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A spacetime cloak, or a history editor MartinW McCall1, Alberto Favaro1, PaulKinsler1 and Allan Boardman


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A spacetime cloak, or a history editor …

A spacetime cloak, or a history editor
MartinW McCall1, Alberto Favaro1, PaulKinsler1
and Allan Boardman

We introduce a new type of electromagnetic cloak, the spacetime cloak (STC), which conceals
events rather than objects. Non-emitting events occurring during a restricted period are never
suspected by a distant observer. The cloak works by locally manipulating the speed of light of
an initially uniform light distribution, whilst the light rays themselves always follow straight
paths. Any ‘perfect’ spacetime cloak would necessarily rely upon the technology of
electromagnetic metamaterials, which has already been shown to be capable of deforming light
in ways hitherto unforeseen—to produce, for example, an electromagnetic object cloak.
Nevertheless, we show how it is possible to use intensity-dependent refractive indices to
construct an approximate STC, an implementation that would enable the distinct signature of
successful event cloaking to be observed. Potential demonstrations include systems that
apparently violate quantum statistics, ‘interrupt-without-interrupt’ computation on convergent
data channels and the illusion of a Star Trek transporter.

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  • 1. Home Search Collections Journals About Contact us My IOPscience A spacetime cloak, or a history editor This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2011 J. Opt. 13 024003 ( View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: The article was downloaded on 16/07/2011 at 00:56 Please note that terms and conditions apply.
  • 2. IOP PUBLISHING JOURNAL OF OPTICSJ. Opt. 13 (2011) 029501 (1pp) doi:10.1088/2040-8978/13/2/029501CorrigendumA spacetime cloak, or a history editorMartin W McCall et al 2011 J. Opt. 13 024003Received 14 September 2010Published 16 November 2010An author was omitted from the author list in reference [7].The correct reference is: McCall M W and Censor D 2007Relativity and mathematical tools: waves moving media Am.J. Phys. 75 1134–40.2040-8978/11/029501+01$33.00 1 © 2011 IOP Publishing Ltd Printed in the UK & the USA
  • 3. IOP PUBLISHING JOURNAL OF OPTICSJ. Opt. 13 (2011) 024003 (9pp) doi:10.1088/2040-8978/13/2/024003A spacetime cloak, or a history editorMartin W McCall1 , Alberto Favaro1 , Paul Kinsler1and Allan Boardman21 Department of Physics, Imperial College London, Prince Consort Road, London SW7 2AZ,UK2 Photonics and Nonlinear Science Group, Joule Laboratory, Department of Physics,University of Salford, Salford M5 4WT, UKE-mail: 14 September 2010, accepted for publication 22 October 2010Published 16 November 2010Online at introduce a new type of electromagnetic cloak, the spacetime cloak (STC), which concealsevents rather than objects. Non-emitting events occurring during a restricted period are neversuspected by a distant observer. The cloak works by locally manipulating the speed of light ofan initially uniform light distribution, whilst the light rays themselves always follow straightpaths. Any ‘perfect’ spacetime cloak would necessarily rely upon the technology ofelectromagnetic metamaterials, which has already been shown to be capable of deforming lightin ways hitherto unforeseen—to produce, for example, an electromagnetic object cloak.Nevertheless, we show how it is possible to use intensity-dependent refractive indices toconstruct an approximate STC, an implementation that would enable the distinct signature ofsuccessful event cloaking to be observed. Potential demonstrations include systems thatapparently violate quantum statistics, ‘interrupt-without-interrupt’ computation on convergentdata channels and the illusion of a Star Trek transporter.Keywords: electromagnetic cloaking, metamaterials, electromagnetic modulation, spacetime,observers and events(Some figures in this article are in colour only in the electronic version)1. Introduction spatial ‘object’ cloaks, which coax electromagnetic radiation around a spatial void invisible to outside observers [2–4].Would it not be amazing if, in an age of ubiquitous These latter developments are extensions of our historical andsurveillance, a method could be devised to somehow remove pre-historical abilities—that light can bend whilst traversinga piece of history that would otherwise be recorded by transparent materials has been known at least since humansa surveying camera? A safe-cracker would be able, for went fishing with spears; even the controlled bending ofa brief time, to enter a scene, open the safe, remove its light in prisms and lenses has been known for centuries.contents, close the door and exit the scene, whilst the record In contrast, the STC utilizes time-dependent nanometre-of a surveillance camera apparently showed that the safe engineered electromagnetic metamaterials to create a temporaldoor was closed all the time. Although this sounds like void within which events can be hidden. Remarkably, ourscience fiction, the lesson from metamaterials research in spacetime ‘event’ cloak leaves light rays undeviated fromthe last decade has taught us that, within certain restrictions, their expected linear spatial trajectories, and instead workssuch speculations are not fantasy. We here show how the by dividing illuminating light into a leading part, which ismagic of editing history can be achieved by introducing the speeded up and passes before a collection of events occur, andconcept of the spacetime cloak (STC). The proposal opens a trailing part, which is slowed down and passes after theyup a new paradigm for electromagnetic cloaking and has have occurred. This process is thus utterly distinct from amajor implications for research in metamaterials, slow light spatial ‘object’ cloak, which instead achieves concealment byand phase modulation. Metamaterials already bend light in bending light around an object. The STC opens a temporaryunconventional ways, producing aberration-free lenses [1] or corridor through which energy, information and matter can be2040-8978/11/024003+09$33.00 1 © 2011 IOP Publishing Ltd Printed in the UK & the USA
  • 4. J. Opt. 13 (2011) 024003 M W McCall et alFigure 1. (a) Conventional spatial cloak based on a planar transformation (x, y) → (x , y ). An observer located to the far right on the x axisdoes not see the object. (b) Spacetime cloak. An analogous coordinate transformation is used, but now in (x, t) rather than (x, y). The cloaknow conceals events near the spacetime origin. The (schematic) intensity distribution for various times is shown on the right, indicating theformation and subsequent evaporation of the intensity null that is fully developed at t = 0. The observer to the right never suspects theoccurrence of any non-radiating events near the spacetime origin and sees a uniform intensity for all time.manipulated or transported undetected. Once the concealed lines to curves, transforms the electromagnetic field so that inpassage has been used, the STC closes by slowing the leading the new systempart of the light, whilst speeding up the trailing part, leaving no ∂Btrace of the cloak, or the concealed events, on the field profile. ∇ ×E =− , ∇ · D = 0, (1) ∂tTo a distant observer, any non-emitting object whose presencepersisted throughout the lifetime of the STC will have had a ∂D ∇ ×H = , ∇ · B = 0. (2)finite interval excised from its history. Any object which emits ∂tlight during STC operation will have that part of its history Since the form of Maxwell’s equations is preserved, thetemporally compressed so that, if not absorbed, it will appear to vacuum relations D = 0 E, B = μ0 H are replaced inthe observer as if occurring in a single instant. We here outline the new coordinate system by appropriate inhomogeneous,the theory and design of the spacetime cloak and describe anisotropic relations D = (r ) · E , B = μ (r ) ·potential applications: hiding events from some observers but H . If instead of transforming coordinates, the originalnot others, using the ‘time gap’ to generate apparent violations coordinate space (r) is filled with a medium (r), μ(r),of quantum statistics, allowing the covert interruption of non- having the same inhomogeneous, anisotropic (and, strictlyinterruptible computation and/or signal processing operations, speaking, instantaneously responding) properties prescribed byand creating the illusion of a Star Trek transporter. (r ), μ (r ), then the linear light paths of vacuum become curved, just as straight lines become curved under r → r . Figure 1(a) shows how a spatial electromagnetic cloak [2] is2. The spacetime cloak concept produced via a time-invariant planar transformation (x, y) → (x , y ) that diverts light rays so as to create a void near theOur method for redacting history is based on applying origin. The polar transformation r = (1 − a/b)r + a , θ = θ ,the principles of transformation optics to produce an z = z distorts linear rays travelling in vacuum from left to rightelectromagnetic cloak working in both space and time, rather to those shown. Equivalently, rays in an anisotropic dielectricthan in just space. Transformation optics relies on mimicking with polar parameters ( , μ)r = (1 − a/r )( r , μr )0 , ( , μ)θ =a coordinate transformation applied to Maxwell’s vacuum (1 −a/r )−1 ( θ , μθ )0 , ( , μ)z = (1 −a/b)−2 (1 −a/r )( z , μz )0equations by means of an inhomogeneous dielectric/magnetic are curved in exactly the same way to produce a cloak aroundmedium. A coordinate transformation r → r , taking straight a cylindrical object of radius a , centred at the origin [4]. 2
  • 5. J. Opt. 13 (2011) 024003 M W McCall et al ct (a) 4 2 0 x -2 -4 -4 -2 0 2 4 (b) 4 U 2 0 4 2 2 0 0 ct -2 -2 x -4Figure 2. (a) Construction of a sub-luminal spacetime cloak. As with figure 1(b) the map (x, t) → (x t ) creates a void near the spacetimeorigin. However, the base space is filled with a medium of refractive index n rather than vacuum, so that prior to the transformation light rays ¯ ¯are straight lines of gradient n . The transformation is a composition of a Lorentz boost, (x, t) → (x, t ), with velocity v = c/n , followed by ¯applying a ‘curtain map’ x = [( δ+|ct | )(x − sgn(x)σ ) + sgn(x)σ ], t = t , followed by an inverse Lorentz transformation, (x , t ) → (x , t ). ¯ δ+nσ ¯ ¯ ¯ ¯ ¯ ¯ ¯(See the appendix for further details.) Shown is the coordinate transformation (x, t) → (x , t ) with σ = 1, n = 2, δ = 0.5 for the curtainmap. The deformed rays all have positive gradients (i.e. propagate forwards in time) and have speed c. The coordinate transformation(x, t) → (x , t ) defines the spacetime transformation that yields equations (1) and (2) with t = t . Alternatively, a material with the required (x, t), μ(x, t), β(x, t), enables the spacetime cloak to be realized in an actual medium. (b) Electromagnetic energy density for various timesfor the map of (a). The hard edges of the curtain map defined above have been softened by the use of mollifier functions (see the appendix),thus avoiding unrealistic instantaneous switching. The intensity is null at events near the spacetime origin. For our spacetime cloak we consider spacetime transfor- light trajectories are actualized when they propagate throughmations (x, t) → (x , t ) and restrict to light propagating a suitable inhomogeneous time-dependent medium, and willforwards along the x axis. In figure 1(b), for example, a then curve around the event occurring at x = 0, t = 0. Asspacetime transformation is carried out in the (x, t) plane that discussed in the appendix, the form of Maxwell’s equationsis analogous to the spatial transformation carried out in the (equations (1) and (2), but now with the addition that t → t )(x, y) plane of figure 1(a). In contrast to the spatial cloak, is again preserved provided the equivalent medium is magneto-where the direction of propagation in the x – y plane is arbitrary, electric with the transverse field components obeyingthe vacuum light rays must follow the straight lines x = ct +const. These rays are then mapped under the transformationto the curved rays shown so that the events within the disc Dy Ey Hz = (x, t) + β(x, t) (3)surrounding the origin are avoided by the new rays. The new Dz Ez −H y 3
  • 6. J. Opt. 13 (2011) 024003 M W McCall et al By −E z Hy = β(x, t) + μ(x, t) . (4) Bz Ey Hz sd 2d sd FastThe medium is thus in general required to be magneto-electric. SlowIn contrast to the spatial cloak, where light curves in the (x, y)plane, ‘curving’ light in the (x, t) plane now refers to different compressed cloparts of the light paths speeding up and slowing down. The aklight proceeds only along the x axis and does not curve in edspace. As a consequence, events in the neighbourhood of theevent at (x, t) = (0, 0) are never suspected by an observerlocated sufficiently far to the right. In fact, all events in shifted Cloakedthe y –z plane that occur near (x, t) = (0, 0) are similarly Uncloaked Rays Raysundetected. The concealment results from various parts of thelight distribution slowing down and speeding up so as to avoid tevents near the spacetime origin. The intensity along the xaxis for different times is shown on the right of figure 1(b),indicating the formation of the intensity null that is fully stretcheddeveloped at t = 0. For later times the null closes up, restoringa uniform intensity distribution so that the observer at x = a ed akrecords a constant intensity for all time. Since the events near clo x(x = 0, t = 0) are never illuminated, the observer to the rightis unaware of their existence. In fact, all events in the (y, z)plane that occur near (x = 0, t = 0) are concealed, so thateffectively a spacetime corridor is opened along which non- Fastradiating events, such as the movement of matter, exchange ofinformation, etc, can occur undetected. The spacetime cloak of figure 1(b) is symmetric, in Slowthat uniform illumination from the right will avoid the samecloaked events for an observer located sufficiently far to theleft. As conceived by the symmetric transformation in thex –t plane of figure 1(b), some rays in the medium will be Figure 3. Spacetime cloak based on refractive index switching. (a)required to have a phase speed exceeding the vacuum speed The base space is a medium of refractive index n with an innerof light (e.g. ray A in the inset) and some rays very close region of width 2d surrounded by two outer regions of extent sdto the cloaked region (e.g. ray B) propagate backwards in identified. For −(1 + s)d x < 0 the transformation is given bytime. Superluminal ray trajectories, which occur in purely x = (s + 2)(s + 1)−1 x + d for t nx and by x = s(s + 1)−1 x − d for t > nx . For 0 x < (s + 1)d the transformation is given byspatial cloaks and in plasma propagation above the plasma x = s(s + 1)−1 x + d for t nx and by x = (s + 2)(s + 1)−1 x − dfrequency, are characterized by the phase velocity exceeding for t > nx . All other regions are unaffected. All rays are sub-luminalthe vacuum speed of light, and are known to be compatible provided n(1 + 1)(s + 2)−1 < 1. Unlike the cloak of figure 2(a), awith special relativity [5]. Waves travelling backwards in time single cloaking operation does not return the system to its originalare precisely the interpretation of waves undergoing negative state. Alternating cloak operations as shown permits periodic restoration of the medium to its original state by interleaving a cloakphase velocity propagation, or negative refraction, where the operating in the reverse direction.reversal of time in the wave’s phase φ = k · r −ωt is equivalentto the reversal of the direction of the wavevector k with respectto electromagnetic flux [6]. However, both superluminal and all rays in figure 2(a) travel slower than the vacuum speed ofnegative phase waves, though accessible through appropriate light. The plots of figure 2(b) show a detailed calculation ofmetamaterials design, can, through similar design ingenuity, the electromagnetic energy density in the equivalent mediumboth be avoided in the spacetime cloak, as we discuss next. defined by the transformation, where the hard edges of the map have been mollified so that the medium is not switched3. Simplified designs instantaneously. The energy null which develops as the cloak is switched on becomes zero at events near the spacetime origin.Referring to figure 2, the coordinate transformation is now Unlike the cloak of figure 1(b), this spacetime cloak onlycarried out against a background of a uniform medium of works for light travelling in the +x direction. An observerrefractive index n > 1, rather than vacuum. The complete to the left of the origin does see the cloaked events, thoughtransformation consists of a Lorentz transformation into a time separations between the cloaked events are first speededframe in which the rays are vertical, opening the void via up, then time-shifted, and then finally slowed down, beforea ‘curtain map’, followed finally by an inverse Lorentz progression returns to normal.transformation back into the medium rest frame (see the The magneto-electric parameter, β(x, t), in equations (3)appendix for further details). The construction is such that and (4) arises almost invariably whenever the transformation 4
  • 7. J. Opt. 13 (2011) 024003 M W McCall et al O (a) (b) 5 with cloak C Fluoresence (a.u.) Occasional 4 without cloak priority bits cloak region 3 A B B 2 1 ... ... Primary computation 0 0 5 10 15 20 D time (ns) (c) B A B O AFigure 4. Realization of the spacetime cloak as a set of addressable planes with metallic inclusions. Applications based on how the centralregion is filled as the intensity null passes over: (a) excited atoms. The exponential fluorescence decay is observed with the emissions duringcloak operation all apparently occurring simultaneously, resulting in an intensity spike, followed by resumption of the exponential decay.(b) ‘Interrupt-without-interrupt’. An occasional signal (channel AB) and a primary computation (channel CD) converge at the same physicalnode. Suppose it is required to process the occasional signal, without interrupting the primary computation. The cloak operation opens atemporal window via which the occasional signal can be processed, whilst the primary computation is reconstituted after the cloak operationto appear as if uninterrupted. (c) A Star Trek transporter. Movement of non-radiating matter whilst the cloak is operating appears to theobserver O to have moved from A to B instantaneously.(x, t) → (x , t ) mixes space and time. This occurs, for and positive for x > 0. The trailing and leading parts ofexample, in a medium of refractive index n moving with the light are thus respectively slowed down and speeded upvelocity v wherein the constitutive relations revert to the to produce the required intermediate null. Non-reciprocal bi-Minkowski form [7], the magneto-electric parameter then anisotropic metamaterials that mimic the constitutive relationsbeing given by for a moving medium have been studied by Tretyakov [8], who has shown that such media can in principle be constructed from v(x, t) n2 − 1 small magnetized ferrite spheres combined with planar-chiral β(x, t) = . (5) c 1 − n 2 v 2 (x, t)/c2 metallic inclusions [9]. A spacetime cloak with a transformation of the form t =The spacetime cloak therefore involves the construction of t , x = x (x, t) is illustrated in figure 3 which only requiresa metamaterial that mimics propagation in a medium with manipulation of (x, t) and μ(x, t). The transformationa velocity v(x, t). The occurrence of the null can then is devised such that the spatio–temporal variation of thesebe understood via the velocity addition formula v (x, t) = parameters matches, so that the impedance is uniform. The[v(x, t)+c/n]/[1+v(x, t)/(cn)], where v (x, t) is the velocity refractive index, however, is switched from high to low beforeof light in the moving medium. Near t = 0, the effective the central region, and from low to high after the centralmedium is arranged so that v(x, t) is negative for x < 0 region. The price of this simplified design is that the index 5
  • 8. J. Opt. 13 (2011) 024003 M W McCall et alFigure 5. Spacetime cloak based on exploiting the nonlinear refractive index properties of optical fibres. The cloak is opened in fibre A, isheld open in fibre B (which contains the core cloak region) and is closed again in fibre C. Two control fields IA and IC modulate the refractiveindex in A (C) from low to high (high to low). The much weaker illumination field experiences the index changes required for the cloakillustrated in figure 3. A signal field injected directly into fibre B is present before, during and after the cloaking from the illumination field.By monitoring the signal as it is coupled out, the spacetime modulation on it from the illumination and the effect of nonlinear refractive indexcan be measured. Changes to the signal field during the cloaking period leave no signature on the illumination field. Timing andsynchronization is only critical in ensuring that the top right apex of the cloaking diamond (figure 5) aligns with the bottom-left corner at anangle given by the average phase speed of the signals. Fibre parameters: silica refractive index ∼1.5; control field intensity set to yield anindex change n = n 2 I ∼ 5 × 10−4 .changes persist in the switched regions, so that, unlike the the cloaked period will all appear to happen simultaneouslyprevious designs, the medium surrounding the spatial origin to the distant observer to the right. The observed spikeis not returned to its original state after the cloak has operated. could be a useful experimental signature, since the requiredPeriodic switching, as shown in figure 3, returns the medium to cloaking period need only be comparable to the spontaneousits original state every other cycle and, moreover, permits bi- lifetime (∼ns), corresponding to a dimension of ∼30 cmdirectionality by alternating the forward cloak with an identical for the cloaked central region. Measurements of theone for light travelling in the −x direction. If the index coherence and statistics of light signals emitted by suchis switched without any impedance matching (as was done a cloaked source will also yield anomalous results, evenfor the first demonstrations of the spatial cloak [4]), then those of a quantum origin such as photon bunching or anti-reflections will occur at the interfaces. The observer to the bunching. Figure 4(b) proposes a specific application inright would consequently notice abrupt changes of the overall signal processing where two channels, proceeding along thefield intensity, but still remain ignorant of the events occurring x and y directions respectively, cross at x = 0. Throughwithin the cloaked period. the operation of the spacetime cloak occasional signals along the y channel can be processed as a priority, whilst4. Demonstrations processing and computation proceeds seamlessly, without interruption, along the x channel, achieving effectively anWhat effects might be observable using the spacetime cloak? ‘interrupt-without-interrupt’. Finally, the spacetime cloakFigure 4 shows a collage of possibilities based around what can achieve the illusion of a matter transporter (figure 4(c)),happens in the y –z plane in the region near x = 0 where/when in which an object moves from (y1 , z 1 ) to (y2 , z 2 ) duringthe cloak operates. Filling this region with excited atoms the cloak’s operation. The observer to the right sees the(figure 4(a)) will cause the standard exponential fluorescence object disappear at (y1 , z 1 ) to then instantaneously reappeardecay to be modified by the cloak, as emissions within at (y2 , z 2 ). 6
  • 9. J. Opt. 13 (2011) 024003 M W McCall et al It is emphasized that all the above demonstrations only optics algorithm in one space and one time dimension. Therequire light propagation in one dimension since producing a result is a cloak that operates in a fundamentally distinct waytemporary null in a collimated wave is all that is necessary to to the purely spatial, or object, cloak, in that it concealsconceal events near x = 0 in the y –z plane for a limited period. events by curving light rays in spacetime rather than in space.However, it is clear that the spacetime cloak concept can be Although this sets new challenges for metamaterials design, weextended to more spatial dimensions where the illuminating have shown how these challenges can be minimized throughlight is not collimated. A spherical wave from a point source, judicious manipulation of the refractive index of the meta-for example, illuminates a set of events lying on a sphere medium. We have proposed a proof-of-principle design basedat radius R from the origin. A spacetime cloak concealing on nonlinear fibre optics that could demonstrate the signaturethese events will do so for distant observers positioned at any of spacetime cloaking, and how the concept could find practicalviewing angle. applications in signal processing. We are sure that there are many other possibilities that are5. Implementation opened up by our introduction of the concept of the spacetime cloak. Whilst the camera is said to never lie, it does not alwaysHow practical is it to build a spacetime cloak? As with record the whole truth.the original proposals for a spatial cloak, we have neglecteddispersion, whose effect in this case will be to constrain therate at which the cloak can be opened and closed. Figure 5 Acknowledgmentsshows a scheme to implement the design of figure 3 usingnonlinear optical fibres. The cloak is opened in fibre A, is Support from EPSRC grant no. EP/E031463/1 and aheld open in fibre B (which contains the core cloak region) and Leverhulme Embedding Emerging Disciplines award isis closed again in fibre C. The primary operation is achieved acknowledged. One of us (MWM) acknowledges a usefulby means of two control fields whose intensity allows the discussion with Professor David Websdale.refractive index in A (C) to be modulated from low (high)to high (low). This is achieved using the nonlinear refractiveindex properties of the fibre (e.g. silica). The illumination field Appendixis affected by the refractive index shifts (i.e. by the cloak), butshould have an intensity modulation far weaker than that used A.1. Calculating the material parameters for the spacetimeby the control fields. The cloak opens up a spacetime gap in cloakthe intensity profile of the illumination field. This gap could be Calculating the required spatial and temporal materialdetected by monitoring leaky spot(s) on fibre B, and looking properties of the spacetime cloak, and the resultant distributionfor the intensity dip(s) in the illumination. Alternatively, of electromagnetic energy, is most easily carried outa signal field could be injected directly into fibre B, and using a covariant approach. We take Cartesian–Lorentzwould therefore always be present, i.e. before, during and afterthe cloaking from the illumination field. By monitoring the coordinates indexed as α = 0–3, i.e. {x α } = (ct, x, y, z).signal as it is coupled out, the spacetime modulation on it The source-free Maxwell equations in a linear mediumfrom the illumination and the effect of nonlinear refractive responding instantaneously to electromagnetic excitation mayindex could be measured. For a brief period, the presence be expressed asof the signal field will be undetectable to the surveilling ∂illumination field and changes to the signal field during the (χ αβμν Fμν ) = 0, (6)cloaking period would leave no signature on the illumination ∂xβfield. The group velocity of the on/off transitions must be no where Fμν is the electromagnetic field tensor given by (in unitsslower than the slow phase velocity and no faster than the fast where c = 1)phase velocity. This ensures that the control/cloaking index ⎛ ⎞transitions always remain inside the cloak. Good dispersion 0 −E x −E y −E zcontrol would be required (probably using photonic crystal ⎜E 0 Bz −B y ⎟ Fμν = ⎜ x ⎝ E y −Bz ⎟, (7)fibres): If the various fields all have distinct frequencies or 0 Bx ⎠polarizations then they can be coupled in and out efficiently Ez By −Bx 0and distinctly. The fibre parameters illustrated in figure 5are optimistic but accessible (though harmonic generation The fourth-rank object χ αβμν relates Fμν to G αβ according toand Raman and Brillouin scattering would also need to beconsidered). Similarly, all coupling is assumed to be perfect, G αβ = χ αβμν Fμν , (8)although slight leakage of control field A into fibres B and Cshould be tolerable. where ⎛ ⎞6. Conclusion 0 Dx Dy Dz ⎜−Dx 0 Hz −H y ⎟ G αβ ⎜ =⎝ ⎟, (9)We have introduced a fundamentally new concept of −D y −Hz 0 Hx ⎠electromagnetic cloaking that exploits the transformation −Dz Hy −Hx 0 7
  • 10. J. Opt. 13 (2011) 024003 M W McCall et al ct ct 4 4 3 3 (a) (b) 2 2 nσ 1 1 0 x 0 x -1 -1 -2 -2 -3 -3 -4 -4 -4 -2 0 2 4 -4 -2 0 2 4 ct′ ct′ 4 4 3 3 2 (d) 2 (c) 1 1 0 x 0 x -1 -1 -2 -2 -3 -3 -4 -4 -4 -2 0 2 4 -4 -2 0 2 4 Figure 6. The detailed construction of the curtain map.i.e. G αβ contains the fields D and H. The fourth-rank object χ 0 2 0 2 = |L α |−1 (− L 0 L 0 + μ−1 L 0 L 0 ) α 0 0 1 1χ αβμν contains the electromagnetic material parameters. In = χ0 3 0 3 ≡ − ⊥ (13)non-covariant notation we have χ = |L α |−1 μ−1 ≡ μ−1 2323 α || (14) D − α −E = , (10) χ 3131 = |L α |−1 (− L 1 L 1 +μ −1 L1 L1 ) H −α T μ−1 B α 0 0 1 1 = χ 1 2 1 2 ≡ μ−1⊥ (15)where T indicates transpose. In fact, χ αβμν is a fourth-rank χ 0 2 1 2 = |L α |−1 (− L 0 L 1 α 0 0 + −1 μ L0 1 L1 1 )tensor density and transforms according to [10] = −χ 0331 =χ 1202 = −χ 3103 ≡ α. (16) β χ α β μ ν = |det(L λ )|−1 L α L β L μ L ν χ αβμν , λ α μ ν (11) The material relations (10) can thus be expressed alternatively aswhere L α = ∂ x α /∂ x α . α Dx = || E x , Bx = μ|| Hx , (17) For a transformation restricted to the (x 0 , x 1 ) plane and Dy Ey Hza base constitutive tensor for which the non-zero elements are =( ⊥ + α 2 μ⊥ ) + αμ⊥ , (18) Dz Ez −H yχ 0101 = χ 0202 = χ 0303 = − ; χ 2323 = χ 3131 = χ 1212 = μ−1we find the non-zero elements of χ α β μ ν to be By −E z Hy = αμ⊥ + μ⊥ . (19)χ 0101 = |L α |−1 (L 0 L1 L0 L1 − L0 L1 L0 L1 Bz Ey Hz α 0 1 0 1 0 1 1 0 Equations (18) and (19) are seen to be of the same form as + L 0 L 1 L 0 L 1 − L 0 L 1 L 0 L 1 )(− ) ≡ − 1 0 1 0 1 0 0 1 || (12) equations (3) and (4). 8
  • 11. J. Opt. 13 (2011) 024003 M W McCall et alA.2. The curtain map in detail The mollified curtain map of equations (24) and (25) is continuous and differentiable at the origin. Points outsideThe curtain map of figure 2 consists of successive spacetime ¯ the rectangle (|x| > σ, |ct | > nσ ) are left unchanged as ¯coordinate transformations (x, t) → (x, t¯) → (x , t ) → ¯ ¯ ¯ ¯out = t¯). To avoid discontinuities at before (i.e. x out = x, t ¯ ¯(x , t ) as shown in figure 6. Light propagating in a uniform the boundary of the box, the maps inside the rectangle andmedium of refractive index n is represented by the straight outside the rectangle are graded into each other by means ofsub-luminal world lines of figure 6(a). The transition from the mollified rectangular ‘top-hat’ function:figures 6(a) to 6(b) is a Lorentz boost with speed v = c/napplied according to 1 ¯ ct + nσ ¯ ct − nσ ¯ ¯ ρ(x, ct ) =tanh − tanh 4 x = (1 − n −2 )−1/2 (x − ct/n), ¯ (20) x +σ ¯ x −σ ¯ × tanh − tanh . (26) t = (1 − n −2 )−1/2 (t − x/nc). ¯ (21) Note that, as → 0, equation (26) becomes a rectangular top-This produces the vertical photon world lines of figure 6(b) hat function. The maps inside and outside the rectangle are ¯wherein the rectangular box (|ct | < nσ, |x| < σ ) is identified. ¯ then graded into each other according toWithin this rectangle a void region is opened up via theoperation of a ‘curtain map’ defined via x = ρ x in + (1 − ρ)x out , ¯ ¯ ¯ (27) ¯ δ + ct ¯ ¯ t = t. (28) x = ¯ x − sgn(x)σ + sgn(x)σ, ¯ ¯ ¯ (22) δ + nσ This procedure ensures that the resultant map (x, t ) → ¯ ¯ (x , t¯ ) defined by combining equations (27) and (28) with ¯ ¯ t¯ = t , (23) equations (24)–(26) is continuous at the boundary of thewith points outside the rectangle being mapped to themselves. rectangle. The complete map (x, t) → (x , t ) is thenThe curtain map is shown for which σ = 1, n = 2, δ = 0.5. specified via the Lorentz transformation of equations (20)Inverse Lorentz-transforming back to the medium rest frame, and (21) sending (x, t) → (x, t¯), the mollified curtain map of ¯where the coordinates are now (x , t ), yields the obliquely equations (27) and (28) sending (x, t¯) → (x , t¯ ), followed by ¯ ¯opened curtain as shown in figure 6(d). The deformed the inverse Lorentz transformation sending (x , t¯ ) → (x , t ). ¯photon world lines in figure 6(d) all have positive gradients The composite map (x, t) → (x , t ), with = 0.08, was used(i.e. propagate forwards in time) and have speeds (determined to generate figure 2 in this paper by calculating numericallyby the gradient of the world line) less than the vacuum speed the transformed material parameters using equations (12)–of light. The hard edges of the curtain map induced by the use (16), and then calculating the electromagnetic energy densityof | · | and sgn(·) functions in equation (22) may be softened by according tothe use of suitable mollifier functions as explained in the next U = 1 (D · E + B · H). 2 (29)section.A.3. Mollifying the hard edges of the curtain map ReferencesAs defined above, the composite map (x, t) → (x, t ) → ¯ ¯ [1] Pendry J B 2000 Negative refraction makes a perfect lens Phys. ¯ ) → (x , t ) is ‘hard-edged’, meaning that sections of Rev. Lett. 85 3966(x , t ¯ [2] Pendry J B, Schurig D and Smith D R 2006 Controllingthe intensity at a given t will change abruptly from zero inside electromagnetic fields Science 312 1780–2the void region to a finite value outside. This would require [3] Leonhardt U and Philbin T G 2006 General relativity inunrealistic instantaneous changes in the material parameters at electrical engineering New J. Phys. 8 247the edge of the cloak. In order to soften these edges we applied [4] Schurig D, Mock J J, Justice B J, Cummer S A, Pendry J B, Starr A F and Smith D R 2006 Metamaterial electromagneticthe following procedure. cloak at microwave frequencies Science 314 977–80 In the vertical ray frame (i.e. figure 6(b)) the functions [5] Brillouin L 1960 Wave Propagation and Group Velocity 1st ednsgn(ξ ) and |ξ | in equation (22) are replaced by tanh(ξ/ ) and (New York: Academic)ξ tanh(ξ/ ), respectively, where is a small parameter. Note [6] Pendry J B 2009 Time reversal and negative refraction Sciencethat, as → 0, tanh(ξ/ ) → sgn(ξ ) and ξ tanh(ξ/ ) → 322 71–3 [7] McCall M W 2007 Relativity and mathematical tools: waves in|ξ |. For points inside the rectangle equations (22) and (23) are moving media Am. J. Phys. 75 1134–40therefore replaced with [8] Tretyakov S A, Nefedov I S and Alitalo P 2008 Generalized ¯ ¯ δ + ct tanh(ct / ) field-transforming metamaterials New J. Phys. 10 115028x in =¯ [x − tanh(x/ )σ ] ¯ ¯ [9] Tretyakov S A 2009 On a possibility to imitate media moving δ + nσ with superluminal velocity 3rd Int. Conf. on Advanced Electromagnetic Materials in Microwaves and Optics + tanh(x/ )σ , ¯ (24) London [10] Post E J 1997 Formal Structure of Electromagneticstin = t¯.¯ (25) (New York: Dover) 9