Metamaterials Demonstrating Focusing

and Radiation Characteristics Applications

A Dissertation Presented

by

Akram Ahmadi

to

The Department of Electrical and Computer Engineering

in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

in the field of

Electrical Engineering

Northeastern University

Boston, Massachusetts

August 2010

Doctoral Committee:

Assistant Professor Hossein Mosallaei, Dissertation Advisor

Professor Anthony Devaney

Professor Carey Rappaport

Assistant Professor Edwin Marengo

Abstract

Metamaterials Demonstrating Focusing and Radiation Characteristics Applications

Akram Ahmadi

Hossein Mosallaei

This dissertation presents theoretical study and numerical evaluation of metama-

terials demonstrating near-field focusing and radiation characteristics. We start with

physical configuration and performance modeling of all-dielectric metamaterials to de-

velop desired (±ε,±µ) by creating electric and magnetic resonant modes. Arraying

these dipole moments can lead to required material properties. Dielectric particles have

the potential to offer both electric and magnetic dipole modes. We examine dielectric

disks and dielectric spheres as the great candidates for establishing the dipole modes

(metamaterial alphabet), and we demonstrate that a structure constructed from unit-

cells of two different spheres (or disks), where one set of them develops electric modes,

and the other set establishes magnetic modes can provide double negative (DNG)

metamaterials. Then some novel applications of metamaterials are investigated. The

concept of high resolution focusing of negative index materials is investigated and their

performance is compared with those for structures made based on the idea of coupled

surface-modes layers. The resonance performance of an electrically small-size radiator

made of Epsilon Negative (ENG) material is studied next. It is demonstrated how

the material polarization can successfully provide resonance radiation at the negative

material constitutive parameters. One of the possible applications of plasmonic ma-

terials is to build antenna devices radiating and receiving electromagnetic energy at

optical frequencies. Design and fabrication of optical antennas with prescribed spatial

patterns is an interesting and challenging task. Based on the concept of scattering

resonance of plasmonic particles, we illustrate the concept of a reflectarray nanoan-

tenna implemented in optics with the use of array of core-shell dielectric-plasmonic

materials, each of them optimized properly to achieve the required phase shift. We

further present several designs of optical nanoantennas arrays composed of parasitic

plasmonic dipoles and loops where they can enhance radiation characteristics and

direct the optical energy successfully.

c©Northeastern University 2010

All Rights Reserved.

i

To my beloved family.

ii

Acknowledgements

I would like to sincerely thank my research advisor Professor Hossein Mosallaei for his

continuous support and encouragement, and for the opportunity he provided for me to

conduct independent research. I would also like to thank my dissertation committee

members, Professor Anthony Devaney, Professor Carey Rappaport, and Professor Ed-

win Marengo for accepting to be on my dissertation committee. My warmest thanks go

to my beloved family for their constant support and endless love. My parents, Soraya

and Asadollah, deserve my deepest appreciation for their selfless support during this

work and in my whole life. I also feel grateful to Professor Mahmoud Shahabadi from

the University of Tehran, who taught me electromagnetics and helped me to continue

my studying and pursue the Ph.D. I would like to thank all the wonderful staff at

the Electrical Engineering Department. In particular my special thanks to Ms. Faith

Crisley, Ms. Sharon Heath, and Ms. Linda Bonda, for their wonderful assistance dur-

ing my graduate studies at Northeastern University. I am indebted to my officemates

and my colleagues at the ECE Department for many interesting discussions and for

providing a stimulating environment. And thanks to all my friends, especially Shirin

and Morteza for their invaluable friendship and support since the very first days I

came to the United States.

iii

Contents

1 Introduction 1

1.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Review of Research Efforts on Nearfield Imaging . . . . . . . . . 3

1.1.2 Review of Research Efforts on Antennas . . . . . . . . . . . . . 4

1.2 Dissertation Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 All-Dielectric Metamaterials: Design and Development 9

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Periodic Photonic Crystals . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Dielectric Disks: Electric and Magnetic Dipole Creation . . . . . . . . . 16

2.4 Metamaterial Realization . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.5 Optical Metamaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.6 Dispersion Diagram Characteristics of Periodic Array of Dielectric Spheres 35

2.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3 Near-Field Focusing 39

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2 Theory and Formulation of Layered Structures . . . . . . . . . . . . . . 41

3.3 Negative Index Material Slab . . . . . . . . . . . . . . . . . . . . . . . 43

3.4 Coupled Surface-Modes Layers . . . . . . . . . . . . . . . . . . . . . . . 46

3.4.1 Analysis of Multiple Thin Film Systems . . . . . . . . . . . . . 48

3.5 FDTD Numerical Analysis of Finite-Size Structure . . . . . . . . . . . 56

3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4 Ellipsoidal Metamaterial Subwavelength Radiator 60

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.2 Resonance Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.3 Calculation of the Lower Bounds on Q . . . . . . . . . . . . . . . . . . 64

4.4 Performance Analysis of ENG Antennas . . . . . . . . . . . . . . . . . 66

iv

4.4.1 Spherical Radiator . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.4.2 Circular Cylindrical Disk Radiator . . . . . . . . . . . . . . . . 69

4.4.3 Circular Cylindrical Rod Radiator . . . . . . . . . . . . . . . . . 73

4.5 MNG Slab Resonance Radiator . . . . . . . . . . . . . . . . . . . . . . 76

4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5 Optical Reflectarray Nanoantenna 80

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.2 Scattering Characteristic of a Core-Shell Nanoparticle . . . . . . . . . . 81

5.3 Optical Reflectarray Nanoantenna . . . . . . . . . . . . . . . . . . . . . 84

5.3.1 Reflection-Phase Synthesis . . . . . . . . . . . . . . . . . . . . . 86

5.3.2 Plasmonic Core-Shells Array Over a Layered Material . . . . . . 87

5.4 Array Design and Scanned-Beam Characteristics . . . . . . . . . . . . . 91

5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6 Optical Nanoloops Array Antenna 97

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.2 Optical Nanodipole Antennas . . . . . . . . . . . . . . . . . . . . . . . 98

6.2.1 Optical Nanodipole Yagi-Uda Antennas . . . . . . . . . . . . . . 100

6.3 Optical Nanoloop Antennas . . . . . . . . . . . . . . . . . . . . . . . . 103

6.3.1 Optical Nanoloops Array Antenna . . . . . . . . . . . . . . . . . 105

6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

7 Conclusions and Recommendations for Future Work 111

7.1 Summary and Contributions . . . . . . . . . . . . . . . . . . . . . . . . 111

7.1.1 Design and Development of All-Dielectric Metamaterials . . . . 111

7.1.2 Novel Applications of Metamaterials . . . . . . . . . . . . . . . 112

7.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

A Photonic Band Gap Calculations Using FDTD Method 117

Bibliography 122

v

List of Figures

1.1 Material classifications [1]. . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1 DNG metamaterial constructed from metallic loops and rods: (a) the

geometry, and (b) its equivalent circuit model. . . . . . . . . . . . . . . 10

2.2 Periodic structure of dielectric slabs: (a) the geometry, and (b) its trans-

mission coefficient. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 Periodic structure of dielectric rods: (a) the geometry, and (b) its trans-

mission coefficient. Note that one layer of dielectric rods does not gen-

erate any band-gap region. . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4 Near-field patterns for Ez in the x-y plane (one unit cell) for five-layer

rods: (a) before band gap (f1 = 2.80GHz), and (b) after band gap

(f2 = 6.60GHz). Note the confinement of dielectric and air modes

inside the dielectric and air regions, respectively. . . . . . . . . . . . . . 15

2.5 Near-field patterns for Ez in the x-y plane (one unit cell) for one-layer

rods at (a) f1 = 2.80GHz, and (b) f2 = 6.60GHz. . . . . . . . . . . . . 16

2.6 Array of all-dielectric disks: (a) the geometry (Λx = Λy = Λz = 1.5cm),

and transmission coefficients for (b) five-layer structure, and (c) one-

layer structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.7 Field distributions inside one unit cell of the one-layer disks array at

f1 = 4.94GHz (HEM11δ mode): (a) E in the x-z plane, and (b) H

in the y-x plane. Near fields are similar to those of a magnetic dipole

oriented along the y direction. . . . . . . . . . . . . . . . . . . . . . . . 19

2.8 Field distributions inside one unit cell of the one-layer disks array at

f1 = 5.97GHz (TM01δ mode): (a) E in the y-z plane, and (b) H in the

y-x plane. Near fields are similar to those of an electric dipole oriented

along the z direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

vi

2.9 Field distributions inside one unit cell of the one-layer disks array at

f3 = 6.08GHz (HEM21δoctupole mode): (a) E, and (b) H in the y-x

plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.10 Array of one-layer all-dielectric spheres: (a) the geometry (Λy = Λz =

2.5cm), and (b) its effective constitutive parameters. . . . . . . . . . . 23

2.11 Transmission coefficient for the all-dielectric spheres depicted in Fig. 2.10(a).

The first and second resonances represent magnetic and electric reso-

nant modes, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.12 Field distributions inside one unit cell of the spheres array: (a) E in

the x-z plane and H in the y-x plane at fm = 4.73GHz, representing

the magnetic dipole moment, and (b) E in the y-z plane and H in the

y-x plane at fe = 6.55GHz, representing the electric dipole moment

(1.5cm× 1.5cm of the unit cell in the y-z directions is plotted). . . . . 25

2.13 Array of three-layer dielectric spheres (Λx = 1.5cm): (a) the geometry,

and (b) its transmission coefficient. . . . . . . . . . . . . . . . . . . . . 25

2.14 Bandwidth enhancement of metamaterial by increasing couplings be-

tween the elements smaller unit-cell size: (a) the geometry, (b) trans-

mission coefficient at the magnetic resonance, and (c) transmission co-

efficient at the electric resonance. The more the couplings the wider the

bandwidth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.15 DNG metamaterial constructed from all-dielectric spheres: (a) the ge-

ometry (Λy = 2.5cm, Λz = 1.5cm), and its equivalent circuit model, and

(b) transmission coefficient. . . . . . . . . . . . . . . . . . . . . . . . . 28

2.16 Phase distribution of the electric field Ez inside the layer of DNG meta-

material [Fig. 2.15(a)] at f = 6.42GHz. The plane wave propagates

from left to the right where the phase is increased in this direction.

The positive slope for the phase in the central part of the layer is a

demonstration of the backward wave generation. . . . . . . . . . . . . . 30

2.17 Field distributions inside one unit cell of the DNG metamaterial [Fig. 2.15(a)]

at f = 6.42GHz: (a) E in the y-z plane, (b) H in the y-x plane, and (c)

E in the x-z plane. Note the creation of electric and magnetic dipole

moments inside the unit cell of the spheres of εr = 40 and εr = 23.8. . . 31

2.18 DNG metamaterial constructed from all-dielectric disks: (a) the geom-

etry (Λy = 2.5cm, Λz = 1.5cm), and (b) its transmission coefficient. . . 32

vii

2.19 Field distributions inside one unit cell of the DNG metamaterial [Fig. 2.21(a)]

at f = 5.97GHz: (a) E in the y-z plane, and (b) H in the y-x plane.

Note the creation of electric and magnetic dipole moments inside the

unit cell of the disks of εr = 60 and εr = 43. . . . . . . . . . . . . . . . 32

2.20 Metamaterial nanostructured spheres: (a) the geometry Λy = Λz =

250nm), and (b) its transmission coefficient. Note the generation of

magnetic and electric resonances. . . . . . . . . . . . . . . . . . . . . . 33

2.21 DNG optical metamaterial constructed from nanostructured dielectric

spheres (operating in magnetic mode) embedded in negative permittiv-

ity host: (a) the geometry, (b) transmission coefficient, and (c) H field

in the y-z plane at f = 529THz. . . . . . . . . . . . . . . . . . . . . . . 34

2.22 (a) The geometry of a 3D array of spheres: Λy/a = Λz/a = 5 and

Λx/a = 3. Dispersion diagram for one-set of dielectric spheres with

permittivity: (b) ε = 40 and, (c) ε = 21. . . . . . . . . . . . . . . . . . 36

2.23 Dispersion diagram for a DNG metamaterial constructed from two-sets

of dielectric spheres with permittivities 40 and 21. Λy/a = Λz/a = 5

and Λx/a = 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.1 The configuration of layered medium . . . . . . . . . . . . . . . . . . . 42

3.2 Source and negative slab metamaterial. . . . . . . . . . . . . . . . . . . 45

3.3 Field profile for the loss-less negative index slab along the (a) propaga-

tion direction (Green shaded region represents the slab,) and (b) lateral

direction at image plane z = −.6λp. Note that the evanescent waves

are amplified through the slab and the fields at the imaging points are

the same as the source point. . . . . . . . . . . . . . . . . . . . . . . . 45

3.4 Field profile for the lossy negative index slab of thickness 4.5d = .45λp:

(a) propagation direction (Green shaded region represents the slab),

and (b) image performance at different image planes of a dipole pair

separated by .2λ0 (d = .1λp). . . . . . . . . . . . . . . . . . . . . . . . . 47

3.5 Effect of loss: (a) transfer function of the slab, and (b) image perfor-

mance at the plane z = −.6λp of a dipole pair separated by .2λ0. It can

be seen that smaller loss provides higher resolution. . . . . . . . . . . . 47

3.6 Single metal-dielectric interface (εi = 1, εm = 1−ω2p/ω

2): (a) geometry,

and (b) dispersion diagram performance. . . . . . . . . . . . . . . . . . 50

3.7 Two-layer ENG coupled surfaces: (a) geometry, and (b) dispersion di-

agram performance when dm = d0 = .05λp. Negative-positive coupled

surfaces demonstrate the forward and backward surface wave branches. 51

viii

3.8 (a) Transfer function for one-layer and two-layer coupled surfaces: Cou-

pling between the layers introduces better evanescent-wave amplifica-

tion for two-layer structure. (b) Field profile along the propagation

direction in two layer ENG (Shaded regions represent the ENG layers.) 52

3.9 Transfer function for two-layer ENG structure with different air gaps.

An optimized distance between the layers provides a smooth transfer

function resulting a higher resolution image. . . . . . . . . . . . . . . . 52

3.10 N-layered ENG-MNG composite (N=9): (a) the electric and magnetic

field profiles along the Propagation direction (Blue-shaded layers repre-

sent ENG and pink-shaded layers are MNG,) and (b) imaging perfor-

mance at different planes (d = .1λp). . . . . . . . . . . . . . . . . . . . 55

3.11 Imaging performance at plane z = −.6λp for different material losses.

Comparing Fig. 3.11 to Fig. 3.5(b) shows that the layered structure has

a better performance than the NIM slab for higher material losses. . . . 56

3.12 FDTD performance: Field profile for the lossy ENG-MNG composite

along the propagating direction; (a) the electric field, and (b) the mag-

netic field. The growth-attenuation behavior of the field is in agreement

with the results obtained from the theory. . . . . . . . . . . . . . . . . 57

3.13 FDTD performance: Field profile along the transverse direction at plane

z = −.6λp. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.1 The geometry of ellipsoid with semi-axes ax, ay and az . . . . . . . . . 62

4.2 Characteristics of the Drude permittivity material. . . . . . . . . . . . 67

4.3 The geometry of the hemisphere radiator constructed from the Drude

dielectric medium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.4 The performance of the hemisphere structure: (a) input impedance, and

(b) return loss and radiation efficiency. . . . . . . . . . . . . . . . . . . 69

4.5 Radiator performance at the resonant frequency, f = 2.36 GHz: (a)

E-field pattern in the y-z plane. Note to the depolarized fields inside

the sphere, and (b) radiation pattern. It presents a dipole mode of the

antenna as expected of the field distribution inside the radiator. . . . . 70

4.6 The geometry of the disk-shaped Drude permittivity radiator. . . . . . 71

4.7 The performance of the disk-shaped radiator: (a) input impedance, and

(b) return loss and radiation efficiency. . . . . . . . . . . . . . . . . . . 72

4.8 E-field pattern in the y-z plane for the disk at the resonant frequency,

f = 3.42 GHz. Note to the strong field depolarization inside the disk

proving large inductive behavior. . . . . . . . . . . . . . . . . . . . . . 72

ix

4.9 The geometry of the rod-shaped Drude permittivity radiator. . . . . . . 73

4.10 The performance of the rod-shaped radiator: (a) input impedance, and

(b) return loss and radiation efficiency. . . . . . . . . . . . . . . . . . . 74

4.11 E-field pattern in the y-z plane for the rod in the y-z plane at the

resonant frequency, f = 1.13 GHz. . . . . . . . . . . . . . . . . . . . . . 75

4.12 Required negative permittivity for radiator resonation versus ellipsoid

aspect ratio. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.13 Slab radiator constructed from the Lorentzian magnetic medium given

by Eq. (4.12): (a) the geometry, and (b) Lorentzian permeability be-

havior. The ground plane is finite with size 22.5mm× 30mm. . . . . . 76

4.14 Magnetic slab radiator: (a) input impedance, and (b) return loss per-

formance. Tuning the feed slot matches the antenna impedance to 50Ω. 77

4.15 (a) Near field in xy-plane, and (b) radiation pattern of the magnetic

slab radiator. Note to the H-field depolarization. The slab generates

magnetic dipole mode radiation performance. . . . . . . . . . . . . . . 78

5.1 A concentric dielectric-plasmonic nanoparticle. . . . . . . . . . . . . . . 82

5.2 Magnitude and phase of the polarizability α of a concentric nanoshell

particle vs: (a) the permittivity of core when b/a = 0.533 and, (b) the

ratio of radii b/a when εcore = 3ε0. Operating wavelength is 357.1 nm

and the shell is made of silver [εshell = (−4.67 + .01i)ε0]. . . . . . . . . 84

5.3 Schematic of the reflectarray nanoantenna structure. . . . . . . . . . . 85

5.4 Resonance performance of a concentric nanoshell particle, b/a = 0.533,

εcore = 3ε0. Close comparison between Mie-theory and FDTD is illus-

trated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.5 FDTD simulated results for different core materials and construction of

phase design curve: (a) reflection amplitude and, (b) reflection phase . 88

5.6 Phase of reflection coefficient vs. the core permittivity at λ0 = 357.1 nm 88

5.7 Radiation pattern in the x-z plane at λ0 = 357.1 nm: (a) θ0 = 15, and

(b) θ0 = 30. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.8 Near-field (Ex) of the reflectarray for 15 beam scanning [Fig. 5.7(a)] in

a plane located at 0.5λ0 above the nanoantenna: (a) magnitude (dB),

and (b) phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.9 Near-field (Ex) of the reflectarray for 30 beam scanning [Fig. 5.7(b)] in

a plane located at 0.5λ0 above the nanoantenna: (a) magnitude (dB),

and (b) phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

x

5.10 FDTD radiation pattern in the x-z plane at λ0 = 357.1 nm: (a) θ0 =

15, and (b) θ0 = 30. Good comparisons compared to dipole-modes

theoretical results (5.7) are observed. . . . . . . . . . . . . . . . . . . . 95

5.11 Radiation patterns in the x-z plane at different frequencies for 30 beam

scanning: (a) f = 0.9f0, (b) f = 0.9f0, (c) f = 0.9f0, and (d) f = 0.9f0

( f0 = 840 THz is the design frequency). . . . . . . . . . . . . . . . . . 96

6.1 A single plasmonic dipole antenna illuminated by an z-polarized electric

field plane wave, W = 30nm, H = 120nm: (a) structure, (b) resonance

performance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

6.2 Directivity (in dB) for resonant plasmonic dipole antenna in plane φ =

0. Maximum directivity is 1.9dB. . . . . . . . . . . . . . . . . . . . . . 100

6.3 3-element nano-optical Yagi-Uda antenna for an operating wavelength

of 760 nm. hr = 130nm, he = 120nm, hd = 105nm, d = 100nm. . . . . . 102

6.4 Directivity (in dB) for the Nano-optical Yagi-Uda antenna shown in

Fig. 6.3in plane φ = 0. Maximum directivity is 3.6dB. . . . . . . . . . . 102

6.5 5-element nano-optical Yagi-Uda antenna for an operating wavelength

of 760 nm. hr = 130nm, he = 120nm, hd = 105nm, d = 100nm. . . . . . 102

6.6 Directivity (in dB) for the Nano-optical Yagi-Uda antenna shown in

Fig. 6.5 in plane φ = 0. Maximum directivity is 4.5dB. . . . . . . . . . 103

6.7 A single plasmonic loop antenna illuminated by an x-polarized electric

field plane wave, l = 85nm, t = 15nm. . . . . . . . . . . . . . . . . . . . 104

6.8 Resonance performance of single plasmonic loop. High scattering occurs

at λ = 1.34µm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6.9 Polarized current on plasmonic loop at resonant wavelength λ = 1.34µm:

(a) normalized |Jx| (dB), and (b) normalized |Jy| (dB). The current

distribution is similar to what one observes in microwave for a rectan-

gular loop antenna with 4l ' λ (The size becomes subwavelength in

optics.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6.10 Far-zone power pattern for single plasmonic loop at the operating wave-

length. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6.11 Directivity (in dB) for resonant plasmonic loop antenna in planes (a)

φ = 0, and (b) φ = π/2 Maximum directivity is 2dB. . . . . . . . . . . 107

xi

6.12 Schematic view of nanoloops antenna array. At operating wavelength

of λ = 1.34µm, the emitter element has the resonant size of 4l1 =

340nm=λ/3.9, and the directors lengths are 4l2 = 4l3 = 260nm. The

reflector spacing is t1 = 125nm, and the directors spacings are t2 =

t3 = 375nm. The emitter and the directors are printed on low dielectric

substrates with εd = 1.5. The silver slab has the thickness of ts =

205nm. A finite-size structure of ls = 500nm in the transverse plane is

considered. The yellow arrow shows the excitation. . . . . . . . . . . . 107

6.13 Directivity (in dB) for parasitic plasmonic loop array antenna in planes:

(a) φ = 0, and (b) φ = π/2. The emission of the coupled system is highly

directed towards upward. Maximum directivity of 8.2dB is established. 108

6.14 Far-zone power pattern for the array antenna. The power is highly

directed towards the upper hemisphere and the back radiation is sup-

pressed. Successful collimation in compared to Fig. 6.10 is illustrated. . 109

6.15 Electric field distribution induced on the nanoloops antenna array at

the operating wavelength of λ = 1.34µm (Normalized and plotted in dB.)109

A.1 An infinite two-dimensional square lattice of circular dielectric cylin-

ders in air: (a) the schematic of structure, (b) Brillouin zone, and (c)

dispersion diagram for TMz polarization (plotted in blue) and for TEz

polarization (in plotted red). . . . . . . . . . . . . . . . . . . . . . . . . 119

A.2 Spectral amplitude at X (kx = π/a, ky = 0) for the infinite two-dimensional

square lattice of circular dielectric cylinders in air. . . . . . . . . . . . . 120

A.3 An infinite two-dimensional triangular lattice of air holes (r/a = 0.48)

in a dielectric (εr = 13): (a) the schematic of structure. The dotted

rectangle shows the unit cell which we use for bang-gap calculation, (b)

Brillouin zone, and (c) dispersion diagram for TEz polarization. . . . . 121

xii

List of Tables

5.1 Core relative permittivity of nanoantenna array elements: (a) θ0 = 15,

and (b) θ0 = 30. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.2 Induced dipoles, pxs and pzs: (a) θ0 = 15, and (b) θ0 = 30. . . . . . . 93

xiii

Chapter 1

Introduction

Metamaterials are receiving increasing attention in the scientific community in recent

years due to their exciting physical properties and novel potential applications [1–5].

In the 1960s, Veselago of Moscow’s P.N. Lebedev Institute of physics examined the

feasibility of media characterized by a simultaneously negative permittivity ε and per-

meability µ [6]. He theoretically concluded that such media are allowed by Maxwell’s

equations and for a uniform plane wave in such a medium the direction of the Poynting

vector is antiparallel to the direction of the phase velocity, contrary to the case of plane

wave propagation in conventional simple media. For this reason, some researchers use

the “backward wave media” to describe this type of media. What is remarkable in

Veslago’s work is his realization that isotropic and homogenous media supporting back-

ward waves ought to be characterized by a negative index. Consequently, when such

media are interfaced with conventional dielectrics, Snell’s Law is reversed, leading to

the negative refraction of an incident electromagnetic plane wave. Such a media can be

called a metamaterial, where the prefix meta, Greek for “beyond” or “after,” suggests

that it possess properties that transcend those available in nature [5].

1

1.1 Background and Motivation

It is well known that the properties of the materials involved in a system determines

the response of the system to the presence of an electromagnetic field. Materials can be

classified based on defining the macroscopic parameters permittivity and permeability

of these materials. The medium classification is illustrated in Fig. 1.1.

Figure 1.1: Material classifications [1].

A medium with both permittivity and permeability greater than zero is called

double-positive (DPS) medium. Most naturally medium, for example dielectrics, fall

under this class. In certain frequency regimes many plasmas exhibit permittivity less

than zero and permeability greater than zero, which are called epsilon-negative (ENG)

medium. In certain frequency regimes some gyrotropic materials exhibit permittivity

greater than zero and permeability less than zero. This class is called mu-negative

(MNG) medium. And finally a medium with both permittivity and permeability less

than zero will be called double-negative (DNG) medium. This class of materials has

only been demonstrated with artificial constructions.

The first published work for providing artificial magnetism can be attributed to

Schelkunoff [7] which was based on the use of resonant loop circuits. He used a

loop circuit inclusion with inductance L terminated to a series capacitor C to achieve

permeability property. Currently, many researchers have used the similar concept to

obtain desired magnetic properties by properly tailoring the loop configuration. The

2

major drawback in using metallic loops is the loss attributed with the conduction loss

in microwave and optical frequencies. In addition, the fabrication of metallic loops at

optical region is very challenging. The promising way to solve these problems is to

use a composite medium constructed of dielectric particles. This structure also has

the potential to offer a wider bandwidth. The first part of this dissertation focuses on

demonstration and development of DNG material by using all dielectric particles.

Metamaterial potential applications are diverse and include remote aerospace ap-

plications, sensor detection and infrastructure monitoring, smart solar power manage-

ment, public safety, radomes, high-frequency battlefield communication and lenses for

high-gain antennas, improving ultrasonic sensors and even shielding structures from

earthquakes. The research in metamaterials is interdisciplinary and involves such fields

as electrical engineering, electromagnetics, solid state physics, microwave and antennae

engineering, optoelectronics, classic optics, material sciences, semiconductor engineer-

ing, nanoscience and others [8]. In the second part of this dissertation the different

applications of metamaterials have been investigated and novel designs for nearfield

imaging and optical nanoantenna have been proposed.

1.1.1 Review of Research Efforts on Nearfield Imaging

Evanescent waves carry subwavelength information of an object. Amplifying these

modes and contributing them into the image plane has been a challenging task in

recent years. There are two sorts of electromagnetic radiation: near field and far

field. The latter propagates as plane waves with a real wave vector, the former has an

imaginary wave vector resulting in an exponential decay and therefore is confined to

the vicinity of the source. Conventional lenses act only on the far field: focusing the

near field requires amplification. Unfortunately for imaging purposes the finer details

of an object are contained in the near field. Based on Veselago’s work [6], Pendry

in [2] showed how a lossless negative index (NI) slab can realize a superlens to focus

3

all the Fourier components of a source. Later on, a series of research started to study

the different aspects of this topic and found out other possible ways to amplify the

evanescent waves [9–18]. Losses are the ultimate limiting factors for resolution and

even a highly conducting metal such as silver has a restricted performance. Redesigning

the lens to minimize absorption will help to attain improved subwavelength resolution.

To use a large magnitude of the real part of ε and to use a layered stack of alternating

negative-positive dielectric layers have been suggested to reduce the effect of losses [19–

21]. This gives a greatly improved performance, but even in these systems losses

eventually limit the resolution. Absorption in the lens materials will always limit the

attainable subwavelength resolution in any implementation. One possibility that arises

in optics is to use optical amplification to overcome absorption and this represents an

interesting option to increase the subwavelength resolution of these superlenses.

1.1.2 Review of Research Efforts on Antennas

The history of antennas dates back to James Clerk Maxwell who unified the theories

of electricity and magnetism, and eloquently represented their relations through a set

of profound equations best known as Maxwell’s Equations [22, 23]. His work was first

published in 1873 [24]. The first wireless electromagnetic system was demonstrated

by Heinrish Rudolph Hertz in 1886 and it was not until 1901 that Guglielmo Marconi

was able to send signals over large distances. From Marconi’s inception through the

1940s, antenna technology was primarily centered on wire related radiating elements

and frequencies up to about UHF. Modern antenna technology was launched while

World War II and beginning primarily in the early 1960s, numerical methods were

introduced that allowed complex antenna system configuration to be analyzed and

designed very carefully.

Antenna engineering has enjoyed a very successful period during the 1940s-1960s.

Although a certain level of maturity has been attained, there are many challenging

4

opportunities and problems to be solved. Integration of new materials into antenna

technology offers many opportunities. Because of the many new applications, the lower

portion of the EM spectrum has been saturated and the designs have been pushed to

higher frequencies, including the millimeter wave frequency bands. Smaller physical

size, wider bandwidth and higher radiation efficiency are three desirable characteristics

of antennas integrated into communication systems. In recent years, considerable

efforts have been devoted towards antenna miniaturization. The challenge is to make

the physical size of the antenna as small as possible along with achieving a wideband

impedance characteristic (Q values close to the lower-bound).

While antenna is a key element in the microwave spectrum to enable wireless data

communication, the extension of this concept into the optics has many applications

and has been a growing research in recent years. Among the technological applica-

tions for optical antennas one can find high-resolution microscopy and spectroscopy,

optical sensors, lasing, solar cells and efficient solid-state light sources, and it has also

become important in biotechnology and medicine. The metals used in antenna de-

signs in microwave/RF frequency domain are highly conductive materials, which in

the theory and the numerical simulation are often modeled as perfect electric conduc-

tors or, sometimes, a high-conductivity surface with certain surface impedance. In

optical domain, the metals behave very differently which means all the concepts and

experiments which have been done in microwave/RF domain cannot be used directly

in optical antenna design and must be re-examined. This opens up a new research

area which is growing so fast these days.

1.2 Dissertation Overview

This dissertation has 5 main chapters along with the Introduction chapter and a con-

clusion statement. We begin from the concept of demonstrating all-dielectric metama-

terial in Chapter 2. Then in following Chapters, we further present several applications

5

of metamaterials in focusing and radiation characteristics. A short description of the

chapters is summarized as below:

Chapter 2: All-Dielectric Metamaterials

In this chapter, physical concept and performance analysis of RF/optical all-dielectric

metamaterials are presented. It is demonstrated that a metamaterial with desired ma-

terial parameters (ε, µ) can be successfully developed by creating electric and magnetic

resonant modes. Dielectric disk and spherical particle resonators are considered as the

great candidates for establishment of dipole moments. A full wave Finite Difference

Time Domain (FDTD) technique is applied to comprehensively obtain the physical

insights of dielectric resonators. Near-field patterns are plotted to illustrate the de-

velopment of electric and magnetic dipole fields. Geometric-polarization control of

the dipole moments allows ε and µ to be tailored to the application of interest. All-

dielectric Double Negative (DNG) metamaterials are designed. Engineering concerns,

such as, loss reduction and bandwidth enhancement are investigated.

Chapter 3: Near-Field Focusing

This chapter reviews the concept of high-resolution imaging of a negative index mate-

rial (NIM) slab and compares its performance with the structure made based on the

idea of coupled surface-modes layers. Fourier-spectrum theoretical model and finite

difference time domain (FDTD) numerical approach are applied to comprehensively

characterize the structures and demonstrate the characteristics. It is highlighted that

if the loss is small, a NIM slab can provide a better performance at a farther distance

than the layered structure with the same thickness. However, considering a realistic

design with relatively large loss, the later will offer a more promising performance

to the loss and the image can be reconstructed in a farther distance from the object

cascading more number of thin-layers.

6

Chapter 4: Ellipsoidal Metamaterial Subwavelength Radiator

The resonance performance and Quality factor of electrically small ellipsoidal radi-

ators made of Epsilon Negative (ENG) material is investigated in this chapter. It

is demonstrated that the material polarization can successfully provide resonance ra-

diation at the negative material constitutive parameters. In principle, arbitrary low

resonant frequencies for a fixed antenna dimension can be achieved. The dependence

of resonant frequency on the shape of the structure is determined. Special attention is

devoted to the sphere, thin disk, and long rod, and physical insights into the radiation

characteristics and Q (or bandwidth) are highlighted.

Chapter 5: Optical Reflectarray Nanoantenna

In this chapter, we study the design of optical nanoantennas and antenna arrays based

on the surface plasmon resonance of plasmonic nanoparticles. We first review the

scattering resonance of plasmonic particles of uniform and concentric structures. Then

using the concept, the design of a reflectarray nanoantenna at optical frequencies whose

elements are nano-sized concentric spherical particles with the core made of ordinary

dielectrics and the shell made of a plasmonic material will be investigated. Modeling

approaches based on finite difference time domain (FDTD) numerical method and Mie

scattering theory are used to characterize and tune the reflectarray design.

Chapter 6: Optical Nanoloops Array Antenna

In this chapter, we create an optical nanoantenna array composed of parasitic plas-

monic loops where they can enhance radiation characteristics and direct the optical

energy successfully. Three metallic loops inspired by the concept of Yagi-Uda antenna

are optimized around the region where they feature high scattering performance to

control the radiation beam. The loop geometry in compared to the dipole configu-

7

ration has the benefit of using the available aperture in an effective way to provide

the higher directivity. The angular emission of the nanoloops array antenna is highly

directive for upward radiation.

Chapter 7: Conclusions and Future works

This chapter concludes this dissertation, summarizes its contributions, and presents

recommendations on future work.

8

Chapter 2

All-Dielectric Metamaterials:

Design and Development

2.1 Introduction

Metamaterials are receiving increasing attention in the scientific community in recent

years due to their exciting physical properties and novel potential applications [1–5].

To achieve a metamaterial with a desired figure of merit, it is required to first cre-

ate appropriate electric and magnetic dipole moments (in small-size scales) utilizing

available materials and then tailor their arrangement to the application of interest.

Basically, the electric and magnetic dipole moments can be envisioned as the alpha-

bet for making metamaterials. For instance, to achieve an artificial magnetism, the

most conventional approach is to implement metallic loops offering magnetic dipole

moments [25]. Conductor rods can be used for producing electric dipole moments [26].

Arrangements of these dipole moments can establish required material parameters, for

instance, a double negative (DNG) metamaterial behavior as depicted in Fig. 2.1.

Most of the metamaterial designs are constructed with the use of metallic elements.

The major drawbacks in using metallic inclusions are their conduction loss and fab-

rication difficulties, especially in the optical frequencies. In addition, they show very

9

Figure 2.1: DNG metamaterial constructed from metallic loops and rods: (a) thegeometry, and (b) its equivalent circuit model.

narrow bandwidth resonant modes. Further, most of the known realizations are highly

anisotropic composites. Recently, a new paradigm for metamaterial development was

introduced by Holloway et al. in Ref. [27], where they used magnetodielectric spheres

for generating required magnetic and electric dipole moments. Later on, Vendik et

al. used the same concept and suggested a more practical approach, such that only

dielectric spheres are involved [28]. Basically, they proposed two sets of spheres hav-

ing the same dielectric materials but different radii. The dielectric material of spheres

is much larger than the host material, such that the wavelength inside the spheres

is comparable to their diameters, and at the same time the wavelength outside the

spheres is large in comparison to the spheres sizes. The electromagnetic fields inside

this composite can be viewed as the superposition of electric and magnetic dipoles

and multipoles of the spheres. Since the permittivity of spheres is much larger than

the host material, the electric and magnetic dipole fields are dominant. Thus, one set

of spheres can offer electric dipole moments, and the other set can provide magnetic

dipole moments. Because of the small-size spheres in terms of host wavelength, one

can successfully assign the effective material parameters (εeff , µeff ) to the bulk com-

posite. The constitutive parameters were formulated originally by Lewin in Ref. [29]

considering the spheres resonate either in the first or second resonant modes of the Mie

series. Then, Jylha et al. improved those formulations by taking into account the elec-

tric polarizabilities of spheres operating in the magnetic resonant modes. In Ref. [30],

10

HFSS software was also used to numerically model the periodic configuration where

the perfect electric conductor (PEC) and perfect magnetic conductor (PMC) surfaces

were located on the periodic sides of the structure. This method is applicable only if

the electric and magnetic fields are polarized normal to the PEC and PMC surfaces,

respectively. In a metamaterial, the electric and magnetic fields can, in general, be

polarized in complex forms inside the unit cell and applying this technique may not

be appropriate.

The advantages of only-dielectric metamaterial in comparison to its metallic coun-

terpart are the better potential for fabrication from RF to optics, and the higher

efficiency because of not having the metallic loss. In addition, one can achieve an

isotropic metamaterial design utilizing spherical geometry inclusions. Further, the di-

electric spheres offer wider bandwidth at the electric and magnetic eigenfrequencies

due to the larger fraction of unit-cell volume that they can occupy.

It is worth noting that if the goal is to achieve a DNG medium at optical frequencies,

one can use only one set of spheres (magnetic resonant mode), and embed them inside a

negative permittivity plasmonic host material, such as metals or semiconductors. This

idea was first proposed by Seo et al. in Ref. [31]. The obtained structure shows more

robust characteristics over the double-spheres lattice design in terms of fabrication

tolerance and bandwidth, although the loss of the host plasmonic medium (the metal)

can be an issue.

The goal of the present work is to provide a comprehensive investigation of dielectric

metamaterials. The physical insights and engineering concerns are addressed. We start

with the periodic photonic band-gap (PBG) crystals, and demonstrate how the band-

gap region is obtained as the result of periodicity along the propagation direction and

diffraction phenomena between the unit cells. The near-field patterns before and after

the gap region are plotted to better understand the PBG behavior. Then, we modify

the geometry of the PBG crystal by considering finite size disks instead of the infinite

rods. The performance is analyzed, and transmission coefficient and near-field patterns

11

are determined. It is illustrated that the dielectric disks can interestingly create electric

and magnetic dipole moments at their resonant modes, which can be successfully used

for the metamaterial development. This process is basically nothing to do with the

periodicity and unit-cell diffractions along the direction of propagation, and allows one

to accomplish a metamaterial with very small-size ingredients. The concept is extended

to spherical particles, and effective constitutive parameters (ε, µ) are presented. A

DNG all-dielectric metamaterial is designed. The dielectric metamaterial is free of

conduction loss and provides a relatively high efficiency. The periodic (or possible

random) arrangement of particles also suppresses the radiation loss that each of the

resonators produces individually. It is shown that by embedding the dielectric particles

close to each other, the couplings between them are increased, and the bandwidth

of a negative permittivity-negative permeability region is effectively enhanced. The

complex metamaterial structures designed in this study are modeled using an advanced

and versatile in-house developed finite difference time domain (FDTD) technique [32–

34].

2.2 Periodic Photonic Crystals

Photonic crystals are a novel class of periodic dielectric structures that by offering

engineered dispersion diagrams effectively manipulate the propagation of EM or op-

tical waves [35, 36]. The discovery of PBG crystals created unique opportunities for

proposing novel devices in both microwave and terahertz frequencies [37–39]. The

main benefit of PBG materials is their construction from all dielectric elements, which

increases their feasibility for fabrication from RF to optics. Although in the begin-

ning the focus was on the utilization of the stop-band region of PBG for controlling

the waves, recently, other applications such as directive emission, negative refraction,

superlensing, etc., with the use of other parts of the PBG dispersion diagram have

been highlighted [40, 41]. One fact that must be carefully considered is that the novel

12

behaviors of the PBG are derived from the unit-cell interactions and periodic dielec-

tric contrasts along the propagation direction, and one needs a specific unit-cell size

to achieve the required diffractions for accomplishing the performance of interest. The

problem is now twofold: first, the unit cell cannot be as small as one is interested in,

and second, the diffraction phenomenon degrades the performance of the PBG in some

specific applications such as directive emission or superlensing devices.

The simplest possible photonic crystal consists of alternating layers of material

with different dielectric constants. Fig. 2.2 depicts the geometry of a one-dimensional

periodic structure of dielectric layers (5-layers along the x) and its transmission coef-

ficient. The periodicity of structure along the x-direction opens up a stop-band region

between the dielectric and air modes.

(a) (b)

Figure 2.2: Periodic structure of dielectric slabs: (a) the geometry, and (b) its trans-mission coefficient.

A two-dimensional photonic crystal is periodic along two of its axes and homoge-

neous along the third. A typical specimen, consisting of a square lattice of dielectric

columns is shown in Fig. 2.3(a). For certain values of the column spacing, this crystal

can have a photonic band gap in the xy-plane. Inside the gap, no extended states

are permitted, and incident light is reflected. But although the multilayer film only

13

reflects light at normal incidence, this two-dimensional photonic crystal can reflect

light incident from any direction in the plane [36].

To show the effects of the unit-cell size and structure periodicity on the PBG

performance, a periodic configuration of dielectric rods with a radius of r = 0.5 cm,

permittivity of εr = 10.2, and a lattice constant of Λ = 1.5 cm is depicted in Fig. 2.3(a).

The rods are infinite along the z direction, and periodic along the y direction. Five lay-

ers are considered in the x direction. The FDTD is applied to obtain the transmission

coefficient for a plane wave with Ez −Hy polarization, propagating through the PBG

structure along the x direction. The result is plotted in Fig. 2.3(b) . The periodicity of

structure along the x direction opens up a stop-band region between the dielectric and

air modes for 0.19 < a/λ0 < 0.29 (-10 dB transmission level) for the electromagnetic

(EM) wave. The size of the unit cell dominantly determines the frequency range of

stop-band performance. The characteristic of the one-layer PBG along the x is also

shown in Fig. 2.3(b). Because of the lack of periodicity and unit-cell diffractions, no

band-gap region in the frequency of interest is observed.

(a)

2.0 3.0 4.0 5.0 6.0 7.0Frequency (GHz)

−40

−35

−30

−25

−20

−15

−10

−5

0

Mag

nitu

de o

f Tra

nsm

issi

on C

oeffi

cien

t (dB

)

5−layer Structure1−layer Structure

(b)

Figure 2.3: Periodic structure of dielectric rods: (a) the geometry, and (b) its transmis-sion coefficient. Note that one layer of dielectric rods does not generate any band-gapregion.

14

Figure 2.4: Near-field patterns for Ez in the x-y plane (one unit cell) for five-layerrods: (a) before band gap (f1 = 2.80GHz), and (b) after band gap (f2 = 6.60GHz).Note the confinement of dielectric and air modes inside the dielectric and air regions,respectively.

The Ez near-field patterns of five-layer PBG before and after the band-gap region

(at f1 = 2.80GHz and f2 = 6.60GHz) are shown in Fig. 2.4. It is observed that at

frequencies before the band gap the electric field is concentrated inside the dielectric

region, giving it a lower frequency, while the mode just above the gap has most of its

power in the air region, so its frequency is raised a bit. This satisfies the electromag-

netic variational theory applied to understand the PBG concept [36]. For the one-layer

PBG nearfield behaviors at f1 and f2 are obtained in Fig. 2.5, and one cannot observe

the similar phenomena as what was obtained for the five-layer case.

Therefore, to achieve a desired performance utilizing the PBG concept (periodic

dielectric contrast), having periodicity and a relatively large size unit cell are essential.

One might be able to reduce the size of the unit cell by increasing the permittivity of

the dielectric rod; however, this will increase the interactions between the unit cells

causing more diffractions along the propagation direction, which might not be suitable

for some applications. In the following sections, we will address how an engineered

dispersion diagram may be successfully tailored using a different concept that is based

15

Figure 2.5: Near-field patterns for Ez in the x-y plane (one unit cell) for one-layer rodsat (a) f1 = 2.80GHz, and (b) f2 = 6.60GHz.

on the creation of dipole modes inside the dielectric resonators. This will introduce a

unique paradigm for the development of functional metamaterials.

2.3 Dielectric Disks: Electric and Magnetic Dipole

Creation

In this section, we introduce the concept of electric and magnetic dipole moments,

and address their potential applications for metamaterial realization. To begin, let us

consider the five-layer PBG structure depicted in the previous section and modify the

geometry by considering finite size disks with thickness L=0.5 cm. The geometry is

shown in Fig. 2.6(a). The FDTD is applied to characterize the structure and obtain

the transmission coefficient. The result is plotted in Fig. 2.6(b). No band-gap region

is observed. Now, we increase the permittivity of dielectric disks to εr = 60 such

that a stopband performance in the frequency range of 4.75 < f(GHz) < 5.10 can be

determined. Interesting enough, that even one layer of this design can also provide

the band-gap phenomenon around the same center frequency (f = 4.94GHz), having

of course a narrower bandwidth, as shown in Fig. 2.6(c).

16

(a)

4.5 5.0 5.5 6.0 6.5Frequency (GHz)

−40

−35

−30

−25

−20

−15

−10

−5

0

5

Mag

nitu

de o

f Tra

nsm

issi

on C

oeffi

cien

t (dB

)

εr=10.2εr=60

(b)

4.5 5.0 5.5 6.0 6.5Frequency (GHz)

−40

−35

−30

−25

−20

−15

−10

−5

0

Mag

nitu

de o

f Tra

nsm

issi

on C

oeffi

cien

t (dB

)

(c)

Figure 2.6: Array of all-dielectric disks: (a) the geometry (Λx = Λy = Λz = 1.5cm),and transmission coefficients for (b) five-layer structure, and (c) one-layer structure.

17

To provide a physical understanding of this phenomenon, the electric and magnetic

field patterns inside one unit cell of the one-layer disks array at f1 = 4.94GHz are

plotted in Fig. 2.7 (snapshot in time). One can observe that the near-field patterns

of the dielectric disk are very similar to those of a magnetic dipole oriented along

the y direction. The near-field patterns at the second and third resonant frequencies

f2 = 5.97GHz and f3 = 6.08GHz are also plotted in Figs. 2.8 and 2.9, respectively.

The disk at the second resonant frequency is almost equivalent to an electric dipole

located along the z axis. The third mode has a resonant frequency very close to the

second mode, and its magnetic field pattern in the equatorial plane exhibits an octupole

characteristic, consisting of two linear quadrupoles rotated by 90 with respect to each

other. Higher order resonant modes can also be generated by the dielectric disks

utilizing the mutipole modes.

Because of the very large permittivity material of the dielectric disk, one can con-

sider the structure as a resonator where most of the fields are localized inside the

medium. Kejfez et al. have performed a comprehensive study of dielectric resonators

in Ref. [42], and clearly illustrated the potential of dielectric cylindrical resonators for

providing electric and magnetic dipole moments. Semouchkina et al. have also noticed

the differences between the field patterns of infinite rods PBG and finite-size cylin-

ders [43]. Peng et al. have also recently illustrated the electric and magnetic mode

development inside the very high permittivity rods [44]. Considering the polarization

of the plane wave excitation, the three resonant frequencies obtained in Fig. 2.6(c)

can be attributed to HEM11, TM01, and HEM21 resonant modes, respectively [42].

The near-field patterns for an isolated finite-size cylinder for the above resonant modes

have been plotted in Ref. [42], and they closely resemble what has been demonstrated

here for the periodic array of the disks. Hence, the stop-band regions in Fig. 2.6(c)

are derived from the resonant modes of the isolated disks, and thus even one layer of

the structure can provide the band-gap property of interest.

The HEM11 mode is sometimes called unconfined mode, because in the limit, as

18

Figure 2.7: Field distributions inside one unit cell of the one-layer disks array atf1 = 4.94GHz (HEM11δ mode): (a) E in the x-z plane, and (b) H in the y-x plane.Near fields are similar to those of a magnetic dipole oriented along the y direction.

Figure 2.8: Field distributions inside one unit cell of the one-layer disks array atf1 = 5.97GHz (TM01δ mode): (a) E in the y-z plane, and (b) H in the y-x plane. Nearfields are similar to those of an electric dipole oriented along the z direction.

19

Figure 2.9: Field distributions inside one unit cell of the one-layer disks array atf3 = 6.08GHz (HEM21δoctupole mode): (a) E, and (b) H in the y-x plane.

εr → ∞, its magnetic field does not vanish on the surfaces of the cavity resonator.

This can be revealed from Fig. 2.7, where the magnetic field is normal to the magnetic

wall boundary of the cavity and cannot be zero in the limiting case. In contrast, the

TM01 mode is of the confined type, since its magnetic field is tangent to the boundary

of the cavity, and in the limit, as εr → ∞, it must be zero along the surface (see

Fig. 2.8). The mode confinement behavior can also be readily seen by looking at the

transmission coefficient plot in Fig. 2.6(c) , where the HEM11 mode (magnetic dipole)

presents a lower Q than the TM01 mode (electric dipole). The octupole performance

of the HEM21 mode represents an inefficient radiator and consequently, its Q factor

is very large. It is worth noting that although each of the disk resonators individually

has some radiation loss, when we arrange them in the periodic fashion, the couplings

between them are increased and the radiation loss is considerably suppressed.

Tailoring the dielectric disks allows one to successfully control the physical perfor-

mance of the design. For example, as mentioned earlier, the resonant frequency of the

HEM21 mode is very close to the electric dipole mode TM01, and if the TM01 mode

is the desired mode of operation, the HEM21 mode may create an undesirable nearby

resonance effect, and one might be interested in suppressing it. This can be simply

20

accomplished by placing a thin wire loop on the end face of the disk resonator where

the electric field has a strong component, or, for instance, since the TM01 mode has a

relatively strong electric field along the axis of rotation, it is possible to tune this mode

by removing the cylindrical center section (leaving a doughnut shape) and replacing

it by a movable dielectric rod.

In summary, the important conclusion of this section is the fact that dielectric

resonators can successfully provide electric and magnetic dipole modes. The dipole

moments can be considered as the alphabet for making metamaterials. For instance,

using an array structure of the magnetic dipole disks (one layer) one can effectively

provide a band-gap medium. The major advantage compared to the PBG concept

is that the unit-cell interaction along the propagation directions is not required for

achieving the functionality of interest. Basically, each of the disks itself provides the

required resonant behavior. In general, by tailoring the electric and magnetic dipole

moments in one unit cell one can make a building-block cell with the figure of merit

of interest. Then, by making a material from these small-size cells, one can claim a

metamaterial design with the homogeneous effective constitutive parameters εeff , µeff .

This will be described in more detail in the next section.

2.4 Metamaterial Realization

The materials presented in the previous section are very helpful in providing a phys-

ical understanding of the dipole modes generation utilizing dielectric resonators. In

this section, we apply this concept to design spherical particle-based metamaterials.

Fig. 2.10(a) shows a periodic array of dielectric spheres having high permittivity εp

embedded inside the nonmagnetic host matrix εh. The structure has an isotropic unit

cell. Using Mie theory, one can express the EM waves of each sphere as an infinite series

of spherical vector functions Mn and Nn. Applying the field transformation between

the nonconcentric spheres, and using the boundary conditions, the array of spheres

21

can be solved analytically [29, 45]. It is assumed that the size of the spheres is compa-

rable to their material wavelength, and small in terms of host material wavelength, so

that the effective material parameters can be accurately defined for the structure. As

demonstrated earlier, the dielectric resonators can offer electric and magnetic dipole

moments, and higher order modes. Indeed, from the Mie series, it clears that the dom-

inant modes (n=1) are TE (magnetic dipole) and TM (electric dipole) waves. Around

the eigenfrequencies of these modes one can assume the existence of only the electric

and magnetic modes and obtain the effective material parameters εeff , µeff for the

periodic spheres as [27]

εeff = εh

(1 +

3νf

εpF (θ)+2εh

εpF (θ)−εh− νf

), (2.1a)

µeff = µ0

(1 +

3νf

F (θ)+2F (θ)−1

− νf

), (2.1b)

where νf is volume fraction of the spheres, and function F (θ) is

F (θ) =2(sin θ − θ cos θ)

(θ2 − 1) sin θ + θ cos θ, (2.2)

with

θ = k0r√

εp,r, (2.3)

where r is the radius of spheres. It is interesting to emphasize that the nonmagnetic

spheres can create magnetism due to the magnetic dipole polarization.

The effective constitutive parameters of the periodic spheres depicted in Fig. 2.10(a),

having dielectric constant εp,r = 40, radius r = 0.5cm, and unit-cell size Λx = 1.5cm,

Λy = Λz = 2.5cm, are plotted in Fig. 2.10(b). The first resonant frequency at

fm = 4.72GHz is associated with the magnetic mode and the second resonance at

fe = 6.61GHz represents the electric mode. As described earlier, the magnetic mode

is an unconfined mode and provides a wider bandwidth. This can be seen from

22

Figure 2.10: Array of one-layer all-dielectric spheres: (a) the geometry (Λy = Λz =2.5cm), and (b) its effective constitutive parameters.

Fig. 2.10(b) and Eq. (2.1), where one can find a larger bandwidth for the TE res-

onance in comparison to the TM resonance by a factor of about εp/εh. It is worth

noting that above the resonant frequencies of magnetic and electric modes, negative

permeability and negative permittivity materials are established, respectively. This

will be used later in this section for the metamaterial realization of DNG behavior.

The FDTD is applied to characterize the structure and obtain the transmission

coefficient for a plane wave propagating through the medium (one layer along x). The

result is shown in Fig. 2.11. Comparing Fig. 2.10(b) with Fig. 2.11, one can observe

that the analytical formulations (2.1) closely estimate the first two resonant frequencies

determined through the FDTD full wave analysis (less than 1% error). However, as

expected, the third resonant frequency at f =6.73 GHz cannot be predicted based on

Eq. (2.1). In practice, the third resonant frequency can set an upper limit on the

frequency band of the second mode where the effective permittivity is defined. It is

interesting to note that the transmission coefficient behavior of the dielectric spheres is

very similar to that of the dielectric disks see Fig. 2.6(c). The near-field distributions

are plotted in Fig. 2.12 for the first two resonant frequencies (fm = 4.73GHz and

fe = 6.55GHz) and clearly validate the existence of magnetic and electric dipole

23

polarizations.

4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5Frequency (GHz)

−40

−35

−30

−25

−20

−15

−10

−5

0

Mag

nitu

de o

f Tra

nsm

issi

on C

oeffi

cien

t (dB

)

Figure 2.11: Transmission coefficient for the all-dielectric spheres depicted inFig. 2.10(a). The first and second resonances represent magnetic and electric reso-nant modes, respectively.

So far, we have described how one can successfully realize a metamaterial with both

electric and magnetic parameters utilizing only-dielectric resonators, fulfilling desired

effective constitutive parameters. The next step is to investigate the possibility of

increasing the bandwidth of the resonant modes. But, first let us clear one issue.

Consider, for instance, the magnetic resonant mode of the one-layer periodic spheres

[Fig. 2.10(a)], having -10 dB bandwidth of about BW = 1.2%. It is well understood

that each of the cavity resonators can be considered as a parallel LC circuit. Cascad-

ing the LC resonant circuits can increase the transmission coefficient bandwidth. In

fact, increasing the number of layers (parallel LC circuits) increases the transmission

coefficient bandwidth. However, it should be noticed that this is nothing to do with

the bandwidth of the metamaterial. The performance of three layers of the spheres

designed in Fig. 2.10(a) is shown in Fig. 2.13. The transmission bandwidth is increased

from 1.2% to about 4.6%; but, both one-layer and three-layer structures have almost

the same µeff given by Eq. (1b), and of course the similar permeability bandwidth.

Increasing the number of layers will simply increase the thickness of the structure.

In this work, a very unique approach for the bandwidth enhancement of metama-

24

Figure 2.12: Field distributions inside one unit cell of the spheres array: (a) E inthe x-z plane and H in the y-x plane at fm = 4.73GHz, representing the magneticdipole moment, and (b) E in the y-z plane and H in the y-x plane at fe = 6.55GHz,representing the electric dipole moment (1.5cm × 1.5cm of the unit cell in the y-zdirections is plotted).

(a)

4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5Frequency (GHz)

−40

−35

−30

−25

−20

−15

−10

−5

0

Mag

nitu

de o

f Tra

nsm

issi

on C

oeffi

cien

t (dB

)

(b)

Figure 2.13: Array of three-layer dielectric spheres (Λx = 1.5cm): (a) the geometry,and (b) its transmission coefficient.

25

terials is presented. Recently, Mosallaei et al. demonstrated how the bandwidth of

the negative permeability medium realized utilizing metallic embedded-loop circuits

can be improved by increasing the couplings between the loop elements [34]. In fact,

based on their circuit model analogy it is shown that the bandwidth of the negative

permeability medium depends strongly on the coupling coefficient κ between the loops,

and can be estimated from the following equation:

∆ω

ωp

=1√

1− κ2− 1, (2.4)

where ωp is the resonant frequency of the loops, and κ < 1. The higher the coupling

coefficient κ the larger the bandwidth. This concept is applied here to the all-dielectric

metamaterial design. Basically, we increase the couplings between the spheres shown

in Fig. 2.10(a), by bringing them closer to each other along the z direction, namely,

assuming Λz = 1.5cm [Fig. 2.14(a)]. Transmission coefficient for the magnetic mode

is plotted in Fig. 2.14(b) illustrating a bandwidth enhancement of more than 100%

compared to the original design Λz = 2.5cm. An almost similar observation for the

electric mode resonance is illustrated in Fig. 2.14(c) (bandwidth is increased from 0.5%

to 1.3%). Basically, when we make the spheres closer to each other, the mode radiation

through the spheres is increased causing the reduction in the Q factor of each of the

spheres, resulting in the bandwidth enhancement of the resonant modes. Slight shifts

in the resonant frequencies due to the coupling effects are also noted.

We will now investigate the development of double negative metamaterials using

dielectric resonators. As highlighted earlier, and can be seen from Fig. 2.10(b), the

periodic array of dielectric spheres can generate both negative effective permeability

and permittivity, however, at different resonant frequencies (fm = 4.72GHz, fe =

6.61GHz). To obtain a DNG behavior around the same resonant frequency, a building-

block unit cell constructed from two spheres having the same size but different dielectric

constants εp1,r = 40 and εp2,r = 23.8, is optimized in Fig. 2.15(a). The set of spheres

26

(a)

4.6 4.7 4.8 4.9 5.0Frequency (GHz)

−40

−35

−30

−25

−20

−15

−10

−5

0

Mag

nitu

de o

f Tra

nsm

issi

on C

oeffi

cien

t (dB

)

Λz=1.5 cmΛz=2.5 cm

(b)

6.3 6.4 6.5 6.6 6.7Frequency (GHz)

−40

−35

−30

−25

−20

−15

−10

−5

0

Mag

nitu

de o

f Tra

nsm

issi

on C

oeffi

cien

t (dB

)

Λz=1.5 cmΛz=2.5 cm

(c)

Figure 2.14: Bandwidth enhancement of metamaterial by increasing couplings betweenthe elements smaller unit-cell size: (a) the geometry, (b) transmission coefficient atthe magnetic resonance, and (c) transmission coefficient at the electric resonance. Themore the couplings the wider the bandwidth.

27

(a)

6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9Frequency (GHz)

−40

−35

−30

−25

−20

−15

−10

−5

0

Mag

nitu

de o

f Tra

nsm

issi

on C

oeffi

cien

t (dB

)

εr=23.8εr=40Double−sphere Struc.Loss Tangent=.001

(b)

Figure 2.15: DNG metamaterial constructed from all-dielectric spheres: (a) the geom-etry (Λy = 2.5cm, Λz = 1.5cm), and its equivalent circuit model, and (b) transmissioncoefficient.

with εp1,r = 40 creates negative effective permittivity about fe = 6.50 GHz, and the

set of spheres with εp2,r = 23.8 generates negative effective permeability about fm =

6.29GHz. Fig. 2.15(b) presents transmission coefficients of both sets, where stop-band

regions are determined in the negative material frequency ranges. It must be mentioned

that in the constructed lattice of both spheres [Fig. 2.15(a)], the electric mode has a

higher Q compared to the magnetic mode, and hence, the coupling effect of the sphere

with dielectric εp2,r = 23.8 on the electric resonance should be larger than that of

the sphere with dielectric εp1,r = 40 on the magnetic resonance. This phenomenon

is carefully explained from another point of view in Ref. [30]; as it is discussed the

electric polarizability of the dielectric sphere operating in the magnetic resonance has

an influence on the electric mode sphere, causing the electric resonance of the double-

sphere lattice to be slightly lower than that of the single-sphere lattice (less than 1

shift). The magnetic resonance stays almost the same (nonmagnetic spheres). Thus,

in Fig. 2.15(b), the electric resonance of the single-sphere lattice should be slightly

shifted down to envision the negative permittivity region of the doublesphere lattice.

28

Considering this, a region with both negative ε and µ is accomplished. Transmission

coefficient for the double-sphere unit cell is shown in Fig. 2.15(b), demonstrating an

almost total transmission in the DNG region, around f=6.42 GHz. The phase of

the field distribution at f=6.42 GHz inside one layer of the metamaterial is shown in

Fig. 2.16. The positive slope for the phase in the central region of the layer clears the

establishment of the DNG medium (backward wave). The electric and magnetic field

intensities inside the unit cell at this frequency are also shown in Fig. 2.17. One can

clearly observe the development of electric and magnetic dipole modes that provide

the required effective material parameters. This also validates the existence of the

dipolar modes assumption, made in the derivation of Eq. (2.1). The effect of the loss

is also studied, by considering spheres with a dielectric loss tangent of tanδ = 0.001.

The result is plotted in Fig. 2.15(b), illustrating less than -1 dB transmission loss in

the DNG region. Utilizing dielectric materials with better loss tangents can of course

provide a higher efficiency.

The same concept can be used to design a DNG metamaterial realized utilizing

dielectric disks, which might be easier for fabrication in some cases. The geometry is

depicted in Fig. 2.18(a). Transmission coefficients and field patterns at f =5.97 GHz

are evaluated in Figs. 2.18(b) and 2.19. Similar observations as the spherical particles

are accomplished.

2.5 Optical Metamaterials

Realization of metamaterials at terahertz frequencies is also of great interest due to the

possibility of designing novel nanoscale devices in the infrared and visible regimes [46–

51]. The concept of all-dielectric metamaterials can be extended to the optical fre-

quencies; however, because of the fabrication limitations one needs to use smaller value

dielectric materials for the resonating inclusions. In this case, larger-size resonators

may be implemented. Fig. 2.20(a) depicts an array of gallium phosphide (GaP) spheres

29

Figure 2.16: Phase distribution of the electric field Ez inside the layer of DNG meta-material [Fig. 2.15(a)] at f = 6.42GHz. The plane wave propagates from left to theright where the phase is increased in this direction. The positive slope for the phasein the central part of the layer is a demonstration of the backward wave generation.

with permittivity 12.25 and a dielectric loss tangent of tanδ = 0.001. The diameter

of spheres is 170 nm. Transmission coefficient performance is shown in Fig. 2.20(b),

where the development of magnetic and electric resonant modes can be observed. One

must notice that because of the low dielectric material of the spheres and their rela-

tively large physical size the couplings between the resonators are increased. This will

generate some difficulty in tuning the DNG medium if two sets of spheres are used.

Although the existing coupling may not be desirable from the fact that the electric

and magnetic resonances are coupled, it can be beneficial from the point that one can

successfully tailor a backward wave using the strong interaction between the spheres.

Work is currently under progress in this direction.

Alternative approaches will be to embed one set of dielectric spheres inside a plas-

monic host medium as obtained by Seo et al. [31]; or to use Drude material coated

spheres as proposed by Wheeler [51]. Here, we investigate the former method by

characterizing the performance of the periodic array of GaP spheres implanted inside

cesium (Cs) host material with a measured plasma wavelength λp = 0.41µm and a

damping constant γ of 51× 1012 [31], shown in Fig. 2.21(a). Note that if one operates

close to the plasma frequency, the index of host material is small, and physically large-

30

(a) (b)

(c)

Figure 2.17: Field distributions inside one unit cell of the DNG metamaterial[Fig. 2.15(a)] at f = 6.42GHz: (a) E in the y-z plane, (b) H in the y-x plane, and (c)E in the x-z plane. Note the creation of electric and magnetic dipole moments insidethe unit cell of the spheres of εr = 40 and εr = 23.8.

31

(a)

5.8 5.9 6.0 6.1 6.2Frequency (GHz)

−40

−35

−30

−25

−20

−15

−10

−5

0

Mag

nitu

de o

f Tra

nsm

issi

on C

oeffi

cien

t (dB

)

εr=43εr=60Double−disc Struc.Loss Tangent=.001

(b)

Figure 2.18: DNG metamaterial constructed from all-dielectric disks: (a) the geometry(Λy = 2.5cm, Λz = 1.5cm), and (b) its transmission coefficient.

Figure 2.19: Field distributions inside one unit cell of the DNG metamaterial[Fig. 2.21(a)] at f = 5.97GHz: (a) E in the y-z plane, and (b) H in the y-x plane.Note the creation of electric and magnetic dipole moments inside the unit cell of thedisks of εr = 60 and εr = 43.

32

(a)

450.0 500.0 550.0 600.0 650.0 700.0 750.0 800.0Frequency (THz)

−40

−35

−30

−25

−20

−15

−10

−5

0

Mag

nitu

de o

f Tra

nsm

issi

on C

oeffi

cien

t (dB

)

(b)

Figure 2.20: Metamaterial nanostructured spheres: (a) the geometry Λy = Λz =250nm), and (b) its transmission coefficient. Note the generation of magnetic andelectric resonances.

size spheres are still electrically small in comparison to the host wavelength. Trans-

mission coefficient of the composite structure is shown in Fig. 2.21(b). The spheres

operate at their magnetic resonant mode and can provide negative effective permeabil-

ity see Fig. 2.21(b). A combination of this with the negative permittivity of cesium

below its plasma resonance offers DNG behavior. In comparison to the double-sphere

resonators design, here only one set of resonators is involved and a wider bandwidth

can be expected. In addition, the spheres operate in their magnetic mode frequency

range, which inherently offers a lower Q than the electric mode. Transmission loss

for this case is about -1.1 dB. The magnetic field pattern at f =529 THz is shown in

Fig. 2.21(c).

33

(a)

475.0 500.0 525.0 550.0 575.0 600.0Frequency (THz)

−40

−35

−30

−25

−20

−15

−10

−5

0

Mag

nitu

de o

f Tra

nsm

issi

on C

oeffi

cien

t (dB

)

Single−sphere Struct.Plasmonic HostDNG Material

(b) (c)

Figure 2.21: DNG optical metamaterial constructed from nanostructured dielectricspheres (operating in magnetic mode) embedded in negative permittivity host: (a) thegeometry, (b) transmission coefficient, and (c) H field in the y-z plane at f = 529THz.

34

2.6 Dispersion Diagram Characteristics of Periodic

Array of Dielectric Spheres

Dispersion diagrams are a useful approach to describe the modal behavior of electro-

magnetic structures. The bandgaps are typically visualized and investigated by com-

puting the dispersion relationship, ω(k), between the temporal and spatial frequencies

of the modes that can propagate in the particular periodic structure of interest. In this

thesis we applied the FDTD numerical method to calculate the dispersion diagrams.

One can find the details about dispersion diagrams and our approach to obtain them

in Appendix A.

In this section, we apply the dispersion diagram modeling tool to characterize the

performance of array of dielectric spheres and explore the development of dielectric

metamaterials. As mentioned earlier, to obtain the backward wave and DNG behav-

iors, appropriate electric and magnetic dipole moments should be created in building-

block unitcells of array configurations. In order to design such a structure, we use a

3D array of two different spheres as a unit-cell; where the spheres have the same sizes

and different materials. The challenge is to establish both the electric and magnetic

dipole resonances around the same frequency band.

Fig. 2.22 shows the geometry and the parameters of the array of spheres. The

performance of the first set described with εr = 40 is plotted in Fig. 2.22(b). In

the dispersion diagram two stop bands (gaps) are observed. The first stop band is

associated with the magnetic resonance where the second band is associated with the

electric resonance. Notice that, the first gap has a wider bandwidth in compared to

the second one. For the other set of sphere with the same geometry but εr = 21, only

the first band gap (magnetic resonance) is shown in the frequency spectrum of interest

in Fig. 2.22(c). As seen by comparing these two diagrams, the electric gap of the first

set and the magnetic gap of the other set are around the same frequency region.

35

Fig. 2.23 shows the dispersion diagram for the two-sets of 3D dielectric spheres.

By combining the two-sets, or basically by uniting the electric and magnetic dipole

modes, a negative slope backward wave behavior is achieved.

(a)

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

β Λx

k Λ

x

(b)

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

β Λx

k Λ

x

(c)

Figure 2.22: (a) The geometry of a 3D array of spheres: Λy/a = Λz/a = 5 andΛx/a = 3. Dispersion diagram for one-set of dielectric spheres with permittivity: (b)ε = 40 and, (c) ε = 21.

2.7 Conclusions

In this study, a comprehensive investigation of all-dielectric metamaterials is addressed.

The FDTD full wave analysis is applied to characterize the interactions of EM-optical

waves with the periodic array of metamaterials, and tailor required designs. Electric

36

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

β Λx

k Λ

x

Figure 2.23: Dispersion diagram for a DNG metamaterial constructed from two-setsof dielectric spheres with permittivities 40 and 21. Λy/a = Λz/a = 5 and Λx/a = 3.

and magnetic near-field patterns are established to highlight the physical insights.

Photonic crystals are searched first, and it is described that the band-gap phe-

nomenon appears as a result of the periodic dielectric contrasts along the propagation

direction. Then, the concept of electric and magnetic dipole modes generation for

metamaterial development is presented. Dielectric disk and spherical particle res-

onators are implemented to create required dipole moments. Arrangements of electric

and magnetic dipole moments in one unit cell tailor the metamaterial to the applica-

tion of interest. The beauty of the developed metamaterial is that even one unit cell of

the structure can provide the figure of merit of interest, and the interactions between

the cells is not essential, as was the case for the PBG. This will allow for making the

small-size unit cell, and defining the effective constitutive parameters accurately. Fur-

ther, a random arrangement might have the potential of offering the similar desired

properties, especially in the spherical resonator case where the unit cell is isotropic.

The physics of electric and magnetic dipole moments are explored. It is shown that

the magnetic resonance is an unconfined mode and has a lower Q than the confined

electric resonant mode. A unique approach is proposed to increase the bandwidth of

the resonant particles. Basically, by bringing the dielectric resonators closer to each

other, the radiation couplings between them are increased, resulting in lowering the

37

Q of each of the resonators. This will establish bandwidth enhancement. Dielectric

metamaterials are free of conduction loss and can provide high efficiency performance.

A DNG metamaterial constructed from two sets of spheres, having the same size

but different materials, is developed. One set of spheres provides negative permittiv-

ity, and the other set offers negative permeability, accomplishing the double negative

metamaterial. One can also design a double-sphere lattice DNG metamaterial using

the spheres of different sizes but having the same materials. Development of negative

index media at terahertz frequencies using plasmonic materials is also studied.

In general, all-dielectric metamaterials appear very promising for addressing some

of the important physical and engineering concerns, such as the loss and bandwidth.

They are quite feasible for fabrication in both microwave and IR-visible spectrums.

38

Chapter 3

Near-Field Focusing

3.1 Introduction

Evanescent waves carry subwavelength information of an object. Amplifying these

modes and contributing them into the image plane has been a challenging task in recent

years. Based on Veselago’s work [6], Pendry in [2] showed how a lossless negative index

(NI) slab can realize a superlens to focus all the Fourier components of a source. Later

on, a series of research started to study the different aspects of this topic and found

out other possible ways to amplify the evanescent waves [9–18]. Basically amplifying

the evanescent waves and tailoring the phase of propagating waves are two important

features which occur in imaging. To figure out the effect of each factor, one can study

the Fourier spectrum analysis to follow the behavior of waves in propagating and

evanescent regions. Other important issues in a realistic negative index material (NIM)

are the effects of material frequency dispersion and the loss that can considerably

degrade the performance. Then, the main engineering concern will be how far from

an object one can reconstruct the image with a high-resolution feature.

The goal of the present work is to provide an engineering investigation of imaging

performance of a NIM slab and compare its performance with coupled layered struc-

tures functioning based on the surface waves amplification. As known, a slab of NIM

39

can only manipulate the image in the transverse plane and a depth reconstruction

cannot be achieved. Further, for transverse pattern resolution, one needs to be close

to the surface to achieve better resolution of the object. Making a bulk material of

NIM with negative permittivity and negative permeability effective materials is also a

real challenge. Besides these, due to relatively large thickness of the slab, material loss

will considerably degrade the image performance of the structure. Therefore, in spite

of many interesting phenomena of NIM, not many practical applications for them have

been realized.

Basically, as amplification of evanescent waves and carrying them to the image

plane is a key to high-resolution imaging, coupled plasmon surface modes concept can

be a great alternative for realizing high-resolution near-field imaging. Decaying fields

can efficiently launch the surface waves along epsilon negative (ENG) or mu negative

(MNG) surfaces. Basically, Pendry in [2] showed how a thin ENG slab can be used for

near field imaging and later Ramakrishna et al discussed in [19–21] how the idea can

be extended to layered structures where the coupling between the surface-modes layers

determines a farther distance recognition. This canalization through the layers was

also studied by P.A. Belov et al in [52] and X. Li et al in [53]. ENG plasmonic surfaces

will respond only to p-polarized waves. To achieve a high-resolution imaging device

functioning properly for both p and s waves, one needs to integrate the ENG layers

with MNG interfaces. The resonance and tunneling performance of pairing an ENG

slab with a MNG slab has been studied by Alu et al in [47, 54] where they define a

conjugate-matched pair to guarantee zero-reflection and total-transmission conditions

for any plane wave impinging on the pair.

The objectives of this chapter are to investigate near field imaging performance of

coupled multi-layered structures of ENG and MNG surfaces, and to provide a com-

parative study with the NIM slab imaging behavior. Theoretical formulations and

dispersion diagram analysis are performed to comprehensively investigate the concept

and demonstrate unique characteristics. An advanced Finite Difference Time Domain

40

(FDTD) technique [32, 34] is also applied to characterize the finite-size composite

ENG-MNG layered validating our theoretical illustrations. The fields inside the layers

and in the transverse planes are obtained carefully. It is demonstrated that for the

layered structure since the thicknesses of the layers are relatively thin, the structure

is not very sensitive to the material loss as is experienced for the NIM configuration.

The construction of an ENG-MNG layered structure can be simpler than a NIM bulky

material, as one can use novel metallic patterns to realize it.

3.2 Theory and Formulation of Layered Structures

To characterize theoretically the performance of a layered structure, a Fourier spectrum

analysis is applied. Let us consider an arbitrary electric dipole oriented along the u

direction (with angle α with respect to z-axis), located at distance d1 in front of an

N-layer structure stacked along the z direction, as depicted in Fig. 3.1. Using the

dyadic Green’s function approach the field due to a Hertzian dipole is given by

E(r) =iωµ(I +∇∇k2

).αIleikr

4πr(3.1a)

H(r) =∇× αIleikr

4πr(3.1b)

where Il is the current moment and k = ω√

µε. When a point source is located next

to a layered medium, it is then best to decompose the field in terms of waves of TM

type and TE type. By expanding the field into plane waves with the Sommerfeld

identity [55, 56],

eik0r

r=

i

2

∫ ∞

−∞dkρ

kρ

kz

H(1)0 (kρρ) eikz |z| (3.2)

the field for the vertical electric dipole (the normal component of the excitation) is

characterized by [56]

41

Figure 3.1: The configuration of layered medium

Eiz =−Il

8πωεi

cos α

∫ ∞

−∞dkρ

k3ρ

k1z

H(1)0 (kρρ) Ai

[eikiz |z| + RTM

i,i+1eikiz(z+2di)

](3.3a)

Hiz =0 (3.3b)

and for the tangential component the field is

Eiz =iIl

8πωεi

sin α cos φ

∫ ∞

−∞dkρk

2ρH

(1)1 (kρρ) Ai

[±eikiz |z| − RTM

i,i+1eikiz(z+2di)

](3.4a)

Hiz =iIl

8πsin α sin φ

∫ ∞

−∞dkρ

k2ρ

k1z

H(1)1 (kρρ) Ai

[eikiz |z| + RTE

i,i+1eikiz(z+2di)

](3.4b)

where kz =√

k20 − k2

ρ for propagating components and kz = i√

k2ρ − k2

0 for evanes-

cent waves which decay exponentially with the z direction. RTMi,i+1 and RTE

i,i+1 are the

reflection coefficients for TM and TE modes, respectively and i stands for the layer

number. In general, for an N-layer medium, the generalized reflection coefficient at

the interface between region i and i + 1 called Ri,i+1, can be obtained as [56],

Ri,i+1 =Ri,i+1 + Ri+1,i+2e

2iki+1,z(di+1−di)

1 + Ri,i+1Ri+1,i+2e2iki+1,z(di+1−di). (3.5)

42

where Ri,i+1 is the Fresnel reflection coefficient for TE and TM modes as,

RTEi,i+1 =

µi+1kiz − µik(i+1)z

µi+1kiz + µik(i+1)z

(3.6a)

RTMi,i+1 =

εi+1kiz − εik(i+1)z

εi+1kiz + εik(i+1)z

(3.6b)

The coefficients Ais are also obtained by applying the boundary conditions through

a recursive process. Then the field can be represented in terms of Fourier spectrum such

that at any plane it is the summation of the propagating and evanescent components

as

Ez =

∫

k2ρ≤k2

0

EPropag.z (kρ, kz)dkρ +

∫

k2ρ>k2

0

EEvan.z (kρ, kz)dkρ (3.7)

Equations (3.3) and (3.4) will provide detailed information about the surface wave

propagation in coupled positive-negative layered structures and their poles manipula-

tion. It will be used in the following sections to successfully tailor imaging character-

istics.

3.3 Negative Index Material Slab

In this section we will review the performance of a NIM slab in propagation and

transverse planes. This will be accomplished with the use of equations derived in the

previous section. The obtained results will then be compared in next section with the

performance of coupled layered surfaces.

The slab material considered here is isotropic and its permittivity and permeability

parameters are assumed to be Drude and Lorentzian models, respectively, given as

below

ε(ω) = ε0

(1− ω2

p

ω(ω + iγp)

)(3.8a)

µ(ω) = µ0

(1− .5

ω2

ω2 − 3/8ω2p + i2γpω

)(3.8b)

43

where ωp is the bulk resonant frequency of the material, and damping factor γp repre-

sents the losses present.

First, we will study a loss-free NIM slab (γp = 0) with thickness of 3d = .3λp

and introducing two surfaces at z = −d1 and −d2 as shown in Fig. 3.2. At operating

frequency ω20 = 0.5ω2

p, the material characteristic parameters are ε = µ = −1; hence

the reflections from the boundaries will be zero and

A2

A1

= ei2k1zd1 ,A3

A1

= e−i2k1z(d2−d1) (3.9)

By substituting Eq. (3.9) into Eq. (3.3) and Eq. (3.4), it can be seen that the

field distribution at |z| = 2d1 and |z| = 2(d2 − d1) will be the same as the field at

source plane (z=0), which means that the source after a double-focusing process will

be reconstructed. Another interesting point which must be highlighted is that in the

evanescent region, the field inside the slab grows exponentially towards the second

surface and then decays after exiting the slab. In Fig. 3.3 these observations are

examined. Fig. 3.3(a) shows the field along the slab axis. To show the performance of

the field in the whole view, the maximum value of kρ in the integration is considered

to be |kmax| = 3k0 and it is obvious that the larger kmax will provide the same behavior

but stronger field amplification at the second surface. The amplification of evanescent

modes introduces the exponentially increase of the field inside the slab. It is worth to

note that the fields at two focusing points |z| = 2d1 = .2λp and |z| = 2(d2−d1) = .6λp

are the same as the field at the source point. In another view, Fig. 3.3(b) shows the

field along the x-direction (transverse direction) at the source and two image planes,

indicating that the field distributions at these planes are exactly the same. In loss free

case, one should note that the decaying profile of the field after the slab will guide one

not to expect the peak-type image along the slab axis which means the details about

the target in depth cannot be detected although a perfect resolution along the surface

direction can be successfully obtained.

44

Figure 3.2: Source and negative slab metamaterial.

(a)

−0.5 0 0.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x (λ0)

Nor

mal

ized

|Dz|

z = 0 (source)z = −2 dz = −6 d

(b)

Figure 3.3: Field profile for the loss-less negative index slab along the (a) propagationdirection (Green shaded region represents the slab,) and (b) lateral direction at imageplane z = −.6λp. Note that the evanescent waves are amplified through the slab andthe fields at the imaging points are the same as the source point.

45

In reality because of causality, the left handed materials should be absorbing.

Therefore let us now assume that the slab has some absorption such that n = −1+ ini.

Then the reflection terms, RTM and RTE, in Eq. (3.3) and Eq. (3.4) will contribute

into the calculations. Let us consider a lossy NIM slab with thickness of 4.5d = .45λp

is illuminated by an electric dipole placed at distance d from the first surface (α =

π/6). Fig. 3.4(a) shows the electric field along the slab axis when the loss factor is

γp = .001ωp. In Fig. 3.4(b) the resolution of images of two dipoles (separated .2λ along

the x-direction) at different planes after the slab has been examined. Because of the

decaying profile of the waves in free space the resolution of image goes away by getting

distance from the slab.

To investigate the effect of the loss, it will be suitable to define a transfer function

for the slab in terms of kρ, given as below

T (kρ) =Ez(kρ, kz)|NIM−slab

z=−d2

Ez(kρ, kz)|free−spacez=−d1

. (3.10)

This is the ratio of the field right after the slab to the field at z = −d1 where there is no

slab. Fig. 3.5(a) shows the transfer function for the NIM slab with different material

losses. The slab with smaller loss transfers and amplifies more components in the

evanescent region that results in higher resolution for the image. We also demonstrate

in Fig. 3.5(b) the images of the electric dipole pair for different losses. As obtained,

one can establish higher resolution for smaller loss tangent, such that for damping

factor larger than γp = .01ωp the dipoles can not been separated in the image plane.

3.4 Coupled Surface-Modes Layers

Despite the simple theory for performing a NIM perfect lens, making a bulk material of

NIM with negative permittivity and negative permeability effective materials is a real

challenge. Besides this, due to relatively large thickness of the slab, material loss will

46

(a)

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.80

0.2

0.4

0.6

0.8

1

x (λ0)

No

rma

lize

d |D

z|

zi = −5.5d

zi = −6d

zi = −7d

zi = −9d

(b)

Figure 3.4: Field profile for the lossy negative index slab of thickness 4.5d = .45λp:(a) propagation direction (Green shaded region represents the slab), and (b) imageperformance at different image planes of a dipole pair separated by .2λ0 (d = .1λp).

0 5 10 15 2010

−15

10−10

10−5

100

105

kρ/k0

| Tra

nsf

er

Fu

nct

ion

|

γp = .001 ω

p

γp = .01 ω

p

γp = .1 ω

p

(a)

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x (λ0)

No

rma

lize

d |D

z|

γp = .001 ω

p

γp = .01 ω

p

γp = .1 ω

p

(b)

Figure 3.5: Effect of loss: (a) transfer function of the slab, and (b) image performanceat the plane z = −.6λp of a dipole pair separated by .2λ0. It can be seen that smallerloss provides higher resolution.

47

considerably degrade the image performance. On the other hand, as the transverse

imaging is the most promising feature of a NIM slab, one can achieve a better practical

subwavelength imaging device (for transverse plane) with the use of coupled layered

surfaces supporting surface waves on their positive-negative boundaries. The positive-

negative layered structures can support surface plasma oscillations (SPO) excited by

evanescent fields. In this section, we will investigate coupled surface wave layers and

compare their performance with the NIM slab.

3.4.1 Analysis of Multiple Thin Film Systems

In general, the presence of surfaces introduces new modes of plasma oscillations in

addition to the bulk mode with different properties and particularly with different

dispersion relations. These modes can be excited by incident electrons, or photons

and can be detected experimentally [57]. To follow the theory for a multiple film

system, we begin with Maxwell’s equations and wish to find solutions which satisfy

Maxwell’s equations with a local current-field relation as follows:

∇.D = 0, (3.11a)

∇.H = 0, (3.11b)

∇.E = −1

c

∂H

∂t, (3.11c)

∇.H =1

c

∂D

∂t, (3.11d)

D = ε(ω)E, (3.11e)

Boundary conditions of continuity of the tangential fields at every boundary should

be applied. Here we are looking for the type of solution corresponds to wave propa-

gation along a direction parallel to the boundary surfaces which separate the different

materials. Here we consider the z axis normal to these surfaces and the x axis is the

direction of wave propagation. Since the magnetic (or TE) waves are purely transverse

48

waves and of no interest to us, we restrict ourselves to the electric (or TM) waves. By

this assumption, there is no y dependence of any of the fields.

The solution for any component of the fields can be represented by the form,

f(x, z, t) = <F (x)ei(ωt−kz) (3.12)

with <k > 0 and =k < 0, so that the wave travels and is attenuated in the positive x

direction. A general solution to 3.11 is a linear combination of the two independent

solutions eKi,mz and e−Ki,mz, where

K2i,m = k2 − ω2εi,m/c2 (3.13)

First, let us examine a single metal-dielectric interface as shown in Fig. 3.6(a). The

solution for Ez is:

Ez = AIeKmz, z < 0 (3.14a)

Ez = AIIe−Kiz, z > 0 (3.14b)

Continuity of the fields across the boundary gives the dispersion relation as,

εiKm + εmKi = 0. (3.15)

The dispersion diagram for an insulator-metal interface when εi = 1, εm = 1−ω2p/ω

2

is plotted in Fig. 3.6(b).

Now, let us investigate the characteristics of a structure constituted from two thin

layers each of them made of a negative permittivity medium. In optics, materials with

negative ε in a limited frequency range are common. The condition for the existence of

a surface plasmon (for p-polarized incident wave) at the surfaces of a layered structure

composed of two ENG layers with thickness dm and separated by d0 in free space (as

49

(a)

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1

k / kp

ω /

ωp

(b)

Figure 3.6: Single metal-dielectric interface (εi = 1, εm = 1 − ω2p/ω

2): (a) geometry,and (b) dispersion diagram performance.

shown in Fig. 3.7(a)), can be obtained by the continuity of these modes across the

boundaries as [57]

1

R4− C

1

R2+ e−4Kmdm = 0 (3.16)

where

R =εmK0 − ε0Km

εmK0 + ε0Km

(3.17a)

C = 2e−2Kmdm + e−2K0d0(1− e−2Kmdm)2 (3.17b)

K2m,0 = k2 − ω2εm,0/c

2 (3.17c)

The dispersion diagram for a structure with dm = d0 = d/2 where d = .1λp is

plotted in Fig. 3.7(b) and demonstrates forward and backward wave branches. To

obtain a better understanding about the performance of the layered structure, the

transmission coefficient T (as defined in Eq. (3.10)) is plotted in Fig. 3.8(a), resembling

a low-pass filter for k-vectors. This can result in field amlification through the layers as

demonstrated in Fig. 3.8(b). This is a consequence of coupling between the positive-

negative surfaces (see Fig. 3.8(b)).

50

The separation between the layers obviously plays an important role to establish

the proper coupling-performance. Hence, one needs to use an optimized separation

between them. This is explored in further details for the transmission coefficient, T,

in Fig. 3.9. As observed, the optimized thickness is d0 = di = .05λp where the effective

ε in transverse plane tends to zero and a smooth transfer function amplifying more k

vectors has been observed.

(a)

0 2 4 6 8 10 12 14 16 180

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

k / kp

ω /

ωp

(b)

Figure 3.7: Two-layer ENG coupled surfaces: (a) geometry, and (b) dispersion diagramperformance when dm = d0 = .05λp. Negative-positive coupled surfaces demonstratethe forward and backward surface wave branches.

Extending the concept of coupled ENG layers to N-layered structure, one can

manipulate the field in a farther distance. This idea has been investigated comprehen-

sively by Ramakrishna et al in [19–21], P. A. Belov et al in [52], and X. Li et al in [53]

for layered periodic structures. Existing TM surface polariton (p-waves) at an inter-

face requires negative dielectric primitivity at one side, whereas the duality supporting

TE surface modes (s-waves) needs negative material permeability. The focus here is

to integrate the ENG layers with the MNG layers making a transverse imaging device

almost independent of the polarization, for instance using alternative ENG and MNG

layers in the structure shown in Fig. 3.1. MNG layers may be utilized by depositing

metallic loop patterns on a coated plasma film or by using coupled ferrite thin-films

51

0 10 20 30 40 50 6010

−12

10−10

10−8

10−6

10−4

10−2

100

102

104

kρ/k

0

| Tra

nsfe

r Fun

ctio

n |

ENG ENG−Air−ENG

(a) (b)

Figure 3.8: (a) Transfer function for one-layer and two-layer coupled surfaces: Cou-pling between the layers introduces better evanescent-wave amplification for two-layerstructure. (b) Field profile along the propagation direction in two layer ENG (Shadedregions represent the ENG layers.)

0 10 20 30 40 50 6010

−20

10−15

10−10

10−5

100

105

kρ/k

0

| T

ran

sfe

r F

un

tion

|

d0 = .02 λ

p

d0 = .05 λ

p

d0 = .1 λ

p

d0 = .2 λ

p

Figure 3.9: Transfer function for two-layer ENG structure with different air gaps. Anoptimized distance between the layers provides a smooth transfer function resulting ahigher resolution image.

52

(i.e. in microwave).

From the effective medium theory, it can be concluded that at positive-negative

material interfaces, a very anisotropic k-dispersion performance is accomplished. The

characteristic of the ith periodic medium can be modeled with uniaxial effective

magneto-dielectric parameters ε and µ tensors as [58]

ε = ε0

εo 0 0

0 εo 0

0 0 εe

; µ = µ0

µo 0 0

0 µo 0

0 0 µe

(3.18)

where subscripts “e” and “o” denote the extraordinary and ordinary waves, respec-

tively. As introduced first by Smith et al in [59], a media with indefinite ε and µ

tensors can provide interesting reflection and refraction behavior. For small periodic-

ity structure, the components of the above tensors are simplified to [58]:

εio =εi1Li

Λi

+ εi2

(1− Li

Λi

)(3.19a)

1

εie

=1

εi1

Li

Λi

+1

εi2

(1− Li

Λi

)(3.19b)

µio =µi1Li

Λi

+ µi2

(1− Li

Λi

)(3.19c)

1

µie

=1

µi1

Li

Λi

+1

µi2

(1− Li

Λi

)(3.19d)

Equations (3.19) reveal that when Λi = 2Li and εi+1 = −εi, µi+1 = −µi, then the

effective ε and µ components in the transverse plane are around zero, where in the

propagation direction are about infinity. This offers the dispersion vector kz = 0 that

implies the Fourier components of the image pass through the structure without any

change in amplitude and phase.

To obtain the field performance in an ENG-MNG layered structure, we use the

theory described in section 3.2. A nine-layer structure is considered whose total thick-

ness is the same as the thickness of NIM slab in the previous section and the layers are

53

composed of ENG and MNG materials with loss factor γp = .001ωp, alternatively. The

thickness of each layer is d/2 (d = .1λp) and the structure is illuminated by a dipole

pair (of separation .2λ0) directed along the u direction (α = π/6) in distance d from

the structure (the same as what we used in section 3.3.) Fig. 3.10(a) represents the

electric and magnetic field distributions along the propagation direction (at ρ = .001λp

and φ = π/4). As expected, the performance for both polarizations is the same and

the image components are transferred successfully providing high-performance sub-

wavelength resolution. The field is amplified inside the layers alternatively and arrives

to the image plane. In Fig. 3.10(b) the image at different planes after the structure is

investigated. Comparing this figure with Fig. 3.4(b), it can be observed that the lay-

ered structure has a better resolution due to the larger enhancement in the k-vectors.

However, since the value of the field at the exiting surface of the layered material is

smaller than that of the NIM slab (for this specific γp), the image reconstruction for

the layered structure degrades more as the distance from the structure output surface

is increased.

However, if the material loss is increased the story is different. Fig. 3.11 demon-

strates the image of the pair of dipoles placed at distance d from the layered structure

for different material losses. The image is observed at z = −6d. As observed, the

layered composite can work successfully up to higher losses around γp = .15ωp and

still can separate the images of two sources, whereas the NIM slab is more sensitive

to the material loss. Basically, the NIM slab for losses larger than γp = .01ωp is un-

able to reconstruct the image (see Fig. 3.5(b)). The large loss of the thick NIM slab

does not allow enough amplification inside the structure where the thin thicknesses of

the positive-negative alternative layered structure allow successful tunneling inside the

medium. By increasing the number of layers the image can be transferred to longer

distances. This behavior is discussed as canalization in the earlier papers [19–21, 52]

for the periodic layered structures.

54

(a)

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.60

0.2

0.4

0.6

0.8

1

x (λ0)

Nor

mal

ized

|Dz|

zi = −5.5d

zi = −6d

zi = −7d

zi = −9d

(b)

Figure 3.10: N-layered ENG-MNG composite (N=9): (a) the electric and magneticfield profiles along the Propagation direction (Blue-shaded layers represent ENG andpink-shaded layers are MNG,) and (b) imaging performance at different planes (d =.1λp).

55

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x (λ0)

Nor

mal

ized

|Dz|

γp = .001 ω

p

γp = .01 ω

p

γp = .1 ω

p

γp = .15 ω

p

γp = ω

p

Figure 3.11: Imaging performance at plane z = −.6λp for different material losses.Comparing Fig. 3.11 to Fig. 3.5(b) shows that the layered structure has a betterperformance than the NIM slab for higher material losses.

3.5 FDTD Numerical Analysis of Finite-Size Struc-

ture

In this section, a Finite Difference Time Domain (FDTD) technique is applied to

characterize the layered composite that has finite-size in the transverse directions. Two

ENG and two MNG layers made of Drude permittivity and Lorentzian permeability

materials, respectively (as defined in Eq. (3.8)) with thickness of d/2 = .05λp and

loss factor γp = .001ωp are stacked alternatively along the z. At operating frequency

ω20 = .5ω2

p, the permittivity of ENG layers and permeability of MNG layers are around

-1 and the slab size in transverse plane is 1.5λ0 × 1.5λ0. An electrical dipole source

making angle π/6 with respect to the z axis is illuminating the structure. Fig. 3.12

represents the electric and magnetic field distributions along the propagation direction

inside the layers. The growth-attenuation behavior of the field is in agreement with

the results obtained from the theory. Fig. 3.13 represents the field distribution along

the transverse direction and for comparison, the field performance when there is no

56

Figure 3.12: FDTD performance: Field profile for the lossy ENG-MNG compositealong the propagating direction; (a) the electric field, and (b) the magnetic field. Thegrowth-attenuation behavior of the field is in agreement with the results obtained fromthe theory.

structure is also plotted. The better image resolution resulting from the evanescent

waves amplifications validates the concept of surface-modes layered design.

−0.2 −0.1 0 0.1 0.2 0.30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x (λ0)

Nor

mal

ized

|Ez|

Layered StructureFree Space

Figure 3.13: FDTD performance: Field profile along the transverse direction at planez = −.6λp.

Note that, the above discussed concept can successfully explain the use of coupled

impedance surfaces to achieve the near-field imaging that has been developed, i.e., by

Tretyakov et al in [60] and Marques et al in [16]. Multilayer coupled impedance surfaces

where each surface includes an optimized metallic pattern, can launch and enhance the

57

surface waves and reconstruct the decaying fields. An optimized coupling between the

surfaces is required to achieve a clear and high resolution image, as described before.

Impedance metasurfaces can be realized in both microwave and optics and will provide

advanced functional components as for instance, is demonstrated by Mosallaei et al

for an antenna substrate design in [61].

3.6 Conclusions

In this study, a comprehensive investigation of high resolution imaging utilizing Fourier

spectrum theoretical model and full-wave FDTD numerical analysis is addressed. To

provide a comparative study between the performance of a NIM slab and coupled

layered structures, first a negative index slab is briefly discussed. It is described that

the high resolution imaging in transverse plane appears as a result of the amplification

of evanescent waves. Then, coupled layered surfaces supporting surface modes are

investigated, enabling high resolution imaging along the transverse direction. The

study of surface-modes layers composed of ENG materials shows that in the region

where the material has negative permittivity, the layered structure supports surface

modes which can be excited by evanescent waves (p-waves). The layers of MNG

materials is a great choice for amplifying surface modes for s-polarized waves. It

is demonstrated that by combining the ENG and MNG layers, a near-field imaging

composite is realized that functions properly for both p and s polarizations.

Since one can make thin layers and cascade them in proper fashion to achieve the

image successfully, one can expect the effect of loss for the layered structure to be much

smaller than that of the bulk NIM slab. For instance, we illustrate using a nine-layer

ENG-MNG structure with loss of γp = .15ωp, the image of two objects separated by

0.2λ can be successfully reconstructed at the distance 0.6λp from the objects; where

with the NIM slab one needs a much smaller material loss of the order of γp = .01ωp. If

the loss is small, the NIM has better performance than the layered structure, however,

58

in a realistic system with reasonable large loss, the layered structure seems to be more

promising.

The coupled layers of ENG and MNG structures may be created with the use of

metallic patterns. It must also be highlighted that increasing the number of layers will

allow a high performance lateral imaging in a longer distance from the object.

59

Chapter 4

Ellipsoidal Metamaterial

Subwavelength Radiator

4.1 Introduction

Smaller physical size, wider bandwidth and higher radiation efficiency are three de-

sirable characteristics of antennas integrated into communication systems. In recent

years, considerable efforts have been devoted towards antenna miniaturization. Funda-

mentally, the ability of any antenna to radiate effectively depends on ka (where k is the

wave number and a is the radius of the smallest sphere enclosing the antenna) [62, 63].

According to Chu [63], the lower bound on the quality factor (Q) of an electrically

small electric or magnetic dipole antenna is inversely related to the radius of the small-

est sphere that can surround it by the formula QChu = 1/(ka)3+1/(ka); so the smaller

the radius, the higher the Q and the narrower the bandwidth. The challenge is to make

the physical size of the antenna as small as possible along with achieving a wideband

impedance characteristic (Q values close to the lower-bound). Best et al. presented

a comprehensive study in [64] for achieving small antennas with low Q performance

with the use of novel topologies, such as spherical or cylindrical folded helix antennas.

Quality factor as low as 1.5 times of the Chu limit was illustrated. Recently, there have

60

been some efforts to produce wideband electrically small resonant antennas by utilizing

negative parameters materials [65–67]. Stuart et al. in [65] excited an epsilon negative

(ENG) sphere with a dipole feed to produce the appropriate polarization required for

the resonance performance. They demonstrated that one can achieve an electrically

small antenna by operating the spherical radiator in a frequency that corresponds to

the region εr = −2. The sphere-shaped structures offer wideband performance close

to the Chu limit. Ziolkowski et al. have also demonstrated other novel small antenna

designs utilizing combinations of epsilon negative (ENG) and mu negative (MNG) or

double negative (DNG) metamaterials [67]. The focus of this work is on study of small

antennas realized by unique materials.

Practically, it is very useful to investigate the effect of the structural shape of the

material of the antenna on the resonant frequency and Q-factor. For example, can

a slab or a long rod of negative permittivity material also radiate efficiently? If so,

what would be the resonant frequencies and associated material indices? How close

would the Q be to the lower bound? The objectives in this study are to investigate

the resonance radiation of metamaterial-based eccentrically shaped structures with

particular emphasis on the bandwidth limitations or quality factors for the antennas.

Thin disks and long rods, as well as the ellipsoids, will be the special cases that allow

us to address the above questions. To simplify the problem, it is assumed that the size

of the antenna-proper is much smaller than the wavelength in both free-space and in

the material. Hence, the time-harmonic quasi-static approximation can be applied to

successfully formulate the problem and predict the physical parameters of the antenna.

A full-wave numerical technique (using CST STUDIO SUITE 2009 [68]) is applied to

comprehensively model the structure and validate the derived theory. We demonstrate

that a volume of negative permittivity material placed on a ground plane and fed by

a coaxial transmission line can produce a small antenna whose operating frequency

depends on material properties and the height to width ratio of the volume. The Q-

factor of the different shaped radiators are numerically studied and compared to the

61

Figure 4.1: The geometry of ellipsoid with semi-axes ax, ay and az

calculated values based on the reported equations in the recent literatures [69–75].

4.2 Resonance Formulation

Fig. 4.1 shows the geometry of an ellipsoid with semi-axes ax, ay, and az located in free-

space and illuminated by an arbitrary polarized electric field E0 = xE0x + yE0y + zE0z.

It is assumed that the ellipsoid has a material permittivity of ε, and a size which

is much smaller than the free-space wavelength (ax, ay, az ¿ λ). A rigorous static

analysis gives the field inside the ellipsoid as [76, 77],

Eint = xE0x[1− (εr − 1)Nx

1 + (εr − 1)Nx

]+ yE0y[1− (εr − 1)Ny

1 + (εr − 1)Ny

]+ zE0z[1− (εr − 1)Nz

1 + (εr − 1)Nz

],

(4.1)

where Ni(i = x, y, z) are depolarization factors determined from

Ni =axayaz

2

∫ ∞

0

ds

(s + a2i )

√(s + a2

x)(s + a2y)(s + a2

z)(4.2)

As can be seen, the depolarization factors Ni play a critical role in determining

the induced electric field. Note that if the applied electric field is initially uniform,

the resultant field within the ellipsoid is also uniform. Another observation is that the

polarization of the induced field can, in general, be different from that of the applied

field.

62

The three depolarization factors for any ellipsoid satisfy

Nx + Ny + Nz = 1 (4.3)

A sphere has three depolarization factors, each equal to 1/3, and the internal field

is aligned with the applied field, either in the same or opposite direction. Other

special cases are an oblate spheroid with ax = ay > az, and a prolate spheroid with

ax > ay = az. Closed-form expressions for the integral (4.2) can be derived for these

cases. For oblate spheroids we have [77],

Nz =1 + e2

e3(e− tan−1(e)) (4.4a)

Nx = Ny =1

2(1−Nz) (4.4b)

where the eccentricity is e =√

a2x/a

2z − 1. For prolate spheroids we have,

Nx =1− e2

2e3(ln

1 + e

1− e− 2e) (4.5a)

Ny = Nz =1

2(1−Nx) (4.5b)

where e =√

1− a2x/a

2z. The practical utility of the spheroidal cases lies in the oblate

spheroid degenerating into a flat disk as az becomes very small (e → ∞); and the

prolate spheroid approaching a rod-shaped structure as az becomes very large (e → 1).

To investigate some of the important physical implications of Eq. (4.1), consider

a metamaterial ellipsoid located in free-space under the influence of a +z-polarized

electric field. For the spherical geometry case, the internal field Eint simplifies to,

Eint = zE0z(1− εr − 1

εr + 2) (4.6)

When the permittivity of the sphere, εr, is larger than the permittivity of the free

63

space in which the sphere is assumed to reside, the sphere is depolarized along the -z

direction, and the total internal field reduces as the permittivity is increased. However,

for the sphere permittivity below the outside material value (vacuum), the sphere can

be polarized along the excitation (+z direction), and near εr = −2, one can establish

a resonance with strong field intensity inside the sphere (independent of the size of

sphere). Below εr = −2, the polarization of induced field switches again from +z to

again -z, and as the permittivity becomes very large negatively, the induced field tends

to cancel the external field producing nearly a zero total field inside the sphere.

Changing the shape of the ellipsoid has some interesting effects on the resonance

performance. For instance, if the geometry deforms from the sphere with polarization

factors (1/3,1/3,1/3) into a flat disk with those of (0,0,1), Eq. (4.1), shows that the

internal field is simplified to Eint = zE0z(1/εr). Thus, the thin disk becomes resonant

at around εr = 0. Hence, altering the antenna shape from that of a sphere to a disk

will shift the required permittivity for resonance from εr = −2 to εr = 0. It is also

found from Eq. (4.1) that by changing the shape of the antenna from a sphere to an

increasingly long rod (along z), the permittivity required to produce a resonance shifts

from εr = −2 to εr → −∞.

4.3 Calculation of the Lower Bounds on Q

So far, we have concentrated on the resonance characteristics of the metamaterial-

based ellipsoid. The bandwidth, which is inversely proportional to the Q of an an-

tenna [69], is another important consideration for practical electrically small antennas.

The concept of lower bounds on the Q of electrically small antennas was first intro-

duced by Wheeler [62] and Chu [63]. According to Chu [63], the minimum Q that one

can achieve for an antenna confined to a spherical volume of radius a obeys the rela-

tionship QChu = 1/(ka)3 + 1/(ka), ka ¿ 1, which means that decreasing the electrical

size of the resonator increases its Q and narrows its bandwidth. Recently, attention

64

has been drawn to the subject of the lower bounds on the Q for antennas confined to

arbitrarily shaped volumes. Gustafsson et al. determined physical bounds on anten-

nas of arbitrary shape [71, 72] using an approach based on fundamental principles of

causality, time-translated invariance, and reciprocity applied to a general set of linear

constitutive relations via a sum rule [72]. More recently, Yaghjian et al. have shown

that the minimum possible Q for an electrically small dipole antenna confined to an

arbitrary volume V will be the Q of a PEC scatterer filling V subject to a uniform

incident electric field. This lower bound Q can be expressed in terms of the direction

of the electric dipole moment (p = p/|p|), and the electrostatic polarizability dyadic

αe of the PEC volume V [73, 74],

Qed,lb =6π

k3

(p.α−1

e .p− V |α−1e .p|2) (4.7)

which for a principal direction of the volume V , becomes

Qed,lb =6π

k3αp

(1− V/αp

)(4.8)

Equations (4.7) and (4.8) apply to linear electric or magnetic dipole antennas whose

exciting sources can be both electric currents and magnetic currents (polarization M)

outside the “antenna-proper.” For electric-dipole antennas confined to an electrically

small volume V excited by electric-currents only, such as the antennas in this study,

the lower bound on the quality factor reduces to [73, 74],

Qeced,lb =

6π

k3p.α−1

e .p (4.9)

which for a principal direction of the volume V , becomes

Qeced,lb =

6π

k3αp

(4.10)

It is often convenient to re-express αp as fsV , where fs is a dimensionless “shape

65

factor.”

In the following sections, we numerically compute the actual Q for simulated spher-

ical, circular-cylindrical-disk, and circular-cylindrical-rod antennas, and then compare

these values of Q to the Q lower bounds given in Eq. (4.10) determined from the shape

factors for these antennas.

4.4 Performance Analysis of ENG Antennas

To provide physical insight into these metamaterial-based antennas, their detailed

performance characteristics will now be investigated. To form an antenna element,

the resonator must be coupled to a transmission line. For each antenna on a ground

plane, (half sphere, half disk, and half rod), the antenna is fed by a 50 Ohm coaxial

transmission line with a small monopole stub. The dimension of the stub is varied in

order to find the optimum impedance match. It is assumed that the metamaterial of

the antenna has a Drude-dispersive permittivity satisfying,

ε(ω) = ε0

(1− ω2

e

ω(ω + iγe)

)(4.11)

where ωe = 2π×4×109(rad/s) is the bulk resonant frequency of the material, and the

damping factor γe = 0.001ωe determines the loss. The Drude permittivity is plotted

in Fig. 4.2. This Drude metamaterial may be constructed in microwave frequencies

with the use of array of subwavelength metallic wires [78], although it features a larger

frequency dispersion than a regular wire medium affecting the desired bandwidth (in

optics, a plasmonic metal can simply provide the Drude dispersive property).

All simulations are performed with the finite integration method using CST [68]

with an absorbing boundary condition implemented at a distance of one wavelength

from the antenna element.

66

0 1 2 3 4 5−20

−15

−10

−5

0

5

10

Rel

ativ

e P

erm

ittiv

ity

Frequency (GHz)

Real−Imaginary

Figure 4.2: Characteristics of the Drude permittivity material.

4.4.1 Spherical Radiator

Fig. 4.3 depicts the geometry of a hemisphere located on a ground plane. For a

large ground plane, the hemisphere can be considered a sphere for modeling purposes.

A probe-feed is used to excite the resonant mode of the sphere. Since the sphere

is assumed to have a very small radius (a = 7.5 mm), compared to wavelength, the

above quasi-static discussions can be applied. The optimum stub length and radius are

determined through simulation to be 4.5 mm and 1.2 mm, respectively. The antenna

input impedance and return loss versus frequency are shown in Fig. 4.4. A resonance

near the frequency which is associated with εr = −2 is determined and a −10dB

impedance matching at f = 2.36 GHz is obtained by tuning the stub (ka = 0.37).

This demonstrates good agreement with the quasi-static prediction for the resonance

frequency at εr = −2. The operating wavelength is 127.1 mm and the stub length

is about λ/29. The diameter of the sphere radiator at the operating frequency is

about λ/8.5. The resonant frequency of the antenna indeed corresponds to that of

the fundamental mode of the negative permittivity sphere explained in the previous

section.

To better understand the physical performance of the sphere, the electric field at

67

Figure 4.3: The geometry of the hemisphere radiator constructed from the Drudedielectric medium.

the resonant frequency is plotted in Fig. 4.5(a) (line-fields at some arbitrary moment in

the y-z plane). Note that the normal component of the electric field has different signs

inside and outside the sphere due to the negative permittivity value. The negative

permittivity sphere acts like an inductor which is in parallel with the dipole-feed

capacitor. The dipole-feed capacitor can be used for tuning the antenna resonant

characteristic. The antenna radiation pattern shown in Fig. 4.5(b) is that of an electric

dipole, as expected from the field distribution inside the radiator. The bandwidth and

Q of the antenna are also determined based on equation (96) of [73]. The 3dB matched

VSWR bandwidth is 6.4% at the operating frequency; therefore, the Q corresponding

to the half-power VSWR bandwidth yields a value of about 31.25 at the resonant

frequency of the antenna. Fig. 4.4(b) shows return loss versus frequency for the same

antenna described above with the lossless material. The length of the stub is tuned

to be 4.4 mm to improve the impedance matching performance. The Q (for 100%

efficiency) corresponding to half-power VSWR bandwidth (5.9%) yields a value of

around 33.9 at the resonant frequency of the antenna, which is about 1.51 times the

Chu lower bound for an antenna with ka = 0.37 (QChu = 22.4). Equation (4.10) also

predicts a Q of 1.5 times the Chu lower bound for a sphere which has a polarizability

of αe = 4πa3 (fs = 3).

Since the material parameters include loss, a reduction in efficiency is expected.

68

The radiation efficiency is plotted versus frequency in Fig. 4.4(b). As observed, near

the operating bandwidth, the efficiency is nearly flat at a value of about 92%.

2 2.1 2.2 2.3 2.4 2.52.5−400

−200

0

200

400

600

800

1000

Frequency (GHz)

Impe

danc

e (O

hms)

ResistanceReactance

(a)

2 2.1 2.2 2.3 2.4 2.52.5−30

−25

−20

−15

−10

−5

0

Frequency (GHz)R

etu

rn L

oss

(d

B)

0

20

40

60

80

100

Ra

dia

tion

Effic

ien

cy (

%)

with lossno loss

(b)

Figure 4.4: The performance of the hemisphere structure: (a) input impedance, and(b) return loss and radiation efficiency.

4.4.2 Circular Cylindrical Disk Radiator

From a practical point of view, one may be interested in the resonance performance

of an ENG disk instead of the sphere, the disk being easier to construct. Fig. 4.6

shows the geometry of a disk with radius of R = 5.31 mm and height of h = 1.77

mm (R/h = 3) located above a ground plane. The disk is composed of the Drude

medium given in (4.11). Using equations (4.1) and (4.4), a resonance with strong field

intensity inside the disk is expected to occur at f = 3.19 GHz, which corresponds to

εr = −0.57 based on the Drude material characteristics. (At this frequency, the ka

of the smallest circumscribing sphere is 0.37.) Here the radius and height of the disk

are approximated by the major and minor axes of an ellipse, respectively. To obtain

an impedance match, the inner stub is given a radius of 0.55 mm and a length of

1.5 mm. The resonance performance with −10dB impedance matching near f = 3.42

GHz is obtained and shown in Fig. 4.7, and the electric field is shown in Fig. 4.8. A

69

(a) (b)

Figure 4.5: Radiator performance at the resonant frequency, f = 2.36 GHz: (a) E-field pattern in the y-z plane. Note to the depolarized fields inside the sphere, and (b)radiation pattern. It presents a dipole mode of the antenna as expected of the fielddistribution inside the radiator.

uniform depolarized field pattern inside the disk is established. It is worth mentioning

that since the normal component Dn of the displacement on the surface of the disk is

continuous, a small value of permittivity inside the radiator which is the case for this

resonant disk produces a very strong internal electric field. The 3dB matched VSWR

bandwidth is about 0.88%.

The lossless case of a 1.5 mm stub with a radius of 0.45 mm is also shown in

Fig. 4.7(b). The 3dB matched VSWR bandwidth of 0.71% at the resonant frequency

gives a Q equal to 281.3. The Q of the disk is much higher than that of a circumscribing

sphere because the disk occupies a much smaller volume than its circumscribing sphere.

To calculate the Q factor of the disk based on Eq. (4.10), one needs to calculate

αe first, which can be estimated from the shape factor fs that computes to 1.97 for a

disk with height to width ratio of 1/3 [75]. From (4.10), this shape factor gives a Q

lower bound of 83.3. Our numerical calculation for the Q of the disk (Q = 281.3) is

much higher than the theoretical lower-bound prediction. This can be mainly because

of the antenna shape and the type of its excitation. In other words, since the electric

flux density Dn must be continuous on the surface of the disk, and since the antenna

goes to the resonance for ε close to zero, the field concentration inside the disk must

70

be very high (Dn = εEn). This will result in determining a very high quality factor

(much larger than the Q lower bound). Special attention should be made for proper

excitation of an antenna with a specific shape to achieve the Q lower bound.

The efficiency curve plotted in Fig. 4.7(b) shows a nearly flat efficiency across the

operating bandwidth at a value of about 72% for the lossy antenna.

By increasing the aspect ratio of the disk (ax/az ≈ R/h), the resonance frequency

will move up toward the region that the permittivity is close to zero. This can be

obtained from (4.1) and (4.4), where the depolarization factor Nz is increased by

increasing the aspect ratio of the disk. Hence, a thin disk is expected to resonate if it

is made out of an epsilon near zero (ENZ) medium.

Basically, as mentioned earlier, the subwavelength structure with negative permit-

tivity can be viewed as an inductor in parallel with the dipole feed capacitor. Since

near the resonant frequency, a strong field depolarization occurs inside the radiator, a

large value of equivalent inductance is produced, thereby providing an inductive input

impedance behavior for the antenna that can be tailored by changing the radiator

shape and optimizing the feeding system to allow successful antenna matching. The

disk like the sphere radiates an electric dipole pattern.

Figure 4.6: The geometry of the disk-shaped Drude permittivity radiator.

71

3.2 3.3 3.4 3.5−4000

−2000

0

2000

4000

6000

Frequency (GHz)

Imp

ed

an

ce (

Oh

ms)

ResistanceReactance

(a)

3.2 3.25 3.3 3.35 3.4 3.45 3.5−30

−25

−20

−15

−10

−5

0

Frequency (GHz)

Re

turn

Lo

ss (

dB

)

0

20

40

60

80

100

Ra

dia

tion

Effic

ien

cy (

%)

with lossno loss

(b)

Figure 4.7: The performance of the disk-shaped radiator: (a) input impedance, and(b) return loss and radiation efficiency.

Figure 4.8: E-field pattern in the y-z plane for the disk at the resonant frequency,f = 3.42 GHz. Note to the strong field depolarization inside the disk proving largeinductive behavior.

72

Figure 4.9: The geometry of the rod-shaped Drude permittivity radiator.

4.4.3 Circular Cylindrical Rod Radiator

The performance of a Drude-material rod-shaped antenna is considered next. The

geometry of a long rod with radius R = 4.23 mm and height h = 12.69 mm located

above a ground plane is shown in Fig. 4.9. Using quasi-static equations (4.1) and (4.5),

a resonance with strong field intensity inside the rod is predicted to occur at f = 1.32

GHz which corresponds to εr = −8.199 at the same ka of 0.37. (The height and

radius of the rod in this calculation are that of an ellipse with these major and minor

axes, respectively.) By optimizing the stub to have a radius of 0.5 mm and length

of 2.35 mm, the antenna is matched to 50 ohm near f = 1.13 GHz . The numerical

performance obtained by CST is illustrated in Figs. 4.10 and 4.11. The 3dB matched

VSWR bandwidth is 2.88%. The return loss for the lossless case of a 2.05 mm stub

(shown in Fig. 10(b)) gives the bandwidth of 2.16% corresponding to a Q of 92.4,

about 2.6 times the Chu lower bound (QChu = 35.5) for an antenna with ka = 0.314

where a is the radius of the smallest sphere enclosing the rod. The lower bound for the

actual rod volume can be found from (4.10) to be 92 after using the computed shape

factor of 10.8 for this rod with height/width = 3 [75]. As observed, the lower-bound

value is close to the actual simulated value for this rod antenna. The internal field is

73

mostly polarized along the axis of the rod. The efficiency of the lossy antenna shown

in Fig. 4.10(b) is nearly flat across the operating bandwidth at a value of about 77%

at the operating frequency of the antenna.

Comparing the near-field pattern of the disk and sphere with that of the rod clearly

reveals that a stronger field is established outside the rod-shaped radiator. This is

related to the higher negative permittivity of the material inside the rod compared to

the disk and sphere cases.

It is very instructive to plot the required negative permittivity for the resonance of

an ellipsoidal radiator in terms of its aspect ratio. This can be accomplished using the

derived quasi-static equations (4.1)-(4.5). The dependence of the required permittivity

on the ellipsoid aspect ratio ax/az (ax = ay) is shown in Fig. 4.12. It is observed that

the sphere resonates at εr = −2, whereas thin disks require small-values of negative

permittivity for establishing the resonance, and the long-rod resonates at a high neg-

ative value of permittivity. The numerical results obtained for the disk, sphere, and

rod are in good agreement with this curve.

1.05 1.075 1.1 1.125 1.175 1.2−200

−100

0

100

200

300

400

500

Frequency (GHz)

Imp

ed

an

ce (

Oh

ms)

ResistanceReactance

(a)

1.05 1.075 1.1 1.125 1.15 1.175 1.21.2−30

−25

−20

−15

−10

−5

0

Frequency (GHz)

Re

turn

Lo

ss (

dB

)

0

20

40

60

80

100

Ra

dia

tion

Effic

ien

cy (

%)

with lossno loss

(b)

Figure 4.10: The performance of the rod-shaped radiator: (a) input impedance, and(b) return loss and radiation efficiency.

74

Figure 4.11: E-field pattern in the y-z plane for the rod in the y-z plane at the resonantfrequency, f = 1.13 GHz.

Figure 4.12: Required negative permittivity for radiator resonation versus ellipsoidaspect ratio.

75

4.5 MNG Slab Resonance Radiator

So far, the concept of subwavelength radiators has been highlighted with the use of

negative permittivity materials. Basically, the major practical issue is that one cannot

achieve a material with negative permittivity in microwave region. Magnetic materials

are more promising in this regard[10]. In fact, one can use a self-biased hexaferrite in

GHz spectrum with negative permeability feature above its resonance, to establish a

small-size radiator. Fig. 4.13(a) depicts the geometry of a subwavelength rectangular

slab made of a magnetic material with Lorentzian permeability function

µ(ω) = µ0(1− κ2 ω2

ω2 − ω2h + i2γhω

), (4.12)

with magnetic resonant frequency ωh = 2π × 2 × 109(rad/s), damping factor γh =

0.001ωh, and coupling coefficient κ = 0.707. The characteristic of medium is plotted

in Fig. 4.9(b). The slab is located above a ground plane containing an aperture to

couple the field from a microstrip line to the system.

(a) (b)

Figure 4.13: Slab radiator constructed from the Lorentzian magnetic medium given byEq. (4.12): (a) the geometry, and (b) Lorentzian permeability behavior. The groundplane is finite with size 22.5mm× 30mm.

The equivalent magnetic current of the aperture excitation can tune the capacitive

76

property of the resonator at the proper Mu Negative (MNG) permeability value. Based

on the quasi-static model a resonant behavior is expected. The FDTD result for the

input impedance is shown in Fig. 4.14(a). The resonant frequency is determined

at f = 2.31GHz associated with µr = −1. The return loss is shown in Fig. 4.14(b)

providing a good impedance matching with the bandwidth of about 0.25%. Optimizing

the resonator shape and feeding system can result in a wider impedance bandwidth.

Magnetic field pattern inside the slab is illustrated in Fig. 4.15(a), representing an

almost uniform depolarized field around the resonance. The radiation pattern is similar

to the field of a magnetic dipole, as obtained in Fig. 4.15(b).

2.295 2.300 2.305 2.310 2.315 2.320 2.325Frequency (GHz)

−100.0

−50.0

0.0

50.0

100.0

Impe

danc

e (O

hms)

Re [Zin]Im [Z in]

(a)

2.295 2.300 2.305 2.310 2.315 2.320 2.325Frequency (GHz)

−35.0

−30.0

−25.0

−20.0

−15.0

−10.0

−5.0

0.0

Ret

urn

Loss

(dB

)

(b)

Figure 4.14: Magnetic slab radiator: (a) input impedance, and (b) return loss perfor-mance. Tuning the feed slot matches the antenna impedance to 50Ω.

4.6 Conclusions

In this study the effects of the shape and material dispersion of epsilon negative (ENG)

radiators on their resonance characteristics in general, and on their quality factor Q in

particular, are investigated. The quasi-static model is applied to theoretically formu-

late the behavior of spherical, disk-shaped, and rod-shaped resonator antennas. It is

77

(a) (b)

Figure 4.15: (a) Near field in xy-plane, and (b) radiation pattern of the magnetic slabradiator. Note to the H-field depolarization. The slab generates magnetic dipole moderadiation performance.

demonstrated that for a spherical geometry the resonance occurs at εr = −2, whereas

a thin disk resonates at smaller negative permittivity and a long rod resonates at a

larger negative permittivity. The full-wave numerical technique using CST software is

applied to fully characterize the antenna radiator and match its input impedance to its

feed line using a monopole stub. Numerically simulated values of the quality factor Q

are compared with the Q lower bounds for these different shaped radiators calculated

from recently published formulas for the Q lower bounds of electric-dipole antennas

confined to an arbitrarily shaped volume. The simulated Qs for the ENG sphere and

rod are almost the same as the theoretical calculation, while that for the disk is about

3.38 times of the calculated value (equation (4.10)). Considering the type of electric

dipole excitation, the sphere and rod use the best opportunity (regarding the excita-

tion and the use of volume) to offer the closest Q to the lower bounds. And obviously,

the sphere provides the minimum Q between all these configurations, although it has

a Q of 1.5 times of the Chu lower bound. The Chu lower bound may be achieved by

an antenna design with proper excitations of both electric and magnetic polarizations

78

and optimal configurations.

We also demonstrate radiation characteristics of small resonators made of negative

permeability materials. It is illustrated how a resonator composed of negative per-

meability medium can successfully establish a small antenna element. The obtained

observations may provide road maps for the future design of metamaterial-based sub-

wavelength antennas.

79

Chapter 5

Optical Reflectarray Nanoantenna

5.1 Introduction

Reflectarray antennas are a class of antennas that combine the features of reflectors

and phased arrays providing a directive beam in a desired scanned angle. The most

important advantages of reflect arrays over phased arrays are the elimination of com-

plexity and losses of the feeding network and the higher efficiency [79]. Also they are

easier for manufacturing in compared to reflector antennas. To design a reflectarray,

the phase of the reflected wave should have a progressive variation over the whole sur-

face such that the total phase delay from the feed to a fixed aperture plane is constant

for all the elements. Then, a critical feature in the design of a reflectarray is the choice

of elements for obtaining the required phase distribution.

In microwave, various potential reflectarray element designs are considered, which

include variable size patches, patches with variable length slot, and patches with fixed

slot fed by variable length stripline [80, 81]. Introducing elements that can work in

optics is of great interest which will be explored in this study for making a reflectarray

nanoantenna.

Recently, Engheta et al suggested a method of realizing nanoantennas system at

optical frequencies by using nanoparticles of concentric structures with cores made of

80

ordinary dielectrics and shell of plasmonic materials [82]. In another study, subwave-

length particles at plasmonic scattering resonance were suggested as antenna elements

for Yagi-Uda antennas at optical frequencies [46, 83]. The scattering resonance of

these concentric structures can be tailored at different wavelength range by adjusting

the core and shell radii or the material properties.

In this study, we illustrate the concept of a reflectarray nanoantenna implemented

in optics with the use of array of core-shell dielectric-plasmonic materials, each of them

optimized properly to achieve the required phase shift. The concept and radiation

performance are investigated. A 3D finite difference time domain (FDTD) technique

is applied to obtain the required reflection phase for a periodic array of a specific

nanoparticle design. Then, the obtained result is integrated into the making a 6 × 6

array of nanoparticles, scanning successfully a narrow beam optical radiation. To

remove the back radiation, the array is pinned on top of a dielectric-silver layer which

removes wave penetration to the other side of the nanoantenna. The radiation pattern

demonstrates a successfully designed optical reflectarray performance.

5.2 Scattering Characteristic of a Core-Shell Nanopar-

ticle

The general solution of the diffraction problem of a single sphere of arbitrary mate-

rial with the frame of electrodynamics was first given by Mie in 1908. He applied

Maxwell’s equations with appropriate boundary conditions in spherical coordinates

using multipole expansions of the incoming electric and magnetic fields. Clusters can

be composed of more than one element forming core-shell particles [84]. Engheta et

al demonstrated the scattering performance of a sphere particle made of plasma and

dielectric materials with the goal of making a Yagi-Uda antenna [46, 83].

Let us assume a concentric spherical particle as we see in Fig. 5.1. Such a concentric

81

structure provides interesting properties when one of the layers is made of plasmonic

particle. Here, we use these interesting properties in design of a reflectarray antenna

for optical frequencies.

Figure 5.1: A concentric dielectric-plasmonic nanoparticle.

The scattering resonate frequency of the core-shell structure depends on material

properties and radii of core and shell. Then, by adjusting these parameters one can

control the resonant performance. A small particle can be modeled as an induced

electric dipole with polarizability α that relates the induced dipole to the incident

field as p = αE and the scattered field by this particle is equivalent to the radiated

field from the induced dipole. The polarizability can be related to the scattering

coefficient of the dipolar term given by Engheta et al as, in [83, 85]

α = −6πε0

k30

icTM1 (5.1)

where k0 is the wavenumber in the surrounding medium and cTM1 is the scattering

coefficient of the TM mode of order 1 in the Mie scattering analysis. cTM1 is given

as [83, 85]

cTM1 = − U1

U1 + iV1

(5.2)

where

82

U1 =

j1(k1b) j1(k2b) y1(k2b) 0

j1(k1b)/ε1 j1(k2b)/ε2 y1(k2b)/ε2 0

0 j1(k2a) y1(k2a) j1(k0a)

0 j1(k2a)/ε2 y1(k2a)/ε2 j1(k0a)/ε0

(5.3)

V1 =

j1(k1b) j1(k2b) y1(k2b) 0

j1(k1b)/ε1 j1(k2b)/ε2 y1(k2b)/ε2 0

0 j1(k2a) y1(k2a) y1(k0a)

0 j1(k2a)/ε2 y1(k2a)/ε2 y1(k0a)/ε0

(5.4)

Here, j1(x) and y1(x) are the first-order spherical Bessel functions of the first and

second kind. j1(x) stands for ∂(xj1(x))/∂x and y1(x) is similarly defined. ε1, ε2 are the

permittivity of the core and of the shell, respectively, while k1, k2 are the wavenumbers

in each respective region. The outer and inner radii of the particle are denoted as a

and b.

If one follows the closed form equation which has been obtained for cTM1 , it will

be seen that by adjusting the ratio b/a or the permittivities of core and shell, the

scattering resonance can be tailored at different wavelength range. In this study, a

Drude material is used to describe the frequency dependence of the permittivity of

silver which is the shell material, i.e.

ε(ω) = ε0

(1− ω2

p

ω(ω + iγp)

)(5.5)

where ωp is the bulk resonant frequency of the material, and damping factor γp repre-

sents the losses present. (In this study, ωp = 2π × 2000THz and γp = 0.001ωp)

Fig. 5.2 shows the magnitude and phase of the polarizability α of a concentric

nanoshell particle for different core materials and for different b/a ratios (see Fig. 5.1(a)

for the geometry of particle). Operating wavelength is 357.1 nm and εshell = (−4.67+

83

.01i)ε0. The outer radius is a = 22.5 nm and in Fig. 5.2(a), b/a is assumed to be 0.533.

Fig. 5.2(b) shows the performance vs b/a when εcore = 3ε0. As seen, due to the shift

in resonant frequency, the phase performance changes and these particles can be used

for a reflectarray antenna. In this study we use the same b/a for all the elements and

change the core permittivity to control the phase (as this will be easier for our FDTD

analysis).

0 1 2 3 4 5 6 70

1

2

3

4

5

6

7

88

εcore

|α| (1

0−

2ε 0

λ03)

0

0.2

0.4

0.6

0.8

11

Arg

(α

) (π

)

Magnitude Phase

(a)

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

8

b/a

|α| (1

0−

2ε 0

λ03)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Arg

(α

) (π

)

PhaseMagnitude

(b)

Figure 5.2: Magnitude and phase of the polarizability α of a concentric nanoshellparticle vs: (a) the permittivity of core when b/a = 0.533 and, (b) the ratio of radiib/a when εcore = 3ε0. Operating wavelength is 357.1 nm and the shell is made of silver[εshell = (−4.67 + .01i)ε0].

5.3 Optical Reflectarray Nanoantenna

To establish an optical reflectarray nanoantenna, one needs to design an array of nano-

radiators where each of them is tailored properly to provide a desired reflection phase,

where as a result the array can re-direct and scan the beam in a specific direction.

Having an array of nanoradiators allows narrowing the radiation beam. Thus, the

reflectarray nanoantenna can successfully scan a directive optical radiation beam. This

will be of significant interest for optical far-field engineering.

84

Figure 5.3: Schematic of the reflectarray nanoantenna structure.

Assuming that the excitation is achieved by a feed located in the far-field of the

array antenna, the phase of the incident wave at each particle is proportional to the

distance dl from the feed (Fig. 5.3). Then, the required phase of the reflected field for

this element to achieve a reflected beam in a given direction (θ0, φ0) is obtained by [79]

Φl = k0[dl − sin θ0(xl cos φ0 + yl sin φ0)] (5.6)

where (xl, yl) is the coordinates of the center of element l and k0 = ω√

µ0ε0. The re-

quired phase can successfully be achieved by optimizing core-shell dielectric-plasmonic

nanoparticles. Basically, one can change the radii of the configuration or its materials

parameters to achieve this importance. Here, for the sake of simplicity in FDTD anal-

ysis, the material core is considered as the variable for obtaining the required reflection

phase.

85

5.3.1 Reflection-Phase Synthesis

In order to characterize the radiating element of an array, the effect of the surrounding

elements needs to be taken into account. The exact method is to do measurement

by placing the element in complete array. For a large array of elements, this will

be costly and time consuming. The waveguide simulator approach provides a simple

and efficient way to determine the performance of the radiating elements in a large

array antenna [81, 86]. To achieve this, one can envision a nanoparticle inside the

array as a periodic configuration and then applied a full wave numerical analysis to

demonstrate the reflection phase from the array configuration. A finite difference time

domain (FDTD) [32–34] approach with periodic boundary condition (PBC) is applied

to characterize the performance of periodic array of nanoparticles.

Let us first highlight the scattering performance of a core-shell nanosphere. Fig. 5.4

illustrates the FDTD simulation of the particle having a = 22.5 nm, b/a = 0.533, silver

as the shell material and εcore = 3ε0. In the FDTD model, the computational domain

is configured with cubical Yee cells with ∆ = 0.75 nm and the core-shell structure is

illuminated by a plane wave having x-polarized electric field and traveling in the -z

direction. The sphere is placed at the center of computational domain (0, 0, 0) and the

scattered field is stored at (0, 0, 100∆) in front of the concentric sphere. The scattered

field is plotted in Fig. 5.4. The polarizability α of the same particle based on Mie

theory is also shown in Fig. 5.4. These two graphs can be compared in regard of the

position of resonances. A good comparison between the full wave numerical analysis

and theoretical model is established. At resonance point (λ0 = 357.1 nm) the error is

less than 4% which can be smaller by choosing a smaller Yee cell.

The array of nanoparticles is investigated next. The objective is to successfully

tailor the reflection phase from a nanoparticle located inside the array to the value

of interest. To accomplish this, the FDTD is applied to characterize one unit-cell of

the periodic structure (array of particles in free-space) and determine the phase of

86

250 300 350 400 450 500 550 600 650 700 7500

1

2

3

4

5

6

7

8

Wavelength (nm)

|α| (

10−2

ε 0 λ03 )

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

|Sca

ttere

d F

ield

|

Mie Theory FDTD

Figure 5.4: Resonance performance of a concentric nanoshell particle, b/a = 0.533,εcore = 3ε0. Close comparison between Mie-theory and FDTD is illustrated.

the reflected field. The unit-cell size is considered to be 150nm × 150nm (in x and

y directions) and the same Yee cell is used. The required phase is manipulated by

changing the core material of the nanoparticle. Fig. 5.5 shows the reflection amplitude

and phase for different core materials (b/a is fixed at 0.533). As observed, the resonance

performance depends on the material properties. Careful selection of the operating

wavelength gives a better control over the reflected phase swing. If we choose operating

wavelength λ = 357.1 nm (f = 840 THz) as the design wavelength, we can realize a

relatively large phase swing of about 115. Fig. 5.6 illustrates the reflection phase

variation in terms of change in the material parameter of core. This is a very useful

curve that will be used in next section to realize a reflectarray of interest.

5.3.2 Plasmonic Core-Shells Array Over a Layered Material

If a non-periodic array of plasmonic core-shells are placed over a layered material,

with the presence of an incident field, each nano-particle can be viewed as an induced

dipole around the scattering resonance of that nano core-shell. The induced dipole

on each nano core-shell is proportional to the total field upon that particle. In this

scenario, the total field upon each nano core-shell can be expressed as the summation

87

300 325 350 375 400 425 450 475 5005000

0.1

0.2

0.3

0.4

0.5

0.6

Wavelength (nm)

Refl

ecti

on

Co

eff

icie

nt

εcore

= 1ε0

εcore

= 2ε0

εcore

= 3ε0

(a) (b)

Figure 5.5: FDTD simulated results for different core materials and construction ofphase design curve: (a) reflection amplitude and, (b) reflection phase

1 1.5 2 2.5 3 3.5 4−120

−100

−80

−60

−40

−20

0

20

εcore

Ref

lect

ion

Pha

se (

deg)

Figure 5.6: Phase of reflection coefficient vs. the core permittivity at λ0 = 357.1 nm

88

of three terms. The first part is associated with the total incident field in the absence

of the nano core-shells (Etotalinc ), the second part is the electric field due to the couplings

between the nano core-shells in the absence of the layered material ( ¯Gldipole(rl, rq)p

q).

Since this term represents the couplings between the nano core-shells, for the lth nano

core-shell we consider the fields of all other nano core-shell except itself (excluding the

field of the lth particle). The last term, is associated with the reflected fields from

the layered substrate ( ¯Glreflected(rl, rq)p

q). Note that for the computing the last two

terms we approximate each nano core-shell with an electric dipole. Hence for the

second term we calculate the dipolar couplings and the Green’s function analysis of

dipoles over layered material is applied for evaluation of the final part [56, 87]. Also,

it is worth mentioning that the fields associated with every nano core-shells (both

couplings and reflected field) is directly proportional to the induced dipole moment,

thus the induced dipole moment for each particle is derived by solving the following

linear system of equations. For l, q ∈ 1, 2, ..., N with N being the total number of

particles, we obtain,

pl = αl

(Etotal

inc (rl) +∑

q,q 6=l

¯Gldipole(rl, rq)p

q +∑

q

¯Glreflected(rl, rq)p

q

), (5.7)

where Etotalinc denotes the sum of the incident field and its reflection from the layered

material in the absence of the nano-spheres. ¯Gldipole is the dyadic Green’s function of

the qth nano core-shell evaluated at the position of the lth particle. ¯Glreflected is the

reflected Greens function of the qth nano-sphere (from the layered material) computed

89

at the location of the lth one, i.e.,

¯Gldipole(rl, rq) =

(k21 + ∂2

∂2x) ∂2

∂x∂y∂2

∂x∂z

∂2

∂x∂y(k2

1 + ∂2

∂2y) ∂2

∂y∂z

∂2

∂x∂z∂2

∂y∂z(k2

1 + ∂2

∂2z)

eik1|rl−rq |

4πε1|rl − rq| , (5.8a)

¯Glreflected(rl, rq) =

Gelxrx(rl − rq) Gelx

ry(rl − rq) Gelxrz(rl − rq)

Gelyrx(rl − rq) Gely

ry(rl − rq) Gelyrz(rl − rq)

Gelzrx(rl − rq) Gelz

ry(rl − rq) Gelzrz(rl − rq)

. (5.8b)

where for example, Gelzrx is Green’s function for the z-directed electric field associated

with an x-directed dipole [56, 87].

The far zone electric field for the array deposited over a layered-substrate can be

evaluated using the conventional steepest decent contour (SDC) technique [56] with

the transformation kρ = k sin θ, where the θ is the spherical angle from the z axis.

Hence, for each dipole (px, py, pz) located at (x0, y0, z0) above a substrate instance, the

upper half-space and lower half-space far-field radiation pattern can be represented

as [88],

E =

∣∣∣∣∣∣∣Eθ

Eφ

∣∣∣∣∣∣∣=

k2J

4πεJ

eikJr

r

∣∣∣∣∣∣∣

(px cos φ + py sin φ

)cos θ Φ2

J − pz sin θΦ1J

−(px sin φ− py sin φ

)Φ3

J

∣∣∣∣∣∣∣. (5.9)

where the index J ∈ [1, N ] is to distinguish between the upper-half (ε1, µ1) and lower-

half (εN , µN). The potential parameters are defined in [87, 88]. Notice that, each

potential is composed of two terms, whereas the first term can be interpreted as the

far zone radiation pattern of a dipole, while the second term can be identified as the

radiation from a dipole located at the image plane weighted by generalized reflection

coefficients.

90

5.4 Array Design and Scanned-Beam Characteris-

tics

In this study, two reflectarray designs (including 6 × 6 elements) for beam scanning

at 15 and 30 are considered. The geometry of reflectarray is depicted in Fig. 5.3.

The center of array is placed at (0, 0, 0). For the first design the feed is located at

(xf = yf = zf ) = (−0.4µm, 0, 2.25µm), and for the second design it is at (xf =

yf = zf ) = (−0.4µm, 0, 4.5µm). The feed excitation is modeled with an infinitesimal

electric dipole which is polarized along the x-direction and operates at λ0 = 357.1nm.

In order to ensure field radiation only in one side of the antenna, the array antenna

is deposited on a silver layer coated by a dielectric material. The silver layer has a

thickness of 35.7nm = 0.1λ0 and the same material as the shell (εrs = −4.67 + 0.01i).

The dielectric material is made of a thin SiO2 film with εrd = 2.2 and thickness of

3.571nm. Considering the thin thicknesses of the substrate layers, they do not have

much effects on the reflection phase, and they will only help to suppress the back

radiation. Hence, one can still use the reflection phase curve demonstrated in Fig. 5.6

for an array of core-shell nanoparticles located in free-space. This will be validated

later by both our theoretical and full-wave numerical models.

Let us first consider the 15 scan angle case (θ0 = 15, φ0 = 0). From Eq. (5.6), one

can first determine the required reflection phases for the array elements. Then, from

Fig. 5.6 the required material parameters for the nanoparticles cores are evaluated, as

given in Table 5.1(a). The cores materials range from εrd = 1.2 to εrd = 3.9. The Mie

theory formulations discussed in section 5.3 is used to determine the equivalent dipole

modes for each of the nanoshells as illustrated in Table 5.2(a). Since the values of pys

are much smaller than those of px and pz dipoles, they are not shown in this table.

The values are normalized to maximum of |p|. The radiation pattern for the array

of dipoles elements located above the layered substrate is obtained in Fig. 5.7(a).

91

Table 5.1: Core relative permittivity of nanoantenna array elements: (a) θ0 = 15, and(b) θ0 = 30.

(a) θ = 15 (b) θ = 30

m=1 m=2 m=3 m=4 m=5 m=6 m=1 m=2 m=3 m=4 m=5 m=6n=1 1.2 2.25 2.7 2.8 2.84 2.8 1.2 2.1 2.7 2.88 3.47 3.91n=2 1.8 2.67 2.87 3.11 3.3 3.04 1.6 2.53 2.77 3.04 3.76 4.62n=3 2.09 2.76 3.0 2.59 3.69 3.51 1.7 2.59 2.81 2.27 3.92 4.77n=4 2.09 2.76 3.0 2.59 3.69 3.51 1.7 2.59 2.81 2.27 3.92 4.77n=5 1.8 2.67 2.87 3.11 3.3 3.04 1.6 2.53 2.77 3.04 3.76 4.62n=6 1.2 2.25 2.7 2.8 2.84 2.8 1.2 2.1 2.7 2.88 3.47 3.91

This validates a successful 15 beam scanning for the reflected field (25 difference

compared to the beam illumination). The half-power beamwidth is 22 which is an

improvement of about 4 times compared to dipole excitation itself (which has a 90

beamwidth). Note that, the obtained beamwidth for reflectarray is in good comparison

with the performance of a uniform array offering 19 beamwidth [23]. Thus, the 6× 6

reflectarray nanoparticles antenna successfully scans a narrow beam optical emission.

One can reduce the beamwidth even much more by simply increasing the number of

array elements [23]. The magnitude (dB) and phase of the x-directed electric field in

an x-y plane located at z = 0.5λ0 above the plane of nano core-shells are also depicted

in Figs. 5.8.

−12

−9

−6

−3

0

60

120

30

150

0

180

30

150

60

120

90 90

(a)

−12

−9

−6

−3

0

60

120

30

150

0

180

30

150

60

120

90 90

(b)

Figure 5.7: Radiation pattern in the x-z plane at λ0 = 357.1 nm: (a) θ0 = 15, and(b) θ0 = 30.

92

Table 5.2: Induced dipoles, pxs and pzs: (a) θ0 = 15, and (b) θ0 = 30.(a) θ = 15

m=1 m=2 m=3 m=4 m=5 m=6

n=10.57e−i0.99π 0.84e−i0.67π 0.63e−i0.41π 0.68e−i0.31π 0.67e−i0.01π 0.56ei0.18π

0.30e−i0.3π 0.25ei0.06π 0.27ei0.45π 0.20ei0.8π 0.20e−i0.88π 0.07e−i0.65π

n=20.79e−i0.97π 0.66e−i0.74π 0.55e−i0.57π 0.56e−i0.48π 0.5e−i0.07π 0.5ei0.07π

0.39e−i0.19π 0.40ei0.14π 0.39ei0.45π 0.30ei0.87π 0.27e−i0.87π 0.08e−i0.59π

n=30.76e−i0.95π 0.61e−i0.78π 0.56e−i0.63π 0.94e−i0.61π 0.33ei0.01π 0.28ei0.15π

0.26e−i0.04π 0.39ei0.29π 0.32ei0.5π 0.33ei0.97π 0.18e−i0.92π 0.03ei0.16π

n=40.76e−i0.95π 0.61e−i0.78π 0.56e−i0.63π 0.94e−i0.61π 0.33ei0.01π 0.28ei0.15π

0.26e−i0.04π 0.39ei0.29π 0.32ei0.5π 0.33ei0.97π 0.18e−i0.92π 0.03ei0.16π

n=50.79e−i0.97π 0.66e−i0.74π 0.55e−i0.57π 0.56e−i0.48π 0.5e−i0.07π 0.5ei0.07π

0.39e−i0.19π 0.40ei0.14π 0.39ei0.45π 0.30ei0.87π 0.27e−i0.87π 0.08e−i0.59π

n=60.57e−i0.99π 0.84e−i0.67π 0.63e−i0.41π 0.68e−i0.31π 0.67e−i0.01π 0.56ei0.18π

0.30e−i0.3π 0.25ei0.06π 0.27ei0.45π 0.20ei0.8π 0.20e−i0.88π 0.07e−i0.65π

(b) θ = 30

m=1 m=2 m=3 m=4 m=5 m=6

n=10.74ei0.86π 0.82e−i0.61π 0.74e−i0.16π 0.41ei0.17π 0.55ei0.55π 0.24e−i0.93π

0.58e−i0.4π 0.39e−i0.05π 0.33ei0.62π 0.25e−i0.89π 0.12e−i0.38π 0.04ei0.46π

n=20.77ei0.9π 0.37e−i0.55π 0.68e−i0.24π 0.30e−i0.01π 0.48ei0.55π 0.12e−i0.95π

0.88e−i0.34π 0.57e−i0.02π 0.41ei0.57π 0.33e−i0.94π 0.14e−i0.42π 0.02ei0.46π

n=30.78ei0.9π 0.51e−i0.5π 0.54e−i0.26π 0.31e−i0.04π 0.53ei0.55π 0.10e−i0.85π

0.71e−i0.34π 0.58ei0.02π 0.42ei0.54π 0.26e−i0.8π 0.12e−i0.19π 0.07ei0.84π

n=40.78ei0.9π 0.51e−i0.5π 0.54e−i0.26π 0.31e−i0.04π 0.53ei0.55π 0.10e−i0.85π

0.71e−i0.34π 0.58ei0.02π 0.42ei0.54π 0.26e−i0.8π 0.12e−i0.19π 0.07ei0.84π

n=50.77ei0.9π 0.37e−i0.55π 0.68e−i0.24π 0.30e−i0.01π 0.48ei0.55π 0.12e−i0.95π

0.88e−i0.34π 0.57e−i0.02π 0.41ei0.57π 0.33e−i0.94π 0.14e−i0.42π 0.02ei0.46π

n=60.74ei0.86π 0.82e−i0.61π 0.74e−i0.16π 0.41ei0.17π 0.55ei0.55π 0.24e−i0.93π

0.58e−i0.4π 0.39e−i0.05π 0.33ei0.62π 0.25e−i0.89π 0.12e−i0.38π 0.04ei0.46π

Figure 5.8: Near-field (Ex) of the reflectarray for 15 beam scanning [Fig. 5.7(a)] in aplane located at 0.5λ0 above the nanoantenna: (a) magnitude (dB), and (b) phase.

93

The similar procedure can be followed to design a 6 × 6 array for 30 scan angle.

Table 5.1(b) illustrates the required core material parameters where they range from

εrd = 1.2 to εrd = 4.8. Induced dipole modes for the nanoparticles are summarized in

Table 5.2(b). Fig. 5.7(b) illustrates the radiation characteristic. The radiation pattern

shows that the main beam is directed along θ0 = 30 whose half-power beamwidth is

about 30. Figs. 5.9 show the distribution of the Ex in an x-y plane, 0.5λ0 above the

nano particles’ plane.

A full-wave FDTD numerical analysis is also applied to validate our dipole-modes

modeling results, and further explore the effects of finite-size substrate on the radiation

characteristics. To accurately model thin silver shells in FDTD, we need to use very

small-size Yee cells, hence characterizing the whole 6 × 6 array would be very huge

and time-consuming. Instead, we model the core-shell structures with their equivalent

induced dipoles (using Mie analytical model), and then we integrate them with FDTD

numerical technique. This is called hybrid FDTD-dipolar mode technique. In the

FDTD simulation, each nanoparticle is modeled by its induced dipoles given in Table

5.2. There will be 6 × 6 sets of dipoles on top of the finite-size layered substrate.

Since the py is about ten times smaller than px and pz, in our simulation we ignored

the pys. At operating wavelength, λ0 = 357.1nm, the slab size in transverse plane is

3λ0× 3λ0. The hybrid FDTD-dipolar modes is applied and the radiation patterns are

demonstrated in Fig. 5.10. Very good comparisons in compared to the full theoretical

model are presented. The side lobes are slightly increased due to the wave diffractions

from the substrate edges. The FDTD results validate successfully the concept and

radiation performance of the optical reflectarray nanoantennas investigated in this

study.

Frequency sensitivity of the reflectarray design is also explored in this study. The

radiation patterns for scanning 30 at different frequencies f = 0.9f0, 0.95f0, 1.05f0, 1.1f0

are plotted in Fig. 5.11. By changing the frequency, both the size and material (for the

silver coatings) parameters of the nanoparticles will be changed affecting the radiation

94

pattern and degrading the performance.

Figure 5.9: Near-field (Ex) of the reflectarray for 30 beam scanning [Fig. 5.7(b)] in aplane located at 0.5λ0 above the nanoantenna: (a) magnitude (dB), and (b) phase.

−12

−9

−6

−3

0

60

120

30

150

0

180

30

150

60

120

90 90

(a)

−12

−9

−6

−3

0

60

120

30

150

0

180

30

150

60

120

90 90

(b)

Figure 5.10: FDTD radiation pattern in the x-z plane at λ0 = 357.1 nm: (a) θ0 = 15,and (b) θ0 = 30. Good comparisons compared to dipole-modes theoretical results(5.7) are observed.

5.5 Conclusions

This study presented the concept of reflectarray nanoantenna implementation in op-

tics for the first time, with the use of array of core-shell nanoparticles. Optimized

95

geometry-material plasmonic nanoparticles determine successfully the required reflec-

tion phases for desired far-field manipulation. Efficient dipole-modes theoretical model

and FDTD full-wave numerical method are applied to demonstrate the physics of ar-

ray of nanoantennas and fully characterize the radiation characteristics. Successful

narrow-beamwidth directive emission is demonstrated. The radiation pattern results

illustrate that the reflectarray nanoantenna is able to very effectively shape the beam

and scan desired directions. Increasing the number of array elements and optimizing

the particles configurations will lead to an entirely new paradigm for efficient wireless

communication in optics.

−10

−5

0

60

120

30

150

0

180

30

150

60

120

90 90

−10

−5

0

60

120

30

150

0

180

30

150

60

120

90 90

−10

−5

0

60

120

30

150

0

180

30

150

60

120

90 90

−10

−5

0

60

120

30

150

0

180

30

150

60

120

90 90

(b)(a)

(c) (d)

Figure 5.11: Radiation patterns in the x-z plane at different frequencies for 30 beamscanning: (a) f = 0.9f0, (b) f = 0.9f0, (c) f = 0.9f0, and (d) f = 0.9f0 ( f0 = 840 THzis the design frequency).

96

Chapter 6

Optical Nanoloops Array Antenna

6.1 Introduction

Antenna is a key element in the microwave spectrum to enable wireless data commu-

nication. The extension of this concept into the optics has many applications and has

been a growing research in recent years [87, 89–96]. Among the technological appli-

cations for optical antennas one can find high-resolution microscopy and spectroscopy,

optical sensors, lasing, solar cells and efficient solid-state light sources, and it has also

become important in biotechnology and medicine.

As we discussed in previous chapters, noble metals in optics offer negative per-

mittivity parameter where a high scattering performance in subwavelength sizes can

be achieved. Arraying subwavelength plasmonic elements in unique configurations

can successfully engineer the optical emission. Recently, optical nanodipoles made of

plasmonic materials and their arrangements in Yagi-Uda definition have been theoret-

ically and experimentally characterized to modify optical emission by various groups

[87, 89–92, 94, 97]. The common Yagi-Uda antenna achieves a high directivity by

placing several scatterers around a resonant feed element. On one side of the feed, the

scatterers are slightly capacitive called directors, and the elements on the other side

are inductive and called reflectors. To use this concept in optics, it has been suggested

97

to place an emitter in an array of properly tuned particles. In recent works, plasmonic

dipoles and core-shell nanospheres have been introduced as the elements of a Yagi-Uda

antenna. In order to achieve a higher directivity, we propose in this chapter to use

loop elements in Yagi-Uda array. Resonant loop antennas are attractive because of

their symmetric radiation patterns and the potential for offering higher directivity. In

this chapter, we first review the optical nanodipole antenna and study the radiation

characteristic of a Yagi-Uda antenna with plasmonic dipoles as the radiating elements.

Then, we investigate the plasmonic nanoloop element and design a highly directed

array antenna for optical frequencies.

6.2 Optical Nanodipole Antennas

Dipole antennas are some of the oldest, simplest and cheapest for many applications

in microwave. In traditional antenna design, characteristic lengths L of antennas are

directly related to the wavelength λ of the incoming (or outgoing) radiation. For ex-

ample, an ideal half-wave dipole antenna is made of a thin rod of length L = λ/2.

However, at optical frequencies an antenna no longer depends to the external wave-

length but to a shorter effective wavelength λeff which depends on the material prop-

erties and the shape of structure [91, 96]. In chapter 4, we studied the effects of

the shape and material dispersion of epsilon negative (ENG) radiators on their reso-

nance characteristics. We demonstrated that the material polarization can successfully

provide resonance radiation at the negative material constitutive parameters. Here,

we consider a small plasmonic particle to study its radiation characteristics and its

performance in an array arrangement.

Let us consider a dipole antenna made of a cubic rod of dielectric material ε(ω),

length H = 120 nm and width W = 30 nm (H/W = 4). Fig. 6.1(a) shows the

geometry of our model. The frequency dependent dielectric function of the metal is

98

described by the Drude model as below,

ε(ω) = ε0

(1− ω2

p

ω(ω + iγp)

), (6.1)

where ωp is the plasma frequency of the material, and damping factor γp represents

the losses present. Here, the dipole antenna is made of Silver (ωp = 2π×2175THz and

γp = 2π × 4.35THz [98]).

An incident plane wave with wavelength λ having z-polarized electric field and

traveling in the x direction illuminates the structure and polarizes the material. The

material polarization provides resonance radiation at a negative material constitutive

parameter. Finite Difference Time Domain technique [34] is applied to simulate the

structure and determine field characteristics. In FDTD model, the computational do-

main is configured with cubical Yee cells with ∆ = 5nm and the structure is illuminated

by a plane wave. The dipole is placed at the center of computational domain (0,0,0)

and the scattered field is stored at (0,0,50∆) in front of the dipole. The scattered

field is plotted in Fig. 6.1(b). A high scattering performance at resonant wavelength

of λ = 760 nm is observed. At resonance, the plasmonic dipole has subwavelength size

of total length of H = λ/6.3.

To study the radiation performance of the resonant dipole, we excite the dipole by

an Ez source placed at the center of dipole (0, 0, 0). To study the angular dependence

of the emission, the angular directivity D(θ, φ) is calculated. The directivity D at a

giving direction (θ, φ) is defined as [23],

D(θ, φ) = 4πU(θ, φ)

Prad

(6.2)

where Prad is the total radiated power of the whole antenna system, U(θ, φ) is the

radiation intensity at the observation angel (θ, φ). The directivity D(θ) in the plane

φ = 0 is plotted in Fig. 6.2. The maximum directivity of D = 1.9dB is obtained.

99

(a)

0.4 0.6 0.8 1 1.20

0.05

0.1

0.15

0.2

0.25

Wavelength (µm)M

agn

itu

de

of

Ez

(b)

Figure 6.1: A single plasmonic dipole antenna illuminated by an z-polarized electricfield plane wave, W = 30nm, H = 120nm: (a) structure, (b) resonance performance.

−3

−1

1

3

5

60

120

30

150

0

180

30

150

60

120

90 90

Figure 6.2: Directivity (in dB) for resonant plasmonic dipole antenna in plane φ = 0.Maximum directivity is 1.9dB.

6.2.1 Optical Nanodipole Yagi-Uda Antennas

In the previous section, we discussed that plasmonic particles at scattering resonance

can be used as optical antenna elements. Arranging the antenna elements in an array

design will enhance radiation characteristics. Yagi-Uda antenna concept is an approach

of designing an endfire-type optical antenna array.

Yagi-Uda antenna is one of the most popular endfire antenna designs in the RF/

microwave domain. It was first developed and described by S. Uda in Japan [99] and

100

later on discussed and made famous by H. Yagi [100]. The Yagi-Uda antenna array

consists of several linear dipole antennas in which only one of them is driven by the

source. The other elements are parasitic radiating elements whose current are induced

by the mutual coupling to the source antenna and to each other. To design a Yagi-Uda

antenna, the impedance of each element on the right hand side of the driving element

should be capacitive and its current leads the inducing emf. The element on the left

hand side is the opposite. The key feature in this design is the length of each element.

The parasitic elements are not of the same length. The elements on the right hand

side of the driving element are a little bit shorter than the resonant length, while the

one on the left is a little longer.

The idea of Yagi-Uda antenna array have been transplanted into optical nanoan-

tenna array design by different authors [82, 97]. To apply the design concept of the

Yagi-Uda array, we begin the design process with three elements: One reflector, one

emitter, and one director. Fig. 6.3 shows the structure. The length of emitter based

on the pervious section is set to be he = 120nm to have the strong resonance at

λ = 760nm but the length of reflector and director have been modified. We declare

the particle at the left side an “inductive” element because the effective induced cur-

rent of the dipole has a phase that lags with respect to the phase of the incident field.

For a similar reason, the particle at the right is a “capacitive” one. The directivity

D(θ) in the plane φ = 0 is plotted in Fig. 6.4. The maximum directivity of D = 3.6dB

is obtained.

In RF/microwave domain, it is discovered that arrays with more than one director

can provide radiation patterns with narrower bandwidth. Fig. 6.5 shows an array of

5 elements and the directivity D(θ) in the plane φ = 0 is plotted in Fig. 6.6. As seen,

the directivity is improved.

101

Figure 6.3: 3-element nano-optical Yagi-Uda antenna for an operating wavelength of760 nm. hr = 130nm, he = 120nm, hd = 105nm, d = 100nm.

−3

−1

1

3

5

60

120

30

150

0

180

30

150

60

120

90 90

Figure 6.4: Directivity (in dB) for the Nano-optical Yagi-Uda antenna shown inFig. 6.3in plane φ = 0. Maximum directivity is 3.6dB.

Figure 6.5: 5-element nano-optical Yagi-Uda antenna for an operating wavelength of760 nm. hr = 130nm, he = 120nm, hd = 105nm, d = 100nm.

102

−3

−1

1

3

5

60

120

30

150

0

180

30

150

60

120

90 90

Figure 6.6: Directivity (in dB) for the Nano-optical Yagi-Uda antenna shown in Fig. 6.5in plane φ = 0. Maximum directivity is 4.5dB.

6.3 Optical Nanoloop Antennas

A microwave metallic loop antenna goes to resonance where βb = 1, 2, 3, ...(b: mean

radius of a circular loop) [101]. This means the required minimum loop diameter to

achieve a high scattering is around 0.3λ. A different scenario is expected in optics

as the incident wave penetrates into the metal and gives rise to oscillation of the

free-electron gas generating surface plasmon (SP) modes [91]. The optical antenna

is scaled down by the effective wavelength of the plasmonic material which depends

on the material property and the shape of the structure [91, 96]. For instance, for a

rectangular loop antenna (constructed from four cubical arms), the higher the aspect

ratio of the metallic arms the higher the operating wavelength and thus the smaller

the size of the resonator.

Here, the physics of the plasmonic loop is integrated into the design of optical

nanoloops array antenna. First, we consider a single plasmonic loop antenna (rectan-

gular) of dielectric material ε(ω), side length l, and thickness t. Fig. 6.7 shows the

parameters used in our model. The frequency dependent dielectric function of the

metal is described by the Drude model Eq. (6.1). Here, the loop antenna is made of

Silver with ωp = 2π × 2175THz and γp = 2π × 4.35THz [98].

103

Figure 6.7: A single plasmonic loop antenna illuminated by an x-polarized electricfield plane wave, l = 85nm, t = 15nm.

An incident plane wave with wavelength λ illuminates the structure and polarizes

the material. The material polarization provides resonance radiation at a negative

material constitutive parameter.

Finite Difference Time Domain technique [34] is applied to simulate the structure

and determine field characteristics. In FDTD model, the computational domain is

configured with cubical Yee cells with ∆ = 5nm and the structure is illuminated by

a plane wave having x-polarized electric field and traveling in the −z direction. The

loop is placed at the center of computational domain (0,0,0) and the scattered field

is stored at (0,0,30∆) in front of the loop. The scattered field is plotted in Fig. 6.8.

A high scattering performance at resonant wavelength of λ = 1.34µm is observed. At

resonance, the plasmonic loop has subwavelength size of total length of 4l = λ/3.9.

The x- and y-components of polarized current distribution are plotted at the resonance

frequency in Figs. 6.9. A successful current circulation is observed.

To study the radiation performance of the resonant loop, we excite the loop by an

Ex source placed at (0, l/2, 0). The 3D power pattern is presented in Fig. 6.10. The

drawing is a series of patterns on planes of constant angles φ. Eφ is almost zero in

the vertical plane φ = 0, π, while Eθ is small in the vertical plane φ = π/2, 3π/2. The

D(θ) in two major planes, φ = 0 and π/2, are plotted in Figs. 6.11. The maximum

104

directivity of D = 2dB is obtained.

0.5 1 1.5 2 2.5 30

0.05

0.1

0.15

0.2

0.25

0.3

Wavelength (µm)

Mag

nit

ud

e o

f E

x

Figure 6.8: Resonance performance of single plasmonic loop. High scattering occursat λ = 1.34µm.

6.3.1 Optical Nanoloops Array Antenna

The next step will be to enhance the directivity of the antenna. To achieve this, first

the back radiation is suppressed by depositing the loop antenna on a silver substrate.

Then we use two more loop antennas above the structure to increase the gain. Fig. 6.12

shows the nanoloop array antenna configuration, inspired by the Yagi-Uda concept.

The antenna consists of one exciter and two directors which are printed on the low

dielectric substrate MgF2 with εd = 1.5 [102]. The silver slab in the back can be

envisioned as the reflector. The size and spacing of the directors are adjusted to

successfully engineer the phase of the scattered fields and enhance the beam radiation

in the upward direction. The design parameters are shown in Fig. 6.12.

Figs. 6.13 shows D(θ) of the array antenna in the planes φ = 0 and π/2. The 3D

power pattern is also plotted in Fig. 6.14. The maximum directivity is about 8.2dB

which is 4.2 times improvement in the power radiation of a single dipole performance.

It is clearly observed that in compared to the single loop antenna (Fig. 6.11) the beam

105

Figure 6.9: Polarized current on plasmonic loop at resonant wavelength λ = 1.34µm:(a) normalized |Jx| (dB), and (b) normalized |Jy| (dB). The current distribution issimilar to what one observes in microwave for a rectangular loop antenna with 4l ' λ(The size becomes subwavelength in optics.)

Figure 6.10: Far-zone power pattern for single plasmonic loop at the operating wave-length.

106

Figure 6.11: Directivity (in dB) for resonant plasmonic loop antenna in planes (a)φ = 0, and (b) φ = π/2 Maximum directivity is 2dB.

Figure 6.12: Schematic view of nanoloops antenna array. At operating wavelength ofλ = 1.34µm, the emitter element has the resonant size of 4l1 = 340nm=λ/3.9, andthe directors lengths are 4l2 = 4l3 = 260nm. The reflector spacing is t1 = 125nm,and the directors spacings are t2 = t3 = 375nm. The emitter and the directors areprinted on low dielectric substrates with εd = 1.5. The silver slab has the thickness ofts = 205nm. A finite-size structure of ls = 500nm in the transverse plane is considered.The yellow arrow shows the excitation.

107

is more collimated and the half-power beamwidths of 71 and 82 are obtained in φ = 0

and φ = π/2 planes, respectively. Increasing the number of array elements can result

in further enhancement in the radiation performance. This can be of great advantage

for enhancing the optical emission with potential integration in many emerging opti-

cal applications such as wireless optical communications, sensing, and molecular and

quantum-dot boosted emissions.

To provide a physical insight of the plasmonic loops nanoantenna, the near field

performance in the antenna array is depicted in Fig. 6.14. In this figure, the magnitude

of electric field in the xz plane is plotted. The Ex-source sets up a strong electric field

in the exciter loop which can induce the surface plasmon modes on the director loops,

and as a result the whole system radiates efficiently and directs the beam successfully.

Figure 6.13: Directivity (in dB) for parasitic plasmonic loop array antenna in planes:(a) φ = 0, and (b) φ = π/2. The emission of the coupled system is highly directedtowards upward. Maximum directivity of 8.2dB is established.

108

Figure 6.14: Far-zone power pattern for the array antenna. The power is highlydirected towards the upper hemisphere and the back radiation is suppressed. Successfulcollimation in compared to Fig. 6.10 is illustrated.

x (µm)

z (µ

m)

−0.3 −0.2 −0.1 0 0.1 0.2 0.3

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0−50

−40

−30

−20

−10

0

Figure 6.15: Electric field distribution induced on the nanoloops antenna array at theoperating wavelength of λ = 1.34µm (Normalized and plotted in dB.)

109

6.4 Conclusions

In summary, we have investigated a highly directive nanoantenna array whose ele-

ments are plasmonic nanoloops. The beam is collimated and high directivity of 8.2dB

is achieved by arraying three loops over a silver substrate where the total size in the z

direction does not exceed 1.125µm ' 0.8λ. The size in transverse direction is 500nm.

The resonant size of the plasmonic loop is scaled down by a factor of about 3.9 in

compared to the size of a microwave loop antenna. The directivity can even be more

enhanced by increasing the number of directors. The loop nanoantenna has the ad-

vantage of offering higher directivity in compared to the traditional dipole antenna

design. The nanoloops array concept proposed in this letter can provide significant

benefits for optical boosted emissions and relevant nanoscale applications.

110

Chapter 7

Conclusions and Recommendations

for Future Work

7.1 Summary and Contributions

This thesis reviewed the concept of Metamaterials and investigated some novel appli-

cations in near-field imaging and antenna design. The following subsections outline

the contributions of this dissertation.

7.1.1 Design and Development of All-Dielectric Metamateri-

als

In this dissertation, a comprehensive investigation of all-dielectric metamaterials is

addressed. The concept of electric and magnetic dipole modes generation for meta-

material development is presented. To achieve a metamaterial with desired figure of

merit, one needs to first create appropriate electric and magnetic dipole moments and

then tailor them to the application of interest. Primarily, the electric and magnetic

dipole moments are the basic foundations for making metamaterials. We implemented

that dielectric resonators can successfully provide electric and magnetic dipole modes.

111

We examined dielectric disk and spherical particle resonators to create required dipole

moments. Using this concept, a DNG metamaterial constructed from two sets of disks

or spheres, having the same size but different materials, is developed.

A very unique approach for the bandwidth enhancement of metamaterials was

presented. We demonstrated that the bandwidth of the resonant modes of the all-

dielectric metamaterials can be improved by increasing the couplings between the

particles. Basically, when we make the spheres closer to each other, the mode radiation

through the spheres is increased causing the reduction in the Q factor of each of the

spheres, resulting in the bandwidth enhancement of the resonant modes.

The concept of all-dielectric metamaterials can be extended to the optical frequen-

cies; however, because of the fabrication limitations one needs to use smaller value

dielectric materials for the resonating inclusions. In this case, larger-size resonators

may be implemented. We realized optical metmaterials by characterizing the perfor-

mance of the periodic array of GaP spheres implanted inside cesium (Cs) host material.

In general, it was discussed that all-dielectric metamaterials appear very promising

for addressing some of the important physical and engineering concerns, such as the

loss and bandwidth. They are quite feasible for fabrication in both microwave and

IR-visible spectrums.

7.1.2 Novel Applications of Metamaterials

After investigation on design and development of metamaterial structures, in the sec-

ond part of the dissertation we explored some of the possible applications of meta-

materials. A comprehensive investigation of high resolution imaging utilizing Fourier

spectrum theoretical model and full-wave FDTD numerical analysis was addressed.

It was described that the high resolution imaging in transverse plane appears as a

result of the amplification of evanescent waves. Then, coupled layered surfaces sup-

porting surface modes were investigated, enabling high resolution imaging along the

112

transverse direction. The study of surface-modes layers composed of ENG materials

showed that in the region where the material has negative permittivity, the layered

structure supports surface modes which can be excited by evanescent waves (p-waves).

It is demonstrated that by combining the ENG and MNG layers, a near-field imag-

ing composite is realized that functions properly for both p and s polarizations. It is

highlighted that since one can make thin layers and cascade them in proper fashion to

achieve the image successfully, one can expect the effect of loss for the layered structure

to be much smaller than that of the bulk NIM slab.

We also presented new designs of optical nanoantennas and nanoarrays. Noble

metals in optics offer negative permittivity parameter where a high scattering perfor-

mance in subwavelength sizes can be achieved. We presented the concept of reflectarray

nanoantenna implementation in optics for the first time, with the use of array of core-

shell nanoparticles. Optimized geometry-material plasmonic nanoparticles determined

successfully the required reflection phases for desired far-field manipulation.

Arraying subwavelength plasmonic elements in unique configurations can success-

fully engineer the optical emission. We investigated a highly directive nanoantenna ar-

ray whose elements are plasmonic nanoloops and we designed an high directed nanoan-

tenna by arraying three loops over a silver substrate. The loop nanoantenna has the

advantage of offering higher directivity in compared to the traditional dipole antenna

design. The nanoloops array concept proposed in this dissertation can provide signif-

icant benefits for optical boosted emissions and relevant nanoscale applications.

The main results presented here were based on our scientific papers listed at the

end of this chapter.

7.2 Future Work

This dissertation addressed a very broad area of research in modern electromagnetics

and optics. The motivations to study optical antennas are obvious, as we discussed

113

the radio frequency spectrum is tightly allocated and the spectrum becomes a rare

resource. Optical antennas will make it possible to benefit from the spectrum resources

in infrared and optical domain which cannot be used currently. In this dissertation

we discussed the possible applications of plasmonic materials to build antenna devices

radiating and receiving electromagnetic energy at optical frequencies. We used the

plasmonic nanoparticles around their surface plasmon scattering resonance as optical

antenna elements and integrated them in array arrangements of reflectarray antennas

and Yagi-Uda antennas. With the success of transplanting the idea of these antennas

from the conventional RF/microwave domain design into the optical design, it would

come next to think what else in the RF/microwave domain can be transferred and be

used in optics.

Antennas have been impervious to the rapidly advancing semiconductor industry.

On the way of using the interesting features of metamaterials, recently there has been

started a new brand of research to incorporate active components into an antenna and

transform it into a new kind of radiating structure that can take advantage of the latest

advances in analog circuit design. The approach for making this transformation is to

make use of non-Foster circuit elements in the matching network of the antenna. Non-

Foster impedance matching is defined as the use of negative inductors and negative

capacitors to manage the transfer of power between a source and a load. A great

deal of efforts is needed to fully investigate this new concept and provide more novel

applications.

114

Publications

Journal Papers

[J1]. Akram Ahmadi and Hossein Mosallaei, “A Plasmonic Nanoloops Array An-

tenna,” submitted to Optics Lett. (2010).

[J2]. Akram Ahmadi, Soheil Saadat, and Hossein Mosallaei, “Resonance and Q

Performance of Ellipsoidal ENG Subwavelength Radiators,” accepted for publication

in IEEE Trans. Antennas and Propagation (2010).

[J3]. Akram Ahmadi, Shabnam Ghadarghadr, and Hossein Mosallaei, “An Optical

Reflectarray Nanoantenna: The Concept and Design,” Optics Express, Vol. 18, No.

1, 123-133 (2009).

[J4]. Akram Ahmadi and Hossein Mosallaei, “On the Image Performance of Nega-

tive Index Slab and Coupled Layered Resonant Surfaces,” Journal of Applied Physics,

Vol. 106, 064502 (2009).

[J5]. Akram Ahmadi and Hossein Mosallaei, “Physical configuration and perfor-

mance modeling of all-dielectric metamaterials,” Physical Review B, Vol. 77, 045104

(2008).

[J6]. Shabnam Ghadarghadr, Akram Ahmadi, and Hossein Mosallaei, “Negative

Permeability-Based Electrically Small Antennas,” IEEE Antennas and Wireless Prop-

agation Letters, Vol. 7 (2008).

Conference Papers

[C1]. Akram Ahmadi and Hossein Mosallaei, “Array of Plasmonic Antennas: The

Concept and Novel Applications,” International Conference on Materials for Energy,

Boston, Oct. (2010).

[C2]. Akram Ahmadi and Hossein Mosallaei, “Near-Field Imaging of Coupled

Surface-Wave Layers,” in Conference on Lasers and Electro-Optics/International Quan-

tum Electronics Conference, OSA Technical Digest (CD) (Optical Society of America),

Baltimore (2009).

[C3]. Akram Ahmadi and Hossein Mosallaei, “Ellipsoidal Negative Parameters

Metamaterial Subwavelength Radiators,” in National Radio Science Meeting, Boulder

(2009).

[C4]. Akram Ahmadi and Hossein Mosallaei, Negative Index Meta-Devices Imaging

and Engineering Concerns, URSI General Assembly, Chicago (2008).

[C5]. Akram Ahmadi and Hossein Mosallaei, All-dielectric metamaterials: double

negative behavior and bandwidth-loss improvement, IEEE Antennas and Propagation

115

International Symposium (2007).

[C6]. Hossein Mosallaei and Akram Ahmadi, From photonic crystals to metama-

terials: physical insights and engineering aspects, IEEE Antennas and Propagation

International Symposium (2007).

[C7]. Hossein Mosallaei and Akram Ahmadi, ”Electric and magnetic dipole mo-

ments of dielectric resonators: An all-dielectric metamaterial design,” International

Congress on Advanced Electromagnetic Materials in Microwaves and Optics, Rome,

Italy, Oct. 22-26 (2007).

[C8]. Hossein Mosallaei and Akram Ahmadi, ”Metamaterial development utiliz-

ing nanoparticle resonators,” URSI National Radio Science Meeting, Ottawa, ON,

Canada, July 22-26 (2007).

Workshops:

[W1]. J. Wu, S. Ghadarghadr, A. Ahmadi, and H. Mosallaei, Metamaterials for Mi-

crowave and Optical Devices, Research and Scholarship Expo, Northeastern University

(2008).

116

Appendix A

Photonic Band Gap Calculations

Using FDTD Method

In the text, we present the band structures for some photonic crystals and explain the

interesting features of each. We use FDTD numerical method to determine the band

structure and here in this appendix we explain how to do so.

Photonic bandgaps are typically visualized and investigated by computing the dis-

persion relationship, ω(k), between the temporal and spatial frequencies of the modes

that can propagate in the particular periodic structure of interest.

To compute the band diagram of a photonic crystal lattice, we use standard FDTD

on the unit cell of that lattice with Bloch or Floquet boundary conditions. Suppose we

have a function f(r) that is periodic on a lattice; that is, suppose f(r) = f(r+R) for

all vectors R that translate the lattice into itself. The discrete translational symmetry

of a photonic crystal allows us to classify the electromagnetic modes with a wave vector

k. The modes can be written in “Bloch” or “Floquet” form, consisting of a plane wave

at arbitrary oblique angles modulated by a function that shares the periodicity of the

lattice. So for the E and H fields on can have:

117

E(r+R, t) = E(r, t)eik.R (A.1a)

H(r+R, t) = H(r, t)eik.R (A.1b)

To implement this, one need to only make sure that the fields that leave one side of

the FDTD model immediately appear on the other side, multiplied by the appropriate

complex number. The computational domain is chosen to be a unit cell of the infinite

crystal. After the initial excitation, fields oscillate in a steady state that is a linear

combination of several eigenstates with the same wave vector k. Frequencies of these

eigenstates can be obtained by a Fourier transformation of the time-domain amplitude

at a given point. The resulting spectrum is composed of a discrete set of peaks, where

each peak corresponds to an eigenfrequency.

Modes in the computational cell are excited using one or several point dipole sources

with Gaussian frequency-profile amplitudes. The oscillation period and the width of

the Gaussian are chosen such that the excitation spectrum covers the frequency range

of interest. In determining the band structure, we use a short pulse in time that

excites a wide frequency range. Both the dipoles and the point where the field is

recorded are placed away from all the symmetry planes, so that modes with different

symmetries can be excited and recorded in one simulation. Instead of exciting several

modes simultaneously using a pulse with a wide spectral range, we can also use a

narrow source (i.e., long duration in time) to selectively excite only one eigenstate at

a specific frequency. The symmetry of the steady state can further be specified by

placing the dipoles in appropriate symmetrical configurations.

Rectangular Photonic Crystal Lattice

As discretization is performed on a rectangular lattice, a natural choice for the compu-

tational domain is rectangular. For example, let us consider an infinite two-dimensional

118

square lattice of circular dielectric cylinders in air as shown in Fig. A.1(a). Fig. A.1(c)

illustrates the band diagram for the structure. The irreducible Brillouin zone for the

structure is a triangle shown in the Fig. A.1(b). Here, a is the center-to-center spacing

of the cylinders and the cylinder relative permittivity is 8.9. In the figure, the vertical

axis represents the normalized temporal frequency ωa/(2πc), where c is the speed of

light. The horizontal axis represents the spatial frequency or k-vector, represented

in three segments to correspond to the edges of the triangular irreducible Brillouin

zone for this lattice. The edges of this Brillouin zone suffice because the modes in the

interior of this zone are, in general, bounded by those on the periphery, and because

once irreducible Brillouin zone is known, then the entire extent of reciprocal space is

known by either symmetry or translation.

(a) (b) (c)

Figure A.1: An infinite two-dimensional square lattice of circular dielectric cylindersin air: (a) the schematic of structure, (b) Brillouin zone, and (c) dispersion diagramfor TMz polarization (plotted in blue) and for TEz polarization (in plotted red).

For each k, a short pulse in time that excite a wide frequency range is applied. For

example, Fig. A.2 shows the magnitude of Ez at one point inside the unit cell. Each

peak corresponds to an eigenfrequency.

119

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8−50

−40

−30

−20

−10

0

10

Normalized Freq. ( ωa/2πc )

Figure A.2: Spectral amplitude at X (kx = π/a, ky = 0) for the infinite two-dimensionalsquare lattice of circular dielectric cylinders in air.

Triangular Photonic Crystal Lattice

For triangular photonic crystal lattices, since in our FDTD code we are using the

cubic Yee cell we prefer to have a cubic lattice cell for band diagram calculation. For

instance, let us consider a triangular lattice of air holes in a dielectric as shown in

Fig. A.3(a). To obtain the dispersion diagram, a cubic unit cell is employed which

contains two primitive cells as shown in Fig. A.3(a). As Eq.A.1 only determines the

phase relations between different cubic cells, the band structure obtained is a folded

version for the underlying lattice. To obtain unfolded band structures we need to

specify the phase relation across different primitive cells. This is achieved by placing a

dipole in each of the two primitive cells. The dipoles are separated by a lattice vector,

the relative phase between them satisfying Blochs theorem.

TE dispersion diagram of the structure (r/a = 0.48) in a dielectric (εr = 13) is

shown in Fig. A.3(c).

120

(a) (b) (c)

Figure A.3: An infinite two-dimensional triangular lattice of air holes (r/a = 0.48) ina dielectric (εr = 13): (a) the schematic of structure. The dotted rectangle shows theunit cell which we use for bang-gap calculation, (b) Brillouin zone, and (c) dispersiondiagram for TEz polarization.

121

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