Cloaked Sensors. Romain Fleury, Jason Soric and Andrea Alù * Dept. of Electrical and Computer Engineering, The University of Texas at Austin

Transcription

1 Physial Bouds o Absorptio ad Satterig for Cloaked Sesors Romai Fleury, Jaso Sori ad Adrea Alù * Dept. of Eletrial ad Computer Egieerig, The Uiversity of Texas at Austi 1 Uiversity Statio C83, Austi, TX 7871, USA *alu@mail.utexas.edu We derive ad disuss geeral physial bouds o the eletromageti satterig ad orptio of passive strutures. Our theory, based o passivity ad power oservatio, quatifies the miimum ad maximum allowed satterig for a objet that orbs a give level of power. We show that there is a fudametal trade-off betwee orptio ad overall satterig suppressio for eah satterig harmoi, providig a tool to quatify the performae of furtive sesors, regardless of the applied priiple for satterig suppressio. We illustrate these fudametal limitatios with examples of light satterig from orbig plasmoi aopartiles ad loaded dipole ateas, evisioig appliatios to the desig of loaked sesors ad orbers with maximized orptio effiiey. PACS: 4.5.Bs, 81.5.Zx, Pt, 4.5.Fx, 4.79.Pw. I. INTRODUCTION 1

2 I the past deade, the suessful desig ad implemetatio of metamaterials with exoti eletromageti properties has opeed several ew veues for maipulatig the propagatio of light ad its iteratio with matter [1]. I partiular, the possibility of iduig ivisibility with passive metamaterial oatigs, the loakig effet, has bee the subjet of itese researh [- 19]. Several experimetally-verified approahes for all-agle eletromageti loakig of threedimesioal objets have bee proposed, iludig (i) the trasformatio method [5-8], whih uses a loak with futioally graded material properties to re-route the power flow aroud the objet, simultaeously brigig its satterig to zero ad isulatig the loaked regio from the outside world; (ii) the satterig aellatio tehique [9-19], i whih a metamaterial oatig with isotropi ad homogeeous ostitutive parameters [9-16], or a thi metasurfae with tailored surfae impedae [17-19], is used to redue the satterig over a give badwidth. I may ases, the study of the loakig performae has bee restrited to ideal situatios ivolvig lossless materials. However, the problem of material losses is a etral issue, espeially i pratial realizatios, i whih the olute loakig redutio may be sigifiatly affeted by orptio losses [3,]. A related problem that has reetly attrated sigifiat attetio is the oe of seeig without beig see [1,]. I may appliatios like o-ivasive sesig ad ommuiatios, beig able to ope our eyes behid a loak, ad extrat iformatio about the outside world while remaiig udetetable, is of primary importae. This possibility has bee demostrated with the satterig aellatio tehique ad the trasformatio approah i the ase of small sesors or power reeivers [3-8], providig exitig veues i a variety of appliatio fields like earfield saig optial mirosopy [9-31], ivisible eletromageti sesors ad photodetetors [3] ad low-observable reeivig ateas [33]. Yet, the idea of beig ivisible while orbig

3 power appears outerituitive, sie ommo sese ad the optial theorem agree o the fat that it is ot possible to orb eergy without reatig a shadow, i.e., ay extitio is assoiated with ozero forward satterig [34,35]. As a osequee, a passive loaked objet beomes eessarily detetable as soo as it starts to orb a portio of the impigig eergy. A better uderstadig of the fudametal limitatios assoiated with loakig orptive objets is ot oly ruial to trasitio from ideal loakig methods to pratial appliatios, but also for uderstadig the physial boudaries to be osidered whe buildig oealed sesors. I this artile, we disuss geeral limitatios o satterig ad orptio from passive objets, stemmig from passivity ad power oservatio. I Setio II, we highlight ad quatify the fudametal trade-off betwee orptio ad loakig, ad provide a aalytial tool to uderstad the behavior of loaked sesors ad low-iterferig power reeivers. I Setio III, we illustrate the geerality of our theory by osiderig pratial examples, iludig optial satterig from lossy aospheres, ore-shell aopartiles ad loaded dipole ateas. II. THEORETICA FORMUATION Cosider the geeral situatio i whih eletromageti waves are sattered off from a passive objet. For simpliity of otatio, i the followig we assume spherial symmetry, but our theory a also be exteded to arbitrarily shaped objets. The satterig problem may be approahed usig the Mie expasio i spherial harmois. The sattered field for plae wave iidee ˆ ikz Ei xee is expressed as a superpositio of spherial harmois [34] 3

4 E E r i r 1 TE 1, (1) sat 1 1 where is the free-spae permeability, m are salar spherial harmois, solutios of the Helmholtz equatio i the spherial oordiate system r,,, ad m 1 due to symmetry, uder a i t e time ovetio. The total satterig ross-setio sat a be expressed as a futio of the Mie satterig oeffiiets TE ad as [34,35], () TE sat ( 1) k 1 where the satterig oeffiiets may be foud for a geeral ore-shell geometry i [36]. The total amout of power extrated from the iidet field is represeted by the total extitio ross-setio ext, whih may be alulated as [34,35] ( 1) Re. (3) TE ext k 1 I additio, the orptio, extitio ad satterig ross-setios are related by eergy oservatio ext sat. (4) We will ow prove that, as a osequee of passivity ( ), the omplex-valued satterig oeffiiets TE ad are restrited to a portio of the omplex plae, amely the losed disk of eter 1/ ad radius 1/. This will be evidet after performig the variable hage 4

5 1 1. (5) TE/ TE/ The expressios ()-(4) for the ross-setios beome: TE ( 1) 1 1, (6) sat sat 1 k 1 TE ( 1) 1 Re 1 Re, (7) ext ext 1 k 1 TE (8) ( 1) 1 1, 1 k 1 where we have itrodued the partial satterig ross-setios assoiated with eah harmoi,, ad sat ( sat ) requires TE/ ext. Ispetig (6), it is evidet that a perfetly loaked objet 1, whih implies i (8). I additio, if the satterer is lossless ( ) the eessarily 1. If losses are preset, passivity ad the TE/ orthogoality of spherial harmois require that, whih traslates ito This proves that, for passive objets, the frequey depedet omplex oeffiiets 1. TE/ are TE / restrited i the omplex plae to the losed uity disk, or equivaletly, the oeffiiets TE/ are restrited to the losed disk of eter 1/ ad radius 1/. A speial ase of iterest is the oe of maximized orptio for the th harmoi. Equatio (8) suggests that is TE/ maximized for. Uder this oditio, we fid the relatioship betwee partial rosssetios 5

6 ext sat 1. (9) k I other words, i the ase of maximized orptio for the th harmoi, the partial orptio ad satterig ross-setios are eessarily equal ad they deped oly o frequey ad o the order, a oditio kow i the atea ommuity as ojugate mathed resoae [37-41]. Figure 1. Partial satterig ross-setios for passive objets, represeted iside the uity disk i the omplex plae. The filled otours represet the levels of sat, while the blak otour lies represet. Both are expressed i ormalized uits of ( 1) / k. 6

7 The above disussio demostrates that the satterig ad orptio ross-setios of ay passive objet are bouded ad fudametally related. To further illustrate this oept, we plot i Figure 1 the partial orptio ad satterig ross-setios of a arbitrary passive objet as a / futio of TE i the losed uity disk of the omplex plae. The satterig ross-setio is show i the filled otour plot, while orptio is represeted with blak otour lies. Both partial ross-setios are expressed i ormalized uits of ( 1) / k, so that the plot remais valid for ay order. This figure is a powerful tool, as it iludes all the available iformatio o orptio, satterig ad extitio for eah harmoi. As expeted, extreme values of the satterig ross-setio are obtaied o the uity irle, for 1, for whih the orptio is zero. This demostrates the importae of miimizig losses whe maximal (strog satterig resoae) or miimal (loakig) satterig is desired. Cloaked sesors lie o the real axis i the rage Re,1, i whih the ratio betwee orptio ad satterig is maximized for a give orptio level, as it a be iferred from the figure.. We defie ow the orptio effiiey for the -th harmoi as the ratio / Combiig (8) ad (6), this quatity may be geerally writte as sat TE/ TE/ 1 TE/ sat 1. (1) For eletrially small objets domiated by dipolar satterig, a ase of partiular iterest i the ase of loaked sesors [1-8], this quatity for 1 oiides with the total orptio effiiey /. I priiple this quatity a be made as large as possible, but, beause of sat 7

8 passivity, TE/ 1 ad the orptio effiiey is fudametally bouded by the total amout of orptio, as we show i Figure, i whih we alulate / TE/ for every admissible value of TE/ of Fig. 1 ito the TE/ / versus sat sat TE/ ad, essetially drawig the image of the uity disk TE/ / ( 1) plae, orrespodig to the blue hahured regio i Figure. This area is iterestigly limited by a fudametal physial boud (solid blak lie) for the orptio effiiey of eah harmoi as a futio of the orptio level. This is osistet with our reet fidigs for ielasti quatum satterig, derived i the otext of desigig loaked sesors for matter waves [4]. Eq. (1) esures that the boudary of the admissible regio (blak solid lie) is obtaied for real values. If orptio is maximized for ay harmoi, the effiiey is 1, TE/ 1 1 osistet with (9), ad the ormalized orptio is 1/ 8, osistet with the most right poit o the blak lie i Fig.. From this poit, it is possible to either derease or irease the orptio effiiey, ad brig it to ay arbitrarily large level of iterest, but oly at the ost of sarifiig part of the orptio. For suffiietly low orptio, it is possible to sigifiatly suppress the satterig, as quatified i Figure. This physial boud desribed here is of primary importae for the desig of passive loaked sesors, as it highlights the ultimate limits of performae for eah satterig harmoi, ad how lose to the optimal satterig redutio we are for a give level of partial orptio oeffiiet. 8

9 Figure. Satterig boud for orptive, passive objets, valid for ay satterig harmoi order. Oly the blue shaded regio o this plae is admissible, yieldig a ultimate limit o the miimum satterig required for a give level of orptio. III. EXAMPES AND DISCUSSION I this setio, we apply the theory developed i Setio II to orete satterig examples. Our goal is to uderlie the geerality of our theory, disuss its impliatios ad show how it may be viewed as a essetial tool to uderstad ad optimize ivisible orbers ad sesors. A. Optial satterig from lossy aospheres As a first example, we osider the satterig from omageti aospheres i the presee of material losses. We show that this simple passive system omplies with our geeral bouds, ad we use our formalism to study ad better explai the geeral relatio betwee satterig ad 9

10 orptio i this example. This satterig problem is solved i detail i the Appedix. A etral questio of iterest is whether or ot this passive system a reah the ultimate bouds derived i Figure, ad uder what oditios. I order to approah this questio aalytially, we first assume that the aosphere is eletrially small, of size x ka 1, a beig the sphere radius. Uder this assumptio, the satterig oeffiiets obtaied from (A8) redue to the quasistati expressio 1 x 4 1 1/ ( 1) 1 i (11) Aordig to the previous setio, i order to lie o the boudary of the allowed regio for the - th harmoi, ad therefore ahieve the most iterestig satterig properties for the give level of orptio, oe eeds to have Im Im 1. Usig Eq. (11), we fid that this oditio is met wheever the permittivity i of the aosphere satisfies (1 ). (1) Assumig that we are able to tailor the sphere permittivity at will, for a fixed level of losses it is possible to reah the fudametal boud for two distit values of as log as 1 1. (13) I the partiular ase of small losses 1, the plus sig solutio i (1) simplifies ito 3 trasparey 1 o( ). (14) 1 1

11 I the limit, this solutio simply overges to the trasparey oditio that miimizes the satterig from the sphere, i.e., 1 o( ). Whe small losses are preset, oditio (14) esures that we hit the boud o the upper portio of the urve, maximizig the orptio effiiey for the give level of orptio. It is worth metioig that this oditio is i geeral ot idetial to the solutio that miimizes the satterig for the hose level of, as it differs from it by a quatity proportioal to o( ). Coversely, i the same low-loss limit the mius sig solutio of Eq. (1) simplifies ito 1 resoae o 1 3 ( ). (15) I the lossless limit, this solutio overges to the oditio that maximizes the satterig ross setio of the sphere, as it oiides with the plasmoi resoae 1/ [43], ad i presee of small losses oditio (15) allows agai hittig the boud. Also i this ase this oditio differs from the oditio to maximize orptio for the give level of by a seod-order term i. The two oditios derived above represet the required values of, for a give level of, to reah the solid blak boudary i Fig.. Note that it is obviously always possible to fid a suitable value of that maximizes the orptio or miimizes the satterig for the hose value of, but oly i the limit these solutios lie o the boud of Fig., aordig to Eqs. (14)-(15). The above quasi-stati aalysis is importat to uveil the omplexity of the satterig problem i relatio to our physial bouds i the geeral dyami ase. To validate our fidigs, we have umerially alulated orptio ad satterig i the fully dyami ase for a aopartile of 11

12 small eletrial size x., for differet values of permittivity. I Figure 3, we show the evolutio of these quatities i the orptio effiiey vs. orptio plae, omparig the obtaied results to the 1 physial boud represeted by the blak solid lie. et us first fous o the solid lies, whih represet the otours obtaied whe sweepig over all real values, while keepig ostat. The blue solid lie is obtaied for a relatively low value of material losses,.5. This value of is sigifiatly smaller tha the ritial value 3/, obtaied by evaluatig (13) for 1, value beyod whih the boud aot be reahed. Therefore we expet the otour to hit the fudametal boud at two distit poits, uder the resoae ad the trasparey oditios. Beause we are i the low-loss limit, the resoae poit orrespods to maximum orptio for the give value of, ad the trasparey poit orrespods to the maximum orptio effiiey. Ideed, the solid blue lie starts i the bottom left orer for large egative values of ad, as grows ad get loser to -, ireases util it reahes the maximum, obtaied for.1 i good agreemet with the quasistati preditio (15). This maximum lies o the physial boud (solid blak lie), osistet with the preditios of our quasi-stati model. As we keep ireasig, mootoially dereases, but yet the orptio effiiey / reahes a maximum o the upper portio of the boud. The assoiated satterig miimum is ideed obtaied for 1, osistet with (14). sat Whe the losses are equal to the ritial value (13), we obtai the red solid lie i Fig. 3. Cosistet with Eq. (1), i this ase the solutios are ow degeerate, therefore we expet the urve to be tagetial to the boud, reahig it at the value 1/ /. Beause the lossless limit assumptio is o loger valid i this ase, we should ot expet the maxima of 1

13 orptio or orptio effiiey to our o the boud. Ideed, these preditios are all verified i the dyami evolutio of the red otour, for whih the orptio ad orptio effiiey maxima our iside the allowed regio, away from the boud. By hoosig be larger tha this ritial value, as i the ase of the solid gree lie ( 3), we do ot hit the physial boud for ay value of. Figure 3 : Satterig ad orptio for a dieletri aosphere of permittivity ' i'' ad eletri size x., for differet searios. The blak solid lies represet the 1 boud. The other solid lies represet otours obtaied whe sweepig over all real values, for.5 (below ritial, blue lie), 1.5 (ritial, red lie) ad 3 (beyod ritial, gree lie). The irular markers represet the dual otours obtaied by sweepig " over all positive values, for a give value of.4 (blue markers), 1 (red markers) ad.84 (gree markers). 13

14 I Figure 3, we also plot dual otours, represeted by irular markers, whih orrespod to the ase i whih is fixed ad is varied over all positive umbers. The blue markers orrespod to.4, the red markers to 1 ad the gree oes to.84. We otie that some of these otours appear to hit the boud, while others stay away from it. To uderstad this behavior, we ivert oditio (1) to express the oditio to iterset the boud for ostat otours, ad fid the uique solutio 1 9 ( 1). (16) Suh a solutio exists oly for 1, i.e., for values of betwee the lossless resoae ad trasparey oditios. This is ideed verified i Figure 3, for whih the.4 (blue) otour ever rosses the boud, while the.84 does. The 1 otour is a limitig ase, whih does ross the boud oly at ifiity i the top left orer of the figure, for whih, osistet with (16). Iterestigly, the 1 otour asymptotially overges to the boud for small values of, but gets away from the boud for suffiietly large values of, overgig asymptotially towards the bottom left orer of the figure. I fat, all the ostat- otours appear to ted to the same oblique asymptote whe losses are suffiietly high. The ostat otours also ted to the same asymptote whe the real part of permittivity is largely positive or egative. This is easily uderstood by rewritig Eq. (1) for the 1 harmoi as sat 8. (17)

15 Due to the fat that 1 is bouded ad well-behaved, for (positive, egative or imagiary) its value asymptotially overges to a ostat ad Eq. (17) maps ito a straight lie o the plae of Fig. 3. I the quasi-stati limit, Eq. (17) a be expliitly writte as 3 ( ) 6 sat x ( 1),,, (18) whih expliitly shows that, i the limit or, we always get the same liear relatioship betwee / ad / : sat sat 3 6 or. (19) x As a result, o the logarithmi sale of Figure 3, all the otours overge i these limits to the asymptote 3 log log 6log x log, () sat for whih the slope is idepedet of the eletrial size of the aosphere, ad the iterept o the vertial axis shifts up for smaller objets. I the followig we refer to this asymptote as the perfet eletri odutor (PEC) limit, for obvious reasos. I the opposite limit of zero material losses, the ostat otours also overge to straight lies i Figure 3, to whih we refer as lossless asymptotes. Their positio deped o the partiular value of, but they are all parallel to the PEC asymptote. This is explaied usig Eq. (18) for. We obtai 15

16 3 6 sat x 1, (1) so that the expressio for the lossless asymptotes is give by 3 log log 6log x log log 1 sat, () ideed desribig lies parallel to the PEC asymptote ad shifted by a amout that depeds o. From (), we reogize that the lossless asymptotes are shifted up with respet to the PEC asymptote whe / / ad dow i the opposite ase. Note that the value equals.5 for the 1 harmoi ad it is idepedet of, f. (1). It is exatly the average of the resoae ad trasparey solutios, regardless of. It turs out therefore ) from the that the PEC asymptote separates the trasparey regio (defied by / resoae regio ( / ). Whe the eletrial size of the sphere varies, the PEC asymptote is aordigly shifted, draggig with it the lossless asymptotes, ad therefore, all otour lies. All these fidigs allows us to fully desribe the dyamis of the urves i Fig. for arbitrary values of or. Figure 4 shows a more omplete diagram for the same size sphere, osiderig may differet values of ' ad '', while fousig o low values of '' for whih the boud a be reahed. Similar to the previous plot, the solid lies are '' -ostat lies, sweepig ' from egative to positive values. The dashed otour lies are istead ' -ostat lies, sweepig '' through positive values. The otours essetially form a urviliear referee system mappig a arbitrary value of omplex permittivity ito the orrespodig level of orptio ad orptio 16

17 effiiey, ad they are foud to spa the etire admissible regio of Fig. 4, implyig that, give the opportuity to arbitrarily vary ' ad '' at the frequey of iterest, we may realize ay allowed level of orptio ad orptio effiiey with just a sigle, dieletri aopartile. Eah otour lie i the figure, for a give value of ' ad '' as idiated i the plot, is formed by segmets of differet olor, ad the solid ad dashed urves iterset oly whe they have the same olor. Figure 4. Similar to Fig. 3, satterig ad orptio for a aopartile of eletri size ka., varyig its permittivity ' i''. The dashed otours are obtaied varyig '', keepig ' ostat at the idiated value. The solid otours are plotted varyig itersetios of urves with same olor. 17 ', keepig '' ostat at the idiated value (i blak). The blak solid lie represets the 1 fudametal boud. Satterig ad orptio for a arbitrary value of omplex permittivity are obtaied at the

18 Figure 5. Same as Figure 4 but for a larger eletrial size, ka.5. As predited by the theory, the PEC asymptote is shifted dow. I Figure 5 we plot the same otours as i Figure 4, but for a bigger eletrial size ( x.5 ), ofirmig the expeted dowward shift of the PEC asymptote osistet with (), whih arries alog all the other otours. These figures sythetially ofirm that the above osideratios, iludig the asymptoti behavior, hold i the geeral dyami ase. Compariso with the physial boud uveils the omplexity of the relatio betwee satterig ad orptio of a aosphere, outliig the role of material losses, size ad permittivity. As a orollary of our fidigs, it is i priiple ot eessary to use a loak to obtai a high level of orptio effiiey, provided that we a arbitrarily vary, as ay poit o the solid blak lie is aessible. It would be ideed suffiiet to have ' 1 to eable the maximum possible orptio effiiey for a give (suffiietly low) level of '', at the trasparey oditio 18

19 (14). Obviously i a realisti seario we do ot have the arbitrary otrol o the omplex permittivity at the frequey of iterest, ad therefore a suitably desiged over may tailor satterig ad orptio with more degrees of freedom, as we disuss ext. B. Furtive optial sesors ad orbers made of ore-shell aopartiles The additio of a loak is useful to hage the dyamis of the satterer respose i the previous plots, whe, as it is usually the ase, we do ot have the flexibility of hagig the permittivity, the material losses, or the size at will. I Figure 6 we osider a ore-shell geometry, as i the plasmoi loakig tehique [9], i whih a lossless shell with radius a ad permittivity.15 overs a ore of radius a with permittivity ' i '', varied over all admissible values. The eletrial size of the objet is ka., for a fixed fillig ratio a/ a.9. The shell permittivity was hose to redue the satterig of a PEC sphere of same size, usig the satterig aellatio method [9]. As visible i the figure, the effet of suh loak desig is ideed to shift up the PEC oblique asymptote, draggig all the otour lies with it. Cosistet with Eq. () ad the ature of the proposed loak, the effet is equivalet to reduig the effetive size of the objet i the PEC limit ad, as a otieable osequee, the resoae regio (below the PEC asymptote) ow iludes a sigifiat portio of the physial boud for whih the orptio effiiey is very large. I other words, the loak opes the iterestig possibility to ahieve plasmoi resoaes i the upper portio of the plot, above the poit of maximal allowed orptio, for whih the orptio effiiey is large ad the satterig is suppressed. These resoat oditios are similar to the oes origially evisioed i [1], for whih orptio ad satterig ross-setios reah a loal maximum at the same frequey, but with large ratios betwee the two. Plasmoi resoaes with high orptio effiiey aot 19

20 be supported by a bare aopartile with similar size, as see i Figure 4. The oly solutio to get a high level of orptio with a bare partile is to exploit the trasparey oditio, haraterized by a satterig dip ad oresoat (flat) orptio ross-setio, osistet with some of the oepts disussed i [41]. A first evidet advatage of usig a suitably desiged loak is the to ope the high-effiiey regio to plasmoi resoaes, eablig resoat sesors with low visibility. Figure 6. Same as Figure 4, for a ore-shell aopartile of eletrial size ka., with.15 ad a.9a. Obviously the presee of the loak a also provide more flexibility to hoose the values of ' for the resoae ad trasparey oditios. I the example of Figure 6 these values are both egative: resoae.7 ad.4, whih may be derived by studyig the ore- trasparey

21 shell struture i the quasi-stati limit, geeralizig the aalysis i the previous setio. For a ore-shell aopartile, the oditios to lie o the boud beome 6 f(, ) 79 g(, ) (3) g(, ) with f (4) (, ) 4 (1 ) 4 ( ) (4 4 ) ad g. (5) (, ) ( ( 1) )( 1 ) As expeted, Eq. (3) ollapses to the bare aosphere ase Eq. (1) whe we osider 1. The oditio to reah the boud ow depeds o ad : low loss limit, the two solutios simplify ito. I the 6 79 g(, ) ( 4( ) (4 4 ) 4( 1 )(1 ) ) ( ( 1 ) )( 1 ) O. (6) This oditio, evaluated for.15 ad a/ a.9, ideed provides.7 ad 1.7, i good agreemet with the values umerially obtaied i Figure 6. I additio, oe a easily verify, followig the same steps as i the previous setio, that i the quasistati ad low-loss limits, the solutio oiides with the plasmoi resoae oditio of a ore-shell partile, ad with the loakig oditio [36]. To uderstad why the PEC asymptote is shifted up i the presee of a plasmoi loak, we write the equivalet of Eq. (18) for a ore-shell struture. I the limit, we obtai that 1

22 the asymptote is a straight lie of the form log( / ) alog( / ) b. The slope a 1, sat as before, ad the iterept of the vertial axis, whih determies the PEC asymptote, is give by b (4(9 x )( 1 ) (9 4 x )( ) 4 ( 9 x )( 1 )) log x ( 1 ). (7) Whe the loak is desiged to ael the satterig of a PEC sphere of same outer radius as the ore-shell aopartile, i.e., uder the oditio [9] 3 1 1, (8) we ideed obtai b, whih elegatly ofirms the behavior of the loaked sesor. This proves that oe a arbitrarily shift upwards the PEC asymptote i the plot by desigig a loak that aels the satterig of a PEC sphere of same outer radius as the osidered partile. This is a importat desig rule for resoat loaked sesors, as it eables peuliar plasmoi resoaes i the high orptio effiiey regio. I Figure 6, for whih.9, we are very lose to the ideal value / 4.94 predited by (8), eablig a sigifiat upward shift of the asymptote. Obviously the fudametal bouds derived i the previous setio are still respeted by loaked partiles, ad the mai effet of addig a loak osists i providig more flexibility i tailorig the dyami relatio betwee orptio ad satterig. I Figure 7, we explore aother loak desig, i.e., oatig the ore with a epsilo-ear-zero (ENZ) shell.1. From Eq. (6) we see that the effet of a ENZ loak is to brig loser the resoae ad trasparey oditios, yieldig whe, osistet with the Fao-like satterig sigatures obtaied with ENZ loaks i [43]. This is also verified i the

23 dyami ase, as see i the figure, i whih resoae ad trasparey poits are brought very lose to eah other,.96 ad.11. Suh desigs may be of iterest to ehae oliear effets for swithig appliatios, exploitig the strog o/off depedey of the satterig ross-setio ad orptio effiiey, extedig the oepts proposed i [45] to the ase of sesors with tuable effiiey. Agai, the physial boud disussed here is ruial to uderstad the limitatios, omplexity ad potetial of these aoswithig devies. Figure 7. Same as Figure 6 but with a differet loak permittivity.1. 3

24 Figure 8. (a) Satterig ad orptio spetrum for a realisti 4m silio partile at optial frequeies. (b) Satterig ad orptio spetrum for the same partile loaked by a 11m plasmoi loak made of silver. () Compariso of the satterig spetrum with the boud for the bare ad loaked ase. 4

25 As a realisti satterig example, we osider the optial satterig from a silio aopartile with radius a m ad ompare it to the oe obtaied whe the same sphere is surrouded by a plasmoi loak made of silver. Material dispersio is take from experimetal data [46-47]. The satterig ad orptio ross-setios of the bare sphere are reported i Figure 8(a) for iidet wavelegths betwee m ad 8 m, ad the otour obtaied whe sweepig frequey is show i blue i Figure 8() ad ompared to the 1 boud. As evidet from these plots, the bare silio aosphere starts orbig sigifiatly oly i the UV rage, due to ireased eletroi orptio proesses at these eergies. The orptio effiiey is lose to uity throughout the optial rage, makig the aosphere a rather ieffiiet orber. I Figure 8(b), we osider the satterig spetrum of the same aopartile embedded i a 11 m silver shell, ad ompared it to the boud i Figure 8() (red lie). As evidet from the figure, the loak ompletely modifies the satterig properties of the struture, eablig it to aess high orptio effiiey values ( 5 ) i the visible rage, ad at the same time reahig the boud. These fidigs show that it is possible to largely maipulate ad optimize the orptio ad orptio effiiey of a aosphere i the frequey rage of iterest by properly oatig it, withi the fudametal bouds derived i the previous setio. C. Tuable ateas with optimal orptio effiiey As a fial example to highlight the breadth of our fidigs, evisio ow a ovetioal radiofrequey sesor, osistig of a eletrially small dipole atea loaded by a impedae Z R ix, as osidered i [41]. The assoiated satterig problem may be aalytially solved, as show i the Appedix, assumig without loss of geerality that the atea is aliged 5

26 i the diretio of polarizatio of the impigig field, by modelig the atea as a dipole with polarizability [41] 3 X X X ir k 3 1 i i i l 4X i X ir 6, (9) where X i is the iput reatae of the dipole atea of half legth l. Also i this ase we hoose a subwavelegth geometry, so that the satterig is domiated by the dipolar otributio. The atea legth is l 3 3 m, with a diameter of 6 μm, ad we operate it at 3 GHz. By varyig the load resistae ad reatae, it is possible to tue the dipolar satterig ad orptio of the objet at will, similar to the previous plots for dieletri aospheres, ad spa the whole admissible regio i Figure 9, i whih we alulate the 1 orptio effiiey ad orptio ross-setios for this loaded dipole, for differet values of Z R ix. I these alulatios, we take the value of iput reatae X i 53.9, alulated for our geometry usig formula (A13), ad we restrit ourselves to R, to esure passivity. The solid otours are geerated by keepig R ostat ad sweepig X from egative (apaitive) to positive (idutive) values. The dashed lies are geerated by keepig X ostat ad sweepig the load resistae through positive values. By itersetig these two sets of otour lies, it is possible to extrat from the figure the orptio ad orptio effiiey for ay omplex value of load impedae. Also here oly otour lies with the same olor iterset. The figure shows similar features as i the previous examples, despite the ompletely differet ature of the satterer. The load resistae of the dipole atea plays a role aalogous to the 6

27 imagiary part of the permittivity of the aosphere, while the load idutae plays the role of the real part of permittivity, ad similar osideratios may be draw as i the previous subsetios. Figure 9. Similar to Figures 3-4, but for a loaded dipole atea, varyig its load impedae R ix. The dashed otours are plotted varyig R, for ostat X at the idiated value. The solid otours are plotted varyig X, for ostat R at the idiated value (i blak). The solid blak lie represets the fudametal limit for the first satterig harmoi. We a explai the dyamis of the otours of Figure 9 by studyig the system i the quasistati limit. Repeatig the steps detailed above i the ase of the aosphere, we fid the two possible solutios to reah the physial boud 7

28 X 1 (5 4 9 Xi R Xi ). (3) Eq. (3) shows that the boud may oly be reahed for suffiietly low values of the load resistae, i.e. satisfyig 3 R Xi. (31) I the limit of a low load resistae R 3 X /, the first oditio simplifies ito i R X X O R, (3) 3 4 i 3X i ad the seod, ito R X X O R. (33) 3 i 3Xi As i the previous examples, it is easy to hek that oditio (3) oiides with a satterig miimum i the low resistae limit, osistet with the fidigs i [41]. This is the trasparey oditio for a small loaded dipole. Coversely, Eq. (33) oiides with the oditio to maximize the orptio for the give value of R i the low resistae limit. This is the atea resoae oditio, aalogous to the plasmoi resoae of the previous setios, for whih the orbed power is maximized. Also i this ase there is a threshold, defied by (31), for the material losses R beyod whih we aot reah the physial boud. As see i Figure 9, our quasi-stati osideratios are fully supported by the dyami alulatios: for fixed (small) load resistae (solid lies), the orptio ross-setio is ideed 8

29 maximized aroud the resoat oditio X Xi, while the orptio effiiey is maximized aroud the trasparey oditio X 4X i, osistet with (3)-(33) ad with the fidigs i [41]. The olute maximum orptio is ahieved uder the ojugate mathed oditio Z Z [37], whih provides uitary orptio effiiey. Similar to the ase of * i aopartiles, the otours of ostat X all overge to the same asymptote whe R or X. Ideed, i both ases we obtai the followig equatio for the asymptote log 16 X, (34) 3 i log log 4 4 sat 3 kl where k ad are respetively the wave umber ad harateristi impedae i free spae, whih represets the ope-iruit asymptote, the aalog of the PEC asymptote i the aosphere ase. Also here, this asymptote shifts up for shorter ateas. As evidet i Figure 9, we also obtai a family of lossless asymptotes for ostat X i the limit R. These asymptotes are all parallel to the ope-iruit asymptote with expressio 16 X ( X X ) log log sat 3 (4 3 i i 4 4 log kl Xi X) (35) ad a upward shift whe X / X ( X X ) /, ad a dowward shift whe i X / X ( X X ) /, with ( X X ) / 5 / i the quasi-stati limit. The ope-iruit i asymptote separates the trasparey ad resoae regios, as disussed for the aosphere seario. This example demostrates the geerality of our aalysis ad of the derived physial bouds. 9

30 ike i the ase of aopartiles, the additio of a loak may be used to effetively redue the size of the atea, shiftig up all otours, osistet with the geometry origially proposed i [1]. These results larify the potetial of the loaked sesor oept ad the reah it may have i maipulatig satterig ad orptio withi the bouds derived here. The loak may allow ahievig large orptio effiieies i the resoat regio, eablig the respose disussed i [1], for whih satterig ad orptio both reah a loal maximum at the desig frequey, with large ratio betwee the two. I this ase, the atea may ot be easily detetable whe out of resoae (sie it almost does ot satter), ad it satters the miimum for the hose level of orptio, resultig i the best ase seario for passive loaked sesors. Our fidigs may eable the desig of optimized loaks for tuable reeivig ateas with high orptio effiiey ad optimal miimum-satterig ateas. We will disuss these issues i further details i a upomig study. IV. CONCUSIONS I this paper, we preseted ad disussed fudametal bouds o satterig ad orptio of passive objets. These limitatios, derived from passivity ad power oservatio, suessfully quatify the miimum ad maximum satterig for a give level of orptio, providig a importat tool to qualitatively ad quatitatively uderstad the limitatios assoiated with loakig orptive objets. We applied our theory to a variety of examples, iludig optial satterig from dieletri aospheres ad ore-shell aopartiles, ad mirowave satterig from a loaded dipole atea. We have explaied the role of the loak i loaked sesor desigs, showig that oe a eable peuliar resoaes for whih both satterig ad orptio reah a 3

31 loal maximum, but with a large ratio betwee them. The derived physial limitatios provide a semial basis i a wide rage of situatios. Our aalysis may be readily applied to bigger objets for whih several harmois otribute to the satterig, as eah of them follows the bouds desribed i this work, a oept that may be used to realize furtive super-orbers [48]. Compariso with the boud preseted here is a relevat figure of merit for ay pratial desig of furtive sesors ad orbers. We believe that these fudametal bouds a be used to formulate a set of desig rules to egieer optimal low-satterig sesors ad orbers i the optial regime, as well as miimum-satterig ateas at radio-frequeies. ACKNOEDGMENTS This works has bee supported by the AFOSR YIP award No. FA ad the DTRA YIP award No. HDTRA APPENDIX I this Appedix, we solve aalytially the geeral problem of satterig of eletromageti waves by a ore-shell geometry loaded by a dipole of polarizability plaed at the eter of a spherial oordiate system ( r,, ) etered with the ore-shell, ad orieted alog ˆx. It is surrouded by the permittivity profile: if r a if a r () r if a r a (A1) 31

32 This geeral geometry iludes all the possible searios aalyzed i the preset paper. ikz A plae wave E xe ˆ e is iidet upo the system, ad we assume a i i t e timedepedee. The total field, the sum of the sattered ad iidet field, may be expaded ito spherial waves ad deomposed ito trasverse eletri (TE) ad trasverse mageti () fields, respetively assoiated with the radial mageti ad eletri vetor potetials [34,49] os 1 A E i r r P si 1 F E i r r P 1 r ( ) (os ) 1 ( 1) TE 1 r ( ) (os ) 1 ( 1), (A) where m P are egedre polyomials ad ( r) / ( r) is the harateristi impedae of the medium, oted, ad i the outside medium, the loak, ad the iside domai, respetively. The wave umber ( r) ( r) will be oted k, k ad k i the outside medium, the loak, ad the iside domai, respetively. The radial futios ( r) i (A) TE/ are solutios of the radial equatio, obtaied by solvig the spherial Helmholtz equatio. Takig ito aout the exitatio field, these futios a be sought, for TE waves, i the form a j ( kr) if r a TE TE TE TE ( r) d j( kr) e y( kr) if a r a TE (1) j( kr) h ( kr) if a r, (A3) where j ad y are spherial Bessel futios of the first ad seod kid, ad (1) h is the spherial Hakel futio of the first kid. The TE radial futios (A3) are uaffeted by the presee of the dipole, sie dipolar radiatio a be desribed as a 1 wave[36]. Therefore, 3

33 the radial futios differ from the expressio i (A3) by the additio of the field radiated by the dipole, yieldig the form a j ( kr) b h ( kr) if r a (1) ( r) d j( kr) e y( kr) if a r a (1) j( kr) h ( kr) if a r (A4) where m is Kroeker's delta futio ad b1 depits the stregth of the radiatio from the dipole atea. The idued dipole momet at the eter is liked with the loal field by the polarizability givig, after some alulatios : p E E a (A5) lo 1. O the other had, the oeffiiet b1 may be related to the stregth p of the dipole, by expressig the usual dipolar radiatio as a Mie series, i.e. [36] 3 k b1 i p. (A6) 6 E The oeffiiet b1 is ow expressed as a futio of a1 ombiig (A5) ad (A6), the the result is iserted ito (A4). Usig (A), (A3) ad (A4), the tagetial fields a be alulated. Eforig the boudary oditios at r a ad r a yields two liear systems, 33

34 3 1 k (1) j ( ka) y( ka) j ( ka) i1 h ( ka) 6 ˆ ˆ a 3 (1) 1 ˆ k ˆ J ( ka) Y ( ka) J ( ka) i 1 H ( ka) ka 6 ka ka d ( ) j k a (1) j ( ka) y( ka) h ( ka) e ˆ J ( ka ) Jˆ k a Yˆ k a Hˆ k a k a k a k a (1) ( ) ( ) ( ) ka (A7) for oeffiiets ad j ( ka) j ( ka) y( ka) ˆ ( ) ˆ ( ) ˆ TE J ka J ka Y ( ka) a ka k TE a ka d (1) TE j ( ka j ( ka) y( ka) h ( ka) e ) ˆ ˆ ˆ (1) TE ˆ J ( ka) Y ( ka) H ( ka) J ( ka ) ka ka ka ka (A8) for TE oeffiiets. We have used the followig otatio: ˆ F ( r ) rf ( r ). (A9) ( r) By solvig the liear systems (A7) ad (A8), the exat solutio for the total field is obtaied. The desired ross-setios a the be alulated usig Eqs. ()-(4). Fially, we give the form of for a loaded dipole atea. For the fudametal physial boud to be respeted, a aurate, power osistet expressio of polarizability must be used. We assume the geeral form: (A1) S i 34

35 If the satterer is lossless, the stati polarizability 1 S is real ad 1 is the radiatio orretio, takig are of power oservatio. By pluggig (A1) ito (A7), assumig 1 S ad imposig the lossless oditio sat ext, we obtai mathematially the eessary oditio, imposed by power oservatio : 3 1 k. (A11) 6 The stati polarizability ase as [41] 1 S is well-kow for dipole ateas, ad is expressed i the geeral 1 S 3 X X iz i i l 4X i iz, (A1) where Z R ix is the omplex impedae loadig the atea ad X i is the egative imagiary part of the iput impedae of the atea, Zi Ri ix i, whih a be approximated for small Hertzia dipoles with the followig futio of the atea half-legth l ad diameter d [37] X i l l 1 d 1. (A13) ta kl The results preseted i this Appedix over all situatios preseted i this artile, takig 1 for uloaked ateas, for ore-shell aopartiles ad, 1 for uloaked spherial aopartiles. The ase of loaked ateas is also iluded by the preset alulatio. 35

36 Referees 1. J.B. Pedry, Phys. Rev. ett. 85, 3966 ().. P. Alitalo ad S.A. Tretyakov, Mat. Today 1, 1 (9). 3. J. Valetie, J. i, T. Zetgraf, G. Bartal, ad X. Zhag, Nature Materials 8, 568 (9). 4. S. Tretyakov, P. Alitalo, O. uukkoe, ad C. Simovski, Phys. Rev. ett. 13, 1395 (9). 5. J.B. Pedry, D. Shurig, ad D.R. Smith, Siee 31, 178 (6). 6. U. eohardt, Siee 31, 1777 (6). 7. D. Shurig, J.J. Mok, B.J. Justie, S.A. Cummer, J.B. Pedry, A.F. Starr, ad D.R. Smith, Siee 314, 977 (6). 8. N. ady ad D.R. Smith, Nature Materials, 11, 5, (13). 9. A. Alù ad N. Egheta, Phys. Rev. E 7, 1663 (5). 1. A. Alù ad N. Egheta, Phys. Rev. ett. 1, (8). 11. A. Alù ad N. Egheta, Phys. Rev. E 78, 456 (8). 1. A. Alù ad N. Egheta, New Joural of Physis 1, (8). 13. A. Alù ad N. Egheta, J. Opt. A 1, 93 (8). 14. P.-Y. Che, J. Sori, ad A. Alù, Adv. Mater. 44, OP81 (1). 15. B. Edwards, A. Alù, M.G. Silveiriha, ad N. Egheta, Phys. Rev. ett. 13, (9). 16. D. Raiwater, A. Kerkhoff, K. Meli, J.C. Sori, G. Moreo, ad A. Alù, New Joural of Physis 14, 1354 (1). 36

37 17. A. Alù, Phys. Rev. B 8, (9). 18. P.-Y. Che ad A. Alù, Phys. Rev. B 84, 511 (11). 19. J.C. Sori, P.Y. Che, A. Kerkhoff, D. Raiwater, K. Meli, ad A. Alù, New J. Phys. 15, 3337 (13).. H. Hashemi, B. Zhag, J.D. Joaopoulos, ad S.G. Johso, Phys. Rev. ett. 14, 5393 (1). 1. A. Alù ad N. Egheta, Phys. Rev. ett. 1, 3391 (9); F.J. Garía de Abajo, Physis, 47 (9).. J.-Z. Zhao, D.-. Wag, R.-W. Peg, Q. Hu, ad M. Wag, Phys. Rev. E 84, 4667 (11). 3. A. Alù ad N. Egheta, Metamaterials 4, 153 (1). 4. G. Castaldi, I. Gallia, V. Galdi, A. Alù, ad N. Egheta, J. Opt. So. Am. B 7, 13 (1). 5. Z. Rua ad S. Fa, J. Phys. Chem. C 114, 734 (1). 6. A. Greeleaf, Y. Kurylev, M. assas, ad G. Uhlma, Phys. Rev. E 83, 1663 (11). 7. X. Che ad G. Uhlma, Opt. Express 19, 518 (11). 8. A. Greeleaf, Y. Kurylev, M. assas, U. eohardt, ad G. Uhlma, PNAS 19, 1169 (1). 9. A. Alù ad N. Egheta, Phys. Rev. ett. 15, 6396 (1). 3. F. Bilotti, S. Triario, F. Pierii, ad. Vegi, Opt. ett. 36, 11 (11). 31. F. Bilotti, F. Pierii, ad. Vegi, Metamaterials 5, 119 (11). 3. P. Fa, U.K. Chettiar,. Cao, F. Afshimaesh, N. Egheta, ad M.. Brogersma, Nature Photois 6, 38 (1). 37

38 33. C.A. Valagiaopoulos ad N.. Tsitsas, Radio Siee 47, (1). 34. C.F. Bohre ad D.R. Huffma, Absorptio ad Satterig of ight by Small Partiles (Joh Wiley & Sos, 8). 35. A. Alu ad N. Egheta, Joural of Naophotois 4, 4159 (1). 36. A. Alù ad N. Egheta, Joural of Applied Physis 97, 9431 (5). 37. C.A. Balais, Atea Theory: Aalysis ad Desig, d Ed (Wiley Idia Pvt. imited, 7). 38. J.B. Aderse ad A. Fradse, IEEE Trasatios o Ateas ad Propagatio 53, 843 (5). 39. D.H. Kwo ad D.M. Pozar, IEEE Trasatios o Ateas ad Propagatio 57, 37 (9). 4. M. Gustafsso, M. Cismasu, ad S. Nordebo, Iteratioal Joural of Ateas ad Propagatio 1, 1 (1). 41. A. Alù ad S. Maslovski, IEEE Trasatios o Ateas ad Propagatio 58, 1436 (1). 4. R. Fleury ad A. Alù, Phys. Rev. B 87, 116 (13). 43. A. Alù ad N. Egheta, IEEE Trasatios o Ateas ad Propagatio 55, 37 (7). 44. F. Motioe, C. Argyropoulos, ad A. Alù, Phys. Rev. ett. 11, (13). 45. C. Argyropoulos, P.-Y. Che, F. Motioe, G. D Aguao, ad A. Alù, Phys. Rev. ett. 18, 6395 (1). 46. E. D. Palilk, Hadbook of Optial Costats of Solids, Aademi Press, Bosto, D.E. Aspes, ad A. A. Studa, Phys. Rev. B 7, 985 (1983). 38

39 48. N. Mohammadi Estakhri ad A. Alù, uder review. 49. C.A. Balais, Advaed Egieerig Eletromagetis (Wiley, 1989). 39