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S. Dai, Q. Ma, T. Andersen, A. S. Mcleod, Z. Fei, M. K. Liu, M. Wagner, [K. Watanabe](https://orcid.org/0000-0003-3701-8119), [T. Taniguchi](https://orcid.org/0000-0002-1467-3105), M. Thiemens, F. Keilmann, P. Jarillo-Herrero, M. M. Fogler, D. N. Basov

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[Subdiffractional focusing and guiding of polaritonic rays in a natural hyperbolic material](https://mdr.nims.go.jp/datasets/575cbed7-c3b9-45f8-89dd-6606652bbd51)

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Subdiffractional focusing and guiding of polaritonic rays in a natural hyperbolic materialARTICLEReceived 24 Dec 2014 | Accepted 19 Mar 2015 | Published 22 Apr 2015Subdiffractional focusing and guiding of polaritonicrays in a natural hyperbolic materialS. Dai1, Q. Ma2, T. Andersen2, A.S. Mcleod1, Z. Fei1, M.K. Liu1,3, M. Wagner1, K. Watanabe4, T. Taniguchi4,M. Thiemens5, F. Keilmann6, P. Jarillo-Herrero2, M.M. Fogler1 & D.N. Basov1Uniaxial materials whose axial and tangential permittivities have opposite signs are referredto as indefinite or hyperbolic media. In such materials, light propagation is unusual leading tonovel and often non-intuitive optical phenomena. Here we report infrared nano-imagingexperiments demonstrating that crystals of hexagonal boron nitride, a natural mid-infraredhyperbolic material, can act as a ‘hyper-focusing lens’ and as a multi-mode waveguide. Thelensing is manifested by subdiffractional focusing of phonon–polaritons launched by metallicdisks underneath the hexagonal boron nitride crystal. The waveguiding is revealed throughthe modal analysis of the periodic patterns observed around such launchers and near thesample edges. Our work opens new opportunities for anisotropic layered insulators in infrarednanophotonics complementing and potentially surpassing concurrent artificial hyperbolicmaterials with lower losses and higher optical localization.DOI: 10.1038/ncomms7963 OPEN1 Department of Physics, University of California, San Diego, La Jolla, California 92093, USA. 2 Department of Physics, Massachusetts Institute of Technology,Cambridge, Massachusetts 02215, USA. 3 Department of Physics, Stony Brook University, Stony Brook, New York 11794, USA. 4 National Institute forMaterials Science, Namiki 1-1, Tsukuba, Ibaraki 305-0044, Japan. 5 Department of Chemistry and Biochemistry, University of California, San Diego, La Jolla,California 92093, USA. 6 Ludwig-Maximilians-Universität and Center for Nanoscience, 80539 München, Germany. Correspondence and requests formaterials should be addressed to D.N.B. (email: dbasov@physics.ucsd.edu).NATURE COMMUNICATIONS | 6:6963 | DOI: 10.1038/ncomms7963 | www.nature.com/naturecommunications 1& 2015 Macmillan Publishers Limited. All rights reserved.mailto:dbasov@physics.ucsd.eduhttp://www.nature.com/naturecommunicationsOne of the primary goals of nanophotonics is concentra-tion of light on scales shorter than the free-spacewavelength l. According to the general principles ofFourier optics, this is only possible provided electromagneticmodes of large tangential momenta kt4o/(2p), normallyevanescent, are nonetheless able to reach the focal plane (thex–y plane). Here o¼ l� 1 is the measure of frequency commonin spectroscopy and kt ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffik2xþ k2yq. In devices known assuperlenses1–6, this requirement is realized via resonanttunneling between the opposite sides of the structure. However,the tunneling is very sensitive to damping, for example, themagnitude of the imaginary part of the permittivity e of thesuperlens material7. The largest characteristic momentum thatcan pass through a superlens of thickness d can be found from therelation Im e � e� ktd . In this regard, hyperbolic media (HM)8,9promise a significant advantage as they support large-khyperbolic polaritons that remain propagating rather thanevanescent, so that the condition on damping is much softer(see below). The unusual properties of hyperbolic polaritons inHM8–20 stem from the dispersion of these modes that is describedby the equation of a hyperboloid:e� 1t k2z þ e� 1z ðk2xþ k2yÞ ¼ 2poð Þ2; ð1Þwhere ez and et�ex¼ ey are the axial and tangential permittivities,respectively. The hyperboloid is single-sheeted if ez40, eto0(type II) and two-sheeted if ezo0, et40 (type I), see Fig 1a,b,respectively. In both cases, the slope of the propagation (groupvelocity) direction, which is orthogonal to the dispersion surface,asymptotically approachestan yðoÞ ¼ iffiffiffiffiffiffiffiffiffiffiffiet oð Þpffiffiffiffiffiffiffiffiffiffiffiez oð Þp : ð2ÞThe condition for achieving super-resolution is Im kzd¼ (ktd)Imtan yB1. Hence, admissible Im ez, Im et scale algebraically ratherthan exponentially with the resolution k� 1t .Directional propagation of hyperbolic polaritons along‘resonance cones’ of apex angle y has been observed in amagnetized plasma21,22, which behaves as a natural HM in themicrowave domain. A major resurgence of interest to HM wasprompted by their discussion in the context of artificial materials(metamaterials)23,24. Examples of such hyperbolic metamaterialsinclude microstrip arrays, where directional propagation andfocusing of hyperbolic polaritons have been experimentallyobserved25,26. Directional optical beams have been studied inplanar25–28 and curved12,29 metamaterials made of alternatinglayers of metals and semiconductors. The work on non-planarstructures12,29 was motivated by theoretical proposals of ahyperlens30–32, a device in which directional beams outgoingfrom a subdiffractional source enable optical magnification.However, improvement over the diffraction limit has so farbeen severely impeded by losses in constituent metals andimperfections of nanofabrication.Recent work33,34 identified hexagonal boron nitride (hBN) as alow-loss natural HM in the mid-infrared domain. This layeredinsulator has emerged as a premier substrate or a spacer for vander Waals heterostructures35,36. Light atomic masses, stronganisotropy and the polar band between B and N yield prominentoptical phonon modes that create two widely separated stop-bands—spectral intervals where one of the principal values of thedielectric tensor is negative33,34,37. The upper band compriseso¼ 1,370–1,610 cm� 1 where the real part of et (the in-planepermittivity) is negative while that of ez is positive. In the lowerband spanning o¼ 746–819 cm� 1, the signs of the permittivitycomponents are reversed. Thus, the out-of-plane crystalvibrations enable the type I hyperbolic response, whereas thein-plane ones accounts for the type II behaviour. Themomentum-frequency dispersion surface for the hyperbolicpolaritons of the upper band resembles a ‘butterfly’ (Fig. 1c)composed of individual hyperbolas sketched in Fig. 1a. It can becontrasted with the flat dispersion surfaces of longitudinalphonons typical for isotropic materials. Effectively, in hBN thelongitudinal phonons are hybridized with the transverse ones byquasi-static Coulomb interaction mediated by large-k photons38.Because the hyperbolic response in hBN originates from theanisotropic phonons, in the following, the large-k hyperbolicpolaritons are referred to as hyperbolic phonon polaritons (HP2).ResultsSubdiffractional focusing and imaging through hBN. In ourexperiments, efficient excitation and detection of HP2 in hBN areaccomplished with the help of optical antenna structures39,40. Theantennas concentrate electric field and bridge the largemomentum mismatch between the free-space photons and theHP2. In our previous work33, we used for this purpose a sharp tipof an atomic force microscope (AFM) incorporated in ourscattering-type scanning near-field optical microscopy (s-SNOM)apparatus (Methods). Here, we additionally demonstrate theantenna and polariton-launching capabilities of Au diskspatterned on a SiO2 substrate. The AFM topography image inFig. 2a depicts Au disks of diameters (top to bottom) 1,000, 500and 200 nm and thickness of about 50 nm. After the subsequentdeposition of hBN crystals of thickness d¼ 100–1,060 nm andlateral sizes up to 10mm, these Au disks become encapsulatedbetween hBN and SiO2. The hBN crystal remains essentially flat,as verified by AFM. Below we present experimental resultsdemonstrating that interaction of these disks with an incidentinfrared beam excites polaritons that travel across hBN andproduce specific contrast patterns at the other surface. We showthat the observed dependence of the near-field images on thefrequency and hBN thickness is the result of directionalpropagation of the polaritons along conical surfaces withfrequency-tunable apex angle given by Equation (2). Thus, hBN−100101,3501,5501,515 cm–1vktvθkzType II�=1,370–1,610 cm–1Type I�=746–819 cm–1ktkz−5−100510 0.50.40.3kt (105  cm–1)� (cm–1)kz (105 cm–1)�/�Figure 1 | Hyperbolic dispersion of hBN. (a) A sketch of the isofrequencycurves for a type II HM, which is realized in the upper stop-band of hBN.The arrow indicates the polariton group velocity. (b) A similar sketch for thetype I case, which is realized in the hBN lower stop-band. (c) The calculateddispersion surface of hBN polaritons. The axes are the tangentialmomentum (kt), the axial momentum (kz) and the frequency (o, rangingfrom 1,370 to 1,515 cm� 1). The colour represents the propagation angle y.The constant-frequency cut o¼ 1,515 cm� 1 is shown by the red line, toemphasize similarity with a. The dispersion of polaritons in a finite-thickness crystal (d¼ 105 nm) is shown by the black lines to clarify theirrelation to Fig. 4a.ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms79632 NATURE COMMUNICATIONS | 6:6963 | DOI: 10.1038/ncomms7963 | www.nature.com/naturecommunications& 2015 Macmillan Publishers Limited. All rights reserved.http://www.nature.com/naturecommunicationsmay emerge as a new standard bearer for mid-infrarednanophotonics by enabling devices for deeply subdiffractionalpropagation, focusing and imaging with tunable characteristics.Representative s-SNOM imaging data are shown in Fig. 2.Figure 2b depicts an s-SNOM scan taken at the top surface ofhBN of thickness d¼ 395 nm at frequency o¼ 1,515 cm� 1(l¼ 6.6 mm). Here we plot the third harmonics of the scatteringamplitude s(o) (Methods). In this image, each Au disk issurrounded by a series of concentric ‘hot rings’ of stronglyenhanced nano-infrared contrast. The diameters of all the disksare much smaller than l (see also Fig. 2a), the smallest one being200 nm¼ l/33. The diameters of the hot rings can be larger,smaller or equal to those of the disks. The spacing betweenadjacent hot rings in the same sample increases with the infraredfrequency but decreases with the sample thickness. We stress thatimages displayed in Fig. 2b could only be detected if the infraredwavelength falls inside the hyperbolic spectral regions. Outside ofthe hBN stop bands, no hot rings can be identified by thes-SNOM. In fact, the entire image is homogeneous, comprisedof nothing but random noise, as illustrated by Fig. 2d foro¼ 1,740 cm� 1 (l¼ 5.7 mm).We now elaborate on the formation of images in Fig. 2recorded with our s-SNOM apparatus with the help of a model ofHP2 propagation through a slab of hBN (Fig. 3a,c,d). Consider aperfectly thin metallic disk sandwiched between a slab of a HM ofthickness d and an isotropic dielectric substrate. The system issubject to a uniform electric field of frequency o and amplitudeE0 in the x direction. An approximate solution for the total fieldin this system can be found analytically (Supplementary Note 1).The corresponding distributions of the z-component of the fieldEz(x, y, z) in the two cross-sections, y¼ 0 (the vertical symmetryplane) and z¼ d� 0 (just below the top surface of the hBN slab),are illustrated in Fig. 3c. These plots are computed for threerepresentative radii of the disk using permittivity values ato¼ 1,515 cm� 1. The plots demonstrate a series of concentrichigh-intensity rings on the top surface, very similar to the data inFig. 2b. The interpretation (Fig. 3a,c) is straightforward: theexternal field polarizes the disk, which perturbs the adjacent HM(hBN in our case) and launches polaritons. The HP2 emissionoccurs predominantly at disk edges due to the high concentrationof electric field therein. Polaritonic rays propagate across the slab,maintaining a fixed angle y with respect to the z axis: the‘resonance cone’ direction18,21,22,25–28. Upon reaching the otherslab surface, they undergo a total internal reflection with thereflected cone extending toward the bottom surface. The processrepeats until eventually the field vanishes because of radialspreading and/or damping. The role of the s-SNOM tip inimaging experiments in Fig. 2 is to out-couple HP2 fields atthe top surface (Fig. 3a). The observed s-SNOM signal isroughly proportional to the amplitude of the electric fieldimmediately above the slab Ez(z¼ d¼ 0). (Note that it isrelated to the field just inside the slab by a constant factor,Ez(z¼ dþ 0)¼ ez(o)Ez(z¼ d� 0).)The above model of image formations via HP2 yields a numberof quantitative predictions that are in accord with our observa-tions. The scenario of oblique propagation implies that upon eachroundtrip across the slab, the excitation front returns to the samesurface displaced radially by the distanced ¼ 2 tanyðoÞ d: ð3ÞAccordingly, the radii of the ‘hot rings’ at the top surface of theslab are given byrn ¼ aþ n� 12� �j d j��������; n¼ 0; � 1; � 2; . . . ð4Þwhere a is the disk radius. The intensity of the rings is expected todecrease with |n|. Consistent with this formula, the smallest ringsin Fig. 3c have the radius r0¼ |a� |d|/2|. Particularly interestingis the case where the innermost ring shrinks to a single brightspot, r0¼ 0. Experimentally, we observed spots of diameter200 nm (the full width at half maximum, see SupplementaryNote 1), which corresponds to l/33 for Fig. 2b (top). Focal spotsof similar size 185–210 nm were observed in all other hBNcrystals, with the thickness up to 1,050 nm (SupplementaryFig. 5).A proposal for focusing of electromagnetic radiation viaresonance-cone propagation in hyperbolic media was theoreti-cally discussed in the context of magneto-plasmas21.Experimental confirmation of this idea in an artificialhyperbolic multi-layer was reported where l/6 focusing wasdeduced from examining the pattern of a polymerized photoresistbehind a two-slit polaritonic launcher26. Here, using a naturalhyperbolic slab (hBN crystal), we demonstrated the l/33 focusingin both spatial directions via out-coupling of polaritons with theinfrared nano-probe. We stress that a distinction should be madebetween ‘focusing’ and ‘imaging.’ Focusing devices can be of bothimaging and non-imaging type41 and both are important inapplications. Our hBN device (Fig. 3a) is an example of the latter.Continuing with the verifiable predictions of our model, wenote that Equations (3) and (4) indicate that the slope tan y of theresonance cone is uniquely related to the radii of the hot rings(Fig. 3a). To test this prediction, we analysed the images collected0 75 nm Min MaxFigure 2 | Sub-diffractional focusing and imaging through an hBN crystal. (a) An AFM image of Au disks defined lithographically on SiO2/Si substratebefore hBN transfer. (b) Near-field amplitude image of the top surface of a 395-nm-thick hBN at infrared laser frequency o¼ 1,515 cm� 1 (l¼ 6.6mm).The observed ‘hot rings’ are concentric with the Au discs. (c) Near-field image of the same sample as in b at o¼ 1,610 cm� 1 (l¼ 6.2mm) where polaritonspropagate almost vertically. (d) Near-field image of the same sample at o¼ 1,740 cm� 1 (l¼ 5.7mm) showing complete homogeneity and lack of anydistinct features. The colour scales for b–d are indicated in d. The scale bars in all panels are 1 mm long.NATURE COMMUNICATIONS | DOI: 10.1038/ncomms7963 ARTICLENATURE COMMUNICATIONS | 6:6963 | DOI: 10.1038/ncomms7963 | www.nature.com/naturecommunications 3& 2015 Macmillan Publishers Limited. All rights reserved.http://www.nature.com/naturecommunicationsfrom samples of different hBN thicknesses and different Au diskdiameters. For each of these, we determined the radius r1 of thefirst-order ring and computed |tan y|¼ (r1� a)/d as a function ofthe infrared frequency (Fig. 3a). As shown in Fig. 3b, all thedata collapse toward a single smooth curve computed fromEquation (2) using optical constants of hBN from ref. 33. Yetanother prediction of the model: the polaritonic rays travel alongthe z axis provided that et(o) and therefore y(o) are vanishinglysmall. This condition is satisfied at o¼ 1,610 cm� 1 (Fig. 2c)where we observe almost 1:1 images of Au disks. Similarbehaviour was observed when instead of the disks morecomplicated metallic shapes were imaged (SupplementaryFig. 5). Thus, the totality of our data establishes the notion ofdirectional propagation of HP2 in hBN over macroscopicdistances with a frequency-tunable slope (Fig. 3b).Real-space imaging of multiple guided polaritons in hBN. Theoutlined real-space picture has a counterpart in its conjugatemomentum space. Mathematically, the resonance cones in thereal space are coherent superpositions of an infinite number ofpolariton modes of a slab. Such modes are characterized byquantized momenta, kz,l¼ (p/d)(lþ a), labelled by integerindex l 33. Here aB1 (in general, o-dependent) quantifies thephase shift acquired at the total internal reflection from the slabsurfaces. Per Equation (1), the tangential momenta of thesemodes are also quantized,kt;lðoÞ ’ cot y oð Þkz;lðoÞ ¼2pd oð Þ ½lþ a oð Þ�: ð5ÞIn the last step, we have applied Equation (3). For illustration, thedispersion curves of such guided modes in the upper stop-band ofhBN of thickness 105 nm are shown in Fig. 1c, where they areoverlaid on the dispersion surface of bulk hBN. The same curvesare replotted as o vs kt in Fig. 4a. In Fig. 4b the dispersion curvesof the guided modes of lower stop-band are shown. An intriguingaspect of these curves is that their slope qo/qkt is positive(negative) in the upper (lower) band. This sign difference is aconsequence of the opposite direction of the group velocity vectorfor the type I and type II cases, cf. Fig. 1a,b. Central to theconnection between the resonance cones in the real space andthe quantized momenta in the k-space is that these momentaform an equidistant sequence of period Dkt¼ kt,lþ 1� kt,l¼ 2p/d.Therefore, if several guided modes are excited simultaneously bya source, their superposition would produce beats with period2p/Dkt in real space. This is precisely the spacing d betweenperiodic revivals of the ‘hot rings’ (Equation (3) and Fig. 2).Thus, the multi-ring images and the existence of higher-orderguided modes are complementary manifestations of the samefundamental physics. In our previous work33, we reported nano-imaging and nano-spectroscopic study of the lowest-momentumguided mode l¼ 0 in hBN crystals. Below we present new resultsdocumenting the first observation of the higher-order (up tothree) guided modes in such materials by direct nano-infraredimaging.To map the dispersion of HP2, we utilized hBN crystals onSiO2 substrate without any intervening metallic disks (Methods).MinMaxxyzE0a�dr1hBNSiO2E0DataTheory012345|tan �|� (cm–1)1,400 1,500 1,600Figure 3 | Image formation. (a) Imaging schematics. Under infrared illumination (green arrow), the polaritons were launched by the Au disk edgesand propagate towards the hBN top surface where the near-field images were recorded via the back-scattered infrared beam (green arrow). Thepropagation angle y can be inferred from the hot ring radius r1, hBN thickness d and disk radius a. (b) The tangent of the propagation angle y derivedfrom imaging data for different hBN samples (symbols) and from Equation (2) (solid line). Squares, triangles, crosses and dots indicate data fromhBN samples with thickness d¼ 395, 984, 270 and 1,060 nm, respectively. (c) The distribution of the z-component of the electric field in the analyticalmodel (see text). The hot rings on the surfaces appear as a result of multiple reflections of polaritons launched at the disk edges. The ratio a/|d|¼0.5, 0.25,0.15 decreases from top to bottom. In the top picture, the smallest ring shrinks to a focal point. The blue arrow indicates the direction of electric field E0 insimulation. (d) Similar to b for a/d¼ 1.12 and (top to bottom) |tan y|¼0.75, 0.375 and 0.01.ab...1,4001,5001,6001,4501,550� (cm–1)1l=02750775800l=0� (cm–1)kt (105 cm–1)0 1 32Figure 4 | Polariton frequency (x) – in-plane momentum (kt) dispersionrelation for hBN. (a) The dispersion curves from Fig. 1c replotted asfrequency (o) versus in-plane momenta (kt). The experimental data(squares) are obtained from the polariton reflection images near the sampleedges (Fig. 5). (b) Same as a for the lower hBN stop-band (SupplementaryFig. 3). Thickness of hBN: 105 nm.ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms79634 NATURE COMMUNICATIONS | 6:6963 | DOI: 10.1038/ncomms7963 | www.nature.com/naturecommunications& 2015 Macmillan Publishers Limited. All rights reserved.http://www.nature.com/naturecommunicationsHere the sharp tip of the s-SNOM serves as both the emitter andthe detector of the polariton waves on the open surface of thehBN. As the tip is scanned toward the sample edge, distinctvariations in the detected scattering amplitude s(o) are observed.Such variations are caused by passing over minima and maximaof the standing waves created by interference of the polaritonslaunched by the tip and their reflections off the sample edges(Fig. 5a). Representative data for the upper stop-band (the type IIhyperbolic region) are shown in Fig. 5b–f, where we plot s(o) atvarious infrared frequencies. Specifically, the image presented inFig. 5b exhibits oscillations with the period B1 mm extendingparallel to the edge of a 31-nm-thick hBN crystal. While theseoscillations are similar to those reported previously33, a high-resolution scan performed very close to the edge (the olivesquare) reveals additional oscillations occurring on a considerablyshorter scale: down to hundreds of nanometres (Fig. 5c–e).Similar results have been obtained using many other samples.For example, Fig. 5f also shows short-scale oscillations near theedges co-existing with longer-range oscillations further awayfrom the edge in the data collected for a thicker hBN crystal(d¼ 105 nm).To analyse the harmonic content of the measured s(o)quantitatively, we used the spatial Fourier transform (FT). Anexample shown in Fig. 5h is the FT of the line trace a fromFig. 5g. The three dominant peaks in the FT are marked with b’(blue), g’ (magenta) and z’ (olive). These peaks have been deemedstatistically significant and their positions kb, kg and kz have beenrecorded for each of the traces studied. We reasoned thatincluding additional weaker peaks into consideration may beunwarranted at this stage. Indeed, the gross features in the real-space trace a exceeding the noise level of B1 a.u. are accountedfor by oscillations in the three partial traces b, g and z, which areobtained by the inverse FT of the shaded regions in Fig. 5h.The remaining step in the analysis is to establish theconnection of thus determined momenta kb, kg and kz and themomenta kt,l of the guided modes, Equation (5). This requiresmore care than in prior studies of single-mode waves in two-dimensional materials33,42–44. The interference patterns near theedge can be created by various combinations of the tip-launchedwaves (labeled by l) and edge-reflected waves (labeled by r). Thetotal momentum of a particular combination is kt,lþ kt,r. If themode index is conserved, l¼ r, the set of possible periods narrowsdown to 2kt,l. This is consistent with our data obtained for severalinfrared frequencies (Fig. 4a), where the symbols indicate kb, kgand kz. These data are in a quantitative agreement with thecalculated dispersion curves for the l¼ 0, 1 and 2 polariton-guided waves in the upper stop-band. The analysis of polaritonpropagation length33 shows that the loss factor is as low asgB0.03 (Supplementary Fig. 4). Dispersion mapping in the lowerband (746–819 cm� 1) where no monochromatic lasers areavailable is discussed in Supplementary Fig. 3. Broad-bandlasers used in an independent study by Li et al. have allowed todemonstrate focusing behaviour of hBN in this challengingfrequency region45.DiscussionData presented in Figs 2–5 demonstrate launching, long-distancewaveguiding transport and focusing of electromagnetic energy inthin crystals of hBN. These phenomena are enabled by directionalpropagation of large-momentum polariton beams in this naturalhyperbolic material. The sharpness of the attained focusing, l/33at distances up to l/6 (Supplementary Fig. 5), in units of the free-space wavelength, surpasses all prior realizations of superlensesand hyperlenses. Remarkably, a simple addition of a circularmetallic launcher transforms an hBN crystal into a powerfulhBN SiO21,420cm–11,410cm–11,390cm–11,400cm–1SiO2051015αβγζ10–1010–1210–1110–131,420cm–1MaxMinζ'F (kt)k t (105 cm–1)0 1 321,400 cm–11,400 cm–1s (�) (a.u.)0 500 1,000 1,500 2,000γ 'L (nm)β'Figure 5 | Imaging of polariton waveguide modes near the hBN edges. (a) Experimental schematic is similar to Fig. 3a except that imaging here isperformed near the edge of an unpatterned sample. (b) Near-field amplitude image measured at 1,420 cm� 1. The olive square indicates the area whoseexpanded view is shown in c–e). (c–e) Near-field image of the area marked in b at several frequencies. hBN thickness in b� e: 31 nm. (f) Near-field imageof 105-nm-thick hBN at 1,400 cm� 1. The cyan dashed lines in b� f indicate the hBN edges. Scale bar in b� f, 300 nm. (g) Line traces perpendicular tothe hBN edge. Trace a was extracted from the image in f. Traces b, g and z were obtained from the Fourier analysis of the trace a as described in thetext. (h) The Fourier transform of trace a in g.NATURE COMMUNICATIONS | DOI: 10.1038/ncomms7963 ARTICLENATURE COMMUNICATIONS | 6:6963 | DOI: 10.1038/ncomms7963 | www.nature.com/naturecommunications 5& 2015 Macmillan Publishers Limited. All rights reserved.http://www.nature.com/naturecommunicationsfocusing19 device! The analysis presented in SupplementaryNote 1 (Supplementary Equation 10) indicates that the size of thefocal spot in our system is limited by the finite thickness B50 nmof Au disks. By using thinner disks, say 20 nm thick, one shouldbe able to achieve focal spots as small as Bl/102, comparableto the spatial resolution of our nano-infrared apparatus.A fundamental advantage of using natural rather than artificialhyperbolic materials is the magnitude of the upper momentumcutoff. In a natural material such as hBN, this cutoff is ultimatelyset by interatomic spacing thus immensely enhancing the spatialresolution. In addition, we have shown that hBN can serve as amulti-mode waveguide for polaritons with excellent figure ofmerit: loss factor as small as gB0.03. These characteristics exceedthe benchmarks46–48 of current metal-based plasmonics andmetamaterials. The physics behind this fundamental advantage ofphonon polaritons over plasmons in conducting media is in theabsence of electronic losses in insulators. Applications of hBN fornon-imaging focusing devices41, subdiffractional waveguides andnanoresonators34 readily suggest themselves45. Combining suchelements together may lead to development of sophisticatednanopolaritonic circuits.MethodsExperimental setup. The nano-imaging and nano-FTIR experiments describedin the main text were performed at UCSD using a commercial s-SNOM(www.neaspec.com). The s-SNOM is based on a tapping-mode AFM illuminatedby monochromatic quantum cascade lasers (QCLs) (www.daylightsolutions.com)and a broad-band laser source utilizing the difference frequency generation(www.lasnix.com)49. Together, these lasers cover a frequency range of700–2,300 cm� 1 in the mid-infrared. The nanoscale near-field images wereregistered by pseudo-heterodyne interferometric detection module with AFMtapping frequency and amplitude around 250 kHz and 60 nm, respectively. Toobtain the background-free images, the s-SNOM output signal used in this work isthe scattering amplitude s(o) demodulated at the nth harmonics of the tappingfrequency. We chose n¼ 3 in this work.Sample fabrication. Silicon wafers with 300-nm-thick SiO2 top layer were used assubstrates for all the samples. The Au patterns of various lateral shapes and 50-nmthickness were fabricated on these wafers by electron beam lithography. The hBNmicrocrystals of various thicknesses were exfoliated from bulk samples synthesizedunder high pressure50. Such microcrystals were subsequently mechanicallytransferred onto either patterned or unpatterned parts of the substrates.References1. Pendry, J. B. Negative refraction makes a perfect lens. Phys. Rev. Lett. 85,3966–3969 (2000).2. Fang, N., Lee, H., Sun, C. & Zhang, X. Sub-diffraction-limited optical imagingwith a silver superlens. Science 308, 534–537 (2005).3. Taubner, T., Korobkin, D., Urzhumov, Y., Shvets, G. & Hillenbrand, R. Near-field microscopy through a SiC superlens. Science 313, 1595 (2006).4. Zhang, X. & Liu, Z. Superlenses to overcome the diffraction limit. Nat. Mater.7, 435–441 (2008).5. Smolyaninov, I. I., Hung, Y. J. & Davis, C. C. 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Nat.Mater. 3, 404–409 (2004).AcknowledgementsD.N.B. acknowledges support from DOE-BES grant DE-FG02-00ER45799 and theGordon and Betty Moore Foundation’s EPiQS initiative through Grant GBMF4533;research on polariton focusing is supported by AFOSR. Work at UCSD is supported bythe Office of Naval Research, AFOSR, NASA and The University of California Office ofthe President. A.S.M. acknowledges support from an Office of Science Graduate ResearchFellowship from U.S. Department of Energy. P.J-H acknowledges support from AFOSRgrant number FA9550-11-1-0225.Author contributionsAll the authors were involved in designing the research, performing the research andwriting the paper.Additional informationSupplementary Information accompanies this paper at http://www.nature.com/naturecommunicationsCompeting financial interests: F.K. is one of the co-founders of Neaspec and Lasnix,producer of the s-SNOM and infrared source used in this work. The remaining authorsdeclare no competing financial interests.Reprints and permission information is available online at http://npg.nature.com/reprintsandpermissions/How to cite this article: Dai, S. et al. Subdiffractional focusing and guiding of polaritonicrays in a natural hyperbolic material. Nat. Commun. 6:6963 doi: 10.1038/ncomms7963(2015).This work is licensed under a Creative Commons Attribution 4.0International License. The images or other third party material in thisarticle are included in the article’s Creative Commons license, unless indicated otherwisein the credit line; if the material is not included under the Creative Commons license,users will need to obtain permission from the license holder to reproduce the material.To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/NATURE COMMUNICATIONS | DOI: 10.1038/ncomms7963 ARTICLENATURE COMMUNICATIONS | 6:6963 | DOI: 10.1038/ncomms7963 | www.nature.com/naturecommunications 7& 2015 Macmillan Publishers Limited. All rights reserved.http://www.nature.com/naturecommunicationshttp://www.nature.com/naturecommunicationshttp://npg.nature.com/reprintsandpermissionshttp://npg.nature.com/reprintsandpermissionshttp://creativecommons.org/licenses/by/4.0/http://www.nature.com/naturecommunications title_link Results Subdiffractional focusing and imaging through hBN Figure™1Hyperbolic dispersion of hBN.(a) A sketch of the isofrequency curves for a type II HM, which is realized in the upper stop-band of hBN. The arrow indicates the polariton group velocity. (b) A similar sketch for the type I case, which is realized i Figure™2Sub-diffractional focusing and imaging through an hBN crystal.(a) An AFM image of Au disks defined lithographically on SiO2solSi substrate before hBN transfer. (b) Near-field amplitude image of the top surface of a 395-nm-thick hBN at infrared las Real-space imaging of multiple guided polaritons in hBN Figure™3Image formation.(a) Imaging schematics. Under infrared illumination (green arrow), the polaritons were launched by the Au disk edges and propagate towards the hBN top surface where the near-field images were recorded via the back-scattered infrare Figure™4Polariton frequency (ohgr) - in-plane momentum (kt) dispersion relation for hBN.(a) The dispersion curves from Fig.™1c replotted as frequency (ohgr) versus in-plane momenta (kt). The experimental data (squares) are obtained from the polariton refl Discussion Figure™5Imaging of polariton waveguide modes near the hBN edges.(a) Experimental schematic is similar to Fig.™3a except that imaging here is performed near the edge of an unpatterned sample. (b) Near-field amplitude image measured at 1,420thinspcm-1. The  Methods Experimental setup Sample fabrication PendryJ. B.Negative refraction makes a perfect lensPhys. Rev. Lett.85396639692000FangN.LeeH.SunC.ZhangX.Sub-diffraction-limited optical imaging with a silver superlensScience3085345372005TaubnerT.KorobkinD.UrzhumovY.ShvetsG.HillenbrandR.Near-field microsc D.N.B. acknowledges support from DOE-BES grant DE-FG02-00ER45799 and the Gordon and Betty Moore FoundationCloseCurlyQuotes EPiQS initiative through Grant GBMF4533; research on polariton focusing is supported by AFOSR. Work at UCSD is supported by the Offi ACKNOWLEDGEMENTS Author contributions Additional information