Roadmap

Roadmap on transformation optics

Martin McCall1, John B Pendry1, Vincenzo Galdi2 , Yun Lai3,S A R Horsley4, Jensen Li5 , Jian Zhu5, Rhiannon C Mitchell-Thomas4,Oscar Quevedo-Teruel6 , Philippe Tassin7, Vincent Ginis8, Enrica Martini9,Gabriele Minatti9, Stefano Maci9, Mahsa Ebrahimpouri6, Yang Hao10 ,Paul Kinsler11 , Jonathan Gratus11,12 , Joseph M Lukens13 ,Andrew M Weiner14 , Ulf Leonhardt15, Igor I Smolyaninov16 ,Vera N Smolyaninova17, Robert T Thompson18, Martin Wegener18,Muamer Kadic18 and Steven A Cummer19

1 Imperial College London, Blackett Laboratory, Department of Physics, Prince Consort Road, LondonSW7 2AZ, United Kingdom2 Fields & Waves Lab, Department of Engineering, University of Sannio, I-82100 Benevento, Italy3National Laboratory of Solid State Microstructures, School of Physics, and Collaborative InnovationCenter of Advanced Microstructures, Nanjing University, Nanjing 210093, People’s Republic of China4University of Exeter, Department of Physics and Astronomy, Stocker Road, Exeter, EX4 4QL UnitedKingdom5 School of Physics and Astronomy, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UnitedKingdom6KTH Royal Institute of Technology, SE-10044, Stockholm, Sweden7Department of Physics, Chalmers University, SE-412 96 Göteborg, Sweden8Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium9Dipartimento di Ingegneria dell’Informazione e Scienze Matematiche, University of Siena, Via Roma, 5653100 Siena, Italy10 School of Electronic Engineering and Computer Science, Queen Mary University of London, London E14FZ, United Kingdom11 Physics Department, Lancaster University, Lancaster LA1 4 YB, United Kingdom12Cockcroft Institute, Sci-Tech Daresbury, Daresbury WA4 4AD, United Kingdom13Quantum Information Science Group, Computational Sciences and Engineering Division, Oak RidgeNational Laboratory, Oak Ridge, TN 37831, United States of America14 School of Electrical and Computer Engineering, Birck Nanotechnology Center, and Purdue QuantumCenter, Purdue University, West Lafayette, IN 47907, United States of America15 Physics of Complex Systems, Weizmann Institute of Science, Rehovot 7610001 Israel16 Institute for systems research, University of Maryland, College Park, MD 20742, United States ofAmerica17Department of Physics, Astronomy and Geosciences, Towson University, Towson, MD 21252, UnitedStates of America18Karlsruhe Institute of Technology, Institut für Angewandte Physik, Wolfgang-Gaede-Straße 1, D-76131Karlsruhe, Germany19 Electrical and Computer Engineering, Duke University, PO Box 90291, Durham, NC 27708, UnitedStates of America

E-mail: m.mccall@imperial.ac.uk

Received 11 September 2017Accepted for publication 26 March 2018Published 22 May 2018

AbstractTransformation optics asks, using Maxwell’s equations, what kind of electromagnetic mediumrecreates some smooth deformation of space? The guiding principle is Einstein’s principle ofcovariance: that any physical theory must take the same form in any coordinate system. This

Journal of Optics

J. Opt. 20 (2018) 063001 (44pp) https://doi.org/10.1088/2040-8986/aab976

2040-8978/18/063001+ 44$33.00 © 2018 IOP Publishing Ltd Printed in the UK1

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requirement fixes very precisely the required electromagnetic medium. The impact of this insightcannot be overestimated. Many practitioners were used to thinking that only a few analyticsolutions to Maxwell’s equations existed, such as the monochromatic plane wave in ahomogeneous, isotropic medium. At a stroke, transformation optics increases that landscapefrom ‘few’ to ‘infinity’, and to each of the infinitude of analytic solutions dreamt up by theresearcher, there corresponds an electromagnetic medium capable of reproducing that solutionprecisely. The most striking example is the electromagnetic cloak, thought to be an unreachabledream of science fiction writers, but realised in the laboratory a few months after the papersproposing the possibility were published. But the practical challenges are considerable, requiringmeta-media that are at once electrically and magnetically inhomogeneous and anisotropic. Howfar have we come since the first demonstrations over a decade ago? And what does the futurehold? If the wizardry of perfect macroscopic optical invisibility still eludes us in practice, thenwhat compromises still enable us to create interesting, useful, devices? While three-dimensional(3D) cloaking remains a significant technical challenge, much progress has been made in twodimensions. Carpet cloaking, wherein an object is hidden under a surface that appears opticallyflat, relaxes the constraints of extreme electromagnetic parameters. Surface wave cloaking guidessub-wavelength surface waves, making uneven surfaces appear flat. Two dimensions is also thesetting in which conformal and complex coordinate transformations are realisable, and thepossibilities in this restricted domain do not appear to have been exhausted yet. Beyondcloaking, the enhanced electromagnetic landscape provided by transformation optics has shownhow fully analytic solutions can be found to a number of physical scenarios such as plasmonicsystems used in electron energy loss spectroscopy and cathodoluminescence. Are there furtherfields to be enriched? A new twist to transformation optics was the extension to the spacetimedomain. By applying transformations to spacetime, rather than just space, it was shown thatevents rather than objects could be hidden from view; transformation optics had provided ameans of effectively redacting events from history. The hype quickly settled into seriousnonlinear optical experiments that demonstrated the soundness of the idea, and it is now possibleto consider the practical implications, particularly in optical signal processing, of having an‘interrupt-without-interrupt’ facility that the so-called temporal cloak provides. Inevitable issuesof dispersion in actual systems have only begun to be addressed. Now that time is included in theprogramme of transformation optics, it is natural to ask what role ideas from general relativitycan play in shaping the future of transformation optics. Indeed, one of the earliest papers ontransformation optics was provocatively titled ‘General Relativity in Electrical Engineering’. Theanswer that curvature does not enter directly into transformation optics merely encourages us tospeculate on the role of transformation optics in defining laboratory analogues. Quite whyMaxwell’s theory defines a ‘perfect’ transformation theory, while other areas of physics such asacoustics are not apparently quite so amenable, is a deep question whose precise, mathematicalanswer will help inform us of the extent to which similar ideas can be extended to other fields.The contributors to this Roadmap, who are all renowned practitioners or inventors oftransformation optics, will give their perspectives into the field’s status and future development.

Keywords: transformation optics, metamaterials, cloaking, antennas, spatial cloaking, spacetimecloaking, spatial dispersion

(Some figures may appear in colour only in the online journal)

Contents

1. Introduction 4

2. Transformation optics and the near field 5

3. Nonlocal and non-Hermitian extensions of transformation optics 7

4. Transformation optics by photonic crystals: a nonlocal route towards the optical frequency regime 9

5. Complex coordinates in optics; conformal maps and analytic continuation 11

6. Extending transformation optics and towards a thin cloak 13

7. Combining curvature and index gradients for surface and guided waves 15

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8. Transformation optics at surfaces 17

9. Metasurface radiation and guidance at microwaves 19

10. Bespoke lenses: an opportunity for tailoring antenna radiation patterns 21

11. Transformation optics for antenna engineering 23

12. Spacetime transformation optics 25

13. Transformations optics with spatial dispersion 27

14. Experimental progress in temporal cloaking 29

15. Cosmology in the laboratory: challenges at the horizon 31

16. Spacetime analogs based on hyperbolic metamaterials 33

17. Transformation optics in general relativity 35

18. Seeking applications in optics and beyond 37

19. Beyond optics: transforming other wave and transport systems 39

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1. Introduction

Martin McCall

Imperial College

A little over a decade ago Pendry et al [1] noted that thelong-known form-invariance of Maxwell’s equations could beinterpreted as an extremely flexible design recipe forachieving virtually arbitrary distortions of electromagneticfields. It is an unusual theory in that the recipe is exact, i.e. allaspects of the electromagnetic field, amplitude, phase andpolarization are in principle relocated by an appropriatematerial design. The method was immediately applied todesign and build a device that guided light around a voidregion in space, thus bringing the celebrated invisibility cloakof science fiction right into the laboratory. Although the cloakis the most striking example, transformation optics (TO) hassince developed many other applications that combine itsdesign ingenuity with parallel developments in metamaterialstechnology. In this Roadmap, several leading practitionersgive their unique personal perspectives of the current status ofthe field, and where the future challenges lie.

Pendry’s article focuses on the near-field and the impact ofTO on surface plasmonics, explaining, for example, the hugeenhancements of local fields around structural singularities, andthe resultant opportunities offered to spectroscopy and nonlinearoptics resulting from the very tight compression of light fields.

TO designs invariably prescribe inhomogeneous materialparameters. A striking departure from this is described by Vin-cenzo Galdi, who examines the use of the TO algorithm in thespectral domain. Such designer dispersion prescribes media withfascinating frequency dependencies, all the while maintaininguniformity in the spatial domain. The theme of spatial dispersionin TO is taken up again in Lai’s article, who discusses howphotonic band-gap engineering is combined with TO to produceomnidirectional impedance matching.

Simon Horsley shows how analytic continuation of thecoordinates into the complex arena permits extensions to mediawith gain and loss, connecting TO closely with so-called PTsymmetric media, and showing how to design reflectionlessmedia. Rather than exploit form invariance under coordinatetransformation, Li et al explain how new possibilities emergeif, instead, form invariance is pursued with respect to trans-formation of the electromagnetic field, leading to, for example,an effective magnetic bending force on light. They also discussstrategies for producing thin cloaks based on metasurfaces.

Rhiannon Mitchell-Thomas reviews geometrical approa-ches confined to curved surfaces, rather than 3D space, showingthat TO designs using isotropic graded index media allow fornovel implementations of, for example, Eaton and Luneburglenses. Discontinuous transformations at surfaces are discussedby Philippe Tassin et al, who show how the latest metasurfacetechnology opens up applications in integrated optics.

Martini et al discuss how a planar version of TO leads tocontrol of surface/plasmonic wave propagation or ‘metasurfing’;while Oscar Quevedo-Tereul shows how TO is being used totailor antenna radiation patterns using bespoke lenses. The themeof antenna engineering is taken up by Yang Hao in reviewinghow TO based radome designs are leading to novel compactphased arrays.

Paul Kinsler and Jonathan Gratus consider generalization ofTO to the spacetime domain. Kinsler’s article discusses space-time cloaking, in which events, not objects, are hidden fromview. Gratus shows that in the broader TO regime with bothtime and space included, spatial dispersion plays a necessary andfundamental role. Lukens et al describe some of the exper-imental efforts in event cloaking, emphasising the role played byspacetime duality in achieving localized shadowing in space-time. They emphasise that applications will likely focus onspecific communication tasks, and offer tantalising connectionsbetween temporal cloaking and quantum information proces-sing.

As Ulf Leonhardt points out in his contribution, followingthe detection of gravitational waves, cosmology has nowentered a golden era, and how electromagnetism might repayits debt to general relativity becomes an issue addressable byTO. Smolyaninov et al discuss, for example, how the mappingbetween metric and material parameters, a key idea in TO,informs cosmology in the optics lab, with inhomogeneoushyperbolic metamaterials directly analogising signature chan-ging events and multiverses. As Thompson makes clear, thedistinction between medium and spacetime contributions inMaxwell’s equations opens up several potential new avenuessince the mapping between distortion and effective medium isactually not unique.

Beyond optics, Martin Wegener takes us from using TO toimprove solar cell efficiency to transformation mechanics,showing how mechanical cloaks can improve the utility ofscaffolds. Steve Cummer also assesses how the TO algorithmcan and has been successfully extended to other disciplinessuch as acoustics, elastodynamics and heat flow.

A recurring theme in all the contributions is the extant gapbetween design aspiration and technological feasibility. The-orists have provided beautiful design tools that can potentiallygive us many very useful and practical devices, but we arefrustrated at often only being able to realise a fraction of thispotential due to the difficulty of producing broad-band meta-material designs requiring sub-wavelength fabrication. Thecompromises currently made on the altar of progress do not dojustice to the original designs. Perhaps the main ‘take-home’message of this Roadmap is to provide a rallying call totechnologists and experimentalists to make the necessary stepchange to meet these challenges and turn, as Cummer notes,simple feasibility demonstrations towards practically usefuldevices.

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2. Transformation optics and the near field

J B Pendry

Imperial College London

Status. Transformation optics (TO) has its origins in Einstein’sgeneral theory of relativity [2]: he showed how Maxwell’sequations are invariant in form under coordinate transformations.Post [3] demonstrated how a coordinate transformation can berepresented as changes to the constitutive parameters, , .e mThese ideas were slow to be adopted by the optics communitybut in 1996 Ward and Pendry [4] faced the challenge of adaptingelectromagnetic finite difference codes to the cylindricalgeometry of a fibre. Originally written in Cartesian coordinatesto tackle photonics crystals the codes did not need to bereprogrammed in cylindrical coordinates, but simply presentedwith a transformed , .e m However the most powerful insightcame with the realisation that under a coordinate transformationlines of force, whether electric or magnetic, remained attachedto the coordinate system. A picture emerged of space as adeformable entity, and the possibility of rearranging the fields bydistorting space, then asking through Einstein’s equations whatvalues of ,e m would send the fields in the desired direction. Thevalue of this insight is that it replaces the intuitive pictures of rayoptics with a way to manipulate field lines, but retaining theinsight offered by a physical picture of what is going on.Furthermore an intuitive picture can be given of how ,e mtransform. Under a simple compression by a of coordinatesalong one axis, components of both ,e m along that axis arereduced by a factor of ;a components perpendicular to the axisare increased by .1a- Any transformation can be represented bya series of compressions and rotations and thus the wholetransformation can be understood in geometrical terms. Areview of these ideas can be found in [5].

These ideas first gained attention when deployed todesign a cloak of invisibility, a challenge that is almostimpossibly difficult using conventional paradigms butridiculously simple for TO. Many cloaks can also beenvisaged in terms of deflected rays, but this picture is notavailable to us for structures that are much less than thewavelength and we are confronted with the so called ‘nearfields’ where the electric and magnetic components are nolonger tightly coupled into the ray picture but lead a moreindependent existence. For example, a cloak for staticmagnetic fields can in no way be envisaged as controlling aset of rays and we are forced instead to think of lines of force.Such a cloak has been realised by Supradeep Narayana andYuki Sato [6] and figure 1 shows their results.

TO has had a strong impact on plasmonic systems. Theoptical response of metals is dominated by the sea of freeelectrons and on the surface the electron sea supportselectromagnetic waves called surface plasmons. Bound to themetal surface and with a wavelength much shorter than that oflight in free space, their behaviour is dominated by the shape ofthe surface. For example, a highly polished silver surface is anexcellent mirror reflecting nearly all incident light. On the other

hand finely divided silver in the form of nanoparticles, such asfound in photographic negatives, absorbs nearly all incidentlight and is black. This is an extreme example of ametamaterial: internal structure dominates the optical response.Understanding the interplay of surface structure with the nearfield has enabled new insights.

Current and future challenges. Surface plasmonics [7] isperhaps the field in which TO has most impact. Plasmonicsburst spectacularly into spectroscopy when Fleischman madethe observation that Raman signals from molecules adsorbed ona rough silver surfaces were many orders of magnitude greaterthan the free space cross sections would predict. The effect canbe explained by huge enhancements of local fields of incidentlight around structural singularities. Heavy computationalstudies confirmed this explanation but provided little insightinto why singularities produce enhancement. It fell to TO toexplain the effect in simple terms.

Consider a structure consisting of a surface containing asharp knife edge. The edge constitutes a singularity and localelectric fields are massively enhanced in its vicinity when thestructure is illuminated. The singularity can be removed by asimple transformation,

x x y

y y x

ln

arctan

2 2¢ = +¢ =

(a)

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-6

No cloak

Cloak

No cloakCloak

sensor 1 sensor 2

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Figure 1. TO was used to control static magnetic fields in thisrealisation of a cloak for magnetism [4]. (a) Magnetic field lines arediverted away from the cloaked region whilst remaining undisturbedoutside the cloak. (b)Measurements of the field with and without thecloak confirm the picture shown in (a).

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where the origin is located on the edge. This transforms theedge to a non singular array of wave guides and thesingularity at the origin of x y, space vanishes to infinity inx y,¢ ¢ space. All singular structures can have their singularitiesremoved in this fashion. Enhancements can be understood asthe compression of an infinite system into a finite one. Thishas the curious consequence that a finite singular structure hasa continuous spectrum typical of an infinite system, and aninfinity in the density of states at the singularity. Though thisresult has to be qualified for real materials where practicalconsiderations of loss and of non locality of their responsecan reduce predicted enhancements. More details can befound in [8].

Another system described in [8] is shown in figure 2. Inthis instance the transformation is an inversion which takes asimple metallic waveguide and transforms it into twotouching cylinders. The transformation is singular at thetouching point which produces intense local fields. Not onlydoes a transformation explain why local fields are so intense,it also provides a means of classifying the spectrum: often it isthe case that the transformed system, lacking the originalsingularity, has a simpler structure. In the examples givenabove the transformed waveguide structure is translationallyinvariant along the x¢ axis allowing modes to be classified asBloch waves, assigned a wave vector, k, and their frequenciescalculated analytically as a function of k.

TO can also be applied in the context of the refractiveindex to remove singularities from a structure as has beenelegantly demonstrated in [9]

Advances in science and technology to meet challenges.Several challenges meet these beautiful theoretical conceptswhen inviting experimentalists into the field. The first is thematerials challenge. Most of the conclusions presented aboveassume an ideal loss free metal described by a local dielectricfunction, .e Locality means that e depends only on frequencyand not on wave vector. Loss absorbs energy and preventsrealisation of ideal field enhancements, so that only relativelyloss free metals such as silver and gold are serious candidatesfor the effects. Non locality has the effect of smearing out anysingularities in the surface structure. In metals non localityarises because the electron sea has limited compressibility andcannot be squeezed into very sharp corners preventingrealisation of an ideal singularity [10–12]. The search is onfor new materials that perform even better than our currentoptimum candidates of silver and gold [13], a challenge mademore difficult by our limited theoretical understanding oflosses and non locality beyond the standards achieved bycurrent best performers.

Beyond that there is the issue of how to sculpt surfaces tothe required geometrical perfection. Metals have very highsurface energy and object to the formation of singularities:sharp edges tend to round off, and singularities formed bytouching curves often result in the fusing of the two surfaces.Engineering on the nanoscale demands expensive apparatusto achieve the precision needed.

Despite these slightly pessimistic observations greatopportunities are to be had. I have alluded to the spectacularenhancements in spectroscopic sensitivity already a reality.Utilising the design tool of TO structures can be optimised togive the best response and single molecule sensitivity canbe had.

It is already possible to compress light into a focus farsmaller than the free space Abbé limit. Compression into asquare nanometre is possible, many orders of magnitude moredense than achieved with conventional lenses thus achievingwith modest power inputs energy densities great enough totrigger nonlinear effects. A cornucopia of new experimentsawaits exploitation of nonlinearity, from simple switching oflight with light to more sophisticated concepts such as phaseconjugation and time reversal.

Concluding remarks. TO is a general tool that provides anew way to visualise electromagnetic phenomena, whilstretaining accuracy at the level of Maxwell’s equations. It hasparticular value for the near field where conventional raypictures are of no use, but where the concept of fields stillapplies. In the above two areas of application are described:DC fields where the entire system exists in the near field, andsurface plasmonics where most of the interesting phenomenaare observed on length scales smaller than the free spacewavelength.

Acknowledgments. This work was supported by the EPSRC(Grant No. EP/L024926/1), and the Gordon and BettyMoore Foundation.

Figure 2. (A) Surface plasmons are excited by light emitted from anexcited atom in a metallic waveguide. (B) Transformation byinversion about the atom centre generates a completely differentstructure of two touching cylinders. (C) The waveguide solutionsfrom (A) map directly to the touching cylinders giving stronglyenhanced fields near the point of contact.

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3. Nonlocal and non-Hermitian extensions oftransformation optics

Vincenzo Galdi

University of Sannio

Status. By interpreting the material effects on wavepropagation as local distortions of a given coordinatereference frame, transformation optics (TO) provides a verypowerful and systematic framework to precisely manipulateelectromagnetic fields [1]. In essence, the conceptual designof a desired response may be conceived by relying primarilyon geometrical intuition based on the geodesic path of lightrays, and is effectively decoupled from the actual materialsynthesis problem, which is eventually posed as a suitableapproximation of some analytically derived ideal constitutive‘blueprints’.

Conventional TO schemes typically rely on local, real-valued coordinate transformations, which inherently implysome limitations in the attainable field-manipulation effects.Thus, typical constitutive blueprints tend to be spatiallyinhomogeneous and anisotropic [1], whereas it is generallynot possible to extrinsically generate nonlinear, nonreciprocal,nonlocal (i.e. spatial dispersion), and non-Hermitian (i.e. lossand/or gain) effects, among others. Several extensions havebeen proposed in order to overcome these limitations. Here,we focus on certain extensions that enable the TO formalismto deal with nonlocal and non-Hermitian effects, which arebecoming increasingly relevant in many metamaterialapplications.

Current and future challenges. In [14, 15], our team laid out thefoundations for a nonlocal TO extension, which can be exploitedto engineer complex spatial-dispersion effects. Such extensionrelies on a reformulation of TO in the frequency-wavevectorphase space, accessed via temporal and spatial Fouriertransforms, and retains the attractive features of conventionalTO in terms of decoupling the actual metamaterial synthesisfrom the conceptual design, with the latter still heavily inspiredby geometrical intuition. As illustrated in figure 3, a broadvariety of interesting dispersion effects can be interpreted in thephase space as geometrical transformations of the conventionalconical dispersion surface of a homogeneous medium. Thus, forinstance, effects such as one-way (nonreciprocal) propagation,additional extraordinary waves, and slow-light can be associatedto analytic properties of the transformation such as non-centersymmetry, multivaluedness, and stationary inflectionpoints, respectively. Similar to the conventional (local) TOapproach, our proposed nonlocal extension systematically yields,in analytic closed-form expressions, the required (nonlocal)constitutive blueprints in terms of wavevector-dependentpermittivity and permeability tensors [14, 15].

By comparison with conventional (local, spatial-domain)TO, we note that the above nonlocal extension inherentlyyields spatially homogeneous constitutive blueprints. In viewof the underlying spatial Fourier transform, the spectralresolution, i.e. the capability to tailor the dispersion surfaces

in the frequency-wavevector phase space, is gained at theexpense of the spatial resolution. On the other hand,conventional TO allows fine tailoring of the spatial response,but completely lacks spectral resolution. Thus, conventionaland nonlocal TO can be viewed as two extremes in the spatialvs. spectral resolution tradeoff. As a future challenge, it wouldbe very intriguing to bridge these two extremes, so as to blendin a desired fashion both spatial and spectral resolution. Fromthe mathematical viewpoint, windowed spatial Fourier trans-forms, such as the Gabor transform, seem to constitute aparticularly suitable, self-consistent framework for such ahybrid TO formulation. In such a scheme, the window widthwould set the spectral vs. spatial resolution tradeoff via anuncertainty relation, and the conventional and nonlocal TOformulations would be recovered as the limits for zero andinfinite window width, respectively. The above-envisionedGabor-type formulations may considerably expand futureprospects for TO applications, encompassing complexscenarios featuring, e.g. nonlocal effects at different spatialscales, such as photonic ‘hypercrystals’. Moreover, they mayalso set the stage for disruptive field-manipulation platformscombining the spatial-routing (e.g. multiplexing, steering)capabilities of conventional TO with the inherent signal-processing capabilities (e.g. convolution, differentiation,integration) enabled by nonlocal TO. This may open upnew intriguing venues in the emerging field of ‘computationalmetamaterials’ [16].

In [17, 18], our team put forward a complex-coordinateTO extension that can naturally handle non-Hermitianmetamaterials featuring spatial modulation of loss and gain.Also in this case, the appealing characteristics of conventional(real-valued) TO in terms of physically incisive modeling andgeometry-driven intuitive design are retained, by combining

Figure 3. Conceptual illustration of the nonlocal TO extension in thefrequency (w)-wavevector k k,x z( ) phase space. (a) Conical disper-sion surface of a homogeneous medium. (b) Hyperbolic dispersion(purely imaginary transformation component). (c) One-way, non-reciprocal propagation (non-centersymmetric transformation). (c)Additional extraordinary wave (multivalued transformation). (d)Frozen mode (stationary inflection point). (f) Dirac-point conicalsingularity (phase-space shift). Reproduced from [15]. CC BY 4.0 ©2016 Optical Society of America.

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https://creativecommons.org/licenses/by/3.0/

the approach with well-established analytic formalisms based,e.g. on the ‘complex-source-point’ (CSP) representation ofbeam-like wave objects, and the ‘leaky-wave’ (complexpropagation constant) modeling of radiating states [17, 18].For instance, as schematized in figure 4, a complex coordinatetransformation relating a ‘source’ and an ‘image’ CSP can beinterpreted in the real, physical space as an input–outputtransformation of an optical beam, induced by a non-Hermitian metamaterial slab, with full control on the waistposition, direction and diffraction length. This represents arather versatile and insightful approach, which brings aboutnew degrees of freedom and may provide new perspectives inthe design of active optical devices and radiating systems,with the use of gain not merely limited to achieve losscompensation.

Metamaterials mixing loss and gain are also receiving asurge of interest as potentially feasible testbeds for studyingphenomena and properties that are distinctive of non-Hermitian systems. These include parity-time symmetry andexceptional points (i.e. degeneracies between complexeigenstates), which are ubiquitous in many disciplines,ranging from quantum mechanics to nuclear physics [19].The application of non-Hermitian TO extensions to thesescenarios is still largely unexplored, and represents one of themost fascinating developments to pursue. Preliminary studieson parity-time-symmetric metamaterials [17] have shownintriguing connections between the onset of spontaneoussymmetry breaking (an example of exceptional point) and thediscontinuous character of the complex-coordinate transfor-mation. This may be indicative of deeper connectionsbetween the analytic properties of the transformations andthe physics of exceptional points, whose exploration mayuncover novel effects and shed new light on the phenomen-ological understanding.

Advances in science and technology to meet challenges.Nonlocal TO extensions crucially rely on nonlocal effectivemodels to approximately match the desired (ideal) constitutiveblueprints with that of a physical metamaterial structure withinappropriate phase-space regions. Therefore, essential to furtheradvances in this field is the availability of a comprehensive‘library’ of effective nonlocal models, building up on general[20] and structure-specific [21] nonlocal homogenizationapproaches. Moreover, the discovery of new structures andregimes where nonlocal effects are enhanced, and possiblyattainable via moderate-permittivity, low-loss materialconstituents may significantly boost the practical applicabilityof nonlocal TO. Finally, the development of full-wave

numerical simulation tools implementing nonlocal constitutiverelationships would also represent a major catalyst.

Non-Hermitian TO, on the other hand, critically relies onthe possibility to precisely tailor the spatial distribution of lossand gain, which is becoming technologically viable. Atmicrowave frequencies, gain may be attained via classicalamplification schemes based on operational amplifiers andGunn diodes, whereas at optical wavelengths active dopantssuch as two-level atoms (or quantum dots) may be exploited.While certain non-Hermitian effects may require unfeasiblyhigh values of gain, it is also important to remark thatcompletely passive (i.e. lossy) structures have been exploitedto experimentally demonstrate pivotal proof-of-concepteffects in the physics of exceptional points (see e.g. [22]).Within this framework, metamaterials with near-zero con-stitutive parameters (real part) [23] appear particularlyattractive in view their well-known capabilities to dramati-cally enhance the effects of relatively low levels of loss and/or gain.

Concluding remarks. In a variety of metamaterialapplications of current interest, spatial dispersion and loss/gain are escalating from second-order nuances to instrumentaleffects. Nonlocal and non-Hermitian extensions of the TOapproach potentially constitute an attractive framework toharness these effects in a versatile, systematic and insightfulfashion.

Figure 4. Schematic representation of the complex-coordinate TOmapping between a ‘source’ and an ‘image’ CSP (Ps̃ and P,ĩrespectively). In the physical, real space the mapping can beinterpreted as a transformation of an input beam into an output beam(blue and red arrows, respectively), induced by a non-Hermitianmetamaterial slab.

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4. Transformation optics by photonic crystals: anonlocal route towards the optical frequency regime

Yun Lai

Nanjing University

Status. Transformation optics (TO) theory was originallyderived in a local medium framework [1, 24]. Sincecoordinate transformation often leads to unusual materialsparameters absent in nature, metamaterials [25] have beenapplied to realize these parameters in novel TO applications,such as cloaking [1, 24, 25] and illusion optics [26]. Most ofthe experimental demonstrations of TO were achieved bymetamaterials. However, at optical frequencies, metamaterialssuffer from a loss in their metallic components. Moreover,optical metamaterials demand nano-scale unit structures,which makes the fabrication of macroscopic samples quitechallenging.

On the other hand, photonic crystals [27] can provide anexcellent low-loss platform for optical devices. Besides thefamous band-gap properties, photonic crystals are also wellknown for exhibiting novel refractive behaviors such asnegative refraction. Interestingly, negative refraction was firstproposed and realized in the field of metamaterials, as a directresult of the double negative parameters. However, negativerefraction was also realized by photonic crystals later. Unlikemetamaterials that use unusual medium parameters to controlthe propagation of waves, photonic crystals utilize theirspatial dispersions, which can be described by the equalfrequency contours (EFCs) in the band structure, to controlthe wave propagation. By designing specific EFCs, therefractive behaviors of TO media and some novel TOapplications can also be simulated by photonic crystals. Forinstance, Urzhumov and Smith first proposed to utilizeelliptical EFCs to simulate the stretched TO media, and thusfurther realize cloaking phenomena [28]. Liang and Liproposed to use photonic crystals to replace the TO mediain bending waveguides [29]. These seminal studies, however,did not explore the possibility of realizing the omnidirectionalimpedance matching condition, which is another distinctsignature of the TO media obtained by coordinate transfor-mation. In fact, if omnidirectional impedance matching canalso be achieved by pure dielectric photonic crystals, therealization of ideal nonreflecting TO applications in theoptical frequency regime is possible.

Current and future challenges. The TO theory can providethe perfect solution of omnidirectional impedance matchingin the local medium framework. However, the solutionsusually require complex permittivity and permeability, whichare difficult to realize. Even in microwave frequencies, theperfect TO parameters are usually reduced significantly toease the fabrication and the property of omnidirectionalimpedance matching is normally sacrificed [25]. This leads toreflection in cloaking experiments, compromising the idealinvisibility.

Interestingly, it is seldom known that, besides the perfectTO solution, there are actually infinite other solutions foromnidirectional impedance matching. But in order to findthese unusual solutions, one has to go beyond the localmedium framework, where the original TO theory is rooted,into the much less explored realm of nonlocal photonicmedia. Such nonlocal media exist in neither natural materialslike dielectrics, nor ordinary metamaterials which function aslocal effective media due to sub-wavelength structures.

Photonic crystals, with microstructures much larger thanthat of metamaterials, can also possess effective permittivityand permeability responses. When the excited bands andeigenmodes are near the Brillouin zone center, photoniccrystals can behave as effective media with zero refractiveindex [30]. While for bands and eigenmodes far away fromthe Brillouin zone center, the effect of inherent nonlocalityappears, which can turn photonic crystals into nonlocalphotonic media with k-dependent parameters. To the surpriseof most people, such nonlocality in photonic crystals can beutilized to realize omnidirectional impedance matching [31].

Advances in science and technology to meet challenges. Infigures 5(a) and (b), we show that the refractive behaviors of aphotonic crystal slab with a ‘shifted’ elliptical EFC areexactly the same as that of a metamaterial slab with an

Figure 5. (a), (b) The refractive behaviors of a metamaterial slab andan ‘ultratransparent’ photonic crystal slab, respectively. (c), (d) Thetransmittance and nonlocal parameters of a designed photoniccrystal. Reprinted with permission from [31] Copyright (2016) bythe American Physical Society.

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elliptical EFC at the Brillouin zone center. It is astonishingthat photonic crystals with such ‘shifted’ EFCs could exhibitthe rare property of omnidirectional impedance matching inTO media. It makes the photonic crystal non-reflective andtotally transparent for incident light from any angle. This isclearly demonstrated in figure 5(c), where a transmittanceT> 99% for all angles within (−89°, 89°) is achieved in aslab of the photonic crystal structure shown in the inset graph,irrespective of the number of layers. In figure 5(d), thenonlocal (k-dependence) nature of the effective permittivityand permeability of this photonic crystal is clearly shown.The retrieved parameters (dots) excellently match with thederived theoretical requirement for omnidirectionalimpedance matching (dashed lines). In a certain sense, theomnidirectional impedance matching may also be understoodas the expansion of the Brewster’s angle from a single angleto all angles. Therefore, a type of ‘ultratransparent’ nonlocalphotonic medium beyond any dielectric is possible [31]. Suchan ultratransparency effect also exists in one-dimensionalphotonic crystals [32]. Moreover, the effect could also bebroadband and polarization-insensitive in some cases [32].

Since the ultratransparent photonic crystals exhibitcontrollable refractive behaviors and omnidirectional impe-dance matching, their functionality is very similar to the idealTO media. In figure 6, we demonstrate an example of a waveconcentrator. The original TO concentrator is discretized intofour layers (shown in figures 6(a) and (b)) and a core. Eachlayer can be replaced by a corresponding type of ultra-transparent photonic crystal with an EFC of the same shape,but ‘shifted’ to the X point in k-space, as shown in figure 6(c).In the simulation results shown in figures 6(d) and (e), onecan see that the concentrating effects of the original TO andphotonic crystal concentrators are almost the same, except fora phase difference of π in the core region. Such a phasedifference is induced by the shift of EFC in k-space. Asshown in figure 5(b), an additional phase of pd is added to thetransmitted waves where p denotes the shift of EFC in k-spaceand d is the thickness of the photonic crystal. In other words,this phase difference is a signature of the nonlocalityeffect [31].

Ultratransparent photonic crystals can provide a puredielectric platform for a large number of novel photonicdevices such as nonlocal TO applications, wide-anglepolarization filter, microwave transparent device, etc. Oneof the most important issues is the design strategy for thenonlocality in such photonic crystals. Besides the EFCs thatcontrol the direction of wave propagation, the effectivesurface impedance should be independently engineered, so asto ensure omnidirectional non-reflection. More optimizationand engineering methods need to be further explored in orderto find out the best strategy as well as the limit of the realizednonlocal photonic media.

Concluding remarks. The original TO theory was establishedin the field of metamaterials with local medium description.Now, pure dielectric photonic crystals with nonlocal effectiveparameters have shown an almost equivalent power incontrolling both the refractive behavior and the impedancematching behavior, for the first time. With the advance in thefabrication techniques of large-scale photonic crystal withmicrometer dielectric structures, this route may lead toexperimental realization of ideal nonreflecting TO applications,such as cloaking in the optical frequency regime.

Acknowledgments. This work has been supported by the StateKey Program for Basic Research of China (Nos.2014CB360505, 2012CB921501), National Natural ScienceFoundation of China (Nos. 11374224, 61671314) and a ProjectFunded by the Priority Academic Program Development ofJiangsu Higher Education Institutions (PAPD).

Figure 6. (a) The design process of a wave concentrator byultratransparent photonic crystals. (b) The parameters of the TOconcentrator. (c) The EFCs of the discretized four layers of the TO(dashed lines) and photonic crystal (solid lines) concentrators.(d), (e) The simulated concentrating effect for the TO and photoniccrystal concentrators, respectively. Reprinted with permission from[31] Copyright (2016) by the American Physical Society.

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5. Complex coordinates in optics; conformal mapsand analytic continuation

S A R Horsley

University of Exeter

Status. Transformation optics is a theory where changingthe coordinate system is used as a tool to solve the difficultproblem of wave propagation in inhomogeneous materials.While we often take these coordinates as real numbers,transformation optics has even more power when we allowthem to take complex values.

Undoubtedly, conformal transformations are the mostwidespread application of complex valued coordinates. Theyhave long been applied to solve Laplace’s equation in twodimensions, in areas such as electrostatics, fluid dynamics,and elasticity [33]. A conformal transformation takes thecomplex numbers z= x+ iy and z∗= x−iy (formed fromthe Cartesian coordinates x and y) to new complex numbers w(z) and w∗(z∗). Such a transformation leaves the form ofLaplace’s equation unchanged, and this form invarianceallows one to map simple solutions in simple geometries toequivalent solutions in more complicated ones. Transforma-tion optics has also made heavy use of conformal transforma-tions to design optical materials. For instance, Leonhardt [24]used a conformal transformation to design an invisibilitycloak; and more recently Pendry et al [8] developed a theoryof plasmonic energy concentrators based on conformaltransformations.

But the utility of complex coordinates is much broaderthan conformal mapping. While a conformal map replaces thetwo-dimensional coordinates x and y with the single complexnumber z and its complex coordinate z∗, one can also treat thex,y coordinates in the wave equation as complex numbersthemselves, in any number of dimensions. Such a ‘complex-ification’ effectively doubles the number of dimensions of thespace.

At the simplest level, this extension of the coordinatesystem allows one to find inhomogeneous materials thatchange the amplitude of a wave without generating anyreflection. As a rough illustration, consider a plane wave in1D, f= exp(inkx) (obeying the Helmholtz equationd2f/dx2+ (nk)2f= 0). Performing a complex rotation ofthe x coordinate x= (a+ i b)x′, we obtain either anexponentially growing or decaying wave, replacing theoriginal real index n with the complex index (a+ ib)n. Asemphasized by Chew [34], the theory of the perfectlymatched layer—nowadays the most common method fortruncating a computational domain—is founded on thisapproach.

Continued interest in transformation optics has led tonew work in this area, applying complex valued spatialcoordinates to design materials with inhomogeneous distribu-tions of loss and gain [35–36]. This has been teamed with therealisation that such optical materials can be used for theexperimental investigation of a family of complex valued(‘PT symmetric’) potentials that have long been a curiosity in

quantum mechanics [37]. For example, it has been found thatreflectionless materials with PT-symmetry can be designedusing a complex coordinate transformation of the Maxwellequations [17].

This ‘complexified transformation optics’ allows for aninteresting way of thinking. When we ordinarily solve thewave equation in real space, we do not consider the analyticcontinuation of the coordinates—the coordinates in the lab arereal numbers! But, were we to look, we would find particularfeatures (e.g. poles or branch cuts) that occur in the wave fieldat complex values of the coordinates (as shown in figure 7).Through introducing an inhomogeneous anisotropic material,a complex coordinate transformation can move these featuresinto real space [35, 17]. Transformation optics with complexcoordinates gives a physical meaning to the analyticcontinuation of a wave field!

Taking seriously the analytic continuation of the waveequation has recently provided some other new and rathergeneral results [35, 36]. These apply to planar media wherethe refractive index (or equivalently the permittivity) varies asa function of one of the coordinates x. If the refractive index n(x) is an analytic function in either the upper or lower halfcomplex x plane, then the material will not reflect wavesincident from one side, whatever the angle of incidence. Inaddition, if n(x) has only poles of order higher than 1 in thecomplex position plane, then the slab will also have the sametransmission coefficient as free space—it will be invisiblefrom one side. The construction of such reflectionless mediais in many cases also practically feasible, and one suchmaterial has recently been experimentally realised [38].

Current and future challenges. If we want to extend thereach of complex analysis in wave propagation problems,there are both mathematical and practical challenges. Here Ishall mention some which I know of, and some questions Ithink are interesting to ask.

I think the practical challenges are quite clear. Take theapplication of conformal transformations to cloaking, or tothe design of plasmonic structures (e.g. gratings). Becauseconformal transformations are a special kind of coordinatetransformation, then there is the question of what happenswhen we deviate from the condition of conformality. Noexperiment will be able to implement a refractive indexprofile accurate to an infinite number of decimal places. Thesame problem is also evident for the analytic profiles given in[36]—no experiment will be able to fabricate a profile that isequivalent to a function that is analytic in the upper halfcomplex position plane. There is always some tolerance tofabrication error. But in many cases, I think it remains achallenge to build the devices we are inventing withtransformation optics (whether using real or complex valuedcoordinates); both to develop a sufficient range of availablematerial parameters, and to make the fabrication error smallenough that the devices function as we intend them to.

Although there are many experiments in this area, I amjust trying to make the point that many of our designs are notyet practically feasible. This is especially true for some of the

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designs where both the real and imaginary parts of thematerial parameters are graded in space.

There are also some interesting mathematical questions toaddress. For instance, we know that making the coordinatescomplex can lead to general results concerning the reflectionand transmission of waves through inhomogeneous media, andthat features in the complex position plane can be given aphysical meaning via transformation optics. But, to date, wehave largely based our understanding on the theory of functionsof a single complex variable. Yet mathematicians havedeveloped the theory of several complex variables—seldomused in optics—with concepts such as ramification taking theplace of the branch cut. An interesting question is whether thetheory of functions of several complex variables can be appliedto derive useful information about the interaction of waves withstructured media. In a similar spirit, it is also interesting to askwhether the theory of complex manifolds (and special casessuch as the Kahler manifold) can be applied to complexcoordinate transformation optics.

Another interesting connection that may be productive toexplore is the connection between complex coordinates and theone-way edge states that are currently attracting interest in thefield of ‘topological photonics’ [39]. This connection does notseem to have been explored, but can be straightforwardlyobserved in the cases where the effective refractive indexvanishes and the field obeys the Laplace equation ∇2f= 0.Because the system only supports interface states that propagatein one direction, the field must become a function of one complexvariable e.g. only z, i.e. f(z)= exp(kz)= exp[k(x+ i y)].

Advances in science and technology to meet challenges.The above mathematical challenges pose questions that do notrequire any new mathematics, but require further time toanswer. In contrast, I think the experimental realisation ofmany of the materials we design does require advances intechnology.

Probably the most important is the ability to preciselyand separately control the reactive and dissipative response ofmaterials, so that refraction, loss and gain can all beseparately graded in space. This could involve the develop-ment of metamaterial elements with a wide range of complexeffective permittivity and permeability values. Gain is anespecially difficult property to control, and the presentimplementations often lead to instabilities.

Although there are cases where both real and imaginaryparts of the material parameters have been separately controlledin space (see e.g. [38]), at the moment this is generally a difficulttask. We should mention that [38] uses a rather ingenious trick,translating the analytic properties of a metamaterial in thefrequency domain to analyticity in the spatial domain, gradingthe resonant properties of metamaterial elements to obtain one ofthe non-reflecting index profiles derived in [36]. Such anapproach may be useful more generally.

Concluding remarks. Complex analysis is an old topic that isbeing given new applications in the field of metamaterials

research. It is used within transformation optics both throughconformal transformations in two dimensions, and throughthe complex transformations and analytic continuation used toinvestigate materials with distributions of loss and gain. In thefuture, there promise to be further applications of themathematics of complex coordinates in optics, which mayenable a deeper understanding of wave propagation incomplex inhomogeneous media.

Acknowledgments. SARH acknowledges funding from theRoyal Society and TATA.

Figure 7. (a) Schematic of a complex coordinate transformation ofthe wave equation, where the real line x= x1 is replaced with acurve C that ventures into the analytically continued wave field. Thisprocess changes both the material parameters and the wave field inthe wave equation; (b) an example of the numerical solution of the1D wave equation in complex coordinates, in this case showing aseries of branch cuts (dashed red lines) emerging from theimaginary axis.

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6. Extending transformation optics and towards athin cloak

Jensen Li, Jian Zhu University of Birmingham

Status. The interest of getting an invisibility cloak has avery long history but its main breakthrough came fromthe establishment of transformation optics (TO). It is thetheoretical framework of TO that gives us a recipe of therequired material profile of a cloak, while metamaterialsprovide us the necessary palette of material parameters. Thefirst metamaterial cloak was realized in the microwave regimewithin half a year from the theory proposal [25]. Figure 8(a)shows the dispersion surface of a TO medium when a lightbends from the freespace to the surface layer of a cloak whilefigure 9(a) shows the simulation of a cylindrical cloak for aplane wave coming from the left. The dispersion surface forboth the TE and TM polarizations become generally ellipticalwith a change in shape to redirect light. The deformation getsbigger near the inner surface of cloak with more extremematerial parameters. It was then quickly recognized thatmetamaterials cannot provide these extreme parameters for acloak to work at optical frequencies due to the limit ofresonance strength. One way out is to make a compromise onthe functionality. If we conceal an object under a carpetinstead of in the middle of air, the transformation involvedcompresses the object to a flat sheet instead of a point. Thisstrategy makes the range of the required material parametersmuch less extreme. Cloaks for visible light, figure 9(b),become realizable (e.g. [40]). Since then, we have beenwitnessing a dynamic period of development of TO in one oftwo directions: one on further simplifying cloaking strategiesto have the ultimate application of cloaks in the real world;another on extending TO as a framework to design opticalcomponents with functionalities beyond the conventionalones. The simplification route pushes the cloak towardshigher frequencies, towards 3D applications, and towardscloaking objects of macroscopic sizes. The generalizationroute applies TO on getting devices beyond a cloak, e.g. toproject an additional virtual object as an optical illusion, todesign silicon photonic components with optimizedperformance, and to apply a transformation approach toother kinds of waves.

A consideration on how to extend the fundamentalprinciple of TO can actually benefit both directions.Coordinate transformation is at the heart of TO to get form-invariance of Maxwell’s equations. A real 3-tuple x y z, ,( ) ismapped to another 3-tuple. One possibility is to generalize thecoordinates to complex numbers. It extends TO to includematerial gain and loss so that active applications like fictitioussource generation and one-way invisibility can be achieved[17]. On the other hand, we can add a fourth coordinate: time.A dynamic event cloak becomes realizable [42]. It turns outthat coordinate transformation is not the only possibility to getform-invariance. The transformation can also happen in thepolarization, an internal degree of freedom, as a gaugetransformation [43]. An example is an abstract rotation in the

Figure 8. Ways to redirect light beam (a) by changing the shape ofdispersion surface from TO-media, (b) by splitting dispersion surfacefrom gauge transform-media. Vertical grey line indicates aninterface. (c) Luneburg lens for getting variable directions from pointsources.

Figure 9. (a), (b) Cylindrical and carpet cloak at visible light fromcoordinate transform. From [40]. Reprinted with permission fromAAAS. (c), (d) Gauge transform to get a cloak and one-way edgestate. (e), (f) 3D carpet cloaks by metasurface [41] and by lens array.From [41]. Reprinted with permission from AAAS.

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J. Opt. 20 (2018) 063001 Roadmap

polarization space for 2D wave propagation:

EiH

x y x yx y x y

EiH

’’

cos , sin ,sin , cos ,

.zz

z

z

f ff f

=-⎛

⎝⎜⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

( ) ( )( ) ( )

The transformation induces a corresponding one on thematerial parameters and forms its own class of materials.Figure 8(b) shows the class generated from such an abstractrotation. The transformed permittivity has additional aniso-tropy xz and .yz Equivalently, the dispersion is split into twopolarizations by shifting the circular one into two oppositedirections in the k-space (versus deforming shape infigure 8(a)). This splitting can be interpreted as a gauge field(pseudo vector potential) for photons. It provides a ‘magnetic’force to redirect light and can be utilized to design a cloak[44]. Figure 9(c) shows the simulations for a cylindrical cloakdesigned with gauge field approach. The rays passing throughthe cloak are squeezed upwards (versus figure 9(a)). Thisasymmetry occurs in opposite ways for the two polarizations.Such an asymmetry with the ‘magnetic’ bending force givesus a route to design TO devices with asymmetric transmissionfunctionality. Figure 9(d) shows an example to generate aone-way edge state, which propagates between two regions ofmetamaterials and is not reflected even when a sharp corner isencountered.

Current and future challenges. Although generalizationsallow us to extend TO applications to new territories, amajor current challenge on designing a cloak using the TOapproach stays behind: to obtain a thin cloak, working forwide angles in 3D, which are essential for a wearable cloak.Taking the cylindrical cloak as example (figure 9(a) with ray-tracing), intuitively the outermost cylindrical layer isresponsible for the light hitting the cloak at a shallow angle,while an inner one, together with the layers outside, areresponsible for the light nearer to normal incidence. Theangular operation thus comes from a finite thickness of thecloak. Unless very extreme material parameters are allowed, asmall thickness and wide-angle operation are difficult torealize at the same time. It also means that we can go furtheron the simplification route to get a thin cloak but with alimited viewing angle.

We have already seen some successful demonstrationsalong this direction. By going to the extreme case of only onesingle layer of metamaterial atoms, as a metasurface, a carpetcloak at the normal incidence can be designed by phasecompensation to get a target reflected plane wave. Surpris-ingly, even though it is designed at normal incidence, theworking angle can be as large as 30 degrees in a demonstrated3D metasurface cloak [41] (figure 9(e)). For the carpet cloak,it is also desirable to go beyond a flat mirror to mimic anarbitrary background image on the floor.

Advances in science and technology to meet challenges. Toenhance the working angle for a thin cloak, bearing in mind

the issues we have discussed earlier, one natural step is tomake a balance between the working angles and thickness ofcloak. Naturally, we can go from a single layer ofmetamaterial atoms to multiple layers or a thicker layer.Recently, the concept of Huygen’s metasurfaces with twolayers of metamaterial atoms have found excellent usage inenhancing transmission efficiency, utilizing both electric andmagnetic response of the metamaterial atoms, with ahomogenized boundary description of the metasurface [45].Such a homogenized boundary description may be utilized toformulate TO instead of using effective medium parametersof the bulk. By designing successive layers of themetasurfaces, the cloaking behavior at a series of viewingangles can be obtained from normal to oblique incidence.

Another approach to get angular operation for a thincloak takes advantage of the concept of integral imaging.Figure 8(c) shows the basic principle in redirecting light by alens array. Unlike TO, in which dispersion surfaces aredefined continuously at any locations inside the device, thedevice is pixelized into an array of lenses and each lenscontrols the direction of the generated light from a pointsource at different foci. Such an array can be used to projectthe desired 3D image you would like the observer to perceivein the case of a cloak. A preliminary demonstration by puttinga 1D array of cylindrical lens on top of a digital display hasachieved a cloak with viewing angle up to 13.4 , varying in asingle direction [46]. Although the viewing angle is limiteddue to paraxial geometrical optics, it gives a practicalapproach for a thin cloak with all three colors, with acompromise on spatial resolution of the image. To get anomnidirectional viewing angle, one can replace the 1Dparaxial lens array by a 2D array of Luneburg lens (asfigure 8(c) depicts). Figure 9(f) shows schematically if weassemble such Luneburg lens arrays into a pyramid, whichprojects the image on the floor consisting of circular-diskpatterns to any angles. Furthermore, to focus light comingfrom different directions to the Luneburg lens array on acommon plane, TO will be very favorable to realize theflattened Luneburg lens [47] for the lens array.

Concluding remarks. Based on the early works of TO togetherwith metamaterials, we now have the basic language indesigning a cloak. We believe there are still hugeopportunities for us to develop TO. A fundamental approachin extending TO’s theoretical framework should be beneficial toboth getting a realistic cloak and designing optical componentswith properties beyond those given by conventionalapproaches. Probably, the wearable cloak for visible light willbe an extension or combination of the TO technique withapproaches such as metasurfaces and integral imaging.

Acknowledgments. We acknowledge support from EuropeanUnion’s Seventh Framework Programme under GrantAgreement No. 630979.

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7. Combining curvature and index gradients forsurface and guided waves

Rhiannon C Mitchell-Thomas1 and Oscar Quevedo-Teruel2

1University of Exeter 2KTH Royal Institute of Technology

Status. Transformation optics has been proven to be a veryversatile and powerful tool. It allows a link between geometryand the properties of a material to be formed, which is exact forwaves propagating through a medium. The method employs acoordinate transformation to predefine the desired wavepropagation characteristics, and this transformation is thenused to calculate the appropriate permittivity and permeabilitywith which to fill space so that this behaviour is achieved. Forwaves confined to a surface, geometrical optics provides asimilar link between geometry and materials [48]. In contrast tothe complex material properties required by transformationoptics, the use of geometrical optics results in purely isotropicindex profiles. These profiles are inhomogeneous, with arotationally symmetric index gradient, meaning that they varyonly along the radial direction of a polar coordinate system, butnot the angular. It also allows for surfaces with non-zerocurvature to be considered, which is not possible withtransformation optics, and this provides new opportunities fordevice design.

Familiar lenses, such as the Luneburg and the Maxwellfish eye lens can be derived using geometrical optics. Theseisotropic, rotationally symmetric index profiles are accurate inthe limit of geometrical optics, and hence the lenses must beelectrically large to operate effectively. Using geometricaloptics, equivalent surfaces for these types of lenses can becalculated [49]. For example, the Maxwell fish eye lens,which has a radially dependent refractive index profile thatvaries from one at the perimeter to two at the centre has anequivalent surface which is a homogeneous hemisphere. Thislens has ray trajectories such that a point source positioned onthe circumference will be imaged on the opposite side of thelens, due to the graded index. Equivalently, rays emitted froma point that are confined to the surface of a hemisphere wouldconverge to a point on the opposite side. In this case, it is thecurvature that accounts for the focusing, as the material of thissurface is homogeneous. The equivalent surfaces for a wholefamily of these rotationally symmetric lenses have beenderived, including those with singularities at their centres, andtwo examples of the latter are shown in figure 10. However,there is a limitation. This is when the index profile increasesat a rate equal to or more rapidly than r1 .2/ When thiscondition is reached, all rays that are incident on the profileare unable to escape and the equivalent surface is no longerclosed.

These equivalent surfaces have found only limitedpractical application [50, 51], and one of the reasons is thatthe shape of the curved surfaces may have undesirablefeatures. For example, the surfaces shown in figure 10 are tallwith respect to the radius, and also have a 90 degree transitionwhen positioned on a flat surface. To overcome this issue, acombination of curvature and index gradients can be used to

achieve the same behaviour, but with more freedom in thesurface shape and index contrast can be limited to achievablevalues [9, 52, 53]. It is in this context that geometrical opticsoffers the most freedom in terms of practical device design,and full advantage can be taken of this versatile technique.

Current and future challenges. This combination ofcurvature and index gradients finds application in a numberof areas, and three different devices will be discussed here.The first of which is a family of lenses that have veryappealing properties, yet are impossible, or very difficult tofabricate in practice. Two examples are shown in figure 10,including the Eaton lens, which retro-reflects all rays that areincident on it, and a monopole lens, which can be used so thatan omnidirectional source only emits energy in one half-spaceof a 2D plane. However, the common feature of these lensesis that they exhibit a singularity in the centre of their refractiveindex profiles. It has been shown that curvature can beemployed to emulate the influence of the singularity using apointed tip at the centre, as shown in figure 10, where thegradient of the tip is defined by the speed with which theprofile approaches infinity [9]. In order to design a practicaldevice, a graded index with a modest contrast of 1.6 iscombined with a curved parallel plate waveguide so that thefield is confined between the plates. The top image infigure 11 shows the electric field in this waveguide, and it canbe seen to accurately retro-reflect the incident input beam sothat the output beam exits the device in the same direction [9].

A further use of geometrical optics is to redesign lensesso that they are conformal to existing surfaces [52]. Theincreasing requirements for ever higher data transfer rates inwireless systems demands improved antennas with higherdirectivity, yet with decreased weight and volume that are lessintrusive in the surrounding environment. One solution is toemploy surface wave lenses, that are thin and lightweight, butwith a large surface area. They can be applied to existingsurfaces of vehicles, for example, but these surfaces may not

Figure 10. Ray tracing in graded index profiles and the equivalenthomogeneous surfaces for the Eaton lens and a monopole lens.

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be flat. In this case, it is known that the curvature of thesurface will degrade the performance of a surface wave lens.However, it is possible to calculate a modified version of agraded index lens that is appropriate for a given surface withcurvature. In the middle image of figure 11, a Luneburg lenson a curved surface is shown. It can be seen that the pointsource excitation is transformed to a plane wave output beam,which could then be leaked into a beam propagating in freespace, creating the basis for a design of a conformalantenna [52].

Finally, an alternative version of a surface wave cloakcan be created, without the use of the transformation opticstechnique [53]. In this case, a curved surface is employed toguide a wave up and over the object to be cloaked. Thiscurvature will cause an undesired distortion to the incidentwave and leave a signature. The route to eliminating anydistortion is to counter the influence of the curvature with useof a graded index profile, so that this combination now has an

equivalent surface that is flat and homogeneous. A surfacecloak is shown in the bottom image in figure 11, where theinput plane wave can be seen to be undistorted afterpropagating through a section of curved waveguide, due tothe rotationally symmetric index which fills the curved region[53]. This type of cloak can be used to avoid scattering thatwould be caused by an obstruction on a surface.

Advances in science and technology to meet challenges. Allof the above devices require an isotropic index gradient to beachieved, for which there are multiple approaches. Dielectricmaterials, engineered so that the relative permittivity can beprecisely controlled are achievable with the combination of alow dielectric host medium loaded with high dielectricparticles. In order to achieve a graded index, usually adiscretization is applied and a number of layers areindividually cast [54]. For waves bound to an interfacebetween a dielectric and free-space, it is also possible toachieve a change in speed of the supported mode simply byvarying the thickness of a homogenous dielectric [55].Alternatively, metasurfaces can be employed, where theindex gradient is achieved by slowly varying the geometricalparameters of the individual meta-elements. This type ofimplementation would limit the operational bandwidth of adevice due to the resonant nature of commonly usedmetasurfaces, but new advances in metasurface design haveshown that these limitations can be considerably relaxed ifhigher symmetries are employed [56].

Rather than rely upon a rapid advance of a certaintechnology to meet requirements, this method of devicedesign is chosen for its suitability with the current state of theart of manufacturing techniques. The fact that no complexanisotropy or simultaneous magnetic and electric propertiesare required is one of the major advantages of this technique.

Concluding remarks. Here, a description of the applicationof geometrical optics to design a new range of devices hasbeen given. This method allows curvature to be employedwhen creating surface or guided wave devices, which onlyrequire very basic material properties. The addition of thisextra degree of freedom, surface curvature, permits thedefinition of purely isotropic materials and can negate theneed for singularities in refractive index profiles. Threeexamples have been shown, including an Eaton lens with afinite and modest refractive index contrast, a conformalLuneburg lens, and a surface cloak, demonstrating theversatility of the technique. All of these devices can befabricated by employing either metasurface or gradeddielectric approaches.

Acknowledgments. The first author was funded by theEngineering and Physical Sciences Research Council(EPSRC), UK under Grand Challenges Grant (EP/N010493/1) ‘SYnthesizing 3D METAmaterials for RF,microwave and THz applications (SYMETA).’

Figure 11. Examples of devices designed using geometrical opticsthat employ curvature. (Top) An Eaton lens with a modest indexcontrast of approx. 1.6. (Middle) A Luneburg lens conformal to acurved surface. (Bottom) A surface cloak where the influence of thecurvature has been removed with an appropriate index profile.

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8. Transformation optics at surfaces

Philippe Tassin and Vincent Ginis

Chalmers University of Technology and Vrije UniversiteitBrussel

Status. About a decade ago, transformation optics establisheda new way to take advantage of the unprecedented properties ofmetamaterials for the design of optical devices. The formalismrelies on the form-invariance of Maxwell’s equations for thepropagation of light in an inhomogeneous medium undercoordinate transformations. This invariance leads to anequivalence relation between the components of a metrictensor and the permittivity and permeability of a medium. Thedesign technique for optical devices then goes as follows: startfrom a Cartesian grid and bend the coordinate lines until theyfollow the desired path. In this way, one defines a metric tensor,which can subsequently be used to calculate the spatiallyvarying permittivity and permeability functions needed for anoptical device in which light rays follow paths along thetransformed coordinate lines.

Transformation optics as a design technique for opticaldevices has three major advantages compared to traditionaldesign techniques. First, it allows dividing the optical designinto two steps. One starts from the desired electromagnetic fielddistributions to determine the material properties of the device.These material properties need then be implemented withnanostructured metamaterials—a challenging problem in itself.The second advantage is that transformation optics is, just asGaussian optics for traditional optical instruments, a completelygeometrical procedure. Finally, transformation optics is notdiffraction-limited. Since it is based on Maxwell’s equation, it isa full-wave technique, and optical devices designed by it do notsuffer from geometrical aberrations.

Current and future challenges. In its original formulationfrom 2006, transformation optics is inherently a three-dimensional method. It relies on continuous transformationsbetween coordinate systems in a three-dimensional space,generating permittivity and permeability functions that varysmoothly as a function of the space coordinates. This methodcreates optical devices with permittivity and permeabilityvarying in all three dimensions.

However, this is not very desirable from a technologicalpoint of view. Metamaterials research has increasinglyfocused on the fabrication of so-called metasurfaces, two-dimensional arrays of meta-atoms, which are easier tofabricate and still allow for versatile transmission andreflection properties. Metasurfaces can now be designed tohave almost any optical property in a flat or curved plane.This makes transformation-optical techniques applicable tometasurfaces very interesting.

When considering transformation optics at surfaces, wemust distinguish between devices where a light beam isscattered by a transformation-optical surface (figure 12) and

devices where a light wave propagates along a transforma-tion-optical surface (figure 13).

In one of the first steps towards transformation optics onsurfaces, a thin metamaterial coating was designed to enhancethe optical force between two traditional optical waveguides[57]. If the metamaterial implements an annihilating trans-formation, then it is decreasing the effective interwaveguidedistance experienced by the light waves in the underlyingelectromagnetic space, thus enhancing the optical forcesbetween the waveguides. Annihilating transformations can beobtained from metasurfaces with a negative index ofrefraction.

Nevertheless, one can make use of discontinuous transfor-mations to describe electromagnetic phenomena that occur at aninterface between a common material (or air) and ametamaterial surface, e.g. the Goos-Hänchen shift [58]. If themetamaterial is the transformation-optical equivalent of acoordinate transformation, then discontinuous transformationsallow us to understand and engineer phenomena at the surfaceof nanophotonic structures in terms of a geometrical framework.

A second reason for wanting to apply transformation opticsto thin two-dimensional optical systems is that optical devicesare increasingly being integrated on photonic integrated circuits.One of the main advantages of integration is of course a smaller

Figure 12. Illustration of a light beam reflected and transmitted at thesurface of a transformation-optical material. With the use of acoordinate transformation, we want to engineer the properties of thescattered beams—for example, direction of propagation, beamshape, and frequency.

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J. Opt. 20 (2018) 063001 Roadmap

footprint. However, with routing of optical signals in suchcircuits being achieved with strip or rib waveguides, bends andprocessing elements become less efficient with furtherminiaturization. A full-wave design technique such as transfor-mation optics could thus advance the miniaturization ofintegrated optical circuits.

A number of research groups have recently madeprogress in designing two-dimensional transformation-opticaldevices for in-plane guided waves. By placing a metamaterialon top of a metal film, it is possible to steer surface plasmonwaves along a bend, focus them, or make them pass over anobstacle [59, 60]. These approaches add a metamaterial on topof a nanophotonic waveguide. By locally modifying theFermi level of graphene, it was also proposed to create a two-dimensional transformational-optical surface that can split orfocus surface waves on graphene. Here, the guided-wave-supporting surface is modified by a transformation [61]. In adielectric waveguide, it is also possible to locally adapt thethickness to achieve a desired spatial distribution of modeindices implementing a transformation optics design [62] atthe expense of backscattering.

Recently, an alternative transformation optics theory wasdeveloped for guided waves on integrated optical waveguides.

Indeed, one problem with transforming integrated opticalwaveguides is that traditional transformations optics imposesmetamaterials in both the core and the cladding of thewaveguide. This results in bulky structures that are difficult tofabricate. However, if one also varies the thickness of thewaveguide, it is possible to transform guided waves inwaveguides where the core is made of a metamaterial, but thecladding is left unaltered [63, 64]. The thickness variation isengineered such that the dispersion relation is unaltered beforeand after the transformation. With this method, beam splittersand beam benders have been designed.

Advances in science and technology to meet challenges.The further development of surface transformation optics atsurfaces will require innovations in theory as well as innanofabrication. For the modification of light beams incidenton a surface, as visualized in figure 12, transformation opticshas been shown to be applicable to homogeneous surfaces,but there is no geometrical theory for inhomogeneoussurfaces. Inhomogeneous metasurfaces were demonstratedto be able to achieve advanced beam manipulation, e.g.generalized refraction [65], but it is currently not completelyunderstood how to describe and engineer this response usingcoordinate transformations.

For the design of guided wave devices, as visualized infigure 13, the theory of transformation optics is now wellunderstood [62–64], but advances in nanofabrication are clearlyrequired. Surface transformation-optical designs require ananisotropic dielectric material layer with varying height. Thismay be achieved with direct laser writing, similarly to how anoptical ground plane cloak was fabricated by creating a matrixof spherical particles of varying size [66]. By replacing thespherical particles with ellipsoidal particles, it is possible to getanisotropic optical properties. At the same time, one caninvestigate how the complex parameters can be further reducedto meet the current state-of-the-art in fabrication.

Concluding remarks. In the past decade, transformationoptics has revolutionized our understanding of andcapabilities to design the interaction between light andthree-dimensional nanostructured devices. However, formany technological applications it is even more importantto be able to understand and design the interaction betweenlight and two-dimensional media. This would allow for themanipulation of free-space travelling beams as well as for in-plane guided waves. In our opinion, further advances in thisfield, both theoretical and fabricational, are needed toaccelerate the widespread use of transformation optics as ageometrical tool to design next-generation optical devices.

Acknowledgments. The authors acknowledge funding fromthe Area of Advance Nanoscience and Nanotechnology, theResearch Foundation of Flanders (Fellowship No. 12O9115N),and COST (Action MP1403 on Nanoscale Quantum Optics).

Figure 13. Illustration of transformation optics in lightwavetechnology. With the use of transformation optics implemented withinhomogeneous metasurfaces or waveguides, photonic functionssuch as steering, wave reshaping, splitting and recombining can beachieved.

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J. Opt. 20 (2018) 063001 Roadmap

9. Metasurface radiation and guidance atmicrowaves

Enrica Martini, Gabriele Minatti, Stefano Maci

University of Siena

Status. Controlling the electromagnetic (EM) field hasbecome a popular research topic in recent years thanks tothe advances done in the framework of metamaterials(MTMs). Metasurfaces (MTSs), i.e. the two-dimensionalequivalent of MTMs, have gained a large interest due to theirengineering applicability to a vast range of frequencies. MTSsconsist of arrangements of electrically small metallic/dielectric inclusions in a regular planar lattice inside or ontop of a thin dielectric layer, which can be manufactured withstandard PCB technologies or additive manufacturing. WhileMTMs characterization is given in terms of equivalentvolumetric constitutive parameters, MTSs characterization isgiven in terms of homogenized boundary conditions (BCs)defined in terms of a space-variable surface impedance.Through a proper spatial modulation of these BCs, one cancontrol surface/plasmonic wave propagation and radiation ina simpler way than volumetric waves in MTMs. The BCs aremodified by spatially changing the characteristics of the smallinclusions in a gradual manner, while maintaining uniformthe periodic lattice.

Depending on the phenomenon we would like to control,we can identify three classes of MTSs:

(i) phase-gradient MTSs for space-wave control;(ii) phase-gradient MTSs for surface-wave (SW) control;(iii) sinusoidally-modulated MTSs for radiation control.

Phase gradient MTSs for space wave control [67] andtheir generalization called ‘Huygens MTSs’ [68, 69] aredesigned with the objective to transform the wavefront andpolarization of space-wave fields. A general wavefront can beobtained by locally controlling the gradient of the phase of thelocal transmission coefficient. Thin lenses, beam deflectorsand beam splitters fall in this class. From an engineering pointof view, these MTSs can be seen as a generalization oftransmitarrays to continuous boundaries, while, from aphysical point of view, they implement the equivalencetheorem.

Phase gradient MTSs for SW control have the objectiveof controlling surface/plasmonic wave propagation path andwavefront. These MTSs mold the wavefront during thepropagation, similar to what happens in transformation optics3D media, but maintaining the wave bounded. The phenom-enon is sometimes called ‘metasurfing’ [70]. Planar lenses,cloaking devices, and general beam forming networkswithout transmission lines belong to this category. Theirbehavior can be described through an approach analogous togeometrical optics in equivalent graded index materials orthrough a planar version of transformation optics [71].

Sinusoidally-modulated MTSs have the objective tolocally change the dispersion properties of a bounded SWso as to transform it into a leaky wave. These MTSs can be

directly used as antennas by exciting them with a coplanarfeeder consisting of an elementary radiator. For centered-point fed MTSs, the impedance BCs are modulated withquasi-periodic functions in the radial direction, thus trans-forming the cylindrical SW excited by the feeder into adesired radiating aperture-field distribution. The feasibility ofthe concept was first demonstrated in [72, 73] and morerecently systematized in [74]. The appeal of these MTSantennas is due to the capability to obtain a complete controlof the aperture field distribution in terms of amplitude, phaseand polarization through an almost analytic design procedure[74]. The possibility of also achieving amplitude controlrenders these antennas much more flexible than reflectarrays;furthermore, a low profile is achieved since the feed is in-plane and not external to the surface. Finally, low losses areobtained thanks to the absence of a beam forming networkand to the non-resonant nature of the MTS inclusions.

Current and future challenges. MTSs have been proven to bean exceptionally effective tool for controlling EM waves.However, several challenges are still open and need to beaddressed for broadening application fields or improvingperformances.

Concerning type (i), a full tailoring of any space field,including reactive field, is still an issue, due to thedependence of the MTS insertion phase on the incidencedirection and due to the limited bandwidth. A relevantchallenge is represented by MTSs that can be used to emulatedifferential or convolution operators, an investigation pio-neered in [16]. A further issue is the one of non-reciprocalMTSs (both gyrotropic and non-gyrotropic). Achieving non-reciprocity requires breaking the time reversal symmetryusing an external force, like ferromagnetic materials; how-ever, this can be also obtained by using active unidirectionalelements in an MTS layer configuration. This opens newinteresting applications like one-way screens, isolatingradomes, radar absorbers and thin cloaks [75]. Several groupshave also progressed in parity-type symmetric non-localMTSs, with the objective to construct super focusinglenses [76].

Concerning phase gradient MTSs for surface andplasmonic waves control (type (ii)), these are still poorlyexplored in the microwave regime. The extension to non-regular grids and to time domain could be among the mostimportant challenges. We can conceive structures able tochange the shape of an impulse during the propagation overthe surface. Hyperbolic planar surfaces may achieve focusingproperties similar to those of hyperbolic media, but in-