10 (a) Leaky-wave radiation region of a NRI metamaterial; (b) mechanism

of radiation by way of phase matching at the interface between the NRI

medium and air...... 27

3.1 (a) 2D TL unit cell and (b) dispersion relation for propagation along a

particular direction in the x-z plane, describing a medium with µ(ω) = µ0

and ²(ω) = ²r²0. The arrows suggest that the group and phase velocities are determined in the direction of increasing frequency. After Ref. 32.

Reprinted with permission of John Wiley & Sons, Inc., copyright °c 2005. 31

xiv 3.2 (a) 2-D dual TL unit cell and (b) dispersion relation for propagation along

a particular direction in the x-z plane, describing a medium with simul-

taneously negative, dispersive parameters µ = −|µ(ω)| and ² = −|²(ω)|.

The arrows suggest that the group and phase velocities are determined

in the direction of increasing frequency. After Ref. 32. Reprinted with

permission of John Wiley & Sons, Inc., copyright °c 2005...... 34

3.3 Polarization due to the application of an electric ﬁeld Ea, modeled as an equivalent electric dipole consisting of charges ±q separated by a distance

lav...... 36

3.4 Determination of the permittivity of a dielectric inserted into a parallel-

plate capacitor by measuring the change in capacitance. The source main-

tains a voltage Va across the plates...... 37

3.5 Magnetization due to the application of a magnetic ﬂux Ba, modeled as an equivalent magnetic dipole consisting of a current I ﬂowing around a

loop of area ds...... 37

3.6 Determination of the permeability of a magnetic material inserted into a

solenoid by measuring the change in inductance. The source maintains a

current Ia in the windings...... 38

3.7 Free charges in a plasma (motion of mobile negative charges generally

represented as opposite motion of both negative and positive charges) drift

so as to lead the applied ﬁeld and enhance the ﬁeld between the plates,

requiring the source to absorb charge in order to maintain a voltage Va. 41

3.8 (a) Generalized magnetic medium for which the elemental particle con-

sists of a loop whose total series impedance is lumped into ZL; (b) Field

orientations for ZL > 0 (inductive loop); (c) Field orientations for ZL < 0 (capacitive loop)...... 42

xv 3.9 TL equivalent lumped circuit representing a medium achieving a negative

permeability through an L–C resonance mechanism...... 47

3.10 (a) 1D array of mutually coupled capacitively loaded inductive loops; (b)

Unit cell for a continuous LH TL, formed by connection of loops in (a). . 50

3.11 Unit cell modeling coupling from free space (represented as a π-network

with impedance Zh and admittance Yh) to a TL metamaterial represented as an array of tightly coupled rings...... 52

3.12 Dispersion diagram showing the solutions to (3.38) when the host TL and

coupled-loop array are decoupled (LMh = 0) but the loops are coupled to

their nearest neighbours, for −L/2LML = p = 1.5 (dark curves – propagation constant; light curves – attenuation constant)...... 55

3.13 Dispersion diagram showing the coupled dispersion curves of the host TL

and coupled-loop array when LMh = 0.125Lh, for −L/2LML = p = 1.5 (dark curves – propagation constant; light curves – attenuation constant). 56

3.14 Dispersion diagram showing the coupled dispersion curves of the host TL

and loop array (LMh = 0.5Lh) when the loop-to-loop mutual inductance

LML = 0. (darkly shaded curves – propagation constant; lightly shaded curves – attenuation constant)...... 57

4.1 (a) 2D series TL unit cell. (b) 2D shunt TL unit cell. Reprinted with

permission from Ref. 60, copyright °c 2006 Optical Society of America. . 61

4.2 Creating a volumetric medium by layering 2D planes. Reprinted with

permission from Ref. 60, copyright °c 2006 Optical Society of America. . 61

xvi 4.3 Array of 2D series TL unit cells with generalized lumped loading. The

lightly shaded region views the unit cell as the series interconnection of four

two-wire lines, and the darkly shaded region views the unit cell as a loaded

ring. The magnetic ﬁeld of the impinging plane wave is normal to the plane

of the page. Reprinted with permission from Ref. 60, copyright °c 2006

Optical Society of America...... 62

4.4 Deﬁnition of the ABCD transfer matrix for a two-port network. Reprinted

with permission from Ref. 60, copyright °c 2006 Optical Society of America. 64

4.5 Periodic analysis of the generalized 2D series TL unit cell. Reprinted with

permission from Ref. 60, copyright °c 2006 Optical Society of America. . 65

4.6 Speciﬁcation of the constituents of the branches of the series TL unit

cell depicted in Fig. 4.5. Reprinted with permission from Ref. 60, copy-

right °c 2006 Optical Society of America...... 66

4.7 Brillouin zone boundary for 2D rectangular lattice indicating high sym-

metry points. Reprinted with permission from Ref. 60, copyright °c 2006

Optical Society of America...... 67

4.8 Representation of transverse-resonance conditions (4.11) describing the

edges of the Γ-point stopbands (s.c. ≡ short circuit; o.c. ≡ open circuit). 70

4.9 Series TL node array loaded in a dual conﬁguration. The arrows indicate

the current directions. Reprinted with permission from Ref. 60, copy-

right °c 2006 Optical Society of America...... 75

4.10 Axial propagation in the 2D Series NRI-TL node array (s.c. ≡ short cir-

cuit). Reprinted with permission from Ref. 60, copyright °c 2006 Optical

Society of America...... 76

xvii 4.11 Representative series NRI-TL dispersion relations for axial propagation

(µp = µ0, ²p = 3²0, g = 0.325, C0 = 1 pF, d = 5 mm): (a) L0 = 3nH;

Impedance-mismatched (open-stopband) case. (b) L0 = 10nH; Impedance- matched (closed-stopband) case. Reprinted with permission from Ref. 60,

4.12 Series TL unit cells: (a) Unloaded. (b) Shunt inductors. (c) Series capac-

itors. (d) Composite series NRI-TL unit cell. Reprinted with permission

from Ref. 60, copyright °c 2006 Optical Society of America...... 81

5.1 (a) A uniplanar array (right) of series-connected NRI-TL sections (left),

as viewed from the top. The arrows indicate current directions. (b) A

volumetric NRI-TL metamaterial constructed by layering the planar arrays

in (a). Reprinted with permission from Ref. 63, copyright °c 2007 IEEE. 84

5.2 Volumetric NRI-TL unit cell for an inﬁnite array excited by TEz-polarized plane wave propagating in the y-direction. Reprinted with permission from

Ref. 60, copyright °c 2006 Optical Society of America...... 86

5.3 Interaction between a normally incident TE-polarized plane wave and the

volumetric layered NRI-TL medium: (a) schematic representation of the

unit cell; (b) equivalent-circuit model of the unit cell. Reprinted with

permission from Ref. 60, copyright °c 2006 Optical Society of America. . 88

5.4 Equivalent-circuit models corresponding to the series unit cell topologies of

Figs. 4.12(a)–4.12(d): (a) Unloaded. (b) Discrete lumped shunt inductors.

cell...... 91

xviii 5.5 Axial dispersion relations corresponding to the series unit cell topologies

of Figs. 4.12(a)–4.12(d) and values reported in Table 5.1, obtained using

the HFSS ﬁnite-element solver (dots) and equivalent circuit model (solid

curves): (a) Unloaded. (b) Discrete lumped shunt inductors. (c) Discrete

lumped series capacitors. (d) Composite series NRI-TL unit cell. Adapted

and reprinted with permission from Ref. 60, copyright °c 2006 Optical

Society of America...... 92

5.6 Axial dispersion relation for volumetric layered NRI-TL structure based on

the design in Table 5.1, but with L0 increased from 5.6nH to approximately 10nH to meet the impedance-matched condition of (5.5); obtained using

the HFSS ﬁnite-element solver (dots) and equivalent circuit model (solid

curve). Reprinted with permission from Ref. 60, copyright °c 2006 Optical

Society of America...... 99

5.7 Fully printed composite series NRI-TL unit cell employing interdigitated

capacitors and strip inductors. Reprinted with permission from Ref. 60,

5.8 Dispersion relations corresponding to the series unit cell topologies of

Figs. 4.12(a)–4.12(d) employing printed elements instead of discrete ele-

ments, obtained using the HFSS ﬁnite-element solver (dots): (a) Unloaded.

(b) Printed lumped shunt inductors. (c) Printed lumped series capacitors.

(d) Composite fully printed series NRI-TL unit cell. Reprinted with per-

mission from Ref. 60, copyright °c 2006 Optical Society of America. . . . 101

5.9 2D reduced Brillouin zone for the NRI band obtained using HFSS. The

dispersion contours indicate that isotropy is achieved in the approach to

the Γ-point (the homogeneous limit). Reprinted with permission from

Ref. 60, copyright °c 2006 Optical Society of America...... 103

xix 5.10 (a) Volumetric layered NRI-TL slab of thickness three cells. (b) Repre-

sentation of inﬁnite slab for illumination by a TE plane wave at normal

incidence using electric and magnetic walls. Reprinted with permission

from Ref. 60, copyright °c 2006 Optical Society of America...... 104

5.11 Printed lumped element design: (a) HFSS transmission phase for slab ar-

rangement of Fig. 5.10(a) (black dots) compared with dispersion of inﬁnite

structure (gray dots); (b) HFSS transmission (S21—black line) and reﬂec-

tion (S11—gray line) magnitudes. Reprinted with permission from Ref. 60, copyright °c 2006 Optical Society of America...... 106

5.12 Discrete lumped element design matched to free space: (a) HFSS trans-

mission phase for slab arrangement of Fig. 5.10(a) (black dots) compared

with dispersion of inﬁnite structure (gray dots) and equivalent circuit mode

(dashed lines); (b) HFSS transmission (S21—black line) and reﬂection

(S11—gray line) magnitudes. Reprinted with permission from Ref. 60, copyright °c 2006 Optical Society of America...... 107

5.13 Possible optical implementations of the series NRI-TL array using plas-

monic nanoparticles for (a) square unit cells, and (b) triangular unit cells.

Reprinted with permission from Ref. 60, copyright °c 2006 Optical Society

of America...... 109

6.1 Fully printed NRI-TL ring employing interdigitated capacitors and mean-

dered inductors. Reprinted with permission from Ref. 63, copyright °c 2007

IEEE...... 114

6.2 (a) Unit cell for volumetric NRI-TL metamaterial consisting of a metal-

lic ring embedded between a 60-mil-thick dielectric (²r = 3) region and

a 120-mil-thick air (²r = 1) region. (b) Dispersion characteristics for axial propagation of a horizontally polarized mode obtained using HFSS.

Reprinted with permission from Ref. 63, copyright °c 2007 IEEE. . . . . 115

xx 6.3 Simulated transmission and reﬂection magnitude and phase for a volu-

metric NRI-TL metamaterial slab of ﬁve-unit-cell thickness obtained us-

ing HFSS (S11–dashed curve, S21–solid curve). Reprinted with permission from Ref. 63, copyright °c 2007 IEEE...... 117

6.4 Magnetic ﬁeld intensities (logarithmic shading and contours) for a single

row of ﬁve unit cells inside the volumetric NRI-TL metamaterial when

illuminated by a normally incident plane wave at 11.36GHz. The ﬁelds

are strongly conﬁned to the layers containing the metallic features, as

highlighted in the inset. Reprinted with permission from Ref. 63, copy-

right °c 2007 IEEE...... 118

6.5 Negative refraction in the volumetric NRI-TL metamaterial (vertical mag-

netic ﬁelds shown) when illuminated by a 15-degree incident plane wave at

10.7GHz, 10.9GHz, 11.1GHz, and 11.3GHz, obtained using HFSS. Arrows

indicate approximate direction of incident and refracted waves...... 119

6.6 Time evolution of backward waves inside the volumetric NRI-TL meta-

material (vertical magnetic ﬁelds shown) at successive source phases, ϕ,

obtained using HFSS. Dotted lines and circles track the progression of

particular phasefronts in both free space and the metamaterial...... 120

6.7 Fabricated volumetric NRI-TL metamaterial (w×h×t = 104.5mm×105.2mm×27.5mm).

The inset shows the printed features comprising each layer. Reprinted with

permission from Ref. 63, copyright °c 2007 IEEE...... 121

6.8 Free-space X-band measurement system consisting of an Agilent network

analyzer, standard gain horn antennas, and Rexolite dielectric lenses.

Reprinted with permission from Ref. 63, copyright °c 2007 IEEE. . . . . 122

xxi 6.9 Measured transmission and reﬂection magnitude and phase for a volumet-

ric NRI-TL metamaterial slab of ﬁve-unit-cell thickness (S11–dashed black

curve, S21–solid black curve). The circles (S11–open gray circles, S21– solid gray circles) represent full-wave simulation data accounting for over-

etching of the printed features. Reprinted with permission from Ref. 63,

6.10 Free-space focusing arrangement consisting of a shielded-loop antenna illu-

minating the front face of the volumetric NRI-TL slab lens and an identical

receiving antenna attached to an xyz-translator (not shown), scanning the

ﬁelds at the rear side of the slab. The antennas are connected to an Ag-

ilent E8364B network analyzer (not shown). Reprinted with permission

from Ref. 63, copyright °c 2007 IEEE...... 125

6.11 Measured ﬁeld magnitude and phase data at the rear side of the volu-

metric NRI-TL slab lens: (a) schematic illustration of focusing and the

measured region (marquee); measured ﬁelds (b) in the PRI frequency re-

gion at 7.00GHz, and (c) in the NRI frequency region at 11.36GHz. The

black curve in (c) is a −3dB contour (normalized to the ﬁeld values at the

focal plane) indicating the width of the beam. Reprinted with permission

from Ref. 63, copyright °c 2007 IEEE...... 126

7.1 (a) NRI-TL free-space metamaterial superlens unit cell. Reprinted with

permission from Ref. 64, copyright °c 2008 American Institute of Physics;

(b) relevant dimensions of CPS TL host medium. Reprinted with permis-

sion from Ref. 119, copyright °c 2009 IEEE...... 132

xxii 7.2 Eﬀective material parameters and transmission/reﬂection magnitudes of

a ﬁve-unit-cell-thick volumetric NRI-TL metamaterial (full-wave simula-

tions – circles, equivalent-circuit model – curves): (a) phase shift per unit

cell βd; (b) relative eﬀective material parameters <{µeff (ω)}/µ0 (dashed

blue curve), <{²eff (ω)}/²0 (dotted green curve); (c) transmission magni-

tude |S21| (solid red curve and circles); reﬂection magnitude |S11| (dashed blue curve and circles). Parts (a) and (b) reprinted with permission from

Ref. 64, copyright °c 2008 American Institute of Physics...... 135

7.3 Equifrequency contour at 2.409GHz for the volumetric NRI-TL meta-

material describing propagation in the layer planes (solid blue curve –

obtained through full-wave FEM simulations) and for propagation in free

space (red squares). Their near-perfect coincidence suggests that the vol-

umetric NRI-TL metamaterial exhibits a nearly isotropic refractive index

of −1 at this frequency. Reprinted with permission from Ref. 119, copy-

right °c 2009 IEEE...... 136

7.4 Full-wave FEM simulation data for volumetric NRI-TL metamaterial slabs

of thickness 3d, 4d, and 5d (d = 7.14mm) represented by diamond-,

square-, and circle-markers, respectively. (a) Phase shift per unit cell βd

(periodic simulation of a single unit cell in an inﬁnitely large array – solid

red curve); inset: relative eﬀective material parameters near design fre-

quency of 2.40GHz (<{µeff (ω)}/µ0 – solid cyan markers, <{²eff (ω)}/²0 –

open green markers); (b) transmission and reﬂection magnitudes (|S21| –

solid red markers, |S11| – open blue markers). Reprinted with permission from Ref. 124, copyright °c 2009 IEEE...... 138

xxiii 7.5 Negative refraction of a 2.39-GHz plane wave incident at various angles

(ψ) on a ﬁve-cell-thick volumetric NRI-TL metamaterial embedded in air

and possessing µeff (ω) = −µ0 and ²eff (ω) = −²0 at 2.39GHz. The arrows indicate the propagation directions of the incident, refracted, and trans-

mitted waves. Reprinted with permission from Ref. 124, copyright °c 2009

IEEE...... 140

7.6 Theoretical resolution enhancement of the volumetric NRI-TL metamaterial

versus frequency. Reprinted with permission from Ref. 124, copyright °c 2009

IEEE...... 141

7.7 (a) Photograph of fabricated NRI-TL lens with inset showing lumped load-

ing of the host CPS TL structure using discrete surface-mount inductors

and capacitors. (w × h × t = 149.9mm×149.5mm×35.7mm, d = 7.14mm);

(b) Measurement arrangement – the marquee indicates the region in which

the ﬁeld data presented in Figs. 7.8 and 7.9 are measured. Reprinted with

permission from Ref. 119, copyright °c 2009 IEEE...... 142

7.8 Raw measured magnitude and phase data for excitation at 2.40GHz with

a single-loop source when (a) the lens is absent and (b) the lens is present.

The black curves trace the half-power contours referenced to the maximum

ﬁeld magnitude at the focal plane (dashed line); (c) A comparison of the

normalized magnitude proﬁles (linear scale) at the focal plane when the

lens is absent (blue circles) and when the lens is present (red squares),

along with the ﬁelds at a distance of t/2 from the source when the lens is

absent (solid black curve). The dotted horizontal line indicates the half-

power levels and shows that the NRI-TL superlens is able to produce an

image of the source with a half-power beam width of 0.18λ0 (minimum

peak-to-null width of 0.16λ0). Reprinted with permission from Ref. 119, copyright °c 2009 IEEE...... 146

xxiv 7.9 Raw measured magnitude and phase data for excitation at 2.40GHz with

two loop sources separated by λ0/3 when (a) the lens is absent and (b) the lens is present. The black curves trace the half-power contours refer-

enced to the maximum ﬁeld magnitude at the focal plane (dashed line);

c) A comparison of the normalized magnitude proﬁles (linear scale) at the

focal plane when the lens is absent (blue circles) and when the lens is

present (red squares), along with the ﬁelds at a distance of t/2 from the

source when the lens is absent (solid black curve). The dotted horizontal

line indicates the half-power levels and shows that the NRI-TL superlens

comfortably diﬀerentiates the sources. Reprinted with permission from

Ref. 119, copyright °c 2009 IEEE...... 149

7.10 Measured normalized transverse ﬁeld proﬁles at the focal plane versus

frequency from 2.00GHz to 2.50GHz (black curve represents the continu-

ous half-power contour). Superresolution evident at 2.40GHz (right inset:

Veselago-Pendry superlensing condition) and 2.08GHz (left inset: perme-

ability resonance). Adapted and reprinted with permission from Ref. 64,

7.11 Measured transverse ﬁeld proﬁles at 2.40GHz (red solid curve) and 2.08GHz

(blue dashed curve), normalized to the maximum ﬁeld value of the former.

Reprinted with permission from Ref. 64, copyright °c 2008 American In-

stitute of Physics...... 152

xxv 8.1 Evolution of the 3D NRI-TL metamaterial: (a) SCN node modeling con-

ventional dielectric media (series TLM node emphasized on one plane); (b)

Unit cell obtained by displacing the SCN by one-half of a period on each

axis (front face emphasized); (c) 3D NRI-TL topology obtained by revers-

ing positions of inductors and capacitors (front face emphasized); (d) Phys-

ical realization of the 3D NRI-TL SCN unit cell in (c) for simulation using

Ansoft HFSS (inset: Brillouin zone contours with high-symmetry points

labelled). Reprinted with permission from Ref. 140, copyright °c 2008

American Institute of Physics...... 157

8.2 3D NRI-TL SCN dispersion characteristics obtained using Ansoft’s HFSS.

The labels on the horizontal axis correspond to the Brillouin-zone high-

symmetry points shown in the inset of Fig. 8.1(d). The NRI passband (be-

low 3GHz) is described by the existence of two modes (square- and circle-

markers) whose dispersions are nearly identical in the eﬀective-medium

regime. Several higher-order modes exist above 3GHz. Reprinted with

permission from Ref. 140, copyright °c 2008 American Institute of Physics. 158

8.3 Dispersion characteristics along the principal propagation directions Γ−X,

Γ − M, and Γ − R superimposed to show that isotropy and polarization-

independence are achieved in the eﬀective-medium regime. In the NRI

passband (below 3GHz), the square- and circle-markers indicate the two

predominant modes. Several higher-order modes exist above 3GHz. Reprinted

with permission from Ref. 140, copyright °c 2008 American Institute of

Physics...... 159

8.4 Transmission and reﬂection magnitudes corresponding to on-axis (Γ − X)

propagation for a ﬁve-unit-cell thick 3D NRI-TL metamaterial slab embed-

ded in vacuum, obtained using Ansoft’s HFSS. Reprinted with permission

from Ref. 140, copyright °c 2008 American Institute of Physics...... 160

xxvi 8.5 Arrangement of plasmonic nanoparticles corresponding to a single unit cell

of the SCN-based 3D NRI-TL metamaterial...... 162

A.1 Interference of the tangential components of the wavevectors of two coher-

ent plane waves incident at angles ±θ...... 172

A.2 Illumination of a lens of ﬁnite aperture size by a point source...... 173

B.1 Volume of space representing region in which ﬁelds may be considered

quasi-static. After Ref. 32. Reprinted with permission of John Wiley &

Sons, Inc., copyright °c 2005...... 180

B.2 Unit cell for a distributed transmission-line network model describing 2D

propagation in a homogeneous medium. Reprinted with permission from

Ref. 17, copyright °c 2002 IEEE...... 182

C.1 Schematic of ATM standard-gain X-band horn antennas (see Table C.1). 186

C.2 Schematic depicting the transmit side of the free-space X-band measure-

ment setup (see Table C.2). The receive side is symmetrically arranged

about the sample...... 188

C.3 Hyperbolic lens geometry; focal length f, refractive index n = 1.59 (Rex-

olite)...... 190

D.1 Ideal balanced loop antenna with circulating current IL and dipolar cur-

rents IV ...... 193

D.2 (a) Loop antenna with coaxial feed line supporting unbalanced current IV ; (b) Current analysis at terminals A and B...... 194

D.3 (a) Shielded-loop antenna; (b) Current analysis at terminals A and B. . . 195

D.4 (a) Unbalanced-loop antenna simulation model; (b) Circulating currents

on loop at 2.4GHz; (c) Current magnitudes at 2.4GHz; (d) Radiation

pattern at 2.4GHz...... 197

xxvii D.5 (a) Shielded-loop antenna simulation model; (b) Circulating currents on

loop at 2.4GHz; (c) Current magnitudes at 2.4GHz; (d) Radiation pattern

at 2.4GHz...... 198

D.6 (a) Fabricated shielded-loop antenna designed for operation at 2.4GHz;

(b) measured E- and H-plane patterns of a shielded-loop antenna designed

for operation at 10GHz...... 199

xxviii Chapter 1

Introduction

Focusing of electromagnetic waves using conventional lenses relies on the collection and interference of propagating waves, but discounts the evanescent waves that decay rapidly from the source. Since these evanescent waves contain the finest spatial details of the source, the image suffers a loss of resolution and is referred to as ‘diffraction-limited’.

Through the theoretical work of V. Veselago [1] and J. B. Pendry [2], it is known that

flat negative-refractive-index (NRI) lenses designed to meet certain strict conditions are able to focus the propagating-wave components of a source without geometric aberration while simultaneously restoring the amplitude of its evanescent-wave components, such that the focal plane contains an exact real image of the source, down to its finest spatial features. As a result, such a lens, appropriately termed a ‘Veselago-Pendry superlens,’ is able to overcome the constraints of the classical diffraction limit, which restricts focusing with conventional lenses to a resolution on the order of half the wavelength of illumination

(see Appendix A).

Although such a lens has been realized using NRI transmission-line (TL) metamaterials (see Chapters 2 and 3) for embedded sources in planar and three-dimensional form [3, 4] and inside waveguide environments [5, 6], it has, so far, eluded practical realization in a form capable of interacting with sources in free space, the form in which it was

1 Chapter 1. Introduction 2

first envisioned. This is largely due to the fact that a true free-space Veselago-Pendry superlens has a number of stringent design requirements: first, the lens must possess isotropic µ = −µ0 and ² = −²0 for the polarization(s) concerned (where µ0 and ²0 are the free-space permeability and permittivity, respectively) in order to be impedance-matched to free space and simultaneously possess a refractive index n = −1, which also renders it aberration-free. Since these materials are necessarily dispersive, achieving these values with adequate precision requires a means of tightly controlling the metamaterial’s frequency response. Second, the lens must be extremely low loss and be adequately thin, since both loss and electrical thickness serve to quickly degrade the resonant evanescent enhancement contributing to subdiffraction imaging [7,8]. A third condition follows from the previous: the unit cells comprising the lens must themselves be deeply subwavelength in size in order to minimize spatial anisotropy and to ensure that the structure possesses the desired bulk response. Last, the transverse dimensions of the lens must be large enough that the lens can be illuminated by a source in free space.

1.1 Motivation

Recent works in the literature have presented experimental demonstrations of subdiﬀrac- tion focusing and imaging in free space using a variety of metamaterial superlenses; however, it is unlikely that any of these metamaterials can be regarded as true Veselago-

Pendry superlenses, in that each contravenes one or more of the above stringent conditions. For example, the plasmonic silver ﬁlm [9], the magneto-inductive lens [10], and the swiss-roll structure [11] recover ﬁne spatial features through evanescent enhancement, but they do not possess a NRI and so cannot focus the propagating wave numbers; unfortunately, this means that sources must be placed very near to, if not directly against, the lens faces, which imposes an extremely short working distance. Although the Veselago-

Pendry superlens also requires that the source be placed in the near ﬁeld of the lens, Chapter 1. Introduction 3

these distances are on the order of λ0/8 (where λ0 is the free-space wavelength), which may be appreciable at RF/microwave frequencies. Subdiffraction imaging phenomena have been successfully extended to the far field using magnifying ‘hyperlenses’ [12–14], but this class of superlenses relies explicitly on anisotropy and sources are, once again, typically applied directly to the hyperlens face; as a result, the working distance on the source side remains limited. Subdiffraction imaging using printed split-ring-resonator-

(SRR-) based structures has been reported in the way of transversely and longitudinally conﬁned subwavelength focal ‘spots’ [15, 16] in spite of their high losses and/or large electrical thickness; however, calculations based on Ref. 7 suggest that the reported loss, lens thickness, and observed resolution ability are inconsistent, if the structures are to be regarded as true Veselago-Pendry superlenses. Indeed, in attempting to explain these inconsistencies, the authors of Refs. 15 and 16 have speculated that anisotropy, rather than subdiﬀraction imaging by way of the restoration of evanescent waves, may be responsible for these phenomena [15].

The ﬁrst ever experimental demonstrations of NRI focusing [17,18] and subdiﬀraction imaging [3] were achieved using NRI-TL metamaterials. However, these phenomena were demonstrated in a microstrip TL environment using embedded microwave sources. It is the goal of this work to extend NRI-TL concepts to the realization of a true Veselago-

Pendry superlens capable of interacting with and manipulating ﬁelds in free space. The

NRI-TL topology inherently supports broadband, low-loss propagation, and its reliance on the periodic lumped loading of a host TL medium can render the unit-cell size deeply subwavelength, facilitating the realization of an adequately thin, low-loss metamaterial superlens. Furthermore, periodically loaded TL networks are easily characterized and may be rapidly fabricated using widely available components and lithographic techniques.

As ﬁrst steps towards the ultimate realization of a free-space Veselago-Pendry superlens, a select few 3D physical realizations of isotropic NRI-TL metamaterials based on the topology of Kron [19] have been proposed for simulation in Refs. 20, 21, and 22. Un- Chapter 1. Introduction 4 fortunately, these 3D topologies are complex, since the pursuit of complete 3D isotropy very often requires 3D symmetry. This requirement severely complicates fabrication and may be responsible for the fact that a true 3D-isotropic, free-space Veselago-Pendry superlens has not yet been realized. However, if it is possible to restrict the polarization and also limit the directions of propagation, practical stratiﬁed structures that can be readily fabricated with prevalent lithographic techniques may become available. Furthermore, such multilayer metamaterials may be based on existing planar NRI-TL metamaterials that are already well understood. Accordingly, the early part of this work examines the possibility of realizing the free-space Veselago-Pendry superlens using a multilayer, or ‘volumetric,’ topology constructed by layering planes of 2D NRI-TL metamaterials.

Such a network, although not 3D-isotropic and also polarization-speciﬁc, would appear isotropic to 2D free-space sources (e.g. line sources) and may be designated an eﬀective

NRI medium for propagation along its constituent NRI-TL planes. A microwave superlens designed in this fashion can be particularly useful for illumination and discrimination of closely spaced buried objects over practical distances by way of back-scattering, for example, in tumour or landmine detection, or for targeted irradiation over electrically small regions in tomography or hyperthermia applications. The latter part of this work describes the application of these ideas to the design of a volumetric NRI-TL metamaterial that experimentally demonstrates, for the ﬁrst time, subdiﬀraction focusing and imaging in free space consistent with the principles of the Veselago-Pendry superlens.

1.2 Objectives

The objectives of the present work are ﬁvefold: (1) the design of a volumetric NRI-

TL metamaterial topology using a multilayer approach, that is capable of interacting with sources in free space; (2) the development of a simple theory to enable accurate design of its effective-medium properties in concert with full-wave simulations; (3) the Chapter 1. Introduction 5 experimental demonstration of free-space focusing using a fully printed metamaterial; (4) the experimental demonstration of free-space subdiffraction focusing and imaging using a multilayer Veselago-Pendry superlens employing discrete lumped elements; and (5) the extension of NRI-TL techniques to alternative 3D-isotropic implementations in the microwave, terahertz (THz) and optical frequency regimes, as well as the investigation of alternate mechanisms of subdiffraction imaging.

1.3 Outline

Chapter 2 provides a background to the rapidly growing ﬁeld of metamaterials research, and begins with a historical perspective that follows metamaterials from their ancestry in the study of artiﬁcial dielectrics to the seminal events that revealed the concepts of negative refraction and the Veselago-Pendry superlens. This is followed by an attempt to capture, albeit not comprehensively, some of the most interesting directions in the literature concerning the varied implementations of metamaterials and potential applications proposed in the electromagnetics and photonics research communities. Chapter 3 is dedicated to a theoretical treatment of TL-based metamaterials from two perspectives: the

first is a TL-network perspective, which establishes the connection between TEM propagation in TL networks of different topologies and plane-wave propagation in equivalent homogeneous dielectrics, including those with negative material parameters; the second is an electrodynamical perspective, which develops arguments for negative permittivity and negative permeability based, respectively, on the behaviour of a parallel-plate capacitor filled with an isotropic plasma, and a solenoid filled with reactively loaded metallic loops. Thereafter, the propagation characteristics of NRI-TL metamaterials are related to those of SRR/wire metamaterials, particularly in the context of exciting an array of tightly coupled resonators from free space, the case which describes propagation in the proposed volumetric NRI-TL metamaterial. Chapter 4 describes a fully unipla- Chapter 1. Introduction 6 nar NRI-TL metamaterial based on a 2D network of reactively loaded coplanar-strip

(CPS) TLs, which serves as the constitutive layer in the proposed volumetric NRI-TL metamaterial. A complete 2D periodic analysis reveals the dispersion properties of the single NRI-TL layer and effective-medium arguments are applied to extract the effective negative permittivity and permeability giving rise to backward-wave propagation and the consequent NRI. Chapter 5 discusses the construction of the volumetric structure from the NRI-TL layer and develops an intuitive two-port equivalent-circuit model that captures the essential dispersion characteristics of the volumetric NRI-TL metamaterial in the effective-medium limit. The model is validated using full-wave simulations illustrating the dispersion and transmission/reflection characteristics of representative volumetric NRI-TL designs as well as the extraction of a complex effective permittivity and permeability. This is followed by a qualitative discussion on how such topologies may be realized at THz and optical frequencies, the latter using plasmonic nanoparticles.

Chapter 6 presents simulations and an experimental demonstration of diﬀraction-limited free-space focusing using a volumetric NRI-TL metamaterial lens realized using fully printed loading elements (interdigitated capacitors and meandered inductors, and without any vias) and illuminated by a free-space source. Chapter 7 presents simulations and an experimental demonstration of free-space imaging with a resolution over three times better than the diﬀraction limit at microwave frequencies using a NRI-TL metamaterial

Veselago-Pendry superlens employing discrete chip inductors and capacitors with high quality factors. Lastly, Chapter 8 proposes a fully 3D isotropic extension of the volumetric NRI-TL topology and presents corroborating full-wave simulations along with a possible optical implementation. Chapter 2

Background

2.1 Metamaterials: Transcendent Artiﬁcial Dielectrics

In the 1950s and 1960s, there was a great impetus towards the realization of ‘artiﬁcial dielectrics’ (see, for example, Refs. 23, 24, and 25, and the list of works referenced in

Ref. 26). These works sought to synthesize effective materials with electromagnetic scatterers, which could be designed to produce a particular macroscopic response at long wavelengths. Recently, artificial dielectrics have experienced a resurgence of interest under the guise of metamaterials. Broadly speaking, metamaterials are artificial dielectrics with transcendent electromagnetic properties. The term ‘artificial’ refers to the fact that the electromagnetic response of these materials is dominated by scattering from periodically or amorphously placed inclusions (e.g. metallic or dielectric spheres, wires, loops) in a natural host medium, and the term ‘transcendent’ suggests that their electromagnetic responses are not found, or not readily found (e.g. at particular frequencies of interest or without other significant drawbacks), in nature. Indeed, this property is the source of the prefix meta, Greek for ‘beyond’ or ‘after’. Most researchers consider metamaterials to be artificial materials that possess a periodicity much smaller than the wavelength of the impinging electromagnetic wave. In this case, the periodic inclusions can be regarded as

7 Chapter 2. Background 8

‘artificial molecules’ that scatter back the impinging electromagnetic fields in a prescribed manner, resulting in a macroscopic response that can be described by ‘effective-medium’ theories and characterized by means of effective material parameters such as permittivity, permeability, and a refractive index. However, the term ‘metamaterial’ has also been used to describe other periodic structures such as electromagnetic bandgap structures or photonic crystals, in which the period is on the order of the wavelength of the impinging electromagnetic wave. The electromagnetic response of such structures is dominated by Bragg-type scattering and/or involves higher-order spatial harmonics (Floquet-Bloch modes). Whereas the prefix ‘meta’ may apply to the intriguing responses of such structures, the large electrical size of their inclusions/perturbations and lattice constants do not readily satisfy the familiarly held notions of a ‘material’, and so the discussions in this work shall be restricted to the ‘effective-medium’ definition of metamaterials.

The artificial dielectrics of the mid-twentieth century sought to reproduce the electromagnetic responses of conventional materials available in nature, and research into artificial dielectrics resulted in a set of techniques that enabled one to design the scatterers – that is, to access the constitutive particles themselves – in order to produce the desired response. This unprecedented freedom in tailoring the effective electromagnetic response of an artificial material prompts the application of artificial-dielectric techniques to the realization of metamaterial properties not available, or not readily available, in nature. All such properties can ultimately be grouped into the macroscopic responses of materials to electric and magnetic fields, which are captured by their permittivity and permeability. Accordingly, it is no surprise that metamaterials research continues to focus on tailoring the effective permittivity and permeability. Although most current metamaterials research aims to synthesize an unusual electromagnetic response in simple media (i.e., linear, homogeneous, and isotropic media), there has also been much work into realizing other properties such as chirality, anisotropy, bianisotropy, and nonlinearity, which, although they are prevalent in nature, can also be regarded as transcendent in this Chapter 2. Background 9 context because of the difficulty of obtaining such responses at microwave frequencies.

2.1.1 Tailoring Permittivity

Early artificial dielectrics composed of arrays of metallic or dielectric particles were suc- cessful in producing a significant electric response that could be captured by an effective relative permittivity, but this work did not explicitly seek to synthesize unusual values of permittivity. However, one of the more exotic (but still conventional) materials examined in the artificial-dielectric community were plasmas. Plasmas can possess an isotropic negative permittivity at frequencies below their plasma frequency. The challenge was to realize plasma-like properties at microwave frequencies using metallic inclusions alone.

In 1954, R. N. Bracewell proposed that propagation in an isotropic plasma could be represented by a transmission-line (TL) model in which an inductor was placed in parallel with the shunt capacitance (the latter representing the free-space permittivity), such that their resonant interaction yields a capacitance at high frequencies (representing a positive permittivity), and an inductance at low frequencies (representing a negative permittivity) [27]. Soon afterwards, artiﬁcial microwave plasmas were realized using arrays of thin, continuous metallic wires (as shown in Fig. 2.1(a) – see, for example, Ref. 28), which can, in the context of Bracewell’s work, be seen to inductively load free space.

Indeed, the importance of the thin-wire medium lies in its recent revival in the context of metamaterials (see Refs. 29 and 30) as well as its relevance to the so-called ‘left-handed

(LH) medium’ (see Sec. 2.2). In the transition from negative to positive values (at the plasma frequency), the eﬀective permittivity assumes a value of zero, a property which also has motivated directions in metamaterials research [31].

2.1.2 Tailoring Permeability

Attempts at producing magnetic responses using artificial dielectrics were few and far between; first, the main purpose of artificial-dielectrics research was often to design spe- Chapter 2. Background 10

(a) (b)

Figure 2.1: (a) Array of thin metallic wires resembling a microwave plasma with ² < 0 for electric ﬁelds polarized as shown; (b) split-ring resonator (SRR) particle yielding µ < 0 for magnetic ﬁelds polarized as shown. Part (b) after Ref. 32. Reprinted with permission of John Wiley & Sons, Inc., copyright °c 2005.

cific refractive indices, of which the simplest property to manipulate was the permittivity; second, the use of simple metallic inclusions could not generate any appreciable magnetic behaviour beyond the typical diamagnetic response. However, one very notable suggestion to create unnaturally large artificial permeabilities was offered by Schelkunoff and

Friis [33], who suggested that a small inductive loop (akin to a magnetic dipole) be loaded in series with a lumped capacitor. The resulting resonant response of the magnetic polar- izability results in arbitrarily large positive permeabilities below the ring resonance. In

1999, Pendry et al. independently introduced the split-ring resonator (SRR), a resonant particle amounting, essentially, to the same capacitively-loaded metallic loop [34]. The

SRR is depicted in Fig. 2.1(b), which also indicates the polarization of the magnetic field that activates it. However, the SRR proposed by Pendry was introduced not only for its strong positive permeability below resonance, but also for its strong negative permeability above resonance (investigated for the purposes of enhancing nonlinear phenomena), which was not explicitly considered by Schelkunoff and Friis. Since the permeability must return to its free-space (positive) value as frequency is increased to infinity, it also possesses a ‘magnetic’ plasma frequency where it assumes a value of zero before crossing Chapter 2. Background 11 once again into positive territory.

The SRR and its variants have also been scaled to produce a magnetic resonance anywhere from THz frequencies [35, 36] to the mid-infrared [37], and magnetism across all colours of the visible spectrum has been realized using plasmonic nanoparticles [38].

Ideas of producing a magnetic response with dielectric-only composites by way of Mie resonances have also been proposed [39–42].

2.1.3 Tailoring the Refractive Index

The sign of the refractive index n is determined by the branch of the square root taken √ in the equation n = ± εµ, where ε and µ represent the permittivity and permeability of the material, respectively (assumed isotropic). Early artificial dielectrics research sought to realize positive refractive indices less than one by tailoring the effective permittivity alone. A negative refractive index is the unusual product of a simultaneously negative permittivity and permeability. This can be shown by introducing a small amount of loss in one of the parameters, for simplicity, and invoking the radiation condition, as discussed in Ref. 43. The NRI metamaterial has, so far, garnered the most attention in metamaterials research for the dramatic phenomena with which it is associated, as well as for its important and far-reaching applications. Accordingly, much of the remainder of this chapter will be devoted to this most interesting and prolific type of metamaterial, which is also known as a ‘left-handed’ (LH) medium.

2.2 The Left-Handed (LH) Medium

In the 1960s, Russian physicist Victor Veselago systematically examined the feasibility of media characterized by a simultaneously negative permittivity ² and permeability µ [1].

Veselago compiled a general and comprehensive argument based on earlier work that such media are allowed by Maxwell’s equations and that plane waves propagating within them Chapter 2. Background 12

k k

S H S H

E E

RH Medium LH Medium

Figure 2.2: Orientation of ﬁeld quantities E, H, Poynting vector S, and wavevector k in RH media and LH media. After Ref. 32. Reprinted with permission of John Wiley & Sons, Inc., copyright °c 2005.

could be described by an electric ﬁeld vector E, magnetic ﬁeld vector H, and wavevector k, forming a left-handed triplet, in opposition to wave propagation in conventional media, in which these three quantities form a right-handed triplet, and accordingly labeled these materials left-handed (LH) media and right-handed (RH) media, respectively. The two arrangements are illustrated in Fig. 2.2. Moreover, although E, H, and k form a left-handed triplet, E, H, and the Poynting vector S maintain a right-handed relationship; thus, in LH media the wavevector k is antiparallel to the Poynting vector S. In the electrical engineering community, this phenomenon is more familiar as the backward wave, and for this reason, some researchers use the term ‘backward-wave media’ to describe LH materials [44]. Certainly, one-dimensional (1D) backward-wave TLs are not new to the electrical engineering community; however, Veselago’s work established the idea that two-dimensional (2D) or three-dimensional (3D) isotropic and homogeneous media supporting backward waves ought to be characterized by a negative refractive index (summarizing similar ideas considered as early as the turn of the 20th century by such notable names as H. Lamb [45], A. Schuster [46], and L. I. Mandel’shtam [47]).

When such media are interfaced with conventional (positive-refractive-index) dielectrics,

Snell’s Law is reversed, leading to the negative refraction of an incident plane wave. This is illustrated in Fig. 2.3. Indeed, all physical phenomena that depend on the refractive Chapter 2. Background 13

RH, nRH>0 LH, nLH<0

θi > 0 θt < 0

Figure 2.3: Negative refraction at an interface between a RH medium and a LH medium. The rays indicate the direction of the wavevector, or phase lag.

RH, nRH>0 LH, nLH<0 RH, nRH>0

s1 f1 f2

source d image

Figure 2.4: Focusing of the rays of a cylindrical or spherical excitation by a LH slab embedded in a RH host medium. The arrows illustrate the focusing eﬀect of propagating (ray or wavevector) components, and the solid curve depicts the restoration of the amplitude of evanescent components.

index acquire a ‘negative’ counterpart. Aside from refraction, Veselago also discussed ideas on a negative Cerenkov radiation principle and even a negative Doppler eﬀect [1].

Harnessing the phenomenon of negative refraction, entirely new refractive devices can be envisioned, such as a ﬂat ‘lens’ without an optical axis, also proposed by Veselago.

This is shown in Fig. 2.4. In general, nonparaxial rays experience geometric aberration; that is, for an arbitrary selection of positive and negative refractive indices, each ray intercepts the principal axis at a diﬀerent point. However, Veselago carefully chose as his example the case of an LH slab with refractive index nLH = −1, embedded in vacuum

(nRH = +1), such that the relative index of refraction is nREL = nLH /nRH = −1. In this Chapter 2. Background 14

special case, |θi| and |θt| become equal, and all the component rays are focused to the same point. Furthermore, for nREL = −1, the slab thickness d, source distance s1, and external focal length f2 in Fig. 2.4 are related through d = s1 + f2, so that the phase lag incurred in the two RH regions (positive optical path lengths) is fully compensated by the phase advance incurred in the LH slab (negative optical path length). Thus, the phase of the source, contributed by its propagating spatial frequency components, is exactly restored at the image plane. However, the finest features of a source are contained in its evanescent spatial frequency components, which decay rapidly and do not make it to the focal plane of conventional lenses. Indeed, the loss of this fine spatial information is the source of the so-called ‘diffraction limit’ (see Appendix A).

In 2000, J. B. Pendry extended the analysis of Veselago’s lens to include evanescent waves and observed that such lenses could overcome the diﬀraction limit [2]. Pendry suggested that Veselago’s lens would allow ‘perfect imaging’ if it were completely lossless, if its refractive index n were exactly equal to −1 relative to the surrounding medium, and if its impedance were matched exactly to that of the surrounding medium. These conditions amount to the requirement that ²LH = −²RH and µLH = −µRH where (²LH ,

µLH ) and (²RH , µRH ) are the real parts of the material parameters in the LH and RH regions, respectively. In addition to focusing propagating waves as would a conventional lens, Pendry showed that such a lens (now commonly referred to as a Veselago-Pendry lens) would additionally support a near-ﬁeld resonant enhancement of evanescent waves that would restore the amplitudes of the decaying evanescent waves emanating from the source when they arrive at the focal plane. Thus, by compensating for both the phase of the propagating-wave components and the amplitude of the evanescent-wave components, the ideal Veselago-Pendry lens is able to reconstruct the complete spectrum of the source at the image plane down to its ﬁnest spatial features. Chapter 2. Background 15

2.3 Metamaterial Implementations

The long inactivity separating the theoretical suggestions of Veselago and the present intense research into metamaterials was due, in large part, to the problem of implementation. However, the last few years have seen many types of LH metamaterials emerge.

Any complete understanding of the macroscopic response of an array of discrete scatterers takes their coupling into account. Nevertheless, when discrete resonant particles are ‘loosely coupled’, these bulk properties are largely determined by the response of the isolated resonators. Their resonant nature often makes them susceptible to loss and they are inherently narrowband; however, they are attractive for the simplicity gained in their analysis and design. The response of ‘tightly coupled’ resonators, on the other hand, is dominated by their mutual interactions, and are often less susceptible to losses and more broadband than their isolated-resonator counterparts. The simplest example is that of the TL, whose lumped-element representation consists of ladder networks of inductors and capacitors. These TLs exhibit no isolated resonator-like characteristics; they are inﬁnitely broadband and less susceptible to losses as a result. In this example, the in-

ﬁnitesimal lumped-element unit cells, which in isolation are discrete L–C resonators, are so tightly coupled that the tendency of a unit cell to transmit energy to its neighbours is the very phenomenon of propagation [48]. The gamut of metamaterials today consists of those that rely on the self-resonant properties of individual scatterers, others that rely on the mutual coupling between scatterers, and still others that exploit both self and mutual interactions. Some of the prevailing metamaterial technologies are discussed below.

2.3.1 The SRR/Wire Metamaterial

The negative-permeability eﬀect associated with an array of SRRs is the macroscopic result of the self-resonant response of the constituent SRRs, each of which produces a local negative magnetization in the vicinity of its resonance. The resonance can be made

Chapter 2. Background 16

Figure 2.5: Depiction of Wire/SRR metamaterial. After Ref. 32. Reprinted with permission of John Wiley & Sons, Inc., copyright °c 2005.

to extend into negative-µ values provided that losses are suﬃciently low.

It was not long after the introduction of the SRR that the thin-wire and SRR arrays, respectively exhibiting a negative permittivity and negative permeability over a particular range of frequencies, were combined into a composite metamaterial by Shelby, Smith, and

Schultz at the University of California at San Diego. Figure 2.5 schematically shows the combined array. Needless to say, their landmark experimental work [49] succeeded in verifying the phenomenon of negative refraction and motivated many other important advances in the ﬁeld of metamaterials.

2.3.2 Magnetic-/Dielectric-Sphere Composite Metamaterials

The SRR and its variants are a class of scatterers constructed from conductors. However, there is another class of structures based on resonant dielectric particles embedded in a host medium. The scatterers (typically spheres) consist of high-permittivity and/or high- permeability materials and are embedded in a low-permittivity/low-permeability host medium. Thus, they are electrically small when compared to the wavelength in the host medium but of resonant dimensions when compared to the wavelength in the scatterer Chapter 2. Background 17

TE011 TM011

Figure 2.6: Spherical resonances in dielectric particles corresponding to negative permeability (TE011) and negative permittivity (TM011). Solid lines: magnetic ﬁelds; dashed lines: electric ﬁelds.

material. Using Mie theory, a resonant eﬀective permittivity/permeability response of the composite medium can be extracted and, much like the SRR response, these Mie resonances can be made to extend into negative territory provided that losses are low.

Various methods have been suggested to simultaneously realize negative permittivity and negative permeability using Mie resonances: in Ref. 40, two sets of particles are used, one employing high-permeability materials to realize a magnetic resonance and the other employing high-permittivity materials to realize an electric resonance; in Ref. 39, two sets of high-permittivity spheres of different radii are used to realize the electric and magnetic resonances. Based on the latter work, the field distributions for the TE011 and TM011 spherical resonances, corresponding to conditions of effective negative permeability and negative permittivity, respectively, are illustrated in Fig. 2.6.

Such composites, whose response is based on the excitation of Mie resonances, are also advantageous in that they can be made fully isotropic. Furthermore, they can be realized at frequencies at which the construction of conductor-based particles would be inordinately diﬃcult [42]. Chapter 2. Background 18

2.3.3 The Negative-Refractive-Index Transmission-Line (NRI-

TL) Metamaterial

Following the seminal and inspiring works of Veselago, Pendry, and the UCSD group, other realizations of the NRI metamaterial were considered. The development of the wire metamaterial and SRR metamaterial was based on concepts in the physics community, largely independent of the work in artiﬁcial dielectrics carried out in the electrical engineering community many decades earlier. Also notable in this context is the work of

Kron, Ramo, and Whinnery in the development of distributed L–C network models for propagation in natural, conventional materials [19,50].

Later in this work, it will be shown that LH media may be modeled simply by ex- changing the positions of the inductance and capacitance in the conventional continuous

TL model, resulting in a ‘dual’ structure with a high-pass topology that successfully negates its eﬀective-medium parameters and produces the correct causal dispersion characteristics associated with LH media [17, 51]. This concept led to the development of the NRI-TL metamaterial, in which the continuous TL model is practically and directly synthesized in periodic form by loading a host TL medium with discrete inductors and capacitors (scatterers) in the dual (high-pass) conﬁguration. A unit cell of this structure is shown in Fig. 2.7(a) for the 2D case. As a result of its distributed TL nature, the NRI-

TL metamaterial exhibits a NRI over a very large bandwidth, and is not as susceptible to losses as other metamaterial implementations. The periodic implementation necessarily diﬀers from the continuous model (see section 3.1), since the host medium contributes a necessary right-handed (RH) component to the dispersion properties (for comparison, this RH component is provided in the wire/SRR metamaterial by the free space separating the inclusions). As with any periodic structure, this results in the formation of stopbands that make the NRI bandwidths ﬁnite; nevertheless, reported NRI-TL fractional bandwidths have exceeded those of the wire/SRR metamaterials by six or seven Chapter 2. Background 19

2⋅⋅⋅C0

L0 2⋅⋅⋅C0

(a) (b)

Figure 2.7: (a) NRI-TL unit cell consisting of a host TL network loaded in a dual con- ﬁguration using lumped inductors and capacitors. (b) Planar NRI-TL lens interfaced with a RH grid containing a voltage source located by the arrow (left) as well as the detecting probe (right). The inset depicts the realized unit cell in microstrip technology. Reprinted with permission from Ref. 18, copyright °c 2003 Optical Society of America.

times.

Balmain et al. have devised anisotropic metamaterials using a TL approach, and have demonstrated negative refraction and focusing by way of a phenomenon akin to the

‘resonance cones’ observed in anisotropic plasmas [52,53]. These metamaterials consist of a 2D periodic L–C grid over ground with series capacitors loading one grid axis and series inductors loading the other, and in some implementations also include shunt reactive elements to ground. Appropriate excitation of the grid forms a resonant path that can be made to scan with frequency.

A comprehensive theory for TL-based metamaterials was developed and both simulations and experiments at microwave frequencies established that such TL networks can be used not only for the modeling of conventional materials but also for the realization of new materials (see Appendix B). Furthermore, since these designs are completely scal- able with frequency, it is entirely foreseeable that metamaterials based on TL-equivalent perspectives can be fabricated in the THz or far infrared range, limited only by the onset Chapter 2. Background 20 of plasmonic resonances in the metals. In fact, even in the optical regime, some recent work has revealed that similar TL concepts can be employed for particular arrangements of plasmonic materials such as silver and gold [54,55].

2.3.4 Plasmonic Materials-Based Metamaterials

The advent of the LH metamaterial and the interest surrounding the Veselago-Pendry lens brought with them ideas of realizing metamaterials at optical frequencies. The ﬁrst microwave and millimetre-wave metamaterials employed conducting inclusions; unfortunately, a number of factors preclude scaling these inclusions for operation at optical frequencies, including inordinately large conductor losses, the elusiveness of the optical magnetic response, and the impracticability of dimensions when the inclusions are required to have features on the nanometre scale. However, some of the properties of interest can be accessed via the principle of plasmonic resonance, which describes the strong interaction between light and nanoparticles made of noble metals in the vicinity of their surface plasmon frequency. In this region, these materials behave like plasmas and can, accordingly, be said to possess a permittivity whose real part is negative. Coupled plasmonic nanoparticles (e.g. pairs of plasmonic rods) can also give rise to circulating currents, yielding an optical magnetic response, as shown in Fig. 2.8(a) [55]. In an array conﬁguration, these particles can be tightly coupled and support the propagation of transverse backward surface waves, without the simultaneous requirement for negative permeability or, indeed, magnetism of any sort [56]. These phenomena have also been considered at interfaces between bulk plasmonic materials and other positive-permittivity materials, as well as plasmonic material slabs [57], as depicted in Fig. 2.8(b). Engheta et al. have proposed TL analogies for plasmonic nanoparticles, suggesting that negative- permittivity particles can be regarded as optical nano-inductors and positive-permittivity particles as nano-capacitors, as shown in Fig. 2.8(c). Viewed in terms of these analogies, the various plasmonic arrangements supporting backward waves, including the plasmonic Chapter 2. Background 21

H > 0 I(z) εεε εεε > 0 d εεε < 0 a εεε > 0

εεε < 0 c b εεε < 0 -I(z) εεε > 0 (a) (b) (c)

Figure 2.8: (a) Depiction of the two-plasmonic-wire system and the ﬁeld and resulting currents producing a magnetic response. After Ref. 55. Reprinted with permission of John Wiley & Sons, Inc., copyright °c 2005; (b) Interfaces of plasmonic (² < 0) layers and conventional dielectrics (² > 0) supporting backward surface waves; (c) interpretation of positive- and negative-permittivity nanoparticles as nanocapacitors and nanoinductors, respectively.

nanoparticle chain and plasmonic slab, can be equated to an optical LH TL [58].

2.4 Metamaterial Applications

Since the performance of all microwave and optical devices depends on the intrinsic parameters of their constituent materials, it is conceivable that implementing these devices using metamaterials could result in improved performance characteristics or altogether new phenomena. Thus, metamaterials provide great potential for application. Many devices employing metamaterials in the microwave or optical regimes have been described in the scientiﬁc literature; some remain as theoretical ideas whereas others have been experimentally validated. This section presents a summary of these works, and in the interest of conciseness, is limited to a review of metamaterials applied to the design of lenses, phase shifters, couplers, dividers, and antennas. Chapter 2. Background 22

2.4.1 Lenses

Pendry’s revelations on the metamaterial superlens motivated a great deal of research into verifying the property of focusing inherent to planar LH slabs, and also the property of superlensing or subwavelength focusing/imaging characteristic of the Veselago-Pendry lens.

TL-based Lenses

The ﬁrst demonstration of NRI focusing by any metamaterial was reported in Refs. 59 and 17 at microwave frequencies using a 2D NRI-TL grid interfaced with a parallel-plate waveguide. The NRI-TL metamaterial was also used to construct the ﬁrst practical

Veselago-Pendry superlens [3], which experimentally demonstrated subwavelength focusing at 1GHz with a resolution three times better than would be produced by a conventional, diﬀraction-limited lens. Anisotropic TL-based metamaterials have shown focusing by way of ‘resonance cones’ into spots on the order of λ/25. However, the imaging ability of all of these lenses is limited to the plane of the NRI-TL structure. To address this limitation, volumetric and 3D TL-based metamaterials able to interact with free-space excitations have recently been proposed and developed [5,20,22,60–62] and in some cases, their focusing properties experimentally veriﬁed [4,63,64].

Optical Lenses

The first optical superlens was constructed from a thin plasmonic silver film [9]; although such plasmonic films cannot be said to support a negative index (they possess only ² < 0) and operate in the extreme near field (quasistatic limit), they do support the resonant enhancement of evanescent waves leading to recovery of the finest spatial features of a source. Indeed, this experiment produced image resolutions six times better (for a uniform grating) and four times better (for an arbitrary pattern) than without the silver superlens. Following this seminal experimental work, other experimental verifications of Chapter 2. Background 23 focusing and negative refraction at optical frequencies using multilayer plasmonic films and novel imaging techniques were reported [65,66].

Wire/SRR-based Lenses

Attempts at focusing using wire/SRR media have produced varied results, including reports of subwavelength focusing without evanescent-wave enhancement, but have often been mired by high losses and narrow bandwidths [10, 15, 67–69]. However, focusing with high intensities has been shown at both microwave frequencies using graded-index

(GRIN) metamaterial lenses. These are constructed using wire/SRR elements whose geometrical features are spatially varied in order to realize eﬀectively inhomogeneous permittivity and permeability proﬁles [70].

Anisotropic Far-Field Lenses

In spite of its intriguing lensing properties, the Veselago-Pendry lens suﬀers from the fact that the image measurement must take place in the near ﬁeld of the lens. It was proposed that a cylindrical or spherical anisotropic lens may be used to transmit (or

‘magnify’) near-field spatial information away from the lens, thereby allowing superresolution in the far field [12, 13]. This transmission occurs by way of channeling the source images radially away from the sources, in a manner akin to that of the resonance cones studied in anisotropic TL metamaterials [52]. The so-called ‘far-field superlens’ or ‘hyperlens’ has been realized and has successfully verified far-field resolution of subwavelength details [14].

2.4.2 Phase Shifters

Among the most unique, but understated, advantages of LH metamaterials are their dispersion properties. Indeed, the propagation constant varies inversely with frequency, such that the effective wavelength describing this propagation varies proportionally with Chapter 2. Background 24 frequency (this is derived explicitly for the TL metamaterial in Chapter 3, where the link to LH, or backward-wave, propagation is clearly established). As a result, a given electrical length can be achieved at lower frequencies with metamaterials. However, there is another implication of this statement: at a given operating frequency, distributed devices can be made physically smaller with metamaterials. One of the first applications of this concept was in the realization of metamaterial phase shifters consisting of alternating sections of conventional TLs and NRI-TL metamaterials [71]. The overall phase shift provided by these structures essentially amounts to the sum of the phase shifts contributed by the RH and LH constituent sections. Thus, these structures benefit from another useful property of metamaterials: that the realized phase shifts can have negative values in the LH region, positive values in the RH region, and a zero value between these two regions. The zero-degree phase shift is particularly useful, since this could otherwise only be achieved using a one-wavelength-long TL. Furthermore, it was shown in these works that the metamaterial phase shifters are generally more broadband than their RH delay-line counterparts. This is due to the fact that the frequency dispersion of the LH sections offsets that of the RH sections. Some zero-degree, coplanar-waveguide- (CPW-) based NRI-TL phase shifters in one, two, four, and eight stages are shown in Fig. 2.9(a) along with a conventional one-wavelength-long CPW line.

2.4.3 Couplers/Dividers

Passive directional couplers and power dividers are staples in microwave and photonic circuit engineering, since these devices form essential building blocks of many practical electromagnetic systems. Both types of devices are distributed in that the manipula- tion of their properties, including power division ratio, matching, bandwidth, isolation, and directivity, relies on their electrical length, and as a result, these structures can be physically large at low frequencies. From the previous discussion on metamaterial phase shifters, it is clear that metamaterials can be applied to the miniaturization of dividers

Chapter 2. Background 25

2 1

NRI MS

4 3 Shorted Interdig. Stubs Capacitor

(a) (b)

Figure 2.9: (a) CPW-based NRI-TL phase shifters (top to bottom: one-stage, two-stage, four-stage, eight-stage), and conventional one-wavelength CPW TL. Reprinted with permission from Ref. 71, copyright °c 2003 IEEE; (b) fully printed NRI-TL coupled-line coupler. Reprinted with permission from Ref. 72, copyright °c 2006 IEEE.

and couplers as well. Metamaterial power dividers for series-fed antenna arrays showed a 97% size reduction compared to conventional designs [73] and the additional control over phase properties using NRI-TL metamaterials inspired the design of novel compact metamaterial baluns [74]. TL-based metamaterial branch-line couplers have also shown significant size reductions [75]. However, metamaterials have also been applied to the design of coupled-line couplers exhibiting very novel and intriguing phenomena. For example, a coupled-line coupler consisting of a conventional microstrip line adjacent to a NRI-TL metamaterial line produces contradirectional power flow when the lines are phase-matched, in stark contrast to the co-directional power flow expected of a similarly designed microstrip-microstrip coupled-line coupler [76]. This intriguing result stems from the fact that the NRI-TL supports backward waves. One such NRI-TL coupler is shown in Fig. 2.9(b). The coupling occurs by a continuous leakage of power from one line into the other, akin to the principle of leakage in conventional leaky-wave antennas, rather than the interference effect exploited by conventional coupled-line couplers. This results in an exponential decay of field strengths along the length of the coupler and, hence, most of the power is coupled over an extremely short length. The same intriguing principle Chapter 2. Background 26 has been proposed for the optical domain using the forward and backward surface waves supported by interfaces between conventional- and plasmonic-material layers [77].

2.4.4 Antennas

Some of the earliest and most intriguing applications of metamaterials concepts were made in the context of antennas. Metamaterials can be used to alter, improve, or otherwise optimize the properties of antennas in a number of ways. For example, their unique dispersive and potentially low-loss properties can be exploited to either reduce beam squinting [43] or facilitate frequency tuning; their phase-compensation properties can be used to design compact antenna feed networks and miniaturize the antenna elements themselves; and their inherent ability to support fast waves (waves with a phase velocity greater than the speed of light in vacuum) enables them to act as leaky-wave antennas with unique radiation properties. Conventional antennas have also benefitted from metamaterials used as frequency-selective surfaces (FSS) or artificial/perfect magnetic conductors (AMC/PMC) patterned into groundplanes to enhance radiation properties by suppressing surface-wave losses or eliminating the quarter-wavelength distance that separates an antenna and its PEC reflector. However, these surfaces are traditionally clas- sified as electromagnetic/photonic bandgap (EBG/PBG) structures and as such, shall be avoided in this discussion in accordance with our original definition of the term ‘material’.

Leaky-Wave Antennas

Radiation by way of leaky modes represents a continuous coupling from a fast-wave structure into free space. Indeed, the upper frequency region of the LH or NRI pass band represents a condition wherein the phase velocity of the guided wave in the metamaterial exceeds that of free space, and power is leaked from the metamaterial. This is illustrated in Fig. 2.10(a). However, metamaterial leaky-wave antennas diﬀer from conventional antennas in that phase matching of the guided backward-wave mode and

Chapter 2. Background 27

sinθθθ=βββ/k0 k 0 θθθ

Radiation Region Air (k0>βββ) NRI βββ S

(a) (b)

Figure 2.10: (a) Leaky-wave radiation region of a NRI metamaterial; (b) mechanism of radiation by way of phase matching at the interface between the NRI medium and air.

the leaky forward-wave mode insists that the latter emerges at a backward (or negative) angle, as shown in Fig. 2.10(b). Backward leaky-wave fan-beam radiation was ﬁrst shown using a 1D CPS-based NRI-TL leaky-wave antenna operating at 15GHz [78]. Further- more, since the dispersion characteristics of these metamaterials can be made to pass directly from LH (NRI) to RH (PRI), the leaky waves follow suit, radiating from negative to positive angles through broadside. This was shown in 1D using both passive and active tunable TL-based leaky-wave antennas [79, 80] and for leaky-wave pencil beams and conical beams using a passive 2D planar NRI-TL metamaterial surface [81,82]. Many of these experimental results are also supported by important theoretical studies establishing the nature of leaky-wave radiation associated with various metamaterials (see, for example, Refs. 83, 84, 85, and 86).

Small Antennas

Small antennas have been a subject of ongoing research primarily because it has been difficult to make such antennas efficient (due both to low radiation resistances and the Chapter 2. Background 28 inability to match such antennas to real sources stemming from the large amount of reactive power stored in the near field) and also to possess a large enough bandwidth to be practical. The principle of phase compensation, which enables large electrical lengths over small physical lengths is, thus, of great importance to these efforts. Indeed, the loading of conventional antennas using LH metamaterials lowers their resonance frequencies; this has been shown for various patch-antenna designs [87–89]. The idea of compact metamaterial series-fed antenna arrays also inspired the design of small ring antennas in which elements are fed in phase using zero-degree NRI-TL metamaterial lines and arranged in a ring of subwavelength dimensions [43,90]. The use of spherical metamaterial shells or even individual metamaterial-inspired reactive elements to resonantly cancel the effects of the near field have also been proposed, and show promise in overcoming the matching problem [91]. Of course, antenna feed networks can also be miniaturized by employing compact metamaterial phase shifters, as shown in Refs. 73 and 92. Chapter 3

Fundamentals of Transmission-Line

Metamaterials

3.1 TL Theory of LH Media

The concept of modeling dielectrics using distributed L–C networks is not new; in fact, it was rigorously studied for conventional dielectrics as early as the 1940s by Kron,

Ramo, and Whinnery [19, 50], when it very eﬀectively generalized the well known low- pass lumped-element distributed representation of the familiar TL in which the series inductance and shunt capacitance represented, respectively, the positive permeability and permittivity of the intrinsic medium. These have become essential techniques in the electrical engineering toolbox (e.g. in the development of the transmission-line matrix

(TLM) method of time-domain modeling [93]) and have also been invaluable to the understanding of metamaterials. In this section, TL techniques are used to model some of the salient features of metamaterials, and these are compared to their conventional counterparts.

Transmission-line modeling represents plane-wave propagation in unbounded natural media (in a particular direction) as TEM guided-wave propagation inside a TL rep-

29 Chapter 3. Fundamentals of Transmission-Line Metamaterials 30

resented by a network of distributed series impedances Z0 and shunt admittances Y 0.

These reactive elements can be directly related to the permittivity and permeability of

the medium ﬁlling the TL through a constant, g, that accounts for the diﬀerence be-

tween the wave impedance in the ﬁlling medium and the characteristic impedance of

the TL, a result of constraining unbounded wave propagation to a particular direction

(see Appendix B for further discussion). The following sections examine 2D TL models

describing the propagation of 2D waves, a representative case for which the notion of a

refractive index is particularly meaningful.

3.1.1 Conventional, Right-Handed (RH) Media

Figure 3.1(a) depicts a lumped-element unit cell (length d) representing 2D propagation

in a conventional (or RH) medium that is isotropic, nonmagnetic, and possesses a rela-

tive permittivity ²r. Retaining their potential frequency dependence in the notation for

0 0 generality, µ(ω) = µ0 and ²(ω) = ²r²0, and accordingly, Z = jωµ0g and Y = jω²r²0/g.

0 0 Here, L = µ0g = L/d (H/m) and C = ²r²0/g = C/d (F/m) are the corresponding distributed quantities, both of which, it should be noted, are positive and real. This model

assumes no losses, but losses can easily been added to the model in the typical TL sense,

i.e., with a resistance in series and a conductance in shunt.

In the continuous limit, with d/λ → 0, the corresponding propagation constant β,

which is obtained from the circuit wave equation,

∂2V ∂2V √ y + y + β2V = 0, β = ± −Z0Y 0 (3.1) ∂x2 ∂z2 y reduces to that of a standard TL, ﬁlled with a nonmagnetic dielectric with relative

permittivity ²r, √ √ 0 0 0 0 √ β = ± −Z Y = ω L C = ω µ0²r²0 = ω/vφ (3.2)

As will be revealed shortly, the choice of the positive root establishes the convention

Chapter 3. Fundamentals of Transmission-Line Metamaterials 31

L/2

L/2 ν g C y ω L/2 0 x ν z p −β +β β d 0 0 (a) (b)

Figure 3.1: (a) 2D TL unit cell and (b) dispersion relation for propagation along a particular direction in the x-z plane, describing a medium with µ(ω) = µ0 and ²(ω) = ²r²0. The arrows suggest that the group and phase velocities are determined in the direction of increasing frequency. After Ref. 32. Reprinted with permission of John Wiley & Sons, Inc., copyright °c 2005.

that the group velocity is positive, which ensures that power ﬂows away from the source.

The resulting dispersion relation, or ω–β curve, reveals the variation of the propagation constant along a particular axis of propagation in the x–z plane as a function of frequency, as shown in Fig. 3.1(b). The magnitude of the phase and group velocities in the medium can also be inferred from the ω–β curve: the phase velocity is deﬁned as the ratio vφ = ω/β, whose magnitude is given by the slope of the line from the origin of the

−1 ω–β curve to a point (ω0, β0), and the group velocity is deﬁned as vg = (∂β/∂ω) , which is the slope of the tangent to the ω–β curve at (ω0, β0). It is evident from Fig. 3.1(b) that the propagation constant of conventional isotropic RH media modeled by a distributed series inductance and shunt capacitance varies proportionally with frequency, as would be expected in conventional dielectrics at low frequencies. It is also clear that the resulting phase and group velocities are parallel and equal (since this topology models Chapter 3. Fundamentals of Transmission-Line Metamaterials 32

a dispersionless medium) and are given by

µ ¶ ω 1 1 ∂β −1 vφ = = √ = √ = = vg (3.3) β L0C0 µ0²r²0 ∂ω

In RH media, a positive phase velocity means that the phase lags in the direction of the

group velocity (in this case, parallel to the Poynting vector). Thus, the refractive index,

which can be deﬁned as the ratio between the speed of light in vacuum and the phase

velocity in the medium, is positive:

√ 0 0 √ c L C µ0²r²0 √ n = = √ = √ = ²r (3.4) vφ µ0²0 µ0²0

Furthermore, the wave impedance of the eﬀective medium can be related to the character-

istic impedance of the distributed network in the continuous limit through the geometrical

constant g, as expected: r r 0 µ0 1 L ηr = = 0 = Z0/g (3.5) ²r²0 g C

3.1.2 Left-Handed (LH) Media

Veselago’s postulation of a negative permittivity and permeability prompts us to ask

whether the L0 and C0 parameters in a network representation can also be made negative.

Naturally, from an impedance perspective, imposing a negative L0 and C0, or equivalently

a negative series impedance −jωL0d and shunt admittance −jωC0d, essentially exchanges

their reactive and susceptive roles, so that the series inductor becomes a series capacitor,

and the shunt capacitor becomes a shunt inductor. The unit cell of the emerging dual

structure is shown in Fig. 3.2(a), and it is easily recognized as having the topology of a

2D high-pass ﬁlter network [17, 59, 94, 95]. The eﬀective permittivity and permeability Chapter 3. Fundamentals of Transmission-Line Metamaterials 33

represented by this topology can be shown to be (see Ref. [32] for a derivation)

1 µ(ω) = − (3.6) ω2C0g g ²(ω) = − (3.7) ω2L0

where the distributed parameters L0 = Ld and C0 = Cd are deﬁned in the peculiar units

[H·m] and [F·m], respectively; their meaning is intuitively clear when the parameters are

instead represented as 1/L0 = (1/L)/d and 1/C0 = (1/C)/d. Contrary to the results for

the RH unit cell, the eﬀective material parameters of the dual network are prominently

negative. However, they are no longer constants, and are, instead, explicit functions of

frequency; in fact, although the continuous limit is an idealized approximation, the par-

ticular dispersive forms of (3.6) and (3.7) ensure that the time-averaged stored electric

and magnetic energies associated with this network are positive, so that the conserva-

tion of energy is not violated [1, 26]. Thus, the simple dual high-pass network, with

distributed series capacitance C0 = Cd (F·m) and shunt inductance L0 = Ld (H·m),

satisﬁes the principal requirement for LH behaviour: the eﬀective material parameters

are simultaneously negative.

The propagation constant associated with the dual structure, found through the ap-

plication of (3.6) and (3.7), boasts a peculiar inverse relationship with frequency,

√ 1 β = − −Z0Y 0 = − √ (3.8) ω L0C0 and the corresponding ω–β curve shown in Fig. 3.2(b). In this case, the phase and group

velocities are antiparallel and are given by

µ ¶ ω √ ∂β −1 v = = −ω2 L0C0 = − = −v (3.9) φ β ∂ω g

where the choice of the negative root in (3.9) has ensured a positive group velocity (in

Chapter 3. Fundamentals of Transmission-Line Metamaterials 34

2⋅⋅⋅C

2⋅⋅⋅C ν g 2⋅⋅⋅C ω L y 0 2⋅⋅⋅C x ν p z −β +β β d 0 0 (a) (b)

Figure 3.2: (a) 2-D dual TL unit cell and (b) dispersion relation for propagation along a particular direction in the x-z plane, describing a medium with simultaneously negative, dispersive parameters µ = −|µ(ω)| and ² = −|²(ω)|. The arrows suggest that the group and phase velocities are determined in the direction of increasing frequency. After Ref. 32. Reprinted with permission of John Wiley & Sons, Inc., copyright °c 2005.

this case also parallel to the Poynting vector), in accordance with the previously adopted convention; that is, in LH media, the phase leads in the direction of positive group velocity, or power ﬂow, a property that reveals the connection between LH propagation and the backward wave. Once again, the reader is reminded that this dispersion relation is derived in the continuous (quasistatic) limit, and is therefore only an approximation to that of a physical periodic structure (see Chapter 4). For example, the approximation is invalidated at high frequencies, where it is evident that the group velocity exceeds the speed of light and the system becomes noncausal.

The relationship between the eﬀective wave impedance and network characteristic impedance is also preserved:

p c − µ(ω)²(ω) 1 n = = = − √ (3.10) √ 2 0 0 vφ µ0²0 ω L C µ0²0 Chapter 3. Fundamentals of Transmission-Line Metamaterials 35

s r µ(ω) 1 L0 η = = = Z /g (3.11) ²(ω) g C0 0

It is noteworthy that the above development has, once again, assumed no losses, but, as before, it may easily be generalized to account for losses.

This conception of the negative LH permeability and permittivity in terms of an equivalent series capacitance and shunt inductance was introduced in Refs. 17 and 59, and since then, it has been applied to the analysis of many diﬀerent metamaterial implementations.

3.2 Negative Parameters: An Electrodynamical Per-

spective

The properties of conventional materials are the macroscopic interpretation of field interactions at the atomic or molecular level. The electrodynamics of materials are quite well understood by classical arguments treating electrons in an ac electric field as driven, damped oscillators; although an understanding of the magnetic properties of materials strictly requires a quantum mechanical treatment, classical models of ac magnetic fields causing electrons to orbit their nuclei, thereby yielding a magnetic moment, may apply under certain circumstances, particularly in the analysis of metamaterials, in which magnetic moments are often produced directly through current loops.

The previous section showed that the reversal of the positions of the series inductor and shunt capacitor in the conventional lumped-element TL model describes backward- wave, or LH, propagation in a material that may be regarded to possess a simultaneously negative permeability and permittivity. In this section, electrodynamical arguments are used to show the reverse: that the behaviour of ﬁelds inside materials synthesized to possess negative permittivity and permeability are akin to the response of a TL model with an inductor in shunt and capacitor in series. This section begins with a review of Chapter 3. Fundamentals of Transmission-Line Metamaterials 36

–q +q

lav

Ea Ea

Figure 3.3: Polarization due to the application of an electric ﬁeld Ea, modeled as an equivalent electric dipole consisting of charges ±q separated by a distance lav.

some of the standard methods of determining permittivity and permeability, followed by a set of analogous arguments that show, intuitively, how negative permittivity and permeability may come about, and how they may be related to the dual TL model.

3.2.1 Determination of Permittivity

The application of an electric ﬁeld to a dielectric medium results in the polarization of its constituent positively- and negatively-charged particles, which can be modeled by an equivalent dipole whose charges +q and −q are displaced by an average distance lav, yielding an electric dipole moment dp = qlav. When all the dipoles in the medium

(volume density of electric dipoles Ne) are aligned, the total dipole moment per unit volume can be expressed as P = Nedp = Neqlav. This is illustrated in Fig. 3.3. One of the standard methods of determining the macroscopic permittivity of homogeneous dielectrics is to measure the change in capacitance that results when a sample of the dielectric is inserted between the plates of a parallel-plate capacitor. Figure 3.4 illustrates the arrangement, which consists of a partially-ﬁlled parallel-plate capacitor, whose plates are maintained at a potential diﬀerence Va by an external voltage source. The purpose of the vacuum region within the plates is to allow the externally applied

ﬁeld, Ea, to be clearly discerned. This applied ﬁeld polarizes the bound charges in the dielectric (which, it is assumed, can be aligned) so that each dipole has an average dipole moment of dp. However, the adjacent polarized charges oppose each other, and so cancel Chapter 3. Fundamentals of Transmission-Line Metamaterials 37

+ + – + – + + + + + + + + + + + + + + + – –

+ – – – – V ρsp a + + + + – dp P Ea – – – – + + + + + + – – – – + – + – + + + + – – – – – – – – – – – – – – – – –

Air Region Dielectric Region

Figure 3.4: Determination of the permittivity of a dielectric inserted into a parallel-plate capacitor by measuring the change in capacitance. The source maintains a voltage Va across the plates.

nˆ Ba Ba ψ

I ds

Figure 3.5: Magnetization due to the application of a magnetic ﬂux Ba, modeled as an equivalent magnetic dipole consisting of a current I ﬂowing around a loop of area ds.

each other in the dielectric region. What remain are the bound charges on the surface

2 of the dielectric, which can be described by a bound surface charge ρsp [C/m ]; this is numerically equal to the net electric polarization P, which is directed with the applied

field: D = ²0Ea + P. Expressed in terms of the applied field, P = ²0χeEa, where the constant of proportionality χe is the electric susceptibility. Thus, the permittivity of the dielectric ² [F/m], defined by D = ²Ea, is ² = ²0(1 + χe). Chapter 3. Fundamentals of Transmission-Line Metamaterials 38

Air Region dm Jms

Magnetic Material

Figure 3.6: Determination of the permeability of a magnetic material inserted into a solenoid by measuring the change in inductance. The source maintains a current Ia in the windings.

3.2.2 Determination of Permeability

The atomic model of a magnetic material can be regarded to consist of a negatively charged electron orbiting a positively charged nucleus, which can, in turn, be perceived as a current flowing around a loop of area ds (in the direction opposite to the electron motion since current is defined as the direction of positive-charge flow), as shown in Fig.

3.5. This appears as a small magnetic dipole with an average magnetic dipole moment dm = Ids. A magnetic flux density Ba, aligns the magnetic dipoles with the fields (i.e., ψ → 0), such that the total magnetic dipole moment is in the direction of the applied field and given by a net magnetization M = Nmdmav = Nm(Ids)av, where dmav = (dm)ˆn. In analogy to the partially dielectric-filled parallel-plate capacitor of Fig. 3.4, consider the solenoid of Fig. 3.6 partially filled with a magnetic material whose permeability is desired to be measured. The current source maintains a current Ia in the solenoid windings; this current establishes the applied magnetic field Ha, which shall be associated with total magnetic fluxes Ba in the vacuum region and Bm in the magnetic-material region. The applied field induces small current loops (magnetic moments dm). However, the adjacent currents between loops travel in opposite directions, and so cancel each other out. The Chapter 3. Fundamentals of Transmission-Line Metamaterials 39 remaining currents are confined to the outer edge of the solenoid, and can be described by a surface current density Jms = M × nˆ|surface [A/m], which is related to a bound

2 volume current density Jm = ∇ × M [A/m ]. The total magnetic flux in the magnetic- material region can be expressed as Bm = µ0(Ha + M). Expressing the proportionality between the net magnetization and the applied field, M = χmHa (χm is the magnetic susceptibility), and defining the permeability of the material µ according to Bm = µHa yields µ = µ0(1 + χm) [H/m].

3.2.3 Positive Permittivity

Since the polarization vector P in polarizable dielectrics in the quasistatic limit is aligned with, and in phase with, the applied field Ea, the surface charges appearing on the upper and lower faces of the dielectric (coming into contact with the plates of the capacitor) neutralize some of the charge on the capacitor plates, which tends to reduce the field in the dielectric region. To maintain a field of Ea, the source Va must deposit more charge on the plates, which implies that the dielectric permits the storage of more energy in the system. The positive permittivity ² = ²0(1 + χe) is guaranteed by the positive electric susceptibility, arising from a positive P aligned with and in phase with Ea. The above expression for the permittivity is obtained by writing Ampère’sLaw in the dielectric in time-harmonic, differential form:

∇ × H = jωD = jω²0Ea + jωP = jω²0Ea + jω²0χeEa = jω²Ea (3.12)

3.2.4 Negative Permittivity

As noted previously, a negative permittivity is not uncommon in nature; plasmas (electrically neutral collections of mobile charged particles) can possess isotropic, naturally negative permittivities below their plasma frequency, ωp. Common examples of plasmas include the Earth’s ionosphere and noble metals at ultraviolet wavelengths. Chapter 3. Fundamentals of Transmission-Line Metamaterials 40

In an electrical plasma, the free electrons drift according to a drift current density

Jdrift. Assuming the plasma is collisionless, and that only the electrons are mobile,

Jdrift = Ne(−e)v (3.13)

where Ne, −e, and v are the electron density, electron charge, and electron drift velocity,

respectively. Now, F = mea = (−e)Ea, where me is the electron mass, and a is the acceleration given by the time derivative of v, a = jωv. This implies that

2 a Ne(−e) F Ne(−e) −eEa Nee Jdrift = Ne(−e) = = = Ea (3.14) jω jω me jω me jωme

So, whereas the displacement current Jdisp = jωD lags Ea in quadrature, the drift

current Jdrift evidently leads Ea in quadrature; thus, the two processes are out of phase with each other. Writing Amp`ere’sLaw in the plasma,

2 µ 2 ¶ Nee Nee ∇ × H = jωDa + Jdrift = jω²0Ea + Ea = jω ²0 − 2 Ea (3.15) jωme ω me

Thus, the eﬀective permittivity inside the plasma is given by

µ ¶ s 2 2 ωp Nee ² = ²0 1 − 2 , ωp = (3.16) ω ²0me

This permittivity is negative for ω < ωp. An intuitive explanation based on Fig. 3.4 can also be made [96], and is shown in Fig. 3.7. The motion of the negative charges alone

in the plasma has been represented more generally in Fig. 3.7 as the opposite motion of

both negative and positive charges. Keeping in mind that the charges in the plasma are

free charges, the 180-degree phase relationship between Jdisp and Jdrift terms suggests that the plasma charges drift opposite to the tendency of the applied ﬁeld to produce Chapter 3. Fundamentals of Transmission-Line Metamaterials 41

+ + + + + + + + + + + + + + + + + + + Va – Ea

– – – – – – – – – – – – – – – – – – –

Air Region Plasma Region

Figure 3.7: Free charges in a plasma (motion of mobile negative charges generally represented as opposite motion of both negative and positive charges) drift so as to lead the applied ﬁeld and enhance the ﬁeld between the plates, requiring the source to absorb charge in order to maintain a voltage Va.

a displacement. However, since the charges are not bound, they will naturally deposit themselves on the capacitor plates, which tends to increase the ﬁeld between the plates.

In order to maintain a ﬁeld of Ea between the plates, the source Va must actually absorb the excess charge, an action that may be interpreted as a reduction in the permittivity

(² < ²0). If the drift current deposits a charge on the plates equal to that initially supplied by the source to maintain Ea, then the source must withdraw all of its charge, since this ﬁeld is now being supported entirely by the plasma. In this case, the source can be removed, and the plasma is in a self-sustained state of oscillation (at its plasma frequency

ωp, where ² = 0). Below its plasma frequency, the plasma supplies more charge to the plates than does the source, and to maintain the ﬁeld Ea, the source must actually supply a negative charge ² < 0). In the last two cases, the medium appears to the source much unlike a capacitor; in fact, it behaves very much akin to an inductor, returning a current to the source under Lenz’s Law when excited, a result that is reminiscent of Bracewell’s conclusion [27]. This is explicitly seen by considering the eﬀect on the TL model lumped shunt admittance Y (that is, the lumped equivalent of a distributed admittance over a Chapter 3. Fundamentals of Transmission-Line Metamaterials 42

Ba Ba Ba

Iind Iind L C M ZL

M (a) (b) (c)

Figure 3.8: (a) Generalized magnetic medium for which the elemental particle consists of a loop whose total series impedance is lumped into ZL; (b) Field orientations for ZL > 0 (inductive loop); (c) Field orientations for ZL < 0 (capacitive loop).

unit cell of length d), when a capacitor (with a parallel-plate area-to-height ratio 1/q) is

ﬁlled with a plasma whose permittivity is given by (3.16). Therefore,

µ 2 ¶ 2 jω²d jω² d ω jω² d ²0ω d Y = = 0 1 − p = 0 + p (3.17) q q ω2 q jωq

in which Bracewell’s parallel shunt connection of a capacitor Cp = ²0d/q (representing

2 free space) and an inductor Lp = q/(ωp²0d) (representing the plasma) is evident.

3.2.5 Positive Permeability

The sign of the magnetic susceptibility χm, hence the sign of the magnetization vector M, decides whether the permeability of a magnetic material is less than or greater than

µ0. Materials in which µ is (typically only slightly) greater than µ0 are either paramagnetic or antiferromagnetic; materials in which µ is less than µ0 are called diamagnetic. For simplicity, consider an artiﬁcial magnetic medium consisting of electrically very small conducting loops, each as shown in Fig. 3.8(a), and imagine that such a medium now partially ﬁlls the solenoid of Fig. 3.6. For generality, the series impedance around each loop (which may include its parasitics and any other lumped loading) has been represented as a single lumped ZL, and indicated in the diagram by a square block. For simplicity, the remainder of this discussion shall assume that ZL is purely imaginary, Chapter 3. Fundamentals of Transmission-Line Metamaterials 43

with XL = Im{ZL}. Now, the magnetic dipole moment of a single loop is dm = Ids. If

2 the loops are assumed circular, then ds = πr0 ˆn, where r0 is the radius and ˆn is a unit vector normal to the plane of the loop deﬁned according to the right-hand rule. Thus, the

2 magnetization produced by a single loop is dm = Iπr0 ˆn = χmHa/Nm = χmBa/(Nmµ0),

2 which can be rewritten as Ba = Nmµ0Iπr0 ˆn/χm. Here, Nm is the total number of loops per unit volume. Faraday’s Law gives

I ZZ d d E · dl = − B · ds = − (B πr2) = V (3.18) dt dt a 0 emf, ind

2 In time-harmonic diﬀerential form, −jωBaπr0 = Vemf,ind = IZL. But, substituting Ba =

2 Nmµ0Iπr0/χm, the following expression for the magnetic susceptibility is produced:

2 2 2 2 −jωµ0Nm(πr0) −jωµ0Nm(πr0) ZL = ⇒ χm = (3.19) χm ZL

This yields a magnetization contribution dm given by

2 2 2 2 dm = χmHa/Nm = −jω(πr0) µ0Ha/ZL = −jω(πr0) Ba/ZL (3.20)

When the total number of loops per unit volume Nm is considered, a total magnetization results:

2 2 M = −jω(πr0) BaNm/ZL (3.21)

This corresponds to an eﬀective permeability given by

2 2 µ = µ0(1 − jωµ0(πr0) Nm/ZL) (3.22)

Here, the negative sign in the second term is indicative of Lenz’s Law in action. However,

since the sign of M · Ba is also dependent on the sign of XL, the following sections shall

examine the cases of XL > 0 and XL < 0 separately. Chapter 3. Fundamentals of Transmission-Line Metamaterials 44

Case 1: Inductive Loop

The condition XL > 0 implies that the loop impedance is inductive. If ZL = jωL, then

−jω(πr2)2B −B dm = 0 a = a (πr2)2 (3.23) jωL L 0

Summing over Nm such loops yields

−B N M = a m (πr2)2 (3.24) L 0

and µ ¶ (πr2)2 µ = µ 1 − µ N 0 (3.25) 0 0 m L

Equation (3.24) suggests that M opposes Ba. In this case, the ﬁelds in the vicinity of

the loop are as shown in Fig. 3.8(b). However, |M| is typically much smaller than |Ba|,

and so µ is only marginally smaller than µ0 as a result of M being negative. This is a typical, and typically weak, diamagnetic response, commonly seen in artiﬁcial dielectrics

with metallic inclusions. In this case, the magnetic surface current produced due to the

magnetization M opposes the source current Ia, and the source must deliver more current to maintain the ﬁeld in the material.

Case 2: Capacitive Loop

The condition XL < 0 implies that the loop impedance is capacitive. If ZL = 1/jωC, then −jω(πr2)2B dm = 0 a = ω2B C(πr2)2 (3.26) 1/jωC a 0

Summing over Nm such loops yields

2 2 2 M = ω BaNmC(πr0) (3.27) Chapter 3. Fundamentals of Transmission-Line Metamaterials 45

and

2 2 2 µ = µ0(1 + ω µ0NmC(πr0) ) (3.28)

Equation (3.27) suggests that M and Ba are co-directed. In this case, the ﬁelds are

as shown in Fig. 3.8(c). Again, |M| is typically much smaller than |Ba|, so µ is only

slightly larger than µ0 due to the positive nature of M. This is typical of a paramagnetic

response, for which the source must absorb the surface current to maintain Ba.

3.2.6 Negative Permeability

It is clear that the permeability typically remains near µ0 in both of the previous cases. Here, following the suggestion of Schelkunoﬀ and Friis [33], it should be interesting to

consider a resonant case:

Case 3: Capacitively Loaded Inductive Loop

Consider now that the loop possesses a total impedance with both inductive and capac-

2 2 itive parts, such that ZL can be expressed as ZL = jωL + 1/jωC = jωL(1 − ω0/ω ), √ where ω0 = 1/ LC is the angular loop resonance frequency. Therefore,

2 2 −jω(πr0) Ba −Ba 2 2 1 dm = 2 2 = (πr0) 2 2 (3.29) jωL(1 − ω0/ω ) L (1 − ω0/ω )

Summing over Nm loops, 2 2 BaNm(πr0) M = − 2 2 (3.30) L(1 − ω0/ω )

and µ 2 2 ¶ µ0Nm(πr0) µ = µ0 1 − 2 2 (3.31) L(1 − ω0/ω )

Now, the sign of M · Ba in (3.30) varies with frequency. For ω >> ω0, M · Ba is negative,

reducing to Case 1, and for ω << ω0, M · Ba is positive, reducing to Case 2. However,

− as ω → ω0 , µ → +∞. That is, M · Ba is strongly positive, and the source is required Chapter 3. Fundamentals of Transmission-Line Metamaterials 46 to supply a nearly inﬁnite current orbiting in the counterclockwise direction in order to

+ maintain the ﬁeld. As ω → ω0 , µ → −∞. In this case, M · Ba is strongly negative, and the source supplies a nearly inﬁnite current orbiting the loop in a clockwise direction.

It should be added that the currents are inﬁnite because of the absence of loss in the development, which is easily lumped directly into ZL.

In the frequency region in which µ < 0, M and Ba are oppositely directed, but

|M| > |Ba|. This implies that the magnitude of the magnetic surface current due to M is greater than the current being supplied by the source. In order to maintain the

ﬁeld, the source must actually generate a stronger positive current; for example, when the surface current produced by M becomes equal to the source current, the only way for the source to maintain Ba is to double its supplied current. In this case, the source appears to ‘charge’ the system. Thus, in the µ < 0 region, the system of loaded loops inside the solenoid behaves like a charging capacitor, wherein the source continuously couples energy into the loops.

3.2.7 Equivalent Circuit

In Bracewell’s equivalent circuit for a plasma, the inductive component of the shunt branch was revealed by replacing the vacuum inside the distributed capacitance representing free-space propagation with a plasma. In analogy, consider the eﬀect of ﬁlling the free-space distributed series inductor with a magnetic material possessing the permeability function of (3.31). Representing the TL equivalent lumped series impedance as

Z = jωµd/q (corresponding to a distributed series impedance lumped over a unit cell of Chapter 3. Fundamentals of Transmission-Line Metamaterials 47

Cs Y

Figure 3.9: TL equivalent lumped circuit representing a medium achieving a negative permeability through an L–C resonance mechanism.

length d), where 1/q is a factor dependent on the geometry of the equivalent TL system,

µ 2 2 ¶ jωµ0d µ0Nm(πr0) Z = 1 − 2 2 q L(1 − ω0/ω ) jωµ d ω2N (µ πr2)2d/q = 0 + m 0 0 q jωL + 1/jωC 2 (jωLM ) = jωLsd − (3.32) ZL

The new symbols introduced in the last line of (3.32) reveal that Z is, in fact, the image

impedance of the free-space lumped series inductance Ls = µ0d/q coupled to the ring p 2 impedance ZL = jωL + 1/jωC via a lumped mutual inductance LM = µ0πr0 Nmd/q. Thus, when this system of loaded rings is placed in free space (with a permittivity represented by the shunt capacitance Cs), and neglecting the electrical response, if any, of the rings, the TL equivalent circuit representing propagation through the system is as shown in Fig. 3.9.

The image impedance describes the process of inductive coupling by which the array of loops is perceived by its environment (e.g., free space or a parallel-plate waveguide).

This impedance possess two distinct resonance frequencies: a ‘ring’ resonance at which Chapter 3. Fundamentals of Transmission-Line Metamaterials 48

the total loop impedance ZL = 0, and a ‘plasma’ resonance at which the total series impedance, or image impedance, Z = 0.

The ring resonance is a property of the loop alone and does not depend heavily on the interaction of the loop with its environment. It represents a transition from negative reactance to positive reactance, and thus, corresponds to a condition of infinite current, since the impedance of a lossless loop seen by a source in the loop is zero at resonance. However, the image impedance perceived by the environment when the loop is at resonance becomes infinite; this can be physically interpreted as energy being coupled from the environment into the loops and trapped. Here, the effective permeability can be interpreted as infinite, since all energy supplied is stored in the resonating loops.

The plasma resonance describes the interaction of the loop with its environment and occurs at a (typically only slightly) higher frequency than the ring resonance of the loop.

At the plasma resonance, the image impedance perceived by the environment is zero. A physical interpretation of this condition may be derived from the observation that a zero image impedance implies a conservation of the ﬁelds across the length of the unit cell.

This is achieved because the coupling of energy from the environment into the loop is accompanied by a commensurate return of energy from the loop to the environment that, furthermore, negates the accumulation of emf simply by virtue of propagation across the length of the unit cell. Here, the eﬀective permeability can be interpreted as zero, since there appears to be no net induced emf (conservative ﬁelds). Chapter 3. Fundamentals of Transmission-Line Metamaterials 49

3.3 TL Metamaterials and the Split-Ring Resonator

Connection

3.3.1 The Recipe for Broadband, Low-loss Left-handedness

Figure 3.9 is a variant of the TL model of the wire/SRR medium presented in Refs. 97 and

98, in which the shunt branch additionally contains an inductive contribution, serving to negate the permittivity. In the last reference, it was shown that the frequency region in which the eﬀective permeability function of (3.32) assumed negative values corresponds the frequency region in which the series branch of Fig. 3.9 is capacitive, as dictated by the TL model of NRI metamaterials. However, as was previously mentioned, this necessary series capacitance is obtained between two (often closely spaced) resonances, which implies that the mechanism by which this is achieved is inherently narrowband and lossy unless steps are taken to maximize the available bandwidth. This is very much unlike the TL metamaterial, which exhibits broadband and low-loss negative material parameters, in spite of its L–C topology, and so it should be interesting to consider how or whether the two can be related.

The TL model of conventional dielectrics possesses a ‘low-pass’ topology, and the dual LH model possess a ‘high-pass’ topology. However, it is clear that neither ideal TL possesses any cutoﬀ frequency; in fact, in this ‘continuous’ limit, they are inﬁnitely broadband, as shown in Sec. 3.1. This attribute is a result of the fact that the individual L–C resonators (unit cells) comprising the lines are so tightly coupled by virtue of electrical connection to their adjacent counterparts that their local resonance property is converted into a propagation phenomenon. To illustrate, consider the 1D periodic array of loops

(period d) shown in Fig. 3.10(a). As before, the loops possess a self-inductance L and are loaded by a lumped capacitance C described by a total loop impedance ZL, and they can, without too great a loss of generality, be regarded as circular loops of radius r0. As dis- Chapter 3. Fundamentals of Transmission-Line Metamaterials 50

ZL ZL ZL

. . . ro . . .

(a)

2C0 2C0 2C0 2C0 2C0 2C0

. . . L0 L0 L0 . . .

d (b)

Figure 3.10: (a) 1D array of mutually coupled capacitively loaded inductive loops; (b) Unit cell for a continuous LH TL, formed by connection of loops in (a).

crete resonators, the loops provide a negative eﬀective permeability within a well-deﬁned

(typically narrow) bandwidth. However, consider the limit in which d → 2r0; in this limit, the loops can no longer be regarded as discrete resonators since their mutual coupling

becomes signiﬁcant. Indeed, as shown in Fig. 3.10(b), it can be appreciated intuitively

that the isolated loops become conjoined and form what appears to be a LH TL, where

the capacitive loading C0 appears in series and the inductance of the loops L0 = L/2 appears in shunt. Thus, it appears that it may be possible to produce left-handedness

from capacitively loaded loops alone, provided that the loops are tightly coupled to each

other. This conclusion was reported in Ref. 48, which derived the following dispersion

equation for the coupled-loop structure of Fig. 3.10(a):

µ 2 ¶ 2 2 L ω0 (πr0) cos(βd) = − 1 − 2 ,Mc = −µ0 3 (3.33) 2Mc ω 2πd

where Mc can be regarded as the mutual inductance linking adjacent loops and the factor Chapter 3. Fundamentals of Transmission-Line Metamaterials 51

p = −L/2Mc can be called the normalized coupling coefficient. It was shown in Ref. 48 that the LH bandwidth becomes infinitely large as p → 1; i.e., as the discrete-resonance property of the rings gives way to their mutual coupling. Conversely, when the loops are completely decoupled (p → ∞), the LH bandwidth is diminished to zero. It was also suggested that the loss associated with a particular Q is minimized as p → 1, the same condition that yielded a maximum bandwidth. Of course, this analysis does not directly account for retardation effects that appear in practical, periodic implementations of the dual TL metamaterial employing a host (RH) TL medium. These effects can be shown to impose further constraints on bandwidth and loss that may be mitigated by minimizing their electrical length (the interested reader is referred to Ref. 48 for further details). The effect of the requisite host medium on the dispersion properties of the dual TL is briefly noted at the end of this chapter and discussed further in Chapter 4. Nevertheless, based on this work, it may be concluded that it is not the absence of resonators that makes

TL metamaterials simultaneously broadband and low-loss; rather, it is the tight coupling between their constituent lumped resonant sections.

This type of dispersion bears interesting similarities to two others described in the literature: the ﬁrst is the dispersion of the magnetoinductive surface wave [99] supported by a chain of coplanar loops, and the second is the dispersion of the continuous 1D dual

TL, which has been described as the limiting case of connected coplanar loops. Both types of dispersion describe LH propagation.

3.3.2 Free-Space Coupling to a TL-Based Metamaterial

The structure shown in Fig. 3.10 describes surface-wave-like propagation along a tightly coupled array of loops, resembling a 1D TL metamaterial, which inherently supports

LH propagation. However, this is limited to 1D and so does not oﬀer any insight into coupling into such a metamaterial from free space. On the other hand, Fig. 3.9 describes coupling between free space and isolated loaded loops resulting in a frequency region Chapter 3. Fundamentals of Transmission-Line Metamaterials 52

ZML ZML

Ir0 Ir−−− Ir+ ZL

I1 Ih ZMh I2

+ Zh +

V1 Yh/2 Yh/2 V2

−−− −−−

Figure 3.11: Unit cell modeling coupling from free space (represented as a π-network with impedance Zh and admittance Yh) to a TL metamaterial represented as an array of tightly coupled rings.

of negative permeability, but requires a negation of the permittivity (e.g., using wires) in order to produce LH propagation. In preparation for the forthcoming discussion of volumetric structures, it should be interesting to consider the propagation characteristics of a bulk mode coupling from free-space into a coupled array of loaded loops, speciﬁcally to determine whether such a system could also produce large LH bandwidths and low losses, and whether additional wires would be needed. For this purpose, consider the unit cell shown in Fig. 3.11. The mutual impedance coupling the host free-space TL

(represented as a π-model with impedance Zh and admittance Yh) to the array of loaded loops is ZMh, and the mutual impedance coupling each loop to its neighbours is ZML. Invoking the Floquet-Bloch boundary conditions shown in (3.34) yields the dispersion of Chapter 3. Fundamentals of Transmission-Line Metamaterials 53 the complex propagation constant γ shown in (3.35):

−γd V2 = V1e

−γd I2 = I1e

+γd Ir− = Ir0e

−γd Ir+ = Ir0e (3.34)

· µ ¶¸ · ¸ Y Z Z Y Z2 cosh γd − 1 + h h cosh γd − L = h Mh (3.35) 2 2ZML 4ZML

The host TL and the loop array may be regarded to be decoupled when ZMh = 0, which renders the right-hand side of (3.35) null. In this case, the dispersion equation is satisﬁed by the existence of two decoupled modes:

Y Z cosh γd = 1 + h h = cosh γ d (3.36) 2 h ZL cosh γd = = cosh γLd (3.37) 2ZML

To properly interpret these results, consider the typical case in which the mutual impedances

ZMh and ZML are inductive. In this case, (3.36) describes propagation through free space, unloaded by the loop array, where the free-space inductance and capacitance embedded in Zh and Yh, respectively, are positive; equation (3.37) is identical to (3.33) in the lossless case for γ = jβ with p = −L/2Mc = 1 and describes LH propagation in the coupled

−1 loops, where ZML can be interpreted as the shunt inductance required to produce a negative permittivity, and where the series capacitance required for a negative permeability is obtained directly from the ring capacitance C embedded in ZL. Hence, (3.35) can be rewritten in a simpler form:

2 YhZMh [cosh γd − cosh γhd] [cosh γd − cosh γLd] = (3.38) 4ZML Chapter 3. Fundamentals of Transmission-Line Metamaterials 54 which illustrates that this is a system of coupled modes in which the free-space bulk mode is forward (RH) and the coupled-loop mode is backward (LH). The right-hand side of (3.38) can be regarded as the coupling constant. The frequency at which the coupling is strongest is obtained from the condition cosh γLd = cosh γhd, which represents a condition of phase-matching (conservation of the wave momentum) between the various spatial harmonics of the two lines.

To test these ideas, let ZMh and ZML correspond to inductances LMh and LML. Furthermore, it shall be insisted that C = 1pF and L = 25.33nH, so that the loop resonance frequency ω0 = 2π × 1GHz, and that the per-unit-length inductance and capacitance of the host TL, Lh and Ch, are equal to the free-space permeability and permittivity, respectively (i.e., Z0 = η0 in this case). The periodic structure shall also be assigned a period of d = 20mm, which is one-ﬁfteenth of the free-space wavelength at the loop resonance frequency. The discussion that follows examines various combinations of the mutual coupling factors LMh and LML.

Case 1: Decoupled System

It was shown that the host TL and loop array may be decoupled by setting ZMh = 0. In this case, one would expect to see a forward-wave (RH) dispersion curve representing free-space propagation and a backward-wave (LH) dispersion curve whose bandwidth depends on the mutual inductance LML between adjacent loops. Accordingly, setting

LMh = 0 and choosing LML from the case −L/2LML = p = 1.5, one obtains the dispersion curves shown in Fig. 3.12 (dark curves – propagation constant; light curves – attenuation constant). Indeed, the backward-wave curve does not interact with the forward-wave dispersion of the host TL. Chapter 3. Fundamentals of Transmission-Line Metamaterials 55

1.5

1 Frequency (GHz) 0.5

0 −3 −2 −1 0 1 2 3 ±αd (light), βd (dark)

Figure 3.12: Dispersion diagram showing the solutions to (3.38) when the host TL and coupled-loop array are decoupled (LMh = 0) but the loops are coupled to their nearest neighbours, for −L/2LML = p = 1.5 (dark curves – propagation constant; light curves – attenuation constant).

Case 2: General Coupled System

A more general case is one in which the host TL and coupled-loop array are coupled by a mutual inductance (chosen to be LMh = 0.125Lh), and once again only the case

−L/2LML = p = 1.5 is considered. The resulting dispersion curves are shown in Fig. 3.13, and possess a number of very interesting features. First, the forward wave and backward wave interact most strongly where their dispersion curves intersect (i.e., where the modes are phase-matched); away from the point of intersection, the two coupled dispersions approach their isolated values, and at any given frequency can be described as either propagating (γ = jβ) or evanescent (γ = α). However, inside the region of interaction, both modal solutions are complex, in that they possess both propagating and evanescent features. Such complex solutions are observed whenever there is coupling between backward and forward waves, as in conventional leaky-wave antennas operating on negative (backward-wave) spatial harmonics [100] and their TL-metamaterial counterparts operating on a fundamental negative spatial harmonic [78, 81], the dispersions of the shielded Sievenpiper high-impedance surface [101], and in the operation of high- Chapter 3. Fundamentals of Transmission-Line Metamaterials 56

1.5

1 Frequency (GHz) 0.5

0 −3 −2 −1 0 1 2 3 ±αd (light), βd (dark)

Figure 3.13: Dispersion diagram showing the coupled dispersion curves of the host TL and coupled-loop array when LMh = 0.125Lh, for −L/2LML = p = 1.5 (dark curves – propagation constant; light curves – attenuation constant).

directivity TL-metamaterial coupled-line couplers [76]. Indeed, the model of Fig. 3.11 can be used to represent any of these systems in the quasistatic limit (a more detailed and accurate treatment could be achieved using multiconductor-TL (MTL) analysis that would account for both inductive and capacitive coupling between the lines as well as their electrical length – for example, see Ref. 102).

Case 3: Isolated L–C Resonator Limit

The other interesting case to consider is that in which ZMh remains ﬁnite, but ZML → 0, suggesting that, although there is coupling between the host TL and individual loops, the loops in the array do not couple to their adjacent neighbours and so appear as isolated

L–C resonators. From (3.37), it is observed that this results in cosh γLd → ∞, and it can be shown that the only propagating solution for γd is as follows:

µ 2 ¶ 1 ZMh cosh γd = 1 + Yh Zh − (3.39) 2 2ZL Chapter 3. Fundamentals of Transmission-Line Metamaterials 57

1.5

1 Frequency (GHz) 0.5

0 −3 −2 −1 0 1 2 3 ±αd (light), βd (dark)

Figure 3.14: Dispersion diagram showing the coupled dispersion curves of the host TL and loop array (LMh = 0.5Lh) when the loop-to-loop mutual inductance LML = 0. (darkly shaded curves – propagation constant; lightly shaded curves – attenuation constant).

Expectedly, this is the dispersion relation corresponding to the unit cell of Fig. 3.9,

2 and the factor Zi = Zh − ZMh/2ZL is the image impedance of the isolated loop seen via the mutual impedance ZMh. It is known that Zi < 0 corresponds to a negative effective permeability. Setting LMh = 0.5Lh and LML = 0 (p → ∞) in (3.38) yields the dispersion curves shown in Fig. 3.14. Here, the solutions in all frequency regions are either propagating or attenuating (as in Fig. 3.12), and the inherent backward-wave bandwidth observed when the loops were tightly coupled has diminished to zero. Instead, the dispersion flattens out at the loop resonance frequency, ω0, leading to a stopband (containing evanescent solutions only) which, according to (3.31), represents a frequency region of negative effective permeability. Propagation is restored at a frequency ωp > ω0 at which the free-space series inductance Lh dominates and the effective permeability is once again positive.

In summary, when the loops are tightly coupled to one another, the propagation characteristics are dominated by the coupled-loop array, which is akin to a dual LH

TL possessing a simultaneous negative permittivity and permeability. Accordingly, a broadband LH region is produced, but it is interrupted by the stopband produced due Chapter 3. Fundamentals of Transmission-Line Metamaterials 58 to coupling with the free-space mode. External wires are required to restore propagation in this region, and the adopted interpretation of their function is that they ‘cut oﬀ’ the free-space mode [5]. On the other hand, weakening the coupling between loops in the array gives way to a dispersion akin to that of an isolated SRR that is described by a negative permeability alone, which represents a condition in which propagation is forbidden. Once again, external wires are required to restore propagation in this region, but their function is now best interpreted as providing a negative permittivity.

In general, it may be said that the dispersion properties of the coupled system are a hybrid between those described by the unit cells in Figs. 3.10 and 3.9, and that the stopband generally contains complex solutions for various degrees of loop-to-loop coupling. Thus, although the coupling phenomena shown in Figs. 3.14 and 3.13 appear to be diﬀerent, the above development proves that they are fundamentally one and the same. This is aﬃrmed by the fact that the frequency of intersection of the forward-wave and backward-wave dispersions (the solution to the condition cosh γLd = cosh γhd) approaches ω0 as LML → 0. Furthermore, it may be concluded that any left-handedness built inherently into the system is necessarily somewhat compromised by the requirement of coupling to such a mode from free space, and external wires are, it seems, unavoidable, particularly in the application of broadband metamaterials for free-space lensing or superlensing, where the frequency of operation is identically equal to the frequency of intersection between the backward-wave and free-space dispersion curves.

3.4 Practical Realization of TL-Based LH Metama-

terials

The unique properties of the dual LH topology described in Section 3.1 were obtained in the continuous limit (d/λ → 0), in which the topology models what may be described as a ‘purely’ LH material. However, moving from modeling to synthesis, any practical Chapter 3. Fundamentals of Transmission-Line Metamaterials 59 realization of such a structure, as in the case of the SRR/wire metamaterial, is a periodic one, and as such, must contain some RH component to its dispersion. In the case of the SRR/wire metamaterial, this RH contribution is provided by the air or dielectric separating the wires and resonant inclusions. Indeed, their eﬀective permittivity and permeability are determined such that they approach the free-space or host-dielectric values at the high-frequency limit. Similarly, in a L–C-based periodic implementation, the RH component is provided by a host TL medium which may be appropriately loaded at regular intervals using lumped inductors and capacitors. This model, known as the

NRI-TL, is a hybrid model that rigorously accounts for the distributed eﬀects of the host medium and the lumped nature of the series-capacitive and shunt-inductive loading. As a result, its validity is not limited to the continuous, or homogeneous, limit, although this is often the regime of greatest interest when seeking eﬀective-medium properties like permittivity, permeability, and refractive index. The following chapter discusses a practical, periodic implementation of the LH medium using a fully uniplanar NRI-TL metamaterial, chosen to facilitate a multilayer implementation that will ultimately be used to demonstrate free-space lensing and superlensing. Chapter 4

Uniplanar Transmission-Line

Metamaterials

The work of Kron, Ramo, and Whinnery in the modeling of conventional dielectrics using distributed TL networks was employed by Johns in his formulation of the TL- matrix (TLM) method of modeling electromagnetic phenomena in the time-domain [103].

Johns identiﬁed the speciﬁc unit cell topologies for such networks for both TE and TM polarizations. The former is known as the series TLM node, and the latter as the shunt

TLM node; they are depicted for the 2D case in Figs. 4.1(a)–4.1(b). In both cases, the capacitive and inductive elements directly determine the constitutive parameters— the desired permittivity and permeability, respectively—of the corresponding eﬀective medium. In realized form using TLs, it is clear that Fig. 4.1(a) is simply the series interconnection of four TLs, whereas Fig. 4.1(b) is a shunt interconnection of four TLs.

Hence, the two topologies shall be referred to as the series TL unit cell and the shunt TL unit cell, respectively.

As noted in Chapter 1, 3D-isotropic TL-based topologies have been proposed [20–22], but they are diﬃcult to fabricate by lithographic techniques alone. By restricting polarization and limiting directions of propagation, practical structures that can be readily

60 Chapter 4. Uniplanar Transmission-Line Metamaterials 61

z z y y

x x (a) (b)

Figure 4.1: (a) 2D series TL unit cell. (b) 2D shunt TL unit cell. Reprinted with permission from Ref. 60, copyright °c 2006 Optical Society of America.

Figure 4.2: Creating a volumetric medium by layering 2D planes. Reprinted with permission from Ref. 60, copyright °c 2006 Optical Society of America.

fabricated with prevalent lithographic techniques may become available. Moreover, certain 3D topologies may, conceivably, be realized by appropriately interlocking these 2D layers in orthogonal planes. Thus, both structures can be based on the 2D topologies already employed for planar phenomena. To this end, consider a 3D network constructed by layering planes of 2D unit cells, as shown in Fig. 4.2. Such a network could, under certain conditions, be designated an eﬀective medium for propagation along the planes, and would best be termed a volumetric layered 2D medium. Although the shunt NRI-

TL topology can be symmetrized by distributing the series inductances across the signal and return paths, it is evident that these two paths must be non-coplanar. As shown in Fig. 4.1(a), this is avoided in the series TL unit cell, making it an ideal topology with which to construct a volumetric layered medium. Consider Fig. 4.3, which depicts Chapter 4. Uniplanar Transmission-Line Metamaterials 62

Figure 4.3: Array of 2D series TL unit cells with generalized lumped loading. The lightly shaded region views the unit cell as the series interconnection of four two-wire lines, and the darkly shaded region views the unit cell as a loaded ring. The magnetic ﬁeld of the impinging plane wave is normal to the plane of the page. Reprinted with permission from Ref. 60, copyright °c 2006 Optical Society of America.

a periodic array of series TL unit cells. In comparing a single cell in the array (lightly shaded region) to Fig. 4.1(a), the reader will notice that the lumped loading has been generalized in preparation for the analysis to follow. Previously, it was noted that this unit cell may be perceived as the series connection of four TL segments. It is also evident from Fig. 4.3 that the unit cell for this array may alternatively be perceived as a ring of impedances connected to adjacent rings via admittances (darkly shaded region). The voltages along the TLs and the opposite currents on the conductors of each two-wire TL segment could alternatively be visualized as a potential difference in the gaps between loops and circulating current induced in each ring by the electric and magnetic fields, respectively, of an impinging plane wave. It will prove useful to refer to both perspectives Chapter 4. Uniplanar Transmission-Line Metamaterials 63 throughout the course of the following analysis. From either perspective, it is evident that the efficacious polarization is that for which the magnetic field lies perpendicular to the ring plane, and the electric field lies within the plane (the propagation vector also lies in the plane), which shall be referred to hereafter as the TE-polarization.

Does the 2D series TL unit cell, like the shunt TL unit cell, also have a ‘dual’ topology capable of producing a LH, or NRI response? Furthermore, can a volumetric NRI medium be created by stacking layers of dual series TL arrays? This chapter deals with the former question by developing the dispersion theory and eﬀective-medium properties of the single-layer series NRI-TL network. The latter question is addressed in Chapter 5, where the single-layer NRI-TL network is used to design a volumetric free-space NRI-TL metamaterial.

4.1 Dispersion Characteristics

This section develops the dispersion characteristics of the single-layer 2D NRI-TL medium based on the series TL network of Fig. 4.3, which will constitute the layers of the proposed volumetric medium. Although this structure may be analyzed in a number of ways, the most simple and instructive employs the TL series junction picture. The following development, a variation of which has also been reported in Ref. 104, is based on a transfer-matrix (ABCD) analysis of periodic networks entirely analogous to that presented for the shunt TL unit cell in Ref. 105. An effective-medium regime will be identified and the constitutive parameters of the effective medium will be determined in this regime (see Appendix B for a discussion on the validity of the effective-medium perspective for TL networks). The transfer matrix definition, which relates the input Chapter 4. Uniplanar Transmission-Line Metamaterials 64

Iin Iout

+++ A B +++ Vin Vout −−− C D −−−

Figure 4.4: Deﬁnition of the ABCD transfer matrix for a two-port network. Reprinted with permission from Ref. 60, copyright °c 2006 Optical Society of America.

and output terminal voltages and currents of a two-port network is shown in Fig. 4.4:        Vin   AB   Vout    =     (4.1) Iin CD Iout

The four-port unit cell for the structure to be analyzed is depicted in Fig. 4.5; in the present case, the transmission matrix orientations and current reference directions have been chosen to exploit symmetry. The input and output voltage and current quantities

(ports {1,2} and {3,4}, respectively) are related through an overall phase shift represented

−jβpd by the complex exponential e , where βp is the Floquet-Bloch wavevector in the direction p = {x, y} and d is the physical cell dimension. That is,

     V1   Vx    =   I1 Ix      V2   Vy    =   I2 Iy      V3   Vx    =   e−jβxd I3 Ix      V4   Vy    =   e−jβyd (4.2) I4 Iy Chapter 4. Uniplanar Transmission-Line Metamaterials 65

+++ V4 −−− I4

4 4 A C

4 4 B D

I1 I3

+++ A1 B1 B3 A3 +++ 1 V1 V3 3 −−− C1 D1 D3 C3 −−−

2 2 B D

2 2 A C y

I2 x

+++ V2 −−−

Figure 4.5: Periodic analysis of the generalized 2D series TL unit cell. Reprinted with permission from Ref. 60, copyright °c 2006 Optical Society of America.

The branches comprising the series TL unit cell are now speciﬁed: each shall consist of a TL of characteristic impedance Z0 and electrical length θ/2 separating a series impedance Z/4 (distributed across both branches of the TL as two elements of value Z/8, and yielding a total series impedance around the unit cell of Z) and a shunt admittance

Y/2 (which, combined with the shunt admittance of the adjacent cell, yields a total admittance of Y ). The cascade of these networks possesses a transmission matrix with

Chapter 4. Uniplanar Transmission-Line Metamaterials 66

Iin Iout Z/8 +++ +++ Vin Y/2 Z0, θθθ/2 Vout −−− −−− Z/8

Figure 4.6: Speciﬁcation of the constituents of the branches of the series TL unit cell depicted in Fig. 4.5. Reprinted with permission from Ref. 60, copyright °c 2006 Optical Society of America.

the following elements:

θ A = cos i 2 Z θ θ B = i cos + jZ sin i 4 2 0 2 Yi θ θ Ci = cos + jY0 sin µ2 2 ¶ 2 µ ¶ ZiYi θ j Z Y θ Di = 1 + cos + + sin (4.3) 8 2 2 2Z0 Y0 2 where it is assumed that all branches are constructed using TLs of the same characteristic impedance and electrical length. It is also clear from (4.3) that the parameters B and

C are concerned solely with the impedance Z and admittance Y , respectively, and A is independent of either. The parameter D may be obtained from the other three parameters using the reciprocity condition.

The analysis to follow considers the case of a uniaxial network consisting of reciprocal branches (AiDi − BiCi = 1); that is,        A1 B1   A3 B3   Ax Bx    =   =   C1 D1 C3 D3 Cx Dx        A2 B2   A4 B4   Ay By    =   =   (4.4) C2 D2 C4 D4 Cy Dy

Chapter 4. Uniplanar Transmission-Line Metamaterials 67

βββyd

Y M

ΓΓΓ X βββxd

Figure 4.7: Brillouin zone boundary for 2D rectangular lattice indicating high symmetry points. Reprinted with permission from Ref. 60, copyright °c 2006 Optical Society of America.

Applying Kirchhoﬀ’s current and voltage laws at the terminal intersections yields four

independent equations, whose solution gives both the Floquet-Bloch propagation con-

stant and the Bloch impedances—the characteristic impedances seen by Floquet-Bloch

waves propagating in the x- and y- directions. The 2D uniaxial dispersion relation can,

therefore, be written as follows:

[AyCy{(1 − cos βxd) + 2BxCx} + AxCx{(1 − cos βyd) + 2ByCy}] = 0 (4.5)

It is also appropriate to derive general expressions for the Bloch impedances:

Ax βxd Z = Vx = j tan Bx Ix Cx 2 A β d Z = Vy = j y tan y (4.6) By Iy Cy 2

The dispersion relations of 2D periodic structures are often described by frequency contours in reciprocal space. The reciprocal space for a square lattice is shown in Fig. 4.7.

The labels in the Figure correspond to the points of high symmetry, and the conditions for each are listed in Table 4.1. These points represent Bragg frequencies—the onset of stopbands produced by the contradirectional coupling between diﬀerent Floquet-Bloch Chapter 4. Uniplanar Transmission-Line Metamaterials 68

Γ βxd = 2nπ βyd = 2nπ X βxd = (2n + 1)π βyd = 2nπ Y βxd = 2nπ βyd = (2n + 1)π M βxd = (2n + 1)π βyd = (2n + 1)π

Table 4.1: Description of the symmetry points of the 2D Brillouin zone for a rectangular lattice (n is an integer corresponding to a particular Floquet-Bloch spatial harmonic). Reprinted with permission from Ref. 60, copyright °c 2006 Optical Society of America.

Cases Propagation Angle Dispersion Relation Isotropic n/a (βd)2 = −8BC 2 Anisotropic φ = 0 (βd) = −4Cx{Bx + By} 2 φ = π/2 (βd) = −4Cy{Bx + By} φ = π/4 (βd)2 = −8CxCy{Bx+By} Cx+Cy

Table 4.2: Dispersion relations for isotropic and anisotropic cases (selected propagation angles). Reprinted with permission from Ref. 60, copyright °c 2006 Optical Society of America, Inc.

spatial harmonics—that can be obtained by substituting the appropriate conditions into

(4.5). Of course, the most interesting propagation characteristics for the present pur-

poses are achieved near Γ, which, at low frequencies, will be shown to correspond to the

eﬀective-medium limit. The vicinity of Γ is described by the conditions

(β d)2 cos(β d) ≈ 1 − x x 2 (β d)2 cos(β d) ≈ 1 − y (4.7) y 2

Representing the propagation directions through an azimuthal propagation angle φ such

that βx = β cos φ and βy = β sin φ, and noting that Ax = Ay, the dispersion equation of (4.5) near Γ yields −4C C {B + B } (βd)2 = x y x y (4.8) 2 2 Cx sin φ + Cy cos φ

Table 4.2 presents the simpliﬁcation of (4.8) in the isotropic case (Bx = By = B,

Cx = Cy = C) and in the anisotropic case for interesting propagation angles. It is Chapter 4. Uniplanar Transmission-Line Metamaterials 69

relevant to note at this point that the eﬀect of the series loading is completely embedded

in the parameter B, whereas that of the shunt loading is embedded in C. When the

network is isotropic, it is clear that all directions of Floquet-Bloch wave propagation

possess the same dispersion characteristics near Γ. The dispersion characteristics for

axial propagation (φ = 0 and φ = π/2) depend on the shunt loading corresponding to

that axis, but depend on the series loading on both axes. Finally, propagation along the

grid diagonals is aﬀected by the series and shunt elements on both axes.

The results are now simplifed for the isotropic case. Substituting the appropriate

expressions for B and C from (4.3) into the isotropic dispersion relation listed in Table 4.2

and factoring yields the complete axial dispersion relation in the vicinity of Γ:

µ ¶ µ ¶ θ Z θ θ Y θ (βd)2 = 4 sin − j cos 2 sin − j cos (4.9) 2 Z0 2 2 Y0 2

The zeros of this function identify the edges of stopbands near Γ, of which there are

an inﬁnite number. The edges of these stopbands are given by the solutions to the

transcendental equations corresponding to each factor:

θ Z tan = j 2 4Z0 θ Y tan = j (4.10) 2 2Y0

These conditions may also be cast as transverse-resonance conditions corresponding to

the TL systems shown in Figs. 4.8(a) and 4.8(b):

Z + Z = 0 4 in,sc Y + Y = 0 (4.11) 2 in,oc

The reader will also note that the TL systems of Figs. 4.8(a) and 4.8(b) may be obtained

from the unit cell shown in Fig. 4.6 by establishing open-circuit conditions and short-

Chapter 4. Uniplanar Transmission-Line Metamaterials 70

Zin,sc Yin,oc

Z/4 Z0, k s.c. Y/2 Z0, k o.c.

d/2 d/2 (a) (b)

Figure 4.8: Representation of transverse-resonance conditions (4.11) describing the edges of the Γ-point stopbands (s.c. ≡ short circuit; o.c. ≡ open circuit).

circuit conditions, respectively, simultaneously on the input and output terminals.

In this regime, the Bloch impedances also have reduced forms:

Aβd Z = j A βxd = j cos φ Bx C 2 2C Aβd Z = j A βyd = j sin φ (4.12) By C 2 2C which suggests the eﬀective Bloch impedance,

q Aβd Z = ± Z2 + Z2 = ±j (4.13) B Bx By 2C so that

ZBx = ZB cos φ

ZBy = ZB sin φ

Thus, it is clear that ZB is also the Bloch impedance for axial propagation (i.e., φ = 0 or φ = π/2). Chapter 4. Uniplanar Transmission-Line Metamaterials 71

4.2 2D Eﬀective-Medium Properties

The per-unit-length (distributed) inductance and capacitance of the medium may be directly obtained from the Floquet-Bloch wavevector β and Bloch impedance ZB as follows:

2 2 0 0 β = ω Leff Ceff 0 2 Leff ZB = 0 Ceff βZ ⇒ L0 = B eff ω 0 β ⇒ Ceff = (4.14) ωZB

The Bloch impedance, along with the isotropic dispersion relation listed in Table 4.2, can be rearranged to express the desired quantities:

4AB L0 = eff jωd 2C C0 = (4.15) eff jωAd

Making these substitutions and inserting the appropriate transmission matrix parameter expressions from (4.3) yields

· ¸ θ 4Z θ jZ L0 = cos2 0 tan − eff 2 ωd 2 ωd · ¸ 2Y θ jY C0 = 0 tan − (4.16) eff ωd 2 ωd

These expressions describe the per-unit-length inductance and capacitance of an ‘eﬀective

TL’ representing propagation in an isotropic TL network near each one of inﬁnitely many

Γ points at successively higher frequencies. They may be related to the constitutive parameters of an ‘eﬀective medium’ when the unit cells are much smaller than the applied Chapter 4. Uniplanar Transmission-Line Metamaterials 72

wavelength. This condition is satisﬁed only at low frequencies, where the dispersion

characteristics are dominated by the lowest order (n = 0) Floquet-Bloch spatial harmonic,

and is known as the eﬀective-medium regime (see Appendix B). At these frequencies, the

interconnecting TLs can be considered to be electrically small so that θ ¿ 1. Applying

this to (4.16) and retaining up to the second-order terms in the Taylor expansion yields

2Z θ jZ L0 = 0 − eff ωd ωd Y θ jY C0 = 0 − (4.17) eff ωd ωd

The host TL parameters may also be written as

√ θ = ω ²pµpd r −1 µp Z0 = Y0 = gp (4.18) ²p

where ²p and µp are the material parameters of the intrinsic medium ﬁlling the host

TL segments, and gp is the positive geometric constant that relates the characteristic impedance of the host TLs to the wave impedance of the intrinsic medium. Inserting

(4.18) into (4.17) yields

jZ L0 = 2µ g − eff p p ωd 0 ²p jY Ceff = − (4.19) gp ωd

where the quantities 2µpgp and ²p/gp are seen to be the per-unit-length inductance and

0 capacitance, respectively, of the host TL. Finally, in this eﬀective-medium regime, Leff

0 and Ceff can also be related to the eﬀective-medium parameters µeff (ω) and ²eff (ω)

through a geometrical factor, which shall be denoted geff . Thus, the eﬀective-medium Chapter 4. Uniplanar Transmission-Line Metamaterials 73 parameters of the network can be written as follows:

0 Leff gp jZ µeff (ω) = g = 2µp − eff geff geff ωd

0 ²pgeff jY geff ²eff (ω) = Ceff geff = − (4.20) gp ωd

Each effective-medium parameter expression consists of two terms: the first is contributed by the (nondispersive) host TL medium alone, while the second is dependent on the loading. Furthermore, it is evident that the effective permeability is determined by the series loading, whereas the effective permittivity is determined by the shunt loading. Revisit- ing Table 4.2, and recalling from the previous discussion that the transmission matrix parameters B and C were respectively associated with the series and shunt loading, it is seen that the effective medium can be made anisotropic only by varying the shunt loading between the axes. However, since the shunt loading on a particular axis determines the effective permittivity for propagation along that axis, it is clear that such a network would be electrically anisotropic, in contrast to the shunt NRI-TL network, for which varying the series loading between the axes yields magnetic anisotropy [106].

In the unloaded case—that is, for Z = Y = 0—the parameters of the unloaded medium are restored. The diﬀerence, however, lies in the factor of two in the eﬀective permeability, resulting from the restrictions in propagation direction in the 2D geometry.

This will be elucidated shortly. For simplicity, it may be assumed that the two geometrical factors are identical—that is, gp = geff = g—which yields

jZ µ (ω) = 2µ − eff p gωd jY g ² (ω) = ² − (4.21) eff p ωd

Essentially, this is equivalent to saying that the ﬁeld mappings associated with propagation in the loaded and unloaded networks are identical; that is, we have taken an Chapter 4. Uniplanar Transmission-Line Metamaterials 74

effective TL filled with 2µp and ²p and substituted the filling medium with µeff (ω) and

²eff (ω). For purely real eﬀective-medium parameters, both Z and Y are required to be

purely reactive. If the lines are loaded with a series inductance L0 and shunt capacitance

C0 (that is, in a low-pass conﬁguration), the following eﬀective-medium parameters are produced:

L µ (ω) = 2µ − j(jωL0) = 2µ + 0 eff p gωd p gd C g ² (ω) = ² − j(jωC0)g = ² + 0 (4.22) eff p ωd p d

Loading in this manner yields positive, nondispersive eﬀective-medium parameters, where

the loading has merely been distributed through the length of the unit cell, d. This

topology can be used to represent conventional, nondispersive, lossless dielectrics, and,

excepting the interconnecting TLs, is identical to the series TLM model. Consider now

loading the lines with a series capacitance C0 and shunt inductance L0 (that is, in a high-pass, or dual, conﬁguration), as shown in Fig. 4.9. The arrows in the ﬁgure indicate

the equal and opposite currents in the TL segments assumed in the preceding analysis.

This perspective also enables one to see the current loops in the ‘ring’ perspective. The

effective-medium parameters for this configuration are strikingly different:

j 1 µeff (ω) = 2µp − = 2µp − g(jωC0)ωd 2 ω gC0d jg g ²eff (ω) = ²p − = ²p − (4.23) (jωL0)ωd 2 ω L0d

It is evident that the second term of each eﬀective-medium parameter is now negative

and strongly dispersive, the latter property consistent with the requirements of causality

[1, 26]. When the parameters are simultaneously negative, this eﬀective medium can be

said to possess a NRI. In anticipation of this result, this array shall hereafter be referred Chapter 4. Uniplanar Transmission-Line Metamaterials 75

Figure 4.9: Series TL node array loaded in a dual conﬁguration. The arrows indicate the current directions. Reprinted with permission from Ref. 60, copyright °c 2006 Optical Society of America.

to as the 2D series NRI-TL network.

It was previously noted that the factor of two in the first term of the effective permeability expression was due to the fact that propagation in the grid is restricted to the two orthogonal grid axes. The effect of the 2D nature of the grid is most easily seen for the case of axial propagation, as shown in Fig. 4.10 for propagation along the x-axis (i.e., with

βy = 0). From (4.6), it may be observed that, if βy = 0, the Bloch impedance ZBy = 0.

As noted in Ref. 104, this implies from (4.6) that Vy = 0, which amounts to short cir- cuiting the dual series NRI-TL node terminals transverse to the direction of propagation.

In this case, the transverse branches of the unit cell act like shorted series TL stubs. In the eﬀective-medium limit, these stubs can be shown to double the eﬀective permeability of the host medium in a manner contrasting, but entirely analogous to, the doubling of

Chapter 4. Uniplanar Transmission-Line Metamaterials 76

s.c.

Figure 4.10: Axial propagation in the 2D Series NRI-TL node array (s.c. ≡ short circuit). Reprinted with permission from Ref. 60, copyright °c 2006 Optical Society of America.

the eﬀective permittivity of the host medium in the 2D shunt NRI-TL grid [17]. For the unloaded medium, whose eﬀective-medium parameters can be obtained from (4.21) by setting Z = Y = 0, it is clear that the phase velocity of propagating Floquet-Bloch √ waves is reduced by a factor of 2 from the intrinsic, a result that is also analogous to that determined for the 2D shunt NRI-TL medium [17,105].

The complete axial dispersion relation for a representative series NRI-TL design is shown in Fig. 4.11(a) in the form of an ω-β diagram. At low frequencies, the interconnecting TLs may be regarded as electrically inﬁnitesimal, and all that remains of the series NRI-TL unit cell are the loading elements. Such a structure behaves as a 2D L–C

filter and, therefore, possesses a Bragg resonance (labelled ωB), below which propagation is forbidden. The fundamental mode supported by this structure, which appears thereafter, clearly possesses the LH or NRI characteristic, and the effective-medium condition is achieved in the approach towards the edge of the first stopband at ωC,1. Propagation is restored beyond ωC,2, where the concavity of the band structure exhibits (conventional) right-handedness.

These dispersion characteristics may also be understood in terms of the material parameter expressions of (4.23). The edges of the stopband shown in Fig. 4.11(a) eﬀectively correspond to the zeroes of the material parameter expressions. Hence, ωC,1 and ωC,2 may Chapter 4. Uniplanar Transmission-Line Metamaterials 77

7 7

6 6

5 5

ω 4 C,2 4

ω 3 C,1 3

Frequency (GHz) Frequency (GHz) ω = ω 2 2 C,1 C,2

1 ω 1 B ω B 0 0 −π −π/2 0 π/2 π −π −π/2 0 π/2 π βd (radians) βd (radians) (a) (b)

Figure 4.11: Representative series NRI-TL dispersion relations for axial propagation (µp = µ0, ²p = 3²0, g = 0.325, C0 = 1 pF, d = 5 mm): (a) L0 = 3nH; Impedance- mismatched (open-stopband) case. (b) L0 = 10nH; Impedance-matched (closed-stopband) case. Reprinted with permission from Ref. 60, copyright °c 2006 Optical Society of America.

be referred to as magnetic and electric ‘plasma frequencies’,

q 1 ωC,1 = (µeff (ωC,1) = 0) 2µpgC0d q g ωC,2 = (²eff (ωC,2) = 0) (4.24) ²pL0d

Below their respective plasma frequencies, the eﬀective-medium parameters are individu-

ally negative. The stopband, therefore, corresponds to the situation in which only one of

the two parameters is negative (the frequency region between ωC,1 and ωC,2), where the eﬀective propagation constant is imaginary. In the frequency range with upper bound

min{ωC,1, ωC,2}, the eﬀective-medium parameters are simultaneously negative, and it is here that the NRI, or LH characteristic, is observed. Chapter 4. Uniplanar Transmission-Line Metamaterials 78

4.3 Closure of the Stopband: The Impedance-Matched

Condition

A similar dispersion relation was obtained for the shunt NRI-TL node [17], in which it was

shown that the stopband may actually be closed by equating the two plasma frequencies,

resulting in an impedance-matched condition. It is evident from (4.24) that there may be

an analogous condition for the series NRI-TL node. Equating the two plasma frequencies

results in the condition

r s L0 2µp √ ωC,1 = ωC,2 ⇒ = g = 2Z0 = Z0,2D (4.25) C0 ²p

√ where Z0 is the characteristic of a single host TL segment, and Z0,2D = 2Z0 is the characteristic impedance seen by a Floquet-Bloch wave propagating in the 2D unloaded

TL grid in the homogeneous limit. Thus, (4.25) states that the stopband may be

closed simply by choosing the loading elements L0 and C0 such that the characteristic impedance of the 2D host medium is equal to that of the underlying purely LH

distributed medium consisting of the loading elements alone, which is identical to the

impedance-matched condition reported for the shunt NRI-TL unit cell in Ref. 17 and

later termed the ‘balanced’ condition [107]. Appropriately adjusting the loading induc-

tance L0 employed in Fig. 4.11(a) to satisfy (4.25) results in the impedance-matched design shown in Fig. 4.11(b). One consequence of the closure of the stopband is the

restoration of a theoretically non-zero group velocity at the point of closure [107]. It

was shown in Ref. 32 that the impedance-matched condition for the shunt NRI-TL node

closed not only the lowest Γ-point stopband, but in fact, closed each one of the inﬁnite

stopbands in the vicinity of Γ, whose edges are described by the solutions of (4.10). The

reader may verify that this is also true of the series NRI-TL node by substituting the

impedance-matched condition (4.25) into (4.10). Chapter 4. Uniplanar Transmission-Line Metamaterials 79

4.4 Practical realization

This section considers a practical realization of the 2D series NRI-TL layers proposed above in preparation for the next section, in which a volumetric layered NRI-TL medium is designed and simulated. In anticipation of using commercially available PCB materials and techniques, the planar host medium shall be constructed from coplanar-strip (CPS)

TLs consisting of two ﬂat, edge-coupled strips (strip width W , separation S) printed on a substrate with height h and relative permittivity ²r. The substrate ground plane is removed and the entire system is embedded in vacuum. The characteristic impedance of the CPS TL is given by [108] 0 η0 K(k ) Z0 = √ (4.26) ²eff K(k) with

0 0 1 K(k ) K(kr) ²eff = 1 + (²r − 1) 2 K(k) K(kr) s µ ¶ S 2 k = 1 − S + 2W √ 0 2 k = s1 − k sinh2(π(S/2)/2h) kr = 1 − sinh2(π(S/2 + W )/2h) p 0 2 kr = 1 − kr (4.27)

where η0 = 377Ω is the free-space wave impedance and K is the complete elliptic integral of the ﬁrst kind. The geometric factor g can also be identiﬁed as follows:

0 Z0 K(k ) g = η = (4.28) √ 0 K(k) ²eff

Wide strips with small gap spacings are employed and placed in a periodic array with period d. As noted previously, such an arrangement may also be viewed as an array of square metallic rings, as shown in Fig. 4.12(a). The unit cell is shown in black. Other Chapter 4. Uniplanar Transmission-Line Metamaterials 80 ring shapes, e.g. circular, triangular, or hexagonal, may also be described by the above theory, provided that appropriate modiﬁcations are made to the geometry-dependent parameters of the host TL medium.

The loading may be achieved either using discrete lumped elements or printed lumped elements, as in the shunt NRI-TL structure. Discrete lumped elements have the beneﬁt that they permit small structures at low frequencies and are easier to model. On the other hand, printed lumped elements allow scalability and lower the cost of fabrication signiﬁcantly. Fig. 4.12(b) shows the placement of the shunt inductive connections between the metallic rings, and Fig. 4.12(c) depicts the series capacitive loading (the individual capacitors on the intersecting CPS lines shown in Fig. 4.9 have been combined in series at the corners of the rings). The resulting planar series NRI-TL unit cell is shown in

Fig. 4.12(d). Chapter 4. Uniplanar Transmission-Line Metamaterials 81

L0/2 S L /2 W 0

L0 L0 L0 L0 2 2 2 2 d

L0/2

(a) (b)

4C0 4C0 4C0 4C0 4C0 4C0 4C0 4C0 L0/2

L0/2 4C0 4C0 4C0 4C0 4C0 4C0 4C0 4C0

L0 L0 L0 L0 2 2 2 2

4C0 4C0 4C0 4C0 4C0 4C0 4C0 4C0 L0/2

L0/2 4C0 4C0 4C0 4C0 4C0 4C0 4C0 4C0

Figure 4.12: Series TL unit cells: (a) Unloaded. (b) Shunt inductors. (c) Series capacitors. (d) Composite series NRI-TL unit cell. Reprinted with permission from Ref. 60, copyright °c 2006 Optical Society of America. Chapter 5

Free-Space Volumetric NRI-TL

Metamaterials

The interest piqued by Veselago’s suggestion of an ideal planar NRI lens able to bring the rays emanating from a source to a focus [1], and Pendry’s suggestion that such a lens would, moreover, be perfect enough to resolve subwavelength details of that source [2], prompted the application of the NRI-TL concept to the veriﬁcation of these phenomena.

Indeed, the ﬁrst experimental demonstrations of focusing and also subwavelength focusing due to an eﬀective NRI were achieved using planar NRI-TL metamaterials [3, 17, 18].

However, as a consequence of their inherently planar form, these experiments were limited to sources and fields that were embedded within the TL network. Veselago, on the other hand, envisioned a LH medium that could interact with and manipulate fields in free space. Such a structure would be necessarily 3D and must also appear isotropic to a source in free space. Although isotropic 3D extensions of the NRI-TL topology have been proposed [20, 22], their prohibitively complex design and implementation requirements have so far prevented experimental verification of their properties for free space excitation.

The implementation described in Ref. 109 was practically realized in Ref. 4 and can be regarded as isotropic for electric ﬁelds perpendicular to the constituent boards, but has

82 Chapter 5. Free-Space Volumetric NRI-TL Metamaterials 83 thus far employed only embedded sources, similar to those in Ref. 3. Various SRR- based structures have been used to show free-space focusing; among them are the lenses reported in Ref. 110, in which the transmission of power is made possible by the coupling between quasimagnetostatic surface waves at the interfaces between the lens faces and air, and in Ref. 15, which employs a SRR/wire topology. The former topology, although very easily constructed, cannot be regarded as a NRI medium: its imaging principle relies only on the amplification of the evanescent Fourier spectral components and not also on the phase compensation of the propagating spectral components via a NRI; consequently, these and similar imaging devices (see, for example, Ref. 111) are limited to operation in the extreme near field. The latter topology, indeed, possesses a NRI, but it requires a complex assembly of orthogonally interleaved printed circuit boards that cannot be automated by any standard fabrication process. In Chapter 4, it was suggested that a simple 3D free-space NRI lens could be constructed by stacking multiple NRI-TL layers, as shown in Fig. 5.1, where each layer employs a ‘series’ TL topology that can be activated by electric and magnetic fields lying parallel and normal to the layers, respectively. The layers are realized in CPS technology (Fig. 5.1(a)), which enables them to be uniplanar, thus avoiding any need for vertical interconnects or vias. In the effective-medium limit, such a lens is isotropic for 2D free-space excitations polarized as above (e.g. an infinite magnetic line source) and can be fabricated in a layer-by-layer fashion using standard lithographic techniques. So as to avoid the suggestion of 3D isotropy, this structure shall hereafter be referred to as a ‘volumetric’ NRI-TL metamaterial.

This chapter presents the design and simulation of practical volumetric metamaterials that exhibit a NRI. An intuitive equivalent-circuit model is developed to aid in the design of volumetric NRI-TL structures and validate their dispersion characteristics as obtained through full-wave simulations. Finally, the possibility of scaling these media for operation at THz frequencies, or employing plasmonic concepts for operation at near infrared or optical frequencies, is discussed.

Chapter 5. Free-Space Volumetric NRI-TL Metamaterials 84

E y H d p

x (a) (b)

Figure 5.1: (a) A uniplanar array (right) of series-connected NRI-TL sections (left), as viewed from the top. The arrows indicate current directions. (b) A volumetric NRI-TL metamaterial constructed by layering the planar arrays in (a). Reprinted with permission from Ref. 63, copyright °c 2007 IEEE.

5.1 Development of a Two-Port Equivalent-Circuit

Model

The series NRI-TL topology depicted in Fig. 5.1(a) employs series-connected junctions of CPS TLs loaded periodically by inductors and capacitors, which can be realized using either discrete (chip) elements or printed (interdigitated or meandered) elements. TL- type currents produced in the CPS lines see the dual loading and result in LH, or NRI, dispersion properties. In Chapter 4, it was shown that these layers could alternatively be viewed as periodic arrays of capacitively loaded metallic rings (period d) connected to each other using inductors and, as also depicted in Fig. 5.1(a), the TL-type currents could then be interpreted as loop-currents orbiting each ring. Such currents can be induced both by a magnetic ﬁeld polarized perpendicular to the ring plane and also by an electric

ﬁeld lying in the ring plane, which would excite the gaps between rings and also the ring capacitors. When the layers are stacked at intervals p and illuminated by a horizontally polarized plane wave at normal incidence, image theory allows the inﬁnite structure to Chapter 5. Free-Space Volumetric NRI-TL Metamaterials 85 be modeled by inserting a single row of unit cells into an ideal parallel-plate waveguide

(PPW) with cross-sectional dimensions d × p, as illustrated in Fig. 5.1(b), for which perfect electric and magnetic walls are placed perpendicular to the electric and magnetic

fields, respectively. In this manner, the coupling between layers is captured by the mutual inductance between the rings and the external PPW; this quantity is dependent on the area occupied by each ring within its unit cell (filling factor), following similar models for the SRR (see, for example, Refs. 34, 97, 98, and 112). When the period d is electrically very small (the ‘effective-medium’ limit), the volumetric structure appears homogeneous and isotropic in the plane of propagation, and the normal incidence constraint can be generalized to all azimuthal angles of incidence.

Although the volumetric NRI-TL topology was based on a multilayer extension of the planar NRI-TL metamaterial, it nevertheless shares a number of its dispersion features with SRR-based structures. However, it is important to note the considerable differences between the two. Foremost, the wire/SRR metamaterial is designed for electric fields polarized vertically along the separated wires and magnetic fields lying in the plane of propagation to activate the SRRs. Consequently, printed-circuit boards (PCBs) bearing the SRRs must be orthogonally interleaved in an egg-crate fashion to produce homogeneity and isotropy in the effective medium limit, preventing them from being implemented in multilayer form for 2D propagation. The NRI-TL metamaterial, on the other hand, is designed for magnetic fields polarized perpendicular to the rings and electric fields lying in the plane of propagation. Therefore, the ring layers are entirely uniplanar and conducive to multilayer implementation; furthermore, the function of the separated wires is satisfied by the integrated loading inductances, which connect the rings via the uniplanar

NRI-TL topology. Lastly, the asymmetrical placement of the capacitive gaps in SRRs makes their response sensitive to their orientation with respect to the incident ﬁeld. This is avoided in the volumetric NRI-TL metamaterial, for which the capacitors are arranged symmetrically around the unit cell by virtue, once again, of the uniplanar NRI-TL topol- Chapter 5. Free-Space Volumetric NRI-TL Metamaterials 86

Electric wall Magnetic wall

z p y

Electric wall

Magnetic wall

Figure 5.2: Volumetric NRI-TL unit cell for an inﬁnite array excited by TEz-polarized plane wave propagating in the y-direction. Reprinted with permission from Ref. 60, copyright °c 2006 Optical Society of America.

ogy. As a result, the volumetric NRI-TL metamaterial allows the realization of isotropic

NRI properties using only lithographic fabrication techniques and requires no vias or vertical interconnections.

It should be noted that, in addition to the bulk LH mode described by the interaction between the NRI-TL rings and external PPW, a single layer of rings also supports a separate, usually narrowband magnetoinductive surface-wave mode, shown also to be a backward wave [99]. However, as discussed in Chapter 3, the strong excitation of this mode requires an extremely tight magnetic coupling between rings, which is not achieved using the dimensions treated in this work and the edge-type coupling aﬀorded by CPS technology. As a result, this mode is not elaborated upon in the present discussion.

By layering series NRI-TL unit cells periodically both in the xy-plane (with horizontal lattice spacing d) and along the z-axis (with vertical lattice spacing p), the volumetric unit cell depicted in Fig. 5.2 is produced. When excited by a TEz-polarized plane wave propagating in the xy-plane, the stacked rings exhibit identical current patterns; consequently, the top and bottom faces of the unit cell of Fig. 5.2 can be regarded as magnetic Chapter 5. Free-Space Volumetric NRI-TL Metamaterials 87 walls. For simplicity, the present study shall be restricted to the highly representative case of axial propagation (normal incidence), in which the electric ﬁeld remains in the x-direction. In this case, the faces of the cube normal to x may additionally be replaced by electric walls.

Based on the assumptions described above, an intuitive equivalent-circuit TL model for the cell depicted in Fig. 5.2 may be developed. The dispersion relation of the volumetric layered NRI-TL medium may then be obtained as that of an inﬁnite periodic structure consisting of such unit cells via a periodic analysis. The interpretation of this equivalent circuit will be made easier by viewing the volumetric unit cell from the top, as shown in Fig. 5.3(a), where the electric walls are as indicated and the magnetic walls are parallel to the page. The orientation of the incident ﬁeld quantities is also indicated. The corresponding equivalent circuit unit cell is shown in Fig. 5.3(b); the following discussion shall justify its constituent elements using the schematic representation in Fig. 5.3(a).

When unloaded by the series NRI-TL layers, the waveguide can be described by a distributed inductance Lwg/d = µ0gwg and capacitance Cwg/d = ²wg²0/gwg, where gwg = d/p and ²wg is the effective permittivity describing the medium filling the waveguide, which is generally inhomogeneous. However, in this chapter, it will be assumed for the sake of simplicity that the waveguide is filled homogeneously with a material with relative permittivity ²wg = ²r. When the layers are introduced, the impinging magnetic field induces currents in the capacitively loaded rings and, thus, alters the inductance of the waveguide. This can be modelled (as described in Refs. 98, 112, and 97 for the

SRR), by coupling Lwg to the rings (described by the total capacitive loading C0 and self-inductance Lr) via a mutual inductance M = LwgF , where F is the eﬀective (area) ﬁlling factor of the rings inside the unit cell. The inductive coupling between the rings and waveguide indirectly describes broadside ring-to-ring coupling via image theory across the magnetic walls; on this point, it is important to recall that the proposed model assumes that edge-coupling between coplanar rings is weak. Chapter 5. Free-Space Volumetric NRI-TL Metamaterials 88

Electric wall

4C0 4C0

L′′′r L′′′r L0 k Lr 2Cxd′′′ H 2 2 2

4C0 4C0

Electric wall d (a)

Z′′′ d

Lr M

Lwg

L0 Cxd′′′

Y′′′ d Cwg L′′′r 4

4C0

d (b)

Figure 5.3: Interaction between a normally incident TE-polarized plane wave and the volumetric layered NRI-TL medium: (a) schematic representation of the unit cell; (b) equivalent-circuit model of the unit cell. Reprinted with permission from Ref. 60, copyright °c 2006 Optical Society of America.

The electric ﬁeld is also perturbed by the introduction of the layers. Currents are produced in the ring when the electric ﬁeld excites the gap between the horizontal CPS lines (through which the electric wall runs), where it experiences the inductive loading

L0. In these regions, the model employs the distributed capacitance of the CPS TLs,

Cx = ²r²0/g (g is as given in (4.28)), in shunt with the loading inductance, L0. However, currents can also be produced when the electric ﬁeld excites the loading capacitors, C0.

0 2 These currents see the eﬀective series inductance Lr = Lr − M /Lwg, which is simply the self-inductance of the ring, less the image inductance due to coupling to the external Chapter 5. Free-Space Volumetric NRI-TL Metamaterials 89

waveguide. In Fig. 5.3(a), the total capacitive loading and eﬀective series inductance have

been divided evenly across the two halves of the ring. Because the impinging electric ﬁeld

simultaneously excites both sides of the ring, Fig. 5.3(b) places them in parallel.

5.2 Volumetric Eﬀective-Medium Properties

The per-unit-length series impedance (Z0) and shunt admittance (Y 0) of Fig. 5.3(b) re-

spectively represent the distributed inductance and capacitance of an eﬀective TL describ-

ing axial propagation in the volumetric layered NRI-TL medium. As with the equivalent

circuit models of the single-layer series and shunt NRI-TL nodes [17, 104], these dis-

tributed parameters describe the magnetic and electrical characteristics of the network,

and therefore, can be related to the eﬀective-medium parameters, µeff (ω) and ²eff (ω), of

the volumetric layered NRI-TL medium via a geometrical factor, geff (see Appendix B). This relationship is established by noting that the eﬀective-medium parameters in the ab-

sence of the 2D layers must reduce to those of the intrinsic medium ﬁlling the waveguide:

µeff (ω) = µ0 and ²eff (ω) = ²r²0, which insists that geff is, in fact, gwg of the waveguide. Thus, the eﬀective-medium parameters of the volumetric layered NRI-TL medium may

be obtained using the following expressions:

jZ0 µeff (ω) = − ωgwg jY 0g ² (ω) = − wg (5.1) eff ω

Substituting the relevant parameters from Fig. 5.3(b) reveals the form of the eﬀective

permittivity and permittivity of the volumetric metamaterial:

µ 2 2 ¶ C0M ω µeff (ω) = µ0 1 + 2 2 Lwg (1 − ω /ωm0) µ 2 2 ¶ 4C0 (1 − ω /ωy) ²eff (ω) = ²0²r 1 + 2 2 2 2 2 2 (5.2) Cwg (1 − ω /ωmp)(1 − ω /ωy) − ω /ωb Chapter 5. Free-Space Volumetric NRI-TL Metamaterials 90

Quantity Symbol Value Horizontal lattice spacing d 5.5 mm Vertical lattice spacing p 4.572 mm (180 mil) Strip width W 1.1 mm Strip spacing S 500 µm Copper thickness tc 17 µm Relative permittivity ²r 3 Total loading inductance L0 5.6 nH Total loading capacitance C0 1 pF

Table 5.1: Design parameters employed for a volumetric layered NRI-TL medium using discrete lumped elements. Reprinted with permission from Ref. 60, copyright °c 2006 Optical Society of America. where

2 1 ωy = 0 L0(Cxd )

2 1 ωm0 = LrC0 2 1 ωmp = 2 (Lr − M /Lwg)C0

2 1 ωb = (5.3) 4L0C0

5.3 Discrete Lumped-Element Design

To gain a full appreciation of the intricacies of this geometry, as well as the ability of the above theory to predict its dispersion features, it should be instructive to examine the axial dispersion characteristics of the volumetric layered medium for the various stages in the evolution of the planar unit cell depicted in Figs. 4.12(a)–4.12(d). The corresponding equivalent-circuit models for each of these unit cells are shown in Figs. 5.4(a)–5.4(d).

Consider a representative design employing discrete lumped elements whose design parameters are listed in Table 5.1. The metal used is copper (σ = 5.8×107 S/m) with ﬁnite thickness tc; the host dielectric possesses a relative permittivity of 3 and also includes material losses through a loss tangent of 0.0013. Ansoft’s HFSS (high-frequency struc- Chapter 5. Free-Space Volumetric NRI-TL Metamaterials 91

Z′′′ d Z′′′ d

Lr Lr M M

Lwg Lwg

Cxd′′′ L0 Cxd′′′

Y′′′ d Y′′′ d Cwg L′′′ r Cwg L′′′ r 4 4

d d (a) (b)

Z′′′ d Z′′′ d

C0 C0

Lr Lr M M

Lwg Lwg

Cxd′′′ L0 Cxd′′′

C Y′′′ d Y′′′ d wg L′′′ r Cwg L′′′ r 4 4

4C0 4C0

d d (c) (d)

Figure 5.4: Equivalent-circuit models corresponding to the series unit cell topologies of Figs. 4.12(a)–4.12(d): (a) Unloaded. (b) Discrete lumped shunt inductors. (c) Discrete lumped series capacitors. (d) Composite series NRI-TL unit cell. Chapter 5. Free-Space Volumetric NRI-TL Metamaterials 92

5 5

4 4

ω ep

3 3 Frequency (GHz) Frequency (GHz)

2 2

1 1

0 0 −π −π/2 0 π/2 π −π −π/2 0 π/2 π βd (radians) βd (radians) (a) (b)

5 5

4 4 ω C,2

ω ω 3 mp 3 C,1

Frequency (GHz) Frequency (GHz) ω ω 0 2 0 2

ω B 1 1

0 0 −π −π/2 0 π/2 π −π −π/2 0 π/2 π βd (radians) βd (radians) (c) (d)

Figure 5.5: Axial dispersion relations corresponding to the series unit cell topologies of Figs. 4.12(a)–4.12(d) and values reported in Table 5.1, obtained using the HFSS finite- element solver (dots) and equivalent circuit model (solid curves): (a) Unloaded. (b) Discrete lumped shunt inductors. (c) Discrete lumped series capacitors. (d) Composite series NRI- TL unit cell. Adapted and reprinted with permission from Ref. 60, copyright °c 2006 Optical Society of America. Chapter 5. Free-Space Volumetric NRI-TL Metamaterials 93 ture simulator) finite-element-method full-wave solver was used to produce the dispersion characteristics of an infinite array of such unit cells, by extracting the eigenfrequencies associated with the appropriate electric-wall, magnetic-wall, and boundary-phase conditions applied to its six exterior faces. In HFSS, this type of simulation is known as an

‘eigenmode simulation’ and yields the eﬀective propagation constant β in a particular direction of propagation as a function of ω (the ‘dispersion diagram’), or β over all directions of propagation at a particular frequency (the ‘isofrequency’ or ‘equifrequency’ surface). The resulting dispersion diagrams corresponding to each case in Figs. 4.12(a)–

4.12(d) are shown in Figs. 5.5(a)–5.5(d) (dots); these data are shown superimposed on the dispersion relations as predicted by the corresponding equivalent circuit models of

Figs. 5.4(a)–5.4(d) (solid curves). Due to the width and varying shape of the CPS conductors, and the resulting non-uniform distribution of current on the rings, it is difficult to define a unique ‘ring size’. Therefore, the rings have been assigned an effective size d0 = 3.66 mm, an empirical value which is employed to estimate the capacitive contribution of the TL segments, as well as the effective filling factor F = (d0/d)2. Although d0 is empirically selected to achieve the best matching between the TL model and full-wave dispersion data, it is, nevertheless, close to the centre-to-centre distance between CPS

TLs across the ring.

5.3.1 Unloaded Case

The dispersion of the unloaded series NRI-TL network, consisting of the host CPS medium alone, is depicted in Fig. 5.5(a). The full-wave dispersion data and the TL model (in the absence of the loading, i.e., L0,C0 → ∞) show good correspondence at low frequencies, after which the latter begins to diverge. Nevertheless, it should be interesting to examine the equivalent unit cell of Fig. 5.4(a) under these conditions. Since currents are induced in the ring by the impinging magnetic ﬁeld, the total series inductance of

2 the waveguide, Lwg, is diminished by a factor (1 − M /(LwgLr)), which remains between Chapter 5. Free-Space Volumetric NRI-TL Metamaterials 94

nil and unity. Since the total per-unit-length series inductance determines µeff (ω), it is clear that the volumetric layered medium consisting of unloaded rings possesses a positive effective permeability, and furthermore, is diamagnetic, a property that is typical of simple artificial dielectrics consisting of metallic inclusions [26]. The shunt branch of the equivalent unit cell consists of the waveguide capacitance Cwg in parallel with the series combination of the CPS capacitance and the effective ring inductance, suggestive of what is seen by the impinging electric field looking at the unloaded rings edge-on. Thus, at low frequencies, the shunt branch appears to describe the shunt capacitive loading of the waveguide, which suggests an effective permittivity greater than unity; this property is also expected of simple artificial dielectrics consisting of disconnected conducting inclusions.

5.3.2 Inductively Loaded Case

When the constituent CPS TL segments are loaded in shunt with inductors, or equivalently, when the rings connected to each other as shown in Fig. 4.12(b), the dispersion relation takes the form shown in Fig. 5.5(b). The forbiddance of propagation at low frequencies can be explained by the fact that the inductive loading creates a continuous conducting path at low frequencies that causes each layer to behave like a ground plane.

Layers of such planes stacked in the z-direction of Fig. 5.2 are unable to support the propagation of a TEz-polarized plane wave. Moreover, from the TL perspective, the inductors can be said to eﬀectively cut oﬀ the TL mode at low frequencies. This is also evident from Fig. 5.4(b), where, in the absence of C0, the shunt branch is essentially shorted by the inductive loading (consisting of L0 as well as the inductive contribution of the ring) at low frequencies. Propagation is eventually restored when the capacitance between the CPS conductors supersedes both the inductive loading and the inductive contribution of the ring—that is, at the resonance frequency of the overall shunt branch in Fig. 5.4(b)—where the total shunt admittance Y 0d = 0 (open circuit). In Fig. 5.5(b), Chapter 5. Free-Space Volumetric NRI-TL Metamaterials 95

this occurs at approximately 3.64GHz, and is labelled ωep. From the effective-medium perspective, this frequency yields the condition ²eff (ωep) = 0 and, thus, corresponds to an effective electric plasma frequency below which the effective permittivity of the medium is negative.

5.3.3 Capacitively Loaded Case

Loading the host TL medium with capacitors as shown in Fig. 4.12(c) yields the dispersion relation of Fig. 5.5(c). This dispersion characteristic may be expected of any capacitively loaded ring structure, and, therefore, is similar to that obtained for arrays of SRRs [34]. Accordingly, the same notation has been employed in identifying the resonance frequencies ω0 = 2.30GHz and ωmp = 2.83GHz. At low frequencies, the series branch of Fig. 5.4(c) is purely inductive (representing a positive eﬀective permeability), while the shunt branch remains capacitive, permitting conventional right-handed propagation. However, the ring resonates at ω0, after which it contributes capacitively to the series branch (strongly negative eﬀective permeability). This capacitance dominates the series branch and forbids propagation until it resonates with the series inductor, which occurs at ωmp. These two frequencies may be given by the following expressions:

1 ω0 = √ LrC0 1 1 ωmp = p = p (5.4) 0 2 LrC0 (Lr − F Lwg)C0

where the relation M = LwgF is used. Thus, it is clear that the region of negative permeability corresponds to the frequency region enclosed by ω0 and ωmp. In fact, ωmp is the resonance frequency of the overall series branch in Fig. 5.4(c), at which the total

0 series impedance Z d = 0 (short circuit). This implies that µeff (ωmp) = 0 and suggests that ωmp can be regarded as an eﬀective magnetic plasma frequency, analogous to ωep in the inductively loaded case. Chapter 5. Free-Space Volumetric NRI-TL Metamaterials 96

5.3.4 Dual L–C Loaded Case: Composite Series NRI-TL

It is worthy to recall here that the inductive loading alone provides a negative effective permittivity below ωep, while the capacitive loading provides a negative effective permeability between ω0 and ωmp. Thus, when the rings are loaded simultaneously using inductors and capacitors, as shown in Fig. 4.12(d), it is reasonable to expect a region of LH propagation enclosed by the range of frequencies in which the effective-medium parameters are simultaneously negative. The dispersion relation for the composite structure takes the form shown in Fig. 5.5(d). Indeed, the most interesting feature of the composite NRI-TL dispersion relation is the restoration of propagation in the region enclosed exactly by the frequencies ω0 and ωC,1 = ωmp, where both the inductively loaded and capacitively loaded cases were previously cut off. As expected, the dispersion relation in this region possesses a LH characteristic. When the permeability once again becomes positive (at ωC,1 = ωmp), a stopband appears and is maintained until ωC,2, after which conventional right-handed propagation is restored. However, the dispersion data in Fig. 5.5(d) also indicate a region of low-frequency propagation below the ring resonance ω0. Previously, it was noted that the inductive loading prevented propagation at low frequencies. However, although the unit cells remain connected by the inductors, the continuous conducting path is now broken by the ring capacitance, C0, which, as previously noted, may also be excited by the impinging electric field. The appearance of C0 in series with the loading inductance in Fig. 5.4(d) ensures that the shunt branch is capacitive at low frequencies and becomes inductive near ωB in Fig. 5.5(d), where the total shunt admittance Y 0d → ∞ (short circuit). Thus, strictly speaking, the effective permittivity of the composite structure can only be regarded as negative between ωB and

ωep.

The correspondence between the equivalent-circuit model and the full-wave simulation results is satisfactory in all of the above cases. The following section discusses some features of the obtained dispersion relations, particularly with reference to the NRI Chapter 5. Free-Space Volumetric NRI-TL Metamaterials 97 bandwidth, and the possibility of designing the volumetric layered medium for a closed stopband.

5.3.5 Bandwidth of the NRI region

The NRI bandwidth, which corresponds to the frequency region in which µeff (ω) and

²eff (ω) are simultaneously negative, can be increased by decreasing ω0 while increasing

ωC,1 = ωmp and ωC,2 = ωep. For the moment, it may be assumed that the inductive loading is fixed. It is seen from (5.4) that the reduction in ω0 is easily achieved by increasing C0; although this results in a commensurate decrease in ωC,1, the latter can be enhanced either by increasing the effective filling factor F (as observed for the SRR [34,

113]) or by increasing the waveguide inductance Lwg. One way to increase Lwg is to increase gwg, which is accomplished either by shrinking the plane separation, p, or by increasing the width of the unit cell, d. These optimizations may cause ωC,1 to overshoot

ωC,2, in which case the upper bound of the NRI region is dictated by the latter, and can be tuned by varying the loading inductance, L0.

5.3.6 Closure of the Stopband

It was previously shown that the stopband for the 2D series NRI-TL network in the homogeneous limit could be closed by equating the expressions for the band edges, or

‘plasma frequencies’ of the eﬀective media, analogous to what was reported for the shunt

NRI-TL node in Ref. 17. This represented an impedance-matched condition in which the characteristic impedance of the network consisting of the loading elements alone was matched to that of the layer in the absence of the loading. In this section, the same concept is applied to the equivalent circuit model of the volumetric layered NRI-

TL medium shown in Fig. 5.3(b). It was shown that ²eff (ωep) = 0 and µeff (ωmp) = 0 represent resonance conditions in which the overall series and shunt branches were,

0 0 respectively, short-circuited and open-circuited. Thus, setting Z (ωp) = Y (ωp) = 0 Chapter 5. Free-Space Volumetric NRI-TL Metamaterials 98

produces a unit cell that appears, at the common resonance frequency ωp, to be a direct, zero-phase connection from input to output, as described in Refs. 114 and 32. Performing the relevant calculations leads to the following condition for closing the stopband:

r s 0 L0 Lr = 0 (5.5) C0 Cwg + Cxd which has been cast in the form of (4.25) for comparison. Alternatively, this may be written as the equality of two resonance frequencies:

1 1 ωmp = p = p = ωx (5.6) 0 0 LrC0 L0(Cwg + Cxd )

Indeed, ωmp is the resonance frequency of the series branch, but ωx is not generally equal to the resonance frequency of the shunt branch, ωep. The origin of ωx is revealed by examination of the shunt branch of Fig. 5.3(b), in which series L–C resonator consisting

0 of Lr/4 and 4C0 (representing the loaded ring) also resonates (i.e., it becomes a short circuit) at ωmp. Thus, when the series branch resonates, the shunt branch reduces to the waveguide capacitance loaded by the CPS capacitance, in shunt with the loading inductance. To put these ideas to the test, the design given in Table 5.1 is adapted to meet the impedance-matched condition of (5.5) by increasing the loading inductance from 5.6nH to approximately 10nH. Fig. 5.6 presents the results of the theory (solid curve) as well as the corresponding full-wave simulation results using HFSS (dots), which demonstrate that the stopband is, indeed, closed.

5.4 Fully Printed Design

It was previously noted that the volumetric layered NRI-TL medium is most easily fabricated in a fully printed form. This section considers a design employing printed lumped capacitors and inductors rather than discrete elements. The shunt inductive loading is Chapter 5. Free-Space Volumetric NRI-TL Metamaterials 99

ω 3 p Frequency (GHz) ω 2 0

ω 1 B

0 −π −π/2 0 π/2 π βd (radians)

Figure 5.6: Axial dispersion relation for volumetric layered NRI-TL structure based on the design in Table 5.1, but with L0 increased from 5.6nH to approximately 10nH to meet the impedance-matched condition of (5.5); obtained using the HFSS ﬁnite-element solver (dots) and equivalent circuit model (solid curve). Reprinted with permission from Ref. 60, copyright °c 2006 Optical Society of America.

realized using a metal strip of width wL connecting the CPS TLs, and the series capacitive loading is realized using interdigitated capacitors with ﬁnger width wf and gap width wg. The resulting geometry of the complete, printed planar unit cell is shown in Fig. 5.7. For comparison with the discrete lumped element geometry, the axial dispersion characteristics of the volumetric medium consisting of layers of fully printed series

NRI-TL arrays is now examined for the various stages in the evolution of the printed planar unit cell (that is, the unloaded ring, shunt loading with metallic strips, series loading with interdigitated capacitors, and composite loading using both metallic strips and interdigitated capacitors). The printed geometry shall employ the design values listed in Table 5.1, with the discrete elements replaced with printed elements employing the values introduced in Table 5.2. The resulting dispersion relations, which should be com- Chapter 5. Free-Space Volumetric NRI-TL Metamaterials 100

wg W

wf wL d

Figure 5.7: Fully printed composite series NRI-TL unit cell employing interdigitated capacitors and strip inductors. Reprinted with permission from Ref. 60, copyright °c 2006 Optical Society of America, Inc.

pared to the corresponding cases in Figs. 5.5(a)–5.5(d), are shown in Figs. 5.8(a)–5.8(d)

(dots). Generally, the dispersion features in each case resemble those of the discrete- element design: the unloaded ring is common to both designs; the inductively loaded structure forbids propagation at low frequencies and restores propagation above a cut- oﬀ frequency ωep; the capacitively loaded case exhibits a stopband between resonance frequencies ω0 and ωmp above and below which propagation is permitted; and the composite fully printed NRI-TL structure shows a backward-wave passband enclosed by the frequencies ω0 and ωC,1 = ωmp, where it can be surmised the eﬀective permittivity and permeability are simultaneously negative. The design parameters listed in Tables 5.1 and

5.2 have been determined through simulations to maximize the NRI bandwidth, which is optimized at 25-30%. Simulations of other such printed designs, not presented here, indicate that the stopband may also be closed in these structures without compromising the NRI bandwidth.

The isotropy of the metamaterial in the eﬀective-medium limit can be ascertained by examining the equifrequency contours in the 2D reduced Brillouin zone corresponding Chapter 5. Free-Space Volumetric NRI-TL Metamaterials 101

18 18 18 18

16 16 16 16

14 14 14 14

ω 12 12 12 12 C,2 ω ep 10 10 10 10 ω ω mp C,1 8 8 8 8 Frequency (GHz) Frequency (GHz) Frequency (GHz) Frequency (GHz) ω ω 0 0 6 6 6 6

4 4 4 4

2 2 2 2

0 0 0 0 −π −π/2 0 π/2 π −π −π/2 0 π/2 π −π −π/2 0 π/2 π −π −π/2 0 π/2 π βd (radians) βd (radians) βd (radians) βd (radians) (a) (b) (c) (d)

Figure 5.8: Dispersion relations corresponding to the series unit cell topologies of Figs. 4.12(a)–4.12(d) employing printed elements instead of discrete elements, obtained using the HFSS ﬁnite-element solver (dots): (a) Unloaded. (b) Printed lumped shunt inductors. (c) Printed lumped series capacitors. (d) Composite fully printed series NRI-TL unit cell. Reprinted with permission from Ref. 60, copyright °c 2006 Optical Society of America.

to the NRI band. These were obtained for the volumetric NRI-TL unit cell using HFSS eigenmode simulations, and are presented in Fig. 5.9; the equifrequency contours are projected to the zero-frequency plane. As expected, the dispersion contours become circular in the approach to the edge of the stopband, or the Γ-point, where homogeneity is achieved, and propagating Floquet-Bloch waves see an isotropic NRI.

Thus, the fully printed realization of the volumetric layered NRI-TL layered medium, like its discrete-loaded counterpart, is capable of producing a region of backward-wave propagation in which the eﬀective refractive index can be said to be negative. However, it is also important to note some of the diﬀerences between the two designs. First, al- Chapter 5. Free-Space Volumetric NRI-TL Metamaterials 102

Quantity Symbol Value Inductor width wL 200 µm Capacitor ﬁnger width wf 100 µm Capacitor gap width wg 100 µm

Table 5.2: Design parameters for interdigitated capacitors and strip inductors employed in the fully printed NRI-TL planar unit cell depicted in Fig. 5.7. Reprinted with permission from Ref. 60, copyright °c 2006 Optical Society of America, Inc.

though the geometry of the host CPS TLs is common to both designs, all of the observed dispersion features in the printed case occur at higher frequencies. This is, undoubtedly, due to the fact that the loading values realized by the printed elements are much smaller than those of the discrete elements listed in Table 5.1. This also aﬀects the electrical size of the unit cells, which is required to be smaller than the wavelength; that is, the eﬀective-medium approximation is valid so long as the unloaded rings are nonresonant.

From Fig. 5.8(a), the first such resonant condition appears to occur near 10GHz. Al- though the NRI band in Fig. 5.8(d) remains below this frequency, their proximity to each other suggests that the effective-medium approximation may be difficult (however, not impossible) to enforce with a fully printed volumetric layered medium. Second, although

ωC,2 corresponds to ωep in the discrete-element design, the printed structure indicates a shift of approximately 10% that may be attributed, at least partially, to the appreciable electrical size of the host medium and printed elements at these frequencies. Third, the stopband separating the low-frequency and NRI bands in the discrete-element design at the edge of the Brillouin zone (i.e., the region enclosed by ωB and ω0 in Fig. 5.5(d)) tends to be much smaller in the printed-element designs. In the present case, which has been optimized for bandwidth, the stopband has been minimized to such an extent that the label identifying the lower cut-oﬀ frequency, ωB, has been omitted. Last, it is observed that that the equivalent circuit-based theory has diﬃculty in accurately predicting the dispersion features of structures in which the loading is weak, as in the printed case, since this does not allow the unit cells to be made electrically very small. Consequently, Chapter 5. Free-Space Volumetric NRI-TL Metamaterials 103

Frequency (GHz) 2

π 0 π/2 −π −π/2 0 0 −π/2 β d (radians) π/2 x π −π d (radians) β y

Figure 5.9: 2D reduced Brillouin zone for the NRI band obtained using HFSS. The dispersion contours indicate that isotropy is achieved in the approach to the Γ-point (the homogeneous limit). Reprinted with permission from Ref. 60, copyright °c 2006 Optical Society of America.

a more reﬁned theory may be required to fully characterize these structures and aid in their design; such a theory would likely describe the CPS NRI-TL layers inside a parallel- plate waveguide as a multiconductor-TL (MTL) problem, and perform a periodic analysis employing capacitance and inductance matrices as opposed to constant values.

The ring capacitance, the reader will recall, helps to determine both the ring resonance frequency ω0 and the magnetic plasma frequency ωmp, the two frequencies that generally decide the bandwidth of the NRI frequency region. Therefore, increasing the ring capacitance depresses both resonances by approximately the same amount, and, thus, lowers the NRI passband. The increase in capacitance could be achieved by using high-permittivity host dielectrics, a possible multilayer metal-insulator-metal (as opposed to the purely coplanar) architecture, or ﬁne interdigitated features, but an easier method may be simply to eliminate some of the capacitors in series. However, such a strategy

Chapter 5. Free-Space Volumetric NRI-TL Metamaterials 104

Magnetic wall

Electric wall Magnetic wall k E Electric wall H (a) (b)

Figure 5.10: (a) Volumetric layered NRI-TL slab of thickness three cells. (b) Representa- tion of inﬁnite slab for illumination by a TE plane wave at normal incidence using electric and magnetic walls. Reprinted with permission from Ref. 60, copyright °c 2006 Optical Society of America.

would render the unit cell of Fig. 5.7 asymmetric, and hence anisotropic even in the eﬀective-medium limit. Alternatively, a unit cell based on a triangular ring may be considered, since such a unit cell reduces the number of required series capacitors to the minimum of three [60].

5.5 Design for Free Space

5.5.1 Transmission characteristics of a volumetric NRI-TL slab

The previous simulations suggest the existence of a NRI, or LH, band for the infinite volumetric layered NRI-TL structure; however, to study transmission properties, it is necessary to consider a structure that is finite in the direction of propagation, or, in other words, possesses a thickness. This ‘slab’ of metamaterial would be inserted into a surrounding medium and illuminated with a plane wave. Figure 5.10(a) illustrates such an arrangement; the volumetric layered NRI-TL metamaterial slab is three unit cells thick and rendered infinite in other directions through the use of appropriate magnetic-wall, Chapter 5. Free-Space Volumetric NRI-TL Metamaterials 105 electric-wall, and boundary-phase conditions. Illuminating the structure by a normally incident TE-polarized plane wave allows the isolation of a unit cell for the finite slab, as shown in Fig. 5.10(b) and make use of image theory to produce the rest of the slab. In the present case, electric walls are placed on the left and right faces to simulate conditions of normal incidence and magnetic walls are placed on the upper and lower faces. The front and back faces are designated port 1 and port 2, respectively. In HFSS, this is known as a

‘driven simulation,’ since a practical source (in this case, a plane wave) drives the system.

The rings are designed as in the previous section, and the medium external to the slab

(from which the plane wave is incident) is a dielectric with relative permittivity ²r = 3 (identical to the filling medium surrounding the 2D series NRI-TL layers). The choice of three unit cells for the slab thickness was influenced by computational limitations resulting from the fine structure in the capacitor interdigitations and inductive strips.

The field transmission (S21) and reflection (S11) coefficients were obtained using HFSS, over roughly the same frequency range shown in Figs. 5.8(a)–5.8(d). Fig. 5.11(a) presents the transmission phase for the finite structure (black dots) superimposed on the dispersion data obtained for the infinite structure (gray dots–taken from Fig. 5.8(d)) and illustrates that a semi-infinite slab of just three cells’ thickness is sufficient to reproduce the general dispersion features of the infinite structure. Fig. 5.11(b) shows the magnitude of the transmission (S21—black line) and reflection (S11—gray line) coefficients. It is evident from Fig. 5.11(b) that this structure is not well matched; in fact, as previously noted, the design was not optimized for matching and instead was focused on improving the NRI bandwidth. This mismatch can, conceivably, be determined as that between the wave impedance of the surrounding space and the effective wave impedance of the volumetric medium. As an example of how the two media can be matched, the following section reverts to a discrete-element design that, in addition to possessing an effective wave impedance equal to that of free space, is designed for a refractive index of −1. Chapter 5. Free-Space Volumetric NRI-TL Metamaterials 106

14 14

ω 12 C,2 12

10 10 ω C,1 8 8 ω 6 0 6 Frequency (GHz) Frequency (GHz) 4 4

2 2

0 0 −π −π/2 0 π/2 π 0 −10 −20 −30 −40 d (radians) S ,S (dB) β 21 11 (a) (b)

Figure 5.11: Printed lumped element design: (a) HFSS transmission phase for slab arrangement of Fig. 5.10(a) (black dots) compared with dispersion of inﬁnite structure (gray dots); (b) HFSS transmission (S21—black line) and reﬂection (S11—gray line) magnitudes. Reprinted with permission from Ref. 60, copyright °c 2006 Optical Society of America.

5.5.2 A Discrete-Element Design Matched to Free Space

This section considers a design based on discrete lumped elements that meets the two conditions for the Veselago-Pendry superlens: (1) the relative refractive index is exactly

−1 in free space; and (2) the wave impedance is matched to that in free space. As a discrete-element design, this metamaterial may be characterized by the eﬀective-medium parameters of (5.1), obtained from the equivalent circuit model of Fig. 5.3(b). Setting

µeff = −µ0 and ²eff = −²0 and retaining as many of the design values in Table 5.1 as possible, it is found that this condition, although quite sensitive to small variations in the design, can be met at a frequency of 2.52GHz simply by increasing the loading inductance

L0 from 5.6nH to 11.37nH. Figure 5.12(a) presents the dispersion characteristics of this design as obtained using the equivalent circuit model (dashed line), HFSS simulations of the inﬁnite periodic structure (gray dots), and HFSS simulations of the slab arrangement Chapter 5. Free-Space Volumetric NRI-TL Metamaterials 107

5 5

4 ω 4 C,2

3 3

ω 2 0 2 Frequency (GHz) Frequency (GHz) ω C,1 ω 1 B 1

0 0 −π −π/2 0 π/2 π 0 −20 −40 S ,S (dB) βd (radians) 21 11 (a) (b)

Figure 5.12: Discrete lumped element design matched to free space: (a) HFSS transmission phase for slab arrangement of Fig. 5.10(a) (black dots) compared with dispersion of inﬁnite structure (gray dots) and equivalent circuit mode (dashed lines); (b) HFSS transmission (S21—black line) and reﬂection (S11—gray line) magnitudes. Reprinted with permission from Ref. 60, copyright °c 2006 Optical Society of America.

of Fig. 5.10(a) employing this design. In this case, the slab is embedded in vacuum. The light cone in vacuum (not shown) is found to intersect the full-wave dispersion data very near the predicted 2.52GHz (between ω0 and ωC,1), where (5.1) simultaneously produces

µeff /µ0 = ²eff /²0 = −1. Figure 5.12(a) shows the transmission (S21—black line) and reflection (S11—gray line) coefficients. The transmission in this frequency region exhibits a peak value of -0.16dB (96%), where the reflection is better than -15dB (3%). Thus, the volumetric NRI-TL medium can be designed to meet the conditions required to realize the Veselago-Pendry superlens for plane-wave components propagating in the plane of the rings. Chapter 5. Free-Space Volumetric NRI-TL Metamaterials 108

5.6 Implementations from THz to Optical Frequen-

cies

This section revisits the idea of scaling such layered metamaterials for operation at THz frequencies and considers plasmonic implementations that may yield a NRI at near infrared or optical frequencies.

At THz frequencies, one is restricted to the use of printed lumped elements; as an example, consider the printed design of Tables 5.1 and 5.2, whose minimum feature size is the ﬁnger width of the interdigitated capacitor (100µm). Current lithographic techniques can produce features that are typically 50 nm across or less. This suggests that the unit cells may be scaled down by a factor of approximately two thousand, implying that this particular design can be fabricated to operate above 15 THz.

The ability to model plasmonic materials/resonances using TL analogies, as described in Refs. 54 and 55, may enable the design of similar printed, volumetric structures at near-infrared or even optical frequencies. It was shown in Ref. 54 that nanometre-scale plasmonic metallic particles, which can exhibit a negative permittivity in the near infrared or visible region, respond to applied ﬁelds in a manner akin to an inductor. Similarly, it is reasonable to identify a capacitance in the fringing ﬁelds between adjacent metallic particles. Using these two concepts, we can envision the optical equivalents of the series

NRI-TL structure formed by replacing the inductive elements with plasmonic nanoparticles, as shown in Fig. 5.13(a) for a square NRI-TL unit cell and in Fig. 5.13(b) for a triangular NRI-TL unit cell. The particles themselves are represented in the diagrams as spheres, but their shapes may be chosen to suit particular unit cell geometries. It is interesting to note that the two resulting lattices of particles, representing the series

NRI-TL topology, resemble those proposed by G. Shvets in Figs. 1(c)–1(d) of Ref. 57 to support LH surface waves and, therefore, exhibit a NRI at optical frequencies. Further support for the feasibility of such arrangements of plasmonic nanoparticles is given in

Chapter 5. Free-Space Volumetric NRI-TL Metamaterials 109

(a)

(b)

Figure 5.13: Possible optical implementations of the series NRI-TL array using plasmonic nanoparticles for (a) square unit cells, and (b) triangular unit cells. Reprinted with permission from Ref. 60, copyright °c 2006 Optical Society of America.

the analytical treatment and ﬁnite-diﬀerence time-domain simulations of a chain of plasmonic spheres by Atwater et al. [56], who have shown that such a structure supports a transverse backward-wave mode (see Fig. 2 of the last reference). It is worthy to note that the arrangements of Figs. 5.13(a)–5.13(b) may possibly be extended to a volumetric structure either by layering these 2D lattices, or by treating them as cross sections of an array of plasmonic cylinders.

On this note, it is evident that the particular arrangement of particles depicted in

Fig. 5.13(a) for the square NRI-TL array appears to be a regular lattice under a 45◦ ro- tation in the plane. Layering such planes in a multilayer fashion similar to that described in this work would result in a rotated simple tetragonal lattice, or cubic lattice if the Chapter 5. Free-Space Volumetric NRI-TL Metamaterials 110 layer spacing is equal to the horizontal lattice spacing. From another perspective, the unit cell of Fig. 5.13(a) corresponds to the face of a face-centered cubic (FCC) lattice, and misaligning alternating layers by half the cell period would result in a 3D FCC lattice of plasmonic nanoparticles, once again elongated in the direction perpendicular to the planes according to the layer spacing. Chapter 6

A Free-Space NRI-TL Lens

6.1 Subwavelength versus Diﬀraction-Limited Focus-

ing

In the previous chapter, it was shown that the volumetric NRI-TL topology could be used to design a flat lens possessing the ‘superresolution’ or ‘subwavelength focusing’ property [60, 62, 115], entitling it to be called a ‘Veselago-Pendry’ superlens. Subwave- length focusing requires a resonant enhancement of the evanescent spectrum containing the subwavelength spatial features of the source, but this is a near-field effect that is extremely sensitive to certain conditions on the size and loss of the metamaterial. In fact, it was shown in Ref. 8 that any lens producing a subwavelength spot must, itself, possess a subwavelength thickness, and its unit cells must, accordingly, be even smaller and possess very low losses. The most critical source of loss, it was reported, was that of the lumped components, represented by their quality factors. In the equivalent-circuit analysis described in Refs. 60, 62, and 115, the lumped loading is assumed low-loss and also strong enough to yield NRI characteristics at low frequencies; this ensures a small unit-cell size and allows the design of electrically thin volumetric superlenses. How- ever, in fully printed implementations employing printed loading elements (such as those

111 Chapter 6. A Free-Space NRI-TL Lens 112 described in the present chapter and those based on SRRs), the strong fields (and consequently large amount of scattering) near the fine metallic features reduce their quality factors significantly. Moreover, these elements cannot typically provide a loading effect strong enough to render the unit cells electrically very small, and so lenses constructed from them tend to be electrically thick, which, according to Refs. 116 and 8, also precludes any sort of practical subwavelength imaging ability. Nevertheless, certain recent work in the literature (see, for example, Refs. 15 and 16) have claimed to observe superresolution using fully printed SRR-based metamaterials in spite of electrically large unit cells and lossy, electrically thick lenses. To address these apparent discrepancies, the next two chapters present two versions of the volumetric NRI-TL metamaterial: this chapter describes a fully printed electrically thick lens employing lumped loading elements

(interdigitated capacitors and meandered inductors) to produce a moderately lossy NRI response at X-band; Chapter 7 describes an electrically thin lens employing discrete, surface-mount lumped loading elements (chip capacitors and inductors) to produce a low-loss NRI response at S-band.

In this chapter, a fully printed volumetric NRI-TL metamaterial is designed and employed as a free-space ﬂat lens. It will be shown experimentally that such a lens does not demonstrate subwavelength focusing or subdiﬀraction imaging, as a result of its electrical thickness and susceptibility to losses. However, although such ‘Veselago lenses’

(in contrast to the term ‘Veselago-Pendry superlenses’) operate within diﬀraction limits, they possess other intriguing phenomena of practical interest, as well as advantages over conventional curved lenses. For example, they produce refraction and focusing without geometric aberrations; they lack a principal axis, and so their ability to focus is invariant to the lateral position of the source; for the same reason, a plane wave passing through a Veselago lens remains unfocused whereas secondary ﬁelds produced, for example, by an illuminated scatterer will backscatter and produce a focus. Furthermore, much could be learned in the fabrication and test of a simple, inexpensive printed lens, prior to the Chapter 6. A Free-Space NRI-TL Lens 113 construction of a more complex superlens based on discrete lumped components.

This chapter presents simulation and measured data conﬁrming the NRI transmission and focusing properties of a volumetric NRI-TL metamaterial Veselago lens designed using fully printed loading elements (interdigitated capacitors and meandered inductors).

A fully printed metamaterial unit-cell design is developed and corresponding full-wave dispersion and transmission simulations are presented; the subsequent sections describe the fabrication of the metamaterial and presents experimental data at X-band showing its transmission properties as well as its free-space focusing ability, arguably the ﬁrst experimental evidence of free-space excitations coupling to a TL-based metamaterial.

6.2 Design

Although the use of discrete (chip) lumped elements aﬀords the ability to precisely tailor the dispersion and transmission features of the volumetric NRI-TL metamaterial (which is a necessity in designing for the stringent conditions associated with superlensing – see

Chapter 7), such implementations require careful placement of the numerous discrete components and can be prohibitively expensive. They also restrict freedom in design: the component self-resonances limit the NRI phenomena to low operating frequencies, and their sizes limit flexibility in choosing the ring size and layer-to-layer spacing. As a result, they may be superfluous for the design of a diffraction-limited metamaterial lens. Indeed, the inherent planar property of the constituent layers in the volumetric

NRI-TL topology suggests for this purpose the use of planar printed lumped elements

(meandered inductors and interdigitated capacitors) that can be integrated into, and fabricated along with, the metallic ring structure. Printed elements, however, often require very small features and typically cannot provide the strong lumped loading of their discrete counterparts; nevertheless, their simplicity makes them attractive for the fabrication of a simple, volumetric NRI-TL metamaterial.

Chapter 6. A Free-Space NRI-TL Lens 114

100µm

5mm 778µm

5.5mm

100µm

Figure 6.1: Fully printed NRI-TL ring employing interdigitated capacitors and meandered inductors. Reprinted with permission from Ref. 63, copyright °c 2007 IEEE.

The NRI-TL ring geometries studied in Ref. 60 and Chapter 5 were designed as square rings with capacitors wrapped around the corners. In the present work, the octagonal ring geometry proposed in Ref. 104 was employed instead. It was decided that the smallest feature size that could reasonably be fabricated, given the available in-house chemical etching facilities, was approximately 100µm; that is, the interdigitated capacitors and meandered inductors were designed using both gaps and lines of 100-µm width. The choice of a minimum feature size also imposes certain constraints on the ring geometry, which must naturally accommodate the area occupied by the printed elements. These and other geometrical attributes were decided through parametric analyses seeking to maximize the bandwidth of the structure, while also keeping ease of fabrication in mind.

For instance, it was observed through simulations that each of the following variations increases the NRI bandwidth: (1) a decrease in the plane spacing, h, resulting in a stronger magnetic coupling between planes; (2) an increase in the permittivity of the surrounding dielectric medium, resulting in an increase in the capacitive loading; and (3) a decrease in the separation between coplanar rings, resulting in a larger ﬁlling factor.

The ﬁnal ring design and its relevant dimensions are depicted in Fig. 6.1. The capacitors were realized using six interdigitated 100-µm ﬁngers, each realizing a capacitance

Chapter 6. A Free-Space NRI-TL Lens 115

5.5mm 5.5mm

180mil

60mil

(a) (b)

Figure 6.2: (a) Unit cell for volumetric NRI-TL metamaterial consisting of a metallic ring embedded between a 60-mil-thick dielectric (²r = 3) region and a 120-mil-thick air (²r = 1) region. (b) Dispersion characteristics for axial propagation of a horizontally polarized mode obtained using HFSS. Reprinted with permission from Ref. 63, copyright °c 2007 IEEE.

on the order of 200fF, and the inductors consisted of a meandered ﬁlament of 100-µm width sunken into the CPS lines to increase its meander length, producing an inductance on the order of 1nH. The rings are made of copper (0.5-oz., σ = 5.8 × 107 S/m) and placed on a Rogers RO3003 (²r = 3, tan δ = 0.0013) dielectric substrate with a thickness of 60mil (1.524mm). The vertical period of the unit cell is 180mil, and the space between layers (120-mil thickness) is air-ﬁlled.

6.3 Simulation

6.3.1 Dispersion Properties

The complete volumetric NRI-TL unit cell is shown in Fig. 6.2(a), and the resulting dispersion data for horizontally polarized modes (magnetic ﬁeld perpendicular to the plane of the rings) are shown in Fig. 6.2(b). Evidently, for these feature sizes, the Chapter 6. A Free-Space NRI-TL Lens 116 corresponding printed element values and unit cell size place the NRI passband region at X-band, between 9.4 and 12 GHz, representing a fractional bandwidth of 24.2%. The light line (not shown) intersects the NRI passband near 11GHz, suggesting that this is also the frequency where the metamaterial possesses a refractive index of −1. Here, as in the structures reported in Ref. 60, the upper and lower edges of the passband are determined primarily by the values of the printed inductors and capacitors, respectively.

The directions of the simulated surface currents on the metallic rings within the passband are also depicted symbolically in Fig. 6.2(a), and are consistent with the loop currents associated with NRI behaviour. However, since the achievable lumped loading is very small, and certainly much smaller than could have been achieved using discrete chip elements, the electrical size of the unit cells at these frequencies is on the order of λ0/5, where λ0 is the wavelength in free space. This allows us to make the eﬀective-medium assumption, albeit cautiously.

6.3.2 Transmission Magnitude and Phase

Simulations of transmission magnitude and phase for a normally incident horizontally polarized plane wave were performed for a slab of five-unit-cell thickness using HFSS, in an arrangement similar to that depicted in Fig. 5.10(b). Infinite transverse dimensions were, once again, modeled using the appropriate boundary-phase conditions, and dielectric and conductor losses, as well as a 2µm-roughness of the metal surface, were included in the simulations. The simulated S21 and S11 magnitudes and phases, shown in Fig. 6.3 (solid and dashed curves, respectively), are in agreement with the dispersion data provided by HFSS, not only in the range of frequencies enclosed by the passband, but also in the slope of the corresponding phase dispersion, which has been verified to follow that of Fig. 6.2(b) closely. The steady-state time evolution of the simulated field en- velopes at frequencies within the passband also indicates a backward wave, the hallmark of left-handedness, which reaffirms the NRI characteristic of the designed volumetric Chapter 6. A Free-Space NRI-TL Lens 117

−10 dB −20

−30 7 8 9 10 11 12 13 Frequency (GHz)

100

0 Degrees −100

7 8 9 10 11 12 13 Frequency (GHz)

Figure 6.3: Simulated transmission and reﬂection magnitude and phase for a volumetric NRI-TL metamaterial slab of ﬁve-unit-cell thickness obtained using HFSS (S11–dashed curve, S21–solid curve). Reprinted with permission from Ref. 63, copyright °c 2007 IEEE.

metamaterial at X-band. The passband exhibits an insertion loss from 3dB at 10GHz

(0.6dB per unit cell) to 9dB at 12GHz (1.8dB per unit cell). Although appreciable, these insertion loss values are indicative of the typically low quality factors of ﬁne-featured printed metallic elements at these frequencies, and can be mitigated using discrete chip lumped elements.

It is worthwhile at this point to consider the distribution of ﬁelds within the metamaterial in the NRI passband. Fig. 6.4 presents the simulated magnetic ﬁeld intensities

(logarithmic shading and contours) for a single row of the metamaterial over three of its faces at 11.36GHz. Although the free-space excitation illuminates the full volume of the

Chapter 6. A Free-Space NRI-TL Lens 118

Figure 6.4: Magnetic field intensities (logarithmic shading and contours) for a single row of five unit cells inside the volumetric NRI-TL metamaterial when illuminated by a normally incident plane wave at 11.36GHz. The fields are strongly confined to the layers containing the metallic features, as highlighted in the inset. Reprinted with permission from Ref. 63, copyright °c 2007 IEEE.

multilayer structure, the metamaterial ﬁelds remain strongly conﬁned to the CPS TL layers (maximum intensity at the meandered inductor sites transverse to the direction of propagation), where they are up to three to four magnitudes higher than the surrounding

ﬁelds. This result aﬃrms the true TL nature of the volumetric NRI-TL metamaterial.

6.3.3 Negative Refraction and Time Evolution of Backward

Waves

Viewing a single row of cells illustrates only the component of the wavevector normal to the interfaces, and so is not conducive to examination of negative refraction, which is based on the matching of the tangential wavevector components. Therefore, the present section shall examine a four-cell region of an inﬁnite slab embedded in air, viewed from the top, and illuminated by a suitably polarized plane wave incident at 15 degrees from the normal. Figure 6.5 shows the steady-state ﬁelds produced by this plane wave (loga-

Chapter 6. A Free-Space NRI-TL Lens 119

10.7GHz 10.9GHz 11.1GHz 11.3GHz

Figure 6.5: Negative refraction in the volumetric NRI-TL metamaterial (vertical magnetic ﬁelds shown) when illuminated by a 15-degree incident plane wave at 10.7GHz, 10.9GHz, 11.1GHz, and 11.3GHz, obtained using HFSS. Arrows indicate approximate direction of incident and refracted waves.

rithmic shading) at several frequencies within the NRI passband, as obtained using HFSS.

The simulations were terminated along the sides using phase-shift boundary conditions consistent with the tangential wavevector component of the incident plane wave. The refraction angle tends to increase as the frequency is increased, suggesting a decreasing

NRI, which is consistent with the frequency dispersion shown previously in Fig. 6.2(b).

This agreement is further aﬃrmed by the fact that the eﬀective wavelength inside the metamaterial increases as the frequency is increased.

The formation of backward waves inside the metamaterial for a normally incident plane wave is merely a special case of the general phenomenon of negative refraction.

The steady-state magnetic ﬁeld magnitudes in the metamaterial region at 10.7GHz are

Chapter 6. A Free-Space NRI-TL Lens 120

ϕϕϕ=0°°° ϕϕϕ=40°°° ϕϕϕ=80°°° ϕϕϕ=120°°°

Figure 6.6: Time evolution of backward waves inside the volumetric NRI-TL metamaterial (vertical magnetic ﬁelds shown) at successive source phases, ϕ, obtained using HFSS. Dot- ted lines and circles track the progression of particular phasefronts in both free space and the metamaterial.

shown in Fig. 6.6 at successive phases in their period. By tracking the progression of the phasefronts, it is evident that the Floquet-Bloch modes prevalent within the volumetric NRI-TL metamaterial are backward waves; that is, whereas the phasefronts of the incident wave travel away from the source, the phasefronts within the metamaterial travel towards the source, and the wave emitted into vacuum at the rear face of the slab resumes its forward nature.

6.4 Experiment

The fabrication of the volumetric structure began with the chemical etching of both the

NRI-TL patterns and ground planes from RO3003 (²r = 3) 60-mil (1.524mm), 0.5-oz

Chapter 6. A Free-Space NRI-TL Lens 121

Figure 6.7: Fabricated volumetric NRI-TL metamaterial (w × h × t = 104.5mm×105.2mm×27.5mm). The inset shows the printed features comprising each layer. Reprinted with permission from Ref. 63, copyright °c 2007 IEEE.

(17µm) copper-clad substrate material using an Advansys ferric chloride PCB etching machine. The layers were then precision-cut using a LPKF Protomat H100 computer- controlled milling machine/circuit-board plotter. A bracket to support the layers at intervals of h=180mil=4.572mm (i.e., with an air gap between layers of 120mil) was fabricated from the same material in the same fashion, and the layers were assembled. The ﬁnal volumetric NRI-TL metamaterial, depicted in Fig. 6.7, comprises 23 layers, each 19 unit cells wide and 5 unit cells thick, with an overall size (w×h×t) of 104.5mm×105.2mm×27.5mm.

6.4.1 Transmission Magnitude and Phase

The experimental testing apparatus consisted of an Agilent E8364B Performance Network

Analyzer (PNA), two X-band standard gain pyramidal horns, and two double-convex hyperbolic 8-inch Rexolite (²r = 2.53) lenses mounted on translating stages aﬃxed to an optical table, similar to the setup employed in Ref. 68. As detailed in Appendix C,

Chapter 6. A Free-Space NRI-TL Lens 122

154 mm 290 mm

8 in

Figure 6.8: Free-space X-band measurement system consisting of an Agilent network analyzer, standard gain horn antennas, and Rexolite dielectric lenses. Reprinted with permission from Ref. 63, copyright °c 2007 IEEE.

the lenses were designed and fabricated to focus the antenna ﬁelds to a beam with a minimum waist diameter contained within the slab face, where the ﬁelds resemble those of a horizontally polarized plane wave (the metamaterial slab was rotated by 90 degrees to match the polarization of the horns). The entire experimental setup is depicted in

Fig. 6.8, along with the relevant dimensions. Following a free-space TRL calibration, the transmission magnitude and phase from 7–13GHz were recorded; these data are presented in Fig. 6.9 (S11–dashed black curve, S21–solid black curve). A passband is visible from approximately 10–12GHz (a fractional bandwidth of approximately 18%), which exhibits a minimum insertion loss of 6dB, or 1.2dB per unit cell, at 11.5GHz.

However, the measured passband is slightly up-shifted and narrower when compared Chapter 6. A Free-Space NRI-TL Lens 123

−10 dB −20

−30 7 8 9 10 11 12 13 Frequency (GHz)

100

0 Degrees −100

7 8 9 10 11 12 13 Frequency (GHz)

Figure 6.9: Measured transmission and reﬂection magnitude and phase for a volumetric NRI-TL metamaterial slab of ﬁve-unit-cell thickness (S11–dashed black curve, S21–solid black curve). The circles (S11–open gray circles, S21–solid gray circles) represent full- wave simulation data accounting for over-etching of the printed features. Reprinted with permission from Ref. 63, copyright °c 2007 IEEE.

to the simulation results of Fig. 6.3; this discrepancy may be attributed primarily to over-etching of the printed features, which was observed in microscope measurements to be non-uniform between the printed features and between layers. To illustrate, the full-wave simulation data shown superposed on Fig. 6.9 (S11–open gray circles, S21–solid gray circles) could be matched to the measured data by incorporating over-etching of

25% for the interdigitated capacitors (shifting up the lower passband edge) and ring structure, and 10% for the meandered inductors (shifting down the upper passband edge); these numbers are in approximate agreement with the typical variations observed Chapter 6. A Free-Space NRI-TL Lens 124 in microscope measurements of the printed features using an Olympus SZX10 research stereo microscope. Simulations not shown in this work indicate that other fabrication tolerances, e.g. slight misalignment (100-200µm) of the layers in the assembly process and the inability to accurately control the spacing between layers due to the ﬂexibility of the substrate material, also aﬀect the passband characteristics, albeit to a much lesser extent.

6.4.2 Focusing

One of the most interesting physical phenomena associated with NRI metamaterials is the ability of a ﬂat slab of such a material to focus the rays emanating from a source. Given the success of planar NRI-TL metamaterials in verifying focusing and subdiﬀraction imaging in a planar microstrip environment, it is of great interest to examine the focusing ability of the designed volumetric NRI-TL metamaterial in free space.

The experimental setup consists of an Agilent E8364B Performance Network Ana- lyzer (PNA) and two small (3mm inner diameter) loop antennas providing the required magnetic-ﬁeld polarization. The loop antennas are fabricated from 1.19mm-diameter semi-rigid 50-Ω coaxial cable and Huber+Suhner connectors, and employ a shielded-loop conﬁguration that minimizes unbalanced currents on the feed (see, for example, Ref. 117).

The operation of the shielded-loop antenna, along with full-wave simulations and measured patterns, is presented in Appendix D. Portions of the measurement apparatus were covered in a metal-backed Emerson and Cuming ECCOSORB AN-74 broadband microwave absorber. As shown in Fig. 6.10, one of the antennas is stationary and illuminates the front side of the slab at a distance of 19mm, while the other, aﬃxed to the linear translation stages of a computer-controlled xyz-translator apparatus (designed by

Newmark Systems, California, USA – not shown), samples the magnetic ﬁelds over a horizontal plane at the rear side of the slab in increments of 1mm (the xyz-translator and PNA were controlled using a custom computer code developed in-house). Although Chapter 6. A Free-Space NRI-TL Lens 125

Figure 6.10: Free-space focusing arrangement consisting of a shielded-loop antenna illuminating the front face of the volumetric NRI-TL slab lens and an identical receiving antenna attached to an xyz-translator (not shown), scanning the ﬁelds at the rear side of the slab. The antennas are connected to an Agilent E8364B network analyzer (not shown). Reprinted with permission from Ref. 63, copyright °c 2007 IEEE.

excitation by single loop differs from the ideal case of an infinite magnetic line source, the strong confinement of the metamaterial fields to the layer planes (see Fig. 6.4), combined with the fact that the single loop may be applied at any of the planes of a sufficiently large lens, suggests that the response to the latter can satisfactorily be obtained as a superposition of the responses to the former; that is, any free-space (in-plane) focusing due to excitation with a single loop should also be evident when the metamaterial is excited by an infinite magnetic line source, and vice versa.

Following an initial SOLT (short-open-load-through) broadband calibration of the measurement system, and averaging over 20 samples to minimize noise, the magnetic-

ﬁeld magnitudes and phases at the rear side of the slab lens were measured. Figure 6.11(a) illustrates the geometry of the focusing setup and the region over which the measurements

Chapter 6. A Free-Space NRI-TL Lens 126

19mm

60mm

Source Detector 96mm

Measured

region

27.5mm

(a)

(b)

(c)

Figure 6.11: Measured field magnitude and phase data at the rear side of the volumetric NRI-TL slab lens: (a) schematic illustration of focusing and the measured region (marquee); measured fields (b) in the PRI frequency region at 7.00GHz, and (c) in the NRI frequency region at 11.36GHz. The black curve in (c) is a −3dB contour (normalized to the field values at the focal plane) indicating the width of the beam. Reprinted with permission from Ref. 63, copyright °c 2007 IEEE. Chapter 6. A Free-Space NRI-TL Lens 127 were taken (marquee). The raw measured data in this region are depicted in Figs. 6.11(b) and 6.11(c) for frequencies of 7.00GHz and 11.36GHz, respectively. The reader will note that the former frequency lies inside the RH or positive-refractive-index (PRI) region of

Figs. 6.2(b) and 6.9; accordingly, there is no evidence of focusing either in the magnitude or in the phase. The latter frequency lies inside the NRI passband of Figs. 6.2(b) and

6.9, and so the corresponding data should suggest the expected focusing phenomena.

Moreover, the observed frequency shift of the passband due to overetching also causes a shift in the frequency at which the refractive index of the metamaterial achieves a value of −1, which is, therefore, expected to occur near 11.36GHz. Indeed, the focal region appears as an elongated beam, whose minimum waist is marked by the formation of two nulls in the transverse direction. The minimum waist occurs at a distance of approximately 8mm from the rear face of the metamaterial. Taken along with the fact that the source is placed 19mm from the front face of the metamaterial, it becomes clear that the observed focusing is consistent with the Veselago lensing condition, which requires that the sum of these distances is equal to the lens thickness (27.5mm in the present case) when the lens possesses a refractive index of −1. The magnitudes of Fig. 6.11(c) have been normalized to the ﬁeld value at the point on the principal axis approximately

15mm from the exit face of the lens, where the beam shows the best confinement (the magnitudes of Fig. 6.11(b) are normalized to the maximum field values, occurring near the exit face of the lens). Also apparent in Fig. 6.11(c) is the expected reversal of the concavity of the phase-fronts at the exit face of the lens and across the focus. Indeed, observing the measured fields at other frequencies in the measured frequency range reveals the evolution of the focal spot from a confined region near 10.7GHz that elongates into a beam towards the end of the NRI passband at 12GHz [17,118].

The −3dB contour plotted on the focal region (black curve) of Fig. 6.11(c) shows that the beam width is approximately 0.55λ0, which is on the order of the minimum width dictated by the diffraction limit (see Appendix A). This result is consistent with the Chapter 6. A Free-Space NRI-TL Lens 128 fact that the volumetric slab lens under study is electrically thick (approximately one free-space wavelength) and moderately lossy, as explained previously. It should be noted that the finite transverse width of the lens (104.5mm) in the above focusing arrangement represents a numerical aperture of 0.94; as a result, the lens is able to collect most of the propagating spectral components of the source and, as such, the finiteness of the metamaterial does not significantly affect the size of the focal spot. Chapter 7

A Free-Space NRI-TL Superlens

As discussed in Chapter 1, the free-space Veselago-Pendry superlens possesses a number of principal requirements: first, µ = −µ0 and ² = −²0, conditions which guarantee both that the real part of its effective refractive index is n = −1 and that it is impedance-matched to free space; second, the lens must be extremely low loss and adequately thin; third, the unit cells constituting the metamaterial must themselves be deeply subwavelength in size; last, the transverse dimensions of the lens must be large enough that the lens can be illuminated by a source in free space. Based on these very stringent constraints, it can be said that a true Veselago-Pendry superlens has not yet been realized for subdiffraction imaging in free space. However, many other varieties of metamaterial have experimentally demonstrated phenomena akin to subdiffraction imaging by associated physical mechanisms. These include the plasmonic silver film [9], the magnetoinductive lens [10], and the swiss-roll structure [11] which, although they do not possess a NRI and so cannot focus the propagating wave numbers, do recover fine spatial features through evanescent enhancement; unfortunately, this means that such lenses impose an extremely short working distance (typically much less than λ0/8, where λ0 is the free-space wavelength) between the source, lens, and image. Subdiffraction imaging phenomena have been successfully extended to the far field using magnifying ‘hyperlenses’ [12–14], but

129 Chapter 7. A Free-Space NRI-TL Superlens 130 this class of superlenses relies explicitly on anisotropy and sources are typically applied directly to the hyperlens face; as a result, the working distance on the source side remains limited. Subdiffraction imaging using printed SRR-based structures has been reported in the way of transversely and longitudinally confined subwavelength focal ‘spots’ [15,16] (it was shown that longitudinal subwavelength resolution is necessarily compromised when transverse subwavelength resolution is achieved [10]), in spite of their high losses and/or large electrical thickness; however, calculations based on Ref. 7 suggest that the reported loss, lens thickness, and observed resolution ability are inconsistent, if the structures are to be regarded as true Veselago-Pendry superlenses. Furthermore, the lenses described in these works are said to possess a refractive index of −1.8, which defies even the basic requirement that the refractive index of a free-space Veselago-Pendry superlens be −1.

Moreover, the use of a higher index reduces the working distance between lens and image.

Thus, the results described in these works do not conform to the imaging principles of the Veselago-Pendry superlens and the requirements outlined above; indeed, in attempting to explain these inconsistencies, the authors of Refs. 15 and 16 have speculated that anisotropy, rather than subdiﬀraction imaging by way of the restoration of evanescent waves, may be responsible for these phenomena [16]. Although these factors preclude their description as Veselago-Pendry superlenses, further research into such structures may reveal other intriguing mechanisms by which subdiﬀraction imaging can be achieved.

In fact, a possible explanation for these results based on anisotropy is suggested at the end of this chapter.

Chapter 6 presented a volumetric NRI-TL free-space metamaterial lens realized using fully printed loading elements (interdigitated capacitors and meandered inductors, and without any vias), which demonstrated diffraction-limited focusing of a free-space magnetic dipole source consistent with the use of large unit cells and an electrically thick lens [63]. It was suggested in that chapter that the unit cells may be made simultaneously low-loss and electrically small by exploiting the strong lumped loading afforded by dis- Chapter 7. A Free-Space NRI-TL Superlens 131 crete chip inductors and capacitors with high quality factors, as in the planar case, which would also enable the realization of an adequately thin NRI-TL metamaterial, free-space superlens. This chapter demonstrates free-space imaging with a resolution over three times better than the diffraction limit at microwave frequencies using a free-space NRI-

TL metamaterial Veselago-Pendry superlens. The use of discrete lumped elements aﬀords precise control over the superlens’ electromagnetic properties and makes it both electrically thin and less susceptible to losses than previous approaches. These results show that the so-far elusive free-space Veselago-Pendry superlens is, indeed, realizable, and arguably represent the ﬁrst such realization. A microwave superlens can be particularly useful for illumination and discrimination of closely spaced buried objects over practical distances by way of back-scattering, for example, in tumour or landmine detection, or for targeted irradiation over electrically small regions in tomography or hyperthermia applications.

7.1 Design

As mentioned in Chapter 3, the NRI-TL approach is known to oﬀer intrinsically large NRI bandwidths and minimize losses through the tight coupling between unit cells [48], which is also related to the electrical size of the unit cells. These features are exploited in the

NRI-TL metamaterial Veselago-Pendry superlens through its reliance on discrete surface- mount chip inductors and capacitors. The strength of the lumped elements loading the host TL medium renders the unit-cell size deeply subwavelength, allows the realization of an adequately thin Veselago-Pendry lens, affords precise control over its effective-medium properties, and mitigates losses. Accordingly, the design process was largely based on the availability of suitable surface-mount components with high quality factors, the desired loading values at the design frequency of 2.40GHz, and sufficiently high self-resonant frequencies. The selected components and their values are listed in Table 7.1. Chapter 7. A Free-Space NRI-TL Superlens 132

Part No. Description Value 0302CS 18NXJL Coilcraft high-Q inductor 19.3nH at 2.4GHz

ATC600L1R2AT200T ATC high-Q capacitor 1.2pF

Table 7.1: Surface-mount inductor and capacitor values.

7.14mm capacitors x

y 1.524mm

1.952mm

inductors (a) (b)

Figure 7.1: (a) NRI-TL free-space metamaterial superlens unit cell. Reprinted with permission from Ref. 64, copyright °c 2008 American Institute of Physics; (b) relevant dimensions of CPS TL host medium. Reprinted with permission from Ref. 119, copyright °c 2009 IEEE.

As before, the TL host medium constituting the layers of the volumetric NRI-TL lens employs a ring geometry based on the series NRI-TL node, which may be realized in a fully uniplanar form using CPS TLs and so obviates the need for vias [60]. The substrate medium was chosen to be 0.5-oz.-copper-clad Rogers RO3003 60-mil (1.524mm) microwave substrate with ² = 3²0 and tan δ = 0.0013, and the particular geometrical features of the ring were decided to accommodate the landing patterns of the selected surface-mount components. A schematic of the complete NRI-TL free-space metamaterial superlens unit cell is shown in Fig. 7.1(a), and the relevant dimensions of the copper traces are shown in Fig. 7.1(b).

One side of the substrate supports the metallic ring geometry onto which the components are to be placed, and the metal on the other side of the substrate is completely removed. Air gaps separating the substrate layers were chosen to be 1.952mm thick, Chapter 7. A Free-Space NRI-TL Superlens 133

Parameter Description Value d Unit cell size 7.1400mm Lwg Parallel-plate waveguide, intrinsic inductance 18.430nH Cwg Parallel-plate waveguide, intrinsic capacitance (57.764 − j0.0526)fF Lr Ring inductance (19.508 + j0.0300)nH C0 Chip capacitance (1.2000 − j0.0018)pF L0 Chip inductance (19.300 − j0.0232)nH Cx CPS-TL intrinsic capacitance per-unit-length (22.023 − j0.0203)pF/m d0 Eﬀective CPS-TL length 6.4304mm M Mutual inductance 13.363nH

Table 7.2: Element values for the equivalent-circuit model describing on-axis propagation through the NRI-TL free-space metamaterial superlens at 2.40GHz.

resulting in a vertical layer spacing p = 3.476mm; these values were decided in concert with the equivalent-circuit model described in Chapter 5, which was slightly adapted to include conductor, dielectric, and component losses. The parameter p provides the

flexibility (by modification of the inductive coupling between rings on adjacent layers) to tune the the effective-medium properties of the metamaterial without having to obtain new surface-mount components or change the layer designs themselves. The remaining parameters in the model, representing the lumped loading and parasitics of the ring topology, were determined directly where values were provided (e.g. the inductance, capacitance, and quality factors of the selected surface-mount components) and otherwise by matching with simulation data (e.g. the mutual inductance between rings on adjacent planes). Table 7.2 summarizes the values employed in the final NRI-TL free-space metamaterial superlens design at the design frequency of 2.40GHz; the reader will note that the losses are represented by the complex nature of the element values, whose imaginary parts can be converted into either resistance or conductance or represented in terms of quality factors. As will be shown in the next section, these values yield effective-medium parameters <{µeff } = −µ0 and <{²eff } = −²0 at an operating frequency of 2.40GHz

(λ0 = 125mm in free space) when illuminated by p-polarized fields propagating in the layer planes (i.e., z-polarized magnetic fields – the polarization to which the NRI-TL lens Chapter 7. A Free-Space NRI-TL Superlens 134 appears isotropic). It is also worthy to note that the chosen unit-cell size, d = 7.14mm square, is nearly λ0/18, and so a lumped description is justified.

7.2 Simulation

Simulations of the designed volumetric NRI-TL topology were performed using Ansoft’s

HFSS [120]. In all cases, the lumped components were modelled by current sheets en- dowed with the appropriate lumped-element boundary conditions; dielectric loss tangents and component losses (as specified in their data sheets in terms of either equivalent series resistance or quality factor) were included, and the metallic features were specified as copper with a bulk conductivity reduced by over 70% from the nominal value to account for surface roughness, by way of HFSS’s ‘finite-conductivity’ boundary condition.

7.2.1 Eﬀective-Medium Properties

The ultimate goal of extracting the eﬀective-medium parameters of a periodic medium requires knowledge of its intrinsic eﬀective TEM propagation constant, β, and wave impedance, η. As described in Chapter 5, the former may be obtained through an eigenmode simulation; the latter may be obtained in a TL environment of known geometry

(e.g. parallel-plate waveguide) by way of its characteristic impedance Z0, through a driven simulation. By associating the obtained S-parameters with a homogeneous slab of equal thickness, its effective homogeneous properties can be obtained by way of an inversion procedure that is approximately valid for electrically thin slabs [121] consisting of electrically very small unit cells; certainly, the volumetric NRI-TL lens falls into this category (a discussion validating these ideas with simulation results follows). The band structure and transmission/reflection magnitudes for propagation of a normally incident plane wave through a five-unit-cell-thick volumetric NRI-TL metamaterial slab are shown in Fig. 7.2. Also shown are the real parts of the relative effective permittivity and per- Chapter 7. A Free-Space NRI-TL Superlens 135

4 2.5 4 (i)

3.5 3.5 2.4

3 3 2.3 −1.5 −1 −0.5 2.2 2.5 (ii) 2.5 2.1 2 2 2 Frequency (GHz) Frequency (GHz) 1.5 Frequency (GHz) 1.5 1.9 −50 0 50

1 1.5 (iii) 1

1 0.5 0.5 0.5

0 0 0 −150 −100 −50 0 50 100 150 −200 0 200 −40 −30 −20 −10 0 µ/µ ε/ε βd (degrees) 0 0 Transmission/Reflection Magnitudes (dB) (a) (b) (c)

Figure 7.2: Effective material parameters and transmission/reflection magnitudes of a five-unit-cell-thick volumetric NRI-TL metamaterial (full-wave simulations – circles, equivalent-circuit model – curves): (a) phase shift per unit cell βd; (b) relative effective material parameters <{µeff (ω)}/µ0 (dashed blue curve), <{²eff (ω)}/²0 (dotted green curve); (c) transmission magnitude |S21| (solid red curve and circles); reflection magnitude |S11| (dashed blue curve and circles). Parts (a) and (b) reprinted with permission from Ref. 64, copyright °c 2008 American Institute of Physics.

meability extracted from the simulated S-parameters. These are represented by circles superposed on the corresponding theoretical curves obtained from the volumetric NRI-

TL equivalent-circuit model and the associated effective-medium parameter expressions of (5.2). These data prove the excellent agreement between the two, especially where the phase shift per unit cell βd is small (corresponding to the effective-medium limit in which effective material parameters ²eff (ω) and µeff (ω) are defined). It should be noted that there are certain frequency regions in which the extracted ²eff (ω), µeff (ω), and β(ω) are omitted; these are frequency regions in which the extraction methods fail due to the resonances that cause large phase shifts per unit cell. The data that remain are generally confined to the region βd < π/4, where the extraction procedure may be regarded as valid.

Strong resonances in the permittivity at 950MHz (Fig. 7.2(b)(iii)) and the perme- Chapter 7. A Free-Space NRI-TL Superlens 136

0 d (degrees) y β

−15

−30 −30 −15 0 15 30 β d (degrees) x

Figure 7.3: Equifrequency contour at 2.409GHz for the volumetric NRI-TL metamaterial describing propagation in the layer planes (solid blue curve – obtained through full-wave FEM simulations) and for propagation in free space (red squares). Their near-perfect coincidence suggests that the volumetric NRI-TL metamaterial exhibits a nearly isotropic refractive index of −1 at this frequency. Reprinted with permission from Ref. 119, copyright °c 2009 IEEE.

ability at 2.08GHz (Fig. 7.2(b)(ii)) ensure that both parameters are negative between

2.08GHz and 2.56GHz, yielding a LH region of propagation with a fractional bandwidth of nearly 21%. A stopband appears once the permittivity enters positive territory at

2.56GHz, and right-handed propagation is restored when the permeability follows suit at 2.95GHz. Figure 7.2(b)(i) shows that the Veselago-Pendry superlensing condition

<{µeff }/µ0 = <{²eff }/²0 = −1 is achieved almost exactly at 2.40GHz. The transmission and reﬂection data of Fig. 7.2(c) verify that the insertion losses near 2.40GHz are limited to 0.2dB per unit cell, which is consistent with the use of low-loss materials and lumped components with high quality factors, and also with the requirements of the

Veselago-Pendry superlens; however, the insertion losses at the permeability resonance at

2.08GHz are expectedly much larger at over 2dB per unit cell. These data also conﬁrm that the metamaterial is well matched to free space at 2.40GHz, where return losses are better than -35dB.

An equifrequency contour at 2.409GHz (a diﬀerence of less than 0.4% from the design Chapter 7. A Free-Space NRI-TL Superlens 137 frequency, and well within the tolerance due to discretization of the FEM mesh) is depicted in Fig. 7.3 and is shown superposed on the equifrequency contour corresponding to propagation in free space (the ‘light cone’). The coincidence of the two curves suggests that the volumetric NRI-TL metamaterial possesses isotropic <{µeff (ω)} = −µ0 and

<{²eff (ω)} = −²0 at this frequency for all directions of propagation within the plane, as required by a true Veselago-Pendry superlens.

7.2.2 Validity of Eﬀective-Medium Interpretation

It has been suggested that eﬀective-medium extraction procedures of the type employed in this work must be applied with great care [122,123], owing to the fact that the procedure is valid only for electrically thin slabs and also that the boundaries of such slabs are, in some cases, loosely deﬁned, particularly when a small number of unit cells is used.

However, the accuracy of this procedure increases as the electrical size of the unit cells, d/λ0, decreases. For the volumetric NRI-TL metamaterial superlens, d = λ0/18 and the total lens thickness is 5d = λ0/3.5, which readily satisfy the conditions of homogeneity and electrical thinness of the lens.

The uniqueness of the extracted parameters may be further affirmed by applying the same extraction procedure to slabs of various thicknesses. Full-wave transmission/reflection simulations of volumetric NRI-TL slabs measuring three, four, and five cells in thickness, are shown in Fig. 7.4. In all plots, the diamonds, squares, and circles represent the three-cell, four-cell, and five-cell cases, respectively. Figure 7.4(a) shows the extracted phase shift per unit cell βd in each case; these markers are nearly coincident and also coincide with the dispersion diagram produced directly through periodic (eigenmode) simulation of a single unit cell in an infinitely large array (solid red curve), suggesting that the dispersion properties of finite NRI-TL slabs as thin as three cells approximate well those of the infinite bulk metamaterial. The inset shows the extracted permittivity (open green markers) and permeability (solid cyan markers), which are also nearly coincident, Chapter 7. A Free-Space NRI-TL Superlens 138

4 4

2.45 3.5 3.5

2.4

3 3 2.35

2.5 2.5 −1.5 −1 −0.5

2 2 Frequency (GHz) Frequency (GHz) 1.5 1.5

1 1

0.5 0.5

0 0 −100 0 100 −40 −30 −20 −10 0 βd (degrees), inset: µ (ω)/µ , ε (ω)/ε eff 0 eff 0 Transmission/Reflection Magnitudes (dB) (a) (b)

Figure 7.4: Full-wave FEM simulation data for volumetric NRI-TL metamaterial slabs of thickness 3d, 4d, and 5d (d = 7.14mm) represented by diamond-, square-, and circle- markers, respectively. (a) Phase shift per unit cell βd (periodic simulation of a single unit cell in an infinitely large array – solid red curve); inset: relative effective material parameters near design frequency of 2.40GHz (<{µeff (ω)}/µ0 – solid cyan markers, <{²eff (ω)}/²0 – open green markers); (b) transmission and reflection magnitudes (|S21| – solid red markers, |S11| – open blue markers). Reprinted with permission from Ref. 124, copyright °c 2009 IEEE.

particularly near the design frequency, 2.40GHz (marked by cross-hairs), verifying that the variation in the extracted material parameters in all cases is minimal. The plot on the right shows transmission (|S21| – solid red markers) and refection (|S11| – open blue markers), and reveals that all three cases of slab thickness produce a passband coinciding with the NRI (backward-wave) frequency region of the dispersion diagram, and exhibit low insertion losses and excellent matching.

In addition to concerns over the mathematical validity of effective-medium descrip- tions, doubts have also been expressed on the premise that metamaterials and other Chapter 7. A Free-Space NRI-TL Superlens 139 artificial materials do not satisfy the intuitive, familiarly held notions of what consti- tutes a material [125, 126] that regard them to be homogeneous and uniform. As in all natural materials (excepting vacuum alone), permittivity and permeability are macroscopic descriptors. The term ‘macroscopic’ in a particular context is largely defined by the wavelength used to ‘view’ the materials in question, which must be sufficiently large compared to the lattice spacing so as not to see the local corrugations between scatterers. For example, natural substances that might satisfy the so-called ‘familiar’ notions of a homogeneous, uniform material at visible frequencies may appear entirely unfamiliar when probed using X-rays. Similarly, the effective-medium properties of periodic structures may be considered valid only in the range of frequencies in which the

Floquet-Bloch wavelengths are much longer than the lattice spacing and also comparable in scale to the free-space (illuminating) wavelengths, so as to ensure that it is the fundamental harmonic (and not higher-order harmonics) that is participating. To illustrate this visually for the case of the volumetric NRI-TL metamaterial superlens, consider the full-wave simulation results presented in Fig. 7.5. These data show negative refraction of a 2.39-GHz plane wave incident at various angles on a ﬁve-cell-thick volumetric NRI-TL metamaterial embedded in air. The metamaterial shown is nearly identical to the design discussed so far in this section, in that it possesses <{µeff } = −µ0 and <{²eff } = −²0 at 2.39GHz using an identical unit cell geometry and loading element values that match those listed in Table 7.2 to within 3%. In all cases, the Floquet-Bloch wavefronts inside the metamaterial are clearly discernible and are comparable in scale to the free-space (illuminating) wavelengths outside the metamaterial, attesting to the validity of the eﬀective-medium interpretation inside the NRI (backward-wave) passband.

Furthermore, these Floquet-Bloch waves are phase-matched to the incident plane wave at both interfaces, and the refractive index of −1 is manifested in the nearly identical absolute values of the incident and refracted angles.

Chapter 7. A Free-Space NRI-TL Superlens 140

ψψψ=0°°° ψψψ=15°°° ψψψ=30°°° ψψψ=45°°°

Figure 7.5: Negative refraction of a 2.39-GHz plane wave incident at various angles (ψ) on a ﬁve-cell-thick volumetric NRI-TL metamaterial embedded in air and possessing µeff (ω) = −µ0 and ²eff (ω) = −²0 at 2.39GHz. The arrows indicate the propagation directions of the incident, refracted, and transmitted waves. Reprinted with permission from Ref. 124, copyright °c 2009 IEEE.

7.2.3 Loss and Expected Resolution

Although the discussion so far has concentrated on real eﬀective permeability and permittivity, the inclusion of loss in the equivalent-circuit model yields the full complex parameters. At the design frequency, the complex eﬀective permeability and permittiv-

0 00 0 00 ity are µeff = µ − jµ = µ0(−1 − j0.057) and ²eff = ² − j² = ²0(−1 − j0.015). It is evident that µ00 is larger than ²00, which raises concerns given that losses quickly degrade the resolution ability of Veselago-Pendry superlenses. However, the volumetric NRI-TL metamaterial operates on the p-polarized ﬁelds, which implies that, in the electrostatic limit, the resolution enhancement R is predominantly susceptible to ²00. Indeed, R can be estimated by inserting ²eff into the resolution enhancement equation of Ref. 116, suitably adjusted for the p-polarization, and reproduced below:

¯ ¯ ¯ ¯ 1 ¯²eff + ²0 ¯ λ0 R = − ln ¯ ¯ (7.1) 2π ²eff − ²0 t Chapter 7. A Free-Space NRI-TL Superlens 141

2.5

1.5 Resolution Enhancement Factor, R

1 2.3 2.35 2.4 2.45 2.5 Frequency (GHz)

Figure 7.6: Theoretical resolution enhancement of the volumetric NRI-TL metamaterial versus frequency. Reprinted with permission from Ref. 124, copyright °c 2009 IEEE.

The resolution enhancement factor R is plotted against frequency in Fig. 7.6, and achieves a peak value of R = 2.71 almost exactly at the design frequency of 2.40GHz. Although these numbers are approximate and rely on estimates of component and material losses, they suggest that the volumetric NRI-TL superlens, as designed, may be able to recover evanescent wavenumbers nearly three times larger than the largest propagating transverse wavenumber k0 = 2π/λ0.

7.3 Experiment

The fabricated multilayer NRI-TL metamaterial Veselago-Pendry lens is shown in Fig. 7.7(a).

The inset depicts the loading on a single layer, from which the reader may make out the

CPS TL, series capacitors (oriented at ±45◦ with respect to the CPS TL axes) and shunt inductors (oriented at 0◦ or 90◦ with respect to the CPS TL axes).

The lens consists of 43 layers, each containing a 21 × 5 array of NRI-TL unit cells.

Since each unit cell measures d = 7.14mm square, the lens measures w = 149.9mm wide by t = 35.7mm thick. The layers were held rigidly in place using a plastic frame designed to maintain the layer period of 3.476mm, resulting in a total lens height of h = 149.5mm. These dimensions are such that a source placed a distance of t/2 from Chapter 7. A Free-Space NRI-TL Superlens 142

y x t

d h

(a)

t/2

t/2 72mm Source

Detector

144mm

Focal plane

Measured region

(b)

Figure 7.7: (a) Photograph of fabricated NRI-TL lens with inset showing lumped loading of the host CPS TL structure using discrete surface-mount inductors and capacitors. (w × h × t = 149.9mm×149.5mm×35.7mm, d = 7.14mm); (b) Measurement arrangement – the marquee indicates the region in which the field data presented in Figs. 7.8 and 7.9 are measured. Reprinted with permission from Ref. 119, copyright °c 2009 IEEE. Chapter 7. A Free-Space NRI-TL Superlens 143 the front face of the lens encounters a numerical aperture of 0.97, which collects most of its propagating spectrum (the restoration of the evanescent spectrum is a function of the lens design and the proximity of the source and lens); thus, the physical size of the NRI-TL does not impose severe restrictions on its imaging ability. The transverse dimensions are approximately 1.2λ0, which provides a sufficient illumination area and minimizes diffraction around the edges.

7.3.1 Fabrication Procedure and Equipment

Layout data for the volumetric NRI-TL layers were used to design a mask for photo- etching of the substrates. Several substrate panels were generously provided by Rogers

Corporation. The mask printing, chemical etching of the substrate material, and contour routing of the boards were done by Saturn Electronics, a PCB manufacturing company based in Romulus, Michigan, USA. In preparation for application of solder and components, the boards were finished using a gold immersion process that involved deposition of approximately 2.5µm of nickel followed by 50-130nm of gold over the copper traces. Sol- der masking was avoided so as to prevent the mask material from affecting the electrical properties of the CPS-TL, to which the effective-medium parameters of the volumetric

NRI-TL metamaterial are sensitively tied. A solder stencil was laser-cut by Stentech of

Markham, Ontario, Canada in preparation for application of the lumped inductors and capacitors. Solder application and component placement were done at George Brown

College, Casa Loma Campus, Toronto, Ontario, Canada, using a Siemens SIPLACE assembly and placement system. Solder was applied to boards via the stencil using a

SIPLACE SP-500 screen printer, and nearly 26000 surface-mount components were precisely placed by a SIPLACE S-27 HM automated pick-and-place machine. The solder was reﬂowed using a Conceptronic HVN2102 high-velocity, forced-convection reﬂow oven. Finally, the boards were cut manually and inserted into a plastic frame made of ABS

(acrylonitrile butadiene styrene) copolymer plastic. The frame was precision-crafted us- Chapter 7. A Free-Space NRI-TL Superlens 144 ing a Dimension BST1200 3D plastic printer supplied with 3D design data. Incidentally, the 10GHz dielectric properties of ABS plastic have been reported to be ²r ≈ 2.79 at 300K, with a loss tangent better than 0.01 [127].

7.3.2 Measurement Facilities and Equipment

The measurement apparatus consisted of a source and detector loop antenna connected to the terminals of an Agilent E8364B Performance Network Analyzer (PNA). The antennas were constructed from a semi-rigid 50-Ω microwave coaxial cable (1.19-mm outer diameter) and Huber+Suhner connectors, and employed a ‘shielded’ topology that provides magnetic-dipole-type fields while simultaneously minimizing unwanted radiation from unbalanced currents on the coaxial feeding structure (see, for example, Ref. 117 and Appendix D). It has been shown that the fields in the volumetric metamaterial remain strongly confined to the layer planes, and so, provided that the fields at the output are measured in the same horizontal plane, the single loop antenna appears to excite the affected layers like an infinite line source [63]. For the superresolution experiments, two identical source antennas were fed coherently using a Mini-circuits 2-way-0-degree power splitter (model #ZN2PD2-50-S+). The source (illuminating) antenna(s) remained fixed at a distance of t/2 = 17.85mm from the front face of the lens, whereas the detector antenna was mounted to the linear stages of an xyz-translator table designed by Newmark

Systems, California, USA and scanned behind the lens in the plane of excitation for field magnitude and phase distributions suggestive of focusing and evanescent decay. The use of a larger loop for the source produces strong fields that are easier to detect, and the use of a small loop for the receiver allows the detection of these fields without disturbing them. Portions of the measurement apparatus were covered in a metal-backed Emerson and Cuming ECCOSORB AN-77 microwave absorber. The xyz-translator and PNA were controlled using a custom computer code developed in-house. Following an initial SOLT

(short-open-load-through) broadband calibration, transmission measurements were taken Chapter 7. A Free-Space NRI-TL Superlens 145

in intervals of 4mm (λ0/31) over the measured region, each swept over 3201 frequencies between 10MHz to 4GHz and averaged 30 times to minimize noise on the measurement.

Figure 7.7(b) shows the measurement arrangement; the marquee identiﬁes the region in which the ﬁelds are sampled and represents the region in which the data of Figs. 7.8 and

7.9 are taken.

7.3.3 Results

Figures 7.8(a) and 7.8(b) present the raw measured ﬁeld magnitude and phase data over the measurement region corresponding to excitation by a single-loop source at 2.40GHz for two cases: Fig. 7.8(a) shows the results of a control experiment in which the ﬁelds in free space are measured before the lens is inserted; Fig. 7.8(b) presents measurements over the same spatial region with the lens in place.

The black curves indicate half-power contours referenced to the maximum field magnitudes at the expected focal plane, indicated in each case by a dashed line. Since the focal plane lies at a distance of 0.57λ0 from the source plane, the evanescent-wave components have all but disappeared in the field magnitude distribution of Fig. 7.8(a), and the half-power contour is approximately 0.80λ0 in width. The nature of the phase fronts in Fig. 7.8(a) also suggests that only propagating fields emanating away from a source located at their phase centre are detected. However, the situation is dramatically different when the lens is inserted: the field magnitudes of Fig. 7.8(b) indicate the formation of a tightly confined focal region whose transverse half-power width at the focal plane is less than 0.18λ0 (minimum peak-to-null width of 0.16λ0), over four times narrower than without the lens and over 3.8 times narrower than that predicted for diffraction-limited images by the Rayleigh criterion (0.61λ0 in free space [128,129] – see Appendix A). The normalized magnitude profiles at the focal plane, as well as the magnitude profile a distance of t/2 from the source are compared in Fig. 7.8(c). The evanescent nature of the

ﬁelds is suggested by the expected decay in the ﬁeld magnitude distribution of Fig. 7.8(b). Chapter 7. A Free-Space NRI-TL Superlens 146

Magnitude (dB) Phase (rad) 0 3 60 60 2 40 −5 40

20 20 1 −10 0 0 0 (mm) −15 (mm) −20 −20 −1 −40 −40 −20 −2 −60 −60 −25 −3 0 20 40 60 0 20 40 60 (mm) (mm) (a)

Magnitude (dB) Phase (rad) 0 3 60 60 2 40 −5 40

20 20 1 −10 0 0 0 (mm) −15 (mm) −20 −20 −1 −40 −40 −20 −2 −60 −60 −25 −3 0 20 40 60 0 20 40 60 (mm) (mm) (b)

0.75 0.707

0.5

0.25 Normalized Field Magnitude

0 −60 −40 −20 0 20 40 60 Focal Plane Location (mm) (c)

Figure 7.8: Raw measured magnitude and phase data for excitation at 2.40GHz with a single-loop source when (a) the lens is absent and (b) the lens is present. The black curves trace the half-power contours referenced to the maximum field magnitude at the focal plane (dashed line); (c) A comparison of the normalized magnitude profiles (linear scale) at the focal plane when the lens is absent (blue circles) and when the lens is present (red squares), along with the fields at a distance of t/2 from the source when the lens is absent (solid black curve). The dotted horizontal line indicates the half-power levels and shows that the NRI-TL superlens is able to produce an image of the source with a half-power beam width of 0.18λ0 (minimum peak-to-null width of 0.16λ0). Reprinted with permission from Ref. 119, copyright °c 2009 IEEE. Chapter 7. A Free-Space NRI-TL Superlens 147

It is also suggested by the phase data of Fig. 7.8(b), which remain nearly constant in the focal region and assume a propagating characteristic further from the focal plane, where the strong evanescent fields have decayed and only the propagating fields remain. These data also confirm the theoretical expectation that subdiffraction focusing manifest itself in the transverse, but not longitudinal, direction [10]. Moreover, the appearance of nulls in the measured fields suggest a confinement of the power distribution to an electrically small transverse region, enabled by the near-field interaction of the evanescent spatial- frequency components (see Appendix A). It is worth noting that the unnormalized peak

ﬁeld intensity of the image measured at the focal plane with the lens in place is 1.7dB higher than that observed over the same free-space distance without the lens in place, attesting to the low-loss nature of the NRI-TL superlens.

Although the equivalent-circuit theory and full-wave simulation data predict that

<{µeff } = −µ0 and <{²eff } = −²0 at 2.40GHz, the existence of appreciable sidelobes in the image of Fig. 7.8(c) indicate that these parameters may deviate slightly from their ideal values at 2.40GHz. Indeed, it has been veriﬁed analytically that a shift of even

0.5% from the operating frequency (12MHz), although it does not severely degrade the resolution ability, results in appreciable sidelobe levels for a lens with infinite transverse dimensions; the finite transverse dimensions of the lens in the present study may serve to further enhance the sidelobe levels through reflections at the edges.

To ensure that the resolution ability of the NRI-TL superlens enables the discrimination of two closely spaced sources, the single source antenna was substituted with two identical shielded-loop antennas (each with a 21-mm diameter) fed coherently using a passive microwave power splitter. Due to mutual coupling, the use of practical sources at close range limits the possible resolution; in fact, the minimum separation of the sources required to resolve them at their half-power levels even at the front face of the lens, where the decaying evanescent spectrum is collected, was experimentally determined to be between λ0/3 and λ0/4. The former separation distance, corresponding to a required Chapter 7. A Free-Space NRI-TL Superlens 148 resolution enhancement of R = 1.83 (as compared to the Rayleigh criterion), was chosen for presentation here to simplify the analysis and discussion of the ﬁndings; nevertheless, the latter represents a resolution enhancement of R = 2.44 which is reasonably close to the maximum theoretical resolution of R = 2.71 shown in Fig. 7.6. Figures 7.9(a) and

7.9(b) present the raw measured ﬁeld data in the absence and presence, respectively, of the NRI-TL superlens.

Once again, the black curves are half-power contours normalized to the maximum

field amplitudes at the focal plane (dashed line). Figure 7.9(c) presents the normalized magnitude profiles of the two sources at the front face of the lens, at the image plane without the lens in place, and at the image plane with the lens in place. It is evident from these data that the NRI-TL superlens is able to recover the fine distinguishing features that are lost when the lens is absent, and nearly reproduces the required resolution enhancement of R = 1.83. This result represents a resolution ability nearly twice as good as that offered by conventional lenses constrained by the diffraction limit, and are arguably the first demonstration of free-space superresolution using a Veselago-Pendry superlens. Further testing of more closely spaced sources promises to reveal an even better resolution ability, possibly in line with that measured for a single source. The evanescent decay of the images from the exit face of the superlens is also evident. It should be noted that the difference in the levels of the two recovered images can be attributed to slight differences in the construction and alignment of the small shielded-loop antennas (which were fabricated by hand), but also to slight asymmetry in their radiation patterns (see

Appendix D). The unnormalized peak ﬁeld intensity of the two images is approximately

1.4dB higher than that observed over the same distance without the lens in place, once again attesting to the low-loss nature of the NRI-TL superlens.

From its very early stages, this work was motivated by the desire to see metamaterial superlenses applied to the imaging of small scatterers at practical focal distances, as in close-range non-invasive tumour detection or land-mine detection. Although the experi- Chapter 7. A Free-Space NRI-TL Superlens 149

Magnitude (dB) Phase (rad) 0 3 60 60 2 40 −5 40

20 20 1 −10 0 0 0 (mm) −15 (mm) −20 −20 −1 −40 −40 −20 −2 −60 −60 −25 −3 0 20 40 60 0 20 40 60 (mm) (mm) (a)

Magnitude (dB) Phase (rad) 0 3 60 60 2 40 −5 40

20 20 1 −10 0 0 0 (mm) −15 (mm) −20 −20 −1 −40 −40 −20 −2 −60 −60 −25 −3 0 20 40 60 0 20 40 60 (mm) (mm) (b)

0.75 0.707

0.5

0.25 Normalized Field Magnitude

0 −60 −40 −20 0 20 40 60 Focal Plane Location (mm) (c)

Figure 7.9: Raw measured magnitude and phase data for excitation at 2.40GHz with two loop sources separated by λ0/3 when (a) the lens is absent and (b) the lens is present. The black curves trace the half-power contours referenced to the maximum field magnitude at the focal plane (dashed line); c) A comparison of the normalized magnitude profiles (linear scale) at the focal plane when the lens is absent (blue circles) and when the lens is present (red squares), along with the fields at a distance of t/2 from the source when the lens is absent (solid black curve). The dotted horizontal line indicates the half-power levels and shows that the NRI-TL superlens comfortably differentiates the sources. Reprinted with permission from Ref. 119, copyright °c 2009 IEEE. Chapter 7. A Free-Space NRI-TL Superlens 150 mental results presented so far pertain to luminous sources, the detection of non-luminous objects by way of backscattered fields may also be facilitated by a metamaterial superlens. For example, it has been proposed that a single luminous source at the front of the lens can be used both to illuminate a scatterer behind the lens and detect it by way of its backscattered secondary fields [130]. Alternatively, it is possible to illuminate the front side of the lens using a normally incident plane wave. Since the Veselago-Pendry lens does not possess a unique principal axis, such a plane wave passes directly through the lens and impinges upon the scatterer, whose backscattered fields are then refocused at the front side of the lens where they may be detected. Since the antennas employed would necessarily be small, matching techniques or time-gating could be employed to isolate the desired backscattered signals.

7.4 Subdiﬀraction Imaging by Other Mechanisms

It was suggested earlier in this chapter that subdiffraction imaging of other types may be obtained by relaxing some of the strict conditions governing Veselago-Pendry superlenses. The examples noted were the plasmonic silver film [9] and the magnetoinductive lens [10], which possess either µ < 0 or ² < 0, but not both, and thus operate on the evanescent spectrum alone. The printed SRR-based structures reported by Aydin et al. have most recently claimed the finest measured free-space superresolution to-date [16], but these structures possess a refractive index of −1.8 at their operating frequency (not

−1), they exhibit a large insertion loss of more than 2dB per unit cell, and the authors have speculated that anisotropy may be directly responsible for the observed phenomena. These attributes are inconsistent with the requirements of a Veselago-Pendry superlens [116], and this type of imaging is more likely related to the highly anisotropic tunneling phenomena observed in previous swiss-roll-/SRR-type metamaterials near the permeability resonance (see, for example, Refs. 131, 132, 133, and 134). In this section, Chapter 7. A Free-Space NRI-TL Superlens 151

Figure 7.10: Measured normalized transverse ﬁeld proﬁles at the focal plane versus frequency from 2.00GHz to 2.50GHz (black curve represents the continuous half-power contour). Superresolution evident at 2.40GHz (right inset: Veselago-Pendry superlensing condition) and 2.08GHz (left inset: permeability resonance). Adapted and reprinted with permission from Ref. 64, copyright °c 2008 American Institute of Physics.

it is shown experimentally that, in addition to superlensing of the Veselago-Pendry type

(produced where its effective material parameters achieve the values <{µeff } = −µ0 and <{²eff } = −²0), the free-space NRI-TL metamaterial superlens exhibits a second type of subdiffraction imaging related to a resonant tunneling effect at the permeability resonance.

Figure 7.10 presents the measured transverse ﬁeld magnitudes at the focal plane versus frequency from 2.00GHz to 2.50GHz when the NRI-TL free-space metamaterial superlens is illuminated by two sources at the source plane that are transversely separated by a distance of λ0/3. The set of magnitudes at a particular frequency are normalized to their maximum value, and the black curve indicates the continuous evolution of the half- Chapter 7. A Free-Space NRI-TL Superlens 152

−2

−4

−6

−8

−10

−12

−14

Normalized Magnitude (dB) −16

−18

−20 −60 −40 −20 0 20 40 60 Transverse Position (mm)

Figure 7.11: Measured transverse field profiles at 2.40GHz (red solid curve) and 2.08GHz (blue dashed curve), normalized to the maximum field value of the former. Reprinted with permission from Ref. 64, copyright °c 2008 American Institute of Physics.

power contour as the frequency is increased. It is expected that superresolution of the two sources be manifested by the formation of two distinct regions of intensity separated at or below their half-power levels. As shown previously in Fig. 7.9, this separation occurs precisely at 2.40GHz, where the Veselago-Pendry superlensing condition (<{µeff } = −µ0 and <{²eff } = −²0) is met (the corresponding simulated effective material parameters are shown in the inset at the lower right-hand corner of Fig. 7.10). The transverse field profile at the focal plane at 2.40GHz is reproduced in Fig. 7.11 (solid curve).

The permeability resonance frequency was suggested by the theory and simulations to occur at 2.08GHz (reproduced in the inset at the lower left-hand corner of Fig. 7.10).

It may be expected from previous works [131–134] that a magnetic excitation at this frequency should result in ﬁelds tunneling through the structure and appearing on the image side as if superresolved. Once again, precisely at 2.08GHz, the measured data indicates the clear discrimination of the two sources. Figure 7.11 presents the transverse

field profile at the focal plane (dashed curve), normalized to the maximum values obtained Chapter 7. A Free-Space NRI-TL Superlens 153 at the design frequency of 2.40GHz. This comparison reveals that the fields produced by resonant tunneling are noticeably less well-defined and also significantly less intense

(by about 10dB, or 2dB per unit cell) than those measured at the Veselago-Pendry superlensing frequency; indeed, the former are subjected to much higher insertion losses at the very narrowband µ-resonance, whereas the latter are produced in a frequency region designed to exhibit low-loss and broadband propagation characteristics using the

NRI-TL approach. Thus, these results also aﬃrm that true Veselago-Pendry superlensing is distinguished from other imaging mechanisms by lower insertion losses.

7.5 A Comment on Bandwidth

Since the principal claim of the volumetric NRI-TL metamaterial superlens is free-space imaging beyond the diffraction limit, it is appropriate to define the bandwidth of the lens as the range of frequencies over which subdiffraction imaging is possible. Subdiffraction performance could, technically, be interpreted as any resolution enhancement R > 1.

From Fig. 7.6, this condition is satisﬁed from approximately 2.34GHz to 2.44GHz, representing a fractional bandwidth of nearly 4.3% in theory. However, since experimental subdiﬀraction imaging data has been obtained, it is more realistic and more practical to evaluate the bandwidth according to these experimental conditions, which demanded a resolution enhancement of R = 1.83. According to Fig. 7.10, resolution of the two images at their −3dB levels is experimentally observed from approximately 2.38GHz to

2.42GHz. This represents a fractional bandwidth of nearly 1.7%, and is approximately in line with the bandwidth for R > 1.83 predicted by the data of Fig. 7.6. It may be noted that this bandwidth is nearly three times larger than the corresponding fractional bandwidth of 0.6% measured near the resonant tunneling frequency. Chapter 8

A 3D NRI-TL Topology for

Free-Space Excitation

Although the volumetric topologies successfully interact with sources in free space, they are, by virtue of their multilayer construction, polarization-speciﬁc. In the pursuit of full 3D isotropic NRI-TL models, it is useful to revisit the 3D TL model of conventional dielectrics proposed by Kron the 1940s [19], which consists of a periodic arrangement of orthogonal, intersecting loops. The inductive loading of the loops and their capacitive interconnections represent the eﬀective material permeability and permittivity, respectively. In the TL-matrix (TLM) method of time-domain electromagnetic modeling, the corresponding unit-cell topology is known as the expanded node (EN) [103, 135], which is derived from Kron’s structure by displacing the planes of periodicity by half a period on each axis. The displaced topology reveals another perspective: the EN can be constructed by placing series TLM nodes of the type shown in Fig. 4.1(a) on the faces of a cube. A completely 3D physical realization of the NRI-TL medium based on the EN topology has been proposed in Ref. 20 simply by reversing the positions of the inductors and capacitors in the conventional topology, and one such design was shown in simulation to produce a NRI in free space with good matching.

154 Chapter 8. A 3D NRI-TL Topology for Free-Space Excitation 155

8.1 Metamaterials Based on the Symmetrical Con-

densed Node (SCN)

Consistent with the principle of duality, there exists a second 3D topology for TLM modeling of conventional dielectrics; this is the symmetrical condensed node (SCN) [136], and its unit cell is shown in Fig. 8.1(a). The SCN can be constructed by placing series TLM nodes on three intersecting orthogonal planes; the constituent series TLM node on one of the three planes has been emphasized in the ﬁgure for clarity. The resulting TL network contains six-ports consisting of four conductors each, which can further be regarded as two pairs of two-conductor TLs supporting orthogonal polarizations (often, this leads to a description of the SCN as a twelve-port network of two-conductor TLs). There have been attempts to capture this form of the SCN using various TL-based metamaterials.

For example, Alitalo et al. [4,109] propose a six-port network in which each port is a two- conductor microstrip TL; however, a necessary polarization conversion within the node precludes its description as isotropic. The realization proposed by Zedler et al. [137] retains the four-conductor topology, but treats the node as two sets of decoupled three- ports, for which each port contains only a two-conductor TL. However, unavoidable coupling between the four conductors constituting each port renders this description of the physical SCN incomplete. This chapter presents simulations verifying the 3D isotropic

NRI properties of a NRI-TL metamaterial based on a true physical realization of the

SCN. Although a full analytical treatment would better be the subject of future work, it is worthy to note that the SCN is most correctly treated using MTL theory, since the four conductors at each port, physically realized, describe a four-conductor TL. As suggested in Chapter 5 for the volumetric NRI-TL metamaterial, the complete dispersion features may be obtained using an appropriate interconnection of MTL matrices and application of the suitable boundary conditions, in a manner analogous to what was done for the planar metamaterials. Chapter 8. A 3D NRI-TL Topology for Free-Space Excitation 156

8.2 Design

Just as the EN topology could alternatively be represented by Kron’s network of conventional dielectrics simply by displacing the unit cells by half a period on each axis, the same translation applied to the SCN yields the topology depicted in Fig. 8.1(b). It is clear that this unit cell may most simply be described as a cube onto whose faces are placed inductively loaded, capacitively interconnected rings, and the corner of each of these faces corresponds to the center of a series NRI-TL node (a ring on one cube face is emphasized for clarity). By replacing the inductive loading of the ring with capacitive loading and the capacitive interconnections with inductive interconnections, the proposed 3D NRI-TL topology of Fig. 8.1(c) is produced. In fact, this 3D topology is a natural extension of the volumetric NRI-TL topology, in which such rings are placed on any two opposite cube faces. It is also reminiscent of earlier suggested 3D metamaterial implementations using split-ring resonators (SRRs) (see, for example, Refs. 34 and 138), in which SRRs placed on the faces of a cube provide a negative eﬀective permeability, and separate conducting wires are required to produce a negative eﬀective permittivity. A recent advance on the 3D SRR topology employs disconnected chiral scatterers to produce a NRI without wires [139]. In contrast, the rings constituting the 3D NRI-

TL metamaterial are inductively connected and therefore, this topology requires neither chirality nor separate conducting wires to produce a NRI.

The practical 3D implementation studied in the present work is depicted in Fig. 8.1(d), and is fully symmetric; as a result, it is also polarization-independent in the homogeneous limit (that is, where the metamaterial behaves as an eﬀective medium). The rings constituting the unit-cell faces are identical in design to those employed in the volumetric

NRI-TL metamaterial described in Chapter 7 (see Fig. 7.1(b)) except that the dielectric medium in the volumetric metamaterial has been replaced in the present 3D metamaterial with vacuum. Chapter 8. A 3D NRI-TL Topology for Free-Space Excitation 157

(a) (b)

M X ΓΓΓ

Figure 8.1: Evolution of the 3D NRI-TL metamaterial: (a) SCN node modeling conventional dielectric media (series TLM node emphasized on one plane); (b) Unit cell obtained by displacing the SCN by one-half of a period on each axis (front face emphasized); (c) 3D NRI-TL topology obtained by reversing positions of inductors and capacitors (front face emphasized); (d) Physical realization of the 3D NRI-TL SCN unit cell in (c) for simulation using Ansoft HFSS (inset: Brillouin zone contours with high-symmetry points labelled). Reprinted with permission from Ref. 140, copyright °c 2008 American Institute of Physics. Chapter 8. A 3D NRI-TL Topology for Free-Space Excitation 158

3.8

3.6

3.4

3.2

2.8 Frequency (GHz)

2.6

2.4

2.2

2 R Γ X M Γ

Figure 8.2: 3D NRI-TL SCN dispersion characteristics obtained using Ansoft’s HFSS. The labels on the horizontal axis correspond to the Brillouin-zone high-symmetry points shown in the inset of Fig. 8.1(d). The NRI passband (below 3GHz) is described by the existence of two modes (square- and circle-markers) whose dispersions are nearly identical in the eﬀective-medium regime. Several higher-order modes exist above 3GHz. Reprinted with permission from Ref. 140, copyright °c 2008 American Institute of Physics.

8.3 Simulation

8.3.1 Dispersion Characteristics

The dispersion characteristics of this 3D unit cell were obtained through ﬁnite-element- method full-wave simulations using Ansoft’s HFSS and are shown in the form of a 3D

Brillouin zone contour in Fig. 8.2; the labels on the horizontal axis correspond to the high symmetry points of the 3D Brillouin zone shown in the inset of Fig. 8.1(d). Following the presentation in Ref. 139, Fig. 8.3 superimposes the data of Fig. 8.2 for the principal propagation directions Γ − X,Γ − M, and Γ − R, in order to aﬃrm isotropy.

The data of Figs. 8.2 and 8.3 resemble those obtained for the volumetric NRI-TL metamaterials based on the series NRI-TL node, and suggest the existence of a backward-wave or NRI frequency region from 2.4GHz to 3.0GHz for propagation along Γ−X and slightly wider for the other two principal directions. This represents a minimum fractional band- Chapter 8. A 3D NRI-TL Topology for Free-Space Excitation 159

ΓX ΓM ΓR 4 ΓX 3.8 ΓM ΓR 3.6

3.4

3.2

2.8 Frequency (GHz) 2.6

2.4

2.2

2 0 π/2 π π√2 π√3 βd (radians)

Figure 8.3: Dispersion characteristics along the principal propagation directions Γ − X, Γ − M, and Γ − R superimposed to show that isotropy and polarization-independence are achieved in the eﬀective-medium regime. In the NRI passband (below 3GHz), the square- and circle-markers indicate the two predominant modes. Several higher-order modes exist above 3GHz. Reprinted with permission from Ref. 140, copyright °c 2008 American Institute of Physics.

width of over 22%. The NRI passband is followed by a small stopband and a series of higher-order forward-wave bands in which the metamaterial can be said to possess a positive refractive index. However, there also exist a series of low-frequency slow-wave modes below 1GHz (not shown in Figs. 8.2 and 8.3) resulting from the discontinuous nature of the metallic traces. Isotropy and polarization-independence in the NRI passband are confirmed by the fact that the two polarization dispersion curves (solid squares and solid circles) are nearly coincident in the vicinity of the Γ−point (where the effective-medium conditions are satisfied and the notion of a refractive index is defined) for all three principal propagation directions. These features are maintained from approximately 2.6GHz to

3GHz and diverge only in the approach towards the Brillouin-zone edges, where spatial anisotropy due to the corrugated structure of the metamaterial is evident. It should also be noted that these dispersion data are devoid of the so-called ‘spurious’ modes observed in the EN-based 3D NRI-TL metamaterial [20]. Chapter 8. A 3D NRI-TL Topology for Free-Space Excitation 160

−2

−4

−6

−8

−10

−12 Magnitude (dB) −14

−16 S 21 −18 S 11 −20 0 0.5 1 1.5 2 2.5 3 3.5 4 Frequency (GHz)

Figure 8.4: Transmission and reﬂection magnitudes corresponding to on-axis (Γ − X) propagation for a ﬁve-unit-cell thick 3D NRI-TL metamaterial slab embedded in vacuum, obtained using Ansoft’s HFSS. Reprinted with permission from Ref. 140, copyright °c 2008 American Institute of Physics.

8.3.2 Transmission Characteristics

Figure 8.4 presents full-wave transmission simulations corresponding to on-axis (Γ − X) propagation for a 3D NRI-TL slab with a five-unit-cell thickness and infinite transverse dimensions embedded in vacuum. A passband corresponding to the NRI frequency region of Figs. 8.2 and 8.3 (Γ − X direction) is distinguished by low insertion losses (less than 0.2dB per unit cell) and a return loss of nearly −20dB at 2.7GHz, which shall be regarded as the 3D NRI-TL metamaterial’s operating frequency. The isotropy of the metamaterial is further corroborated by the fact that the unit-cell size at 2.7GHz is d = 7.14mm ≈ λ0/16, which readily satisfies the effective-medium condition. Although not presented here, the steady-state time-domain behaviour of the simulated fields in the

ﬁve-cell structure indicate the clear progression of a backward wave in the NRI passband.

It was noted that the 3D NRI-TL metamaterial dispersion is marked by a NRI passband as well as a set of low-frequency modes. In fact, these two regions of propagation are separated by a 3D electromagnetic bandgap spanning over 2GHz (fractional bandwidth of over 67%). From the transmission data of Fig. 8.4, it is evident that this bandgap Chapter 8. A 3D NRI-TL Topology for Free-Space Excitation 161 is marked by strong reactive attenuation (better than 40dB). A 3D bandgap possessing these remarkable properties can be eminently useful in applications in which spatial

ﬁltering or isolation are required.

8.4 Optical Implementation

Chapter 5 proposed optical implementations of the series NRI-TL array and volumetric

NRI-TL metamaterial using analogous arrangements of plasmonic nanoparticles. Apply- ing the same concept to the SCN-based 3D NRI-TL metamaterial yields the unit cell shown in Fig. 8.5. Once again, the plasmonic spheres represent the rings and their inductive interconnections, and the intervening gaps support fringing ﬁelds that provide the necessary ring capacitance. It is clear that this arrangement contains a total of three plasmonic spheres (twelve quarters) per unit cell. Shifting this unit cell by half a period in each axis reveals that the twelve quarter-spheres may also be arranged as six half-spheres placed on each face of the cube. This topology may simply be described as a face-centered lattice, akin to the face-centered cubic (FCC) lattice with the corner elements removed. It is also interesting to note that the plasmonic counterparts of both the SCN- and EN-based 3D NRI-TL metamaterials are identical, which aﬃrms the notion that both topologies produce equivalent, albeit dual, representations of the same isotropic metamaterial characteristics. Chapter 8. A 3D NRI-TL Topology for Free-Space Excitation 162

Figure 8.5: Arrangement of plasmonic nanoparticles corresponding to a single unit cell of the SCN-based 3D NRI-TL metamaterial. Chapter 9

Conclusions

9.1 Summary

The realization of a true Veselago-Pendry superlens capable of interacting with and manipulating fields in free space remained elusive, largely due to the difficulty of meeting its stringent design constraints and also to the problem of realizing a full 3D isotropic, polarization-independent implementation. This work has presented a new class of volumetric metamaterials based on 2D NRI-TL layers that, although polarization-specific, may be easily constructed, and thereby extends the benefits of the NRI-TL approach to free-space realizations. The volumetric NRI-TL topology requires no vertical interconnects or vias and can be easily fabricated using standard lithographic techniques in either fully printed form or using discrete lumped elements. An equivalent circuit model was developed to enable accurate design of its dispersion and transmission characteristics, including those associated with Veselago-Pendry superlensing.

A volumetric free-space NRI-TL metamaterial lens employing fully printed loading elements was designed and fabricated to exhibit NRI properties at X-band. Transmission magnitude and phase measurements were obtained using a free-space X-band measurement system and showed good correspondence with simulations. The metamaterial was

163 Chapter 9. Conclusions 164 then employed as a ﬂat slab lens and demonstrated free-space focusing within its NRI passband, evidenced by a focal region with a half-power width of 0.55λ0 and the corresponding reversal of the concavity of the phasefronts at the lens face and across the focus.

These experimental results established the ability of volumetric NRI-TL metamaterials to interact with excitations in free space and also affirmed the theoretical expectation that high losses and large electrical thickness preclude the observation of subdiffraction focusing and imaging, the phenomena associated with the Veselago-Pendry superlens. It was determined that these constraints could be overcome using discrete lumped elements with high quality factors rather than printed elements, and the developed equivalent-circuit model was used to predict and design its dispersion and transmission characteristics in concert with full-wave simulations. The use of discrete lumped elements affords precise control over the material parameters of the lens and is much less susceptible to losses than other implementations. These ideas were applied to the realization of a volumetric

NRI-TL Veselago-Pendry superlens designed to possesses a refractive index of −1 and a wave impedance equal to that of vacuum at a frequency of 2.40GHz. The experimental results reveal free-space focusing of a single source to a minimum peak-to-null beamwidth of less than one-sixth of a wavelength and a free-space resolution of two sources displaced transversely by a distance of one-third of a wavelength over a bandwidth of nearly

2%, both well below the classical diffraction limit. These results are also used to offer insight into other mechanisms that may explain subdiffraction-imaging results in the literature derived from metamaterials not meeting the strict conditions associated with the

Veselago-Pendry superlens.

The volumetric NRI-TL implementation is attractive at microwave frequencies because its constituent layers may be easily and rapidly fabricated using existing PCB fabrication techniques and facilities. A microwave superlens designed in this fashion can be particularly useful for illumination and discrimination of closely spaced buried objects over practical distances by way of back-scattering, for example, in tumour or landmine Chapter 9. Conclusions 165 detection, or for targeted irradiation over electrically small regions in tomography or hyperthermia applications. The possibility of scaling the volumetric layered medium to THz frequencies and possible plasmonic implementations at near-infrared or optical frequencies were also discussed in this work. Finally, a fully isotropic, polarization-independent

3D metamaterial structure related to the volumetric NRI-TL structure was proposed.

Topological simpliﬁcations, along with ongoing developments in fabrication techniques, should facilitate the realization of each of these implementations and encourage their application to imaging problems in biomedicine, microelectronics, and defense.

9.2 Contributions

This section lists book chapters, refereed journal and conference papers, and other aca- demic contributions made during the course of this thesis work.

9.2.1 Book Chapters

[B4] Invited: A. K. Iyer and G. V. Eleftheriades, “Metamaterials,” in Wiley-IEEE

Encyclopedia of Electrical and Electronics Engineering, Wiley-IEEE Press, under

review.

[B3] Invited: A. K. Iyer and G. V. Eleftheriades, “Fundamentals of Transmission-

Line Metamaterials,” in Handbook on Artiﬁcial Materials, F. Capolino, ed., CRC

Press, accepted.

[B2] Invited: A. K. Iyer and G. V. Eleftheriades, “Negative-Refractive-Index Transmission-

Line (NRI-TL) Metamaterial Lenses and Superlenses,” in Handbook on Artiﬁcial

Materials, F. Capolino, ed., CRC Press, accepted.

[B1] A. K. Iyer and G. V. Eleftheriades, “Negative-Refractive-Index Transmission-

Line Metamaterials,” in Negative-Refraction Metamaterials: Fundamental Princi- Chapter 9. Conclusions 166

ples and Applications, G. V. Eleftheriades and K. G. Balmain, eds. Wiley-IEEE

Press (Toronto: 2005), ISBN: 0-471-60146-2.

9.2.2 Journal Papers

[J5] A. K. Iyer and G. V. Eleftheriades, “Free-space imaging beyond the diﬀraction

limit using a Veselago-Pendry transmission-line metamaterial superlens,” accepted

for publication in the IEEE Trans. Antennas Propagat., 2009.

[J4] A. K. Iyer and G. V. Eleftheriades, “A three-dimensional isotropic transmission-

line metamaterial topology for free-space excitation,” Appl. Phys. Lett., vol. 92,

no. 26, p. 261106, Jul. 2008.

[J3] A. K. Iyer and G. V. Eleftheriades, “Mechanisms of subdiﬀraction free-space

imaging using a transmission-line metamaterial superlens: An experimental veri-

ﬁcation,” Appl. Phys. Lett., vol. 92, no. 13, p. 131105, Mar. 2008.

[J2] A. K. Iyer and G. V. Eleftheriades, “A multilayer negative-refractive-index transmission-

line (NRI-TL) metamaterial free-space lens at X-Band,” IEEE Trans. Antennas

Propagat., vol. 55, no. 10, pp. 2746–2753, Oct. 2007.

[J1] Invited: A. K. Iyer and G. V. Eleftheriades, “Volumetric layered transmission-

line metamaterial exhibiting a negative refractive index,” J. Opt. Soc. Am. B.

(JOSA-B), Focus Issue on Metamaterials, vol. 23, pp. 553–570, Mar. 2006.

9.2.3 Conference Papers and Abstracts

[C10] A. K. Iyer and G. V. Eleftheriades, “Eﬀective-medium properties of a free-space

transmission-line metamaterial superlens,” 2009 International Symposium on An-

tenna Technology and Applied Electromagnetics (ANTEM) and the Canadian Ra-

dio Sciences Meeting (URSI/CNC), (Banﬀ, AB, Canada), February 2009. Chapter 9. Conclusions 167

[C9] Invited: G. V. Eleftheriades, A. K. Iyer, and A. Wong, “Transmission-line

metamaterial lenses and metascreens for free-space superlensing,” to appear in

2nd International Congress on Advanced Electromagnetic Materials in Microwaves

and Optics Digest (Pamplona, Spain), September 2008.

[C8] A. K. Iyer and G. V. Eleftheriades, “Free-space sub-diﬀraction imaging using a

transmission-line superlens,” IEEE Antennas and Propagation Society 2008 Inter-

national Symposium Digest, (San Diego, CA, USA), July 2008.

[C7] Invited: A. K. Iyer and G. V. Eleftheriades, “Negative-refractive-index transmission-

line lenses for free-space focusing,” in International URSI Commission B – 2007

Electromagnetic Theory (EMTS2007) Symposium Digest (Ottawa, ON, Canada),

July 2007.

[C6] A. K. Iyer and G. V. Eleftheriades, “Characterization of a volumetric negative-

refractive-index transmission-line (NRI-TL) metamaterial for incident waves from

free space,” in Proceedings of the 2006 European Conference on Antennas and

Propagation (Nice, France), Nov. 2006.

[C5] A. K. Iyer and G. V. Eleftheriades, “Characterization of a multilayered negative-

refractive-index transmission-line (NRI-TL) metamaterial,” in IEEE Microwave

Theory and Techniques Society 2006 International Microwave Symposium Digest

(San Francisco, CA, USA), pp. 428–431, June 2006.

[C4] A. K. Iyer and G. V. Eleftheriades, “A volumetric negative-refractive-index meta-

material based on uniplanar transmission-line layers,” Proc. URSI North Ameri-

can Radio Science Meeting (Boulder, CO, USA), Jan. 2006.

[C3] Invited: A. K. Iyer and G. V. Eleftheriades, “Leaky-wave radiation from planar

negative-refractive-index transmission-line metamaterials,” in IEEE Antennas and

Propagation Society 2004 International Symposium Digest, (Monterey, CA, USA),

pp. 1411–1414, June 2004. Chapter 9. Conclusions 168

[C2] Invited: A. K. Iyer, K. G. Balmain, and G. V. Eleftheriades, “Dispersion

analysis of resonance cone behaviour in magnetically anisotropic transmission-

line metamaterials,” IEEE Antennas and Propagation Society 2004 International

Symposium Digest, (Monterey, CA, USA), pp. 3147–3150, June 2004.

[C1] Invited: A. K. Iyer and G. V. Eleftheriades, “Leaky-wave radiation from a two-

dimensional negative-refractive-index transmission-line metamaterial,” in Proceed-

ings of 2004 URSI EMTS International Symposium on Electromagnetic Theory

(Pisa, Italy), pp. 891–893, May 2004.

9.3 Future Work

There is no doubt that metamaterials research has experienced an explosive growth over the last decade. This is evidenced by the ever-increasing number of publications, meetings, and research thrusts dedicated to the subject in the microwave and optics communities, as well as a growing recognition of their commercial value in industry.

However, the true legacy of metamaterials may well be their role in reviving and building upon the design techniques developed in the artiﬁcial-dielectrics community in the mid- twentieth century, which established that the notions of a ‘material’ and its properties are open to interpretation based on the wavelength with which they are ‘viewed’ and the materials and methods with which the constituent inclusions are designed. This is a most useful perspective in a modern research context, since any phenomenon, technique, or device based on the dielectric and magnetic properties of materials therefore becomes open to re-examination, for it may possess a tangible ‘metamaterial’ counterpart with useful implications, intriguing properties, or improved performance.

Much of metamaterials research has evolved to a point of maturity wherein the principal phenomena have been experimentally verified using proof-of-concept prototypes. This is particularly true of NRI-TL metamaterials, which have given rise to novel couplers, Chapter 9. Conclusions 169 dividers, phase shifters, and antennas, and demonstrated intriguing physical phenomena like negative refraction and, as in this work, free-space subdiffraction imaging. What remains is a need to apply the knowledge gained in these areas to the solution of real-world problems. In particular, there remains a need to investigate methods for superlensing in heterogeneous environments using bulk 3D metamaterials, particularly if the technology is to be adopted for imaging of human tissues (e.g., for focused detection, tomography, and/or targeted hyperthermia of tumours over electrically small regions), soil (e.g., for detection of landmines and other small buried features, which may be invisible to conventional ground-penetrating radar), or behind barriers (e.g., through walls/doors or as an alternative to X-ray-based security scanners). Such bulk metamaterials could also be explored for their potential in manipulating/enhancing the radiation characteristics of nearby antennas. In addition to passive structures, active 3D metamaterials whose properties can be electronically tuned may prove useful for these applications, either to enhance matching to the surrounding environment, to improve the bandwidth of operation of superlenses, or to enable beam scanning of metamaterial-enhanced antennas. This will require parallel efforts in both design and practical realization of 3D-isotropic metamaterial topologies, such as the one suggested in this work. The design of both volumetric and 3D-isotropic NRI-TL structures may be facilitated by employing MTL techniques along with standard periodic microwave network analysis, as suggested in Chapters 5 and

8; indeed, the dispersion properties of the former may be obtained through the application of suitable boundary conditions on the latter. As explored in this work, there are also opportunities to design and fabricate superlenses at optical frequencies through the use of plasmonic nanocircuit concepts in concert with established NRI-TL metamaterial techniques.

As structures that inherit their properties from fine spatial features, subwavelength inclusions, and sensitive geometrical constraints, metamaterials and other artificial materials require precise, repeatable, and robust fabrication techniques, and this is particularly Chapter 9. Conclusions 170 true for implementations in the terahertz or optical regimes. Moreover, it is necessary to minimize both the complexity and the cost of fabrication in order to encourage their acceptance and commercial viability in industry. Thus, there is a need to establish channels of interdisciplinary collaboration between electrical engineers, materials scientists, chemical engineers, and physicists in order to exchange ideas on the fabrication materials and techniques offering the greatest potential for success, particularly at terahertz and optical frequencies. Appendix A

The Diﬀraction Limit

The spatial features of a source in vacuum may be represented by a spectrum of wavenumbers (spatial frequencies). In rectangular coordinates, the consistency condition in vacuum is given in terms of these wavenumbers as

2 2 2 2 2 kx + ky + kz = k0 = ω ²0µ0 (A.1)

For simplicity, we may limit our discussion to a two-dimensional case in which kx = 0, and we may further identify ky and kz as the longitudinal and transverse components, respectively. Therefore, ky may be written as

q 2 2 ky = ± k0 − kz (A.2)

From (A.2), it is evident that ky is real for |kz| ≤ k0 and imaginary for |kz| > k0; the former condition describes propagating waves that incur phase with progression in the spatial direction y, and the latter condition describes evanescent waves whose amplitudes decay exponentially in y.

The imaging of such a source implies the reconstruction of this spectrum and the accuracy of this reconstruction (or ‘resolution’) is, expectedly, constrained by the limits

171 Appendix A. The Diffraction Limit 172

k0 k0 θθθ θθθ

kz kz

Figure A.1: Interference of the tangential components of the wavevectors of two coherent plane waves incident at angles ±θ.

of the imaging apparati (e.g. lenses and filters). However, even the use of optically perfect apparati does not permit infinite resolution, and this is due to a more fundamental constraint known as the ‘diffraction limit’, which is a consequence of the seemingly unavoidable loss of some of the wavenumbers in the reconstruction of a source’s spectrum at a displaced image plane. Therefore, the diffraction limit may be loosely defined as the best resolution achievable given the loss of these components, in a given imaging arrangement.

A.1 Imaging: An Interference Phenomenon

Imaging, as it is referred to in this work, amounts to the interference of the spectral components of a source conveyed to the image plane by a collimating device (e.g. a lens).

The production of an interference pattern by coherent light is illustrated in Fig. A.1, which depicts two plane waves with intrinsic wavenumber k0 impinging from diﬀerent directions at an angle θ on the plane y = 0. The wavevector components tangential to the image plane, kz = k0 sin θ, produce a magnitude distribution proportional to cos(kzz) with peaks and nulls occurring at values kzz = 2nπ and kzz = (2n + 1)π, respectively (n

Appendix A. The Diffraction Limit 173

k0 k0

θθθmax θθθmax

Figure A.2: Illumination of a lens of ﬁnite aperture size by a point source.

is an integer). The resolution may be defined according to the most rapid transition in intensity, which corresponds to the peak-to-null distance; according to this definition, the interference of tangential components kz results in a resolution ∆ = π/kz = λ0/(2 sin θ), where λ0 is the free-space wavelength. Of course, particular free-space sources/geometries (e.g. point sources, apertures) are described by the presence of a number of interfering plane-wave components and so different limits have been suggested for different source cases. For example, the Rayleigh criterion treats incoherent illumination by two point sources and is often given as ∆ = 1.22λ0/(2 sin θ), where the numerical factor is chosen to define resolution arbitrarily as the minimum distance at which the centre of the diffraction pattern of one object coincides with the first null of that of the second object [128,129].

The Abb´elimit/barrier, deﬁned in a similar way for coherent illumination by two point sources, is ∆ = 1.64λ0/(2 sin θ) [128]. It may be noted that the common features of all of these limits are that they are of the same general order and that resolution may only be increased through the participation of higher tangential wavenumbers in the image reconstruction. Appendix A. The Diffraction Limit 174

A.2 Imaging in the Far Field

Imaging in the far-ﬁeld with a conventional lens is constrained by a number of factors, even if it is assumed that the lens is lossless and free of geometric aberrations. One of these is its ﬁnite aperture size. Adopting a ray-optics picture, as shown in Fig. A.2, it is evident that the rays impinging on a thin lens at an angle in excess of θmax are not collected, and the spatial information they carry is not conveyed to the image plane. According to the previous discussion, this limits the maximum available resolution of the system to ∆ =

π/(k0 sin θmax), from which it appears that the resolution may be enhanced by maximizing the lens aperture. However, we may consider an ultimate theoretical limit when the lens aperture becomes inﬁnite, or equivalently, when θmax → π/2 (grazing incidence). In this case, the maximum available resolution remains limited to ∆ = π/k0 = λ0/2.

A.3 Subdiﬀraction Imaging

The above argument has implicitly assumed that a conventional far-field imaging arrangement implicates only the propagating spectral components; indeed, the far-field diffraction limit ∆ = π/k0 implies that only propagating wavenumbers participate in the image reconstruction. This is reasonable, since the amplitudes of the evanescent spectral components decay rapidly from the source within distances on the order of a fraction of a wavelength and are, therefore, unavailable to participate in the image reconstruction in the far field. If imaging optically in the near field, some of the more slowly-decaying evanescent components of the source may have appreciable magnitudes, and a high resolution is possible. In other words, the availability of evanescent transverse wavenumbers implies that the diffraction limit of such a system ∆ = π/kz corresponds to a better resolution than that available in a far-field setup. This is the principle by which near-field scanning optical microscopes (NSOMs) produce high-resolution images, although such a method imposes extremely short working distances and deals with surfaces only. Never- Appendix A. The Diffraction Limit 175 theless, it is clear that the diffraction limit at which resolution becomes possible is also determined by the distance between the object and image. Armed with this notion, we shall define free-space ‘subdiffraction’ imaging for the purposes of this work as imaging over a particular distance with a resolution better than that observable in free space over the same distance.

A.4 The Veselago-Pendry Superlens

If it is possible to restore the phase of the propagating components of a source (i.e., by focusing them without geometric aberrations) and simultaneously restore the amplitude of its evanescent components at the image plane, then it is theoretically possible to optically reconstruct an object exactly. Veselago suggested that a ﬂat slab of material with a relative permittivity and permeability simultaneously equal to −1 would also possess a refractive index of −1 [1]. Such a slab embedded in vacuum would, by virtue of negative refraction, focus the rays emanating from a source without geometric aberration to a single point both inside and outside the slab, as shown in Chapter 2 (see the arrows in Fig. 2.4). Pendry extended these conditions to evanescent waves, and showed that such a slab also captures the entire decaying evanescent source spectrum and restores its amplitude by resonantly enhancing these ﬁelds within the slab [2]. The enhanced

fields decay to their original levels in vacuum, precisely at the image plane (see the solid curves in Fig. 2.4). Thus, the image produced by such a ‘Veselago-Pendry superlens’ is, theoretically, an exact replica of the source object. However, in order to practically detect the evanescent fields decaying from the source, it is clear that the source must be in the near field of the slab. Furthermore, it was shown that the ‘superlensing’ effect may only be practically observed if the slab is very low loss and electrically thin [7,8], and experimental demonstrations to-date using true Veselago-Pendry superlenses, including that presented in this work, have typically employed quarter-wavelength-thick slabs and object-to-image Appendix A. The Diffraction Limit 176 distances of approximately half a wavelength [3,64]. Specifically, the free-space volumetric

NRI-TL Veselago-Pendry superlens produced a focus with a minimum peak-to-null width of 0.16λ0 ≈ λ0/6 over a distance of 0.57λ0 (λ0 = 125mm). This performance, when compared to the Rayleigh criterion, represents a resolution enhancement of over 3.8 times, and suggests a recovery of evanescent wavenumbers in the range |kz| < 3.8k0. Of course, the Rayleigh criterion is a far-field diffraction limit, and so this comparison would only be valid provided that imaging over the same distance in free space is approximately diffraction-limited. To illustrate that evanescent contributions become negligible over the source-to-image distance of 0.57λ0 = 71.4mm in free space alone, consider the decay lengths of evanescent waves corresponding to the highest transverse wavenumber kz =

3.8k0, which contributes the ﬁnest spatial features of the measured image. According to (A.2), the evanescent decay constant in the longitudinal direction can be obtained p √ 2 2 from ky = ± k0 − (3.8k0) = ±jk0 13.44 ≈ j3.67k0. Thus, the decay length is on the order of 1/|ky| = 5.4mm. Conversely, over the full source-to-image distance in free space, these evanescent components decay by well over 100dB. Thus, imaging over a distance of 0.57λ0 in free space alone may be considered to be approximately diﬀraction-limited, which validates the superlensing performance of the NRI-TL Veselago-Pendry superlens over the same distance. Appendix B

Transmission-Line Networks as

Artiﬁcial Materials

B.1 Artiﬁcial Materials and the Long-Wavelength

Limit

The roughly two decades including and following the second World War saw a flurry of research into the realization and characterization of ‘artificial dielectrics’, low-loss and lightweight structures designed to mimic the macroscopic electromagnetic response of natural dielectrics. These structures consisted of discrete electromagnetic scatterers (e.g. electrically small metallic inclusions in spherical or wire-like form) arranged into periodic, ordered arrays inside a host, or background, medium like vacuum. The term ‘artificial dielectric’ was coined by Winston Kock [23–25], who formulated a comprehensive and general theory illustrating the direct analogies that artificial dielectrics shared with natural crystalline media. He noted that the observable properties of all materials, natural or artificial, are tied to the wavelengths of light with which they are viewed and suggested that artificial dielectrics “respond to radio waves just as a molecular lattice responds to light waves” [23]. For example, at wavelengths on the order of the lattice constant these

177 Appendix B. Transmission-Line Networks as Artificial Materials 178 structures, like natural solids, exhibit diffraction effects. Such short wavelengths perceive the structural properties of the lattice and the microscopic response of the scatterers. The macroscopic perspective is achieved by those wavelengths that are much longer than the lattice constant; such long wavelengths do not possess sufficient spatial resolution to see the corrugations distinguishing scatterers in the array and instead observe their collec- tive electromagnetic response. It is in this regime, known as the long-wavelength limit or the effective-medium limit, that artificial materials appear to be uniform, homogeneous

‘eﬀective’ media possessing an eﬀective permittivity, permeability, and refractive index.

B.2 The Lumped-Element Transmission-Line Model

As demonstrated in Chapter 3, the effective permittivity of an artificial material may be approximated by placing the scatterer into an electrically very small parallel-plate capacitor otherwise filled with vacuum. The resulting effect on the charge on the plates

(or voltage across the plates) may be interpreted as a change in the capacitance of the system stemming from a change in the permittivity of the filling medium. Similarly, the effective permeability may be approximated by placing the scatterer into an electrically very thin solenoid otherwise filled with vacuum. The resulting effect on the current in the windings may be interpreted as a change in the inductance of the system stemming from a change in the permeability of the filling medium. This is, in fact, a homogenization procedure, since the electromagnetic response of the scatterer is lumped uniformly into the background medium. Moreover, the measured voltages and currents are terminal characteristics that cannot distinguish the response of the scatterer from that of a homogeneous material possessing the equivalent uniform effective permittivity and permeability.

The validity of this homogenization depends on the enforcement of quasistatic field conditions (which are akin to the conditions enforced by the long-wavelength limit) across the length of the system. Under quasistatic conditions, the above systems may be re- Appendix B. Transmission-Line Networks as Artificial Materials 179 garded to constitute a lumped ‘unit cell’ which, when cascaded, may be used to describe propagation in a homogeneous medium possessing the determined effective permittivity and permeability. This process describes the familiar relationship between the propagation of a plane wave through a homogeneous medium and the propagation of a TEM wave through a planar transmission line (TL) filled with a homogeneous medium. The intrinsic parameters of the former are the permeability, µ, and permittivity, ², of the filling √ medium (uniquely specifying the propagation constant, β = ω ²µ, and wave impedance, p η = µ/²), whereas the intrinsic parameters of the latter are the per-unit-length inductance, Lx = µg, and capacitance, Cx = ²/g (uniquely specifying the propagation √ √ p constant, β = ω LxCx = ω ²µ, and characteristic impedance, Z0 = Lx/Cx = gη). The per-unit-length inductance and capacitance may be represented more generally as a per-unit-length series impedance, Z = jωLx, and per-unit-length shunt admittance,

Y = jωCx, respectively. It is worth noting that all of these parameters are directly related to the permeability and permittivity of the ﬁlling medium through a constant, g, accounting for the geometry of the TL. A lumped-element model of the homogeneously

filled TL describes these per-unit-length quantities in terms of lumped impedance and admittance over an infinitesimal section. The enforcement of quasistatic electric and magnetic fields across this infinitesimal section enables the definition of voltages and currents that are unique at a given terminal (over a single transverse plane) and differential between terminals (i.e., across the length of the unit cell). The derivation of this TL model, which links the field quantities describing plane-wave propagation in a homogeneous bulk medium to the circuit quantities describing TEM propagation in a TL, is the subject of the remainder of this section, and is adapted from Ref. 32.

Consider a volume of space represented as a three-dimensional unit cell of dimensions

∆x, ∆y, and ∆z over which quasistatic field conditions exist. As depicted in Fig. B.1, the unit cell is homogeneously filled with a material possessing a permittivity ²(ω) and permeability µ(ω), the (generally dispersive) effective parameters that we wish to model Appendix B. Transmission-Line Networks as Artificial Materials 180

∆∆∆x ∆∆∆z

(εεε,µµµ) ^y ∆∆∆y x^ ^z (x,y,z)

Figure B.1: Volume of space representing region in which ﬁelds may be considered quasi-static. After Ref. 32. Reprinted with permission of John Wiley & Sons, Inc., copyright °c 2005.

using an equivalent TL network. To simplify the problem for 2D lossless propagation, it is appropriate to assume that there is no ﬁeld variation in the y-direction and restrict the development for the quasi-TMy case alone, in which the predominant electric and magnetic ﬁeld components are Ey, Hx, and Hz. Spatial discretization of Maxwell’s equations over this cube results in the following expressions:

Ey(x0 + ∆x, z0) − Ey(x0, z0) = −jωµ(ω)Hz∆x (B.1)

Ey(x0, z0 + ∆z) − Ey(x0, z0) = +jωµ(ω)Hx∆z (B.2) and

[Hx(x0, z0 + ∆z) − Hx(x0, z0)] ∆x

− [Hz(x0 + ∆x, z0) − Hz(x0, z0)] ∆z = +jω²(ω)Ey(x0, z0)∆x∆z (B.3) Appendix B. Transmission-Line Networks as Artificial Materials 181

The definitions of potential difference and current using field quantities are as follows:

Z a0 Va0 − Va = − E · dl (B.4) I a I = H · dl (B.5) C where a − a0 is any path connecting the bottom and top faces of the cube, and C is a suitably chosen closed contour intersecting its bottom or top face. Since the ﬁelds are quasistatic within the volume of the unit cell, the integrals degenerate into the simple products Vy = Ey∆y (assuming the bottom face of the unit cell is taken as the zero reference potential), Iz = −Hx∆x and Ix = Hz∆z. Furthermore, deﬁning the impedance and admittance quantities as

Zx = jωµ(ω)∆x∆y/∆z (B.6)

Zz = jωµ(ω)∆y∆z/∆x (B.7)

Y = jω²(ω)∆x∆z/∆y (B.8) and rearranging, (B.1)–(B.3) reduce to

Vy(x0 + ∆x, z0) − Vy(x0, z0) = −ZxIx (B.9)

Vy(x0, z0 + ∆z) − Vy(x0, z0) = −ZzIz (B.10) and

[Iz(x0, z0 + ∆z) − Iz(x0, z0)] + [Ix(x0 + ∆x, z0) − Ix(x0, z0)] = −YVy(x0, z0) (B.11)

The first (second) equation suggests that the potential difference between the front and back (left and right) faces of the cube of Fig. B.1 results from a current drawn by an effective impedance Zx (Zz). The third equation suggests that the potential difference

Appendix B. Transmission-Line Networks as Artificial Materials 182

Vy+dVy Ix+dIx (Zx/2) (Zz/2) Vy Vy+dVy Iz Iz+dIz (Zz/2)

(Zx/2) Y y V Vy x Ix

Figure B.2: Unit cell for a distributed transmission-line network model describing 2D propagation in a homogeneous medium. Reprinted with permission from Ref. 17, copyright °c 2002 IEEE.

between the top and bottom faces is given by the current drawn by an admittance Y .

Evidently, these are Kirchhoﬀ’s voltage and current laws for the per-unit-length lumped- element model of the symmetric 2D transmission-line unit cell shown in Fig. B.2.

It is clear from (B.6)–(B.8) that the lumped impedance and admittance quantities, as well as their corresponding per-unit-length quantities, are directly related to ²(ω) and µ(ω) through a constant term given by the geometry of the unit cell. From an analysis perspective, this result suggests that any homogeneous and lossless dielectric can be modeled at a particular frequency ω0 by discrete unit cells containing only reactive elements, apportioned such that the per-unit-length reactances, through the geometry of the unit cell, represent propagation in a homogeneous medium possessing effective material parameters ²(ω0) and µ(ω0). It is worth noting that the equivalence between the field and circuit quantities is established at well-defined ‘terminals’ external to the unit cell, but collectively represent the homogenized effect of all scattering internal to the unit cell. Appendix B. Transmission-Line Networks as Artificial Materials 183

B.3 From Modeling to Synthesis: The NRI-TL Meta-

material

From a synthesis perspective, it would appear that, if such a TL network of reactances could be realized, and quasistatic conditions satisﬁed, then waves propagating within the network with angular frequency ω0 could be said to experience the eﬀective material parameters ²(ω0) and µ(ω0), which may be extracted from the corresponding per-unit- length series impedance and shunt admittance by way of equations (B.6)–(B.8). This is of particular interest in the quest to synthesize exotic material parameters; for example, the negative-refractive-index (NRI) metamaterial is described by a simultaneously negative permittivity and permeability. From (B.6)–(B.8), it is evident that a TL unit cell designed to mimic propagation in such a medium requires Zx < 0, Zz < 0, and Y < 0. The required capacitive per-unit-length series reactance and inductive per-unit-length shunt admittance are, indeed, the foundations for the development of the NRI-TL metamaterial.

In the spirit of the classical body of work in artiﬁcial dielectrics, the NRI-TL metamaterial concept attempts to physically realize this model by periodically loading a host

TL medium using lumped series capacitors and shunt inductors at deep-subwavelength intervals; here, the host TL medium is the background medium, and the lumped loading represents the desired response at the scatterering sites. The small electrical distance separating the unit cells (or, equivalently, the small phase shifts from terminal to terminal) enforces the quasistatic, or long-wavelength, condition required for the definition of effective-medium properties. The periodic nature of the structure supports the propagation of a spectrum of backward- and forward-wave spatial harmonics from terminal to terminal whose wavenumbers may be determined through a rigorous periodic analysis (see Chapter 4); however, the fundamental harmonic in NRI-TL metamaterials is a backward-wave harmonic. When this fundamental backward harmonic is made to be dominant (as is generally achieved in the long-wavelength limit), propagation in the Appendix B. Transmission-Line Networks as Artificial Materials 184 loaded TL structure can be likened to propagation in a homogeneous medium possessing an ‘effective’ negative permittivity and permeability. Therefore, the terminal characteristics achieved by periodically loading the host TLs constituting the NRI-TL metamaterial would be identical to the terminal characteristics of the same host TLs if filled with the determined effective permittivity and permeability. Thus, the scattering by the periodic inclusions is effectively homogenized into the background medium. This was illustrated for an NRI-TL metamaterial employing a parallel-plate host medium, series capacitive gaps, and shunt inductive sheets in Ref. 21. Appendix C

Free-Space Measurement Apparatus

Design

This Appendix presents the design of a free-space transmission/reﬂection measurement system at X-band used to determine the scattering parameters of the fully printed volumetric NRI-TL metamaterial slab lens described in Chapter 6.

In the experimental determination of the transmission/reflection properties of materials, it is necessary to ensure that the sample under test is placed in the far field of the illuminating antennas and to minimize diffraction around the sample edges. This can be accomplished using both horns and collimating lenses, which produce Gaussian-like beams whose transverse phase profile resembles that of a plane wave at the beam waist.

This is particularly useful if the results are to be interpreted by an effective-medium homogenization theory to extract the effective-medium parameters of the sample. For relatively small samples (whose largest transverse dimension is less than a wavelength, for example), the absence of collimating lenses would require illuminating antennas in the far-field along with absorbers placed around the sample to capture the extraneous components. In fact, such methods were also attempted in the course of this work; however, it was determined that these measurements were very sensitive to the arrangement

185 Appendix C. Free-Space Measurement Apparatus Design 186

c A

d b E

Figure C.1: Schematic of ATM standard-gain X-band horn antennas (see Table C.1).

of the absorbers and, therefore, produced ambiguous results. Thus, it was decided to proceed with the design of dielectric collimating lenses that could collect the standard-gain horn-antenna ﬁelds and produce a Gaussian beam illumination at the sample resembling a plane wave. Similarly, the ﬁelds transmitted through the sample are collected by a symmetrical arrangement on the opposite side.

C.1 Standard-Gain Horn Antenna Speciﬁcations

The pyramidal standard-gain X-band horn antennas used in this work were produced by ATM (Advanced Technical Materials, 49 Rider Ave., Patchoque, NY, 11772 USA).

Figure C.1 and Table C.1 indicate the relevant dimensions and properties (most of which can be found online at http://www.atmmicrowave.com).

C.2 Bulk Rexolite Speciﬁcations

The dielectric lenses were fabricated from bulk Rexolite available in an eight-inch-diameter cylinder stock. The nominal dielectric properties of Rexolite at 10GHz are ²r = 2.53, tan δ = 0.00066 − 0.004 (www.sdplastics.com). To ensure an ability to fabricate the lens Appendix C. Free-Space Measurement Apparatus Design 187

Parameter Description Value ∆f Frequency range 8.2GHz–12.4GHz S/N Serial number G098603-{01,02} ◦ φE E-plane -3dB beam angle 29.3 ◦ φH H-plane -3dB beam angle 29.0 a Mouth width 67.6mm b Mouth height 49.5mm c Throat width (WR90 connector) 22.9mm d Throat height (WR90 connector) 10.2mm rh Taper length 111.0mm (measured)

Table C.1: Properties and dimensions of ATM standard-gain X-band horn antennas depicted schematically in Fig. C.1

Parameter Description Value wi Lens illumination radius 76.2mm ys Sample size 104.5mm w1 Minimum beam waist at horn 24.8mm w2 Minimum beam waist at sample 45.0mm z1 Beam range from horn to lens 186.8mm z2 Beam range from lens to sample 289.8mm zh Beam waist oﬀset 33.5mm

Table C.2: Dimensions of measurement setup depicted schematically in Fig. C.2 proﬁles accurately, it was decided that the area of illumination of the lens by the antenna should be limited to a maximum diameter of six inches (152.4mm). The exact curvature proﬁle of the lenses required to meet this constraint shall be described after the optimal dimensions of the experimental setup are determined.

C.3 Measurement Setup and Design

The transmit side of the free-space measurement setup is depicted schematically in

Fig. C.2, and the corresponding values are presented in Table C.2. The receive side is, of course, identical and symmetrically arranged about the sample. It is now shown how the values reported in Table C.2 are arrived at; these calculations are based on a Appendix C. Free-Space Measurement Apparatus Design 188

Horn Lens Sample

2w1

2wi 2w2

z1 z2

Figure C.2: Schematic depicting the transmit side of the free-space X-band measurement setup (see Table C.2). The receive side is symmetrically arranged about the sample.

procedure described in Ref. 141. All values are calculated relative to an operating frequency of 10GHz (λ = 30mm), which lies at the center of the X-band. It is important to note that the following design procedure is approximate but valid throughout the X- band, and is robust enough to handle asymmetry of the Gaussian beams between the two transverse axes and inevitable measurement tolerances.

The radiated ﬁelds near a horn closely resemble a Gaussian beam. For horn dimensions a × b, optimal coupling to the Gaussian beam occurs for thin waists of w1x =

0.35a = 23.7mm and w1y = 0.5b = 24.8mm. The Rayleigh range z0 for a minimum waist w0 is given by

2 z0 = πw0/λ (C.1)

For the minimum waists w1x and w1y, (C.1) yields Rayleigh ranges of z0,1x = 58.6mm and z0,1y = 64.2mm. The range z1 at which a Gaussian beam with Rayleigh range z0 Appendix C. Free-Space Measurement Apparatus Design 189

and minimum waist w0 achieves the desired illumination radius (waist wi) is given by s µ ¶2 wi z1 = z0 − 1 (C.2) w0

Therefore, the ranges corresponding to the x- and y-dimensions of the beam waist are, respectively, z1x = 179.5mm and z1y = 186.8mm, and it is clear that the beam is asymmetric. To increase the available working distance, z1 = z1y = 186.8mm is chosen; however, the choice is arbitrary since the diﬀerence of just over 7mm between the two values lies well within the quantity min{z0,1x, z0,1y}, over which the Gaussian-beam phase fronts are roughly planar. Nevertheless, from now on, only the values corresponding to the y-direction will be determined.

Based on the -3dB beam widths of the standard-gain horns (see Table C.1), the half-power main-lobe levels of the horn-antenna patterns illuminate a diameter of only

96.5mm at a distance of z1; this implies that the lens comfortably collects most of the power radiated by the horns. It should also be noted that this choice of z1 results in a waist width w1x of 79mm instead of the desired 76.2mm, which corresponds to a larger waist at the sample plane as well. However, it was determined that these values are only slightly larger than the desired values and so, once again, do not signiﬁcantly aﬀect the design.

Now, it is necessary to determine the distance z2 to focus the waist wi at the lens to a minimum waist of w2y = 45mm. The Rayleigh range corresponding to this waist is z0,2y = 212.0mm, and according to (C.2), the required range is z2 = 289.8mm.

The ﬁeld phase and magnitude at the mouth of a horn are approximately equal to those produced by a point source emanating spherical waves from the horn apex (point

A in Fig. C.1). As a result, the aperture ﬁelds do not represent those of a Gaussian beam at its minimum waist, where the phase front would be ﬂat, resembling a plane wave.

However, it is possible to determine a ‘waist oﬀset,’ zh, that would cause the aperture Appendix C. Free-Space Measurement Apparatus Design 190

f z

n=1.59

Figure C.3: Hyperbolic lens geometry; focal length f, refractive index n = 1.59 (Rexolite).

phase of the spherical wave to approximately coincide with that of the Gaussian beam at the mouth of the horn. The waist oﬀset is determined from the quadratic equation

2 2 2 zh − Rhzh + (πw0/λ) = 0 (C.3)

where Rh is separately calculated for the E- and H-planes as the total length from the horn apex to the mouth along the taper. Based on this calculation, it is determined that a reasonable range within which to choose the waist oﬀset is 24mm ≤ zh ≤ 43mm. Thus, the average value zh = 33.5mm is chosen.

C.4 Lens Design

The equation for the waist of a Gaussian beam versus its range is the equation for a vertically-oriented hyperbola whose asymptotes deﬁne an origin. From a geometrical optics perspective, this origin may be said to correspond to the focus of rays on the principal axis of a lens. Based on this intuitive argument, it was decided to design a Appendix C. Free-Space Measurement Apparatus Design 191

biconvex lens with focal lengths corresponding to the previously determined z1 and z2. To minimize geometric aberrations, the lens is designed to have a hyperbolic proﬁle.

Figure C.3 shows the proﬁle and focal length of a single side, and its hyperbolic proﬁle is given by [142] p y = ± (n2 − 1)z2 + 2(n − 1)fz (C.4)

The quantity n = 1.59 is the refractive index of the Rexolite material.

The complete setup consisted of two biconvex lenses (one for the transmit side and the other for the receive side), each of which was constructed by adjoining two single-convex lenses. The lenses were cut to speciﬁcations by Peter C. Kremer at the University of

Toronto using a lathe and cutting tool mounted to micrometer stages that were adjusted manually. Finally, the lenses were polished and two styrofoam frames were designed to hold the assembly together. Appendix D

The Shielded-Loop Antenna

The ideal loop antenna is modeled as a perfectly conducting loop supporting a circulating current. For an infinitesimal loop, this condition produces a radiation pattern with a null in the directions perpendicular to the plane of the loop. However, for small loops of practical size, it is possible to support a linear (dipolar) current in addition to a circulating current, as shown in Fig. D.1 [143]. In the receive mode, these currents can be produced by an impinging plane wave whose magnetic field is oriented perpendicular to the loop, and whose electric field lies in the plane of the loop and also parallel to the feed. Whereas the former produces oppositely directed currents on opposite sides of the loop (circulating current IL), the latter produces co-directed currents (dipolar current, denoted IV /2). This notation, as well as some of the following concepts, are borrowed from the excellent development in Ref. 143. In the transmit mode, the dipolar currents may contribute an electric-dipolar component to the radiation pattern and may entirely obscure the magnetic-dipolar characteristics produced by the circulating current; reciprocally, in the receive mode, the dipolar currents may contribute to the received signal and obscure the sensitive detection of magnetic fields, the principal purpose for which such small loop antennas are employed. For simplicity, the remainder of the discussion shall consider the small loop antenna in its receive mode.

192 Appendix D. The Shielded-Loop Antenna 193

½IV IL IL ½IV

½IV IL ½IV

+ V -

Figure D.1: Ideal balanced loop antenna with circulating current IL and dipolar currents IV .

D.1 Unbalanced Currents

The co-directed currents on opposite sides of a loop induced by an impinging electric

field are spatially separated and, therefore, generally result in a net unbalanced current that produces a voltage V across the terminals of the antenna load, affecting the received signal. However, a condition of current balance may be restored using structural symmetry in the design of the loop antenna; complete symmetry of the induced currents with respect to the antenna feed line and generator/load minimizes the effect of the dipolar currents on the load voltage and ensures that the detected signal is predominantly a function of the impinging magnetic field [117]. It would seem that the simplest way to build a practical loop antenna using a coaxial feed is as shown in Fig. D.2(a). The outer conductor and insulating diectric are stripped over a certain length (starting at point

A), the exposed inner conductor is shaped into a loop, and ﬁnally the loop is attached to the outer conductor of the coaxial line (at point B). However, this method produces strong unbalanced currents on the feed. The net unbalanced current on the exterior surface of the outer conductor is denoted Iim, and can be determined simply by analyzing the currents over the cross section at terminals A and B, as shown in Fig. D.2(b). The Appendix D. The Shielded-Loop Antenna 194

½IV IL IL ½IV ½IV – IL

½IV + IL

B A B A

IV Iim

½IV – IL

+ V - (a) (b)

Figure D.2: (a) Loop antenna with coaxial feed line supporting unbalanced current IV ; (b) Current analysis at terminals A and B.

1 loop current 2 IV − IL on the inner conductor of the coaxial line sets up an identical, oppositely directed current on the inner surface of the outer conductor. The satisfaction

of Kirchhoﬀ’s current law at node B therefore implies that

1 1 I + I + I − I = I = I (D.1) 2 V L 2 V L im V

Therefore, the net unbalanced current on the feed is equal to the dipolar current IV .

D.2 Balancing Currents with the Shielded-Loop

One way to achieve complete structural symmetry using a coaxial feed is to use a

‘shielded-loop’ topology, one of many of which is shown in Fig. D.3(a). The shielded-loop

antenna maintains the shielded coaxial structure except in the vicinity of a small gap

over which the outer conductor is removed, and the exposed inner conductor is connected

to the outer shield. The balancing eﬀect of this topology becomes clear when one realizes Appendix D. The Shielded-Loop Antenna 195

B A

½IV IL ½IV ½IV + IL ½IV – IL B A

I Solid S IS Conductor

+ V - (a) (b)

Figure D.3: (a) Shielded-loop antenna; (b) Current analysis at terminals A and B.

that it may be synthesized from the topology of Fig. D.2(a) simply by extending points

A and B around the loop such that the gap and feed line are symmetrically arranged.

In doing so, an impinging wave sets up circulating and dipolar currents on the outside of the coaxial shield; however, only the circulating currents contribute to a uniform ﬁeld produced across the gap, and this ﬁeld drives a transmission-line mode inside the coaxial line. Once again, an argument based on the conservation of current can be made in the vicinity of the gap, as shown in Fig. D.3(b). The current in the inner conductor is denoted IS and is determined as follows:

µ ¶ 1 1 I = − I − I = I + I ⇒ I = 0 ⇒ I = I (D.2) S 2 V L 2 V L V S L

Thus, the ‘antenna’ in the shielded-loop architecture is the shield itself, since it is the

currents established on the outer surface of the outer coaxial conductor that drive the

transmission-line mode inside the feed line. Appendix D. The Shielded-Loop Antenna 196

D.3 Simulations

To verify the arguments presented above, finite-element-method full-wave simulations using Ansoft’s HFSS [120] were performed for the unbalanced- and shielded-loop topologies of Fig. D.2(a) and Fig. D.3(a). The transmission line on which the topologies are based is a Huber + Suhner 50-Ω semi-rigid coaxial cable consisting of silver-plated inner and outer conductors (diameters of 0.29mm and 1.19mm, respectively) and filled with Teflon

(diameter 0.93mm).

The unbalanced-loop antenna is shown in Fig. D.4(a) and is excited through the feed line at 2.4GHz. The circulating currents on the antenna are shown in Fig. D.4(b), and the current magnitudes of Fig. D.4(c) indicate the current distribution on the feedline.

It is evident, particularly from Fig. D.4(b), that there is a strong unbalanced current generated on the outer surface of the coaxial feed line. Figure D.4(d) shows the radiation pattern of this antenna corresponding to the orientation of the unbalanced loop as shown in Fig. D.4(a). The pattern resembles that of an electric dipole oriented parallel to the feed line, which is consistent with the existence of an unbalanced current (the skewing of the pattern to one side is a result of the asymmetric connection of the loop to the outer conductor). Moreover, the absence of a null perpendicular to the loop plane suggests that the contribution of the loop current to the radiation pattern is overshadowed by that of the unbalanced current.

The shielded-loop antenna is shown in Fig. D.5(a) and it is also excited through its feed line at 2.4GHz. The circulating currents on the antenna are shown in Fig. D.5(b) and the current magnitudes are shown in Fig. D.5(c). It is evident that the unbalanced currents on the feed line are nearly eliminated. Accordingly, the radiation pattern of the shielded- loop antenna shown in Fig. D.5(d) clearly resembles the pattern of a small magnetic dipole with a null in the direction perpendicular to the loop plane (it is evident from the slight non-uniformities of the pattern that the small asymmetries remaining in the shielded- loop topology may still have an eﬀect, albeit minor). Indeed, the simulated shielded-loop Appendix D. The Shielded-Loop Antenna 197

(a) (b)

Figure D.4: (a) Unbalanced-loop antenna simulation model; (b) Circulating currents on loop at 2.4GHz; (c) Current magnitudes at 2.4GHz; (d) Radiation pattern at 2.4GHz.

antenna is virtually identical to the receiving antenna fabricated and employed in the testing of the volumetric NRI-TL superlens in Chapter 7.

D.4 Radiation Pattern Measurements

Several shielded-loop antennas of various sizes were constructed during the course of this work. For example, the transmitting shielded-loop antenna used in the experimental veriﬁcation of superlensing at 2.4GHz described in Chapter 7 is shown in Fig. D.6(a).

Figure D.6(b) is a rectangular plot of the measured E- and H-plane patterns of of a similar shielded-loop antenna at 10GHz, which was employed as a transmitting antenna in the free-space Veselago lens arrangement described in Chapter 6. The null at θ = 0 in the H-plane pattern and the relatively constant gain in the E-plane as φ is swept are Appendix D. The Shielded-Loop Antenna 198

(a) (b)

Figure D.5: (a) Shielded-loop antenna simulation model; (b) Circulating currents on loop at 2.4GHz; (c) Current magnitudes at 2.4GHz; (d) Radiation pattern at 2.4GHz.

indicative of a true magnetic-dipole-like radiator. Appendix D. The Shielded-Loop Antenna 199

−5

−10

−15 dB (normalized to max) −20 E−plane H−plane

−150 −100 −50 0 50 100 150 Angles φ, θ (degrees) (a) (b)

Figure D.6: (a) Fabricated shielded-loop antenna designed for operation at 2.4GHz; (b) measured E- and H-plane patterns of a shielded-loop antenna designed for operation at 10GHz. Bibliography

[1] V. G. Veselago, “The electrodynamics of substances with simultaneously negative

values of ² and µ,” Sov. Phys. Usp, vol. 10, no. 4, pp. 509–514, 1968, translation

based on the original Russian document, dated 1967.

[2] J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett., vol. 85,

no. 18, pp. 3966–3969, Oct. 2000.

[3] A. Grbic and G. V. Eleftheriades, “Overcoming the diﬀraction limit with a planar

left-handed transmission-line lens,” Phys. Rev. Lett., vol. 92, p. 117403, Mar. 2004.

[4] P. Alitalo, S. Maslovski, and S. Tretyakov, “Experimental veriﬁcation of the key

properties of a three-dimensional isotropic transmission-line superlens,” J. Appl.

Phys., vol. 99, p. 124910, Jun. 2006.

[5] S. M. Rudolph and A. Grbic, “Volumetric negative-refractive-index medium ex-

hibiting broadband negative permeability,” J. App. Phys., vol. 102, p. 013904,

2007.

[6] M. Stickel, F. Elek, J. Zhu, and G. V. Eleftheriades, “Volumetric negative-

refractive-index metamaterials based upon the shunt-node transmission-line con-

ﬁguration,” J. App. Phys., vol. 102, p. 094903, Jul. 2007.

200 Bibliography 201

[7] D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J.

B. Pendry, “Limitations on subdiﬀraction imaging with a negative refractive index

slab,” App. Phys. Lett., vol. 82, pp. 1506–1508, Mar. 2003.

[8] A. Grbic and G. V. Eleftheriades, “Practical limitations of sub-wavelength resolu-

tion using negative-refractive-index transmission-line lenses,” IEEE Trans. Anten-

nas and Propagat., vol. 53, pp. 3201–3209, Oct. 2005.

[9] N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diﬀraction-limited optical imaging

with a silver superlens,” Science, vol. 308, no. 5721, pp. 534–537, 2005.

[10] F. Mesa, M. J. Freire, R. Marqu´es,and J. D. Baena, “Three-dimensional superres-

olution in metamaterial slab lenses: Experiment and theory,” Phys. Rev. B, vol. 72,

no. 23, p. 235117, 2005.

[11] M. C. K. Wiltshire, “Radio-frequency (RF) metamaterials,” Phys. Status Solidi B,

vol. 244, no. 4, pp. 1227–1236, Mar. 2005.

[12] A. Salandrino and N. Engheta, “Far-ﬁeld subdiﬀraction optical microscopy using

metamaterial crystals: Theory and simulations,” Phys. Rev. B, vol. 74, p. 075103,

2006.

[13] Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: Far-ﬁeld imaging

beyond the diﬀraction limit,” Opt. Express, vol. 14, no. 18, pp. 8247–8256, 2006.

[14] Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-ﬁeld optical hyperlens mag-

nifying sub-diﬀraction-limited objects,” Science, vol. 315, no. 5819, p. 1686, 2007.

[15] K. Aydin, I. Bulu, and E. Ozbay, “Focusing of electromagnetic waves by a left-

handed metamaterial ﬂat lens,” Opt. Express, vol. 12, no. 22, pp. 8753–8759, 2005.

[16] ——, “Subwavelength resolution with a negative-index metamaterial superlens,”

App. Phys. Lett., vol. 90, no. 25, p. 254102, Jun. 2007. Bibliography 202

[17] G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, “Planar negative refractive index

media using periodically L–C loaded transmission lines,” IEEE Trans. Microwave

Theory Tech., vol. 50, no. 12, pp. 2702–2712, Dec. 2002.

[18] A. K. Iyer, P. C. Kremer, and G. V. Eleftheriades, “Experimental and theoretical

veriﬁcation of focusing in a large, periodically loaded transmission line negative

refractive index metamaterial,” Opt. Express, vol. 11, pp. 696–708, Apr. 2003.

[Online]. Available: http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-

7-696

[19] G. Kron, “Equivalent circuit of the ﬁeld equations of maxwell,” Proc. IRE, vol. 32,

no. 5, pp. 289–299, May 1944.

[20] A. Grbic and G. V. Eleftheriades, “An isotropic three-dimensional negative-

refractive-index transmission-line metamaterial,” J. Appl. Phys., vol. 98, p. 043106,

Aug. 2005.

[21] ——, “Super-resolving negative-refractive-index transmission-line lenses,” in

Negative-Refraction Metamaterials: Fundamental Principles and Applications, G.

V. Eleftheriades and K. G. Balmain, Eds. New York, NY: Wiley-IEEE Press, Jul.

2005, pp. 93–170.

[22] W. J. Hoefer, P. P. So, D. Thompson, and M. M. Tentzeris, “Topology and design of

wideband 3D metamaterials made of periodically loaded transmission line arrays,”

in IEEE MTT-S International Microwave Symp. Digest, Long Beach, CA, Jun.

12–17 2005.

[23] W. E. Kock, “Radio lenses,” Bell Lab. Rec., pp. 177–216, May 1946.

[24] ——, “Metallic delay lenses,” Bell Syst. Tech. J., vol. 27, pp. 58–82, Jan. 1948. Bibliography 203

[25] ——, “Metal lens antennas,” in Proceedings, IRE and Waves and Electrons, Nov.

1946, pp. 828–836.

[26] R. E. Collin, Field Theory of Guided Waves, 2nd ed. Toronto, ON: Wiley-IEEE

Press, 1990.

[27] R. N. Bracewell, “Analogues of an ionized medium: Applications to the iono-

sphere,” Wireless Eng., vol. 31, pp. 320–326, Dec. 1954.

[28] W. Rotman, “Plasma simulation by artiﬁcial dielectrics and parallel-plate media,”

IRE Trans. Antennas Propag., vol. AP-10, no. 1, pp. 82–85, Jan. 1962.

[29] J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency

plasmons in metallic mesostructures,” Phys. Rev. Lett., vol. 76, no. 25, pp. 4773–

4776, Jun. 1996.

[30] D. F. Sievenpiper, M. E. Sickmiller, and E. Yablonovitch, “3D wire mesh photonic

crystals,” Phys. Rev. Lett., vol. 76, no. 14, pp. 2480–2483, Apr. 1996.

[31] M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through

subwavelength channels and bends using ²-near-zero materials,” Phys. Rev. Lett.,

vol. 97, p. 157403, Apr. 2006.

[32] A. K. Iyer and G. V. Eleftheriades, “Negative-refractive-index transmission-line

metamaterials,” in Negative-Refraction Metamaterials: Fundamental Principles

and Applications, G. V. Eleftheriades and K. G. Balmain, Eds. New York, NY:

Wiley-IEEE Press, Jul. 2005, pp. 1–52.

[33] S. A. Schelkunoﬀ and H. T. Friis, Antennas: Theory and Practice. New York,

NY: John Wiley and Sons, Inc., 1952, pp. 584–585. Bibliography 204

[34] J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from

conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory

Tech., vol. 47, no. 11, pp. 2075–2084, Nov. 1999.

[35] T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D.

N. Basov, and X. Zhang, “Terahertz magnetic response from artiﬁcial materials,”

Science, vol. 303, no. 5663, pp. 1494–1496, 2004.

[36] S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis,

“Magnetic response of metamaterials at 100 terahertz,” Science, vol. 306, no. 5700,

pp. 1351–1353, 2004.

[37] S. Zhang, W. Fan, B. K. Minhas, A. Frauenglass, K. J. Malloy, and S. R. J.

Brueck, “Midinfrared resonant magnetic nanostructures exhibiting a negative per-

meability,” Phys. Rev. Lett., vol. 94, p. 037402, 2005.

[38] W. Cai, U. Chettiar, H. Yuan, V. de Silva, A. Kildishev, V. Drachev, and V.

Shalaev, “Metamagnetics with rainbow colors,” Opt. Express, vol. 15, pp. 3333–

3341, 2007.

[39] O. G. Vendik and M. S. Gashinova, “Artiﬁcial double negative (DNG) media com-

posed by two diﬀerent dielectric sphere lattices embedded in a dielectric matrix,” in

Proc. of the 34th European Microwave Conference, Amsterdam, The Netherlands,

Oct. 2004, pp. 1209–1212.

[40] C. L. Holloway, E. F. Kuester, J. B. Baker-Jarvis, and P. Kabos, “A double neg-

ative (DNG) composite medium composed of magnetodielectric spherical particles

embedded in a matrix,” IEEE Trans. Ant. and Propagat., vol. 5, no. 10, pp. 2596–

2603, 2003. Bibliography 205

[41] V. Yannopapas and A. Moroz, “Negative refractive index metamaterials from in-

herently non-magnetic materials for deep infrared to terahertz frequency ranges,”

J. Phys.: Cond. Matt., vol. 17, p. 3717, 2005.

[42] M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Three-dimensional array of

dielectric spheres with an isotropic negative permeability at infrared frequencies,”

Phys. Rev. B, vol. 72, p. 193103, 2005.

[43] G. V. Eleftheriades, “Microwave devices and antennas using negative-refractive-

index transmission-line metamaterials,” in Negative-Refraction Metamaterials:

Fundamental Principles and Applications, G. V. Eleftheriades and K. G. Balmain,

Eds. New York, NY: Wiley-IEEE Press, Jul. 2005, pp. 53–91.

[44] I. V. Lindell, S. A. Tretyakov, K. I. Nikoskinen, and S. Ilvonen, “BW media—media

with negative parameters, capable of supporting backward waves,” Microwave Opt.

Tech. Lett., vol. 31, no. 2, pp. 129–133, Oct. 2001.

[45] H. Lamb, “On group-velocity,” in Proc. London Math. Soc., vol. 1, 1904, pp. 473–

479.

[46] A. Schuster, An Introduction to the Theory of Optics. London, UK: Edward

Arnold, 1904.

[47] L. I. Mandel’shtam, “Group velocity in a crystal lattice,” Zh. Eksp.

Teor. Fiz., vol. 15, pp. 475–478, 1945, english translation by E. F.

Kuester (also available is a relevant lecture of Prof. Mandel’shtam dis-

cussing the concept of negative refraction a year earlier in 1944:

http://ece-www.colorado.edu/ kuester/mandelshtam1944.pdf). [Online]. Avail-

able: http://ece-www.colorado.edu/ kuester/mandelshtam1945.pdf Bibliography 206

[48] G. V. Eleftheriades, “Analysis of bandwidth and loss in negative-refractive-index

transmission-line (NRI-TL) media using coupled resonators,” IEEE Microwave and

Wireless Comp. Lett., vol. 17, no. 6, pp. 412–414, 2007.

[49] R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental veriﬁcation of a negative

index of refraction,” Science, vol. 292, pp. 77–79, Apr. 2001.

[50] J. R. Whinnery and S. Ramo, “A new approach to the solution of high-frequency

ﬁeld problems,” Proc. IRE, vol. 32, no. 5, pp. 284–288, May 1944.

[51] A. Sanada, C. Caloz, and T. Itoh, “Planar distributed structures with negative

refractive index,” IEEE Trans. Microwave Theory Tech., vol. 52, no. 4, pp. 1252–

1263, Apr. 2004.

[52] K. G. Balmain, A. A. E. L¨uttgen,and P. C. Kremer, “Power ﬂow for resonance cone

phenomena in planar anisotropic metamaterials,” IEEE Trans. Antennas Propa-

gat.: Special Issue on Metamaterials, vol. 51, no. 10, pp. 2612–2618, Oct. 2003.

[53] ——, “Using resonance cone refraction for compact RF metamaterial devices,” in

Proc. of the International Conference on Electromagnetics in Advanced Applica-

tions (ICEAA’03), Torino, Italy, Sep. 8–12 2003, pp. 419–422, ISBN 88-8202-008-8

(on CD, ISBN 88-8202-009-6).

[54] N. Engheta, A. Salandrino, and A. Al`u,“Circuit elements at optical frequencies:

nanoinductors, nanocapacitors, and nanoresistors,” Phys. Rev. Lett., vol. 95, p.

095504, Aug. 2005.

[55] A. K. Sarychev and V. M. Shalaev, “Plasmonic nanowire materials,” in Negative-

Refraction Metamaterials: Fundamental Principles and Applications, G. V. Eleft-

heriades and K. G. Balmain, Eds. New York, NY: Wiley-IEEE Press, Jul. 2005,

pp. 313–338. Bibliography 207

[56] S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal

nanoparticle chain waveguides,” Phys. Rev. B, vol. 67, no. 20, p. 205402, May 2003.

[57] G. Shvets, “Photonic approach to making a material with a negative index of

refraction,” Phys. Rev. B, vol. 67, no. 3, p. 035109, Jan. 2003.

[58] A. Al`uand N. Engheta, “Optical nanotransmission lines: synthesis of planar left-

handed metamaterials in the infrared and visible regimes,” J. Opt. Soc. Am. B,

vol. 23, pp. 571–583, 2006.

[59] A. K. Iyer and G. V. Eleftheriades, “Negative refractive index metamaterials sup-

porting 2-D waves,” in IEEE MTT-S International Microwave Symp. Digest, vol. 2,

Seattle, WA, Jun. 2–7 2002, pp. 1067–1070.

[60] ——, “Volumetric layered transmission-line metamaterial exhibiting a negative re-

fractive index,” J. Opt. Soc. Am. B.: Focus Issue on Metamaterials, vol. 23, pp.

553–570, Mar. 2006.

[61] M. Zedler, C. Caloz, and P. Russer, “Circuital and experimental demonstration

of a 3D isotropic LH metamaterial based on the rotated TLM scheme,” in IEEE

MTT-S International Microwave Symp. Digest, Honolulu, HI, Jun. 3–8 2007.

[62] A. K. Iyer and G. V. Eleftheriades, “Characterization of a multilayered negative-

refractive-index transmission-line (NRI-TL) metamaterial,” in IEEE MTT-S Int.

Microwave Symp. Dig., San Francisco, CA, Jun. 11–16 2006, pp. 428–431.

[63] ——, “A multilayer negative-refractive-index transmission-line (NRI-TL) meta-

material free-space lens at X-band,” IEEE Trans. Antennas Propagat., vol. 55,

no. 10, pp. 2746–2753, Oct. 2007. Bibliography 208

[64] ——, “Mechanisms of subdiﬀraction free-space imaging using a transmission-line

metamaterial superlens: An experimental veriﬁcation,” Appl. Phys. Lett., vol. 92,

no. 13, p. 131105, Mar. 2008.

[65] H. J. Lezec, J. A. Dionne, and H. A. Atwater, “Negative refraction at visible

frequencies,” Science, vol. 316, no. 5823, pp. 430–432, 2007.

[66] D. Korobkin, Y. A. Urzhumov, C. Zorman, and G. Shvets, “Far-ﬁeld detection of

the super-lensing eﬀect in the mid-infrared: theory and experiment,” J. Modern

Opt., vol. 52, no. 16, pp. 2351–2364, 2005.

[67] A. A. Houck, J. B. Brock, and I. L. Chuang, “Experimental observations of a left-

handed material that obeys snells law,” Appl. Phys. Lett., vol. 90, no. 16, p. 137401,

2003.

[68] R. B. Greegor, C. G. Parazzoli, J. A. Nielsen, M. A. Thompson, and and M. H.

Tanielian, “Simulation and testing of a graded negative index of refraction lens,”

Appl. Phys. Lett., vol. 87, p. 091114, 2005.

[69] I. Bulu, H. Caglayan, and E. Ozbay, “Experimental demonstration of subwave-

length focusing of electromagnetic waves by labyrinth-based two-dimensional meta-

materials,” Opt. Lett., vol. 31, no. 6, pp. 814–816, 2006.

[70] T. Driscoll, D. N. Basov, A. F. Starr, P. M. Rye, S. Nemat-Nasser, D. Schurig, and

D. R. Smith, “Free-space microwave focusing by a negative-index gradient lens,”

App. Phys. Lett., vol. 88, no. 8, p. 081101, 2006.

[71] M. A. Antoniades and G. V. Eleftheriades, “Compact linear lead/lag metamaterial

phase shifters for broadband applications,” Ant. Wireless. Propagat. Lett., vol. 2,

pp. 103–106, 2003. Bibliography 209

[72] R. Islam and G. V. Eleftheriades, “Printed high-directivity metamaterial ms/nri

coupled-line coupler for signal monitoring applications,” Microwave and Wireless

Comp. Lett., vol. 16, no. 4, pp. 164–166, 2006.

[73] M. A. Antoniades and G. V. Eleftheriades, “A broadband series power divider us-

ing zero-degree metamaterial phase-shifting lines,” Microwave and Wireless Comp.

Lett., vol. 15, no. 11, pp. 808–810, 2005.

[74] ——, “A broadband wilkinson balun using microstrip metamaterial lines,” Anten-

nas and Wireless Propagat. Lett., vol. 4, pp. 209–212, 2005.

[75] R. Islam and G. V. Eleftheriades, “Phase-agile branch-line couplers using meta-

material lines,” Microwave and Wireless Comp. Lett., vol. 14, no. 7, pp. 340–342,

2004.

[76] R. Islam, F. Elek, and G. V. Eleftheriades, “Coupled-line metamaterial coupler

having co-directional phase but contra-directional power ﬂow,” Electronics Lett.,

vol. 40, no. 5, pp. 315–317, 2004.

[77] Y. Wang, R. Islam, and G. V. Eleftheriades, “An ultra-short contra-directional

coupler utilizing surface plasmon-polaritons at optical frequencies,” Opt. Express,

vol. 14, no. 16, pp. 7279–7290, 2006.

[78] A. Grbic and G. V. Eleftheriades, “Experimental veriﬁcation of backward-wave

radiation from a negative refractive index metamaterial,” J. Appl. Phys., vol. 92,

no. 10, pp. 5930–5935, 2002.

[79] L. Liu, C. Caloz, and T. Itoh, “Dominant mode leaky-wave antenna with backﬁre-

to-endﬁre scanning capability,” Electronics Lett., vol. 38, no. 23, pp. 1414–1416,

Nov. 2002. Bibliography 210

[80] S. Lim, C. Caloz, and T. Itoh, “Metamaterial-based electronically controlled

transmission-line structure as a novel leaky-wave antenna with tunable radiation

angle and beamwidth,” IEEE Trans. Microwave Theory Tech., vol. 53, no. 1, pp.

161–173, Jan. 2005.

[81] A. K. Iyer and G. V. Eleftheriades, “Leaky-wave radiation from a two-dimensional

negative-refractive-index transmission-line metamaterial,” in Proc. of 2004 URSI

EMTS International Symp. on Electromagnetic Theory, Pisa, Italy, May 26 2004,

pp. 891–893.

[82] C. A. Allen, C. Caloz, and T. Itoh, “Leaky-waves in a metamaterial-based two-

dimensional structure for a conical beam antenna application,” in IEEE MTT-S

International Microwave Symp. Digest, vol. 1, Fort Worth, TX, Jun. 1–6 2004, pp.

305–308.

[83] P. Burghignoli, G. Lovat, F. Capolino, D. R. Jackson, and D. R. Wilton, “Directive

leaky-wave radiation from a dipole source in a wire-medium slab,” IEEE Trans.

Antennas and Propagat., vol. 56, no. 5, pp. 1329–1339, May 2008.

[84] G. Lovat, P. Burghignoli, F. Capolino, and D. R. Jackson, “High directivity in low-

permittivity metamaterial slabs: Ray-optic vs. leaky-wave models,” Microwave and

Opt. Tech. Lett., vol. 48, no. 12, pp. 2542–2548, 2006.

[85] G. Lovat, P. Burghignoli, F. Capolino, D. R. Jackson, and D. R. Wilton, “Analysis

of directive radiation from a line source in a metamaterial slab with low permit-

tivity,” IEEE Trans. Antennas and Propagat., vol. 54, no. 3, pp. 1017–1030, Mar.

2006.

[86] T. Kokkinos, C. D. Sarris, and G. V. Eleftheriades, “Periodic FDTD analysis of

leaky-wave structures and applications to the analysis of negative-refractive-index Bibliography 211

leaky-wave antennas,” IEEE Trans. Microwave Theory Tech., vol. 54, no. 4, pp.

1619–1630, Jun. 2006.

[87] A. Al`u,F. Bilotti, N. Engheta, and L. Vegni, “Subwavelength, compact, reso-

nant patch antennas loaded with metamaterials,” IEEE Trans. Antennas Propa-

gat., vol. 55, no. 1, pp. 13–25, Jan. 2007.

[88] S. F. Mahmoud, “A new miniaturized annular ring patch resonator partially loaded

by a metamaterial ring with negative permeability and permittivity,” Antennas and

Wireless Propagat. Lett., vol. 3, pp. 19–22, 2004.

[89] K. Z. Rajab, R. Mittra, and M. T. Lanagan, “Size reduction of microstrip antennas

using metamaterials,” in IEEE Antennas and Propagat. Soc. International Symp.

Digest, vol. 2B, Washington, DC, Jul. 3–8 2005, pp. 296–299.

[90] F. Qureshi, M. A. Antoniades, and G. V. Eleftheriades, “A compact and low-

proﬁle metamaterial ring antenna with vertical polarization,” Antennas and Wire-

less Propagat. Lett., vol. 4, pp. 333–336, 2005.

[91] R. W. Ziolkowski and A. Erentok, “At and below the Chu limit: passive and active

broad bandwidth metamaterial-based electrically small antennas,” IET Microwave

Antennas Propagat., vol. 1, no. 1, pp. 116–128, 2007.

[92] M. A. Antoniades and G. V. Eleftheriades, “A metamaterial series-fed linear dipole

array with reduced beam squinting,” in IEEE Antennas and Propagat. Soc. Inter-

national Symp. Digest, Albuquerque, NM, Jul. 9–14 2006, pp. 4125–4128.

[93] C. Christopoulos, The Transmission-Line Modeling Method: TLM. Piscataway,

NJ: IEEE Press, 1995.

[94] C. Caloz, H. Okabe, H. Iwai, and T. Itoh, “Transmission line approach of left-

handed materials,” in USNC/URSI National Radio Science Meeting Digest. Bibliography 212

[95] A. A. Oliner, “A periodic-structure negative-refractive-index medium without res-

onant elements,” in USNC/URSI National Radio Science Meeting Digest.

[96] H. G. Booker, Cold Plasma Waves. Boston, MA: Martinus Nijhoﬀ Publishers,

1984, pp. 26–27.

[97] G. V. Eleftheriades, O. Siddiqui, and A. K. Iyer, “Transmission line models for

negative refractive index media and associated implementations without excess

resonators,” IEEE Microwave Wireless Components Lett., vol. 13, no. 2, pp. 51–

53, Feb. 2003.

[98] J. D. Baena, J. Bonache, F. Martin, R. Marqu´esSillero, F. Falcone, T. Lopetegi, M.

A. G. Laso, J. Garc´ıa-Garc´ıa,I. Gil, M. F. Portillo, and M. Sorolla, “Equivalent-

circuit models for split-ring resonators and complementary split-ring resonators

coupled to planar transmission lines,” IEEE Trans. Microwave Theory Tech.,

vol. 53, no. 4, pp. 1451–1461, Apr. 2005.

[99] E. Shamonina, V. Kalinin, K. H. Ringhofer, and L. Solymar, “Magnetoinductive

waves in one, two, and three dimensions,” J. Appl. Phys., vol. 92, no. 10, pp.

6252–6261, Nov. 2002.

[100] A. Hessel, “General characteristics of traveling-wave antennas,” in Antenna Theory,

R. E. Collin and F. J. Zucker, Eds. New York, NY: McGraw-Hill, 1969, vol. 1,

pp. 151–258.

[101] F. Elek and G. V. Eleftheriades, “Dispersion analysis of the shielded sievenpiper

structure using multiconductor transmission-line theory,” IEEE Microwave Wire-

less Components Lett., vol. 14, no. 9, pp. 434–436, Sep. 2004.

[102] ——, “Simple analytical dispersion equations for the shielded sievenpiper struc-

ture,” in IEEE MTT-S International Microwave Symposium Digest, San Francisco,

CA, Jun. 11–16 2006, pp. 1651–1654. Bibliography 213

[103] C. R. Brewitt-Taylor and P. B. Johns, “On the construction and numerical solution

of transmission-line and lumped network models of maxwell’s equations,” Int. J.

Numerical Methods in Eng., vol. 15, pp. 13–30, 1980.

[104] F. Elek and G. V. Eleftheriades, “A two-dimensional uniplanar transmission-line

metamaterial with a negative index of refraction,” New J. Phys., vol. 7, no. 163,

Aug. 2005.

[105] A. Grbic and G. V. Eleftheriades, “Periodic analysis of a 2-D negative refractive

index transmission line structure,” IEEE Trans. Antennas Propagat.: Special Issue

on Metamaterials, vol. 51, no. 10, pp. 2604–2611, Oct. 2003.

[106] A. K. Iyer, K. G. Balmain, and G. V. Eleftheriades, “Dispersion analysis of reso-

nance cone behaviour in magnetically anisotropic transmission-line metamaterials,”

in 2004 IEEE Antennas and Propagation Society International Symposium Digest,

Monterey, CA, Jun. 23 2004, pp. 3147–3150.

[107] A. Sanada, C. Caloz, and T. Itoh, “Characteristics of the composite right/left-

handed transmission lines,” IEEE Microwave Wireless Components Lett., vol. 14,

no. 2, pp. 68–70, Feb. 2004.

[108] R. Simons, Coplanar Waveguide Circuits, Components, and Systems. Toronto,

ON: Wiley, 2001.

[109] P. Alitalo, S. Maslovski, and S. Tretyakov, “Three-dimensional isotropic perfect

lens based on LC-loaded transmission lines,” J. App. Phys., vol. 99, p. 064912,

Mar. 2006.

[110] M. J. Freire and R. Marqu´es,“Planar magnetoinductive lens for three-dimensional

subwavelength imaging,” Appl. Phys. Lett., vol. 86, p. 182505, Apr. 2005. Bibliography 214

[111] S. Maslovski, S. Tretyakov, and P. Alitalo, “Near-ﬁeld enhancement and imaging

in double-planar polariton-resonant structures,” J. App. Phys., vol. 96, no. 3, pp.

1293–1300, 2004.

[112] M. Shamonin, E. Shamonina, V. Kalinin, and L. Solymar, “Resonant frequencies of

a split-ring resonator: analytical solutions and numerical simulations,” Microwave

and Opt. Tech. Lett., vol. 44, no. 2, pp. 133–136, Jan. 2005.

[113] D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Com-

posite medium with simultaneously negative permeability and permittivity,” Phys.

Rev. Lett., vol. 84, no. 18, pp. 4184–4187, May 2000.

[114] A. Al`uand N. Engheta, “Pairing an epsilon-negative slab with a mu-negative slab:

Resonance, tunneling, and transparency,” IEEE Trans. Antennas Propagat.: Spe-

cial Issue on Metamaterials, vol. 51, no. 10, pp. 2558–2571, Oct. 2003.

[115] A. K. Iyer and G. V. Eleftheriades, “Characterization of a volumetric negative-

refractive-index transmission-line (NRI-TL) metamaterial for incident waves from

free space,” in Proc. 2006 European Conference on Ant. and Propagat., Nice,

France, Nov. 6–11 2006.

[116] D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J.

B. Pendry, “Limitations on subdiﬀraction imaging with a negative refractive index

slab,” Appl. Phys. Lett., vol. 82, no. 10, pp. 1506–1508, 2003.

[117] R. W. King, “The loop antenna for transmission and reﬂection,” in Antenna The-

ory, R. E. Collin and F. J. Zucker, Eds. New York, NY: McGraw-Hill, 1969, vol. 1,

pp. 478–480.

[118] R. W. Ziolkowski, “Gaussian beam interactions with double-negative (DNG) meta-

materials,” in Negative-Refraction Metamaterials: Fundamental Principles and Ap- Bibliography 215

plications, G. V. Eleftheriades and K. G. Balmain, Eds. New York, NY: Wiley-

IEEE Press, Jul. 2005, pp. 171–211.

[119] A. K. Iyer and G. V. Eleftheriades, “Free-space imaging beyond the diﬀraction limit

using a Veselago-Pendry transmission-line metamaterial superlens,” IEEE Trans.

Antennas Propagat., accepted for publication, 2009.

[120] (2008) Ansoft corporation. [Online]. Available:

http://www.ansoft.com/products/hf/hfss/

[121] D. R. Smith, S. Schultz, P. Mark˘os,and C. M. Soukoulis, “Determination of ef-

fective permittivity and permeability of metamaterials from reﬂection and trans-

mission coeﬃcients,” Phys. Rev. B, vol. 65, no. 19.

[122] E. Saenz, P. M. T. Ikonen, R. Gonzalo, and S. A. Tretyakov, “On the deﬁnition of

eﬀective permittivity and permeability for thin composite layers,” J. Appl. Phys.,

vol. 101, p. 114910, 2007.

[123] C. R. Simovski and S. A. Tretyakov, “Local constitutive parameters of metama-

terials from an eﬀective-medium perspective,” Phys. Rev. B, vol. 75, p. 195111,

2007.

[124] A. K. Iyer and G. V. Eleftheriades, “Eﬀective-medium properties of a free-space

transmission-line metamaterial superlens,” in 2009 International Symposium on

Antenna Technology and Applied Electromagnetics (ANTEM) and the Canadian

Radio Sciences Meeting (URSI/CNC), Banﬀ, AB, Canada, Feb. 15–19 2009.

[125] S. C. Cripps, “Metamaterialism,” IEEE Microwave Magazine, vol. 7, no. 6, pp.

32–37, Dec. 2006.

[126] R. Mittra, “A critical examination of the issue of validity of the eﬀective medium

approach to characterizing metamaterials,” in Proc. International Congress on Ad- Bibliography 216

vanced Electromagnetic Materials in Microwaves and Optics, Pamplona, Spain,

Sep. 24 2008.

[127] B. Riddle, J. Baker-Jarvis, and J. Krupka, “Complex permittivity measurements

of common plastics over variable temperatures,” IEEE Trans. Microwave Theory

and Tech., vol. 51, no. 3, pp. 727–733, Mar. 2003.

[128] M. Born and E. Wolf, Principles of Optics, 7th ed. Cambridge, UK: Cambridge

University Press, 1999, pp. 465–471.

[129] E. Hecht, Optics, 3rd ed. Don Mills, ON: Addison Wesley Longman, Inc., 1998,

pp. 461–465.

[130] G. Wang, J. R. Fang, and X. T. Dong, “Refocusing of backscattered microwaves in

target detection by using lhm ﬂat lens,” Opt. Express, vol. 15, no. 6, pp. 3312–3317,

Mar. 2007.

[131] M. C. K. Wiltshire, J. V. Hajnal, J. B. Pendry, D. J. Edwards, and C. J.

Stevens, “Metamaterial endoscope for magnetic ﬁeld transfer: near ﬁeld imaging

with magnetic wires,” Opt. Express, vol. 11, no. 7, pp. 709–715, 2003. [Online].

Available: http://www.opticsexpress.org/abstract.cfm?URI=oe-11-7-709

[132] J. D. Baena, L. Jelinek, R. Marqu´es,and F. Medina, “Near-perfect tunneling and

ampliﬁcation of evanescent electromagnetic waves in a waveguide ﬁlled by a meta-

material: Theory and experiments,” Phys. Rev. B, vol. 72, no. 7, p. 075116, 2005.

[133] A. N. Lagarkov and V. N. Kissel, “Near-perfect imaging in a focusing system based

on a left-handed-material plate,” Phys. Rev. Lett., vol. 92, no. 7, p. 077401, Feb.

2004. Bibliography 217

[134] Pavel A. Belov and Constantin R. Simovski, “Subwavelength metallic waveguides

loaded by uniaxial resonant scatterers,” Phys. Rev. E, vol. 72, no. 3, p. 036618,

2005.

[135] J.S. Nielsen and W. J. R. Hoefer, “Generalized dispersion analysis and spurious

modes of 2-D and 3-D TLM formulations,” IEEE Trans. Microwave Theory Tech.,

vol. 41, no. 8, pp. 1375–1384, Aug. 1993.

[136] P. B. Johns, “A symmetrical condensed node for the tlm method,” IEEE Trans.

Microwave Theory Tech., vol. 35, no. 4, pp. 370–377, Apr. 1987.

[137] M. Zedler, C. Caloz, and P. Russer, “A 3-D isotropic left-handed metamaterial

based on the rotated transmission-line matrix (TLM) scheme,” IEEE Trans. Mi-

crowave Theory Tech, vol. 55, no. 12, pp. 2930–2941, Dec. 2007.

[138] T. Koschny, L. Zhang, and C. M. Soukoulis, “Isotropic three-dimensional left-

handed metamaterials,” Phys. Rev. B, vol. 71, no. 12, p. 121103, 2005.

[139] L. Jelinek, R. Marqu´es,F. Mesa, and J. D. Baena, “Periodic arrangements of

chiral scatterers providing negative refractive index bi-isotropic media,” Phys. Rev.

B, vol. 77, no. 20, p. 205110, 2008.

[140] A. K. Iyer and G. V. Eleftheriades, “A three-dimensional isotropic transmission-

line metamaterial topology for free-space excitation,” Appl. Phys. Lett., vol. 92,

no. 26, p. 261106, Jul. 2008.

[141] P. F. Goldsmith, Quasioptical Systems: Gaussian Beam Quasioptical Propagation

and Applications. New York, NY: IEEE Press/Chapman & Hall Publishers, 1998,

pp. 174–175. Bibliography 218

[142] D. G. Bodnar, “Lens antennas,” in Antenna Engineering Handbook, 3rd ed., R. C.

Johnson and H. Jasik, Eds. New York, NY: McGraw-Hill Professional, 1992, pp.

16–4.

[143] R. W. P. King, H. R. Mimno, and A. H. Wing, Transmission Lines, Antennas, and

Waveguides. New York, NY: Dover Publications, Inc., 1965, pp. 231–235.