Determining the Effective Parameters of Metamaterials

By

Jonathan Woodley

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

Copyright © 2012 by Jonathan Woodley Determining of the Effective Parameters of Metamaterials Jonathan Woodley Doctor of Philosophy Department of Electrical and Computer Engineering University of Toronto, 2012 Abstract

In this dissertation the proper determination and allowable signs of the effective parameters of metamaterial structures will be examined. First, a method that was commonly used to determine the presence of a negative index of refraction will be discussed. It will be shown that this method, which relies on the appearance of transmission peaks in the region where the real parts of the effective permittivity and permeability are expected to be negative, does not provide sufficient evidence that a negative index exists. Two alternate methods will then be presented that can be used to properly determine the sign of the index. Then, the form of the index in media that exhibit backward wave propagation will be examined from a purely three dimensional wave propagation point of view. It will be shown that in an isotropic medium backward wave propagation requires that the index be negative and in an anisotropic medium it requires that the index be negative along at least one of the three principal axes. In short, the necessary and sufficient condition for the negative index of refraction is the existence of the backward wave. Next, a technique commonly used to retrieve the effective parameters in metamaterials from transmission and reflection data will be considered. It will be shown that this retrieval technique can lead to unphysical claims that the imaginary parts of the effective permittivity or permeability can be negative even though the medium remains passive. By comparing the effective parameters obtained analytically and from the retrieval technique it will be shown that these unphysical claims are the result of error in the numerical simulations. The concepts of causality and analyticity will also be discussed by considering the Lorentzian model and it will be shown that this model does not allow the imaginary parts of the permittivity or permeability to be negative in the metamaterials consisting of split ring resonators and split wires.

ii Acknowledgements

I would like to thank my supervisor Professor Mojahedi for his advice and guidance. I would also like to thank my fellow graduate students for all the many helpful conversations and walks. Finally, no acknowledgments section would be complete without a mention of the author’s family. I would like to thank them above all else.

iii Table of Contents

Chapter 1. Introduction and Background...... 1

1.1 Introduction...... 1 1.1.1 Veselago and Left-Handed Media ...... 1 1.1.2 Negative Refraction ...... 3 1.1.3 Propagation in a Negative Index Medium ...... 6 1.2 The First Negative Index Metamaterials...... 7 1.2.1 The Array of Wires ...... 7 1.2.2 The Array of Split Ring Resonators...... 9 1.2.3 The First Negative Index Medium...... 11 1.2.4 Experimental Verification of Negative Refraction...... 13 1.3 Conclusion and Thesis Goals...... 15 1.4 Chapter Summary ...... 16

Chapter 2. Left-Handed and Right-Handed Metamaterials...... 21

2.1 Introduction...... 21 2.2 The Transmission Magnitude ...... 23 2.3 The Transmission Phase ...... 26 2.4 Dispersion Diagrams...... 29 2.5 The Effects of Unit Cell Size...... 32 2.6 Conclusion ...... 34

Chapter 3. Backwards Waves...... 37

3.1 Introduction...... 37 3.2 “Perfect” Backwards Waves...... 38 3.3 Backward Wave Transmission Line ...... 43

iv 3.4 “Imperfect” Backward Waves ...... 44 3.5 The Requirements of Dispersion IN negative Index Metamaterials...... 46 3.6 Anisotropic Media ...... 47 3.7 Conclusion ...... 55

Chapter 4. The Retrieval Technique...... 57

4.1 Introduction...... 57 4.2 The Retrieval Technique...... 58 4.2.1 An Array of Split Wires...... 61 4.2.2 An Array of Split Ring Resonators...... 65 4.2.3 Continuous Strip Wires and Split Ring Resonators...... 68 4.3 Conclusion ...... 72

Chapter 5. The Imaginary Parts of the Effective Permittivity and Permeability.....75

5.1 Introduction...... 75 5.2 The Lorentzian Model for Metamaterials...... 77 5.3 An Array of Dielectric Spheres...... 81 5.3.1 Mie Theory...... 82 5.3.2 Numerical Simulations...... 89 5.4 Discussion...... 90 5.5 Examining the Sensitivity of the Retrieval Technique ...... 96 5.6 Retrieval on a Dispersionless Dielectric Slab ...... 99 5.7 Other Homogenization Techniques ...... 103 5.8 Conclusions...... 106

Chapter 6. Conclusion ...... 111

List of Contributions...... 114

v Appendix A. The 6 Family Model ...... 115

A.1 Introduction...... 115 A.2 Symmetry Operations ...... 116 A.3 The 6 Families...... 117 A.4 The 6 Families in the 4-Quadrant Model...... 121 A.5 Conclusion ...... 127

Appendix B. The Derivation of Heat Evolution ...... 129

B.1 Heat evolution of Non-Monochromatic Waves ...... 129 B.2 Heat Evolution of Monochromatic Waves...... 131

Appendix C. Mie Theory and the Array of Dielectric Spheres ...... 133

C.1 Introduction ...... 133 C.2 Field patterns resulting from the sphere resonances ...... 133 C.3 Obtaining the expressions for the effective permittivity and permeability...137

vi List of Tables

Chapter 3. Backwards Waves

3.1 Index seen by different polarizations in the one- and two-sheeted hyperboloids...... 52

Appendix A. The 6 Family Model

A.1 Signs of u and v in each of the 6 Families...... 121 A.2 The values of m and l, the signs of u and v and the form of the index in each quadrant of the u-v plane for the passive case...... 124 A.3 The values of m and l, the signs of u and v and the form of the index in each quadrant of the u-v plane for the active case...... 126

vii List of Figures

Chapter 1. Introduction

1.1 Incident, reflected, and transmitted wave vectors at (a) an RHM/RHM interface and (b) an RHM/LHM interface...... 4 1.2 A periodic structure of thin conducting wires...... 9 1.3 The split ring resonator and an array of resonators...... 10 -2 -3 1.4 Real and imaginary parts of μeff for ρ = 2000 Ω/m, a = 10 m, c = 10 m, d = 10-4 m, l = 2x10-3 m, and r = 2x10-3 m ...... 11 1.5 The first left handed medium studied by Smith et al. at the University of California San Diego. The dimensions of the rings are a = 10mm, c = 0.8mm, d = 0.2mm, r = 1.5mm. The radius of the wires was 0.4mm...... 12 1.6 Transmission measured by Smith et. al. for propagation through an array of circular SRRs (solid line) and an LHM composed of circular SRRs and metal posts (dashed line) ...... 12 1.7 Smith’s two dimensional LHM [5]. The ring dimensions used were w = 2.62mm, c = 0.25mm, d = 0.3mm, g = 0.46mm. The lattice spacing was 5mm and the strips were 0.5mm wide...... 14 1.8 Smith’s results for refraction experiments on Teflon and a LHM...... 14

Chapter 2. -Handed and Right-Handed Metamaterials

2.1 SRR used in the OS and SS structures. The dimensions of the rings are r = 0.506 mm, c = 0.124 mm, d = 0.15 mm, and g = 0.114 mm...... 23 2.2 (a) Opposite side (OS) structure and (b) same side (SS) structure. In each case the SWs are 0.5 mm wide and the substrate is 0.5mm thick with a dielectric constant of 3.02. The dimensions of the unit cell in the OS configuration are 2.5×2.5×2.5mm and in the SS configuration they are 4×2.5×2.5mm...... 24

viii 2.3 Simulated transmission magnitude for propagation through 4 unit cells of (a) the OS structure and (b) the SS structure. Solid (dashed) vertical lines indicate the points at which the structures make transitions from RHM (LHM) to LHM (RHM) behavior...... 25 2.4 Transmission phase for propagation through 1, 2, 3, and 4 unit cells of (a) OS structure and (b) SS structure. Solid (dotted) vertical lines mark transitions from RHM (LHM) to LHM (RHM) behavior. Arrows mark the locations of the transmission peaks from the corresponding OS or SS plots in Fig. 2.3...... 28 2.5 Dispersion plots for (a) OS structure and (b) SS structure. In the OS configuration the unit cell size was 2.5×2.5×2.5mm. The unit cell size was 4×2.5×2.5mm for the SS configuration...... 31 2.6 Magnetic fields resulting from the currents in the SRR and SW for (a) OS configuration, (b) SS configuration...... 32 2.7 Dispersion curves for the SS structure. In each case the y- and z-dimensions of the unit cell were 2.5mm and the x-dimension was varied...... 34

Chapter 3. Backwards Waves

3.1 Phase index given in (7). Note that b = -1 in the negative index case (branch-I) and b = 1 in positive index case (branch-II)...... 42 3.2 Dispersion relation for the phase index given by (7). Note that b = -1 in branch-I and b = 1 in branch-II...... 42

2 3.3 Index given in (10) normalized by a factor of co . Note that b = -1 in the negative index case (branch-I) and b = 1 in positive index case (branch-II)...... 43 3.4 General one dimensional transmission line model in the phasor domain...... 44 3.5 (a) 3D and (b) 2D k–Space diagrams for RHM and LHM two-sheeted

hyperboloids. In the RHM case the parameters are εs = 1, εz = -2, and µr = 1. The

parameters in the LHM case are εs = -1, εz = 2, and µr = -1. The k-surfaces for the RHM and LHM cases are identical. The angle between the phase velocity and group velocity for the RHM and LHM cases are shown on the 2D plots...... 49

ix 3.6 (a) 3D and (b) 2D k–Space diagrams for RHM and LHM one-sheeted hyperboloids.

In the RHM case the parameters are εs = -1, εz = 2, and µr = 1. The parameters in

the LHM case are εs = 1, εz = -2, and µr = -1. The k-surfaces for the RHM and LHM cases are identical. The angle between the phase velocity and group velocity for the RHM and LHM cases are shown on the 2D plots...... 51 3.7 Calculated angle between the phase and group velocity vectors for the RHM and LHM hyperboloidal k-surfaces considered. (a) Two-Sheeted hyperboloid and (b) one-Sheeted hyperboloid...... 53

Chapter 4. The Retrieval Technique

4.1 Unit cell for (a) the split wire, (b) the SRR and (c) a continuous SW and SRR...... 60

4.2 (a) S11 and (b) S12 for propagation through an array of split wires infinite in the transverse plane (x-y) and one unit cell thick in the propagation direction...... 62 4.3 Real and imaginary parts of (a) effective index of refraction and (b) characteristic impedance of an array of split wires...... 63 4.4 Real and imaginary parts of (a) effective permittivity and (b) effective permeability of an array of split wires...... 64 4.5 Dimensions of the SRR used in the simulations were w = 1.3 mm, c = 0.124 mm, d = 0.15 mm, g = 0.288 mm, and r = 0.506 mm...... 66

4.6 (a) S11 and (b) S12 for propagation through an array of SRRs infinite in the transverse plane and one unit cell thick in the propagation direction. The dimensions of the SRRs are given in Fig. 5...... 66 4.7 Real and imaginary parts of the effective (a) index of refraction and (b) characteristic impedance of an array of SRRs. The dimensions of the SRRs are given in Fig. 5...... 67 4.8 Real and imaginary parts of the effective (a) permittivity and (b) permeability of an array of SRRs. The dimensions of the SRRs are given in Fig. 5...... 68

x 4.9 (a) S11 and (b) S12 for propagation through an array of continuous SWs and SRRs infinite in the transverse plane and one unit cell thick in the propagation direction. The width of the wires is 0.5 mm and the dimensions of the SRRs are given in Fig. 5...... 69 4.10 Real and imaginary parts of the effective (a) index of refraction and (b) characteristic impedance of an array of continuous SWs and SRRs. The width of the wires is 0.5 mm and the dimensions of the SRRs are given in Fig. 5...... 70 4.11 Real and imaginary parts of the effective (a) permittivity and (b) permeability of an array of continuous SWs and SRRs. The width of the wires is 0.5 mm and the dimensions of the SRRs are given in Fig. 5...... 71

Chapter 5. The Imaginary Parts of the Effective Permittivity and Permeability

5.1 An array of dielectric spheres...... 83 5.2 Real (a) and imaginary (b) parts of the effective index of refraction obtained analytically (red dotted line) and from using the retrieval technique on S- parameters obtained from FDTD (blue solid line) and FEM (green dashed line) simulations...... 84 5.3 Real (a) and imaginary (b) parts of the effective characteristic impedance obtained analytically (red dotted line) and from using the retrieval technique on S-parameters obtained from FDTD (blue solid line) and FEM (green dashed line) simulations...... 85 5.4 Real (a) and imaginary (b) parts of the effective permittivity obtained analytically (red dotted line) and from using the retrieval technique on S- parameters obtained from FDTD (blue solid line) and FEM (green dashed line) simulations...... 86 5.5 Real (a) and imaginary (b) parts of the effective permeability obtained analytically (red dotted line) and from using the retrieval technique on S- parameters obtained from FDTD (blue solid line) and FEM (green dashed line) simulations...... 87

xi 5.6 (a) S11 and (b) S12 calculated analytically for propagation through a 10μm

thick slab (red dotted line) and determined from FDTD (blue solid lines) and FEM (green dashed lines) simulations for propagation through a sheet of dielectric spheres infinite in the transverse plane and one unit cell thick in the propagation direction...... 88 5.7 Magnitude of the terms zn '' ' and zn ''' calculated (a) from FEM simulations and (b) from FDTD simulations and (c) from the analytical calculations...... 94 5.8 (a) Real and (b) imaginary parts of the effective permittivity obtained from the retrieval technique after adding a 1% error (dotted red lines) in the

magnitude of S12 for the analytical results. The solid blue lines represent the analytical results obtained without introducing error...... 97 5.9 (a) Real and (b) imaginary parts of the effective permeability obtained from

the retrieval technique after adding a 1% error in the magnitude of S12 for the analytical results. The solid blue lines represent the analytical results obtained without introducing error...... 98

5.10 (a) S11 and (b) S12 calculated analytically and numerically for propagation through an 8mm slab with ε = 15 and μ = 2 ...... 100 5.11 (a) Real part of the effective index and (b) imaginary part of the effective impedance calculated analytically and obtained using retrieval for propagation through an 8mm slab with ε = 15 and μ = 2 ...... 101 5.12 (a) Imaginary part of the permittivity and (b) imaginary part of the permeability calculated analytically and obtained using retrieval for propagation through an 8mm slab with ε = 15 and μ = 2 ...... 102 5.13 SRR used by Wheeler’s point dipole homogenization technique. The dimensions are w = 9.6mm, c = 0.9mm, d = 0.6mm and g = 2.4mm...... 104 5.14 Effective permittivity obtained for an array of SRRs using Wheeler’s point dipole homogenization technique in the (a) x-direction and (b) y-direction.....105 5.15 Effective permeability obtained for an array of SRRs using Wheeler’s point dipole homogenization technique in the z-direction...... 106

xii Appendix A. The 6 Family Model

A.1 Placement of the 6 Families on the u-v plane. Blue text indicates Families with a negative index of refraction, red text indicates Families with a positive index of refraction and purple text indicates Families that can have either sign for the index...... 125

Appendix C. Mie Theory and the Array of Dielectric Spheres

C.1 Electric field patterns on a surface concentric to the spheres but outside the spheres...... 134

C.2 Real and imaginary parts of (a) a1 and (b) a2 for a single dielectric sphere .....135

C.3 Real and imaginary parts of (a) b1 and (b) b2 for a single dielectric sphere.....136

xiii List of Acronyms

LHM...... Left-Handed Medium OS ...... Opposite Side PEC...... Perfect Electric Conductor PMC ...... Perfect Magnetic Conductor RHM ...... Right-Handed Medium SRR ...... Split Ring Resonator SRRSW ...... Split Ring Resonator and Strip Wire SS ...... Same Side SW ...... Strip Wire

List of Important Symbols

co ...... Speed of light in a vacuum ω ...... Radial frequency φ ...... Phase advance ko ...... Free space wave vector k ...... Wave vector v p ...... Phase velocity vg ...... Group velocity

S11 ...... Reflection scattering parameter

S12 ...... Transmission scattering parameter

ε o ...... Permittivity of free space

μo ...... Permeability of free space ε ...... Electric permittivity μ ...... Magnetic permeability

xiv n ...... Index of refraction z ...... Characteristic impedance

xv Chapter 1

Introduction and Background

1.1 Introduction

Media with a negative index of refraction have received much attention in recent years. Because of their unique properties a great deal of time has been spent formulating new designs and envisioning new applications for these artificial materials. Because a negative index of refraction is a relatively new subject, many problems and misconceptions have arisen regarding the behavior and properties of media that exhibit this phenomenon. In the following chapters some of these problems and misconceptions will be discussed along with their solutions. In this chapter some background on negative index media will be provided. First, the original theories concerning negative index media will be outlined. These theories suggest how a negative index can be obtained and what type of phenomena could be expected in such a medium. Then, the construction and experimental verification of the first negative index medium will be discussed followed by a description of some of the earliest negative index media. Finally, the chapter will close with a summary of the following chapters in this thesis.

1.1.1 Veselago and Left-Handed Media

In 1967 a Russian physicist, Victor Veselago, theorized that a medium where the real parts of the electric permittivity, ε , and magnetic permeability, μ , were simultaneously negative would exhibit unusual behavior [1]. First, he posed that such a medium would be characterized by a left-handed relationship between the electric field

1 v v v vector E , magnetic field vector H , and wave vector k . To see this consider Maxwell’s curl equations for a linear, isotropic, homogeneous, source-free medium

v v ∂B ∇ E −=× (1.1) ∂t v v ∂D ∇ H =× (1.2) ∂t

v v v v where D and B are related to E and H by

v v D = εE (1.3) v v B = μH . (1.4)

For a plane wave propagating through this medium with fields of the form v vv − j v−• ωtrk )( = EE oe equation (1.1) can be written as

v v v k =× ωμHE . (1.5)

Similarly (1.2) can be written as

v v v H =× ωεEk . (1.6)

In a medium where ε and μ are real (no losses) and positive, (1.5) and (1.6) describe the v v v typical right-handed relationship between E , H and k . Such a medium will be referred to as right-handed medium (RHM). However if ε and μ are real and negative, (1.5) and (1.6) become

v vv −=× μω HEk (1.7) v v v −=× εω EkH (1.8)

2

which describes a left-handed relationship among these vectors. Because of this relationship media that exhibit this behavior have been dubbed left-handed media (LHM). One of the implications of this left-handed relationship, backward wave propagation, will be discussed further in Chapter 3.

1.1.2 Negative Refraction

In his paper Veselago also posed that media with negative real parts of ε and μ would be of interest as they would exhibit phenomenon such as negative refraction, reversed Doppler shift, and focusing. Negative refraction can be understood by considering the two interfaces shown in Fig. 1.1. In each case the medium to the left of the interface is an RHM with n1 > 0 . In Fig. 1.1(a) the medium to the right of the interface is a RHM with n2 > 0 while in Fig. 1.1(b) the medium to the right of the interface is a LHM with n2 < 0 . The absolute value of the index of the medium to the right of the interface is the same in each case, only the sign changes. Assuming a wave incident on the interface from the left and polarized in the y-direction the continuity of the electric field at the interface gives the following relationship between the incident, reflected, and transmitted waves:

v v v v v v 1 −•− ωtrkj )( 1 −•− ωtrkj )( 2 −•− ωtrkj )( ieE + r eE = t eE (1.9)

with

ω v v kl = nl , and l +=• lzlx zkxkrk , (1.10) co

where l = 2,1 , and co is the speed of light in vacuum. Equation (1.9) can be simplified by dividing out the time dependency and placing the interface at z = 0 in the x-y plane, which implies

3

− ix xkj )( − rx xkj )( − tx xkj )( i + r = t eEeEeE (1.11)

with

ω klx = n sinθll . (1.12) co

Fig. 1.1. Incident, reflected, and transmitted wave vectors at (a) an RHM/RHM interface and (b) an RHM/LHM interface1.

Since (1.11) must be valid at all points on the interface it follows that the phase terms

must be equal ( == kkk txrxix ). From (1.12) this has two important consequences. First it implies that the incident and reflected angles must be equal

θ θ ri == θ1 (1.13)

and second, it gives the following relationship

1 Note: although the configuration shown is that of an s-polarized wave a similar argument applies to p- polarized waves.

4

sin θ = nn sin θ 2211 (1.14)

where θ t = θ 2 . These are Snell’s laws of reflection (1.13) and refraction (1.14). At the interface between the RHM and LHM (1.14) becomes

n sinθ −= n sinθ2211 . (1.15)

Solving for θ 2 and using the fact that sine and arcsine are odd functions gives

n1 n1 θ 2 = arcsin( θ1 −= arcsin()sin( θ1 )sin( . (1.16) − n2 n2

Since n1 > 0 it follows that the angles of refraction in the RHM and LHM media must have opposite signs. In other words, the wave incident on the RHM/LHM interface refracts on the other side of the normal as compared to the wave incident on the RHM/RHM interface (as shown in Fig. 1.1). Since the angle refracted in the RHM case

is defined as positive it follows that θ 2 < 0 . Another interesting consequence of the negative index is that the propagation vector of the refracted wave in the LHM in Fig. 1.1(b) points towards the interface. To understand this consider the propagation vector of the refracted wave at a point away from the interface (z ≠ 0)

ω t tztx zkxkrk =+=• sin( θ ˆ + cosθ2222 znxn ˆ) . (1.17) co

In the RHM case where n2 > 0 and θ 2 > 0 the wave points in the positive x and z directions as expected; but in the LHM case where n2 < 0 and θ 2 < 0 equation (1.17) becomes

5 ω t rk n2 θ2 )sin(( ˆ nx 2 −−−−=• θ2 )cos( zˆ) co ω = 2 θ2 )sin(( ˆ − 2 θ2 )cos( znxn ˆ) (1.18) co

π π where the fact that θ > 0)cos( for θ <<− has been used. Here, the component of 22 the wave tangential to the interface (the x component) has not changed sign and is still preserved across the interface. On the other hand the component of the wave normal to the interface (the z component) has changed sign and is responsible for the unusual behavior of the wave.

1.1.3 Propagation in a Negative Index Medium

It is important to note that if in a simple lossless medium just one of the parameters ε or μ is negative then the wave will not propagate. For example, assuming ε < 0 and μ > 0 , the index in the medium is given by

n με ))(())(( −=±−±=±±= j μεμε . (1.19)

Note that the index in (1.19) is written according to the engineering convention. In this convention the negative sign must be chosen in front of the root in (1.19) to ensure v vv •− rkj v passivity. Applying (1.19) to a forward propagating wave with the form = oeEE (ignoring the time dependency) gives

ω ω ω j ˆ•− rkn v j εμ )( ˆ•−− rkj v − με )( ˆ•rk v vv co v co v co = o = oeEeEE = oeE (1.20)

so that the wave is attenuated. On the other hand, if both parameters are negative (ε < 0 and μ < 0 ) the index as given by

6

n με j 2 ))(())(( −=−±−±=−±−±= μεμε (1.21)

is real and the wave propagates. Note that since ε and μ have the same form (both are real and negative) the same choice is made for the ± in front of each root in (1.21). The chosen signs cancel and the negative sign on the right side of (1.21) results from j 2 . Unfortunately, at the time no materials with either a negative permittivity or negative permeability were available that could conveniently be combined to test Veselago’s theories experimentally and it was not for 30 years that such materials became available.

1.2 The First Negative Index Metamaterials

1.2.1 The Array of Wires

The first of these materials was introduced in 1996 by Pendry et al. at the Imperial College in London [2]. This group showed that an array of thin metallic rods could be made to display a plasma frequency at microwave frequencies. Because of the low value of the plasma frequency, this structure (shown in Fig. 1.2) could be used to produce a negative real part of the permittivity at low frequencies, without the large losses resulting from the corresponding imaginary part. It was shown that the effective permittivity of the array of wires can be approximated by

ω 2 ε 1−= p (1.22) eff 2 − jγωω

with

2π c2 ω 2 = o (1.23) p a a2 )ln( r

7

and

a ωε 22 γ = po (1.24) πr 2σ

where ω p is the plasma frequency, γ is the damping constant, σ is the conductivity of the metal in the wires and r and a are given in Fig. 1.2. Equation (1.22) is an example of

2 a Lorentzian (with ωo = 0 ). This type of expression will be discussed further in Chapter 5. There are two important points that should be noted when examining (1.22). First, the

22 2 real part of the permittivity will be negative for ωω p −< γ and second, ω p and γ only depend on the dimensions of the structure. Hence, this structure can be customized to produce a negative real part of the permittivity at a desired frequency. It should be noted that the proper function of this structure requires that the electric field be polarized parallel to the wires so that the currents in the wires are excited. In addition it is necessary that the radius of the wires be much smaller than the lattice spacing ( r << a ). If this isn’t the case, the natural logarithm in (1.23) would approach unity and the plasma wavelength would be approximately twice the lattice spacing resulting in Bragg diffraction effects. Finally, this structure also requires that the wavelength of the propagating wave is much larger than the lattice spacing (λ >> a ) so that the propagating wave sees the structure as a continuous medium and not a series of lumped elements. Media such as this, which consist of periodically placed elements whose lattice spacing is much smaller than the wavelength of light, will be referred to as metamaterials. Although many negative real permittivity media were known at the time, it is this structure that would later be used to corroborate Veselago’s predictions.

8

Fig. 1.2. A periodic structure of thin conducting wires from [2].

1.2.2 The Array of Split Ring Resonators

In 1999 the same group led by Pendry introduced a medium that could produce a negative real part of the permeability at microwave frequencies [3]. This medium is an array of split ring resonators (SRRs) which, even though it is composed of non-magnetic materials, exhibits a negative real part of the permeability in the region between the resonance and plasma frequencies. Figure 1.3 shows the SRR and the dimensions of the array. Note that in order to excite the SRRs the magnetic field must be polarized in the direction perpendicular to the plane of the SRR so that currents are excited in the rings. In their paper the authors derived the following expressions to approximate the effective permeability of the array of SRRs

Fω 2 μeff 1−= 22 (1.25) o −− jωγωω

with

9 π r 2 F = , (1.26) a2 3lc2 ω 2 = o , (1.27) o 2c πε ln( ) r 3 r d

and

2lρ γ = (1.28) rμo

where r, c, a and d are given in Fig. 1.3, ρ is the resistance of a unit length of the metal

in the rings, ε r is the dielectric constant of the substrate on which the rings are printed, and l is the spacing between consecutive substrates. Note that (1.25) has a Lorentzian- like appearance except for the ω 2 term in the numerator. From (1.27) it is important to note that the resonance frequency of an SRR scales inversely with its dimensions. Put another way, if the size of the SRR doubles, the wavelength at which it resonates also

doubles. Figure 1.4 shows the real and imaginary parts of μeff determined using (1.25).

From Fig. 1.4 it is clear that this medium produces a negative real part of μeff . As with the array of wires, the array of SRRs also requires λ >> a so that the propagating wave sees a continuous medium and not the fine structure of the rings.

Fig. 1.3. The split ring resonator and an array of resonators.

10

-2 -3 Fig. 1.4. Real and imaginary parts of μeff for ρ = 2000 Ω/m, a = 10 m, c = 10 m, d = 10-4 m, l = 2x10-3 m, and r = 2x10-3 m [3].

1.2.3 The First Negative Index Medium

Soon after, in 2000, Smith et al. at the University of California San Diego built an array of interspersed metal posts and SRRs (Fig. 1.5) [4]. The dimensions of the posts were chosen to generate a negative real ε eff at frequencies below 12 GHz so that propagation below this frequency would be cutoff as expected from (1.19). The dimensions of the SRRs were then chosen to generate a negative real μeff in a frequency band centered approximately around 4.85 GHz (the resonance freq uency). The transmission through an array of SRRs by themselves is shown in Fig. 1.6 (solid line). As expected from (1.19) the transmission is low in the resonance region where the real part of μeff is negative. For the combined metal post and SRR array, the goal was to measure the transmission through the structure and look for a transmission peak in the region where both parameters were expected to be negative. In this region the index would have the form given by (1.21) and the wave would propagate. Figure 1.6 shows the transmission

11

Fig. 1.5. The first left handed medium studied by Smith et al. at the University of California San Diego. The dimensions of the rings are a = 10mm, c = 0.8mm, d = 0.2mm, r = 1.5mm. The radius of the wires was 0.4mm. From Ref. [4].

Fig. 1.6. Transmission measured by Smith et. al. for propagation through an array of circular SRRs (solid line) and an LHM composed of circular SRRs and metal posts (dashed line) [4].

12 for propagation through the array of posts and SRRs (dashed line). Clearly, a

transmission peak occurs in the region where the real parts of ε eff and μeff are expected to be negative. However, because of material losses (both ohmic and dielectric) and scattering losses the transmission peak is only -30dB. As will be discussed in Chapter 2 the appearance of a transmission peak in the region where the real parts of ε eff and μeff are expected to be negative is not sufficient evidence that a negative real part of the index exists. Fortunately, Smith corroborated these results using another experiment as will be discussed in the next section.

1.2.4 Experimental Verification of Negative Refraction

In 2001 the same group led by Smith used a modified version of the metal post and SRR structure to show negative refraction experimentally [5]. In the new structure (Fig. 1.7) the metal posts were replaced by strip wires (SWs) and the circular SRRs were replaced by square SRRs. The advantage of using SWs over posts is that they can easily be printed on the dielectric along with the SRRs (on the opposite side in this case). The use of square SRRs can be understood by considering the following. In (1.25) the resonator strength of the SRR is proportional to F, the ratio of the area contained inside the SRR (π r 2 for the circular case) to the area of the cross section of the unit cell ( a 2 ). Since the unit cell is cubic using a square SRR results in a larger F than using a circular SRR so that the resonator strength is increased. The propagating wave was guided by metal plates on the top and bottom of the structure. Figure 1.8 shows the result of two refraction experiments. The first was performed on a sample of Teflon (dashed line) and shows a positive refraction with a peak at approximately 27°. The second was performed on the LHM (solid line) in Fig. 1.7 and shows a negative refraction with a peak at approximately − 61° . Note that the peaks are normalized so that their magnitudes were unity. This experiment provided the first definite proof of negative refraction and confirmed Veselago’s predictions.

13

Fig. 1.7. Smith’s two dimensional LHM [5]. The ring dimensions used were w = 2.62mm, c = 0.25mm, d = 0.3mm, g = 0.46mm. The lattice spacing was 5mm and the strips were 0.5mm wide.

Fig. 1.8. Smith’s results for refraction experiments on Teflon and a LHM [5].

Once Veselago’s theory that a medium with negative real parts of ε and μ would exhibit negative refraction was corroborated numerous new media were designed and proposed that exhibited a negative index of refraction at microwave [6-10], infrared [11] and optical [12 - 14] frequencies.

14 1.3 Conclusion and Thesis Goals

As is often the case with new areas of research there was a great deal of excitement and momentum driving the field forward in the search for new metamaterials and applications for them. The result was that new metamaterials were developed before sufficient time was spent understanding the fundamentals of these media and the phenomena they exhibit. For example, starting as early as Veselago’s original paper conflicting statements were made about the sign of the group velocity in a negative index metamaterial stating that it was necessarily positive [15] or negative [1, 4, 16]. Finally it was shown that the group velocity in a negative index metamaterial could only be negative in an anomalous dispersion region which occurs in the transmission stopband [17]. Over the years many other misconceptions have surfaced. For instance, one of the methods commonly used to determine the sign of the index in metamaterials, such as the combined SW and SRR structure, could lead to false positives (or negatives, as the case may be). As a result better and more reliable methods were required to properly determine the sign of the index. Also, it was shown that a negative index of refraction implied backward wave propagation. However, the reverse case was not considered. In other words, is a negative index a necessary and sufficient condition for backward wave behavior? Finally, a new technique became popular for determining the effective parameters of metamaterials. However, the use of this technique led to claims that some metamaterials could exhibit negative imaginary parts of the effective permittivity or permeability (implying active behavior), even though the media were composed of passive elements. It is the goal of this thesis to clear up these misconceptions regarding metamaterials and the signs of their effective parameters. The problems that will be looked at are summarized as follows:

• The problem of determining the sign of the real part of the index of refraction in metamaterials will be considered and three methods to determine this sign will be discussed.

15 • The relationship between backward wave propagation and the sign of the real part of the index of refraction in isotropic and anisotropic media will be examined. • The validity of negative imaginary parts of the effective permittivity or permeability in passive media will be examined and the technique which predicts them will be analyzed.

1.4 Chapter Summary

In Chapter 2 one of the early methods used to confirm the existence of a negative index of refraction in metamaterials will be discussed. This method, which was introduced in section 1.2.3, assumes the existence of a negative index based on the

emergence of a transmission peak in the region where the real parts of ε eff and μeff are expected to be negative. By comparing the results of using this technique on two structures composed of both SWs and SRRs it will be shown that this technique can lead to false positives. As a result two alternate methods that correctly determine the sign of the index will be proposed. The first proposed method determines the sign of the index by comparing the phase accrued for propagation through different lengths of the medium and the second method determines the sign of the index from the slope of the dispersion diagrams for the medium. In Chapter 3 the phenomenon of backward wave propagation in negative index metamaterials will be discussed. Although it is known that all negative index media exhibit this behavior the question as to whether the reverse is also true will be examined. That is: do all backward wave media have a negative index of refraction? To answer this question two types of backward waves will be introduced; “perfect” and “imperfect” backward waves. The first of these, the “perfect” backward wave, refers to propagation v in an isotropic medium where the angle between the phase velocity, v p , and group v velocity, vg , is exactly 180° . The second, the “imperfect” backward wave, refers to propagation in an anisotropic medium where the dot product of the group and phase v v velocity ( v p • vg ) is negative, but the vectors are not necessarily anti-parallel. In the isotropic case it will be shown that all backward wave media do indeed have a negative

16 index of refraction and in the anisotropic case it will be shown that backward wave propagation requires that the index to be negative along at least one axis in the medium. A popular technique used to determine the effective parameters of metamaterials will be derived and discussed in Chapter 4. This technique, which will be referred to as the retrieval technique, obtains ε eff , μeff , neff and the effective characteristic impedance,

zeff , of a medium from its reflection and transmission data (S-parameters) [18]. A discussion of this technique is necessary as its use has led to claims that some

metamaterials can exhibit negative imaginary parts of ε eff or μeff while remaining passive [9, 10, 12, 13, 19-24]. To demonstrate this, the results of using the retrieval technique for an array of SWs, an array of SRRs, and a combined array of SWs and SRRs will be presented and it will be shown that in these cases negative imaginary values for

ε eff or μeff are also predicted. Unfortunately, as no accurate analytical expressions for the ε eff and μeff of the SWs and SRRs are available [(1.22) and (1.25) are approximations] there is no way to test the validity of these results. In Chapter 5, the validity of negative imaginary parts of ε and μ will be examined. First, this will be done from the point of view of causality and analyticity by considering the Lorentzian model which is commonly used to characterize metamaterials [as seen in (1.22) and (1.25)] and it will be shown that this model does not allow the imaginary parts of either ε or μ to be negative in the metamaterials considered (SWs and SRRs). Then, the validity of the retrieval technique will be examined by considering a structure for which the effective parameters can be calculated both analytically and by using the retrieval technique. This structure, which has a resonant behavior similar to that of the SWs and SRRs, is the array of dielectric spheres. By comparing the analytical and retrieved effective parameters of the array of dielectric spheres it will be shown that the negative values for the imaginary parts of the effective parameters predicted by the retrieval technique are the result of error in the numerical simulations. Finally, the sensitivity of the retrieval technique to errors in the S-parameters will be analyzed and it will be shown that even small errors can cause the prediction of negative imaginary parts

of ε eff or μeff . In Chapter 6 our final thoughts and conclusions will be summarized.

17 In the appendix a new system will be presented that characterizes the behavior of a medium from the signs of ε ' , ε '', μ' , and μ '' without any need for information on their magnitudes. It will be shown that the 16 sign combinations of ε ' , ε '', μ' , and μ'' that must be considered when calculating n can be reduced to 6 Families, each exhibiting a specific behavior. Hence, once the signs of ε ' , ε '', μ' , and μ '' are known the medium can be assigned to one of the 6 Families giving useful insight on its behavior and possible restrictions on the sign of its index without the need for calculation. In particular, it will be shown that of the 16 cases 4 are restricted to n'> 0 and another 4 to n'< 0 for both passive ( n > 0'' ) and active ( n '' < 0 ) media. Not only does this system categorize the cases where the sign of the index should be obvious but it also restricts the possible sign of the index in the more unusual cases (such as those where the parameters ε ' and μ' have opposite signs).

References

[1] V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of ε and µ", Sov. Phys. Usp. 10, 509 (1968). [2] J. B. Pendry, A. J. Holden, W. J. Stewart, I. Youngs, "Extremely low frequency plasmons in metallic mesostructures", Phys. Rev. Lett 76, 4773 (1996). [3] J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena", IEEE Trans. on Microwave Theory and Techniques 47, 2075 (1999). [4] D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permittivity and permeability", Phys. Rev. Lett. 84, 4184 (2000). [5] R. A. Shelby, D. R. Smith, S. Schultz, "Experimental verification of negative index of refraction", Science 292, 77 (2001). [6] A. Grbic, G. V. Eleftheriades, “Experimental verification of backward-wave radiation from a negative refractive index metamaterial”, J. of Appl. Phys. 92, No. 10 (Nov. 2002)

18 [7] G. V. Eleftheriades, A. K. Iyer, P. C. Kremer, "Planar negative refractive index media using periodically LC loaded transmission lines", IEEE Trans. On Microwave Theory and Techniques 50, 2702 (2002). [8] C. Caloz, H. Okabe, H. Iwai, and T. Itoh, "Application of the transmission line theory of left-handed (LH) materials to the realization of a microstrip LH line", IEEE AP-S/URSI Internatianal Symp., San Antonio, TX, (2002). [9] J. Zhou, L. Zhang, G. Tuttle, T. Koschny, C. M. Soukoulis, “Negative index materials using simple short wire pairs”, Phys. Rev. B 73, 041101 (2006). [10] J. Wang, S. Qu, Z. Xu, H. Ma, S. Xia, Y. Yang, X. Wu, Q. Wang, C. Chen, “Normal-incidence left-handed metamaterials based on symmetrically connected split-ring resonators”, Phys. Rev. E 81, 036601 (2010). [11] M. S. Wheeler, J. S. Aitchison, M. Mojahedi, “Coated nonmagnetic spheres with a negative index of refraction at infrared frequencies”, Phys. Rev. B 73, 045105 (2006). [12] U. K. Chettiar, A. V. Kildishev, T. A. Klar, V. M. Shalaev, “Negative index metameterials combining magnetic resonators with metal films”, Opt. Express 14, No. 17, 7872 (2006). [13] S. Zhang, W. Fan, K. J. Malloy, S. R. J. Brueck, “Demonstration of metal- dielectric negative-index metamaterials with improved performance at optical frequencies”, J. Opt. Soc. Am. B 23, No. 3 (2006). [14] V. M. Shalaev, W. Cai, U. K. Chettiar, H.-K. Yuan, A. K. Sarychev, V. P. Drachev, A. V. Kildishev, “Negative index of refraction in optical metamaterials”, Opt. Lett. 30, No. 24 (2005). [15] D. R. Smith, N. Kroll, “Negative Refractive Index in Left-Handed Materials”, Phys. Rev. Let. 85, No. 14 (Oct. 2000). [16] R. A. Shelby, D. R. Smith, S. C. Nemat-Nasser, S. Schultz, “Microwave Transmission Through a Two-Dimensional, Isotropic, Left-Handed Metamaterial”, Appl. Phys. Let. 78, No. 4 (Jan. 2001). [17] J. F. Woodley, M. Mojahedi, "Negative group velocity and group delay in meft- handed media", Phys. Rev. E 70 (2004).

19 [18] D. R. Smith, S. Schultz, P. Markos, C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients”, Phys. Rev. B 65, 195104 (2002). [19] P. Markos, C. M. Soukoulis, “Transmission properties and effective electromagnetic parameters of double negative metamaterials”, Optics Express 11, 649-661 (2003). [20] T. Koschny, P. Markos, D. R. Smith, and C. M. Soukoulis, “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials”, Phys. Rev. E 68, 065602 (2003). [21] L. Zhen, J. T. Jiang, W. Z. Shao, C. Y. Xu. “Resonance-antiresonance electromagnetic behavior in a disordered dielectric composite”, App. Phys. Lett. 90, 142907 (2007). [22] T. Lepetit, E. Akmansoy, J. -P. Ganne, “Experimental evidence of resonant effective permittivity in a dielectric metamaterial”, J. of App. Phys. 109, 023115 (2011). [23] C. Rockstuhl, T. Zentgraf, H. Guo, N. Liu, C. Etrich, I. Loa, K. Syassen, J. Kuhl, F. Lederer, H. Giessen, “Resonance of split-ring resonator metamaterials in the near infrared”, App. Phys. B 84, 219 (2006). [24] L. Fu, H. Schweizer, H. Guo, N. Liu, H. Giessen, “Synthesis of transmission line models for metamaterial slabs at optical frequencies”, Phys. Rev. B 78, 115110 (2008).

20 Chapter 2

Left-Handed and Right-Handed Metamaterials

2.1 Introduction

In 2001 the first metamaterial with a negative real part of the index of refraction, n < 0' , was presented [1]. This metamaterial was composed of an array of metal wires which exhibit a negative real part of the permittivity, ε < 0' [2], and an array of split ring resonators (SRRs) which exhibit a negative real part of the permeability, μ < 0' [3]. On their own, each of these arrays produces a propagation stopband in the negative parameter (ε < 0' or μ < 0' ) regions where the index

jnnn ''' −=−±=−= j μεμε , (2.1)

is imaginary. As stated in Chapter 1, the index in (2.1) is written in the engineering

v v convention. In this convention a forward travelling wave has the form e •− rkj so that the negative sign in front of the root in (2.1) must be chosen to ensure that the medium remains passive. By combining these structures it was hoped that a region with n < 0' would appear, where the ε < 0' stopband of the wires and the μ < 0' stopband of the SRRs overlapped. This n < 0' region would manifest itself with the appearance of a passband in the combined structure which was indeed the case [1]. Later the presence of a negative index of refraction was corroborated by an experiment where microwaves were negatively refracted when propagating through a similar metamaterial [4]. Since then, many new negative index metamaterials have been proposed and constructed [5-9]. Because the electric field vector, magnetic field vector and wave vector form a left- handed triplet in these negative index metamaterials they are also referred to as left- handed media (LHM).

21 In general LHM were constructed by combining two media that are known to exhibit ε < 0' and μ < 0' , respectively. Then, to determine if the combined structure had a negative index of refraction the transmission through the structure was measured and if a transmission peak (passband) appeared in the region where ε ' and μ' of the constituent media were expected to be negative the structure was assumed to exhibit LHM behavior [1, 6, 10]. This was often considered to be conclusive evidence that the medium was left- handed (LH) and no additional proof, such as the refraction experiments mentioned above, was used to corroborate the results. Unfortunately, the presence of a transmission peak is not sufficient evidence that a medium exhibits left-handed behavior. One example of this can be seen by considering an array of metal strip wires (SW) and an array of SRRs. When isolated each of these arrays produce the expected negative behavior (ε < 0' or μ < 0' ) and it was assumed that their behavior did not deviate significantly from their isolated responses when put in close proximity to each other in the combined structure. However, when they are combined into a single structure their field patterns overlap and can interfere significantly. In fact, depending on the relative positioning of the SWs and SRRs and the geometry of the unit cell this interference can be such that the observed transmission remains but the sign of the index (or handedness of the structure) changes. Therefore, the emergence of a transmission peak in the region where ε ' and μ' are expected to be negative is not sufficient evidence that a medium exhibits a negative index of refraction (or left-handed behavior) and in order to correctly determine the sign of the index (or handedness) other techniques such as calculating or measuring the insertion phase or examining the dispersion diagrams must be employed. It should be noted that since it was shown that the presence of a transmission peak cannot be used to determine the sign of the index, other methods such as parameter retrieval have been employed [11]. This method, however, can lead to other problems and is discussed in chapters 4 and 5. In this chapter two structures composed of SWs and SRRs will be introduced and the transmission magnitude for propagation through them will be examined. The sign of the index in these structures will be determined by considering the propagation phase and dispersion diagrams. It will be shown that even though both structures exhibit a transmission peak in the region where ε ' and μ' are expected to be negative one

22 structure is right-handed and the other is left-handed. Finally, the effect of the relative position of the SWs and SRRs and the size of the unit cell on the sign of the index will be discussed.

2.2 The Transmission Magnitude

Consider two structures composed of SRRs and SWs. In the first structure the SRRs and SWs will be printed on opposite sides of a dielectric substrate and in the second they will be printed on the same side of the substrate. In both cases the substrate is 0.5mm thick with a dielectric constant of 3.02. The dimensions of the SWs and SRRs in both structures are the same with r = 0.506 mm, c = 0.124 mm, d = 0.15 mm, g = 0.114 mm, w = 1.3 mm, s = 2.5 mm and t = 0.5 mm (Fig. 2.1). The unit cells for these structures, which will be referred to as the opposite side (OS) and same side (SS) structures, are shown in Fig. 2.2. In the OS case the SRR is located at the center of the unit cell. In the SS case the SRR is offset 0.425mm from the center of the unit cell in the y-direction and the space between the SWs and SRRs is 0.35 mm. The dimensions of the SRRs were chosen to be exactly half of those in [12] doubling their operating frequency.

Fig. 2.1. SRR used in the OS and SS structures. The dimensions of the rings are r = 0.506 mm, c = 0.124 mm, d = 0.15 mm, g = 0.114 mm, w = 1.3 mm, s = 2.5 mm and t = 0.5 mm.

23

Using Ansoft HFSS, a commercial full wave finite element method simulation package, the transmission characteristics for propagation through the OS and SS structures were numerically simulated. In each case perfect electric conductor (PEC) boundary conditions were assigned to the faces of the unit cell perpendicular to the z- direction and perfect magnetic conductor (PMC) boundary conditions were assigned to the faces perpendicular to the x-direction. These boundary conditions were used so that a TEM wave propagates through the cell in the y-direction with the polarization necessary to excite the negative parameter behavior of the SWs and SRRs. In the propagation direction the structures were 4 unit cells thick and the unit cell size in the OS case was 2.5×2.5×2.5 mm and in the SS case it was 4×2.5×2.5 mm (the reason for the difference in the x-dimension will be discussed in section 2.5). The simulated transmission magnitudes for propagation through the OS and SS structures are shown in Fig. 2.3(a) and Fig. 2.3(b), respectively. On both plots the region between the vertical lines indicates a negative index region (these lines and how they are obtained will be discussed in section 2.3).

(a) (b)

z z

x y x y

Fig. 2.2. (a) Opposite side (OS) structure and (b) same side (SS) structure. In each case the SWs are 0.5 mm wide and the substrate is 0.5mm thick with a dielectric constant of 3.02. In the SS case the space between the SWs and SRRs is 0.35 mm. The dimensions of the unit cell in the OS configuration are 2.5×2.5×2.5 mm and in the SS configuration they are 4×2.5×2.5 mm.

24 OS 0 (a) -10

-20

-30

-40

-50

Transmission Magnitude (dB) -60 20 21 22 23 24 Frequency (GHz)

SS 0 (b)

-10

-20

-30

Transmission Magnitude (dB) -40 21 22 23 24 Frequency (GHz)

Fig. 2.3. Simulated transmission magnitude for propagation through 4 unit cells of (a) the OS structure and (b) the SS structure. Solid (dashed) vertical lines indicate the points at which the structures make transitions from RHM (LHM) to LHM (RHM) behavior.

From Fig. 2.3 transmission peaks appear in the OS structure at 22.6 GHz and in the SS structure at 22.75 GHz. In both cases these transmission peaks (the reason for the difference in the magnitude of the peaks will be discussed in section 2.4) occur in the region where ε ' and μ' are expected to be negative and, hence, it would be tempting to conclude that both structures exhibit left-handed behavior. In fact the transmission

25 characteristics for propagation through SW and SRR configurations such as the OS and SS structures have been simulated and measured experimentally by many researchers [6, 12, 13] and it was often inferred that these media exhibited left-handed behavior based on the appearance of a transmission peak in the region where ε ' and μ' where expected to be simultaneously negative. However, as will be shown in the next two sections, this is not sufficient evidence that left-handed behavior has occurred.

2.3 The Transmission Phase

For a medium with index of refraction n ω )( the phase accrued by a wave propagating through a length L of the medium is given by

ω −= ωφ )( Ln (2.2) co

where co is the speed of light in vacuum. If the wave propagates through different lengths of the medium, given by L1 and L2 , the phase difference is given by

ω ωφ −−=Δ LLn 12 ))(( . (2.3) co

Using equation (2.3) the sign of the index in the medium can be determined since, for

> LL 12 , the phase difference will be negative in RHM [ n ω) > 0( ], and positive in LHM [ n ω)( < 0 ]. In other words, in the case of RHM the insertion phase of the longer structure will lie below that of the shorter structure on the phase diagrams, whereas the opposite is true for the LHM. Figures 2.4(a) and 2.4(b) show the transmission phase simulated in HFSS for propagation through 1, 2, 3, and 4 unit cells of the OS and SS structures, respectively. The region between the vertical lines is a negative index region. From Fig. 2.4 it is immediately clear that for both the OS and SS structures there are regions where the phase lines cross each other. From (2.3) these crossings indicate

26 regions where the sign of the index changes. For the OS structure in Fig. 2.4(a) the phase lines cross at 20.7 GHz and 22.85 GHz for the two, three, and four unit cell cases. Vertical lines are placed at these frequencies to mark the transitions. The transition from RHM to LHM at 20.7GHz is marked by a solid vertical line and the transition from LHM to RHM at 22.85GHz is marked by a dashed vertical line so that the region between these lines is a n < 0' region. For the SS structure in Fig. 2.4(b) the phase lines cross at 22.4 GHz and 22.65GHz, for the two, three, and four unit cell cases. The transition from RHM to LHM at 22.4GHz is marked by a solid vertical line and the transition from LHM to RHM at 22.65GHz is marked by a dashed vertical line so that the region between these lines is a n < 0' region. For the one unit cell OS case the phase line crosses at 20.7 GHz and 22.95 GHz so that n < 0' in the region between these frequencies. For the one unit cell SS case the phase line crosses at 22.57 GHz and 22.65 GHz so that n < 0' in this region between these frequencies. The difference in behavior of the single and multiple unit cell cases is caused by the interactions between neighboring SRRs. In the one unit cell case, the resonance region is due to an individual SRR which results in a single peak. In the two, three, or four unit cell cases both the individual SRR resonances and the interactions between neighboring SRRs contribute so that there is a second peak in the phase plots and their LHM regions differ slightly from the single unit cell cases. The location of the transmission peaks in Fig. 2.3 for the OS and SS structures are marked in their corresponding phase plots in Fig. 2.4 by vertical arrows. What is important to note is that in the OS case the transmission peak is inside the n < 0' region whereas in the SS case the transmission peak is outside the n < 0' region. Hence, in the passband, the OS structure is LHM and the SS structure is RHM. From the above analysis it is clear that the presence of a transmission peak is not sufficient evidence that a medium is exhibiting left-handed behavior in the passband. In this section it was shown that the transmission phase can be used to properly determine the sign of the index in a structure. Unfortunately, this analysis requires the phase for propagation through at least two structures with different lengths which may be inconvenient. In the next section another technique to determine the sign of the index will be considered which uses the dispersion diagrams.

27

OS 250 (a) 200 1 Cell 2 Cells 150 3 Cells 4 Cells 100

50

0 Transmission Phase (degrees) -50 20 21 22 23 24 Frequency (GHz)

50 SS (b) 1 Cell 25 2 Cells 3 Cells 4 Cells 0

-25

-50

Transmission(degrees) Phase -75 21 22 23 24 Frequency (GHz)

Fig. 2.4. Transmission phase for propagation through 1, 2, 3, and 4 unit cells of (a) OS structure and (b) SS structure. Solid (dotted) vertical lines mark transitions from RHM (LHM) to LHM (RHM) behavior. Arrows mark the locations of the transmission peaks from the corresponding OS or SS plots in Fig. 2.3.

28

2.4 Dispersion Diagrams

The dispersion diagrams of the OS and SS structures were simulated using HFSS and are shown in Figs. 2.5(a) and 2.5(b), respectively (blue solid lines). Periodic boundary conditions were used on all sides of the unit cells shown in Fig. 2.2 with no phase variation on the faces perpendicular to the x- and z-directions. In the y-direction (the propagation direction) the phase difference between the faces was swept between 0 and 180 degrees. The light line was also calculated and is shown in Fig. 2.5 (red dashed lines). In both the OS and SS cases when the mode resulting from the combined SRR and SW structure (this mode will be referred to as the SRRSW mode) intersects the light line it couples to the light line. Hence, for comparison the dispersion plot for the OS case was also calculated using an equivalent transmission line model which does not take this coupling into account [green dotted line on Fig. 2.5(a)]2. In this model series and shunt RLC resonators which mimic the behavior of the SWs and SRRS are added to a transmission line. The values of the lumped elements in the resonators are derived from

the parameters F, ωo , ω p and γ in the expressions for the permittivity (1.22) and permeability (1.25). In this case the values used for the series RLC resonator were

7 −10 −13 Rseries ×= 1097.6 Ω , Lseries = 39×10.3 H and Cseries ×= 1044.1 F and the values

3 −9 used for the shunt RLC resonator were Rshunt = 1095.8 Ω× , Lshunt = .1 ×1012 H and

6 Cshunt ×= 1095.8 F . Note that because of the complex nature of the interactions between the SRRs and SWs in the SS case (as will be discussed in the following sections) no simple equivalent transmission line model exists for this structure. Since the dispersion diagrams are symmetric with respect to the origin the question arises as to which branch to use. In the one dimensional case considered here the group velocity is given by the local derivative of the dispersion diagrams. Since, in the passband, the group velocity is the same as the energy velocity the branch must be

2 This was done using a technique similar to that in [14].

29 chosen that predicts positive energy propagation [13]. The branches with a positive group velocity are marked with a (I) in Fig. 2.5(a) and 2.5(b). The phase velocity at any point on the dispersion diagrams is given by a line connecting that point to the origin. From Fig. 2.5 this means that in the OS structure the phase velocity of Branch-I is negative and in the SS structure the phase velocity of Branch-I is positive. Since Branch-I is defined as the positive group velocity branch this means that the OS structure exhibits backward wave behavior and the SS structure exhibits forward wave behavior. In other words, n < 0' for Branch-I of the OS structure and n > 0' for Branch-I of the SS structure. These results corroborate those obtained in section 2.3: that the passband of the OS structure is a LHM and the passband of the SS structure is a RHM. The nature of the coupling in Fig. 2.5 also provides information about the sign of the index in the OS and SS structures. In the OS case, when the light line intersects and couples with the SRRSW mode a band gap appears, indicative of contra-directional coupling where SRRSW mode and light line have counter-propagating wave vectors. In other words, while the light line is a forward traveling wave the SRRSW mode is a backward travelling wave. This type of behavior was also shown for coupling between RHM and LHM modes in a metamaterial coupler [15]. In the SS structure, on the other hand, when the SRRSW mode and light line intersect there is no band gap, indicative of co-directional coupling where the wave vectors of the SRRSW mode and light line are propagating in the same direction. To understand why the OS and SS structures behave differently the field configurations of the SWs and SRRs in each structure must be examined. Figure 2.6 gives a visual representation of the magnetic fields set up by the currents in the SWs and SRRs in a plane perpendicular to the z-axis (see Fig. 2.2) and cutting through the center of the unit cell. In the OS structure [Fig. 2.6(a)] when the SW and SRR are printed on opposite sides of the dielectric substrate their fields do not overlap significantly and consequently their individual behavior is preserved. On the other hand, in the SS structure [Fig. 2.6(b)], when the SW and SRR are printed on the same side of the dielectric substrate there is considerable overlap and their individual field patterns are not preserved. A similar argument was presented in [16].

30

OS 24.5 (a) I II 24 HFSS Light Line 23.5 TLM model

23 Frequency (GHz)

22.5 -180 -120 -60 0 60 120 180 kd (degrees)

SS

(b) 24.5 II I

23.5 HFSS Light Line Frequency (GHz) Frequency

22.5 -180 -120 -60 0 60 120 180 kd (degrees)

Fig. 2.5. Dispersion plots for (a) OS structure and (b) SS structure. In the OS configuration the unit cell size was 2.5×2.5×2.5mm. The unit cell size was 4×2.5×2.5mm for the SS configuration.

31

Fig. 2.6. Magnetic fields resulting from the currents in the SRR and SW for (a) OS configuration, (b) SS configuration.

The interactions between the SWs and SRRs also explain why the peak for the OS structure in Fig. 2.3(a) was more pronounced than the peak for the SS structures in Fig. 2.3(b). In the OS case the peak results from propagation through an LHM passband where both ε < 0' and μ < 0' . On the other hand, in the SS case since the bandwidth of the μ < 0' region is reduced (because of the interactions between the SWs and SRRs) the peak results from propagation through an RHM band with ε < 0' and μ > 0' resulting in low transmission. Note that in this case even though ε ' is negative its absolute value is small so the wave isn’t strongly attenuated. These results agree with those from section 2.3 that, in the passband, the OS structure is a LHM and the SS structure is a RHM. However, as shown in Fig. 2.4(b) the SS structure still exhibits LHM behavior in the region to the left of the transmission peak (i.e. the stopband). The question then arises as to whether it is possible to extend this behavior to the passband. This question will be addressed in the next section by considering the effect of the unit cell size on the behavior of the SS structure.

2.5 The Effects of the Unit Cell Dimensions

Consider an array of SWs. If the unit cell size of the array is reduced the metal strips in the array move closer together. As this happens the volume fraction of air in the unit cell is reduced and the volume fraction of metal increases. The medium comes closer and closer to the case of a bulk metal and consequently, the plasma frequency of

32 the medium increases. Another way to look at this is to consider the following: as strips move closer together the strength of their collective resonant response increases (the number of resonators per unit volume increases). The same observation can be made for the array of SRRs. If the unit cell size of the array is reduced, i.e. bringing the rings closer together, the strength of their resonant responses increases. In both cases increasing the resonator strength means the value of the negative parameters (ε < 0' or μ < 0' ) also increases. This does not change when the SWs and SRRs are combined in the OS or SS structures. When the unit cell size is reduced the strength of SWs and SRRs responses still increases even though the SW and SRR field patterns may interfere with each other. The question then arises as to whether decreasing the unit cell size of the SS structure could strengthen the resonant response such that the negative index region seen in Fig. 2.4(b) expands to include the passband. To investigate this problem the band diagrams for several SS structures with different unit cell sizes in the x-direction [Fig. 2.2(b)] were calculated and are shown in Fig. 2.7. The green dotted line in Fig. 2.7 shows the SS structure considered in the previous sections with unit cell size 4×2.5×2.5 mm. As mentioned previously to ensure that the energy velocity is positive Branch-I must be selected. Since the phase velocity is also positive for this branch this is a RHM ( n > 0' ) band. In this case the passband has a bandwidth of approximately 730MHz. The red dashed line in Fig. 2.7 corresponds to a SS structure for which the x-dimension of the unit cell has been reduced from 4 mm to 2.5 mm. In this case the slope of the band is negative at low phase values and slightly positive at high phase values so that it is not clearly either LHM or RHM. In this case the passband is nearly flat with a bandwidth of only 130 MHz. The blue solid curve gives the band diagram for an SS structure with a unit cell size of 1.5 mm in the x-direction. To ensure positive energy propagation Branch-II must be selected so that this is a LHM ( n < 0' ) band with a bandwidth of 250 MHz. Therefore, by reducing the x-dimension of the unit cell the SS structure can be made to transition from RHM behavior (4 mm case) to LHM behavior (1.5 mm case) and the transition occurs when the x-dimension is approximately 2.5 mm (where the passband was nearly flat). The transition from RHM to LHM behavior can also be seen in the coupling between the SRRSW mode and the light line. In the 4mm SS structure the coupling does

33 not result in a band gap since the wave vectors of the SRRSW mode and the light line propagate in the same direction (co-directional coupling). On the other hand in the 2.5mm and 1.5mm SS structures, where LHM behavior is observed, a band gap appears since the wave vectors of the SRRSW mode and the light line are counter-propagating (contra-directional coupling).

25.5 1.5 mm 2.5 mm 24.5 4 mm

23.5

Frequency (GHz) II I

22.5 -180 -120 -60 0 60 120 180 kd (degrees)

Fig. 2.7. Dispersion curves for the SS structure. In each case the y- and z-dimensions of the unit cell were 2.5mm and the x-dimension (legend) was varied.

Therefore, both the OS and SS structures can produce LHM behavior but in the SS case a smaller unit cell size (i.e. a stronger resonance) is required to compensate for the effects of the interference between the SW and SRR field patterns.

2.6 Conclusion

In this chapter the behavior of two SRR and SW configurations was investigated and it was shown that the presence of a transmission peak in the region where ε ' and μ' are expected to be simultaneously negative is not sufficient evidence of LHM behavior. In order to properly determine the sign of the index two methods were presented. In the

34 first method the sign of the index is calculated from the phase difference for propagation through several lengths of the medium. In the second method the sign of the index is determined by examining the band structure. In this case either the slope of the band diagrams or the nature of the coupling to the light line can be used. The relative placement of the SW and SRR in the unit cell was also discussed. It was shown that in the OS structure there was little overlap between the fields produced by the SW and SRR so that the medium was clearly a LHM. On the other hand, in the SS structure there was significant overlap so that the medium could be either a RHM or LHM, depending on the unit cell size. The results presented in this chapter were published in Physical Review E in 2005 [17].

References

[1] D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permittivity and permeability", Phys. Rev. Lett. 84, 4184 (2000). [2] J. B. Pendry, A. J. Holden, W. J. Stewart, I. Youngs, "Extremely low frequency plasmons in metallic mesostructures", Phys. Rev. Lett 76, 4773 (1996). [3] J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena", IEEE Trans. on Microwave Theory and Techniques 47, 2075 (1999). [4] R. A. Shelby, D. R. Smith, S. Schultz, "Experimental verification of negative index of refraction", Science 292, 77 (2001). [5] G. V. Eleftheriades, A. K. Iyer, P. C. Kremer, "Planar negative refractive index media using periodically LC loaded transmission lines", IEEE Trans. On Microwave Theory and Techniques 50, 2702 (2002). [6] M. Bayindir, K. Aydin, E. Ozbay, P. Markos, C. M. Soukoulis, "Transmission properties of composite metamaterials in free space", Appl. Phys. Lett., 81 (2002). [7] O. F. Siddiqui, M. Mojahedi, G. V. Eleftheriades, "Periodically loaded transmission line with effective negative refractive index and negative group velocity", IEEE Trans. On Antennas and Propagation 51, 2619 (2003).

35 [8] O. F. Siddiqui, S. J. Erickson, G. V. Eleftheriades, "Time-domain measurement of group delay in negative refractive index transmission line metamaterials", IEEE Trans. on Microwave Theory and Techniques 52, 1449 (2004). [9] C. Caloz, H. Okabe, H. Iwai, and T. Itoh, "Application of the transmission line theory of left-handed (LH) materials to the realization of a microstrip LH line", IEEE AP-S/URSI Internatianal Symp., San Antonio, TX, (2002). [10] R. W. Ziolkowski, "Design, fabrication, and testing of double negative metamaterials", IEEE Trans. on Antennas and Propagation 51, 1516 (2003). [11] D. R. Smith, S. Schultz, P. Markos, C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients”, Phys. Rev. B 65, 195104 (2002). [12] R. A. Shelby, D. R. Smith, S. C. Nemat-Nasser, and S. Schultz, “Microwave Transmission Through a Two-Dimensional, Isotropic, Left-Handed Metamaterial”, App. Phys. Lett., 78, 489 (2001). [13] J. F. Woodley, M. Mojahedi, "Negative group velocity and group delay in left- handed media", Phys. Rev. E 70 (2004). [14] G. V. Eleftheriades, O. Siddiqui, A. K. Iyer, "Transmission line models for negative index media and associated implementations without excess resonators", IEEE Microwave and Wireless Comp. Lett. 13, 51 (2003). [15] R. Islam, F. Elek, G. V. Eleftheriades, “Coupled-line metamaterial coupler having co-directional phase but contra-directional power flow”, Electron. Lett. 40, No. 5 (2004). [16] C. M. Soukoulis, Photonic Crystals and Light Localization in the 21st Century (Kluwer Academic Publishers, Netherlands, 2001). [17] J. F. Woodley, M. Mojahedi, "Left-handed and right-handed metamaterials composed of split ring resonators and strip wires", Phys. Rev. E 71, 066605 (2005).

36 Chapter 3

Backwards Waves

3.1 Introduction

Soon after the first medium with a negative index of refraction was constructed and experimentally verified [1] there was an explosion of publications on the subject [2 - 5]. An important characteristic of negative index media is that the electric field vector, v v v E , magnetic field vector, H , and wave vector, k , form a left-handed triplet, a relationship which has caused them to be dubbed left-handed media (LHM). Since the v Poynting vector, S , in these media given by

v 1 vv * Re[( ×= HES )] (3.1) 2

v v v v v implies a right-handed relationship between E , H and S it follows that k and S must be anti-parallel for an isotropic negative index medium. In other words, media with a negative index of refraction are necessarily backward wave media. The question then arises as to whether the opposite is also true. That is, does an anti-parallel relationship v v between k and S necessarily imply the presence of a negative index of refraction? And if so, what would be the functional form of this index? In order to investigate this problem the backwards wave phenomenon will be examined from a three dimensional wave propagation point of view. To perform this investigation it is important to consider two points:

37 v v (i) Only the directions of k and S are relevant since we are concerned with whether the vectors are parallel (normal wave propagation) or anti-parallel (backwards wave propagation), the magnitudes of the vectors can be ignored. (ii) Propagation will be considered through a transmission passband (i.e. outside any regions of anomalous dispersion).

v v From (i) the phase velocity, v p , can be substituted for k since the two vectors are always v v parallel. From (i) and (ii) the group velocity, vg , can be substituted for S since these vectors are always parallel in the passband. To examine the relationship between backwards wave behavior and the sign of the index we will begin by deriving the index of refraction necessary to produce a v v medium where the angle between v p and vg is exactly 180º. The propagation in such a v v medium will be referred to as “perfect” backwards wave propagation since v p and vg are “perfectly” anti-parallel. It will be shown that in such a medium the index of refraction is v v necessarily negative and isotropic. The case where v p and vg are no longer “perfectly” anti-parallel but have at least one anti-parallel component will then be considered. In this case propagation will be referred to as “imperfect” backwards wave propagation and it will be shown that the medium must be anisotropic with a negative index of refraction along at least one axis. In both the “perfect” and “imperfect” cases constraints on the derived forms of the index are obtained using the fact that the group velocity must be v v positive in the passband [6, 7]. Finally, the relationship between v p and vg in several anisotropic uniaxial right-handed media (RHM) and LHM will be examined.

3.2 “Perfect” Backwards Waves

To investigate the backwards wave phenomenon we begin from a three dimensional wave propagation point of view. Consider an arbitrary medium with phase index n(k). The question is, without making any assumptions about the sign of the index,

38 what functional form must this index have to result in backward wave propagation? To v v find this functional form we examine v p and vg in the medium as given by

c v = o kˆ (3.2) p kn )( kc vv ∇= o (3.3) kg kn )(

ˆ where co is the speed of light in vacuum and k is the unit vector in the direction of phase propagation. In an arbitrary medium these vectors can be related as follows

gzgygx = vdvdvdvvv pzzpyypxx ),,(),,( (3.4)

where the di (i=x, y, z) are arbitrary constants that relate the individual components of the vectors. Any differences in the directions of the vector components are accounted for by

the signs of the di. Note that if any of the v pi are zero in equation (3.4) the corresponding

vgi will also be zero which may be incorrect. In such a case a new coordinate system can

always be defined in which none of the v pi are zero. In this section we will consider the v v simplest backward wave medium. That is, a medium where v p and vg are "perfectly" anti-parallel and the angle between them is exactly 180°. Looking at equation (3.4) this v v has two requirements. First, to ensure that v p and vg are co-linear the di must be chosen v v such that == ddd zyx = d . Second, to ensure that v p and vg are anti-parallel d must be negative and will be written as d −= d . Using these requirements (3.4) becomes

vv g −= vdv p . (3.5)

Finally, plugging (3.2) and (3.3) into (3.5) gives

39 kc c o −=∇ d o kˆ . (3.6) k kn kn )()(

Any phase index, n(k), that satisfies (3.6) will result in backward wave propagation at all frequencies. Setting d =1 for simplicity the solution to the vector differential equation in (3.6) is

)( = kbkn 2 (3.7)

where b is an arbitrary constant that has units m2. The phase index in (3.7) is plotted in Fig. 3.1 using b = 1 (red curve) for the positive index case and b = −1 (blue curve) for the negative index case. The dispersion relation corresponding to the phase index in (3.7) is given by

kc c ω k)( o == o . (3.8) kn )( kb

The dispersion relation given by (3.8) is plotted in Fig. 3.2 where two branches are labeled branch-I and branch-II. Note that although (3.8) can give negative values for ω k)( these solutions are unphysical so that only solutions with ω k > 0)( are considered. From equation (3.8) the requirement that ω k > 0)( puts constraints on the sign of b in each branch. In branch-I b must be negative since k < 0 and in branch-II b must be positive since k > 0 . In each branch the phase velocity has the same sign as the wave v v vector so that v p < 0 in branch-I and v p > 0 in branch-II. The group velocity can be determined by plugging (3.7) into (3.3) which gives

c v −= o kˆ (3.9) g bk 2

40 v v v v so that vg > 0 in branch-I and vg < 0 in branch-II. Therefore, v p and vg are "perfectly" anti-parallel everywhere in the dispersion diagram and the phase index derived from (3.6) results in backwards wave behavior at all frequencies as expected. As mentioned in point (ii) the analysis in this chapter assumes propagation through a transmission passband (i.e. away from any anomalous dispersion regions) where the group velocity and energy velocity are the same. Assuming that the waves are generated by a source at rv = 0 only waves propagating away from the source (i.e. waves with a positive group velocity) present a valid solution [6, 7]. Therefore, in this case, branch-II is unphysical and only branch-I should be considered. As mentioned, since k < 0 in Branch-I, b must be negative to ensure that ω k > 0)( . However, b < 0 is also the condition for obtaining a negative phase index in (3.7) [Fig. 3.1]. Therefore, for the case considered, a negative phase index is a necessary condition for obtaining backwards wave behavior. The index necessary to obtain backward wave behavior can be obtained as a function of frequency using (3.7) and (3.8) giving

c2 n ω)( = o . (3.10) bω 2

This index is plotted in Fig. 3.3 where the positive index (b > 0 ) and negative index (b < 0 ) branches are shown. It is important to note that the expressions for the index and dispersion relation derived in (3.7), (3.8) and (3.10) are purely mathematical results and that no physical medium can exhibit these at all frequencies. One way to see this is to consider the following: at high frequencies the phase index should approach unity. However since k approaches zero at high frequencies the phase index as given by (3.7) also approaches zero. Hence it is not possible to create a medium that exhibits backwards wave behavior at all frequencies.

41 10 Branch-II 5 b > 0

0 n(k) Branch-I -5 b < 0

-10 -3 -2 -1 0 1 2 3 k

Fig. 3.1. Phase index given in (3.7). Note that b = -1 in the negative index case (branch-I) and b = 1 in positive index case (branch-II).

Fig. 3.2. Dispersion relation for the phase index given by (3.7). Note that b = -1 in branch-I and b = 1 in branch-II.

42 20 Branch-II 10 b > 0 )

ω 0 n( Branch-I -10 b < 0

-20 0123 ω

2 Fig. 3.3. Index given in (3.10) normalized by a factor of co (the speed of light squared). Note that b = -1 in the negative index case (branch-I) and b = 1 in positive index case (branch-II).

3.3 Backward Wave Transmission Line

The index in (3.10) has the same form as the effective index for a transmission line consisting of a series capacitance and shunt inductance, a structure that is well known to support backward wave propagation [8-11]. To see this consider the one dimensional transmission line shown in Fig. 3.4. The propagation constant for this transmission line is given by

βαγ =+= ZYj (3.11)

For a lossless transmission line consisting of a series capacitance ( = 1 ωCjZ ) and shunt

inductance ( = 1 ωLjY ) the propagation constant becomes

1 1 j 1 jβγ == ( )( ) −= . (3.12) LjCj ωωω LC

43 Comparing the propagation constant in (3.12) to that in a free space medium

(γ = ω cnj o ) gives

c 1 n −= o . (3.13) ω 2 LC

Since equation (3.13) is just (3.10) with −= o LCcb and since all the parameters in (3.13) are positive the medium must support backward wave propagation.

Fig. 3.4. General one dimensional transmission line model in the phasor domain.

3.4 “Imperfect” Backwards Waves

In section 3.2 the simplest case of a backward wave medium was considered v v where v p and vg were "perfectly" anti-parallel. In this section the more general case will be considered where the angle between these vectors is no longer exactly 180° but where at least one of their components is anti-parallel. In this case the angle between the two vectors will be in the range 90° < θ < 270° or, stated another way, the dot product between the two vectors will always be negative. This situation will be referred to as "imperfect" backwards wave propagation. Beginning with (3.4), the relationship between v v v p and vg is given by

44 gzgygx = vdvdvdvvv pzzpyypxx ),,(),,( (3.14)

where the di (with i = x, y, z) are constants that account for differences in magnitude and v v direction between the components of v p and vg . The difference between each v v component of v p and vg can be written in the simplified form

= vdv piigi (3.15)

Substituting (3.2) and (3.3) into (3.15) gives

1 k ∂ i kn )( ki co ki co ( − 2 ) = di (3.16) i kn i kn )()( ∂ki k i kn )( k which, with some manipulation can be written as

− di )1( k ∂ i kn )( = 2 . (3.17) i kn i kn )()( ∂ki

The solution to the differential equation in (3.17) is given by

1−di i )( = αikkn (3.18)

where αi is an arbitrary constant. Using (3.18) the index along each axis can be v v calculated from the di relating the components of v p and vg . If these di differ, this corresponds to the case of an anisotropic medium implying that "imperfect" backwards waves are only possible in this type of medium. In section 3.6 several anisotropic media will be considered and it will be shown that for these cases the "imperfect" backward wave behavior requires the index to be negative along at least one axis. From (3.18) this means that α i must be negative along at least one of the axes. It should be noted that the

45 simple isotropic case considered in section 3.2 can be recovered from (3.18) by setting

i dd −== 1.

3.5 Dispersion Requirements in Negative Index Metamaterials

In the previous sections the functional form of the index was derived from a purely three dimensional wave propagation point of view by considering the relationship v v between v p and vg . Further constraints on the form of the index can be determined by v considering the implications of point (ii) on vg : that for propagation in the passband the v group velocity must be positive. Equation (3.3) for the vg can be written as

c kc ∂ kn )( vv ( o −= o )kˆ . (3.19) g kn kn )()( 2 ∂k

v Any solution for the phase index must give a positive vg when substituted into (3.19) in order to be considered a valid solution. From (3.7) and (3.18) a general form for the phase index can be written as

)( = γ kkn p (3.20)

where γ and p are arbitrary constants. Substituting (3.20) into (3.19) gives

− pc )1( v = o ˆ v −= pvk )1( . (3.21) g γk p p

v v From the right side of (3.21) v p and vg have the same sign (forward waves) if p < 1 and v they have opposite signs (backward waves) if p > 1. Since = kk is positive the v requirements for a positive vg in (3.21) are

46 p > 1 for γ < 0 , (3.22a) or

p < 1 for γ > 0 . (3.22b)

Note that the phase index (3.7) for the isotropic case corresponds to (3.22a) with p = 2 > 1 and γ = b < 0. This can be seen by substituting (3.7) into (3.19) giving

c v o >−= 0 (3.23) g kb 2 which is just equation (3.9). Condition (3.22a) corresponds to backward wave v v propagation through the passband of a LHM ( v p < 0 and vg > 0 ). On the other hand, condition (3.22b) corresponds to normal propagation through the passband of a RHM v v ( v p > 0 and vg > 0 ). In the anisotropic case equation (3.17) and the constraints in (3.22a) must be applied in each direction for which there is a passband. Therefore, equations (3.17) and (3.22a) provide a framework for generating a phase index which will exhibit backwards wave behavior for either the isotropic or anisotropic cases. Finally, it should be noted that (3.20) and (3.22a) demonstrate one of the defining characteristics of negative index media: for a medium to have a negative index of refraction (γ < 0) it must necessarily be dispersive (p > 1).

3.6 Anisotropic Media

In the previous sections the functional form of the index that would result in backward wave propagation was derived for both the "perfect" and "imperfect" cases. In the "perfect" case it was shown that the phase index is necessarily isotropic and negative and in the "imperfect" case it was shown that the index was necessarily anisotropic but no

47 constraints were placed on the sign of the index. In this section the constraints on the sign of the index for the anisotropic case will be investigated. For simplicity, we consider a uniaxial medium whose permittivity and permeability are given by

⎡ε s 00 ⎤ = εε ⎢ ε 00 ⎥ (3.24) o ⎢ s ⎥ ⎣⎢ 00 ε z ⎦⎥

μ = μ μ ro (3.25)

where ε o and μo are the free space permittivity and permeability, respectively. In this uniaxial medium the optical axis is directed along the z axis. In this case any wave whose v D vector is polarized in the x-y plane corresponds to the ordinary wave and, as such, has an isotropic character. Since the isotropic case was already considered in section 3.2, the v ordinary wave will not be discussed here. A wave whose D vector is polarized in the plane containing both the optical axis and the propagation direction is referred to as an extraordinary wave. Using the technique outlined in [12] the dispersion relation for the extraordinary wave in the medium can be derived and is given by

2 2 1 kz 2 11 2 11 = μω r k y (2[ kx () ++++ ) 2 s zs εεεεε zs

2 2 2 2 2 kz k y kx kz 2 11 2 11 2 k (4 2 k y (2() kx () +++++++−± ])) . (3.26) s zs zs s zs εεεεεεεεεε zs

The k-space diagram of the extraordinary wave for propagation through a medium with εs = 1, εz = -2, and µr = 1 is shown in Fig. 3.5. It should be noted that the k-space surfaces in this section are calculated at a fixed frequency value, ω . However, the value of ω only scales the plots and does not change their shapes. Therefore, since we are mainly interested in the shapes of the plots a convenient value of ω = 1 was chosen.

48

(b)

θLHM Vg,LHM

θRHM Vp Vg,RHM ky

Vp . Vg,LHM < 0

Vp . Vg,RHM > 0

kz

Fig. 3.5. (a) 3D and (b) 2D k–Space diagrams for RHM and LHM two-sheeted hyperboloids. In

the RHM case the parameters are εs = 1, εz = -2, and µr = 1. The parameters in the LHM case are

εs = -1, εz = 2, and µr = -1. The k-surfaces for the RHM and LHM cases are identical. The angle between the phase velocity and group velocity for the RHM and LHM cases are shown on the 2D plots.

Consider the extraordinary wave propagating along each of the three principal axes. Let us begin with propagation in the x-direction. Since the definition of the v extraordinary wave requires that D be polarized in the plane containing the propagation v v direction (the x axis in this case) and the optical axis (z-axis) and since ⊥ kD this wave will be polarized along the optical axis (z-axis). For this polarization the extraordinary

49 wave will see an index of n z με r ==−== j 2)1)(2( and will be cutoff so that the k- surface shown in Fig. 3.5(a) has a null along the line (kx, 0, 0). Similarly, the extraordinary wave travelling in the y-direction, which will also be polarized along the z- axis (optical axis), will be cutoff resulting in a null along the line (0, ky, 0). For a wave propagating in the z-direction the definition of an extraordinary wave no longer applies v since k and the optical axis (z-axis) are collinear and do not form a plane. This is the case of an ordinary wave polarized in the x-y plane. This wave propagates since it sees an index n s με r ==== 1)1)(1( which is real and positive. It is interesting to note that if the signs of the effective parameters are reversed so that εs = -1, εz = 2, and µr = -1 the shape of the k-surface is unaffected. In this case the waves travelling in the x- and y-directions are still cutoff and the wave travelling in the z- direction will still propagate but the index will be real and negative n (ε s μr −=−=−== 1)1)(1 (LHM). The k-surface in Fig. 3.5 is an example of a two- sheeted hyperboloid. This type of surface is produced in a uniaxial medium when the extraordinary wave only propagates in the direction of the optical axis. An example of a one-sheeted hyperboloid is shown in Fig. 3.6. For this type of v surface, only a wave whose D vector is polarized in the direction of the optical axis v propagates. As a result there is no ordinary wave in this case since its D vector must be perpendicular to the optical axis. The k-surface in Fig. 3.6 was calculated for a medium with εs = -1, εz = 2, and µr = 1 in the RHM case and with εs = 1, εz = -2, and µr = -1 in the LHM case. Contrary to the two-sheeted hyperboloid the extraordinary waves travelling in the x- and y-directions, and polarized along the z-axis, propagate seeing an index n = 2 in the RHM case and n −= 2 in the LHM case. The extraordinary waves travelling in the z-direction are cutoff ( n = j ). The behavior of the extraordinary wave in the one- and two-sheeted hyperboloids is summarized in Table 3.1.

As mentioned above, if the signs of the effective parameters εs, εz, and µr are reversed the k-surfaces remain unchanged so that the RHM surfaces are indistinguishable from the LHM surfaces. As a result, although the k-surface plots provide useful information about the wave propagation, they give no information about the sign of the

50 index. In order to investigate the sign of the index in these media we return to our v v analysis of the relationship between the v p and vg .

(b) Vg,RHM θRHM

θLHM

Vp Vg,LHM ky Vp . Vg,LHM < 0

Vp . Vg,RHM > 0

kz

Fig 3.6. (a) 3D and (b) 2D k–Space diagrams for RHM and LHM one-sheeted hyperboloids. In

the RHM case the parameters are εs = -1, εz = 2, and µr = 1. The parameters in the LHM case are

εs = 1, εz = -2, and µr = -1. The k-surfaces for the RHM and LHM cases are identical. The angle between the phase velocity and group velocity for the RHM and LHM cases are shown on the 2D plots.

51 Table 3.1. Index seen by different polarizations in the one- and two-sheeted hyperboloids. k-Surface 2-Sheeted 1-Sheeted z z Polarization x-y plane x-y plane (optical axis) (optical axis) Index ±1 j 2 j ± 2

v v For both k -surfaces the angle between v p and vg was calculated as a function of kz along a cut taken in the k y - kz plane. The resulting angles are shown for the two- sheeted and one-sheeted hyperboloids in Fig. 3.7. In the RHM case for both surfaces v v (blue solid lines) the angle between v p and vg begins at 0° at the onset of propagation

( k z = 1 for the two-sheeted case and k z = 0 for the one-sheeted case) and approaches 90° asymptotically from below as kz increases. In the LHM case (red dashed lines) the angle begins at 180° at the onset of propagation and approaches 90° asymptotically from above v v as kz increases. Hence, the dot product of v p and vg is always positive in the RHM case and negative in the LHM case. This shows that the LHM versions of both the one- and two-sheeted hyperboloids exhibit “imperfect” backwards wave behavior. It should be v v noted that for the two-sheeted hyperboloid the y components of v p and vg were parallel and their z components were anti-parallel in the LHM case. On the other hand, for the v v one-sheeted hyperboloid the y components of v p and vg were anti-parallel and their z components were parallel in the LHM case. Hence, LHM behavior was the result of v v oppositely directed components in v p and vg along only one axis emphasizing that “imperfect” backwards waves do not require all of the components of these vectors to be anti-parallel. Looking at equation (3.4) this has a useful implication. If the coordinate v system is rotated so that v p lies along one of the principal axes (in this example the x axis will be used) then equation (3.4) becomes

v gzgygx = pxx )0,0,(),,( = vdvdvvv px (3.27) so that

52

) v gx = vdxv px . (3.28)

In this case the condition for backward wave behavior becomes d x < 0 and only the x v component of vg needs to be considered, a problem which is mathematically the same as the isotropic case considered in (3.5).

Two-Sheeted Hyperboloid 180

150 (a) RHM 120 LHM 90

60

Angle (degrees) Angle 30

0 012345 kz

One-Sheeted Hyperboloid 180 (b) 150 RHM 120 LHM 90

60

Angle (degrees) Angle 30

0 012345 kz Fig. 3.7. Calculated angle between the phase and group velocity vectors for the RHM and LHM hyperboloidal k-surfaces considered. (a) Two-Sheeted hyperboloid and (b) one-Sheeted hyperboloid.

53 In the above analysis equi-frequency surfaces were used to investigate the v v relationship between v p and vg in anisotropic media. In other words, these surfaces were calculated for a fixed frequency and ε and μ were constant. The question then arises: how would including temporal dispersion affect the above analysis? To answer this we must consider the following two points.

a) The “geometry” of the k-surface (an ellipsoid, a one-sheeted hyperboloid, a two- sheeted hyperboloid, or a null surface for which no propagation is allowed) is determined by the signs of the parameters ε and μ. b) The “dimensions” of the k-surfaces (i.e. the vertices of the ellipsoid or hyperboloid) are determined by the magnitudes of ε and μ.

If the frequency is no longer fixed, a new k-surface will be generated for each frequency value considered, resulting in a family of surfaces. In any frequency region where the parameters do not change sign the k-surfaces will all share the same “geometry” (i.e. ellipsoid, hyperboloid, no propagation) so that the sign of the dot v v product between v p and vg in this region, and whether the medium is RHM or LHM, will not change. If one of the parameters changes signs the “geometry” changes (for example from a one-sheeted hyperboloid to a two-sheeted hyperboloid) and the v v relationship between v p and vg changes. This relationship is then maintained until one of the parameters changes signs again. Therefore, for the purposes of this investigation, it is sufficient to examine the general behavior of the four geometries (a single ellipsoid, a one-sheeted hyperboloid, a two-sheeted hyperboloid, and a null surface) in the absence of temporal dispersion since the effects of the dispersion are only significant in that they can result in a transition from one geometry to another. The general case of an ellipsoid (a spheroid) was considered in section 3.2 and the one- and two-sheeted hyperboloids were examined in this section.

54 3.7 Conclusion

The problem of backward wave propagation was studied from a purely three dimensional wave propagation point of view and two different types of backward waves were defined: “perfect” and “imperfect”. In a “perfect” backward wave the angle v v between v p and vg is exactly 180° and in an “imperfect” backward wave the dot product between the phase and group velocity vectors is negative (excluding the case where the angle between them is 180°). The forms of the indices necessary to produce these types v v of backward waves were obtained by examining the relationship between v p and vg . In both cases, it was shown that backward wave behavior could only be the result of propagation through a negative index medium. In the case of the “perfect” backward wave the index was necessarily negative and isotropic while for the “imperfect” backward wave, the medium was anisotropic possessing a negative index along at least one of the three principal axes. Constraints on the functional form of the indices were also obtained using the fact that the group velocity must be positive in a transmission passband. This work was published in the Journal of the Optical Society of America B in 2006 [13].

Referrences

[1] D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneous negative permeability and permittivity”, Phys. Rev. Lett. 84, 4184-4187 (2000). [2] T. Grbic, and G. V. Eleftheriades, “Experimental verification of backward-wave radiation from a negative refractive index metamaterial”, J. of Appl. Phys. 92, 5930-5935 (2002). [3] J. Woodley, M. Mojahedi, “Negative group velocity and group delay in left- handed media”, Phys. Rev. E 70, 046603 (2004). [4] D. R. Smith, D. Schurig, J. B. Pendry, “Negative refraction of modulated electromagnetic waves”, Appl. Phys. Lett. 81, 2713-2715 (2002).

55 [5] V. Lindell, S. A. Tretyakov, K. I. Nikoskinen, S. Ilvonen, “BW Media with negative parameters, capable of supporting backward waves”, Microwave and Opt. Tech. Lett. 31, 129-133 (2001) [6] M. Mojahedi, E. Schamiloglu, K. Agi, and K. J. Malloy, “Frequency-Domain detection of superluminal group velocity in a distributed Bragg reflector” IEEE J. of Quant. Elec. 36, 418-424 (2000). [7] E. L. Bolda, and R. Y. Chiao, “Two theorems for the group velocity in dispersive media”, Phys. Rev. A 48, 3890-3894 (1993). [8] S. Ramo, Fields and Waves in Communication Electronics, (John Wiley & Sons, Inc., New York, 1994). [9] A. Grbic, G. V. Eleftheriades, “Experimental verification of backward-wave radiation from a negative refractive index metamaterial”, J. of Appl. Phys. 92, No. 10 (Nov. 2002) [10] A. K. Iyer, G. V. Eleftheriades, “Negative refractive index metamaterials supporting 2-D waves”, IEEE MTT-S International Symposium Digest 2, 1067- 1070 (2002). [11] C. Caloz, H. Okabe, H. Iwai, and T. Itoh, , “Application of the transmission line theory of left-handed (LH) metamaterials to the realization of a microstrip “LH line” ”, IEEE AP-S/URSI Internatianal Symp., San Antonio, TX, 412-415 (2002). [12] J. A. Kong, “Electromagnetic Wave Theory”, (John Wiley & Sons, Inc., Toronto, 2nd edition, 2000). [13] J. F. Woodley, M. Mojahedi, “Backward wave propagation in left-handed media with isotropic and anisotropic permittivity tensors”, Josa B 23, num. 11, 2377 (2006).

56 Chapter 4

The Retrieval Technique

4.1 Introduction

In this chapter the signs of the imaginary parts of the effective permittivity, ε , and permeability, μ , in metamaterials will be examined. Such an examination is necessary as there have been recent claims that some metamaterials can exhibit ε < 0'' or μ < 0'' while remaining passive [1-9]. For example, it was claimed that an array of split wires, a medium that exhibits ε < 0' , also exhibits μ < 0'' in the resonance region [2]. Conversely, it has also been claimed that an array of split ring resonators (SRRs), a medium that exhibits μ < 0' , also exhibits ε < 0'' in the resonance region [2]. Finally, it was put forward that a medium composed of both strip wires (SWs) and SRRs, a medium that exhibits n < 0' , also exhibits ε '' < 0 in the magnetic resonance region [1]3. In each of these cases the negative value in the imaginary part is accompanied by an anti- resonance in the real part. These results were obtained using a technique that retrieves the effective parameters (n, z, ε and μ ) of a medium from simulated or experimental S- parameters [10]. Hereon, this technique will be referred to as the retrieval technique. In this chapter the derivation of the retrieval technique will be discussed. Then the technique will be used to obtain the effective parameters of some common metamaterials: an array of split wires, an array of SRRs and a combined array of SWs and SRRs. For each of these metamaterials it will be shown that the technique predicts either ε < 0'' or μ < 0'' , which seems to be unphysical. Hence, the results presented in this chapter serve to demonstrate the critical need for an examination of the validity of the retrieval technique. If accurate analytical expressions existed to describe the behavior

3 The reason why this combined medium exhibits ε < 0'' and not μ < 0'' will be explained in Chapter 5.

57 of these structures it would be a simple matter to test the validity of these results. Unfortunately, because these metamaterials are composed of complex inclusions no analytical expressions exist to accurately determine their effective parameters. Because of this, the results of the retrieval technique cannot be corroborated or repudiated in the cases considered. Therefore, in the next chapter another medium will be considered for which the validity of the retrieval technique can be examined.

4.2 The Retrieval Technique

As mentioned above the claims that either the imaginary part of the permittivity or permeability can be negative in certain metamaterials have been based on the results of a homogenization method referred to as the retrieval technique. In this technique, it is assumed that the metamaterial in question can be replaced by an isotropic, homogeneous slab of finite thickness that produces the same S-parameters as the metamaterial. The parameters of this slab are then referred to as the effective parameters of the metamaterial. To calculate these effective parameters consider the S-parameters for propagation through a slab of thickness d in free space [10]

j 1 ω (z − nd]sin[) SS == 2 cz , (4.1) 2211 ω j 1 ω nd]cos[ (z ++ nd]sin[) c 2 cz 1 SS == . (4.2) 12 21 ω j 1 ω nd]cos[ (z ++ nd]sin[) c 2 cz

The effective index of refraction and normalized effective characteristic impedance of the slab can be obtained by inverting (4.1) and (4.2) and are given by

1 −1 1 2 2 2πm n = [cos 1( 11 SS 12 )] ++− (4.3) o 2Sdk 12 o dk

58 2 2 11 ++ SS 12 ))1( z = 2 2 (4.4) 11 +− SS 12 ))1(

where ko is the free space wave vector and m is an integer that accounts for the different possible branch cuts of the inverse cosine function. It is then straightforward to find the effective permittivity and permeability using

n ε = , (4.5) z μ = nz . (4.6)

Although this is the most commonly used form of the retrieval technique many variations have been derived including variations for isotropic [13], inhomogeneous [14], bianisotropic [15, 16] and magnetoelectic [17] media. All of these variations have also predicted ε < 0'' or μ < 0'' . It should be noted that the length, d, of the effective slab can be chosen to be any value and that the physical length (in the propagation direction) of the metamaterial is generally used. From equation (4.4) we see that the chosen length has no effect on the retrieved effective impedance but has an inverse relationship with the effective index. In order to demonstrate the type of behavior predicted by the retrieval technique three common metamaterial structures are considered. These are an array of split wires (a ε < 0' medium), an array of SRRs (a μ < 0' medium) and a composite structure composed of SWs and SRRs (a n < 0' medium). The unit cell of each structure is shown in Fig. 4.1. The S-parameters for propagation through a single unit cell of the structures were determined using Ansoft HFSS, a commercially available software simulator, and the effective parameters were obtained from the S-parameters using equations (4.3) – (4.6). In each case the following boundary conditions were used. On the top and bottom faces of the unit cell (perpendicular to the y-direction) perfect electric conductor (PEC) boundary conditions were used. These boundary conditions force the tangential component of the electric field to be zero so that the electric field is polarized in the y- direction. On the front and back faces (perpendicular to the x-direction) perfect magnetic

59 boundary conditions (PMC) were used. These boundary conditions force the tangential component of the magnetic field to be zero ensuring that the magnetic field is polarized along the x-direction. In addition the PEC and PMC boundary conditions act as mirrors so that the medium appears infinitely periodic in the x and y directions and the propagating wave is a TEM wave. On the side faces (perpendicular to the z-direction) wave ports where used so that the wave propagates in the z-direction.

Fig. 4.1. Unit cell for (a) the split wire, (b) the SRR and (c) a continuous SW and SRR.

60 4.2.1 An Array of Split Wires

The array of split wires was proposed by Smith and Soukoulis in (2003) as an alternative to the array of continuous wires [2]. The advantage of the split wire is that by careful selection of the length of the wire segments and the width of the split the value of the resonance frequency and the strength of the resonance can be tuned whereas in the case of the continuous wire the resonance frequency is always at 0 Hz. Figure 4.2 shows the simulated S-parameters for propagation through one unit cell of Smith’s array of split wires immersed in air and infinite in the transverse plane (x-y). The dimensions of the unit cell are 3.33mm x 7.33mm x 3.66mm. The cross section of the wire was square with 0.33mm sides, the length of the wire is 7mm, the split is 0.33mm and the material of the wires was PEC. From Fig. 4.2 the peak in the reflection ( S11 ) and sharp drop in the transmission ( S12 ) indicate that the resonance frequency is at 24.2GHz. The real and imaginary parts of the retrieved effective index of refraction and normalized characteristic impedance were obtained from the S-parameters using (4.3) and (4.4) and are shown in Fig. 4.3(a) and 4.3(b). Note that the sharp features of the plots are due to the fact that there are no material losses (the wires are in air and composed of PEC). From Fig. 4.3 the medium has a positive real effective index of refraction and both the imaginary part of the effective index and the real part of the effective impedance are positive indicating that the medium behaves passively. Note that the imaginary part of the effective impedance is negative so that this medium behaves capacitively in the resonance region, the significance of this will be discussed in Chapter 5. The effective permittivity and permeability were obtained using (4.5) and (4.6) and are shown in Fig. 4.4(a) and 4.4(b) respectively. Figure 4.4(a) shows that the real part of the effective permittivity is negative (as expected) and that its imaginary part is positive indicating passive behavior. From Fig. 4.4(b) the real part of the permeability is positive but exhibits an anti-resonance while its imaginary part is negative indicating active behavior. Because there are no material losses in the simulations the resonances aren't dampened resulting in large values of n and ε . In fact the index is large enough that the Bragg condition λ = 2dn is satisfied in the re sonance region of the periodic array. As a result the index is bounded and reaches a maximum value given by

61

c π n = o (4.7) max ωd

where co is the speed of light in vacuum and = 66.3 mmd is the thickness of the unit cell in the pro pagation direction. The index given by (4.7) is plotted in Fig. 4.3(a) [purple dashed curve] where it is clear that the retrieved index does not exceed nmax in the resonance region.

0 0

-2 S11 -10 S12 -4 -20

-6 -30 dB

-8 -40

-10 -50

-12 -60 5 1015202530 Frequency (GHz)

Fig. 4.2. (a) S11 and (b) S12 for propagation through an array of split wires infinite in the transverse plane (x-y) and one unit cell thick in the propagation direction (z).

It is argued in [2] that the anti-resonance in μ occurs because the index is bounded. In other words since μ is calculated from

n2 μ = max (4.8) ε

62 and nmax is bounded, μ must be anti-resonance when ε is resonant. However, this anti- resonant behavior has been predicted by the retrieval technique in many other metamaterials in which the index was not bounded (see the following sections and [3-9]).

n = n' - jn'' 4 3.5 (a) 3 2.5 2 1.5 n' 1 n''

0.5 n max 0 5 1015202530 Frequency (GHz)

z = z' + jz'' 0.7 1 (b) 0.6 0 0.5 -1 0.4 -2 0.3 -3 0.2 z'

0.1 z'' -4

0 -5 5 1015202530 Frequency (GHz)

Fig. 4.3. Real and imaginary parts of (a) effective index of refraction and (b) normalized characteristic impedance of an array of split wires.

63 ε = ε' - jε'' 50 (a) 40

30 ε'

20 ε''

10

0

-10 5 1015202530 Frequency (GHz)

μ = μ' - jμ'' 5 0.5 (b) 0 4

μ' -0.5 3 μ'' -1 2 -1.5

1 -2

0 -2.5 5 1015202530 Frequency (GHz)

Fig. 4.4. Real and imaginary parts of (a) effective permittivity and (b) effective permeability of an array of split wires.

One of the best examples is given in [18] (which is written by the authors of [2]) where anti-resonant behavior is predicted by the retrieval technique in a medium operating at λ = 25d . Hence, this argument does not seem to explain the origin of this behavior. This is just one example of the type of behavior predicted by the retrieval

64 technique. In the following section the results of using this technique to find the effective parameters of an array of SRRs will be discussed.

4.2.2 An Array of Split Ring Resonators

The SRRs that will be examined here are the same as those used in Chapter 2. The dimensions of the SRRs (shown in Fig. 4.5) are r = 506.0 mm, c = 0.124 mm, d = 0.15 mm, g = 0.114 mm and w = 1.3 mm. The unit cell is cubic with 2.5mm sides and filled with air. The SRRs are flat and made of copper. Figure 4.6 shows the S- parameters for propagation through an array of SRRs infinite in the transverse plane and one unit cell thick in the propagation direction. The resonance frequency of the rings is 30.75GHz. The real and imaginary parts of the effective index of refraction and normalized characteristic impedance of the SRRs were obtained using (4.3) and (4.4) and are shown in Fig. 4.7(a) and 4.7(b), respectively. Similar to the case of the split wires the real part of the effective index is positive and both the imaginary part of the effective index and real part of the effective impedance are positive, so that the medium behaves passively. On the other hand, contrary to the case of the split wires, the medium behaves inductively in the resonance region where the imaginary part of the effective impedance is positive (this will be discussed further in Chapter 5). The real and imaginary parts of the permittivity and permeability of the array of SRRs were then obtained using equation (4.5) and (4.6) and are shown in Fig. 4.8, respectively. In this case the real part of the effective permittivity exhibits an anti-resonance which is accompanied by a negative imaginary part indicating an active electric response. On the other hand, the real part of the permeability is negative (as expected) and its imaginary part is positive indicating a passive magnetic response. Note that the results for the SRRs don’t have the sharpness exhibited by the split wires because in this case material losses were included in the simulations. However, in terms of the behavior of the permittivity and permeability this structure seems to be the reverse of the split wire structure seen in Section 4.2.1 (i.e. the signs of the real and imaginary parts of the permittivity and permeability have switched). In the next section a structure composed of a combination of wires and SRRs will be considered.

65

Fig. 4.5. The SRR used in the simulations with w = 1.3 mm, c = 0.124 mm, d = 0.15 mm, g = 0.288 mm, and r = 0.506 mm.

0

-5 S11

-10 S12

-15 dB

-20

-25

-30 28 30 32 34 Frequency (GHz)

Fig. 4.6. (a) S11 and (b) S12 for propagation through an array of SRRs infinite in the transverse plane and one unit cell thick in the propagation direction. The dimensions of the SRRs are given in Fig. 4.5.

66 n = n' - jn'' 2 (a)

1.5

1

n' 0.5 n''

0 28 29 30 31 32 33 34 Frequency (GHz)

z = z' + jz'' 3 (b) 2.5 z' 2 z'' 1.5

1

0.5

0 28 29 30 31 32 33 34 Frequency (GHz)

Fig. 4.7. Real and imaginary parts of the effective (a) index of refraction and (b) normalized characteristic impedance of an array of SRRs. The dimensions of the SRRs are given in Fig. 4.5.

67 ε = ε' - jε'' 3 0 (a) 2.5 -0.4 2 ε'

1.5 ε'' -0.8

1 -1.2 0.5

0 -1.6 28 29 30 31 32 33 34 Frequency (GHz)

μ = μ' - jμ'' 6 (b)

4 μ'

μ'' 2

0

-2 28 29 30 31 32 33 34 Frequency (GHz)

Fig. 4.8. Real and imaginary parts of the effective (a) permittivity and (b) permeability of an array of SRRs. The dimensions of the SRRs are given in Fig. 4.5.

4.2.3 Continuous Strip Wires and Split Ring Resonators

The continuous SW and SRR structure considered here is the same as that introduced in Chapter 2. The unit cell is square with 2.5mm long sides and is filled with

68 air. The strip material is copper and the strips are flat and 0.5mm wide. The SWs and SRRs are printed on opposite sides of a 0.5mm thick substrate with a dielectric constant of ε = 02.3 . Figure 4.9 shows the S-parameters for propagation through the combined array. The real and imaginary parts of the effective index of refraction and normalized characteristic impedance were obtained using equation (4.3) and (4.4) and are shown in Fig. 4.10(a) and 4.10(b), respectively. In this case the combined structure exhibits a negative real part of the effective index of refraction (as expected) and both the imaginary part of the effective index and the real part of the effective impedance are positive indicating passive behavior. The imaginary part of the effective impedance is negative in the resonance region indicating that the SW resonance dominates and the medium behaves capacitively (this will be discussed further in Chapter 5).

0 -3 -6 -9 -12

dB -15 -18 -21 S11 -24 S12 -27 21 21.5 22 22.5 23 Frequency (GHz)

Fig. 4.9. (a) S11 and (b) S12 for propagation through an array of continuous SWs and SRRs infinite in the transverse plane and one unit cell thick in the propagation direction. The width of the wires is 0 .5 mm and the dimensions of the SRRs are given in Fig. 4.5.

69 n = n' - jn'' 0 3 (a) 2.5 -0.5

2 -1 1.5 -1.5 n' 1 -2 n'' 0.5

-2.5 0 21 21.5 22 22.5 23 Frequency (GHz)

z = z' + jz'' 4 0 3.5 (b) -0.5 3 -1 -1.5 2.5 -2 2 -2.5 1.5 z' -3 1 z'' -3.5 0.5 -4 0 -4.5 21 21.5 22 22.5 23 Frequency (GHz)

Fig. 4.10. Real and imaginary parts of the effective (a) index of refraction and (b) normalized characteristic impedance of an array of continuous SWs and SRRs. The width of the wires is 0.5 mm and the dimensions of the SRRs are given in Fig. 4.5.

70 ε = ε' - jε'' 0 0.05 0 -0.2 (a) -0.05 -0.1 -0.4 -0.15 -0.2 -0.6 -0.25 ε' -0.3 -0.8 ε'' -0.35 -0.4 -1 -0.45 21 21.5 22 22.5 23 Frequency (GHz)

μ = μ' - jμ'' 8 12 (b) 6 10 4 8 2 6 0 μ' 4 -2 μ'' -4 2 -6 0 21 21.5 22 22.5 23 Frequency (GHz)

Fig. 4.11. Real and imaginary parts of the effective (a) permittivity and (b) permeability of an array of continuous SWs and SRRs. The width of the wires is 0.5 mm and the dimensions of the SRRs are given in Fig. 4.5.

Using equation (4.5) and (4.6) the real and imaginary parts of the effective permittivity and permeability were obtained and are shown in Fig. 4.11(a) and 4.11(b), respectively. From Fig. 4.11(a) the real part of the effective permittivity is negative and

71 exhibits an anti-resonance in the magnetic resonance region at 22GHz. This anti- resonance is accompanied by a negative imaginary part indicating active behavior. The real part of the effective permeability in Fig. 4.11(b) is also negative but accompanied by a positive imaginary part indicating passive behavior. This immediately raises the question as to why the structure composed of both SWs and SRRs exhibits a negative imaginary effective permittivity (as seen with the SRRs) but not a negative imaginary permeability (as seen with the wires). This will be discussed in Chapter 5. Considering the results of using the retrieval technique on these three common metamaterials one realizes that there is a critical need for analytical expressions that give exact values for their effective parameters. If such expressions existed it would be a simple matter to either corroborate or repudiate the retrieved results. Unfortunately, due to the complexity of these structures no exact analytical expressions have yet been derived to predict their behavior (although attempts have been made [11, 12]). Therefore, it would be immensely illustrative to consider a metamaterial that behaves similarly to the wires or SRRs but for which we do posses analytical expressions that can be used to exactly or at least accurately predict their behavior. In this case, the effective parameters could be determined both analytically and through the retrieval technique and valuable insight would be gained as to the validity of this technique. In the following chapter such a metamaterial will be considered.

4.3 Conclusion

In this chapter a popular technique used to retrieve the effective parameters of metamaterials was presented and examples of its use were given. This technique retrieves the effective parameters of a medium from its S-parameters assuming that the medium can be approximated by an isotropic, homogeneous slab. An examination of the retrieval technique is necessary because its use has led to claims that some metamaterials can exhibit either ε < 0'' or μ < 0'' while remaining passive. The results of using the retrieval technique on several common metamaterials (an array of split wires, an array of SRRs and a combined array of SWs and SRRs) were demonstrated and it was shown that in each case either ε < 0'' or μ < 0'' was predicted. Unfortunately, since no analytical

72 expressions exist to accurately determine the effective parameters of these metamaterials these results cannot be corroborated. Therefore, in the next chapter the validity of the retrieval technique will be discussed.

References

[1] P. Markos, C. M. Soukoulis, “Transmission properties and effective electromagnetic parameters of double negative metamaterials”, Optics Express 11, 649-661 (2003). [2] T. Koschny, P. Markos, D. R. Smith, and C. M. Soukoulis, “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials”, Phys. Rev. E 68, 065602 (2003). [3] C. Rockstuhl, T. Zentgraf, H. Guo, N. Liu, C. Etrich, I. Loa, K. Syassen, J. Kuhl, F. Lederer, H. Giessen, “Resonance of split-ring resonator metamaterials in the near infrared”, App. Phys. B 84, 219 (2006). [4] J. Zhou, L. Zhang, G. Tuttle, T. Koschny, C. M. Soukoulis, “Negative index materials using simple short wire pairs”, Phys. Rev. B 73, 041101 (2006). [5] U. K. Chettiar, A. V. Kildishev, T. A. Klar, V. M. Shalaev, “Negative index metameterials combining magnetic resonators with metal films”, Opt. Express 14, No. 17, 7872 (2006). [6] L. Zhen, J. T. Jiang, W. Z. Shao, C. Y. Xu. “Resonance-antiresonance electromagnetic behavior in a disordered dielectric composite”, App. Phys. Lett. 90, 142907 (2007). [7] L. Fu, H. Schweizer, H. Guo, N. Liu, H. Giessen, “Synthesis of transmission line models for metamaterial slabs at optical frequencies”, Phys. Rev. B 78, 115110 (2008). [8] J. Wang, S. Qu, Z. Xu, H. Ma, S. Xia, Y. Yang, X. Wu, Q. Wang, C. Chen, “Normal-incidence left-handed metamaterials based on symmetrically connected split-ring resonators”, Phys. Rev. E 81, 036601 (2010).

73 [9] T. Lepetit, E. Akmansoy, J. -P. Ganne, “Experimental evidence of resonant effective permittivity in a dielectric metamaterial”, J. of App. Phys. 109, 023115 (2011). [10] D. R. Smith, S. Schultz, P. Markos, C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients”, Phys. Rev. B 65, 195104 (2002). [11] R. Marques, F. Medina, R. Rafii-El-Idrissu, “Role of bianisotropy in negative permeability and left-handed metamaterials”, Phys. Rev. B. 65, 144440 (2002). [12] A. Ishimaru, “Generalized constitutive relations for metamaterials based on the quasi-static lorentz theory”, IEEE. Trans. Antennas Propag. 51, No. 10, 2550 (2003). [13] X. Chen, T. M. Grzegorcyk, B.-I. Wu, J. Pachero, and J. A. Kong, "Robust method to retrieve the constitutive effective parameters of metamaterials", Phys. Rev. E 70, 016608 (2004). [14] D. R. Smith, D. C. Vier, Th. Koschny, and C. M. Soukoulis, "Electromagnetic parameter retrieval from inhomogeneous metamaterials", Phys. Rev. E 71, 036617 (2005). [15] X. Chen, B.-I. Wu, J. A. Kong, and T. M. Grzegorcyk, "Retrieval of the effective constitutive parameters of bianisotropic metamaterials", Phys. Rev. E 71, 046610 (2005). [16] Z. Li, K. Aydin, and E. Ozbay, "Determination of the effective constitutive parameters of bianisotropic metamaterials from reflection and transmission coefficients", Phys. Rev. E 79, 026610 (2009). [17] D. R. Smith, "Analytic expressions for the constitutive parameters of magnetoelectric metamaterials", Phys. Rev. E 81, 036605 (2010). [18] Th. Koschny, P. Markos, E. N. Economou, D. R. Smith, D. C. Vier, and C. M. Soukoulis, "Impact of inherent periodic structure on effective medium description of left-haned and related metamaterials", Phys. Rev. B 71, 245105 (2005).

74 Chapter 5

The Imaginary Parts of the Effective Permittivity and Permeability

5.1 Introduction

In the previous chapter a technique often used to retrieve the effective parameters of metamaterials from their S-parameters was derived. Using this technique, which has been dubbed the retrieval technique [1], the effective parameters of several common metamaterials were obtained and it was shown that the retrieval technique predicted either ε < 0'' or μ < 0'' in these metamaterials even though they remained passive ( n < 0'' ). These predictions are consistent with those obtained by other authors employing this technique [2-10]. In this chapter the validity of ε < 0'' or μ < 0'' in a passive medium and of the retrieval technique will be discussed. Techniques such as the retrieval technique are used because in many of the cases considered the metamaterials are composed of complex inclusions for which it is very difficult to obtain exact analytical expressions for effective parameters. Unfortunately, in cases where there are no exact analytical expressions available [such as the split wire and Split Ring Resonator (SRR)] there is no easy way to corroborate the validity of the retrieved results. In order to test the validity of this technique then it would be instructive to consider a metamaterial that behaves similarly to the SRR or split wire but for which accurate analytical expressions describing its behavior are available. The analytical results could then be compared to the retrieved results and some insight into the validity of the retrieval technique could be gained. In [1, 11] it is argued that ε < 0'' or μ < 0'' does not violate passivity because the rate at which energy is dissipated into heat is positive in the system. To understand this argument consider the heat dissipation in a medium [12] given by

75

2 2 = + μεω HEQ )''''( 4. (5.1)

In the case of a passive medium the energy of the wave is absorbed and dissipated into the surrounding medium in the form of heat so that Q is positive. Conversely, if Q < 0 then the medium is seen to take energy from its surrounding and feed it to the propagating wave so that the wave is amplified and the medium is considered active. One example of such a system is a laser in which energy is pumped into a medium in order to amplify a beam of light. It should be noted that such a process can only occur if energy is supplied by a source. Indeed if such a process where to happen spontaneously in nature it would be a violation of the third law of thermodynamics, i.e. the entropy of the system would decrease. It is for this reason that ε < 0'' or μ < 0'' is troubling in a medium such as an SRR or wire where, with the exception of the incoming excitation, no additional energy is supplied to the system. However in the SRR (or wire) medium although the retrieved ε '' (or μ '' ) is negative it turns out that the accompanying μ '' (or ε '' ) is large and positive so that Q is still positive (Q > 0). As a result it is unclear if passivity has been violated in these structures and it is more useful to consider the principles of causality and analyticity instead (see section 5.2). Note that it has been suggested that the existence of ε '' < 0 or μ < 0'' in a medium could be tested by placing v v it in a capacitor or solenoid where one of the fields ( E or H ) dominates [13]. In this case Q would necessarily be negative (the medium would exhibit gain) but to date no data has been presented to support this. In this chapter the concepts of causality and analyticity will be discussed by considering the Lorentzian model, a model commonly used to characterize and predict the behavior of many metamaterials. It will be shown that the Lorentzian model does not allow ε < 0'' or μ < 0'' in the metamaterials considered. Then, the effective parameters of an array of dielectric spheres will be calculated both analytically and by using the retrieval technique. By comparing the analytical and retrieved results the source of the

4 The derivation of Q is discussed in Appendix B.

76 negative imaginary parts predicted by the retrieval technique can be brought to light so that the validity of this technique can be put to scrutiny.

5.2 The Lorentzian Model for Metamaterials

In 1996 Pendry et al. developed a negative permittivity metamaterial consisting of an array of continuous wires [14]. In this work they showed that the permittivity of this array could be approximated analytically by the Lorentzian model

ω 2 ε 1−= pe 22 −− i ωγωω eoe (5.2)

where ω pe is the electric plasma frequency, ωoe is the electric resonance frequency and

γ e is the electric damping constant. Then in 1999 the same group introduced the first negative permeability metamaterial which was composed of an array of SRRs [15]. In this work the authors showed that the permeability of an array of SRRs could also be approximated analytically by a Lorentzian model

ω 2 μ 1−= pm eff 22 −− i ωγωω om m (5.3)

where ω pm is the magnetic plasma frequency, ωom is the magnetic resonance frequency and γ m is the magnetic damping constant. Note that both of these structures are discussed in more detail in Chapter 1. Since then many new metamaterials have been proposed and the Lorentzian model is commonly used to approximate and describe their behavior. A question then arises: what information does the Lorentzian model give us regarding the validity of negative imaginary parts of the permittivity or permeability in metamaterials. To consider this, expand the permittivity into its real and imaginary parts according to

77 222 2 pe − ωωω oe )( ωωγ pee jεεε ''' =−= 222 2 − j 222 2 . (5.4) oe +− eωγωω oe +− eωγωω )()()()(

Note that (5.4) is written according to the engineering convention where a forward v v v •− rkj v propagating wave has the form = oeEE . Examining (5.4), for positive frequencies and real values of γ e and ωoe , the imaginary part of the permittivity can only be negative

2 under two conditions: γ e < 0 or ω pe < 0 . Let us consider each case separately.

Case 1: γ < 0 e

To consider this case we look at the definition of the permittivity: the v v permittivity,ε , relates the displacement vector, D , and the electric field vector, E . In the time domain this relationship is expressed by the well known result [16]

⎧ ∞ ⎫ = ε + − ),()(),(),( dtrEGtrEtrD τττ (5.5) 0 ⎨ ∫ ⎬ ⎩ ∞− ⎭ where G τ )( is the susceptibility kernel given by

1 ∞ ωε )( G(τ ) = ( − )1 jωτ de ω . (5.6) 2π ∫ ε ∞− o

Substituting (5.2) into (5.6) and performing the integration the susceptibility kernel becomes

τ −γ e τν )sin( G )( = ωτ 2 e 2 o Θ τ )( (5.7) pe ν o

78 γ 2 where Θ τ )( is the step function and ν ω 22 −= e . Substituting γ < 0 into (5.7) the oeo 4 e susceptibility kernel takes the form

τ γ 2 e 2 oτν )sin( )( = ωτ pe eG Θ τ )( (5.8) ν o which is unbounded as τ ∞→ . As a result the susceptibility kernel, and hence the permittivity, is not analytic in this case. By invoking the Titchmarsh theorem this implies that the permittivity is also not causal [17]. Therefore, the case with γ e < 0 is unphysical and cannot be considered as a possible candidate causing the imaginary part of the permittivity to be negative.

2 Case 2: ω pe < 0

To consider this case let us revisit the derivation of the Lorentzian dispersion relations using a harmonic electron oscillator model. To derive the Lorentzian dispersion we begin by considering the effect of an incident electric field, E , on the electrons in a medium. The equation of motion for such an electron is given by

•• • 2 e ωγ oe −=++ eExmxmxm (5.9) where x is the position of the electron from equilibrium, e is the charge of an electron, m is its mass and the prime indicates differentiation with respect to time. Looking at the left

•• side of equation (5.9) the first term on the left, xm , is Newton’s second law of motion,

• the second term, γ e xm , describes the damping of the electron and the third term ωoe xm describes the restoring force (similar to a spring) acting on the electron by the nucleus of

ωtj ωtj the atom. Assuming time harmonic fields, we can write = oeEE and = xx oe so that (5.9) can be written as

79

− eEo xo = 2 2 . (5.10) e ++− mjmm ωωγω oe

Using equation (5.10), the Polarization induced by the incident electric field can be written as

2 ENe o NexP o =−= 2 2 (5.11) oe ++− jmmm eωγωω where N is the number of atoms per unit volume. Using another expression for the polarization, = ε o χEP , and the definition for the permittivity ε = ε o + χ)1( we can write

2 mNe ε 1−= . (5.12) 22 −− i ωγωω eoe

The above derivation for the permittivity in (5.12) was performed in the classical domain. If we consider quantum mechanically considerations the expression would also include the oscillator strength, f, that relates the energy difference between two electron states in the atom and is given by

2mω 2 f = o jxi (5.13) h where ħ is the Planck constant, and i and j are two energy states of the atom [18]. Finally, introducing the oscillator strength into equation (5.12) we get the Lorentzian form of the permittivity

2 mfNe ε 1+= 22 (5.14) oe −− j eωγωω

80 where

2 fNe ω 2 = . (5.15) pe m

Let us now return to the problem of a negative imaginary part of the permittivity. Since N, e, and m are all defined as real and positive the only way for the plasma

2 frequency, ω pe , to be negative is if the oscillator strength, f, is negative. However, f can only be negative in a medium undergoing population inversion [19, 20]. Since the metamaterials in question are not undergoing population inversion the plasma frequency

2 in these media cannot be purely imaginary and we cannot consider ω pe < 0 . By careful consideration of the Lorentzian model it has been shown here that this model does not allow the imaginary part of the permittivity to be negative in the metamaterials considered. As a final note it should be pointed out that all of the results for the permittivity in this section can be similarly shown to apply for the permeability.

5.3 An Array of Dielectric Spheres

As mentioned above one of the greatest obstacles to metamaterial research is that due to the complex structures involved there is great difficulty in determining exact analytical expressions for the effective parameters. It is for this reason that homogenization procedures such as the retrieval technique are so often used. Unfortunately, using such techniques presents a double edged knife. They are necessary because no analytical expressions are available, but then their questionable results cannot be tested due to the lack of these expressions. In simple cases, such as the analytical case of a slab with Lorentzian permittivity and permeability [equations (5.2) and (5.3)], the retrieval technique returns the correct result but this does not mean it will stand up in more complex cases involving simulated and measured data. For this reason it is critical to test the technique using simulated or measured data of a resonant structure that behaves similarly to the split wires or SRRs but for which accurate or exact analytical expressions exist. By comparing the analytical and retrieved results invaluable insight

81 could be gained into the validity of the retrieval technique. In this section a medium will be considered for which by using Mie theory and the Claussius-Mossoti equation accurate analytical expressions describing its effective parameters can be obtained. This structure is an array of dielectric spheres.

5.3.1 Mie Theory

To determine the effective parameters of an array of dielectric spheres we begin with a single sphere. Using Mie theory the electric and magnetic fields scattered by a single sphere of radius r due to an incident plane wave can be calculated and are proportional to 5

' ' m m − ψψψψ mm nxxxnxn )()()()( am = ' ' (5.16) m m − ψξξψ mm nxxxnxn )()()()(

' ' m m − ψψψψ mm nxxnxnx )()()()( bm = ' ' (5.17) m m − ψξξψ mm nxxnxnx )()()()(

where = rkx o , ψ m x)( and ξ m x)( are the spherical Riccati – Bessel functions, the primes indicate differentiation with respect to the argument, m indicates the term in the multipole expansion and n is the index of the spheres. The dielectric spheres are immersed in air and ko is the free space wave vector. The scattering coefficients given in (5.16) and (5.17) present exact solutions to Maxwell’s equations in the long wavelength limit (effective medium approximation). In Appendix C it is shown that in the case considered here only the dipole terms a1 (corresponding to the electric dipole) and b1 (corresponding to the magnetic dipole) make a significant contribution so that the higher order multipole terms can be neglected. Once the fields for a single sphere are known the effective permittivity and permeability of an array of spheres [Fig. 5.1] can be calculated using the Claussius-Mossoti equation and are given by

5 A more detailed description of Mie theory and the array of dielectric spheres is presented in appendix C.

82 3 o + 4π V aiNk 1 ε an = 3 (5.18) o − 2π V aiNk 1

3 o + 4π V biNk 1 μan = 3 (5.19) o − 2π V biNk 1

where NV is the volume density of the spheres. Note that using the Claussius- Mossoti expression requires that three assumptions be made. First, that the spheres behave like dipoles so that the Lorentz local field approximation is valid. This is justified since it was shown in Appendix C that only the dipole terms make a significant contribution to the scattered fields. Second, that the array behaves like a dilute gas. In other words, although the spheres do interact with each other they are not tightly packed together. And third, that the Lorentz local field is the same at each lattice point. This is justified by the periodicity of the lattice.

Fig. 5.1. An array of dielectric spheres.

83 n' 3 (a) 2.5 Retrieval (FDTD) Analytical 2 Retrieval (FEM) 1.5

1

0.5

0 12345 Frequency (THz)

n'' 2.5 (b) 2 Retrieval (FDTD) Analytical 1.5 Retrieval (FEM)

1

0.5

0 12345 Frequency (THz)

Fig. 5.2. Real (a) and imaginary (b) parts of the effective index of refraction obtained analytically (red dotted line) and from using the retrieval technique on S-parameters obtained from FDTD (blue solid line) and FEM (green dashed line) simulations.

84 z' 1.4 (a) 1.2 Retrieval (FDTD) 1 Analytical 0.8 Retrieval (FEM) 0.6 0.4 0.2 0 12345 Frequency (THz)

z'' 1.2 (b) 1 Numerical (FDTD) 0.8 Analytical

0.6 Numerical (FEM)

0.4

0.2

0 12345 Frequency (THz)

Fig. 5.3. Real (a) and imaginary (b) parts of the normalized effective characteristic impedance obtained analytically (red dotted line) and from using the retrieval technique on S-parameters obtained from FDTD (blue solid line) and FEM (green dashed line) simulations.

85 ε' 3.5 (a) 3 2.5 2

1.5 Retrieval (FDTD) Analytical 1 Retrieval (FEM) 0.5 Field Averaging (FEM) 0 12345 Frequency (THz)

ε'' 1.2 (b) Retrieval (FDTD) 0.8 Analytical Retrieval (FEM) 0.4 Field Averaging (FEM)

0 12345 -0.4

-0.8 Frequency (THz)

Fig. 5.4. Real (a) and imaginary (b) parts of the effective permittivity obtained analytically (red dotted line), using the field averaging technique on the FEM simulations (purple dash-dotted line), and using the retrieval technique on S-parameters obtained from FDTD (blue solid line) and FEM (green dashed line) simulations.

86 μ' 4 (a) 3 Retrieval (FDTD) Analytical 2 Retrieval (FEM)

1

0 12345 -1

-2 Frequency (THz)

μ'' 5 (b) 4

Retrieval (FDTD) 3 Analytical 2 Retrieval (FEM) 1

0 12345 Frequency (THz)

Fig. 5.5. Real (a) and imaginary (b) parts of the effective permeability obtained analytically (red dotted line) and from using the retrieval technique on S-parameters obtained from FDTD (blue solid line) and FEM (green dashed line) simulations.

87 S11 0

-5 (a)

-10

-15 dB FDTD -20 Mie -25 FEM

-30 12345 Frequency (THz)

S12 0

-2 (b)

-4

dB FDTD -6 Mie -8 FEM

-10 12345 Frequency (THz)

Fig. 5.6. (a) S11 and (b) S12 calculated analytically for propagation through a 10μm thick slab (red dotted line) and determined from FDTD (blue solid lines) and FEM (green dashed lines) simulations for propagation through a sheet of dielectric spheres infinite in the transverse plane and one unit cell thick in the propagation direction.

Once the effective permittivity and permeability of the array of spheres is known the effective index of refraction and characteristic impedance can be calculated using

88

nan = με anan (5.20)

μ an zan = . (5.21) ε an

The dielectric spheres considered here have a radius of = 4μmr with a dielectric constant of ε −= j10200 . The array of spheres is immersed in air and the unit cell size is a = 10μm . The real and imaginary parts of the effective permittivity, permeability, index of refraction and characteristic impedance of the array of dielectric spheres were calculated using equations (5.18) – (5.21) and are shown in Figs. 5.2, 5.3, 5.4 and 5.5, respectively (dotted red lines). The S-parameters for propagation through a homogeneous isotropic slab with = 10μmd were also calculated and are shown in Fig. 5.6 (dotted red lines).

5.3.2 Numerical Simulations

The S-parameters were also simulated for propagation through an array of spheres infinite in the transverse plane and one unit cell thick in the propagation direction using both Ansoft HFSS, a finite element method (FEM) full field solver, and Lumerical, a finite difference time domain (FDTD) solver, and are shown in Fig. 5.6. In both cases PEC conditions were used on the boundaries perpendicular to the electric field and PMC boundary condition were used on the surfaces perpendicular to the magnetic field. The effective parameters of the array of spheres where then calculated from these S- parameters using the retrieval technique [equations (4.3) - (4.6)]. The effective index of refraction and normalized characteristic impedance are shown in Fig. 5.2 and 5.3, respectively and the effective permittivity and permeability are shown in Fig. 5.4 and 5.5, respectively (dashed green lines for the FEM results and solid blue lines for the FDTD results). The effective permittivity was also calculated from the FEM simulations using the field averaging homogenization technique proposed by Acher [21]. In this technique the

89 effective permittivity is calculated from the averaged electric field and displacement vectors using

v D z S 0εε eff = v (5.22) E z V where signifies an average over the surface of the unit cell perpendicular to the z- S direction and signifies an average over the volume of the unit cell. The effective V permittivity obtained using this technique is shown in Fig. 5.4.

5.4 Discussion

Comparing Figs. 5.2 – 5.6 the shapes of the curves for the S-parameters, real and imaginary parts of the effective index, characteristic impedance and permeability match. On the other hand, the curves for the permittivity don’t match. In particular Fig. 5.4 shows the appearance of an anti-resonance in the real part of the permittivity accompanied by a negative imaginary value at 2.56THz in the retrieved results. Neither the analytical results nor the results obtained using the field averaging technique show this behavior. It is important to emphasize two points. First, the green dashed line and purple dash-dotted lines for the permittivity in Fig. 5.4 are obtained from the same FEM simulations implying that the problem is not with the simulations but with one of the homogenization techniques. Second, the index of refraction, impedance and permeability obtained analytically and from the retrieval technique match implying that the problem occurs only when calculating the permittivity. Indeed, if there was any behavior in the array of spheres that either the analytical technique or numerical methods failed to predict this would also manifest in the plots for the index and impedance, not just the permittivity. This type of anti-resonant and negative imaginary behavior was also predicted by the retrieval technique in the case of the SRRs in section 4.2.2. The difference between this case and that of the SRRs, however, is that now accurate analytical calculations

90 describing the behavior of the spheres are available to us and they do not corroborate the results of using the retrieval technique. The question then arises, what is the source of this discrepancy? To understand the source of this difference it is helpful to further examine equation (4.5) which is used in the retrieval technique and can be expanded as

+inn ''' −+ izzinn )''')('''( + + − znzniznzn )''''''('''''' iεε ''' =+ = 22 = 22 (5.23) +izz ''' +zz ''' +zz ''' .

It is clear from equation (5.23) that when using the retrieval technique the sign of the imaginary part of the effective permittivity is determined by the sign of the term

− znzn '''''' . (5.24)

Since passive media are being considered the passivity conditions are enforced

n > 0'' and

z > 0' (5.25) so that equation (5.24) becomes

− znzn '''''' . (5.26)

Therefore, the sign of the imaginary part of the permittivity in the retrieval technique is determined by the sign of n' (the sign of the index of refraction), the sign of z '' (whether the medium is inductive or passive) and the relative magnitudes of the terms zn ''' and zn ''' . For the array of dielectric spheres Figs. 5.2 and 5.3 show that the index of refraction is positive, n'> 0 , and that the medium behaves inductively, z > 0'' so that equation (5.26) can be written as

91

− znzn '''''' (5.27)

and the sign of the imaginary part of the permittivity in an array of spheres as given by the retrieval technique is determined solely by the relative magnitudes of the terms zn ''' and zn ''' . These terms are plotted for both the FEM [Fig. 5.7(a)] and the FDTD results

[Fig. 5.7(b)]. For the sake of comparison zn '' ' and '' zn ' obtained from the analytical calculations are also plotted in Fig. 5.7(c). Comparing the results for each method in Fig. 5.7 something becomes immediately clear. Whereas the effective index of refraction, n , and the characteristic impedance, z , for the analytical and retrieved results match well (Fig. 5.2 and 5.3) the plots for n' z'' and n '' z' do not. In the analytical case [Fig. 5.7(c)] the two curves are nearly indistinguishable but close inspection shows that '' zn ' is always slightly larger than zn ''' so that the imaginary part of the permittivity is nearly zero, but always positive. However, in both the FEM [Fig. 5.7(a)] and FDTD [Fig. 5.7(b)] cases these terms diverge in the resonance region and zn ''' becomes larger than n '' z' so that the imaginary part of the permittivity is negative. This demonstrates that even though there was only a slight mismatch between the effective index and characteristic impedance determined analytically and by retrieval (due to numerical error in the FEM and FDTD simulations) this slight mismatch results in a large divergence in the values for zn ''' and zn ''' . In addition, because we have znzn ≈− 0'''''' in the resonance region

[as seen in Fig. 5.7(c) for the analytical results] this means that any small deviation from the expected value can have a significant effect on the sign of the imaginary part of the effective permittivity. Therefore, even if the results for the effective index of refraction and characteristic impedance match well between the three methods, the nature of the calculations used by the retrieval technique introduce a significant sensitivity to even small numerical errors present in the FEM and FDTD methods.

92 3 (a) 2.5 n'z'' Num. (FEM) 2 n''z' Num. (FEM) 1.5

1

0.5

0 22.53 Frequency (THz)

2.5 (b) 2 n'z'' Num. (FDTD) n''z' Num. (FDTD) 1.5

1

0.5

0 22.53 Frequency (THz)

93 2.5 (c) 2 n''z' Analytic

n'z'' Analytic 1.5

1

0.5

0 22.53 Frequency (THz)

Fig. 5.7. Magnitude of the terms zn ''' and zn ''' calculated (a) from FEM simulations and (b) from FDTD simulations and (c) from the analytical calculations.

In this examination the question arises as to why the permittivity exhibited an anti-resonance and a negative imaginary part but the permeability still matched well between the three methods. It is reasonable to ask this question especially since in section 4.2.1 it was shown that this type of behavior was not isolated to the permittivity but could also occur in the permeability, as was the case with the array of split wires. To understand this let us look at equation (4.6) and expand it

μ μ ++=+ = − + + znzniznznizzinni )''''''('''''')''')('''(''' . (5.28)

Once again enforcing the passivity conditions and recalling that for the array of dielectric spheres n > 0' and z > 0'' the imaginary part of the permeability is given by

μ += znzn '''|'''|'' . (5.29)

Expression (5.29) shows that the imaginary part of the permeability, as determined by the retrieval technique, must be positive for the array of dielectric spheres. In fact,

94 comparing equation (5.23) and (5.28) for the effective permittivity and permeability we see something interesting. The imaginary parts of these in a passive medium are determined by the terms

− znzn )''''''( ε '' = and μ += znzn '''''''' (5.30) 22 +zz '''

Upon examining equation (5.30) it is clear that depending on the signs of n' and z '' in a particular medium only one of the two effective imaginary parts can be negative when using the retrieval technique and the other must always be positive. Let us look at some examples. Consider the array of cut wires examined in section 4.2.1, in this structure the effective index of refraction was positive and the medium behaved capacitively so that n > 0' and z < 0'' . Therefore in this case equations (5.30) become

+ znzn )''''''( ε '' = 22 and μ −= znzn '''''''' (5.31) +zz ''' so that the imaginary part of the permittivity must be positive and the imaginary part of the permeability can be negative, which matches the results seen for this structure in Section 4.2.1. For the combined array of continuous wires and SRRs examined in section 4.2.3 the index of refraction was negative and the medium behaved capacitively so that n < 0' and z < 0'' . Equation (5.30) then becomes

− znzn )''''''( ε '' = 22 and μ += znzn '''''''' (5.32) +zz ''' so that the imaginary part of the effective permittivity can be negative but the imaginary part of the effective permeability must always be positive, as was the case observed in section 4.2.3. It should be noted that the array of SRRs examined in section 4.2.2 has the same behavior as that of the array of dielectric spheres considered in this section (i.e. n > 0' and z > 0'' ). Considering the examples above the retrieval technique will be

95 problematic when obtaining one of the effective parameters, the effective permittivity or the effective permeability, and which will be problematic can be determined by the signs of the real part of the effective index of refraction and the imaginary part of the effective characteristic impedance.

5.5 Examining the Sensitivity of the Retrieval Technique

In order to further investigate the sensitivity of the retrieval technique to small errors in the S-parameters let us introduce such errors in the analytical expressions and attempt to reproduce the predicted behavior. Such an analysis will give further insight into the tolerance of the retrieval technique to errors. From Fig. 5.6 it is clear that when comparing the analytically (Mie theory) determined S-parameters to those simulated numerically (FEM and FDTD) there is both a magnitude difference and a frequency shift present between the two sets of results. In order to reproduce the behavior predicted by the retrieval technique let us introduce a small magnitude difference into the analytical S- parameters and then use this technique to obtain the effective parameters. Figures 5.8 and 5.9 show the effective permittivity and permeability, respectively, obtained by adding an imaginary frequency shift of j10GHz (which corresponds approximately to a 1% change in magnitude) to the analytical S12 . The analytically obtained effective permittivity and permeability of the spheres obtained without adding error are also shown for comparison. Figure 5.8 shows that adding just a 1% error in the magnitude of S12 results in the anti-resonant behavior and negative imaginary part of the permittivity seen in the previous section when using the retrieval technique on the simulated S-parameters for the spheres. On the other hand, the permeability in Figure 5.9 is not greatly affected by this change, as expected from equation (5.29).

96 ε' 2.6 (a)

2.2

1.8 Analytical 1.4 Analytical (Error)

1 12345 Frequency (THz)

ε'' 0.8 (b)

Analytical 0.4 Analytical (Error)

0

-0.4 12345 Frequency (THz)

Fig. 5.8. (a) Real and (b) imaginary parts of the effective permittivity obtained from the retrieval technique after adding a 1% error (dotted red lines) in the magnitude of S12 for the analytical results. The solid blue lines represent the analytical results obtained without introducing error.

97 μ' 4 (a) 3 Analytical 2 Analytical (Error) 1

0

-1

-2 12345 Frequency (THz)

μ'' 5 (b) 4

3 Analytical 2 Analytical (Error) 1

0 12345 Frequency (THz)

Fig. 5.9. (a) Real and (b) imaginary parts of the effective permeability obtained from the

retrieval technique after adding a 1% error in the magnitude of S12 for the analytical results. The solid blue lines represent the analytical results obtained without introducing error.

98 5.6 Retrieval on a Dispersionless Dielectric Slab

The sensitivity of the retrieval technique to error in the numerical simulations can also be seen when retrieving the effective parameters of a simple medium such as a dispersionless slab. The S-parameters for propagation through an 8mm thick slab with ε = 15 and μ = 2 were calculated analytically and using Ansoft HFSS (FEM). The slab was immersed in air, PEC and PMC boundaries were used to ensure plane wave propagation and the dimensions were chosen to match those used in a similar simulation presented in [22]. In the FEM simulations the transmission through the slab was simulated 4 times with increasing levels of refinement (the system was modeled using 2167, 3670, 7069 and 9486 tetrahedra). Figure 5.10 shows the magnitude of the S- parameters. The effective index, impedance, permittivity and permeability were then calculated from the S-parameters using the retrieval technique (4.3) – (4.6). The real part of the index, n' , and imaginary part of the impedance, z '' , are shown in Fig 5.11(a) and (b), respectively and the imaginary parts of the permittivity, ε '' , and permeability , μ'', are shown in Fig 5.12(a) and (b), respectively. From Fig. 5.12 both the retrieved ε '' and μ'' for the dispersionless slab exhibit negative imaginary parts (although they are never both negative at the same time). As in the cases of the SWs, split strip wires, SRRs, and dielectric spheres the parameter that will be problematic (ε'' or μ '' ) can be determined from equation (5.30) and the signs of n' and z '' (Fig. 5.11). This is emphasized by the fact that each sign change in ε '' and μ'' corresponds to a sign change in z '' . Examining Fig. 5.12 another important point can be made. As the level of refinement in the simulations increases the magnitude and bandwidth of the ε '' < 0 and μ '' < 0 regions decreases. In other words, as the error in the S-parameters decreases ε '' and μ'' approach their actual values (ε '' = 0 and μ' = 0' ) emphasizing that the negative values are the result of error in the simulations. Note that the magnitudes of the S-parameters and retrieved index presented here match those given in [22] but the retrieved permittivity and permeability do not. This is because the negative values of the imaginary parts of the permittivity and permeability result from errors in the numerical simulations which cannot be exactly reproduced.

99

S11 Magnitude 0 (a) -10 -20 -30 Analytical 2167 dB -40 3670 7069 -50 9486 -60 -70 6.5 8.2 9.9 11.6 13.3 15 Frequency (GHz)

S12 Magnitude 0 (b) -1

-2

-3 Analytical -4 2167 3670 -5 7069 9486 -6 6.5 8.2 9.9 11.6 13.3 15 Frequency (GHz)

Fig. 5.10. (a) S11 and (b) S12 calculated analytically and numerically for propagation through an 8mm slab with ε = 15 and μ = 2 .

100 n' 5.48 (a)

5.475

5.47 Analitical 5.465 2167 3670 7069 5.46 9486

5.455 6.5 8.2 9.9 11.6 13.3 15 Frequency (GHz)

z'' 0.02 (b) Analytical 2167 0.01 3670 7069 9486 0

-0.01

-0.02 12 13 14 15 Frequency (GHz)

Fig. 5.11. (a) Real part of the effective index and (b) imaginary part of the effective impedance calculated analytically and obtained using retrieval for propagation through an 8mm slab with ε = 15 and μ = 2 .

101 ε'' 0.24 (a) 0.16

0.08

0 Analytical -0.08 2167 3670 -0.16 7069 9486 -0.24 12 13 14 15 Frequency (GHz)

μ'' 0.24 (b) 0.16

0.08

0 Analytical -0.08 2167 3670 -0.16 7069 9486 -0.24 12 13 14 15 Frequency (GHz)

Fig. 5.12. (a) Imaginary part of the permittivity and (b) imaginary part of the permeability calculated analytically and obtained using retrieval for propagation through an 8mm slab with ε = 15 and μ = 2 .

102 5.7 Other Homogenization Techniques

Although the retrieval technique is arguably the most popular homogenization technique currently in use to determine the effective parameters of metamaterials it is not the only technique available. In addition to Acher’s field averaging technique presented earlier in this chapter another homogenization technique proposed by Ishimaru has also been used to determine the effective parameters of an array of SRRs. In this technique a single inclusion (in this case a SRR) is excited by a quasi-static electric or magnetic field v v and the resulting current density, J e or J m , at the site of the inclusion is calculated numerically. The effective electric and magnetic dipole moments of the inclusion are then calculated from these current densities. Using the effective dipole moments the ×66 polarizability tensor [α ] can be calculated using

v v ⎡ p⎤ ⎡El ⎤ ⎢ v ⎥ = []α ⎢ v ⎥ (5.33) ⎣m⎦ ⎣Dl ⎦ where the subscript l indicates the Lorentz local fields. The structure is excited by either a quasi-static electric or magnetic field along one axis at a time until the polarizability tensor is filled out. Once the polarizability tensor is complete the effective parameter tensors can then be determined using the Claussius-Mossoti expression. Using this technique Ishimaru determined the effective parameters of an array of circular SRRs and showed that the real part of the permittivity exhibits a normal resonance and the imaginary part is positive everywhere [23]. A variation of Ishimaru’s technique was developed by Wheeler at the University of Toronto. In this method the effective electric and magnetic dipole moments are calculated from the fields scattered by the inclusion instead of from the volume currents. The advantage of this technique is that only the fields scattered at all angles around the inclusion are needed, no information about the nature of the inclusion itself is required. Using this technique the effective parameters of an array of SRRs (Fig 5.13) with lattice spacing a = 11mm and dimension w = 9.6mm, c = 0.9mm, d = 0.6mm and g = 2.4mm were calculated. The effective permittivity of the array is shown in Fig 5.14 and the

103 effective permeability is shown in Fig. 5.15. The wave propagated in the y-direction and its electric field was polarized in the x-direction. Note that this technique accounts for the bianisotropy of the SRR and that neither the effective permittivity in the x- or y-directions exhibits an anti-resonance or negative imaginary part in Fig. 5.14. Another homogenization technique was presented by Alu in 2011 in which the effective parameters of an array of dielectric spheres was calculated and the imaginary part of the permittivity was positive [24]. Unfortunately, because the homogenization techniques mentioned in this section can be tedious or even difficult to implement the retrieval technique remains a more popular method for determining the effective parameters of metamaterials.

Fig. 5.13. SRR used by Wheeler’s point dipole homogenization technique. The dimensions are w = 9.6mm, c = 0.9mm, d = 0.6mm and g = 2.4mm.

104 εxx 3 (a) 2.5

2

1.5 ε' 1 ε''

0.5

0 036912 Frequency (GHz)

εyy 15 (b) 10 ε' ε''

5

0 036912 -5

-10 Frequency (GHz)

Fig. 5.14. Effective permittivity obtained for an array of SRRs using Wheeler’s point dipole homogenization technique in the (a) x-direction and (b) y-direction.

105 μzz 10

6 μ' 2 μ''

-2

-6

-10 036912 Frequency (GHz)

Fig. 5.15. Effective permeability obtained for an array of SRRs using Wheeler’s point dipole homogenization technique in the z-direction.

5.8 Conclusion

By considering the concepts of causality and analyticity it has been shown that the Lorentzian model, which is often used to describe and predict the dispersive characteristics of SWs and SRRs, does not allow ε < 0'' or μ < 0'' in these media. In addition, using the retrieval technique on a resonant metamaterial for which accurate analytical expressions are available (i.e. an array of dielectric spheres) it has been shown that the anti-resonant behavior in the real part and the negative imaginary part in an effective parameter is the result of small error in the numerical simulations compounded by the form of the equation used in the retrieval technique. In addition it was shown that the anti-resonant and negative imaginary behavior predicted by the retrieval technique can be reproduced in the analytical calculations by introducing a small error

(approximately 1%) in the magnitude of S12 further demonstrating the sensitivity of this technique. The sensitivity was also demonstrated in the case of propagation through a simple dispersionless slab and it was shown that the retrieval technique predicts both ε < 0'' and μ < 0'' regions (although they are never both negative at the same time).

106 Also, it was shown that other available homogenization techniques presented by Acher, Ishimaru and Wheeler do not predict anti-resonant real parts or negative imaginary parts in either the permittivity or permeability, it is only the retrieval technique that makes such predictions. In fact, when using both Acher’s homogenization technique and the retrieval technique on the same FEM simulations Acher’s technique does not predict this behavior. Based on the results of this section the retrieval technique can be used to provide good results for the effective index of refraction and characteristic impedance. Then, depending on the signs of n' and z '' , it can be used to provide results for one of the parameters, the effective permittivity or permeability. The work presented in this chapter was published in the Journal of the Optical Society of America B in 2010 [25]. The results of this chapter are summarized below:

• The Lorentzian model does not allow the metamaterials considered (SWs, split strip wires, SRRs, dielectric spheres) to have ε < 0'' or μ < 0'' . • When other homogenization techniques are used to determine the effective parameters of SWs, SRRs or spheres they do not predict ε < 0'' or μ < 0'' . Only the retrieval technique predicts this type of behavior. • Acher’s homogenization technique and the retrieval technique are used on the same FEM simulations but only the retrieval technique predicts ε < 0'' . • In the case of the spheres, n, z and μ obtained from Mie theory and the retrieval technique match indicating that the trouble occurs specifically when calculating ε. This trouble occurs because the retrieval technique is very sensitive to small variations in n or z. • The parameter that will be problematic can be discerned from the signs of n' and z '' without any information about the structure being simulated or the types of simulations used (FEM, FDTD…) emphasizing that the problem originates from how ε and μ are calculated in the retrieval technique. • The retrieval technique predicts ε < 0'' or μ < 0'' in the case of a simple dispersionless homogeneous slab.

107 • The anti-resonance in the real part and negative imaginary part of ε in the spheres can be reproduced in the analytical results by introducing a small error (1%).

References

[1] D. R. Smith, S. Schultz, P. Markos, C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients”, Phys. Rev. B 65, 195104 (2002). [2] P. Markos, C. M. Soukoulis, “Transmission properties and effective electromagnetic parameters of double negative metamaterials”, Optics Express 11, 649-661 (2003). [3] T. Koschny, P. Markos, D. R. Smith, and C. M. Soukoulis, “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials”, Phys. Rev. E 68, 065602 (2003). [4] C. Rockstuhl, T. Zentgraf, H. Guo, N. Liu, C. Etrich, I. Loa, K. Syassen, J. Kuhl, F. Lederer, H. Giessen, “Resonance of split-ring resonator metamaterials in the near infrared”, App. Phys. B 84, 219 (2006). [5] J. Zhou, L. Zhang, G. Tuttle, T. Koschny, C. M. Soukoulis, “Negative index materials using simple short wire pairs”, Phys. Rev. B 73, 041101 (2006). [6] U. K. Chettiar, A. V. Kildishev, T. A. Klar, V. M. Shalaev, “Negative index metameterials combining magnetic resonators with metal films”, Opt. Express 14, No. 17, 7872 (2006). [7] L. Zhen, J. T. Jiang, W. Z. Shao, C. Y. Xu. “Resonance-antiresonance electromagnetic behavior in a disordered dielectric composite”, App. Phys. Lett. 90, 142907 (2007). [8] L. Fu, H. Schweizer, H. Guo, N. Liu, H. Giessen, “Synthesis of transmission line models for metamaterial slabs at optical frequencies”, Phys. Rev. B 78, 115110 (2008). [9] J. Wang, S. Qu, Z. Xu, H. Ma, S. Xia, Y. Yang, X. Wu, Q. Wang, C. Chen, “Normal-incidence left-handed metamaterials based on symmetrically connected split-ring resonators”, Phys. Rev. E 81, 036601 (2010).

108 [10] T. Lepetit, E. Akmansoy, J. -P. Ganne, “Experimental evidence of resonant effective permittivity in a dielectric metamaterial”, J. of App. Phys. 109, 023115 (2011). [11] V. A. Markel, “Can the imaginary part of permeability be negative?”, Phys. Rev. E 78, 026608 (2008). [12] L. D. Landau, E. M. Lifshitz, “Electrodynamics of Continuous Media”, Butterworth Heinemann, 2002. [13] A. L. Efros, “Comment II on “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials” ”, Phys. Rev. E 70, 048602 (2004). [14] J. B. Pendry, A. J. Holden, W. J. Stewart, I. Youngs, “Extremely low frequency plasmons in metallic mesostructures”, Phys. Rev. Lett 76, 4773 (1996). [15] J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena”, IEEE Trans. on Microwave Theory and Techniques 47, 2075 (1999). [16] J. D. Jackson, Classical Electrodynamics, John Wiley and Sons, Inc., 1999. [17] G. B. Arfken, H. J. Weber, F. Harris, Mathematical Methods for Physicists: A Comprehensive Guide, Sixth Ed., Academic Press, 2005. [18] R. Y. Chiao, J. Boyce, M. W. Mitchell, “Superluminality and parelectricity: The ammonium maser revisited”, App. Phys. B 60, 259-265 (1995). [19] R. Y. Chiao, “Superluminal (but causal) propagation of wave packets in transparent media with inverted atomic populations.”, Phys. Rev. A 48, No. 1, 48 (1993). [20] P. W. Milonni, J. H. Eberly, Lasers, Wiley-Interscience, 1988. [21] O. Acher, A. L. Adenot, F. Duverger, “Fresnel coefficients at an interface with a lamellar composite material”, Phys. Rev. B 62, No. 20 (2000). [22] V. Tyagi, E. Semouchkina, “Sensitivity analysis of the effective parameters extraction procedure for metamaterial applications”, Microwave Opt. Technol. Lett. 51, No. 2, 1013-1017 (2009). [23] A. Ishimaru, S.-W. Lee, Y. Kuga, V. Jandhyala, “Generalized constitutive relations for metamaterials based on quasi-static Lorentz theory”, IEEE Trans. On Antennas and Prop. 51, No. 10 (2003).

109 [24] A. Alu, “Restoring the physical meaning of metamaterial constitutive parameters”, Phys. Rev. B 83, 08112(R) (2011). [25] J. F. Woodley, M. Mojahedi, “On the signs of the imaginary parts of the effective permittivity and permeability in metamaterials”, J. Opt. Soc. Am. B 27, No. 5, 1016 (2010).

110 Chapter 6

Conclusion

In this dissertation the proper determination and allowable signs of the effective parameters of metamaterial structures was investigated. In Chapter 2 a method that was often used to determine the sign of the index in metamaterials was considered. This method predicts the presence of a negative index based on the appearance of a transmission peak in the region where ε ' and μ' are expected to be both negative. By investigating the behavior of two SRR and SW configurations it was shown that the presence of a transmission peak is not sufficient evidence of a negative index. In order to properly determine the sign of the index two alternate methods were presented. In the first method the sign of the index is calculated from the phase difference for propagation through several lengths of the medium and in the second method the sign of the index is determined from the band structure. One advantage of the second method is that the sign of the index manifests itself in two distinct ways on the band diagrams: through the slope of the band or through the presence of contra-directional coupling when the band intersects the light line. The form of the index necessary to produce backward wave propagation was studied from a purely three dimensional wave propagation point of view in Chapter 3 and two different types of backward waves were defined. The first type, the “perfect” backward wave, corresponds to the case of an isotropic medium in which the angle v v between v p and vg is exactly 180°. In this case it was shown that the index is necessarily negative. The second type, the “imperfect” backward wave, corresponds to v v an anisotropic medium where the dot product between v p and vg is negative (excluding the case where the angle between them is 180°). In this case it was shown that the index along at least one of the principle axes must be negative. In other words, in both cases the propagation of backward waves implies the presence of a negative index of refraction,

111 and vice versa. Constraints on the functional form of the index were also obtained using the fact that the group velocity must be positive in a transmission passband of a negative index medium. In Chapter 4 the retrieval technique commonly used to obtain the effective parameters of metamaterials from simulated or experimentally measured S-parameters was derived and discussed. Using the retrieval technique the effective parameters of several common metamaterial structures (an array of split strip wires, an array of SRRs and a combined array of SWs and SRRs) were obtained. In each case the parameters obtained from the retrieval technique exhibited either ε < 0'' or μ < 0'' even though the metamaterials in question remained passive ( n > 0'' ). These results are just a few examples of the type of behavior often predicted by this technique. Unfortunately, since no analytical expressions exist to accurately determine the effective parameters of these metamaterials these results cannot be corroborated. The validity of ε < 0'' or μ < 0'' in passive media was examined in Chapter 5. First, the concepts of causality and analyticity were discussed by considering the Lorentzian model, a model commonly used to characterize and predict the behavior of metamaterials. It was shown that, for the metamaterials in question, this model does not allow ε < 0'' or μ '' < 0 . Then, in order to test the validity of the retrieval technique another metamaterial, an array of dielectric spheres, was considered. This metamaterial has a similar resonant behavior to that of the SWs and SRRs but, unlike the SWs and SRRs, its effective parameters can be calculated analytically. By comparing the analytical and retrieved effective parameters for the array of spheres it was shown that the negative imaginary part predicted by the retrieval technique is the result of errors in the simulated S-parameters. The sensitivity of the retrieval technique to errors in the S- parameters was further investigated and it was shown that even small errors

(approximately 1%) in the magnitude of S12 can cause the retrieval technique to predict a negative imaginary part of ε . This sensitivity, however, does not affect the retrieved values of n and z . In addition, depending on the signs of n' and z '' only ε or μ will exhibit this sensitivity so that the retrieval technique can still be used to obtain n, and z .

112 In the appendix a new system was introduced which characterizes the behavior of any medium based only on the signs of its effective parameters but not their magnitudes. Using this system the 16 possible sign combinations of the parameters ε ' , ε '' , μ' , and μ '' that must be considered when calculating n are reduced to 6 Families, each with a predetermined behavior. Considering both passive ( n > 0'' ) and active (n '' < 0 ) media it is shown that 4 out of the 16 cases must have n'> 0 and another 4 must have n'< 0 . Not only does this system categorize the cases where the sign of the index should be obvious (Families 1 and 5) but it also restricts the possible sign of the index in the more unusual cases (such as Families 4 and 6).

113 List of Contributions

Refereed Journal Papers

[1] J. F. Woodley, M. Mojahedi, "Left-handed and right-handed metamaterials composed of split ring resonators and strip wires", Phys. Rev. E 71, 066605 (2005). [2] J. F. Woodley, M. Mojahedi, “Backward wave propagation in left-handed media with isotropic and anisotropic permittivity tensors”, Josa B 23, num. 11, 2377 (2006). [3] J. F. Woodley, M. Mojahedi, “On the signs of the imaginary parts of the effective permittivity and permeability in metamaterials”, J. Opt. Soc. Am. B 27, No. 5, 1016 (2010).

Refereed Conference Papers

[1] J. F. Woodley, M. Mojahedi, “The signs of the imaginary parts of the effective permittivity and permeability in metamaterials”, AP-S International Symposium and USNC/URSI National Radio Science Meeting, Toronto, July 11-17, 2010. [2] J. F. Woodley, M. Mojahedi, “Determining the sign of the index in metamaterials composed of split ring resonators and strip wires using dispersion diagrams or the insertion phase,” Photonics North: The International Society for Optical Engineering (SPIE), Toronto, Canada, September 12-14, 2005. [3] J. F. Woodley, M. Mojahedi, “Backwards waves in anisotropic left-handed media,” Antennas, Radar, and Wave Propagation Conference (IASTED and IEEE AP-S), Banff, Alberta, Canada, July 19-21, 2005. [4] J. F. Woodley, M. S. Wheeler, and M. Mojahedi, “Left-handed and right-handed metamaterials composed of Split Ring Resonators and Strip Wire” AP-S International Symposium and USNC/URSI National Radio Science Meeting, Washington DC, July 3-8, 2005.

114 Appendix A

The 6 Family Model

A.1 Introduction

As mentioned in previous chapters soon after the first metamaterials were constructed and experimentally verified [1, 2] these media became the topic of much research. In the following years many new metamaterials were conceived of [3-7] and, based on simulated or experimentally data, many claims were made regarding the sign of the real part of the index in these media. Unfortunately, as pointed out in Chapter 2, some of the criteria used to determine the sign of the index in a medium (such as the emergence of a passband) were not reliable. This demonstrated the need for a better understanding of the conditions that produce a negative index and the need for better tools to properly determine the sign of the index. In chapter 2 methods to properly determine the sign of the real part of the index from simulated and experimental data (using dispersion diagrams and transmission phases) were discussed. Here the problem will be examined from a purely mathematical standpoint. In this chapter a new system will be developed to characterize media entirely from the signs of the electric permittivity, ε , and magnetic permeability, μ , without any need for information on their magnitudes. It will be shown that the 16 possible sign combinations of ε ' , ε '' , μ' , and μ '' that must be considered when calculating the index of refraction can be reduced to 6 Families, each of which exhibits a specific behavior. In fact some of these Families will be exclusive to a specific sign for the real part of the index of refraction. With this system, once the signs of ε ' , ε '', μ' , and μ'' are known the particular combination can immediately be assigned to one of the Families so that useful insight on its behavior and possible restrictions on the sign of its index can be obtained without the need for calculation. This system also shows some of the more

115 peculiar cases that can lead to a negative index of refraction such as a medium where only one of the parameters ε ' or μ' is negative.

A.2 Symmetry Operations

Starting with the expression for the index of refraction in terms of ε and μ we write

n 2 = εμ . (A.1)

Expanding this expression in terms of the real and imaginary parts of ε ε −= jε ''' and μ μ −= jμ ''' gives

n 2 −−= j ε '( +ε '' μμμεμε )''''''''' . (A.2)

For reasons that will become clear let us use the variables ξ and η instead of ε and μ where ξ = ε (or μ ) and η = μ (or ε ). In this case (A.2) becomes

n2 ηξηξ j '( ηξ '' ηξ )''''''''' +=+−−= jvu (A.3) where

u ξ η −= ξ η '''''' (A.4) v ξ η −−= ξ η'''''' . (A.5)

Here u and v represent the real and imaginary parts of n 2 . Looking at (A.3) two important points can be noted. These are:

116 (i) if the signs of all the parameters ξ ', ξ '' , η', and η '' are changed (A.3) is unaffected and (ii) if ξ and η are interchanged (A.3) is again unaffected.

Equation (A.3) is also unchanged if both (i) and (ii) are applied consecutively. Since each parameter ξ ', ξ '' , η', and η '' can be either positive or negative there are 16 possible combinations to consider in (A.3). However, because (A.3) is unaffected by operations (i) [changing the signs] and (ii) [interchanging ξ and η ], which will be dubbed symmetry operations, the number of cases that must be considered can be reduced from 16 individual cases to 6 Families of cases. Each member within a Family will be subject to the same set of restrictions regarding the allowable signs of the real part of the index. Because of this the Families are also dubbed behavioral Families.

A.3 The 6 Families

To see how the 6 behavioral Families are determined consider the 16 sign combinations in (A.3):

Case 1: ξ > 0' , ξ > 0'' , η > 0' , η > 0'' Case 2: ξ > 0' , ξ > 0'' , η > 0' , η < 0'' Case 3: ξ > 0' , ξ > 0'' , η < 0' , η > 0'' Case 4: ξ > 0' , ξ > 0'' , η < 0' , η < 0'' Case 5: ξ > 0' , ξ < 0'' , η > 0' , η > 0'' Case 6: ξ > 0' , ξ < 0'' , η > 0' , η < 0'' Case 7: ξ > 0' , ξ < 0'' , η < 0' , η > 0'' Case 8: ξ > 0' , ξ < 0'' , η < 0' , η < 0'' Case 9: ξ < 0' , ξ > 0'' , η > 0' , η > 0'' Case 10: ξ < 0' , ξ > 0'' , η > 0' , η < 0'' Case 11: ξ < 0' , ξ > 0'' , η < 0' , η > 0''

117 Case 12: ξ < 0' , ξ > 0'' , η < 0' , η < 0'' Case 13: ξ < 0' , ξ < 0'' , η > 0' , η > 0'' Case 14: ξ < 0' , ξ < 0'' , η > 0' , η < 0'' Case 15: ξ < 0' , ξ < 0'' , η < 0' , η > 0'' Case 16: ξ < 0' , ξ < 0'' , η < 0' , η < 0''

In Case 1 performing the symmetry operation (i) (changing the signs of all the parameters) on this case gives

(i)∗ (ξ > 0' , ξ > 0'' , η > 0' , η > 0'' ) ⇒ ξ < 0' , ξ < 0'' , η < 0' , η < 0'' which is Case 16. Note that the symbol ∗ is used to indicate that the symmetry operations (i) or (ii) is acting on one of the cases. Performing symmetry operation (ii) (interchanging ξ and η ) on Case 1 gives

(ii)∗ (ξ > 0' , ξ > 0'' , η > 0' , η > 0'' ) ⇒ ξ > 0' , ξ > 0'' , η > 0' , η > 0'' which is just Case 1 again. Finally, performing (i) and (ii) on Case 1 consecutively [the order in which (i) and (ii) are applied is not important] yields

(i)∗ (ii)∗ (ξ > 0' , ξ > 0'' , η > 0' , η > 0'' ) ⇒ ξ < 0' , ξ < 0'' , η < 0' , η < 0'' which is Case 16 again. Using these results Cases 1 and 16 can be grouped together in a Family:

Family 1

Case 1: ξ > 0' , ξ > 0'' , η > 0' , η > 0'' Case 16: ξ < 0' , ξ < 0'' , η < 0' , η < 0'' .

118 Considering Case 2, performing symmetry operation (i) gives

(i)∗ (ξ > 0' , ξ > 0'' , η > 0' , η < 0'' )⇒ ξ < 0' , ξ < 0'' , η < 0' , η > 0'' which is Case 15. Performing (ii) on Case 2 gives

(i)∗ (ξ > 0' , ξ > 0'' , η > 0' , η < 0'' )⇒ ξ > 0' , ξ < 0'' , η > 0' , η > 0'' which is Case 5 and performing (i) and (ii) on Case 2 consecutively gives

(i)∗ (ii)∗ (ξ > 0' , ξ > 0'' , η > 0' , η < 0'' )⇒ ξ < 0' , ξ > 0'' , η < 0' , η < 0'' which is Case 12. Using these results the next Family becomes

Family 2

Case 2: ξ > 0' , ξ > 0'' , η > 0' , η < 0'' Case 5: ξ > 0' , ξ < 0'' , η > 0' , η > 0'' Case 12: ξ < 0' , ξ > 0'' , η < 0' , η < 0'' Case 15: ξ < 0' , ξ < 0'' , η < 0' , η > 0'' .

The other Families can be determined similarly and are given below

Family 3

Case 3: ξ > 0' , ξ > 0'' , η < 0' , η > 0'' Case 8: ξ > 0' , ξ < 0'' , η < 0' , η < 0'' Case 9: ξ < 0' , ξ > 0'' , η > 0' , η > 0'' Case 14: ξ < 0' , ξ < 0'' , η > 0' , η < 0''

119 Family 4

Case 4: ξ > 0' , ξ > 0'' , η < 0' , η < 0'' Case 13: ξ < 0' , ξ < 0'' , η > 0' , η > 0''

Family 5

Case 6: ξ > 0' , ξ < 0'' , η > 0' , η < 0'' Case 11: ξ < 0' , ξ > 0'' , η < 0' , η > 0''

Family 6

Case 7: ξ > 0' , ξ < 0'' , η < 0' , η > 0'' Case 10: ξ < 0' , ξ > 0'' , η > 0' , η < 0'' .

Within each of these Families (A.3) has the same form no matter which Family member is used. For example consider Family 2. Plugging Case 2 (the first member of this Family) into (A.3) gives

n2 j +−−+= ηξηξηξηξ |)'''||'''|(|''''||''| . (A.6)

If any of the other members from Family 2 are plugged into (A.3) the result will always be (A.6). But what does this means in terms of the parameters u and v defined in (A.4) and (A.5)? Recall that u and v represent the real and imaginary parts of n 2 . Looking at equation (A.6) it is clear that the real part of n 2 will always be positive, regardless of the magnitudes of ξ ', ξ '' , η', and η '' . On the other hand the imaginary part can be positive or negative depending on the magnitudes of ξ ', ξ '' , η', and η '' . This means that for Family 2 u > 0 and v can be positive or negative. Similar results can be determined for each Family and are summarized in table A.1.

120 Table A.1. Signs of u and v in each of the 6 Families. Family u v 1 (2 members) +/- - 2 (4 members) + +/- 3 (4 members) - +/- 4 (2 members) +/- + 5 (2 members) +/- + 6 (2 members) +/- -

From Table A.1 it is clear that in the Families with 4 members (Families 2 and 3) the sign of u is fixed and v can have any sign while in the Families with 2 members (Families 1, 4, 5, and 6) the sign of v is fixed and u can have any sign. In terms of the cases themselves this means that in 8 cases the sign of u is fixed and in the other 8 cases the sign of v is fixed.

A.4 The 6 Families in the 4-Quadrant Model

In order to see significance of fixing the signs of u and v consider equation (A.3). This equation can be rewritten in polar form to give

2 =+= aejvun jθ (A.7) where

+= vua 22 v θ = ArcTan )( + mπ . (A.8) u

Here m arises due to the degeneracy in the Tangent function for values of the argument in the 1st and 3rd quadrants and in the 2nd and 4th quadrants. The values of m are

121 determined by the signs of u and v and are given in Table A.2. In order to take the root of (A.7) to obtain the index we note that the roots of a complex number are given by

1 2 l 2πθπθ l = i Cosaz (( jSin() +++ )) (A.9) i i i i where l = 0, 1, 2, ... i-1. Taking the square root of (A.7) with i = 2 gives

1 θ θ = 2 (( π () +++ πljSinlCosan )). (A.10) 2 2

θ θ θ θ Using the fact that ( πlCos −=+ l Cos()1() ) and ( πlSin −=+ l Sin()1() ) (A.10) 2 2 2 2 becomes

1 θθ −= l 2 Cosan (()1( + jSin() )) . (A.11) 2 2

The term − )1( l represents the ± that results when taking the square root. In the case of the index, the choice of l is determined by whether the medium is passive or active. That is, in that case of a passive medium l is selected so that the imaginary part of the index is preceded by a negative sign ( = ± − jnnn ''' ) and in the case of an active medium l is selected so that the sign is positive ( = ±nn '+ jn '' ). Since this analysis will be concerned only with the signs of the real and imaginary parts of the index, the positive constant

1 1 2 = (ua 2 + v 2 ) 4 can be ignored. Plugging (A.8) into (A.11) gives

1 v mπ 1 v mπ n −∝ l Cos(()1( ArcTan )( ++ jSin() ArcTan )( + )) (A.12) 2 u 2 2 u 2 which can also be written as

122

mπ 1 v 1 v j n −∝ l Cos(()1( ArcTan + jSin())( ArcTan )))( e 2 . (A.13) 2 u 2 u

The argument inside the Cosine and Sine will always have values in the range π 1 v π ≤− ArcTan )( ≤ . In this range the Cosine will always be positive and the Sine 4 2 u 4 will always have the sign of its argument. In addition the Arctangent will also always have the same sign as its argument. Using these points equation (A.13) becomes

mπ v j l +−∝ jSgnCn ))(()1( eS 2 (A.14) u where

1 v = CosC (| ArcTan |)( 2 u 1 v = SinS (| ArcTan |)( . (A.15) 2 u

Here Sgn is the sign function. From equation (A.14) the signs of the real and imaginary parts of the index are determined by the signs of u and v, the sign of m (which is also determined by the signs u and v ) and by l (i.e. the passivity of the medium). Assuming a passive medium the signs of the real and imaginary parts of the index are determined completely by the signs of u and v. The usefulness of writing the index in the form given by (A.14) can be demonstrated by making a plot with u on the x-axis and v on the y-axis (Fig. A.1). In the u - v plane the sign of the real part of the index of refraction in each quadrant is fixed. This can be shown by evaluating (A.14) in each quadrant. For example, in quadrant 1 u > 0, v > 0 and m = 0 . Plugging these in (A.14) gives

123 l +−∝ jSCn )()1( . (A.16)

Since we are assuming a passive medium l must be chosen so that the index has the form −±= jnnn ''' . In this case l = 1 must be chosen and (A.16) becomes

−−∝ jSCn . (A.17)

Recall that C and S are always positive. Table A.2 summarizes the values of m and l, the signs of u and v, and the form of the index determined from (A.14) in each quadrant, for the passive case. Note that in quadrants 1 and 2 the n < 0' while in quadrants 3 and 4 n > 0' .

Table A.2. The values of m and l, the signs of u and v and the form of the index in each quadrant of the u-v plane for the passive case. Quadrant u v m l (passive) n n' 1 + + 0 1 -C-jS - 2 - + -1 0 -S-jC - 3 - - 1 1 S-jC + 4 + - 0 0 C-jS +

Using Table A.1 each Family has been placed onto the u-v plane (Fig. A.1). From Fig. A.1 it is immediately clear that each Family spans 2 quadrants and that Families 1 and 6 (which are restricted to quadrants 3 and 4) will always have n > 0' and Families 4 and 5 (which are restricted to quadrants 1 and 2) will always have n < 0' . Families 2 and 3, on the other hand, can have either n > 0' or n < 0' and in these cases some knowledge of the magnitudes of ξ ', ξ '' , η', and η '' is necessary to determine the sign of n' . The common element in Families 2 and 3 is that each of their members consists of a sign combination where 3 of the parameters (ξ ', ξ '' , η', or η '' ) have the same sign so that only one of the parameters stands out. Every other combination is restricted to a particular sign for n' . Looking at Families 2 - 4 there are some peculiar sign

124 combination that can lead to n < 0' . In particular Family 3 (excluding cases 8 and 14 for reason that will be discussed below) demonstrates that n < 0' is possible with only one of the parameters ξ ' or η' negative.

Fig. A.1. Placement of the 6 Families on the u-v plane. Blue text indicates Families with a negative index of refraction, red text indicates Families with a positive index of refraction and purple text indicates Families that can have either sign for the index.

Special attention should be given to cases 6, 8, 14, and 16. In these cases ξ < 0'' and η < 0'' so that it is unrealistic to consider these to be passive [as shown in equation (5.1)]. However these cases are included here for the sake of completion since this analysis can easily be adapted to consider active media. To do so would only require l to be changed in each quadrant from 0 to 1 or vice versa (i.e. choosing the opposite sign in

125 front of the root). In this situation Quadrants 1 and 2 have n > 0' and quadrants 3 and 4 have n < 0' but the Families do not move on the u-v plane. Table A.3 summarizes the active case.

Table A.3. The values of m and l, the signs of u and v and the form of the index in each quadrant of the u-v plane for the active case. Quadrant u v m l (passive) n n' 1 + + 0 Even C+jS + 2 - + -1 Odd S+jC + 3 - - 1 Even -S+jC - 4 + - 0 Odd -C+jS -

Finally it should be noted that it is also possible to perform a similar analysis for the characteristic impedance of a medium given by

ξ η ξ η ++ j ξ η −ξ η )''''''('''''' z 2 = = ea jθz . (A.18) +ξξ ''' 22 z with

+ vu 22 a = zz z +ξξ ''' 22

vz θ z = ArcTan )( + mzπ u z

u z ξ η += ξ η ''''''

vz ξ η −= ξ η '''''' . (A.19)

In this case the passivity is determined by the choice of z > 0' (for a passive medium) or z < 0' (for an active medium) and the quadrants on the u z - vz plane are divided into regions with z > 0'' or z < 0'' .

126

A.5 Conclusions

In this chapter a new system was introduced which characterizes the behavior of any medium based only on the signs of its effective parameters but not their magnitudes. This system, developed entirely from mathematical considerations reduces the 16 sign combinations that must be considered when calculating n from ε ' , ε '' , μ' , and μ '' to 6 Families each with a specified behavior. It was shown that out of the 16 cases 4 are restricted to n > 0' and another 4 to n < 0' for both passive and active media. Not only does this system categorize the cases where the sign of the index should be obvious (Families 1 and 5) but it also restricts the possible sign of the index in the more unusual cases (such as Families 4 and 6).

References

[1] D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permittivity and permeability", Phys. Rev. Lett. 84, 4184 (2000). [2] R. A. Shelby, D. R. Smith, S. Schultz, "Experimental verification of negative index of refraction", Science 292, 77 (2001). [3] G. V. Eleftheriades, A. K. Iyer, P. C. Kremer, "Planar negative refractive index media using periodically LC loaded transmission lines", IEEE Trans. On Microwave Theory and Techniques 50, 2702 (2002). [4] M. Bayindir, K. Aydin, E. Ozbay, P. Markos, C. M. Soukoulis, "Transmission properties of composite metamaterials in free space", Appl. Phys. Lett., 81 (2002). [5] O. F. Siddiqui, M. Mojahedi, G. V. Eleftheriades, "Periodically loaded transmission line with effective negative refractive index and negative group velocity", IEEE Trans. On Antennas and Propagation 51, 2619 (2003). [6] O. F. Siddiqui, S. J. Erickson, G. V. Eleftheriades, "Time-domain measurement of group delay in negative refractive index transmission line metamaterials", IEEE Trans. on Microwave Theory and Techniques 52, 1449 (2004).

127 [7] C. Caloz, H. Okabe, H. Iwai, and T. Itoh, "Application of the transmission line theory of left-handed (LH) materials to the realization of a microstrip LH line", IEEE AP-S/URSI Internatianal Symp., San Antonio, TX, (2002).

128 Appendix B

The Derivation of the Heat Evolution

B.1 Heat Evolution for Non-Monochromatic Waves

In this appendix the heat evolution, Q, in an isotropic, homogeneous medium with permittivity, ε ω)( , and permeability, μ ω)( , will be derived. To do this we first note that the electric field and displacement vectors can be written in the time domain as

∞ v 1 v ωtj )( = ∫ )( deEtE ωω (B.1) 2π ∞− and

∞ v 1 v ωtj tD )( = ∫ )()( deE ωωωε (B.2) 2π ∞− and that the time derivative of (B.2) is given by

v ∞ ∂ tD )( jω v ωtj = ∫ )()( deE ωωωε . (B.3) ∂t 2π ∞−

The rate of change of energy in a unit volume of the medium is given by the divergence of the Poynting vector

v v v vvv ∂ tD )( v ∂ tB )( tEtHtEtS )())()(()( •=ו−∇=•∇− tH )( •+ (B.4) ∂t ∂t

129

v v v v v v where the vector identity ×∇•−×∇•=ו∇ BAABBA )()()( and Maxwell’s equations were used to obtain the right side of the (B.4). The total real and imaginary power entering a point in the medium is given by integrating (B.4) over time

∞ ∞ v v v v ∂ tD )( v ∂ tB )( ∫∫ tEtS )(( •=∂•∇− tH )( •+ )dt . (B.5) ∞− ∞− ∂t ∂t

v v Substituting (B.1) and (B.3) [and similar equations for tH )( and tB )( ] into (B.5) gives

v ∫ tS =∂•∇−

∞ ∞ ∞ j v vv v +ωω )'( tj E • HE •+ ωωωμωωωεω ')]'()'()()'()'()([ ωω 'dtddeH . 2 ∫∫∫ π )2( ∞− ∞− ∞− (B.6) Performing the integration with respect to time on the right side of (B.6) gives the delta

∞ function ∫ +ωω )'( tj dte += ωωπδ '(2 ) so that (B.6) becomes ∞−

∞ v j ∞ ∞ v vv v ∫ dtS =•∇− ∫∫ E • HE •+ H + dd ωωωωδωωμωωωεωω ')'()]'()'()()'()'()([' ∞− 2π ∞− ∞−

(B.7) and integrating over ω' gives

∞ v j ∞ v vv v dtS ∫∫ E HE −−•+−−•−=•∇− )]()()()()()()[( dH ωωωμωωωεωω . ∞− 2π ∞− (B.8)

130 v From (B.2) we know that tD )( is real. Hence, it follows that

v v ** E =−− E ωωεωωε )()()()( . Substituting this into (B.8) gives

∞ ∞ v j * vv * * vv * ∫ dtS −=•∇− ∫ EE +• • )]()()()()()([ dHH ωωωωμωωωεω . (B.9) ∞− 2π ∞−

Since the terms ωε ω)(' and ωμ ω)(' are odd they disappear in the integration and only the terms with ωε ω)('' and ωμ ω)('' survive. The total real power dissipated, or heat evolution Q, is then

∞ ∞ ∞ v j v 2 v 2 Qdt ∫∫ dtS −=•∇−= ∫ + ]|)(|)(''|)(|)(''[ dHjEj ωωωμωωεω ∞− ∞− 2π ∞−

∞ 1 v 2 v 2 = ∫ E + ]|)(|)(''|)(|)(''[ dH ωωωμωωεω (B.10) 2π ∞−

B.2 Heat Evolution for Monochromatic Waves

In the case of a monochromatic wave the derivation is simpler. In this case the electric field and displacement vectors can be written as

v v ωtj 1 v ωtj v * − ωtj = ω ])(Re[)( = ω + ω eEeEeEtE ))()(( (B.11) 2 and

v v ωtj 1 v ωtj v ** − ωtj tD = ωωε eE ])()(Re[)( = eE + ωωεωωε eE ))()()()(( (B.12) 2 with

131 v ∂ tD 1)( v ωtj v ** − ωtj = ωωωε )()(( − ωωωε eEjeEj ))()( . (B.13) ∂t 2

v Substituting (B.11) and (B.13) into (B.4) gives (using similar expressions for tH )( and v tB )( )

v 1 v ωtj v * − ωtj 1 v ωtj v ** − ωtj )( =•∇− ω + ω ))()(( • ωωωε )()(( − ωωωε eEjeEjeEeEtS ))()( 2 2

1 v ωtj v * − ωtj 1 v ωtj * v * − ωtj + ω + ω ))()(( • ωωωε )()(( − ωωωε eHjeHjeHeH ))()( 2 2

(B.14) Averaging (B.14) with respect to time gives

jω * v 2 * v 2 Q = − E −+ H ωωμωμωωεωε ])())()(()())()([( . 4

ω v 2 v 2 = E + H ωωμωωε )|)(|)(''|)(|)(''( . (B.15) 2

Since every term in (B.15) is real it represents the time averaged real energy dissipation, or time averaged heat evolution, Q, in the medium. More information on the derivations presented in this Appendix can be found in [1].

References

[1] L. D. Landau, E. M. Lifshitz, “Electrodynamics of Continuous Media”, Butterworth Heinemann, 2002.

132 Appendix C

Mie Theory and the Array of Dielectric Spheres

C.1 Introduction

In this appendix the behavior of an array of dielectric spheres will be discussed. First, the scattering of an incoming plane wave by a single sphere will be considered. In this case the expressions for the scattered electric and magnetic fields obtained using Mie theory will be given and the transverse electric fields of the dipole and quadrupole modes will be plotted. Then, using the scattering information of a single sphere, the effective parameters of an array of spheres will be determined using the Claussius-Mossoti equation.

C.2 Field patterns resulting from the sphere resonances

Consider a single dielectric sphere immersed in air with radius r and index n.

v − jkz Assuming an incident plane wave of the form = 0 yeHH ˆ , the electric and magnetic fields scattered by the sphere are given by 6

' ' m m − ψψψψ mm nxxxnxn )()()()( am = ' ' (C.1) m m − ψξξψ mm nxxxnxn )()()()(

' ' m m − ψψψψ mm nxxnxnx )()()()( bm = ' ' . (C.2) m m − ψξξψ mm nxxnxnx )()()()(

6 Equations (C.1) and (C.2) are identical to (5.16) and (5.17) from Chapter 5. They are repeated here for convenience.

133 where = rkx o , ψ m x)( and ξ m x)( are the spherical Riccati – Bessel functions, the primes indicate differentiation with respect to the argument, m indicates the term in the multipole expansion and n is the index of the spheres. The dielectric spheres are immersed in air and ko is the free space wave vector.

Fig. C.1. Electric field patterns on a surface concentric with the spheres but outside the spheres [1, 2(p.98)].

Figure C.1 shows the transverse (non-radial) components of the electric field on an imaginary surface concentric to the sphere but with a larger radius for the a1 , a2 , b1 and b2 modes. Because the a1 and a2 have no radial magnetic field components they are referred to as transverse magnetic (TM) modes. Similarly, the b1 and b2 are referred to as

134 transverse electric (TE) modes because they have no radial electric field components.

From Fig. C.1, the a1 and b1 modes behave like electric and magnetic dipoles, respectively. On the other hand the a2 and b2 modes behave like electric and

a 1 0.008 0 (a) -0.01 0.006 Real -0.02 Imaginary 0.004 -0.03

-0.04 0.002 -0.05

0 -0.06 12345 (THz)

a 2 8.00E-05 0.00E+00 (b) -1.00E-04 6.00E-05 Real Imaginary -2.00E-04 4.00E-05 -3.00E-04

2.00E-05 -4.00E-04

0.00E+00 -5.00E-04 12345 (THz)

Fig. C.2. Real and imaginary parts of (a) a1 and (b) a2 for a single dielectric sphere.

135

b1 0.05 (a) 0.04

0.03 Real 0.02 Imaginary

0.01

0 12345 -0.01

-0.02

-0.03 (THz)

b2 0.0008 (b) 0.0006 Real 0.0004 Imaginary

0.0002

0 12345 -0.0002

-0.0004 (THz)

Fig. C.3. Real and imaginary parts of (a) b1 and (b) b2 for a single dielectric sphere. magnetic quadrupoles, respectively. In each case the “poles” are regions where the transverse fields are zero and the fields are purely radial (the radial fields do not show up on the imaginary surface since they are perpendicular to this surface).

136 In general, the field patterns excited on the surfaces of the spheres are a superposition of the modes shown in Fig. C.1 and of higher order modes. However, in the frequency range of interest in Chapter 5 (1-5 GHz) only the a1 and b1 modes make a significant contribution. To see this, the real and imaginary parts of a1 and a2 were calculated from (C.1) and are plotted in Fig. C.2. Similarly, the real and imaginary parts of b1 and b2 were calculated from (C.2) and are plotted in Fig. C.3. For both the electric and magnetic cases the magnitude of the dipole terms ( a1 and b1 ) are several orders of magnitude larger than those of the quadrupole terms ( a2 and b2 ) so that the quadropole terms do not make a significant contribution and can be neglected.

C.3. Obtaining the expressions for the effective permittivity and permeability

Since, in the frequency region of interest, only the dipole terms need to be considered, it is convenient to compare the expressions for the fields excited on the spheres to the standard expression for dipole radiation. Consider the b1 mode which is the first mode excited in the spheres (the lowest frequency mode). For this mode the scattered magnetic field is given by [3]

v 3 j e− orjk H sca = obH 1 ˆ ( ×× yrr ˆˆ ) . (C.3) 2 o rk

For a magnetic dipole the radiated fields are given by

3 − orjk v ko e ˆ H dipole −= ˆ (ˆ×× lrr ) . (C.4) 4π o rk

v v v where l is the magnetic moment and is given by incm == αα om yHHl ˆ . Note that in both (C.3) and (C.4) only far field contributions are being considered. Comparing (C.3) and (C.4) the effective polarizability of a single sphere can be written as

137 3 m −= 6πα 1 kbj o . (C.5)

In the long wavelength limit, the effective permeability of an array of spheres can be related to the polarizability using the Claussius-Mossoti expression

3 μan −1 α m = ( ) (C.6) NV μan + 2 where N is the number of spheres per unit volume. Solving (C.6) for the effective permeability gives

3 o − 4π V bjNk 1 μ an = 3 (C.7) o + 2π V bjNk 1 which is equation (5.19) from Chapter 5. Equation (5.18) can be derived similarly.

References

[1] G. Mie, “Beitrage zur optic truber medien speziell kolloidaler metallosungen”, Ann. Phys. 25, 377 (1908). [2] C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small particles, John Wiley & Sons Inc., 1998. [3] M. S. Wheeler, M. Mojahedi, “Three dimensional array of dielectric spheres with an isotropic negative permeability”, Phys. Rev. B 72, 193103 (2005).

138