A New Era of Exotic Electromagnetism

GENERAL ¨ ARTICLE A New Era of Exotic Electromagnetism

K Porsezian and Ancemma Joseph

Nature hasitsownlimitsand naturally-occurring materials exhibit restricted electromagnetic properties bound within those limits. However, mod- ern dayresearchers have developed arti¯cially structured materialcomposites,called `metamaterials', that possess signi¯cant potentialtopro- (left) K Porsezian works at vide electromagnetic properties that are quite the Department of Physics, unusual and are not found in nature. This ar- Pondicherry Central University, Puducherry, ticle is aimed atgivingan introductory review India. He is investigating of this class of `designer materials' and their su- solitons and modulation perior properties that are not present in their instability in nonlinear constituent components. optics, Bose-Einstein condensates, metamaterials, 1. Metamaterial { Advent photo-refractive materials, photonic crystal fibers and Metamaterials are arti¯cially constructed structures ex- also integrability aspects of hibiting unconventionalcharacteristics thathave never nonlinear partial differential been observed in nature before. This factgivesthem equations. their name `metamaterials' { materials beyond (natural) (right) Ancemma is working materials. Unlike naturalmedia,theseare materials towards her doctoral degree which derive their properties from their structure rather in Pondicherry University, Puducherry, India. Her thandirectlyfrom their composition as shown in Fig- primary research interests ure 1. Though the notion of metamaterials wasorigi- are based on studying the nally proposed in regard to the idea of a negative re- nonlinear pulse propagation fractive index, or negative-index materials (NIMs), to- through nonlinear periodic daythevarious interpretations of the term have led to structures and negative index materials. their versatile terminologies like double negative material(DNG), left-handed material(LHM),backward phase material(BPM), negative phase velocity (NPV)material, etc. The vital decisive factor responsible for theunusual be- havior of these materials is the e®ective negative refrac- Keywords tive index resulting from the negative realpart of the Metamaterial, negative refraction, left-handedness.

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Figure 1. Conventional and metamaterial composition.

magnetic permeability and negative realpart of electric permittivity. The dielectric permittivity and magnetic permeability are the key factors responsible for all the electric and magnetic characteristics exposed by materials. While some materials possess negative permittivity (like for metals below their plasma frequency), no naturalmaterials are seen with negative permeability. Tracing back the history of metamaterialconcept,itwas the Soviet physicistVictorVeselago's genius which postulated the physicalpermissibility of materials possessing simultaneous negative permittivity and permeability back in the late 1960s. He hadenvisioned the unusual electrodynamics of double negative materials in his fa- mous seminalpaper in Soviet Physics Uspekhi. This pu- tative concept which laylatent for almost three decades got resurrected with the pioneering e®ort of John Pendry of ImperialCollege, London who proposed a blueprint It was the Soviet structure exhibiting simulataneous negative permeabil- physicist Victor ity and negative permittivity. Veselago’s genius which postulated the These materials, which are still in their infancy, have exposed a varietyof fascinating e®ects such as reversal physical permissibility of Snell's law, reversed Doppler e®ect, Cerenkov radi- of materials ation, sub-di®raction imaging, photon tunneling, back- possessing simul- ward wave antennas,phase combination, electrically taneous negative small resonators, and are expected to satisfy a multi- permittivity and tude of other requirements in complex environments. In permeability back in the next section, wepeepinto each of the intriguing and the late 1960s. exciting traits of metamaterials.

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2. Metamaterial { UnusualTraits From the foundations of macroscopic 2.1 Left-handedness electrodynamics From the foundations of macroscopic electrodynamics given by J C Maxwell given by J C Maxwell (1831{1879 ), it iswell known (1831–1879), it is well that the propagation of electromagnetic waves in a known that the medium is determined by dielectric permittivitiy and propagation of magnetic permeability of the medium. Accordingly, the electromagnetic e®ective refractive index of a medium can be written as waves in a medium is n = p²¹.Now,asperVeselago's notion, if both ² and determined by ¹ take up negative values in a given wavelengthrange, dielectric permittivitiy we can write ¹ = ¹ ei¼ and ² = ² ei¼. Then one can j j j j and magnetic easily see thattherefractive index permeability of the medium. n = ² ¹ e2i¼ = ² ¹ pe2i¼ = ² ¹ : (1) j jj j j jj j ¡ j jj j This impliesp thattherepfractive index opf the medium with simultaneous negative ² and ¹ must be also negative. Now what could be the consequences of this fact? For a monochromatic plane wave, the Maxwell's equations can be written as ~k E~ = ! ¹H~ and ~k H~ = £ c £ ! ²E~ . ¡ c Interestingly, one canseethat, while for positive values of ² and ¹, E~ , H~ and ~k form a right-handed triplet, but for Veselago's case, they form the left-handed triplet. ~ c ~ ~ While Poynting vector S = 4¼ E B is always directed away from the source for both£ left and right-handed cases, the wave vector surprisingly takes uptwodirec- tions, `away from the source' for the right-handed and `towards the source' for the left-handed case. One can see this fact well depicted in Figure 2.

Figure 2. (a) Right-handed media with H >0,P>0. (a) (b) (b) Left-handed media with H<0,P<0.

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Light that enters 2.2 Negative Refraction from an ordinary Astrikingfeatureof the negative index materialisthe medium to the reversalof one of the fundamentalprincipleof optics, metamaterial the Snell's law. Light thatentersfrom anordinary medium will medium to the metamaterial medium will undergo re- undergo refraction fraction opposite to that occuring in the conventional opposite to that ordinary medium. How could this be explained? Con- occuring in the sider the refraction phenomena occurring when a beam conventional travels from one medium into another. ordinary medium. Suppose if the two media areof positive index with ²>0 and ¹>0, then we will have anordinary refraction case. But for the case when the second mediumisof negative index with ²<0 and ¹<0, we have a beamgoing from anordinary medium into a negative medium. One should rememberherethat, for any case, the boundary conditions requiretangential components of E~ and H~ , and normalcomponents of D~ and B~ to be continuous at the interface, i.e.,

Et1 = Et2;Ht1 = Ht2;

and ²1En1 = ²2En2;¹1Hn1 = ¹2Hn2: (2)

Now, it is clearthat x and y components of the ¯eld are not changed at transition from medium 1 to 2, re- gardless of the signs of ² and ¹.Asfor the normal z components of the¯eld, they preserve their directions if ² and ¹ preserve their signs in both media.Otherwiseit is obvious that they change their directions and hence the negative refraction as shown in Figure 3.

Figure 3. Negative refraction.

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2.3 Reversed Doppler Shift An apparent change in the frequency of a Thinking of oneself standing atthecrossroads and hear- wave when there is a ing an approaching ambulance will immediately bring to our mind the `Doppler e®ect'. As the vehicle races to- relativemotion wards us, the pitch of the siren becomes higher, and between the observer as it recedes, it gets lowered. This well-known fact of and the source of an apparent change in the frequency of a wave when waves is termed as there is a relative motion between the observer and the Doppler effect. source of wavesistermedas Doppler e®ect. As the object emitting the wave approaches the observer,thein- tervals between the waves diminish; in other words, the radiation gets squeezed, resulting in increased frequency and a blue shift. As the object recedes away from the observer, the wave gets stretched leading to a red shift. Now how could this e®ect get reversed in metamaterials? The apparent frequency measured by the observer when the source is moving inside the medium at velocity v is

!0 = °(! + ~k:v)with~k = n!=c: (3) j j 2 2 1=2 Taking up the relativistic factor ° =(1 v =c )¡ and ¡ considering emission along the direction of the motion of the source in the ordinary (n =1)ormetamaterial medium (n = 1), we get the apparent frequency to be ¡ c + v !0 = ! for right-handed medium; c v r ¡ c v and !0 = ¡ !; for left-handed medium: (4) c + v r

Thus one canseethat the Doppler e®ects gets exactly reversed in a metamaterial medium due to the reversed phase vector.

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The original principle 2.4 Reversal of Fermat's Principle of Fermat states that a beam of light The originalprinciple of Fermatstates that a beamof chooses such an light chooses such an actualpath between two points which is the one traversed in the least time. It was actual path between further reformulated as the route chosen by the light two points which is to be of minimum optical length. However, in the case the one traversed in of metamaterial the term `minimum' (both for path or the least time. time) is not apt. Let us consider the case of light propagating between two points sayAand B. First, asinCase 1 of Figure 4(a), choose pointAtobeina medium of positive refractive index n1, and pointBtobeina medium of positive refractive index n2.Inthiscase of positive n1 and n2,theray will travel along AO1B with the incident and refractive angles (' and Ã respectively) obeying Snell's law. Note that for Snell's law to be valid, the opticalpath length must vanish and here the variation of the opticallength n1 AO1 + n2 O1B will be minimum through AO1B. If boththemediums n1 and n2 are con- sidered to be negative, the propagation will be along the same path, but thewave vector will be opposite to the

(a)

(b) Figure 4. (a) The possible trajectories for the beam to pass between two points A and B between two different mediums (b) Cerenkov radiation.

168 RESONANCE ¨February 2012 GENERAL ¨ ARTICLE direction of rays leading to negative opticallengthand The proper it turns out to be maximum for the actual path AO1B. formulation would be that light choses to Consider a situation when n = n2=n1 is negative, i.e., light is propagating from anordinary mediumto a meta- travel along the path materialmediumasshowninCase 2 of Figure 4a.Here that corresponds to a there is an ambiguity in the actualpath being a mini- local extremum in the mum or a maximum. The actualpath traversedbythe spatial derivative of the total travel time light will be along AO3B, with incident and refracting angles obeying the Snell's law, but this time for the neg- through all possible paths. ative value of Ã.Thevirtualpaths AO2BorAO4Bwill not be chosen by the light even though they areshorter or longer in terms of thetimeof travel. Hence the proper formulation would be thatlightchoses to travel along the path that corresponds to a localex- tremum in the spatialderivative of the totaltravel time through all possible paths. An extremum could be de- ¯ned asvariation in the opticalpath becoming zero, i.e., A B n(r)ds =0.Thusfor the case of a beamof light trav- eling from right- to left-handed metamaterialmedium, R the variation in the path AO3B is minimum. Unlike the Fermat's version of theterm`minimumopticalpath- way', it is an extremum which could be minimum, maximum or a point of in°ection. The phenomenon of Cerenkov radiation 2.5 Reversed Cerenkov Radiation discovered by Soviet The phenomenon of Cerenkov radiation discovered by Physicist Pavel A Soviet Physicist Pavel A Cerenkov is the emission of a Cerenkov is the cone of electromagnetic radiation when a charged par- emission of a cone of ticle (usually electrons) passes through a medium at a electromagnetic speed greater than the speed of light in thatmedium. radiation when a One canexplain the physics behind this as the response charged particle of the medium to the transit of charged particle through (usually electrons) it. As the charged particle travels, it disrupts the local passes through a electromagnetic ¯eld, displacing and polarising the elec- medium at a speed trons in the medium. After the transit, the electrons greater than the restore themselves to equilibrium with the emission of speed of light in that photons which would constructively interfere to give the medium.

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For a metamaterial observed intensity.Nowthe angle of theconeof ra- medium, the diation in a normalmediumisacute while thatina refractive index n metamaterialmedium is obtuse. has a negative To explain this reversal, consider Figure 4b, where the value, and therefore charged particle situated at the leftcorner attimet =0, this angle becomes traverses to the right corner with velocity Vp equalto°c obtuse, i.e., the in time t.Thedistance traversed will beequaltoVp t. particle will radiate The refractive index of the medium being n, the cone from a cone behind c of radiation would have traversed a distance n t.Hence itself, rather than in the acute angle of this cone will be front of it. V 1 cos µ = p = : (5) Vem n° Now for a metamaterialmedium,therefractive index n has a negative value, and therefore this angle becomes obtuse, i.e., the particle will radiate from a cone behind itself,rather than in front of it. 2.6 Reversed Goos{HÄanchen Shift Consider a collimated beam incident on a rarer medium from a denser medium at an angle greater thanthecritical angle; the phenomenon one would expect to emerge is `totalinternalre°ection'. Here the incident beamre- °ects back into theincident medium, butitdoessofrom a point which is displaced from the point atwhichit struck the plane. This lateralshiftwasdiscoveredbyF Goos and H HÄanchen and hence the name of this e®ect. Consider a collimated beam incident on a To have a mathematical picture of this phenomenon, consider a wave impinging onto a rarer medium at an rarer medium from a angle greater thanthecritical angle. The re°ection co- denser medium at an e±cient is a complex number with unit amplitude and angle greater than the phase Á,i.e., critical angle; the phenomenon one Re°ection coe±cient R =exp(2IÁ); would expect to 1 2 1 ¹0 kx kt emerge is ‘total where Á = tan¡ ¡ : (6) ¹ k internal reflection’. ¡ Ã p t z !

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Figure 5. (a) Goos–Hänchen (a) (b) shift, (b) Casimir effect in metamaterial.

Note that kt denotes the transmitted wavenumber and kx;kz are the components of the incident wave-vector. The phase Á introduces a spatialshiftinthere°ection of a ¯nite beam, which is termed the Goos{HÄanchen shift. Considering the case of a beamincidentfrom a positive index medium with refractive index n1 on anordinary positive index medium with refractive index n2 > 0 and when the angles of incidence are su±ciently away from the critical angle and the grazing angle, the displace- ment of the beamwasshowntobe = @Á asdepicted @kx in ¯rst case of Figure 5a.Butastheinci4 ¡dent beamen- counters a metamaterial medium with negative index of refraction, i.e., with n2 < 0, the phase Á takesupthe valueoppositeinsigntothatof positive index materials as ¹t takes up negative value. This in turn leads to the Goos-HÄanchen shift in the direction opposite to that of materials with positive indices of refraction asdepicted in second ¯gure of Figure 5a. The Goos-Hänchen shift is in the 2.7 Reversal of Casimir Force of Attraction direction opposite to In 1948, HBG Casimir predicted the attraction between that of materials a pair of neutral, parallel conducting plates, the e®ect with positive indices being purely due to the e®ect of quantum mechanical of refraction.

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Theattractive vacuum modes. An amazing feature is thatthisfun- Casimir force in damental attraction can be turned into repulsion if a transformed space metamaterialissandwiched between theparallel plates. turns into a It canbeexplained from Figure 5b asthetransformation repulsive force in of Casimir cavity of size a in physical x space into a cav- physical space. ity in x0 of size a0.Thecavityinphysicalspace increases when the transformedcavity decreases. Consequently, the attractive Casimir force in transformed space turns into a repulsive force in physicalspace. This could lead to the achievement of quantum levitation. 3. Metamaterial { Realization While Veselago hadhypothesized the concept of metamaterials with ²<0 and ¹<0, this notion remained unappreciated fordecades asnometamaterialseem-ed to exist naturally. One mayeasily ¯nd naturalmaterials with ²<0(e.g., gold, silver and other metals) up to The boom in the the visible frequencies; but no magnetic materials with metamaterial research ¹<0 are known to exist atoptical frequencies till to- came with the exciting day. The boom inthe metamaterialresearch came with suggestion made by the exciting suggestionmade by Pendry et al.touti- Pendry et al to utilize lize split-ring resonators to achieve negative magnetic split-ring resonators to permeability. achievenegative The negative permittivity of a metamaterial structure magneticpermeability. canberealized by considering an arrayof long, parallel, thin metal wires embedded in a dielectric medium (Fig- ure 6a). With such an arrayof unit cell length a and

(a) (b)

Figure 6. Constituents of metamaterial: (a) An array of metallic wires; (b) A split-ring resonator.

172 RESONANCE ¨February 2012 GENERAL ¨ ARTICLE the radius of single wire r, the plasma frequency for the As one applies a longitudinalplasma mode is time-varying magnetic field perpendicular to 2 2¼c the ring’s surface, the !p = ; with the e®ective dielectric sa2 ln a=r magnetic field !2 produced by the permittivity ² =1 p : (7) ¡ !2 currents induced in dependence on the At frequencies less thanplasma frequency ! , one achie- p resonant properties of ves negative permittivity. the structure, either A highly conductive split-ring resonator structure (Fig- oppose or enhance ure 6b) with the capacitancebetweentherings balancing the incident field, thus the inductance provides the desired negativemagnetic resulting in positive or permeability. As one applies a time-varying magnetic negative effective P. ¯eld perpendicular to the ring's surface, the magnetic ¯eld produced by the currents induced in dependence on the resonant properties of the structure, eitheroppose or enhance the incident ¯eld, thus resultinginpositive or negative e®ective ¹.Forthecirculardouble split-ring resonator, the e®ective permeability is ¼r2=a ¹ =1 : (8) ¡ 1+(2¾i=!r¹ ) (3d=(¼2¹ !2² ²r3)) 0 ¡ 0 0 The thin wire{split-ring resonator structure is the vital building block for achieving negative e®ective ² and ¹ and negative refractive index but only in THz frequencies. Many di®erent structures like `cut-wires sep- arated by dielectric spaces', `¯shnet structures', `paired nanorods' are fabricated these days to realize metamaterials in the optical regime. 4. Metamaterial { Exciting Future Out of the exotic electromagnetic applications which bring forth the concept of metamaterial into the fore- Anexciting front of the research arena, the prominent exciting ap- application of plication of metamaterial achieved practically is super, metamaterial hyper lenses.These lenses could be better thanconven- achieved practically is tional lenses in the sense that, anordinary lens cannot super, hyper lenses.

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The evanescent field focus light down to more than about half its wavelength, gets enhanced inside the di®raction limit. Whatdoesthismean? As one con- the metamaterial lens siders a source and a lens placed along the z axis, the and gets restored to electromagnetic ¯eld emanating from the source could givesub-diffraction be written as a superpositionof plane waves which could resolution. be written as

E(x; y; z;t)= A(KxKy)expi(Kzz + Kxx K K Xx y !2 K y !t ; K K2 K2 ; + y ) where z = 2 ( x + y ) (9) ¡ §r c ¡ with corresponding to ordinary andmetamaterial § 2 medium respectively. In ordinary materials, when (Kx + 2 !2 Ky ) < c2 ,then it corresponds to propagating modes 2 2 with long-range information, and when (Kx + Ky ) > !2 c2 ,thenKz becomes imaginary, leadingtoevanescent ¯elds decaying exponentially with z. These modes ac- tually contain theinformation about thehighfrequency (minute scale) features of the source being imaged.The maximum possible resolutionthatcan beobtained from ! 2¼ such a case would be c = ¸ .Inother words, if the lens is at a distance larger thantheoperating wavelength ¸,then Kz component will not be seen; thus ¸ remains' the basic resolution limit in far-¯eld approx- imation. Now in the case of metamaterial, with Kz = !2 (K2 + K2), the evanescent modes containing ¡ c2 ¡ x y highq frequency features will grow insteadof decaying exponentially. Thus the evanescent ¯eld gets enhanced inside the metamateriallens and gets restored to give sub-di®raction resolution. Shielding an object Another novel application of negative refraction which is with a metamaterial like magic could be `invisibility cloaking'. Shielding an cloak would render object with a metamaterialcloak would render light to light to curve around curve around the object andreturntoitsoriginalpath the object and return on the farside. The detector or the observers of the to its original path on object would be completely unaware of the presence of the far side. such anobject{ even its shadow will not be visible.

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Apart from all these important and exciting linearef- The nonlinear fects and their consequences, there is yet another very effects of important window, the nonlinear e®ects of metamate- metamaterials has rials, thathasremained vastly unexplored. One can remained vastly achieve the prospect of nonlinearelectromagnetic re- unexplored. sponses, like cubic or quadratic nonlinear responses, in metamaterial structures by insertion of nonlinearelements within them. It could be done by embedding the split-ring resonators in a Kerr-type dielectric or by in- cluding certain nonlinearelements(e.g., diodes) in the split-ring resonators' paths. A few groups have started exploring this avenue with results like exhibition of second harmonicgeneration, opticalbistability, parametric ampli¯cation, soliton formation, etc., in metamaterials. The surface of non- linearmetamaterialmediumacts as a mirror re°ect- ing o® energy in the form of second harmonics when there is anexact phase matching between a backward- propagating wave of the fundamental frequency and the forward-propagating wave. The phenomenon of bistability which enables a system to exhibit two steady trans- mission states for the same input intensity has also been predicted of a metamaterialwhenembedded in a Kerr- nonlinearmediumleading to e®ective nonlinear medium switching on and o® of the magnetic permeability. A mechanism of nonlinearcouplerwithpositive index and negative index channel could result in the bistability due to the backward coupling between the modes propagat- A mechanism of ing in the two channels. Periodic metamaterial structure nonlinear coupler with metal-dielectric (Kerr-type nonlinear) slabs show with positive index evidence of self focusing and discrete soliton formation and negative index due to three-fold interplaybetweenperiodicity, nonlin- channel could result earity and surface plasmon tunnelling. Parametric am- in the bistability due pli¯cation and four-wave mixing kind of nonlinear e®ects to the backward have also been predicted. Even a novel amalgamation of metamaterial and ¯ber optics, a `meta-¯ber', is fore- coupling between the seen where the duo { `conventional and surface plasmon modes propagating in the two channels.

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waveguiding' { is expected to emerge with many poten- tial applications. The bottomline is thattheseexoticmetamaterials may not be speci¯cally oriented towards one or a few areasof physics, but theirunusual properties will o®er bene¯ts over a wide rangeof applications. Acknowledgements The author wishes tothanktheGovernmentof India agencies, (CSIR, IFCPAR, DST-DFG andDST-Ramanna Fellowship), for ¯nancial support. The authors also thank Dr. S Sivaprakasam, Pondicherry University, for criticalreading of the manuscript.