Illusion optics: The optical transformation of an object into another object. Abstract

Transcription

1 Illusion opics: The opical ransformaion of an objec ino anoher objec Yun Lai,* Jack Ng,* HuanYang Chen, DeZhuan Han, JunJun Xiao, Zhao- Qing Zhang and C. T. Chan Deparmen of Physics The Hong Kong Universiy of Science and Technology Clear Waer Bay, Kowloon, Hong Kong, China Absrac We propose o use ransformaion opics o generae a general illusion such ha an arbirary objec appears o be like some oher objec of our choice. This is achieved by using a remoe device ha ransforms he scaered ligh ouside a virual boundary ino ha of he objec chosen for he illusion, regardless of he profile of he inciden wave. This ype of illusion device also enables people o see hrough walls. Our work exends he concep of cloaking as a special form of illusion o he wider realm of illusion opics. * These auhors conribued equally o his work. -mail: To whom correspondence should be addressed. -mail: phzzhang@us.hk (Z. Q. Zhang); phchan@us.hk (C. T. Chan) 1

2 As he saying goes, seeing is believing. Throughou hisory, winessing wih he eyes has been used as proof of exisence or as evidence. On he oher hand, he effecs of illusions, such as mirages, have been well known o lead people o draw incorrec conclusions, someimes wih dire consequences. Recenly, he rapid developmen of ransformaion opics [1-22] has enabled he design of new maerials ha can seer ligh along arbirary curves and he implemenaion is made possible by a new kind of manmade maerials called meamaerials [23-27]. Among various novel applicaions, he mos fascinaing is a cloaking device designed o bend ligh around a concealed region, rendering any objec inside he region invisible [1-10]. Cloaking can be regarded as creaing an illusion of free space. In his paper, we discuss a more generalized concep of illusion: making an objec of arbirary shape and maerial properies appear exacly like anoher objec of some oher shape and maerial makeup. Using ransformaion opics, we design an illusion device consising of wo disinc pieces of meamaerials, which are called he complemenary medium and he resoring medium. The complemenary medium concep, which was firs proposed by Pendry e al. o make focusing lenses [28, 29], is applied here o cancel a piece of space opically, including he objec [21, 22]. Then, he resoring medium resores he cancelled space wih a piece of he illusion space ha is embedded wih he oher objec chosen for he illusion. Regardless of he profile and he direcion of he inciden ligh, he illusion device can ransform he scaered ligh ouside a virual boundary ino ha of he second (illusion) objec; i herefore creaes a sereoscopic illusion for any observer ouside he virual boundary. The principle behind his illusion device is no ligh bending, bu raher he exac cancellaion and resoraion of he opical pah of ligh wihin he virual boundary. Unlike previous ligh-bending cloaking devices [1-10], he consiuive parameers of he illusion device do no have a complex spaial disribuion or any singulariies. More surprisingly, he illusion device works a a disance from he objec. An ineresing implicaion of his remoe feaure is he abiliy o open a virual aperure in a wall so ha one can peep hrough walls in a noninvasive manner. By making an illusion of a hole in a wall, one can see hrough he wall as if he wall has acually had a hole, and for his purpose, monochromic funcionaliy is sufficien. 2

3 A simple schemaic diagram illusraing our idea is shown in Fig. 1. In Fig. 1(a), an illusion device is placed nex o a domain ha conains a man (he objec). The passive device causes any observer ouside he virual boundary (he dashed curves) o see he image of a woman (he illusion) inside he illusion space depiced in Fig. 1(b). We will show ha we can design such an illusion device, which makes he elecromagneic fields ouside he virual boundary in boh he real and illusion spaces exacly he same, irrespecive of he profile of he inciden waves. A blueprin for he device is shown in Fig. 1(c), in which here are wo regions. Region 2 includes he complemenary medium used o annihilae he opical signaure of he man and region 1 includes he resoring medium ha creaes he image of he woman. Boh media are designed using ransformaion opics [1-4]. The complemenary medium is formed by a coordinae ransformaion of folding region 3, which conains he man, ino region 2. The resoring medium is formed by a coordinae ransformaion of compressing region 4 in Fig. 1(d), which conains he illusion, ino region 1. The permiiviy and permeabiliy ensors of boh media in he illusion device can be expressed as: ( 2) ( 3) T μ = Aμ A /dea, ( 1) ( 4) T ε = Bε B /deb and ( 1) ( 4) ( 2) ( 3) T ε = Aε A /dea, T μ = Bμ B /deb, where ε ( i) and ( i) μ are he permiiviy and permeabiliy ensors in region i, A and B are he Jacobian ransformaion ensors wih componens ( 2) ( 3) A = x x and ij i j ( 1) ( 4) B = x x, ij i j corresponding o he coordinae ransformaions of folding region 3 ino region 2 and compressing region 4 ino region 1, respecively. The elecromagneic fields in he complemenary and he resoring media can also be obained from ransformaion opics [1-4] as : ( 2) ( Τ = A ) 1 ( 3), ( 2) Τ 1 ( 3) H = ( A ) H, ( 1) Τ 1 ( 4) = ( B ) and ( 1) Τ 1 ( 4) H = ( B ) H, where ( i) and ( i) H are he elecric and magneic fields in region i, respecively. From he maching of he boundary condiions on surface a (he red solid curve) beween he complemenary medium and he resoring medium, 2 1 we have a = a and ( a) 2 1 ( a) H = H, where subscrip indicaes ransverse componens along he surface. Boh he folding ransformaion, A, and compression ransformaions, B, map one par of he virual boundary, i.e. surface c (he red dashed curves), o surface a. If his one-o-one mapping from c o a is he same for boh A 3

4 and B, hen we can obain from ransformaion opics ha ( c) ( 3 ) ( 4 H ( c) = H ) ( c) on surface c. In addiion, we also have ( d) ( d) 1 4 ( d) 3 4 ( c) = and 1 4 ( d) = and H = H on he oher par of he virual boundary, i.e., surface d (he blue dashed curves), as long as d is no changed during ransformaion B. Therefore, he angenial componens of he elecromagneic fields on he whole virual boundary (including c and d ) are exacly he same in he real and illusion spaces, and, consequenly, by he uniqueness heorem, he elecromagneic fields ouside are also exacly he same. Any observer ouside he virual boundary will see elecromagneic waves as if hey were scaered from he illusion objec (he woman and nohing else), and hus an illusion is creaed. A deailed proof is provided in he Auxiliary Maerial [30]. In he following, we describe full wave simulaions using a finie elemen solver (Comsol Muliphysics) o demonsrae he explici effec of an illusion device ha ransforms a dielecric spoon of ε o = 2 ino a meallic cup of ε i = 1 in wo dimensions. The elecromagneic waves can be decoupled ino T waves ( along he z direcion) and TM waves ( H along he z direcion); we show only he T resuls for breviy (he parameers can be uned o work for boh T and TM waves). Figs. 2(a) and 2(c) plo, respecively, he scaering paerns of he dielecric spoon and he meallic cup, under he illuminaion of a T plane wave (propagaing from lef o righ) of wavelengh λ = 0.25 uni. In Fig. 2(b), an illusion device is placed beside he spoon. The scaering paern around he spoon and he illusion device is alered in such a way ha i appears as if here is only a meallic cup. This can be clearly seen by comparing he field paerns of he spoon plus he illusion device shown in Fig. 2(b) wih ha of he meallic cup shown in Fig. 2(c). The field paerns are indeed idenical ouside he virual boundary. Inside he virual boundary, he field paerns in Figs. 2(b) and 2(c) are compleely differen. The fields beween he spoon and he illusion device are srong due o he exciaion of surface resonances induced by he muliple scaering of ligh beween he spoon and he illusion device. We noe ha he illusion effec is a seady sae phenomenon ha akes some ime o esablish. More simulaion resuls under differen kinds of inciden waves can be found in he Auxiliary Maerial [30]. 4

5 The illusion device in Fig. 2(b) is composed of four pars. The lower rapezoidal par is he complemenary medium formed by a simple coordinae ransformaion of y ( 2) ( 3) = y 2. I is composed of a negaive index homogeneous medium of ε =, z2 2 μ ( 2) x = 2 and ( 2) μ = 0.5, wih an embedded ani-objec of he dielecric spoon wih y ε ( 2) oz 2 2 = 4 and μ μ. The upper lef riangular par, he upper righ riangular par, o = and he upper middle recangular par collecively consiue he resoring medium. The upper lef and righ riangular pars are composed of an homogeneous medium wih ( 1) ε = 4, μ = 4, μ = 20.5 and μ z xx yy () 1 ( 4) 3( ± 0.6) = 1 4 3( ± 0.6) xy = ± 9, formed by he coordinae ransformaions of y x y x, respecively. The upper middle recangular par is composed of an homogeneous medium of z1 ( 1) ε = 4, μ x = 4 and iz ( 1) μ = 0.25, wih an embedded compressed version of he meallic cup illusion of ( 1) ( 1) ( 1) ε = 4 and μ μ, formed by he coordinae ransformaion of y () 1 ( 4 ) ( y ) y i = 0.6 = I is imporan o noe ha he permiiviy and permeabiliy of he illusion device are boh composed of simple homogeneous media and his simpliciy is a consequence of he simple coordinae ransformaions applied here. They do no bend sraigh ligh pahs ino curved ones as in ligh-bending cloaking devices [1-10]. The complemenary medium is obained from he ransformaion opics of folded geomery (see, for example, Leonhard e al. [10]). I is composed of lef-handed meamaerials wih simulaneously negaive permiiviy and permeabiliy. The medium can be isoropic if we apply a ransformaion of y ( 2) = y ( 3) insead of y ( 2) ( 3) = y 2. This kind of meamaerial has been exensively sudied in he applicaion of he superlens [28], and i has been fabricaed by various resonan srucures a various frequencies [23-27]. The oher key componen of he illusion device is he resoring medium, which projecs he opical illusion of he meallic cup. I is composed of he homogeneous medium wih posiive bu anisoropic permeabiliy. This kind of medium can be designed from layer-srucured meamaerials [15]. We noe ha some special illusion ricks by image projecion using ransformaion opics have been discovered, such as he shifed-posiion image of an objec inside a 5

6 meamaerial shell [16], he cylindrical superlens [17], he superscaerer [18], he reshaper [19] and he super absorber [20]. Recenly, we proposed an approach o realize cloaking a a disance by using an ani-objec [21, 22]. Here, by combining he ani-objec cloaking funcionaliy and he image projecion funcionaliy, we achieve a general form of illusion opics such ha an objec can be disguised ino somehing else and he illusion device iself is invisible. This general form of illusion opics wih arbirary shape and generalized opology is proved mahemaically as i is designed wih ransformaion opics and he funcionaliy is also demonsraed numerically. From a muliple scaering poin of view, he illusion opics is in fac raher remarkable as i is by no means obvious ha he ani-objec cancelling and he image projecion funcionaliy do no obsruc or inerfere wih each oher. Anoher ineresing applicaion of our illusion device is ha i enables people o open a virual hole in a wall or obsacle. As our illusion device works a a disance from he objec, i is capable of ransforming only one par of an objec ino an illusion of free space, hus rendering ha par invisible while leaving he res of he objec unaffeced. By making one par of he wall invisible (i.e., making an illusion of a hole ), we can hen see hrough he wall and obain informaion from he oher side. In Fig. 3(a), we see ha a wall of ε o = 1 wih a widh of 0.2 unis is capable of blocking mos of he energy of he T elecromagneic waves radiaing from a poin source of λ = 0.25 uni placed a ( 0.7,0). When he illusion device is placed on he righ side of he wall, as shown in Fig. 3(b), he elecromagneic waves can penerae hrough he wall as well as he illusion device and arrive on he righ side. This effec can also be undersood as he unneling of elecromagneic waves via he high-inensiy surface waves localized a he inerface beween he wall and he complemenary medium. The phase informaion is accuraely correced by he resoring medium in he illusion device, such ha he ransmied field paerns on he righ side become he same as hose of he elecromagneic waves peneraing hrough a real hole, as shown in Fig. 3(c). Thus, an observer on he righ side can peep hrough he virual hole as if he/she is peeping hrough a real hole a he working frequency of he illusion device. The consiuive pars of he illusion device are similar o ha in Fig. 2(b) and described in deail in he Auxiliary Maerial [30]. Similarly, an objec hidden in a conainer can be compleely revealed by using he illusion opics o 6

7 change he conainer ino an illusion of free space. This is also demonsraed in he Auxiliary Maerial [30]. In principle, he illusion opics allows us o remoely change he opical response of an objec ino ha of any oher objec chosen for illusion a a seleced frequency, wihou he need o change he consiuens and shape of he rue objec or even cover is surface. This opens up ineresing possibiliies. For insance, an illusion waveguide or phoonic crysal would allow he conrol of ligh propagaion in acual free space; an illusion ip migh perform near-field scanning opical microscopy wihou physically approaching a surface. However, he heoreical foundaion of he illusion device is ransformaion opics and, as such, our heory relies on he validiy and accuracy of a linear coninuous medium ha describes he homogenized elecromagneic fields in meamaerials. This requiremen is crucial in he inerface beween he complemenary medium and he cancelled objec due o he high-inensiy local fields as well as rapid oscillaions here. The range of he virual boundary also plays an imporan role. When i is large, he field a he boundary will be large as well. Anoher issue ha we have no considered is loss, which will degrade he illusion effec unless he objec is close o he device. If hese issues and challenges can be solved wih advances in meamaerial echnologies, we should be able o harness he power of ransformaion opics o creae illusions. This work was suppored by Hong Kong Cenral Allocaion Gran No. HKUST3/06C. Compuaion resources are suppored by Shun Hing ducaion and Chariy Fund. We hank Dr. KinHung Fung, ZhiHong Hang, Jeffrey ChiWai Lee and HuiHuo Zheng for helpful discussions. 7

8 References 1. U. Leonhard, Science 312, 1777 (2006). 2. J. B. Pendry, D. Schurig, and D. R. Smih, Science 312, 1780 (2006). 3. A. Greenleaf, M. Lassas, and G. Uhlmann, Physiol. Meas. 24, 413 (2003). 4. G. W. Milon, M. Briane, and J. R. Willis, New J. Phys. 8, 248 (2006). 5. D. Schurig, J. J. Mock, B. J. Jusice, S. A. Cummer, J. B. Pendry, A. F. Sarr, and D. R. Smih, Science 314, 977 (2006). 6. W. Cai, U. K. Cheiar, A. V. Kildishev, and V. M. Shalaev, Na. Phoon. 1, 224 (2007). 7. J. Li and J. B. Pendry, Phys. Rev. Le. 101, (2008). 8. R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui and D. R. Smih, Science 323, 366 (2009). 9. U. Leonhard and T. Tyc, Science 323, 110 (2009). 10. U. Leonhard and T. G. Philbin, New J. Phys. 8, 247 (2006). 11. A. Alu and N. nghea, Phys. Rev. Le. 100, (2008). 12. M. Rahm, D. Schurig, D. A. Robers, S. A. Cummer, D. R. Smih, and J. B. Pendry, Phoon. Nanosruc.: Fundam. Applic. 6, 87 (2008). 13. A. V. Kildishev and.. Narimanov, Op. Le. 32, 3432 (2007). 14. M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smih, Phys. Rev. Le. 100, (2008). 15. H. Y. Chen and C. T. Chan, Phys. Rev. B 78, (2008). 16. Y. Luo, J. J. Zhang, H. Chen, B.-I. Wu, and J. A. Kong, arxiv: (2008). 17. M. Yan, W. Yan, and M. Qiu, Phys. Rev. B 78, (2008). 18. T. Yang, H. Y. Chen, X. D. Luo, and H. R. Ma, Op. xpress 16, (2008). 19. H. Y. Chen, X. H. Zhang, X. D. Luo, H. R. Ma, and C. T. Chan, New J. Phys. 10, (2008). 20. J. Ng, H. Y. Chen, and C. T. Chan, Op. Le. 34, 644 (2009). 21. Y. Lai, H. Y. Chen, Z. Q. Zhang, and C. T. Chan, Phys. Rev. Le. 102, (2009). 22. G. A. Zheng, X. Heng, and C. H. Yang, New J. Phys. 11, (2009). 23. D. R. Smih, J. B. Pendry, and M. C. K. Wilshire, Science 305, 788 (2004). 24. C. M. Soukoulis, S. Linden, and M. Wegener, Science 315, 47 (2007). 8

9 25. H. J. Lezec, J. A. Dionne, and H. A. Awaer, Science 316, 430 (2007). 26. J. Valenine, S. Zhang, T. Zengraf,. Ulin-Avila, D. A. Genov, G. Baral, and X. Zhang, Naure 455, 376 (2008). 27. J. Yao, Z. W. Liu, Y. M. Liu, Y. Wang, C. Sun, G. Baral, A. M. Sacy, and X. Zhang, Science 321, 930 (2008). 28. J. B. Pendry, Phys. Rev. Le. 85, 3966 (2000). 29. J. B. Pendry and S. A. Ramakrishna, J. Phys.: Condens. Maer 14, 8463 (2002); 15, 6345 (2003). 30. See Auxiliary Maerial for a deailed mahemaical proof of illusion opics, more simulaions of illusion opics under differen inciden waves as in Fig. 2(b), and a deailed descripion of he illusion device in Fig. 3(b). 9

10 Fig. 1 (color online). The working principle of an illusion device ha ransforms he sereoscopic image of he objec (a man) ino ha of he illusion (a woman). (a) The man (he objec) and he illusion device in real space. (b) The woman (he illusion) in he illusion space. (c) The physical descripion of he sysem in real space. The illusion device is composed of wo pars, he complemenary medium (region 2) ha opically cancels a piece of space including he man (region 3), and he resoring medium (region 1) ha resores a piece of he illusion space including he illusion (region 4 in (d)). Boh real and illusion spaces share he same virual boundary (dashed curves). 10

11 Fig. 2 (color online). A numerical demonsraion of ransforming he sereoscopic image of a dielecric spoon of ε o = 2 (he objec) ino ha of a meallic cup of ε i = 1 (he illusion) hrough an illusion device, under an inciden T plane wave from he lef. (a) The scaering paern of he dielecric spoon. (b) The scaering paern of he dielecric spoon is changed by he illusion device. Ouside he virual boundary, he scaering paern becomes he same as ha of he meallic cup, which is shown in (c). 11

12 Fig. 3 (color online). The illusion device can creae he illusion of a hole so ha people can see hrough a wall a a seleced frequency. (a) The elecromagneic radiaion from a T poin source on he lef side is blocked by a slab of ε o = 1. (b) When an illusion device is aached o he wall, he elecromagneic radiaion can now unnel hrough he wall o he righ side. The far field radiaion paern is exacly he same as ha of he radiaion hrough a real hole, which is shown in (c). 12

13 Auxiliary maerial of Illusion opics: The opical ransformaion of an objec ino anoher objec Yun Lai*, Jack Ng*, Huanyang Chen, DeZhuan Han, JunJun Xiao, Zhao-Qing Zhang and C. T. Chan Deparmen of Physics The Hong Kong Universiy of Science and Technology Clear Waer Bay, Kowloon, Hong Kong, China Par A: A rigorous proof of he illusion opics in 3D by ransformaion opics We shall prove here ha by using he complemenary medium and he resoring medium designed from ransformaion opics, we are able o ransform an objec ino an illusion of anoher objec of our choice. Boh he objec and he illusion can be anisoropic and/or inhomogeneous. Consider he configuraion depiced in Fig. A1. The real space is divided ino four domains: regions 1, 2, 3, and he region ouside surfaces c and d. The illusion space is divided ino wo domains: region 4 and he region ouside surfaces c and d. Under ligh illuminaion, here will be a soluion in each of hese regions. Our ask is o prove ha under arbirary ligh illuminaion, he soluion ouside surfaces c and d is he same for boh he real space and he illusion space, such ha any ouside observer would hink ha he/she has seen he illusion while wha are really here are he objec and he illusion device. () i () i () i We parameerize region i by he generalized curved coordinaes ( u, v, w ), as depiced in Fig. A2. The permiiviy and permeabiliy ensors in region i are respecively denoed as () () () () ε i ( u i, v i, w i ) and μ ( u, v, w ), and he elecric and magneic fields () i () i () i () i of region i are respecively denoed as () i () i and H. Region 3 is a piece of space embedded wih he objec ha we wan o ransform ino somehing else. Region 2 is composed of he complemenary medium of region 3, whose dielecric properies are obained by he coordinae ransformaion of folding region 3 ino region 2: 13

14 ( 2) ( 3) T ε = Aε A /de A, ( 2) ( 3) T μ = Aμ A /de A, (1) wih each poin on surface c being mapped o a poin on surface a in a one-o-one and coninuous manner, and each poin on surface b being mapped back o iself. Here, (2) (2) (2) (3) (3) (3) u v w (2) (2) (2) A = (3) (3) (3) u v w (2) (2) (2) (2) w w w (3) (3) (3) w is he Jacobian ransformaion ensor of he folding ransformaion. From ransformaion opics, he elecromagneic fields of region 2 and 3 are relaed by ( 2) ( 3) Τ A =, (3) Τ ( 2) ( 3) AH = H. I can be shown ha he boundary condiions on surface b are fulfilled. By exploiing our freedom o selec he parameric coordinae w such ha surface b is a consan level surface of (2) w and (3) w, we have on surface b: (2) (2) w w = = 0. (4) (3) (3) Furhermore, since each poin on surface b is being mapped back o iself, he parameric (2) (2) (3) (3) coordinae ( u, v ) can be chosen o exacly coincide wih (, ) which gives on surface b u v on surface b, = = 0, = = 1. (2) (2) (3) (3) (2) (2) (3) (3) Subsiuing qs. (4) and (5) ino q. (2), we obain, on surface b, (5) 14

15 (2) 1 0 (3) w (2) A ( b b) = 0 1 (3) w. (6) (2) w 0 0 (3) w Using qs. (3) and (6), i can be shown ha he boundary condiions on surface b are fulfilled: (3) (2) (3) (2) (3) = (2), (3) = (2), v v u u (7) (3) (2) (3) (2) H = H, H = H. (3) (2) (3) (2) v v u u We nex consider he mapping of surface c o surface a in real space. On surface a and on he side of region 2, we can again exploi our freedom o choose he parameric coordinae such ha surface a is a consan level surface of (3) is a consan level surface of w, such ha, on surface a, w The ransformaion Jacobian is hen w (2) (2) = = 0 (3) (3) (2) w, and similarly surface c. (8) (2) (2) (2) (3) (3) (3) u v w (2) (2) (2) A ( c a) = (3) (3) (3) u v w. (9) (2) w 0 0 (3) w Using qs. (3) and (9), we obain he relaions of he angenial fields on surface c in real space and surface a as: (2) (2) (3) (2) (2) (3) = (3) (2) + (3) (2) u u v (2) (2) (3) (2) (2) (3) = (3) (2) + (3) (2) v u v and he expressions of he magneic fields are similar. and On he oher hand, region 1 is composed of he resoring medium wih soluions (1) H. Since (1) and,, (10) (1) (1) H are he soluions in real space, hey mus saisfy he 15

16 boundary condiion on surface a. Accordingly, he ransverse componen of equals ha of (2) and (2) H on surface a, respecively: =, =, H H, H H. (1) (2) (1) (2) (1) v (2) v (1) u (2) u (1) (2) (1) (2) (1) = (2) (1) = (2) v v u u Subsiuing q. (11) ino q. (10), we obain (2) (2) (3) (1) (1) (3) = (3) (1) + (3) (1) u u v (2) (2) (3) (1) (1) (3) = (3) (1) + (3) (1) v u v,, (1) and and he expressions of he magneic fields are similar. We noe ha he dielecric properies of region 1 are deermined by he coordinae ransformaion of compressing region 4 in he illusion space ino region 1: ( 1) ( 4) T ε = Bε B /deb (13) () 1 ( 4) T μ = Bμ B /deb wih each poin on surface c being mapped o a poin on surface a in a one-o-one and coninuous manner, and each poin on surface d being mapped back o iself. Here (11) (12) (1) (1) (1) (4) (4) (4) u v w (1) (1) (1) B = (4) (4) (4) u v w (14) (1) (1) (1) w w w (4) (4) (4) w is he Jacobian ransformaion ensor of he compressing ransformaion. The elecromagneic fields in he resoring medium can also be obained from ransformaion opics: ( 1) ( 4) Τ B =, Τ () 1 ( 4) BH = H. On surface a and on he side of region 1, he ransformaion Jacobian is (15) (1) H 16

17 (1) (1) (1) (4) (4) (4) u v w (1) (1) (1) B ( c a) = (4) (4) (4) u v w, (16) (1) w 0 0 (4) w where we have again chosen he parameric coordinae such ha surface a is a consan level surface of a we have (1) w and surface c is a consan level surface of (4) w such ha, on surface (1) (1) w w = = 0. (17) (4) (4) Using qs. (15) and (16), he relaions of he angenial fields on surface c in illusion space and surface a are given by (1) (1) (4) (1) (1) (4) = (4) (1) + (4) (1) u u v (1) (1) (4) (1) (1) (4) = (4) (1) + (4) (1) v u v By comparing qs. (12) and (18), i is clear ha on surface c, if and only if on surface a: = (4) (3) (4) u (3) u = (4) (3) (4) v (3) v (2) (1) (2) (1) =, =, (3) (4) (3) (4) (2) (1) (2) (1) =, =. (3) (4) (3) (4) Since boh A and B map surface c o a, we can always choose A and B such ha hey map he same poin on surface c o he same poin on surface a. Accordingly, q. (20) can be fulfilled. Wih ha, we have proved ha he angenial fields on surface c are he same for boh he real space and he illusion space. For he angenial field on surface d, since B maps each poin of surface d back o iself, similar o he case of boundary condiion maching on surface b, i can be easily seen ha he angenial fields on surface d are exacly he same for boh he real space and he illusion space. Since surfaces c and,,., (18) (19) (20) 17

18 d ogeher form a closed surface, and boh fields on surfaces c and d are he same for boh he real space and he illusion space, by he uniqueness heorem, he field ouside surfaces c and d for boh he real space and he illusion space are exacly he same. Wih ha, we have disguised he objec ino he illusion and hus compleed our proof. We noe ha while our proof here is for hree-dimensional geomeries, i can be easily generalized o wo dimensions. Moreover, i can also be generalized o he case in which he illusion device does no share a par of is boundary wih he virual boundary, i.e. surface d, as Fig. A3 shows. In his case, he resoring medium is compleely surrounded by he complemenary medium. 18

19 Fig. A1. The working principle of an illusion device ha ransforms he sereoscopic image of he objec (a man) ino ha of he illusion (a woman). (a) The man (he objec) and he illusion device in real space. (b) The woman (he illusion) in he illusion space. (c) The physical descripion of he sysem in real space. The illusion device is composed of wo pars, he complemenary medium (region 2) ha opically cancels a piece of space including he man (region 3), and he resoring medium (region 1) ha resores a piece of he illusion space including he illusion (region 4 in (d)). Boh real and illusion spaces share he same virual boundary (dashed curves). Fig. A2. An illusraion of an arbirary curved coordinae sysem. 19

20 Fig. A3. Anoher opology of illusion device, in which he resoring medium (region 1) is compleely surrounded by he complemenary medium (region 2). The boundary of region 3 (curve c) is he virual boundary. 20

21 Par B: Numerical demonsraion of he illusion opics by using he sysem in Fig. 2(b) under various kinds of inciden waves o show ha he device funcionaliy is independen of he form of he inciden waves. Fig. B1. A T plane wave of wavelengh 0.25 uni inciden from below. Fig. B2. A T poin source of wavelengh 0.25 uni placed a (-0.8, -0.6). Fig. B3. A T poin source of wavelengh 0.25 uni placed a (0.8, 0.9). From hese numerical simulaion resuls, i can be clearly seen ha he illusion opics effec is independen of he inciden angle and profile of he inciden waves. 21

22 Par C: Descripion of he illusion device demonsraed in Fig. 3(b), and a numerical simulaion of revealing an objec hidden inside a conainer. The illusion device in Fig. 3(b) is composed of four pars. The lef rapezoidal par in conac wih he wall is he complemenary medium wih ε z2 = 2, ( 2) μ =, formed by a coordinae ransformaion of y 2 x 2 x 3 2 ( 2) μ = 0.5 and x =. Here, he complemenary medium is only negaive in permeabiliy because he cancelled wall is negaive in permiiviy (i.e., meallic). The upper and lower riangular pars and he middle recangular par on he righ consiue he resoring medium. The upper and lower riangular pars are composed of a medium wih ε z1 = 4, ( 1) ( 1) μ = 9.25, xx ( 1) μ = 4 and μ = 6, formed by he coordinae ransformaions of xy () 1 ( 4) ± 2( 0.5) = 1 4 ± 2( 0.5) x y x y, respecively. The middle recangular par is composed of a medium of ε z1 = 4, ( 1) 1 μ = 0.25 and μ = 4, formed by he coordinae x ransformaion of () 1 ( 4 x 0.2 = 1 4 x ) 0.2. Since he aim is o creae a piece of free space in his case, here is no compressed version of any illusion objec inside he resoring medium. This super-vision illusion device does no require a broad bandwidh and hus can be consruced by resonan meamaerials designed a a single seleced working frequency. y yy 22

23 Fig. C1. A numerical demonsraion of revealing an objec hidden inside a conainer by using illusion opics. (a) An objec of ε = 5 is hidden inside a circular shell of ε = 1 (meallic), such ha a T plane wave inciden from he lef canno see he objec. (b) A circular illusion device consising of an inner circular layer of complemenary medium ha opically cancels he shell, and a circular layer of resoring medium ha resores a circular layer of free space, is placed ouside he shell. I is clearly seen ha he scaering paern ouside he device is now changed ino exacly he same paern as he scaering paern of he objec iself, as is shown in (c). 23