Analytical solution of the almost-perfect-lens problemR. Merlin

Citation: Applied Physics Letters 84, 1290 (2004); doi: 10.1063/1.1650548 View online: http://dx.doi.org/10.1063/1.1650548 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/84/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Study of plasmonic crystals using Fourier-plane images obtained with plasmon tomography far-field superlenses J. Appl. Phys. 110, 083109 (2011); 10.1063/1.3654001 Integrated plasmonic lens photodetector Appl. Phys. Lett. 94, 083501 (2009); 10.1063/1.3086898 Subwavelength imaging by metallic slab lens with nanoslits Appl. Phys. Lett. 91, 201501 (2007); 10.1063/1.2811711 Near-field imaging in the megahertz range by strongly coupled magnetoinductive surfaces: Experiment and abinitio analysis J. Appl. Phys. 100, 063105 (2006); 10.1063/1.2349469 Erratum: “Analytical solution of the almost-perfect-lens problem” [Appl. Phys. Lett.84, 1290 (2004)] Appl. Phys. Lett. 85, 2144 (2004); 10.1063/1.1789572

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Analytical solution of the almost-perfect-lens problemR. Merlina)

FOCUS Center and Department of Physics, The University of Michigan, Ann Arbor, Michigan 48109-1120

~Received 10 November 2003; accepted 7 January 2003!

The problem of imaging for a slab of a lossless left-handed material with refractive indexn52(12s)1/2 is solved analytically forusu!1. The electromagnetic field behavior is determinedlargely by singularities arising from the excitation of surface polaritons with wave vectorq→6`.Depending on the sign ofs, the near-field is either odd or even with respect to the lens middle plane.Consistent with other nonanalytical studies, the resolution depends logarithmically onusu. Withminor alterations, these results apply as well to the electrostatic limit. ©2004 American Instituteof Physics. @DOI: 10.1063/1.1650548#

In the 1870’s Abbe proved1 that the smallest feature alens can image is limited by diffraction to;l/2n wherel isthe wavelength of light andn is the refractive index. Despitemany attempts to circumvent this barrier2–4 significantprogress has remained elusive. Recently, Pendry5 argued thata slab of a left-handed~LH! substance withe5m521 shouldbehave as a perfect lens~e andm are, respectively, the per-mittivity and the magnetic permeability!. The terms opticalleft and right handedness were introduced by Veselago6 todistinguish substances with bothe,0 and m,0 and, thus,n,0 from conventional, right-handed~RH! n.0 media. Fol-lowing Pendry’s work5 and the experimental demonstrationof negative refraction at microwave frequencies,7 LH sub-stances have attracted a great deal of interest along withsome contention.8–27 While recent experiments26,27 have putto rest concerns regarding the far field behavior of negative-refraction slabs, the question of near-field focusing has re-mained highly controversial.17–25In this letter, we provide ananalytical answer to this problem.

We consider the propagation of electromagnetic wavesfrom vacuum to a LH medium occupying the half spacez.0, and we assume that Im~e!5Im~m!50. The casee5m521 ~Ref. 5! will be referred to as ideal refraction. LetH and E be the magnetic and the electric field, andv thefrequency of light. The transverse magnetic solutions toMaxwell’s equations, are of the formHy5h(z)exp(2ivt1iqx), Hx5Hz50 ~with few modifications, arguments simi-lar to those discussed below apply as well to transverse elec-tric modes!. From the expression forH, we can obtain theelectric field usingE52( ic/ev)“3H. For z.0, we haveh5M 1 exp(1kz)1M2exp(2kz) where

k5H iAemv2/c22q2 q2,emv2/c2

Aq22emv2/c2 q2.emv2/c2 ~1!

while, for vacuum,h5A1 exp(1k0z)1A2 exp(2k0z) withk05k~e5m51!. We observe thatk5k0 for e5m521 andalso that, sinceHy and (]Hy /]z)/e must be continuous atthe boundary,A25M 1 andA15M 2 for an ideal interface.Hence, refraction causes a reversal in the sign of the expo-nent for both propagating (q2,emv2) and evanescent (q2

.emv2) waves. In a slightly modified form, this featureaccounts for the unusual optical properties of LH substances

and, in particular, for the remarkable converging lens perfor-mance of planar RH/LH interfaces.6 The latter effect can beunderstood by considering a two-dimensional source atz52, for which the radiative component in vacuum can begenerally written as

HyR5E

2v/c

1v/c

H~q!eiqx1 iAv2/c22q2uz1,udq, ~2!

then, for an ideal interface Eq.~2! is also the solution forz,0. Forz.0, we readily obtain

HyR5E

2v/c

1v/c

H~q!eiqx2 iAv2/c22q2~z2, !dq ~3!

which exhibits aberration-free focusing atz5,. As first dis-cussed by Veselago,6 the ideal vacuum-LH interface is a par-ticular case of the problem of refraction at a RH/LH bound-ary. Veselago6 showed that LH materials generally behave asoptical media with negative refractive indexnL52(em)1/2

so that a flat interface connecting such a medium to a RHsubstance, with refractive indexnR , acts as a converginglens with focal length given bynL,/(nL2nR) ~images arefree of aberrations only for the ideal casenL /nR521!.

The above results apply only to radiative modes and,thus, to length scales>l. Features of smaller sizes are con-tained in the near-field5

HyNF5E

uqu.v/cH~q!eiqx2Av2/c22q2uz1,udq. ~4!

Because evanescent waves cannot be amplified in conven-tional refraction~this can be attained in some sense withmirrors!, the dimensions of the focal spot are at best of orderl.1 However, for ideal RH–LH refraction, amplificationseems possible given that exp(2k0z) connects to exp(k0z) forA25M 1. Thus, one might be led to believe that evanescentmodes focus atz5, and, therefore, that a perfectly resolvedimage can be obtained. It is immediately obvious that thisargument poses a problem since physically sound solutionscannot grow away from the interface. As indicated byHaldane22 and others,23–25the absence of a well-behaved so-lution is due to resonant excitation of surface plasmons or,more generally, polaritons causing the field to become infi-nitely large ate521 ~this problem does not affect the farfield!. The dispersion of these modes obeysk/k052e28,29

and, thus, the frequency at whiche521 is always the solu-tion for q→6` where the density of states diverges. Wea!Electronic mail: merlin@umich.edu

APPLIED PHYSICS LETTERS VOLUME 84, NUMBER 8 23 FEBRUARY 2004

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observe that this singularity can be avoided by adding a dis-sipative term, and that other approaches for introducing aqcutoff have been proposed.22,23

The considerations for a single boundary can be easilyextended to two interfaces and, in particular, for a negative-refraction slab occupying the region 0,z,d and sand-wiched by vacuum. With the source, as previous, atz52,and providedd.,, it can be shown that there are now twofar-field images which are aberration free fore5m521. Thefirst image is inside the medium, atz5,, and the second is atz52d2,.6 Notably, and different from the single interface,the slab geometry admits an acceptable solution for evanes-cent modes ate5m521 since the exponential that growswith z inside the slab can be matched to a decaying expo-nential. This is the celebrated Pendry’s solution which leadsto a perfect image of the source, with infinite resolution, atz52d2,.5 Similar to the single interface, however, Pend-ry’s solution for evanescent modes is not free of polaritonproblems. For a slab in vacuum, the polariton dispersion,given by (k2k0e)/(k1k0e)56exp(kd),28 also has the so-lution e521 for q→6`. As will be discussed resonant ex-citation of such modes leads to a divergence of the field forcertain intervals ofz.

To avoid the singularities associated with high-q polari-tons, we takee5211s ~but keepm521! and solve theevanescent-mode problem for a lossless LH slab in the limitusu!1. The refractive index isn52(12s)1/2. Note that LHmaterials must necessarily exhibit dispersion, i.e.,s gener-ally depends on frequency. For calculating the Green’s func-tion, the relevant two-dimensional source is a uniformly dis-tributed line of dipoles which, for simplicity, we place atz52d/2 ~the images are atz5d/2 and 3d/2). The currentdensity is j x5pd(x)d(z1,)e2 ivt, j y5 j z50, andH(q)52sgn(z1d/2)p/c.11 Adding Eqs.~2! and ~4!, and integrat-

ing, we obtain the following expression for thesourcefield:

HyS5

ppv

c23

uz1d/2u

A~z1d/2!21x2H1

~1!@vA~z1d/2!21x2/c#

~5!

containing both propagating and evanescent terms;H1(1) is a

Hankel function. Forz.d, we write h5B2 exp(2k0z) anduse the boundary conditions atz50 andz5d to obtainB2.Explicitly, for z.d the contribution of evanescent modes tothe field is

HyNF52

p

c Euqu.v/cF~q!eiqx2k0zdq, ~6!

where25

F~q!54kk0eek0d/2

~k0e1k!2ekd2~k0e2k!2e2kd. ~7!

As shown by Pendry5 using a different method,F(q)5exp(3k0d/2) for s[0. Hence, an ideal slab provides a per-fect image of theuqu.c/v components of the source atz53d/2. By adding the near- and far-field contributions, itcan be shown more generally that the total refracted field forz.3d/2 is exactly given byHy

S(z22d). However, noticethat Hy

NF diverges in the intervald,z,3d/2 if s[0. Thelimit s→0 is considered in the following.

Since the singularities are atq56`, we calculate thefield by dividing the integral@Eq. ~6!# into two regions:~i!v/c,uqu,Q and ~ii ! uqu.Q. HereQ is an auxiliary vari-able satisfyingv/c!Q!d21 lnusu21 ~the final expressionbelow does not depend onQ). In the first region, we sets50whereas, in the second region, we deal with the singularityusing the approximationF(q)e2k0z'e2uqu(z1d/2)/(e22uqud

2s2/4). Keeping terms.s2 and replacingu5z23d/2, weobtain

c

pHy

NF'p

2d H cotF p

2d~u2 ix !G S s2

4 D ~u2 ix !/2d

1cotF p

2d~u1 ix !G S s2

4 D ~u1 ix !/2dJ

15 22e2vu/c~u cosvx/c2x sinvx/c!

u21x2u,0

pN1@vA~u21x2!/c#vu/c

A~u21x2!1E

2v/c

1v/c

cosqx cos@~v2/c22q2!1/2u#dq u.0

~8!

whereN1 is a Neumann function. A typical field profile isshown in Fig. 1~a!. Consistent with the previous discussion,the real part of the exponent ofs is such that, fors→0, thenear-field diverges ifz,3d/2 (u,0) while the term that de-pends ons vanishes ifz.3d/2 (u.0). Accordingly, thelength scale of the interference pattern shown in Fig. 1~a!evolves fromd for z,3d/2 to l for z.3d/2. Figure 1~b! isa high resolution image of the region delineated by the rect-angle in Fig. 1~a!, with the focal point at its center. Thecalculated magnetic field, its derivatives and, hence,E are allcontinuous at the focal point. We emphasize that Eq.~8! isvalid for z.d. Using the same procedure, the induced mag-netic field can be gained for arbitraryz. Inside the slab, i.e.,

for 0,z,d, we get approximately2sgn(s)HyNF(z1d)

1HyNF(2d2z) whereas, forz,0, we have2sgn(s)Hy

NF

(2z1d). Here,HyNF(z) is the field forz.d as defined in

Eq. ~8!. The two solutions are shown in Fig. 1~c! for x50.This result is not unexpected since the polariton dispersionexhibits two branches for which the associated fields have awell-defined parity. These findings are consistent with thetime-domain studies of Go´mez-Santos.24 For a time-varyingperturbation with a spectrum that is symmetric and centeredat the frequencyV for which s[0, only the interface atz5d becomes excited due to cancellation between the oddand even solutions; see Fig. 1~c!. We further note that at the

1291Appl. Phys. Lett., Vol. 84, No. 8, 23 February 2004 R. Merlin

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interfaces, where the field is largest,HyNF}usu21/2; this is an

important result. Sinces}~v2V!, this shows that the fieldinduced by anonmonochromaticsource, of arbitrary fre-quency spectrum, is an integrable function ofv for all x andz.

The behavior of the total field at the image plane is ofparticular interest. Adding the radiative component,Hy

R, wehave atz53d/2 (u50):

c

p~Hy

NF1HyR!'

p

dcothS xp

2d D sinF x

2dln~s2/4!G , ~9!

accordingly, the resolution length is

LR522pd

lnus/2u. ~10!

This expression is identical to that obtained by Smithet al.25

using a back-of-the-envelope argument, and is also consis-tent with the analysis of Go´mez-Santos.24 Furthermore, Eq.~9! supports Pendry’s claim of perfect imaging in thatc/p(Hy

NF1HyR)→24pd(x) for s→0. However, as already

noted in Ref. 25, the resolution is severely limited by thelogarithmic dependence of Eq.~10! and, moreover, by thefact that the field exhibits a saddle point atx50, z53d/2 sothat the depth of focus is poorly defined; see Fig. 1~b!.

In the electrostatic limit (l@d) the behavior of an idealLH slab is closely related to that of a medium withe[21and arbitrarym.5 Interestingly, the field pattern in Fig. 1bears a strong resemblance to the numerical studies reportedin Ref. 16 for the near-field of a slab of SiC. Some reflectionshows that our analysis and, in particular, Eq.~8! also appliesto the electrostatic case provided we make the substitution

2p(d/l)5Ausu ln(2/usu). Using this expression and Eq.~10! we can easily calculate the lens’ resolution. The depen-dence ofLR on l/d is shown in Fig. 2.

This work was supported by the AFOSR under ContractNo. F49620-00-1-0328 through the MURI Program and bythe NSF Focus Physics Frontier Center. The author wouldlike to thank J. Wahlstrand for help with the calculations andG. Shvets for useful discussions.

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FIG. 1. ~Color! ~a! Calculated contour plot of the magnitude of the magneticnear field, HNF ~logarithmic scale!. The focal point is atx50 and z53d/2. Parameters ares51023 and d5l/5p. The green arrow indicatesthe slab-vacuum interface;~b! high resolution image of the near-focal pointregion. The scale foruHNFu is linear;~c! dependence of the induced field onthe direction perpendicular to the slab atx50. The top~antisymmetric! andbottom~symmetric! solutions correspond, respectively, to positive and nega-tive s. The slab is represented by the red rectangle.

FIG. 2. Electrostatic limit; dependence of the resolution of the lens,LR , onl/d for e521 and arbitrary magnetic permeability.

1292 Appl. Phys. Lett., Vol. 84, No. 8, 23 February 2004 R. Merlin

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