A complementary media invisibility cloak that can cloak objects at a distance outside the cloaking shell. Abstract

Transcription

1 A cmplementary media invisibility clak that can clak bjects at a distance utside the claking shell Yun Lai, Huanyang Chen, Zha-Qing Zhang and C. T. Chan Department f Physics The Hng Kng University f Science and Technlgy Clear Water Bay, Kwln, Hng Kng, China Abstract Based n the cncept f cmplementary media, we prpse an invisibility clak perating at a finite frequency that can clak an bject with a pre-specified shape and size within a certain distance utside the shell. The clak cmprises f a dielectric cre, and an anti-bject embedded inside a negative index shell. The claked bject is nt blinded by the claking shell since it lies utside the clak. Full-wave simulatins in tw dimensins have been perfrmed t verify the claking effect. 1

2 Recently, great prgress has been made in bth the thery and experiment f invisibility clak [1-18]. One apprach t achieve the invisibility clak is t emply the transfrmatin ptics cncept t exclude electrmagnetic waves in certain regins and smthly fit them t fields utside the device [1-13]. The permittivity and permeability f such a clak are determined by the crdinate transfrmatin [1, 2] f expanding a pint r a line int a finite vlume f claked space. Such a transfrmatin media claking device can hide any bject inside the claked dmain. The bject hidden inside the claked dmain has t be blind, since n utside electrmagnetic waves can reach int the claked space. Anther apprach f claking is t reduce the lwest rder scattering crss sectin f an bject by cvering it with layers f metamaterials [14, 15]. In this apprach, the prperties f the claking device depend n the bject t be claked. In the appraches mentined abve, the claking shell enclses the bject t be hidden. It is als pssible t have claking effects n an bject which lies utside the claking shell. In particular, it has been mathematically prven that a plarizable line diple in a matrix f dielectric cnstant ε m becmes invisible in the vicinity f a cated cylinder with cre dielectric cnstant ε c ε m and surface cating dielectric cnstant ε s ε m in the quasistatic limit [16-18]. The rigin f this external claking effect is the anmalus lcalized resnance f the cated cylinder that cancels any induced mment. In this paper, we prpse a new recipe f an invisibility clak that can hide an bject that is external t the clak itself. The clak depends n the bject, but the bject can have arbitrary shape. The idea is based n cmbining the cncept f cmplementary media and transfrmatin ptics [19-23]. It is knwn that cmplementary media can ptically cancel a certain vlume f space at a certain frequency. This cncept has imprtant implicatins, mst ntably in the perfect lens [19] and nvel imaging devices [20], and has mtivated varius applicatins, such as the superscatterer [24, 25], the cylindrical superlens [26], the anti-clak [27], and an alternative strategy f invisibility clak (which is different frm ur apprach here) [13]. Cmplementary media f electrns have als been discussed [21]. Here, we shw that a new type f invisibility clak that emplys an anti-bject embedded in a negative index shell can make an bject that lies utside the claking shell invisible. The wrking principle can be described in tw steps. First, the bject as well as the surrunding space is ptically 2

3 canceled by using a cmplementary media layer with an embedded cmplementary image f the bject, which is referred as the anti-bject hereafter. Then, the crrect ptical path in the canceled space is restred by a dielectric cre material. As a result, the ttal system is effectively equal t a piece f empty space fitted int the cancelled space, and invisibility is achieved. The key idea behind this design is the cmplementary media [19-22] which can be regarded as a special kind f transfrmatin media. Accrding t the crdinate transfrmatin thery, when a space is transfrmed int anther space f different shape and size, the permittivity ε and permeability μ in the transfrmed space x are given by T T ε = AεA /deta and μ = AμA /deta, where ε and μ are the permittivity and permeability in the riginal space x, and A is the Jacbian transfrmatin tensr with cmpnents Aij = x i xj. The transfrmatin media exhibits tw interesting prperties. First, the ptical path in the transfrmatin media is exactly the same as that in the riginal space. Secnd, the transfrmatin media is reflectinless as lng as the uter bundary crdinates befre and after a crdinate transfrmatin is unchanged [22, 23]. Based n these principles, a kind f cmplementary media can be designed under a special kind f crdinate transfrmatin, i.e. flding a piece f space int anther. When a wave crsses the flding line frm the riginal space int the flded space, it starts t experience a negative ptical path as that in the riginal space. A gd example f the cmplementary media is the perfect lens [19], as is shwn in Fig. 1(a). A perfect lens is a slab f ε = 1 and μ = 1, frmed under a crdinate transfrmatin f x = x fr 0 < x < L, i.e. flding a slab f 0 < x < L int the slab f L< x < 0. Here L is the slab width. In this case, the ptical phase at x same, and the regin ( L, L) appears t be nnexistent. = L and x = L are exactly the Nw cnsider the simple scheme as shwn in Fig. 1(b). An bject with permittivity ε and permeability μ is placed at the right (in a layer f air). A slab f cmplementary media can still be designed by the crdinate transfrmatin x = x fr 0 < x < L, which results in a slab f ε = 1 and μ = 1 with an embedded cmplementary image bject with permittivity ε and permeability μ. Then, similar t the case f perfect 3

4 lens, the ptical phase at ± L are exactly the same. The bject and cmplementary media are ptically canceled by each ther. Plane wave will pass thrugh withut scattering. Since a slab f cmplementary media is required t be infinitely lng in the y axis, it wuld be interesting t cnstruct cmplementary media with a finite vlume. In Fig. 1(c), we shw the scheme f a circular layer f cmplementary media with parameters ε and μ that ptically cancels an uter circular layer f air. The cmplementary media can be btained by a crdinate transfrmatin f flding the layer f air (b< r < c) int the layer f cmplementary media ( a< r < b). Here a, b and c are the cre radius, the uter radius f cmplementary layer, and the uter radius f the canceled air layer, respectively. Cnsider a general crdinate transfrmatin f the frm r = f ( r ) [23], in which f ( r ) is a cntinuus functin f r that satisfies f ( b) and μ f the cmplementary layer can be written as r ε θ = μ θ = f f r ( ) ( r ) and ( ) ( ) ( ) ( ) ( ) f r ε z = μ z = f r ( r ) = b and f ( a) ε = μ = r r = c. The ε ( ) f r 1 r f r ( ). Fr a simple linear functin f r = r b c b a b + b, ε and μ are anistrpic. But fr the specific chice, f r = b r and f ( ) 2 2 c= b a, we btain istrpic parameters ε and μ : μ r μ ε z b r 4 4 = θ = 1, = fr TE waves, and ε r ε μ z b r 4 4 = θ = 1, = fr TM waves. We nte in particular that the radial and tangential cmpnents are cnstants. If we let the cre material t be a perfect electric cnductr (PEC), the system becmes the s-called superscatterer [24, 25] with an effective PEC bundary at radius c. Here, in rder t 2 2 restre the ptical path, we cnsider a cre material with μ r = μ θ = ε z a c = 1 fr TE waves and transfrmatin f r ε r ε μ z a c 2 2 = θ = = 1 fr TM waves, which are btained by crdinate = c a r, i.e. cmpressing a large circle f air with radius c int a small circle with radius a. With such a chice f dielectric cre material, the wave experiences the same ptical path as that in an circle f air with a radius c. In this way, the whle system, including the uter air layer, the cmplementary media layer and the 4

5 cre material, is ptically equal t a circle f air with radius c, and is thus invisible tgether t any frm f external illuminatin. Nw we g ne step further. Suppse an bject f permittivity ε and permeability μ is added in the uter circular layer f air, and we want t make it invisible. Then, accrding t the transfrmatin media thery, it is always pssible t include a cmplementary image bject with parameters ε and μ : ε r μ f r ( r ) 1 = =, ε μ r f r ε μ r ε f r θ θ = = θ μ θ ( ) ( r ) f and ε μ ε z z = = z μz layer, such that the bject f ε and ( ) f r r ( r ) f r r ( ) in the cmplementary media μ is ptically cancelled, as shwn in Fig. 1(d). Thus, the bject als becmes invisible t any incident waves. This recipe pens up a new way t cnceal an bject frm electrmagnetic waves within a specific distance utside the clak. In the fllwing, we carry ut full wave simulatins using a finite element slver (Cmsl Multiphysics) t demnstrate the functinality f the external cmplementary media invisibility claks. We cnsider cases f TE plarizatin (E alng z-directin), and impse an incident plane wave frm the left r a pint suce with wavelength λ = 1unit. First, we demnstrate the scheme shwn in Fig. 1(c), i.e. a circular layer f cmplementary media ( 0.5 < r < 1) and a cre material ( r < 0.5 ) that is ptically equal t a circle f air ( 2 f r = r, we r < ). Under a linear transfrmatin mapping ( ) 3 2 btain the parameters f the cmplementary media: μ ( 1.5) [ 2, 0.5] θ = r ( r 1.5) [ 2, 0.5] and ε ( 4 6) [ 8, 2] μ r = r r, z = r r. In Fig. 2(a), the calculated electric fields are shwn. The absence f scattered waves clearly verifies the invisibility f the whle system. Next, we demnstrate the scheme shwn in Fig. 1(d), the claking f an bject by cmplementary media. A dielectric curved sheet f thickness 0.3 with parameters ε = 2 and μ = 1 t be claked is psitined between the circles f r = 1.5 and r = 1.8. In Fig. 2(b), the scattering pattern f such a single dielectric curved sheet is shwn. In rder t make the bject invisible, we mdify the cmplementary media layer in Fig. 2(a) t include a cmplementary image bject, i.e. the anti-bject, with 5

6 parameters ε = 2ε and μ = μ, psitined between the circles f r = 0.6 and z z r = The clak is cmpsed f the mdified cmplementary layer embedded with the anti-bject and a cre material. In Fig. 2(c), we shw the calculated electric fields which clearly demnstrate the external claking effect. We nte that the invisibility clak des nt cver the bject surface here, and the claking effect cmes frm the ptical anti-bject embedded in the negative index shell. The scattering effect f the bject and the anti-bject cancel each ther. We emphasize that there is n shape r size cnstraint n the bject t be claked, as lng as it fits int the regin bunded between b<r<c. In Fig. 2(d), we demnstrate the claking f tw curved sheets. The sheet n the left bunded between 1.2 < r < 1.5 has an anistrpic permeability f μ = 1 and μ θ = 1. The sheet n the dwnside bunded between 1.5 < r < 1.8 has a linearly changing permittivity f ( r) μ = μ, ε z ε z θ θ ε = In this case, an anti-bject f μ r = μ r, ( ) = and an anti-bject f ε ε ( ) = and μ = μ, are z z r embedded at the crrespnding image psitins in the negative index shell. Perfect claking effect is demnstrated with a pint surce psitined at (2,-2). We will illustrate the claking scheme with anther example. We will clak a dielectric shell f ε = 2 and μ = 1 bunded between 1.5 < r < 1.8, as shwn in Fig. 3(a), which als shws its scattering pattern under a plane wave illuminatin. In this case, the anti-bject is an image shell with ε = 2ε and μ = μ, psitined between the z z circles f r = 0.6 and r = 0.75 inside the negative index cmplementary shell. In Fig. 3(b), we shw the calculated electric fields after the dielectric shell f 1.5 < r < 1.8 is claked by a shell cmpsed f a cmplementary media 0.5 < r < 1 with anti-bject and a cre material r < 0.5. Please nte again that the dielectric shell is utside the clak. The perfect plane wave pattern manifests the claking effect. In Fig. 3(c), we cnsider anther circular shell f ε = 1 and μ = 1. The scattering pattern fr such a shell, which is shwn in Fig. 3(c) is similar t that f a metal shell. In this case, the antibject is the cmplementary image shell with ε z = ε z and μ = μ. In Fig. 3(d), the r calculated electric fields after claking are shwn. Again, an excellent claking effect is manifested by the perfect plane wave pattern. 6

7 We wuld like t cmpare this anti-bject clak t resnance clak f Miltn et al. While bth can clak an bject external t the claking shell, Miltn s clak claks a pint-like bject (crss sectin is nn-zer fr finite-sized bject) in the electrstatic limit. Our design wrks fr finite size bjects in a finite frequency. In cnclusin, we prpse a perfect invisibility clak that can make an bject utside its dmain invisible. The crucible element is a cmplementary anti-bject embedded inside a shell with flded gemetry. This is different frm the riginal perfect clak [2], in which the hidden bject is blind since it sits inside the claking shell. Our bject sits utside the device, at the expense that the clak must be custm-made as it has t carry an anti-bject. This scheme can be extended t three dimensins, and is applicable t any bject that can be described by a given permittivity ε and permeability μ. This wrk was supprted by the Central Allcatin Grant frm the Hng Kng RGC thrugh HKUST3/06C. We thank Dr. JunJun Xia, DeZhuan Han, Jack Ng, KinHung Fung, ZhiHng Hang and Jeffrey ChiWai Lee fr helpful discussins. 7

8 References 1. U. Lenhardt, Science 312, 1777 (2006). 2. J. B. Pendry, D. Schurig, and D. R. Smith, Science 312, 1780 (2006). 3. A. Greenleaf, M. Lassas, and G. Uhlmann, Physil. Meas. 24, 413 (2003). 4. D. Schurig, J. J. Mck, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, Science 314, 977 (2006). 5. S. A. Cummer, B.-I. Ppa, D. Schurig, D. R. Smith, and J. B. Pendry, Phys. Rev. E 74, (2006). 6. Z. Jacb, L. A. Alekseyev, and E. Narimanv, Opt. Express 14, 8247 (2006). 7. W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, Nat. Phtn. 1, 224 (2007). 8. M. Rahm, D. Schurig, D. A. Rberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, Phtn. Nanstruct: Fundam. Applic. 6, 87 (2008). 9. U. Lenhardt and T. G. Philbin, New J. Phys. 8, 247 (2006). 10. H. Y. Chen and C. T. Chan, Appl. Phys. Lett. 90, (2007). 11. Z. Ruan, M. Yan, C. W. Neff, and M. Qiu, Phys. Rev. Lett. 99, (2007). 12. H. Chen, B.-I. Wu, B. Zhang and J. A. Kng, Phys. Rev. Lett. 99, (2007). 13. Y. Lu, J. J. Zhang, H. Chen, B.-I. Wu, J. A. Kng, arxiv: A. Alu and N. Engheta, Phys. Rev. E 72, (2005). 15. A. Alu and N. Engheta, Phys. Rev. Lett. 100, (2008). 16. G. W. Miltn and N. A. P. Nicrvici, Prc. R. Sc. A 462, 3027 (2006). 17. N. A. P. Nicrvici, G. W. Miltn, R. C. McPhedran, and L. C. Btten, Opt. Express 15, 6314 (2007). 18. G. W. Miltn, N. P. Nicrvici, R. C. McPhedran, K. Cherednichenk, and Z. Jacb, arxiv: J. B. Pendry, Phys. Rev. Lett. 85, 3966 (2000). 20. J. B. Pendry and S. A. Ramakrishna, J. Phys.: Cndens. Matter 14, 8463 (2002); 15, 6345 (2003). 21. K. Kbayashi, J. Phys.: Cndens. Matter 18, 3703 (2006). 22. W. Yan, M. Yan and M. Qiu, arxiv: A. V. Kildishev and E. E. Narimanv, Opt. Lett. 32, 3432 (2007). 24. T. Yang, H. Y. Chen, X. D. Lu and H. R. Ma, Opt. Express 16, (2008). 25. H. Y. Chen, X. H. Zhang, X. D. Lu, H. R. Ma and C.T. Chan, t appear in New J. 8

9 Phys. (2008), arxiv: M. Yan, W. Yan and M. Qiu, Phys. Rev. B 78, (2008). 27. H. Y. Chen, X. D. Lu, H. R. Ma and C. T. Chan, Opt. Express 16, (2008). 9

10 ε = 1 μ = 1 Air ε = 1 μ = 1 Air (a) (b) ε, μ ε, μ -L 0 L Air Air ε, μ ε, μ (c) c ε, μ ε, μ a b (d) ε, μ ε, μ Fig. 1 (clr nline). (a) A cmplementary media slab L< x < 0 that ptically cancels a slab f air 0 < x < L. (b) A cmplementary media slab with an embedded cmplementary image f ε and μ that ptically cancels an bject f ε and μ in air. (c) A system cmpsed f a circular layer f air (b< r < c), a circular layer f cmplementary media f ε, μ ( a< r < b) and a cre material f ε, μ ( r < a) that is ptically equal t a large circle f air ( r < c). (d) A scheme t clak an bject f ε, μ by placing a cmplementary image f the bject, i.e. the anti-bject f ε, μ, in the cmplementary media layer f a< r < b. 10

11 Fig. 2 (clr nline). Snapshts f the ttal electric fields under an incident TE plane wave frm the left ((a)-(c)) and a pint surce ((d)). (a) A circular cmplementary media layer f 0.5 < r < 1 and a cre material f r < 0.5. (b) A curved sheet f thickness 0.3, with permittivity ε = 2. (c) The curved sheet in (b) is claked by an invisibility clak cmpsed f a circular layer f cmplementary media with an embedded anti-bject and a cre material. (d) A curved sheet n the left with anistrpic permeability μ r = 1 and μ = 1, and anther curved sheet n the dwnside with permittivity θ ( r) ε = , are bth claked by an invisibility clak with tw crrespnding anti-bjects embedded in the cmplementary media layer. 11

12 Fig. 3 (clr nline). Snapshts f the ttal electric fields under an incident TE plane wave frm the left. (a) A dielectric circular shell with permittivity ε = 2. (b) The circular shell in (a) is claked by an invisibility clak cmpsed f a circular layer f cmplementary media with an embedded anti-bject shell and a cre material inside the shell. (c) A circular shell with permittivity ε = 1. (d) The circular shell in (c) is claked by a similar invisibility clak f that in (b). 12