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NADER ENGHETA, PhD, is the H. Nedwill Ramsey Professor in the Department of Electricaland Systems Engineering, University of Pennsylvania. A Guggenheim Fellow, a recipient of the IEEE Third Millennium Medal and the NSF Presidential Young Investigator Award, a Fellow of the IEEE, and a Fellow of the Optical Society of America, he is an Associate Editorof IEEE Antennas and Wireless Propagation Letters and was an IEEE Antennas and Propagation Society Distinguished Lecturer from 1997–1999.

RICHARD W. ZIOLKOWSKI, PhD, is a professor in the Department of Electrical and Computer Engineering with a joint appointment in the College of Optical Sciences, University of Arizona. He was elected by the faculty to be the first Kenneth von Behren Chaired Professor and has been a recipient of the Tau Beta Pi Professor of the Year Award and the IEEE and Eta Kappa Nu Outstanding Teaching Award. Professor Ziolkowski is a Fellow of the IEEE and a Fellow of the Optical Society of America. He was the 2005 president of the IEEE Antennas and Propagation Society.

De la contraportada

Leading experts explore the exotic properties and exciting applications of electromagnetic metamaterials

Metamaterials: Physics and Engineering Explorations gives readers a clearly written, richly illustrated introduction to the most recent research developments in the area of electromagnetic metamaterials. It explores the fundamental physics, the designs, and the engineering aspects, and points to a myriad of exciting potential applications. The editors, acknowledged leaders in the field of metamaterials, have invited a group of leading researchers to present both their own findings and the full array of state-of-the-art applications for antennas, waveguides, devices, and components.

Following a brief overview of the history of artificial materials, the publication divides its coverage into two major classes of metamaterials. The first half of the publication examines effective media with single (SNG) and double negative (DNG) properties; the second half examines electromagnetic band gap (EBG) structures. The book further divides each of these classes into their three-dimensional (3D volumetric) and two-dimensional (2D planar or surface) realizations. Examples of each type of metamaterial are presented, and their known and anticipated properties are reviewed.

Collectively, Metamaterials: Physics and Engineering Explorations presents a review of recent research advances associated with a highly diverse set of electromagnetic metamaterials. Its multifaceted approach offers readers a combination of theoretical, numerical, and experimental perspectives for a better understanding of their behaviors and their potentialapplications in components, devices, and systems. Extensive reference lists provide opportunities to explore individual topics and classes of metamaterials in greater depth.

With full-color illustrations throughout to clarify concepts and help visualize actual results, this book provides a dynamic, user-friendly resource for students, engineers, physicists, and other researchers in the areas of electromagnetic materials, microwaves, millimeter waves, and optics. It equips newcomers with a basic understanding of metamaterials and their potential applications. Advanced researchers will benefit from thought-provoking perspectives that will deepen their knowledge and lead them to new areas of investigation.

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Metamaterials

John Wiley & Sons

Chapter One

1.1 INTRODUCTION

To the best of our knowledge, the first attempt to explore the concept of "artificial" materials appears to trace back to the late part of the nineteenth century when in 1898 Jagadis Chunder Bose conducted the first microwave experiment on twisted structures—geometries that were essentially artificial chiral elements by today's terminology. In 1914, Lindman worked on "artificial" chiral media by embedding many randomly oriented small wire helices in a host medium. In 1948, Kock made lightweight microwave lenses by arranging conducting spheres, disks, and strips periodically and effectively tailoring the effective refractive index of the artificial media. Since then, artificial complex materials have been the subject of research for many investigators worldwide. In recent years new concepts in synthesis and novel fabrication techniques have allowed the construction of structures and composite materials that mimic known material responses or that qualitatively have new, physically realizable response functions that do not occur or may not be readily available in nature. These metamaterials can in principle be synthesized by embedding various constituents/inclusions with novel geometric shapes and forms in some host media (Fig. 1.1). Various types of electromagnetic composite media, such as double-negative (DNG) materials, chiral materials, omega media, wire media, bianisotropic media, linear and nonlinear media, and local and nonlocal media, to name a few, have been studied by various research groups worldwide.

As is well known, in particulate composite media, electromagnetic waves interact with the inclusions, inducing electric and magnetic moments, which in turn affect the macroscopic effective permittivity and permeability of the bulk composite "medium." Since metamaterials can be synthesized by embedding artificially fabricated inclusions in a specified host medium or on a host surface, this provides the designer with a large collection of independent parameters (or degrees of freedom)—such as the properties of the host materials; the size, shape, and composition of the inclusions; and the density, arrangement, and alignment of these inclusions—to work with in order to engineer a metamaterial with specific electromagnetic response functions not found in each of the individual constituents. All of these design parameters can play a key role in the final outcome of the synthesis process. Among these, the geometry (or shape) of the inclusions is one that can provide a variety of new possibilities for metamaterials processing.

Recently, the idea of complex materials in which both the permittivity and the permeability possess negative real values at certain frequencies has received considerable attention. In 1967, Veselago theoretically investigated plane-wave propagation in a material whose permittivity and permeability were assumed to be simultaneously negative. His theoretical study showed that for a monochromatic uniform plane wave in such a medium the direction of the Poynting vector is antiparallel to the direction of the phase velocity, contrary to the case of plane-wave propagation in conventional simple media. In recent years, Smith, Schultz, and their group constructed such a composite medium for the microwave regime and demonstrated experimentally the presence of anomalous refraction in this medium.

For metamaterials with negative permittivity and permeability, several names and terminologies have been suggested, such as "left-handed" media; media with negative refractive index; "backward-wave media" (BW media); and "double-negative (DNG)" metamaterials, to name a few. Many research groups all over the world are now studying various aspects of this class of metamaterials, and several ideas and suggestions for future applications of these materials have been proposed.

It is well known that the response of a system to the presence of an electromagnetic field is determined to a large extent by the properties of the materials involved. We describe these properties by defining the macroscopic parameters permittivity ε and permeability µ of these materials. This allows for the classification of a medium as follows. A medium with both permittivity and permeability greater than zero (ε > 0, µ > 0) will be designated a double-positive (DPS) medium. Most naturally occurring media (e.g., dielectrics) fall under this designation. A medium with permittivity less than zero and permeability greater than zero (ε < 0, µ > 0) will be designated an epsilon-negative (ENG) medium. In certain frequency regimes many plasmas exhibit this characteristic. For example, noble metals (e.g., silver, gold) behave in this manner in the infrared (IR) and visible frequency domains. A medium with the permittivity greater than zero and permeability less than zero ( ε > 0, µ < 0) will be designated a munegative (MNG) medium. In certain frequency regimes some gyrotropic materials exhibit this characteristic. Artificial materials have been constructed that also have DPS, ENG, and MNG properties. A medium with both the permittivity and permeability less than zero (ε < 0, µ < 0) will be designated a DNG medium. To date, this class of materials has only been demonstrated with artificial constructs. This medium classification can be graphically illustrated as shown in Figure 1.2.

While one often describes a material by some constant (frequency-independent) value of the permittivity and permeability, in reality all material properties are frequency dependent. There are several material models that have been constructed to describe the frequency response of materials. Because the magnetic field of an electromagnetic wave is smaller than its electric field by the wave impedance of the medium in which it is propagating, one generally focuses attention on how the electron motion in the presence of the nucleus and, hence, the basic dipole moment of this system are changed by the electric field. Understanding this behavior leads to a model of the electric susceptibility of the medium and, hence, its permittivity. On the other hand, there are many media for which the magnetic field response is dominant. One can generally describe the magnetic response of a material in a fashion completely dual to that of the electric field using the magnetic susceptibility and, hence, its permeability. While the magnetic dipoles physically arise from moments associated with current loops, they can be described mathematically by magnetic charge and current analogs of the electric cases.

One of the most well-known material models is the Lorentz model. It is derived by a description of the electron motion in terms of a driven, damped harmonic oscillator. To simplify the discussion, we will assume that the charges are allowed to move in the same direction as the electric field. The Lorentz model then describes the temporal response of a component of the polarization field of the medium to the same component of the electric field as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)

The first term on the left accounts for the acceleration of the charges, the second accounts for the damping mechanisms of the system with damping coefficient Γ_L, and the third accounts for the restoring forces with the characteristic frequency f₀ = ω₀/2π. The driving term exhibits a coupling coefficient χ_L. The response in the frequency domain, assuming the engineering exp(+jωt) time dependence, is given by the expression

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)

With small losses Γ_L/[ω₀ << 1 the response is clearly resonant at the natural frequency f₀. The polarization and electric fields are related to the electric susceptibility as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3)

The permittivity is then obtained immediately as ε_Lorentz(ω) = ε₀[1 + χ_e,Lorentz(ω)].

There are several well-known special cases of the Lorentz model. When the acceleration term is small in comparison to the others, one obtains the Debye model:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.4)

When the restoring force is negligible, one obtains the Drude model:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.5)

where the coupling coefficient is generally represented by the plasma frequency χ_D = ω²_p. In all of these models, the high-frequency limit reduces the permittivity to that of free space.

Assuming that the coupling coefficient is positive, then only the Lorentz and the Drude models can produce negative permittivities. Because the Lorentz model is resonant, the real part of the susceptibility and, hence, that of the permittivity become negative in a narrow frequency region immediately above the resonance. On the other hand, the Drude model can yield a negative real part of the permittivity over a wide spectral range, that is, for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Similar magnetic response models follow immediately. The corresponding magnetization field components M_i and the magnetic susceptibility χ_m equations are obtained from the polarization and electric susceptibility expressions with the replacements E_i -> H_i, P_i/ε₀ -> M_i. The permeability is given as µ(ω) = µ₀[1 + χ_m(ω)].

Metamaterials have necessitated the introduction of generalizations of these models. For instance, the most general second-order model that has been introduced for metamaterial studies is the two-time-derivative Lorentz metamaterial (2TDLM) model:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.6)

This 2TDLM model incorporates all the standard Lorentz model behaviors including the resonance behavior at 0 but allows for additional driving mechanisms that are important when considering time-varying phenomena. It satisfies a generalized Kramers–Krönig relation and is causal if χ_γ > -1. It has the limiting behaviors [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The high-frequency behavior has the peculiar property that if -1 < χ_γ < 0, then 0 < lim_ω->∞ ε(ω) < 1, which leads to the interesting but still controversial transvacuum-speed (TVS) effect.

1.2 WAVE PARAMETERS IN DNG MEDIA

One must exercise some care with the definitions of the electromagnetic properties in a DNG medium. Ziolkowski and Heyman thoroughly analyzed this concept mathematically and have shown that in DNG media the refractive index can be negative. In particular, in a DNG medium where ε < 0 and µ < 0, one should write for small losses:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.7)

accounting for the branch-cut choices. This leads to the following expressions for the wavenumber and the wave impedance, respectively:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.8)

where the speed of light c = 1/ [square root of ε₀µ₀] and the free-space wave impedance η₀ = [square root of ε₀µ₀]. One sees that the index of refraction

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.9)

has a negative real part. Its imaginary part is also negative corresponding to the passive nature of the DNG medium.

The index of refraction of a DNG metamaterial has been shown theoretically to be negative by several groups (e.g.,), and several experimental studies have been reported confirming this negative-index-of-refraction (NIR) property and applications derived from it, such as phase compensation and electrically small resonators, negative angles of refraction (e.g.,), sub-wavelength waveguides with lateral dimension below diffraction limits, enhanced focusing, backward-wave antennas, Cerenkov radiation, photon tunneling, and enhanced electrically small antennas. These studies rely heavily on the concept that a continuous-wave (CW) excitation of a DNG medium leads to a NIR and, hence, to negative or compensated phase terms.

1.3 FDTD SIMULATIONS OF DNG MEDIA

In this chapter and in Chapter 2, we present several finite-difference time-domain (FDTD) simulation results for wave interactions with DNG media, in addition to analytical descriptions. Consequently, we briefly discuss some of the features of the FDTD simulator specific to the DNG structures. It should be emphasized that the use of this purely numerical simulation approach does not involve any choices in defining derived quantities to explain the wave physics, for example, no wave vector directions or wave speeds are stipulated a priori. In this manner, it has provided a useful approach to studying the wave physics associated with DNG metamaterials.

As in, lossy Drude polarization and magnetization models are used to simulate the DNG medium; specifically the permittivity and permeability are described in the frequency domain as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.10)

where ω_pe, ω_pm and Λ_e, Λ_m denote the corresponding plasma and damping frequencies, respectively. These models are implemented into the FDTD scheme by introducing the associated electric and magnetic current densities and the equations that govern their temporal behavior.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.11)

The choices of the space and time locations of the discretized electric and magnetic currents, as well as the polarization and magnetization fields, are made self-consistently following the conventional FDTD method. The simulation space is truncated with a metamaterial-based absorbing boundary condition. The FDTD cell size in all of the cases presented here was λ₀/100 to minimize the impact of any numerical dispersion on the results.

Although in some of the analytical and numerical studies, as well as experiments, considered by other groups (e.g.,) the Lorentz model and its derivatives have been used, here the Drude model is preferred for the FDTD simulations for both the permeability and permittivity functions because it provides a much wider bandwidth over which the negative values of the permittivity and permeability can be obtained. This choice is only for numerical convenience and it does not alter any conclusions derived from such simulations; that is, the negative refraction is observed in either choice. However, choosing the Drude model for the FDTD simulation also implies that the overall simulation time can be significantly shorter, particularly for low-loss media. In other words, the FDTD simulation will take longer to reach a steady state in the corresponding Lorentz model because the resonance region where the permittivity and permeability acquire their negative values would be very narrow in this model.

1.4 CAUSALITY IN DNG MEDIA

As for the causality of signal propagation in a DNG medium, we note that if one totally ignores the temporal dispersion in a DNG medium and considers carefully the ramifications of a homogeneous, nondispersive DNG medium and the resulting NIR, one will immediately encounter a causality paradox in the time domain, that is, a nondispersive DNG medium is noncausal. However, a resolution of this issue was uncovered in by taking the dispersion into account in a time-domain study of wave propagation in DNG media. The causality of waves propagating in a dispersive DNG metamaterial was investigated both analytically and numerically using the one-dimensional (1D) electromagnetic plane-wave radiation from a current sheet source in a dispersive DNG medium. A lossy Drude model of the DNG medium was used, and the solution was generated numerically with the FDTD method. The basic 1D geometry is shown in Figure 1.3. The signal direction of propagation (D.O.P.) is from left to right.

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