Left-handed metamaterial coatings for subwavelength-resolution imaging

Transcription

1 99 J. Ot. Soc. Am. A / Vol. 9, No. 9 / Setember Zaata-Rodríguez et al. Left-handed metamaterial coatings for subwavelength-resolution imaging Carlos J. Zaata-Rodríguez,, * David Pastor, Luis E. Martínez, and Juan J. Miret Deartment of Otics, University of Valencia, Dr. Moliner 5, 46 Burjassot, Sain Deartment of Otics, Pharmacology and Anatomy, University of Alicante, P.O. Box 99, Alicante, Sain *Corresonding author: carlos.zaata@uv.es Received May 7, ; revised July, ; acceted July 8, ; osted July 3, (Doc. ID 6838); ublished August 3, We reort on a rocedure to imrove the resolution of far-field imaging by using a neighboring high-index medium that is coated with a left-handed metamaterial. The resulting lot can also exhibit an enhanced transmission by considering roer conditions to retract backscattering. Based on negative refraction, geometrical aberrations are considered in detail since they may cause a great imact in this sort of diffraction-unlimited imaging by reducing its resolution ower. We emloy a standard aberration analysis to refine the asymmetric configuration of metamaterial suerlenses. We demonstrate that low-order centrosymmetric aberrations can be fully corrected for a given object lane. For subwavelength-resolution imaging, however, high-order aberrations become of relevance, which may be balanced with defocus. Not only the oint sread function but also numerical simulations based on the finite-element method suort our theoretical analysis, and subwavelength resolution is verified in the image lane. Otical Society of America OCIS codes:.99, 6.398,... INTRODUCTION An asymmetric flat suerlens is a film made of a left-handed metamaterial (LHM) that is deosited on a smooth, transarent body such as glass with ositive dielectric constant. Therefore the object sace has an index of refraction (IR) different from that in the image sace, leading to an asymmetric arrangement. Originally, this idea was conceived because using a solid substrate, this imaging device will be mechanically much more stable than a layer sustained in free sace []. There, only asymmetric silver suerlenses were analyzed in detail, for which amlification of evanescent waves sustained by surface waves is more favorable if the real art of the dielectric constant of the metal and the substrate matches excet for its sign. In the case that the lensing flat slab shows effectively a negative ermeability, negative refraction allows imaging mainly using homogeneous waves. Moreover, if the IR of the outut medium is higher than that IR corresonding to the medium surrounding the object, some evanescent waves emitted by the source become homogeneous after assing through the lens. This fact allows the formation of far-field images with subwavelength resolution. Unfortunately there is no erfect image lane in the asymmetric arrangement and the image suffers from aberrations. Note that the root of aberrations in symmetric suerlenses is diverse and may be caused by materials having an imedance (and IR) not matched to free sace [,3], material losses [4,5], because the equifrequency curve is slightly deformed from an ideal sherical shae articularly for large angular comonents [6,7], and caused by the anisotroic effect from nonmagnetic anisotroic media [8,9]. Here we focus on the IR mismatching that comes naturally in the asymmetric configuration. Moreover, Seidel aberrations have been discussed in different kinds of imaging nanostructures like metallodielectric hotonic crystals (PCs) [,], negative-refractive lenses fabricated out of a silicon-on-insulator PC slab [], graded PC lenses [3], and sherical lenses comosed of LHMs [4,5]. In the revious examles, nonaertured suerlenses are usually considered and oblique aberrations may be disregarded. Finally, in order to avoid this awkward situation, a transformation design of otical elements that erform imaging has been roosed, free from geometric aberrations [6], though it seems far from being exerimentally demonstrated. In this aer we investigate the effects of rimary sherical aberration (SA) and higher-order SA in LHM asymmetric lenses. In Section, rimary SA is corrected for a given object lane; however, some residual aberrations for nearby object lanes remain. This effect is not observed in erfect symmetric lenses since it is caused by the breaking of the shift invariance along the otic axis driven by asymmetry of this new imaging roblem. In Section 3 we show that the unbalanced arrangement allows roer conditions to retract backscattering. The diffractive behavior of SA-corrected metamaterial coatings are disclosed in Section 4 by using electric diole fields. Here we demonstrate subwavelength caabilities in far-field imaging. In Section 5 we rovide the oint sread function (PSF) of such antireflection suerlenses and we estimate the limit of resolution unambiguously. Finally, the main conclusions are outlined in Section 6.. SPHERICAL ABERRATION IN FLAT LENSES Let us consider the asymmetric flat lens shown in Fig.. A oint object O is susended at a distance s from the front face of the suerlens made of a material exhibiting negative IR, n <. Assuming that the object sace is characterized by //999-7$5./ Otical Society of America

2 Zaata-Rodríguez et al. Vol. 9, No. 9 / Setember / J. Ot. Soc. Am. A 993 n n n 3 P h O σ Q O Q O 3 σ σ 3 h s d s 3 P s3 /d Fig.. (Color online) Schematic reresentation of an asymmetric flat lens of refractive index n and width d. a6 d 5 a4 d 3 an IR n >, and that the width d of the lens is sufficiently large, a real Gaussian image O is formed inside the LHM [7]. Traveling through the lens exit surface we reroduce the secondary, outlying image at O 3 in a semi-infinite dielectric of IR n 3. Provided the IR in the image sace turns out to be ositive, n 3 >, the image oint O 3 is also real, which is located at a distance s 3. Note that we emloy oriented axial distances, i.e., s < and s 3 > for a real object and a real image, resectively. Also the lens width d>. For a nonaertured setu, the chief ray joins the oints O, O and the Gaussian image O 3 by means of the same straight line. Let us evaluate the aberration of a ray assing through the oint P, which is laced on the exit surface at a height h, with resect to the chief ray. This aberration is estimated by the otical-ath difference of both light rays, i.e., W n O P O Q n P P Q Q n 3 P O 3 Q O 3. Note that O Q s, Q Q d, and Q O 3 s 3. Finally, this ray aberration reads aroximately [8] W h a h a 4h 4 a 6h 6 : () The aberration terms a, a 4, and a 6 are attributed to defocus, rimary SA, and fifth-order SA, resectively. These aberration coefficients are evaluated by using the geometrical relations tan σ h s and tan σ h h d, and the Snell law n sin σ n sin σ : () In articular, the Gaussian image lane is given under the condition a, which yields s s 3 n 3 d : (3) n n Therefore an axial dislacement of the object oint O changing s leads to an image shift following a direct roortion, as shown in Fig.. Note that a real araxial image O 3 is attained with the condition d s n n, that is, if the secondary araxial image O is also a real image. At the Gaussian image oint O 3, where Eq. (3) is satisfied, the aberration coefficient for rimary SA gives a 4 n n n 3 n n 3 d n3 n 3 n s 8n 3 n d n s 4 : (4) Equation (4) is obtained from Eq. () by using a Taylor exansion of W around h. This result is slightly different if the aberration is calculated surface by surface, and the aberration -4. s /d Fig.. (Color online) Geometrical imaging for a flat lens of n sandwiched between dielectric media of indices of refraction n and n 3 4. Gaussian imaging based on Eq. (3). Red line reresents rimary SA given by Eq. (4) and blue line reresents fifth-order SA given by Eq (6). of the flat lens is obtained by adding the aberration contributions of all its surfaces [8]. Note that rimary SA cannot be totally corrected for s < when n n 3 excet for the erfect lens, where additionally n n 3. This is a wellknown case where high-order aberration coefficients also vanish leading to stigmatic imaging. Also a lane-arallel asymmetric late may be corrected of rimary SA. Provided the equation a 4 is satisfied, we obtain a linear relationshi between the lens width d and the on-axis object distance s n3 n 3 n n 3 n 3 d; (5) n in terms of the IRs of the media involved. A given flat lens cannot be corrected of rimary SA for more than one object lane, as shown in Fig., and therefore images originated from scatterers that fail to kee Eq. (5) suffer from SA. Furthermore, the rimary SA coefficient a 4 diverges for the limiting case s 3, exceting when n n, leading to erfect geometric imaging. Therefore quality of the (real) image imroves as the (real) object oint O come close to the inut surface. In Fig. 3, we lot a ray tracing for a flat metamaterial lens of n surrounded by object and image media of IRs n and n 3 4. Fixing the lens width d, Eqs. (3) and (5) rovide the values s 3.6d and s.d, resectively. The corresonding ray tracing is shown in Fig. 3. Stigmatic imaging may roduce a convergent focused beam of numerical aerture n 3 sin α n that leads to an angular semiaerture α 4.5. In our case, the numerical aerture is slightly reduced down to an effective value α eff.9 caused by noncorrected high-order aberrations. To insect the deterioration of the image due to SA effects, we also resent in Fig. 3 the ray tracing for a oint object laced at s.4d further from the lens entrance surface. We observe a ray distribution that is barely confined around the Gaussian image oint O 3, reresented as a green dot in the image sace. As a limiting case, we lot in Fig. 3 the trajectories of rays emerging from a oint that is located at s.5d that leads to j a 4 j. We oint out that Eq. (5) gives a negative value of s rovided that the IR n 3 in the image lane is either higher.6

3 994 J. Ot. Soc. Am. A / Vol. 9, No. 9 / Setember Zaata-Rodríguez et al. LHM deosited on to of the transarent substrate. The coefficient of reflection (r) and the coefficient of transmission (t) evaluated from the object lane in front of the asymmetric layered lens to the image lane are [] r r ; r ;3 ex iβ d r ; r ;3 ex iβ d ex iβ s ; (7a) LHM t t ;t ;3 ex iβ s iβ d iβ 3 s 3 : (7b) r ; r ;3 ex iβ d Here the roagation constant is q β i σ i ε i μ i k k k ; (8) LHM Fig. 3. (Color online) Ray tracing for an object oint located at s.d, which is corrected of rimary SA, s.4d, and s.5d from the front surface of a flat lens of width d. Indices of refraction are the same as in Fig.. Traces corresonding to araxial (sloe lower than 3 ) and nonaraxial rays are drawn in different colors. Small light-colored stars reresent conjugated oints. or lower than n and jn j simultaneously. In order to achieve a subwavelength effect, we aim for transforming evanescent waves emitted by the source O into homogeneous wave modes in the image sace. In this case it is referable for a high-index transarent medium n 3 >n to register the image. The aberration coefficient a 6 for the fifth-order SA may be estimated analytically. Provided Eqs. (3) and (5) are satisfied, this high-order SA coefficient gives a 6 n3 n n 3 n n 3 6 6n n4 3 n n 5 s 5 : (6) Therefore, assuming n n 3, a 6 vanishes only for the following trivial solutions: () imaging under mirror-symmetry negative refraction on the outut surface when n 3 n and s, and () incidence of a collimated bundle of rays for which s tends to infinity. Exceting these secial cases, fifth-order SA cannot be corrected. In Fig. we resent j a 6 j for a given numerical examle. To conclude this section, let us make an exlanatory remark concerning the IRs of the media involved on the analysis of the image formation. The ratio n 3 n rovides the relative enlargement of satial bandwidth corresonding to evanescent waves in the object sace that are transformed into homogeneous lane waves in the image sace. This is clearly a subwavelength effect, which has been exloited elsewhere [9]. In image formation, this hysical henomenon leads to a suerresolving effect, which will be develoed in Section 5. On the other hand, we oint out that the value of n is arbitrarily chosen rovided it takes a negative value. In fact, this is a degree of freedom that may be rofited at the time of imosing an additional constraint of interest. 3. CONTROL OF REFLECTION LOSSES In order to take a suitable choice for the value of n, we considered the reflection and transmission roerties of light that iminges obliquely onto the LHM thin film, which has been where σ i for the dielectrics and σ for the LHM, k π λ is the wavenumber in vacuum, and ε i and μ i stand for the relative ermittivity and ermeability of the media involved, resectively. Also k k x ;k y is the transverse wave vector, i.e., the rojection of the wave vector of the incident field over a lane that is arallel to each flat-lens interface. The Airy s formulae (7) deend on the reflection coefficient at a single interface, r i;j μ jβ i μ i β j μ j β i μ i β j ; (9) which is valid for s-olarized waves. For -olarized waves, r i;j ε j β i ε i β j ε j β i ε i β j alies directly to the transverse magnetic field. In all cases, they also deend on the transmission coefficient t i;j r i;j. A LHM layer with β d m π ; () for m ; ; can be used to eliminate the reflection of light comletely, which is intrinsically a disersive henomenon deending uon k. This is commonly denominated an antireflecting coating. For that urose we additionally imose r ; r ;3 in Eq. (7a), leading to r. This condition is satisfied if β μ μ 3 μ β β 3 ; () assuming that the wave fields are s-olarized. For normally incident light, i.e., k, Eq. () is simly Z Z Z 3, where Z i is the intrinsic imedance of the medium i. Obviously, this equation for zero reflectance is a result that alies for all states of olarization. Moreover, assuming that μ i σ i, we finally obtain a condition n n n 3 involving the IRs of all media. Note that the latter equation is held in simulations shown in Fig., and it is well-known in the theory of antireflecting films when its IR is jn j. Finally, a quarter-wave layer satisfying Eq. () with d λ 4 mλ for m, being λ λ n, is of interest. In Figs. 4 4, we show the transmission coefficient t that has been evaluated for s-olarized waves and suerlenses of different widths. Otimum geometrical conditions are assumed under all circumstances, where Eqs. (3) and (5) are satisfied. We observe that jtj.5 for k in all cases

4 Zaata-Rodríguez et al. Vol. 9, No. 9 / Setember / J. Ot. Soc. Am. A k /k 8.7 m=3 m=. k /k 6.5 m= t t. k /k k /k 8 (e) m=3 k /k 6 (f) m= k /k and, therefore, reflection is extinguished (Transmittance for s-olarized waves is T s jtj n 3 n ). However, reflectance might have a certain significance in higher satial frequencies. We also observe a flat variation of the argument of t for k <k, which is a consequence of eliminating rimary SA. For ultrathin slabs, however, jtj is of relevance at higher frequencies. By considering a boundless medium of IR n, note that there is no time-averaged ower flow for k >k. The field intensity within this sectral domain, in the resence of the suerlens, is by no means zero, and transmittance might reach values higher than unity, as seen for m. In these cases, the time average of the ower flow in the object sace is suorted artially by evanescent waves that, in rincile, can contribute to the far field. For them, the hase in the satial sectrum changes by far and, therefore, aberrated images are exected in the Gaussian image lane. In Figs. 4(d) 4(f), we also resent the transmission coefficient t in amlitude and hase for -olarized waves. Finally, from Figs. 4 and 4(d) we observe that those satial frequencies surassing 4k have a small contribution to the image formation; note that the wave field also falls off fast in the transit from the outut lane of the lens toward the image lane, thus frustrating a three-dimensional (3D) focusing []. For a slab width much higher than the wavelength, the evanescent waves emitted by the source oint O cannot reach the entrance face and homogeneous waves satisfying k k contribute effectively to the transmitted field in the image sace, as shown in Figs. 4 and 4(f) for d 5.5λ. Decreasing d down to values close to λ leads to the conversion of evanescent waves in the medium to homogeneous waves in the medium. In Figs. 4 and 4(e), we observe a critical articiation of waves with transverse satial frequencies k <k < k for a lens width d.875λ. In the limit d.5λ associated with m, we include the satial bandwidth into the interval k <k < 4k involving evanescent waves in media and, which are transformed into homogeneous waves in the image sace, as seen in Figs. 4 and 4(d). For that reason such a sectral stretching allows a subwavelength-resolution effect in the formation of far-field images. (d) m= Fig. 4. (Color online) Transmission coefficient (modulus and argument) for s-olarized waves and (d) (f) -olarized waves in a suerlens of μ i. and ε 4 i.. Surrounding transarent media have again indices of refraction n and n 3 4. We consider different widths for the LHM flat lens: and (d) d.5λ ; and (e) d.875λ ; and (f) d 5.5λ. Note that all the horizontal scales are not the same. 4. IMAGING ELECTRIC DIPOLE FIELDS It is commonly acceted that electric diole fields, due to their high satial confinement, are electromagnetic sources aroriate for the examination of the limit of resolution in nearfield suerlenses [,3]. For that urose we use the field distribution generated by an infinite line source reducing 3D calculations to a simler two-dimensional (D) roblem. The orientation of the line emitter lies along the y-axis, which is arallel to the inut and outut surfaces of the thin LHM coating. The electric diole field results from elementary oint dioles with diole moments,, which are resumed to be aligned also in the y direction, as shown in Fig. 5. In an unbounded transarent medium of ermittivity ε and ermeability μ, the electric field may be written as [4] E i k 4ε H k R ŷ; () where k k ε μ, the unit vector ŷ is oriented along the y- axis, and R x z denotes the distance from the line source to the oint under observation within the lane xz. Also H is the Hankel function of the first kind, which may be written in an integral reresentation as H k R Z ex ik x x iβ z dk π β x ; (3) for z>. Note that k y here. If we consider the electric field that is transmitted through the thin LHM suerlens shown in Fig., assuming that the line diole is laced at O, it finally yields E i k Z ŷ t k 4πε x ex ik xx iβ 3 z dk β x : (4) In this equation we emloy the transmission coefficient t evaluated from the object lane to the image lane, which is given in Eq. (7b), for s-olarized waves. Therefore z stands for the image lane. In Fig. 6, we resent the modulus of the electric field jej that is emitted by an electric line diole and that is transmitted through a LHM lens of μ i. and ε 4 i. and different widths. The flat lens is sandwiched between media of IR n and n 3 4. The object oint O is laced at a distance s.d from the suerlens, following Eq. (5)to minimize rimary SA. We comute the scattered field within the interval z s 3 constituting the real image sace. The I Fig. 5. (Color online) Electric line diole comosed of a continuous distribution of oint dioles that are oriented in the y direction and simulating a current I flowing along the y-axis. E H

5 996 J. Ot. Soc. Am. A / Vol. 9, No. 9 / Setember Zaata-Rodríguez et al. x/λ - - z/s3 - z/s3 - z/s3 AMPLITUDE x=.69 - x/λ - x/λ - x/λ Fig. 6. (Color online) Modulus of the electric field emitted by a line electric diole and transmitted through a negative-index slab with μ i. and ε 4 i. and different widths: d.5λ, d.875λ, and d 5.5λ. In all cases we resent the field within z s 3. The density lots are normalized to unity at the araxial image oint x; z ;. The dashed line indicates oints where amlitude falls off. The thin vertical line marks the Gaussian image lane. The wave fields corresonding to the Gaussian image lane for,, and are lotted in (d), (e), and (f), resectively. numerical simulations were erformed using a finite-element method and also emloying Eq. (4) to verify the validity of our results. If the suerlens has a width d below the wavelength, as used in Fig. 6, one would exect to achieve suerresolution. In this case, however, the FWHM of the modulus of the electric field in the Gaussian image lane yields Δ x.69 in units of λ, even exceeding the limit of resolution alied to diffraction-limited systems. In order to understand such a behavior, we analyze the transmission coefficients lotted in Fig. 3. We observe that the hase is stabilized for jk x j <k, but it has a fast decreasing variation for higher satial frequencies. In this sense, the effective bandwidth in the transmission coefficient for frequencies associated with inhomogeneous waves in the medium is, in ractical terms, three times larger than the bandwidth for homogeneous waves; therefore the unbalanced contribution of the different satial frequencies will make the image recovery difficult. A simle defocus rovoked by a shift of the image lane toward the LHM lens serves to diminish the hase variation and aberration effects in the image. Exactly at the exit surface of the suerlens, the FWHM of the electric field is Δ x.86, leading to a subwavelength resolution. A different behavior is exected for d λ. Figure 6 illustrates the diffraction behavior of a thin LHM film in the case that d.875λ ; that is, m 3 in Eq. (). The limit of resolution has decreased substantially in the Gaussian image lane, where Δ x.38. Taking in mind the results shown in Fig. 3, this suerresolving resonse is attributed to evanescent waves in medium that are converted into homogeneous waves in medium, which belong to the sectral range k < jk x j < k. In this sectral band, however, the coefficient of transmission resents some strong variations in its hase, reventing us from the observation of an aberration-free image. Moreover, the hase of the transmission coefficient increases with k x so that a defocus is exected to balance high-frequency aberrations. Contrarily to the revious case, (d) (e) (f) x=.38 x=.56 the on-axis shift must be erformed moving far from the LHM lens in order to achieve the minimum sot size. In ractice, the resolution imrovement that is attainable with defocus may be considered negligible. Finally, if d 5.5λ, as shown in Fig. 6, the FWHM of the central lobe in the araxial image lane yields Δ x.56 in units of λ, which is close to the diffraction limit, λ. In this case, the deth of focus is significantly short, which allows the evaluation of the FWHM along the z-axis. This gives Δ z 9.8, also in units of λ. We conclude that the focused wave field is localized much stronger in the transverse direction than on axis. The analysis that we have carried through in Section 4 so far is essentially for s-olarization. We oint out that a similar -olarization analysis is also ossible based on magnetic dioles aligned along the y-axis. For that urose, now we consider a line source with a uniform distribution of elementary magnetic diole moment m, which is laced at O, and for which the magnetic field in the image sace is simly H i k m Z ŷ t k 4πμ x ex ik xx iβ 3 z dk β x : (5) Making use of the duality theorem [4], here t from Eq. (7b)is evaluated for -olarized waves. Again z stands for the image lane. In Fig. 7, we resent the modulus of the magnetic field as it is transmitted through the same LHM lens of μ i. and ε 4 i.. By comaring the wave field in Fig. 7 with the scattered field reresented in Fig. 6, for different widths of the LHM slab, we conclude that discreancies are areciable, in general, which are clearly attributed to deartures in the coefficient of transmission t for both olarizations. Also we find that the sot size of the wave field is lower for magnetic dioles. For convenience, let us leave in Section 5 the discussion concerning how olarization imacts uon the limit of resolution of LHM coatings. 5. PSF AND LIMIT OF RESOLUTION In order to estimate the limit of resolution unambiguously, we follow a different aroach that is based on the imulse x/λ - - z/s3 - z/s3 - z/s3 AMPLITUDE x=.36 - x/λ - x/λ - x/λ Fig. 7. (Color online) j Hj from Eq. (5) as it is evaluated in the image sace of a LHM flat lens with μ i. and ε 4 i. and different widths: d.5λ, d.875λ, and d 5.5λ. Again, the fields in the Gaussian image lane for,, and are lotted in (d), (e), and (f), resectively. (d) (e) (f) x=.38 x=.5

6 Zaata-Rodríguez et al. Vol. 9, No. 9 / Setember / J. Ot. Soc. Am. A 997 resonse of the otical system, also known as the PSF. This rocedure may be carried out assuming that the imaging device under consideration is a linear and shift invariant (LSI) system. Note that a scattered field transmitted through a stack of dielectric-lhm layers remains unchanged in shae and magnitude if the scatterers are dislaced in any direction that is arallel to the surfaces. Therefore the asymmetric LHM flat lens that we are considering is a LSI system in D. Moreover, the imulse resonse of the LHM suerlens deends on the state of olarization of the wave field and, therefore, we may evaluate the PSF for -olarized and s-olarized waves, indeendently. This is discussed thoroughly in [5], and here we only give a brief summary. Based on the angular sectrum reresentation of the scattered field, for s-olarized waves, the electric field E in the image sace may be exressed as a D convolution, E R ;z E sc R h R ;z ; (6) where E sc is the wave field at the object lane and R x; y stands for a sace-domain D vector that is erendicular to the unit vector ẑ. Strictly seaking, Eq. (6) is not restricted to scatterers that are located at the object lane. The scalar 3D PSF is h R ;z ZZ π t k ex i k R iβ 3 z d k ; (7) and is derived by using the transmission coefficient that corresonds to the object lane and its conjugate image lane as given in Eq. (7b). Note that h R ;z> for t reresents the roagator of the first Rayleigh Sommerfeld integral, and it is related with a divergent wave whose focus is laced in the center of the image lane z [6]. In fact, for symmetric flat lenses where n n 3, Eq. (6) may be set as a 3D convolution by virtue of its roerty of shift invariance along the z-axis [5]. However, asymmetric flat lenses are not invariant under dislacements on the otic axis. This is in agreement with our discussion in Section concerning the correction of rimary SA, which is achieved in a unique object lane. In order to comare the PSF with the electric-field resonse of the LHM lens over a line electric diole, calculated from Eq. (4), it is more aroriate to derive the PSF in D; that is, h x; z Z t k π x ex ik x x iβ 3 z dk x : (8) In Fig. 8, we resent the D PSF for the same suerlens considered, for instance, in Fig. 6. One more time, the object lane is laced at a roer distance s to comensate rimary SA. We find again that the diffractive behavior of a slab width d below the wavelength differs substantially from that lens with d λ. Additionally, the imulse resonse is notably different for s-olarized waves and -olarized waves. Note that the PSF for -olarized waves is comuted by using in Eq. (8) the corresonding coefficient of transmission t. The FWHM of the PSF central lobe for s-olarized waves takes higher values than those evaluated for olarization, esecially in ultrathin LHM layers. For instance, if d.5λ, x/λ - - z/s3 - z/s3 - z/s3 (d) (e) (f) m= m=3 m= x/λ x,s=.83 m= x,s=.8 m=3 x,s=.68 AMPLITUDE m= - x/λ - x/λ - x/λ shown in Fig. 8(g), the FWHM of the PSF in the Gaussian image lane yields Δ x;s.83 in units of λ, for s-olarized waves, which is much higher than the FWHM encountered for -olarized waves, Δ x;.3. However, differences derived by the state of olarization are negligible in the case d 5.5λ, as shown in Fig. 8(i). From Figs. 8 and 8(f), we conclude that this is true not only in the Gaussian image lane but also in out-of-focus lanes. Note that the FWHM from the PSF is slightly greater than that obtained in Section 4 from a line diole. It is worthy to oint out that a similar effect has been reorted when comaring the PSF and the image of a subwavelength Gaussian beam in metal-dielectric multilayers [7]. In both cases, the exlanation is nevertheless not difficult. The broader PSF has an irregular hase variation, not shown in Fig. 8, which is of critical relevance in the convolution given in Eq. (6). In a similar manner, fast changes in the hase of an incident wave field may lead to severe distortions in the image sace. Obviously, hases of inut fields and hases of PSFs would not lay a role if the undulatory suerosition (6) were fully incoherent. As a consequence, subwavelength signals transmitted by LHM coatings occasionally yield anomalous localized distributions whose FWHMs surass the limit of resolution determined by the PSF. 6. CONCLUSIONS m=3 - - z/s3 - z/s3 - z/s3 (g) (h) (i) x,=.3 x,=.46 We analyzed LHM suerlenses in an asymmetric arrangement, focusing on rimary aberrations and backscattering effects. Both are unwanted effects derived from imedance mismatch at boundaries. The IR of the LHM satisfying the antireflection coating condition n n n 3 minimizes backscattered light. Provided that d m λ 4n and assuming that x,=.6 m= m= Fig. 8. (Color online) Modulus of the D PSF jh j for a LHM flat lens with μ i. and ε 4 i. for different states of olarization: alies for -olarized waves and (d) (f) for s olarization. In (g) (i), we chart the data for the Gaussian image lane. The slab width is also varied: d.5λ for subfigures laced in the left column, d.875λ for subfigures in the central column, and d 5.5λ for lots on the right.

7 998 J. Ot. Soc. Am. A / Vol. 9, No. 9 / Setember Zaata-Rodríguez et al. losses in the metamaterial are negligible, reflection of light at normal incidence is comletely eliminated. To avoid rimary SA, we show that the object lane will be laced at an aroriate distance from the front interface of the suerlens. Nevertheless residual aberrations come out in the Gaussian conjugate lane, esecially for slabs with a subwavelength width. Proximity of the source from the entrance surface of the lens and of the image from the exit interface, as it is comared with the wavelength, are crucial in order to exhibit a subwavelength resolution. Under these circumstances, coating suerlenses may recover subwavelength information from the scattered wave field. Balancing residual aberrations may lead to focal shifts to achieve an imulse resonse of least sot size. Both finite-element analysis and PSF are used to estimate the limit of resolution for s-olarized and -olarized waves. ACKNOWLEDGMENTS This research was funded by the Sanish Ministry of Economy and Cometitiveness under the roject TEC REFERENCES. S. A. Ramakrishna, J. B. Pendry, D. Schurig, D. R. Smith, and S. Schultz, The asymmetric lossy near-erfect lens, J. Mod. Ot. 49, ().. P. Loschialo, D. Forester, D. Smith, F. Rachford, and C. Monzon, Otical roerties of an ideal homogeneous causal left-handed material slab, Phys. Rev. E 7, 3665 (4). 3. Z. Lin and Y. Zou, Low-order aberration corrections of multilayer flat lenses using negative-index materials, Al. Ot. 45, (6). 4. P. Valanju, R. Walser, and A. Valanju, Wave refraction in negative-index media: always ositive and very inhomogeneous, Phys. Rev. Lett. 88, 874 (). 5. P. Loschialo, D. Smith, D. Forester, F. Rachford, and J. Schelleng, Electromagnetic waves focused by a negative-index lanar lens, Phys. Rev. E 67, 56 (3). 6. T. Matsumoto, S. Fujita, and T. Baba, Wavelength demultilexer consisting of hotonic crystal suerrism and suerlens, Ot. Exress 3, (5). 7. T. Matsumoto, K.-S. Eom, and T. Baba, Focusing of light by negative refraction in a hotonic crystal slab suerlens on silicon-on-insulator substrate, Ot. Lett. 3, (6). 8. T. Dumelow, J. da Costa, and V. Freire, Slab lenses from simle anisotroic media, Phys. Rev. B 7, 355 (5). 9. H. Luo, Z. Ren, W. Shu, and F. Li, Construction of a olarization insensitive lens from a quasi-isotroic metamaterial slab, Phys. Rev. E 75, 66 (7).. I. Bulu, H. Caglayan, and E. Ozbay, Negative refraction and focusing of electromagnetic waves by metallodielectric hotonic crystals, Phys. Rev. B 7, 454 (5).. J. Li and C. T. Chan, Imaging using nano metallic films: from evanescent wave lens to resonant tunnelling lens, arxiv:hysics/77v (7).. T. Asatsuma and T. Baba, Aberration reduction and unique light focusing in a hotonic crystal negative refractive lens, Ot. Exress 6, (8). 3. Q. Wu, J. M. Gibbons, and W. Park, Graded negative index lens by hotonic crystals, Ot. Exress 6, (8). 4. D. Schurig and D. Smith, Negative index lens aberrations, Phys. Rev. E 7, 656 (4). 5. J. Chen, C. Radu, and A. Puri, Aberration-free negativerefractive-index lens, Al. Phys. Lett. 88, 79 (6). 6. D. Schurig, J. B. Pendry, and D. R. Smith, Transformationdesigned otical elements, Ot. Exress 5, (7). 7. V. G. Veselago, The electrodynamics of substances with simultaneously negative values of ϵ and μ, Physics-Usekhi, (968). 8. V. N. Mahajan, Otical Imaging and Aberrations. Part I. Ray Geometrical Otics (SPIE, 998). 9. N. Calander, Surface lasmon-couled emission and Fabry- Perot resonance in the samle layer: a theoretical aroach, J. Phys. Chem. B 9, (5).. P. Yeh, Otical Waves in Layered Media (Wiley, 988).. R. Marques, M. J. Freire, and J. D. Baena, Theory of threedimensional subdiffraction imaging, Al. Phys. Lett. 89, 3 (6).. B. Wood, J. B. Pendry, and D. P. Tsai, Directed subwavelength imaging using a layered metal-dielectric system, Phys. Rev. B 74, 56 (6). 3. T. Hakkarainen, T. Setälä, and A. T. Friberg, Subwavelength electromagnetic near-field imaging of oint diole with metamaterial nanoslab, J. Ot. Soc. Am. A 6, 6 34 (9). 4. C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, 989). 5. C. J. Zaata-Rodríguez, D. Pastor, and J. J. Miret, Threedimensional oint sread function and generalized amlitude transfer function of near-field flat lenses, Al. Ot. 49, (). 6. M. Nieto-Veserinas, Problem of image suerresolution with a negative-refractive-index slab, J. Ot. Soc. Am. A, (4). 7. R. Kotyński and T. Stefaniuk, Multiscale analysis of subwavelength imaging with metaldielectric multilayers, Ot. Lett. 35, ().