PHYSICAL REVIEW B 68, 045115 ~2003!

Subwavelength imaging in photonic crystals

Chiyan Luo,* Steven G. Johnson, and J. D. JoannopoulosDepartment of Physics and Center for Materials Science and Engineering, Massachusetts Institute of Technology,

Cambridge, Massachusetts 02139, USA

J. B. PendryCondensed Matter Theory Group, The Blackett Laboratory, Imperial College, London SW7 2BZ, United Kingdom

~Received 5 December 2002; revised manuscript received 14 February 2003; published 29 July 2003!

We investigate the transmission of evanescent waves through a slab of photonic crystal and explore therecently suggested possibility of focusing light with subwavelength resolution. The amplification of near-fieldwaves is shown to rely on resonant coupling mechanisms to surface photon bound states, and the negativerefractive index is only one way of realizing this effect. It is found that the periodicity of the photonic crystalimposes an upper cutoff to the transverse wave vector of evanescent waves that can be amplified, and thus aphotonic-crystal superlens is free of divergences even in the lossless case. A detailed numerical study of theoptical image of such a superlens in two dimensions reveals a subtle and very important interplay betweenpropagating waves and evanescent waves on the final image formation. Particular features that arise due to thepresence of near-field light are discussed.

DOI: 10.1103/PhysRevB.68.045115 PACS number~s!: 78.20.Ci, 42.70.Qs, 42.30.Wb

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I. INTRODUCTION

Negative refraction of electromagnetic waves, initiaproposed in the 1960s,1 recently attracted strong researinterest2–16 and generated some heated debate.17–29 In par-ticular, much attention was focused on the intriguing posbility of superlensingsuggested in Ref. 5: a slab of uniform‘‘left-handed material’’ with permittivitye521 and perme-ability m521 is capable of capturing both the propagatiand evanescent waves emitted by a point source placefront of the slab and refocusing them into aperfect pointimage behind the slab. While the focusing effect of propgating waves can be appreciated from a familiar picturerays in geometric optics, it is amazing that perfect recovof evanescent waves may also be achieved viaamplifiedtransmission through the negative-index slab. Some ofdiscussion of this effect relies on an effective-mediumodel6 that assigns a negativee and a negativem to a peri-odic array of positive-index materials: i.e., a photonicrystal.30 Such an effective-medium model holds for largscale phenomena involving propagating waves, providedthe lattice constanta is only a small fraction of the free-spacwavelength in the frequency range of operation. Howeverthe phenomenon of superlensing, in which the subwalength features themselves are of central interest,effective-medium model places severe constraints on thetice constanta: it must be smaller than the subwavelengdetails we are seeking to resolve. The question of wheand to what extent superlensing would occur in the mgeneral case of photonic crystals still remains unclear.

Recently, we showed that all-angle negative refract~AANR! is possible using a photonic crystal with a positiindex, and we demonstrated imaging with a resolution beor on the order of wavelength in this approach.12,15 In thispaper, we investigate the possibility of photonic-crystalperlensing in detail by studying the transmission of evancent waves through a slab of such a photonic crystal. I

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important to note that the transmission considered herefers fundamentally from its conventional implication of eergy transport, since evanescent waves need not carry enin their decaying directions. Thus, it is possible to obtatransmission amplitudes for evanescent waves greatlyceeding unity without violating energy conservation. Hewe discuss two mechanisms linking amplification of evancent waves to the existence of bound slab photon staThese bound states are decoupled from the continuumpropagating waves, thus our findings are distinct fromeffect of Fano resonances31 in electromagnetism, which werrecently studied in the context of patterned periodstructures32–36 and surface-plasmon-assisted enertransmission.37–39As for the problem of negative refractionwe have found that the concept of superlensing doesnot ingeneral require a negative refractive index. Moreover,argue that the surface Brillouin zones of both photonic crtals and periodic effective media provide a natural upper coff to the transverse wave vector of evanescent wavescan be amplified, and thus no divergences exist at large trverse wave vectors24 in photonic crystals or in effectivemedia.27 As with effective media, the lattice constant of phtonic crystals must always be smaller than the details toimaged. We derive the ultimate limit of superlens resolutiin terms of the photonic-crystal surface periodicity and inthat resolution arbitrarily smaller than the wavelength shobe possible in principle, provided that sufficiently high delectric contrast can be obtained. We pursue these idearealistic situations by calculating the bound photon statecarefully designed two-dimensional~2D! and 3D slabs ofAANR photonic crystals and present a comprehensivemerical study of a 2D superlensing structure. A subtle avery important interplay between propagating waves aevanescent waves on image formation is revealed, whmakes the appearance of the image of a superlens subtially different from that of a real image behind a convetional lens. These numerical results confirm the qualitat

©2003 The American Physical Society15-1

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LUO, JOHNSON, JOANNOPOULOS, AND PENDRY PHYSICAL REVIEW B68, 045115 ~2003!

discussions in this paper and can be readily compareexperimental data.

This paper is organized as follows. Section II is a genediscussion for amplified transmission of evanescent wathrough a photonic-crystal slab. Section III considersimplementation of superlensing in photonic crystals andultimate resolution limit and, also, describes qualitativelyexpected appearance of the image of a superlens. Sectiopresents numerical results for superlensing in a model ptonic crystal. Section V discusses further aspects of sulensing in photonic crystals, and Sec. VI summarizespaper.

II. ORIGIN OF NEAR-FIELD AMPLIFICATION

Let us first consider the transmission of a light wathrough a slab of lossless dielectric structure@Fig. 1~a!#, pe-riodic in the transverse direction, at a definite frequencyv~with free-space wavelengthl52pc/v) and a definitetransverse wave vectork. Let the slab occupy the spatiaregion2h,z,0, and in this paper we also assume a mirsymmetry with respect to the planez52h/2 at the center ofthe dielectric structure. The transmission through a finite-sstructure can be conceptually obtained by first considethe transmission through a single interface between airthe photonic crystal@Fig. 1~b!# and then summing up all thcontributions as light bounces back and forth inside the sThe incident fieldFin , the reflected fieldFre f l , and the trans-mitted fieldFtrans are related to each other by the transmsion coefficientt, defined formally byFtrans5tFin , and thereflection coefficientr, defined formally byFre f l5rFin .Here, all fields are understood to be column vectorspressed in the basis of all the eigenmodes of the corresping medium, with transverse wave vectors differing fromkonly by a reciprocal lattice vector of the surface. In this wt and r are matrices represented in these basis modes.

FIG. 1. Illustration of light-wave transmission through a phonic crystal.~a! Transmission through a slab of photonic crystal. Tincident fieldFin and the reflected fieldFre f l are measured at thleft surface of the slab (z52h), and the transmitted fieldFtrans ismeasured at the right surface of the slab (z50). ~b! Transmissionthrough a boundary surface between air and a semi-infinite photcrystal. The incident fieldFin , the reflected fieldFre f l , and thetransmitted fieldFtrans are all measured at the air/photonic-crysinterface. In both figures,k is the transverse wave vector of lighandas indicates the surface periodicity.

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overall transmission coefficient through the slab structcan be written as

t5tp-a~12Tk,pr p-aTk,pr p-a!21Tk,pta-p . ~1!

In Eq. ~1!, ta-p and tp-a are the transmission coefficienthrough the individual interfaces from air to the photoncrystal and from the photonic crystal to air, respectively,r p-ais the reflection coefficient on the photonic-crystal/air bounary, andTk,p is the translation matrix that takes the fieldfrom z52h to z50 inside the photonic crystal. Whenh isan integral multiple of the crystal’sz period,Tk,p is diagonalwith elementseikzh for a crystal eigenmode of Bloch wavvector k1kzz, with Im kz>0. In what follows, we discussthe possibility of amplification int, beginning with the diag-onal element that describes the zeroth-order transmissioi.e., transmission of waves withk in the first surface Bril-louin zone, referred to ast00.

Generally speaking, Eq.~1! describes a transmitted wavthat is exponentially small for large enoughuku. This can beseen in the special case of a slab of uniform material wpermittivity e and permeabilitym, where all the matrices inEq. ~1! can be expressed in the basis of a single plane wand thus reduce to a number. In particular,Tk,p5exp(ikzh)5exp(ihAemv2/c22uku2), which becomes exponentiallsmall asuku goes aboveAemv/c. Equation~1! then becomesa familiar elementary expression

t005tp-ata-peikzh

12r p-a2 e2ikzh

. ~2!

Equation~2! has an exponentially decaying numerator, whfor fixed r p-a the denominator approaches 1. Thus, wavwith large enoughuku usually decay during transmission iaccordance with their evanescent nature.

There exist, however, two separate mechanisms by whthe evanescent waves can be greatly amplified through trmission, a rather unconventional phenomenon. Themechanism, as proposed in Ref. 5, is based on the factunder appropriate conditions, the reflection and transmisscoefficients through individual interfaces can becomediver-gentand may thus be called asingle-interface resonance. Forexample, under the condition of single-interface resonanta-p ,tp-a ,r p-a→`, and in the denominator of Eq.~2! theterm r p-a

2 exp(2ikzh) dominates over 1. In this limit, Eq.~2!becomes

t005tp-ata-p

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The divergences in Eq.~3! cancel each other, and the nresult is that for largeuku, t005exp(2ikzh), leading to ampli-fication of exactly the right degree to focus an image. Tsame arguments can be applied to the general case of Eq~1!,as long askz is regarded as thez component of the corresponding eigenmode’s wave vector with thesmallestimagi-nary part (Imkz), which produces the dominant term veikzh. As elements ofr p-a grow sufficiently large, the matrixproductTk,pr p-aTk,pr p-a dominates over the identity matrix

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SUBWAVELENGTH IMAGING IN PHOTONIC CRYSTALS PHYSICAL REVIEW B68, 045115 ~2003!

Since, in this case, the matrix under inversion in Eq.~1!scales as exp(22 Imkzh) and the rest scales aexp(2Im kzh), the amplification behavior is still present igeneral, in each element oft. The transmissiont00 can berepresented by Eq.~3! with the coefficients to the exponential replaced by smooth functions ofv andk.

The second mechanism for enhancement of evaneswaves relies on a direct divergence in the overall transmsion: i.e., anoverall resonance. This is clear from the uni-form medium case~2!, whose denominator becomes zewhen 12r p-a

2 exp(2ikzh)50, which is the condition for transverse guiding via total internal reflection. An evanescentcident wave can satisfy this condition exactly. It thus hothat a direct divergence can exist in the overall transmissof evanescent waves without any accompanying sininterface resonance, and therefore finite and strong ampcation of evanescent waves results when the incidence wdoes not exactly satisfy but is sufficiently close to this codition of overall resonance. In this case, there is no uplimit on the amplification and the transmission can evenceed that prescribed by Eq.~3!; hence there is the potential tform an image provided that the correct degree of amplifition is induced. These arguments are also valid in the gencase ~1!, at or near the singular points of2Tk,pr p-aTk,pr p-a whose inverse occurs in the transmissioIf we write the relation between the light frequencyv andwave vectork at these singular points asv5v0(k), thenclose to such a resonance the transmissiont00 is described by

t005C0~v,k!

v2v0~k!, ~4!

whereC0(v,k) is some smooth function ofv andk. For agivenv, the issue is then to design photonic crystals withappropriate dispersion relationv0(k) so that Eq.~4! approxi-mates the required amount of amplification.

Both mechanisms of evanescent-wave amplificationscribed here involve some divergences in the transmisprocess. Physically, such a divergence means that enerbeing pumped indefinitely by the incident wave into both ttransmitted and reflected fields, whose amplitudes increastime without limit. Equivalently, a finite field inside thstructure can be produced by zero incident field; i.e., it ibound ~guided! electromagnetic mode. In other words,bound photon mode on the air/photonic-crystal surface~asurface state! leads to a single-interface resonance, anbound photon state inside the slab leads to an overall rnance. In the overall resonance case, the dispersion relof the bound photon mode is justv5v0(k) in Eq. ~4!. Asimilar equation can also be used to represent the zerorder term intp-a close to a single-interface resonance:

tp-a,005Cp-a~v,k!

v2vp-a~k!, ~5!

with the single-interface bound photon dispersion relatvp-a(k) and a smooth functionCp-a(v,k). BothCp-a andC0here represent the coupling strength between the incidewave and the respective bound photon state.

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It is instructive to compare these two amplificatiomechanisms by their applicable ranges. As shown in Refin an ideal material slab withe(vsp)521 and m(vsp)521, every evanescent wave is amplified by a singinterface resonance at the surface plasmon frequencyvsp .For such a slab,r p-a diverges for any incidentk and nooverall resonance happens at the plasmon frequency. Hever, bothe andm are necessarily dispersive,7 and detuningfrom the single-interface resonance frequency we can sathe guiding condition

12r p-12 ~v6!e2ikzh50 ~6!

at two separate frequenciesv6 , above and belowvsp , re-flecting the fact that the surface photon states on theinterfaces of the slab interact with each other, forming symetric and antisymmetric combinations. In the general caEq. ~6! can be satisfied and bound photon states insideslab form even without the prior existence of interfastates—i.e., withoutr p-a diverging. Thus both mechanismmay be available to amplify evanescent waves. To havsingle-interface resonance in Eq.~1!, it is required that theterm associated with single-interface reflections dominover 1. This can be expressed as

uv2vp-a~k!u!uCp-aue2Im kzh. ~7!

For an overall resonance, the condition becomes

uv2v0~k!u'uC0ue2Im kzh ~8!

in order to produce an amplification magnitude similarthat in Eq.~3!. We note thatCp-a and C0 are on the sameorder of magnitude if the bound photon modes insideslab are constructed from combinations of the surface phostates. It is thus clear that in the general case, amplificaof evanescent waves requires operation much closer toexact resonance in the single-interface resonance mechathan in the overall resonance mechanism. In addition,like the situation near a surface photon state discussedviously, the overall resonance can in principle happen nebulk-guided mode that is not evanescent inside the photocrystal slab. Put in another way, in general amplificationmore easily achieved using an overall resonance than usisingle-interface resonance. In the following, therefore,make primary use of the second resonance mechanismrealize amplification of evanescent waves in the mannercussed here.

Amplification thus arises due to the coupling betweenincident evanescent field and bound photon states of infilifetime, which usually exist below the light line.40 In a pe-riodic structure, the range of wave vector region belowlight line is limited by the boundary of the surface Brillouizone, due to Bragg scattering. What happens to the transsion of evanescent waves whose wave vectors are so lthat they lie beyond the first surface Brillouin zone and bcome folded back into the light cone? In this case, the asciated slab photon resonance mode changes from aboundstate to aleaky state, and its frequencyv0(k) becomescomplex—i.e.,v0(k)→v0(k)2 ig(k), with v0 andg beingreal. This situation is actually described by the diagonal e

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LUO, JOHNSON, JOANNOPOULOS, AND PENDRY PHYSICAL REVIEW B68, 045115 ~2003!

ment tnn of t with n5” 0 if we usen to index the surfacereciprocal lattice vectors and assumek to be in the first sur-face Brillouin zone. When the incidence evanescent wavsufficiently close to a leaky photon mode the transmissbecomes

tnn5C0~v,k!

v2v0~k!1 ig~k!, ~9!

which always has a finite magnitude. In principle,utnnu canalso reach values larger than unity provided thatg is smallenough. However, asn goes away from 0 the spatial variation in the incident wave becomes more rapid. The leaphoton state, on the other hand, always maintains a confield profile with variations on a fixed spatial scale, roughthat of each component in a cell of the crystal. Hence, fonsufficiently far from 0,C0 in Eq. ~9!, determined by theoverlap between the incident wave and the slab phomodes, must always approach zero and so mustutnnu. Notethat this is not true in the case of bound photon modes wg50: amplified transmission may still occur if the incidewave is close to an exact overall resonance. The numeresults presented later in this paper indicate that, forstructure considered here, the transmission for evaneswaves coupling to leaky modes withn5” 0 is alwayssmalland the possible amplification effect can be ignored.

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It should be clear from this discussion that an amplifitransmission of evanescent waves at a givenv andk is notrestricted to materials withe,0 or m,0 only and can beachieved by coupling to bound photon states in general.other interesting feature of this approach is that, as showRef. 5, with the single-interface resonance-amplificatmechanism, the reflection coefficientr can vanish, but in theoverall-resonance mechanism here an amplified transmisprocess also implies an amplified reflected evanescentin general. Since the latter mechanism is used in our numcal calculations below, most of the effects that arise duethe transmitted evanescent waves should also be expectthe reflected waves as well. These might lead to nontrivconsequences—for example, a feedback on the emitsource. In this paper, we assume that the source field iserated by some independent processes and ignore the ptial influences of these effects.

III. PHOTONIC CRYSTAL SUPERLENSES

We now consider the problem of superlensing at a givfrequencyv in photonic crystals. An ideal point source emia coherent superposition of fieldsFsource(k) of differentwave vectorsk, with uku,v/c being propagating waves anuku.v/c being evanescent waves. We place such a psource on thez axis ~the optical axis! at z52h2u and ob-serve the image intensityI image in z>0 to be

I image~r !5U E dk„ . . . ,tkexp~ ikn•r t!, . . . …•Tk,a~z!t~k!Tk,a~u!Fsource~k!U2

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stands for transverse coordinates,kn5k1Gn , with Gn beingthe reciprocal vectors of the surface, andkn is the transversewave vectors in the air basis that we have been using.brackets represent a row of polarizations and phases obasis, and its dot product with the column vector of the tramitted field produces the complex field amplitude. The ingral is carried out over the first surface Brillouin zone, andthe case of a uniform material it is over allk’s in the trans-verse plane.

Conventional lenses only image the portion of the indent field withuku,kM for kM,v/c, limited by the numeri-cal aperture. Aperfect lens,5 made of left-handed materialwith e521 andm521, not only focuses all propagatinwaves with negative refraction, but also amplifies all evancent waves withuku.v/c, so that all Fourier components othe source field reappear perfectly in the image plane of gmetric optics. One unusual aspect in such a scenario isthe fields in certain spatial regions becomedivergentand theassociated energy density becomes infinite if the lossesignored.24 However, this observation merely reflects the fathat infinite-resolution imaging requires an infinite time iterval to reach its steady state, and the original proposal5 still

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remains valid in the sense that arbitrarily fine resolution isprinciple possible. Here, we use the termsuperlensto denotea negative-refractive slab that not only focuses all propaging waves by negative refraction without limitation of finitaperture, but also amplifies at leastsomeevanescent waves ina continuous range beyond that of the propagating wavesthis context,superlensingrefers to the unconventional imaging effects due to the presence of the additional near-filight. The phenomena considered here thus contain the mfeatures of Ref. 5. In general, however, the magnitudetransmission will not reproduce exactly that for perfect image recovery, and the resulting image will beimperfectandpossess quantitative aberrations. Below, we focus our attion on 2D situations where most of the quantities cantreated as scalars. We choose the transverse coordinatex and use unity for everyGn component ofFsource(k) for allk, appropriate for a point source in 2D.

Our starting point here is the focusing by negative refrtion of all propagating waves withk,v/c ~AANR! ~Ref.12!; a brief discussion of the behaviors outside the frequerange of AANR will be presented in Sec. V. With AANR, apropagating waves can be transmitted through the photocrystal slab with transmittance of order unity and producintensity maximum behind the slab; i.e., they focus into

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SUBWAVELENGTH IMAGING IN PHOTONIC CRYSTALS PHYSICAL REVIEW B68, 045115 ~2003!

real image there. Superlensing requires amplified transmsion for an additional range of transverse wave vectov/c,k,kM . In Ref. 5, kM5`. The important differencefor superlensing with a photonic crystal is thatkM is in gen-eral finite. This is clear from the discussion in Sec. II, whea finite high cutoff to the transmission spectrum results aconsequence of Bragg scattering of light to leaky phomodes. This finitekM makes the image reconstruction prcess through a photonic-crystal superlens no longer divereven in the lossless case. Physically, the amplification of enescent waves requires near-resonance coupling and thsulting growth of an approximate bound photon state durtransmission. Amplification of largerk components thus requires exponentially higher energy density in the bound pton mode and an exponentially longer time interval to reaa steady state. Our numerical results below are mainlyculated in the frequency domain and therefore represensteady-state behavior after the transients have died awafinite time.

To actually realize amplification of evanescent wavesv/c,k,kM , one must design the photonic-crystal structucarefully so that Eq.~8! holds for all evanescent waves ithis range. For largek, kz51Av2/c22k2 has a large posi-tive imaginary part, and therefore the operation frequencvshould be very close to the resonance frequencyv0(k) of abound photon state. This means that for largek, the disper-sion relationsv0(k) must approach a ‘‘flat’’ line nearvwithin the AANR range, as shown in Fig. 2. In general, theare two classes of bound photon modes within a photocrystal slab. One consists of those guided by the slabwhole, similar to the guided modes in a uniform dielectslab. The other class includes those guided by the air/surfaces; i.e., they are linear combinations of surface stat30

FIG. 2. Schematic illustration of amplification of evanescewaves and superlensing in photonic crystals. The light-shadedgions are the light cone, and the dark shaded region is the AAfrequency range. The curve markedv0(k) outside the light cone isa band of bound photon states inside a slab of photonic crystal.dashed line markedv indicates the operation frequency. Amplification requires thatuv2v0(k)u be small for allk.v/c, especiallythe largek’s, which in turn requires thev0(k) curve be ‘‘flat.’’ Inthis repeated zone scheme, a bound photon state may only exthe rangev/c,k,2p/as2v/c, which imposes an upper cutofor superlensing using photonic crystals.

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which decay exponentially both in air and in the slab awfrom the surface. Although both classes of bound modesbe employed to achieve amplification for a givenk, most ofthe wave vector regionv/c,k,p/as within the AANR fre-quency range involves a partial photonic band gap andonly accommodate the surface photon states. Furthermthe surface states of photonic crystals are known to depon the fine details of the surface structure—e.g., the surftermination position in a given direction—and can be tunto be any frequency across the band gap at least for a sik point.30 Thus, the slab surface photon states are attraccandidates for achieving flat bound photon bands withinAANR range for superlensing. In our numerical example,give one 2D design that meets this goal by simply adjustthe termination position of the crystal surface.

What is the ultimate limit to the imaging resolution ofphotonic-crystal superlens? Following Ref. 5, we may intpret the image of a point source as an intensity peak witthe constant-z plane of AANR focusing, and we can measuthe resolution of such a peak by the distance betweennearest minima of this intensity peak. Taking into accouthe decay of the evanescent light as it travels in air fromsource plane to the incidence surface of the slab and fromexit surface of the slab to the image plane, we can estimthe image field using a simplified model. In this model, wassume unit total transmission from the source to the imfor uku,kM and zero transmission foruku.kM . The intensityprofile on the image plane then reads

I image~x!5U E2kM

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which has a peak amplitude atx50 with a transverse sizeD52p/kM—i.e., the distance between the first zeroaround the peak. This image size is zero in a material we521 andm521, sincekM5`, leading to the interpretation of a ‘‘perfect’’ image. At present in a photonic crystaquantitative estimates of the minimum possibleD ~i.e., maxi-mum possiblekM) may be obtained by looking at Fig. 2. Ithe best situation, all the surface modes withk,0.5(2p/as) can be used directly for amplification if thesatisfy ~8!,41 which giveskM>0.5(2p/as). Since this esti-mate ignores the strong Bragg-scattered wave componenthe surface states near the surface Brillouin zone edge, itconservative estimate. From Fig. 2, we also deduce the mmum wave vector of a ‘‘flat’’ surface band below the lighline that can be coupled to at frequencyv to be (12vas/2pc)(2p/as)52p/as2v/c, which is an overesti-mate. Putting these results together, we thus obtain themate resolution limit of a photonic-crystal superlens to be

asl

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According to the Rayleigh criteria, the minimum feature sithat can be resolved by such a device isD/2. Thus, the reso-lution of a photonic-crystal superlens at a single frequenconly limited by its surface periodas instead of the wave-length of operationl; i.e., superlensing is in general possib

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LUO, JOHNSON, JOANNOPOULOS, AND PENDRY PHYSICAL REVIEW B68, 045115 ~2003!

FIG. 3. ~Color! Bound photon modes with TE~magnetic field perpendicular to the plane! polarization in a 2D photonic crystal slab.~a!Left panel: the actual photonic crystal used in calculation. The parameters areas5A2a, b150.5as , b250.2as , andh54.516as . Rightpanel: the calculated band structure of bound photon modes~black curves!, plotted on top of the photonic band structure projected alongsurface direction~the red-filled region!. The blue-filled region indicates the light cone. The green range is the frequency range for AAthis photonic crystal.~b! Distribution of the magnetic field perpendicular to the plane for the surface photonic modesk50.45 (2p/as). Left and right panels represent odd and even symmetries with respect to the mirror plane at the center of the strucand blue indicate positive and negative values of the magnetic field.

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in a positive-index photonic crystal. Our considerations agive a guideline for designing high-resolution superlensfor a given wavelength, the smalleras—i.e., the lower in theband structure one operates with AANR—the better the relutions will be. In principle, by using sufficiently large delectric constants in its constituents, a photonic-crystal sulens can be designed to operate at a wavelength arbitrlarger thanas . A similar superlensing trend42 is also achiev-able using localized plasmon polariton resonances in metphotonic crystals,43,44 in accordance with known results ithe left-handed materials. If, furthermore, a sufficiently flsurface band is achieved in the AANR frequency rangesuch a photonic-crystal slab by manipulating its surfastructures, imaging arbitrarily exceeding the diffraction limis possible. Therefore, there is no theoretical limit to suplensing in photonic crystals in general. In practice, of couravailable materials, material losses, and unavoidable imfections in surface structures will limit the performancesuch superlenses.

It must be noted that the image of a superlens consideabove is substantially different from the conventional rimage of geometric optics. Conventional real optical imagalways correspond to an intensity maximum: i.e., a peakthe field amplitude distribution both inx and z directions.When only the propagating waves are transmitted throthe superlens, they similarly produce an intensity maximin z.0: i.e., the image of AANR. The position of this imag

04511

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may be simply estimated by paraxial geometric optaround thez axis. However, when evanescent waves arecluded, they bring distortions to the image and the resultintensity maximum is no longer at the position of the AANimage. A simple illustration of the image pattern is providby our simplified cutoff model. The full expression of thimage in this model with a high cutoffkM.v/c can bewritten as

I image~x,z!5U E2v/c

v/c

eikx1 iAv2/c22k2(z2z0)dk

1S E2kM

2v/c

1Ev/c

kM D eikx2(Ak22v2/c2)(z2z0)dkU2

,

~13!

wherez5z0 is assumed to be the focusing plane of AANInside the absolute value sign of Eq.~13!, the first term hasconstructive interference atz5z0 and represents an intensitmaximum there, but the second term always displays asmetric amplitude distributions inz acrossz5z0. Thus, forkM.v/c the overall intensity distribution no longer hasmaximum atz5z0. The detailed image pattern has a sentive dependence on the interplay between the propagaand evanescent waves.

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SUBWAVELENGTH IMAGING IN PHOTONIC CRYSTALS PHYSICAL REVIEW B68, 045115 ~2003!

For kM slightly abovev/c, the strength of evanescenwaves is comparable to that of propagating waves. An intsity maximum still exists in the regionz.0, but is shiftedaway fromz5z0 toward the lens. This intensity maximumthus appears as a real image similar to conventional opNote that in general it will have a transverse size that ilittle bit smaller than 2p/kM—i.e., the transverse size of thpeak in the planez5z0. Since this situation is forkM slightlyabovev/c, it may be called themoderate subwavelengtlimit, and the transverse size of an intensity maximumalways limited by a fraction of the wavelength.

When kM exceeds a certain threshold, the evanescwaves begin to dominate the image pattern. In this situatan intensity maximum completely disappears in the regz.0, where the optical intensity becomes monotonicallycreasing. We estimate thekM threshold for this behavior tobe aboutkM ,th51.35v/c, with a transverse size of the intensity maximum about half a wavelength at this threshold,ing the simplified model~13!. This crude qualitative estimatwill be confirmed in our numerical calculations. The casesuperlensing withkM.kM ,th can thus be called theextremesubwavelength limit. From another viewpoint, atv/c!kMwe havel@as , and if we also assume that the slab thickneh is small compared tol, then the system may be regardto be in the near-static limit. The absence of an intensmaximum in z.0 may then be understood simply by thelementary fact that in electrostatics and magnetostaticstentials can never reach local extrema in a sourceless spregion. Thus, in the extreme subwavelength limit the iming effect of a superlens is strictly in the transverse directonly. Compared to a conventional lens, which generally hapower-law decaying intensity distribution away from its image, the superlens has a characteristic region betweensuperlens and the image where an exponentially growingtensity distribution exists.

A related effect of evanescent waves on the image is tfor kM.v/c, the image intensity is generally larger than thfor kM<v/c. This occurs in the simplified model~13! due tothe addition of evanescent-wave components. Note alsoin this model, no exact resonant divergence is present intransmission and the intensity enhancement effect is pronent in the region between the slab and image plane of gmetric optics. A more general situation occurs when theerating frequencyv falls inside the narrow frequency rangof the bound photon states, so that a distinct numberbound mode with near-zero group velocities can be excon exact resonance. The contribution from one such eresonance pole atv5v0(k0) to the transmission may bestimated as

E dkC0~v,k!

v0~k0!2v0~k!'

1

~]v/]k!uk5k0

E dkC0~v,k!

k02k.

~14!

The integral overk, though not suitable for analytical evaluation in general, can usually be regarded as having a fiprincipal value and depends on the detailed behaviorC0(v,k). The influence of each pole on the transmitted iage can be thus seen to be inversely proportional to the g

04511

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velocity of the bound photon state at the resonance anstrongest for modes with the smallest group velocities. Whoperating on exact resonance to the flat surface bandsvery small group velocities, the field pattern in a wide sparegion inz.0 that extends beyond the image may be domnated by the surface states, often with extraordinary strenThus, we can call this regime of superlensingenhanced sur-face resonance. On the one hand, such an effect mightuseful in applications where a large field amplitude is dsired. On the other hand, since a surface resonance is ade-localized field distribution, there is very little informationcontained within the intensity distribution about the tranverse location of the source in this case. This is a subtle pto be avoided in imaging applications.

In many experimental situations, light intensity is th

FIG. 4. Frequency spectrum of transmission and links to boslab photon modes.~a! Zeroth-order transmission (ut00u2) throughthe photonic-crystal slab in Fig. 3 for various transverse wave vtors, plotted on a logarithmic scale vs frequency.~b! The transmis-sion curves in~a! are plotted on the bound photon band structurethe photonic-crystal slab@Fig. 3~a!#. The arrows indicate the transverse wave vector for each transmission curve. The shaded regithe light cone.

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ntosition of

LUO, JOHNSON, JOANNOPOULOS, AND PENDRY PHYSICAL REVIEW B68, 045115 ~2003!

FIG. 5. ~Color! Transmission and intensity distribution in the image space forv50.193 (2pc/a), for a photonic crystal slab in Fig. 3illuminated with an ideal point source.~a! Zeroth-order transmission@ ut00u2 for k,0.5(2p/as)] plotted on a logarithmic scale vs the incidetransverse wave vector. The gray curve indicates the transmission value for perfect image reconstruction at the AANR focusing pthe present slab. The effective high cutoff of the transverse wave vectorkM is marked out in dashed lines.~b! The intensity distribution inreal space to the right of the slab forz.0. The right surface of the slab is atz50. ~c! The data in~b! plotted in the plane ofz50.6as

50.16l—i.e., through an intensity maximum.~d! The data in~b! plotted withx50—i.e., on thez axis.

caerinf sinis

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quantity that is responsible for most physical effects andbe measured directly. Both the subwavelength transvresolution and the spatial region of extraordinarily hightensity can thus serve as direct experimental evidence operlensing. From the viewpoint of applications, imagingthe transverse direction alone below the diffraction limitsufficient and desirable for many situations, such as senand detecting or strong focusing for active phenomena.considerations indicate the possibility of a variety of imapatterns impossible in conventional geometric optics inimage of a superlens, based on the interplay between nfield and far-field light. With a photonic crystal, a flexibsuperlens may be constructed in which all of these physeffects are readily observable.

IV. NUMERICAL RESULTS

In this section, we present numerical results that confithe above discussion on superlensing in photonic crysThe main crystal structure we choose to study is a squlattice of two overlapping rectangular air voids in alossless

04511

nse-u-

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dielectric e512, oriented along the~11! direction and withthe various sizes specified in Fig. 3~a!. The lattice constant ofthe square lattice isa, and the surface lattice constant isas

5A2a. This configuration possesses a similar infinite-crysband structure to the one studied in Ref. 12. Since it ilayered structure along the~11! direction, this crystal struc-ture also allows for efficient numerical computationstransmission through a finite slab in the frequency domaFor brevity, we assume a TE~magnetic field perpendicular tothe 2D plane! polarization in all our calculations in this paper, and similar results can be expected for TM modeswell.

A. Surface band structure

The band structure of the bound photon states onphotonic-crystal slab suspended in air with~11! surface ter-mination is calculated by preconditioned conjugate-gradiminimization of the block Rayleigh quotient in a plane-wabasis45 using the supercell approach. The results are psented in Fig. 3~a!. Below the light cone and inside the re

5-8

SUBWAVELENGTH IMAGING IN PHOTONIC CRYSTALS PHYSICAL REVIEW B68, 045115 ~2003!

FIG. 6. ~Color! Numerical results similar to those shown in Fig. 5 forv50.192 (2pc/a). ~c! is a plot in the plane ofz50.6as

50.16l ~not through an intensity maximum!.

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gion of projected band structure of an infinite crystal, tmodes are bound photon states guided by the slabwhole; the modes inside the partial photonic band gap aresurface states guided around the air/slab interfaces inphotonic crystal. The field profiles of the symmetric and atisymmetric combinations of the surface modes on thesurfaces with respect to the mirror symmetry plane are ashown in Fig. 3~b!. Deep in the gap where the confinementstrong, the splitting between these two bands becomes sand the two bands merge into one curve in Fig. 3~a!. Thecrystal thicknessh and its associated surface termination psition are chosen so that the frequencies of the surface mlie inside the AANR frequency region with little dependenon the transverse wave vectork—i.e., two flat, nearly degenerate bound photon bands near the frequencyv50.192(2pc/a). This situation thus approximately realizethat in Fig. 2 and is well suited to achieve superlensing.

B. Transmission spectrum

The transmission calculations for arbitrary frequencyvand wave vectork have been performed in the scatterinmatrix approach under Bloch periodic boundary conditiosuch as that of Whittaker and Culshaw for patterned laphotonic structures.46 To compare the results with those o

04511

aheis-oo

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tained from eigenmode computation by plane-wave expsion, we fix the incident wave vector and calculate the fquency spectrum of the transmission. The transmissionpresented on a logarithmic scale in Fig. 4~a!. The pro-nounced peaks in the transmission indicate resonant extion of the bound photon states by the incident radiation, athey approach infinity in the limit of continuous numericsampling points in frequency. From the comparison betwthe transmission peaks and the surface band structure in4~b!, we find excellent agreement between the two numermethods. Near each resonance, the transmission of evacent waves reaches large amplification values well exceeunity, providing the basis of superlensing.

An unusual feature to notice in the transmission spectrunot expected in a uniform negative-index slab, is the prence of zeros in the transmission when the frequency ithe photonic band gap. These zeros occur as sharp dipthe spectrum, and the number of these dips increases withslab thickness in a manner suggestive of Fabry-Perot inferences. However, they cannot be explained by the convtional Fabry-Perot interference, which only occurs in the dnominator of Eq.~2! and never leads to a zero. Instead, thoccur as a result of interference between the different enescent eigenmodes of the photonic crystal in Eq.~1!,coupled together through the phase matrixTk,p . Zeros occur

5-9

LUO, JOHNSON, JOANNOPOULOS, AND PENDRY PHYSICAL REVIEW B68, 045115 ~2003!

FIG. 7. ~Color! Numerical results similar to those shown in Fig. 5 forv50.191 (2pc/a). ~c! is a plot in the plane ofz5as50.27l ~notthrough an intensity maximum!.

se

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because inside a photonic band gap, several eigenmodehave equal amplitude and cancel each other exactly. Thzeros are apparently quantitative new features in the trmission spectrum, but since they are only at a discrete nber of frequency points, they do not change the qualitastructure of imaging. The quantitative effects of transmisszeros are included in the numerical results of superlensinthe next section.

C. Image patterns of a superlens

We calculate the transmission as a function of incidwave vectork at a fixed frequencyv close to that of thesurface modes, using the method of Sec. IV B. The comptransmission data for these plane waves are then summeachz values as in Eq.~10! to obtain the image pattern froma point-dipole source placed on thez axis at z52h2u.Hereu50.1as is used in these calculations, and thek sam-pling points range fromk525(2p/as) to k55(2p/as) insteps of 0.001(2p/as), to model the continuous range o2`,k,`. This finite resolution roughly corresponds tofinite transverse overall dimension of the structure of 10periods and is sufficient for illustrating the various appeances of the image.47 Our results are summarized in Figs.6, and 7. The frequency of operation is shifted by on0.001(2pc/a) from one figure to the next. In the all case

04511

canses--

enin

t

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here, the transmission for propagating waves is nearlysame and close to unity. However, large differences infield patterns forz.0 can be observed in the results, indcating that a fine control over the transmission of evanescwaves is possible.

For v50.193(2pc/a) ~Fig. 5!, the operation frequency isoutside the frequency range of the ‘‘flat’’ surface bands. Ttransmission results in Fig. 5~a! show smooth behaviothroughout the range of wave vectors and exhibit zerosnoted in Sec. IV B. Notice that the magnitude of transmsion oscillates around order unity for the evanescent wafor 0.27(2p/as),k,0.73(2p/as), but for k.0.73(2p/as)the transmission drops precipitously to a low level below31023. This confirms our previous expectation that t‘‘flat’’ bound photon band below the light cone (0.2,kas/2p,0.73) should lead to amplified transmissionevanescent waves and that the amplification effect shodisappear when the evanescent wave is coupled back intocontinuum. In the calculated image shown in Fig. 5~b!, aclear intensity maximum atx50,z50.6as50.16l in freespace can be observed. Quantitative cross sections ofmaximum in both thex and thez axes are shown in Figs. 5~c!and 5~d!, respectively. The transversex size of this peak is0.66l,l, demonstrating that the contribution of evanescwaves to imaging is comparable to that of propagat

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SUBWAVELENGTH IMAGING IN PHOTONIC CRYSTALS PHYSICAL REVIEW B68, 045115 ~2003!

waves. This situation, however, still possesses an intenmaximum and is therefore in the moderate subwavelenregime. The imaging pattern is similar to what we obtainpreviously using the finite-difference time-domain~FDTD!method,12 in which an intensity maximum was identifiewith the location of a real image. Meanwhile, in the prescase, the geometric image location of AANR calculated frconstant-frequency contours of this photonic crystal is az51.4as50.38l. In the constant-z plane of the AANR image,the intensity distribution inx is similar to that in Fig. 5~c!,with a transverse sizeD50.71l. This value corresponds tan effective high cutoffkM'1.4v/c50.38(2p/as), which isclose to the thresholdkM value in the simplified model obtained in Sec. III. For the present position of the AANimage, we plot the amplification required to restore tsourceperfectlyin gray lines in Fig. 5~a!, which can be com-pared to the actual transmission data and the effective cukM . Although the imaging is not perfect, the rangev/c,k,kM roughly indicates the interval in which the actual tranmission follows the behavior in the ideal case. SincekM isactually larger than the wave vectors of several transmiszeros in the spectrum, we have confirmed that these zeronot have a significant influence on superlensing.

If v is decreased slightly tov50.192(2pc/a) ~Fig. 6!,the frequency falls inside the narrow range of the surfmode frequencies. The transmission increases dramaticand pairs of peaks in the transmission spectrum occur,resenting excitation of surface mode combinations of eand odd parity. These surface modes have large amplituas evidenced by the compressed color table in plotting6~b! and the exponential decay of intensity alongz axis inFig. 6~d!, and they now completely dominate the image.accordance with our discussion before, the focusing effecpropagating waves becomes insignificant against this strbackground. If the field distribution in a plane of constanz

FIG. 8. The detailed surface band structure and its influencesubwavelength imaging. The solid circles are the divergence pin the calculated transmission through the structure in Fig. 4.dark-shaded area in the upper left corner is the light cone.frequency range for imaging with none or moderate subwavelencontribution is from 0.1928 to 0.1935. The frequency rangeextreme subwavelength superlensing is from 0.1905 to 0.1911.region between them is the region of the flat surface bandsenhanced surface resonance. The sequence of these three freqranges here are due to the particular shape of the surface bandcan be different in other systems.

04511

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is measured, an example shown in Fig. 6~c!, many closelyspaced, near-periodic strong peaks occur, in striking contto the familiar appearance of a focused optical image. Hthis pattern of intensity distribution persists for increasingzin the near field and even appears on the focusing planAANR z51.1as50.31l. Due to the exponential decay ointensity along thez axis and thedelocalizedfield distribu-tion in the transverse direction, neither thez coordinate northe transverse location of the source can be easily retriefrom this image pattern. The present image field patternhence undesirable for imaging purposes and should insbe exploited in situations where enhanced intensity inextended spatial region is preferred. We infer the effectcutoff wave vector by the width of the central peakD51.8as50.49l on the plane of AANR image, and obtaikM50.56(2p/as), as marked out on Fig. 6~a! where thetransmission curve forperfect image reconstruction is alsplotted. It is evident that the actual transmission deviasignificantly from the ideal case, which explains the noncused image pattern.

An image pattern with intermediate behavior betwethese two situations can occur, for example, if we takev tobe v50.191(2pc/a) ~Fig. 7!. This frequency is outside the‘‘flat’’ surface band frequency range, and consequentlytransmission becomes smooth again. Amplified evaneswaves are still present in the image space, which createexponentially decaying intensity profile along thez axis asshown in Fig. 7~d!. In contrast to the case in Fig. 6, a distinintensity peak can now appear within a plane of constanzshown in Fig. 7~c!, with a size significantly smaller than thwavelength. Here we have actually achievedD50.45l at z5as50.27l, approximately the same location as predictby AANR. This image size is in accordance with the geneprediction of Eq.~12! whereasl/(l2as)50.37l and 2as50.54l for the present photonic crystal. We infer the higcutoff wavelength kM in this case to bekM52.2v/c50.6(2p/as), which corresponds to the extreme subwavlength limit. In this limit, there is no intensity maximum iz.0, consistent with our previous expectation, and the cculated transmission fork,kM also displays roughly thesame trend ink dependence as the ideal transmissionperfect image recovery. We have therefore found a superling image pattern quite similar to that considered in toriginal perfect lens proposal,5 in the present case with aupper cutoff, without requiring negative-index materials.

To summarize, these computational results establishsuperlensing is possible with carefully designed photocrystals and exhibit large modifications to the image fiedistribution due to the presence of evanescent light. To pserve an intensity maximum in the image space, only a mest amount of evanescent waves can be included in the merate subwavelength regime, and the resolution ofresulting intensity maximum is limited by a fraction of thwavelength~Fig. 5!. Alternatively, it is also possible to enhance the transmission of evanescent waves to such an ethat they dominate over propagating waves~Fig. 6!. In thiscase of enhanced surface resonance, a focused image dpears completely and the field distribution becomes thathe resonantly excited, delocalized surface modes. Furtmore, a scenario qualitatively similar to that of a left-hand

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t

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LUO, JOHNSON, JOANNOPOULOS, AND PENDRY PHYSICAL REVIEW B68, 045115 ~2003!

FIG. 9. ~Color! Numerical re-sults of the imaging for variousfrequencies throughout the firsphotonic band for the structure inFig. 3. ~a! Intensity distributionalong the transverse directioncommonly measured atz50.5as

for several frequencies shown ainsets. Thisz value is chosen forexhibition of large near-field ef-fects at certain frequencies@e.g.,v50.145(2pc/a)]. The trans-verse intensity distribution alarger z values has a similar-shaped background but weakenear-field modulations.~b! Inten-sity distribution along thez axisfor the shown frequencies. In botpanels the inset numbers are thfrequencies corresponding to eaccurve, in units of (2pc/a).

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material withe521 andm521 can be obtained~Fig. 7!,with an evanescent field distribution containing a localizintensity peak in the focusing plane of AANR. The minimusize of this peak in this extreme subwavelength regimelimited only by the surface periodicity of the photonic crysand not by the wavelength of light and is in principle onlimited by the refractive index. Finally, our calculations idicate that in the present structure, each of these imageterns can appear in a narrow frequency range inside thaAANR, as indicated in the detailed surface band structureFig. 8. We conclude that the intensity distribution of the otical image formed by a superlens depends sensitively ondetailed balance between propagating and evanescent wand can be tuned with great flexibility in photonic crystal

V. DISCUSSION

For completeness, we show in Fig. 9 the calculated nfield intensity distributions inz.0 for a point source of vari-

04511

d

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ous frequencies throughout the first photonic band, withother parameters the same as those in Sec. IV C. It is cthat for frequencies lower than the AANR range„v5(0.050,0.100,0.145)(2pc/a)…, since most of the propagating waves do not experience negative refraction and arefocused, a broad background peak is always present intransverse direction. An interesting feature to observe isv50.145(2pc/a) is close to the band edge where there amany flat bands of guided photon bound modes that canresonantly excited. Consequently, significant subwavelensurface resonance features appear on the broad backgrbehind the slab. However, the overall resolution is ndetermined by the background, which is spatially broand does not correspond to a subwavelength imaging efFor frequencies above the AANR range„v50.195,0.210(2pc/a)…, since some of incident propagatinradiation from air will experience total external reflectiothe transverse resolution is always limited to be larger thor equal to the operating wavelength. All these can be co

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SUBWAVELENGTH IMAGING IN PHOTONIC CRYSTALS PHYSICAL REVIEW B68, 045115 ~2003!

FIG. 10. ~Color! Calculated transverse intensity distribution for imaging with lossy photonic crystals. Each inset number correspthe permittivity of the dielectric host for the curve of the same color. The crystal and point source are otherwise identical to those iThe intensity is plotted in the planez5as at the frequencyv50.191(2pc/a).

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pared tov50.193(2pc/a) where the extraordinary supelensing enhancement in both the imaging resolution andtensity is shown. From this analysis, we conclude thattheonly frequencies at which one can observe superlensinginside the AANR range and close to a flat surface band.

The above discussion has focused on ideal situationsno material absorption of light or structural imperfections.practice, material losses are always present, which methat no transmission considered here will be truly infinite.general, appreciable material losses will impose severe ltations on the transmission coefficient of evanescent wain a manner similar to that of the intrinsic energy leakarate of a crystal mode above the light line, which in tureduces the superlensing effect. However, it is also expethat, in the limit of extremely small material loss, in thsense implied by the original proposal of perfect lens,5 ourfindings about the image of a superlens will remain valid.an example, we show the calculated focusing effectslightly lossy photonic crystals in Fig. 10. The lossesmodeled as a positive imaginary part on the permittivitye ofthe dielectric host, and results are calculated at the extrsubwavelength frequencyv50.191(2pc/a) for e startingfrom e51210.01i up to 1210.05i . As the losses increasethe strength of the transmitted near fields is attenuated,the subwavelength features in the central image peak grally disappear. It is clear that a resolution at or belowD50.5l for a localized intensity peak inx is still achievable ife<1210.01i . The effects of surface imperfections on suwavelength imaging can also be qualitatively analyzed.consider these defects to occur only on a length scale th

04511

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smaller than a lattice constant and, thus, much smaller tthe operating wavelength, with correspondingly little inflence on propagating waves. Since the transmission ofnescent waves depends sensitively on the bound surfaceton states, which in turn depend sensitively on the surfstructure, imperfections are expected to be most influenon the crystal surface. Their effects may thus be minimizby improving the surface quality. Another kind of structurimperfection is a finite lateral size of the crystal. We haapplied the FDTD method to such finite systems and fouthat, for a 20-period-wide slab, a focusing resolution arouD50.6l can be still obtained. These considerations suggthat the effects described in this paper should be observin realistic situations.

Our discussion on the image of a 2D superlens can beto experimental verification in a manner similar to that sugested in Ref. 12. Because the superlensing occurs in thephotonic band, it should also be directly applicable to 2photonic-crystal systems suspended in 3D.48 A more interest-ing extension of these phenomena would be to a fullsystem, which requires much more intensive computationnumerical modeling. For example, in 3D the resolutionfocusing with infinite aperture but without evanescent wavis still limited by the wavelengthl, while the surface peri-odicity discussed in Eq.~12! should be replaced by the reciprocal of the minimum radius of the surface Brillouin zonWe show here in Fig. 11 the results of the computed bouphoton modes of a slab of a 3D photonic crystal proposeRef. 15. As discussed in detail there, this photonic crysenables AANR in full 3D and is most effective for wave

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LUO, JOHNSON, JOANNOPOULOS, AND PENDRY PHYSICAL REVIEW B68, 045115 ~2003!

polarized along (101). The surface band structure alongGKandGM computed here, complicated as it may seem at fisight, bears a striking similarity to the TE and TM slab plariton bands of a dispersive negative-index materials calated by Ruppin49,50 when the direction of light polarizationis taken into account. For the particular surface terminatshown in Fig. 11, it is possible to obtain surface states witthe AANR range of this photonic crystal. Since there is sa vast amount of freedom in tuning the fine details of

*Electronic address: chiyan@mit.edu1V.G. Veselago, Sov. Phys. Usp.10, 509 ~1968!.2J.B. Pendry, A.J. Holden, W.J. Stewart, and I. Youngs, Phys. R

Lett. 76, 4773~1996!.3H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamu

T. Sato, and S. Kawakami, Phys. Rev. B58, 10 096~1998!.4J.B. Pendry, A.J. Holden, D.J. Robbins, and W.J. Stewart, IE

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FIG. 11. Bound photon modes and projected band structurea 3D photonic crystal capable of AANR. The solid lines are bouphoton modes, and the dashed lines are the outlines for projephotonic band structure on the surface Brillouin zone. The ligshaded region is the light cone. The dark-shaded range is the AAfrequency range. A cross section of the crystal and the surfacelouin zone are shown as insets. The thickness of the photocrystal slab ish53.47a.

04511

st-u-

nnle

crystal surface structure without breaking the periodicitycan be further expected that particular designs exist whlead to flat surface bands and can enable superlensing in3D. This tunability and flexibility in our approach shoulmake photonic crystals a powerful and beautiful candidatemanipulating and focusing light on subwavelength scaespecially in the optical regime.

VI. CONCLUSIONS

In conclusion, we have explained the principles of tamplified transmission of evanescent waves and superlenin general photonic crystals and presented specific designsuperlenses based on AANR in photonic crystals as wellcomprehensive numerical study of their subwavelengthaging properties in 2D. Special emphasis is given to thecusing resolution and image patterns of these devices,our studies demonstrate that the interplay between propaing and evanescent waves can lead to various image beiors not possible with conventional lenses in geometrictics. We hope that this work should clarify the underlyinphysics of near-field imaging and stimulate interests ofperimental studies of superlensing.

ACKNOWLEDGMENT

This work was supported in part by the Material ReseaScience and Engineering Center program of the NatioScience Foundation under Grant No. DMR-9400334 andDepartment of Defense~Office of Naval Research! Multidis-ciplinary University Research Initiative program under GraNo. N00014-01-1-0803.

v.

,

E

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z,

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