DOI: 10.1126/science.1220600, 549 (2012);337 Science et al.J. B. Pendry

Transformation Optics and Subwavelength Control of Light

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Transformation Optics andSubwavelength Control of LightJ. B. Pendry,1* A. Aubry,2 D. R. Smith,3 S. A. Maier1

Our intuitive understanding of light has its foundation in the ray approximation and isintimately connected with our vision. As far as our eyes are concerned, light behaves like astream of particles. We look inside the wavelength and study the properties of plasmonic structureswith dimensions of just a few nanometers, where at a tenth or even a hundredth of the wavelengthof visible light the ray picture fails. We review the concept of transformation optics thatmanipulates electric and magnetic field lines, rather than rays; can provide an equally intuitiveunderstanding of subwavelength phenomena; and at the same time can be an exact descriptionat the level of Maxwell’s equations.

The field of optics is not just confined toimaging but extends to communication,sensing, cancer treatment, and even weld-

ing of automobile parts. Manipulation of light isvital to exploitation of its potential, but light isdifficult to control on length scales less than thewavelength, where diffraction causes convention-al optical components to fail.This is the length scale of atomsand molecules, living cells, elec-trons, computer microchips, andmicromechanical devices. Sub-wavelength control demands newoptical materials, and effortshave turned to metals such asgold and silver, where the plas-mon collective excitations of theconduction electrons couple tolight and can compress the cap-tured energy into just a few cu-bic nanometers. We review thelatest developments in transfor-mation optics applied to plas-monic systems whereby Snell’slaw, the traditional design toolof optics, is being replaced bytransformation optics, a new toolthat is fully compatible with thewave nature of light as describedby Maxwell’s equations.

Snell’s law tells how light is refracted bytransparent media. It gives a picture of light’sprogress in terms of rays, which can be thoughtof as streams of photons. This simple, intuitivepicture is a vital component of the design process

and explains why the law is still widely used de-spite its neglect of reflection at interfaces anddiffraction by small features. OnlyMaxwell’s equa-tions give a precise description of all the classicalfeatures of light, but they lack the visual intuitionprovided by the ray picture. Transformation opticsaims to give a picture equally intuitive to that of

ray optics by using instead the raw elements ofMaxwell’s equations—the electric and magneticfield lines introduced by Faraday—and gives rulesfor how these lines can bemanipulated almost atwill by a suitable choice of material.

To understand how transformation opticsworks, imagine a uniform electric displacementfield in free space. The location of the field linesis recorded on a system of coordinates (Fig. 1). Ifwe postulate that the field lines are fixed to thecoordinates so that the coordinate system carriesthe field lines with it under a distortion, then wecan shape the trajectories of the field simplyby distorting the coordinates. It has long beenknown that writingMaxwell’s equations in a newcoordinate system does not change the form but

changes only the values of permittivity and per-meability as follows

e′i′j′ ¼ ½detðLÞ�−1Li′i L

j′j e

ij

m′i′j′ ¼ ½detðLÞ�−1Li′i L

j′j m

ij

where e and m are the permittivity and the per-meability, respectively, in the original coordinateframe and e′ and m′ are the corresponding quan-tities in the transformed frame (1, 2).L is given bythe first derivatives of the coordinate transformation

Lj′j ¼

∂x j′

∂x j

The transformed values of e′ and m′ ensure thatMaxwell’s equations are obeyed by the new con-figuration of the field lines.

Transformation optics comes about fromthe realization that the field lines are glued to thecoordinate system; this insight gives rise to theintuitive picture that we seek as a replacement forSnell’s law. Any conserved electromagnetic fieldthat can be represented by a field line has theproperty of adhesion to the coordinate frame.Thus, the Poynting vector also transforms in thisfashion. The field lines of the Poynting vector

can, in fact, be thought of as amore precise definition of raysof light.

Early applications of transfor-mation optics involved adaptingcomputer codes fromCartesian tocylindrical geometries (3). How-ever, with the advent of meta-materials, the realization emergedthat transformation optics couldbe more than simply a computa-tional tool and was subsequentlyapplied as a means of determin-ing thematerial properties neededto reshape the perfect lens (4),allowing it tomagnify objects (5).Detailed reviews can be foundin (6, 7).

We note that transformationoptics can be applied to otherequations of physics, such asthe Helmholtz equation, an ap-

proximation to Maxwell’s equations (8) and toacoustics (9).

Transformation optics can render trivial thedesign of optical structures that would be other-wise difficult or seemingly impossible (10, 11). Forexample, transformation optics was used to designa cloak of invisibility, whose material propertiesflowed naturally from a relatively simple trans-formation (8, 12). The technique is ideally suitedto the task of steering field lines away from a hiddenregionwhile leaving them undisturbed in the vicin-ity of the observer. We exploit the general appli-cability of the technique to treat subwavelengthfields occurring in plasmonic nanosystems, givingus a precise design tool. Although the transforma-tion of near-fields is an extremely versatile tool

REVIEW

1The Blackett Laboratory, Department of Physics, Imperial Col-lege London, London SW7 2AZ, UK. 2Institut Langevin, ÉcoleSupérieure de Physique et de Chimie Industrielles de la Ville deParis, ParisTech, CNRS UMR 7587, 1 rue Jussieu, 75005 Paris,France. 3Center for Metamaterials and Integrated Plasmonicsand Department of Electrical and Computer Engineering, DukeUniversity, Box 90291, Durham, NC 27708, USA.

*To whom correspondence should be addressed. E-mail:j.pendry@imperial.ac.uk

A B

yv

ux

Fig. 1. (A) Field lines of a uniform electric displacement field. (B) Distortion of spaceas if it were a rubber sheet bends the field line, and the coordinate system records thedistortion.

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applicable for the study of numerous phenomena,we focus on the light-harvesting properties ofnanoparticles to illustrate the techniques.

Harvesting LightIf a photon is to have a strong interaction witha molecule, its energy must somehow be con-centrated into the same dimensions as those ofthe molecule. This harvesting process is wellknown in nature, for example, in the chloro-phyll complex through which light is collectedand delivered to the reaction center to create sug-ars from carbon dioxide. We show how to mimicnature by exploiting surface plasmons. Ideally,the structure should harvest a broad spectrumof radiation.

Raman spectroscopy of rough silver sur-faces first revealed the potential for enhancedphoton-molecule interaction (13). Raman sig-nals can be spectacularly enhanced, sometimesby a factor on the order of a million, when theinteraction takes place at a rough metal surface.It is now believed that the enhancement is dueto the concentration of the electric field at sin-gularities in the geometry of the surface. Manycomputer simulations of Maxwell’s equationshave confirmed this picture (14–21). Early workshowed the importance of geometrical singular-ities in producing enhancement (22–24). Physicalinsight has been provided by the hybridization

model (25). A review of light concentration isgiven in (26).

We start from a simple configuration that hasthe required broadband harvesting properties,at first paying little attention to the geometry be-cause we know that the geometry can be changedby a transformation, leaving the spectrum unal-tered (27–30). Figure 2A shows a waveguidecomprising a planar cavity between two silversurfaces. The system is infinite in directions par-allel to the surfaces, and therefore the spectrumis continuous and is formed by the hybridizationof surface plasmons on opposing surfaces. Thelower band extends from zero up to the surfaceplasmon frequency, and the upper band, fromthe surface plasmon frequency to the bulk plas-mon frequency,wp.We shall mainly be concernedwith the lower band, which is usually muchbroader than the upper band and therefore moreflexible in its applications.

A dipole source, such as an excited atom,placed in the waveguide now excites surfaceplasmon modes, which transport energy away toinfinity. Although this is harvesting of a sort, it isnot particularly useful. To remedy this, we makea radical transformation of the geometry.

First, we assume that the electric fields dom-inate so that we neglect retardation and that thefields are oriented in the xy plane and are invar-iant along the out-of-plane dimension. Because

the field pattern is effectively two-dimensional (2D),we are able to apply the theory of optical confor-mal transformations to the geometry (5, 8). Theconformal transformation ensures that all mate-rial properties are maintained as isotropic. Con-sider the conformal transformation defined by

z′ ¼ 1=z

where z ¼ xþ iy and the origin is taken to be atthe center of the dipole. This transformation takesplanes into cylinders, points at infinity to the neworigin, and points at the origin to the new infinity.A dipole can be represented as two very largeopposite charges very close together. Under thistransformation, the new dipole consists of twovery large charges at infinity, which create auniform electric field at the origin. In other words,in the new coordinate system (Fig. 2), the twosemi-infinite planes become two kissing cylin-ders, and excitations are created by a uniformelectric field such as might be imposed by anincident plane wave. Our assumption that theelectric field dominates is valid provided that thecylinder radii are much less than the wavelengthof light. Another transformation optics–based ap-proach is found in (31).

Conformal transformations in 2D have theproperty that in-plane components of the permit-tivity tensor are unchanged in the new coordinatesystem. This is not true of conformal transfor-mations in 3D, although transformation opticshas nevertheless been successfully applied to 3Dsystems (32).

The original system is excited by a dipole andharvests its energy to infinity, but the transformedsystem is excited by the uniform electric field of aplane wave and the energy harvested is focusedto the origin. Just as energy never reaches infinityin the original, so it never reaches the origin in thenew system. As the excited waves travel withslower and slower group velocity, their energydensity increases, giving rise to a huge field en-hancement (Fig. 2C). Note also the compressionof the surface plasmon wavelength. In an idealloss-free system, energy densitywould rise remorse-lessly to infinity (the waves are never reflected),but in practice losses intervene and eventuallyterminate the increase in energy density for re-alistic values of the silver permittivity. Despitethe presence of loss, very large enhancements areseen: A field enhancement of 104 would enhancethe sensitivity to Raman signals by a factor of1016. As we shall see, loss is not the ultimatelylimiting factor. Nevertheless, in practice, verysubstantial enhancements can be expected forsingular structures.

Physical Insight and Hidden SymmetryIt has long been known that singular structuresgreatly enhance the energy density of radiation,and indeed sophisticated computer simulationsreveal many of the details (14–21). So what doestransformation optics add to the story?

Take the kissing cylinder configuration shownin Fig. 2: Computer simulations variously work

A

C

By

x Surface plasmons

Dipole

Invert aboutdipole center

Externalelectricfield

Kissingcylinders

Ex/E

0

176

–104

+104

0

178 180θ(deg)

θ

Fig. 2. (A) A mother structurecomprising a 2D cavity sand-wiched between two flat silversurfaces. Light is captured froma dipole source located insidethe cavity, and the energy trans-ported to infinity. (B) Transfor-mation of the left-hand structurevia an inversion about the cen-ter of the dipole gives rise to adaughter structure that is finitebut inherits the same broadbandspectrum. (C) The field enhance-ment at the surface of two touch-

ing silver cylinders of equal radii at a frequency 0:7wsp, wherewsp is the surface plasmon frequency of anisolated cylinder. The blue curve is calculated by using the experimentally measured permittivity; the redcurve shows the effect of doubling the imaginary part of the permittivity. The incident electric fieldorientation Ex is shown by the black arrows.

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either by quantizing the fields on a mesh or bymaking an expansion in many cylindrical har-monics. The picture that emerges is a complexrepresentation of the fields. Transformation op-tics reveals that this complexity is not intrinsic tothe spectrum but is imposed by the transforma-tion from a simple system to a more complex one.The kissing cylinders have little intrinsic sym-metry: just an axis of rotation and a couple ofmirror planes. However, the original planar wave-guide has translational symmetry that enables usto classify all eigenstates by a Bloch wave vec-tor, k. Although this symmetry is hidden by thetransformation, it continues as a good quantumnumber for the eigenstates of the new system,offering valuable insight into their nature (27).

In computational studies, each new structurerequires computation to begin again. In transfor-mation optics, our “mother” structure is relatedto an infinity of “daughter” structures, giving usthe ability to classify a whole category of struc-tures as belonging to the same family. In general,mother structures that are infinite, as is the case inFig. 2, give rise to continuous spectra. Transfor-mations to finite structures give rise to singu-larities because infinity is mapped into somethingthat is finite. In Fig. 3, left, we show a series ofsingular structures obtained by transforming planarwaveguides, and therefore all of them have con-tinuous spectra and show large field enhance-ments in the vicinity of the singularities.

On the other hand, starting from a finite pla-nar structure means that the spectrum is discrete,as is the case for all nonsingular finite structures.The transformed structures of Fig. 3, right, derivefrom planar mother structures that are finite;

in consequence they inherit the discrete spectra,and their geometry is free of the singularitiesthat would be essential were the spectra to becontinuous.

The various transformations used to generatethese structures can be seen in (33–35).

Radiative CorrectionsSo far we have assumed a very small system andneglected radiative corrections. However, radi-ative scattering increasingly competes with theharvesting process as system size increases andultimately limits how large a system can be andstill have useful harvesting capability.

Radiation is a loss mechanism and can beapproximated by introducing an absorbing me-dium outside a large cylinder that encases thewhole system (Fig. 4B) (36). If only dipole ra-diation is appreciable, then this approximationis exact. Applying transform optics in reverse takesthe large hollow cylinder into a small cylindercontaining the absorbing material (Fig. 4A). As ithappens for all systems for which harvesting issubstantial, dipole radiation dominates and this isa very accurate approximation. Figure 4C showsthe reduction in enhancement resulting from ra-diative corrections calculated by using the dipoleabsorber approximation and by direct computersimulation. Radiative corrections degrade thisharvester’s performance. For systems of 200-nmdimensions and larger, most of the incident ra-diation is scattered rather than harvested.

Nonlocal CorrectionsWe have seen how radiative corrections spoil theharvesting process for large systems. However,

there is a limit to how small wecan make our system and stillhave it function efficiently. Quiteapart from difficulties of con-structing nanoscale systems, non-local effects become substantialfor systems smaller than a fewnanometers in diameter (37–39).

Nonlocality arises becauseof quantum effects in the metal’sconduction electrons. If we ap-ply an electric field to a metalsurface, electrons migrate to thesurface and screen the interior ofthemetal from the field. Becausethe electrons are not infinitelycompressible, the screening elec-trons are not located precisely atthe surface but are distributed ina layer whose thickness is of theorder of the FermiThomas screen-ing length, or about 0.1 nm in atypical metal.

The local approximationamounts to assuming that thescreening electrons are confinedto an infinitely thin surface layer.This is a good approximation un-less the system dimensions are

very small indeed; unfortunately, that is the casefor harvesting systemswhere the enhanced fieldsare found for surfaces that approach very closely.As a result, nonlocal effects ultimately limit theenhancement. Technically, the charge distributioninside a metal is represented by the longitudinalmodes: the bulk plasmons. For most purposes inoptics, bulk plasmons play no role: Their fre-quency is assumed independent of wave vector,q, and they cannot be excited by external radiation.But for small systems, their dispersion, wpðqÞ,becomes appreciable and has to be taken intoaccount. If w > wpðq ¼ 0Þ, bulk plasmons havefinite group velocity and carry energy away fromthe surface into the bulk. Ifw < wpðq ¼ 0Þ, bulkplasmon modes decay as expð−dzÞ, the exponentd defining a decay length representing the finitesize of the nonlocal surface charge. In contrast, thetransverse modes for w < wpðq ¼ 0Þ represent

Infinite planar structurecontinuous spectrum

Singular structuresnon-singular

or 'blunted' structures

Finite planar structurediscrete spectrum

Fig. 3. A single mother structure, such as the planar waveguideshown in Fig. 2, can be transformed into an infinite variety of daughterstructures; that is, they all have identical spectra. If themother planarstructure is infinite, the spectrum is continuous and the daughterstructures are singular, that is, possessed of sharp features; if finite,the spectrum is discrete and the daughter structures nonsingular,that is, no sharp features.

A

B

C

Dipole

absorber

Scattered field

Scattered field

Incident electric field

Ex/E

0

–2000

2000

0

175 180

θ(deg)

D = 200nm

Fig. 4. (A) A small lossy dielectric sphere trans-forms into (B) a large hollow dielectric absorber.In this way, radiation losses can be taken into ac-count. (C) Field enhancement as a function of an-gle around the cylinder. Blue line, no radiativecorrection; red line, dipole absorber correction;and green line, computer simulation includingretardation.

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the decay of the fields into the bulk. Ultimatelyfor very small particles, the dielectric descriptionbreaks down entirely as the electron levels arequantized, and electron tunneling between com-ponents of the system comes to dominate. Thedebate is ongoing as to how small a system mustbe for a dielectric description to fail, but unpub-lished experiments indicate that, even for par-ticle separations of less than a nanometer, thenonlocal theory works well.

Transformation optics has beengeneralized to deal with these non-local effects (40). Figure 5 showsthe theory applied to a system oftwo touching silver nanocylinders.Figure 5B shows calculated crosssections for differing degrees of non-locality, the green curve for d–1 =0.16 nm being the appropriate valuefor silver. Evidently in the nonlocalsystem, the continuous spectrum isreplaced by a series of discrete lev-els. This comes about because sur-face plasmons are no longer preventedfrom reaching the touching point,and, when they meet, their phasesmust match, imposing quantization.The effect grows more evident withincreasing nonlocality. In Fig. 5C,we show the enhancement at thetouching point as a function of fre-quency and cylinder radius. As dis-cussed, large cylinders show poorenhancement because of competi-tion from radiative losses. Small cyl-inders show a pronounced blue shiftand reduced enhancement. The op-timum choice of radius seems to bein the range 35 nm < R < 80 nm.

ConclusionsOur aim has been to showcase themost recent application of trans-formation optics, to the design ofmassively subwavelength systems,with particular examples chosen fromplasmonic devices designed to con-centrate light. Transformation op-tics has now found applications tomany fields: initially to adaptationof computer programs to varying ge-ometries; then to negative refractiondevices andperfect lenses, enormous-ly expanding the potential applica-tions; to the problem of invisibility;and now to subwavelength structures.

The key ingredients of trans-formation optics are the electric andmagnetic field lines. These replacethe rays of light that appear in theapproximations of Snell’s law andray optics but retain the visual in-sight while fully satisfyingMaxwell’sequations. Indeed physical insightis the key benefit of this approach to

electromagnetism. Our expectation is that trans-formation optics will be the design tool of choicein electromagnetic theory.

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Acknowledgments: D.R.S. and J.B.P. were partially supportedby the U.S. Army Research Office (grant no. W911NF-09-1-0539);A.A. by the European Community project PHOME (ContractNo. 213390); J.B.P. and S.A.M. by the Leverhulme Trust;and S.A.M. by the Engineering and Physical SciencesResearch Council.

10.1126/science.1220600

A

B

C

Transformed system

σ abs (

nm)

R (

nm)

Smearing varies

Physicalsystem

Constantcharge

smearing

160

10-1

100

102

120

40

0.2 0.4

0.5

δ–1 = 0.016 nm

δ–1 = 0.160 nm

δ–1 = 1.600 nm

1.0

0.6 0.8 1.0ϖ/ϖSP

ϖ/ϖSP

EX

E0

Fig. 5. (A) (Left) Nonlocality smears the screening charge sothat in effect the surfaces no longer touch, and therefore fieldenhancement is reduced. (Right) Transformation optics distortsthe length scales in the transformed system, and as a result thesmearing of the surface charge varies with position. (B) Theabsorption cross section for cylindrical dimers of radius 10 nmfor various values of the nonlocal decay length, d−1. Full linesshow the transformation optics result; dots are computed byusing comsol. The gray line shows the local result for the dimers;the black line, the local results for a single cylinder; and thedashed line, the nonlocal results for a single cylinder. (C) Ab-solute value of the field enhancement at the touching point forcylinders of various radii (R).

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