Achieving transparency with plasmonic and metamaterial coatings

Transcription

1 Phys. Rev. E, 72, 6623 (9 pages), 25 DOI:.3/PhysRevE Ahievig trasparey with plasmoi ad metamaterial oatigs Adrea Alù,2, ad Nader Egheta,* Dept. of Eletrial ad Systems Egieerig, Uiversity of Pesylvaia, Philadelphia, 94 PA, USA 2 Dept. of Applied Eletrois, Uiversity of Roma Tre, Rome, Italy Abstrat: The possibility of usig plasmoi ad metamaterial overs to drastially redue the total satterig ross setio of spherial ad ylidrial objets is disussed. While it is ituitively expeted that ireasig the physial size of a objet may lead to a irease i its overall satterig ross setio, here we see how a proper desig of these lossless metamaterial overs ear their plasma resoae may idue a dramati drop i the satterig ross setio, makig these objet early ivisible or trasparet to a outside observer a pheomeo with obvious appliatios for low-observability ad o-ivasive probe desig. Physial isights ito this pheomeo ad some umerial results are provided. PACS umbers: a, e, 42.5.Gy, 33.2.Fb A. Alù, ad N. Egheta, Physial Review E, 72, 6623, 25. Copyright (25) by the Ameria Physial Soiety. INTRODUCTION Ahievig ivisibility or low observability has bee the subjet of extesive studies i the physis ad egieerig ommuities for deades. The use of absorbig srees (e.g., []) ad ati-refletio oatigs (e.g., [2]) to dimiish the satterig or the refletio from objets, for example, are ommo i several appliatios. If the former require absorptio, ad therefore power dissipatio, the latter are suitable primarily for plaar or early-plaar trasparet objets. I this paper, however, we disuss how low-loss (i the limit eve o-loss) passive overs might be utilized i order to drastially redue the satterig from spherial ad ylidrial objets without requirig high dissipatio, but relyig o a ompletely differet mehaism. For this purpose, we osider here the use of materials with egative or low eletromageti ostitutive parameters, e.g., ertai metals ear their plasma frequey or metamaterials with egative parameters. Several oble metals ahieve this requiremet for their eletrial permittivity at the ifra-red (IR) or visible

2 Phys. Rev. E, 72, 6623 (9 pages), 25 DOI:.3/PhysRevE regimes, eve with reasoably low losses [3]-[4]. At lower frequeies, moreover, artifiial materials ad metamaterials may be properly sythesized to meet similar requiremets at the desired frequey (see referees i [5]). At mirowaves, for istae, the tehology of artifiial materials with uusual eletromageti properties has a log traditio i the egieerig ommuity. Materials with low-positive, ear-zero or egative relative permittivity have bee sythesized by embeddig arrays of thi wires i a host material [6]-[8], ad their properties have bee exploited i differet mirowave appliatios over the years, e.g., for highly diretive ateas [9]-[] or more reetly for aomalous tuelig ad trasmissio (e.g., []-[2]). By a similar priiple, arrays of resoatig loops or split-rig resoators affet i a aalogous way the permeability of the bulk material ad have bee studied i the last deade [3]-[4]. The ombiatio of these ilusios i omposite metamaterials with a egative idex of refratio [5] has reetly attrated a great deal of attetio i the sietifi ommuity. The idea of utilizig low-permittivity oatigs to idue ivisibility has bee suggested i the quasi-stati limit for spherial objets (i.e., for the ase of a sphere beig small ompared with the operatig wavelegth) ad for the higher-order terms i power series expasio [6, p. 49-5], [7]-[9]. Here, however, we explore fully the ase of plasmoi ad metamaterial overs with egative or low-permittivity ad/or permeability materials i the dyami full-wave satterig ase (i.e., the sphere does ot eed to be eletrially very small), providig some ew physial isights ad possibility of optimized desigs for reduig the total satterig ross setio of spherial objets whose dimesios are omparable with the wavelegth of operatio. THEORETICAL ANALYSIS Let us osider a spherial satterer of radius a, omposed of a homogeeous material with permittivity ε ad permeability μ, surrouded by free spae (with ostitutive parameters ε ad μ ). (As will be metioed later, i the limit this sphere a also be a perfet eletri odutor.) Would it be possible to over suh a objet with a lossless (or low-loss) spherial shell (with ε, μ ad radius a > a) i suh a way that the total satterig ross setio drastially dimiishes, eve though its overall physial size has * To whom orrespodee should be addressed, egheta@ee.upe.edu 2

3 Phys. Rev. E, 72, 6623 (9 pages), 25 DOI:.3/PhysRevE evidetly ireased? Here we explore the oditios uder whih this may be possible. The geometry of the problem is depited i Fig.. Let the illumiatig wave be represeted by a i t e ω moohromati plae wave. All the ostitutive parameters may i geeral be omplex at the frequey of iterest ω, takig ito aout possible losses. Adoptig a stadard Mie expasio, oe a write the sattered field from a plae wave as the well-kow disrete sum of spherial harmois with omplex amplitudes TE ad ( beig a iteger), respetively for the TE ad spherial waves (see e.g. [2]). Their expressios for the problem at had may be writte i the oveiet form [2]: U TE TE = TE TE U + iv, U = U + iv, () where the futios U ad V are real valued i the limit of o ohmi (material) losses, ad i the polarizatio are give by the determiats: U = ( ) ( ) ( ) j ka j k a y k a kaj ka kaj ka kay ka ( ) / ε ( ) / ε ( ) / ε j ( k a ) y ( k a ) j ( k a ) ka j( ka ) / ε ka y( ka) / ε ka j( ka) / ε, (2) V = ( ) ( ) ( ) j ka j k a y k a kaj ka kaj ka kay ka ( ) / ε ( ) / ε ( ) / ε j ( k a ) y ( k a ) y ( k a ) ka j( ka) / ε ka y( ka ) / ε ka y( ka ) / ε. (3) Aalogous expressios for the TE polarizatio are obtaied after substitutig ε with μ i (2) ad (3). I these formulas k ω εμ, k ω εμ ad k ω ε μ are the wave umbers i the three regios ad (). j, (). y are spherial Bessel futios. The satterig oeffiiets are related to the total satterig ross setio through the formula: 2π Q = + +. (4) TE 2 2 ( 2 )( ) s 2 k = Geerally speakig, the satterig ross setio Q s is determied by the satterig oeffiiets up to the order = Nmax, sie the suessive satterig oeffiiets beyod max N will be egligible [6]. The value 3

4 Phys. Rev. E, 72, 6623 (9 pages), 25 DOI:.3/PhysRevE of N N usually ireases with the physial/eletrial size of the satterer (a rule of thumb is that max ( k a ) [6, p. 26]), ad that is oe of the reasos that larger objets geerally have a wider max satterig ross setio. Followig this observatio, oe may expet, ituitively, that overig a objet would irease its satterig ross setio, together with its physial/eletrial size, sie more satterig oeffiiets otribute i the summatio. However, as will be show below, this statemet is ot always true, ad with judiiously hoosig the over material ad its size, oe may be able to make a give homogeous (or eve metalli) sphere less detetable to a exteral observer. Let us first oetrate o the ase i whih the satterer is small eough to osider oly the dipolar term ( = ) i the Mie expasio. This ase is osistet with what was doe before i [6]-[9]. I this ase, i.e., whe ka, ka, ka, the expressios i (2) ad (3) are redued to the followig [2]: where U + π ( ka ) ( + / ) ε ( + / ) ε / ε, (5) 4 ( 2+ )( /2 )! γ γ γ / ε γ / ε / ε V ( + ) + ( + ) ( + ) ( + / ) ε ( + / ) ε / ε ( k a ), (6) γ γ γ ε γ ε ε + / / / γ a/ a is the ratio of ore-shell radii. These losed-form expressios reveal iterestig properties for eletrially small satterers. Firstly, as expeted, for small satterers the value of U is small ad teds to zero as a. This is osistet with the fat that usually a small satterer has a very low sattered field. I fat, osiderig stadard dieletris far from their plasma frequey (all permittivities greater or equal to TE ε, all permeabilities equal to μ ) ad the sum i the deomiator of is domiated by V, yieldig the followig approximate expressio for : 2+ ( ) ( ) j k a f γ, (7) where f ( γ ) are positive real futios of γ. This ofirms that for suh tiy dieletri satterers the satterig properties are domiated by the first-order term, orrespodig to the radiatio from a eletri dipole. This also shows that overig this struture with a dieletri shell may result i ireasig its 4

5 Phys. Rev. E, 72, 6623 (9 pages), 25 DOI:.3/PhysRevE satterig ross setio together with its physial/eletrial size (i.e., a ). However, there is oe possibility for whih addig a over may lead to a drasti redutio of the total satterig ross setio from a small sphere, ad that is whe the over parameters are hose suh that the determiat U i (5) may approah zero, resultig i, whih is the domiat term i (4), to beome zero. I the limit for whih (5) applies, its value will be idetially zero for a give order, if γ satisfies the followig oditio: γ ( ) ( ) ( ) ( ) ε ε + ε + ε = 2+ T ε ε + ε+ ε. (8) where T is a shorthad for the ( ) 2 + -st root. Clearly, i order for γ to have physial meaig, its value should satisfy γ. This result is osistet with the fidigs i the quasi-stati ase i [6]-[9]. I Fig. 2 the values of permittivities for whih (8) a be fulfilled are reported i a otour plot, with lighter regios idiatig higher values of T. Whe γ = T for a give, the orrespodig satterig oeffiiet beomes idetially zero (i the small-sphere approximatio). Similar results may be obtaied for the TE polarizatio by replaig the permittivities with the orrespodig permeabilities. Oe may otie that if both materials are stadard materials far from their plasma frequeies (i.e., with ostitutive parameters higher tha those of free spae) it is ot possible to ahieve suh a oditio for ay value of γ. If however we are allowed to use overs with permittivity or permeability lower tha the oe i free spae, or with egative values, whih might represet metals lose to their plasma frequey or alteratively low-permittivity, low-permeability, or egative-parameter metamaterials, the followig Fig. 2 a proper hoie of the ratio of ore-shell allows brigig to zero oe partiular. If the ore sphere (the oe to be hidde) is dieletri (ad eletrially small, i order to apply (8)), the the eletri dipole term is the domiat term i satterig, ad all the other terms i (4) are egligible. I this ase, from (8) the oditio for the over radius beomes: a = 3 ( ε ε )[ 2ε+ ε] ( ε ε)[ 2ε + ε ] a, (9) whih is osistet with the formula derived i [6, p. 5] i the stati ase. This applies, of ourse, oly whe the pair of permittivities ε, ε falls i allowable regios i Fig. 2. As a speial ase of iterest, we 5

6 Phys. Rev. E, 72, 6623 (9 pages), 25 DOI:.3/PhysRevE may osider the ase i whih the ore sphere is perfetly odutig, for whih get a 3 ( ε ε )/( 2ε ε ) = + a, whih may be satisfied whe ε i. I this ase, we ε is hose i the iterval < ε < ε. It may be oted here that with some other ombiatios of parameters it is possible to derive aother ratio γ for whih the determiat V i (6) beomes zero. As already disussed i our previous work [2]-[22], this ratio would idue a aomalous plasmoi resoae i the ore-shell system, ausig a opposite effet, amely a highly ireased satterig from a eletrially small ore-shell system. Whe the size of the sphere ireases, expressio (9) is o more effetive i determiig the exat radius of the outer ore to ahieve the trasparey oditio. I this ase, however, it is still possible to ahieve aother oditio for whih U =, whih happes at the partiular γ that makes the determiat i (2) zero. This oditio is effetive for bigger dimesios of the sphere for whih the quasi-stati limit does ot apply, eve though this does ot eessarily imply that the satterig from the sphere will beome zero, sie U = is for the dipolar term to be zero, ad for too large spheres higher order terms i (4) may beome more sigifiat tha the first-order term. I this ase, however, it may be possible to exploit a similar approah with a multi-layer over, whih should provide more degrees of freedom, ad thus a proper desig might make simultaeously zero some of the higher-order terms that otribute otieably to the satterig of the partile. This is oe of the subjets i this area we are urretly ivestigatig. It is iterestig to ote that it may be possible to derive aalogous theoretial results for other aoial geometries. I the ylidrial ase, for istae, followig similar steps it may be show that the trasparey oditio for the TE (with respet to the ylider axis, i.e., with the mageti field parallel with the ylider axis) polarizatio beomes: γ γ ( ε ε )( ε+ ε) ( ε ε)( ε + ε ) = 2 μ μ μ μ 2 = = 2 for for, () where γ represets agai the ratio of ore-shell radii, ad the formulas may be derived by duality. Other geometrial shapes ad referee systems would allow obtaiig aalogous oditios, whih are valid i the quasi-stati approximatio, ad may be similarly exteded to the full-wave aalysis. 6

7 Phys. Rev. E, 72, 6623 (9 pages), 25 DOI:.3/PhysRevE NUMERICAL RESULTS AND DISCUSSION A importat questio is how large the radius a may beome i the geeral full-wave satterig ase while oe would still be able to ahieve a drasti redutio of satterig ross setio with a sigle over, aelig just the first order term. For this purpose, Fig. 3 shows the full-wave total satterig ross setio of a dieletri partile ( ε = 4ε ) overed by a plasmoi or metamaterial over with ε = 3ε, i terms of the ratio a/ a, for differet sizes of the ore-shell partile. For this ombiatio of parameters the value of T i (8) for = is give by T.6. As a be see from this Figure, whe the outer radius of the overed partile is extremely small ( a = λ /, λ = 2 π /k beig the free-spae wavelegth), its satterig ross setio goes to ear zero very lose to that ratio. Ireasig the partile size ( a = λ / ), the ross setio a still beome very small, makig the objet early ivisible, (sie the higher order satterig oeffiiets are still egligible), eve though the quasi-stati solutio is o more adequate ad the trasparey oditio should be obtaied diretly by equatig Eq. (2) to zero, shiftig dowwards the required value of γ for the trasparey i this ase. Eve with a larger outer radius ( a = λ /5 ) we may get a very low satterig, but of ourse the value of γ for the miimum satterig has bee shifted dowwards agai. I order to give a physial isight ito this trasparey pheomeo, oe may thik of it i the quasi-stati approximatio ad with the goal of simply reduig the dipolar term. As is well kow, the dipolar satterig is the mai respose related to the polarizatio vetor P= ( ) ε ε E idued loally by the loal field as a result of exitig eletri field. A regular dieletri sphere would show a sattered dipolar field due to the total idued eletri dipole momet, whih i the quasi-stati limit orrespods to the itegral of the vetor P over the volume of the sphere. A over with ε < ε, however, would show a loal polarizatio vetor P= ( ε ε ) E i the over, beig ati-parallel to the loal eletri field i the over, whih after beig itegrated over the shell volume may ael the origial dipole momet of the ore itself. With the proper hoie of radius a for the over this aellatio may be omplete, ad this is what formula (9) 7

8 Phys. Rev. E, 72, 6623 (9 pages), 25 DOI:.3/PhysRevE heuristially represets. This is summarized i Fig. 4 (a similar ituitive explaatio has bee provided i [23] for the stati seario). We ote that this aellatio pheomeo does ot rely o a resoat pheomeo, ulike the opposite pheomeo of the satterig resoae by suh a ore-shell system (for whih V = ) [2]-[22]. This is a iterestig poit, sie it esures: a) that this pheomeo is ot strogly sesitive to losses or to the spherial symmetry of the partile, ad therefore ohmi losses or imperfetios i the partile shape will ot preipitously alter this pheomeo, ad b) that the pheomeo may ot be strogly sesitive to the ratio oditio (8) (i.e., it has a larger ratio badwidth ), as a also be see i the examples of Fig. 3. Usig a similar argumet, we may apply the trasparey pheomeo to a plasmoi partile ( ε = 3ε, a = λ / ) by oatig it with a dieletri material ( ε = ε ). For this ombiatio T.825 ad therefore the required radius for the over is a.2a i order to ahieve trasparey. This is show i Fig. 5, where the satterig ross setio (a) ad the otributios from the several satterig oeffiiets (b) are depited varyig the over radius. It is iterestig to otie how lose to the ear-zero-satterig ratio we fid a peak i the satterig ross setio (Fig. 5a), whih is learly due to the resoae of the 2 satterig oeffiiet (Fig. 5b). This is expeted whe V 2 = i Eq. (3), whih i the quasi-stati approximatio is expeted for a.5a. It a also be otied that the trasitio i whih approahes zero i Fig. 5b is a relatively smooth oe (very differet form the satterig peak, as disussed i [2]), implyig that the rage of the ratio of radii i Fig. 5a for whih the satterer has a very low satterig ross setio is relatively broad. As was just poited out, this is aother idiatio of the o-resoat behavior of this pheomeo. Fig. 6 reports the same plots as i Fig. 5, but for a larger partile ( a = λ /5 ). The results show that eve for a partile far from the quasi-stati oditio, the satterig ross setio may be drastially redued with the same priiple. (Oe may appreiate how the higher order terms i (4) are sigifiat i Fig. 6b.) The effetiveess of the over is due to the fat that the higher order satterig terms are also i geeral affeted whe suh a omplemetary over is employed (Fig. 6b). It should be uderlied that the ombiatios of parameters hose i Figs. 4, 5 ad 6 also allow the 8

9 Phys. Rev. E, 72, 6623 (9 pages), 25 DOI:.3/PhysRevE presee of the resoae peaks desribed i [2] (at whih the differet V may go to zero) for other ratios of radii (they are learly see i Figs. 5b ad 6b). For larger partiles the presee of this peak worses the trasparey performae of these plasmoi ad metamaterial overs: the miimum dip we see i the plot of satterig ross setio is losely surrouded by the regios where the resoat maxima are preset, ad as a result a good trasparey oditio may ot be effetively ahieved for larger partiles (at least with a sigle-layer homogeeous over with egative-parameter metarials). Therefore, i order to ahieve better trasparey performae for partiles that do ot satisfy the quasi-stati approximatio, the trasparey ratio T ad the high-satterig resoat ratios should be as far apart as possible. There is ideed a rage of material parameters i Fig. 2 where this oditio is possible: whe oe of the two materials has a positive permittivity, but lower tha the permittivity of the outside regio (e.g., free spae). I this rage of material parameters, it a ideed be show that o resoae peak may arise i the quasi-stati approximatio (the determiat i (6) aot be zero for material parameters i this rage [2]). Fig. 7, as a example, shows the results for a sphere with ε = ε ad a = λ / overed with a material with ε =.5ε, as a futio of a / a. I this ase, as a be see, the redutio of total satterig ross setio a also be ahieved, ad it is learly evidet from Fig. 7b that o high-satterig resoat peak is aroud this trasparey poit. The total satterig ross setio of larger objets may be suessfully redued by further reduig the positive permittivity of the over. Fig. 8 shows some results for the same ier material as i Fig. 7, but twie its radius (ad so eight times its volume). The over utilized ow has ε =.ε. We ote how the satterig ross setio has bee drastially redued at the optimum radius for whih =. The residual satterig ross setio at the miimum dip i Fig. 8a is maily due to the otributio of the TE mageti dipole otributio, sie this is ot diretly affeted by the low permittivity of the over, as a be see from Fig. 8b. Provided that we may fid (or sythesize, as disussed i the itrodutio) a material whose permeability may be brought dow towards zero at the same frequey, however, oe might be able to further redue the satterig from the objet usig a similar priiple. Choosig a over TE permeability as μ =.25μ, for whih U = at the same a =.2a where we get the miimum i Fig. 8a, we are able to make the objet eve less visible, as show i Fig. 9. 9

10 Phys. Rev. E, 72, 6623 (9 pages), 25 DOI:.3/PhysRevE Fig. shows the otour plots of the magitude of the radial ompoet of the ear-zoe sattered eletri field for the ase of a dieletri sphere with ε = ε, μ = μ ad a = λ /5 i four differet ases: o over (Fig. a), a dieletri over with ε = 5ε ad a =.2a (Fig. b), the same over as i Fig. 8 (Fig. ) ad the same over as i Fig. 9 (Fig. d). I all these ases the system is assumed to be exited by a plae wave propagatig alog the ẑ diretio with a uit-amplitude eletri field polarized alog ˆx, ad the plots are i the x-y plae, where the eletri multipole otributio is maximum. We ote that the otour plots are all ormalized to the same value, so we a ompare the field amplitudes i the differet figures. As a be easily see, i the first ase without over (Fig. a) the eletri field idues a strog eletri dipole momet iside the sphere, whih osequetly geerates a dipolar sattered field i the outside regio. This is the usual situatio. Addig a over made of a regular dieletri with arbitrarily seleted ε = 5ε the sattered field is further ehaed (Fig. b), as expeted. The sattered radiatio beomes eve less symmetri due to the otributio of the quadrupole ad higher order multipoles. However, whe the over used i Fig. 8 with ε =.ε, μ = μ ad a =.2a is used (Fig. ), a sattered field omparable with the previous ases (a ad b) is observed iside the objet, but the over almost aels the sattered radiatio outside. The residual sattered radiatio (whih is very weak ompared with the sattered radiatio i Fig. a or b) is learly due to the quadrupolar ad otupolar terms, but it has bee drastially redued i magitude, osistet with what was show i Fig. 8a. Whe the over of Fig. 9 (with ε =.ε, μ =.25μ ad a =.2a) is employed, the sattered field is eve further redued, eve though the effet of this over is maily see i the mageti field plots (Fig. ), sie it has bee desiged to redue the mageti dipole radiatio. Fig. shows the orrespodig plots similar to Fig., but for the magitude of the radial ompoet of the ear-zoe sattered mageti field, i the y-z plae where the mageti multipoles otribute maximally to the radial ompoets. Fig. a shows the radial ompoets of the mageti field sattered by a sigle dieletri sphere, ad it is domiated by the mageti dipole otributio. Ireasig the size by addig aother dieletri over, as show i Fig. b, produes a irease i the sattered field, ad ow the mageti quadrupole beomes also otieable. Whe the over of Fig. is used here, istead, the mageti ear field is similar to the oe obtaied without ay over, as expeted from the results of Fig. 8. However,

11 Phys. Rev. E, 72, 6623 (9 pages), 25 DOI:.3/PhysRevE addig the low-permeability oditio to the over, as show i Fig. d, the mageti dipole otributio to the sattered field is essetially aeled. Now the objet is almost ivisible i both the E ad H plaes, i.e., i the whole spae. Fig. 2 shows the total time-averaged Poytig vetor distributio i the x-z plae for the same four ases. Whe the trasparey overs are used (Figs. 2, 2d), oe otes how the iomig plae wave seems to pass through the sphere without beig affeted otieably by the sattered field. I other words, a observer seated outside the overed sphere, eve i the ear field, would ot be able to essetially sese the presee of the sphere (sie there is almost o sattered field outside), although the fields are otieably haged iside the sphere ad its over. If a ovetioal satterer is plaed behid suh a trasparet sphere, a observer would iterestigly otie the presee of that satterer without essetially seeig the trasparet sphere betwee him/her ad the satterer itself. Fig. 3 reports the similar results as i Fig., but for a smaller sphere ( a = λ /2 ). I this ase, the sattered fields from the sphere aloe ad from the sphere overed with a stadard dieletri (Figs. 3a ad 3b) are domiated by the eletri dipole term, ad the mageti dipole otributio is egligible. Coverig it with the proper over (sie we are i this quasi-stati ase we may simply use the approximate expressio (9), whih yields for the radius a =.9a), the trasparey is effetively ahieved. Due to its small dimesios, there is o eed to employ low- μ materials, sie the H-plae field otributio is already egligible. The related plots therefore are ot reported here. We eed to poit out the sesitivity/robustess of the results preseted here to some realisti parameters. As disussed previously, the trasparey pheomeo is ot due to a resoae, but just relies o the overall aellatio of the multipolar satterig fields (see Fig. 4). This is why this trasparey effet has a relatively broader rage for seletio of ratio of radii ad over sizes (we are ot restrited to oe sigle value for the over, but there is a relatively broader rage of values for whih a over with egative polarizability may hide a dieletri sphere). By the same toke, this pheomeo may be robust with the variatio of the material losses. Fig. 4, as a example, shows how the results of Fig. 7a ad Fig. 8a are modified by the presee of losses i the over material. As a be see, the trasparey effet is ot affeted muh by the presee of reasoable

12 Phys. Rev. E, 72, 6623 (9 pages), 25 DOI:.3/PhysRevE ohmi (material) losses. For the dispersio with the Drude or Loretzia models, i the frequey regios ear the plasma frequeies where the ostitutive parameters have values lose to zero (whih we have show to be more effetive i iduig this trasparey effet) there are o sharp resoaes i the medium dispersio, ad therefore, followig the Kramers-Kroig relatios, we expet less ohmi losses tha ear the resoat frequeies [4]. We speulate that this setup is ot too sesitive to possible imperfetios i the ostrutio ad geometry of the over. Beig a o-resoat pheomeo, we do ot expet substatial hages i the results reported here for quasi-spherial or quasi-ylidrial objets with small imperfetios o their surfae. The fat that a egative polarizability material would effetively ael oe or more multipolar otributios of the sattered field is ideed ot a property of the spherial or the ylidrial geometry, ad it is preditable that formulas aalogous to (9) ad () may be derived as well for several other geometries ad for more omplex objets. As a fial remark, we should poit out that this trasparey pheomeo may somehow be related to the issues of o-radiatig soures i the iverse satterig ad soure problems i eletromagetis ad aoustis [24]-[25]. We are urretly ivestigatig this poit. SUMMARY I this otributio we have studied how it is possible to drastially redue the satterig ross setio of spherial ad ylidrial objets usig lossless plasmoi or metamaterial overs. We have provided physial isights ad umerial examples of how a proper desig of these metamaterial overs ear their plasma resoae may idue a dramati drop of the satterig ross setio (eve i the absee of material loss) makig the objet early trasparet to a exteral observer. We have also disussed how this effet is isesitive to the possible losses or other imperfetios i strutures. This pheomeo a provide umerous potetial appliatios i the desig of low-observable targets, low-ouplig i desely-paked devies, ad o-ivasive field ao-probes. These issues are urretly uder ivestigatio. ACKNOWLEDGEMENTS This work is supported i part by the U.S. Defese Advaed Researh Projets Agey (DARPA) Grat Number HR-4-P-42. Adrea Alù has bee partially supported by the 24 SUMMA Graduate 2

13 Phys. Rev. E, 72, 6623 (9 pages), 25 DOI:.3/PhysRevE Fellowship i Advaed Eletromagetis. We thak Dr. W. Ross Stoe for poitig us to the referees [24] ad [25] ad the topi of iverse satterig problems ad Dr. Alexey Viorgradov for brigig to our attetio Ref [7]. REFERENCES [] R. L. Fate, ad M. T. MCormak, IEEE Tras. Ateas Propag., 3, 443 (968). [2] J. Ward, Vauum, 22, 369 (972). [3] P. B. Johso, ad R. W. Christy, Phys. Rev. B, 6, 437 (972). [4] L. Ladau, ad E. M. Lifshitz, Eletrodyamis of otiuous media (Elsevier, 984). [5] R. W. Ziolkowski, ad N. Egheta (guest editors), IEEE Tras. Ateas Propag., Speial Issue o Metamaterials, 58 (23). [6] W. Rotma, IRE Tras. Ateas Propag.,, 82 (962). [7] J. B. Pedry, A. J. Holde, D. J. Robbis, ad W. J. Stewart, J. Phys.: Cod. Matter,, 4785 (998). [8] A. N. Lagarkov, S. M. Matytsi, K. N. Rozaov, ad A. K. Saryhev, J. Appl. Phys., 84, 386 (998). [9] J. Brow, Pro. IEE, IV, 5, (953). [] I. J. Bahl, ad K. C. Gupta, IEEE Tras. Ateas Propag., 22, 9 (974). [] A. Alù, ad N. Egheta, IEEE Tras. Ateas Propag., 5, 2558 (23). [2] A. Alù, F. Bilotti, N. Egheta, ad L. Vegi, Pro. of IEE Semiar o Metamaterials for Mirowave ad (Sub) millimetre Wave Appliatios, Lodo, UK, pp. /-/6, November 24, 23. [3] J. B. Pedry, A. J. Holde, D. J. Robbis, ad W. J. Stewart, IEEE Tras. Mirowave Theory Teh, 47, 275 (999). [4] P. Gay-Balmaz ad O. J. F. Marti, Appl. Phys. Lett., 8, 939 (22). [5] R. A. Shelby, D. R. Smith, ad S. Shultz, Siee, 292, 77 (2). [6] C. F. Bohre, ad D. R. Huffma, Absorptio ad Satterig of Light by Small Partiles (Wiley, New York, 983). [7] Z. Hashi, ad S. Shtrikma, J. Meh. Phys. Solids,, 27 (963). [8] M. Kerker, J. Opt. So. Am., 65, 376 (975). [9] H. Chew, ad M. Kerker, J. Opt. So. Am., 66, 445 (976). 3

14 Phys. Rev. E, 72, 6623 (9 pages), 25 DOI:.3/PhysRevE [2] J. A. Stratto, Eletromageti Theory (MGraw-Hill Comp., 94). [2] A. Alù, ad N. Egheta, J. Appl. Phys., 97, 943 (2 pages) (25). [22] A. Alù, ad N. Egheta, Pro. of 24 IEEE Radar Coferee, Philadelphia, USA, April 26-29, 24, pp [23] M. Kerker, Aerosol Siee ad Tehology,, 275 (982). [24] N. Bleistei ad J. Cohe, J. Math. Phys. 8, 94 (977) [25] B. J. Hoeders, i Iverse Soure Problems i Optis, H. P. Baltes (ed.), Vol. 9 of Topis i Curret Physis, Spriger, Berli, 4 (978) FIGURES Figure Cross setio of a spherial satterer omposed of two oetri layers of differet isotropi materials i a suitable spherial referee system. Figure 2 Rages of material permittivities for whih the oditio (8) admits physial solutios ad otour plot for the values of the ratio T ( polarizatio). 4

15 Phys. Rev. E, 72, 6623 (9 pages), 25 DOI:.3/PhysRevE Figure 3 Normalized satterig ross setio for a spherial partile with ε = 4ε, μ = μ, ad the over with ε = 3ε, μ = μ, versus the ratio a/ a for three differet sizes of the outer radius of the over. Figure 4 Heuristi iterpretatio of the trasparey pheomeo: aellatio of the overall dipole 5

16 Phys. Rev. E, 72, 6623 (9 pages), 25 DOI:.3/PhysRevE momet through a idued egative polarizatio vetor. Figure 5 (a) Normalized total satterig ross setio ad (b) otributios of the several (o-zero) satterig oeffiiets, i terms of the over radius for a fixed ier ore ( ε = 3ε, ε = ε, μ = μ = μ, a = λ / ). 6

17 Phys. Rev. E, 72, 6623 (9 pages), 25 DOI:.3/PhysRevE Figure 6 The same as Figure 5, but a = λ /5 ad higher order terms beome sigifiatly higher. But still, a sigifiat redutio of the satterig ross setio may be ahieved. 7

18 Phys. Rev. E, 72, 6623 (9 pages), 25 DOI:.3/PhysRevE Figure 7 (a) Normalized total satterig ross setio ad (b) otributios of the several (o-zero) satterig oeffiiets, i terms of the over radius for a fixed ier ore ( ε = ε, ε =.5ε, μ = μ = μ, a = λ / ). 8

19 Phys. Rev. E, 72, 6623 (9 pages), 25 DOI:.3/PhysRevE Figure 8 (a) Normalzied total satterig ross setio ad (b) otributios of the several (o-zero) satterig oeffiiets, i terms of the over radius for a fixed ier ore ( ε = ε, ε =.ε, μ = μ = μ, a = λ /5 ). 9

20 Phys. Rev. E, 72, 6623 (9 pages), 25 DOI:.3/PhysRevE Figure 9 (a) Normalized total satterig ross setio ad (b) otributios of the several (o-zero) satterig oeffiiets, i terms of the over radius for a fixed ier ore ( ε = ε, ε =.ε, μ = μ, μ =.25μ, a = λ /5 ). Figure Cotour plots of the distributio of the magitude of radial ompoet of the sattered eletri field i the x-z plae, idued by a plae wave travelig alog the z diretio with a eletri field polarized alog the x axis: (a) for a sphere with ε = ε, μ = μ, a = λ /5 ; (b) for the same 2

21 Phys. Rev. E, 72, 6623 (9 pages), 25 DOI:.3/PhysRevE sphere, but overed with ε 5ε same as (), but with μ =.25μ. =, μ = μ, a =.2a; () the same as (b), but with ε =.ε ; (d) the Figure Cotour plots of the distributio of the radial ompoet of the sattered mageti field i the y-z plae, aalogous to Fig.. Figure 2 Total time-averaged Poytig vetor distributios i the x-z plae for the ases of Figs. -. 2

22 Phys. Rev. E, 72, 6623 (9 pages), 25 DOI:.3/PhysRevE Figure 3 Cotour plots of the distributio of the radial ompoets of the sattered eletri field i 22

23 Phys. Rev. E, 72, 6623 (9 pages), 25 DOI:.3/PhysRevE the x-z plae, idued by a plae wave travelig alog the z axis with a eletri field polarized alog the x axis: (a) for a sphere with ε = ε, μ = μ, a = λ /2 ; (b) for the same sphere, but overed with ε = 5ε, μ = μ, a =.9a; () the same as (b), but with ε =.ε. Figure 4 (a) The same as Fig. 7a; ad (b) the same as Fig. 8a, osiderig ohmi losses i the over materials. 23