UNCLASSIFIED //FO R O FFICIAL H EE O NLY D efense Intelligence Reference D ocument A cquisition Threat S upport March 2010 : 1 Decem ber 2009 ^08-1003 -001 Invisibility Cloaking: Theory and Experiments UNCLASSI FIED //FO R O FFICIAL WEE O NLY UNCLASSIFIED //FUH O FFICIAL USE H WEI Invisibility Cloaking: Theory and Experiments P repared by: Acquisition Support D ivision (D WO -3) D efense Warning O ffice D irectorate for Analysis D efense Intelligence Agency Author: A A P Person 68 Administrative Note COPYRIGHT WARNING: Further dissemination of the photographs in this publication is not authorized. This product is one in a series of advanced technology reports produced in FY 2009 under the Defense Intelligence Agency, Defense Warning Office's Advanced Aerospace Weapon System Applications (AAWSA) Program. Comments or questions pertaining to this document should be addressed to|AAP Person 1 J, AAWSA Program Manager, Defense Intelligence Agency, ATTN: CLAR/DWO-3, Bldg 6000, Washington, DC 2034 0-5100. ii UNCLASSIFIED //EfiB^SH 6MUU6E-&Nfe¥' UNCLASSI FI ED / / FUR UhH llAL USE UNL! Contents Introduction................................................................................................................v Camouflage......................................................................................... 1 Transparency.......................................................................................................—> 3 Cloaking......................................................................................................................5 Metamaterials............................................................................... 9 O ptical Metamaterials...................................................................... 11 Fundamental P roblem..............................................................................................15 Curved Space....................................... 17 Broadband Invisibility............................................................................. 19 Implementation........................................................................................................20 O ptical Cloaking............................................................................... 22 Summary................................... 23 References......................................................................... 23 Figures Figure 1. B2 Stealth Bomber........................................... 1 Figure 2. O ptical Camouflage.......................................... 2 Figure 3. The Secret O ptical Camouflage......................... 2 Figure 4. H . G . Wells's Th e In visib le M an : Invisibility by Transparency................ 3 Figure 5 . Complementary Media Invisibility Cloak.................... 4 Figure 6. Cloaking Shell............................................................................. 5 Figure 7 . The Invisible Woman.............................................................................. 6 Figure 8. Fermat's P rinciple .......................................................................................7 Figure 9. An O ptical Material D eforms the Coordinates of Space..............................8 Figure 10. Light Waves at Cloaking D evice.......................................... 9 Figure 11. Cloaking D evice for Microwaves...................................... 10 iii UNCLASSI FIED //P O R O FFICIAL USE O hL¥ UNCLASSI FI ED //FO R O FFICIAL USE O NLY Figure 12. Lycurgus Cup (British Museum; AD Fourth Century). .............. 11 Figure 13. Idea for a Cloaking D evice for Visible Light...........................................12 Figure 14. Advances in Metamaterials............................................. 13 Figure 15 . D emonstration of Negative Refraction With "Bulk" O ptical Metamaterials Made of Nano-Fishnets.. 14 Figure 16. D emonstration of Negative Refraction With "Bulk" O ptical Metamaterials Made of Nanowires.......................................15 Figure 17 . Fundamental P roblem of Transformation-Based Cloaking D evices.....16 Figure 18. Wave P ackets are Made by Combining Waves With D ifferent Frequencies.......................................................... 17 Figure 19. Stereographic P rojection........................................................ 18 Figure 20. Non-Euclidean Cloaking D evice in Two D imensions...............................19 Figure 21. Three-D imensional Cloaking............................................ 20 Figure 22. Coordinate Transformation Implemented by a G round-P late Cloak......21 Figure 23. Implementation of the G round-P late Cloak...........................................21 iv UNCLASSI FIED //FO R O FFICIAL USE O NLY UNCLASSIFIED //FO R O mCIAL USE O MUY Invisibility Cloaking: Theory and Experiments Introduction The idea of invisibility has fascinated people for millennia, inspiring many myths, novels, and films. Invisibility cloaking has recently become a subject of science and technology. This paper describes the important current theoretical and experimental developments and tries to project into the future. v UNCLASSI FIED //FO R O FFICIAL UCE O NEY UNCLASSIFIED //rQR O FFICIAL UO E O Nte1!1 Camouflage Invisibility m ay be achieved through three principal m ethods: cam ouflage, transparency, and cloaking. Many anim als and som e plants use cam ouflage to disguise them selves from predators— for exam ple, by assum ing the shapes and colors of objects in their surroundings. The m ilitary has long used form s of cam ouflage; a recent m ilitary application of cam ouflage is stealth technology. Stealth planes have aerodynam ically unusual, edgy shapes and are coated with a special m aterial. Both features serve the sam e purpose: to m ake the plane "invisible" to radar. How does it work? In radar, electrom agnetic m icrowaves are em itted by a source, and their reflection by an object— an airplane, for exam ple— is detected. From the direction and the tim e delay of the reflected waves, the direction and distance of the object are inferred. If the object does not reflect the electrom agnetic m icrowaves back to the source, it will not appear on the radar. This is precisely what stealth technology achieves. Owing to the edgy shape of the stealth plane, m ost of the incident Figure 1. B2 Stealth Bomber electrom agnetic waves are reflected in different directions; the coating of the plane absorbs the rest. In this way, the stealth plane has becom e com pletely black in the spectral range of radar. As for radar waves, the sky is black, not blue, and the plane has assum ed the color of the background: the stealth plane is cam ouflaged. 1 UNCLASSI FIED / / FO R O FFICIAL USE O Mte¥ UNCLASSIFIED / /«QMFH €MMSMNfe¥ Another recent exam ple of cam ouflage is optical cam ouflage, developed by the U niversity of Tokyo's Tachi Laboratory. Figure 2 shows an exam ple of optical cam ouflage. A cam era captures the background scene behind the person. The im age is processed and projected onto the person via a sem itransparent m irror. The person wears an "invisibility cloak" m ade of a retroreflective m aterial that reflects light back in the direction whence it cam e, like cats-eye-reflectors do. As the cloak carries the projected im age of the background, the person seem s to disappear, but surely the equipm ent standing around the person is clearly visible. In addition, optical cam ouflage works only in one direction; seen from the side, the person is visible. N evertheless, optical cam ouflage m ay becom e a useful tool in som e situations Figure 2. O ptical Camouflage (Tachi Laboratory, Tokyo) where obstacles are in the way of sight— for exam ple in surgery, where the surgeon's hands and instrum ents m ay obstruct the view. Figure 3 shows how optical cam ouflage works. Figure 3. The Secret O ptical Camouflage (Tachi Laboratory, Tokyo) 2 U N CLASSI FI E D / /fQB^EEJCJALUSMNM UNCLASSIFIED //TO W O FFICIAL USE O H LY Transparency H. G. W ells's novel The Invisible M an represents another strategy for becom ing invisible: transparency. In W ells's novel, the invisible m an, a disgruntled college professor, invents a substance that som ehow changes the refractive index1 of his body. Most transparent substances, like glass, air, or water, m odify the speed of light, because the atom s or m olecules of these substances absorb and re-em it light, which takes tim e. The delay caused by the atom s and m olecules results in a reduced speed of light and hence in a refractive index larger than 1. It is, however, also possible to achieve a refractive index sm aller than 1, although only in narrow bands of the spectrum . In these cases, the atom s or m olecules advance the wave fronts of light because they are excited such that their electron 1 The refractive index is the ratio between the speed of light in vacuum and the speed of light in a m aterial. Figure 4. H . G . Wells's Th e In visib le M an : Invisibility by Transparency clouds oscillate ahead of the light. If the refractive index is uniform in a m aterial, light is reflected and refracted at the boundary but otherwise is traveling straight through. On the other hand, if the refractive index varies, light is scattered at the index inhom ogeneities and gets lost. Most white substances appear white because of such scattering. Milk, for exam ple, consists of m inuscule oily droplets— fat— in water. The refractive index of the droplets differs from water, and hence light is scattered at them ; it does not penetrate the substance, and the diffused light appears as white. N ow, hum an bodies are visible, because they absorb light. Most of the absorption is due to the scattering of light in biological tissue, in the cells of which the bodies are m ade. If the refractive indices of a person's cells could som ehow be changed to the refractive index of air, the person would becom e transparent and disappear from view— like the Invisible Man. Som e anim als (for exam ple, som e jellyfish) are transparent, but the cells of higher order anim als are usually m uch too com plex and diverse for transparency to becom e a serious option for disguise. E xceptions are the transparent parts of the body, m ost notably eye lenses, which consist of uniform cells kept in a state between life and death. If this balance is upset, the lenses becom e opaque as a cataract develops. Transparency is the idea behind som e proposed form s of invisibility by technology. For exam ple, in plasm onic covering,1 a particle should be surrounded by layers m ade of m etals and transparent substances, such as glass. The layers are designed such that they cancel the scattering of light at the particle, hence m aking both the particle and the layers transparent— that is, invisible. Another proposal2 exploits the resonance of the particle with a negative-refractive m aterial that cancels out scattering. In a m aterial with negative refraction, the wave fronts of light appear to m ove in the opposite 3 UNCLASSI FIED / / EQB O FFICIAL USE O NL^ UNCLASSIFIED //rO R O FFICIAL UD E O NLY direction from the propagation. The clearest and m ost advanced form of this concept is the com plem entary m edia invisibility cloak.3 Here, an optical antiobject is placed beside the object one wishes to m ake disappear. The antiobject should be m ade of a negatively refractive m aterial that exactly com pensates the optical appearance of the object. An im age is contained in the deform ations of light-wave fronts caused by the im aged object. If these deform ations are reversed, the im age disappears, and the object becom es transparent— that is, invisible. The optical antiobject m ust be tailored to the object and placed precisely at the correct distance— that is, the distance where it is m ade to cancel the im age of the object. The m ore com plex the object is, the m ore com plex the antiobject m ust be for reversing all the scatterings of light. Such cloaking at the distance cannot be instantaneous, as the light scattered by both object and antiobject m ust settle to a stationary state where it becom es synchronized. A stationary light field has only one color. So, in practice, these form s of transparency will work only for sm all objects and for sm all parts of the spectrum and not for large objects in m any colors. a) e '= -1 H = -1 -L 0 c) Figure 5 . Complementary Media Invisibility Cloak: (a) The slab of em pty space x with 0 < x < L is optically canceled by a slab of negative-index m aterial in -L < x < 0. (b) The sam e cancellation effect works with an object in 0 < x < L if the negative-index slab contains an antiobject, (c) A spherical shell b < r < c is optically canceled by a negative-index shell a < r < b. If the core r < a Is optically equivalent to a sphere of radius c, then this device Is invisible, (d) The sam e as (c), but with an object in the canceled shell b < r < c. The object is cloaked: both it and the cloaking sphere are Invisible. 4 UNCLASSI FIED //FO K O FFICIAL USE O NLY UNCLASSIFIED //P O R O FFICIAL USE O NL¥ Cloaking Cloaking4’5 is a universal strategy for invisibility that works for objects of arbitrary com positions and shapes within a given size. In cloaking, the hidden object is enclosed by the cloaking device, a transparent shell that guides light around the object as if the light would propagate through em pty space. In this way, both the interior of the cloaking device is hidden and the act of hiding is concealed. Figure 6. Cloaking Shell6 How does one find the right design for such a cloaking device? As the Invisible Man sym bolizes transparency as a strategy for invisibility, inspiration for cloaking m ay com e from the Invisible W om an, a cartoon figure from the Fantastic Four. The Invisible W om an is said to create a m ysterious force field around her that bends space. Light follows the curved space such that it sm oothly flows around the Invisible W om an, like water in a stream flowing around an obstacle. The key idea here is the concept of curved space used for invisibility. The idea that turns the Invisible W om an with her fabled force field from a fictitious character into som ething close to reality is the insight that no force field is needed, that light-refracting m aterials like glass or water appear as curved spaces by them selves. 5 UNCLASSI FIED //FO R QEEIG iAE USE O NhY UNCLASSIFIED //TO W O FFICIAL USE UNL! This idea com es from Ferm at's principle of the shortest optical path.7 Suppose that light travels from A to B. According to Ferm at's principle, light follows the path that takes the shortest tim e. In refractive m aterials, the speed of light is m odified— in m ost cases reduced— by the refractive index. The tim e it takes for light to pass through an infinitesim al elem ent of space is proportional to the refractive index. So, if the index varies in an optical m aterial or in the boundaries between m aterials, the optical m easure of path length varies, which is the defining feature of a curved geom etry. To provide a sim ple exam ple, a lens focuses parallel light rays into a point; the parallels m eet there, which is the hallm ark of a non-E uclidian geom etry. So som ething as fam iliar as a lens creates som ething as fantastic as a curved geom etry. Figure 7 . The Invisible Woman, Invisibility by bending light, an inspiration for cloaking. 6 UNCLASSI FIED //FO R O FFICIAL UD E O Nfe¥ UNCLASSIFIED //P O W O FFICIAL USE O NLY Figure 8. Fermat's P rinciple. The grey level indicates the refractive index. In traveling from A to B, light follows the path that takes the shortest tim e, traveling, if possible, through regions of low index. The bending of light in m aterials or, equivalently, the change of the spatial geom etry by optical m aterials is the cause of m ost optical illusions. For instance, the figure used to illustrate Ferm at's Principle shows the path of light in a m irage. The air above a hot surface, hot tarm ac, or desert sand is hotter than the air farther above. As hot air is thinner than cooler air, the refractive index is lower above the hot surface. Light rays are bent upwards, conjuring up im ages of water in the distance that, in reality, are im ages of the sky. Such optical geom etries m ay also be em ployed for creating the ultim ate illusion: invisibility. Im agine an optical m aterial changing the geom etry of space, as shown in Figure 9 below. 7 UNCLASSI FIED //MR O FFICIAL USE O NLY UNCLASSIFIED //FO R O FFICIAL USE O NLY B Figure 9. An O ptical Material D eforms the Coordinates of Space. A: virtual space, B: real space. The m aterial acts like a transform ation of space. In the device, each point in real space is put to a new location in a virtual space (outside of the device, space rem ains the sam e). The device creates the illusion that light propagates along straight lines through the virtual space, whereas in reality it is bent. Suppose the device encloses a hollow region that is not part of the virtual space or, to be m ore precise, a region with a surface that, in virtual space, has becom e a single point. The red circle in Figure 9 B is reduced to the red dot in Figure 9 A. As the hollow region is not part of the virtual space where light propagates, everything inside it has becom e invisible. As the coordinates of virtual space sm oothly go over into the coordinates of real space at the outer surface of the device, light rays are not distorted. Any object inside the cloaking device is hidden, and so is the act of hiding itself. Moreover, the cloaking device not only would bend light rays but would m odify the entire structure of light waves in such a way that detecting the hidden object is im possible. Light waves would advance around the hidden core of the device, engulfing it, as Figure 10 shows.8 UNCLASSI FIED //BO B QEEICIAk USE O Nb* UNCLASSIFIED //rO R O FFICIAL USE O NLY Figure 10. Light Waves at Cloaking D evice. W ave propagation in (a) virtual space and (b) real space. This idea of cloaking by coordinate transform ations was put forward by two independent groups. Their theories appeared in S cience Express on 25 May 2006 and were published back to back in S cience m agazine later on. The first paper9 only considered isotropic m aterials, optical m aterials where the speed of light at each point is the sam e in each direction but m ay vary from point to point. Most natural optical m aterials, except for som e crystals and all liquid crystals, are isotropic. In isotropic m aterials, invisibility is perfect only for light rays, but the cloaking device m ay cause dislocations of light waves. In addition to the optical im plem entation of coordinate transform ations, som e other tricks are required. The second paper10 considered anisotropic m aterials for cloaking or other m anipulations of electrom agnetic waves. Here, perfect invisibility is possible in principle (but not in practice, as is discussed later). In October 2006, the first cloaking device was dem onstrated,11 for m icrowaves. S cience m agazine regarded cloaking as one of the top 10 science breakthroughs of the year (it was top in physics and engineering). S cientific A m erican listed the inventors of cloaking devices— Sir John Pendry, David Sm ith, David Schurig, and U lf Leonhardt— am ong the top 50 policy, business, and research leaders of the year. The first paper12 on cloaking had initially been rejected by m ost m ajor science and physics journals before it finally appeared in S cience, but since 2006, cloaking has becom e a m ainstream subject on which about a thousand papers have been published so far. Metamaterials The first prototype13 of a cloaking device was designed to operate in the m icrowave region of the electrom agnetic spectrum , for a wavelength of about 3 cm . The device consists of 10 rings of flexible circuit board. The copper of the circuit board has been etched away, apart from characteristic structures of about 3 -m m size, so-called split­ ring resonators. 9 UNCLASSI FIED //BO B nEEIFTAI USE O NI Y UNCLASSIFIED //FO R O FFICIAL UD E O NLY The split-ring resonators are electrom agnetic circuits; they respond to the electrom agnetic field of m icrowave radiation. Their response depends on their shapes. For exam ple, in the cloaking device shown in Figure 11, the double stripes in the m iddle of the split­ ring resonators vary from ring to ring. As these stripes form an electric capacitor, the capacitance of the resonators also varies. The colored curves show how the electrom agnetic functions change over the distance from the center of the cloaking device as a result of the varied capacitance. As they are always positive, negative refraction is not required for cloaking. At the inner ring, the red curve reaches zero, defining the boundary of the cloaking device. The rings with their split-ring structures are designed to perform an approxim ation of the Figure 11. Cloaking D evice for Microwaves14 coordinate transform ation shown and explained in the previous section. How is this possible? The split-ring resonators act like the atom s or m olecules of a norm al optical or electrom agnetic m aterial: they absorb electrom agnetic waves and re-em it them with a phase delay or advance that depends on their electrom agnetic response. Like atom s or m olecules, they are m uch sm aller than the electrom agnetic wavelength— 3 -m m cell size versus 3 -cm wavelength in the case of the m icrowave cloaking device15— such that the waves cannot resolve them individually but, rather, react to them as if they were a bulk m aterial with electrom agnetic properties that m ay differ from point to point. U nlike atom s or m olecules, the electrom agnetic response of each split-ring resonator is tailor- m ade because it depends on the shape of the resonator that can be easily m odified. For exam ple, in the case of the m icrowave-cloaking device,16 the electrom agnetic response depends on the capacitance that is varied by changing the length of the double stripes in the resonators. An unstructured circuit board reacts com pletely differently to the m icrowave radiation: it would sim ply reflect it like the m esh in the window of a m icrowave oven. A m aterial with electrom agnetic or optical properties that depends on structures m uch sm aller than the wavelength is called a m etam aterial. Metam aterials per se are nothing new; the ancient Rom ans invented the first optical m etam aterial: ruby glass. The Rom ans probably did not know it, but their recipe for ruby glass contained one crucial ingredient:17 tiny gold droplets, typically 5-60 nanom eters (nm ) in size. These gold particles color the glass in an extraordinary way, as dem onstrated by the exquisite Lycurgus Cup shown in Figure 12. In daylight, the cup appears a greenish color, but illum inate it from the inside, and it glows ruby. The gold particles act like the split-ring resonators of the m icrowave-cloaking device,18 but here on light, not on m icrowave radiation. Light consists of electrom agnetic waves as well, but with significantly sm aller wavelengths of around 500 nm . The gold particles are thus m uch sm aller than the wavelength of light, and they turn out to be resonators as well: in them , electric currents flow in a way that is dictated by their shapes and sizes. W hen the light wave hits the resonance of the gold particle, m ost of its energy is 10 UNCLASSI FIED //BO B O FFICIAL USE O MbY UNCLASSI FI ED //FO R O FFICIAL USE O NLY converted into the electrom agnetic oscillation of the gold particle in m uch the sam e way a tuning fork responds to sound of the right frequency. However, the electrom agnetic oscillation is dam ped out by the electric resistance in the m etal, and its energy is absorbed and ultim ately turned into heat. The color of light corresponds to the frequency or wavelength. If one of the frequencies is absorbed, the corresponding color is m issing in the spectrum . For the gold particles in the Lycurgus Cup, this color is green; a spectrum with green m issing appears red, which produces the cup's exquisite color. (Light is also scattered in the m aterial, and this scattering is enhanced near the green of the resonance; hence the greenish color of the cup seen in daylight.) W hat is new about m etam aterials is the degree of control on their structures achieved by applying m odern technology and the level of theoretical Figure 12. Lycurgus Cup (British Museum; fourth century AD ). This Rom an cup is m ade of ruby glass. W hen viewed in reflected light— for exam ple, in daylight— it appears green. However, when a light is shone into the cup and transm itted through the glass, it appears red. The cup illustrates the m yth of King Lycurgus. He is seen being dragged into the underworld by the Greek nym ph Am brosia, who is disguised as a vine. understanding of their workings. The Rom ans m ost probably never understood why ruby glass is neither golden like gold nor transparent like glass, its ingredients, but ruby. They did not know that light is an electrom agnetic wave, nor did they know the basic laws of electrom agnetism . And they would not have had the technological tools to use this knowledge in the design of novel optical m etam aterials. O ptical Metamaterials As light is sim ply an electrom agnetic wave with shorter wavelengths than m icrowave radiation, one could im agine an optical cloaking device as the m icrowave cloak but with m uch sm aller cells, fitted to the sm aller wavelength. However, this sim ple idea is too sim ple, for two different reasons. One is that m etals like the copper of the circuit board or the gold of ruby glass are m ore electrically resistant to currents oscillating with the frequency of visible light than to currents in the m icrowave range of the spectrum . Second, and m ore im portant, the cells of a m etam aterial also em it electrom agnetic radiation in an incoherent way, not just as a coherent response to the incom ing electrom agnetic wave, sim ilar to the spontaneous em ission of light by atom s and m olecules. The spontaneous em ission is significantly stronger in the optical range of the spectrum . In short, m etam aterials do not scale; they m ust be designed differently for visible light, and the loss of light by absorption and incoherent scattering usually is greater for visible light than for m icrowaves. Figure 13 below illustrates the idea19 for an optical cloaking m etam aterial. Instead of split-ring resonators, nano-scale m etal wires are em bedded in a transparent host m aterial, for exam ple glass. The wires replace the split-ring resonators on the circuit board of the m icrowave-cloaking device. They act sim ilarly to the gold particles em bedded in ruby glass; their optical properties 11 UNCLASSI FIED //FO R O FFTCTAI iiffomiv UNCLASSI FI ED //F8R O FFICIAL USE O NLY depend on their lengths and on their arrangem ent, which, in principle, can be tailor- m ade and controlled using the tools of m odern nanotechnology. The thin wires will have lower electric losses than split-ring resonators, and their radiation losses by the equivalent of spontaneous em ission are reduced as well. Such optical cloaking devices do not yet exist, but one can gauge the progress in the required technology by considering the progress in negatively refracting optical m aterials. Figure 13. Idea for a Cloaking D evice for Visible Light. Metal nanowires replace the split-ring resonators of the m icrowave cloaking device. Coordinate transform ation and structure of the optical cloak, a: The coordinate transform ation that com presses a cylindrical region r < b into a concentric cylindrical shell a < r < b. There is no variation along the vertical direction. The radii ri and r2 define the internal and external radius of a fraction of the cylindrical cloak, b: A sm all fraction of the cylindrical cloak. The wires are all perpendicular to the cylinder's inner and outer interfaces, but their spatial positions do not have to be periodic and can be random .20 Figure 14 below21 illustrates the route toward achieving negative refraction in the visible range of the spectrum . Losses typically are a greater problem for negatively 12 UNCLASSI FIED //MR O FFICIAL USE O NLY UNCLASSIFIED //FeR O FFICIAL UO E O NLY refractive m aterials than for the m etam aterials of cloaking devices. So the graph indicates the possible progress toward optical cloaking. 1000 1 pm 100 10 um& 100 M H 2 0.0b 10 cm 2000 2001 2002 2003 20062004 2005 1 cm 2007 Figure 14. Advances in Metamaterials. The solid sym bols denote m aterials with negative refraction; the open sym bols denote optical m aterials with negative m agnetic response. Orange: data from structures based on the double split-ring resonator (SRR); green: data from U -shaped SRRs; blue: data from pairs of m etallic nanorods; red: data from the "fishnet" structure. The four insets give pictures of fabricated structures in different frequency regions?2 W avelength On the other hand, cloaking devices require bulk m etam aterials with varying cell structures. The first m oderately bulk negatively refracting m aterials23 '24 were m ade only recently. The m ost severe practical problem of the currently discussed cloaking devices is not the technology for m anufacturing and structuring the required m etam aterials but a problem at the core of their principal design.25 13 UNCLASSI FIED //FO R O FFICIAL USS O NLY UNCLASSI FI ED //F8R O FFICIAL USE 8NL¥ C Lens 1 S am ple Lens 2 C am era Figure 15 . D emonstration of Negative Refraction With "Bulk" O ptical Metamaterials Made of Nano­ Fishnets26 Im ag e 14 U N CLASSI FI E D //FO P O P H CIAL UP C O NCT UNCLASSI FIED / / FO R O FFICIAL USE O NLY Figure 16. D emonstration of Negative Refraction With "Bulk" O ptical Metamaterials Made of Nanowires27 Fundamental P roblem The dem onstrated m icrowave-cloaking device only works correctly for m icrowave radiation of a specific frequency (wavelength), and the proposed optical cloaking device would also work for just one frequency; that is, for only one color. Light of different colors would be severely distorted. So, to see things disappear in a cloaking device, one should wear tinted glasses of the required color, which of course com pletely defeats the purpose. This design flaw is inevitable,28 no m atter how m uch progress is m ade in the technology of m etam aterials, for the following reason: the device is designed such that light waves traveling around the object enclosed by the cloaking device are com pletely indistinguishable from light waves propagating through em pty space. This is achieved by im plem enting the coordinate transform ation shown below in Figure 17.29 15 UNCLASSI FIED / / FO R O FriCML USE O NEY UNCLASSI FIED / / FUR O FFICIAL USE UNLT A Figure 17 . Fundamental P roblem of Transformation-Based Cloaking D evices’0 The device creates the illusion of the em pty virtual space A where light travels along straight lines, whereas in reality light rays are curved by the coordinate transform ation from virtual space to real space B. If the light waves are indistinguishable from light propagating through em pty space, the speed of light in the cloaking device m ust be larger than the speed of light in the surrounding m aterial— air, for instance— to m ake up for the longer path on the detour through the cloak. To m ake m atters worse, the speed of light m ust be infinitely large at the inner lining of the invisibility cloak. To understand this, consider a light ray that just straddles the red point in the virtual space shown in A. In real space, B, this point is enlarged to a finite volum e that contains the hidden core of the cloaking device. N ow, if for light propagation virtual space and real space are indistinguishable, the light ray should pass the extended path along the inner lining in precisely the tim e it takes to pass a single point, zero tim e. Consequently, the speed of light m ust approach infinity near the core of the cloaking device. The following argum ent shows that this is possible in principle, but also that such devices would be com pletely useless as a cloaking device in practice. In wave propagation, one distinguishes between the phase velocity and the group velocity. The phase velocity is the velocity at which the phase fronts of waves appear to m ove. For light, the wave fronts are the features of oscillations across space and tim e; by them selves they do not transport energy or inform ation. On the other hand, the phase fronts are orthogonal to the paths of light rays; if they are tilted, rays are refracted. Therefore, the refraction of light, the bending of light rays, is controlled by the phase velocity. The refractive index that enters Ferm at's principle of the shortest optical path is the phase index, the ratio between the speed of light in vacuum and the phase velocity in the m aterial. The group velocity is the speed at which wave packets, pulses, and m ost inform ation travels; it is the velocity of a wave group. Such a group consists of a range of single-frequency waves that, by their interference, establish the group, the wave packet, as Figure 18 below shows. 16 UNCLASSI FIED //FO R O FFICIAL UO E O H LY UNCLASSIFIED //rO R O FFICIAL 1190 O NLY Figure 18. Wave P ackets are Made by Combining Waves With D ifferent Frequencies. The picture shows the sim plest exam ple: two waves (A and B) that add up to the wave packet (C), Suppose that the phase velocity varies for different frequencies, what is called dispersion. In this case, the wave group m ade by the constructive interference of the single-frequency waves m oves at a different speed than the phase velocity: group and phase velocities differ. So, in dispersive m aterials, the phase velocity m ay approach infinity without violating the principles of relativity, but only for a single frequency, because otherwise the group velocity would tend to infinity as well. The cloaking of electrom agnetic waves of fixed frequency is possible, as the successful dem onstration of the m icrowave-cloaking device has confirm ed, but the cloaking of wave packets carrying inform ation is im possible. It turns out3 1 that the group velocity actually tends to zero at the inner lining of such cloaking devices; wave packets would get stuck there instead of traveling around. Turning invisibility from a tantalizing idea into a practical device requires a new paradigm .3 2 Curved Space Light rays are curved in m aterials with varying refractive index. In conventional cloaking devices, the rays are curved because the m aterial perform s a transform ation to curved coordinates. However, the curvature of a space does not depend on coordinates; curved coordinates create the illusion of curvature, but the space they describe is still flat. A flat space obeys the axiom s of E uclidean geom etry, in particular the parallel axiom : through each point outside out of straight line goes exactly one parallel line; parallels never m eet. The light rays focused by a lens clearly violate the parallel axiom , because parallel light rays m eet at the focus of the lens. Optical m aterials establish non-Euclidean geom etries in general; the E uclidian geom etries of cloaking devices are rather the exceptions. The advantage of E uclidean spaces is that one can easily visualize them ; curved space is difficult to com prehend, in particular three-dim ensional curved space. However, two-dim ensional curved spaces can be visualized as surfaces of three-dim ensional curved objects. These surfaces are the virtual spaces that are im plem ented, by the optical m aterial, in physical space. The sim plest exam ple is the sphere. On the surface of the sphere, the equivalent of straight lines, the geodesic lines, are the great circles. The great circles originating from one point m eet again at the antipodal points, which shows that the surface of the sphere establishes a non-Euclidean geom etry. To im plem ent this geom etry in the two- 17 UNCLASSI FIED //FO R O FFTCTA! IIFFO MIV UNCLASSI FI ED //FO A O FFICIAL USE UNL! dim ensional plane, one can use the stereographic projection, the central ingredient of the Mercator projection in cartography that is used to m ap the surface of a round object, the E arth, onto a flat sheet of paper. A line drawn from the N orth Pole of the sphere through a point on the surface intersects the equatorial plane at one point. This point is the stereographic projection of the point on the sphere. Figure 19 shows that the stereographic projection of a circle on the sphere is a circle on the plane with a different radius. The stereographic projection distorts the m easure of space, but the distortion around any given point is the sam e in all directions, because otherwise circles would be deform ed. Therefore, an optical m aterial that im plem ents the geom etry of the sphere via the stereographic projection m ust be isotropic. One can read off the required refractive index from the drawing as follows: in virtual space, on the sphere, light propagates at the speed of light in vacuum from a point to its infinitesim ally close neighbor; in physical space the distance between the two infinitesim al neighbors is m odified— the speed of light is changed by the refractive index that is given by the ratio between infinitesim al distances in virtual space and the corresponding distances in physical space. For the stereographic projection, the refractive index is sm aller than 1 for points on the northern hem isphere and larger than 1 on the southern hem isphere. This device is known as Maxwell's fish­ eye lens.3 3 In this lens, light follows the great circles, light goes around in circles, and light rays m eet at antipodal points; the fish-eye m akes a perfect lens (although a fairly near-sighted one). It is possible to extend these ideas to three-dim ensional curved spaces. For exam ple, the surface of the four-dim ensional sphere is a three-dim ensional curved space, and the device im plem enting this hyperspace object is just a three­ dim ensional fish-eye. Hyperspace is not out of this world; it can be built, and it turns out to be practically useful for invisibility.3 4 Figure 19. Stereographic P rojection 18 UNCLASSIFIED //rO R O FFICIAL USE O NLY UNCLASSIFIED //FO R O FFICIAL USE O NLY Broadband Invisibility To understand why non-Euclidean geom etry com es to the rescue of invisibility, consider the following two-dim ensional exam ple.3 5 Im agine a virtual space m ade of a flat space, a sheet of paper, and a curved space, the surface of a sphere. The two spaces touch at one line. Consider the fate of light rays in this two-dim ensional virtual world. Light rays would either pass the sphere or enter, through the connecting line, the surface of the sphere, whereupon, after one loop, they would continue in the sam e direction as they entered, as if the tour on the sphere had never happened. The sphere is invisible; it shows only as a tim e delay of the light ray. Although the sphere is invisible, it does not m ake som ething else invisible yet. However, this is easily arranged. Im agine a m irror around the equator of the sphere. The ray bounces off at the m irror, but, after another bounce, is back on track. A m irror in this curved space reflects light back to itself! The m irror creates the illusion that the light perform s a full great circle, whereas in reality it stays on one hem isphere. The other hem isphere is hidden. Alternatively, som e lines on the sphere are never crossed by light rays. Such lines can be opened like an eye; the space they enclose is hidden from sight. Figure 20. Non-Euclidean Cloaking D evice in Two D imensions. The device creates the illusion shown in A: light propagates through a virtual space that consists of a plane and the surface of a sphere, a curved space, which touch along a line. Som e incident light rays venture from the plane to the sphere; they return after one loop and continue in the sam e direction. N ote that the rays never cross the red zigzag line on the sphere. Plane and sphere carry a coordinate grid that is m apped onto physical space B. The m agenta circle defines the boundary of the device. Its interior has been expanded to m ake space for the grid of the sphere. In particular, the line where plane and sphere touch has been opened like an eye (thick black lines) to include the sphere. This is not a cloaking device yet, but one could place a m irror around the equator of the virtual sphere C, m aking the northern 19 UNCLASSI FIED //BO R O FFICIAL UO E O Nt* UNCLASSIFIED //F8R O FFICIAL USE O ML¥ hem isphere invisible and creating the sam e illusion as shown in A. Alternatively (D), one could expand the red line that light never crosses to create a hidden space. W hy are such curved optical spaces of any practical advantage? They seem m ore com plicated, but the distortion of space in such spaces is always finite, never infinite as in the conventional E uclidean cloaking devices. As the spatial distortions directly correspond to the required refractive indices, the required optical properties are never infinite, and hence such devices can, in principle, operate in a broad band of the spectrum . Curved space is m ore practical than flat space, although the theory is m ore com plicated. These ideas can be extended from the two-dim ensional toy m odel to the three-dim ensional world, but they can no longer be visualized. Figure 21 below shows som e ray trajectories in three-dim ensional non-E uclidean cloaking devices. Figure 21. Three-D imensional Cloaking. One can extrapolate the ideas illustrated in the previous figure to three-dim ensional space, replacing the plane by flat space and the sphere by a hypersphere. The lentil-shaped object indicates the hidden interior of the device; the partly shaded grid, the boundary of the invisibility device. For better contrast, light rays are shown in red. A: Rays are bent around the invisible region. B: In three dim ensions, som e rays turn out to perform two loops in hyperspace that appear in physical space as light wrapped around the invisible interior. Such non-E uclidean cloaking devices are im perfect because they delay the light traveling through the cloak. W ith sensitive tim ing or wave-front sensing one could, in principle, detect the presence of the cloaking device. Perfect cloaking is im possible, but as long as tim e delays and wave-front dislocations are of no concern, invisibility could becom e reality. Implementation N on-E uclidean cloaking devices do not have an obvious sym m etry like the E uclidean m icrowave-cloaking device.3 6 They require m aterials with an electrom agnetic response that varies from cell to cell and is anisotropic. Most probably, such cloaking structures can be m ade for m icrowaves. A precursor of the necessary technology is the recently dem onstrated ground-plate cloak.3 7 This device im plem ents the coordinate transform ation shown below (that already appeared in the first paper on cloaking by 20 UNCLASSI FIED //FO R O FFICIAL UG C O Nfe¥ UNCLASSIFIED //F8R O FFICIAL USE O NLY coordinate transform ations). This is not a cloaking device: it com presses an extended region of space to a reflecting plate; the interior is hidden behind the plate, but the plate is clearly visible. Such a device conceals the extension and shape of the hidden region but not the fact of hiding itself. B Figure 22. Coordinate Transformation Implemented by a G round-P late Cloak In order to im plem ent the ground-plate cloak, thousands of cells with split-ring resonators with individual, tailor-m ade electrom agnetic properties were designed, as Figure 23 shows. a{m m | Figure 23. Implementation of the G round-P late Cloak38 21 UNCLASSI FIE D //6O B O FFICIAL USE O NLY UNCLASSIFIED //«JR QFH MAL USE O NI Y O ptical Cloaking Cloaking in the optical range of the spectrum poses several challenges. The present design of non-E uclidean cloaking devices still requires m aterials where, in som e parts of the device, the speed of light is larger than in the environm ent of the device, which, in practice, m eans larger than the speed of light in vacuum . Most probably, this problem can be circum vented by inventing new designs and new geom etrical form s of suitable curved spaces, because there is no m athem atical reason why non-E uclidean cloaking should be lim ited in this way. However, solving this problem takes im agination and m athem atical creativity; it cannot be planned by a clear roadm ap, but it can be encouraged and stim ulated. It could take 1 or 2 years or a m uch longer tim e until such designs are invented; truly im aginative research is unpredictable. This research takes a specific m indset, clear m athem atical thinking com bined with playfulness and physical intuition, a stim ulating environm ent, and freedom . The greatest challenge for turning invisibility from an idea into a workable device is not technology but im agination. The only way to solve this problem is to follow the Solom onic advice to invest in the right people. The technology for cloaking will depend on the design of such advanced cloaking devices. Probably they will require highly anisotropic m aterials, but perhaps liquid crystals could be sufficient. Maybe m etam aterials are not needed after all. In this case, invisibility could becom e a feasible technology within a generation. If optical m etam aterials are needed, they will rely on structuring on extrem ely short scales, possibly on sub-nanom eter distances. The technology for m aking such structures will be developed because the silicon-electronics industry will need them ; but whether large- scale devices with sub-nanom eter structures can be m ade rem ains to be seen. Another practical challenge is im pedance m anaging. Ideal cloaking devices require m aterials with equal electric and m agnetic response because they im plem ent geom etries and geom etries are universal— they act on both the electric and the m agnetic fields of electrom agnetic waves like light. In practice, broadband optical m aterials m ostly respond to the electric field but not to the m agnetic one. Optical m agnetism has been dem onstrated with m etam aterials,3 9 but only in narrow regions of the spectrum . If the electric response differs from the m agnetic response, the electrom agnetic im pedance is m ism atched, which results in reflections. One could reduce such reflections by using sm ooth refractive-index profiles as appropriate antireflection coatings. Most probably, cloaking devices will be rigid shells; to m ake them flexible like wearable invisibility cloaks poses a significant challenge. The reason is that their optical properties m ust be adjusted to their geom etrical shapes, as the refractive-index profile of a cloaking device depends on its shape. If the shape changes, the index-profile m ust follow suit. The required optical properties should be calculated in real tim e, and the m aterial should change accordingly. Liquid crystals could adjust their optical properties, but controlling a large, com plicated array of liquid crystals with possibly several layers appears to be difficult, despite the progress m ade in liquid-crystal displays. 22 UNCLASSI FIED / / FO R O FFICIAL USE O NLY UNCLASSI FI ED //FO R O FFICIAL UO C O NL¥ Summary A cloaking device is a passive device m ade of a transparent m aterial that guides light around any object in its interior as if the light has passed through em pty space. 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