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1 2 3 4 Roadmap on Transformation 5 Martin McCall 1,*, John B Pendry 1, Vincenzo Galdi 2, Yun Lai 3, S. A. R. Horsley 4, Jensen Li 5, Jian 6 Zhu 5, Rhiannon C Mitchell-Thomas 4, Oscar Quevedo-Teruel 6, Philippe Tassin 7, Vincent Ginis 8, 7 9 9 9 6 10 8 Enrica Martini , Gabriele Minatti , Stefano Maci , Mahsa Ebrahimpouri , Yang Hao , Paul Kinsler 11 11,12 13 14 15 9 , Jonathan Gratus , Joseph M Lukens , Andrew M Weiner , Ulf Leonhardt , Igor I. 10 Smolyaninov 16, Vera N. Smolyaninova 17, Robert T. Thompson 18, Martin Wegener 18, Muamer Kadic 11 18 and Steven A. Cummer 19 12 13 14 Affiliations

15 1 16 Imperial College London, Blackett Laboratory, Department of , Prince Consort Road, 17 London SW7 2AZ, 18 19 2 & Waves Lab, Department of , University of Sannio, I-82100 Benevento, Italy 20 21 3 College of Physics, Optoelectronics and & Collaborative Innovation Center of Suzhou Nano 22 and , Soochow University, Suzhou 215006, China 23 24 4 University of Exeter, Department of Physics and , Stocker Road, Exeter, EX4 4QL United 25 26 Kingdom 27 5 28 School of Physics and Astronomy, University of Birmingham, Edgbaston, Birmingham, B15 2TT, 29 United Kingdom 30 31 6 KTH Royal Institute of Technology, SE-10044, Stockholm, Sweden 32 33 7 Department of Physics, Chalmers University , SE-412 96 Göteborg, Sweden 34 35 8 Vrije Universiteit Brussel Pleinlaan 2, 1050 Brussel, Belgium 36 37 9 Dipartimento di Ingegneria dell'Informazione e Scienze Matematiche, University of Siena, Via Roma, 38 39 56 53100 Siena, Italy 40 10 41 School of and Computer Science, Queen Mary , 42 London E1 4FZ, United Kingdom 43 44 11 Physics Department, Lancaster University, Lancaster LA1 4 YB, United Kingdom 45 46 12 Cockcroft Institute, Sci-Tech Daresbury, Daresbury WA4 4AD, United Kingdom. 47 48 13 Information Science Group, Computational and Engineering Division, Oak Ridge 49 National Laboratory, Oak Ridge, Tennessee 37831, USA 50 51 14 School of Electrical and Computer Engineering, Birck Nanotechnology Center, and Purdue 52 53 Quantum Center, Purdue University, West Lafayette, Indiana 47907, USA 54 15 55 Physics of Complex Systems, Weizmann Institute of Science, Rehovot 7610001 Israel 56 57 16 Institute for systems , University of Maryland, College Park, MD 20742, USA 58 59 17 Department of Physics, Astronomy and Geosciences, Towson University, Towson, MD 21252, 60 USA AUTHOR SUBMITTED MANUSCRIPT - draft Page 2 of 59

1 2 3 18 Karlsruhe Institute of Technology, Institut für Angewandte Physik, Wolfgang-Gaede-Straße 1, D- 4 5 76131 Karlsruhe, Germany

6 19 7 Electrical and Computer Engineering, Duke University, PO Box 90291, Durham, NC 27708, USA 8 9 *Guest editor of the roadmap 10 11 Email: [email protected] 12 13 14 15 Abstract 16 asks ’s what kind of electromagnetic medium recreate some 17 smooth of space. The guiding principle is Einstein’s principle of covariance: that any 18 19 physical must take the same form in any . This requirement fixes very 20 precisely the required electromagnetic medium. 21 22 The impact of this insight cannot be overestimated. Many practitioners were used to thinking that only 23 a few analytic solutions to Maxwell’s equations existed, such as the monochromatic plane wave in a 24 homogeneous, isotropic medium. At a stroke, Transformation Optics increases that landscape from 25 ‘few’ to ‘infinity’, and to each of the infinitude of analytic solutions dreamt up by the researcher, 26 corresponds an electromagnetic medium capable of reproducing that solution precisely. 27 28 The most striking example is the electromagnetic cloak, thought to be an unreachable dream of 29 science fiction writers, but realised in the laboratory a few months after the papers proposing the 30 possibility were published. But the practical challenges are considerable, requiring meta-media that 31 32 are at once electrically and magnetically inhomogeneous and anisotropic. How far have we come 33 since the first demonstrations over a decade ago? And what does the future hold? If the wizardry of 34 perfect macroscopic optical still eludes us in practice, then what compromises still enable 35 us to create interesting, useful, devices? 36 37 While 3D cloaking remains a significant technical challenge, much progress has been made in 2- 38 . Carpet cloaking, wherein an object is hidden under a surface that appears optically flat, 39 relaxes the constraints of extreme electromagnetic parameters. Surface wave cloaking guides sub- 40 41 surface waves, making uneven surfaces appear flat. Two dimensions is also the setting in 42 which conformal and complex coordinate transformations are realisable, and the possibilities in this 43 restricted domain do not appear to have been exhausted yet. 44 45 Beyond cloaking, the enhanced electromagnetic landscape provided by Transformation Optics has 46 shown how fully analytic solutions can be found to a number of physical scenarios such as plasmonic 47 systems used in energy loss spectroscopy (EELS) and cathodoluminescence (CL). Are there 48 further fields to be enriched? 49 50 A new twist to Transformation Optics was the extension to the space-time domain. By applying 51 transformations to space-time, rather than just space, it was shown that events rather than objects 52 could be hidden from view; Transformation Optics had provided a means of effectively redacting 53 54 events from history. The hype quickly settled into serious nonlinear optical experiments that 55 demonstrated the soundness of the idea, and it is now possible to consider the practical implications, 56 particularly in optical processing, of having an ‘interrupt-without-interrupt’ facility that the so- 57 called temporal cloak provides. Inevitable issues of dispersion in actual systems have only begun to 58 be addressed. 59 60 Now that time is included in the programme of Transformation Optics, it is natural to ask what role ideas from can play in shaping the future of Transformation Optics. Indeed, one of Page 3 of 59 AUTHOR SUBMITTED MANUSCRIPT - draft

1 2 3 the earliest papers on Transformation Optics was provocatively titled ‘General Relativity in Electrical 4 5 Engineering’. The answer that curvature does not enter directly into transformation optics merely 6 encourages us to speculate on the role of Transformation Optics in defining laboratory analogues. 7 8 Quite why Maxwell’s theory defines a ‘perfect’ transformation theory, while other areas of physics 9 such as are not apparently quite so amenable, is a deep question whose precise, 10 mathematical answer will help inform us of the extent to which similar ideas can be extended to other 11 fields. 12 13 The contributors to this roadmap review, who are all renowned practitioners or inventors of 14 Transformation Optics, will give their perspectives into the field’s status and future development. 15 16 17 18 Contents 19 20 1. Introduction 21 22 23 2. Near Field Transformation Optics 24 25 3. and Spectral Domain Transformation Optics 26 3.1 Nonlocal and non-Hermitian extensions of transformation optics 27 3.2 Transformation optics by photonic crystals: a nonlocal route towards the optical 28 29 regime 30 31 4. Complexification in Transformation Optics 32 33 5. Surface Transformation Optics 34 35 5.1 Extending transformation optics and towards a thin cloak 36 5.2 Combining curvature and index for surface and guided waves 37 5.3 Transformation optics at surfaces 38 39 6. Antennas 40 41 6.1 Metasurface and Guidance at 42 6.2 Bespoke lenses: An opportunity for tailoring radiation patterns 43 6.3 Transformation Optics for Antenna Engineering 44 45 7. Cloaking 46 7.1 Spacetime Transformation Optics 47 48 7.2 Transformations optics with spatial dispersion 49 7.3 Experimental progress in temporal cloaking 50 51 8. Transformation Optics for Analogue Cosmology 52 8.1 Cosmology in the laboratory: challenges at the horizon 53 54 8.2 Spacetime analogs based on hyperbolic 55 8.3 Transformation optics in general relativity 56 57 9. Optics and Beyond 58 9.1 Seeking applications in optics and beyond 59 9.2 Beyond Optics: Transforming Other Wave And Transport Systems 60

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1 2 3 . They also discuss strategies for producing thin 4 1. Introduction – Martin McCall 5 Imperial College cloaks based on metasurfaces. 6 7 A little over a decade ago Pendry et al noted that the 8 long-known form-invariance of Maxwell’s equations Rhiannon Mitchell-Thomas reviews geometrical 9 could be interpreted as an extremely flexible design approaches confined to curved surfaces, rather than 3- 10 recipe for achieving virtually arbitrary distortions of D space, showing that TO designs using isotropic 11 electromagnetic fields. It is an unusual theory in that graded index media allow for novel implementations 12 13 the recipe is exact, i.e. all aspects of the of, for example, Eaton and Luneburg lenses. 14 , amplitude, phase and Discontinuous transformations at surfaces are 15 polarization are in principle relocated by an discussed by Philippe Tassin et al, who show how the 16 appropriate material design. The method was latest metasurface technology opens up applications in 17 immediately applied to design and build a device that integrated optics. 18 guided light around a void region in space, thus 19 20 bringing the celebrated invisibility cloak of science 21 fiction right into the laboratory. Although the cloak is Martini et al discuss how a planar version of 22 the most striking example, transformation optics has Transformation Optics leads to control of 23 since developed many other applications that combine surface/plasmonic wave propagation or ‘metasurfing’; 24 its design ingenuity with parallel developments in while Oscar Quevedo-Tereul shows how TO is being 25 metamaterials technology. In this Roadmap review used to tailor antenna radiation patterns using bespoke 26 27 several leading practitioners give their unique personal lenses. The theme of antenna engineering is taken up 28 perspectives of the current status of the field, and by Yang Hao in reviewing how TO based radome 29 where the future challenges lie. designs are leading to novel compact phased arrays. 30 31 Pendry’s article focuses on the near-field and the impact of TO on surface plasmonics, explaining, for 32 Paul Kinsler and Jonathan Gratus consider example, the huge enhancements of local fields around 33 generalization of TO to the spacetime domain. 34 structural singularities, and the resultant opportunities Kinsler’s article discusses spacetime cloaking, in wich 35 offered to spectroscopy and nonlinear optics resulting events, not objects, are hidden from view. Gratus 36 from the very tight compression of light fields. 37 shows that in the broader TO regime with both time 38 and space included, spatial dispersion plays a 39 necessary and fundamental role. Lukens et al describe Transformation Optics designs invariably prescribe 40 some of the experimental efforts in event cloaking, 41 inhomogeneous material parameters. A striking emphasising the role played by space-time duality in 42 departure from this is described by Vincenzo Galdi, achieving localized shadowing in space-time. They 43 who examines the use of the TO algorithm in the 44 emphasise that applications will likely focus on spectral domain. Such designer dispersion prescribes 45 specific tasks, and offer tantalising media with fascinating frequency dependencies, all the 46 connections between temporal cloaking and quantum while maintaining uniformity in the spatial domain. 47 information processing. 48 The theme of spatial dispersion in TO is taken up again 49 in Lai’s article, who discusses how photonic band-gap As Ulf Leonhardt points out in his contribution, 50 engineering is combined with TO to produce following the detection of gravitational waves, 51 omnidirectional impedance matching. cosmology has now entered a golden era, and how 52 might repay its debt to GR becomes 53 Simon Horsley shows how analytic continuation of the an issue addressable by TO. Smolyaninov et al discuss, 54 coordinates into the complex arena permits extensions 55 for example, how the mapping between metric and to media with gain and loss, connecting TO closely 56 material parameters, a key idea in TO, informs with so-called PT symmetric media, and showing how 57 cosmology in the optics lab, with inhomogeneous 58 to design reflectionless media. Rather than exploit hyperbolic metamaterials directly analogising signature 59 form invariance under coordinate transformation, Li et changing events and multiverses. As Thompson makes 60 al explain how new possibilities emerge if, instead, clear, the distinction between medium and spacetime form invariance is pursued with respect to contributions in Maxwell’s equations opens up several transformation of the electromagnetic field, leading to, for example, an effective magnetic bending on Page 5 of 59 AUTHOR SUBMITTED MANUSCRIPT - draft

1 2 3 potential new avenues since the mapping between 4 5 distortion and effective medium is actually not unique. 6 7 8 Beyond optics, Martin Wegener takes us from using 9 TO to improve efficiency to transformation 10 , showing how mechanical cloaks can 11 improve the utility of scaffolds. Steve Cummer also 12 13 assesses how the TO algorithm can and has been 14 successfully extended to other disciplines such as 15 acoustics, elastodynamics and heat flow. 16 17 18 A recurring theme in all the contributions is the extant 19 20 gap between design aspiration and technological 21 feasibility. Theorists have provided beautiful design 22 tools that can potentially give us many very useful and 23 practical devices, but we are frustrated at often only 24 being able to realise a fraction of this potential due to 25 the difficulty of producing broad-band 26 27 designs requiring sub-wavelength fabrication. The 28 compromises currently made on the altar of progress 29 do not do justice to the original designs. Perhaps the 30 main ‘take-home’ message of this review is to provide 31 a rallying call to technologists and experimentalists to 32 make the necessary step change to meet these 33 34 challenges and turn, as Cummer notes, simple 35 feasibility demonstrations towards practically useful 36 devices. 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 AUTHOR SUBMITTED MANUSCRIPT - draft Page 6 of 59

1 2 3 4 2. Near Field Transformation Optics – JB 5 Pendry 6 Imperial College London 7 8 Status 9 10 Transformation optics (TO) has its origins in Einstein’s 11 general : he showed how Maxwell’s 12 equations can be written in a non-Cartesian space and 13 how changes to the coordinate system can be 14 represented as changes to the constitutive parameters, 15 . These ideas were slow to be adopted by the 16 optics community but in 1996 Ward and Pendry [1] 17 faced the challenge of adapting electromagnetic finite 18 difference codes to the cylindrical geometry of a fibre. 19 20 Originally written in Cartesian coordinates to tackle photonics crystals the codes did not need to be 21 Figure 1 TO was used to control magnetic fields in 22 reprogrammed in cylindrical coordinates, but simply this realisation of a cloak for [3]. 23 presented with a transformed . However the most (a) lines are diverted away from the 24 powerful insight came with the realisation that under a cloaked region whilst remaining undisturbed outside the 25 coordinate transformation lines of force, whether cloak. (b) measurements of the field with and without the 26 electric or magnetic, remained attached to the cloak confirm the picture shown in (a). 27 coordinate system. A picture emerged of space as a 28 deformable entity, and the possibility of rearranging TO has had a strong impact on plasmonic systems. The 29 the fields by distorting space, then asking through optical response of metals is dominated by the sea of 30 free and on the surface the electron sea 31 Einstein’s equations what values of would send supports electromagnetic waves called surface 32 the fields in the desired direction. The value of this 33 insight is that it replaces the intuitive pictures of ray . Bound to the metal surface and with a 34 optics with a way to manipulate field lines, but wavelength much shorter than that of light in free 35 retaining the insight offered by a physical picture of space, their behaviour is dominated by the shape of the 36 what is going on. Furthermore an intuitive picture can surface. For example, a highly polished silver surface 37 be given of how transform. Under a simple is an excellent mirror reflecting nearly all incident 38 light. On the other hand finely divided silver in the compression by of coordinates along one axis, 39 form of nanoparticles, such as found in photographic 40 components of both along that axis are reduced negatives, absorbs nearly all incident light and is black. 41 by a factor of ; components perpendicular to the axis This is an extreme example of a metamaterial: internal 42 are increased by . Any transformation can be structure dominates the optical response. 43 Understanding the interplay of surface structure with 44 represented by a series of compressions and rotations the near field has enabled new insights. 45 and thus the whole transformation can be understood in 46 geometrical terms. A review of these ideas can be had 47 in reference [2]. 48 Current and Future Challenges These ideas first gained attention when deployed to 49 Surface plasmonics [4] is perhaps the field in which 50 design a cloak of invisibility, a challenge that is almost impossibly difficult using conventional but TO has most impact. Plasmonics burst spectacularly 51 into spectroscopy when Fleischman made the 52 ridiculously simple for TO. Many cloaks can also be 53 envisaged in terms of deflected rays, but this picture is observation that Raman from 54 not available to us for structures that are much less adsorbed on a rough silver surfaces were many orders 55 than the wavelength and we are confronted with the so of magnitude greater than the free space cross sections 56 called ‘near fields’ where the electric and magnetic would predict. The effect can be explained by huge 57 components are no longer tightly coupled into the ray enhancements of local fields of incident light around 58 picture but lead a more independent existence. For structural singularities. Heavy computational studies 59 example a cloak for static magnetic fields can in no confirmed this explanation but provided little insight 60 way be envisaged as controlling a set of rays and we into why singularities produce enhancement. It fell to are forced instead to think of lines of force. Such a transformation optics to explain the effect in simple cloak has been realised by Supradeep Narayana and terms. Yuki Sato [3] and figure 1 shows their results. Page 7 of 59 AUTHOR SUBMITTED MANUSCRIPT - draft

1 2 3 Consider a structure consisting of a surface containing effect of smearing out any singularities in the surface 4 5 a sharp knife edge. The edge constitutes a singularity structure. In metals non locality arises because the 6 and local electric fields are massively enhanced in its electron sea has limited compressibility and cannot be 7 vicinity when the structure is illuminated. The squeezed into very sharp corners preventing realisation 8 singularity can be removed by a simple transformation, of an ideal singularity [7,8,9]. The search is on for 9 new materials that perform even better than our current 10 optimum candidates of silver and gold [10], a 11 challenge made more difficult by our limited 12 theoretical understanding of losses and non locality 13 where the origin is located on the edge. This beyond the standards achieved by current best 14 transforms the edge to a non singular array of wave performers. 15 guides and the singularity at the origin of space 16 17 vanishes to infinity in space. All singular 18 structures can have their singularities removed in this 19 fashion. Enhancements can be understood as the 20 compression of an infinite system into a finite one. 21 This has the curious consequence that a finite singular 22 structure has a continuous spectrum typical of an 23 infinite system, and an infinity in the density of states 24 at the singularity. Though this result has to be qualified 25 26 for real materials where practical considerations of loss 27 and of non locality of their response can reduce 28 predicted enhancements. More details can be had in 29 reference [5]. 30 Another system described in [5] is shown in figure 2. 31 In this instance the transformation is an inversion 32 33 which takes a simple metallic and Figure 2 (A) Surface plasmons are excited by light 34 transforms it into two touching cylinders. The emitted from an excited in a metallic waveguide. 35 transformation is singular at the touching point which (B) Transformation by inversion about the atom centre 36 produces intense local fields. Not only does a generates a completely different structure of two 37 transformation explain why local fields are so intense, touching cylinders. (C) The waveguide solutions from 38 it also provides a means of classifying the spectrum: (A) map directly to the touching cylinders giving 39 often it is the case that the transformed system, lacking strongly enhanced fields near the point of contact. 40 the original singularity, has a simpler structure. In the 41 examples given above the transformed waveguide 42 structure is translationally along the axis Beyond that there is the issue of how to sculpt surfaces 43 allowing modes to be classified as Bloch waves, to the required geometrical perfection. Metals have 44 very high surface energy and object to the formation of 45 assigned a wave vector, , and their 46 calculated analytically as a function of . singularities: sharp edges tend to round off, and singularities formed by touching curves often result in 47 TO can also be applied in the context of the refractive the fusing of the two surfaces. Engineering on the 48 index to remove singularities from a structure as has nanoscale demands expensive apparatus to achieve the 49 been elegantly demonstrated in reference [6] 50 precision needed. 51 Advances in Science and Technology to Meet Despite these slightly pessimistic observations great 52 Challenges 53 opportunities are to be had. I have alluded to the 54 Several challenges meet these beautiful theoretical spectacular enhancements in spectroscopic sensitivity 55 concepts when inviting experimentalists into the field. already a . Utilising the design tool of TO 56 The first is the materials challenge. Most of the structures can be optimised to give the best response 57 conclusions presented above assume an ideal loss free and single sensitivity can be had. 58 metal described by a local function, . It is already possibility to compress light into a focus 59 Locality means that depends only on frequency and 60 far smaller than the free space Abbé limit. not on wave vector. Loss absorbs energy and prevents Compression into a square nanometre is possible, realisation of ideal field enhancements, so that only many orders of magnitude more dense than achieved relatively loss free metals such as silver and gold are with conventional lenses thus achieving with modest serious candidates for the effects. Non locality has the power inputs energy densities great enough to trigger AUTHOR SUBMITTED MANUSCRIPT - draft Page 8 of 59

1 2 3 non linear effects. A cornucopia of new experiments 4 5 awaits exploitation of nonlinearity, from simple 6 switching of light with light to more sophisticated 7 concepts such as phase conjugation and time reversal. 8 Concluding Remarks 9 10 TO is a general tool that provides a new way to 11 visualise electromagnetic phenomena, whilst retaining 12 accuracy at the level of Maxwell’s equations. It has 13 particular value for the near field where conventional 14 ray pictures are of no use, but where the concept of 15 fields still applies. In the above two areas of 16 application are described: to DC fields where the entire 17 system exists in the near field, and surface plasmonics 18 19 where most of the interesting phenomena are observed 20 on length scales smaller than the free space 21 wavelength. 22 Acknowledgments This was supported by the 23 EPSRC (Grant No. EP/L024926/1), and the Gordon 24 and Betty Moore Foundation. 25 26 27 References 28 [1] Ward AJ, and Pendry JB, 1996 Refraction and 29 Geometry in Maxwell’s Equations 30 Journal of Modern Optics, 43 773-93 31 [2] Pendry JB, Luo Y, and Zhao Rongkuo 2015 32 Transforming the optical landscape Science 348 33 521-4 34 [3] Supradeep Narayana and Yuki Sato 2012 DC 35 Magnetic Cloak Advanced Materials 24 71-74 36 [4] Maier SA 2007 Plasmonics: Fundamentals and 37 Applications (Springer, New York) 38 [5] Pendry JB, Aubry A, Smith DR, and Maier SA 2012 39 Transformation optics and subwavelength control of 40 light Science 337, 549-52. 41 [6] Horsley SAR, Hooper IR, Mitchell–Thomas RC and 42 Quevedo–Teruel O 2014 Removing singular 43 refractive indices with sculpted surfaces Scientific 44 Reports 4 4876 45 [7] Mortensen NA, Raza S, Wubs M, Søndergaard T 46 and Bozhevolnyi SI 2014 A generalized non-local 47 optical response theory for plasmonic nanostructures 48 5, 3809. 49 [8] Zhu W, Esteban R, Borisov AG, Baumberg JJ, 50 Nordlander P, Lezec HJ, Aizpurua J and Crozier KB 51 2016 Quantum mechanical effects in plasmonic 52 structures with subnanometre gaps Nature 53 Communications 7, 11495. 54 [9] Luo Y, Fernandez-Dominguez AI, Wiener A, Maier 55 56 SA, and Pendry JB 2013 Surface Plasmons and 57 Nonlocality: A Simple Model Physical Review 58 Letters, 111, 093901 59 [10] Brown AM, Sundararaman R, Narang P, 60 Goddard III WA, and Atwater HA 2016 Nonradiative Decay and Hot Carrier : Effects of Phonons, Surfaces, and Geometry ACS Nano 10 957–966 Page 9 of 59 AUTHOR SUBMITTED MANUSCRIPT - draft

1 2 3 associated to analytic properties of the transformation 4 3. Spatial Dispersion and Spectral Domain such as non-centersymmetry, multivaluedness, and 5 Transformation Optics 6 3.1 Nonlocal and non-Hermitian extensions of stationary inflection points, respectively. Similar to the conventional (local) TO approach, our proposed 7 transformation optics Vincenzo Galdi – nonlocal extension systematically yields, in analytic 8 University of Sannio closed-form expressions, the required (nonlocal) 9 10 constitutive blueprints in terms of wavevector- 11 Status dependent and permeability [2,3]. 12 13 By interpreting the material effects on wave propagation 14 as local distortions of a given coordinate reference 15 frame, transformation optics (TO) provides a very 16 powerful and systematic framework to precisely 17 manipulate electromagnetic fields [1]. In essence, the 18 conceptual design of a desired response may be 19 conceived by relying primarily on geometrical intuition 20 based on the path of light rays, and is 21 effectively decoupled from the actual material synthesis 22 23 problem, which is eventually posed as a suitable 24 approximation of some analytically derived ideal 25 constitutive “blueprints”. 26 27 Conventional TO schemes typically rely on local, real- 28 valued coordinate transformations, which inherently Figure 1 – Conceptual illustration of the nonlocal TO 29 imply some limitations in the attainable field- extension in the frequency ( )-wavevector ( ) phase 30 manipulation effects. Thus, typical constitutive 31 space (see [3]). (a) Conical dispersion surface of a blueprints tend to be spatially inhomogeneous and homogeneous medium. (b) Hyperbolic dispersion (purely 32 anisotropic [1], whereas it is generally not possible to 33 imaginary transformation component). (c) One-way, 34 extrinsically generate nonlinear, nonreciprocal, nonreciprocal propagation (non-centersymmetric transformation). (c) Additional extraordinary wave 35 nonlocal (i.e., spatial dispersion), and non-Hermitian (multivalued transformation). (d) Frozen mode (stationary 36 (i.e., loss and/or gain) effects, among others. Several inflection point). (f) Dirac-point conical singularity (phase- 37 extensions have been proposed in order to overcome space shift). 38 these limitations. Here, we focus on certain extensions 39 that enable the TO formalism to deal with nonlocal and By comparison with conventional (local, spatial- 40 non-Hermitian effects, which are becoming domain) TO, we note that the above nonlocal extension 41 increasingly relevant in many metamaterial inherently yields spatially homogeneous constitutive 42 applications. blueprints. In view of the underlying spatial Fourier 43 transform, the spectral resolution, i.e., the capability to 44 Current and Future Challenges tailor the dispersion surfaces in the frequency- 45 46 wavevector phase space, is gained at the expense of the 47 In [2,3], our team laid out the foundations for a nonlocal spatial resolution. On the other hand, conventional TO 48 TO extension, which can be exploited to allows fine tailoring of the spatial response, but 49 complex spatial-dispersion effects. Such extension completely lacks spectral resolution. Thus, conventional 50 relies on a reformulation of TO in the frequency- and nonlocal TO can be viewed as two extremes in the 51 wavevector phase space, accessed via temporal and spatial vs. spectral resolution tradeoff. As a future 52 spatial Fourier transforms, and retains the attractive challenge, it would be very intriguing to bridge these 53 features of conventional TO in terms of decoupling the two extremes, so as to blend in a desired fashion both 54 actual metamaterial synthesis from the conceptual spatial and spectral resolution. From the mathematical 55 design, with this latter still heavily inspired by viewpoint, windowed spatial Fourier transforms, such as 56 the Gabor transform, seem to constitute a particularly 57 geometrical intuition. As illustrated in Fig. 1, a broad suitable, self-consistent framework for such a hybrid TO 58 variety of interesting dispersion effects can be 59 interpreted in the phase space as geometrical formulation. In such a scheme, the window width would 60 transformations of the conventional conical dispersion set the spectral vs. spatial resolution tradeoff via an surface of a homogeneous medium. Thus, for instance, uncertainty relation, and the conventional and nonlocal effects such as one-way (nonreciprocal) propagation, TO formulations would be recovered as the limits for additional extraordinary waves, and slow-light can be zero and infinite window width, respectively. The above-envisioned Gabor-type formulations may AUTHOR SUBMITTED MANUSCRIPT - draft Page 10 of 59

1 2 3 considerably expand future prospects for TO novel effects and shed new light on the 4 phenomenological understanding. 5 applications, encompassing complex scenarios 6 featuring, e.g., nonlocal effects at different spatial 7 scales, such as photonic “hypercrystals”. Moreover, 8 they may also set the stage for disruptive field- 9 manipulation platforms combining the spatial-routing 10 (e.g., multiplexing, steering) capabilities of 11 conventional TO with the inherent signal-processing 12 capabilities (e.g., convolution, differentiation, 13 integration) enabled by nonlocal TO. This may open up 14 new intriguing venues in the emerging field of 15 “computational metamaterials” [4]. 16 17 18 In [5,6], our team put forward a complex-coordinate TO 19 extension that can naturally handle non-Hermitian 20 metamaterials featuring spatial modulation of loss and 21 gain. Also in this case, the appealing characteristics of 22 conventional (real-valued) TO in terms of physically Figure 2 – Schematic representation of the complex- 23 incisive modeling and geometry-driven intuitive design coordinate TO mapping between a “source” and an “image” 24 are retained, by combining the approach with well- CSP ( and , respectively). In the physical, real space the 25 established analytic formalisms based, e.g., on the 26 mapping can be interpreted as a transformation of an input “complex-source-point” (CSP) representation of beam- beam into an output beam ( and arrows, respectively), 27 like wave objects, and the “leaky-wave” (complex 28 induced by a non-Hermitian metamaterial slab. ) modeling of radiating states [5,6]. 29 For instance, as schematized in Fig. 2, a complex Advances in Science and Technology to Meet 30 Challenges 31 coordinate transformation relating a “source” and an 32 “image” CSP can be interpreted in the real, physical 33 space as an input-output transformation of an optical Nonlocal TO extensions crucially rely on nonlocal 34 beam, induced by a non-Hermitian metamaterial slab, effective models to approximately match the desired 35 with full control on the waist position, direction and (ideal) constitutive blueprints with that of a physical 36 diffraction length. This represents a rather versatile and metamaterial structure within appropriate phase-space 37 insightful approach, which brings about new degrees of regions. Therefore, essential to further advances in this 38 freedom and may provide new perspectives in the field is the availability of a comprehensive “library” of 39 design of active optical devices and radiating systems, effective nonlocal models, building up on general [8] 40 with the use of gain not merely limited to achieve loss and structure-specific [9] nonlocal homogenization 41 compensation. approaches. Moreover, the discovery of new structures 42 and regimes where nonlocal effects are enhanced, and 43 possibly attainable via moderate-permittivity, low-loss 44 Metamaterials mixing loss and gain are also receiving a material constituents may significantly boost the 45 surge of interest as potentially feasible testbeds for 46 studying phenomena and properties that are distinctive practical applicability of nonlocal TO. Finally, the 47 of non-Hermitian systems. These include -time development of full-wave numerical simulation tools 48 and exceptional points (i.e., degeneracies implementing nonlocal constitutive relationships would 49 between complex eigenstates), which are ubiquitous in also represent a major catalyst. 50 many disciplines, ranging from to 51 [7]. The application of non-Hermitian Non-Hermitian TO, on the other hand, critically relies 52 TO extensions to these scenarios is still largely on the possibility to precisely tailor the spatial 53 distribution of loss and gain, which is becoming 54 unexplored, and represents one of the most fascinating developments to pursue. Preliminary studies on parity- technologically viable. At frequencies, gain 55 may be attained via classical amplification schemes 56 time-symmetric metamaterials [5] have shown 57 intriguing connections between the onset of based on operational amplifiers and Gunn , 58 spontaneous symmetry breaking (an example of whereas at optical active dopants such as 59 exceptional point) and the discontinuous character of two-level (or quantum dots) may be exploited. 60 the complex-coordinate transformation. This may be While certain non-Hermitian effects may require indicative of deeper connections between the analytic unfeasibly high values of gain, it is also important to properties of the transformations and the physics of remark that completely passive (i.e., lossy) structures exceptional points, whose exploration may uncover have been exploited to experimentally demonstrate Page 11 of 59 AUTHOR SUBMITTED MANUSCRIPT - draft

1 2 3 pivotal proof-of-concept effects in the physics of 4 5 exceptional points (see, e.g., Ref. [10]). Within this 6 framework, metamaterials with near-zero constitutive 7 parameters (real part) [11], appear particularly attractive 8 in view their well-known capabilities to dramatically 9 enhance the effects of relatively low levels of loss and/or 10 gain. 11 12 Concluding Remarks 13 14 In a variety of metamaterial applications of current 15 interest, spatial dispersion and loss/gain are escalating 16 from -order nuances to instrumental effects. 17 Nonlocal and non-Hermitian extensions of the TO 18 approach potentially constitute an attractive framework 19 to harness these effects in a versatile, systematic and 20 insightful fashion. 21 22 Further perspectives on nonlocal and non-Hermitian TO 23 extensions are discussed in Sections 3.2, 7.2 and Section 24 4, respectively. 25 26 References 27 [1] Pendry J B, Schurig D and Smith D R 2006 28 Controlling electromagnetic fields Science 312 29 1780–1782 30 [2] Castaldi G, Galdi V, Alù A and Engheta N 2012 31 Nonlocal transformation optics Phys Rev Lett 108 32 063902 33 [3] Moccia M, Castaldi G, Galdi V, Alù A and Engheta 34 N 2016 Dispersion engineering via nonlocal 35 transformation optics Optica 3 179–10 36 [4] Silva A, Monticone F, Castaldi G, Galdi V, Alù A 37 and Engheta N 2014 Performing mathematical 38 operations with metamaterials Science 343 160–163. 39 40 [5] Castaldi G, Savoia S, Galdi V, Alù A and Engheta N 41 2013 PT metamaterials via complex-coordinate 42 transformation optics Phys Rev Lett 110 173901 43 [6] Savoia S, Castaldi G and Galdi V 2016 Complex- 44 coordinate non-Hermitian transformation optics J 45 Opt 18 1–13 46 [7] Heiss W D 2012 The physics of exceptional points J 47 Phys A 45, 444016. 48 [8] Silveirinha M G 2007 Metamaterial homogenization 49 approach with application to the characterization of 50 microstructured composites with negative parameters 51 Phys Rev B 75 115104 52 [9] Chebykin A V, Orlov A A, Vozianova A V, 53 Maslovski S I, Kivshar Y S and Belov P A 2011 54 Nonlocal effective medium model for multilayered 55 metal-dielectric metamaterials Phys Rev B 2011 84 56 115438 57 [10] Doppler J, Mailybaev A A, Böhm J, Kuhl U, Girschik 58 A, Libisch F, Milburn TJ, Rabl P, Moiseyev N and 59 Rotter S 2016 Dynamically encircling an exceptional 60 point for asymmetric mode switching Nature 537 76– 79 [11] Liberal I and Engheta N 2017 Near-zero photonics Nat 11 149–158 AUTHOR SUBMITTED MANUSCRIPT - draft Page 12 of 59

1 2 3 3.2 Transformation optics by photonic crystals: a 4 5 nonlocal route towards the optical frequency 6 regime – Yun Lai 7 Soochow University 8 9 Status 10 11 Transformation optics (TO) theory was originally 12 derived in a local medium framework [1,2]. Since 13 coordinate transformation often leads to unusual 14 materials parameters absent in nature, metamaterials 15 [3] have been applied to realize these parameters in 16 novel TO applications, such as cloaking [1-3] and 17 [4]. Most of the experimental 18 demonstrations of TO were achieved by metamaterials. 19 However, at optical frequencies, metamaterials suffer 20 from the loss in their metallic components. Moreover, 21 optical metamaterials demand nano-scale unit 22 23 structures, which makes the fabrication of macroscopic 24 samples quite challenging. 25 On the other hand, photonic crystals [5] can provide an 26 excellent low-loss platform for optical devices. Besides 27 the famous band-gap properties, photonic crystals are 28 also well known for exhibiting novel refractive 29 30 behaviors such as . Interestingly, Figure 1 – (a), (b) show the refractive behaviors of a 31 negative refraction was first proposed and realized in metamaterial slab and an “ultratransparent” photonic 32 the field of metamaterials, as a direct result of the crystal slab, respectively. (c), (d) show the transmittance 33 double negative parameters. However, negative and nonlocal parameters of a designed . 34 refraction was also realized by photonic crystals later. Reprinted with permission from Ref. [9] Copyright 35 Unlike metamaterials that use unusual medium (2016) by the American Physical Society. 36 parameters to control the propagation of waves, medium framework. However, the solutions usually 37 photonic crystals utilize their spatial dispersions, which require complex permittivity and permeability which 38 can be described by the equal frequency contours are difficult to realize. Even in microwave frequencies, 39 (EFCs) in the band structure, to control the wave the perfect TO parameters are usually reduced 40 propagation. By designing specific EFCs, the refractive 41 significantly to ease the fabrication and the property of behaviors of TO media and some novel TO 42 omnidirectional impedance matching is normally applications can also be simulated by photonic crystals. 43 sacrificed [3]. This led to reflection in cloaking For instance, Urzhumov and Smith first proposed to 44 experiments, compromising the ideal invisibility. 45 utilize elliptical EFCs to simulate the stretched TO 46 media, and thus further realize cloaking phenomena Interestingly, it is seldom known that beside the perfect 47 [6]. Liang and Li proposed to use photonic crystals to TO solution, there are actually infinite other solutions 48 replace the TO media in bending [7]. for omnidirectional impedance matching. But in order 49 These seminal studies, however, didn’t explore the to find these unusual solutions, one has to go beyond 50 possibility of realizing the omnidirectional impedance the local medium framework, where the original TO 51 matching condition, which is another distinct signature theory is rooted, into the much less explored realm of 52 of the TO media obtained by coordinate nonlocal photonic media. Such nonlocal media exist in 53 transformation. In fact, if omnidirectional impedance neither natural materials like , nor ordinary 54 55 matching can also be achieved by pure dielectric metamaterials which function as local effective media 56 photonic crystals, the realization of ideal nonreflecting due to sub-wavelength structures. TO applications in the optical frequency regime is 57 Photonic crystals, with microstructures much larger possible. 58 than that of metamaterials, can also possess effective 59 permittivity and permeability responses. When the 60 Current and Future Challenges excited bands and eigenmodes are near the Brillouin The TO theory can provide the perfect solution of Zone center, photonic crystals can behave as effective omnidirectional impedance matching in the local media with zero refractive index [8]. While for bands and eigenmodes far away from the Brillouin Zone Page 13 of 59 AUTHOR SUBMITTED MANUSCRIPT - draft

1 2 3 center, the effect of inherent nonlocality appears, words, this phase difference is a signature of the 4 nonlocality effect [9]. 5 which can turn photonic crystals into nonlocal photonic media with k-dependent parameters. To the surprise of 6 Ultratransparent photonic crystals can provide a pure most people, such nonlocality in photonic crystals can 7 dielectric platform for a large number of novel be utilized to realize of omnidirectional impedance 8 photonic devices such as nonlocal TO applications, matching [9]. 9 wide-angle polarization filter, microwave transparent 10 device, etc. One of the most important issues is the 11 Advances in Science and Technology to Meet 12 design strategy for the nonlocality in such photonic 13 Challenges crystals. Besides the EFCs that control the direction of 14 In Figs. 1(a) and 1(b), we show that the refractive wave propagation, the effective surface impedance 15 behaviors of a photonic crystal slab with a “shifted” should be independently engineered, so as to ensure 16 elliptical EFC are exactly the same as that of a omnidirectional non-reflection. More optimization and 17 metamaterial slab with an elliptical EFC at the engineering methods need to be further explored in 18 Brillouin Zone center. It is astonishing that photonic order to find out the best strategy as well as the limit of 19 20 crystals with such “shifted” EFCs could exhibit the 21 rare property of omnidirectional impedance matching 22 in TO media. It makes the photonic crystal non- 23 reflective and totally transparent for incident light from 24 any angle. This is clearly demonstrated in Fig. 1(c), 25 where a transmittance T>99% for all angles within (- 26 89 ͦ, 89 )ͦ is achieved in a slab of the photonic crystal 27 structure shown in the inset graph, irrespective of the 28 number of layers. In Fig. 1(d), the nonlocal (k- 29 dependence) nature of the effective permittivity and 30 permeability of this photonic crystal is clearly shown. 31 The retrieved parameters (dots) excellently match with 32 33 the derived theoretical requirement for omnidirectional 34 impedance matching (dashed lines). In a certain sense, 35 the omnidirectional impedance matching may also be 36 understood as the expansion of the Brewster’s angle 37 from a single angle to all angles. Therefore, a type of 38 “ultratransparent” nonlocal photonic medium beyond 39 any dielectric is possible [9]. Such an ultratransparency 40 effect also exist in one-dimensional photonic crystals 41 [10]. Moreover, the effect could also be broadband and 42 polarization-insensitive in some cases [10]. 43 44 Since the ultratransparent photonic crystals exhibit Figure 2 – (a) shows the design process of a wave 45 controllable refractive behaviors and omnidirectional concentrator by ultratransparent photonic crystals. (b) 46 impedance matching, their functionality is very similar shows the parameters of the TO concentrator. (c) shows 47 to the ideal TO media. In Fig. 2, we demonstrate an the EFCs of the discretized four layers of the TO (dashed 48 example of a wave concentrator. The original TO lines) and photonic crystal (solid lines) concentrators. (d) 49 concentrator is discretized into four layers (shown in and (e) show the simulated concentrating effect for the TO and photonic crystal concentrators, respectively. 50 Figs. 2(a) and 2(b)) and a core. Each layer can be 51 Reprinted with permission from Ref. [9] Copyright replaced by a corresponding type of ultratransparent 52 (2016) by the American Physical Society. 53 photonic crystal with an EFC of the same shape, but 54 “shifted” to the point in k-space, as shown in Fig. the realized nonlocal photonic media. 55 2(c). In the simulation results shown in Figs 2(d) and 56 2(e), one can see that the concentrating effects of the Concluding Remarks 57 original TO and photonic crystal concentrators are 58 almost the same, except for a phase difference of π in The original TO theory was established in the field of 59 the core region. Such a phase difference is induced by metamaterials with local medium description. Now, 60 the shift of EFC in k-space. As shown in Fig. 1(b), an pure dielectric photonic crystals with nonlocal additional phase of pd is added to the transmitted effective parameters have shown an almost equivalent waves where p denotes the shift of EFC in k-space and power in controlling both the refractive behavior and d is the thickness of the photonic crystal. In other the impedance matching behavior, for the first time. AUTHOR SUBMITTED MANUSCRIPT - draft Page 14 of 59

1 2 3 With the advance in the fabrication techniques of 4 5 large-scale photonic crystal with micrometer dielectric 6 structures, this route may lead to experimental 7 realization of ideal nonreflecting TO applications such 8 as cloaking in the optical frequency regime. 9 Acknowledgments – This work has been supported by 10 the State Key Program for of China 11 12 (No. 2014CB360505, No. 2012CB921501), National 13 Foundation of China (No. 11374224, 14 No. 61671314) and a Project Funded by the Priority 15 Academic Program Development of Jiangsu Higher 16 Education Institutions (PAPD). 17 18 References 19 [1] Pendry JB, Schurig D, and Smith DR 2006 20 Controlling electromagnetic fields Science 312 21 1780-1782 22 [2] Leonhardt U 2006 Optical conformal mapping, 23 Science 312 1777-1780 24 [3] Schurig D, Mock JJ, Justice BJ, Cummer SA, 25 Pendry JB, Starr AF, and Smith DR 2006 26 Metamaterial electromagnetic cloak at microwave 27 frequencies, Science 314 977-980 28 [4] Lai Y, Ng J, Chen H, Han D, Xiao J, Zhang Z, and 29 Chan CT 2009 Illusion optics: The optical 30 transformation of an object into another object Phys 31 Rev Lett 102 253902 32 [5] Joannopoulos JD, Johnson SG, Winn JN, and Meade 33 34 RD 2008 Photonic Crystals: Molding the Flow of 35 Light (Princeton University Press, Princeton, USA, 36 2 ed.) 37 [6] Urzhumov YA, and Smith DR 2010 Transformation 38 optics with photonic band gap media Phys Rev Lett 39 105 163901 40 [7] Liang Z, and Li J 2011 Scaling two-dimensional 41 photonic crystals for transformation optics Opt. 42 Express 19 16821-16829 43 [8] Huang X, Lai Y, Hang ZH, Zheng H, and Chan CT 44 2011 Dirac cones induced by accidental degeneracy 45 in photonic crystals and zero-refractive-index 46 materials Nat Mater 10 582-586 47 [9] Luo J, Yang YT, Yao ZQ, Lu WX, Hou B, Hang 48 ZH, Chan CT, and Lai Y 2016 Ultratransparent 49 Media and Transformation Optics with Shifted 50 Spatial Dispersions Phys Rev Lett 117 223901 51 [10] Yao ZQ, Luo J, and Lai Y 2016 Photonic crystals 52 with broadband, wide-angle and polarization- 53 insensitive transparency Opt Lett 41 5106-5109 54 55 56 57 58 59 60 Page 15 of 59 AUTHOR SUBMITTED MANUSCRIPT - draft

1 2 3 4 4. Complexification in Transformation 5 Optics –S. A. R. Horsley 6 University of Exeter 7 8 Status 9 10 Transformation optics is a theory where changing the 11 coordinate system is used as a tool to solve the difficult 12 problem of wave propagation in inhomogeneous 13 materials. While we often take these coordinates as real 14 numbers, transformation optics has even more power 15 when we allow them to take complex values. 16 17 Undoubtedly, conformal transformations are the most 18 widespread application of complex valued coordinates. 19 They have long been applied to solve Laplace’s 20 in two dimensions, in areas such as 21 , , and elasticity [1]. A 22 conformal transformation takes the complex numbers 23 z=x+iy and z*=x-iy (formed from the Cartesian 24 coordinates x and y) to new complex numbers w(z) and 25 w*(z*). Such a transformation leaves the form of 26 27 Laplace’s equation unchanged, and this form invariance 28 allows one to map simple solutions in simple 29 geometries, to equivalent solutions in more complicated 30 ones. Meanwhile, a conformal transformation of the 31 Helmholtz equation maps wave propagation in one 32 refractive index profile, to propagation in another 33 perhaps much more intricate one. Transformation 34 optics has made heavy use of conformal transformations 35 to design optical materials. For instance, Leonhardt [2] Figure 1: (a) Schematic of a complex coordinate 36 used a conformal transformation along with some clever transformation of the , where the real line 37 analysis of a branch cut to design one of the first x=x1 is transformed to a curve C that ventures into the 38 invisibility cloaks; and more recently Pendry and others analytically continued wave field. This process changes 39 both the material parameters and the wave field in the 40 [3] developed a theory of plasmonic energy concentrators based on conformal transformations. wave equation; (b) an example of the numerical solution 41 of the 1D wave equation in the complex plane, in this 42 But the utility of complex coordinates is much broader case showing a series of branch cuts (dashed red lines) 43 than conformal mapping. While a conformal map emerging from poles along the imaginary axis. 44 replaces the two-dimensional coordinates x and y with 45 the single z and its complex conjugate truncating a computational domain - is founded on this 46 z*, one can also treat the x,y coordinates in the wave approach. 47 48 equation as complex numbers themselves. Such a ‘complexification’ effectively doubles the number of 49 50 dimensions of the space. 51 At the simplest level, this extension of the coordinate Continued interest in transformation optics has led to 52 system allows one to find inhomogeneous materials that new work in this area, applying complex valued spatial 53 coordinates to design materials with inhomogeneous 54 change the amplitude of a wave without generating any reflection (through a transformation like that shown in distributions of loss and gain [5-7]. This has been 55 teamed with the simultaneous realisation that such figure 1a). As a rough illustration consider a plane wave 56 ‘lossy and gainy’ optical materials can be used for the 57 in 1D, ϕ=exp(inkx) (obeying the Helmholtz equation 2 2 2 experimental investigation of a breed of complex valued 58 d ϕ/dx +(nk) ϕ=0). Performing a complex rotation of 59 the x coordinate x=(a+i b)x’, we obtain either an (‘PT symmetric’) potentials that have long been a 60 exponentially growing or decaying wave, replacing the curiosity in quantum mechanics [8]. For example it has original real index n with the complex index (a+ib)n. As been found that reflectionless materials with PT- emphasized by Chew [4], the theory of the perfectly symmetry can be designed using a complex coordinate transformation of the Maxwell equations [6]. matched layer – nowadays the most common method for

AUTHOR SUBMITTED MANUSCRIPT - draft Page 16 of 59

1 2 3 This ‘complexified transformation optics’ allows for an both to develop a sufficient range of available material 4 5 interesting way of thinking. When we ordinarily solve parameters, and to make the fabrication error small 6 the wave equation in real space we don’t consider the enough that the devices function as we intend them to. analytic continuation of the coordinates - the 7 Although there are many experiments in this area, I am coordinates in the lab are real numbers! But, were we 8 just trying to make the point that many of our designs to look more carefully at the solution to the wave 9 remain very difficult to put into practice are not yet equation we would find particular features (e.g. poles or 10 practically feasible. This is especially true for some of 11 branch cuts) that occur in the wave field at complex the designs where both the real and imaginary parts of 12 values of the coordinates (see figure 1b). Through the material parameters are graded in space. 13 introducing an inhomogeneous anisotropic material, a 14 complex coordinate transformation can move these There are also some interesting mathematical questions 15 features into real space [5,6]. Transformation optics to address. For instance, we know that making the 16 with complex coordinates gives a physical meaning to coordinates complex can lead to general results 17 concerning the reflection and transmission of waves 18 the analytic continuation of a wave field! through inhomogeneous media [5,7], and that features 19 Taking seriously the analytic continuation of the wave in the complex position plane can be given a physical 20 equation has recently provided some other new and meaning via transformation optics [6]. But to date we 21 rather general results [5,7]. These apply to planar have largely based our understanding on the theory of 22 media where the refractive index (or equivalently the 23 functions of a single complex variable. Yet permittivity) varies as a function of one of the 24 mathematicians have developed the theory of several coordinates x. If the refractive index n(x) is an analytic 25 complex variables – seldom used in optics - with function in either the upper or lower half complex x 26 concepts such as ramification taking the place of the plane then the material will not reflect waves incident 27 branch cut. An interesting question is whether the from one side, whatever the angle of incidence. In 28 theory of functions of several complex variables can be addition, if n(x) has only poles of order higher than 1 in 29 applied to derive useful information about the 30 the complex position plane then the slab will also have interaction of waves with structured media. In a similar 31 the same transmission coefficient as free space – it will spirit, it is also interesting to ask whether the theory of 32 be invisible from one side. The construction of such complex (and special cases such as the Kahler 33 reflectionless media is in many cases also practically ) can be applied to complex coordinate 34 feasible, and one such material has recently been transformation optics. 35 experimentally realised [9]. 36 Another interesting connection that may be productive 37 to explore, is the connection between complex 38 39 Current and Future Challenges coordinates and the one-way edge states that are currently attracting interest in the field of ‘topological 40 If we want to extend the reach of complex analysis in photonics’ [10]. This connection does not seem to have 41 wave propagation problems there are both mathematical been explored, but can be straightforwardly observed in 42 and practical challenges. Here I shall mention some the cases where the effective refractive index vanishes 43 which I know of, and some questions I think are 44 so that the field obeys the Laplace equation ∇2ϕ=0. interesting to ask. 45 Because the system only supports interface states that 46 I think the practical challenges are quite clear. Take the propagate in one direction, the field must become a 47 application of conformal transformations to cloaking, or function of a single complex variable e.g. only z, i.e. 48 to the design of plasmonic structures (e.g. gratings). ϕ(z)=exp(kz)=exp[k(x+i y)]. 49 Because conformal transformations are a special kind of 50 coordinate transformation, then there is the question of 51 52 what happens when we deviate from the condition of Advances in Science and Technology to Meet 53 conformality. No experiment will be able to implement Challenges a refractive index profile accurate to an infinite number 54 The above mathematical challenges pose questions that of decimal places. The same problem is also evident for 55 do not require any new mathematics, but require further the analytic profiles given in [7] – no experiment will be 56 time to answer. In contrast, I think the experimental 57 able to fabricate a profile that is exactly equivalent to a realisation of many of the materials we design does 58 function that is analytic in the upper half complex require advances in technology. 59 position plane. There is always some tolerance to error, 60 but the experiment needs to be within this tolerance. But Probably the most important is the ability to precisely In many cases I think it remains a challenge to build the and separately control the reactive and dissipative devices we are inventing with transformation optics response of materials, so that refraction, loss and gain (whether using real or complex valued coordinates); can all be separately graded in space. This could involve Page 17 of 59 AUTHOR SUBMITTED MANUSCRIPT - draft

1 2 3 the development of metamaterial elements with a wide [8] Ruter C E, Makris K G, El-Ganainy R, Christodoulides D 4 N, Segev M and Kip D 2010 Observation of parity-time 5 range of complex effective permittivity and permeability values. Gain is an especially difficult symmetry in optics Nat. Phys. 6 192-195 6 [9] Jiang W, Ma Y, Yuan J, Yin G, Wu W and He S 2017 7 property to control, and present implementations often lead to instabilities. Deformable broadband engineered 8 with an analytical spatial Kramers-Kronig permittivity profile 9 Although there are cases where both real and imaginary Laser Phot. Rev. 11 1600253 10 parts of the material parameters have been separately [10] Lu L, Joannopoulos J D and Soljacic M 2014 11 controlled in space (see e.g. [9]), at the moment this is Topological Photonics Nat. Phot.. 8 821-829 12 13 generally a difficult task. We should mention that [9] 14 grades the resonant properties of metamaterial elements 15 to obtain one of the non-reflecting index profiles derived 16 in [7], using the frequency domain Kramers-Kronig 17 relations to guarantee the required distribution of 18 material properties. Such an approach may be useful 19 more generally. 20 21 22 Concluding Remarks 23 24 Complex analysis is an old topic that is being given new 25 practical relevance in the field of metamaterials 26 research. It is used within transformation optics both 27 through conformal transformations in two dimensions, 28 and through the complex transformations and analytic 29 continuation used to investigate materials with 30 distributions of loss and gain. In the future there 31 promise to be further applications of the mathematics of 32 33 complex coordinates in optics, which may enable a 34 deeper understanding of wave propagation in complex 35 inhomogeneous media. 36 37 38 Acknowledgments – SARH acknowledges financial 39 support from a Royal Society TATA University 40 Research Fellowship (RPG-2016-186). 41 42 43 References (separate from the two page limit) 44 [1] Schinzinger R S and Laura P A A 1991 Conformal 45 Mapping: Methods and Applications, Dover (New York) 46 [2] Leonhardt U 2006 Optical Conformal Mapping Science 47 312 1777-1780 48 [3] Pendry J B, Aubry A, Smith D R and Maier S A 2012 49 Transformation Optics and Subwavelength Control of Light 50 Science 337 549-552 51 [4] Chew W C and Weedon W H 1994 A 3D perfectly 52 matched medium from modified Maxwell’s equations with 53 stretched coordinates Micro. Opt. Tech. Lett. 7 599-604 54 [5] Horsley S A R, King C G and Philbin T G 2016 Wave 55 propagation in complex coordinates J. Opt. 18 044016 56 [6] Castaldi G, Savoia S, Galdi V, Alu A and Engheta N 2013 57 PT Metamaterials via Complex-Coordinate Transformation 58 Optics Phys. Rev. Lett. 110 173901 59 [7] Horsley S A R, Artoni M and La Rocca G C 2015 Spatial 60 Kramers-Kronig relations and the reflection of waves Nat. Phot. 9 436-439 AUTHOR SUBMITTED MANUSCRIPT - draft Page 18 of 59

1 2 3 4 5. Surface Transformation Optics 5 5.1 Extending transformation optics and towards a 6 thin cloak – Jensen Li, Jian Zhu, University of 7 Birmingham 8 9 Status 10 11 The interest of getting an invisibility cloak has a very 12 long history but its main breakthrough came from the 13 establishment of transformation optics (TO). It is the 14 theoretical framework of TO to give us a recipe of the 15 required material profile of a cloak while 16 metamaterials provide us the necessary palette of 17 material parameters. The first metamaterial cloak was 18 realized in the microwave regime within half a year 19 20 from the theory proposal [1]. Figure 1(a) shows the 21 dispersion surface of a TO medium when a light bends 22 from the freespace to the surface layer of a cloak while Figure 1 Ways to redirect light beam (a) by changing the 23 Fig. 2(a) shows the simulation of a cylindrical cloak shape of dispersion surface from TO-media, (b) by 24 for a plane wave coming from the left. The dispersion splitting dispersion surface from gauge transform-media. 25 surface for both the TE and TM polarizations become Vertical grey line indicates an interface. (c) Luneburg lens for getting variable directions from point sources. 26 generally elliptical with a change in shape to redirect 27 light. The deformation gets bigger near the inner loss so that active applications like fictitious source 28 surface of cloak with more extreme material generation and one-way invisibility can be achieved 29 parameters. It was then quickly recognized that [3]. On the other hand, we can add a fourth coordinate: 30 metamaterials cannot provide these extreme parameters time. A dynamic event cloak becomes realizable [4]. It 31 for a cloak to work at optical frequencies due to the turns out coordinate transformation is not the only 32 possibility to get form-invariance. The transformation 33 limit of resonance strength. One way out is to make a can also happen in the polarization, an internal degree 34 compromise on the functionality. If we conceal an 35 object under a carpet instead of in the middle of air, the of freedom, as a gauge transformation [5]. An example 36 transformation involved compresses the object to a flat is an abstract rotation in the polarization space for 2D 37 sheet instead of a point. This strategy makes the range wave propagation: 38 of the required material parameters much less extreme. ( , ) ( , ) 39 Cloaks for visible light, Fig. 2(b), become realizable = . ′ ( , ) ( , ) 40 (e.g. Ref [2]). Since then, we have been witnessing a 𝑬𝑬𝒛𝒛 𝐜𝐜𝐜𝐜𝐜𝐜 𝝓𝝓 𝒙𝒙 𝒚𝒚 − 𝐬𝐬𝐬𝐬𝐬𝐬 𝝓𝝓 𝒙𝒙 𝒚𝒚 𝑬𝑬𝒛𝒛 41 � ′ � � � � � dynamic period of development of TO in one of two The transformation𝒊𝒊𝑯𝑯𝒛𝒛 𝐬𝐬𝐬𝐬𝐬𝐬 𝝓𝝓 induces𝒙𝒙 𝒚𝒚 a corresponding𝐜𝐜𝐜𝐜𝐜𝐜 𝝓𝝓 𝒙𝒙 𝒚𝒚 one𝒊𝒊𝑯𝑯 𝒛𝒛on the 42 directions: one on further simplifying cloaking material parameters and forms its own class of 43 strategies to have the ultimate application of cloak in materials. Figure 1(b) shows the class generated from 44 such an abstract rotation. The transformed permittivity 45 the real world; another on extending TO as a 46 framework to design optical components with has additional anisotropy and . Equivalently, functionalities beyond the conventional ones. The the dispersion is split into two polarizations by shifting 47 𝒙𝒙𝒙𝒙 𝒚𝒚𝒚𝒚 48 simplification route pushes the cloak towards higher the circular one into two opposite𝝐𝝐 directions𝝐𝝐 in the k- 49 frequencies, towards 3D applications, and towards space (vs. deforming shape in Fig. 1(a)). This splitting 50 cloaking objects of macroscopic sizes. The can be interpreted as a gauge field (pseudo vector 51 generalization route applies TO on getting devices potential) for photon. It provides a “magnetic” force to 52 beyond a cloak, e.g. to project an additional virtual redirect light and can be utilized to design a cloak [6]. 53 object as an optical illusion, to design silicon photonic Figure 2(c) shows the simulations for a cylindrical 54 components with optimized performance, and to apply cloak designed with gauge field approach. The rays 55 transformation approach to other kinds of waves. passing through the cloak are squeezed upwards (vs. 56 Fig. 2(a)). This occurs in opposite ways for 57 A consideration on how to extend the the two polarizations. Such an asymmetry with the 58 fundamental principle of TO can actually benefit both “magnetic” bending force gives us a route to design 59 directions. Coordinate transformation is at the heart of TO devices with asymmetric transmission 60 TO to get form-invariance of Maxwell’s equations. A functionality. Figure 2(d) shows an example to real 3-tuple ( , , ) is mapped to another 3-tuple. One generate a one-way edge state, which propagates possibility is to generalize the coordinates to complex numbers. It extends𝒙𝒙 𝒚𝒚 𝒛𝒛 TO to include material gain and Page 19 of 59 AUTHOR SUBMITTED MANUSCRIPT - draft

1 2 3 between two regions of metamaterials and is not angles and thickness of cloak. Naturally, we can go 4 from a single layer of metamaterial atoms to multiple 5 reflected even when a sharp corner is encountered. layers or a thicker layer. Recently, the concept of 6 Current and Future Challenges 7 Huygen’s metasurfaces with two layers of 8 Although generalizations allow us to extend TO metamaterial atoms have found excellent usage in 9 applications to new territories, a major current enhancing transmission efficiency, utilizing both 10 challenge on designing a cloak using the TO approach electric and magnetic response of the metamaterial 11 stays behind: to obtain a thin cloak, working for wide atoms, with a homogenized boundary description of 12 angles in 3D, which are essential for a wearable cloak. the metasurface [8]. Such a homogenized boundary 13 Taking the cylindrical cloak as example (Fig. 2(a) with description may be utilized to formulate TO instead of 14 ray-tracing), intuitively the outermost cylindrical layer using effective medium parameters of the bulk. By 15 is responsible for the light hitting the cloak at a shallow designing successive layers of the metasurfaces, the 16 angle, while an inner one, together with the layers cloaking behavior at a series of viewing angles can be 17 outside, are responsible for the light nearer to normal 18 obtained from normal to oblique incidence. incidence. The angular operation thus comes from a 19 Another approach to get angular operation for finite thickness of the cloak. Unless very extreme 20 a thin cloak takes advantage of the concept of material parameters are allowed, a small thickness and 21 imaging. Figure 1(c) shows the basic principle in wide-angle operation are difficult to realize at the same 22 redirecting light by a lens array. Unlike TO, in which time. It also means that we can go further on the 23 dispersion surfaces are defined continuously at any 24 simplification route to get a thin cloak but with limited locations inside the device, the device is pixelized into 25 viewing angle. 26 an array of lenses and each lens controls the direction 27 We have already seen some successful of the generated light from a point source at different 28 demonstrations along this direction. By going to the foci. Such an array can be used to project the desired 29 extreme case of only one single layer of metamaterial 3D image you would like the observer to perceive in 30 atoms, as a metasurface, a carpet cloak at the normal the case of a cloak. A preliminary demonstration by 31 incidence can be designed by phase compensation to putting a 1D array of cylindrical lens on top of a digital 32 get a target reflected plane wave. Surprisingly, even display has achieved a cloak with viewing angle up to 33 though it is designed at normal incidence, the working . °, varying in a single direction [9]. Although the 34 angle can be as large as 30 degrees in a demonstrated viewing angle is limited due to paraxial geometrical 35 3D metasurface cloak [7] (Fig. 2(e)). For the carpet optics,𝟏𝟏𝟏𝟏 𝟒𝟒 it gives a practical approach for a thin cloak 36 cloak, it is also desirable to go beyond a flat mirror to with all three colors, with a compromise on spatial 37 mimic an arbitrary background image on the floor. 38 resolution of the image. To get an omnidirectional 39 Advances in Science and Technology to Meet viewing angle, one can replace the 1D paraxial lens 40 Challenges array by a 2D array of Luneburg lens (as Fig. 1(c) 41 depicts). Figure 2(f) shows schematically if we 42 assemble such Luneburg lens arrays into a pyramid, 43 which projects the image on the floor consisting of 44 circular-disk patterns to any angles. Furthermore, to 45 focus light coming from different directions to the 46 Luneburg lens array on a common plane, TO will be 47 very favorable to realize flattened Luneburg lens [10] 48 for the lens array. 49 50 Concluding Remarks 51 52 Based on the early works of TO together with 53 metamaterials, we have now the basic language in 54 designing a cloak. We believe there are still huge 55 opportunities for us to develop TO. A fundamental Figure 2 (a) and (b): cylindrical and carpet cloak at approach in extending TO’s theoretical framework 56 visible light from coordinate transform (from Ref.[2]). 57 (c) and (d): gauge transform to get a cloak and one-way should be beneficial to both getting a realistic cloak 58 edge state. (e) and (f) are 3D carpet cloaks by and designing optical components with properties 59 metasurface (from Ref. [7]) and by lens array. beyond those given by conventional approaches. 60 Probably, the wearable cloak for visible light will be an To enhance the working angle for a thin cloak, bearing extension or combination of the TO technique with in mind the issues we have discussed earlier, one approaches such as metasurfaces and integral imaging. natural step is to make a balance between working AUTHOR SUBMITTED MANUSCRIPT - draft Page 20 of 59

1 2 3 Acknowledgments – We acknowledge support from 4 5 European Union’s Seventh Framework Programme 6 under Grant Agreement No. 630979. 7 8 References (separate from the two page limit) 9 [1] Schurig D, Mock J J, Justice B J, Cummer S A, 10 Pendry J B, Starr A F, and Smith D R 2006 11 Metamaterial Electromagnetic Cloak at Microwave 12 Frequencies Science 314 977-980 13 [2] Gharghi M, Gladden C, Zentgraf T, Liu Y, Yin X, 14 Valentine J and Zhang X 2011 A carpet cloak for 15 visible light Nano Lett. 11, 2825–2828 16 [3] Castaldi G, Savoia S, Galdi V, Alù A and Engheta N 17 2013 PT Metamaterials via Complex-Coordinate 18 Transformation optics. Phys. Rev. Lett. 110, 173901 19 [4] McCall M W, Favaro A, Kinsler P, and Boardman 20 A 2010 A spacetime cloak, or a history editor J. 21 Opt., 13, 024003 22 [5] Liu F and Li J 2015 Gauge field optics with 23 anisotropic media Phys. Rev. Lett. 114 103902 24 [6] Liu F, Horsley S A R and Li Jensen 2017 25 Invisibility cloaking using pseudomagnetic field 26 for photon Phys. Rev. B 95 075157 27 [7] Ni X et al 2015 An ultrathin invisibility skin cloak 28 29 for visible light Science 349 1310-1314 30 [8] Pfeiffer C and Anthony G 2013 Metamaterial 31 Huygens’ surfaces: tailoring wave fronts with 32 reflectionless sheets Phys. Rev. Lett. 19 197401 33 [9] Choi J S and Howell J C 2016 Digital integral 34 cloaking Optica 3 536 35 [10] Hunt J, Tyler T, Dhar S, Tsai Y-J, Bowen P, 36 Larouche S, Jokerst N M and Smith D R 2012 37 Planar, flattened Luneburg lens at 38 wavelengths Opt. Exp. 20 1706-1713 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Page 21 of 59 AUTHOR SUBMITTED MANUSCRIPT - draft

1 2 3 5.2 Combining curvature and index gradients for 4 5 surface and guided waves − 6 Rhiannon C Mitchell-Thomas1 and Oscar Quevedo- 7 Teruel2 8 1University of Exeter 9 2KTH Royal Institute of Technology 10 11 Status 12 13 Transformation optics has been proven to be a very 14 versatile and powerful tool. It allows a link between 15 geometry and the properties of a material to be formed, 16 which is exact for waves propagating through a 17 medium. The method employs a coordinate 18 transformation to predefine the desired wave 19 propagation characteristics, and this transformation is 20 21 then used to calculate the appropriate permittivity and Figure 1 – in graded index profiles and the 22 permeability with which to fill space so that this equivalent homogeneous surfaces for the Eaton lens and 23 behaviour is achieved. For waves confined to a surface, a Monopole lens. 24 provides a similar link between 25 geometry and materials [1]. In contrast to the complex incident on the profile are unable to escape and the 26 material properties required by transformation optics, equivalent surface is no longer closed. 27 the use of geometrical optics results in purely isotropic These equivalent surfaces have found only limited 28 index profiles. These profiles are inhomogeneous, with practical application [3,4], and one of the reasons is 29 a rotationally symmetric index , meaning that that the shape of the curved surfaces may have 30 they vary only along the radial direction of a polar undesirable features. For example, the surfaces shown 31 coordinate system, but not the angular. It also allows 32 in figure 1 are tall with respect to the radius, and also for surfaces with non-zero curvature to be considered, 33 have a 90 degree transition when positioned on a flat 34 which is not possible with transformation optics, and surface. To overcome this issue, a combination of 35 this provides new opportunities for device design. curvature and index gradients can be used to achieve 36 Familiar lenses, such as the Luneburg and the Maxwell the same behaviour, but with more freedom in the 37 fish eye lens can be derived using geometrical optics. surface shape and index contrast can be limited to 38 These isotropic, rotationally symmetric index profiles achievable values [5,6,7]. It is in this context that 39 geometrical optics offers the most freedom in terms of 40 are accurate in the limit of geometrical optics, and hence the lenses must be electrically large to operate practical device design, and full advantage can be 41 taken of this versatile technique. 42 effectively. Using geometrical optics, equivalent 43 surfaces for these types of lenses can be calculated [2]. Current and Future Challenges 44 For example, the Maxwell fish eye lens, which has a This combination of curvature and index gradients 45 radially dependent refractive index profile that varies 46 from one at the perimeter to two at the centre has an finds application in a number of areas, and three 47 equivalent surface which is a homogeneous different devices will be discussed here. The first of 48 hemisphere. This lens has ray trajectories such that a which is a family of lenses that have very appealing 49 point source positioned on the circumference will be properties, yet are impossible, or very difficult to 50 imaged on the opposite side of the lens, due to the fabricate in practice. Two examples are shown in 51 graded index. Equivalently, rays emitted from a point figure 1, including the Eaton lens, which retro-reflects 52 all rays that are incident on it, and a Monopole lens, 53 that are confined to the surface of a hemisphere would converge to a point on the opposite side. In this case, it which can be used so that an omnidirectional source 54 only emits energy in one half-space of a 2D plane. 55 is the curvature that accounts for the focusing, as the material of this surface is homogeneous. The However, the common feature of these lenses is that 56 they exhibit a singularity in the centre of their 57 equivalent surfaces for a whole family of these refractive index profiles. It has been shown that 58 rotationally symmetric lenses have been derived, 59 including those with singularities at their centres, and curvature can be employed to emulate the influence of 60 two examples of the latter are shown in figure 1. the singularity using a pointed tip at the centre, as However, there is a limitation. This is when the index shown in figure 1, where the gradient of the tip is profile increases at a rate equal to or more rapidly than defined by the speed with which the profile approaches infinity [5]. In order to design a practical device, a . When this condition is reached, all rays that are graded index with a modest contrast of 1.6 is combined 𝟐𝟐 𝟏𝟏⁄𝒓𝒓 AUTHOR SUBMITTED MANUSCRIPT - draft Page 22 of 59

1 2 3 with a curved parallel plate waveguide so that the field 4 5 is confined between the plates. The top image in figure 6 2 shows the in this waveguide, and it can 7 be seen to accurately retro-reflect the incident input 8 beam so that the output beam exits the device in the 9 same direction [5]. 10 A further use of geometrical optics is to redesign lenses 11 12 so that they are conformal to existing surfaces [6]. The 13 increasing requirements for ever higher data transfer 14 rates in systems demands improved antennas 15 with higher directivity, yet with decreased weight and 16 volume that are less intrusive in the surrounding 17 environment. One solution is to employ surface wave 18 lenses, that are thin and lightweight, but with a large 19 surface area. They can be applied to existing surfaces 20 of , for example, but these surfaces may not be 21 flat. In this case, it is known that the curvature of the 22 surface will degrade the performance of a surface wave 23 lens. However, it is possible to calculate a modified 24 25 version of a graded index lens that is appropriate for a 26 given surface with curvature. In the middle image of 27 figure 2, a Luneburg lens on a curved surface is shown. 28 It can be seen that the point source excitation is Figure 2 – Examples of devices designed using 29 transformed to a plane wave output beam, which could geometrical optics that employ curvature. Top: An 30 then be leaked into a beam propagating in free space, Eaton lens with a modest index contrast of approx. 1.6. 31 creating the for a design of a conformal antenna Middle: A Luneburg lens conformal to a curved surface. 32 [6]. Bottom: A surface cloak where the influence of the 33 curvature has been removed with an appropriate index 34 Finally, an alternative version of a surface wave cloak profile. can be created, without the use of the transformation 35 individually cast [8]. For waves bound to an interface 36 optics technique [7]. In this case, a curved surface is between a dielectric and free-space, it is also possible 37 employed to guide a wave up and over the object to be 38 cloaked. This curvature will cause an undesired to achieve a change in speed of the supported mode 39 distortion to the incident wave and leave a signature. simply by varying the thickness of a homogenous 40 The route to eliminating any distortion is to counter the dielectric [9]. Alternatively, metasurfaces can be 41 influence of the curvature with use of a graded index employed, where the index gradient is achieved by 42 profile, so that this combination now has an equivalent slowly varying the geometrical parameters of the 43 surface that is flat and homogeneous. A surface cloak individual meta-elements. This type of implementation 44 would limit the operational bandwidth of a device due 45 is shown in the bottom image in figure 2, where the input plane wave can be seen to be undistorted after to the resonant nature of commonly used metasurfaces, 46 but new advances in metasurface design have shown 47 propagating through a section of curved waveguide, due to the rotationally symmetric index which fills the that these limitations can be considerably relaxed if 48 higher symmetries are employed [10]. 49 curved region [7]. This type of cloak can be used to 50 avoid that would be caused by an obstruction Rather than rely upon a rapid advance of a certain 51 on a surface. technology to meet requirements, this method of 52 device design is chosen for its suitability with the Advances in Science and Technology to Meet 53 current state of the art of manufacturing techniques. Challenges 54 The fact that no complex anisotropy or simultaneous 55 All of the above devices require an isotropic index magnetic and electric properties are required is one of 56 gradient to be achieved, for which there are multiple 57 the major advantages of this technique. approaches. Dielectric materials, engineered so that the 58 Concluding Remarks 59 can be precisely controlled are 60 achievable with the combination of a low dielectric Here, a description of the application of geometrical host medium loaded with high dielectric particles. In optics to design a new range of devices has been given. order to achieve a graded index, usually a This method allows curvature to be employed when discretization is applied and a number of layers are creating surface or guided wave devices, which only Page 23 of 59 AUTHOR SUBMITTED MANUSCRIPT - draft

1 2 3 require very basic material properties. The addition of 4 5 this extra degree of freedom, surface curvature, permits 6 the definition of purely isotropic materials and can 7 negate the need for singularities in refractive index 8 profiles. Three examples have been shown, including 9 an Eaton lens with a finite and modest refractive index 10 contrast, a conformal Luneburg lens, and a surface 11 cloak, demonstrating the versatility of the technique. 12 All of these devices can be fabricated by employing 13 either metasurface or graded dielectric approaches. 14 15 Acknowledgments – The first author was funded by 16 the Engineering and Physical Sciences Research 17 Council (EPSRC), UK under Grand Challenges Grant 18 (EP/N010493/1) ‘‘SYnthesizing 3D METAmaterials 19 for RF, microwave and THz applications (SYMETA).” 20 21 References 22 [1] Leonhardt U and Philbin T G, 2010 Geometry and 23 Light: The Science of Invisibility Dover, New 24 York 25 26 [2] Sarbort M and Tyc T 2012 Spherical media and 27 geodesic lenses in geometrical optics Journal of 28 Optics 14 075705 29 [3] Kunz K S 1954 Propagation of Microwaves 30 between a Parallel Pair of Doubly Curved 31 Conducting Surfaces Journal of 32 25 642 33 [4] DuFort E C, Uyeda H A 1984 Geodesic Dome- 34 Lens Antenna US Patent 4,488,156 35 [5] Horsley S A R, Hooper I R, Mitchell–Thomas R 36 C and Quevedo–Teruel O 2014 Removing 37 Singular Refractive Indices with Sculpted 38 39 Surfaces Scientific Reports 4 4876 40 [6] Mitchell-Thomas R C, Quevedo-Teruel O, 41 McManus T M, Horsley S A R and Hao Y 2014 42 Lenses on Curved Surfaces Optics Letters 39 43 3551-3554 44 [7] Mitchell-Thomas R C, McManus T M, Quevedo- 45 Teruel O, Horsley S A R and Hao 2013 Y Perfect 46 Surface Wave Cloaks Physical Review Letters 47 111 213901 48 [8] Quevedo-Teruel O, Tang W, Mitchell-Thomas R 49 C, Dyke A, Dyke H, Zhang L, Haq S and Hao Y 50 51 2013 Transformation Optics for Antennas: Why 52 limit the bandwidth with 53 Metamaterials? Scientific Reports 3 1903 54 [9] Mitchell-Thomas R C, Quevedo-Teruel O, 55 Sambles J R and Hibbins A P 2016 56 Omnidirectional surface wave cloak using an 57 isotropic homogeneous dielectric coating 58 Scientific Reports 6 30984 59 [10] Quevedo-Teruel O, Ebrahimpouri M, Ng Mou 60 Kehn M 2016 Ultra Wide Band Metasurface Lenses Based on Off-Shifted Opposite Layers IEEE Antennas and Wireless Propagation Letters 15 484 AUTHOR SUBMITTED MANUSCRIPT - draft Page 24 of 59

1 2 3 property in a flat or curved plane. This makes 4 5.3 Transformation optics at surfaces – Philippe transformation-optical techniques applicable to 5 Tassin and Vincent Ginis 6 Chalmers University of Technology and Vrije metasurfaces very interesting. 7 Universiteit Brussel 8 9 Status 10 11 About a decade ago, transformation optics established 12 a new way to take advantage of the unprecedented 13 properties of metamaterials for the design of optical 14 devices. The formalism relies on the form-invariance 15 of Maxwell’s equations for the propagation of light in 16 an inhomogeneous medium under coordinate 17 transformations. This invariance leads to an 18 equivalence relation between the components of a 19 metric and the permittivity and permeability of a 20 medium. The design technique for optical devices then 21 goes as follows: start from a Cartesian grid and bend 22 23 the coordinate lines until they follow the desired path. 24 In this way, one defines a , which can 25 subsequently be used to calculate the spatially varying 26 permittivity and permeability functions needed for an 27 optical device in which light rays follow paths along 28 the transformed coordinate lines. 29 30 Transformation optics as a design technique for optical

31 devices has three major advantages compared to traditional design techniques. First, it allows dividing Figure 1 – Illustration of a light beam reflected and 32 transmitted at the surface of a transformation-optical 33 the optical design into two steps. One starts from the material. With the use of a coordinate transformation, we 34 desired electromagnetic field distributions to determine want to engineer the properties of the scattered beams—for 35 the material properties of the device. These material example, direction of propagation, beam shape, and 36 properties need then be implemented with frequency. 37 nanostructured metamaterials—a challenging problem 38 in itself. The second advantage is that transformation When considering transformation optics at surfaces, we 39 optics is, just as Gaussian optics for traditional optical must distinguish between devices where a light beam is 40 instruments, a completely geometrical procedure. scattered by a transformation-optical surface (Figure 1) 41 42 Finally, transformation optics is not diffraction-limited. and devices where a light wave propagates along a 43 Since it is based on Maxwell’s equation, it is a full- transformation-optical surface (Figure 2). wave technique, and optical devices designed by it do 44 In one of the first steps towards transformation optics not suffer from geometrical aberrations. 45 on surfaces, a thin metamaterial coating was designed 46 Current and Future Challenges to enhance the optical force between two traditional 47 optical waveguides [1]. If the metamaterial implements 48 In its original formulation from 2006, transformation an annihilating transformation, then it is decreasing the 49 optics is inherently a three-dimensional method. It effective interwaveguide distance experienced by the 50 relies on continuous transformations between light waves in the underlying electromagnetic space, 51 coordinate systems in a three-dimensional space, thus enhancing the optical between the 52 generating permittivity and permeability functions that waveguides. Annihilating transformations can be 53 vary smoothly as a function of the space coordinates. 54 obtained from metasurfaces with a negative index of This method creates optical devices with permittivity 55 refraction. 56 and permeability varying in all three dimensions. Nevertheless, one can make use of discontinuous 57 However, this is not very desirable from a transformations to describe electromagnetic 58 technological point of view. Metamaterials research phenomena that occur at an interface between a 59 has increasingly focused on the fabrication of so-called common material (or air) and a metamaterial surface, 60 metasurfaces, two-dimensional arrays of meta-atoms, e.g., the Goos-Hänchen shift [2]. If the metamaterial is which are easier to fabricate and still allow for versatile the transformation-optical equivalent of a coordinate transmission and reflection properties. Metasurfaces transformation, then discontinuous transformations can now be designed to have almost any optical Page 25 of 59 AUTHOR SUBMITTED MANUSCRIPT - draft

1 2 3 allow us to understand and engineer phenomena at the well as in nanofabrication. For the modification of light 4 beams incident on a surface, as visualized in Figure 1, 5 surface of nanophotonic structures in terms of a 6 geometrical framework. transformation optics has been shown to be applicable 7 to homogeneous surfaces, but there is no geometrical 8 A second reason for wanting to apply transformation theory for inhomogeneous surfaces. Inhomogeneous 9 optics to thin two-dimensional optical systems is that metasurfaces were demonstrated to be able to achieve 10 optical devices are increasingly being integrated on advanced beam manipulation, e.g., generalized 11 photonic integrated circuits. One of the main refraction [9], but it is currently not completely 12 advantages of integration is of course a smaller understood how to describe and engineer this response 13 footprint. However, with routing of optical signals in using coordinate transformations. 14 such circuits being achieved with strip or rib 15 waveguides, bends and processing elements become 16 17 less efficient with further miniaturization. A full-wave 18 design technique such as transformation optics could 19 thus advance the miniaturization of integrated optical 20 circuits. 21 22 A number of research groups have recently made 23 progress in designing two-dimensional transformation- 24 optical devices for in-plane guided waves. By placing a 25 metamaterial on top of a metal film, it is possible to 26 steer surface plasmon waves along a bend, focus them, 27 or make them pass over an obstacle [3,4]. These 28 29 approaches add a metamaterial on top of a 30 nanophotonic waveguide. By locally modifying the 31 Fermi level of graphene, it was also proposed to create 32 a two-dimensional transformational-optical surface that 33 can split or focus surface waves on graphene. Here, the 34 guided-wave-supporting surface is modified by a 35 transformation [5]. In a dielectric waveguide, it is also Figure 2 – Illustration of transformation optics in lightwave 36 possible to locally adapt the thickness to achieve a technology. With the use of transformation optics 37 desired spatial distribution of mode indices implemented with inhomogeneous metasurfaces or 38 implementing a transformation optics design [6] at the waveguides, photonic functions such as steering, wave 39 expense of backscattering. reshaping, splitting and recombining can be achieved. 40 41 Recently, an alternative transformation optics theory For the design of guided wave devices, as visualized in 42 was developed for guided waves on integrated optical 43 Figure 2, the theory of transformation optics is now waveguides. Indeed, one problem with transforming well understood [6-8], but advances in nanofabrication 44 integrated optical waveguides is that traditional 45 are clearly required. Surface transformation-optical transformations optics imposes metamaterials in both designs require an anisotropic dielectric material layer 46 the core and the cladding of the waveguide. This 47 with varying height. This may be achieved with direct 48 results in bulky structures that are difficult to fabricate. laser writing, similarly to how an optical ground plane 49 However, if one also varies the thickness of the cloak was fabricated by creating a of spherical 50 waveguide, it is possible to transform guided waves in particles of varying size [10]. By replacing the 51 waveguides where the core is made of a metamaterial, spherical particles with ellipsoidal particles, it is 52 but the cladding is left unaltered [7,8]. The thickness possible to get anisotropic optical properties. At the 53 variation is engineered such that the dispersion relation same time, one can investigate how the complex 54 is unaltered before and after the transformation. With parameters can be further reduced to meet the current 55 this method, beam splitters and beam benders have state-of-the-art in fabrication. 56 been designed. 57 58 Concluding Remarks 59 Advances in Science and Technology to Meet 60 In the past decade, transformation optics has Challenges revolutionized our understanding of and capabilities to The further development of surface transformation design the interaction between light and three- optics at surfaces will require innovations in theory as dimensional nanostructured devices. However, for AUTHOR SUBMITTED MANUSCRIPT - draft Page 26 of 59

1 2 3 many technological applications it is even more 4 5 important to be able to understand and design the 6 interaction between light and two-dimensional media. 7 This would allow for the manipulation of free-space 8 travelling beams as well as for in-plane guided waves. 9 In our opinion, further advances in this field, both 10 theoretical and fabricational, are needed to accelerate 11 the widespread use of transformation optics as a 12 geometrical tool to design next-generation optical 13 devices. 14 15 Acknowledgments – The authors acknowledge 16 17 funding from the Area of Advance Nanoscience and 18 Nanotechnology, the Research Foundation of Flanders 19 (Fellowship No. 12O9115N), and COST ( 20 MP1403 on Nanoscale ). 21 22 23 References 24 25 [1] Ginis V, Tassin P, Soukoulis CM and 26 Veretennicoff I 2013 Enhancing optical gradient 27 forces with metamaterials, Phys. Rev. Lett. 110, 28 057401 29 [2] Lambrechts L, Ginis V, Danckaert J and Tassin P 30 31 2017 Transformation optics for surface phenomena: 32 Engineering the Goos-Hänchen effect, Phys Rev B 33 95, 035427 34 [3] Huidobro PA, Nesterov M, Martin-Moreno L and 35 Garcia-Vidal FJ 2010 Transformation Optics for 36 Plasmonics, Nano Lett 10, 1985–1990 37 [4] Liu Y, Zentgraf T, Bartal G and Zhang X 2010 38 Transformational Plasmon Optics, Nano Lett 10, 39 1991–1997 40 [5] Vakil A and Engheta N, Transformation Optics 41 Using Graphene 2011 Science 332, 1291–1294. 42 [6] Gabrielli LH, David L, Johnson SG and Lipson M 43 2012 On-chip transformation optics for multimode 44 waveguide bends, Nature Commun 3, 1217 45 [7] Viaene S, Ginis V, Danckaert J and Tassin P 2016 46 Transforming two-dimensional guided light using 47 nonmagnetic metamaterial waveguides, Phys Rev B 48 93, 085429 49 [8] Viaene S, Ginis V, Danckaert J and Tassin P, 50 Mitigating optical singularities in coordinate-based 51 metamaterial waveguides 2017 Phys Rev B 95, 52 155412 53 [9] Yu N, Genevet P, Kats MA, Aieta F, Tetienne J-P, 54 Capasso F and Gaburro Z 2011 Light Propagation 55 with Phase Discontinuities: Generalized Laws of 56 Reflection and Refraction, Science 334, 333–337 57 [10] Ergin T, Stenger N, Brenner P, Pendry JB and 58 Wegener M 2010 Science 328, 337–339 (2010). 59 60 Page 27 of 59 AUTHOR SUBMITTED MANUSCRIPT - draft

1 2 3 sometimes called “Metasurfing” [4]. Planar lenses, 4 6. Antennas cloaking devices, and general beam forming networks 5 6.1 Metasurface Radiation and Guidance at 6 Microwaves– Enrica Martini, Gabriele Minatti, without transmission lines belong to this category. 7 Stefano Maci, University of Siena Their behavior can be described through an analogous 8 to geometrical optics in equivalent graded index 9 Status materials or through a planar version of 10 Controlling the electromagnetic (EM) field has become Transformation Optics [5]. 11 a popular research topic in the recent years thanks to 12 Sinusoidally-modulated MTSs have the objective to the advances done in the framework of metamaterials locally change the dispersion properties of a bounded 13 (MTMs). Metasurfaces (MTSs), i.e. the two- 14 SW so as to transform it into a leaky wave. These dimensional equivalent of MTMs, have gained a large 15 MTSs can be directly used as antennas by exciting 16 interest due to their engineering applicability to a vast them with a coplanar feeder consisting of an 17 range of frequencies. MTSs are constituted by elementary radiator. For centered-point fed MTSs, the 18 arrangements of electrically small metallic/dielectric impedance BCs are modulated with quasi-periodic 19 inclusions in a regular planar lattice inside or on top of functions in the radial direction, thus transforming the 20 a thin dielectric layer, which can be manufactured with cylindrical SW excited by the feeder into a desired 21 standard PCB or additive manufacturing. radiating aperture-field distribution. The feasibility of 22 While MTMs characterization is given in terms of the concept was first demonstrated in [6] and [7] and 23 equivalent volumetric constitutive parameters, MTSs more recently systematized in [8]. The appeal of these 24 characterization is given in terms of homogenized 25 MTS antennas is due to the capability to obtain a boundary conditions (BCs) defined in terms of a space- complete control of the aperture field distribution in 26 variable surface impedance. Through a proper spatial 27 terms of amplitude, phase and polarization through an modulation of these BCs one can control 28 almost analytic design procedure [8]. The possibility of 29 surface/plasmonic wave propagation and radiation in a achieving also amplitude control renders these 30 simpler way than volumetric waves in MTMs. The antennas much more flexible than reflectarrays; 31 BCs are modified by spatially changing the furthermore, a low profile is achieved since the feed is 32 characteristics of the small inclusions in a gradual in-plane and not external to the surface. Finally, low 33 manner, while maintaining uniform the periodic lattice. losses are obtained thanks to the absence of a beam 34 Depending on the phenomenon we would like to forming network and to the non-resonant nature of the 35 MTS inclusions. 36 control, we can identify three classes of MTSs: 37 Current and Future Challenges 38 (i) phase-gradient MTSs for space-wave control; 39 (ii) phase-gradient MTSs for surface-wave (SW) MTSs have been proven to be an exceptionally 40 control; effective tool for controlling EM waves. However, 41 (iii) sinusoidally-modulated MTSs for radiation several challenges are still open and need to be 42 control. addressed for broadening application fields or 43 improving the performances. 44 Phase gradient MTSs for space wave control ([1]) and 45 Concerning type (i), a full tailoring of any space field, their generalization called “Huygens MTSs” ([2], [3]), including reactive field, is still an issue, due to the 46 are designed with the objective to transform the 47 dependence of the MTS phase discontinuity on the wavefront and polarization of space-wave fields. A 48 incidence direction and due to the limited bandwidth. 49 general wavefront can be obtained by locally A relevant challenge consists on MTSs that can be 50 controlling the gradient of the phase of the local used to emulate differential or convolution operators, 51 transmission coefficient. Thin lenses, beam deflectors, investigation pioneered in [9]. A further issue is the 52 beam splitters fall in this class. From an engineering one of non-reciprocal MTSs (both gyrotropic and non- 53 point of view, these MTSs can be seen as a gyrotropic). Achieving non-reciprocity requires 54 generalization of transmitarrays to continuous breaking the time reversal symmetry using an external 55 boundaries, while from a physical point of view, they force, like ferromagnetic materials; however, this can 56 implement the equivalence theorem. be obtained also by using active unidirectional 57 58 Phase gradient MTSs for SW control have the elements in an active MTS layer configuration. This 59 objective of controlling surface/plasmonic wave opens new interesting applications like one-way 60 propagation path and wavefront. These MTSs mold the screens, isolating radomes, radar absorbers and thin wavefront during the propagation, similarly to what cloaks [10]. Several groups have also progressed in happens in transformation optics 3D media, but parity-type symmetric non-local MTSs, with the maintaining the wave bounded. The phenomenon is objective to construct super focusing lenses [11]. AUTHOR SUBMITTED MANUSCRIPT - draft Page 28 of 59

1 2 3 Concerning phase gradient MTSs for surface and each small inclusion of the MTS would allow for a 4 local control of the BCs, leading to the possibility of 5 plasmonic waves (type (ii)), these are still poorly 6 explored in the microwave regime. The extension to dynamically configuring the MTS behavior. Hyper 7 non-regular grids and to time domain could be among density of elements and complex biasing networks are 8 the most important challenges. We can conceive the major challenges. Micro electromechanical devices 9 structures able to change the shape of an impulse (MEMS) seem to be inappropriate for this scope due to 10 during the propagation over the surface. Hyperbolic the relevant problems of reliability. On the other hands, 11 planar surfaces may achieve focusing properties tunable materials like liquid crystals, being developed 12 similar to those of hyperbolic media, but in-plane. for optical scopes, suffer at microwave frequencies of 13 These typologies include graphene and new low significant losses and slow commutation times for 14 dimensional materials [12]. certain applications. Among the phase changing 15 materials, the vanadium dioxide (VO2), despite its Concerning sinusoidally-modulated MTSs for radiation 16 interesting properties of exhibiting thermal phase 17 control (type (iii)), the most relevant current challenge transition that can be exploited by optical pumping, is 18 concerns the increase of the operational bandwidth, difficult to be controlled in the operational 19 especially when the antenna pattern is associated with environment, and it is still at a low technological 20 shaping or dual polarization capability. The bandwidth readiness levels. Optical pumping associated with 21 problem is related to the dispersive nature of the MTSs printed on silicon or gallium arsenide (GaAs) is 22 solutions currently adopted to implement the artificial instead presently more mature. 23 BCs and may actually limit the applicative fields. 24 25 Multilayer MTSs have not been exploited yet in this Another interesting development from both modeling 26 field and may lead to significant benefits in terms of and realizations is the one towards conformal MTSs, 27 bandwidth. Polarization independent surfaces (and appealing for instance for devices on board of vehicles 28 therefore dual polarization antennas) is another current or aircrafts. research challenge. Some work has already been done 29 Concerning non-reconfigurable MTSs, the main 30 in this direction; a polarization independent MTS technological challenges are related to their fabrication. 31 antenna has been designed in a narrow bandwidth [13]. One the one hand, high precision processes are needed 32 In [14], polarization insensitive MTSs are proposed, to move towards higher frequencies. On the other 33 which work in a wide band, but the absence of a hand, the fast development of 3D technologies 34 ground plane reduces the efficiency in their usage as opens a range of opportunities for the realization of 35 antennas. 36 new, low cost structures. 37 Future challenges are concerned with the Finally, advances in science and technology are mainly 38 39 generalization and refinement of modelling tools. related to the implementation of dynamically adaptive 40 Indeed, as a MTS-based device usually involves MTSs. Technological solutions that have been 41 several thousands of unit cells, possibly containing proposed up to now have indeed severe limitations. 42 some small geometrical details, a full wave analysis of Tunable materials suffer of high losses, low speed, 43 the whole structure is not feasible with general-purpose large temperature sensitivity or small tunability range. 44 software tools. It is therefore necessary to develop Active standard components are characterized by non- 45 effective, ad hoc analysis and design tools. negligible losses and high complexity of the bias 46 Furthermore, additional efforts are required to make network. Electromechanically devices have low 47 the modelling tools faster, for the purpose of reliability. 48 embedding them into proper optimization procedures. Concluding Remarks 49 Also, incorporating electronic devices or tunable 50 materials in the MTSs would require an update of the Encompassing with imagination, the future MTS could 51 52 modelling tools. be interpreted as a conformal surface whose characteristics can be adapted time-to-time to the 53 Advances in Science and Technology to Meet needs, changing the role of the surface or of its 54 Challenges 55 subparts from guiding structures to radiative devices. 56 Among the future challenges in the microwave regime The research framework on MTSs is wide, still in 57 for the typology (iii) - also sometimes common to (i) progress, and potentially not limited to the ICT world. 58 and (ii) – the most relevant one is probably concerned

59 with the design of dynamically adaptive and References 60 reconfigurable MTSs, namely surfaces that can change [1] N. Yu et al., “Flat Optics: Controlling Wavefronts their configuration subjected to an external with Optical Antenna Metasurfaces,” in EEE programmable control, or ultimately, energized by the Journal of Selected Topics in Quantum , incoming wave. Associating an active component to Page 29 of 59 AUTHOR SUBMITTED MANUSCRIPT - draft

1 2 3 vol. 19, no. 3, pp. 4700423-4700423, May-June [14] M. Li, S. Xiao, J. Long and D. F. Sievenpiper, 4 “Surface Waveguides Supporting Both TM Mode 5 2013. 6 [2] C. Pfeiffer, A. Grbic, “Metamaterial Huygens’ and TE Mode With the Same Phase Velocity,” in 7 Surfaces: Tailoring Wave Fronts with IEEE Transactions on Antennas and Propagation, 8 Reflectionless Sheets,” Phys. Rev. Lett., vol. 110, vol. 64, no. 9, pp. 3811-3819, Sept. 2016. 9 no. 19, pp. 197401, May 2013. 10 [3] A. Epstein and G. V. Eleftheriades, “Passive 11 Lossless Huygens Metasurfaces for Conversion of 12 Arbitrary Source Field to Directive Radiation,” 13 in IEEE Transactions on Antennas and 14 Propagation, vol. 62, no. 11, pp. 5680-5695, Nov. 15 2014. 16 [4] S. Maci, G. Minatti, M. Casaletti and M. 17 18 Bosiljevac, “Metasurfing: Addressing Waves on 19 Impenetrable Metasurfaces,” in IEEE Antennas and 20 Wireless Propagation Letters, vol. 10, no., pp. 21 1499-1502, 2011. 22 [5] E. Martini, M. Mencagli, D. González-Ovejero and 23 S. Maci, “Flat Optics for Surface Waves,” in IEEE 24 Transactions on Antennas and Propagation, vol. 25 64, no. 1, pp. 155-166, Jan. 2016. 26 [6] B. H. Fong, J. S. Colburn, J. J. Ottusch, J. L. Visher 27 and D. F. Sievenpiper, “Scalar and Tensor 28 Holographic Artificial Impedance Surfaces,” in 29 IEEE Transactions on Antennas and Propagation, 30 31 vol. 58, no. 10, pp. 3212-3221, Oct. 2010. 32 [7] G. Minatti, F. Caminita, M. Casaletti and S. Maci, 33 “Spiral Leaky-Wave Antennas Based on Modulated 34 Surface Impedance,” in IEEE Transactions on 35 Antennas and Propagation, vol. 59, no. 12, pp. 36 4436-4444, Dec. 2011. 37 [8] G. Minatti et al., “Modulated Metasurface Antennas 38 for Space: Synthesis, Analysis and Realizations,” in 39 IEEE Transactions on Antennas and Propagation, 40 vol. 63, no. 4, pp. 1288-1300, April 2015. 41 [9] A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. 42 Alù and N. Engheta, “Performing Mathematical 43 44 Operations with Metamaterials,” Science, vol. 343, 45 no. 6167, pp. pp. 160-163, 2014. 46 [10] S. Taravati; B. A. Khan; S. Gupta; K. Achouri; C. 47 Caloz, “Nonreciprocal Nongyrotropic Magnetless 48 Metasurface,” in IEEE Transactions on Antennas 49 and Propagation, doi: 10.1109/TAP.2017.2702712. 50 [11] F. Monticone, C. Valagiannopoulos, A, Alù, 51 “Parity-Time Symmetric Nonlocal Metasurfaces: 52 All-Angle Negative Refraction and Volumetric 53 Imaging”, Phys. Rev. X., Vol. 6 no. 4, pp. 41018- 54 41031, Oct. 2016. 55 [12] A. Vakil and N. Engheta, “Transformation optics 56 57 using graphene,” Science, vol. 332, no. 6035, pp. 58 1291–1294, 2011. 59 [13] A. Tellechea Pereda et al., “Dual Circularly 60 Polarized Broadside Beam Metasurface Antenna,” in IEEE Transactions on Antennas and Propagation, vol. 64, no. 7, pp. 2944-2953, July 2016. AUTHOR SUBMITTED MANUSCRIPT - draft Page 30 of 59

1 2 3 6.2 Bespoke lenses: An opportunity for tailoring 4 Phase 5 antenna radiation patterns – Oscar Quevedo-Teruel, front #1 6 Mahsa Ebrahimpouri Bespoke 7 KTH Royal Institute of Technology lens #1 8 9 Status 10 Flat phase The concept of transformation optics was defined in Phase 11 front 12 2006 [1]. Since then, the electromagnetic engineering front #2 13 community has been working on how to use this 14 concept to improve the conventional properties of 15 microwave circuits and antennas [2]. However, Bespoke 16 found a major limitation in the first lens #2 17 implementations based on analytical transformations: 18 narrow bandwidth of operation. This narrow band is Figure 1 Bespoke lens concept: Two given phase 19 – related to the need for highly anisotropic materials with fronts (left) are transformed into a desired front phase, 20 permittivity and permeability components lower than 21 which in this case is flat (right). the unity [3]. 22 One common requirement for an antenna is to produce 23 To overcome this initial limitation, two a planar front phase in a determined direction, which is 24 implementations were proposed: Non-Euclidean [4,5] a flat phase front as represented in Fig. 2, right side. A 25 and quasi-conformal transformations [6-8]. Both bespoke lens will be the quasi-conformal transformed 26 techniques demonstrated that fully dielectric and lens that can match the original front phase to the 27 isotropic materials are possible for the design of 28 desired pattern, in this case, a flat one. However, this optically transformed lenses. The non-Euclidean 29 theory is not limited exclusively to achieve a plane 30 technique does not assume any approximation, but it is wave. It can be extended to other desired radiation 31 limited to two-dimensional designs. On the other hand, patterns. For example, the radiation pattern may have a 32 the quasi-conformal technique assumes broader beam, or multiple beams in given directions. In 33 approximations, but it can be employed to produce that case, the required phase will not be a flat surface, 34 three-dimensional lenses. If the employed materials but the required front phase for obtaining the desired 35 have low losses, these lenses can be used to produce a radiation pattern. 36 new kind of broadband directive antennas [7]. 37 One of the advantages of this technique is that the 38 However, here again, there was a theoretical limitation. transformation will only require isotropic materials, 39 The existing implementations of quasi-conformal since it is quasi-conformal. Additionally, if we assume 40 transformations did not take into account the properties a slight degradation of the results, simply removing the 41 of the feeding antenna. This difficulty was not noticed regions in which the refractive indexes are smaller than 42 when an original lens, for example a Luneburg lens, unity [6], the lens can be implemented with fully 43 was re-shaped into a different coordinate system [6]. dielectric materials. If these dielectric materials are 44 However, when the transformation was employed to carefully chosen, our antenna can have inherently low 45 directly producing a lens [8], the technique was limited losses, and an ultra wide band response [7]. 46 only to omnidirectional feeding sources. 47 In order to illustrate the potential of this technique, in 48 Recently, the original theory of quasi-conformal Fig. 2, we show the results of a bespoke lens design. 49 transformations has been extended to produce lenses This figure illustrates the phase front in three 50 ad-hoc to a given feeding antenna [9]. This is the orthogonal planes after applying the ‘bespoke’ concept 51 concept of ‘bespoke lenses’. Let’s assume a specific to a holey spiral feeding antenna. More details about 52 feeding, for example, a , a monopole, or a this implementation can be found in [9]. These results 53 microstrip patch antenna. All these sources are not 54 demonstrate that even in the case of a circular omnidirectional, and they have certain radiation 55 polarized antenna, it is possible to achieve a constant 56 properties, including a given directivity. These phase front for a given feeding. In terms of radiation 57 radiation properties are mainly given by the far-field pattern, the bespoke lens has contributed to increase 58 phase distribution of the electric and magnetic fields. significantly the directivity of a given spiral at the 59 For example, on the left side of Fig. 1, we have broadside direction. These results are independent of 60 included two possible cases of phase distribution. the frequency, as far as the dielectric constant of the These phase distributions will produce a certain far- employed materials are stable with frequency, and the field radiation pattern that can be derived analytically lens size is large enough to produce an increase of the or numerically. aperture of the antenna. Page 31 of 59 AUTHOR SUBMITTED MANUSCRIPT - draft

1 2 3 Current and Future Challenges 4 5 As we have previously mentioned, the concept of 6 bespoke lenses has been very recently introduced in 7 [9]. However, it has been only demonstrated through 8 simulations. In [9], bespoke lenses were designed for 9 an aperture feeding, a spiral with circular polarization, 10 and an ultra wide band slot. In all these cases, 11 12 simulated results demonstrated the advantages of using 13 this approach. 14 Although the benefits of bespoke lenses are clear, the 15 manufacturing complexity is increased with respect to 16 conventional lenses. In order to implement these 17 lenses, a discretization is needed. The discretization 18 19 level will depend on the complexity of the changes 20 required for the desired phase. Then, a number of low 21 loss dielectric materials must be synthesized. Finally, 22 these materials must be drilled or molded to obtain the 23 designed shapes. Although these shapes are commonly 24 rounded, their cost of manufacturing is typically higher Figure 2 – Phase (in radians) of the electric field outside 25 than a conventional hemi-spherical or elliptical lens. of the bespoke lens in three orthogonal planes. 26 27 Therefore, the challenge is to produce dielectric Therefore, these metasurfaces could be employed to 28 materials with low losses that could be synthesized at produce ultra wide band lenses. low cost with relative freedom in terms of molded 29 The second technological advance is the emergent 30 shapes. expansion of 3D printers. In their first 31 32 Advances in Science and Technology to Meet implementations, 3D printers were conceived to build 33 Challenges mechanical objects. However, nowadays it is already possible to acquire 3D printers that make use of low 34 A few years ago, the scientific opinion was that lens loss dielectric materials. With the fast progress of 3D 35 antennas were reminiscent of the 80s, which were 36 already proven to be unsuccessful. Most of the printers, the practical implementation of low cost and 37 low loss bespoke lenses is becoming a reality. researchers in the field of antennas accepted that arrays 38 and reflectors were the only two techniques that could However, producing high resolution and smooth 39 shapes is still a challenge. If the contours of the be employed to produce efficient and directive 40 materials have roughness due to the low resolution of 41 antennas. However, expert opinion has drastically changed in the last few years. the printer, the quality of the overall antenna will be 42 drastically affected at high frequencies. To overcome 43 Most of the new wireless applications are expected to this problem, and to reduce the time of manufacturing 44 be allocated at higher frequency bands. At those 45 that increases drastically the cost, in the future, it will 46 frequencies, arrays are expensive due to the cost of be necessary to employ at least two resolutions to 47 their feeding networks. Additionally, the losses of construct the diverse parts of the lens. these networks are extremely high, even when the 48 Concluding Remarks 49 structures are fully metallic. On the other hand, 50 reflectors can only produce steerable angles with Here, we have explained the concept of bespoke lens 51 mechanical engines. In this situation, lenses started to antennas. The ‘bespoke’ concept is a promising 52 be considered as a suitable technique to produce low technique that can be employed to produce lenses ad- 53 cost and highly directive antennas at high frequencies. hoc for a given feeding antenna. Therefore, the overall 54 antenna system is more directive than the original 55 Additionally, two other technological advances have contributed to increasing the popularity of lens radiator. Furthermore, this technique can also be 56 employed to tailor a radiation pattern. Practical 57 antennas. The first one is the growing development of demonstrators have not yet been built, but with the 58 metasurfaces. Metasurfaces can be employed to development of new 3D printers, which make use of 59 achieve equivalent refractive indexes and, therefore, to microwave dielectric materials and higher symmetry 60 implement two-dimensional lens antennas. Recent discoveries have demonstrated that when these metasurfaces, bespoke lenses have a propitious future. metasurfaces are generated with higher symmetries, their equivalent response is low dispersive [10].

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1 2 3 References 4 5 [1] U. Leonhardt, “Optical conformal mapping,” 6 Science, vol.312, no.5781, pp.1777-1780, 2006. 7 [2] D. Kwon and D. H. Werner, “Transformation 8 Electromagnetics: An Overview of the Theory and 9 Applications,” IEEE Antennas Propagation 10 Magazine, vol. 52, pp.24, 2010. 11 [3] D. A. Roberts, N. Kundtz, and D. R. Smith, “Optical 12 lens compression via transformation optics”, Optics 13 Express, vol.17, no.19, pp.16535, 2009. 14 [4] R. C. Mitchell-Thomas, O. Quevedo-Teruel, T. M. 15 McManus, S. A. R. Horsley, and Y. Hao, “Lenses 16 on Curved Surfaces”, Optics Letters, vol.39, 17 pp.3551-3554, 2014. 18 [5] S. A. R Horsley, I. R. Hooper, R. C. Mitchell– 19 Thomas, and O. Quevedo–Teruel, “Removing 20 Singular Refractive Indices with Sculpted Surfaces”, 21 Scientific Reports, vol.4, pp.4876, 2014. 22 23 [6] N. Kundtz and D. R. Smith, “Extreme-angle 24 broadband metamaterial lens”, , 25 vol.9, no.2, pp.129, 2010. 26 [7] O. Quevedo-Teruel, W. Tang, R. C. Mitchell- 27 Thomas, A. Dyke, H. Dyke, L. Zhang, S. Haq, and 28 Y. Hao, "Transformation Optics for Antennas: Why 29 limit the bandwidth with Metamaterials?" Scientific 30 Reports, vol.3, pp.1903, 2013. 31 [8] Q. Wu, Z. H. Jiang, O. Quevedo-Teruel, J. P. 32 Turpin, W. Tang, Y. Hao, and D. H. Werner, 33 “Transformation Optics Inspired Multibeam Lens 34 Antennas for Broadband Directive Radiation”, IEEE 35 Transactions on Antennas and Propagation, vol.61, 36 no.12, pp.5910-5922, 2013. 37 [9] M. Ebrahimpouri and O. Quevedo-Teruel “Bespoke 38 Lenses Based on Quasi Conformal Transformation 39 Optics Technique”, IEEE Transactions on Antennas 40 and Propagation, vol. 65, no. 5, pp. 2256-2264, 41 May 2017. 42 [10] O. Quevedo-Teruel , M. Ebrahimpouri, M. Ng Mou 43 Kehn, “Ultra Wide Band Metasurface Lenses Based 44 on Off-Shifted Opposite Layers”, IEEE Antennas 45 and Wireless Propagation Letters, vol. 15, pp. 484- 46 487, 2016. 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Page 33 of 59 AUTHOR SUBMITTED MANUSCRIPT - draft

1 2 3 6.3 Transformation Optics for Antenna 4 5 Engineering – Yang Hao 6 Queen Mary University of London 7 8 Status 9 10 Antennas have played an important role in many 11 modern technological innovations ranging from 12 Marconi's first transatlantic wireless transmission 13 through Sir Tizard's radar to modern cellular 14 15 communications. Now enabled with two recent 16 developments - transformations optics and the design 17 and fabrication of novel electromagnetic materials, 18 antenna engineers have been equipped with new design 19 tools which provide entirely fresh solutions to classical g 20 problems restricted by fundamental physics such as the 21 Chu-Harrington Limit in electrically small antennas, 22 and enable new ways to manipulate the emission, 23 propagation and absorption of EM radiation. This goes 24 far beyond what can be accomplished with traditional 25 materials in the form of lenses and mirrors, requiring Figure 1. A selection of TO based antennas for space and 26 surface wave [4, 5]. both nano-composites and also those with properties 27 that do not exist in nature (i.e., metamaterials [1]). TO 28 behind the design of carpet cloaks, has offered an has emerged as a new for EM design, 29 alternative to TO antenna designs using all dielectric providing equivalent material properties through a 30 materials [5]. The paper published in IEEE well-chosen change of coordinates, in order to achieve 31 Transactions on Antennas and Propagation introduces unprecedented wave manipulation [2, 3]. This is 32 a robust engineering design approach supported with 33 essential for the development of conformal antennas or design examples of flat reflector and lens antennas [3]. 34 flat panel antennas for both SATCOM and aerospace In this paper, benefits of applying all dielectrics in 35 applications. The required material properties are antenna implementations have been fully quantified in 36 complex (both permittivity and permeability are order to offer broad operational bandwidths and such 37 generally anisotropic and spatially varying). TO is at ideas have been further developed into the design of 38 the heart of exciting ideas such as shaped reflectors and 39 novel reflect-arrays. Further work has shown that lens [4] with beam scanning and collimation 40 future antennas may be made from structures similar to capabilities while keeping low profiles and small RCS. 41 “LEGO” brick boxes and arbitrary radiation beams can Traditional phased arrays have limitations in wide- 42 be synthesized by dynamic changes of material angle beam-steering while TO based radome designs 43 properties of each building block. This concept have opened up new possibilities to quest for low-cost 44 complements so-called “coding metamaterials”, and compact phased arrays. Earlier work of TO based 45 “digital metamaterials” and “programmable 46 antennas utilized fully benefits of metamaterials, which metamaterials”, which find numerous applications in 47 contain both electric and magnetic material properties radar, imaging and communications with the aid of 48 being anisotropic and frequency-dispersive. Peculiar advanced signal processing techniques such as 49 radiation performances can be achieved with the “compressive sensing”. 50 sacrifices in antenna gain as well as the bandwidth. 51 Approximations can be made in several engineering 52 Planar antennas remain to be a robust solution to many 53 oriented designs by restricting the use of no-resonating and magnetic metamaterials. The approach has led to applications, including airborne radars not only for the 54 design of phased arrays but also as surface wave 55 the emergence of several novel lens antenna designs, notably a flat Luneburg lens [4]. Flat Luneburg lens radiators. To this end, TO provides a general method to 56 achieve antennas and any other EM devices for surface 57 antennas arise from industrial challenges on highly waves, with the aim of linking the geometry and 58 conformal and directive antennas, which are 59 broadband, beam-steerable and possess low sidelobes material coatings either isotropic or anisotropic over an 60 under high power operations. The design methodology arbitrarily, curved surface for all angles of incidence. has consequently been applied to demonstrate antennas The theory of TO for surfacewaves was presented in at high frequencies ranging from millimeter wave to the design of cloak [2] and later experimentally optics. Quasi-conformal transformation optics, the idea demonstrated. This is a general theory that can be utilized for the design of conformal surface wave- AUTHOR SUBMITTED MANUSCRIPT - draft Page 34 of 59

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3 based antenna devices, which are retrofitted to vehicles 4 5 or airborne platforms, where perturbations of the 6 surface are necessary for structural or aerodynamic 7 reasons, but create scattering of the surface waves, 8 which is detrimental to their performance. The concept 9 has been consequently extended to the design of 10 plasmonic metasurfaces and 2D materials for optical 11 applications. 12 13 To make TO antennas with required exotic materials in 14 large quantities, modern fabrication techniques will be 15 needed, including the use of nano-composites and Figure 2 – A photo of EPSRC QUEST programme grant 16 research team including some prominent members of 17 graded-index coatings [3, 8-10]. The permittivity scientific and industrial advisory team. 18 values have been achieved using alternative dielectric 19 mixtures with differing volume fractions and particle sizes ranging from nanometer to micron ranges. 20 aircraft fuselage, which are constructed using advanced Proprietary titanate based ceramic materials were used 21 materials. This concept presents a highly innovative 22 and the device was fabricated using a series of novel but challenging objective since solutions must also be 23 techniques, which can be broken down into three manufacturable at reasonable cost while meeting 24 distinct stages: particulate filler preparation, composite structural and system functionalities. In addition, the 25 production and a multi-cast, sequential layer effects of new functionalities on strict mechanical and 26 fabrication [3]. 27 safety performance must be considered, and MRO- 28 based repair and maintenance must remain possible. 29 Current and Future Challenges Engineering challenges relevant to the development of 30 a product from its initial concepts to manufacturing 31 Recent UK-led breakthroughs in the theory of TO, will be addressed. This requires the translation of novel 32 such as the possibilities concerning cloaking and academic research into practical tools that can be easily 33 invisibility, have caught both the scientific and popular adopted by industry. 34 imagination, and have stimulated a huge growth in 35 related research around the world. The potential of the 36 Advances in Science and Technology to Meet underlying TO approaches however have much wider Challenges 37 applicability than cloaking alone, in arguably more 38 important applications that span communications, 39 An adaptive approach to design is therefore required to energy transfer, sensors and security. Under an EPSRC 40 optimize across (i) electromagnetic performance, (ii) funded QUEST Programme Grant, a cross-disciplinary 41 aerodynamic performance based on realistic loads and team was set up to include theorists, modellers, 42 non-linear vibrations and (iii) manufacturability, manufacturers and engineers who have been able to 43 particularly drawing on latest 3D additive approaches work together and bridge theory to manufacture and 44 to embedded functional materials. We aim to develop a 45 testing, with a clear focus on the reduction to practice novel computational tool-set that will be based on 46 and demonstration of potentially radical new concepts recent scientific advances in electromagnetics, 47 devices including antennas. 48 atomistic-scale material and data-driven modeling both at the functional/structural dimensions and over 49 There is an industry-wide expectation that in order to 50 the multi-scale geometric complexity. This deployment 51 meet the challenges of future aircraft communications will provide robust design methodologies that can 52 systems radically novel design approaches are needed. minimize the cost during the prototyping stage by 53 Current designs of communication system are based on providing results in a realistic time frame and will lead 54 non-conformal solutions and/or mechanically steerable to optimal engineering designs in relation to aircraft 55 antenna systems using mechanically steerable that are ad hoc at best and heuristic at worst [6, 7]. 56 parabolic dishes (bulky) or phased arrays (expensive), 57 both covered with excessive radomes. These solutions Acknowledgments – The author would like to thank 58 are protuberant, increasing aerodynamic drag, fuel the Engineering and Physical Sciences Research 59 consumption, visibility, and degrading handling Council (EPSRC), UK under a Programme Grant 60 qualities. (EP/I034548/1) ‘‘The Quest for Ultimate Electromagnetics using Spatial Transformations To enable this step change, seamlessly embedded (QUEST)’’ for the funding, all researchers and antennas are needed so that they become a part of the Page 35 of 59 AUTHOR SUBMITTED MANUSCRIPT - draft

1 2 3 industrial partners in the project team and all advisory 4 5 board members for their contributions. 6 7 References 8 9 [1] Hao Y, Mittra R. FDTD modeling of metamaterials: 10 Theory and applications. Artech house, 2008. 11 [2] RC Mitchell-Thomas, TM McManus, O Quevedo- 12 Teruel, SAR Horsley, Y Hao, “Perfect Surface 13 Wave Cloaks”, Physical Review Letters 111 (21), 14 213901, 2013. 15 [3] Wenxuan Tang, Christos Argyropoulos, Efthymios 16 Kallos, Wei Song, Yang Hao. “Discrete Coordinate 17 Transformation for Designing All-Dielectric Flat 18 Antennas.” IEEE Transactions on Antennas and 19 Propagation vol. 58, (12) 3795-3804, 2010. 20 [4] Mateo-Segura C, Dyke A, Dyke H, Haq S, Hao Y. 21 “Flat Luneburg Lens via Transformation Optics for 22 Directive Antenna Applications”. IEEE 23 Transactions on Antennas and Propagation, 24 Volume:62, Issue: 4. 2014 25 26 [5] O. Quevedo-Teruel, W. Tang, R. C. Mitchell- 27 Thomas, A. Dyke, H. Dyke, L. Zhang, S. Haq, Y. 28 Hao, Transformation Optics for Antennas: Why 29 limit the bandwidth with Metamaterials?, Scientific 30 Reports (Nature Publishing Group), vol. 3, pp. 1903, 31 2013. 32 [6] T. P. Runarsson and X. Yao, “Stochastic Ranking 33 for Constrained Evolutionary Optimization,” IEEE 34 Transactions on Evolutionary Computation, vol. 4, 35 no.3, pp.284-294, 2000. 36 [7] B. Vial and Y. Hao, Topology optimized all- 37 dielectric cloak: design, performances and modal 38 picture of the invisibility effect, Opt. Express, vol. 39 23, pp. 23551-23560, 2015. 40 [8] D.V. Isakov, , Q. Lei, F. Castles, C.J. Stevens, 41 C.R.M. Grovenor, P.S. Grant, 3D printed 42 anisotropic dielectric composite with meta-material 43 features, Materials & Design, vol. 93, pp. 423, 2016. 44 [9] F. Castles, D. Isakov, A. Lui, Q. Lei, C. E. J. 45 Dancer, Y. Wang, J. M. Janurudin, S. C. Speller, C. 46 R. M. Grovenor, P. S. Grant, Microwave dielectric 47 characterisation of 3D-printed BaTiO3 ABS 48 polymer composites, Scientific Reports 6, Article 49 number: 22714 (2016). 50 [10] P. S. Grant, F. Castles, Q. Lei, Y. Wang, J. M. 51 Janurudin, D. Isakov, S. Speller, C. Dancer, C. R. 52 M. Grovenor, Manufacture of electrical and 53 magnetic graded and anisotropic materials for novel 54 55 manipulations of microwaves, Phil. Trans. R. Soc. 56 A, vol. 373, pp 20140353, 2015. 57 58 59 60 AUTHOR SUBMITTED MANUSCRIPT - draft Page 36 of 59

1 2 3 easily advanced enough to get things done, as 4 7. Spacetime Cloaking demonstrated by experiments that produce either 5 7.1 Spacetime Transformation Optics – 6 Paul Kinsler individual or (more usefully) periodic streams of 7 Lancaster University cloaks, and that even work at data 8 rates. Further, some quite general explorations of 9 Status future possibilities for STTO have been written [8,9]. 10 But what next? 11 Space-Time Transformation Optics (STTO) is a 12 relatively new field of opportunity for electromagnetics 13 researchers. It was initiated in 2011 with the first 14 proposal for a STTO cloak, also described as an “event 15 cloak” or “history editor” [1]. It was followed 16 remarkably quickly by experimental demonstration [2] 17 using the technologies of nonlinear optics and 18 photonics. An STTO device is necessarily dynamic, 19 20 but the modulated material properties were 21 implemented not using exotic time-addressable 22 metamaterial structures, but instead with so-called 23 “time lenses” made from dispersion controlled optical 24 fibres. 25 The easiest design principle for event cloaks, or indeed 26 any STTO device, is that of speed control of the 27 Figure 1: A “speed cloak” implementation of a stream of 28 illumination. By ensuring none of this background light illuminates the selected event, but guaranteeing alternating space-time cloaks, designed using a medium 29 which can be controllably switched between a fast and 30 that it nevertheless departs the device it as if it would slow propagation speed. Adapted, with permission, from 31 have, the space-time cloak carries out its history [1], Copyright 2011 IOP Publishing Ltd. 32 editing trick. Most simply, in the 1+1D example shown

33 in figure 1, if earlier (later) parts of the illuminating 34 wave travel faster (slower) than normal, a dark region 35 of “shadow” opens up, in which un-illuminated - and Current and Future Challenges 36 therefore unseen - events can take place. Then, with the 37 speed modulation reversed, the gap can be closed. Given the current existence of space-time cloaking 38 apparatus operating at telecommunications data rates 39 Experimental schemes for space-time cloaking distinct [3], one might wonder whether that, from the point of a 40 from the first example also exist: the most , the challenges to the field are now firmly 41 applications-oriented being based on either the time- within the scope of engineering. However, despite 42 domain Talbot effect [3], or the recent application what engineers might have planned, this would be an 43 using temporal Fraunhofer diffraction [4]. Competing over-optimistic assessment. From the point of view of 44 approaches have involved accelerating wave packets, 45 physics, there remain two significant challenges to the Fourier analysis, or polarization bypass, and indeed 46 field. As it stands, current implementations of event other schemes, although in some cases the definition of 47 cloaks are basically “speed cloaks” [8], which, whilst what space-time cloaking means has perhaps been 48 working perfectly well, implement an approximate over-stretched. 49 form of the cloaking transformation. 50 Beyond the idea of cloaking as an end in itself, in A more correct treatment of STTO [10] that allows for 51 ordinary spatial transformation optics we also have 52 the effect of dispersion strongly pushes us towards the carpet cloaking and exterior cloaking [5]. Work on 53 use quite sophisticated mathematics; i.e. the language space-time carpet cloaking already exists [6], but it 54 of differential forms. This is something of a barrier to demands independent bi-directional material 55 wider investigation of STTO, since these mathematical properties, which are hard to implement. Relativistic 56 skills are less prevalent in the transformation optics movement of cloaks, movement being the most 57 community, which more usually relies on vector or 58 straightforward of space-time transformations, has also tensor-based formalisms. In such standard approaches, 59 been addressed [7]. But it remains an interesting particularly the vectorial one, it can be difficult to 60 question as to whether a space-time exterior cloak can avoid implicit or hidden assumptions. Nevertheless, even make sense as a concept. there is still significant scope for STTO design based We can therefore see that the current state of play has on the simple speed-modulation analogies, it is worth the theory of STTO being well understood, and is noting that space-time transformations require more Page 37 of 59 AUTHOR SUBMITTED MANUSCRIPT - draft

1 2 3 thought than simply building a device with a dynamic optically tunable metamaterials cells. However, it is 4 5 and controllable response. not clear how these might be integrated into something with the precise timing requirements of an event cloak. 6 Beyond the mathematical demands of understanding 7 STTO, there are the technological necessities of Generally, STTO implementations are of the approx 8 building devices. For the implementation of a “speed cloak” sort, and ignore the complications of the 9 metamaterial-based space-time cloak we need transform's effects on temporal and spatial dispersion. 10 metamaterial unit cells that can not only have their Estimates [10] show that in the optical range, these 11 12 response modified dynamically [11], but which are effects are likely to be very small, since bandwidths 13 easily addressable and can be controlled within precise tend to be narrow and the is very fast. 14 timing requirements. Given that the unit cell responses Here acoustic implementations may again be useful, by 15 have to be modulated along with the illuminating light enabling greater scope for experimental probing of 16 as it propagates, this is particularly challenging for these subtleties. 17 electromagnetic signals. There remain other suggestions from the original paper 18 19 Another puzzle is what STTO will actually be useful that as yet lack experimental implementation [1]: 20 for – especially when we decide to think beyond the perhaps most notably the teleport illusion, and the 21 scope of straightforward research demonstrators and signal processing ``interrupt-without-interrupt’’ idea. 22 towards engineering applications. Perhaps some Both do not obviously require a great improvement in 23 already exist – the time lenses used in the STTO experimental technology, but do need to replace a 24 experiment could simply be rebadged as STTO series of one-off (disconnected) events with an ordered 25 elements, if the time lens operation was just recast as stream of related events to manipulate. With such a 26 being designed from an appropriate transformation. stream of events, it would even be possible to try to 27 However, we will only advance our understanding and build the editor [8] extension of the history 28 technology by adding something new and concrete, editor concept [1]. 29 rather than by claiming progress based on a mere 30 The mathematics required to treat approximate models renaming process, however aptly chosen it might seem. 31 of STTO devices is neither too abstract or too difficult 32 Meeting the Challenges in comparison to typical theoretical approaches, 33 although particular calculations might contain many The first ST cloak paper [1] proposed an optical fibre 34 terms, or features difficult to simplify. However, a implementation, suggesting that a dynamically 35 rigorous treatment is far more demanding, as modulated refractive index profile controlled by 36 demonstrated recently by Gratus et al [10]. Further, 37 nonlinear refractive index manipulation. Of course, this although adding the consideration of static curvature 38 was a theorists’ suggestion, so it should be no surprise into such calculations is not a great complication, if 39 that when the actual experiments appeared, they did one were to attempt a STTO design in a full dynamical 40 things more elegantly, with dispersion engineered spacetime – i.e. a general relativistic formulation – 41 time-lenses [2]. In terms of nonlinear optics then. One item that needs to be addressed is therefore 42 implementations, it seems that to a large extent the how to make the advanced mathematics either more 43 technology is more than up to current challenges. 44 palatable or more accessible to the wider research 45 However, traditional TO implementations typically use community. 46 metamaterials, engineered unit cells assembled in Concluding Remarks 47 arrays to set up the necessary spatial properties, and 48 sometimes modulation properties. Is there scope for In this section I have reflected on some of the recent 49 this kind of approach to building space-time cloaks? developments in space-time transformation optics. The 50 The timing challenges alone would be tremendous, promising start, where the second experimental paper 51 52 although would be easier if built in a high index noted operation of devices with "telecoms data rates" 53 background context. Given those difficulties, an looks set to continue. However, the arrivals of 54 acoustic ST cloak, with its lower frequencies and interesting new implementations have yet to become a 55 slower wave speeds may be the correct angle to take, flood, so it seems clear at the moment that there 56 since EM/electronic signaling speeds far outstrip the remains a lot of scope for progress in research, as well 57 speed of sound; or maybe there is scope for working at as development of real-world applications. 58 longer wavelengths, in the microwave regime, in order

59 to simplify fabrication. 60 Acknowledgments – I would like to acknowledge A recent review reports [11] electrically tunable discussions with MW and JG; along with funding from metamaterials from the into the STFC (the Cockcroft Institute ST/G008248/1) and infrared, at modulations up to THz frequencies; and also success with mechanically, thermally, and AUTHOR SUBMITTED MANUSCRIPT - draft Page 38 of 59

1 2 3 EPSRC (the Alpha-X project EP/J018171/1 and 4 5 EP/N028694/1.) 6 7 8 References 9 [1] McCall M W, Favaro A, Kinsler P, Boardman A 10 D, 2011, A spacetime cloak, or a history editor 11 J. Opt. 13 024003 12 [2] Fridman M, Farsi A, Okawachi Y and Gaeta A 13 L, 2012, Demonstration of temporal cloaking 14 Nature 481 62-65 15 16 [3] Lukens J M, Leaird D E, Weiner A M, 2013, A 17 temporal cloak at data rate 18 Nature 498 205-208 19 [4] Zhou F, Dong J, Yan S, Yang T, 2017, 20 Temporal cloak with large fractional hiding 21 window at telecommunication data rate, Opt. 22 Comm. 388 7783 23 [5] Lai Y, Chen H, Zhang Z-Q, Chan C T, 2009, 24 Complementary media invisibility cloak that 25 cloaks objects at a distance outside the cloaking 26 shell, Phys. Rev. Lett. 102 093901 27 28 [6] Kinsler P, McCall M W, 2014, Transformation 29 devices: carpets in space and time, Phys. Rev. A 30 89 063818 31 [7] Halimeh JC, Thompson RT, Wegener M 32 Invisibility cloaks in relativistic , Phys. 33 Rev. A 93 013850 34 [8] Kinsler P, McCall M W, 2014, Cloaks, editors, 35 and bubbles: applications of spacetime 36 transformation theory, Ann. Phys. (Berlin) 526 37 38 51-62 39 [9] Kinsler P, McCall M W, 2015, The futures of 40 transformations and metamaterials, Photon. 41 Nanostruct. Fundam. Appl. 15 10-23 42 [10] Gratus J, Kinsler P, McCall M W, 43 Thompson R T, 2016, On spacetime 44 transformation optics: temporal and spatial 45 dispersion New J Phys 18 123010 46 47 [11] Fan K, Padilla W J, 2015, Dynamic 48 electromagnetic metamaterials Materials Today 49 18 39-50 50 51 52 53 54 55 56 57 58 59 60 Page 39 of 59 AUTHOR SUBMITTED MANUSCRIPT - draft

1 2 3 4 7.2 Transformation optics with spatial dispersion – 5 Jonathan Gratus, Lancaster University and Cockcroft 6 Institute. 7 8 Status 9 10 Combining spatial dispersion (also known as nonlocal 11 effects) with transformation optics is relatively new 12 area of research and there is not much literature to date. 13 In [1,2] the authors consider plasmons on touching and 14 nearly touching nano wires and use conformal 15 transformation optics. They need to consider spatial 16 dispersion, since it limits the maximum fields one may 17 see in such a scenario. In [3] the authors consider the 18 fields due to a nano sphere in contact with a surface in 19 which surface plasmons are transmitted. In all these 20 cases spatial dispersion arises naturally due to the 21 limited comprehensibility of the electrons. In [4] the 22 authors consider creating a spatially dispersive media 23 24 in order to create nonreciprocal media, leading to ultra- 25 transparent material. In [5,6] the authors consider Figure 1: The support (red) of the integral kernel homogeneous media and use spatial dispersion in order 26 associated with a temporally dispersive medium, inside the 27 to manipulate the dispersion relations. In [7] the backward lightcone (grey). The Polarisation 28 authors consider the transformation optics of a ( , ) depends on the electric field ( , ) in the its 29 spacetime cloak and spatial dispersion arises naturally past, at the point only.

30 as the result of the transformation on a temporally 𝑷𝑷 𝑡𝑡 𝒙𝒙 𝑬𝑬 𝑡𝑡′ 𝒙𝒙 31 dispersive pulse. By its very nature𝒙𝒙 transformation optics leads to media 32 with constitutive relations which depend on position. Spatial dispersion is usually referred to when the 33 Together with spatial dispersion, this requires 34 constitutive relations (permittivity, permeability and prescribing a constitutive relation which depends on 35 magnetoelectric effects) depend on the wavevector in both and . Since these are Fourier conjugate 36 addition to the frequency . It occurs when the variables, it appears to be inconsistent to depend on 37 polarization at one point depends not only on𝒌𝒌 the both. 𝒙𝒙There are𝒌𝒌 a number of ways handling this: In 38 electric field at but also on𝝎𝝎 the electric field in a most work [1-3] various scale-dependent 39 neighbourhood𝑷𝑷 of . Some𝒙𝒙 authors also talk about 40 approximations are used to decouple and , so that it weak spatial dispersion,𝒙𝒙 which is the result of 41 is reasonable for the constitutive relation to depend on reformulating a 𝒙𝒙 non spatially dispersive 42 both. For example we might assume 𝒌𝒌that 𝒙𝒙( ) magnetoelectric medium. Here we are concerned with 43 where is the size over which significant changes in strong spatial dispersion and consider magnetoelectric 44 the constitutive relations occur. By contrast𝒌𝒌 ⋅, others𝜹𝜹𝜹𝜹 ≪ [4𝟏𝟏- effects separately. 45 6] consider𝜹𝜹𝜹𝜹 constitutive relations which depend only on

46 In metamaterials spatial dispersion is often considered ( ) , . Thus they consider homogeneous regions of 47 a nuisance and occurs when the size of the space. In general the solutions in each region will 48 microcells starts becoming comparable with the 49 require 𝝎𝝎 𝒌𝒌 additional boundary conditions to match [8]. wavelength . As a rule of thumb, one may say𝑳𝑳 that the 50 A more accurate method is to write the constitutive 51 constitutive relations of metamaterial are temporally dispersive when𝝀𝝀 / , and is both spatially and relations either as an integral kernel [9,10] or as a 52 [7]. Example kernels associated 53 temporally dispersive when / / . For > 54 the use of effective𝑳𝑳 ≾ 𝛌𝛌 𝟏𝟏𝟏𝟏media approximations is no with local and nonlocal constitutive relations are given 55 longer valid. Note that for𝛌𝛌 𝟏𝟏𝟏𝟏reasons≾ 𝑳𝑳 ≾ 𝛌𝛌of 𝟐𝟐causality,𝑳𝑳 in figures 1 and 2. The hydrodynamic model becomes ( ) 56 spatially𝛌𝛌 dispersive implies temporally dispersive. a differential equation for the polarization , 57 Therefore saying a medium is spatially dispersive 58 + + = 𝑷𝑷 (𝒕𝒕 ,𝒙𝒙 ) means both. The simplest constitutive relation, which 𝟐𝟐 𝟐𝟐 59 incorporates spatial dispersion is the hydrodynamic 𝟐𝟐 𝟐𝟐 𝟐𝟐 𝝏𝝏 𝑷𝑷 𝝏𝝏𝝏𝝏 𝟎𝟎 𝒙𝒙 𝝏𝝏 𝑷𝑷 𝑷𝑷 60 Lorentz model which has = and In both𝟐𝟐 of 𝜸𝜸cases the𝝎𝝎 𝑷𝑷constitutive− 𝜷𝜷 𝟐𝟐 relations𝝎𝝎 𝑬𝑬 𝒕𝒕 𝒙𝒙can be generalised,𝝏𝝏𝒕𝒕 so𝝏𝝏𝒕𝒕 that they can𝝏𝝏𝒙𝒙 be interpreted as 𝟎𝟎 depending both on and . Thus in the differential ( , ) = + 𝝁𝝁 𝝁𝝁 + +𝟐𝟐 + 𝝎𝝎𝑷𝑷 𝒌𝒌 𝒙𝒙 𝜺𝜺 𝝎𝝎 𝒌𝒌 𝜺𝜺𝟎𝟎 𝟐𝟐 𝟐𝟐 𝟐𝟐 𝟐𝟐 −𝝎𝝎 𝒊𝒊𝒊𝒊𝒊𝒊 𝝎𝝎𝟎𝟎 𝜷𝜷𝒙𝒙 𝒌𝒌𝒙𝒙 AUTHOR SUBMITTED MANUSCRIPT - draft Page 40 of 59

1 2 3 equation above , and may depend on and relationship between the dispersion relation and the 4 , refractive index. The transmission and reflection 5 . 𝝎𝝎𝟎𝟎 𝜸𝜸 𝝎𝝎𝑷𝑷 𝜷𝜷 𝒕𝒕 6 The use of general relativity, inherent in transformation 7 𝒙𝒙 optics, implies that the tools can be easily extended to 8 spacetime transformations, as required for spacetime 9 cloaking. These leads to constitutive relations which 10 11 depend on which event in spacetime is chosen. In this 12 case the constitutive relations need to depend of ( , )

13 and their Fourier conjugates ( , ). Further, when 14 generalising the PDE above, one has to decide whether𝒕𝒕 𝒙𝒙 15 to use Lie derivatives or covariant𝝎𝝎 𝒌𝒌derivatives. In [7] 16 the authors chose to use the . This 17 should be considered as part of the constitutive 18 relations, and would have to be determined 19 experimentally. This will require a moving medium or 20 a strong , since both types of Figure 2: The support (red) of the intergral kernel 21 derivatives reduce to the same partial derivatives in the associated with a temporally and spatially dispersive 22 medium, inside the backward lightcone (grey). The 23 static case. Polarisation ( , ) depends on the electric field ( , ) 24 In [7] the authors derive the full constitutive relations in the its past lightcone. 25 in terms of differential equations, and then make the 𝑃𝑃 𝑡𝑡 𝒙𝒙 𝐸𝐸 𝑡𝑡′ 𝒙𝒙′ 26 “slowly varying envelope approximation” and the coefficients can then be used to calculate the 27 “gradual transformation approximation” in order to 28 impedance, so together one can uniquely determine the find a similar expression. By contrast others [1-3] use 29 permittivity and permeability. By contrast, if there is an approximation scheme from the outset. A 30 spatial dispersion no such calculation can be made. theoretical challenge therefore would be to show that 31 Consequently, the dispersion relation can give such an ab initio approximation scheme is valid for all 32 definitive proof that spatial dispersion is present. For systems by deriving it from the full system. 33 example A: if the equi-frequency plots are not 34 Current and Future Challenges symmetric, or B: if there are multiple wavevectors 35 36 Up to now spatial dispersion has usually been which solve the dispersion relation, which have the same frequency, direction and polarization. However𝒌𝒌 37 considered an annoyance. have tried to 38 manufacture materials which minimise its effects. The the dispersion relations cannot tell us what the 39 next major step will require the manufacturing of constitutive relations are. For example, for a linearly 40 materials in which the desired spatial dispersion is polarised wave travelling in a homogeneous isotropic ( , ) = 41 designed into the material. spatial dispersive medium and and no 42 magnetoelectric effect, then Maxwell's equations give 43 Devices that implement the approximation where the dispersion relation𝜺𝜺 𝝎𝝎 𝒌𝒌 = 𝝁𝝁( ) 𝝁𝝁𝟎𝟎 where 44 ( ) are foreseeable. One challenge therefore is ( ) ( ( ), ) = . It is easy to see that one 45 to design microcells which result in a prescribed spatial𝒌𝒌 ⋅ 𝝎𝝎 𝜴𝜴 𝒌𝒌 can 𝟐𝟐replace ( , ) with 𝟐𝟐 ( , ) + ( , ) 46 dispersion,𝜹𝜹𝜹𝜹 ≪ 𝟏𝟏 for example by including a wire medium. It 𝜴𝜴 𝒌𝒌 𝝁𝝁𝟎𝟎 𝜺𝜺 𝜴𝜴 𝒌𝒌 𝒌𝒌 𝒌𝒌 47 𝟐𝟐 is not unreasonable to consider these structures to be ( , ) without changing the dispersion 48 𝜺𝜺 𝝎𝝎 𝒌𝒌 𝜺𝜺 𝝎𝝎 𝒌𝒌 𝒂𝒂 𝝎𝝎 𝒌𝒌 �𝒌𝒌 − dynamic, enabling the creation of spacetime cloaks 𝟐𝟐 49 relation.𝟎𝟎 Thus a major advance would be to develop a 50 which can hide dispersive pulses. piece𝝎𝝎 𝝁𝝁 of𝜺𝜺 𝝎𝝎software,𝒌𝒌 � in which one inputs a unit cell, and it 51 A longer term challenge will be to create materials outputs the effective constitutive relations, including 52 which implement the approximated constitutive any magnetoelectric effects. 53 relations. This is likely to be an order of magnitude A further challenge is to see if one can find the 54 harder. 55 additional boundary conditions needed to implement 56 In terms of numerical simulations there is a major an edge to the medium. This will probably involve the 57 challenge in calculating effective spatially dispersive combination of a homogeneous metamaterial with a 58 constitutive relations for a given unit cell. Most metasurface, representing the interface, both of which 59 existing software can use a frequency domain solver may be spatially dispersive. 60 for finding the dispersion relation and a time domain Concluding Remarks solver for finding the transmission and reflection coefficients. If a medium is not spatially dispersive Using spatial dispersion in transformation optics gives (and not magnetoelectric) there is a one to one many exciting opportunities, both to understand better Page 41 of 59 AUTHOR SUBMITTED MANUSCRIPT - draft

1 2 3 scenarios where nonlocal effects occur naturally and 4 5 also to introduce them to create new applications. 6 There are significant theoretical, numerical and 7 manufacturing challenges, some of which we have 8 highlighted. 9 10 11 Acknowledgments – The author is grateful for the 12 support provided by STFC (the Cockcroft Institute 13 ST/G008248/1 and ST/P002056/1) and EPSRC (the 14 Alpha-X project EP/J018171/1 and EP/N028694/1). 15 16 References 17 18 19 [1] Fernández-Domínguez AI, Zhang P, Luo Y, Maier 20 SA, García-Vidal FJ and Pendry JB. 21 22 Transformation-optics insight into nonlocal effects 23 in separated nanowires. Physical Review B, 24 86(24):241110, 2012. 25 [2] Fernández-Domínguez AI, Wiener A, García-Vidal 26 FJ, Maier SA and Pendry JB 2012 Transformation- 27 optics description of nonlocal effects in plasmonic 28 nanostructures Physical review letters 29 108(10):106802, 2012. 30 [3] Cirací C, Hill RT, Mock JJ, Urzhumov Y, 31 Fernández-Domínguez AI, Maier SA, Pendry JB, 32 Chilkoti A and Smith DR 2012 Science, 33 337(6098):1072--1074 34 [4] Luo J, Yang Y, Yao Z, Lu W, Hou B, Hang ZH, 35 Chan CT and Lai Y Ultratransparent media and 36 transformation optics with shifted spatial dispersions 37 2016 Physical review letters 117(22):223901 38 [5] Castaldi G, Galdi V, Alù A and Engheta N 2012 39 Nonlocal transformation optics Physical review 40 letters 108(6):063902 41 [6] Moccia M, Castaldi G, Galdi V, Alù A and Engheta 42 N 2016 Dispersion engineering via nonlocal 43 transformation optics Optica 3(2):179--188 44 [7] Gratus J, Kinsler P, McCall MW and Thompson RT 45 2016 On spacetime transformation optics: temporal 46 and spatial dispersion New Journal of Physics 47 18(12):123010 48 [8] Pekar SI 1958 The theory of electromagnetic waves 49 in a crystal in which excitons are produced Sov. 50 Phys. JETP 6(4):785 51 52 [9] Agranovich VM and Ginzburg V 2013 Crystal 53 optics with spatial dispersion and excitons volume 54 42 Springer Science & Business Media 55 [10] Gratus J and Tucker RW 2011 Covariant 56 constitutive relations and relativistic inhomogeneous 57 plasmas Journal of 58 52(4):042901 59 60 AUTHOR SUBMITTED MANUSCRIPT - draft Page 42 of 59

1 2 3 4 7.3 Experimental progress in temporal cloaking – 1 2 5 Joseph M Lukens and Andrew M Weiner 1 6 Oak Ridge National Laboratory 7 2Purdue University 8 9 Status 10 11 It is certainly not surprising that experiment has trailed theory for the exotic spacetime cloaks introduced Figure 1 – Temporal ray diagram for first temporal cloak 12 demonstration. Reprinted by permission from Macmillan earlier in this roadmap. Nonetheless, experiments have 13 Publishers Ltd: Nature [2], copyright (2012). 14 advanced with remarkable speed, with several temporal 15 cloaks already demonstrated and exciting possibilities chirp discontinuities. Combined with the original time 16 ready to be explored. Though not essential to cloaking cloak [2], these two proof-of-principle experiments 17 as a concept, spacetime duality [1] has proven an form the foundation on which subsequent temporal 18 important catalyst in the rapid development of cloaks can build. 19 temporal cloaks. In contrast to the use of “spacetime” Current and Future Challenges 20 in “spacetime cloak” as an integration of spatio- 21 temporal characteristics, spacetime duality Despite the brisk pace of the foundational experiments, 22 distinguishes space and time in a useful way, by noting time cloaking remains far from ideal, particularly in its 23 24 the formal correspondence between spatial Fourier restriction of cloakable probe fields and events. For example, both initial cloaks [2,3] accept only a 25 optics and narrowband temporal dispersion. In the case 26 of temporal cloaking in particular, such duality helps monochromatic probe which—while an excellent 27 demystify the notion of hiding transient events by benchmark case, containing no initial time gaps—is 28 supplying more tangible spatial equivalents. Indeed, certainly not arbitrary. Similarly, time gaps 29 the first demonstrated temporal cloak can be approaching the full modulation period are essential for 30 represented by the temporal ray diagram in Fig. 1 [2]. expanding the range of events which can be cloaked. 31 A monochromatic probe field receives a sharp chirp On a technical side, recent developments have brought 32 discontinuity via nonlinear optical wave mixing with a improvements to the cloaking window question. In [4], 33 tailored pump pulse; by propagating through a a time gap approaching 90% was obtained using highly 34 dispersive element, the frequencies separate and leave nonlinear fiber for spectral broadening. This approach 35 represents an interesting hybrid of the first two 36 a time hole with zero intensity, wherein any event has 37 no impact; then finally, moving through the matched temporal cloaks, combing the electro-optic modulation 38 system returns the probe to its initial undisturbed state, of [3] for high data rates with the optical nonlinearity 39 with no trace of the event. This seminal experiment of [2] for short-pulse compression. On another front, 40 succeeded in cloaking a four-wave mixing signal from the cloaking window was increased not by pulse 41 the probe field and confirmed, within just one year compression for a fixed period, but by repetition-rate 42 from its initial proposal, that time cloaking is possible reduction for fixed pulse width, using the extended 43 in a fiber-optic system. Talbot effect—similar to pulse picking but with no 44 intrinsic loss [5]. These experiments indicate promise 45 Shortly thereafter emerged an alternative temporal in realizing fractional cloaking windows approaching 46 cloaking scheme based on electro-optic modulation 100%, although we anticipate diminishing returns for 47 rather than nonlinear mixing [3]. Whereas both each step toward this limit, with small improvements 48 nonlinear optical mixers and electro-optic modulators requiring significant increases in system complexity. 49 50 implement the same physical operation—namely, Finding new technological approaches may be required 51 temporal phase modulation—they do so in very to achieve such limits. 52 distinct ways in terms of execution and performance. 53 Accordingly, this telecom-compatible, electro-optic The second outstanding challenge relates to the 54 approach was able to boost the repetition rate by arbitrariness of the probe fields which a temporal cloak 55 several orders of magnitude, into the gigahertz regime, can manipulate. Up to this point, the experimental 56 and produced time gaps equal to roughly one half- focus has been on continuous-wave inputs at a given 57 period; together these enabled cloaking of high-speed frequency; by spacetime duality, the spatial equivalent 58 optical data streams for the first time, extending time would be a cloak which hides an object from a plane 59 cloaking to the realm of telecommunications—a fitting wave at one angle of illumination only. Just as a 60 application space for clandestine cloaks. Interestingly, unidirectional cloak is only a penultimate realization of this demonstration also enlisted spacetime duality via spatial cloaking, temporal cloaks which can expand the temporal Talbot effect, using interference to from single-frequency to more arbitrary inputs produce temporal gaps at high speeds without any represent a crucial goal in the coming years. Closely Page 43 of 59 AUTHOR SUBMITTED MANUSCRIPT - draft

1 2 3 related is an even more fundamental question: How 4 5 can we make current temporal cloaks viable in tackling 6 practical problems? For example, the standard 7 paradigm described above does indeed enable secret 8 disruption of communication, but offers no positive 9 way to transmit the cloaked information. Nonetheless, 10 important first steps addressing these broad challenges 11 have been realized. Building on the same platform as 12 [3], an improved setup exploited the spectrally periodic Figure 2 – Simultaneous cloaking and data transmission 13 nature of the temporal Talbot effect to accept multiple for wavelength-multiplexed channels. Cloaking along 14 probe wavelengths; as shown in Fig. 2, one channel 15 the short-wavelength (blue) channel (left) or along the can be cloaked, while the other faithfully conveys the long-wavelength (red) channel (right). Reproduced with 16 data stream [6]. Moreover, this configuration 17 permission of OSA from [6]. 18 succeeded in taking a modulated probe input and 19 transmitting it through an event unscathed, We note one intriguing connection for improving time 20 demonstrating a novel form of tampering resistance. cloaking, deriving from a somewhat unlikely source: 21 These experiments not only reveal cloaking quantum information processing (QIP). QIP utilizes 22 effectiveness with polychromatic inputs, but they also quantum states, coherent manipulations, and 23 provide concrete examples of time cloaks used to entanglement in order to realize certain informational 24 improve, rather than interrupt, communication. We tasks more effectively than possible in classical 25 also note an experiment employing polarization systems. Though typically not associated with cloaks 26 switching to hide or transmit high-speed data [7]. designed for classical electrodynamics, photonic QIP 27 While not temporal cloaking in the true sense (opening requirements share much in common. For example, a 28 gaps in time), this polarization switch suggests an successful QIP system necessitates arbitrary and 29 important principle when designing time cloaks for unitary optical mode transformations; likewise, the 30 transformation for an ideal cloak directs arbitrary 31 applications: the experimental means is secondary to 32 the application—that is, one should focus on achieving inputs away from a particular cloaked region, and then 33 a given result, rather than adhering to a particular transforms back to the input, all unitarily (i.e., without 34 theoretical pattern. In light of this second wave of loss). In time-frequency space, the degrees of freedom 35 experiments, for the near term we see temporal relevant to temporal cloaking, recent research has 36 cloaking best suited to specific optical communication theoretically developed universal quantum computing 37 tasks, on an application-by-application basis. The more based on frequency-encoded , pulse shapers, 38 challenging, long-range trajectory toward (semi-) and electro-optic phase modulators [8]—the very 39 universal communication cloaks then can be pursued in spectro-temporal phase operations on which previous 40 parallel. linear temporal cloaks have relied. The same 41 operations form the basis for new entanglement tests 42 Finally, the greatest—and by far most uncertain— for frequency-entangled photons [9]. And since 43 universal unitary operations encompass those needed 44 challenge for the time cloak experimentalist is uniting 45 the aforementioned techniques with metamaterials, in for time cloaking, this advance in photonic QIP 46 order to realize a true spacetime cloak. All temporal presents a concrete tool for cloak design. While these 47 cloaks thus far can be classified as spatially one- QIP results apply specifically to discrete optical mode 48 dimensional, creating time gaps in a single spatial spaces (like wavelength-multiplexed networks), they 49 mode. And no explicit proposal for integrating spatial should generalize to continuous spaces as well; 50 and temporal cloaks has been offered, at least in the developing such a paradigm in more detail is an 51 form of a prescription fulfillable with current important area for future work. 52 technology. This is not horribly surprising, given the Concluding Remarks 53 technological dissimilarity between metamaterial 54 approaches (for spatial cloaks) and high-speed fiber Temporal cloaking experiments have progressed 55 optics (for temporal cloaks). Yet if the history of steadily following their initial proposal, with fiber- 56 optic instantiations attaining high-speed data cloaking, 57 transformation optics has taught us anything, it is never and important progress in communication-enabling as 58 to eliminate fantastic possibilities, for one never knows 59 when an unexpected breakthrough might turn science well as communication-thwarting systems. Technical 60 fiction into reality. improvements to current cloak designs are ongoing; expanding time cloaks to more general probe Advances in Science and Technology to Meet waveforms is promising; and experimental realization Challenges of a full spacetime cloak represents an exciting though AUTHOR SUBMITTED MANUSCRIPT - draft Page 44 of 59

1 2 3 still largely indefinite prospect. Yet regardless of how 4 5 the field progresses in the coming years, time cloak 6 experiments have already revealed valuable 7 connections between transformation optics, optical 8 communications, and even quantum information— 9 links which bring new insights into all fields involved 10 and suggest unique opportunities for improving real- 11 world applications. 12 13 Acknowledgments – This work was performed in part 14 at Oak Ridge National Laboratory (ORNL), operated 15 by UT-Battelle for the U.S. Department of Energy 16 under contract no. DE-AC05-00OR22725. JML is 17 funded by a Wigner Fellowship at ORNL. AMW 18 recognizes support from the National Science 19 Foundation under grants ECCS-1407620 and ECCS- 20 1509578. 21 22 References 23 [1] Kolner B H 1994 Space-time duality and the 24 theory of temporal imaging IEEE J. Quantum 25 26 Electron. 30 1951-63 27 [2] Fridman M, Farsi A, Okawachi Y, and Gaeta A L 28 2012 Demonstration of temporal cloaking Nature 29 481 62-5 30 [3] Lukens J M, Leaird D E, and Weiner A M 2013 A 31 temporal cloak at telecommunication data rate 32 Nature 498 205-8 33 [4] Zhou F, Dong J, Yan S, and Yang T 2017 34 Temporal cloak with large fractional hiding 35 window at telecommunication data rate Opt. 36 Commun. 388 77-83 37 [5] Li B, Wang X, Kang J, Wei Y, Yung T, and Wong 38 39 K K Y 2017 Extended temporal cloak based on the 40 inverse temporal Talbot effect Opt. Lett. 42 767-70 41 [6] Lukens J M, Metcalf A J, Leaird D E, and Weiner 42 A M 2014 Temporal cloaking for data suppression 43 and retrieval Optica 1 372-5 44 [7] Bony P-Y, Guasoni M, Morin P, Sugny D, Picozzi 45 A, Jauslin H R, Pitois S, and Fatome J 2014 46 Temporal spying and concealing process in fibre- 47 optic data transmission systems through 48 polarization bypass Nature Commun. 5 4678 49 [8] Lukens J M and Lougovski P 2017 Frequency- 50 51 encoded photonic qubits for scalable quantum 52 information processing Optica 4 8-16 53 [9] Imany P, Jaramillo-Villegas J A, Odele O D, Han 54 K, Qi M, Leaird D E, and Weiner A M 2017 55 Demonstration of frequency-bin entanglement in 56 an integrated optical microresonator CLEO 57 JTh5B.3 [postdeadline] 58 59 60 Page 45 of 59 AUTHOR SUBMITTED MANUSCRIPT - draft

1 2 3 4 8. Transformation Optics for Analogue 5 Cosmology 6 8.1 Cosmology in the laboratory: challenges at the 7 horizon – Ulf Leonhardt 8 Weizmann Institute of Science 9 10 11 Status 12 An important inspiration for transformation optics has 13 been the connection between electromagnetism in 14 media and general relativity [1]. This connection is 15 based on the long-known mathematical fact [2] that 16 Maxwell’s equations in curved coordinates or curved 17 18 space-time geometries (Fig. 1) are equivalent to 19 Maxwell’s equations in certain impedance-matched 20 -electric media. One can therefore make use of some of the concepts of general relativity in 21 22 electromagnetism. In particular, the concept of 23 coordinate-transformations and coordinate invariance Figure 1 Curved spaces correspond to dielectric media 24 has been rather fruitful in what became known as for electromagnetic fields. The figure visualizes 25 transformation optics. Invisibility cloaking by spatial Riemann’s curvature tensor that quantifies the degree of 26 transformations [1], for example, is simply a curvature in (reproduced from [1]). 27 consequence of coordinate invariance: as Maxwell’s 28 equations are coordinate-invariant, the effect of a should quantum mechanics reign over general 29 relativity? This is perhaps how a philosopher might coordinate-transformation – implemented with an 30 argue . A physicist would ask the question what the 31 appropriately-designed medium – is invisible, evidence for quantum physics in general relativity is. 32 provided the transformation does not affect the 33 observer. The question is whether macroscopic There is no experimental evidence for quantum 34 electromagnetism can pay back some the debt it owes whatsoever. But is there some theoretical evidence, 35 to general relativity. Thanks to the recent detection of some hint hidden in the theory that suggests a quantum 36 gravitational waves [3] and to other advances in origin for general relativity? In fact, there is one: the 37 instrumentation and technique [4], cosmology has principle of least action. General relativity obeys the 38 entered a golden era where precise scientific data has action principle – Einstein’s field equations follow 39 40 become available. Cosmologists are no longer ‘often from Hilbert’s action. An action principle is 41 in error, but seldom in doubt’ (L D Landau). incomprehensible in : how would, for 42 Cosmology has become a hard science very well worth example, a mechanical particle ‘know’ in advance that 43 contributing to. What are the challenges and problems when it starts at point A and ends at point B it should 44 in cosmology where the concepts of macroscopic minimize its action along the trajectory? How should 45 electromagnetism can become useful? space-time ‘know’ in advance its evolution? Quantum 46 mechanics resolves this puzzle: the mechanical particle Current and Future Challenges 47 takes all paths, the paths interfere with each other with 48 One of the major problems of is the the action as phase, and then the most probable one is 49 of the underlying theory of cosmology and the path of minimal action. Similarly, in the quantum 50 gravity, general relativity, with quantum mechanics. 51 picture all space-time geometries should coexist as 52 One may wonder whether there are deeper reasons why quantum possibilities, and then their interference 53 quantum mechanics and general relativity seem singles out the most probable one minimizing Hilbert’s 54 incompatible. Philosophically, quantum mechanics and action of Einstein’s equations. The quantum picture 55 general relativity belong to different categories: would explain Hilbert’s action, even if we have not 56 quantum mechanics in its general form is a metatheory found the precise picture yet. 57 concerned with the transition from potentiality to In my opinion, the action principle is the strongest 58 reality, general relativity is the theory of space and argument for . So, if quantum gravity 59 time. Quantum mechanics does not exist in real space – does not exist, what would replace the principle? Is 60 it exists in Hilbert space, so why should general there an alternative, equally fundamental derivation of relativity reside in Hilbert space as well? As the Einstein’s equations? Yes there is: Jacobson has found philosophical categories of quantum mechanics and that Einstein’s equations follow from general relativity are different and of equal rank, why AUTHOR SUBMITTED MANUSCRIPT - draft Page 46 of 59

1 2 3 as an equation of state [5]. For this, Jacobson I also believe that advances in both theory and 4 experiment of the forces of the quantum [10] 5 considered a causal horizon (a light cone) and 6 associated an entropy with the horizon area and a 7 temperature. These ideas came from Bekenstein’s 8 concept of black-hole thermodynamics [6]. Jacobson 9 has generalized them from the horizons of black holes 10 to arbitrary causal horizons. There, the role of the 11 temperature plays the Unruh temperature [7] seen by 12 an accelerated observer that probes the causal horizon. 13 Yet the Unruh temperature is still a concept derived 14 from quantum mechanics. Moreover, the Unruh 15 radiation of accelerated observers has never been 16 observed in an experiment. There, I believe, is room 17 18 for insights and techniques from macroscopic 19 electromagnetism and optics. 20 Another major puzzle of cosmology is the observed 21 fact [4] that the is accelerating at a rate far 22 exceeding its visible content (Fig. 2). Some Figure 2. Expansion of the universe. The observed 23 expansion is dominated by unobserved dark energy 24 form of ‘dark energy’ – for want of a better term – (black) and (olive ). Both represent a 25 with repulsive gravity seems to contribute to the lion’s mystery, in particular dark energy, that appears to 26 share of the expansion, accounting for about 70% of generate repulsive gravity. Figure reproduced from 27 the universe’s mass. Additionally, about a quarter of https://www.nobelprize.org/ popular material to the 2011 28 the mass appears to be made up by ‘dark matter’; only . 29 some 5% of the matter of the universe is of known 30 physics (Fig. 2). Some form of ‘dark energy’ of a 31 will shed some light on ‘dark energy’. There we have different strength may also have driven the observed theoretical tools available that are testable in modern 32 rapid of the universe shortly after the big 33 high-precision experiments and may thus have a basis bang. It was speculated that the ‘dark energy’ 34 in fact. Using the analogy between vacuum forces in 35 originates from the fluctuation energy of the quantum media and vacuum forces in space-time geometries we 36 vacuum in the universe. However, a simple estimation can then extrapolate this known laboratory physics to 37 of this energy, using a cut-off at the Planck scale, gives the physics of space. Whether this idea gives the right 38 a result that disagrees with the observed rate by 120 order of magnitude for the expansion is not clear yet, 39 orders of magnitude [8]. Yet ideas of cosmology in the but at the very least we will know whether the quantum 40 laboratory have already come to the rescue of the vacuum can be made responsible – or not – for the 41 theory: Volovik [9] developed a thermodynamical expansion of the universe. 42 argument – based on the analogy of cosmology with 43 Concluding Remarks 44 the physics of superfluid Helium – that the ‘dark 45 energy’ should be close to zero. The actual value is Transformation optics has been inspired by the analogy 46 indeed close to zero, but not quite. It remains to between electromagnetism in media and in the space- 47 calculate the finite part. Apart from Volovik’s idea [9], time of general relativity [1]; it is time to see whether 48 not much progress has been made in solving this macroscopic electromagnetism and optics can make 49 mystery. conceptual contributions to cosmology. We discussed 50 two possible areas where this might happen: the Unruh 51 Advances in Science and Technology to Meet effect and quantum forces in cosmology. There 52 Challenges transformation optics has the potential to contribute to 53 I believe that experimental techniques from quantum fundamental science. 54 optics will progress to the stage where an observation 55 Acknowledgments 56 of the Unruh effect becomes possible. Such techniques 57 are based in the analogy between moving media and I thank Yael Avni, David Bermudez, Yehonathan 58 space-time geometries [1]. With manipulations of light Drori, Mathias Fink, Emmanuel Fort, Itay Griniasty, 59 fields one could create the accelerations required. Such Mordehai Milgrom, Sahar Sahebdivan, Ephraim 60 experiments will clarify the physical essence of the Shahmoon, William Simpson, and Yana Zilberg for Unruh effect and hence give a better justification of stimulating discussions. This work is supported by the Jacobson’s thermodynamical argument for the Einstein European Research Council and the Israel Science equation as an equation of state. Foundation, a research grant from Mr. and Mrs. Louis Page 47 of 59 AUTHOR SUBMITTED MANUSCRIPT - draft

1 2 3 Rosenmayer and from Mr. and Mrs. James Nathan, 4 5 and the Murray B. Koffler Professorial Chair. 6 7 References 8 9 [1] Leonhardt U and Philbin T 2010 Geometry and 10 Light: The Science of Invisibility (Dover, 11 Mineola) 12 [2] Plebanski J 1960 Electromagnetic Waves in 13 Gravitational Fields Phys. Rev. 118 1396-1408 14 [3] Abbott B P et al. 2016 Observation of 15 Gravitational Waves from a Binary 16 Merger Phys. Rev. Lett. 116 061102 17 [4] Perlmutter S, Smith B P and Riess A G 2012 18 19 Nobel lectures Rev. Mod. Phys. 84 1127-1175 20 [5] Jacobson T 1995 Thermodynamics of Spacetime: 21 The Einstein Equation of State Phys. Rev. Lett. 22 1260-1263 23 [6] Bekenstein J D 1974 Generalized second law of 24 thermodynamics in black-hole physics Phys. Rev. 25 D 9 3292-3300 26 [7] Unruh W G 1976 Notes on black-hole 27 evaporation Phys. Rev. D 14 870-892 28 [8] Brumfiel G 2007 Unseen Universe: A constant 29 problem Nature 448, 245-248 30 [9] Volovik G E 2003 The Universe in A Helium 31 32 Droplet (Clarendon Press, Oxford) 33 [10] Simpson W M R and Leonhardt U editors 2015 34 Forces of the Quantum Vacuum (World 35 Scientific, Singapore) 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 AUTHOR SUBMITTED MANUSCRIPT - draft Page 48 of 59

1 2 3 8.2 Spacetime analogs based on hyperbolic 4 1 5 metamaterials Igor I. Smolyaninov and Vera N. Smolyaninova2 6 1 7 University of Maryland 2 8 Towson University 9 Status 10 Modern developments in gravitation research indicate 11 12 that classic general relativity is an effective 13 macroscopic theory, which needs to be replaced with a 14 more fundamental description based on yet unknown 15 microscopic degrees of freedom. However, our ability 16 to obtain experimental insights into the future 17 fundamental theory is strongly limited by low energy 18 scales available to terrestrial physics. The emergent 19 analogue spacetime program offers a promising way 20 around this difficulty. Looking at such systems as 21 superfluid helium and atomic Bose-Einstein 22 condensates, discover how macroscopic 23 field arise from well-studied atomic degrees of 24 Figure 1 Experimental model of world line behaviour in 25 freedom. Recent introduction of metamaterials and an “expanding universe” using a plasmonic hyperbolic 26 transformation optics appear to be an exciting new metamaterial [4]: Optical (a) and AFM (b) images of the 27 development in this field. Metamaterial optics is not metamaterial. The plasmon source is shown by arrows. 28 limited by the properties of atoms and molecules given (c) Plasmonic rays or “world lines” increase their spatial 29 to us by nature. “Artificial atoms” used as building separation as a function of “timelike” radial coordinate. 30 blocks in metamaterial design offer much more For the sake of clarity, edge scattering is partially 31 freedom in constructing analogues of various exotic blocked. (d) Schematic view of world lines behaviour near the . 32 spacetime metrics, such as black holes [1], 33 [2], cosmic strings [3], and even the metric of the Big monochromatic extraordinary light distribution in a 34 Bang [4]. Explosive development of this field promises hyperbolic metamaterial along some spatial direction 35 new insights into the fabric of spacetime, which cannot may model the “flow of time” in a three dimensional 36 be gleaned from any other terrestrial experiments. In (2+1) effective Minkowski spacetime [4]. If an 37 38 addition, compared to the standard general relativity, interface between two metamaterials is engineered so 39 metamaterial optics gives more freedom to design an that the effective metric changes signature across the 40 effective space-time with very unusual properties. interface, two possibilities may arise. If the interface is 41 Light propagation in all static general relativity perpendicular to the time-like direction z, this 42 situations can be mimicked with positive εik = µik [5], coordinate does not behave as a “timelike” variable 43 while the allowed parameter space of the metamaterial any more, and the continuous “flow of time” is 44 optics is broader. Thus, flat Minkowski space-time interrupted. This situation (which cannot be realized in 45 with the usual (-,+,+,+) signature does not need to be a classic general relativity) may be called the “end of 46 starting point. Other effective signatures, such as the time”. It appears that optics of metamaterials near the 47 “two times” physics (-,-,+,+) signature may be realized “end of time” event is quite interesting and deserves a 48 [6]. Metric signature change events (in which a phase detailed study. For example, in the lossless 49 approximation all the possible “end of time” scenarios 50 transition occurs between say (-,+,+,+) and (-,-,+,+) 51 space-time signature) are being studied in Bose- lead to field divergencies, which indicate quite 52 Einstein condensates and in some modified gravitation interesting linear and nonlinear optics behaviour near 53 theories (see ref.[7], and the references therein). It is the “end of time”. On the other hand, if the 54 predicted that a residing on a metamaterial interface is perpendicular to the space- 55 spacetime undergoing a signature change reacts like direction of the effective (2+1) Minkowski 56 violently to the imposition of the signature change. spacetime, a Rindler horizon may be observed (Rindler 57 Both the total number and the total energy of the metric approximates spacetime behaviour near the 58 particles generated in a signature change event are black hole event horizon [8]). 59 formally infinite. While optics of bulk hyperbolic Current and Future Challenges 60 metamaterials provides us with ample opportunities to Experimental realization of various metamaterial observe metric signature transitions [6], even more spacetime analogs described above requires interesting physics arise at the metamaterial interfaces. engineering of low loss hyperbolic metamaterials Very recently it was demonstrated that mapping of Page 49 of 59 AUTHOR SUBMITTED MANUSCRIPT - draft

1 2 3 having spatially-dependent uniaxial anisotropic 4 5 dielectric permittivity tensors with opposite signs of 6 their diagonal components εx= εy= ε1>0 and εz = ε2<0. 7 The wave equation, which describes propagation of 8 extraordinary light in such metamaterials is formally 9 equivalent to a 3D Klein-Gordon equation describing a

10 massive ϕω : 11 12 ∂ 2ϕ 1  ∂ 2ϕ ∂ 2ϕ  ω 2 m*2 c 2 ω  ω ω  0 (1) 13 − 2 + 2 + 2 = 2 ϕω = 2 ϕω ε ∂z ε  ∂x ∂y  c  14 1 2   15 in which the spatial coordinate z=τ behaves as a 16 17 “timelike” variable, and εz = ε2 plays the role of a time- 18 dependent scale factor. Therefore, eq.(1) describes 19 world lines of massive particles which propagate in a flat (2+1) Minkowski spacetime. When a metamaterial 20 Figure 2 Magnified image of one of the Minkowski 21 is built and illuminated with a coherent extraordinary domains in a illustrates inflation-like 22 CW laser beam, the stationary pattern of light expansion of the effective spacetime near the domain 23 propagation inside the metamaterial represents a wall. The plot compares measured and theoretically 24 complete “history” of a toy (2+1) dimensional calculated dependencies of the spacetime scale factor –εz 25 spacetime populated with particles of mass m*. This on the effective time. The calculations are based on 26 “history” is written as a collection of particle world analysis of the measured stripe contrast in the image 27 lines along the “timelike” z coordinate, as illustrated in 28 cosmology [7], natural emergence of a large number of Fig.1. While such linear “static” models are interesting 29 Minkowski , and inflation. Moreover, it also to fabricate and study, it is clear that non-linear 30 appears that due to variations in size and local “dynamic” self-assembled hyperbolic metamaterial 31 magnetic field, individual Minkowski domains in the systems in which the effective spacetime configuration 32 ferrofluid exhibit different “laws of physics”, such as is defined by temperature, external fields and various 33 different strength of effective gravity, and different 34 physical interactions of the constituent parts will radiation lifetimes due to variations in the local Purcell 35 provide a much more interesting playground for the factor. Thus the ferrofluid-based model may be used to 36 emergent analogue spacetime paradigm. 37 illustrate the fine-tuning mechanism in cosmology. It 38 Advances in Science and Technology to Meet is remarkable that all these effects may be studied via 39 Challenges direct microscopic observations, as illustrated in Fig.2. In addition, the ferrofluid-based macroscopic self- 40 The latter challenge has been met in recent studies of 41 assembled 3D metamaterials may also exhibit reach ferrofluid-based self-assembled hyperbolic 42 physics associated with microscopic topological metamaterials [9]. Such fluid-based metamaterials 43 defects of the effective Minkowski spacetime. As was exhibit strong nonlinearities, so that nonlinear light 44 pointed out recently by Mielczarek and Bojowald [10], propagation through the ferrofluid may be described in 45 the properties of self-assembled magnetic nanoparticle- a similar fashion as in general relativity. Moreover, 46 based hyperbolic metamaterials exhibit strong 47 when the ferrofluid is subjected to a modest external similarities with the properties of some microscopic 48 magnetic field, the nanoparticles inside the ferrofluid quantum gravity models, such as loop quantum 49 form small hyperbolic metamaterial domains (as cosmology. 50 illustrated in Fig.2), which from the electromagnetic 51 standpoint behave as individual “Minkowski Concluding Remarks 52 universes” inside a Euclidean background: the metric Despite these very interesting features of the 53 signature transition in a ferrofluid leads to separation 54 hyperbolic metamaterial-based spacetime models, we of the effective spacetime into a multitude of 55 should emphasize that the described analogy between intermingled Minkowski and Euclidean domains, 56 the extraordinary light propagation inside the ferrofluid giving rise to a picture of “metamaterial multiverse” 57 and the dynamics of massive particles in Minkowski [9]. As illustrated in Fig.2, inflation-like behaviour 58 spacetime is far from being perfect. The main difficulty appears to be generic within the individual Minkowski 59 comes from the cross-coupling between extraordinary domains. Thus, ferrofluid-based self-assembled 60 and ordinary light inside the ferrofluid, which may be metamaterial geometry captures many features of caused by domain interfaces and internal defects. Since several cosmological models of the multiverse, such as ordinary light does not obey the same wave equation metric signature transition scenario in loop quantum (1), such a cross-coupling breaks the effective Lorentz AUTHOR SUBMITTED MANUSCRIPT - draft Page 50 of 59

1 2 3 symmetry of the system. In addition, such a model is 4 5 necessarily limited to 2+1 spacetime dimensions. 6 Nevertheless, despite these limitations the developed 7 metamaterial model of the cosmological multiverse 8 appears to be quite interesting, since it is able to 9 replicate many of its hypothesized features in the 10 laboratory setting. 11 12 13 References 14 [1] Smolyaninov I 2003 Surface plasmon toy-model of 15 a rotating black hole New Journal of Physics 5, 147 16 [2] Greenleaf A, Kurylev Y, Lassas M, Uhlmann G 17 2007 Electromagnetic wormholes and virtual 18 magnetic monopoles from metamaterials Phys. Rev. 19 Lett. 99, 183901 20 [3] Mackay T, Lakhtakia A 2010 Towards a 21 metamaterial simulation of a spinning cosmic string 22 Phys. Lett. A 374, 2305-2308 23 [4] Smolyaninov I, Hung Y 2011 Modeling of time with 24 metamaterials JOSA B 28, 1591-1595 25 [5] Landau L, Lifshitz E The Classical Theory of Fields 26 (Elsevier, Oxford 2000). 27 [6] Smolyaninov I, Narimanov E. 2010 Metric signature 28 transitions in optical metamaterials Phys.Rev. Lett. 29 105, 067402 30 [7] White A, Weinfurtner S, Visser M 2010 Signature 31 change events: A challenge for quantum gravity? 32 Class. Quantum Gravity 27, 045007 33 [8] Smolyaninov I, Hwang E, Narimanov E 2012 34 Hyperbolic metamaterial interfaces: Hawking 35 radiation from Rindler horizons and spacetime 36 signature transtions Phys. Rev. B 85, 235122 37 [9] Smolyaninov I, Yost B, Bates E, Smolyaninova V 38 2013 Experimental demonstration of metamaterial 39 “multiverse” in a ferrofluid Optics Express 21, 40 14918-14925 41 42 [10] Bojowald M, Mielczarek J, 2015 Some implications 43 of signature-change in cosmological models of loop 44 quantum gravity J. Cosmology Astroparticle Phys. 45 08 052 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Page 51 of 59 AUTHOR SUBMITTED MANUSCRIPT - draft

1 2 3 configuration. The dispersionless, impedance-matched 4 8.3 Transformation optics in general relativity – 5 Robert T. Thompson vacuum is transformed to dispersionless, impedance- 6 Institute of Applied Physics, Karlsruhe Institute of matched media, and we cannot expect to generate 7 Technology nonlinearities from linear processes. Thus, as it stands, 8 TO is not very good at modelling or controlling these 9 Status real-world phenomena. 10 Progress in TO, and the ability to draw upon ideas from 11 The similarity of the light-bending property of curved 12 vacuum to that of refractive media was one general relativity and apply them to optics, is predicated 13 of the earliest predictions of general relativity. Although on 1) advancing the spacetime-covariant, tensorial 14 Eddington’s observation of the deflection of starlight formulation of electrodynamics in media that is 15 passing near the sun confirmed this prediction, he also compatible with the curved manifolds of relativity, and 16 remarked that the same effect could be achieved in 2) expanding the transformation concept to control these 17 otherwise flat space if the region around the sun were advanced features. 18 filled with an appropriately refracting medium [1]. 19 Progress to date has been achieved through covariant, Gordon then reversed the argument by asking whether a tensorial descriptions of real-world phenomena in 20 refracting medium could be identified with a curved 21 media. For example, recent developments improve the spacetime, and found the optical metric of an isotropic 22 accounting of properties like dispersion and 23 medium residing in a curved background spacetime [2]. nonlinearities – but the ability to control or engineer 24 While studying the propagation of electromagnetic them through a transformation procedure is limited. 25 waves in gravitational fields, Plebanski obtained 26 Utility has been gained by restricting the set of 27 = = , = (1) transformations, e.g. conformal or quasi-conformal, but 0𝑗𝑗 expanding the transformation concept is more 28 √−𝑔𝑔 𝑖𝑖𝑖𝑖 𝑔𝑔 challenging. It may be tempting to try generalizing 29 𝜀𝜀as the𝜇𝜇 effective− 𝑔𝑔00 𝑔𝑔constitutive𝛾𝛾 − 𝜖𝜖relatio𝑖𝑖𝑖𝑖𝑖𝑖 𝑔𝑔00ns for curved 30 spacetimes [3]. Leonhardt and Philbin’s virtual space by interpreting it 31 as a truly curved virtual manifold, thereby defining a 32 Since the actual relative permeability and permittivity of “projective spacetimes TO” as an analog spacetimes 33 free space are always identically ε = μ = 1, with γ = 0, type projection into a medium living in flat spacetime, 34 Plebanski’s result should be understood as identifying but such an approach requires caution. 35 curved spacetime solutions of Maxwell’s equations with 36 solutions inside a refracting medium in otherwise flat For analog spacetimes, Eq. (1) stems from a particular 37 spacetime, thereby mimicking in some way the light- choice of projection map between curved and flat 38 deflecting properties of the vacuum. This recipe for a spacetimes (one with Jacobian matrix Λ=I) [6]. But 39 laboratory-accessible dielectric analog representation of there is no natural identification of curved spacetime 40 a curved spacetime was exploited by de Felice to with a medium in flat spacetime, and thus no canonical 41 choice of projection exists. As a result, analog 42 describe the dielectric analog of a black hole [4], residing in flat spacetime. spacetimes are manifestly non-covariant since 1) any 43 spacetime can be represented by an infinite number of 44 The geometric, or spacetime, picture of transformation 45 physically inequivalent media, which must be optics (TO) established by Leonhardt and Philbin [5] interpreted relative to the chosen projection, and 2) two 46 turns out to take the same form as Eqs. (1), but arises 47 different coordinate representations of the same curved from a slightly different conceptual basis than analog 48 vacuum do not correspond to two different coordinate 49 spacetimes. representations of the same analog [8]. 50 Both can be described in terms of diffeomorphisms Furthermore, this type of projection connects non- 51 between manifolds, but whereas analog spacetimes are 52 isometric spaces and therefore the medium cannot a projection from curved to flat spacetimes [6], TO maps simultaneously mimic all aspects of light propagation in 53 a given spacetime to itself [5,7]. For TO in flat 54 the curved spacetime, in much the same way that any 55 spacetime, one finds the transformation medium to be flat map of Earth always introduces distortions to some 56 given by Eqs. (1), but where one uses the transformed surface features [9]. The Plebanski map preserves some 57 metric idea of the coordinate description of a ray trajectory, but 58 it does not simultaneously preserve physically = . (2) 59 meaningful and measurable quantities like the evolution 60 𝜇𝜇 𝜈𝜈 Current𝜇𝜇′𝜈𝜈′ and𝜇𝜇′ Future𝜈𝜈′ 𝜇𝜇 𝜇𝜇Challenges of the cross sectional area of a beam, i.e. the focus, 𝑔𝑔 𝛬𝛬 𝛬𝛬 𝑔𝑔 instead distorting these features [9]. In other words, The transformations involved in TO are linear given projection φ, observers in each manifold, making operations that preserve the symmetry of the initial measurements on φ-related congruences with their own AUTHOR SUBMITTED MANUSCRIPT - draft Page 52 of 59

1 2 3 respective metrics, will report different results for some an expression would enhance our understanding of light 4 5 measurements. in media. 6 Advances in Science and Technology to Meet In the geometric optics limit, first and second order 7 Challenges kinematics of congruences can provide generalizations 8 of the Raychaudhuri and geodesic devaition equations Despite both the longstanding, and recent TO-inspired, 9 in terms of the optical metric – standard tools in general interest in the analogy between spacetimes and 10 relativity that describe the evolution of congruences – 11 refracting media, relatively little has been done on the that will enhance analysis and ray tracing of beams in 12 covariant, tensorial formulation of electrodynamics in media. 13 refracting media within a curved background spacetime. 14 At the next order of expansion beyond ray optics, the Early work by Post established a tensorial formulation 15 transport equations provide information on the of electrodynamics in media [10], but the crucial aspect 16 evolution of the polarization, and chirality. A fully 17 lacking from Post’s analysis is a distinction between the covariant formulation of the transport equations in 18 spacetime and medium contributions. Such an media would enable a rigorous understanding of how 19 extension may be made for linear media in an arbitrary these features behave in TO, and may result in a 20 spacetime by writing Maxwell’s equations as [7] 21 methodology that allows control independently of the 22 = 0, = , = (3) ray trajectory. 23 Impedance matching is used as a quantifier of the “no 24 where the Hodge star * contains all information about reflection” concept and has clear relevance to TO, but it 25 the𝑑𝑑𝑑𝑑 background𝑑𝑑𝑑𝑑 spacetime,𝐽𝐽 𝐺𝐺 ∗ 𝜒𝜒 while𝜒𝜒 the tensor χ, whose 26 vacuum value χvac is uniquely defined such that the is so far unclear how this arises in the context of 27 vacuum is a trivial dielectric, incorporates complete spacetime covariant electrodynamics in media. 28 information about the linear medium, i.e. permeability, The inclusion of dispersion in TO is discussed in section 29 permittivity, and magnetoelectric couplings. 7.2, and has primarily been achieved by adopting a 30 differential operator approach [11]. An integral kernel 31 By explicitly separating contributions of the spacetime 32 and the medium, TO is understood as an automorphism approach is potentially more general, but so far has only 33 of the manifold, where the spacetime metric and been studied by considering the simplified case where 34 coordinates are fixed once and for all and the the is assumed to be that of a background 35 transformation acts directly on the fields [7]. This Minkowski spacetime, which neglects both the 36 formulation explicitly recognizes the unalterability of possibility of a curved background spacetime and the 37 the background spacetime, which need not be flat, and modification to the propagator concordant with the 38 the transformation has a physical interpretation as the modified lightcone within the medium. Ultimately, an 39 insertion of χ into the system. It has also been shown understanding of the covariant Green tensor in media 40 that this formulation is fully covariant [8], and since it within curved spacetimes is necessary for a full-wave 41 is isometric by construction it does not the suffer the analysis and scattering matrix calculations. At this 42 infidelity of “projective spacetimes TO.” level, electromagnetic waves can reflect off the 43 spacetime curvature and self-interfere – important 44 Inside refracting media, the lightcone does not coincide 45 effects that are not currently modeled in conjunction with that of the background spacetime, and light is in with ordinary refraction in flat spacetimes. 46 general both non-geodesic and non-null with respect to 47 the background spacetime. Instead, the lightcone is Concluding Remarks 48 defined by an emergent structure called the optical 49 By developing a fully spacetime covariant, tensorial 50 metric that contains both background metric and theory for electrodynamics in media within curved 51 medium contributions. Physically, an observer makes spacetimes, conceptual issues can be resolved and new 52 measurements with respect to the background spacetime features can be included in TO. A few of the possible 53 metric, so the optical metric should be understood as an next steps in this programme have been outlined here. 54 additional structure rather than a replacement of the 55 background metric. Ideally, we should eventually be able to discuss all of 56 classical optics in media within a fully spacetime- 57 Since the optical metric controls many aspects of light covariant formalism, which could have applications 58 propagation, a better understanding of it could lead to beyond TO. For example, almost everything we know 59 improvements in the construction of transformation about the universe comes from observations of 60 media. A fully covariant, tensorial expression for the electromagnetic waves, and curvature effects like optical metric of a general linear medium in an arbitrary gravitational lensing provide important information. spacetime, that distinguishes medium and spacetime But the universe is not empty; light also propagates contributions, does not currently exist. Obtaining such through potentially refractive dust, gas clouds, and Page 53 of 59 AUTHOR SUBMITTED MANUSCRIPT - draft

1 2 3 accretion disks around massive objects, the proper 4 5 accounting of which could shed additional light on the 6 universe. 7 8 9 10 11 References (separate from the two page limit) 12 [1] A.S. Eddington, Space, Time, and Gravitation, 13 Cambridge, University Press (1920). 14 W. Gordon, “Zur Lichtfortpflanzung nach der 15 [2] 16 Relativitätstheorie,” Annalen der Physik 72 421 17 (1923). 18 [3] J. Plebanski, “Electromagnetic waves in gravitational 19 fields,” Physical Review 118 1396 (1960). 20 [4] F. de Felice, “On the gravitational field acting as an 21 optical medium,” General Relativity and Gravitation 22 2 347 (1971). 23 U. Leonhardt and T.G. Philbin, “General relativity in 24 [5] 25 ,” New Journal of Physics 8 247 26 (2006). 27 [6] R.T. Thompson and J. Frauendiener “Dielectric 28 analog spacetimes,” Physical Review D 82 124021 29 (2010). 30 [7] R.T. Thompson, S.A. Cummer, and J. Frauendiener, 31 “A completely covariant approach to transformation 32 optics,” Journal of Optics 13 024008 (2011). 33 34 [8] R.T. Thompson and M. Fathi, “Shrinking cloaks 35 in expanding space-times: The role of coordinates 36 and the meaning of transformations in 37 transformation optics,” Physical Review A 92 38 013834 (2015). 39 [9] M. Fathi and R.T. Thompson, “Cartographic 40 distortions make dielectric spacetime analog models 41 imperfect mimickers,” Physical Review D 93 124026 42 (2016). 43 [10] E.J. Post, Formal Structure of Electromagnetics, 44 North Holland (1962). 45 46 [11] Gratus, et. al. “On spacetime transformation 47 optics: temporal and spatial dispersion,” New J. of 48 Physics 18 123010 (2016). 49 50 51 52 53 54 55 56 57 58 59 60 AUTHOR SUBMITTED MANUSCRIPT - draft Page 54 of 59

1 2 3 Metal contacts on solar cells can cover up to 10% of 4 9. Optics and Beyond the usable area. We have designed and realized three- 5 9.1 Seeking applications in optics and beyond – 6 Martin Wegener and Muamer Kadic dimensional broadband graded-index cloaks [1] via 7 Schwarz-Christoffel conformal maps [2]. But this 8 Karlsruhe Institute of Technology (KIT) approach is overkill because the phase of light or, 9 equivalently, the time of arrival of light plays no role 10 for this application. Furthermore, a solar cell is a dead- 11 Status end street because all the light hitting the active area 12 We view transformation optics as a design tool – a ideally gets absorbed. These aspects allow for a 13 very intuitive and mathematically intriguing design modified and simpler approach in which we again start 14 from a spatial coordinate transformation (in fact, a 1D 15 tool. To design what? The fascinating idea of macroscopic omni-directional broadband free-space version of Pendry’s transformation of a point to a 16 circle/sphere). Rather than mapping it onto a material- 17 invisibility cloaks for electromagnetic waves has 18 served as a catalyzing example. However, it is now parameter distribution, we map it onto a dielectric free- 19 clear that fundamental bandwidth restrictions imposed form surface, i.e., onto the shape of a surface of a bulk 20 by relativity (i.e., the impossibility of superluminal dielectric such as or a polymer [2] (see Figure 1). 21 energy transport) mean that invisible people walking 22 on the streets will forever remain a matter of science 23 fiction. 24 25 So where should this field go? What real-world devices 26 can we design? After all, researchers have designed 27 optical devices such as complex lens systems by means 28 other than transformation optics for centuries already. 29 We still see opportunities and challenges in visible- 30 frequency optics, but even more so in areas beyond 31 optics, for which the above fundamental restrictions 32 due to relativity do not apply. 33 34 Current and Future Challenges 35 Figure 1 – Electron micrograph of a fabricated cloak for a Electrical contacts on optical devices are ubiquitous. 36 metal contact (20 µm width) on a silicon surface. The 37 Examples include solar cells, photodetectors, and structure has been designed by mapping a 1D coordinate 38 large-area organic light-emitting diodes (OLEDs) as transformation onto a dielectric (polymer) free-form surface, 39 future wall paper. On the one hand, metal contacts on which refracts the incident light such that it avoids the metal 40 top of these device are wanted electrically, e.g., to contact. Reproduced with permission of OSA from 41 avoid losses by serial resistances. On the other hand, Reference [2]. 42 they are unwanted optically because the contacts cast For 1D arrays of long metal wires, this approach works 43 shadows, thereby reducing conversion efficiency in amazingly well for all angles, polarizations, and colors 44 solar cells, reducing effective quantum efficiency in of incident sun light, even for area filling fractions of 45 detectors, or leading to spatially inhomogeneous light the metal contacts as large as 20%. Furthermore, after 46 emission in OLED wall paper. Transparent contacts 47 making a master by state-of-the-art 3D laser (such as indium-tin oxide films in touch screens) are 48 nanolithography, the resulting structures can be mass- often a solution, but not always. In these cases, one 49 replicated by established imprinting techniques, such needs a device on top of the contact that guides the 50 that real-world applications in terms of solar cells on light around the contact. 51 your roof appear possible [3]. Many solar cells, 52 At first sight, this task sounds just like any ordinary however, do have additional wider bus bar lines 53 invisibility cloak. While one even wants operation for a orthogonal to the metal fingers discussed. This more 54 55 broad range of visible colors, these invisible-contact demanding cloaking problem has not been solved yet. problems are simpler in various ways. One needs to 56 The cloaking of metal contacts on OLEDs is a distinct distinguish between the ballistic and the diffusive 57 problem. In other words: If you just use a free-form regime of light propagation. The ballistic regime 58 surface like above, designed for a solar cell, it will not applies to solar cells or photodetectors, whereas 59 work [4]; it will even make the shadow cast by the 60 OLEDs are Lambertian light emitters and often OLED contact more pronounced. The reason lies in the diffusive light scattering layers are added on top of different angular distributions of light rays. them. Nevertheless, one can use a different spatial transformation that leads to a different free-form Page 55 of 59 AUTHOR SUBMITTED MANUSCRIPT - draft

1 2 3 surface. This spatial transformation, however, turns out shear forces. Another challenge is to extend this 4 approach to the wave regime. 5 to be an implicit one; the spatial transformation depends on the result, requiring an iterative solution. 6 Yet another avenue is based on the fact that the While our corresponding results are encouraging 7 elasticity equations do become form invariant for a (unpublished), we emphasize that again only the 1D 8 more general class of elastic solid, e.g., for so-called problem of isolated wires or arrays of parallel metal 9 Cosserat materials, which exhibit additional wires has been solved. The problem of cloaking 10 (rotational) degrees of freedom. One opportunity and 11 hexagonal or square lattices of contact wires, remains challenge of the field lies in designing and 12 open. 13 experimentally realizing such generalized elastic solids 14 As an alternative, we have shown that the idea of in microstructured form to then use them for general 15 cloaking based on the macroscopic Maxwell equations transformation mechanics. 16 can be translated to the regime of light diffusion, 17 essentially because the stationary version of Fick’s 18 diffusion equation is mathematically analogous to 19 electrostatics [4-7]. Here, the light diffusivity is 20 controlled by the density of scattering particles. On this 21 basis, we have successfully cloaked contacts [4]. The 22 remaining challenge lies in the 3D micro- 23 manufacturing of such structures in a manner suitable 24 25 for mass products. Again, the problem of hexagonal or 26 square lattices of contact wires has not been solved so 27 far. 28 Together with the above free-form surface approach, in 29 regard to metal contacts on Lambertian OLEDs, the 30 two above possibilities compete against each other, 31 Figure 2 – Photograph of a fabricated and characterized two- 32 while we are presently not aware of other competing approaches. dimensional static mechanical cloak, designed by using the 33 direct-lattice-transformation approach. The lattice constant 34 Let us consider an example from mechanics. In civil of the hexagonal lattice in the surrounding of the cloak is 35 engineering, one often needs some sort of light-weight about a = 7 mm. Reproduced with permission from 36 scaffold structure. Suppose you need to punch a hole Reference [10]. 37 into this structure, e.g., for obtaining a feedthrough. 38 Advances in Science and Technology to Meet 39 Obviously, this hole will weaken the support structure, Challenges 40 raising the question whether we can build something 41 around the void to make the overall arrangement Generally, approaches solving the inverse problem of 42 appear in all respects as without the hole being how to get from the material parameters derived from 43 punched in. In other words: We aim at a mechanical transformation optics (or counterparts thereof) to 44 cloak. In 2006, Milton showed that the continuum- concrete metamaterial microstructures need to be 45 mechanics elasticity equations are not form-invariant developed. 46 under general spatial transformations, neither in the In regard to cloaking of general elastic waves by 47 static nor in the dynamic case (which is a show stopper 48 microstructured cloaks (compare Figure 2), perfectly for the ideas of transformation optics) – at least not for matched layers (PMLs) at simulation domain 49 the elasticity tensors of ordinary materials. In selected 50 boundaries will need to be worked out – otherwise special cases, such as 2D flexural waves one gets away 51 such cloaks cannot even be tested numerically on the 52 with ordinary solids [8] or for 3D pressure waves with computer. 53 pentamode mechanical metamaterials [9]. The latter 54 can be seen as solids approximating liquids in that their The corresponding fabrication technologies are almost 55 shear modulus is small compared to their bulk there. However, the field would certainly benefit from 56 modulus. advances in 3D laser nanoprinting in regard to speed, 57 spatial resolution, precision, and from the availability 58 Another option in mechanics (and in other areas) is to of a wider range of ingredient materials. 59 use direct spatial transformations of discrete lattices Concluding Remarks 60 [10]. This simple and direct approach works amazingly well in two dimensions and in the static case (also see Real-world applications of transformation optics and Figure 2), but so far lacks a sound theoretical invisibility cloaking such as cloaked contacts on solar justification in regard to why it also takes care of the cells or on OLEDs have come into reach. Further AUTHOR SUBMITTED MANUSCRIPT - draft Page 56 of 59

1 2 3 application possibilities arise beyond optics, e.g., in 4 5 mechanics. There, however, much further basic work 6 needs to be done. 7 Acknowledgments – We acknowledge the 8 contributions of all of the members of the KIT group 9 involved in this work. We also acknowledge support 10 by the Helmholtz program Science and Technology of 11 12 Nanosystems (STN), by Deutsche Forschungs- 13 gemeinschaft (DFG) through program DFG SPP 1839 14 “Tailored Disorder”, and by the Hector Fellow 15 Academy. 16 References 17 18 [1] Ergin T, Stenger N, Brenner P, Pendry JB, and 19 Wegener M 2010 Three-Dimensional Invisibility 20 Cloak at Optical Wavelengths Science 328, 337–339 21 [2] Schumann MF, Wiesendanger S, Goldschmidt JC, 22 23 Bläsi B, Bittkau K, Paetzold UW, Sprafke A, 24 Wehrspohn R, Rockstuhl C, and Wegener M 2015 25 Cloaked contact grids on solar cells by coordinate 26 transformations: Designs and prototypes Optica 2, 27 850–853 28 [3] Schumann MF, Langenhorst M, Ding K, Paetzold 29 UW, and Wegener M 2017 All-angle invisibility 30 cloaking of contact fingers on solar cells by 31 refractive free-form surfaces Adv. Opt. Mater., 32 submitted 33 34 [4] Mayer F, Schittny R, Egel A, Niemeyer A, Preinfalk 35 J, Lemmer U, and Wegener M 2016 Cloaking 36 contacts on large-area organic light-emitting diodes 37 Adv. Opt. Mater. 4, 740–745 38 39 [5] Schittny R, Kadic M, Bückmann T, and Wegener M 40 2014 Invisibility Cloaking in a Diffusive Light 41 Scattering Medium Science 345, 427–429 42 [6] Schittny R, Niemeyer A, Kadic M, Bückmann T, 43 Naber A, and Wegener M 2015 Diffuse-light all- 44 solid-state invisibility cloak Opt. Lett. 40, 4202– 45 4205 46 47 [7] Schittny R, Niemeyer A, Mayer F, Naber A, Kadic 48 M, and Wegener M 2016 Invisibility cloaking in 49 light scattering media Laser Photon. Rev. 10, 382– 50 408 51 [8] Stenger N, Wilhelm M, and Wegener M 2012 52 53 Experiments on elastic cloaking in thin plates Phys. 54 Rev. Lett. 108, 014301 55 [9] Bückmann T, Thiel M, Kadic M, Schittny R, and 56 Wegener M 2014 An elasto-mechanical unfeelability 57 cloak made of pentamode metamaterials Nat. 58 Commun. 5, 4130 59 60 [10] Bückmann T, Kadic M, Schittny R, and Wegener M 2015 Cloak design by direct lattice transformation Proc. Natl. Acad. Sci. USA 112, 4930–4934 Page 57 of 59 AUTHOR SUBMITTED MANUSCRIPT - draft

1 2 3 9.2 Beyond Optics: Transforming Other Wave And 4 5 Transport Systems – Steven A. Cummer, Duke 6 University 7 8 Status 9 10 Transformation optics was an astonishing theoretical 11 breakthrough. Beyond bringing invisibility into the 12 realm of the feasible, it remains one of few general tools 13 for solving the electromagnetic design and synthesis 14 problem. One can define a particular geometric 15 operation—such as stretching, twisting, and 16 displacing—to be performed on electromagnetic fields, 17 and the transformation optics framework provides exact 18 material parameters to implement that operation. 19 Following this important discovery, it was 20 naturally of great interest (and importance) to know 21 whether the coordinate transformation framework could 22 23 be applied to control fields in other wave and transport 24 systems. The peculiarities of coordinate Figure 1 – Illustration of some of the wide range of 25 transformations in electromagnetics, embodied in physical systems that admit transformation-type 26 relativity theory, suggested that the answer might be no. solutions and the effective material parameters that must 27 However, it was soon shown that the transformation be controlled in each case. 28 framework could be applied to acoustic waves in fluids question is obvious: are there more systems that can be 29 without steady flow [1,2]. It soon became clear that the 30 controlled using the transformation approach, and what concept could be applied to other linear wave and are the limits of applicability? This is a largely 31 transport systems, including the Schrodinger equation 32 theoretical question that is waiting to be tackled, should [3] and heat conduction and diffusion [4]. It should be additional wave and transport systems of interest be 33 mentioned that the existence of transformation solutions 34 identified. Past work in this area has followed one of to transport had been discovered previously [5]. 35 two general approaches: identifying a mapping of the 36 The ability to control transport and wave system in question to one known to admit 37 propagation in nearly arbitrary ways in a wide range of transformation-type solutions, or the demonstration of 38 physical systems offers amazing possibilities for both the coordinate transformation invariance of the dynamic 39 exploring wave behavior in complex environments and equations themselves. 40 developing devices of practical value. The key step in One research direction enabled by 41 transitioning transformation theory to physical transformation design is the experimental 42 implementation in these systems is the ability to create 43 implementation of interesting wave and transport the medium effective properties needed to control the physics by analogue. The similarity of the 44 transport or wave propagation in each system. In 45 transformation approach across different wave systems acoustics, for examples, these properties are mass creates connections that enable experiments in one 46 density and bulk modulus (compressional stiffness), 47 domain to have much broader implications. One while in heat conduction, these are the specific heat and 48 example is exploiting the theoretical analogy between 49 thermal conductivity. Understanding the theoretical light propagation near a black hole and wave 50 capabilities and limits of the transformation approach in propagation in a moving fluid to explore difficult-to- 51 each of these physical systems, developing new access phenomena such as Hawking radiation [6, and 52 approaches to control the essential material properties references therein]. Other universal wave phenomena, 53 effectively and efficiently, and transferring knowledge such as propagation in parity-time symmetric systems, 54 and results from one physical system to another all can be accessed and experimentally explored with the 55 remain active research areas with critical unanswered most convenient wave system yet the implications are 56 questions. 57 broadly applicable. A second direction is the development of 58 Current and Future Challenges 59 materials and devices for controlling waves and 60 The applicability of the transformation design approach transport in these different domains, including to at least some non-electromagnetic wave and transport acoustics, elastodynamics, and heat flow. Interestingly, systems has opened the door to new research directions many of the challenges in implementing transformation and raises some important scientific questions. One materials and devices in all of these domains are similar AUTHOR SUBMITTED MANUSCRIPT - draft Page 58 of 59

1 2 3 to those in optics and electromagnetics. The One practically important wave system that 4 remains relatively unexplored is solid elastodynamics. 5 transformation design approach typically yields 6 continuum material parameters that are anisotropic and Part of the challenge is that the transformation design 7 smoothly inhomogeneous, and these properties need to approach leads to solid materials that exhibit so-called 8 be controlled with a precision not normally available in Willis coupling [10]. Although continuum solids do not 9 natural materials. have this property, microstructured materials can, 10 The most successful solution to this challenge provided they have the right kind of internal asymmetry 11 has been engineered metamaterials. Metamaterial and or lattice structure. The practical importance of waves 12 related design approaches have successfully been and vibrations in solids suggests that further theoretical 13 applied in a variety of wave and transport systems to and experimental developments in this area may have 14 experimentally demonstrate the feasibility of practical substantial impact. 15 implementations of transformation designs [e.g. 7]. 16 That said, improvements to metamaterial design and Concluding Remarks 17 fabrication are still needed to carry these concepts 18 The power of the transformation design framework is beyond simple feasibility demonstrations towards 19 not limited to optics and electromagnetics. It appears to practically useful devices. 20 be applicable, to at least some degree, to most if not all

21 wave and transport systems. These include acoustic Advances in Science and Technology to Meet 22 waves in fluids, heat conduction, elastodynamic waves 23 Challenges in solids, and even quantum mechanical matter waves. 24 25 The instrincally multiscale design, simulation, and Anisotropic and inhomogeneous effective material 26 fabrication of metamaterials remains a major hurdle for properties are the critical component in experimentally 27 the design of transformation-based devices for any wave realizing the level of wave control the transformation 28 system. Subwavelength scales are needed to mimic framework promises. Designing and fabricating 29 continuum materials, while the device-scale artificial metamaterials with subwavelength structure 30 inhomogeneity is typically many orders of magnitude that spans tens to hundreds (or more) wavelengths in 31 larger. Additive manufacturing is a promising path, but total size remains a primary technical challenge in all of 32 most the capabilities of most implementations at present these areas. When this is possible, truly arbitrary wave 33 are not sufficiently multiscale. and transport control become feasible. 34 Another challenge across all transformation- 35 based designs is the physical realization of the required References 36 range of continuum material properties. Different 37 [1] Cummer S and Schurig D 2007 One path to acoustic 38 background materials impose different practical cloaking New J. Phys 9 45 39 constraints. In airborne acoustics, for example, it is [2] Chen H and Chan C 2007 Acoustic cloaking in three 40 difficult for nonresonant structures to behave as though dimensions using acoustic metamaterials Appl. Phys. 41 they are lighter or more compressible than air. In 42 contrast, for underwater acoustics, it is challenging to Lett. 91 183518 43 create structures that are more than a few times more [3] Zhang S, Genov D, Sun C and Zhang X 2008 44 dense or stiff than water. The range of material Cloaking of matter waves Phys. Rev. Lett. 100 45 parameters needed can be controlled to some degree 123002 [4] Guenneau S, Amra C and Veynante D 2012 46 through the transformation framework, and developing 47 Transformation thermodynamics: cloaking and design approaches that merge transformation theory concentrating heat flux Opt. Exp. 20 8207–18 48 with practical material fabrication limits will likely be 49 [5] Greenleaf A, Lassas M and Uhlmann G 2003 essential for successful devices in any wave system. 50 Anisotropic conductivities that cannot be detected by 51 A particular advantage that acoustics offers to EIT Physiol. Meas. 24 413–9 52 transformation-based design lies in the relatively slow [6] Leonhardt, U 2015 On cosmology in the laboratory 53 propagation velocity and relatively long time scales of Phil. Trans. R. Soc. A 373 20140354 54 frequencies of interest. The [7] Zigoneanu L, Popa B and Cummer S 2014 Three- 55 response can be controlled actively at microsecond to dimensional broadband omnidirectional acoustic 56 millisecond timescales [8], and this enables the ability ground cloak Nat. Mat. 13 352–5 57 to create local material behavior not easily implemented [8] Popa B, Zigoneanu L and Cummer S 2013 Tunable 58 in other wave systems, including gain. One example active acoustic metamaterials Phys. Rev. B. 88 59 that exploits this is demonstrating the ability to control 024303 60 exceptional points in parity-time symmetric systems [9]. [9] Shi C, Dubois M, Chen Y, Cheng L, Ramezani H, The development of new active approaches to control Wang Y and Zhang X Accessing the exceptional local acoustic metamaterial response should enable points of parity-time acoustics Nat. Comms. 7 11110 acoustic analogues of very complex wave systems. Page 59 of 59 AUTHOR SUBMITTED MANUSCRIPT - draft

1 2 3 [10] Milton G, Briane M and Willis J 2006 On cloaking 4 for elasticity and physical equations with a 5 transformation invariant form New J. Phys. 8 248 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60