Abstract
Uniaxial materials whose axial and tangential permittivities have opposite signs are referred to as indefinite or hyperbolic media. In such materials, light propagation is unusual leading to novel and often nonintuitive optical phenomena. Here we report infrared nanoimaging experiments demonstrating that crystals of hexagonal boron nitride, a natural midinfrared hyperbolic material, can act as a ‘hyperfocusing lens’ and as a multimode waveguide. The lensing is manifested by subdiffractional focusing of phonon–polaritons launched by metallic disks underneath the hexagonal boron nitride crystal. The waveguiding is revealed through the modal analysis of the periodic patterns observed around such launchers and near the sample edges. Our work opens new opportunities for anisotropic layered insulators in infrared nanophotonics complementing and potentially surpassing concurrent artificial hyperbolic materials with lower losses and higher optical localization.
Introduction
One of the primary goals of nanophotonics is concentration of light on scales shorter than the freespace wavelength λ. According to the general principles of Fourier optics, this is only possible provided electromagnetic modes of large tangential momenta k_{t}>ω/(2π), normally evanescent, are nonetheless able to reach the focal plane (the x–y plane). Here ω=λ^{−1} is the measure of frequency common in spectroscopy and . In devices known as superlenses^{1,2,3,4,5,6}, this requirement is realized via resonant tunneling between the opposite sides of the structure. However, the tunneling is very sensitive to damping, for example, the magnitude of the imaginary part of the permittivity ɛ of the superlens material^{7}. The largest characteristic momentum that can pass through a superlens of thickness d can be found from the relation . In this regard, hyperbolic media (HM)^{8,9} promise a significant advantage as they support largek hyperbolic polaritons that remain propagating rather than evanescent, so that the condition on damping is much softer (see below). The unusual properties of hyperbolic polaritons in HM^{8,9,10,11,12,13,14,15,16,17,18,19,20} stem from the dispersion of these modes that is described by the equation of a hyperboloid:
where ɛ_{z} and ɛ_{t}≡ɛ_{x}=ɛ_{y} are the axial and tangential permittivities, respectively. The hyperboloid is singlesheeted if ɛ_{z}>0, ɛ_{t}<0 (type II) and twosheeted if ɛ_{z}<0, ɛ_{t}>0 (type I), see Fig 1a,b, respectively. In both cases, the slope of the propagation (group velocity) direction, which is orthogonal to the dispersion surface, asymptotically approaches
The condition for achieving superresolution is Im k_{z}d=(k_{t}d)Im tan θ∼1. Hence, admissible Im ɛ_{z}, Im ɛ_{t} scale algebraically rather than exponentially with the resolution .
Directional propagation of hyperbolic polaritons along ‘resonance cones’ of apex angle θ has been observed in a magnetized plasma^{21,22}, which behaves as a natural HM in the microwave domain. A major resurgence of interest to HM was prompted by their discussion in the context of artificial materials (metamaterials)^{23,24}. Examples of such hyperbolic metamaterials include microstrip arrays, where directional propagation and focusing of hyperbolic polaritons have been experimentally observed^{25,26}. Directional optical beams have been studied in planar^{25,26,27,28} and curved^{12,29} metamaterials made of alternating layers of metals and semiconductors. The work on nonplanar structures^{12,29} was motivated by theoretical proposals of a hyperlens^{30,31,32}, a device in which directional beams outgoing from a subdiffractional source enable optical magnification. However, improvement over the diffraction limit has so far been severely impeded by losses in constituent metals and imperfections of nanofabrication.
Recent work^{33,34} identified hexagonal boron nitride (hBN) as a lowloss natural HM in the midinfrared domain. This layered insulator has emerged as a premier substrate or a spacer for van der Waals heterostructures^{35,36}. Light atomic masses, strong anisotropy and the polar band between B and N yield prominent optical phonon modes that create two widely separated stopbands—spectral intervals where one of the principal values of the dielectric tensor is negative^{33,34,37}. The upper band comprises ω=1,370–1,610 cm^{−1} where the real part of ɛ_{t} (the inplane permittivity) is negative while that of ɛ_{z} is positive. In the lower band spanning ω=746–819 cm^{−1}, the signs of the permittivity components are reversed. Thus, the outofplane crystal vibrations enable the type I hyperbolic response, whereas the inplane ones accounts for the type II behaviour. The momentumfrequency dispersion surface for the hyperbolic polaritons of the upper band resembles a ‘butterfly’ (Fig. 1c) composed of individual hyperbolas sketched in Fig. 1a. It can be contrasted with the flat dispersion surfaces of longitudinal phonons typical for isotropic materials. Effectively, in hBN the longitudinal phonons are hybridized with the transverse ones by quasistatic Coulomb interaction mediated by largek photons^{38}. Because the hyperbolic response in hBN originates from the anisotropic phonons, in the following, the largek hyperbolic polaritons are referred to as hyperbolic phonon polaritons (HP^{2}).
Results
Subdiffractional focusing and imaging through hBN
In our experiments, efficient excitation and detection of HP^{2} in hBN are accomplished with the help of optical antenna structures^{39,40}. The antennas concentrate electric field and bridge the large momentum mismatch between the freespace photons and the HP^{2}. In our previous work^{33}, we used for this purpose a sharp tip of an atomic force microscope (AFM) incorporated in our scatteringtype scanning nearfield optical microscopy (sSNOM) apparatus (Methods). Here, we additionally demonstrate the antenna and polaritonlaunching capabilities of Au disks patterned on a SiO_{2} substrate. The AFM topography image in Fig. 2a depicts Au disks of diameters (top to bottom) 1,000, 500 and 200 nm and thickness of about 50 nm. After the subsequent deposition of hBN crystals of thickness d=100–1,060 nm and lateral sizes up to 10 μm, these Au disks become encapsulated between hBN and SiO_{2}. The hBN crystal remains essentially flat, as verified by AFM. Below we present experimental results demonstrating that interaction of these disks with an incident infrared beam excites polaritons that travel across hBN and produce specific contrast patterns at the other surface. We show that the observed dependence of the nearfield images on the frequency and hBN thickness is the result of directional propagation of the polaritons along conical surfaces with frequencytunable apex angle given by Equation (2). Thus, hBN may emerge as a new standard bearer for midinfrared nanophotonics by enabling devices for deeply subdiffractional propagation, focusing and imaging with tunable characteristics.
Representative sSNOM imaging data are shown in Fig. 2. Figure 2b depicts an sSNOM scan taken at the top surface of hBN of thickness d=395 nm at frequency ω=1,515 cm^{−1} (λ=6.6 μm). Here we plot the third harmonics of the scattering amplitude s(ω) (Methods). In this image, each Au disk is surrounded by a series of concentric ‘hot rings’ of strongly enhanced nanoinfrared contrast. The diameters of all the disks are much smaller than λ (see also Fig. 2a), the smallest one being 200 nm=λ/33. The diameters of the hot rings can be larger, smaller or equal to those of the disks. The spacing between adjacent hot rings in the same sample increases with the infrared frequency but decreases with the sample thickness. We stress that images displayed in Fig. 2b could only be detected if the infrared wavelength falls inside the hyperbolic spectral regions. Outside of the hBN stop bands, no hot rings can be identified by the sSNOM. In fact, the entire image is homogeneous, comprised of nothing but random noise, as illustrated by Fig. 2d for ω=1,740 cm^{−1} (λ=5.7 μm).
We now elaborate on the formation of images in Fig. 2 recorded with our sSNOM apparatus with the help of a model of HP^{2} propagation through a slab of hBN (Fig. 3a,c,d). Consider a perfectly thin metallic disk sandwiched between a slab of a HM of thickness d and an isotropic dielectric substrate. The system is subject to a uniform electric field of frequency ω and amplitude E_{0} in the x direction. An approximate solution for the total field in this system can be found analytically (Supplementary Note 1). The corresponding distributions of the zcomponent of the field E_{z}(x, y, z) in the two crosssections, y=0 (the vertical symmetry plane) and z=d−0 (just below the top surface of the hBN slab), are illustrated in Fig. 3c. These plots are computed for three representative radii of the disk using permittivity values at ω=1,515 cm^{−1}. The plots demonstrate a series of concentric highintensity rings on the top surface, very similar to the data in Fig. 2b. The interpretation (Fig. 3a,c) is straightforward: the external field polarizes the disk, which perturbs the adjacent HM (hBN in our case) and launches polaritons. The HP^{2} emission occurs predominantly at disk edges due to the high concentration of electric field therein. Polaritonic rays propagate across the slab, maintaining a fixed angle θ with respect to the z axis: the ‘resonance cone’ direction^{18,21,22,25,26,27,28}. Upon reaching the other slab surface, they undergo a total internal reflection with the reflected cone extending toward the bottom surface. The process repeats until eventually the field vanishes because of radial spreading and/or damping. The role of the sSNOM tip in imaging experiments in Fig. 2 is to outcouple HP^{2} fields at the top surface (Fig. 3a). The observed sSNOM signal is roughly proportional to the amplitude of the electric field immediately above the slab E_{z}(z=d=0). (Note that it is related to the field just inside the slab by a constant factor, E_{z}(z=d+0)=ɛ_{z}(ω)E_{z}(z=d−0).)
The above model of image formations via HP^{2} yields a number of quantitative predictions that are in accord with our observations. The scenario of oblique propagation implies that upon each roundtrip across the slab, the excitation front returns to the same surface displaced radially by the distance
Accordingly, the radii of the ‘hot rings’ at the top surface of the slab are given by
where a is the disk radius. The intensity of the rings is expected to decrease with n. Consistent with this formula, the smallest rings in Fig. 3c have the radius r_{0}=a−δ/2. Particularly interesting is the case where the innermost ring shrinks to a single bright spot, r_{0}=0. Experimentally, we observed spots of diameter 200 nm (the full width at half maximum, see Supplementary Note 1), which corresponds to λ/33 for Fig. 2b (top). Focal spots of similar size 185–210 nm were observed in all other hBN crystals, with the thickness up to 1,050 nm (Supplementary Fig. 5).
A proposal for focusing of electromagnetic radiation via resonancecone propagation in hyperbolic media was theoretically discussed in the context of magnetoplasmas^{21}. Experimental confirmation of this idea in an artificial hyperbolic multilayer was reported where λ/6 focusing was deduced from examining the pattern of a polymerized photoresist behind a twoslit polaritonic launcher^{26}. Here, using a natural hyperbolic slab (hBN crystal), we demonstrated the λ/33 focusing in both spatial directions via outcoupling of polaritons with the infrared nanoprobe. We stress that a distinction should be made between ‘focusing’ and ‘imaging.’ Focusing devices can be of both imaging and nonimaging type^{41} and both are important in applications. Our hBN device (Fig. 3a) is an example of the latter.
Continuing with the verifiable predictions of our model, we note that Equations (3) and (4) indicate that the slope tan θ of the resonance cone is uniquely related to the radii of the hot rings (Fig. 3a). To test this prediction, we analysed the images collected from samples of different hBN thicknesses and different Au disk diameters. For each of these, we determined the radius r_{1} of the firstorder ring and computed tan θ=(r_{1}−a)/d as a function of the infrared frequency (Fig. 3a). As shown in Fig. 3b, all the data collapse toward a single smooth curve computed from Equation (2) using optical constants of hBN from ref. 33. Yet another prediction of the model: the polaritonic rays travel along the z axis provided that ɛ_{t}(ω) and therefore θ(ω) are vanishingly small. This condition is satisfied at ω=1,610 cm^{−1} (Fig. 2c) where we observe almost 1:1 images of Au disks. Similar behaviour was observed when instead of the disks more complicated metallic shapes were imaged (Supplementary Fig. 5). Thus, the totality of our data establishes the notion of directional propagation of HP^{2} in hBN over macroscopic distances with a frequencytunable slope (Fig. 3b).
Realspace imaging of multiple guided polaritons in hBN
The outlined realspace picture has a counterpart in its conjugate momentum space. Mathematically, the resonance cones in the real space are coherent superpositions of an infinite number of polariton modes of a slab. Such modes are characterized by quantized momenta, k_{z,l}=(π/d)(l+α), labelled by integer index l^{33}. Here α∼1 (in general, ωdependent) quantifies the phase shift acquired at the total internal reflection from the slab surfaces. Per Equation (1), the tangential momenta of these modes are also quantized,
In the last step, we have applied Equation (3). For illustration, the dispersion curves of such guided modes in the upper stopband of hBN of thickness 105 nm are shown in Fig. 1c, where they are overlaid on the dispersion surface of bulk hBN. The same curves are replotted as ω vs k_{t} in Fig. 4a. In Fig. 4b the dispersion curves of the guided modes of lower stopband are shown. An intriguing aspect of these curves is that their slope ∂ω/∂k_{t} is positive (negative) in the upper (lower) band. This sign difference is a consequence of the opposite direction of the group velocity vector for the type I and type II cases, cf. Fig. 1a,b. Central to the connection between the resonance cones in the real space and the quantized momenta in the kspace is that these momenta form an equidistant sequence of period Δk_{t}=k_{t,l+1}−k_{t,l}=2π/δ. Therefore, if several guided modes are excited simultaneously by a source, their superposition would produce beats with period 2π/Δk_{t} in real space. This is precisely the spacing δ between periodic revivals of the ‘hot rings’ (Equation (3) and Fig. 2). Thus, the multiring images and the existence of higherorder guided modes are complementary manifestations of the same fundamental physics. In our previous work^{33}, we reported nanoimaging and nanospectroscopic study of the lowestmomentum guided mode l=0 in hBN crystals. Below we present new results documenting the first observation of the higherorder (up to three) guided modes in such materials by direct nanoinfrared imaging.
To map the dispersion of HP^{2}, we utilized hBN crystals on SiO_{2} substrate without any intervening metallic disks (Methods). Here the sharp tip of the sSNOM serves as both the emitter and the detector of the polariton waves on the open surface of the hBN. As the tip is scanned toward the sample edge, distinct variations in the detected scattering amplitude s(ω) are observed. Such variations are caused by passing over minima and maxima of the standing waves created by interference of the polaritons launched by the tip and their reflections off the sample edges (Fig. 5a). Representative data for the upper stopband (the type II hyperbolic region) are shown in Fig. 5b–f, where we plot s(ω) at various infrared frequencies. Specifically, the image presented in Fig. 5b exhibits oscillations with the period ∼1 μ m extending parallel to the edge of a 31nmthick hBN crystal. While these oscillations are similar to those reported previously^{33}, a highresolution scan performed very close to the edge (the olive square) reveals additional oscillations occurring on a considerably shorter scale: down to hundreds of nanometres (Fig. 5c–e). Similar results have been obtained using many other samples. For example, Fig. 5f also shows shortscale oscillations near the edges coexisting with longerrange oscillations further away from the edge in the data collected for a thicker hBN crystal (d=105 nm).
To analyse the harmonic content of the measured s(ω) quantitatively, we used the spatial Fourier transform (FT). An example shown in Fig. 5h is the FT of the line trace α from Fig. 5g. The three dominant peaks in the FT are marked with β’ (blue), γ’ (magenta) and ζ’ (olive). These peaks have been deemed statistically significant and their positions k_{β}, k_{γ} and k_{ζ} have been recorded for each of the traces studied. We reasoned that including additional weaker peaks into consideration may be unwarranted at this stage. Indeed, the gross features in the realspace trace α exceeding the noise level of ∼1 a.u. are accounted for by oscillations in the three partial traces β, γ and ζ, which are obtained by the inverse FT of the shaded regions in Fig. 5h.
The remaining step in the analysis is to establish the connection of thus determined momenta k_{β}, k_{γ} and k_{ζ} and the momenta k_{t,l} of the guided modes, Equation (5). This requires more care than in prior studies of singlemode waves in twodimensional materials^{33,42,43,44}. The interference patterns near the edge can be created by various combinations of the tiplaunched waves (labeled by l) and edgereflected waves (labeled by r). The total momentum of a particular combination is k_{t,l}+k_{t,r}. If the mode index is conserved, l=r, the set of possible periods narrows down to 2k_{t,l}. This is consistent with our data obtained for several infrared frequencies (Fig. 4a), where the symbols indicate k_{β}, k_{γ} and k_{ζ}. These data are in a quantitative agreement with the calculated dispersion curves for the l=0, 1 and 2 polaritonguided waves in the upper stopband. The analysis of polariton propagation length^{33} shows that the loss factor is as low as γ∼0.03 (Supplementary Fig. 4). Dispersion mapping in the lower band (746–819 cm^{−1}) where no monochromatic lasers are available is discussed in Supplementary Fig. 3. Broadband lasers used in an independent study by Li et al. have allowed to demonstrate focusing behaviour of hBN in this challenging frequency region^{45}.
Discussion
Data presented in Figs 2, 3, 4, 5 demonstrate launching, longdistance waveguiding transport and focusing of electromagnetic energy in thin crystals of hBN. These phenomena are enabled by directional propagation of largemomentum polariton beams in this natural hyperbolic material. The sharpness of the attained focusing, λ/33 at distances up to λ/6 (Supplementary Fig. 5), in units of the freespace wavelength, surpasses all prior realizations of superlenses and hyperlenses. Remarkably, a simple addition of a circular metallic launcher transforms an hBN crystal into a powerful focusing^{19} device! The analysis presented in Supplementary Note 1 (Supplementary Equation 10) indicates that the size of the focal spot in our system is limited by the finite thickness ∼50 nm of Au disks. By using thinner disks, say 20 nm thick, one should be able to achieve focal spots as small as ∼λ/10^{2}, comparable to the spatial resolution of our nanoinfrared apparatus. A fundamental advantage of using natural rather than artificial hyperbolic materials is the magnitude of the upper momentum cutoff. In a natural material such as hBN, this cutoff is ultimately set by interatomic spacing thus immensely enhancing the spatial resolution. In addition, we have shown that hBN can serve as a multimode waveguide for polaritons with excellent figure of merit: loss factor as small as γ∼0.03. These characteristics exceed the benchmarks^{46,47,48} of current metalbased plasmonics and metamaterials. The physics behind this fundamental advantage of phonon polaritons over plasmons in conducting media is in the absence of electronic losses in insulators. Applications of hBN for nonimaging focusing devices^{41}, subdiffractional waveguides and nanoresonators^{34} readily suggest themselves^{45}. Combining such elements together may lead to development of sophisticated nanopolaritonic circuits.
Methods
Experimental setup
The nanoimaging and nanoFTIR experiments described in the main text were performed at UCSD using a commercial sSNOM (www.neaspec.com). The sSNOM is based on a tappingmode AFM illuminated by monochromatic quantum cascade lasers (QCLs) (www.daylightsolutions.com) and a broadband laser source utilizing the difference frequency generation (www.lasnix.com)^{49}. Together, these lasers cover a frequency range of 700–2,300 cm^{−1} in the midinfrared. The nanoscale nearfield images were registered by pseudoheterodyne interferometric detection module with AFM tapping frequency and amplitude around 250 kHz and 60 nm, respectively. To obtain the backgroundfree images, the sSNOM output signal used in this work is the scattering amplitude s(ω) demodulated at the nth harmonics of the tapping frequency. We chose n=3 in this work.
Sample fabrication
Silicon wafers with 300nmthick SiO_{2} top layer were used as substrates for all the samples. The Au patterns of various lateral shapes and 50nm thickness were fabricated on these wafers by electron beam lithography. The hBN microcrystals of various thicknesses were exfoliated from bulk samples synthesized under high pressure^{50}. Such microcrystals were subsequently mechanically transferred onto either patterned or unpatterned parts of the substrates.
Additional information
How to cite this article: Dai, S. et al. Subdiffractional focusing and guiding of polaritonic rays in a natural hyperbolic material. Nat. Commun. 6:6963 doi: 10.1038/ncomms7963 (2015).
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Acknowledgements
D.N.B. acknowledges support from DOEBES grant DEFG0200ER45799 and the Gordon and Betty Moore Foundation’s EPiQS initiative through Grant GBMF4533; research on polariton focusing is supported by AFOSR. Work at UCSD is supported by the Office of Naval Research, AFOSR, NASA and The University of California Office of the President. A.S.M. acknowledges support from an Office of Science Graduate Research Fellowship from U.S. Department of Energy. P.JH acknowledges support from AFOSR grant number FA95501110225.
Author information
Affiliations
Department of Physics, University of California, San Diego, La Jolla, California 92093, USA
 S. Dai
 , A. S. Mcleod
 , Z. Fei
 , M. K. Liu
 , M. Wagner
 , M. M. Fogler
 & D. N. Basov
Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02215, USA
 Q. Ma
 , T. Andersen
 & P. JarilloHerrero
Department of Physics, Stony Brook University, Stony Brook, New York 11794, USA
 M. K. Liu
National Institute for Materials Science, Namiki 11, Tsukuba, Ibaraki 3050044, Japan
 K. Watanabe
 & T. Taniguchi
Department of Chemistry and Biochemistry, University of California, San Diego, La Jolla, California 92093, USA
 M. Thiemens
LudwigMaximiliansUniversität and Center for Nanoscience, 80539 München, Germany
 F. Keilmann
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Contributions
All the authors were involved in designing the research, performing the research and writing the paper.
Competing interests
F.K. is one of the cofounders of Neaspec and Lasnix, producer of the sSNOM and infrared source used in this work. The remaining authors declare no competing financial interests.
Corresponding author
Correspondence to D. N. Basov.
Supplementary information
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Supplementary Information
Supplementary Figures 15, Supplementary Notes 12 and Supplementary References.
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Further reading

1.
Light: Science & Applications (2018)

2.
Ultraconfined surface phonon polaritons in molecular layers of van der Waals dielectrics
Nature Communications (2018)

3.
npj 2D Materials and Applications (2017)

4.
Nanoimaging of resonating hyperbolic polaritons in linear boron nitride antennas
Nature Communications (2017)

5.
Ultralowloss polaritons in isotopically pure boron nitride
Nature Materials (2017)
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