Planar refraction and lensing of highly confined polaritons in anisotropic media

ARTICLE Planar refraction and lensing of highly confined polaritons in anisotropic media J. Duan1,2,12, G. Álvarez-Pérez 1,2,12, A. I. F. Tresguerres-Mata 1, J. Taboada-Gutiérrez 1,2, K. V. Voronin 3, A. Bylinkin 4,5, B. Chang6, S. Xiao 7, S. Liu8, J. H. Edgar 8, J. I. Martín1,2, V. S. Volkov 3,9, R. Hillenbrand 10,11, J. Martín-Sánchez 1,2, A. Y. Nikitin 5,10 & P. Alonso-González 1,2✉ Refraction between isotropic media is characterized by light bending towards the normal to the boundary when passing from a lowto a high-refractive-index medium. However, refraction between anisotropic media is a more exotic phenomenon which remains barely investigated, particularly at the nanoscale. Here, we visualize and comprehensively study the general case of refraction of electromagnetic waves between two strongly anisotropic (hyperbolic) media, and we do it with the use of nanoscale-confined polaritons in a natural medium: α-MoO 3 . The refracted polaritons exhibit non-intuitive directions of propagation as they traverse planar nanoprisms, enabling to unveil an exotic optical effect: bending-free refraction. Furthermore, we develop an in-plane refractive hyperlens, yielding foci as small as λ p /6, being λ p the polariton wavelength (λ 0 /50 compared to the wavelength of free-space light). Our results set the grounds for planar nano-optics in strongly anisotropic media, with potential for effective control of the flow of energy at the nanoscale. https://doi.org/10.1038/s41467-021-24599-3 OPEN 1Department of Physics, University of Oviedo, Oviedo, Spain. 2Center of Research on Nanomaterials and Nanotechnology, CINN (CSIC-Universidad de Oviedo), El Entrego, Spain. 3Center for Photonics and 2D Materials, Moscow Institute of Physics and Technology, Dolgoprudny, Russia. 4CIC nanoGUNE BRTA, Donostia–San Sebastian, Spain. 5Donostia International Physics Center (DIPC), Donostia/San Sebastián, Spain. 6National Centre for Nano Fabrication and Characterization, Technical University of Denmark, Lyngby, Denmark. 7DTU, Fotonik, Department of Photonics Engineering and Center for Nanostructured Graphene, Technical University of Denmark, Lyngby, Denmark. 8Tim Taylor Department of Chemical Engineering, Kansas State University, Manhattan, KS, USA. 9GrapheneTek, Skolkovo Innovation Center, Moscow, Russia. 10 IKERBASQUE, Basque Foundation for Science, Bilbao, Spain. 11 CIC nanoGUNE BRTA and Department of Electricity and Electronics, UPV/EHU, Donostia - San Sebastian, Spain. 12 These authors contributed equally: J. Duan, G. Álvarez-Pérez. ✉email: [email protected] NATURE COMMUNICATIONS | (2021) 12:4325 | https://doi.org/10.1038/s41467-021-24599-3 | www.nature.com/naturecommunications 1 1234567890():,; Hyperbolic electromagnetic waves arise as a consequence of the intrinsic anisotropy of the crystal lattice in natural media1–3and of the artificially engineered anisotropic dielectric environment in metamaterials4–12, which leads to a metallic-like response (negative permittivity) along one (two) of the principal axes in such media and a dielectric-like response (positive permittivity) along the other two (one). Despite their fundamental interest and their potential for the development of new optical applications, these exotic waves are still scarcely explored. In particular, refraction of hyperbolic waves has only been studied for the case in which the incident beam comes from an isotropic medium, typically free space5–7. As such, the general case of refraction, involving hyperbolic waves in which both the incident and the refracted waves exhibit non-collinear wavevector kand energy flux S, remains experimentally unexplored, particularly at the nanoscale, where the specific case of negative refraction of highly confined polaritons has only recently been theoretically proposed13,14. The study of the general case of refraction could extend our capabilities to control the flow of light. Importantly, the recent discoveries of phonon polaritons (PhPs) in van der Waals crystals15,16 with hyperbolic dispersion, such as h-BN17–19,α-MoO 3 20–25, and α-V 2 O 5 26, have provided unique material platforms to study optical phenomena27,28 within strongly anisotropic natural media. In particular, PhPs in αMoO 3 feature in-plane hyperbolic propagation, ultra-low losses, and strong confinement, offering the possibility to visualize refraction directly on the crystal surface and at the nanoscale, which can open new routes in planar nano-optics29. Here, we theoretically and experimentally demonstrate the general case of refraction at the interface between two strongly anisotropic (hyperbolic) media. Importantly, we do it at the nanoscale and in a low-loss natural medium by visualizing the propagation of polaritons as they traverse planar nanoprisms tailored on the surface of α-MoO 3 . Our images show nonintuitive directions of propagation and strong confinement of the refracted waves, enabling to unveil an exotic optical effect: bending-free refraction, which extends the current capabilities to control the propagation of light at the nanoscale. Furthermore, we develop an in-plane refractive hyperlens, yielding foci as small as λ p /6, being λ p the polariton wavelength (λ 0 /50 with respect to the wavelength of light in free space). Our findings provide fundamental knowledge and an effective strategy for the manipulation of polaritons in anisotropic media, paving the way for integrated flat subwavelength optics. Results and Discussion Theory of refraction between hyperbolic media. The unique properties of polaritons in hyperbolic media can be better understood by analyzing their isofrequency curve (IFC), a slice of the polariton dispersion surface in momentum-frequency space (kx,ky;ω) by a plane of constant frequency ω0. The IFCs of polaritons in two different hyperbolic media are illustrated in Fig. 1a, b. For convenience, we consider these two media to be defined by the same hyperbolic slab (with representative permittivity εx=−5, εy=1, εz=5, see ‘Methods’) placed on two different dielectric substrates (with permittivities εsub =1 and εsub =5). In both cases, the IFCs describe open hyperbolas (black and gray curves, respectively). As a result, not all wavevectors k are allowed in these media, which implies that polaritons cannot propagate along all in-plane directions in real space. In fact, propagation is only allowed within the sectors jtanðkx=kyÞj<ffiffiffiffiffiffiffiffiffiffiffiffiffiffi εy=εx qlimited by the asymptotes of the hyperbola in the ðkx;kyÞplane (see Fig. 1a, b). Additionally, the Poynting vector S—which determines the propagation direction of the polariton and is normal to the IFC30,31—is not in general collinear with k, as indicated in Fig. 1a (they are collinear only along the x-axis in Fig. 1a). As such, the properties of propagating polaritons in hyperbolic media are different to those in isotropic media, where the IFCs are circular (see dashed cyan curve in Fig. 1a) and polaritons, as is well-known, are allowed to propagate along all in-plane directions in real space with the same absolute value of the wavevector, k, which is always collinear to S. Importantly, such properties of polaritons propagating in hyperbolic media have a dramatic effect when they refract at a boundary between two different hyperbolic media. Particularly, momentum conservation at the boundary implies that the projection kkof the incident and refracted wavevectors ( kin and kout, respectively) must be conserved (with kk¼kinsinφ, where φis the angle of the boundary as shown in Fig. 1a), giving rise to the Fig. 1 Schematics of refraction of polaritons between two hyperbolic media. a Isofrequency curves of polaritons propagating in a hyperbolic slab (with ε x =−5; ε y =1; ε z =5) placed on two different semi-infinite substrates with εsub =1 (black curve) and εsub =5 (gray curve) that define two different hyperbolic media (medium 1 and 2, respectively). The incident wave in medium 1 is characterized by collinear kin and Sin (as in an isotropic medium, indicated by a dashed cyan circle). Upon refraction into medium 2, momentum conservation at the boundary (orange line), kjj (kk¼kin sinφ, where φis the angle of the boundary), is fulfilled by non-collinear kout and Sout. The dashed orange lines represent the normal to the boundary. bThe general case of refraction between two hyperbolic media is represented by an incident wave from medium 1 with non-collinear kin and Sin (normal to the isofrequency curve). When the wave refracts into medium 2, momentum conservation at the boundary (orange line) is fulfilled by noncollinear kout and Sout. The dashed orange lines represent the normal to the boundary. cReal-space illustration of refraction between two hyperbolic media shown in awhere the incident wave exhibits collinear kin and Sin, i.e. θink¼θinS, giving rise to non-collinear kout and Sout, i.e. θoutS≠θoutk.d Real-space illustration of the general case of refraction between two hyperbolic media shown in bwhere both the incident and the outgoing wave exhibits non-collinear kand S, i.e. θink≠θinSand θoutk≠θoutS. The tangents parallel to both hyperbolas give rise to bending-free refraction, i.e. θinSθoutS. The orange dashed lines in c,drepresent the normal to the boundary. The white and gray regions in c,dcorrespond to α-MoO 3 /air and α-MoO 3 /SiO 2 , respectively. ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-021-24599-3 2NATURE COMMUNICATIONS | (2021) 12:4325 | https://doi.org/10.1038/s41467-021-24599-3 | www.nature.com/naturecommunications generalized Snell’s law32: kin sin θink  ¼kout sin θoutk  ;ð1Þ where θinkand θoutkare the angles that kin and kout form with the normal to the boundary, respectively (see Fig. 1c, d). The propagation directions of the incident and refracted waves are then given by Sin and Sout, respectively, i.e. the directions normal to the hyperbolic IFCs for each case, which are in general noncollinear with kin and kout (Fig. 1b–d), and thus refraction can occur at angles θoutSthat can be different from θoutk. This behavior is in stark contrast to that in isotropic media, in which momentum conservation at the boundary implies that refraction occurs always at θoutk=θoutS. A particular case of refraction between hyperbolic media for an incident wave with collinear kin and Sin is shown in Fig. 1a. We observe that the refracted wave (Sout) bends away from the direction of Sin towards the boundary (Fig. 1c), in contrast to what is expected in isotropic media for a wave passing from a low refractive index to a high refractive index, where Sout bends towards the normal to the boundary (see Supplementary Fig. 8). Also, the modulus of kout is much larger than that of kin, showing a strong wavelength reduction. The general case of refraction occurs when kin and Sin of the incident wave are not collinear. This case, which has not been tackled experimentally to date, is sketched in Fig. 1d, where a wave impinges at a tilting angle θinSon the boundary between two hyperbolic media. The boundary is also tilted a given angle with respect to the crystal axes. Interestingly, due to the similar shapes of the IFCs in the considered hyperbolic media (α-MoO 3 / air and α-MoO 3 /SiO 2 , black and gray curves in Fig. 1b, respectively), the Poynting vectors of the incident and refracted waves are parallel for almost any kin and boundary angle φ, especially in the region where the arms of both hyperbolic IFCs are straight. Hence, the refracted wave propagates almost parallel to the incident wave (i.e. θinSθoutS), as if the incident wave had been transmitted directly without any change in its propagation direction (black and blue arrows in Fig. 1d). This feature opens the door to the realization of bending-free refraction at arbitrary incident angles in anisotropic media, which is not possible in isotropic media. Note, however, that the modulus and direction of kout is very different from the modulus and direction of kin, which opens the door to direct engineering of the wavelength and wavefront without modifying the direction of propagation of the wave. Nanoimaging of refraction of in-plane hyperbolic polaritons. We experimentally demonstrate and comprehensively study the characteristics of refraction of polaritons propagating in hyperbolic media. This provides the demonstration of these effects at the nanoscale and/or in a natural medium. To do so, we design and fabricate planar prisms (“Methods”) in a slab of the naturally hyperbolic van der Waals crystal αMoO 3 . We then visualize the propagation of hyperbolic phonon polaritons (HPhPs) passing through them by polariton wavefront mapping using a scattering-type scanning near-field optical microscope (s-SNOM, see “Methods”). To define the prisms, we etch away triangular regions in a silica (SiO 2 ) substrate on top of which we place a 160-nm-thick αMoO 3 slab, thus forming a region α-MoO 3 /air with a different refractive index, and thus different polaritonic dispersion, with respect to the region α-MoO 3 /SiO 2 . The different polaritonic dispersions in these two regions are clearly corroborated by the near-field image of Fig. 2a, taken at an incident wavelength λ0=11.3 μm. Specifically, we observe HPhPs launched inside the prism (highlighted by white dashed lines) by the flake edge (see “Methods”and Supplementary Fig. 4), which propagate with collinear kin and Sin and longer wavelength λin (whitearrow),i.e.smaller wavevector kin =2π/λin (black arrow), than outside the prism (λp,red arrow), thus revealing the lower refractive index of the prisms. We also observe in the same near-field image that when the HPhPs reach a boundary of the prism tilted at an angle θin~ 55°, they refract into the αMoO 3 /SiO 2 region (note that reflection is expected to be very weak, see Supplementary Note 8) along a different direction (Soutexp , blue arrows) with respect to which their wavefronts are tilted (koutexp, green arrows). Thus, the refracted energy flux and wavevector are not collinear, in agreement with our predictions for hyperbolic refraction in Fig.1a. Importantly, while the wavevector, koutexp, refracts towards the normal (note that polariton launching by the prism boundary can be ruled out, see Supplementary Note 3), the energy flux, Soutexp,bendsawayfromit. This is in stark contrast to what is expected for a wave propagating from a lower refractive index region to a higher refractive index region in isotropic media. In addition, the modulus of the refracted wavevector koutexp (~6.48 μm−1) is considerably larger than that of the incident wavevector kin (~2.09 μm−1) and that of the polariton wavevector along the x-direction outside the prism kp(~4.08 μm−1), revealing a strong wavelength reduction resulting from refraction in hyperbolic media. These findings are in perfect agreement with the case predicted in Figs. 1a–c for hyperbolic refraction considering collinear kin and Sin . To unambiguously verify the features of refraction between hyperbolic media, we carry out full-wave numerical simulations which mimic our experiments33 (see “Methods”). The resulting spatial distribution of the out-of-plane component of the electric field, Re Ezx;y  ;is plotted in Fig. 2b, clearly showing refraction of both the energy flux (Sout) and the wavevector (kout), in excellent qualitative and quantitative agreement ( kout ~6.35μm−1)withthe experimental image in Fig. 2a. In addition to numerical simulations, we also validate our experimental results by performing analytical calculations34 (Fig. 2c) analogous to those shown in Fig. 1.Namely, we calculate the IFCs of HPhPs in α-MoO 3 /air (gray curve), and αMoO 3 /SiO 2 (black curve) regions at λ0=11.3 μm, and, applying the condition of momentum (wavevector) conservation at the boundary (orange line), we extract the Poynting vector and wavevector of the refracted polaritons. Again, we observe refraction of the energy flux (Sout, blue arrow) with a tilted wavevector (kout,greenarrow)in excellent agreement with the experiment, as well as with the fullwave numerical simulations. To further analyze refraction in hyperbolic media, we also perform experiments at a different illuminating wavelength λ0 (different tilting angles of the prism boundary θare also shown in Supplementary Fig. 2), as shown in Fig. 2d–f for λ0=11.1 μm. Interestingly, we observe that, in this case, both the angular separation between koutexp and Soutexp and the confinement effect are larger, being the refracted wave (Soutexp) almost parallel to the prism boundary and the modulus of koutexp (12.47 μm−1) about four times larger than that of kin (~3.14 μm−1) and 2.1 times larger than that of kp(~5.92 μm−1). Altogether, these results demonstrate the efficient refractive nature of our planar prisms, enabling us to visualize in real-space three important features of highly confined polaritons refracted at the boundary between two hyperbolic media: (i) large tilting of their wavefronts (given by koutexp) with respect to their propagation direction (given by Soutexp), (ii) counter-intuitive directions of propagation, and (iii) subwavelength confinement (with respect to polaritons along the same crystal axis in the same medium). Sub-diffractional planar lensing of hyperbolic polaritons. Such unique features of refracted polaritons in naturally in-plane hyperbolic media open the door to focus ultra-confined NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-021-24599-3 ARTICLE NATURE COMMUNICATIONS | (2021) 12:4325 | https://doi.org/10.1038/s41467-021-24599-3 | www.nature.com/naturecommunications 3 polaritons in a planar geometry. To demonstrate this possibility, we design and fabricate a planar lens in α-MoO 3 (Fig. 3a). As noted above (Fig. 2), when HPhPs with collinear kin and Sin (red arrow in Fig. 3b) refract at a boundary with another hyperbolic medium with higher refractive index (such as when passing from α-MoO 3 /air to α-MoO 3 /SiO 2 ), they bend away from the normal to the boundary (blue arrow in Fig. 3b). This means that, in the case of considering a prism of triangular shape, as that shown in Fig. 3a, all the refracted polaritons (blue arrows) can converge into a single spot, and thus the prism acts as a focusing lens for highly confined polaritons (see Supplementary Note 9 for the analytical description of general shapes of lenses focusing hyperbolic waves). More importantly, as such a lens is based on refraction of HPhPs, the refracted waves can potentially feature infinitely large wavevectors when the boundary is perpendicular to the asymptote of the hyperbolic IFC, which would yield deeply sub-diffractional foci sizes. However, as HPhPs decay exponentially, the intensity at a distant focus can be weak due to propagation losses, which would be more notable for large wavevectors approaching the asymptote of the hyperbolic IFC. Consequently, we designed our triangular lens looking for a compromise between a large refracted wavevector and a long propagation length of the refracted polaritons. According to our theoretical calculations (Supplementary Fig. 7), this compromise is obtained for an angle of the in-plane wavevector of about 62°. Therefore, we fabricate a triangular prism with boundaries perpendicular to this angle (Fig. 3a). The experimental and simulated near-field images for this lens design upon illumination at 11.16 μm are shown in Figs. 3c, d, respectively. In both images, we observe refraction of incident HPhPs with collinear kin and Sin (red arrows) at the lens boundaries (black dashed contour) resulting in HPhPs with non-collinear kout and Sout (green and blue arrows, respectively) propagating along directions almost parallel to the boundaries (blue arrows), which eventually converge, resulting in a focus. This result is in stark contrast to that observed in a similar lens based on refraction of highly confined polaritons in an isotropic material, such as h-BN, in which refracted polaritons bend towards the normal (Supplementary Fig. 8), making all them to diverge (Fig. 3e). The near-field images of our hyperbolic lens reveal that the wavevector kout of the refracted HPhPs is much larger than the wavevector kin of the incident HPhPs (note that in the experimental image there is a contribution of tiplaunched HPhPs with two different kin wavevectors). More importantly, the wavevector kout is also much larger than the wavevector kpof HPhPs propagating along the x-direction in the flake on top of SiO 2 (black arrows). In order to avoid the influence from the permittivity of substrate, we evaluate the focusing resolution by comparing the full-width at half-maximum (FWHM) with the wavelength of polaritons propagating in MoO 3 /SiO 2 (λp¼2π=jkpj). Since the focus shows a FWHM of ~240 nm (red dots and gray curve in Fig. 3f, for experimental and simulated line profiles, respectively), we obtain a focus that is much smaller than the polariton wavelength (λp) along the xdirection, or the free-space illumination (λ0), namely of ~λp=6, or λ0=50. This resolution reveals that a diffraction-limited optical system in hyperbolic media can show a focus that is much smaller than the incident polaritonic wavelength, which again reflects the unique behavior of electromagnetic waves in hyperbolic media. Moreover, this result confirms our planar lens based on refraction of HPhPs as a nano-optical element that greatly exceeds the focusing resolution of any lens based on refraction of highly confined polaritons in isotropic media35–38. Visualization of the general case of refraction. So far, we have visualized refraction in hyperbolic media for the case in which 14 7 0 -7 -14 k y (μm -1 ) -14 -7 0 7 14 k x (μm -1 ) a λ0~11.1μm λ0~11.3μm 2μm θin -30 -15 0 15 30 20 10 0 -10 -20 k y (μm-1) 2μm kinǁSin kinǁSin kout kin kout kin Experiment Simulation Theory + − Exp. Re(σ3) + − Sim. Re(Ez) 2μm kinǁSin Sp Sout kout 2μm Sout kout Sp kout-exp kout-exp Sout-exp Sout-exp Sout Sout λp λp λp λp λin λin Boundary t Boundary x y kp θin kinǁSin kp bc de f Fig. 2 Real-space visualization of refraction between two anisotropic media using highly confined polaritons with collinear incident k and S. a,b Experimental Reðσ3ðx;yÞÞ (a) and simulated ReðEzðx;yÞÞ (b) near-field images of HPhPs propagating in a 160-nm-thick α-MoO 3 flake at λ0=11.3 μm. The white and black dashed lines mark triangular prisms fabricated by etching an air cavity on the SiO 2 substrate below the α-MoO 3 flake. Sin and kin display the direction of propagation and the wavevector of incident polaritons in α-MoO 3 /air, respectively. Horizontal propagation of non-refracted polaritons is marked as kpand Sp. Upon refraction at a boundary of the prism with an angle θin ~ 55°, HPhPs bend away from the normal, Soutexp (blue arrow), with a tilted wavevector koutexp (green arrow). Compared to non-refracted HPhPs, indicated by λp, the refracted HPhPs are stronger confined (with a wavelength about 1.6 times shorter. cAnalytic IFCs of α-MoO 3 /SiO 2 (black hyperbolas) and α-MoO 3 /air (gray hyperbolas) effective media in a, b, and considering momentum conservation at the boundary (orange line), the extracted wavevector and direction of the refracted polaritons, kout and Sout, respectively, are in good agreement with both experiment and simulation. d,eExperimental Reðσ3ðx;yÞÞ (d) and simulated ReðEzðx;yÞÞ (e) near-field images of HPhPs propagating in a 160-nm-thick α-MoO 3 flake at λ0=11.1 μm. The refracted HPhPs propagate almost parallel to the boundary with a wavelength 2.1 times smaller than λp.fAnalytic IFCs of α-MoO 3 /SiO 2 (black hyperbolas) and α-MoO 3 /air (gray hyperbolas) effective media in d,e. The orange dashed lines in a–frepresent the normal to the boundary. ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-021-24599-3 4NATURE COMMUNICATIONS | (2021) 12:4325 | https://doi.org/10.1038/s41467-021-24599-3 | www.nature.com/naturecommunications 0.01.02.03.0 Distance ( μm ) )mron( y t isnetnI f FWHM~240 nm ~λp/6 ~λ0/50 Exp Sim Boundary bSout kout 02040-20-40 0 15 30 -15 -30 k x (μm -1 ) k y (μm -1 ) kinǁSin Sout MoO3/Air Focus -MoO 3 SiO 2 α akout kinǁSin x y + - Exp. Re(σ3) Focus kpSp c kout-exp Sout-exp 1μm MoO3/Air kinǁSin x y Focus kpSp d Sim. Re(Ez) kout Sout 1μm MoO3/Air -MoO 3 SiO 2 α kinǁSin x y + - kpSp e h-BN/Air 1μm h-BN SiO 2 kout Sout Diverging kinǁSin Sim. Re(Ez) + - Fig. 3 Sub-diffractional planar lens based on refraction of HPhPs. a Schematics of a refractive hyperbolic lens fabricated by etching a triangular air cavity on the SiO 2 substrate below the α-MoO 3 flake. The top and bottom boundaries of the lens have the same slope as the boundary in b. Upon refraction at the boundaries (orange contour line), polaritons bend far away from the normal, Sout (blue arrows), with a tilted wavevector kout (green arrows), converging at a focal spot (red dot). Sin and kin display the direction of propagation and the wavevector of incident polaritons in α-MoO 3 /air, respectively. bAnalytic isofrequency curves (IFCs) of polaritons propagating in α-MoO 3 /SiO 2 (black hyperbola) and α-MoO 3 /air (gray hyperbola). When the boundary (orange line) is nearly perpendicular to the asymptote of the open hyperbolic IFC, the refracted polaritons propagate (Sout, blue arrows) almost parallel to the boundary with large non-collinear wavevector (kout, green arrows). cExperimental near-field image of the refractive planar hyperlens (black dashed line) for polaritons in a 170-nm-thick α-MoO 3 slab, at λ0=11.16 μm. The polaritons converge upon refraction at the triangular boundary. Compared to nonrefracted polaritons, indicated by kpand Sp(black arrows) the refracted polaritons, Soutexp (blue arrows), propagate nearly parallel to the boundary. d Simulated near-field image of the refractive planar hyperlens (black dashed line) considered in band visualized in c.e, Simulated near-field image of a refractive lens for in-plane isotropic polaritons in a 170-nm-thick h-BN slab, at λ0=6.5 μm. Upon refraction at the triangular boundary (black dashed line), polaritons bend towards the normal, Sout (blue arrow), with collinear wavevector kout (green arrow), yielding a diverging effect. Horizontal propagation of non-refracted polaritons is marked as kpand Sp.fNear-field intensity profiles (gray line and red dots) extracted through the focus spot along the vertical direction in dand c, respectively. Both curves are normalized to the near-field intensity far away from the lens and flake edges. Confinement factors as large as ~λp/6 and ~λ0/50 are obtained with respect to the polariton and free-space light wavelengths. NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-021-24599-3 ARTICLE NATURE COMMUNICATIONS | (2021) 12:4325 | https://doi.org/10.1038/s41467-021-24599-3 | www.nature.com/naturecommunications 5 collinear kin and Sin refract into polaritons with non-collinear kout and Sout. However, the general phenomenon of refraction involves the incident polaritons exhibiting non-collinear kin and Sin (as sketched in Fig. 1b, d). In the following, we study this fundamental phenomenon in hyperbolic media (Fig. 4). To do this, we again fabricate prisms in α-MoO 3 (following the same structure design as in Fig. 2) and visualize (by s-SNOM) the propagation of HPhPs refracting upon them. As shown in the near-field image of Fig. 4a, we observe that HPhPs launched by the edge of the flake (black arrows) refract at boundary-1 (Fig. 4c), and the outcoming HPhPs propagate with non-collinear kR and SRinside the prism (orange arrows). As such, these polaritons can now be used to visualize the general case of refraction at another boundary of the prism. However, to carry out this experiment successfully, we need to ensure that: (i) the angle of the boundary allows refraction of polaritons according to momentum conservation, and (ii) the HPhPs can reach this boundary within a reasonable propagation distance. We fulfill these conditions by considering a horizontal boundary (boundary-2) followed by a vertical boundary (boundary-3) in the triangular prism, as shown in the near-field image of Fig. 4a. In particular, we observe that HPhPs with non-collinear kRand SR are reflected at boundary-2 (Fig. 4d), yielding polaritons with non-collinear kin and Sin (green arrows) propagating directly towards boundary-3, and reaching it within a reasonably short distance. Consequently, these polaritons refract at boundary-3, which results in polaritons with non-collinear kout and Sout (violet arrows), as predicted by momentum conservation (Fig. 4e), and a -10 0 10 -10 0 10 + - Exp. Re(σ3) + - Sim. Re(Ez) k out S out k R SRkin Sin kout Sout kR 1μm k in S out k out 1μm Boundary-2 Boundary-1 Boundary-3 Boundary-2 Boundary-1 Boundary-3 Non-collinear incident and refracted k and S (at boundary-3) S R S in kS kS x y b S k k y (μm-1) Sin SR cde -10 0 10 k x (μm -1 ) -10 0 10 k x (μm -1 )k x (μm -1 ) Refraction at Boundary-1 Reflection at Boundary-2 Refraction at Boundary-3 k R S R k in S in kin kR Fig. 4 Real-space visualization of the general case of refraction between two anisotropic media using nanoscale-confined HPhPs passing through a bending-free planar prism. a Experimental Re σ3x;yðÞ  near-field images of polaritons propagating in a 231-nm-thick α-MoO 3 flake at λ0=11.0 μm. The black contour line marks a triangular prism fabricated by etching an air cavity on the SiO 2 substrate below the α-MoO 3 flake. A first refraction takes place at boundary-1 (orange solid line) for incident polaritons with collinear kand S(black arrows), yielding refracted polaritons with non-collinear kRand SR (orange arrows). These polaritons then reflect at boundary-2 (green solid line), yielding polaritons with non-collinear kin and Sin (green arrows). A second refraction at boundary-3 (violet solid line) yields polaritons with non-collinear kout and Sout (violet arrows). bSimulated ReðEzðx;yÞÞ near-field images of HPhPs for the case shown in a. Dashed lines in experimental and simulated near-field images indicate the wavefronts of polaritons as they pass through the prisms. c–eAnalytic IFCs of HPhPs in MoO 3 /air (gray curve) and MoO 3 /SiO 2 (black curve), predicting the directions of refraction or reflection of HPhPs at boundary-1 (c), boundary-2 (d), and boundary-3 (e) based on momentum conservation. The orange (c), green (d), and purple (e) solid lines represent the boundary-1, boundary-2, and boundary-3, while the corresponding dashed lines represent the normal to the boundary. ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-021-24599-3 6NATURE COMMUNICATIONS | (2021) 12:4325 | https://doi.org/10.1038/s41467-021-24599-3 | www.nature.com/naturecommunications in good agreement with numerical simulations mimicking the experiment (Fig. 4b). This result unambiguously demonstrates refraction of waves whose energy flux and wavevector directions are non-collinear, and thus constitutes the real-space visualization of the general case of refraction between two hyperbolic media, which, in addition, we demonstrate at the nanoscale and in a natural medium. Furthermore, we highlight that, since kin and kout are close to the asymptotes of the IFCs, where the tangents to both hyperbolas are parallel, Sin and Sout are almost parallel after refraction at boundary-3. As a result, polaritons refract upon boundary-3 behave as if they had been directly transmitted without any change in their direction of propagation, despite the wavevector does change upon this refraction phenomenon. Therefore, refraction upon this prism is bending-free, in excellent agreement with our theoretical prediction shown in Fig. 1d. Such bending-free refractive prism showed in our work will open possibilities to engineer polaritonic wavefronts at the nanoscale without the need of changing their direction of propagation. Conclusions In summary, our work explores the character of refraction between two strongly anisotropic media, which, despite its fundamental importance, has remained elusive up to now. Our observations of refraction of strongly confined polaritons in a hyperbolic biaxial van der Waals crystal reveal an exotic optical effect: bending-free refraction, which opens the door to ondemand steering of light at the nanoscale and in natural media. Furthermore, our demonstration of a subwavelength planar lens at the nanoscale based on hyperbolic refraction paves the way for the development of planar nano-optical elements in anisotropic media. Altogether, our results open new avenues for integrated flat optics, directional energy transfer, and heat management applications, as well as for mid-infrared (bio) sensing. Methods Sample fabrication and characterization. The α-MoO 3 and h-BN nanometerthick flakes were obtained by mechanical exfoliation using a Nitto tape (Nitto Denko Co., SPV 224P) from commercial α-MoO 3 bulk crystals and isotopically enriched h-BN (10B) crystals grown via the atmospheric pressure flux growth method19. First, the bulk crystals were thinned down employing the Nitto tape and transferred on an optically transparent polydimethylsiloxane (PDMS) stamp. Second, selected flakes with sharp edges were identified by an optical microscope and transferred on the target substrate by the dry-transfer technique, which allows a precise positioning and alignment of the flakes on top of the triangular air regions (2D prisms) fabricated on a 500-nm-thick SiO 2 layer grown by wet oxidation on a Si substrate. For an efficient transfer, the substrate was heated up to 200 °C39. The triangular air regions were fabricated in a two-steps process: (i) the air structured patterns were defined using a direct writing system (MLA 100, Heidelberg Instrument) equipped with a 365 nm LED light source on a 1.5-µmthick positive tone photoresist (AZ MiR 701, MicroChemicals) previously deposited on the SiO 2 /Si substrate; (ii) part of the SiO 2 layer was etched away by a fluorine-based plasma using an inductively coupled plasma system (Advanced Oxide Etcher, SPTS), generating gaps with a depth of around 350 nm into the SiO 2 layer. To get rid of photoresist residues and contaminants, the etched substrates were cleaned with a plasma ashing system (PVA TePla 300, PVA TePla AG) before transferring α-MoO 3 and h-BN flakes. Infrared near-field nanoimaging. Infrared nanoimaging was performed with a commercial scattering-type scanning near-field optical microscope (s-SNOM)40,41 from Neaspec GmbH. A tunable DRS Daylight Solutions quantum cascade laser (from 880 to 1100 cm−1) was used as excitation source by focusing the light onto a metal-coated (Pt/Ir) AFM (atomic force microscope) tip oscillating at a tapping frequency of ∼280 kHz with a tapping amplitude ∼100 nm. We illuminate the αMoO 3 flakes with s-polarized infrared light and keep the polarization direction of the incident electric field (E inc ) perpendicular to the flake edges to launch HPhPs. The flakes were raster scanned and the tip-scattered field E sca was recorded with a pseudo-heterodyne Michelson interferometer35 and detected using a liquid nitrogen cooled HgCdTe (MCT) detector. To suppress far-field background signals, the detected signal was demodulated at the nth harmonics of the tip oscillating frequency (n¼3 in our work), yielding the complex-valued near-field signals σn¼sneiφn, with snbeing the near-field amplitude and φnbeing the near-field phase. Throughout the manuscript, we show the real part of the near-field signal, Re(σ3ðx;yÞ), as a function of the tip position ðx;yÞ. Edge-launched polaritons yield fringes with spacing λp, where λpis the polaritonic wavelength, while tip-launched polaritons produce fringes with period of λp=2, existing close to the flake edges (see Figs. 2and 3in the main text). Full-wave numerical simulations using finite-element methods. We simulated s-SNOM near-field images using the finite-element-method numerical software COMSOL MULTIPHYSICS.α-MoO 3 slabs were placed on top of SiO 2 substrates where we defined the same geometry of the air structures as in the experiments. The thickness of the slabs was set to the value extracted from AFM measurements of the α-MoO 3 flakes in the corresponding experimental s-SNOM images. The whole system was illuminated with plane waves with s-polarization perpendicular to the flake edges. Given that the s-SNOM signal can be approximated by the vertical component of the electric field42, we calculated the real part of zcomponent the of the near-field signal at 50 nm above the slab surface, Re Ezx;y  . Meshing types and sizes were optimized to ensure a good convergence of simulated results. The experimental s-SNOM images were well reproduced by our simulated images (see Figs. 2–4in the main text). Yet, since our numerical simulations do not consider the influence of the tip, only the wavefront of edge-launched polaritons is seen (note the difference with the measurements close to the flake edges, where dense fringes appear due to tip-launched polaritons). The dielectric permittivity for α-MoO 3 and isotopically enriched h-BN were taken from references19,33. The real part of the dielectric function of SiO 2 used throughout this work was extracted by fitting the values reported in references43–45 to experimental HPhPs dispersions with a Lorentzian model. HPhPs dispersion and IFC from analytical calculations. The analytically calculated dispersion for polaritons in a biaxial slab embedded between two semi-infinite media34 is given by kðωÞ¼ρ darctan ε1ρ εz  þarctan ε3ρ εz  þπl  ;l¼0;1;2¼; where kis the in-plane momentum, i.e. k¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 xþk2 y q,ε1and ε3are the permittivity of the superstrate and substrate, respectively, dis the thickness of biaxial slab, ρ¼iffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi εz=ðεxcos2αþεysin2αÞ q, and where αis the angle between the x-axis and k. Based on the equation above, we calculate the IFC of HPhPs by varying αfrom 0° to 360° for a fixed incident frequency ω. In all cases we show propagating modes, i.e. those for which jRe kðÞj>jIm kðÞj. In addition, since the polaritonic wavevector depends on ε3(ε1=1 all throughout this work as the superstrate is air), different substrate permittivities can give rise to different effective hyperbolic media. For the general case discussed in Fig. 1in the main text, we select representative permittivity values: εx¼εy¼εz¼−3 for the isotropic slab and εx¼5;εy¼ 1;εz¼5 for the hyperbolic slab, respectively, since at least one negative component is necessary for the existence of polaritons. The slab thickness is set to 100 nm in both cases. The permittivity of the substrates ε3are set to 1 and 5, for the medium in which polaritons are incident and refracted, respectively (white and gray regions in Fig. 1c, d). For the specific case discussed in Figs. 2–4we calculate the ICFs in α-MoO 3 slabs on top of air and SiO 2 using the permittivity values described in the section “Full-wave numerical simulations using finite-element methods”. 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S.X. acknowledges the support from Independent Research Fund Denmark (Project No. 9041-00333B). B.C. acknowledges the support from VILLUM FONDEN (No. 00027987). The Center for Nanostructured Graphene is sponsored by the Danish National Research Foundation (Project No. DNRF103.) K.V.V. and V.S.V. gratefully acknowledge the financial support from the Ministry of Science and Higher Education of the Russian Federation (Agreement No. 075-15-2021-606). J.M.-S. acknowledges financial support through the Ramón y Cajal Program from the Government of Spain (RYC2018-026196-I). A.Y.N. and J.I.M. acknowledge the Spanish Ministry of Science, Innovation and Universities (national projects MAT201788358-C33-R and PID2019-104604RB/AEI/10.13039/501100011033). R.H. acknowledges financial support from the Spanish Ministry of Science, Innovation and Universities (national project RTI2018-094830-B-100 and the project MDM-2016-0618 of the Marie de Maeztu Units of Excellence Program) and the Basque Government (grant No. IT1164-19). A.Y. N. also acknowledges the Basque Department of Education (grant no. PIBA-2020-10014). P.A.-G. acknowledges support from the European Research Council under starting grant no. 715496, 2DNANOPTICA and the Spanish Ministry of Science and Innovation (State Plan for Scientific and Technical Research and Innovation grant number PID2019-111156GB-I00). Author contributions P.A.-G. and J.D. conceived the study. P.A.-G. supervised the project. J.D. carried out the near-field imaging measurements with the help of A.B. J.D., A.I.F.T.-M., S.X., B.C., and J.I.M. contributed to sample fabrication. S.L. and J.H.E. provided the isotopically enriched boron nitride. G.Á.-P. and K.V.V. carried out the analytical calculations and A.I.F. T.-M. performed the numerical simulations with the help of G.Á-P. and supervised by J.M.-S. and A.Y.N. P.A.-G., J.D., G.Á.-P., J.T.-G., V.S.V., A.Y.N., R.H., and J.M.-S. participated in data analysis. P.A.-G., J.D., and G.Á.-P. wrote the manuscript with input from the rest of authors. Competing interests R.H. is cofounder of Neaspec GmbH, a company producing scattering-type near-field scanning optical microscope systems, such as the one used in this study. The remaining authors declare no competing interests. Additional information Supplementary information The online version contains supplementary material available at https://doi.org/10.1038/s41467-021-24599-3. Correspondence and requests for materials should be addressed to P.A.-G. Peer review information Nature Communications thanks the anonymous reviewers for their contribution to the peer review of this work. Peer reviewer reports are available. 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