Abstract
Polaritons – coupled excitations of photons and dipolar matter excitations – can propagate along anisotropic metasurfaces with either hyperbolic or elliptical dispersion. At the transition from hyperbolic to elliptical dispersion (corresponding to a topological transition), various intriguing phenomena are found, such as an enhancement of the photonic density of states, polariton canalization and hyperlensing. Here, we investigate theoretically and experimentally the topological transition, the polaritonic coupling and the strong nonlocal response in a uniaxial infraredphononic metasurface, a grating of hexagonal boron nitride (hBN) nanoribbons. By hyperspectral infrared nanoimaging, we observe a synthetic transverse optical phonon resonance (strong collective nearfield coupling of the nanoribbons) in the middle of the hBN Reststrahlen band, yielding a topological transition from hyperbolic to elliptical dispersion. We further visualize and characterize the spatial evolution of a deeply subwavelength canalization mode near the transition frequency, which is a collimated polariton that is the basis for hyperlensing and diffractionless propagation.
Introduction
Uniaxial metasurfaces are thin layers of engineered subwavelength structures, whose inplane effective permittivity tensor components are unequal (ε_{eff,x} ≠ ε_{eff,y}), and thus support different types of inplane anisotropic polaritons^{1,2,3,4,5,6,7,8,9,10}. When both ε_{eff,x} and ε_{eff,y} are negative but with different absolute values, the polaritons propagating along the metasurface exhibit an ellipticallyshaped dispersion diagram, i.e., the polariton momentum k describes an elliptical isofrequency contour (IFC) in kspace^{6,9,11}. On the other hand, when ε_{eff,x} and ε_{eff,y} have opposite signs, polaritons possess a socalled hyperbolic dispersion (k describes hyperbolic IFCs in kspace)^{1,4,5,6,7,8,9,12,13}, exhibiting increased polariton confinement and raylike anisotropic propagation along the surface. These two types of anisotropic metasurfaces can be applied, for example, to enhance optical birefringence^{14,15}, to control light polarization^{16}, for nanoscale directional polariton guiding^{5,8,9,17,18}, and for subwavelengthscale optical imaging^{9,19}.
A particularly interesting regime arises when uniaxial metasurfaces exhibit a topological transition of the IFCs upon frequency variation^{8,9,20}, changing from hyperbolic to elliptical. It offers unique opportunities in nanophotonics, for example, for enhancing the local photonic density of states^{8,9,20} and for supporting deeply subwavelength canalization modes^{17,21}. These canalization modes can exhibit extremely anisotropic inplane polariton momenta, which results in nanoscale and nearly diffractionfree electromagnetic energy transport with applications in hyperlensing^{8,9,17} and control of nearfield heat transfer^{9}. It has been shown theoretically that the topological transition and the canalization modes are determined by the polaritonic nearfield coupling of subwavelength elements comprising the metasurfaces^{7,9,17}. However, so far, the polaritonic coupling governing the topological transition has not been experimentally demonstrated. Only few experimental studies at microwave frequencies have visualized weakly confined canalization modes at 10 GHz^{12}. Here, we demonstrate that the strong collective nearfield coupling of subwavelength elements in an infraredphononic metasurface (a hBN nanograting) yields a synthetic optical phonon resonance and subsequently a topological transition. By hyperspectral nanoimaging, we are able to observe the topological transition and the strong coupling of the nanoribbons (the metasurface elements in our case) in spatial and spectral domains. We also provide realspace images of deeply subwavelength canalization polaritons, which experimentally demonstrate that these modes are the consequence of the strong collective polaritonic nearfield coupling of the nanoribbons and the associated strong nonlocal response of the metasurface.
Results
Nonlocal effective medium theory for a hBN metasurface
Boron nitride exhibits a negative and isotropic inplane permittivity (ε_{hBN,t} = ε_{x} = ε_{y} < 0) and a positive outofplane permittivity ε_{hBN,z} in its upper midinfrared Reststrahlen band (the frequency region between transversal and longitudinal optical phonon frequencies, TO and LO, respectively), where phonon polaritons (PhPs) exist^{5,22,23,24,25,26,27,28,29,30}. Patterning a thin hBN flake into a periodic array of nanoribbons (nanograting) creates an infrared metasurface with strong inplane anisotropy, which can support inplane hyperbolic phonon polaritons when the nearfield coupling between the ribbons is weak^{5} (Fig. 1a, d, e, ribbon width w = 70 nm, gap size g = 30 nm, thickness h = 20 nm). By considering now strong polaritonic nearfield coupling of the ribbons and the subsequent strong nonlocal effect induced by the periodic structuring (the effective permittivity depends on the polariton momentum k that is controlled by the grating period L, see discussions in ref. ^{8,9}), the effective anisotropic permittivity (ε_{eff,x}, ε_{eff,y,} ε_{eff,z}) of this metasurface needs to be described by a modified effective medium model^{8,9}, yielding
where \(\xi = \frac{g}{L}\) is the filling factor. Importantly, ε_{c} is a nonlocal correction parameter capturing the polaritonic nearfield coupling of adjacent ribbons and the corresponding nonlocal correction^{8,9}, which depend on grating period, thickness, and filling factor.
We note that strong polaritonic nearfield coupling of the hBN ribbons and the associated nonlocal response cannot be captured by a standard effective medium model based on the Maxwell–Garnett (MG) approximation. This is because the MG model assumes that each ribbon is polarized only by the external field (illumination). However, the polarization of each ribbon in the grating also depends on the fields generated by the adjacent ribbons, implying that the grating needs to be treated as a spatially dispersive and thus nonlocal medium when the interribbon coupling is strong and dependent on the polariton momentum k. Nonlocal modeling of polaritoncoupled nanoribbons has recently been demonstrated with densely packed twodimensional (2D) graphene nanoribbon arrays^{8,9}. To that end, the nonlocal correction parameter \(\sigma _{\mathrm{c}} =  i\frac{{2\omega \varepsilon _0L}}{\pi }\ln \left[ {\csc \left( {\frac{\pi }{2}\xi } \right)} \right]\) was introduced to calculate the effective conductivity of the graphene ribbon array according to \(\sigma _{{\mathrm{eff}},x} = \left[ {(1  \xi )/\sigma _{\mathrm{g}} + \xi /\sigma _{\mathrm{c}}} \right]^{  1}\) and \(\sigma _{{\mathrm{eff}},y} = \left( {1  \xi } \right){\upsigma}_{\mathrm{g}}\), with σ_{g} being the graphene conductivity. We employ this model to obtain the corrected permittivity model of our hBN metasurface of finite thickness h. To that end, we use the relation \(\sigma \cong  i\omega h\varepsilon _0\varepsilon\) to obtain the nonlocal correction parameter \(\varepsilon _{\mathrm{c}} = \frac{{2L}}{{\pi h}}\ln \left[ {\csc \left( {\frac{\pi }{2}\xi } \right)} \right]\) that is used in Eq. 1. Treating the hBN metasurface as a 2D conductivity sheet is justified since the hBN grating thickness h and the grating period L are much smaller than the polariton wavelength (i.e., λ_{p} » h, L) (ref. ^{6,31}). Note that in our previous work^{5}, we used the standard MG permittivity model (i.e. without the parameter ε_{c}) to describe inplane hyperbolic phonon polariton on the same hBN metasurface. This is possible as the polaritonic coupling of the hBN ribbons is negligible within the inplane hyperbolic frequency range.
The effective permittivity ε_{eff,x} calculated according to Eq. 1 (red line in Fig. 1b) reveals the emergence of a new TO phonon frequency at ω_{STO} = 1478 cm^{−1}, perpendicular to the ribbons. It results from the strong collective nearfield coupling of the dipolar PhP resonance of the individual hBN ribbons (Supplementary Figs. 1–3), analog to the TO phonon in polar crystals. For this reason, we name this collective mode a synthetic TO phonon (STO)^{32}. Note that the standard MG effective medium model also predicts the existence of the STO resonance (dashed gray line in Fig. 1b). However, it is shifted by about 50 cm^{−1} to higher frequencies because polaritonic nearfield coupling and nonlocal effects induced by the grating geometry are not considered. As shown below, the STO predicted by our modified effective medium model (Eqs. 1 to 3) is in excellent agreement with both numerical and experimental results (see Fig. 1c, Fig. 3 and Supplementary Figs. 1 and 4).
Numerical verification of anisotropic polaritons in the hBN metasurface
As a result of the STO, the photonic local density of states (PLDOS) on the metasurface differs dramatically from the one of natural hBN flakes, as confirmed by the calculated PLDOS spectra shown in Fig. 1c. In the simulations a point dipole source is placed at the height of 200 nm above the surface (see Methods). The PLDOS spectrum of the hBN layer of h = 20 nm thickness exhibits a strong peak around ω =1450 cm^{−1} (dashed gray line), due to the excitation of a fundamental PhP “waveguide” mode in the hBN slab^{22,26}. In contrast, the metasurface (modeled as a h = 20 nm thick layer of an effective medium according to Eqs. 1–3) exhibits two PLDOS peaks located on either side of the STO (at 1430 and 1480 cm^{−1}, blue line). These two peaks are verified by a fullwave numerical simulation using a real 20nmthick grating structure (red line) and indicate that two distinct PhP modes are excited on the metasurface (Supplementary Fig. 3). This result further corroborates the validity of the modified effective medium theory described by Eqs. 1–3 (in contrast to the standard MG theory, which does not account for the strong nearfield coupling of polariton modes and fails in quantitative prediction of the STO frequency).
Below the STO frequency, the dipole excites PhPs possessing an inplane hyperbolic dispersion^{5} (because Re(ε_{eff,x}) > 0 and Re(ε_{eff,y}) < 0), which are formed by nearfield coupling of the waveguide polaritons propagating along individual nanoribbons (note that this coupling is weak and thus yields only a positive value for ε_{eff,x}). The propagation of these inplane hyperbolic PhPs (HPhPs) is highly anisotropic along the metasurface, exhibiting the typical ray pattern of hyperbolic polaritons (Fig. 1d). Fourier transform (FT) of the nearfield distribution E_{z} indeed yields a hyperbolic IFC describing the polariton momentum k (Fig. 1e) in momentum space at fixed frequency.
Above the STO frequency, the dipoleexcited PhPs have extremely elliptical inplane dispersion, owing to ε_{eff,x} and ε_{eff,y} being negative but with largely different absolute values. The elliptical PhPs (EPhPs) are due to the strong collective nearfield coupling (yielding a negative ε_{eff,x}) of individual nanoresonators (i.e. the nanoribbons exhibiting Fabry–Pérot polariton resonances, see our verifications in Fig. 3 and Supplementary Fig. 7), which are visualized by the simulation shown in Fig. 1f, where the dipole source launches a collimated polariton beam with lateral confinement of about 150 nm (∼λ/45). The FT of the nearfield distribution confirms the polaritons´ extreme elliptical IFC in kspace (Fig. 1g; note that the IFC is not a perfectly closed ellipse owing to polariton damping by intrinsic material losses).
Although it could be expected that EPhPs near the STO may suffer from the large imaginary part of ε_{eff,x} (Supplementary Fig. 5), their absolute propagation length is comparable to the one of HPhPs (Fig. 1d and Supplementary Fig. 6). This can be explained by the large negative real part of ε_{eff,x} near the STO, which actually reduces the field confinement inside the material, repelling the fields and hence reducing the absorption^{17}. By increasing the frequency, the EPhPs become more confined and decay faster (Fig. 1h, more simulations shown in Supplementary Fig. 4), exhibiting a weaker ellipticity (Fig. 1i) owing to the decreasing figure of merit Re(ε_{eff,x})/Re(ε_{eff,y}). The highly collimated EPhP modes described in Fig. 1f, h are also known as canalization modes^{17,21}. They have been explored in bulk metamaterials^{21,33} or twodimensional metasurfaces^{17} for various applications, including hyperlensing^{9,17,34} and subwavelength focusing^{33}. Theory also has been predicting the canalization of plasmon polaritons on metallic metasurfaces (graphene^{17}, black phosphorus^{13}, or metals^{7}) at optical (visible and infrared) frequencies, which, however, has not been experimentally demonstrated. In our work we do not only theoretically predict the canalization of lowloss deeplyconfined phonon polaritons but also demonstrate them experimentally via infrared nanoimaging.
Polaritoninterferometric nanoimaging of HPhPs and EPhPs
To image the canalization of EPhP modes, we fabricated a hBNmetasurface (w = 75 nm and g = 25 nm) on a 20nmthick flake of monoisotopic lowloss hBN^{5,28,35,36} (schematics are shown in Fig. 2a; for details see Methods). We first performed polaritoninterferometric nanoimaging on the metasurface with a scatteringtype scanning nearfield optical microscope (sSNOM)^{37,38}. The metallized tip of an atomic force microscope (AFM) is illuminated by a ppolarized infrared laser beam, operating as an infrared nanoantenna that concentrates the incident field at its sharp apex, yielding a nanoscale nearfield spot for launching the polaritons. The polaritons propagate away and are reflected at the boundaries of the metasurface. They propagate back and interfere with the polariton field below the tip, forming interference fringes (with spacing equals to half the polariton wavelength, λ_{p}/2), which are visualized by recording the tipscattered field as a function of tip position^{37,38}.
Figure 2b, c present polaritoninterferometry images (amplitude signals, s) taken at frequencies within the HPhP and EPhP regions, respectively at ω = 1415 and ω = 1510 cm^{−1}. On the grating area, we observe only horizontal fringes in the HPhP spectral range (Fig. 2b), and only vertical fringes in the EPhP spectral range (Fig. 2c). The two distinct fringe orientations reveal the different propagation directions of the polaritons, as predicted in our simulations shown in Fig. 1d, h, and h. They provide a first experimental indication for the existence of EPhPs above STO.
Hyperspectral nanoimaging of polariton evolution in the hBN metasurface
To explore the frequency range and dispersion of the two types of polaritons, we recorded nearfield spectroscopic line scans (along the lines marked in Fig. 2b, c). In the line scan parallel to the ribbons (Fig. 3a), we observe a horizontal feature around ω = 1400 cm^{−1}, matching well the TO phonon of hBN (see Fig. 3i for a comparison of experimental and simulated nearfield spectra). Above the TO, we observe a series of fringes (indicated by dashed black curves), whose spacing is reducing with increasing frequencies. They reveal the inplane HPhPs propagating parallel to the ribbons, whose wavelength is shrinking at higher frequencies. Around ω = 1500 cm^{−1} we observe a broad horizontal (nondispersive) feature that fits well the STO (see Fig. 3i).
In the line scan perpendicular to the grating (Fig. 3b), we again observe the horizontal features corresponding to the TO phonon and the STO, respectively (see also Supplementary Fig. 7). In the whole spectral region between TO and STO (the HPhP range) we do not observe fringes, indicating the absence of polariton propagation perpendicular to the grating. More interestingly, we see an interference fringe (marked by a blue dashed curve) above STO. Its distance to the boundary decreases with increasing frequency, corroborating that the fringes in Fig. 2c indeed reveal a polariton propagating perpendicular to the grating. We note that the fringe intensity is modulated by the ribbons. Inside the gaps between the ribbons, the fringe intensity is higher, as here the tip more efficiently launches the polariton propagating perpendicular to the ribbons. The tiplaunched polariton is reflected at the boundary, giving rise to the observed fringe (illustration in Fig. 3h). This propagating polariton mode is actually caused by the polaritonic nearfield coupling of neighboring ribbons, similar to energy transport in plasmonic particle chains^{39}. We further observe a horizontal series of bright spots at ω = 1570 cm^{−1}. However, this feature is not accompanied by interference fringes at higher frequencies, indicating a purely localized mode. A zoomin image and analysis (Supplementary Fig. 7) indeed show that the bright spots correspond to a localized secondorder transverse polariton resonance of the ribbons, as illustrated in Fig. 3g.
Numerical simulations of the spectroscopic line scans (Fig. 3c, d; a dipole source was scanned above the grating and the field below the dipole was recorded, see Methods) confirm our experimental results, particularly the interference fringe (dashed blue curve) above the STO and the localized ribbon resonance around ω = 1570 cm^{−1} (marked by red arrow). We repeated the simulations for a metasurface described by the nonlocal effective medium theory described by Eqs. 1 to 3 (Fig. 3e, f), reproducing well the results of Fig. 3c, d. However, the signal modulation introduced by the grating is absent (due to spatial homogenization of the metasurface), thus yielding a clearer map. The good agreement of the different simulations confirms the validity of our nonlocal effective medium model, which is particularly important to properly capture the properties of the canalization polaritons near the topological transition.
Altogether, Fig. 3a, b experimentally verify two spectral regions (separated by the STO) within the hBN Reststrahlen band, in which two different types of polaritons exist. Their different propagation directions provide experimental evidence that the IFC of the polariton momentum undergoes a topological transition across the STO. To demonstrate tunability of the STO resonance experimentally (theoretical calculations shown in Supplementary Fig. 2), we show in the Supplementary Fig. 8 that the STO can be tuned from 1480 cm^{−1} (for the metasurface shown in Fig. 3 with ribbon width w = 75 nm and gap size g = 25 nm) to 1460 cm^{−1} by fabricating a metasurface with a different filling ratio (w = 220 nm and g = 40 nm).
Realspace imaging of canalization polariton modes
To experimentally visualize the canalization mode near the STO, predicted in Fig. 1f, h, we imaged the polariton emitted from an infrared antenna (a gold rod) on another metasurface (fabricated together with the one in Fig. 2 on the same flake, topography in Fig. 4a). The antenna concentrates the midinfrared illumination to nanoscale spots at its antenna extremities, acting as a nanoscale source for launching the polaritons. Figure 4b–d presents the experimental images of the antennalaunched polaritons propagating and decaying along the metasurface (see also Supplementary Fig. 9). In the grating area, periodic horizontal bright lines are observed. As explained in Fig. 3, they correspond to the strong nearfields inside the gaps, because of enhanced tippolariton coupling. More importantly, we indeed observe the deepsubwavelength canalization EPhP — a collimated polariton beam (with a lateral confinement of 310 nm, ∼λ/22, see Fig. 4j; see also the FT results in Supplementary Fig. 10) emitted from the antenna extremity. At ω = 1495 cm^{−1} it is collimated over at least five ribbons (Fig. 4b). At higher frequencies, the polaritons are more confined and thus decay faster (Fig. 4c, d), but they still extend farther than the antenna fields decaying on a dielectric substrate (ε_{hBN} ≈ 1 at ω = 1735 cm^{−1}, Fig. 4h). In a control experiment, we imaged the polaritons launched by the antenna on the unpatterned hBN (Fig. 4g, topography in Fig. 4f), showing radial propagation along the hBN, in striking contrast to the canalization modes (Fig. 4b–d).
We numerically verified the antennalaunched canalization polaritons by simulating the electromagnetic nearfield distribution around the antenna on the grating structure (Fig. 4e). On the other hand, the simulated canalization mode propagates longer than the experimental one, which can be explained by stronger damping in the experiment caused by fabrication uncertainties and material damage from etching (see discussion section below).
Discussion
An intriguing result of our experiments (Fig. 4b–d) is the direct visualization of energy flow transported along a chain of coupled polaritonic nanoresonators^{39}. More precisely, we use the antenna to locally illuminate the first ribbon. Energy flows directionally to the next ribbons owing to the polaritonic nearfield coupling and the extreme inplane anisotropy of the canalization mode, which avoids energy spreading in other directions. The electricfield decay length for this process is quantified to be about 220 nm (blue line in Fig. 4i, background subtracted, see Supplementary Fig. 11) by fitting the nearfield profile (along the vertical dashed blue line in Fig. 4b) with an exponential decay (dashed red line in Fig. 4i). This value is much larger than the one of antenna fields on the bare dielectric substrate (green line in Fig. 4i, decay length < 50 nm). Our results therefore provide a direct realspace observation of energy flow through coupled infrared phononpolaritonic nanoresonators separated by nanoscale air gaps, with important consequences for the development of infrared photonic and thermal devices based on nearfield polaritonic coupling. These results also confirm the important role of strong coupling between neighboring resonators to achieve extreme anisotropy and canalization. The consequent nonlocality, well captured by our homogenized metasurface model, plays an important role in the physics demonstrated in this paper.
We finally discuss the lifetime of the canalization polaritons. According to simulations (Supplementary Fig. 6), the canalization polaritons on the metasurface exhibit a lifetime that is comparable to that of phonon polaritons in hBN slabs that have the same thickness as the metasurface. In the experiment, however, the measured propagation length is about 2.5 times shorter. We explain this finding by additional polariton damping caused by polariton scattering and absorption at inhomogeneities and eventually material damage at the ribbon edges induced by etching. The simulations also reveal that the polariton propagation lengths and lifetimes can be increased at least by a factor of 2 by removing the relatively lossy SiO_{2} substrate (by suspending the metasurface) or by replacing it by a lowloss substrate such as CaF_{2}. It thus can be expected that improving the fabrication process and employing lowloss substrates can enhance the propagation length of the canalization polaritons in potential future applications.
STOs (strong collective coupling of the metasurface elements) and topological transitions may also be envisioned in other types of metasurfaces, for instance based on strongly coupled graphene (or black phosphorus) nanoresonators^{8,9}, which may lead to electricallytunable collective resonances and canalization polaritons for sensing and thermal emission applications at infrared and THz frequencies. The demonstrated deepsubwavelength canalization polaritons hold promise for many exciting applications, including inplane hyperlensing^{8,9}, onchip collimated polariton emitting, waveguiding, and focusing^{8,9,17}.
Methods
Nanoimaging
We used a commercial sSNOM system (from Neaspec GmbH) based on an atomic force microscope (AFM). The Ptcoated AFM tip (oscillating vertically at a frequency Ω ≈ 270 kHz) was illuminated by light from a wavelengthtunable continuouswave quantum cascade laser. The backscattered light was collected with a pseudoheterodyne interferometer^{40}. To suppress background contribution in the tipscattered field, the interferometric detector signal was demodulated at a higher harmonic nΩ(n ≥ 2), yielding nearfield amplitude s_{n} and phase φ_{n} images. Figures 2 and 4 show amplitude s_{3} images.
Spatiospectral nearfield observation
For spatiospectral observation shown in Fig. 3, the sSNOM tip and sample were illuminated with a broadband midinfrared laser. The tipscattered signal was analyzed with an asymmetric Fourier transform spectrometer (based on a Michelson interferometer), in which tip and sample were located in one of the interferometer arms^{25,26}. An interferogram was measured by recording the demodulated detector signal (the harmonic 3Ω for background suppression) as a function of the position of the reference mirror, at a fixed tip position. Subsequent Fourier transform of the recorded interferogram yields a complexvalued nearfield point spectrum^{25,26}. We scanned the tip parallel or perpendicular to the hBN ribbons, respectively. At each tip position, we recorded a complexvalued nearfield point spectrum. By plotting the recorded nearfield amplitude s_{3} as a function of the tip position and the operation frequency, we obtained the images shown in Fig. 3a, b.
Sample preparation
For experiments we used isotopically (^{10}B) enriched hBN (details of the growing process can be found in refs. ^{35,36}), which exhibits ultralowloss phonon polaritons^{28}. We fabricated infrared metasurfaces by the etching process reported in ref. ^{5}.
Numerical simulations
We used a finiteelementmethod based software (COMSOL Multiphysics) for simulations. In the simulations, the permittivity of the isotopically enriched hBN was taken from ref. ^{5}. Simulations of the real grating metasurface (referred to as grating and/or MS) consider the real threedimensional geometry (given by w, g, L, and h) and hBN permittivity. Simulations of the homogenized metasurface (referred as to effm) consider a homogeneous slab of thickness h=20 nm with effective permittivities ε_{eff,x}, ε_{eff,y,} and ε_{eff,z} described in Eqs. 1–3. Further details of the simulations are provided in the Supplementary Note 1.
Data availability
The data that support the findings of this study are available from the corresponding author on reasonable request.
Code availability
All codes used to evaluate the data are available from the corresponding author upon request.
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Acknowledgements
The authors acknowledge financial support from the Spanish Ministry of Science, Innovation and Universities (national projects MAT201788358C3, MAT201565159R, MAT201565525R, RTI2018094830B100, RTI2018094861B100, and the project MDM20160618 of the Maria de Maeztu Units of Excellence Program) and the Basque Government (PhD fellowship PRE 2018 2 0253 and grant No. IT116419). Further, support from the Materials Engineering and Processing program of the National Science Foundation, award number CMMI 1538127, the II−VI Foundation, the Air Force Office of Scientific Research MURI program, the Vannevar Bush Faculty Fellowship, and the Office of Naval Research is greatly appreciated. P.L. acknowledges the startup funding from Huazhong University of Science and Technology. C.W.Q. acknowledges financial support from A*STAR Pharos Program (grant number 15270 00014, with project number R263000B91305).
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P.L. and R.H. conceived the study. Sample fabrication was performed by I.D. and S.V., coordinated by S.V., and supervised by F.C. and L.E.H. P.L. performed the experiments and simulations. G.H., M.T., and F.J.A.M. contributed to the nonlocal modeling and the simulations. S.L. and J.H.E. grew the isotopically enriched boron nitride. C.W.Q, A.A., and R.H. coordinated and supervised the work. P.L. and R.H. wrote the manuscript with the input of all other coauthors.
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R.H. is cofounder of Neaspec GmbH, a company producing scatteringtype scanning nearfield optical microscope systems, such as the one used in this study. The remaining authors declare no competing interests.
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Li, P., Hu, G., Dolado, I. et al. Collective nearfield coupling and nonlocal phenomena in infraredphononic metasurfaces for nanolight canalization. Nat Commun 11, 3663 (2020). https://doi.org/10.1038/s41467020174259
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