Met a Materials- A New Direction in Materials

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ISSN 10876596, Glass Physics and Chemistry, 2010, Vol. 36, No. 5, pp. 521553. Pleiades Publishing, Ltd., 2010.Original Russian Text A.A. Zhilin, M.P. Shepilov, 2010, published in Fizika i Khimiya Stekla.

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INTRODUCTION

In the last decade, metamaterialsartificial materials designed for the use in the filed of electromagnetism, including optical applicationshave beenintensively developed all over the world. Metamaterials have been engineered from structural elements

with the type and mutual arrangement that can bespecified during the fabrication. If the size of structuralelements in a metamaterial and the characteristic distance between them are considerably smaller than theradiation wavelength, the metamaterial can be considered as a continuous medium characterized by thepermittivity and the permeability. In this case, thestructural element of the metamaterial is an analog ofan atom, a molecule, or a unit cell of a conventionalmaterial.

Modern technologies, including nanotechnologies, and prospects of their development have openedup wide possibilities for producing metamaterials with

a controlled (preliminarily calculated) structure thatcan exhibit unusual or even unique electromagneticproperties.

It should be noted that the theory plays a key role inthe design of these metamaterials: initially, theoreticalprinciples have been used for choosing the type ofstructures that could ensure necessary electromagnetic properties, then, quantitative characteristics ofthe structure have been selected and optimized usingthe simulation of the electromagnetic response, and,

thereafter, the material has been fabricated. The preparation of metamaterials with controlled properties

without preliminary theoretical calculations seems tobe unrealistic.

Important milestones on the way of the development of this new direction in materials science were anidea of the creation of superlenses based on metamaterials (2000) and the experimental preparation of thenegative refractive index metamaterial for radiation

with a frequency of 10.5 GHz (2001). The latter facthas demonstrated the possibility of engineeringmetamaterials with properties that are not observed fornatural materials. At present, the works concerning anincrease in the operating frequency of negativerefractive index metamaterials have been continued.

Progress in the fabrication of metamaterials hasallowed one to raise the question as to the possibility ofproducing invisible objects. An intensive discussion ofthe invisibility problem, which was started in 2005, has

been carried out in two directions. In the firstapproach to the solution of this problem, the case inpoint is objects with sizes that are smaller than or of theorder of the electromagnetic radiation wavelength.The second approach considers the possibility of creating invisible cloaks that conceal the objects contained in them from an external observer. The cloaks inthe second approach should be completely preparedand the objects in the first approach should be partiallyfabricated from materials with unusual electromag

Metamaterials: A New Direction in Materials Science

A. A. Zhilin and M. P. Shepilov*

Scientific Research and Technological Institute of Optical Materials Science, AllRussia Science CenterS. I. Vavilov State Optical Institute, ul. Babushkina 36, St. Petersburg, 192171 Russia

*email: [email protected], [email protected]

Received February 10, 2010

AbstractA review is presented of materials science of metamaterials, i.e., the area of research that has beenextensively developed over the last decade. Metamaterials are artificial materials consisting of structural elements, whose type and mutual arrangement can be specified during the fabrication. The development ofmetamaterials was associated initially with the idea of the design of electromagnetic media with a negativerefractive index and, more recently, with the prospect for fabrication of superlenses, invisible objects, andother optical devices. As a result, there appeared a new branch of materials science and an intimately relatednew branch of opticstransformation optics. This review has discussed the main directions of research inthis field and problems encountered in this way. It is noted that metamaterials cannot be designed withoutinvoking the most modern tools for performing numerical simulation and that their implementation requiresthe use of highlevel technologies. The idea of metamaterials is important not only for optics and electromagnetism but also for acoustics.

Key words: metamaterials, negative refractive index, superlens, invisibility cloaks, transformation optics,nanostructuring

DOI: 10.1134/S1087659610050019


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netic properties and, generally speaking, can be produced only with the use of metamaterials.

The second approach to the solution of the invisibility problem is based on the transformation of coordinates (mapping method). In recent two or three

years, this method has been used for solving a numberof problems of theoretical optics. We can argue that anew direction in optics, i.e., the socalled transformation optics, has been formed to date. Devices that have

been theoretically developed in the framework of thisdirection also cannot be implemented without use ofmetamaterials.

This paper represents a review of the literature onthe problem of metamaterials and related problems. Itshould be noted that, in a number of works, the termsrefractive index, invisible, transparent, etc.have been used for electromagnetic radiation with a

wider spectral range as compared to the visible lightregion, namely, for radiation with wavelengths fromseveral centimeters (gigahertz frequencies) to severalhundred nanometers (visible light). We will use thesame extended version of the terms.

NEGATIVE REFRACTIVEINDEX METAMATERIALS

Hypothetical LeftHanded Media(Media with Negative Refractive Index)

A light ray passing from medium 1 to medium 2changes its direction: it is refracted. The refraction law(Snells law) relates the angle of incidence (the angle

between the normal to the interface and the incident

ray in medium 1) and the angle of refraction (theangle between the normal and the refracted ray inmedium 2) so that the incident and refracted rays lieon different sides of the normal to the interface [1]:

(1)

Here, ni

is the refractive index of the ith medium,which is expressed in terms of the permittivityi andpermeabilityi (as a rule, it is assumed that i = 1 inthe optical range of frequencies [2]; however, for thepurpose of further presentation, we will considerthe general case i 1). Figure 1 taken from [3] showsthe refraction of the ray at the interface between twomedia. The incident and refracted rays are designatedas 1 and 4, respectively, and it is assumed that themedia are transparent and that medium 2 is opticallydenser; i.e., n2 > n1 1.

Mandelstam [4] was first to consider and Veselago[3] thoroughly analyzed the case of the socalled neg

ative refraction. Veselago examined the propagation ofelectromagnetic waves in a hypothetical medium withnegative values of the permittivity < 0 and permeability < 0. He noted that, for a plane monochromatic

wave propagating in this medium, the electric fieldvector E, the magnetic field vector H, and the wavevector k form a lefthanded vector triple, whereasthese vectors in conventional (righthanded) mediaform a righthanded vector triple. These hypotheticalmedia were referred to by Veselago as the lefthandedmedia.

In the lefthanded medium, the energy flux densityvector (Poynting vector) and the group velocity vector(coinciding with the former vector in direction) are

antiparallel to the wave vector k and, hence, to thephase velocity vector [3]. It should be noted that consideration of a negative phase velocity goes back to the

works by Lamb [5] (in hydrodynamics) and Schuster[6] (in optics), as well as to the more detailed work byMandelstam [7] (solid state physics).

Veselago [3] showed that the Snells law (1) holdstrue for any combinations of righthanded and lefthanded media in the use of the generalized definitionof the refractive index:

(2)

i.e., under the assumption that the refractive indices oflefthanded media are negative. It should be notedthat, in media with < 0 and > 0 or > 0 and < 0,

waves attenuate rapidly [8].

In passage from the conventional (righthanded)medium with the refractive index n1 1 to the lefthanded medium with the refractive indexn2 < 0, theincident and refracted rays lie on the same side of thenormal to the interface (Fig. 1, rays 1, 3) in contrast to

sinsin

n2

n1, ni( ii i, 1 2 ).,= = =

n1 pi ii, pi 1 i 0 i 0>,>( ),= =

pi 1 i 0 i 0


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METAMATERIALS: A NEW DIRECTION IN MATERIALS SCIENCE 523

the case of conventional refraction (rays 1, 4). In theconstruction of Fig. 1, it was assumed that |n2| > n1.

Veselago [3] noted that, if light passes from themedium with 1 > 0 and 1 > 0 to the medium with 2 =1 and 2 = 1, the reflected ray is absent. The case

where the first medium is vacuum (1 = 1 and 1 = 1)and 2 = 2 = 1, i.e., the refractive index of the second medium is n = 1, is most interesting. This case is

associated with one of the possible applications ofmedia with a negative refractive index. Simple geometric considerations demonstrate [3] that, in vacuum, a plate from a lefthanded material with a thickness dand refractive indexn = 1 focuses radiation ofa point source located at a distance l< dfrom the plateat a point. In other words, this plate is a lens in whichthe image formation is clearly illustrated in Fig. 2taken from [9]. As was theoretically showed by Pendry[10], this lens is perfect (superlens) in the sense thatits use makes it possible to overcome the diffractionresolution limit associated with the wave nature oflight. The Pendry idea of the superlens, in addition to

the prediction of the structure and fabrication of negative refractive index metamaterials (see below),resulted in a sharp intensification of investigations inthis field. However, the superlens concept came undercriticism [1113]. The answers to the questions formulated in [1113] were given in [1417], where thesuperlens concept and its limitations were refined. Thesuperlens was numerically simulated in [18, 19].Moreover, the problem regarding the superlens wasalso discussed in [8, 9, 20, 21]. The possibility of overcoming the diffraction resolution limit with the use ofa planeparallel plate with a negative refractive index(superlens) was first experimentally proved for photonic crystals in [22] and metamaterials in [23].

Finally, reasoning from the requirement of a positive field energy, Veselago [3] made the inferencethat, first, the values of and can be simultaneouslynegative only in the presence of a frequency dispersion and, second, the frequency dependences of thepermittivity and permeability should satisfy the conditions

(3)

where is the angular frequency.

Here, it is pertinent to note one more limitation. As

was noted in [8], there are no fundamental objectionsagainst the negativity of the real parts of the quantities and . At the same time, the absorbed energy in athermodynamically equilibrium medium should bepositively definite, which leads to negative values ofthe imaginary parts of the quantities and . In a thermodynamically nonequilibrium medium (gainmedium), this requirement fails.

In conclusion of this section, we note the reviews[24, 25] that contain interesting information on the

( )

0, ( )

0,>>

history of the appearance of ideas associated with thelefthanded media.

Development of Samplesof Negative Refractive Index Metamaterials

The permittivity and permeability characterizea macroscopic response to applied electric and magnetic fields. These are macroscopic parameters thatare determined in the course of macroscopic measurements as a result of the averaging over fairly large timeintervals and spatial volumes. For conventional materials, the characteristic scale of the spatial averaging

should considerably exceed the scale of the materialinhomogeneity, which is determined by the size of thevolume per atom (molecule) or the unit cell size in thecase of crystalline solids.

Now, we explain why it is impossible to produceconventional materials with a negative refractive indexin the nearIR and visible spectral ranges [8]. Theappearance of negative values of the permittivity andpermeability is associated with the resonance phenomena in the response of the medium to the electromagnetic action. As in the case of any resonance, theresponse follows in phase with the applied field at frequencies below the resonant frequency. In some rangeof frequencies exceeding the resonant frequency, theresponse follows in antiphase with the applied field,

which provides the possibility of observing the negativepermittivity or permeability if the resonance is sufficiently sharp. There are conventional materials characterized by a strong electric response at almost anyspecified frequency in the range from radio frequencies to frequencies of UV radiation. However, themagnetic response in the majority of conventionalmaterials is limited by low frequencies of the micro

wave range. Therefore, the permeability for these

Fig. 2. Image formation by a planeparallel plate producedfrom a material with the refractive indexn = 1 [9].


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materials almost does not differ from unity already atfrequencies substantially lower than the optical frequencies [2, p. 374]. The magnetic polarization, as arule, is caused either by unpaired electron spins ororbital electron currents, and collective excitations ofthis type usually occur at low frequencies [8]. Someferromagnetic, ferrimagnetic, and antiferromagneticmaterials exhibit a specific magnetic activity at frequencies up to several hundred gigahertz [2631].However, these materials are rare and have narrow

bands of the magnetic activity. At the same time, thematerials artificially structured on microscales andnanoscales, i.e., metamaterials, can have a magneticactivity at almost any frequencies in the range fromseveral hertz [32] to frequencies of the nearIR range[33], including radio frequencies [34, 35], microwavefrequencies [36, 37], and frequencies from severalunits [38] to several hundred terahertz [39]. In regardto optical magnetism, it was noted in [40] that, atpresent, its implementation [41, 42] is accompanied

by a high level of losses.

In this respect, artificial materials of the new type,i.e., metamaterials, have been treated in the lastdecade as promising materials for the observation ofthe negative refractive index; in this case, the responseto the electromagnetic field is determined by the

behavior of structural elements forming the metamaterial and the size of the structural elements significantly exceeds atomic sizes. In order for the notions ofthe permittivity and permeability can be applied tometamaterials, it is necessary to assume that the size ofstructural elements of the metamaterial should beconsiderably smaller than the radiation wavelength. Itis these materials characterized by a negative refractive

index that below will be called the negative refractiveindex metamaterials in contrast to photonic crystalline structures in which there can occurs a negativerefraction but the radiation wavelength is of the orderof the lattice constant of the photonic crystal (i.e., thenotions of the permittivity and permeability in a con

ventional sense are inapplicable to consideration ofnegative refraction in photonic crystals) [8].

Extensive investigations of metamaterials tracesback to the Pendry ideas of structured artificial materials that can have a negative permittivity [43, 44] or anegative permeability [36].

The permittivity of metals is determined by theresponse of a freeelectron gas that is characteristic ofa plasma and is negative in the UV spectral range [8].These materials can be characterized by numerousresonance states localized on their surface and referredto as surface plasmons [45, 46]. An analysis demonstrates [8, 46] that surface plasmons cannot be excitedon a perfect flat surface with the use of propagatinglight modes; i.e., in order to excite these plasmons, it is

necessary to provide coupling mechanisms, such as thesurface roughness, lattice structures, or dielectric connection to vacuum (hemisphere, prism). The interaction of surface plasmons with radiation on structuredsurfaces offers new possibilities in plasmonics.

In order to obtain a negative permittivity in thegigahertz spectral range, Pendry et al. [43, 44] proposed to use twodimensional and threedimensionalstructures consisting of cylindrical metal wires. In thethreedimensional structure, the wires form edges of asimple cubic lattice. In the twodimensional structure,the parallel wires are arranged so that their cross sections by a perpendicular plane form a square lattice

(Fig. 3). The frequency dependences of the permittivity were analytically calculated and numerically simulated [43, 44] and measured experimentally [44] forthese structures. The corresponding results allowedone to make the inference that the frequency of plasmons in these systems lies in the gigahertz range for a

wire diameter of the order of a micron and a characteristic interwire distance of the order of several millimeters. For example, for the twodimensional systemshown in Fig. 3, the analytical expression for the frequency of the longitudinal plasmon mode p (theelectric field vector is directed along the wires) has theform

(4)

where ris the wire diameter,a is the constant of the twodimensional lattice (see Fig. 3), and c is the speed oflight in free space. Forr= 1 m and a = 5 mm, relationship (5) gives p = 8.2 GHz, which is in good agreement

with the frequencyp = 8.3 GHz determined by thenumerical simulation [44]. The wavelength of this radiation is p 4 cm, and the condition for the applicability

p2 2c2

a2

a/r( )ln,=

Fig. 3. Twodimensional system of metal cylinders [43, 44]for which the frequency of the longitudinal plasmon mode(expression (4)) is determined by the geometric parameters of the structure.


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of the effectivemedium approximation pa is fulfilled. The example of the experimental implementa

tion of the system of this type at higher frequencies isdescribed in [47]. In this case, r = 15 m and a =120 m, and the frequency p = 0.7 THz calculatedfrom relationship (4) agrees well with the observed frequency and the condition for the applicability of theeffectivemedium approximation is also satisfied(p/a 3.6). Therefore, a change in the geometricparameters of this structured system makes it possible tocontrol the position of the spectral range of the negativepermittivity.

A negative permeability in the gigahertz frequencyrange can be obtained in a structured system that is

built up from nonmagnetic conductors, i.e., resona

tors each of which consists of two planar split rings[36]. Two possible variants of the splitring resonatorare depicted in Fig. 4. The alternating magnetic fielddirected perpendicular to the ring plane excites theelectric current in the rings, which are inductancecapacitance elements: the rings themselves are inductances and the gap between the rings and cuts in therings form capacitors. The current excitation has a resonant character, and the resonant frequency of this LCresonator depends on the geometric parameters.

Initially, Pendry et al. [36] considered a system ofidentical cylindrical resonators with parallel axes. Thepoints of intersection of the resonator axes with the

plane perpendicular to these axes form a square latticewith the period a. Each resonator consists of two conducting coaxial cylindrical layers that have differentdiameters and are cut along the lines parallel to theaxis so that the cross section of the resonator by theplane is a splitring resonator shown in Fig. 4a. Anapproximate formula was derived for the permeabilityof this structured system when the magnetic field isoriented along the resonator axes under the assumption that the gap between the rings d(Fig. 4a) is small

as compared to the ring radius r. The effective permeability has the resonance form

(5)

where

(6)

Here, a is the period in the arrangement of resonators;0 and 0 are the permittivity and permeability of freespace, respectively; is the relative permittivity of thecompound in the gap between rings (in [36], = 1);and is the resistance of the cylindrical resonator per

unit square of its surface. The factorfis the filling factor of the material.

The dependence eff() (relationship (5)) has aresonant character with the resonant frequency0; inthis case, the resonance is associated with the inductancecapacitance resonance in the system. For asmall attenuation (i.e., in the case of a low resistivity

when 0), the response does not coincide in phasewith the applied magnetic field at frequencies > 0 andthe quantityeff is negative in the range m > > 0,

where m is the magnetic plasma frequency, which, bydisregarding the resistivity of the material, can be represented in the form [36]

(7)

Formula (7) indicates that the filling factor plays adecisive role in the width of the range m0 in whicheff () < 0. The inclusion of the resistivity of thematerial broadens the resonance, and, in the case ofthe material with a high resistivity, the resonance issuppressed so that the range of negative values of thereal part of the permeability eff() disappears [8].

ef f ( ) 1f2

02 2 i

,+=

r2/a2, 03d

002

r3

1/2, 2

0r.= = =

m 3d1 f( )00

2r

3

1/2.=

Fig. 4. Two possible variants of the splitring resonator (see text for explanation).


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This statement is illustrated by Fig. 5. This figure wasconstructed with the geometric parameters used forthe structure in [36, relationships (26)]. For small val

ues of ( 1 ), the system is characterized by thenegative real part of the permeability in the frequencyrange 2.95 GHz < = /(2) < 4.16 GHz with thelimits determined by the values of0 (expression (6)) andm (expression (7)). This range narrows with an increasein the resistivity in the range 1 < < 6.8 . Finally,at > 6.8 , the real part of the permeability is positivefor any frequencies ( = 7 for curve 3in Fig. 5). Itshould also be noted that, in the frequency range 6.6 cm; i.e., it is larger than the period a = 5 mm

by a factor of more than ten, and, hence, the conditionfor the effectivemedium approximation is satisfied.

This system of cylindrical resonators does notexhibit a magnetic activity when the magnetic field isdirected perpendicular to the axes of the cylinders;i.e., the metamaterial is uniaxial. In order to overcomethis limitation, Pendry et al. [36] proposed to replacethe cylindrical resonator under consideration by a setof planar ring resonators (Fig. 4). The centers of thering resonators are equidistantly arranged on the axis

with the period l, their planes are perpendicular to theaxis, and the ring orientation is identical and corresponds to the orientation of the cross sections of the

cylindrical resonator. By replacing the cylindrical resonators in the above system by sets of ring resonatorsand assuming that l< r, the authors derived the relationship for the permeability of this system in thedirection of the axes and demonstrated that the permeability can have negative values. The authors alsonoted that the continuous electrical path that ischaracteristic of cylindrical resonators is absent in thesystem of ring resonators. Like the initial system ofcylindrical resonators, the obtained system of ring resonators is uniaxial; however, the use of ring resonatorsallows one to construct the isotropic metamaterial, for

which the unit cell is shown in Fig. 6 [36]. It should be

noted that the equalityl= a is valid for this metamaterial and the inequalityl< rused to derive the formulafor the permeability does not hold true, because thering diameter should be smaller than the lattice spacing; i.e., r< a/2. Other inequalities used for derivingthe formula can also be invalid in real systems. However, the authors believe that the nonfulfillment of theinequalities has an effect only on the accuracy of thederived relationships rather than on the functional possibilities of the structures under consideration. Additional details of the electromagnetic response of splitring resonators were theoretically investigated in [48].

The use of splitring resonators made it possible to

fabricate metamaterials with the magnetic activity atfrequencies of approximately 1 GHz [49], 5 GHz [37],10 GHz [5052], 100 GHz [53], 1 THz [38], and30 THz [54]. The magnetic resonance at higher frequencies of100 THz [39] was observed for systems ofsingle rectangular rings (in contrast to double ringsshown in Fig. 4) with sizes of 320 nm. For a similarsystem containing rings with a smaller size (200 nm),the magnetic resonance was observed at frequencies of200 THz ( = 1.5 m) in the case where the light

5

0

5

432

10

5

0432

123

123

Frequency, GHz

Im(eff)

Re(eff)

(a)

(b)

Fig. 5. Frequency dependences of the (a) real and (b) imaginary parts of the permeability eff() (relationship (5)).The calculations were performed for r = 2 mm, d =0.1 mm, a = 5 mm, = 1, and three values of the resistivity = (1) 0.2, (2) 1.0, and (3) 7.0 .

Fig. 6. Unit cell of the isotropic metamaterial fabricatedfrom splitring resonators [36].


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wave vector was directed along the normal to the ringplane and at frequencies of370 THz ( = 800 nm) inthe case of oblique incidence of light [55]. The minimum permeability was observed at the wavelength =1.67 nm and was = 0.25. It should also be noted thatthe structures characterized by the magnetic activity inthe radio frequency range were fabricated using morecomplex elements, i.e., the socalled Swiss rolls [35],

which were previously studied theoretically in [36].

The first metamaterials with negative values of thepermittivity and permeability for radiation with a specific frequency, i.e., negative refractive index metamaterials, were prepared by combining the structure ofparallel wires (Fig. 3) with periodic arrays of splitringresonators (Fig. 4). The simplest metamaterial of thistype (Fig. 7) was proposed and experimentally investigated by Smith et al. [37]. The authors made the inference that the metamaterial is characterized by negative values of the permittivity and permeability withrespect to radiation with a specific frequency andpolarization (see the caption of Fig. 7). However, in

view of its structure this metamaterial is substantiallyanisotropic.

More recently, Shelby et al. [50] proposed themetamaterial that was isotropic in two dimensions andconsisted of copper splitring resonators and copper

wires (strips) (Fig. 8). The unit cell size a was equal to5 mm. The array of parallel wires (vertical wires inFig. 8) provides a negative permittivity in the gigahertzfrequency range in the case where the electric field isdirected along the wires. The array of splitring resonators ensures a negative magnetic permeability in thecase where the magnetic field vector lies in the plane

perpendicular to the wires (the horizontal plane inFig. 8), so that the permeability in this plane does notdepend on the direction of the magnetic field vector inthe framework of the effectivemedium approximation (twodimensional isotropy of the system). Consequently, this metamaterial should have a negativerefractive index for electromagnetic radiation with the

wave vector oriented perpendicular to the wires (or, toput it differently, radiation polarized parallel to the

wires), and, for this orientation, the refractive index istwodimensional isotropic. By using the prism fromthis metamaterial, Shelby et al. [50] thoroughly investigated the refraction of radiation at the frequency =10.5 GHz (the wavelength 3 cm is six times largerthan the lattice constant a) and with the wave vectororiented perpendicular to the wire direction and led tothe conclusion that the Snells law (1) holds true andthe refractive index of the metamaterial is

n = 2.7 0.1. (8)

Reasoning from the results of the experimental investigations of the frequency dependence of the refractive index [50, Fig. 4], the authors assumed that the

refractive index is negative in the frequency rangefrom approximately 10.2 to 10.8 GHz; i.e., in thisrange, both the permittivity and the permeability arenegative.

The metamaterial studied in [51] differs from themetamaterial investigated in [37] (see Fig. 7) by thepresence of the additional wire (in the unit cell) withthe same direction as that shown in Fig. 7, the squareshape of rings, and sizes of elements. In [51], theexperimental investigation and calculations were per

E

k

H

E

k

H

Fig. 7. Metamaterial [37] for which the permittivity and permeability are negative for radiation with a frequency of5 GHz, theelectric vector parallel to wires, the magnetic vector perpendicular to the planes of ring resonators, and the wave vector parallel tothe ring plane and perpendicular to wires. Shown is the righthanded vector triple (E, H, k) for the incident wave (waves in empty

space). In the medium, the vectors have opposite directions and form the lefthanded triple. The lattice constant is a = 8 mm, thewire radius is 0.8 mm, the inner radius of the inner ring is r= 1.5 mm, the ring width is w = 0.8 mm, and the distance between therings is d= 0.2 mm (see designations in Fig. 4a).


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formed for radiation with a specified orientation of the

wave vector and the electric and magnetic field vectors, which is shown in Fig. 7 (correspondingly, theauthors characterize the metamaterial as onedimensional). The refraction experiments showed that, forthe radiation with the above orientation, the refractiveindex in the frequency range 12.613.2 GHz is negative and the real part of the refractive index at a frequency of 12.6 GHz is equal to 1.05. The experimental values of the refractive index are in good agreement

with the values obtained by numerically solving theMaxwells equations. It should be noted that the inference made in [51] regarding the negative refractiveindex is based on the direct experiments on the refraction, whereas Smith et al. [37] used the indirect data

on the transmission. Gokkavas et al. [53] fabricatedthe metamaterial of the same type but with three wiresin the unit cell with the refractive indexn 5 in thefrequency range 98104 GHz.

Planar structures in which the unit cells have a sizeof approximately 100 m and consist of a splitringresonator (Fig. 4a) and a wire piece were prepared byMoser et al. [56]. The authors studied the transmissionspectra and made the inference that these structuresare electromagnetic metamaterials for radiation with afrequency of the order of 1 THz. The properties ofthese structures were not investigated in more detail.

We do not know works in which double or single

splitring resonators were used for fabricating negativerefractive index metamaterials at frequencies higherthan those achieved in [53].

However, there exists one more approach to thedesign of metamaterials characterized by a magneticactivity with the use of localized plasmon resonances

without classical splitring resonators [57]. Let us consider this approach. Lagarkov and Sarychev [58] notedthat a pair of noble metal nanorods can exhibit a largeparamagnetic response, and Podolskiy et al. [59] the

oretically demonstrated that this pair can also be char

acterized by a magnetic response at a wavelength of1.5 m. The fact that a set of these pairs can have anegative permeability at a wavelength of1.5 m wasexperimentally confirmed in [60]. In [59], it was predicted for the first time that this pair of rods can have anegative value of the real part of the refractive index inthe visible spectral range, and the problem under consideration was discussed in more detail in [6163]. Forexample, the calculations showed that a pair of parallelcylindrical gold rods 162 nm in length, 32 nm in diameter, and with a distance of 80 nm between the axes hasnegative electric and magnetic responses in the wavelength range 500600 nm in the case where the electric vector of the incident plane wave is directed alongthe rods and the magnetic vector is perpendicular tothe plane of the rods. The qualitative aspects of theappearance of resonances in this system were discussed by Klar et al. [57].

The system consisting of pairs of gold nanorods wasexperimentally investigated by Shalaev et al. [64]. Thetwodimensional regular array of pairs of nanorods ona glass substrate was prepared using electronbeamlithography, so that the plane containing a pair of rods

was perpendicular to the substrate. The sample, itspreparation, and the technique for measuring therefractive index are described in detail in [42, 65]. In

[64], the inference was made that, for light that is incident perpendicular to the substrate and polarizedidentically to that described in the above calculations[63] (see the previous paragraph), the refractive indexof the metamaterial is negative in the wavelength range1.41.6 m. At a telecommunication wavelength of1.5 m, the refractive index was equal to 0.3 0.1.The results of the measurements of the spectral dependence of the refractive index are in agreement with theresults of the performed numerical simulation.

z

yx

(a) (b)

Fig. 8. (a) Twodimensional metamaterial [50] characterized by the negative refractive index n = 2.7 0.1 at a frequency of10.5 GHz and (b) its fragment. The lattice constant a is 5 mm.


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However, here, we should note the following. In[64], the refractive index was determined from the dataon the complex reflection and transmission coefficients of a thin metamaterial layer (i.e., by the indirectmethod [66]) in the case where the light flux was perpendicular to the substrate on which pairs of nanorods

were located. Therefore, the radiation passes throughthe metamaterial that is not a bulk layer. This layer can

be treated as monomolecular because a pair of nanorods is a molecule of the metamaterial. In thiscase, the losses, which can be evaluated from theexperimental data on the transmission and reflection[64, Fig. 2a] are no less than 15% in the spectral range1.41.6 m, in which the real part of the refractiveindex is negative. For example, the losses at a wavelength of 1.5 m are more than 30%. This means thatthe bulk material under consideration would beopaque and the possibility of directly measuring itsrefractive index seems to be problematic. This corresponds to large values of the imaginary part of thecomplex refractive index; for example, n 0.3 + i2.7at a wavelength of 1.5 m [64, Fig. 3b].

Grigorenko et al. [67] fabricated systems consistingof pairs of gold nanodots (objects with sizes ofapproximately 100 nm) on a substrate and investigatedtheir interaction with polarized light incident perpendicular to the substrate. Systems of this type also represent a monomolecular layer of the metamaterial.The authors established that these systems can havenegative values of the real part of the permeability inthe visible spectral range. Moreover, they can havesimultaneously negative values of the real parts of thepermittivity and permeability; for example, in one ofthe cases under consideration, these values were estimated to be Re() 0.7 and Re() 0.3 at some

wavelength in the green range [67, p. 20]. Nonetheless, the authors noted that the negative refractioncannot be observed as a result of the large values of theimaginary parts (Im() 1i).

By using the results of the numerical simulation,Zhang et al. [68] predicted and prepared [69, 70] onemore type of metamaterials with a negative refractiveindex in the IR spectral range. The authors considereda threelayer planar structure (Fig. 9) with a system ofholes that form a twodimensional lattice (fishnetstructure). The central layer is a dielectric (Al2O3)coated by gold films on both sides. The thickness of thedielectric layer was equal to 60 [68, 69] and 75 nm[70], and the thickness of each coating was 30 nm. Thesizes and the period in the arrangement of holes were ofthe order of several hundred nanometers. The calculations [68] carried out for four structures with rectangular holes and characteristic sizes ax= ay = 801 nm, dx=500 nm, and dy = 100, 200, 300, and 500 nm (seeFig. 9) demonstrated that these structures are characterized by the negative refractive index for the radiation with a wavelength of approximately 2 m, whichis incident perpendicular to the layer and has a polarization shown in Fig. 9. According to the calculations,

the figure of merit that is defined as the ratio (takenwith the opposite sign) between the real (negative)and imaginary parts of the refractive index of themetamaterial

= Re(n)/Im(n) (9)

can reach values of 26 for the structures under consideration, which significantly exceed 0.1 obtainedin [64] (see above). For example, the maximum valueof = 6 was observed at 1.9 m for two structures

with sizes dy = 100 and 200 nm.

The experimentally fabricated threelayer fishnetstructure with circular holes [69] has a refractive indexn 2.5 + i4 ( 0.6) for the radiation with a wavelength of approximately 2 m, which is incident normal to the layer and has a particular polarization. As

was noted in [70], this structure is characterized by alow transmission (


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studied in [64] and [6870] are inverse with respect toeach other: instead of pairs of metal rods in one structure [64], pairs of holes in two metal films are used inother structures [6870], which illustrates the use ofthe Babinet principle in the development of metamaterials [71].

Dolling et al. [72] fabricated the metamaterial withthe geometric structure similar to the structure of themetamaterial theoretically considered in [68] (seeFig. 9). The central layer was prepared from the MgF2dielectric with the thickness s = 30 nm, the coatings

with the thickness t= 45 nm were produced from silver, and the system of holes was characterized by thesizes ax= ay = 600 nm, dx= 316 nm, and dy = 100 nm.In [72], the authors noted that the attenuation for sil

ver thin films in a wavelength range of1.5 m is fourtimes lower than that for gold films. The prepared fishnetmaterial is characterized by a negative refractive indexand a low level of losses in the spectral range 1.351.45 m; in particular, the maximum figure of merit 3 (relationship (9)) is observed at a wavelength ofapproximately 1.4 m for Re(n) 1. A decrease inthe characteristic sizes of the structure of this type to

s = 17 nm, t= 40 nm, ax= ay = 300 nm, dx= 102 nm,and dy = 68 nm [73] made it possible to decrease the

operating wavelength of the metamaterial to 780 nm(Re(n) 0.6); however, the loss level in this case wasmore significant ( 0.5). Finally, Dolling et al. [74]experimentally demonstrated that a similar metamaterial can be simultaneously characterized by negative

values of group and phase velocities.

Zhang et al. [75] calculated the electromagneticproperties of bulk metamaterials in the form of stacksof threelayer fishnet structures for polarized radiation

with a specified orientation (Fig. 10) with the wave

length in the range 1.52.2 m. The thickness of theair gap between structures was equal to 5 nm, and theparameters of the threelayer structures (the dielectricthickness is equal to 60 nm, the thickness of gold filmsamounts to 30 nm, ax = ay = 801 nm, dx = 500 nm,dy = 200 nm) (see Fig. 9) were chosen to be identicalto those in the optimum variant of the threelayerstructure considered in [68]. In [75], the calculations

were performed for the stacks consisting of 1, 2, 5, 6,100, and 200 threelayer fishnet structures. The unitcell size for this metamaterial in the direction of propagation of radiation (the thickness of the threedimensional structure plus the thickness of the air gap) isequal to 125 nm; i.e., this size is substantially smallerthan the radiation wavelength under consideration,

whereas the thickness of the stacks consisting of 100and 200 structures considerably exceeds the wavelength. The calculations showed that, for one threelayer fishnet structure, the real part of the refractiveindex is negative in the wavelength range 1.772.18 m.The changeover to the stack consisting of two and morestructures leads to a shift in the range of negative values

toward the shortwavelength range (1.52.0 m). Inthis case, the results of calculations rapidly converge

with respect to the number of structures in the stack,so that the calculated values of the real and imaginaryparts of the refractive index are almost identical for thestacks containing five and ten structures. The imaginary parts of the refractive index are small (smallerthan 0.1 in the range 1.52.0 m), whereas the realparts of the refractive index noticeably exceed in magnitude the values obtained for the single structure. Asa result, the figure of merit of the stack (i.e., the bulkmaterial) is considerably larger than that of the singlestructure [68] and reaches values of 25 in the wavelength range of approximately 1.7 m. Therefore,Zhang et al. [75] theoretically showed that the stacksof fishnet structures are promising metamaterials forthe observation of the negative refractive index in theIR spectral range.

Moreover, Zhang et al. [75] also noted that theremoval of air gaps between the structures in the stackdepicted in Fig. 10 does not lead to a substantialchange in the result of the calculations. In this case,the metamaterial is a layered structure in which metaland dielectric layers alternate with each other andthere is a regular system of holes with axes perpendicular to the layer planes. The bulk material of this type(Fig. 11) with the negative refractive index was

described in [76]. The authors prepared the layeredstructure in which silver layers 30 nm thick alternated

with magnesium fluoride (MgF2) layers 50 nm thick.The initial sample consisted of 21 layers. Focused ionetching was used to produce a regular system of rectangular holes with the period ax= ayp = 860 nm andsizes dxa = 565 nm and dyb = 265 nm. It should benoted that, in view of the specificity of the procedurefor producing holes, their size decreased with anincrease in the depth of the material (Fig. 11b). The

Fig. 10. Images of a fragment of the metamaterial in theform of a stack of threelayer fishnet structures [75].


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sample was used to prepare the prism by treating oneof its faces, and the angle between the untreated andtreated faces (the base angle of the prism) is designatedas (Fig. 12). Two prisms with the angles = 5.0 and4.7 were prepared. Light was incident perpendicularto the untreated face (Fig. 12b) and does not change itsdirection in the metamaterial, i.e., propagated according to the scheme shown in Fig. 10. In this case, theangle of incidence on the treated face of the prisminside the metamaterial was equal to the angle . The

measurement of the angle of refraction for theescape from the metamaterial (Fig. 12b) allowed oneto calculate the refractive index of the metamaterialdirectly from the Snells law (1). If the refractive indexof the metamaterial is negative, the refracted ray islocated at the right of the normal to the surface shown

by the dashed line in Fig. 12b. The measurements performed in the spectral range 1.2001.775 m demonstrated that, in this range, the real part of the refractiveindex monotonically decreases with an increase in the

wavelength from n = 0.63 0.05 at = 1.2 m to n =1.23 0.34 at = 1.775 m; in this case, n < 0 for

wavelengths from 1.540 to 1.775 m (the range withlonger wavelengths was not studied in the work). Inthis case, the authors noted that the period of themetamaterial in the direction of radiation propagation(the sum of the thicknesses of the silver and magnesium fluoride layers) is equal to 80 nm, i.e., considerably smaller than the radiation wavelength. The resultsof the experimental determination of the real part of

the refractive index are in satisfactory agreement withthe results of the simulation. However, the experimental and theoretical evaluations of the figure of merit differ substantially. At > 1.54 m, the experimentallydetermined values of increase monotonically withan increase in the wavelength and the maximum valueof 3.5 is achieved for the longest wavelength =1.775 m under investigation. The calculated dependence () exhibits a maximum 18 at 1.72 m,and, at = 1.775 m, the simulation leads to 17.

(a)

1 m

ab

p

AgMgF2

ba

p

(b)

Fig. 11. Metamaterial in the form of a layered structure inwhich silver layers (light) alternate with magnesium fluoride (MgF2) layers (dark) with a system of holes [76]:(a) schematic drawing and (b) scanning electron microscope image of the real structure.

5 m

(a)

(b)

f2n = 1

n < 0

Prism

Quartz

Fig. 12. (a) Prism prepared from the layered metamaterialand (b) scheme of experiments on the study of refraction ofIR radiation (according to [76]).

5 m


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The authors explained this substantial discrepancybetween the experimental and theoretical values of by the imperfection of the prepared metamaterial.However, they noted that the obtained value of 3.5is among the best experimentally determined valuesand express their hope that an improvement of theprocedure for fabricating the metamaterial will allowone to approach the theoretical value of 20.

It should be noted that, not long before the publishing of the work by Valentine et al. [76], the prismmade from the metamaterial that represents a stack offishnet structures was used by NavarroCia et al. [77]to experimentally demonstrate the negative refractiveindex for the radiation at a frequency of approximately55 GHz ( 5.5 mm). Fifteen metal plates 0.5 mmthick with a regular system of circular holes 2.5 mm indiameter were arranged in a stack not densely but withair gaps of 1 mm, which can be considered an analog ofdielectric interlayers investigated in [76]. The system ofholes in the plate [77] represented a rectangular planarlattice with the periods ax= 3 mm and ay = 5 mm. The

prism prepared from this metamaterial with the angle = 26.6 was used in the scheme for measuring therefractive index, which is similar to the schemedepicted in Fig. 12b. The results of the measurementsagree well with the results of the calculations. It wasshown that the refractive index is close to zero at thefrequencyf 58 GHz, decreases with a decrease in thefrequency, and is equal to 1 atf 53.5 GHz. Therefore, the authors in [77] presented the bulk metamaterial with the negative refractive index; however, like theother bulk metamaterial of this type [76], it is anisotropic and characterized by the obtained refractive indexonly for a specific direction of propagation and a particular polarization of radiation.

The negative refractive index metamaterials in the visible spectral range were prepared by Lezec et al.[78]. The authors designed and fabricated the planarmetamaterial based on nanosized metaldielectric

waveguide structures. They directly observed the isotropic negative refraction in two dimensions over a

wide range 50 nm in the vicinity of the wavelength 500 nm. For example, the produced material at the wavelength = 514 nm was characterized bythe refractive indexn 5. The figure of merit wasestimated to be high ( 4). The authors suggestedthat blocks from this metamaterial can be used fordesigning metamaterials characterized by negative

refraction in three dimensions. As far as we know, the problem of fabrication of

bulk isotropic metamaterials with a negative refractiveindex has remained unsolved to date. Molecules ofthis metamaterial should form a macroscopicallyhomogeneous and isotropic system, for example, belocated at sites of a threedimensional cubic lattice.Systems of this type that consist of fourfold split conducting rings and crossing wires and are characterized

by a negative refractive index were theoretically con

sidered in [79]. More complex metal structures withcubic symmetry that are intended for observing a negative refractive index in the IR spectral range were calculated in [80]. Finally, as was proposed in [78] andnoted above, bulk isotropic metamaterials with a negative refractive index can be prepared from blocks oftwodimensional metamaterials based on waveguidestructures. However, the practical implementation of

the aforementioned and similar structures is a complex problem, especially in the nanometer range ofsizes.

From our point of view, the simplest way of fabricating bulk isotropic metamaterials with a negativerefractive index consists in producing twophase systems with a random or regular distribution of phases,

where one of the phases provides a negative electricresponse and the other phase ensures a negative magnetic response in a specified frequency range [81, 82].

Kussow et al. [81] theoretically investigated themetamaterial that consists of monodisperse sphericalparticles randomly arranged in a matrix (or regularly

located at sites of a simple cubic lattice). The authorstreated the polycrystalline magnesium diboride MgB2as one of the phases (matrix) and the polaritonic material SiC as the second phase (particles). The interaction of the system of particles with electromagneticradiation, which is described in terms of the Mie theory [83] and the effectivemedium approximation in aparticular form (see, for example, [81, 84]), has a resonant character, and there exists a frequency range in

which the effective permeability of the metamaterial isnegative. As was demonstrated in [81], the plasmonresponse of the matrix provides a negative permittivityin another frequency range. By properly choosing the

volume fraction (f 0.3) and the particle radius (r100 nm), the authors in [81] succeeded in theoreticallypredicting that these ranges can be partially overlap;i.e., they predict the possibility of designing themetamaterial of the above type with a negative refractiveindex in the visible spectral range. The figure of meritdepends on the material parameters used in the calculations and reaches rather large values of 210. In

view of its structure (polycrystalline matrix, sphericalparticles with a random arrangement or arrangementat sites of the cubic lattice), the metamaterial of thistype is a bulk isotropic optical material.

In [82], the authors considered a random mixture ofspherical particles of two types, namely, SiC particlesthat provide a negative effective permeability of themixture and gold particles that ensure a negative permittivity. The results of the calculations performed forthe particles with radii rSiC = 65 nm and rAu = 10 nm and

volume fractionsfSiC = 0.3 andfAu = 0.46 demonstratedthat, in the wavelength range with a width of 2040 nm near a wavelength of 370 nm, this bulk isotropicmetamaterial should have a negative refractive indexn = 14 and be characterized by high values of thefigure of merit 2.5. It should be noted that the sys


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tems consisting of spherical particles of two types(dielectric and metallic) were previously investigatedtheoretically [84] under conditions that particles ofeach type form a simple cubic lattice. The fulfillmentof this condition in practice seems to be complicated.

Therefore, at present, the problem associated withthe design of bulk isotropic metamaterials with a negative refractive index has been on the agenda. An

increase in the figure of merit of metamaterials is alsoan acute problem. In conclusion, we note that we didnot concern the methods for fabricating the aforementioned metamaterials. These methods are described inthe review [85].

Nowadays, the problems of materials science ofmetamaterials have gone far beyond the problemregarding the preparation of negative refractive indexmetamaterials. In particular, the problems consideredin the next section require the fabrication of spatiallyinhomogeneous anisotropic metamaterials with controlled electromagnetic properties. In a number ofcases, these metamaterials have been produced using

traditional techniques, such as the incorporation ofmetallic elements (for example, splitring resonators)into the unit cell of the metamaterial. In addition, nontraditional approaches have also been proposed fordesigning metamaterials. As an example, Smolyaninovet al. [86] proposed to emulate anisotropic metamaterials by waveguides with a variable cross section.

METAMATERIALS AND THE PROBLEMOF IMPLEMENTATIONOF INVISIBLE OBJECTS

A human imagination has long appealed to the ideaof the possibility of creating invisible objects. The casein point was both objects that do not perturb the lightpropagation (for example, invisible man) anddevices that cloak objects and make them invisible(cap of darkness).

The implementation of invisibility or lowobservability of objects has been the subject of extensive physical and engineering investigations for manydecades. In order to reduce the observability of anobject in scattered or reflected light, absorbing screens(see, for example, [87]) or antireflection films (see, forexample, [1, 88]) have been used in a number of applications.

The possibility of controllably changing the properties of metamaterials (see the previous section)resulted in the appearance of two approaches to thesolution of the problem of invisibility of objects. Thefirst approach [89] considers objects that have sizessmaller than or of the order of the wavelength and are

built up from conventional materials and metamaterials so that the action of this object on the propagationof the electromagnetic field would be minimum(transparent objects). According to the secondapproach [90, 91], metamaterials should be used for

fabricating a cloak that would bypass electromagneticradiation around an object and retain a constancy ofthe field outside the cloak.

Ideas of the Design of Transparent Objectswith Sizes Smaller than or of the Order

of the Wavelength

Al and Engheta [89] considered the possibility ofapplying plasmonic or metamaterial coatings in orderto considerably reduce radiation scattering fromspherical objects with diameters smaller than the radiation wavelength. It should be noted that this workmarked the beginning of an active discussion of thepossibility of using metamaterials to solve the problemof the design of invisible objects.

Before proceeding to the discussion of [89], wenote that some ideas proposed in this work were previously put forward by other authors. First, let us dwellon earlier works concerning the problem of invisibilityof objects without pretending to the completeness of a

review. Initially, these works were associated with theinvestigations of anomalous light scattering in glasses.

An analysis of investigations in the review [92]revealed an anomalous character of light scattering inphaseseparated glasses, i.e., glasses that undergophase separation: in a number of cases, the dependence of the scattering coefficient sc on the wavelength has the form sc8 (and differs from theRayleigh dependence sc 4) and the scatteringindicatrix is directed backward.

The qualitative explanation of the anomaly in theangular dependence of the scattered radiation at theearly stages of the diffusionlimited phase transforma

tion was proposed by Walker and Guinier [93]. It wasassumed that the average refractive index of the systemconsisting of a particle and a diffusion zone around itis equal or close to the refractive index of the matrix,

which decreases the scattering intensity in the limit ofsmall scattering angles (small scattering vectors orlarge wavelengths). For larger angles (smaller wavelengths), the interference of fluxes scattered by different volume elements leads to an increase in the scattering intensity. In other words, in the case where thelight wavelength considerably exceeds the particlesize, the observability of the composite particle (particle and diffusion shell) in the matrix should be significantly lower than that of a homogeneous particle.

The quantitative theoretical analysis of light scattering from a system consisting of a particle plus a shellin a matrix was performed in [94] (see also the review[92, formulas (12)(15)]). In [94], the author considered a homogeneous particle with the radius R1 and ahomogeneous surrounding concentric spherical layer

with the outer radius R2 in a homogeneous matrix. Itwas assumed that the permittivities of the particle 1,the spherical layer2, and the matrixm obey the inequality1 > m > 2 (or the inequality1 < m < 2). The


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use of the RayleighGans approximation and the balance condition (compensation condition) [92, 94]

(10)

for fairly long wavelengths ( > 4 R2) leads to thespectral dependence of the scattering coefficient sc8 and backward scattering indicatrix. This behavioris associated with the interference of waves scattered

by the particle and the shell, which results in adecrease in the light scattering intensity in the range oflong wavelengths due to the suppression of the dipolecomponent (4) in the scattering. In this case, thescattering coefficient for the particle in the shell is substantially smaller than that for the particle directlyembedded in the matrix. Therefore, the shell that satisfies the compensation condition (10) considerablydecreases the observability of the particle in thematrix. In the RayleighGans approximation, thecompensation condition is the condition of weakobservability and determines the values of the param

eters for which the effective refractive index of the particle in the shell is equal to the refractive index of thematrix (we do not assign a rigorous physical meaningto the notion of the effective refractive index).

Shatilov [94] used the simplest (rectangular) profile of the dependence of the permittivity (refractiveindex) on the distance between the particle center andthe point of observation. Goldstein [95] considered amore complex case when the refractive index in theshell is determined by the concentration profile of thecomponent diffusing to the particle or from the particle.The conclusions obtained in [95] for the character oflight scattering at early stages of phase separation are in

agreement with those obtained in [94].The RayleighGans approximation was used in

[94, 95]. One of the conditions for its applicability, i.e.,the closeness of the refractive indices of the matrix,particle, and shell [83, Chapter 6], frequently does nothold true. In this respect, the study of light scatteringfrom particles in shells was continued; in this case, theterm invisible was brought into use [96].

In the electrostatic approximation, Kerker [96]derived the relationship for the intensity of radiationscattered by a small nonabsorbing ellipsoid that consists of an ellipsoidal core and a confocal shell and isembedded in a nonabsorbing matrix. By using this

relationship, the author found the condition underwhich the scattered field disappears; i.e., the ellipsoid becomes invisible. In a particular case of a particleconsisting of a spherical core in a spherical shell, thecondition for the absence of the scattered field (thecondition for the invisibility of the particle) hasthe form

(11)

R13 1 2( ) R2

3 m 2( )=

m

R1

R2

3 m 2( ) 1 22+( )

m 22+( ) 1 2( )=

(see formulas (28), (25), and (26) in [96] or Section5.4 in the monograph [83]), where the used designations were introduced above. This relationship can beused in the cases where the field of the incident wavecan be considered uniform within the particle; i.e., ifthe particle is fairly small as compared to the wavelength. The quantitative condition of the smallnessfor the composite particle was not formulated in

[83, 96], whereas this condition for the nonabsorbinghomogeneous particle with the refractive indexn andthe radius R in the matrix with the refractive indexnmis reduced to the fulfillment of two inequalities (seeSection 5.2 in [83]): x 1 and mx 1, where x=2R/m = 2Rnm/ is the diffraction parameter of theparticle; m = n/nm is the relative refractive index; and and m are the light wavelengths in vacuum andmatrix material, respectively.

It should be noted that, when the permittivities ofthe particle, shell, and matrix are close to each other,formulas (10) and (11) lead to close ratios R1/R2. For asufficiently small particle characterized by a signifi

cant contrast of the permittivities of the particle, shell,and matrix, it is necessary to use condition (11), whichin actual fact is the condition for the vanishing of thepolarizability of the particle in the matrix (see Section5.2 in [83]).

It is also interesting to note that, by analyzing thepossibility of satisfying the conditions for the invisibilityof the particle in vacuum (formula (11) with m = 1),Kerker [96] considered a wide range of permittivities ofthe core and shell [, ], including the untraditionalrange of negative values and values smaller than unity.

In [97], the authors analyzed the cross section oflight scattering from a spherical particle that consists

of a core and a shell and cannot obey the smallness criterion. It was noted that, in the electrostatic approximation [96], this small composite particle is treated asa dipole so that the core and the shell are characterized

by different polarizations. When condition (11) is fulfilled, the polarizations of the core and the shell areequal in magnitude and opposite in sign; i.e., thedipole moment becomes equal to zero and the particlein the dipole approximation does not scatter light(is invisible). In [97], the discussion is based on theexpansion of the scattering cross section in powers ofthe wave vector, which is valid for any finite particlesize. The numerical calculations were carried out forthe values ofnm = 1.00, n1 = 1.97, and n2 = 0.66 (the

refractive indices of the medium, core, and shell,respectively) and two diffraction parameters of theparticlex2 = 2R2nm/ = 0.3 and 1.6, which characterize the particle size with respect to the wavelength. Thescattering cross section was calculated as a function ofthe volume fraction of the core in the particle f =(R1/R2)

3.

Forx2 = 0.3, the scattering cross section exhibits asharp minimum atf1 0.415. The minimum scatteringcross section for the composite particle is six orders of


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magnitude smaller than that for the homogeneousparticle that has the same size and consists of thematerial of the core or the shell. Therefore, the smallcomposite particle for some ratio between the parameters becomes almost invisible. The obtained value of

f1 for which the scattering cross section is minimumalmost coincides with the value off0 0.414 calculatedin the electrostatic approximation (11) and depends

only on the quantities nm, n1, and n2.In the case ofx2 = 1.6, the scattering cross section

is minimum atf2 0.31, which differs noticeably fromf0. Moreover, the minimum in this case is not so deep;the cross section decreases by approximately twoorders of magnitude as compared to the case of thehomogeneous particle.

Furthermore, in [97], it was also demonstratedthat, when condition (11) is satisfied, the dipole components in scattering are absent and the residual scattering cross section is proportional to 8 for fairlylong wavelengths. In this case, the predominant backscattering takes place. These inferences correlate with

those previously obtained in the RayleighGansapproximation [94].

The most significant qualitative conclusion following from the results obtained in [97] is that the substantial decrease in the observability is possible onlyfor small composite particles with sizes smaller than orof the order of the wavelength.

In [98], the rigorous solution [99] to the problem oflight scattering from a spherical particle consisting of ahomogeneous core and a homogeneous shell was usedto examine the cases in which the real part of the permittivity of the shell or the core is negative. As a resultof the rigorous solution [99] (see also [83, Chapter 8]),

the scattering efficiency (the ratio between the scattering cross section and the crosssectional area of theparticle) can be represented in the form of the series

(12)

in which the terms correspond to different multipolecomponents of scattered radiation, beginning with thedipole component (n = 1), an and bn are the scatteringcoefficients, andx2 = 2R2nm/ is the size parameterintroduced above.

It was noted in [98] that, for small particles (at

x2 < 0.3), all scattering coefficients, except for thecoefficient a1 can be ignored in expansion (12). Inturn, the coefficient a1 can be represented in the formof the expansion in powers ofx2 with the main term

(13)

Qsc 2/x2( )2

2n 1+( ) an2

bn2

+( ),n 1=

=

a12

3 ix2

3

2 m( ) 1 22+( ) q

322 m+( ) 1 2( )+

2 2m+( ) 1 22+( ) q3

22 2m( ) 1 2( )+

,

where we used the aboveintroduced designations forthe permittivities and the geometric factorq = R1/R2(0 q 1), which represents the ratio between the radiiof the core and the shell. In this case, the scatteredfield corresponds to the field of the oscillating dipole

with the polarizability proportional to a1 (electrostaticapproximation). The condition for anomalously lowscattering, i.e., the vanishing of the numerator in

expression (13) [98], is equivalent to the abovederivedcondition (11).

In the examples considered in [98], it was assumedthat the particle is in vacuum (m = 1). In these examples, the authors calculated the efficiencies of scattering and absorption from the rigorous formulas (formula (12) for the scattering efficiency) with exactexpressions for the scattering coefficients. The mostinteresting is a series of the calculations with the use ofthe experimental data on 2 for a silver shell containinga dielectric core (1 > 1). Apart from the variation inthe geometric factorq = R1/R2 in these calculations,the authors considered different wavelengths, different

particle sizes, and different permittivities of the core.In particular, the scattering and absorption efficiencies were calculated as a function of the wavelength forparticles with a specified geometry and composition.

Although, in a number of cases, the scattering efficiency for small particle can reach very low values (forexample, at a specific wavelength for particles with aspecified geometry), the absorption efficiency in thesecases considerably (by several orders of magnitudes)exceeds the scattering efficiency. Consequently, underspecific conditions, the scattering of monochromaticlight from small particles that consist of a dielectriccore and a silver shell in vacuum can be reduced considerably, whereas the absorption of light by the same

particles remains at a high level.The results obtained in [98] demonstrate that, in

order to fabricate small particles invisible in vacuum,it is necessary to develop the shell material characterized by the negative real part and fairly small imaginarypart of the permittivity. In [98], the authors also noted(but not considered) the possibility of using a material

with the permittivity lower than unity.

Therefore, the modern work by Al and Engheta[89] on the problem of invisibility of small particles in

vacuum was preceded by a number of works, at least, apart of which we cited above. The investigation performed in [89] is based on the analysis of light scatter

ing from a particle that consists of a core and a shell inthe cases where (meta)material of the core and/or theshell is characterized by negative permittivity and/or anegative permeability. This analysis was carried out in[100] (see also Erratum to this work [101]), where it

was noted, in particular, [100, p. 2] that the solutionobtained previously in [99] for a particle from conventional materials is also applicable in the case underconsideration when the signs of the expressions

and are chosen correctly. It should bejj j/j


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noted that this choice in the case ofj< 0 and j< 0 wasfirst discussed by Veselago [3] and the light scatteringin the case where only one of the permittivities of theparticle (1 or2) is negative was studied earlier in [98].

Al and Engheta [89] used a scattering series for aparticle consisting of materials and/or metamaterials[99] that, with due regard for the remarks made in theprevious paragraph, is equivalent to series (12). Firstand foremost, the authors in [89] derived the approximate relationship for the coefficients of the scatteringseries in the case of a small particle and demonstratedthat, if the core of the particle is a dielectric, the maincontribution to the scattering is made by the TMdipole mode (dipole mode of the electric type [83,

Section 4.3.2]) with the scattering coefficient a1 or

according to the designations of formula (12) and thework [89], respectively. By equating this coefficient tozero, the authors in [89] obtained the invisibility condition equivalent to relationship (11).

Then, the authors performed the numerical calcu

lations of the scattering series as a function of the geometric factorq = R1/R2 for a number of values of theparameters R2/ (orR1/), j, and j(j= 1 for the coreand 2 for the shell) under the condition that one of thecomponents of the particle (the core or the shell) is aconventional dielectric (j > 1, j = 1) and the othercomponent is an unconventional material or ametamaterial (j' < 1, j' 1). In the graphical representation of the results of the calculations, apart fromthe total scattering cross sections, the values of indi

vidual terms of the scattering series are also presentedin [89], which allows one to judge the contributions ofdifferent normal modes [83, Section 4.3.2] to the scattering.

In [89], the results of the numerical calculationsshowed that the particle with a fairly small size (R2/ a is

determined by the expression n(z) = |1 (a/z)2

|, i.e.,differs from unity for any finite value ofz.

Leonhardt [90] called attention that the imperfection of the device is associated with the partial reflection of light from the surface|z| = a and with the timedelay of rays that bend around the object. He assumedthat the reflection can be reduced by introducing atransition layer instead of a sharp boundary or usingantireflection films. In his opinion, the imperfectionsof the invisibility device in rays can be made as small as


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ZHILIN, SHEPILOV

the inaccuracy of geometrical optics, whereas theobject cannot be completely hidden from waves.

Some aspects of the invisibility device of the abovetype, including the problem of the time delay, are considered in more detail in [111]. The method proposedin [90] did not receive wide use. However, the ideasformulated in [90] were used to develop the model[112] in which the authors eliminated the imperfections of the device (time delay and reflection effects).

Unlike Leonhardt, who discussed the invisibilityproblem in the geometrical optics approximation,Pendry et al. [91] used the Maxwells equations in theirapproach to the theoretical treatment of the invisibility cloak. These authors were based on the resultsobtained in [113] (they are reproduced in the electronic appendix to [91]), according to which the Max

wells equations for monochromatic radiation in theabsence of free charges and currents are invariant withrespect to spatial deformations, i.e., transformationof coordinates, if this is accompanied by the corresponding renormalization of the permittivity andpermeability. It should be noted that, in the generalcase, this renormalization leads to coordinatedependent tensors of these quantities, even though the per

mittivity and permeability in the initial coordinateswere homogeneous and isotropic. If any region of thespace is not deformed, i.e., the initial and transformedcoordinates of points in this region coincide with eachother, the permittivities, permeabilities, and electromagnetic fields of the initial and transformed systemsalso coincide within this region. Generally speaking,the electromagnetic fields in the initial and transformed systems differ and are related by particularexpressions.

In order to more clearly illustrate these statements,we demonstrate them using a simple transformation asan example. Let us write the Maxwells equations

(14)

in free space ( = = 1, the initial system) in thespherical coordinate system (r, , ) [114]. For example, for the component of the first equation in system(14), we obtain (this component is chosen as the mostcompact in the writing)

(15)

Now, we perform the transformation of the coordinates R = R(r) that does not include the angles, whereR(r) is the onetoone function and

(16)

By changing over to the variable R in Eq. (15) withallowance made for the differentiation rule ()/r=(dR/dr)()/R and performing simple manipulations, we obtain the following equation for the transformed system:

(17)

Equations (15) for the initial system and (17) for thetransformed system are identical in the form but differin the values of the component of the permittivity:

H 0E/t, E 0H/t==

1

r rH[ ]

rHr

0

Et .=

R r( ) rr

.

1

R

R R

r

RH

1

dR /drHr

= 01

dR /dr

t

r

RE

.

Fig. 13. Trajectories of rays in the Leonhardt invisibility device [90] in the cases of propagation of the incident flux (a) in the direction of thexaxis and (b) at an angle with respect to this axis. The black circle at the center of the figure indicates the invisibilityregion.


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= 1 for the initial system and = (dR/dr)1 for thetransformed system. Equation (17) also indicates thatthe field components in the transformed system arerepresented by the expressions enclosed in parentheses, i.e., expressed through the corresponding components in the initial system.

The transformation of other components of the

first and second equations of system (14) can be considered in a similar manner. The equations for thetransformed system are the Maxwells equations in themedium with the permittivities and permeabilities

(18)

In this case, the field components in the transformed

system and the initial system (H, E) are

related by the expressions

(19)

(the electric field components are transformed in asimilar way). Formulas (18) and (19) suggest that, as

was noted above, the permittivities, permeabilities,and electromagnetic fields in the initial and transformed systems coincide in the spatial regions that donot undergo deformation (R = r).

In [91], the authors considered the transformation

(20)

which corresponds to compression of a sphere ofradius R2 into a concentric spherical layer with theouter radius R2 and the inner radius R1. This transformation represents a particular case of the transformationR = R(r) considered above (see relationships (14)(19)).Upon transformation (20), the spatial region r> R2 isnot deformed (R = r); i.e., as follows from the formulas given in [113] or formulas (18) and (19), the permittivities, permeabilities, and electromagnetic fieldsare identical for the initial and transformed system atR = r> R2. For example, if the plane wave propagates in

the initial system, the field in the transformed systemoutside the sphere R > R2 will be also described by theplane wave. Then, the region R < R1 in the transformedsystem has no analog in the initial system, the permittivities and permeabilities in this region can be specified arbitrarily, and the external field, i.e., the field ofthe initial system does not penetrate into this region.Finally, by using the formulas given in [113], as wasdone in [91], or the simpler relationships (18) and(19), we can derive the following expressions for the

'

R' R'r

2

R2

dR

dr,= =

' ' ' '1

dR /dr .= = = =

H,( E )

H R1

dR /drHr, H

r

RH, H

r

RH= = =

R R1 r R2 R1( )/R2 r R2 R1 and R < R1, which are not

electromagnetically related to each other for the radiation with the frequency under consideration.

Pendry et al. [91] noted that the cloak material atthe outer boundary (r = R2) obeys the relationships

= = 1/ and = = 1/ which ensure

the absence of reflection at the boundary. It shouldalso be noted that the anisotropy and spatial inhomogeneity of the permittivities and permeabilities are fundamentally important properties of the cloak materialand can be provided for metamaterials [115117].

According to [91], the possibility of obtaining small values of the quantities ' and ' seems to be moreproblematic.

In order to demonstrate the operation of thiscloak, the authors considered two examples. In thefirst example, it was assumed that the radiation wavelength is small R2 and the path of rays that are incident on the cloak from an infinitely distant source andcharacterize the behavior of the Poynting vector is calculated in the geometrical optics approximation. Figure 14 presents the results of these calculations in theplane that is parallel to the direction of incident fluxand passes through the center of the sphere (Fig. 14a)and in the threedimensional representation(Fig. 14b). It can be seen from this figure that the raysenter into the cloak and bend around the cloaked

region R < R1, so that the initial direction of ray propagation is restored at the exit of the cloak; i.e., the content of the cloaked region and the cloak itself are invisible for the external observer. In the second example,the authors considered the case R2 (nearfieldlimit), placed a point charge in the vicinity of thecloak, and used the electrostatic approximation forconstructing the field lines of the electric induction

vector (Fig. 15). It can be seen from Fig. 15 that thefield does not penetrate into the cloaked region and

R' R'R2

R2 R1

R R1

R

2

,= =

' ' ' 'R2

R2 R1.= = = =

' ' R' ' ' R' ,


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that the field outside the cloak is identical to that in its

absence.

Finally, the authors asked the question as towhether the cloaking effect occurs for some frequencyrange or only for one frequency. In the example underconsideration, the case in point was one fixed frequency. It should be noted that, in terms of geometrical optics (Fig. 14), the rays entering into the cloakpropagate along curved, i.e. longer, trajectories ascompared to those propagating in free space. However,at the exit of the cloak, their phase should be identicalto that for rays propagating in free space, because thecloak does not perturb the external fields. This implies

that the phase velocity of rays in the cloak is higher

than that in free space and this statement violates nophysical laws. By assuming that the dispersion isabsent, the phase and group velocities should be identical, whereas the group velocity cannot exceed the

velocity of light. Therefore, the dispersion should takeplace and the cloak can efficiently operate only at onefrequency.

The theorem of the uniqueness of the solution tothe inverse scattering problem [109] was not mentioned in [91], and, naturally, the fact that the resultsobtained contradict this theorem was not commented.The corresponding comment was done in more recent

work [118] and reduced to the fact that the uniqueness

theorem [109] was proved for an isotropic media andinapplicable to the case of the cloak from a substantially anisotropic material [91].

According to Rozanov [119], the anisotropy of thecloak material is of little significance for the violationof the uniqueness theorem. This author proposedanother explanation for the violation of this theorem,

which is based on the fact that the permittivity andthe permeability tend to infinity at the inner boundary of the cylindrical cloaking cloak (see below formulas (23)). This point of view seems to be incorrect.

Actually, in the case of the spherical cloak, the permittivities and permeabilities (relationships (21)) remain

finite and the only existing explanation is associated withthe anisotropy of the cloak material. Furthermore, it isimprobable that the factors responsible for the violationof the uniqueness theorem in the threedimensional andtwodimensional cases are different. Finally, reasoningfrom formulas (18) and expressions (22) derived below,

we can predict invisible optical inhomogeneities withsmoothly varying finite (but anisotropic) permittivitiesand permeabilities in both the threedimensional andtwodimensional cases. For these inhomogeneities,

Fig. 14. Trajectories of rays in the invisibility cloak proposed in [91] and characterized by the permittivities and permeabilitiesdefined by expressions (21) forR2 = 2R1. Trajectories are shown (a) in the plane that is parallel to the direction of the incidentflux and passes through the center of the cloak and (b) in the threedimensional space.

Fig. 15. Point charge in the vicinity of the invisibility cloak[91]. Field lines of the electric induction vector are shownunder the assumption that R2 (nearfield limit). Thefield does not penetrate into the cloaked region and, outside the cloak, is identical to that in the absence of thecloak.


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which are not invisibility cloaks, the uniqueness theorem is also violated. In this respect, the anisotropy ofthe material should be considered the fundamentalfactor responsible for the violation of the uniquenesstheorem.

Let us briefly compare the invisibility cloak modelsproposed in [90, 91]. The approach proposed byLeonhardt [90] can be directly used only in the two

dimensional case, whereas the approach by Pendry etal. [91] developed for the threedimensional case can

be easily extended to the twodimensional case (see below). The cloaks considered in [90] can strictlyensure invisibility only in the geometrical optics limitand require additional measures for the elimination ofreflection effects at the boundaries of the outer andinner regions, whereas the cloaks proposed in [91] arefree of these limitation and complication. For thecloaks proposed by Leonhardt [90], the size of theregion in which the refractive index differs noticeablyfrom unity, i.e., the outer size of the cloak, is considerably larger than the size of the invisible region,

whereas, for the cloak proposed by Pendry et al. [91],there is the only limitation, namely, the cloak radiusshould be larger than the radius of the invisibilityregion. The cloaks proposed in [90] do not possess acircular symmetry and consist of an isotropic material,and the cloaks proposed in [91] are spherically symmetric structures from an anisotropic material. On the

whole, the fabrication of the cloaks proposed in [91]seems to be more promising, which is reflected in thesubsequent development of the invisibility idea in theliterature. The approach by Leonhardt [90] is of interest in that it shows the possibility of creating the invisibility device with the use of the isotropic material, atleast, for the twodimensional case in the geometricaloptics approximation.

It is interesting that, already in 1961, Dolin published the paper [120] in which he very closely came tothe modern solution to the invisibility problem. First,Dolin investigated the invariance of the Maxwellsequations with respect to a specific type of transformations of the space metric and permittivities and permeabilities of a medium, which was also done in [113] butconsiderably later, and proposed to use this invariancefor the solution to some electrodynamic problems.Second, as one of the applications, Dolin consideredthe transformation of free space upon changing overfrom the spherical coordinates r, , and to the coordinates R(r), , and , where R = R(r) satisfies condition (16), and found the type of the inhomogeneitythat does not perturb the external electromagnetic

field, i.e., is invisible for the external observer.1

Inorder to predict the invisibility cloak in its modernmeaning, he was short of one step, namely, the use of

1 It should also be noted that the permittivities and permeabilitiesobtained in [120] are inverse with respect to those followingfrom formulas (18).

the transformation R(r) that transforms the completespace into its part, such as transformation (20).

The ideas formulated in [91] were subsequentlyused in [121, 40] for the theoretical development ofsimpler (simplified) cloaks. In this case, for simplification, the authors examined th

Met a Materials- A New Direction in Materials

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