Cloaking mechanism with antiphase plasmonic satellites

Transcription

1 Univerity of Pennylvania ScholarlyCommon Departmental Paper (ESE) Department of Electrical & Sytem Engineering Cloaking mechanim with antiphae plamonic atellite Mário G. Silveirinha Univerity of Coimbra Andrea Alù Univerity of Pennylvania, Nader Engheta Univerity of Pennylvania, Follow thi and additional work at: Recommended Citation Mário G. Silveirinha, Andrea Alù, and Nader Engheta, "Cloaking mechanim with antiphae plamonic atellite",. November 28. Copyright American Phyical Society. Reprinted from: Cloaking mechanim with antiphae plamonic atellite Mario G. Silveirinha, Andrea Alu, and Nader Engheta, Phy. Rev. B 78, 2519 (28), DOI:1.113/PhyRevB Publiher URL: Thi paper i poted at ScholarlyCommon. For more information, pleae contact repoitory@pobox.upenn.edu.

2 Cloaking mechanim with antiphae plamonic atellite Abtract In thi work we theoretically demontrate the poibility of cloaking a given object by urrounding it with a finite collection of uitably dimenioned, dicrete "antiphae" plamonic catterer. It i hown that the total cattering from the object may approximately be canceled by the current induced on a finite number of the plamonic "atellite" catterer, effectively making the whole ytem inviible to an external oberver. Unlike other approache, the propoed olution allow one to cloak a given object in a noninvaive manner ince the antiphae atellite, being finite in number, do not need to fully cover and be in direct contact with the cloaked Keyword genetic algorithm, plamon, TRANSPARENCY, METAMATERIAL Comment Copyright American Phyical Society. Reprinted from: Cloaking mechanim with antiphae plamonic atellite Mario G. Silveirinha, Andrea Alu, and Nader Engheta, Phy. Rev. B 78, 2519 (28), DOI:1.113/PhyRevB Publiher URL: Thi journal article i available at ScholarlyCommon:

3 PHYSICAL REVIEW B 78, Cloaking mechanim with antiphae plamonic atellite Mário G. Silveirinha, 1,2 Andrea Alù, 1,3 and Nader Engheta 1, * 1 Department of Electrical and Sytem Engineering, Univerity of Pennylvania, Philadelphia, Pennylvania 1914, USA 2 Department of Electrical Engineering Intituto de Telecomunicaçõe, Univeridade de Coimbra, 33 Coimbra, Portugal 3 Department of Electrical and Computer Engineering, Univerity of Texa at Autin, Autin, Texa , USA Received 2 May 28; revied manucript received 14 October 28; publihed 12 November 28 In thi work we theoretically demontrate the poibility of cloaking a given object by urrounding it with a finite collection of uitably dimenioned, dicrete antiphae plamonic catterer. It i hown that the total cattering from the object may approximately be canceled by the current induced on a finite number of the plamonic atellite catterer, effectively making the whole ytem inviible to an external oberver. Unlike other approache, the propoed olution allow one to cloak a given object in a noninvaive manner ince the antiphae atellite, being finite in number, do not need to fully cover and be in direct contact with the cloaked DOI: 1.113/PhyRevB PACS number : Sq, 41.2.Jb, e I. INTRODUCTION When an electromagnetic wave illuminate an object, part of the energy of the incoming beam i cattered in the whole pace, making it preence detectable to neighboring oberver. In many application, uch a remote ening, radar, radome, and medical or biological meaurement, it may be deirable to dratically reduce thi cattering, ideally independent of the direction of the impinging wave and the oberver poition. Thi general problem of cloaking an object or a region of pace with a uitable cover that induce inviibility ha received ignificant attention recently. In particular, our group ha demontrated, analytically and numerically, that when a given object i covered with a properly deigned plamonic hell then, depite the fact that it phyical ize increae, it total cattering cro ection may dratically drop. 1 Other cloaking technique baed on coordinate tranformation, 2 5 optical conformal mapping, 6 and anomalou localized reonance 7 have alo been recently uggeted. Thee technique are baed on quite different principle from the one dicued in thi work, and in general require the availability or contruction of complex, highly aniotropic reonant and nonuniform material with electrical and/or magnetic repone. On the contrary, in the catteringcancellation technique 1 the cloak may be homogeneou and iotropic, in general independent of the wave polarization and it direction of incidence. A review and comparion of different cloaking technique employing metamaterial and plamonic wa recently reported in Ref. 8. The phenomenon decribed in Ref. 1 i a conequence of the out-of-phae polarizability of plamonic material, which, under uitable condition, may compenate for and annihilate the cattering from the cloaked Thi effect enable the tunneling of the electromagnetic wave through and/or around the cloaking ytem and the poibility for an oberver to ee through and beyond the cloaked object at the deign frequency. A related tunneling effect may alo be oberved in other more complex cenario involving cloed geometrie. 9 In Ref. 1 and 11 we alo demontrated how thi tranparency technique may be applied at frequencie for which plamonic material are not directly available in nature, by deigning the cover a a parallel-plate metamaterial with metallic implant. In Ref. 12, moreover, we proved that the propoed tranparency technique i robut againt variation in the deign parameter, uch a variation in frequency around the deign frequency or modification of ize and hape of the cloaked In Ref. 13 we extended thee concept to collection of particle and larger In Ref. 14, finally, we howed that multilayered cloak may enure multifrequency operation and further improved performance. Here, following a imilar principle, we dicu and analyze the idea of uing antiphae catterer placed near an object to achieve imilar effect without the need for totally covering the object to be cloaked. Indeed, under ome circumtance it may not be ideal or practically poible to fully cover the deired object with a uniform cover, and thu the technique propoed here may provide more flexibility and degree of freedom in thee ituation. Moreover, it may often be difficult to realize the required plamonic cover when the object of interet ha a complex hape. Since the tranparency technique propoed in Ref. 1 i hown to be baed on an integral effect rather than on a reonance phenomenon, 12 it i expected that cattering cancellation may till be achieved uing a limited number of concentrated, properly deigned atellite provided that we can induce an analogou negative cattering in them. Analogou to Ref. 1, thi negative cattering may be provided by material with real part of the permittivity le than that of free pace or that of the background medium. A dicued in Ref. 1 and 12, uch material exhibit a negative polarizability; i.e., the local induced dipole moment i 18 out of phae with the impinging electric field. Thi provide the poibility of canceling part, if not mot, of the cattered power. The aim of thi paper i to demontrate, analytically and with full-wave imulation, that urrounding an object with a uitable agglomerate of antiphae catterer yield an alternative, viable, and arguably imple, noninvaive cloaking mechanim. The antiphae atellite are deigned in uch a way that the collective uperimpoition of the cattered wave interfere detructively, making the total ytem inviible to radiation /28/78 2 / The American Phyical Society

4 SILVEIRINHA, ALÙ, AND ENGHETA PHYSICAL REVIEW B 78, z y x 1 1 of atellite E inc obj d H inc 4.1 FIG. 1. Color online Geometry of the problem: a ytem formed by a dielectric cylindrical object and antiphae plamonic catterer o-called atellite illuminated by an incoming plane wave. The poition and the material parameter of the atellite are optimized uing, for example, the genetic algorithm GA to reduce the total cattering cro ection from the whole ytem, inducing an effective cloaking mechanim. II. THEORETICAL ANALYSIS To begin with, we will develop a theoretical model that decribe approximately the cattering propertie of a ytem formed by a given object and an agglomerate of atellite ee Fig. 1 for a repreentative geometry. A explained in Sec. I, the object i urrounded by a et of antiphae catterer, which hould be deigned in uch a way that the total cattering from the overall ytem i made very mall due to detructive interference at all viible angle. For implicity, it i aumed that the geometry i invariant to tranlation along the z direction, and that the incoming wave propagate on the x-y plane, o that the problem i effectively two dimenional 2D. The electric field i aumed to be polarized along the z direction. In order to obtain an analytical olution that may enable a qualitative decription of the main feature of the ytem, we will conider the cae in which both the object and the atellite are electrically mall o that their interaction may be treated a an electric dipole-type interaction. In Sec. III, we will preent a general deign procedure baed on a full-wave electromagnetic analyi that take into account all the interaction between the particle and the effect of higher dipole moment, and doe not require that the particle be treated a electric dipole. Let e i be the electric polarizability per unit of length pul along the direction of the ymmetry axi of a given particle the ith particle. The electric dipole moment pul induced on the particle along the z direction i p i e = i e E i loc, i where E loc i the local field at the particle ite r i, and i given by the uperpoition of the incident wave and the field cattered by the remaining particle in the ytem. Within the dipole approximation, the field cattered by the ith particle i R E i r k 2 r r i p e i, where r= x,y repreent a generic point of pace, k = i the free-pace wave number, and r r i = i/4 H 1 k r r i i the fundamental olution of Helmholtz equation, with H n a the Hankel function of firt 1 kind and order n. Suppoe that the conidered ytem of particle i illuminated by an incoming wave with electric field E inc r. In thee condition, it i clear that the local field at the ith particle ite i i = E inc r i + E loc j i 2 C i,j p e j, C i,j = k 2 r i r j, 3 i.e., the local field depend on the incident field a well a on the electric dipole moment induced in all the other particle of the ytem. Subtituting the above expreion into Eq. 1, i it i found that the induced dipole moment p e i =1,...,N verify the linear ytem 1 e i p e i j i j p e i C i,j = E inc.1 FIG. 2. Color online Total cattering linewidth fora2d ytem formed by a circular object and four antiphae atellite ee Fig. 1, a a function of the real part of the complex permittivity of the atellite, = +i, and different value of the lo tangent. Solid line: full-wave reult; dahed line: analytical model. i normalized to the cattering linewidth of the object tanding alone in free pace,. The object ha permittivity obj =4. and diameter D= /6. The radiu of the atellite i R =.5D and the ditance between the center of the object and the center of the atellite i d=.6d. i =1,...,N, where we et E i inc =E inc r i. By olving the above ytem it i poible to calculate the induced dipole moment, and in thi way characterize the cattering propertie of the ytem. In fact, uing Eq. 2 and the aymptotic behavior of the Hankel function for large r, it i clear that the total cattered field in the far zone i

5 CLOAKING MECHANISM WITH ANTIPHASE E tot r = k 2 2 k r ei k r /4 e ik rˆ r p e j j j k 2 2 k r ei k r /4 p e,tot, 5 where rˆ = r/ r i the direction of obervation, and p e,tot = j p e j i the total electric dipole moment of the ytem. The lat identity in Eq. 5 i valid when all the particle are within a relatively hort ditance a compared to the freepace wavelength from the origin, a will be aumed in what follow. In thee condition, the radiation pattern aociated with the cattered field may be aumed uniform and the higher-order multipole produced by the arbitrary ditribution of polarization current may be neglected. Conequently the total cattering linewidth of the ytem for planewave incidence analogou to the total cattering cro ection for two-dimenional problem i uch that PHYSICAL REVIEW B 78, p e,tot 2. Thi analytical theory i completely general and it could be applied to a generic ytem with an arbitrary geometry and arbitrary number of dipole-type particle a long a their patial ditribution i over an electrically mall cro ection. We will relax thi aumption in Sec. III, where we apply a full-wave analyi. In what follow, we retrict our attention to the configuration depicted in Fig. 1 formed by an object urrounded by four ymmetrically poitioned identical atellite. The ditance between the object and the atellite i d. The electric polarizability of the object i e,obj, wherea the electric polarizability of the atellite i e,. Auming that the incoming field i a plane wave propagating along the x direction, E inc =E e ik x, and taking into account all the ymmetrie of the ytem, it i traightforward to olve the 5 5 linear ytem Eq. 4 in cloed analytical form. Detailed calculation how that the total electric dipole moment of the ytem i given by 6 p e,tot 2 1 = E e,obj 1 + co k d + 1 e, 2C 1 C 2 +2C 3 + co k d 4C e,obj 2C 1 + C 2 1 1, 7 e,obj e, where C =k 2 i/4 H 1 k d, C 1 =k 2 i/4 H 1 k 2d, and C2 =k 2 i/4 H 1 k 2d. In order to make the ytem effectively tranparent to the incoming wave, it i neceary that p e,tot = ee Eq. 6. Thu, Eq. 7 ugget that the condition of total tranparency may be achieved provided the electric polarizability of the atellite i choen uch that 1 e, = 2 1 e,obj 1 + co k d +2C 1 + C 2 2C 3 + co k d. 8 Notice that the right-hand ide of the above equation depend excluively on the electric polarizability of the object, on the frequency of operation, and on the ditance d. To have a better feeling of the phyical meaning of the tranparency condition, let u conider the particular cae in which both the atellite and the object have circular cro ection. The electric polarizability along the axi of ymmetry of a cylindrical dielectric object with circular cro ection and relative permittivity and radiu R i ee, for example, Ref. 15 e 1 = i k 4 1 J k m R k m 1 J 1 k m R H 1 1 k R + k H 1 k R, where k m =k and Jn i the Beel function of the firt kind and order n. When the particle are very ubwavelength and k R 1 and k m R 1, it i poible to approximate the above expreion by e = 1 R 2 i 4 k 2 + k +ln 2 k R, where i the Euler contant. Thu, in the tatic limit e 1/ 1 R 2. It i alo imple to verify that in the tatic limit the interaction contant vanih: C =C 1 =C 2 =. Hence, in thee condition tranparency condition 8 reduce to R = tatic limit, 11 obj 1 R obj where R and obj R obj are the permittivitie radii of the atellite and object, repectively. Notice that Eq. 11 may alo be rewritten a 4 1 R 2 + obj 1 R 2 obj =, which ha a very intuitive meaning: it etablihe that the tranparency condition i achieved when the patial average electric uceptibility of all particle in the ytem vanihe. Thi reult i conitent with the dicuion in our earlier work, 1,1 where it wa hown that the tranparency phenomenon tem from the annihilation of the electric dipole moment induced in the ytem. Neverthele, it hould be emphaized that Eq. 11 i only valid near the tatic limit, and even for relatively mall catterer the frequency correction are not necearily negligible. Thee correction may be taken into account by olving with repect to the exact tranparency condition Eq. 8 with the electric polarizability given by Eq. 9. One important point i that in general the required permittivity for the atellite may be complex, even if the permittivity of the object i purely real. Thi effect i a conequence of the finite

6 SILVEIRINHA, ALÙ, AND ENGHETA PHYSICAL REVIEW B 78, of atellite (dip. approx.) 1.2 obj y D.1 z x R FIG. 3. Color online Total cattering linewidth fora2d ytem formed by a circular object and four antiphae atellite ee the inet a a function of the real part of the complex permittivity of the atellite, = +i, and different value of the lo tangent. i normalized to the cattering linewidth of the object tanding alone in free pace,. The dahed vertical line repreent the olution obtained uing the GA. The dahed red line repreent the reult obtained uing the dipole approximation in the lole cae. electrical ize of the particle and related field retardation from Eq. 11, it i clear that in the tatic limit thi ituation cannot occur. From the ign of the imaginary part in Eq. 1, it turn out that for relatively mall configuration uch a thoe conidered here, ome aborption lo in the atellite may yield further reduction in the cattering, and annihilate the total cattering from the ytem. In order to validate the theory developed in thi ection, we have calculated the total cattering linewidth of a ytem formed by an object with permittivity obj =4. and diameter D= /6 urrounded by four atellite i the wavelength in free pace at the operating frequency. The radiu of the atellite i R =.5D and the ditance between the center of the object and the center of the atellite i d=.6d. In Fig. 2 the total cattering linewidth i hown a a function of the real part of the complex permittivity of the atellite = +i and for different value of lo tangent. The olid line were calculated uing a dedicated full-wave methodof-moment MoM code, and fully take into account all the detail of the tructure and the effect of higher-order mode. The dahed line were obtained by uing the propoed analytical formalim Eq. 6 and 7. Notice that i normalized to the cattering linewidth of the object when the atellite are removed. It i een that both et of reult concur well and predict that when the real part of the permittivity of the atellite i near 13, the cattering from the total ytem i dratically reduced. The mall difference between the analytical and the numerical reult which are exacerbated by the logarithmic cale are due to the effect of higher-order mode, which are neglected by the analytical model. It may be een that the propoed cloaking mechanim i quite robut againt the introduction of lo, and that the cattering trength of the combined objectatellite ytem remain much le than that characteritic of the iolated object, even when a ignificant amount of lo i added to the ytem. In thi example, Eq. 8 predict that the tranparency condition i achieved with = i. FIG. 4. Color online a : Ditribution of the real part of the Poynting vector when a cylindrical object tanding alone in free pace i illuminated by a plane wave. b : Similar to a but the object i cloaked with four antiphae atellite, whoe geometry and material parameter have been optimized uing the GA. Thu, the addition of a very mall amount of lo to the ytem may improve the performance of the cloaking, effectively compenating for the dynamic coupling among the particle. Thi poibility will be demontrated in Sec. III for a different, more general etup. It i intereting to note that the quaitatic tranparency condition Eq. 11 predict that the optimal permittivity of the atellite i = 74, and thu obviouly underetimate the value of the permittivity of the atellite. A dicued before, thi happen becaue Eq. 11 i only valid very near the tatic limit, while in thi example the diameter of the object, D= /6, and the correponding field retardation are non-negligible. The agreement between the analytical model and the fullwave imulation improve if the electrical ize of the ytem i made maller. On the other hand, when the ize of either the atellite or the object become comparable to the freepace wavelength, the analytical model become ignificantly le accurate and one ha to reort to full-wave numerical method in order to determine a uitable configuration that may enable reducing the pecific cattering of a given In Sec. III, we explore thee poibilitie employing a genetic algorithm GA

7 CLOAKING MECHANISM WITH ANTIPHASE PHYSICAL REVIEW B 78, III. OPTIMIZATION USING THE GENETIC ALGORITHM FIG. 5. Color online Ditribution of the amplitude of the total electric field for the ame cenario a in Fig. 4. Here we apply an optimization technique to calculate the relative poition, ize, and electromagnetic propertie of the antiphae catterer, in order to caue the inviibility of the overall ytem at the wavelength of interet. More pecifically, for the configuration in Fig. 1, the objective i to calculate the permittivity, the radiu R, and the diplacement, which minimize the total cattering linewidth of the ytem. Unlike in Sec. II, the olution of the electromagnetic problem i full wave, and thu the method doe not require anymore that the interaction between the different particle i limited to the dipole approximation. Moreover, a full-wave olution enable taking into account the contribution to the cattered field of all the multipole, and thu the parameter of the antiphae atellite are tuned in order to minimize not only the cattering from the total electric dipole moment, but alo the cattering from higher-order multipole. In order to obtain a nearly iotropic repone independent of the direction of the incoming plane wave, we till uppoe that the atellite are placed ymmetrically around the For implicity and without lo of generality, we will alo aume that all the antiphae atellite are identical and have circular cro ection, which i arguably the mot intereting cenario from the point of view of a poible experimental demontration of thi phenomenon for geometrie with high ymmetry. The concept propoed here may be eaily generalized to the cae in which the atellite are not identical, which may even provide further degree of freedom for uppreing the poible reidual cattering from the ytem, particularly in the cae for which the object to be cloaked i nonymmetrical. The GA i an adaptive earch algorithm baed on the evolutionary idea of natural election and biology. 16 Due to it veratility and generality, the GA i well uited to determine in an efficient manner the optimal olution of the problem (a) no atellite (b) no atellite =-75. =1.4,opt opt. opt. config config =-125. =1.1,opt =2.,opt FIG. 6. Color online Radiation pattern in decibel aociated with cattered field for different configuration of the conidered ytem. The radiation pattern for the optimal configuration blue line and for an object iolated in free pace red line are hown in both part. a : Radiation pattern for the cae where the atellite are at the optimal ditance from the object, but have nonoptimal dielectric contant = 125. and = 75.. b : Radiation pattern for the cae where the atellite have an optimal dielectric contant, but are located at ditance from the object different from the optimal ditance,opt

8 SILVEIRINHA, ALÙ, AND ENGHETA PHYSICAL REVIEW B 78, of atellite obj 4. D Y DX 4 R.1.2 FIG. 7. Color online Total cattering linewidth fora2d ytem formed by a kite-haped object and four antiphae atellite ee the inet a a function of the real part of the complex permittivity of the atellite, = +i, and different value of the lo tangent. i normalized to the cattering linewidth of the object tanding alone in free pace,. The dahed vertical line repreent the olution obtained uing the GA. under tudy. The GA approach i baed on the evolution of an initial randomly generated population of candidate olution individual of the optimization problem. The variable to be optimized are coded into chromoome which define the genetic code of each individual. A fitne function i ued to ort the individual in uch a way that the betranking individual are thoe that yield better olution of the optimization problem. The initial population undergoe a election proce that mimic natural election, enuring the reproductive ucce of the mot fit individual croover and poibly allowing for a low rate of mutation of the genetic code. In each generation, the wort-ranking part of population i replaced with offpring. The proce i repeated iteratively, o that after ome generation the initial population ha evolved toward better olution of the problem. We have applied the GA to find an adequate configuration for the anticattering atellite that bring down the cattering linewidth of the overall ytem. To thi end, the parameter, r, and were coded into 32 bit and aociated with a chromoome, where by definition r =R / r 1. Each individual in the population correpond to a very pecific value of the parameter, R, and. The optimal olution wa earched in a pace of element uch that 2 1,.1D D, and.5 r.95, where D i the diameter of the object to be cloaked. For implicity, we aumed in the optimization that the atellite are lole. The fitne function wa choen equal to 1/, where i the total cattering linewidth of the ytem for plane-wave incidence. wa computed numerically uing a dedicated method-of-moment code. The initial population had 18 individual and wa generated randomly. The mutation percentage wa et equal to 5%. Typically, convergence wa reached or at leat no further improvement were oberved after 4 5 generation. In the firt example, it wa aumed that the diameter of the object i D=.25 and the permittivity of the object i obj =4. Fig. 1. It i worth noting that although the diameter of the object i a fraction of the wavelength, i.e., D FIG. 8. Color online Similar to Fig. 4 but for a kite-haped =.25, the dielectric object can till catter ignificantly due to it ize being near a reonance ince the diameter i about half of the wavelength in the dielectric. Therefore, deigning a cloak to dratically reduce thi cattering would be ueful for everal application. The GA yielded the following optimized parameter for the atellite:,opt = 12,,opt =.1D, and R,opt =.94D. It i intereting to note that,opt i very cloe to the lower bound of the conidered interval of admiible olution for, wherea R,opt i very cloe to the upper bound of the interval of admiible olution for R. A imilar trend wa oberved in all the example that we tudied. It implie that the mot advantageou configuration for the antiphae atellite i the one in which they are clutered very near to the The phyical jutification for thi property tem from the fact that the optimal configuration for the antiphae atellite i uch that the overall ytem remain a compact a poible. Thi enure that the phyical ize of the overall ytem i kept comparable to that of the original object, and in particular minimize the contribution of puriou higher-order multipolar component to the cattered field. To give an idea of the variation in the cattering linewidth with the permittivity of the atellite and alo indirectly to tudy the bandwidth of the cloaking phenomenon, in Fig. 3 we plot normalized to the cattering linewidth of the object when the atellite are removed, a a function of, for different value of the lo tangent. It i een that coni

9 CLOAKING MECHANISM WITH ANTIPHASE PHYSICAL REVIEW B 78, of atellite D Y D X R FIG. 1. Color online Total cattering linewidth fora2d ytem formed by an elliptical object and eight antiphae atellite ee the inet a a function of the real part of the complex permittivity of the atellite, = +i, and different value of the lo tangent. i normalized to the cattering linewidth of the object tanding alone in free pace,. The dahed line vertical repreent the olution obtained uing the GA. FIG. 9. Color online Similar to Fig. 5 but for a kite-haped tent with the olution of the GA, the cattering linewidth ha a ignificant and broad dip at,opt = 12. Thi demontrate that the antiphae atellite effectively cloak the object, making it nearly inviible to an external oberver. The broadne of the dip in the cattering linewidth confirm the nonreonant nature of thi cloaking mechanim, conitent with the finding in Ref. 12. In accordance with the reult in Sec. II, it i een that the cloaking effect ha a good tolerance to the effect of aborption lo. We have alo plotted in Fig. 3 dahed line the repone yielded by the analytical model derived in Sec. II Eq. 6 and 7. It i clear that the reult predicted by the analytical model are too coare and fail to decribe accurately the propertie of the ytem. Thi happen becaue the electrical ize of the conidered object i fairly large D=.25 =.5 d, where d i the wavelength in the dielectric. To further validate the GA reult, we have ued CST MI- CROWAVE STUDIO Ref. 17 to imulate the full-wave electromagnetic field produced by plane-wave incidence at the frequency of operation. The ditribution of the computed real part of the Poynting vector and the amplitude of the total electric field for an incident plane wave propagating along the x direction are preented in Fig. 4 and 5, repectively. Part a of the figure repreent the cae in which the object tand alone in free pace, wherea b correpond to the cae of a cloak formed by antiphae atellite. The contrat between the two part i triking: while in a the incoming wave i trongly cattered by the object, in b the electric field and the Poynting vector are nearly uniform away from the cloaking ytem, confirming that the antiphae atellite effectively cloak the object from the external field. In Fig. 6 we depict the radiation pattern aociated with the cattered field for different configuration of the conidered ytem. In part a, we tudy how the directivity and magnitude of the radiation pattern varie a the permittivity of the added atellite change around the optimal value, auming that the ditance between the atellite and the object i the one obtained with the GA. A expected, the radiation pattern ha typically a maximum along the x axi the direction of the incoming wave. In part b we tudy the enitivity of the radiation pattern when the ditance between the antiphae catterer and the object i made different from the optimal value,opt, keeping the dielectric contant of the atellite equal to the optimum value: =,opt.it i een that the radiation pattern may vary ignificantly when i changed, and that the cattering trength may increae very ignificantly when the atellite are diplaced away from the Thi effect i caued by the increae in the overall phyical ize of the ytem, which enhance the effect of the higher-order multipole and completely detune the ytem. In order to demontrate the generality of the propoed cloaking mechanim and the fact that the decribed phenomenon i not pecific for cylindrical object with circular cro ection, we have analyzed more complex geometrie, uch a the kite -haped object depicted in Fig. 7. The permittivity of the object i the ame a in the previou example, and the diameter of the object along the y axi i D y =.14. Uing the GA it wa determined that the mot advantageou parameter for the antiphae atellite are,opt = 25.4,,opt =.12D y, and R,opt =.11D y. The plot of the cattering linewidth a a function of i hown in Fig. 7, for different

10 SILVEIRINHA, ALÙ, AND ENGHETA PHYSICAL REVIEW B 78, FIG. 11. Color online Similar to Fig. 4 but for an ellipticalhaped value of the lo tangent. uite interetingly, it i een that in thi example the effect of aborption lo in the atellite may help in reducing the cattering linewidth of the ytem. In fact, a dicued in Sec. II, in general the optimal value for the permittivity of the atellite i complex valued. We have verified numerically that at = 25.4 the cattering linewidth decreae monotonically with increaing in the interval.1 not hown here for brevity. The minimum i reached at =.11. For.11 the cattering linewidth increae monotonically with but remain maller than the value of for = a long a.23. Again, thee reult demontrate the robutne of the propoed cattering-cancellation mechanim againt the addition of lo, conitent with Ref. 12, which may, a demontrated by thi example, even further enhance the inviibility effect. The repreentation of the real part of the Poynting vector and of the total electric field are hown in Fig. 8 and 9, repectively, for propagation along the x direction lole cae. It i clear that depite the complexity of the conidered object, the antiphae atellite can effectively annihilate the cattering from the object, making it tranparent to the incoming field. In a final example, we have conidered a needle-type object with permittivity obj =4. and elliptical cro ection, a illutrated in the inet of Fig. 1. The principal axe of the FIG. 12. Color online Similar to Fig. 5 but for an ellipticalhaped ellipe are uch that D y =.25 and D x =.3D Y. The object i now urrounded by eight identical antiphae catterer. The atellite are poitioned at the point D x /2+,nD y /4, n= 1,,1, and, D y /2+, where the center of the coordinate ytem i coincident with the geometrical center of the Uing the GA, it wa found that a uitable et of parameter for the cloak i,opt = 1.,,opt =.6D y, and R,opt =.55D y. Not urpriingly, due to the increae in the number of atellite, the overall required negative permittivity for each atellite i lower in magnitude compared to the previou cenario. The plot of the cattering linewidth a function of i depicted in Fig. 1, demontrating a good tolerance to variation in the permittivity of the antiphae atellite around the optimal value and to the preence of material aborption. In Fig. 11 and 12 it i confirmed that the negative cattering provided by the atellite enable an overall reduction in the total cattering cro ection of the ytem, and that the field ditribution of the electric field and of the real part of the Poynting vector are nearly uniform, coincident with thoe of the incoming wave. IV. CONCLUSIONS In thi work we have demontrated that the cloaking mechanim preented in Ref. 1 may be generalized to the

11 CLOAKING MECHANISM WITH ANTIPHASE cae in which the cloak i formed by a et of uitable dicrete catterer deignated a atellite urrounding the It ha been verified that uch conceptually imple and noninvaive cloaking olution may yield a dramatic drop in the cattering cro ection for moderately ized object and it ha a good tolerance to the effect of lo. The relative poition, ize, and electromagnetic propertie of the antiphae catterer have been optimized uing the GA. A larger number of antiphae catterer may even provide more degree of freedom for tailoring the way in which the cloaking phenomenon take place for more complex-haped The large PHYSICAL REVIEW B 78, number of degree of freedom and the potential of the GA technique may provide a viable way to cancel imultaneouly multiple-cattering order, creating an opportunity to extend thi cloaking mechanim to electrically larger object and collection of ACKNOWLEDGMENTS M.G.S. wa partially upported by Fundação para a Ciência e Tecnologia during hi tay at Univerity of Pennylvania. *Author to whom correpondence hould be addreed; engheta@ee.upenn.edu 1 A. Alù and N. Engheta, Phy. Rev. E 72, J. B. Pendry, D. Schurig, and D. R. Smith, Science 312, D. Schurig, J. J. Mock, B. J. Jutice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, Science 314, S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. Pendry, Phy. Rev. E 74, W. Cai, U. K. Chettiar, A. V. Kildihev, and V. M. Shalaev, Nat. Photonic 1, U. Leonhardt, Science 312, G. W. Milton and N. A. Nicorovici, Proc. R. Soc. London, Ser. A 462, A. Alù and N. Engheta, J. Opt. A, Pure Appl. Opt. 1, M. G. Silveirinha and N. Engheta, Phy. Rev. Lett. 97, M. G. Silveirinha, A. Alù, and N. Engheta, Phy. Rev. E 75, M. G. Silveirinha, A. Alù, and N. Engheta, Phy. Rev. B 78, A. Alù and N. Engheta, Opt. Expre 15, A. Alù and N. Engheta, Opt. Expre 15, A. Alù and N. Engheta, Phy. Rev. Lett. 1, M. G. Silveirinha, Phy. Rev. E 73, Y. R. Sammi and E. Michielen, Electromagnetic Optimization by Genetic Algorithm, Wiley Serie in Microwave and Optical Engineering Wiley, New York, CST MICROWAVE STUDIO 5., CST of America, Inc.,