A near-perfect invisibility cloak constructed with homogeneous materials

Transcription

1 A nea-pefet invisibility loak onstuted with homogeneous mateials Wei Li, Jianguo Guan, * Zhigang Sun, Wei Wang, and Qingjie Zhang 1 State Key Lab of Advaned Tehnology fo Mateials Synthesis and Poessing, Wuhan Univesity of Tehnology, Wuhan 437, China guanjg@whut.edu.n Abstat: A nea-pefet, non-singula ylindial invisibility loak with diamond oss setion is ahieved by a two-step oodinate tansfomation. A small line segment is stethed and then blown up into a diamond spae, and finally the loak onsisting of fou kinds and eight bloks of homogeneous tansfomation media is obtained. Numeial simulations onfim the well pefomane of the loak. The opeation bandwidth of the loak is also investigated. Ou sheme is pomising to eate a simple and well-pefomed loak in patie. OCIS odes: ( ) Metamateials; (3.35) Invisibility loaks; (16.119) Anisotopi optial mateials; (6.11) Eletomagneti optis. Refeenes and links 1. J. B. Pendy, D. Shuig, and D. R. Smith, "Contolling Eletomagneti Fields," Siene 31, (6).. S. A. Cumme, B. I. Popa, D. Shuig, D. R. Smith, and J. Pendy, "Full-Wave Simulations of Eletomagneti Cloaking Stutues," Phys. Rev. E 74, 3661 (6). 3. F. Zolla, S. Guenneau, A. Niolet, and J. B. Pendy, "Eletomagneti Analysis of Cylindial Invisibility Cloaks and the Miage Effet," Opt. Lett. 3, (7). 4. H. Chen, B. I. Wu, B. Zhang, and J. A. Kong, "Eletomagneti Wave Inteations with a Metamateial Cloak," Phys. Rev. Lett. 99, 6393 (7). 5. D. Shuig, J. J. Mok, B. J. Justie, S. A. Cumme, J. B. Pendy, A. F. Sta, and D. R. Smith, "Metamateial Eletomagneti Cloak at Miowave Fequenies," Siene 314, (6). 6. W. Cai, U. K. Chettia, A. V. Kildishev, V. M. Shalaev, and G. W. Milton, "Nonmagneti Cloak with Minimized Satteing," Appl. Phys. Lett. 91, (7). 7. W. Yan, M. Yan, and M. Qiu, "Non-Magneti Simplified Cylindial Cloak with Suppessed Zeoth Ode Satteing," Appl. Phys. Lett. 93, 199 (8). 8. B. I. Popa and S. A. Cumme, "Cloaking with Optimized Homogeneous Anisotopi Layes," Phys. Rev. A 79, 386 (9). 9. W. Cai, U. K. Chettia, A. V. Kildishev, and V. M. Shalaev, "Optial Cloaking with Metamateials," Nat. Photon. 1, 4-7 (7). 1. B. L. Zhang, H. S. Chen, and B. I. Wu, "Limitations of High-Ode Tansfomation and Inident Angle On Simplified Invisibility Cloaks," Opt. Expess 16, (8), M. Yan, Z. Ruan, and M. Qiu, "Satteing Chaateistis of Simplified Cylindial Invisibility Cloaks," Opt. Expess 15, (7), 1. G. Isi, R. Gaji, B. Novakovi, Z. V. Popovi, and K. Hingel, "Radiation and Satteing Fom Impefet Cylindial Eletomagneti Cloaks," Opt. Expess 16, (8), M. Yan, Z. Ruan, and M. Qiu, "Cylindial Invisibility Cloak with Simplified Mateial Paametes is Inheently Visible," Phys. Rev. Lett. 99, 3391 (7). 14. W. X. Jiang, T. J. Cui, X. M. Yang, Q. Cheng, R. Liu, and D. R. Smith, "Invisibility Cloak without Singulaity," Appl. Phys. Lett. 93, 1941 (8). 15. G. X. Yu, W. X. Jiang, and T. J. Cui, "Invisible slab loaks via embedded optial tansfomation," Appl. Phys. Lett. 94, 4194 (9). 16. M. Rahm, S. A. Cumme, D. Shuig, J. B. Pendy, and D. R. Smith, "Optial Design of Refletionless Complex Media by Finite Embedded Coodinate Tansfomations," Phys. Rev. Lett. 1, 6393 (8). 17. J. S. Li and J. B. Pendy, "Hiding Unde the Capet: A New Stategy fo Cloaking," Phys. Rev. Lett. 11,391 (8). 18. E. Kallos, C. Agyopoulos, and Y. Hao, "Gound-plane quasiloaking fo fee spae", Phys. Rev. A 79, 6385 (9).

2 19. K. Guven, E. Saenz, R. Gonzalo, E. Ozbay, and S. Tetyakov, "Eletomagneti loaking with anonial spial inlusions", New. J. Phys. 1, (8).. H.Y. Chen and C. T. Chan, "Eletomagneti wave manipulation by layeed systems using the tansfomation media onept", Phys. Rev. B 78, 544 (8). 1. Intodution Based on the oodinate tansfomation method, Pendy et al. eently povided a theoy to design a pefet invisibility loak [1]. The devie makes the loaked egion totally uneahable fo eletomagneti (EM) waves and simultaneously neve distubs the EM field outside the loak, thus an abitay objet may be hidden in it. The pefet loaking effet had been onfimed by the full wave numeial simulation [-3] and futhe theoetial analysis [4]. An expeimental attempt [5] had also been made to demonstate the piniple of suh loaks. Howeve, patial fabiation of the full-paamete invisibility loaks is not so fa ahievable, fo the neessay mateials ae usually highly anisotopi, inhomogeneous and even singula. Theefoe, geat effots have been devoted to simplifying the onstitutive paametes of the mateials omposing the loaks [, 5-9]. Unfotunately, most of the simplifiations ae pefomed in a plane [1-13] and the esulted loaks even satte moe stongly than the bae objet when the EM wave is inident obliquely onto them [1]. The non-singula elliptial loak [14] has simple onstitutive paametes and good pefomane without simplifiations. But the mateial paamete tensos of the elliptial loak as well as the simplified loaks in [1-13] ae still ontinuously hanging in the adial dietion. The loaking shell has to be ut into disete layes when we patially fabiate the loaks. The loaking pefomane is thus inevitably edued o even signifiantly deeased with espet to the pefet ones. Fo minimizing the degadation of the loaking pefomane in patial implementation, the ontinuous loaking shell must be ut into adequate layes whih have diffeent onstitutive paametes. Howeve, involving too many kinds of mateials in the loaks will again geatly ompliate the fabiation. Theefoe, it is still a hallenge to povide a simple loak with well loaking pefomane. Reently, Yu et al. [15] poposed and analyzed a lass of invisible slab loaks based on an "embedded optial tansfomation method" [16]. The slab loak is made of homogeneous mateials, whih ae easy to ealize in patie. Howeve, the slab seems to be infinite long aoding to the tansfomation. Thus the appliation of the slab loaks is limited. Anothe big pogess on the implementation of the loaks is the "gound-plane loaks" [17] whih need only taditional, isotopi and eadily aessible mateials. The fist vesion of the goundplane loak has to be embedded in a medium suh as glass, and then the woking bakgound is extended to ai o vauum by a simplifiation [18]. Howeve, unlike the invisibility loaks whih aim to ende the loaked objet totally invisible, the gound-plane loaks hide an objet behind a onduting sheet o a mio. Theefoe, in many ases, the invisibility loaks annot be eplaed by the gound-plane loaks. In this wok, a ylindial invisibility loak with a diamond-shaped oss setion is poposed based on the tansfomation optis. The tansfomation whih does not expand a point as usual, but a line segment into a loaked egion thus avoids singulaity in mateial paametes [14]. As a esult, the loak is equivalent to a small line segment in the EM fields. If the line segment is shot enough, a nea pefet loak will be obtained. On the othe hand, beause we only ompess o steth the spae in two othogonal dietions homogeneously in the tansfomation as we will see in the following, the esultant diamond shaped loak onsists of eight bloks of homogeneous mateials and only fou kinds of diffeent mateials ae involved in. Thus, without any futhe simplifiations, a nea pefet loak omposed of few bloks of homogenous mateials is obtained. Compaing with the simplified loaks, the asobtained loak is simple and has bette pefomane even in oblique inidene ase sine the loak equivalent to a small line segment whih an be fa smalle that the loaked objet. Ou loak also has the potential to be a boadband one beause the mateials ae non-singula.

3 . Deivation of the mateial paametes of the diamond shaped loak Now let us deivate the mateial paametes of the diamond shaped loak. Fo simpliity, we estit ouselves in two-dimensional (D) ases. We stat fom the oodinate tansfomation whih expands a line segment to a diamond spae. The tansfomation poeeds in two steps. Fig. 1(a)-() show the oiginal Catesian spae (x, y, z), the tansitional Catesian spae (x', y', ), and the final tansfomed Catesian spae (x'', y'', '), espetively. In the oiginal spae, a line segment with a length of a and midpoint at oigin is paallel to the x axis. The big diamond aea depited in Fig. 1a outlines the spae to be tansfomed. The fist step is aim to steth the line segment fom a to b. Fo this pupose, a small diamond spae of whih the line segment is one of the diagonal lines is stethed in x dietion by b/a times, while the spae between the two diamonds is ompessed homogeneously in x dietion. As shown in Fig. 1(b), the tansfomed big diamond in spae (x', y', ) is divided into eight pats I-VIII by the stethed small diamond and the oodinate axes. Fo eah pat, we an give the tansfomation equations. Take the two pats (egion I and II) in the fist quadant as examples, in egion I: in egion II: ( a b) b ( a b) x ' = y + x +, y' (1) ( a ) d a a b x ' = x, y' () a The tansfomation equations of the othe egions an also be given similaly. In fat, due to the axial symmety of the defomation with espet to the oodinate axes, the tansfomation equations ae symmetial too. Fo example, egions IV and III ae symmetial to egions I and II, espetively. The tansfomation equations of them ae simila to eah othe in fom: in egion IV: in egion III: ( a b) b ( a b) x ' = y + x, y' (3) ( a ) d a a b x ' = x, y' (4) a Fig. 1. The oodinate tansfomation fo a diamond shaped loak. (a) The oiginal diamond shaped egion with a line segment a in it. (b) The line segment and the small diamond spae in (a) is stethed in x dietion by b/a times. () The stethed line segment is tansfomed into a diamond shaped spae, and the egions II, III, VI and VII ae ompessed. Consequently, a diamond loak is obtained.

4 So fa, we have stethed the given shot line segment to a long one. In the seond step, the stethed line segment is futhe expanded into a hombi spae, and the egions between the expanded hombi spae and the small diamond bounday, i.e., egions II, III, VI and VII, ae ompessed homogeneously in y dietion. Fig. 1() shows the final tansfomed spae. Finally, the tansfomation fom the tansitional Catesian spae (x', y', ) to the final tansfomed Catesian spae (x'', y'', ') in egion II is expessed as: d e e x '' = x', y'' = y' x' + e, (5) d b The tansfomation in egion III, VI and VII ae simila to (5). In the othe egions, no tansfomation is poeeded in the seond step so we have x''=x' and y''=y'. At the end of the tansfomation, a loak ontaining eight bloks of mateial is obtained whose oute bounday and inne bounday ae both diamond shaped. Combining the tansfomations of the two steps, the total tansfomation in eah blok is thus obtained. We still take the two egions in the fist quadant as examples, in egion I: in egion II: ( a b) b ( a b) x '' = y + x +, y'' ' (6) ( a ) d a a b d e e x '' = x, y'' = y x + e, ' (7) a d a It should be noted that b,, d and e ae oodinate values elated to the shape of the loak (see Fig. 1()), and a is half of the length of the equivalent line segment. These paametes satisfy the inequalities >b>a> and d>e>. One we get the tansfomation equations, we an ompute the Jaobian tansfomation matix J eadily and then wok out the elative mateial paametes of the loak by (suppose the oiginal spae is fee spae) [3] T ε = µ = J J det(j ) (8) Fo the two egions in the fist quadant, the non-zeo omponents of the mateial paamete tensos ae, in egion I: in egion II: ε xx = µ xx = ( A + B )/ A ε xy = ε yx = µ xy = µ yx = B / A ε yy = µ yy = 1/ A ε zz = µ zz = 1/ A (9) ε xx = µ xx = M / K ε xy = µ xy = MT / K ε yy = µ yy = K / M + MT ε zz = µ zz = 1/ MK (1) / K b ( b a) whee A=, B=, M=b/a, T=e/b and K=1-e/d. The mateial paametes of the est a ( a ) d pats of the loak an also be obtained similaly. Inteestingly, the paametes in all egions

5 ae onstants one the shape of the loak and the equivalent line segment has been fixed, beause no othe vaiables ae involved in the mateial paametes tensos in this ase. Clealy, thee is also no singulaity in the mateial paametes. Fom Fig. 1(), one an lean that the loak is onstuted with eight bloks of mateials. Moeove, only fou kinds of mateials ae needed indeed beause the loak is symmetial with espet to the oigin, i.e., the onstitutive paametes in egions I, II, III and IV ae idential to that of egions V, VI, VII and VIII, espetively. The above haateistis of the loak highly simplify the onstution of the devie. 3. Numeial simulations and disussion In this setion, finite elements simulations ae aied out to show the popeties of the diamond loak. We only onside the TM (tansvese magneti) polaization plane wave ase in the simulations. In fat, simila esults would be obtained in TE (tansvese eleti) polaization plane wave ase. The plane wave is.15 GHz in fequeny. Fo TM polaization, only ε xx, ε xy, ε yx, ε yy and µ zz omponents of the mateial paametes tenso ae elevant. Due to the symmety of the mateial paametes tenso, we have ε xy =ε yx. So we only need ε xx, ε xy, ε yy and µ zz omponents fo the simulations. The onstants defining the shape of the diamond loak ae set as b = 1 m, = m, d = 4 m and e = 1 m. The half length of the equivalent line segment a is hosen to be a small value of.15 m. A pefet eleti onduting (PEC) ylinde is fitted into the loaked egion. Fig. (a) shows the magneti field distibution when a TM polaized plane wave is impinging on the loak fom left to ight. The wave fonts ae bent aound the loaked aea in the left pat, and then estoed thei oiginal dietions by the ight pat, whih is simila to most of the loak dedued by the oodinate tansfomation method. The fields outside the loak ae almost unpetubed exept tiny satteing due to the impefetion of the loak. If the loaked objet is not a PEC ylinde whih was just fitted into the loaked egion, the EM field will penetate into the oe of the devie. In this ase, the loaking pefomane should vay with the popeties suh as the size, position and mateial paametes of the loaked objet. To avoid this, a PEC lining is usually added to the inne wall of the loak shell. Notie that all the mateials we used in the simulation ae lossless. In patie, the mateials ae inevitably lossy, so it is impotant to see what will happen when loss is intodued into the loak. Hee we simulate the ase when loss tangent of pemeability and pemittivity ae both.1. The esult is demonstated in Fig. (b), whih shows stong fowad satteing but hadly affets the field distibution in othe dietions. This esult is simila to that of the peviously studied ylindial [] and spheial loaks [4]. Fig.. Magneti field distibution fo (a) the nea pefet diamond loak without loss, and (b) the diamond loak omposed of lossy mateials with loss tangent of.1 fo both pemittivity and pemeability when a TM plane wave is inident fom left to ight. Sine the diamond loak equivalents to a line segment instead of a point, it will have vaiant pefomanes when the EM wave is inident fom diffeent dietions. Speifially, when the inident wave is pependiula to the equivalent line segment, the loak will show the stongest satteing. Numeial simulation has also been aied out to examine the pefomane of the loak in this ase. Fig. 3(a) gives the magneti field distibution of the esult whih shows a tiny satteing. Howeve, if the equivalent line segment is too long,

6 stong satteing will be indued by the loak along the dietion pependiula to the line segment. Fig. 3(b) shows the ase when a =.5 m. To futhe qualify these esults, the total satteing oss setions vesus a ae plotted in Fig. 4(a) when the inident wave is pependiula, paallel and at a 45 degee angle to the +x axis espetively, Fom Fig. 4(a) we an lean that the longe the equivalent line segment, the stonge satteing the loak will indue. With the ineasing of the wave inident angle, the total satteing also ineases quikly. Howeve, the satteing an be kept at a vey low level so long as the equivalent line segment is shot. Fig. 3. Magneti field distibution when a TM plane wave is impinging on the loak with an effetive line segment length a = (a).15 m and (b).5 m fom top to bottom. As disussed above, the length of the equivalent line segment deides the loaking pefomane when the inident wave is not paallel to the line, but it does not ompliate the onstution of the loak, i.e., the loak is still made up of a few bloks of homogenous mateials. In this sense, we an ahieve an unompliated loak abitaily nea the pefet one by hoosing a small value of a. Howeve, it should be notied that an exteme small value of a may also lead to exteme mateial paametes, whih ae had to be ealized, aoding to the mateial paametes equations suh as Eq. (9) and (1). With egad to the othe paametes, namely, b,, d and e, they ae ielevant to the pefomane of the loak in piniple. Howeve, they ae losely elated to the mateial paametes of the loak. We an adjust them to set the mateial paametes within a desiable ange at the pie of ineasing the thikness of the loak. Finally, onsideing fabiating the loak, it is pefeable to denote the mateial paametes in thei eigenbasis, whee the pemittivity and pemeability tensos ae diagonal. Fotunately, due to the symmety of the tensos, one an always do this. Fo example, the mateial paametes of egion I (see Eq. (9)) an be expess as follow: (1 + A + B ε = µ = + A + B 1 ) / A (1 + A + B A + B 1 ) / A (11) 1/ A An expeimental implementation of the devie may be ahieved by anisotopi metamateials suh as split-ing esonatos (SRRs) [5], anonial spials [19] at miowave fequenies, and photoni ystals at optial fequenies. Howeve, metamateials ae not always needed. By using the effetive medium theoy (EMT) one an also onstut the anisotopi devies with layeed isotopi mateials []. A elated expeimental wok is now being onsideed. The bandwidth has been a big issue fo taditional tansfomation optis based loaks, whih wok only at a single fequeny beause of the singulaities in the mateial paametes of them. The as-obtained diamond-shaped loak is non-singula, thus it is expeted to opeate at boad bandwidth. Hee we investigate the loaking pefomane of the diamond-shaped loak in the ange of.1-. GHz when a plane wave is inident fom top to bottom. Note that the devie is dispesive like most of the epoted loaks. To intodue the dispesion into the loak, we assume that the mateial paametes of the loak follow the standad Loentz model:

7 η j ( j j j ω ω) = 1+ [ η ( ω ) 1] f ( ω)/ Re[ f ( )] (1) whee η j epesents j omponent of pemittivity o pemeability tenso, and f j (ω)=ω p /(ω aj -ω - iωγ) ae the Loentzian dispesive funtions, whee γ is the damping fato, ω p is the plasma fequeny, ω aj is the atom esonated fequeny. We set ω = π.15 GHz, ω p = 4ω, γ =.1ω, ω aj =.6ω when η j < 1 and ω aj = 1.4ω when η j 1. Fig. 4(b) shows that the nomalized total satteing oss setion vaies with the fequeny. At fequeny of.15 GHz, the total satteing is quite small. Howeve, as the EM wave fequeny deviates fom that point, the total satteing ineases dastially beause of the dispesion and the losses bought by the dispesion. In spite of that, the loak is found to pefom well in a etain ange of fequenies. In the ange of GHz, the total satteing oss setion an be edued 5% at least. The esult suggests that the diamond-shaped loak do wok at multiple fequenies othe than a single fequeny like the singula ones, though the bandwidth is limited due to the dispesion and losses of the loak. It is imaginable that diffeent dispesion paametes o diffeent mateials as well as the paametes of the loak suh as the length of the equivalent line segment may lead to diffeent bandwidth, whih we will disuss elsewhee. Fig. 4. Nomalized total satteing oss setion (nomalized to the diamond PEC ylinde) as a funtion of (a) the half length of the equivalent line segment a at.15 GHz when the inident wave is pependiula (blak line), paallel (blue tiangle) and at a 45 degee angle to the +x axis (ed etangula) espetively fo the non-dispesive diamond-shaped loak, and (b) fequenies when a=.15 m and the inident wave is pependiula to +x axis fo the dispesive diamond-shaped loak. 4. Summay In summay, we have pesented a nea-pefet diamond-shaped loak based on a two-step oodinate tansfomation. Sine the tansfomation expands a line segment instead of a point into the loaked egion, singulaities ae avoided in the mateial paametes of the devie. Moeove, due to homogeneously stething and ompessing at two othogonal dietions duing the tansfomation, the esulted tansfomation media ae homogeneous. Numeial simulations eveal that the loak has a good loaking pefomane fo all the inident dietions of plane waves and an wok at multiple fequenies. The loak is simply onstuted by only fou kinds of non-singula homogeneous mateials, whih make the devie easy to ealize in patie. Aknowledgements This wok was suppoted in pat by the National High-Tehnology Reseah and Development Pogam of China unde Gant No. 6AA3A9, in pat by the National Defense Basi Reseah pojet unde Gant No. D146157, in pat by the Young Teahe Gant fom Fok Ying Tung Eduation Foundation unde Gant No and in pat by the Ministy of eduation of China unde Gant No. PCSIRT644.