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Terahertz (THz) electromagnetic radiation is key to access collective excitations such as magnons (spins), plasmons (electrons), or phonons (atomic vibrations), thus bridging topics between optics and solid-state physics. Confinement of THz light to the nanometer length scale is desirable for local probing of such excitations in low-dimensional systems, thereby circumventing the large footprint and inherently low spectral power density of far-field THz radiation. For that purpose, phonon polaritons (PhPs) in anisotropic van der Waals (vdW) materials have recently emerged as a promising platform for THz nanooptics. Hence, there is a demand for the exploration of materials that feature not only THz PhPs at different spectral regimes but also host anisotropic (directional) electrical, thermoelectric, and vibronic properties. To that end, we introduce here the semiconducting vdW-material alpha-germanium(II) sulfide (GeS) as an intriguing candidate. By employing THz nanospectroscopy supported by theoretical analysis, we provide a thorough characterization of the different in-plane hyperbolic and elliptical PhP modes in GeS. We find not only PhPs with long lifetimes (τ > 2 ps) and excellent THz light confinement (λ0/λ > 45) but also an intrinsic, phonon-induced anomalous dispersion as well as signatures of naturally occurring, substrate-mediated PhP canalization within a single GeS slab., Parts of this research were carried out at ELBE at the Helmholtz-Zentrum Dresden - Rossendorf e. V., a member of the Helmholtz Association. T.N., M.O., L.W., S.C.K., L.M.E., and T.V.A.G.O. acknowledge the financial support by the Bundesministerium für Bildung und Forschung (BMBF, Federal Ministry of Education and Research, Germany, Project Grant Nos. 05K16ODA, 05K16ODC, 05K19ODA, and 05K19ODB) and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy through Würzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum Matter - ct.qmat (EXC 2147, project-id 390858490). F.H. and L.M.E. gratefully acknowledge financial support by the DFG through the project CRC1415 (ID: 417590517). A.Y.N. acknowledges the Spanish Ministry of Science and Innovation (Grants MAT201788358-C3-3-R and PID2020-115221GB-C42) and the Basque Department of Education (Grant PIBA-2020-1- 0014). G.Á .-P. acknowledges support through the Severo Ochoa Program from the Government of the Principality of Asturias (Grant No. PA-20-PF-BP19-053). 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 No. PID2019- 111156GB-I00). We are grateful for the fruitful discussions with Flávio H. Feres (Laboratório Nacional de Luz Síncrotron, Brazil).

7 pags., 4 figs., Negative reflection occurs when light is reflected toward the same side of the normal to the boundary from which it is incident. This exotic optical phenomenon is not only yet to be visualized in real space but also remains unexplored, both at the nanoscale and in natural media. Here, we directly visualize nanoscale-confined polaritons negatively reflecting on subwavelength mirrors fabricated in a low-loss van der Waals crystal. Our near-field nanoimaging results unveil an unconventional and broad tunability of both the polaritonic wavelength and direction of propagation upon negative reflection. On the basis of these findings, we introduce a device in nano-optics: a hyperbolic nanoresonator, in which hyperbolic polaritons with different momenta reflect back to a common point source, enhancing the intensity. These results pave way to realize nanophotonics in low-loss natural media, providing an efficient route to control nanolight, a key for future on-chip optical nanotechnologies., G.Á.-P. and J.T.-G. acknowledge support through the Severo Ochoa Program from the Government of the Principality of Asturias (grant numbers PA-20-PF-BP19-053 and PA-18-PF-BP17-126, respectively). Q.O. acknowledges support from the Australian Research Council (ARC; CE170100039 and DE220100154). This work was performed in part at the Melbourne Centre for Nanofabrication (MCN) in the Victorian Node of the Australian National Fabrication Facility (ANFF). hBN crystal growth was supported by the National Science Foundation award number CMMI 1538127. V.G. acknowledges the Ministerio de Ciencia, Innovación y Universidades through the grant MELODIA (PGC2018-095777-B-C21). V.G. thanks the “ENSEMBLE 3–Centre of Excellence for nanophotonics, advanced materials and novel crystal growth-based technologies” project (GA no. MAB/2020/14) carried out within the International Research Agendas program of the Foundation for Polish Science cofinanced by the European Union under the European Regional Development Fund and the European Union’s Horizon 2020 research and innovation programme Teaming for Excellence (GA no. 857543) for support of this work. 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 María de Maeztu Units of Excellence Program) and the Basque Government (grant no. IT1164-19). J.M.-S. acknowledges financial support from the Ramón y Cajal Program of the Government of Spain and FSE (RYC2018-026196-I) and the Spanish Ministry of Science and Innovation (State Plan for Scientific and Technical Research and Innovation grant number PID2019-110308GA-I00). A.Y.N. acknowledges the Spanish Ministry of Science and Innovation (grants MAT201788358-C3-3-R and PID2020-115221GB-C42) and the Basque Department of Education (grant PIBA-2020-1-0014). 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). Q.B. acknowledges the support from Shenzhen Nanshan District Pilotage Team Program (LHTD20170006).

Optical nanoantennas are of great importance for photonic devices and spectroscopy due to their capability of squeezing light at the nanoscale and enhancing light–matter interactions. Among them, nanoantennas made of polar crystals supporting phonon polaritons (phononic nanoantennas) exhibit the highest quality factors. This is due to the low optical losses inherent in these materials, which, however, hinder the spectral tuning of the nanoantennas due to their dielectric nature. Here, active and passive tuning of ultranarrow resonances in phononic nanoantennas is realized over a wide spectral range (≈35 cm−1, being the resonance linewidth ≈9 cm−1), monitored by near-field nanoscopy. To do that, the local environment of a single nanoantenna made of hexagonal boron nitride is modified by placing it on different polar substrates, such as quartz and 4H-silicon carbide, or covering it with layers of a high-refractive-index van der Waals crystal (WSe2). Importantly, active tuning of the nanoantenna polaritonic resonances is demonstrated by placing it on top of a gated graphene monolayer in which the Fermi energy is varied. This work presents the realization of tunable polaritonic nanoantennas with ultranarrow resonances, which can find applications in active nanooptics and (bio)sensing., J.M.-S. acknowledges financial support from the Ramón y Cajal Program of the Government of Spain and FSE (Grant No. RYC2018-026196-I) and the Spanish Ministry of Science and Innovation (State Plan for Scientific and Technical Research and Innovation Grant Number PID2019-110308GA-I00). 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). G.Á.-P. and J.T.-G. acknowledge support through the Severo Ochoa Program from the Government of the Principality of Asturias (Grant nos. PA20-PF-BP19-053 and PA-18-PF-BP17-126, respectively). A.Y.N. acknowledges the Spanish Ministry of Science and Innovation (Grant Nos. MAT201788358-C3-3-R and PID2020-115221GB-C42) and the Basque Department of Education (Grant No. PIBA-2020-1-0014) J.H.E. acknowledges support for h-BN crystal growth from the National Science Foundation, Award Number CMMI-1538127. R.H. acknowledges financial support from the Spanish Ministry of Science, Innovation and Universities (National Project Grant No. RTI2018-094830-B-100 and the Project Grant No. MDM-2016-0618 of the Marie de Maeztu Units of Excellence Program), the Basque Government (Grant No. IT1164-19), and the European Union's Horizon 2020 research and innovation programme under the Graphene Flagship (Grant Agreement Numbers 785219 and 881603, GrapheneCore2 and GrapheneCore3). I.D. acknowledges the Basque Government (Grant No. PRE_2019_2_0164). Work at MIT was partly supported through AFOSR Grant No. FA9550-16-1-0382, through the NSF QII-TAQS program (Grant No. 1936263), and the Gordon and Betty Moore Foundation EPiQS Initiative through Grant No. GBMF9643 to P.J.-H.

Phonon polaritons (PhPs)─lattice vibrations coupled to electromagnetic fields─in highly anisotropic media display a plethora of intriguing optical phenomena, including ray-like propagation, anomalous refraction, and topological transitions, among others, which have potential for unprecedented manipulation of the flow of light at the nanoscale. However, the properties of these PhPs are intrinsically dictated by the anisotropic crystal structure of the host material. Although in-plane anisotropic PhPs can be steered, and even canalized, by twisting individual crystal slabs in a van der Waals (vdW) stack, active control of their propagation via external stimuli presents a significant challenge. Here, we report on a technology in which anisotropic PhPs supported by biaxial vdW slabs are actively tuned by simply gating an integrated graphene layer. Excitingly, we predict active tuning of optical topological transitions, which enable controlling the canalization of PhPs along different in-plane directions in twisted heterostructures. Apart from their fundamental interest, our findings hold promises for the development of optoelectronic devices (sensors, photodetectors, etc.) based on PhPs with dynamically controllable properties., G.Á.-P. acknowledges support through the Severo Ochoa Program from the government of the Principality of Asturias (Grant No. PA-20-PF-BP19-053). K.V.V. and V.S.V. acknowledge financial support from the Ministry of Science and Higher Education of the Russian Federation (Agreement No. 075-15-2021-606). 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 No. PID2019-111156GB-I00). A.Y.N. acknowledges the Spanish Ministry of Science and Innovation (Grants MAT201788358-C3-3-R and PID2020-115221GB-C42) and the Basque Department of Education (Grant PIBA-2020-1-0014).

The Functional Linear Model with Functional Response (FLMFR) is one of the most fundamental models to assess the relation between two functional random variables. In this paper, we propose a novel goodness-of-fit test for the FLMFR against a general, unspecified, alternative. The test statistic is formulated in terms of a Cram\'er-von Mises norm over a doubly-projected empirical process which, using geometrical arguments, yields an easy-to-compute weighted quadratic norm. A resampling procedure calibrates the test through a wild bootstrap on the residuals and the use of convenient computational procedures. As a sideways contribution, and since the statistic requires a reliable estimator of the FLMFR, we discuss and compare several regularized estimators, providing a new one specifically convenient for our test. The finite sample behavior of the test is illustrated via a simulation study. Also, the new proposal is compared with previous significance tests. Two novel real datasets illustrate the application of the new test., Comment: 24 pages, 2 figures, 10 tables. Suplementary material: 2 pages, 1 figure

Phonon polaritons (PhPs)—light coupled to lattice vibrations—with in-plane hyperbolic dispersion exhibit ray-like propagation with large wave vectors and enhanced density of optical states along certain directions on a surface. As such, they have raised a surge of interest, promising unprecedented manipulation of infrared light at the nanoscale in a planar circuitry. Here, we demonstrate focusing of in-plane hyperbolic PhPs propagating along thin slabs of α-MoO3. To that end, we developed metallic nanoantennas of convex geometries for both efficient launching and focusing of the polaritons. The foci obtained exhibit enhanced near-field confinement and absorption compared to foci produced by in-plane isotropic PhPs. Foci sizes as small as λp/4.5 = λ0/50 were achieved (λp is the polariton wavelength and λ0 is the photon wavelength). Focusing of in-plane hyperbolic polaritons introduces a first and most basic building block developing planar polariton optics using in-plane anisotropic van der Waals materials., J.M.-S. acknowledges financial support from the Ramón y Cajal Program of the Government of Spain and FSE (RYC2018-026196-I) and the Spanish Ministry of Science and Innovation (State Plan for Scientific and Technical Research and Innovation grant number PID2019-110308GA-I00). 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). J.T.-G. acknowledges support through the Severo Ochoa Program from the Government of the Principality of Asturias (PA-18-PF-BP17-126). G.A.-P. acknowledges support through the Severo Ochoa Program from the Government of the Principality of Asturias (PA-20-PF-BP19-053). K.V.V. and V.S.V. acknowledge the financial support from the Ministry of Science and Higher Education of the Russian Federation (agreement no. 075-15-2021-606). A.Y.N. acknowledges the Spanish Ministry of Science, Innovation, and Universities (national projects MAT2017-88358-C3-3-R and PID2020-115221GB-C42) and the Basque Department of Education (PIBA-2020-1-0014). R.H. acknowledges financial support from the Spanish Ministry of Science, Innovation, and Universities (national project number RTI2018-094830-B-100 and project number MDM-2016-0618 of the Marie de Maeztu Units of Excellence Program) and the Basque Government (grant number IT1164-19).

Polaritons with directional in-plane propagation and ultralow losses in van der Waals (vdW) crystals promise unprecedented manipulation of light at the nanoscale. However, these polaritons present a crucial limitation: their directional propagation is intrinsically determined by the crystal structure of the host material, imposing forbidden directions of propagation. Here, we demonstrate that directional polaritons (in-plane hyperbolic phonon polaritons) in a vdW crystal (α-phase molybdenum trioxide) can be directed along forbidden directions by inducing an optical topological transition, which emerges when the slab is placed on a substrate with a given negative permittivity (4H–silicon carbide). By visualizing the transition in real space, we observe exotic polaritonic states between mutually orthogonal hyperbolic regimes, which unveil the topological origin of the transition: a gap opening in the dispersion. This work provides insights into optical topological transitions in vdW crystals, which introduce a route to direct light at the nanoscale., G.Á.-P. and J.T.-G. acknowledge support through the Severo Ochoa Program from the government of the Principality of Asturias (grant nos. PA20-PF-BP19-053 and PA-18-PF-BP17-126, respectively). K.V.V. and V.S.V. acknowledge the Ministry of Science and Higher Education of the Russian Federation (no. 0714-2020-0002). J. M.-S. acknowledges financial support through the Ramón y Cajal Program from the government of Spain and FSE (RYC2018-026196-I). A.Y.N. acknowledges the Spanish Ministry of Science, Innovation and Universities (national project no. MAT201788358-C3-3-R), and the Basque Department of Education (PIBA-2020-1-0014). P.A.-G. acknowledges support from the European Research Council under starting grant no. 715496, 2DNANOPTICA.

Refraction between isotropic media is characterized by light bending towards the normal to the boundary when passing from a low- to 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-MoO3. 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 λp/6, being λp the polariton wavelength (λ0/50 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., G.Á.-P. and J.T.-G. acknowledge support through the Severo Ochoa Program from the Government of the Principality of Asturias (nos. PA-20-PF-BP19-053 and PA-18-PF-BP17-126, respectively). 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-C3-3-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-2020-1-0014). 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).

Layered materials in which individual atomic layers are bonded by weak van der Waals forces (vdW materials) constitute one of the most prominent platforms for materials research. Particularly, polar vdW crystals, such as hexagonal boron nitride (h-BN), alpha-molybdenum trioxide (&alpha, MoO3) or alpha-vanadium pentoxide (&alpha, V2O5), have received significant attention in nano-optics, since they support phonon polaritons (PhPs)―light coupled to lattice vibrations― with strong electromagnetic confinement and low optical losses. Recently, correlative far- and near-field studies of &alpha, MoO3 have been demonstrated as an effective strategy to accurately extract the permittivity of this material. Here, we use this accurately characterized and low-loss polaritonic material to sense its local dielectric environment, namely silica (SiO2), one of the most widespread substrates in nanotechnology. By studying the propagation of PhPs on &alpha, MoO3 flakes with different thicknesses laying on SiO2 substrates via near-field microscopy (s-SNOM), we extract locally the infrared permittivity of SiO2. Our work reveals PhPs nanoimaging as a versatile method for the quantitative characterization of the local optical properties of dielectric substrates, crucial for understanding and predicting the response of nanomaterials and for the future scalability of integrated nanophotonic devices.

Layered materials in which individual atomic layers are bonded by weak van der Waals forces (vdW materials) constitute one of the most prominent platforms for materials research. Particularly, polar vdW crystals, such as hexagonal boron nitride (h-BN), alpha-molybdenum trioxide (α-MoO3) or alpha-vanadium pentoxide (α-V2O5), have received significant attention in nano-optics, since they support phonon polaritons (PhPs)―light coupled to lattice vibrations― with strong electromagnetic confinement and low optical losses. Recently, correlative far- and near-field studies of α-MoO3 have been demonstrated as an effective strategy to accurately extract the permittivity of this material. Here, we use this accurately characterized and low-loss polaritonic material to sense its local dielectric environment, namely silica (SiO2), one of the most widespread substrates in nanotechnology. By studying the propagation of PhPs on α-MoO3 flakes with different thicknesses laying on SiO2 substrates via near-field microscopy (s-SNOM), we extract locally the infrared permittivity of SiO2. Our work reveals PhPs nanoimaging as a versatile method for the quantitative characterization of the local optical properties of dielectric substrates, crucial for understanding and predicting the response of nanomaterials and for the future scalability of integrated nanophotonic devices.

Phonon polaritons—light coupled to lattice vibrations—in polar van der Waals crystals are promising candidates for controlling the flow of energy on the nanoscale due to their strong field confinement, anisotropic propagation and ultra-long lifetime in the picosecond range1,2,3,4,5. However, the lack of tunability of their narrow and material-specific spectral range—the Reststrahlen band—severely limits their technological implementation. Here, we demonstrate that intercalation of Na atoms in the van der Waals semiconductor α-V2O5 enables a broad spectral shift of Reststrahlen bands, and that the phonon polaritons excited show ultra-low losses (lifetime of 4 ± 1 ps), similar to phonon polaritons in a non-intercalated crystal (lifetime of 6 ± 1 ps). We expect our intercalation method to be applicable to other van der Waals crystals, opening the door for the use of phonon polaritons in broad spectral bands in the mid-infrared domain., J.T.-G. and G.Á.-P. acknowledge support through the Severo Ochoa Program from the Government of the Principality of Asturias (nos. PA-18-PF-BP17-126 and PA-20-PF-BP19-053, respectively). J.M.-S. acknowledges finantial support from the Clarín Programme from the Government of the Principality of Asturias and a Marie Curie-COFUND grant (PA-18-ACB17-29) and the Ramón y Cajal Program from the Government of Spain (RYC2018-026196-I). K.C., X.P.A.G., H.V. and M.H.B. acknowledge the Air Force Office of Scientific Research (AFOSR) grant no. FA 9550-18-1-0030 for funding support. I.E. acknowledges financial support from the Spanish Ministry of Economy and Competitiveness (grant no. FIS2016-76617-P). A.Y.N. acknowledges the Spanish Ministry of Science, Innovation and Universities (national project no. MAT2017-88358-C3-3-R) and the Basque Government (grant no. IT1164-19). Q.B. acknowledges the support from Australian Research Council (grant nos. FT150100450, IH150100006 and CE170100039). R.H. acknowledges support from the Spanish Ministry of Economy, Industry, and Competitiveness (national project RTI2018-094830-B-100 and the Project MDM-2016-0618 of the María de Maeztu Units of Excellence Program) and the Basque Goverment (grant no. IT1164-19). P.A.-G. acknowledges support from the European Research Council under starting grant no. 715496, 2DNANOPTICA.

Functional linear models are one of the most fundamental tools to assess the relation between two random variables of a functional or scalar nature. This contribution proposes a goodness-of-fit test for the functional linear model with functional response that neatly adapts to functional/scalar responses/predictors. In particular, the new goodness-of-fit test extends a previous proposal for scalar response. The test statistic is based on a convenient regularized estimator, is easy to compute, and is calibrated through an efficient bootstrap resampling. A graphical diagnostic tool, useful to visualize the deviations from the model, is introduced and illustrated with a novel data application. The R package goffda implements the proposed methods and allows for the reproducibility of the data application.

Electromagnetic field confinement is crucial for nanophotonic technologies, since it allows for enhancing light–matter interactions, thus enabling light manipulation in deep sub-wavelength scales. In the terahertz (THz) spectral range, radiation confinement is conventionally achieved with specially designed metallic structures—such as antennas or nanoslits—with large footprints due to the rather long wavelengths of THz radiation. In this context, phonon polaritons—light coupled to lattice vibrations—in van der Waals (vdW) crystals have emerged as a promising solution for controlling light beyond the diffraction limit, as they feature extreme field confinements and low optical losses. However, experimental demonstration of nanoscale-confined phonon polaritons at THz frequencies has so far remained elusive. Here, it is provided by employing scattering-type scanning near-field optical microscopy combined with a free-electron laser to reveal a range of low-loss polaritonic excitations at frequencies from 8 to 12 THz in the vdW semiconductor α-MoO3. In this study, THz polaritons are visualized with: i) in-plane hyperbolic dispersion, ii) extreme nanoscale field confinement (below λo ⁄75), and iii) long polariton lifetimes, with a lower limit of >2 ps., T.V.A.G.O., T.N., L.W., M.O., S.C.K., and L.M.E. acknowledge the financial support by the Bundesministerium für Bildung und Forschung (BMBF, Federal Ministry of Education and Research, Germany, Project Grant N°s 05K16ODA, 05K16ODC, 05K19ODA, and 05K19ODB) and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy through Würzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum Matter - ct.qmat (EXC 2147, project-id 390858490). G.Á-P. and J.T.-G. acknowledge the support through the Severo Ochoa Program from the Government of the Principality of Asturias (grants PA-20-PF-BP19-053 and PA-18-PF-BP17-126, respectively). P.A-G. acknowledges support from the European Research Council under Starting Grant 715496, 2DNANOPTICA. A.Y.N. acknowledges the Spanish Ministry of Science, Innovation and Universities (national project N° MAT201788358-C3-3-R) and the Basque Department of Education (grant PIBA-2020-1-0014). E.J.H. Lee acknowledges financial support from the Spanish Ministry of Science, Innovation and Universities, through the “María de Maeztu” Programme for Units of Excellence in R&D (CEX2018-000805-M) and the Ramón y Cajal grant RYC-2015-17973.

Recent discoveries have shown that, when two layers of van der Waals (vdW) materials are superimposed with a relative twist angle between them, the electronic properties of the coupled system can be dramatically altered. Here, we demonstrate that a similar concept can be extended to the optics realm, particularly to propagating phonon polaritons–hybrid light-matter interactions. To do this, we fabricate stacks composed of two twisted slabs of a vdW crystal (α-MoO3) supporting anisotropic phonon polaritons (PhPs), and image the propagation of the latter when launched by localized sources. Our images reveal that, under a critical angle, the PhPs isofrequency curve undergoes a topological transition, in which the propagation of PhPs is strongly guided (canalization regime) along predetermined directions without geometric spreading. These results demonstrate a new degree of freedom (twist angle) for controlling the propagation of polaritons at the nanoscale with potential for nanoimaging, (bio)-sensing, or heat management., J.T.-G. and G.Á.-P. acknowledge support through the Severo Ochoa Program from the Government of the Principality of Asturias (Nos. PA-18-PF-BP17-126 and PA-20-PF-BP19-053, respectively). J.M.-S. acknowledges financial support through the Ramón y Cajal Program from the Government of Spain (RYC2018-026196-I). A.Y.N. acknowledges the Spanish Ministry of Science, Innovation and Universities (national Project No. MAT2017-88358-C3-3-R). P.A.-G. acknowledges support from the European Research Council under starting Grant No. 715496, 2DNANOPTICA.

The biaxial van der Waals semiconductor α-phase molybdenum trioxide (α-MoO

Phonon polaritons-light coupled to lattice vibrations-in polar van der Waals crystals are promising candidates for controlling the flow of energy on the nanoscale due to their strong field confinement, anisotropic propagation and ultra-long lifetime in the picosecond range

Phonon polaritons (PhPs) have attracted significant interest in the nano-optics communities because of their nanoscale confinement and long lifetimes. Although PhP modification by changing the local dielectric environment has been reported, controlled manipulation of PhPs by direct modification of the polaritonic material itself has remained elusive. Here, chemical switching of PhPs in α-MoO3 is achieved by engineering the α-MoO3 crystal through hydrogen intercalation. The intercalation process is non-volatile and recoverable, allowing reversible switching of PhPs while maintaining the long lifetimes. Precise control of the intercalation parameters enables analysis of the intermediate states, in which the needle-like hydrogenated nanostructures functioning as in-plane antennas effectively reflect and launch PhPs and form well-aligned cavities. We further achieve spatially controlled switching of PhPs in selective regions, leading to in-plane heterostructures with various geometries. The intercalation strategy introduced here opens a relatively non-destructive avenue connecting infrared nanophotonics, reconfigurable flat metasurfaces and van der Waals crystals., Phonon polaritons hold promises for nanophotonic applications but external control of phonon polaritons remains challenging. Here, the authors achieve reversible and non-volatile switching of phonon polariton by modifying crystal structure and lattice vibrations via hydrogenation.

Anisotropic crystals have recently attracted considerable attention because of their ability to support polaritons with a variety of unique properties, such as hyperbolic dispersion, negative phase velocity, or extreme confinement. Particularly, the biaxial crystal α−MoO3 has been demonstrated to support phonon polaritons, light coupled to lattice vibrations, with in-plane anisotropic propagation and unusually long lifetime. However, the lack of theoretical studies on electromagnetic modes in biaxial crystal slabs impedes a complete interpretation of the experimental data, as well as an efficient design of nanostructures supporting such highly anisotropic polaritons. Here, we derive the dispersion relation of electromagnetic modes in biaxial slabs surrounded by semi-infinite isotropic dielectric half-spaces with arbitrary dielectric permittivities. Apart from a general dispersion relation, we provide very simple analytical expressions in typical experiments in nano-optics: the limits of short polaritonic wavelength and/or very thin slabs. The results of our study will allow for an in-depth analysis of anisotropic polaritons in novel biaxial van der Waals materials., A.Y.N. acknowledges the Spanish Ministry of Science, Innovation and Universities (national Project No. MAT2017-88358-C3-3-R) and Basque Government (Grant No. IT1164-19). P.A.-G. acknowledges support from the European Research Council under Starting Grant No. 715496, 2DNANOPTICA. K.V.V. and V.S.V. acknowledge support from the Russian Science Foundation, Grant No. 18-19-00684.

The biaxial van der Waals semiconductor α‐phase molybdenum trioxide (α‐MoO3) has recently received significant attention due to its ability to support highly anisotropic phonon polaritons (PhPs)—infrared (IR) light coupled to lattice vibrations—offering an unprecedented platform for controlling the flow of energy at the nanoscale. However, to fully exploit the extraordinary IR response of this material, an accurate dielectric function is required. Here, the accurate IR dielectric function of α‐MoO3 is reported by modeling far‐field polarized IR reflectance spectra acquired on a single thick flake of this material. Unique to this work, the far‐field model is refined by contrasting the experimental dispersion and damping of PhPs, revealed by polariton interferometry using scattering‐type scanning near‐field optical microscopy (s‐SNOM) on thin flakes of α‐MoO3, with analytical and transfer‐matrix calculations, as well as full‐wave simulations. Through these correlative efforts, exceptional quantitative agreement is attained to both far‐ and near‐field properties for multiple flakes, thus providing strong verification of the accuracy of this model, while offering a novel approach to extracting dielectric functions of nanomaterials. In addition, by employing density functional theory (DFT), insights into the various vibrational states dictating the dielectric function model and the intriguing optical properties of α‐MoO3 are provided., G.Á-P. and J.T.-G. acknowledge support through the Severo Ochoa Program from the I of the Principality of Asturias (grants No. PA-20-PF-BP19-053 and PA-18-PF-BP17-126, respectively). I.E. acknowledges support from the Spanish Ministry of Economy and Competitiveness (FIS2016-76617-P). J.M.-S. acknowledges support through a Clarín Marie Curie-COFUND grant from the Government of the Principality of Asturias and the EU (PA18-ACB17-29), and the Ramón y Cajal Program (RYC2018-026196-I) from the Government of Spain. Q. B. acknowledges support from Australian Research Council (ARC, FT150100450, IH150100006 and CE170100039). J.D.C. was supported by the National Science Foundation (U.S.A.) under grant number U0048926. A.Y.N. acknowledges the Spanish Ministry of Science, Innovation and Universities (national project MAT2017-88358-C3-3-R) and Basque Government (grant No. IT1164-19). P.A-G. acknowledges support from the European Research Council under Starting Grant 715496, 2DNANOPTICA.