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- Herausgeber: John Wiley & Sons
- Kategorie: Fachliteratur
- Sprache: Englisch
This book addresses artificial materials including photonic crystals (PC) and metamaterials (MM). The first part is devoted to design concepts: negative permeability and permittivity for negative refraction, periodic structures, transformation optics. The second part concerns PC and MM in stop band regime: from cavities, guides to high impedance surfaces. Abnormal refraction, less than one and negative, in PC and MM are studied in a third part, addressing super-focusing and cloaking. Applications for telecommunications, lasers and imaging systems are also explored.
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Seitenzahl: 515
Veröffentlichungsjahr: 2012
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Table of Contents
Introduction
PART 1. A FEW FUNDAMENTAL CONCEPTS
Chapter 1. Definitions and Concepts
1.1. Effective parameters of materials
1.2. Terminology of artificial materials
1.3. Negative refraction: stakes and consequences
1.4. Bibliography
Chapter 2. The Metamaterial Approach — Permeability and Permittivity Engineering
2.1. Background history
2.2. An imbricated lattice approach
2.2.1. Permittivity engineering: the Drude model
2.2.2. Permeability engineering: hyperbolic or Lorentz model
2.2.3. Negative refraction index
2.2.4. Microwave virtual prototype
2.3. Cell approach.
2.3.1. Extraction procedures of effective parameters in periodic structures.
2.3.2. A combined latticeapproach: the Ω particle
2.4. Alternative approach: Mie resonances
2.5. Bibliography
Chapter 3. Photonic Crystal Approach — Band Gap Engineering
3.1. Historical background
3.2. Study tool: band structure
3.2.1. A few considerations about band gaps and pass bands
3.2.2. Photonic crystal modeling
3.3. 2D ½ photonic crystals
3.3.1. Dimensionality issues
3.3.2. A parametric study
3.3.3. Extraction of effective parameters
3.4. A few words on three-dimensional photonic crystals
3.5. Conclusion: metamaterials or photonic crystals?
3.6. Bibliography
Chapter 4. Transformation Optics
4.1. Context
4.2. Method description
4.2.1. Conceptual approach
4.2.2. Attainable devices
4.2.3. A few considerations on impedance and transformation optics
4.3. Bibliography
PART 2. MATERIALS USEDIN A BAND GAP REGIME
Chapter 5. Point and Extended Defects in Photonic Crystals
5.1. Context
5.2. Defect zoology
5.3. Selectivity of photonic crystal microcavities
5.3.1. Factors of merit: potential performances
5.3.2. Modal analysis of the H1 cavity
5.4. Waveguiding in photonic crystals
5.4.1. Operating principles
5.4.2. Waveguide losses: light lines and leaky modes
5.4.3. Three-dimensional analysis of propagation losses
5.5. Slowing down light
5.6. Bibliography
Chapter 6. Routing Devices made from Photonic Crystals
6.1. The building brick: the add/drop filter
6.2. A few photonic crystal approaches
6.2.1. Pseudo-gallery mode coupler
6.2.2. Coupler using a low group velocity guided mode
6.2.3. Codirectional coupling with a higher-order mode
6.3. Interference-based couplers
6.3.1. A “single branch” approach
6.3.2. Multi-branch coupler
6.4. Conclusion
6.5. Bibliography
Chapter 7. Single Negative Metamaterials
7.1. Context
7.2. ENGs: negative permittivity materials
7.2.1. A few notions about plasmons
7.2.2. Photon sieves
7.2.3. Plasmonic lenses
7.3. MNGs: negative permeability materials
7.3.1. Split-ring resonator lattices and frequency increase
7.3.2. Alternatives for negative permeability in optics
7.4. What of frequency-selective surfaces?
7.5. Bibliographyc
PART 3. MATERIALSIN AN ABNORMAL REFRACTION REGIME (N < 1 ANDN < 0)
Chapter 8. Two-dimensional Microwave Balanced Composite Prism
8.1. Why use a microwave prism?
8.2. Conception and sizing of a balanced composite lattice
8.2.1. Unit cell sizing
8.2.2. Effective parameter sensitivity: towards the notion of tuning
8.2.3. Analysis in terms of electric equivalent circuits
8.3. Two-dimensional prism
8.3.1. Design and simulation of a two-dimensional prism
8.3.2. Fabrication
8.3.3. Angular characterization
8.4. Bibliography
Chapter 9. Metal-dielectric Materials – from the Terahertz to the Visible
9.1. From the terahertz to the infrared
9.2. A backward propagation line at terahertz frequency
9.2.1. Concepts
9.2.2. Sizing and design
9.2.3. Characterization
9.3. From “nano”-resonators to “fishnets”
9.3.1. From “micro”-resonators to “nano”-resonators
9.3.2. Subwavelength apertures for the terahertz
9.3.3. Toward the infrared and the visible
9.4. Three-dimensional metamaterials
9.5. Bibliography
Chapter 10. Abnormal Refraction in Photonic Crystals
10.1. Context
10.2. (An)isotropy in photonic crystals
10.2.1. Study tool: isofrequency curves
10.2.2. Descriptive parameters of isofrequency curves
10.2.3. Geometry and isotropy
10.3. Exploiting anisotropy
10.3.1. Autocollimation
10.3.2. Ultra-refraction, negative refraction and the superprism effect
10.4. Focalization and negative refraction: looking for isotropy
10.4.1. Some notes of caution concerning vocabulary
10.4.2. In the first band
10.4.3. In the second band
10.5. Bibliography
Chapter 11. A Photonic Crystal Flat Lens at Optical Wavelength
11.1. A bit of background
11.2. How to define a typical prototype at optical wavelengths
11.3. Lens optimization: impedance and resolution
11.3.1. Engineering cavity modes
11.3.2. Defect engineering
11.3.3. Interface engineering
11.4. Experiments
11.4.1. Nanofabrication
11.4.2. Optical characterization in the near field
11.5. Reverse engineering: from a two-dimensional prototype to three-dimensional reality
11.6. Conclusion
11.7. Bibliography
Chapter 12. Wave-controlling Systems — Towards Bypass and Invisibility
12.1. “Transformation optics” or “dispersion engineering”
12.2. Component approaches for controlling waves
12.2.1. Gradient-index lenses
12.2.2. Optical mirages
12.2.3. Photon traps
12.2.4. Magic carpets
12.2.5. 2D and 3D invisibility capes
12.2.6. Hyperlenses and associated components
12.3. Invisibility at terahertz frequencies: Mie resonances
12.3.1. Determination of parameters using transformation optics
12.3.2. Metamaterials with Mie resonators
12.4. An alternative with the photonic crystal: the butterfly
12.4.1. Description and how it works
12.4.2. Performances of the system
12.5. Perspectives
12.6. Bibliography
PART 4. MOVING TOWARD APPLICATIONS
Chapter 13. Guiding, Filtering and Routing Electromagnetic Waves
13.1. Context
13.2. Guiding: propagation lines and tunable phase shifters
13.2.1. In a “photonic crystal” regime
13.2.2. In a “metamaterial” regime
13.3. Filtering
13.3.1. The “metamaterial” approach: substrate engineering
13.3.2. “Photonic crystal” approach: cavity engineering
13.4. Metamaterial-based routing
13.5. Conclusion
13.6. Bibliography
Chapter 14. Antennas
14.1. Towards the miniaturization of transmission/reception systems
14.2. Directivity engineering
14.2.1. Antennas and electromagnetic crystals
14.2.2. Antenna in the cavity
14.2.3. Subwavelength cavities
14.3. Subwavelength sizing
14.3.1. Patch antennas
14.3.2. Monopole and dipole antennas
14.3.3. The utility of double negative materials
14.4. Conclusion
14.5. Bibliography
Chapter 15. Optics: Fibers and Cavities
15.1. Optical issues: the privileged domain of photonic crystals
15.2. Microstructured optical fibers
15.2.1. Principle of guiding in microstructured fibers
15.2.2. Some properties of microstructured fibers
15.2.3. A vast field of applications
15.3. Toward zero threshold lasers
15.3.1. Controlling spontaneous emissions
15.3.2. Application
15.3.3. Photonic laser crystals performance
15.4. Bibliography
Chapter 16. Detection, Imaging and Tomography Systems
16.1. From detection to imaging
16.2. Terahertz sensors
16.3. Direct approach for imaging
16.4. Detection and image reconstruction
16.4.1. An approach inspired by tomography in reflection
16.4.2. Performances and constraints
16.5. A vast field to explore
16.6. Bibliography
Conclusion
Index
First published 2012 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:
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© ISTE Ltd 2012
The rights of Olivier Vanbésien to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.
Library of Congress Cataloging-in-Publication Data
Vanbesien, Olivier.
Artificial materials / Olivier Vanbesien.
p. cm.
Includes bibliographical references and index.
ISBN 978-1-84821-335-7
1. Materials. 2. Synthetic products. I. Title.
TA403.V376 2012
620.1--dc23
2011052452
British Library Cataloguing-in-Publication Data
A CIP record for this book is available from the British Library
ISBN: 978-1-84821-335-7
Introduction
In the field of electromagnetic and/or optical waves, the concept of “artificial material” covers any fabricated device, be it metallic, dielectric or metal-dielectric, enabling a varyingly complex control of wave propagation in a chosen wavelength range. A common point to all approaches is that the structural scale of the building blocks of a new material is proportional to its operating wavelength. From this first principle stem two relatively distinct research paths, “photonic crystals” (or photonic/electromagnetic materials with a band gap) and “metamaterials”.
The chosen structure scale of “photonic crystals” is within the range of the wavelength (λ). The characteristic dimensions of the networks typically range from λ/3 to λ/4. To describe the photonic crystals, we shall use the notion of “band structure” to utilize the diffraction and refraction properties of the pass and/or stop bands. For metamaterials, the most common approach is to consider very small structure scales compared to the wavelength, typically smaller than λ/10. In this “metamaterial” regime, we shall try to shape classical material parameters, such as permittivity and permeability, for example, to obtain values that are negative at the same wavelengths for these two quantities. We will thus obtain original properties that are nonexistent in nature, hence the use of the “meta” prefix, signifying “beyond”.
Following this first definition, we should not separate the two types of materials created. This is because it would often be greatly tempting, and justified, to define the equivalent parameters of permittivity and permeability for the photonic crystals or to use dispersion diagrams for metamaterials. When we study the physical effects linked to these artificial materials, such as “negative refraction”, we shall notice that the two approaches are closely linked and inseparable in the concepts used, although they each have their own particularity.
This notion of artificial material has led to negative comments in the scientific communities it concerns (broadly speaking, electromagnetics and optics). This started in the 1990s for “electromagnetic or photonic crystals” and carried on in the 2000s with “metamaterials”. The latest decades have produced abundant literature on such one-, two- or three-dimensional devices. Many physics and applied physics researchers has viewed this as a more or less clumsy grooming of old themes reintroduced with appropriate terminology. Although this critique is understandable when it comes to certain works concerning one-dimensional devices, it is clear that “artificial materials”, and specifically the ones with two dimensions, have brought a new impetus and help us consider a control of the propagation of waves that was previously unimaginable. We now have ultimate control of propagation with a possible verification of these effects on almost the entire electromagnetic and optical spectrum, thanks to visible progress in fabricating technology, especially on the nanometric scale. Exploration of the third dimension is now possible and allows us to hope for a huge field of potential applications, which could keep generations of researchers busy.
It would be unjust to write such a book without quoting the few researchers who have published the seminal articles in this field in the introduction. These researchers include:
– V. Veselago for his theoretical work in electromagnetism and for popularizing the concepts of left-handed materials and negative refraction a long time after details about them were first published;
– E. Yablonovitch for photonic crystals; and
– Sir J. Pendry for establishing the conditions enabling the fabrication of these materials with simultaneously negative permittivity and permeability.
Many other authors, who have focused on the experimental demonstration of these effects across the electromagnetic spectrum, should obviously be listed here but to keep the list short they are mentioned throughout the chapters of this book.
The thematic field covered by artificial materials is extremely vast. The choice of an outline is arbitrary. Instead of establishing a hierarchy of themes according to the structural scale of the artificial material or field of the intended wavelength, we have decided on an approach starting from general concepts and ending with targeted applications, going through the detailed study of “physical effects” demonstrations. One of our ambitions is to bring together the scientific communities working on microwaves, electromagnetism and optics, around a few uniting concepts, forgetting about the “artificial” borders that some have tried to establish between “metamaterials” and “photonic crystals”, for example.
This book will revolve around four major axes:
– In Part 1, the fundamental concepts are described. After some definitions, the metamaterial approach and the associated permittivity and permeability engineering will be presented. The photonic crystal and band gap approach will follow, and the common ground between the different approaches will be underscored. Finally, the transformation optics that generalizes all of the aforementioned concepts, leading to a localized engineering of parameters, will be mentioned.
– In Part 2, we shall present a detailed study of photonic crystals and/or metamaterials in a regime of band gaps. Cavities and extended defects for devices in charge of locating, guiding and routing waves will be studied.
– Part 3 will focus on artificial materials in an abnormal refraction regime, which present positive refraction indexes that are still inferior to one or more negative refraction indexes. We shall also analyze the origin of backward wave propagation, autocollimation and ultra-refraction effects. This will take place across the electromagnetic spectrum through generic examples: a two-dimensional prism in micro-waves, a retrograde propagation line at terahertz frequency, a flat photonic crystal lens at optical wavelengths and finally optical bypass or invisibility devices conceivable at various wavelengths.
– Part 4 focuses as closely as possible on the potential applications of the effects illustrated in the first three parts. Beyond the demonstrators of “physical” effects, we shall evaluate the relevancy and feasibility of original prototypes by expanding our field of investigation, not only to double negative materials (with a negative refraction index) but also to single negative materials (permittivity or permeability). The field of filtering and/or routing of waves, the field of antennas, the field of lasers as well as the field of imaging systems and tomography will be mentioned.
Without pretending to be exhaustive, which is almost impossible as the field of artificial materials is still evolving and has not reached maturity, especially when it comes to tridimensional structures, the point of this book is to use established concepts to enable new ideas to take shape and develop in the future.
This book aims to target a broad audience, from Masters students to early-stage researchers to any scientist curious to discover a world close to theirs. Individuals who specialize in or are very knowledgeable about one of this book’s themes might find his or her centers of interest too superficially broached, for which I apologize in advance.
Concerning the bibliography, for nearly 30 years now, literature first on photonic crystals, then on metamaterials and finally transformation optics has been growing and growing. Without forgetting that the seminal and precursory works in the field of microwaves for the control of electromagnetic wave propagation in the wider sense appeared even earlier. It is therefore almost impossisble to cite all works of value (without forgetting one)! In this introduction, I will start with a set of PhD theses passed on this theme by students from the DOME (Dispositifs Opto- et Micro-électroniques quantiques) team, supervised and/or co-supervised by its head, Didier Lippens, and myself at IEMN, Lille University, France. Other PhD theses are in progress, their manuscripts are on their way, but their work has also fueled this book. Without them, this work would not have seen the light of day…
A selection of review articles will also be proposed at the end of each chapter. The list is not exhaustive and the reader can refer to the cited works to inform himself more completely on a particular topic.
Theses
[AKA 02] AKALIN T., Dispositifs de propagation, de filtrage et de rayonnement électromagnétique basés sur les structures périodiques, Lille University of Science and Technology, 2002.
[BOR 09] BORJA A.L., Engineering applications of metamaterials in electromagnetics: filters and antennas, Universidad Politecnica de Valencia, 2009.
[CAR 98] CARBONELL J., Electromagnetic analysis of active and passive devices for space applications, Lille University of Science and Technology, 1998.
[CRO 09] CROENNE C., Contrôle de la propagation et du rayonnement électromagnétique par les métamatériaux, Lille University of Science and Technology, 2009.
[DAN 01] DANGLOT J., Dispositifs microondes et optiques à base de matériaux à gaps de photon, Lille University of Science and Technology, 2001.
[DEC 04] DECOOPMAN T., Multiplicateurs de fréquence et métamatériaux en technologie finline, Lille University of Science and Technology, 2004.
[FAB 08] FABRE N., Matériaux main gauche et cristaux photoniques pour l’optique: approche diélectrique, Lille University of Science and Technology, 2008.
[FAS 05] FASQUEL S., Propriétés optiques de structures guidantes en cristal photonique, Lille University of Science and Technology, 2005.
[HOF 10] HOFMAN M., Composants optiques à base de cristaux photoniques pour applications à l’imagerie infrarouge, Lille University of Science and Technology, 2010.
[HOU 09] HOUZET G., Déphasage composite accordable et routage partiel par la technologie des métamatériaux, Lille University of Science and Technology, 2009.
[ZHA 09] ZHANG F., Technologies des métamatériaux électromagnétiques en volume: application aux éléments de guidage et de rayonnement, Lille University of Science and Technology, 2009.
I must make special mention of Eric Lheurette, his thesis manuscript was also a priceless aid:
[LHE 09] LHEURETTE E., Métamatériaux à indice négatif de réfraction: des micro-ondes aux fréquences térahertz, Habilitation à diriger des recherches, Lille University of Science and Technology, 2009.
PART 1
A Few Fundamental Concepts
In Part 1, we present the two most common approaches to describe artificial materials: permittivity and permeability engineering for “metamaterials” and band gap engineering for “photonic crystals”. We shall focus on the similarities between these approaches. The principles of “transformation optics” shall also be broached to generalize and go beyond the aforementioned concepts.
Chapter 1
Definitions and Concepts
1.1. Effective parameters of materials
Among all the properties necessary to describe materials macroscopically, we shall deliberately limit ourselves to a reduced yet sufficient set of parameters in order to consider the phenomena of electromagnetic or optical wave propagation. When working with passive and linear materials, we are led to define the effective parameters of the continuous medium, inasmuch as they shall represent the properties of the material on a large scale in front of their own structure.
These material parameters come into play in the so-called “constitutive” equations of the medium. According to the previously mentioned hypotheses, for a given angular frequency, ω, these equations can be written as follows:
[1.1]
[1.2]
with and respectively being the electric induction, magnetic induction electric field and magnetic field at the point of vector position and angular frequency ω.
Working from considerations about the symmetry of the built materials or about the theorem of reciprocity, we can show that many of the terms of these tensors are null. In the most common cases that are studied in the field of artificial materials, it is generally possible to limit ourselves to diagonal permittivity and permeability tensors and to disregard the bianisotropic tensors. Even if the structures designed present unit cells with a strong bianisotropy, we shall ensure they are positioned or laid out so as to minimize this effect. Let us also mention that each of these parameters is by definition a complex quantity, and its imaginary part translates a phenomenon of wave attenuation.
In the case of diagonal tensors of permittivity and permeability, in a given (x, y, z) orthonormal coordinate system, we will often be led to consider the propagation of a plane wave, for example in direction z with an electric field direction towards x and a magnetic field towards y. In this case, using a permittivity quantity εeff (direction x: εexx) and a permeability quantity μeff (direction y: μyy) will be sufficient to describe propagation in the material. This ultimate simplification to two parameters will be virtually systematic for all the materials presented from now on in this book, except for devices based on transformation optics.
Starting with the two parameters εeff and μeff, we can define two additional effective parameters: the optical effective index neff and the effective medium impedance zeff, defined by:
[1.3]
We should note, as previously mentioned, that εeff and μeff are complex quantities. So the passage of εeff and μeff to neff and zeff is not immediate and in the extraction of parameters requires us to take into account physical considerations to validate the calculations. For example, to respect the principle of causality the real part of impedance for passive materials must stay positive; the same is true for the imaginary part of the optical index.
[1.4]
where c is the speed of light.
Effective impedance translates the relation between electric and magnetic fields and the plane wave in the materials and has a direct effect on the calculation of the coefficient of reflection at the interface. We shall see the crucial importance of this parameter when we address the factors of merit of devices made from artificial materials. If the qualitative demonstration of original physical effects depends very lightly on its value, we shall see that disregarding its optimization can very swiftly “kill” the development of any performing application.
In the case of the artificial materials we shall subsequently study, recourse to these effective parameters will be very frequent. It is, however, essential to remember that they originally helped describe the average behavior of heterogeneous mediums structured on a microscopic scale in a homogeneous and macroscopic manner. For the phenomena of propagation in artificial materials, we shall remember that the domain of validity of these parameters will be limited by the structure scale d and that a rough estimate in relation to the wavelength λ will be sufficient to consider the material homogeneous: d<λ/10. We shall, however, see that for reasons of simplification the use of these material parameters is practical and common even when the ratio between d and λ exceeds this limit.
1.2. Terminology of artificial materials
The vocabulary used to differentiate artificial materials is often complex and depends on their applications. However, the borders between the significations of the terms employed are sometimes blurry and can lead to noticeable states of confusion. When should the term metamaterial be used? Or the notion of a left-handed material? Or that of double negative medium? Or the terminology of photonic and/or electromagnetic crystal? Is the photonic crystal a metamaterial? There are so many questions that are sometimes hard to answer, since opinions are very much divided.
One natural starting point for this “terminology” is to use the parameters of permittivity and permeability defined in the previous section. This was Veselago’s approach in his seminal paper in 1967 that the entire scientific community quotes today as soon as it tackles the field of artificial materials. This first classification is based on the respective signs of the real parts of permittivity (ε) and permeability (μ). Figure 1.1 is an illustration of this approach.
Figure 1.1.Propagation of electromagnetic waves according to the permittivity and permeability of the medium
The ordinary case is when ε > 0 and μ > 0. In this case, according to Maxwell’s equations the propagation of a plane wave is possible. The electric () and magnetic () fields are orthogonal and the wave vector () is perpendicular to the previous plane and indicates the propagation direction. In this case, the (, , ) trihedral is a direct trihedral and the Poynting vector () is collinear to and has the same direction. We can then talk, if necessary, of “right-handed material”. For any wave resolved on the basis of propagative and evanescent waves, by definition we only keep the propagative components of the wave after a certain propagation distance. In this case the group velocity and phase velocity are of the same sign.
When one of these quantities, either e or µ, becomes negative, the medium becomes opaque to electromagnetic waves and there is no propagation. The waves become evanescent. We will then have a “single negative medium”. While this holds little interest on its own, we shall see that such a medium can be used as a surface, as a thin or nanostructured film to create resonance effects between propagative mediums, for example.
The most original case corresponds to the fourth quadrant for which e and µ are simultaneously negative. Let us mention that in principle no “natural” material is endowed with this property. The only way to obtain this is to artificially “fabricate” the material. In this case, the analysis of the propagation equations shows that the medium becomes transparent once more and allows the propagation of waves. In such a “double negative medium”, however, the propagation becomes retrograde and the wave vector takes on the shape of an indirect trihedral (given by the left hand) with electric and magnetic fields. We then talk of a “left-handed material”. Group velocity and phase velocity then have opposite signs. If we keep a positive sign for the direction of energy propagation, we have a negative phase velocity. We shall return to the main consequence of this double negative medium, which provides us with the possibility to utilize the notion of negative refraction, in section 1.3. Moreover, these materials can theoretically amplify the evanescent components of any wave, which is a perilous and sometimes even contested subject, to which we shall also return.
Initially, the notions of “left-handed” materials and “metamaterials” were often mistaken for each other. These two notions now tend to be differentiated. A “metamaterial” is any artificial material used in a regime in which the wavelength is large compared to the internal structure of the material. This will preferably be called a “metamaterial regime”. So a single negative material can be considered to be like a metamaterial even if it is not left handed. In this category of “metamaterials”, we can find periodic or non-periodic artificial structures. Periodicity is not a requirement, even if periodic structures will generally be studied and developed. This is the point where we can introduce the notion of periodic artificial materials, also known as “photonic crystals”, “electromagnetic crystals” or even “photonic band gap materials”. We should note that, under the impulse of Yablonovitch in the eighties, these materials were designed and studied for their ability to block the propagation of light compared to semi-conductor crystals and their gap which forbids the propagation of electrons. It is only more recently that the scientific community has also looked into different modes of propagation in these crystals, with the discovery of possible ultra-refraction or negative refraction effects. In general, photonic crystals are periodic structures characterized by their unit cell parameter a, and are utilized for their stop and pass bands on wavelengths such as λ/10 < a <λ/2. They are thus not really in a metamaterial regime, even though the designation is sometimes improperly used.
Let us also mention that another usual difference between “photonic crystals” and “metamaterials” is the nature of the unit cell making up the periodic lattice. The unit cell in itself is infinitely more complex in the latter. A second difference is that, when it comes to metamaterials, the unit cell has the properties enabling us to obtain negative e and µ simultaneously. The lattice array only amplifies the phenomenon (while also minimizing the bianisotropic effects). In photonic crystals, it is the lattice effect and the periodicity that enable us to obtain the targeted properties to the point of obtaining a double negative material with a photonic crystal. The design and engineering of the values targeted, however, will be more difficult in this case compared to “real” metamaterials.
Before going on to discuss the different ways by which we can approach genuine engineering of material parameters in Chapters 2, 3 and 4, it is necessary to go into more detail about the physical phenomenon of “negative refraction” and its various consequences. This subject was at the root of the explosion of works on metamaterials, and has expanded to the field of transformation optics, yet is still the source of serious debate among the optic and electromagnetic communities.
1.3. Negative refraction: stakes and consequences
Negative refraction is a direct consequence of the existence of “double negative materials”. In this case, it was shown that in order to respect the integrity of Maxwell’s equations when trying to introduce the notion of equivalent refraction index in lieu of the terms of permittivity and permeability (generally for isotropic mediums in the x and y field directions), we must use the solution:
[1.5]
where εeff and μeff are both negative. The zeff definition remains unchanged and still has a real positive part.
Figure 1.3. Flat lens concept based on a homogeneous material with a negative refraction index: (a) propagative components; (b) amplification of evanescent waves; and (c) surface mode excitation
Research to create such a “superlens” continued at the same time as the theoretical studies continues through the use of metamaterials and photonic crystals. Some devices that have been skillfully designed have enabled researchers to experimentally reach a resolution with a wavelength much smaller than the Rayleigh criterion. None of the artificial materials used corresponds to the “ideal criteria” defined above and the precise origin of this super-resolution is still subject to very critical scrutiny.
Research to find a superlens is not the only new branch to develop from artificial material research, but it is without doubt one of the most striking. This is why it is the subject of a more in-depth focus. We shall, however, see in this book that the field of investigation has expanded well beyond the superlens. In Chapters 2 to 4, we shall focus on the fundamentals of the three research paths that are most commonly used:
– the metamaterial approach: permittivity and permeability engineering;
– the photonic crystal approach: band structure engineering; and
– the transformation optics approach: localized engineering.
1.4. Bibliography
[BEN 03] BEN-ARYEH Y. “Nonclassical high resolution effects produced by evanescent waves”, J. Opt. B: Quantum Semiclassical Opt., vol. 5, no. 6, pp. S557, 2003.
[EFR 07] EFROS A.L., LI C., “Electrodynamics of left-handed medium”, Solid State Phenomena, vol. 121, no. 3, pp. 1065-68, 2007.
[GAR 02] GARCIA N., NIETO-VESPERINAS M. “Left-handed materials do not make a perfect lens”, Phys. Rev. Lett., vol 88, no.20, 207403, 2002.
[MAY 04] MAYSTRE D., ENOCH S., “Perfect lenses made with left-handed materials: Alice’s mirror?”, J. Opt. Soc. Am. A, vol. 21, no. 1, pp. 122-31, 2004.
[PEN 00] PENDRY J.B., “Negative refraction makes a perfect lens”, Phys. Rev. Lett., vol. 85, no. 18, pp. 3966-69, 2000.
[PEN 02] PENDRY J.B., ANANTHA RAMAKRISHNA S., “Near-field lenses in two dimensions”, J. Phys. Condens. Matter., vol. 14, no. 36, pp. 8463, 2002.
[PEN 03] PENDRY J.B., ANANTHA RAMAKRISHNA S., “Focusing light using negative refraction”, J. Phys. Condens. Matter., vol. 15, no. 37, pp. 6345, 2003.
[PER 05] PERRIN M., FASQUEL S., DECOOPMAN T., MÉLIQUE X., VANBÉSIEN O., LHEURETTE E., LIPPENS D. “Left-handed electromagnetism obtained via nanostructured metamaterials: comparison with that from microstructured photonic crystals”, J. Opt. A: Pure Appl. Opt., vol. 7, no. 2, pp. S3, 2005.
[SMI 00] SMITH D.R., KROLL N., “Negative refraction Index in Left-handed materials”, Phys. Rev. Lett., vol. 85, no. 14, pp. 2933-36, 2000.
[VES 68] VESELAGO V.G., “The electrodynamics of substances with simultaneously negative of ε and μ”, Sov. Phys-Usp, vol. 10, pp. 509-14, 1968.
Chapter 2
The Metamaterial Approach – Permeability and Permittivity Engineering1
2.1. Background history
As we mentioned in Chapter 1, Veselago’s works were very rich in potential applications but were left by the wayside for a few decades because it was difficult to imagine how to design a material with effective negative permeability. The notion of artificial magnetism only appeared in the wake of Sir Pendry’s works at the end of the 1990s.
The metamaterial adventure only really started under the impulse of Sir Pendry, in the middle of the 1990s, first by proposal that took into consideration the specific properties of wire lattices. It is known that metal is opaque to electromagnetic waves whose frequencies are lower than its plasma frequency, the latter being in the range of a few thousand terahertz (THz). We can consider this opacity as equivalent to a negative permittivity of the corresponding medium (see the Drude model in section 2.2.1). In this case, in the range of microwaves (a few GHz) or optics (a few hundred THz), effective permittivity is indeed negative… but its amplitude is too great and thus unusable from an application point of view.
How can we lower the plasma frequency while keeping a permittivity behavior similar to that of metal? Sir Pendry suggested the use of a three-dimensional lattice of very fine wires, as shown in Figure 2.1a. Using the conduction currents generated over the surface of the metals when illuminated by an incident wave at a given frequency, it was shown that for even small quantities of material, depending on the lattice geometries used, it was possible to significantly influence the electromagnetic field. Thus the values of the effective parameters of the artificial materials fabricated. This only works if the size and/or periodicity of the structures used are small compared to the operating wavelength. In this way, an equivalent plasma frequency can be defined through geometric considerations, for example, and be positioned at will in the electromagnetic spectrum, from microwaves to optics (even if the use of metals in optics is still subject to control of losses). For example, as in Sir Pendry’s seminal paper, this plasma frequency could be lowered to a few GHz by using a lattice of micronic wires in a cubic mesh (with a period of a few millimeters). The whole set could also be embedded into a different dielectric from air whose own permittivity would have to be taken into account in the evaluation of the new plasma frequency.
Figure 2.1.Diagrams of: (a) a three-dimensional lattice of continuous wires; and (b) of a double split-ring resonator (according to J.B. Pendry et al. [PEN 99]). Copyright American Physical Society and IEEE
Following this, Sir Pendry was also the first to find a way to reach negative permeability. A material’s given permeability versus angular frequency usually follows a hyperbolic model (resonant) defined by two characteristic angular frequencies. There is a band of frequency between these two in which equivalent permeability can become negative. As for the method used for permittivity, we now have to try to position these two frequencies in the band in which we want to use the artificial material. One possible solution was suggested by the notion of a “split-ring resonator”, illustrated in Figure 2.1b. Let us spend some time explaining its mode of operation.
Take a set of identical closed metal rings placed in a dielectric matrix. If the rings have small dimensions compared to the wavelength they are placed so that their axis of symmetry is parallel to the exciting magnetic field, nothing happens and the artificial material we have defined has no significant magnetic activity. If the rings are split, however, a resonant frequency appears for which the conduction current may be important. At this frequency, the presence of split rings has a strong influence on the global magnetic field, which in terms of effective parameters results in a resonating permeability that is potentially negative within a certain frequency band. The notion of a double resonator, as shown in Figure 2.1, can amplify the targeted magnetic activity. On a fundamental level, this activity can be modeled using a Lorentz law (see section 2.2.2), using the angular frequency parameters linked to the geometric dimensions of the resonators (diameter, size of the split, width of the metal strip, gaps between the rings, etc.).
The degree of freedom achieved by the conception of these two lattices made it possible to conceive two artificial materials, one showing a negative permittivity, the other a negative permeability, with a common frequency band in which the two parameters were negative and “reasonable” (of a few units).
Can the “mix” of these two materials lead to the creation of a material with a negative index? A positive answer to this question was given by D. Smith’s team in the United States, with the first demonstration in microwaves of the transmission of electromagnetic energy through such a material at the beginning of the 2000s. This experiment has since been replicated many times in various laboratories using very different prototypes. These prototypes have also enabled researchers to go beyond a mere indirect demonstration of transmission, as we will see in more detail in the following chapters. We have, for example, proof of negative refraction at a long distance thanks to a prism (the wave is transmitted on the same side of the line normal to the object as the incident wave), as we will see in more detail in the chapter 8.
Starting with this experience, researchers’ imaginations led to the emergence of numerous ideas to obtain materials with negative indexes, some of which are shown in Figure 2.2. Due to the mutual influences of lattices with simultaneously negative permittivity and permeability, two conception strategies appeared that we shall explore:
– “imbricated lattices” (shown in structures (a) and (g) in Figure 2.2) for which permittivity and permeability are studied separately, followed by the mixing conditions;
– “combined lattices” (shown in structures (d), (e), (f), (h) and (i) in Figure 2.2) for which a unit particle (or cell) presenting both an electric and a magnetic activity is conceived.
Figure 2.2.Diagrams of various metallic structures to obtain effective negative parameters (according to (a) R.A Shelby et al., [SHE 01b]; (b) J. Garcia-Garcia et al., [GAR 05]; (c) D. Schurig et al., [SCH 06]; (d) C.R. Simovski, Institute of Fine Mechanics and Optics, St Petersbourg; (e) H. Chen et al. [CHE 04]; (f) H. Chen et al., [CHE 05];(g) T. Koschny et al. [KOS 05]; (h) V.M. Shalaev et al., [SHA 05]; (i) X. Zhou et al., [ZOU 07]. Structure (c) only manifests an electric activity; structure (b) a magnetic one. For the other structures, the two effective parameters are negative in a given frequency band. (a) and (g) use the notion of imbricated lattices; (d), (e), (f), (h) and (i) the notion of combined lattices. Copyright American Institute of Physics, Optical Society of America and American Physical Society
Finally, at the moment of conception, special attention will be paid, in terms of sensitivity, to the incidence of the material’s incoming wave (for an isotropic medium), according to the type of applications aimed for and the required dimensionality of the device (one-, two- or three-dimensional). We shall encounter the semantic debate on whether or not the use of the notion of metamaterial is relevant again, especially in the case of one-dimensional structures.
2.2. An imbricated lattice approach
This approach can be presented almost analytically. It uses the classical Drude model and hyperbolic Lorentz model to describe the effective parameters of materials, permittivity and permeability respectively, to deduce an effective refraction index by multiplying both terms. Experimentally, this is the same as mixing, or imbricating two lattices, but does not systematically give us information on the physical manner in which these lattices must be geometrically mixed to offer an experimental solution enabling us to obtain negative refraction.
2.2.1.Permittivity engineering: the Drude model
The electric activity, also called effective permittivity, can be described in a metallic medium by a classical Drude model whose frequency dependence answers the following equation:
[2.1]
with ωp being the plasma frequency of the material and γ a loss term.
Let us remember that for a bulk metal, the plasma frequency is in the range of a few thousand THz. A reduced plasma frequency can, however, be obtained in the case of metallic wire lattices and should reach any value in the microwave range.
We should also note in this model that the term of losses (the imaginary part of permittivity) is all the more important as the angular frequency is low. In reality, however, other loss terms will have to be taken into account that will have a tendency to increase alongside the frequency, as we would intuitively expect.
2.2.2.Permeability engineering: hyperbolic or Lorentz model
In a similar manner, by using a hyperbolic model we can study the evolution of the effective permeability of a material presenting a resonance.
In this case, μ is expressed as follows:
[2.2]
This model produces a frequency band situated between ω0 and ωp for which μ is negative. As was the case for permittivity, there is a great level of dispersion and the amplitudes obtained strongly depend on the term of losses introduced in the modeling. If γ is equal to 0, permeability can evolve between +∞ and −∞. The greater γ is, the smaller the variation will be. In some cases, permeability will remain positive throughout the spectrum.
Moreover, we can see that the imaginary part of permeability peaks around ω0. It would thus not be very interesting to work in the vicinity of this angular frequency. Instead we will use the area around ωp where the term of losses diminishes considerably. This unfortunately also reduces the frequency range which should be usable in terms of negative parameters.
2.2.3. Negative refraction index
In an ideal setting, working from equations [2.1] and [2.2], by correctly extracting the root of the multiplication of ε by μ (and by a positive imaginary part of this extraction keeping as reference point), we could find the complex index of refraction of such a virtual material that combines wire and ring resonators. This result is given in Figure 2.5.
Figure 2.5.Refractive index of a virtual metamaterial obtained by mixing the lattices of lattice 2.3 (Drude model) and Figure 2.4 (hyperbolic model)
Let us mention that the gray section has no physical meaning in this case as only one of the two parameters (in this case ε) is negative and the medium is opaque to electromagnetic waves. Beyond the permeability resonance, we find a dispersive and negative index of reasonable value ranging from -2 to 0 that becomes positive after the plasma frequencies of ε and μ are considered equal in the example.
We shall note that beyond these equal plasma frequencies, we obtain a nonnegligible frequency band in which n, ε and μ have real parts that are lower than 1, describing a medium with interesting properties that could be used in transformation optics in Chapter 4.
2.2.4. Microwave virtual prototype
To illustrate this approach, let us consider a prototype similar to those conceived, designed and characterized by D. Smith with microwaves, based on the ideas of Sir Pendry at the beginning of the 2000s. These first prototypes, innovative but imperfect, were later optimized to obtain pass band performances with acceptable loss levels compatible with the requirement of radio frequency (RF) circuits. Here, the strategy is to “mix” a wire lattice with electric activity and a resonator lattice with magnetic activity.
Figure 2.6.Building materials: (a) wire lattice with negative permittivity and (b) associated insertion losses according to the number of cells present in the direction of propagation; (c) magnetic split-ring resonator for the synthesis of a negative permeability and (d) associated insertion loss for a unit cell
Figure 2.6 shows the two constitutive “materials” used to design the “metamaterial” with the associated parameters. The characteristic dimensions are fixed to obtain a left-handed band around 12 GHz. In the wire lattice shown in Figure 2.6a, we can see that it is opaque to frequencies lower than 13 GHz in Figure 2.6b, which is equivalent to an effective negative permittivity. A pass-band is obtained beyond. The ripples observed correspond to cavity modes from the multiple unit cells in the direction of propagation. The double split-ring resonator shows an almost absolute reflectivity around 12 GHz, illustrating the rejecting aspect of the resonance and the existence of an important magnetic activity (μ resonance).
When these two lattices are mixed, as in Figure 2.7a, we hope that the electric and magnetic properties of each will add up without mutually influencing each other. These two effects act on perpendicular field values, limiting coupling effects. This can be verified in Figure 2.7b, where the bandwidth is regained between 12 and 13 GHz, yet taken singly both lattices are opaque. The level of transmission obtained is around -2 dB for a prototype presenting five cells in the direction of propagation. This result takes into account the metallic losses and mainly reflects the impedance mismatch between entrance–exit ports and the “metamaterial”.
Figure 2.7.Double negative material or “left-handed material”: (a) diagram of principle; (b) associated insertion losses; and (c) display of the inversion of the wave vector in relation to the Poynting vector
To check numerically that the band obtained is left-handed, the field maps for different phases of the incident field are given in Figure 2.7c. They highlight the inversion of the wave vector in relation to the Poynting vector by following the phase fronts in the “metamaterial” and outside it.
Let us mention that the designed structure, like many “metamaterials”, is highly anisotropic and that the desired properties are only valid for one given direction of the incident field. The nature of the imbrication of the lattices itself forbids us from obtaining isotropic properties. The desire to group the properties on a single particle, which we shall look into in section 2.3, aims in part to find a solution to this intrinsic limitation. We will also show a certain number of inversion techniques that enable us to extract dispersion curves as well as effective values of index, impedance, permittivity and permeability from the constitutive parameters.
2.3. Cell approach
2.3.1. Extraction procedures of effective parameters in periodic structures
2.3.1.1. Inversion relation
The previous approach, based on the use of analytical formulae for ε and μ, can be avoided in some cases, notably for periodic structures. In that case we will use the notion of the dispersion diagram instead, even if it means returning to effective parameters later on.
In this case, we must be able to describe the metamaterial in terms of a unit cell which, correctly studied, should enable us to know all the properties of the material when it is built as an infinite lattice. We will focus in detail on such techniques, used in the case of dielectric photonic crystals, later on. Here we will look at a conception method close to experimental reality, which will help us to efficiently characterize the fabricated prototypes.
As we have seen previously, transmission information is insufficient to prove that a material is left-handed. More information is necessary, such as the evolution of the phase of the transmission coefficient versus frequency, to which we will return, or the study of the dispersion diagram. Different strategies can be developed to ascertain information depending on the nature of the unit cell of the lattice and there are few methods in the case of metallic or metal-dielectric structures.
To illustrate this aspect, let us imagine that we are able to define a unit cell enabling us to work on both permittivity and permeability, such as an Ω-shaped structure (see Figure 2.2d) or an S-shaped structure (see Figure 2.2e). Basing our work on this cell, we must be able to retrieve the complex propagation constant γ of the infinite material created by the periodic arrangement of these “particles” in the three directions of space. Indeed, the three-dimensional reality is usually extremely complex and, to extract a set of useable effective parameters we need to choose a configuration to address the material’s properties.
So, in general, we pick a propagation direction and a wave polarization to calculate (or measure) the scattering matrix (S matrix) that provides us with the reflection and transmission coefficients under matching conditions. This calculation can be drawn out by studying the complete solution of Maxwell’s equations (using commercial or non-commercial software for finite element simulation, for example). We must impose appropriate boundary conditions for the field perpendicular to propagation in order to simulate an infinite number of cells (periodic conditions). The S matrix obtained is converted into a chain matrix (or ABCD matrix):
[2.3]
If the device can be considered as a monomodal homogeneous propagation medium, the chain matrix can be written as:
[2.4]
Using the term of the chain matrix A, we can write:
[2.5]
Beyond this dispersion diagram, the chain matrix enables us to return to the effective parameter couples (n, z) and (εeff, μeff). The refraction index n is obtained with:
[2.6]
while the impedance can be inferred as follows:
[2.7]
Finally,
[2.8]
The previous parameters are complex numbers and the extraction itself is not immediate. From a numerical point of view, many branches are found and we must choose those with a physical meaning. For example, we must obtain negative imaginary parts to describe the notion of propagation loss in these passive mediums for which no amplification phenomenon can be reached.
Let us also remember that the parameters we have obtained are only representative of one polarization and one direction of propagation. The procedure must be reproduced for other polarizations and/or other privileged directions of the unit cell according to its level of symmetry.
2.3.1.2. Other methods of extracting effective parameters
If the previous method has the advantage of giving the whole set of structure parameters, then as long as we have the numerical (or experimental) means of characterizing in modulation and in phase the transmission and reflection coefficients for a unit cell of the material, the literature provides us with other approaches. Some are simplified and enable us to obtain part of the effective parameters under certain hypotheses, while others are more rigorous and in some cases free us from the delicate choice of physical solution in a series of numerical solutions.
Among the simplified solutions, let us mention the case of low-loss materials for which we can directly link the complex propagation constant and the transmission coefficient independently from the medium impedance. In the same way, there are two methods we can use based on a differential approach. Here, we characterize two materials of different thicknesses (without having to limit ourselves to the unit cell) and can free ourselves from all the “accesses” to materials in order to focus on the core states.
Thanks to the first method, based on the evolution of the transmission coefficient phase, we can trace back the real part of the refraction index through the following equation:
[2.9]
where d2 and d1 represent the respective thicknesses of the two test samples, k0 is the wavelength in the vacuum and Δφ is the phase difference between the two transmission coefficients.
The determination of n will be more accurate when the material impedance is closer to that of the medium of the incident wave. This approach means that, no matter what the impedances of the mediums, we can know whether the material is left-handed (Δφ positive or phase advancement for the most narrow material) or right-handed (the opposite).
The second method, called the Bianco and Parodi method, comes from microwaves and also enables us to return to the complex propagation constant thanks to the properties of the scattering matrix through the calculation or measurement of the scattering matrices of two materials of different thicknesses. This means that results obtained by the extraction method are not dependant on the type of access, which is an advantage but also prevents the researcher from accessing impedance information.
[2.10]
Another method that is more rigorous but which requires more calculations enables us to access effective parameters ε and
