Abstract
Extreme anisotropy in some polaritonic materials enables light propagation with a hyperbolic dispersion, leading to enhanced lightmatter interactions and directional transport. However, these features are typically associated with large momenta that make them sensitive to loss and poorly accessible from farfield, being bound to the material interface or volumeconfined in thin films. Here, we demonstrate a new form of directional polaritons, leaky in nature and featuring lenticular dispersion contours that are neither elliptical nor hyperbolic. We show that these interface modes are strongly hybridized with propagating bulk states, sustaining directional, longrange, subdiffractive propagation at the interface. We observe these features using polariton spectroscopy, farfield probing and nearfield imaging, revealing their peculiar dispersion, and – despite their leaky nature – long modal lifetime. Our leaky polaritons (LPs) nontrivially merge subdiffractive polaritonics with diffractive photonics onto a unified platform, unveiling opportunities that stem from the interplay of extreme anisotropic responses and radiation leakage.
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Introduction
Several natural materials, including van der Waals materials, semiconductors, and dielectric polar crystals, have been attracting significant attention in recent years for their structural, electronic, and optical properties, forming an excellent platform to engage strong interactions between light and dipoleactive material excitations^{1,2,3,4,5}. Their strong coupling with light supports the emergence of exotic halflight halfmatter particles, known as polaritons, across a broad range of wavelengths from the midinfrared to the visible and ultraviolet regimes. When polaritonic materials have principal elements of their permittivity tensor with opposite signs, they support hyperbolic modes, recently observed in a variety of natural materials, including graphite^{6}, tetradymite (Bi_{2}Te_{2}S)^{7}, hexagonal boron nitride (hBN)^{8}, alphaphase molybdenum trioxide (αMoO_{3})^{9}, alphaphase vanadium pentoxide (αV_{2}O_{5})^{10}, and hybrid phononplasmon polaritonic materials^{11,12}. Their extreme optical anisotropy, stemming from exciton or phonon resonances polarized in specific directions as a function of the underlying lattice geometry, enables enhanced lightmatter interactions, nanoscale localization of electromagnetic energy, and highly directional propagation beyond the diffraction limits. These features are directly associated with the open topology of their hyperbolic dispersion contours^{6}. In hBN, such extreme anisotropy is observed with respect to the materials interface normal, implying that hyperbolic polaritons travel in the bulk, making their direct observation and manipulation challenging^{8,13,14,15,16,17,18,19}. On the contrary, αMoO_{3} features a lattice configuration supporting extreme anisotropy in the interface plane, leading to hyperbolic polaritons in thin films, more easily accessible through nanoscale objects close to the interface with air^{9,20,21,22,23,24}, and providing opportunities for twisted nanooptics^{25,26,27,28}. Calcite (calcium carbonate)^{29,30}, as another remarkable example, features extreme anisotropy whose optical axis (OA) orientation can be controlled with respect to the material interface with air, leading to ghost hyperbolic polaritons (gHPs) bound to the interface but with tilted phase fronts entering the bulk^{31}. Further lowering the symmetry, monoclinic crystals like βGa_{2}O_{3} have been recently shown to support hyperbolic shear polaritons, in which nonorthogonal phonon resonances sustain hyperbolic propagation with asymmetric loss profiles and an OA rotating as a function of frequency^{32}. The growing family of polaritonic materials has been opening unique opportunities for lightmatter nanoscale routing, transport, and manipulation.
So far, all polaritons and the associated exciting features have been strictly bound to the material they live in, either in the form of volumeconfined modes or of surface waves traveling along the interface planes, evanescently decaying away from the surface and exhibiting subdiffractive features^{33}. In the limit of negligible material loss, these guided modes feature realvalued inplane momenta larger than the one in the surrounding materials, associated with the bound nature of their propagation. In conventional refractive optics, it is common to consider leaky waves as a different class of guided modes at an interface, featuring a complex inplane momentum living within the light cone of at least one of the materials forming the interface and hence associated with radiation leakage^{34,35,36,37}.
In this work, we merge these fields of research and demonstrate that natural anisotropic crystals can support a regime for polariton propagation inherently leaky in nature, which is at the same time bound to the interface with air and hybridized with refractive bulk modes. This class of polaritons features a form of isofrequency contours (IFCs) of dispersion with a peculiar lenticular shape, forming a cusp or a wedge at a highsymmetry point in momentum space. Interestingly, despite featuring IFCs with a closed topology, i.e., limited to a finite region in momentum space, these leaky polaritons (LPs) support longdistance and directional propagation along the material interface with large Qfactors. These features are remarkably different from conventional leakywaves.
Results
Eigenmode analysis with complexvalued wavenumber
To reveal their features, we study the exemplary case of polaritons at the interface between air and calcite. Calcite’s OA can be slanted with respect to the interface by an angle \(\theta\) [Fig. 1a]. Polaritons at the aircalcite interface have been recently explored in the upper Reststrahlen band^{31}, which are associated with a typeII hyperbolic dispersion of their phonon polaritons [see material dispersion in Supplementary Information, Fig. S1]. When \(\theta\, \ne \, {0}^{\circ }\), the broken symmetry between the phonons and the interface tilts the polariton’s wave vectors with respect to the interface but are still bound to the interface, leading to the emergence of gHPs, as shown in Fig. 1a for \(\omega=1470\,{{{{{{\rm{cm}}}}}}}^{1}\). The Fourier transform (FT) of the inplane fields (Fig. 1b) is associated with a hyperbolic IFC, shown in Fig. 1c. In contrast, the lower Reststrahlen band, for which \({\epsilon }_{\parallel }\, < \, 0\) and \({\epsilon }_{\perp }\, > \, 0\), supports typeI hyperbolic phonon polaritons in the bulk. This frequency range, as well as the neighboring εnearzero transparent regime where \({\epsilon }_{\perp }\, > \,0\) and \(0\, < \,{\epsilon }_{\parallel }\, < \,{\epsilon }_{{{{{\rm{air}}}}}}\), both support the emergence of LPs at the interface with air. When excited by a localized emitter in the hyperbolic regime at \(\omega=887{{{{{{\rm{cm}}}}}}}^{1}\) (Fig. 1a), the LP field distribution is remarkably directional in the interface plane (Fig. 1d), showing four emission lobes. The corresponding FT, however, is associated with an IFC with a peculiar lenticular shape (Fig. 1e), with a closed topology despite the directionality of emission.
These dispersion features can be analyzed by solving sourcefree Maxwell’s equations at the aircalcite interface, for the moment neglecting material absorption. The polariton dispersion can be computed by enforcing the continuity of the tangential fields at the interface: we generally assume that the inplane wavenumbers in the x/y directions \({k}_{x,y}\) are realvalued, and obtain
where \(\hat{{{{{{\rm{D}}}}}}}\) is a \(4\times 4\) matrix (see Supplementary Information Section I). The inplane dispersion of gHPs in the typeII hyperbolic regime at \(\omega=1470\,{{{{{{\rm{cm}}}}}}}^{1}\) is shown as a red dashed curve in Fig. 1c, in agreement with the FT of the nearfields. In this regime, calcite only supports the propagation of extraordinary waves in bulk, and their IFC is a hyperboloid aligned with the OA (generally oriented at an angle \(\theta\) with respect to the interface, as shown in Fig. 1f). Around the origin, we also show the spherical IFC of freespace radiation modes. When projected onto the interface plane, three different regions emerge for the inplane momentum: (1) the free space light cone (FSLC) (white circle), (2) the light cone of extraordinary waves in calcite (green hyperbolae), and (3) the remainder of the inplane kspace (gray) where no bulk modes are supported at either side of the interface. Surface hyperbolic polaritons (sHPs) \((\theta={0}^{\circ })\) and gHPs (\(\theta\, \ne \,{0}^{\circ }\)) guided at the interface and with real inplane wave numbers are supported in the region (3), with their dispersion depicted by red lines in Fig. 1f, bound to the interface, carrying power along it and evanescently decaying away from it.
In contrast, LPs result from the hybridization of surfacebound modes with propagating bulk waves. They naturally emerge in the typeI hyperbolic regime, e.g., at \(\omega=887\,{{{{{{\rm{cm}}}}}}}^{1}\) for calcite, in which bulk propagation is now associated with ordinary waves with isotropic dispersion in kspace, described by the outer sphere in Fig. 1g (top panel), as well as extraordinary waves with hyperbolic dispersion, described by the two purple hyperboloids with major axis parallel to the OA. The projected dispersions onto the k_{x}–k_{y}(interface) plane show a more complex geometry, with four distinct regions: regions (1–3) (R1–R3) resemble the ones in the typeII hyperbolic regime. In addition, R(4) outside the FSLC (blue ring) supports the propagation of ordinary bulk waves. Thus, while the extraordinary interface mode with inplane momentum in R(1) can radiatively leak to both sides of the interface, in R(4) the interface modes are confined on the air side but leak into a refractive ordinary wave in calcite (see Supplementary Information Section I). As we tilt the OA with respect to the interface, \(\theta \ne {0}^{^\circ }\) (bottom panels of Fig. 1g) a new region, R(5), appears due to the skewed hyperboloid projection overlapping with R(4), thus supporting radiation leakage into ordinary and extraordinary waves. The inplane momentum diagram in Fig. 1g implies that the interface modes can become compatible with radiation towards both sides of the interface [R(1)] or into calcite [R(4)], hence we cannot generally expect guided modes with realvalued \(({k}_{x},{k}_{y})\), even in the limit of negligible material dissipation. When searching for polariton modes at the air–calcite interface in this regime, we need to look for complexvalued wavenumbers satisfying Eq. (1) in the form \(q={q}_{{{{{{\rm{r}}}}}}}+{{{{{\rm{i}}}}}}{q}_{{{{{{\rm{i}}}}}}}\), for each realvalued polar angle \(\phi\) in the k_{x}–k_{y} plane (see Supplementary Information Section IV for the reason why \(\phi\) needs to be realvalued, a good approximation even when material loss is considered). As a result, Eq. (1) becomes complexvalued, and the choice of appropriate branch cuts and the sign of complex wave vectors become crucial to identify physical solutions^{35} (see Supplementary Information Section I). The resulting IFCs at \(\omega=887{{{{{{\rm{cm}}}}}}}^{1}\) are shown as red lines in Fig. 1g for different values of \(\theta\). Interestingly, we observe the emergence of LPs, supported in R(1), R(4) of the inplane kspace, with a lenticular dispersion featuring a closed topology, consistent with the numerical prediction in Fig. 1e. As we increase the angle \(\theta\) between the OA and the interface, the extraordinary wave hyperboloids tilt, and hence their projections in the k_{x}–k_{y} plane move towards the center, forming R(5) and pushing the LP dispersion towards the origin in the \({k}_{x}\) direction, until the critical angle \({\theta }_{{{{{{\rm{c}}}}}}}\) at which the projected hyperbolic contours enter the FSLC, R(4) is split into two isolated regions separated by hyperbolic contours R(5), and hence LPs cannot be supported any longer.
The complexvalued solutions of Eq. (1) shown in Fig. 1e, g constitute the LP spectrum, which effectively hybridizes bulk ordinary waves in calcite with confined extraordinary surface modes, leading to a new form of polariton. Its hybrid features strongly depend on the inplane momentum: for (k_{x}, k_{y}) parallel or adjacent to the direction of the OA, LPs display surface polariton (\(\theta={0}^{^\circ }\)) or gHP like \((\theta\, \ne \,{0}^{^\circ })\) features, with power flow nearly parallel to the interface and smaller radiation leakage, implying four lobes of directional emission into the bulk. When the inplane wave vector rotates away from the OA, the LPs are more strongly coupled to the bulk ordinary wave with skewed wavefronts. Hence their radiation leakage is larger and \({q}_{{{{{{\rm{i}}}}}}}\) increases outside the FSLC (see Supplementary Information Sections I and III for an extended discussion of the azimuthal dependence of LPs and their eigenmodal field and Poynting vector distributions). The highsymmetry points k_{y} = 0 (ϕ = 0°) and k_{x} = 0 (ϕ = 90°) are particularly interesting, as they correspond to the two extremes of the LPs hybrid nature: at k_{y} = 0, the LP is purely bound to the interface, while at k_{x} = 0 it is purely refractive and coupled to the ordinary bulk mode, corresponding to Brewster mode with ppolarization (see Supplementary Information Section II). Notably at these two extremes, the imaginary part of the wavevector q_{i} vanishes. LPs can emerge also in the transparent regime, for example around ω = 890 cm^{−1}. In this scenario, the IFCs follow a similar lenticular dispersion, but they are discontinuous when the direction of the wave vector comes close to the OA, because their dispersion intersects the extraordinary wave light line, which is now elliptical in shape (see Supplementary Information Fig. 6).
Directionality of LPs
As shown in Fig. 1a(right panel), d, LPs support peculiar directional features in the interface plane, despite being associated with closed contours. The lenticular contours are directly responsible for the directional inplane propagation of LPs, providing a much stronger overall directionality than elliptical or circular dispersion contours. In order to explain this unexpected property of the lenticular dispersion, we analyze fields and Poynting vectors of LPs on the aircalcite interface plane for \(\theta={23.3}^{^\circ }\) in Fig. 2. In particular, we compare the magnitude of the electric field distribution of gHPs (Fig. 2a) and LPs (Fig. 2b, c) in real space, and overlay the FT from their \({{{{{{\rm{E}}}}}}}_{z}\) fields with their IFCs obtained from Eq. (1) (Fig. 2d–f), considering realistic calcite loss. The emission directionality is associated with the Poynting vector direction for all inplane momenta, which is normal or nearly normal to the IFCs for surface polaritons. As shown in Fig. 2d, the Poynting vectors of gHPs for each hyperbolic branch are mostly aligned to each other, due to the open topology of their dispersion, leading to highly directional propagation in real space. LPs, due to their peculiar lenticular dispersion, also feature power flows that are uncommonly parallel to each other in the same quadrant, exhibiting directional propagation in real space (Fig. 2b, c). This is surprising since their IFC as shown in Fig. 2e, f has a closed or nearly closed topology for \(\omega=887\,{{{{{{\rm{cm}}}}}}}^{1}\) and \(\omega=890\,{{{{{{\rm{cm}}}}}}}^{1}\), corresponding to the typeI hyperbolic and transparent regimes, respectively. The brighter spectra outside the FSLC of the FT in Fig. 2e, f indicate the power flow in the plane is mainly associated with polaritons outside the FSLC, which indeed have Poynting vectors (white color in Fig. 2e, f) well aligned across the dispersion curve, far more than they would be with a conventional circular or elliptic dispersion. This directionality is fostered by the fact that the LP propagation length is almost constant along the IFC outside the FSLC, despite the increasing radiation leakage for increasing \(\left\phi \right\). This effect arises because the effective material loss of the hybridized mode simultaneously decreases for increasing \(\left\phi \right\), leading to enhanced directionality (see Supplementary Information Section III for an extended discussion and Section VII for a quantitative study of directionality).
Polariton spectroscopy and farfield reflectance spectroscopy
To experimentally observe LPs and map their peculiar dispersion in and out of the FSLC, we performed both polariton spectroscopy and farfield reflectance spectroscopy. First, we probed the LP dispersion in the frequency domain using Ottotype prism coupling^{32,38} for an air–calcite (100) interface with the OA slanted by \(\theta=23.3^\circ\) (see Methods for details). We placed a prism made of dielectric Thallium Bromoiodide (KRS5) above the aircalcite interface with a fixed air gap \({d}_{{{{{{\rm{gap}}}}}}}=4\,{{{{{\rm{\mu }}}}}}{{{{{\rm{m}}}}}}\), and acquired reflectance spectra at the prism backsurface for several azimuthal angles \(\phi\) between the incidence plane and the OA, see Fig. 3a. By fixing the incidence angle \(\beta\), we select a specific value of inplane momentum \(\frac{{q}_{{{{{{\rm{r}}}}}}}}{{k}_{0}}\) of the excited evanescent waves. The resulting reflectance spectra for \(\beta=27^\circ\), corresponding to \(\frac{{q}_{{{{{{\rm{r}}}}}}}}{{k}_{0}}=1.075\) at \(\omega=887\) cm^{−1}, are shown in Fig. 3b. We observe a highquality polariton resonance at \(\phi=0^\circ\) (OA parallel to the incidence plane), which blueshifts and quickly broadens as we rotate the sample in either direction about its surface normal. The experimental data are in good agreement with transfer matrix simulations (see Methods) shown in Fig. 3c. In order to map the inplane dispersion of LPs, we acquired the azimuthal dependence for several incidence angles (see Supplementary Information section V for the full data set), and extracted the resonance positions, as well as their quality (Q)factors. The retrieved resonance frequency map shown in Fig. 3d, in excellent agreement with the simulated data in Fig. 3e, displays the peculiar lenticular LP dispersion. This is confirmed in the retrieved IFCs from these data (see Methods), shown in Fig. 3f, which provide direct experimental proof of the lenticular inplane dispersion of LPs. Further experiments (shown in Supplementary Information, Figs. S9 and S10) confirm similar features for \(\theta=48.5^\circ\) using calcite and for \(\theta=0^\circ\) using αQuartz.
The experimental and simulated Qfactors of LPs in calcite are shown in Fig. 3g, h, respectively. We retrieved Qfactor values along the IFCs at multiple frequencies, as shown in Fig. 3i. Remarkably, we are generally able to observe very large Qfactors, exceeding 300 at \(\phi=0^\circ\) and Q ~ 150 at \(\left\phi \right =20^\circ\), demonstrating a long lifetime for LPs, despite their leaky nature, a consequence of the low damping of the associated optical phonon polarized in the OA direction^{39}, leading to unprecedented directional radiation leakage. Strikingly, the Qfactors rapidly decrease for increasing \(\left \phi \right,\) for all momenta (Fig. 3h) and at all frequencies (Fig. 3i). Hence, the LPs experience a steep increase in optical loss as the polarization plane is rotated away from the OA. Nonetheless, lowloss LP propagation also in directions away from the OA is wellsupported since the decreasing Qfactor is compensated by an increasing group velocity (see Supplementary Information IX). Thus, the azimuthal dependence of the LP Qfactor is indeed consistent with the realspace propagation patterns predicted in Fig. 2b, c, where offOA directions dominate the propagation patterns, as experimentally verified in Fig. 4.
Before discussing these nearfield measurements, we additionally performed farfield experiments to verify the LP dispersion also inside the FSLC. For this purpose, we acquired freespace reflectance spectra of calcite (100) for a series of large incidence angles \(\beta\) and azimuthal angles \(\phi\), see Fig. 3j for the experimental arrangement. The reflectance data allows for extraction of isofrequency reflectance maps, as shown in Fig. 3k, l (see Supplementary Information X for the full data set), in excellent agreement with the simulations shown in Fig. 3m, n. In these data, the reflectance minima mark the LP, which here—inside the FSLC—can be understood as a Brewsterlike mode with strong azimuthal dispersion. It is this component of the LPs that allows freespace coupling and directional thermal emission, absent in any other polariton platform.
Nearfield optical microscopy
In order to visualize the directional excitation and longrange propagation of LPs in real space, we performed nearfield imaging of LPs on an air–calcite interface with \(\theta=23.3^\circ\). A gold nanodisk (diameter \(D\) = 4 μm) was deposited on the interface as an optical antenna that concentrates ppolarized midinfrared illumination (with an oblique incident angle \(\beta\) = 30° along z) into a localized hotspot, thus enabling the launching of LPs (as shown in Fig. 4a and Supplementary Information Fig. S11). The launched field E_{p} propagates along the surface, interfering with the incident field E_{in}, and the interference field is mapped by a sharp metallic tip of scatteringtype scanning nearfield optical microscope (sSNOM) with nanoscale resolution. Figure 4a presents the nearfield measured amplitude of disklaunched LPs at \(\omega=883\,{{{{{{\rm{cm}}}}}}}^{1}\), revealing LPs with an arctype wavefront. The interference between LPs and the incident field renders the wavenumber of LPs slightly enlarged in \({k}_{x}\) by the interference factor \(\Delta={k}_{0}*\sin \beta\)^{31}. The desired signaltonoise ratio is calculated (Supplementary Information Fig. S16) to estimate the quality of the experimental results. Figure 4b presents the FT of the nearfield distribution in Fig. 4a, confirming the peculiar lenticular shape of IFC and its closed topology in kspace (details in Supplementary Information Figs. S11 and S12). The yellow dashed line indicates the FSLC and the red solid line presents the theoretical dispersion, exhibiting good agreement between theoretical and experimental results. At frequency \(\omega=887\,{{{{{{\rm{cm}}}}}}}^{1}\) the measured LP fields exhibit higher directionality, consistent with the more pronounced flattening of the IFC shown in Fig. 4d, again agreeing well with our theoretical predictions. The directional propagation of LPs observed here is consistent with our numerical simulations at oblique illumination (Supplementary Information S13). Figure 4e, f shows (e) the spatial field distribution of LPs and (f) the corresponding IFC at \(\omega=890\,{{{{{{\rm{cm}}}}}}}^{1}\) in the transparent regime, in which the directionality of propagation is further enhanced, now being strongly dominated by the flat parts of the lenticular IFC. Both arrows estimated from simulation in Fig. 4c, e are closely aligned with the confined directional fields, demonstrating how the direction of polariton propagation can be finetuned by detuning the illumination frequency. More results from additional measurements and comparisons between the theoretical and experimental results can be found in Supplementary Information Figs. S14 and S15. Directional propagation of LPs is also sustained by tuning the OA orientation of calcite (Supplementary Information Fig. S16), consistent with our theoretical prediction. The real space imaging of LPs and recorded dispersion further validate the observation of a new form of polaritons, directional and leaky in nature and significantly distinct from hyperbolic surface and volume polaritons. To quantify the propagation damping of LPs, we extract the lines of the nearfield signal along the left dashed line in Fig. 4e; the decay of the fringes away from the Au antenna can be attributed to the damping of LPs and geometrical spreading^{40}. The relative propagation length of the measured LPs is estimated to be 13.3, which is larger than the estimated value of 6.4 for gHPs, thus indicating a lower propagation loss for LPs. The detailed comparison between LPs and gHPs can be found in Supplementary Information Fig. S17.
Discussion
In this work, we have unveiled with theoretical investigations, kspace, and realspace nearfield measurements, a new form of polariton, living at the interface of highly anisotropic materials, leaky in nature and with unique spectral features stemming from the hybridization between refractive bulk modes and surfacebound polaritons. We experimentally demonstrated its exotic dispersion features in and out of FSLC and directional propagation properties, probing via an Ottotype prism coupling geometry and in real space via nearfield imaging. The observed LPs feature both lenticular dispersion and associated inplane directionality, enabling us to bridge the concepts of leaky waves and anisotropic surface polaritons. Distinct from conventional leaky wave phenomena occurring in structured interfaces with subwavelength, carefully tailored geometries or isotropic material, our work demonstrates the existence of these hybrid modes in natural, unstructured, and anisotropic crystals.
We emphasize that the fundamental nature of LPs and gHPs is very different. LPs stem from the hybridization between surface polaritons and propagating bulk modes. Therefore, their inplane momentum is generally complex, even in the absence of material loss. As a result, LPs possess both tilted wavefronts and Poynting vectors canted away from the interface, whose magnitude can grow as they move away from the interface because of their forward leakywave nature. Ghost polaritons, on the contrary, have a complex outofplane momentum component, but their inplane component is realvalued. In fact, gHPs are surface polaritons that have slanted phase wavefronts, but their Poynting vector is parallel to the interface. Hence, the power flow necessarily decays away from the interface like conventional surface waves. In addition, gHPs usually emerge in uniaxial media like calcite when the OA is tilted with respect to the interface. LPs, as our complex eigenvalue analysis shows, can exist in a wide range of natural anisotropic materials as long as a leaky mode can hybridize with guided modes at the interface. The formation of LPs can hence be generalized to a wide range of natural and engineered photonic systems, including hyperbolic and zeroindex metamaterials^{41,42,43}. Our additional experiments on αquartz (Supplementary Information Fig. S10) provide direct evidence of LPs in other naturally hyperbolic materials. While their propagation along the interface is subdiffractive, LPs are complementary to conventional hyperbolic polaritons, with more limited confinement, which may be traded for easier excitation and longer lifetimes with Qfactors exceeding 300 and longer propagation lengths exceeding 13, as we explicitly demonstrate in Fig. 3i and Fig. 4g.
Despite emerging at the surface of an anisotropic medium, LPs display confinement factors that may not be as large as conventional hyperbolic polaritons because of the closer distance of their dispersion to FSLC. However, the hyperbolicity of the medium and associated strong anisotropy enables the peculiar dispersion features, directional inplane propagation, and, interestingly, directional radiation leakage in the farfield (Fig. S20), a phenomenon not achievable in other platforms. Furthermore, highly confined volume hyperbolic modes have small group velocities, such that it is challenging to leverage those strongly confined modes for energy transport in photonic platforms. In this context, LPs can provide reasonable group velocities, long propagation distances, and high Qfactors, offering unprecedented opportunities for nanophotonic integration. Thus, we anticipate that the directional LPs with lenticular dispersion, combined with wavefront engineering, may have important implications to further expand the role of polaritonics in nanoscale applications from midinfrared to farinfrared imaging, biosensing, and nanobeam guiding on integrated platforms.
Methods
Materials and fabrication
We use commercially available bulk calcite single crystals (Kejing, a Chinese company) with the characteristic plane (100) and (104), corresponding to the cases of OA with θ = 23.3° and 48.5° to surfaces, and an acut αQuartz sample (used for Ottotype prism coupling experiments, Supplementary Information Fig. S10) were purchased from MaTeck GmbH (Germany). For sSNOM sample preparation, the electron beam resist (PMMA) and then a conductive polymer were spincoated on the calcite surface. An electron beam lithography system was used to pattern the gold disk antennas. The standard liftoff procedure was put to use after electronbeam evaporation of a Cr/Au (3 nm/40 nm) layer on the developed resist.
Ottotype prism coupling measurements
An Ottotype prism coupling experiment is employed to optically excite nonradiative surface waves, see Fig. 3a. In the Otto geometry, the sample is brought in close vicinity to the prism. Incident light enters the prism at an angle β greater than the critical angle for total internal reflection producing evanescent waves on the back side of the prism that can couple to surface waves on the sample. By measuring the intensity of the reflected light over a certain frequency range of the incident radiation, the polariton resonances are detected as absorption dips in the reflectance spectrum. The Otto geometry enables experimental control over the excitation efficiency via tunability of the air gap width \({d}_{{{{{{\rm{gap}}}}}}}\). In the experiment, the prismsample spacing was kept fixed at \({d}_{{{{{{\rm{gap}}}}}}}\, \approx \,4{{{{{\rm{\mu }}}}}}{{{{{\rm{m}}}}}}\). The air gap was measured by whitelight interferometry, which can determine gap sizes from 500 nm up to the highest resolvable coherence length of 60 µm.
As a light source, we employ a midinfrared free electron laser (FEL) that provides high power, tunable radiation (3–50 µm) with a narrow bandwidth (\(\sim 0.3\%\)), further detail on the FHI FEL are reported elsewhere^{40}. In particular, we focus on the spectral range between \(850\, {{{{{\rm{and}}}}}}\, 920\,{{{{{{\rm{cm}}}}}}}^{1}\), corresponding to the lower reststrahlen band of calcite, where LPs are supported. The FEL allows us to perform narrowband spectroscopy, i.e. to scan the frequency, providing the full reflectance spectra, at a selected combination of inplane momentum \(\frac{{q}_{{{{{{\rm{r}}}}}}}}{{k}_{0}}\) and azimuth angle \(\phi\). The two parameters \(\frac{{q}_{{{{{{\rm{r}}}}}}}}{{k}_{0}}\) and \(\phi\) correspond to magnitude and propagation direction of the inplane momentum of the incident beam, respectively. Different inplane momenta are experimentally accessed by varying the incidence angle \(\beta\) (indicated in Fig. 3a), which is achieved by rotating the entire Otto geometry, whereas changes in the azimuth angle are obtained by rotating the sample.
For the azimuthal dispersion shown in Fig. 3b, the measurements were taken at a fixed incidence angle of \(27^\circ\) (corresponding to \(\frac{{q}_{{{{{{\rm{r}}}}}}}}{{k}_{0}}=1.075\)), whilst the resonance frequency maps depicted in Fig. 3d (and the relative Qfactors in Fig. 3g) were obtained by varying both parameters. In the latter case, the Otto reflectance spectra were taken for five different azimuth angles ϕ, at β = 26°, 28°, 30°, 32°, and 34° corresponding to inplane momenta of \(\frac{{q}_{{{{{{\rm{r}}}}}}}}{{k}_{0}}\, \approx \,1.04,\,1.11,\,1.18,\,1.25,\) and \(1.32\) (at about \(900\,{{{{{{\rm{cm}}}}}}}^{1}\)). The reflectance minima (representing the polariton resonances) and their Qfactors were extracted, as displayed in Fig. 3d, g, respectively. These maps enabled us to additionally calculate the inplane dispersion curves (IFCs) at multiple frequencies (Fig. 3f) by interpolating the momentum for a given frequency in the frequency–momentum dispersion for each measured azimuth angle. The corresponding Qfactors (plotted in Fig. 3i) were derived by interpolating the FWHM of each reflectance dip for the inplane momenta obtained from the previous interpolation in the FWHMmomentum dependence. To define the optical axis, the absolute azimuth offset \(\triangle \phi\) of the sample was computed by minimizing the sum of squared deviations of the experimentally derived IFCs (circles in Fig. 3f) from the simulated curves (lines in Fig. 3f), resulting in a rotation of the optical axis with respect to the xaxis of the laboratory coordinate system of \(\Delta \phi=1.6^\circ\). The experimental data (circles) in Fig. 3f, i have been shifted accordingly.
Farfield reflectance measurements
A homebuilt double goniometer is employed for β–2β scanning of the incidence angle by rotating the sample by β and the reflectance detector by 2β, respectively. The FEL beam is mildly focused (spot size ~500 µm) onto the sample under grazing incidence. The reflected intensity is detected by a pyroelectric detector, whose signal is divided by that of an identical reference detector. Twodimensional scans of the FEL wavenumber and incidence angle, respectively, are acquired for each sample azimuthal angle. Spectral drift of the FEL is accounted for by online monitoring of the singleshot FEL spectrum and interpolation of the data onto a wavenumberincidence angle grid during postprocessing. Notably, absolute referencing of the reflected intensity is difficult in this arrangement due to finite sample size (resulting in beam clipping at large incidence angles approaching 90°) and sample wobble. To minimize these effects on the measured intensities, we optimized the beam alignment for each azimuthal angle, thereby rendering an absolute reflectance referencing using a reference reflector unreliable. It is, in practice, also impossible to achieve identical absolute intensities despite alignment optimization. We thus normalized the 2D wavenumberincidence dataset for each azimuth according to the predicted global maximum from the equivalent simulated dataset.
sSNOM measurements
For nearfield imaging, a commercial sSNOM system (Neaspec GmbH, Munich, Germany) based on an atomic force microscope (AFM) was used. A metalized AFM tip (silicon coated with platinum) oscillates vertically with an amplitude of about 70 nm at a frequency of f ≈ 270 kHz and is illuminated by a ppolarized midinfrared quantum cascade laser with operation frequency ranging from 870 to 950 cm^{−1}. The signaltonoise ratio of the SNOM measurement is relatively low as the illuminated frequency approaches the lower frequency limit of the laser. The sharp apex of the tip offers the nanoscale resolution for recording LPs. The scattered signals are collected with a pseudoheterodyne interferometer for suppressing background signals. All nearfield signals are demodulated at a higher harmonic nΩ (n ≥ 3), yielding both amplitude s_{n} and phase φ_{n} images. Figure 4 shows amplitude s_{3} images of LPs.
Transfer matrix technique
The calculations of all simulated data shown in Fig. 3 were performed using a \(4\times 4\) transfer matrix formalism^{44}. The formalism allows for the calculation of reflection and transmission coefficients in any number of stratified media with arbitrary dielectric tensor, which allows to account for the anisotropy of our samples.
Numerical simulations
COMSOL version 5.6 was used for simulating point dipole excitation of both gHPs and LPs at the interface of calcite and air. A point dipole was placed 200 nm above the surface of a semispherical calcite, where spherical scattering boundary conditions were used on the spherical boundaries to absorb all outgoing radiation. The radius of semispherical calcite was sufficiently large \(R=80\) µm such that the wave is sufficiently damped when it reaches the boundary and thus has little impact on the results. The dielectric function and parameters of calcite were adopted from ref. ^{31}.
Data availability
The prism coupling and farfield reflectance experimental data and analysis scripts can be accessed at https://doi.org/10.5281/zenodo.7804555. All other data featured in the figures of this paper and other findings of this study are available from the corresponding author upon reasonable request.
Code availability
The codes that support the findings of this study are available from the corresponding authors upon reasonable request.
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Acknowledgements
This work was partially supported by the Office of Naval Research, a Vannevar Bush Faculty Fellowship, the Simons Foundation, and the Air Force Office of Scientific Research MURI program. P.L. acknowledges the support from the National Natural Science Foundation of China (Grant no. 62075070) and the support from the National Key Research and Development Program (Grant no. 2021YFA1201500). GC, SW, MW, and AP thank Wieland Schöllkopf and Sandy Gewinner (FHI Berlin) for operating the freeelectron laser. AP thanks Sebastian Mährlein (FHI Berlin) for carefully reading the paper.
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X.N., A.A., E.M.R., and E.G. developed the theory. X.N. and W.M. performed the COMSOL simulations. G.C. and S.W. performed the prism coupling experiments and analyzed the data. G.C., A.P., and E.M.R. performed the transfer matrix simulations. W.M. fabricated the calcite samples. W.M. and P.L. performed the sSNOM measurements and analyzed the data. X.N., A.A., G.C., A.P., W.M., and P.L. wrote the paper, with input and comments from all authors. A.A., A.P., and P.L. conceived the idea and oversaw the project.
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Ni, X., Carini, G., Ma, W. et al. Observation of directional leaky polaritons at anisotropic crystal interfaces. Nat Commun 14, 2845 (2023). https://doi.org/10.1038/s41467023383267
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DOI: https://doi.org/10.1038/s41467023383267
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