Hyperbolic materials exhibit sub-diffractional, highly directional, volume-confined polariton modes. Here we report that hyperbolic phonon polaritons allow for a flat slab of hexagonal boron nitride to enable exciting near-field optical applications, including unusual imaging phenomenon (such as an enlarged reconstruction of investigated objects) and sub-diffractional focusing. Both the enlarged imaging and the super-resolution focusing are explained based on the volume-confined, wavelength dependent propagation angle of hyperbolic phonon polaritons. With advanced infrared nanoimaging techniques and state-of-art mid-infrared laser sources, we have succeeded in demonstrating and visualizing these unexpected phenomena in both Type I and Type II hyperbolic conditions, with both occurring naturally within hexagonal boron nitride. These efforts have provided a full and intuitive physical picture for the understanding of the role of hyperbolic phonon polaritons in near-field optical imaging, guiding, and focusing applications.
The propagation of sub-diffractional waves in hyperbolic media1 enables many unusual optical possibilities such as hyperlensing2,3,4, negative refraction5,6, enhanced quantum radiation7, nanolithography8 and sub-diffractional resonators9,10. Very recently, it was demonstrated that the highly directional propagation of volume-confined, hyperbolic polaritons (HPs) is key for these sub-diffractional phenomena8,10,11. Their directionality derives from the sign and magnitude of the two principal (in- and out-of-plane) components of the dielectric-permittivity tensor (), which have opposite signs in hyperbolic materials. The propagation angle θ (for example, the angle between the Poynting vector and the z axis) of the HPs in hyperbolic media can be roughly approximated as10,
where ɛt=ɛxx=ɛyy and ɛz=ɛzz are the in- and out-of-plane dielectric permittivities of the hyperbolic medium, respectively. Therefore, by controlling the ratio of the two principal dielectric components, the propagation direction of the HPs can be tuned.
Until very recently, hyperbolic media have been explored through man-made hyperbolic metamaterial (HMM) structures, such as metal-dielectric multilayers4,5, nanowire6,12 or nanopyramid arrays9 embedded within a dielectric medium. In HMMs, the effective dielectric permittivities are determined by the geometric parameters of their subwavelength unit cells1. As such, the maximum wavevector k that can be induced to propagate through the material is limited by the size of the artificial unit cell. This in turn limits the degree of optical confinement and spatial resolution that can be realized. Furthermore, the high losses associated with noble metals13,14,15 used in man-made HMM structures result in short propagation lengths, quite broad resonance linewidths and in terms of hyperlens designs, low transmission efficiency.
During the search for better plasmonic materials14,15, polar dielectrics capable of supporting phonon-polaritons such as silicon carbide16,17,18,19,20 and hexagonal boron nitride (hBN)10,11,21 have been demonstrated as superior alternatives to metals at mid-infrared to THz frequencies. Interestingly, many phonon-resonant materials such as quartz22, zinc oxide23, calcite24 and hBN10,11,21 are natural hyperbolic materials25,26,27. These natural hyperbolic materials support hyperbolic phonon-polariton modes within homogeneous crystals with atomic-scale unit cells, thus the upper limit on the highest propagating wavevectors k associated with artificial metal-dielectric HMMs is no longer an issue. Instead, photonic confinement within tiny volumes in the few nanometre range becomes possible. This was recently demonstrated by Dai et al.11 where surface phonon polariton propagation within a three monolayer (<1 nm) thin flake of hBN was reported. It is the propagation of such high-k fields that are scattered off or launched from deeply sub-diffractional objects that is at the heart of super-resolution imaging. These benefits are also coupled with a drastic reduction in the optical losses compared with HMMs, which results in improved performance, that is, higher field confinement10,11 and improved image resolution.
In contrast to HMMs reported to date, hBN offers the additional functionality of sub-wavelength imaging in different spectral regions through the presence of two separate spectral bands (Supplementary Fig. 1) that exhibit inverted hyperbolic response, making this an ideal material for exploring the basic phenomenon of HPs. These two regimes are referred to as the lower and upper Reststrahlen bands10,11, where this term refers to the spectral range between the longitudinal and transverse optic phonons of a polar crystal where a negative real part of the dielectric function is observed. The presence of two bands results from the highly anisotropic crystal structure of hBN, where a, b and c axis oriented optic phonons are supported and are widely separated in frequency28. These two bands not only exhibit hyperbolic behaviour, but the crystal axis featuring negative real permittivity is inverted, thus the lower and upper bands offer Type-I (Re(ɛt) > 0 and Re(ɛz) < 0 in ∼760 < ω< 825 cm−1) and Type-II (Re(ɛt) < 0 and Re(ɛz) > 0 in ∼1,360 < ω< 1,610 cm−1) hyperbolic response, respectively. The inversion of the signs of the dielectric function results in unusual behaviour, such as a negative (positive) z-component of the group velocity in the upper (lower) Reststrahlen bands. This results in unique phenomenon such as higher order resonance modes occurring at lower (higher) frequencies. Until now, due to a lack of a homogeneous material exhibiting both types of hyperbolicity, a comprehensive study of the impact of these two unique regimes for nanoimaging and super-resolution focusing have not been experimentally probed, with Caldwell et al.10 providing the only prior study comparing the unique behaviours of these two regimes, albeit within the context of three-dimensionally confined cavities. In addition to providing a homogeneous medium exhibiting both Type I and II hyperbolicity, hBN also exhibits much lower losses (higher efficiencies) than plasmonic materials, with the imaginary part of the dielectric function, Im(ɛt) ∼ 0.1 for Re(ɛt)=−1 at ω=1573, cm−1 and Im(ɛz) ∼ 0.1 for Re(ɛz)=−1 at ω=809 cm−1, which is crucial for realizing extended propagation and detection of high-k polariton modes (deeply sub-diffractional confinement)10,11,17.
The dielectric permittivities are also highly dispersive in these two regions, giving rise to a frequency-dependent ratio between ɛt and ɛz. Thus, because of the presence of both Type I and II hyperbolicity, from equation (1) we can predict that the propagation angle θ will be an increasing (decreasing) function of the frequency in the lower (upper) Reststrahlen bands. As shown in Fig. 1a,b, as the frequency is increased, θ increases from about 16° to 87° (Δθ≈71°) within the Type I lower Reststrahlen band, while it decreases from θ≈83° to θ≈2°(Δθ ≈ 81°) in the Type II upper Reststrahlen band.
Here we experimentally verify this inversion of the propagation angle within the two spectral bands, providing near-field imaging of sub-diffractional objects under a thin slab of hBN. This investigation also enabled the experimental demonstration of the unusual imaging properties of both Type I and II hyperbolic media, thereby providing the physical description necessary to realize and optimize near-field imaging using natural hyperbolic media.
Theory and simulations
This frequency-dependent tuning of the angular HP propagation in hBN and the suitability of this material for near-field imaging can be easily quantified and visualized via two-dimensional (2D) numerical calculations. For this, we consider a 0.3 μm wide gold stripe on a Si substrate with an hBN cover layer (the thickness h=1 μm) illuminated by a p-polarized plane wave incident from the top. Simulated electric-field distributions (|Ez|) at six typical operation frequencies are presented in Fig. 1c–h, and demonstrate the directional nature of the HPs. Here the high-k fields scattered from the edges of the embedded gold stripe are induced to propagate within the hBN flake (as marked in Fig. 1d), with the angle of propagation being directly dependent upon the frequency of operation. As stated previously1,2,3,4, in the absence of the hyperbolic dispersion, such high-k modes would be evanescent (that is, decay rapidly) within the medium. Each edge of the Au stripe excites two sub-diffractional HPs that propagate at the angles ±θ. This frequency dependent propagation angle quantitatively agrees with the analytical predictions based on equation (1) (see Fig. 1a,b), therefore the propagation angle can be predicted via the simple ratio of the extraordinary (z axis) and ordinary (x–y plane) components of the anisotropic dielectric function of hBN.
The anticipated super-resolution imaging performance of the hBN slab is directly tied to the propagation angle, as demonstrated in Fig. 1c-h, which as noted above is directly dependent on the type of hyperbolicity. For instance, at wavelengths with very low propagation angles (Fig. 1c,f), the image of the sub-diffraction Au stripe is nearly perfectly restored on the top surface of the hBN, similar to near-field superlensing29,30,31. Namely, the restored width d′ is nearly identical to the actual width d of the investigated stripe (for example, d′≈309 nm, around λ/40-resolution at ω=778.2 cm−1 or λ=12.85 μm). However, as the angle is increased (Fig. 1d,e for the lower and Fig. 1g,h for the upper Reststrahlen bands), an enlarged outline of the object image is obtained with the width D(ω)= d+2h tanθ(ω). Note that due to the inversion of the dependence of θ on ω between the two spectral bands, superlensing-type response is observed at low and high frequencies within the lower and upper Reststrahlen bands, respectively, while the enlarged imaging behaviour is observed at high and low frequencies.
This unusual enlargement can be clearly observed in the more practical three-dimensional (3D) cases. As shown in Fig. 2a, the 0.6-μm-diameter gold disc is once again perfectly restored at very shallow angles; however, at larger propagation angles a double concentric ring-like field distribution is recorded in the near-field (Fig. 2b), rather than a direct replication of the original field distribution of the object. This enlarged pattern results from the frequency-dependent propagation angle of the HPs (cone-like shape for 3D case, see the sketch in Fig. 2c). This unexpected phenomenon is also found for other shapes (like a Au square and stripe) as shown in Fig. 2d–i. Intriguingly, the trace of the HP cones reconstructs the enlarged and slightly distorted pattern that is still able to identify the outline (shape) information of the object. Quantitatively, through recording the HP-reconstructed outline (D(ω)), we can extract the geometric size and shape from the strict relationship of D(ω)=d+2h tanθ(ω). These results suggest an interesting HP-based imaging scenario, which will be verified and visualized by our experiments below.
Experimental results in Type I hyperbolic band
The predicted imaging mechanism is verified by our experiments first in the Type I hyperbolic band, as presented in Fig. 3. A schematic of the experimental setup is provided in Fig. 3a where the images restored by the hBN layer are recorded using a scattering-type scanning near-field optical microscope (s-SNOM). We use a 0.15-μm-thick exfoliated hBN flake to image the underlying, 30 nm tall, gold nanodiscs with 0.3-μm diameter and 1.3-μm centre-to-centre separation. The metallic tip of the s-SNOM is illuminated by a home-built, tunable broadband infrared laser32,33 with the peak position of the laser spectrum (inset in Fig. 3a) matched to the lower, Type-I hyperbolic region of hBN (760 cm−1 <ω<825 cm−1). Both the optical and topographic information at the top surface of the hBN layer are collected simultaneously (details in Methods). In the obtained topographic image (Fig. 3b), the gold discs are masked by the covering hBN layer, while in the broadband-SNOM image (Fig. 3c), all three nanodiscs are clearly resolved with bright contrast (that is, high signal-to-background ratio). To ensure that the imaging is indeed due to the hyperbolic nature of the hBN slab, a control image was also collected at a frequency outside of the lower Reststrahlen band using a CO2 laser at ω=952 cm−1, where both components of the dielectric function are positive (Re(ɛt)=8.8 and Re(ɛz)=2). As shown in Fig. 3e, in contrast to the hyperbolic case, only weak features of the discs are observed through the thin hBN layer in the control experiment. This is further demonstrated by the line profiles taken across the two discs for both cases presented in Fig. 3f, with a marked enhancement in the imaging efficiency observed in the hyperbolic regime. Such an effect would be further amplified within thicker hBN slabs, whereby any structural morphology would be totally lost, but the near-field imaging properties retained. The hyperbolic images also provided a narrower FWHM (full width at half maximum) of about 0.5 μm in addition to the markedly improved contrast (signal-to-background ratio). Considering the resolved deeply subwavelength optical FWHM (∼λ/24), this comparison clearly confirms the improved near-field imaging by the hBN layer.
Although the broadband s-SNOM image shown in Fig. 3c is able to resolve the discs, it does not reflect the actual field distribution of the image due to its detection scheme33. Because of the broadband light source used, the detection is not monochromatic as in the simulations presented in Figs 1 and 2. Instead, the collected images are the superposition of all the different frequency components of the broadband laser and therefore the concentric-ring field distributions are not directly observed. Further, depending on the chosen interference phase difference (constructive or destructive phases, involving the position of a reference mirror in our setup, see Methods), it can also show reversed imaging contrasts (see ref. 33). Therefore, the broadband s-SNOM image alone cannot directly reveal the actual image of the field distributions of the discs and the frequency dependence of the hBN layer. For this, monochromatic s-SNOM images would be required; however, currently such sources are not readily available within this spectral range.
To determine the actual imaging response through the hBN layer at each individual wavelength, we performed Fourier transform infrared nanospectroscopy (nano-FTIR)11,33 along a line scan (dashed line in Fig. 3c) across two discs. At each pixel, nano-FTIR delivers a full infrared spectrum recorded at the spatial resolution of the probing tip (∼50 nm). Thus we obtain s-SNOM signals s2(ω, x) as a function of the frequency ω and the spatial position x11,16,33. This hyperspectral imaging allows the extraction of detailed spatial line profiles at various frequencies as shown in Fig. 3g. For frequencies within the lower Reststrahlen but ω > 780 cm−1, two peaks are observed for each disc (see the typical case at ω=783 cm−1). These peaks (with the width d′ ∼ 0.3 μm, Supplementary Figs 2 and 3) correspond to the edge-excited HPs. The distance D between the two edge-launched peaks increases from about 0.4 to ∼1.25 μm (for the right disc) when changing the frequency from 780 to 807 cm−1, with this distance being directly related to the frequency-dependent HP propagation angle. At frequencies ω<780 cm−1, the two edge-launched peaks are not resolved by the hBN layer due to the very small directional angle of HPs, leading to one single broad peak observed for each disc. On the basis of the smallest peak-to-peak separation in our results being ∼0.4 μm at ω=780 cm−1 (λ=12.8 μm), this corresponds to a sub-diffractional resolution of about λ/32 (also see Supplementary Fig. 3).
For the comparison of the theoretically predicted propagation angles (Fig. 1a) and those experimentally derived from the s-SNOM measurements, we plot the experimentally determined D (blue curve and data) and corresponding directional angles θ (red curve and data) of the HPs as functions of ω in Fig. 3h. The theoretical result (from D(ω)=d+2h tanθ(ω)) is also shown for comparison. Good quantitative agreement is found between the experiments and theory. This verifies the predicted tuning range of the propagation angle of the HPs to be about 35°−70° and the tunable ratio (D/d) to range from 1.39 to 4.2. However, certain discrepancies are still found in this comparison, and we do not observe multiple peaks originating from the multireflection of the HPs within the hBN slab.
Experimental results in Type II hyperbolic band
To extract the near-field frequency-dependent imaging and HP propagation angles, near-field imaging experiments were also performed in the Type II hyperbolic band by using a line-tunable quantum cascade laser (spanning from 1,310 to 1,430 cm−1). This monochromatic laser allows us to present the HP-reconstructed imaging phenomenon in a more intuitive way compared to the case with the broadband laser. First, we performed the s-SNOM measurements to image a Au stripe (about 1-μm long, 100-nm wide, as sketched in Fig. 4a) that is covered by the 0.15-μm hBN layer (atomic force microscopy (AFM) topography in Fig. 4b). In comparison to the circular discs, this rectangular object can avoid the potential confusion caused by the HP cones (circular cross section) for understanding the imaging results. Because the deeply sub-diffractional width (<λ/70) of the stripe, optical information cannot be recorded at frequencies outside the hyperbolic band, as shown in Fig. 4c. In contrast, we clearly observe enlarged optical patterns formed by the HPs (Fig. 4d–f). Qualitatively, these elliptical, rectangular patterns carry and reflect the shape information of the original object. Quantitatively, from the measured width and length of the patterns, in conjunction with the known HP propagation angle, we can extract the geometric size of the investigated object. Using this approach, we estimate the width of the stripe is about 0.12 μm, and the estimated length is about 1.1 μm, which is consistent with SEM measurements of the stripe prior to hBN exfoliation. We note that for frequencies where HPs have smaller diffractional angles (namely, the HP cone with the very small diameter), the reconstructed image will be similar to the one-to-one near-field superlensing29,30,31. Therefore, in such HP-based imaging phenomenon, it has two frequency-dependent operation modes: the enlarged reconstruction and the one-to-one superlensing.
As previously discussed, we also experimentally demonstrated that the diameter of the HP cone increases with decreasing frequency, which is inverted with respect to the Type I band. These two distinct frequency dependences, found in two spectral regimes through imaging of the sub-diffractional objects, verify again that the two types of hyperbolic dispersion exist in the hBN crystal, without the need for fabricating HMM structures.
In addition to the isolated structures, we also investigated periodic arrays of nanostructures. We imaged the arrays of the Au nanodiscs (5 × 5 array, sketched in Fig. 5a) with different separations. The diameter of the discs was fixed at 0.3 μm. The gap separation between two discs varied from g=0.1 μm to g=1 μm. The experimental results are shown in Fig. 5d–f. We observed the complicated overlapping of the HP cone launched by each disc. However, with these sub-diffractional HPs, we have the opportunity to distinguish the deeply sub-diffractional structures that are not readily seen in either AFM topography or near-field imaging at frequencies outside of the hyperbolic regime (see the comparison of Fig. 5b–d).
HPs for sub-diffractional focusing and selective waveguiding
In addition to the already proven high-resolution imaging with tunable enlargement of the outline of sub-diffractional objects using only a simple slab of hBN, the highly directional nature of HPs can also result in sub-diffractional focusing behaviour. When imaging the nanodiscs with a large diameter of 0.75 μm, we observe the focusing spot with the width of about 0.175 μm (∼λ/40) at ω=1,420 cm−1 (Supplementary Fig. 4). This is because all the HP cones launched by the nanodisc superimpose into the centre point, leading to the concentration of the light. We also investigate this focusing effect with the broadband laser in the Type I hyperbolic band. However, the obtained results are also the superposition of all the frequency components, which are not intuitive. We also note that this super-focusing effect is independently shown and discussed in ref. 34, which was performed concurrently with the work discussed here.
Both the enlarged imaging and the super-focusing are based on the volume-confined, frequency-dependent propagation angle of HPs in hBN. We can envision other potential disruptive technologies based on the great potential of the highly directional HPs. First, this frequency-selective waveguiding could be useful for photonic switching or computing, infrared filtering, or various other nanophotonic applications (Supplementary Fig. 5a, b). Another potential application is realized in the form of an ultracompact subwavelength spectrometer (Supplementary Fig. 5c,d). A natural hBN layer should allow for the spatial separation or filtering of incoming broadband light into different wavelength channels, much like a grating, which could then be detected by subwavelength infrared detector pixels. This particular spectrometer configuration could also be used for chemical and biological detection schemes, in the form of spatially resolved infrared spectroscopy. Under broadband mid-infrared illumination the HPs could carry the vibration (or absorption) information of molecules in contact with the surface, dispersing the spectral information at different angles, enabling them to be spatially resolved by a near-field intensity detector (like the s-SNOM tip) without the need of spectrometers.
Our work along with that of ref. 34, demonstrate the complete hyperbolic imaging response of hBN and its potential for improving the near-field imaging of deeply embedded objects35,36 in both the Type I lower and Type II upper Reststrahlen bands, respectively. More specifically, we reveal the hyperbolic nature of the hBN layer for near-field waveguiding, imaging, focusing and its dependence on the operational frequency. Although all our results are restricted in the near field, we also expect that these intriguing findings will benefit far-field imaging by introducing specific geometric designs, such as circular or wedge-shaped hyperlenses2,3. Furthermore, as a van der Waal’s crystal37, hBN lends itself to incorporation on non-planar and flexible substrates more amenable to true hyperlensing methodologies4. The realization of a naturally occurring, Type I and II hyperbolic media enable various opportunities for nanophotonics that go beyond sub-diffractional near-field imaging and potential hyperlensing. Due to the similar material anisotropy present in other polar dielectric van der Waals crystals37, such as MoS2 or WS2, the natural hyperbolic response of hBN may be general to the entire class of polar 2D crystals, thus expanding the potential spectral range of this behaviour from the mid-infrared into the single-digit terahertz spectral region17,24.
The gold nanostructures used for the near-field experiments were fabricated on a 1-μm-thick intrinsic silicon substrate using electron beam lithography into a bilayer PMMA (poly(methylmethacrylate)) resist. The nanostructures varied in size from 0.2–1 μm in diameter and in arrays with 0.1–1 μm edge-to-edge gaps. A standard liftoff procedure was used following the thermal evaporation of Cr (5 nm)/Au (30 nm) metallization.
hBN crystals were grown using the high-pressure/high-temperature method38,39. The standard exfoliation process was used to randomly deposit hBN flakes of various thicknesses onto a PMMA/PMGI (polydimethyl glutarimide) bilayer spun on a separate silicon substrate. Here the PMMA layer played the role of the flake carrier membrane and the PMGI served as a sacrificial lift-off layer later dissolved by tetramethyl ammonium hydroxide (TMAH) solution (MICROPOSIT MF-319). AFM was utilized to select specific flakes with both sufficient thickness and lateral size for the imaging experiments. The PMMA carrier membrane with an appropriate hBN flake was lifted-off from the substrate and put onto the supportive metal ring held by a home-made micromanipulator. With the help of the micromanipulator, the hBN flake was aligned and transferred face down onto the predefined gold nanostructure by releasing the carrier membrane from the metal ring. Following the transfer, the sample with the carrier membrane on was heated to 130 °C for about 10 min to soften the PMMA membrane and improve the adhesion of hBN to the underlying nanostructures and silicon substrate. After that the carrier membrane was dissolved in acetone leaving the hBN flake covering the entire array of nanostructures. To improve the adhesion, an ultrasonic clean in acetone and isopropyl alcohol with subsequent oxygen plasma clean was performed on the silicon substrate before the hBN transfer. More details of this transfer technique are given in ref. 40.
Infrared s-SNOM measurements
An s-SNOM (commercially available, Neaspec GmbH) system was used to simultaneously measure the optical near fields and topography. The laser system used in Type I hyperbolic band was developed by the Fraunhofer ILT32,33. It consists of a commercially available picosecond-laser as the pump source and two subsequent nonlinear converter steps to cover the mid-infrared range. The peak wavelength is continuously tunable from ω=625 cm−1 (∼16 μm) to ω=1,100 cm−1 (∼9 μm) with bandwidths of some tens to more than hundred wavenumbers. At a repetition rate of 20 MHz and pulse duration of 10 ps, the system provides an average power of up to 10 mW. To address the lower Type-I hyperbolic region of the hBN, the peak position of the laser spectrum in our measurements was set to be at around ω= 790 cm−1 (12.7 μm) with a FWHM of about 90 cm−1. To suppress the far-field background contribution and solely measure the near-field contribution, the optical signal was demodulated at higher harmonics (nΩ) of the oscillation frequency (Ω∼270 kHz) of the cantilever (in our cases, n=2 for the broadband s-SNOM). Nano-FTIR spectra were obtained by constantly moving the mirror in the reference arm of the Michelson interferometer, recording the resulting interferograms and their corresponding complex Fourier transformation11,33. In contrast to conventional far-field FTIR, this setup allows to record spectral information with a spatial resolution of down to several tens nm. For 2D imaging, the position of the reference mirror was fixed to be at the position about λ/8 away from the maximum of the interference signals. This allows for a visualization of even small spectral changes31. The extracted line profiles shown in Fig. 3g were numerically smoothed by using a fast Fourier transform smoothing with three adjacent pixels. This smoothing does not improve the resolution, but rather leads to a conservative estimation of resolution (details in Supplementary Figs 2 and 3). The monochromatic quantum cascade laser used in the Type II band (Figs 4 and 5) is commercially available from Daylight Solutions. The demodulation order n = 3 is chosen for the monochromatic s-SNOM imaging. Near-field optical amplitude (sn) and phase (ϕn) are separated with an interferometric setup11,19.
2D simulations (Fig. 1) were carried out by the finite-element software COMSOL Multiphysics. A plane-wave illumination was set by using scattering boundary condition. The surrounding boundaries used perfectly matched layer absorbing boundary conditions. 3D simulations (Fig. 2) were done by using CST Microwave Studio. Open boundary conditions were used. We also checked different mesh sizes to make sure that all the simulations reach proper convergence. The dielectric data of hBN used in all the simulations are extracted from far-field FTIR measurements10.
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This work was supported by the Excellence Initiative of the German federal and state governments, the Ministry of Innovation of North Rhine-Westphalia, the DFG under SFB 917 and the Korean Defense Acquisition Program Administration and the Agency for Defense Development as a part of a basic research program under the contract UD110099GD. Funding for J.D.C. was provided by the Office of Naval Research and administered by the NRL Nanoscience Institute and was carried out at the University of Manchester through the NRL Long-Term Training (Sabbatical) Program. A.K. and K.S.N. acknowledge support from the Engineering and Physical Sciences Research Council (UK), The Royal Society (UK), European Research Council and EC-FET European Graphene Flagship. P.L. and Th.T. thank G. von Plessen and D. Chigrin for valuable comments and the proofreading.
The authors declare no competing financial interests.
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Li, P., Lewin, M., Kretinin, A. et al. Hyperbolic phonon-polaritons in boron nitride for near-field optical imaging and focusing. Nat Commun 6, 7507 (2015). https://doi.org/10.1038/ncomms8507
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