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Physica B 338 (2003) 329–332

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doi:10.1016

Refining the perfect lens

J.B. Pendry*, S. Anantha Ramakrishna

The Blackett Laboratory, Imperial College, Prince Consort Road, London SW7 2BU, UK

Abstract

Some time ago it was shown that a slab of material with

e1 ¼ �1; m1 ¼ �1

and suspended in vacuo has the ability to focus a perfect image: both the near-field and far-field components are

delivered to the image plane with the correct amplitude and phase reproducing every detail in the original source [Phys.

Rev. Lett. 85 (2000) 3966]. Real materials fall short of this ideal particularly with respect to losses which manifest

themselves as imaginary components to e1 and m1: In this talk we shall examine how to minimise the restrictions by

structuring the lens into a series of thin slices. This results in a novel mapping onto an anisotropic system which behaves

like an optical fibre bundle, but operating on the near field.

r 2003 Elsevier B.V. All rights reserved.

Keywords: 78.20.Ci; 42.30.Wb; 78.45.+h

It is common knowledge that a lens will notproduce an image better resolved than thewavelength of light, l; even though the objectradiating light contains details much finer than l:The reason for this is not hard to see. Suppose theaxis of the lens is the z-axis, so that light wavesleaving the object have the form

EðrÞ ¼Xkx;ky

E0ðkx; kyÞ exp ðikxx þ ikyy þ ikzz � iotÞ;

ð1Þ

where

kz ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffio2=c20 � k2

x � k2y

q: ð2Þ

The wave vectors kx; ky represent a Fourierdecomposition of the object, so that larger values

onding author.

ddress: jpendry@imperial.ac.uk (J.B. Pendry).

- see front matter r 2003 Elsevier B.V. All rights reserve

/j.physb.2003.08.014

of these variables correspond to finer details.However, kz only takes real values if

o2

c20¼2pl

> k2x þ k2

y; ð3Þ

otherwise kz is imaginary and corresponds towaves which decay exponentially away from theobject and make little or no contribution to theimage. Hence the finest details of the object areeliminated from the image. For this reason thesedecaying components are referred to as ‘the nearfield’.This limit to the resolving power of a lens was

thought to be an immutable law of nature, butrecent work has shown that this is not the case. Ifwe can amplify the Fourier components whichnormally decay exponentially, they also can maketheir contribution to the image. It was shown thata parallel-sided slab of material with the property

d.

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Fig. 1. Resonant excitations at a silver surface when e1 ¼ �1:(a) silver supports surface plasmon excitations on a free surface;

(b) this antisurface plasmon state cannot exist in isolation

because the wave field diverges at infinity; (c) however, surface

and antisurface plasmon combine to give the resonant

amplifying state when two surfaces and an object are present.

J.B. Pendry, S.A. Ramakrishna / Physica B 338 (2003) 329–332330

of negative refractive index implied by

e ¼ �1; m ¼ �1 ð4Þ

will do the job of amplifying the near field [1], aswell as focussing the far field [2], and hence will intheory form a perfect image. In practice thiscondition can never be completely achievedbecause of imperfections in real materials whichwill always absorb some radiation resulting in asmall imaginary component of both e and m: Anegative refractive index has been demonstratedfor the far field [3,4]. A popular account can befound in Ref. [5]. Many papers have appeared onthis topic, amongst which [6–13].In the extreme near-field limit,

o2

c205k2

x þ k2y; ð5Þ

electric and magnetic fields become independent ofone another so that for an image comprising solelyelectric fields we need only require that e ¼ �1 torefocus them. This condition is approximated bysilver at a frequency in the optical region of thespectrum and we can use a slab of silver as anapproximation to the perfect lens for objects muchsmaller than the wavelength.In fact we can understand the amplification

process as the excitation of surface plasmonresonances on the surface of the silver. Incidentfields couple to the surface plasmons to justthe right degree to compensate for the exponentialdecay. Fig. 1 shows the situation. Fig. 1(a) showsa surface plasmon wave field at a free surface.Note that the electric fields are equal and oppositeon either side of the surface. Fig. 1(b) showsanother situation where the fields match at thesurface, but which does not correspond to a truesurface plasmon because the fields diverge expo-nentially at infinity. However, this second set ofwave fields is vital to the functionality of the lensbecause they form a bridge between the decayingfields of the object and the surface resonance onthe far surface. Fig. 1(c) shows a slab of silver witha near-field source exciting a surface plasmonvia an ‘anti-plasmon’. As a result the fieldsare amplified to their correct values in theimage plane.

In the near-field limit the amplification by theslab is given by

limk2xþk2y-N

TP ¼

4e1exp �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2

x þ k2y

qd

� �

ðe1 þ 1Þ2 � ðe1 � 1Þ2exp �2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2

x þ k2y

qd

� �: ð6Þ

In the ideal situation when e ¼ �1; all waves areamplified exponentially

limk2xþk2y-Ne-�1

TP ¼ exp þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2

x þ k2y

qd

� �; ð7Þ

but if there is some absorption present the best wecan do is

e1 ¼ �1þ id; ð8Þ

then

TPEexp �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2

x þ k2y

qd

� �

d2 þ exp �2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2

x þ k2y

qd

� � ð9Þ

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J.B. Pendry, S.A. Ramakrishna / Physica B 338 (2003) 329–332 331

and only the part of the wave field for which

doexp �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2

x þ k2y

qd

� �ð10Þ

is amplified. This limits the resolution to

D ¼2p

kmaxE

2pd

jln dj: ð11Þ

There is a variant on the slab configuration whichgreatly reduces the impact of absorption. ConsiderFig. 2, on the left-hand side in 2(a) we see theoriginal lens in operation. The lens has a thicknessd and the total distance between object and imageis 2d. On the right-hand side in 2(b) we see the lenscut into several pieces and the air gap equallydistributed between these pieces. Provided that thematerial is lossless this makes no difference to theimage, because each slablet makes its owncontribution to amplification resulting in the sameamplitude in the image plane and hence a wellfocussed image.However the distribution of the wave field is

very different in the two cases. For the distributedlens system nowhere does the amplitude becomeextreme. In contrast the original lens may producesome very large amplitudes indeed. This makes ussuspect that the distributed lens may be lesssusceptible to absorption: see Refs. [11–13]. Thisis in fact the case.We can consider the limit of an infinitely finely

divided lens cut into many thin slices. In this limitwe can treat the slices as an effective medium.Considering the average electric field leads us tocalculate the effective dielectric function for adirection normal to the slices, ez: If e1 is the

Fig. 2. An ideal lossless lens can be divided into several slices and will

presence of losses the performance of distributed lens degrades much

field reduces the effect of loss.

dielectric function of the slices and e2 that for thespaces between, then

e�1z ¼ 12ðe�11 þ e�12 Þ ¼ 0; ez ¼ N: ð12Þ

Likewise considering the displacement field paral-lel to the slices gives

ejj ¼ 12e1 þ e2ð Þ ¼ 1

2�1þ 1ð Þ ¼ 0: ð13Þ

The same result would have been achieved byembedding a set of very fine infinitely conductingwires in an insulating matrix with e ¼ 0: Put likethis it is not hard to see why the system behaveslike a lens: the potential of an object on onesurface is conducted point by point to the othersurface as though they were connected by wires.Fig. 3 makes the point graphically.As long as the system is loss free, the slices are

equivalent to the slab: both refocus the light over adistance 2d. However, the systems are quitedifferent in the presence of losses. Suppose thate1 ¼ �1þ id then

e�1z ¼ 12

1

�1þ idþ 1

� �E� i

2d; ezE2id�1

ejj ¼ 12�1þ idþ 1ð Þ ¼ i

2d: ð14Þ

In the near-field limit dispersion of a wave in ananisotropic system is given by

kz ¼ kJ

ffiffiffiffiffiffiffiffi�eJez

rE i

2kJd; ð15Þ

therefore in the absence of absorption kz ¼ 0; thefield propagates through the slices without changeof amplitude or phase, almost like light passingthrough an optical fibre bundle, except that thedetails of the image as now much less wavelength.

still produce a focussed image distance 2d from the object. In the

less rapidly because the generally smaller amplitude of the wave

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Fig. 4. Layering a lens into slices reduces the impact of losses

and enhances resolution. The object comprises two slits of 5 nm

width and a peak-to-peak separation of 45 nm; dashed

curve:single slab of silver, e1 ¼ �1þ 0:4i; of thickness d ¼40 nm, full curve: layered stack comprising 8� 5mm of silver

(i.e. the same total thickness).

Fig. 3. On the left we see a perfect lens cut into many thin slices, each with e1 ¼ �1: This anisotropic system behaves in the limit of very

thin slices like a set of very highly conducting wires embedded in a medium for which e ¼ 0: Hence we can understand how imaging

comes about: the equivalent wires conduct the potential point by point from object to image.

J.B. Pendry, S.A. Ramakrishna / Physica B 338 (2003) 329–332332

In the presence of absorption this is no longer true,and the image suffers some attenuation by

exp ð2ikzdÞEexp ð�kJddÞ; ð16Þ

giving an effective cutoff in resolution at

Dslice ¼2p

kJmaxE2pdd: ð17Þ

Comparison of Dslice and Dslab shows that Dslice hasa much more favourable dependence on d andtherefore the effects of absorption are much lessfor the equivalent slice system. We illustrate this

point in Fig. 4 where we compare the effect ofabsorption on slabs and slices.So to conclude, the original perfect lens makes

heavy demands on the properties of materials,particularly the need for low losses. Howevermodifying the structure of the lens by cutting intothin slices considerably alleviates the effects ofabsorption and greatly enhances resolution.

Acknowledgements

S.A.R. would like to acknowledge the supportfrom DoD/ONR MURI grant N00014-01-1-0803.

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