Introduction

Recently, symmetry engineering of electromagnetic modes in metamaterials has gained a lot of attention as an effective approach to tailor optical properties and produce novel responses and functionality1,2. This methodology has been especially valuable in the context of topological photonics, where spatial symmetries can be used to generate photonic pseudo-spin. This produces structured light on a chip that can be trapped and guided directionally along topological interfaces depending on the helicity of the field structure3,4,5,6. More recently, crystalline symmetries have also been utilized to generate higher-order topological phases and new classes of topological defects, facilitating the trapping of radiation to zero-dimensional boundaries to form nanoscale topological photonic cavities7,8,9. In fact, topological boundary modes have been used across frequency domains and spatial dimensions, to enable fascinating applications, from nonlinear optics10 and lasers11,12 to light-matter interactions13,14,15,16,17,18. Despite recent successes, the structure of optical fields on a nano-structured polaritonic chip is not well-understood and therefore, remains underutilized to control light-matter waves. Complete control of optical field structure would make it possible to design selection rules for interactions of light and matter excitations. Some degree of chiral light-matter interactions in topological systems integrating van der Waals materials has been reported for guided topological edge modes19,20,21, but such studies on higher-order and topological defect modes are still lacking. Here we present a hybrid polaritonic cavity design based on a zero-dimensional (0D) topological defect in a planar metasurface. This design is capable of trapping point-like higher-order topological (HOT) polariton states. We also experimentally demonstrate the chiral structure of these states stemming from their topological nature with spiral polarization texture and selective one-way coupling to topological interface modes.

Results

Hybrid metasurface design and effective model

We present the design of a hybrid topological metasurface which consists of a patterned silicon (Si) slab covered with a thin layer of hexagonal boron nitride (hBN). hBN is a van der Waals material that supports in-plane transverse optical vibrational modes at mid-IR frequencies22,23,24,25,26. Strong coupling between photonic and phononic degrees of freedom produces hybrid polaritonic states.

The photonic structure is composed of hexamers of triangular holes in a silicon slab. This pattern represents a deformed honeycomb lattice, also known as a Kekulé pattern. The degree of deformation is quantified by the strength of shrinkage or expansion of the distance between 6 nearest neighboring elements in the hexagonal unit cells compared to a regular graphene-like honeycomb array. Previous studies have leveraged this geometry for its waveguiding properties at interfaces between topologically dissimilar domains formed by expanded and shrunken hexamers. We introduce topological defects by stitching 3 domains with the same deformation type. Next, we combine the two approaches to demonstrate a heterogeneous platform that allows both guiding 1D edge modes and trapping 0D topological defect modes in the same device. The 3 offset sectors of the shrunken hexamers meet at a single core point, as illustrated in Fig. 1a, and produce a point defect and line defects along the 3 rays emanating from the core point. The core unit cells can be delineated differently in three domains, marked by\(\,\lambda=1\), \(2\), \(3\). As discussed in the Supplementary Note IIB, the emergence of 0D modes at such topological defects can be explained from the Wannierization with charge centers analysis, which allows classifying them as HOT states27.

Fig. 1: Topological defects in a hybrid metasurface.
figure 1

a Schematics of a topological point defect created by stitching domains with three different choices of the unit cell denoted as \(\lambda={\mathrm{1,2,3}}\); b zigzag domain wall between expanded and shrunken hexamers vs the line defect separating the off-set domains. c Numerically simulated electromagnetic energy spectra revealing frequency bands and differing dispersion of edge states at the zig-zag domain wall and line defect interfaces. Magenta and white dotted lines show edge-state dispersion calculated based on the effective Dirac model. Magenta solid lines trace the spectral position of the phonon mode in hBN. The dark-gray line marks the frequency of the bound state at the point defect, whose simulated spatial structure is depicted in d squared absolute value of the out-of-plane magnetic field component distribution \({\left|{H}_{z}\right|}^{2}\) in a silicon slab is overlaid with a transparent contour plot \({|u|}\) obtained from the Dirac model within the continuum approximation. Bar insert visualizes photonic and phononic fractions in the HOT point defect mode. e Optical and SEM images of the fabricated metasurface (the triangle side is \(s=1668\) nm, a spatial period is \(a=4220\) nm, the clustering parameter \({{\rm{C}}}=\frac{3{{\rm{R}}}}{{{\rm{a}}}}=0.935\), where \(R\) is the distance between the centers of the nearest triangles. f, g Experimental real-space images of the fields confined to the line defect (f) and the point defect (g) at wavelengths 6.80 μm and 6.95 μm respectively.

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When coupled to hBN, the physics behind photon-phonon interactions and the formation of hybrid polariton states in the system can be captured by an effective Dirac-like Hamiltonian acting on the six-component wave function \(\psi={({u}_{a},\,{u}_{b},\,{v}_{a},\,{v}_{b},\,{p}_{1},\,{p}_{2})}^{T}\) comprised of photonic (\({u}_{a},\,{u}_{b},\,{v}_{a},\,{v}_{b}\)) and phononic (\({p}_{1},{p}_{2}\)) degrees of freedom, corresponding to the electric fields of the photonic metasurface and in-plane vibrations of TO phonons in hBN, respectively. The indices \(a\) and \(b\) denote two triangular sublattices of the honeycomb lattice. The (non-interacting) photonic sector represents the massive Dirac model with a gapped doubly degenerate spectrum in the vicinity of the Γ point of the Brillouin zone28,29,30,31,32,33,34,35. The modes of the perforated slab are quasi-transverse-electric (quasi-TE) polarized: the electric field is polarized mostly in-plane (aligned within the \({xy}\) plane of Bloch-wave propagation with Floquet periodicity), while the magnetic field component is primarily oriented along the \(z\) direction. These modes are then conveniently classified according to their in-plane symmetry of the predominant out-of-plane component of the magnetic field \({H}_{z}\). The photonic band structure across the whole Brillouin zone (along the high-symmetry directions ΓKMΓ) and field profiles of the eigenmodes are shown in Supplementary Fig. S3, alongside the results for the corresponding tight-binding model. In the spectral range of interest, it is sufficient to retain only four bands of circularly-polarized dipolar \({p}_{\pm }\) and quadrupolar \({d}_{\pm }\) electromagnetic modes supported by the metasurface, which are pairwise degenerate at the Γ point. Consistent with refs. 31,32,34,35, the validity of such approximation is justified in Supplementary Note IA. The two opposite helicities \(\pm\) are assigned to pseudo–spin-up and pseudo–spin-down states. The rigorous definition of the pseudospin through the total angular momentum of light in the medium follows ref. 36. The pseudospin is generated by the fermionic-type pseudo-time-reversal symmetry \({T}^{2}=-1\), which in turn, as follows from the group-theoretical analysis, stems from the rotational \({C}_{6}\) symmetry of the crystal37. Representative computations for the patterned metasurfaces were reported in refs. 21,34. According to these simulations, in the far field, the total angular momentum is entirely defined by the spin (polarization) component, whereas in the near field, it is dominated by the orbital momentum, following the conversion of spin-to-orbital angular momentum. The lower-frequency bulk modes are dipolar (quadrupolar) in the shrunken (expanded) lattices, allowing a topological band inversion in the mode orbital ordering28,29,30,31,32,33,34,35. In our metasurface, they couple to nondispersive mid-infrared TO phonons in layered hBN, where the helical structure of the coupling is dictated by the circularly polarized nature of the dipolar \({p}_{\pm }\) and quadrupolar \({d}_{\pm }\) electromagnetic modes20,21. The two phononic degrees of freedom represent two phonon modes in the 2D plane, which are degenerate due to the in-plane isotropy of hBN. These modes are almost nondispersive in the photonic momentum range, because the lattice constant of the atomic lattice structure, which supports phonon modes is orders of magnitude smaller than that of the photonic structure. With an appropriate basis choice, this formalism leads to the effective Hamiltonian within the 2D approximation [see Supplementary Note I]

$$\hat{H}=\left(\begin{array}{cccccc}{\omega }_{0} & {\Delta }^{*} & 0 & \tilde{\Lambda }\left({-k}_{x}-i{k}_{y}\right) & {\Delta }_{2} & -{\Delta }_{1}\\ \Delta & {\omega }_{0} & \tilde{\Lambda }\left({k}_{x}+i{k}_{y}\right) & 0 & {\Delta }_{2}^{*} & {\Delta }_{1}^{*}\\ 0 & \tilde{\Lambda }\left({k}_{x}-i{k}_{y}\right) & {\omega }_{0} & \Delta & -{\Delta }_{1}^{*} & {\Delta }_{2}^{*}\\ \tilde{\Lambda }\left(-{k}_{x}+i{k}_{y}\right) & 0 & {\Delta }^{*} & {\omega }_{0} & {\Delta }_{1} & {\Delta }_{2}\\ {\Delta }_{2}^{*} & {\Delta }_{2} & -{\Delta }_{1} & {\Delta }_{1}^{*} & {\omega }_{{{\rm{p}}}} & 0\\ -{\Delta }_{1}^{*} & {\Delta }_{1} & {\Delta }_{2} & {\Delta }_{2}^{*} & 0 & {\omega }_{{{\rm{p}}}}\end{array}\right),$$
(1)

where \({\omega }_{{{\rm{p}}}}\) is the phonon resonant frequency detuned from the midgap frequency \({\omega }_{0}\) of a photonic subsystem, or equivalently the Dirac point of the spectrum in the undeformed lattice. The mass term \(\Delta\) is generated by the Kekulé deformation, and it is given by \(\Delta=-{\Lambda {{\rm{e}}}}^{i\Theta }\), where the phase \(\Theta \left({{\rm{\lambda }}}\right)=(2\lambda -1)2\pi /3\) depends on the choice of unit cell and wraps to \(2\pi\) upon a cyclic rotation: \(\Theta \left({{\rm{\lambda }}}=1\right)=2\pi /3\), \(\Theta \left({{\rm{\lambda }}}=2\right)=2\pi\), \(\Theta \left({{\rm{\lambda }}}=3\right)=4\pi /3\). \(\tilde{\Lambda }\) in front of the wavevector components is the renormalized Dirac velocity. The interaction between phonons and photons is described by \({\Delta }_{1,2}={{q}_{{\mathrm{1,2}}}e}^{i\Theta }/\sqrt{2}\), where \({q}_{{\mathrm{1,2}}}\) are the coupling strengths of flat-band phonon resonances to the dipolar and quadrupolar modes, respectively. Specifically, we consider the shrunken lattice and assume that the phonon line intersects the lower-frequency photonic bulk band only, which is dipolar in nature. Hence, we can neglect the coupling to the spectrally remote quadrupolar band \({q}_{2}=0\). The expressions for the coefficients \(\tilde{\Lambda }\) and \(\Lambda\) can be deduced from TBM, as detailed in Supplementary Note I. In fact, the \({H}_{{{\rm{z}}}}\) component can be linked to the discretized wavefunction in the 2D TBM approach. Respective parameter estimates were obtained by fitting our model to full-wave numerical simulations, using a three-dimensional finite-element-method (FEM) solver COMSOL Multiphysics (see “Methods” and Supplementary Note I).

The effective Dirac-like model (1) provides an intuitive analytical model of the system in the continuum limit. The dispersion of bulk modes in the regions \({{\rm{\lambda }}}=1\), \(2\), \(3\) is the same, and is found as a function \(\omega \left({{\boldsymbol{k}}}\right)\) from the eigenvalue problem \(\omega {{\boldsymbol{\psi }}}=\hat{H}{{\boldsymbol{\psi }}}\). Replacing momenta with spatial derivatives in Eq. (1), \({k}_{x,y}=-i{\partial }_{x,y}\), we can search for eigenmodes in the continuous Dirac medium with a non-uniform spatial distribution of \(\Delta\). We first focus on line defects formed by stitching two domains, as illustrated in Fig. 1b, along the line \(y=0\). In this scenario, we impose periodic boundary conditions along the \(y\) axis. Symmetry breaking at the interface is linked to an abrupt change of the parameter \(\Theta \left(y\right)\) at the interface, corresponding to two domains with unit cells \({{\rm{\lambda }}}=1\) and \({{\rm{\lambda }}}=2\), respectively. Neglecting for a moment the coupling to phonons, for the photonic modes bound to the interface of the form \({\propto e}^{i{k}_{x}x-{\kappa }_{y}{|y|}}\), we obtain the analytical dispersion relation \({\omega }_{{{\rm{e}}}.{{\rm{s}}}.}({k}_{x})={\omega }_{0}\pm \sqrt{{\Lambda }^{2}+4{\tilde{\Lambda }}^{2}{k}_{x}^{2}}\) and a decay rate \({\kappa }_{y}=|\sqrt{3}\Lambda /2\tilde{\Lambda }|\). We then numerically calculate the dispersion of the hybrid edge polariton modes by applying Susskind discretization38,39 (see Supplementary Note II). Due to the photon-phonon interactions, the dispersion is gapped, and a shift from \({\omega }_{0}\) of the gap center is observed at \(k=0\). The results are plotted in Fig. 1c together with the spectra obtained from full-wave simulations in the Floquet-periodic supercell geometries (see “Methods”). Crucially, the gap in the line-defect edge state spectrum is significantly (about 5 times) larger, when compared to the one of the armchair domain wall traditionally employed for zero-dimensional (0D) trapping via corner states40. In such a gap, well-localized lower-dimensional (higher-order) topological defect modes can arise. At the same time, the edge states at the zigzag domain wall have linearly crossing branches separated by a very small gap, which can be neglected in the continuum description (1).

The stepwise angular dependence of \(\Theta (\theta )\) with the polar angle \(\theta\) in our sectorial design enables trapping hybrid states at the point defect. Instead of a continuous vortex modulation combined with a radial potential wall41,42,43,44, we apply a discrete modulation that makes fabrication easier and allows building networks of topological interfaces. We look for the solution that resides at one sublattice, i.e., we assume an eigenvector of the form \({{{\boldsymbol{\psi }}}}_{{{\rm{p}}}.{{\rm{d}}}.}={(0,\,{u}_{b},\, 0,\,{v}_{b},\, {p}_{1},\,{p}_{2})}^{T}\equiv {(0,\, u,\, 0,\,{u}^{*},\,{p}_{1},\,{p}_{2})}^{T}\). Solving the system of equations for the components of \({{{\boldsymbol{\psi }}}}_{{{\rm{p}}}.{{\rm{d}}}.}\) with the continuity conditions at the three rays, we obtain the frequency of the 0D HOT mode, \({\omega }_{{{\rm{d}}}}=\left({\omega }_{{{\rm{p}}}}+{\omega }_{0}\right)/2 + \sqrt{{\left({\omega }_{{{\rm{p}}}}-{\omega }_{0}\right)}^{2}+{2q}_{1}^{2}}/2\), and the spatial profile of this defect mode (see Supplementary Note II). The relation between photonic and phononic contributions is found to be \(\left|{v}_{b}/{p}_{1}\right|=\left|{u}_{b}/{p}_{2}\right|={q}_{1}/\{\sqrt{2}\left({\omega }_{{{\rm{d}}}}-{\omega }_{0}\right)\}\). This ratio can be altered by detuning the position of the midgap frequency with respect to the phonon resonance. The point-defect mode is spatially localized around the defect, it populates only one of the two triangular sublattices, and the rotation of \(\Theta (\theta )\) determines the sublattice. Consistently, our first-principles simulations for the three-domain structure of Fig. 1a reveal a high-quality in-gap state bound to the core of the structure and hosted in the ~0.5 μm-wide band gap at the wavelength of 6.93 μm. The topological origin of the point-defect-induced mode appeals to the general mechanism discovered by Jackiw and Rebbi45,46, according to which a Dirac insulator can host a topologically-protected zero-mode trapped on a topological defect in the Dirac mass47. Regardless of whether the Dirac insulator is trivial or nontrivial, this is a local property in the proximity of a real-space defect. However, from a more rigorous real-space perspective, these modes can be interpreted to originate from non-vanishing (1/2-quantized) Wannier polarization of the line defects, with one of the Wannier orbitals localized at the junction point of the three domains (Supplementary Note IIB), thus justifying their classification as HOT states.

To experimentally confirm the presence of these polaritonic point-defect modes, we fabricated a hybrid metasurface consisting of a silicon photonic crystal slab patterned with the design discussed above and covered by a 15 nm layer of hBN. The thickness of the hBN layer effectively controls the oscillation strength of phonons, as well as the overlap with optical modes, thus defining coupling strength between photons and TO phonons. A thickness in the range of 10 to 25 nm was found to be the most suitable for sufficiently strong coupling, and at the same time relatively low loss in the resultant polaritonic system. The metasurface was formed by patterning the silicon-on-sapphire substrate using electron beam lithography and following anisotropic plasma etching (see “Methods”). A thin hBN layer was exfoliated from a bulk crystal and transferred on top of the metasurface to cover the area with interconnected line defects.

A custom-built mid-IR imaging setup was used for the characterization of the sample in both real- and Fourier space (see “Methods” and Supplementary Note III). The setup is equipped with a thermal imaging camera and a quantum cascade laser with operation range from 6.5 to 7.6 μm and 5-nm spectral resolution. Along with recording the spatial distribution of field intensity near the surface of the sample, this setup allows constructing 3D Fourier maps by consecutively acquiring momentum-space images by varying the frequency of the quantum cascade laser and then stitching the obtained iso-frequency contours.

Firstly, we confirmed that transferring of the hBN film leads to strong coupling between photonic and phononic modes, which gives rise to the formation of upper and lower polariton branches, – the effect that can be clearly seen in the experimentally extracted band structures of Fig. 2 when compared to the metasurface without hBN layer.

Fig. 2: Polariton strong coupling.
figure 2

Bulk band diagrams obtained from experimental measurements and numerical simulations for (a) photonic and (b) hybrid polaritonic metasurfaces with shrunken hexamers patterning. The lower-frequency bands in (a) possess dipolar character, while the higher-frequency bands are quadrupolar. In (b), the dashed gray line marks the spectral position of the hBN phonon mode.

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We then probed the defect modes of the polaritonic system by direct real-space imaging. Our experimental results demonstrate the presence of spatially localized trapped modes in the bandgap of the metasurface. The spatial profiles of 1D line and 0D point (HOT) defect modes are tightly trapped to the introduced line and point topological defects, respectively. 0D HOT mode is spectrally located inside a gap of the line-defect spectrum, matching simulation results, that evidences its relation to Dirac physics and topological origin. In agreement with our numerical simulations, we observed a point defect mode at the wavelength of 6.95 μm.

Far-field imaging of the point-defect-localized HOT modes

Next, we characterized the far-field properties of the polaritonic HOT modes, their radiation pattern and polarization texture. In general, the radiative nature was shown to offer the opportunity to probe topological phases in various photonic systems31,48. For comparison, we theoretically considered structures with both types of lattice deformations, shrunken and expanded. We first retrieved the far-field mode profiles from 3D modeling in COMSOL (see Fig. 3a). The modes exhibit narrow radiation patterns with a beam divergence of about 15°, which host a singularity in the normal direction. Their spiral polarization texture of the 0D mode can be described by the far-field electric field

$${E}_{{FF}}\propto {{\rm{| }}}{S}_{\pm } > \,\propto {e}^{i\varphi }\left|{LCP} > \pm i{e}^{-i\varphi }\right|{RCP} > ,$$
(2)

i.e., the field can be represented as a superposition of two optical fields bearing orthogonal polarization states, left- \({|LCP} > \) and right- \({|RCP} > \) circular polarizations, and different phases. The upper/lower sign corresponds to expanded/shrunken lattices, which imparts opposite spiral rotations, see Fig. 3a. In fact, the same angular dependence can be theoretically inferred by analyzing the TBM solution for the core-bound state in a finite lattice, or the above-mentioned analytical solution of the continuum Dirac model. In order to show this, we move to the basis of circularly polarized dipolar \({p}_{\pm }\) and quadrupolar \({d}_{\pm }\) states (see Supplementary Note IV). Only the dipolar components lead to far-field radiation, therefore we can reconstruct the vector field and its Stokes parameters using the Fourier-transformed electric field projections \({E}_{x,y}\) defined as \({E}_{x}{\propto {p}_{+}+p}_{-}\), \({E}_{y}\propto {i}({p}_{+}-p_{-})\). In this way, we can obtain both the intensity distributions and polarization structures, fully consistent with the results of 3D modeling (see Fig. 3a, bottom row). Remarkably, the orientations of the three-lobe patterns for the case of shrunken and expanded hexamers bear opposite chirality to each other.

Fig. 3: Radiation characteristics of the HOT point-defect modes.
figure 3

a Far-field radiation patterns (top row) and polarization textures (middle row) of the 0D HOT defect mode in metasurfaces based on shrunken (left) and expanded (right) hexamers obtained in 3D full-wave simulations. The far-field intensity distributions host a singularity at the normal direction, feature three lobes in directivity and chiral spirals in polarization, in agreement with the vector field maps theoretically reconstructed using the continuum Dirac model (a, bottom row). b Momentum-space experimental images for the shrunken-lattice case taken with a polarization of excitation beam falling oriented at 0° (top) and 45° (middle), and the superimposed image (bottom) as taken for unpolarized light excitation. The white dashed-line circle corresponds to the 10° angular distribution of emitted light. Black lines depict the experimentally reconstructed distribution of the state of polarization.

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We experimentally studied the radiation patterns and polarization textures of the fabricated polaritonic metasurface by characterizing it with back focal plane imaging. Since the quantum cascade laser source emits linearly polarized light, we collected radiation profiles for different positions of a half-wave plate, which rotates the angle of polarization of incident light. For each half-wave plate position, the emission pattern appears as two unequal lobes, which rotate following the changing angle of the polarization of excitation (as shown in Fig. 3b), implying the expected three-lobe spiral polarization distribution of the far-field radiation pattern. Superimposing images of different polarization orientations confirms that far-field directivity of the 0D HOT mode features a pattern with three lobes, and a singularity in the normal direction. Acquired momentum-space images of the vicinity of the point defect reveal that the directivity of the lobes corresponds to angles below 10°, in good agreement with the results of numerical simulations.

We additionally implemented a Stokes polarimetry measurement setup to acquire the full set of Stokes parameters for the higher-order topological defect mode and extract its polarization structure (Supplementary Note III). For this purpose, we performed six intensity measurements in momentum space: four measurements for different orientations of the linearly polarized excitation beam and two measurements for circularly polarized excitation. The calculated Stokes parameters allowed us to reconstruct the momentum-space distribution of the state of polarization for the 0D mode (Fig. 3b, bottom row), which clearly shows a spiral texture, fully consistent with our theoretical results.

Selective mode excitation in heterogeneous junctions

Next, in order to demonstrate compatibility of the proposed 0D modes with the conventional pseudo-spin-locked 1D boundary modes, we designed a trapezoid-shaped geometry made of heterogeneous interconnects formed by separated line defects and domain walls with the zig-zag cut created by properly interfacing domains of shrunken and expanded hexamers, as shown in Figs. 4a, b. The distance between the domain wall and the line defect was chosen to be large enough to avoid strong coupling between the corresponding eigenmodes, see Supplementary Note V. The vertices of the trapezoid accommodate the point defects, being chiral partners of each other. A spin-polarized edge wave with a given helicity is excited by the circularly-polarized source at the zig-zag domain wall21 and directionally guided along the interface between trivial and nontrivial domains towards the corner junction. If driven at the point-defect resonant frequency, directional propagation of the edge wave along the zig-zag domain wall leads to the excitation of the hybrid topological defect state confined at one of the junctions, as exemplified in simulations in Figs. 4c, d.

Fig. 4: Hybrid topological landscape.
figure 4

a Top view of the trapezoidal geometry made of interconnected line defects and zig-zag domain walls. Areas of shrunken (gray and yellow, C = \(0.935\)) and expanded (blue, C = 1.05) hexamers are made distinguished by color. The circle arrows illustrate circularly-polarized driving sources of opposite helicities (\(\sigma \pm\)) positioned at the domain wall, which excite counter-propagating edge modes (visualized by straight arrows) guided in the opposite directions. b Optical image of the experimental sample. Red and blue lines illustrate line defects and zig-zag domain walls respectively. c, d Results of the numerical simulations for right- (left-) circularly-polarized source excitation, labeled as \(\sigma+\) and \(\sigma -\), correspondingly, at the resonant frequency of the HOT point-defect mode. Shown is the squared magnitude of the out-of-plane magnetic field component \({\left|{H}_{z}\right|}^{2}\). e, f Real-space experimental images of the intensity field distributions at the wavelengths at and off the HOT point-defect resonance for right-(e) and left-(f) circularly-polarized light excitation. Dashed lines illustrate interconnected line defects and dotted lines illustrate zig-zag domain walls.

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To experimentally demonstrate the selective excitation of polaritonic HOT defect modes, we fabricated a polaritonic structure with trapezoid-shaped domain walls induced by both topological line defect and zig-zag cut. A layer of hBN was transferred on top of the silicon photonic metasurface covering both types of domain walls and their corner junctions (see Fig. 3b). Our experimental results show that the excitation of edge modes from the center of the zig-zag domain wall with left- and right-circularly polarized light leads to unidirectional propagation of spin-up and spin-down modes towards the corresponding corner junctions. When the frequency of excitation matches the resonance frequency of point HOT defect modes, a directionally propagating edge state gives rise to the excitation of the polaritonic HOT mode, spatially confined around interconnections of topological defect domain walls (Figs. 4e, f). Excitation at a frequency off the point defect resonance shows only unidirectional propagation of edge modes without the excitation of defect modes, as expected.

Discussion

In summary, we demonstrated a hybrid topological metasurface incorporating hBN and capable of efficiently trapping higher-order topological (HOT) phonon-polaritonic modes at the 0D topological defects created by stitching three mutually displaced Kekulé patterns. Thanks to the discrete parameter modulation, the proposed heterogeneous interface design was used for mutual transformation of topological hybrid states of different dimensionality, from boundary modes to 0D defect sates, within the unified polaritonic platform. Our design approach opens new avenues for engineering topological networks in the form of extended regular patterns—topological polaritonic super-crystals, supporting topological states made of topological modes themselves. It also heralds the possibility for topological phase transitions driven by tuning the strength of light-matter interactions relative to the characteristics of the photonic band structure. A versatile control over propagating and trapped modes in strong light-matter coupling regimes opens a new direction for mid-infrared applications. Specifically, the system considered offers manipulation of half-light, half-matter excitations of mixed light and in-plane phonons in hBN via guiding, resonant trapping to 0D HOT defect states and emission in the form of structured modes into the far field. Thus, our proposed platform can serve as an interconnect between mechanical/vibrational modes of materials and structured electromagnetic radiation, opening new opportunities for a broad range of optical technologies. We also envision the possibility of selective excitations of structured vibrational modes in such a hybrid platform by incident mid-IR Gaussian vortex beams, which will bring optical control over phononic modes to an unparalleled level.

Methods

Numerical simulations

We used commercial software COMSOL Multiphysics for the first-principles simulations presented in this work. The structural parameters were estimated from SEM images of the fabricated samples. We set the dielectric constant of silicon to 11.69, and the sapphire (Al2O3) substrate permittivity to \(2.05\). The thickness of the hBN on top of the patterned silicon metasurface was 15 nm. The pattern geometry was generated in COMSOL Multiphysics via the LiveLink interface for MATLAB. The undeformed unit cell of the uniform honeycomb lattice satisfies the ratio \(a/R=3\), where \(a\) is a lattice constant, and \(R\) is the distance between the center of unit cell to the centroid of each triangular hole. Keeping a fixed, shrunken hexamers are made by reducing \(R,\) that is described by the clustering parameter \({{\rm{C}}}=\frac{3R}{a}=0.935 < 1\). The expanded hexamers are made, by contrast, with \({{\rm{C}}}=1.05 > 1\). The transition from shrunken to expanded deformations, governed by changing С, is accompanied by the band inversion at small values of the in-plane wave vector that switches the band structure topology from trivial to nontrivial. For the two types of interfaces, zig-zag-shaped domain wall and line defect, we performed 3D supercell simulations driven by plane wave excitation in the frequency domain. The supercell consisted of 2 domains that formed an interface boundary aligned with the \(x\) direction. Each domain comprised 8 unit cells along the \(y\) axis and 2 unit cells along the \(x\) axis. Periodic Floquet-type boundary conditions were imposed in the x direction. All other boundaries were surrounded by perfectly matched layers. Band diagrams were recovered by scanning over the source frequencies and Floquet wavenumbers and color mapping the time-averaged energy density integrated over the unit cells adjacent to the interface.

Sample fabrication

Silicon photonic metasurface was fabricated by means of electron beam lithography. An array of triangular-shaped holes was imprinted into a silicon-on-sapphire (SOS) substrate with 1 μm of silicon and 500  μm of sapphire. The pattern with an isolated defect junction and no zigzag domain walls was used for the experiments presented in Figs. 1–3, while the trapezoid geometry incorporating both the defects and bent domain walls was employed in Fig. 4. At the first stage, e-beam resist ZEP520A-7 (300 nm thick) was spin-coated on the substrate and baked at 180 °C for 4 min. Next, a film of anti-charge agent (DisCharge, DisChem Inc.) was spin-coated on top of the e-beam resist to prevent charging. Electron-beam lithography exposure (Elionix ELS-G100) was followed by removing the discharging film by rinsing it in water. The exposed resist was developed in n-amyl acetate at 0 °C for 1 min. After that, silicon was etched for the full 1 μm depth by inductively coupled plasma (Oxford PlasmaPro System). A recipe based on C4F8/SF6 gases with an optimized etching rate of about 2.5 nm/s at 5 °C was used for the anisotropic etching process. During the last stage, the residue of the unexposed resist was removed by immersion of the sample into N-methyl-2-pyrrolidone (NMP) solution heated to 60°С. A thin hBN layer of 15 nm thickness was exfoliated from a bulk crystal (HQ Graphene) onto a thick PDMS stamp using the standard tape technique and precisely transferred on top of the fabricated silicon metasurface to cover the region of interest by the home-built transfer system. An atomic force microscope (Bruker FastScan) was used to confirm the thickness of the hBN layer (see Supplementary Note III).

Optical characterization

For the optical characterization of the fabricated hybrid polaritonic metasurfaces, we employed a custom-built setup designed for both real-space and back focal plane imaging of samples in the reflection geometry in the mid-IR region. The scheme of the setup is shown in Supplementary Fig. S9. The setup is equipped with a quantum cascade laser (QCL) Daylight MIRcat-QT and microbolometer INO MicroXcam-384-i camera. A 0.56 NA Thorlabs Black Diamond molded aspheric lens (Thorlabs C028TME-F) was used as an objective. The back focal plane imaging technique was used for the characterization of radiation patterns of the observed modes. This technique allows probing the angular distribution of the sample’s radiation by imaging the back focal plane of the objective collecting this radiation. Thus, momentum-space far-field directionality distributions of scattering from the fabricated structures were recovered from images of the objective’s back focal plane recorded at fixed wavelengths of QCL (also known as iso-frequency contours). We used 4 f configuration with two CaF2 lenses (20 cm focal length) to bring the back focal plane of the operated objective onto the camera’s plane. Moving the second lens to another position allowed us to perform direct real-space imaging of the samples at the same setup. We acquired angular and spatial profiles of the modes of the fabricated structures by collecting real-space images and iso-frequency contours while sequentially tuning the QCL wavelength in the spectral range of 6.5–7.6 μm. Modes of the structure were probed for linearly polarized light in a cross-polarization regime. A wire grid polarizer was used for implementing a crossed polarization configuration. Rotation of the angle of polarization of the excitation beam was performed by rotating a tunable half-wave plate (Alphalas). Circularly polarized light excitation for selective mode excitation in heterogeneous junctions and Stokes polarimetry measurements was implemented by placing a tunable quarter-wave plate (Alphalas) into the incident beam path after the QCL.