REVIEW 1 ,i.e.,a Journal of n Vol.3, 1–30, 2006 ∗ demonstrated in 4 1 Computational and Theoretical Nanoscience and Christian Hafner 3 by the general consideration of the electrodynamics 2 REFRACTION PHENOMENA their pioneering workartificial that material it with electrodynamiccan is be characteristics possible described that by to aNegative Index negative fabricate index Metamaterial of (NIM).At an refraction thatcept time, of the negative con- index ofand refraction itself unusual, seemed to although be1968 it new had been introduced already in properties of the materials with simultaneously negative 2.THEORY OF NEGATIVE INDEX In 2000 Smith, Schultz and coworkers not observed before inexample natural media.The of most not prominent materials previously with observed properties ative are negative Index index meta- Metamaterials of (NIMs)Handed refraction, that Materials i.e., are (LHMs).After also Nega- metamaterials first called attracted experiments, Left much these interest andanalyzed.Despite were of intensively theirprovide conceptional rich simplicity, NIMs andwell surprising understood in phenomena detail.In and theous following theoretical are we aspects present still together vari- withfocus not fabrication on issues and thestructure main calculations, trends and in numericalare experimental simulations, necessary studies, that for all materials. band- the practical implementation of NIM Valery Shklover, 2 1546-198X/2006/3/001/030 doi:10.1166/jctn.2006.002 119991 Moscow, Russia Leonid Braginsky, 1 Laboratory of Crystallography, ETH Zürich, CH-8093 Zürich, Switzerland 3 Negative Refractive Index, Metamaterials, Left-HandedNumerical Materials, Methods, Photonic Crystals, Nanotechnology. Institute of Semiconductors’ Physics, 13 Lavrent’eva Pr., 630090 Novosibirsk, Russia Negative Refractive Index Materials 2 Laboratory of Magnetic Materials, A. M. Prokhorov Institute of General Physics, 38 Vavilov Street, 1 Victor Veselago, Laboratory for Electromagnetic Fields and Microwave Electronics, ETH Zürich, CH-8092 Zürich, Switzerland 4 Keywords: The main directions ofor studies metamaterials, of are materials reviewed. with First,history negative the index of physics of this of refraction, scientific the alsoexhibit branch phenomenon called negative of are left-handed refraction negative outlined. refraction are Thennegative and recent discussed. the index results In material of the configurations studiesare third of and part presented. photonic of numerical crystals The metamaterials methods that Finally, advantages details that for of and exhibit the the the negative fabrication simulationite index of shortages of different metamaterials, properties kinds of photonic of existing crystals, metamaterialsnamely computer are and given. radio transmission packages This line frequencies, includes areexamples compos- metamaterials analyzed. microwaves, of for practical terahertz, different applications wavelengths infrared, of metamaterials and are visible presented. light. Furthermore, some Copyright © 2006 AmericanAll Scientific rights Publishers reserved Printed in the United States of America Configurations with NIMs ...... 14 Configurations with NIMs ...... 16 Refraction ...... 20 NIMs for Visible Light ...... 11 References ...... 29 ..vial Maxwell4.1.Available Solvers ...... 4.2. Time-Domain 12 Solutions of 4.3. Frequency-Domain Solutions of 4.4. Design of Metamaterials with Negative Index of 4.5.State of the Art ...... 22 5.1.Composite Metamaterials ...... 5.2.Photonic Crystals 22 ...... 5.3.Applications 25 ...... 25 3.1.Self-Collimating in Photonic Crystals3.2...... Inclusions of Plasmonic 10 Nanospheres; Author to whom correspondence should be addressed. ∗ J. Comput. Theor. Nanosci. 2006, Vol. 3, No. 2 1.INTRODUCTION Artificial media, i.e., metamaterialsthat may are exhibit properties much moreral pronounced materials than or those they found may even in exhibit natu- properties that were 4.Numerical Modeling and Simulations ...... 11 5.Fabrication of NIMs ...... 22 6.Conclusion and Outlook ...... 28 CONTENTS 1.Introduction ...... 2.Theory of 1 Negative Index3.Negative Refraction Refraction Phenomena in Photonic ...... Crystals ...... 1 8 REVIEW utemr,h svc-himno h hsc eto fteSpeeAtsainCmiteo usa(A)and (VAK) Russia of Committee Attestation Supreme Research. Russia.” the in Humanitarian of “Investigated for section journal Foundation scientific physics Russian electronic as the the active the and of is of Research (2002).He vice-chairman vice-editor Fundamental Federation is for Russian Foundation the he Russian of Furthermore, Scientist the Honoured in also expert is an (2004).He prize V.A.Fock academician the o n eaieidxo ercin eal-ilcrcpooi rsas n cnignafil pia microscopy. with optical materials nearfield including scanning metamaterials and doped crystals, photonic composite metallo-dielectric optics, refraction, integrated of ultra-dense index on negative is and low focus research current His patents. projects 5 industry-oriented owns several he sensors.Currently, of coatings, manager protection is nanostructures, journals.He of international field in the and in Russian in publications peer-reviewed eaieRfatv ne Materials Index Refractive Negative 2 aotutrs unu os n htnccytl;pyia hnmn na interface, an on phenomena physical phenomena. crystals; optical superlattices, photonic and at theory, microstructures, transport and Professor systems: matter dots, dimensional Assistant condensed quantum low and include of nanostructures, is properties Physics interests 1999.He optical Semiconductor research and in of University.His electronic (Novosibirsk) Institute State Physics the Novosibirsk condensed Semiconductor in at the degree of Researcher Ph.D. Institute Senior a the and a 1978, from in physics University State mater Kyrgyz from physics theoretical endBraginsky Leonid of winner a and (1976), USSR of science for LHM). Prize Materials, State Left-Handed the called electrodynamics of (so considered winner index time refraction a first of is V.G.Veselago the value at negative was with electrodynam- 1966–1972, material and in of is physics, published state he areas papers, solid main now his Technology.The magnetism, and ics.In until are V.G.Veselago solid Physics 1980 of of of interest investigation Institute.From Institute scientific P.N.Lebedev the Moscow of at for at physics 1974 both applied in fields, of spectra physics) professor magnetic molecular state of high (solid investigation in Science Physics.He radiospectropy of states the General Doctor for of degree 1959 the Institute the in and A.M.Prokhorov of degree head the Ph.D. is at his he received materials 1983 magnetic 1983.Since to of 1952 laboratory from Moscow, Institute, Physical P.N.Lebedev Georgievich Victor Veselago ae neouinr taeis eei loihs eei rgamn,admn more. many and programming, genetic algorithms, genetic optimizers mode strategies, numerical evolutionary various domain, for on developed package frequency based also software and optics.He MaX-1 and the time electromagnetics and in computational award techniques, Cray estimation solvers Seymour parameter difference the techniques, of finite matching Multipole prize computing—, second Multiple the scientific the by 1990 for namely in solvers, awarded Waves com- was Maxwell for (MMP)—that Electromagnetic method Program various Academy.Christian semi-analytical for and Electromagnetics (a and electromagnetics) Laboratory Technique the ETH Multipole putational of the Generalized the member at the at developed a Group Professor Hafner is is Optics Electronics.He he Microwave Computational 1999 and respectively.Since the 1987 of and head 1980, 1975, in Zurich, Hafner Christian 120 Office than Swiss more the of author for is projects photonics, coatings.He storage nanostructures, protection chemistry, and and solid-state structures, harvesting are organic-inorganic energy interests hybrid V.Shklover scientific on 1995 and main ETH Crystallogra- Energy.His 1991 at of the Zurich.Between of than ETH Laboratory at the the Bern, of working at of Materials scientist was University of senior the Department is the he at of 1991 first his phy since Switzerland, were and made in dissertations Zürich, and six of working 1974 University supervision, is his in he 1981.Under Sciences 1989 in prepared.Since of 1971.He doctor) of Russia Nesmeyanov-Institute Academy in (in from Russian Technology dissertation chemistry second the Chemical physical of in Fine candidate) Compounds of Russia Organo-Element Institute (in Ph.D. Moscow his at earned electronics for materials Shklover Valery opee ohudrrdaeadgaut tde negneigof engineering in studies graduate and undergraduate both completed eevdhsdpoa i hD ere n ei eed rmteETH the from legendi venia and degree, Ph.D. his diploma, his received a oni rne(ygztn n15.ercie i M.S.degree in his received 1956.He in (Kyrgyzstan) Frunze in born was rdae rmMso nvriyi 92 n a with was and 1952, in University Moscow from graduated .Cmu.Ter aoc.3 1–30, 3, Nanosci. Theor. Comput. J. eeaoe al. et Veselago 2006 REVIEW 3 (6) (5) and Anti- 5 are both and even 2 depends on n are simultane- and and of the continuity and (path 1–4) and for negative gr n = √ and and , we must accept that the index ph Negative Refractive Index Materials =± sin sin n 7 More on the history of anti-parallel and 4 is a scalar, not depending on coordinates and Later on, this question was considered more in n 6 The appearance of a negative factor of refraction Snellius law for positive 8 , the ray propagates along the way 1–3.This unusual (path 1–3). Here, the positive sign is used for the usual case, If we want to keep the usual notation of Snellius’ law All the previous considerations imply that the index of the ray propagates along the way 1–4 through the inter- earlier. parallel phase andSnellius’ law group as illustrated velocities in Figure immediately 1.For positive affect propagation of thedirection ray of is the a vectors consequence of the opposite face between two media.If oneand of the media has negative the frequency.Indeed, if the dependency(frequency on dispersion) the would frequency be absent, the energy of the whereas the negative sign is used when of the tangentialinterface components between of the the twoof media.Such wave waves unusual vector refraction was onSchuster. discussed the probably for the first time by velocity vectors may be found on the internet. detail by Mandelstam. and Fig. 1. work of Pafomov. negative. refraction time.It is necessary take into account that also for negative requires rewriting of (1) in more general way of refraction is negative when ously negative or when thevelocities vectors are of anti-parallel.Exactly the in phasetive and this index group of way, refraction the was nega- introduced in 1968 . ph (3) (1) (4) (2) was and does not complies . are possi- .There are are in prin- and gr and and and and : and . n and magnetic per- differ from the elec- 2006 E B and t .Thus, it is obvious that H c D and in more detail in the H S t c 1 c 3 and × c 1 √ E = =− =− = = = E H E S H rot k rot k for negative values of gr .The direction of the phase velocity H , but a left-handed one for negative form a right-handed triple of vectors for pos- and E . . k are simultaneously negative.The inverse state- ph and , and and H Remember that the Pointing vector One can immediately see from Eq.(2) that the vectors The main question now is how the electrodynamics of For uniform plane waves we obtain It is important to notice that the antiparallelity of the It is easy to show that the answer 3 is correct.Let us The choice of a negative sign for both and , , whereas the direction of the group velocity (2) Simultaneously negative values of (1) There areinvariant with regard no to theof differences, simultaneous change i.e., the signs electrodynamics is ciple impossible becauseprinciples. this conflicts with(3) some Simultaneously negative basic valuesble, but of the electrodynamics ofelectrodynamics such for materials the differs case from of positive with the direction of the vector the phase and the group velocitiesment are holds: antiparallel When when an the isotropic phase medium and areacterized the antiparallel, by group the negative medium velocities values of is of char- always forms a right-handed triplethe of vectors vectors together with For this reason,Materials NIMs (LHMs). are frequently called Left-Handed itive E the materials with negative J. Comput. Theor. Nanosci. 3, 1–30, of the wave coincides withk the direction of the wave vector Veselago et al. values of themeability dielectric permittivity vectors already described by Sivukhin consider the Maxwell curl equations three possible answers to this “rhetoric” question: trodynamics of materials with positive cause mathematical contradictions; innot particular change the this classical does expression for REVIEW fisra atcags ocaiyti,w write we this, clarify sign To the changes? when part changes, real vector its wave of the of part imaginary W eaieRfatv ne Materials Index Refractive Negative field neetn correlation interesting euto rsneo h mgnr at nteexpressions the in parts a imaginary for is the of part presence imaginary media, of complex.Its result lossy becomes in vector sign.However, wave negative the a obtains vector aeil otemtraswt eaierefraction. negative with usual materials from the transition possible to negative the materials tran- to amplifiers.This the connected quantum not with of is sition case material the a in as to absorption of absorption parts positive imaginary the with vector, at wave sign the the of change part to necessary imaginary is the it of sign the change 4 hneo h ino h elpart real the of the sign that follows the it of this change small.From is dissipation the when od.o h pca case special the holds.For for equations dispersion al hnetesg fteiaiaypart imaginary the of sign the change cally r eaieWe rqec iprineit,teenergy the W exists, dispersion frequency negative.When are and o h ae nmdawt eaieidx h wave the index, negative with media in waves the For n a aiyseta h aenumber wave the that see easily can One hsepeso spstv o eybodcasof class broad very a for positive is expression This eoe oiiewhen positive becomes utb rte nadfeetmanner: different a in written be must Fresnel hscllwEuto o omgei prahCretequation Correct approach nonmagnetic for Equation Cherenkov Doppler, Snellius, law Physical I. Table rwtrangle Brewster eeto ofcetfrnra alo light of fall normal for coefficient Reflection W < ntebre ewe w media two between border the on and Ti orsod oatasto rmamaterial a from transition a to corresponds .This = k n E = = 0 = hneo oepyia asfrom laws physical some of Change S,teqeto rssi h ino the of sign the if arises question the .So, k 2 c √
than + W + = → H jk =
n< 1
+ /z 2
= 0 ol engtv,when negative, be would c = j ph 1 c + √ = =
+ / r / A E 1 1 2 j n 2 − − 2 c gr = = +
A A √ + =
A and =
2 2 2 2 + 2 j (9) 2 → 8 aiylast the to leads easily (8) , k
+ n
osntautomati- not does = H j √
2 +
Frexample, .For k = j k
I re to order .In 1to fsin if is
= and 1. (10) (11) (8) (7) r ⊥ / = = sin n a htdatclycag for change drastically that las odto o h bec frflcino ih naflat a I). Table on (see light considered of the is reflection media when two of example, between absence border for the equations, for incorrect condition to lead may nteefrua,tevalue group. the second formulas, the these In to belong formulas, Fresnel’s particular, for obtained also is sign negative ysvrlidpnetgroups independent of several validity by negative the with verified metamaterial Eq.(5) a the such of made prism to is a,DplradCeekvefcsaei h first the in expression are the effects formulas, Cherenkov their and group.In Doppler law, lius’ the from approach turning when netic change that laws physical situation. permeability corre- with materials optics nonmagnetic and case negative.Many the electrodynamics is to of spond question equations this and to laws answer the general, In when n valid, electrody- sciences of technical formulas n related and and laws optics, the namics, all are extent what To result. orcgieta trmispstv o eaievalues negative for positive important is remains it.It it in of light that of recognize speed the to as feature medium, unique each a of is impedance wave impedance.The wave eghadtewoesse a ecaatrzdb a by characterized of be values effective may negative with system model wave- elements whole macroscopic the these the than smaller and of are length size them between field.The interact- distances antennas the interact magnetic and the the that are rings with antennas and ing the field, 5).The are electric wires, the Chapter essence, with (see straight in order wires, and geometric straight strict rings meta- a copper in first disposed of research.The NIM consisted for materials breakthrough of a values negative vided by characterized be could elcdby replaced n n 1 1 1 Adrc measurement direct .A →− = tg − h arcto ftecmoiemetamaterials composite the of fabrication The h aso eeto n ercino ih n,in and, light of refraction and reflection of laws The soecnsefo al ,teeaetregop of groups three are there I, Table from see can one As questions: important very poses NIMs of discovery The cos cos sngtv?Cnw lassml hnetesign the change simply always we Can negative? is n = n = 1.The and 2 = n /n 21 + − √ n ntg = n n 1 2 2 s o xml,i h aeo nlis law? Snellius’ of case the in example, for as, + cos cos A osqec,the consequence, a .As n omgei approach nonmagnetic n u to but 2 2 / = r 1 qain oteeatexpressions.Snel- exact the to equations .Cmu.Ter aoc.3 1–30, 3, Nanosci. Theor. Comput. J. √ √ .When / 9 n sin = fteageo ercinfra for refraction of angle the of Teeprmn a repeated was experiment .The n / 1 = r 10–12 /z ⊥ sin and √ = = where , = = n z z z . omgei approach nonmagnetic ihtesm positive same the with = .al ulnsthe outlines I 1.Table 2 2 2 hudb hne not changed be should ed omn formu- many to leads cos cos − n 2 1 r ohngtv,a negative, both are 21 z 1 /z = 2 2 + − z n 2 1 = z z = 2 − − 1 1 + cos cos eeaoe al. et Veselago 2 √ √ z 1 1 and 2 1 / / 2 1 nonmag- 1 utbe must / 1 9 sthe is 1 2006 pro- that and REVIEW 5 = n (15) and one .Case 2 B n is negative. : Light travels 3 2 is introduced 1 0 (14) /n .Furthermore, n 1 and AO 2 /n n = 1 2 n n = n n B& = 3 and are negative, the prop- n are virtual ways illustrating 1 2 O n turns out to be negative, 2 B .Case n 4 n B ndl L 1 AO B + A and AO 0. 3 1 (this value is really an eiconal) and that the real way of light corre- Negative Refractive Index Materials n = > B 1 L L between the points A and B in most 2 AO 1 /n in Eq.(4) can be negative, it is clear L AO 2 n n $%n a priori = = it will be a maximum.Note that both cases n $L B 1 .The ways B AO 3 Light propagation from point A to point B through a flat inter- 0: The light travels along AO > 2 The situation greatly changes if The optical length In the case, when both Since the value This expression changes when a negative /n 1 obtains n that the optical length A RHM is thenLight on one will side then and a propagate LHM along on the the other way side. agation of thecase, ray but with will onethe be important wave the vector difference.In in same thei.e., both first as from media case, A in is tois directed the B, directed along but previous against the into rays, the the A.Thus, direction second the case of opticaland the the length for wave rays, vector i.e., from B the real way fromspreading time, A i.e., to the Bby formulation is the of not Fermat’s time thecorrect principle of formulation shortest is spreading in based terms iscal on of length: not the “The extremum correct ofto real in the way the opti- general.The of localterm spreading extremum “local” light of takes corresponds thepossible into optical optical account ways that length.” that fulfill there Using the can the equations be (2)common several and case—when (3). thefrom index point of to point—is refraction equal is to changed the integral along correspond to Fermat’s principle. Fig. 2. face between two media with refraction indices for the LHM.Therefore, theoptical length” condition is “extremum valid.However, ofsible in the this to case confirm itsponds is exactly to impos- the maximumof or the exactly to optical the way.The minimum the type geometry of the and extremum on depends the on values . 1 n N) (12) = , and sin B 2 is zero: 1 n = $L AO &fuzzy 0 (13) = = sin 1 respectively.If n 2 B B& 1 n 1 2006 O O 2 is 2 and n is minimum and positive. n 1 L + in the expression under the n B + 1 1 1 AO → AO to exact formulas, for example, the AO 1 1 for n L = $%n and satisfy Snellius’ law L = Fermat%60s%20principle&ct and considerably change when turning from = $L change simultaneously.Note that the formula → n and are both positive, the ray travels along 2 and n Concerning Table I, we would like to add a considera- Note that the term “minimum” (of way or time) should This is confirmed by Figure 2 that illustrates the possible It is not difficult to show that Snellius’ law (5) is valid Finally, there is a third group of equations that strongly Here, the length 1.The light spreadsone from along one the point shortestthe of path.Here, minimum term space of time “shortest” to for implies the passing another “Physical this way.For encyclopedic example, in clopedia, dictionary,” M., 1983 Soviet one finds: Ency- in The space ray between of twothe light points always time along spreads of thatthat way, its connect along passing these which is points. less2.The than light along spreadsone any from on other one a way point paththe of that optical space corresponds way to toin the (minimum another minimum optical the length length).For of Britishsearch?query example, encyclopedia (http://www.britannica.com/ one finds: Light traveling betweensuch two that points the seeks number a ofthe path waves points) (the is optical length equal,neighboring between paths. in the first approximation, to that in angles square root. tion of a veryas the important Fermat part principle.This ofdifferent principle versions geometric is in optics, described the in known literature: two be replaced insometimes some by cases “extremum.” byattention To the to clarify term the this, “maximum”Obviously, we two that and now both formulations pay passes formulations of through are Fermat’s usual, correct principle. Right-Handedthey when Material both light (RHM), but arethrough not a RHM correct, and partly when through the a LHM. lightways of partly a passes raywith the crossing indices a of flat refraction surface between two media for the Brewster anglepolarization in of Table the Ithe light.For formula corresponds can perpendicular to be parallel changing polarization, obtained from that given in Table I by The optical length of this way if and only if the variation of the optical length J. Comput. Theor. Nanosci. 3, 1–30, and formula for the Brewster angle.Itthe is expression important under to the notefor square that the root Brewster inof the angle exact does formula not change when the signs nonmagnetic approach depend on Veselago et al. REVIEW akadwvsi rnmsinlnswr lostudied. also were ago, lines time transmission long in of obtained.A waves phenomena not backward the be devices can refraction such negative example.In devices typical electronic a neg- refraction.The vacuum are in by of waves described backward index known be negative well cannot and that permeability systems ative many in wavesexist backward refraction.Moreover, negative hereby and oee,teueo aeil ihngtv ausof values negative with materials of material of values isotropic use negative the an by However, in characterized appears waves is refraction backward that negative scheme have that specified.The we presupposes also when 2 are Figure elements in publications scheme first the the of dates of the in and outlined authors is 3.The Figure refraction” “negative and waves” “backward of values positive.negative always is receiver, the velocity, to group source the the that be from mind would directed in velocity” bearing phase appropriate, “negative velocity.In more term group the negative opinion, or our waves,” “backward so- negative to for called corresponds condition orientation essential antiparallel an refraction.Such is velocities group and media. dispersive group in known, different well are is velocities it phase velocity.As and group the then rather i.3. Fig. Materials Index Refractive Negative iso iesrcue bandfo Ccircuits. trans- LC 3D for from and obtained typical 2D structures proposed are line recently that mission in phenomena uniaxial present are these main LHMs in the obtained but be structures, not can refraction Negative index the on depends that wave the refraction of of is wind that phase slab total NIM 4. the is Chapter i.e., in image detail lens, its more LHM and such in flat object considered zero.Exactly an a to between for equal length observed optical is zero special length be only.In a will optical LHM length through the this passes cases, value.So, light any the and if negative sign any have can eaierfato.h eainbtenbcwr waves backward between relation refraction.The negative 6 and , h oia ceelnigtentos“simultaneously notions the linking scheme logical The phase the of vectors the of orientation antiparallel The h ocp fteotcllnt scnetdwt the with connected is length optical the of concept The iue3ilsrtstehsoyo akadwvsadof and waves backward of history the illustrates 3 Figure al tde ftengtv ercinphenomenon. refraction negative the of studies Early n sntteol a ooti akadwaves backward obtain to way only the not is n hc enstepaevlct flight, of velocity phase the defines which , and ”“eaiefco frefraction,” of factor “negative ,” , and , 1 2 n 1 . , tv ercinmyocrwtotangtv ne of index negative a Ye, without Zhen Following occur refraction! may refraction ative olwn rae ento fdrc n akadwaves the used: 5.However, backward also and Figure is direct on of shown definition is broader following it as antiparallel, are boml(eaie ercinRpitdwt emsinfo [19], from permission al., with et Y.Zhang refraction.Reprinted (negative) abnormal ahrte w-o he-iesoa ei ihnegative with media three-dimensional or two- crystals— then photonic rather to structures—equivalent mechanical i.4. Fig. n14.ohSutradMnesa eerdt earlier to referred Lamb. by by Mandelstam time work and first Shuster 1944.Both the in for shown was Shuster refraction negative and antb ecie yasaa ercinindex refraction permittivity scalar properties a the material by the described in but be propagation refraction, cannot ray negative the of to case propa- corresponds the ray crystal observed the 4.The in negative Figure gation with in materials shown in as and refraction, crystals coincide anisotropic not do in velocities well- both group of and directions and phases, the rays, that widespread refracted known well very is refraction: negative crystals.It a provide Anisotropic that materials mention of class to explored important is waves. backward provided that model ru eoiisi positive is velocities group aeW s h em“akadwv”we h vectors the when wave” “backward S term the use care.We refraction.” “quasi-negative nitreit aewe h vectors has the one when that obvious case is wave.It intermediate backward an a has one hold and wave forward as defined is when wave corresponding the , and ocrigteraiaino eaierfato,it refraction, negative of realization the Concerning h em“akadwv”sol lob ade with handled be also should wave” “backward term The and , k ih rpgto naYVO a in propagation Light 6 n10,adte nmr ealb Mandelstam by detail more in then and 1904, in ad codnl,tepaeadgopvelocity) group and phase the accordingly, (and, n As Poklington .Also 20 hs e.Lett. Rev. Phys. 16 hntesaa rdc ftepaeand phase the of product scalar the When .Cmu.Ter aoc.3 1–30, 3, Nanosci. Theor. Comput. J. adltmadLm osdrdlinear considered Lamb and Mandelstam ta defines —that ph ph 1 157404 91, · · 17 gr gr 4 tde iermechanical linear a studied < > 19 irsa.oml(oiie and (positive) bicrystal.Normal n (17) (16) 0 0 satno.hs neg- tensor.Thus, a is - hscs a ecalled be may case this (2003) ©2003. .© ph eeaoe al. et Veselago and n because gr 2006 are 7 REVIEW 7 (18) (21) (19) (20) increases z k is the trans- axis, and this value can not d z x k only.In material 26 x k + , 2 x 2 k in the free space: = 0 − + k 2 2 2 c c ≥ = is the direction of the propaga- d d>, 0 Negative Refractive Index Materials x z k = z
0 1 1 0 1 1 0 1 1
– – – It is remarkable that negative illumination is observed Y=(0,1,0) Y=(0,1,0) Y=(0,1,0) tom: Circular endings, lossthese tangent plots.The 0.0001. wavelength is Arbitrary 1 units (arbitrary are unit). used for absolute value of the intensity.Froma this, strong one absorption can is seetangents that obtained of even only for 0.01.minimum rather Since to small the the distance loss centerloss of peak the tangent, is first it also local isslab much decays obvious affected with that by increasing the the loss resolution tangent. of thefor NIM some intervals.In these intervals,minated the from focus the line wrong isflow side, illu- from which the indicates thatline lateral and energy sides then turns of backit towards the the again slab slab in crosses through such theside.The a the way focus question focus that line, nowmum” this is of time whether the from the illuminationmaximum the “first near wrong of the local the main mini- the negative peak zero illumination is illumination, or the when local the the resolution location is of defined using Fig. 23. lation of a truncatedtom 2D side.The electric NIM field slab isOnly polarized illuminated the perpendicular by left to half a the of planeshape, point the shown. (symmetric) loss source structure tangent to is 0.01 shown.Top: the equal Rectangular (complex to relative permittivity and permeability 2006 is applied.This code also provides features for Illumination (arbitrary units) along the focus line obtained for 89 88 Figure 23 shows the time average of the Poynting vector For more extensive studies on the NIM slab resolution J. Comput. Theor. Nanosci. 3, 1–30, Fig. 22. domain methods as longcated as geometry models and with linear notered and material very especially compli- properties when are highfollowing consid- accuracy results is the desired.For MMP the code solver contained in the MaX-1 the NIM configurationdifferent shown loss tangents in ranging Figure from 0.1 21.FEMLAB to simulation 0.0001. for Veselago et al. advanced post processing. field for a NIMcular slab endings.Since the with field rectangular alongstrong shape the even and NIM far with boundaries away is the cir- from shape the of source, the itstrongly ending affect is its and not resolution.As the trivial onethe width that can of endings see, the the strongly field NIMdistance near depends do it on not is thethe almost shape, field not but intensity affected.One along atstrong also the some when NIM can the boundary see lossthe becomes that tangent field very of near thethan the near NIM the boundary is focus may point.This small.Then, encountered gives even in a be the hint to FEMLAB much the simulation. problems stronger one best analyzes the illuminationthe along time the average focus of line, the i.e., dicular Poynting vector to component a perpen- linepoint.Figure parallel 24 to shows the the NIMwith illumination slab for a trough different single the cases pointsources, focus source for and the withsection electric two plane symmetric field (shown point perpendicular intion Fig.23).For (magnetic to the field the othercurves perpendicular), cross because polariza- one the obtains relativeative permittivity identical is permeability equal for to bothfree the the space.As rel- one NIM can andalmost see, the no the surrounding role shapeentirely) of (the and the corresponding the endings width curves play half of overlap the without almost NIM a may significantthe be change reduced focus.Obviously, to the of one NIM the loss illumination drastically near affects the REVIEW m r bevdat observed are ima peak.When broad itne ewe h ymti ore are sources symmetric the between distances ostnetrdcdt .01 re,mgna le aea e u 2 but but locations red red as different as Same with blue: Same points magenta, blue: source Green, half.Light 0.0001. symmetric one to to reduced reduced tangent green: width loss ending.Dark but circular source, red with as single black Same shape, as Rectangular 0.01.Red:Same 23.Black: tangent loss Figure in shown structure 24. Fig. Materials Index Refractive Negative sicesd h luiainpa eoe ekrand weaker becomes peak illumination broader.For the increased, is the i.e., for value, value bigger better the a take to we When therefore resolution. and distance mini- smaller leads local a obviously nearest illumination to the zero the to of peak location main mum.The the from distance the o h I ihls agn .1adpttosymmet- two put and at 0.01 sources tangent point loss ric with NIM the for aino h I lbi )ta ol eepce for expected function be nonlinear would a 1)—that has is lens”—one “perfect slab a NIM the of cation h itnebtentesuc points source the between distance the h aiu filmnto n h rtlclmini- distance local illumination the of also first that maxims source.Note observed the point between and single a between illumination for distance mum of the resolution than maximum slab reasonable NIM the more the much of are that definitions for ways different hr r w ore mgdb h I,bttedistance the but points NIM, image the the by between imaged sources two are there 20 prahsteln fpretimaging perfect of line the approaches nta falna dependence linear a of Instead inta hr sasnl onssuc only.When source impres- points the single a obtains is one there line that focus sion the of a point illumination exhibits center the clearly the at that maximum line global focus single the along illumination an bet rmn on ore smsedn eas fthe of because misleading is sources point many or objects hti hw nFgr 5I a ese that seen be can 25.It Figure in shown is that increasing
f(x) 1E1 0 0 luiain(rirr nt)aogtefcsln o h NIM the for line focus the along units) (arbitrary Illumination x source x source Tecre hw nFgr 5offer 25 Figure in shown curves .The circular ending,singlesource,halfwidth circular ending,singlesource,losstangent0.00001 circular ending,2sources,a=0.28 circular ending,2sources,a=0.3625 circular ending,2sources,a=0.18125 circular ending,singlesource rectangular, singlesource a = x sfrhricesd w qa max- equal two increased, further is x =± 0 5 =± 28 x d d source x image max = . 1 0.5 x x Ti hnidctsthat indicates then .This =± max 0 = 5 56 2 = d a x = x =± max x n ban very a obtains one d source x 2 max a =± 102 . source x d/ snteulto equal not is = = a temagnifi- (the o extended for ,w obtain we 2, Nt htthe that .Note ,ie,from i.e., 0, x x x d = source max max = 2 x x x 0 source 5 x source source 3635 with source ) ) . i.25. Fig. n emtiiy oersnneefc utb present. be must in effect atoms resonance of some role permittivity, permeability the and negative both play obtaining metamateri- that materials.For assemble atoms” natural to “artificial ways from many als are there general, In Negative with Metamaterials of Design 4.4. sources the NIM. between 2 the interactions for and by Fig.25 caused in points) (illustrated source properties imaging nonlinear uv)A nFgr 3 rirr nt r used. (brightest are 2.5 units arbitrary to 23, curve) Figure (darkest in 0.5 curve).As from ranging wavelengths different ilmnto ek ntefcsln)frteNMsa hw in shown at slab placed NIM monopoles the symmetric for two line) by ± focus illuminated 23, the Figure on peaks (illumination x source ne fRefraction of Index TeLcto n mltd r ie sfntosof functions as given are amplitude and Location .The Location x .Cmu.Ter aoc.3 1–30, 3, Nanosci. Theor. Comput. J. max lf)adamplitude and (left) a max rgt fteimages the of (right) eeaoe al. et Veselago x source 2006 x for = REVIEW 21 Later, a FDTD 1 However, since Swiss Then, the FDTD code 22 From the theoretical point 111 112 was applied. Negative Refractive Index Materials 57 was used in conjunction with the Transfer 55 Split ring resonators consist of two metallic micro strip For the analysis of a single split ring resonator, first of view, existing FITDtain and implementations FDTD of codes periodic thatcannot boundary accurately do conditions handle not thin and con- metalthe strips, are first certainly not choice.Brute-forceFDTD computations codes provide with very limitedficient standard accuracy for that the is accurate notbased analysis suf- on and arrays optimization of ofCurrently, split NIMs one ring therefore resonatorsNIMs and can may wire be still antenna. designed by hopealthough proper these numerical that optimizations, are much expected to better be extremely demanding. 4.4.2. Transmission Line Networks NIMs made of split ringbe resonators considered and wire as antennasAn arrays may alternative of is resonating to antenna fabricate structures. arrays of resonating circuits Fullwave rolls mayfrequencies—which be leads used tomental in setup—split a ring large practice resonators andcuit size similar only structures printed of cir- at operating theattractive. at relatively experi- microwaves low are much more rings with a slit.Theycular may and have very square different rings shape.Cir- configurations are most have simple.Even several these parametersof simple the (width micro and strips, height slit, distance between material the properties rings,rounding size of medium) of the the thatthat rings, need the substrate, to desired and bequency.The NIM tuned sur- split property in ring is such resonatorsmall antenna may a observed that be way at istivity.A considered some second responsible as (wire) fre- for antenna a must thealso be negative obtaining combined permit- with negative for permittivity.Thiseral contributes additional sev- modelare parameters.Because resonant, both the antennas the NIM design becomes and very fabrication narrow need banded to be and highlythe accurate. FITD code MAFIA code was appliedsquare to split the rings simulation with of wires. planar stacks of Matrix Method (TMM) for acular similar analysis and with square both cir- split rings. resolution might bein obtained this with case, a thenot thin slab exhibit silver a is negative film, noization index longer but of plays a refraction an metamaterial, (thereforecloser important polar- it to role), does contact and lithographya the than lens. configuration to is optical imaging with 4.4.1. Split Ring Resonators and SwissThe Rolls accurate numericalMaxwell analysis solver of Swiss wouldrequires rolls be extremely fine with extremely discretization.Fortunately, any approx- imate difficult solutions because may it be derived. In 80 that super 98 97 2006 For obtaining a NIM one may 22 Such structures are only used at rel- 22 or periodic boundary conditions. 81 Since NIMs at microwaves have periodic, i.e., crystal- Since the NIM slab performs the better the thinner it like structure, it(PhCs) is with respect natural to negative toimportant index properties.Here, to explore it note is Photonic thatvery the Crystals small cell compared size with of theeffects a wavelength.Thus, occur PhC near in usually field the is vicinityare of not a not PhC surface.These obtained effects the from NIM a by macroscopic simplifiedmaterials.This material model is properties that like very ordinary NIM important describes structures.For for example, the the resolutionis analysis of of the a NIM thin better slab distances that of thinner the theslab sources slab are. and is of and the the focus shorteris, line the one from finally the with can the consider wavelength.In a thissolution case, slab is a that sufficient.In quasi-static is quasi-static nearand solutions, thin field magnetic the compared field electric componentssufficient decouple.Therefore, for it the is electrostaticpermittivity only.It solution is to well known havedescribed that a by ordinary a negative metals negative are permittivitynear at plasmon optical resonance.It frequencies, has been shown, order to obtain isotropicat metamaterial least properties on three needs dicular “artificial directions atoms” for oriented each lattice in cell. three perpen- atively low frequencies.At microwave frequencies,itors capac- and inductors usuallytechnology, are i.e., fabricated with as microboards.A metallic famous strip microwave structures structure for onis obtaining the printed NIMs split-ring circuit resonator. assemble many Swiss rolls,general micro split-ring strip resonators structures orcial that atoms.” more Theoretically, play this should the bethe role in orientations such of a and “artifi- way that locationsrandomized of when the one atomsthat wants may are to be easily somehow obtain bea described an scalar by isotropic a permeability.This is scalar NIM permittivity done,ral and for metamaterials example, when are chi- fabricatedoriented by spiral suspending antennas arbitrarily cal in analysis some of dielectric.Thedemanding.Fortunately, such numeri- NIM isotropic metamaterials metamaterials usuallysist is con- extremely of “artificialcubic) atoms” lattice located likenumerical crystals.This on analysis drastically because ative only reduces cell) regular needs one the (usually to latticeical be cell code modeled—provided that (primi- canGreen’s the functions handle numer- periodic structures using periodic J. Comput. Theor. Nanosci. 3, 1–30, Veselago et al. Resonances are typicallythat observed are for not “artificialshrinking the small atoms” size comparedadvantage of of with the techniques “artificial the usedobtaining atoms” in wavelength.For one small electrical engineering may lumped for inductors.For take elements example, such one assisting can capacitors of roll and two in thinso-called a metallic Swiss rolls. foils.This capacitor essentially con- leads to REVIEW eaieRfatv ne Materials Index Refractive Negative lbilmntdb w ymti on ore,spatial sources, point symmetric 0.4 two resolution by PhC illuminated a wavelength, considering slab the resolution.By than the smaller limits certainly much which not is that constant tice section. this in described software commercial any and lines. transmission of network a in n ueia ehdfrsligMxelsequations with Maxwell’s simulated solving be for can method numerical configurations any such loss, low FDTD. but with using involved, simulated PhCs then was the slab of PhC diagrams finite band the the of analysis iuain eepromdt nlz h lbwt all- refraction. with slab negative PhC a angle analyze to performed were simulations nw,btteptnilo Isfrpatclapplications practical for NIMs well of are potential effects the elementary are but few known, structures a NIM only 3D even missing.Thus, or still 2D of studies extensive and ulcto,tePEapproach this PWE simulation.In the numerical sur- publication, from PhC and demonstrated source was between face distance short a of restriction ti otntrlt iuaesc icisuigTLM using circuits such simulate to natural codes course, most resonators.Of is ring split it printed for of line as technology circuits transmission same microwave corresponding the method essentially the TLM using the fabricate networks using to NIMs also simulate but to one allow only as(hs eerpre ae nanmrclanalysis numerical method. a matrix on transfer based a reported with were Crys- (PhCs) Photonic of tals properties refraction anomalous 2000, In Crystals ring Photonic split 4.4.3. for as same the are resonators. optimization structure of fafwNMsrcue nml,pim,sas n simple and slabs, prisms, (namely, lenses structures NIM few models a 2D to of and simplified and permittivity mainly this, structures negative of NIM both permeability.Despite analyze with to metamaterials used design be may codes commer- and cial methods numerical different many Currently, Art the of State 4.5. metals. of dispersion the with exhibit they problems because fewer favorable fre- are investigations, methods domain such analyzed quency candidates.For be NIM should as PhCs carefully also metallic-dielectric free,” and “for metallic frequencies optical provide of exten- at metals require resolution permittivity optimizations.Since negative would the improved.This and much be simulations how may sive 0.5 slab known limit PhC optical not the a is from away it far really Currently, not is it aging, stoi aepoaaini bevdi h PhC. the in observed is propagation than shorter wave rather was channeling isotropic that constant.Therefore, slab lattice PhC the the from than distance a at was source oecrflstudy careful more 22 h anpolmo hsfrspresn stelat- the is superlensing for PhCs of problem main The ic tnadPC oss ftosml dielectrics simple two of consist PhCs standard Since 106 108 ihafwseicprmtr eepublished were parameters specific few a with ) uha MEFiSTo-3D. as such , a reported. was 115 uelnigo h ihu the without PhC a of superlensing 113 114 115 37 65 93 nteesmltos the simulations, these In ae,2 n DFDTD 3D and 2D Later, lhuhti sencour- is this Although eieti h problems the this Beside a sal sdfrthe for used usually was 102 105 hsde not does This 46 na In 116 , . meitl ed oangtv ne frefraction of index negative a to leads immediately edslesta r atadacrt ttesm ieare time same the at accurate and required. fast solver.Thus, are field that the solvers dif- field of inaccuracies of by thousands disturbed heavily of are procedures analysis optimization structures.Furthermore, numerical ferent the optimizations require obtain- metamaterials.Numerical may for quality required opti- high be worse.Here, would ing even models 3D is based situation with mizations the of structures software properties, metamaterial commercial of NIM with design experience the maybe of scientists.For and lack studies, advanced the extensive of also for requirement routines processing the effortpost structures, computational high complicated the all. are for at this explored for been reasons not main The has geometry complex more with ie a rpsdb edyadcwresi 1998. in group coworkers same and the Pendry 1999, In by metallic proposed from was frequencies wires neg- microwave with at materials permittivity fabricate ative to concept experimental The NIMS OF 5.FABRICATION eaieidxo ercinfr1D for refraction of with index CMMs Transmission negative of (b) a fabrication of and configurations.The substrate (TL) combinations Line a (a) on properties: elements conductive NIM exhibiting CMMs itive 0 l CM)adPooi rsas(hs.Msexhibit permittivity (PhCs).CMMs negative Crystals simultaneously MetaMateri- Photonic and Composite (CMMs) refraction: als negative a exhibiting NIMs. to way per- the negative obtain opened also meability.This to so-called (SRRs), elements, Resonators conductive Split-Ring of non-magnetic (resonance) in motion collective electrons of response inductive the elzdfrfeunisaon 0Gz( GHz first 10 were around fields, frequencies for capable electromagnetic LC realized external plates, of with circuit interact arrays electronic to designed on mounted artificially oscillators, of consisting CMMs Metamaterials some Composite 5.1. at characteristics dispersion of frequencies. peculiarities of bility eetdrcin nsaeadpstoe nacbclattice, cubic permeability effective a with on metamaterial positioned a and dif- space a three in provides in directions ferent oriented one SRRs bigger many the capacitance.When large inside a smaller ring at the oriented resonance and diameter oppositely provide ring rings than larger the in much shown wavelength in cir- is splits SRRs copper 26.The the printed of Figure geometry using specific with fabricated very cuits.The combined were SRRs wires the straight reported.First, been has gation Deeti hsaecmoe fmtraswt pos- with materials of composed are PhCs .Dielectric w arcto ehd r sdfrtems fthe of most the for used are methods fabrication Two urnl,w a itnus w id fmaterials of kinds two distinguish can we Currently,
and
< 0
u xii eaierfato because refraction negative exhibit but , .Cmu.Ter aoc.3 1–30, 3, Nanosci. Theor. Comput. J. ihnacranfeunyrange.This frequency certain a within 22 rpsdt aeavnaeof advantage take to proposed 1
n 2D and < , 0 ≈ eff eeaoe al. et Veselago n permea- and cm). 3 sobtained. is 9 44 propa- 117 n 2006
< 21 REVIEW × = 118 23 d (32) 44 524 mm 5 d comprise 0 1 1 g L 2 = is much smaller 40 cells, 105 h d × − 105 mm).The square P for the 2D transmission × d = of height .© 2003, Optical Society of N 94 5 (2003) 2 are discussed in the paper. 21 cells, 105 shown in Figure 28 is an exam- × = Negative Refractive Index Materials r d 119 0 11, 696 /g C 2 1 are the intrinsic material parameters of describe negative frequency dependent and P 0 − L P and m wide microstrip lines with separations of = and Optic Express P Prototype of large periodically loaded TL CMM, containing 0 C N 400 The alternative principle of the Transmission Line (TL) = where the reduction offrequencies, SRR the dimensions placement for ofof X-band wire the microwave strips SRR, behind2 substrate increase per of unit theprinting cell.Photolithography density the technique of SRRs was and theof used the the thin wire for substrate wires plates.It elements is on to and important the the to two note wire that sides elements theand must SRRs negative provide permittivity negative at permeability overlapping frequencies. contributions.The NIM unit cellgrid loaded consists with of surface-mounted capacitors aembedded and in microstrip inductors the substrate.The wholeon device a was ceramic mounted substrate ple of thiscan approach.The be equivalent obtained material from parameters Eqs.(32): The large periodic array Fig. 28. two regions with200 mm) negative and index PRIinset properties of shows (21 scheme refraction of theline (21 unit NIM cell material.The of theinsets.The real size near-field unit detection cell probeconsidered is structure as also is homogeneous shown.This shown media ifthan on may an the external be the unit wavelength.Reprinted with round cell permissionIyer from constant et [119], A.K. al., America. Other modifications compared to the first study approach andparameters the corresponding equivalent material the host transmission line mediumitive (that parameters contributes of to the pos- sions NIM material).The reactive inclu- and consists ofof regions refraction.Both with negative mediaw and are positive built index on a square grid of 5 mm.The media may be considered as homogeneous if = 44 d 9 (31) 5 mm. 5 1 4.845 GHz = 25 mm, 5 ≈ r 0 Phys. Rev. Lett. = c 2 mm, 5 0 iw1 = 2006 22 2 d + 2 0 F − 8 mm, 2 5 0 .© 2001, American Institute of Physics. = 62 mm.Unit cell forms lattice constant of − 5 c 2 1 = (2001) = w eff is the dissipation factor.For obtaining a 1 78, 489 46 mm, .© 2000, American Physical Society. 5 0 .Reprinted with permission from [44], R.A.Shelby et al., is the fractional area of the unit cell occupied = one obtains approximately: Design of the unit cell of SRR used for construction of 2D Design and resonance curve of individual cupper split-ring res- g (2000) F eff Microwave scattering measurements were performed 30 mm, 5 84, 4184 Reprinted with permission from [1], D.R.Smith et al., with a 2D squarements array fabricated of with SRRs acially combined period available materials.The with of 2D wire 5.0 isotropy ele- ofachieved mm this using by NIM was commer- placingally the perpendicular SRR directions as elements illustrated along in Figure two 27. mutu- where by the SRR, NIM the SRRs are combinedprovides with negative a permittivity. thin wire medium that Appl. Phys. Lett. J. Comput. Theor. Nanosci. 3, 1–30, Fig. 27. CMM.Resonance at GHz was observed.Dimensions: Fig. 26. For onator (SRR) for construction of 1D CMM.Resonance at Veselago et al. was observed.Dimensions: 0 5.0 mm and consists ofthick) 6 mounted copper SRR on and 0.25long 2 and thick copper located wire fiberglass strips on G10 (0.25angle the mm substrate. other of side Wires 90 of are substrate.Substrate 1 plates cm make an REVIEW wavelength). z iae ihL eoac at resonance LC with 400 ricated a on microstructures eaieRfatv ne Materials Index Refractive Negative 24 split- (closed of CSRRs suggested. spectra was and transmission resonators) SRRs the ring using of fabricated study permittiv- comparison CMMs negative a and by permeability ity of negative of inset regions the in shown as cor- 28. the Figure structure, for properties infinite MHz) in predicted responding properties, the (960 NIM with frequency exhibiting agreement Bragg good GHz, from 2.5 prop- approximately range backward-wave to the of in region agation well-defined characteristics a dispersion revealed measured wavelength.The constant nal cell unit the n togmgei ciiyadngtv emaiiyin permeability negative demonstrat- and array mid-IR. activity nanostructured applied magnetic a fre- strong successfully of optical ing was fabrication visible the to design even for down or CMM scaled infrared new at be quency.A NIM not the probably achieve may support dielectric 8 1 in resonance of size observed range gold cell experimentally the a than unit above smaller substrate.The layer much dielectric a ZnS on a layer on staples gold sized eil ihL eoac at resonance LC with terials capacitance. the increase further to aeyb h fetv permittivity effective approxi- the characterized by lattice—are electron mately 3D the constrained along structures—with transport wire SRRs.Thin properties. active enable to frequency) resonance at field eidcsrcuewt atc osati h sub- the in constant lattice a ( range with wavelength structure periodic a oprlyr nacrutbadwith board circuit a on layers copper inadtikeso h ie htfr h lattice. the form that wires concentra- the the of thickness varying and by tion region microwave and infrared hr h lsafrequency plasma the where igesltrn eoaoswt size with resonators split-ring single ic o10n,acmaidb eoac hf to shift resonance a the by of accompanied reduction nm, The 180 to studied: transparency pitch were 10 its structure and to designed 2 grating due the between pitch chosen lengths a was wave with ZnS for staples nm.The Au 600 of of growth the for applied iearcaglrltiewt atc constants lattice with lattice rectangular a vide d 5 5 mm, 8 30 ietosrespectively. directions = sn ht-rlfrtdpoes h lnrSRR planar the process, photo-proliferated a Using nitrsigapoc o h eicto fthe of verification the for approach interesting An npatc,temtli R lmn tutrso a on structures element SRR metallic the practice, In aoarcto a sdi re ofbiaemetama- fabricate to order in used was Nanofabrication qain(1 rvdsteefciepreblt for permeability effective the provides (31) Equation 1 5 6m eearne ntefloigwyt pro- to way following the in arranged were mm 66 ,adtecag ftedeeti aeil(SiO material dielectric the of change the and m, 123 a z hsnwCMcnit fary fnano- of arrays of consists CMM new This = 121 6 = 5 h 3 The m ,1,ad1 nt along units 18 and 15, 5, mm: 5 4–7 ,/ 7, d .nefrmti ihgah was lithography m.Interferometric eff hc uSReeet formed elements SRR Cu thick m , smc mle hna exter- an than smaller much is = stewvlnt fexternal of wavelength the is 1 ≈ hc usrt eefab- were substrate thick m − 0 H ( THz 100 120 p , p 2 R nt aeo thin of made units SRR 2 a erdcdt the to reduced be can ≈ H ( THz 1 .oictosof m.Modifications = s 4 ≈ = 5 a n thickness and 4 3 = 2 mand nm 320 m). 0 mis nm 600 ≈ a x x 300 117 , = y 21 and , Gold a (33) , 122 y = = m 2 ) ei edculn ihteL eoac assnegative causes resonance mag- LC permittivity. the the with where coupling normal range field at frequency netic with resonance the capaci- spectra LC indicate the transmission the simulations the with of of excitation field an incidence.Comparisons external to the leads tor of component the trical than smaller of wavelength were resonance elements elements).All SRR 3136 raswt atc constants lattice with arrays auePbihn Group. Publishing Nature thickness epnea pia rqece.erne ihpriso rm[124], from permission al., with et frequencies.Reprinted A.N.Grigorenko optical at response 29. Fig. a emtiswr lofbiae.h emtywas geometry fabricated.The also were geometries lar red at ( resonance light plasmon provides geometry cell sen.This coupling. light of incident h constant efficient lattice more a to At pil- leads non-cylindrical shape a lar and range. calculations, process visible numerical microfabrication the the preliminary in of NIMs optimization reso- the of of lithography After fabrication current number the to smaller way for the a methods opens to also but (due modes), reduction nant at loss used to only geometry lead not essential may split-ring geometry an resonator double is simple microwaves.Such pillars the which Au resonators, of short basic of simplification pairs as a used by (Fig.29) created was resonance plas- mon asymmetric material.The non-magnetic intrinsically emltorpy hyfbiae ei nasraearea surface a on media S fabricated they lithography, beam aesceddt aea motn tptwrsNIMs towards frequencies. step important visible an at make co-workers to and succeeded in extremely Grigorenko have namely result losses.Nevertheless, visi- could technologically big of which too aside, interaction is material the with light approx- on which ble to questions down leaving nm, features demanding, critical to 10 of down size cell imately the unit the and of nm size 100 the scaling requires quencies ≈ = h elzto fPnr’ idea Pendry’s of realization The . mm 0.1 09 maddaee of diameter and nm 80–90 , E atr fmdu aryo uplas ihmagnetic with pillars) Au of (array medium of pattern SEM ≈ d 7 m.utemr,NM ihohrpil- other with NIMs nm).Furthermore, 670 2 = ihngtv emaiiyi iil using visible in permeability negative with 0n eearne nproi square periodic on arranged were nm 20 .Cmu.Ter aoc.3 1–30, 3, Nanosci. Theor. Comput. J. .Ptoi.Nature Petrovic. J. 124 a , sn ihrslto electron- high-resolution Using = = 0 m uplaso height of pillars Au nm, 400 a 3 = .opigo h elec- the of m.Coupling 5–0 m(56 nm 450–900 d 22 ≈ 3,335 438, tvsbelgtfre- light visible at 0 mwr cho- were nm 100 (2005) eeaoe al. et Veselago ©2005, .© × 56 2006 = REVIEW = 25 The a that the 132 (1) Dual- 119 600 nm.An 133 For the mea- = rods (dielectric 132 129 3 a O 125 2 131 , the PhC was arranged g The PhC pattern was fabri- rods.It was noted 130 3 175 and 0.35 was fabricated for O 5 m. 2 Negative Refractive Index Materials 0 125 nm) at the telecommunication = = 55 5 r 1 r/a = which opens the way to a variety of well , 9, radius 0.316 mm, height 1.25 mm) on a A 2D unit cell consists of resonant structure 128 =
122 125 4 RH TLs of Branch Line Couples (BLCs) and ring cou- 2 and 1D or 2D (depending on the unit cell structure) The experimental demonstration of the light focusing The concept of composite right/left-handed (CRLH) An interesting PhC consisting of Al mode propagation, between 8.6and and between 11 GHzexperimental, 6.4 band-structure, (TM and and mode) simulationsachieved, 9.8 results were GHz (TE mode). Extensive transmission line theoryrials (TL) can approach be forat CRLH demonstrated least mate- by simulated microwave several applications: already realized or important point is thatat the the external resonance free-space frequencyscale wavelength is of much the larger unit-cellsider than dimensions.This the the allows designed length one materialwhich to as is con- effectively not homogeneous— theability case for in usual these PhCs.Thesimulations. negative nanomaterials perme- is confirmed by numerical 5.3. Applications Many NIM concepts wererealized suggested up but to now.One only ofculty a the of few obvious the reasons were fabrication is ofhomogeneity.Current artificial the materials diffi- NIMs with rarely sufficient to achieve structure a unit wavelength ratio better than 10:1. negative refraction in this PhC proceedsbands in and higher does (valence) nottransmission interfere efficiency. with the Bragg reflection and due to negative refractionusing at a infrared PhC with frequency airindex was holes contrast done on InP/GaInAsP/InP a slab hexagonal lattice (lattice in constant a low- 480 nm, hole radius wave length of square lattice with measurements of lowering group velocity. as parallel plate waveguidetric with field TM parallel mode to and the Al with elec- band components capable to reducenents the in number wireless communication of systems.By compo- ,/ replacing the plers with CRLH TLs, thelimitation harmonic can operational be frequency overcome.(2) Neworder antennas resonance include zero (ZOR) antennas,/ of physical size less than cated using electron-beam lithographying.The and thickness chemical of etch- layer the is InP 200 nm top andis layer 420 a nm and respectively.Another PhC example intermediate composednanoscale. of metallic resonators structured on a surements of the group velocity constant metamaterials was suggested by Caloz and Itoh. with square Ag columns of side length tailored PhC structures.Thewere advantages reported of to be metallic highestuation, PhC dielectric and constants, the low possibility atten- of focusing. = r ratio Trans- Further- r/a 126 are positive 124 63 cm were 5 0 = range (for exam- and r , 2006 –M direction and indi- 26 cm and radius 5 1 Two techniques, manual 1 3 at the green resonance), = 5 127 0 h is the lattice constant) forming a ≈− 2(
5 0 within the same This PhC crystal consists of cylindri-
= 60 cm and outer radius 125 r/a 7 and = 5 125 0 h has a rather large imaginary component that pre- ≈−
and negative Negative refraction in a metallic PhC with hexagonal Negative refraction at microwave frequencies was 63 cm, with
5 arranged on aof triangular 0.2. Negative lattice refraction with was found the for same both TM and TE more, a remarkable optical impedanceobserved, matching which effect was is characterizedof by the the reflection totalwith from suppression different refraction the indices.Thisibility interface leads of between to the total two structuredpolarization invis- of media films the at incident green light. frequencies at TM 5.2. Photonic Crystals The negative refractiontonic can crystals (PhC) be that—ininhomogeneous realized contrast media also to with theble with a to CMMs—are pho- the lattice wavelength.Although constants both compara- in dielectric PhCs,refraction typical and superresolution NIM canliarities phenomena be expected of of from the pecu- negative The dispersion main characteristics advantage of ofthey PhCs certain can over be PhCs. CMMs more currently easilyfrequencies. scaled is to that 3D and adapted to visible lattice acting aslens) a at flat microwave frequencies lensfor was without reported TM optical at mode. 10.4 axis GHz (Pendry assembly of aluminaused rods in this andgated study rapid in for the phototyping, fabricating wavetive low-loss were range refraction PhCs, from was investi- 26 showntransmission both GHz mode in to experiments FDTD 60 on calculations GHz.Nega- a and 2D lattice. mission measurements confirmed negative refractionthe using interfaces of thecated the PhC maximum in angular the range13.7 of GHz. negative refraction 2D at were and used 3D for the PhCs demonstrationand consisting of millimeter NIM of in wave alumina the ranges. microwave rods cal Cu rods of the height a triangular lattice.Furthermore,the cylindrical height Cu tubes of J. Comput. Theor. Nanosci. 3, 1–30, observed in both dielectricple, using and a metallic square PhCs, array for of exam- alumina rods in air. Veselago et al. optimized to provide efficientwithin electromagnetic the interaction pillar pair.Theresonances symmetric for and individual anti-symmetric be treated pillars as and plasmonreflection resonances pillar spectra in pairs nanoparticles.In atpolarizations, the could “green” normal and incidence “red”during the for resonances rotation are of TM the observed ture and samples, of dependent the TE on fabricated theout, struc- media.As though the the authors fabricated have structures pointed exhibit both negative ple but vented the observation of negative refraction. 0 REVIEW togcia fettan effect chiral strong 30. Fig. Materials Index Refractive Negative I einadla oteetnino eaierefrac- negative of extension the field. to tion lead and the simplify design could NIM polarization.This one only of refraction plates. RH RH metamate- with parallel CRLH possible 2D two placing not between by rials are realized that be and can index materials frequency refractive certain negative a curved a and at exhibit flat that Both point.(3) lenses transition microwave LH/RH the veloc- group at non-zero ity TL with CRLH wavelength in infinite because forward an radiation and supports broadside region with region LH RG the the in backward CRHL radiates LWA scanning.The capa- (LWAs) backfire-to-endfire continuous Antennas of Wave ble Leaky scanned frequency bemros euiyiaig imlclrfingerprint- sensing. biomolecular remote imaging, for ing, security used tun- mirrors, be lenses, adaptive can able cavities, frequencies compact as optical applications and such THz at response r led sdi h H optics. THz wavelength the a in order used of already dimensions are the with grids and wires erne ihpriso rm[3] J.B.Pendry, [131], (2004) from permission with Reprinted etfruainaewrhmninn eas hymay devices, they electromagnetic new because principally mentioning to lead worth are formulation cept 30. Figure in shown is structure chiral nant ea iefitr n hs hfesfo irwvsto microwaves from shifters phase phase frequencies. and and optical filters including group time applications the delay new of interesting provides directions velocities in opposite propagation with wave NIMs backward waves.The evanescent of enhancement backward-wave and diffraction, for sub-wavelength medium propagation, homogeneous effectively an 26 h nrdcino hrlt ih euti negative in result might chirality of introduction The h arcto fmtmtraswt magnetic with metamaterials of fabrication The e da nNM htaesilo h ee fcon- of level the on still are that NIMs on ideas New ©20,SinePrisosDepartment. Permissions Science 2004, .© osbedsg o eoatcia tutr.odto of structure.Condition chiral resonant for design Possible 131 h xml fpatclraiaino reso- of realization practical of example The ? 129 ≈ 121 +r/, ifrn lmnslk metallic like elements Different 0 at r , 0 ed osalvle of values small to leads Science 132 0,1353 306, namely, ? . i w ifrn hs eemaue experimentally. measured were PhCs) different two (in mg ihsz uhsalrthan the another. smaller transfer to much space can size that with device image matching optical some rather tcnitdo ealcwrsadSR sebe on assembled (Fig.31). SRRs structure GHz. and cell 14.7 wires periodic at metallic operating a of was CMM consisted PhC.The geometry It and curved CMM with date: to lens studied NIM both of using designed types were range microwave the in refraction of enough. width small the is when that slab observed NIM is remembering the only and effect mind lens” in “super the this keeping lens,” “super h a ecie nScin52 ale,slow Earlier, 5.2. Section in described reported was of those PhC to arrays similar was parallel (WSs) previously. interleaved strips wire of and SRRs consisting CMM, The euto fthe of reduction etoe bv n spitdoterirb n fthe of one by the paper earlier this out to appropriate.As of pointed back really authors as is come and lens” above we “super mentioned 7, term the Figure if in question illustrated lens, that NIM shifters phase materials. conventional and with filters possible line not delay are veloc- are group transmission materials low low of ity with applications PhCs important and most loss.The CMMs for velocities expected group are lower PhC).Even 2.5:1 polystyrene to colloidal compared PhC in fabricated in (9:1 contrast dielectric esfor lens lens for plano-convex a achieved the was by lens.Focusing this for obtained ratio and the ture with lattice square a on arranged ae htnccrystals. photonic based otedfiiino esa nisrmn fgeomet- of instrument an as lens ( a optics of rical definition the to td steiaigo a-edmcoaeradiation Fig.32). microwave (PhC, crystal far-field photonic of dielectric imaging a the using is study reported. was measurement and simulation the between ftePCbn tutr steflteigo h adat band the of flattering where the edges, is band structure the band PhC the of re fc3hdbe bevdi Si/SiO in observed been had c/3 of order edtrie rmtebn structure. can band that the dispersion wave from the determined of be character the different by with GHz explained PhCs 7–8.5 for CCM, GHz for 7.7–8.5 GHz and 9.9–10.3 of window frequency n 2c eeipeetdTecrepnigPhC Al of corresponding array periodic 17.5, implemented.The a 13.5, were of of consists cm radii 22 curvature with and Lenses observed. index was refraction negative frequency-dependent ceitc)adcrepnigyrdcdaerto nthe in aberration reduced correspondingly and of acteristics) value any for curvature equation Using o ru eoiiso /0(naCM n c/10 and CMM) a (in c/50 of velocities group Low h uvdpaocnaelne ihngtv ne of index negative with lenses plano-concave curved The eoedsusn h xeietlrslso h flat the of results experimental the discussing Before h eodeapeo neprmna I lens NIM experimental an of example second The n< f 9 stefcllength), focal the is 44 0, a>,>$ h ple arcto ehdo h Al the of method fabrication applied The R> n 136 .Cmu.Ter aoc.3 1–30, 3, Nanosci. Theor. Comput. J. = n> g ntefloig eaeuigteterm the using are we following, the In 1 .I esshv agrrdu of radius larger a have lenses 0.NIM 129 sdet togmdlto fthe of modulation strong to due is − 1, 134 h I lbde o correspond not does slab NIM the R/f , g R< a 135 n< sepce ob low.Strong be to expected is = ( R h esrdlow measured The ,adfraplano-concave a for and 0, b srne ouigchar- focusing (stronger 0 sterdu ftecurva- the of radius the is n + 2 O =− c 3 .h spres is “superlens” ).The 137 , os( rods 0 129 rmoepitof point one from 5 t92 H is GHz 9.25 at 4 odagreement Good r/a h e feature key The 2 n AlGaAs- and eeaoe al. et Veselago r/a = ai)cnbe can ratio) 8 < n 5 = )i air, in 9) g g i the (in 0 fthe of 138 5 2006 175. 2 O 129 A 0 3 REVIEW 27 The d> 0.6) to can be 141 55 THz 0 = k ∼ 4—because 0.85). ,/ = is the depth of Cylindrical and $ 6or 139 the sub-diffraction- / The development of 0 650 nm, NA , 110 140 = , , where 2 405 nm, NA = , Negative Refractive Index Materials (NA) is of great importance: When ,/ d = 2 value corresponds to the diameter of the max $ 365 nm, the image of 60 nm nanowires on In the second example, = sin 123 is refraction index of media, and NA is numerical , n ,/n The difficulties in NIM design in infrared (IR) and The phenomenon of superlensing becomes very promis- Progress in the design of graded negative index of ≈ 120 nm pitch was demonstrated, or necessity to use shorter wavelengthsdensity to increase can the be storage illustrated$ by the diffraction limit problem: (mid-IR). these fabrication methods wasority called for as device-based fundamentalorganized research pri- for in radio, thelattice millimeter engineering.Only field wave a and oftal few photonic materials results of super could successful begold experimen- mentioned.One example pattern is with aa periodic lattice ZnS constant dielectric 600 layer, nm showing separated resonance by at aperture.The smallest resolvable mark.But the imageventional resolution optical in storage con- is limited by the diffraction limit focus, the index of the PMMA,1.5. lens, Note and photoresist that was aphotoresist close second to was PMMA missing.The spacer calculatedof between this transfer lens silver function superlensing and a system range allows of to evanescent estimate, component that from 2 to 4 limited imaging withsurface a plasmon silver excitation super was reported.Thesystem lens superlensing in slab the assisted second50 by example nm has thick the patterned following Crwave) structure: structure over (illuminated by a athick 40 plane Ag nm filmdirectly thick (playing is the PMMA placed on role spacer alight of 120 on nm of the thick a photoresist.Using “super UV 35 lens”) that nm more complexusing spherical a GRIN NIM17 lenses GHz.Ray slab were tracing calculations withsimulations fabricated were and used graded MicroWave to Studio determine index the refraction of gradient. refractionvisible light at essentiallylimited limit success the in NIMstressed the applications.The also fabrication in of theresults press-release optical in “List LHMs ofof the was recent excellence METAMORPHOSE. area research of materials” by the EU network enhanced and recovered by plasmon excitations.Theness thick- of silver layer 40 nm, the enhancement isAlthough damped promising by results material were absorption. achieved,are many open: questions How can theinated? observed significant What losses is beon elim- the the dependence thickness ofspacers? of the the transfer function silver layer anding, on when the considering the PMMA age needs of density increased opticalstorage.These in stor- trends digital arement obvious recording from in “red” and (normal the DVD, in recent develop- the computer “blue” optical discs (BD, refraction (GRIN) lenses was reported. x 86, 201108 .© 2003, American 2006 (2003) Appl. Phys. Lett. 138 84, 3232 .The lens was fabricated in a cylindrical y Appl. Phys. Lett. (a) 0.251 cm 138 Scheme of microwave focusing device (plano-concave lens) Composite metamaterial (CMM) lens operating as optical ele- .© 2005, American Institute of Physics. It is important to note that the bandwidth of the PhC (b) Institute of Physics. J. Comput. Theor. Nanosci. 3, 1–30, Fig. 32. based on photonicmicrowave source, crystal was (PhC).An keptsurface.The at sensor X-band was a waveguide, mounted distance tothe of used a electric-field 150 XY component as translation cm of stage,with a from the scanning permission the relevant for from flat microwave [138], range.Reprinted lens P.Vodo et al., lens (2 GHz,larger or than the 22.7% bandwidth of ofdispersion. the the CMM operation lens, due range) to weaker is much image compared withNIM positive lens index has lenses.Besides, muchof lower the weight the than same a focalspace conventional lens applications.Notice length that what the could refractiveable be index in achiev- advantageous a for PhCfocal allows length for further anddimensions controlled could of change the lead of optical the to systems. further reduction of the (2005) Fig. 31. geometry.(a) Unit cell structureture of of lens the based.(b) lens constructed Plane-concaveradius from of struc- flat curvature is unit 12 cells.LensParazzoli cm.Reprinted aperture with et is permission 20.3cm, from al., [137], C.G. ment with negative indexand of varied refraction.Arrangement thickness of along 4 cells along Veselago et al. REVIEW eaieRfatv ne Materials Index Refractive Negative o ueNM u xii eaierfato n similar are and refraction that negative we crystals exhibit properties but photonic NIMs NIM of differ- pure provide not structures to that considered addition structures also present.In of be types are may that ent NIMs equations when and description laws applied mathematical physical materi- proper fundamental index a of negative including histor- of (NIMs), theoretical, aspects als important experimental most and the ical, outlined have We OUTLOOK AND 6.CONCLUSION nanocrystalline thin AgO in (aperture) effects to image-forming important plasmon is surface layer.It that silver stress nanocrystalline thin a using paper the limitation high-density in diffraction considered for no first effects with was recording lens” progress. near-field “super in optical is of technologies application recording The new for search and ftersntrmtra.eea prahst solve selection to suggested, the approaches were in material.Several problem difficulties this resonator to the due of problems basic ters was slab the and pinhole Ag nm, 150 the between separation The akdprilscnann nidcdmgei moment resonance, multipole magnetic high-order induced exhibiting and an containing particles packed loetnieystudied. extensively also case). this in nm 100 1 of at NIM of fabrication 10 around iae fatraigZSa ihrfatv-ne mate- high-refractive-index as ZnS rial alternating of tilayer tbe u edl n ucl rsbe d Single-phase be (d) material. to erasable, the have quickly of character and marks readily recorded readout but (c) K), stable, beam), (700–800 laser erasing (low-power regimes: K), transitions three all (900–1000 state of functionality of Recording solid (b) reversible, material (a) be to the including: have layer, for recording requirements the important sev- equally crystalliza- are eral there and deformation.Incidentally, large melting without showing tion BeFeO, (d) fcc tunable recording), transitions showing exhibiting FePtAg films alloys nanocrystalline (c) alloys CdSe photoluminescence, change nanocrystalline reversible phase (b) transforma- GeSbTe tions, phase (a) crystal-amorphous reversible them exhibiting among gested, n oi Iscmoe fsltrn eoaoswt nano- with resonators split-ring dimensions, of composed NIMs tonic a ecniee spoiigmaterials. materials promising as related near-field considered be and can super-resolution discs.SiC in (super-RENS) structure already realized cessfully ac.h atapoc a elzdexperimentally realized reso- was plasmon approach surface last thin nance.The exhibiting (d) nanospheres, layers plasmonic nanocrystalline resonating in inclusions 28 2 h arcto fNM totclfeunisencoun- frequencies optical at NIMs of fabrication The aeil o eesbeotclsoaercrigare recording storage optical reversible for Materials = n 1 1 = 5 38 , 2 5 a aidbten10ad40nm. 450 and 120 between varied was 144 35 sraeplrtn)adcudb sdfor used be could and polaritons) (surface m a sdAfcslk fetwsobserved. was effect focus-like used.A was b aoaeil ossigo closely of consisting nanomaterials (b) n MgF and 2 141 tecl a ob ftesize the of be to has cell (the m slwrfatv-ne material low-refractive-index as eea aeil eesug- were materials Several 143 x 142 aes(5n)wr suc- were nm) (15 layers → mn hm a Pho- (a) them: among hr eidcmul- periodic a where ergnl(magnetic tetragonal 146 i has SiC 145 c nano- (c) < 145 0 a rqece,wdntefeunybn n euethe reduce and opti- band towards frequency losses. range the frequency widen the frequencies, size cal push the structures, reduce to the important of is metamaterials it these lossy.Therefore, third, very and are band frequency narrow only a properties for NIM larger. exhibit much metamaterials be these may Secondly, frequen- structures metamaterial microwave the at where out cies carried were both experiments all, these with of permeability.First negative metamaterials and of permittivity fab- negative the characterization by and confirmation rication experimental initial Veselago’s found although already importance idea practical future.This high the of in available is become will that NIMs expect begin- better can much very we the manufacturing.Therefore, at NIM of still ning are fast we the nanotechnology, of of promising.Despite struc- progress most and currently optical are techniques that at the tures namely outlined optimization, have and frequencies.We design more NIM the for than need a still is software. reliable there and applied, efficient commercial be progress.Although may the packages for play will role design important may NIM an structures for computer-aided NIM complex, NIMs highly promising desired be because from and away currently use far practical still the are structures.Since the NIMs NIM for available improved essential numeri- are of solvers appropriate design Maxwell with such a NIMs.Linked optimizers, establish first cal were that the that structures resonators in ring the split used and of impor- wires an example, design for play the NIM, also for analysis role NIM tant numerical the solvers for that Maxwell used is configuration application.The therefore NIM slab prominent and most limit NIM the diffraction the the overcome on to focus promises problems, special various structures to a applied with of and compared analysis were numerical methods detailed corresponding geometry.The numer- the complicated by with for NIMs.Theoreti- complemented simulations often in currently ical observed are also studies cal are that phenomena eea eaaeil hc ol edt e promising new more to on lead research could the which expect intensify metamaterials will we general research this, NIM storage.Beside the data that to opportunities optical lead new could density super- up limit—which high diffraction open optical the could as overcoming slabs for considered NIM be thin lenses, cannot wavelength the materials. focus composite a of it with properties research time electromagnetic material same on in the activities NIMs.At strong quality the requires high improves finding significantly of This chance shape. semiconductors, different with metals, etc.) (dielectrics, materials differ- natural new of ent composed of metamaterials design i.e., the structures, by artificial widened be may properties magnetic costs. fabrication the urnl,teNMfbiaini uhmr demanding more much is fabrication NIM the Currently, lhuhNMsaswt iebge oprdwith compared bigger size a with slabs NIM Although electro- desired with materials available of range The 147 ial,i ilb motn odatclyreduce drastically to important be will it Finally, .Cmu.Ter aoc.3 1–30, 3, Nanosci. 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